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Efficiency/equity analysis of water resources problems--a game theoretic approach

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Efficiency/equity analysis of water resources problems--a game theoretic approach
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Ng, Elliot Kin, 1950-
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English
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xi, 160 leaves : ill. ; 28 cm.

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Subjects / Keywords:
Cost allocation ( jstor )
Cost efficiency ( jstor )
Cost functions ( jstor )
Cost of equity ( jstor )
Cost savings ( jstor )
Game theory ( jstor )
Linear programming ( jstor )
Minimization of cost ( jstor )
Water resources ( jstor )
Water use efficiency ( jstor )
Dissertations, Academic -- Environmental Engineering Sciences -- UF
Environmental Engineering Sciences thesis Ph. D
Information storage and retrieval systems -- Water resources development ( lcsh )
Water resources development -- Cost effectiveness -- Data processing ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1985.
Bibliography:
Bibliography: leaves 150-158.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Elliot Kin Ng.

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EFFICIENCY/EQUITY ANALYSIS OF WATER RESOURCES
PROBLEMSA GAME THEORETIC APPROACH
By
ELLIOT KIN NG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985


To my parents
and
my wife, Eileen,
and children, Matthew, Michelle, Michael


ACKNOWLEDGMENTS
I would like to thank my chairman, Dr. James P. Heaney,
for the many hours spent guiding this research. His
encouragement, support, and friendship during my three years
at the University of Florida have been invaluable. I would
also like to thank the other members of my supervisory
committee, Dr. Sanford V. Berg, Dr. Donald J. Elzinga,
Dr. Wayne C. Huber, and Dr. Warren Viessman, for their time
and support. In addition, I wish to thank the U.S. Air
Force for giving me the opportunity to pursue the Ph.D.
degree.
Thanks are also due to several fellow students who
have made my program enjoyable and memorable. In particu
lar, I wish to thank Mr. N. Devadoss, Mr. Mun-Fong Lee, and
Mr. Robert Ryczak. I would also like to give special thanks
to Mr. Robert Dickinson for keeping an extra copy of the
LP-80 and Mrs. Barbara Smerage for doing such an excellent
job typing this manuscript.
I am extremely grateful to my parents for instilling in
me a desire to seek further eduction. Furthermore, I am
especially thankful to my wife, Eileen, for typing initial
drafts of this manuscript and for her love, encouragement,
and sacrifices throughout my program. We will miss the
iii


croissants, pizzas, and hoagies that supplemented my late
night studies. Finally, I wish to thank my children,
Matthew, Michelle, and Michael, for their love and under
standing during the countless times I have chased them out
of my study.
IV


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iii
LIST OF TABLES vii
LIST OF FIGURES ix
ABSTRACT x
CHAPTER
1 INTRODUCTION 1
2 LITERATURE REVIEW 4
Efficiency Analysis 4
Equity Analysis 5
Conclusions 8
3 EFFICIENCY ANALYSIS 10
Introduction 10
Partial Enumeration Techniques 12
Total Enumeration Techniques 15
Modeling Network Problems as Digraphs 16
The Total Enumeration Procedure 21
Computational Considerations 30
Summary 3 8
4 EQUITY ANALYSIS 39
Introduction 39
Cost Allocation for Regional Water
Networks 4 0
Criteria for Selecting a Cost
Allocation Method 45
Ad Hoc Methods 4 8
Defining Identifiable Costs as Zero... 49
Defining Identifiable Costs as
Direct Costs 54
v


Cooperative Game Theory 64
Concepts of Cooperative Game Theory... 65
Unique Solution Concepts 75
Empty Core Solution Concepts 87
Cost Allocation in the Water Resources
Field 88
Separable Costs, Remaining Benefits
Method 9 0
Minimum Costs, Remaining Savings
Method 9 5
Allocating Cost Using Game Theory Concepts. 99
The k Best System 99
The Dummy Player 108
Comparing Methods 115
Summary 119
5 EFFICIENCY/EQUITY ANALYSIS 120
Introduction 120
Maximum Cost 122
Minimum Cost 129
Fairness Criteria 132
Summary 13 3
6 CONCLUSIONS AND RECOMMENDATIONS 135
APPENDIX EFFICIENCY/EQUITY ANALYSIS OF A THREE-
COUNTY REGIONAL WATER NETWORK WITH
NONLINEAR COST FUNCTION 142
REFERENCES 15 0
BIOGRAPHICAL SKETCH 159
vi


LIST OF TABLES
Table Page
3-1 Example of Total Enumeration Procedure for
3-Node Digraph 27
3-2 The Number of Independent Calculations to
Find the Costs of Spanning Directed Trees
for All Possible Subdigraphs 31
3-3 Summary of Computational Effort for Digraphs
Shown in Figure 3-4 34
3-4 Efficiency Analysis of a Three-User Water
Supply Network with Nonlinear Cost Function
Using Lotus 1-2-3 37
4-1 Projected Population Growth and Projected
Average Per Capita Demand 41
4-2 The Costs and Percent Savings for All
Options 44
4-3 Cost Allocation Matrix 50
4-4 Cost Allocation of Optimal Network Based on
Population 52
4-5 Cost Allocation of Optimal Network Based on
Demand 5 3
4-6 Cost Allocation of Optimal Network with Use
of Facilities Method 56
4-7 Cost Allocation for the Use of Facilities
Method 5 7
4-8 Cost Allocation of Optimal Network with
Direct Costing/Equal Apportionment of
Remaining Costs Method 60
4-9 Cost Allocation for Direct Costing/Equal
Apportionment of Remaining Costs Method 61
vii


4-10 Core Geometry for Three-Person Cost Game
Example 7 3
4-11 Cost Allocation for Three-County Example
Using the Shapley Value 80
4-12 Cost Allocation for Three-County Example
Using the Nucleolus 85
4-13 Empty Core Solution Methods 89
4-14 Cost Allocation for Three-County Example
Using the SCRB Method 94
4-15 Cost Allocation for Three-County Example
Using the MCRS Method 98
4-16 Nominal Versus Actual Core Bounds for
Optimal Network Game 100
4-17 Cost Allocations for the Optimal Network and
the Second Best Network ($) 103
4-18 Cost Allocation for Option 3 as a Two-Person
Game Using the SCRB Method 10 9
4-19 Comparing Cost Allocations for Option 3 as
Two-Person Game and Three-Person Game Using
the SCRB and MCRS Methods Ill
4-20 Core Bounds for Option 3 as a Three-Person
Game 112
4-21 Core Bounds for Option 3 as a Three-Person
Game with County 1 as a Dummy Player 114
4-22 Core Bounds for Option 3 as a Two-Person
Game 116
4-23 Comparison of Methods Discussed for
Allocating Costs of Water Resources Projects. 117
5-1 Using Independent Calculations from the
Total Enumeration Procedure to Find c(i),
c(S), and c(N) for the Three-County Regional
Water Network Problem 121
5-2 Efficiency/Equity Analysis of the Optimal
Network 12 6
viii


LIST OF FIGURES
Figure Page
3-1 Types of Cost Functions 13
3-2 Example Digraph Representing a Regional
Water Network Problem for Three Users 18
3-3 Flow Diagram of Total Enumeration Procedure
for n-Node Digraph 23
3-4 Examples of 3,4,5-Node Digraphs 33
4-1 Lengths of Interconnecting Pipelines 43
4-2 Geometry of Core Conditions for Three-Person
Cost Game Example 71
4-3 Core for the Optimal Network Game
(C (N ) = $4,556,409 ) 101
4-4 Core for the Second Best Network Game
( C (N) = $4,556,826 ) 102
4-5 Reduction in. Core as c(N) Increases from
c X(N) to c (N) 107
IX


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
EFFICIENCY/EQUITY ANALYSIS OF WATER RESOURCES
PROBLEMSA GAME THEORETIC APPROACH
By
Elliot Kin Ng
August, 1985
Chairman: James P. Heaney
Major Department: Environmental Engineering Sciences
Successful regional water resources planning involves
an efficiency analysis to find the optimal system that maxi
mizes benefits minus costs, and an equity analysis to appor
tion project costs. Traditionally, these two problems have
been treated separately. This dissertation incorporates
efficiency analysis and equity analysis into a single
regional water resources planning model.
A reliable total enumeration procedure is used to find
the optimal system for regional water network problems.
This procedure is easy to understand and can be implemented
using readily available computer software. Furthermore, the
engineer can use realistic cost functions or perform detailed
cost analysis and, also, examine good suboptimal systems. In
addition, this procedure finds the optimal system for each
individual and each subgroup of individuals; hence, an equity
x


analysis can be accomplished using the theory of the core
from cooperative n-person game theory.
Game theory concepts are used to perform an equity
analysis on the optimal system as well as good suboptimal
systems. For any system, an equitable cost allocation exists
if a core exists. However, if a game is not properly
defined, even a cost allocation in the core may be
inequitable.
A rigorous procedure using core conditions and linear
programming is described to determine the core bounds. An
individual's lower core bound and upper core bound unambigu
ously measure the individual's minimum cost and maximum cost,
respectively. Traditional approaches for quantifying minimum
cost and maximum cost assume that either a regional system
involving the grand coalition is built or all the individuals
will go-it-alone. However, this rigorous procedure accounts
for the possibility that a relatively attractive system
involving subgroups may form. Furthermore, this rigorous
procedure gives a general quantitative definition of marginal
cost and opportunity cost. Once the minimum cost and maximum
cost for each individual are determined, a basis for
equitable cost allocation is available.
Finally, efficiency analysis and equity analysis are
not separable problems but are related by the economics of
all the opportunities available to all individuals in a
project.
xi


CHAPTER 1
INTRODUCTION
In situations where multiple purposes and groups can
take advantage of economies of scale in production and/or
distribution costs, a regional water resources system is an
attractive alternative to separate systems for each purpose
and each group. However, a regional system imposes complex
economic, financial, legal, socio-political, and organiza
tional problems for the water resources professionals. This
dissertation examines two problems associated with regional
water resources planning that are typically treated
separately, yet are closely related.
The first problem involves performing an efficiency
analysis to determine the economically efficient or optimal
regional system that maximizes benefits minus costs. Once
the optimal regional system is determined, a major task
still remains to allocate project costs; therefore, an
equity analysis must be performed to apportion project costs
in an equitable manner. This second problem is viewed from
the perspective of each purpose and each group because they
must each be convinced that the optimal regional system is
their best alternative; otherwise, voluntary participation
will be difficult. No doubt, each purpose's and each
1


2
group's decision to participate in the optimal regional
system depends on its allocated cost, and not necessarily on
what is best for the region.
The prevailing belief is that efficiency analysis and
equity analysis are separate problems and, therefore,
research has either focused entirely on efficiency analysis
or equity analysis. Research on efficiency analysis has
mainly been on the application of partial enumeration tech
niques to find optimal regional systems, while research on
equity analysis has continued to explore the application of
concepts from cooperative game theory to allocate project
costs. The purpose of this dissertation is to integrate
efficiency analysis and equity analysis into a single
regional water resources planning model characterized by
economies of scale. The model to be presented incorporates
a total enumeration procedure along with concepts from
cooperative game theory for efficiency/equity analysis. The
specific application is to determine the least cost regional
water supply network and to determine a "fair" allocation of
costs among the multiple users.
Chapter 2 reviews selected works on efficiency analysis
and equity analysis of water resources problems. Chapter 3
presents a reliable total enumeration procedure for effi
ciency analysis of regional water supply network problems.
However, unlike traditional partial enumeration techniques
used for efficiency analysis that give only the optimal


3
solution, this procedure also gives all the optimal solu
tions for each user and each subgroup of users which are
necessary information to perform an equity analysis using
concepts from cooperative game theory. In addition, this
procedure gives all the suboptimal solutions. Chapter 4
shows how the information from the total enumeration
procedure is used to perform an equity analysis of not only
the optimal solution, but also "good" suboptimal solutions.
Chapter 5 reveals how efficiency analysis and equity
analysis are related. Finally, Chapter 6 summarizes the
results and conclusions.


CHAPTER 2
LITERATURE REVIEW
Efficiency Analysis
During the past two decades, the problem of finding the
economically efficient or optimal regional water system has
been extensively modeled as a mathematical optimization
problem. A review of selected works on efficiency analysis
of regional water systems that includes Converse (1972),
Graves et al. (1972), McConagha and Converse (1973), Yao
(1973), Joeres et al. (1974), Bishop et al. (1975), Jarvis
et al. (1978), Whitlatch and ReVelle (1976), Brill and
Nakamura (1978), and Phillips et al. (1982) indicates a
variety of partial enumeration techniques, e.g., nonlinear
programming, for finding optimal regional systems. These
optimal regional systems can be a least cost system or a
system that maximizes benefits minus costs. Generally,
regional water resources planning problems exhibit economies
of scale in cost and, therefore, involve nonlinear concave
cost functions. Consequently, to a great extent, the
selection of the partial enumeration optimization technique
to apply to a particular problem depends on the characteri
zation of the nonlinear concave cost functions. For
instance, linear programming can be applied if the nonlinear
4


concave cost functions are represented by linear
approximations.
5
Equity Analysis
Unfortunately, successful regional planning is not
merely knowing the optimal regional system but must also
include an equity analysis to find an acceptable allocation
of costs among the participants. Otherwise, the optimal
system will be difficult to implement. Of the publications
cited in the preceding paragraph, only McConagha and
Converse (1973) dealt with both efficiency and equity in
regional water planning. In addition to presenting a
heuristic procedure for finding the least cost regional
wastewater treatment facility for seven cities, they evalu
ated the equity of several cost allocation procedures.
Although they recognized that an equitable cost allocation
should not charge any city or subgroup of cities more than
the cost of an individual treatment facility, they did not
include the possibility of subgroup formation in their
analysis.
Giglio and Wrightington (1972) introduced concepts from
cooperative game theory as a way to consider the possibility
of subgroup formation in allocating costs of water
projects. However, their treatment of cooperative game
theory was incomplete. Therefore, they concluded that the
game theory approach rarely yields a unique cost allocation


6
and proceeded to recommend the separable costs, remaining
benefits (SCRB) method or methods based on measure of pollu
tion. Shortly thereafter, several researchers applied
popular unique solution concepts from game theory like the
Shapley value and the nucleolus to allocate the costs of
regional water systems. Heaney et al. (1975) applied the
Shapley value to find an equitable cost allocation of common
storage units for storm drainage for pollution control among
competing users. Suzuki and Nakayama (1976) applied the
nucleolus to assign costs for a water resources development
along Japan's Sakawa and Sagami Rivers. Loehman et
al. (1979) used a generalization of the Shapley value to
allocate the costs of a regional wastewater system involving
eight dischargers along the lower Meramec River near St.
Louis, Missouri.
Subsequently, Heaney (1979) established that the fair
ness criteria used for allocating costs in the water
resources field and the concepts used in cooperative game
theory are equivalent. Moreover, Straffin and Heaney (1981)
showed that a conventional method for allocating costs used
by water resources engineers is identical to a unique solu
tion concept used by game theorists. More recently, Young
et al. (1982) compared proportionality methods, game
theoretic methods, and the SCRB method for allocating cost
and concluded that the game theoretic methods may be too
complicated while the SCRB method may give inequitable cost


7
allocations. Meanwhile, Heaney and Dickinson (1982)
revealed why the SCRB method may fail to give equitable cost
allocations and proposed a modification of the SCRB method
that uses game theory concepts along with linear programming
to insure an equitable cost allocation can be found if one
exists.
The possibilities of using concepts from cooperative
game theory as a basis for allocating costs of water
projects continue to develop. In fact, concepts from coop
erative game theory are gaining acceptance in other fields
as well. Researchers in accounting are looking toward coop
erative game theory as a possible solution to the arguments
by Thomas (1969, 1974) that any cost allocation scheme in
accounting is arbitrary and hence not fully defensible.
Recent works by Jensen (1977), Hamlen et al. (1977, 1980),
Callen (1978), and Balachandran and Ramakrishnan (1981)
applied concepts from cooperative game theory to evaluate
the equity of existing and proposed cost allocation schemes
in accounting. Meanwhile, in economics, concepts from
cooperative game theory are frequently used as a basis for
evaluating subsidy-free and sustainable pricing policies for
decreasing cost industries, e.g., the work of Loehman and
Whinston (1971, 1974), Faulhaber (1975), Sorenson et
al. (1976, 1978), Zajac (1978), Panzar and Willig (1977),
Faulhaber and Levinson (1981), and Sharkey (1982b).


8
Conclusions
Three conclusions can be made from reviewing the
literature on efficiency analysis and equity analysis of
regional water resources planning. First, there is a gap in
the research to jointly examine efficiency and equity in
regional water resources planning. In spite of a continual
effort to find economically efficient regional water systems
and equitable cost allocation procedures, no published work
incorporates both efficiency analysis and equity analysis in
a single regional water resources planning model using
realistic cost functions. Heaney et al. (1975) and Suzuki
and Nakayama (1976) used linear cost models while Loehman et
al. (1979) used conventional cost curves. Secondly, the
cost allocation literature in the water resources field has
consistently allocated the costs of treatment and piping
together even though federal guidelines suggest that piping
cost be allocated separately from treatment cost to the
responsible users (Loehman et al., 1979; U.S. Environmental
Protection Agency, 1976). Finally, the cost allocation
literature has dealt with allocating the cost of the optimal
system. However, situations in practice may require that
"good" suboptimal systems be considered; therefore, an
acceptable cost allocation procedure should be able to
allocate the costs of several systems under consideration in
an equitable manner. These three conclusions formed the
basis for the research undertaken in this dissertation.


Chapter 3 begins integrating efficiency analysis and
equity analysis by searching for a computational procedure
9
to simultaneously perform an efficiency analysis and
calculate all the necessary information to perform an equity
analysis using concepts from cooperative game theory.


CHAPTER 3
EFFICIENCY ANALYSIS
Introduction
The importance of both efficiency analysis and equity
analysis in planning regional water resources systems is
well recognized. Over the years, researchers have applied
methods ranging from simple cost-benefit analysis to sophis
ticated mathematical programming techniques to search for
economically efficient or optimal regional water resources
systems. Yet, the implementation of regional systems is
difficult unless an equitable financial arrangement is found
to allocate project costs among individuals (or partici
pants) in a project. Until recently, a theoretically sound
basis for allocating costs has eluded the water resources
professional. However, there is increasing interest in
using the theory of the core from cooperative n-person game
theory as a basis for allocating costs, e.g., see Suzuki and
Nakayama (1976), Bogardi and Szidarovsky (1976), Loehman et
al. (1979), Heaney and Dickinson (1982), and Young et
al. (1982). The theory of the core is based on principles
of individual, subgroup, and group rationality. This means
that no individual or subgroup of individuals should be
allocated a cost in excess of the cost of nonparticipation,
10


11
while total cost must be apportioned among all individuals.
The cost of nonparticipation is simply the cost that each
individual and each subgroup of individuals must pay to
independently acquire the same level of service by the most
economically efficient means. As a result, to evaluate
efficiency/equity for a regional system with n individuals,
it is necessary to determine 2n-l optimal solutions.
Although the close association between efficiency
analysis and equity analysis is recognized, there have been
few attempts to incorporate these two analyses in regional
water resources planning. A typical efficiency analysis
usually ends with determining the optimal solution for a
problem without addressing cost allocation, and a typical
equity analysis begins by assuming the 2n-l optimal solu
tions are available to accomplish the cost allocation. This
disjointed approach to efficiency/equity analysis is
fostered by a belief that these two problems are independent
(James and Lee, 1971; Loughlin, 1977). Furthermore,
reliable techniques for finding the 2n-l optimal solutions
to accomplish an efficiency/equity analysis of most problems
encountered in actual practice are unavailable.
This chapter begins by evaluating the applicability of
partial and total enumeration techniques for finding the
2n-l optimal solutions for problems with different types of
cost functions. Subsequently, a computational procedure is
described to examine a regional water supply network problem


12
wherein we need to find the economic optimum and a "fair"
allocation of costs among the individuals in the project.
In order to do the cost allocation we need to find the
costs of the optimal systems for each individual and each
subgroup of individuals since these costs are going to be
the basis for cost allocation.
Partial Enumeration Techniques
The difficulty of finding the optimal solution for a
particular problem depends on the nature of the cost func
tions. Generally, a cost function can be classified as
either linear, convex, concave, S-shape, or irregular (see
Figure 3-1). To find the optimal solution for problems with
either linear or convex cost functions is straightforward
using readily available and reliable linear programming
codes. Accordingly, a vast body of overlapping theoretical
results is available from classical economics and operations
research, e.g., convex programming, for finding the optimal
solution to problems with convex cost functions. However,
problems with linear and convex cost functions are unable to
characterize the economies of scale in cost typically
encountered in regional water resources planning.
The concave cost function is generally used to
represent economies of scale, and several partial enumera
tion techniques are available for dealing with this cost


13
Figure 3-1. Types of Cost Functions.


14
function. One approach surveyed by Mandl (1981) is
separable programming which takes advantage of readily
available linear programming codes by using a piecewise
linear approximation of the concave cost function.
Unfortunately, this approach is rather tedious to use and
guarantees only a local optimal solution. A second approach
is to retain the natural concave cost function and apply a
general nonlinear programming code. However, according to
surveys by Waren and Lasdon (1979) and Hock and Schittkowski
(1983), general nonlinear programming codes may converge to
local optima and may be subject to other failures, e.g.,
termination of code. A final approach used by Joeres et
al. (1974) and Jarvis et al. (1978) is to approximate the
concave cost function with several fixed-charge cost
functions and apply a mixed-integer programming code. This
approach guarantees a globally optimal solution, but
standard mixed-integer programming codes are expensive to
use. More importantly, unresolved problems remain as to
how to properly define a fixed charge problem. If the fixed
charge formulation is used because it is computationally
expedient, then the resulting cost estimates may distort the
cost allocation procedure. Given the current status of
partial enumeration techniques for finding the optimal
solutions to perform efficiency/equity analysis for problems
with concave cost functions, one can conclude that other
methods must be used. Obviously, this conclusion applies


15
to problems with S-shape and irregular cost functions as
well.
Total Enumeration Techniques
Total enumeration techniques can be used to find the
optimal solution for a problem regardless of the types of
cost functions involved. The ability to handle irregular
cost functions is especially important because this type of
cost function is frequently used by state-of-the-art cost
estimating models like CAPDET, i.e., Computer Assisted
Procedure for Design and Evaluation of Wastewater Treatment
Systems (U. S. Army Corps of Engineers, 1978) and MAPS,
i.e., Methodology for Areawide Planning Studies (U. S. Army
Corps of Engineers, 1980). For example, in MAPS, the cost
function for constructing a force main is composed of
separate cost functions for pipes, excavation, appurten
ances, and terrain. Furthermore, each of these cost func
tions is based on site-specific conditions. For instance,
the cost function for pipe includes the cost of purchasing,
hauling, and laying the pipe and depends on the material,
diameter, length, and maximum pressure. No doubt, the
composite site-specific cost function for a force main may
be nonlinear, nonconvex, multimodel, and discontinuous.
Another advantage with a total enumeration technique is
that it presents and ranks all of the alternative solu
tions. Unlike partial enumeration techniques which only


16
present the optimal solution for consideration, total
enumeration techniques allow examination of suboptimal
solutions which may be preferable when factors other than
cost are considered. For example, proven engineering design
or socio-political values are difficult to incorporate into
an optimization model even if the problem is well defined,
so the optimal solution may be so unrealistic that another
solution must be selected.
Depending on the size of the problem, a possible
drawback with total enumeration techniques may be the compu
tational effort to enumerate all possible solutions.
However, for some problems, total enumeration may be the
only meaningful approach. For these problems, the challenge
with using a total enumeration approach is to find ways to
reduce the computational effort by applying mathematical
techniques or engineering considerations. After a discus
sion on modeling network problems as digraphs, a total
enumeration procedure that does not require extensive compu
tational effort to find the least cost network for each
individual and each group of individuals is presented.
Modeling Network Problems as Digraphs
Consider a situation wherein an existing water supply
source, S, is going to serve n users with demands of Q^, Q2,
. . Q respectively. Assume that the water source is
able to supply the total demand by the n users without


17
facility expansion except for a new regional water network.
Furthermore, consider a particular system with three users
that can be served directly by the source, and engineering
considerations, e.g., gravity flow, have determined that it
is feasible to send water from user 1 to both user 2 and
user 3, and from user 2 to user 3. For this particular
system, assume the total cost function for constructing a
pipeline is rather simple. From Sample (1983), the total
cost function for constructing a pipeline is characterized
by economies of scale and can be expressed as a linear
function of distance and a nonlinear function of flow; or
C = aQbL (3-1)
where C = total cost of pipeline, dollars
Q = quantity of flow, mgd
L = length of pipeline, feet, and
a, b = parameters, 0 Given this situation, the objective of the regional water
authority is to determine the least cost water network for
each user and each group of users in order to perform
efficiency/equity analysis.
This problem can be modeled as a digraph or directed
graph (see Figure 3-2) consisting of nodes to represent the
source and users, and directed arcs to represent all


18
Figure 3-2.
Example Digraph Representing a Regional Water
Network Problem for Three Users.


19
possible interconnecting pipelines. If water can be sent in
either direction between two users, then the pipeline is
represented by two oppositely directed arcs. Consequently,
any regional water network problem can be modeled by a
digraph.
Before continuing, a few brief definitions and concepts
are necessary since the nomenclature used in the network
and graph theory literature is not standardized. A digraph
or directed graph, D(X,A), consists of a finite set of
nodes, X, and a finite set of directed arcs, A. A directed
arc is denoted by (i,j) where the direction of the arc
(shown by an arrow) is from node i to node j; node i is
called the initial node and node j is called the terminal
node. A subdigraph of D(X,A) has a set of nodes that is a
subset of X but contains all the arcs whose initial and
terminal nodes are both within this subset. A path from
node i to node j is simply a sequence of directed arcs from
node i to node j. An elementary path is a path that does
not use the same node more than once. A circuit is an
elementary path with the same initial and terminal node. A
directed tree or an arborescence is a digraph without a
circuit for which every node, except the node called the
root, has one arc directed into it while the root node has
no arc directed into it. A spanning directed tree of a
digraph is a directed tree that includes every node in the
digraph. If a cost, C(i,j) is associated with every arc


20
(i,j) of a digraph, then the cost of a directed tree is
defined as the sum of the costs of the arcs in the directed
tree. Finally, a minimum spanning directed tree of a
digraph is the spanning directed tree of the digraph with
the least cost. For the reader desiring more information
regarding networks and graphs, numerous texts are available,
e.g., Christofides (1975), Minieka (1978), and Robinson and
Foulds (1980 ) .
The problem of finding the least cost water network for
each user and each group of users is the same as finding the
minimum spanning directed tree rooted at node S for all
possible subdigraphs as well as the digraph shown in Figure
3-2. In general, not every digraph has a spanning directed
tree; however, for a realistic problem one can assume a
pipeline is available to serve all individuals participating
in a regional system. Thus, a spanning directed tree exists
for digraphs representing realistic regional water network
problems.
Although algorithms are found in Gabow (1977) and
Camerini et al. (1980a, 1980b) for finding the minimum
spanning directed tree or the K best spanning directed
trees, these algorithms assume a linear cost model in which
the cost on each arc is given prior to initiating the
algorithm. As a result, these algorithms are not applicable
to problems with nonlinear costs on each arc. That is,
the cost along each arc cannot be determined in advance


21
because the cost is a function of the quantity of flow along
the arc; yet, the quantity of flow along the arc is a
function of the path in which the arc belongs.
The Total Enumeration Procedure
The procedure for enumerating and calculating the costs
of all the spanning directed trees for all possible sub
digraphs as well as the digraph is based on recognizing that
a large number of spanning directed trees of a digraph can
be constructed from specific spanning directed trees of
subdigraphs. These specific spanning directed trees are
characterized by one arc emanating from the root node and
are referred to as "essential spanning directed trees." In
contrast, "inessential spanning directed trees" are charac
terized by more than one arc emanating from the root node.
The procedure sequentially calculates the costs of essential
spanning directed trees for subdigraphs with increasing
number of nodes, until the costs of essential spanning
directed trees are calculated for all possible subdigraphs
and for the digraph. Meanwhile, the cost of each inessen
tial spanning directed tree for all possible subdigraphs as
well as the digraph is calculated simply by summing the
costs of essential spanning directed trees of subdigraphs
that are associated with each arc emanating from the root
node of the inessential spanning directed tree. That is,
each arc emanating from the root node belongs to an


22
essential spanning directed tree of a subdigraph. By apply
ing this procedure the costs of all the spanning directed
trees can be systematically enumerated for all possible
subdigraphs as well as the costs of all the spanning
directed trees for the digraph. As a result, the least cost
network for each user and each group of users is found.
In the following discussion, "n-node" means the number
of nodes, not including the root node, is n; e.g., an i-node
digraph or subdigraph consists of i+1 nodes if the root node
is counted. The total enumeration procedure for the n-node
digraph is summarized by the flow diagram shown in Figure 3-3.
Step 1 begins the procedure for evaluating all subdi
graphs consisting of the root node and one other node, i.e.,
the 1-node subdigraphs.
Step 2 initializes a count of the number of combina
tions of i-node subdigraphs evaluated.
Step 3 generates all possible combinations of i-node
subdigraphs from the n-node digraph. The number of possible
combinations is (^). For example, the 3-node digraph shown
3
in Figure 3-2 has (or three possible 2-node subdigraphs,
i.e., subdigraphs consisting of the following sets of nodes
{S,1,2 }, S,1,3), and (S,2,3>.
Step 4 selects one i-node subdigraph not previously
selected and enumerates all of its spanning directed trees.
A spanning directed tree may not exist in a case where a
path does not exist from the root node to every node in the


23
Figure 3-3. Flow Diagram of Total Enumeration Procedure
for n-Node Digraph


not every node in the i-node
i-node subdigraph, i.e.,
subdigraph has an arc directed into it.
Actually, only the essential spanning directed trees
need to be enumerated. The enumeration of inessential
spanning directed trees is simply done by finding all
possible combinations of i-node digraphs from the entire set
of essential spanning directed trees enumerated previously,
i.e., all essential spanning directed trees for all possible
subdigraphs of the i-node subdigraph. This process
substantially reduces the effort involved in enumerating all
the spanning directed trees for an i-node subdigraph because
a large number of spanning directed trees are inessential.
If the i-node subdigraph is unusually large and dense,
algorithms are available in Chen and Li (1973), Christofides
(1975), and Minieka (1978) for generating spanning directed
trees.
If necessary, a procedure in Chen (1976) can be used to
compute the number of spanning directed trees of an i-node
subdigraph or an n-node digraph. A directed tree matrix, M,
is defined for a digraph, where equals the number of
arcs directed into node i and nm ^ is equal to the negative
of the number of arcs in parallel from node i to node j.
The number of spanning directed trees rooted at node S for
the digraph defined by M is given by the determinant of the
minor submatrix resulting from deleting the Sth row and


25
column of M. Applying this procedure to the 3-node digraph
in Figure 3-2 gives the following directed tree matrix.
S
1
2
3
S 1 2 3
0 -1 -1 -1
0 1-1-1
0 0 2 -1
0 0 0 3
The determinant of the minor submatrix resulting from delet
ing the Sth row and column is six, so there are six spanning
directed trees rooted at node S for this digraph.
Step 5 calculates the cost of each spanning directed
tree enumerated in Step 4. The cost for each essential
spanning directed tree is calculated independently. How
ever, the cost for each inessential spanning directed tree
is simply calculated by summing the costs of essential
spanning directed trees of subdigraphs calculated previously
that are associated with the arcs emanating from the root
node. For inessential spanning directed trees the costs can
be calculated along with the enumeration process described
in Step 4.
Step 6 ranks all the spanning directed trees for the
i-node subdigraph according to cost. The minimum spanning
directed tree is the least cost network for the users
associated with the set of nodes for the i-node subdigraph.


