- Email: [email protected]
Reference point approach for multiple decision makers
European Journal of Operational Research 164 (2005) 785–791 www.elsevier.com/locate/dsw
Decision Aiding
Reference point approach for multiple decision makers Risto Lahdelma a, Kaisa Miettinen a
b,*,1
, Pekka Salminen
c
Department of Information Technology, University of Turku, Lemmink€aisenkatu 14 A, FIN-20520 Turku, Finland b Helsinki School of Economics, P.O. Box 1210, FIN-00101 Helsinki, Finland c School of Business and Economics, P.O. Box 35 (MaE), FIN-40014 University of Jyv€askyl€a, Finland Received 28 April 2003; accepted 27 January 2004 Avilable online 24 March 2004
Abstract We consider multiple criteria decision-making problems where a group of decision-makers wants to find the most preferred solution from a discrete set of alternatives. We develop a method that uses achievement functions for charting subsets of reference points that would support a certain alternative to be the most preferred one. The resulting descriptive information is provided to the decision-makers in the form of reference acceptability indices and central reference points for each decision alternative. Then, the decision-makers can compare this information with their own preferences. We demonstrate the use of the method using a strategic multiple criteria decision model for an electricity retailer. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Multiple criteria analysis; Multiple criteria decision making; Discrete optimisation; Group decision making; Achievement function
1. Introduction A fundamental problem in choosing a decision aid method for a real-life problem is that different methods may provide different results with the same data, and there is usually no way to objectively identify the best alternative or the best method. The practical relevance of decision support methods can be measured, for example, by
*
Corresponding author. Fax: +358-9-431-38-535. E-mail addresses: kaisa.miettinen@hkkk.fi, [email protected].fi (K. Miettinen). 1 Her research was supported by the Academy of Finland, grant #65760.
the understandability of preference modelling used, how well a number of decision makers (DMs) are supported and whether inaccurate information can be handled (Goicoechea et al., 1982; Hobbs et al., 1992). The majority of multiple criteria decision support methods apply some decision model to aggregate together different criteria measurements and subjective preference information from the DMs (see, e.g., Keeney and Raiffa, 1976; Matsatsinis and Samaras, 2001). A common way to represent the DMs’ preferences is to use weighting coefficients for the criteria. However, the use of weights has many drawbacks and difficulties. In real-life problems it may be impossible to obtain exact and reliable weight information from the DMs. Besides, the meaning
0377-2217/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.01.030
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R. Lahdelma et al. / European Journal of Operational Research 164 (2005) 785–791
of the weights depends on the decision model and it may be unrealistic to assume the DMs to understand very complex decision models. Even though several weight elicitation methods have been suggested, they tend to provide different weights for the same problem (see, e.g., Schoemaker and Waid, 1982; Weber and Borcherding, 1993). Furthermore, when multiple DMs are involved, there may not be any jointly acceptable way to combine weights from multiple DMs. One approach to overcome problems with missing, imprecise or conflicting weight information is to use so-called preference information free methods. These include, for example, the comparative hypervolume criterion by Charnetski and Soland (1978), the overall compromise criterion method by Bana e Costa (1986), and stochastic multicriteria acceptability analysis (SMAA) methods by Lahdelma et al. (1998, 2003) and Lahdelma and Salminen (2001). These methods are based on inverse weight space analysis to reveal what kinds of preferences favour each alternative. Instead of weights, a more straightforward technique to represent the DMs’ preferences in a decision model is through goals or reference points. This means that the DMs specify aspiration levels, that is, desirable or preferable values for each criterion. Goal programming (see, e.g., Charnes and Cooper, 1961) aims at finding a solution that minimises the distance to the specified goal and Wierzbicki (1982, 1986) presented the idea of achievement (scalarizing) functions as a way to overcome some weaknesses of traditional goal programming. Achievement functions are derived on the basis of reference points to project an arbitrary reference point to the set of nondominated attainable solutions. The achievement function is constructed in such a way that if the reference point is dominated, the optimisation will advance past the reference point to a nondominated solution. This means that the drawback of measuring the distance between a reference point and the set of feasible solutions is avoided. Thus, nondominated solutions can be characterized by achievement functions. An achievement function is a function sz : Z ! R, where z 2 Rk is an arbitrary reference point and Z Rk is the set of the alternatives. Here we have k
criteria to be considered simultaneously and they are all to be maximized. Thus, each alternative is a k-dimensional vector. Furthermore, we assume that the DMs prefer more to less. By a nondominated solution z 2 Z we mean a solution (alternative) where none of its components can be improved without impairing at least one of the others. For further details see, for example, Miettinen (1999). Under certain assumptions, the achievement function produces nondominated solutions for any z 2 Rk and different nondominated solutions can be found by moving the reference point only. An example of achievement functions is sz ðzÞ ¼ min ½wi ðzi zi Þ þ q i¼1;...;k
k X
wi ðzi zi Þ;
ð1Þ
i¼1
