Economic efficiency and cost allocation for water resource projects with economies of scale

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Economic efficiency and cost allocation for water resource projects with economies of scale
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Publication No. 80


Economic Efficiency and Cost Allocation for Water

Resource Projects with Economies of Scale


by


Bonnie Walker Proefke


Department of Environmental Engineering Sciences
University of Florida


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Publication No. 80


ECONOMIC EFFICIENCY AND COST ALLOCATION FOR WATER
RESOURCE PROJECTS WITH ECONOMIES OF SCALE




BY




BONNIE WALKER PROEFKE


A RESEARCH PROJECT PRESENTED TO
THE DEPARTMENT OF ENVIRONMENTAL ENGINEERING SCIENCES
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF MASTER OF ENGINEERING








UNIVERSITY OF FLORIDA


1984
















ACKNOWLEDGEMENTS


I would like to express my appreciation to the follow-

ing individuals who have offered their help and encourage-

ment throughout this study: my graduate advisor Dr. James

P. Heaney for his guidance during both the investigative and

writing phases of this project, Elliot Ng for his valuable

thoughts and input, and Robert Dickinson for his help with

the mathematical programming codes. Finally I would like to

thank my husband Richard for his love and patience through-

out this work.

















TABLE OF CONTENTS


ACKNOWLEDGEMENTS . . . .

LIST OF TABLES . . . .

LIST OF FIGURES . . . .

INTRODUCTION . . . . .

SECTION I LITERATURE REVIEW . .

Natural Monopoly . .
Optimization . . .
Cost Allocation . .
Summary . . .

SECTION II METHODOLOGY . . .

Project Evaluation Steps .
Discussion and Wastewater Reuse
Example . . .

CONCLUSIONS . . . . .

APPENDIX . . . . .

REFERENCES . . . . .


iii


Page

ii

iv

v

1

3

4
13
24
38

40

40

41

103

105

108

















LIST OF TABLES



Table Page

1 Cost Functions for Wastewater Reuse
Example . . . . .. 47

2 Optimal Resource Allocation for Case 1 . 76

3 Charges for Case 1 . . 80

4 Optimal Resource Allocation for Case 2 86

5 Charges for Case 2 . ... . .. 87

6 Optimal Resource Allocation for Case 3 . 92

7 Charges for Three-Person Game for Case 3 98
















LIST OF FIGURES



Figure Page

1 Example of an unsustainable natural
monopoly . . . . 9

2 Conditions for economic efficiency . .. 15

3 Selection of price under decreasing average
cost conditions . . . 26

4 Selection of price under increasing average
cost conditions . . . 26

5 System for wastewater reuse example .. 44

6 Case 1 benefit and cost relationships for
site 1 . . . . . 60

7 Priority ranking example for direct cost
considerations for case 1 . . .. 62

8 Case 2 benefit and cost relationships for
site 1 . . . . . 66

9 Example of shift in priority for case 2 67

10 Case 3 benefit and cost relationships for
site 2 . . . . 69

11 Net benefits for rank order determination
for case 1 . . ... . 75

12 Core and charges for three person game for
case 1 . . . . 82

13 Net benefits for rank order determination
for case 2 . . . . 85

14 Core and charges for three-person game for
case 2 . . . . 90

15 Core and charges for three-person game for
case 3 . . . . 97
















INTRODUCTION


Many economic optimization techniques are available for

evaluating various water resource project designs. Also,

numerous methods have been proposed or are now used to

apportion costs of a water resource project among partici-

pants and/or purposes. However, no extensive research has

been directed towards evaluating the relationship between

the economic optimization and cost allocation phases of

planning water resource projects. The purpose of this study

is to provide a basis for establishing and evaluating the

relationship between economic efficiency and project cost

allocation. First, a review of some recently developed

theories on the formation and sustainability of natural

monopolies is presented. Several important implications

concerning the efficiency and fairness of project coalition

formation may be derived from these economic theories. To

provide a basis for quantifying the implications derived

from economic theory, methods for evaluating economic

efficiency and for apportioning the cost among participants

of a joint water resource project are reviewed. Finally,

through the simple example of a small wastewater reuse

project, several of these methods are applied to demonstrate

how efficiency and equity might be related in order to





2



evaluate various joint project designs. In particular,

implications for the special case of economies of scale in

production for three different demand relationships are

examined.
















SECTION I

LITERATURE REVIEW



Some insight into the current dilemma regarding the

interaction of efficiency and equity can be gained by

reviewing the work of Loughlin (1977) and Rossman (1978).

Loughlin examines the efficiency of three cost allocation

schemes based on a set of efficiency and equity criteria.

He contends that economic efficiency and cost allocation are

independent of each other since costs allocated to a parti-

cipant or project purpose are not the costs to consider in

deciding project feasibility, optimal scale of development,

or which participants to include in the project. Rossman

notes that by judging the cost allocation methods in terms

of economic evaluation criteria, Loughlin contradicts the

argument that cost allocation is not required for economic

evaluation. In fact Zajac (1978) proposes that from the

viewpoint of either efficiency or equity, pricing and

entry-exit (from the joint project) must be considered

jointly, not in isolation. This section examines the

theoretical basis for these two differing viewpoints. The

theory of natural monopoly provides the basis of the argu-

ment for joint consideration of efficiency and equity, while

the principles of engineering cost allocation often dictate









independent financial and economic analyses. In addition to

these conceptual views of the problem, methods for quanti-

fying efficiency and equity are reviewed.



Natural Monopoly

Recent developments in the theory of natural monopoly

provide some important implications for the roles of eco-

nomic optimization and fairness in coalition formation of

joint projects. A brief overview of some of the important

concepts on natural monopolies will provide an understanding

of these implications.

A competitive market prevails if the number of sellers

is large enough so that no one seller is able to influence

the market price by a unilateral change in output. Compet-

itive equilibrium is assumed to occur where marginal costs

equal marginal benefits, i.e. where the supply and demand

curves intersect. A competitive equilibrium has the prop-

erty known as Pareto optimality where net benefits are

maximized so that no other feasible distribution of outputs

can improve the welfare (measured in terms of net benefits)

of one individual without reducing the welfare of another.

However, as demonstrated in the next section, maximum net

benefits do not always correspond to marginal conditions.

Therefore maximum net benefits is a more generally appli-

cable condition for economic optimality.

In contrast to the competitive market is the monopo-

listic market with only one active seller (or firm) in the










market. Sharkey (1982a) describes conditions for the

formation and stability of a natural monopoly which results

when a single firm can produce at lower cost than any

combination of two or more firms in a market where firms are

able to compete on an equal basis. Entry of firms in a

natural monopoly could only reduce social welfare (measured

in terms of cost of producing a given output) by raising

total costs of production. Assuming all firms in the market

have the same cost function C, a single firm is more effic-

ient than two or more firms if



C(q) < C(q ) + + C(qk) (1)



where q = vector of outputs in a particular market =

(ql' n)
1 k
q ,. q = output vectors that sum to q.,



If inequality (1) holds for all feasible disaggregations of

q, then C is subadditive at q and the market is a natural

monopoly. Indeed, a natural monopoly exists if and only if

the cost function is subadditive, that is if



C(q) < C(x) + C(q-x) (2)


for any x such that 0 < x < q.









Economies of scale exist where average costs fall with

increasing output or where

C(Xq) < X C(q) (3)

for all A such that 1 < X < 1 + E where E is a small posi-

tive number. Economies of scale are sufficient but not

necessary conditions for subadditivity of single output cost

functions. However, in a multiple output market, subaddi-

tivity holds only if there are also economies of joint

production. One such measure is economy of scope which

exists if a single firm can produce any vector of outputs

more efficiently than two or more specialty firms for a

constant level of production of each output. While economy

of scope is a useful intuitive concept, it is not particu-

larly helpful in deriving actual sufficient conditions for

subadditivity. Therefore, Sharkey presents other specific

properties of the cost function which may be empirically

verified. See Sharkey (1982a) for a discussion of these

properties.

Sharkey (1982a) observes that economies of scale in the

production of a private good are closely related to the

concept of public goods which are consumed collectively

rather than individually. A fixed cost in the production of

such a good has the property of collectiveness since it is

consumed and must be paid by the consumers collectively.

Economists have long recognized that collective goods cannot

be efficiently allocated through the market system. As a

result, various forms of public control have evolved for









financing and allocating collective goods. Therefore,

natural monopolies (such as public utilities) are often

referred to as regulated firms in the economics literature.

Given that under subadditive cost conditions, the

natural monopoly is the least cost (or maximum welfare)

market configuration, the question remains as to how to set

regulated prices. Since the competitive equilibrium is

economically efficient, one alternative might be to regulate

prices to equal marginal costs for a given level of produc-

tion. However, in many regulated industries where there are

often large fixed costs, marginal cost is below average

cost. With prices set at marginal costs, total revenues

fall short of total costs. A regulated firm would be able

to survive only if subsidized by revenues raised in other

parts of the economy. Such subsidies might be distributed

by the government. Alternatively, owners of a regulated

firm might receive compensation by the firm's customers for

the use of the owner's capital. The legal precept in the

United States requires that regulated firms operate at a

zero profit, where all costs are considered. The problem of

determining regulated, economically efficient prices is

addressed in many theoretical models. Zajac (1978) con-

siders several simple models such as Ramsey prices, two-part

tariffs, and self-selecting two-part (block) tariffs. As

Zajac (1978) observes, economic theories on regulated firm

pricing have focused mainly on the question of economic









efficiency with very little emphasis on issues of justice

and fairness.

Zajac (1978) further observes that even if the cost

structure is subadditive and a single supplier can provide

all services cheaper than some combination of separate

suppliers, the possibility may exist that some group of

services can be provided cheaper by a separate supplier.

This then is the focus of the modern theory of natural

monopoly in which models by Zajac (1972) and Faulhaber

(1972, 1973) demonstrate some of the inherent contradictions

involved in setting prices that are both economically

efficient and sustainable (no cross-susidization). Fol-

lowing is a brief discussion of these concepts of efficiency

and cross-subsidization and how they relate to issues of

economic analysis and cost allocation of joint projects.

Although natural monopoly is by definition the most

efficient form of production, there may be no price that

simultaneously satisfies total market demand, provides

revenues to cover total cost, and discourages entry by other

firms. This situation is illustrated in Figure 1 for the

case of a single output natural monopoly. To satisfy demand

and revenue requirements, the price is set at P > P m

However, to discourage entry the price must be set so that

P = P < P This natural monopoly is therefore unsustain-
o m
able. A natural monopoly for which an entry deterring price

exists is termed a sustainable natural monopoly. This

concept was first defined by Baumol, Bailey, and Willig






























Price m
$/unit p
o


Average
Cost -. ,


Demand


Output, units


Figure 1. Example of an unsustainable natural monopoly.








(1977) and Panzar and Willig (1977). When an unsustainable

multiple output natural monopoly operates under conditions

where entry by rival firms is barred, prices must be such

that one output is subsidizing another. The theory of

cross-subsidization is proposed by Faulhaber (1975), Zajac

(1972, 1978), Sharkey (1982b), and Sandberg (1975).

Sharkey (1982a) describes conditions under which both

single output and multiple output natural monopolies are

sustainable by two approaches; the intuitive concept of

sustainability and the theory of cooperative games.

Applying the former approach, Sharkey (1982a) defines

conditions for sustainability based on various cost function

properties including declining average costs which are

extended to the concept of the supportable cost function.

In the cooperative game approach, buyers in a market are

viewed as players in a cooperative game. The objective is

to form coalitions for which production and distribution of

outputs are most favorable. A natural monopoly exists if

the coalition of all buyers achieves a better outcome than

any partition of buyers into subcoalitions. A necessary

condition for natural monopoly is that the characteristic

function be superadditive (revenue maximization) or sub-

additive (cost minimization). The characteristic function

is given by the outcomes in terms of costs, benefits, or

savings for the various subcoalition combinations. The

characteristic function C for a cost game is subadditive if









C(S) + C(T) > C(S U T) SnT = c (4)



where S and T are any two subsets of the grand coalition N

and S U T # 1. The core of the game defines stable outcomes

for which no player or subcoalition of players would benefit

by leaving the grand coalition. Therefore, if the core

exists, the market will be a stable natural monopoly.

Sharkey (1982a) defines two constraints. If p =

(p pn) is the price vector, q = (ql qn) is the

output vector, and N represents the set of all buyers, then




E p.q = C(N) (5)
ieN



This constraint requires prices such that total revenues

cover total costs. In addition, to assure that individual

players or subcoalitions cannot be served more cheaply, the

price vector must satisfy the constraint



E p.q. < C(S) (6)
iES



for all subsets S C N. Sharkey then proves several theorems

that guarantee the existence of the core. These conditions

include properties of the cost function that are essentially

the same as the sufficient conditions for subadditivity and

sustainability. As will be seen in a later discussion,









these same cooperative game theoretic results have been

directly applied to several current cost allocation methods.

Finally, Sharkey (1982a) provides two warnings with

respect to the cooperative game model. First, the model

assumes that coalition formation is costless. However, real

coalitions form with an additional transaction cost (Heaney,

1983). Secondly, the cooperative game model is derived from

the underlying noncooperative game where the monopolistic

firm and rival firms are players. As a result, important

aspects of the interaction between players may be lost.

Zajac (1978) provides some additional, very important

warnings. First, for games involving more than just a few

players, the number of possible groupings that must be

evaluated may be formidable. Perhaps most importantly, the

approach generally implies consumption of prespecified

quantities of goods or services. This assumption might be

valid where all buyer demands are completely insensitive to

price variations. However, this is rarely the case.

Therefore, analyses must examine variations in quantities

demanded with varying prices. As price and quantity vary so

do total revenues and costs, resulting in important implica-

tions for the efficiency of various stable price vectors.

For example, if efficiency is defined in terms of maximizing

overall benefits (derived from demand) minus costs (derived

from supply), the optimal single firm or grand coalition

configuration allocates output for an implied price vector.









This price vector need not be stable even though a set of

stable price vectors may exist. Any of these other price

vectors would not encourage efficient output consumption and

so would yield a net benefit necessarily less than that

implied by strict optimization. Herein lies the inherent

conflict between efficiency and equity in regulated firm

pricing. From the game theoretic approach, this concept may

be extended to the case where costs must be apportioned

among groups in a joint project.



Optimization

Evaluation of the efficiency of an endeavor implies

optimization which involves the determination of a highest

or lowest value over some range. Jelen and Black (1983)

group optimization problems into three categories. The

first, preferential optimization is subject to preference

and taste only. The second, mathematical or physical

optimization is not subject to these considerations. The

third category is economic optimization which is a combina-

tion of preferential and mathematical optimization where

preference elements are expressed quantitatively. Econo-

mists define economic efficiency in terms of Pareto opti-

mality where resources are allocated so as to most effec-

tively serve consumers' tastes and preferences. Efficiency

in the choice of quantities of different outputs requires

that for each output, marginal costs of production equal

marginal benefits to each buyer. As noted, the competitive









equilibrium automatically satisfies this condition. How-

ever, as noted earlier marginal conditions do not always

correspond to maximum net benefits. When the marginal cost

curve lies above the marginal benefit curve for lower output

levels, the point where the two curves meet actually corres-

ponds to minimum net benefits. In addition, where produc-

tion capacity constraints restrict output, marginal

conditions may never even be reached. Here, economic

efficiency can be defined only in terms of the difference

between total benefits and total costs with the optimum

corresponding to maximum net benefits. This situation is

examined more closely for a special case in the wastewater

reuse example problem. For benefits and costs measured in

terms of dollar worth, the conditions for economic effi-

ciency are defined as in Figure 2.

Application of these economic principles to the evalua-

tion of engineering projects requires analysis of the demand

for project output and project costs. Marginal costs

represent supply while marginal benefits depict the demand

curve. Project evaluation, then, requires determination of

costs and benefits as a function of the various outputs

followed by maximization of their difference. Determination

of project costs and benefits involves application of

various cost engineering principles and techniques (e.g.,

see James and Lee, 1971). Rather, this study focuses on

efficiency and equity issues for previously determined














Marginal
Values
$/unit


Costs



Benefits


p I--


Output, units
(a)


Total
Values
$


Qm Q
Output, units
(b)



Figure 2. Conditions for economic efficiency: (a) marginal
benefits equal marginal costs for output Q
correspond to (b) maximum net benefits for
output Q. (Q corresponds to optimal output
conditions wen production is limited to Q ).









project costs and benefits. For a given set of cost and

benefit information, various mathematical results and

numerical techniques of optimization theory may be applied

to project evaluation. An overview of some of the more

important results and techniques for engineering applica-

tions is presented next.

Optimization theory addresses problems requiring

minimization or maximization of a real-valued objective

function subject to satisfying a number of equality and/or

inequality constraints. The objective function and con-

straints are expressed in terms of the decision variables

(output quantities). Linear programming (LP) defines a

class of optimization problems in which the objective

function and constraints are linear. The problem is solved

by means of an iterative procedure. The most widely used is

the simplex method which proceeds by moving from one fea-

sible solution to another in such a way so as to improve the

value of the objective function. For a more complete

discussion, see any of several texts including Beightler et

al. (1979), McMillan (1975), and S.P. Bradley et al. (1977).

With the general availability of several commercial

software packages, linear programming provides an effective

means of solving many classes of engineering problems. In

particular, network models which involve the distributions

of products from plants to consumer markets exhibit a

special LP structure that can be exploited in developing an

efficient solution algorithm. S.P. Bradley et al. (1977)









describe a general solution approach derived from speciali-

zing the rules of the simplex method. Many water resource

problems are conveniently modeled as network problems and

are readily solved using appropriate solution algorithms.

Mandl (1981) surveys the state-of-the-art in network models

and algorithms that can be applied to planning of irrigation

and wastewater systems. For the case where total treatment

plant and pipe costs are linear functions of capacity and

all capacities and fixed demands are known, the problem is

formulated so that the objective is to minimize costs

subject to conservation of flow constraints. To solve this

problem G.H. Bradley et al. (1977) and Maurras (1972) have

developed codes that are specializations of the simplex

algorithm. While both algorithms find the global optimum,

the suitability of a linear objective cost function restricts

application of the model. Furthermore, the codes are not

generally available and can only be obtained from the

authors. Integer programming models can incorporate fea-

tures to form a more complex network model than the strict

LP model. The technique has generally been applied to the

fixed-charge model which assumes a linear cost function but

with fixed project costs. However, integer programming may

be used to approximate nonlinear functions by means of

linear equations coupled with logical restrictions (integer

variables). This piecewise linearization method is analogous

to the delta-method applied in separable programming as

described later in this paper. Using a special branch-and-









bound algorithm by Rardin and Unger (1976), Jarvis et al.

(1978) apply integer programming as a fixed-charge model for

design of wastewater systems exhibiting economies of scale.

Standard integer programming codes are commercially avail-

able, e.g. the integer programming code by IBM Corporation.

Although many real world problems are nonlinear, the

availability of powerful linear programming methods provides

a major incentive for approximating nonlinear problems in

linear form. Therefore, one common approach to the general

problem is to replace nonlinearities with linear approxima-

tions and solve the resulting linear program. Reklaitis et

al. (1983) explore three basic strategies for employing

linear approximations in solving nonlinear problems. All of

these methods approximate a nonlinear function f(x) in the

vicinity of a point X0 by a Taylor series expansion where

higher order terms are ignored. The point X0 is called the

linearization point. The three basic linearization methods

differ primarily in the manner and frequency with which the

linearizations are updated. In the direct successive LP

approach, the LP solution defines the direction for a line

search. Two successive LP algorithms, the Frank-Wolfe

algorithm.for linearly constrained problems, and the

Griffith and Stewart method for nonlinear constraints, are

presented in Reklaitis et al. (1983) and S.P. Bradley

(1977). The Frank-Wolfe algorithm forms a linear approxima-

tion at the point X by replacing the objective function

with its current value plus a linear correction term. The









resulting LP solution is used to define a search direction.

The search direction is given by the line segment joining

the LP solution and the linearization point. Therefore, the

successive LP approach may be viewed as an alternating

series of LP and line-search subproblems. To maintain

feasibility in the nonlinearly constrained problem, the

Griffith and Stewart method bounds the step size for each

intermediate LP solution. Since this algorithm proceeds

rather slowly towards a solution, the successive LP approach

is most appropriate for problems with only a few nonlinear

terms.

The idea of the cutting plane approach is to succes-

sively improve the linear approximations to the constraint

boundary in the region near the solution as the solution is

approached from outside the feasible region. Gottfried and

Weisman (1973) describe Kelley's algorithm in which linear-

ized constraints (cutting-planes) are introduced one at a

time to successively eliminate portions of the previous

approximation of the feasible region. The resulting series

of LP subproblems can be solved using specialized LP methods.

However, this approach is not applicable for equality-

constrained problems. In addition, convergence is ensured

only for convex problems. In spite of these difficulties

this strategy has been effectively used to solve some

specially structured problems.

The third linearization technique is separable program-

ming which utilizes piecewise linear approximations of









separable nonlinear functions over the full range of the

problem variables. The method is applicable to problems of

the form



n
Maximize or (Minimize)
E fj(xj)
j=l 3





n
Subject to E g. (x) < (= or >)0 (i = 1,2,...m)
j=1 J



x. > 0 (j = 1,2,...n)



Because decision variables appear separately in each func-

tion f. and gij, the objective and constraint functions are

separable. Instead of solving this nonlinear problem

directly, an effective strategy is to make appropriate

approximations so that linear programming can be used.

Hadley (1964) and S.P. Bradley et al. (1977) examine two

commonly used approximation techniques, the delta (6)-method

and the lambda (X)-method. Although ensuring only a locally

optimal solution, this approach is particularly suited to

large network models since the separable programming mode of

commercially available LP codes can efficiently solve very

large problems. This model frequently arises in engineering

applications particularly water resource planning formula-

tions.









Currently, most mathematical programming applications

to water resource problems incorporate linearization tech-

niques and linear programming solutions. The linearization-

based algorithms often cannot give acceptable estimates for

the boundaries of the feasible region or the objective

function. Rather than rely on inaccurate linearization to

define the location of a point, linear approximations might

be used only to determine a good local direction for search.

Examination of values of the original objective and con-

straint functions can then yield the optimal point along the

search direction. This strategy is analogous to uncon-

strained gradient search methods such as the conjugate

gradient and quasi-Newton methods where linear approximation

is used to determine a good search direction and actual

function values guide the search along this. direction. In

the constrained case directions must be chosen to yield

feasible points. Reklaitis et al. (1983) provide an excel-

lent discussion of these direction-generation methods.

The group of direction-generation techniques called

feasible direction methods requires solution of an LP

subproblem to determine a direction that is both a descent

(or ascent) direction and feasible. The objective of the LP

subproblem is to solve for the direction that maximizes the

increase (or decrease) in the objective function subject to

feasibility constraints. This method has disadvantages of a

slow rate of convergence and inability to directly accommo-

date nonlinear equality constraints.









Another group of methods known as generalized reduced

gradient methods (GRG) simply solves a set of linear equa-

tions instead of the LP solution to determine a favorable

search direction. This method uses all nonbasic variables

to define the direction and so is a generalization of the

convex simple method, the direct analog to the linear

simplex method. Incorporation of conjugate gradient or

quasi-Newton strategies accelerates convergence of this

method. The method may also be extended to accommodate

nonlinear constraints. A number of GRG codes are currently

available. Reklaitis et al. (1983) survey the major compar-

ative studies which examine the relative merits of the

various methods and codes.