26
Step 7 checks the counter to see if all possible
combinations of i-node subdigraphs have been evaluated. If
not, Step 8 advances the counter by one before returning to
Step 4 to evaluate another i-node subdigraph. If all of the
possible combinations of i-node subdigraphs have been
evaluated, the procedure goes to Step 9 and begins the
evaluation of subdigraphs with i+1 nodes.
Step 10 checks if the n-node digraph has been evalu
ated. If not, the procedure returns to Step 2 and proceeds
to evaluate the subdigraphs with i+1 nodes; otherwise, the
procedure terminates.
The total enumeration procedure is illustrated in Table
3-1 using the regional water network problem modeled by the
3-node digraph shown in Figure 3-2.
During the first iteration all combinations of
1-node subdigraphs are evaluated. For this simple case
3
three combinations, i.e., (^) = 3, are evaluated. Further
more, each combination has only one spanning directed tree,
and the one spanning directed tree is essential. As a
result, the cost of the spanning directed tree for each
combination must be calculated. Obviously each spanning
directed tree is the least cost network for the associated
user. During the second iteration, three combinations,
3
i.e., (2) =3, of 2-node subdigraphs are evaluated. In
this case, each combination has two spanning directed trees,
but the cost of only one spanning directed tree needs


27
Table 3-1. Example of Total Enumeration Procedure for
3-Node Digraph
Iteration i-Node
i Subdigraphs
Spanning Directed
Trees for i-Node
Subdigraph
Are Spanning
Directed Trees
Essential?
i = l
(S, 1}
-

Yes
(S, 2}
-
Yes
(S, 3}
Yes
i=2 {S,l,2}
(D-^ Yes
{S, 1,3 }
No
Yes
{S 2,3 }
No
Yes
No


28
Table 3.1. Continued.
Iteration i-Node
i Subdigraphs
Spanning Directed Are Spanning
Trees for i-Node Directed Trees
Subdigraph Essential?


29
to be calculated. The cost of the inessential spanning
directed tree is simply found by summing the costs of the
corresponding essential spanning directed trees calculated
during the first iteration. The minimum spanning directed
tree for each combination is the least cost network for the
associated group of users. Finally, for the third itera
tion, i.e., i=n, the 3-node digraph is being evaluated. This
3-node digraph has six spanning directed trees, and these
six spanning directed trees can be enumerated by inspection.
The four inessential spanning directed trees can be
enumerated by simply finding all possible combinations of
3-node digraphs from the essential spanning directed trees
generated during the first and second iterations. Thus,
only two independent calculations are necessary to find
the costs of the essential spanning directed trees. Mean
while, the cost of the four inessential spanning directed
trees is calculated simply by summing the costs of essential
spanning directed trees for subdigraphs previously calcu
lated during the first two iterations. For example, in
Table 3-1, the cost for the inessential spanning directed
tree consisting of the set of arcs {(S,3), (S,l), (1,2)} is
determined by summing the costs of the two essential
spanning directed trees consisting of the sets of arcs
{(S,3) } and {(S,l), (1,2)} associated with the two sub
digraphs consisting of the sets of nodes S,3} and {S,l,2},
respectively. Therefore, eight independent calculations are


30
necessary to find the costs of the six spanning directed
trees for the digraph, and only two of the six spanning
directed trees are essential. In fact, the eight indepen
dent calculations enable us to find all 2n-l or seven
optimal solutions necessary to perform efficiency/
equity analysis. Table 3-2 shows that the number of
independent calculations necessary to find the cost of all
the spanning directed trees for all possible subdigraphs is
simply equal to the number of independent calculations to
find the cost of all the spanning directed trees for the
digraph less the number of essential spanning directed trees
for the digraph. Consequently, for our 3-node digraph, six
independent calculations are necessary to find the optimal
solution for each user and each subgroup of users. For the
balance of this chapter, the optimal solution for each user
and each subgroup of users will be referred to as the 2n-2
optimal solutions. Finally, all suboptimal solutions are
enumerated for all possible subdigraphs as well as for
the digraph.
Computational Considerations
Although the number of independent calculations neces
sary to find the costs of all the spanning directed trees
for all possible subdigraphs as well as the digraph is
uniquely determined by the configuration of the digraph, we
can get a sense of the computational effort by examining the


31
Table 3-2. The Number of Independent Calculations to Find
the Costs of Spanning Directed Trees for All
Possible Subdigraphs.
Independent
Calculation
Is Independent Calcu
lation Used to Find
the Costs of Spanning
Directed Trees for the
Digraph?
Is Independent Calcu
lation Used to Find
the Costs of Spanning
Directed Trees for All
Possible Subdigraphs?
-*
Yes
Yes
0H0)
Yes
Yes
(D-KD
Yes
Yes
Yes
Yes
Yes
Yes
--
Yes
Yes
dX
Yes
No
Ql
v )
Yes
No
Total Number
of Yes
8
6


32
three digraphs shown in Figure 3-4. For the 3-node digraph,
six independent calculations are necessary to find the costs
of the four spanning directed trees for the digraph, and
only one of the four spanning directed trees is essential.
More importantly, 12 calculations are necessary to find the
seven optimal solutions, but only 6 of the 12 calculations
(50%) are independent. Furthermore, only five independent
calculations are necessary to find the 2n-2 optimal solu
tions. For the 4-node digraph, 10 independent calculations
are necessary to find the cost of the eight spanning
directed trees for the digraph, and only one of the eight
spanning directed trees is essential. For this digraph, 33
calculations are necessary to find the 15 optimal solutions,
but only 10 of the 33 calculations (30%) are independent.
Moreover, only nine independent calculations are necessary
to find the 2n-2 optimal solutions. Finally, for the 5-node
digraph, 19 independent calculations are necessary to find
the costs of the 24 spanning directed trees for the digraph,
but only 2 of the 24 spanning directed trees are essential.
In this case, 109 calculations are necessary to find the 31
optimal solutions, but only 19 of the 109 calculations (17%)
are independent. From these 19 independent calculations,
only 17 are necessary to find the 2n-2 optimal solutions.
As we can see, summarized in Table 3-3, a large number of
the spanning directed trees of a digraph are inessential.


33
gur 3~4
£*
aPles
f 3
,4,5~Uocl
e Di
9riPhi


Table 3-
3. Summary
of Computational Effort
for Digraphs
Shown in Figure
3-4.
Digraph
2n-l
Optimal
Solutions
Number of
Spanning
Directed
Trees
Number of
Inessential
Spanning
Directed
Trees
Number of
Calculations
to Find
2n-l Optimal
Solutions
Number of
Independent
Calculations
to Find
2n-l Optimal
Solutions (%)
Number of
Independent
Calculations
to Find
2n-2 Optimal
Solutions
3-node
7
4
3
12
6 (50%)
5
4-node
15
8
7
33
10 (30%)
9
5-node
31
24
22
109
19 (17%)
17
u>


35
Also, the percentage of independent calculations decreases
as the number of nodes for a digraph increases.
The 5-node digraph in Figure 3-4 shows that the actual
number of independent calculations necessary to determine
the 31 optimal solutions to perform efficiency/equity analy
sis of a regional water network problem involving five
users is rather small. In fact, a regional water network
serving five users may be considered a fairly large
network. As larger systems form, increases in transactions
costs because of multiple political jurisdictions, growing
administrative complexity, etc., may eventually offset
the gains from a regional system. In any case, real
regional water network problems probably involve fairly
small and sparse networks. That is, large networks can
usually be broken down into smaller networks for analysis
based on natural geographical and hydrological features,
political boundaries, etc. Also, in actual problems there
may not be that many choices for routing pipelines. Thus,
the number of independent calculations necessary to
calculate the 2n-l optimal solutions for a realistic
regional water network should not be unreasonable.
One of the advantages of using this total enumeration
procedure is that it can be accomplished on a personal
computer using readily available software. Thus, decision
makers involved with planning and negotiating a regional
water network can have easy access to information to aid the


36
decision-making process. For instance, the procedure can
be implemented using the extremely "user friendly" Lotus
1-2-3 spreadsheet software package. Lotus 1-2-3 has the
mathematical functions to handle calculations involving
nonlinear cost functions or involving detailed cost
analysis. A sample Lotus 1-2-3 printout is shown in Table
3-4 for a hypothetical water network problem modeled by the
3-node digraph shown in Figure 3-2. This printout should be
self-explanatory. The top portion of the printout contains
the data for the problem, and the bottom portion is the
calculations associated with the total enumeration pro
cedure. The sorting capabilities of Lotus 1-2-3 allow
automatic ranking of all the feasible solutions according to
cost. Moreover, the Lotus 1-2-3 electronic spreadsheet
automatically recalculates all values associated with a
formula whenever a new value is entered or an existing value
is changed. This automatically gives the total enumeration
procedure the capability for sensitivity analysis. For
example, the set of all feasible solutions ranked according
to cost can be evaluated as the economies of scale, as
represented by the value of b in equation (3-1), is varied
over a specific range of values. Thus, for a regional
network problem of realistic size, all the feasible
solutions can be enumerated using a spreadsheet software
package.


37
Table 3-4. Efficiency Analysis of a Three-User Water
Supply Network with Nonlinear Cost Function
Using Lotus 1-2-3.
Sara
Distance : L(i,j) is the distance in feet from i to j
L (S, 1) =
L(S,2) =
17320 L(S,3)=
26000 L(1,2)=
3025k L (1,3) =
13130 L (2,3) =
19673
15500
Demand : Q(i) is the demand in mgd for user i
Q(1)= 1 Q(2) =
Cost Function: a(Q~b)L a=
5 Q (3) =
38 b=
3.51
Calculations With
C(i..j)[x]= Cost of network [x] for
Total Enumeration
i..j ; C(i..j)= L
Procedure
east cost
C(1)[S1]= 646000 C(2)[S2]=2463343.
C(3)[S3]=2012935.
C(12)[SI,12]= 2984140.
C (12)[SI;S2]= 3109348.
C(12)=
2934140.
C(13)[SI,13]= 2618975.
C(13)[S1;S3]= 2658986.
C(13)=
2618975.
C(23) [S2,23]= 4061294.
C(23)[32;S3]= 4476835.
C(23)=
4061294.
C(123)[SI,12,23]= 4648439.
C (123)[SI,12;S3]= 4997126.
C(123) [SI,12,13]= 4640756.
C (123)[SI,13;S2]= 5032324.
C (123) [SI;S2,23]= 4737294.
C(123)[SI;S2;S3]= 5122335.
C(123)=
4640756.
Sort C(123) in ascending order
Paths Cost
C(123)[SI,12,13]= 4643755.
C (123) [SI,12,23]= 4548439.
C(123)[S1;S2,23]= 4707294.
C (123)[SI,12;S3]= 4997126.
C(123)[S1,13;S2]= 5082824.
C(123) [S1;S2;S3]= 5122835.
BEST
C(123)=
4540756


38
Summary
A total enumeration procedure for finding the optimal
solutions necessary for efficiency/equity analysis of
realistic regional water network problems is presented. The
procedure can be easily understood and applied by engineers
with little knowledge or experience in operations research
techniques. Furthermore, the procedure allows the engineers
to handle all problems regardless of the types of cost
function involved or to perform detailed cost analysis.
Finally, if the optimal solution is impractical for
implementation, all suboptimal solutions ranked according to
cost are readily available for consideration.


CHAPTER 4
EQUITY ANALYSIS
Introduction
Proposed regional water resources systems involve
multiple purposes and groups who must somehow share the cost
of the entire project. The project may focus on construc
tion of a large dam which serves numerous purposes such as
water supply, flood control, and recreation. Also, canals
from the dam direct the water to nearby users. A signifi
cant portion of the total cost of this project may involve
elements which serve more than one purpose and/or group.
These costs are referred to as joint or common costs. In
such cases, it is possible to find the optimal or the most
economically efficient regional system, i.e., the one that
maximizes benefits minus costs. However, a major effort
remains to somehow apportion the project cost in an
equitable manner. In fact, the importance of the financial
analysis to apportion project cost is not limited to the
optimal system but includes any other integrated systems
being considered for implementation as well.
This chapter examines principles of cost allocation
using concepts from cooperative n-person game theory. An
39


40
example regional water network is used to illustrate these
principles.
Cost Allocation for Regional Water Networks
A hypothetical situation similar to options contained
in the West Coast Regional Water Supply Authority's master
plan for Hillsborough, Pasco, and Pinellas counties in
Florida (Ross et al., 1978) is now considered. Phase I
(1980-1985) of the plan recommends the use of groundwater
from existing and newly developed well fields to satisfy
water demands in the tri-county area. For this hypothetical
problem, assume that an existing well field is the most high
quality and cost effective water supply source (S) available
for three counties (1, 2, and 3) with projected demands of
1, 6, and 3 million gallons per day (mgd), respectively.
The demand for each county is based on projected population
growth and average per capita demand over a period of 5
years (see Table 4-1). Assume that the existing well field
is currently operating below its capacity of 20 mgd and can
satisfy the additional 10 mgd demanded by the three
counties. In addition, assume that no facility expansion is
required except for a new regional water network. Further
more, each county can be served directly by the well field,
and engineering considerations, e.g., gravity flow, have
determined that water can be sent from county 1 to both
county 2 and county 3, and from county 2 to county 3. The


41
Table 4-1. Projected Population Growth and Projected
Average Per Capita Demand.
County
Projected
Population Growth
Projected Average
Per Capita Demand
(gal/cap-day)
Projected
Additional
Demand
(mgd)
1
8,000
125
1
2
40,000
150
6
3
18,750
160
3
Total
66,750

10
Weighted
Average
...
150
..


42
lengths of all possible interconnecting pipelines are shown
in Figure 4-1. For our hypothetical problem, assume that
the total cost of constructing a pipeline has strong
economies of scale and is C = 38Q'^L, where C is total cost
of pipeline in dollars, Q is quantity of flow in mgd, and L
is the length of pipeline in feet.
Given the problem just described, the cost of a pipe
line serving county 1 alone is $646,000; the cost of a
pipeline serving county 2 alone is $2,420,095; and the cost
of a pipeline serving county 3 alone is $1,990,992. The
total cost for three individual pipelines is $5,057,087.
However, when the costs for all the options available to
these three counties are enumerated using the procedure
outlined in the preceding chapter, we see that the counties
can do better by cooperating (see calculations in Appendix A
using Lotus 1-2-3). There may be a slight difference
between the numbers used in the text and the numbers in
Appendix A because of rounding off. Also, cost data are
only significant to the nearest thousand dollars.
If the three counties cooperate, they can construct the
least cost or optimal network consisting of pipelines from
the well field to county 1, from county 1 to county 2, and
from county 2 to county 3 (see Table 4-2). This optimal
network costs $4,556,409 and represents a savings of 9.9% or
$500,678 when compared with the cost for three individual
pipelines. Obviously, constructing the optimal network is


&
3 TC^
-V
l>e
,rv<3
f
te*
c
,t^'
ec
O-
y<5
f i-Q
ue


44
Table 4-2. The Costs and Percent Savings for All Options.
Option
(Rank)
Cost ($)
Savings (%)
1
4,556,409
9.90
2
4,556,826
9.89
3
4,630,177
8.44
4
4,919,503
2.72
5
5,006,734
1. 00
6
5,057,087
0


45
in the best interest of the three counties, but to implement
this least cost network, an equitable way to allocate the
cost among the three counties must be found. This financial
problem is known as a cost allocation problem. The complex
ity is introduced because the counties share common pipes.
Criteria for Selecting a Cost Allocation Method
Several sets of criteria for selecting a cost alloca
tion method are found in the literature. For the water
resources field, criteria for allocating costs date back to
the Tennessee Valley Authority (TVA) project in 1935 when
prominent authorities were brought together to address the
cost allocation problem. They developed the following set
of criteria for allocating costs (Ransmeier, 1942,
pp. 220-221) :
1. The method should have a reasonable logical
basis. It should not result in charging any
objective with a greater investment than the fair
capitalized value of the annual benefit of this
objective to the consumer. It should not result
in charging any objective with a greater invest
ment than would suffice for its development at an
alternate single purpose site. Finally, it should
not charge any two or more objectives with a
greater investment than would suffice for
alternate dual purpose or multiple purpose
improvement.
2. The method should not be unduly complex.
3. The method should be workable.
4. The method should be flexible.
5. The method should apportion to all purposes
present at a multiple purpose enterprise a share
in the overall economy of the operation.


46
This set of criteria developed for the water resources
field is similar to the following set of criteria proposed
by Claus and Kleitman (1973) for allocating the cost of a
network:
1. The method must be easy to use and under
standable to users. They must be able to predict
the effects of changes in their service demands.
2. The method must have stability against system
breakup. It should not be an advantage to one or
more users to secede from the system. Thus, there
are limits to which a method can subsidize one
user or class of user at the expense of others.
3. It is desirable, though not necessary, that the
costing be stable under evolutionary changes in
the system or under mergers of users.
4. It is again desirable that the method should
preserve the substance and appearance of non
discrimination among users.
5. If the method represents a change from present
usage it is desirable that transition to the
new method be easy.
From these two sets of criteria, the most important
criterion for selecting a method to allocate the cost of a
regional water network is the method's ability to ensure
stability or prevent breakup of the network. That is, the
method should not allocate cost in a manner whereby an
individual or a subgroup of individuals can acquire the same
level of service by a less expensive alternative. Other
wise, the individual or subgroup of individuals will con
sider their allocated cost inequitable or unfair and secede
from the regional network for a less expensive alternative.


Heaney (1979) has expressed these fairness criteria for an
equitable cost allocation mathematically as follows:
1) x(i) < minimum [b(i), c(i)] VieN
(4-1)
where
x (i)
cost allocated to individual i
b (i )
benefit of individual i
c(i) = the alternative cost to individual i
of independent action, and
N
set of all individuals; i.e.,
N = {1,2 . ,n }.
r r
This criterion simply means that individual i should not be
charged a cost greater than the minimum of individual i's
benefit and alternative cost for independent action.
2) Z x(i) _< minimum [b(S), c(S)] V Scn
i eS
(4-2)
where c(S) = alternative cost to subgroup S of
independent action, and
b(S) = benefit of subgroup S.
This second criterion extends the first criterion to include
subgroup of individuals as well. These two fairness
criteria are now used to evaluate some simple and seemingly
fair cost allocation schemes for our regional water network
problem. Throughout this chapter, we will assume for our
regional water network problem that each county's and each


48
subgroup of counties1 alternative cost of independent action
is less than or equal to each county's and each subgroup of
counties' benefits, respectively; i.e.,
c(i) = minimum [b(i), c(i)] V ieN, and (4-3)
c(S) = minimum [b(S), c(S)] V ScN.
Ad Hoc Methods
Over the years, many ad hoc methods have been proposed
or used to apportion the costs of water resources projects
(Goodman, 1984). In general, ad hoc methods used in the
water resources field for allocating costs can be described
as follows: allocate certain costs that are considered
identifiable to an individual directly and prorate the
remaining costs, i.e., total project cost less the sum of
all identifiable costs, among all the individuals in the
project by some physical or nonphysical criterion. Mathe
matically, this can be expressed as follows:
x ( i ) = x(i)id + iMi)*rc
(4-4)
where
x (i)
x (i)
id
ip (i)
rc
cost allocated to individual i,
costs identifiable to
individual i,
prorating factor for individual i, and
remaining costs, i.e.,
c(N)


49
Furthermore, the requirement that Z iM i) = 1.0 should be
ieN
obvious.
James and Lee (1971) summarize 18 ways for allocating
the costs of water projects depending on the definition of
identifiable costs and the basis for prorating the remaining
costs (see Table 4-3). Basically, the differences among
these 18 methods are the following three ways of defining
identifiable costs: 1) zero, 2) direct or assignable costs,
or 3) separable costs; and the following six ways of
prorating remaining costs: 1) equal, 2) unit of use,
3) priority of use, 4) net benefit, 5) alternative cost, or
6) the smaller of net benefit or alternative cost. The next
two sections analyze the effects of defining identifiable
costs as either zero or direct costs. A detailed treatment
of separable costs, i.e., the difference between total
project costs with and without an individual, is given in
the section on the separable costs, remaining benefits
method.
Defining Identifiable Costs as Zero
The simplest way to allocate costs is to define identi
fiable costs as equal to zero and prorate total project cost
by some physical or nonphysical criterion. For example,
population and demand are two ways to prorate total project


50
Table 4-3. Cost Allocation Matrix.
Definition of
Identifiable
Cost
Basis for Prorating
Remaining Costs
A.
Zero
B.
Direct
Cost
C.
Separable
Cost
a.
Equal
Aa
Ba
Ca
b.
Unit of Use
Ab
Bb
Cb
c.
Priority of Use
Ac
Be
Cc
d.
Net Benefit
Ad
Bd
Cd
e.
Alternative Cost
Ae
Be
Ce
f .
Smaller of d.
or e.
Af
Bf
Cf
Source: Modified from James and Lee, 1971, p. 533.


51
cost (Young et al., 1982). Using these two ways to prorate
the cost of the optimal network for our regional water
network problem gives the following cost allocations (see
calculations in Table 4-4 and Table 4-5):
Proportional to Population
County 1 $ 546,769
County 2 2,733,845
County 3 1,275,795
$4,556,409
Proportional to Demand
County 1 $ 455,641
County 2 2,733,845
County 3 1,366,923
$4,556,409
Although these cost allocations are simple to calculate and
easy to understand, they fail to implement the optimal
network because county 2 considers these cost allocations
unfair. In contrast to counties 1 and 3, county 2 loses
money by being allocated a cost in excess of its go-it-alone
costs using either of these two methods. Consequently,
county 2 would rather acquire a pipeline by itself than
cooperate with counties 1 and 3 to construct the optimal
network. The principal failure with these proportionality


Table 4-4
Cost Allocation of Optimal Network Based on Population.
County i
Population
Percent of
Total Population
Allocated
Cost ($)
x (i)
Go-It-Alone
Cost ($)
c (i )
Is
x ( i ) < c ( i ) ?
1
8,000
12
546,769
646,000
Yes
2
40,000
60
2,733,845
2,420,095
No
3
18,750
28
1,275,795
1,990,992
Yes
Total
66,750
100
4,556,409
5,057,087

Ln
N)


Table 4-5. Cost Allocation of Optimal Network Based on Demand
County i
Demand
(mgd)
Percent of
Total Demand
Allocated
Cost ($)
x (i)
Go-It-Alone
Cost ($)
c (i )
Is
x(i) < c(i)?
1
1
10
455,641
646,000
Yes
2
6
60
2,733,845
2,420,095
No
3
3
30
1,366,923
1,990,992
Yes
Total
10
100
4,556,409
5,057,087

U1
OJ


54
methods is that they do not recognize explicitly each
individual's contribution to total project cost.
Defining Identifiable Costs as Direct Costs
A way to recognize each individual's contribution to
total project cost is by defining identifiable costs as
those costs that can be directly assigned, and prorating the
remaining costs by some physical or nonphysical criterion
such as use or number of individuals; i.e.,
x (i) = x(i)direct + 4(i)-re (4-5)
where x(i)-,. = direct cost or assignable cost
direct -7
to individual 1.
Although this direct costing approach intuitively seems
fair, inequitable and unpredictable cost allocations can
result. To illustrate, two direct costing methods are
applied to our regional water network problem.
A common approach to allocating remaining costs is by
some physical measure of each individual's use of the common
facilities; this method is generally referred to as the use
of facilities method (Loughlin, 1977; Goodman, 1984). This
traditional method is easy to understand and apply because
quantitative information on a physical measure of use is
generally available. In the water resources field, use
can be measured in terms of the storage capacity and/or the


55
quantity of water flow provided by the common facilities.
For our regional water network problem, the flow to each
county is the obvious measure of use to apportion the costs
of common pipelines since the assumed cost function depends
on the flow. In the case of the optimal network, the only
direct cost is the cost of the pipeline from county 2 to
county 3 serving county 3, and the use of facilities method
gives the following cost allocation (see calculations in
Table 4-6).
$ 204,283
2,221,299
2,130,827
$4,556,409
County 1
County 2
County 3
Total
Unfortunately, this cost allocation does not implement the
optimal network because county 3 can do substantially better
by going alone, i.e., $1,990,992 versus paying $2,130,827.
In addition to giving an inequitable cost allocation
for the optimal network, the use of facilities method can
promote noncooperation if other networks are also being
considered. Table 4-7 shows the cost allocations for all
possible options available to the three counties using the
use of facilities method. Suppose the "second best" network
or option 2 is also being considered by the counties. The
second best network consists of the pipelines from the well
field to county 1, from county 1 to county 2, and from
county 1 to county 3. This second best network costs


Table 4-6. Cost Allocation of Optimal Network with Use of Facilities Method
Pipeline
S-l
1-2
2-3
Total
Cost ($)
Go-It-Alone
Cost ($)
c (i )
Length
(f t)
17,000
13,100
15,500


Q
(mgd)
10
9
3


Pipeline
Cost
($)
2,042,832
1,493,400
1,020,177
4,556,409

Cost for
County 1
($)
Q=1 mgd
204,283
0
0
204,283
646,000
Cost for
County 2
(?)
Q=6 mgd
1,225,699
995,600
0
2,221,299
2,420,095
Cost for
County 3
($)
Q=3 mgd
612,850
497,800
1,020,177
2,130,827
1,990,992
Ln


57
Table 4-7. Cost Allocation for the Use of Facilities Method.
Cost Allocation to
($)
County i
Zx (i)
Is Cost
Allocation
Option
(Rank)
County 1
x(l)
County 2
x (2 )
County 3
x (3 )
($)
Equitable?
1
204,283
2,221,299
2,130,827
4,556,409
No
x(3)>c(3 )
2
204,283
2,445,055
1,907,488
4,556,826
No
x(2)>c(2 )
3
646,000
1,976,000
2,008,177
4,630,177
No
x(3)>c(3 )
4
244,165
2,684,346
1,990,992
4,919,503
No
x(2)>c(2)
5
323,000
2,420,095
2,263,639
5,006,734
No
x(3)>c(3)
6
646,000
2,420,095
1,990,992
5,057,087

(4)
(5)
(6)


58
$4,556,826 or $417 more than the optimal network; so, both
networks are essentially comparable in cost, and either
network might be considered the least cost network. In
fact, the second best network becomes the optimal network if
the economies of scale or the value of b in the cost
function is .51 instead of .50 (see Table 3-4). Never
theless, applying the use of facilities method to this
second best network gives the following cost allocation.
$ 204,283
2,445,055
1,907,4 88
$4,556,826
County 1
County 2
County 3
In this case, the cost allocation fails to implement
the second best network because county 2 is better off going
alone, i.e, paying $2,420,095 rather than $2,445,055.
Furthermore, if we examine the cost allocation for the
optimal network and the second best network, another problem
is evident. Although the costs for the two networks are
$417 apart, the difference in costs between the two networks
for county 2 and county 3 is enormous. Consequently, this
cost allocation method imposes another obstacle for the
counties to cooperate and implement either one of the two
networks. County 2 strongly opposes the second best network
because of its substantially higher cost while county 3
strongly opposes the optimal network for the same reason.
This problem is even more serious when more options are
considered by the counties. Table 4-7 indicates tremendous


59
differences in allocated cost for each county depending on
the network, thereby making cooperation very difficult.
This situation shows the danger for individuals to simply
accept the least cost network without carefully examining
all of their options if the use of facilities method for
allocating costs is chosen.
Another simple way to prorate the remaining costs is to
divide it equally among the individuals associated with the
common facilities (see calculations for optimal network in
Table 4-8). Table 4-9 shows the cost allocations using this
egalitarian approach and indicates that none of the cost
allocations for options with savings are equitable. At
first glance, the cost allocation for option 5 appears
equitable because each county is charged a cost less than
or equal to its go-it-alone cost. However, closer examina
tion reveals that counties 1 and 2 can do better as a
coalition. They can construct a pipeline from the well
field to county 1 and from county 1 to county 2, i.e.,
option 4, for $2,928,511 rather than pay the sum of their
costs for option 5, i.e., $3,066,095. Unfortunately, a
transition from option 5 to option 4 causes county 1 to lose
money, i.e., $854,577 for option 4 versus $646,000 for
option 5. To further complicate matters, option 5 only
gives a 1% savings and requires county 1 to cooperate with
county 3 to build a pipeline without getting any savings.


