PB-208 154
MATHEMATICAL MODELING AND COMPUTER SIMULATION FOR
DESIGNING MUNICIPAL REFUSE COLLECTION AND HAUL SERVICES
S. Wersan, et al
Northwestern University
Evanston, Illinois
1971
DISTRIBUTED BY:
KFDi
National Technical Information Service
U. S. DEPARTMENT OF COMMERCE
5285 Port Royal Road, Springfield Va. 22151
This document has been approved for public release and sale.
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EPA-SW-6RG-71
PB 208 154
MATHEMATICAL MODELING AND COMPUTER SIMULATION
FOR DESIGNING MUNICIPAL REFUSE COLLECTION AND HAUL SERVICES
This final report (SW-6rg) on work performed under
solid waste management research grant no. UI-00699
to the Northwestern University was written by
S. WERSAN, J. QUON, and A. CHARNES
and is reproduced as received from the grantee.
U.S. ENVIRONMENTAL PROTECTION AGENCY
1971
Reproduced by
NATIONAL TECHNICAL
INFORMATION SERVICE
Springfield, Vi. 22151
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BIBLIOGRAPHIC DATA
SHEET
1. Report No.
EPA-SW-6RG-71
3. Recipient's Accession No.
4. Tide and Subtitle
Mathematical Modeling and Computer Simulation for Designing
Municipal Refuse Collection and Haul Services
5- Report Date
1971
6.
7. Author(s)
S. Wersan, J. Quon, and A. Charnes
8. Performing Organization Kept.
No.
9. Performing Organization Name and Address
Northwestern University
Evanston, Illinois 60201
10. Project/Task/Work Unit No.
ll.X3SXOOM»/Grant No.
Research Grant
UI-00699
12. Sponsoring Organization Name and Address
U.S. Environmental Protection Agency
Office of Solid Waste Management Programs
Rockville, Maryland 20852
13. Type of Report & Period
Covered
Final Report
14.
15. Supplementary Notes
16. Abstracts The first part of the report concentrates on the haul operation through a study
of mathematical models for the location or selection of disposal sites and the alloca-
tion of collection territories thereto. In considering "location-allocation" problems,
several disposal sites are located anywhere in a plane with simultaneous allocation of
refuse sources to disposal sites. The measure of effectiveness is minimization of ag-
gregate haul distance. For "selection-allocation" problems, the set of eligible dis-
posal site locations is known and one is to pick a proper subset of these and a refuse
source allocation that minimizes the aggregate haul distance or cost. These are charac-
terized as mixed integer programming problems with a coupled network analog and an
approximating algorithm. The second part is concerned with an analysis of the collec-
tion and haul operations combined, done with the use of two computer simulation models.
The first model is based en the daily route method of refuse collection practiced in the
Village of Winnetka, Illinois. The second model is based on constant length workday
rules found in Chicago, Illinois. Tables and graphs are presented that show the numeri-
cal bounds on the usefulness of the daily route method as it is affected by certain
factors; and how constant workday rules influence quality of service and cost effec-
tiveness.
17. Key Words and Document Analysis. 17o. Descriptors
*Refuse disposal, *Mathematical models, *Computerized simulation, *Collection,
*Selection, *Hauling, Cost analysis, Systems analysis
17b. Identifiers/Open-Ended Terms
*Solid waste disposal, Haul distance, Haul cost, Winnetka (Illinois), Chicago (Illinois)
17c. COSATI Field/Group 13B
18. Availability Statement
Release to public
19.. Security Class (This
Report)
yNCLASSIFIEI
LIED.
(Thii
20. Security Class (This
Page
UNCLASSIFIED
21- No. of Pages
446
22. Price
FORM NTIS-35 (10-70)
USCOMM-DC 40329-P71
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FOREWORD
Current solid waste collection practices do not differ significantly
from those used at the turn of the century. This lag in the development
of new technologies has had a marked economic impact, because it costs
four times as much to collect residential solid wastes and transport them
to a disposal site as it does to actually dispose of them. Ways must be
found, therefore, to maximize efficiency while minimizing costs.
An important objective of the U.S. Environmental Protection Agency
is to aid in developing economic and efficient solid waste management
practices. As authorized under the Solid Waste Disposal Act of 19&5
(P. L. 89-272) and the Resource Recovery Act of 1970 (P. L. 91-512), the
Agency awards research grants to nonprofit institutions in this effort to
stimulate and accelerate the development of new or improved ways of
handling the Nation's discarded solids. The present document reports on
work completed under one of these research grants.
The first part of the report concentrates on the haul operation
through a study of mathematical models for the location or selection of
disposal sites and the allocation of collection territories thereto. The
second part is concerned with an operational analysis of the collection
and haul operations combined, done with the use of two computer programs
especially written for the simulation of the operation of refuse systems.
11
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Publication of this report does not signify that the contents necessarily
reflect the views and policies of the U.S. Environmental Protection Agency,
nor does mention of commercial products constitute endorsement or
recommendation for use by the U.S. Government.
--SAMUEL HALE, JR.
Deputy Assistant Administrator
for Solid Waste Management
111
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CONTENTS
Page
ABSTRACT xi
PREFACE 1
Section A 1
An Overview of Applications of System-Analytic Methods
to Sanitary Engineering 1
Section B 6
The Nature of the Refuse System 6
Section C 10
General Outline of This Dissertation 10
PART I 1-1
On the Location of Refuse Disposal Sites and the Allocation
of Service Territories . 1-1
Chapter 1 1-2
Location Problems: A Review of the Literature 1-2
Chapter 2 1-19
Location-Allocation Models .... 1-19
Section A 1-19
Mathematical Definition and Assumptions 1-19
Section B 1-30
Solution of a One Sink Problem 1-30
Section C 1-34
Area Exclusions and the One Sink Problem 1-34
Section D 1-45
Multiple Sink Problems as Integer Programming Problems . . . 1-45
Section E 1-51
A Theory of Median Sets 1-51
Section F 1-64
An Approximating Algorithm Based on the Theory of
Median Sets 1-64
IV
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Section G 1-73
Finding Initial Solutions 1-73
Chapter 3 1-78
Selection-Allocation Models 1-78
Section A v. . 1-78
Mathematical Definition and Assumptions 1-78
Section B 1-81
Characterization as a Network Flow Model 1-81
Section C 1-88
An Approximating Algorithm: The Minimum Elimination-
Gain Algorithm (MEGA) 1-88
Section D 1-97
Applications of MEGA to Location-Allocation and
Other Problems 1-97
Chapter 4 1-98
Extensions of Selection-Allocation Models 1-98
Section A . , 1-98
S-A Problems with Weighted Sources 1-98
Section B 1-103
S-A Problems with Site Acquisition and Processing Costs . . 1-103
Section C . . . 1-107
Extended Models for Considering the Development and
Use of a Selected Site 1-107
Chapter 5 1-116
Summary and Conclusions 1-116
Figures
1-1 Median Solution and an Area Exclusion 1-35
1-2 Area Exclusion of a Central Business District 1-39
1-3 Structure of Constraint Matrix of (2.22) 1-50
l-4a Paths from Sources to Sinks Unavoidably Crossed .... 1-60
l-4b Reallocation to Cause Uncrossing 1-61
1-5 Initial Solution for Example 1 1-69
1-6 Result of First Exchange in Example 1 1-70
1-7 Result of Second Exchange in Example 1,
Local Optimum Solution 1-71
1-8 Horizontal Problem Corresponding to Example 1,
Figures 1-5 to 1-7 1-75
v
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Page
1-9 Vertical Problem Corresponding to Example 1,
Figures 1-5 to 1-7 1-75
l-10a P-System, N=7, S=4 1-82
l-10b A-R-System. S=4, m=2 1-82
1-lla Acceptance System Network for Extended Selection Model . 1-110
1-llb Physical Network of Extended Selection Model 1-111
Tables
1-1 Source Locations for Example 1 1-68
1-2 Comparison of Results £j vs. E-metric 1-72
1-3 Coincidence Table for Horizontal and Vertical
Solutions of Figures 1-8 and 1-9 1-77
l-4a Selection-Allocation Problem, N=7, S=5. Initial
Allocation 1-90
l-4b Second Tableau, N=7, S=5 1-90
1-5 Illustration of Tie-Breaking 1-93
l-6a MEGA with Weights on a Problem of Maranzana. Initial
Tableau 1-101
l-6b 7th Tableau 1-102
l-6c Final Tableau for M=2 1-102
PART II 2-1
On the Simulation of Refuse Collection and Haul Operations .... 2-1
Chapter 1 2-2
Introduction 2-2
Section A 2-2
Solid Wastes Literature 2-2
Section B 2-13
The Nature and Limitations of Simulation 2-13
Section C 2-33
The Simulation of Refuse Collection and Haul Operations . . 2-33
Chapter 2 2-36
Rational Methods for Design of Refuse Collection/Haul
Systems 2-36
Section A 2-38
An Overview of the University of California Report 2-38
VI
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Page
Section B 2-44
Factors Influencing the Pickup and Haul Operations 2-44
Section C 2-50
Development and Use of Rational Design Formulas 2-50
Section D 2-62
An Evaluation of the University of California Report .... 2-62
Chapter 3 2-66
Development of the Simulation Program 2-66
Section A 2-66
Sources of Variability in Refuse Collection/Haul
Operations 2-66
Section B 2-75
Some Concepts Underlying the Simulation Programs 2-75
Chapter 4 2-90
Simulation Studies of Refuse Collection and Haul under the
Daily Route Method 2-90
Section A 2-93
Invariants of the Reference Systems 2-93
Section B 2-95
Variable Parameters of the Reference System 2-95
Section C 2-99
Results and Analysis 2-99
Appendix to Chapter 4: Notation 2-129
Chapter 5 2-132
Simulation Studies of Alternative Policies in a Constant
Length Workday System 2-132
Section A 2-134
The Reference System under Constant Length Workday 2-134
Section B 2-138
Description of Alternate Policies 2-138
Section C 2-141
Results and Analysis 2-141
Appendix to Chapter 5: Notation 2-154
Vll
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Page
Chapter 6 2-159
Extensions and Modifications of Simulation Models 2-159
Chapter 7 2-164
Summary and Conclusions 2-164
Figures
2-1 Flowchart for Simulating Buffon's Needle Experiment . . . 2-19
2-2 Flowchart of One Hand of the Game of "Clock" 2-22
2-3 Tree of Random Walk Moves 2-24
2-4 Probability Plot of Refuse Production, Winnetka,
Illinois, March-April, 1958 2-68
2-5 Verbal Flowchart of the Movement of a Single Truck.
Daily Route Method, Winnetka, Illinois 2-82
2-6 Verbal Flowchart of the Movement of a Single Truck.
Constant Length Workday, Chicago, Illinois 2-85
2-7 Truck Capacity Utilization and the Number of Trips
per Day vs. the Coefficient of Variability 2-105
2-8 Overall Collection Efficiency and Length of Workday
vs. the Coefficient of Variability 2-106
2-9 Haul Efficiency and Haul Time vs. the Coefficient
of Variability 2-107
2-10 Truck Capacity Utilization and the Number of Trips
per Day vs. the Assignment 2-108
2-11 Overall Collection Efficiency and Length of Workday
vs. the Assignment 2-109
2-12 Haul Efficiency and Haul Time vs. the Assignment .... 2-110
2-13 Overall Collection Efficiency and Length of Workday vs.
One-way A!-metric Haul Distance 2-111
2-14 Haul Efficiency and Haul Time vs. One-way Hi-metric
Haul Distance 2-112
2-15 Correlation of Overall Collection Efficiency with the
Pickup Coefficient 2-113
2-16 Queuing Problem at the Disposal Site: Waiting Time
per Day vs. the Number of Unloading Docks 2-114
2-17 Hypothetical Collection Area 2-137
Vlll
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Page
Tables
2-1 Statistics on Refuse Production. Winnetka, Illinois,
March-April, 1958 2-69
2-2 Values of Input Parameters 2-96
2-3 Example Output Information from Simulation 2-100
2-4a Distribution of Percent of Trips vs. Percent Capacity
Used. Coefficient of Variability 0/y = 0.1 2-102
2-4b Distribution of Percent of Trips vs. Percent Capacity
Used. Coefficient of Variability a/y = 0.3 and 0.6 . 2-103
2-4c Contingency Table Analysis of E Frequency
Histogram--A = 2.04, 0/y = 0?1 2-104
2-5 Frequency Distribution of Length of Workday and Settings
of Input Values of 25 Cases 2-119
2-6 Cumulative Distribution of E for a/y = 0.3, 0.6 with
A = 3.06 ? 2-122
2-7 Summary of Values of Input Parameters 2-136
2-8 Summary of Operational Policies Considered 2-139
2-9 Example Output Information from Simulation Model .... 2-142
2-10 Example Output Histogram of Length of Workday 2-143
2-11 Example Output Histogram of Truck Capacity Utilization . 2-144
2-12 Effect of Relay Policy on the Quality and Cost of
Service Provided 2-146
2-13 Effect of Overtime Policy on the Quality and Cost of
Service Provided 2-148
2-14 Effect of Assignment Policy on the Quality and Cost of
Service Provided 2-150
2-15 Cost and Quality of Service for Twelve Operational
Policies 2-151
Appendix 2-A 2-171
Figures
A.I Haul Speed vs. Round Trip Distance 2-179
A.2 Verbal Flowchart of Outer Program 2-184
A.3 Verbal Flowchart of Event Clock 2-186
A.4 PUT-Subroutine for Storing Events 2-189
A.5 Event Type 1: Truck Arrives at Disposal Site .... 2-191
A.6 Event Type 2: Collection Unit Entered for First Time. 2-194
A.7 Event Type 3: Truck Leaves Disposal Site 2-196
IX
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Event Type 4: Collection Unit is PROP(H) Finished
Page
2-199
A.8
A.9 Event Type 5: Partially Done Collection Unit is
Entered 2-201
A. 10 Event Type 6: Truck Arrives at Garage 2-203
A.11 CMPJOB: Subroutine to Determine Refuse Amount and
Collection Time
TRATIM: Subroutine to Compute Haul Time and Distant . 2-204
A.12 CAPRUL: Capacity Rule Subroutine
FINEXU: Find Next CU Subroutine 2-209
A.13 HISTOG: General Histogram Reader 2-212
A. 14 SOUP: General Statistics Accumulator 2-214
A.15 PWFHST: Frequency Histogram Packer 2-216
A.16 UNPWFH: Frequency Histogram Unpacker 2-219
Tables
A.I Example of Input Card Deck 2-235
A.2a Sample of Printed Output, First Page 2-237
A.2b Sample of Printed Output, Second Page 2-238
A.3a Listing of Punched Card Output 2-243
A.3b Listing of Punched Card Output (Cont.) 2-244
REFERENCES 2-282
x
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ABSTRACT
Part I of this dissertation begins with consid-
eration of problems called "location-allocation" problems
in which several disposal sites are to be physically
located with simultaneous allocation of refuse sources
to disposal sites. The disposal site may be located
anywhere in a plane and the measure of effectiveness is
minimization of aggregate haul distance. By measuring
distance with the Hi-metric three things are accomplished:
(1) It is possible to reduce these Ai-metric
location-allocation problems to mixed
integer programming problems.
(2) Using the mixed integer programming formu-
lation, one may add constraints which
proscribe certain areas for the placement
of disposal sites.
(3) The single site location problem has a
closed-form solution related to the geo-
metric median of the source points.
Using this property, a Theory of Median
Sets is established and an alternating
location-allocation algorithm is
constructed for the multi-site problem.
XI
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A second class of problems called "selection-
allocation" problems is defined. Here the set of
eligible disposal site locations is known and one is to
pick a proper subset of these and a refuse source
allocation which minimizes the aggregate haul distance
or cost. These problems are characterized as mixed
integer programming problems with a coupled network
analog and an approximating algorithm, the Minimum
Elimination Gain Algorithm is established. This latter
algorithm is extended to consider weighted sources and
site acquisition costs. The network analog is extended
to consider site acquisition, facility development and
operation costs as well as aggregate haul cost.
Part II of this dissertation traces the develop-
ment of two computer simulation models. The first model
is based on the daily route method of refuse collection
practiced in the Village of Winnetka, Illinois. Using
data relevant to Winnetka, a series of simulation runs
was made to delineate the interdependencies of parameters
involved in the functioning of a refuse collection system.
In particular, the percent of truck capacity utilized,
the number of daily trips, the overall collection
efficiency, the length of workday, the haul efficiency,
and the haul time as—percentage of total time were
Xll
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measured as functions of the coefficient of variability of
refuse production, the refuse assignment and one-way haul
distance. A further set of runs studied the effect of
the number of unloading platforms on average and maximum
waiting times at the disposal site. The interpretation
of the results presented in tables and graphs allows
numerical bounds on the usefulness of the daily route
method to be established regarding the coefficient of
variability, the refuse assignment and haul distance.
The second model is based on constant length
workday rules found in Chicago, Illinois. Using data
relevant to Chicago, a series of runs was made to measure
the quality of service and cost effectiveness of different
combinations of overtime, last-load relay and assignment
policy. Here assignment means both the average daily
number of truck loads and the average daily number of
eight hour shifts. The results presented in a series of
v
tables show as one example that the use of overtime is
most cost effective when combined with a reasonable time
assignment (less than one).
Xlll
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PREFACE
Section A
AN OVERVIEW OP APPLICATIONS OP SYSTEM-
ANALYTIC METHODS TO SANITARY ENGINEERING
The past decade has seen a dramatic growth of
awareness and concern in the general public about the
environments that surround us. Public apathy toward the
pollution of our rivers and the poisoning of our atmos-
phere is gone. We have become concerned about rebuilding
our cities, beautifying our roadsides, preserving our
remaining wilderness areas, and showing that the
twentieth century and de Tocqueville's America [Pr. 15] —
are not wholly incompatible.
This movement is a natural outgrowth of the con-
servation movement which was born in the early years of
this century and is a reaction to our 40$ increase in
population (1939-1964) and our increasing urbanization,
suburbanization, and "automobilification" .
In the growth of this awareness, matters which
were once universally ignored became the crusades of a
foresighted few, then the expected duties of local and
state governments, and finally, of the federal government
If Numerals in brackets indicate references. See page 2-282.
1
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which today aids local governments in the construction of
sewage treatment facilities and sponsors research to
improve the technologies and augment the basic scientific
foundations of the disciplines involved. One of the
research philosophies and bodies of methodology that have
been adopted in some of this research is the "systems"
approach. As expressed by W. R. Lynn [Pr. 11]:
"It is obvious that satisfactory solutions
to this problem [disposition of solid wastes]
cannot be obtained by viewing each operation
within the processes of refuse collection as an
independent unrelated function. Rather, in order
to achieve efficient solutions to these problems
effort must be directed toward viewing the prob-
lem in its entirety as an interconnected system
of component operations and functions.
Unfortunately, until comparatively recently,
solutions to this class problem have been
unavailable because of the difficulty in making
a systematic analysis of large complex problems.
However, the advent of new mathematical tech-
niques in systems analysis and operations
research coupled with the availability and facili-
ty of high speed computing devices will now permit
investigation of this type."
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In actual practice, applications of these tech-
niques to the field of sanitary engineering have been
pursued in two directions. These are: To better under-
stand and design component processes and to synthesize
such optimal components into optimal systems. The aim of
this dissertation is to follow both of these trends in
that field of sanitary engineering which is concerned
with the disposal of solid wastes, but it is not irrele-
vant to review briefly the results that have been accom-
plished in other areas of sanitary engineering by the use
of those methods. It will be seen that these results are
largely concentrated in the area of water pollution and
the closely related area of water resources management.
W. S. Galler [Pr. 3] (see also [Pr. 4]) used non-
linear programming to determine the optimal configuration
(radius, depth, and recirculation ratio) of biological
(trickling) filters, an important sewage and industrial
wastes treatment process. By optimal was meant minimum
operating and construction cost subject to the constraint
that the pollutional load (biochemical oxygen demand or
BOD) be reduced to a prescribed level.
Dr. Lynn's dissertation [Pr. 10] (see also [Pr. 8]
and [Pr. 9])represents a middle ground proceeding in the
direction of increasing synthesis. Assuming the exist-
-------
ence of optimally designed components capable of operat-
ing at prescribed levels, Lynn constructed a mathematical
model of the flow of sewage through a treatment plant.
The object was to select the types of treatment processes,
their operating levels, and their order of application to
achieve for a treatment plant as a whole the same objec-
tive stated above regarding the biological filter. A
second, dynamic model investigated the elements of timing
and financing in the face of population growth.
Proceeding further in the direction of synthesis
are four works which deal with the pollution problems and
the water resources management of entire river basins.
A 1962 book of Maass, et_.£LL. entitled Design of
Water Resources Systems [Pr. 12] discussed the theoreti-
cal, methodological, economic, and political foundations
underlying such system designs. No specific river
system was analyzed.
This book laid the groundwork for the later work
of Hufschmidt and Piering [Pr. 5], Simulation Techniques
for Design of Water Resources Systems, in which computer
simulation techniques were used to determine the optimal
placement of six multipurpose dams on the Lehigh River in
Pennsylvania.
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Linear programming models of pollution abatement
in an entire but largely hypothetical river basin were
constructed by R. A. Deininger [Pr. 2]. One such model
sought minimal treatment cost subject to prescribed
levels of BOD reduction. A second model incorporated a
dual criterion: BOD to be reduced and dissolved oxygen
to be augmented. Several other models were formulated
but not pursued. Perhaps the most interesting of these
is a model which considers the cost of water purification
as well as sewage treatment on the realistic premise that
the river is used subsequently downstream for water
supply.
A later work by Loucks [Pr. 6] (see also [Pr. ?])
analyzed the cost and effectiveness of treatment from the
viewpoint of probability theory. The aim here was to
determine the cost of maintaining a prescribed proba-
bility level on the minimum dissolved oxygen content of a
river under the random variation in pollutional load and
other relevant factors.
A discussion of published work and work in
progress on the application of system analytic methods to
the field of solid wastes will be deferred until the
introduction to Part II, i.e. Chapter 1.
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Section B
THE NATURE OF THE REFUSE SYSTEM
It is appropriate and logical at this point to
develop some feeling for the nature of refuse systems and
the complex inter-relation of its parts that impels the
system analytic approach.
Aesthetic and economic considerations aside, a
city is — to one responsible for the planning and opera-
tion of a municipal refuse collection service — a
conglomeration of people together with their residences,
businesses, factories, and institutions set down in a
maze of streets, highways, and traffic. In the normal
course of events, these people produce residential,
commercial, manufacturing, and institutional wastes that
must be gathered by men and equipment, transported over
streets and highways by men and equipment, and finally
disposed of by some means, again by men and equipment.
Overlying and governing these three major areas
of collection, transport and disposal is the system of
policy, planning, finance, and administration that pays
the men, buys the equipment, acquires the disposal
facilities, creates an intelligent working entity from
all of these, and acts as a buffer between those served
-------
and those serving.
The preceding has been a bare physical and insti-
tutional outline of a refuse system. We may flesh-out
this skeleton by noting that a refuse system functions in
the environment of a particular city: This city has its
own particular climate, topography, seasonal variations,
growth patterns, and population with its own habits,
traditions, standards of living, and political institu-
tions — all of which affect a refuse system. Moreover,
a refuse system does not stand still in time. Long term
trends such as the Increased use of prepared and conven-
ience foods reduce the putresclble organic content of
refuse, but yield more containers to throw away while
the Introduction of garbage grinders produces similar
changes.
The most important driving force behind a refuse
system is its workload which is largely determined by the
following set of decisions:
(1) Classes of residences and other sites to be
served, e.g.»are businesses required to hire
a private hauler?
(2) The type(s) of refuse to be collected, e.g.
are grinders to be used? Shall yard refuse
and leaves be accepted?
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(3) The frequency of collection. This may
depend on the season and the amount of
putrescible matter present.
(4) The degree of cooperation required from
those served, more specifically with regard
to
(a) Separation of refuse types into differ-
ent containers,
(b) Yard, alley or curb pickup, and
(c) Standardization of container size.
Items (1), (2), and (3) have the most profound
effects on the entire system for they dictate via the
workload the numbers of men and the cubic yards of truck
capacity that will be required. Item (^1) enters the
picture largely by affecting the efficiency of collection
as measured for example in man-minutes per ton of refuse
collected.
The amounts of refuse, their types, and the
degree of separation may also affect decisions on the
best method of ultimate disposal. The method of disposal,
in turn, bears heavily on where the disposal site(s) may
be located, and the location of the disposal site(s)
determines the proportion of the workday spent on the
haul operation. Finally, the length of the haul operation
-------
Is a significant factor in setting work force size and
purchasing equipment.
Thus, there is here truly a system which even-
tually must be viewed "... in its entirety as an inter-
connected system of operations and functions". [Pr. 11]
-------
Section C
GENERAL OUTLINE OF THIS DISSERTATION
This dissertation is divided into two distinct
parts. In the first part we isolate and concentrate on
the haul operation through a study of mathematical
models for the location or selection of disposal sites
and the allocation of collection territories thereto.
The second part of this dissertation is concerned with an
operational analysis of the collection and haul operations
combined and this is done with the use of two especially
written computer programs for the simulation of the
operation of refuse systems.
Among the long-term secular changes which cities
face is the urbanization of the land separating them;
much has been written lately about the growth of
"megalopoli". This has two effects on the planning of
refuse disposal. First, it means that the land required
for disposal operations is becoming more expensive, or
alternatively land at the right price is retreating
further from the city. Secondly, it means that the
component governmental units of megalopoli must sooner or
later join together in a regional approach to their
common problem. As expressed by the Northeastern Illinois
Metropolitan Planning Commission:
10
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11
"That refuse disposal is a metropolitan
problem is evident in the frequent shipment of
refuse across county and municipal boundaries.
Large tracts of land are needed to accomodate
refuse, and the difficulty of obtaining such
tracts often requires refuse disposal sites to
be located far from the collection sources."
[Pr. 13]
For economic perspective we may note that the
haul operation— accounts for some 25-35% [Pr. 1] of
municipal refuse budgets and these aggregate to some
four-billion dollars a year nationally.
Thus motivated, we are led to investigate the
questions:
(1) Where is the best place to locate one or
more refuse disposal sites given the
locations of the areas to be served?
(2) Which areas should be served by which
disposal site?
- Roughly, the haul operation may be defined as that
part of the refuse collection-disposal activity which
takes place between the last container of a load and the
first container of the next load. It is here extended to
include travel to and from the garage.
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12
The assumption lying behind these questions is
that we are free to place a disposal site anywhere in a
large, unrestricted area. The discussion above makes it
apparent that this is an unrealistic assumption. Hence,
building on the insights gained from answering questions
(1) and (2) we next ask:
(3) What happens to the choices of location and
allocation of- collection sources when
certain areas are Ineligible for the
location of disposal sites?
It is clear that we get even closer to reality
when the process of declaring certain areas Ineligible
culminates in the situation wherein only a small number
of land parcels remain eligible. Now we ask:
(4) Given a definite number of eligible sites
and their physical locations, what is the
2/
best way to pick a proper subset— of them
to be disposal sites and to allocate
collection sources to them?
The mathematical models that yield answers to
these questions are investigated in Part I of this
dissertation. The models arising from questions (1),
(2), and (3) are termed location-allocation problems and
~ Meaning some but not all. We shall later restrict
this to mean also > 2 .
-------
13
are discussed in Chapters 1 and 2 . The models
arising from question (4) are termed selection-allocation
problems and are discussed in Chapters 1 , 3 and
4 . The selection-allocation model lends itself readily
to considerations such as site acquisition cost and
processing costs (per unit of refuse processed) at the
selected sites. Selection-allocation models embodying
these features are considered in Chapter 4 .
The matter of processing cost naturally generates
a more general question:
(5) Having selected sites on the basis of
reasonable system-wide transportation and
acquisition costs, what disposal processes
should be used • at these sites?
In Chapter 4 models to answer this question are
discussed, it being pointed out that mature engineering
judgment has already limited the set of eligible sites to
those suitable to the disposal processes deemed feasible
on the basis of other considerations, thus this sort of
model is really concerned with choices of_ capacity and
processing rate. In such models we are presented with a
total amount of refuse to be processed and the points
where the refuse originates. This refuse is to be taken
to a certain number of sites selected from a list of
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14
eligible sites at "minimum" aggregate transit cost. For
each site selected we pay a one-time acquisition cost,
and we select a processing capacity and rate from a list
of alternatives appropriate to the given site. This
choice requires the payment of a one-time development
cost, and imposes on us a capacity and processing rate
which when taken together with those from all of the
other sites must be adequate to process the refuse on
hand. It is to be noted that the total cost is made up
of transit, site-acquisition, site-development, and
operation costs (suitably discounted for time disparities)
and thus the overall optimum program may not really have
minimal transit cost which was the only criterion in the
first of these models. *
It is clear upon reflecting on this sequence of
increasingly complex models that further increases in
complexity buy very little further in predictive powers
and insight. The chief criticism, might be phrased as
follows:
The methods of Part I concentrate heavy guns upon
a frozen sliver of time taken from a world undergoing
short term random variation, seasonal fluctuation, and
long-term (secular) changes. How can methods which study
static situations be used to understand a dynamic reality?
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15
Or more fundamentally, what methods are available to study
dynamic systems? To answer these questions we submit the
following brief list of methods:
1. Empirical Approach: In the context of
refuse collection and haul systems, this
means study of dynamics in the field by
experimenting with actual crews,
equipment, etc.
2. Analogy: Planning is done by comparison
of a given city to another and using an
analogous solution.
3. Mathematical Analysis: An example of
such analysis is contained in the models
of Part I. Others will be discussed in
Part II.
^' Simulated Experimentation: In this age,
this generally means a simulated experi-
ment conducted by the use of a computer
program.
Part II of this dissertation is concerned with
the design, use, and interpretation of results from such
a computer simulation model hence we defer until Chapter
1 , a more detailed discussion of the nature and limita-
tions of simulation and its proper place with respect to
the other methodologies just listed.
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16
In Chapter 2 we enlarge upon certain rational
methods which have been developed for the design of
entire static refuse collection/haul systems. Chapter
3• first examines the sources of variability selected
as being of greatest importance in a refuse system and
then turns to a discussion of some of the concepts used
to render the real world physical entities of men, trucks,
and refuse amenable to simulation by a digital computer
program. Two programs are described here. The first
simulates the daily route method of refuse collection as
practiced in the Village of Winnetka, Illinois. The
description of this program is supplemented by a detailed
description in Appendix 2-A . The second program,
derived from the first, simulates the fixed-length work-
day method of refuse collection as practiced in certain
parts of the City of Chicago. Only the specific points
of difference are described.
Chapters 4 and 5 present the main results
of Part II. They are concerned with the results obtained
from using the two simulation models on particular subsets
of input data. Each chapter discusses the inputs used,
the results obtained, and the practical implications of
these results to the planners and operators of refuse
collection/haul systems.
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17
Chapter 6 proposes extensions of the simulation
models and draws upon shortcomings discovered therein to
delineate characteristics desirable in future models.
Finally, Chapter 7 summarizes and states the results
of Part II.
-------
PART I
ON THE LOCATION OP REFUSE DISPOSAL SITES
AND THE
ALLOCATION OF SERVICE TERRITORIES
1-1
-------
CHAPTER 1
Section 1
Location Problems : A Review of the Literature
The location-allocation and selection-allocation
models to be investigated in Part I have their roots in
mathematical antiquity. The germinal problem for the
location-allocation model is the generalized Weber
problem discussed immediately below. Selection-
allocation problems arise out of location-allocation
problems in the manner suggested in Section C of the
Preface.
The generalized Weber problem is
Given a set of fixed points {(a., b . ) , i » l , 2 , n)
and weights (w.^ > 0} , determine (x, y) that
minimizes
i=n
/
, y) = wi[(x - a±)2 + (y - bi)2]12 (l.i)
1=1
1-2
-------
1-3
i.e. the weighted sum of the distances from the fixed
points to the minimum point.
Up until 1962, the only things known about the
solution to (1.1) were: a) for special values of
n (n < 4) ; b) for certain degenerate cases such as
colinearity [all the points on a straight line yielding
x = I w. a./£ w. , y - Z w. b./Z w. (the means)] and n-
i11!1 i11!1
fold symmetry of the (a., b.) and the w. [e.g., the
apices of an n-gon where n is even and opposite apices
have equal weights =»(x, y) at center of n-gon] ;
c) for general n , the theorems of Tedenat and Steiner
[A 5] which cover the case of equal weights.
Tedenat's Theorem (1810): A necessary condition
that (x, y) be the minimum points of (1.1) with all
w. equal is that the sum of the cosines of the angles
between any arbitrary line In the plane and the set of
lines connecting (a±, b^ to (x, y) be zero.
Steiner's Theorem (1837): A necessary and suffi-
cient condition that (x, y) be the minimum point of
(1.1) with all w.^ equal is that the sum of the cosines
and the sines of said angles be zero. (See the reviews of
results in Cooper [A 5] and Eisemann [A 10].)
-------
Alfred Weber's classic Theory of the Location of
Industries EA 13] (reviewed in [A 5]) is apparently the
first place that the problem appears in an economic con-
text as opposed to a pure mathematical abstraction. The
economic context makes natural the first apparent intro-
duction of weights into the problem (hence the name),
although one wonders if the weighted problem may not be
buried in the early literature of physics inasmuch as the
weighted (or equal weights) situation is easily repre-
sented by a physical analog which Weber and several
others since have done (see [A 10] and [A 27]).
It should be noted first that the problem is not
easily amenable to solution by standard methods of elemen-
tary calculus, and except for the cases n < 4 the
results cited above are characterizations of the solution
not directions for educing it. Secondly, and sadly, one
has the feeling that articles concerning this problem
(1.1) will appear in the literature until the end of time,
each with scant reference to what has gone before. For
example, it stretches credulity beyond bounds to believe
that no one had proved a theorem on existence and unique-
ness prior to 1961 (see Palermo [A 27] and Tideman [A 30]).
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1-5
The first reference (1962) to an algorithm
for the solution of (1.1) is found in Kuhn and Kuenne
[A 22] . A second algorithm (1963) is that of Cooper
[A 5] which we discuss in detail.
Having proved the existence of a minimum (but
not uniqueness), i.e.
> 0, > 0
8x2 3y2
and the Lagrange condition
92P 32F ,32P N2 ^ n
— - v 1 > u ,
Cooper introduces the following recursive scheme:
Let D£ - [(xk - a±)2 + (f- bj^)2]1/2 where k denotes
steps of the recursion process.
Then differentiating (1.1) and solving for x ,
y we have
-------
1-6
with
k+1
X =
y =
colinear
solution
.2)
The iteration is maintained until
k+1
- xj < e and |y - y
No proof of convergence is given. Since the
solution is unique (in the absence of colinearity) one
need worry only whether {| x j } -»• °° or whether it might
oscillate. Cooper has had no such ill fortune in hundreds
of uses of this algorithm reported here and in a
subsequent article [A 6] .
-------
3-7
Finally, Bellman (1965) reduced the problem to a
dynamic programming problem [A 2] .
As a last Interesting note on (1.1), it should be
noted that it may be converted into an unweighted problem
simply by replication of fixed points, i.e.
Let M > 0 be 3 Mw. is an integer Vi and
let N = Z Mw, , then
Min P(x, y) = Min G(x, y)
where (1.3)
G(x, y) = 2^ C(x - a±)2 + (y - b±)2]1/2
1=1
is equivalent to the original problem.
We turn now to the thither side of the hyphen,
allocation. To understand this we shall first cite sev-
eral examples of location-allocation problems and then
adopt some terminology which will be used consistently
throughout Part I.
Location-allocation problems arise in every social
and economic context where an activity is to be operated
-------
1-8
on such a universal scale that it is neither possible nor
desirable to have the center of this activity at a single
location. The classic example in the area of location
economics is the choice of manufacturing sites to satisfy
the demands of a population distributed over a plane, i.e.
the problem of determining "market areas". Other examples
are the placement and division of network among telephone
exchanges, the location of schools and the setting of
school district boundaries, and from electrical engineer-
ing, the placement of electronic modules in the backplane
of a computer to minimize wire length. Our present
example is, of course, the determination of refuse dis-
posal sites and the allocation of collection territories
thereto.
To keep in step with this purpose we adopt the
following terminology: we shall call the fixed points the
"sources" (of refuse) and the minimal points "sinks".
(This is the reverse of Cooper's [A 5], [A 6] terminology
as his work is based on a distributive rather than an
ingathering process.) We shall, furthermore, coin the
term "sinkmates" or "sinkmate groups™ when we are speaking
of sources allocated to a given sink. Thus, using this
terminology,we are able to say that location-allocation
problems are concerned simultaneously with the optimal
-------
1-9
location of sinks and the optimal determination of sink-
mate groups. A more exact mathematical characterization
is deferred to Chapter i.
The earliest work in this area was that of
L. Kantorovitch, the distinguished Russian mathemati-
cian, in 19^2, but not available in English translation
until 1958 [A 14] . This- is a highly abstract mathemati-
cal discussion, the full complexity of which, is beyond
our present scope to discuss. We shall, however, approach
it by citing an example with an infinite number o_f sources
and an infinite number of sinks whose solution is charac-
terized by this paper. The characterization covers also
the finite source - finite sink problems with which we
shall be concerned.
Kantorovitchfs example is as follows:
Suppose we have the topographical or relief map of a cer-
tain region R before and after a cut and fill operation,
i.e.
z = fv(x, y) is the relief before
z = fQ(x, y) is the relief after .
a.
-------
1-10
Assuming conservation of volume
'(x, y) dx dy = // f0(x, y) dx dy ,
— v JJ
R R
and given a continous function C(x, y, x", y') > 0
which is the cost of moving a unit volume from (x, y)
to (x", y') , find a translocation function
^ (x, y, x% y') which tells how much earth to move from
(x, y) to (x', y') which minimizes the total earth
moving expense.
The gist of Kantorovitch's characterization is
based on the following definition: A translocation is
said to be potential if there exists a function U(x, y)
such that
|U(x, y) - U(x', y')| < C(x, y, x', y')
and
U(x, y) - U(x', y') = C(xa y, x', y')
when
(x, y) •* (x-, y').
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1-11
Kantorovltch then proves that a translocatlon is minimal
if and only if it is potential.
The identical result was rediscovered and pub-
lished as an abstract in 1962 by Mcllroy [A 23] • The
underlying assumptions and method of proof- are
nowhere nearly as general as those of Kantorovitch, but
to be fair, Mcllroy does mention having produced a
solution algorithm.
Kantorovitch1s work also anticipates the potential
characterization of W. Prager [A 28] writing before
Kantorvitch's work was translated.
In isard's well known Location and Space-Economy
[A 18] (also reviewed in [A 5] )especially Chapters 7 and
11, he considers and formulates problems concerned with
many sources (our sinks) and destinations (our sources),
but actually confines his attention to marketing areas
each served by its own source. This is, as we shall see,
the equivalent of several location problems of the Weber
type discussed above and ducks the issue of simultaneous
determination of both location and allocation.
For other general and specialized works in loca-
tion theory and application see Marshak [A 15] and the
comprehensive bibliography [D 2] .
— As understood from attending the symposium where the
paper was given.
-------
1-12
Cooper's 1963 paper entitled Location Allocation
Problems (op. clt.) tackles the problem of simultaneous
determination of location and allocation by letting the
source locations themselves represent a set of eligible
sink locations. This yields (^) = m!(n- m)! solu~
tions to be evaluated as follows:
1) Assign each of the n - m non-sink sources
to the nearest sink.
2) Sum the distances.
When this has been done the smallest such allocation, or
some set of the smallest such allocation can be refined
by the "exact location method" of (1.2) . We note as
does Cooper that this procedure does not guarantee opti-
mality and although time consuming (since (n) can be a
very large number) is preferable to complete enumeration
, k=m ,
of sinkmate groups since S(n, m) = ^~ I (, )(-!) (m-k)n
m' k=o k
is much larger.
To this we add our notes that this method of pro-
cedure renders the problem a selection-allocation problem,
i.e.,the sinks are to be chosen from amongst a set of
known points. Hence the methods of Chapter 3 are
applicable as will be shown. Second,we note that allow-
ing sinks to be located at sources will render some of the
D's equal to zero. In a personal communication Dr.
