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Solving the discrete multiple criteria problem using linear prospect theory
146
European Journal of Operation al Research 72 (1994) 146-154 North-Rolland
Theory and Methodology
Solving the discrete multiple criteria problem using linear prospect theory Pekka Salminen Department of Economics and Management, University of lyviiskylii, PL 35, 40351 lyviiskylii, Finland Received October 1991; revised January 1992
Abstract: Prospect theory developed by Kahneman and Tversky is a popular model of choice in decision problems under uncertainty. Prospect the ory has recently been extended to multiple criteria choice problems. In this paper, an interactive method for solving discrete multiple criteria decision problems, based on prospect theory type value functions, has been developed. Piecewise linear marginal value functions are assumed to approximate the S-shaped value functions of prospect the ory. Therefore, the proposed procedure is valid only for convex preferences. Keywords: Multiple criteria; Decision theory
1. Introduction
Tversky [8] has developed a choice the ory, in which alternatives are evaluated based on comparisons of criterion-wise differences between decision alternatives (the additive difference mode!). Later Kahneman and Tversky [1] developed prospect the ory, where outcomes are expressed as positive or negative deviations (gains or losses) from a reference alternative. Although value functions differ among individu ais, Kahneman and Tversky propose that they are commonly S-shaped, concave ab ove the reference point and convex below it. Furthermore, value functions are commonly assumed steeper for losses th an for gains. Prospect theory was originally developed for single criterion problems, but the ideas have been extended to multiple criteria decision probCorrespondence to: Dr. P. Salminen, Department of Economies and Management, University of Jyvaskyla, PL 35, 40351 Jyvaskyla, Finland.
lems as weIl (Korhonen, Moskowitz and WaIlenius [3]). In this chapter we propose an interactive method for solving discrete deterministic multiple criteria de ci sion problems, assuming prospect theory type of value functions for the decisionmakers. The method builds on existing work, namely Zionts [9], Korhonen, Wallenius and Zionts [5], and Korhonen, Moskowitz and WaIlenius [2]. The proposed method makes use of pairwise comparisons of attainable decision alternatives to identify the best alternative. We assume m deterministic decision alternatives, p criteria, and one decision-maker, whose choice behavior is modeled using (Iinear) prospect theory (Kahneman and Tversky [1]). We further assume that aIl alternatives can be compared. At most m - 1 pairwise questions are required to identify the best alternative. For most problems, considerably fewer comparisons are required. The method initially chooses an arbitrary set of positive multipliers (or equal weights) for the
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P. Salminen / Soluing the discrete multiple criteria problem
criteria, and generates a composite linear value function using these multipliers. Then the alternative that maximizes this function is found. Next, one or more adjacent nondominated alternatives are identified and presented to the DM for his/her evaluation. The DM is asked to choose between the current alternative and the adjacent nondominated alternatives. Based on the DM's answer(s), inequalities are generated and used to eliminate alternatives which will not be preferred to the current alternative. The procedure employs a modification of the simplex method of linear programming, in which the criterion values are divided into gains (multiplier À +) and losses (multiplier À -), with respect to a reference alternative. As prospect theory suggests, losses carry more weigth th an gains (À +.:s; À -). Once an adjacent nondominated alternative is found that is preferred to the current alternative, we select it to be the current reference point. The process is then repeated by identifying adjacent nondominated alternatives to the currently best alternative, asking new questions of the DM, and so on. Convergence to an overall best alternative is assured, given that the DM's choice behavior is consistent with the underlying theory. Alternatives less preferred to the current reference alternative are dropped. The problem of eliminating alternatives based on DM's implied preferences (convex cones) plays an importnat role in our procedure. This problem was originally considered by Zionts [9], Zionts and WaIlenius [10], and Korhonen, WaIlenius and Zionts [5]. In Zionts and WaIlenius [10], a method for solving the continuous problem is presented, and in Zionts [9] (see the Appendix) the discrete problem with a linear underlying value function is considered. In Korhonen et al. [5] a procedure is developed for solving discrete problems in which the underlying value function is assumed to be quasi concave. In this chapter we develop a method for solving discrete problems in which the DM's choice behavior is based on differences with respect to a reference point. We use piecewise linear difference functions. The use of adjacent nondominated alternatives relaxes the form of the value function aIlowing them to be quasiconcave. In our case, the cone of dominated alternatives st arts from the more preferred point, instead of the less preferred point as in Korhonen et al. [5]. This increases the power of the
147
elimination procedure. However, from the point of prospect theory, only piecewise linear difference functions are considered; generaIly S-shaped difference functions need not result in convex preferences.
2. Sorne theory 2.1. Prohlem description
We assume a single decision-maker, a set {Xl' x 2 ,···, xml of m deterministic decision alternatives, and p criteria, which define an m X p decision matrix X whose elements are denoted by x ij ' i El = {l, 2, ... , m} and j E J =
{l, 2, ... , pl. Also, we assume that the DM's preferences can be modeIled using piecewise linear (implicit) difference functions for each criterion. The differences are measured from a DM's subjective reference point (unknown), which may be his/her current alternative or vector of aspiration levels for each criterion. 2.2. Nondominated adjacent alternatives
Our algorithm is based on comparing nondominated adjacent alternatives against the current alternative. A nondominated adjacent alternative is defined as foIlows (Korhonen and WaIlenius [4]): consider the current alternative x r and an alternative Xi; Xi is a nondominated adjacent alternative for x" if 1) Xik>X rk for sorne k= 1, 2, ... ,p; 2) there is no alternative Xl with 1 =F i,r such that
for aIl j
=
1, 2, ... , p.
(1)
In Figure 1 we illustrate this with a simple twocriteria case. Alternative Xi is a nondominated adjacent alternative for x" if there exists no real decision alternatives in the shaded region. 2.3. Consistency with linear prospect theory
FoIlowing Korhonen et al. [3], a linear programming formulation can be used to test whether the subjects' choices are consistent with linear prospect theory. By assuming piecewise linear
P. Salminen / Solving the discrete multiple criteria problem
148
2.4. Reference point
x;
X.
1
x
Figure 1. Alternative
Xi
Here we assume that the current alternative or the vecor of the DM's aspiration levels are possirefers ble reference alternatives. The notation to the DM's real reference alternative. The notation X r is used to indicate the currently best alternative. When the current alternative is the real reference alternative, x; = xr' In the case where x r is not the real reference alternative, we assume that X r converges to
r
is a nondominated adjacent alternative for x r
x;.