Table 4-8. Cost Allocation of Optimal Network with Direct Costing/Equal
Apportionment of Remaining Costs Method.
Pipeline
S-l
1-2
2-3
Total
Cost
($)
Go-It-Alone
Cost($)
c (i )
Length
(ft)
17,000
13,100
15,500


Q
(mgd)
10
9
3


Pipeline
Cost
($)
2,042,832
1,493,400
1,020,177
4,556,409

Cost for
County 1
($)
Q=1 mgd
680,944
0
0
680,944
646,000
Cost for
County 2
($)
Q=6 mgd
680,944
746,700
0
1,427,644
2,420,095
Cost for
County 3
($)
Q=3 mgd
680,944
746,700
1,020,177
2,447,821
1,990,992
o


Table 4-9. Cost Allocation for Direct Costing/Equal
Apportionment of Remaining Costs Method
61
Cost Allocation to County i
Option
(Rank)
County 1
x (1)
($)
County 2
x (2 )
County 3
x (3)
f x ( i )
($)
Is Cost
Allocation
Equitable?
1
680,944
1,427,644
2,447,821
4,556,409
No
X(1)>c(1)
x(3)>c(3)
2
680,944
1,900,300
1,975,582
4,556,826
No
x(l)>c(l)
3
646,000
1,482,000
2,502,177
4,630,177
No
x(3)>c(3 )
4
854,577
2,073,934
1,990,992
4,919,503
No
X(1)>c(1)
5
646,000
2,420,095
1,940,639
5,006,734
No
x(1)+x(2)>
c (12 )
6
646,000
2,420,095
1,990,992
5,057,087
(4)
(5)
(6)


62
Given these observations, the stability of option 5 as a
regional water network is at best questionable. Again, if
the allocated costs for counties 2 and 3 for the optimal
network are compared to the second best network, a similar
situation like the one discussed for the use of facilities
method exists. That is, counties 2 and 3 face substantially
different costs for these two networks with comparable
costs.
Thus, assigning direct costs does not help eliminate
inequitable cost allocations. In fact, direct costing
methods can impose additional obstacles to cooperation.
This occurs because the assignment of direct costs depends
on the configuration of the facilities. For instance, the
cost of the pipeline from county 2 to county 3 for our
regional water network problem can be a direct cost or a
joint cost depending on the network. The cost of the pipe
line is a direct cost for county 3 if the second best
network, i.e., option 2, is being considered; yet, the cost
of the pipeline is a joint cost for counties 2 and 3 if the
optimal network, i.e., option 1, is being considered. These
changes in the cost classification for the pipeline from
county 2 to county 3 contribute to the tremendous difference
in the cost allocations for counties 2 and 3 for the two
comparable cost networks. This situation indicates an
additional criterion not addressed by Claus and Kleitman
(1973) for selecting a procedure to allocate network cost.


63
The cost allocation procedure should be independent of
network configuration; otherwise, the cost allocation pro
cedure can promote noncooperation if more than one network
is being considered.
In summary, two approaches for allocating costs in the
water resources field have been examined: 1) allocate total
project cost in proportion to a physical or nonphysical
criterion; or 2) allocate assignable costs directly and
prorate the remaining costs by a physical or nonphysical
criterion. In general, these two approaches are simple to
apply and easy to understand. In fact, these two approaches
are currently accepted cost allocation methods used in
accounting (Kaplan, 1982). However, these two approaches
are unable to consistently provide an equitable cost
allocation when an equitable cost allocation exists, i.e.,
sometimes these methods work and sometimes they fail.
Furthermore, methods attempting to assign costs directly may
be influenced by the configuration of the facilities and may
discourage cooperation when more than one configuration is
being considered. This is particularly evident for our
regional water network problem. For a theoretically sound
method that is able to find an equitable cost allocation if
one exists and is not influenced by the configuration of the
facilities, concepts from cooperative n-person game theory
are necessary.


64
Cooperative Game Theory
Game theory has been with us since 1944 when the first
edition of The Theory of Games and Economic Behavior by John
Von Neumann and Oskar Morgenstern appeared. In particular
we are interested in games wherein all of the players
voluntarily agree to cooperate because it is mutually bene
ficial. Furthermore, games are studied in three forms or
levels of abstraction. The extensive form requires a com
plete description of the rules of a game and is generally
characterized by a game tree to describe every player's
move. A game in normal form condenses the description of a
game into sets of strategies for each player and is
represented by a game matrix. However, most efforts in
cooperative game theory have been with games in charac
teristic function form whereby the description of a game is
in terms of payoffs rather than rules or strategies. The
characteristic function form appears to be the most
appropriate for studying coalition formation which is an
essential feature in cooperative games. Also, cooperative
games can be of three types depending on whether the game is
defined in terms of costs, savings, or values. To keep the
notation as simple as possible, only cost games will be
discussed. Introductory and intermediate material on coop
erative game theory can be found in Schotter and Schwodiauer
(1980), Jones (1980), Luce and Raiffa (1957), Lucas (1981),
Rapoport (1970), Shubik (1982), and Owen (1982).


65
Concepts of Cooperative Game Theory
Let N = {1,2,...,n} be the set of players in the game.
Associated with each subset of S players in N is a charac
teristic function c, which assigns a real number c(S) to
each nonempty subset of S players. For cost games, the
characteristic function, c(S), can be defined as the least
cost or optimal solution for the S-member coalition if the
N-S member or complementary coalition is not present.
However, depending on how the problem is defined, alterna
tive definitions for c(S) may be required. For example,
Sorenson (1972) presents the following four alternative
definitions for the characteristic cost function:
c^S)
value to coalition if S is given preference
over N-S.
C2(S) = value of coalition to S if N-S is not
present,
c^iS) = value of coalition in a strictly competitive
game between coalition S and N-S, and
c^(S) = value of coalition to S if N-S is given
preference.
If c(S) can be defined as the least cost solution for
coalition S if N-S is not present, then the cost game is
naturally subadditive; i.e.,
c(S) + c(T) > c(SUT)
ShT = 0, S,TcN
(4-6)


66
where 0 is the empty set; and S and T are any two disjoint
subsets of N. Subadditivity is a natural consequence of
c(S) because the worst S and T can do as a coalition is the
cost of independent action; i.e.,
c(S) + c(T) = c(SUT) SOT = 0, S,TCN. (4-7)
A coalition in which the players realize no savings from
cooperation is said to be inessential.
General reasons why subadditivity exists are discussed
by Sharkey (1982a). The primary reason why subadditivity
exists for our regional water network problem is because of
the economies of scale in pipeline construction cost. For a
single output cost function, C(q), economies of scale is
defined by
C(Aq) < Ac(q) (4-8)
where q = output level, and for all A such that
1 < A £ 1 + e, e is a small positive number.
This definition means that the average costs are declining
in the neighborhood of the output q. From Sharkey (1982a),
economies of scale is sufficient but not a necessary condi
tion for subadditivity. Subadditivity is a more general


67
condition which allows for both increasing marginal cost
and increasing average cost over some range of outputs.
Solution concepts for cooperative cost games involve
the following three general axioms of fairness (Heaney and
Dickinson, 1982; Young et al., 1982);
1) Individual Rationality: Player i should not pay
more than his go-it-alone cost, i.e.,
x(i) < c(i), ¥ ieN, (4-9)
where x(i) is the allocated cost or the charge to player i.
2) Group Rationality: The total cost of the grand
coalition, c(N), must be apportioned among the N players;
i.e.,
Ex(i)=c(N). (4-10)
ieN
3) Subgroup Rationality: This final axiom extends the
notion of individual rationality to include subgroups, i.e.,
no subgroup or subcoalition S should be apportioned a cost
greater than its go-it-alone cost, or
1 x(i) < c(S), V SCN. (4-11)
is S
The set of solutions or charges satisfying the first two
axioms is called the set of imputations, while the


68
additional restriction of the third axiom defines what is
known as the core of the game. For subadditive cost games
the set of imputations is not empty, but the core may be
empty. Shapley (1971) has shown that the core always exists
for convex games. A cost game is convex if
c(S) + c(T) > c(SUT) + c(ShT) SOT i 0, V S,TcN (4-12)
or equivalently, convexity can be written as
c(SUi) c(S) > c(TUi) c(T) ScTcN {i}, ieN. (4-13)
Convexity simply means the incremental cost for player i to
join coalition T is less than or equal to the incremental
cost for player i to join a subset of T. This notion of
convexity is analogous to economies of scale and implies the
game has a particular form of increasing returns to scale in
coalition size. As will be shown, the more attractive the
game, i.e., larger savings in project costs, the greater the
chance that the game is convex; whereas, if the game is less
attractive, i.e., lower savings in project costs, the poten
tial for a nonconvex game or an empty core game is greater.
To illustrate the concept of the core, assume a three-
person cost game with the following characteristic function
values:


69
c(l) =
35
c (2 )
= 45
c ( 3 ) =
50
c(12 ) =
66
c (13 )
= 75
c(23 ) =
87
c(12 3 )
= 100
This game is subadditive so each player has an incentive to
cooperate; i.e.,
c(l) + c(2)
+
c ( 3 )
>
c(12 3 )
c(l)
+
c (23 )
>
c(123 )
c (2 )
+
c (13 )
>_
c (123)
c ( 3 )
+
c (12 )
>
c(123)
c (1)
+
c (2 )
>
c (12 )
c(l)
+
c (3 )
>
c (13 )
c (2 )
+
c (3 )
>
c ( 23 ) .
Furthermore, this game is convex; i.e.,
c (12 )
+
c (13 )
>
c(12 3 )
+
c (1)
c (12 )
+
c (23 )
>
c(12 3 )
+
c (2 )
c (13 )
+
c ( 23 )
>
c(12 3 )
+
c ( 3 ) .
Using the three general axioms of fairness, the core
conditions are as follows:
x(l) < 35
x ( 2) < 45
x ( 3 ) < 50
x(l) + x(2) < 66
x(l) + x(3) < 75
x(2) + x(3) < 87
x(l) + x(2) + x(3) = 100.


70
The first three conditions determine the upper bounds on
x(i), i = 1,2,3, while the last four conditions determine
bounds
on x(i),
i = 1,
r 2 ,
r 3
. e.
t
c(123 )
- c(23)
= 13
<
x(l)
<
35
= c(l)
c(123 )
- c(13 )
= 25
<
x (2 )
<
45
= c ( 2 )
c(12 3 )
- c(12 )
= 34
<
x (3)
<
50
= c ( 3 )
For a three-person game, graphical examination of the
core conditions and the nature of the charge vectors is
possible using isometric graph paper (Heaney and Dickinson,
1982). As shown on Figure 4-2, each player is assigned a
charge axis. The plane of triangle ABC, with vertices
(100,0,0), (0,100,0), and (0,0,100), represents points
satisfying group rationality (axiom 2); whereas, the smaller
triangle abc represents the set of imputations satisfying
both individual rationality (axiom 1) and group rationality
(axiom 2). The vertices a, b, and c represent the charge
vectors: [35, 15, 50], [5, 45, 50], and [35, 45, 20],
respectively. Line ab represents the upper bound for player
3, i.e., x(3) = c(3), where c(123) c(3) is allocated
between players 1 and 2. As we move along line ab from
point a to point b, the allocation to player 1 decreases
from c(l) to c(123) c(2) c(3), i.e., from 35 to 5, while
the allocation to player 2 increases from c(123) c(l) -
c(3) to c(2), i.e., from 15 to 45. Similar explanations can
be given for lines be and ac. A more restrictive set of
solutions satisfying subgroup rationality (axiom 3),


71
x( 3)
Figure 4-2.
Geometry of Core Conditions for Three-
Person Cost Game Example.


72
the shaded area on triangle abc, is the core for this game.
The geometry of the core for this convex game is a hexagon.
Line de represents the lower bound for player 2 or the set
of charges where c(13) is allocated between player 1 and
player 3 with the remainder, c(123) c(13), going to player
2. Similar explanations can be given for lines fg and hi
which are the lower bounds for players 1 and 3,
respectively; and for lines id, gh, and ef which are the
upper bounds for players 1, 2, and 3, respectively.
If an allocation lies outside the core, an inequitable
situation prevails. For instance, point Z in Figure 4-2
allocates player 2 a cost less than its lower bound,
c(123)-c(13), which means c(13) increases or the cost
allocated to players 1 and 3 increases. Clearly, player 1
and player 3 can do better by forming their own two-person
coalition rather than subsidizing player 2.
As mentioned earlier, the convexity of a game and its
attractiveness are related. This relationship is illustrated
in Table 4-10. When the costs for the two-person coalitions
progressively decrease, there is less incentive for forming
the grand coalition so the core becomes progressively smaller
and the game becomes progressively more nonconvex. As a
consequence of the core conditions for a three-person sub
additive cost game, a condition can be derived to determine
if a core exists. From subgroup rationality and group
rationality, we have the following conditions:


73
Table 4-10. Core Geometry for Three-Person Cost Game
Example.
Characteristic Function
c(1) = 35, c(2) = 45,
c(3) = 50, c(123) = 100 Geometry
Zc(ij) of Core
c (12) c(13 ) c(23 )
66
75
87
228
< }
Hexagon
61
73
86
220
A>
Pentagon
59
71
85
215
ZA
Trapezoid
58
70
80
208
A
Triangle
56
68
76
200

Point
55
65
72
192
x (2 )
x(1) //
-\rx<3)
//y Empty
Source: Modified from Fischer and Gately, 1975, p. 27a.


7 4
x(l) + x(2) < c(12)
x(l) + x(3) < c(13)
x ( 2 ) + x ( 3 ) < c ( 2 3 )
x(l) + x(2) + x(3) = c(12 3)
(4-14)
Summing the subgroup rationality conditions gives
2[x(1) + x(2) + x ( 3 ) ] < c(12) + c(13) + c(23). (4-15)
If the group rationality conditions are substituted into the
above equation, then we have the following condition to
determine if a core exists:
2 c(12 3) < c(12 ) + c(13 ) + c(2 3 ) .
(4-16)
Therefore, in Table 4-10, the core exists as long as the sum
of the two-person coalitions is greater than 200 or twice
the value of the grand coalition. When the sum of the
two-person coalitions equals 200, the core reduces to a
unique vector, i.e., X = [24, 32, 44]. Finally, when the
sum of the two-person coalition is less than 200, then the
core is empty. Unfortunately, for larger games there is no
simple condition for checking the existence of a core;
however, as we will see later, a check can be made using
linear programming.


75
Unique Solution Concepts
The three axioms of fairness defining the core of the
game significantly reduce the set of admissible solutions.
Unless the core is empty or is a unique vector, an infinite
number of possible equitable charge vectors remain to be
considered, so additional criteria are needed to select a
unique charge vector. Numerous methods are available for
selecting a unique charge vector; but the two most popular
methods discussed in the literature are the Shapley value
(Shapley, 1953; Heaney, 1983b; Shubik, 1962; Heaney et al.,
1975; Littlechild, 1970) and the nucleolus (Schmeidler,
1969; Kohlberg, 1971; Suzuki and Nakayama, 1976).
Shapley value. The Shapley value for player i is
defined as the expected incremental cost for the coalition
of adding player i. Thus, each player pays a cost equal to
the incremental cost incurred by the coalition when that
player enters. Since the coalition formation sequence is
unknown, the Shapley value assumes an equal probability for
all sequences of coalition formation, i.e., the probability
of each player being the first to join is equal, as are the
probabilities of joining second, third, etc. For an n
person game there are n! orderings. The six sequences of
coalition formation for a three-person game are as follows:
(123) (213) (231)
(132)
(312)
(321)


7 6
Therefore, the Shapley value or the cost to player 1 for a
three-person game is
$(1) = 1/3 c(1) + 1/6 [c(12) c(2)] + 1/6 [c(13 c(3)]
+ 1/3 [c(12 3) c(23)].
(4-17)
Player 1 has 1/3 probability of entering the coalition as
the first player and 1/3 probability of entering the
coalition as the last player. In addition, player 1 has 1/6
probability of entering the coalition after player 2 and 1/6
probability of entering the coalition after player 3.
Notice that [c(S+i) c(S)] is the incremental cost of
adding player i to the S coalition.
The general formula for the Shapley value for player i
is
4> ( i ) = I a. (S) [ c {S) c ( S { i } ) ]
Scn 1
(4-18)
where
(s-i)! (n s)
rf!
s is the number of players in coalition S,
n! is the total number of possible sequences
of coalition formation,
(s-1)! is the number of arrangements for those
players before S, and
(n-1)! is the number of arrangements for those
players after S.


77
For example, for i = 1, n = 3:
a1(l)
0121/3!
= 1/3
ax(12) =
1111/31
= 1/6
a1(13) =
1111/3!
= 1/6
a1(123) =
2101/31
= 1/3
Total
1.0
Note that
2 i(i) = c(N). (4-19)
i eN
Furthermore, if the game is convex, the Shapley value lies
in the center of the core (Shapley, 1971).
The Shapley value is criticized for several reasons.
It may fall outside the core for nonconvex games, and it may
be computed even when the core does not exist (Hamlen,
1980). Furthermore, the Shapley value is computationally
burdensome for large games. For an n-person game, the
Shapley value for each player requires the computation of
n 1
2 coefficients and incremental costs. For example, an
eight player game requires 128 coefficients and incremental
costs to calculate the charge for each player.


78
Loehman and Whinston (1976) attempted to reduce the
computational burden of the Shapley value by relaxing the
assumption that all sequences of coalition formation are
equally likely. This generalized Shapley value allows using
a priori information to eliminate impossible sequences of
coalition formation. Unfortunately, when Loehman et
al. (1979) applied the generalized Shapley value to an
eight-player regional wastewater management problem, they
got a solution outside the core (Heaney, 1983a).
Littlechild and Owen (1973) developed the simplified
Shapley value for games wherein the characteristic function
is a cost function with the property that the cost of any
subcoalition is equal to the cost of the largest player in
the subcoalition. Although Littlechild and Thompson (1977)
demonstrated the computational ease of the simplified
Shapley value in their case study of airport landing fees
consisting of 13,572 landings by 11 different types of
aircraft, the use of the simplified Shapley value is
restricted to games with these special properties.
Before calculating the Shapley value for our regional
water network problem, the total enumeration procedure
described in the preceding chapter is used to find the
following characteristic cost function values (see Appendix
A) :
c(1) = 646,000 c(2 ) = 2,420,095 c(3 ) = 1, 990,992
c(12) = 2,928,511 c(13)
2,586,638
c ( 2 3 )
3,984,177


79
and
1
c
(12
3)
4,
556,
409
2
c
(12
3)
4,
556,
826
3
c
(1,
23) =
4,
630,
177
4
c
(12
,3) =
4,
919,
503
5
c
(13
,2) =
5,
006,
734
w
c
dr
2,3) =
5,
057,
087
k t h
where c (hi,j) is the cost of the k best regional water
network consisting of pipelines from the well field to
county h, from county h to county i, and from the well field
. w
to county j. Also, c (1,2,3) is the cost for each county to
t h
go-it-alone. The cost allocation associated with the k
t h
best regional water network, i.e., the kc network game, is
simply found by setting c(N) equal to c (N).
The Shapley values for all options available to the
three counties are calculated in Appendix A and summarized
in Table 4-11. Appendix A also checks whether each Shapley
value satisfies core conditions. All of the network games
in this example are nonconvex. Table 4-11 shows that the
cost allocations for the optimal and the second best
networks, i.e., the first two options, satisfy all core
conditions; therefore, these cost allocations are in the
core and are considered equitable. Furthermore, unlike the
cost allocations using the direct costing methods discussed
earlier, the cost allocations for these two comparable cost


80
Table 4-11. Cost Allocation for Three-County Example Using the
Shapley Value.
Cost Allocation to County i Is Cost
($) Allocation
Zx(i) In Core?
Option County 1 County 2 County 3 ($) (From Ap-
(Rank) x(l) x(2) x(3) pendix A)
1
590,087
2,175,905
1,790,417
4,556,409
Yes
2
590,226
2,176,044
1,790,556
4,556,826
Yes
3
614,677
2,200,494
1,815,006
4,630,177
No
4
711,119
2,296,936
1,911,448
4,919,503
No
5
740,196
2,326,013
1,940,525
5,006,734
No
6
646,000
2,420,095
1,990,992
5,057,087



81
networks are nearly identical. The Shapley value divided
the additional $417 for the second best network equally
among the counties. Option 3 illustrates the failure of the
Shapley value to consistently give a core solution for
nonconvex games. As shown in Appendix A, the cost alloca
tion for option 3 fails to satisfy subgroup rationality for
the coalition consisting of county 2 and county 3; i.e.,
x(2) + x(3) > c(23 ) (4-20)
Moreover, options 4 and 5 illustrate Shapley values for
games with an empty core. The nonexistence of the core for
network games with options 4 and 5 can be determined by
using other game theory methods, e.g., nucleolus. A close
examination of the core conditions for network games with
options 4 and 5 reveals these games are no longer subaddi-
tive. By defining c(N) as c (N), c(N) is no longer the
least cost or optimal solution for the grand coalition.
t h
Consequently, the k best network game is not naturally
subadditive even though c (N) may be less expensive than
w
c (N). In any event, because network games with options 4
and 5 are not subadditive, there is no incentive to
cooperate. Therefore, options 4 and 5 no longer need to be
considered by the counties.
Nucleolus. The other popular method to obtain a unique
charge vector is to find the nucleolus. For a cost game,


82
the fairness criterion used by the nucleolus is based on
finding the charge vector which maximizes the minimum
savings of any coalition.
For each imputation in the core of a cost game, a
2n
vector in R is defined. The components of this vector are
arranged in increasing order of magnitude and are defined by
e(S) = c(S) z x(i) V ScN. (4-21)
ieS
2 n
The imputation whose vector in R is lexicographically the
largest is called the nucleolus of the cost game. Given two
vectors, X = (x^,...,x ) and Y = (y^,...,y ), X is lexi
cographically larger than Y if there exists some integer k,
1 < k < n, such that x^ = y^ for 1 _< j < k and x^. > y^
(Owen, 1982). Basically, e(S) represents the minimum
savings of coalition S with respect to charge vector X.
Obviously, the coalition with the least savings objects to
charge vector X most strongly, and the nucleolus maximizes
this minimum savings over all coalitions.
The nucleolus can be found by solving at most n-1
linear programs (Kohlberg, 1972; Owen, 1974, 1982), where
the first linear programming problem is
maximize e(l)
subject to
e(l) + x(i) < c(i)
Â¥ ieN
(4-22)


83
e (1) + £ x ( i ) < c (S) ¥ S N
ie S
£ x ( i ) = c (N)
ie N
x ( i ), e (1) > 0
The nucleolus is calculated by sequentially solving for
e(l), then e(2), e(3), etc., where e(i) is the ifc^ smallest
savings to any coalition.
Unlike the Shapley value, the nucleolus always is in
the core for games with nonempty core. In fact, the
nucleolus is always unique. However, the nucleolus is
criticized because it cannot be written down in explicit
form (Spinetto, 1975), and that it is difficult to compute
and use in practice (Gugenheim, 1983). Probably the most
difficult problem with using the nucleolus is the acceptance
of its notion of fairness as opposed to other prevailing
notions of fairness without generating unending
controversies and debates. The nucleolus is generally
considered to be analogous to Rawls' (1971) welfare
criteria: the utility function of the least well off
individual is maximized. Other notable notions of fairness
include (1) Nozick's (1974) procedural approach to justice,
and (2) Varian's (1975) or Baumol's (1982) definition of


84
equitable distribution whereby no one prefers the
consumption bundle of anyone else.
Calculating the nucleolus for our regional water
network problem using the linear programming problem (4-22)
gives the results summarized in Table 4-12. Equitable cost
allocations are given for the first three options, and the
cost allocations for the optimal and the second best
networks are essentially the same. The additional $417 for
the second best network is apportioned as follows:
County
1
$209
County
2
104
County
3
104
Total
$417
No cost allocations are given for options 4 and 5 because
these network games have empty cores. That is, the linear
programming problem (4-22) is infeasible. Finally, Table
4-12 reveals that each of the three counties has an
incentive to cooperate in order to implement the cheapest
regional water network.
Propensity to disrupt. Another unique solution concept
worth mentioning because of its intuitive appeal is the
concept of an individual player's "propensity to disrupt."
Gately (1974) defined an individual player i's propensity to
disrupt as a ratio of what the other players would lose if
player i refused to cooperate over how much player i would
lose by not cooperating. Mathematically, player i's


85
Table 4-12. Cost Allocation for Three-County Example Using
the Nucleolus.
Cost Allocation to
(?)
County i
lx (i )
Is Cost
Allocation
Option
(Rank)
County 1
x(l)
County 2
x (2 )
County 3
x (3 )
($)
In
Core?
1
609,116
2,144,583
1,802,710
4,556,409
Yes
2
609,325
2,144,687
1,802,814
4,556,826
Yes
3
4
646,000
2,163,025
1,821,152
4,630,177
Yes
5
6
646,000
2,420,095
1,990,992
5,057,087
--


86
propensity to disrupt, d(i), a charge vector, X =
[x(1),...,x(n)], which is in the core is
c(N-i) £ x(j )
d ( i ) = 21 (4-23)
c ( i ) x(i )
The higher the propensity to disrupt, the greater a player's
threat to the coalition; e.g., d(i) = 10 implies player i
could impose a loss of savings to the other players 10 times
as great as the loss of savings to player i. As an
illustration, the propensity to disrupt is calculated for
each of the counties using the nucleolus for the optimal
network of our regional water network problem: X =
[609,116; 2,144,583; 1,802,710].
d (1)
c(23 x(2) x(3)
c (1) x (1)
= 1.0
d (2 )
_ c(13 ) x (1) x(3 )
= .63
c ( 2 ) x ( 2 )
d (3 )
_ c(12 ) x(1) x ( 2 )
= .93
c ( 3 ) x ( 3 )
The calculations show that none of the counties have a
strong threat against the other two counties with the
nucleolus charge vector. County 1 could impose a loss
to the other two counties which equals the loss imposed on
itself, while, county 2's or county 3's departure would hurt
the departing county more than it would hurt the remaining
two counties.


87
Gately suggested equalizing each player's propensity to
disrupt as a final cost allocation solution. Subsequently,
Littlechild and Vaidya (1976) have generalized Gately's
concept of an individual player's propensity to disrupt to
include a coalition S's propensity to disrupt. That is, a
coalition S's propensity to disrupt is defined as the ratio
of what the complementary coalition, N-S, stands to lose
over what the coalition S itself stands to lose for a given
charge vector. More recently, Chames et al. (1978 ) and
Chames and Golany (1983) refined these propensity to
disrupt concepts into a unique solution concept which
appears to have some empirical support. Finally, Straffin
and Heaney (1981) have shown that Gately's propensity to
disrupt is exactly the alternative cost avoided method first
proposed during the TVA project in 1935. The alternative
cost avoided method is discussed in the section on the
separable costs, remaining benefits method.
Empty Core Solution Concepts
Examining games with an empty core is an active area of
research. An empty core implies that no equitable cost
allocation exists, and results from games wherein the addi
tional savings from forming the grand coalition is rela
tively small. That is, the savings resulting from forming
smaller coalitions are almost as much as the savings from
forming the grand coalition. Therefore, proposed solution


88
concepts generally seek to relax the bounds on subgroup
rationality until a "quasi" or "anti" core is created.
Table 4-13 lists four methods for finding a charge vector
for games with an empty core.
In any case, given the modest amount of economic gain
for games with an empty core, it may be more advantageous to
forego the grand coalition in favor of smaller coalition
formations as suggested by Heaney (1983a). Furthermore,
engineering projects tend to have a large proportion of the
costs common to all participants; consequently, one would
expect these games to be very attractive and games with an
empty core to be fairly rare. Nevertheless, the game theory
approach does alert us that a problem exists in allocating
costs for such cases.
Cost Allocation in the Water Resources Field
Straffin and Heaney (1981) showed that the criteria of
fairness as expressed by equations (4-1) and (4-2) associated
with cost allocation proposed by the TVA experts in the 1930's
paralleled the development of the concepts of individual and
subgroup rationality found in cooperative game theory. Given
that full costs have to be recovered, the core conditions are
equivalent to the fairness criteria for allocating cost
originally proposed by the TVA experts. Therefore, current
practice for allocating costs in the water resources field
should require the solution be in the core of a game.