-------
1-13
Cooper has explained that terms for which D. = 0 are
to be dropped from the summations of (1.2) .
In a subsequent article, Cooper [A 6] provided
theoretical bounds on the value of the objective function
F through the following theorems:
Theorem 1_ (Cooper) : For the n source, one
sink location (-allocation) problem
k — y» 0 mtlr H
11 A/ jr\. ~_L.
I T-*
d, „ < Min F
n - 1
k=2 £=1
where
2Tl/2
C(ak - a,)2 + (bk - .,,)«]
Theorem 2_ (Cooper) : For the n source, m
sink location-allocation problem if there are at least
three sources in each sinkmate group and e, < e? < e < •••
- em n are the f
being ranked then
- em n are the factors dk£ ^see tneorem 1) after
p=l
-------
Theorem 3. (Cooper) :
i=n
Y^ C(a± - x)2 + (b± - y)2]1/2 > Min P
1=1
where
l=n l=n
±
x = / . a^/n. v =7 . b4/n .
1=1 1=1
Further, he described and compared the following heuristic
algorithms:
A. Destination Subset Algorithms: This is the
algorithm of the first article.
B. Random Destination Algorithm: This is
algorithm A except that the sink subsets are generated
randomly on the supposition that luck may get us to or
near the optimum sooner than total enumeration of the
(JJ) possibilities.
C. Successive Approximation Algorithm: In this
algorithm one solves first the two sink problem by
algorithm A. We next place a third sink at each one of
the other sources choosing thus the best one. This is
-------
1-15
repeated for a fourth sink, etc., until the requisite
number of sinks m Is reached.
D. Alternate Location and Allocation Algorithm:
Starting with an initial allocation, the "exact location
method" of (1.2) is used to locate the sinks. Then each
source is examined to see if it is nearer any other sink
than its own. If not, we are done. If the allocation
changes, the process is repeated.
Cooper notes that if D, R, A are the sets of
solutions generated by algorithms A, B, and D respec-
tively and if s is the single solution generated by
algorithm C then R c D , s e D but D n A = 0 ,
R H A = 0 and s ¥ A.
When algorithms A and B are supplemented by
refining the best solutions by the "exact location method
of (1.2) it is found that algorithm A is most accurate
but by far slowest. Algorithms B and D are tied for
accuracy with D, three to four times as fast.
In support of the notion that economically a very
good solution may be as good as the optimal solution,
Cooper adduces evidence to the relative insensitivity of
F to large changes in the sink locations with correct
or nearly correct allocations, i.e.,location-allocation
problems are typified by having many solutions close to
the optimum.
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1-16
Selection-allocation problems have been touched
on briefly above and in section C of the Preface. The
nearest approach to a problem of this sort in the litera-
ture is in the work of Hakimi [A 15], Prank [A 12], and
Maranzana [A 24] . Actually, Hakimi and Prank's work
stands between the area selection problems discussed
above and selection-allocation problems where the eligible
sinks are a set of known points. In these works, the
minimum point must be a point lying on a network of
paths connecting weighted vertices .
For Hakimi, the weights are known while for Prank
the weights are probablistic, but in neither work is "the
problem of simultaneous sink selection and allocation
considered. Maranzana [op. cit . ] poses a problem which
is actually a selection-allocation problem. Define P.
as a center of gravity (C.G.) of a set of weighted
nodes Q by :
D(pr V ± wk D(pi> V
where D(P., Pk) is the minimal distance on network
paths from P. to P,
J
-------
1-17
The algorithm proceeds essentially as Cooper's algorithm
D with the C.G. (1.4) replacing the "exact location
method" (1.2). The method is monotonic but can be shown
to converge occasionally to a local optimum which is not
a global optimum (see Chapter 4).
For other works of related interest see
Efroymson and Ray [A 8], Kuhn and Kuenne [op. cit . ] ,
Armour and Buff a [A 1], Garett and Plyter [A 14], Kuehn
and Hamburger [A 21], Baumol and Wolfe [A 3] , Koopmans
and Beckmann [A 20], Eilon and Deziel [A 9], Hillier
[A 17] , and Drysdale and Sandiford [A 7]. A elementary,
tutorial approach to several kinds of location problems is
provided by the second chapter of Noble's recent book
[A 26].
A selection of the electrical engineering litera-
ture pertaining to the backboard wiring problems is in-
cluded in the bibliography for Part I (section B).
The approach taken in this dissertation to location-
allocation problems is two- fold. The Euclidean metric
(distance function) (x - a±)2 + (y - b±)2 is replaced
by the J^-metric variously termed rectangular distance
or the "metropolitan" metric wherein the distance from
(x, y) to (ai, b±) is given by |x - a^, | + |y - b^J .
This is a valid approach in its own right as well as an
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1-1B
excellent approximation to the Euclidean distance problems
of Cooper. This altered objective allows the solution of
the pure location problems to be stated in closed form
rather than as a recursive algorithm and permits a par-
ticularly efficient version of Cooper's algorithm D to
be constructed. It also permits the resultant location-
allocation problem to be expressed as a linear program
in integer variables.
The first use of the a,-metric in such problems
is due to Prager [A 28]. The median approach was first
recognized by Charnes, Quon and Wersan [A 4] , [A 34]
and subsequently rediscovered by Francis [A 11]. See
also Hilliar [17].
The approach presented here to selection-
location problems is felt to be unique and without
precedent in the literature.
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CHAPTER 2
LOCATION-ALLOCATION MODELS
Section A
Mathematical Definition and Assumptions
Location-allocation problems in their most general
form are typified by the following statement:
Given; (i) The location in Cartesian co-ordinates of
n points called sources. Note that the
dimensionality of the space containing
these points need not be specified although
we assume two dimensions hereafter.
(2) The amount of a homogeneous commodity of
flow present at each source.
(3) A number m < n of points called sinks.
The determination of m is touched upon
by Cooper [A 53. This is a matter of
economic trade-off analysis between the
saving in transportation costs resulting
from many sinks and the capital and operat-
ing expenses incurred in obtaining them.
See Chapter 4.
1-19
-------
1-20
(4) Either
(1) A set of n functions each giving the
cost of moving a unit of the commodity
from the i-th source to a point
(x, y) .
or
(ii) A set of m functions each giving the
cost of moving a unit of the commodity
from an arbitrary point x, y to the
j-th sink.
The distinction between these alternate
schemes is that the latter is the more
natural formulation if the cost depends on
the total amount received at the J-th sink.
(5) A set of capacity constraints which limit
the amount of commodity each sink can or
may receive. Likewise, there may be floors
under these amounts.
To Determine:
(1) The location in Cartesian co-ordinates of
the m sinks (location variables).
(2) The amount sent from the i-th source to the
j-th sink, and thereby, the total amount
received at each sink (allocation variables)
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1-21
Such that; The aggregate weighted transportation cost is
minimized.
It is obvious that very little can be done with
so general a formulation until some specification and
simplification is done. The first such choice we make
is to assume
(Al) Unit transportation cost shall be independent
of the amount received at the j-th sink.
This is a reasonable assumption in the con-
text of refuse disposal sites. This may not
be so for the processing costs thereat which
are not considered in this chapter.
In terms of the above,
Let:
(a., b.) be the location of the sources
VJL > 0 be the commodity present at
the i-th source
i * 1,2,...,n
x., y.) be the locations of the sinks
L., U. be the lower and upper capacity
bounds at the sinks
= 1,2,. . . ,m
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1-22
be the allocation variables
L, b±, x., y.) be the cost
functions
i = 1,2}...,n
0 V(±. .
(2.1)
Theorem 1^: In the absence of capacity constraints,
there is an optimal solution of (2.1) obtained by sending
all of the commodity V at a source to the costwise
nearest sink.
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1-23
Proof: Let the optimal solution of (2.1) without
the capacity constraints be given by
J-l
V ' {(X' y
* * *.
and let ty±* = ^>(a±t b±, x., y,) . Then
« It
r VU *i;
r j i j
Let i|/lk = Min ty^. , then
J
* v-^ . * *
- L (vik^i
i
E* v^ ^c~™* * . *
vi *ik *
i
* # «
but v. . > 0 V v and ipi. - \J»lk > 0 by definition
-------
hence
*
p
i.e. one does no worse by sending all of V. to the k-th
sink which is by definition the costwise nearest.
On the other hand if there are capacity con-
straints it may happen:that the quantity
V± is either > U£ or <
in which case the capacity constraints are not obeyed,
Thus, we may let a.. = v../V. and assuming
(A2) There are no capacity constraints at the
sinks.
-------
1-25
We rewrite (2.1) as
i=n J=ro
Minimize / ^ ^ a. . i/>. .
1=1 J=l
(2.2)
=m
S.T. *"
a. . = 0 or 1
Alternatively, we may take the same course that
was taken in converting (1.1) to (1.3)» i.e.
Let M > 0 be such that T. = MV. is an Integer
and define N * 2 T. . Then we may replicate each
1 1
source an integral number of times T. and assume
that the commodity of flow comes in indivisible
units, say truckloads, i.e.
i-N
1=1 J=l
(2.3)
j=m
J=l
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1-26
a = l if the i-th source sends its
J truckload to the J-th sink
= 0 otherwise.
In this scheme, then, one may if one desires
reintroduce capacity constraints without feeling that one
is trampling on a theorem. We take this latter course,
but we shall hold off on capacity constraints until
Section C below.
It remains only to say something more specific
about the functions ^(a. , b. , x y ) . Let us then ask
what common sense dictates for the properties of these
functions.
First, in any realistic situation we would expect
that ^ . . > 0 . (No subsidized routes.)
^- J ~*
Second, although there are economic situations in
which the costs are not the same in both directions, we
are concerned with the movement of refuse in the direction
of the disposal sites hence we accept the notion that
V(a±, b±, Xj, yj) = *(Xj, y ^ , a±, b±) as a useful
simplification.
Third, we would expect that when any sink moves
close to a source that ij>. . -*• 0 .
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1-27
Lastly, given a trip origin and destination,
common sense tells us that it should cost more to make
the trip by way of a detour off the direct path than by
way of the direct path.
There is a class of functions having these sensi-
ble properties and these are the metrics or distance
measuring formulae. One such metric is the Euclidean
metric which we have seen in Chapter 1:
dE(r, r') =[(x - x')2 + (y - y')2]1/2
where (2.4A)
r = (x, y) , r' = (x', y')
Thus, we shall assume
(A3) The cost is proportional to distance, but
the metric we shall use shall be the L , or
rectangular distance metric
(r, r') - |x - x'| + |y - y'| (2.4B)
The name *, comes from the name * -metrics which are a
1 P
family of metrics
-------
1-28
d^ (r, r-) = [|x - x'|p + |y - y'|p]1/p
P
(2.5)
p > 0 is an integer
of which the Euclidean metric can be recognized as the
second member.
There are several reasons that Justify the choice
of the I,-metric. The direct paths of the fc,-metric
are in general L-shaped and conform to the experience of
pedestrians and traffic in most North American cities;
the streets are laid out in a rectangular grid and to go
from hither to thither one must follow an L-shaped path
up the avenues and down the streets.
Even in cities and metropolitan areas where there
are many diagonal streets (e.g. Washington, D.C.,
Indianapolis, Indiana) it is precisely these streets which
one desires to keep free of truck traffic since they are
generally crowded with other kinds of traffic.
There are, moreover, many cities which having
irregular street patterns are nevertheless polarized as
regards speed of travel. Thus, for example, north-
south traffic moves at a faster pace in Manhattan Island
than does east-west traffic. In such situations the
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1-29
problem can be posed in time proportional co-ordinates.
Finally, as we shall see, we can often obtain
the correct E-metric allocation for Cooper's problems
by using the i,-metric, i.e. the £,-metric provides a
useful approximation to the solution of E-metric problems.
Summing up these we may state our problem thus:
Given points called sources located at (a., b.)
1=1, 2, ...,n. These sources are the points
where refuse loads will depart for disposal
sites. There are T. trucks at each source. If
we are concerned with capacity constraints at the
sinks we define N = Z T. ; otherwise N = n .
We must determine the locations (x., y.)
J J
J = 1, 2, ..., m and the allocation of sources
to sinks on an indivisible basis to
i=N j=m
Minimize ^ Jj °ij (' xj ~ aj + lvj ~ bil>
(2.6)
S.T. a = 1 and a = 0 or 1 .
-------
Section B
Solution of a One Sink Problem
We begin our investigation of the solution of
(2.6) by considering the case n = 1 . Under this
assumption, the allocation problem disappears and we
have
1-N
Min ^ (|x - a±| + |y - b±|) (2.7)
1=1
We note that there are no constraints that relate
x and y hence (2.7) can be written as two separate
minimization problems
i=N
Min ^ |x - a1|
1=1
and (2.8)
i=N
Min y - b±
1=1
We concentrate, therefore, on only one of these.
1-30
-------
1-31
The solution of (2.8) is well known, the answer
is given by
Theorem 2_: x = Median {a.}
i.e. for N odd , x = a>(N+1)/2
(2.9)
for N even , a.j^/2 < x < ^,
AN ^ £i ' J.
where a^" < a« < ••• < a' and {&'} = {a.} .
Some authors define the median in terms of the
minimization property of (2.8), but we shall for the sake
of completeness prove the result here.
Proof:
i=N i=k i=N
f(x) = V^ |x - a^| = kx - 2^ ai - (N - k)x + T^ sf±
1=1 i=l i=k+l
i=k i=N
= (2k - N)x - ^ sC± + V^ aT±
1=1 i=k+l
where a^ < a' < • • • < a' < x < a,'., < • • • < a,"T .
-------
1-32
Thus, f(x) is a linear function of x within
each closed interval Caj,» av+l^ °^ s^°Pe 2k - N and
is continuous for all x .
N
Hence, for k < •=• , the slope is negative and
for k > 73- , the slope is non-negative.
The minimum of f(x) occurs, therefore, at
x = a''/:T.nx/ for N odd and at any point in,
(N+D/2
for N even (slope is zero in this
closed interval).
Thus, the pure location problem (2.7) which is
the £,-equivalent of the pure location problem (1.3)
has the closed-form solution:
x = Median {ai> , y = Median {b1> (2.10)
and for the multi-sink problems it follows that
Given an allocation, the optimal locations for
that allocation is gotten by using (2.10) m
times where each median is to be taken over the
appropriate group of sinkmates.
-------
1-33
For brevity we shall refer to this as the Median
Principle in the sequel.
-------
Section C
Area Exclusions and the One Sink Problem
It may happen that the solution gifren by (2.10)
allows the minimal point (x, y) to be located in an
area which other considerations exclude from eligibility.
This means that a constraint must be added to (2.7)
expressing the area exclusion. We illustrate with an
example (Figure 1-1).The median solution to the six-
source, single-sink location problem illustrated is any
point on the heavy line AB . Suppose, however, that
we are now told that no sinks are allowed to the right of
the oblique line PQ . Our intuition suggests that the
best location for the sink under this area exclusion is
that point in permitted territory which is £n-closest to
v J-
the line AB . This is the point C which lies on the
^-circle (CDEP) of smallest radius (AC) with center
(A) on the line AB .
Since intuition is a better source of inspiration
than truth, we need a constructive way of solving the
problem with area exclusion. To that end, we first state
the problem and then convert it to a form amenable to
standard methods of solution. To add some generality, we
may imagine that there are K straight lines partici-
-------
1-35
' y
cli—* - ^
B
Figure 1-1
Median Solution and an
Area Exclusion
-------
1-36
pating in the area exclusion so that the point (x, y)
must
1=N
Minimize Y^ (| x - a11 + |y - b±|)
1=1
S.T. ckx--+ dky < ek (2.11)
k = 1, 2, ..., K
and x, y > 0
The addition of the non-negativity constraints
x > 0 , y > 0 will be explained shortly. We now Intro-
duce the concept of the positive and negative parts of a
function, which is
Suppose we have a function f(x) which may
assume any real value (-00 < f(x) < °°) . Then
the positive and negative parts of f(x) are
f (x) and f~(x) respectively and are defined
as
-------
1-37
"+<*> • n
f"(x)
as f(x)
as f(x)
(2.12)
Prom this we may note that
a)
b)
f+(x) - f~(x) = f(x)
f+(x) + f"(x)
(2.13)
Using this concept we define variables (u., u~) and
(v., v~) as the positive and negative parts respectively
of the functions x - a.^ and y - b. respectively.
Thus, (2.11) becomes
Minimize
u
S.T. u. - u. = x - a.
r± - v± = y - b±
ckx
^ ek
1 = 1, 2, ..., N
k = 1, 2, . . ., K
(2.1*0
all variables > 0
-------
1-38
Thug we have removed the absolute value brackets
from the problem and replaced (2.11) by the minimization
of a linear function subject to linear constraints, i.e.
(2.14) is a linear programming problem and as such can be
solved by the simplex algorithm (see [C 6] for example).
This algorithm produces non-negative answers only and
this is why we added the constraints x, y > 0 to (2.11)
This is a situation which can alwayi
be arranged by
shifting the origin sufficiently fair to the southwest.
There is another case of area exclusion of inter-
est to planners. Suppose there is a simply connected
(i.e.,all in one piece) zone called the central business
district (CBD) whose boundaries are parallel to the axes,
and we are forbidden to place a disposal site therein.
(See Figure 1-2) . Then the area exclusion constraints take
the form
(x < A,) or (A~ < x) or (A, < x < A0 and y < Bn)
— _L —— c. _ -L — — c. — _L
or (A, < x < A0 and y > B0) (2.15)
X ~ ~~ C. —• c.
This is an "exclusive-or" not a "logical-or" so
that there is no territory, for example, that slmultane-
-------
1-39
SINK AT A->C
Figure 1-2 Area Exclusion of a
Central Business District
-------
1-40
ously obeys x < A, and x > Ap. This area exclusion
literally puts a hole in the solution space and renders
the problem "non-convex". This means that the set of
eligible points (those outside the CBD) are not a con-
vex set in that every straight line connecting two points
in the set is not wholly inside the set; line UV is an
example. This is a property that must be possessed by
the sets of eligible points (solution space) In order to
be solvable by linear programming.
This non-convexity can be repaired by resorting
to Indicator variables, i.e. variables taking the values
0 or 1 only. We have met this type of variable in
(2.6) in the °-jils which expressed the indivisibility
of a truckload.
Thus, let M be a very large number (say
100 Max{|Ap |, | Bp|}) and denote the four disjoint areas
described In (2.15) as RI} Rp, R_, R. respectively.
Then we define
Xk = 1 if (x, y) e Rk\
>k = 1, 2, 3, 4 (2.16)
- 0 otherwise I
-------
Then if we define the variables x^, yk, k= 1, 2, 3, 4
by
X ~ X -i "» X ^ • X^ • ^)i
(2.17)
ysv +v +v 4-v^
«/l Jo "/D O'W
the following set of constraints purchases for us the
exclusion of the CBD
a)
b) A2 X2 < x2 < M\2 (R2)
c) A.,^ \3 < x3 < A2 X (R3) (2.18)
0 < y3 < B-j^ X3
d) A^^ X^ < x^ < A2 X^ (R^)
B2 xn < y^ < MX^
e) X, + X2 + X_ + Xjj = 1 (Disjointness)
-------
Summing up we write the CBD model as
i=N
/ + . - . +
Min
1=1
.^ - u. = x, + x2 + x-. +
V — V~ =V +V +V + Vi — b
Vj Vj y -, T _y _ -i y _ T ,y i. u.
•1=1, 2, ..., N (2.19A)
(2.18 a - e)
All variables > 0
XR = 0 or 1 only , k = 1, 2, 3, ^
This is a mixed-integer programming problem, i.e.
it is a linear program some of whose variables may take
arbitrary non-negative values while others may take only
non-negative integer values, here further specialized to
the pair 0 or 1 . A separate listing of appropriate
references to works in the area of integer linear pro-
gramming is included in the bibliography for Part I
(Section C).
-------
Section D
Multiple Sink Problems as_ Integer
Programming Problems
Once we advance to two or more sinks to be deter-
mined simultaneously, the allocation variables
a*4( = 0 or 1) re-enter the problem. We have seen that
given an allocation our problem becomes m (the number
of sinks) separate location problems which can be solved
by the methods of sections B and C . On the other
hand, given the locations of the sinks, the allocation is
clear: each source to the nearest sink. The complexity
of location-allocation models is in the required simul-
taneity of determination.
Applying the principles of (2.12) and (2.13) to
(2.6) we have
i-N J=m
Minimize
1=1 J-l
+
- a
i • 1, 2, ..., N
- 1, 2, ..., m
1 i = 1, 2, ..., N
all variables >0,a..=0,1.
*"
-------
1-46
This is a problem with linear constraints (except
for a.. = 0, 1) and a non-linear objective function.
•^ J
We show now how to linearize the objective function.
First, we return to absolute value notation, and
defining M to be a very large number, say
M > Max |a.| + Max |b.| , we introduce auxiliary
~ i 1 i 1
variables t.. as shown:
i=N J=m
1=1 J=l
J=m
S.T.
±J
(i)
0 < t±J < M(l - a±J) (ii) (2.21)
0 < t±j < |xj| + |yj| E x. +Yj (iii)
all variables non-negative.
The identity in the last constraint is based on
the assumption that the non-negativity of all the x. ,
J
y. has been assured by a suitable placement of the origin
-------
1-4?
Note that a = 1 =» t± . =0 and x. + y^ > 0
(redundant) and the corresponding term of the objective
function becomes |x, - a^ + |y, - b.jj ; whereas
ai1 = ° "* *11 - M (redundant) and tji 5 x-» + y-t » but
since x. + y. « M and since the process of minlmiza-
J J
tion will force t^, in the absence of any other
constraints to be as large as possible, we have t
..
i j
x. + y. and the corresponding term of the objective
J J
function vanishes. Thus at the points a . . = 0 or 1
^- J
the objective function behaves like that of (2.6).
This process should more accurately be called
a "quasilinearization" since it rests on the non-linear
requirement a. . = 0 or 1 only. Thus, it is important
to note that whereas there is a natural interpretation in
(2.6) for 0 < c^ < 1 , such values of a. in (2.21) may
produce ludicrous results. For example, the choice
ot^ = 1/m Vl Vj reduces to the unconstrained minimiza-
tion of
i=N J=N 0 b
(
*i
XJ -m-
i
m
i=l .1=]
-------
1-48
which bears no relation to the physical situation, i.e. ,
|x, - a./mf fights for a central value for x. while
-x. fights for as large a positive value as possible.
Returning to (2.21) we may remove absolute value
brackets in the same manner as before to get
i=N j=m
Min
i=l J-l
(2.21 (1) - (ill))
and (2.22)
- bi
with all variables non-negative.
This is a mixed-integer problem with 4mN + N
constraints in 6mN + 2m variables ignoring the slack
variables required to render the inequalities equations
This means prodigious problem sizes for even moderate
values of m and N (e.g. m = 3 , N = 50 gives an
array 650 x 906) . There are, however, reasons for
-------
believing that the use of integer programming techniques
may be feasible for solving large location-allocation
problems. These reasons are:
(1) Sparsity and regularity of structure. The
density of the structure part of the array
of (2.22) is 7/((ton + 1)(3N + D) x 100$
sparse. The structure is quite regular and
the columns or rows could be generated by
algorithm one at a time as needed. See
Figure 3.
(2) Great recent advances in methods for solving
mixed-integer and integer programs (see
bibliography).
(3) The solution process may be accelerated (and
often be simply a confirmation of optimality)
by use of procedures such as those to be
discussed in section P for finding good
local optima to be used as advanced starts
in the mixed-integer programming formulation.
-------
1-50
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-------
Section E
A Theory o_f Median Sets
In this section, we are going to develop more
fully the ideas begun in section B concerning the median
nature of the solution of pure ^-location problems,
i.e. the Median Principle. According to this principle,
when an allocation is given, the £,-metric location-
allocation problem becomes m separate location problems
whereof the solutions are the medians of the m sinkmate
groups.
In Section P below we are going to develop an
algorithm which is like Cooper's Alternate Location and
Allocation Algorithm (see page 1-15), i.e. starting with
an initial allocation we find points which were better
placed in different sinkmate groups, perform the exchange
and resolve the location problems. Accordingly, we must
know what happens to the location of the median of a sink-
mate group when a source is ^anDend d^ ' To ^hat end» we
sharpen our definition of "median" and develop the
appropriate theorems.
Definition: Let ^a^K > ^bi^1 be the abscissae
and ordinates respectively of the sources allocated to the
1-51
-------
1-52
J-th sink. Let
A. = Median fa.}.
and
B, = Median
according to the definition of median given by (2.9).
Note that A. and B. are thus either points or closed
J J
intervals independently of one another. By the median
set M. of the sources allocated to the j-th sink we
J
shall mean
(2.23)
which is short-hand for
M. = set of all points (x., y.) such that
x, is taken from the point set A. and
y. is taken from the point set B. .
-------
1-53
Thus M, is either a point, a closed interval parallel
j
to an axis, or a rectangle with sides parallel to the
axes.
Theorem 3: Suppose a source is
a set of k{~~} sources. Let the median sets before and
>2
Jc Mk+1
after be denoted by M^ and { . ,} respectively. Then
M*'1
the intersection { . . , } is not empty and the point in
^"1
V< further L»> ^ 'Sieved^
this intersection.
Proof: The proof will be given only for the
adjoining part of the theorem.
Let Mk = Ak x Bk
Mk+1 = Ak+1 x Bk+1
where A , B are the one-dimensional median sets of the
definition (2.23). Further, let the co-ordinates of
the adjoined point be (a', b') and assume
al - a2 - '" - ak ; bl - b2 - "* - bk '
-------
Case 1: k is odd =» Ak = a, .,
KT!
(i) a' < a^ . Ak+1
2
(ii) ak-1 < a' < ak+1 , Ak+1 . [a' ,
2 2 ~2
(ill) ak< a^ < a- Ak+1 - [ak+1 , a'l
(iv)
2
Thus, A H A = ak+, and the closeness
~2~
condition holds in the x-direction.
Case. 2: k is even =» Ak = [ak/2 , ak/2+1D
(i) *' < ak/2 .A = a* Akn
(ii) ak/2 <
-Ak+1 = a'. Ak n Ak+1 = a'
-------
1-55
Vr-fl
(111) a < a' -A* = ak/2+1
.k n .k+1
A ° A = ak/2+l '
Thus A H A is not empty and the closeness
condition holds in the x-direction.
Similarly, Bk n Bk+1 is not empty and the close-
ness condition holds in the y-direction (k even or
odd).
Now (Ak 0 Ak+1) x (Bk 0 Bk+1) = (Ak x Bk)
H (Ak+1 x B) . (This is a self-evident proposition
from set theory which will not be proved here.) But
since the Cartesian product of two non-empty sets cannot
be empty we have
H jj t 0 .
Moreover, the distance from the adjoined point
(a', b') to Mk H M*01"1 is the sum of distances from
a' to Ak n A1^1 and from b' to Bk n Bk+1 . These
component distances have been shown to have the closeness
property, hence their sum has it likewise.
We proceed now to the main prop of our algorithm.
To facilitate its presentation we adopt the following
-------
1-56
notation:
p. : a source located at (a., b.) .
I. : {i|a.. = 1) i.e. the j-th sinkmate group,
M, : Median Set of {p±|i e I,> .
S. : A sink located at (x., y.) € M.
J J J J
d(Pl, Sj) = |xj - a±| + |yj -bj .
In terms of this notation, the Median Principle
stated above becomes
Min d(P> x) " d(p' S} (2.24)
X
-------
1-57
Max d(p., S.)
s i J
d, , « Mln d(p, , S.) .
— 1J q J- J
(2.25)
Then a necessary and sufficient condition that
the solution [{I. , S.}] be a local optimum obtained
J J
by the exchange of_ single sources is that for all pairs
{(r, q)| r t q , 1 < r , q < m , Mp 0 M = 0}
dir - -iq f0r each
That is, the value of the objective function can decrease
if and only if the largest distance from a source p. to
its own median set (M ) exceeds the smallest distance
to a point in any other median set (M ) .
Proof; Assume the condition of the theorem holds,
but the solution is not a local optimum obtainable in the
manner indicated. Then there exists a source p, ,
h e I such that if p, is allocated to some other sink
S e M , the value of the function will decrease.
4 T.
-------
1-58
Let i; = Ir - h = Ir 0 C(h) (complimentation)
and let Shr e Mr be such that d(p, S) =
h, hr
be such that
Further, let F , F' be the values of the objec
tive function before and after the re-allocation of p
to Sq . Then
i>V + 2- d(pi>
w
r
iel
q
16
by assumption
-------
1-59
The question now is whether on shifting sinks from
Mr and S^ e Mq to s; * M/ and S^ « Mq
respectively, we will by the Median Principle reduce F'
below F . The answer is no, because by theorem 3
S~hr e Mp 0 Mr and S e M N M' . Hence, F is opti-
mal (locally) and the condition is sufficient.
On the other hand, if the condition does not hold
for some source p. , h e I , then p, may by theorem 3
be re-allocated to S. for a net decrease in F of
—hq
d. - d, . Hence, the original value of F was not
optimal (locally) and the condition is necessary. Q.E.D.
An auxiliary device for reducing the value of F
is provided by the following result.
Theorem 5.: If £, -paths connecting a pair of dis-
tinct sources to their respective, distinct sinks are
unavoidably crossed then either
(a) The solution in which this occurs is not
optimal, or
(b) If it is optimal, there exists an alternate,
uncrossed optimum.
Proof: Suppose for m > 2 that two sources p, ,
P2 are connected to a pair of sinks S, and S*
respectively and their paths are crossed. (See Figure 4a.)
-------
1-60
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-------
1-61
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-------
1-62
Let x denote the point of crossing. Then
either
d(x, S-j^) > d(x, S2) ,
d(x, S1) < d(x, S2) ,
or d(x, S..) = d(x, Sp) .
In the first case, re-allocate p, to S2 ,
improving the solution by at least d(x, S,) - d(x, S2) .
In the second case (Figure 4a), re-allocate P2 to S^
improving the solution by at least d(x, Sp) - d(x, S,) .
This re-allocation is illustrated in Figure 4b.
In both of these cases, an improvement being
possible, neither could have been optimal. In the third
case, the solution may or may not be optimal. If it was
not optimal, either of the above re-allocations can cause
at worst an alternate solution with the same value of F
which does not have a crossed path. If the solution was
optimal, either re-allocation causes an alternate, un-
crossed optimum.
It should be noted that many paths may be crossed
so that the process given above may have to be repeated.
-------
1-63
Moreover, the re-allocation may cause paths previously
uncrossed to become crossed. Therefore, the term
"uncrossed" as used in the preceding discussion should be
taken to refer only to the given crossing. It is clear,
however, that iterated use of the above procedure will
finally lead to a totally uncrossed solution, optimal
or not. Q.E.D.
In view of this theorem, it is reasonable to
require that any starting procedure for an algorithm
produce uncrossed starting solutions.
-------
Section F
An Approximating Algorithm Based
on the Theory of Median Sets
As stated above, we have been erecting a founda-
tion for a single point exchange algorithm. We may pause,
though, before stating this algorithm to consider the
possibility of multi-point exchanges.
When more than one point is removed from or
adjoined to a group, the possible movements of the altered
median set become quite complicated. For example,
Mk H ji^1 ^ 0 and Mk+1 H Mk+2 t 0 do not imply that
M fl M ? 0 . Hence we lose the prop of our theory
which depends on knowing where the median sets go.
Another approach to multi-point methods is had via
graphical means. Starting with some initial solution we
draw about each sink the smallest &,-circle (a diamond,
see Figure 1) that Just covers all its sources. The
inclusion of alien sources in such circles can thus make
profitable exchanges of several points .quite obvious,
visually. The difficulty with this method is that when
the median set is not a unique point, the arbitrary selec-
tion of a point as the center of the circle may be the
wrong choice. As a practical matter, the method has
1-64
-------
1-65
proved in several instances incapable of improving
verified local optima.
The single-point exchange algorithm consists of
the following steps:
(1) Make an initial allocation of sources to
sinkmate groups and determine their median
sets. We shall say more about this in Sec-
tion G below and in Chapter 1.3-D.
(2) Set r = 1 (sink number) and k = 0
(exchange counter).
(3) For each i e I compare d. vs d. ,
q t r .
(4) If all d~ir < d. go to step 7; otherwise
go to step 5«
(5) For some h e l^ , d"hr > d. : move S^ to
S. , S to S, and re-allocate h to
I . Increase k by 1 .
(6) Go to step 3.
(7) If r / m , increase r by 1 and return to
step 3-
If r = m and k / 0 , return to step 2.
If r = m and k = 0 , the algorithm ends
with a local optimum determined.
This version of the algorithm removes all "better
elsewhere" points from a sinkmate group before proceeding
-------
1-66
to the next group. Thus, it is not inconceivable that a
source may switch back and forth several times before
settling down.
Another approach to an algorithm would be to go
through all points and choose that exchange for which
d, - d. is largest. Furthermore, for m > 4 two or
more such exchanges might be sought. While such measures
may in general lead to fewer solution iterations, there is
no guarantee that the solution it reaches will have any
lower a value of P than the algorithm stated.
There are two additional comments apropos at this
point. First, the algorithm as presently constituted, has
nothing to say about ties, i.e., when some d, = d, and
no d, > d . Second, it is clear that any sinkmate
group which consists of a single point (a singleton) may
be augmented by transfer from other sinkmate groups, but
will never be put into another group. This means that an
initial solution should probably try to avoid singletons.
To finish this discussion of the algorithm, we
illustrate it by an example taken from Cooper [A 5], who
gives eight such examples each having n = 7 and m = 2 .
The example used here is the first of these. The results
on all eight examples are compared with Cooper's E-metric
results in Table 2.
-------
1-67
The source locations are shown in Table 1-1. We
take our initial guess at an allocation to be
= {1, 2, 3, 4}
I2 =
as illustrated in Figure 1-5. Figures 1-6 and 1-7 show
the balance of the solution with the computations
required appearing therein.
Further investigation shows that the solution of
Figure 1-7 is actually the global optimum. That the algo-
rithm can stall at non-global local optima is apparent
from Table 1-2. Despite the possibility, Table 1-2 shows
that the alternating location-allocation or single point
exchange algorithm is an excellent means for determining
an E-metric allocation. Moreover, we observe that
F0 > F., . This is a good upper bound on F^ but the
*, — Ei Ct
result cannot be proved, i.e. there may be problems where
Fjl < PE slnce tne optimal J^ and optimal E allocations
are not necessarily identical.
-------
1-68
•
1
1
2
3
4
5
6
7
ai
15
5
10
16
25
31
22
bj
15
10
27
8
14
23
29
Table 1-1 Source Locations for Example 1
-------
1-69
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-------
1-72
CASE 1
SOURCE COORDINATES:
ALLOCATIONS:
/, - ALTERNATE ALLOCATIONS:
VALUE OF OBJ FUNCTION:
CASE 2
SOURCE COORDINATES:
ALLOCATIONS:
/, - ALTERNATE ALLOCATIONS
VALUE OF OBJ FUNCTION:
CASE 3
SOURCE COORDINATES:
ALLOCATIONS:
/, - ALTERNATE ALLOCATIONS:
VALUE OF OBJ FUNCTION:
CASE 4^
SOURCE COORDINATES:
ALLOCATIONS:
/, - ALTERNATE ALLOCATIONS:
VALUE OF OBJ FUNCTION:
CASE 5
SOURCE COORDINATES:
ALLOCATIONS:
/, - ALTERNATE ALLOCATIONS:
VALUES OF OBJ FUNCTION:
CASE 6
SOURCE COORDINATES:
ALLOCATIONS:
/, - ALTERNATE ALLOCATIONS:
VALUE OF OBJ FUNCTION:
CASE 7^
SOURCE COORDINATES:
ALLOCATIONS:
/, - ALTERNATE ALLOCATIONS:
VALUE OF OBJ FUNCTION:
CASE 8
-------
Section G
Finding Initial Solutions
We return now to the problem of providing an
initial allocation for getting the algorithm started.
There is no doubt that the best method for small problems
is the use of the human eye and educated guessing. For
larger problems there are two methods that are relevant
both to the ^,-metric and the E-metric cases. The
more powerful of these methods is based on the minimum
elemination gain algorithm (MEGA) which is discussed in
the next chapter. Hence, we defer discussion of this
method until then.
We present here a method for finding an initial
solution which is based on the use of the single point
exchange algorithm on two simpler problems derived from
the original problem. These problems, called the
horizontal and the vertical problems, are one-dimensional
problems gotten by projecting the sources on the abscissa
and the ordinate respectively. If two points project to
the same abscissa, they retain their separate identities.
Thus, the horizontal problem corresponding to Example 1,
Figures 1-5 to 1-7, is shown In 1-8,and the vertical
problem in Figure 1*9.
1-73
-------
We note that these one-dimensional problems are
easily solved because
(a) The tests of the algorithm must be performed
on only two points as regards a pair of
sinks. These are the "boundary" points, e.g.
Pjj and p7 in Figure 1-8 and PI and pg
in Figure 1-9.
(b) For m > 3 , the test need only be performed
on neighboring sinkmate groups.
The optimal horizontal and vertical solutions are
illustrated in Figures 1-8 and 1-9.
To make use of these solutions for the two-
dimensional problem we construct an N x N table of
coincidences as follows :
If points p, and p^ appear in the same sink-
mate group in the {h°1} soiutlon> i-e-
are " sinkmates» enter a (} in the
k , i box of the table. The result of this
operation is shown in Table 1-3.
We reason now as follows: If a pair of points
have both an h and a v in their table entry then they
are a favored combination and should appear in the same
sinkmate group in the Initial solution; If there is
-------
1-75
1 I
f •
1
t
1
I4
•
7
5
_6
*
h-slnkmates
20 v 30
s • •• ^*« — '
h-sinkmates
40
Figure 1-8 Horizontal Problem Corresponding to
Example 1, Figures 1-5 to 1-7
5 1
6 37
v-sinkmates
20 30
v-sinkmates
40
Figure 1-9 Vertical Problem Corresponding to
Example 1, Figures 1-5 to 1-7
-------
1-76
neither an h nor a v we shall see that this pair do
not become sinkmates; If there is either an h alone
or a v alone we shall be arbitrary.
Accordingly, we may interpret Table 1-3 to favor
an initial allocation of {1, 2, ^} , {6, 7> with the
placement of p.~ and p^ yet to be decided. Using the
arbitrary selection,
I1 = {1, 2, 3, 4}
I2 =.(5, 6 7)
we have the initial solution of Figure 1-5- If we had
made the opposite choice I, = (1, 2, 4, 5) ,
IP = (3, 6, 7) , we would have had the optimal solution
of Figure 1-7.
The method presented here will not work if all the
sources p. are scattered about a horizontal or verti-
cal line. If this occurs the horizontal or vertical
solution (respectively) should be used As the initial
solution.
-------
1-77
?-* 2
I
hv
a
h
h
3
hv
hv
h
4
V
V
V
5
V
h
6
V
h
hv
Table 1-3 Coincidence Table for Horizontal and Vertical
Solutions of Figures 1-8 and 1-9
-------
CHAPTER 3
SELECTION-ALLOCATION MODELS
Section A
Mathematical Definition and Assumptions
In the Introduction (section C) and In Chapter 1
we have alluded briefly to the manner In which selection-
allocation problems arise from the location-allocation
models of Chapter 2. We shall discuss the origin, nature,
characterizations and methods of solving these problems
more fully in this chapter and the next.
It may be noted that section C of Chapter 2 which
dealt with area exclusions and the one sink problem was
not succeeded in due course by a section dealing with area
exclusions and multiple sink problems. It is clear from
section C and D of Chapter 2 that the model (2.22) can be
augmented by an appropriate set of area-excluding linear
constraints to achieve this end. Further reflection on
the real world indicates, however, that area exclusions for
aesthetic, political, zoning, or economic reasons are so
widespread that the area available for location o_f sinks
i§. likely t£ dwindle t_o a few parcels whose locations
are known. On the other hand, Cooper simplifies the
1-78
-------
1-79
solution of location-allocation problems by using the
sources themselves as a restricted solution space for the
sink locations. Finally, in setting up regional economic
plans for product distribution it frequently is desirable
to restrict the choice of distribution centers (our sinks)
to cities, excluding the countryside (e.g. , Maranzana
[A 2^*]).
