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Multicriteria compensability analysis Article in European Journal of Operational Research · February 2001 DOI: 10.1016/S0377-2217(00)00198-3 · Source: RePEc
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European Journal of Operational Research 133 (2001) 190±209
www.elsevier.com/locate/dsw
Theory and Methodology
Multicriteria compensability analysis Al®o Giarlotta
*
Facolt a di Economia, Istituto di Matematica, Universit a di Catania, Corso Italia 55, 95129 Catania, Italy Received 3 November 1998; accepted 13 July 2000
Abstract Passive and Active Compensability Multicriteria ANalysis (PACMAN) is a multicriteria methodology based on a decision maker (DM)-oriented concept of compensation, called compensability. The preliminary stage of PACMAN is compensability analysis. This is the procedure that translates information about intercriteria compensability provided by the DM into analytical form. In this paper, a theoretical modelization of compensability analysis is proposed. Codi®cation of information is carried out by building the so-called compensatory functions for each ordered pair of criteria, distinguishing the compensating or active criterion from the compensated or passive one. Some technical features of these functions are analyzed and a methodology for their construction is presented. On the basis of compensability analysis the compensatory strength of each criterion can be evaluated by computing its active and passive compensatory powers, thus obtaining a partial preorder on the set of criteria. Moreover, this type of approach to intercriteria relations allows one to estimate the DM's global aptitude to compensate among criteria. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Multiple criteria analysis; Compensability analysis; Pairwise criterion comparison approach; Compensation; Weights
1. Introduction Within Multiple Criteria Decision Aid (MCDA), a basic step in preference modeling is the determination of intracriteria and intercriteria relations (Roy, 1985; Vincke, 1992). In this paper we concentrate our attention on the estimation of intercriteria relations, and we address this topic by
*
Tel.: +39-095-375344; fax: +39-095-370574. E-mail address:
[email protected] (A. Giarlotta).
using a new notion of compensation, called compensability. The concept of compensation has recently received much attention in the literature, being regarded as a fundamental feature in the modelization of intercriteria relations. Compensation among criteria is related to the possibility that a large improvement with regard to one or more criteria can oset a small worsening with regard to one or more other criteria, or vice versa. Therefore, the idea of compensation is naturally linked to other concepts that express a possible relation among criteria, such as those of weight,
0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 0 ) 0 0 1 9 8 - 3
A. Giarlotta / European Journal of Operational Research 133 (2001) 190±209
lexicographic ordering, interaction, synergy, redundancy, etc. Several papers have examined the concept of compensation, both per se and with respect to related concepts. Some authors (Bouyssou and Vansnick, 1986; Bouyssou, 1986; Fishburn, 1976; Matarazzo, 1991; Vansnick, 1986) have de®ned a formal notion of compensation, classifying multicriteria methodologies as either compensatory or noncompensatory. Noncompensatory preference structures and aggregation procedures have been the object of careful study, since (totally) compensatory models employed in classical Multiple Criteria Decision Making (MCDM) do not usually allow a faithful modelization of many decision problems. Other authors (Grabisch, 1996; Marichal and Roubens, 1998; Podinovski, 1994) have introduced a formal notion of the importance of criteria. In some cases, the relationship between importance and interaction among criteria (and, speci®cally, compensation) has been analyzed (Roubens, 1996; Roy and Mousseau, 1996). However, the literature on the topic has been mainly concentrated on the study of multicriteria methodologies, aggregation procedures and preference structures on the basis of compensation and related concepts. Therefore, de®nition and usage of compensation have essentially been methodoriented, since this concept has been regarded as a theoretical device of classi®cation. On the contrary, the notion of compensation examined in this paper, namely compensability, is aimed at capturing the behavior of a decision maker (DM) towards the possibility to compensate among criteria. In our approach, intercriteria compensability remains somehow ``the possibility that an advantage on one criterion can oset a disadvantage on another one'', but as it is determined by a DM and not by a method. Therefore, being more or less compensatory is not regarded here as the characteristic of a multicriteria methodology or of an aggregation procedure. Instead, it is an intrinsic feature of a DM. In this sense, we speak of a DM-oriented usage of the concept of compensation. Our approach is named Passive and Active Compensability Multicriteria ANalysis (PAC-
191
MAN). The de®nition of compensability uses the Pairwise Criterion Comparison Approach (PCCA; Matarazzo, 1990), also distinguishing the compensating (or active) criterion from the compensated (or passive) one. Therefore, compensability is determined for each ordered pair
i; j of evaluation criteria on the basis of ad hoc information provided by the DM. We de®ne i . j-bicriteria compensability as the DM's degree of con®dence about the possibility that a positive dierence of evaluations on the active criterion i has to compensate a negative dierence of evaluations on the passive criterion j. Compensability analysis is the procedure that makes this notion operative via a strict interaction between the DM and the decision aider (DA) on the basis of data provided by the DM. In this paper we present a possible modelization of compensability analysis. The development of the topic is purely theoretical, since our goal is to establish a general framework for the treatment of the concept of compensability. In a successive paper we will describe how compensability analysis can be implemented eectively in PACMAN, and we will discuss both possible applications of this theory to existing methodologies and several practical examples. The paper is organized as follows. In Section 2 we illustrate the goal of our approach by pointing out what the notion of compensability allows one to do in the realm of intercriteria relations. In Section 3 we give a brief outline of the three stages of PACMAN. The next two sections are devoted to an implementation of the ®rst stage of PACMAN, compensability analysis, via construction of compensatory functions. Speci®cally, in Section 4 we study some properties of these functions, whereas in Section 5 we propose a way of constructing them and we apply our procedure to model a lexicographic ordering and a trade-o approach. Further developments of compensability analysis are the subject of the following two sections. In fact, in Section 6 we associate with each criterion a pair of indices expressing its active and passive compensatory strength, and on this basis we obtain a partial preorder on the set of criteria. In Section 7 we build an index that estimates the DM's aptitude to compensate among
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A. Giarlotta / European Journal of Operational Research 133 (2001) 190±209
criteria. Section 8 groups observations and future directions of research. 2. Motivation for the approach The purpose of this section is to illustrate why and how our notion of compensability arises naturally in the context of modeling intercriteria relations. Speci®cally, we ®rst point out the theoretical advantages of the concept of compensation over other related concepts. Successively we show how an eective notion of compensation, as that of bicriteria compensability, could enable one to cope with some typical problems that arise in the modelization of intercriteria relations. Let us observe that our notion of compensability is only an operative version of the concept of compensation. As a consequence, its possible advantages have their basis in the fact that compensation in itself is ab origine a more powerful theoretical tool than other related concepts. In fact, modelization of intercriteria relations is more ¯exible whenever an eective notion of compensation is employed. For instance, a trade-o between criteria is determined necessarily for a couple of criteria and assumes by hypothesis a perfect compensability between them. On the other hand, compensation can be potentially determined for every subset of criteria, and all possible interactions among criteria (synergic eects, redundancy, etc.) can be explicitly taken into account (Grabisch and Roubens, 1997; Mousseau, 1993). Our approach to compensation aims at introducing some tools to study the decision procedures from this alternative point of view. We try to capture information not usually considered in the existing methodologies by stimulating the DM to analyze his aptitude towards compensability. Successively, available information is reordered in a form which is suitable for an easy interpretation and modi®cation, thus providing the basis for a feedback process. An example should clarify the matter. Suppose that a DM wants to buy a new car and he bases his decision on ®ve criteria: maximum speed