Full Text



EFFICIENCY/EQUITY ANALYSIS OF WATER RESOURCES
PROBLEMS—A GAME THEORETIC APPROACH
Bv
ELLIOT KIN NG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1985

To my parents
and
my wife, Eileen,
and children, Matthew, Michelle, Michael

ACKNOWLEDGMENTS
I would like to thank my chairman, Dr. James P. Heaney,
for the many hours spent guiding this research. His
encouragement, support, and friendship during my three years
at the University of Florida have been invaluable. I would
also like to thank the other members of my supervisory
committee, Dr. Sanford V. Berg, Dr. Donald J. Elzinga,
Dr. Wayne C. Huber, and Dr. Warren Viessman, for their time
and support. In addition, I wish to thank the U.S. Air
Force for giving me the opportunity to pursue the Ph.D.
degree.
Thanks are also due to several fellow students who
have made my program enjoyable and memorable. In particu¬
lar, I wish to thank Mr. N. Devadoss, Mr. Mun-Fong Lee, and
Mr. Robert Ryczak. I would also like to give special thanks
to Mr. Robert Dickinson for keeping an extra copy of the
LP-80 and Mrs. Barbara Smerage for doing such an excellent
job typing this manuscript.
I am extremely grateful to my parents for instilling in
me a desire to seek further eduction. Furthermore, I am
especially thankful to my wife, Eileen, for typing initial
drafts of this manuscript and for her love, encouragement,
and sacrifices throughout my program. We will miss the
iii

croissants, pizzas, and hoagies that supplemented my late
night studies. Finally, I wish to thank my children,
Matthew, Michelle, and Michael, for their love and under¬
standing during the countless times I have chased them out
of my study.
IV

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iii
LIST OF TABLES vii
LIST OF FIGURES ix
ABSTRACT x
CHAPTER
1 INTRODUCTION 1
2 LITERATURE REVIEW 4
Efficiency Analysis 4
Equity Analysis 5
Conclusions 8
3 EFFICIENCY ANALYSIS 10
Introduction 10
Partial Enumeration Techniques 12
Total Enumeration Techniques 15
Modeling Network Problems as Digraphs 16
The Total Enumeration Procedure 21
Computational Considerations 30
Summary 3 8
4 EQUITY ANALYSIS 39
Introduction 39
Cost Allocation for Regional Water
Networks 4 0
Criteria for Selecting a Cost
Allocation Method 45
Ad Hoc Methods 4 8
Defining Identifiable Costs as Zero... 49
Defining Identifiable Costs as
Direct Costs 54
v

Cooperative Game Theory 64
Concepts of Cooperative Game Theory... 65
Unique Solution Concepts 7 5
Empty Core Solution Concepts 87
Cost Allocation in the Water Resources
Field 88
Separable Costs, Remaining Benefits
Method 9 0
Minimum Costs, Remaining Savings
Method 9 5
Allocating Cost Using Game Theory Concepts. 99
The k Best System 99
The Dummy Player 108
Comparing Methods 115
Summary 119
5 EFFICIENCY/EQUITY ANALYSIS 120
Introduction 120
Maximum Cost 122
Minimum Cost 129
Fairness Criteria 132
Summary 13 3
6 CONCLUSIONS AND RECOMMENDATIONS 135
APPENDIX EFFICIENCY/EQUITY ANALYSIS OF A THREE-
COUNTY REGIONAL WATER NETWORK WITH
NONLINEAR COST FUNCTION 142
REFERENCES 15 0
BIOGRAPHICAL SKETCH 159
vi

LIST OF TABLES
Table Page
3-1 Example of Total Enumeration Procedure for
3-Node Digraph 27
3-2 The Number of Independent Calculations to
Find the Costs of Spanning Directed Trees
for All Possible Subdigraphs 31
3-3 Summary of Computational Effort for Digraphs
Shown in Figure 3-4 34
3-4 Efficiency Analysis of a Three-User Water
Supply Network with Nonlinear Cost Function
Using Lotus 1-2-3 37
4-1 Projected Population Growth and Projected
Average Per Capita Demand 41
4-2 The Costs and Percent Savings for All
Options 44
4-3 Cost Allocation Matrix 50
4-4 Cost Allocation of Optimal Network Based on
Population 52
4-5 Cost Allocation of Optimal Network Based on
Demand 5 3
4-6 Cost Allocation of Optimal Network with Use
of Facilities Method 56
4-7 Cost Allocation for the Use of Facilities
Method 57
4-8 Cost Allocation of Optimal Network with
Direct Costing/Equal Apportionment of
Remaining Costs Method 60
4-9 Cost Allocation for Direct Costing/Equal
Apportionment of Remaining Costs Method 61
vii

4-10 Core Geometry for Three-Person Cost Game
Example 7 3
4-11 Cost Allocation for Three-County Example
Using the Shapley Value 80
4-12 Cost Allocation for Three-County Example
Using the Nucleolus 85
4-13 Empty Core Solution Methods 89
4-14 Cost Allocation for Three-County Example
Using the SCRB Method 94
4-15 Cost Allocation for Three-County Example
Using the MCRS Method 98
4-16 Nominal Versus Actual Core Bounds for
Optimal Network Game 100
4-17 Cost Allocations for the Optimal Network and
the Second Best Network ($) 103
4-18 Cost Allocation for Option 3 as a Two-Person
Game Using the SCRB Method 10 9
4-19 Comparing Cost Allocations for Option 3 as
Two-Person Game and Three-Person Game Using
the SCRB and MCRS Methods Ill
4-20 Core Bounds for Option 3 as a Three-Person
Game 112
4-21 Core Bounds for Option 3 as a Three-Person
Game with County 1 as a Dummy Player 114
4-22 Core Bounds for Option 3 as a Two-Person
Game 116
4-23 Comparison of Methods Discussed for
Allocating Costs of Water Resources Projects. 117
5-1 Using Independent Calculations from the
Total Enumeration Procedure to Find c(i),
c(S), and c(N) for the Three-County Regional
Water Network Problem 121
5-2 Efficiency/Equity Analysis of the Optimal
Network 12 6
viii

LIST OF FIGURES
Figure Page
3-1 Types of Cost Functions 13
3-2 Example Digraph Representing a Regional
Water Network Problem for Three Users 18
3-3 Flow Diagram of Total Enumeration Procedure
for n-Node Digraph 23
3-4 Examples of 3,4,5-Node Digraphs 33
4-1 Lengths of Interconnecting Pipelines 43
4-2 Geometry of Core Conditions for Three-Person
Cost Game Example 71
4-3 Core for the Optimal Network Game
(C (N ) = $4,556,409 ) 101
4-4 Core for the Second Best Network Game
( C (N) = $4,556,826 ) 102
4-5 Reduction in. Core as c(N) Increases from
c X(N) to c (N) 107
IX

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
EFFICIENCY/EQUITY ANALYSIS OF WATER RESOURCES
PROBLEMS—A GAME THEORETIC APPROACH
By
Elliot Kin Ng
August, 1985
Chairman: James P. Heaney
Major Department: Environmental Engineering Sciences
Successful regional water resources planning involves
an efficiency analysis to find the optimal system that maxi¬
mizes benefits minus costs, and an equity analysis to appor¬
tion project costs. Traditionally, these two problems have
been treated separately. This dissertation incorporates
efficiency analysis and equity analysis into a single
regional water resources planning model.
A reliable total enumeration procedure is used to find
the optimal system for regional water network problems.
This procedure is easy to understand and can be implemented
using readily available computer software. Furthermore, the
engineer can use realistic cost functions or perform detailed
cost analysis and, also, examine good suboptimal systems. In
addition, this procedure finds the optimal system for each
individual and each subgroup of individuals; hence, an equity
x

analysis can be accomplished using the theory of the core
from cooperative n-person game theory.
Game theory concepts are used to perform an equity
analysis on the optimal system as well as good suboptimal
systems. For any system, an equitable cost allocation exists
if a core exists. However, if a game is not properly
defined, even a cost allocation in the core may be
inequitable.
A rigorous procedure using core conditions and linear
programming is described to determine the core bounds. An
individual's lower core bound and upper core bound unambigu¬
ously measure the individual's minimum cost and maximum cost,
respectively. Traditional approaches for quantifying minimum
cost and maximum cost assume that either a regional system
involving the grand coalition is built or all the individuals
will go-it-alone. However, this rigorous procedure accounts
for the possibility that a relatively attractive system
involving subgroups may form. Furthermore, this rigorous
procedure gives a general quantitative definition of marginal
cost and opportunity cost. Once the minimum cost and maximum
cost for each individual are determined, a basis for
equitable cost allocation is available.
Finally, efficiency analysis and equity analysis are
not separable problems but are related by the economics of
all the opportunities available to all individuals in a
project.
xi

CHAPTER 1
INTRODUCTION
In situations where multiple purposes and groups can
take advantage of economies of scale in production and/or
distribution costs, a regional water resources system is an
attractive alternative to separate systems for each purpose
and each group. However, a regional system imposes complex
economic, financial, legal, socio-political, and organiza¬
tional problems for the water resources professionals. This
dissertation examines two problems associated with regional
water resources planning that are typically treated
separately, yet are closely related.
The first problem involves performing an efficiency
analysis to determine the economically efficient or optimal
regional system that maximizes benefits minus costs. Once
the optimal regional system is determined, a major task
still remains to allocate project costs; therefore, an
equity analysis must be performed to apportion project costs
in an equitable manner. This second problem is viewed from
the perspective of each purpose and each group because they
must each be convinced that the optimal regional system is
their best alternative; otherwise, voluntary participation
will be difficult. No doubt, each purpose's and each
1

2
group's decision to participate in the optimal regional
system depends on its allocated cost, and not necessarily on
what is best for the region.
The prevailing belief is that efficiency analysis and
equity analysis are separate problems and, therefore,
research has either focused entirely on efficiency analysis
or equity analysis. Research on efficiency analysis has
mainly been on the application of partial enumeration tech¬
niques to find optimal regional systems, while research on
equity analysis has continued to explore the application of
concepts from cooperative game theory to allocate project
costs. The purpose of this dissertation is to integrate
efficiency analysis and equity analysis into a single
regional water resources planning model characterized by
economies of scale. The model to be presented incorporates
a total enumeration procedure along with concepts from
cooperative game theory for efficiency/equity analysis. The
specific application is to determine the least cost regional
water supply network and to determine a "fair" allocation of
costs among the multiple users.
Chapter 2 reviews selected works on efficiency analysis
and equity analysis of water resources problems. Chapter 3
presents a reliable total enumeration procedure for effi¬
ciency analysis of regional water supply network problems.
However, unlike traditional partial enumeration techniques
used for efficiency analysis that give only the optimal

3
solution, this procedure also gives all the optimal solu¬
tions for each user and each subgroup of users which are
necessary information to perform an equity analysis using
concepts from cooperative game theory. In addition, this
procedure gives all the suboptimal solutions. Chapter 4
shows how the information from the total enumeration
procedure is used to perform an equity analysis of not only
the optimal solution, but also "good" suboptimal solutions.
Chapter 5 reveals how efficiency analysis and equity
analysis are related. Finally, Chapter 6 summarizes the
results and conclusions.

CHAPTER 2
LITERATURE REVIEW
Efficiency Analysis
During the past two decades, the problem of finding the
economically efficient or optimal regional water system has
been extensively modeled as a mathematical optimization
problem. A review of selected works on efficiency analysis
of regional water systems that includes Converse (1972),
Graves et al. (1972), McConagha and Converse (1973), Yao
(1973), Joeres et al. (1974), Bishop et al. (1975), Jarvis
et al. (1978), Whitlatch and ReVelle (1976), Brill and
Nakamura (1978), and Phillips et al. (1982) indicates a
variety of partial enumeration techniques, e.g., nonlinear
programming, for finding optimal regional systems. These
optimal regional systems can be a least cost system or a
system that maximizes benefits minus costs. Generally,
regional water resources planning problems exhibit economies
of scale in cost and, therefore, involve nonlinear concave
cost functions. Consequently, to a great extent, the
selection of the partial enumeration optimization technique
to apply to a particular problem depends on the characteri¬
zation of the nonlinear concave cost functions. For
instance, linear programming can be applied if the nonlinear
4

concave cost functions are represented by linear
approximations.
5
Equity Analysis
Unfortunately, successful regional planning is not
merely knowing the optimal regional system but must also
include an equity analysis to find an acceptable allocation
of costs among the participants. Otherwise, the optimal
system will be difficult to implement. Of the publications
cited in the preceding paragraph, only McConagha and
Converse (1973) dealt with both efficiency and equity in
regional water planning. In addition to presenting a
heuristic procedure for finding the least cost regional
wastewater treatment facility for seven cities, they evalu¬
ated the equity of several cost allocation procedures.
Although they recognized that an equitable cost allocation
should not charge any city or subgroup of cities more than
the cost of an individual treatment facility, they did not
include the possibility of subgroup formation in their
analysis.
Giglio and Wrightington (1972) introduced concepts from
cooperative game theory as a way to consider the possibility
of subgroup formation in allocating costs of water
projects. However, their treatment of cooperative game
theory was incomplete. Therefore, they concluded that the
game theory approach rarely yields a unique cost allocation

6
and proceeded to recommend the separable costs, remaining
benefits (SCRB) method or methods based on measure of pollu¬
tion. Shortly thereafter, several researchers applied
popular unique solution concepts from game theory like the
Shapley value and the nucleolus to allocate the costs of
regional water systems. Heaney et al. (1975) applied the
Shapley value to find an equitable cost allocation of common
storage units for storm drainage for pollution control among
competing users. Suzuki and Nakayama (1976) applied the
nucleolus to assign costs for a water resources development
along Japan's Sakawa and Sagami Rivers. Loehman et
al. (1979) used a generalization of the Shapley value to
allocate the costs of a regional wastewater system involving
eight dischargers along the lower Meramec River near St.
Louis, Missouri.
Subsequently, Heaney (1979) established that the fair¬
ness criteria used for allocating costs in the water
resources field and the concepts used in cooperative game
theory are equivalent. Moreover, Straffin and Heaney (1981)
showed that a conventional method for allocating costs used
by water resources engineers is identical to a unique solu¬
tion concept used by game theorists. More recently, Young
et al. (1982) compared proportionality methods, game
theoretic methods, and the SCRB method for allocating cost
and concluded that the game theoretic methods may be too
complicated while the SCRB method may give inequitable cost

7
allocations. Meanwhile, Heaney and Dickinson (1982)
revealed why the SCRB method may fail to give equitable cost
allocations and proposed a modification of the SCRB method
that uses game theory concepts along with linear programming
to insure an equitable cost allocation can be found if one
exists.
The possibilities of using concepts from cooperative
game theory as a basis for allocating costs of water
projects continue to develop. In fact, concepts from coop¬
erative game theory are gaining acceptance in other fields
as well. Researchers in accounting are looking toward coop¬
erative game theory as a possible solution to the arguments
by Thomas (1969, 1974) that any cost allocation scheme in
accounting is arbitrary and hence not fully defensible.
Recent works by Jensen (1977), Hamlen et al. (1977, 1980),
Callen (1978), and Balachandran and Ramakrishnan (1981)
applied concepts from cooperative game theory to evaluate
the equity of existing and proposed cost allocation schemes
in accounting. Meanwhile, in economics, concepts from
cooperative game theory are frequently used as a basis for
evaluating subsidy-free and sustainable pricing policies for
decreasing cost industries, e.g., the work of Loehman and
Whinston (1971, 1974), Faulhaber (1975), Sorenson et
al. (1976, 1978), Zajac (1978), Panzar and Willig (1977),
Faulhaber and Levinson (1981), and Sharkey (1982b).

8
Conclusions
Three conclusions can be made from reviewing the
literature on efficiency analysis and equity analysis of
regional water resources planning. First, there is a gap in
the research to jointly examine efficiency and equity in
regional water resources planning. In spite of a continual
effort to find economically efficient regional water systems
and equitable cost allocation procedures, no published work
incorporates both efficiency analysis and equity analysis in
a single regional water resources planning model using
realistic cost functions. Heaney et al. (1975) and Suzuki
and Nakayama (1976) used linear cost models while Loehman et
al. (1979) used conventional cost curves. Secondly, the
cost allocation literature in the water resources field has
consistently allocated the costs of treatment and piping
together even though federal guidelines suggest that piping
cost be allocated separately from treatment cost to the
responsible users (Loehman et al., 1979; U.S. Environmental
Protection Agency, 1976). Finally, the cost allocation
literature has dealt with allocating the cost of the optimal
system. However, situations in practice may require that
"good" suboptimal systems be considered; therefore, an
acceptable cost allocation procedure should be able to
allocate the costs of several systems under consideration in
an equitable manner. These three conclusions formed the
basis for the research undertaken in this dissertation.

Chapter 3 begins integrating efficiency analysis and
equity analysis by searching for a computational procedure
9
to simultaneously perform an efficiency analysis and
calculate all the necessary information to perform an equity
analysis using concepts from cooperative game theory.

CHAPTER 3
EFFICIENCY ANALYSIS
Introduction
The importance of both efficiency analysis and equity
analysis in planning regional water resources systems is
well recognized. Over the years, researchers have applied
methods ranging from simple cost-benefit analysis to sophis¬
ticated mathematical programming techniques to search for
economically efficient or optimal regional water resources
systems. Yet, the implementation of regional systems is
difficult unless an equitable financial arrangement is found
to allocate project costs among individuals (or partici¬
pants) in a project. Until recently, a theoretically sound
basis for allocating costs has eluded the water resources
professional. However, there is increasing interest in
using the theory of the core from cooperative n-person game
theory as a basis for allocating costs, e.g., see Suzuki and
Nakayama (1976), Bogardi and Szidarovsky (1976), Loehman et
al. (1979), Heaney and Dickinson (1982), and Young et
al. (1982). The theory of the core is based on principles
of individual, subgroup, and group rationality. This means
that no individual or subgroup of individuals should be
allocated a cost in excess of the cost of nonparticipation,
10

11
while total cost must be apportioned among all individuals.
The cost of nonparticipation is simply the cost that each
individual and each subgroup of individuals must pay to
independently acquire the same level of service by the most
economically efficient means. As a result, to evaluate
efficiency/equity for a regional system with n individuals,
it is necessary to determine 2n-l optimal solutions.
Although the close association between efficiency
analysis and equity analysis is recognized, there have been
few attempts to incorporate these two analyses in regional
water resources planning. A typical efficiency analysis
usually ends with determining the optimal solution for a
problem without addressing cost allocation, and a typical
equity analysis begins by assuming the 2n-l optimal solu¬
tions are available to accomplish the cost allocation. This
disjointed approach to efficiency/equity analysis is
fostered by a belief that these two problems are independent
(James and Lee, 1971; Loughlin, 1977). Furthermore,
reliable techniques for finding the 2n-l optimal solutions
to accomplish an efficiency/equity analysis of most problems
encountered in actual practice are unavailable.
This chapter begins by evaluating the applicability of
partial and total enumeration techniques for finding the
2n-l optimal solutions for problems with different types of
cost functions. Subsequently, a computational procedure is
described to examine a regional water supply network problem

12
wherein we need to find the economic optimum and a "fair"
allocation of costs among the individuals in the project.
In order to do the cost allocation we need to find the
costs of the optimal systems for each individual and each
subgroup of individuals since these costs are going to be
the basis for cost allocation.
Partial Enumeration Techniques
The difficulty of finding the optimal solution for a
particular problem depends on the nature of the cost func¬
tions. Generally, a cost function can be classified as
either linear, convex, concave, S-shape, or irregular (see
Figure 3-1). To find the optimal solution for problems with
either linear or convex cost functions is straightforward
using readily available and reliable linear programming
codes. Accordingly, a vast body of overlapping theoretical
results is available from classical economics and operations
research, e.g., convex programming, for finding the optimal
solution to problems with convex cost functions. However,
problems with linear and convex cost functions are unable to
characterize the economies of scale in cost typically
encountered in regional water resources planning.
The concave cost function is generally used to
represent economies of scale, and several partial enumera¬
tion techniques are available for dealing with this cost

13
Figure 3-1. Types of Cost Functions.

14
function. One approach surveyed by Mandl (1981) is
separable programming which takes advantage of readily
available linear programming codes by using a piecewise
linear approximation of the concave cost function.
Unfortunately, this approach is rather tedious to use and
guarantees only a local optimal solution. A second approach
is to retain the natural concave cost function and apply a
general nonlinear programming code. However, according to
surveys by Waren and Lasdon (1979) and Hock and Schittkowski
(1983), general nonlinear programming codes may converge to
local optima and may be subject to other failures, e.g.,
termination of code. A final approach used by Joeres et
al. (1974) and Jarvis et al. (1978) is to approximate the
concave cost function with several fixed-charge cost
functions and apply a mixed-integer programming code. This
approach guarantees a globally optimal solution, but
standard mixed-integer programming codes are expensive to
use. More importantly, unresolved problems remain as to
how to properly define a fixed charge problem. If the fixed
charge formulation is used because it is computationally
expedient, then the resulting cost estimates may distort the
cost allocation procedure. Given the current status of
partial enumeration techniques for finding the optimal
solutions to perform efficiency/equity analysis for problems
with concave cost functions, one can conclude that other
methods must be used. Obviously, this conclusion applies

15
to problems with S-shape and irregular cost functions as
well.
Total Enumeration Techniques
Total enumeration techniques can be used to find the
optimal solution for a problem regardless of the types of
cost functions involved. The ability to handle irregular
cost functions is especially important because this type of
cost function is frequently used by state-of-the-art cost
estimating models like CAPDET, i.e., Computer Assisted
Procedure for Design and Evaluation of Wastewater Treatment
Systems (U. S. Army Corps of Engineers, 1978) and MAPS,
i.e., Methodology for Areawide Planning Studies (U. S. Army
Corps of Engineers, 1980). For example, in MAPS, the cost
function for constructing a force main is composed of
separate cost functions for pipes, excavation, appurten¬
ances, and terrain. Furthermore, each of these cost func¬
tions is based on site-specific conditions. For instance,
the cost function for pipe includes the cost of purchasing,
hauling, and laying the pipe and depends on the material,
diameter, length, and maximum pressure. No doubt, the
composite site-specific cost function for a force main may
be nonlinear, nonconvex, multimodel, and discontinuous.
Another advantage with a total enumeration technique is
that it presents and ranks all of the alternative solu¬
tions. Unlike partial enumeration techniques which only

16
present the optimal solution for consideration, total
enumeration techniques allow examination of suboptimal
solutions which may be preferable when factors other than
cost are considered. For example, proven engineering design
or socio-political values are difficult to incorporate into
an optimization model even if the problem is well defined,
so the optimal solution may be so unrealistic that another
solution must be selected.
Depending on the size of the problem, a possible
drawback with total enumeration techniques may be the compu¬
tational effort to enumerate all possible solutions.
However, for some problems, total enumeration may be the
only meaningful approach. For these problems, the challenge
with using a total enumeration approach is to find ways to
reduce the computational effort by applying mathematical
techniques or engineering considerations. After a discus¬
sion on modeling network problems as digraphs, a total
enumeration procedure that does not require extensive compu¬
tational effort to find the least cost network for each
individual and each group of individuals is presented.
Modeling Network Problems as Digraphs
Consider a situation wherein an existing water supply
source, S, is going to serve n users with demands of Q^,
. . . , Q , respectively. Assume that the water source is
able to supply the total demand by the n users without

17
facility expansion except for a new regional water network.
Furthermore, consider a particular system with three users
that can be served directly by the source, and engineering
considerations, e.g., gravity flow, have determined that it
is feasible to send water from user 1 to both user 2 and
user 3, and from user 2 to user 3. For this particular
system, assume the total cost function for constructing a
pipeline is rather simple. From Sample (1983), the total
cost function for constructing a pipeline is characterized
by economies of scale and can be expressed as a linear
function of distance and a nonlinear function of flow; or
C = aQbL (3-1)
where C = total cost of pipeline, dollars
Q = quantity of flow, mgd
L = length of pipeline, feet, and
a, b = parameters, 0 Given this situation, the objective of the regional water
authority is to determine the least cost water network for
each user and each group of users in order to perform
efficiency/equity analysis.
This problem can be modeled as a digraph or directed
graph (see Figure 3-2) consisting of nodes to represent the
source and users, and directed arcs to represent all

18
Figure 3-2.
Example Digraph Representing a Regional Water
Network Problem for Three Users.

19
possible interconnecting pipelines. If water can be sent in
either direction between two users, then the pipeline is
represented by two oppositely directed arcs. Consequently,
any regional water network problem can be modeled by a
digraph.
Before continuing, a few brief definitions and concepts
are necessary since the nomenclature used in the network
and graph theory literature is not standardized. A digraph
or directed graph, D(X,A), consists of a finite set of
nodes, X, and a finite set of directed arcs, A. A directed
arc is denoted by (i,j) where the direction of the arc
(shown by an arrow) is from node i to node j; node i is
called the initial node and node j is called the terminal
node. A subdigraph of D(X,A) has a set of nodes that is a
subset of X but contains all the arcs whose initial and
terminal nodes are both within this subset. A path from
node i to node j is simply a sequence of directed arcs from
node i to node j. An elementary path is a path that does
not use the same node more than once. A circuit is an
elementary path with the same initial and terminal node. A
directed tree or an arborescence is a digraph without a
circuit for which every node, except the node called the
root, has one arc directed into it while the root node has
no arc directed into it. A spanning directed tree of a
digraph is a directed tree that includes every node in the
digraph. If a cost, C(i,j) is associated with every arc

20
(i,j) of a digraph, then the cost of a directed tree is
defined as the sum of the costs of the arcs in the directed
tree. Finally, a minimum spanning directed tree of a
digraph is the spanning directed tree of the digraph with
the least cost. For the reader desiring more information
regarding networks and graphs, numerous texts are available,
e.g., Christofides (1975), Minieka (1978), and Robinson and
Foulds (1980 ) .
The problem of finding the least cost water network for
each user and each group of users is the same as finding the
minimum spanning directed tree rooted at node S for all
possible subdigraphs as well as the digraph shown in Figure
3-2. In general, not every digraph has a spanning directed
tree; however, for a realistic problem one can assume a
pipeline is available to serve all individuals participating
in a regional system. Thus, a spanning directed tree exists
for digraphs representing realistic regional water network
problems.
Although algorithms are found in Gabow (1977) and
Camerini et al. (1980a, 1980b) for finding the minimum
spanning directed tree or the K best spanning directed
trees, these algorithms assume a linear cost model in which
the cost on each arc is given prior to initiating the
algorithm. As a result, these algorithms are not applicable
to problems with nonlinear costs on each arc. That is,
the cost along each arc cannot be determined in advance

21
because the cost is a function of the quantity of flow along
the arc; yet, the quantity of flow along the arc is a
function of the path in which the arc belongs.
The Total Enumeration Procedure
The procedure for enumerating and calculating the costs
of all the spanning directed trees for all possible sub¬
digraphs as well as the digraph is based on recognizing that
a large number of spanning directed trees of a digraph can
be constructed from specific spanning directed trees of
subdigraphs. These specific spanning directed trees are
characterized by one arc emanating from the root node and
are referred to as "essential spanning directed trees." In
contrast, "inessential spanning directed trees" are charac¬
terized by more than one arc emanating from the root node.
The procedure sequentially calculates the costs of essential
spanning directed trees for subdigraphs with increasing
number of nodes, until the costs of essential spanning
directed trees are calculated for all possible subdigraphs
and for the digraph. Meanwhile, the cost of each inessen¬
tial spanning directed tree for all possible subdigraphs as
well as the digraph is calculated simply by summing the
costs of essential spanning directed trees of subdigraphs
that are associated with each arc emanating from the root
node of the inessential spanning directed tree. That is,
each arc emanating from the root node belongs to an

22
essential spanning directed tree of a subdigraph. By apply¬
ing this procedure the costs of all the spanning directed
trees can be systematically enumerated for all possible
subdigraphs as well as the costs of all the spanning
directed trees for the digraph. As a result, the least cost
network for each user and each group of users is found.
In the following discussion, "n-node" means the number
of nodes, not including the root node, is n; e.g., an i-node
digraph or subdigraph consists of i+1 nodes if the root node
is counted. The total enumeration procedure for the n-node
digraph is summarized by the flow diagram shown in Figure 3-3.
Step 1 begins the procedure for evaluating all subdi¬
graphs consisting of the root node and one other node, i.e.,
the 1-node subdigraphs.
Step 2 initializes a count of the number of combina¬
tions of i-node subdigraphs evaluated.
Step 3 generates all possible combinations of i-node
subdigraphs from the n-node digraph. The number of possible
combinations is (^). For example, the 3-node digraph shown
3
in Figure 3-2 has (or three possible 2-node subdigraphs,
i.e., subdigraphs consisting of the following sets of nodes
{S,1,2 }, ÍS,1,3), and (S,2,3>.
Step 4 selects one i-node subdigraph not previously
selected and enumerates all of its spanning directed trees.
A spanning directed tree may not exist in a case where a
path does not exist from the root node to every node in the

23
Figure 3-3. Flow Diagram of Total Enumeration Procedure
for n-Node Digraph

not every node in the i-node
i-node subdigraph, i.e.,
subdigraph has an arc directed into it.
Actually, only the essential spanning directed trees
need to be enumerated. The enumeration of inessential
spanning directed trees is simply done by finding all
possible combinations of i-node digraphs from the entire set
of essential spanning directed trees enumerated previously,
i.e., all essential spanning directed trees for all possible
subdigraphs of the i-node subdigraph. This process
substantially reduces the effort involved in enumerating all
the spanning directed trees for an i-node subdigraph because
a large number of spanning directed trees are inessential.
If the i-node subdigraph is unusually large and dense,
algorithms are available in Chen and Li (1973), Christofides
(1975), and Minieka (1978) for generating spanning directed
trees.
If necessary, a procedure in Chen (1976) can be used to
compute the number of spanning directed trees of an i-node
subdigraph or an n-node digraph. A directed tree matrix, M,
is defined for a digraph, where equals the number of
arcs directed into node i and iru ^ is equal to the negative
of the number of arcs in parallel from node i to node j.
The number of spanning directed trees rooted at node S for
the digraph defined by M is given by the determinant of the
minor submatrix resulting from deleting the Sth row and

25
column of M. Applying this procedure to the 3-node digraph
in Figure 3-2 gives the following directed tree matrix.
S
1
2
3
S 1 2 3
0 -1 -1 -1
0 1-1-1
0 0 2 -1
0 0 0 3 /
The determinant of the minor submatrix resulting from delet¬
ing the Sth row and column is six, so there are six spanning
directed trees rooted at node S for this digraph.
Step 5 calculates the cost of each spanning directed
tree enumerated in Step 4. The cost for each essential
spanning directed tree is calculated independently. How¬
ever, the cost for each inessential spanning directed tree
is simply calculated by summing the costs of essential
spanning directed trees of subdigraphs calculated previously
that are associated with the arcs emanating from the root
node. For inessential spanning directed trees the costs can
be calculated along with the enumeration process described
in Step 4.
Step 6 ranks all the spanning directed trees for the
i-node subdigraph according to cost. The minimum spanning
directed tree is the least cost network for the users
associated with the set of nodes for the i-node subdigraph.

