Stochastic goal programming: A mean–variance approach

European Journal of Operational Research 131 (2001) 476±481

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Theory and Methodology

Stochastic goal programming: A mean±variance approach Enrique Ballestero

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Dpto. Economia y Ciencias Sociales Agrarias, Escuela Tecnica Superior de Ingenieros Agr onomos, Ciudad Universitaria, s/n, 28040 Madrid, Spain Received 4 March 1999; accepted 9 February 2000

Abstract We propose a stochastic goal programming (GP) model leading to a structure of mean±variance minimisation. The solution to the stochastic problem is obtained from a linkage between the standard expected utility theory and a strictly linear, weighted GP model under uncertainty. The approach essentially consists in specifying the expected utility equation corresponding to every goal. Arrow's absolute risk aversion coecients play their role in the calculation process. Once the model is de®ned and justi®ed, an illustrative example is developed. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Expected utility theory; Goal programming

1. Introduction In its stochastic formulations, this paper refers to a prevalent GP model instead of the complete spectrum of goal programming (GP) approaches. The model considered (Charnes and Cooper, 1961, and a huge body of literature) is structured as a system of linear goals with positive and negative deviations between every goal and its respective target or aspiration level. The achievement function to minimise is the weighted sum of the deviational absolute values. As the weighted linear achievement function is essential in the proposed model, other minimisation forms such as lexico-

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Fax: +34-1-3365797. E-mail address: [email protected] (E. Ballestero).

graphic GP (Charnes and Cooper, 1961; Ijiri, 1965; Lee, 1972; Ignizio, 1982), minmax GP (Flavel, 1976) or Chebishev minimisation, and polynomial GP are not considered in this paper. Our stochastic approach substantially di€ers from others in the GP literature. Indeed antecedents are few. Apart from Chance-Constrained programming (Charnes and Cooper, 1959) and its applications to GP (e.g., De et al., 1982) see Contini (1968) and Stancu-Minasian and Tigan (1988) as well as Stancu-Minasian (1997). In Romero's survey, the topic of stochastic GP is only brie¯y mentioned (cf. Romero, 1990). This scarcity of references on stochastic GP also ensues in the recent survey of Tamiz et al. (1998). Consider a decision-maker with several goals such as costs, pro®ts, sales revenue, etc. given by the linear equations

0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 0 ) 0 0 0 8 4 - 9

E. Ballestero / European Journal of Operational Research 131 (2001) 476±481

gj ˆ

s X

nij xi ;

j ˆ 1; 2; . . . ; q;

…1†

iˆ1

where nij is the random variables with mean values, variances and covariances accurately speci®ed. xi is the decision variables. The mean value of the goal gj will be denoted as gj . For example, suppose a case of farm management where the xi decision variables are land allocated to the dry farming crops i ˆ 1; 2; . . . ; s. The goal g1 (given by Eq. (1) for j ˆ 1) refers to the sales revenue that the farmer receives for the crops. For each crop, we have a certain per-acre sales revenue ni1 which is a random variable as depending on the rain levels and their e€ects on the i crop. Market prices also a€ect the revenues. From weather historical series and price indexes, the decision-maker can obtain reliable information on mean values E…ni1 †, variances r2 …ni1 † and covariances. Beyond farm management many technological and economic problems now stated in terms of non-stochastic GP (e.g., network programming, see (Heeseok et al., 1994)) might be reformulated in stochastic terms with fruitful results. Assumptions. There is no loss of generality if it is assumed that every goal behaves such that ``more is better''. In fact, suppose a ``more is worse'' goal g such as production costs. This goal can converted into ``more is better'' by changing g1 ˆ g0

g;

…2†

477

the standard properties of increasing utility and decreasing marginal utility. Moreover, Von Neumann and Morgenstern's EU key property holds, namely, that the expected utility EUj …gj † is an appropriate variable to rank combinations of random alternatives such as (1). Justi®cation. The properties just assumed are commonly accepted in the utility literature. See, e.g., Copeland and Weston (1988, pp. 80, 81). Assumption 2. The decision-maker behaves in accordance with the principles of ``satis®cing'' logic such as accepted in GP. In particular, the decisionmaker has a target for the expected utility EUj …gj †. Justi®cation. ``Satis®cing'' logic is widely defended in the GP literature cited above. 2. The model From Assumption 1, the goals gj will be rewritten in terms of expected utility as follows: Gj ˆ EUj …gj †;

