A linear implementation of PACMAN

European Journal of Operational Research 205 (2010) 401–411

Contents lists available at ScienceDirect

European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Decision Support

A linear implementation of PACMAN Silvia Angilella a,*, Alfio Giarlotta a, Fabio Lamantia b a b

Department of Economics and Quantitative Methods, Faculty of Economics, University of Catania, Corso Italia 55, I-95129 Catania, Italy Department of Business Sciences, Faculty of Economics, University of Calabria, Via Pietro Bucci 3C, I-87036 Arcavacata di Rende (CS), Italy

a r t i c l e

i n f o

Article history: Received 26 September 2008 Accepted 6 January 2010 Available online 22 January 2010 Keywords: Multiple criteria analysis Pairwise criterion comparison approach Compensation Compensability analysis Compensatory function Sensitivity analysis

a b s t r a c t PACMAN (Passive and Active Compensability Multicriteria ANalysis) is a multiple criteria methodology based on a decision maker oriented notion of compensation, called compensability. A basic step of PACMAN is the construction of compensatory functions, which model intercriteria relations for each pair of criteria on the basis of compensability. In this paper we examine a simplified version of PACMAN, which uses the so-called linear compensatory functions and consistently reduces the overall complexity of its implementation in practical cases. We use MathematicaÒ to develop a computer-aided graphical interface that eases the interaction among the actors of the decision process at each stage of PACMAN. We also propose the possibility to perform a sensitivity analysis in this simplified version of PACMAN as a nonlinear optimization problem. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Passive and Active Compensability Multicriteria ANalysis (PACMAN) is a multiple criteria methodology developed within the framework of the Pairwise Criterion Comparison Approach (PCCA) [18]. A PCCA decision procedure is essentially composed of three phases, aimed at evaluating the binary relation existing between any two feasible alternatives. Specifically, for each couple of distinct alternatives a and b, these phases can be described as follows: (i) Comparison of a and b by means of each (ordered or unordered – depending on the methodology employed) pair of criteria. This analysis is performed by constructing and evaluating binary elementary indices related to a and b. (ii) Comparison of a and b by means of the whole set of criteria. This is obtained by aggregating (in one or more stages) the elementary indices related to a and b into global preference indices related to a and b. (iii) Determination of the (crisp or fuzzy) binary relation existsing between a and b. This relation can be one of strong preference, weak preference, indifference or incomparability. The recommendations obtained in phase (iii) are exploited to create a (possibly weighted) graph, which has all feasible alternatives as vertices and the binary relation existing between each pair * Corresponding author. E-mail addresses: [email protected] (S. Angilella), [email protected] (A. Giarlotta), [email protected] (F. Lamantia). 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.01.007

of them as edges. This graph represents a partial pre-order on the set of feasible alternatives. Finally, the partial pre-order is used by the DA (Decision Aider) to help the DM (Decision Maker) in analyzing and possibly ‘‘solving” the decision problem under consideration (cf. [16]). A peculiar feature of PACMAN is compensability analysis: this is a stage preliminary to phases (i)–(iii), which has the goal to evaluate the DM’s aptitude to compensate among criteria. This evaluation is done on the basis of a notion of compensation, called compensability, which – contrary to what is usually done in the literature (see, e.g., [4,5]) – is designed to be a DM’s characteristic and not a feature of a methodology. In fact, compensability is a numerical evaluation of ‘‘the possibility that an advantage on one criterion can offset a disadvantage on another criterion” [9,10]. In this evaluation procedure, we distinguish the compensating (or active) criterion from the compensated (or passive) one. It follows that compensability is, in general, a non-symmetric notion. The idea of distinguishing active and passive effects (of evaluations, criteria, etc.) is not new in the literature. Several multi-criteria methodologies have implemented this distinction using different approaches. Within the class of ELECTRE methods [20,21], the relevance of each criterion is determined by two sets of parameters: the importance coefficients (the active effects) and the veto thresholds (the passive effects). In order to determine for each couple of alternatives a; b whether to accept or refuse the outranking relation ‘‘a is at least as good as b”, each ELECTRE method performs a concordance test and a discordance test. For example, in ELECTRE I the concordance index is computed by summing the importance weights associated to the criteria that

402

S. Angilella et al. / European Journal of Operational Research 205 (2010) 401–411

are in favor of the above assertion. In a second step, the discordance index is computed on each criterion that opposes the outranking relation and is compared to the relative veto threshold. Therefore, the importance weights – which express an intrinsic weight of each criterion [22] – are somehow related to the active contribution of each criterion to the acceptance of an outranking relation. On the other hand, the veto thresholds express the passive resistance of each criterion to the acceptance of an outranking relation. At the end of the procedure, (some of) ELECTRE methods – similarly to PACMAN – establish a fundamental system of preferences on the set of alternatives. The PROMETHEE methods [2] also deal with this distinction between active and passive effects, even if they do that in a symmetric way. In fact, the so-called outranking character and outranked character of each alternative are evaluated by the same type of procedure, as a result of an aggregation of partial preference indices and weights. In PROMETHEE all partial indices take into account the value of each difference of evaluations, using a suitable type of function that is chosen by the DM among six predetermined types. (Note that in PACMAN the compensatory functions are fully shaped by the DM.) Finally – differently from PACMAN – PROMETHEE methods establish a partial ranking (PROMETHEE I) or a complete ranking (PROMETHEE II) of alternatives. An idea of compensation among criteria, which does not distinguish between active and passive effects, is also present in some UTA-like methods in the framework of the functional approach. For example, in GRIP [7] the authors propose a decision model in which the DM provides several pieces of information of an ordinal/qualitative type. Specifically, the DM is expected to provide the following preference information: (i) a partial pre-order on  the set of reference alternatives A ; (ii) a partial pre-order on the   set A  A , comparing global intensities of preference for some pairs of alternatives; (iii) a partial pre-order on the set A  A for each criterion, comparing partial intensities of preference for some pairs of alternatives. This yields a set of linear constraints, which are used to build a set of additive value functions compatible with the DM’s preference information. This construction is performed on the basis of an ordinal regression, similarly to what happens in UTA [13]. Then, GRIP builds two binary relations on the set of alternatives: the necessary weak preference and the possible weak preference. These relations are successively exploited obtaining, respectively, the necessary ranking (a partial pre-order) for all compatible value functions and the possible ranking (strongly complete and negatively transitive) for at least one value function. Moreover, GRIP obtains necessary and possible (comprehensive and partial) intensities of preference relations on the set of ordered pairs of alternatives. The main feature that distinguishes PACMAN from all multi-criteria methodologies mentioned above is that active and passive effects of (differences of) evaluations are used to determine intercriteria compensability via the construction of a compensatory function for each ordered pair of distinct criteria. This allows one to meaningfully model situations in which, for example, a criterion g i has a strong active power of compensation but a weak passive resistance to compensation. This is indeed the case when a large difference of evaluations on a particular criterion g i between two feasible alternatives a and b might strongly enhance the arguments to accept an overall preference of a over b (strong active compensatory power) but not contribute too much in preventing b from being overall preferred to a (weak passive compensatory power). Recently, we have introduced the notion of an ‘‘implementation” of PACMAN, with the goal of allowing a more effective use of the methodology [1]. Much attention has been devoted to an axiomatization of the properties that each implementation has to satisfy, paying particular attention to some special implementations (regular, co-symmetric, lexicographic, etc.). In the process of

designing and describing the main properties of implementations of PACMAN, a basic step is the description of an effective procedure to construct compensatory functions, which are the tools used to numerically translate the DM’s information about intercriteria compensability. This phase of PACMAN is very delicate, because it requires a strict and continuous interaction between the DA and the DM. The aim of this paper is twofold. First, we present a simplified version of PACMAN, whose implementation is easier and more feasible in practical cases. In fact, we analyze in great detail the construction of a specific form of compensatory functions, which will be called linear. The advantage of dealing with linear compensatory function lies in the simplicity of the procedure to assess them, since the DM has only to determine two special points (where the compensability is, respectively, total and null). To illustrate how to construct a compensatory function and how to obtain a partial pre-order on the set of feasible alternatives, we provide some simple yet clarifying examples. The second purpose of this paper is to propose a sensitivity analysis of the results with respect to changes in compensatory functions. In particular, we consider the problem of finding the smallest modification of linear compensatory functions that forces a change in the DM’s preference ordering of alternatives (e.g., passing from strong preference to indifference or incomparability). The paper is organized as follows. In Section 2 we describe the main features of a linear implementation of PACMAN. Section 3 contains a didactic example, in which all aspects of the three phases of PACMAN are carefully described and discussed. In Section 4 we state a nonlinear programming problem to describe a sensitivity analysis of PACMAN with respect to modifications in linear compensatory functions. Some final remarks about this implementation of PACMAN are presented in Section 5.

