- Email: [email protected]
SMAA - Stochastic multiobjective acceptability analysis
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER
European Journal of Operational Research 106 (1998) 137-143
Theory and Methodology
S M A A - Stochastic multiobjective acceptability analysis Risto Lahdelma
a, *, J o o n a s H o k k a n e n
h, P e k k a S a l m i n e n c
a Systems Analysis Laboratory, Helsinki University of Technology, Otakaari IM, FIN-02150 Espoo, Finland b Paavo Ristola Consulting Engineers Ltd, VfiinSnkatu 6, FIN-40100 JyvfiskyliJ, Finland c School of Business and Economics, University of Jyvfiskylfi, P.O. Box 35, FIN-40351 JyviJskyliJ, Finland
Received 13 May 1996; accepted 24 March 1997
Abstract
Stochastic multiobjective acceptability analysis (SMAA) is a multicriteria decision support technique for multiple decision makers based on exploring the weight space. Inaccurate or uncertain input data can be represented as probability distributions. In SMAA the decision makers need not express their preferences explicitly or implicitly; instead the technique analyses what kind of valuations would make each alternative the preferred one. The method produces for each alternative an acceptability index measuring the variety of different valuations that support that alternative, a central weight vector representing the typical valuations resulting in that decision, and a confidence factor measuring whether the input data is accurate enough for making an informed decision. © 1998 Elsevier Science B.V. Keywords: Decision support systems; Decision theory; Multi criteria analysis; Utility theory
1. Introduction
We consider practical discrete multiobjective decision-making problems with multiple decisionmakers (DMs). In the literature the decision support process is often presented as a search for the best alternative or a ranking order of all the alternatives, based on preference information obtained from the DMs (Keeney and Raiffa, 1976; Saaty, 1980; Steuer, 1986; Vincke, 1992). According to our experiences, the DMs in public political decision situations prefer methods which do not require them to express their preferences explicitly, but rather describe the potential actions and their consequences in an appropriate
* Corresponding author. Fax: (+358) 9-451-3096; e-mail: [email protected].
form, in order to allow the final decision to be made by themselves. In this paper we develop a method for exploring the weight space based on an assumed utility function. The idea is built on the work of Bana e Costa (1986, 1988). Bana e Costa computes an acceptability index for each alternative, measuring the variety of different preferences (weight combinations) as a volume in the 3-dimensional weight space. The earlier paper presented a general n-dimensional framework, but the computational formulae and example were developed only for the 3-dimensional case. We extend the methodology by considering the criterion values as n-dimensional stochastic probability distributions. S M A A has been developed in the context of aiding public environmental multicriteria decision making to support a large group of DMs. In Finland,
0377-22t7/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PH S0377-2217(97)00163-X
R. Lahdehna et a l . / European Journal o f Operational Research 106 (1998) 137-143
138
such decision processes are regulated by the Environmental Impact Analysis legislation (Ministry of Environment, 1995) and they are normally performed in the following steps: (1) identification of all stakeholders, (2) compilation of all potential alternatives, (3) construction of the set of jointly accepted criteria, (4) measurement of the criteria, and (5) the comparison stage, where multicriteria decision models and methods are applied and the results are presented to the DMs for further evaluation. We will present the SMAA - stochastic multiobjective acceptability analysis technique - in two phases. First we present the deterministic case, in which the exact criterion values for each alternative are known. Then we generalise the technique for the case where the criterion values are stochastic variables and they are represented by probability distributions.
basic favourable weight vectors for alternative i; central weight vector for alternative i; set of feasible weight vectors; set of favourable weight vectors for alternative i; confidence factor for alternative i.
¢
wi
W
3. The deterministic case
In the deterministic case the values gij of each criterion j for each alternative i are given. Any type of utility function, jointly accepted by the DMs, could be used in SMAA. For simplicity, we use in this paper the utility additive form. The criterion values are thus mapped into the range [0,1 ] by partial utility functions u j(.) (1)
Uij = Uj( g i j ) "
2. Notation
The overall utility for each alternative is then expressed as a convex combination of the criteria utilities using some unknown normalised weights wj
2.1. Constants
u,=
Ewju,j,
weW,
(2)
J
m number of alternatives; n number of criteria.
2.2. Indices i ~ {1. . . . . m} index for alternatives; j ~ {1 . . . . . n} index for criteria.
W=(wE~n:w>_OA Z w j ~ I } .
