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A multivariate utility function approach to stochastic capacity planning
3
Engineering Costs and Production Economics, 12 (1987) 3-13 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
A MULTIVARIATE UTILITY FUNCTION APPROACH TO STOCHASTIC CAPACITY PLANNING* Dan B. Rinks 1, Jeffrey L. Ringuest 2 and Michael H. Peters 1 lOuantitative Business Analysis Department, Louisiana State University, Baton Rouge, LA 70803 (U.S.A) 2Administrative Sciences Department, Boston College, Chestnut Hi//, MA 02167 (U.S.A.)
ABSTRACT
The problem being considered is the expansion of a single capacity when the installation cost is large and there are neither obsolescence nor spatial effects on the expansion. Demand is assumed to follow an evolving process," in particular, the demand increments are normally distributed with linearly increasing mean and variance in time (i.e., a Wiener process). Selecting the optimum size and timing of capacity additions in the face of uncertain demand forecasts involves both the minimization of cost as well as the minimization of risk. For the situation where demand acts as a Wie-
ner process, Kang and Park have derived the expectation of the "equivalent cost rate." Using the standard deviation of the equivalent cost rate as a measure of the risk of expansion, Kang and Park then minimized an objective function that was a linear combination of cost and risk. The purpose of this paper is to extend this line of research. Using standard decision analysis procedures, a multiattribute utility function is constructed that reflects the decision maker's tradeoff for cost and risk. A significant advantage of the utility function approach is that it allows for nonlinear trade-offs of cost and risk.
INTRODUCTION
tional to size); (2) there is a continuing (possibly changing) need for the facilities; and (3) the facilities added are durable. Examples of capacity decisions that reasonably meet these criteria include communication networks, electric power generation and transmission, manufacturing facilities, and public facilities such as schools, water systems, roads, etc. Actual capacity expansion problems are invariably complex with many subtle nuances that resist rigorous mathematical modelling. Nevertheless, economists and others have studied the capacity expansion problem extensively by making simplifying assumptions.
The size and timing of capacity is an important decision problem for both private industry and government. Because there are typically large amounts of resources involved and strategic consequences, the concomitant benefits, costs and risks of such decisions require careful analysis. The classic capacity expansion problem, as noted by Friedenfelds [ 1 ], is characterized by: (1) the cost of added capacity exhibits economies-of-scale (i.e., cost is less than propor*Presented at the Fourth International Working Seminar on Production Economics, Igls, Austria, Feb. 17-21, 1986.
0167-188X/87/$03.50
© 1987 Elsevier Science Publishers B.V.
4 Under the assumption that growth in d e m a n d is deterministic, Chenery [2] developed an optimizing model for predicting investment behavior. Subsequently, Manne [ 3] generalized the results for probabilistic growth. Numerous others (e.g., [4], [5], and [ 6 ] ) h a v e contributed to the understanding of capacity expansion by formulating m o d e l s that capture certain aspects of real capacity expansion decision making. In the face of uncertain d e m a n d forecasts, selecting the o p t i m u m size and timing of capacity additions involves both the minimization of cost as well as the minimization of risk. For the situation where d e m a n d acts as a Wiener process, Kang and Park [7] have derived the expectation of the "equivalent cost rate." Using the standard deviation of the equivalent cost rate as a measure of the risk of expansion, Kang and Park then minimized an objective function that was a linear combination of cost and risk. The purpose of this paper is to extend this line of research. Using standard decision analysis procedures a multiattribute utility function is constructed that reflects the decision maker's trade-offfor cost and risk. The significant advantage of the utility function approach is that it allows for non-linear trade-offs of cost and risk. The remainder of the paper is organized in the following manner. In the next section, the model that Kang and Park developed is presented. After this, the general five step procedure of Keeney and Raiffa for assessing multidimensional utility functions is reviewed, and then applied to stochastic capacity expansion. A numerical example follows. In conclusion, the pros and cons of a decision analysis approach to stochastic capacity expansion are discussed. KANG A N D PARK MODEL
The problem being considered is the expansion of a single capacity when the installation cost is large and there are neither obsolescence nor spatial effects on the expansion. D e m a n d
is assumed to follow an evolving process; in particular, the d e m a n d increments are normally distributed with linearly increasing mean and variance in time. This assumption on d e m a n d describes a Wiener process [ 3 ]. The analysis of Kang and Park includes both backlogging and no backlogging. Only capacity expansion without backlogging is considered in this paper; consequently, only that portion of the model development dealing with the no backlogging case is presented herein. Let u(t;x) represent the probability density (p.d.f.) that d e m a n d first exceeds the initial level by x at time t when d e m a n d increments are normally distributed with linearly increasing mean ~t and variance a2t. u(t;x)=
x_.
x/(27ct72t 3)
e~_(x_¢02/2~20}
(1)
The installation costs that result from a single capacity increment of size x are assumed to be given by a cost relationship of the following form. G(x) =kx ~ (2) where k is a constant and a is the economies of scale factor, k > 0 and 0 < a < 1. In addition, the cost of installing the capacity at regenerative times t~, i = 1,2 ..... discounted to the time origin t~ is G(t~;x) where
(3)
G(t,;x)=G(x) e ....
and r is the discount rate. The equivalent cost rate (ECR) for G(t~;x), denoted by V(t~;x), is defined by V(t,;x) =
~G(x)
ifr~O
G(x)
if r=0
{1 r
(4 )
where filling time t is an independent and identically distributed random variable with p.d.f, u(t). Since V(t~;x) is independent of t,-, it is more simply represented as V(x). The ECR can be interpreted as a discounted time average of the total investments on the infinite horizon, where the expectation of ECR is co
~rk'x"~=oe~"~_ E{V(x)} = LE{I/t} kx~
ifr~O ifr=O
(5)
5
ASSESSING A STOCHASTIC CAPACITY EXPANSION UTILITY FUNCTION
and ,l, = ~ [1 -,/(1 + 2nra2 /~2) ]
(6)
oo
E{f( t ) } = f f ( t )u( t;x)dt
Keeney and Raiffa Assessment Procedure
•
o
In considering the riskiness of a project, Kang and Park argue that one of the major sources of risk is the uncertainty of the installation cost itself; this is more likely to be a reasonable assertion if the capacity installation costs are very large. Next, it is assumed that the standard deviation of the ECR, S{ V(x)}, can be used to represent the risk of expansion where s{ V(x)} =
{
rkx~{~(n+l)e~"x-[~e~"~]2} .=o
½
r#O
.=o
k.xa[ E(1/t 2) _ {E(1/t ) } 2] ~
(7) r=0
An objective function that is a linear combination of expected cost and risk is then minimized. m i n i m i z e [ a E { V( x ) } + (1 - a) S{ V( x ) } ]
(8)
subject to x > 0 for a given value of a, 0 < a < 1. Obviously a, the risk aversion factor, controls the weight given to risk minimization versus cost minimization. Whereas this procedure does provide a method for explicitly incorporating risk in the capacity expansion problem, it does so in a simple fashion. Specifically, only linear tradeoffs between cost and risk can be considered with eqn. (8). In addition, Kang and Park made no mention of how a should be determined for a specific decision problem. It is intended, one supposes, that decision analysis procedures would be used to assess a. This being the case, we propose enlarging the role of decision analysis to include the construction of a multiattribute utility function that reflects the decision maker's preference for capacity expansion alternatives that involve cost and risk. This utility function, rather than optimizing eqn. (8), is used for evaluating capacity expansion.