ms, safety
sf , horse power
hp, price
pr and fuel consumption
fc; the ®rst three cri-
teria are to be maximized, the last two minimized. Furthermore assume that he gives the following piece of preferential information about intercriteria relations: he is willing to pay $1000 more for an increase of 20 hp in horse power. In our language, this means that a positive dierence of 20 hp totally compensates a negative dierence of $1000. Instead, in a traditional MultiAttribute Value Theory (MAVT) setting, this piece of information is used to establish a precise value for the trade-o between the two criteria pr and hp. In other words, the given data are automatically extended as follows: there is a perfect compensability between a dierence of $1000 in price and a dierence of opposite (in the obvious sense, since hp is to be maximized, pr minimized) sign of 20 hp in horse power, regardless of the role (active or passive) that each criterion plays. Now, a number of questions naturally arise: 1. Does this piece of information imply that the DM is willing to pay $100 more to get an increase of 2 hp? Or, using the terminology of compensation, does a positive dierence of 2 hp perfectly compensate a negative dierence of $100 in price? 2. Is it necessarily true that the DM is willing to accept a reduction of 20 hp versus a decrease of $1000 in price? That is, in our language, does a positive dierence of $1000 perfectly compensate a negative dierence of 20 hp? 3. What about 17 hp instead of 20 hp, keeping ®xed the dierence in price? Does this positive dierence in horse power still totally compensate a $1000 negative dierence in price? Or, more likely, is its compensatory eect only partial? And, in the latter case, to what extent? The ®rst type of question is related to the linearity of the trade-o. This means that a proportional variation of the two dierences does not aect the value of the trade-o, which depends upon their ratio only. Since linearity of the trade-o is postulated in many MAVT models, the answer to questions of this kind is always positive in these cases. Questions of the second type are connected to the symmetry of the trade-o. Thus, it is not relevant as to which is the compensating criterion and the compensated one. In our example, the given
A. Giarlotta / European Journal of Operational Research 133 (2001) 190±209
piece of information was originally related to the possibility that a positive dierence of 20 hp can completely oset a negative dierence of $1000. Owing to the property of symmetry, this was also extended to the possibility that a positive dierence of $1000 can totally compensate a negative dierence of 20 hp. Since a trade-o is always symmetric, again the answer to this kind of questions is positive. Finally, questions of the third type are even meaningless in a trade-o context, since there is no possibility of partial compensation. It is possible to approach the problem of determining the intercriteria relations in a dierent way, using tools that address all the questions above and that give the possibility of answering in a way which depends on the DM's aptitude to compensate among criteria. In fact, the trade-o approach has the great advantage of being easily understandable and handy, but it does not provide the DM with the capability of choosing an answer to the natural questions raised above. Indeed, it is true that the process leading to the determination of a trade-o is a complex one and possibly requires several evaluations and assessments from the DM. On the other hand, all this information is, at the end of the process, collapsed into a single output, the trade-o, which is related to an ordered pair of values only. And it should be treated as such. In our opinion, assumptions as those related to linearity and symmetry are very useful in order to simplify modelization, but they probably add some arbitrariness to it. Furthermore, a tradeo approach does not provide the DM and the DA with the possibility of dealing with cases of partial compensation, as those illustrated in (3). In our approach we try to address all these problems constructively. In fact, we synthesize the relationship between any two criteria in an analytical way, by building two functions that examine the compensatory eect of each criterion over the other one. Speci®cally, we construct, for each couple of criteria i and j, two compensatory functions: CFi.j (the compensatory function of i over j) and CFj.i (the compensatory function of j over i). The function CFi.j evaluates the compensatory eect of any positive dierence on the active cri-
193
terion i over any negative dierence on the passive criterion j. Therefore, linearity is not assumed, it may only result as a consequence of the information provided by the DM, after being reordered in the form of compensatory function. Note that the symmetric compensatory eect, obtained by considering j as the active criterion and i as the passive one, is measured by a dierent function, CFj.i . Thus, symmetry is not a necessary feature of the approach, even if it might be a highly desirable property in some particular cases. Finally, these functions range over the closed interval 0; 1, where 0 corresponds to absence of compensation and 1 to total compensation. All intermediate values express the possibility of a partial compensation, thus addressing the third type of questions. Of course, there are many drawbacks to our approach, the ®rst being the amount and precision of information required for constructing all compensatory functions. We will address these problems later on (see Sections 4, 5 and 8). 3. PACMAN In this section we give a brief outline of PACMAN (for a detailed description, see Giarlotta, 1998). We start with some preliminary de®nitions related to a quantitative discrete multicriteria decision problem. 3.1. Preliminary de®nitions Let A fa; b; . . .g be a ®nite set of m P 2 alternatives, and G fgj : j 2 J g; J f1; . . . ; ng, be a family of n P 2 criteria of evaluation, where gj : A ! R is an interval scale of measurement (Roberts, 1979) representing the jth criterion according to a nondecreasing preference. In the following we shall identify criterion gj with its index j, and the set of criteria G with its index set J as well. In this way we can refer to a generic alternative a 2 A as an n-tuple
a1 ; . . . ; an , where the jth component is the evaluation of a 2 A by means of criterion j 2 J , i.e., aj gj
a. Without loss of generality, we assume that on each criterion there
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A. Giarlotta / European Journal of Operational Research 133 (2001) 190±209
are at least two levels that evaluate a particular attribute (or class of attributes). Let aj and bj be two reference evaluations provided by the DM on criterion j 2 J , such that aj < bj . These evaluations are usually ®xed a priori by the DM according to the decision problem at hand. In the following, aj and bj will be, respectively, the minimum and the maximum values that can be assumed on j. Thus, aj 2 aj ;bj 8j 2 J ; 8a 2 A. The dierence of the two bounds determines, for each j 2 J , the admissible range rj bj aj . Let dj : A A ! R, de®ned by dj
a; b aj bj 8
a; b 2 A A; be the dierence function on criterion j 2 J . The normalized dierence function Dj : A A ! R on criterion j 2 J is then de®ned by Dj
a; b dj
a; b=rj . Obviously, 1 6 Dj
a; b 6 1 8j 2 J 8
a; b 2 A A. Normalized dierence functions will be the building blocks of compensability analysis. In fact, construction of compensatory functions is based on the comparison of normalized dierences on two distinct criteria. Notice that the hypothesis that criteria are interval scales plays a fundamental role in this setting. Indeed, only under this assumption dierence functions provide a meaningful tool for comparing criteria. 3.2. The three stages of PACMAN PACMAN is composed of three successive steps: (1) compensability analysis, the procedure aimed at modeling intercriteria relations by means of compensability; (2) construction, at several levels of aggregations of binary indices (related to pairs of alternatives), expressing the degree of active or passive preference of an action over another one; (3) determination of a binary relation of strict preference, weak preference, indierence or incomparability for each couple of alternatives, on the basis of two valued relations of compensated preference. Since in this paper we deal extensively with the ®rst step, in the sequel we will brie¯y sketch the other two stages.
