Construction of nonreciprocal fuzzy preference relations with the use of preference functions

Construction of nonreciprocal fuzzy preference relations with the use of preference functions

Roberta Parreiras^*; Petr Ekel

ABSTRACT

In order to model the preferences of a decision-maker (DM) by means of fuzzy preferencerelations, a DM can utilize different preference formats (such as ordering of the alternatives, utility values, multiplicative preference relations, fuzzy estimates, and reciprocal as well as nonreciprocal fuzzy preference relations) to express his/her judgments. Afterward, the obtained information is utilized to constructfuzzy preference relations. Here we introduce a procedure that allows the use of so-called preference functions (which is a preference format utilized in the methods of PROMETHEE family) to construct nonreciprocal fuzzy preference relations. With diverse preference formats being offered, a DM can select the onethat is the most convenient to articulate his/her preferences. In order to demonstrate the applicability of theproposed procedure a multicriteria decision-making problem related to the site selection for constructing anew hospital is considered here.

Keywords: multicriteria decision-making, fuzzy preference relations, preference functions.

1 INTRODUCTION

The approaches to dealing with multicriteria (multiattribute) decision-making problems, which, for instance, are discussed in Orlovsky (1978), Chiclana et al. (1998), Ekel et al. (1998), Chiclana et al. (2001), are directed at processing the individual preferences as a pair (X, R), X = {X₁, X₂,..., X_n} represents a finite and discrete set of alternatives, which are to be evaluated, compared, ordered, and/or prioritized under the consideration of a set of fuzzy nonstrictpreference relations R = {R₁, R₂,..., R_q}, given in accordance with a set of criteria F = {F₁, F₂,..., F_q} being considered.

A fuzzy nonstrict preference relation (Orlovsky, 1978) consists of a binary fuzzy relation (BFR), which is a fuzzy set with bi-dimensional membership function R_p(X_k , X_l) : X × X →[0, 1].In essence, the membership function of the pth fuzzy preference relation indicates in the unit interval the degree to which the alternative X_kis at least as good as X_l, when the criterion F_p, is considered by a particular decision-maker (DM). In a somewhat loose sense R_p(X_k , X_l) also represents the degree of truth of the statement "X_kis at least as good as X_l" (Kulshreshtha & Shekar, 2000).

In dealing with (X, R) models, a fundamental question arises on how one can construct fuzzy preference relations to reflect the preferences of a DM. In practice, a DM can use different formats to establish preferences for the alternatives under consideration (Herrera-Viedma et al.,2002; Zhang et al., 2007; Pedrycz et al., 2011), including, the ordering of the alternatives, theutility values, multiplicative preference relations, fuzzy estimates, and fuzzy preference relations (additive reciprocal and nonreciprocal). In real world applications, several factors may lead aDM to select a different way for expressing his/her preferences about each criterion. Amongthese factors we can list the following (Pedrycz et al., 2011):

• each criterion comes with its significance (a fundamental feature which provides significance to the difference between two degrees evaluated on this criterion). Depending whether this significance has a qualitative or quantitative character, the use of certain preference formats can make the preference elicitation process easier and also more reliable;

• each criterion is associated with information arising from different sources and with information having different levels of uncertainty;

• a DM may find that his/her preference strengths can be better reflected or quantified by aspecific preference format;

• a fact that a DM may posses previous knowledge or experience in expressing a specific preference format can motivate him/her to choose using it again.

The information captured by those different preference formats can be utilized to construct fuzzy preference relations by means of applying adequate transformation functions (Chiclana et al.,1998, 2001; Herrera-Viedma et al., 2002; Pedrycz et al., 2011). Alternatively, a DM can directlyassess the fuzzy nonstrict preference relation by determining the preference strength of one alternative over another as a number in the unit interval. Current literature contains different encoding schemes which can be utilized in the construction of fuzzy preference relations (refer to (Wang,1997) for instances of fuzzy preference relations based on different encoding schemes). In thispaper, we consider a particular encoding scheme which can be utilized to construct a nonreciprocal fuzzy preference relation (NRFPR) (Orlovsky, 1978; Fodor & Roubens, 1994a; Ekel &Schuffner Neto, 2006; Pedrycz et al., 2011). Such encoding scheme allows one to construct a NRFPR which is compatible with:

• a rational approach (Ekel et al., 1998) for deriving fuzzy preference relations, which isbased on the comparison of the fuzzy estimates provided by a DM to evaluate each alternative (a discussion of the advantages of such rational approach can be found in (Pedryczet al., 2011));

• the notion of fuzzy nonstrict preference relation belonging to fuzzy preference structureswith incomparability relation (Fodor & Roubens, 1994a). In this way, the use of a NRFPR permits one to capture the preference degree of each pairwise comparison and the confidence degree of each pairwise comparison as well. A DM should assign a low credibility degree to a paired comparison whenever he/she can not define with conviction the preference degree for one alternative over another. The lack of conviction may be associated with missing information or to the occurrence of contradictory information about one alternative or both alternatives belonging to the pair (Fodor & Roubens, 1994a,b).

Motivated by a demand for different preference formats to choose it as comfortable as possiblefor a DM to articulate his/her preferences (Zhang et al., 2007; Pedrycz et al., 2011), here we introduce a procedure that allows the use of preference functions to construct NRFPRs. In themethods of PROMETHEE family (Brans & Vincke, 1985), a DM must define the shape of the preference functions in accordance with his/her preferences so that they can be utilized to compare the alternatives and to construct outranking relations for each criterion being considered in the multicriteria analysis. Here, the preference functions are implemented as part of a generalframework for preference modeling in a fuzzy environment, which offers six preference formatsfor preference articulation. They are: the ordering of the alternatives, the utility values, the multiplicative preference relations, the fuzzy estimates, the fuzzy preference relations (reciprocal andnonreciprocal), and the preference functions. With the availability of several preference formats,the choice of the most suitable one is a prerogative of a DM. The applicability of the preference functions to construct NRFPRs, within such general framework for preference modeling, isdemonstrated through the solution of a decision-making problem related to the site selection forconstructing a new hospital (Vahidnia et al., 2009).

2 JUDGMENTS OF STRICT PREFERENCE, INDIFFERENCE, ANDINCOMPARABILITY

Let us consider that a DM is asked to compare two alternatives X_k∈ X, X_l∈ X for a givencriterion and determine which, between these two, he/she prefers. One of the following answersis expected (Fodor & Roubens, 1994a):

• X_kand X_lare indifferent;

• X_kis strictly better than X_l;

• X_lis strictly better than X_k;

• X_kand X_lare incomparable (a DM may not be able to compare the alternatives due tomissing or uncertain information or as a consequence of the existence of conflicting information).

Accordingly, in order to realistically characterize this comparison between two alternatives, threemain types of judgments can be distinguished, namely indifference, strict preference, and incomparability. Given the pth criterion, these judgments can be modeled by means of three specificBFRs, namely the fuzzy indifference relation I_p, the fuzzy strict preference relation P_p, and the fuzzy incomparability relation J_p, in such a way that the membership function of each BFR quantifies in the interval [0, 1], the credibility or the intensity of the observed judgment.

In fuzzy preference models based on BFRs, given De Morgan triplet (T, S, N), it is possible to define I_p, P_p, and J_pexclusively in terms of the fuzzy nonstrict preference relation R_p. It is important to indicate that the fuzzy nonstrict preference relation, by definition, can be expressed as (Fodor & Roubens, 1994a):

By considering that I_pcorresponds to all pairs of alternatives that simultaneously satisfy R_p(X_k , X_l) and R_p(X_l , X_k), the indifference relation can be stated as

where R_p^-1 is the inverse relation of R_p, that is R^-_p ¹(X_k , X_l) = R_p(X_l , X_k) (Orlovsky, 1978, 1981).

Similarly, as P_p(X_k , X_l) implies that R_p(X_k , X_l) and N (R_p(X_l , X_k)), the strict preference can be specified as

where R^d corresponds to the dual relation of R_p, that is

(Fodor & Roubens, 1994a).

Finally, as the relation J_p(X_k , X_l) implies that N (R_p(X_k , X_l)) and N (R_p(X_l , X_k)), the incomparability relation is given by

where corresponds to the complementary relation of R_pthat is

(Fodor & Roubens, 1994a).