Keeney [9 ] and Keeney and Raiffa [ 10] present a general five step procedure for assessing multidimensional utility functions. The first step in the procedure is to familiarize the decision maker with the framework to be used in the succeeding steps. This includes identifying the person whose utilities are to be assessed, the issues to which the needed utilities are relevant, the attributes to be evaluated and the relevant dimensions of the attributes to be ovaluated. In the second step of the assessment procedure the relevant utility independence assumptions are ascertained. This step will determine the form of the multidimensional function, i.e. additive, multiplicative or multilinear. These assumptions are verified by presenting the decision maker with an appropriate set of multiattributed lotteries. The assessments obtained will either support or reject the independence assumptions. Step three in the Keeney procedure requires that a single attribute conditional utility function is assessed for each attribute. Typically this assessment begins by identifying the relevant qualitative characteristics of the unidimensional utility function. This includes determining if the function should be monotonic and determining the decision maker's risk posture with respect to the particular attribute. It is then necessary to quantify points on the curve by presenting the decision maker with a set of uniattributed lotteries. These assessed points are used to fit an appropriate functional form (i.e., one that exhibits the observed qualitative characteristics). Finally, it is necessary to check the consistency of the assessed conditional functions. This is usually accomplished by presenting the decision maker with an additional set of uniattributed lotteries and comparing the
elicited responses with those obtained from the assessed function. Obviously, any observed inconsistencies will require some retracing of steps in the overall procedure. In the previous step, each conditional utility function is assessed on an arbitrary scale. Thus, in step four, it is necessary to rescale each single attribute function to a c o m m o n origin and a c o m m o n unit of measure. This is done by determining two levels of each attribute that are indifferent, implying equal utility, and determining the appropriate linear transformation. Once the conditional utility functions are consistently scaled, it is a simple matter to determine the parameters of the multiattribute utility function. The fifth and last step in the procedure is a consistency check. Here, the decision maker is presented with pairs of multiattributed outcomes and asked to identify the more preferred outcome. The consistency of the elicited preferences is then compared with those implied by the multiattribute utility function.
be used to estimate parameters for specific functional forms. Klein et al. [ 8 ] suggest that a s u m m e d exponential of the form: U( xi) : ai + bie a''x; + c;e "~2'x',
(9)
can be used in most cases. This expression is robust in that it will model all of the most comm o n risk attitudes (i.e. constant risk aversion, increasing risk aversion, decreasing risk aversion, constant risk proneness, decreasing risk proneness, increasing risk proneness, risk proneness for small attribute values and risk aversion for large attribute values, risk aversion for small attribute values and risk proneness for large attribute values) depending on the values of the five parameters. Step four involves determining the scaling constants for the multiattribute utility function. If attributes Xl and x2 are mutually utility independent (multiplicative form), then their two-attribute utility function is given by U(x~ ,x2) =k, U~(x~) + k2 U2(x2) +klk2k3U,(xl)U2(x2)
Application to Stochastic Capacity Expansion In the first step of the K - R procedure, cost and risk would be identified as the relevant issues, and a plausible range of values for the two attributes would be ascertained. At least three attributes are needed to distinguish between multiplicative and multilinear utility function forms. Since the problem as posed contains only two dimensions, cost and risk, the multilinear form is eliminated as an alternative. Furthermore, the multiplicative form reduces to the additive form as a special case. Thus, in step two, we will assume the multiplicative form is appropriate for capareity expansion. In step three, quantifying points on a univariate utility function can be accomplished several different ways, such as a five-point assessment procedure (e.g., Daellenbach et al. [ 11 ], p. 307). Statistical curve fitting can then
(10)
The scaling constants kl, k2 and k3 can be inferred from two indifference statements (see Daellenbach et al. [ 11 ], p. 640). This procedure is illustrated in the next section.
NUMERICAL EXAMPLE In this section we will illustrate by way of a numerical example the multiattribute utility function approach to stochastic expansion. The development of the multiattribute utility function for the example will follow the Keeney and Raiffa procedure. The derived utility function is then used to select an o p t i m u m capacity and various sensitivity analyses are performed to evaluate the sensitivity of the solution to changes in model parameters. Basic model parameters for this example are taken from a generalized problem presented in Kang and Park: ~ = 1; a2=0.5; k = 2 0 ; a = 0 . 6 ; r=0.15.
7
Step 1 Let xt = the expected value of future discounted costs; and x2 = the variance of future discounted costs.
Figure 1 illustrates how different values of 2~ effect the univariate utility function for the expected value of future discounted cost. Similarly, with 2 2 = - 0 . 5 0 , the univariate utility function for the variance for future discounted costs is U2(x2) = 1 . 0 0 6 7 8 3 7 - 0 . 0 0 6 7 8 3 7 e °'5°x2
Assume that the relevant ranges ( m i n i m u m , m a x i m u m ) for x~ and x2 are (10,30) and (0,10), respectively.
Step 2 Assume the multiplicative form is appropriate.