(2) The goal of PACMAN is to determine the relation between any two alternatives a; b 2 A by using the results of compensability analysis. This is accomplished by constructing some binary indices related to a and b, aiming at an estimation of active and passive compensatory eects. The basic idea is that any positive normalized dierence Di
a; b has a double eect: active, since it gives some contribution to the (possible) overall preference of a over b; passive, since it determines the resistance to the (possible) overall preference of b over a. Therefore, a partial preference of a over b on criterion i 2 J may enlarge both the set of arguments to accept a global preference of a over b and the set of arguments to reject a global preference of b over a (Roy, 1985, 1990; Roy and Mousseau, 1996). Estimation of active and passive eects is done using the PCCA approach. In fact, construction of the binary indices that evaluate these eects is done in two stages: modelization on pairs of criteria ®rst, and aggregation of these partial results successively. A detailed description of the procedure can be found in Giarlotta (1998). (3) The last step is the determination of a relation between any two alternatives, which could be strict preference P, weak preference Q, indierence I or incomparability R. In PACMAN, the construction of a relational system of preferences is shaped by a relation of compensated preference. This valued binary relation is determined, for each pair of alternatives
a; b 2 A A, from the values of the two reciprocal global net indices related to a and b. In Fig. 1, how to determine the relation between a and b from the values of the two reciprocal global net indices P
a; b and P
b; a is shown. The result is a fundamental system of preferences (Roy, 1985). 4. Compensability analysis The ®rst step in PACMAN aims at estimating intercriteria compensability on the basis of information given ad hoc by the DM. Providing this type of data often presents several diculties to the DM. In fact, as shown in psychological studies (Payne et al., 1993), the human mind is able to
A. Giarlotta / European Journal of Operational Research 133 (2001) 190±209
Fig. 1. Determination of a relation between the two alternatives a and b on the basis of the values of global net indices.
process only few conceptual units at a time. In order to partially overwhelm this type of diculties, we ask the DM to evaluate intercriteria compensability on the basis of comparisons among few objects. Speci®cally, intercriteria relations are estimated via bicriteria compensability, taking into consideration two criteria at a time and examining their reciprocal behavior with respect to compensation. The output of this process is the construction of a compensatory function CFi.j for each ordered pair of criteria
i; j 2 J 2 . A precise estimation of compensatory eects for every possible active and passive dierence is clearly a dicult task to accomplish, both for the amount and the sensitivity of information required to obtain a proper and complete modelization. Therefore, we build CFi.j as a fuzzy function (Zimmermann, 1996), associating with any pair of normalized dierences of the type
Di ; Dj 2
0; 1 1; 0 the degree of credibility that the positive dierence Di totally compensates the negative dierence Dj . Successively, extending CFi.j in frontier by continuity, we obtain a fuzzy compensatory function, de®ned on the closed square 0; 1 1; 0 and with values in 0; 1. Thus, the output of this construction is a function which gives a pointwise estimation of the level of compensability of a nonnegative dierence on the ac-
195
tive criterion i over a nonpositive dierence on the passive criterion j. De®ning CFi.j also on the frontier of the domain implies that compensability takes place also when the active and/or the passive dierences are equal to zero, that is, in the zones where, in principle, it is meaningless to speak of compensability. Nevertheless, this adds no inconsistency to the modelization, since it does not aect the value of the binary indices built in the second stage of PACMAN (Giarlotta, 1998). On the other hand, de®nition of compensatory functions also in these limit cases eases their assessment, by allowing a better (graphic) DM/DA interaction. Note also that we are using interchangeably the expressions ``level of compensability'' and ``degree of credibility of total compensability'' in relation to the membership grade of a compensatory function. In fact, according to the three semantics of fuzzy sets (Dubois and Prade, 1997), it shares characteristics of the degree of preference semantic and of the degree of uncertainty semantic. Now we examine some conditions about the possible behavior of compensatory functions. Note that these properties concern compensatory functions related to pairs of distinct criteria only, since we put, by de®nition, CFi.i 0 8i 2 J . The ®rst two conditions A1 and A2, analyzed in Section 4.1, are basic assumptions and must hold for every compensatory function. Further properties A3, A4 and A5 that compensatory functions may verify are examined in Sections 4.2 and 4.3. 4.1. Basic axioms: Weak monotonicities and continuity For each
i; j 2 J 2
i 6 j, the following two conditions must hold for the compensatory function CFi.j : (A1) Weak monotonicities 0 6 x1 6 x2 6 1 ^
16y 60
) CFi.j
x1 ; y 6 CFi.j
x2 ; y; 0 6 x 6 1 ^ 1 6 y1 6 y2 6 0 ) CFi.j
x; y1 6 CFi.j
x; y2 :
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A. Giarlotta / European Journal of Operational Research 133 (2001) 190±209
(A2) Continuity
4.2. Compensability for small and large dierences
CFi.j is continuous everywhere on 0; 1 1; 0:
The ®rst property related to a single function is the following:
The assumptions in (A1) are rather natural. They state that the compensatory eect is an increasing function of both the active and passive dierence (recall that the passive dierence is negative). Note that these conditions do not exclude the possibility of a constant compensatory function, even if this should be regarded as a very unlikely situation. Condition (A2) might seem a less natural assumption at ®rst sight. Nevertheless, continuity of compensatory functions is more than an ad hoc conjecture introduced only to simplify the assessment of technical parameters. Indeed, the ratio of this assumption is purely theoretical, since our goal is to ensure that the quanti®cation of the compensatory eect is done without gaps. To this aim we should require that compensatory functions have a Darboux-type property (i.e., they satisfy the intermediate value property on the range). For the sake of simplicity, we postulate the (slightly stronger) continuity condition. On the other hand, requiring (A2) does not reduce the range of applicability of our approach, since every Lebesgue measurable real-valued function de®ned on a compact Hausdor space can be approximated by a continuous function to any degree of precision (for an exact formulation of this principle in its full generality, see Lusin's Theorem, Folland, 1984, p. 211). Finally, note that no smoothness condition is required. Some compensatory functions may satisfy additional conditions as well. These properties are related either to a single function (see (A3)) or to a couple of functions (see (A4) and (A5)). Condition (A3) postulates a typical behavior of a single compensatory function in a small neighborhood of the points
1; 0 and
0; 1 of the domain. On the other hand, properties (A4) and (A5) take into account couples of compensatory functions, examining the relationship existing between two reciprocal compensatory functions (of i over j, and of j over i).
(A3.1) Total compensability for large over small dierences 9d > 0 j x 2
1
d; 1 ^ y 2
d; 0
) CFi.j
x; y 1: This condition states that large positive dierences on the active criterion i completely oset small negative dierences on the passive criterion j. By the continuity axiom, this property implies CFi.j
x; 0 1 8x 2
1 d; 1: Therefore, whenever (A3.1) holds, there is a basic neighborhood of the point
1; 0 in the subspace topology of 0; 1 1; 0 R2 , where the degree of credibility of total compensability is 1. Condition (A3.1) represents the most likely relationship within any pair of criteria. Nevertheless, it might happen that some pairs of criteria show a dierent behavior. Therefore, we can associate with each pair (i; j) of distinct criteria a value M M
i; j 2 0; 1 that estimates the compensatory eect of i over j for large active dierences over small passive dierences: (A3.1M ) M-compensability for large over small differences 9M 2 0; 1 9d > 0 j x 2
1
d; 1 ^ y 2
d; 0
) CFi.j
x; y M: The last condition states that, for some pairs of criteria, compensability is usually constant in a small neighborhood of the point
1; 0, but the compensatory eect needs not to be total. In these cases, a value of M less than 1 needs to be interactively assessed. Two examples of compensatory functions satisfying (A3.1M ) (for M 1 and M 0:8) are given in Fig. 2. Some remarks are in order. First, (A3.1) is a particular case of (A3.1M ), obtained for M 1. We stated (A3.1) beforehand because it is the most likely instance of (A3.1M ). In fact, a value of M less than 1 is possible only in speci®c cases, in which the active criterion is very weak and/or the passive
A. Giarlotta / European Journal of Operational Research 133 (2001) 190±209
197
4.3. Symmetry and complementarity
Fig. 2. Two compensatory functions satisfying A3.1M and A3.2m for
M; m
1; 0:3 and
M; m
0:8; 0.
