Systems Analysis Laboratory, Helsinki University of Technology,. Otakaari 1M, FIN-02150 Espoo, Finland. PEKKA SALMINEN. School of Business and ...
University of Jyväskylä Department of Mathematics Laboratory of Scientific Computing Report 8/1997
IDENTIFYING COMPROMISE ALTERNATIVES IN GROUP DECISION-MAKING BY USING STOCHASTIC MULTIOBJECTIVE ACCEPTABILITY ANALYSIS
Risto Lahdelma
Pekka Salminen
UNIVERSITY OF JYVÄSKYLÄ JYVÄSKYLÄ 1997
Table of Contents 1. INTRODUCTION.................................................................................................................1 2. THE PROBLEM ...................................................................................................................2 2.1. Notation................................................................................................................................2 2.2. Stochastic Multicriteria Decision Problem ..........................................................................3 3. STOCHASTIC MULTIOBJECTIVE ACCEPTABILITY ANALYSIS .........................4 3.1. The Original SMAA.............................................................................................................4 3.2. Why Should the Original SMAA Be Extended ?.................................................................7 4. THE SMAA-2 METHOD.....................................................................................................8 4.1. Computing Rank Acceptabilities .........................................................................................8 4.2. Holistic evaluation of rank acceptabilities .........................................................................11 4.3. Adding Partial Preference Information..............................................................................12 4.4. A Step-by-Step Procedure for the SMAA-2 Method .........................................................14 5. DISCUSSION ......................................................................................................................14 6. CONCLUSIONS .................................................................................................................15 7. REFERENCES....................................................................................................................16
IDENTIFYING COMPROMISE ALTERNATIVES IN GROUP DECISION-MAKING BY USING STOCHASTIC MULTIOBJECTIVE ACCEPTABILITY ANALYSIS RISTO LAHDELMA1 Systems Analysis Laboratory, Helsinki University of Technology, Otakaari 1M, FIN-02150 Espoo, Finland PEKKA SALMINEN School of Business and Economics, University of Jyväskylä, P.O.Box 35, FIN-40351 Jyväskylä, Finland. Acknowledgment: This research is supported, in part, by the Academy of Finland, the Wihuri Foundation, and the Technology Development Centre Finland. Abstract: Stochastic multiobjective acceptability analysis (SMAA) is a multicriteria decision support method for multiple decision makers in discrete problems. In SMAA, the decision makers need not express their preferences explicitly or implicitly. Instead, the method is based on exploring the weight space in order to describe the valuations that would make each alternative the preferred one. Inaccurate or uncertain criteria values are represented by probability distributions from which the method computes confidence factors describing the reliability of the analysis. In this report we present the SMAA-2 method, which extends the original SMAA by considering all ranks in the analysis. In situations, where the ‘elitistic’ SMAA may assess large acceptability only for extreme alternatives without sufficient majority support, the more holistic SMAA-2 analysis can be used to identify good compromise candidates. We consider also situations, where partial preference information is available. Key words: decision support systems, decision theory, multicriteria analysis, utility theory AMS subject classifications: 90B50, 90A30, 90A05, 92H10, 92K10
1.
INTRODUCTION
A variety of multicriteria decision methods use weights for describing the relative importance of different criteria. Several weight elicitation methods have been suggested in order to find the actual weights of a decision maker (DM). The suggested procedures, however, tend to provide different weights for the same problem (e.g., Borcherding et al., 1991, Schoemaker and Waid 1982, Stillwell et al., 1987, Weber and Borcherding, 1993). Also, in a problem with multiple DMs, reaching consensus about exact weights may be difficult. Sometimes partial weight information can be obtained in form of intervals or a priority ordering of the different criteria. Often the DMs do not want to express weights at all. The latter may be attributed to
1
Leave of absence from: Laboratory of Scientific Computing, University of Jyväskylä, P.O.Box 35, FIN-40351 Jyväskylä, Finland.