(3) J The normalisation constraints (3) define the set of feasible weight vectors W which is an ( n - 1 ) dimensional simplex in the n-dimensional weight space. In the three-criteria case W is a two-dimensional area as illustrated in Fig. 1. The feasible
2.3. Other symbols w2
~/ij ai
gij ui uj(gij)
Uij w
w~
stochastic value of criterion j for alternative i; acceptability index for altemative i; joint probability distribution for the criterion values; value of criterion j for alternative i; overall utility of alternative i; utility function of criterion j into the range [0,1]; utility of criterion j for alternative i; weight vector [w I. . . . . w.]; weight for criterion j;
Fig. I. The set of feasible weight vectors W in the three-criteria case.
R. Lahdelma et al./ European Journal of Operational Research 106 (1998) 137-143
weight vectors represent different possible valuations of the decision makers. With a given weight vector, the multicriteria decision problem is solved by computing the u i values and choosing the alternative with the largest overall utility. The weights are, for various reasons, not always known. In group decision making the decision makers often have different opinions about what weights should be used for each criterion. Even a single decision maker may be unable or unwilling to specify the weights. For this reason, we choose a different approach. For each alternative i we determine the set of weight vectors W; which makes the overall utility of alternative i greater than or equal to the utility of any other alternative. We call W i the set offavourable weight vectors for alternative i, because alternative i becomes the best choice (not necessarily unique) with any w ~ W~. The set of favourable weight vectors is a subset of the set of feasible weight vectors satisfying the linear constraints u ~ > u k,
k = 1. . . . . m ; k ~ : i .
(4)
We solve the system formed by constraints ( 2 - 4 ) and an arbitrary objective function as a linear programming (LP) problem, e.g., max s.t.
139
Fig. 2. The set of favourable weight vectors Wi for alternative i and the central weight vector w~ in the three-criteria case. In the three-criteria case W i is a polygon-shaped area as illustrated in Fig. 2. Depending on the linear constraints (4), the polygon may also be degenerate or Wi may be empty. Next we perform a more elaborated analysis on the sets of favourable weight vectors of the efficient alternatives. There is potentially an infinite number of favourable weight vectors. However, because the set W i is a convex polytope, it can be represented as a convex combination of its vertices, the basic favourable weight vectors w~ as
0
]~.,wjuij>~_~wjukj, J
b
k=l ..... m;k-~i,
J
Ewj= l, J
wj>0.
(5) Solving the problem has two possible outcomes. Finding an optimal solution indicates that at least one weight vector makes alternative i better or at least as good as the other alternatives. If the problem is infeasible, alternative i is dominated by one or more of the other alternatives. Such inefficient alternatives can be eliminated from the set of alternatives to be considered, because strictly better alternatives exist. In some cases the DMs might want to consider inefficient alternatives, if they are not much worse than the other alternatives. The stochastic technique presented in the next section will in fact include such alternatives in the analysis.
b
The vertices of this polytope are equivalent with the basic feasible solutions of the above LP problem. Special LP techniques can be used for generating all such solutions. The volume of W i is then computed as the (n 1)-dimensional integral vol(Wi) = L O w .
(7)
The central weight vector for alternative i is defined as the centre of gravity of this polytope,
f. w w/f Wi
/
Wi
We use a uniform, random distribution of weight vectors when computing the integrals. We argue that without prior knowledge about the D M s ' valuations, the central weight vector is the best single vector representation of the valuations of a typical DM who claims to prefer alternative i. When partial prefer-
140
R. Lahdelma et al. / European Journal of Operational Research 106 (1998) 137-143
ence information is available, this can be modelled by using appropriate weight distributions. With the assumed weight distribution, the central weight vector is an unbiased estimate for w under the condition that alternative i is preferred. Another nice property of w[ is that u~i can be scaled arbitrarily. Other 'typical' weight vectors could also be defined but without some or all of the above properties. For example, •
the vector with minimal maximum distance to
w,, •
the centre of the sphere or ellipsoid inscribed in
•
or vectors based on non-uniform weight distributions.
The acceptability index for alternative i is defined as the ratio between the volumes of Wg and the feasible weight space W: ai = vol(Wi)/vol(W).
of use for the decision maker. They can be easily obtained from the basic favourable weights min wb < w/< max w/~. b
When the number of criteria is large, the LP problem may have a very large number of basic optimal solutions, and computing them may be very time consuming. Also computing the volume of an multidimensional convex polytope involves considerable effort. We can use more efficient techniques for obtaining most of the above information; e.g. the ranges for the favourable weights can be computed by solving LP problems with the objectives of minimising and maximising the value of each weight in turn, i.e. solving the problems min (max)
wi
s.t.