In addition to strategies for exploiting linear approx-

imations to nonlinear problem formulations are methods that

use higher order approximations, specifically quadratic

approximations. Essential to the strategy for employing

quadratic approximations is the method of quadratic program-

ming. Quadratic programming involves use of a simplex-like

algorithm to obtain a solution for a problem consisting of a

quadratic objective function and linear constraints. Since

the partial derivatives of a quadratic function are linear,

a modified linear programming problem is generated by

application of Kuhn-Tucker conditions (and thus forming

partial derivatives). The method yields a local optimum

which is guaranteed globably optimal only for strictly

concave or convex functions. Hadley (1964) discusses








several computational techniques for solving quadratic

programming problems.

Reklaitis et al. (1983) examine several approaches

using full quadratic approximation of objective and con-

straint functions which prove no easier to solve than the

general nonlinear problem. In addition, formulation of

quadratic programming subproblems (quadratic objective

function and linear constraints) results in no significant

improvement over the successive LP approaches. However,

formulation of a subproblem objective function with the

quadratic term as the second derivative of the Lagrangian

function does provide the basis for an efficient algorithm

for generating good search directions. The difficulty of

providing second derivatives for the problem functions is

resolved using quasi-Newton methods which only require

differences of gradients of the Lagrangian function to

approximate and update the second derivative. The result is

a sequential quadratic programming (SQP) algorithm in which

the solution to the quadratic programming subproblem defines

the search direction. Reklaitis et al. (1983) survey some

of the major studies which examine the relative merits of

the various quadratic approximation methods and codes. The

survey also includes studies which compare various nonlinear

programming methods and codes based on criteria such as

efficiency, global convergence, ability to solve various

types of problems, and ease of use. In general, nonlinear

programming techniques have not been applied to water








resource problems. However, results of the comparative

studies indicate that GRG and SQP have features that are

important to engineering optimization.



Cost Allocation

Economic efficiency in many water resource projects may

be achieved by taking advantage of (1) economies of scale in

production and distribution facilities, (2) the assimilative

capacity of the receiving environment, (3) excess capacity

in existing facilities, (4) multipurpose opportunities,

and/or (5) multigroup cooperation (Heaney and Dickinson,

1982). The results of economic analysis often indicate that

participants should be combined into a cooperative joint

venture. Like natural monopolies and public goods, the

market system may not efficiently allocate the benefits (or

costs) of joint water resource projects. Therefore, given

an economically efficient joint project design the task of

distributing the economic impacts among all of the partici-

pants remains. The objectives of cost allocation include

(1) satisfaction of the financial requirement that project

revenue equals project cost; and (2) satisfaction of econo-

mic requirements so as to encourage optimum use of project

output. As noted previously, financial and economic

requirements are satisfied simultaneously under pure com-

petition. The market reaches equilibrium where marginal

cost equals marginal benefits. At this intersection margi-

nal cost also equals average cost so that the resulting









price vector P and output vector Q satisfy both economic and

financial requirements.

The absence of such a competitive market system

requires some form of public control to administer prices

which properly allocate resources. For example, when

average costs are decreasing, a price based on efficiency

does not satisfy the financial requirement. This can be

seen in Figure 3 where the optimal output Q and corres-

ponding marginal price P are still given by the inter-

section of marginal cost and marginal benefit. However,

average cost exceeds marginal cost so that a price P is
a-

required so that revenues will just cover costs. Unfor-

tunately, average cost pricing at P restricts use and

forces a suboptimal situation.

In contrast to economies of scale, conditions of

increasing average cost result in an optimum pricing scheme

that more than satisfies the financial requirement.

Figure 4 shows that for increasing average costs, marginal

cost lies above average cost so that marginal pricing

produces a net revenue of Q(Pm P ). In water resource
m a
applications this presents a problem of what to do with the

net revenue. According to project financial requirements,

only the cost legally obligated during project construction

and operation must be recovered.

With the competitive market as a model for a "first

best" pricing scheme, a common pricing approach is to

pretend that a market exists for project outputs. The

















Price
$/unit


Output,units


Figure 3. S
ci












Price
$/unit


election of price under decreasing average
ost conditions.


Average
Cost



Demand


Figure 4.


Output, units


Selection of price under increasing average
cost conditions.









objective of the resulting market analysis is. to determine a
"second best" allocation of resources given that prices are

constrained to deviate from marginal cost. This strategy

has dominated regulated firm pricing schemes where

decreasing average cost conditions are predominant (Zajac,

1978). James and Lee (1971) describe three basic approaches

for this pricing strategy as applied in water resources

planning. One approach is to employ price discrimination to

capture some of the consumer surplus. The objective in

price discrimination is to charge those users receiving

marginal benefits greater than Pm (refer to Figure 3) an

extra fee to satisfy the financial requirement. Even though

prices above marginal cost reduce full economic utilization,

the net effect may be a reasonable compromise between the

economic and financial requirements. A second approach is

to maintain marginal pricing while raising the additional

required revenue through a fixed, nonmarginal charge. Each

user would pay a flat fee plus a per-unit charge equal to

marginal cost. A final approach is to set price equal to

marginal cost while supplementing the resulting revenue

through subsidies. Such subsidies are usually distributed

by the government and involve redistribution of income from

those paying taxes to those receiving subsidies.

Despite the potential of marginal cost pricing, actual

practice often dictates application of charges based on some

measurable unit called a charging vehicle. James and Lee

(1971) describe three such charging vehicles which include









(1) ability to pay, (2) benefit received, and (3) quantity

of output or average cost. While schemes involving each of

these vehicles can be formulated to satisfy financial

requirements, a theoretical difficulty centers on economic

grounds. Economists argue that unless price equals marginal

cost, users are not given the proper incentive to balance

the value they derive from use of incremental output against

its marginal opportunity cost. Indeed, there has been

continued interest in the application of marginal cost

pricing techniques to water resource planning. For example,

Hanke and Davis (1973) report significant potential for

marginal cost pricing for municipal water services, indus-

trial and municipal sewage treatment, navigation, and flood

control. Guariso et al. (1981) present an iterative algor-

ithm for determining optimal water supplies and demands in a

regional network where marginal cost equals marginal bene-

fits. The authors propose use of resulting marginal cost

prices as reference points for evaluating prices determined

by conventional methods.

Despite continued interest in marginal cost pricing,

most water resource planners and governing agencies rely on

numerous (some ad hoc) methods of dividing project costs

among participants. Loughlin (1977) expresses the pre-

vailing view that economic and financial analysis are

independent of each other. The rationale is that joint

costs which cannot be directly attributed to a participant

are not marginal and so do not influence optimal design









(except for total project justification). Since joint costs

must be paid, financial analysis is required to allocate

them among participants. Consequently, economic efficiency

in water resources planning is usually defined by the

maximum positive difference between total benefits and total

costs. Cost allocation, then, involves apportionment of the

costs of the economically optimal solution among partici-

pants. Notably absent from this concept of economic effi-

ciency is the complete notion of Pareto optimality which

requires that for an economically efficient allocation of

resources, no other feasible distribution of output can

benefit one participant without harming another. In prac-

tice, the use of an incomplete definition for economic

efficiency results in cost allocation schemes that imply

prespecified demand quantities relatively insensitive to

price. This model may or may not be appropriate depending

on the actual supply and demand conditions.

The objective of the cost allocation phase of water

resources planning is to determine a vector of charges that is

acceptable to all project participants. If charges are not

acceptable to all, some may choose not to participate

resulting in a suboptimal final project design. In other

words, the objective is to define an equitable vector of

charges for which no individual or group of participants

would benefit by leaving the joint project. This exactly

corresponds to the concept of sustainability of natural

monopoly pricing which can be described using cooperative









N-person game theory as in Sharkey (1982a). In fact,

several of the conventional or proposed procedures for

allocating costs directly incorporate cooperative game

theoretic concepts.

Heaney (1979a) describes the evolution of current

practice beginning with the Tennessee Valley Authority (TVA)

studies in the 1930's and including the Federal Inter-Agency

Studies in the 1950's. Comparison of conventional method

criteria and game theoretic concepts reveals important

similarities including the requirement for full recovery of

costs and the notion that no participant should be charged

more than he would pay if he acted independently. These

requirements constitute the set of imputations of a coopera-

tive game. Heaney and Dickinson (1982) propose an extension

of current practice to require that charges satisfy the full

core constraints so that no group of participants is charged

more than it would pay if acting alone. Following is a

brief description of some commonly used conventional and

proposed allocation methods.

James and Lee (1971) identify cost allocation methods

by the definition of cost used, the cost directly assigned

to each participant, and the cost allocation vehicle. The

result is a matrix of 18 possible ways to apportion costs.

Three of the most commonly used cost allocation methods

include the use of facilities, alternative justifiable

expenditure (AJE), and separable costs remaining benefits

(SCRB) methods.









Loughlin (1977) describes the use of facilities method

which allocates joint costs in proportion to some measure of

the relative use of the central project facilities by each

participant. Joint costs are determined by subtracting

either direct costs or separable costs. Direct costs are

defined to be the costs of the elements of the project which

are used solely by that participant. Separable costs are

the differences between the total multigroup project cost

and the cost of the project without the participant. They

include direct costs as well as the incremental costs of

changing the size of joint cost elements. The use of

facilities method is generally considered acceptable only

where joint use is clearly determinable on a comparative

basis (Loughlin, 1977). The method does not incorporate any

of the stability criteria included in other conventional

methods.

Recommendations by the Federal Inter-Agency River Basin

Committee extend the notion of equity in cost allocation by

requiring that participants receive a proportional share of

the savings resulting from joint projects. Consequently,

the Committee recommends use of the SCRB method and the AJE

method when the effort or expense to obtain information for

the SCRB method is prohibitive (Loughlin, 1977). Heaney

(1979a) describes the AJE method for which each participant

is assigned his direct cost plus a share of the remaining

joint costs (total cost less all direct costs) in proportion

to his alternative costs avoided. Alternative costs avoided









are defined as the difference between the participant's

stand alone cost and direct cost. The method recognizes

that the benefit to each individual participant could be

less than the go-it-alone cost. In addition, the

alternative justifiable expenditure method is often much

easier to calculate than the SCRB method described below.

The method has been favored by the Tennessee Valley

Authority as well as the U.S. Environmental Protection

Agency.

In contrast to the AJE method, the SCRB method uses

separable costs rather than direct costs to determine joint

costs. The SCRB procedure assigns to each participant his

separable costs plus a share in the remaining joint costs in

proportion to the remaining benefits (as limited by alterna-

tive costs). For convex games the SCRB solution lies in the

center of the core. A game is convex if



C(S) + C(T) > C(S U T) + C(S n T) S nT # (7)



for all S and T subsets of the grand coalition N and C

defined previously as the characteristic cost function. The

Federal Power Commission, the U.S. Army Corps of Engineers,

and the Bureau of Reclamation have relied on the SCRB method

almost exclusively. All water resource agencies are

applying the SCRB method for multiobjective and multipurpose

federally assisted reservoir projects (Loughlin, 1977).

Although conventional methods such as the SCRB method

incorporate fairness criteria similar to some of those









employed in cooperative game theory, corrections are

required to ensure that the resulting vector of charges is

sustainable (lie within the core if the core exists).

Heaney (1979b) proposes that incorporation of individual

benefits in the SCRB method should be extended to include

subcoalition benefits. In addition, Heaney and Dickinson

(1982) propose a generalization of the SCRB method to

incorporate full core constraints. The rationale here is

that for games where the separable costs which are the

incremental costs for each participant (player) to join the

coalition last are not the lowest incremental costs, the

SCRB method prorates joint costs based on upper and lower

bounds that are not in the core. Consequently, for noncon-

vex games the SCRB solution does not lie in the center of

the core and may not lie within the core at all for extreme

cases. The proposed method involves solution of a system of

linear programs to delineate the core bounds followed by

proration of joint costs based on these actual core bounds.

For the case where no core exists, Heaney and Dickinson

(1982) suggest relaxation of intermediate coalition con-

straints to determine a compromise solution. This genera-

lized SCRB procedure is called the minimum cost remaining

savings (MCRS) method.

Analogous to the "fair solutions" of the conventional

SCRB and MCRS methods are some of the unique solution

concepts used in game theory. Two of the most popular

unique solution notions are the Shapley value and the









nucleolus. The idea of the Shapley value is that each

participant should pay the incremental cost of adding him to

the coalition. With the assumption that all coalition

formation sequences are equi-likely, the Shapley value

assigns to each player the incremental cost he brings to

coalitions expected over all coalition formation sequences.

The resulting charge to the ith participant is



X(i) = Z a.(S) [C(S) C(S {i})l (8)
SCN
S C 1N



where ac(S) = (S -) (n-S)!
1 n!



and n is the total number of participants. If the game is

convex, the Shapley value is in the center of the core.

However, for nonconvex games the Shapley value may fall

outside of the core. In addition, computations for projects

involving more than just a few participants are quite

tedious. Littlechild and Owen (1973) present a simplified

Shapley value for application in a special class of multi-

purpose projects where the purpose with the largest cost of

separate action in a coalition determines the characteristic

function cost for the entire coalition. Littlechild and

Thompson (1977) demonstrate the advantages of the simple

Shapley value for determining aircraft landing fees. Heaney

(1979a) demonstrates how the simple Shapley value might be

applied to multipurpose water resource projects through an









example for pollutant pricing in a wastewater treatment

plant.

In addition to the computational problems, Loehman et

al. (1979) find another deficiency in application of the

Shapley value to real situations. Loehman et al. (1979)

argue that it may be unrealistic to assume that all orders

of users are equi-likely as assumed for the Shapley value.

Loehman and Whinston (1976) have developed a generalized

Shapley value that represents the expected incremental cost

where all orders are not equi-likely. Loehman et al. (1979)

apply the general Shapley value to an eight-city regional

wastewater treatment system in which coalition sequences

that do not occur are identified and their probabilities set

to zero in Shapley value computations. In the eight-city

example, "impossible" coalition formation sequences are

given as those that are not economically viable. This

application of the general Shapley value is not correct for

the conventional definition of the characteristic cost

function. The characteristic function C(S) is usually

defined as the optimal solution for that coalition. At

worst, no lower cost results in coalition formation in which

case the coalition is said to be inessential, that is



C(S) + C(T) = C(S U T) S n T = t (9)



for all subsets S and T in N. By setting to zero the

probabilities of inessential coalitions, weak players that









acutally contribute the smallest savings to the grand

coalition become stronger players and are undercharged while

actual strong players are overcharged. This distorts the

game and may lead to very unfair charges. If the criteria

for identifying impossible coalition sequences are

restricted to include only political or strictly physical

considerations, application of the general Shapley value

might be appropriate.

The nucleolus maximizes the minimum savings of any

coalition and requires N-1 linear programs. The solution

satisfies all of the core constraints, always exists, and is

unique. See Heaney (1979a) or Lucas (1981) for more infor-

mation on computing the nucleolus.

At this point in the discussion, two particular points

warrant further consideration. First, the matter of

defining an appropriate characteristic function presents

some problems. Throughout much of the game theory and cost

allocation literature the characteristic function is assumed

given with little consideration of how it is derived.

Conventional and game theory concepts and methods are of

little value unless the required characteristic function

values can be unambiguously defined. Sorenson (1972)

defines 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









C3(S) = value of coalition in a strictly competitive
game between coalition S and N-S

C4(S) = value of coalition to S if N-S is given
preference



As will be demonstrated in this study, alternative defini-

tions can be used depending on how the problem is defined.

The second consideration involves the notion of strict

optimization in engineering design. Often economic require-

ments call for a large or complex project design that may be

difficult if not impractical to implement. The preferred

solution may be a relatively efficient design that is easier

to implement. The argument for good suboptimal solutions is

founded on several grounds. Heaney (1983) observes that as

the size of a regional project increases, transaction costs

might be expected to increase at the margin due to multiple

political jurisdictions, growing administrative costs and

shifting of environmental impacts. Although most economic

analyses ignore these transaction costs, Heaney (1983)

estimates that they may run from 2 to 10 percent of total

costs. When transaction costs are high an intermediate but

simpler solution might be preferred. Unfortunately, little

work has been done examining viable intermediate economic

solutions. One reason for this is that the availability of

powerful optimization techniques such as linear programming,

reduced gradient, and sequential quadratic programming

techniques ignore suboptimal solutions that may be close

enough to the optimum to be satisfactory. Wilde (1978)









develops the concept of "satisfactory design" as an approach

to convert relatively difficult optimization problems into

the construction of easy to compute bounds and estimates of

the ideal optimum. Finally, the viability of suboptimal

designs is especially apparent given the uncertainty of

physical and economic data. With regard to wastewater

treatment systems, cost estimates are usually based on power

function approximations of available data. Consequently,

first order cost estimates which merely specify treatment

process types such as primary sedimentation or activated

sludge may be in error by nearly 60 percent. Even more

detailed second order estimates involving specific component

cost estimates such as filtration media, backwash pumping,

surface washing facilities may err by more than thirty

percent (Clark and Dorsey, 1982). Likewise, performance

estimates are often highly variable. Clearly, emphasis on

strict economic optimization may be inappropriate when

uncertainty of data is high. Suboptimal solutions may be

obtained with less effort and expense while achieving a

satisfactory result. Given this wide variability in cost

estimates, it is not surprising that practicing profes-

sionals do not seem to concern themselves with the average

cost/ marginal cost controversy.



Summary

Several important observations may be extracted from

this literature review. First, economic theory addresses









only the case where the optimal design corresponds to

marginal benefits equal to marginal costs. Economists argue

that given this optimal condition, the proper pricing scheme

in a monopolistic (economies of scale) market is one that

produces the least damaging deviation from strict marginal

pricing. Economic theory fails to address the problem that

arises when the economic optimum does not correspond to

marginal benefits equal to marginal costs. While recent

theories on both natural monopoly pricing and engineering

cost allocation incorporate the concept of sustainability,

current cost allocation methods typically ignore marginal

cost considerations. Instead marginal considerations are

viewed as strictly a part of the economic evaluation phase

of project planning and therefore independent of the cost

allocation phase. A major point of discussion in this paper

concerns the applicability of these two conflicting view-

points. Finally, with regard to economic optimization, most

water resource applications involve linear programming

solutions to the linear cost and fixed-charge problems.

Although economies of scale in project construction and

operation are typical for many water resource projects,

incorporation of concave cost functions has generally been

avoided due to conceptual and computational problems.

However, recently developed codes may provide a means of

formulating more realistic models for engineering applica-

tion.
















Section II

METHODOLOGY



The purpose of this section is to develop a general

approach for examining the relationship between the economic

and financial objectives in water resources planning.

Following is a suggested sequence of steps that provides a

means of describing and evaluating this relationship. The

steps are listed below in general terms and are followed by

a discussion of possible results and conclusions for a

simple wastewater reuse example. The discussion includes

three different cases defined according to problem objec-

tives and corresponding cost and benefit relationships.



Project Evaluation Steps

1. Examine overall project objectives.

2. Define the system for study.

3. Formulate the problem in terms of specific objectives

and constraints for the system defined in step 2.

4. Examine how the problem objectives and constraints

relate to theory and available methodology, e.g.

economic theory, optimization techniques, and cost

allocation methods.









5. Apply appropriate methodologies based on the analysis

in step 4.

6. Examine implications for economic efficiency and equity

in terms of tradeoffs and possible compromise solu-

tions.



Discussion and Wastewater Reuse Example



Step 1. Examine Overall Project Objectives

This first step is required to formulate the problem.

Often the motivation behind a proposed water resource

project involves political, social, and environmental

considerations as well as economic and financial objectives.

Clear understanding of the major and underlying objectives

is essential to the evaluation of tradeoffs and viability of

possible compromise solutions. Donovan et al. (1980)

discuss several of the possible considerations involved in

wastewater reuse planning. One objective might be to reduce

water supply costs to a group of users by taking advantage

of economies of scale in a wastewater reuse project. Other

objectives might include increasing available water during

drought by decreasing demand on the community's existing

water supplies. In this way potable water supplies can be

protected for more valuable use. Other goals may require a

reuse system to provide social benefits such as development

of municipal recreational facilities. Still other motiva-

ting factors might involve improvement or shifting of









environmental quality impacts. An underlying issue for any

objective is the question of how best to apportion project

costs (or benefits). The objective of the resulting finan-

cial analysis might be an equitable but simple allocation of

project costs (or benefits). Inherent tradeoffs may develop

between these two financial objectives (equity and

simplicity) as well as among financial and economic, social,

or political objectives. Finally, planning objectives must

incorporate any legal or institutional constraints.

The objective for the example wastewater reuse study is

to determine the system design that minimizes total water

supply costs to potential wastewater users and to fairly

allocate these costs among participants.



Step 2. Define the System for Study

Before the problem can be properly formulated for

analysis, the system under consideration must be carefully

defined. A proper definition includes identification and

description of technical, economic, social, legal, and

political relationships. Although this is an extremely

important phase of water resources planning, it is often

quite complex. See Sample (1983) for detailed treatment of

this phase in an actual wastewater reuse study for the South

Florida Water Management District. Detailed consideration

of how the system is determined for this example might

detract from the emphasis of this paper. Therefore, the

final system for the example problem is simply given as









shown in Figure 5. The system consists of a secondary

treatment plant as the single wastewater supply source and

four irrigation sites. Locations of pipes connecting the

treatment plant to each site are fixed by right-of-way

restrictions.

Costs for consideration in subsequent economic and

financial analysis include the cost of additional treatment

at the plant and transportation costs. Additional informa-

tion includes the demand schedule for each user. This is

derived from water use patterns or from alternative water

supply costs. Alternative water supply costs may be

referred to as on-site costs while treatment and trans-

portation costs may be termed off-site costs. In actual

practice, costs for consideration may also include costs for

water quality monitoring, replumbing, storm water runoff

control, and future capacity expansions.

Application of cost engineering principles results in

estimates for treatment and transportation costs. For

planning studies, cost estimates are often derived from

statistical cost equations corrected with appropriate

updating and localizing factors. The general form for the

equations is a power relationship of the major input and

output variables for construction and operation and mainte-

nance of the various project components and unit processes.

For the wastewater reuse example, a reasonable representa-

tion for treatment costs and on-site costs is given by a



















Site 4


4 = 2.5 MGD


Site 1


Treatment


Q2 = 2 MGD
m,2


Site 3


Qm,3 = 1.5 MGD


Figure 5. System for wastewater reuse example.


M,l1 = 3 MGD









power function of the flow variable. In contrast, trans-

portation costs which include pipe and pumping costs are

best estimated from static head, pipe diameter, and. pipe

length as well as flow. For a given pipeline path, the pipe

length is fixed and static head may be reasonably estimated.

As a result, cost estimates for the example transportation

costs may be reduced to functions of flow and pipe diameter.

In addition, for a given flow a tradeoff exists between pipe

costs which increase with increasing pipe diameter and

pumping costs which decrease with increasing diameter. Deb

(1978) presents an optimization model which selects the

optimal pipe diameter to minimize total transportation

costs. Sample (1983) modifies this procedure to include a

more detailed analysis of transportation component costs.

The result is a scheme to compute optimum pipe diameter as a

function of flow so that total transportation costs may be

presented as strictly dependent on treated wastewater flow.

All costs are estimated by a power function of the form



C = a Qb



where

C = annual cost, 10 $3

Q = treated wastewater flow, MGD

a,b = constant function parameters









All estimates include construction and operation and mainte-

nance costs and are assumed to be properly updated with

local information.

The specific equations for the example problem are

given in Table 1. Cost relationships for off-site costs

exhibit economies of scale as indicated by the exponent

parameters which are less than one for both treatment and

transportation. This is consistent with conditions found in

actual practice. For example, Sample (1983) found signifi-

cant economies of scale in both treatment and transportation

costs in the South Florida study. The example off-site

relationships are consistent with these general results.