where q > 0 is a sufficiently small scalar and w is a fixed positive scaling vector. Usually, wi is set to be equal to the inverse of the difference between the best and the worst value for each criterion i. The indifference curves of this function are shifted widened positive orthants where the linear component multiplied by q widens the angles of the indifference curves. Note that the reference point does not affect the shape of the indifference curves, only their position. Also, note that all the reference points on the line zi ¼ zi þ t= wi (i ¼ 1; . . . ; k, t 2 R) correspond to the same family of indifference curves and identical preference order among the alternatives. For more examples of achievement functions, see, for example, Wierzbicki (1986). As mentioned, when compared to weights, reference points provide a more direct way for the DMs to express their desires and, thus, to affect the solution. Reference point approaches model satisfying behaviour instead of optimising (see, e.g., Simon, 1958). Reference point approaches do not, of course, solve all problems in obtaining preference information. Still, these methods may bring the DMs closer to understanding what they really want. However, reference point approaches do not apply directly to group decision-making. A group of DMs may have very different reference points and there is no generally accepted method how to combine this information. For this reason, we
R. Lahdelma et al. / European Journal of Operational Research 164 (2005) 785–791
suggest a new multiple criteria group decisionsupport method for discrete problems based on multiple reference points. The new Ref-SMAA method (SMAA based on reference points) is based on inverse analysis in the reference point space. The reference point space is a region that includes the reference points of individual DMs. The method generates random reference points from the reference point space and evaluates the decision alternatives based on an achievement function. Alves and Clımaco (2001) have recently presented similar ideas about exploring the reference point space, but in the context of multiobjective pure integer programming problems with deterministic criteria measurements. The Ref-SMAA method provides descriptive information of the sets of reference points that favour each alternative. To be more precise, socalled reference acceptability indices measure the variety of reference points that make an alternative most preferred. This information can be used for classifying the alternatives into more or less acceptable ones, and into those that are not acceptable. If the acceptability index is large, the alternative is likely to be acceptable for a large variety of DMs having different preferences when compared to an alternative supported by few reference points only. In addition, the method proposes a typical reference point supporting a certain alternative. This information gives the DMs the possibility to evaluate whether such a reference point could reflect their preferences.
2. The Ref-SMAA method We consider a cooperative group of D DMs who have a set of m alternatives Z ¼ fz1 ; z2 ; . . . ; zm g, from which one is to be chosen. The alternatives are evaluated in terms of k criteria. The suggested method can even easily handle problems where the criteria values are not precisely known. If uncertain or imprecise criteria values are included, they are represented by stochastic variables nij (i ¼ 1; . . . ; m, j ¼ 1; . . . ; k) with an assumed or an estimated joint probability distribution and a density function f ðnÞ in the space Z Rmk .
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The idea of the method is to chart reference points that support each alternative to be the most preferred one. The method generates random reference points z from the reference point The reference point space is a subset of space Z. Rk , and it is specified so the criterion space, Z that it includes the reference points of individual DMs. The reference point space can be specified so that it includes also the possible inaccuracies in DMs’ reference points. With totally missing information about the DMs’ reference points, the reference point space can be chosen, for example, so that it includes all the alternatives. When some information about the reference points is available, the reference point space can be defined accordingly. The achievement function used in Ref-SMAA may be selected in different ways. It is used to map stochastic criteria and reference point distributions into new distributions sz ðni Þ. The method determines for each alternative zi the set of favourable i ðnÞ defined as reference points Z z ðni Þ P sz ðnj Þ; j ¼ 1; . . . ; mg: i ðnÞ ¼ fz 2 Zjs Z i ðnÞ makes the overall Any reference point z 2 Z i preference of z greater than or equal to the preference of any other alternative. All further analysis is based on the properties of the sets of favourable reference points. In deterministic two-criterion models and using an achievement function of type (1), the sets of favourable reference points can be determined graphically, as illustrated in Fig. 1. This can be done by searching for reference points that make two adjacent alternatives equally good. Such a reference point z is found between each pair of adjacent alternatives by sliding the indifference curve of the achievement function to a position where it intersects the two alternatives. Because all reference points on the line w1 ðz1 z1 Þ ¼ w2 ðz2 z2 Þ correspond to the same set of indifference curves, such lines slice the reference point space into sets of favourable reference points for each alternative. In Fig. 1, we have four alternatives (A, B, C and D) denoted by black dots in a problem involving two deterministic criteria. The indifference curves corresponding to two different reference points (white dots) are shown as dashed
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B (0.1,0.9)
C
Criterion 2
A (0,1)
(0.75,0.75)
D (0.9,0.1) Criterion 1 Fig. 1. Sets of favourable reference points of four deterministic alternatives (A, B, C and D).