Therefore, once we have at hand the values of R_p(X_k , X_l) and R_p(X_l , X_k), the estimation of I_p, P_p, and J_pis realized on the basis of (2), (3), and (4), respectively. As one can note, those three expressions require the selection of a t-norm operator. Unfortunately, as it has been discussed,for instance, in De Baets & Fodor (1997), it is not simple to select a t-norm to implement (2)-(4),if we want to preserve certain desirable properties of a fuzzy preference structure. Indeed, anegative result demonstrated in Alsina (1985) indicates that, if a De Morgan triplet is utilized to represent the complement, the intersection and the union of BFRs, then the equality

is not satisfied for any binary fuzzy relations Z and W . If we consider Z as being R_pand W as R^d_p, then it implies that the relationships (2), (3), and (1) are inconsistent for any reflexive fuzzy binary relation, if we use the same intersection operator to implement the two intersection operations in (5).

Among the admissible t-norms to be utilized in this context (Fodor & Roubens, 1994b), weselected the min operator to implement the intersection in (2) and in (4) and the Lukasiewicz t-norm to implement the intersection in (3), as given by the following expressions:

It should be indicated that the definition of the fuzzy strict preference relation as given by (7) is in conformity with the Orlovsky choice procedure (Orlovsky, 1978), which is described in Section 5.

3 DIRECT ASSESSMENT OF NONRECIPROCAL FUZZY NONSTRICTPREFERENCE RELATIONS

In real applications, a DM is usually asked to pick up the values in the unit interval that reflectthe level of credibility or just the strength of his/her nonstrict preference for one alternative over the other. The encoding scheme considered here is reflected by a NRFPR which verifies thefollowing conditions (Pedrycz et al., 2011):

• if R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = 1, then X_kis indifferent to X_l;

• if R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = 0 , then X_kis strictly preferred to X_l;

• if R_p(X_k , X_l) = 0 and R_p(X_l , X_k) = 1, then X_lis strictly preferred to X_k;

• if R_p(X_k , X_l) = 0 and R_p(X_l , X_k) = 0, then X_kand X_l are not comparable;

• the entries of the main diagonal are filled with 1, due to the reflexivity of R_p(X_k , X_l).

Intermediate judgments among the situations described above are also allowed. They can beinterpreted as follows:

• if 0 < R_p(X_k , X_l)< 1 and R_p(X_l , X_k) = 1, then X_lis weakly preferred to X_k;

• if R_p(X_k , X_l) = 1 and 0 < R_p(X_l , X_k)< 1, then X_kis weakly preferred to X_l;

• if 0 < R_p(X_k, X_l)< 1 and R_p(X_l , X_k) = 0, then Xk is weakly preferred to X_land, at the same time, X_kand X_lare to a degree considered incomparable;

• if R_p(X_k , X_l) = 0 and 0 < R_p(X_l , X_k)< 1, then X_lis weakly preferred to X_kand, at the same time, X_kand X_lare to a degree considered incomparable.

It is important to indicate that relations (6) and (7) determine a clear relationship between thevalues of R_p(X_k , X_l) and R_p(X_l , X_k). If (6) and (7) are utilized for defining the fuzzy indifference relation and the fuzzy strict preference relation, respectively, then the difference between R_p(X_k , X_l) and R_p(X_l , X_k) reflects the level of strict preference between the alternatives X_kand X_l. Besides, the minimum value between R_p(X_k , X_l) and R_p(X_l , X_k) reflects the level of indifference between the alternatives X_kand X_l.

With reference to the consistency of NRFPRs, the weak transitivity, which is given by (Herrera-Viedma et al., 2004).

If R_p(X_k , X_j) > R_p(X_j , X_k) and R_p(X_j , X_l) > R_p(X_l , X_j), then

leads to the minimum requirement that should be satisfied by rational preference judgments. Itimplies that if someone says that X_kis preferred to X_jand that X_jis preferred to X_l, then X_kispreferred to X_l, without considering the strength of the preferences. It is also the minimum requirement for the application of decision-making methods based on the Orlovsky choice function (Orlovsky, 1978), if one wants some rational properties to be attained (Sengupta, 1998).