1
several reasons: time pressure limits the DMs’ attention to the problem, the DMs are not able to assign any weights due to the difficulty of the problem, or the DMs do not want to restrict themselves to some weights, and therefore refuse to provide any input for a multicriteria model. The DMs may, for example, expect that their weights will change over time (Kornbluth, 1992). The Stochastic Multiobjective Acceptability Analysis (SMAA) method was developed for support in discrete group decision making problems, in which weight information is absent (Lahdelma et al., 1996). This problem has also been considered by Bana e Costa (1986), Nijkamp et al. (1990), and Voogd (1983). The SMAA method explores the n-dimensional weight space based on an assumed utility function (or value function) and stochastic criteria values. In fact, the SMAA-approach can be adapted to any multicriteria model using weights as input. For example, the SMAA-3 method (Hokkanen et al., 1997) uses a double threshold model as in the ELECTRE III decision aid (Roy, 1996). The original SMAA method provides descriptive information of the sets of weight vectors supporting each alternative as the preferred one. In particular, SMAA computes acceptability indices, which measure the variety of different preferences that give each alternative the best rank. This information can be used for classifying the alternatives into more or less acceptable ones, and into those which are not acceptable. However, SMAA ignores information about the other ranks. This can make it difficult to identify good compromise alternatives in particular when the acceptability is distributed among extreme alternatives, which each obtain the best rank according to a few weights, but obtain a very bad rank through other weights. The potential compromise alternatives are then those which yield a good but not necessarily the best rank through many different weights. In this report we develop the SMAA-2 method which extends the analysis to the sets of weight vectors for any rank from best to worst for each decision alternative. The obtained rank acceptabilities can be examined graphically. The rank acceptabilities can then be combined using so called meta-weights into holistic acceptability indices, describing the overall acceptability of each alternative. Different candidates for meta-weights are considered. The structure of this paper is as follows. We first define the discrete stochastic multicriteria group decision making problem. Then in Section 3 we present a slightly generalized version of the original SMAA and discuss some difficulties that may occur when interpreting the results. In Section 4, we present the extended SMAA-2 method, and consider situations in which partial weight information is available. This is followed by a discussion and conclusions.
2.
THE PROBLEM
2.1.
Notation D m n
number of DMs number of alternatives number of criteria 2
2.2.
d i j r
index over DMs 1,…,D index over alternatives 1,…,m index over criteria 1,…,n index over ranks 1,…,m
A ai ahi aki αr bri f (w) f(ξ) fij(ξij) pci pki ui(w) u(xi,w) uj(ξij) W Wi(ξ) Wri(ξ) w wci wk i X xi xij ξij
set of alternatives {x1,...,xm} acceptability index holistic acceptability index kbr acceptability index meta-weight rank r acceptability index joint density function for DMs’ weights joint density function for criteria values density function for ξij confidence factor kbr confidence factor expected utility as function of weights w utility function partial utility function for criterion j set of feasible weight vectors set of favorable weight vectors set of rank r weight vectors weight vector [w1,...,wn] central weight vector central kbr weight vector stochastic criteria space the ith alternative deterministic value of criterion j for alternative xi stochastic value of criterion j for alternative xi
Stochastic Multicriteria Decision Problem
Consider a cooperative group of D DMs who have a set of m alternatives A = {x1, x2,..., xm}, from which one is to be chosen. The alternatives are evaluated in terms of n criteria. It is assumed that the DMs’ preference structure can be represented by a real-valued utility function u(xi,w), which maps the different alternatives to utility values ui(w) = u(xi,w)
(1)
using a weight vector w for each DM to quantify their subjective preferences. If criteria values were precisely known and a single weight vector could be commonly agreed on, the problem would be trivially solved by evaluating the utility function for each alternative and choosing the alternative with the largest utility ui(w)≥uk(w), k=1,…,m. However, in our problem setup neither criteria values nor weights are precisely known.
3
Uncertain or imprecise criteria values are represented by stochastic variables ξij with assumed or estimated joint probability distribution and density function f(ξ) in the criteria space X. Often the criteria values can be treated as independent stochastic variables, and their joint density function can be expressed as a product f(ξ) = ∏i,j fij(ξij).