…3†

where the decision-maker's utility function Uj satis®es the standard properties of increasing utility and decreasing marginal utility. According to well-known developments in the theory (see Pratt, 1964) Eq. (3) turns into 1 Gj ˆ Uj …gj † ‡ Uj00 …gj †r2 …gj † ‡ o…gj †; 2

…4†

where g0 is a referential cost suciently high (possibly, production cost now). The proposed stochastic model relies on Von Neumann and Morgenstern's utility axioms, say, comparability, transitivity, strong independence, measurability and ranking (cf. Von Neumann and Morgenstern, 1953). These axioms lead to the following EU key property: expected utility is an appropriate variable to rank combinations of random alternatives. See proof in Copeland and Weston (1988, pp. 80, 81).

where Uj00 …gj † is the second derivative at point gj and o…gj † is a small variable. According to Arrow's normalisation, both sides of Eq. (4) are normalised by Uj0 …gj †, namely, the ®rst derivative at point gj (cf. Arrow, 1965, p. 94). Thus we obtain h i 1 0 RAj …gj †r2 …gj † GN j ˆ Uj …g j †=Uj …g j † 2 h i ‡ o…gj †=Uj0 …gj † ; …5†

Assumption 1. The decision-maker's utility Uj …gj † corresponding to each random value of gj satis®es

aversion coecient for random changes in the goal level at point gj (cf. Arrow, 1965, p. 94).

whereh GN goal while RAj …gj † ˆ j is the normalised i 00 0 … 1† Uj …gj †=Uj …gj † is Arrow's absolute risk

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E. Ballestero / European Journal of Operational Research 131 (2001) 476±481

Moreover,   Uj …gj †=gj > Uj0 …gj †

…6†

since the average utility is greater than the marginal utility due to the standard properties of the decision-maker's utility function (see Assumption 1). From Eq. (6) we get h i Uj …gj †=Uj0 …gj † > gj : …7† From Assumption 2 and inequality (7), we rewrite Eq. (5) as follows: 1 RAj …gj †r2 …gj † GN j ˆ aj ‡ bj 2 h i ‡ o…gj †=Uj0 …gj † i h 0 ˆ GN ˆ a ‡ b †=U …g † ‡ o…g j j j j j j 1 RAj …gj †r2 …gj † ˆ aj ‡ pj 2

…8†

nj

with the condition g j P aj ;

…9†

where aj is the target or aspiration level for the GN j normalised expected utility of the gj goal, bj is a positive variable, pj ˆ bj ‡ o…gj †=Uj0 …gj † is the positive deviational variable of the goal, nj ˆ 12 RAj …gj †r2 …gj † is the negative deviational variable of the goal. Notice that the nj negative deviational variables are the only unwanted variables since the ``more is worse'' goals have been converted into ``more is better''. Hence, we must minimise the weighted sum of the negative deviational variables subject to conditions (9). In symbols, we have Min

q X jˆ1

aj RAj …gj †r2 …gj †;

for all j

3. Illustrative example In an area, the available land can be allocated to the following activities: crops (x1 acres), livestock (x2 acres) and forest (x3 acres). The decisionmaker focuses on two random linear goals, pro®t …g1 † and environmental bene®ts …g2 †. In Table 1 the mean values and variances for the nij random variables are shown. Notice that the random variable ni1 is the per acre pro®t corresponding to crops (if i ˆ 1), livestock (if i ˆ 2) and forest (if i ˆ 3). For the sake of simplicity and ease of presentation all covariances are taken to be equal to zero. The targets established by the decision-maker for pro®t and environmental bene®ts as well as Arrow's absolute risk aversion coecients and the decision-maker's preference weights attached to the deviational variables in the achievement function appear in Table 2. All these preferences and attitudes can be speci®ed through computerised dialogues. See Geo€rion et al. (1972), Zionts and Wallenius (1976) and Olson (1992). Arrow's coef®cients speci®cation is explained below. Regarding Table 1 Stochastic GP for allocating land: Numerical information Pro®t

Crops Livestock Forest

Environmental bene®ts

Mean value

Variance

Mean value

Variance

0.82000 0.75000 0.60000

0.02690 0.02250 0.00810

0.14000 0.65000 0.95000

0.00020 0.00423 0.00903

…10†

where aj is the preference weight for the j deviational variable. Minimisation (10) is subject to g j P aj

quadratic programming straightforwardly provides the xj decision variables.