Step 3 Assume that a special case of functional form (9), namely: U(Xi) = ai - bie-~'x'; b, > 0, 2, > 0
is appropriate. This assumption implies that the decision maker is constantly risk averse (or nearly so) over the relevant range of the attributes. Since, for this example, the ranges on both x~ and x2 are relatively narrow this will likely be a reasonable assumption. We may arbitrarily set U~(10)=I.0, U , ( 3 0 ) = 0 . 0 , U 2 ( 0 ) = I . 0 , and U2(10)=0.0. Substituting these values into (9) results in a, - b t e - i o a , = 1.0 a n d at - bj e-30~, = 0 . 0
Step 4 Again, without loss of generality, we arbitrarily, set U ( 1 0 , 0 ) = I . 0 and U ( 3 0 , 1 0 ) = 0 . Next, the DM would be asked to specify a value xi so that he or she is indifferent between the following two outcomes: A l = [ X l at specified level, x 2 at least preferred value] A2= [xl at least preferred value, x2 at most preferred value] [xt = ? , x 2 = 10] ~ [x, = 30,x2 = 0 ]
Suppose the answer is Xl = 23.0. Then, we know that Uj ( x t = 2 3 . 0 ) = 1 . 1 5 6 5 2 5 - 0 . 0 5 7 5 8 0 e °'t°(z3"°~ = 0 . 5 8 2 2
and U ( 2 3 . 0 , 1 0 ) ~ U ( 3 0 , 0 ) . have
From (10), we
=k~ U~(30)+k2Uz(O)+k~k2k3Ut(30)Uz(O)
= 1.0
Now fix 2t at a particular value, say 2~ = - 0 . 1 0 . This is equivalent to determining the decision maker's propensity for riskiness. When 2i < 0, the DM is risk-averse; ,~i= 0, the DM is neutral; and 2i > 0, the DM is risk-prone. Solving for a~ and b~, with 2 1 = - 0 . 1 0 , one obtains the values 0.057580 and 1.156525, respectively, and the univariate utility function for the expected value of future discounted cost is Ut(xt ) = 1 . 1 5 6 5 2 5 - 0 . 0 5 7 8 5 0 e °'x°x'
Figures 2(a) and 2(b) depict the univariate utility function for the variance of future discounted costs for values of 22 of - 0.30, - 0.50, and - 0.70.
k~ Ul(23.0)+kzUz(lO)+klk2k3Ut(23.0)Uz(lO)
Simple algebra yields bj [ e -~°~' - e - i o ~ , ]
(12)
(11 )
Substituting in the known utility values, this last expression simplifies to k2 = 0.5822kl. For the second indifference statement, we ask for an outcome [xl,x2] which is indifferent to a 50-50 reference lottery involving the most preferred and the least preferred outcomes, i.e. [ 10,0] and [30,10]. This choice can be made easier for the DM by fixing x~ to its 0.50 individual utility value, namely x, =24.338 (see Fig. 2(a)). Suppose that the answer to this reference lottery is x2=6.0. Thus, U(24.338,6.0)=0.50
1,0 0.9 0.8 0.7 0 o
0.6
ft.
0 >,. I-
0.5
"i 0.4 I':::3 0.3 0.2 0.1
~0
I 12
I 14
I 16
I 18
EXPECTED
I
20
VALUE
I 22
14
2
FUTURE DISCOUNTED
I 26
I 28
COST
Fig. 1. Univariate utility functions for the expected value of future discounted costs. 1.0 0.9 0,8 0.7
7- o.8 ~ 0.5 >-
~. J o.4 ~0.3
0.2 0.1 I
I
I
I
I
0 STANDARD
DEVIATION
OF F U T U R E
DISCOUNTED
COST
Fig. 2(a). Univariate utility functions for the standard deviation of future discounted costs. [24.388, x2=?]
A,~ ,~.,JA 2
[lo, o]
.50
Fig. 2(b). Indifference statement involving a 50-50 reference lottery.
[3o.10]
30
and from (12) U2 (6.0) = 1.0067837 0.0067837e °'5°~6"°) = 0.8705. Consequently, from eqn. (10) we have the relationship U (24.338,6.0) = K~U1 (24.338) + K2U2(6.0) + K~K2K3U~ (24.338) U2(6.0) = 0.50. Substituting in the known utility values, this last expression simplifies to k3-
0.50-- 1.0120kl 0.25602kl 2
Now, we find kl from evaluating (10) with U(10,0) = 1.0 and the known utility values. k~(1) + 0.5882kt (1) + k~(0.5882k~)[(0.50 -1.0120kt 2](1)(1)=1.0 kj =0.20186
Since expressions have been found for k2 and in terms of k~, their values may now be determined. k3
k~=0.11873 k3 =28.34664
Therefore, the joint utility function for cost and risk under mutual utility independence is U(xl ,x2) = 0.20186 [ 1.15625 -- 0.057580e °'l°x'] + 0.11873 [ 1.0067837e °5°x' ] + (0.20186)(0.11873)(28.34664) U~(x~) U2(x2)
In order to obtain the utility for a given capacity expansion size x, the cost (x~) and risk (x2) for the given size are first evaluated by eqns. (5) and (7). Next, eqn. (13) is used to determine the utility. Figure 3 depicts the utility for differing capacity expansion sizes. The " o p t i m u m " size is between 8.0 and 8.5 (the utility is practically the same). Several different kinds of sensitivity analyses are appropriate. For instance, one interest is to determine the sensitivity of the o p t i m u m capacity expansion size to the decision maker's propensity for risk. Suppose it is decided that the variance of future discounted cost is very important, but the decision maker is unsure of his/her ability to articulate responses in the univariate assessment procedure. In this
case, the effect of different values for 22 could be investigated. Figure 4 illustrates the utility for capacity expansion size for different values of 22. The relative flatness of the utility function in the neighborhood of the o p t i m u m capacity expansion size illustrates that a wide range of sizes have nearly the same utility (for the given problem parameters: ~= 1, a2= 0.5, k = 2 0 , a = 0 . 6 , and r = 0 . 1 5 ) . To more fully investigate the sensitivity of the o p t i m u m capacity size, several additional analyses were performed for different parameter values. Specifically, the discount rate, r, was investigated for additional values of 0.10 and 0.20; the economies of scale factor, a, for additional values of 0.65 and 0.70; and the constant of proportionality, k, for additional values of 15 and 25. In each analysis, a single parameter value was changed from the original parameter values (the base case); the joint utility function for (13) with 2 1 = - 0 . 1 0 and 2 2 = - 0 . 5 0 was employed. The utility curves that resulted from this procedure are depicted in Figs. 5, 6, and 7. In considering the results of sensitivity analyses such as these, the decision maker is left to draw his/her own conclusions about what capacity expansion should actually be implemented. Our philosophy is that models like these are best employed in a decision support context. In this vein, we believe model construction and implementation should facilitate sensitivity analyses in order to assist the DM in his/her search for a capacity expansion size that satisfies the goals and constraints of the situation. A N ALTERNATIVE A P P R O A C H
The decision maker may be unaccustomed to thinking in terms of the attributes x~ = E [ V(x)] and x2=S [ V(x)]. Thus, it may be difficult to assess the conditional utility functions required to use eqn. (10). Alternatively, it may be possible to obtain a set of isopreference curves. Each isopreference curve is used to represent a set of consequences that are equally desirable to the decision maker.