(2)
Deterministic criteria values xij can be treated as a special case of independent stochastic values with density functions fij(ξij)=δ(ξij-xij),
(3)
where δ is Dirac’s delta. Similarly, the DMs’ unknown or partially known preferences are represented by a weight distribution with density function f(w) in the set of feasible weights W defined as W = {w∈Rn: w ≥ 0 and ∑j wj = 1}.
(4)
The set of feasible weights is thus an (n-1)-dimensional simplex. The utility function is then used to map stochastic criteria and weight distributions into utility distributions u(ξi,w). Total lack of knowledge about weights is represented by a uniform weight distribution in W. This distribution has density function f(w) = 1/vol(W),
(5)
where the n-1 -dimensional volume of the feasible weight simplex is vol(W) = n½/(n-1)! .
(6)
We will discuss some possibilities for modeling partial preference information in Section 4.3.
3.
STOCHASTIC MULTIOBJECTIVE ACCEPTABILITY ANALYSIS
3.1.
The Original SMAA
The original SMAA is based on the sets of favorable weights Wi(ξ) defined as Wi(ξ) = {w∈W: u(ξi,w) > u(ξk,w), k=1,…,m}.
(7)
Any weight w∈Wi(ξ) assigns alternative xi the best rank. All further analysis is based on the properties of these sets. The first descriptive measure is the acceptability index ai which is defined as the expected (n1)-dimensional volume of the favorable weights. The acceptability index is a measure of the
4
variety of different valuations making the studied alternative preferred, and it is computed as a multidimensional integral over the criteria distributions ξ and favorable weight space as ai = ∫X f(ξ) ∫Wi(ξ) f(w) dw dξ.
(8)
The acceptability index can be used for classifying the alternatives into more or less acceptable ones (ai>0), and those which are not acceptable (ai zero or near-zero). A zero acceptability index indicates that the alternative is inefficient with respect to the assumed utility function. In the two-criteria case with deterministic criteria values and a linear utility function, the favorable weights and acceptability indices can be illustrated graphically as in Figure 1.
a 1 =33%
W1 a 2 =38%
x1
W2
(3,9)
x2 (7,7)
W3
a 3 =29%
x3 (9,2)
Figure 1. Favorable weights and acceptability indices in deterministic 2-criteria case with linear utility function (criteria values and weight vectors are not in the same scale). The alternatives are drawn as points in the criterion plane. The feasible weights are represented by normalized vectors in the first quadrant, and the favorable weights for each alternative are represented by the gray area between the extreme favorable vectors at each alternative. With a uniform weight distribution in W the acceptability index is proportional to the distance between the end points of the extreme vectors and it is most conveniently expressed as a percentage. Precisely speaking, given the slopes s1, s2 of the extreme vectors for an alternative, the acceptability index is given by 1/(1+s2)-1/(1+s1). The central weight vector wci is defined as the expected center of gravity of the favorable weight space. The central weight vector is computed as an integral of the weight vector over the criteria and weight distributions by wci = ∫X f(ξ) ∫Wi(ξ) f(w) w dw dξ.
5
(9)
With the assumed weight distribution, the central weight vector is the best single vector representation of the preferences of a typical DM supporting xi. The central weights can be presented to the DMs in order to help them understand how different weights correspond to different choices. The confidence factor pci is defined as the probability for the alternative to be the preferred one if the central weight vector is chosen. The confidence factor is computed as an integral over the criteria distributions ξ by pci = ∫ξ: ui(ξi,wci) > uk(ξk,wci) f(ξ) dξ.
(10)
The confidence factor measures whether the criteria data is accurate enough to discern the alternatives when the central weight vector is used. The confidence factor can in a similar manner be calculated for any given weight vector and alternative. The confidence factor can be described as the proportion of the fuzzy criterion space which makes the alternative the best with the given weight vector. This is illustrated in Figure 2, where the criteria values for x1 and x3 are deterministic, the fuzzy criteria values for x2 are represented by a rectangular area, and wc is an arbitrary weight vector.
w
c
x1 p c2
x2 x3
Figure 2. Confidence factor as a proportion of the fuzzy criteria space of x2. The above formulation of SMAA allows very flexible and detailed modeling of uncertain criteria values, and partial or absent preference information. The multi-dimensional integrals must in the general case be computed using numerical techniques. Monte-Carlo simulation, for example, is suitable. In the case where the distribution of different DMs’ weights equals that used in the computations, the acceptability index describes the portion of DMs supporting the alternative considered. This will, of course, not generally be true with the uniform weight distribution in W.