…11†

together with other possible linear constraints as well as the non-negativity conditions. The above

Table 2 Stochastic GP for allocating land: Targets, Arrow's coecients and preference weights

Target (aj ) Arrow's coecients (RAj ) Preference weights (aj )

Pro®t

Environmental bene®ts

0.67000 0.64000 0.66000

0.50000 0.41000 0.33000

E. Ballestero / European Journal of Operational Research 131 (2001) 476±481

the targets, remember these aspiration levels a1 and a2 refer to the decision-maker's expected utility for pro®t and environmental bene®ts, respectively. Therefore the precise knowledge of the decision-maker's utility functions can help the determination of suitable targets. For this purpose, see the Bayesian methodology in French (1988, Ch. 5) with detailed examples of interactive dialogues which highlight the way to put them in practice. Let us now apply quadratic programming (10) and (11) with the numerical data from Tables 1 and 2. First, we have a1 RA1 ˆ 0:66  0:64 ˆ 0:4224; a2 RA2 ˆ 0:33  0:41 ˆ 0:1353: By introducing the above coecients into the achievement function (10) we write Min ……0:4224  0:02690 ‡ 0:1353  0:00020†  x21 † ‡ ……0:4224  0:02250 ‡ 0:1353  0:00423†  x22 † ‡ ……0:4224  0:00810 ‡ 0:1353  0:00903†  x23 † subject to constraints (11), that is 0:82  x1 ‡ 0:75  x2 ‡ 0:60  x3 P 0:67; 0:14  x1 ‡ 0:65  x2 ‡ 0:95  x3 P 0:50; together with x1 ‡ x2 ‡ x3 ˆ 1: The last constraint indicates that the available land is 1 acre. By solving the above programming, we obtain x1 ˆ 0:218; x2 ˆ 0:247; x3 ˆ 0:535:

4. Specifying Arrow's coecients The precise question to estimate Arrow's coef®cients through a dialogue between the analyst and the decision-maker is explained elsewhere (see

479

Ballestero, 1997; also Ballestero and Romero, 1998). However, an alternative technique which is easier to apply, can be implemented as follows. Only for the purpose of specifying Arrow's absolute risk aversion coecients and keeping matters as simple as possible, we here assume exponential utility, namely, Uj …gj † ˆ 1

e

bj gj

;

where bj is a positive constant (see Kallberg and Ziemba, 1983). Then, Arrow's absolute risk aversion coecient becomes h i … 1†b2j RAj …gj † ˆ … 1† Uj00 …gj †=Uj0 …gj † ˆ … 1† bj ˆ bj : We can now estimate values proportional to bj by using Saaty's paired comparison technique (cf. Saaty, 1994). In the decision maker's judgement on risk aversion for random changes in the goal level at point gj , this risk aversion for the gj goal is compared to risk aversion for the gh goal considering all possible …j; h† pairs. The results of such comparisons are represented in a square matrix as usual. For the judgements or comparisons, the analyst employs the so-called ``fundamental scale'' 1, 3, 5, 7 and 9 proposed by Saaty. Suppose, e.g., we are comparing RA1 ±RA2 . If the decision-maker's judgement slightly favours the ®rst risk aversion over the second one, then, the score corresponding to this ``intensity of importance'' is equal to 3 (See Saaty, 1994, p. 26). 5. Applicability Finally, let us comment on the applicability of the proposed model to larger, more realistic problems. First, we should explain as to just how our model di€ers from the mean±variance approach to portfolio optimisation. This ®nancial standard approach is only a particular case of that considered in our stochastic GP model. In fact, looking at constrained minimisation (10) and (11), one can easily see that the standard mean±variance approach

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E. Ballestero / European Journal of Operational Research 131 (2001) 476±481

Min r2 …g1 †

…12†

subject to g 1 P a1 ; g1 ˆ

s X

…13† ni1 xi ;

…14†

iˆ1 s X

xi ˆ 1

…15†

iˆ1

is a special case of (10) and (11) with q ˆ 1. However, this standard mean±variance model is based on unrealistic assumptions such as the absence of personal income tax. This means that the investor is indi€erent to the form (dividends or capital gains) in which the return on the investment is received (see Elton and Gruber, 1984, p. 275). By applying our stochastic GP model, such a diculty is surmounted. Indeed, we can introduce the following goals: g1 ˆ

s X

ni1 xi ;

…16†

ni2 xi ;

…17†

iˆ1

g2 ˆ

s X iˆ1

where ni1 and ni2 are random capital gains and random dividends, respectively. Together with these goals, the proposed model could allow for other interesting goals (for example, weighted trading of assets in the stock market). Generally, the goal variance r2 …gi † is given by r2 …gj † ˆ Xj  fCovgj  XjT ;

…18†

where Xj is the vector of the decision variables while fCovgj represents the covariance matrix. Therefore, the proposed model leads to a quadratic objective function with linear constraints. This quadratic programming can be straightforwardly solved by using Lingo or other appropriate software tools.