For these reasons we are brought to consider
selection-allocation problems which are stated thusly:
Given; (1) A number n of points called sources, whose
locations are known, but are not germane
to the problem.
(2) The amount of a homogeneous commodity of
flow present at each source.
(3) A number S of points called "eligible sink
sites" whose locations are likewise known,
but are also not germane to the problem.
(4) A number 1 < m < S which is the number of
sinks to be selected from the set of eligible
sink sites.
(5) n-S numbers d^, representing the cost of
sending a unit of the commodity from the i-th
source to the J-th sink.
-------
1-80
(6) Capacity ceilings and floors (+°°, 0
respectively if need be) at each sink.
T£ Determine:
(1) The set of sinks to be selected from amongst
the set of eligible sink sites.
(2) The amount of commodity sent from each source
to each selected sink.
Such that: The total weighted transportation cost is
minimized within the capacity constraints (if
any) .
We may as we have done for location-allocation
problems show that in the absence of capacity ceilings (or
floors) there is an all-or-nothing solution at least as
good as any "continuous" solution (Theorem 1). Thus we
list our assumptions:
(1) Each source has an indivisible unit amount to
send. We achieve this, if necessary by using
the replication process indicated in Chapter
2.
(2) Capacity ceilings or floors do not apply.
(3) The numbers (not functions) d.. are given,
-*• J
are independent of receipts at a sink, may or
may not be proportional to distance, and if
they are ,no choice of metric is Implied.
-------
Section B
Characterization As_ a Network Flow Model
The easiest characterization of a selection-
allocation problem is perhaps as a network flow model
To understand this we define the following:
x,, = 1 if the i-th source sends its
1J unit to the J-th sink
= 0 otherwise
y. = the amount received at the
3 J-th sink.
a. =1 if the J-th sink is selected
J (accepted).
= 0 otherwise
p. » 1 if the J-th sink is rejected
* 0 otherwise
The model consists of two network systems. These
are shown }n Figures l-10a and l-10b. Figure l-10a repre-
sents the "physical" system. Shown on the left are the
s6urces. Each source is connected to every one of the eli-
gible sink sites which are represented as nodes in the
center. Each source receives a unit' inflow which is
divided up into flews x.. to the eligible sinks. The flow
1-81
-------
1-82
SOURCES
E.S. SITES
E.S. SITES
Figure l-10a
P-System, N»7, S=
(Not all- lines for
•x.., shown)
Figure l-10b
A-R-System, S=*J, |n=2
-------
1-83
received at each sink is collected as flows y, which are
J
sent to a figurative summing node (Z) on the right. The
outflow N from the I node exactly matches the inflow
to the sources.
The equations which represent this network are
obtained from the Kirchoff node conditions. These
equations are
1-N
Z
1=1
-L. —H
Z xu = yj (11) (3-2)
2J yj - N (in)
J-i
Equation (i) expresses the idea that the total
outflow from any source is equal to the total inflow which
is 1. Equation (ii) says that the receipts at a sink
equals the amount sent to the sink. Equation (ill) says
that the amount received at all sinks equals the amount
received at all sources.
-------
1-84
Note that this physical (P) system says nothing
about the notion of selecting some m out of S eligi-
ble sink sites and rejecting the other S - m . This
idea is expressed by the accept-reject (AR) system of
Figure lOb. The entities represented here are except for
the eligible sink sites contrivances that fit the idea of
acceptance-rejection and have no real physical meaning.
Two nodes, the accept (A) node and the reject
(R) node, receive quantities m and S - m of "accept"
and'reject"substance respectively. The accept node
divides its m units of accept-substance into S flows
01 going to the eligible sink sites. Similarly, the
reject node divides its S - m units of reject-substance
into S flows p. going to the eligible sink sites. At
the eligible sink sites the accept and reject flows are
received and siphoned out of the system, one unit of some-
thing escaping from each eligible sink site. The equations
of flow for the A - R system are
J^
£ «, - m (i)
J=l
.1=3
J^ Pj = S - m (11) (3.3)
J=l
a, + p, * 1 (ill)
-------
1-85
Thus far, there is nothing which links the two
systems, yet we realize that in the P-system a receipt
y, cannot be >0 if the corresponding eligible sink site
has been rejected (a. = 0, p. »
express this mathematically as
has been rejected (a. = 0, p. = 1) in the A-R system. We
0 < y1 < No. •
(3.4)
These constraints are termed coupling conditions and are
the only constraints containing variables from both the
P and A-R systems. The total model is then
i-N J-S
Minimize
.1=3
S.1
1 - 1, 2, .... N
J-l
1-N
x.
- 0 J = 1, 2, ,.., S P (3.5)
1=1
J-l
-------
m
J=s
= S — m
J=i
1-86
A.R.
ot + P = 1 J = 1, 2, .. . , S
y, < Not, j = 1, 2, ..., S Coupling
1J »
* PJ =
or
only
y. > 0 integer.
Several things should be noted about this (all)
integer linear programming problem:
(1) As we have noted, both the P and A-R systems
are representable as network flows. There-
fore, either system alone solves easily and
without extra effort with all-integer
answers. It is the imposition of the coup-
ling, constraints that requires special
integer methods on the system as a whole.
-------
1-87
(2) The constraint matrix of (3.5) Is block
angular, i.e., the form is:
Such problems have been
termed "multi-copy" or
"multi-page"programs and
extensive work has been
p
0
0
A-R
0
done on efficient algorithms to capitalize
on this structure (see [C 6] for non-
integer problems).
(3) The formulation of (3.5) makes it apparent
that a selection-allocation problem is
equivalent to a transportation problem (see
[C 6]) which has the added stipulation that
some of the warehouses must be shut down.
It is extremely easy to generate solutions
to the problem: pick any m of the eligi-
ble sink sites and assign each source to
the nearest of these.
-------
Section C
ATI Approximating Algorithm: The Minimum
Elimination-Gain Algorithm (MEGA)
Instead of pursuing the solution of selection-
allocation problems via the path of integer programming,
we now turn our attention to an approximating algorithm
which renders excellent loeal optima.
The idea behind this algorithm is very simply
stated: Let each and every eligible sink be chosen, in
violation of the stated goal of having only m f S sinks.
Then we may reduce this violation by eliminating one sink,*
The eliminant is the eligible sink whose elimination
causes the value of the objective function
1 J
to increase the least amount, thus the name minimum
elimination-gain algorithm. This process is repeated on
S - 1, S - 2, ... eligible sinks until S - m sinks have
been eliminated.
To make this process clearer we shall illustrate
it with an example in which no ties occur between elimi-
nants. Then we shall illustrate an iteration of another
1-f
-------
1-89
example in which tie-breaking is required.
Suppose we have seven sources and five eligible
sink sites and the table of d,. values is as shown in
Table l-4a. The initial allocation of sources to sinks is
shown by the circle around the minimum d., in each row.
We may read the table to see that
Source 1 goes to sink 1
Source 2 goes to sink 2
Source 3 goes to sink 3
Sources 4, 5, 6 go to sink 4
Source 7 goes to sink 5 •
Moreover, the value of the objective function is the sum
of the circled elements or F = 26 .
We now reason as follows: If any given column is
eliminated with a loss of circled elements, then the cir-
cle must move to the next smallest element, of the row in
which the circle was lost. Thus, for example, if column
4 were to be eliminated three circled elements would be
lost and we would have
Source 4 would go to sink 3 for a net gain of 3 = 13 - 10
Source 5 would go to sink 5 for a net gain of 16 = 17 - 1
Source 6 would go to sink 1 for a net gain of 12 = 19 - 7
Total gain from eliminating column 4 = 31
-------
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-------
1-91
This is shown in Table 1-Ma by subscripting each
circled element by the gain in P which its loss would
invoke. Obviously, the total gain in P resulting from
the elimination of a column is the sum of these subscripts
over the column. These sums, the elimination-gains, are
shown at the bottom of the table labelled G, . By what
we have said above, we see that the first eliminant is
eligible sink (column) 5. The result of this elimination
is shown in Table l-4b where we note tbat the elimination
of a column also manifests its effect on the next table
by taking away some second-bests.
Without prolonging the example we may look ahead
to see that the final solution for m = 2 will be:
Source 2 to sink 2 .
All else to sink 4 .
Two kinds of ties occur. The first kind, that
between elements of a row is of no consequence. When a
source is indifferent to a group of sinks, choose one
arbitrarily since the contribution to G. will be zero
J
no matter where the circle is until only one of this
group is left.
-------
1-92
The second kind of tie is between G.'s and is
J
important. This situation is handled by the following
addition to the rules (which will be restated more
formally anon):
In each column for which G, is tied for mini-
mality tag those elements which are second best
of their respective rows. Compute a sum analogous
to G. based on the differences between third-
bests and these second-bests. Use these sums in
place of G. to break the tie. If any second
generation G.fs are still tidd we repeat the
process with third and fourth-bests, etc. until
either the tie is broken or we exhaust the number
of columns left. At this point we may try to
break the tie on the basis of the largest sum of
d..'s . If this fails, we choose arbitrarily.
An illustration of this tie breaking procedure is
given in Table 1-5 which is a symmetric distance matrix.
The minimal elements are again circled while the second-
best elements are enclosed in squares. The subscripts on
the circles are as before while the subscripts on the
squares are the differences, third-best minus second-
best. Note that the original and the tie-breaking sums
are differentiated by a superscript.
-------
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1-93
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1-94
Before giving a precise, formal description of
MEGA we pause to make the following observations:
(1) Note that a solution is given for every m",
m < m' < S , hence the MEGA provides some
insight into the cost of reducing the number
of sinks, i.e. the G.'s may be regarded as
J
analogous to the shadow prices of linear
programming. The minimal G. used in reduc-
ing the number of sinks from k + 1 to k
is the added transportation cost invoked by
this reduction. This must, of course, be
weighed against the capital and operating
costs involved in having a k + 1-st sink.
(2) A procedure which might produce even better
solutions would be to allow an eliminant to
replace a column (sink) still in any itera-
tion numbered two more than the iteration on
which it was originally removed. Thus in
Table 5 we would consider trading column 4
for another column when the number of columns
had been reduced to 5.
We now state the MEGA more formally by making the
following definitions:
-------
1-95
Let J * (j|j has not been eliminated} ,
{6i} = {dij'J e J}
but
6i - 6i - "* - 6i
where p is the number of J's in J , and ties for
precedence are broken on the basis of the smaller value
of J .
Secondly, by the statement
= 6i
we shall understand that 6^ is not merely equal to d..
but is the identical element. This is an important dis-
tinction when more than one d.. in a row has the same
ij
value.
Define g = « - 6 if ^
otherwise .
-------
1-96
Finally, let
The Steps of MEGA
1. Set p = S , J * (1, 2, ..., S} .
2. Set t = 1, T = J .
' 3. For each J e T compute G. .
J
4. If (G1|j € T} has a unique minimum go to step
J
10; otherwise go to step 5.
5. t + 1 ->• t ; if t = p go to step 8; otherwise
go to step 6.
6. Redefine {T = (JIG*'1 is minimal} .
7. Go to step 3.
8. For each j e T compute a^ =/^ d.,., and define
A by a^ = Max ffj . If i ls not unique, define
J •!•
I = Min{j|j € T , cjj is Maximal} .
9. Go to step 11.
10. Define 4 by G^ = Min G^ .
J6T J
11. J-
12• If p = m quit; otherwise go to step 2.
-------
Section D
Applications p_f MEGA to Location-Allocation
and Other Problems
We have noted above that Cooper [A 5], [A 6]
solves location-allocation problems by an algorithm called
the Destination Subset Algorithm. In this algorithm the
locations of the sinks are restricted to the sources
(Cooper's destinations) themselves. Cooper suggested that
the ( ) possible solutions be generated.
It is apparent that picking the best m sinks
from amongst N sources is a selection-allocation prob-
lem and is thus solvable by the MEGA. Table 1-5 is, as a
matter of fact, derived from Cooper's second example (see
Table 1-2) by using H-L rather than Euclidean-metric dis-
tances. Application of MEGA to Table 1-5 results in the
^-allocations shown in Table 1-2 one of which is identical
with the E-allocatlon. It should be reiterated that the
tableau handled by MEGA need not be square nor need it be
symmetric.
1-97
-------
CHAPTER 4
EXTENSIONS OP SELECTION-ALLOCATION MODELS
Section A
S-A Problems with Weighted Sources
In the location-allocation models and the
selection-allocation models which we have thus far dis-
cussed sources have represented single truckloads of
refuse. For many applications outside of the field of
solid wastes it may be desirable to allow different
amounts of the commodity of flow be present at the various
sources. Within the field of solid wastes, advantages
may be gained by taking many sources and representing them
as a single source. Without committing oneself to a speci-
fic method for doing so, it can be seen that such
aggregation allows larger areas to be analyzed; simplifi-
cation is purchased at the price of an accuracy which may
be inappropriate.
The mathematical statement of a weighted
selection-allocation problem without capacity constraints
is given by (3-5) with the following changes to the
objective function, the P-system, and the coupling:
1-98
-------
1-99
Minimize
1-N J=S
S.T. > . x.. = 1 (no change)
J-l
i=N
1=1
J-S
J-l
J -
S>P
Coupling
where v. = weight at the i-th source
i=N
and V =
1-1
There is no change to the A-R system.
The MEGA may be extended to cope with such a
weighted problem. The only change required is in the
key definition (3.6):
-------
1-100
This Is made more concrete by the following
example which is presented by Maranzana [A 24] as an
example which causes his algorithm to end with a local
optimum. The initial and the last two tableux are shown.
Note that MEGA can be used to reflect analysis upon the
weights themselves. For example, the next to the last
tableau shows that eligible sink 7 becomes competitive
with number 5 when the sum of the 6-th through the 9-th
weights equals 20.
It should be noted that the example is especially
constructed to have a self-evident optimum (2 and 5) and
the local optimum found by Maranzana's algorithm (see page
1-16) is 5 and 7.
-------
1-101
10
100
10
10
100
1
1
1
1
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10
20
20
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140
130
140
140
100
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10
20
130
120
130
130
1000
20
10
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20
30
140
130
140
140
100
20
10
20
©
^10
30
140
130
140
140
100
30
20
30
30
(6)
^0
110
100
110
110
2000
140
130
140
140
110
©
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120
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140
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F = 0
Table l-6a MEGA with Weights on a Problem of
Maranzana. Initial Tableau
-------
1-102
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Section B
S-A Problems with Site Acquisition
and Processing Costs
In section C of Chapter 1-2 we gave an example of
a fixed charge problem in the context of a problem in
area exclusion. Having now passed to the ultimate in area
exclusion, to wit, restriction to a finite set of points
of known location, we turn to consideration of methods for
making these models more realistic economically.
Thus far the only costs considered by the
selection-allocation models have been the costs of trans-
portation. This is given route-by-route in the coeffici-
ents d.. (see page 1-79) with the aggregate transporta-
tion cost given by
1-N J-S
P = X) ViZ dlj XU (4'3)
1=1 J-l
in the case of weighted sources. There are, however, two
additional broad categories of cost which must enter into
the economic design of a refuse system. These are the
capital costs involved in acquiring and developing a refuse
disposal site and secondly the costs of operating and
1-103
-------
1-104
maintaining the site. (It is often considered expedient
if not appropriate to lump other system costs with the
latter costs.)
To include these three categories of cost in the
same analysis one must ascertain the proper scaling to be
used amongst them so that one is not talking about one
day's transportation cost versus one year's operation and
maintenance. Assuming that the proper discounting factors
have been computed and lumped into the coefficients about
to be defined we let
K, = Capital costs of acquiring and developing
the j-th eligible sink site.
P. = Cost per unit refuse processed at the j-th
site when developed.
We may now include these cost considerations in
the selection-allocation formulation of (4.1) and (3-5)
by using the following objective function:
i=N j=S J=S
P =E vi E dij XiJ + Z (KJ aj * Pd V <*.*>
i=l j=l J=l
Note that the capital cost K. is not invoked unless the
J
-------
1-105
J-th site is accepted (a. = 1) and is invoked en toto or
J ~~~" "~~~~~
not at all. In such a context, the A-R system is not as
fictional as first described. Note also that the process-
ing cost, P. is brought to bear on the full amount of
refuse y. arriving at the j-th site; allowance for
partial operation may be included in the computation of
PJ •
We end this section by showing how the MEGA may
be extended to include the fixed charge or capital costs.
The extension to account for operation and maintenance
costs is not so easily made and is harder to justify.
This will not be done here. The extension of MEGA to
capital cost is accomplished by redefining
JUJ-J
The rationale for this definition harks back to the
v
intended meaning for the G 's as originally defined,
namely the amount by which F will increase if site j
is eliminated, on the supposition that at each step all
sites are in operation.
By way of illustration we may turn back to Table
6b arid ask ourselves what K, K and K must be for
-------
1-106
site 5 to be preferable to site 7. Accordingly we have
by (4.5):
o = 2600 + Ke + K,,
£ 0 I
Gc = 2000 + K0 + K7
I? £ l
and G? - 400 + K2 + K .
Thus, we desire that
2600 + K5 + K? > 2000 + K2 + K?
and 400 + K2 + K5 > 2000 + K2 + K7
or 600 > K~ -
5
and K5 - K? > 1600 .
Thus, for example if (K2, K5, K?) = (1300, 2500, 800)
we have G2 = 5900, G,_ = 4100, and G_ = 4200 and site 5
is preferred over site 7.
-------
Section C
Extended Models for Considering the
Development and Use of a
Selected Site
Once a selection-allocation problem has been
solved by an exact algorithm or the MEGA, all we have
done is find a set of refuse disposal sites and an alloca-
tion of service areas (represented by source points)
thereto which minimizes the aggregate transportation cost,
Using the extensions of the previous section it is possi-
ble to bring the costs of acquiring, developing and
operating the site into the picture, but on closer exami-
nation, the manner in which this was done (see equation
(4.4)) was a very rough cut at the economic analysis one
desires in doing such planning. Clearly, the cost of
acquiring a site and the cost of developing the site are
separate items; lumping them together is made necessary
by the limitations of the selection-allocation model.
What we seek then is a model or a. class of models
which, taking cognizance of the use to which a site will
be put, is able to separate acquisition from development
cost. It is clear that both development cost and operat-
ing costs depend upon the type and operating level of the
facility developed at a site.
1-10?
-------
1-108
The problem of deciding what type of facility
imposes an unwarranted and needless complication upon the
analysis. The process of sifting available locations to
determine the set of eligible sites to present to the
analytic model rightly includes consideration of how any
site may be used. That is, the analyst using his
seasoned engineering Judgment has probably decided
already how any given site will be used. This is a
qualitative Judgment which may weigh Intangibles impossi-
ble or difficult to quantify and model.
On the other hand, given the type of facility to
be erected on a site, there are latitudes of choice open
in .determining how large, how fast, how expandable, etc.
to make a facility at any given site. This Is an area of
quantitative decision making where computer-oriented
analyses can aid human Judgment.
If, for example, a site is suitable for an
incinerator, the site development choices may represent
several combinations of capacity and processing rate of
interest to the planner. The planner has the responsi-
bility for developing a reasonable number of realistic
alternatives which covers the range of practical interest
for the site, locality, etc.
-------
1-109
Here, again, we are opting for simplicity by
laying a equitable burden upon the planner. The alterna-
tive is to consider continuous variation of design
parameters. This is possible in the design of a single
facility, but may be exceedingly complex in a system of
interrelated facilities.
A model for the analysis of the acquisition,
development, and operation of multiple sites is illus-
trated in Figures 1-lla and 1-llb. Here we are selecting
two sites (m = 2) out of four eligible sites (S = 4).
Sites 1 and 2 are each developable in three ways while
sites 3 and 4 are developable in four ways each
(w^ = 3, W2 = 3, W3 = 4, W^ = 4) . Figure 1-lla presents
the analog of the A-R system for the selection-allocation
model. The left half of the network is familiar with
the A node, nodes 1, 2, 3, and 4 and the variables
ct,(j = 1, 2, 3, 4) retaining their former meaning, but
J
the R part of the network has been replaced by a system
of doubly-indexed flows 6., representing choice of the
k*-th development of the J-th eligible site. Figure 1-llb
represents che physical or P system. Here again, the
flow from an eligible site is split into different flows
on an all or nothing basis amongst the different develop-
ment possibilities.
-------
1-110
LABELS OF
DEVELOPMENTS
ELIGIBLE SINK SITES
COST PERTAINING
TO A FLOW
Figure 1-lla Acceptance System Network for
Extended Selection Model
-------
1-111
DEVELOPMENT LABELS
V2
V5-
(notalUjj shown)
Figure 1-llb Physical Network of Extended Selection Model
-------
1-112
To make this more precise, we define the follow-
ing variables and constants:
V. = Amount of refuse at the i-th source.
v.. = Flow sent from the i-th source to the
^-J
J-th site. We are reverting to a con-
tinuous variable formulation since
capacity constraints may be relevant in
the choice of a development.
d., = Cost of transporting one unit (e.g. ton)
of refuse from the i-th source to the
J-th sink.
y.. = Amount of refuse processed by the k-th
J K
development at the J-th site.
p.v = Processing cost per unit refuse using
J «•
the k-th development at the J-th site.
(Operation and maintenance, etc.).
6., = 1 if the k-th development at the J-th
site is undertaken
= 0 otherwise.
D., = Cost of undertaking the k-th development
JK
at the J-th site.
a. =1 if the J-th site is chosen
u
= 0 otherwise.
-------
1-113
A. » Cost of acquiring the J-th site.
J
We are assuming here that the various costs are
suitably scaled (discounted) with respect to one another
so that the component parts of the objective function can
be added realistically. The overall model is then
Minimize dv + (Pjkyjk + Djk6Jk)
i J j k
+
J
subject to
S
Vi
k i • k
J k i i J
i V vi
-------
1) All variables non-negative and a. , 6. = 0
J J •K
or 1 only.
2) m = Number of sites to be selected.
3) All summations over i are from 1 to N ,
the number of sources. All summations over
J are from 1 to S the number of eligible
sink sites. All summations over k are from
1 to W. the number of ways to develop the
J-th site.
In addition to the constraints shown in (4.6) it
is quite likely that capacity constraints would be
relevant. A simple capacity constraint of the form
is easily accomodated, but what is more likely is a com-
pound capacity constraint which says
If 0 < > v . . < C, , use development 1
- £,^j 1J ~ J 1
±
If C. < > v.. < C.. use development 2 .
J * - 4^1 J-J - J 2
etc.
-------
1-115
Such a system of capacity constraints can be imposed by
altering the coupling constraints. For example, if a set
of capacity constraints of this nature were to be imposed
on the first site of the example of Figure 11, the
coupling constraints might read
< Cii6ii
C11612 < *12 S C12612
C12613 1 y!3 - °13613
-------
CHAPTER 5
SUMMARY AND CONCLUSIONS
The motivation for and the course taken by the
Investigations of Part I have revolved about the answers
to two questions:
1) Where is the best place to locate one or
more refuse disposal sites given the loca-
tions of the areas to be served?
2) Which areas should be served by which
disposal site?
We began our answering of these questions by
making several assumptions. We assumed, most fundamen-
tally, that one plan for locating the disposal sites and
allocating the service areas would be .distinguished from
another such plan on the criterion of aggregate trans-
portation distance of service areas to disposal sites.
We assumed, further, that there would be no capacity con-
straints at any disposal site and that distances would be
measured by the A,-metric.
Under these assumptions we were able to formulate
the location-allocation problem as a linear program
involving Integer variables. Then, taking advantage of
1-116
-------
1-117
the median nature of pure location problem solutions
when using the £-,-metric, we constructed a theory of
median sets which was used to support an algorithm for
the approximate solution of location-allocation
problems. This algorithm operated by alternately locating
and allocating.
Up to this point we had been assuming that the
disposal sites could be located anywhere in the plane.
Thus, we turned to the question:
3) What happens to the choices of disposal
site location and the allocation of
service areas when certain areas are
ineligible for the location of disposal
sites?
For simple cases, we demonstrated both graphical
and analytic solutions. For a problem involving the
location of a single disposal site, the best location is
that location in permitted territory A^-closest to the
unrestricted optimum location.
Instead of pursuing this question for the problem
of locating several disposal sites, we passed question 3)
to the limit: the process of declaring certain areas
ineligible culminates in the situation wherein only a
small number of land parcels remain eligible. The
-------
1-118
question then becomes:
4) Given a definite number of eligible sites
and their physical locations, what is the
best way to choose a proper subset (i.e.
some but not all) of them to be disposal
sites and to allocate service areas to them?
Whereas the former models were called location-
allocation problems, we termed the models resulting from
question 4) "selection-allocation" problems. The common
features of the two classes of models are the problem of
allocation and the goal of minimizing aggregate transpor-
tation distance. In comparing the two classes of models,
one may conclude that the location-allocation problem is
likely to be more of an interesting mathematical abstrac-
tion than a useful technique. The real-world situation
facing the planner bears a stronger resemblence to the
situation underlying the selection-allocation problem.
Again, we were able to characterize the selection-
allocation problem as a linear programming problem with
Integer variables. The transition to a linear program
was aided by a network-flow analog and did not depend
on any particular metric for giving the distances. Thus,
assumption of Si -metric distances (which was necessary
for the linearization of the location-allocation problem)
-------
1-119
was dropped. As before, an approximating algorithm was
developed. This algorithm, termed the minimum elimination
gain algorithm (MEGA), assumes that ali eligible sites
are used to start. These are eliminated one-by-one until
the requisite number remain. The basis for elimination
is: eliminate that site whose elimination causes the
minimum gain (increase) in the aggregate distance
function.
The final section of Part I attempted to exploit
the flexibility of the selection-allocation model to
introduce further elements of realism. The first such
extension allowed the service areas to be weighted
according to relative activity. It was found that the
MEGA was useful in this extension. Next, and
more important, we investigated the extension of
selection-allocation models to cover site acquisition,
development and operating costs. Several models were
proposed as integer-linear programs. Only the isolated
problem of site acquisition cost could be handled by the
MEGA.
To extend the remarks made above concerning com-
parison between the location-allocation models and
selection-allocation models, the writer ventures to specu-
late that the extended models which encompass acquisition,
-------
1-120
development, and operating costs as well as aggregate
transportation cost will prove to be valuable planning
tools when fully and intelligently exploited. It is
appropriate to repeat here that intelligent exploitation
of mathematical models puts a great burden on the planner
to make those choices and eliminate those alternatives
which are within the realm of mature engineering judgment,
Intelligent exploitation of mathematical models regards
the latter as an aid to, not a substitute for, such
judgment.
-------
PART II
ON THE SIMULATION OP REFUSE COLLECTION
AND HAUL OPERATIONS
2-1
-------
CHAPTER 1
INTRODUCTION
Section A
Solid Wastes Literature
In this section It is our intent to review the
scope and accomplishments of work done in the field of
solid wastes. More particularly we shall concentrate on
investigative methods that are quantitative, mathematical
and/or analytic for the most part and on work of the last
thirty years. It would not be proper, however, to dis-
miss preceding work without at least a cursory sketch.
The solution to the problems of refuse collection
and disposal was simple in the civilizations of antiquity
and remained essentially the same into modern times where
populations were rural, agrarian and sparse. The solution
consisted ©f selecting a place removed from the village
(presumably downwind) and dumping the refuse there on the
open ground. Such dumping grounds have proved the
treasure troves of modern archeologists. The responsi-
bility for refuse removal was on the individual family in
small settlements and we may assume that some sort of
2-2
-------
2-3
organized activity manned by slaves was used in the cities
of antiquity.
A modern literature on refuse begins sometime in
the nineteenth century with the growth of organized muni-
cipal services in all areas affecting the public health.
Such literature was and still is the literature produced
by and for day-to-day practitioners in the field of
public work or by engineers reporting on specific solu-
tions to specific problems. Modern examples of such
literature are found in publications such as American Gas
Monthly (articles concerning use of gas as an auxiliary
fuel in refuse incineration), Engineering News (design of
incinerators and other facilities), Power Engineering,
Gas Age, Combustion, Plant Engineering, Publie Works
(by and for public works officials and functionaries),
Refuse Removal Journal (the same, only more specifically
toward those in solid wastes), Sewage and Industrial
Wastes (concerned with solid wastes produced by indus-
tries), and finally Water Works Engineering (where solid
wastes interface with water supply and treatment). No
specific works from these Journals are cited herein.
Probably the cornerstones of recent work in the
field of solid wastes are two volumes published by the
American Public Works Association, Refuse Collection
-------
2-4
Practice [5] first published in 19^1 and revised in 1958,
and Municipal Refuse Disposal [4], 1961. The first of
these deals exhaustively and systematically with all
aspects of refuse collection starting with a statement of
the problem, an examination of the types and amounts of
substances aggregatively called refuse, the role of the
householder (or other producers) in preparation of refuse
for collection, analysis of the costs of refuse collec-
tion, survey of methods of collection and types of equip-
ment, and going on to discuss financing and multifarious
aspects of operating and administering such systems. We
shall have more to say anon on methods of planning refuse
collection systems described by this book.
Municipal Refuse Disposal concentrates more on
the description and analysis of different methods of ulti-
mate disposal, their advantages and disadvantages and
their interaction with various aspects of the collection
system. Both of these books contain excellent selected
bibliographies.
The thing to be noted about both of these books
is that all of the identifiable subcomponents of the
total system are discussed and analyzed separately.
Frequently, this analysis involves equations or formulas,
but when it comes to describing the interaction of these
-------
2-5
parts, the description is more likely to be the qualita-
tive description of tendencies than the quantitative
prediction of formulas.
The first large-scale effort to combine the work
study method of industrial engineering with an overall
quantitative rational analysis of an entire refuse collec-
tion and haul system was undertaken by the Sanitary Engi-
neering Research Project of the University of California
(Berkeley). [1]— . Among the factors entering this
analysis were pickup efficiency measured inversely by man
minutes per ton, crew size, haul distance to the disposal
site (from the center of the city), percent of pickup made
from curb versus percent from rear of house (assuming
blocks bisected by alleys), etc. A more extensive listing,
definition and discussion of these factors and an examina-
tion and critique of these methods of analysis will be
2/
undertaken in the next chapter,— but it should be noted
x
here that this work established the definitions and con-
cepts upon which this and much present-day work is based.
Indeed, the Sanitary Engineering Research Laboratory of
the University of California, (Richmond) is still a primary
-This is also reference Pr. 1 of the Preface.
2/
- A synopsis of the data and analytical techniques devel-
oped in this report are included as Appendix D to Refuse
Collection Practice [5].
-------
2-6
source of research and trained personnel In the area of
solid wastes. (See below).
It has often been remarked that a great deal of
what's new in America starts on the West Coast. This
certainly seems to be true when applied to the growing
awareness of such things as air pollution and ghetto pol-
lution (Watts). In the area of solid wastes, the seed
planted by the Richmond group took root in the Los Angeles
area.
In the late 50's, early 60's, Los Angeles was
faced with massive planning and design problems imposed by
vast growth over a huge geographical area and compounded
by the need to end backyard incineration of refuse in
order to ameliorate the growing problem of air pollution.
Various techniques such as statistical analysis, queuing
theory and more recently, following lead of the research
reported hereinunder, Monte Carlo simulation have been
applied to problems such as the location and design of
transfer stations and fleet maintenance reserve [?]. The
Los Angeles group is also pioneering in the design of
automated data gathering mechanisms to be installed on
refuse collection vehicles^- ; the underlying need of all
techniques is good data.
Personal communication.
-------
2-7
Another application of system analytic technique
that can be described in more detail concerns the City
of Vancouver, B.C. [8]. The acquisition of new equipment
of larger capacity necessitated the coalescence of the
more numerous, smaller collection routes ("beats" in the
language of the report) into fewer, larger ones. The
study began with the gathering of detailed data on each
block of the city. This data was used to get a regres-
sion formula relating time of collection to such factors
as volume of refuse, numbers of containers, number of
container groups, travel distance around the block, etc.
Then the transportation algorithm of linear programming
was used to balance the workload between collection
routes. The analog of the suppliers was found in truck-
less or overloaded routes, the role of the receivers was
taken by underworked trucks, while the commodity being
"transported" was city blocks. The objective function
being minimized reflected the number of changes from the
existing routing, i.e. the number of blocks previously on
the same route now on different routes. Thus,the overall
objective was to minimize the disruption needed to achieve
an equitable balance of workload within the constraints
of volume, length of workday and physical compactness.
-------
2-8
Among other work preceding 1965 we mention also
the pioneer work of Charnes, Quon, and Wersan ([12] -
[18]) part of which is extended in this dissertation.
And as evidence of growing awareness and desire to do
something we have already noted the work of the North-
eastern Illinois Metropolitan Area Planning Commission
(Reference Pr. 13 of the Preface) and the A.P.W.A.'s
Solid Waste Research Needs (Reference Pr. 11 of the
Preface) which is in effect an agenda of the research now
in progress in many places.
Research in solid wastes increased immensely in
the years 1961 - 1965 because of an increasing willing-
ness of the United States Public Health Service and the
National Institutes of Health to sponsor and support such
research-* . Nonetheless, the year 1965 represents a
cleavage between two research methodologies. Prior to
1965 one saw first the gradual emergence of the problem,
pursued initially without urgency or pressure. Then a
few researchers became concerned and began pioneer inves-
tigative work on limited funds and with scant communica-
tion so that the work that emerged was marked by large
The work underlying this dissertation was so supported
and this is acknowledged elsewhere.
-------
2-9
gaps and some duplication. By the early 1960's the prob-
lem of solid wastes had become so universally recognized
that funds became available, but the need and the clamor
were for Immediate solutions, not research. In order to
provide for an orderly direction of future research,
Congress passed the Solid Waste Disposal Act of 1965- It
was hoped that this act would buttress a more sophisti-
cated research methodology: one in which the environmen-
tal objectives of wastes management are first delineated
in public policy. Then the problems which must be solved
to meet these objectives are tackled in a coordinated
manner and there exists a yardstick against which to
measure proposed research.
A summary of research which has been performed
largely under the aegis of this act was provided by a con-
ference held in Milwaukee, Wisconsin in July 1967 [6]^» .
A brief description of this work follows:
Working with the City of Berkeley, California,
C. G. Golueke is developing an overall "solid wastes sys-
tem evaluation model" [6-A1]. Although not spelled out
in detail in the conference proceedings, a part of this
work entails the development of a refuse generation model.
- The 59 papers are numbered A1-A11, B1-B12, etc. For the
sake of brevity we shall list all the relevant papers under
the same bibliographical heading and refer to them as
[6-A1], [6-D4], etc.
-------
2-10
This means being able to predict the amount of refuse
produced (or to generate a random variable representing
same) given all the relevant factors which affect refuse
production. Since a refuse generation model is part of
the work to be described hereinunder we shall defer
further discussion of the subject to Chapter 3» but
it is apropos to note that the problem is of wide concern
as evidenced by papers [6-A7, Dl, D2, Dk, D5, F3, F4, F5»
P6].
Other investigative groups seeking to create
overall systems evaluation models are the Aerojet-General
Corp. group [6-A3] which is putting its emphasis on devel-
opment of measures of effectiveness, a group at the
University of Southern California [6-A11], and finally
the City of Los Angeles which is developing a management
Information system aimed at the supervision of operations
and which as noted previously has used and is using such
techniques as linear programming,queuing theory and Monte
Carlo simulation [6-Dl]. In the City of Baltimore,
Maryland a group at The John Hopkins University is develop-
ing a Monte Carlo model for the design of a refuse col-
lection system employing transfer stations [6-D2] while at
Harvard the use of shipborne incinerators is being investi-
gated by the use of a simulation model [6-D5].
-------
2-11
In summary, work in progress falls into the
following categories:
1. Investigation and improvement of the tech-
nology of current disposal techniques.
2. Investigation of proposed new disposal tech-
niques (shipborne incineration, refuse pipe-
line, etc. ).
3. Creation of an improved data base (refuse
generation models, etc.).
4. Creation of a management and planning science
for solid wastes through the use of computer
and system analytic techniques (which is not
confined to this category).
This appears to be an appropriate point to inject
some personal comment. This author has come to feel that
a part of the research money that the Federal government
is putting into the problems of solid wastes would be
more wisely spent on investigation of the economic and
social incentives needed to
(a) Decrease refuse production.
(b) Salvage useful materials from refuse.
Point (a) is based on the real observation (cited
below in its proper place) that people tend to produce
more refuse when more frequent and more inclusive (of
refuse types) service is provided. Paced with a more
-------
2-12
restrictive service, old clothes get saved for Goodwill,
newspapers for the Boy Scouts, etc. Thus, the economic
problem of trying to salvage such material from refuse
Is solved by providing an incentive that keeps them out
of the refuse container in the first place. Another
example that readily springs to mind is the proliferation
of no-return beverage containers. If man is to learn to
live cooperatively with his environment rather than
disasterously, he may have to aacrifice many such waste-
ful conveniences. In the end, it boils down to whether
government is to forever pander to unbridled demands for
bigger, better and more by spending more and more money
less and less effectively. No one denies that the
Federal government or the several states now have the
power to take such strong measures as banning beer cans,
but Americans have a nortorious habit of not doing any-
thing until a gun is against their temple.
Point (b) is based on recent reports of success
in coking automobile tires and domestic refuse to produce
large quantities of fuel gas and tar-like chemicals which
can provide the raw materials for plastics, drugs, etc.
Such methods, if they can be exploited without air
pollution hazard, combined with incentives and regula-
tions which reduce the amounts of refuse can well be the
key to the future in this area.
-------
Section B
The Nature and Limitations of_ Simulation
As we have noted above, Monte Carlo simulation
has become a widely used technique since it was first
reported in the works of Charnes, Quon, Elsen, Tanka, and
Wersan ([12]-[l8]) hereinunder extended. We have also
noted that simulation is one of a variety of techniques
intended to provide insight and predictive power in
dynamic, random, changing environments. By way of con-
trast and Introduction, then, we propose first to expand
upon the list of study methods proposed in the Preface,
turning thence to an extensive examination of the variety
of techniques which collectively have come to be called
"Monte Carlo methods".
First among investigative methods that might have
an intuitive appeal to a planner wishing to ask a variety
of "what if" questions would be to perform experiments
with the entities involved. To a limited extent this has
been, is being and will continue to be done. Thus, for
example, in seeking ways to control the fly breeding
problem in refuse awaiting collection, Ecke, ejb al. per-
formed actual experiments with two different frequencies
of collection against two different containerizatlon
2-13
-------
2-14
methods: (once weekly, twice weekly) x (cans, suspended
paper bags) [6-F6]. Moreover, the traditional method of
balancing workload over collection routes has been by
trial and error experimentation with crews and equipment
([5], p. 203), but the scope of questions that can be
answered by experimentation is limited. And these limits
exclude arbitrary purchase of equipment, large capital
expenditures, hiring or firing of personnel or massive
tinkering with the public's patience in varying frequency
and/or site of collection.
The next approach on some scale of intuitive
appeal is to use someone else's experience to represent
an experiment on our part; i.e. ,reasoning by anology.
One finds a city of similar population, occupation, etc..
and notes that what was good for it may be good for us.
A little more tenuous than this is the use of historic
analogy; failing to find a contemporary analog, one seeks a
city which was some years ago in a situation similar to
our present situation. Probably, the best place for
"data" of this kind is as incidental backup for plans
underwritten by more solid forms of analysis, or perhaps
as a goad to civic pride when bond referenda are at stake.
When used as the primary means of analysis, too much time
is later spent figuring out why the analogy didn't hold
after all.
-------
2-15
The third general category of methods used for
studying dynamic systems is a large one subsuming a wide
variety of mathematical techniques. Proceeding from simple
to complex we have first mathematical methods of classi-
cal engineering analysis such as statistical analysis (n-
year flood, n-year storm) for the study of both long
and short term variation and the marriage of the classi-
cal engineering methods of analysis via differential
equations with probability theory which has come to be
known as queuing theory (study of short term random demand
on waiting line facilities). Here also we have the use
of current demand, population prediction models, economic
factors and over-design factors to yield facilities
designed for a certain "design life". Certain rational
methods of this type have been developed for the design
of refuse collection systems and these will be discussed
more fully in the next chapter.