lines. The four sets of favourable reference points are illustrated using different patterns. The descriptive measure reference acceptability index ai (for i ¼ 1; . . . ; m) is defined as the ratio between the expected volume of the set of favourable reference points and the volume of the reference point space. The reference acceptability index is computed as a multi-dimensional integral over the criteria value distributions and the favourable reference point space as Z Z ai ¼ f ðnÞ f ðzÞ dz dn: Z
zi ðnÞ
The reference acceptability index is a measure of the variety of different reference points making the studied alternative the most preferred one. The bigger the index, the more different reference points exist supporting that alternative as the most preferred one. In practice, we generate criterion values from their distributions and reference and points from a uniform distribution f ðzÞ in Z record the share of reference points supporting each alternative as the most preferred one. The reference acceptability index can be used for classifying the alternatives into more or less acceptable ones ðai > 0Þ and into those that are not acceptable (ai is zero or near-zero). Alternatives with large reference acceptability indices are robust candidates particularly in group decision making. It is important to note that such alterna-
tives are likely to remain good solutions also in the future, subject to changing preferences, new stakeholders, and changing or more accurate criteria. If the reference points used represented exactly the reference points of the DMs, the reference acceptability index could be understood as the share of votes supporting each alternative as the most preferred one. However, in a general case the reference points generated by our method cannot be assumed to represent the preference distribution of the real DMs and, thus, reference acceptability indices should not be used for absolute ranking of the alternatives. The central reference point zi (for i ¼ 1; . . . ; mÞ is defined as the expected centre of gravity of the set of favourable reference points. The central reference point is computed as a multi-dimensional integral of the reference point vector z over the criteria value distributions and the favourable reference point space as Z Z zi ¼ f ðnÞ f ðzÞz dz dn=ai : z
zi ðnÞ
The central reference point represents the preferences of an average DM supporting zi . When choosing among different robust (widely acceptable) alternatives, the DMs can compare their own preferences against the central reference points. In a real-life decision problem, central reference points may differ more or less from the actual reference points of the DMs but they can be presented to the DMs in order to help them understand how different reference points correspond to different choices. (In practice, the multi-dimensional integrals are computed through Monte– Carlo simulation by generating random numbers for criteria values and reference points from their distributions and collecting statistics about the alternatives they support.) In all, the descriptive measures of Ref-SMAA, reference acceptability indices and central reference points, are all related to reference points. Thus, the measures as well as the original alternatives all belong to the criterion space. This means that they are easily understandable for the DMs because they have a straightforward meaning in relation with the personal aspiration levels of the DMs. The DMs do not need to adopt and use
R. Lahdelma et al. / European Journal of Operational Research 164 (2005) 785–791
artificial concepts (like weights), which decreases cognitive burden and makes the approach easy to understand. One should note that the reference point space can be defined in many ways and it is not necessarily a subset of the space formed by the alternatives. Potential problems may occur if the reference point space is extended outside that space because the sets of favourable reference points for different alternatives do not grow in the same proportion. The same problems apply also with weights in weight space analysis. However, with weights this problem is alleviated by the fact that it is commonly accepted to use non-negative normalized weights.
3. Example As an example, we consider a strategic decision problem of an electricity retailer. The retailer aims to maximize its long and short-term profit and to increase its market share both in regular and
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‘green’ electricity while managing the risks involved. We have nine alternatives and the following four criteria in the planning period of three years: long-term profit (the profit in millions of euros accumulated during the planning period), short-term profit (the profit in millions of euros from the first year), market share (share (%) at the end of the planning period) and green market share (market share (%) of green electricity at the end of the planning period). The criteria values and their uncertainties are presented in Table 1. We next analyse the problem using the RefSMAA method and the achievement function (1) with equal weights and q ¼ 0:1. Prior to the analysis, the expected values of all the criteria are scaled into the range ½0; 1. Yet, the central reference points are presented in the original scales. We assume that there is no information about the DMs’ reference points. Therefore, we use as the ¼ ½0; 1k . reference point space the hyper-cube Z Table 2 presents the reference acceptability indices ai (in percentage terms) and the components of the central reference points zi for each alternative i.