6
Any type of utility function jointly accepted by the DMs can be used with SMAA. In practice, the utility additive form might be appropriate. In this form the overall utility is expressed as a convex combination of partial utility functions u(ξi,w) =
∑
n j =1
wj uj(ξij),
w∈W.
(11)
The additive utility function along with deterministic criteria values and a uniform weight distribution allows the computations to be performed using linear programming and related techniques (Lahdelma et al., 1996). When the partial utility functions are affine transformations, a linear utility function results. Advantages with the linear type of utility function is that it is simple to handle both theoretically and computationally, and easy to understand by the DMs. Also, with a linear type of utility function the weights can be interpreted as ‘price-coefficients’ for the different criteria, or equivalently, ratios between weights can be interpreted as trade-off ratios between criteria.
3.2.
Why Should the Original SMAA Be Extended ?
The original SMAA method is not intended for directly ranking the alternatives. Instead, the acceptability index is used for coarse classification of the alternatives into more or less acceptable ones, and those which are not acceptable. The original SMAA is based entirely the sets of weights assigning an alternative the first rank. This ‘elitistic’ treatment may cause three kinds of problems that make it difficult to distinguish between widely acceptable and not so acceptable alternatives based on the acceptability index: 1. extreme alternatives may obtain excessively high acceptability, 2. neighboring alternatives decrease each others acceptability, and 3. potential compromise alternatives (alternatives which are rarely ranked best, but often second, third, etc.) may obtain too small an acceptability. The problems are illustrated in Figure 3. Here x4 is an extreme alternative with the best value on criterion 1 and the worst value on criterion 2. Alternative x4 obtains a substantial acceptability from weight vectors that emphasize criterion 1. To some extent this effect could be alleviated by appropriate scaling or by introducing preference information in form of weight constraints, if available. Neighboring alternatives x1 and x2 reduce each others acceptability. They may jointly receive a very large acceptability, but their closeness causes them to divide their mutual acceptability space in some proportion. To some extent this problem could be handled by aggregating neighboring alternatives together, or by including only one neighbor into the analysis at a time. The difficulty naturally lies in defining and identifying neighbors in a large decision problem and justifying their special treatment. Alternative x3 is nearly convex dominated by x2 and x4, and thus hardly ever the most preferred alternative, obtaining near-zero acceptability. Alternative x2 is, however, never the worst. Alternative x2 would thus be a very promising candidate for a compromise when it is
7
not possible to reach agreement between the x1-x2 group and x4. This might be the case even if x2 were convex dominated with zero acceptability.
a1 =20% a2 =23% x1 x2
a3 =3%
x3 a5 =55% x4 Figure 3. Extreme alternative x4, neighboring alternatives x1 and x2, and almost convex dominated ‘potential compromise’ alternative x3.
4.
THE SMAA-2 METHOD
For the reasons described, we develop in the following the SMAA-2 method by extending the acceptability analysis to the sets of weight vectors for each rank from best to worst for each decision alternative. The obtained rank acceptabilities can then be used for identifying potential compromise alternatives in situations where the original acceptability indices fail to resolve the problem. Then we show how the rank acceptabilities can be combined using so called meta-weights into holistic acceptability indices, describing the overall acceptability of each alternative.
4.1.
Computing Rank Acceptabilities
We define the rank of each alternative as an integer from the best rank (=1) to the worst rank (=m) by means of a ranking function rank(ξi,w) = 1 +
∑
m k =1
ρ(u(ξk,w)>u(ξi,w)),
(12)
where ρ(true) = 1 and ρ(false) = 0. The SMAA-2 method is based on analyzing the sets of rank r weights Wri(ξ) defined as 8
Wri(ξ) = {w∈W: rank(ξi,w) = r}.