2.5. Convex cones of dominated alternatives
value functions, the following inequality is generated for each choice Xi' i El, preferred to the reference point X r :
(2) and for the reference point X r preferred to the available choice Xi' an inequality of the following type is generated:
(3) where e is a scalar variable, and vectors z are
zi
t
and
if Xij -X rj 2 0, if Xij -X rj < 0, if Xij -X rj < 0, if Xij -X rj 2 0, and where A+ is a vector of weights corresponding to the gains and A- is a vector corresponding to the losses with respect to the reference point xr' To estimate A, the following linear program is solved: Max S.t.
e
+ LAj-
Aj 2At 20,
2.5.1. Ziont's discrete method Ziont's discrete method [9] initially selects an arbitrary set of positive multipliers for the criteria, and generates a composite linear value functian these multipliers. Next, one or more adjacent efficient alternatives are presented to the DM for his/her evaluation. Based on the DM's answers, inequalities are generated and used to eliminate alternatives which will not be preferred to the current alternative. This is done using linear programming. Elimination in this method is based on linear value functions. If alternative h is preferred to alternative k, a constraint LAj(Xhj -X kj ) 2 e
(2), (3), and LAt
An essential feature of the method described in this chapter is the use of the DM's pairwise judgments to generate regions in which preferred decision alternatives cannot be located. Next, we briefly de scribe two decision models, which closely relate to the method suggested in this paper. They are the Zionts' discrete method [9], in which a linear value function is assumed, and the convex cone method of Korhonen, Wallenius and Zionts [5], in which the form of the DM's value function is relaxed to be quasiconcave.
(A)
j
=
1,
j=l, ... ,p,
(4)
where the last set of inequalities (At::;; Aj-) forces the value functions to be steeper for losses than for gains. If max e > 0, the DM's preferences are consistent with the theory presented. Otherwise they are not.
is added to the linear program, where e is a small positive constant. If k is the preferred alternative, an inequality LAj(Xkj -X hj ) 2 e
(B)
j
is written. Then a linear programming problem is formulated, where e is maximized over the above
149
P. Salminen / Solving the discrete multiple criteria problem
constraints «A) and (B». The maximum of E ~ 0, if the comparisons are based on linear value function. The idea is to test whether sorne unexamined alternative x * can be preferred to the currently best alternative. If this is not possible with multipli ers À which are consistent with previous answers (constraints), the alternative x * can be dropped. The method continues by considering every efficient alternative h adjacent to k, that is, alternatives h for which LÀ / x hj - X kj) can be positive without violating any of the previously generated inequalities on the weights Ài. If there are no such alternatives, the oldest responses are deleted until h is found. If aIl the alternatives have aIready been dropped, the procedure stops. 2.5.2. The canvex cane method of Korhonen, Wallenius and Zionts In the method developed by Korhonen et al. [5], the ide a is fundamentaIly similar to that of the Zionts method [9]. However, they relax the form of the DM's value function aIlowing it to be quasiconcave (= convex indifference curve). This has the effect of not permitting the use of the preferred alternative h in constraints (A) and (B) since it also eliminates alternatives which are convex dominated by h and k Ca convex dominated alternative may be the best alternative). Instead, the constraints that eliminate alternatives must be generated by using the less preferred alternatives. If the unexamined alternative x * cannot be preferred to k, th en neither can it be preferred to h (h is preferred to k). The method initially chooses an arbitrary set of positive multipliers by means of which it generates a composite linear value function. An alternative that maximizes the value function is chosen. The DM chooses between this alternative and the most preferred alternative found so far. Whichever is preferred is denoted as xc. Subsequently every efficient alternative x h adjacent to Xc is considered; that is, for which LjÀj(Xhj - cc) can be positive with aIl other sums LjÀ/Xij - xc) ~ O. If there are no adjacent efficient solutions, the procedure stops. Otherwise the DM compares solution Xc with an adjacent efficient solution x h . If he prefers Xh' a constraint LjÀ/Xhjxc) ~ E is generated, whereas if he prefers xc, a constraint LjÀ/Xcj - Xh) ~ s results. On the basis of responses by the DM to questions during an
iteration where the least preferred solution can be uniquely determined, one or more cones are generated. The elimination of alternatives is performed as foIlows: x* can be eliminated if E ~ 0 and f(y)
s
s.t.
lLi(Y-X)-E~X*-y,
lLi~O,
for aIl (x, y)sP. Rather than solving a smaIl linear programming problem for each x *, Korhonen et al. use a modified efficiency pro gram for quickly identifying efficient vectors Csee [5]). A new set of consistent multipliers are then generated which are consistent with the generated inequalities. An alternative that maximizes this linear value function is then identified and the DM chooses between the resulting alternative and the most preferred alternative so far. The preferred alternative will be denoted as xc' and the process continues until m - 1 alternatives have been eliminated. Obviously, the assumption of quasiconcave value functions is more realistic th an linear value functions. However, the quasiconcave procedure is not as efficient in eliminating inferior alternatives as the linear procedure. 2.5.3. The suggested method The method proposed in this paper is a blend of methods described above. In addition, we assume prospect theory type of value functions for the DM. We illustrate the method with a simple two-criteria example. In Figure 2a we consider the situation where x r is the real reference alternative (x r is preferred to x J By assumed piecewise linearity (i.e. linear indifference curve starting at x r ), it foIlows that aIl alternatives on or dominated by alternatives along the line segment starting at x r are not preferred to xr. Figure 2b illustrates the case where x r is not the real reference point: again x r is preferred to Xi. The assumed piecewise linear difference functions are concave; the sum of two or more concave functions is also concave. By (quasi)concavity it follows that aIl alternatives on or dominated by alternatives along the line segment starting at Xi are not preferred to Xi' and thus not preferred to x r (Korhonen et al. [5]). Because of the use of
P. Salminen / Solving the discrete multiple criteria problem
150
and the previous x r to Xi. If max E ~ 0 for sorne x *, it can be dropped from further analysis; it cannot be preferred to x r • If the current alterna-
x.
1.
a)
b)
x
... r
x
x;,
r
Figure 2. Sorne illustrations
nondominated adjacent alternatives, there exist no real alternatives that would be convex dominated by X r and Xi. Therefore the cases above are equivalent in eliminating dominated alternatives. Linear mathematics is used, although in the latter case (x r is not the real reference alternative), in fact, alternatives are eliminated based on (quasi)concavity. Notice that when we assume piecewise linear value functions, the best alternative can be convex dominated. In this case the real reference point is also convex dominated. With the postulated assumptions, an alternative Xi which is equal to cannot be eliminated in the process. We consider two different cases: 1) the DM's real reference point (x;) equals the currently best alternative, 2) do es not equal the currently best alternative; xr is not preferred to x;, which is assumed to be stable; X r converges towards We assume that in both cases the preferences over the x/s (the current alternatives) form a transitive system. Case 1: x; = the current alternative. Gains and losses can be directly assessed, x r is assumed to be known. An unexamined alternative x * can be eliminated from further evaluation if max E ~ 0 in the foHowing linear program:
x;
x;
x;.
Max s.t.