10 hO
0.9
z o z Q. X 14J
0.8
n~
>- 0.7 k._1 0.6
0.5
I
I
I
I
I
I
I
I
I
2
4
6
8
I0
12
14
16
18
20
EXPANSION SIZE
CAPACITY
Fig. 3. Multiattribute utility function for stochastic capacity expansion. 1.0
0.9
0.8
0.7
0.6
0.5 2.5
4.15
615
815 CAPACITY
,~5
,;5
'
14.5 EXPANSION SIZE
I
I
16.5
18.5
Fig. 4. Utility for capacity e~pansion size for different values of~.2.
In order to assess an isopreference curve the decision maker is asked a series of indifference questions. A typical indifference question would be as follows: Compare a particular capacity expansion having attributes X I : 1 0 ( c o s t ) and X 2 : 6 (risk) with a second capacity expansion having a cost of x~ = 12. What value of risk (x2) in the second expansion option would result in you having no preference between the expansion alternatives?
[Xl= 10, X2=6] Reference capacity expansion size
~ [X1=12,
X2:?
]
Second capacity expansion option
By repeating the indifference question with the same reference capacity expansion size and varying the second expansion option, a series of these points can be assessed and then connected with a smooth curve. Different isopreference curves are assessed by varying the reference capacity size. Figure 8 presents three isopreference curves
11 1.0 r=.lO 0.9 z 01
0.8 n,,o
u. 0.7 >i--
..I I-
0.6
0.5
. . .4 .
2
6
8
I
'o
I
'2
14 '
16 '
I
20
CAPACITY EXPANSION
Fig. 5. Sensitivity of utility for expansion to changes in the discount rate, r. 0.91
O.S Z
ow Z
~O.7 tO Ix
o i, 0.6 J
~0.5
20 CAPACITY EXPANSION Fig. 6. Sensitivity o f u t i l i t y f o r expansion to changes in the economies-of-scale factor, a.
for the decision maker in the previous example. That is, these contours are consistent with eqn. (13 ). If the decision maker faces the situation where he or she must choose from among a finite set of expansion alternatives, the coordinates of these alternatives can be plotted over the assessed contours. In Fig. 8 the coordinates of expansions of size 5, 10, 15 and 20 are displayed. The xl and x2 coordinates for these expansions are computed from (5) and 6) respectively. The choice from among these expansion alternatives is obvious. Clearly, the expansion of size 10 lies on the most preferred
lsopreferennce curve. For the case where any size expansibn is possible, additional points could be computed and plotted. A smooth curve would then be drawn through these points. The optimal capacity expansion size can then be estimated from this curve. While this approach may appear as cumbersome as the previous analysis, it has certain advantages. No independence assumptions are required in order to estimate isopreference curves. In addition, this approach is not bound to any specific conditional utility function forms. Thus, the isopreference curve approach
12 1.0 0.9 0.8 0,7 0.6 0.5 0.4 0.3 0.2
I 4
0
I 6
I I I I 8 I0 12 t4 CAPACITY EXPANSION
I 16
I IS
20
Fig. 7. Sensitivity of utility for expansion to changes in the constant of proportionality, k. 9
8s U
~7
>~
~
0 CapaOity Expansion Size Of 15 " ~PacliliEPx °n'i°n$izeOfZO
i 0
Direction Of
I0
i ....
i
i
\
I /
i
i
F i
i
ii
/ i
12 14 16 18 EXPECTED VALUE FUTURE DISCOUNTED COSTS
e Ii
20
Fig. 8. Isopreference contours for cost and risk.
is more general than directly assessing the decision maker's utility function. Furthermore, the decision maker may find indifference questions easier to understand than lotteries. CONCLUSIONS There are several advantages to a multiattribute utility function approach to the capacity expansion problem. To the extent that the multiattribute utility function can be accurately assessed, multidimensional alternatives
can be evaluated consistent with the decision maker's preference structure. Modelling of nonlinear trade-offs between attributes is possible. Although we have illustrated the procedure for just two attributes, cost and risk, other dimensions might also be appropriate. More than two attributes do not present any problems conceptually. Sensitivity analyses are easily accomplished with this approach and can be presented in a manner comprehensible to the decision maker. However, the difficulties associated with assessing, constructing, and using multiattri-
13
bute utility functions should not be overlooked. Many decision makers find it difficult to articulate preferences and trade-offs. Furthermore, the technique itself is likely to be unfamiliar to the DM and consequently viewed with suspicion. Similar to other techniques that require the aid of analysts, the overall success of multiattribute assessment is closely related to the skill of the decision analyst assisting with the assessments. Despite all of these implementation obstacles, we believe that the multiattribute utility function approach has much to offer in the analysis of problems as important as capacity expansion.
2 3 4 5 6 7 8
9 10
REFERENCES Friedenfelds, J. (1981). Capacity Expansion: Analysis of Simple Models with Applications. Elsevier North Holland, New York.
11
Chenery, H.B. (1952). Overcapacity and the acceleration principle. Econometrica (January). Manne, A.S. (1961). Capacity expansion and probabilistic growth. Econometrica, 29(4): 632-649. Edenkotter, D. (1977). Capacity expansion with imports and inventories. Manage. Sci., 23(7): 694-702. Giglio, R.J. (1970). Stochastic capacity models. Manage. Sci., 17(3): 174-184. Tapiero, C.S. (1979). Capacity expansion of a deteriorating facility under uncertainty. R.A.I.R.O., 3(1): 55-66. Kang, H.W. and Park, S.J. (1983). An efficient curve in stochastic capacity expansion. OMEGA: Int. J. Manage. Sci., 11(2): 147-153. Klein, G., Moskowitz, H., Mahesh, S. and Ravindran, A. (1985). Assessment ofmultiattribute measurable value and utility functions via mathematical programming. Decision Sci., 16(3): 309-324. Keeney, R.L. (1972). An illustrative procedure for assessing multiattributed utility functions, Sloan Manage. Rev., 14(1): 37-50. Keeney, R.L. and Raiffa, H. (1975). Decisions with Multiple Objectives: Preferences and Value Tradeoffs. New York: John Wiley & Sons, New York. Daellenbach, H.G., George, J.A. and McNickle, D.C. (1983). Introduction to Operations Research Techniques, 2rid edn. Allynand Bacon, Newton, MA.