On-site equations for the example problem are also given in

Table 1 for three cases including economies and diseconomies

of scale as well as linear on-site costs. These three cases

are examined to demonstrate different results and implica-

tions for economic and financial analysis.

A final consideration for defining the system involves

specification of maximum demand and plant capacity con-

straints. Two different plant capacity limits are consid-

ered corresponding to the case where supply does not limit

system design (plant capacity, Qm,t = 10 MGD) and, alterna-

tively, to the case where system design is supply limited

(plant capacity, Q m,t= 6MGD). This analysis assumes that

each potential wastewater user's current quantity of total

water use is equal to the maximum demand fixed by physical

requirements. However, each user's demand for recycled















Table 1. Cost Functions for Wastewater Reuse Example


Cost Component
7
On-site Treatment, Ct 60(Q of + Q0 2 + Qof3 + Q )


Site, i
1 2 3 4

Transportation, C 8(Qof, 5(Qof 13 (Qof) 17 (Qof,4) 6
pi of,1 of,2 of,3 of,4

On-site, C
on, i
Case 1
Linear Costs 80Qon,l 70Q on,2 90Qon,3 100Qon,4

Case 2 9 8
Economies of Scale 100(Qon,l) 80(Q on,2) 100(Qon,3)' 120(Qon,4) .7

Case 3 1.2
Diseconomies of Scale 25(Q on,)1.7 40(Q on,21.4) 45(Q0 n)1' 70(0 on,41.1
n, on,2 o n,3 .on, 4)









water is price dependent as given by on-site cost relation-

ships. Therefore, the demand constraint is that total water

supplied to each user must equal the fixed demand specified

in Figure 5. However, the quantities obtained on-site and

off-site are determined by the economic and financial

analysis in the steps that follow.



Step 3. Formulate the Problem in Terms of Specific
Objectives and Constraints

As stated in step 1 the objectives of the wastewater

reuse study are twofold. The first objective involves

economic efficiency for which the desired design is one that

minimizes total irrigation water costs for the defined

system. The second objective involves financial considera-

tions for which an equitable apportionment of final project

costs is desired. The economic objective and corresponding

constraints are specified in terms of the system defined in

step 2. The resulting economic optimization problem is to



4
Minimize Z = E (C + C .) + C
on,i p,' t


subject to Qoni + Qofi = i = 1, 2, 3, 4



4
Qofi m m,t
i=1









where


C
on,i
C =
p,I


Ct =

on,i =

Gof,i =
Qm,i

Qm,t =


on-site water supply cost for site i, 10 3$/year

transportation cost for pipeline to site i,

103$/year

treatment cost 103 $/year

on-site water supply to site i, MGD

off-site water supply to site i, MGD

maximum water demand for site i, MGD

treatment plant capacity, MGD.


This cost minimization problem is equivalent to the net

benefit maximization problem for benefits defined as alterna-

tive (on-site) costs avoided. In general, a correct specifi-

cation of core constraints incorporates individual and

subcoalition benefits as well as costs. For this example,

the problem is defined so as to include benefits implicitly

as alternative costs avoided.

With regard to financial objectives, recall that if the

core exists it contains the set of outcomes for which no

coalition would be better off not participating. This seems

a reasonable set of minimum criteria for an equitable

solution. Since the subsequent analysis shows that the core

does exist for each of the three cases in this example, we

can justifiably eliminate any allocation result not in the

core. The core may be represented by the following set of

constraints.









X(i) < C(i) i = 1, 2, 3, 4 (10)


E X(i) < C(S) S CN (11)
ieS


Z X (i) = C(N) (12)
iEN



where

C(S) = the characteristic cost function for

coalition S

X(i) = charge to player i

N = grand coalition of players.



Step 4. Examine How the Problem Objectives and Constraints
Relate to Theory and Available Methodology

This step involves examining how economic and cost

allocation theory apply to the problem as defined in step 3.

The water resource planning and pricing literature often

defines problems where the total potential demand lies

within the economies of scale range of project cost func-

tions. This is analogous to assuming that the project

design has no production capacity limit or that the capacity

limit is not binding. For the wastewater reuse problem, the

size of the project is limited by either total maximum

demand or treatment plant capacity. In addition, most

conventional cost allocation and game theory solutions do

not consider demand. Charges determined by methods such as

SCRB or the Shapley value can be stable in the economic

sense only for inelastic user demand. This assumption may








be valid when potential participants must satisfy a fixed

demand with no alternatives other than joining the project

coalition or constructing an independent project with the

same cost functions) as the joint project. Where there are

economies of scale, benefits grow as coalition size in-

creases, and the analysis becomes a problem of determining

how the benefits (costs) are to be distributed. A similar

effect results in the wastewater reuse example but is

complicated by production limits and demand considerations.

For the example we will examine three types of demand

relationships and the implications of production limits on

economic efficiency and cost allocation. As will be demon-

strated later in this section, economic optimization for the

wastewater reuse problem may be achieved rather easily for

certain types of cost relationships. However, in other

cases mathematical programming techniques may be required to

solve the general nonlinear optimization problem. Three

optimization methods: separable programming, generalized

reduced gradient, and sequential quadratic programming are

considered mainly on the basis of the availability of

computer codes.

Recall that separable programming is a linearization

technique which utilizes piecewise linear approximations of

separable nonlinear functions. This technique is applicable

to the example problem since all problem decision variables

appear separately in the objective and constraint equations.









Separable programming is contained within the linear program-

ming procedures of the Mathematical Programming System

(MPS/360) available through the IBM Corporation (1971). The

separable programming procedure employs the delta-method

described by Hadley (1964) in which the separable nonlinear

functions are approximated by means of linear equations

coupled with logical restrictions. The solution is given by

a simplex procedure modified to incorporate the required

logical restrictions. One caution in applying this tech-

nique is that the solution may converge to a local optimum.

This possibility may be reduced by solving both the dual and

primal problem or solving the problem for different initial

values for the decision variables to verify results.

Application of the MPS/360 code proved rather tedious

for the wastewater reuse example. The user is required to

construct appropriate piecewise linearizations for each

separable function. The example problem objective function

contains several different separable functions that must be

approximated. Furthermore, each set of computer input data

is very specific to the particular problem definition.

Relatively small changes in the problem definition often

require extensive alterations in the computer input.

Therefore, this method may not be practical for planning

studies involving many different nonlinear cost relation-

ships or studies requiring flexibility in defining these

relationships and problem constraints. This is the case for

the wastewater reuse example where the problem is defined









for various demand and constraint conditions. As a result,

this technique was abandoned in favor of a more flexible

code.

One such code is NPSOL by Gill et al. (1983) which uses

a sequential quadratic programming algorithm in which the

search direction is the solution of a quadratic programming

subproblem. The algorithm requires that the user define the

problem in terms of the specific objective and constraint

functions and their partial derivatives. Compared to

separable programming, this code requires much less prelim-

inary work by the user and easily accommodates changes in

problem definition. However, this code is very sensitive to

problem conditioning and is best suited to small, dense

problems containing nonlinear constraints. Attempts to

apply NPSOL to the example problem resulted in problems of

ill-conditioning not easily remedied.

The third code, GRGA by Abadie (1975), employs the

generalized reduced gradient method. As with NPSOL the user

must define the problem in terms of the specific objective

and constraint functions as well as partial derivatives.

Comparative studies indicate that this code is among the

most efficient and least sensitive to problem condition (see

Reklaitis et al., 1983). Unfortunately the only available

documentation for this code is in French. Additionally, the

program generates all output in French. However, applica-

tion of the code to the example results in no conditioning









problems. Therefore, results presented for the example

problem are those determined from GRGA.

Financial analysis for the example problem involves

evaluation of various cost allocation schemes including the

Shapley value and MCRS method as well as marginal and

average cost pricing strategies. As the financial objective

is to determine a set of charges that satisfy core con-

straints, the characteristic cost function must be defined.

In conventional cost allocation theory, the characteristic

cost function is defined under an implicit assumption of

inelastic demand. That is, the quantity of output demanded

by each player is assumed fixed. The player, then, must

satisfy his fixed demand through either independent action

or cooperation in a joint project. The characteristic

function is computed for fixed quantities using fixed cost

relationships. However, it is often unrealistic to assume

an inelastic demand. In the wastewater reuse example, the

total quantity of water required for irrigation at each site

is assumed fixed, but the demand for wastewater is price

sensitive since each user has an alternative water supply

option with its own cost relationship. The player may

satisfy his demand through varying degrees of participation

in a joint project as opposed to the usual all or nothing

restriction. Independent action, then, may involve a

quantity of project output different from the quantity

assigned for joint project cooperation. Determination of

the characteristic function requires specification of









appropriate quantities for independent action. For the

wastewater reuse example, these quantities may be taken as

the off-site supply corresponding to the minimum cost

combination of on-site and off-site water supply for each

player. However, where potential demand exceeds production

capacity, it is necessary to define how the restricted

supply is to be apportioned not only for the final project

but for independent action as well. Therefore, to determine

the optimal go-it-alone solution for coalition S, it is

necessary to apportion an output quantity to S by defining

the behavior of the complementary coalition N-S.

Recall that Sorenson (1972) suggests four possible

definitions for the characteristic function. We will

examine each of these definitions to determine which might

be the most appropriate for application to the example

problem. Sorenson defines C1(S) as the value to the coali-

tion if S is given preference over N-S. If S has preference

over N-S, S gets all the wastewater it wants while entering

the project after N-S. This way S would get all of the

highest savings from off-site economies of scale without

having to share them with other groups. However, this would

result in a game that is not subadditive. Coalition S would

never want to cooperate with other groups and have to share

savings.

A second definition, C2 (S), is the value to the coali-

tion if N-S is not present. This would require that S has








the option to build and operate its own facility or be

allowed to use the central facility up to its willingness to

pay (as given by alternative costs) its maximum demand, or

plant capacity, whichever comes first. This may or may not

lead to a subadditive game depending on actual relative

costs and capacity limitations, e.g. S may gain more

savings by independent action as defined by C2(S) than by

joining a coalition where its level of activity may be

restricted.

A third definition, C3(S)k involves a strictly competi-

tive game between S and N-S. Since joint cost economies of

scale imply a basic cooperative attitude among participants,

C3(S) is certainly not an appropriate definition for this

example.

As a final definition, C4(S) is the value of coalition

S if N-S is given preference. Giving preference to N-S

means letting N-S go last so that it can realize highest

marginal savings. There are at least two ways C4(S) can be

computed using this definition. First, given that Q is the

economically efficient output allocation vector for the

grand coalition N, C(S) may be computed using the go-it-

alone cost for its allocated portion of Q N This is equiva-

lent to making S go first to receive its allocated quantity

of resource. This implies a cost to N-S equal to

C(N) C(S). With economies of scale C(S) will always be

more than the cost in a larger coalition for a given QN









thus ensuring subadditivity. However, this method is not

really in keeping with the concept of independent action

since it depends on results of economic analysis involving

N. However, it does provide a means of ensuring that all

members of the grand coalition will have a right to some of

the potential savings whether or not they cooperate. In

effect, this method of computing C(S) uses the results of

economic analysis to establish ownership. Perhaps a better

approach is the idea of letting the members of N-S go ahead

and form their own coalition and least cost solution. If

the plant capacity limit is reached, S will get nothing and

C(S) is just S's alternative (on-site) cost. In this way

not everyone is guaranteed some of the potential savings.

This result may be appropriate if S is such a weak player

that it contributes savings only when joining relatively

large coalitions. If there were some remaining capacity, we

would define C(S) as the value to S of the least cost

solution for the remaining supply. The implied cost to N-S

is again C(N) C(S). Coalition S might object since N-S

gets some additional savings by having S go first. On the

other hand, coalition N-S could argue that to charge S less

would require some degree of cooperation by S. This last

definition is probably the best for this particular applica-

tion since it ensures game subadditivity and is in keeping

with notions of independent action.









Before discussing the various specific cases for the

example problem, a few comments are in order to clarify some

possible points of confusion. First, in the analysis that

follows it is often easier to discuss benefits or net

benefits rather than alternative costs avoided or cost

savings. This interchangeability of terms is a result of

equivalence in problem formulations. Recall that defining

the economic objective as a cost minimization problem is

equivalent to defining it as a net benefit maximization

problem. Benefits are computed from on-site cost as fol-

lows:

B(i) = C .n(Q .) C .o(Q, Q ofi) (13)
on,i m,i on,i m,i of,i


where B(i) = benefits to user i, 10 3/year



and C on,i Qm and Qof, are as defined previously in

Table 1.

An additional point involves inclusion of direct costs

in the analysis. It may seem reasonable to allocate direct

costs separately from joint costs. However, since each

participant in the optimal project benefits from the inclu-

sion of all the other participants, each participant should

be willing to include direct transportation costs in the

total cost that must be allocated. In particular, if direct

costs are not included, the relatively high direct costs

assigned to some users might discourage their cooperative

participation, resulting in a suboptimal final project









design. Inclusion of direct costs in the analysis means

that, in general, a single cost function will not apply. A

single cost function often applies only to joint project

components. Usually, different cost relationships apply to

different cost elements. This is the case for the waste-

water reuse problem where estimating functions for trans-

portation costs are different for each site.



Case 1. Single-Step Demand Function

A single-step user demand is derived from linear

on-site costs which in turn imply a constant unit water

price. This situation may be common in actual practice

where a user's current unit cost for water is the only

information easily available. The single-step demand curve

for site 1 in the wastewater example is shown in Figure 6(a).

The flow level Q m, corresponds to the maximum demand flow

for site 1. The unit cost of $80,00 per MGD represents the

average (or marginal) benefit to the site and corresponds to

a linear benefit curve as shown in Figure 6(b).



Economic Optimization

First we consider joint project components, for the

moment ignoring direct cost components. This corresponds to

consideration of central facility treatment costs while

ignoring transportation costs. For this simple case, the

potential user (player) with the highest average benefit

gets priority, i.e. Qofi = Min[Qm,i' remaining plant
Qf Min[Qm j





















80

103 $/MGD


Demand=Marginal Benefits









M,1
Om, 1 = 3 MGD
Qof,l (MGD)
(a)


240



103 $


Qof,l (MGD) Qof, (MGD)
(b) (c)






Figure 6. Case 1 benefit and cost relationships for
site 1: (a) marginal benefits, (b) total
benefits, and (c) net benefits.


240



103 $









capacity]. This is true because net benefits are contin-

uously increasing as shown in Figure 6(c) for site 1. If

some plant capacity remains, the user with the next highest

average benefit receives output up to his maximum demand or

plant capacity. This rank order assignment of wastewater

supply continues until either potential demand or the plant

capacity limit is reached. The result for this simple case

is a bandwagon effect for which the order that players join

the bandwagon has important implications for economic

optimization. In particular, if plant capacity is less than

total potential demand, some players may be partially or

totally excluded from the project. The excluded players

contribute lower benefits and so are given lower priority in

joining the project.

Results are not quite so easily seen when direct costs

are added to the analysis. Since direct cost components

such as the transportation costs here usually involve cost

relationships which differ from joint costs and differ from

other direct costs, the player with the highest average

benefit may not contribute the greatest net benefit to the

project. As shown in Figure 7 player A's average benefit is

less than player B's average benefit, but A also has rela-

tively low direct costs so that his net benefits are

actually greater than those for B for a given level of

output Q To allow for this possibility, analyses which

include direct costs must consider total net benefits

instead of just average benefits. The result is similar to

























Total
Net
Benefits


Q, MGD Q, MGD


Player A


Player B


Figure 7. Priority ranking example for direct cost
considerations for case 1.


103$


103 $









the rank order effect for joint costs. To maximize total

net benefits, the player with the highest total net benefits

for any arbitrary output level receives highest priority.

This result is valid even when each player's benefits are

not strictly greater than costs since priority ranking is

determined by total net benefits. An important result for

case 1, then, is that the economic optimum is achieved by

assigning output to each user according to a priority

ranking based on comparison of total net benefits. The net

benefits a user contributes to the project are limited only

by maximum demand or plant capacity constraints since net

benefits are continuously increasing. In step 5 we apply

the general result for case 1 to the wastewater reuse

example.



Cost Allocation

With the above general results for case 1, we next

consider several cost allocation schemes in search of an

appropriate strategy to satisfy financial objectives. The

methods for consideration include marginal pricing from

economic theory, the Shapley value from game theory, and

MCRS (as a generalization of SCRB) and average cost pricing

methods from conventional practice. First, we completely

eliminate the possibility of applying marginal pricing

strategies. Clearly, where net benefits are continuously

increasing within the range of maximum demand and plant

capacity limits, marginal costs will equal marginal benefits









only at a point of minimum net benefits. Consequently, the

concept of marginal pricing is not applicable for case 1.

To evaluate the remaining methods, we must first define

the core for the final project design in characteristic

function form as defined by C4(S). When considering joint

cost functions such as central facility treatment costs,

economies of scale guarantee the existence of a core and in

particular a strictly convex core (Tschirhart, 1975). Note

that the direct costs have no bearing on overall convexity.

Cost allocation solutions for a convex game exhibit certain

properties. Namely, the Shapley value will be in the center

of the core and extremely close to the MCRS solution (Heaney

and Dickinson, 1982). Also, for a convex game the MCRS and

SCRB solutions are identical. Furthermore, since the

marginal cost curve falls below a decreasing average cost

curve, a vector of charges based on average cost pricing

also satisfies core constraints. Computations presented in

step 5 demonstrate these general results for the wastewater

reuse example. In step 6 we examine some important implica-

tion of these results.



Case 2. Upward Sloping Demand Function

An upward-sloping demand for reclaimed water is derived

from decreasing on-site marginal costs. This situation

might correspond to economies of scale in on-site pumping

costs or a schedule of declining unit or block prices for

purchased water. A sketch of the on-site cost curve and









corresponding upward-sloping demand is shown in Figure 8 for

site 1. Again, the flow level Q ml corresponds to the

maximum demand for the site. The resulting total benefit

curve is also shown in Figure 8.

Economic optimization

The situation here is very similar to that in case 1.

As demonstrated in Figure 8, net benefits are continuously

increasing for a given cost function. Therefore, the same

general results apply in which the economic optimum is

achieved through priority assignment of output based on

comparison of net benefits for each user. However, unlike

the previous case, the output level for comparison of net

benefits may not be strictly arbitrary where the net benefit

curve for one player intersects the net benefit curve for

another player. The output level corresponding to an

intersection implies a switch in priority from one player to

another. This is demonstrated in Figure 9. At flow levels

less than Qk player A dominates while at flow levels greater

than Qk' player B dominates. Therefore, priority between A

and B depends on quantity of output available to the players.

In general, then, priority among players in a subcoalition S

may not be given by the priority ranking in the grand

coalition. On the other hand, if the net benefit curves do

not intersect within the range of interest, priority among

players is determined by comparison of net benefit contribu-

tions for any arbitrary flow level within the range of













of, 1, MGD

269r -


On-Site
Costs
103$


Qm, 1
-on, 1,MGD
(a)


Off-Site
Benefits
103 $


= 3 MGD


103 $/MGD

80.6


of 1, MGD
(b)


Qof MGD
(c)


Figure 8. Case 2 benefit and cost relationships for
site 1: (a) total on-site costs, (b) marginal
benefits, and (c) net benefits.


269




103 $































Total Net
Benefits,$


Qk
Q, MGD


Figure 9. Example of shift in priority for case 2.









interest. The resulting rank order priority holds for all

subcoalitions as well as the grand coalition.

Cost allocation

The discussion regarding financial analysis for case 1

is applicable to case 2 as well. General conclusions for

financial analysis are identical for the two cases as

demonstrated in steps 5 and 6.



Case 3. Downward-Sloping Demand Function

A downward-sloping wastewater demand is derived from

increasing on-site marginal costs. This situation might

correspond to diseconomies of scale in on-site operating

costs or increasing unit or block prices for purchased

water. The on-site cost curve and corresponding demand and

benefit curves are sketched in Figure 10 for site 2.

Economic optimization

As for the first two cases, the least cost solution

results when each user is assigned wastewater supply on the

basis of his contribution to total net benefits. In con-

trast to cases 1 and 2, if case 3 benefits are compared to

off-site costs, say treatment costs, their difference is not

strictly increasing. As demonstrated in Figure 10 benefits

increase for lower flow levels, reaching a maximum at Q*

before beginning to decrease. The flow level Q* corresponds

to the point where marginal costs equal marginal benefits.

Therefore, when a user's demand is downward sloping, his
















Qof ,2, MGD


105

On-Site
Costs
103 $


Qon, 2, MGD
(a)


73.9


103 $/MGD


105




103 $


of,2,
(b)


Off-Site
Benefits
103 $

.05
, = 2 MGD


Off-SiteN
Costs ; '\ xrnr
osts i \ NMaximum
f Net
I^ 1 Benefits
Total
Benefits

0 2

Qof,2, MGD
(c)


Figure 10. Case 3 benefit and cost relationships for
site 2: (a) total on-site costs, (b) marginal
benefits, and (c) net benefits.









contribution to the project is limited by marginal consider-

ations as well as maximum demand and plant capacity limita-

tions. For single-step or upward-sloping demand conditions,

the only possibility for a mix of on-site and off-site

supply to a participant occurs when the plant capacity limit

is reached before the last ranking participant fulfills his

maximum demand. For downward-sloping demand conditions, the

result may be a mix of supply to any or all of the project

participants.

Wastewater supply is still allocated according to

priority based on net benefit contributions, but that

priority is not so easily determined. As for case 2 the

quantities for comparison of net benefit contributions

cannot be arbitrarily chosen. In particular, the player

contributing the greatest net benefits at a given flow level

may not contribute the greatest net benefits at another

larger flow level. This is true since a player's marginal

benefits decrease with increasing quantity of acquired

output. Therefore, as a high priority player's marginal

benefits decrease with increasing supply, the potential

contributions of other players become increasingly attrac-

tive perhaps reaching an output level where a new player

gains priority. This output level corresponds to a point

where one player's net benefit curve intersects another

player's net benefit curve. This is analogous to the

situation in case 2. Consequently, priority among players









in any coalition S depends upon the quantity of resource

available to S as defined by C4(S). The result is that

determining priority among players requires additional

information regarding relative net benefits at various

output levels and quantities corresponding to marginal

conditions (marginal benefits equal to marginal costs).

For this example, the nonlinear nature of the problem and

inclusion of direct as well as joint costs may preclude

simple solution by hand calculation in favor of an appro-

priate mathematical programming code. Such is the case in

step 5 where we apply the GRGA reduced gradient code to

solve the economic optimization problem for case 3.

Cost allocation

General conclusions for cases 1 and 2 regarding the

nature of the core and various cost allocation methods are

not applicable to this case. First, since the optimal design

may include quantities corresponding to marginal conditions,

a charge system based at least in part on marginal pricing

might be viable. However, recall that strict marginal

pricing does not satisfy total cost requirements.

To evaluate the remaining methods, we again define the

core for the final project design in characteristic function

form .from C4(S). Intuitively, one might expect that the

resulting cost allocation game is convex as in the previous

two cases. However, as we will see, economies of scale

guarantee convexity only for C(S) defined for fixed quanti-

ties for an assumed inelastic demand or for quantities given









by C4(S) for single-step or upward-sloping demand, i.e.

cases 1 and 2. For case 3 the C4(S) definition ensures

subadditivity since the quantity of resource a player

receives in each subcoalition does not exceed the quantity

he receives in the grand coalition. However, with dis-

economies of scale in on-site costs, the largest incremental

savings from avoided costs correspond to the first units of

acquired output. At the same time, incremental savings due

to off-site economies of scale are greatest at larger output

levels. If on-site diseconomies are relatively strong, the

incremental savings from avoided on-site costs at low output

levels may be greater than the incremental off-site savings

at higher output levels. So, even though a player receives

less output and less total savings by joining smaller

coalitions he may realize greater incremental savings than

by joining a larger coalition. The result is that the cost

allocation game may not be convex. That is, a player or

group of players could have higher incremental savings (or

lower incremental costs) by joining the project earlier, say

second or third, instead of last. Recall that in cases 1

and 2 savings from off-site economies of scale and alterna-

tive costs avoided both increase with output so that the

highest incremental savings occurs at the highest output

levels, i.e. by being the last player to join the grand

coalition.