E
;\+(x*-x r)+ +;\-(x*-Xr)-~E, ;\+(xr-xJ+ +;\-(xr-XJ-~E, ;\+~;\-,
";\++";\:-=1 k..] k..] ,
tive x r is preferred to the initial real reference point we assume that the current alternative forms the new real reference point. Case 2: x r is nat equal ta x;. We assume that gains and losses are assessed with respect to the unknown x r'. When x r is preferred to x·P and x'r is preferred to x" the foUowing constraint is obtained:
(5)
where x r is preferred to Xi and Xi consists of aH alternatives considered not preferred to x r" That is, we consider every preferred alternative (one by one) to be the reference point. If Xi in sorne comparison is preferred to x" we switch Xi to x r
;\+(xr-x;)+ +;\-(xr-x;)-;\+(xi-x;)- -;\-(xi-xir~o,
(6)
which is concave when ;\ + ~ ;\ -, since in that case the single difference functions are concave. Because Xi is a nondominated adjacent alternative to x r' no real alternatives can be convex dominated by x r and Xi. Therefore the linear program (5) can be applied also in this situation to define the status of the x * 's. However, while in the first case it is possible to define exactly the region where x * cannot be preferred to x" in the latter case we ob tain only a subregion of it (Figure 3). In Figure 3, the method eliminates alternatives located in the shaded regions. The dotted lines describe possible real (unknown) indifference curves. Notice that when x r is convex nondominated, ;\ + = ;\ -, which results in a linear value function. We would usuaUy assume that we do not know in advance which of the above cases occurs. Therefore, although we do not need nondominated adjacent alternatives in case 1, we recommend them to be used in this procedure. The use of old preference information deserves a comment. We may use the foUowing earlier preference relations: (a) the current x r is preferred to aH Xi less preferred so far (in case 1 and 2), or (b) aU gains and losses among old comparisons are arranged to the linear program (5) in the spirit of constraint (6) with respect to the current reference alternative (in case 1). In the latter arrangement the relations between less preferred alternatives are also taken into account. This may easily cause the model to become inconsistent in case 2; we need to check its consistency. If max E ~ 0 in the linear program (5), with x * dominating the current x the com-
"
P. Salminen / Solving the discrete multiple criteria problem
151
\
\
al
bl
\
,
Figure 3. The difference between known and unknown reference point in eliminating dominated alternatives. (a) x, is the reaI reference point x;, x, is preferred to Xi and Xk' A+ < A- (b) x; is preferred to x" x, is preferred to Xi and Xk' A+ < A-
parisons are not consistent with assumed behavior. In this case we suggest to delete the oldest responses until consistency is obtained.
and continue the process. Keep the earlier preference relations in the linear program (e.g., x, is preferred to aIl x) and check the consistency of the model.
3. A step-by-step statement of the method 4. An example Step 1. Choose an arbitrary set of weights wj > o for the criteria, j = 1, ... ,p (for convenience, set LWj = 1). These weights are used only in Step 1. Later we use different multipliers for gains and losses. Step 2. Choose the decision alternative that maximizes LWjX ij considering every alternative. CalI this alternative the first reference point (x,). Step 3. Consider the nondominated decision alternatives x i adjacent to the current x, one by one. (a) For Xi not preferred to x" write an inequality A+(x,-x i )+ +A-(x,-xi)-;:::s,
and solve the linear program (5) separately for each unexamined alternative. Drop alternatives for which max s ::; 0 Gnferior ones). (b) For an adjacent nondominated decision alternative Xi preferred to x" move to Xi' and switch Xi with X,. Write an inequality A+(x,-xJ+ +A-(x,-xJ-;:::s.
Substitute this new X, for the old x, in constraints generated in (a) (if they exist) and solve the linear program (5) separately for each unexamined alternative. Drop inferior alternatives. Step 4. If no adjacent nondominated decision alternatives remain, x, is the best alternative; stop. If more than one unexamined adjacent nondominated alternative remain, return to Step 3,
Let us consider a two-criteria problem (both criteria to be maximized). Due to the simplicity of the example, we do not need old preference information. Therefore there is no operational difference between cases 1 and 2. The set of available decision alternatives is: A =(4, 4), B = (3,4.2) C = (2,5.5), D = (5,3.7), E = (5.5,3), F = (6,2). For a starting point (with equal weights for Xl and x 2 ) we obtain D (the first x,). The nondominated adjacent alternatives are A and E. Let us assume that A is preferred to D (A is the second x r ). From this we obtain the folIowing linear program: Max
s
S.t.
Ai(x* -A)+ +Al(x* -A)+Ai(x*-A)+ +Az(x*-A)-;:::s, Ai(A-D)+ +A 1 (A-D)+Ai(A-D)+ +Az(A-D)-;:::s,
I>t + I>; =
1,
P. Salminen / Solving the discrete multiple criteria problem
152
where
X
* represents the unexamined alternatives
B, C, D, E, and F. These alternatives are tested one by one. If max e ~ 0 for an unexamined
c
alternative, if cannot be preferred to A. For the unexamined alternative Ethe linear program is: Max
e
S.l.
A:(5.5 - 4) + AI(O) + A;(O) +Ai(3 - 4)
~
•
F
e, Figure 4. The elimination of E and F
~e,
this case, the best alternative (A) was found after 3 pairwise comparisons.
A: -Al ~ 0, A; -Ai
~
0,
A: +AI +A; +Ai
=
1,
A ~O. Clearly max e cannot be positive when At ~ Ai(max e = -0.072, with A: = Al = 0.171 and A; = A2 = 0.329). Alternative E can be dropped from further analysis. Similarly, alternative F can be dropped as weIl. Only the status of Band C remains unknown, because for the se alternatives max e is positive. Alternative D is dropped because A is preferred to D. This is illustrated in Figure 4. The only nondominated adjacent alternative to A (the currently best alternative) is B. Let us assume that A is preferred to B (A is still x). The linear program for C is (in a two-criteria problem we do not need earlier preference relations: A is preferred to D, E, and F): Max S.t.
A
~
A:(O) +AI(4-5) +A;(4-3.7) +Ai(O)
B
•
e A:(O) + AI (2 - 4) + A;(5.5 - 4) +Ai(O) ~ e,
A:(4-3) +AI(O) +A;(O) +Ai( 4 - 4.5) A: -Al
~
0,
A; -Ai
~
0,
~
e,
A: +AI +A; +Ai = 1, A ~O. In this case max e is positive (max e = 0.05, with A: = Al = 0.2 and A; = Ai = 0.3) and C cannot be dropped. The final comparison is made between A and C. Suppose A is preferred to C. In
5. Discussion In the proposed method we use piecewise linear value functions and nondominated adjacent alternatives. Compared to Zionts' method, the efficiency of the method does not decrease through these modifications, but the problems of the linearity assumption are avoided. For exampIe, Zionts [9] discusses the problem of ordinal variables. In that case, depending upon the scales used, the same alternative may be either convex dominated or not. In our method, the convex dominated alternatives would be dropped from further evaluation without the use of nondominated adjacent alternatives. Linear value functions can be incorporated in the framework of linear prospect theory. If the difference functions are linear for gains and losses, and A+ = A-, linear value functions result. This can be explained as follows using the reference point: If the real reference point domina tes aH decision alternatives, or if the reference point is dominated by the decision alternatives, only losses or grains, respectively, are obtainable. In these cases we only observe comparisons based on a part of the difference functions (either los ses or grains, but not both). If the difference functions are S-shaped, the method is not generaHy valid. The method works in theory, when the sum of the difference functions is quasiconcave. This is true, for example, when gains and losses are treated similarly with for gains and respect to each criterion (e.g. for losses, where a > 1).