2. A simplified version of PACMAN In this section we describe a linear implementation of the PACMAN methodology. Specifically, in the first subsection we introduce some basic notations and outline the three phases of PACMAN. In the second subsection we examine the main features of phase I of PACMAN – compensability analysis – emphasizing the algorithmic construction of the so-called linear compensatory functions. In the last two subsections we describe phases II and III of this simplified version of PACMAN. For a full analysis of the general set-up of the PACMAN decision procedure, the reader is referred to [9,10] for the methodology and its properties, and to [1] for the notion of implementation of PACMAN and some related theoretical results. All definitions given in this paper are obtained as particular cases of the general framework. However, for the sake of simplicity, we describe them directly, often avoiding any explicit reference to the general case. 2.1. Preliminaries Let A ¼ far : r 2 Rg, with R ¼ f1; . . . ; mg, be a set of m P 2 alternatives. These alternatives are evaluated on the basis of a consistent [20] family of n P 2 criteria G ¼ fg j : j 2 Jg, where J ¼ f1; . . . ; ng. We assume that each criterion g j : A ! R in G is an interval scale of measurement [19]. We denote by MðAÞ ¼ ðarj Þ the m  n matrix that collects the evaluations of the alternatives in A by means of the criteria in G, i.e., arj :¼ g j ðar Þ for each j 2 J and r 2 R. We slightly abuse notation and identify each alternative ar 2 A with the vector of its evaluations by the criteria in G, namely, ar ¼ ðar1 ; . . . ; arn Þ. The range of each criterion g j 2 G is assumed to be a non-trivial compact interval ½aj ; bj  # R, where the minimum aj and the maximum bj of the range of g j are usually fixed a priori by the DM.

403

S. Angilella et al. / European Journal of Operational Research 205 (2010) 401–411

Therefore, for each j 2 J and r 2 R, we have aj 6 arj 6 bj . The evaluations of the alternatives in A are normalized using these parameters, namely, arj :¼ ðarj  aj Þ=ðbj  aj Þ. Thus, we have arj 2 ½0; 1 for each j 2 J and r 2 R. Differences of evaluations of two distinct alternatives ar ; as 2 A by a criterion g j 2 G are normalized using the same technique. Specifically, we set Dj ðar ; as Þ :¼ ðarj  asj Þ= ðbj  aj Þ. Note that, for each j 2 J and r; s 2 R, we have Dj ðar ; as Þ ¼ Dj ðas ; ar Þ 2 ½1; 1. Since in PACMAN criteria are interval scales, normalized differences provide us with a meaningful tool to compare alternatives. Admittedly, the choice of the parameters used to normalize evaluations might theoretically affect the final result. However, these changes appear to be minimal in practical cases. The set of ordered pairs of different alternatives (respectively, criteria) is denoted by CA (respectively, CG ). Thus, we have CA :¼ A  A n fðar ; ar Þ : r 2 Rg (respectively, CG :¼ G  G n fðg j ; g j Þ : j 2 Jg). Note that jCA j ¼ m ðm  1Þ and jCG j ¼ n ðn  1Þ. Normalized differences with respect to each criterion g j 2 G can be arranged in an m  m antisymmetric1 matrix MðDj Þ, where the generic element in the rth row and the sth column is Dj ðar ; as Þ. The (general) PACMAN decision procedure is composed of three successive phases: (I) Compensability analysis, which is the procedure aimed at evaluating intercriteria relations via the construction of compensatory functions for each ordered pair of distinct criteria. (II) Pairwise comparison of alternatives via the construction of active binary indices and passive binary indices at several levels of aggregation. These indices, which express the degree of active and passive preference of each alternative over each of the others, are built by directly taking into account the results of compensability analysis. (III) Determination of a fundamental system of preferences on the set of alternatives, comparing the active and passive global indices obtained at the end of phase (II). Comparing phases (I), (II) and (III) of PACMAN with phases (i), (ii) and (iii) of a generic PCCA decision procedure, we observe the following.  Phase (I) of PACMAN is absent from a generic PCCA procedure. We further note that most PCCA methodologies use an exogenous notion of weight of criteria, whereas in PACMAN the determination of a notion of compensatory power of criteria (which is the local equivalent to the notion of weight) is endogenous to the methodology.  Phase (II) of PACMAN collects phases (i) and (ii) of a PCCA decision procedure. This is done only for convenience, since in PACMAN the number of indices and aggregations is higher than in other methodologies, also because we explicitly separate active indices from passive indices.  Phase (III) of PACMAN is phase (iii) of a PCCA procedure. This phase explicitly includes the possibility of incomparability between two feasible alternatives. In what follows we describe these three phases in a simplified implementation of PACMAN. 2.2. Phase I: Linear compensability analysis

MAN is the subject of [10], so here we only give a brief outline. On the other hand, we carefully examine the procedure to construct a so-called linear compensatory function, since this is the main feature of the simplified approach described in this paper. Let ar and as be two distinct alternatives in A. If ar is evaluated better than as by criterion g j 2 G (i.e., arj > asj ), then the positive normalized difference Dj ðar ; as Þ gives a rough measure of the local (i.e., with respect to criterion g j ) strength of ar over as . At a global level (i.e., with respect to the whole set G), this positive normalized difference interacts with all the other normalized differences of evaluations, showing a double effect: active, since it gives some contribution to the (possible) overall preference of ar over as ; passive, since it determines a resistance to the (possible) overall preference of as over ar . Therefore, a partial preference of ar over as on criterion g j may enlarge both the set of arguments to accept a global preference of ar over as and the set of arguments to reject a global preference of as over ar . In PACMAN, for each ðg i ; g j Þ 2 CG the DM determines the degree of confidence that a positive normalized difference Di 2 ð0; 1 on the active criterion g i compensates a negative normalized difference Dj 2 ½1; 0Þ on the passive criterion g j . This evaluation is translated into a numerical form via the construction of a continuous compensatory function CFi.j , which associates to each pair of normalized differences ðDi ; Dj Þ 2 ð0; 1  ½1; 0Þ the level of credibility that the positive difference Di totally compensates the negative difference Dj . Successively, this function is extended in frontier by continuity, thus obtaining a compensatory function CFi.j : ½0; 1 ½1; 0 ! ½0; 1, which is continuous over its whole domain. By definition, we set CFj.j to be identically equal to 0 for each j 2 J. In [10] (Section 5 and Appendix A) the author describes in full detail a procedure to build a compensatory function. In this paper we adopt a less theoretical approach and analyze algorithmically the construction of a simplified form of compensatory functions, termed linear. Formally, the domain of a linear compensatory function CFi.j is a disjoint union

½0; 1  ½1; 0 ¼ Ni.j [P _ i.j [T _ i.j ; where Ni.j ; P i.j ; T i.j denote the areas of null, partial and total compensability, respectively. The following conditions must hold:  CFi.j restricted to N i.j is identically equal to 0;  CFi.j restricted to T i.j is identically equal to 1;  CFi.j restricted to P i.j is linearly increasing in both arguments. Analytically, a linear compensatory function (LCF for short) is the unique continuous extension to ½0; 1  ½1; 0 of a function defined as follows:

8 0 if > > > > > < D þD kð0Þ þ1 i j ij CFi.j ðDi ; Dj Þ :¼ if ð1Þ ð0Þ 2kij kij > > > > > : 1 if ð0Þ

ð0Þ

ð1Þ

kij  1 < Di þ Dj < 1  kij ;

Di þ Dj < 1  kð1Þ ij

ð1Þ

ð1Þ

 kij þ kij 6 2;  if

ð0Þ kij

ð1Þ

ð0Þ

ð1Þ

þ kij ¼ 2, then either kij ¼ 0 or kij ¼ 0. ð0Þ

1

Recall that a square matrix A is antisymmetric if A ¼ AT .