Ewjuo>_Zwju~j, J
k=l
.....
m;k4~l,
J
Y ' . ~ ) = I, J
(9)
The acceptability index is a measure for the variety of different valuations (represented by feasible weight vectors) which allow for that alternative to be chosen. The acceptability index can also be interpreted as the probability for a certain alternative to become best, assuming the weight distribution used in the computations. A zero acceptability index indicates that the polytope is degenerate (of dimension lower than n - 1), and in order to make that alternative the best, at least two weights have to depend linearly from the others (the first dependency being the normalisation condition (3)). A decision maker choosing an alternative with zero acceptability index must in fact be able to justify why the weights must satisfy such additional dependencies. In addition, there must exist at least one other alternative with a nonzero acceptability index, for which the same weight vector is favourable. The decision maker must also be able to justify why not to choose the equally good but 'more acceptable' alternative instead. The choice of an alternative with a zero acceptability index cannot thus be motivated without referring to criteria external to the model. The ranges for the favourable weights can also be
(10)
b
wj>__0,
(ll) for each criteria j. If the upper and lower bounds for some weight coincide, the corresponding weight volume and acceptability index must be zero, and a more expensive computation is avoided. Approximate values for a i and w~ can also be computed efficiently using numerical methods.
4. T h e stochastic case
In the stochastic case the values Yij of each criterion j for each alternative i are stochastic variables. The joint probability distribution of "Yij is specified by a density function f ( y ) . In the special case, where the criterion values are independent stochastic variables, the density function can be expressed as a product f(Y) = l~f/j(Yii).
(12)
i,j
The deterministic case is obtained as a special case with f,.i(yij) = 6(3,ii - g,7), where 6 is Dirac's delta. We use the utility functions u j(.) to map the crite-
R. Lahdelma et al. / European Journal of Operational Research 106 (1998) 137-143 rion distributions into a utility distribution. We could compute the expected values of the utilities by uij
-- f f(y)ui(yu) dy
(13)
and then continue as in the deterministic case. This is indeed useful as a preliminary analysis of the problem, but for a more thorough analysis we must extend the method to the stochastic case. We express the overall utility ug(y~,w) as a convex combination of the criteria utility distributions:
ui(Yi,W) = ~_,wjuj(yij),
w~W.
Similar to the deterministic case, we define the set of favourable weight vectors Wi(Y) as = {w~W:
bli('Yi,W ) ~ l,lk( V k , W ) ,
k = l .... m ; k ~ i } .
(16)
The central weight vector for alternative i is defined as the expected centre of gravity of this volume
f wdwf dwld,. +w~(~) /
\+w~(~)
(,7)
The stochastic acceptability index for alternative i is defined as the ratio between the expected volume of Wi(y) and the volume of the feasible weight space W:
ai = E(vol(Wg(y)))/vol(W).
(18)
In the deterministic case the central weight vector makes an alternative with a nonzero acceptability index better than the other alternatives with certainty. In the stochastic case we calculate the confidence factor for each alternative as the probability for the alternative to be the preferred one if the central weight vector is chosen. The confidence factor is expressed as an integral over the distributions for criteria values y as pt =
f,
Alternative
gi
g2
g3
g4
I I1 I11 IV V VI A
5 2.5 3 3 2 4 +0.5
- 1.5 - 3.8 -2.8 - 3.2 - 6.7 - 3.4 + 10%
15 25 10 16 0 30 5:10
4.0 2.5 2.8 3.2 1.0 3.5 5:0.5
The confidence factor measures whether the criteria data is accurate enough to discern between alternatives when the central weight vector is used. Similarly, the confidence factor can be calculated for any given weight vector and alternative.
(15)
The expected value of volume of Wi(Y) is computed as a multi-dimensional integral over the weights and the probability distributions of the criteria values
E(vol(Wi(Y)))=fyf(y)fw,,v)dwdy.