Table 1 Expected values and standard deviations for the four criteria Alternative
Long-term profit (L)
Short-term profit (S)
Market share (M)
Green market share (G)
S1 S2 S3 S4 S5 S6 S7 S8 S9
439 ± 72 426 ± 62 264 ± 67 444 ± 62 605 ± 58 449 ± 63 449 ± 53 457 ± 63 453 ± 53
163 ± 18 159 ± 16 104 ± 16 163 ± 16 220 ± 16 166 ± 16 164 ± 14 165 ± 16 163 ± 14
12.1 ± 0.2 12.1 ± 0.2 13.1 ± 0.2 12.1 ± 0.2 11.0 ± 0.2 12.1 ± 0.2 12.1 ± 0.2 12.1 ± 0.2 12.1 ± 0.2
9.3 ± 3.2 14.8 ± 3.2 9.3 ± 3.2 9.3 ± 3.2 9.3 ± 3.2 4.3 ± 3.2 9.3 ± 3.2 9.3 ± 3.2 14.8 ± 3.2
Table 2 Reference acceptability indices and central reference points Alt
ai (%)
ziL
ziS
ziM
ziG
S1 S2 S3 S4 S5 S6 S7 S8 S9
8 20 1 7 10 1 8 11 34
437 415 323 441 483 461 440 448 428
165 156 121 165 183 174 165 164 157
12.2 12.2 13.0 12.2 11.3 12.3 12.2 12.2 12.1
8.8 10.4 8.8 8.9 8.8 6.6 8.7 8.6 10.1
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Alternative S9 obtains the largest reference acceptability index of 34%. This means that the largest subset of the reference point space supports this alternative as the best one. The central reference point of S9 is near the centre of the reference point space, indicating that the potential supporters of this alternative require reasonably high criteria values for all the criteria. In addition, a fairly high proportion (20%) of the reference points supports S2. The central reference point related to S2 is very similar to that of S9s. In contrast, alternatives S3 and S6 both obtain an acceptability of only 1%. They are somewhat extreme alternatives. The central reference point related to S3 indicates that its potential supporters are satisfied with fairly low long-term and shortterm profit while emphasising the market share whereas DMs who are not interested in a high green market share would support alternative S6. Note that the alternatives should not in general be ranked based on the reference acceptability index. This index should be used for coarse classification of the alternatives into more or less acceptable ones, and those that are not acceptable.
desired levels that the DMs want to reach as the aspiration levels do. Also, the cognitive limits of the human mind may be less restrictive in conjunction with reference points where the DM can define her/his aspiration level for each criterion separately, while in weight-based methods (s)he must consider trade-offs between two or more criteria simultaneously. In Ref-SMAA, the theoretical problem of combining conflicting preferences of multiple DMs is avoided. It can be used for identifying good compromise alternatives that are acceptable from several points of view. The method computes a reference acceptability index for each alternative, describing the variety of different reference points (in the reference point space) that favour the alternative. The method also computes the central reference point for each alternative, corresponding to the typical reference point of a DM preferring that alternative. The computations can be implemented efficiently through stochastic simulation, where criteria values and reference points are drawn from their distributions. The most important assumption is that the DMs jointly accept the achievement model to be used in the analysis.
4. Conclusions References We have introduced the Ref-SMAA method for supporting discrete multiple criteria decisionmaking involving multiple DMs. The method can be applied for problems where both criteria data and preference information is uncertain or inaccurate (or preference information is totally missing). The uncertain criteria are represented by stochastic distributions. The preference structure of the DMs’ is based on aspiration levels forming reference points in the reference point space and the preferences are represented by achievement functions. The reference point space is defined to reflect the DMs’ possible reference points. Reference points can be expected to be easier to understand for the DMs than, for example, weighting vectors. A reference point describes what the DM wants in a certain situation, while weights do not have such a clear and direct interpretation. Weights only give indication of the trade-offs between criteria, but do not specify the
Alves, M.J., Clımaco, J., 2001. Indifference sets of reference points in multi-objective integer linear programming. Journal of Multi-Criteria Decision Analysis 10, 177–189. 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. Charnes, A., Cooper, W.W., 1961. Management Models and Industrial Applications of Linear Programming, vol. 1. John Wiley, New York. Charnetski, J.R., Soland, R.M., 1978. Multiple-attribute decision making with partial information: The comparative hypervolume criterion. Naval Research Logistics Quarterly 25, 279–288. Goicoechea, A., Hansen, D., Duckstein, L., 1982. Introduction to Multiobjective Analysis with Engineering and Business Applications. John Wiley, New York. Hobbs, B.F., Chankong, V., Hamadeh, W., Stakhiv, E.Z., 1992. Does choice of multicriteria method matter? An experiment in water resources planning. Water Resources Research 28 (7), 1767–1779. Keeney, R.L., Raiffa, H., 1976. Decisions with Multiple Objectives: Preferences with Value Tradeoffs. John Wiley, New York.