ð1Þ

where kij ; kij 2 ½0; 2 are two non-negative parameters assessed by the DA for each ði; jÞ 2 J2 . These parameters must satisfy the following conditions: ð0Þ

The first phase of PACMAN is compensability analysis. This is the procedure that allows the DA to evaluate analytically the DM’s aptitude to compensate among criteria. This phase of PAC-

Di þ Dj < kð0Þ ij  1

ð1Þ

The mathematical meaning of the parameters kij and kij is ð0Þ ð1Þ rather simple, since the larger kij (respectively, kij ) is, the larger the area N i.j (respectively, T i.j ) is.

404

S. Angilella et al. / European Journal of Operational Research 205 (2010) 401–411 ð0Þ

ð1Þ

þ   Observe that pþ i ; pi 2 ½0; 1. The closer ðpi ; pi Þ to the pair of val ues (1, 1) is, the stronger criterion g i is; dually, the closer ðpþ i ; pi Þ to (0, 0), the weaker the criterion. Our approach allows one to consider criteria that have a strong active power but a weak passive  power. For example, if ðpþ i ; pi Þ ¼ ð0:74; 0:21Þ, then criterion g i 2 G will be very active in contributing to a possible preference of an alternative over another one, but not so resistant in opposing such a preference. Dually, it is also possible to have a criterion g i 2 G with a weak active power but a strong passive resistance, e.g.,  ðpþ i ; pi Þ ¼ ð0:16; 0:87Þ. (See Section 6 in [10] for an example, which better illustrates the notions of active and passive compensatory power of a criterion.) Note that in the case when each CFi.j is the  reference compensatory function, then we have ðpþ i ; pi Þ ¼ ð0:5; 0:5Þ for all i 2 J. Compensability analysis also enables the DA to obtain an estimation of the DM’s aptitude to compensate among the criteria in G. We define a DM compensability index as follows:

Note that in the two limiting cases, namely, kij þ kij 2 f0; 2g, ð0Þ the function CFi.j has a very simple form. Specifically, if kij þ ð0Þ kij ¼ 2, then CFi.j is constant on its whole domain, being identically ð0Þ equal to 0 in case that kij ¼ 2, and identically equal to 1 in case that ð1Þ kij ¼ 2. Constant compensatory functions are used in very specific cases, such as in a lexicographic implementation of PACMAN (see [1, ð0Þ ð1Þ Section 8]). On the other hand, if kij ¼ kij ¼ 0 (i.e., N i.j is the singleton fð0; 1Þg and T i.j is the singleton fð1; 0Þg), then we obtain the so-called reference compensatory function [10]. ð0Þ ð1Þ In the non-limit cases (i.e., 0 < kij þ kij < 2) at least one the areas of null and total compensability is an infinite set, namely, either a triangle or a pentagon. For example, in Fig. 1 we describe such a function: on the left hand side we represent the partitioned ð0Þ ð1Þ domain of an LCF such that 1 < kij < 2 and 0 < kij < 1, whereas ð0Þ on the right hand side we draw its graph. (Note that kij > 1 implies that the area of null compensability N i.j is a pentagon.) Observe that in general, an LCF always has the following features:

P

 the areas of null and total compensability are either a point or a right-angle isosceles triangle or a pentagon (obtained as the restriction of a right-angle isosceles triangle);  in the area of partial compensability its graph is the restriction of the unique plane that makes it continuous over its whole domain. ð0Þ

cDM ðGÞ :¼

pþi :¼

j¼1

pi :¼ 1 

0

1

CFi.j ðDi ; Dj ÞdDi dDj

ð1Þ

n1

0

1

CFi.j ðDi ; Dj ÞdDi dDj

 :

nðn  1Þ

2.3. Phases II: Linear binary preference indices Phase II of PACMAN consists of the construction of binary preference indices related to ordered pairs of distinct alternatives. These indices are first built at an elementary level, considering two criteria at a time in accordance with a PCCA philosophy [17,18]. Successively the elementary indices are aggregated in a suitable way, in order to finally obtain global preference indices. At each stage of the procedure, active and passive indices are computed separately. More specifically, for each ordered pair of distinct alternatives, we construct:



; n1   R R Pn 1 0 CFj.i ðDj ; Di ÞdDj dDi j¼1 0 1

R R 1 0

Note that cDM ðGÞ 2 ½0; 1. In particular, we say that a DM is totally compensatory if cDM ðGÞ ¼ 1, totally non-compensatory if cDM ðGÞ ¼ 0, and half-compensatory if cDM ðGÞ ¼ 1=2. For example, a DM is halfcompensatory in the following cases: (i) all compensatory functions are equal to the reference compensatory function; (ii) under a lexicographic implementation of PACMAN; (iii) more generally, under a co-symmetric implementation of PACMAN (see [1] for more details).

As usual, we collect all parameters kij and kij into two square ð0Þ ð1Þ matrices of order n, namely, Kð0Þ ¼ ðkij Þ and Kð1Þ ¼ ðkij Þ, where all ð0Þ ð1Þ elements kjj and kjj on the main diagonal are equal to zero by definition. Compensability analysis allows the DA to evaluate the compensatory strength of each criterion over the other criteria in G according to the DM’s preference structure. This notion of strength of a criterion somehow refines the notion of importance/weight, since it explicitly separates active and passive relevance of a criterion. For each criterion g i 2 G, we define its active compensatory power  pþ i and its passive compensatory power pi as follows [9,10]:

Pn R 1 R 0

i;j2J

(i) active and passive elementary indices (related to each ordered pair of distinct criteria); (ii) active and passive partial indices (related to each criterion);

:

1 0.8 0.6 0.4 0.2 0 0

Fig. 1. LCF obtained for

ð0Þ kij

−0.2

¼

9 7

−0.4

and

−0.6

ð1Þ kij

¼

−0.8

1 . 4

−1

0

0.2

0.4

0.6

0.8

1

S. Angilella et al. / European Journal of Operational Research 205 (2010) 401–411

(iii) active and passive global indices (related to the whole set of criteria); (iv) a net global index. In the sequel, we outline the construction of these indices for an arbitrary ordered pair of distinct alternatives ðar ; as Þ 2 CA . (i) First of all, for each ðg i ; g j Þ 2 CG , we evaluate the active elementary index Pþ i.j ðar ; as Þ and the passive elementary index Pi.j ðar ; as Þ. We recall that for each couple of distinct alternatives far ; as g # A and couple of distinct criteria fg i ; g j g # G, there are eight elementary indices to be computed, four active (having a ‘‘+” as a superscript) and four passive (having a ‘‘” as a superscript). Four of these elementary indices are related to the ordered pair ðar ; as Þ 2 CA , namely, Pþi.j ðar ; as Þ; Pþj.i ðar ; as Þ; Pi.j ðar ; as Þ; Pj.i ðar ; as Þ. The other four are related to the ordered pair ðas ; ar Þ 2 CA , and they are Pþi.j ðas ; ar Þ; Pþj.i ðas ; ar Þ; Pi.j ðas ; ar Þ; Pj.i ðas ; ar Þ. Here we only give the definition of two of them, both related to the ordered pair ðar ; as Þ, and specifically: the active elementary index Pþ i.j ðar ; as Þ, where g i is the active criterion and g j the passive one; the passive elementary index P j.i ðar ; as Þ, where g j is the active criterion and g i the passive one. The definition of the other six indices related to the couple of alternatives far ; as g is, mutatis mutandis, the same. In [1] we state all axioms that elementary indices have to satisfy and propose several definitions of these indices (ordinal, cardinal and mixed) satisfying these axioms. Here we limit our analysis to those elementary indices associated to an ordinal and a cardinal choice. In the case of an ordinal choice,  the two elementary indices Pþ i.j ðar ; as Þ and Pj.i ðar ; as Þ are defined as follows (the symbols Di and Dj stand, respectively, for Di ðar ; as Þ and Dj ðar ; as Þ):

8 if Di > <1 _ þ ðar ; as Þ :¼ CFi.j ðDi ; Dj Þ if Di P i.j > : 0 if Di 8 > <1 _  ðar ; as Þ :¼ 1  CFj.i ðDj ; Di Þ P j.i > : 0

> 0 and Dj P 0 > 0 and Dj < 0 ;