Table 1 The criteria values o f the alternatives a n d their uncertainties
(14)
J
Wi('~)
141
(19)
5. Example We use as an example a restricted set of criteria and alternatives from a real-life municipal general plan application (Hokkanen et al., 1997). The example consists of m = 6 alternatives and n = 4 criteria. Table 1 shows the (uncertain) criteria values. The values for the criteria are assumed to be uniformly distributed in the range [ g u - AU, gu + AU]" The criteria g], g3 and g4 are assumed to be known with a certain absolute accuracy and the values for gz with a certain relative accuracy as specified in the last row of Table 1. In this example, linear partial utility functions are chosen to scale the interval between the worst and best value of each criterion to the range [0,1]. This transformation could, of course, be done in many different ways. For example, non-linear partial utility functions could be used, or the scaling could be done based on the interval between 'anti-ideal' and 'ideal' criteria values. The results of the SMAA analysis are shown in Table 2. The table shows the acceptability indices (ai), confidence factors (pC), central weights (w C) and bounds (w rain, w r~ax) for the weights. All numbers are rounded and expressed as percentages. Only alternatives I, II and VI obtain clearly nonzero acceptability indices. Alternative V is not ac-
R. Lahdelma et al. / European Journal of Operational Research 106 (1998) 137-143
142
Table 2 Acceptability indices, confidence factors, bounds for the weights (%) Alternative a i pC w i 28 I 74 99 w ¢ Wmm 0 wmaX 97 1I
1
12
wc W mm wmaX
11I
0
0
we Wmm
wmax IV
0
1
wc w rain wmax
V
0
central weights and w2
w3
w4
28 0 96
17 0 94
26 0 97
9 0 48
15 0 58
65 35 96
11 0 53
2 0 4
51 34 65
35 34 38
12 1 28
6 0 26
15 0 56
55 17 98
24 0 79
17 0 84
16 0 72
46 0 98
21 0 95
w¢ wmin
wmaX
VI
25"
71
wc Wmln
wmaX
ceptable with any set of weights and the central weights cannot thus be determined. Alternatives III and IV obtain near-zero acceptability. This means that they cannot be accepted unless the DMs' preferences fall into a very small space around w c. In addition, the confidence factors for alternatives III and IV indicate a near-zero probability that these alternatives would (considering the uncertainty of the criteria values) be preferred even with their central weight vectors. We thus eliminate them. The remaining alternatives I, II and VI should be evaluated by the DMs. Most likely the DMs will drop alternative II, because the acceptability index is very small, the confidence factor is very small, and possibly because of the relatively high central and minimum weights on the criterion g3The choice between alternatives I and VI must be based on the DMs preferences. The central weight on criterion g3 is much higher for VI (46%) than I (17%). This indicates that DMs considering g3 important might support alternative VI while others might support alternative I. The confidence factors (99%, 71%) for both alternatives are reasonably high for making an 'informed' decision.
A linear utility function was assumed in this example. A linear utility function allows interpreting ratios between criteria weights as trade-off ratios. An interesting research topic would thus be to use the SMAA technique for reconstructing trade-off ratios from past decision problems.
6. Conclusions
We have presented the SMAA-technique for supporting multiobjective decision making for multiple decision makers. The method makes it possible to analyse n-dimensional multiobjective problems with stochastic criterion values. One advantage with the SMAA-technique is, that it does not require the decision makers to express their preferences explicitly or implicitly. This property is particularly useful in real-life public political decision making processes. The SMAA-technique determines a stochastic acceptability index for each alternative, describing the variety of different valuations (weight combinations) that support the preference of that alternative. The method also computes the central weight vector for each alternative, corresponding to the typical valuations of a decision maker preferring that alternative. In addition, the method determines a confidence factor for each altemative, measuring the preference probability of the altemative considering the uncertainty of the criterion values. The confidence factor can be used for measuring the sufficient accuracy of the input data for making an informed decision. The method can be extended to consider preferences either as exact weight values or stochastic probability distributions. The confidence factors provide then a ranking of the alternatives considering the uncertainty of the input data. The most important,assumption is that the DMs jointly accept the shape of the utility function used. The shape and scaling of the utility functions will affect the resulting acceptability index values.
Acknowledgements
This work was partly supported by the Academy of Finland, and the Wihuri Foundation.