criterion is very strong. In particular, it is rather unlikely that some pairs of criteria verify the other limit case of condition (A3.1M ), obtained for M 0. Nevertheless, some existing modelizations do assume (A3:10 ) implicitly for many pairs of criteria (cf. Section 5.3 on the lexicographic approach). Note that if (A3:10 ) holds, CFi.j is identically equal to zero, because of the monotonicity axioms. Finally, observe that there can be compensatory functions that do not satisfy (A3.1M ) for any M 2 0; 1, e.g., a function strictly increasing in at least one argument. Next, we examine a condition which is dual to (A3.1). It describes a possible (and natural) behavior of a compensatory function in a small neighborhood of the point
0; 1: (A3.2) Null compensability for small over large dierences 9d > 0 j x 2
0; d ^ y 2 1; 1 d ) CFi.j
x; y 0: By the continuity condition, CFi.j
0; y 0 8y 2 1; 1 d. More generally, we can associate with each pair
i; j 2 J 2 of distinct criteria a number m m
i; j 2 0; 1 evaluating the compensatory eect of small active dierences on i over large passive dierences on j. (A3.2m ) m-compensability for small over large differences
In general, it is not requested that two reciprocal compensatory functions CFi.j and CFj.i be connected by a functional relation, since they are determined on the basis of dierent information given by the DM. Nevertheless, in some cases (see Section 5.3 for some illustrative examples) these functions could be linked to each other. In the following, we examine some theoretical conditions related to their reciprocal behavior. Our ®rst goal is to associate with each couple fi; jg J of distinct criteria two indices: a symmetry index (measuring the degree of similarity of CFi.j and CFj.i ) and a complementarity index (measuring the degree to which CFi.j and CFj.i are complementary to each other). To this aim we build two auxiliary functions related to these two compensatory functions, namely: · SymDif ij : 0; 12 ! R, de®ned 8
x; y 2 0; 12 by SymDif ij
x; y jCFi.j
x; y
· ComplSumij : 0;1 ! R, de®ned 8
x;y 2 0; 12 by ComplSumij
x; y CFi.j
x; y CFj.i
y; x: One can easily check that these functions satisfy the following properties: Lemma 4.1. For every i, j 2 J ; i 6 j, and for every x; y 2 0; 1 it results: (a) 0 6 SymDif ij
x; y 6 1; (a0 ) 0 6 ComplSumij
x; y 6 2, (b) SymDif ij
x; y SymDif ji
x; y; (b0 ) ComplSumij
x; y ComplSumji
y; x. Now, de®ne, for each pair of distinct criteria
i; j 2 J 2 , their symmetry index symm
i; j and their complementarity index compl
i; j as follows: Z 1Z 1 SymDif ij
x; y dx dy; symm
i; j 1 0
9m 2 0; 1 9d > 0 j x 2
0; d ^ y 2 1; 1 d ) CFi.j
x; y m: See Fig. 2 for two examples of functions satisfying (A3.2m ).
CFj.i
x; yj;
2
compl
i; j 1 1
Z
1 0
0
Z 0
1
ComplSumij
x; y dx dy :
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A. Giarlotta / European Journal of Operational Research 133 (2001) 190±209
The next result collects some obvious facts about these indices. Proposition 4.2. For every i; j 2 J ; i 6 j, it results: (a) symm
i; j 2 0; 1; (a0 ) compl
i; j 2 0; 1, (b) symm
i; j symm
j; i, (b0 ) compl
i; j compl
j; i, (c) symm
i; j 0 () either
CFi.j 0 and CFj.i 1 or
CFi.j 1 and CFj.i 0, (c0 ) compl
i;j 0 () either CFi.j 0 CFj.i or CFi.j 1 CFj.i , (d) symm
i; j 1 () CFi.j CFj.i , (d0 ) compl
i; j 1 ( CFi.j
x; y 1 CFj.i
y; x 8x; y 2 0; 1. Note that in (d0 ) the condition on the right-hand side is sucient but not necessary to be compl
i; j 1. In some decision problems, many reciprocal compensatory functions might show only a partial degree of symmetry and of complementarity. In particular, with respect to the latter, a couple fi; jg of criteria could be ipocomplementary or ipercomplementary in one of the following ways: Weak ipocomplementarity [ipercomplementarity] 9x; y 2 0; 1 j CFi.j
x; y CFj.i
y; x < 1 > 1: Normal ipocomplementarity [ipercomplementarity] Z 1Z 0 CFi.j
x; y dx dy 0
1
Z
1 0
Z
0 1
CFj.i
x; y dx dy < 1
> 1:
Reciprocal compensatory functions will usually be either ipocomplementary or ipercomplementary in some sense. Returning to our example of cars (see Section 2), assume that the DM likes sports cars, but at the same time he wants a very high level of safety. Then, all couples of criteria containing sf will probably be (at least normally) ipocomplementary couples, whereas fhp; msg and ffc; prg will most likely be two (at least normally) ipercomplementary couples. The ideas of iper- or ipocomplementarity are strictly related to the concepts of redundancy and synergy among criteria, respectively. The latter notions have been recently introduced in the multicriteria literature (Roubens, 1996). In some cases (see Section 5.3), there might be couples of criteria that do satisfy equality constraints, such as: (A4) Perfect symmetry CFi.j
x; y CFj.i
x; y
8x; y 2 0; 1:
(A5) Perfect complementarity CFi.j
x; y CFj.i
y; x 1
8x; y 2 0; 1:
Note that a couple of criteria ful®lling one of these conditions will show a limit degree of symmetry and/or complementarity. In fact, by Proposition 4.2, (A4) holds i symm
i; j 1, whereas (A5) implies compl
i; j 1. Couples of criteria satisfying (A4) and (A5) will be called, respectively, symmetric and complementary. In the example of cars, fhp; msg and ffc; prg could be symmetric couples. Fig. 3 shows a complementary couple of criteria.
Strong ipocomplementarity [ipercomplementarity] 8x; y 2 0; 1 j CFi.j
x; y CFj.i
y; x < 1 > 1: Clearly, strong implies normal, which in turn implies weak, for both ipo- and ipercomplementarity. Couples of criteria satisfying these conditions will be called (weakly, normally, strongly) ipocomplementary and ipercomplementary, respectively.
Fig. 3. A complementary couple fi; jg of criteria.