ð2Þ

60 if Di > 0 and Dj P 0 if Di > 0 and Dj < 0 : if Di 6 0: ð3Þ

The logic underlying an ordinal framework is similar to that of non-compensatory outranking methods (in particular, the ELECTRE methods [6,21]). In fact, if the differences of evaluations of ar and as by criteria g i and g j have the same sign (concordance case, first law), then both active and passive effects are constantly equal to 1. On the other hand, in case of difference of evaluations having an opposite sign (discordance case, second law), the two effects depend only on the degrees of confidence CFi.j and CFj.i , respectively. Note that the active (respectively, passive) elementary index has by definition the value zero in case that the active (respectively, passive) difference Di is non-positive, since there cannot be any active (respectively, passive) compensatory effect of Di over Dj .The logic underlying a cardinal choice is similar to that of the MAUT approach [8,14]. The two cardinal elementary indices € þ ðar ; as Þ and P €  ðar ; as Þ are defined as follows (again, Di P j.i i.j and Dj stand, respectively, for Di ðar ; as Þ and Dj ðar ; as Þ):

_ þ ðar ; as Þ and € þ ðar ; as Þ ¼ Di ðar ; as Þ P P i.j i.j €  ðar ; as Þ ¼ Di ðar ; as Þ P _  ðar ; as Þ; P j.i j.i

405

teristic of a cardinal choice is that active (respectively, passive) compensatory effects are a certain percentage of the active (respectively, passive) difference Di ðar ; as Þ. (Note that in the following we simplify notation and use the symbol _ and P € , since the type of (ordinal or cardinal) P in place of P implementation will be clear from the context.) Active and passive elementary indices are collected in 2jCA j square matrices of order n. The two matrices associated to the pair  ðar ; as Þ are denoted by MðPþ . ðar ; as ÞÞ and MðP. ðar ; as ÞÞ; their generic element in the ith row and jth column is, respectively, Pþi.j ðar ; as Þ and Pi.j ðar ; as Þ. Note that all entries in the main diagonal of these two matrices are equal to zero, since we  set by definition Pþ j.j ðar ; as Þ ¼ Pj.j ðar ; as Þ :¼ 0 for each j 2 J. (ii) In the second stage of phase II, we separately aggregate all active elementary indices and all passive elementary indices related to ðar ; as Þ. For each i 2 J, the ith row of MðPþ . ðar ; as ÞÞ contains all active elementary indices related to ðar ; as Þ such that criterion g i is active. We aggregate these indices into an active partial index Pþ i ðar ; as Þ, which estimates the active contribution of criterion g i to the relation of compensated preference of ar over as . The index Pþ i ðar ; as Þ is obtained as a simple average2 of the n  1 elements in the ith row of þ MðPþ . ðar ; as ÞÞ such that j – i (the index Pi.i ðar ; as Þ ¼ 0 is not taken into account to compute this average value). Thus, overall we obtain n active partial indices Pþ i ðar ; as Þ related to ðar ; as Þ, which are collected in an n-dimensional column vector VðPþ i ðar ; as ÞÞ. Similarly, the ith column of MðP . ðar ; as ÞÞ contains all passive ða ; a Þ, where the criterion g i is paselementary indices P r s j.i sive. As before, we take the simple average of all the elements  in the ith column of MðP . ðar ; as ÞÞ except Pi.i ðar ; as Þ ¼ 0. This  yields a passive partial index Pi ðar ; as Þ, indicating the passive resistance of g i 2 G to the relation of compensated preference of as over ar . All passive partial indices P i ðar ; as Þ are then collected into an n-dimensional column vector VðP i ðar ; as ÞÞ. (iii) Next, we obtain an active global index Pþ ðar ; as Þ and a passive global index P ðar ; as Þ by aggregating the elements of, respec tively, VðPþ i ðar ; as ÞÞ and VðPi ðar ; as ÞÞ. This aggregation is done by using again a simple average operator2. We collect all active and passive global indices related to the pair ðar ; as Þ 2 CA in two m  m matrices MðPþ Þ and MðP Þ, whose generic element in the rth row and the sth column is, respectively, Pþ ðar ; as Þ and P ðar ; as Þ. Again, all entries in the main diagonal of these two matrices are equal to zero. (iv) Finally, we compute a net global index Pðar ; as Þ :¼ Pþ ðar ; as Þ  P ðas ; ar Þ, which evaluates the net result of all arguments supporting a compensated preference of ar over as versus all arguments opposing this preference. Net global indices related to all ordered pairs of distinct alternatives are collected in an m  m matrix MðPÞ ¼ MðPþ Þ  MðP ÞT . 2.4. Phase III: Modelling of preferences Phase III of PACMAN establishes – on the basis of the evaluation of net global indices – a binary relation within each couple of distinct alternatives. This relation can be either strong preference P, or weak preference Q, or indifference I, or incomparability R. The set fP; Q ; I; Rg is a relational system of preferences [20]. Let e 2 ½0; 1Þ be a suitable sensitivity threshold fixed by the DM. Using e, we transform each net global index Pðar ; as Þ into an integer index Wðar ; as Þ as follows:

ð4Þ

_ denote the corresponding ordinal elewhere the symbols P mentary indices defined by formulas (2) and (3). The charac-

2 In the general set-up of PACMAN, active and passive (partial and global) aggregation functions are operators that are idempotent, monotone and bounded. In a linear implementation of PACMAN, we always use the simple average.

406

S. Angilella et al. / European Journal of Operational Research 205 (2010) 401–411

Table 1 Evaluations of alternatives.

Car a Car b Car c

1. Price (€)

2. Speed (km/hour)

3. Consumption (km/l)

16,000 15,000 12,000

180 160 130

15 16 25

Table 2 Normalized evaluations.

Car a Car b Car c

1. Price

2. Speed

3. Consumption

1 0.75 0

1 0.6 0

0 0.1 1

8 if Pðar ; as Þ 2 ðe; 1 > <1 Wðar ; as Þ :¼ 0 if Pðar ; as Þ 2 ½e; e > : 1 if Pðar ; as Þ 2 ½1; eÞ: All indices Wðar ; as Þ are aggregated into a m  m matrix W, whose entries can only be either 1, 0 or 1. (The advantage of dealing with W instead of MðPÞ is only computational.) Then, we evaluate a relation of compensated preference of ar over as on the basis of the value of Wðar ; as Þ: this relation is accepted if Wðar ; as Þ ¼ 1, doubtful if Wðar ; as Þ ¼ 0 and rejected if Wðar ; as Þ ¼ 1. Finally, a relational system of preferences fP; Q ; I; Rg is established as follows:

ar Pas () Wðar ; as Þ ¼ 1 and Wðas ; ar Þ ¼ 1 ar Qas () ðWðar ; as Þ ¼ 1 and Wðas ; ar Þ ¼ 0Þ or ðWðar ; as Þ ¼ 0 and Wðas ; ar Þ ¼ 1Þ ar Ias () ðWðar ; as Þ ¼ 0 and Wðas ; ar Þ ¼ 0Þ or ðWðar ; as Þ ¼ 1 and Wðas ; ar Þ ¼ 1Þ ar Ras () Wðar ; as Þ ¼ 1 and Wðas ; ar Þ ¼ 1: 3. An illustrative example Let us consider a multi-criteria decision problem of ranking some cars on the basis of a set of three criteria G ¼ fg 1 ; g 2 ; g 3 g: price g 1 (evaluated in €), maximum speed g 2 (in km/hour) and fuel consumption g 3 (in km/l). The criteria evaluations of three alternatives are displayed in Table 1; we take price with negative sign, in order to deal with a maximization problem for all criteria. On each criterion g j , the normalization of evaluations is done by taking as minimum aj and maximum bj of the range of g j , respectively, the minimum and the maximum evaluation of the three cars at hand. The corresponding normalized evaluations are displayed in Table 2. Observe that price and fuel consumption evaluate the three cars in the same ranking, namely, a1 b1 c and a3 b3 c. On the other hand, maximum speed reverses their order, since we have c2 b2 a. In the sequel we describe the interactive procedure3 that leads to the construction of linear compensatory functions for this specific example. After computing all normalized differences (using the same limit values of the range as in the normalization of evaluations), we have to assess six compensatory functions. Below we describe the procedure to assess CF1.2 , the compensatory function of price (active criterion) over maximum speed (passive criterion). On the x-axis, we have price in terms of absolute (positive) differences, so ranging from € 0 to € 4000 (where € 4000 is the difference between the