R. Lahdelraa et a l . / European Journal of Operational Research 106 (1998) 137-143
References Bana e Costa, C.A., 1986. A multicriteria decision aid methodology to deal with conflicting situations on the weights. European Journal of Operational Research 26, 22-34. Bana e Costa, C.A., 1988. A methodology for sensitivity analysis in three-criteria problems: A case study in municipal management. European Journal of Operational Research 33, 159-173. Hokkanen, J., Lahdelma, R., Miettinen, K., Salminen, P., 1997. Determining the Implementation Order of a General Plan by Using a Multicriteria Method. Paper submitted for discussion at the 45th Meeting of the European Working Group 'Multi-
143
criteria Aid for Decisions', CelS.kovice, 20-21-3-1997, Czech Republic. Keeney, R.L., Raiffa, H., 1976. Decision with Multiple Objectives: Preferences and Values Tradeoffs. Wiley, New York. Ministry of Environment, 1995. Waste Act. Ministry of Environment, Helsinki, Finland. Saaty, T., 1980. The Analytic Hierarchy Process. McGraw-Hill, New York. Steoer, R., 1986. Multiple Criteria Optimization: Theory, Computation, and Application. John Wiley and sons, New York. Vincke, Ph., 1992. Multicriteria Decision-Aid. John Wiley and sons, New York.
European Journal of Operational Research 106 (1998) 137-143
Theory and Methodology
S M A A - Stochastic multiobjective acceptability analysis Risto Lahdelma
a, *, J o o n a s H o k k a n e n
h, P e k k a S a l m i n e n c
a Systems Analysis Laboratory, Helsinki University of Technology, Otakaari IM, FIN-02150 Espoo, Finland b Paavo Ristola Consulting Engineers Ltd, VfiinSnkatu 6, FIN-40100 JyvfiskyliJ, Finland c School of Business and Economics, University of Jyvfiskylfi, P.O. Box 35, FIN-40351 JyviJskyliJ, Finland
Received 13 May 1996; accepted 24 March 1997
Abstract
Stochastic multiobjective acceptability analysis (SMAA) is a multicriteria decision support technique for multiple decision makers based on exploring the weight space. Inaccurate or uncertain input data can be represented as probability distributions. In SMAA the decision makers need not express their preferences explicitly or implicitly; instead the technique analyses what kind of valuations would make each alternative the preferred one. The method produces for each alternative an acceptability index measuring the variety of different valuations that support that alternative, a central weight vector representing the typical valuations resulting in that decision, and a confidence factor measuring whether the input data is accurate enough for making an informed decision. © 1998 Elsevier Science B.V. Keywords: Decision support systems; Decision theory; Multi criteria analysis; Utility theory
1. Introduction
We consider practical discrete multiobjective decision-making problems with multiple decisionmakers (DMs). In the literature the decision support process is often presented as a search for the best alternative or a ranking order of all the alternatives, based on preference information obtained from the DMs (Keeney and Raiffa, 1976; Saaty, 1980; Steuer, 1986; Vincke, 1992). According to our experiences, the DMs in public political decision situations prefer methods which do not require them to express their preferences explicitly, but rather describe the potential actions and their consequences in an appropriate
* Corresponding author. Fax: (+358) 9-451-3096; e-mail: [email protected].
form, in order to allow the final decision to be made by themselves. In this paper we develop a method for exploring the weight space based on an assumed utility function. The idea is built on the work of Bana e Costa (1986, 1988). Bana e Costa computes an acceptability index for each alternative, measuring the variety of different preferences (weight combinations) as a volume in the 3-dimensional weight space. The earlier paper presented a general n-dimensional framework, but the computational formulae and example were developed only for the 3-dimensional case. We extend the methodology by considering the criterion values as n-dimensional stochastic probability distributions. S M A A has been developed in the context of aiding public environmental multicriteria decision making to support a large group of DMs. In Finland,
0377-22t7/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PH S0377-2217(97)00163-X
R. Lahdehna et a l . / European Journal o f Operational Research 106 (1998) 137-143
138
such decision processes are regulated by the Environmental Impact Analysis legislation (Ministry of Environment, 1995) and they are normally performed in the following steps: (1) identification of all stakeholders, (2) compilation of all potential alternatives, (3) construction of the set of jointly accepted criteria, (4) measurement of the criteria, and (5) the comparison stage, where multicriteria decision models and methods are applied and the results are presented to the DMs for further evaluation. We will present the SMAA - stochastic multiobjective acceptability analysis technique - in two phases. First we present the deterministic case, in which the exact criterion values for each alternative are known. Then we generalise the technique for the case where the criterion values are stochastic variables and they are represented by probability distributions.
basic favourable weight vectors for alternative i; central weight vector for alternative i; set of feasible weight vectors; set of favourable weight vectors for alternative i; confidence factor for alternative i.