4 INDIRECT ASSESSMENT OF NONRECIPROCAL FUZZY NONSTRICT PREFERENCE RELATIONS

In individual (as well as in group decision-making), when different preference formats are utilized by a DM to express his/her preferences, the information must be made uniform under adequate transformation functions, before being analyzed. These transformation functions can convert heterogeneous preference information, which may be qualitative or quantitative, two-valuedor fuzzy, ordered or non-ordered, ordinal or cardinal into fuzzy preference relations, which forma more general preference model (Chiclana et al., 1998, 2001; Herrera-Viedma et al., 2002; Pedrycz et al., 2011). As it was indicated above, the five main formats, that can be utilize by a DM to articulate his/her preferences, in addition to the NRFPRs, are:

• Vector with the ordering of all alternatives. The ordering of alternatives from best to worst can be represented as an array O_p=[O_p(X₁), . . . , O_p(X_n)], with O_p(X_k) beinga permutation function, which returns the position of alternative X_kamong the integer values {1, 2,..., n} (Chiclana et al., 1998).

• Vector of the utility values. A measurable utility function U_p(x) : X →[0, 1], foundedon differences in preference strengths (Belton, 1999; Farquhar & Keller, 1989), allows a DM to assign a utility value from the unity interval [0, 1] to each alternative belonging to X.

• Multiplicative preference relation. The multiplicative preference relation can be represented as a n × n positive reciprocal matrix M_preflecting the preference intensity ratio between the alternatives in accordance with the Analytic Hierarchy Process approach (Saaty, 1980). Each entry M_p(X_k , X_l) of this reciprocal matrix represents a preference intensity ratio and can be interpreted as "X_kis M_p(X_k , X_l) times more dominant than X_l" (Saaty, 1980) or as "X_kis M_p(X_k , X_l) times as good as X_l" (Chiclana et al., 2001). Under the condition of multiplicative reciprocity, once a DM provides M_p(X_k , X_l), the value of M_p(X_l , X_k) is automatically inferred as M_p(X_l , X_k) = 1/M_p(X_k , X_l).

• Fuzzy estimates. The elements of X can be evaluated with the use of fuzzy estimates L_p={l_p(X₁), . . . , l_p(X_n)}, being l_p(X_k) the fuzzy estimate associated with alternative X_kfrom the point of view of a given criterion F_p. The fuzzy estimate l_p(X_k) refers toa fuzzy number that can be directly specified by a DM or indirectly expressed by meansof linguistic terms from some set S such as, for instance, S(F_p) ={low quality, average quality, high quality}. In the latter case, the linguistic terms must be converted into fuzzyestimates, as required to perform the analysis of the problem.

• Additive reciprocal fuzzy preference relation (ARFPR). An ARFPR consists of a specific type of fuzzy preference relation which satisfies the additive reciprocity condition:

where RR_prepresents the ARFPR (here, NRFPR is denoted as R_pand ARFPR is denoted as RR_p).

The encoding scheme usually associated with ARFPRs can be summarized as the following rules:

• RR_p(X_k , X_l) = 0.5 means that X_kis indifferent to X_l;

• 0 < RR_p(X_k , X_l) < 0.5 means that X_lis preferred to X_k;

• 0.5 < RR(X_k , X_l) < 1 means that X_kis preferred to X_l;

• the entries of the main diagonal are filled with 0.5, as each element is equal to itself and,as a result, indifferent to itself.

In (Chiclana et al., 1998), one can find different transformation functions for converting a vector with the ordering of all alternatives or a vector of the utility values associated with each alternative into an ARFPR. A transformation function for converting multiplicative preference relations into an ARFPR is proposed in (Chiclana et al., 2001). In (Pedrycz et al., 2011), newtransformation functions are proposed. They convert preference information expressed in thosefive different formats described above into NRFPRs. Some of the transformation functions proposed in (Queiroz, 2009; Pedrycz et al., 2011) are summarized in Table 1. It should be indicated that they are defined for the case of maximization criteria (if the criterion being considered hasa minimization character, then the direction of the signs of all the inequalities appearing in thetransformation functions should be reversed).