3 This procedure is supported by a graphical interface, developed by the authors using MathematicaÒ.

cheapest and the most expensive car). On the y-axis, we represent absolute (negative) differences on maximum speed, ranging from 0 km/hour to 50 km/hour (which is the difference of maximum speed between the slowest and the fastest car). The DA starts asking the DM some specific questions, which aim at determining the smallest difference in price that totally compensates a given difference in maximum speed. For example, assume that the DM declares that an active difference of € 3500 totally compensates a passive difference of 10 km/hour; in other terms, the DM judges that a loss in maximum speed of 10 km/hour is perfectly compensated by a decrease in price of € 3500. In this case, the procedure will start from the point (3500, 10), where by definition CF1.2 is equal to 1. Now keeping constant the passive difference of 10 km/hour, the DA tries to ascertain whether the DM would indicate a difference in price smaller than € 3500, which nonetheless totally compensates the same loss of maximum speed. For this reason the DA will diminish the active difference in price down to a point where the DM states, for example, that a difference in price of less than € 2400 begins to partially but not totally compensate a passive difference of 10 km/hour. It follows that (the normalization of) the point T  ð2400; 10Þ lies on the line segment that separates the area of total compensability from the area of partial compensability. In the same way, the DA asks the DM to determine an active difference of price that has no compensatory effect whatsoever on a certain fixed passive difference of maximum speed. For example, let us assume that the DM states that a decrease in price of € 300 cannot compensate at all a loss in maximum speed of 40 km/ hour. Now taking constant the passive difference of 40 km/hour, the DA will increase the active difference of price up to a point elicited by the DM, say € 600, where the bigger price reduction starts to partially compensate a loss of 40 km/hour in maximum speed. It follows that (the normalization of) the point N  ð600; 40Þ lies on the line segment that separates the area of null compensability from the area of partial compensability. After the detection of these two switching points T and N, the xaxis D1 and the y-axis D2 are re-scaled in terms of normalized differences, with D1 ranging from 0 to 1, and D2 from 1 to 0. The normalized points corresponding to T and N are, respectively, T   ð0:6; 0:2Þ and N   ð0:15; 0:8Þ. To fully determine the area of total compensability T 1.2 (respectively, the area of null compensability N 1.2 ), now it suffices to draw the segment in the square ½0; 1  ½1; 0 that passes through the point T  (respectively, N  ) and has slope 45°. On the left hand side of Fig. 2, the areas of total and null compensability are given by, respectively, the triangle with vertices M  ð1; 0Þ; R  ð1; 0:6Þ; S  ð0:4; 0Þ, and the triangle with vertices O  ð0; 1Þ; P  ð0; 0:65Þ; Q  ð0:35; 1Þ. ð0Þ ð1Þ Finally, we set the two parameters k12 and k12 . If the area of null compensability N 1.2 is a triangle (as in this example), then we let ð0Þ k12 be equal to the first coordinate of the point ðx0 ; 1Þ 2 @N 1.2 \ @D. (By @N 1.2 and @D we denote, respectively, the perimeter of the area of null compensability and the perimeter of the square ½0; 1  ½1; 0.) In our example, this is the point Q  ð0:35; 1Þ, ð0Þ hence k12 ¼ 0:35. On the other hand, in the case that N 1.2 is a pentagon (see, e.g., Fig. 1 in Section 2.2), if ðx0 ; 0Þ is one of the two points in the boundary among the area of null compensability and the perimeter of the square ½0; 1  ½1; 0, then we set ð0Þ k12 :¼ x0 þ 1. ð1Þ To determine k12 , we proceed similarly. Namely, if the area of total compensability T 1.2 is a triangle (as in this example), then ð1Þ we let k12 be equal to the opposite of the second coordinate of the point ð1; y1 Þ 2 @T 1.2 \ @D. In our example, this is the point ð1Þ R  ð1; 0:6Þ and hence k12 ¼ 0:6. In the case that T 1.2 is a pentað1Þ gon, the parameter k12 is determined in a way dual to that used ð0Þ to fix k12 . Analytically, the compensatory function CF1.2 : ½0; 1 ½1; 0 ! ½0; 1 is defined by

407

S. Angilella et al. / European Journal of Operational Research 205 (2010) 401–411

1 0.8 0.6 0.4 0.2 0 0 −0.5 −1 0 ð0Þ

0.2

0.4

0.8

0.6

1

ð1Þ

7 Fig. 2. Construction of CF1.2 , obtained for k12 ¼ 20 and k12 ¼ 35.

CF1.2 ðDi ; Dj Þ :¼

8 > <1

if Di þ Dj 2 ½0:4; 1

0:65þDi þDj > 1:05

:

0

if Di þ Dj 2 ð0:65; 0:4Þ : if Di þ Dj 2 ½1; 0:65

Note that in the area of partial compensability, we have CF1.2 ðDi ; Dj Þ ¼ z, where ðDi ; Dj ; zÞ is a point of the plane passing through the four co-planar points P0  ð0; 0:65; 0Þ; Q 0  ð0:35; 1; 0Þ; R0  ð1; 0:6; 1Þ; S0  ð0:4; 0; 1Þ. The graph of CF1.2 is displayed on the right hand side of Fig. 2. This interactive procedure to assess compensatory functions has to be carried out for every ordered pair of distinct criteria ðg i ; g j Þ 2 CG . Thus, in our example, in order to define six compensatory functions, we have to determine 12 parameters, which are stored in the matrices Kð0Þ and Kð1Þ . Let us suppose that the results of a linear compensability analysis related to the problem at hand produce the following matrices:

0

ð0Þ

K

1

0 0:35 0:5 B C ¼ @ 0:4 0 0:2 A; 0:1 0:2 0

0

K

ð1Þ

1

0 0:6 0:7 B C ¼ @ 0:8 0 0:2 A: 0:5 0:2 0

ð5Þ

In this case, as for the compensatory strength of each criterion, we þ þ    have4 ðpþ 1 ; p1 Þ ¼ ð0:414; 0:642Þ; ðp2 ; p2 Þ ¼ ð0:422; 0:546Þ; ðp3 ; p3 Þ ¼ ð0:436; 0:539Þ. Thus, fuel consumption has the strongest active power and the weakest passive power, whereas the opposite holds for speed; price appears to be in an intermediate position. Note that our DM is less than half-compensatory, being cDM ðGÞ ¼ 0:424. In the following subsections, we continue to examine this example in order to get further insight into it by carrying out different implementations of PACMAN. 3.1. A cardinal framework To start, we consider a cardinal implementation of PACMAN, in which all elementary binary indices are cardinal. Normalized differences from Table 2 can be arranged into matrices MðDi Þ; i ¼ 1; 2; 3, whose rows and columns refer to cars a, b, c (in this order). We fix e :¼ 0:05 ¼ 5% as a sensitivity threshold. Cardinal elementary indices related to each ordered pair of alternatives ðx; yÞ 2 CA can be easily calculated by formulas (4). Then, they are arranged in six 3  3 matrices MðPþ . ðx; yÞÞ and six 3  3 matrices MðP . ðx; yÞÞ. Successively, we compute the entries of six  column vectors VðPþ i ðx; yÞÞ and six column vectors VðPi ðx; yÞÞ

4

All computations are rounded off to the third decimal digit.

of order 3, whose elements are, respectively, the active indices Pþi ðx; yÞ and the passive indices Pi ðx; yÞ. For example, if we consider alternatives a and b, then we have VðPþ i ða; bÞÞ ¼ ! ! 1 MðPþ . ða; bÞÞ 13 , where 1p is the unitary column vector of order 2 p P 1. Finally, we calculate the active global index related to each ordered pair of distinct alternatives ðx; yÞ by averaging all active indices Pþ i ðx; yÞ related to that pair. For example, if we consider the pair ða; bÞ 2 CA , then we have

Pþ ða; bÞ ¼

T ! !T ! 1 1 VðPþi ða; bÞÞ 13 ¼ 1n MðPþ. ða; bÞÞ 1n : 3 nðn  1Þ

The analogous computation of P ða; bÞ is carried out aggregating by columns, the details being left to the reader. All global active and passive indices are collected in the following matrices:

0

0

B MðPþ Þ ¼ @ 0:073

0:075 0:125 0

1

C 0:037 A;