(13)
A weight w∈Wri(ξ) assigns utilities for the alternatives in such a fashion so that alternative xi obtains rank r. The rank acceptability index bri is then defined as the expected volume of the set of rank weights. The rank acceptability index is a measure of the variety of different valuations granting alternative xi rank r, and is computed as a multidimensional integral over the criteria distributions and the rank weights using bri = ∫X f(ξ) ∫Wri (ξ) f(w) dw dξ.
(14)
Evidently, the first rank acceptability indices b1i are equivalent to the original acceptability indices of SMAA. The rank acceptability indices can be examined graphically in order to compare how different varieties of weights support each rank for each alternative. Candidates for most acceptable alternatives should be those with high acceptabilities for the best ranks. When seeking compromises, alternatives with large acceptabilities for the worst ranks should be avoided. The rank weights and rank acceptabilities are illustrated in Figure 4 using the same twocriteria case as in Section 3.1. The figure shows the favorable first rank weights in light gray as previously, bordered by the favorable second rank weights in dark gray.
b
1 1
=33% b 21 =21% b
x1
1 2
=38% b 22 =62%
(3,9)
x2 (7,7) b 23 =18% b
1 3
=29%
x3 (9,2)
Figure 4. First and second rank acceptabilities in deterministic 2-criteria case with linear utility function. Based on the first rank acceptability alone, it is not possible to claim the superiority of any one alternative over the others. In a conflict situation (assuming that the used weight distribution matches the preferences of the DMs) no alternative would gain majority. When seeking compromises by accepting both first and second rank alternatives, it is easy to see that the 9
middle alternative is a very strong candidate, because it can only obtain first or second rank by any weight vector (b12+b22=100%), while only 54% would support alternative x 1 and 47% x3. Note that x2 remains the strongest compromise candidate regardless of the weight distribution used in the computations. The rank acceptabilities for all ranks are shown in Table 1 and the corresponding rank acceptability profiles are drawn as a 3-dimensional graph in Figure 5. It is easy to see that all alternatives in this case have roughly equal first rank acceptabilities, but alternative x2 has the largest second rank acceptability and zero acceptability for the worst rank. Table 1. Rank acceptabilities (%) for all alternatives and ranks. Alternative
b1
b2
b3
x1
33
21
46
x2
38
62
0
x3
29
18
54
0.70
0.50 0.40 0.30 0.20
Acceptability
0.60
0.10 0.00 b3 x3 b2 x2
Rank b1
Alternative x1
Figure 5. Rank acceptability profiles for alternatives. Considering other rank acceptabilities brings new information into the acceptability analysis. This additional information can be used for handling the problems listed in the previous section: 1) Extreme alternatives, which may obtain a very high first rank acceptability do not obtain good overall evaluation unless they also receive other high ranks frequently. 2) The problem of neighbors reducing each other’s first rank acceptability is compensated by the fact that they increase each other’s acceptability for subsequent ranks. 3) Potential compromise 10
alternatives, which may obtain zero or near-zero first rank acceptability can still be granted good overall evaluation if they obtain other good ranks frequently. The rank acceptabilities can be used directly in the multicriteria evaluation of the alternatives. For large problems, we suggest an iterative process, where the k best ranks (kbr) acceptabilities aki =
∑
k r =1
bri
(15)
are analyzed at each iteration k. The kbr acceptability aki is a measure of the variety of different valuations that assign alternative xi any of the k best ranks. Note that a1i= ai. The analysis proceeds until one or more alternatives reach a sufficient majority of the weights. The weight space corresponding to the k best ranks for an alternative can also be described by means of the central kbr weight vector wki defined as wki = ∫X f(ξ)
∑
k
∫Wri (ξ) f(w) w dw dξ.
r =1
(16)
With the assumed weight distribution, the central kbr weight vector is the best single vector representation for the preferences of a typical DM who assigns an alternative any rank from 1 to k. Obviously, w1i= wci. The central kbr weights can be used in a similar manner as the central weight vectors in SMAA. The kbr confidence factor pki is defined as the probability that the alternative receives any rank from 1 to k if the central kbr weight vector is chosen. The kbr confidence factor is computed as an integral over the criteria distributions in X by pki = ∫ξ: rank(ξi,wki)≤k f(ξ) dξ.