Step 5. Apply Appropriate Methodologies



Case 1. Single-Step Demand Function

Economic optimization

The optimization problem for case 1 is to



Minimize Z = 80Qon,1 + 70Qon,2 + 90Qon,3 + 100Qon,4

4 13 35
+ 8(Qof,l)4 + 5(Qof,2)' + 13(Qof,3)


6
+ 17(Q )of,4) + 60(Q of + Qof,2 + Qof,3



+ Qof,4) 7


S.T. Qon,l + Qof,1 = 3 MGD

on,2 + Qof,2 = 2 MGD

Qon,3 + of,3 = 1.5 MGD

Qon,4 + of,4 = 2.5 MGD
Qof,l + Qof,2 + Q of,3 + Q of4 10 MGD

6 MGD



To demonstrate the proposed general results for the

case of single-step demand functions, we first examine

potential contributions from each player and establish the

priority ranking that minimizes total water supply costs.

Referring to Table 1 and equation 13 we compute off-site









costs and benefits for any arbitrary flow level. We then

rank each player according to his net benefit contribution.

As shown in Figure 11, site 4 receives highest priority

followed in rank order by sites 3, 1 and 2.

Since the total potential demand for wastewater is only

9 MGD, the problem constraint for a treatment plant capacity

limit of 10 MGD is not tight. The optimal project design

includes all sites contributing a positive net benefit to

the project. Results of the rank order calculations indi-

cate that all sites are viable and should be included to

achieve the optimal design. The resulting project consists

of all sites each receiving a total wastewater supply equal

to its maximum demand quantity. Project costs total

$343,000 per year, a 55 percent savings over alternative

on-site costs. Results are summarized in Table 2.

For treatment plant capacity equal to 6 MGD, the

maximum supply of wastewater cannot satisfy total water

supply requirements for the four sites. The wastewater

supply must be allocated among the sites so as to achieve

overall minimum water supply costs. As stated previously,

wastewater supply is allocated according to a rank order

procedure. Using the rank order established above we assign

wastewater as given in Table 2. As the highest ranking

player, site 4 receives a quantity of wastewater equal to

its maximum water demand. With next highest priority,

site 3 receives a quantity also equal to its maximum demand.

At this point, there are only 2 MGD of wastewater left. As

























100


80


Total Net
Benefits
103 $


Site 1


Site 4


Q C MGD










Figure 11. Net benefits for rank order determination
for case 1.












Table 2. Optimal Resource Allocation for Case 1







Qm,t Qm,i of,i
Site MGD MGD MGD

1 10 3.0 3.0

2 10 2.0 2.5

3 10 1.5 1.5

4 10 2.5 2.5



Total 9.0 9.0



1 6 3.0 2.0

2 6 2.0 0.0

3 6 1.5 1.5

4 6 2.5 2.5



Total 9.0 6.0









the third ranking player, site 1 receives all of this

remaining supply to satisfy two-thirds of its total irri-

gation requirement. The resulting project design completely

excludes site 2. The annual final system cost is $346,000.

Cost allocation

To apply and evaluate the various cost allocation

methods we first compute the characteristic function for

both plant capacity limits. Recall that C(S) is defined by

C4(S) which gives N-S priority over S. For the 10 MGD

capacity limit, giving, coalition N-S priority does not

restrict the quantity of wastewater available to coali-

tion S. Coalition S receives wastewater flow equal to its

total maximum demand but is assigned the cost of supplying

the quantity alone, without benefit of the economies of

scale from including N-S. The resulting characteristic

cost functions are



-C(l) = 142 C(2) = 104 C(3) = 96 C(4) = 143

C(12) = 204 C(13) = 200 C(14) = 240 C(23) = 166

C(24) = 208 C(34) = 204 C(123) = 257 C(124) = 294

C(134) = 292 C(234) = 262 C(1234) = 343



where the corresponding subadditive game is convex as

verified in the Appendix.

The resulting minimum fairness criteria are given by

the following core constraints.









X(1)

X(2)





X(1) + X(2)

X(1)

X(1)

X(2)

X(2)


X(1) +

X(1) +

X(l1)--


X(2)

X(2)


X(2)

X(1) + X(2)


< 142

< 104

X(3) < 96

X(4) < 143

< 204

+ X(3) < 200

+ X(4) < 240

+ X(3) < 166

+ X(4) < 208

X(3) + X(4) < 204

+ X(3) < 257

+ X(4) < 294

+ X(3) + X(4)-< 292

+ X(3) + X(4) < 262

+ X(3) + X(4) = 343


Since the game is convex, the first four and last five

conditions stipulate the upper and lower bounds on X(i).

That is, a vector of charges within the core must satisfy


81 < X

51 < X

49 < X

86 < X

X(1) + X(2)


(1) < 142

(2) < 104

(3) < 96

(4) < 143

+ X(3) + X(4) = 343









The resulting Shapley, MCRS (equivalent to SCRB for a convex

game), and average cost pricing solutions are presented in

Table 3. As expected the Shapley value and MCRS solutions

are in the center of the core. Although the set of charges

derived from average cost is positioned away from the core

center in favor of the highest priority sites, 3 and 4, we

see that the shift is not extreme and that each site is

still awarded large savings. This particular result has

important implications for the final selection of an appro-

priate pricing scheme.

For the 6 MGD capacity limit, giving N-S priority

does restrict the quantity of wastewater available for

coalition S. The available supply must be allocated among

the players in S according to the established priority

order. For the 6 MGD three-site system, priority among

sites 1, 3 and 4 in any subcoalition is the same as that

already established, i.e. site 4 has highest priority,

site 3 second, and site 1 lowest. A sample calculation of

the characteristic function is shown below.



C(1): let (3,4) go first, Qof,3 + Qof,4 = 4 MGD

6 4 = 2 MGD left

C(1) = 8(2)0.4 + 60(2)0.7 + 80 = 188

Similarly,

C(3) = 136 C(4) = 201

C(13) = 251 C(14) = 292 C(34) = 258

C(134) = 346












Table 3. Charges for Case 1


Total cost, 103 $/year

Qm,t Average Cost
Site MGD Shapley MCRS Charge Savings

1 10 102.92 102.27 114.33 52%

2 10 69.25 69.48 76.22 46%

3 10 64.53 65.39 57.17 58%

4 10 106.25 105.87 95.28 62%




Total 343 343 343


1 6 126.33 125.82 148.30 38%

3 6 83.33 85.07 74.14 45%

4 6. 136.33 135.11 123.56 51%



Total 346 346 346









The resulting subadditive cost game is convex since

C(13) + C(14) > C(134) + C(1)

251 + 292 > 346 + 188 OK


C(34) > C(134) + C(3)

258 > 346 + 136



C(34) > C(134) + C(4)

258 > 346 + 201


The core bounds are given by


X11)


X(3)


X(1)

X{1)


+ X(3)


X(3)

X(1) + X(3)


X(4) <



+ X(4) <

+ X(4) <

+ X(4) =


88 < X(l) < 188

54 < X(3) < 136

95 < X(4) < 201

X(1) + X(3) + X(4) = 346


The Shapley, MCRS, and average cost pricing solutions

are shown in Table 3 and in Figure 12. Again, as expected,


C(13)

251



C(14)

292


188

136

201

251

292

258

346






























AVERAGE 1"--- SHAPLEY
COST / \ CORE X(4) > 95




13


C(134)= 346







Figure 12. Core and charges for three-person game for
case 1.









the Shapley value and MCRS solution are in the center of the

core. Again the average cost solution shifts away from the

core center in favor of the strongest players. However,

even as the lowest ranking player, site 1 still receives a

significant savings over exclusive on-site water supply.



Case 2. Upward-Sloping Demand Function

Economic optimization

The optimization problem for case 2 is to


9 9 8
Minimize Z = 100(Q on,l) + 80(Q on,2) + 100(Q on,3)
7 .4 3
+ 120(Q on,4) + 8(Qof, ) + 5(Qof,2)
on,4 ^of,1 of,2
+ 13(Qof,3)5 + 17lQof,4'6

+ 60(Qof,1 + Qof,2 + Qof,3 + Qof,4)7



S.T. Qon,l + Qof,l = 3 MGD

Qon,2 + Qof,2 = 2 MGD

Qon,3 + Qof,3 = 1.5 MGD

Qon,4 + Qof,4 = 2.5 MGD

Qof,! + Qof,2 + Qof,3 of,4 -10 MGD
6 MGD



To determine the least cost solution for the case of

upward-sloping demand functions, we apply the principles

established for case 1. Note that in the range of interest

(0 to 9 MGD) only two net benefit curves intersect. The net

benefits for sites 1 and 3 are equal for a flow level of









1.38 MGD. The consequences of this shift in priority are

demonstrated in Figure 13 where net benefit contributions

are compared for flow levels in the range of interest.

Priority between sites 1 and 3 depends on whether the supply

available to the two sites is greater than or less than

1.38 MGD.

Again the constraint for a plant capacity of 10 MGD is

not tight. For wastewater supply equal to maximum demand,

each site contributes positive net benefits. Therefore, the

optimal project includes all four sites, each receiving an

output quantity equal to its maximum demand. This result is

identical to case 1.

For a plant capacity of 6 MGD the limited supply is

allocated according to the priority ranking established in

Figure 13. The most efficient resource allocation is given

in Table 4. Site 4 is completely excluded from the project

so that the final design consists of only the three highest

ranking sites. Sites 1 and 3 each receive off-site supply

quantities which satisfy total water demand, while site 2

must settle for a mix of on- and off-site supply. The

project represents a 48% savings that must be allocated

among the three participants.

Cost allocation

For the 10 MGD capacity, the characteristic function

and cost allocation solutions are identical to those for

case 1 as shown in Table 5. For the 6 MGD capacity limit,

























120


100


Total Net
Benefits,$


Site 1


Site 4
Site 2


Site 3 /


Qof, MGD








Figure 13. Net benefits for rank order determination
for case 2.













Table 4. Optimal Resource Allocation for Case 2






Qm,t Om,i Qof,i
Site MGD MGD MGD

1 10 3.0 3.0

2 10 2.0 2.0

3 10 1.5 1.5

4 10 2.5 2.5



Total 9.0 9.0


1 6 3.0 3.0

2 6 2.0 1.5

3 6 1.5 1.5

4 6 2.5 0.0



Total 9.0 6.0













Table 5. Charges for Case 2








Total Cost, 103 $/year

Qm,t Average Cost
Site MGD Shapley MCRS ,Charge Savings


1 10 102.92 102.27 114.33 52%

2 10 69.25 69.48 76.22 46%

3 10G 64.58 65.39 57.17- 58%

4 10 106.25 105.87 95.28 62%



Total 343 343 343


1 6 126.33 125.66 132.40 51%

2 6 78.33 78.45 88.30 41%

3 6 82.33 82.89 66.23 52%



Total 287 287 287









the available supply is allocated according to the estab-

lished priority ranking. The characteristic cost function

for player 1 is computed by


C(1): let (2,3)



C(1) = 8(2.5) 0.4


go first, Qof,2 + Qof,3 = 3.5 MGD

6 3.5 = 2.5 MGD left

+ 60(2.5)0.7 + 100(0.5) 0.9 =179


Similarly,


C(2) =

C(12) = 233


128 C(31 = 130

C(13) = 239

C(123) = 287


C(23) = 194


This game is convex since


C(13) + C(12)

239 + 233



C(12) + C(23)

233 + 194



C(13) + C(23)

239 + 194


> C (123)

> 287



> C(123)

> 287



> C(123)

> 287


The core bounds are given by


+ C(1)

+ 179



+ C(2)

+ 128



+ C(3)

+ 130









X(1)


X(2)


X(1)

X(1)


+ X(2)


X(2)

X(1) + X(2)


< 179

< 128

X(3) <, 130

< 233

X(3) < 239

X(3) < 194

X(3) = 287


93 < X(1) < 179

48 < X(2) < 128

54 < X(3) < 130

X(l) + X(2) + X(3) =


287


The Shapley value, MCRS, and average cost pricing solutions

are given in Table 5 and in Figure 14. They demonstrate the

applicability of important case 1 general results to case 2.



Case 3. Downward-Sloping Demand Function

Economic optimization

The optimization problem for case 3 is to


Minimize Z =

+

+

+


1.7 14 1 2
25(Q on) 7 + 40(Q on2)14 + 45(Qn,)

70(Qon,4)'1 + 8(Q of, )'4 + 5(Qof,2)'3

13(Q 3) + 17(Qof,)6

60 (Qof,l + Qof,2 + Qof,3 + Qof,4)7








































AVERAGE ____,
COST CORE




C(123)= 287














Figure 14. Core and charges for three-person game for
case 2.









S.T. Qon,l + QofI = 3 MGD

Qon,2 + Qof,2 = 2 MGD

Qon,3 + of,3 = 1.5 MGD

Qon,4 + Qof,4 = 2.5 MGD

Qof,1 + Qof,2 +of,3 + Qof4 -10 MGD
6 MGD



Application of the GRGA code to determine the minimum

cost solution for the case of a downward sloping demand

function yields the optimal project designs given in

Table 6. The role of marginal considerations in determining

optimal resource allocation is evident. Although plant

capacity is not a limiting factor for project size in the

first design, the optimal output level of 8.41 MGD does not

correspond to the 9 MGD potential demand. Only site 4

receives its maximum demand. That the remaining sites are

assigned quantities less than their maximum demand results

from the effect of marginal conditions in which the quantity

assigned to the site corresponds to an output level where

marginal costs equal marginal benefits. The results of

marginal considerations and shifting priorities are also

evident in the 6 MGD capacity design. Here, the capacity

constraint is tight, and the final project includes only

three participants.












Table 6. Optimal Resource Allocation for Case 3






Qm,t m,i of,i
Site MGD MGD MGD

1 10 3.0 2.56

2 10 2.0 1.89

3 10 1.5 1.46

4 10 2.5 2.5




Total 9.0 8.41


1 6 3.0 1.99

2 6 2.0 1.51

3 6 1.5 0.0

4 6 2.5 2.5



Total 6.0 6.0









Cost allocation

Since analysis of either of the above project designs

can be used to effectively demonstrate all relevant results

and conclusions, we arbitrarily select the three-site design

for computational ease only. To determine the characteris-

tic function for this design, we apply GRGA to establish the

optimal output level for coalition N-S. Using any remain-

ing capacity as the new maximum supply constraint, we then

apply the code to determine the optimal "go-it-alone" output

quantity for coalition S. Computations are summarized

below.


C(1): let (2,4) go first, Qof,2 +

6 4.32 =

C(1) = 8(1.68)0.4 + 60(1.68)0.7 +



C(2): let (1,4) go first, Qof, +

6 4.95 =

C(2) = 5(1.05)0.3 + 60(1.05)0.7 +



C(4): let (1,2) go first, Qof, +

6 4.24 =

C(4) = 17(1.76)0.6 + 60(1.76)0.7 +


Qof,4 = 4.32 MGD

1.68 MGD left

25(1.32)1.7 =136



Qof,4 = 4.95 MGD

1.05 MGD left

40(0.95)1.4 =104



Qof,2 = 4.24 MGD

1.76 MGD left

70(0.74)1.1 = 163









C(12): let (4) go first, Qof,4 -2.5 MGD

6 2.5 = 3.5 MGD left

C(12) = 8(1.99)0.4 + 5(1.51)0.3 + 60(3.5) 0.7

+ 25(1.01)1.7 + 40(0.49)1.4 = 200


C(14) :



C(14) =


let (2) go first, Qof,2 = 1.64 MGD

6 1.64 = 4.36 MGD left

8(1.88)0.4 + 17(2.48)0.6 + 60(4.36)0.7

+ 25(1.12)1.7 + 70(0.02)1.1 =


239.


C(24): let (1) go first, Qof1, = 2.24 MGD

6 2.24 = 3.76 MGD left

C(24) = 5(1.29)0.3 + 17(2.47)0.6 + 60(3.76)07

+ 40(0.71)1.4 + 70(0.03)1.1 = 212



C(124) = 8(1.99)0.4 + 5(1.51)0.3 + 17(2.5)0.6

+ 60(6)0.7 + 25(1.01)1.7

+ 40(0.49)1.4 = 296


We then check the appropriate conditions for game convexity.


C(12) +

200 +



C(12) +

200 +


C(14)

239



C(24)

212


> C(124) + C(1)

> 296 + 136



> C(124) + C(2)

> 296 + 104




Full Text

PAGE 1

WATER IiRESOURCES researc center Publication No. 80 Economic Efficiency and Cost Allocation for Water Resource Projects with Economies of Scale by Bonnie Walker Proefke Department of Environmental Engineering Sciences University of Florida UNIVERSITY OF FLORIDA

PAGE 2

Publication No. 80 ECONOMIC AND COST ALLOCATION FOR WATER RESOURCE PROJECTS WITH ECONOMIES OF SCALE BY BONNIE WALKER PROEFKE A RESEARCH PROJECT PRESENTED TO THE DEPARTMENT OF ENVIRONMENTAL ENGINEERING SCIENCES OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF OF ENGINEERING UNIVERSITY OF FLORIDA 1984

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ACKNOWLEDGEMENTS I would like to express my appreciation to the following individuals who have offered their help and encouragement throughout this study: my graduate advisor Dr. James P.Heaney for his guidance during both the investigative and writing phases of this project, Elliot Ng for his valuable thoughts and input, and Robert Dickinson for his help with. the mathematical programming codes. Finally I would like to -------------tliank my liusband Rl.cnara:-fornis love anapa"EI"ence through-out this work. ii

PAGE 4

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii LIST OF TABLES . . --. iv LIST OF FIGURES INTRODUCTION SECTION I SECTION II CONCLUSIONS APPENDIX REFERENCES . . . LITERATURE REVIEW Natural Monopoly Optimization Cost Allocation SUIIunary METHODOLOGY .' Project Evaluation Steps Discussion and Wastewater Reuse Example . . iii v 1 3 4 13 ;:-------vr----------38 40 40 41 .103 105 108

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Table 1 2 3 4 5 6 7 LIST OF TABLES Cost Functions for Wastewater Reuse Example .. Optimal Resource Allocation for Case 1 .. ." Charges for Case 1 Optimal Resource Allocation for Case 2 Charges for Case 2 Allocation for Case 3 .. Charges for Three-Person Game for Case 3 iv Page 47 76 80 86 87 92 98

PAGE 6

LIST OF FIGURES Figure Page 1 Example of an unsustainable natural monopoly 9 2 Conditions for economic efficiency 15 3 Selection of price under decreasing average cost conditions _. 26 4 Selection of price under ihcreasing average cost conditions 26 5 6 7 8 9 10 11 12 13 14 15 System for wastewater reuse example Case 1 benefit and cost relationships for site 1 Priority ranking example for direct cost considerations for 1 Case 2 benefit and cost relationships for site 1 Example of shift in priority for case 2 Case 3 benefit and cost relationships for site 2 . Net benefits for rank order determination for case 1 Core and charges for three person game for case 1 Net benefits for rank order determinati.on for case 2 o. Core and charges for three-person game for case 2 Core and charges for three-person game for case 3 . v 44 60 62 66 67 69 75 82 85 90 97

PAGE 7

INTRODUCTION lv1any economic optimization techniques are available for evaluating various water resource project Also, numerous methods have been proposed or are now used to app'ortion costs a water resource project among participants and/or purposes. However, no extensive research has been directed towards evaluating the relationship between the economic optimization and cost allocation phases of planning water resource is to provide a basis for establishing' and evaluating the relationship between economic efficiency and project cost allocation. First, a review of some recently developed theories on the formation and s"J.stainabili ty of natural monopolies is presented. Several important implications concerning the efficiency and fairness of project coalition formation may be derived from these economic theories. To provide a basis for quantifying the implications derived from economic theory, methods for evaluating economic efficiency and for apportioning the cost among participants of a joint water resource project are reviewed. Finally, through the simple example of a small wastewater reuse project, several of these methods are applied to demonstrate how efficiency and equity might be related in order to 1

PAGE 8

'. evaluate various joint project designs. In implications for the special case of economies of scale in production for three different demand relationships are examined. 2

PAGE 9

SECTION I LITERATURE REVIEW Some insight into the current dilemma regarding the interaction of efficiency and equity can be gained by reviewing the work of Loughlin (1977)-and Rossman (1978). Loughlin examines the efficiency of three cost allocation schemes based on a set of efficiency and equity criteria. He contends that economic efficiency and cost allocation are of eacn other srnce costs cipant or project purpose are not the costs to consider in deciding project feasibility, optimal scale of development, or which participants to include in the project. Rossman notes that by judging the cost. allocation metllods in of economic evaluation criteria, Loughlin contradicts the argument that cost allocation is not required for economic evaluation. In fact Zajac (1978) proposes tha.t from the viewpoDit of either efficiency or equity, pricng and entry-exit (from the joint project) must be considered jointly, not in isolation. This section examines the theoretical. basis for these two differing viewpoints. The theory of natural monopoly provides the basis of the argu-ment for joint consideration of efficiency and equity, while the principles of engineering cost allocation often dictate 3

PAGE 10

, 4 independent financial and economic analyses. In addition to these conceptual views of the problem, methods for quanti-fying efficiency and equity are reviewed. Natural Monopoly Recent developments in the theory of natural monopoly provide some important implications for the roles of eco-nomic optimization and fairness in coalition formation of joint projects. A'brief of some of the important concepts on natural monopolies will provide an understanding of these implications. A competitive market prevails if the number of sellers __________ no one seller is able to influence the market price by a unilateral change, in output. Compe,t-itive equilibrium is assumed to occur where marginal costs -equal marginal benefits, i.e. where the supply and demand curves intersect. A competitive equilibrium has the prop-erty known as Pareto optimality where net benefits are maximized so that no other feasible distribution of outputs can improve the welfare (measured in terms of net benefits) of one individual without reducing the welfare of another. However, as demonstrated in the next section, maximum net benefi ts do not always ,correspond to marginal conditions. Therefore maximum benefits is a more generally appli-cable condition for economic optimality. In contrast to the competitive market is the rnonopo-listic market with only one active seller (or firm) in the

PAGE 11

5, market. Sharkey (1982a) describes conditions for the formation and stability of a natural monopoly which results when a single firm can produce at lower cost than any combination of two or more firms in a market where firms are able to compete on an equal basis. Entry of firms in a natural monopoly could only reduce social welfare (measured in terms of cost of producing a given output) by raising total costs of production. Assuming all firms in the market have the same cost function C, a single firm is more effic-ient than, two or more firms if 1 k C(q) < C(q ) + + C(q ) (1) where q = vector of outEuts in a particular market = (q1', qn) 1 k q .'. q = output vectors that sum to q,. If inequality (1) holds for all feasible disaggregations of q, then C is subadditive at q and the market is a natural monopoly. Indeed, a natural monopoly exists if and only, if the cost function is subadditive, that is if C(q) < C(x) + C(q-x) (2) for any x such that 0 < x < q.