The models of Part I, more especially of Chapter
1-4, are an attempt to provide a system model which ties
together a great many factors simultaneously, but as we
have noted these models are simplifications of a dynamic
world. Some insight into the problems engendered by long-
term change can be gotten by solving these models repeat-
edly with the cost and refuse production data inputs that
-------
2-16
we expect will hold at various times in the future (sensi-
tivity analysis). The problem here, of course, is that
much is yet to be learned about the creation of adequate
models of both population growth and present refuse pro-
duction hence models of future refuse production are
doubly hazardous. (See [11] for a discussion of demo-
graphic models.)
An alternative to running a time-fixed model many
times is to create a model which allows planning over a
time horizon, I.e., we have a large model whose subcom-
ponents are the time-fixed models. These are coupled by
constraints representing intertemporal conservation of
matter (etc.) and imposition of common sense. Thus a
variable which previously specified whether or not to
acquire a given site is now represented by the sum of
time subscripted variables which specify when to acquire
that site (if at all). The coupling constraints stipulate
such rules as "if acquired, it is acquired only once,"
"if acquired and/or operating in time period t it must
operate in subsequent periods (unless filled up as a
sanitary landfill site)", etc. The shortcomings of such
an approach are first its overweening complexity and
probable exhaustion of computer capacity, and second, it
also requires the use of present-day guesses about
-------
2-17
tomorrow's data. In the case of site selection this
entails the assumption that sites available now will
continue to be or requires us to surmise which sites will
become availabe that aren't now, etc. A part of the
problem of future data can be obviated by running models
with future data at low and high levels to study the
range of outcomes.
Although used by some to study long term growth
and change, the greatest use of simulation has been and
is to study short term random and cyclical variation. In
this day simulation has come to be conducted chiefly by
the use of computer programs, but we may note that simula-
tion is really an old technique and probably one of the
oldest learning exercises known to man. ("Let's play
house.") In the field of technology, simulation has such
antecedents as basin testing of physical models of ships,
wind-tunnel testing of airframes and airfoils, static
loading models of structures and the use of analog
computers. (See, for example, references A9, A10 and A27
of Part I.) But as applied to modern research in the
social and physical sciences, simulation has come to mean:
"the construction and manipulation of an operat-
ing model, that model being a physical or symbolic
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2-18
representation of some or all aspects of a complex
system or process." [10]—
Such operating models are "representations of behaving
systems that attempt to reproduce the processes ln_ action"
[10]— . To Illustrate these ideas and to motivate fur-
ther abstract discussion let us examine several situations
which might be studied by simulation.
Example 1: Suppose we lay out a sheet of paper
with parallel lines of spacing W and begin dropping a
needle of length L onto it. Each time we do so we tally
up the trial number and whether or not the needle crossed
7/
one of the lines on the paper. After many trials- , the
ratio of the number of crossings to total trials may be
taken to approximate the probability that the needle will
&/
cross the line.- The flowchart for a program to simu-
late this experiment is given in Figure 2-1. It should
be noted that in this example the computer is being used
as a device that quickly generates random numbers-^/* there
-/Italics added.
7/
-1 Just how many is a matter of statistical analysis and
familiarity with a particular model. This is usually a
matter of art rather than science.
o/
-^ is the classical problem known as Buffon's Needle.
For a judicious choice of the ratio L/W , the experiment
can be used to compute ir .
About which much has been written.
-------
2-19
r
READ IN
DATA
N,W, L
K =0
COUNTER FOR NUMBER
OF CROSSINGS
DO LOOP
1-1
YES
NO
COMPUTE
AND
PRINT
RESULT
P - K/N
GENERATE TWO INDEPENDENT
RANDOM NUMBERS u,, u2
FROM UNIFORM DENSITY
ON to; u
NO
NO CROSSING
1
Y=Wu,
S = SIN (•§ us)
CROSSES
Y IS THE LOCATION OF
THE MIDPOINT OF THE
NEEDLE
0 IS THE ACUTE ANGLE
BETWEEN THE NEEDLE AND
CLOSER OF THE RULED LINES
tr
Figure 2-1 Flowchart for Simulating Buffon's
Neddie Experiment
-------
2-20
IS nothing In the problem that precludes manual simulation
or theoretical analysis.
Example 2: In contrast to this simple problem,
we might consider the solitaire-like card game known as
"clock". In this game 13 piles of four cards each are
dealt out, face down, 12 of them corresponding to the
hours on the clock and the other at the center. A card
is turned up on the center pile and is placed on the pile
corresponding to its value (A - 1, J = 11, Q = 12,
K = 13; K goes to center pile) and the top card of that
pile is then turned up, etc. The game stops when all
four kings are on the center pile and the "penalty" is
the number of unturned cards remaining.
Suppose now that we limit our inquiry to finding
the value of P[Kd = 0] where Kd is the number of
remaining cards. A small amount of reflection shows:
1) P[Kd = 0] is so small that an extraordinarily large
number of games would have to be played manually to
determine it, 2) the determination of P[Kd =0] is
nearly impossible by theoretical means. But we do note
that the event Kd = 0 is impossible if there isn't at
least one king on the bottom of a pile as originally
dealt. Thus, if
Kb = Event that at least one king
is on the bottom of a pile .
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2-21
P[Kd - 0] - P[Kd - 0 0 Kb] - P[Kd - 0|Kb] P[Kb]
Thus, the analysis via a simulation program can be sharp-
ened (meaning fewer trials) if we compute instead
P[Kd = 0|Kb] and multiply it by P[Kb] which can be
computed theoretically^ Thus, in the flowchart shown
in Figure 2-2, the initial box marked "DEAL" is a pro-
cess of randomly placing a king on the bottom of a pile
and then randomly placing the other 51 cards. This deal
is referred to in the flowchart as the array (Njj) with
four rows and thirteen columns with the values
1 < N.. < 13 . We note next that the game may be likened
to a treasure hunt in which there are four clues at each
of thirteen caches. Each clue tells which cache contains
the next clue. The rules stipulate that a clue may be
used only once, one may not peek at a clue out of sequence
and the game ends when we seek a fifth clue from the
initial cache. Thus, in simulating "clock" it is imma-
terial which pile is considered initial (we have used the
ace pile), and second, we do not need to simulate the
physical movement of cards; a card stays where it was
originally dealt but is marked as used.
P[Kb] = .6961 .
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2-22
DEAL
(CREATE
±
Kd = 52
M = 1
L. = 0
J = M
i » L, + 1
Yes
r
Game Ends
Kd = Number of Cards Remaining
L. = Number of Cards Used
J in J-th Pile
Kd = Kd - 1
No
*See text
Figure 2-2 Flowchart of One Hand of the
Game of "Clock"
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2-23
This example indicates that a simulation need
not be a slavish imitation of every detail of the situa-
tion being modeled. On the other hand, the particular
situation being modeled here is rather static; once the
cards are dealt there can be but one outcome. Thus the
function of the program is to discover this outcome
swiftly.
Example 3: Suppose at time t = 0 that a drunk
is supporting a lamp post located at (0,0) in a
coordinate system. At times t - 1, 2, 3> ... he makes
a random move of one unit (step) north, east, south or
west with equal probabilities of 1/4 from the position
occupied at time t - 1. We now have a situation which
is thoroughly random; there is no deterministic deal at
the beginning of the game. There is rather, a "game
tree" with each node allowing four branches (Figure
2-3). Of course, many sequences such as + x , - x and
+ y , - y lead one back to the starting position. Among
«
the things we might want to know is: given a bar at
(m, n) i* (0, 0) how long will it take on the average for
the drunk to get help for the lamppost? This is a simple
example of a class of problems called "random walk prob-
lems" the solutions to which are used in such diverse
fields as solving partial differential equations. The
-------
+x
-X
Figure 2-3 Tree of Random Walk Moves
-------
2-25
particular random walk described here can be solved ana-
lytically, but we can easily conjur up simple variations
for which the analysis gets sticky:
a) There are two drunks moving randomly: given
their starting positions how long does it
take them to meet?
b) There are many particles uniformly spread
in 3-space. (A random step can now also be
up or down.) When two randomly walking
particles meet they coalesce and a two par-
ticle system moves at half the speed of one,
etc. After deciding what to do about
collisions that do not occur at lattice
points, we ask: What does the particle size
distribution look like as t -»•«>?
In the last variation one is forced for the first
time to consider events taking place in time. While it
is true that the Buffon's Needle experiment, the game of
"clock" and the simple random walk take place in time,
the program is indifferent to whether the time period is
seconds, hours or centuries. In the last variation of
random walk, time is an intrinsic variable since now
events are occuring at different rates and being initiated
asynchronous^. Time no longer passes in discrete steps
-------
2-26
of equal size. Thus, we may classify simulation programs
as static or dynamic and the latter as synchronous or
asynchronous. The simulation programs developed herein
are dynamic, asynchronous models and we are now ready to
describe the implementation and exercise of such a model:
"... the problem or system under study is first
described as the sequence of individual operations
to be performed.[E.g., move truck from garage
to first collection area, fill truck, go to dis-
posal site, etc.] This may be called the 'Plow
Model1 of the system. It is then necessary to
have data indicating how the individual operations
are interrelated and to have the frequency distri-
bution of elapsed time [and refuse production,
work rates, etc.] for each individual operation
for the different conditions to be explored.
"Then inputs of such items as manpower,
scheduling methods, or amounts of equipment,
facilities, etc., are systematically varied.
By consulting the time data mentioned above in a
random manner [i.e., sampling from the frequency
distributions], the [random] overall time for the
sequence of operations can be determined. This
process performed over and over simulates opera-
-------
2-27
tion of the system and permits accruing such total
system data as average equipment and manpower
utilization ... . Such outputs are then used to
evaluate the desirability of the given input under
test, and in effect a simulated experiment has
thus been conducted. [10] ==•'
12/
Thus, the major problems for which Monte Carlo—
simulation Is used are those entailing complicated cas-
cades of random events taking place (usually) in time
and (usually) asynchronously. This is an apt description
of a refuse collection system in operation.
Having dwelt at some length on the limitations of
other methods of analysis, we would be remiss if we did
not point out the weaknesses that are inherent in simula-
tion.
First, simulation along with other methods of
analysis has the problem of securing reasonably accurate
data. When certain Items of data are not available, one
must either recast the analysis to avoid their use,
secure the data or run the simulation with a reasonable
range of values bracketing the supposed true value. But
with simulation, the data problem is even more acute; one
—/Italics added.
127
— So called due to the intimate connection existing
between the calculus of probabilities and games of chance.
-------
2-28
needs to know not only normative (average) values for cer-
tain parameters, but since many of these parameters are
really random variables one needs to know the form of the
associated probability density functions (or higher mom-
ents). A further difficulty one faces with data when
preparing a simulation is the problem of the independence
of random variables. To some extent this is taken care
of in the flow model of the system, when one allows cer-
tain events to occur only when others have first occurred
but this is not the whole story.
In the field of solid wastes one of the first
places one confronts the data problem is in providing a
refuse generation model. We have defined this above as
being a rule or procedure for predicting the amount of
refuse generated (random variable) given all the relevant
factors. Present refuse generation models are crude, and
this is one of the most active areas in the field of solid
wastes today.
A second major limitation on simulation enters
when the simulation entails human behavior. A point in
the flow model may be reached where a decision or action
Is required by a human: how can we simulate the actual
decision process? Some of the alternatives are:
a) Always do the same thing.
-------
2-29
b) Provide a list of possible actions and the
associated probabilities (based on observa-
tion) of their being invoked. Are we sure
that we have observed enough cases to be
aware of all alternatives?
c) Build some feedback or "learning" into the
program so that successful responses are
reinforced and unsuccessful ones punished
or inhibited.
d) Try to simulate the human mind's process of
scanning the present states of the system
(including perhaps its propensity to err and
to tire) to weigh the possibilities before
acting. Here, one has replaced the one
large decision by a host of smaller ones,
and one may not have achieved anything there-
by.
A refuse collection system is, of course, a system
in which human decision processes enter into the final
result. A prime example is the decision as to when a
collection truck is sufficiently full to return to the
disposal site. This problem will be discussed more fully
in Chapter 3, where we discuss some of the assumptions
and concepts that went into the present simulation
programs.
-------
2-30
A third major limitation on simulation Is again
one that is common to other methods of analysis. This is
the problem of detail versus aggregation, of accuracy
versus practical computation times, etc. Here, as else-
where, the guiding principle has to be: what am I going
to use the model for? The physicist builds a model of
mechanics that assumes frictionless planes, weightless
levers, etc; engineers must reintroduce the physical
reality of friction. The navigator gets along with a
Mercator projection, an analyst of satellite trajectories
favors a globe while a cosmologist can get along with a
point mass; all are models of the Earth. We recognized
this principle above when we simulated Buffon's Needle
by generating random numbers, not by considering the
mechanical problem of a needle-shaped object falling
through a viscous medium under the force of gravity. We
recognized this when we simulated the game of "clock" by
leaving the cards in place. So too in simulating the
actions of refuse collectors there is detail which exists,
but consideration of which may be neglected in order to
answer the sort of questions for which the simulation
model is intended. Thus, for example, we Ignore such
realities as coffee breaks and lunch hours.
-------
2-31
A fourth and final limitation on the efficiency
of simulation (and intimately related to the immediately
preceding) is the set of restrictions Imposed by the
computer itself (speed, capacity), computer languages
and the current state-of-the-art in simulation (which is
recognized as a valid special interest area by the
Association for Computing Machinery).
When all is said and done, one has in hand a pro-
gram exhibiting, hopefully, the best advantages and
avoiding the worst disadvantages. A necessary step
ignored in the quotation above is the process of verify-
ing the model. This means, for example, that when
probability densities having mass = 1 concentrated at
one point (the average) are used, results predictable in
the absence of random variation should be achieved. Then
when the necessary data are secured concerning a specific
area, the answers generated should be in reasonable agree-
ment with observation. Then one is ready to use the
simulation model to extrapolate as suggested above. Every
run must be_ regarded as_ a trial o_f validity and results
clearly out of line with reason must bring either the
model or the line of reasoning Into question.
The number of problems, research programs, etc.,
using simulation techniques has multiplied in recent
-------
2-32
years far out of proportion to the growth in the number of
computers on which to run them, hence it is neither
possible nor desirable to provide a comprehensive biblio-
graphy on simulation. However, we do wish to mention
three works which have had influence on the development
of this simulation program. The first is simulation in
social science: readings [sic] edited by Harold Guetzkow
[10] which we have quoted above. The second work is
Mlpreanalysis o_f Socioeconomic Systems: A Simulation
Study [11] which contains among other models a demographic
model attempting to simulate the growth of populations
from such elementary considerations as birth, death,
marriage, propensity to remarry, etc. The third work,
one not generally available, is a collection of mimeo-
graphed course notes from the University of Michigan [3].
-------
Section C
The Simulation of Refuse Collection
and Haul Operations
The desire to experiment with the use of Monte
Carlo simulation for the analysis of refuse collection
and haul operations grew out of an awareness of the suc-
cessful use of the method in many other fields (as cited
above), out of a study of the rational methods developed
by the University of California [1], and finally out of
an awareness that the day-to-day operation of a refuse
collection and haul service is subject to many random
variations which are glossed over by these rational
methods. The rational methods use average refuse produc-
tion for an entire city, average collection efficiency
for all crews and distance of the center of the city to
the disposal site to represent average one-way haul dis-
tance; this is macroanalysis. Simulation allows one to
consider all of the inhomogeneities of time, place, and
circumstance to whatever degree of picayune detail one
can tolerate. Can such microanalysis yield anything that
the macroanalysis cannot? As stated by Quon, et_ al:
"The rational method has proven adequate for the
purposes of providing an initial selection of
2-33
-------
2-34
equipment and operating conditions and an evalua-
tion of the annual cost of the collection system,
where basic data and experience pertaining to
systems of a similar character are available."[13]
What the rational method cannot provide which simu-
lation can is some indication of how the rationally planned
system actually performs according to such criteria as
length of workday, efficiencies of equipment utilization,
etc. The averages may be fine, but there are real opera-
tional headaches involved in large fluctuations above or
below the average. Moreover, experimentation in the field
can yield adjustments that improve the rationally planned
system, but the scope of this experimentation la limited.
Thus, simulation may be regarded as a means for rapidly
performing experimental changes on a system to delineate
the Interdependences of the various parameters. Such
results, combined with mature engineering Judgment can
select the most promising combination of changes. (Great-
est desired change per dollar is one obvious criterion.
It can be stated with some assurance that the parameters
of primary importance are not the same in all places at all
times.
The simulation programs reported here were based on
the methods of collection of the Village of Winnetka,
-------
2-35
Illinois using data collected there in 1958 and on the
City of Chicago. Among the kinds of results seemingly
yielded by the simulation program in these circumstances
is the idea that the daily route method—^ is unsuitable
when the ratio a/u (standard deviation/mean) for refuse
l^/
production exceeds 0.15—.
The significance of this particular result is
large because (no matter the actual value) it gives a
quantitative bound on the usefulness of a particular
collection discipline where none existed before. A
systematic management science for solid wastes can only
be built out of such pieces.
At one time it would have been appropriate to
extrapolate from the above to a speculation that simula-
tion would be an important tool in the future of solid
wastes. This is no longer necessary; the future has
arrived and as we have noted, simulation is being used
widely.
—See Chapter 3.
— 'See Chapter 4.
-------
CHAPTER 2
RATIONAL METHODS FOR DESIGN OP REFUSE
COLLECTION/HAUL SYSTEMS
As stated in the previous chapter, the first
comprehensive scientific effort to develop rational
methods and design criteria for refuse collection/haul
systems was developed by the University of California
Sanitary Engineering Research Project In the early 1950's.
The results of this work are reported in An Analysis of
Refuse Collection and Sanitary Landfill Disposal ,
-l c/
Technical Bulletin No. 8, Series 37 (December 1952)—x .
Because of the intrinsic Importance of this work
and because many of its concepts and results have been
borrowed In the development of the present simulation
program, it is necessary and proper to devote this chapter
to a detailed exposition and discussion of the report.
To accomplish this, the balance of this chapter will
follow this outline:
A. An overviev; of the University of California
Report.
B. Factors Influencing the Pickup and Haul
— "The report" for the balance of this chapter.
2-36
-------
2-37
Operations.
C. Development and U&e of Rational Design
Formulas.
D. An Evaluation of the University
of California Report.
-------
Section A
An Overview of the University
of California Report
The motivation for the study itself and for its
methodology is best expressed by the following quotations
from the report:
"The rational design of a refuse collection system
is an engineering problem. The efficiency of a
collection system depends on the judicious combi-
nation of the size of the collection vehicle,
number of men per vehicle, the type of refuse
pickup, and the number of trips per day for a
given haul distance to the disposal site."
"The principal reason for resolving refuse
collection into the unit operations ...-pickup,
haul, off-route, and at the disposal site - is
to permit a rational analysis and design of the
collection system."
"An economic analysis of refuse collection
systems entails the evaluation of the two princi-
pal components of cost: the labor cost and the
operating expense."
An analytic scheme with such aims must eventually
have numerical inputs and some knowledge of factors in the
2-38
-------
2-39
environment which influence these inputs. Accordingly, a
field survey was conducted on thirteen California cities
during 1950 and 1951. The results of this data gathering
are reported in three tables for which a summary follows:
Table I: Presents unit refuse production in each
of the thirteen cities in two volumetric and two gravi-
metric units: yds3/service/week, ftVcapita/day,
Ibs/service/week, and Ibs/capita/day. Volumetric units
are not used after their initial appearance here. This
is because observations cover both open-body and
mechanical-packer trucks, so that the compaction ratios
have to be known in the case of the latter. These are
hard to determine since first of all they depend on the
composition and the moisture content of the refuse, and
second Inefficient use of the packers was observed to be
common. The use of Ibs/capita/day also vanishes, no
doubt because in conducting a field survey it is easier
to count the houses than the people on a block.
The refuse production figures are presented as
city-wide averages for all trips observed within a given
city. The cities are segregated into three groups on the
basis of frequency of collection. The first group (eight
cities) with primarily once-a-week collection showed an
-------
2-40
"average" refuse production of 32.3 lbs/service/weeki£'
The second group with primarily twice-a-week collection
(combined refuse in both groups) showed an average of
47.0 Ibs/service/week. The third group represents two
cities having separate collections of garbage (including
commercial swill — primarily restaurant wastes) and
trash at different frequencies. The ratio of the group
II to group I average is ^1.4. This propensity to set
out more refuse when more frequent service is provided
though not supported by tests of statistical validity
has been observed by Quon and Tanaka [14] and more
recently by Smallwood and Gallerii/ This figure, -1.4, is
used in a curious computation on manpower requirement
which will be enlarged on below.
— Quotes have been used on the word average because this
author Is dubious of the way they are computed. Rates,
percentages, etc. unless based on equal numbers of obser-
vations should be done by weighting, i.e. if p. = refuse
production (Ibs/svc/wk) and S, = number of services in
the 1-th city the average should be computed
as
PI SI * Z si
_ T
not p = — Z p.
* n vi
where the summation is over n cities . A similar criti-
cism applies to many of the other averages displayed
throughout the report. Having no detailed information
from which to recompute them, we will continue to use
them without further comment.
17/
— - Personal communication.
-------
2-41
Table II: Presents pickup time measured in man-
minutes per ton for the commercial districts of seven
cities. Adflitional information presented for each city
includes crew size (men), commercial pickups as percent
of total, and tons per trip. The average pickup time
is reported as 136 man-min/ton.
Table III: Presents pickup time measured in man-
minutes per ton for primarily residential areas of eleven
cities (19 observations = 4 cities both 1950 and 1951 and
7 cities either 1950 or 1951). The additional information
provided includes type of truck (packer or open), crew
size, reatf of house pickups as percent of total, residen-
tial pickups as percent of total, service density
(services per mile), and a breakdown of the pickup opera-
tion by pereentage of time spent in sub-tasks such as on
truck, on property, loading, etc. The average residential
pickup time is 1^8 man-min/ton. The discrepancy between
the commercial and residential times can be ascribed
chiefly to greater container density for the former.
The basic data presented, the report then turns
to an analysis of factors affecting refuse production,
pickup time and haul time. This part of the report will
be discussed in greater detail in the next section.
-------
2-42
The analytic models are based on the division of
the total operation into the unit operations of pickup,
haul, off-route, and at disposal site. These are defined
as follows:
Pickup: Net time consumed during actual collec-
tion from the first stop with an empty truck to the last
stop before proceeding to the disposal site.
Haul: Net time required for the round trip to
the disposal site. This begins with the last container
of a given route and ends either with arrival at the first
container of the succeeding route or at the garage or city
yard at day's end. It excludes time spent at the disposal
site unloading or waiting to unload.
Off-route: A catchall for time spent resting,
smoking, eating, on coffee breaks, contacting residents
or supervisors, or not otherwise ascribable to the other
three activities. This has been set at a flat 15 minutes
per trip throughout the balance of the report. No allow-
ance for this factor is made in the simulation programs.
At_ Disposal Site; Unloading or waiting to unload.
This is set at a flat 5 minutes per trip for the rest of
the report. It is treated as a random variable in the
simulation with waiting time dependent on the number of
trucks competing for the unloading spaces.
-------
2-43
The total labor component of the cost is built
out of these unit operations In the following word
formula.
Total Man-Minutes _ Pickup . Haul Off-route At Site
per Trip ~ Time Time Time Time
One may divide this formula by average tons of refuse per
trip to get a formula expressed In man-minutes per ton.
The latter formula when multiplied by the average wage
rate in dollars per man-minute yields the labor cost of
the operation in dollars per ton.
The operating expense component of the cost is
obtained from the formula
Total Operating Annual Average Operation &
Cost per Year = Depreciation + Annual + Maintenance
per Truck (Str.Line) Interest per Year
This is divided by total tons collected per year per
truck to yield operating expense in dollars per ton which
is commensurate with the labor cost.
A final section of this report is devoted to using
the formulas to design a system for a hypothetical city.
We shall review this example in section C of this chapter.
-------
Section B
Factors Influencing the Pickup
and Haul Operations
A significant part of the report is concerned
with establishing quantitative measures of the effect of
various environmental and system design factors on refuse
production, the pickup operation, and on the haul opera-
tion. In this section, we shall examine these findings
and comment briefly on how they relate to the simulation
programs.
We have noted the effect of frequency of collec-
tion on refuse production. A similar effect was noted on
pickup time "other potential variables being constant".
The ratio of pickup times for once and twice
weekly collections was observed to be 1.2.
This is accounted for by noting that in each
of the two twice-a-week collections the crew must cover
the same amount of walking on the property and container
handling as in the one once-a-week collection and all
this for seven-tenths of the base amount of refuse. Using
the 1.2 ratio stated the time requirement for the twice-
a-week collection is .7(1.2) = .84 or 16% less than that
for a once-a-week collection . (See the comment on the
-------
2-45
use of this ratio (1.2) In section D of this chapter.)
A part of the field survey noted only cursorily above
consisted of breaking the pickup operation down Into
component operations and observing the percent of time
devoted to each. These component operations and the
average percentages (Table III) are:
(a) On Truck 24.9
(b) On Street 15.5
(c) On Property 17.6
(d) At Container 14.1
(e) Loading 20.1
(f) Waiting 3.7
(g) Resting 4.2
Time studies such as these can be useful when data are
available to indicate how variations in the collection
methods, equipment, etc. affect them. Then formulas may
be devised to predict the change in overall pickup time
which in turn can be plugged into the formulas for
rational design,the development of which we shall trace In
the next section. Or such predicted changes can be used
as input to the simulation program.
-------
2-46
The frequency ratios 1.4 and 1.2 which we have be
been discussing do not enter into the simulation program
except insofar as the user of the program uses them to
alter the input data.
A second factor which was readily observable by
the field survey was the effect of container location.
Each pickup that was observed was classified as either a
"rear-of-house" pickup or an "alley or curb" pickup,
and the resultant trip was noted as to its percentage of
the former. (Zero percent ROH = 100% AOC, etc.) Rear-
of-house implies that the collectors must enter upon the
property and either transfer the refuse to larger "tote"
containers or bring the resident's containers out to the
18/
street and (in some areas) return them when emptied.—
Alley collection implies that there are alleys bisecting
the blocks (going out of fashion) and the "on property", "at
container'^ and"on street"components of the pickup are
reduced, the latter because the alley makes both sides of
1 fi /
—In Redlands, California, rear-of-house collection is
practiced with the collector driving a skip-loader trac-
tor up the driveway to the container. He then empties the
container into the hydraullcally operated bucket on the
front of his vehicle. The bucket is set at a convenient
loading height. After several such visits, he finds the
packer truck and uses the hydraulic system to empty his
load directly into the compactor. More than one such
tractor is assigned to a given truck. This system
obviates some of the drawbacks of rear-of-house collection
as well as some of the inefficiencies of mechanical packer
trucks.
-------
2-47
a block accessible simultaneously. Curb collection
requires the resident to put his containers on the curb
on the appointed day(s) and remove them when emptied.
It is possible to spot two cities using 1005? or nearly
so alley or curb pickup in the data of Table III. These
are Burbank and Long Beach with zero % of time in each
spent on property or at container and on street compon-
ents of only 5 and 6% respectively. The effect of con-
tainer location on pickup time is shown by the trend
line (least square?) of Chart 1. Zero percent ROH yields
104 man-min/ton on the line while 1005? ROH yields 162
man-min/ton. (See remarks about this chart in section D
of this chapter).
Container location per se does not enter into the
simulation program except as it is employed by the pro-
gram user to alter input data.
A third major influence on pickup time observed
in the field survey was the use of open-body vs. mechani-
cal packer trucks. The comparisons between truck types
is made in Tables IV and V. While we have expressed some
doubt about these comparisons in section D below, taking
them at face value we find that open-body trucks are more
efficient with an average pickup time of 135 man-min/ton
while mechanical packers take 159 man-min/ton. In the
-------
2-48
analytic sections of the report, the discrepancy is taken
to be IQ%. The chief mechanism for this difference as
discerned from Table V seems to be the waiting component
of the pickup, i.e. time spent at the compactor with con-
tainer to be emptied while the compactor digests material
just put into it. For open-body trucks this comes to 1.9%
of pickup time (or .019 x 135 = 2.57 man-min/ton) while
for mechanical packers it is 7.0% of pickup time (or
.07 x 159 = 11.13 man-min/ton for a net difference of 8.56
man-min/ton). Again, truck type influences the simulation
program only via data input.
While the use of open-body trucks may be a moot
point as we enter the 70's, this choice as well as
frequency of collection and container location together
with decisions on separation of refuse types and types
acceptable for collection exemplify the general consid-
eration "level of service expectation" which was discussed
in the Preface.
Turning to the haul operation, the basic determi-
nant is, of course, distance, but comparisons of effici-
ency can be made If the labor time for the haul expressed
in man-minutes per ton is normalized by the distance to
yield haul efficiency in man-minutes per ton-mile. The
-------
2-49
observations range from a low (best) of 1.75 man-min/
ton-mi at Lodi for a 21.9 mile trip with an average 4.34
tons per trip to a high of 7.63 man-min/ton-mi at
Orovllle for a 7.8 mile trip with an average of 1.83
tons per trip.
The significant observation about the haul is
that the average speed of the collection vehicles
increases with the distance travelled (Chart 5) hence
if a • crew size, c « tons carried per trip, T = time
for trip in minutes, d « distance in miles and S = speed
in miles per minute we have for E = haul efficiency.
E - £ I , _a_T . _a_ (2a)
c d c ST c S
or
<
dd 2 dd
since we have noted that 5d > ° • This is seen in Chart
4.
Haul efficiency is an output of the simulation
and is as we shall see subject to wild distortion by small
partial loads. Chart 5 is used by the simulation program
to provide the mean speed of the refuse vehicles.
-------
Section C
Development and Use of Rational
Design Formulas
The word formulas which have been introduced above
can be rendered quantitative by the introduction of the
following variables. The symbols used are those given by
the report itself and differ from those used in discuss-
ing the results of the simulation runs. In the use of the
formulas, the report assumes, as noted, a flat 15 minute
per trip off-route and 5 minutes per trip at dispoal site
time.
a - Number of men per truck (crew size).
b = Pickup time in man-minutes per ton.
c = Average tons of refuse per trip.
d = Round-trip haul distance in miles.
e = Average (Inverse) haul speed in minutes per
mile.
f » Total man-minutes per trip.
g = Number of trips per day.
h = Initial cost of truck in dollars.
i = Interest rate on capital.
k = Total number of trips per year.
s = Useful life of truck in years.
2-50
-------
2-51
m » Total mileage per trip » d + route mileage.
0 « Operation and maintenance of truck, dollars
per mile.
q » Total operating cost per truck per year.
x - Total man-minutes per ton.
y = Total operating cost (fixed charges, operation
and maintenance) in dollars per ton of refuse
collected.
Accordingly, the formula for total man-minutes per trip
becomes
f = be + ade + 15a + 5a (2.3)
where the components on the right are respectively pickup
time, haul time, off-route time, and at disposal site
time per trip. Dividing both sides by c we get
x = £ - b + |(de + 20) (2.4)
C C
total labor time in man-minutes per ton for all unit
operations. Finally, if we let (these symbols are not
used in report)
-------
2-53
w » Average wage rate in dollars per man-
minute.
CL = Total labor cost in dollars per ton
CL = wx (2.5)
which gives the total labor cost in dollars per ton.
Formula (2.4) is solved in the report by the con-
struction of nomographs, Chart 6 for long hauls and Chart
7 for short hauls. Here, of course, we shall solve (2.4)
directly. The proposed use of formulas (2.4) or (2.5) is
to gauge the efficiency of existing systems. Thus, if
we have
d = 16 miles.
e = 3 min/mile (20 mph).
a = 3 men.
c = 3 tons/trip.
b =150 man-min/ton.
w = .025 $/man-min ($1.50/hr).
= .025C150 + |(16?3 + 20)] = .025(218) = $5.45/ton
which may be compared to the existing situation by
-------
2-53
reducing the annual refuse budget by 10% (to account for
administration and employee benefits), subtracting off
the fixed and operation maintenance costs of the fleet,
and dividing by total tonnage collected to get the com-
parison figure.
Operating costs per year per truck are obtained
from the second formula of section B as follows:
£ + |j(s + 1) + km0 (2.6)
where the three terms on the right are respectively
annual straight line depreciation,average annual interest,
and operation and maintenance per year per truck. If this
is now divided by ck * total tons per year per truck we
get
the total operating cost in dollars per ton. In formula
(2.7), the two terms on the right are respectively fixed
charges and operation and maintenance.
Continuing the above example, if
m = 20 miles/trip.
g = 3 trips/day.
-------
2-54
h - $2300.
3=6 years.
0 = .16 $/mile.
1 = .04.
We have with the additional Information that the trucks
work 6 days per week, k = 52'6»3 = 936 trips/yr/truck
and
$.15/ton + $1.07/ton
As before, nomographs are provided to facilitate solution
(but restricted to I = .04).
Thus, the total collection cost for this system
is $6.67/ton exclusive of administration and employee
benefits ($5.45/ton + $1.22/ton).
In considering formula 2.7 it should be noted
that other methods for computing depreciation and average
interest are sometimes used, that salvage value of the
truck at the end of its useful life is ignored, that the
factor 0 Itself consists of fixed and variable items,
— The report erroneously introduces $1.82/ton into subse-
quent computations after first finding $1.21/ton from the
nomographs.
-------
2-55
and that In considering operating expense on a per truck
basis economies of scale are Ignored. It is Interesting
also to speculate on whether formula (2.7) can be used to
answer such a question as: How much more would It be
worth to put into 0 if doing so would purchase an extra
year of useful life? Proceeding on the naive assumption
that this is possible, one finds the break-even point to
be 2-tenths of a cent per mile or 20 x 1136 x .002 * $37.
per truck per year. This seems too good a bargain to be
true.
Turning from the evaluation of existing systems to
the design of a total system from the ground up, and bas-
ing such designs on a fixed length of_ work day of 8 hours
(480 minutes) and fixed number of trips per day we obtain
from formula (2.3)
* be + a(de + 20) (2.8)
g
or dividing by a
480
. + de + 20 . (2.9)
Formula (2.9) is solved in four nomographs, Charts 10 - 13
for (g = 2 and 3) x (e = 3 and 3.5) .
-------
2-56
The example used to illustrate the use of formula
(2.9) is the following (slightly simplified). In the
following the example we shall concentrate on the design
selected as most practical and instead of tabulating all
the inputs, we shall introduce them where first needed.
In performing this sample design, an eight-fold
table (Table VI) is constructed: (g = 2 and 3) x
(a = 2 and 3) x (ppen-body and packer-body). Of these,
the most economical proves to be impractical because it
requires too large a load (tons/trip). The computations
for the second best design are then
b = 147 man/mln/ton.
g = 2 trips/day.
a = 2 men.
d = 4 miles round trip to incinerator.
e = 3-5 min/mile.
Determine c from (2.9)
-ifa-20 -1(3.5)
or c = 2.80 tons/trip, and from (2.4) and (2.9)
a 480 2 480 ,„., . .,
x = — • —— = -5—75- • —^— = 171 man-min/ton
C K c. * O cl
-------
2-57
Thus, If w « .025 $/man-min
C, = .025(171) - $4.28/ton ($4.27 in report) .
LI
Finding the operation expense is not as simple as
the report purports since the route mileage (I) which must
be added to d to get m must be found:
Assuming that the truck works 6 days a week,
52 weeks a year, the total yearly tonnage
for a truck is
6 days 2 trips x 52 wks x 2.8 tons _ 17^7 tons
week day year trip.truck" yr.truck
But we know that residential refuse production is
i?-i*n «„„« x 32 wks 1 ton 48 Ibs _ 15175 tons
12160 svcs x __ x 200Q ton * __ _
tons
JOJ-fP
Hence there must be
yr. truck
in the fleet (for residential pickup).
Thus, with residential density at 90 svcs/miie
and collection frequency at 2 trips/route/week
20/
— (covered quotient)
-------
2-58
12160 eves x 2 trips/route/week x 1 ^ 2.51 mi
9 trucks 2 trips 5 days 90 svcs *" route
day truck week mile
Finally, m=d+I=4+2.51=6.51 or 7 to the
nearest mile.
In computing the operating expense we assume
h = $7500.
k = 624 trips/yr(= 6 x 2 x 52) .
1 = .04.
s=6 years .
0 = .16 $/mile.
Thus,
I1 =
Thus, assuming an incineration cost of $2.15/ton we have
a total cost of 4.28 + 1.20 + 2.15 = $7.63/ton ($7.62/
ton in report) .
As a sample of the questions which this rational
design method can answer, consider:
If sanitary landfill disposal costs $.66/ton, how
far can the refuse be hauled at the same total cost
-------
2-59
(exclusive of administration, etc.) as that Just computed
for incineration?
In order to answer this question and to eliminate
the need for trial and error with the nomographs, we
propose here to extend these rational methods by develop-
ing a formula which relates the economic capacity c to
the round-trip haul distance. We shall assume
(1) 52 weeks operation per year.
(2) 20 minutes/trip off-route and at disposal
site. (The place to change this in the
formula is obvious.)
(3) e(the inverse speed, in min/mile) will be
treated as a constant rather than as a
function of distance.
(*O In computing the number of trucks we shall
Ignore the need for a covered quotient.
We shall need the following additional notation:
D = Days worked per week.
P - Shorthand for J- [1+ ~(s +1)].
S c.
i = Frequency of collection (trips/route/week).
J * Job, i.e. number of services.
Ki * Total cost for collection, haul and disposal
by incineration ($/ton).
Kg = Cost for disposal (only) by sanitary landfill.
-------
2-60
P » Refuse production (Ibs/service/week).
p » Service density (services/mile).
T = Number of trucks in the collection fleet.
Repeating some of the computations for I In general form
Total Yearly 2 (2>10)
Tons per truck J &
Total Yearly -- TP
Refuse Production = ^~£r (2.11)
tons/year 200°
T = - = (2 12)
(2.10) 2000 Dgc ^.1^
J<( - 2000
p
Finally, noting that k = 52 Dg
or
H (de + 20) + _h d . 2000
-------
2-61
whence
[K± - K8 - wb - 2°g° *0] c = [wae + 0]d + 20 wa +
(2.l4a)
Inserting all known values except d and c and using
e = 3 we have for a » 3 , g = 2
3.137 c - .385 d + 3.784 (2.l4b)
Thus, If trucks of 4 tons capacity are a practical stand-
ard size (costing $7500), a haul of 22.8 miles (one way
11.4) would be practical. In fact, the method of trial
and error yield d = 18, c = 3-38 which lies on the
line (2.l4b). Considering the compaction ratio this
yields an odd-size truck which is rounded up to 15 yards.
-------
Section D
An Evaluation of the University
of California Report
The basic method for supplying scientific insight
and predictive power in an area where little or none has
been used before is to observe, to devise an unambiguous
terminology to describe the phenomena, to define useful
units for measuring those entities which can be measured,
and to provide a conceptual framework for viewing the
interrelations of component parts. In that this has been
accomplished by the University of California report, it
is praiseworthy. If there is anything of a major nature
to criticize (besides the method of taking averages) it
Is the over-reliance on the use of averages as a means of
comparing group characteristics without any statistical
tests on the validity of those comparisons.
The data gathered in the field survey phase of
the project fairly cry out for the use of analysis of
variance for the separation of effects and the assessment
of interactions. Thus, for example, in assessing the
determinants of pickup time, such factors as service
density, open vs. packer, percent alley-curb vs rear-of-
house, crew size, city, and year data was gathered are
2-62
-------
2-63
all valid classifications for a nearly textbook analysis
of variance model. This lack of statistical sophistica-
tion is reflected in Table IV where we are told that
mechanical packers average 159 man-minutes per ton to
135 for open-body trucks despite a glaring difference of
33 containers per mile between the two groups. Again,.
in Chart 1 we note that of the 1? points plotted to give
a trend line on the pickup time as a function of percent
rear-of-house pickups, five of the six open-body points
are above the line and seven of the eleven packer points
are below the line; doesn't this Justify the use of two
lines?