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Finally, a criterion c 2 J is said to be absolutely complementary i all the doubletons in J containing c are complementary, i.e., ComplSumch 1 8h 2 J n fcg. In the following (see Sections 6 and 7), we will denote by Compl
J the subset of J containing all absolutely complementary criteria. 5. Construction of compensatory functions In Section 5.1 we state some properties of compensatory functions that are forced by basic axioms and other conditions. A possible procedure to build compensatory functions is sketched in Section 5.2 (and completed in Appendix A). Finally, we present two illustrative examples in Section 5.3. 5.1. Some properties Recall that CFi.j
Di ; Dj measures the degree of credibility that the active dierence Di completely compensates the passive dierence Dj . Thus, we can distinguish three levels of compensability for each CFi.j , corresponding to total, partial and null degrees of credibility. They are de®ned, respectively, as follows: toti.j CFi.j1
f1g;
parti.j CFi.j1
0; 1;
nulli.j CFi.j1
f0g: The family ftoti.j ; parti.j ; nulli.j g is a partition (possibly improper, since some of the classes may be empty) of the domain 0; 1 1; 0 of CFi.j . More generally, let Mij and mij , 0 6 mij 6 Mij 6 1, be the maximum and minimum values taken by CFi.j (see axioms (A3.1M ) and (A3.2m )). Then, we can consider the related partition fmaxi.j ; inti.j ; mini.j g of 0; 1 1; 0, where max CFi.j1
fMij g; i.j
inti.j CFi.j1
mij ; Mij ;
min CFi.j1
fmij g i.j
correspond, respectively, to maximum, intermediate and minimum levels of compensability of i over j. Note that, unless Mij mij (i.e., CFi.j is con-
199
stant), there will be three classes in the partition, because of axiom (A2). In order to assess a fuzzy function CFi.j , the DM is required to determine only the areas where the level of compensability is maximum and minimum, since in the area of intermediate compensability the behavior of CFi.j is adapted by the DA in the way we shall soon discuss. Acceptance by the DM of the basic axioms and of some other conditions simpli®es the assessment of compensatory functions, as it is shown by the following facts (for all unde®ned notions, see Massey, 1967; Engelking, 1989): Proposition 5.1. For each i; j 2 J
i 6 j it results: (a) if (A1) holds for
i; j, then maxi.j and mini.j are nonempty simply connected subsets of 0; 1 1; 0; (b) if (A2) holds for
i; j, then maxi.j and mini.j are nonempty compact subsets of 0; 1 1; 0. Proof (for maxi.j ). (a) We claim that maxi.j is starlike with respect to the point
1; 0, i.e., for any other point
x; y 2 maxi.j the line segment joining
x; y and
1; 0 lies entirely in maxi.j . Obviously, axiom (A1) yields that
1; 0 2 maxi.j . Now ®x
x; y 2 maxi.j nf
1; 0g, and consider the subset of the domain de®ned by R
x; y f
w; z 2 0; 1 1; 0 : w P x ^ z P yg: Then, R
x; y maxi.j by (A1), whence the line segment joining
x; y and
1; 0 lies entirely in maxi.j . This proves the claim. Now our result follows readily from the following two facts (see Massey, 1967, pp. 66±67): if X Rn is starlike with respect to the point x0 2 X , then X is contractible to the point x0 ; if a topological space X is contractible to a point, then X is simply connected. (b) Since maxi.j is obviously bounded, it suces to show that it is closed. This is an immediate consequence of the hypothesis. In particular, we have: Corollary 5.2. For each i; j 2 J
i 6 j it results:
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(a) if (A1) holds for
i; j, then toti.j and nulli.j are simply connected subsets of 0; 1 1; 0; (b) if (A2) holds for
i; j, then toti.j and nulli.j are compact subsets of 0; 1 1; 0; (c) if (A1), (A2) and (A3.1), [A3.2] hold for
i; j, then toti.j nulli.j is a nonempty simply connected compact subset of 0; 1 1; 0.
functions need to be constructed. In fact, the existence of a symmetric or a complementary couple fi; jg J allows one to compute CFi.j from CFj.i (or vice versa).
In the sequel, by inner boundary of maxi.j mini.j we mean the planar curve that bounds maxi.j mini.j towards the interior of the square 0; 1 1; 0. A further property of maxi.j is the following:
Assessment of compensatory functions for each ordered pair of distinct criteria is a rather complex procedure. A continuous and careful interaction between the DM and the DA is necessary to provide a faithful representation of intercriteria relations by means of compensability. In this respect, we can distinguish two successive phases in the procedure used to construct a compensatory function: (1) Determination (by the DM) of the two subsets of the domain corresponding to maximum and minimum levels of compensability. (2) Assessment (by the DA) of the behavior of the function in the zone of intermediate compensability. Even if the ®rst step is mainly required of the DM and the second required of the DA, both of them rely on a strict DM/DA dialogue. In the following we sketch the two phases of the procedure to assess CFi.j . (i) First of all, the DM has to determine the limit levels of credibility for compensability of i over j, namely, maximum level M Mij and minimum level m mij . In most circumstances maximum will be total (i.e., M 1) and minimum will be null (i.e., m 0), but other cases are theoretically possible. Successively, the DM will decide whether conditions (A3.1M ) and (A3.2m ) hold or not. If they are acceptable, the areas of maximum
maxi.j and minimum
mini.j compensability need to be determined. Due to monotonicities, it suces to assess only the inner boundary of these areas. This could be achieved in several ways, via a strict DM/ DA interaction. For instance, the DM could provide information in order to determine two points on each boundary, and then the DA could adapt (according to the DM's scheme of preferences) two plane curves (linear, circular, etc.) passing through these points.
Proposition 5.3. Let i; j 2 J
i 6 j be such that (A1) holds for
i; j. If 9M 2 0; 1 such that (A3.1M ) holds for
i; j, then any line with positive slope lying in the Di Dj -plane intersects the inner boundary of maxi.j in at most one point. Proof. Assume that there is a line with slope r > 0 intersecting the inner boundary curve of maxi.j in at least two points, say
x; y and
w; z. Since r > 0, we can assume without loss of generality x < w and y < z. But then (A1) implies again that R
x; y maxi.j , whence
w; z should be in the interior of maxi.j . A contradiction. Of course, the same result holds, mutatis mutandis, for mini.j . The geometrical property stated in the last proposition will be determinant to construct CFi.j in the most dicult cases (see Section 5.2 and Appendix A). It is not unlikely that most compensatory functions do satisfy, besides the basic conditions of monotonicity (A1) and continuity (A2), also those of large over small total compensability (A3.1) and small over large null compensability (A3.2). In these cases, Corollary 5.2 (c) and Proposition 5.3 ensure that the areas of the domain corresponding to total and null degrees of credibility have a rather regular shape, thus simplifying the assessment of the relative compensatory functions. Note that the conditions of perfect symmetry and perfect complementarity turn out to be helpful assumptions as well. Indeed, if (A4) or (A5) hold for some couples of criteria, fewer compensatory
5.2. The two phases of the construction
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(2) In the next step, the behavior of CFi.j in the area inti.j of intermediate compensability has to be determined by the DA. In this respect, there are several ways of extending by continuity the de®nition of the compensatory function to all its domain. Our choice is motivated by the following remark. Suppose that the DM is not able (or does not want) to give any information about i . j-compensability. Is there any function that is a natural candidate for representing the level of compensation of an active dierence on i over a passive dierence of j? Owing to total absence of information, we conjecture that the compensatorial behavior of i over j can be approximated by the ®rst function represented in Fig. 4. That is, the restriction to 0; 1 1; 0 of the plane passing through the points
1; 0; 1,
1; 1; 1=2 and
0; 1; 0. This will be our reference function. This function will play the main role in the construction of compensatory functions. In fact, the principle underlying its construction will be used to model compensatory functions in the cases of partial or total lack of information from the DM. Speci®cally, in the process of completing the de®nition of compensatory functions, we will employ the following guideline: extend consistently pieces of information already provided by the DM, minimizing introduction of arbitrary information. The conditions (described below) that we require CFi.j to satisfy are technical instances of this general principle. The ®rst condition that any compensatory function must ful®ll is due to the total lack of information from the DM in the are inti.j : CFi.j has
Fig. 4. Absence or partial lack of information: Reference compensatory function and a possible variation.
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to be strictly increasing in both arguments. This reasonable condition should minimize arbitrariness in the process of quantifying the compensatory eect for intermediate values of the normalized dierences. The second condition aims at respecting as faithfully as possible form and extension of the areas (if any) of minimum and maximum compensability, as chosen by the DM in the preceding step. This is obtained by modeling the level curves of CFi.j (i.e., its sections with planes parallel to the Di Dj -plane) via a continuous transformation of the inner boundary of mini.j into the inner boundary of maxi.j . Of course this procedure cannot be applied whenever both maxi.j and mini.j degenerate into the singletons f
1; 0g and f
0; 1g, respectively. In this limit case, a simplifying condition of linearity is assumed for the assessment of CFi.j within inti.j . More precisely, the following ®ve cases may occur: (i) both maxi.j and mini.j are singletons; (ii) maxi.j is a singleton and the inner boundary of mini.j is a segment; (iii) mini.j is a singleton and the inner boundary of maxi.j is a segment; (iv) both inner boundaries are segments and they are coplanar; (v) all the other cases. In case (i), the compensatory function is completely determined by linearity: CFi.j will be the restriction to 0; 1 1; 0 of the plane passing through the points
1; 0; Mij ;
0; 1; mij and
1; 1;
Mij mij =2. Note that, for Mij 1 and mij 0, this is our reference function. In Fig. 4, the second graph is that of a compensatory function CFi.j for which maxi.j f
1; 0g; mini.j f
0; 1g, and the limit values of compensability are Mij 0:7 and mij 0:1. In all other cases the faithfulness condition applies, and assessment of CFi.j in inti.j is determined on the basis of the form and extension of the limit areas mini.j and maxi.j . Transition from mini.j to maxi.j is achieved via a suitable transformation of the interior boundary of these areas, that is, by modifying continuously the shape of the inner curve that limits mini.j until reaching the inner boundary of maxi.j . The linearity condition
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eases the assessment of CFi.j within inti.j in cases (ii), (iii) and (iv). In these situations, CFi.j is completed by taking the restriction to inti.j of a uniquely determined plane: the plane containing the point and the segment in cases (ii) and (iii), the plane containing the two segments in case (iv). The construction of CFi.j within inti.j is more complicated in case (v). Basically, completion is achieved by using a suitable homotopic transformation between an extension of the limit curve of mini.j and an extension of the limit curve of maxi.j . See Appendix A for a careful description of the technical details of the construction.