R. Lahdelma et al. / European Journal of Operational Research 164 (2005) 785–791 Lahdelma, R., Salminen, P., 2001. SMAA-2: Stochastic multicriteria acceptability analysis for group decision making. Operations Research 49 (3), 444–454. Lahdelma, R., Hokkanen, J., Salminen, P., 1998. SMAA–– stochastic multiobjective acceptability analysis. European Journal of Operational Research 106 (1), 137–143. Lahdelma, R., Miettinen, K., Salminen, P., 2003. Ordinal criteria in stochastic multicriteria acceptability analysis. European Journal of Operational Research 147 (1), 117–127. Matsatsinis, N.F., Samaras, A.P., 2001. MCDA and preference disaggregation in group decision support systems. European Journal of Operational Research 120, 414–429. Miettinen, K., 1999. Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston.
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Schoemaker, P.J., Waid, C.C., 1982. An experimental comparison of different approaches to determining weights in additive value models. Management Science 28, 182–196. Simon, M., 1958. Administrative Behavior. MacMillan, New York. Weber, M., Borcherding, K., 1993. Behavioral influences on weight judgments in multiattribute decision making. European Journal of Operational Research 67, 1–12. Wierzbicki, A.P., 1982. A mathematical basis for satisficing decision making. Mathematical Modelling 3 (25), 391– 405. Wierzbicki, A.P., 1986. On the completeness and constructiveness of parametric characterizations to vector optimization problems. OR Spektrum 8, 73–87.
Decision Aiding
Reference point approach for multiple decision makers Risto Lahdelma a, Kaisa Miettinen a
b,*,1
, Pekka Salminen
c
Department of Information Technology, University of Turku, Lemmink€aisenkatu 14 A, FIN-20520 Turku, Finland b Helsinki School of Economics, P.O. Box 1210, FIN-00101 Helsinki, Finland c School of Business and Economics, P.O. Box 35 (MaE), FIN-40014 University of Jyv€askyl€a, Finland Received 28 April 2003; accepted 27 January 2004 Avilable online 24 March 2004
Abstract We consider multiple criteria decision-making problems where a group of decision-makers wants to find the most preferred solution from a discrete set of alternatives. We develop a method that uses achievement functions for charting subsets of reference points that would support a certain alternative to be the most preferred one. The resulting descriptive information is provided to the decision-makers in the form of reference acceptability indices and central reference points for each decision alternative. Then, the decision-makers can compare this information with their own preferences. We demonstrate the use of the method using a strategic multiple criteria decision model for an electricity retailer. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Multiple criteria analysis; Multiple criteria decision making; Discrete optimisation; Group decision making; Achievement function
1. Introduction A fundamental problem in choosing a decision aid method for a real-life problem is that different methods may provide different results with the same data, and there is usually no way to objectively identify the best alternative or the best method. The practical relevance of decision support methods can be measured, for example, by
*
Corresponding author. Fax: +358-9-431-38-535. E-mail addresses: kaisa.miettinen@hkkk.fi, [email protected].fi (K. Miettinen). 1 Her research was supported by the Academy of Finland, grant #65760.
the understandability of preference modelling used, how well a number of decision makers (DMs) are supported and whether inaccurate information can be handled (Goicoechea et al., 1982; Hobbs et al., 1992). The majority of multiple criteria decision support methods apply some decision model to aggregate together different criteria measurements and subjective preference information from the DMs (see, e.g., Keeney and Raiffa, 1976; Matsatsinis and Samaras, 2001). A common way to represent the DMs’ preferences is to use weighting coefficients for the criteria. However, the use of weights has many drawbacks and difficulties. In real-life problems it may be impossible to obtain exact and reliable weight information from the DMs. Besides, the meaning
0377-2217/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2004.01.030
786
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of the weights depends on the decision model and it may be unrealistic to assume the DMs to understand very complex decision models. Even though several weight elicitation methods have been suggested, they tend to provide different weights for the same problem (see, e.g., Schoemaker and Waid, 1982; Weber and Borcherding, 1993). Furthermore, when multiple DMs are involved, there may not be any jointly acceptable way to combine weights from multiple DMs. One approach to overcome problems with missing, imprecise or conflicting weight information is to use so-called preference information free methods. These include, for example, the comparative hypervolume criterion by Charnetski and Soland (1978), the overall compromise criterion method by Bana e Costa (1986), and stochastic multicriteria acceptability analysis (SMAA) methods by Lahdelma et al. (1998, 2003) and Lahdelma and Salminen (2001). These methods are based on inverse weight space analysis to reveal what kinds of preferences favour each alternative. Instead of weights, a more straightforward technique to represent the