0 1 0 0:096 0:187 B C 0 0:079 A: MðP Þ ¼ @ 0:087 0:542 0:496 0 0

0:480 0:444

These matrices evaluate the strength of the arguments, respectively, supporting and opposing a compensated preference of each action over each other. For instance, entry (3, 1) in MðPþ Þ is the value of the index Pþ ðc; aÞ, which synthesizes all the arguments supporting a compensated preference of car c over car a. On the other hand, entry (1, 3) in MðP Þ is the value of the index P ða; cÞ, which synthesizes all the arguments opposing a compensated preference of c over a. Since the difference Pðc; aÞ ¼ Pþ ðc; aÞ  P ða; cÞ ¼ 0:480  0:187 ¼ 0:293 is greater than the chosen sensitivity threshold e ¼ 0:05, then we get Wðc; aÞ ¼ 1, i.e., the relation of compensated preference of c over a is accepted. On the other hand, we have Pða; cÞ ¼ Pþ ða; cÞ  P ðc; aÞ ¼ 0:125  0:542 < 0:05 ¼ e, hence Wða; cÞ ¼ 1, i.e., the relation of compensated preference of a over c is rejected. It follows that c is strongly preferred to a (see Section 2.4). From the matrix of net global indices MðPÞ ¼ MðPþ Þ  MðP ÞT , we obtain the corresponding f1; 0; 1g-valued 0 1 0 0 1 matrix W ¼ @ 0 0 1 A, which yields the relational system of 1 1 0 preferences aIb, cPa, cPb. Incidentally, we observe that the binary relation between a and b depends on the chosen sensitivity threshold. Specifically, we

408

S. Angilella et al. / European Journal of Operational Research 205 (2010) 401–411

have incomparability aRb if e 2 ½0; 0:012Þ, weak preference aQb if e 2 ½0:012; 0:023Þ, and indifference aIb if e 2 ½0:023; 1Þ. On the other hand, the preference of c over a and b is quite stable with respect to variations of the sensitivity threshold e, since we get cPa and cPb as long as e 2 ½0; 0:293Þ. Sensitivity of these results with respect to a (small) change in some compensatory functions is investigated in Section 4. 3.2. An ordinal framework Next, we examine the same example using an ordinal implementation of PACMAN, in which all elementary binary indices are ordinal. Since the only difference with respect to a cardinal implementation is the definition of elementary indices, we omit all calculations that lead to the following matrix of net global indices:

0

0

B MðPÞ ¼ @ 0:184 0:294

0:292 0:417 0 0:403

1

C 0:542 A: 0

If we set, as in the cardinal case, e :¼ 0:05, then the corresponding matrix W yields bPa; cPa; cPb. Some remarks are useful at this point. First we observe that for the same sensitivity threshold e as in the cardinal case, we have bPa in place of bIa. This change of relation is not surprising in view of the philosophy underlying an ordinal framework. In fact, contrary to what happens whenever cardinal elementary indices are used, in an ordinal implementation the numerical difference in evaluations on each criterion is relevant only indirectly, being used exclusively to obtain the value of the corresponding compensatory function (cf. Eq. (4) in Section 2.3). Thus, the preference of b over a obtained here basically depends on the fact that b is preferred to a by the majority of criteria (two out of three), since this yields a larger amount of ordinal arguments in favor of a preference of b over a. More generally, we observe that an ordinal implementation is always very sensitive to the number of criteria that favor one alternative over another one, but not so sensitive to the magnitude of the preference on each criterion. Further, note that since a linear implementation employs the simple average as an aggregation operator at each stage, then the binary preferences obtained using a ordinal implementation cannot be reversed (passing, e.g. from bPa to aPb) regardless of the definition of all compensatory functions. This fact is a particular case of a general statement, which links ordinal implementations and aggregation operators satisfying certain properties to a situation in which one alternative is preferred to another one by means of most criteria (see Remark 1 below). Owing to these shortcomings of an ordinal implementation, we believe that a cardinal framework is more appropriate to deal with decision problems as the one illustrated here. On the other hand, the use of an ordinal implementation might be more suited in case we choose different aggregation functions: for example, we could always take weighted averages (that depend directly on active and passive compensatory powers), or separate active aggregations (for which we choose average-like operators) from passive aggregations (for which we choose maximum-like operators), etc. Remark 1. As already observed, in the given set-up it is impossible to reverse the preference relation from bPa to aPb, no matter how we carry out the compensability analysis. This fact is an instance of a more general statement, which holds under a linear ordinal implementation of PACMAN. In this respect, let us examine a particular case. Let x; y 2 A be two feasible alternatives to be evaluated by means of a set of n P 3 criteria. Assume that x and y are such that x is preferred to y on one criterion, whereas y is preferred to x in the remaining n  1 criteria. It is easy to check that

the binary relation between x and y can only be one of the following: yPx or yQx or yIx. To conclude we observe that the relation aIb can be obtained by increasing the sensitivity threshold, e.g., e 2 ½0:292; 1Þ. However, it is not possible to obtain aPb, since the inequality Pða; bÞ 6 Pðb; aÞ holds for any choice of compensatory functions (not necessarily linear). By a similar argument, one can show that in an ordinal framework, the relations cPa and cPb hold for any choice of compensatory functions, provided that we aggregate using a simple average. 4. Sensitivity analysis in PACMAN As previously shown, an assessment of linear compensatory functions (LCFs) requires a strict interaction between the DA and the DM, and induces a partial pre-order on the set of alternatives. In this section we consider the problem of finding the smallest modification of LCFs that induces a change in the previous ordering. ð0Þ ð1Þ Let W ð0Þ ¼ ðwij Þ and W ð1Þ ¼ ðwij Þ be two n  n matrices of ð0Þ ð0Þ e ð1Þ :¼ Kð1Þ þ W ð1Þ e thresholds. Further, let K :¼ K þ W ð0Þ and K denote the new matrices of parameters for modified LCFs, where Kð0Þ and Kð1Þ are the matrices associated to the original LCFs. Let us suppose that according to the original LCFs and to a cardinal implementation of PACMAN, we have ar Pas , i.e., Pðar ; as Þ > e and Pðas ; ar Þ < e. To find the smallest modification in the parameters of LCFs that forces this preference to be reversed, we perform a sensitivity analysis in the spirit of Mareschal and Wolters [15]. However, contrary to what they do in their study, we do not determine the consequences of a modification of the aggregating weights on a given ranking, although this analysis could also be done in PACMAN. Instead, we concentrate on a sensitivity analysis with respect to changes in the LCFs, the aim being to show the flexibility of this methodology even in the apparently restricted situation in which we use the simple average as a unique aggregation operator. Summing up, we face the following nonlinear programming problem (NLP):

min

n X n  2  2  X ð0Þ ð1Þ s:t: wij þ wij i¼1

j¼1

e ðas ; ar Þ P e; P ð0Þ wij ð0Þ

þ

ð1Þ wij

e ðar ; as Þ 6 e; P

<2

ð0Þ

wij P kij ;

ð0Þ kij ð1Þ



ð1Þ kij ; ð1Þ

wij P kij ;

ð6aÞ ð6bÞ ð6cÞ

e ð:; :Þ is the net global inwhere i; j 2 f1; . . . ; ng are such that i – j; P dex calculated according to modified LCFs obtained from the new e ð1Þ , and e is a new sensitivity e ð0Þ and K matrices of parameters K threshold. The choice to minimize a quadratic function is quite natural in this setting, as we want to remain as adherent as possible to ð0Þ the original LCFs elicited by the DM. (Note that variables wij and ð1Þ wij can also take negative values.) As for the constraints (6a)–(6c), those in (6a) force the relation as Par to hold for every choice of the sensitivity threshold e 2 ð0; eÞ. If the NLP is unfeasible due to these constraints, we can argue that alternative as is never preferred to alternative ar for any modification of LCFs (with fixed aggregation functions). Observe that it is also possible to reformulate constraints (6a) to impose that two specific alternatives are in a relation different from strong preference, or to determine the sensitivity of a certain pre-order of alternatives to changes in (some or all) LCFs, etc. All other constraints in (6b) and (6c) are imposed so that in each e ð1Þ is non-negative and that the e ð0Þ and K LCF every element of K ð0Þ ð1Þ k ij < 2 hold for each ðg i ; g j Þ 2 CG . Note that by inequalities e k ij þ e