4.2.
(17)
Holistic evaluation of rank acceptabilities
The problem of comparing the alternatives in terms of their rank acceptabilities can be seen as a ‘second-order’ multicriteria decision problem. This leads us into a complementary approach of combining the rank acceptabilities into holistic acceptability indices ahi for each alternative using ahi =
∑
m
r =1
αr bri,
(18)
where αr are so called meta-weights. There are many possible choices for meta-weights. The acceptability index of SMAA can be seen as a special case of the holistic acceptability index with meta-weights α=(1,0,…,0). The kbr acceptabilities correspond to meta-weights with k ones and m-k zeros. Using the rank numbers α=(1,2,…,m) would be equal to computing the expected rank for each alternative. It would seem that introducing the meta-problem would make things more complicated: instead of struggling with incomplete or absent weight information, we now have to decide how to obtain meta-weight information, as well. 11
However, the criteria of the meta-problem are not equally important, but a complete priority order between them is well justified. Natural requirements for meta-weights are thus that they be non-negative, normalized and non-increasing when the rank increases (i.e., ∑rαr=1 and α1≥α2≥…≥αm≥0). The ideal set of meta-weights should thus be chosen from the continuum between the extremely ‘elitistic’ w=(1,0,…,0) which results in the original acceptability index and the equalizing w=(1/m,1/m,…,1/m) failing to discern between the alternatives. Possible candidates satisfying these requirements could be linear weights αr=2(m-r)/(m(m-1)), inverse weights αr=1/(r ∑i =11 / i ), and centroid weights αr= (1/m) ∑i=r 1 / i . m
m
Figure 6 illustrates the differences of linear, inverse and centroid weights for 8 ranks. We observe that linear weights give less importance to the best ranks and more to the mediocre ranks. Inverse and centroid weights emphasize the best ranks. Compared with centroid weights, inverse weights distribute weight more evenly over the worst ranks. Thus, aggregation by inverse weights is insensitive to the order among the worst ranks. Results of Barron and Barrett (1996) support using centroid weights for modeling exactly this kind of normalized weights with complete priority order (but in the context of criteria weights, not metaweights). Of course, it is also possible to repeat the SMAA procedure here in order to find out which kind of meta-weights favor each alternative. 0.4 0.35 0.3 0.25 centroid linear
0.2
inverse 0.15 0.1 0.05 0 1
2
3
4
5
6
7
8
Figure 6. Meta-weights for combining rank acceptability indices.
4.3.
Adding Partial Preference Information
In certain decision problems with multiple DMs it is indeed possible to obtain more or less specific weight information. In principle there are two mechanisms for including preference information into SMAA and SMAA-2. The more general mechanism is to model the preferences using an appropriate density function f(w) for the weights. A simpler and 12
intuitively more easy to understand technique is to add constraints to the set of feasible weights W while restricting the uniform distribution in the restricted weight space W’. The density function is then f’(w)=1/vol(W’) when w∈W’ and f’(w)=0 when w∈W\W’. In particular the following types of restrictions on the weight space can be handled (ordered by increasing complexity): 1. Partial (or complete) ranking of criteria (wj ≥ wk for some j,k). 2. Intervals for weights (wj∈[wminj, wmaxj]). 3. Intervals for weight ratios (trade-offs) (wj/wk∈[wminjk, wmaxjk]). 4. Linear inequality constraints for weights (Bw≤c). 5. Nonlinear inequality constraints for weights (g(w)≤0). The preference information from multiple DMs can then be combined using for instance one of the following techniques: I.