¢
wi
W
3. The deterministic case
In the deterministic case the values gij of each criterion j for each alternative i are given. Any type of utility function, jointly accepted by the DMs, could be used in SMAA. For simplicity, we use in this paper the utility additive form. The criterion values are thus mapped into the range [0,1 ] by partial utility functions u j(.) (1)
Uij = Uj( g i j ) "
2. Notation
The overall utility for each alternative is then expressed as a convex combination of the criteria utilities using some unknown normalised weights wj
2.1. Constants
u,=
Ewju,j,
weW,
(2)
J
m number of alternatives; n number of criteria.
2.2. Indices i ~ {1. . . . . m} index for alternatives; j ~ {1 . . . . . n} index for criteria.
W=(wE~n:w>_OA Z w j ~ I } .
(3) J The normalisation constraints (3) define the set of feasible weight vectors W which is an ( n - 1 ) dimensional simplex in the n-dimensional weight space. In the three-criteria case W is a two-dimensional area as illustrated in Fig. 1. The feasible
2.3. Other symbols w2
~/ij ai
gij ui uj(gij)
Uij w
w~
stochastic value of criterion j for alternative i; acceptability index for altemative i; joint probability distribution for the criterion values; value of criterion j for alternative i; overall utility of alternative i; utility function of criterion j into the range [0,1]; utility of criterion j for alternative i; weight vector [w I. . . . . w.]; weight for criterion j;
Fig. I. The set of feasible weight vectors W in the three-criteria case.
R. Lahdelma et al./ European Journal of Operational Research 106 (1998) 137-143
weight vectors represent different possible valuations of the decision makers. With a given weight vector, the multicriteria decision problem is solved by computing the u i values and choosing the alternative with the largest overall utility. The weights are, for various reasons, not always known. In group decision making the decision makers often have different opinions about what weights should be used for each criterion. Even a single decision maker may be unable or unwilling to specify the weights. For this reason, we choose a different approach. For each alternative i we determine the set of weight vectors W; which makes the overall utility of alternative i greater than or equal to the utility of any other alternative. We call W i the set offavourable weight vectors for alternative i, because alternative i becomes the best choice (not necessarily unique) with any w ~ W~. The set of favourable weight vectors is a subset of the set of feasible weight vectors satisfying the linear constraints u ~ > u k,
k = 1. . . . . m ; k ~ : i .
(4)
We solve the system formed by constraints ( 2 - 4 ) and an arbitrary objective function as a linear programming (LP) problem, e.g., max s.t.
139
Fig. 2. The set of favourable weight vectors Wi for alternative i and the central weight vector w~ in the three-criteria case. In the three-criteria case W i is a polygon-shaped area as illustrated in Fig. 2. Depending on the linear constraints (4), the polygon may also be degenerate or Wi may be empty. Next we perform a more elaborated analysis on the sets of favourable weight vectors of the efficient alternatives. There is potentially an infinite number of favourable weight vectors. However, because the set W i is a convex polytope, it can be represented as a convex combination of its vertices, the basic favourable weight vectors w~ as
0
]~.,wjuij>~_~wjukj, J
b
k=l ..... m;k-~i,
J
Ewj= l, J
wj>0.
(5) Solving the problem has two possible outcomes. Finding an optimal solution indicates that at least one weight vector makes alternative i better or at least as good as the other alternatives. If the problem is infeasible, alternative i is dominated by one or more of the other alternatives. Such inefficient alternatives can be eliminated from the set of alternatives to be considered, because strictly better alternatives exist. In some cases the DMs might want to consider inefficient alternatives, if they are not much worse than the other alternatives. The stochastic technique presented in the next section will in fact include such alternatives in the analysis.
b
The vertices of this polytope are equivalent with the basic feasible solutions of the above LP problem. Special LP techniques can be used for generating all such solutions. The volume of W i is then computed as the (n 1)-dimensional integral vol(Wi) = L O w .
(7)
The central weight vector for alternative i is defined as the centre of gravity of this polytope,
f. w w/f Wi
/
Wi
We use a uniform, random distribution of weight vectors when computing the integrals. We argue that without prior knowledge about the D M s ' valuations, the central weight vector is the best single vector representation of the valuations of a typical DM who claims to prefer alternative i. When partial prefer-
140
R. Lahdelma et al. / European Journal of Operational Research 106 (1998) 137-143
ence information is available, this can be modelled by using appropriate weight distributions. With the assumed weight distribution, the central weight vector is an unbiased estimate for w under the condition that alternative i is preferred. Another nice property of w[ is that u~i can be scaled arbitrarily. Other 'typical' weight vectors could also be defined but without some or all of the above properties. For example, •
the vector with minimal maximum distance to
w,, •
the centre of the sphere or ellipsoid inscribed in
•
or vectors based on non-uniform weight distributions.