409

S. Angilella et al. / European Journal of Operational Research 205 (2010) 401–411

requiring that the last inequality is strict, we exclude de facto that – for the problem at hand – a compensatory function can be constant (i.e., identically equal to zero or one). This causes no great loss of generality, since such a modelling of compensatory functions is employed only in very special cases, e.g., a lexicographic ordering of criteria [1,10]. The NLP problem considered here has 2ðn2  nÞ variables ð0Þ ð1Þ wij ; wij , and 3ðn2  nÞ þ 2 constraints; its numerical implementation is quite easy. All calculations presented in this paper are carried out employing a standard simulated annealing algorithm. 4.1. An example Let us reconsider the example presented in Section 3.1, in case that a cardinal implementation of PACMAN is employed. For the previously defined LCFs, if e 2 ½0:012; 0:023Þ, then the global matrix MðPÞ obtained before yields the f1; 0; 1g-valued matrix

0 B

0

0 1

1 C

W ¼ @ 1 0 1 A; 1

1

0

whence car a is weakly preferred to car b. In Fig. 3, we set e :¼ 0:018 and represent the relation aQb by a filled circle. Our first goal is to transform this relation of weak preference into one of strong preference of a over b. We try to obtain aPb by employing a NLP, where e ðb; aÞ 6 e, and e ða; bÞ P e and P the first two constraints become P the other constraints are kept unchanged. We take e :¼ 0:023, since this is the least upper bound of the set of sensitivity thresholds that guarantees aQb under the elicited LCFs (see the matrices Kð0Þ and Kð1Þ indicated in (5), Section 3). At the optimal solution the objective function takes the value ð0Þ k 21 ¼ f1 ¼ 0:097. The only modification in LCFs are given by e ð0Þ ð1Þ ð1Þ e e e all remaining 0:695; k 23 ¼ 0:168; k 21 ¼ 0:678; k 23 ¼ 0:287, entries of the two matrices of parameters being as in (5). From f PÞ, we obtain, for all the new matrix of global indices Mð e 2 ½0; 0:023Þ, the following f1; 0; 1g-valued matrix

0

1

0 1 1 C e ¼B W @ 1 0 1 A: 1

1

Table 3 Comparison of compensatory powers after a sensitivity analysis.

1 2 3

Fig. 3. Preference relations between a and b resulting from sensitivity analysis with e ¼ 0:018.

e pþ i

b pþ i

p i

e p i

b p i

0.414 0.422 0.436

0.414 0.515 0.436

0.547 0.39 0.481

0.642 0.546 0.539

0.568 0.546 0.521

0.667 0.369 0.547

the relation of incomparability is now empty, because it is ruled out by the feasibility of the first two constraints of the NLP. Some further considerations are useful at this stage. Since the only modified compensatory functions are CF2.1 and CF2.3 , it follows that only one active compensatory power changes, namely, the active compensatory power of criterion two (maximum speed), eþ increasing from pþ 2 ¼ 0:422 to p 2 ¼ 0:515. For the same reason, we    e e ¼ p , but now p ¼ 0:568 < p have e p 2 2 1 1 and p 3 ¼ 0:521 < p3 (see Table 3). The interpretation of the switching from aQb into aPb is rather intuitive in the light of compensability analysis. Indeed, since car a dominates car b with respect to maximum speed but is dominated with respect to price and consumption, we find that a is strongly preferred to b by either (i) strengthening the active power of speed, or (ii) weakening the passive resistance of price and consumption. In this way we have that a higher value of maximum speed is very appreciated by the DM, who is also not too sensitive to a price cut and/or a lower fuel consumption. We observe that even with modified LCFs the DM would still be less than c DM ðGÞ ¼ 0:455. half-compensatory, being e Next, we try to reverse the preference relation, in order to obtain a relation of strong preference bPa of b over a. The first two b ða; bÞ 6 e, where b ðb; aÞ P e and P constraints in the NLP become P again we take e :¼ 0:023, and LCFs are obtained from (5) as in the previous example. In this case the objective function at an optimum solution has value f2 ¼ 0:321, and we obtain

0

0

c PÞ ¼ B Mð @ 0:023 0:448

0:023 0:454

1

C 0:475 A;

0 0:474

0

with

0

0

Thus, a is strongly preferred to b (whereas the relations cPa and cPb remain unchanged). In Fig. 3, we represent the relation aPb by a filled square. It is easy to verify that for any choice of e 2 ½0; 1Þ,

pþ i

b ð0Þ K

0 B ¼ @ 0:354 0:1

0:77 0 0:326

1 0:5 C 0:168 A; 0

0

b ð1Þ K

1 0 0:306 0:7 B C ¼ @ 0:885 0 0:215 A: 0:5

0:019

0

Therefore, now we have bPa, yet cPa and cPb continue to hold. In Fig. 3, we represent the relation bPa by a filled triangle. Observe that this time the changes in LCFs induce a change of the active and passive compensatory powers for all criteria (see Table 3). In particular, compensatory powers of price and fuel consumption increase, whereas the opposite happens for maximum speed. The explanation of this fact is similar to that given for the previous example. It is possible to reverse the relation aPb into bPa by (i) strengthening the overall importance of criteria where b dominates a, and (ii) weakening the overall importance of the unique criterion where a dominates b. Observe that in order to obtain a switch in the preference relation from aPb to bPa, we need more changes than those necessary to pass from aQb to aPb, namely, both strengthen a (suitable) active power and weaken (suitable) passive powers. Further, note that the changes in the structure of LCFs, which force the relation of weak preference aQb to become a relation of strong preference aPb, are fewer than those required to switch from aPb to bPa. In fact, this statement can be made more precise by comparing the values f1 < f2 of the objective function at an optimum in the two cases. The DA could, for instance, ask the DM whether the changes in LCFs in the first sensitivity analysis are

410

S. Angilella et al. / European Journal of Operational Research 205 (2010) 401–411

‘‘tolerable” or not, in order to infer whether car a is strongly preferred or weakly preferred to car b. On the other hand, substantial changes in the elicited LCFs (hence in compensatory powers and in the DM’s aptitude to compensate among criteria) are required in order to obtain a relation of strong preference bPa. Therefore, in this example it is reasonable to deduce that the DM’s scheme of preferences goes in the direction of a (weak or strong) preference of car a over car b. Here are some final remarks. First, note that despite the fact that in our example cars a and b are alternatives having very similar evaluation profiles, we have shown that it is possible to obtain any type of binary relation between them (indifference, incomparability, weak preference or strong preference) by suitably (and meaningfully) manipulating the parameters of the model. Second, these results are obtained within a very simplified version of PACMAN, where we employ only linear compensatory functions and a unique aggregation operator. Therefore, the possibility to discern between alternatives might certainly be enhanced if we were to employ other types of implementations, or adopt different aggregating functions, or drop the assumption of linearity.

5. Conclusions In this paper we have proposed a simplified version of PACMAN, which has several advantages from the computational point of view, compared to the general features of the original PACMAN methodology. In fact, the implementation described here substantially reduces both the amount and the technical content of the information to be provided by the DM. Further, this approach enables the DA to set-up a handful interactive procedure to estimate all the parameters of the model. We have encoded the whole decision procedure using MathematicaÒ, building a computer-aided interface that makes the interaction between DM and DA easy and understandable at each stage. The effective implementation of PACMAN is simplified by the usage of linear compensatory functions on one hand and the simple average as unique aggregation operator on the other hand. The construction of linear compensatory functions is algorithmically simple, since it only requires the assessment of two parameters per function. Further, these parameters have an immediate economic interpretation. On the other hand, the usage of the simple average as an aggregation operator seems to be the most suitable whenever a cardinal implementation is employed, also because it eases the interpretation of the aggregation procedure by the DM. Our approach is certainly not free from problems. Some drawbacks of the original PACMAN decision procedure have already been pointed out in previous papers [9,10]. These problems are related to the following issues: (i) large amount and technical content of the information to be provided by the DM; (ii) assumptions underlying the construction of evaluation criteria; (iii) interpretation of the results in the light of the parameters of the model. Problem (i) might become rather serious in real-life applications that require the consideration of a large number of evaluation criteria. In fact, the most relevant difficulty in applying PACMAN is the construction of compensatory functions. The combinatorial explosion of the amount of data to be provided by DM in case that there is a large number of criteria may certainly affect the feasibility of the procedure. From this point of view, one should either concentrate on few evaluation criteria that are considered relevant for the decision problem at hand, or use some simplifying assump-