6. Concluding remark The solution to stochastic weighted GP just obtained seems to be suciently general and applicable to real world problems. The proposed approach to strictly linear, weighted GP model under uncertainty is related to Von Neumann and Morgestern basis of expected utility theory. In this way, the model is a departure from other less general approaches in the literature, and therefore it seems to make a signi®cant contribution. The computing process does not require cumbersome calculations, as it is reducible to quadratic programming with linear constraints. References Arrow, K., 1965. Aspects of the Theory of Risk Bearing. Academic Book Store, Helsinki. Ballestero, E., 1997. Utility functions: A compromise approach to speci®cation and optimization. Journal of Multi-Criteria Decision Analysis 6, 11±16. Ballestero, E., Romero, C., 1998. Multiple Criteria Decision Making and its Applications to Economic Problems. Kluwer Academic Publishers, Dordrecht. Charnes, A., Cooper, W.W., 1959. Chance-constrained programming. Management Science 6, 73±79. Charnes, A., Cooper, W.W., 1961. Management Models and Industrial Application of Linear Programming. Wiley, New York. Contini, B., 1968. A stochastic approach to goal programming. Operations Research 3, 576±586. Copeland, T.E., Weston, J.F., 1988. Financial Theory and Corporate Policy. Addison-Wesley, Reading, MA. De, P.K., Acharya, D., Sahu, K.C., 1982. A chance-constrained goal programming model for capital budgeting. Journal of the Operational Research Society 33, 635±638. Elton, E., Gruber, M., 1984. Modern Portfolio Theory and Investment Analysis. Wiley, New York. Flavel, R.B., 1976. A new goal programming formulation. Omega 4, 731±732. French, S., 1988. Decision theory: An introduction to the mathematics of rationality. Ellis Horwood Limited, Chichester, England. Geo€rion, A.M., Dyer, J.S., Feinberg, A., 1972. An interactive approach for multi-criterion optimisation, with an application to the operation of an academic department. Management Science 19, 357±368. Heeseok, L., Shi, Y., Stolen, J., 1994. Allocating data ®les over a wide area network: Goal setting and compromise design. Information & Management 26, 85±93. Ignizio, J.P., 1982. Linear Programming in Single & MultipleObjective Systems. Prentice-Hall, Englewoods Cli€s, NJ.

E. Ballestero / European Journal of Operational Research 131 (2001) 476±481 Ijiri, Y., 1965. Management Goals and Accounting for Control. North-Holland, Amsterdam. Kallberg, J.G., Ziemba, W.T., 1983. Comparison of alternative utility functions in portfolio selection problems. Management science 29 (11), 1257±1275. Lee, S.M., 1972. Goal Programming for Decision Analysis. Auerbach publishers, Philadelphia. Olson, D.L., 1992. Review of empirical studies in multiobjective mathematical programming; subject re¯ection of nonlinear utility and learning. Decision Science 23, 1±20. Pratt, J.W., 1964. Risk aversion in the small and in the large. Econometrica 32 (1±2), 122±136. Romero, C., 1990. Handbook of Critical Issues in Goal Programming. Pergamon Press, Oxford. Saaty, T., 1994. How to make a decision: The analytic hierarchy process. Interfaces 24, 19±43.

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Stancu-Minasian, I.M., Tigan, St., 1988. A stochastic approach to some linear fractional goal programming problems. Kybernetika 24 (2), 139±149. Stancu-Minasian, I.M., 1997. Fractional Programming, Theory, Methods and Applications. Kluwer Academic Publishers, Dordrecht. Tamiz, M., Jones, D., Romero, C., 1998. Goal programming for decision making: An overview of the current state-ofthe-art. European Journal of Operational Research 111, 569±581. Von Neumann, J., Morgenstern, O., 1953. Theory of Games and Economic Behaviour, third ed. Princeton University Press, Princeton, NJ. Zionts, S., Wallenius, J., 1976. An interactive programming method for solving the multiple criteria problem. Management Science 22, 652±663.