A curiosity in the report is the following compu-
tation based on the factor 1.4 = ratio of average twice-
a-week production to once-a-week production and the fac-
tor 1.2 = ratio of twice-a-week pickup time to once-a-
week pickup time:
"Thus, if three men were required to provide once
weekly pickup service to a given area, theoreti-
cally 3 x 1.2 plus 3(1.4 - 1.0) = 3.6 + 1.2 or
4.8 men would be required to supply the same area
with twice weekly service."
The actual answer depends on the assumption one
makes as to how much time it should take to perform the
-------
2-64
two collections: Assume base data of 1 ton refuse pro-
duction from a given area and a pickup time of 150 man-
min/ton collected once-weekly.
1. Require that the two collections be done in the
same time (Tx) as the singleton collection
150 x IT
3M = 50 min
1 80 — x 1 4T
L u ^ -1 '
= 5.04 men (= 1.2 x 1.4 x 3)
2. Decide that pickup time is proportional to the
amount of refuse and therefore require the Job to
be done in 1.4Tj minutes
i Rn — x i 4T
/, j, v J.OU -=• X 1. qi
} = 3'6 men (= 1'2 x 3) •
3. Decide that the '"terrain" is the main influence
on pickup time hence require the job to be done
in 2T minutes
-------
2-65
— x i
'
= 2.52 men
Each of these is correct under the assumptions
stated. The question is which is the correct assumption?
This author believes that the correct assumption is a
mixture between the extremes of 2. and 3., although as
we shall see below, 3. is more nearly correct for the
Village of Winnetka, i.e., the physical requirements of the
collection procedure (walking, toting, etc.) dominate
refuse production as the determinant of pickup time.
It should be pointed out also that the unit
operations defined here do not fit a situation in which
the collection personnel do not accompany the driver and
truck to the disposal site. One such collection disci-
pline, the relay or shuttle method, as described by
Refuse Collection Practice [5 - pp. 153-15*1] has a driver
with an empty truck meet the collectors at the point
where the previous truck left off. A modified relay
method is evaluated in Chapter 5.
-------
CHAPTER 3
DEVELOPMENT OP THE SIMULATION PROGRAM
Section A
Sources of Variability in Refuse
Collection/Haul Operations
In the Preface, we noted that the significant
parameters of a refuse collection system are subject to
variability or fluctuation. We identified these types of
variation as being long-term or secular variations,
seasonal or cyclical variations, and short-term random
fluctuations. This work concentrates on the latter two
types of variation, and to these we add, by way of con-
trast to the use of average values only, the real inhomo-
geneities of physical placement, 1. e. ,distances travelled
by collection vehicles are measured from actual points of
departure for the disposal site. We add, also, variable
amounts of waiting time at the disposal site.
Moreover, in using simulation to study the effects
of such variability on the operation of a refuse system
we need, as we inaicated in Chapter 1 of Part 2, know-
ledge of the probability laws governing the observed
randomness. The use of random number generators to
2-66
-------
2-6?
produce random inputs according to some probability law
is concealed in every crevice of the simulation program.
It is, therefore, the purpose of this section to reveal
these places, the probability laws used, and the Justifi-
cation fcr their use.
Determination of Refuse Amounts: A survey of the
Village of Winnetka in 1958 provided data which when
plotted on arithmetic probability (graph) paper by indi-
vidual trucks showed that the probability dis-
tribution of refuse production is reasonably close to
normal. One such plot for truck PW43 is shown as Figure
2.4. For all five trucks, the mean (M), standard
deviation (a), and number of trips in the sample (n) are
shown in Table 2.1. The same data revealed, to a high
degree of statistical confidence (Chi-square test), that
in Winnetka's twice-a-week collection the refuse amounts
produced when four days had elapsed since last collection
were 4/3 the amounts produced when three days had elapsed.
This is another way of saying that refuse production
appears to be uniform over the refuse collection oycle.
A further major source of variation in refuse production
is seasonal variation. In most areas the increased
availability of fresh produce in the summer months com-
bined with lawn and garden trimmings raises the refuse
-------
2-68
£ _ , ; _p ( r"T~~
\ '
..
• 'T " i 4_"t . M _,.
j i_ i-
1 O LrrLii L riJimi HJL iJ
^S"|£^^^Tfr"fl
20 :p=i=B=="= ;fF|f^:^:-=j:=:^== J::==
—I -T ( 1 A- — i
*JA .Jd •! * a^ ^L •&
! I T
1 j ,
3 , 1 J
40 "T^-i •fpT--fr-"+'~l "-p
h- T^ri r~EiS-'-i:-^S^
50 ^"T^-^-^f TTBi±tfr "1
I F J .
iT i2T t
~T~ 1 ^'j. ^ !
«c— i ^"^ -I
TA -....M.--.. .... -L ,;..,,. -i-f -- ,'tfi ........ —i-1- ..« -
^} ...... J 1 ^ -I- -L. -l
80==:^ = i:::^!::z::=^^=|::r:
=f ^"J !::i"£t:::=ffi^~J
" T" jir ' ' "1 i ' '
90 £"-^z i +-S-T-
TV _j — ^f. ^ 1--1-4- — j
4 ^
A k \
os ^-i ., ..J.L. , , , ,LL.!L_.,J,J._..
1 !
f I Tii
98 1 I i ..... ).., iLll-
~r~r '•*
i ,•'
^ ^
'3
31 rf '
!.--! -M _
==5: = |S = EE|:| = iJEE:|SEE;
::;S;:;:g::j:=: = :::±::: = i:;: go ^
3jr Tn"t
*-l* IT «
:~ ~"iL::":::4±::i-i:::::::::ii o
j * j Q£
±; ::~::::::: : :±::::: ::i:i::::: «-»
( F , J O
-u_X-i-- . T T. ou Z
i ^ui.-i-'t -- +•-- **
::::ilafflt::£::::::::±=:::±:: 50 £
T *"
_:i_:^oi::»: ::::::»;:: ::::^;m: 40 K!
... _j
»«•
_ _._ ... .4. - . ^
. _. ~* 3i..;_. u
j_ . _)__ Qt
^^^^^^^^^20 ^
E£i£E ~"? —
r tf'T " 10
..A. 1...1 i. 2
65 70 75 80 85 90 95
REFUSE PRODUCTION- LiS/SiRViCE/WiEK
100
Figure 2-4 Probability Plot of Refuse Production,
Winnetka, Illinois, March - April, 1958
-------
2-69
Truck No.
PW 3
PW 12
PW 43
PW 47
PW 54
Mean(n)
78.5
75.1
80.6
84.0
76.1
Std.Dev.(a)
13-1
19.9
10.0
5.6
15.3
a/n
0.167
0.265
0.12U
0.067
0.201
Number of
Trips (N)
7
8
8
7
7
Table 2-1 Statistics on Refuse Production.
Winnetka, Illinois, March-April,1958.
-------
2-70
production of these months above that of the winter
months. An example of this can be seen In Refuse Collec-
tion Practice [5] in Figure 14 (p. 34) which shows the
weekly production in tons for a full year in Cincinnati,
Ohio. The magnitude of the summer peaks is in the
neighborhood of 1.3-1-5 times the winter and spring
levels. Provision for the influence of seasonal varia-
tion has been made in the simulation program. The
detailed description of the "refuse-generator" is included
in Appendix 2-A.
Determination of* Pickup Time: Linear regression
methods were used to relate the time of refuse collection
(minutes) to the amount of refuse collected and the num-
ber of services. It was found that for WInnetka the
effect of the amount of refuse Is statistically insignifi-
cant so that the refuse term is dropped from the regres-
sion formula. The explanation for this is simply that
almost all residential refuse in Winnetka is collected
from the rear of the house in a city virtually without
alleys. Thus the subcomponents of the pickup operation
such as "on property" and "at container" are dominant in
the pickup operation and much the same time is required
no matter what the amount of refuse.
-------
2-71
The resultant regression formula takes the form
6" = Time (minutes) = aS + y (3-D
where S = number of services and
a, -y are the regression
coefficients
This formula has been modified to include the effect of
route mileage.(See Appendix 2-A.)
Common sense decrees that any activity conducted
out-of-doors must be affected by the weather. A means
for incorporating weather severity as a factor in the
time computation is described in Appendix 2-A.
Determination of Distance: Each point in the
collection area which could conceivably be the departure
point for a trip to the disposal site (only one disposal
site) is known by its physical coordinates on the city
map. Thus, when a truck is full (see next section) the
distance is computed from where it actually is. The
method for distance computation is by the 2-,-or metro-
politan metric which has been described in Part I . If
the amount of refuse collected is a random variable then
-------
2-72
it follows that the point at which a truck of given
capacity is fully loaded is also a random variable.
Determination of Haul Speed: The basic model for
the determination of haul speed as a function of haul
distance was derived from Chart 5 of the condensation of
[1] appearing in Refuse Collection Practice [5] (Figure
29 of original report). The detailed description of
this model is contained in Appendix 2-A. The haul speed
derived from this model is used as the mean of a normal
distribution with a standard deviation of 0.1 x the mean
speed. There is no empirical justification for imposing
random variation on haul speed in this manner except that
it appears reasonable and is preferable to no randomness
whatever.
Time Spent a_t Disposal Site: The disposal site
was modeled after the incinerator plant of Evanston,
Illinois. Here, there are a limited number of unloading
docks which are used by the trucks on a first-come-first-
served basis. Thus, a variable wait-to-unload time may
be generated. This time may be eliminated simply by
setting the number of unloading docks equal to or greater
than the number of trucks.
Once an unloading dock has been secured, the
actual unloading time is generated as a random variable
-------
2*73
uniformly distributed between 5 and 9 minutes. The
choice of these limits and of the uniform (rectangular)
density was based on conversations with Winnetka offi-
cials. For the simulation model reported in Chapter
2-5, based on the City of Chicago, the limits used were
5 to 15 minutes and 2 to 6 minutes.
Determination of_ Capacity; For the simulation
program reported in Chapter 4 , based on the daily route
method (see section B below) of the Village of Winnetka,
randomness was used as a method for simulating the
behavior of the crews in determining when the truck was
sufficiently full to require a trip to the disposal site.
The capacity rule (see subroutine CAPRUL in Appendix 2-A)
used was briefly:
1) If this was the last CU (see section B below)
of the day, permit up to 5% overload.
2) If this was not the last CU of the day,
generate a random pseudo-capacity = Truck
capacity (tons) times a random variable from
the normal density N(.95, .05). Measure
this pseudo-capacity against refuse already
loaded plus the refuse in the CU.
To quote from an early report on this simulation program:
"The motivation for 'pseudo-capacity' is that it
-------
reflects the actual behavior of refuse collecters
we've observed more closely than would Insisting
that the truck be filled to capacity each time.
Refuse collectors don't stop, for example, when
half a container has been loaded onto the truck
or when they are in the middle of a street block.
They have a 'feel' for when the truck is getting
full (usually determined by the way the truck
handles or 'lugs') and stop collecting when
they've reached a 'good cut-off point'." [16]
It will be observed, nonetheless, that the results
reported in Chapter 2-4 exhibit the effects of small
partial loads which are not prevented by this capacity
rule. An alternative capacity rule applicable to the
daily route method, gained by the wisdom of hindsight,
will be discussed in Chapter 2-6. A look-ahead rationale
is used in the constant length workday simulation of the
City of Chicago reported in Chapter 2-5. This will be
discussed in the next section under differences between
the two programs.
-------
Section B
Some Concepts Underlying the Simulation Programs
As has been noted above, there are two simulation -
models under discussion. The first is based on the daily
route method of refuse collection as practiced in the
Village of Winnetka, Illinois. The second model was pro-
duced by making modifications to the first and is based on
A
the constant length workday method of refuse collection
as practiced in the City of Chicago.
In this section we shall present some of the con-
cepts and organizational methods that underline both
models. Detailed information on the first model and on
the common portions of the second may be found in Appendix
2-A. The specific differences in concept and organization
incorporated into the second model will be discussed here.
Collection Units: The fundamental organizational
concept underlying both models is the collection unit
(CU henceforth). A collection unit is, ideally, the
smallest, compact piece of collection territory serviced
by a single truck for which certain data is available.
It represents a compromise between the homogenizing aggre-
gation of the city as a whole (e.g. the rational methods)
and the cumbersome detail of considering the city service
2-75
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2-76
by service. The input information required for each CU
are:
1) Average refuse production in pounds per
service per day (y).
2) The standard deviation of same (a).
3) The regression coefficients a and y mentioned
in section A above.
4) The days of the week the CU is to be
serviced.
5) The number of services (S).
6) The service density in services per mile (P ).
S
Internal mileage m. is gotten from dividing
services by service density.
7) Physical location: (x, y) coordinates in
miles.
In actual practice, items 1) - 3) are properties
of uP to 10 "districts" which need not be physically com-
pact, and these properties are conveyed to the CU's by
stating to which district they belong. One may note that
the data provided are based on observation. One might
have provided more fundamental data on income, percent
alley pickup, etc. and created a refuse generating program
to predict refuse production based on these fundamental
determinants.
-------
2-77
The important property required in designating an
area to be a CU is that errors arising from considering it
to be homogeneous be negligible. The meaning of homo-
geneity, insofar as the simulation models are concerned,
is that a CU is proportionally divisible. Thus, if a
truck can only load -^ of the refuse found in a CU, when
it returns (as it must under the daily route method) the
2 2
remaining •=• of the refuse will be collected in -^ of the
p
time and the truck will travel =• of the miles (m. ) origi-
nally computed for the whole CU. Moreover, the location
of the CU ((x, y) coordinates) apply to all such opera-
tionally defined subdivisions of the CU for purposes of
computing distances to and from the disposal site or
garage. Both simulation models assume that the string of
CU's which comprise the day's work for the truck which
services them are physically contiguous so that neither
time nor mileage are expended in going from a completed
CU to the next CU on the given day's schedule. A total
of 1000 CU's is allowed.
Disposal Site and Garage; A single disposal site
and garage are provided for. They are specified by their
physical locations (which need not be the same) in a
coordinate system consistent with that used to locate
-------
2-78
the CU's. In addition, the number of unloading docks
('berths, platforms, etc.) available at the disposal site
must be specified.
Trucks: Trucks are specified as to their number
(up to 6k allowed), the crew size K , and capacity in
tons (C).
Time Organization: The time organization of both
models is based on the World Calendar quarter of 91 days
(13 weeks). All summary data are printed out at the end
of a quarter's operation. The next largest time unit is
the month. A World Calendar quarter consists of three
?-\ /
months of 31, 30, and 30 days in that order.— Weather
induced slowdowns in work rate (man-minutes per ton) and
seasonally related adjustments to refuse production are
changed on a monthly basis. The customary seven day week
is the basis for scheduling the work of the trucks. A
given CU may be serviced by its truck up to six times per
week. The next time unit is the day which is the time
unit during which simulation takes place. Within a day
are variable length entities called events.
21/
— The 365-th and 366-th days (when required) are placed
between December 30 and January 1. These are called
World Day 1 and World Day 2 and do not correspond to one
of the 7 days of an ordinary week. Thus, each quarter
begins on Sunday and each year's calendar is the same as
another's except for the number of World Days.
-------
2-79
Program Organization: The actual simulation pro-
ceeds on a daily basis. This daily simulation is called
the inner program. The inner program is governed by an
outer program which regulates the counting of days into
weeks, months, and quarters, and which tallies dally
results into quarterly counters and prints results at the
end of each quarter.
The day begins with the dispatching of each truck
that is to work on a given day of the week to its first
CU of the day and ends when all trucks have returned to
the garage. In between lies a variable sequence of
events of minimal length zero. These events (listed in
the order of their implementation as computer programs)
are:
1) Truck arrives at_ disposal site: Here a truck
may encounter delay if no unloading dock is
open. Once an unloading dock is secured, a
variable random unloading time is incurred.
2) CU is_ entered for first time: The refuse
amount, time and travel are computed and the
capacity rule is invoked to determine whether
the entire amount of refuse can be loaded.
3) Truck leaves disposal site: For the garage
or to resume collecting.
-------
2-80
4) Collection unit Is partially completed: This
event Is created when event type 3 or 5
(below) determine that there Is too much
refuse in a CU to fit the remaining capacity
of the truck.
5) Truck enters partially collected CU: This
event is created when event type 1 or a
previous event type 5 has occurred without
finishing the CU.
6) Truck arrives at_ garage; The day ends when
all working trucks perform this event.
Full program details and interactions of these
events may be obtained from Appendix 2-A.
Within the inner program is an event scheduler
called the event clock or the time status record (TSR).
The function of the TSR is as follows: When events are
created, they are set to occur at a certain number of
minutes from the time of creation. At the same time, all
pending events are stacked in a list telling how many
minutes from now they will occur. The TSR takes the newly
created event and places it in the list so that all these
times are in ascending order. Thus, the earliest occur-
ring (next) event is on the top of the list. When an
-------
2-81
event is to be executed, this top-of-the-list event is
removed and the number of minutes from now when it was
to have occurred is subtracted from the occurrence time
of the other pending events. This same time is added to
the time of the day counter — thus, time passes in
variable length segments — asynchronously. For full
details see Appendix 2-A.
Daily Workflow of a Single Truck; The simulation
models are best understood in terms of a workflow diagram,
of a single truck's operation for one day. It must be
pointed out that in such a workflow diagram, the workings
of the TSR and the actual program operation are lost.
The first workflow diagram pertains to the daily
route method of refuse collection as practiced in Winnetka,
Under this method, each crew is assigned one or more work
areas which are to be serviced at some prescribed weekly
frequency. The area assigned for any given day must be
completely serviced on that day. Each crew operates
independently, helped by and helping no other crew and
quitting early or late as the workload demands. The work-
flow for this method of collection is shown in Figure
2-5.
Before considering the daily workflow of a truck
operating under the constant length workday rules of
-------
2-82
ARRIVE AT UNSERVICED CU
GENERATE AMOUNT OF REFUSE
THEREIN, TIME TO COLLECT
IT ALL, AND PROPORTION
THAT WILL FIT INTO UN-
USED CAPACITY REMAINING
IN TRUCK.
CALCULATE TRAVEL
TIMETOFIRSTCU
OF THE DAY.
LEAVE'GARAGE
LOAD THAT PROPORTION
SELECT NEXT
CUTOBE
VISITED.
WAS IT ALL
LOADED?
IS THIS THE LAST
CU FOR TODAY?
CALCULATE TRAVEL TIME
TO DISPOSAL SITE.
ARRIVE AT DISPOSAL SITE
IS AN UNLOADING DOCK
AVAILABLE?
CALCULATE DUMPING TIME
ARRIVE AT PARTIALLY SERVICED CU
REDUCE REFUSE BY AMOUNT ALREADY
COLLECTED, COLLECTION TIME BY AMOUNT
ALREADY SPENT. DETERMINE PROPORTION
OF REMAINDER THAT WILL FIT INTO
EMPTY TRUCK.
STAY ON QUEUE UNTIL
ONE. IS VACATED.
CALCULATE TRAVEL
TIME TO NEXT CU.
SELECT NEXT CU TO
BE VISITED.
WAS CU COMPLETELY
Figure 2-5 Verbal Flow Chart of the
Movement of a Single Truck.
Dally Route Method,
Winnetka, Illinois
-------
2-83
Chicago, we must Introduce some new terminology and spell
out the ways in which this second model differs from the
first.
The Constant Length Workday Mode:!: In contrast
to the dally route method, crews working on a constant
length workday may leave over for servicing tomorrow any
/ work which they are unable to finish today within the
allotted workday. Any such work which is left over and
then is not done the next day may be counted as not done.
Thus, percentages of services serviced on time, one day
late, and two or more days late provide an index to the
quality of service being afforded. A model based on the
constant length workday must have built into it a set of
decision criteria simulating human Judgment in answering
the question: "Is there enough time left before quitting
time for us to undertake the next task?"
Since this model is designed also to evaluate a
last load relay policy, and since the use or non-use of
this policy affects the way the criteria are computed, we
first define this relay policy. The last load relay
policy is simply to have the driver and the crew return
directly to the garage with the last load of the day.
From there, an auxiliary driver takes the loaded truck to
the disposal site, the regular driver and crew being
-------
allowed to quit within the allot ted workday. This policy
has, of course, cost implications in the wages of the
auxiliary driver, but these will be discussed in Chapter
2-5-
Following the verbal flowchart of the operation
of a single truck under constant length workday rules,
Figure 2-6 , the first place where such a criterion is
examined is just before beginning to service a fresh CU.
The question asked is: "Is estimated loading time
available greater than CLT 1?" (constant length time 1).
That is, if we take the number of minutes presently left
until quitting time and if we subtract from this the
average amount of time we shall have to spend traveling
and the average amount of time spent unloading, will
there be enough time (CLT 1) left to make it worthwhile
to start servicing this CU? To express this as a formula,
assume that we are using an 8 hour workday (480 minutes)
and that
T * time elapsed up to now.
T_TTr. = average time to travel from this CU to
\j\JD
the disposal site.
T__ = average time to travel from the disposal
site to the garage.
-------
2-85
IS ESTIMATED
LOADING TIME AVAILABLE
GREATER THAN CLT17
GENERATE QUANTITY OF
REFUSE THEREIN. TIME
TO COLLECT IT, AND
PROPORTION THAT WILL
FIT INTO UNUSED CA
PACITY REMAINING IN
TRUCK AND CAN BE
LOADED WITHIN LOAD-
ING TIME AVAILABLE
IS TIMF SPENT
GREATER THAN
CLT3?
IS THIS LAST LOAD'
LOAD THAT PORTION
IS REFUSE ON
BOARD GREATER
THAN HIRI . CAPACITY >
WAS ALL REFUSE
IN CD LOADED?
IS ESTIMATED
LOADING TIME
AVAILABLE
GREATER THAN
CLT2?
IS THIS
LAST LOAD?
IS THIS
LAST LOAD?
IS
TIME SPENT
GREATER THAN
CLT3?
Figure 2-6 Verbal Flowchart of the Movement of a Single
Truck.
Constant Length Workday, Chicago, Illinois
-------
2-86
= average time to travel from this CU
to the garage.
TDS = average time spent at the disposal site.
TV = estimated time for non-loading activi-
ties .
ETL = estimated time left for loading.
Then for the non-relay case and for all but the last load
of the day under the relay policy
TCUD + TDG + TDS
while for the last load of the day under relay
TV = T (3.2b)
Using whichever of these definitions of TV is appropriate
ETL = 480 - T - TV (3-3)
and ETL is to be tested against CLT 1. In the examples
discussed in Chapter 2-5, the value of CLT 1 = 15 mln.
and T_o =24 mln.
JJo
-------
2-87
The second place where this sort of look-ahead
criterion is employed is at the disposal site when it is
known that the previous load was not the last load of the
day. The question, "Is estimated loading time available
greater than CLT 2?", translates as "Is there enough tirte
left to make it worthwhile going after the next load?"
For the non-relay case and for all but the last
load of the day under relay
2 TCUD + TDG + TDS (3'4a)
while for the last load of the day under relay
TV = TCUD + TCUG '
In the former of these two formulas for TV, T~,™ is the
average time to travel from the disposal site to the
first CU of the next load (which may have already
been started). Since we do not know, in general whether
this is the same CU that will be being serviced at the
end of the next load, we use 2 T_Tm as an approximation
UUJJ
for the round trip time. ETL is computed as before, but
CLT 2 is computed as a function of TV:
-------
CLT 2 = 6 + e-TV (3-5)
where 6 = 20 min. and e = 0.2.
Thus, the truck and crew return to an unserviced
or incompletely serviced CU only when the net estimated
time left for loading is greater than 20 minutes plus 20%
of the estimated time for other activities.
The criterion CLT 3, queried when it has been
determined that a non-empty truck is on its last load of
the day, governs the use of relay. A value of CLT 3
greater than 500 means relay is not being used while a
value of zero implies the use of relay for the last load
of the day.
It should be noted that the number of trips per
day is fixed in this model whereas there was no such
limit in the dally route model.
Further differences between the two models include
the capacity rule and the introduction of randomness into
the refuse pickup rate (man-minutes per ton) in a differ-
ent manner. The capacity rule used in the constant
length workday model is simply to test refuse already on
board the truck against an input parameter, HIRI, times
the maximum capacity. Thus, for capacity at 5.5 tons and
-------
2-89
HIRI = 0.9, a load of 4.9^ tons would allow further
loading to proceed, but only as far as 5.5 tons.
In the daily route model, the daily weather
severity, WESEV, was derived as a random multiplicative
constant from a month-dependent histogram. When 9 ,
the pickup time for S services had been derived from
the formula (3.1) •
= otS
the actual value was found from
9 = 6-WESEV
(3.6)
In the constant length workday model, 0 is found by
8 - 6'U(0.8, 1.2)
(3.7)
where U(0.8, 1.2) is a random variable uniformly
distributed between 0.8 and 1.2.
-------
CHAPTER 4
Simulation Studies of Refuse Collection and
Haul under the Pally Route Method
The aim of the studies to be presented below Is
to delineate the interdependencles of several significant
parameters involved in the functioning of a refuse
collection system. The method of study is the use of a
Monte Carlo or mathematical simulation technique imple-
mented through a computer program. Monte Carlo techniques
in general and this simulation program in particular have
been discussed in Chapter 2 and 3 and Appendix 2-A.
We adopt the following point of view in under-
taking these studies. First, we suppose that we have
applied the rational methods discussed in Chapter 2
to the solution of the design problem of a situation
akin to that of the Village of Winnetka. Next, we
note that at many points where we made use of average
in this design process, there is actually considerable
day-to-day fluctuation and seasonal variation. We decide
therefore, to study the effects of such variation in our
initial design, but in doing so, we are not willing to
2-90
-------
2-91
explore these effects on too many variations of the base
design. That is, we have exercised some mature engineer-
ing Judgment in deciding that certain parts of the system,
for example — crew size, shall remain Invariant. We are,
thus, using the simulation program for a set of limited,
economical experiments in order to discover without actual
field trials both the effects of variability and which
changes indicated by these effects will be most amenable
or promising for field trial. We are not relying upon
the results of these simulation studies to provide exact
answers to questions arising from the everyday operation
of a refuse collection system. We believe, rather, that
the results from this method of analysis can be relied
upon to describe the interaction of the variables involved
and to differentiate between variables of primary and
secondary importance.
The outline for the balance of this chapter is as
follows: In section A we shall discuss the variables
which we have held invariant in these studies. In section
B we shall present the ranges of the variables allowed to
change. In section C we present the results of some
twenty-five simulation runs (each one run twice) through
a series of graphs and tables. These results are inter-
preted, insofar as possible, on a physical basis. The
-------
2-92
conclusions reached apply, of course, only to the refer-
ence system. These conclusions may be made applicable to
other similar systems by running the program on data
applicable to those systems.
-------
Section A
Invariants of the Reference Systems
Invariant parameters are, as we have noted, dic-
tated by initial judgments on the relative importance of
each relevant parameter; that is, the formulation of a
tractable but meaningful problem entails decisions as to
which variables to hold fixed and which to vary. The
invariance of other parameters is dictated by dependency
relationships between these parameters to output para-
meters or to a particular computational algorithm employed
at the point or points where these variables impact on the
system. Thus, for example, crew size is an invariant
parameter if we do not (as we have not) build into the
simulation program a rule relating crew size to pickup
efficiency in a non-linear manner. That is, if two men
collect refuse at 120 man-minutes per ton, and we assume
through our computational rules that three men would
collect the same refuse in 2/3 of the time, then pickup
time is dependent on crew-size in a trivial linear manner
and we gain nothing in varying both the pickup time and
the crew size unless we pair the variations. The point to
keep in mind is that simulation cannot generate new infor-
mation; it is simply a method for making quick and economi-
cal computations of a complex nature on available
2-93
-------
2-94
information.
To summarize, the parameters held invariant are:
a) the daily route method discussed in
Chapter 3i
b) crew size of two men, including the driver;
c) twice weekly collection frequency with each
truck working four days a week, i.e., 52
days in a 13 week quarter;
d) no unloading dock bottleneck time at the
disposal site.
-------
Section B
Variable Parameters of the Reference System
The input parameters which are allowed to vary
are summarized in Table 2-2. (A glossary of the symbols
used to represent these parameters as well as the Invari-
ants and the various results or output parameters are
given as an appendix to this chapter.) The values in
Table 2-2 which are underlined are those which represent
the base system. Consideration of Table 2-2 shows that
the Cartesian product of the variables would result in
1388 different combinations. Rather than run all these
cases many of which would be either ridiculous or super-
fluous and which would in any case require too much com-
puter time and require excessive time for digestion and
analysis, we have adopted a method of cross-sectioning in
which only one variable at a time is varied with all other
variables held at the base (underlined) values. The
exceptions to this are y , the refuse production in potmds
per service per day, which is run vs. all the entabled
values of the coefficient of variability a/v ; and the
truck capacity C (tons) which is also run against all
values of the coefficient of variability. The resulting
twenty-five cases are summarized on the bottom of Table
2-95
-------
2-96
Input Parameter Values
Number of Services
per Collection Unit, S 200 330 500
Service Density, p ,
services/mile s 22 70 100
Haul Distance, m, , miles 2 if 8 12
Quantity of Refuse, n, 5 10 15
Ibs/service/day
Coefficient of Variability, CT/M, 0.1 0.3 0.6
Pickup Time Coefficient, a, 0.5 0.7 l.o 1.3
min/service
Truck Capacity, C, tons 3^5
Table 2-2 Value of Input Parameters
-------
2-97
2-5 • (Note that A Is not one of the Input variables;
A will be defined and discussed in section C.)
Closely related to the selection criteria for
selecting these twenty-five cases is the decision to run
the system in an uncoupled, single season manner; that is,
as presently constituted, the only place where there is
interaction between trucks is when they are competing for
unloading docks at the disposal site, and it was therefore
decided to treat the study of such competition as a separ-
ate special case. The other twenty-five cases are run as
single truck runs. Moreover, for these twenty-five cases,
the problem of seasonal fluctuation in refuse production
has been handled not by running through a year using the
seasonal factor mechanism of the model, but instead
separate single production level quarters have been run
with different levels of refuse production, number of
services, etc. The study of the variability of work rate
has not been handled via the weather severity mechanism of
the model, but instead by running different quarters with
changes in a the time per service coefficient discussed
in Chapter 3.
The queuing problem at the disposal site, i.e.,
the waiting time at the disposal site as a function of the
number of unloading docks provided, was studied for a
fleet of ten 3-2 ton, two 4 ton, and one 2.5 ton truck
-------
2-98
(thirteen trucks). The first eleven trucks had 3 man
crews, the last 2 men. The garage was located at coordi-
nate (in miles) (5, 5) ; the disposal site at (10, 10) ;
and the CU's were centered at regular points in a grid
having both x and y increments of 0.2 miles with
most CU's containing 600 services at a density of 60
services per mile (served once a week). A few CU's had as
many as 1200 services at a density of 90 services per mile
and were served twice a week. Refuse production varia-
bility ranged from a/y = 0.125 to 0.500 and the pickup
rate varied between 98 and 300 man-minutes per ton with
most crews collecting at 120 man-minutes per ton.
-------
Section C
Results and Analysis
The results generated by the simulation program
are reported on a quarterly basis, truck by truck and
pertain to all the work assigned to that truck in that
quarter. These results are divided into three major cate-
gories: Time Distribution, Crew Performance, and Truck
Utilization. Additionally, frequency histograms are
generated for the length of workday (H ) and for truck
capacity utilization (£) . A digest of the simulation
results for the base cases 7 and 7R (repeat run) is
shown in Table 2-3 . The results shown in this table
demonstrate the consistency of the results from run to run
although it might be argued that this consistency results
from the low coefficient of variability of refuse produc-
tion o/y = 0.1 . An inspection of cases 9 and 9R
which differ from cases 7 and 7R only in having
a/u = 0.6 shows maximum discrepancies of 8% in the average
column. The discrepancies in the minimum and maximum
columns are, as one would expect, larger and these will be
apparent from the graphs of the results which appear
below.
2-99
-------
2-100
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-------
2-101
The frequency histogram results are shown in
Tables 2-4a, 2-4b , and 2-4c. The capacity utilization
histogram Is gathered into twelve percentile groups
(originally four percentile groups) and the results are
shown in percent of trips rather than in absolute numbers
of trips. These will be discussed at greater length
below.
The results, other than the frequency histograms,
are exhibited as a series of graphs, Figure 2-7 through
2-16. A concise presentation of results is obtained by
confining our attention to three main parameters as the
abscissae of these graphs. These are a/y , the coeffi-
cient of variability of refuse production; A (called
A in Chapter 5, the refuse assignment, i.e.,
the average quantity of refuse assigned to a crew per day,
expressed in truck loads. A may be computed by the
following formula for twice-a-week collection
A = Lavg. 3-5 y S (k .
* f .1;
2000 C
-------
2-102
1 Percent of Capacity
0 s E < 12%
c
12 s E < 24/o
c
24 s Ec < 36/0
36 5 EC < 48$
48 * EC < 60$
60 £ E < 72%
72 s EC < 84$
84 ^ E < 96$
96 s E < 108$
108 s E <: 120$
O
4
1.02
3.2
16.8
12.9
0.
0.
0.
8.4
4l.9
16.8
0.
12
1.17
2.4
4.2
10.1
18.4
3.0
0.
0.
30.4
30.9
0.6
20
1.23
^.5
5.7
4.0
12.5
13.6
0.6
1.7
21.0
36.4
0.
17
1.53
0.5
1.4
9.1
12.0
2.9
6.3
10.1
35.6
21.6
0.5
1, 10, 11
14- 16,
23-25
2.04*
0.3
1.2
^.5
7.0
5-3
6.9
8.1
39.7
26.8
0.2
13
2.92
0.
0.8
2.2
6.6
9.1
5.2
3.9
42.3
29.7
0.3
1
3.06
0.
0.5
i.l
2.5
3.6
5.5
11.8
46.9
27.2
0.6
* All cases include repeat run.
* No statistically significant difference was found in this group
before summation. See Table 2.4c
Table 2-4a Distribution of Percent of Trips vs. Percent Capacity Used.
Coefficient of Variability cr/ti = 0.1
-------
2-103
Case*.
Assignment
•2T
22
TF
1 Percent of Capacity
1.02
1.23
1.53
2.04
3.06
0 s E < 12$
c
1.1
12 <: E <
c
10.0.
24
3.0
36
48 5 E <
c
3.3
60 s E <
4.6
72
14.2,
84 s Ec < 96$
42.1
96 <; E < 108$
C
17.6
22.8
"26.9
30.8,
'34.7
108 s E < 120$
0.
0.
0.
* All cases include repeat run.
Table 2-4b Distribution of Percent of Trips Vs. Percent Capacity
Used. Coefficient of Variability CT/M, =
OV3/
. 7o.(
-------
2-104
Group*
o- 36$
36 - 48$
48 - 60$
60 - 72$
72 - 84$
84 - 96$
96 - 120$
N
•J
7
11
19
13
21
17
117
62
260
10
16
23
12
26
16
109
60
262
11
13
11
24
18
22
103
69
260
14
13
23
13
15
14
115
67
260
15
16
20
13
14
22
99
77
261
16
21
14
8
18
27
90
82
260
23
18
15
16
17
23
99
73
26i
24
16
20
13
13
32
91
76
261
25
17
19
12
19
17
108
68
260
Ni.
141
164
124
161
190
931
634
2345
= N
Table 2~4c Contingency Table Analysis of E Frequency
c
Histogram — A = 2.0k, oyV =0.1
Legend for Table 2.4c
Notes: l) First three and last two groups of Tables 2. k a and b are
lumped to assure each n.. s 5.
2) Null hypothesis: H : For each row there exists a p.
such that E.p. = 1 and 3 the probability of a result
belonging to the i-th group = p. for all cases.
O ^
3) Define X (sample) = S.E.[(N.. - p.N..) /p.N..] where
-L J -*-J i J 1 J
p. = N../N..(maximum liklihood estimate of p.), and N.. =
row sum, N. . = column sum, N.. = grand total.
4) x (sample) = 55-71 for (9-1)(7-1) = 48 degrees of freedom
2 2
and P (x > X (sample)] = 0.2074.
2 2
5) x (sample) thus falls far to the left of x for any
reasonable region of rejection of H .(P [x~ > )C,] = .05
is usual.) Therefore, we accept H
-------
2-105
N
•- O
1 1
O)
o
9—I
X
O
£
P
O
Cf
O
00
O
V
CO
O
CO
p.
•H
(H
•P
-------
2-106
O
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i
10 O
S
o
<->
O
O
o
o
O
•H
£ rH
^ -H
MXJ
C CO
c; c
O
05
OO
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(U
M
•H
uoj/uiLu uouj 'g
-------
2-107
3
3
8 §
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d
o
•p
c
0)
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x:
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0) Cti
w >
c
CTJ
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0)
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rt
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f-,
3
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2-108
IM
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w
a
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EH
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-------
2-109
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CO
§
CM
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8
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8
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2-110
©
et
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2-111
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rt
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to
> 0)
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^ W
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rH
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cti
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C
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CM
hD
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-------
2-113
O
O
CM
E.manmin/ton
O
O
O
3
O
CM
o
00
0)
x:
-P
-p
•H
S
O
C
O
«M
O
o
•H
-P
OJ
O
O
in
i
CM
QJ
faO
•H
E, man min/ton
-------
2-114
12345
NUMBER OF UNLOADING DOCKS
Figure 2-16
Queuing Problem at the Disposal Site
Waiting Time per Day vs. the Number
of Unloading Docks
-------
2-115
The other derived parameters appearing in the graphs are:
1) E , truck capacity utilization expressed as
c
percent. This is related to the parameters
of Table 2-3 by the formula
E = WC 1005? (4.2)
c*
where "•" stands for minimum, maximum or average with the
same choice to appear for each "•" in any use of the
formula.
2) T = H f , used as the denominator in comput-
ing haul time as a percentage of total time
worked per day.
Hw. ' a S * * +Th- + N- x td= T (4.3)
where a, y have been explained in Chapter 3
and all other quantities appear in Table 2-3 with the
"dot" convention the same as immediately above.
The simulated behavior of six performance para-
meters as a function of the coefficient of variability of
refuse production (daily fluctuation") for the reference
system described in sections A and B is shown in Figures
-------
2-116
2-7 through 2-9- The value of a/y for Wlnnetka, a
largely commuter community, ranges from 0.067 to 0.265
as shown in Table 2-1. Thus, the lower halves of these
graphs represent to some extent the behavior of these
parameters in Winnetka while the upper halves represent
investigation of the auitability of the daily route
method for higher values of a/y . The only previous
investigation of the Influence of a/y appears in [131
of which this dissertation is an extension.
Figure 2-7a shows that truck capacity utilization
is not greatly influenced by changes in a/y . Truck
capacity utilization is defined as the ratio of average
refuse carried per trip (£) to the truck's capacity
both in tons (formula (4.2)). In performing this sort of
analysis on a fleet of trucks where capacity is expressed
in cubic yards, the compaction ratios of the different
trucks must be taken into account. The minimum truck
capacity utilization curve (E m4n) was interpreted in
[13] to mean "that small loads are unavoidable to a large
extent and cannot be reduced •• by a change in the
boundaries of the collection units to effect an increase
or decrease in a/y . Small loads can be avoided only
for a/y < 0.1." With the added evidence obtained from
the consideration of the frequency histograms of capacity
-------
2-11?
utilization in Tables 2~4a and 2-4b, we must amend this
conclusion. Considering constant values of the assign-
ment A , we see that small loads are present at all
levels of o/y with a pronounced tendency to become
bimodal (two peaked) as either
-------
2-118
emphasizes that working with average values sheds no light
on the lows and highs of effort required to provide a
given level of service to a community.