PAGE 12

6 Economies of scale exist where average costs fall with increasing output or where CO-g) < A Ceq} (3) for all A such that 1 < A < 1 + e: where e: is a small. positive number. Economies of scale are sufficient but not necessary conditions for subadditivity of single output cost functions. However, in a multiple output market, subadditivity holds only if there are also economies of joint production. One such measure is economy of scope which exists if a single firm can produce any vector of outputs more efficiently than two or more specialty firms for a constant level of production of each output. Whil.e economy of scope is a useful intuitive concept, it is not particularly helpful in deriving actual sufficient conditions for subadditivity. Therefore, Sharkey presents other specific properties of the cost function which may be empirically verified. See Sharkey (1982a) for a discussion of these properties. Sharkey (1982a) observes that economies of scale in the production of a private good are closely related to the concept of public goods which are consumed collectively rather than individually. A fixed cost in the production of such a good has the property of collectiveness since it is consumed and must be paid by the consumers collectively. Economists have long recognized that collective goods cannot be efficiently allocated through the market system. As a result, various forms of public control have evolved for

PAGE 13

7 financing and allocating collective-goods. Therefore, natural monopolies (such as public utilities) are often referred to as regulated firms in the economics literature. Given that under subadditive cost conditions, the natural monopoly is the least cost (or maximum welfare) market configuration, the question remains as to how to set regulated prices. Since the competitive equilibrium is economically efficient, one alternative might be to regulate prices to equal marginal costs for a given level of produc-tion. However, in many regulated industries where there are often large fixed costs, marginal cost is below average cost. with prices set at marginal costs, total revenues ____ short of total costs. A reguJ.atedfirm woulg be aQle to survive only if subsidized by revenues raised in other parts of the economy. Such subsidies might be distributed by the government. Alternatively, owners of a regulated firm might receive compensation by the firm's customers for the use of the owner's capital. The legal precept in the United States requires that regulated firms operate at a zero profit, where all costs are considered. The problem of determining regulated, economically efficient prices is addressed in many theoretical models. Zajac (1978) con-siders several simple models such as Ramsey prices, two-part tariffs, and self-selecting two-part (block) tariffs. As Zajac (1978) observes, economic theories on regulated firm pricing have focused mainly on the question of economic

PAGE 14

efficiency with very little emphasis on issues of justice and fairness. Zajac (1978) further observes that even if the cost structure is subadditive and a single supplier can provide all services cheaper than some combination of separate suppliers, the possibility may exist that some group of services can be provided cheaper bya separate supplier. This then is the focus of the modern theory of natural monopoly in which models by Zajac (1972) and Faulhaber 8 (1972, 1973) demonstrate some of the inherent contradictions involved in setting prices that are both economically efficient and sustainable (no cross-susidization). Fol-lowing is a brief discussion of these concepts of efficiency and cross-subsidization and how they relate to issues of economic analysis and cost allocation of joint projects. Although natural monopoly is by definition the most efficient form of production, there may be no price that simultaneously satisfies total market demand, provides revenues to cover total cost, and discourages entry by other firms. This situation is illustrated in Figure 1 .for the case of a single output natural monopoly. To satisfy demand and revenue requirements, the price is set at P > P m However, to discourage entry the price must be set so that p = Po < Pm This natural monopoly is therefore unsustainable. A natural monopoly for which an entry deterring price exists is termed a sustainable natural monopoly. This concept was first defined by Baumol, Bailey, and Willig

PAGE 15

Price $/unit I I I I I Demand 9 --------------------I .\ I I Q Output, units Figure 1. Example of an unsustainable naturalmonopoly.

PAGE 16

i -10 (1977) and Panzar and Willig (1977). When an unsustainable multiple output natural monopoly operates under conditions where entry by rival firms is barred,prices must be such that one output is subsidizing another. The theory of cross-subsidization is proposed by Faulhaber (1975), Zajac (1972, 1978), Sharkey (1982b), and Sandberg (1975). Sharkey (1982a) describes conditions under which both output and multiple output natural monopolies are sustainable by two approaches; the intuitive concept of sustainability and the theory of cooperative games. Applying the former approach, Sharkey (l982a) defines conditions for sustainability based on various cost function properties including declining average costs which are extended to the concept of the supportable cost function. In the cooperative game approach, buyers in a market are viewed as players in a cooperative game. The objective is to form coalitions for which production and distribution of outputs are most favorable. A natural monopoly exists if the coalition of all buyers achieves a better outcome than any partition of buyers into subcoalitions. A condition for natural monopoly is that the characteristic function be superadditive (revenue maximization) or sub-additive (cost minimization). The characteristic function is given by the outcomes in terms of costs, benefits, or savings for the various subcoalition combinations. The characteristic function C for a cost game is subadditive if

PAGE 17

11 C(S) + C(T) > C(S UT) SnT= CP (4 ) where Sand T are any two subsets of the grand coalition N and S U T =f.
PAGE 18

12 these same cooperative game theoretic results have been directly applied to several current cost allocation methods. Finally, Sharkey (1982a) provides two warnings with respect to the cooperative game model. First, the model assumes that coalition formation is costless. However, real coalitions form with an additional transaction cost (Heaney, 1983). Secondly, the cooperative game model is derived from the underlying noncooperative game where the monopolistic firm and rival firms are players. As a result, important aspects of the interaction between players may.be lost. Zajac (1978) provides some additional, very important warnings. First, for games involving more than just a few players, the number of possible groupings that must be evaluated may be formidable. Perhaps most importantly, the ap.proach generally implies consumption pre specified quantities of goods or services. This assumption might be valid where all buyer demands are completely insensitive to price variations. However, this is rarely the case. Therefore,analyses must examine variations in quantities demanded with varying prices. As price and quantity v'ary: so. do total revenues and costs, resulting in important implica-tions for the efficiency of various stable price vectors. For example, if efficiency is defined in terms of maximizing overall benefits (derived from demand) minus costs (der.ived from supply), the optimal single firm or grand coalition configuration allocates output for an implied price vector.

PAGE 19

13 This price vector need not be stable even though a set of stable price vectors may exist. Any of these other price vectors would not encourage efficient output consumption and so would yield a net benefit necessarily 1ess than that implied by strict optimization. Herein lies the inherent conflict between efficiency and equity in regulated firm pricing. From the game theoretic approach, this concept may be extended to the case where costs must be apportioned among groups in a joint project. Optimization Evaluation of the efficiency of an endeavor implies optimization which involves the determination ____ or lowest value over some range. Jelen and Black (1983) group optimization problems into three categories The first, preferential optimization is subject to preference and taste only. The second, mathematical or physical optimization is not subject to these considerations. The third category is economic optimization which is a combina-tion of preferential and mathematical optimization where preference elements are expressed quantitatively. Econo-mists define economic efficiency in terms of Pareto opti-mality where resources are allocated so as to most effec-tively serve consumers' tastes and preferences. Efficiency in the choice of quantities of different outputs requires that for each output, marginal costs of production equal benefits to each buyer. As noted, the competitive

PAGE 20

14 equilibrium automatically satisfies this condition. How-ever, as noted earlier marginal conditions do not always correspond to maximum net benefits. When the marginal cost curve lies above the marginal benefit curve for lower output levels, the point where the two curves meet actua'lly corres-ponds to minimum net benefits. In addition, where produc-tion capacity constraints restrict output, marginal conditions may never even be reached. Here, economic efficiency can be defined only in terms of the difference between total benefits and total costs with the optimum corresponding to maximum net benefits. This situation is examined more closely for a special case in the wastewater reuse example problem. For benefits and costs measured in terms of dollar worth, the conditions for economic effi-ciency are defined as in Figure 2. Application of these economic principles to the evalua-tion of engineering projects requires analysis of the demand for project output and project costs. Marginal costs represent supply while marginal benef.its depict the demand curve. Project evaluation, then, requires determination of costs and benefits as a function of various outputs followed by maximization of their difference. Determination of project costs and benefits involves application of various cost engineering principles and techniques (e.g., see James and Lee, 1971). Rather, this study focuses on efficiency and equity issues for previously determined

PAGE 21

1-1arginal Values $/unit Total Values $ p Costs Benefit$ Q Output, units (a) Total Benefits Qrn Q Output, units (b) Co.sts 15 Figure 2. Conditions for economic efficiency: (a) marginal benefits equal marginal costs for output Q correspond to (h) maximum net benefits for output Q. (Q corresponds to optimal output conditions wWen production is limited to Q ). m

PAGE 22

16 project costs and benefits. For a given set of cost and benefit information, various results and numerical techniques of optimization theory maybe applied to project evaluation. An overview of some of the more important results and techniques for engineering applica-tions is presented next. Optimization theory addresses problems requiring minimization or maximization of a real-valued objective function subject to satisfying a number of equality and/or inequality constraints. The objective function and con-stra.ints are expressed in terms of the decision variables (output quantities). Linear programming (LP) defines a class of optimization problems in which the objective function and constraints are linear. The problem is solved by means of an iterative procedure. The most widely used is the simplex method which proceeds by moving from one fea-sible solution to another in such a way so as to improve the value of the objective function. For a more complete discussion, see any of several texts including Beightler et ale (1979), McMillan (1975), and S.P. Bradley et ale (1977). With the general availability of several commercial software packages, linear programming provides an effective means of solving many classes of engineering problems. In particular, network models which involve the distributions of products from plants to consumer markets exhibit a special LP structure that can be exploited in developing an efficient solution algorithm. S.P. Bradley et al. (I977)

PAGE 23

17 describe a general solution approach derived from speciali-zing the rules of the simplex method. Many water resource problems are conveniently modeled as network problems and are readily solved using appropriate solution algorithms. Mandl (1981) surveys the state-of-the-art in network models and algorithms that can be applied to planning of irrigation and wastewater systems. For the case where total treatment plant and pipe costs are linear functions of capacity and all capacities and fixed demands are known, the problem is formulated so that the objective is to minimize costs subject to conservation of flow constraints. To solve this problem G.H. Bradley et al. (1977) and Maurras (1972) have developed codes that are specializations of the simplex algorithm. While both algorithms find the global optimum, the suitability of a linear objective cost function restricts -application of the model. Furthermore, the codes are not generally available and can only be obtained from the authors. Int"eger programming models can incorporate fea-tures to form a more complex network model than the strict LP model. The technique has generally been applied to the fixed-charge model which assumes a linear cost function but with fixed project costs. However, integer programming may be used to approximate nonlinear functions by means of linear equations coupled with logical restrictions (integer variables). This piecewise linearization method is analogous to the delta-method applied in separable programming as described later in this paper. Using a special branch-and-

PAGE 24

18 bouhd algorithm by Rardin and Unger (1976), Jarvis et ale (1978) apply integer programming as a fixed-charge model for design of wastewater systems exhibiting economies of scale. Standard integer programming codes are commercially avail-able, e.g. the integer programming code by IBM Corporation. Although many real world problems are nonlinear; the availability of powerful linear programming methods provides a major incentive for approximating nonlinear problems in linear form. Therefore, one common approach to the general problem is to replace nonlinearities with linear approxima-tions and solve the resulting linear program. Reklaitis et ale (1983) explorethree basic strategies for employing linear approximations in solving nonlinear problems. All of these methods approximate a nonlinear function f(x} in the vicinity of a point X by a Taylor series expansion where o higher order terms are ignored. The point Xo is called the linearization point. The three basic linearization methods differ primarily in the manner and frequency with which the linearizations are updated. In the direct successive LP approach, the LP solution defines the direction for a line search. Two successive LP algorithms, the Frank-Wolfe algorithm for linearly constrained problems, and the Griffith and Stewart method for nonlinear constraints, are presented in Reklaitis et ale (1983) and S.P. Bradley (1977). The Frank-Wolfe algorithm forms a linear approxima-tion at the point X by replacing the objective function o with its current value plus a linear correction term. The

PAGE 25

19 resultingLP solution is used to define a search The search direction is given by the line segment joining the LP solution and the linearization point. Therefore, the successive LP approach may be viewed as an alternating series of LP and line-search subproblems. To maintain feasibility in the nonlinearly constrained problem, the Griffith and Stewart method bounds the step size for each intermediate LP solution. Since this algorithm proceeds rather slowly towards a solution, the successive LP approach is most appropriate for problems with only a few nonlinear terms. The idea of the cutting plane approach is to succes-sively improve the linear approximations to the constraint boundary in the region near the solution as the solution is approached from outside the feasible region. Gottfried and Weisman (1973) describe Kelley's-algorithm in which linear-ized constraints (cutting-planes) are introduced one at a time to successively eliminate portions of the previous approximation of the feasible region. The resul.ting series of LP subproblems can be solved using specialized LP methods. However, this approach is not applicable for equality-constrained problems., In addition, convergence is ensured only for convex problems. In spite of these difficulties this strategy has been effectively used to solve some specially structured problems. The third linearization technique is separable program-ming which utilizes piecewise linear approximations of

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20 separable nonlinear functions over the full range of the problem variables. The method is applicable to problems of the form n Maximize or (Minimize) E j==l f. (x.) JJ Subject to n E j==l g .. (x.) < (= or 2-) 0 J x. > 0 (j == 1,2, n) J (i = 1,2, m) Because decision variables appear separately in each func-tion f. and g .. the objective and constraint functions are J separable. Instead of solving this nonlinear problem directly, an effective strategy is to make appr.opriate approximations so that linear programming can be used. Hadley (1964) and S.P. Bradley et ale (1977) examine two commonly used approximation techniques, the delta (o}-method and the (A)-method. Although ensuring only a locally optimal solution, this approach is particularly suited to large network models since the separable programming mode of commercially available LP codes can efficiently solve very large problems. This model frequently arises in engineering applications particularly water resource planning formula-tions.

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21 Currently, most mathematical programming applications to water resource problems incorporate linearization tech-niques and linear programming solutions. The linearization-based algorithms often cannot give acceptable estimates for the boundaries of the feasible region or the objective function. Rather than rely on inaccurate linearization to define the location of a point, linear approximations might be used only to determine a good iocal direction for search. Examination of values of the original objective and con-straint functions can then yield the optimal point along the search direction. This strategy is analogous to uncon-strained gradient search methods such as the conjugate gradient and quasi-Newton methods where linear approximation ---------------------------------------------------------------------------------------------------------------------------------' is used to determine a good search direction and actual function values guide the search along this. direction. In the constrained case directions must be chosen to yield feasible points. Reklaitis et ale (1983) provide an excel-lent discussion of these direction-generation methods. The group of direction-generation techniques called feasible direction methods requires solution of an LP subproblem to determine a direction that is both a descent (or ascent) direction and feasible. The objective of the LP subproblem is to solve for the direction that maximizes the increase (or decrease) in the objective function subject to feasibility constraints. This method has disadvantages of a slow rate of convergence and inability to directly accommo-date nonlinear equality constraints.

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Another group of methods known as qeneralized reduced gradient methods (GRG) simply solves a set of linear equations instead of the LP solution to determine a favorable search direction. This method uses all nonbasic variables to define the direction and so is a generalization of the convex simple method, the direct analog to the linear simplex method. Incorporation of conjugate gradient or quasi-Newton strategies accelerates convergence of this method. The method may also be extended to accommodate 22 nonlinear constraints. A number of GRG codes are currently available. Reklaitis et ale (1983) survey the major compar-ative studies which examine the relative merits of the various methods and codes. In addition to strategies for exploiting linear approximations to nonlinear problem formulations are methods that use higher.order approximations,-specifically quadratic approximations. Essential to the strategy for employing quadratic approximations is the method of quadratic programming. Quadratic -programming involves use of a simplex-like algorithm to obtain a solution for a problem consisting of a quadratic objective function and linear constraints. Since the partial derivatives of a quadratic function are linear, a modified linear programming problem is generated by application of Kuhn-Tucker conditions (and thus forming partial derivatives). The method yields a local optimum which is guaranteed globably optimal only for strictly concave or convex functions. Hadley (1964) discusses

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23 several computational techniques for solving quadratic programming problems. Reklaitis et ale (1983) examine several approaches using full quadratic approximation of objective and con-straint functions which prove no easier to solve than the general nonlinear problem. In addition, formulation of quadratic programming subproblems (quadratic objective function and linear constraints) results in no significant improvement over the successive LP approaches. However, formulation of a subproblem objective function with the quadratic term as the second derivative of the Lagrangian function does provide the basis for an efficient algorithm for generating good search directions. The difficulty of providing second derivatives for the problem functions is resolved using quasi-Newton methods which only require differences of gradients of the Lagrangian function to approximate and update the second derivative. The result is a sequential quadratic programming (SQP) algorithm in which the solution to the quadratic programming subproblem defines the search direction. Reklaitis et al. (1983) survey some of the major studies which examine the relative merits of the various quadratic approximation methods and codes. The survey also includes studies which compare various nonlinear programming methods and codes based on criteria such as efficiency, global convergence, ability to solve various types of problems, and ease of use. In general, nonlinear programming techniques have not been applied to water

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24 resource problems. However, results of the comparative studies indicate that GRG and SQP have features that are important to engineering optimization. Cost Allocation Economic efficiency in many water resource projects may be achieved by taking advantage of (1) economies of scale in production and distribution facilities, (2) the assimilative capacity of the receiving environment, (3) excess capacity in existing facilities, (4) mUltipurpose opportunities, and/or (5) multigroup cooperation (Heaney and Dickinson, 1982). The results of economic analysis often indicate that participants should be combined into a cooperative joint venture. Like natural monopolies and public goods, the market system may not efficiently allocate the benefits (or costs) of joint water resource projects. Therefore, given an economically efficient joint project design the task of distributing the economic impacts among all of the participants-remains. The objectives of cost allocation include (1) satisfaction of the financial requirement that project revenue equals project cost; and (2) satisfaction of econo-mic requirements so as to encourage optimum use of project output. As noted previously, financial and economic requirements are satisfied simultaneously under pure com-petition. The market reaches equilibrium where marginal cost equals marginal benefits. At this intersection margi-nal cost also equals average cost so that the resulting

PAGE 31

25 price vector P and output vector Q satisfy both economic and financial requirements. The absence of such a competitive market system requires some form of public control to administer prices which properly allocate resources. For example, when average costs are decreasing, a price based on efficiency does not satisfy the financial requirement. This can be seen in Figure 3 where the optimal output Q and corres-ponding marginal price P are still given by the inter-m section of marginal cost and marginal benefit. However, average cost exceeds marginal cost so that a price P is a required so that revenues will just cover costs. Un for-tunately, average cost pricing at P restricts use and a forces a suboptimal situation. In contrast to economies of scale, conditions of increasing average cost result in an optimum pricing scheme that more than satisfies the financial requirement. Figure 4 shows that for increasing average costs, marginal cost lies above average cost so that marginal pricing produces a net revenue of Q(P -P ). In water resource m a applications this presents a problem of what to do with the net revenue. According to project financial requirements, only the cost legally obligated during project construction and operation must be recovered. With the competitive market as a model for a nfirst best" pricing scheme, a common pricing approach is to pretend that a market exists for project outputs. The

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, j. >' ':.-Price $/unit Demand Q Output ,units Figure 3. Selection of price under decreasing average cost conditions. 'Price $/unit Surplus Q Output, units Demand Figure 4. Selection of price under increasing average cost conditions. 26

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l 27 objective of the resulting market analysis is. to determine a "second best" allocation of resources given that prices are constrained to deviate from marginal cost. This strategy has dominated regulated firm pricing schemes where decreasing average cost conditions are predominant (Zajac, 1978). James and Lee (1971) describe three basic approaches for this pricing strategy as applied in water resources planning. One approach is to employ price discrimination to capture some of the consumer surplus. The objective in price discrimination is to charge those users receiving marginal benefits greater than P (refer to Figure 3) an m extra fee to satisfy the financial requirement. Even though prices above marginal cost reduce full economic utilization, the net effect may be a reasonable compromise between the economic and financial requirements. A second is to maintain marginal pricing while.raising the additional required revenue through a fixed, nonmarginal charge. Each user would pay a flat fee plus a per-unit charge equal to marginal cost. A final approach is to set price equal to marginal cost while supplementing the resulting revenue through subsidies. Such subsidies are usually distributed by the. government--and involve redistribution of income from those paying taxes to those receiving subsidies. Despite the potential of marginal cost pricing, actual practice often dictates application of charges based on some measurable unit called a charging vehicle. James and Lee (1971) describe three such charging vehicles which include

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28 (1) ability to pay, (2) benefit received, and (3) quantity of output or average cost. While schemes involving each of these vehicles can be formulated to satisfy financial requirements, a theoretical difficulty centers on economic grounds. Economists argue that unless price equals marginal cost, users are not given the proper incentive to balance the value they derive from use of incremental output against its marginal opportunity cost. Indeed, there has been continued interest in the application of marginal cost pricing techniques to water resource planning. For example, Hanke and Davis (1973) report significant potential for marginal cost pricing for municipal water services, indus-trial and municipal sewage treatment, navigation, and flood control. Guariso et ale (1981) present an iterative algor-ithm for determining optimal water supplies and demands in a -regional network where marginal cost equals marginal bene-fits. The authors propose use of resulting marginal cost prices as reference points for evaluating prices determined by conventional methods. Despite continued interest in marginal cost pricing, most water resource planners and governing agencies rely qn numerous (some ad hoc) methods of dividing project costs among participants. Loughlin (1977) expresses the pre-vailing view that economic and financial analysis are independent of each other. The rationale is that joint costs which cannot be directly attributed to a participant are not marginal and so do not influence optimal design

PAGE 35

29 (except for total project justification). Since joint costs must be paid, financial analysis is required to allocate them among participants. Consequently, economic efficiency in water resources planning is usually defined by the maximum positive difference between total benefits and total costs. Cost allocation, then, involves apportionment of the costs of the economically optimal solution among participants. Notably absent from this concept of economic effi-ciency is the complete notion of Pareto optimality which requires that for an economically efficient allocation of resources, no other feasible distribution of output can benefit 'one participant without harming another. In prac-tice, the use of an incomplete definition for economic efficiency results in cost allocation schemes that imply prespecified demand quantities relatively insensitive to price. This model mayor may not be appropriate depending on the actual supply and demand conditions. The objective of the cost allocation phase of water resources planning is to determine a vector of charges that is acceptable to all project participants. If charges are not acceptable to all, some may choose not to participate resulting in a suboptimal final project design. In other words, the objective is to define an equitable vector of charges for which no individual or group of participants would benefit by leaving the joint project. This exactly corresponds to the concept of sustainability of natural monopoly pricing which can be described using cooperative

PAGE 36

N-person game theory as in Sharkey (1982a). In fact, several of the conventional or proposed procedures for allocating costs directly incorporate cooperative game theoretic concepts. 30 Eeaney (1979a) describes the evolution of current practice beginning with the Tennessee Valley Authority (TVA) studies in the 1930 's and including the Federal Inter-Agency Studies in the 1950's. Comparison of conventional method criteria and game theoretic concepts reveals important similarities including the requirement for full recovery of costs and the notion that no participant should be charged more than he would pay if he acted independently. These requirements constitute the set of imputations of a cooperative game. Heaney and Dickinson (1982) propose an extension of current practice to require that charges satisfy the full core constraints so that no group of participants is charged more than it would pay if acting ala-ne. Following is a brief description of some commonly used conventional and proposed allocation methods. James and Lee (1971) identify cost allocation methods by the definition of cost used, the cost directly assigned to each participant, and the cost allocation vehicle. The result is a matrix of 18 possible ways to apportion costs. Three of the most commonly used cost allocation methods include the use of facilities, alternative justifiable expenditure (AJE) and separable costs remaining benefits (SeRB) methods.