DMs’ preferences in a decision model is through goals or reference points. This means that the DMs specify aspiration levels, that is, desirable or preferable values for each criterion. Goal programming (see, e.g., Charnes and Cooper, 1961) aims at finding a solution that minimises the distance to the specified goal and Wierzbicki (1982, 1986) presented the idea of achievement (scalarizing) functions as a way to overcome some weaknesses of traditional goal programming. Achievement functions are derived on the basis of reference points to project an arbitrary reference point to the set of nondominated attainable solutions. The achievement function is constructed in such a way that if the reference point is dominated, the optimisation will advance past the reference point to a nondominated solution. This means that the drawback of measuring the distance between a reference point and the set of feasible solutions is avoided. Thus, nondominated solutions can be characterized by achievement functions. An achievement function is a function sz : Z ! R, where z 2 Rk is an arbitrary reference point and Z Rk is the set of the alternatives. Here we have k
criteria to be considered simultaneously and they are all to be maximized. Thus, each alternative is a k-dimensional vector. Furthermore, we assume that the DMs prefer more to less. By a nondominated solution z 2 Z we mean a solution (alternative) where none of its components can be improved without impairing at least one of the others. For further details see, for example, Miettinen (1999). Under certain assumptions, the achievement function produces nondominated solutions for any z 2 Rk and different nondominated solutions can be found by moving the reference point only. An example of achievement functions is sz ðzÞ ¼ min ½wi ðzi zi Þ þ q i¼1;...;k
k X
wi ðzi zi Þ;
ð1Þ
i¼1
where q > 0 is a sufficiently small scalar and w is a fixed positive scaling vector. Usually, wi is set to be equal to the inverse of the difference between the best and the worst value for each criterion i. The indifference curves of this function are shifted widened positive orthants where the linear component multiplied by q widens the angles of the indifference curves. Note that the reference point does not affect the shape of the indifference curves, only their position. Also, note that all the reference points on the line zi ¼ zi þ t= wi (i ¼ 1; . . . ; k, t 2 R) correspond to the same family of indifference curves and identical preference order among the alternatives. For more examples of achievement functions, see, for example, Wierzbicki (1986). As mentioned, when compared to weights, reference points provide a more direct way for the DMs to express their desires and, thus, to affect the solution. Reference point approaches model satisfying behaviour instead of optimising (see, e.g., Simon, 1958). Reference point approaches do not, of course, solve all problems in obtaining preference information. Still, these methods may bring the DMs closer to understanding what they really want. However, reference point approaches do not apply directly to group decision-making. A group of DMs may have very different reference points and there is no generally accepted method how to combine this information. For this reason, we
R. Lahdelma et al. / European Journal of Operational Research 164 (2005) 785–791
suggest a new multiple criteria group decisionsupport method for discrete problems based on multiple reference points. The new Ref-SMAA method (SMAA based on reference points) is based on inverse analysis in the reference point space. The reference point space is a region that includes the reference points of individual DMs. The method generates random reference points from the reference point space and evaluates the decision alternatives based on an achievement function. Alves and Clımaco (2001) have recently presented similar ideas about exploring the reference point space, but in the context of multiobjective pure integer programming problems with deterministic criteria measurements. The Ref-SMAA method provides descriptive information of the sets of reference points that favour each alternative. To be more precise, socalled reference acceptability indices measure the variety of reference points that make an alternative most preferred. This information can be used for classifying the alternatives into more or less acceptable ones, and into those that are not acceptable. If the acceptability index is large, the alternative is likely to be acceptable for a large variety of DMs having different preferences when compared to an alternative supported by few reference points only. In addition, the method proposes a typical reference point supporting a certain alternative. This information gives the DMs the possibility to evaluate whether such a reference point could reflect their preferences.
2. The Ref-SMAA method We consider a cooperative group of D DMs who have a set of m alternatives Z ¼ fz1 ; z2 ; . . . ; zm g, from which one is to be chosen. The alternatives are evaluated in terms of k criteria. The suggested method can even easily handle problems where the criteria values are not precisely known. If uncertain or imprecise criteria values are included, they are represented by stochastic variables nij (i ¼ 1; . . . ; m, j ¼ 1; . . . ; k) with an assumed or an estimated joint probability distribution and a density function f ðnÞ in the space Z Rmk .