5.3. Two illustrations In this section we apply compensability analysis to model two cases of well-known types of intercriteria information: the lexicographic ordering and the trade-o approach. The lexicographic model (Fishburn, 1974) has been extensively studied in the literature and has been classi®ed as a typical example of totally noncompensatory model. Recall that, in a (pure) lexicographic aggregation procedure, the set of criteria J is totally ordered in such a way that if i precedes j in this order (notation: i j, then any dierence on i cannot be compensated at all by any dierence on j. In other words, only the ordinal rank of criteria is relevant in establishing preferences among alternatives. Compensability analysis can be used naturally as a tool to model a lexicographic ordering of criteria. In order to assess compensatory functions, we note that if i precedes j in the corresponding total order on J , then not only any negative difference on i cannot be compensated at all by a positive dierence on j, but also any positive difference on i compensates totally a negative dierence on j. In other words, for each i; j 2 J such that i j, the related reciprocal compensatory functions are of the following types: CFj.i 0
and
CFi.j 1:
Consequently, all compensatory functions satisfy, besides (A1) and (A2), either total compensability
for large over small dierences (A3.1) or null compensability for small over large dierences (A3.2). Moreover, while perfect symmetry is strongly violated by all reciprocal pairs, they do satisfy perfect complementarity, i.e., J Compl
J . In a MAVT approach, intercriteria information is always represented in the form of trade-os (Keeney and Raia, 1976). Formally, a trade-o sij between two criteria i; j 2 J (to be maximized) is de®ned as sij dj =di , where dj denotes the increment in the evaluation on criterion j necessary to compensate a decrement di in the evaluation on criterion i, all the evaluations on the other criteria being kept constant. In practice, for the sake of simplicity, in a lot of decision models the trade-os are assumed to be both linear and constant. Linear means that a decrement of h times di will be perfectly compensated by an increment of h times dj for all meaningful values of a positive h. Constant means that sij depends only on the ratio between di and dj , and not on the level of the evaluations on i and j. Note that in the classical additive model (Keeney and Raia, 1976) these conditions are implicitly assumed. In the case of linear and constant trade-os, we can model intercriteria relations using compensability analysis as follows. After normalization of evaluations, the trade-o between criteria i and j ensures a perfect compensability between the two positive normalized dierences D0i (on criterion i) and D0j (on criterion j), without distinguishing active and passive criteria. Therefore, the two reciprocal compensatory functions will be given the value 1 in the following points: CFi.j
D0i ; D0j 1 CFj.i
D0j ; D0i : Now, due to the linearity of trade-o, both compensatory functions will assume the value 1 also along a segment: CFi.j in the points of the domain lying on the line passing through
0; 0 and
D0i ; D0j ; CFj.i in those along the line containing
0; 0 and
D0j ; D0i . Finally, the monotonicity axioms force CFi.j 1 and CFj.i 1 in the two areas (right triangles or trapezoids) having these two segments, respectively, as inner boundary. See Fig. 5.
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Fig. 5. Partial de®nition of compensatory functions in a tradeo case.
Therefore, information about the trade-o between i and j determines exactly toti.j and totj.i . Now, the de®nition of CFi.j and CFj.i in the whole domain could be completed in any way. For instance, using the linearity condition, we can complete them by taking the restriction of the planes passing through the above segments and the point
0; 1 (cf. case (iii) in Section 5.2). Thus, any couple of compensatory functions will always satisfy (beside basic axioms) at least the total compensability for large over small dierences (A3.1). Furthermore, it is immediate to prove that, regardless of how the de®nitions of compensatory functions are completed, all couples of criteria will be at least normally ipercomplementary in this case (cf. Fig. 5). 6. Compensatory strength of criteria The concept of weight of criteria has been extensively studied in multicriteria decision analysis, since it represents a leading aspect in the determination of intercriteria relations. Unfortunately, its interpretation is not straightforward, being strictly linked to the way in which it is used. In a MAVT context, weights are in fact tradeos between criteria, i.e., scale constants allowing homogeneous comparisons on dierent dimensions. Conversely, in an outranking approach, weights represent importance of criteria, since they are connected to the strength of possible coalitions of criteria (Roy, 1985; Vincke, 1992). Several approaches have been proposed in the literature to deal with the dicult problem of estimating weights of criteria. Recall, among the
203
others, Saaty (1980), Bana e Costa (1986), Solymosi and Dombi (1986), Takeda et al. (1987), Vansnick (1986). Other authors (Mousseau, 1993; Roy and Mousseau, 1996) have developed a theoretical framework to analyze the notion of importance of criteria. Within this approach, they also establish links between importance and compensation. A dierent formalization of the concept of importance is given in Podinovski (1994). Finally, a notion of weight has been developed also within the framework of game theory, linking it to the concept of interaction among criteria (Grabisch, 1996; Roubens, 1996; Grabisch and Roubens, 1997; Marichal and Roubens, 1998). In PACMAN intercriteria relations are modeled on the basis of bicriteria compensability only. Therefore, we cannot aim at estimating the weight of a criterion in a traditional way. Nevertheless, a meaningful measure of the strength of a criterion can be determined by using compensability analysis. Since we distinguish active from passive compensability of a criterion, we shall have two indices of its compensatory strength: capability to compensate actively the other criteria of J on one hand, and capability to resist passively the remaining criteria of J on the other hand. For each i 2 J we de®ne the active compensatory power of i by Pn R 1 R 0 CF
D ; D dD dD i.j i j i j j1 0 1 pi ; n 1 and the passive compensatory power of i by Pn R 1 R 0 CF
D ; D dD dD j.i j i j i j1 0 1 pi 1 : n 1 Note that pi ; pi 2 0; 1 8i 2 J . Clearly, the higher these indices, the stronger the criterion. A large value of pi is typical of a criterion that can possibly give a great contribution to the relation of (compensated) preference. In the example of sports cars, horse power and maximum speed should be of this type. On the other side, a criterion with a high pi can most likely be a strong opponent to the relation of (compensated) preference, i.e., it might act as a veto criterion (Roy,
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1985). In the considered example, safety could play this role. A pair of values close to
1; 0 will be typical of a criterion giving important active contribution to preference but scarce passive resistance to it, whereas pairs close to
0; 1 will characterize criteria that are not very active but are very resistant. The possible presence of a dominant criterion, that is, i 2 J such that CFi.j 1 and CFj.i 0 for each j 2 J n fig, is codi®ed by
pi ; pi
1; 1. Formally, the compensatory strength of each criterion is given by the map Pow : J ! 0; 1 0; 1, de®ned by Pow
i
pi ; pi 8i 2 J . Then, if we take the simple product order on 0; 1 0; 1 (see Just and Weese, 1996, p. 25), the map Pow induces a partial order 6 p on J by
Note that 0 6 cDM
J 6 1 (recall that by de®nition CFi.i 0 8i 2 J ). A DM is said to be ipocompensatory i cDM
J 2 0; 1=2, ipercompensatory i cDM
J 2
1=2; 1, halfcompensatory i cDM 1=2. The (unlikely) extreme situations, namely, cDM
J 0 and cDM
J 1, are typical of a totally noncompensatory DM and of a totally compensatory DM, respectively. There are some relationships between the existence in J of absolutely complementary criteria and the value of the DM compensability index. First result is that the presence within J of criteria that are absolutely complementary causes a progressive restriction in the range of possible values of cDM
J . Its easy proof is omitted.