The acceptability index for alternative i is defined as the ratio between the volumes of Wg and the feasible weight space W: ai = vol(Wi)/vol(W).
of use for the decision maker. They can be easily obtained from the basic favourable weights min wb < w/< max w/~. b
When the number of criteria is large, the LP problem may have a very large number of basic optimal solutions, and computing them may be very time consuming. Also computing the volume of an multidimensional convex polytope involves considerable effort. We can use more efficient techniques for obtaining most of the above information; e.g. the ranges for the favourable weights can be computed by solving LP problems with the objectives of minimising and maximising the value of each weight in turn, i.e. solving the problems min (max)
wi
s.t.
Ewjuo>_Zwju~j, J
k=l
.....
m;k4~l,
J
Y ' . ~ ) = I, J
(9)
The acceptability index is a measure for the variety of different valuations (represented by feasible weight vectors) which allow for that alternative to be chosen. The acceptability index can also be interpreted as the probability for a certain alternative to become best, assuming the weight distribution used in the computations. A zero acceptability index indicates that the polytope is degenerate (of dimension lower than n - 1), and in order to make that alternative the best, at least two weights have to depend linearly from the others (the first dependency being the normalisation condition (3)). A decision maker choosing an alternative with zero acceptability index must in fact be able to justify why the weights must satisfy such additional dependencies. In addition, there must exist at least one other alternative with a nonzero acceptability index, for which the same weight vector is favourable. The decision maker must also be able to justify why not to choose the equally good but 'more acceptable' alternative instead. The choice of an alternative with a zero acceptability index cannot thus be motivated without referring to criteria external to the model. The ranges for the favourable weights can also be
(10)
b
wj>__0,
(ll) for each criteria j. If the upper and lower bounds for some weight coincide, the corresponding weight volume and acceptability index must be zero, and a more expensive computation is avoided. Approximate values for a i and w~ can also be computed efficiently using numerical methods.
4. T h e stochastic case
In the stochastic case the values Yij of each criterion j for each alternative i are stochastic variables. The joint probability distribution of "Yij is specified by a density function f ( y ) . In the special case, where the criterion values are independent stochastic variables, the density function can be expressed as a product f(Y) = l~f/j(Yii).
(12)
i,j
The deterministic case is obtained as a special case with f,.i(yij) = 6(3,ii - g,7), where 6 is Dirac's delta. We use the utility functions u j(.) to map the crite-
R. Lahdelma et al. / European Journal of Operational Research 106 (1998) 137-143 rion distributions into a utility distribution. We could compute the expected values of the utilities by uij
-- f f(y)ui(yu) dy
(13)
and then continue as in the deterministic case. This is indeed useful as a preliminary analysis of the problem, but for a more thorough analysis we must extend the method to the stochastic case. We express the overall utility ug(y~,w) as a convex combination of the criteria utility distributions:
ui(Yi,W) = ~_,wjuj(yij),
w~W.
Similar to the deterministic case, we define the set of favourable weight vectors Wi(Y) as = {w~W:
bli('Yi,W ) ~ l,lk( V k , W ) ,
k = l .... m ; k ~ i } .
(16)
The central weight vector for alternative i is defined as the expected centre of gravity of this volume
f wdwf dwld,. +w~(~) /
\+w~(~)
(,7)
The stochastic acceptability index for alternative i is defined as the ratio between the expected volume of Wi(y) and the volume of the feasible weight space W:
ai = E(vol(Wg(y)))/vol(W).
(18)
In the deterministic case the central weight vector makes an alternative with a nonzero acceptability index better than the other alternatives with certainty. In the stochastic case we calculate the confidence factor for each alternative as the probability for the alternative to be the preferred one if the central weight vector is chosen. The confidence factor is expressed as an integral over the distributions for criteria values y as pt =
f,
Alternative
gi
g2
g3
g4
I I1 I11 IV V VI A
5 2.5 3 3 2 4 +0.5
- 1.5 - 3.8 -2.8 - 3.2 - 6.7 - 3.4 + 10%
15 25 10 16 0 30 5:10
4.0 2.5 2.8 3.2 1.0 3.5 5:0.5
The confidence factor measures whether the criteria data is accurate enough to discern between alternatives when the central weight vector is used. Similarly, the confidence factor can be calculated for any given weight vector and alternative.
(15)
The expected value of volume of Wi(Y) is computed as a multi-dimensional integral over the weights and the probability distributions of the criteria values
E(vol(Wi(Y)))=fyf(y)fw,,v)dwdy.