Union: W’ = ∪d Wd, f’(w) = 1/vol(W’)
II. Intersection: W’ = ∩d Wd, f’(w) = 1/vol(W’) III. Averaged density: W’ = W, f’(w) = 1/D
∑
D d =1
fd(w)
In group decision-making, the weight constraints can be obtained through one of the following ways. a) Asking the DMs to provide partial rankings for the criteria. The different DMs’ rankings can be handled as type 1 information and combined through union or intersection. b) Requiring the DMs to specify upper or lower bounds or both for each weight. The DMs’ weight interval information is represented as type 2 information and aggregated together through union or average density. Intersection could easily lead to empty W’. c) Specifying trade-off ratios or intervals for the criteria. The different DMs’ trade-off intervals can be represented as type 3 information and combined through union or average density. d) Asking exact weights from the DMs. This kind of weight information can be combined, for instance, by forming the convex hull spanned by the DMs’ weights and representing it as type 4 information. Alternatively, a discrete or continuous weight distribution f’(w) can be derived and used in the analysis. e) Asking the DMs to make pair-wise comparisons between alternatives. With a linear utility function this can be handled as type 4 information and with non-linear utility functions as type 5 information. The DMs preferences can be combined using the average density.
13
As a remark, we question the prescriptive use of holistic evaluation techniques (such as pairwise comparisons between alternatives) for eliciting weights. Decomposing a decision problem into assigning weights for criteria is supposed to make the DMs task easier by allowing them to focus on individual criteria or pairs of criteria at a time. If the DMs are able to make holistic evaluations, what is the decomposed criteria weight model needed for?
4.4.
A Step-by-Step Procedure for the SMAA-2 Method
The following procedure incorporates both the iterative kbr method and the use of the holistic acceptability index. 1) Define the problem in terms of alternatives and criteria. 2) Choose the shape and scaling of the utility function jointly accepted by the DMs. 3) Include any possible weight information pertinent to the analysis. 4) Calculate the rank acceptability indices for each alternative and each rank (bri). 5) Calculate the kbr acceptabilities, central weights and confidence factors (aki, wki, pki). 6) Calculate the holistic acceptability indices ahi for each alternative. 7) Present the results from steps 4) to 6) to the DMs. They may choose which measures they find most useful: rank acceptabilities, kbr measures, or holistic indices. 8) The DMs decide whether the process is completed or should continue. If the DMs are able to provide some weight information, return to step 3). Alternatively the process may be repeated from step two or one onwards.
5.
DISCUSSION
In a real-world decision problem with multiple DMs there may not be several realistic alternatives for the utility function used with the method. Although the theory of the method allows any type of function to be used, the form to be assumed will probably be linear or some concave function. In the type of real problems for which SMAA and SMAA-2 have been developed, it is not possible to study the DMs preference structure. Prospect theory type functions (Kahneman and Tversky, 1979) could also be considered due to their descriptive power observed in several studies, however, the difficult choice of the reference point with multiple DMs may result in an unreliable analysis. The main point in the choice of the utility function for the method, anyway, is that the DMs accept it as the basis of the analysis; without this, the results are of no use. In particular when no preference information is available, the scaling of criteria values may greatly influence the results given by the method. When weight information is expressed, the DMs must also consider the scaling and understand how their weights affect the overall utilities. Scaling is generally done by using ideal and anti-ideal criteria values. The choice to 14
be made here is whether these values are taken from the set of alternatives at hand, or whether some information outside the problem is taken into account when defining these ideals (von Nitzsch and Weber, 1993). The success of the earlier versions of this type of decision-aiding tools (SMAA and SMAA-3) in real applications (Lahdelma et al., 1995, Salminen et al., 1996, Hokkanen et al., 1997) leads us to believe that the developed version will be useful in practical decision-making. Several environmental political decision-making situations for example, fulfill the ‘requirements’ for applying the method; a lack of preference information, no exact measurement of the criteria, and the presence of several DMs. So as to obtain useful results with this technique it is important that the method be very well explained to the DMs. Otherwise, the descriptive information, which may sometimes be difficult to understand, becomes useless. In particular, the DMs may require additional explanations as to how uncertainty is handled through stochastic techniques. Future research in this direction needs comparisons with different types of preference models. These include, for example, the utility (or value) function based versions, ELECTRE based versions (e.g., Roy, 1996), and PROMETHEE (e.g., Brans and Vincke, 1985). All of these procedures have been successfully applied in real decision-making with explicit weights. Now, it would be interesting to study these models in the SMAA-2 sense; how useful information is handled through each technique, how the results differ, and what the DMs’ actual comments are. Another important research topic is the use of different distributions in the analysis.