26
Step 7 checks the counter to see if all possible
combinations of i-node subdigraphs have been evaluated. If
not, Step 8 advances the counter by one before returning to
Step 4 to evaluate another i-node subdigraph. If all of the
possible combinations of i-node subdigraphs have been
evaluated, the procedure goes to Step 9 and begins the
evaluation of subdigraphs with i+1 nodes.
Step 10 checks if the n-node digraph has been evalu¬
ated. If not, the procedure returns to Step 2 and proceeds
to evaluate the subdigraphs with i+1 nodes; otherwise, the
procedure terminates.
The total enumeration procedure is illustrated in Table
3-1 using the regional water network problem modeled by the
3-node digraph shown in Figure 3-2.
During the first iteration all combinations of
1-node subdigraphs are evaluated. For this simple case
3
three combinations, i.e., (^) = 3, are evaluated. Further¬
more, each combination has only one spanning directed tree,
and the one spanning directed tree is essential. As a
result, the cost of the spanning directed tree for each
combination must be calculated. Obviously each spanning
directed tree is the least cost network for the associated
user. During the second iteration, three combinations,
3
i.e., (2) =3, of 2-node subdigraphs are evaluated. In
this case, each combination has two spanning directed trees,
but the cost of only one spanning directed tree needs

27
Table 3-1. Example of Total Enumeration Procedure for
3-Node Digraph
Iteration i-Node
i Subdigraphs
Spanning Directed
Trees for i-Node
Subdigraph
Are Spanning
Directed Trees
Essential?
i = l
ÍS, 1}
Yes
(S, 2}
®-
— Yes
(S, 3}
®—
— Yes
i=2 {S,l,2}
(D-^®— Yes
{S, 1,3 }
No
Yes
{S , 2,3 }
No
Yes
No

28
Table 3.1. Continued.
Iteration i-Node
i Subdigraphs
Spanning Directed Are Spanning
Trees for i-Node Directed Trees
Subdigraph Essential?

29
to be calculated. The cost of the inessential spanning
directed tree is simply found by summing the costs of the
corresponding essential spanning directed trees calculated
during the first iteration. The minimum spanning directed
tree for each combination is the least cost network for the
associated group of users. Finally, for the third itera¬
tion, i.e., i=n, the 3-node digraph is being evaluated. This
3-node digraph has six spanning directed trees, and these
six spanning directed trees can be enumerated by inspection.
The four inessential spanning directed trees can be
enumerated by simply finding all possible combinations of
3-node digraphs from the essential spanning directed trees
generated during the first and second iterations. Thus,
only two independent calculations are necessary to find
the costs of the essential spanning directed trees. Mean¬
while, the cost of the four inessential spanning directed
trees is calculated simply by summing the costs of essential
spanning directed trees for subdigraphs previously calcu¬
lated during the first two iterations. For example, in
Table 3-1, the cost for the inessential spanning directed
tree consisting of the set of arcs {(S,3), (S,l), (1,2)} is
determined by summing the costs of the two essential
spanning directed trees consisting of the sets of arcs
{(S,3) } and {(S,l), (1,2)} associated with the two sub¬
digraphs consisting of the sets of nodes ÍS,3} and {S,l,2},
respectively. Therefore, eight independent calculations are

30
necessary to find the costs of the six spanning directed
trees for the digraph, and only two of the six spanning
directed trees are essential. In fact, the eight indepen¬
dent calculations enable us to find all 2n-l or seven
optimal solutions necessary to perform efficiency/
equity analysis. Table 3-2 shows that the number of
independent calculations necessary to find the cost of all
the spanning directed trees for all possible subdigraphs is
simply equal to the number of independent calculations to
find the cost of all the spanning directed trees for the
digraph less the number of essential spanning directed trees
for the digraph. Consequently, for our 3-node digraph, six
independent calculations are necessary to find the optimal
solution for each user and each subgroup of users. For the
balance of this chapter, the optimal solution for each user
and each subgroup of users will be referred to as the 2n-2
optimal solutions. Finally, all suboptimal solutions are
enumerated for all possible subdigraphs as well as for
the digraph.
Computational Considerations
Although the number of independent calculations neces¬
sary to find the costs of all the spanning directed trees
for all possible subdigraphs as well as the digraph is
uniquely determined by the configuration of the digraph, we
can get a sense of the computational effort by examining the

31
Table 3-2. The Number of Independent Calculations to Find
the Costs of Spanning Directed Trees for All
Possible Subdigraphs.
Independent
Calculation
Is Independent Calcu¬
lation Used to Find
the Costs of Spanning
Directed Trees for the
Digraph?
Is Independent Calcu¬
lation Used to Find
the Costs of Spanning
Directed Trees for All
Possible Subdigraphs?
©-*©
Yes
Yes
0—H0)
Yes
Yes
(D-KD
Yes
Yes
Yes
Yes
Yes
Yes
©-»©-»©
Yes
Yes
dX ©
Yes
No
Ql
©v © )
Yes
No
>(?/
Total Number
of Yes
8
6

32
three digraphs shown in Figure 3-4. For the 3-node digraph,
six independent calculations are necessary to find the costs
of the four spanning directed trees for the digraph, and
only one of the four spanning directed trees is essential.
More importantly, 12 calculations are necessary to find the
seven optimal solutions, but only 6 of the 12 calculations
(50%) are independent. Furthermore, only five independent
calculations are necessary to find the 2n-2 optimal solu¬
tions. For the 4-node digraph, 10 independent calculations
are necessary to find the cost of the eight spanning
directed trees for the digraph, and only one of the eight
spanning directed trees is essential. For this digraph, 33
calculations are necessary to find the 15 optimal solutions,
but only 10 of the 33 calculations (30%) are independent.
Moreover, only nine independent calculations are necessary
to find the 2n-2 optimal solutions. Finally, for the 5-node
digraph, 19 independent calculations are necessary to find
the costs of the 24 spanning directed trees for the digraph,
but only 2 of the 24 spanning directed trees are essential.
In this case, 109 calculations are necessary to find the 31
optimal solutions, but only 19 of the 109 calculations (17%)
are independent. From these 19 independent calculations,
only 17 are necessary to find the 2n-2 optimal solutions.
As we can see, summarized in Table 3-3, a large number of
the spanning directed trees of a digraph are inessential.

33
3
gure ^~4
• Ex
*mPles
°f 3
,4,5~Nocl
e o¿
9r¿iPhi

Table 3-
3. Summary
of Computational Effort
for Digraphs
Shown in Figure
3-4.
Digraph
2n-l
Optimal
Solutions
Number of
Spanning
Directed
Trees
Number of
Inessential
Spanning
Directed
Trees
Number of
Calculations
to Find
2n-l Optimal
Solutions
Number of
Independent
Calculations
to Find
2n-l Optimal
Solutions (%)
Number of
Independent
Calculations
to Find
2n-2 Optimal
Solutions
3-node
7
4
3
12
6 (50%)
5
4-node
15
8
7
33
10 (30%)
9
5-node
31
24
22
109
19 (17%)
17
u>
4^

35
Also, the percentage of independent calculations decreases
as the number of nodes for a digraph increases.
The 5-node digraph in Figure 3-4 shows that the actual
number of independent calculations necessary to determine
the 31 optimal solutions to perform efficiency/equity analy¬
sis of a regional water network problem involving five
users is rather small. In fact, a regional water network
serving five users may be considered a fairly large
network. As larger systems form, increases in transactions
costs because of multiple political jurisdictions, growing
administrative complexity, etc., may eventually offset
the gains from a regional system. In any case, real
regional water network problems probably involve fairly
small and sparse networks. That is, large networks can
usually be broken down into smaller networks for analysis
based on natural geographical and hydrological features,
political boundaries, etc. Also, in actual problems there
may not be that many choices for routing pipelines. Thus,
the number of independent calculations necessary to
calculate the 2n-l optimal solutions for a realistic
regional water network should not be unreasonable.
One of the advantages of using this total enumeration
procedure is that it can be accomplished on a personal
computer using readily available software. Thus, decision
makers involved with planning and negotiating a regional
water network can have easy access to information to aid the

36
decision-making process. For instance, the procedure can
be implemented using the extremely "user friendly" Lotus
1-2-3 spreadsheet software package. Lotus 1-2-3 has the
mathematical functions to handle calculations involving
nonlinear cost functions or involving detailed cost
analysis. A sample Lotus 1-2-3 printout is shown in Table
3-4 for a hypothetical water network problem modeled by the
3-node digraph shown in Figure 3-2. This printout should be
self-explanatory. The top portion of the printout contains
the data for the problem, and the bottom portion is the
calculations associated with the total enumeration pro¬
cedure. The sorting capabilities of Lotus 1-2-3 allow
automatic ranking of all the feasible solutions according to
cost. Moreover, the Lotus 1-2-3 electronic spreadsheet
automatically recalculates all values associated with a
formula whenever a new value is entered or an existing value
is changed. This automatically gives the total enumeration
procedure the capability for sensitivity analysis. For
example, the set of all feasible solutions ranked according
to cost can be evaluated as the economies of scale, as
represented by the value of b in equation (3-1), is varied
over a specific range of values. Thus, for a regional
network problem of realistic size, all the feasible
solutions can be enumerated using a spreadsheet software
package.

37
Table 3-4. Efficiency Analysis of a Three-User Water
Supply Network with Nonlinear Cost Function
Using Lotus 1-2-3.
Sara
Distance : L(i,j) is the distance in feet from i to j
L (S, 1) =
L(S,2) =
17320 L(S,3)=
26000 L(1,2)=
3325k L(1,3)=
13130 L(2,3)=
19673
15500
Demand : Q(i) is the demand in mgd for user i
Q(1)= 1 Q(2)=
Cost Function: a(Q~b)L a=
6 Q (3) =
38 b=
7
3.51
Calculations With
C(i..j)[x]= Cost of network [x] for
Total Enumeration
i..j ; C(i..j)= L
Procedure
east cost
C(1)[S1]= 646000 C(2)[S2]=2463343.
C(3) [S3]=2012935.
C(12)[SI,12]= 2984140.
C (12)[SI;S2]= 3109348.
C(12)=
2934140.
C(13)[SI,13]= 2618975.
C(13)[S1;S3]= 2658986.
C(13)=
2518975.
C(23) [S2,23]= 4061294.
C(23)[32;S3]= 4476835.
C(23)=
4061294.
C(123)[SI,12,23]= 4648439.
C (123)[SI,12;S3]= 4997126.
C(123) [SI,12,13]= 4640756.
C (123)[SI,13;S2]= 5032324.
C (123) [SI;S2,23]= 4737294.
C(123)[SI;S2;S3]= 5122335.
C(123)=
4640756.
Sort C(123) in ascending order
Paths Cost
C(123)[SI,12,13]= 4643755.
C (123) [SI,12,23]= 4548439.
C(123)[SI;S2,23]= 4707294.
C (123)[SI,12;S3]= 4997126.
C(123)[S1,13;S2]= 5082824.
C(123)[S1;S2;S3]= 5122835.
BEST
C(123)=
4540756

38
Summary
A total enumeration procedure for finding the optimal
solutions necessary for efficiency/equity analysis of
realistic regional water network problems is presented. The
procedure can be easily understood and applied by engineers
with little knowledge or experience in operations research
techniques. Furthermore, the procedure allows the engineers
to handle all problems regardless of the types of cost
function involved or to perform detailed cost analysis.
Finally, if the optimal solution is impractical for
implementation, all suboptimal solutions ranked according to
cost are readily available for consideration.

CHAPTER 4
EQUITY ANALYSIS
Introduction
Proposed regional water resources systems involve
multiple purposes and groups who must somehow share the cost
of the entire project. The project may focus on construc¬
tion of a large dam which serves numerous purposes such as
water supply, flood control, and recreation. Also, canals
from the dam direct the water to nearby users. A signifi¬
cant portion of the total cost of this project may involve
elements which serve more than one purpose and/or group.
These costs are referred to as joint or common costs. In
such cases, it is possible to find the optimal or the most
economically efficient regional system, i.e., the one that
maximizes benefits minus costs. However, a major effort
remains to somehow apportion the project cost in an
equitable manner. In fact, the importance of the financial
analysis to apportion project cost is not limited to the
optimal system but includes any other integrated systems
being considered for implementation as well.
This chapter examines principles of cost allocation
using concepts from cooperative n-person game theory. An
39

40
example regional water network is used to illustrate these
principles.
Cost Allocation for Regional Water Networks
A hypothetical situation similar to options contained
in the West Coast Regional Water Supply Authority's master
plan for Hillsborough, Pasco, and Pinellas counties in
Florida (Ross et al., 1978) is now considered. Phase I
(1980-1985) of the plan recommends the use of groundwater
from existing and newly developed well fields to satisfy
water demands in the tri-county area. For this hypothetical
problem, assume that an existing well field is the most high
quality and cost effective water supply source (S) available
for three counties (1, 2, and 3) with projected demands of
1, 6, and 3 million gallons per day (mgd), respectively.
The demand for each county is based on projected population
growth and average per capita demand over a period of 5
years (see Table 4-1). Assume that the existing well field
is currently operating below its capacity of 20 mgd and can
satisfy the additional 10 mgd demanded by the three
counties. In addition, assume that no facility expansion is
required except for a new regional water network. Further¬
more, each county can be served directly by the well field,
and engineering considerations, e.g., gravity flow, have
determined that water can be sent from county 1 to both
county 2 and county 3, and from county 2 to county 3. The

41
Table 4-1. Projected Population Growth and Projected
Average Per Capita Demand.
County
Projected
Population Growth
Projected Average
Per Capita Demand
(gal/cap-day)
Projected
Additional
Demand
(mgd)
1
8,000
125
1
2
40,000
150
6
3
18,750
160
3
Total
66,750
—
10
Weighted
Average
...
150
..

42
lengths of all possible interconnecting pipelines are shown
in Figure 4-1. For our hypothetical problem, assume that
the total cost of constructing a pipeline has strong
economies of scale and is C = 38Q'^L, where C is total cost
of pipeline in dollars, Q is quantity of flow in mgd, and L
is the length of pipeline in feet.
Given the problem just described, the cost of a pipe¬
line serving county 1 alone is $646,000; the cost of a
pipeline serving county 2 alone is $2,420,095; and the cost
of a pipeline serving county 3 alone is $1,990,992. The
total cost for three individual pipelines is $5,057,087.
However, when the costs for all the options available to
these three counties are enumerated using the procedure
outlined in the preceding chapter, we see that the counties
can do better by cooperating (see calculations in Appendix A
using Lotus 1-2-3). There may be a slight difference
between the numbers used in the text and the numbers in
Appendix A because of rounding off. Also, cost data are
only significant to the nearest thousand dollars.
If the three counties cooperate, they can construct the
least cost or optimal network consisting of pipelines from
the well field to county 1, from county 1 to county 2, and
from county 2 to county 3 (see Table 4-2). This optimal
network costs $4,556,409 and represents a savings of 9.9% or
$500,678 when compared with the cost for three individual
pipelines. Obviously, constructing the optimal network is

3 tt<3d
-1 â– 
1>e
.rvQ
at-
c°
ec
.*<3
?VP€
f i-Q
Xii(

44
Table 4-2. The Costs and Percent Savings for All Options.
Option
(Rank)
Cost ($)
Savings (%)
1
4,556,409
9.90
2
4,556,826
9.89
3
4,630,177
8.44
4
4,919,503
2.72
5
5,006,734
1. 00
6
5,057,087
0

45
in the best interest of the three counties, but to implement
this least cost network, an equitable way to allocate the
cost among the three counties must be found. This financial
problem is known as a cost allocation problem. The complex¬
ity is introduced because the counties share common pipes.
Criteria for Selecting a Cost Allocation Method
Several sets of criteria for selecting a cost alloca¬
tion method are found in the literature. For the water
resources field, criteria for allocating costs date back to
the Tennessee Valley Authority (TVA) project in 1935 when
prominent authorities were brought together to address the
cost allocation problem. They developed the following set
of criteria for allocating costs (Ransmeier, 1942,
pp. 220-221) :
1. The method should have a reasonable logical
basis. It should not result in charging any
objective with a greater investment than the fair
capitalized value of the annual benefit of this
objective to the consumer. It should not result
in charging any objective with a greater invest¬
ment than would suffice for its development at an
alternate single purpose site. Finally, it should
not charge any two or more objectives with a
greater investment than would suffice for
alternate dual purpose or multiple purpose
improvement.
2. The method should not be unduly complex.
3. The method should be workable.
4. The method should be flexible.
5. The method should apportion to all purposes
present at a multiple purpose enterprise a share
in the overall economy of the operation.

46
This set of criteria developed for the water resources
field is similar to the following set of criteria proposed
by Claus and Kleitman (1973) for allocating the cost of a
network:
1. The method must be easy to use and under¬
standable to users. They must be able to predict
the effects of changes in their service demands.
2. The method must have stability against system
breakup. It should not be an advantage to one or
more users to secede from the system. Thus, there
are limits to which a method can subsidize one
user or class of user at the expense of others.
3. It is desirable, though not necessary, that the
costing be stable under evolutionary changes in
the system or under mergers of users.
4. It is again desirable that the method should
preserve the substance and appearance of non¬
discrimination among users.
5. If the method represents a change from present
usage it is desirable that transition to the
new method be easy.
From these two sets of criteria, the most important
criterion for selecting a method to allocate the cost of a
regional water network is the method's ability to ensure
stability or prevent breakup of the network. That is, the
method should not allocate cost in a manner whereby an
individual or a subgroup of individuals can acquire the same
level of service by a less expensive alternative. Other¬
wise, the individual or subgroup of individuals will con¬
sider their allocated cost inequitable or unfair and secede
from the regional network for a less expensive alternative.

Heaney (1979) has expressed these fairness criteria for an
equitable cost allocation mathematically as follows:
1) x(i) < minimum [b(i), c(i)] VieN
(4-1)
where
x (i)
cost allocated to individual i
b (i )
benefit of individual i
c(i) = the alternative cost to individual i
of independent action, and
N
set of all individuals; i.e.,
N = {1,2 , . . . ,n }.
r • • • r
This criterion simply means that individual i should not be
charged a cost greater than the minimum of individual i's
benefit and alternative cost for independent action.
2) Z x(i) _< minimum [b(S), c(S)] V Scn
i eS
(4-2)
where c(S) = alternative cost to subgroup S of
independent action, and
b(S) = benefit of subgroup S.
This second criterion extends the first criterion to include
subgroup of individuals as well. These two fairness
criteria are now used to evaluate some simple and seemingly
fair cost allocation schemes for our regional water network
problem. Throughout this chapter, we will assume for our
regional water network problem that each county's and each

48
subgroup of counties1 alternative cost of independent action
is less than or equal to each county's and each subgroup of
counties' benefits, respectively; i.e.,
c(i) = minimum [b(i), c(i)] V ieN, and (4-3)
c(S) = minimum [b(S), c(S)] V ScN.
Ad Hoc Methods
Over the years, many ad hoc methods have been proposed
or used to apportion the costs of water resources projects
(Goodman, 1984). In general, ad hoc methods used in the
water resources field for allocating costs can be described
as follows: allocate certain costs that are considered
identifiable to an individual directly and prorate the
remaining costs, i.e., total project cost less the sum of
all identifiable costs, among all the individuals in the
project by some physical or nonphysical criterion. Mathe¬
matically, this can be expressed as follows:
x ( i ) = x(i)id + iMi)*rc
(4-4)
where
x (i)
x (i)
id
ip (i)
rc
cost allocated to individual i,
costs identifiable to
individual i,
prorating factor for individual i, and
remaining costs, i.e.,
c(N)

49
Furthermore, the requirement that Z (i) = 1.0 should be
ieN
obvious.
James and Lee (1971) summarize 18 ways for allocating
the costs of water projects depending on the definition of
identifiable costs and the basis for prorating the remaining
costs (see Table 4-3). Basically, the differences among
these 18 methods are the following three ways of defining
identifiable costs: 1) zero, 2) direct or assignable costs,
or 3) separable costs; and the following six ways of
prorating remaining costs: 1) equal, 2) unit of use,
3) priority of use, 4) net benefit, 5) alternative cost, or
6) the smaller of net benefit or alternative cost. The next
two sections analyze the effects of defining identifiable
costs as either zero or direct costs. A detailed treatment
of separable costs, i.e., the difference between total
project costs with and without an individual, is given in
the section on the separable costs, remaining benefits
method.
Defining Identifiable Costs as Zero
The simplest way to allocate costs is to define identi¬
fiable costs as equal to zero and prorate total project cost
by some physical or nonphysical criterion. For example,
population and demand are two ways to prorate total project

50
Table 4-3. Cost Allocation Matrix.
Definition of
Identifiable
Cost
Basis for Prorating
Remaining Costs
A.
Zero
B.
Direct
Cost
C.
Separable
Cost
a.
Equal
Aa
Ba
Ca
b.
Unit of Use
Ab
Bb
Cb
c.
Priority of Use
Ac
Be
Cc
d.
Net Benefit
Ad
Bd
Cd
e.
Alternative Cost
Ae
Be
Ce
f.
Smaller of d.
or e.
Af
Bf
Cf
Source:
Modified from James and Lee, 1971, p. 533.

51
cost (Young et al., 1982). Using these two ways to prorate
the cost of the optimal network for our regional water
network problem gives the following cost allocations (see
calculations in Table 4-4 and Table 4-5):
Proportional to Population
County 1 $ 546,769
County 2 2,733,845
County 3 1,275,795
$4,556,409
Proportional to Demand
County 1 $ 455,641
County 2 2,733,845
County 3 1,366,923
$4,556,409
Although these cost allocations are simple to calculate and
easy to understand, they fail to implement the optimal
network because county 2 considers these cost allocations
unfair. In contrast to counties 1 and 3, county 2 loses
money by being allocated a cost in excess of its go-it-alone
costs using either of these two methods. Consequently,
county 2 would rather acquire a pipeline by itself than
cooperate with counties 1 and 3 to construct the optimal
network. The principal failure with these proportionality

Table 4-4. Cost Allocation of Optimal Network Based on Population.
County i
Population
Percent of
Total Population
Allocated
Cost ($)
x (i)
Go-It-Alone
Cost ($)
c (i )
Is
x ( i ) < c ( i ) ?
1
8,000
12
546,769
646,000
Yes
2
40,000
60
2,733,845
2,420,095
No
3
18,750
28
1,275,795
1,990,992
Yes
Total
66,750
100
4,556,409
5,057,087
—
Ul
ro

Table 4-5. Cost Allocation of Optimal Network Based on Demand
County i
Demand
(mgd)
Percent of
Total Demand
Allocated
Cost ($)
x (i)
Go-It-Alone
Cost ($)
c (i )
Is
x(i) < c(i)?
1
1
10
455,641
646,000
Yes
2
6
60
2,733,845
2,420,095
No
3
3
30
1,366,923
1,990,992
Yes
Total
10
100
4,556,409
5,057,087
—
U1
u>

54
methods is that they do not recognize explicitly each
individual's contribution to total project cost.
Defining Identifiable Costs as Direct Costs
A way to recognize each individual's contribution to
total project cost is by defining identifiable costs as
those costs that can be directly assigned, and prorating the
remaining costs by some physical or nonphysical criterion
such as use or number of individuals; i.e.,
x (i) = x(i)direct + 4»(i)-re (4-5)
where x(i),. = direct cost or assignable cost
ireC to individual i.
Although this direct costing approach intuitively seems
fair, inequitable and unpredictable cost allocations can
result. To illustrate, two direct costing methods are
applied to our regional water network problem.
A common approach to allocating remaining costs is by
some physical measure of each individual's use of the common
facilities; this method is generally referred to as the use
of facilities method (Loughlin, 1977; Goodman, 1984). This
traditional method is easy to understand and apply because
quantitative information on a physical measure of use is
generally available. In the water resources field, use
can be measured in terms of the storage capacity and/or the

55
quantity of water flow provided by the common facilities.
For our regional water network problem, the flow to each
county is the obvious measure of use to apportion the costs
of common pipelines since the assumed cost function depends
on the flow. In the case of the optimal network, the only
direct cost is the cost of the pipeline from county 2 to
county 3 serving county 3, and the use of facilities method
gives the following cost allocation (see calculations in
Table 4-6).
$ 204,283
2,221,299
2,130,827
$4,556,409
County 1
County 2
County 3
Total
Unfortunately, this cost allocation does not implement the
optimal network because county 3 can do substantially better
by going alone, i.e., $1,990,992 versus paying $2,130,827.
In addition to giving an inequitable cost allocation
for the optimal network, the use of facilities method can
promote noncooperation if other networks are also being
considered. Table 4-7 shows the cost allocations for all
possible options available to the three counties using the
use of facilities method. Suppose the "second best" network
or option 2 is also being considered by the counties. The
second best network consists of the pipelines from the well
field to county 1, from county 1 to county 2, and from
county 1 to county 3. This second best network costs

Table 4-6. Cost Allocation of Optimal Network with Use of Facilities Method
Pipeline
S-l
1-2
2-3
Total
Cost ($)
Go-It-Alone
Cost ($)
c (i )
Length
(f t)
17,000
13,100
15,500
—
—
Q
(mgd)
10
9
3
—
—
Pipeline
Cost
($)
2,042,832
1,493,400
1,020,177
4,556,409
—
Cost for
County 1
($)
Q=1 mgd
204,283
0
0
204,283
646,000
Cost for
County 2
($)
Q=6 mgd
1,225,699
995,600
0
2,221,299
2,420,095
Cost for
County 3
($)
Q=3 mgd
612,850
497,800
1,020,177
2,130,827
1,990,992
Ln

57
Table 4-7. Cost Allocation for the Use of Facilities Method.
Cost Allocation to
($)
County i
Zx (i)
Is Cost
Allocation
Option
(Rank)
County 1
x(l)
County 2
x (2 )
County 3
x (3 )
(?)
Equitable?
1
204,283
2,221,299
2,130,827
4,556,409
No
x(3)>c(3 )
2
204,283
2,445,055
1,907,488
4,556,826
No
x(2)>c(2 )
3
646,000
1,976,000
2,008,177
4,630,177
No
x(3)>c(3 )
4
244,165
2,684,346
1,990,992
4,919,503
No
x(2)>c(2)
5
323,000
2,420,095
2,263,639
5,006,734
No
x(3)>c(3)
6
646,000
2,420,095
1,990,992
5,057,087
—
(4)
(5)
(6)

58
$4,556,826 or $417 more than the optimal network; so, both
networks are essentially comparable in cost, and either
network might be considered the least cost network. In
fact, the second best network becomes the optimal network if
the economies of scale or the value of b in the cost
function is .51 instead of .50 (see Table 3-4). Never¬
theless, applying the use of facilities method to this
second best network gives the following cost allocation.
$ 204,283
2,445,055
1,907,4 88
$4,556,826
County 1
County 2
County 3
In this case, the cost allocation fails to implement
the second best network because county 2 is better off going
alone, i.e, paying $2,420,095 rather than $2,445,055.
Furthermore, if we examine the cost allocation for the
optimal network and the second best network, another problem
is evident. Although the costs for the two networks are
$417 apart, the difference in costs between the two networks
for county 2 and county 3 is enormous. Consequently, this
cost allocation method imposes another obstacle for the
counties to cooperate and implement either one of the two
networks. County 2 strongly opposes the second best network
because of its substantially higher cost while county 3
strongly opposes the optimal network for the same reason.
This problem is even more serious when more options are
considered by the counties. Table 4-7 indicates tremendous

59
differences in allocated cost for each county depending on
the network, thereby making cooperation very difficult.
This situation shows the danger for individuals to simply
accept the least cost network without carefully examining
all of their options if the use of facilities method for
allocating costs is chosen.
Another simple way to prorate the remaining costs is to
divide it equally among the individuals associated with the
common facilities (see calculations for optimal network in
Table 4-8). Table 4-9 shows the cost allocations using this
egalitarian approach and indicates that none of the cost
allocations for options with savings are equitable. At
first glance, the cost allocation for option 5 appears
equitable because each county is charged a cost less than
or equal to its go-it-alone cost. However, closer examina¬
tion reveals that counties 1 and 2 can do better as a
coalition. They can construct a pipeline from the well
field to county 1 and from county 1 to county 2, i.e.,
option 4, for $2,928,511 rather than pay the sum of their
costs for option 5, i.e., $3,066,095. Unfortunately, a
transition from option 5 to option 4 causes county 1 to lose
money, i.e., $854,577 for option 4 versus $646,000 for
option 5. To further complicate matters, option 5 only
gives a 1% savings and requires county 1 to cooperate with
county 3 to build a pipeline without getting any savings.