In this paper, those transformation functions summarized in Table 1 are utilized to create a general framework for constructing NRFPRs on the basis of preference information which may be expressed in different formats. Given a set X of alternatives, a DM can express his/her preferences by ordering the alternatives from best to worst, by constructing a utility function or by performing pairwise comparisons between the alternatives to construct a multiplicative preference relation, a ARFPR or a NRFPR. At the same time, in order to increase the flexibility of such general framework, the results of (Ekel et al., 1998) are utilized to derive NRFPRs from the fuzzy estimates given by a DM to evaluate the alternatives. In this way, once a DM provides fuzzy estimates F_p(X_k), k = 1, 2,..., n to evaluate all the alternatives for a criterion F_pwhichcan be measured on a numerical scale, if the essence of preference is coherent with the natural order (>) along the axis of measurable values of F_p, then the following expressions can be utilized to construct a NRFPR:

where ƒ_p(Xk) and ƒ_p(X_l) are real numbers reflecting the evaluation of the attribute F_p(with a maximization character) for the alternatives X_kand X_l; F_p( ƒ_p(X_k)) and F_p( ƒ_p(X_l)) represent the membership functions of the fuzzy sets F_p(X_k) and F_p(X_l) evaluated at ƒ_p(X_k) and ƒ_p(Xl),respectively. It should be indicated that (11) and (12) are defined for maximization criteria. The direction of the signs of the inequalities in (11) and (12) should be reversed, if a minimizationcriterion is being considered.

5 CONSTRUCTION OF NONRECIPROCAL FUZZY PREFERENCE RELATIONS WITH THE USE OF PREFERENCE FUNCTIONS

The results of (Brans & Vincke, 1985) permit one to apply preference functions as an additional format that can be used by a DM to express his/her preferences. In particular, in the family of PROMETHEE methods, the preferences being restricted to a single criterion F_p, are modeled through a preference function s_p(X_k , X_l), in such a way that it reflects the preference level of X_kover X_l, according to the following rules:

• if s_p(X_k , X_l) = 0, both alternatxves are considered indifferent to each other;

• if s_p(X_k , X_l) = 1, X_kis strictly preferred to X_l;

• if 0 < s_p(X_k , X_l)< 1, X_kis weakly preferred to X_l.

The preference functions are usually defined in terms of the difference

in such a way that they transform the difference in the evaluations of two alternatives into apreference intensity between 0 and 1. The preference functions can have different shapes, as long as they are nondecreasing functions of the difference d_p(X_k , X_l) (Brans & Vincke, 1985). The methods PROMETHEE I and PROMETHEE II admit six different generalized models for the preference function, which cover most part of the scenarios encountered in real applications(Brans & Vincke, 1985). They are: the usual criterion, the quasi-criterion, the level-criterion, the linear criterion, the linear criterion with indifference region, and the Gaussian criterion. Withthe availability of those generalized functions, it may be easier for a DM to define a preferencefunction reflecting his/her preferences. Next, we describe how one can construct NRFPRs basedon each one of those preference functions. It should be indicated that, here we consider thecriteria as being of maximization type (without loss of generality).

Let us begin by considering three preference functions that are particularly easy to define: the usual criterion, the quasi-criterion, and the level-criterion. Figure 1 shows the graphical representation of the usual criterion, which may be considered as being the simplest type of preference function, since it does not require any parameter to be set by a DM. As it can be seen in Figure 1, the level of strict preference of X_kover X_lis equal to 1 for d_p(X_k , X_l)> 0 and is null for d_p(X_k , X_l)< 0. if a dm wants to construct a nrfpr based on the usual criterion, the following rules must be applied to the comparison of each pair of alternatives:

• if d_p(X_k , X_l) < 0, then R_p(X_k , X_l) = 0 and R_p(X_l , X_k) = 1;

• if d_p(X_k , X_l) = 0, then R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = 1;

• if d_p(X_k , X_l) > 0, then R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = 0.