j 6 p i () pj 6 pi
Proposition 7.1. Let jJ j nP 2 and jCompl
J j p P 1: Then,
and
pj 6 pi :
Note that 6 p is only a partial preorder on J , and it is very likely that many criteria will be incomparable with respect to it. A total order can be obtained in very particular cases, e.g., in a lexicographic ordering. In fact, in the latter case, if J f1; 2; . . . ; ng, then the total order 1 2 n is induced by the map Pow de®ned by n i n i Pow
i ; 8i 2 J : n 1 n 1 As a consequence, the dominant criterion will have the maximum compensatory strength, being Pow
1
1; 1, whereas the dominated criterion will have the minimum compensatory strength, being Pow
n
0; 0. 7. Compensatoriness of the DM It is possible to obtain from compensability analysis an estimation of the DM's aptitude to compensate within pairs of criteria of J . We de®ne the DM compensability index (with respect to the set of criteria J , jJ j n P 2) by R R P 1 0 CF
D ; D dD dD i.j i j i j i;j2J 0 1 cDM
J : n
n 1
n
1
n 61
2
n n
n 1
n
1
n
p
6 cDM
J
2
n n
n 1
p
:
In particular, we have: Corollary 7.2. If all criteria of J are absolutely complementary, then the DM is halfcompensatory. A second observation is that being ipocompensatory, ipercompensatory or halfcompensatory turns out to be a DM's characteristic that does not depend on the criteria of J that are absolutely complementary. To prove this fact, we need the following preliminary result: Lemma 7.3. Let jJ j n P 3. Then the following equality holds 8c 2 Compl
J : ncDM
J
n
2cDM
J n fcg 1:
Proof. Let c 2 Compl
J : By de®nition (here we need n P 3)
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cDM
J n fcg P
i;j2J nfcg
R R 1 0 0
1
n
CFi.j
Di ; Dj dDi dDj 1
n
2
:
Furthermore, for all h 2 J n fcg; CFh.c CFc.h ComplSumch 1. Thus R R P 1 0 CF i.j i;j2J nfcg 0 1 cDM
J n
n 1 P R1 R0 CF CF h.c c.h h2J 0 1 n
n 1 n 2 1 c
J n fcg ; n DM n which implies the thesis.
Then, we have: Proposition 7.4. Let jJ j n P 3. Then, for all c 2 Compl
J : (a) cDM
J < 1=2 ) cDM
J n fcg < 1=2; (b) cDM
J > 1=2 ) cDM
J n fcg > 1=2; (c) cDM
J 1=2 ) cDM
J n fcg 1=2. Proof. We prove (a), the other cases being similar. Let c 2 Compl
J . From the previous lemma, we obtain cDM
J n fcg
1 n
2
ncDM
J
1:
Now the result follows from the hypothesis.
Proposition 7.4 shows that ipocompensatoriness, ipercompensatoriness and halfcompensatoriness of the DM are invariants of Compl
J . The following consequence completes the picture: Corollary 7.5. If J0 Compl
J is such that jJ n J0 j P 2, then: (a) cDM
J < 1=2 ) cDM
J n J0 < 1=2; (b) cDM
J > 1=2 ) cDM
J n J0 > 1=2; (c) cDM
J 1=2 ) cDM
J n J0 1=2:
205
Proof. To prove (a), let J0 Compl
J be such that jJ0 j m, where jJj m P 2 by hypothesis. Now apply part (a) of the last proposition m times for each c 2 J0 : Parts (b) and (c) are proved similarly. Note that the restriction jJ n J0 j P 2 cannot be dropped, since jJ n J0 j 6 1 implies J Compl
J , whence cDM
J 1=2 by Corollary 7.2. Our last remark summarizes the relationship between DM compensability index, compensatory powers and compensatory functions in the limit cases of a totally noncompensatory or a totally compensatory DM. Its proof is immediate. Proposition 7.6. Let jJ j P 2. Then: (a) cDM
J 0 () Pow
i
0; 1 8i 2 J () CFi.j 0 8i; j 2 J
i 6 j; (b) cDM
J 1 () Pow
i
1; 0 8i 2 J () CFi.j 1 8i; j 2 J
i 6 j. As an example, we now examine the possible values of cDM
J in the two cases considered in Section 5.3, namely, the lexicographic ordering and the trade-o modelization. The following result, whose validity is ensured by the discussion below, summarizes the situation. Proposition 7.7. A lexicographic DM is always halfcompensatory, whereas a trade-off DM is always ipercompensatory. As already said, in a lexicographic ordering intercriteria relations are modeled via compensability analysis by requiring that i j i CFi.j 1 and CFj.1 0 8i; j 2 J ; i 6 j. Hence, by Corollary 7.2, a lexicographic DM will be halfcompensatory. Note that this result is not in contrast with the classi®cation of the lexicographic aggregation procedure as a totally noncompensatory model, since we are looking at the problem from the perspective of the DM and not of the model. Recognition and distinction of the two faces of compensability, active and passive, determine this situation. Indeed, if criterion i precedes criterion j in the lexicographic order, then the compensatory eect of j over i is null, but the compensatory eect
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of i over j is total. As a consequence, the DM is partially compensatory. For what concerns modelization of (constant and linear) trade-os by means of compensability analysis, we have already observed at the end of Section 5.3 that every couple of criteria will be (at least) normally ipercomplementary. Then, the de®nition of cDM
J yields immediately ipercompensatoriness of the DM in this case. 8. Conclusions In this paper we have examined the ®rst stage of PACMAN, namely, how the notion of compensability can be employed to model intercriteria relations. Compensability analysis is the procedure used to reorganize intercriteria information provided by the DM in a meaningful and transparent way, through the construction of fuzzy compensatory functions. We have concentrated our attention on the theoretical aspects of this modelization, with the declared goal of establishing a theory before dealing with its applications. We are currently working on further developments of the whole PACMAN procedure, aiming at illustrating the approach through links with existing methodologies and several practical examples. Our approach is not free of problems, the most relevant being the large amount of information requested to model intercriteria relations by means of compensability. In fact, we need to build compensatory functions for each ordered pair of distinct criteria, that is n
n 1 functions. Moreover, the construction of each function requires a careful interaction between the actors of the decision process. As a consequence, our approach can be reasonably pursued only whenever few evaluations of criteria are considered. Nevertheless, this limitation is not always unpleasant. Indeed, forcing the DM to concentrate his attention on a few aspects of the decision problem at hand requires the identi®cation of criteria whose importance is worth the time needed for a very careful modelization. In this respect, construction of compensatory functions sheds light on the DM's behavior by reorganizing his
perceptions about intercriteria compensability in an understandable and transparent way. Another possible criticism is that we use compensability as a unique tool to model intercriteria relations. Even if this might seem a restrictive choice, there are some advantages as well. First of all, no technical parameters other than those (easily understandable and representable) related to the construction of compensatory functions are required for the modelization of intercriteria relations. Second, no weighting of criteria is needed to model intercriteria relations, thus avoiding the appeal to an outer procedure not perfectly integrated with the whole methodology. Indeed, as already observed, a peculiar weighting procedure is even intrinsic to compensability analysis. Third, all information provided by the DM with respect to compensability is directly used in our approach, as can be seen in the construction of the binary indices of PACMAN (Giarlotta, 1998). Finally, compensability analysis stimulates a fruitful interaction between the DM and the DA. Indeed, one of the main features of our approach is that intercriteria compensability can be modeled with respect to real scenarios, treating each pair of criteria in a particular way. The complexity and length of the related procedure are the prices to pay for the attempt to represent a DM's preferential information in a faithful, transparent and readable way. Future research will be oriented in two directions. First, as already said, we are examining the possible relationships of PACMAN with existing methodologies. Secondly, we are studying a more eective implementation of the approach. Speci®cally, we are considering the possibility of integrating into the approach ordinal information about compensatory strength of criteria, in order to ensure a more faithful representation of DM's beliefs. At the same time, we are exploring a way to deal with cases of incomplete information about compensability, with the aim of reducing the excessive amount of data required from the DM. Technically, a possible approach to situations of incomplete data is to use the DM compensability index as an invariant of the DM's beliefs about compensability, so as to make up for some lack of information. In these cases, suitable vari-