Engineering Costs and Production Economics, 12 (1987) 3-13 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
A MULTIVARIATE UTILITY FUNCTION APPROACH TO STOCHASTIC CAPACITY PLANNING* Dan B. Rinks 1, Jeffrey L. Ringuest 2 and Michael H. Peters 1 lOuantitative Business Analysis Department, Louisiana State University, Baton Rouge, LA 70803 (U.S.A) 2Administrative Sciences Department, Boston College, Chestnut Hi//, MA 02167 (U.S.A.)
ABSTRACT
The problem being considered is the expansion of a single capacity when the installation cost is large and there are neither obsolescence nor spatial effects on the expansion. Demand is assumed to follow an evolving process," in particular, the demand increments are normally distributed with linearly increasing mean and variance in time (i.e., a Wiener process). Selecting the optimum size and timing of capacity additions in the face of uncertain demand forecasts involves both the minimization of cost as well as the minimization of risk. For the situation where demand acts as a Wie-
ner process, Kang and Park have derived the expectation of the "equivalent cost rate." Using the standard deviation of the equivalent cost rate as a measure of the risk of expansion, Kang and Park then minimized an objective function that was a linear combination of cost and risk. The purpose of this paper is to extend this line of research. Using standard decision analysis procedures, a multiattribute utility function is constructed that reflects the decision maker's tradeoff for cost and risk. A significant advantage of the utility function approach is that it allows for nonlinear trade-offs of cost and risk.
INTRODUCTION
tional to size); (2) there is a continuing (possibly changing) need for the facilities; and (3) the facilities added are durable. Examples of capacity decisions that reasonably meet these criteria include communication networks, electric power generation and transmission, manufacturing facilities, and public facilities such as schools, water systems, roads, etc. Actual capacity expansion problems are invariably complex with many subtle nuances that resist rigorous mathematical modelling. Nevertheless, economists and others have studied the capacity expansion problem extensively by making simplifying assumptions.
The size and timing of capacity is an important decision problem for both private industry and government. Because there are typically large amounts of resources involved and strategic consequences, the concomitant benefits, costs and risks of such decisions require careful analysis. The classic capacity expansion problem, as noted by Friedenfelds [ 1 ], is characterized by: (1) the cost of added capacity exhibits economies-of-scale (i.e., cost is less than propor*Presented at the Fourth International Working Seminar on Production Economics, Igls, Austria, Feb. 17-21, 1986.
0167-188X/87/$03.50
© 1987 Elsevier Science Publishers B.V.
4 Under the assumption that growth in d e m a n d is deterministic, Chenery [2] developed an optimizing model for predicting investment behavior. Subsequently, Manne [ 3] generalized the results for probabilistic growth. Numerous others (e.g., [4], [5], and [ 6 ] ) h a v e contributed to the understanding of capacity expansion by formulating m o d e l s that capture certain aspects of real capacity expansion decision making. In the face of uncertain d e m a n d forecasts, selecting the o p t i m u m size and timing of capacity additions involves both the minimization of cost as well as the minimization of risk. For the situation where d e m a n d acts as a Wiener process, Kang and Park [7] have derived the expectation of the "equivalent cost rate." Using the standard deviation of the equivalent cost rate as a measure of the risk of expansion, Kang and Park then minimized an objective function that was a linear combination of cost and risk. The purpose of this paper is to extend this line of research. Using standard decision analysis procedures a multiattribute utility function is constructed that reflects the decision maker's trade-offfor cost and risk. The significant advantage of the utility function approach is that it allows for non-linear trade-offs of cost and risk. The remainder of the paper is organized in the following manner. In the next section, the model that Kang and Park developed is presented. After this, the general five step procedure of Keeney and Raiffa for assessing multidimensional utility functions is reviewed, and then applied to stochastic capacity expansion. A numerical example follows. In conclusion, the pros and cons of a decision analysis approach to stochastic capacity expansion are discussed. KANG A N D PARK MODEL
The problem being considered is the expansion of a single capacity when the installation cost is large and there are neither obsolescence nor spatial effects on the expansion. D e m a n d
is assumed to follow an evolving process; in particular, the d e m a n d increments are normally distributed with linearly increasing mean and variance in time. This assumption on d e m a n d describes a Wiener process [ 3 ]. The analysis of Kang and Park includes both backlogging and no backlogging. Only capacity expansion without backlogging is considered in this paper; consequently, only that portion of the model development dealing with the no backlogging case is presented herein. Let u(t;x) represent the probability density (p.d.f.) that d e m a n d first exceeds the initial level by x at time t when d e m a n d increments are normally distributed with linearly increasing mean ~t and variance a2t. u(t;x)=
x_.
x/(27ct72t 3)
e~_(x_¢02/2~20}
(1)
The installation costs that result from a single capacity increment of size x are assumed to be given by a cost relationship of the following form. G(x) =kx ~ (2) where k is a constant and a is the economies of scale factor, k > 0 and 0 < a < 1. In addition, the cost of installing the capacity at regenerative times t~, i = 1,2 ..... discounted to the time origin t~ is G(t~;x) where
(3)
G(t,;x)=G(x) e ....
and r is the discount rate. The equivalent cost rate (ECR) for G(t~;x), denoted by V(t~;x), is defined by V(t,;x) =
~G(x)
ifr~O
G(x)
if r=0
{1 r
(4 )
where filling time t is an independent and identically distributed random variable with p.d.f, u(t). Since V(t~;x) is independent of t,-, it is more simply represented as V(x). The ECR can be interpreted as a discounted time average of the total investments on the infinite horizon, where the expectation of ECR is co
~rk'x"~=oe~"~_ E{V(x)} = LE{I/t} kx~
ifr~O ifr=O
(5)
5
ASSESSING A STOCHASTIC CAPACITY EXPANSION UTILITY FUNCTION
and ,l, = ~ [1 -,/(1 + 2nra2 /~2) ]
(6)
oo
E{f( t ) } = f f ( t )u( t;x)dt
Keeney and Raiffa Assessment Procedure
•
o
In considering the riskiness of a project, Kang and Park argue that one of the major sources of risk is the uncertainty of the installation cost itself; this is more likely to be a reasonable assertion if the capacity installation costs are very large. Next, it is assumed that the standard deviation of the ECR, S{ V(x)}, can be used to represent the risk of expansion where s{ V(x)} =
{
rkx~{~(n+l)e~"x-[~e~"~]2} .=o
½
r#O
.=o
k.xa[ E(1/t 2) _ {E(1/t ) } 2] ~
(7) r=0
An objective function that is a linear combination of expected cost and risk is then minimized. m i n i m i z e [ a E { V( x ) } + (1 - a) S{ V( x ) } ]
(8)
subject to x > 0 for a given value of a, 0 < a < 1. Obviously a, the risk aversion factor, controls the weight given to risk minimization versus cost minimization. Whereas this procedure does provide a method for explicitly incorporating risk in the capacity expansion problem, it does so in a simple fashion. Specifically, only linear tradeoffs between cost and risk can be considered with eqn. (8). In addition, Kang and Park made no mention of how a should be determined for a specific decision problem. It is intended, one supposes, that decision analysis procedures would be used to assess a. This being the case, we propose enlarging the role of decision analysis to include the construction of a multiattribute utility function that reflects the decision maker's preference for capacity expansion alternatives that involve cost and risk. This utility function, rather than optimizing eqn. (8), is used for evaluating capacity expansion.