Figure 2 shows a graphical representation of the preference function corresponding to the quasicriterion and of the NRFPR based on the quasi-criterion. It is worth noting that the usual criterion can be seen as a particular case of the quasi-criterion where the bounds of the indifference interval [-a_p, a_p] are null, that is a_p= 0. If a DM wants to construct a NRFPR coherently with the quasi-criterion, the pairwise comparisons must attain the following conditions:

• if d_p(X_k , X_l) < - a_p, then R_p(X_k , X_l) = 0 and R_p(X_l , X_k) = 1;

• if -a_p< d_p(X_k , X_l) < a_p, then R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = 1;

• if d_p(X_k , X_l) > a_p, then R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = 0.

The level-criterion allows discriminating an indifference interval [-a_p, a_p] and a weak preference interval [a_p, b_p] (and [-b_p, -a_p]). The pairwise comparisons must satisfy the following conditions in order to construct a NRFPR coherently with the level-criterion:

• if d_p(X_k , X_l) < - b_p, then R_p(X_k , X_l) = 0 and R_p(X_l , X_k) = 1;

• if -b_p< d_p(X_k , X_l) < -a_p, then R_p(X_k , X_l) = 0.5 and R_p(X_l , X_k) = 1;

• if -a_p< d_p(X_k , X_l) < a_p, then R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = 1;

• if a_p< d_p(X_k , X_l) < b_p, then R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = 0.5;

• if d_p(X_k , X_l) > b_p, then R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = 0.

The preference functions of the linear criterion, the linear criterion with an indifference region, and the Gaussian criterion present a smooth transition between indifference and strict preference, which permits a DM to make judgments at different levels of weak preference. The graphical representations of those preference functions are shown in the Figures 4, 5, and ⁶, respectively.

In the visualization of the linear criterion (refer to Fig. 4), the slope of the preference function depend on the value of the preference threshold b_p. Thus, in order to define a NRFPR coherently with the linear criterion, the pairwise comparisons must be in accordance with the following conditions:

• if d_p(X_k , X_l) < - b_p, then Rp(X_k , X_l) = 0 and R_p(X_l , X_k) = 1;

• if -b_p< d_p(X_k , X_l) < 0, then R_p(X_k , X_l) = 1 -|d_p(X_k , X_l)|/b_pand R_p(X_l , X_k) = 1;

• if 0 < d_p(X_k , X_l) < - b_p, then R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = 1 -|d_p(X_k , X_l)|/b_p;

• if d_p(X_k , X_l) > b_p, then R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = 0.

The Linear criterion with indifference region can be considered an extension of the linear criterion to deal with judgments of indifference when the pair of alternatives have similar evaluations and their difference lies in an indifference interval [-a_p, a_p]. A NRFPR is coherent with thisdefinition of linear criterion with indifference region if it is in accordance with the following conditions (refer to Fig. 5):

• if d_p(X_k , X_l) <| - b_p, then R_p(X_k , X_l) = 0 and R_p(X_l , X_k) = 1;

• if -b_p< d_p(X_k , X_l) < -a_p, then R_p(X_k , X_l) = 1 -|d_p(X_k , Xl)|/b_pand R_p(Xl , Xk) = 1;

•if -a_p< d_p(X_k , X_l) < a_p, then R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = 1;

• if a_p< d_p(X_k , X_l) < b_p, then R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = 1 -|d_p(X_k , X_l)|/b_p;

• if d_p(X_k , X_l) > b_p, then R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = 0.

Finally, in the case of the Gaussian criterion, σ _p is the distance between the origin and the inflexion point of the curve s_p(X_k , X_l). The Gaussian Criterion allows a transition (without discontinuities) between judgments of indifference and strict preference. In order to define a NRFPR coherently with the Gaussian criterion, the pairwise comparisons must be in accordance with the following conditions:

• if d_p(X_k , X_l) < 0, then R_p(X_k , X_l) = exp(-|d_p(X_k , X_l)|/2σ_p²) and R_p(X_l , X_k) = 1;

• if d_p(X_k , X_l) > 0, then R_p(X_k , X_l) = 1 and R_p(X_l , X_k) = exp(-|d_p(X_k , X_l)|/2σ_p²),

where σ_p is the distance between the origin and the inflexion point of the preference function.