Figure 2-8a reveals that the over-all efficiency,
E, in jnan-jninutes per ton is affected by a/y . A wide
variation in E is seen at large values of a/y and
this suggests that as in Winnetka, the daily route method
is not suited to situations with a/y in excess of 0.3-
As an example, cutting the curves of Figure 2-8a at
a/y = 0.15 shows a range of E from 95 to 205 man-
minutes per ton. The implications of this range can be
gauged from Figure 2-8b showing the variation in length
of workday with a/y . If operational policy requires
that the length of the workday not vary by more than 30
minutes above or below the average then one is advised
not to use the daily route method unless a/y < 0.15 as
indicated on the figure. Further light is shed on the
subject of length of workday by considering Table 2-5
showing the average frequency histogram (run and repeat
run) of length of workday for all twenty-five cases.
It is clear that the effects of a/y on the spread of
the histogram are most pronounced when the assignment A
is high. This suggests that high assignments do not go
well together with high variability under the daily route
-------
2-119
?
05
i
&
p*
Case (Includes
CO
£!
I'
IA
CM
a
PO
CM
r-4
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8
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00
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-------
2-120
method if controls of this sort are desired.
The variation of haul efficiency, E, , measured
in man-minutes per ton-mile is shown in Figure 2-9a.
It is clear that the average and minimum values of E.
are unaffected by variations in a/y . The maximum
curve of E, reflects via its appearance in the denomi-
nator the effects of small loads. Contrary to the
conclusion reached by [13] the apparent hump at a/y = 0.3
is not some special property of a/y =0.3 but is simply
caused by the random appearance of a disproportionate
number of elevated E, figures for the cases having
n j max
a/y = 0.3- If> instead of computing
T
h ,max
h,max 2L
n • max
we take the readings of E directly from the computer
printouts and eliminate those E, which are clearly
n. 3 rricix
out of line and take a weighted average where the weight
is the number of trips for the given run, we get a curve
of the appearance of max> in Figure 2-9a • (This has
not been done for the minimum and average curves.) It
should also be pointed out that there is no evidence in
the partial load histograms of Tables 2-4a and 2-^b to
-------
2-121
support a special status for a/y = 0.3.
Figure 2 9b depicts the relationship between haul
time as a percentage of total time (daily) and a/y . One
expects and finds these curves to be similar to those of
Figure 2-7b, i.e., they closely reflect the behavior of
the daily number of trips under changes in a/y . The
analysis of this data in [13] stated that the nondepen-
dency of the average curve on a/y "suggests that the
number of partial loads is the same for all a/y > 0.1 ."
The evidence available from Table 2-.4b suggests that
while this is not precisely so, there are enough partial
loads at both a/y =0.3 and 0.6 to account for this
effect. At the highest level of assignment A = 3.06,
the cumulative distributions for both a/y =0.3 and
a/y =0.6 are pretty close (Table 2-6 ).
We turn next to analysis of the effects of the
assignment A which has been defined above. Figure 2-10a
shows the relationship between E and A . This graph
c
taken together with Tables 2-4a and 2-4b indicates that
increasing the assignment up to A = 2 (particularly at
low levels of a/y) increases capacity utilization by
shifting upward and smoothing out the initially bimodal
density. The underlying mechanism causing this shift is
not apparant from the simulation results. Above A = 2 ,
the distribution of E continues to flatten but not as
C
-------
2-122
o
H-
00
•
r*"
CO
Q
LU
^^
p
<
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c$
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*
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•
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^
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p.
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Pn
-------
2-123
dramatically. Thus, A is not the cause of small loads.
Again, we must point at the daily route method itself and
our capacity rule.
Figure 2-10b shows the increase in the number of
trips necessary to service the assigned area as the quan-
tity of refuse expressed in truckloads increases. The
linear behavior of the average curve is, of course,
expected. The variations above and below average due to
variable refuse production are also expected. What is not
expected is the lack of variation at A = 1.5. Changes in
A are due to changes in the number of services, changes
in the refuse production rate or changes in the size of
the truck. In the present instance, all the cases at
A = 1.5 coincide with truck capacity C = 4. It is
possible that this capacity bears some felicitous relation
to the other parameters of the base case that enable it to
"swallow" the effects of variation. There is nothing about
Table 2-lJa t b that would seem to shed light on this and
the matter warrants further investigation.
The relationship between E and A shown in
Figure 2-lla is predictable on theoretical grounds. Using
a daily basis and letting the crew size be denoted by K
we may write a "rational" formula for E as
-------
2-124
(.8 + Y) t fV + V
CA CV €
whence
[ E - (- + t)] A = (a S + y) (4.5)
which Is a rectangular hyperbola with asymptotes
E = K(2m,/V + t,)/C and A = 0 . Thus, we note that
beginning at A = 1.2 some other mechanism of the system
causes the curves to be depressed below the curve predict-
ed by (4.5). As A increases from 2.0 to 3.0 the
effect of this mechanism abates and the approach of the
curve to the asymptote is reestablished. For the average
curve, (4.5) becomes (E - 25) A = 207 so that for
A = 1.0, 1.2, 1.5, 2.0, 3.0 we should have E = 232
(close), 198 (low), 163 (low), 128 (close), 94 (close).
Again, the underlying mechanism of this anomaly seen in
all three curves is unclear and warrants further investi-
gation. It is likely that this behavior Is related to the
behavior of A = 1.5 noted above in Figure 2-10b and is
related to the constriction of the minimum, average, and
maximum curves of H as a function of A seen in Figure
w
-------
2-125
2-lib. The cause put aside, one may still conclude that
for the reference system an assignment of A = 1.5 pro-
vides the best system performance in terms of constancy
of the number of trips per day, length of workday and
lowest over-all collection efficiency. This conclusion
is confirmed by the results shown in Figures 2-12a and b.
The former (with the maximum curve not corrected for
small loads) shows that A > 1.2 provides lowest and
nearly constant haul efficiency; the latter shows that
for A = 1.5, the minimum, average, and maximum values
of the proportion of haul time are nearly constant.
We turn next to four graphs which exhibit the
behavior of E, HW, Eh, and Th/T with changes in nt ,
the one-way haul distance from the collection unit to the
disposal site. It is to be recalled that distances in
both simulation models are computed by the £,-metrlc
(L-shaped paths). The changes in m, are made while
holding all other parameters at the base values as shown
in Table 2-2 . Figures 2-13a and b show that both E,
the over-all collection efficiency, and H , daily time
inf
worked, increase with the haul distance. The smooth
linear appearance of the average curves conforms to the
linear dependence of the underlying rational formulas on
m . The divergence of the minimum and maximum curves can
-------
2-126
only be ascribed to random elements Introduced Into the
haul speed as all points In these figures correspond to
cr/y = 0.1 and A = 2.0 . We note, also, from Figure 2-13b
that for the reference system, the requirement that the
length of workday not vary by more than 30 minutes above
or below the average indicates that the one-way haul dis-
tance should not exceed 6 miles. Some additional light is
shed on the effect of m, on H by returning to Table
2-5 and considering cases 14, 7, 15, and 16 in that
order. We note that not only does the workday length
Increase with m, , but that the distribution of the
length of workday spreads out and becomes pronouncedly
blmodal with almost all days appearing equally divided
between the two end intervals. We suggest that the reason
for this is that the actual inequitable division of work-
load between the two collection days of the schedule (one
getting 3 day's worth and the other 4 day's worth of
refuse) is masked by the smaller haul distances. The
lower hump represents the 3 day collection, the upper hump
represents the 4 day collection and the separation between
the humps represents the increasing proportion of the
workday devoted to the haul operation. This latter effect
is confirmed by Figure 2-l4b.
Figure 2-l4a (with the maximum curve not corrected
for small loads) shows in the average and minimum curves
-------
2-127
that haul efficiency tends to level out for m. > ^ miles
since the increase in haul speed with distance compensates
for the longer haul distance. The curve of speed vs.
distance is discussed in Appendix 2-A. There it will be
seen that m, = 2 is handled differently than the other
values of m, appearing in Figures 2-13 and 2-14 . The
maximum curve shows again, the effect of small loads.
There is nothing in the capacity utilization histogram
data to indicate any alteration of distribution under
changes in m, .
Figures 2-15s. and b reflect the dependence of T
on a and y via the equation (3-D of Part 2
T = a S + y . These graphs indicate that a (service
factor) has a stronger influence on E than does y
(density factor). This is merely a function of the way
the model was constructed to emulate a situation
(Winnetka) in which refuse amount Had nothing to do with
pickup time, but the number of services was dominant.
The three curves in Figure 2-15a meet at the E-axis
at the common value Y = 33- The anomalous behavior of
the three curves of Figure 2-15b at Y = ^0 warrants
further investigation. Whatever function is relevant to
describe T as a function of observable factors,
P
results such as those of Figures 2-15a and 2-15b allow
-------
2-128
estimates to be made of the value of various innovations
in technology and/or policy via their predictable effects
on the coefficients a and Y or whatever expanded set
of coefficients is relevant.
Figure 2-16 shows total quarterly waiting time
for a fleet of 13 trucks (described in section B) as a
function of the number of unloading docks For the system
being simulated, a ratio of one unloading dock to every
3 or 4 trucks seems to bring the waiting time down to an
acceptable level. This is reflected, likewise, by the
plot of the maximum waiting time per truck per trip.
This curve shows that the maximum wait is reduced from
10 minutes with 3 unloading docks to about 7 minutes by
the addition of a fourth dock. Considering the fleet
total, the fourth unloading dock saves 237 man hours per
year.
-------
2-129
APPENDIX TO CHAPTER 4
NOTATION
A « the assignment, quantity of refuse assigned
per day expressed in number of full truck-
loads per day;
C » truck capacity, tons;
C » average effective capacity of the truck;
tons;
CU = collection unit;
E = overall collection efficiency, man-min per
ton;
E = average capacity of the truck utilized, %\
G
E, = average haul efficiency, man-min per ton-mij
E = pickup efficiency, man-min per ton ;
P = frequency of collection per service per
week ;
H * length of workday, min or hrs ;
Yt
K = crew size, number of men per truck ;
L = quantity of refuse collected per day, tons ;
LQ = the load, quantity of refuse collected per
d
day expressed in number of full truckloads
per day ;
-------
2-130
I - quantity of refuse collected per trip, tons;
M. = internal mileage travelled per day, mi;
1^ = haul mileage per day, mi;
m. = internal mileage travelled per trip, mi;
m , = mileage between garage and disposal site, ml;
m = one way distance (A,-metric) from a collec~
tion unit to the disposal site, ml >
N = number of trips per day '»
ZN = total number of trips in a quarterj
S = number of services j
T = haul time per day, min ;
T = pickup time per day, min;
t, = time at the disposal site, per trip, minJ
t, = waiting time at the disposal site per trip,
min '>
t = pickup time per trip, minJ
£T, = total time at the disposal site in a quarter,
min ;
IIV = total haul time in a quarter, min;
h
ET = total pickup time in a quarter, min '>
V = average speed of the truck during haul, mph;
a = a constant characteristic of the time needed
to collect refuse from a service, min/service ;
-------
2-131
V = a constant characteristic of the time spent
in traveling within the collection unit,
min;
M - refuse production, Ib/service/day»
P_ * service density, number of services per
s
mile;
a = standard deviation of the refuse production
6 » pickup time for S services, min*
-------
CHAPTER 5
Simulation Studies of Alternative Policies
227
In a Constant Length Workday System —
In this chapter we present studies made on a
constant length workday refuse collection system based
on the City of Chicago. The dally workflow and the
chief differences between this model and the model
used in the previous chapter has been discussed in
Section B of Chapter 3.
The aim of these studies is to examine system per-
formance in terms of its quality under varying policies .
These policies are last load relay as defined above (yes
or no), allowance of overtime (yes or no, but see below
for clarification), and assignment (six levels). We are
concerned here with two kinds of assignment. The first is
the refuse assignment A which was used under the symbol
A In the previous chapter. This is the average daily
amount of refuse expressed as number of truckloads . The
second kind of assignment is the time assignment A,
U
which is the average daily hours assigned expressed as
number of 8-hour periods. (A glossary of notation used in
[15].
2-132
-------
2-133
this chapter appears at the end of the chapter as an
appendix.)
In the previous chapter we were performing para-
metric studies so that a great many input parameters were
varied. Here almost all variables that can be varied are
held to a single input value and the changes range instead
over the policy choices.
-------
Section A
The Reference System under
Constant Length Workday
The invariant parameters of the system simulated
are
1) A constant length workday.
2) Collection crew consisting of three loaders
and a driver.
3) Collection units are serviced once a week.
The work week is Monday through Friday.
Each truck services two CU's per day (ten
per week) and each crew is limited to a
maximum of two trips per day. Work may be
left over to tomorrow or beyond.
4) The number of services per CU is 180 and
the service density, p , is 90 services
s
mile.
5) The average quantity of refuse produced, y ,
is 8 pounds per service per day with a stan-
dard deviation a = 6.4 pounds per service
per day. The normal density is truncated at
the left for a minimum refuse production of
4 pounds per service per day. On a weekly
2-134
-------
2-135
basis, the mean, standard deviation and
minimum are respectively 56 (" 8 x 7) »
17(= 6.4 x /7) and 28( = 4x7). These
values are characteristic of high population
density, low per capita income areas [6-D2].
6) The time spent at the disposal site is uni-
formly distributed between 5 and 15 minutes
or 2 and 6 minutes. The queuing time at the
disposal site is handled as in the previous
model.
Pickup time computation and the decision criteria
were discussed in section B of Chapter 3 of Part II, and
the rest of the inputs are summarized in Table 2-7 . The
geographical layout of the CU's and the disposal site is
depicted in Figure 2-17. Each day begins and ends at the
garftgft and travel speeds are computed as shown in
Appendix 2-A.
-------
2-136
Input Parameter Value
Total number of services 1800
Service density, p , in services per mile 90
s
Haul distance, m,, in miles (see Fig. 2.17)
Average quantity of refuse, M,, in pounds per
service per day 8
Standard deviation for quantity of refuse, a, in
pounds per service per day 6.4
Pickup time coefficient, a, in min per service O.Y
Pickup time coefficient, |3, dimensionless 1.0
Pickup time coefficient, Y, in min 33
Pickup time distribution parameter, (b - a), in min 0.4
Maximum truck capacity, C, in tons 5.5
Decision criteria at collection unit
CLT 1, in min 15
HIRI, as fraction of C 0.9
Decision criterion at disposal site, CLT 2
constant, 6, in min 20
coefficient, e, fraction 0.2
Table 2-7 Summary of Values of Input Parameters
-------
2-137
Garage
(4.00,5.()d
I
(5.00,5.00)
1
Disposal Site
(0,0)
8
,250)
I
(6.67,5.00)'
E
«K
CN
10
(6.67,2.95)-*
ttmi,
Figure 2-17 Hypothetical Collection Area
-------
Description of Alternate Policies
Table 2-8 summarizes the alternate policies of
relay, overtime and assignment under investigation and
introduces the symbols for them to be used in the subse-
quent tables and discussion.
The last load relay policy as noted previously is
governed by the setting of the decision criterion CLT 3-
under this policy, the last load of the day is delivered
to the garage whence the relay driver takes It to'the
disposal site allowing the three loaders and regular
driver to quit within the allotted workday (assignment
A.). Relay policy can be and is hereinunder combined
O
with the overtime policies. The overtime policy is
specified by three overtime parameters OT1, OT2 and
OT3 denoting respectively number or hours overtime per-
mitted to service yesterday's (or earlier), today's, and
tommorrow's refuse assignment. The two policies used
here are P, : no overtime in any category; and P? :
two hours allowed for yesterday's and today's collection,
but no overtime allowed for living units (LU's) scheduled
for tomorrow.
2-138
-------
2-139
Policy
Relay
Rl
R2
Overtime
Pl
P2
Assignment
Al
A2
A3
\
A5
A6
Information Associated with Policy
Relay
no
yes
OT1 OT2 OT3
000
220
F Ar At
7 2.28 1.16
7 2.28 1.10
8 2.00 1.05
8 2.00 0.98
9 1.78 0.96
9 1.78 0.90
Table 2-8 Summary of Operational Policies Considered
-------
2-140
The collection and haul costs relevant to these
policies were based on hourly wage rates of $4.00 for the
regular driver,$3.75 for each of the three loaders, $4.00
for the relay driver, and on a daily truck rental of
$32.00. An eight hour day is charged for regular members
of the crew with overtime billed at time-and-a-half for
the actual time involved. The relay driver is billed on
a straight time basis.
The assignment policies are chosen to center
about a refuse assignment, A , of 2 and about a time
assignment, A. , of 1. These are respectively the
t
expected refuse amount in truckloads and the expected
workday length in units of eight hours. In the cases run,
the assignment was varied by changing the fleet size, P ,
from 7 through 9 trucks. As will be seen, both the refuse
and the time assignments affect the quality of service and
the need for overtime.
-------
2-141
Section C
Results and Analysis
The types of outputs provided by the simulation
model shown in Table 2-9 are quite similar to those pro-
vided by the first model (Table 2-3 ). What has been
added is ZT , the time in minutes spent in overtime work
over the quarter; En , the number of days during the
quarter (out of a fixed total of 65 days) when overtime
work was required; and an entirely new section entitled
"Quality of Service Provided." This section reports the
percentage of collection units serviced on time (Q, ), the
percentage of collection units serviced one day late (Q2)»
and the percentage of collection units serviced one day
early (QO. The latter category generally appears only at
zero level in the results reported here since OT3 is
kept at zero in both overtime policies investigated. The
percentage of collection units not serviced (i.e., serviced
two or more days late), Q , can be obtained by subtracting
Q! + Q2 + Qg from 1005?.
Further insight is obtained from frequency histo-
grams of the length of workday and capacity utilization.
Examples are shown in Tables 2-10 and 2-11 . This is the
same type of information which was given in the previous
chapter in Tables 2-4a, 2-4b and 2-5.
-------
2-1112
Value or mnimaL Maximum Standard
Average Deviation
Time Distribution
total pickup time, ET , in %
total haul time, ET. , in %
total ti-je at disposal site,
ET,, in "L
d
pickup time per trip, t , in min
haul time per day, T , in min
time at disposal site per trip,
t,, in min
total overtime, £T , in min
Crew Performance
hours worked per day, H , in hours
w
pickup efficiency, E , man-min
per ton **
over-all collection efficiency,
E, man-min per ton
refuse collected per day, L, in
tons
number of days worked overtime,
o
Truck Utilization
total trips in quarter, EN
trips per day, N
haul speed, 60V, mph
effective truck capacity, C ,
in tons
refuse collected per trip, £,
in tons
internal distance traveled per
trip, m.
Quality of Service Provided
% of collection units serviced
on time, Q1
% of collection units serviced
one day late, Qg
% of collection units serviced
one day early, Q
82.0
15.1
2.9
191 93 312 1+2
70 71 83 5
1^ 6 52 7.3
26
7.77
153
188
9-95 6'9b 11.0 l.lk
130
2
19.0
5.00
4.98 1.86 5.50 0.91+
3.98 3-31 4.2U 0.21+
76.2
23.8
0.0
Table 2-9 Example Output Information from Simulation Model8"
^ime Span: 13 weeks; Operating Policies R^, P.,, A, ; truck no. 1
-------
2-143
Length of Workday,
Upper Limit, hrs.
^.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
Number of Days
In Interval.
0
0
0
0
0
3
11
29
22
0
0
0
0
0
0
Table 2-10 Example Output Histogram of Length of Workdaya
o
. Time Span: 13 weeks; Operating Policies Eg, P.., A, ; truck no. 1
-------
2-144
Capacity
Utilized
Upper Limit.
4
8
12
16
20
2k
28
32
36
40
44
48
52
56
60
Table 2-11
Number of Trips °!'?f?lt^ Number of Trips
In Interval „ ^... In Interval
Upper Limit
0
0
0
0
0
0
0
0
2
0
1
1
3
4
0
Example Output Histogram of
64
68
72
76
80
84
88
92
96
100
104
108
112
116
00
Truck
0
10
1
1
3
5
0
4
4
91
0
0
0
0
0
a
Capacity Utilization
Time Span: 13 weeks; Operating Policies R^, P , A, ; Truck No. 1
-------
2-145
Costs of the service provided are computed from
the wages and rentals cited in section B on three bases:
the total cost of collection, C. , in dollars per week;
the cost in dollars per living unit (service)per week
($/LU/wk), C, ; and cost in dollars per ton of refuse
collected C~ •
Table 2-12 shows the effect of relay policy when
no overtime is used and the assignment policies are A-
and Aj.—*• . Without allowance for overtime, the effect
of relay as compared to no relay is to increase the total
cost, C. s and the unit cost , C, . This increase is due
to the added cost of the relay driver. The unit cost C~
is seen to decline, but this may be ascribed to random
fluctuation in the amount of refuse produced. There is
no such random fluctuation in the number of living units
so that the unit cost, C, , is proposed as a more reliable
index of the cost effectiveness of policy changes.
At first sight, the difference of .11 $/LU/wk
between relay and non-relay seems insignificant, but this
can amount to .57 $/LU/yr or 1026 $/trmck/yr based on the
1800 living units served by a truck in this example.
There is, of course, a great deal of difference between
the quality of service provided and a citizen might well
— The relay effect dominates the assignment effect here.
-------
2-146
Operating Policies P ,
Relay Policy
Item
Cost of Service
Total cost, C , $/wk 6,150 6,300
Unit cost, C , $/LU/Wk 0.42? 0.4-38
Unit cost, Cp, $/ton 16.7 15.9
A
Quality of Service
Percentage of collection units
serviced on schedule, Q 6.2 76.2
Percentage of collection units
serviced one day late, Q^ 83.8 23.8
Percentage of collection units
serviced one day early, Q 0.0 0.0
Percentage of collection units
not serviced, Q 10.0 0.0
Time Span: 13 weeks
Table 2-12 Effect of Relay Policy on the Quality
and Cost of Service Provided
-------
-2-147
consider 57 cents a year a bargain for having his refuse
picked up on time three times out of four rather than
(roughly) one time in twenty (and very late one time In
ten).
The major factors involved here are the amount of
time available for collection (increased by relay), the
average refuse production CM) and the variability of
refuse production (a) . The no-relay policy, R-^ ,
yields poor performance because the 8-hour time assignment
leaves insufficient slack to absorb the shocks of varia-
tions of refuse production above the average. (Recall
that we are using a large ratio a/y =0.8 and the
effect of truncation at the left is to skew the average
to the right.) There are of course other variations in
performance (pickup rate, wait at the disposal site,
travel time, etc.) for which the provision of slack time
would be useful. It is apparent that the extra time
provided by relay under policy R? is a good dose of
slack time. Another factor relevant here is the capacity
of two loads per day of 11 tons is not sufficient to
accommodate the variations In refuse production without
the use of relay.
The effect of overtime on the cost and quality of
services is shown in Table 2-13. The availability of
-------
2-11» 8
Operating Policies R2, A,
Item
Overtime Policy
Pl ' P2
Cost of Service
Total cost, C , $/wk 6,300 6,3^0
0
Unit cost, C , $/LU/wk 0.1*38 0.^38
Unit cost, C2, $/ton 15-9 i6-1
o
Quality of Service
Percentage of collection units
serviced on schedule, Q ?6.2 58.5
Percentage of collection units
serviced one day late, Q^ 23.8 0.0
Percentage of collection units
serviced one day early, Q 0.0 41.5
Percentage of collection units
not serviced, Q 0.0 0.0
Q
Time Span: 13 weeks
Table 2-13 Effect of Overtime Policy on the Quality
and Cost of Service Provided
-------
slack provided by overtime is not needed since the no-
overtime system already has the slack provided by relay.
All the relay plus overtime system can find to do with
the superfluous time is service many collection units a
day early. Very little overtime is actually used and the
unit cost,comes out nearly the same. A better comparison
on overtime is shown by cases 5 and 6 of Table 2-15 •
There, an improvement in quality is purchased at a large
increase in the unit' cost (C») .
The cost effectiveness of improvement in the
quality of service is shown by Table 2-14 . Here we see
that going from assignment A2 to A^ we get an &I%
improvement in on-time service for an increase of total
cost per week of $800. Going from assignment A^ to A/-
we pay an additional $800 per week but only get an addi-
tional 12% improvement in on-time service: a prime
example of the "law of diminishing returns".
Twelve of the twenty-four possible combinations
of policy were run and the results arranged by decreasing
assignment are shown in Table 2-15 . The table includes
unit costs, quality of service (plus an over-all rating),
and report of time worked. The values of H are
W
average daily hours worked over the 65 days of a 13 week
period; ZTQ and ZnQ are for the same period the total
-------
2-150
Operating Policies Rp, P
Item
Cost of Service
Total cost, C , $/wk
"C
Unit cost, C , $/LU/wk
Unit cost, Cp, $/ton
Q
Assignment Policy
A2 A4
5,510 6,300
0.386 0.438
15.8 15.9
A6
7,090
0.494
16.5
Quality of Service
Percentage of collection units
aT
serviced on schedule, Q 5.4 76.2
Percentage of collection units
serviced one day late, Qp 83.0 23.8
Percentage of collection units
serviced one day early, Q 0.0 0.0
Percentage of collection units
not serviced, Q 11.6 0.0
ime Span: 13 weeks
87.7
12.3
0.0
0.0
Tatle 2-14 Effect of Assignment Policy on the Quality
and Cost of Service Provided
-------
2-151
£°
T) W
.M
fn O
3* ^"J
§
EH K
t- _=(- 0
US CM
OO rH (A
• • •
0 r-H O
VO
C— J- IA
t-— CO t—
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0) -p
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CO O
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ft
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t>> OO
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'r-t
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-p
w H
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•H 0
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S1
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ft ft ft
CO VO VD
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^
-------
2-152
hours of overtime and the number of days overtime was
required.
One observes that as the assignment decreases
going from case 1 to case 12, the quality of service
improves and the unit cost, C^ , increases. The unit
cost, C , as previously noted is not a reliable index
U
since the quantities of refuse produced are subject to
wide variations.
Comparing even- with even- and odd- with odd-
numbered runs, we note that except for run 12, decreasing
the time assignment causes an improvement in the quality
of service. We note also that comparing odd- with even-
numbered runs, the use of overtime always improves the
quality of service. If we look at the unit costs, however,
we note that the improvement gained by use of overtime
raises the unit cost, C, , when A. < 1 and leaves it the
same or nearly so when A, > 1 . We conclude that the
sagacious use of small amounts of overtime provides
improvement in service at slight increase in unit cost
when the total time assignment is less than one. It is
also apparent from inspection of Table 2-15 that the
refuse assignment, A , cannot be wholly compensated
either by overtime or relay when it is set too large.
-------
2-153
The cheapest and next cheapest "good" cases are
8 and 6. Operationally, case 6 may be preferable to the
cheaper 8 under a system in which the residents set out
their refuse containers on appointed days. Under such a
system, early pickup may actually be an inconvenience.
-------
APPENDIX TO CHAPTER 5
NOTATION
The following symbols were adopted for use in
this chapter.
A = designation for different assignment policies;
A = the average quantity of refuse assigned to a
crew per day, in number of truckloads per
day;
A. = the average length of workday assigned to a
crew, in eight hour units;
(b - a) = the difference between upper and lower limits
of the normalized uniform distribution used
in determing pickup time;
C = the maximum truck capacity, in tons;
Cg = average effective capacity of the truck, in
tons;
C. = total cost of collection and haul, in $/week;
C, = unit cost of collection and haul, in $/LU/
week;
Cp = unit cost of collection and haul, in $/ton;
CLT 1 = decision criterion for the operation of a
truck;
2-15*1
-------
2-155
CLT 2 = decision criterion for the operation of a
truck;
CLT 3 = decision criterion for the operation of a
truck;
CU = symbol for collection unit;
DS = symbol for disposal site;
E = over-all collection efficiency, in man-
min per ton;
E = pickup efficiency, in man-min per ton;
P = the number of trucks in a fleet;
G = symbol for garage or parking lot;
H = average length of workday, in min or hrs;
w
HIRI = decision criterion for the determination of
remaining truck capacity at the completion
of each CU;
k = constant, zero to one;
L = quantity of refuse collected per day, in
tons;
LU = symbol for living unit;
£ = quantity of refuse collected per trip, in
tons;
mi = internal mileage traveled per trip, in miles;
m^ = one-way distance (£,-metric)from a collection
unit to the disposal site, in miles;
-------
2-156
mf = (2m. - 5-5), in miles;
N = designation for different policy on the
maximum number of trips per day;
N = the maximum number of trips per day;
HlclX
ZN = total number of trips in a quarter (13 week
period);
En = the number of days in a quarter in which a
crew worked overtime;
OT l"
the amount of overtime permitted when ser-
OT 2
vicing collection units, in hrs;
OT 3
P = designation for different overtime policies;
Q - the percentage of collection units not
receiving service;
Q-, = the percentage of collection units receiving
service on-time;
QP = the percentage of collection units receiving
service one day late;
QO - the percentage of collection units receiving
service one day early;
R = designation for different relay policies;
S = number of services, or living units;
T = haul time per day, in min;
-------
2-157
t, * time at the disposal site, per trip, includ-
ing time spent in queue, in min per trip;
t = pickup time per trip, in min;
IT, = total time at the disposal site in a quarter,
in min;
ET, = total haul time in a quarter; in min;
ET = total overtime in a quarter, in hrs;
ZT = total pickup time in a quarter, in min;
TV = estimated time needed for activities other
than loading, in min;
V = speed of travel, in mph;
a = a constant characteristic of the time needed
to collect refuse from a service, in min per
service;
Y - a constant characteristic of the time spent
in traveling within the collection unit, in
min per trip;
6 = a constant;
e = a constant;
6 = pickup time for S services, in min;
6 = the average pickup time for S services, in
min;
M = refuse production, in pounds per service per
day;
-------
2-158
p = service density, number of services per mile;
s
and
a = standard deviation of the refuse production,
in pounds per service per day.
-------
CHAPTER 6
Extensions and Modifications
ojF Simulation Models
The purpose of thJL chapter is to exercise the
wisdom which grows out of hindsight. Experience with
the simulation programs has indicated several areas where
improvement should be sought. We shall list chese and
then discuss them.
1. Data base improvement.
2. Improvements in generating refuse
variability .
3. Small loads and the capacity rule problem.
4. Data reduction, graphics and other computer-
oriented improvements .
"Data base" is a buzz-word which has become
fashionable in the late 1960's. It means, simply, the
large amounts of systematized quantitative and qualita-
tive information and the methods for keeping these current
which must underlie any successful effort to apply system
analytic methodologies. Efforts of the type conducted by
2V
the University of California — need to be repeated
— See Chapter 2 of Part 2.
2-159
-------
2-160
for current conditions in many places. Such studies
should have the active participation of persons experi-
enced in the design of experiments at an early stage, and
as noted in Chapter 2, methods of regression analysis and
analysis of variance and covariance, etc., should be
applied to the data gathered to assess the relevance of
different factors on refuse production and collection
efficiency.
Closely related to these ideas is the need for more
definite knowledge of the probability laws that underlie
refuse production (point 2 above). We have indicated on
slim evidence that refuse production is normally distri-
257
buted—- In both simulation models. Disturbed by the
possibility that we might generate negative amounts of
refuse, we have truncated the left tail of the normal
density at some arbitrary minimum refuse production.
There may be probability laws that better describe
refuse production and these should be sought.
Supposing that we have done this, and we have a
clear idea how refuse production varies on the basis of
pounds per service, per day, some valid questions may be
257
-^ Log normality has been observed in Raleigh, N.C.
(Personal communication.)
-------
2-161
raised as to how to turn this micro-analytic fact into a
valid refuse production figure for a large number of
services (a CU). We have relied on the central limit
theorem of the theory of probability. This theorem states
that the average value of a large number of observations
tends to be normally distributed no matter what the prob-
ability law governing the individual observation. Thus,
we have gotten a CU's refuse production by sampling from
N(DP, /D~a) where D is the number of days since the last
collection. This value, if greater than the le.^t-tail
truncation value, is then multiplied by the number of
services. One may very well ask, and be answered by
embarrassed silence, why the refuse production wasn't
P /• /
generated form N(SD|j, /SDa ) .—x One may also note that it
seems unlikely that on any given day one CU will be
extremely high in refuse production while another will be
low. We should seek, then, a scheme for generating refuse
that causes groups of CU's of like nature to move together,
One might also argue that refuse production may behave
more like the Dow Jones Average than a Brownian particle,
i.e., instead of up and down every day we expect several
days of consecutive ups, some uncertain plateaus and some
consecutive days of downs with occasional rallies.
— This was done in the case of the simulation model in
Raleigh, N.C., with the log normal density.
-------
2-162
Closely allied to these problems are the need for more
realistic ways of Introducing seasonal variation in
refuse production, the effects of weather on collection
efficiency (and haul speed), and better knowledge of how
haul speed varies both as a random variable and as a
function of haul distance.
The ideas sketched in the previous paragraphs
are important for the problem of small loads and capacity
rules. We have noted previously that the introduction of
some look ahead rationality and a very simple capacity
rule produced fewer and larger small loads under the con-
stant length work day rules where only two trips per day
were permitted. Since the daily route method is still
appropriate to many communities, we may ask if the prolif-
eration of small loads, not observed to the same extent in
practice in Winnetka as in simulation can be reduced by
some means. The answer is yes: If groups of like CU's
tend to go up and down together, if variation goes up and
down in spells of several days, if collectors can read the
calendar and see lawns beginning to sprout, then they can
anticipate the amounts of refuse to be collected and base
their capacity decision on a convenient number of loads
of approximately equal size. Thus, if they anticipate a
ten-ton day and the truck has a capacity of three tons,
-------
2-163
they aim for three loads of 3 5- tons each instead of two
loads of 4 and one of 2 tons. This is probably much
easier to state than to implement.
The last category of improvement is largely a
matter of making better USP of computer resources.
Instead of making many runs and performing the necessary
analysis by hand and slide i'ule as was done here, future
programs will be written to store the results of many
runs and then perform the analysis, and plot the graphs.
Looking further ahead, but not too much, one may antici-
pate simulation programs where the investigator sits at
a remote console cathode ray tube (CRT) and poses his
questions and receives his visible output on the CRT in
short order.
Mechanics, data and so forth settled, what sorts
of problems in solid wastes should models of this sort
be extended to? The answer is to every conceivable type
of situation. We have considered two very simple situa-
tions. There are many more, and problems of growing
importance in such areas such as supplemental transporta-
tion, methods of disposal, volume reduction, etc., can be
furthered by the use of simulation.
-------
CHAPTER 7
Summary and Conclusions
Simulation models were developed for a daily
route method system of refuse collection based on the
Village of Winnetka, Illinois, and for a constant length
work day based on the City of Chicago. The development
of these models is discussed in Chapter 3 and Appendix
2-A.
The daily route method model was exercised to
delineate the interdependencies of several significant
parameters involved in the functioning of a refuse collec-
tion system. In particular the model was used to find
maximum, average and minimum value plots of
E = Percent of capacity utilized
N = Number of daily trips
E = Overall collection efficiency (man-min/ton)
H = Work day length (minutes)
E, = Haul efficiency (man-min/ton-mile)
T,/T = Percent of total time taken up by haul
operation
2-16/1
-------
2-165
against the coefficient of variability of refuse produc-
tion, o/u , and the refuse assignment, A, the average
daily amount of refuse in truckload units . The last four
of the above list were also exhibited as functions
(minimum, average, and maximum values) of m. , the one-
way haul distance (Jl, -metric) . These graphs are supple-
mented by frequency histograms of truck capacity utiliza-
tion (E ) and length of workday (H ) arranged by value
of a/M and value of A . Also provided are plots that
correlate system behavior (E) with changes in a a^d y,
the coefficients of the function which relates pickup
time to the number of services (S) and the internal route
mileage (m. ) . Finally, a plot is provided for a com-
posite system of thirteen trucks to illustrate the behav-
ior of the average and maximum queuing times at
the disposal site as a function of the number of unloading
docks available.
It is to be reiterated that the particular
numerical values cited in the conclusions listed below
pertain to the reference system described in sections A
and B of Chapter 4. The values that would pertain to any
other system would have to be derived by exercising the
simulation model on data from that system.
-------
2-166
1) Inspection of the frequency histograms of
E shows that small partial loads are
c
present at all values of a/y and A with
a pronounced tendency to have a widely
separated blmodal density at low values of
both. Small loads are a function of the
capacity rule and the demands of the daily
route method as simulated in the present
model.
2) The overall collection efficiency, E, is
affected by changes in a/y . For the
reference system, the daily route method does
not seem to be desirable for values of
a/y > 0.15. This is the value of a/y for
which the variation in the length of the
work day amounts to approximately 30 minutes
above and below the average.
3) Consideration of the frequency histograms of
length of work day shows that the effect of
a/y on the spread of the density are great-
est for large values of A . We conclude
that under the daily route method large A
and large a/y should not go together.
-------
2-167
4) For the reference system, Increasing the
assignment up to A = 2, particularly at
low values of a/\i increases capacity
utilization by shifting upward and smooth-
ing out the initially bimodal density.
Above A = 2, the density continues to
flatten, but not as dramatically. A is
not the cause of small loads.
5) For the reference system, an assignment of
A = 1.5 appears to provide an optimum time
buffer in a system requiring 2-3 loads a day
to effectively swallow the effects of random
variation in refuse production. This results
in essentially equal H for the minimum,
maximum and average curves when A = 1.5.
The underlying mechanism is unexplained.
6) An assignment of A = 1.5 also provides the
overall best system performance in terms of
constancy of number of loads per day and
lowest collection efficiency.
7) The length of work day (H ), overall collec-
tion efficiency (E) and proportion of haul
fclriie (Th/T) are proportional to one-way haul
distance, mh. The haul efficiency is
-------
2-168
essentially constant for m, = 4 to 12
miles because distance Is offset by greater
haul speed.
8) For the reference system, a distance m, < 6
miles Is desirable in that the variation in
the length of the workday does not exceed
thirty minutes above or below the average
for such distances.
9) The length of workday frequency histogram
spreads out and becomes bimodal as the dis-
tance ITL increases. This reflects disparity
between 3 and 4 day loads in a twice-a-week
collection system.
10) The overall collection efficiency (E)
responds to changes in a and y that
reflect the manner in which these are used
in the program to determine pickup time.
11) Quarterly waiting time for a thirteen truck
fleet is reduced exponentially by increases
in the number of unloading docks. A fourth
unloading dock saves 237 man-hours per year
over three docks. A ratio of one dock to
every three or four trucks appears to be
indicated.
-------
2-169
Alternate policies of last load relay, overtime
and assignment (time assignment and refuse assignment)
were tested for twelve of twenty-four possible combina-
tions in a simulation of a constant length workday
collection system. The significant findings are
12) Unit cost with units dollars per living
unit per week are a more accurate index of
the effect of a policy as the number of
living units serviced per week is essentially
constant while unit cost based on tons of
refuse fluctuates due to variable refuse
production. The use of the former unit cost
is indicated and used below.
13) The use of a relay driver to haul the last
load of the day generally improves the
quality of service as measured by on-time
collection. This Is bought at an increased
unit cost.
lt|) When relay and overtime are used together
too much slack may be provided for the shocks
of variation. This may result in ahead-of-
time collections (not always beneficial).
15) Decreasing the assignment results in better
service at a higher unit cost. The law of
-------
2-170
diminishing returns is clearly at work
here with earlier decreases in the assign-
ment more cost effective than later ones.
16) The line A, = 1 (where A, is the expected
v b
daily time assignment in 8-hour units) pro-
vides a dividing line on the cost effective-
ness of overtime. At all levels of A, ,
U
some improvement in quality of service is
found with the use of overtime. For A, > 1
this improvement results in increased unit
cost; for A. < 1, the improvements in
quality results in essentially no increase
in unit cost.
17) Overtime cannot wholly compensate for too
large a refuse assignment A . Good
service at low unit cost is secured with
A = approximately 2 and A, less than 1.
X \~,
-------
APPENDIX 2-A
I. Introduction
The purpose of thin appendix is to provide a
detailed manual for the understanding, use, and eventual
further extension of the s/stem of programs that has been
written to simulate the operations of a rufuse collec-
tion and haul system.
Some of the general material appearing in Section
II below has already appeared in Chapter 2, Section 3.
We repeat it here in a more succinct form to allow all
material pertaining to the daily route simulation model
to appear in one place.