-ara
ra
P. Salminen / Solving the discrete multiple criteria problem
We have assumed a stable unknown reference point (until xr is preferred to x;, then the preferred alternative forms the new reference point), or a reference point that equals the currently best alternative. If the unknown reference point changes during comparisons to the extent that preferences for the current reference points, x" do not form a transitive system wh en they conthen verge towards the real reference point, we are not able to use aIl old preference relations. Only the current x r is preferred to the current Xi. However, we need to test the consistency of the performed comparisons to find out wh ether the assumed choice behavior is consistent with respect to one reference point. If in the linear program (5) max E cannot be positive with x * dominating x" the behavioral assumptions are no longer valid. In this case we suggest to delete oldest constraints until the consistency of the model is obtained.
x;,
6. Conclusion In this paper we have developed a method for solving discrete multiple criteria problems by blending ideas from several sources. The method is based on the use of pairwise judgments of the DM to identify his/her best decision alternative. It is assumed that the DM's choice behavior can be described with linear prospect theory. An essenti al feature of the method is the use of the DM's pairwise judgments to generate cones to eliminate inferior solutions. The best solution so far is used (technically) as the reference alternative. The method uses adjacent nondominated alternatives, against which the current reference alternative is compared. The preferences need not to be linear; the method works when the preferences result in convex indifference curves (quasiconcave value function). Based on earlier studies of the methods that use pairwise preference information, we can draw conclusions regarding the power of this method. With a small number of criteria, the number of judgments required will be relatively smaIl. If the number of criteria exceeds four, the efficiency of the method is likely to decrease rapidly [5,7]. The method improves on the method of Zionts [9] by relaxing the behavioral assumptions. Through the
153
use of linear mathematics the method is as powerful in eliminating dominated alternatives as Zionts' method [9] (discounting the cost of additional se arch for nondominated adjacent alternatives). In the suggested method the cone of dominated solutions is formed in the same spirit as in Korhonen et al. [5], but through the use of nondominated adjacent solutions, the domination cone starts always at least as close to the preferred alternative as is the case in their method, or closer to it. Therefore the elimination procedure will be more effective. GeneraIly, the algorithm can be used to solve any discrete problem, where the DM's underlying value function is quasiconcave. The reason is that we use a concave implicit value function. A concave function is also quasiconcave and we base our elimination of alternatives on quasiconcavity. With nonlinear piecewise difference functions (e.g., S-shaped), the resulting preferences need not be convex; therefore the procedure is not valid under such circumstances.
Appendix. A step-by-step statement of Zionts' discrete method [9] Step 1. Choose an arbitrary set of weights wj > 0, for the criteria j = 1, ... , p (p = number of criteria). Step 2. Choose the decision that maximizes Ewjx ij considering every decision i. CalI this the new maximizing decision. Step 3. (This step is to be omitted the first time.) Ask if the new maximizing decision is preferred to the old one. If yes, designate as solution k the new maximizing decision. If no or 1 don't know, designate as solution k a solution preferred to the old solution k. Generate a constraint (as in Step 5) based on the response. Step 4. Consider every efficient decision h adjacent to k, that is, for which EÀj (x hj - Xk) can be positive with aIl other sums E\.(x ij - x k ) ~ 0, Àj ~ 0, j = 1, ... , p, and consistent with previous responses (i.e., consistent with aIl previously generated inequalities on the weights À). If there are no such decisions, go to Step 10. Step 5. For each decision h found in Step 4, ask: "Which do you prefer, h or k? .
P. Salminen / Solving the discrete multiple criteria problem
154
(a) If h is preferred, add a constraint LÀj( X hj - Xkj)
~e
j
(e
> 0 and small). (b) If k is preferred, add a constraint
LÀj(Xkj-Xhj)
(e
~e
> 0 and small).
(c) For answers of l don't know, add no constraints. Step 6. If no h is preferred to k in Step 5, go to Step 10. Step 7. Find a set of weights satisfying aIl previously generated contraints and Àj ~ e. The purpose of e is to assure a strict inequality À j > O. Step 8. If no such solution exists, delete the oldest generated constraints and go to Step 7 above. Otherwise go to Step 9. Step 9. Let wj = Àj' j = 1, ... , p, found in Step 7 and go to Step 2. Step 10. Delete the oldest responses generated in Steps 3 or 5 and go to Step 4. If aIl responses have been dropped, rank order the alternatives and stop.
References [1) Kahneman, D., and Tversky, A, "Prospect theory: An analysis of decisions under risk", Econometrica 47 (1979) 262-291.
[2) Korhonen, P., Moskowitz, H., and Wallenius, J. (1986), "A progressive algorithm for modelling and solving multiple criteria decision problems", Operations Research 34 (1986) 726-731. [3) Korhonen, P., Moskowitz, H., and Wallenius, J., "Choice behavior in interactive multiple criteria decision making", Annals of Operations Research 23 (1990) 161-179. [4) Korhonen, P., and Wallenius, J., "Malli usean paatoksentekijan diskreetin neuvotteluongelman ratkaisemiseksi", in: C. Carlsson (ed.), Proceedings of Management Science in Finland, (MASC), Stiftelsen for Abo Akademi, 1980. [5) Korhonen, P., Wallenius, J., and Zionts, S., "Solving the discrete multiple criteria problem using convex cones", Management Science 30 (1984) 1336-1345. [6) Salminen, P., "Validating Kahneman-Tversky's prospect theory in a multiple criteria decision-making context", unpublished manuscript, 1990. (7) Salminen, P., Korhonen, P., and Wallenius, J., "Testing the form of a decision-maker's multiattribute value function based on pairwise preference information", Journal of the Operational Research Society 40 (1989) 299-302. [8) Tversky, A, "Intransitivity of preferences", Psychological Review 76 (1969) 31-45. [9) Zionts, S., "A multiple criteria method for choosing among discrete alternatives", European Journal of Operational Research 7 (1981) 143-147. [10) Zionts, S., and Wallenius, J., "An interactive multiple objective linear programming method for a class of underlying nonlinear utility functions", Management Science 29 (1983) 519-529.