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31 Loughlin (1977) describes the use of facilities method which allocates joint costs in proportion to some measure of the relative use of the central project facilities by each participant. Joint costs are determined by subtracting either direct costs or separable costs. Direct costs are defined to be the costs of the elements of the project which are used solely by that participant. Separable costs are the differences between the total rnultigroup project cost and the cost of the project without the participant. They include direct costs as well as the incremental costs of changing the size of joint cost elements. The use of facilities method is generally considered acceptable only where joint use is clearly determinable on a comparative basis (Loughlin, 1977). The method does not incorporate any of the stability criteria included in other conventional methods. Recommendations by the Federal Inter-Agency River Basin Committee extend the notion of equity in cost allocation by requiring that participants receive a proportional share of the savings resUlting from joint projects. Consequently, the Committee recommends use of the SCRB method and the P-+JE method when the effort or expense to obtain information for the SCRB method is prohibitive (Loughlin, 1977). Heaney (1979a) describes the AJE method for which each participant is assigned his direct cost plus a share of the remaining joint costs (total cost less all direct costs) in proportion to his alternative costs avoided. Alternative costs avoided

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32 are defined as the difference between the participant's stand alone cost and direct cost. The method recognizes that the benefit to each individual participant could be less than the go-it-alone cost. In addition, the alternative justifiable expenditure method is often much easier to calculate than the SCRB method described below. The method has been favored by the Tennessee Valley Authority as well as the u.s. Environmental Protection Agency. In contrast to the AJE method, the seRB method uses separable costs rather than direct costs to determine joint -costs. The SCRB procedure assigns to each participant his proportion to the remaining benefits (as limited by alternative costs). For convex games the SeRB solution lies in the center of the core. A game is convex if C(S) + C(T) > C(S U T) + C(S n T) (7) for all Sand T subsets of the grand coalition Nand C defined previously as the characteristic cost function. The Federal Power Commission, the u.S. Army Corps of Engineers, and the Bureau of Reclamation have relied on the seRB method almost exclusively. All water resource agencies are applying the SCRB method for multiobjective and mUltipurpose federally assisted reservoir projects (Loughlin. 1977). Although conventional methods such as the SCRBmethod incorporate fairness criteria similar to some of those

PAGE 39

employed in cooperative game theory, corrections are required to ensure that the resulting vector of charges is sustainable (lie within the core if the core exists). Heaney (1979b) proposes that incorporation of benefits in the SCRB method should be extended to include subcoalition benefits. In addition, Heaney and Dickinson (1982) propose a generalization of the SCRB method to 33 -incorporate full core constraints. The rationale here is that for games where the separable costs which are the incremental costs for each participant (player) to join the coalition last are not the lowest incremental costs, the _SCRB method prorates joint costs based on upper and lower bounds that are not in the core. Consequently, for nonconvex games the SCRB solution does not lie in the center of the core and may not lie within the core at all for extreme cases. The proposed me-thod inv6lves solution of a system of linear programs to delineate the core bounds followed by proration of joint costs based on these actual core bounds. For the case where no core exists, Heaney and Dickinson (1982) suggest relaxation of intermediate coalition constraints to determine a compromise solution. This generalized SCRB procedure is called the minimum cost remaining savings (MCRS) method. Analogous to the "fair solutions" of the conventional SCRB and MCRS methods are some of the unique solution concepts used in game theory. Two of the most popular unique solution notions are the Shapley value and the

PAGE 40

34 nucleolus. The idea of the Shapley value that each participant should pay the incremental cost of adding him to the coalition. With the assumption that all coalition formation sequences are equi-likely, the Shapley value assigns to each player the incremental cost he brings to coalitions expected over all coalition formation sequences. The resulting charge to the ith participant is XCi) = l: a. (S) [C (S) -C (S -{ i })} S eN]. (8) where (S -1) (n-S) = n! and n is the total nu:mber of participants. If the game is convex, the Shapley value is in the center of the core. However, for nonconvex games the Shapley value may fall outside of the core. In addition, computations for involving more than just a few participants are quite tedious. Littlechild and Owen (1973) present a simplified Shapley value for application in a special class of multi-purpose projects where the purpose with the largest cost of separate action in a coalition determines the characteristic function cost for the entire coalition. Littlechild and Thompson (1977) demonstrate the advantages of the simple Shapley value for determining aircraft landing fees. Heaney (1979a) demonstrates how the simple Shapley value might be applied to mUltipurpose water resource projects through an

PAGE 41

example for pollutant pricing in a 'wastewater treatment plant. 35 In addition to the computational problems, Loehman et ale (1979) find anotherdeiiciency in application of the Shapley value to real situations. Loehman et al (1979) argue that it may be unrealistic to assume that all orders of users are equi-likely as assumed for the Shapley value .. Loehrnan and Whinston (1976) have developed a generalized Shapley value that represents the expected incremental cost where all orders are not equi-likely. Loehman et ale (1979) apply the general Shapley value to an eight-city regional wastewater treatment system in whicbcoalition sequences that do not occur are identified and their 2robabilities set to zero in Shapley value computations. In the eight-city example, lIimpossible" coalition formation sequences are given as those that are not economically viable. This application of the general Shapley value is not correct for the conventional definition of the characteristic cost function. The characteristic function C(S) is usually defined as the optimal solution for that At worst, no lower cost results in coalition formation in which case the is said to be inessential, that is C(S) + C(T) = C(S U T) S n T = ( 9-) for all subsets Sand T in N. By setting to zero the probabilities of inessential coalitions, weak players that

PAGE 42

36 acutally contribute the smallest savings to'the grand coalition become stronger players and are undercharged while actual strong players are overcharged. This distorts the game and may lead to very unfair charges. If the criteria for identifying impossible coalition sequences are restricted to include only political or strictly physical considerations, application of the general Shapley value might be appropriate. The nucleolus maximizes the minimum savings of any coalition and requires N-I linear programs. The solution satisfies all of the core constraints, always exists, and is unique. See Heaney (1979a) or Lucas (1981) for more infor-mation on computing the nucleolus. At this point in the discussion, two particular points warrant further consideration. First, the matter of defining an appropriate characte-ristic function presents some problems. Throughout much of the game theory and cost allocation literature the characteris.tic function is assumed given with little consideration of how it is derived. Conventional and game theory concepts and methods are of little value unless the required characteristic function values can be unambiguously defined. Sorenson (1972) defines the following four alternative definitions for the characteristic cost function. = value to coalition if S is given preference over N-S = value of coalition to S if N-S is not present

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= value of coalition in a strictly competitive game between coalition Sand N-S = value of coalition to S if N-S is given preference As will be demonstrated in this !=Otudy, alternative defini-tions can be used depending on how the problem is defined. 31 The second consideration involves the notion of strict optimization in engineering design. Often economic ments call for a large or complex project design that may be difficult" if not impractical to implement. The preferred solution may be a relatively efficient design that is easier to implement. The argument for good suboptimal solutions is founded on several grounds. Heaney (1983) observes that as the size of a regional project increases, transaction costs might be expected to increase at the margin due to multiple political jurisdictions, growing-administrative costs and shifting of environmental impacts. Although most economic analyses ignore these transaction costs, Heaney (1983) estimates that they may run from 2 to 10 percent of total costs. When transaction costs are high an intermediate but simpler solution might be preferred. Unfortunately, work has been done examining viable intermediate economic solutions. One reason for this is that the availability of powerful optimization techniques such as linear programming, reduced gradient, and sequential quadratic programming techniques ignore suboptimal solutions that may be close enough to the optimum to be satisfactory. Wilde (1978)

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,. 38 develops the concept of "satisfactory design" as an approach to convert relatively difficult optimization problems into the construction of easy to compute bounds and estimates of the ideal optimum. Finally, the viability of suboptimal designs is especially apparent given the uncertainty of physical and economic data. With regard to wastewater treatment sys.tems, cost estimates are usually based on power function approximations of available data. Consequently, first order cost estimates which merely specify treatment process types such as primary sedimentation or activated sludge may be in error by nearly 60 percent. Even more detailed second order estimates involving specific component __ .such as filtration media, backwash __ "---_u __ surface washing facilities may err by more than thirty percent (Clark and Dorsey, 1982). Likewise, performance estimates are often highly variable. Clearly, emphasis on strict economic optimization may be inappropriate when uncertainty of data is high. Suboptimal solutions may be obtained with less effort and expense while achieving a satisfactory result. Given this wide variability in cost estimates, it is not surprising that practicing profes-sionals do not seem to concern themselves with the average cost/ marginal cost c6ntroversy. Summary Several important observations may be extracted from this literature review. First, economic theory addresses

PAGE 45

39 only the case where the optimal design corresponds to marginal benefits equal to marginal costs. Economists argue that given this optimal condition, the proper pricing scheme in a monopolistic (economies of scale}. market is one that produces the least damaging deviation from strict marginal pricing. Economic theory fails to address the problem that arises when the economic optimum does not correspond to marginal benefits equal to marginal costs. While recent theories on both natural monopoly pricing and engineering cost allocation incorporate the concept of sustainability, current cost allocation methods typically ignore marginal cost considerations. Instead marginal considerations are viewed as strictly a part of the economic evaluation phase of project planning and therefore independent of the cost allocation phase. A major point of discussion in this paper concerns the appiicabili ty of th-ese two conflicting viewpoints. Finally, with regard to economic optimization, most water resource applications involve linear programming solutions to the linear cost and fixed-charge problems. Although economies of scale in project construction. and operation are typical for many water resource projects, incorporation of concave cost functions has generally been avoided due to conceptual and computational problems. However, recently developed codes may provide a means of formulating more realistic models for engineering application.

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Section II METHODOLOGY The purpose of this section is to develop a general approach for examining the relationship between the economic and financial objectives in water resources planning. Following is a suggested sequence of steps that provides a means of describing and evaluating this relationship. The steps are listed below in general terms and are followed by a discussJ.on o-f possible Fesults-and a -simple wastewater reuse example. The discussion includes three different cases defined according to problem objectives and corresponding cost and benefit relationships. Project Evaluation Steps 1. Examine overall project objectives. 2. Define the system for study. 3. Formulate the problem in terms of specific objectives and constraints for the system defined in step 2. 4. Examine how the problem objectives and constraints relate to theory and available methodology, e.g. economic theory, optimization techniques, and cost allocation methods. 40

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5. Apply appropriate methodologies based on the anal.ysis in step 4. 41 6. Examine implications for economic e.fficiency and equity in terms. of tradeoffs and possible compromise solutions. Discussion and Wastewater Reuse Example Step 1. Examine Overall Project Objectives This first step is required to formulate the problem. Often the motivation behind a proposed resource project involves political, social, 'and environmental considerations as well as economic and financial objectives. Clear understanding of the major and underlying objectives is essential to the evaluation of tradeoffs and viability of possible compromise solutions. Donovan. et al. (1980) discuss several of the possible considerations involved in wastewater reuse planning. One objective might be to reduce water supply costs to a group of users byt.akinq advantage of economies of scale in a wastewater reuse Other objectives might include increasing available water during drought by decreasing demand on the community's existing water supplies. In this way potable water supplies can be .protected for more valuable use. Other goals may require a reuse system to provide social benefits such as development of municipal recreational facilities. Still other motivating factors might involve improvement or shifting of

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---42 environmental quality impacts. An underl.ying issue for any objective is the question of how best to apportion project costs (or benefits). The objective of the financial analysis might be an equitable but simpl.e allocation of project costs (or benefits). Inherent tradeoffs may develop between these two financial objectives (equity and. simplicity) as well as among financial and economic, social, or political objectives. Finally, planning objectives must incorporate any legal or institutional constraints. The objective for the example wastewater reuse study is to determine the system design that minimizes total water supply costs to potential wastewater users and to fairly allocate these costs among participants. Step 2. Define the System for Study Before the problem can be properly formulated for analysis, the system under consideration must be carefully defined. A proper definition includes identification and description of technical, economic, social, legal, and political relationships. Although this is an extremely important phase of water resources planning, it is often quite complex. See Sample (1983) for detailed treatment of this phase in an actual wastewater reuse study for the South Florida Water Management District. Detailed consideration of how the system is determined for this example might detract from the emphasis of this paper. Therefore, the final system for the example problem is simply given as

PAGE 49

43 shown in Figure 5. The system consists of a secondary treatment plant as the single wastewater supply source and four irrigation sites. Locations of pipes connecting the treatment plant to each site are fixed by right-of-way restrictions. Costs for consideration in subsequent economic and financial analysis include the cost of additional treatment at the plant and transportation costs. Additional informa-tion includes the demand schedule for each user. This is derived from water use patterns or from alternative water supply costs. Alternative water supply costs may be referred to as on-site costs while treatment and trans-portation costs may be termed off-site costs. In actual practice, costs for consideration may also include costs for water quality monitoring, replumbing, storm water runoff control, and future capacity expansions. Application of cost engineering principles results in estimates for treatment and transportation costs. For planning studies, cost estimates are often derived from statistical cost equations corrected with appropriate updating and localizing factors. The general form for the equations is a power relationship of the major input and output. variables for construction and operation and mainte-nance of the various project components and unit processes. For the wastewater reuse example, a reasonable representa-tion for treatment costs and on-site costs is given by a

PAGE 50

44 Site 4 = 2.5 MGD o -3MGD -m,l Q =2 l-iGD -m,.2 = 1.5 MGD Figure 5. System for wastewater reuse.example.

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45 power function of the flow In contrast, trans-portation costs which include pipe and pumping costs are best estimated from static head, pipe diameter, and. pipe length as well as flow. For a given pipeline path, the pipe length is fixed and static head may be reasonably estimated. As a result, cost estimates for the example transportation costs may be reduced to functions of flow and pipe diameter. In addition, for a given flow a tradeoff exists between pipe costs which increase with increasing pipe diameter and pumping costs which decrease with increasing diameter. Deb (1978) presents an optimization model which selects the optimal pipe diameter to minimize total transportation costs. Sample (1983) modifies this procedure to include a more detailed analysis of transportation component costs. The result is a scheme to compute optimum pipe diameter as a function of flow so that total transportation costs may be presented as strictly dependent on treated wastewater flow. All costs are estimated by a power function of the form where 3 C = annual cost, 10 $ Q = treated wastewater flow, MGD a,b = constant function parameters

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'. 46 All estimates include construction and operation and mainte-nance costs and are assumed to be properly updated with local information. The specific equations for the example problem are given in Table 1. Cost relationships for off-site costs exhibit economies of scale as indicated by the exponent parameters which are less than one for both treatment and transportation. This is consistent with conditions found in practice. For example, Sample (1983) found signifi-cant economies of scale in both treatment and transportation costs in the south Florida study. The example relationships are consistent with these general results. ___________________ .Qn-site __ __ for the_exam2le 2roblem are also given in __________ #<. Table 1 for three cases including economies and diseconomies of scale as well as linear on-site costs. These three cases are examined to demonstrate different results and implica-tions for economic and financial analysis. A final consideration for defining the system involves specification of maximum demand and plant capacity con-straints. Two different plant capacity limits are consid-ered corresponding to the case where supply does not limit system design (plant capacity, Q t = 10 MGD) and, alternam, tively, to the case where system design is supply limited (plant capacity, Q t= 6 MGD). This analysis assumes that m, each potential wastewater user's current quantity of total water use is equal to the maximum demand fixed by physical requirements. However, each user's demand for recycled

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Table 1. Cost Functions for Wastewater ',euse Example Cost Component On-site Treatment, C t 60(Qof,1 + Qof + Q + Q ).7 2 'of,3 of,4 'Site, i 1 3 Transportation, C .4 i .3 "13 (0 ) .5 p,l 8(QOf,1) iCQOf,2) :of ,3 I On-site, Co I n,l Case 1 I Linear Costs 80Qon,1 71Qon,2 90Q 3 on, Case 2 100(Qon 100 (Q ') 8 Economies of Scale 8Q(Q ).9 I on,2' on,3 Case 3 Diseconomies of Scale 25(0 )1.7 4Q(Q )1.4 4S(Q )1.2 on,l' i on,2 'on,3 ; 4 17(Q ).6 of,4 100Q 4 on, 120(Qon,4) 7Q(Qon,4)1.1 .r::

PAGE 54

48 water is price dependent as given by on-site cost relation-ships. Therefore, the demand cons'traint is that total water supplied to each user must equal the fixed demand specified in Figure 5. However, the quantities obtained on-site and off-site are determined by the economic and financial analysis in the steps that follow. Step 3. Formulate the Problem in Terms of Specific Objectives and Constraints As stated in step 1 the objectives of the wastewater reuse study are twofold. The first objective involves economic efficiency for which the desired design is one that minimizes total irrigation water costs for the defined second obJective .involves flnanc.rar-considera";" tions for which an equitable apportionment of final project costs is desired. The economic _objective and corresponding constraints are specified in terms of the system defined in step 2. The resulting economic optimization problem is to !lrinimize z = sub-ject to 4 E (C + C .) + Ct i=l on,l p,l 4 1: i=1 Qof,i < i = 1, 2, 3, 4

PAGE 55

where 49 C = on-site water supply cost, for site i, l03$/year on,l. C = transportation cost for pipeline to site i, p,l. C t = treatment cost lO3$/year, Qon,i = on:"'site water supply to site, i, MGD' Qof,i = off-site water supp.ly to site i, MGD Qm,i = maximum water demand for site i, MGD Qm,t = treatment plant capacity, MGD. This cost minimization problem is equivalent to the net benefit maximization problem for benefits defined as alterna-tive (on-site) costs avoided. In general, a correct specifi-cation of core constraints incorporates individual and subcoalition benefits as well as costs. For this example, the problem is defined so as to -include benefits implicitly as alternative costs avoided. With regard to financial objectives, recall that if the core exists it contains the set of outcomes for which no coalition would be better off not participating. This seems a reasonable set of minimum criteria for an equitable solution. Since the subsequent analysis shows that the core does exist for each of the three cases in this example, we can justifiably eliminate any allocation result not in the core. The core may be represented by the following set of constraints.

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where X(i) < C(i) E XCi) i C(S) iES E X (i) = C (N) iEN i = 1, 2, 3, 4 seN C(S) = the characteristic cost function for coalition S XCi) = charge to player i N = grand coalition of players. so (10) (11) (12) Step 4. Examine How the Problem Objectives and Constraints -Re-late-to-Theo-ry l-ahle----Me thodology This step involves examining how economic and cost allocation theory apply to the problem as defined in step 3. The water resource planning and pricing literature often defines problems where the total potential demand lies within the economies of scale range of project cost func-tions. This is analogous to assuming that the project design has no production capacity limit or that the capacity limit is not binding. For the wastewater reuse problem, the size of the project is limited by either total maximum demand or treatment plant capacity. In addition, most conventional cost allocation and game theory solutions do not consider demand. Charges determined by methods such as SCRB or the Shapley value can be stable in the economic sense only for inelastic user demand. This assumption may

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51 be valid when potential participants must satisfy a fixed demand with no alternatives other than joining the project coalition or constructing an independent project with the same cost function(s) as the joint project. Where there are economies of scale, benefits grow as coalition size increases, and the analysis becomes a problem of determining how the benefits (costs) are to be distributed. A similar effect results in the wastewater reuse example but is complicated by production limits and demand considerations. For the example we will examine three types of demand relationships and the implications of production limits on economic efficiency and cost allocation. As will be demonstrated later in this section, economic optimization for the wastewater reuse problem may be achieved rather easily for certain types of cost relationships. However, in other cases mathematical programming t-echniques may be required to solve the general nonlinear optimization problem. Three optimization methods: separable programming, generalized reduced gradient, and sequential quadratic programming are considered mainly on the basis of the availability of computer codes. Recall that separable programming is a linearization technique which utilizes piecewise linear approximations of separable nonlinear functions. This technique is applicable to the example problem since all problem decision variables appear separately in the objective and constraint equations.

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52 Separable progranuning is contained within the linear programming procedures of the Mathematical Programming System (MPS/360) available through the IBM Corporation (1971). The separable programming procedure employs the delta-method described by Hadley (1964) in which the separable nonlinear functions are approximated by means of linear equations coupled with logical restrictions. The solution is given by a simplex procedure modified to incorporate the required logical restrictions. One caution in applying this technique is that the solution may converge to a local optimum. This possibility may be reduced by solving both the dual and primal problem or solving the problem for different initial values for the decision variables to verify results. Application of the MPS/360 code proved rather tedious for the wastewater reuse example. The user is required to construct appropriate piecewise linearizations for each separable function. The example problem objective functi.on contains several different separable functions that must be approximated. Furthermore, each setof computer input data is very specific to the particular problem definition. Relatively small changes in the problem definition often require extensive alterations in the computer input. Therefore, this method may not be practical for planning studies involving many different nonlinear cost relationships or studies requiring flexibility in defining these relationships and problem constraints. This is the case for the wastewater reuse example where the problem is defined

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,.53 for various demand and constraint conditions. As a result, this technique was abandoned in favor of a more flexible code. One 'such code is NPSOL by Gill et a1. (1983) which uses a sequential quadratic programming algorithm in which the search direction is the solution of a quadratic prog+amming subproblem. The algorithm requires that the user define the problem in terms of the specific objective and constraint functions and their partial derivatives. to separable programming, this code requires much less prelim-inary work by the user and easily accommodates changes in problem definition. However, this code is very sensitive to _________ _is __ suited ____________ problems containing nonlinear const'raints. Attempts to apply NPSOL to the example problem resulted in problems of -ill-conditioning not easily remedied. The third code, GRGA by Abadie (1975), employs the generalized reduced gradient method. As with NPSOL the user must define the problem in terms of the specific objective and constraint functions as well as partial derivatives. Comparative studies indicate that this code is among the most efficient and least sensitive to problem condition (see Reklaitis et al., 1983). Unfortunately the only avai1able documentation for this code is in French. Additionally, the program generates all output in French. However, application of the code to the example results in no conditioning

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;, 54 problems. Therefore, results presented for the example problem are those determined from GRGA. Financial analysis for the example problem involves evaluation of various cost allocation schemes including the Shapley value and MCRS method as well as marginal and average cost pricing strategies. As the financial objective is' to determine a set of charges that satisfy core con-straints, the characteristic cost function must be defined. In conventional cost allocation theory, the characteristic cost function is defined under an implicit assumption of inelastic demand. That is, the quantity of output demanded 'by each player is assumed fixed. The player, then, must satisfy his fixed demand through either independent action or cooperation in a joint project. The characteristic function is computed for fixed quantities using fixed cost relationships. However, it is often unrealistic to assume an inelastic demand. In the wastewater reuse example, the total quantity of water required for irrigation at each site is assumed fixed, but the demand for wastewater is price sensitive since each user has an alternative water supply option with its own cost, relationship. The player may satisfy his demand through varying degrees of participation in a joint project as opposed to the usual all or nothing restriction. Independent action, then, may involve a quantity of project output different from the quantity assigned for joint project cooperation. Determination of the characteristic function requires specification of

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appropriate quantities for independent action. For the wastewater reuse example, these quantities may be taken as the off-s.ite supply corresponding to the minimum cost combination of on-site and off-site water supply for each 55 player. However, where potential demand exceeds production capacity, it is necessary to define how the restricted supply is to be apportioned not only for the final project but for independent action as well. Therefore, to determine the optimal go-it-alone solution for coalition S, it is necessary to apportion an output quantity to S by defining the behavior of the complementary coalition N-S. Recall that Sorenson (1972) suggests four possible definitions for the characteristic function. We will examine each of these definitions to determine which might be the most appropriate for application to the example problem. Sorenson defines C 1 (S) as the value to. the coalition if S is given preference over N-S. If S has preference over N-S, S gets all the wastewater it wants while entering the project after N-S. This way S would get all of the highestsavings from off-site economies of scale without having to share them with other groups However, this would result in a game that is not sub additive. Coalition S would never want to cooperate with other groups and have to share savings. A second definition, C 2(S), is the value to thecoalition if N-S is not present. This would require that S has

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56 the option to build and operate its own facility or be allowed to use the central facility up to its wil.lingness to pay (as given by alternative costs), its maximum demand, or plant capacity, whichever comes first. This mayor may not lead to a subadditive game depending on actual relative costs and capacity limitations, e.g. S may gain more savings by independent action as defined by C 2(S) than by joining a coalition where its level of activity may be restricted. A third definition, C 3(S)k involves a strictly competitive game between Sand N-S. Since joint cost economies ofscale imply a basic cooperative attitude among participants, C 3 (S} is certainly not an appropriate definition for this example. As a final definition, C 4 (S} is the value of coalition S if N-S is given preference. Giving preference to N-S means letting N-S go last so that it can realize-highest marginal savings. There are at least two ways C4(S) can be computed using this definition. First, given that ON is the economically efficient output allocation vector -for the grand coalition N, C(S) may be computed using thego-it-alone cost for its allocated portion of ON-This .l.sequiva-lent to making S go first to receive its allocated quantity of resource. This implies a cost to N-S equal to C(N} -C (S) With economies of scale C(S) will always be more than the cost in a larger coalition for a given QN'

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57 thus ensuring subadditivity. However, this method is not really in keeping with the concE?pt of independent action since it depends on results of economic analysis involving N. However, it does provide a means of ensuring that all members of the grand coalition will have a right to some of the potential savings whether or not they cooperate. In effect, this method of computing C(S) uses the results of economic analysis to establish ownership. Perhaps a better approach is the idea of letting the members of N-S go ,ahead and form their own coalition and least cost solution.. If the plant capacity limit is reached, S will get nothing and C(S) is just SIS alternative (on-site) cost. In this way not everyone is guaranteed some of the potential savings. This result may be appropriate if S is such a weak player that it contributes savings only when joining relatively large coalitions. If there were-some remaining capacity, we would define C(S) as the value to S of the least cost solution for the remaining supply. The implied cost to N-S is again C(N) -C(S). Coalition S might object since N-S gets some additional savings by having S go first. On the other hand, coalition N-S could argue that to charge SIess would require some degree of cooperation by S. This last definition is probably the best for'this particular applica-tion since it ensures game subadditivity and is in keeping with notions of independent action.