It is not difficult to demonstrate that NRFPRs constructed coherently with those six types of preference functions always satisfy weak transitivity (refer to expression (9)). For this demonstration, it is worth noting that R_p(X_k , X_l) > R_p(X_l , X_k) only if s_p(X_k , X_l) > 0 (which the reader caneasily verify for each type of preference function) and that a nondecreasing preference function satisfies s_p(X_k , X_l) > 0 only if d_p(X_k , X_l) > 0. In this way, if R_p(X_k , X_j) > R_p(X_j , X_k) and R_p(X_j , X_l) > R_p(X_l , X_j), then d_p(X_k , X_j) > 0 (which means that f_p(X_k) > ƒ_p(X_j)) and, at the same time, d_p(X_j , X_l) > 0 (which means that ƒ_p(X_j) > ƒ_p(X_l)). It is also possible to infer that ƒ_p(X_k) > ƒ_p(X_j)> ƒ_p(X_l), which results in d_p(X_k , X_l) > 0, s_p(X_k , X_l) > 0,and R_p(X_k , X_l)> R_p(X_l , X_k).

6 A METHOD FOR MULTICRITERIA DECISION MAKING BASED ON THE ORLOVSKY CHOICE PROCEDURE

The Orlovsky choice procedure makes use of a fuzzy strict preference relation given by (7) to carry out the choice of alternatives (Orlovsky, 1978, 1981). Considering the pth criterion, as P_p(X_l , X_k) describes the set of all alternatives X_kthat are strictly dominated by X_l, its compliment corresponds to the set of alternatives that are not dominated by X_l. Therefore, in order to meet the set of alternatives from X that are not dominated by any other alternative, it suffices to obtain for each alternative X_kbelonging to X the intersection of for all X_lbelonging to X. This intersection is the set of nondominated alternatives with the membership function

which reflects the level of nondominance of each alternative X_k.

A natural choice for a monocriteria problem based on this model should be the alternatives providing:

It is worth emphasizing that the alternatives satisfying

are actually nonfuzzy nondominated and can be considered as the nonfuzzy solution for thechoice problem (Orlovsky, 1978, 1981).

Expressions (7), (14), and (15) may be used to solve choice or ranking problems not only with asingle criterion, but also with multiple criteria. In particular, several procedures that allow one to include multiple criteria in the analysis of a decision making problem are discussed in Orlovsky(1981), Ekel & Schuffner Neto (2006). Let us consider one of them.

Having at hand nonstrict preference relations for each criterion, one possible procedure for solving multicriteria problems consists in obtaining a global relation through the intersection of those relations as follows:

The use of intersection to aggregate all criteria is suitable, when it is a necessary condition thata good alternative X_kmust simultaneously satisfy F₁ and F₂ and ... and F_q. Among the t-norm operators, the use of the min operator, as proposed in (Orlovsky, 1981), allows one to constructthe global fuzzy nonstrict preference relation

under a completely non-compensatory approach for multicriteria decision-making, in the sensethat the high satisfaction of some criteria does not relieve the remaining ones from the requirement of being satisfied (there is no compensation among the criteria). Such approach is alsoconsidered pessimistic, since it gives emphasis to the worst evaluations of each alternative.

Having at hand the global fuzzy nonstrict preference relation, equations (7), (14) and (15) canbe subsequently applied directly to G and the result corresponds to a fuzzy set of nondominatedalternatives fulfilling the role of a Pareto set (Orlovsky, 1981).

However, in case of not being possible to distinguish two or more alternatives, the contractionof (18) is possible by differentiating the importance of R_p, p = 1,..., q with the use of the weighted sum:

where λ_p, p = 1,..., q are importance factors of the corresponding criteria, defined as w_p ∈ [0, 1] and

The construction of T (X_k , X_l), allows one to obtain the membership function NDT (X_k) ofthe fuzzy set of nondominated alternatives by subsequently applying (7) and (14) to (19). The intersection

provides us with a set of alternatives with the highest level of nondominance

7 APPLICATION EXAMPLE

The multicriteria decision-making problem related to the site selection for constructing a new hospital is studied in Vahidnia et al. (2009). Here, we consider a simplified version of this problem, in which six sites are to be ranked, taking into account the following four criteria:

1. Distance from the arterial routes (minimization criterion);

2. Land cost (minimization criterion);

3. Population density (maximization criterion);

4. Average travel time to the nearest existing hospital (maximization criterion).

Hospitals should be located close to main public transport routes. Here, criterion F₁ allows one toevaluate the distance from each site to the closest arterial route. Besides, for social convenience, it is important to guarantee that every person lives in an area with accessible hospital service. Criterion F₄ evaluates the average time one would take to travel between the site and the nearestexisting hospital. The fifth criterion which could be considered is associated with the pollution level present at each site. However, as the sites considered here do not significantly differ with respect to this criterion, the DM decided to eliminate it from the subsequent multicriteria analysis. Figure 7 presents the fuzzy scores utilized for evaluating the third criterion, F₃-population density. Table 2 shows the evaluation matrix of the alternatives.

When the criterion F₁ was considered by the DM, he decided not to infer the degrees of his preferences directly from those distances listed in Table 2. The DM felt more comfortable with utilizing the multiplicative preference relation to articulate his preferences as follows:

Indeed, it is worth noting in (22) that the degrees of preference of the DM can not be directly inferred from the distances from arterial streets. For instance, although the distance of X₄ is ten times larger than the distance of X₅, X₅ is not considered ten times better than X₄. At the same time, the distance of X₂ is higher than the distance of X₆. However, the DM judged X₂ and X₆ as being indifferent to each other. Besides, it is important to indicate that although (22) does not satisfy the multiplicative transitivity, the NRFPR

constructed with the use of H₃ (refer to Table 1) satisfies the weak transitivity and this is a satisfactory level of consistency for the application of the decision-making method to be used.

In the case of criterion F₂, the DM decided to use a preference function corresponding to a Linear criterion with b₂ = 5($/m²). The obtained NRFPR is as follows:

The population density of each alternative, which is related to criterion F₃, was estimated by him,by the linguistic terms shown in Table 2. By applying the expressions (11) and (12) to constructa NRFPR from the comparison of fuzzy estimates, the following NRFPR is obtained:

Finally, in the case of criterion F₄, a preference relation corresponding to a Quasi-criterion, with a₄ = 3 min. The obtained NRFPR is as follows:

Having at hand the NRFPRs (23)-(26), the multicriteria analysis begins by applying (18) to obtain a global relation under a pessimistic approach through the intersection:

Afterward, expression (7) provides the strict preference relation

Finally, with the use of (14), we obtain the nondominance degrees of each alternative

We can order them from best to worst as (X₁ ≈ X₃ ≈ X₆) X₂ X₅ X₄. In order to distinguish alternatives X₁, X₃, and X₆, a subsequent analysis is performed by applying (19) and (20). Considering that the DM defined the weights of the criteria as λ₁ = λ2 = 0.2 and λ3 = λ4 = 0.3, the convolution (19) provides the following matrix of pairwise comparisons:

In this way, by processing (30) with the use of (7) and (14), it is possible to obtain the followingfuzzy set of nondominated alternatives:

Finally, the intersection of (29) and (30), obtained in accordance with (20), makes it possible tocontract the fuzzy set of nondominated alternatives as follows:

The final ranking of the alternatives can be defined on the basis of (32) as

CONCLUSIONS

In this paper we showed how the preference functions can be utilized to construct NRFPRs. The preference functions correspond to a well established preference format which has been utilized to model the preferences of a DM in the methods PROMETHEE I and PROMETHEE II. The results of this paper allow a more flexible input of preferences in the decision-making process based on the analysis of (X, R) models (now, it is possible to offer to a DM the following formats: the ordering of the alternatives, the utility values, the multiplicative preference relations, the fuzzy estimates, the reciprocal as well as the nonreciprocal fuzzy preference relations and thepreference functions). The applicability of the proposed procedure is demonstrated through the solution of a multicriteria decision-making problem related to the site selection for constructing a new hospital.

ACKNOWLEDGMENTS

This research is supported by the National Council for Scientific and Technological Developmentof Brazil (CNPq) - grants PQ: 307406/2008-3 and PQ: 307474/2008-9.

Received October 22, 2010

Accepted December 12, 2011

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Publication Dates

• Publication in this collection
  09 Aug 2013
• Date of issue
  Aug 2013

History

• Received
  22 Oct 2010
• Accepted
  12 Dec 2011