European Journal of Operation al Research 72 (1994) 146-154 North-Rolland
Theory and Methodology
Solving the discrete multiple criteria problem using linear prospect theory Pekka Salminen Department of Economics and Management, University of lyviiskylii, PL 35, 40351 lyviiskylii, Finland Received October 1991; revised January 1992
Abstract: Prospect theory developed by Kahneman and Tversky is a popular model of choice in decision problems under uncertainty. Prospect the ory has recently been extended to multiple criteria choice problems. In this paper, an interactive method for solving discrete multiple criteria decision problems, based on prospect theory type value functions, has been developed. Piecewise linear marginal value functions are assumed to approximate the S-shaped value functions of prospect the ory. Therefore, the proposed procedure is valid only for convex preferences. Keywords: Multiple criteria; Decision theory
1. Introduction
Tversky [8] has developed a choice the ory, in which alternatives are evaluated based on comparisons of criterion-wise differences between decision alternatives (the additive difference mode!). Later Kahneman and Tversky [1] developed prospect the ory, where outcomes are expressed as positive or negative deviations (gains or losses) from a reference alternative. Although value functions differ among individu ais, Kahneman and Tversky propose that they are commonly S-shaped, concave ab ove the reference point and convex below it. Furthermore, value functions are commonly assumed steeper for losses th an for gains. Prospect theory was originally developed for single criterion problems, but the ideas have been extended to multiple criteria decision probCorrespondence to: Dr. P. Salminen, Department of Economies and Management, University of Jyvaskyla, PL 35, 40351 Jyvaskyla, Finland.
lems as weIl (Korhonen, Moskowitz and WaIlenius [3]). In this chapter we propose an interactive method for solving discrete deterministic multiple criteria de ci sion problems, assuming prospect theory type of value functions for the decisionmakers. The method builds on existing work, namely Zionts [9], Korhonen, Wallenius and Zionts [5], and Korhonen, Moskowitz and WaIlenius [2]. The proposed method makes use of pairwise comparisons of attainable decision alternatives to identify the best alternative. We assume m deterministic decision alternatives, p criteria, and one decision-maker, whose choice behavior is modeled using (Iinear) prospect theory (Kahneman and Tversky [1]). We further assume that aIl alternatives can be compared. At most m - 1 pairwise questions are required to identify the best alternative. For most problems, considerably fewer comparisons are required. The method initially chooses an arbitrary set of positive multipliers (or equal weights) for the
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P. Salminen / Soluing the discrete multiple criteria problem
criteria, and generates a composite linear value function using these multipliers. Then the alternative that maximizes this function is found. Next, one or more adjacent nondominated alternatives are identified and presented to the DM for his/her evaluation. The DM is asked to choose between the current alternative and the adjacent nondominated alternatives. Based on the DM's answer(s), inequalities are generated and used to eliminate alternatives which will not be preferred to the current alternative. The procedure employs a modification of the simplex method of linear programming, in which the criterion values are divided into gains (multiplier À +) and losses (multiplier À -), with respect to a reference alternative. As prospect theory suggests, losses carry more weigth th an gains (À +.:s; À -). Once an adjacent nondominated alternative is found that is preferred to the current alternative, we select it to be the current reference point. The process is then repeated by identifying adjacent nondominated alternatives to the currently best alternative, asking new questions of the DM, and so on. Convergence to an overall best alternative is assured, given that the DM's choice behavior is consistent with the underlying theory. Alternatives less preferred to the current reference alternative are dropped. The problem of eliminating alternatives based on DM's implied preferences (convex cones) plays an importnat role in our procedure. This problem was originally considered by Zionts [9], Zionts and WaIlenius [10], and Korhonen, WaIlenius and Zionts [5]. In Zionts and WaIlenius [10], a method for solving the continuous problem is presented, and in Zionts [9] (see the Appendix) the discrete problem with a linear underlying value function is considered. In Korhonen et al. [5] a procedure is developed for solving discrete problems in which the underlying value function is assumed to be quasi concave. In this chapter we develop a method for solving discrete problems in which the DM's choice behavior is based on differences with respect to a reference point. We use piecewise linear difference functions. The use of adjacent nondominated alternatives relaxes the form of the value function aIlowing them to be quasiconcave. In our case, the cone of dominated alternatives st arts from the more preferred point, instead of the less preferred point as in Korhonen et al. [5]. This increases the power of the
147
elimination procedure. However, from the point of prospect theory, only piecewise linear difference functions are considered; generaIly S-shaped difference functions need not result in convex preferences.
2. Sorne theory 2.1. Prohlem description
We assume a single decision-maker, a set {Xl' x 2 ,···, xml of m deterministic decision alternatives, and p criteria, which define an m X p decision matrix X whose elements are denoted by x ij ' i El = {l, 2, ... , m} and j E J =
{l, 2, ... , pl. Also, we assume that the DM's preferences can be modeIled using piecewise linear (implicit) difference functions for each criterion. The differences are measured from a DM's subjective reference point (unknown), which may be his/her current alternative or vector of aspiration levels for each criterion. 2.2. Nondominated adjacent alternatives
Our algorithm is based on comparing nondominated adjacent alternatives against the current alternative. A nondominated adjacent alternative is defined as foIlows (Korhonen and WaIlenius [4]): consider the current alternative x r and an alternative Xi; Xi is a nondominated adjacent alternative for x" if 1) Xik>X rk for sorne k= 1, 2, ... ,p; 2) there is no alternative Xl with 1 =F i,r such that
for aIl j
=
1, 2, ... , p.
(1)
In Figure 1 we illustrate this with a simple twocriteria case. Alternative Xi is a nondominated adjacent alternative for x" if there exists no real decision alternatives in the shaded region. 2.3. Consistency with linear prospect theory
FoIlowing Korhonen et al. [3], a linear programming formulation can be used to test whether the subjects' choices are consistent with linear prospect theory. By assuming piecewise linear
P. Salminen / Solving the discrete multiple criteria problem
148
2.4. Reference point
x;
X.
1
x
Figure 1. Alternative
Xi
Here we assume that the current alternative or the vecor of the DM's aspiration levels are possirefers ble reference alternatives. The notation to the DM's real reference alternative. The notation X r is used to indicate the currently best alternative. When the current alternative is the real reference alternative, x; = xr' In the case where x r is not the real reference alternative, we assume that X r converges to
r
is a nondominated adjacent alternative for x r
x;.