787
The idea of the method is to chart reference points that support each alternative to be the most preferred one. The method generates random reference points z from the reference point The reference point space is a subset of space Z. Rk , and it is specified so the criterion space, Z that it includes the reference points of individual DMs. The reference point space can be specified so that it includes also the possible inaccuracies in DMs’ reference points. With totally missing information about the DMs’ reference points, the reference point space can be chosen, for example, so that it includes all the alternatives. When some information about the reference points is available, the reference point space can be defined accordingly. The achievement function used in Ref-SMAA may be selected in different ways. It is used to map stochastic criteria and reference point distributions into new distributions sz ðni Þ. The method determines for each alternative zi the set of favourable i ðnÞ defined as reference points Z z ðni Þ P sz ðnj Þ; j ¼ 1; . . . ; mg: i ðnÞ ¼ fz 2 Zjs Z i ðnÞ makes the overall Any reference point z 2 Z i preference of z greater than or equal to the preference of any other alternative. All further analysis is based on the properties of the sets of favourable reference points. In deterministic two-criterion models and using an achievement function of type (1), the sets of favourable reference points can be determined graphically, as illustrated in Fig. 1. This can be done by searching for reference points that make two adjacent alternatives equally good. Such a reference point z is found between each pair of adjacent alternatives by sliding the indifference curve of the achievement function to a position where it intersects the two alternatives. Because all reference points on the line w1 ðz1 z1 Þ ¼ w2 ðz2 z2 Þ correspond to the same set of indifference curves, such lines slice the reference point space into sets of favourable reference points for each alternative. In Fig. 1, we have four alternatives (A, B, C and D) denoted by black dots in a problem involving two deterministic criteria. The indifference curves corresponding to two different reference points (white dots) are shown as dashed
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R. Lahdelma et al. / European Journal of Operational Research 164 (2005) 785–791
B (0.1,0.9)
C
Criterion 2
A (0,1)
(0.75,0.75)
D (0.9,0.1) Criterion 1 Fig. 1. Sets of favourable reference points of four deterministic alternatives (A, B, C and D).
lines. The four sets of favourable reference points are illustrated using different patterns. The descriptive measure reference acceptability index ai (for i ¼ 1; . . . ; m) is defined as the ratio between the expected volume of the set of favourable reference points and the volume of the reference point space. The reference acceptability index is computed as a multi-dimensional integral over the criteria value distributions and the favourable reference point space as Z Z ai ¼ f ðnÞ f ðzÞ dz dn: Z
zi ðnÞ
The reference acceptability index is a measure of the variety of different reference points making the studied alternative the most preferred one. The bigger the index, the more different reference points exist supporting that alternative as the most preferred one. In practice, we generate criterion values from their distributions and reference and points from a uniform distribution f ðzÞ in Z record the share of reference points supporting each alternative as the most preferred one. The reference acceptability index can be used for classifying the alternatives into more or less acceptable ones ðai > 0Þ and into those that are not acceptable (ai is zero or near-zero). Alternatives with large reference acceptability indices are robust candidates particularly in group decision making. It is important to note that such alterna-
tives are likely to remain good solutions also in the future, subject to changing preferences, new stakeholders, and changing or more accurate criteria. If the reference points used represented exactly the reference points of the DMs, the reference acceptability index could be understood as the share of votes supporting each alternative as the most preferred one. However, in a general case the reference points generated by our method cannot be assumed to represent the preference distribution of the real DMs and, thus, reference acceptability indices should not be used for absolute ranking of the alternatives. The central reference point zi (for i ¼ 1; . . . ; mÞ is defined as the expected centre of gravity of the set of favourable reference points. The central reference point is computed as a multi-dimensional integral of the reference point vector z over the criteria value distributions and the favourable reference point space as Z Z zi ¼ f ðnÞ f ðzÞz dz dn=ai : z
zi ðnÞ
The central reference point represents the preferences of an average DM supporting zi . When choosing among different robust (widely acceptable) alternatives, the DMs can compare their own preferences against the central reference points. In a real-life decision problem, central reference points may differ more or less from the actual reference points of the DMs but they can be presented to the DMs in order to help them understand how different reference points correspond to different choices. (In practice, the multi-dimensional integrals are computed through Monte– Carlo simulation by generating random numbers for criteria values and reference points from their distributions and collecting statistics about the alternatives they support.) In all, the descriptive measures of Ref-SMAA, reference acceptability indices and central reference points, are all related to reference points. Thus, the measures as well as the original alternatives all belong to the criterion space. This means that they are easily understandable for the DMs because they have a straightforward meaning in relation with the personal aspiration levels of the DMs. The DMs do not need to adopt and use
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artificial concepts (like weights), which decreases cognitive burden and makes the approach easy to understand. One should note that the reference point space can be defined in many ways and it is not necessarily a subset of the space formed by the alternatives. Potential problems may occur if the reference point space is extended outside that space because the sets of favourable reference points for different alternatives do not grow in the same proportion. The same problems apply also with weights in weight space analysis. However, with weights this problem is alleviated by the fact that it is commonly accepted to use non-negative normalized weights.