II. An Overview of Concepts Used in this Program
1. General Description
A. Physical Organization of_ the City: For
the purposes of this simulation, the city is divided into
"districts." The district is meant to convey characteris-
tics of neighborhood type, and as such, a given district
need not be physically contiguous. Such characteristics
of neighborhood type, important to the design of a
refuse collection system, might include presence or
2-171
-------
2-172
absence of alleys, one and two-family or multiple-
family dwellings, income, size of families, etc. The data
to be provided for each district are:
a) Average refuse production in pounds per
service per day.
b) The standard deviation of same.
c) The parameters of a function relating refuse
pickup time to the number of services, the
amount of refuse, service density, and
internal mileage of sub-areas of some combi-
nation of somo or all of these factors.
The data provided in a), b), and c) are applica-
ble homogeneously in all subdivisions of the district.
The simulation system allots up to 10 districts.
Each district is further subdivided into "collec-
tion units" (CU's henceforth) which are small, convenient
physically contiguous areas lying wholly within their
respective districts. A given CU is serviced by only one
truck.
The data to be provided for each CU includes:
a) An indication of which days of the week it
is to be serviced.
b) Which district it belongs to.
c) The number of services.
-------
2-173
d) Service density.
e) Its physical location.
The data provided in a) - e) are applicable homo-
geneously in all portions of a CU . While there is no
named entity smaller than a CU, this assumption of homo-
geneity is important when a truck does not finish a CU
before returning to the disposal site, in which case a
partial CU remains to be serviced. The simulation system
provides for a maximum of 1000 CU's in the entire city.
A single disposal site and a garage are provided
for. They may be at different physical locations.
The physical locations of the CU's, the garage,
and the disposal site are given by their Cartesian
coordinates. The origin of the coordinate system is
immaterial, but must be internally consistent throughout
a problem.
B. Method of Work Organization: This simulation
system simulates refuse collection according to the daily
route method. Under this method of work organization,
each crew is assigned one or more work areas (zones,
beats) which are to be serviced some fixed number of times
per week. The area assigned for any given day of the
week must be completely serviced on that day. Each crew
operates independently, helped by and helping no other
-------
2-174
crew and quitting early or late as the work load demands.
Each truck (crew) is given a string of consecu-
tively numbered CU's. Within this string, those to be
serviced on any given day do not have to be adjacent,
but day-mate CU's should be arranged so that the higher
numbered CU's are progressively nearer the disposal site.
The data to be provided for each truck are:
a) The number of the first CU and the total
number of CU's in its zone.
b) Truck capacity.
c) Crew size.
The simulation provides for up to 64 trucks to be
operating simultaneously.
C. Time Organization: The largest time period
recognized by the simulation system is a World Calendar
quarter of 91 days (13 weeks). The quarter is the basic
reporting period of the simulation system; data accumu-
lated on a trip or daily basis Ls printed and/or punched
out for each truck at the end of each simulated quarter.
The next largest time unit is the month. A World
Calendar quarter consists of three months of 31, 30 and
30 days in that order. Weather induced slowdowns and
seasonal adjustments in refuse production are changed on
a monthly basis.
-------
2-175
The customary 7-day week Is the basis for
scheduling the work of the trucks. A given CU may be
serviced from 1 to 6 times per week.
The actual simulation proceeds on a daily basis
governed by an outer program regulating the counting of
days into weeks, months and quarters. The day begins
with the dispatching of each truck that is to work on the
given day of the week to its first CU of the day and ends
when all trucks have returned to the garage.
Within a day are variable length entities called
"events" of minimal length zero. The day's operation of
a truck is composed of a string of events: leave garage,
arrive at first CU, finish first CU, ..., finish load,
arrive at disposal site, leave disposal site, ...,
arrive at garage. The setting of events is governed by
the string of events that has already occurred. The
sequencing of events is governed by a section of the main
program called the time status record (TSR or event
clock). A trip is coincidental with the event: "truck
arrives at disposal site with refuse."
Closely related to events is an entity called
status. An event tells the program what a truck will do
do next; status tells what the truck is now doing.
Status is not used in the simulation model at present, but
-------
2-176
Is Included to facilitate future inclusion of (a) truck
breakdown event(s). The interpretation of the status
code is given in the Glossary.
&• 0" Some Computational Methods and Assump-
tions : While most of the following Information will be
repeated later in greater detail, a general description
of the simulation system requires a few general remarks
about some of the basic determinants of the simulated
results.
(a) Petermination p_f Refuse Amounts :
Refuse production is assumed to be normally distributed
with the left tail of the normal curve truncated at 1
pound per service for any CU. The mean and standard
deviation of refuse production given for the CU's district
are modified according to the number of days since the
last collection by the following scheme:
Let y and a be the mean and standard deviation
respectively of refuse production in a given CU (pounds
per service per day);
Let D be the number of days since this CU was
last serviced;
Let S be the number of services in this CU;
And let F be the seasonal factor pertaining
to the current month.
-------
2-177
Then, the refuse (in tons) to be collected from
this CU is given by a random variable R where
R ~ NCvi'D, a'/D]'S'F/2000
(See Figure 2-4 and page 2-67 -)
(b) Pet erminat ion of pickup t ime : Linear
regression analysis of 43 trips in Winnetka, Illinois
(March-April, 1958) suggests that the pickup time is gov-
erned by the number of services; the effect of the amount
of refuse is statistically insignificant. The regression
equation is
Time (minutes) = ot S + Y
where S = nimber of services
We have modified this formula on the assumption
that Y is at least partially explained by mileage trav-
elled during collection. Thus, the formula used in the
simulation system is
Time (minutes) = a S + y' m,
where m^ - internal mileage
Y' » Y/6
3 = mean internal mileage of the
CU's of the regression analysis.
-------
2-178
(c) Det ermlnatIon of distances: The compu-
tation of distances is based on the so-called "metropoli-
tan metric" (£-, -metric) in which the -path from one point
to another is an L-shaped path:
d(p, p') " |x - x'| + |y - y'|
where d(p, p') = distance from point p = (x, y) to point
P" = (x", y') and M| |" = absolute value.
The use of this metric is Justified by the rectan-
gular grid arrangment of many North American street plans.
(d) Petermination of speeds of traveJL times :
The speed of trucks during their hauls is determined as a
function of the distance to be traveled as computed above.
The shape of the curve relating speed to distance is shown
by Figure A.I. The equations for this curve are also
indicated therein. These equations are based on a least
squares fit of the curve given in Figure 29 of "An
Analysis of Refuse Collection and Sanitary Landfill Dis-
posal," University of California, 1952.
(e) The Effect o_f Weather and Seasonal Vari-
ation: At the beginning of each day, a weather severity
factor is generated from a frequency distribution apropos
to the month of operation. This weather severity factor
-------
2-179
in IM
iri tri
VI I
II (I I
I>T>
tn
iri
I
C0
OS
duu
INVERSE HAUL SPEED V1 (min./mi.)
»- O
0)
o
c
CO
-P
CO
•H
O
D,
•H
C
3
o
0)
D,
CO
cti
<
0)
bO
•H
-------
2-180
is multiplied into the collection time for each collection
unit serviced during the day. A weather severity factor
of 1 means that today's weather is normal; greater than 1
implies a weather-induced slowdown.
Seasonal variation in refuse production ±s like-
wise handled by a multiplicative factor. Taking some
month, say January, as the base month, the seasonal
factors of the other months are the ratios of average
refuse production in those months to the average refuse
production in January. No provision is made to alter the
variability of refuse production seasonally, and the
seasonal factor is the same for all districts and CU's.
III. Detailed Description of the Simulation System
The simulation system consists of a main program,
RCS002 (Refuse Collection Simulation), and a series of
subroutines. Most of these subroutines are called only by
the main program, a few are called by both the main pro-
gram and one or more subroutines, while others are called
only by other subroutines.
The main program contains the basic structure of
the simulation and consists of three sections: the outer
program which reads in'the input data, passes time by
days, weeks, months, and quarters, presets the proper
-------
2-181
initial conditions peculiar to each time period, and calls
for the quarterly output of results; the event clock or
time status record which sequences the events of all
operating trucks and passes time within the daily simula-
tion; and finally, the events themselves which are the
basic representations of the movements and operations of
the trucks, and which give rise to the occurrence of
further events to be processed by the event clock. In
the following description, the subroutine "PUT" which is
an integral part of the event clock, is described with
the event clock rather than with the subroutines.
The subroutines are divided into three groups:
data generating subroutines, two in number, which gener-
ate refuse amounts, collection times, pickup mileages,
haul times and haul distances; auxiliary subroutines
which perform such functions as determining when a truck
is full, finding the next CU, accumulating statistics,
etc; and miscellaneous subroutines which generate random
numbers, round off decimals, etc.
In the following descriptions, flowcharts of two
types are presented. The first type is a verbal flow-
chart avoiding the use of subscripts and actual program
terminology, and giving a general understanding of the
purposes and the functions of the program section under
-------
2-182
discussion. The second type is a detailed computer pro-
gram flowchart.
1. The Main Program (RCS002)
A. The Outer Program; The outer program
performs the following functions:
a) Reads input data, checks them for
internal consistency and stores them where they will be
needed.
b) Sets up and keeps track of the
various time counters (quarters, months, weeks, days) and
takes the following appropriate actions:
(1) End of quarter: Prints quar-
terly results and returns either to seek new data, termi-
nate all runs, or to set up conditions for the next
quarter's simulation. The latter consists of setting
monthly, weekly, and daily time counters to their Initial
values, and initializing all quarterly output data
accumulators.
(2) End of week: Sets day of week
counter (TODAY) back to the number of the first day of the
week on whieh any truck works.
(3) Beginning of day: Sets up
first job for each truck that is working today and selects
today's random weather event.
-------
2-183
End of_ day: Advances day
counter and all other time counters to check for the end
of the larger time periods.
(5) Beginning of month: Selects
new monthly seasonal factor.
The following verbal flowchart (Figure A.2) out-
lines the actions of the outer program. The event clock
and the events are each represented by a single box which
will be expanded below.
The variable names used in the detailed flowchart
are explained in the Glossary of this Appendix.
B. The Event Clock and "PUT" Subroutines: As
discussed above, the day starts with a group of trucks
leaving their garage at the same time. As the day pro-
gresses, differing refuse amounts, work rates and
distances, etc. quickly put an end to synchronism between
trucks. Thus, an orderly procedure is required to keep
track of what each truck is doing. This will be all the
more important as further areas of truck-to-truck inter-
action are added to the progrsm (e.g. relay methods).
The event clock is a section of the main program
which utilizes a set of lists called collectively the
"time status record" (TSR) to properly sequence pending
events and cause the passage of time.
-------
C
START
SET TO
FIRST
QUARTER
PUNCH AND
PRINT
QUARTERLY
RESULTS
DECREASE REPEAT
COUNTER BY ONE
I
INITIALIZE
TIME AND DATA
COUNTERS
ADVANCE DAY
COUNTER
ADVANCE TO
NEXT QUARTER
RESET END OF
MONTH SIGNAL
NO
MAIN LOOP OF SIMULATION
ADVANCE TO
NEXT
MONTH
I
SET UP DAY
OF WEEK
£
SIMULATE ONE DAY
(INNER PROGRAM)
FIND FIRST JOB
FOR EACH TRUCK
GENERATE
WEATHER SEVERITY
Figure A.2 Verbal Flowchart of Outer Program
-------
2-185
In its simplest form, the TSR consists of two
lists, a list of pending events (PEVLST) and a list of
the time of their occurrence (TIMLST), the latter in
monotonlc non-decreasing order. By "event" is meant
a subroutine or as in this program a section of the main
program that simulates a certain kind of system action
and through whose action the future occurrence of other
events may be initiated. By "time of occurrence" is
meant the time from "right now" which must pass until the
event will occur.
The TSR used in this program differs only in that
a third list is carried (PTRNO) which specifies the num-
ber of the truck which will be involved in the pending
event. The operation of the simulation through the event
clock takes place as follows (see Figure A.3).
1) The number of events counter (NOEV) is test-
ed. If NOEV = 0, the simulation ends;
otherwise go to step 2 .
2) Pick up the first occurring pending event
and its associated time and truck number.
By the ordering of the times, these will
be the first items on their respective lists.
-------
2-186
O
rH
O
W
-------
2-18?
27/
3) If the event has been scratched~~~ (In the
sense of horse-racing) the incremental time
is set to 0; otherwise the incremental time
is the time associated with the event in
step 2 .
4) All lists are shoved up one place, i.e.,
2nd-*- 1st, 3rd •> 2nd, etc., and simultane-
ously each time is reduced by the incremental
time as determined in steps 2 and 3 •
5) The number of events counter (NOEV) is
reduced by 1.
6) If the event has been scratched, return to
step 1 ; otherwise go to step 7 .
7) The master daily time counter (TELAPS) is
increased by the incremental time.
8) The appropriate event program is entered.
With one exception (see Event Type 4, below) all
event programs terminated by returning to step 1 of the
event clock.
When one of these events generates the occurrence
of a future event, this event together with its truck
number and time of occurrence are entered into the TSR
by the PUT subroutine which is called as: CALL PUT (TIME,
27 /
—'-See Event Type 6 below.
-------
2-188
EVENT) (the truck number is carried in a COMMON section).
The PUT subroutine (see Figure A. 4) using the
counter NOEV searches down TIMLST for the first entry
thereof which exceeds "TIME". If all entries of
TIMLST < TIME, then TIME, EVENT and truck number are
placed beyond the end of TIMLST, PEVLST and PTRNO,
respectively and NOEV + 1 replaces NOEV. Otherwise,
denoting K < NOEV as the index of the first entry such
that TIMLST(k) > TIME, the entries of all three lists are
moved up (NOEV - th + NOEV + 1 - st, ... Kth •»• K + 1 - st)
and the new entries are inserted at the Kth position and
finally NOEV + 1 replaces NOEV.
The present version of the program allows for the
stacking of 500 pending events.
C. The Events: The events described hereinunder
are called event type 1, 2, etc. This numbering has no
relation to their typical order of occurrence, but merely
reflects the historical acoident of the order in which
they were written.
These program segments are best understood in
terms of their detailed flowcharts (A.5 - A.10) and the
Glossary of Notation. We give here verbal descriptions
with a minimum of frills and terminology.
-------
2-189
TIMLST(I+I)=TIMIST(L)
PEVLST(n-l)nTIMlST(L)
PTRNO(H-I)=PTRNO(L]
L=NOEV+J
-M
TIMIST(K) =T
PEVIST(K) = i
PTRNO(K)=H
NOEVsNOEV-fl
RETURN
Figure A.4 PUT-Subroutlne for Storing Events
-------
2-190
a) Event Type !_ . . . Truck Arrives at Dis-
posal Site (Figure A.5): At the beginning of the program
a logical variable (SKPDK) is set to .TRUE. if the num-
ber of unloading docks NLD > the number of trucks and to
.FALSE, otherwise. This is used to shunt around a great
deal of tne logic in this section if no unloading bottle-
neck is possible.
1) Set the time at the disposal site
counter for this particular truck to minus the value of
the master daily time counter (TELAPS). This is done
so that when the truck leaves the disposal sites (Event
Type 3j below), the time at the D.S. can be gotten simply
*
by adding the value of TELAPS at that time.
2) If SKPDK = .TRUE, go to step 8' .
If SKPDK , .FALSE., go to itep 3 .
3) Locate an unused unloading dock.
If one can be found, denote its number by II and go to
step 8 . If all unloading docks are busy, go to step k .
4) Locate the first occurring event
type 3 in the TSR. This tells when some other truck will
leave the disposal site freeing an unloading dock. Call
the associated time TIMLST(IJ).
5) PUT a new event type 1 in the TSR
at TIMLST(IJ). This forces the truck to arrive again at
-------
2-191
0)
-P
to
o
a
CO
•H
Q
to
O
-------
2-192
the disposal site; in effect, the truck is placed on a
queue waiting for a free unloading dock.
6) If this truck was arriving at the
disposal site fresh (i.e., not on the queue) then its
unloading dock bottleneck time counter UDBNT (truck #)
will be zero. At this point, if UDBNT(truck #)= 0, set
UDBNT(truck #) = -TELAPS(soe remark in step 1 above),
otherwise leave it as is.
7) Go to step 1 of the event clock.
8) Set unloading dock #11 to busy
status .
8') Accumulate the haul time and haul
distance generated in reaching the disposal site.
9) If UDBNT(truck #) = 0, go to step
11 . If < 0, go to step 10 .
10) Compute unloading bottleneck time
= UDBNT (truck #) + TELAPS and generate statistics (see
subroutine SOUP). Reset UDBNT (truck #) to zero.
11) Accumulate statistics (SOUP) on
haul efficiency (man • minutes per ton • mile). Increase
trip counter (truck #) by 1.
12) Generate a random unloading time
uniformly distributed between 5 and 9 minutes and PUT an
event type 3 in the TSR to occur at this time.
-------
2-193
13) Go to step 1) of the event clock.
b) Event Type 2^ ... Collection Unit Entered
for First Time (Figure A.6): The net effect of this event
is to set up the job to be done on a fresh CU and to
determine how much can be taken aboard the truck (the
truck may have serviced ot»ier CU's before starting this
one). This event sets up the job continuation with an
event type 4.
1) Accumulate the haul time and haul
distance that were generated in reaching this CU.
2) Call subroutine CMPJOB(compute
Job) to generate the random amount of refuse (RFPCU) in
this CU, the time to collect this refuse (TTFPCU), and
the internal distance (DINPCU) to be traveled while
collecting. Each of these quantities is indexed by
truck #.
3) Call subroutine CAPRUL (capacity
rule) to generate the proportion PROP of the refuse
present in this CU (PROP is indexed by truck #) that can
be loaded.
4) PUT an event type 4 (CU is parti-
ally finished) in the TSR to occur at time = PROP x
TTFPCU.
5) Go to step 1 of the event clock.
-------
U=NPCU(H)
CULTIM(H) = CULTIM(H)+HULTIM(H)
CULDIS(H) = CULDIS(H)+HULDIS(H)
( CMPJOB )
CAPRUL
PUT
STATUS(H)=S
11)13
Figure A.6 Event Type 2: Collection Unit Entered
for First Time
-------
2-195
c) Event T^pe 3. ... Truck Leaves Disposal
Site (Figure A.7). The function of this event is to
accumulate trip statistics and to continue the job
either
i) Re-entering an unfinished CU
ii) Going to an unstarted CU
111) Going to the garage (all CU's
serviced.
1) If 3KPDK (see Event Type 1, above)
= .TRUE, go to step 3 . 0 ;herwise go to step 2 .
2) Set an unloading dock to not-busy
status.
3) Commute proportion of capacity
utilized = Tonnage of refuse aboard / capacity of truck
in tons and update histogram.
4) Accumulate statistics on trip
counters of tonnage, time, and distance by adding in trip
counters thereof.
5) Accumulate statistics on trip
counters of tonnage, time, and distance (see SOUP sub-
routine) .
6) Bring time spent at disposal site
counter up-to-date for this truck by adding TELAPS to it
(see Event Type 1, step 1 ), and accumulate statistics
(SOUP).
-------
-------
2-197
7) If the proportion completed at the
last CU serviced by this truck (PROP) is less than 1, go
to step 8 , otherwise go to Step 11 .
8) Entering here indicates there is a
CU which has not been completely serviced. Compute the
haul distance and haul tim? from the disposal site to
this CU (subroutine TRATIM).
9) PUT an event type 5 (partially
done CU entered) into the TSR at time = haul time just
computed.
10) Go to step 1 of the event clock.
11) Determine whether truck is through
for the day. This is done with subroutine FINEXU (find
next unit) and is indicated by an inability to find a
successor unit. If the truck is through for the day, go
to step 12 , otherwise go to step 15 .
12) Compute the haul distance and haul
time from the disposal sit? to the garage (TRATIM).
13) PUT an event type 6 (truck arrives
at garage) into the TSR at time = haul time just computed.
14) Go tD step 1 of the event clock.
15) If the truck is not through for the
day: Compute the haul time and haul distance (TRATIM)
from the disposal site to the next CU found by FINEXU.
\
-------
2-198
16) PUT an event type 2 (collection
unit entered for first time) Into the TSR at time =
haul time just computed.
17) Go to step 1 of the event clock.
d) Event Type 4_ . . . Collection Unit is_
Partially Finished (Figure A.8). This event is the
successor to the two CU entry events (2 above and 5
below). In this event trip tonnage, time, and distance
measurements are accumulated, and it is determined
whether the truck must now go 1 o the disposal site or
whether it has enough empty sp;ice to permit it to go on
to the next CU.
1) Accumulate trip tonnage, and dis-
tance measurements.
2) If the proportion (PROP) of this
CU collected is less than 1, go to step 3 , otherwise go
to step 6 .
3) Compute haul time and haul distance
from this CU to the disposil site (subroutine THATIM).
^) PUT an event type 1 (truck arrives
at disposal site) into the TSR at time = haul time just
computed.
5) Go to step 1) of the event clock.
-------
2-199
U=NPCU(H)
CULTIM(H)=CULTIM(H)+HULTIM(H)
CULDIS (H) = CULDIS (H) + HULDIS (hi)
F=1-PROP(H)
RFPCU(H)=F*RFPCU(H)
TTFPCUH;SF*TTFPCU(H)
DINPU(H>F»DINPCU(H)
STATUS(H)=3
/ PUT
<(PROP(H)*TTFPCU
\
-K CAPRUL
Figure A.8 Event Type 4: Collection Unit
is PROP(H) Finished
-------
2-200
6) Determine whether truck is through
for the day (see step 11 in event type 3 above). If yes,
go to step 3 above, otherwise go to step 7 •
7) Set up parameters for going direct-
ly to the next GU and transfer directly to step 1 of
event type 2 without going through the event clock. This
assumes that there is_ negligible haul time and distance
between successive CUfs of a. truck's route.
e) Event Type 5. ... Partially Done CU is_
Entered (Figure A.9). This event occurs only as a result
of a truck returning to collection after a visit to the
disposal site, but it is not the only event that may
succeed such a visit,
1) Accumulate the haul time and haul
distance generated in reaching this CU from the disposal
site .
2) Let P = 1 - PROP where PROP =
the proportion of this CU that has already been serviced.
Then, let RPPCU * F replace RFPCU = refuse present CU;
TTFPCU x F replace TTPPCU = time to finish p_resent CU;
DINPCU x F replace DINPCU = distance i_nside p_resent CU.
These latter quantities having been computed originally
in event type 2 by subroutine CMPJOB.
-------
2-201
U = NPCU(H)
TRPT0N(H) TRPT0N(H) + PR^P(H) * RFPCU(H)
TRPTIM(H) = TRPTIM(H) + PR<>P(H) * TTFPCU(H)
TRPDIS(H) = TRPDIS(H) + PR0P(H) • DINPCU(H)
YES
(FINEXU \
(K, I, J) /
K = 2
K=1
/ TRATIM(2) \
/PUT \
\(HULTIM(^,1)/
DSINCE(H) = J
NPCU(H) = I
HULTIM(H) = 0.
HULDIS(H) = 0.
STATUS(H) = 4
STATUS(H) = 5
Figure A. 3
Figure A.9 Event Type 5: Partially Done Collection
Unit is Entered
-------
2-202
3) Enter subroutine CAPRUL (capacity
rule) to determine proportion (PROP) of remainder thus
computed that will now fit into empty truck.
4) PUT an event type 4 (Collection
Unit is Partially Finished) into TSR at time = PROP x
TTFPCU.
5) Go to step 1) of the event clock.
f) Event Type 6_ . .. Truck Arrives at
Garage (Figure A.10). This event is the termination of
the day's work for each truck. Statistics on daily ton-
nage, time, and distance are accumulated and a histogram
record of hours worked is updated. Any pending events
pertaining to this truck are scratched. This latter task
is superfluous in the present program, but is included for
the sake of flexibility.
1) Accumulate statistics on daily
tonnage, time, and distance, and on haul time, haul dis-
tance, and trip count.
2) Update hours-worked histogram.
3) Scratch any pending everts in TSR
pertaining to this truck.
4) Go to step 1 of the event clock.
-------
2-203
SOUP
]DAYTON (H).3),
/ SOUP >.
UDAYTIM(H),5)\
I
/ SOUP \
/(DAYDIS (H). 100
/SOUP \
{(CULTIM (HI + \
\HULTIM(H),6)/
CULTIM (H) -0.0
/ SOUP
((CULOIS
\HULOI
iOUP \
US (H)+ \
is(H>, \\y
I
CULOIS (Hi =0.0
I
SOUP
((TRPCNriH).
XsTElAPS/4o
/ PWFHST \
(X.4) / \ (X.3) /
I6«J
'l6>NOEV?
16-16+1
STATUS (HMO
PEVLST (16) =
-PEVLSTII6)
Figure A.10 Event Type 6: Truck Arrives at Garage
-------
2-204
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-------
2-205
2. Data Generating Subroutines
A. Job Computation (CMPJOB) (Figure A. 11)
This subroutine is used to compute the
job to be done on a newly entered CU. The job is speci-
fied in terms of the three quantities:
RFPCU(H) = Refuse in present CU (in tons)
DINPCU(H) » Distance inside p_resent CU
(in miles)
TTFPCU(H) » Time to finish present CU
(in minutes) where H = index
of the Hth truck.
DINPCU, mileage traveled while collect-
ing, is computed from the quotient services in the
CU/service density therein in services per mile.
TTFPCU, Ls in the present version of
this subroutine, computed from a regression formula which
is discussed above in section II» l.D(a).
For further details, see detailed flow-
chart .
USAGE: CALL CMPJOB: all inputs and
outputs are in COMMON and labeled COMMON sections.
-------
B- Travel Time and Distance (TRATIM)
(Figure A.11).
This subroutine is used in each event
program section where the possibility of truck movement
exists. It is also used in the outer program when setting
up the initial trip of a truck from the garage to its
first CU of the day.
USAGE: CALL TRATIM(J)
where J = 1 denotes garage to CU
J = 2 denotes disposal site to
CU or vice versa, and
J = 3 denotes disposal site to
garage.
All other inputs and outputs are carried
in COMMON and labeled COMMON sections.
METHOD: (See also section II, l,D.(d) of
this appendix.)
Let Y = Distance to be traveled. The
source cited above gives a speed vs. distance curve based
on total haul distance to and from the disposal site. The
distance Y is a one-way distance in all three cases (J)
above. For J = 1, 3 there is no round trip at all and
for case J = 2, it appeared an unwarranted complication
to have to reach ahead to find out where the truck might
be going next. Therefore, to approximate round
2-206
-------
2-20J
trip distance, Y is doubled m, = 2-Y) in all cases and
the computation of speed is based on m, .
For Distances
1) D - 0, the final results
HULDIS(H) = 0 (haul distance Hth
truck-miles)
HULTIM(H) » 0 (haul time Hth truck-
minutes)
are returned.
2) 0 < m^ < 5.5, let V~l = 6 minutes per mile
3) 5-5 < n^, let Dx = n^ - 5-5 and let V
c D"
b ~ 0.44-48 f 0.2646 D'mlnutes Per mlle-
In cases 2) and 3) S is further modified by
letting V""1 • N(l, 0.1) replace V"1 .
Finally, HULDIS(H) = Y(miles) and HULTIM(H) =
Y • V" (minutes). Note that distances are computed by
the metropolitan metric as described in Section II,
l.D.(c) of this appendix.
For further details, see flowchart.
-------
2-208
3. Auxiliary Subroutines
A- Capacity Rule (CAPRUL) (Figure A.12)
This subroutine is used to test the
refuse present (either just generated or remaining to be
collected) in a given CU against the amount of refuse
already collected by the truck and the capacity of the
truck. The quantity returned is PROP(H), the proportion
of the present CU that the Hth truck can load. The steps
of the determination .are as follows:
1) Call the PINEXU (Find Next Unit)
subroutine to determine whether this is or is not the
last unit for today.
2) a) If this is_ the last unit for
today, determine if the sum of the refuse in this collec-
tion unit and the refuse already loaded exceed the truck's
capacity by more than 5%. If yes, proceed to step 3; if
no, set PROP(H) = 1.0.
b) If this is not the last unit
for today, proceed to step 3 •
3) Generate a Pseudo-capacity = truck
capacity * r where r is a random variable ^N(.95, .05)•
4) Set PROP(H) = ratio of unused
pseudo-capacity to refuse present in this collection
-------
2-209
0)
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-------
2-210
unit, I.e.
PSEUDO-CAPACITY - REFUSE LOADED
REFUSE IN CU
5) If PROP(H) now exceeds 1.0,
reduce it to 1.0.
USAGE: CALL CAPRUL (all inputs and outputs
in COMMON).
B. Find Next Unit (FINEXU) (Figure A.12)
This subroutine, as its name suggests,
finds the next unit to be serviced by a given truck on
a given day, or it indicates that the truck is now work-
ing on its final CU and there is no next unit.
The inputs of this subroutine are:
H = Truck number
NPCU(H) = Number of present CU for truck H
NCUZ(H) = Number of collection units in the
zone of truck H
DOW(U) = Numbar of the schedule controlling
the servicing of collection unit
unit U.
ISCHED(TODAY,DOW(U)) = An irray entry either = 0, or > 0
If = 0, this CU(U) is not serviced TODAY.
-------
2-211
If > 0, this CU(U) is serviced today and this
number is the number of the days since
the last collection.
The inputs are sent to the subroutine
via unlabeled and labeled COMMON statements.
USAGE: CALL PINEXU (IR, IU, ID)
The items of the calling sequence
are the only outputs returned by the
subroutine. These are:
IR : = 2 This is the last unit of the day
for this truck today. Ignore IU
and ID.
= 1 There is a next unit, and
IU : = The number of the next unit found
ID : = The number of days since this unit
was last serviced.
C. General Histogram Reader (HISTOG)
(Figure A.13)
This subroutine is used only once in
the present version of the program. It is available,
however, for any future expansion to return a random
variable from any arbitrary probability density function.
The p.d.f. is presented as a vector of densities repre-
senting a histogram. The sum of the entries must equal
-------
2-212
= RAND0M(RN)
1 = 1 + 1
YES
NO
Y =
K =
C
RETURN
NO _^ \. YES
K = I
Figure A.13 HISTOG: General Hist
ogram Reader
-------
2-213
1.0. The item returned to the calling program is the
Index K of the histogram which has been chosen arbi-
trarily. The main program may then use K in a formula
to determine the random argument (equally-spaced) or
use K to look up the argument from a list of (unequally-
spaced) arguments.
USAGE: CALL HISTOG (A, N, K)
A is the vector of densities
N is the length of A
K is the index as described above
METHOD: Generate a random 0 < u < 1 .
Find largest index K such that
K
E
1=1
D. General Statistics Accumulator (SOUP)
(FigureA.14)
This subroutine accumulates the minimum,
maximum, sum, and sum of squares of (at present) 12
different variables for up to 64 different trucks:
USAGE: CALL SOUP (X, K)
X is variable to be accounted for
-------
2-214
A(H, 3, K) = A(H, 3, K) + X
A(H,4, K) = A(H, 4, K) + X2
YES
A(H, I, K) =X
YES
A(H, 2, K) = X
INDEX K:
LEGEND
INDEX H: TRUCK NUMBER
SECOND INDEX: 1..MIN. (X)
2.. MAX. (X)
3..2X
4..2X2
1
2
3
4
5
6
7
8
9
10
11
12
VARIABLE X CATEGORY
TRIP COUNT
TRIP TONNAGE
DAILY TONNAGE
TRIP TIME
DAILY TIME
HAUL TIME
DISPOSAL SITE (DUMP) TIME
UNLOADING WAIT TIME
TRIP DISTANCE
DAILY DISTANCE
HAUL DISTANCE
HAUL EFFICIENCY
Figure
SOUP: General Statistics Accumulator
-------
2-215
K is the variable type (1 < K < 12)
H is the truck number is contained in
unlabeled COMMON.
At the end of appropriate time periods,
the sums and sums of squares may be used for computing
averages and second moments.
The 12 variables represented by the
index K are listed in Figure A-14.
E. Report Waiter (REPORT)
This subroutine is the largest in the
program, and it is neither feasible nor desirable to dis-
cuss it here in detail. The end result of its operation
may be inferred from a perusal of a sample printout and
the description thereof which appear below.
F. Frequency Histogram Packer (PWFHST)
(Figure A.15)
This program is used to create frequency
histograms of arbitrary performance characteristics.
USUAGE: CALL PWFHST (X, I)
where X » a particular value of the perform-
ance characteristic
I = the number of the histogram type in
which X will be counted
-------
2-216
(PWFHST (X,
NHF =
X-
r-cijm
Car II)
NHF = MAX0 < O, M1N0 (l4,NHF+l)
MIST (NW,H,J^=MIST(NW,H,H)-I-INCR(NF)
RETURN
Figure A. 15 PWFHST: Freciuency Histogram Packer
-------
2-217
H = truck number (transferred in blank
COMMON).
The program provides for up to 18 such histograms (index
I above) for each of 64 trucks. Each such histogram
consists of 15 "slots" as follows:
Given for each histogram type (I), two
constants C, and C? ...
SLOT # is used to count x such that
C < x < C + C
1 - x 1 + 2
15
13C
Jl + 13C2 - X < ""l"°°" '
The storage requirements, if each slot
occupied a complete computer word for 18 histogram types
-------
2-218
for 64 trucks would be 17280 (=18 x 64 x 15) words. This
has been cut down to 3458 (= 18 x 64 x 3) words by pack-
ing slots at five seven-bit fields per word. Thus, this
routine will work on any computer having 35 or more binary
bits per word exclusive of the sign bit. This routine
will not work on a decimal computer.
It is the program user's responsibility
to design his runs in such a manner that the count in any
given slot does not exceed 127 (= 1 111 111 binary).
This can be accomplished in part by splitting the range
that is expected for x so that different parts of the
range belong to different histogram types. Thus, in the
program histogram types 1 and 2 describe the frequency
distribution of capacity over the range 0 - 60% and 60 -
<*>%, respectively while histogram types 3 and 4 describe
the frequency distribution of length of work day over the
ranges 0 - 11.5 hours and 11.5 - +°° hours respectively.
For information on adding new histogram types,
see "Block Data Section (BDHST)" below (4.C).
G. Frequency Histogram Unpacker (UNPWFH)
(Figure A.16)
This is a companion subroutine to the
immediately foregoing subroutine. The function is simply
to take the 3 five-slot words pertaining to a certain
-------
2-219
UNPWFH(H,NAMHST)
1=1+1
YES
/ RETURN J
NO
(I,H)
YES
NO
1=1
LB=6-L
K = J
= K/128
IV(LB,I)=K-128*J
Figure A.16 UNPWPH: Frequency Histogram Unpacker
-------
2-220
truck and store the 15 slots in 15 consecutive computer
words, so that they may be printed out by REPORT if IHISTO
= 0 (option). As above, this routine will not work on
a decimal computer.
The histograms are stored in an array called MIST
which is dimensioned either 192 x 18 or 3 x 64 x 18 con-
tained in a labeled common section HIST.
USAGE: CALL UNPWPH (H, NAMHST, IV)
H(integer) = truck number
NAMHST = MIST (1, J) for Jth histo-
gram when calling program
has MIST dimensioned 192 x
18; = MIST (1, 1, J) when
calling program has MIST
dimensioned 3 x 64 x 18.
IV = Name of 15 word vector for
receiving unpacked histogram
"slots".
4• Miscellaneous Subroutines
A. Random Number Generators
a) RANDOM
USAGE: Y = RANDOM (RNO)
RNO is an integer used as the "seed" for generating
pseudo-random integers. Y, the value returned, is a
-------
2-221
floating point (real) variable : 0 < Y < 1 .
This routine ls_ not provided with the program
since it is too highly-machine-dependent. It is possible
to write a FORTRAN version that will operate, say, on a
35 bit (plus sign) computer having an 8-bit excess-128
exponent and a 27 bit mantissa, but although the program
might be FORTRAN, compiled and run on any other type of
computer, it would produce meaningless results.
This should prove little hinderence since pseudo-
random number generators are available for almost any
computer. Thus, if a given computer has a routine called
Y = UNIDRV(X) (uniformly distributed random variable),
the following program can be quickly written:
FUNCTION RANDOM (X)
RANDOM = UNIDRV (X)
RETURN
END
b) RANDND
USAGE: Z = RANDND(XMU, SIGMA, RNO)
Returns.Z drawn from N(y, a) where
y = XMU
a = SIGMA
-------
2-222
RNO is the seed used to generate two uniformly distributed
random variables Ul, U2 e(0, l) using the function sub-
program RANDOM described above.
METHOD: Z = SIGMA • Y + YMU
where Y ^ N(0, 1)
On odd entries generates:
XI = (-2 log Ul)1/2sin 2TTU2
(Natural log)
X2 = (-2 log Ul)1/2cos 2TTU2
On odd entries sets Y = XI in formula above.
On even entries sets Y = X2 left from prior odd entry
and uses formula above.
B> Three DecJmal Place Rounder (RND3PL)
This rout .ne is used to round a floating
point number to 3 decimal places. This can be done in
many cases simply by adding 0.0005, or by printing the
number with a format "FW.3." On the other hand, if
several such numbers are to be added and their sum is to
reflect accurately the sum of its rounded components, the
decimal places beyond the third must also be removed.
-------
2-223
The most accurate way to do this is in machine
language. The FORTRAN routine provided here is a compro-
mise and is for positive numbers only.
C. Block Data Section (BDHST)
This is a non-executed program segment
that allows constants and alphanumeric labels to be com-
piled into the labeled comri on block, /LHST/ used by
PWPHST and REPORT.
The items loaded are:
NHIST = The number of histograms currently generated
by the program. This is presently set = *J,
but may be increased up to 18 .
MXAHST - A 15 x 18 array containing the alphanumeric
labeling for the 15 intervals of up to 18
histograms.
LHN = A 3 x 18 array each column of which contains
the alphanumeric column heading appearing on
the print-out of the arrays. Each column
heading is contained in 3 computer words
(= 18 characters).
Cl = An 18 place vector whose elements are the
upper bounds of the first intervals of each
histogram type.
-------
2-224
C2 = An 18 place vector whose elements are the
common interval lengths for each histogram
type.
IV. Preparation of_ Data for Input
The input data deck makes use of the FORTRAN IV
features, NAMELIST, which the user is advised to read.
Runs (i.e., different problems, other variations of
a given problem) may be stacked up one behind the other.
For all runs after the initial one, only that data need be
entered which differs from the data of the immediately pre-
ceeding run. Of the sever different data sections to be
discussed below, only the first must be encountered on
every run.
All data cards must be blank in column 1. Each data
section is read under the control of a different NAMELIST
the name of which is given on the title line in paren-
theses, and if read, must be read in the order given here.
1. Run Setup Data (RUNSET)
This data section may contain the following
items:
a) FOLLOW: This is a 6 position list of indi-
cators which are denoted below as f..
f . = 0 means that the i data section ±s_
not included in this run.
-------
2-225
t 0 the i data section i_s_ included
in this run.
The index i corresponds to the
following data sections
i Data Section (NAMELIST name)
1 WSPDAT
2 TRKDAT
3 DISDAT
4 CUDATA
5 SKEDAT
6 GDSDAT
Example: FOLLOW = 0, 1, 1, 1, 0, 0, ... or FOLLOW = 0,
3*1, 0, 0, ... indicates that WSFDAT (weather and seasonal
factor data), SKEDAT (scheduling data), and GDSDAT (garage
and disposal site data) will not be read (i.e., will
remain unchanged from previous run) for this case.
b) PROBID: This is a case number for identi-
fication purposes. Example: PROBID = 10115.
c) NQTRS: These are four Integers signifying
the number of times each c uarter of the year will be
simulated.
Example: NQTRS = 2, 3*0, ... indicates that the first
quarter (January through March) will be simulated twice
and the remaining quarters not at all.
-------
2-226
d) IPUNCH = 0 Do not punch results on
cards
= 1 Do punch results on cards.
This refers to all results (In a raw form as explained
below), Including generated histograms.
e) IHISTO = 0 Print histogram results on
quarterly report
- 1 Do not print histograms on
quarterly report.
f) RNO = seed from which random numbers
will be generated.