Table 4-8. Cost Allocation of Optimal Network with Direct Costing/Equal
Apportionment of Remaining Costs Method.
Pipeline
S-l
1-2
2-3
Total
Cost
($)
Go-It-Alone
Cost($)
c (i )
Length
(ft)
17,000
13,100
15,500
—
—
Q
(mgd)
10
9
3
—
—
Pipeline
Cost
($)
2,042,832
1,493,400
1,020,177
4,556,409
—
Cost for
County 1
($)
Q=1 mgd
680,944
0
0
680,944
646,000
Cost for
County 2
($)
Q=6 mgd
680,944
746,700
0
1,427,644
2,420,095
Cost for
County 3
($)
Q=3 mgd
680,944
746,700
1,020,177
2,447,821
1,990,992
o

Table 4-9. Cost Allocation for Direct Costing/Equal
Apportionment of Remaining Costs Method
61
Cost Allocation to County i
Option
(Rank)
County 1
x (1)
($)
County 2
x (2 )
County 3
x (3)
f x ( i )
($)
Is Cost
Allocation
Equitable?
1
680,944
1,427,644
2,447,821
4,556,409
No
X(1)>c(1)
x(3)>c(3)
2
680,944
1,900,300
1,975,582
4,556,826
No
x(l)>c(l)
3
646,000
1,482,000
2,502,177
4,630,177
No
x(3)>c(3 )
4
854,577
2,073,934
1,990,992
4,919,503
No
X(1)>c(1)
5
646,000
2,420,095
1,940,639
5,006,734
No
x(1)+x(2)>
c (12 )
6
646,000
2,420,095
1,990,992
5,057,087
(4)
(5)
(6)

62
Given these observations, the stability of option 5 as a
regional water network is at best questionable. Again, if
the allocated costs for counties 2 and 3 for the optimal
network are compared to the second best network, a similar
situation like the one discussed for the use of facilities
method exists. That is, counties 2 and 3 face substantially
different costs for these two networks with comparable
costs.
Thus, assigning direct costs does not help eliminate
inequitable cost allocations. In fact, direct costing
methods can impose additional obstacles to cooperation.
This occurs because the assignment of direct costs depends
on the configuration of the facilities. For instance, the
cost of the pipeline from county 2 to county 3 for our
regional water network problem can be a direct cost or a
joint cost depending on the network. The cost of the pipe¬
line is a direct cost for county 3 if the second best
network, i.e., option 2, is being considered; yet, the cost
of the pipeline is a joint cost for counties 2 and 3 if the
optimal network, i.e., option 1, is being considered. These
changes in the cost classification for the pipeline from
county 2 to county 3 contribute to the tremendous difference
in the cost allocations for counties 2 and 3 for the two
comparable cost networks. This situation indicates an
additional criterion not addressed by Claus and Kleitman
(1973) for selecting a procedure to allocate network cost.

63
The cost allocation procedure should be independent of
network configuration; otherwise, the cost allocation pro¬
cedure can promote noncooperation if more than one network
is being considered.
In summary, two approaches for allocating costs in the
water resources field have been examined: 1) allocate total
project cost in proportion to a physical or nonphysical
criterion; or 2) allocate assignable costs directly and
prorate the remaining costs by a physical or nonphysical
criterion. In general, these two approaches are simple to
apply and easy to understand. In fact, these two approaches
are currently accepted cost allocation methods used in
accounting (Kaplan, 1982). However, these two approaches
are unable to consistently provide an equitable cost
allocation when an equitable cost allocation exists, i.e.,
sometimes these methods work and sometimes they fail.
Furthermore, methods attempting to assign costs directly may
be influenced by the configuration of the facilities and may
discourage cooperation when more than one configuration is
being considered. This is particularly evident for our
regional water network problem. For a theoretically sound
method that is able to find an equitable cost allocation if
one exists and is not influenced by the configuration of the
facilities, concepts from cooperative n-person game theory
are necessary.

64
Cooperative Game Theory
Game theory has been with us since 1944 when the first
edition of The Theory of Games and Economic Behavior by John
Von Neumann and Oskar Morgenstern appeared. In particular
we are interested in games wherein all of the players
voluntarily agree to cooperate because it is mutually bene¬
ficial. Furthermore, games are studied in three forms or
levels of abstraction. The extensive form requires a com¬
plete description of the rules of a game and is generally
characterized by a game tree to describe every player's
move. A game in normal form condenses the description of a
game into sets of strategies for each player and is
represented by a game matrix. However, most efforts in
cooperative game theory have been with games in charac¬
teristic function form whereby the description of a game is
in terms of payoffs rather than rules or strategies. The
characteristic function form appears to be the most
appropriate for studying coalition formation which is an
essential feature in cooperative games. Also, cooperative
games can be of three types depending on whether the game is
defined in terms of costs, savings, or values. To keep the
notation as simple as possible, only cost games will be
discussed. Introductory and intermediate material on coop¬
erative game theory can be found in Schotter and Schwodiauer
(1980), Jones (1980), Luce and Raiffa (1957), Lucas (1981),
Rapoport (1970), Shubik (1982), and Owen (1982).

65
Concepts of Cooperative Game Theory
Let N = {1,2,...,n} be the set of players in the game.
Associated with each subset of S players in N is a charac¬
teristic function c, which assigns a real number c(S) to
each nonempty subset of S players. For cost games, the
characteristic function, c(S), can be defined as the least
cost or optimal solution for the S-member coalition if the
N-S member or complementary coalition is not present.
However, depending on how the problem is defined, alterna¬
tive definitions for c(S) may be required. For example,
Sorenson (1972) presents the following four alternative
definitions for the characteristic cost function:
c.(S) = value to coalition if S is given preference
over N-S.
C2(S) = value of coalition to S if N-S is not
present,
c^iS) = value of coalition in a strictly competitive
game between coalition S and N-S, and
c^(S) = value of coalition to S if N-S is given
preference.
If c(S) can be defined as the least cost solution for
coalition S if N-S is not present, then the cost game is
naturally subadditive; i.e.,
c(S) + c(T) > c(SUT) ShT = 0, S,TcN
(4-6)

66
where 0 is the empty set; and S and T are any two disjoint
subsets of N. Subadditivity is a natural consequence of
c(S) because the worst S and T can do as a coalition is the
cost of independent action; i.e.,
c(S) + c(T) = c(SUT) SOT = 0, S,TCN. (4-7)
A coalition in which the players realize no savings from
cooperation is said to be inessential.
General reasons why subadditivity exists are discussed
by Sharkey (1982a). The primary reason why subadditivity
exists for our regional water network problem is because of
the economies of scale in pipeline construction cost. For a
single output cost function, C(q), economies of scale is
defined by
C(Aq) < Ac(q) (4-8)
where q = output level, and for all A such that
1 < A £ 1 + e, e is a small positive number.
This definition means that the average costs are declining
in the neighborhood of the output q. From Sharkey (1982a),
economies of scale is sufficient but not a necessary condi¬
tion for subadditivity. Subadditivity is a more general

67
condition which allows for both increasing marginal cost
and increasing average cost over some range of outputs.
Solution concepts for cooperative cost games involve
the following three general axioms of fairness (Heaney and
Dickinson, 1982; Young et al., 1982);
1) Individual Rationality: Player i should not pay
more than his go-it-alone cost, i.e.,
x(i) _< c(i), ¥ ieN, (4-9)
where x(i) is the allocated cost or the charge to player i.
2) Group Rationality: The total cost of the grand
coalition, c(N), must be apportioned among the N players;
i.e.,
Ex(i)=c(N). (4-10)
ieN
3) Subgroup Rationality: This final axiom extends the
notion of individual rationality to include subgroups, i.e.,
no subgroup or subcoalition S should be apportioned a cost
greater than its go-it-alone cost, or
1 x(i) < c(S), V SCN. (4-11)
ie S
The set of solutions or charges satisfying the first two
axioms is called the set of imputations, while the

68
additional restriction of the third axiom defines what is
known as the core of the game. For subadditive cost games
the set of imputations is not empty, but the core may be
empty. Shapley (1971) has shown that the core always exists
for convex games. A cost game is convex if
c(S) + c(T) > c(SUT) + c(ShT) SOT i 0, V S,TcN (4-12)
or equivalently, convexity can be written as
c(SUi) - c(S) > c(TUi) - c(T) ScTcN - {i}, ieN. (4-13)
Convexity simply means the incremental cost for player i to
join coalition T is less than or equal to the incremental
cost for player i to join a subset of T. This notion of
convexity is analogous to economies of scale and implies the
game has a particular form of increasing returns to scale in
coalition size. As will be shown, the more attractive the
game, i.e., larger savings in project costs, the greater the
chance that the game is convex; whereas, if the game is less
attractive, i.e., lower savings in project costs, the poten¬
tial for a nonconvex game or an empty core game is greater.
To illustrate the concept of the core, assume a three-
person cost game with the following characteristic function
values:

69
c(l) =
35
c (2 )
= 45
c ( 3 ) =
50
c(12 ) =
66
c (13 )
= 75
c(23 ) =
87
c(12 3 )
= 100
This game is subadditive so each player has an incentive to
cooperate; i.e.,
c(l) + c(2)
+
c ( 3 )
>
c(12 3 )
c(l)
+
c (23 )
>
c(123 )
c (2 )
+
c (13 )
>
c (123)
c ( 3 )
+
c (12 )
>
c(12 3 )
c (1)
+
c (2 )
>
c (12 )
c(l)
+
c (3 )
>
c (13 )
c (2 )
+
c (3 )
>
c ( 23 ) .
Furthermore, this game is convex; i.e.,
c (12 )
+
c (13 )
>
c(12 3 )
+
c (1)
c (12 )
+
c (23 )
>
c(12 3 )
+
c (2 )
c (13 )
+
c ( 23 )
>
c(12 3 )
+
c ( 3 ) .
Using the three general axioms of fairness, the core
conditions are as follows:
x(l) < 35
x ( 2) < 45
x ( 3 ) < 50
x(l) + x(2) < 66
x(l) + x(3) < 75
x(2) + x(3) < 87
x(l) + x(2) + x(3) = 100.

70
The first three conditions determine the upper bounds on
x(i), i = 1,2,3, while the last four conditions determine
bounds
on x(i),
i = 1 i
-2,
r 3 , Í
. e.
â–º t
c (123 )
- c(23)
= 13
<
x(l)
<
35
= c(l)
c(123)
- c(13 )
= 25
<
x (2 )
<
45
= c ( 2 )
c(12 3 )
- c(12 )
= 34
<
x ( 3 )
<
50
= c ( 3 )
For a three-person game, graphical examination of the
core conditions and the nature of the charge vectors is
possible using isometric graph paper (Heaney and Dickinson,
1982). As shown on Figure 4-2, each player is assigned a
charge axis. The plane of triangle ABC, with vertices
(100,0,0), (0,100,0), and (0,0,100), represents points
satisfying group rationality (axiom 2); whereas, the smaller
triangle abc represents the set of imputations satisfying
both individual rationality (axiom 1) and group rationality
(axiom 2). The vertices a, b, and c represent the charge
vectors: [35, 15, 50], [5, 45, 50], and [35, 45, 20],
respectively. Line ab represents the upper bound for player
3, i.e., x(3) = c(3), where c(123) - c(3) is allocated
between players 1 and 2. As we move along line ab from
point a to point b, the allocation to player 1 decreases
from c(l) to c(123) - c(2) - c(3), i.e., from 35 to 5, while
the allocation to player 2 increases from c(123) - c(l) -
c(3) to c(2), i.e., from 15 to 45. Similar explanations can
be given for lines be and ac. A more restrictive set of
solutions satisfying subgroup rationality (axiom 3),

71
x( 3)
Figure 4-2.
Geometry of Core Conditions for Three-
Person Cost Game Example.

72
the shaded area on triangle abc, is the core for this game.
The geometry of the core for this convex game is a hexagon.
Line de represents the lower bound for player 2 or the set
of charges where c(13) is allocated between player 1 and
player 3 with the remainder, c(123) - c(13), going to player
2. Similar explanations can be given for lines fg and hi
which are the lower bounds for players 1 and 3,
respectively; and for lines id, gh, and ef which are the
upper bounds for players 1, 2, and 3, respectively.
If an allocation lies outside the core, an inequitable
situation prevails. For instance, point Z in Figure 4-2
allocates player 2 a cost less than its lower bound,
c(123)-c(13), which means c(13) increases or the cost
allocated to players 1 and 3 increases. Clearly, player 1
and player 3 can do better by forming their own two-person
coalition rather than subsidizing player 2.
As mentioned earlier, the convexity of a game and its
attractiveness are related. This relationship is illustrated
in Table 4-10. When the costs for the two-person coalitions
progressively decrease, there is less incentive for forming
the grand coalition so the core becomes progressively smaller
and the game becomes progressively more nonconvex. As a
consequence of the core conditions for a three-person sub¬
additive cost game, a condition can be derived to determine
if a core exists. From subgroup rationality and group
rationality, we have the following conditions:

73
Table 4-10. Core Geometry for Three-Person Cost Game
Example.
Characteristic Function
c(1) = 35, c(2) = 45,
c(3) = 50, c(123) = 100 Geometry
Zc(ij) of Core
c (12) c(13 ) c(23 )
66
75
87
228
< }
Hexagon
61
73
86
220
A>
Pentagon
59
71
85
215
ZA
Trapezoid
58
70
80
208
A
Triangle
56
68
76
200
•
Point
55
65
72
192
x (2 )
x(1) //
-\rx<3)
//y Empty
Source: Modified from Fischer and Gately, 1975, p. 27a.

7 4
x(l) + x(2) < c(12)
x(l) + x(3) < c(13)
x ( 2 ) + x ( 3 ) < c ( 2 3 )
x(l) + x(2) + x(3) = c(12 3)
(4-14)
Summing the subgroup rationality conditions gives
2•[x(1) + x(2) + x ( 3 ) ] < c(12) + c(13) + c(23). (4-15)
If the group rationality conditions are substituted into the
above equation, then we have the following condition to
determine if a core exists:
2 * c(12 3) < c(12 ) + c(13 ) + c(2 3 ) .
(4-16)
Therefore, in Table 4-10, the core exists as long as the sum
of the two-person coalitions is greater than 200 or twice
the value of the grand coalition. When the sum of the
two-person coalitions equals 200, the core reduces to a
unique vector, i.e., X = [24, 32, 44]. Finally, when the
sum of the two-person coalition is less than 200, then the
core is empty. Unfortunately, for larger games there is no
simple condition for checking the existence of a core;
however, as we will see later, a check can be made using
linear programming.

75
Unique Solution Concepts
The three axioms of fairness defining the core of the
game significantly reduce the set of admissible solutions.
Unless the core is empty or is a unique vector, an infinite
number of possible equitable charge vectors remain to be
considered, so additional criteria are needed to select a
unique charge vector. Numerous methods are available for
selecting a unique charge vector; but the two most popular
methods discussed in the literature are the Shapley value
(Shapley, 1953; Heaney, 1983b; Shubik, 1962; Heaney et al.,
1975; Littlechild, 1970) and the nucleolus (Schmeidler,
1969; Kohlberg, 1971; Suzuki and Nakayama, 1976).
Shapley value. The Shapley value for player i is
defined as the expected incremental cost for the coalition
of adding player i. Thus, each player pays a cost equal to
the incremental cost incurred by the coalition when that
player enters. Since the coalition formation sequence is
unknown, the Shapley value assumes an equal probability for
all sequences of coalition formation, i.e., the probability
of each player being the first to join is equal, as are the
probabilities of joining second, third, etc. For an n
person game there are n! orderings. The six sequences of
coalition formation for a three-person game are as follows:
(123) (213) (231)
(132)
(312)
(321)

7 6
Therefore, the Shapley value or the cost to player 1 for a
three-person game is
$(1) = 1/3 c(1) + 1/6 [c(12) - c(2)] + 1/6 [c(13 - c(3)]
+ 1/3 [c(12 3) - c(23)].
(4-17)
Player 1 has 1/3 probability of entering the coalition as
the first player and 1/3 probability of entering the
coalition as the last player. In addition, player 1 has 1/6
probability of entering the coalition after player 2 and 1/6
probability of entering the coalition after player 3.
Notice that [c(S+i) - c(S)] is the incremental cost of
adding player i to the S coalition.
The general formula for the Shapley value for player i
is
4> ( i ) = I a. (S) [ c {S) - c ( S - { i } ) ]
Scn 1
(4-18)
where
(s-i)! (n - s)Í
rf!
s is the number of players in coalition S,
n! is the total number of possible sequences
of coalition formation,
(s-1)! is the number of arrangements for those
players before S, and
(n-1)! is the number of arrangements for those
players after S.

77
For example, for i = 1, n = 3:
a1(l)
0121/3!
= 1/3
ax(12) =
1111/31
= 1/6
a1(13) =
1111/3!
= 1/6
a1(123) =
2101/31
= 1/3
Total
1.0
Note that
2 i(i) = c(N). (4-19)
i eN
Furthermore, if the game is convex, the Shapley value lies
in the center of the core (Shapley, 1971).
The Shapley value is criticized for several reasons.
It may fall outside the core for nonconvex games, and it may
be computed even when the core does not exist (Hamlen,
1980). Furthermore, the Shapley value is computationally
burdensome for large games. For an n-person game, the
Shapley value for each player requires the computation of
n 1
2 coefficients and incremental costs. For example, an
eight player game requires 128 coefficients and incremental
costs to calculate the charge for each player.

78
Loehman and Whinston (1976) attempted to reduce the
computational burden of the Shapley value by relaxing the
assumption that all sequences of coalition formation are
equally likely. This generalized Shapley value allows using
a priori information to eliminate impossible sequences of
coalition formation. Unfortunately, when Loehman et
al. (1979) applied the generalized Shapley value to an
eight-player regional wastewater management problem, they
got a solution outside the core (Heaney, 1983a).
Littlechild and Owen (1973) developed the simplified
Shapley value for games wherein the characteristic function
is a cost function with the property that the cost of any
subcoalition is equal to the cost of the largest player in
the subcoalition. Although Littlechild and Thompson (1977)
demonstrated the computational ease of the simplified
Shapley value in their case study of airport landing fees
consisting of 13,572 landings by 11 different types of
aircraft, the use of the simplified Shapley value is
restricted to games with these special properties.
Before calculating the Shapley value for our regional
water network problem, the total enumeration procedure
described in the preceding chapter is used to find the
following characteristic cost function values (see Appendix
A) :
c(1) = 646,000 c(2 ) = 2,420,095 c(3 ) = 1, 990,992
c(12) = 2,928,511 c(13)
2,586,638
c ( 2 3 )
3,984,177

79
and
1
c
(12
3)
4,
556,
409
2
c
(12
3)
4,
556,
826
3
c
(1,
23) =
4,
630,
177
4
c
(12
,3) =
4,
919,
503
5
c
(13
,2) =
5,
006,
734
w
c
dr
2,3) =
5,
057,
087
k t h
where c (hi,j) is the cost of the k best regional water
network consisting of pipelines from the well field to
county h, from county h to county i, and from the well field
. w
to county j. Also, c (1,2,3) is the cost for each county to
t h
go-it-alone. The cost allocation associated with the k
t h
best regional water network, i.e., the kc network game, is
simply found by setting c(N) equal to c (N).
The Shapley values for all options available to the
three counties are calculated in Appendix A and summarized
in Table 4-11. Appendix A also checks whether each Shapley
value satisfies core conditions. All of the network games
in this example are nonconvex. Table 4-11 shows that the
cost allocations for the optimal and the second best
networks, i.e., the first two options, satisfy all core
conditions; therefore, these cost allocations are in the
core and are considered equitable. Furthermore, unlike the
cost allocations using the direct costing methods discussed
earlier, the cost allocations for these two comparable cost

80
Table 4-11. Cost Allocation for Three-County Example Using the
Shapley Value.
Cost Allocation to County i Is Cost
($) Allocation
Zx(i) In Core?
Option County 1 County 2 County 3 ($) (From Ap-
(Rank) x(l) x(2) x(3) pendix A)
1
590,087
2,175,905
1,790,417
4,556,409
Yes
2
590,226
2,176,044
1,790,556
4,556,826
Yes
3
614,677
2,200,494
1,815,006
4,630,177
No
4
711,119
2,296,936
1,911,448
4,919,503
No
5
740,196
2,326,013
1,940,525
5,006,734
No
6
646,000
2,420,095
1,990,992
5,057,087
—

81
networks are nearly identical. The Shapley value divided
the additional $417 for the second best network equally
among the counties. Option 3 illustrates the failure of the
Shapley value to consistently give a core solution for
nonconvex games. As shown in Appendix A, the cost alloca¬
tion for option 3 fails to satisfy subgroup rationality for
the coalition consisting of county 2 and county 3; i.e.,
x(2) + x(3) > c(23 ) . (4-20)
Moreover, options 4 and 5 illustrate Shapley values for
games with an empty core. The nonexistence of the core for
network games with options 4 and 5 can be determined by
using other game theory methods, e.g., nucleolus. A close
examination of the core conditions for network games with
options 4 and 5 reveals these games are no longer subaddi-
tive. By defining c(N) as c (N), c(N) is no longer the
least cost or optimal solution for the grand coalition.
t h
Consequently, the k best network game is not naturally
subadditive even though c (N) may be less expensive than
w
c (N). In any event, because network games with options 4
and 5 are not subadditive, there is no incentive to
cooperate. Therefore, options 4 and 5 no longer need to be
considered by the counties.
Nucleolus. The other popular method to obtain a unique
charge vector is to find the nucleolus. For a cost game,

82
the fairness criterion used by the nucleolus is based on
finding the charge vector which maximizes the minimum
savings of any coalition.
For each imputation in the core of a cost game, a
2n
vector in R is defined. The components of this vector are
arranged in increasing order of magnitude and are defined by
e(S) = c(S) - z x(i) V ScN. (4-21)
ieS
2 n
The imputation whose vector in R is lexicographically the
largest is called the nucleolus of the cost game. Given two
vectors, X = (x^,...,x ) and Y = (y^,...fy ), X is lexi¬
cographically larger than Y if there exists some integer k,
1 < k < n, such that x^ = y^ for 1 _< j < k and x^. > y^
(Owen, 1982). Basically, e(S) represents the minimum
savings of coalition S with respect to charge vector X.
Obviously, the coalition with the least savings objects to
charge vector X most strongly, and the nucleolus maximizes
this minimum savings over all coalitions.
The nucleolus can be found by solving at most n-1
linear programs (Kohlberg, 1972; Owen, 1974, 1982), where
the first linear programming problem is
maximize e(l)
subject to
e(l) + x(i) < c(i)
Â¥ ieN
(4-22)

83
e (1) + £ x ( i ) < c (S) ¥ S N
ie S
£ x ( i ) = c (N)
ie N
x ( i ), e (1) > 0
The nucleolus is calculated by sequentially solving for
e(l), then e(2), e(3), etc., where e(i) is the ifc^ smallest
savings to any coalition.
Unlike the Shapley value, the nucleolus always is in
the core for games with nonempty core. In fact, the
nucleolus is always unique. However, the nucleolus is
criticized because it cannot be written down in explicit
form (Spinetto, 1975), and that it is difficult to compute
and use in practice (Gugenheim, 1983). Probably the most
difficult problem with using the nucleolus is the acceptance
of its notion of fairness as opposed to other prevailing
notions of fairness without generating unending
controversies and debates. The nucleolus is generally
considered to be analogous to Rawls' (1971) welfare
criteria: the utility function of the least well off
individual is maximized. Other notable notions of fairness
include (1) Nozick's (1974) procedural approach to justice,
and (2) Varian's (1975) or Baumol's (1982) definition of

84
equitable distribution whereby no one prefers the
consumption bundle of anyone else.
Calculating the nucleolus for our regional water
network problem using the linear programming problem (4-22)
gives the results summarized in Table 4-12. Equitable cost
allocations are given for the first three options, and the
cost allocations for the optimal and the second best
networks are essentially the same. The additional $417 for
the second best network is apportioned as follows:
County
1
$209
County
2
104
County
3
104
Total
$417
No cost allocations are given for options 4 and 5 because
these network games have empty cores. That is, the linear
programming problem (4-22) is infeasible. Finally, Table
4-12 reveals that each of the three counties has an
incentive to cooperate in order to implement the cheapest
regional water network.
Propensity to disrupt. Another unique solution concept
worth mentioning because of its intuitive appeal is the
concept of an individual player's "propensity to disrupt."
Gately (1974) defined an individual player i's propensity to
disrupt as a ratio of what the other players would lose if
player i refused to cooperate over how much player i would
lose by not cooperating. Mathematically, player i's

85
Table 4-12. Cost Allocation for Three-County Example Using
the Nucleolus.
Cost Allocation to
($)
County i
lx (i )
Is Cost
Allocation
Option
(Rank)
County 1
x(l)
County 2
x (2 )
County 3
x (3 )
($)
In
Core?
1
609,116
2,144,583
1,802,710
4,556,409
Yes
2
609,325
2,144,687
1,802,814
4,556,826
Yes
3
4
646,000
2,163,025
1,821,152
4,630,177
Yes
5
6
646,000
2,420,095
1,990,992
5,057,087
—

86
propensity to disrupt, d(i), a charge vector, X =
[x(1),...,x(n)], which is in the core is
c(N-i) - £ x(j )
d ( i ) = 3ZÍÍ (4-23)
c ( i ) - x(i )
The higher the propensity to disrupt, the greater a player's
threat to the coalition; e.g., d(i) = 10 implies player i
could impose a loss of savings to the other players 10 times
as great as the loss of savings to player i. As an
illustration, the propensity to disrupt is calculated for
each of the counties using the nucleolus for the optimal
network of our regional water network problem: X =
[609,116; 2,144,583; 1,802,710].
d (1)
c(23 - x(2) - x(3 )
c (1) - x (1)
= 1.0
d (2 )
_ c(13 ) - x (1) - x(3 )
= .63
c ( 2 ) - x ( 2 )
d (3 )
_ c(12 ) - x(1) - x ( 2 )
= .93
c ( 3 ) - x ( 3 )
The calculations show that none of the counties have a
strong threat against the other two counties with the
nucleolus charge vector. County 1 could impose a loss
to the other two counties which equals the loss imposed on
itself, while, county 2's or county 3's departure would hurt
the departing county more than it would hurt the remaining
two counties.

87
Gately suggested equalizing each player's propensity to
disrupt as a final cost allocation solution. Subsequently,
Littlechild and Vaidya (1976) have generalized Gately's
concept of an individual player's propensity to disrupt to
include a coalition S's propensity to disrupt. That is, a
coalition S's propensity to disrupt is defined as the ratio
of what the complementary coalition, N-S, stands to lose
over what the coalition S itself stands to lose for a given
charge vector. More recently, Chames et al. (1978 ) and
Chames and Golany (1983) refined these propensity to
disrupt concepts into a unique solution concept which
appears to have some empirical support. Finally, Straffin
and Heaney (1981) have shown that Gately's propensity to
disrupt is exactly the alternative cost avoided method first
proposed during the TVA project in 1935. The alternative
cost avoided method is discussed in the section on the
separable costs, remaining benefits method.
Empty Core Solution Concepts
Examining games with an empty core is an active area of
research. An empty core implies that no equitable cost
allocation exists, and results from games wherein the addi¬
tional savings from forming the grand coalition is rela¬
tively small. That is, the savings resulting from forming
smaller coalitions are almost as much as the savings from
forming the grand coalition. Therefore, proposed solution

88
concepts generally seek to relax the bounds on subgroup
rationality until a "quasi" or "anti" core is created.
Table 4-13 lists four methods for finding a charge vector
for games with an empty core.
In any case, given the modest amount of economic gain
for games with an empty core, it may be more advantageous to
forego the grand coalition in favor of smaller coalition
formations as suggested by Heaney (1983a). Furthermore,
engineering projects tend to have a large proportion of the
costs common to all participants; consequently, one would
expect these games to be very attractive and games with an
empty core to be fairly rare. Nevertheless, the game theory
approach does alert us that a problem exists in allocating
costs for such cases.
Cost Allocation in the Water Resources Field
Straffin and Heaney (1981) showed that the criteria of
fairness as expressed by equations (4-1) and (4-2) associated
with cost allocation proposed by the TVA experts in the 1930's
paralleled the development of the concepts of individual and
subgroup rationality found in cooperative game theory. Given
that full costs have to be recovered, the core conditions are
equivalent to the fairness criteria for allocating cost
originally proposed by the TVA experts. Therefore, current
practice for allocating costs in the water resources field
should require the solution be in the core of a game.

89
Table 4-13. Empty Core Solution Methods.
Method
Approach
Source
1. Least Core or
8-Core
Relax c(S) Shapley and Shubik,
1973; Young et al.,
1982; Williams, 1982,
1983
2. Weak Least Core Relax c(S)
or a-Core
Shapley and Shubik,
1973; Young et al.,
1982; Williams, 1982,
1983
3. Minimum Cost, Relax c(S)
Remaining Savings
Heaney and Dickinson,
1982
4 .
Relax c(N)
Chames and Golany,
1983
Homocore

90
Separable Costs, Remaining Benefits Method
As discussed previously, ad hoc methods generally used
in the water resources field allocate certain costs that are
considered identifiable costs directly, and prorate the
remaining costs by some criterion. The primary difference
among these ad hoc methods is how identifiable costs are
defined. We have already shown that defining identifiable
costs as either zero or direct costs does not insure an
equitable or core solution. We will now discuss several
methods whereby identifiable costs are defined as separable
costs.
The recommended cost allocation method in the water
resources field in the United States is the separable costs,
remaining benefits (SCRB) method (Federal Inter-Agency River
Basin Committee, 1950; Loughlin, 1977, 1978; Rossman, 1978;
Heggen, 1980; Goodman, 1984). This method assigns each
individual (or purpose) in a joint venture its separable
costs and a share of the remaining costs in proportion to
the remaining benefit, i.e., the minimum of the benefit or
alternative cost less separable costs. Separable costs for
individual i are defined as the difference between the cost
of the joint venture with and without individual i.
Separable costs include both the direct cost attributable to
the entering individual and the incremental costs associated
with a larger project because of the inclusion of another
individual. Mathematically, separable costs are

91
sc (i)
c(N) - c(N-i) ¥ isN
where
sc (i)
separable costs to individual i,
C (N )
least cost system associated with
group N, and
c(N-i) =
least cost system associated with
subgroup N-i.
After the separable costs for each individual have been
allocated, the remaining costs to be assigned are called
nonseparable costs (ncs), or
nsc = c(n) - £ sc(i). (4-25)
i£N
For the SCRB method the nonseparable costs are prorated on
the basis of the remaining benefits; therefore, this
prorated share is
[min [b(i), c(i)] - sc(i)]
^i£N ^min c(i)] - sc(i)}}
where b(i)
benefit of individual i
c(i) = alternative cost of individual i if
i acts independently, and
6 (i)
prorating factor for the SCRB
method.