2.5. Convex cones of dominated alternatives
value functions, the following inequality is generated for each choice Xi' i El, preferred to the reference point X r :
(2) and for the reference point X r preferred to the available choice Xi' an inequality of the following type is generated:
(3) where e is a scalar variable, and vectors z are
zi
t
and
if Xij -X rj 2 0, if Xij -X rj < 0, if Xij -X rj < 0, if Xij -X rj 2 0, and where A+ is a vector of weights corresponding to the gains and A- is a vector corresponding to the losses with respect to the reference point xr' To estimate A, the following linear program is solved: Max S.t.
e
+ LAj-
Aj 2At 20,
2.5.1. Ziont's discrete method Ziont's discrete method [9] initially selects an arbitrary set of positive multipliers for the criteria, and generates a composite linear value functian these multipliers. Next, one or more adjacent efficient alternatives are presented to the DM for his/her evaluation. Based on the DM's answers, inequalities are generated and used to eliminate alternatives which will not be preferred to the current alternative. This is done using linear programming. Elimination in this method is based on linear value functions. If alternative h is preferred to alternative k, a constraint LAj(Xhj -X kj ) 2 e
(2), (3), and LAt
An essential feature of the method described in this chapter is the use of the DM's pairwise judgments to generate regions in which preferred decision alternatives cannot be located. Next, we briefly de scribe two decision models, which closely relate to the method suggested in this paper. They are the Zionts' discrete method [9], in which a linear value function is assumed, and the convex cone method of Korhonen, Wallenius and Zionts [5], in which the form of the DM's value function is relaxed to be quasiconcave.
(A)
j
=
1,
j=l, ... ,p,
(4)
where the last set of inequalities (At::;; Aj-) forces the value functions to be steeper for losses than for gains. If max e > 0, the DM's preferences are consistent with the theory presented. Otherwise they are not.
is added to the linear program, where e is a small positive constant. If k is the preferred alternative, an inequality LAj(Xkj -X hj ) 2 e
(B)
j
is written. Then a linear programming problem is formulated, where e is maximized over the above
149
P. Salminen / Solving the discrete multiple criteria problem
constraints «A) and (B». The maximum of E ~ 0, if the comparisons are based on linear value function. The idea is to test whether sorne unexamined alternative x * can be preferred to the currently best alternative. If this is not possible with multipli ers À which are consistent with previous answers (constraints), the alternative x * can be dropped. The method continues by considering every efficient alternative h adjacent to k, that is, alternatives h for which LÀ / x hj - X kj) can be positive without violating any of the previously generated inequalities on the weights Ài. If there are no such alternatives, the oldest responses are deleted until h is found. If aIl the alternatives have aIready been dropped, the procedure stops. 2.5.2. The canvex cane method of Korhonen, Wallenius and Zionts In the method developed by Korhonen et al. [5], the ide a is fundamentaIly similar to that of the Zionts method [9]. However, they relax the form of the DM's value function aIlowing it to be quasiconcave (= convex indifference curve). This has the effect of not permitting the use of the preferred alternative h in constraints (A) and (B) since it also eliminates alternatives which are convex dominated by h and k Ca convex dominated alternative may be the best alternative). Instead, the constraints that eliminate alternatives must be generated by using the less preferred alternatives. If the unexamined alternative x * cannot be preferred to k, th en neither can it be preferred to h (h is preferred to k). The method initially chooses an arbitrary set of positive multipliers by means of which it generates a composite linear value function. An alternative that maximizes the value function is chosen. The DM chooses between this alternative and the most preferred alternative found so far. Whichever is preferred is denoted as xc. Subsequently every efficient alternative x h adjacent to Xc is considered; that is, for which LjÀj(Xhj - cc) can be positive with aIl other sums LjÀ/Xij - xc) ~ O. If there are no adjacent efficient solutions, the procedure stops. Otherwise the DM compares solution Xc with an adjacent efficient solution x h . If he prefers Xh' a constraint LjÀ/Xhjxc) ~ E is generated, whereas if he prefers xc, a constraint LjÀ/Xcj - Xh) ~ s results. On the basis of responses by the DM to questions during an
iteration where the least preferred solution can be uniquely determined, one or more cones are generated. The elimination of alternatives is performed as foIlows: x* can be eliminated if E ~ 0 and f(y)
s
s.t.
lLi(Y-X)-E~X*-y,
lLi~O,
for aIl (x, y)sP. Rather than solving a smaIl linear programming problem for each x *, Korhonen et al. use a modified efficiency pro gram for quickly identifying efficient vectors Csee [5]). A new set of consistent multipliers are then generated which are consistent with the generated inequalities. An alternative that maximizes this linear value function is then identified and the DM chooses between the resulting alternative and the most preferred alternative so far. The preferred alternative will be denoted as xc' and the process continues until m - 1 alternatives have been eliminated. Obviously, the assumption of quasiconcave value functions is more realistic th an linear value functions. However, the quasiconcave procedure is not as efficient in eliminating inferior alternatives as the linear procedure. 2.5.3. The suggested method The method proposed in this paper is a blend of methods described above. In addition, we assume prospect theory type of value functions for the DM. We illustrate the method with a simple two-criteria example. In Figure 2a we consider the situation where x r is the real reference alternative (x r is preferred to x J By assumed piecewise linearity (i.e. linear indifference curve starting at x r ), it foIlows that aIl alternatives on or dominated by alternatives along the line segment starting at x r are not preferred to xr. Figure 2b illustrates the case where x r is not the real reference point: again x r is preferred to Xi. The assumed piecewise linear difference functions are concave; the sum of two or more concave functions is also concave. By (quasi)concavity it follows that aIl alternatives on or dominated by alternatives along the line segment starting at Xi are not preferred to Xi' and thus not preferred to x r (Korhonen et al. [5]). Because of the use of
P. Salminen / Solving the discrete multiple criteria problem
150
and the previous x r to Xi. If max E ~ 0 for sorne x *, it can be dropped from further analysis; it cannot be preferred to x r • If the current alterna-
x.
1.
a)
b)
x
... r
x
x;,
r
Figure 2. Sorne illustrations
nondominated adjacent alternatives, there exist no real alternatives that would be convex dominated by X r and Xi. Therefore the cases above are equivalent in eliminating dominated alternatives. Linear mathematics is used, although in the latter case (x r is not the real reference alternative), in fact, alternatives are eliminated based on (quasi)concavity. Notice that when we assume piecewise linear value functions, the best alternative can be convex dominated. In this case the real reference point is also convex dominated. With the postulated assumptions, an alternative Xi which is equal to cannot be eliminated in the process. We consider two different cases: 1) the DM's real reference point (x;) equals the currently best alternative, 2) do es not equal the currently best alternative; xr is not preferred to x;, which is assumed to be stable; X r converges towards We assume that in both cases the preferences over the x/s (the current alternatives) form a transitive system. Case 1: x; = the current alternative. Gains and losses can be directly assessed, x r is assumed to be known. An unexamined alternative x * can be eliminated from further evaluation if max E ~ 0 in the foHowing linear program:
x;
x;
x;.
Max s.t.