Keeney [9 ] and Keeney and Raiffa [ 10] present a general five step procedure for assessing multidimensional utility functions. The first step in the procedure is to familiarize the decision maker with the framework to be used in the succeeding steps. This includes identifying the person whose utilities are to be assessed, the issues to which the needed utilities are relevant, the attributes to be evaluated and the relevant dimensions of the attributes to be ovaluated. In the second step of the assessment procedure the relevant utility independence assumptions are ascertained. This step will determine the form of the multidimensional function, i.e. additive, multiplicative or multilinear. These assumptions are verified by presenting the decision maker with an appropriate set of multiattributed lotteries. The assessments obtained will either support or reject the independence assumptions. Step three in the Keeney procedure requires that a single attribute conditional utility function is assessed for each attribute. Typically this assessment begins by identifying the relevant qualitative characteristics of the unidimensional utility function. This includes determining if the function should be monotonic and determining the decision maker's risk posture with respect to the particular attribute. It is then necessary to quantify points on the curve by presenting the decision maker with a set of uniattributed lotteries. These assessed points are used to fit an appropriate functional form (i.e., one that exhibits the observed qualitative characteristics). Finally, it is necessary to check the consistency of the assessed conditional functions. This is usually accomplished by presenting the decision maker with an additional set of uniattributed lotteries and comparing the
elicited responses with those obtained from the assessed function. Obviously, any observed inconsistencies will require some retracing of steps in the overall procedure. In the previous step, each conditional utility function is assessed on an arbitrary scale. Thus, in step four, it is necessary to rescale each single attribute function to a c o m m o n origin and a c o m m o n unit of measure. This is done by determining two levels of each attribute that are indifferent, implying equal utility, and determining the appropriate linear transformation. Once the conditional utility functions are consistently scaled, it is a simple matter to determine the parameters of the multiattribute utility function. The fifth and last step in the procedure is a consistency check. Here, the decision maker is presented with pairs of multiattributed outcomes and asked to identify the more preferred outcome. The consistency of the elicited preferences is then compared with those implied by the multiattribute utility function.
be used to estimate parameters for specific functional forms. Klein et al. [ 8 ] suggest that a s u m m e d exponential of the form: U( xi) : ai + bie a''x; + c;e "~2'x',
(9)
can be used in most cases. This expression is robust in that it will model all of the most comm o n risk attitudes (i.e. constant risk aversion, increasing risk aversion, decreasing risk aversion, constant risk proneness, decreasing risk proneness, increasing risk proneness, risk proneness for small attribute values and risk aversion for large attribute values, risk aversion for small attribute values and risk proneness for large attribute values) depending on the values of the five parameters. Step four involves determining the scaling constants for the multiattribute utility function. If attributes Xl and x2 are mutually utility independent (multiplicative form), then their two-attribute utility function is given by U(x~ ,x2) =k, U~(x~) + k2 U2(x2) +klk2k3U,(xl)U2(x2)
Application to Stochastic Capacity Expansion In the first step of the K - R procedure, cost and risk would be identified as the relevant issues, and a plausible range of values for the two attributes would be ascertained. At least three attributes are needed to distinguish between multiplicative and multilinear utility function forms. Since the problem as posed contains only two dimensions, cost and risk, the multilinear form is eliminated as an alternative. Furthermore, the multiplicative form reduces to the additive form as a special case. Thus, in step two, we will assume the multiplicative form is appropriate for capareity expansion. In step three, quantifying points on a univariate utility function can be accomplished several different ways, such as a five-point assessment procedure (e.g., Daellenbach et al. [ 11 ], p. 307). Statistical curve fitting can then
(10)
The scaling constants kl, k2 and k3 can be inferred from two indifference statements (see Daellenbach et al. [ 11 ], p. 640). This procedure is illustrated in the next section.
NUMERICAL EXAMPLE In this section we will illustrate by way of a numerical example the multiattribute utility function approach to stochastic expansion. The development of the multiattribute utility function for the example will follow the Keeney and Raiffa procedure. The derived utility function is then used to select an o p t i m u m capacity and various sensitivity analyses are performed to evaluate the sensitivity of the solution to changes in model parameters. Basic model parameters for this example are taken from a generalized problem presented in Kang and Park: ~ = 1; a2=0.5; k = 2 0 ; a = 0 . 6 ; r=0.15.
7
Step 1 Let xt = the expected value of future discounted costs; and x2 = the variance of future discounted costs.
Figure 1 illustrates how different values of 2~ effect the univariate utility function for the expected value of future discounted cost. Similarly, with 2 2 = - 0 . 5 0 , the univariate utility function for the variance for future discounted costs is U2(x2) = 1 . 0 0 6 7 8 3 7 - 0 . 0 0 6 7 8 3 7 e °'5°x2
Assume that the relevant ranges ( m i n i m u m , m a x i m u m ) for x~ and x2 are (10,30) and (0,10), respectively.
Step 2 Assume the multiplicative form is appropriate.