92
The total charge to the individual is
x (i )
sc(i) + 8 (i) • nsc.
(4-27)
The total charge, c(N), is
c(N)
lx (i)
Isc(i) + Z 6 (i) • nsc (4-28)
where
IB(i)
1.0.
The alternative justifiable expenditure method and the
alternative cost avoided method are two variants of the SCRB
method frequently mentioned in the water resources litera¬
ture. The alternative justifiable expenditure method is
recommended when data are not available for separable
costs. Then, separable costs are defined as direct costs,
and the nonseparable costs equal total project cost less the
sum of all direct costs and are distributed in proportion to
the remaining benefits. The alternative justifiable
expenditure method is equivalent to the Louderback-Moriarity
method recently proposed by Balachandran and Ramakrishnan
(1981) in the accounting literature. The alternative cost
avoided method is equivalent to Gately's propensity to
disrupt with separable cost defined as before, but the
nonseparable costs are distributed in proportion to the

93
alternative cost avoided, i.e., the difference between the
alternatiave cost for the single purpose project and the
separable costs.
The SCRB method is now applied to our regional water
network problem. The SCRB calculations are in Appendix A,
and the results are summarized in Table 4-14. Like the
nucleolus, the SCRB method gives equitable cost allocations
for the first three options, and the three counties have an
incentive to cooperate to implement the cheapest regional
water network. Furthermore, Appendix A shows the SCRB
method gives cost allocations for games without a core,
i.e., solutions for network games for options 4 and 5 just
like the Shapley value. Again, the cost allocations for the
optimal network and the second best network are essentially
the same. The additional $417 for the second best network
follows:
County 1
$211
County 2
89
County 3
117
Total
$417
For our regional water network problem, each county's
and each subgroup of counties' alternative cost of inde¬
pendent action is less than or equal to each county's and
each subgroup of counties' benefits; therefore, the SCRB
method is identical to the alternative cost avoided method
and Gately's propensity to disrupt method. For example, if

94
Table 4-14. Cost Allocation for Three-County Example Using the
SCRB Method.
Cost Allocation to
($)
County i
£ x ( i )
Is Cost
Allocation
In Core?
Option
(Rank)
County 1
x(l)
County 2
x (2 )
County 3
x (3 )
($)
(From Ap¬
pendix A)
1
604,369
2,165,958
1,786,082
4,556,409
Yes
2
604,580
2,166,047
1,786,199
4,556,826
Yes
3
4
646,000
2,178,677
1,805,500
4,630,177
Yes
5
6
646,000
2,420,095
1,990,992
5,057,087
—
(3)
(4)
(6)

95
the SCRB charge vector for the optimal network is used: X =
[604,369; 21,165,958; 1,786,082], the propensity to disrupt
is d(i) = .77, i = 1,2,3. Since d(i) < 1, i = 1,2,3, each
county is only a weak threat to the other two counties.
Minimum Costs, Remaining Savings Method
Although the SCRB method is recommended practice, it
ignores subgroup rationality for projects with more than
three individuals (Giglio and Wrightington, 1972; Young
et al., 1982). The SCRB method only considers information
on coalitions of size 1, (N-l), and N. As a result, the
SCRB solution may be outside the core. Recently, Heaney and
Dickinson (1982) showed the SCRB method may use infeasible
upper bounds for apportioning the nonseparable costs in
addition to ignoring information on subgroup rationality.
As an improvement they proposed the minimum costs, remaining
savings (MCRS) method as a generalized SCRB method. With
the inclusion of all available information on subgroup
rationality, the MCRS method uses linear programming to
determine the minimum and maximum feasible costs for each
individual. The feasible costs are then used as bounds to
prorate the nonseparable costs just like the SCRB method;
however, the feasible bounds now ensure the core conditions
are met if the core is not empty. If the game is convex,
the MCRS and the SCRB methods are identical.

96
Mathematically, the MCRS method can be stated as
x (i)
x(i) + 8 ( i ) • nsc
min
(4-29)
where nsc =
c (N ) - 1 x (i ) .
. min
i eN
6 ( i ) =
[x(i) -x(i) . ]
max min
t x(i) -x(i) . ]}
. „ max min J
ieN
and x(i) . and x(i) are found by solving the following
min max 1 v
2n linear programs:
max or min x(i)
subject to x(i) _< c(i) ¥ isN (4-30)
1 x(i) < c(S) ¥ ScN
ie S
1 x ( i ) = c (N)
ieN
x(i ) >0 ¥ ieN
In addition, the MCRS method can be used to determine if a
game has a core by checking whether the linear programming
problem (4-30) is feasible. If the linear programming
problem (4-30) is infeasible, the game has an empty core.
To find a unique charge vector to games with an empty core

97
using the MCRS method, the bounds or characteristic cost
function values for the S-member coalitions are relaxed
until a core appears. This procedure can be formulated by
the following linear programming problem:
minimize 0
subject to x(i) £ x(i) - 0c(S) < c(S) ¥ ScN (4-31)
ieS
£ x ( i ) = c (N )
ieN
x ( i ) > 0
The MCRS method is now applied to our regional water
network problem. The calculations are contained in Appendix
A and the results are summarized in Table 4-15. Table 4-15
shows equitable cost allocations for the first three
options. Again, the cost allocations for the optimal
network and the second best network are essentially the
same. The additional $417 for the second best network is
allocated as follows:
County 1 $217
County 2 1
County 3 199
Total
$417

98
Table 4
-15. Cost
MCRS
Allocation for Three-County Example Using the
Method.
Cost Allocation to County i Is Cost
($) Allocation
I x ( i ) In Core?
Option
(Rank)
County 1
x(l)
County 2 County 3 (?) (From Ap-
x(2) x(3) pendix A)
1
606,861
2,151,206
1,798,342
4,556,409
Yes
2
607,078
2,151,207
1,798,541
4,556,826
Yes
3
4
646,000
2,163,025
1,821,152
4,630,177
Yes
5
6
646,000
2,420,095
1,990,992
5,057,087
—

99
Furthermore, the MCRS cost allocations in Table 4-15
encourage the counties to cooperate to construct the cheap¬
est regional water network possible because the cost for
each county progressively decreases with decreasing total
project cost.
Since none of the network games are convex, the MCRS
solutions are different from the SCRB solutions because the
MCRS method uses actual core bounds rather than nominal core
bounds to apportion the nonseparable cost. For instance,
the actual core bounds and the nominal core bounds for the
optimal network game, shown in Table 4-16, illustrate the
major difference between the MCRS method and the SCRB
method. That is, the SCRB method distorts the allocation of
the nonseparable cost by using infeasible bounds.
Allocating Cost Using Game Theory Concepts
The k Best System
The cores for the optimal network game and the second
best network game of our regional water network problem is
shown in Figures 4-3 and 4-4 along with charge vectors for
some of the cost allocation methods we discussed. By com¬
paring Figures 4-3 and 4-4, we can see summarized in Table
4-17 that the SCRB and the game theory methods not only give
equitable cost allocations for the two comparable cost
networks, but each of these methods also gives almost

100
Table 4-16.
Nominal Versus Actual Core
Optimal Network Game.
Bounds for
Nominal Core Bounds
Player i
Lower Bound =
c(N) - c(N-l)
Upper Bound =
c (i )
1
572,232
646,000
2
1,969,771
2,420,095
3
1,627,898
1,990,992
Actual Core Bounds
Player i
Lower Bound From
LP (4-30) with
Objective Function
Min x(i)
Upper Bound From
LP (4-30) with
Objective Function
Max x(i)
1
572,232
646,000
2
1,969,771
2,356,279
3
1,627,898
1,990,992

101
A Proportional to Population
^ Proportional to Demand
• Use of Facilities
o Direct Costing/Equal Apportionment of
Remaining Costs
x Shapley Value, Nucleolus, SCRB (Gately's
Propensity to Disrupt), MCRS
Figure 4-3.
Core for the Optimal Network Game
(C(N) = $4,556,409).

102
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Figure 4-4.
Core for the Second Best Network Game
(C(N) = $4,556,826).

Table 4-17. Cost Allocations for the Optimal Network and the Second Best Network ($).
Method
Network
County 1
County 2
County 3
1. Optimal
590,087
2,175,905
1,790,417
Shapley
2. Second Best
590,226
2,176,044
1,790,556
Difference
(2 - 1)
139
139
139
1. Optimal
609,116
2,144,583
1,802,710
Nucleolus
2. Second Best
609,325
2,144,687
1,802,814
Difference
(2 - 1)
209
104
104
1. Optimal
604,369
2,165,958
1,786,082
SCRB
(Gately's
2. Second Best
604,580
2,166,047
1,786,199
Propensity
to Disrupt)
Difference
(2 - 1)
211
89
117
1. Optimal
606,861
2,151,206
1,798,342
MCRS
2. Second Best
607,078
2,151,207
1,798,541
Difference
(2 - 1)
217
1
199
103

104
identical cost allocations for the two comparable cost
networks.
The game theory approach is able to give equitable
solutions for the two comparable cost networks because the
cost of independent action for each county and each subgroup
of counties is recognized by the core conditions, i.e., the
values of the characteristic cost function. Furthermore,
the game theory approach is able to give nearly identical
cost allocations for the two comparable cost networks
because the core conditions for the two comparable cost
networks are essentially the same except for the group
rationality condition or the value of c(N), i.e.,
Core Conditions for the Optimal
Network Game
x(l)
<
646,000
X (2 )
<
2,420,095
x (3 )
£
1,990,992
x(l)
+
x (2 )
<
2,928,511
x(l)
+
x (3 )
<
2,586,638
X (2 )
+
x (3 )
<
3,984,177
x (1) + x ( 2 )
+
x (3 )
=
4,556,409
Core Conditions f
or •
the Second
Best Network Game
x(1) < 646,000
x(2 ) < 2,420,095
x ( 3 ) < 1,990,992

105
x (1) + x ( 2 )
x (1 ) + x ( 3 )
x ( 2 ) + x ( 3 )
x(l) + x(2) + x(3)
< 2,928,511
< 2,586,638
< 3,984,177
= 4,556,826
The second best network naturally has a higher value for
1 2
c(N), i.e., c (N) < c (N); consequently, the second best
network game is less attractive than the optimal network
game in terms of cost. Therefore, the core of the second
best network game is naturally smaller and more nonconvex
than the core of the optimal network game, but this "reduced
core" is a subset of the core for the optimal network game.
This is shown by examining the following actual core bounds
for the two comparable cost network games:
Actual Core bounds for Optimal Network
572,232 < x(1) < 646,000
1,969,771 < x(2) < 2,356,279
1,627,898 < x(3) < 1,990,992
Actual Core Bounds for Second Best Network
572,649 < x(l) < 646,000
1,970,188 < x(2) < 2,355,862
1,628,315 < x(3) < 1,990,992
The core for the second best network has slightly higher
lower bounds and slightly lower upper bounds compared with
the core for the optimal network.

106
This natural reduction in the core as c(N) increases
k-1 k
from c (N) to c (N) is also shown in Figure 4-5 by examin¬
ing nominal core bounds. Since the individual rationality
t h
core conditions are identical for the (k-1)1" best network
t h
game and the k best network game, the nominal upper core
bounds (NUB) for both of these network games are identical,
i.e., NUB^ ^ (i ) = NUB^ (i) = c(i), V ieN. Furthermore,
since the subgroup rationality core conditions are identical
t h t h
for the (k-1) best network game and the kc best network
game, the values of c(N-i), ¥ i£N, for both of these network
games are identical. Consequently, as c(N) increases from
k-1 k
c (N) to c (N), the nominal lower core bounds (NLB), i.e.,
c(N) - c(N-i), are increasing by the same value for all
individual i. Thus, Figure 4-5 shows that the increasing
values of the nominal lower core bounds as c(N) increases
k ” 1 k
from c (N) to C (N) is responsible for the reduction in
the core.
th
The financial viability of the k best system can now
be determined. As the value of c(N) increases progressively
t h
for the k best system, the core progressively reduces
t h
until the core possibly becomes empty. Therefore, all k
best systems associated with games with a core can be con¬
sidered financially viable since an equitable cost alloca-
t h
tion can be found. However, for all k best systems
associated with games with an empty core, either an empty
core cost allocation procedure is necessary, or these

107
Upper
Bound
NLB: Nominal
Lower
Bound
Figure 4-5. Reduction in. Core as c(N) Increases from
ck'x(N) to cK(N).

108
systems should not be considered because of the minimal
economic gain or the loss of subadditivity.
The Dummy Player
In allocating the cost for option 3 (see Table 4-2) of
our regional water network problem, there may be a tempta¬
tion to simply treat this network as a two-person game
involving counties 2 and 3 rather than a three-person game
that also includes county 1. The cost allocation for two-
person games is very simple. As shown by Heaney and
Dickinson (1982), the core for a two-person game is a line
and always exists. Furthermore, the two-person game is
always convex, so the SCRB solution and the MCRS solution
are identical and are located at the center of the core.
If option 3 is treated as a two-person game, the cost
allocation is trivial. County 1 simply pays its go-it-alone
cost while counties 2 and 3 share the saving equally between
themselves (see calculations in Table 4-18), i.e.,
County 1 $ 646,000
County 2 2,206,640
County 3 1,777,537
$4,630,177
Although this cost allocation satisfies core conditions
regardless of whether option 3 is treated as a two-person
game or a three-person game, there is a considerable differ¬
ence between this solution and the solutions by the SCRB and

Table 4-18.
Cost Allocation for
Method.
Option 3
as a Two-Person
Game Using the SCRB
County i
SC(i)=C(N)-c(N-i)
6( i ) nsc
=c(N) - Isc(i)
Allocated Cost ($)
x(i)=sc(i)+8(i) * nsc
2
1,993,185
1/2
426,910
2,206,640
3
1,564,082
1/2
426,910
1,777,537
c(2) = 2,420,095
c(3 ) = 1,990,992
c(23 ) = 3,984,177
109

110
the MCRS methods when option 3 is treated as a three-person
game. This is shown in Table 4-19. Whether option 3 can be
properly treated as a two-person game is important to know
since county 2 and county 3 face substantially different
costs.
To determine whether county 1 can be excluded in the
cost allocation for option 3, the core for option 3 as a
three-person game is examined. The core conditions for
option 3 as a three-person game are as follows:
LI:
x(l)
<
646,000
L2 :
x ( 2 )
<
2,420,095
L3 :
x (3 )
<
1,990,992
L4 :
x(l)
+
x (2 )
<
2,928,511
L5 :
x (1)
+
x (3 )
<
2,586,638
L6 :
x (2 )
+
x (3 )
<
3,984,177
L7 :
x (1) + x ( 2 )
+
x (3 )
=
4,630,177
The core bounds for this three-person game are found by
using linear programming problem (4-30). As expected, Table
4-20 shows county 1 simply pays its go-it-alone cost, i.e.,
$646,000. However, Table 4-20 indicates core constraint L4
is binding for x(2) and core constraint L5 is binding for
3 max
x(3) . Therefore, constraints L4 and L5 involving
max
subcoalitions with county 1 are essential for determining
the core for option 3 as a three-person game. Conse¬
quently, county l's participation cannot be ignored when

Table 4-19. Comparing Cost Allocations for Option 3 as Two-Person Game and
Three-Person Game Using the SCRB and MCRS Methods.
Approach
(Method)
Cost Allocation to County i ($)
County 1 County 2 County 3
x(l) x(2) x(3)
Is Cost
Allocation in
Core of Two-
Person Game?
Is Cost
Allocation in
Core of Three
Person Game?
Two-Person
(SCRB or
MCRS)
646,000
2,206,640
1,777,537
Yes
Yes
Three-Person
(SCRB)
646,000
2,178,677
1,805,500
Yes
Yes
Three-Person
(MCRS)
646,000
2,163,025
1,821,152
Yes
Yes

112
Table 4-20. Core Bounds for Option 3 as a Three-Person
Game.
Shadow Price
for Core
Constraint
from
LP (4-30)
Core Bound
LI
L2 L3
L4
L5 L6
L7
x (1)
max
646,000
+1
x ( 2 ) =
max
2,282,511
+ 1
+ 1
-1
x (3 )
max
1,940,638
+ 1 +1
-1
x(l) . =
min
646,000
-1
+ 1
x (2 )
min
2,043,539
-1
+ 1
x ( 3 ) . =
mm
1,701,666
-1
+ 1

113
determining the cost allocation for counties 2 and 3 for
option 3.
Suppose county 1 is a dummy player who contributes no
savings to any coalition. This means subcoalitions (12) and
(13) are inessential; i.e.,
c(12) = c(1) + c(2) = 3,066,095, and
c(13) = c(l) + c(3) = 2,636,922.
If county 1 is a dummy player, the core conditions for
option 3 as a three-person game can be rewritten as
follows:
LI:
x(l)
<
646,000
L2 :
x (2 )
<
2,420,095
L3 :
x ( 3 )
<
1,990,992
L4 :
x(l)
+
x (2 )
<
3,066,095
L5 :
x(l)
+
x (3 )
<
2,636,992
L6 :
x ( 2 )
+
x (3 )
<
3,984,177
L7:
x (1) + x ( 2 )
+
x ( 3 )
-
4,630,177
Table 4-21 shows the core bounds for this game by solving
linear programming problem (4-30). Again, county 1 simply
pays its go-it-alone cost. However, Table 4-21 indicates
core constraints L4 and L5 involving subcoalitions with
county 1 are no longer binding for x(2) and x(3) ,
respectively. The binding core constraint for x(2)max is L2
and for x(3) is L3. Although Table 4-21 indicates core

114
Table 4-21. Core Bounds for Option 3 as a Three-Person Game
with County 1 as a Dummy Player.
Shadow Price
for Core
Constraint
from
LP (4-30)
Core Bound
LI
L2
L3
L4
L5 L6
L7
x(l)
max
=
646,000
+ 1
x (2 )
max
=
2,420,095
+ 1
x (3 )
max
=
1,990,992
+ 1
x(l) .
mm
=
646,000
-1
+ 1
x ( 2 ) .
mm
=
1,993,185
-1
+ 1
x ( 3 ) .
mm
=
1,564,082
-1
+ 1

115
constraints L4 and L5 are binding for x(2) . and x(3) . ,
^ min mm
respectively, L4 and L5 are identical to L2 and L3, respec¬
tively, because x(l) = 646,000. Consequently, when county 1
is a dummy player, the three-person game can be reduced to a
two-person game with the following core conditions:
L2 :
x (2 )
_<
2,420,095
L3 :
x (3 )
<
1,990,992
L6 :
x ( 2 ) + x ( 3 )
=
3,984,177
Comparing Table 4-21 and Table 4-22 reinforces this result
because the core bounds for the two-person game are
identical to the core bounds for the three-person game with
county 1 as a dummy player.
In summary, an n-person game can be reduced to an
(n-1)-person game only if the player removed from the game
is a dummy player, i.e., a player who contributes no savings
to any coalition. Otherwise, the cost allocation may be
distorted even if it satisfies the core conditions for both
the n-person game and the (n-1)-person game.
Comparing Methods
The methods we discussed are compared in Table 4-23.
For any particular problem, any of these methods may suc¬
cessfully find equitable cost allocation. However, the
most suitable method for allocating cost in the water
resources field appears to be the MCRS method. Although the
game theory and the SCRB methods all give equitable cost

116
Table 4-22. Core Bounds for Option 3 as a Two-Person
Game.
Shadow Price for
Core
Constraint from LP
(4-30)
Core Bound
L2
L3
L6
x (2 )
max
2,420,095
+ 1
x (3 )
max
1,990,992
+ 1
x ( 2 ) . =
min
1,993,185
-1
+ 1
x ( 3 ) . =
mim
1,564,082
-1
+ 1

Table 4-23. Comparison of Methods Discussed for Allocating Costs of Water Resources
Projects.
Method
Feature
Propor¬
tionality
Direct
Costing
Shapley
Value
Nucleolus
SCRB
MCRS
1. Will always get core
solution for convex game
No
No
Yes
Yes
Yes
Yes
2. Will always get core solu¬
tion for noncovex game
No
No
No
Yes
No
Yes
3. Applicable to empty core
game
No
No
No
No
No
Yes
4. Can determine if core
exists
No
No
No
Yes
No
Yes
5. Calculations easy for one
or more systems
Yes
Yes
No
No
Yes
Yes
6. Independent of system
configuration
Yes
No
Yes
Yes
Yes
Yes
7. Similar to methods used in
water resources field
Yes
Yes
No
No
Yes
Yes
8. Easy to understand
Yes
Yes
Yes
No
Yes
Yes
9. Currently accepted
accounting method
Yes
Yes
No
No
No
No
Total Number of Yes
5
4
3
4
5
8
117

118
allocation if the game is convex, this convexity check
may be burdensome for large games. In any case, both the
Shapley value and the SCRB method may break down when the
game is nonconvex. This is especially undesirable when
suboptimal systems are evaluated since these games will
naturally increase in nonconvexity with increasing value
of c(N). In contrast, both the nucleolus and the MCRS
methods give core solutions regardless of the degree of
nonconvexity of a game. However, the MCRS method has an
advantage over the nucleolus method in the water resources
field because the MCRS method extends the presently
recommended SCRB method. This avoids controversies over
the acceptance of a different fairness criterion with using
the nucleolus method. Moreover, while both the MCRS and
nucleolus methods require solving multiple linear
programming problems, the MCRS method is much easier to
solve. The constraint sets for each of the 2n linear
programming problems for the MCRS method are identical,
whereas the nucleolus requires changing the constraint set
for each of the possible n-1 linear programming problems.
This makes the calculation of the nucleolus much more
complex, especially when several systems are involved.
However, even if several systems are involved, the sets of
constraints are essentially unchanged except for the value
of c(N) when using the MCRS method. Finally, the MCRS
method is easier to understand and explain to the eventual

decision makers which is also a criterion for selecting a
cost allocation method.
119
Summary
Several intuitively appealing ad hoc methods for
allocating cost fail to give an equitable solution when an
equitable solution exists. Moreover, in situations where
several facility configurations are being considered, some
ad hoc methods encourage noncooperation because these
methods are not independent of the configuration of the
facility. In order to overcome these shortcomings with ad
hoc methods, concepts from cooperative game theory are
t h
necessary. The basis for determining whether the kc best
system is financially viable is the existence of the core.
This is because an equitable cost allocation exists to
implement the system. However, a core solution may be
"inequitable" if caution is not taken to include all
nondummy players. Several game theoretic methods for
allocating cost are examined, but the most suitable method
for allocating cost in the water resource field appears
to be the MCRS method. This conclusion is based on
1) reliability of finding an equitable cost allocation;
2) simplicity of computing the cost allocation for one or
more systems; 3) adaptability to recommended methodology;
and 4) ease of understanding.

CHAPTER 5
EFFICIENCY/EQUITY ANALYSIS
Introduction
The cost allocation literature is replete with
terminologies like opportunity cost, alternative cost, and
marginal cost to represent the maximum or minimum amount
that an individual should be charged. However, the precise
meaning of these terms is obscured because no procedure for
measurement is usually given. This chapter outlines a
rigorous procedure to unambiguously quantify an individual's
maximum cost and minimum cost for equity analysis. Further¬
more, efficiency analysis and equity analysis are shown to
be related by the costs of all opportunities available to
all individuals in a project.
Before the results in this chapter are discussed,
recall that the computational effort used by the total
enumeration procedure described in Chapter 3 to find c(N) is
concurrrently used to find c(i) and c(S) with little addi¬
tional effort. That is, the independent calculations are
not only used to find c(N) but are also used to find c(i)
and c(S). This important aspect of the procedure is
illustrated in Table 5-1 using our three-county regional
water network problem.
120

121
Table 5-1. Using Independent Calculations From the Total
Enumeration Procedure to Find c(i),c(S), and
C(N) for the Three-County Regional Water
Network Problem.
Independent Calculation:
(SI); (S 2); (S3); (SI,12); (SI,13); (S2,23);
(SI,12,23); (SI,12,13)
Efficiency Analysis for c(N):
c(123) = minimum [(SI,12,23); (SI,12,13);
(SI,12) + (S3); (SI,13) + (S2);
(SI) + (S 2,2 3); (SI) + (S2) + (S3)]
Efficiency Analysis for c(i) and c(S):
c(l) = (SI) c(2) = (S2) c(3) = (S3)
c(12) = minimum [(SI,12); (SI) + (S2)]
c(13) = minimum [(SI,13); (SI) + (S3)]
c(23) = minimum [(S2,23); (S2) + (S3)]
Note: (Si,ij) represents the cost of the water network
consisting of pipelines from the well field to
county i and from county i to county j.

122
Maximum Cost
Opportunity cost (or alternative cost) is a concept
used in economics to define the true cost of any action and
is measured by the cost of the next best alternative that
must be forgone when an action is taken (Nicholson, 1983).
As a result, there is an opportunity cost for each
individual associated with a regional water project because
each individual must forego the opportunity to acquire the
same level of service by either going-it-alone or joining a
subcoalition. Although the costs for an individual to
go-it-alone and for each subcoalition an individual can join
to acquire the same level of service can be measured, an
individual's opportunity cost cannot be determined just from
these costs. This is because there is no way of specifying
an individual's next best alternative without also knowing
the individual's cost of joining each subcoalition. This
means the cost of each subcoalition an individual can join
must also be allocated. To further complicate matters, an
individual's opportunity for joining a subcoalition depends
on the opportunities available to the other individuals in a
regional water project as well. For example, in a three-
person game not every individual can join a two-person
subcoalition. Kaplan (1982) correctly states that oppor¬
tunity costs are extremely important for decision making,
yet difficult to measure because opportunity costs arise
from transactions not executed. However, since opportunity

123
cost is the cost of an individual's next best alternative,
the principle of individual rationality requires that the
opportunity cost be the limit on how much an individual can
be charged for joining the regional system. Otherwise, the
individual can obviously do better by paying for his next
best alternative rather than joining the regional system.
Consequently, the maximum cost for individual i is
equivalent to individual i's opportunity cost, and the
maximum charge for individual i can be stated mathematically
as follows:
x (i ) < x ( i )
maximum
¥ i£N
(5-1
where x(i)
x (i )
maximum
N
= cost allocated to individual i,
= maximum cost for individual i given
individual i's opportunity to go-it-
alone or join a subcoalition, and
= set of all individuals; i.e.,
N = (l, 2,...,n ).
This condition simply means individual i should be charged a
cost less than or equal to individual i's cost of going-it-
alone or joining a subcoalition. The maximum cost for
individual i is now unambiguously quantifiable as the upper
core bound for individual i, and is found by using the
procedure outlined for the MCRS method to find the upper
core bound for individual i, i.e.,

124
x(i)
maximum
maximize x(i)
(5-2)
subject to
x(i) £ x(i) < c(S) V ScN
ie S
z x(i)
c(N)
ieN
x ( i ) > 0
V ieN
The interpretation of the upper core bound is now
clear. The upper core bound for individual i is found by
considering all the opportunities available to all
individuals participating in a regional system. These
opportunities, represented by the core conditions, include
the possibility of all the individuals going-it-alone or
some combination of subcoalition formation. Without knowing
what will eventually take place, the core of a game accounts
for all possibilities by representing all possible feasible
solutions. These feasible solutions are found by taking the
convex combination of the feasible core bounds. The
feasible upper core bound determined from linear programming
problem (5-2) represents the maximum cost for individual i.
Any charges exceeding x(i)
for individual i represent
maximum
solutions outside the core, and, consequently, individual i
can do better by going-it-alone or joining a subcoalition.
The upper core bounds for the optimal network game of
our regional water network problem are now examined. This

125
is a nonconvex game whereby at least one subcoalition forma¬
tion is relatively attractive in comparison to the grand
coalition. From Table 5-2, the upper core bound for county
2 indicates county 2's maximum cost is $2,356,279 which is
not simply the cost for county 2 of independent action,
i.e., $2,420,095. The difference between the value of
county 2's maximum cost, x(2) . , and go-it-alone cost,
c(2), is due to the consideration by the core of not only
county 2's opportunities for acquiring the same level of
service by going-it-alone or joining a subcoalition, but
also similar opportunities for county 1 and county 3 as
well. To find which opportunities are relevant in determin¬
ing x(2) . , linear programming problem (5-2) can be
solved. Table 5-2 indicates core constraints L4 and L6
representing subcoalitions (12) and (23), respectively, are
binding when solving for x(2) . This result means the
opportunities of forming subcoalitions (12) and (23) limit
how much county 2 can be charged. However, when linear
programming problem (5-2) is solved for x(l) and
x(3) . , Table 5-2 indicates that the only binding core
maximum J 3
constraints are Ll and L3, respectively. Therefore, county
l's and county 3's maximum costs are their respective
go-it-alone costs. Linear programming problem (5-2) is an
unambiguous and rigorous procedure to refute any claims by
county 1 and/or county 3 that their maximum cost is less
than c(l) and c(3), respectively, because of opportunities

Table 5-2. Efficiency/Equity Analysis of the Optimal Network
Efficiency Analysis
Equity Analysis
Binding Constraint
Network
Savings
Cost ($)
Core Condition
m
max
min
x(l)
x (2 )
x ( 3 )
x(l )
x ( 2 )
x (3 )
c(123 )
9.9
c(123 ) = 4,556,409
LI: x(l) < c(1)
X
c (1,23 )
8.4
c(1)+c(23) = 4,630,177
L2: x(2) < c(2)
c (12,3 )
2.7
c(3)+c(12) = 4,919,503
L3: x(3) < c(3)
X
c (13,2 )
1.0
c{2)+c(13) = 5,006,734
L4: x(1)+x(2) < c(12)
X
X
c ( 1,2,3 )
0
c(1)+c(2)+c(3) = 5,057,087
L5: x(1)+x(3) < c(13)
X
L6: x(2)+x(3 ) < c(23 )
X
X
L7 : x(1)+x(2)+x(3 ) = c(123)
X
X
X
Note: x(l) = 646,000; x(2) = 2,356,279
max max
c(l)
= 646,000- c(2)
X(3)max= *'990,992; x(1) =572,232; x(2 ) = 1,969,771; x (3 ) . = 1,627,098
max min min min
= 2,420,095; c(3)
= 1,990,992; sc(l) =572,232; sc<2) = 1,969,771; sc(3)
1,627,890