E
;\+(x*-x r)+ +;\-(x*-Xr)-~E, ;\+(xr-xJ+ +;\-(xr-XJ-~E, ;\+~;\-,
";\++";\:-=1 k..] k..] ,
tive x r is preferred to the initial real reference point we assume that the current alternative forms the new real reference point. Case 2: x r is nat equal ta x;. We assume that gains and losses are assessed with respect to the unknown x r'. When x r is preferred to x·P and x'r is preferred to x" the foUowing constraint is obtained:
(5)
where x r is preferred to Xi and Xi consists of aH alternatives considered not preferred to x r" That is, we consider every preferred alternative (one by one) to be the reference point. If Xi in sorne comparison is preferred to x" we switch Xi to x r
;\+(xr-x;)+ +;\-(xr-x;)-;\+(xi-x;)- -;\-(xi-xir~o,
(6)
which is concave when ;\ + ~ ;\ -, since in that case the single difference functions are concave. Because Xi is a nondominated adjacent alternative to x r' no real alternatives can be convex dominated by x r and Xi. Therefore the linear program (5) can be applied also in this situation to define the status of the x * 's. However, while in the first case it is possible to define exactly the region where x * cannot be preferred to x" in the latter case we ob tain only a subregion of it (Figure 3). In Figure 3, the method eliminates alternatives located in the shaded regions. The dotted lines describe possible real (unknown) indifference curves. Notice that when x r is convex nondominated, ;\ + = ;\ -, which results in a linear value function. We would usuaUy assume that we do not know in advance which of the above cases occurs. Therefore, although we do not need nondominated adjacent alternatives in case 1, we recommend them to be used in this procedure. The use of old preference information deserves a comment. We may use the foUowing earlier preference relations: (a) the current x r is preferred to aH Xi less preferred so far (in case 1 and 2), or (b) aU gains and losses among old comparisons are arranged to the linear program (5) in the spirit of constraint (6) with respect to the current reference alternative (in case 1). In the latter arrangement the relations between less preferred alternatives are also taken into account. This may easily cause the model to become inconsistent in case 2; we need to check its consistency. If max E ~ 0 in the linear program (5), with x * dominating the current x the com-
"
P. Salminen / Solving the discrete multiple criteria problem
151
\
\
al
bl
\
,
Figure 3. The difference between known and unknown reference point in eliminating dominated alternatives. (a) x, is the reaI reference point x;, x, is preferred to Xi and Xk' A+ < A- (b) x; is preferred to x" x, is preferred to Xi and Xk' A+ < A-
parisons are not consistent with assumed behavior. In this case we suggest to delete the oldest responses until consistency is obtained.
and continue the process. Keep the earlier preference relations in the linear program (e.g., x, is preferred to aIl x) and check the consistency of the model.
3. A step-by-step statement of the method 4. An example Step 1. Choose an arbitrary set of weights wj > o for the criteria, j = 1, ... ,p (for convenience, set LWj = 1). These weights are used only in Step 1. Later we use different multipliers for gains and losses. Step 2. Choose the decision alternative that maximizes LWjX ij considering every alternative. CalI this alternative the first reference point (x,). Step 3. Consider the nondominated decision alternatives x i adjacent to the current x, one by one. (a) For Xi not preferred to x" write an inequality A+(x,-x i )+ +A-(x,-xi)-;:::s,
and solve the linear program (5) separately for each unexamined alternative. Drop alternatives for which max s ::; 0 Gnferior ones). (b) For an adjacent nondominated decision alternative Xi preferred to x" move to Xi' and switch Xi with X,. Write an inequality A+(x,-xJ+ +A-(x,-xJ-;:::s.
Substitute this new X, for the old x, in constraints generated in (a) (if they exist) and solve the linear program (5) separately for each unexamined alternative. Drop inferior alternatives. Step 4. If no adjacent nondominated decision alternatives remain, x, is the best alternative; stop. If more than one unexamined adjacent nondominated alternative remain, return to Step 3,
Let us consider a two-criteria problem (both criteria to be maximized). Due to the simplicity of the example, we do not need old preference information. Therefore there is no operational difference between cases 1 and 2. The set of available decision alternatives is: A =(4, 4), B = (3,4.2) C = (2,5.5), D = (5,3.7), E = (5.5,3), F = (6,2). For a starting point (with equal weights for Xl and x 2 ) we obtain D (the first x,). The nondominated adjacent alternatives are A and E. Let us assume that A is preferred to D (A is the second x r ). From this we obtain the folIowing linear program: Max
s
S.t.
Ai(x* -A)+ +Al(x* -A)+Ai(x*-A)+ +Az(x*-A)-;:::s, Ai(A-D)+ +A 1 (A-D)+Ai(A-D)+ +Az(A-D)-;:::s,
I>t + I>; =
1,
P. Salminen / Solving the discrete multiple criteria problem
152
where
X
* represents the unexamined alternatives
B, C, D, E, and F. These alternatives are tested one by one. If max e ~ 0 for an unexamined
c
alternative, if cannot be preferred to A. For the unexamined alternative Ethe linear program is: Max
e
S.l.
A:(5.5 - 4) + AI(O) + A;(O) +Ai(3 - 4)
~
•
F
e, Figure 4. The elimination of E and F
~e,
this case, the best alternative (A) was found after 3 pairwise comparisons.
A: -Al ~ 0, A; -Ai
~
0,
A: +AI +A; +Ai
=
1,
A ~O. Clearly max e cannot be positive when At ~ Ai(max e = -0.072, with A: = Al = 0.171 and A; = A2 = 0.329). Alternative E can be dropped from further analysis. Similarly, alternative F can be dropped as weIl. Only the status of Band C remains unknown, because for the se alternatives max e is positive. Alternative D is dropped because A is preferred to D. This is illustrated in Figure 4. The only nondominated adjacent alternative to A (the currently best alternative) is B. Let us assume that A is preferred to B (A is still x). The linear program for C is (in a two-criteria problem we do not need earlier preference relations: A is preferred to D, E, and F): Max S.t.
A
~
A:(O) +AI(4-5) +A;(4-3.7) +Ai(O)
B
•
e A:(O) + AI (2 - 4) + A;(5.5 - 4) +Ai(O) ~ e,
A:(4-3) +AI(O) +A;(O) +Ai( 4 - 4.5) A: -Al
~
0,
A; -Ai
~
0,
~
e,
A: +AI +A; +Ai = 1, A ~O. In this case max e is positive (max e = 0.05, with A: = Al = 0.2 and A; = Ai = 0.3) and C cannot be dropped. The final comparison is made between A and C. Suppose A is preferred to C. In
5. Discussion In the proposed method we use piecewise linear value functions and nondominated adjacent alternatives. Compared to Zionts' method, the efficiency of the method does not decrease through these modifications, but the problems of the linearity assumption are avoided. For exampIe, Zionts [9] discusses the problem of ordinal variables. In that case, depending upon the scales used, the same alternative may be either convex dominated or not. In our method, the convex dominated alternatives would be dropped from further evaluation without the use of nondominated adjacent alternatives. Linear value functions can be incorporated in the framework of linear prospect theory. If the difference functions are linear for gains and losses, and A+ = A-, linear value functions result. This can be explained as follows using the reference point: If the real reference point domina tes aH decision alternatives, or if the reference point is dominated by the decision alternatives, only losses or grains, respectively, are obtainable. In these cases we only observe comparisons based on a part of the difference functions (either los ses or grains, but not both). If the difference functions are S-shaped, the method is not generaHy valid. The method works in theory, when the sum of the difference functions is quasiconcave. This is true, for example, when gains and losses are treated similarly with for gains and respect to each criterion (e.g. for losses, where a > 1).