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58 Before discussing the various cases for the example problem, a few comments are in order to clarify some possible points of confusion. First, in the analysis that follows it is often easier to discuss benefits or net benefits rather than alternative costs avoided or cost savings. This interchangeability of terms is a result of equivalence in problem formulations. Recall that defining the economic objective as a cost minimization problem is equivalent to defining it as a net benefit maximization problem. Benefits are computed from on-site cost as fol-lows: where 3 B(i) = benefits to user i, 10 Iyear and C ., Q ., and Q f are as defined previously in on,l m,l 0 ,1 Table 1. An additional point involves inclusion of direct costs in the analysis. It may seem reasonable to allocate direct costs separately from joint costs. However, since participant in the optimal. project benefits from the inclu-sion of all the other participants, each participant should be willing to include direct transportation costs in the total cost that must be allocated. In particular, if direct costs are not included, the relatively high direct costs assigned to some users might discourage their cooperative participation, resulting in a suboptimal final project

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59 design. Inclusion of direct costs in the analysis means that, in general, a single cost function will not apply. A single cost function often applies only to joint project components. Usually, different cost relationships apply to different cost elements. This is the case for the waste-water reuse problem where estimating functions for trans-portation costs are different for each site. Case 1. Single-Step Demand Function A single-step user demand is derived from linear on-site costs which in turn imply a constant unit water price. This situation may be common in actual practice where a user's current unit cost for water_is the_only information easily available. The single-step demand curve for site 1 in the wastewater example is shown in Figure 6(a}. -The flow level Q 1 corresponds to the maximum demand flow m, for site 1. The unit cost of $80,00 per MGD represents the average (or marginal) benefit to the site and corresponds to a linear benefit curve as shown in Figure 6(b). Econ?mic optimization First we consider joint project components, for the moment ignoring direct cost components. This corresponds to consideration of central facility treatment costs while ignoring transportation costs. For this simple case, the potential user (player) with the highest average benefit gets priority, i.e. Q f = Min[Q remaining plant o ,1 m,l

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l03$/MGD Demand=Marginal Benefits 8.0 Q f 1 (MGD) 0, (a) Q = 3 MGD m,l 60 _____________________ 4 0 -=--=-_'--_--=-_=-=-_ _________ _________ Qof,l (MGD) (b) 3 Total I Benefits Net Benefits Qaf,l (MGD) (e) 3 Figure 6. Case 1 benefit and cost relationships for site 1: (a) rnarginalbenefits, (b) total benefits, and (c) net benefits.

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61 capacity]. This is true because net benefits are contin-uously increasing as shown in Figure G(c) for site 1. If some plant capacity remains, the user with the next highest average benefit receives output up to his maximum demand or plant capacity. This rank order assignment of wastewater supply continues until either potential demand or the. plant capacity limit is reached. The result for this simple case is a bandwagon effect for which the order that players join the bandwagon has important implications for economic optimization. In particular, if plant capacity is less than total potential demand, some players may be partially or totally excluded from the project. The excluded players __ Qontril:mte and so are giyen_Iower __ priori tY' in joining the project. Results are not quite so easily seen when direct costs are added to the analysis. Since direct cost components such as the transportation costs here usually involve cost relationships which differ from joint costs and differ from other direct costs, the player with the highest average benefit may not contribute the greatest net benefit to the project. As shown in Figure 7 player A's average penefit is less than player B's average benefit, but A also has rela-tively low direct costs so that his net benefits are actually greater than those for B for a given level of output Q. To allow for this possibility, analyses which p include direct costs must consider total net benefits in$tead of just average benefits. The result is similar to

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Q, MGD Player A 103$ ITotal INet IBenefits 62 I Net I Benefits I I Direct I ---.---Costs----L-___________ ... I Q, MGD Player B Figure 7. Priority ranking example for direct cost considerations for case 1.

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63 the rank order effect for joint costs. To maximize net benefits, the player with the highest tota1 net benefits for any arbitrary output level receives highest priority. This result is valid even when each player's benefits are not strictly greater than costs since priority ranking is determined by total net benefits. An important result for case 1, t.hen, is that the economic optimum is achieved by assigning output to each user according to a priority ranking based on comparison of total net benefits. The net benefits a user contributes to the project are 1imited on1y by maximum demand or plant capacity constraints since net benefits are continuously increasing. In step 5 we apply .... ______ _____ __ to __ __ reuse __ __________ example. cost Allocation With the above general results for case 1, we consider several cost allocation schemes in search of an appropriate strategy to satisfy financial objectives. The methods for consideration include marginal pricing from economic theory, the Shapley value from game t.heory, and MCRS (as a generalization of SCRB) and average cost pricing methods from conventional practice. First, we completely eliminate the possibility of applying marginal pricing strategies. Clearly, where net benefits are continuously increasing within the range of maximum demand and plant capacity limits, marginal costs will equal marginal benefits

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64 only at a point of minimum net benefi-ts. Consequently, the concept of marginal pricing is not applicable for case 1. To evaluate the remaining methods, we must first define the core for the final project design in characteristic function form as defined by C 4(S). When considering joint cost functions such as central facility treatment costs, economies of scale guarantee the existence of a core and in particular a strictly convex core (Tschirhart, 1975). Note that the direct costs have no bearing on overall. convexity. Cost allocation solutions for a convex game exhibit certain properties. Namely, the Shapley value will_be in the center of the core and extremely close to the MCRS sol.ution (Heaney and Dickinson, 1982). Also, for a convex game the MCRS and SCRB solutions are identical. Furthermore, since the marginal cost curve falls below a decreasing average cost curve, a vector of charges based on average cost pricing also satisfies core constraints. Computations presented in step.5 demonstrate these general results for the wastewater reuse example. In step 6 we examine some important implica-tion of these results. Case 2. -Sloping Demand Function An upward-sloping demand for reclaimed water is derived from decreasing on-site marginal costs. This situation might correspond to economies of scale in on-site pumping costs or a schedule of declining unit or block prices for purchased water. A sketch of the on-site cost curve and

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corresponding upward-sloping demand is shbwn in 8 for site 1. Again, the flow level Q 1 corresponds to the m, maximum demand for the site. The resulting total benefit curve is also shown in Figure 8. Economic optimization The situation here is very similar to that in case 1. As demonstrated in Figure 8, net benefits are continuously' increasing for a given cost function. Therefore, the same general results apply in which the economic optimum is achieved through priority assignment of output based on comparison of net benefits for each user. However, unlike the previous case, the output level for comparison of net benefi ts may not be strictly arbitrary where the net benefi t curve for one player intersects the net benefit curve for another player. The output level corresponding to an intersection implies a switch irr priority from one player to another. This is demonstrated in Figure 9. At flow levels less than Q k player A dominates while at flow levels greater than' Qk' player B dominates. Therefore, priority between A and B depends on quantity of output available to the players. In general, then, priority among players in a subcoalition S may not be given by the priority ranking in the grand coalition. On the other hand, if the net benefit curves do not intersect within the range of interest, priority among players is determined by comparison of net benefit contributions for any arbitrary flow level within the range of

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269 On-site Costs 103$ Marginal Benefits 80.61;....----I 3 0 Qof,l, MGD (b) Q of,l, ------Q on,l, (a) 3 1-1GD 0 If l Of -S:Lte I Benefits : 103 $ I I 1269 ... Qm,l = 3 MGD l-1GD 26 9 ------------' -Qof,l, MGD (e) Net Benefits Figure 8. Case 2 benefit and cost relationships for site 1: (a) total on-site costs, (b) marginal benefits, and (c) net benefits. 66

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Total Net Benefits, $. I 67' --------------------------------------------------------------------------------I I I I I Qk Q, MGD Figure 9. Example of shift in priority for case 2.

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interest. The resulting rank order priority holds for all subcoalitions as well as the grand coalition. Cost allocation 68 The discussion regarding financial analysis for case 1 is applicable to case 2 as well. General conclusions for financial analysis are identical for the two cases as demonstrated in steps 5 and 6. Case 3. Downward-Slopi,ng Demand Function A downward-sloping wastewater demand is derived from inc"reasing on-site marginal costs. This situation might correspond to diseconomies of scale in on-site operating ________________ __ !ncreasiE_g uni block ____ ______ water. The on-site cost curve and corresponding demand and benefit curves are sketched in Figure 10 for site 2. Economic optimization As for the first two cases, the least cost solution results when each user is assigned wastewater supply on the basis of his contribution to total net benefits. In contrast to cases 1 and 2, if case 3 benefits are compared to off-site costs, say treatment costs, their difference is not strictly increasing. As demonstrated in Figure 10 benefits increase for lower'flow levels, reaching a maximum at Q* before beginning to decrease. The flow level Q* corresponds to the point where marginal costs equal marginal benefits. Therefore, when a userls demand is downward sloping, his

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105 on-Site! Costs 103$ 2 Q f 2 MGD o Q MGD on,2, (a) I I IOff-site I Benefits I 103$ I '105 Q =2MGD in, 2 ------------------.--------c--------------------105 73.9 103$/MGD 2 Q f 2 MGD o (b) J Maximum Net : Benefits I Q 2 Q MGD of,2, ee) Figure 10. Case 3 benefit and cost relationships for site 2: (a) total on-site 'costs, (b) marginal benefits, and (c) net benefits.

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. 70 contribution to the project is limited by marginal considerations as well as maximum demand and plant capacity limita-For single-step or upward-sloping demand conditions, the only possibility for a mix of on-site and off-site supply to a participant occurs when the plant capacity limit is reached before the last ranking participant fulfills his maximum demand. For downward-sloping demand conditions, the result may be a mix of supply to any or all of the project Wastewater supply is still allocated according to priority based on net benefit contributions, but that priority is not so easily determined. As for case 2 the quantities for comparison of net benefit contributions cannot be arbitrarily chosen. In particular, the player contributing the greatest net benefits at a given flow level may not contribute the greatest net benefits at another larger flow level. This is true since a player's marginal benefits decrease with increasing quantity of acquired output. Therefore, as a high priority player's marginal benefits decrease with increasing supply, the potential contributions of other players become increasingly attrac-tive perhaps reaching an output level where a new player gains priority. This output level corresponds to a point where one player's net benefit curve intersects another player's net benefit curve. This is analogous to the situation in case 2. Consequently, priority among players

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71 in any coalition S depends upon the quantity of resource available to S as defined by C 4(S). The result is that determining priority among players requires additional information regarding relative net benefits at various' output levels and quantities corresponding to marginal conditions (marginal benefits equal to marginal costs). For this example, the nonlinear nature of the problem and inclusion of direct as well as joint costs may preclude simple solution by. hand calculation in favor of an .appropriate mathematical programming code. Such is the case in; step '5 where we apply the GRGA reduced gradient code to solve the economic optimization problem for cas-e 3. Cost allocation General conclusions for cases land 2 regarding the nature of the core and various cost allocation methods are not applicable to this case. First, since the optimal design may include quantities corresponding to marginal conditions,. a charge system based at least in part on marginal pricing might be viable. However, recall that strict marginal pricing does not satisfy total cost requirements. To evaluate the remaining methods, we' again defin'e the core for the fina-l project design in characteristic function form .from C 4(S). Intuitively, one might expect that the resulting cost allocation game is convex as in the previous two cases. However, as we will see, economies of scale guarantee convexity only for C(S) defined for fixed quantities for an assumed inelastic demand or for quantities given

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72 by C 4(S) for single-step or upward-sloping demand, i.e. cases 1 and 2. For case 3 the C 4(S) definition ensures subadditivity since the quantity of resource a player receives in each subcoalition does not exceed the quantity he receives in the grand coalition. However, with diseconomies of scale in on-site costs, the largest incremental savings from avoided costs correspond to the first units of acquired output. At the same time, incremental savings due to off-site economies of s,cale are greatest at larger output levels. If on-site diseconomies are relatively strong, the incremental savings from avoided on-site costs at low output levels may be greater than the incremental off-site savings at higher output levels. So, even though a player receives less output and less total savings by joining smaller coalitions he may realize greater incremental savings than by joining a larger coalition. The result is that the cost allocation game may not be convex. That is, a player or group of players could have higher incremental savings (or lower incremental costs) by joining the project earlier, say second or third, instead of last. Recall that in cases 1 and 2 savings from off-site economies of scale and tive costs avoided both increase with output so that the highest incremental savings occurs at the highest output levels, i.e. by being the last player to join the grand coalition.

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73 Step 5. Apply Appropriate Nethodologies Case 1. Single-Step Demand Function Economic optimization The optimization problem for case 1 is to Minimize Z = 80Q 1 + 70Q 2 + 90Q 3 + 100Q 4 on, on on on I + 8(Q ).4 + 5(Q )e3 + 13(Q ).5 of,l of,2 of,3 + 17(QOf,4) + 60(Qof,1 + 0of,2 + 0of,3 S.T. Qon,l + Qof,l = 3 MGD Qon,2 + Qof,2 = 2 MGD Qon,3 + Qof,3 = 1.5 MGD Qon,4 + Qof,4 = 2.5 MGD Qof,l + Qof,2 + Qof,3 + Qof,4 < 10 MGD 6 MGD To demonstrate the proposed general results for the case of single-step demand functions, we first examine potential contributions from each player and establish the priority ranking that minimizes total water supply costs. Referring to Table 1 and equation 13 we compute off-site

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. \, 74 cos.ts and benefits for any arbitrary flow level. We then rank each player according to his net benefit contribution. As shown in Figure 11, site 4 receives highest priority followed in rank order by sites 3, 1 and 2. Since the total potential demand for wastewater is only 9 MGD, the problem constraint for a treatment plant capacity limit of 10 MGD is not tight. The optimal project design includes all sites contributing a positive net benefit to the project. Results of the rank order calculations indicate that all sites are viable and should be included to achieve the optimal The resulting project consists of all sites each receiving a total wastewater supply equal to its maximum demand quantity. Project costs total $343,000 per year, a 55 percent savings over alternative on-site costs. Results are summarized in Table 2. For treatment plant capacity equal to 6 MGD, the maximum supply of wastewater cannot satisfy total water supply requi+ements for the four sites. The wastewater supply must be allocated among the sites so as to achieve overall minimum water supply costs.. As stated previously, wastewater supply is allocated according to a rank order procedure. Using the rank order established above ,.;e assign wastewater as given in .Table 2. As the highest ranking player, site 4 receives a quantity of wastewater equal to its maximum water demand. With next highest priority, site 3 receives a quantity also equal to its maximum demand. At this point; there are only 2 MGD of wastewater left. As

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Site 1 100 80 Total Net I Benefits 60 t 103$ I J I 2' 4-1 I 20 I t .1 I I 1.5 2 Qof, MGD Figure 11. Net benefits for rank order determination for case 1.

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i, Table 2. Optimal Resource Allocation for Case 1 Site 1 2 3 4 -1 2 3 4 Qm,t MGD 10 10 10 10 Total 6 6 6 6 Total Qm,i MGD 3.0 2.0 1.5 2.5 9.0 3.0 2.0 1.5 2.5 9.0 Q f o MGD 3.0 2.5 1.5 2 .5 -9.0 2.0 0.0 1.5 2.5 6.0 76,

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77 the third ranking player, site I receives all of this remaining supply to satisfy two-thirds of its total irrigation requirement. The resulting project design completely excludes site 2. The annual final system cost is $346.000. Cost allocation ---To apply and evaluate the various cost allocation methods we first compute the characteristic function for both plant capacity limits. Recall that C(S) is defined by C,(S) which gives N-S priority over S. For the 10 MGD capacity limit, giving. coalition N-S priority does not restrict the quantity of wastewater available to coalition S. Coalition "S receives wastewater flow equal to. its total maximum demand but is assigned the cost of supplying the quantity alone, without benefit of the economies of scale from including N-S. The resulting characteristic cost functions are -C(l) = 142 C(12) = 204 C(24) = 208 C(2) = 104 C(l3) = 200 C(34) = 204 C(3) = 96 C(4) -143 C(14) = 240 e(23) -166 C(123) = 257 C(124) -294 C(134) = 292 C(234) = 262 C(1234) = 343 where the corresponding. subaddi ti ve game is convex as verified in the Appendix. The resulting minimum fairness criteria are given by the following core constraints.

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x (1) < 142 X (2) < 104 X (3) < 96 X (4) < 143 X (I) + x (2) < 204 X (1) + X(3) < 200 X (1) + X(4) < 240 X(2) + X(3) < 166 X(2) + X(4) < 208 X (3) + x (4) < 204 X (1) + x (2) + X(3) < 257 X (1) + x (2) + X(4) <. 294 X-{-1) +X (-J-}-+-X-( 4 ) < 292 X (2) + X(3) + X(4) < .262 X (1) + x (2) + X(3) + X(4) = 343 Since the game is convex, the first four and las,t five conditions stipulate the upper and lower bounds on XCi).' That is, a vector of charges within the core must satisfy 81 < X (1) < 142 51 < X (2) < 104 49 < X (3) < 96 86 < X (4) < 143 X(l) + X(2) + X(3) + X(4) = 343 78

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The resulting Shapley, MCRS (equivalent to SCRB for a convex game), and average cost pricing solutions are presented in Table 3. As expected the Shapley value and MCRS solutions are in the center of the core. Although the set of charges derived from average cost is positioned away from the core center in favor of the highest priority sites, 3 and 4, we see that the shift is not extreme and that each site is still awarded large savings. This particular result has important implications for the final selection of an appro-. priate pricing scheme. For the 6 MGD capacity limit, giving N-S priority does restrict the quantity of wastewater available for coalition S. The available supply must be allocated among the players in S according to the established priority order. For the 6 MGD three-site system, priority among sites 1, 3 and 4 in any subcoalition is the same as that already established, i.e. site 4 has highest priority, site 3 second, and site 1 lowest. A sample calculation of the characteristic. function is show.n below. C(1): let (3,4) go first, Qof,3 + 0of,4 = 4 MGD 6 -4 = 2 MGD left C(1) = 8(2) + 60(2) + 80 = 188 Similarly, C(3) = 136 C(4) = 201 C(13) = 251 C(14) = 292 C(34) -258 C(134) = 346

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80 Table 3. Charges for Case 1 Total cost, 3 10 $/year Q -m,t Average Cost Site I1GD Shapley MCRS Charge Savings --1 10 102.92 102.27 114.33 52% 2 10 69.25 69.48 76.22 46% 3 10 64.53 65.39 57.17 58% -4 10 106.25 105.87 95.28 62% Total 343 343 343 1 6 126.33 125.82 148,,30 38% 3 6 83.33 85.07 74.14 45% 4 6-136.33 135.11 123.56 51% ----Total 346 346 346 '----

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. The resulting subadditive cost game is convex since e(13) + C(14) > C(134) + C(l) 251 + 292 > 346 + 188 OK t(13) + C(34) > C(134) + C(3) 251 + 258 > 346 + 136 OK C(14) + C(34) l C(134) + C(4) 292 + 258 > 346 + 201 OK The core bounds are given by X-1L} X (3) < 136 X(4) < 201 X(1) + X(3) < 251 X(l) + X (4) < 292 X (3) + X (4) < 258 X (1) + X(3) + X (4) = 346 or 88 X(1) < 188 54 (3) < 136 < 201 X (1) + X (3) + X (4) = 346 81 The Shapley, MCRS, and average cost pricing solutions are shown in Table 3 and in Figure 12. Again, as expected,

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X(4) < 201 AVER AGEL-_--!---\r-,o COST X(4) 95 1 L-______________ .. 3' C(134) = 346 Figure 12. Core and charges f,or three-person game for case 1. 82

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-. : .. 83 the Shapley value and MCRS solution are in the center of the core. Again the average cost solution shifts away from the core center in favor of the strongest players. However, even as the lowest player, site 1 still receives a significant savings over exclusive on-site water supply. Case 2. Upward-Sloping Demand Function Economic optimization Th.e optimization problem for case 2 is to Minimize Z = 100 (Oon, 1) .9 + 80 (0 ) .9 on,2 + 100{Q ).8 on,3 + 120(0 on,4 ) .7 + 8(0 ).4 of,l + 5(Q ).3 of,2 + 4) .6 T 60(00f,1 + Qof,2 + 0of,3 + Of,4) S.T. Qon,l + of,l = 3 MGD Qon,2 + of,2 = 2 MGD on,3 + of,3 = 1.5 MGD on,4 + of,4 = 2.5 MGD Qof,l + Qof, 2 + Qaf, 3 + Qof 4 < 10 MGD 6 MGD To determine the least cost solution for the case of upward-sloping demand functions, we apply the principles established for case 1. Note that in the range of interest (0 to 9MGD) only two net benefit curves intersect. The net benefits for sites 1 and 3 are equal for a flow level of

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84 1.38 MGD. The consequences of this shift in priority are demonstrated in Figure 13 where net benefit contributions are compared for flow levels in the range of Priority between sites 1 and 3 depends on whether the supply available to the two sites is greater than or less than 1. 38 MGD. Again the constraint for a plant capacity of 10 MGD is not tight. For wastewater supply equal to maximum demand. each site contributes positive net benefits. Therefore, the optimal project includes all four sites; each receiving an output quantity equal to its maximum demand. This result is. identical to case 1. For a plant caQacity of 6 MGD the limited supply is allocated according to the priority ranking estab.lished in Figure 13. The most efficient resource allocation is given in Table 4. Site 4 is completely excluded from the project so that the final design consists of only the three highest ranking sites. Sites 1 and 3 each receive off-site supply quantities which satisfy total water demand,. while site 2 must settle for a mix of on-and off-site The project represents a 48% savings that must be allocated among the three participants. Cost allocation For the 10 MGD capacity, the characteristic function and cost allocation solutions are identical to those for case 1 as shown in Table 5. For the 6 MGD capacity limit,.