Table 1 The criteria values o f the alternatives a n d their uncertainties
(14)
J
Wi('~)
141
(19)
5. Example We use as an example a restricted set of criteria and alternatives from a real-life municipal general plan application (Hokkanen et al., 1997). The example consists of m = 6 alternatives and n = 4 criteria. Table 1 shows the (uncertain) criteria values. The values for the criteria are assumed to be uniformly distributed in the range [ g u - AU, gu + AU]" The criteria g], g3 and g4 are assumed to be known with a certain absolute accuracy and the values for gz with a certain relative accuracy as specified in the last row of Table 1. In this example, linear partial utility functions are chosen to scale the interval between the worst and best value of each criterion to the range [0,1]. This transformation could, of course, be done in many different ways. For example, non-linear partial utility functions could be used, or the scaling could be done based on the interval between 'anti-ideal' and 'ideal' criteria values. The results of the SMAA analysis are shown in Table 2. The table shows the acceptability indices (ai), confidence factors (pC), central weights (w C) and bounds (w rain, w r~ax) for the weights. All numbers are rounded and expressed as percentages. Only alternatives I, II and VI obtain clearly nonzero acceptability indices. Alternative V is not ac-
R. Lahdelma et al. / European Journal of Operational Research 106 (1998) 137-143
142
Table 2 Acceptability indices, confidence factors, bounds for the weights (%) Alternative a i pC w i 28 I 74 99 w ¢ Wmm 0 wmaX 97 1I
1
12
wc W mm wmaX
11I
0
0
we Wmm
wmax IV
0
1
wc w rain wmax
V
0
central weights and w2
w3
w4
28 0 96
17 0 94
26 0 97
9 0 48
15 0 58
65 35 96
11 0 53
2 0 4
51 34 65
35 34 38
12 1 28
6 0 26
15 0 56
55 17 98
24 0 79
17 0 84
16 0 72
46 0 98
21 0 95
w¢ wmin
wmaX
VI
25"
71
wc Wmln
wmaX
ceptable with any set of weights and the central weights cannot thus be determined. Alternatives III and IV obtain near-zero acceptability. This means that they cannot be accepted unless the DMs' preferences fall into a very small space around w c. In addition, the confidence factors for alternatives III and IV indicate a near-zero probability that these alternatives would (considering the uncertainty of the criteria values) be preferred even with their central weight vectors. We thus eliminate them. The remaining alternatives I, II and VI should be evaluated by the DMs. Most likely the DMs will drop alternative II, because the acceptability index is very small, the confidence factor is very small, and possibly because of the relatively high central and minimum weights on the criterion g3The choice between alternatives I and VI must be based on the DMs preferences. The central weight on criterion g3 is much higher for VI (46%) than I (17%). This indicates that DMs considering g3 important might support alternative VI while others might support alternative I. The confidence factors (99%, 71%) for both alternatives are reasonably high for making an 'informed' decision.
A linear utility function was assumed in this example. A linear utility function allows interpreting ratios between criteria weights as trade-off ratios. An interesting research topic would thus be to use the SMAA technique for reconstructing trade-off ratios from past decision problems.
6. Conclusions
We have presented the SMAA-technique for supporting multiobjective decision making for multiple decision makers. The method makes it possible to analyse n-dimensional multiobjective problems with stochastic criterion values. One advantage with the SMAA-technique is, that it does not require the decision makers to express their preferences explicitly or implicitly. This property is particularly useful in real-life public political decision making processes. The SMAA-technique determines a stochastic acceptability index for each alternative, describing the variety of different valuations (weight combinations) that support the preference of that alternative. The method also computes the central weight vector for each alternative, corresponding to the typical valuations of a decision maker preferring that alternative. In addition, the method determines a confidence factor for each altemative, measuring the preference probability of the altemative considering the uncertainty of the criterion values. The confidence factor can be used for measuring the sufficient accuracy of the input data for making an informed decision. The method can be extended to consider preferences either as exact weight values or stochastic probability distributions. The confidence factors provide then a ranking of the alternatives considering the uncertainty of the input data. The most important,assumption is that the DMs jointly accept the shape of the utility function used. The shape and scaling of the utility functions will affect the resulting acceptability index values.
Acknowledgements
This work was partly supported by the Academy of Finland, and the Wihuri Foundation.
R. Lahdelraa et a l . / European Journal of Operational Research 106 (1998) 137-143
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