Step 3 Assume that a special case of functional form (9), namely: U(Xi) = ai - bie-~'x'; b, > 0, 2, > 0
is appropriate. This assumption implies that the decision maker is constantly risk averse (or nearly so) over the relevant range of the attributes. Since, for this example, the ranges on both x~ and x2 are relatively narrow this will likely be a reasonable assumption. We may arbitrarily set U~(10)=I.0, U , ( 3 0 ) = 0 . 0 , U 2 ( 0 ) = I . 0 , and U2(10)=0.0. Substituting these values into (9) results in a, - b t e - i o a , = 1.0 a n d at - bj e-30~, = 0 . 0
Step 4 Again, without loss of generality, we arbitrarily, set U ( 1 0 , 0 ) = I . 0 and U ( 3 0 , 1 0 ) = 0 . Next, the DM would be asked to specify a value xi so that he or she is indifferent between the following two outcomes: A l = [ X l at specified level, x 2 at least preferred value] A2= [xl at least preferred value, x2 at most preferred value] [xt = ? , x 2 = 10] ~ [x, = 30,x2 = 0 ]
Suppose the answer is Xl = 23.0. Then, we know that Uj ( x t = 2 3 . 0 ) = 1 . 1 5 6 5 2 5 - 0 . 0 5 7 5 8 0 e °'t°(z3"°~ = 0 . 5 8 2 2
and U ( 2 3 . 0 , 1 0 ) ~ U ( 3 0 , 0 ) . have
From (10), we
=k~ U~(30)+k2Uz(O)+k~k2k3Ut(30)Uz(O)
= 1.0
Now fix 2t at a particular value, say 2~ = - 0 . 1 0 . This is equivalent to determining the decision maker's propensity for riskiness. When 2i < 0, the DM is risk-averse; ,~i= 0, the DM is neutral; and 2i > 0, the DM is risk-prone. Solving for a~ and b~, with 2 1 = - 0 . 1 0 , one obtains the values 0.057580 and 1.156525, respectively, and the univariate utility function for the expected value of future discounted cost is Ut(xt ) = 1 . 1 5 6 5 2 5 - 0 . 0 5 7 8 5 0 e °'x°x'
Figures 2(a) and 2(b) depict the univariate utility function for the variance of future discounted costs for values of 22 of - 0.30, - 0.50, and - 0.70.
k~ Ul(23.0)+kzUz(lO)+klk2k3Ut(23.0)Uz(lO)
Simple algebra yields bj [ e -~°~' - e - i o ~ , ]
(12)
(11 )
Substituting in the known utility values, this last expression simplifies to k2 = 0.5822kl. For the second indifference statement, we ask for an outcome [xl,x2] which is indifferent to a 50-50 reference lottery involving the most preferred and the least preferred outcomes, i.e. [ 10,0] and [30,10]. This choice can be made easier for the DM by fixing x~ to its 0.50 individual utility value, namely x, =24.338 (see Fig. 2(a)). Suppose that the answer to this reference lottery is x2=6.0. Thus, U(24.338,6.0)=0.50
1,0 0.9 0.8 0.7 0 o
0.6
ft.
0 >,. I-
0.5
"i 0.4 I':::3 0.3 0.2 0.1
~0
I 12
I 14
I 16
I 18
EXPECTED
I
20
VALUE
I 22
14
2
FUTURE DISCOUNTED
I 26
I 28
COST
Fig. 1. Univariate utility functions for the expected value of future discounted costs. 1.0 0.9 0,8 0.7
7- o.8 ~ 0.5 >-
~. J o.4 ~0.3
0.2 0.1 I
I
I
I
I
0 STANDARD
DEVIATION
OF F U T U R E
DISCOUNTED
COST
Fig. 2(a). Univariate utility functions for the standard deviation of future discounted costs. [24.388, x2=?]
A,~ ,~.,JA 2
[lo, o]
.50
Fig. 2(b). Indifference statement involving a 50-50 reference lottery.
[3o.10]
30
and from (12) U2 (6.0) = 1.0067837 0.0067837e °'5°~6"°) = 0.8705. Consequently, from eqn. (10) we have the relationship U (24.338,6.0) = K~U1 (24.338) + K2U2(6.0) + K~K2K3U~ (24.338) U2(6.0) = 0.50. Substituting in the known utility values, this last expression simplifies to k3-
0.50-- 1.0120kl 0.25602kl 2
Now, we find kl from evaluating (10) with U(10,0) = 1.0 and the known utility values. k~(1) + 0.5882kt (1) + k~(0.5882k~)[(0.50 -1.0120kt 2](1)(1)=1.0 kj =0.20186
Since expressions have been found for k2 and in terms of k~, their values may now be determined. k3
k~=0.11873 k3 =28.34664
Therefore, the joint utility function for cost and risk under mutual utility independence is U(xl ,x2) = 0.20186 [ 1.15625 -- 0.057580e °'l°x'] + 0.11873 [ 1.0067837e °5°x' ] + (0.20186)(0.11873)(28.34664) U~(x~) U2(x2)
In order to obtain the utility for a given capacity expansion size x, the cost (x~) and risk (x2) for the given size are first evaluated by eqns. (5) and (7). Next, eqn. (13) is used to determine the utility. Figure 3 depicts the utility for differing capacity expansion sizes. The " o p t i m u m " size is between 8.0 and 8.5 (the utility is practically the same). Several different kinds of sensitivity analyses are appropriate. For instance, one interest is to determine the sensitivity of the o p t i m u m capacity expansion size to the decision maker's propensity for risk. Suppose it is decided that the variance of future discounted cost is very important, but the decision maker is unsure of his/her ability to articulate responses in the univariate assessment procedure. In this
case, the effect of different values for 22 could be investigated. Figure 4 illustrates the utility for capacity expansion size for different values of 22. The relative flatness of the utility function in the neighborhood of the o p t i m u m capacity expansion size illustrates that a wide range of sizes have nearly the same utility (for the given problem parameters: ~= 1, a2= 0.5, k = 2 0 , a = 0 . 6 , and r = 0 . 1 5 ) . To more fully investigate the sensitivity of the o p t i m u m capacity size, several additional analyses were performed for different parameter values. Specifically, the discount rate, r, was investigated for additional values of 0.10 and 0.20; the economies of scale factor, a, for additional values of 0.65 and 0.70; and the constant of proportionality, k, for additional values of 15 and 25. In each analysis, a single parameter value was changed from the original parameter values (the base case); the joint utility function for (13) with 2 1 = - 0 . 1 0 and 2 2 = - 0 . 5 0 was employed. The utility curves that resulted from this procedure are depicted in Figs. 5, 6, and 7. In considering the results of sensitivity analyses such as these, the decision maker is left to draw his/her own conclusions about what capacity expansion should actually be implemented. Our philosophy is that models like these are best employed in a decision support context. In this vein, we believe model construction and implementation should facilitate sensitivity analyses in order to assist the DM in his/her search for a capacity expansion size that satisfies the goals and constraints of the situation. A N ALTERNATIVE A P P R O A C H
The decision maker may be unaccustomed to thinking in terms of the attributes x~ = E [ V(x)] and x2=S [ V(x)]. Thus, it may be difficult to assess the conditional utility functions required to use eqn. (10). Alternatively, it may be possible to obtain a set of isopreference curves. Each isopreference curve is used to represent a set of consequences that are equally desirable to the decision maker.