-ara
ra
P. Salminen / Solving the discrete multiple criteria problem
We have assumed a stable unknown reference point (until xr is preferred to x;, then the preferred alternative forms the new reference point), or a reference point that equals the currently best alternative. If the unknown reference point changes during comparisons to the extent that preferences for the current reference points, x" do not form a transitive system wh en they conthen verge towards the real reference point, we are not able to use aIl old preference relations. Only the current x r is preferred to the current Xi. However, we need to test the consistency of the performed comparisons to find out wh ether the assumed choice behavior is consistent with respect to one reference point. If in the linear program (5) max E cannot be positive with x * dominating x" the behavioral assumptions are no longer valid. In this case we suggest to delete oldest constraints until the consistency of the model is obtained.
x;,
6. Conclusion In this paper we have developed a method for solving discrete multiple criteria problems by blending ideas from several sources. The method is based on the use of pairwise judgments of the DM to identify his/her best decision alternative. It is assumed that the DM's choice behavior can be described with linear prospect theory. An essenti al feature of the method is the use of the DM's pairwise judgments to generate cones to eliminate inferior solutions. The best solution so far is used (technically) as the reference alternative. The method uses adjacent nondominated alternatives, against which the current reference alternative is compared. The preferences need not to be linear; the method works when the preferences result in convex indifference curves (quasiconcave value function). Based on earlier studies of the methods that use pairwise preference information, we can draw conclusions regarding the power of this method. With a small number of criteria, the number of judgments required will be relatively smaIl. If the number of criteria exceeds four, the efficiency of the method is likely to decrease rapidly [5,7]. The method improves on the method of Zionts [9] by relaxing the behavioral assumptions. Through the
153
use of linear mathematics the method is as powerful in eliminating dominated alternatives as Zionts' method [9] (discounting the cost of additional se arch for nondominated adjacent alternatives). In the suggested method the cone of dominated solutions is formed in the same spirit as in Korhonen et al. [5], but through the use of nondominated adjacent solutions, the domination cone starts always at least as close to the preferred alternative as is the case in their method, or closer to it. Therefore the elimination procedure will be more effective. GeneraIly, the algorithm can be used to solve any discrete problem, where the DM's underlying value function is quasiconcave. The reason is that we use a concave implicit value function. A concave function is also quasiconcave and we base our elimination of alternatives on quasiconcavity. With nonlinear piecewise difference functions (e.g., S-shaped), the resulting preferences need not be convex; therefore the procedure is not valid under such circumstances.
Appendix. A step-by-step statement of Zionts' discrete method [9] Step 1. Choose an arbitrary set of weights wj > 0, for the criteria j = 1, ... , p (p = number of criteria). Step 2. Choose the decision that maximizes Ewjx ij considering every decision i. CalI this the new maximizing decision. Step 3. (This step is to be omitted the first time.) Ask if the new maximizing decision is preferred to the old one. If yes, designate as solution k the new maximizing decision. If no or 1 don't know, designate as solution k a solution preferred to the old solution k. Generate a constraint (as in Step 5) based on the response. Step 4. Consider every efficient decision h adjacent to k, that is, for which EÀj (x hj - Xk) can be positive with aIl other sums E\.(x ij - x k ) ~ 0, Àj ~ 0, j = 1, ... , p, and consistent with previous responses (i.e., consistent with aIl previously generated inequalities on the weights À). If there are no such decisions, go to Step 10. Step 5. For each decision h found in Step 4, ask: "Which do you prefer, h or k? .
P. Salminen / Solving the discrete multiple criteria problem
154
(a) If h is preferred, add a constraint LÀj( X hj - Xkj)
~e
j
(e
> 0 and small). (b) If k is preferred, add a constraint
LÀj(Xkj-Xhj)
(e
~e
> 0 and small).
(c) For answers of l don't know, add no constraints. Step 6. If no h is preferred to k in Step 5, go to Step 10. Step 7. Find a set of weights satisfying aIl previously generated contraints and Àj ~ e. The purpose of e is to assure a strict inequality À j > O. Step 8. If no such solution exists, delete the oldest generated constraints and go to Step 7 above. Otherwise go to Step 9. Step 9. Let wj = Àj' j = 1, ... , p, found in Step 7 and go to Step 2. Step 10. Delete the oldest responses generated in Steps 3 or 5 and go to Step 4. If aIl responses have been dropped, rank order the alternatives and stop.
References [1) Kahneman, D., and Tversky, A, "Prospect theory: An analysis of decisions under risk", Econometrica 47 (1979) 262-291.
[2) Korhonen, P., Moskowitz, H., and Wallenius, J. (1986), "A progressive algorithm for modelling and solving multiple criteria decision problems", Operations Research 34 (1986) 726-731. [3) Korhonen, P., Moskowitz, H., and Wallenius, J., "Choice behavior in interactive multiple criteria decision making", Annals of Operations Research 23 (1990) 161-179. [4) Korhonen, P., and Wallenius, J., "Malli usean paatoksentekijan diskreetin neuvotteluongelman ratkaisemiseksi", in: C. Carlsson (ed.), Proceedings of Management Science in Finland, (MASC), Stiftelsen for Abo Akademi, 1980. [5) Korhonen, P., Wallenius, J., and Zionts, S., "Solving the discrete multiple criteria problem using convex cones", Management Science 30 (1984) 1336-1345. [6) Salminen, P., "Validating Kahneman-Tversky's prospect theory in a multiple criteria decision-making context", unpublished manuscript, 1990. (7) Salminen, P., Korhonen, P., and Wallenius, J., "Testing the form of a decision-maker's multiattribute value function based on pairwise preference information", Journal of the Operational Research Society 40 (1989) 299-302. [8) Tversky, A, "Intransitivity of preferences", Psychological Review 76 (1969) 31-45. [9) Zionts, S., "A multiple criteria method for choosing among discrete alternatives", European Journal of Operational Research 7 (1981) 143-147. [10) Zionts, S., and Wallenius, J., "An interactive multiple objective linear programming method for a class of underlying nonlinear utility functions", Management Science 29 (1983) 519-529.