This is needed only on first case.
Example: RNO =30517578125 (an Integer).
The above inputs are combined as follows (out of order to
show that order Is unimportant within a section):
1 2
$RUNSET RNO = 30517578125, IHISTO = 1,
IPUNCH = 0, NQTRS = 2,3*0, PROBID = 10115,
FOLLOW = 0, 3*1, 0, 0
2. Weather and. Seasonal Factor Data (WSFDAT)
This data section may contain the following
items:
a) SEASF: This is a 12 place list giving
the relative refuse production in each of the 12 months
-------
2-227
vs. a user selected base month. For example, SEASP (1) =
1.0, 0.96, ..., 1.09 indicates that February has 96%
and December, 109% the refuse production of January.
b) XLIST: This is a 4 place list of
relative collection time factors which will be selected
randomly according to the iveather severity distribution
given by the next item. An example will be given below.
c) WEDATA: This is a triply subscripted
array, 4x3x4. The first subscript corresponds to
the elements of XLIST, the second to the month of the
quarter, and the third to the quarter. The elements are
probabilities of occurrence of a slowdown corresponding
to XLIST.
Example: Suppose that analysis of the effect of weather
on pickup efficiency shows that we may categorize most
weather events into four groups with relative work times
of 1.00, 1.05, 1.10, 1.15- Then these four entries in
the given order are the elements of XLIST. Suppose
further that the following data applies to the first
three months of the year.
-------
2-228
PROBABILITIES OP OCCURRENCE OF
GIVEN PERCENTAGES OP NORMAL WORK RATE
Percent
Pickup
of Normal
Efficiency
100
105
110
115
Jan.
.50
.25
.15
.10
Month
Feb.
.35
• 35
.25
.05
Mar.
.40
.30
.20
.10
SUM 1.00 1.00 1.00
Then a part of WSFDAT might read
1 2
$WSFDAT XLIST = ]., 1.05, 1.1, 1.15,
WDATA = .5, .25, .15, .1, .35, ..., .2, .1,
3. Truck Data (TRKDAT)
This data section may contain the following
items:
a) NUMTRK: The number of trucks whose
operation will be simulated: 0 < NUMTRK < 64 .
b) NFCTJ: Number of first CU in the zone
served by each truck. This is a string of up to 64
integers.
c) NCUZ: Number of CU's in the zone served
by each truck. This is a string of up to 64 integers.
-------
2-229
d) CAPCTY: The capacity of each truck in
tons. This is a string of up to 64 numbers: whole
numbers will be converted to floating point.
e) CREW: Tie number of men associated with
each truck. This is a string of up to 64 whole numbers.
Example:
1 2
$TRKDAT NUMTRK = 5, NPCU = 1, 5, 9, 13, 17,
NCUZ = 5*4, CAPCTY = 1, 2, 3-, 4., 4,
CREW = 2, 2, 3, 3, 4
4. District Data (DISDAT)
This data section may include the following
items:
a) NDIST: Number of districts: 0 < NDIST
< 10.
b) APROD: Average refuse production in
pounds per service per day. These are the y's described
in Section II, l.D.(a) of this appendix. Input as a string
of up to 10 numbers.
c) VPROD: Standard deviation of refuse
production in like units. These are the a's of Section
II, l.D.(a). Input as a string of up to 10 numbers.
d) ALPHA Each of these is a string of up
BETA to 10 numbers. These are the
GAMMA regression constants discussed
in Section II, l.D.(b).
-------
2-230
Example:
1 2
$DISDAT NDIST = 2, APROD = 5, 10.,
VPROD = 1.5, 3.333, ALPHA = 2*10,
BETA = 2*5, GAMMA = 2*33
5. Collection Unit Data (CUDATA)
The number of collection units, NUNITS,
Is not entered as Input to the program. The quantity
NUNITS is computed as
NUMTRK
NUNITS = V* NCUZ
h
where NUMTRK and the (NCUZ, } are entered in section
h
TRKDAT described above. It is the user's responsibility
to see that data entered in this section correspond
properly with TRKDAT. In particular, it is required that
*
NUMTRK < NUNITS < 1000 .
The items of data that may be entered in
this section include the following strings of up to 1000
numbers:
-------
2-231
a) DOW: Integers which give the number of
the individual schedules (column indices of arrays ISCHED
(6, 8). These schedules which are discussed below, dic-
tate which day of the week a CU is to be serviced.
b) DIST: The number of the district with-
in which the CU is wholly contained.
c) SERV: The number of services in each
CU.
d) DNSTY: The service density in services
per mile.
XCOOR The loca'fcion of tne cu as dis-
e) . cussed in Section II, l.A. of
YCOOR' this appendix distances in miles
Example: (In accord with example in TRKDAT above.)
1 2
$CUDATA DOW = 1, 1, 2, 2, 1, 1, 2, 2, ..., 1, 1, 2, 2,
DIST = 10*1, 10*2, SERV = 20*500, DNSTY = 10*90, 10*140,
XCOOR = 1, 1.5, 1, .75, ..., 2.3, YCOOR = 1, ..., 5 .
6. Scheduling Data (SKEDAT)
This data section may contain the following
items:
a) MSCHED: This is a string of seven inte-
gers that governs the advancement of the tday counter (DAY)
and the value !of TODAY (= 1 Monday, ..., = 6 Saturday).
The following examples illustrate the construction of a
-------
2-232
MSCHED string.
Example 1: Suppose that at least one truck works
on Tuesdays, Thursdays and Fridays and no trucks work on
any other day.
Step 1: Construct a list in ascending order of
the "TODAY" values of the days worked. Put a zero at the
head of the list and 7 plus the first number after zero
at the end.
LIST + 02459
+ 7
Step 2: Compute first differences (elements):
LIST -> 0 2 4 5 9
FIRST DIP. -»- 2214
Step 3: Above list write between each pair the
sum 1 + the left member of the pair. These are the list
indices of the elements:
POSITION •*• 1356
LIST -»• 0 2 4 5 9
FIRST DIP. ->• 2214
-------
2-233
Thus the resultant MSCHED = 2, 0, 2, 0, 1, 4, 0. The
positions not computed as above, are filled with zeros.
The program does not use these elements.
Example 2: Trucks service some of their CU's on
Mondays and Thursdays and the rest on Tuesdays and
Fridays .
Step 1: 012458
Step 2: 012458
11213
Step 3: 12356
012458
11213
MSCHED = 1, 1, 2, 0, 1, 3, 0 .
b) ISCHED: This is an array 6 x 8 each
column of which may constitute a schedule of collection
for a CU. Thus in Example 2 above, Monday-Thursday CU's
have a schedule 400300. A zero indicates that the
CU is not serviced on the corresponding day of the week.
A non-zero conveys the opposite and the value given i.e.
-------
2-234
4 and 3 in the example, are the number of days since
the last collection.
Example: (Corresponds to Example 2 above).
$SKEDAT MSCHED = 1, 1, 2, 0, 1, 3, 0,
ISCHED = 4, 0, 0, 3, 0, 0,
ISCHED (1, 2) = 0, 4, 0, 0, 3, 0
Referring back to section CUDATA, CD's with a
Monday-Thursday collection should have DOW = 1 and
those with a Tuesday-Friday collection should have DOW =
2 , for this example.
7- Garage and Disposal Site Data (GDSDAT)
The following items may be entered in this
data section:
a) NLD: The number of goading docks, i.e.
positions at which a truck may dump its load at the
disposal site.
XCOO^D
b) YCOORD The locati°n o? the disposal
site with respect to same origin used for the location of
CU's. Distance in miles.
x XCOORG _. . . 4 _ , .
c) YCOORG location of the garage.
See remarks in b) immediately above.
-------
2-235
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-------
2-236
V. Description of Output
1. Printed Output
One or two pages of output are printed for
each truck at the end of each simulated quarter. A
sample of this printed output is given in Table A.2a and b
The first page of the output and all items
thereon are printed, i.e.,they are not subject to
supression via an option entered on the RUNSET card of
the input (see above). This option is the input variable
IHISTO. The second page reports certain frequency
histograms described below and is suppressed by setting
IHISTO ? 0 and permitted if IHISTO = 0.
A. First Page Format
The first line reports truck number,
truck capacity, crew size, the number of the quarter (1,
2, 3, or 4) and the 6-placc integer problem identification
(PROBID).
The next section, entitled "The Job
Undertaken by This Truck," gives a brief recapitulation
of the input data related to this truck and is readily
understood in terms of the description of the input given
above.
The third section cf the first printed page is
-------
2-237
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-------
2-239
called "Digest of Main Results." This consists of three
columns of titles and four columns of figures. The
meaning of these is as follows:
a) First column of titles:
Days worked — this may be differ-
ent for different trucks and should be checked against
the scheduling input data. Always a multiple of 13.
Number p_f trips — the total num-
ber of loads carried to the disposal site in the number
of days worked. Travel to and from the garage is not
included in this count.
Minimum, maximum and average trips
per day — self explanatory.
b) Second column of titles: Two
figures accompany all but the last title in this column.
Of the paired columns of figures, the figures on the left
are total time in minutes over the quarter, and the
figures on the right are percentages of total time spent
by this truck in the following categories of time utili-
zation.
1) Pickup Time - total time spent
in actual refuse pickup.
2) Haul Time - total time spent
in travelling to and from the disposal site and the garage
-------
2-240
5) Dump Time - total time spent
at the disposal site discharging loads, exclusive of any
time spent waiting for an unloading dock to be vacated.
^) Walt Time - total time spent
at the disposal site waiting for an unloading dock to be
vacated.
5) Total Time - The sum of 1)
through 4).
The final (unpaired) figure in this
column gives the average hours worked per day and is com-
puted by dividing 5) above by 60 x days worked.
c) Third column o£ titles:
AGGR COLRATE - The average value
of the aggregate collection rate in man-min per ton.
This is CREW x Pickup Time (b.5 above)/total tons
collected.
PKUP COLRATE - The average value
of the collection rate in man-min per ton during collec-
tion only. This is CREW x Pickup Time (b.l above)/total
tons collected.
AVG SPEED - The average speed of
this truck in miles per hoar during all travelling opera-
-------
2-241
EFF CAPACITY — The effective
capacity, i.e. average tons per load.
PCT. CAP. UTIL — Average percent
capacity utilization = 100 * (effective capacity / truck
capacity).
AVG H-EFFCY — The average haul
efficiency in units of man- min per ton mile.
The fourth section of the first page is
entitled "Table of Detailed Results" and gives the mini-
mum, maximum, average, and standard deviation of the
items listed to the left. All times are in minutes per
day or per trip.
B. Second Page Format
The second page of printed output is under
control of the input option IHISTO in input section
RUNSET (= 0 allows printing; ^ 0 suppresses printing).
The contents of this page are two frequency
histograms of two columns each. The first of these
histograms gives the trip distribution of percent capacity
utilization. The range of percentages covered is from
0 to 116$ in 4$ intervals (closed on the left and open on
the right). The final interval covers the range 116$ to
+ infinity. The sum of the frequencies equals the total
number of trips made by this truck.
-------
2-242
The second histogram is the day distribution
of daily hours worked (length of workday). The range of
time is covered as follows: An initial interval going
from 0 to 4.5 hours, 28 intervals going from 4.5 to 18.5
hours in half hour increments, and a final interval going
from 18.5 hours to + infinity. All intervals are closed
on the left and open on the right. The sum of the
frequencies equals the number of days worked by this
truck.
Many of ,he upper range limits are
omitted due to space restrictions. When such a limit is
missing, it is to be interpreted as being equal to the
lower limit of the next interval.
The program may be altered to allow up
to 14 more such histogram columns to be printed on this
page.
2. Punched Card Output
Three main groups of cards are punched and
(see sample in Table A.3a and b). The last of these is
miscellaneous and will be disposed of forthwith. This
group is a single card and takes the form RNO =11 ...
I $. This allows re-entry of the random number seed. Its
-------
2-243
lOOOul
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4121
1000
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RNO=-l4790672812
Table A.3a Listing of Punched Card Output
-------
2-24 U
100001
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Table A.3b Listing of Punched Card Output (Cont.)
-------
output is not under the control of an option, and one
such card is punched for each quarter simulated.
A- Detailed Data Cards
The first group of cards consists of
twelve cards per truck per quarter. The format of these
twelve cards is as follows:
CARD COLUMNS
1-6 7-9 10-12 13-15 16-30 31-15 46-60 61-75
PRGBID QTR TRKNO CARDNO ITMIN ITMAX ITSUM ITSSQ
FORMAT 16 13 13 13 E15-8 E15.8 E15-8 E15-8
PROBID - same as on input section RUNSET (q.v.).
QTR - the number of the quarter of the year.
TRKNO - the number of the truck = 1, 2, ...,
NUMTRK.
CARDNO - the card number in this series of 12 =
1, 2, ..., 12.
ITMIN - the minimum value of the item.
ITMAX - the maximum value of the item.
ITSUM - the sum of the values of the item (not
the average).
ITSSQ - the sum of the squares of the values of
the item.
-------
2-2H6
Before enumeration of the items, note that when finding a
minimum one sets the location where it is to be stored
to a very large number (in this case, 99,999,999) and
replaces it each time a smaller value is found. If no
positive value of the item is over generated, we punch
out ITMIN = 99,999,999 and ITMAX through ITSSQ = 0.
The items are as follows:
Card #
1 The trip count.
2 Tonnage on a trip basis.
3 Tonnage on a daily basis.
4 Pickup time on a trip basis.
5 Pickup time on a daily basis.
6 Haul time on a daily basis.
7 Dump time on a trip basis.
8 Wait time' on a trip basis.
9 Pickup mileage on a trip basis.
10 Pickup mileage on a daily basis.
11 Haul mileage on a daily basis.
12 Haul efficiency (man-min per ton-mile)
on a round trip basis.
-------
2-247
B. Frequency Histogram Cards
The second group consists of four cards
per truck per quarter. These contain the frequencies of
fifteen intervals per card.
CARD COLUMNS
1-6 7-9 10-12 13-15 16 - 75
PROBID QTR TRKNO CARDNO FIFTEEN CONSECUTIVE
FREQUENCIES
FORMAT 16 13 13 13 15 or 15
PROBID, QTR, TRKNO are as above.
CARDNO - the card number in this series of 4 = 1, 2, 3,
4 .
FREQUENCIES - are as described above in the second page
of printed output. Cards 1 and 2 are the
capacity utilization histogram and cards 3
and 4 are the length of workday histogram.
-------
2-248
DETAILED LISTING OP SIMULATION PROGRAM
-------
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GLOSSARY OF NAMES AND TERMS
FOR APPENDIX 2-A
A.
Variables
APROD
CREW
CULDIS
CULTIM
DAYDIS
DAYTIM
DAYTON
DINPCU
DIST
DNSTY
DSINCE
DUMP
HULDIS
HULTIM
mean of refuse production in
Ibs/serv/day.
number of men on truck.
cumulative haul distance.
cumulative haul time.
total distance during day.
total time during day.
total tons hauled during day.
distance inside present CU.
number of district to which CU
belongs.
service density in services/mile.
days since last collection of CU
NPCU(I).
temporary counter to count time
(min.) spent by truck at disposal
site.
haul distance.
haul time.
2-279
-------
2-28Q
LDBI
NOEV
NPCU
PEVLST
PROP
PSCAP
PTRNO
RPPCU
SEASP
SERV
SKPDK
TELAPS
TRPCNT
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UDBNT
VPROD
tells whether unloading berth Is
being used.
number of events.
number of collection units
presently being serviced.
list of pending events.
proportion of refuse present in
this CU that can be loaded.
pseudo-capacity Of truck.
number of truck involved in
pending event.
refuse in present CU.
seasonal factor (month).
number of services.
logical variable (skip dock) see
page 2-190.
time elapsed (master daily time
counter).
trip count.
time to finish present CU.
unloading dock bottleneck time.
standard dev. of refuse production
in Ibs/serv/day
-------
2-281
B.
WESEV
XCOORN
YCOORN
Status Code
1
2
weather severity.
physical coordinates of
N = blank: CU's
N = D : Disposal site
N = G Garage
10
Truck not working today.
Truck enroute from jar age to firr.t
CU of day.
Truck collecting refuse (f^om
NPCU(D) .
Truck finished with collection unit
and is proceeding to next.
Truck has finished a load and is
traveling to disposal.
Truck is at disposal site and is
dumping.
Truck is at disposal site and is
waiting for vacant unloading dock.
Truck has dumped load and is
traveling to next-load area.
Truck has dumped last load of the
day and is enroute to garage.
Truck at garage and through for day.
-------
REFERENCES FOR PREFACE
[Pr 1] An Analysis of Refuse Collection and
Sanitary Landfill Disposal, Sanitary
Engineering Research Project, Techni-
cal Bulletin No. 8, University of
California, Berkeley, December 1952.
[Pr 2] Deininger, R. A., Water Quality Management :
The Planning of Economically; Optimal
Pollution Control Systems, Ph.D.
Dissertation, Northwestern University,
Evanston, Illinois, June
[Pr 3] Galler, W. S., Optimization Analysis for
Biological Filters, Ph.D. Dissertation,
Northwestern University, Evanston,
Illinois, June 1965.
[Pr 4] _ , and Gotaas, H. B., "Opti-
mization Analysis for Bilogical Filter
Design," J. Sanitary Engr. Div. A.S.C.E,
V. 92, No. SA 1, pp. 163-182, February
1966.
[Pr 5] Hufschmidt, M. M. and Fiering, M. B., Simu-
lation Techniques for the Design of
Water Resources Systems , Harvard
University Press, Cambridge, 1966.
[Pr 6] Loucks, D. P. , A Probabllstic Analysis of
Waste Water Treatment Systems, Ph.D.
Dissertation, Cornell University,
Ithaca, N. Y. , September 1965.
[PR 7] _ , ReVelle, C. S., and Lynn,
W. R. , linear Programming Models for
Water Pollution Control," Management
Science, V. 14, No. 4, pp. B166-B181,
December 1967.
[Pr 8] Lynn, W. R., Logan, J. A., and Charnes, A.,
"Systems Analysis for Planning Waste-
water Treatment Plants," J. Water
Pollution Control Federation, V. 3^,
pp. 565-581, June 1962.
2-282
-------
2-283
v __ •
[pr 9] __ _ __ j and Chai-nes , A., ''The Dyadic
Computation of a r -neralized Network
Model of a Sewage Treatment System/'
ONR Research Memorunau:n 19, Northwestern
Universlt.ys Tee/v/oic^l .:c,~ Inst . ^
October,, 19>^.
[Pr 10] _ — — - __________ -' ^h.D. Oist-rrtaticr
,
^LMP- 131ci "'\2a!iqlnj< of Se_wage_ Treatment
wo££s».No:.?! -".westen'- "university,~Evanston,
Illlnoib, -Tune 1953.
[Pr 11] __, App-.Tic'.i - D of So rid Wastes
Researcit lic-_eds_, APWA "\erearch Founda-
tion, Project No ,113, Chicago, Illinois,
May 1962
[Pr 12] Maas, A. et_ al_. , D-i-si^n o? W_a_t_er_ Qesources
Systems^ "Harvard" ifniVersity pfcss,
Cambridge, Massachusetts, 1962
[Pr 13] Sheaffer, J. ri, e^ao.,., li«i£use DiiiP2^al_
Nggcis and P_rac_t_-"oes~i_n Northeastern
Illinois, Techni'cal Re'port" No. 3,
Northeas'tern Illinois Metropolitan Area
Planning Commission, Chicago, Illinois,
June 1963,
[Pr 14] Smallwood, C., Jr., nailer, -,v, S., and
Anderson, C. A«. "lvaluetjon of Alter-
natives in Refuy-:-.- ^xsno?a.«'-Progress
Report, Fir^i; Year ^JViuajy 1 - December
31, 1967)," USPHS Demonstration Grant
1-DOl-Ul-OOSO-Cl, North Carolina State
University &t Raleigh, Raleigh, January
1968,
[Pr 15] de Tocquevllle, Alexis, La Derno_cratie_ e_n
Amerij^ue., 1835. This book is a journal
and series oT essays reporting on the
travels in America of this distinguished
French statesman and man of letters. In
it he comn.enzs on vhe so eta' and politi-
cal ir.stltu-'.oni- .N" ;.,;•: yc .DI;T, republic
and desrr>it-;-.d j/ grea" ae:all the un-
spoiled wonders oT uve American v.-llder-
ness .
-------
REFERENCES FOR PART I
A. Mathematics, Economics, and Location Theory
[Al] Armour, G. C., and Buffa, E.S., "A Heuristic
Algorithm and Simulation Approach to Relative
Location of Facilities," Man. Scl., V. 9,
pp. 29^-309, (1963).
[A2] Bellman, R., "An Application of Dynamic Program-
ming to Location-Allocation Problems,"
SIAM Review, V. 7, pp. 126-128, (1965).
[A3] Baumol, W. J., and Wolfe, P., "A Warehouse Loca-
tion Problem," Op. Res., V. 6, pp. 252-263,
. (1958).
[A1!] Charnes, A., Quon, J. E., and Wersan, S. J.,
"Location-Allocation Problems in the £,-
Metric," Joint CORS-ORSA 25th Nat'l
Meet., Montreal, Que., May 1964.
[A5] Cooper, L., "Location-Allocation Problems," Op.
Res., V. 11, pp. 331-3^3, (1963).
[A6] , "Heuristic Methods for Location-
Allocation Problems," SIAM Review, V. 6,
PP. 37-53, (1964) .
[A7] Drysdale, J. K., and Sandiford, P. J., "A Method
for Locating Warehouses," 30th Nat'l Meet.
ORSA, Durham, N. C., October 1966.
[A8] Efroymson, M. A., and Ray, T. L., "A Branch-Bound
Algorithm for Plant Location," Op. Res.,
V. 14, pp. 361-368, (1966).
[A9] Eilon, S., and Deziel, D. P., "Siting a Disti'lhu
tion Centre, an Analogue Computer
Application," Man. Sci., V. 12, pp. B245-
B254, (1966).
2-284
-------
2-285
[A10]
[All]
[A12]
[A13]
[All]
[A15]
[A16]
[A17]
[A18]
[A19]
[A20]
Eisemann, K., "The Optimum Location of a Center,"
Problem No. 62-11 In "Problems and Solu-
tions," M. Klamkln, Ed,, SIAM Review, V. 4.,
pp. 39^-395, (1962).
Francis, R. L. , "On the Location of Multiple New
Facilities with Respect to Existing Facili-
ties," J. Ind. Kng., V, 15, pp. 106-107,
Frank, H., "Optimum Locations on a Graph with
Probabilistic Demands,." Op. Res . , V. 14,
pp. 409-^21, (1966).
Friedrich, C. J., ^lIlM vJeber's Theory of_ the
Location of_ Industrie^, U. Chicago Press,
Chicago, 111., 1929.
Garett, J. W. , and Plyter, N. V., "The Optimal
Assignment of Facilities to Locations by
Branch and Bound," Op. Res . , V. 14, pp. 210-
252, (1966).
Hakimi, S. L., "Optimum Locations of Switching
Centers and the Absolute Centers and Medians
of a Graph," Op. Res., V. 12, pp. 450^-459,
(1964). -
Heller, I., "On the Problem of" Shortest Path
Between Points ," I & II (Abstract), Bull. A.
Math. S_£c.s 59, 6, (November, 1953).
Hillier, F. S., "Quantitative Tools for Plant
Layout Analysis," J. Ind. Eng. , V. 1*4,
pp. 33-40, (1963).
Isard, W., Location and Space Economy, Technology
Press, M.I.T., Cambridge, Mass., 1956.
Kantorovitch, L., "On the Trans location of Masses,"
Man. Sci. , V. 5, pp. 1-4, (1958).
Koopmans, T. C. and Beckmann, M. , "Assignment
Problems and the Location of Economic
Activity," Econometrics., V. 25, pp. 53-76,
(1957) .
-------
2-286
[A21] Kuehn, A. A., and Hamburger, M. J., "A Heuristic
Program for Locating Warehouses," Man. Sci.,
V. 9, PP. 643-666, (1963).
[A22] Kuhn, H. W., and Kuenne, R. E., "An Efficient
Algorithm for the Numerical Solution of the
Generalized Weber Problem in Spatial Econom-
ics," J. Reg. Sci., V. 4, pp. 21-33, (1962).
[A23] Mcllroy, M, D., "Transportation Problems with
Distributed Loads," abstract In Recent
Advances in Mathematical Programming, R. L.
Graves and P. Wolfe, Eds., McGraw-Hill, New
York, (196-3). " '
[A24] Maranzana, P. E., "On the Location of Supply
Points to Minimize Transportation Costs,"
IBM Systems J., pp. 129-135, June 1963.
[A25] Marschak, T., "Centralization and Decentralization
in Economic Organizations," Econometrlca,
V. 27, PP. 399-429, (1959).
[A26] Noble, B., "Optimum Location Problems," ch. 2 of
Applications of Undergraduate Mathematics in
Engineering, pp. 13-33> Macmillan Co., New
York, (1967).
[A27] Palermo, P. P., "A Network Minimization Problem,"
IBM J., V. 5, PP. 335-337, (1961). See [A30],
[A28] Prager, W. , "A Generalization of Hitchcock's Tran
Transportation Problem," J. Math.Physics,
V. 36, pp. 99-106, (1957-195FT7" "
[A29] , "Numerical Solution of the
Generalized Transportation Problem," Naval
Res. Log. Q., V. 4, pp. 253-261, (1957) .
[A30] Tideman, M., "Comment on 'A Network Minimization
Problem'," IBM J., V. 6, pp. 259, (1962),
[A31] Tutte, W. T., "On Hamiltonian Circuits," London
Math. Soc. J., XXI, Part 2, 82, pp. 98-101,
-------
2-287
[A32] Verblinsky, S., "On the Shortest Path Through a
Number of Points," P£.OC_. Am. Math. Spc.,
II, 6 (Dec. 1951).
[A333 Vergin, R. C. and Rogers, J. D., "An Algorithm
and Computational Procedure for Locating
Economic Facilities," Man. Scl., V. 13,
pp. B240-B251, (1967).
[A34] Wersan, S. J., Quon, J. E., and Charnes, A.,
"Systems Analysis of Refuse Collection and
Disposal Practices," 1962 Yearbook A.P.W.A.,
Chicago, Illinois, (1
B. Computer Technology: Backplane Wiring
[Bl] Glaser, R. H., "A Quasi-Simplex Method for Design-
ing Suboptimum Packages of Electronic Build-
ing Blocks," 1959 Computer Applications
Symposium, Armour Research Foundation,
Illinois Institute of Technology, Chicago,
Illinois.
[B2] Kurtzberg, J. M. , "Algorithms for Backplane Formu-
lation," Proc. O.N.R. SyjnppsjAim cm Micro-
electronics and Large Systems, Washington,
D. C. , 18 November 19 6T.
[B3] _ , "Backboard Wiring Algorithms for
the Placement and Connection Order Problems,"
Burroughs Corp. Te ch . Rep. TR60-40 (Company
Confidential)," 2$ June I960.
[B4] __ , "Computer Mechanization of Design
Procedures," Proc. 6th Annual Conf . Am. Inst .
Ind. Eng. , Detroit, Michigan, 15 October 1964.
[B5] _ , and Seward, J., "Program for Star
Cluster Wiring of Backplanes," Burroughs
Corp. Internal Report s January
[B6] Loberman, M. , and Weinberger, A., "Formal Proce-
dures for Connecting Terminals with a Minimal
Total Wire Length," J.A.C.M. V. 4, pp. 428-
427, (1957).
-------
2-288
[B7] Rutman, R. A., "An Algorithm ror Placement of
Interconnected Elements Based on Minimum
Wire Length," AFIPS Proc., 1964 Spring
Joint Computer Conf., April 1964.
[B8] Steinberg, L., "The Backboard Wiring Problem:
A Placement Algorithm," SIAM Review V. 3>
pp. 37-42, (1961).
C. Integer Programming
[Cl] Balas, E., "An Additive Algorithm for Solving
Linear Programs with Zero-one Variables,"
OJD. Res., V. 13, pp. 517-546, (1965).
[02] Balinski, M. L., "Integer Programming: Methods,
Uses, Computation," Man. S.ci., V. 12,
pp. 253-313, (1965).
[C3] Beale, E. M. L., "A Method of Solving Linear Pro-
gramming Problems When Some but Not all of
the Variables Must Take Integral Values,"
Statistical Techniques Res e ar ch Group, Tech-
nical Report No. 19, Princeton University,
TT95F). ~
[C4] Ben Israel, A., and Charnes, A., "On Some Problems
of Diophantine Programming," Cahlers du
Centre 1 'Etudes de Re"cherce Operationelle.
V. 4, pp. 215-280, Brussels, 1962.
[C5] Benders, J. P., Catchpole, A. R., and Kuiken, L.
C., "Discrete Variable Optimization Problems,"
Paper presented to the Rand Symposium on
Mathematical Programming, Santa Monica, 1959.
[C6] Charnes, A., and Cooper, W. W., Management Models
and Industrial Applications oj[ Linear Pro-
gramming, John Wiley, New York, 19151.
-------
2-289
[07]
[08]
[09]
[010]
[Oil]
[012]
[013]
[015]
[016]
Dantzig, 0. B., "Discrete Variable Extremum
Problems," Op_. Res., V. 5, pp. 266-277,
(1957).
Elmaghraby, S. E., "An Algorithm for the Solution
of the Zero-One Problem of Integer Linear
Programming," Doctoral Dissertation,
Department of Industrial Administration, Yale
University, May, 1963.
Pleishmann, B., "Computational Experience with the
Algorithm of Balas," Op_. Res., V. 15,
pp. 153-155, (1967).
Gilmore, P. C., and Gomery, R. E., "A Linear
Programming Approach to the Cutting-Stock
Problem," p_p_. Res., V. 9, pp. 849-859, (1961),
and
" A Linear
Programming Approach to the Cutting-Stock
Problem, Part II," Op. Res., V. 11, pp. 863-
888, (1963).
Glover, P., "A Bound Escalation Method for the
Solution of Integer Programs," Cahiers
(See [04].), V. 6, (1964).
"A Multiphase Dual Algorithm for
the Zero-One Integer Programming Problem,"
Op_. Res., V. 13, pp. 879-919, (1965).
, and Zionts, S., "A Note on the
Additive Algorithm," Op_. Res. , V. 13,
pp. 546-549, (1965).
Gomery, R. E., "Outline of an Algorithm for Inte-
ger Solutions to Linear Programs," Bull.
A.M.S., 64, 3(1958).
, "Essentials of an Algorithm for
Integer Solutions to Linear Programs," in
Recent Advances In Mathematical Programming,
R. L. Graves and P. Wolfe, Eds., McGraw-Hill,
New York, 1963.
-------
2-290
[C17] , "An All-Integer Programming
Algorithm," Ch. 13 of Industrial Scheduling,
J. R. Muth and G. L. Thompson, Eds.,
Prentice-Hall, Englewood Cliffs, 1963.
[C18] , and Hoffman, A. J., "On the Con-
vergence of an Integer Programming Process,"
Naval Research Logistics Quarterly, V. 10,
pp. 121-124, (1963).
[C19] Haldi, J., and Isaacson, L. M., "A Computer Code
for Integer Solutions to Linear Programs,"
"p_£. Res., V. 13, pp. 9^6-959, (1965).
[C20] Land, A. H., and Dolg, A. G., "An Automatic
Method of Solving Discrete Programming
Problems," Econometrica, V. 28, pp. 497-520,
(I960).
[C21] Lemke, C. E., and Spielberg, K., "Direct Search
Zero-One Integer and Mixed Integer Program-
ming," IBM 39.008, Report Np_. 3, June 1966.
[C22] Markowitz, H. M., and Manne, A. S., "On the
Solution of Discrete Programming Problems,"
Econometrlca, V. 25, pp. 84-110, (1957).
[C23] Martin, G. T., "An Accelerated Euclidean
Algorithm for Integer Linear Programming,"
in Recent Advances (See [C15].).
[C24] Petersen, C.Q., "Computational Experience with
Variants of the Balas Algorithm Applied to
the Selection of R & D Projects," Man. Sci.,
V. 13, pp. 736-745, (1967).
[C25] Thompson, G. L., "The Stopped Simplex Method,
Part I," Revue Francaise de_ Recherche
Opgrationelle, V. 8, pp. 159-182, (196 ).
[C26] , "The Stopped Simplex Method,
Part II," Revue Francaise de Recherche
Opgrationelle, V. 9, pp. ,(1965).
[C27] Tucker, A. W., "On Directed Graphs and Integer
Programs," Princeton-IBM Mathematical
Research Projects. Technical Report, I960-
-------
2-291
[C28] Walker, R. J., "An Enumerative Technique for a
Class of Combinatorial Problems, Combinator-
ial Analysis, Proc. of Symposia in Appl .
Math . , R. Bellman and M. Hall, Jr., Eds.,
V. 10, pp. 91-9^, A.M. A., Providence, R.I.,
I960.
[C29] Wersan, S. J., "Computer Implementation of Balas1
Algorithm for Zero-One Linear Programs,"
presented at ^CM-SIGMAP Colloquium on
Branch and Bound Methods , Princeton, N. J.,
January 29-31,
[C30] _ , "BALGOR: A Program for Solving
Linear Programs in Zero-One Variables,"
Aerospace Corporation Report No. TR-0158
(S 3816-10") -1, San Bernardino, California,
September 196?.
[C31] Young, R. D., "A Primal (All-Integer) Integer
Programming Algorithm," J. of Res . of the
N.B.S. - B. Mathematics and Mathematical
Physics, V. 69B, No. 3, PP • 213-250,
(July-September 1965).
[C32] Zionts, S., Thompson, G. L. , and Tonge, F. M. ,
"Techniques for Removing Nonbinding Con-
straints and Extraneous Variables from
Linear Programming Problems," Carnegie
Inst. of Technology, Graduate School of Ind.
Admin., November, 1964.
D. Bibliographies and Encyclopedias
[Dl] Encyklopadie der Mathematischen Wissenschaffen,
V. 3, Part 1, B. G. Teuber, Leipzig, 190?.
[D2] Selected Bibliography on Location Theory,
(mimeographed; /Northwestern University,
Transportation Center Library, Evanston,
Illinois, 10 December 1963.
[D3] Industrial Location Bibliography, Real Estate
Research Program, Division of Research,
Graduate School of Business Administration,
University of California, Los Angeles,
July 1959.
-------
REFERENCES FOR PART II
AH Analysis of Refuse Collection and Sanitary
Landfill, Technical Bulletin No. 8, Series
37, Sanitary Engineering Research Project,
University of California, Berkeley, December,
1952.
[2] "Fresno Region Solid Waste Management Study,"
Vols. I, II, and III, a report to California
Department of Public Health, Report No. 3^13>
by Aerojet General Corporation and Engineer-
ing Science, Inc., Azusa, California, June,
1967.
[3] Modeling and Simulation in Operations Research,
compendium of lecture notes for two week
course of same name, Engineering Summer Con-
ferences, University of Michigan, Ann Arbor,
Michigan, I960.
[4] Municipal Refuse Disposal, prepared by the Commit-
tee on Refuse Disposal, American Public
Works Association, A.P.W.A. Research Founda-
tion, 1961.
[5] Refuse Collection Practice, second edition, pre-
pared by the Committee on Refuse Collection,
American Public Works,Association, A.P.W.A.,
Research Foundation, 1958.
[6] Solid Waste Research and Development, proceedings
of the Engineering Foundation Research
Conference, University School, Milwaukee,
Wisconsin, July 2*J-?3, 1967. The individual
papers referred to In this dissertation
are:
McGaubey,, P. H.5 "Goals of the Conference,"
Keynote speech. Contrasts re
methodologies.
2-292
-------
2-293
Weaver, L. , "Solid Wastes: A New Test for
Research Engineering," Keynote speech.
Statement of crisis; significance of
Solid Wastes Disposal Act of 1965.
'Al: Golueke, C. G., "Comprehensive Studies of
Solid Waste Management."
A2 Spradlin, B. C., "The System of Dynamics of
Solid Waste Management and Control."
A3 Mitchell, R. E., Bowerman, P. R., and Walsh,
T. E., "A Systemized Approach to Urban-
Rural Solid Waste Management."
A7 Galler, W. S., "Study and Investigation of
Solid Waste Disposal, City of Raleigh,
North Carolina."
All Dudley, R. H., and Hekemian, K. K., A Systems
Approach to Solid Waste Management."
Dl Betz, J. M., "Application of EDP and OR
Techniques ."
D2 Kruse, C. W., Liebman, J. C., and Truitt,
M. M., "Optimal Policies for Solid
Waste Collection."
D4 Zandi, I., "Solid Waste Pipeline."
D5 First, M. W., et. al., "Shipborne Municipal
Incineration.1|r~
P3 Hickman, H. L., Jr., and Kennedy, J. C. ,
"Studies of Characteristics of Municipal
and Residential Solid Waste."
FH , and
"Seasonal Variation in Municipal Solid
Waste Output."
F5 Summary of [14].
F6 Ecke, D. H., Linsdale, D. W., Rogers, P. A.,
and Bellenger, G., "Fly and Economic
Evaluation of Urban Garbage Systems."
-------
2-294
[?] Bowerman, F. R. , "Municipal Refuse Transfer
Stations," 1962 Yearbook of the American
Public Works Association, Chicago, Illinois,
December 1962, pp. 191-194. Presented at
A.P.W.A. Congress, New Orleans, Louisana,
October, 1962.
[8] Curtis, W. H., and Martin, R. M. , "An Information
Retrieval System for Urban Areas, (Refuse
Scheduling Analysis)," Technical Report,
City of Vancouver, B. C., February, 19 65.
[9] Golueke, C. G., and McGauhey, P. H., "Comprehen-
sive Studies of Solid Wastes Management,"
Sanitary Engineering Research Laboratory,
Reports No. 17-3.6 , University of California,
Berkeley, California.
[10] Guetzkow, H., Ed., simulation in social
sciences : readings, Prentice-Hall, Englewood
Cliffs, N. J. , 1963-
[11] Orcutt, G., et . al. , Microanalysis of Socionomic
Systems : A Simulation Study, Harper and Row,
New York,
[12] Quon, J. E., Charnes , A., and Wersan, S. J.,
"Location of Disposal Sites," First Progress
Report under Research Grant EF 00355-01
(Division of Environmental Engineering and
Food Protection Bureau of State Services
U.S.P.H.S.,) Northwestern University Techno-
logical Institute, May 20, 1963.
[13] ___ , ________________ , _
"Simulation and Analyses of A Refuse Collec
tion System," Journal of the Sanitary Enj^l-
neering Division, ASCE, Vol. 91, No. oA5,
Proc. Paper 4491, October, 1965, pp. 17-36 •
Presented at ORSA Symposium on One re.4" tons
Research and the Quality of the Environment}
New York, N. Y., May 30, 1967,
[lit] , Tanaka, M. . and Char-;ec. '
''Refuse Quantities a:^ the .Jreqacncy ji
Service," Journal of_ the Sanitary Engineer-
ing Division. ASCE, Vol. 94, No. SA2, Proc.
Paper 5917, April, 1968, pp. 40-s—! ',.
-------
I
2-295
' [15] • _ _ , _ , and Wersan, S.
"
_
J. , "Simulation Model of Refuse Collection
Policies," Journal of the Sanitary Engi-
neering Division, ASCE, Vol. 95, SA3,
PP» 575-592, June 1969.
[16] Wersan, S. J., Eisen, R., and Charnes, A.,
"Computer Simulation of a Refuse Collection
System," Systems Research Memorandum No.
101, Northwestern University, April, 19%^.
[17] _ _ , Quon, J. E., and Charnes, A.,
"Location-Allocation Prq.blems in the
J^-Metric," presented at the CORS/ORSA
Meeting, Montreal, Quebec, May 28, 1964.
[18] > > and
"Systems Analysis of Refuse Collection and
Disposal Practices," 1962 Yearbook of the
American Public Works Association, Chicago,
Illinois, December 1962, pp. 195-211.
Presented at A.P.W.A. Congress, New Orleans,
Louisana, October 3, 1962.
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