6.
CONCLUSIONS
Stochastic Multiobjective Acceptability Analysis (SMAA) is a method intended to assist multicriteria decision-making with multiple DMs in situations where little or no weight information is available, and the criteria values may be inaccurate. SMAA computes the acceptability of an alternative as a volume in the weight space, measuring the variety of different favorable preferences that yield the alternative the best rank. We have developed the SMAA-2 method by extending the acceptability analysis to all ranks. The rank acceptabilities can be efficiently used for identifying potential compromise alternatives; alternatives which do not often receive the best rank, but are widely acceptable for their otherwise good ranks. The method can also be used for pin-pointing particularly bad compromises; i.e. those extreme alternatives that obtain only a very good or very bad rank according to most weights. The rank acceptabilities can be examined individually, but SMAA-2 provides also holistic acceptability indices by aggregating rank acceptabilities together using so called meta-weights. The SMAA methods can be used with or without preference information. We have suggested a procedure whereby preference information is iteratively added, in order to focus the weight space analysis to more accurately correspond to the actual preferences of the DMs. A particular strength of SMAA-2 is that a maximal amount of automatic processing on the
15
decision problem can be done before each interaction phases with the DMs. Also, we expect a very small amount of iterations (1-2) to suffice in most cases.
7.
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, pp. 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, pp. 159-173. Barron F.H., Barrett B.E., 1996: Decision Quality Using Ranked Attribute Weights, Management Science, Vol. 42, No. 11, pp. 1515-1523. Borcherding K., Eppel T., von Winterfeldt D., 1991: Comparison of Weighting Judgments in Multiattribute Utility Measurement, Management Science, 37, 12, pp. 1603-1619. Brans J.P., Vincke Ph., 1985: A Preference Ranking Organization Method, Management Science 31 (6), pp. 647-656. 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 “Multicriteria Aid for Decisions”, Celákovice, 20.-21.3.1997, Czech Republic. Kahneman D., Tversky A., 1979: Prospect Theory: An Analysis of Decisions Under Risk, Econometrica 47, pp. 262-291. Kornbluth J.S.H., 1992: Dynamic Multi-Criteria Decision Making, Journal of Multi-Criteria Decision Analysis, Vol. 1, pp. 81-92. Lahdelma R., Hokkanen J., Salminen P., 1995: Stochastic Multi-objective Acceptability Analysis for Development of Helsinki Harbour, Proc. NOON-to-NOON Seminar on Decision Science and Applications, Åbo Academy University, December 12.-13., Turku, pp. 57-75. Lahdelma R., Hokkanen J., Salminen P., 1996: SMAA - Stochastic Multiobjective Acceptability Analysis, European Journal of Operational Research, (to appear). Nijkamp P., Rietveld P., Voogd H., 1990: Multicriteria Evaluation in Physical Planning, North-Holland. Roy B., 1996: Multicriteria Methodology for Decision Aiding, Kluwer Academic Publishers, Dordrecht. Salminen P., Hokkanen J., Lahdelma R., 1996: Multicriteria Decision Making Project on Environmental Problems, University of Jyväskylä, Department of Mathematics, Laboratory of Scientific Computing, 5/1996, 50 p. 16
Schoemaker P.J., Waid C.C., 1982: An Experimental Comparison of Different Approaches to Determining Weights in Additive Value Models, Management Science, 28, 2, pp. 182-196. Stillwell W.G., von Winterfeldt D., John R.S., 1987: Comparing Hierarchical and Nonhierarchical Weighting Methods for Eliciting Multiattribute Value Models, Management Science, 33, 4, pp. 442-450. von Nitzsch R., Weber M., 1993: The Effect of Attribute Ranges on Weights in Multiattribute Utility Measurements, Management Science, 39, 8, pp. 937-943. Voogd H., 1983: Multicriteria Evaluation for Urban and Regional Planning, Pion Ltd., London. Weber M., Borcherding K., 1993: Behavioral Influences on Weight Judgments in Multiattribute Decision Making, European Journal of Operational Research, 67, pp. 1-12.
17