127
to join a subcoalition. For example, even though subcoali¬
tion (13) is an essential coalition, Table 5-2 indicates
that subcoalition (13) is never a factor in determining the
maximum cost for any individuals in the game.
Another aspect of properly quantifying maximum cost
is in calculating the amount of savings from a regional
water project. Normally, the amount of savings is based on
each individual's go-it-alone cost; i.e.,
Savings (%) = 100 - [ C[Nl x 100] V ieN. (5-3)
2 c ( i )
ieN
However, equation (5-3) assumes that either the regional
water project involving the grand coalition is built or all
the individuals will go-it-alone and, unfortunately, does
not consider the possibility that a relatively attractive
regional water project involving subcoalitions may be
formed. To account for the possibility of relatively
attractive subcoalition formations, the amount of savings
from a regional water project should be defined in terms of
maximum cost as defined by linear programming problem (5-2);
i.e.,
Savings (%)
100
[-
i
C (N)
2 x ( i )
EN
x 100]
maximum
V ieN. (5-4)

128
For our regional water network problem, the amount of sav¬
ings from the optimal network is 9.9% if equation (5-3) is
used and is 8.7% if equation (5-4) is used. This result
indicates failure to consider the possibility of attractive
subcoalition formations can lead to an overestimation of
maximum cost and, consequently, an overestimation of the
amount of savings from a regional water project.
For convex games, an individual's maximum cost is
simply the individual's go-it-alone cost. A game is convex
if none of the subcoalitions are attractive relative to the
grand coalition. As discussed in Chapter 4, the nominal
core bounds and the actual core bounds are identical if a
game is convex. As a result, the maximum cost for each
individual in a convex game is simply the individual's
go-it-alone cost; i.e., x(i) . = c(i). This means in a
^ maximum
convex game none of the subcoalition core constraints for
linear programming problem (5-2) are binding; therefore,
none of the individuals in a convex game can claim a maximum
cost less than their go-it-alone cost because of opportuni¬
ties to join a subcoalition.
In summary, individual i's upper core bound represents
individual i's maximum cost. If a game is convex,
x(i) is simply equal to individual i's go-it-alone
maximum ^ 1 ^ ^
cost, i.e., c(i); but, if a game is nonconvex, x(i)
must be determined from linear programming problem (5-2).
Whether a game is convex or nonconvex can be determined

either by a convexity check using equation (4-12) or (4-13),
or by comparing the nominal core bounds with the actual core
bounds determined from linear programming problem (4-30).
If a game is large, convexity check using equation (4-12) or
(4-13) can be burdensome, and using linear programming
problem (4-30) is more practical. In any event, determining
whether a game is convex or nonconvex requires knowing the
least cost system or characteristic cost function for each
subcoalition, i.e., c(S). Furthermore, if a game is
nonconvex, the importance of each subcoalition for equity
analysis cannot be determined until linear programming
problem (5-2) is solved. As we can see, efficiency analysis
and equity analysis are related because the maximum costs
which determine the maximum charges for the individuals in a
regional project are found by considering the economics of
all opportunities available to all individuals in the
project.
Minimum Cost
A corresponding interpretation of an individual's lower
core bound can be given. The lower core bound represents
the minimum cost assignable to individual i for joining the
grand coalition based on considering all the opportunities
for all individuals in a project. Therefore, the minimum
charge for individual i can be expressed mathematically as
follows:

130
x (i)
>
X (i )
minimum
V ieN
(5-5)
where x(i) = cost allocated to individual i,
x (i )
minimum
= minimum cost to individual i to join
the grand coalition, and
N = set of all individuals; i.e.,
N = (1,2,...,n }.
This condition simply means individual i should be charged a
cost greater or equal to individual i's cost of joining the
grand coalition. The minimum cost for individual i is found
by using the procedure outlined for the MCRS method to find
the lower core bound for individual i, i.e.,
x(i) . . = minimize x(i) (5-6)
minimum
subject to x(i) _< c(i) V ieN
£ x(i) < c(S) V ScN
ie S
£ x ( i ) = c (N)
i£N
x ( i ) > 0 ¥ ieN
Any charges less than x(i) . . for individual i mean
3 minimum
individual i is not paying for its minimum cost to join the

131
I
grand coalition and represent solutions outside the core
whereby individual i is being subsidized.
The accepted practice of using separable cost or
marginal cost as the minimum cost is based on the margin-
ality principle that every individual should be charged at
least the additional cost of being served (Young et al.,
1982) and assumes that the minimum cost to join a coalition
is last. The practicability of this assumption is troubling
because not every individual can join a coalition last. In
fact, a coalition may form without any clear understanding
of the sequence of formation. Furthermore, Heaney (1979)
has shown that the assumption is only true if a game is
convex; i.e.,
c(N) - c(N - {i}) < c(S) - c[(S) - {i}] V SCN. (5-7)
Fortunately, linear programming problem (5-6) eliminates any
ambiguities in defining or quantifying the minimum cost for
each individual in a project. Table 5-2 indicates that the
minimum cost for each county of the optimal network for our
regional water network problem is equal to each county's
separable cost, i.e., the binding core constraints for
x(i) . . is c(N) and c(N - {i}).
minimum
In summary, individual i's lower core bound represents
individual i's minimum cost. If a game is convex,

132
x(i) . . is simply equal to individual i's separable
minimum
cost or marginal cost, i.e.,
x(i) . . = sc(i) = c(N) - c(N - {i}) V ieN. (5-8)
minimum
For nonconvex games, x(i) . . may or may not be identical
to the separable cost or marginal cost for individual i;
therefore, linear programming problem (5-6) must be solved.
Fairness Criteria
The fairness criteria for an equitable cost allocation
expressed by equations (4-1) and (4-2) can now be simply and
explicitly stated in terms of the core bounds, i.e.,
x(i) . < x(i) < x(i)
min — — max
V ieN.
(5-9)
This condition embodies the fairness criteria expressed by
equations (4-1) and (4-2), yet clearly defines the range of
costs that each individual i can be charged without violat¬
ing individual rationality and/or subgroup rationality.
Moreover, the lower and upper core bounds for each
individual can be unambiguously quantified and interpreted
as each individual's minimum cost and maximum cost,
respectively. Finally, equation (5-9) can simplify the cost
allocation procedure because the minimum cost for each
individual is already determined, and only the remaining

costs need to be allocated. Equation (4-4) can now be
rewritten as follows:
133
x ( i ) = x ( i ) .
min
+ ip ( i ) • rc
V ieN
(5-10)
where
x(i) = cost allocated to individual i
= minimum x(i) from linear
programming problem (5-6)
^(i) = prorating factor for individual i, and
rc
remaining costs
i. e. ,
c (N) - £
i eN
Therefore, any cost allocation procedure that is agreeable
to the individuals participating in a regional project can
be used to apportion the remaining costs as long as
inequalities (5-9) are satisfied. For example, the remain¬
ing costs might be prorated in proportion to a measure of
use.
Summary
The interpretation of the core bounds is now clear.
The lower core bound is a measure of minimum cost, while
the upper core bound is a measure of maximum cost. The
procedure for unambiguously measuring these costs is the
same procedure used in the MCRS method for finding the core
bounds. However, to determine the core bounds, an
efficiency analysis is necessary to find the costs of all

134
opportunities available to all individuals in a project.
Once the core bounds are found, they can be used as simple
guidelines for allocating costs.

CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS
The motivation for this dissertation is based on the
following three conclusions from reviewing the literature on
efficiency analysis and equity analysis of regional water
resources planning. First, no published work has
incorporated efficiency analysis and equity analysis into a
single regional water resources planning model using
realistic cost functions. Secondly, the allocation of
piping cost has not been examined separately from treatment
cost. Thirdly, the cost allocation literature has not dealt
with situations whereby good suboptimal systems are
considered along with the optimal system.
The first conclusion establishes the primary purpose of
this dissertation. That is, to integrate efficiency analy¬
sis and equity analysis into a single water resources plan¬
ning model using realistic cost functions. The selection of
a regional water network problem is obviously based on the
second conclusion. Finally, the third conclusion helped
initiate a search for a reliable computational procedure to
find good suboptimal systems that ultimately led to develop¬
ing a total enumeration procedure for efficiency/equity
analysis of regional water network problems.
135

136
A major task with integrating efficiency analysis and
equity analysis is finding a computational procedure. The
principles of individual, subgroup, and group rationality
from cooperative game theory give a theoretically sound
basis for equity analysis. Consequently, successful
efficiency/equity analysis depends on having a reliable
method for finding not only the optimal regional system, but
also the optimal system for each individual and each sub¬
group of individuals. Basically, either a partial enumera¬
tion or a total enumeration approach can be used to find
these optimal solutions. Reliable partial enumeration
techniques can be used for problems with well defined cost
functions; but, for the types of cost function generally
encountered in actual practice a total enumeration technique
must be used.
A reliable total enumeration procedure for finding the
least cost water supply network for each individual, each
subgroup of individuals, and the region is described. This
procedure is easy to understand and use, and allows the
engineer to use realistic cost functions or to perform
detailed cost analysis. More importantly, the computational
effort used by this procedure to find the optimal regional
system can be concurrently used to find the optimal system
for each individual and each subgroup of individuals with
little additional effort. Furthermore, this procedure
naturally gives all the suboptimal systems; therefore, good

137
suboptimal regional systems can be examined when factors
other than cost are considered.
Once a reliable method is available to find the optimal
solutions, equity analysis can be accomplished using
concepts from cooperative game theory. The financial
viability of any system is based on the existence of a core
because an equitable cost allocation can be found if a core
exists. As the costs of suboptimal systems increase, the
core naturally reduces in size until the core possibly
becomes empty. Any system with an empty core is considered
not financially viable because of the minimal economic gain
or the loss of subadditivity.
In comparing several ad hoc and game theory methods for
allocating costs, the MCRS method appears to be the most
suitable method for the water resources field. More
importantly, the MCRS method gives a procedure for finding
the core bounds by simply using the core conditions along
with linear programming.
The lower core bound and the upper core bound for an
individual provide unambiguous measures of the individual's
minimum cost and maximum cost, respectively. These costs
are found by considering all the opportunities, represented
by the core conditions, available to all individuals in a
project. If the core conditions do not account for all
the opportunities for each individual in a project to form
an essential subcoalition, then the core bounds may be

138
distorted. In such cases, even a cost allocation in the
core may be inequitable.
Whether a game is convex or nonconvex is essential to
how minimum cost, maximum cost, and savings are determined.
The traditional approach is to assign minimum cost as the
cost to join a coalition last, to assign maximum cost as the
go-it-alone cost, and to calculate savings with respect to
go-it-alone cost. These traditional approaches are
acceptable only if a game is convex. For nonconvex games,
these traditional approaches overlook opportunities to form
good subcoalitions and, therefore, may distort the analysis
by overestimating minimum cost, maximum cost, and savings.
If a game is nonconvex, the minimum cost should be found by
using linear programming problem (5-6), and the maximum cost
should be found by using linear programming problem (5-2).
Moreover, savings should be calculated with respect to the
maximum cost as defined by linear programming problem (5-2).
By knowing each individual's minimum cost and maximum
cost, a basis for finding an equitable cost allocation
is available. Since the minimum cost for each individual is
already determined, decision makers simply need to agree on
a method to apportion remaining costs without exceeding any
individual's maximum cost.
In summary, efficiency analysis and equity analysis in
regional water resource planning are not separable
problems. An efficiency analysis is necessary to find an

139
optimal or a good suboptimal system, but to implement this
desirable system an equity analysis must be accomplished.
Yet, accomplishing an equity analysis depends on an effi¬
ciency analysis to find the optimal system for each
individual and each subgroup of individuals to account for
all opportunities available to all individuals in the
project. Otherwise, each individual's minimum cost and
maximum cost cannot be properly determined. Therefore, an
efficiency analysis is incomplete unless it also provides
the necessary information to accomplish an equity analysis.
Finally, some thoughts for further research generated
during the course of this dissertation are listed.
1. If different cost functions are used for different
individuals in a regional project, a consistent set of
accounting procedures is necessary to insure the cost
functions are based on a comparable set of cost data. What
is a consistent set of accounting procedures?
2. In any practical application, total project costs
are not known precisely until after the project has been
completed; therefore, a legitimate concern is how cost
overruns are to be allocated.
3. Determine mathematical procedures to compute the
total number of calculations to enumerate all 2n-l optimal
solutions for any given digraph and to compute how many of
these calculations are independent calculations.

140
4. Set up a computer model for detailed cost analysis,
e.g., MAPS, and apply the procedures described in this
dissertation to perform an efficiency/equity analysis of a
real regional water network problem.
5. After a regional network is in place, how should
the cost of expanding the network to serve new users be
allocated.
6. A method to quantify transactions cost is necessary
to better evaluate the financial gains of a regional system.

APPENDIX
EFFICIENCY/EQUITY ANALYSIS OF A THREE-COUNTY
REGIONAL WATER NETWORK WITH NONLINEAR
COST FUNCTION

Appendix A contains the efficiency/equity calculations for tn
three-county cost game example using Lotus 1-2-3. The results a
presented as follows:
Table 1 Efficiency Calculations With Total Enumeration Procedure
Table 2 Cost Allocation for Option 1
Table 3 Cost Allocation for Option 2
Table 4 Cost Allocation for Option 3
Table 5 Cost Allocation for Option 4
Table 6 Cost Allocation for Option 5
Table 7 Template Used for Calculations
Data
Distance : L(i,j) is the distance in feet from i to j
L (S,1)= 17000 L(S,3)= 30250 L(l,3)= 19670
L (S, 2) = 26000 L(1,2)= 13100 L(2,3) = 15500
Demand : Q(i) is the demand in mgd for player i
Q(1)= 1 Q(2)= ‘ 5 Q(3)= 3
Cost Function: a(Q~b)L a= 38 b= 3.5
Table 1
Calculations With
Total Enumeration
Procedure
C(i.-j)[x]= Cost of network [x] for
i..j ; C
(i..j)= Least cost
C(1)[S1]= 646000
C(2) [S2]=2420095.
C(3) [S3] =
=1990992.
C(12)[SI,12]=
2923511.
C(12)=
2928511.
C(12)[S1;S2]=
3066095.
C (13) [SI,13] =
2586638.
C(13)=
2586638.
C (13) [S1;S3] =
2636992.
C(23) [S2,23] =
3984177.
C(23)=
3934177.
C (23)[S2;S3] =
4411088.
C (123) [SI,12,23] =
4556409.
3
C(123)[S1,12;S3]=
4919503.
0
C (123)[SI,12,13] =
4556826.
Of
C(123)=
4556409.
C(123) [S1,13;S2] =
5006734.
0
C (123)[S1;S2,23] =
4630177.
0
C (123)[S1;S2;S3] =
5057083.
o
Sort C(123) in ascending order (Yes=
=l,Uo=0):
Paths
Cost
Convex
C (123)[SI,12,23] =
4556409.
0
C(123) [SI,12,13] =
4556826.
0
BEST
C(123)[SI;S2,23]=
4630177.
0
C(123)=
4556409
C(123)[SI,12;S3]=
4919503.
0
C (123) [SI,13;S2] =
5006734.
0
C(123)[S1;S2;S3]=
5057083.
0
142
flj n

143
Table 2 Cost Allocation For C(123) = 4556409
Calculate the Shapley value (SV) for tine best C(123):
SV(1)= 593037.3' SV(2)= 2175904. SV(3)= 1790416.
Sura of SV(1)+SV(2)+SV(3)= 4556409
Check core conditions: (core test valid for subadditive gane only)
IS core empty (yes=l,no=0)? 0
Does the Shapley value satisfy the core conditions?
SV(1) SV(2) SV(3) SV(1)+SV(2) SV(1)+SV(3) SV(2)+SV(3) Calculate the SCRB value (X) for the best C(123):
Nominal Core Bounds:
572231.0 1969770. 1627897. X(1)= 604369.0 X(2)= 2165957. X(3)= 1736032.
Sura of X(l)+X (2)+X(3)= 4556409
IS core empty (yes=l,no=0)? 0
Does the SCR3 value satisfy the core conditions?
X(1) X(2) X(3) X (1) +X (2) X(1)+X(3) X(2)+X(3) Calculate the MCRS value (M) for the best C(123):
Actual Core Bounds Determined From LP:
572232 1969771 1627893 M (1)= 6O6860.3 M(2)= 2151206.
Sum of M(l)+M(2)+M(3)= 4556409
646000
2356279
1990992
M(3) = 1793342.
IS core empty (yes=l,no=0)?
0
Does the MCRS value satisfy the core conditions?
M(1) M(2) M(3) M(1)+M(2) M(l)+M(3) M(2)+M(3)
Table 3
Cost Allocation For C(123) = 4556323
Calculate the Shapley value (SV) for the best C(123):
SV(1)= 590226.3 5V(2)= 2176043. SV(3)= 1790555.
Sun of SV{1)+SV(2)+SV(3) = 4555826
Check core conditions: (core test valid for subadditive gane only)
IS core empty (yes=l,no=0)? 0
Does the Shapley value satisfy the core conditions?
SV(1) SV(2) SV(3) SV(1)+SV(2) S V( 1)+SV (3) SV(2)+SV(3) Calculate the SCRB value (X) for the best C(123):
Nominal Core Bounds:
572648.0 1970187. 1628314. X (1)= 504580.4 X(2)= 2166046. X(3)= 1736199.
Sun of X(1)+X(2)+X(3)= 4556826
IS core empty (yes=l,no=0)? 0
Does the SCRB value satisfy the core conditions?
X(1) X (2) X (3) X (1) +X (2) X (1) +X (3) X(2) +X(3) Calculate the MCRS value (M) for the best C(123):
Actual Core Bounds Determined From LP:
572649 1970188 1628315 M(l)= 607077.0 M(2)= 2151237. M(3)= 1793541.
Sum of M(1)+M(2)+M(3)= 4556826
IS core empty (yes=l,no=0)? 0
Does the MCRS value satisfy the core conditions?
M(1) M(2) M(3) N(l) +M(2) M(l) +M(3) M(2)+M(3) 1
1
1

145
Table 4 Cost Allocation For C(123) = 4630177
Calculate the Shapley value (SV) for
SV(1)= 614676.6’ SV(2)= 2200494.
Sum of SV(1)+SV(2)+S V(3)= 4630177
the best C(123):
SV(3)= 1315006.
Check core conditions: (core test valid for subadditive game
IS core empty (yes=l,no=3)?
3
Does the Shapley value satisfy tine core conditions?
SV(1) 1
SV(2) 1
SV(3) 1
SV(1)+SV(2) 1
SV(1)+SV(3) 1
SV(2) +SV (3) 3
Calculate the SCRB value (X) for the
Nominal Core Bounds:
best C(123):
645999.0 546000
2043538. 2420095.
1731665. 1990992.
X(l)= 645999.4 X(2)= 2178677.
Sum of X(1)+X(2)+X(3)= 4630177
X (3)= 1835499.
IS core empty (yes=l,no=0)?
0
Does the SCRB value satisfy the core
conditions?
X(1) 1
X (2) 1
X (3) 1
X (1) +X (2) 1
X (1) +X (3) 1
X (2) +X (3) 1
Calculate the MCRS value (M) for the
Actual Core Bounds Determined From LP
645000 2043539 1701666 M (1)= 646000 M(2)= 2163325
Sum of M(1)+M(2)+M(3)= 4630177
best C(123):
i •
646300
2282511
1940638
V(3)= 1821152
IS core empty (yes=l,no=0)?
0
Does the VCRS value satisfy the core
conditions?
M(1) 1
M(2) 1
M (3) 1
M(l)+M(2) 1
M(l) +M(3) 1
M(2)+M(3) 1

146
Table 5 Cost Allocation For C(123) = 4919503
Calculate the Shapley value (SV) for the best C(123):
SV(1)= 711118.6 SV(2)= 2296936. SV(3)= 1911448.
Sum of SV(1)+SV(2)+SV(3)= 4919503
Check core conditions: (core test valid for subadditive game only)
IS core empty (yes=l,no=0)? 1
Does the Shapley value satisfy the core conditions?
SV(1) SV(2) SV(3) SV(1) +SV(2) SV(1)+SV(3) SV(2)+SV(3) Calculate the SCR3 value (X) for the best C(123):
Nominal Core Bounds:
935325.0 2332864. 1990991. X (1)= 449026.7 X(2)= 2479433. X(3)= 1990992.
Sum of X(1)+X(2)+X(3)= 4919503
IS core empty (yes=l,no=0)? 1
Does the SCRB value satisfy the core conditions?
X(1) X(2) X(3) X(1)+X(2) X (1) +X (3) X(2)+X(3) 1
0
0
1
1
Table 6 Cost Allocation For C(123) = 5006734
Calculate the Shapley value (SV) for the best C(123):
SV(1)= 740195.6 SV(2)= 2326013. SV(3)= 1940525.
Sum of SV(1)+SV(2)+SV(3)= 5006734
Check core conditions: (core test valid for subadditive game only)
IS core empty (yes=l,no=0)? 1
Does the Shapley value satisfy the core conditions?
SVdXC(l) 0
SV(2) SV(3) SV(1)+SV(2) SV(1)+SV(3) SV(2)+SV(3)
147
Calculate the SCRB value (X) for the best C(123):
Nominal Core Bounds:
1022556. 2420095. 2078222. X(l) = 505116.4 X(2)= 2420095. X(3)= 1981521.
Sum of X(1)+X(2)+X(3)= 5006734
IS core empty (yes=l,no=0)? 1
Does the SCRB value satisfy the core conditions?
X (1) 1
X (2) 0
X(3) 1
X (1) +X (2) 0
X (1) +X (3) 1
X(2)+X(3) 0
Table 7 Template
Used for Calculations
A 3
C D E F
G
H
Data
203
204
Distance : L(i,j)
is the distance in feet from i to j
205
L (S, 1) = 17000
L'S,3) = 30250 L(1,3)= 19670
205
L (S,2)= 26000
L(1,2)= 13100 L(2,3)= 15500
207
Demand : Q(i) is the demand in mgd for player i
2G3
Q(l) =
1 Q (2) = 6 Q (3) =
3
209
Cost Function: aCQ^bJL a= 33 b=
0.5
210
Calculations With Total Enumeration
Procedure
C(i..j)[x]= Cost of network [x] for i..j ; C(i..j)= Least cost for
i..j
C(1)[S1]= 646000
C(2)[S2]=2423095. C(3)[S3]=1990992.
215
C (12)[SI,12] =
2928511. C(12)=
2923511.
217
C (12)[S1;S2] =
3066095.
218
C (13)[SI,13] =
2586633. C(13)=
2585533.
220
C(13)[SI;S3]=
2636992.
221
C(23)[S2,23]=
3984177. C(23)=
3934177.
223
C(23)[S2;S3]=
4411038.
224
C(123)[SI,12,23] =
4556409. 0
226
C(123)[SI,12;S3]=
4919503. 0
227
C(123)[SI,12,13]=
4556326. 0 C(123)=
4556409.
228
C(123)[SI,13;S2]=
5006734. 0
229
C(123)[S1;S2,23]=
4630177. 0
230
C(123)[SI;S2;S2]=
5057088. 0
231

148
Sort C(123) in ascending order (Yes=
1,'-10=0) :
233
Paths
Cost
Convex
234
C(123)[SI,12,23]=
4556409.
0
235
C(123)[SI,12,13]=
4556826.
0 BEST
236
C(123)[SI;S2,23]=
4633177.
0 C(123)= 4556409
237
C(123)[SI,12;S3]=
4919503.
0
238
C(123)[SI,13;S2]=
5006734.
0
239
C(123)[S1;S2;S3]=
5057038.
0
240
241
Cost Allocation For C(123) = 4556439
243
Calculate the Shapley value
(SV) for
the best C(123):
245
SV(1)= 590087.3 SV(2)=
2175904.
SV(3)= 1790416.
245
Son of SV(1)+SV(2)+SV(3)=
4556409
247
Check core conditions:
(core test valid for subadditive game
only)
IS core empty (yes=l,no=3)?
0
250
Does the Shapley value satisfy the core conditions?
251
SV(1) 1
252
SV(2) 1
253
SV(3) 1
254
SV(1) +SV(2) 1
255
SV(1)+SV(3) 1
256
SV(2)+SV(3) 1
257
253
Calculate the SCRB value (X) for the
best C(123):
260
Nominal Core Bounds:
261
572231.0
646000
262
1969770.
2420095.
263
1627897.
1990992.
264
X (1)= 604369.fi X(2) =
2165957.
X (3)= 1736032.
265
Sum of X(l)+X(2)+X(3)=
4556409
266
IS core empty (yes=l,no=0)7
â–º
0
268
Does the SCR3 value satisfy the core
conditions?
269
X(1) 1
270
X (2) 1
271
X (3) 1
272
X(1)+X(2) 1
273
X (1) +X (3) 1
274
X(2)+X(3) 1
275
276
Calculate the MCRS value (N) for the
¡ best C(123):
278
Actual Core Bounds Determined From LP:
279
572232
646000
230
1969771
<.M (2 )<
2356279
231
1627893
1990992
232
M(l)= 606860.3 M(2)=
2151206.
M(3)= 1793342.
233
Sum of M(1)+M(2)+M(3)=
4556409
234

149
IS core empty (yes=l,no=0)?
0
235
Does the MCRS value satisfy
the core conditions?
287
M(1) 1
288
M(2) 1
239
M(3) 1
290
M (1) +M (2) 1
291
M(l) +M (3) 1
292
M(2)+M(3) 1
293
294
Partial Listing of Cell Formulas
Total Enumeration Procedure:
B215: +$E$210* ($C$209''$G$21C) *$B$206
D215: +$E$210*($E$209~$G$210)*$3$207
F215: +$E$210*($G$209~$G$210)*$D$206
C217: +$E$210*(($C$209+$E$209)~$G$210)*$S$206+$E$210*($E$209~$G$210)*$D
$207
C217: @MIN ($C$217.. $C$218)
C21S: +$B$215+$D$215
D226: +$E$210*(($C$209+$E$209+$G$209) ~$G$210)*$B$206+$E$210*(($E$209+$G
$209) ~$G$210*$D$207+$E$210* ($G$209~$G$210) *$F$207
D227: +$C$217+$F$215
D223: +$E$210*(($C$209+$E$209+$G$209r$G$210)*$5$205+$E$210*($E$2S9~$G$
210)*$D$207+$E$210*($G$209~$G$210)*$F$206
D229: +$C$220+$D$215
D230: +$B$215+$C$223
D231: +$3$215+$D$215+$F$215
E235: §IF(+$G$217+$G$220>=+$D$226+$S$215#ANDjf+$G$217+$G$223>=$D$226+$D$
215#AND#+$G$220+$G$223>=$D$226+$F$215
Shapley value:
B246:
D247:
E250:
E252:
E253:
E254:
E255:
E256:
E257:
SCRB:
C262:
E262:
B255:
MCRS:
B233:
l/3*$3$215+l/6*($G$217-$D$215)+1/6*($G$220-$F$215)+1/3*($C$237-$G
$223)
+$B$246+$D$246+$F$246
9IF(+$G$217+$G$220+$G$223>=2*$G$237,0,1)
0IF(+$B$246<=+$B$215,1,0)
@IF(+$D$246<=+$D$215,1,0)
@IF(+$F$246<=+$F$215,1,0)
0IF(+$B$246+$D$246<=+$G$217,1,0)
0IF(+$B$246+$F$245<=+$G$220,1,0)
0IF(+$D$24o+$F$246<=+$G$223,1,0)
+$G$237-$G$223
+$B$215
(($BS215-$C$262)/($B$215-SC$262+$D$215-$C$253+$F$215-$C$264))*($G
$237-$C$262-$C$263-$C$264)+$C$262
(($E$280-$C$280)/($E$280-$C$280+$E$281-$C$2S1+$E$282-$C$282))*(SG
$237-$C$280-$C$231-$C$232)+$C$280

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BIOGRAPHICAL SKETCH
Elliot Kin Ng was born August 9, 1950, in San
Francisco, California, the son of Wah Hin Ng and Kit Har
Yan. In 1968, he graduated from Lowell high school in San
Francisco. He entered the University of California,
Berkeley, in 1968 and received a Bachelor of Science in
electrical engineering in 1972 and a Master of Science in
electrical engineering in 1974. Subsequently, he spent two
years working for Bechtel Incorporated, San Francisco,
California, as an electrical engineer for the chemical and
refinery division. He was commissioned in the U.S. Air
Force as a captain in 1976. He held assignments at USAF
School of Aerospace Medicine, Brooks AFB, Texas (Environ¬
mental, Safety and Facility Manager), USAF Occupational and
Environmental Health Laboratory, Brooks AFB, Texas (Con¬
sultant, Water Resources Engineer), and USAF Hospital,
Wurtsmith AFB, Michigan (Bioenvironmental Engineer). During
his assignments at Brooks AFB, Texas, he received a Master
of Science degree in environmental management from the
Univerity of Texas, San Antonio. He was selected by the Air
Force Institute of Technology (AFIT) in 1981 for doctoral
training in environmental engineering. In 1982, he entered
the University of Florida to pursue the Doctor of Philosophy
159

160
degree. He was promoted to major in 1984. He is a
registered professional engineer in the state of
California. He and his wife, Eileen, have three children,
Matthew Elliot (age 4), Michelle Eileen (age 2), and Michael
Elliot (age 1).

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
ames P. Heaney, Chairman
Professor of Environmental
Engineering Sciences
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Sanford V. Berg
Associate Professor^of
Economics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Donald J. Elzinga
Professor of Industrial and
Systems Engineering

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Wayne C. Huber
Professor of Environmental
Engineering Sciences
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Warren Viessman
Professor of Environmental
Engineering Sciences
This dissertation was submitted to the Graduate Faculty of
the College of Engineering and to the Graduate School and
was accepted as partial fulfillment of the requirements for
the degree of Doctor of Philosophy.
August, 1985
C*~
Dean, College of Engineering
(J
Dean, Graduate School

UNIVERSITY OF FLORIDA
3 1262 08554 1349






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