A. Giarlotta / European Journal of Operational Research 133 (2001) 190±209
ations of the reference function could be employed as related compensatory functions. Speci®cally, in the case of a complete absence of information on a pair of criteria, we could use a variation of the reference function that keeps cDM
J constant; in the case of a partial lack of information, we could use another variation of the reference function, namely, one that keeps cDM
J unchanged and is determined according to the existence and the form of possible constant areas. Acknowledgements The author wishes to thank Angelo Coco, Silvestro Lo Cascio, Benedetto Matarazzo, Riccardo Re and Piero Ursino for the helpful discussions and comments. He also thanks two anonymous Referees for many valuable suggestions and remarks, which led to a more eective and clear presentation of the topic. Appendix A Here we give a more detailed description of the second phase in the construction of a compensatory function. Namely, we examine the procedure used to extend CFi.j by continuity within inti.j , once the two constant areas mini.j and maxi.j have been determined by the DM in the ®rst phase of the process. Since simple cases have been already treated in Section 5.2 and some other cases can be easily derived, we shall concentrate on a particular circumstance, which requires a more careful analysis. Let us consider the case of a compensatory function CFi.j satisfying the following conditions: · 0 < mij < Mij < 1; · both maxi.j and mini.j are not singletons; · at least one between maxi.j and mini.j has a nonlinear inner boundary; · the points of intersection between the inner boundary of maxi.j and the boundary of 0; 1 1; 0 are
a; 0 and
1; b with 0 < a < 1 and 1 < b < 0; · the points of intersection between the inner boundary of mini.j and the boundary of
207
0; 1 1; 0 are
c; 1 and
0; d with 0 < c < 1 and 1 < d < 0. Thus, after the ®rst phase of the construction, the partial graph of CFi.j might look like the one on the left in Fig. 6, where the points A; B; C and D have the following coordinates: A
a; 0; Mij , B
1; b; Mij , C
c; 1; mij , D
0; d; mij . Then, we need to extend by continuity CFi.j to all of its domain. The de®nition of CFi.j is completed in two steps: (1) extension, in a suitable way, of the two boundary curves of maxi.j and mini.j on the lateral faces of the cube, until they form two paths having the same initial and ®nal points (indicated with P and Q in Fig. 6); (2) de®nition of CFi.j within inti.j , using a certain homotopic transformation between the two extensions obtained in step 1. Let us examine these two steps in more detail. (1) We ®rst determine the coordinates p; q 2
0; 1 of the two points P
1; 1; p and Q
0; 0; q on the vertical edges of the cube (see the picture on the right in Fig. 6) by the following formulas: p h
1
c=
2
b
c;
q h
1
d=
2
a
d;
where h Mij mij > 0. These two formulas have an easy interpretation, being obtained by linear interpolation, between B and C for the point P , between A and D for Q. In Fig. 7 it is shown geometrically how P is determined from B and C. Successively, we extend the boundary curve of maxi.j on the two lateral faces of the cube (where it
Fig. 6. Partial de®nition of a possible compensatory function, as determined after the ®rst phase of the construction (on the left) and after the ®rst step of the second phase (on the right).
208
A. Giarlotta / European Journal of Operational Research 133 (2001) 190±209
Fig. 7. Extending the boundary curves to the faces of the cube: Determination of the point P on the vertical edge.
already lies) by taking the two segments PB and AQ. Similarly, the boundary curve of mini.j is extended by taking the two segments PC and DQ. Finally, the required extensions will be the (oriented) paths (in R3 ) PBAQ and PCDQ (see Fig. 6). Note that if we join these two curves and project them onto 0; 1 1; 0, we get the boundary of inti.j . (2) De®nition of CFi.j within inti.j is obtained by using a suitable homotopic transformation between the two curves PBAQ and PCDQ. Again, since this construction is not uniquely determined, we have to make some choices, trying to force uniqueness in a rather natural way. We ®rst need a parameterization in R3 of the two space curves PBAQ and PCDQ. In the following, we denote PBAQ by cM and PCDQ by cm ; we also denote by c0M and c0m the plane curves that are the projections onto the Di Dj -plane of the space curves cM and cm , respectively. Note that, by geometrical properties of c0M and c0m (see Proposition 5.3), any line in a pencil Dj rDi s (with ®xed slope r > 0) intersects each curve c0M and c0m in at most one point. In particular, if we choose a pencil with r 1 and let s vary in 2; 0, we get exactly one point of intersection per curve (as shown in Fig. 8). Therefore, the above parameter s 2 2; 0 can be used for a simultaneous parameterization of c0M and c0m . Since there is a bijective corespondence between cM and c0M (and between cm and c0m as well) given by the projection, this parameterization extends to the space curves cM and cm . See Fig. 8. Next, we need a suitable homotopic transformation between cM
s and cm
s. The condition of
Fig. 8. Completing the construction of the compensatory function: Parameterization of the boundary space curves, creation of the straight line homotopy, and determination of the function in the intermediate zone.
linearity (see section 5) yields the selection of the straight line homotopy C : 2; 0 0; 1 ! R3 de®ned 8
s; t 2 2; 0 0; 1 by C
s; t tcm
s
1
tcM
S:
Last, we have to de®ne CFi.j within inti.j by using this homotopic transformation. Geometrically, the graph of the function CFi.j within inti.j is determined as the image in R3 of the above homotopic transformation C. It is a surface that is the union of the straight segments [cm
s; cM
s]. In the right picture of Fig. 8, two points F and G lying on the extended boundary space curves cM
s and cm
s are connected by a segment on the basis of the straight line homotopy. The analytical representation of CFi.j within inti.j is then given by writing the third component of C as a function of its ®rst two components. Note that this analytical expression can always be deduced from the knowledge of the expressions for cM
s and cm
s. Therefore this procedure is eective. References Bana e Costa, C., 1986. A multicriteria decision aid methodology to deal with con¯icting situations on the weights. European Journal of Operational Research 26 (1), 22±34. Bouyssou, D., 1986. Some remarks on the notion of compensation in MCDM. European Journal of Operational Research 16, 150±160. Bouyssou, D., Vansnick, J.C., 1986. Noncompensatory and generalized noncompensatory preference structures. Theory and Decision 21, 251±266. Dubois, D., Prade, H., 1997. The three semantics of fuzzy sets. Fuzzy Sets and Systems 90 (2), 141±150.
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