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I Total Net Benefits, $ 120 100 80 4Q 20 .', f Figure 13. Site Site 3 Q f MGD o 1 Net benefits for rank order determination for case 2. 85

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Table 4. Optimal Resource Allocation for Case 2 site 1 2 3 4 1 2 3 4 Qm,t MGD -10 10 10 10 Total 6 6 6 6 Total Qm,i MGD 3.0 2.0 1.5 2.5 .0 3-.0 2.0 1.5 2.5 9.0 Qof,i MGD 3 0 2.0 .. 1.5-2.5 9.0 3.0 1.5 1.5 0.0 6.0

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/' I ." Table 5. Charges for Case 2 Total Cost, Qm,t Site MGD Shapley MCRS 1 10 102.92 102.27 2 10 69.25 69.48 :3 4 10 106.25 105.87 -Total 343 343 1 6 126.33 125.66 2 6 78.33 78.45 3 6 82.33 82.89 [ Total 287 287 87 103$/year Average Cost Charge Savinqs 114.33 52% -76.22 46% 5J;-:t,.--S--sr 95.28 62% 343 132.40 51% 88.30 41% 66.23 52% 287

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the available supply is allocated according to the established priority ranking. The characteristic cost function for player 1 is computed by C(l): let (2,3) go first, Qof,2 + Qof,3 = 3.5 MGD 6 -3.5 = 2.5 MGD left C(l) = 8(2.5) + 60(2.5) +100(0.5) =179 Similarly, C(2) = 128 C(3) = 130 C (12) = 233 C(13) = 239 C(123) = 287 C(23) = 194 This game is convex since C(13) + C(12) C(123) + C(l) 239 + 233 > -287 + 179 OK C(12) + C(23) C(123) + C(2) 233 + 194 > 287 + 128 OK C(13) + C(23) > C(123) + C(3) 239+ 194 > 287 + 130 OK The core bounds are given by 88

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.. or x (1) < 179 x (2) < 128 X(3) <.130 X(1) + X(2) < 233 X(1) + X(3) < 239 X(2) + X(3} < 194 X(l) + X(2) + X(3) = 287 93s.. X(l) s.. 179 48 s.. X(2) s.. 128 54 s.. X(3) s.. 130 X(I) + X(2) + X(3) = 287 89 The Shapley value, MCRS, and average cost pricing solutions are given in Table 5 and in Figure 14. They demonstrate the -applicability of important case 1 general results to case 2. Case 3. Downward-Sloping Demand Function Economic optimization The optimization problem for case 3 is to Minimize Z = 25{Q )1.7 on,1 + 40(Q }1.4 + on,2 45(Q )1.2 on,3 + 70(Q on,4 ) 101 + 8 (Q ) .4 of,1 + 5(Q ).3 of,2 + 13(Q )05 + 17(Q ).6 of,3 of,4 + 60(Qof,1 + Qof,2 + Qof,3 + QOf,4)

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54 AVERAGE __ COST C(123)= 287 Figure 14. Core and charge$ for three-person game for case 2.

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91 S.T. Qon,l + QOf,'l = 3 I1GD Qon,2 + Qof,2 = 2 MGD Q on,3 + Qof,3 = 1.5 MGD Qon, 4 + Qo f 4 = 2.5 MGD Qof,! + Qof,2 + Qof,3 + of,4 < 10 MGD 6 MGD Application of the GRGA code to determine the minimum cost solution for the case of a downward sloping demand function yields the optimal project designs given in Table 6. The role of marginal considerations in determining optimal resource allocation is evident. Although plant capacity is not a limiting factor for Broject size first design, the optimal output level of 8.41 MGD does not correspond to the 9 MGD potential demand. Only site 4 receives its maximum demand. That the remaining sites are assigned quantities less than their maximum demand results from the effect of marginal conditions in which the quantity assigned to the site corresponds to an output level where IDarginal costs equal marginal The results of marginal considerations and shifting priorities are also evident in the 6 MGD capacity design. Here, the capacity constraint is tight, and the final project includes only three participants.

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Table 6. Optimal Resource Allocation for Case 3 Q m,t Site MGD 1 10 2 10 3 10 4 10 --Total 1 6 2 6 3 6 4 6 Total -Qm,i MGD 3.0 2.0 1.5 2w5 -. 9.0 3.0 2.0 1.5 2.5 6.0 Qof,i MGD 2.56 1.89 1.46 2.5 8.41 1.99 1.51 0.0 2.5 6.0 92 Ii Ii

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93 Cost allocation Since analysis of either of the above project designs can be used to effectively demonstrate all relevant results and conclusions, we arbitrarily select the three-site design for computational ease only. To determine the characteristic function for this design, we apply GRGA to establish the optimal output level for coalition N-S. Using any remain-ing capacity as the new maximum supply constraint, we then apply the code to determine the optimal "go-it-aloneu output quantity for coalition S. Computations are summarized below. C(1): let (2,4) go + Qaf,4 = 4.32 MGD_ 6 -4.32 = 1.68 MGD left C(l) = 8(1.68) + 60(1.68) + 25(1.32,1.7 =136 C (2) : let (1,4) go first, Qof,l + Qof,4 = 4.95 MGD 6 -4.95 = 1.05 MGD left C (2) = 5(1.05) + 60(1.05) + 40(0.95)1.4 =104 C (4) : let (1,2) go first, Qof,l + Qof,2 = 4.24 MGD 6 -4.24 = 1. 76 MGD left C (4) = 17(1.76) + 60(1.76) + 70(0.74)1.1 = 163

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C(l2) : let ( 4) go first, Qof 4 == "2.5 MGD 6 -2.5 = 3.S MGD left C (12) = 8(1.99) + 5(1.51) + 60(3.S)O.7 25(1.01)1.7 40(0.49)1.4 = + + C(14) : let (2) go first, Qof,2 = 1.64MGD 6 -1.64 = 4.36 MGD left C (14) = 8(1.88) + 17(2.48) + 60(4.36) + 25(1.12)1.7 + 70CO.02)I.l = C(24): let (1) go first, Qof,1 = 2.24 MGD 6 -2.24 = 3.76 MGD 1eft C(24) = 5(1.29) + 17(2.47) + 60{3.76)0.7 200 239. + 40(0.71)1.4 + 70(0.03)1.1 = 212 C(124) = + 5 (1.5-1)0.3 + 17 {2 5} () .6 + 60(6) + 25(1.01)1.7' + 40(0.49)1.4 = 296 94 We then check the appropriate conditions for game convexity C(12) + C(14) C(124) + C(I) 200 + 239 > 296 + 136 OK C(12) + C(24) C(124) + C(2) 200 + 212 > 296 + 104 OK

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95 C(14) -t-C(24)' > C(124) + C(4) 239 + 212 > 296 + 163 NO Failure of this last condition indicates that the core is nonconvex. Sites 1 and 2 would each rather join the grand coalition second after site 4. The core constraints are x (1) < 136 X (.2) < 104 X (4) 163 X(1) + X (2) < 200 X (1) + X (4) < 239 X (2) + X(4) < 212 i XU) + X (2) + X (4) = 296 -Nominal bounds for the vector of charges are given by separable cost considerations, i.e. incremental costs of joining the grand coalition last. Nominal bounds correspond to actual bounds only for convex games. Actual maximum and. minimum charges that satisfy all core constraints may be determined from the solution of a system of six linear programs (See Heaney and Dickinson, 1982). A comparison of nominal and actual core bounds reveals only a relatively small difference in the upper bound for X(4).

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nominal bounds, 84 < X(I) < 136 57 < X(2) < 104 96 < X(4) < 163 X(I) + X(2) + X(4) = 296 actual bounds, 84 i X(1) < 136 57 i X(2) < 104 96 i X(4) < 155 X(l) + X(2) +_ XC4} = 296--96 Even though the core is nonconvex it is still relatively close to convexity. In general the proximity of the core to convexity depends on the speciffc cost relationships. The Shapley value, MCRS, and average cost solutions are presented in Table 7 and in Figure 15. All three solutions satisfy core constraints but only the MCRS solution is in the center of the core. The average cost solution is close to the lower core bound for site 4, relatively far from the core center. As discussed in the final step of analysis, the implications here are not so clear as for the first two cases.

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X (4) $155 AVERAG..:::;;E ____ COST CORE ________________________ C(124) = 296 Figure 15. Core and charges for three-person game for case 3. 97

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Table 7. Charges for Three-Person Game for Case 3 Total CO$t, Qm,t Average Cost Site MGD Sha MCRS Chare Savin s 1 6 102-.00 103.41 118 .. 40 5% 2 6 72.50 74.58 78.93 25% 4 6 121.50 ll8.0J. 98.67 48% ----Total 296 296 296

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Step 6. Examine Implications of Economic and Financial Analysis 99 In this step we examine implications of results for the three cases. Recall that economic optimization for econo-mies of scale in production with nondecreasing demand functions (cases 1 and 2) results in a strict rank order priority among potential participants which mayor may not vary with output. Under these circumstances, proper econo-mic analysis guarantees a convex set of sustainable charges. The Shapley value, MCRS, and average cost solution vectors are contained within this convex set. Although each of these pricing solutions satisfies the game theory stability criteria recognized by both economic and cost allocation ---theory, economists might argue-that only those schemes based on marginal cost pricing will encourage economically effi-cient use of. available resources_. As demonstrated, marginal cost pricing is not applicable for single-step or upward-sloping demand relationships. Furthermore, where production capacity constraints are not tight each player receives his maximum demand quantity of output. Therefore, any sustain-able charge vector encourages this economically efficient maximum participation. Where the project design is supply limited, any sus-tainable charge vector encourages efficient resource use for all but the last ranking player "cut offlt at the capacity output level. At a unit price corresponding to his assigned charge, this player wants to buy additional output to satisfy his maximum demand. In addition, any player

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excluded during the optimization phase of analysis might want to participate arguing that he shoul.d have an opportunity to share in the savings. It might be reasonable to expect that an appropriate governing agency coul.d enforce 100 consumption restrictions to ensure optimal resource allocation. This is in contrast to the public goods "and natural. monopoly pricing problem where there are many consumers as opposed to a relatively small group of potential buyers. Restrictions on how much of a public good or natural mono-poly product consumers may purchase are generally impractical and often inappropriate. Of the unique solution methods considered, only the Shapley value and MCRS solution are guaranteed to be in the center of the core for cases land 2. The notion that the core center is the most equitable solution persists in game theory but is not universally accepted in water resource applications. In addition, uncertainties in technical and cost estimation data may indicate that emphasis on the center of the core is inappropriate. A suitable alternative cost allocation method might be average. cost pricing which has several advantages. First, it is computationally 'easy, avoiding the problem of defining and computing the characteristic function." ,This advantage is especially apparent for projects involving more than just three or four groups. In addition to computational ease average cost pricing has the advantage of being easily understood and accepted by parti-cipants and administrators. As a case in point, Loehman et

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ale (1979) observe that decision makers involved in a proposed regional wastewater treatment system for the Meramec River Basin prefer uniform unit prices based on average costs to nonuniform charges even at the expense of economic efficiency. 101 Finally, we discuss implications for the third case where demand relationships are downward sloping. Rather than a strict rank order based on continuously increasing net benefits, priority among players depends on marginal considerations as well as plant capacity and maximum demand constraints. As a result, marginal pricing might be a viable cost allocation strategy. Any marginal pricing scheme requires some means of recovering total project costs and is further complicated by the fact that not all allocated quantities are necessarily determined from marginal conditions. Some allocated quantities may be the result of production capacity or maximum demand considerations. The problem is how to apply marginal cost pricing strategies where such strategies are applicable to only a portion of the total system price vector. With regard to the Shapley value, MCRS, and average cost solutions, we examine the nature of the resulting cost allocation game. Recall that the game is subadditive but not necessarily convex. Only the MCRS solution is guaranteed to be in the core. However, except for extreme conditions the Shapley value will probably satisfy core constraints and may even be relatively close to the core center as in the wastewater reuse example.

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102 The location of the average cost solution is less predic-table and may lie outside of the core even for relatively attractive games that are barely nonconvex. Therefore, for a nonconvex game, other approaches such as the Shapley value and particularly the MCRS method may be more appropriate than average cost pricing. '.:-: ...

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CONCLUSIONS Results of this analysis indicate several important implications regarding the relationship between economic optimization and cost allocation for joint water resource projects. These implications may be generalized for various supply and demand relationships. This paper examines the special case of economies of scale in production for three types of demand relationships. Two groups of general results are derived. E'1.rst, where the nature of tlie and demand curves are such that net benefits are continu-ously increasing, the resulting cost allocation game is convex. Furthermore, optimal resource allocation is achieved through a rank order assignment of output to each participant. The resource quantity allocated to each participant is limited only by demand or production capacity cons"traints. Consequently, economic optimization and cost allocation may indeed be independent of each other. A" simple average cost pricing scheme is not only sustainable but is easily determined and implemented. However, if efficient resource utilization is not enforceable for supply limited projects, a suboptimal solution may result. Where net benefits are not strictly increasing, the relationship between economic efficiency and cost allocation 103

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S 104 is not so clear. Joint consideration of efficiericy and equity analysis might be warranted in some instances. In particular, where marginal considerations are determining factors for efficient resource allocation, marginal cost pricing strategies may be appropriate. Thus, results from economic analysis are applied to financial analysis so that the, two are directly related to each other. Where efficient resource allocation can be enforced, other charging schemes including the Shapley value, MCRS, and average pricing solutions may be appropriate. However, since the cost allocation may be nonconvex, only the MCRS solution is guaranteed to be sustainable. Again, where consumption restrictions are not enforced, implementation of a nonmargi--------------------nal cost price vector may result in a suboptimal final system. Herein lies the tradeoff between efficiency and equity. Evaluation of this tradeoff is an important topic for future research.

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APPENDIX Convexity Conditions for Four-Person Game for Cases 1 and 2 C(12) + C(13) l C(123) + Cel) 204 + 200 > 257 +. 142 OK C(14) + C(12) l C(124) + C(l) 04 > 294 + 147 OK C(13) + C(14) > C(134) + C(l) --200' + 240 > 292 + 147 OK C(12) + C(23) l C(123) + C(2) 204 + 166 > 257 + 104 OK C(24} + C(l2) l C(l24) + C(2) 208 + 204 > 294 + 104 OK C(23) + C(24) l C(234) + C(2) 166 + 208 > 262 + 104 OK 105

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106 C(13) + C(23) C(123) + C(3) 200 + 166 > 257 + 96 OK C(34) + C(13) C(134) + C(3) 204 + 200 > 292 + 96 OK C(23) + C(34) C(234) + C(3) 166 + 204 > 262 + 96 OK + C(24) C(124) + C(4) 240 '+ 208 > 294 + 143 OK C(34) + C(14) C(134) + C(4) 204 + 240 > 292 + 143 OK C(24) + C(34) 2 C(234) + C(4) 208 + 204 > 262 + 143 OK C(123) + C(14) C(1234) + C(l) 257 + 240 > 343 + 142 OK C(124) + C(13) 2 C(1234) + C(l} 294 +200.2 343 + 142 OK C(134) + C(12) C(1234) + C(l) 292 + 204 > 343 + 142 OK

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107 C(123) + C(24) C(1234) + C(2} 257 + 208 > 343 + 104 OK C(124) + C(23) C(1234) + C(2) 294 + 166 > 343 + 104 OK Ct234) + C(12) C(1234) + C(2) 262 + 204 > 343 + 104 OK C(123) + C(34} C(1234) + C(3} 343 + 96 OK C(134L + C(23} > __ 292 + 166 > 343 + 96 OR C(234) + C(13) > C(1234} + C(3) 262 + 200 > 343 + 96 OK C(124) + C(34} C(1234} + C(4) 294 + 204 > 343 + 143 OK C(134) + C(24) C(123A} + C(4) 292 + 208 > 343 + 143 OK C(234) + C(14) C(1234) + C(4, 262 + 240 > 343 + 143 OK

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REFERENCES Abadie, j. (1975). Methods du Gradient Reduit Generalise: Le Code GRGA. Paris, France. Note HI 1756/00, Electricite de France, Baumol,. W.J., Bailey, E.E., and Willig, and Invisible-Hand Theorems on the Prices in a Multiproduct Monopoly. Review, 67, 350-356. R.D. (1977). Weak Sustainability of American Economic Beightler, C.S., Phillips, D.T., and Wilde, D.J. (1979). Foundations of Optimization. Prentice-Hall, Inc., Englewood Cliffs, New Jersey. Bradley, G.H., Brown, G.G., and Graves, G.W. (1977) __ ___ Design and Implementation of Large Scale Primal Transshipment Algorithms. Management Science, 24, 1-34. Bradley, S.P., flax, A.C., and Magnanti, T.L. (1977). Applied Mathematical Addison-Wesley Publishing Company, Reading, Massachusetts. Clark, R.M., and Dorsey, P. (1982). A Model of Costs for Treating Drinking Water. Journal of American Water' Works Association, .1 618-626. Deb, A.K. (ed.) (1978). Dual water Supply Seminar and Workshop. National Science Foundation, West Chester, Pennsylvania. Donovan, J.F., Bates, J.E., and Powell, C.H. {1980). -Guidelines for Water Reuse. Under U.s. EPA Contract Number 68-03-2686', Camp, Dresser & McKee, Inc. Faulhaber, G.R.(1972).. On Subsidization: Some Observations and Tentative Conclusions. Proceedings of a Conference on Communication Policy 'Research, Washington, D.C. Faulhaber, G. R. (1975). Subsidization: Pricing in Public Enterprises. American Economic Review, 65, 966-977. 108

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109 Gill, P.E., Murray, Saunders, M.A., and (1983). User's Guide for SOL/NPSOL: A Fortran Package for Nonlinear Programming. Technical Report SOL 83-12, Stanford University, Stanford, California. Gottfired, B.S., and Weisman, J. (1973). Introduction to Optimization Theory. Prentice-Hall, Inc., Englewood Cliffs, New Jersey. Guariso, G., Maidrnent, D., Rinaldi, S., and Soncini-Sessa, R. (1981). Supply and Demand Coordination in Water Resources Management. Water Resources Research, .!1, 776-782 Hadley, G. (1964). Nonlinear and Dynamic Programming. Addison-Wesley Publishing Company, Inc., Reading, Massachusetts. Hanke, S.H., and Davis, R.E. (1973). Potential for Marginal Cost pricing in Wa,ter Resource Management. Water Resources Research, 2, 808-825. Heaney, J.P. (1979). Economic/Financial Ana1ysis of Urban Water Quality Management Problems. EPA Grant Number R 802411, Cincinnati, Ohio. Heaney, J.P. ,(1979). Efficiency/Equity Analysis of Environmental Problems--A Game Theoretic Perspective. Applied Game Theory, S.J. Brams, A. Schotter, and G .. Schwodiaver, (eds.), Physica-Verlag, Wurzburg, Germany. Heaney, J.P., and Dickinson, R.E. (1982). Nethods for Apportioning Cost of a Water Resource Project. Water RE'sources Research, .!.!!., 476-482. Heaney, J.P. (1983). Coalition Formation and the Size of Regional Pollution Control Systems. Buying a Better Environment: Cost Effective Re ulation Throu h Permi.t Treading, E.F. Joeres, and M.H. Davi (eds.), UnJ.versity of Wisconsin Press, Madison, Wisconsin IBM Corporation. (l971). Mathematical Programming System/ 360--User's Manual. James, L.D., and Lee, R.R. (1971). Economics of Water Resources Planning. McGraw-Hill Book Company, New York. Jarvis, J.J., Rardin, R.L., Unger, V.E., Moore, R.W., and Schimpeler, C.C. (1978). Optimal Design of Regional Wastewater Systems: A Fixed-Charge Network Flow Model. 9perations Research, 26, 538-550.

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110 Jelen, F.C., and Black, J.H. (1983). Cost and Optimization Engineering. McGraw-Hill Book Company, New York. Littlechild, S.C., and Owen G. (1973). A Simple Expression for the Shapley Value in a Special Case. Management Science, 370-372. Littlechild, S.C., and Thompson, G.F. (1977). Aircraft Landing Fees: A Game Theory Approach. Bell. Journal of Economics, 18, 186-204. Loehman, E., and Whinston, A. (1976). A Generalized Cost Allocation Scheme. Theory and Measurement of Economic Externalities, A. Stevens and Y. Lin, (eds.), Academic, New York. Loehman, E., Orlando, J Tschirhart, J., and Whinston, A. (1979). Cost allocation for a Regional Wastewater Treatment System. Water Resources Research, 15, 193-202. Loughlin, J.C. (1977). The Efficiency and Equity of Cost Allocation Methods for Multipurpose Water Projects. Water Resources Research, ll, 8-14. --IJucas-,-V] .-F. (-1-981:). The-Mul ttpurpose Ccmperative-Garnes. Game Theory and Its Applications, W.F. Lucas, (ed.), American Mathematical Society, Providence, Rhode Island. Mandl, C.E. (1981). A Survey of Mathematical Optimization Models and Algorithms for Designing and Extending Irrigation and Wastewater Networks. Water Resources Research, lI, 769-7/5. Maurras, J.F. (1972). Optimization of the Flow Through Networks with Gains. Mathematical Programming, l, 135-144. McMillan, C., Jr. (1970). Mathematical Programming. John Wiley & Sons, Inc., New York. Panzar, J.C., and Willig, R.D. (1977). Sustainability of Natural Monopoly. Economics, !, 1-22. Free Entry and the Bell Journal of Rardin, R.L., and Unger, V.E. (1976). Solving Fixed Charge Network Problems with Group Theory Based Penalties. Naval Research Logistics Quarterly, 23, 67-84. Reklaitis, G.V., Ravindran, A., and Ragsdell, K.M. (19B3). Engineering Optimization: Methods and Applications. John Wiley & Sons, New York.

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111 Rossman, L.A. (1978). Comrnent on iEfficiency and Equity of Cost Allocation Methods for Multipurpose Water Projects' by James C. Loughlin. Water Resources Research" li, 1195-1196. Sample, D.J. (1983). An Engineering, Economic and Financial Feasibility Analysis of Wastewater Reuse in Eastern Palm Beach County, Florida. Master's Thesis, University of Florida, Gainesville, Florida. Sandberg, I.W. (1975). Two Theorems on a Justification of the Multiservice Regulated Company. Bell Journal of Economics, 346-356. Sharkey, W.W. (1982). The Theory of Natural Monopolies. Cambridge University Press, Cambridge. Sharkey, W. w. (1982). Suggestions for a Game Theoretic Approach for Public Utility Pricing and Cost Allocation. Bell Journal of Economics, 13, 57-58. Sorenson, S.W. (1972). A Mathematical Theory of Coalitions and Competition in Resource Development. Ph.D. Dissertation, University of Texas, Austin, Texas. Tschirhart, J .-T -. -(1975-) A Game Theoretic Appro-ach to Pricing in Decreasing Cost Industries. Ph.D. Dissertation, Purdue University, West Lafayette, Indiana. Wilde, D.J. (1978). Globaliy Optimal Design. John Wiley and Sons, New York. Zajac, E.E. (1972). Some Preliminary Thoughts on Subsidizati0n. Proceedings of a Conference on Communication Policy Research, Washington, D.C. Zajac, E.E. (1978). Fairness or Efficiency: An Introduction to Public Utility Pricing. Ballinger Publishing Company,