3. Example As an example, we consider a strategic decision problem of an electricity retailer. The retailer aims to maximize its long and short-term profit and to increase its market share both in regular and
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‘green’ electricity while managing the risks involved. We have nine alternatives and the following four criteria in the planning period of three years: long-term profit (the profit in millions of euros accumulated during the planning period), short-term profit (the profit in millions of euros from the first year), market share (share (%) at the end of the planning period) and green market share (market share (%) of green electricity at the end of the planning period). The criteria values and their uncertainties are presented in Table 1. We next analyse the problem using the RefSMAA method and the achievement function (1) with equal weights and q ¼ 0:1. Prior to the analysis, the expected values of all the criteria are scaled into the range ½0; 1. Yet, the central reference points are presented in the original scales. We assume that there is no information about the DMs’ reference points. Therefore, we use as the ¼ ½0; 1k . reference point space the hyper-cube Z Table 2 presents the reference acceptability indices ai (in percentage terms) and the components of the central reference points zi for each alternative i.
Table 1 Expected values and standard deviations for the four criteria Alternative
Long-term profit (L)
Short-term profit (S)
Market share (M)
Green market share (G)
S1 S2 S3 S4 S5 S6 S7 S8 S9
439 ± 72 426 ± 62 264 ± 67 444 ± 62 605 ± 58 449 ± 63 449 ± 53 457 ± 63 453 ± 53
163 ± 18 159 ± 16 104 ± 16 163 ± 16 220 ± 16 166 ± 16 164 ± 14 165 ± 16 163 ± 14
12.1 ± 0.2 12.1 ± 0.2 13.1 ± 0.2 12.1 ± 0.2 11.0 ± 0.2 12.1 ± 0.2 12.1 ± 0.2 12.1 ± 0.2 12.1 ± 0.2
9.3 ± 3.2 14.8 ± 3.2 9.3 ± 3.2 9.3 ± 3.2 9.3 ± 3.2 4.3 ± 3.2 9.3 ± 3.2 9.3 ± 3.2 14.8 ± 3.2
Table 2 Reference acceptability indices and central reference points Alt
ai (%)
ziL
ziS
ziM
ziG
S1 S2 S3 S4 S5 S6 S7 S8 S9
8 20 1 7 10 1 8 11 34
437 415 323 441 483 461 440 448 428
165 156 121 165 183 174 165 164 157
12.2 12.2 13.0 12.2 11.3 12.3 12.2 12.2 12.1
8.8 10.4 8.8 8.9 8.8 6.6 8.7 8.6 10.1
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Alternative S9 obtains the largest reference acceptability index of 34%. This means that the largest subset of the reference point space supports this alternative as the best one. The central reference point of S9 is near the centre of the reference point space, indicating that the potential supporters of this alternative require reasonably high criteria values for all the criteria. In addition, a fairly high proportion (20%) of the reference points supports S2. The central reference point related to S2 is very similar to that of S9s. In contrast, alternatives S3 and S6 both obtain an acceptability of only 1%. They are somewhat extreme alternatives. The central reference point related to S3 indicates that its potential supporters are satisfied with fairly low long-term and shortterm profit while emphasising the market share whereas DMs who are not interested in a high green market share would support alternative S6. Note that the alternatives should not in general be ranked based on the reference acceptability index. This index should be used for coarse classification of the alternatives into more or less acceptable ones, and those that are not acceptable.
desired levels that the DMs want to reach as the aspiration levels do. Also, the cognitive limits of the human mind may be less restrictive in conjunction with reference points where the DM can define her/his aspiration level for each criterion separately, while in weight-based methods (s)he must consider trade-offs between two or more criteria simultaneously. In Ref-SMAA, the theoretical problem of combining conflicting preferences of multiple DMs is avoided. It can be used for identifying good compromise alternatives that are acceptable from several points of view. The method computes a reference acceptability index for each alternative, describing the variety of different reference points (in the reference point space) that favour the alternative. The method also computes the central reference point for each alternative, corresponding to the typical reference point of a DM preferring that alternative. The computations can be implemented efficiently through stochastic simulation, where criteria values and reference points are drawn from their distributions. The most important assumption is that the DMs jointly accept the achievement model to be used in the analysis.
4. Conclusions References We have introduced the Ref-SMAA method for supporting discrete multiple criteria decisionmaking involving multiple DMs. The method can be applied for problems where both criteria data and preference information is uncertain or inaccurate (or preference information is totally missing). The uncertain criteria are represented by stochastic distributions. The preference structure of the DMs’ is based on aspiration levels forming reference points in the reference point space and the preferences are represented by achievement functions. The reference point space is defined to reflect the DMs’ possible reference points. Reference points can be expected to be easier to understand for the DMs than, for example, weighting vectors. A reference point describes what the DM wants in a certain situation, while weights do not have such a clear and direct interpretation. Weights only give indication of the trade-offs between criteria, but do not specify the
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