tions to reduce both the amount and the technical content of the information to be provided. Since the former type of solution has to be evaluated on a case-to-case basis, in this paper we have attempted to propose a solution in the latter sense. Concerning (ii), in PACMAN we explicitly assume that criteria are interval scales (i.e., full significance of differences of evaluations regardless of the magnitude of the two evaluations). Of course, this is a fairly limiting condition, as we have already emphasized. Nevertheless, there are several multi-criteria methodologies that explicitly or implicitly make this assumption, e.g., MAPPAC [17], MACHBETH [3] and all multi-criteria methods based on the Choquet integrals [11,12]. Furthermore, the interval scale hypothesis is not necessarily a strong limitation in certain real-life situations. In fact, if the problem at hand is to select an item within a certain segment of the market, then the range of each criterion might be small enough to make this hypothesis meaningful. For example, in the process of selecting a small sized family car, it only yields a limited amount of arbitrariness to assume that a difference in fuel consumption of, say, 2 km=l has the same meaning at any level of consumption. Finally, for what concerns (iii), we are aware of the fact that PACMAN is a fairly complicated multi-criteria methodology, which may be regarded as too sophisticated (even presenting some blackbox effects) to be fully understood by a DM. On the other hand, we believe that its flexibility – which lies in the presence of several parameters to be assessed and aggregation functions to be chosen – is also a tool to shape the methodology and make it as much adherent as possible to the DM’s scheme of preferences. Specifically, the declared objective of this flexibility is an attempt to capture several features of the DM’s aptitude in relation to intercriteria compensability. From this point of view, PACMAN may be regarded as a DM-oriented multi-criteria decision procedure; accordingly, we talk of a compensatory or non-compensatory DM, instead of compensatory or non-compensatory methodology [9,10]. That is the reason why all previous papers on the subject (especially the most recent one [1]) have been partially devoted to emphasize the flexibility of the procedure and its theoretical capability to reproduce some well-known models in the literature, such the lexicographic, the semi-lexicographic, etc. Admittedly, an excessive amount of flexibility fires back at a practical level, especially if the parameters to be evaluated/chosen are too many and require a rather large cognitive effort. If this is the case, a DM would probably be discouraged from using the whole methodology. In particular, concerning the choice of aggregation functions, in this paper we have decided to use the simple average (and not, for example, a weighted average), only because it appears to be the less intrusive way of aggregating evaluations. Of course, this does not imply that other types of (meaningful) aggregation operators can be chosen in particular cases [1]. Other types of drawbacks might be in direct relation to the simplified version of PACMAN described in this paper. In fact, an apparent constraint of our approach is that we only allow linear compensatory functions. This appear to be a limitation in the modelization of intercriteria relationships, since in some cases different types of compensatory functions might fit better the decision problem at hand (e.g., a compensatory function that is everywhere differentiable). Nevertheless, the limits connected to the assumption of linearity seem to be overwhelmed by its computational advantages. In fact, the assumption of linearity has an effect on the output that is far smaller than one might believe. Indeed, once that the areas of null and total compensability have been determined by the DM, to assume a linear or a differentiable behavior in the area of partial compensability has very little impact on most of the features of the methodology, and in particular on the final relational system of preferences. Therefore, the axiom of linearity only yields a fairly limited amount of arbitrariness.

S. Angilella et al. / European Journal of Operational Research 205 (2010) 401–411

On the other hand, the assessment of linear compensatory functions is extremely easy and understandable if we use the interactive procedure developed in this paper. The notable simplification in practical implementations that derives from it quite compensates the loss of flexibility in modeling different types of compensatory functions. From this point of view, one might think of the linearity of the approach simply as the first (necessary) step toward a much more interesting implementation of compensatory functions. Furthermore, the assumption of linearity makes it simple to formulate a sensitivity analysis related to the obtained results as a (nonlinear) programming problem, in order to explore how some preference relations within couples of actions are stable with respect to changes in the LCFs elicited by the DM. In fact, we have observed that the modified parameters of linear compensatory functions induce meaningful changes in active and passive compensatory powers of criteria, which can be easily interpreted from the decision point of view. Acknowledgments The authors thank the Editor and some anonymous referees for their comments and suggestions, which yielded a decisive improvement in the quality of the paper. The authors also thank Professor B. Matarazzo for some useful suggestions, and Professor N.H. March for a careful reading of the manuscript. References [1] S. Angilella, A. Giarlotta, Implementations of PACMAN, European Journal of Operational Research 194 (2) (2009) 474–495. [2] J.P. Brans, B. Mareschal, Ph. Vincke, How to select and rank projects: The PROMETHEE method, European Journal of Operational Research 21 (1986) 251–266. [3] C.A. Bana e Costa, J.C. Vansnick, MACBETH: An interactive path towards the construction of cardinal value functions, International Transactions in Operational Research 1 (4) (1994) 387–500. [4] D. Bouyssou, Some remarks on the notion of compensation in MCDM, European Journal of Operational Research 16 (1986) 150–160.

411

[5] D. Bouyssou, J.C. Vansnick, Noncompensatory and generalized noncompensatory preference structures, Theory and Decision 21 (1986) 251–266. [6] J. Figueira, V. Mousseau, B. Roy, ELECTRE methods, in: J. Figueira, S. Greco, M. Ehrgott (Eds.), Multiple Criteria Decision Analysis: State of the Art Surveys, Springer, Berlin, 2005, pp. 134–162. [7] J. Figueira, S. Greco, R. Słowin´ski, Building a set of additive value functions representing a reference preorder and intensities of preference: GRIP method, European Journal of Operational Research 195 (2) (2008) 460–486. [8] P.C. Fishburn, Utility Theory for Decision-Making, John Wiley & Sons, New York, 1970. [9] A. Giarlotta, Passive and Active Compensability Multicriteria ANalysis (PACMAN), Journal of Multi-Criteria Decision Analysis 7 (1998) 204–216. [10] A. Giarlotta, Multicriteria compensability analysis, European Journal of Operational Research 133 (2001) 190–209. [11] M. Grabisch, Ch. Labreuche, The Choquet integral for the aggregation of interval scales in multicriteria decision making, Fuzzy Sets and Systems 137 (2003) 11–26. [12] M. Grabisch, Ch. Labreuche, Fuzzy measures and integrals in MCDA, in: J. Figueira, S. Greco, M. Ehrgott (Eds.), Multiple Criteria Decision Analysis: State of the Art Surveys, Springer Science + Business Media, Inc., 2005, pp. 345–407. [13] E. Jacquet-Lagrèze, Y. Siskos, Assessing a set of additive utility functions for multicriteria decision-making, the UTA method, European Journal of Operational Research 10 (1982) 151–164. [14] R.L. Keeney, H. Raiffa, Decision with Multiple Objectives – Preferences and Value Tradeoffs, Wiley, New York, 1976. [15] B. Mareschal, W.T.M. Wolters, Novel types of sensitivity analysis for additive MCDM methods, European Journal of Operational Research 81 (1995) 281– 290. [16] J.M. Martel, B. Matarazzo, Other outranking approaches, in: J. Figueira, S. Greco, M. Ehrgott (Eds.), Multiple Criteria Decision Analysis: State of the Art Surveys, Springer, Berlin, 2005, pp. 197–262. [17] B. Matarazzo, Multicriterion analysis of preferences by means of pairwise action and criterion comparisons (MAPPAC), Applied Mathematics and Computation 18 (2) (1986) 119–141. [18] B. Matarazzo, A pairwise criterion comparison approach: The MAPPAC and PRAGMA methods, in: C.A. Bana e Costa (Ed.), Readings in Multiple Criterion Decision Aid, Springer, Berlin, 1990, pp. 253–273. [19] F.S. Roberts, Measurement Theory with Applications to Decision-Making, Utility and the Social Sciences, Addison-Wesley, London, 1979. [20] B. Roy, Méthodologie Multicritére d’Aide àla Décision, Economica, Paris, 1985. [21] B. Roy, The outranking approach and the foundations of ELECTRE methods, in: C.A. Bana e Costa (Ed.), Readings in Multiple Criteria Decision Aid, Springer, Berlin, 1990. [22] B. Roy, V. Mousseau, A theoretical framework for analyzing the notion of relative importance of criteria, Journal of Multi-criteria Decision Analysis 5 (1996) 145–159.