10 hO
0.9
z o z Q. X 14J
0.8
n~
>- 0.7 k._1 0.6
0.5
I
I
I
I
I
I
I
I
I
2
4
6
8
I0
12
14
16
18
20
EXPANSION SIZE
CAPACITY
Fig. 3. Multiattribute utility function for stochastic capacity expansion. 1.0
0.9
0.8
0.7
0.6
0.5 2.5
4.15
615
815 CAPACITY
,~5
,;5
'
14.5 EXPANSION SIZE
I
I
16.5
18.5
Fig. 4. Utility for capacity e~pansion size for different values of~.2.
In order to assess an isopreference curve the decision maker is asked a series of indifference questions. A typical indifference question would be as follows: Compare a particular capacity expansion having attributes X I : 1 0 ( c o s t ) and X 2 : 6 (risk) with a second capacity expansion having a cost of x~ = 12. What value of risk (x2) in the second expansion option would result in you having no preference between the expansion alternatives?
[Xl= 10, X2=6] Reference capacity expansion size
~ [X1=12,
X2:?
]
Second capacity expansion option
By repeating the indifference question with the same reference capacity expansion size and varying the second expansion option, a series of these points can be assessed and then connected with a smooth curve. Different isopreference curves are assessed by varying the reference capacity size. Figure 8 presents three isopreference curves
11 1.0 r=.lO 0.9 z 01
0.8 n,,o
u. 0.7 >i--
..I I-
0.6
0.5
. . .4 .
2
6
8
I
'o
I
'2
14 '
16 '
I
20
CAPACITY EXPANSION
Fig. 5. Sensitivity of utility for expansion to changes in the discount rate, r. 0.91
O.S Z
ow Z
~O.7 tO Ix
o i, 0.6 J
~0.5
20 CAPACITY EXPANSION Fig. 6. Sensitivity o f u t i l i t y f o r expansion to changes in the economies-of-scale factor, a.
for the decision maker in the previous example. That is, these contours are consistent with eqn. (13 ). If the decision maker faces the situation where he or she must choose from among a finite set of expansion alternatives, the coordinates of these alternatives can be plotted over the assessed contours. In Fig. 8 the coordinates of expansions of size 5, 10, 15 and 20 are displayed. The xl and x2 coordinates for these expansions are computed from (5) and 6) respectively. The choice from among these expansion alternatives is obvious. Clearly, the expansion of size 10 lies on the most preferred
lsopreferennce curve. For the case where any size expansibn is possible, additional points could be computed and plotted. A smooth curve would then be drawn through these points. The optimal capacity expansion size can then be estimated from this curve. While this approach may appear as cumbersome as the previous analysis, it has certain advantages. No independence assumptions are required in order to estimate isopreference curves. In addition, this approach is not bound to any specific conditional utility function forms. Thus, the isopreference curve approach
12 1.0 0.9 0.8 0,7 0.6 0.5 0.4 0.3 0.2
I 4
0
I 6
I I I I 8 I0 12 t4 CAPACITY EXPANSION
I 16
I IS
20
Fig. 7. Sensitivity of utility for expansion to changes in the constant of proportionality, k. 9
8s U
~7
>~
~
0 CapaOity Expansion Size Of 15 " ~PacliliEPx °n'i°n$izeOfZO
i 0
Direction Of
I0
i ....
i
i
\
I /
i
i
F i
i
ii
/ i
12 14 16 18 EXPECTED VALUE FUTURE DISCOUNTED COSTS
e Ii
20
Fig. 8. Isopreference contours for cost and risk.
is more general than directly assessing the decision maker's utility function. Furthermore, the decision maker may find indifference questions easier to understand than lotteries. CONCLUSIONS There are several advantages to a multiattribute utility function approach to the capacity expansion problem. To the extent that the multiattribute utility function can be accurately assessed, multidimensional alternatives
can be evaluated consistent with the decision maker's preference structure. Modelling of nonlinear trade-offs between attributes is possible. Although we have illustrated the procedure for just two attributes, cost and risk, other dimensions might also be appropriate. More than two attributes do not present any problems conceptually. Sensitivity analyses are easily accomplished with this approach and can be presented in a manner comprehensible to the decision maker. However, the difficulties associated with assessing, constructing, and using multiattri-
13
bute utility functions should not be overlooked. Many decision makers find it difficult to articulate preferences and trade-offs. Furthermore, the technique itself is likely to be unfamiliar to the DM and consequently viewed with suspicion. Similar to other techniques that require the aid of analysts, the overall success of multiattribute assessment is closely related to the skill of the decision analyst assisting with the assessments. Despite all of these implementation obstacles, we believe that the multiattribute utility function approach has much to offer in the analysis of problems as important as capacity expansion.
2 3 4 5 6 7 8
9 10
REFERENCES Friedenfelds, J. (1981). Capacity Expansion: Analysis of Simple Models with Applications. Elsevier North Holland, New York.
11
Chenery, H.B. (1952). Overcapacity and the acceleration principle. Econometrica (January). Manne, A.S. (1961). Capacity expansion and probabilistic growth. Econometrica, 29(4): 632-649. Edenkotter, D. (1977). Capacity expansion with imports and inventories. Manage. Sci., 23(7): 694-702. Giglio, R.J. (1970). Stochastic capacity models. Manage. Sci., 17(3): 174-184. Tapiero, C.S. (1979). Capacity expansion of a deteriorating facility under uncertainty. R.A.I.R.O., 3(1): 55-66. Kang, H.W. and Park, S.J. (1983). An efficient curve in stochastic capacity expansion. OMEGA: Int. J. Manage. Sci., 11(2): 147-153. Klein, G., Moskowitz, H., Mahesh, S. and Ravindran, A. (1985). Assessment ofmultiattribute measurable value and utility functions via mathematical programming. Decision Sci., 16(3): 309-324. Keeney, R.L. (1972). An illustrative procedure for assessing multiattributed utility functions, Sloan Manage. Rev., 14(1): 37-50. Keeney, R.L. and Raiffa, H. (1975). Decisions with Multiple Objectives: Preferences and Value Tradeoffs. New York: John Wiley & Sons, New York. Daellenbach, H.G., George, J.A. and McNickle, D.C. (1983). Introduction to Operations Research Techniques, 2rid edn. Allynand Bacon, Newton, MA.