Improving Consistency in Fuzzy Preference ... - Semantic Scholar
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New Trends in Software Methodologies, Tools and Techniques H. Fujita et al. (Eds.) IOS Press, 2014 © 2014 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-61499-434-3-194
Improving Consistency in Fuzzy Preference Relations with an Allocation of Information Granularity Francisco Javier CABRERIZO a,1 , Witold PEDRYCZ b,c,d , Francisco CHICLANA e , and Enrique HERRERA-VIEDMA c,f a Dept. of Software Engineering and Computer Systems, Universidad Nacional de Educaci´on a Distancia, Spain b Dept. of Electrical & Computer Engineering, University of Alberta, Canada c Dept. of Electrical & Computer Engineering, King Abdulaziz University, Saudi Arabia d Systems Research Institute, Polish Academy of Sciences, Poland e School of Computer Science and Informatics, De Montfort University, UK f Dept. of Computer Science and Artificial Intelligence, University of Granada, Spain Abstract. An important issue to bear in mind in Group Decision Making situations is that of consistency. However, the expression of consistent preferences is often a very difficult task for the decision makers, specially in decision problems with a high number of alternatives and when decision makers use fuzzy preference relations to provide their opinions. It leads to situations where a decision maker may not be able to express all his/her preferences properly and without contradiction. To overcome this problem, we propose the concept of the information granularity being regarded as an important and useful asset supporting the goal to reach consistent fuzzy preference relations. To do so, we develop a concept of granular fuzzy preference relation where each pairwise comparison is formed as a certain information granule instead of a single numeric value. As being more abstract, the granular format of the preference model offers the required flexibility to increase the level of consistency. Keywords. Consistency, granularity of information, particle swarm optimization, fuzzy preference relation
1. Introduction A Group Decision Making (GDM) problem is usually understood as a decision problem which consists in finding the best alternative(s) from a set of feasible alternatives, X = {x1 , . . . , xn }, according to the preferences provided by a group of decision makers, E = {e1 , . . . , em }, characterized by their experience and knowledge. To do this, decision makers have to convey their preferences or opinions by means of a set of evaluations over a set of possible alternatives. 1 Corresponding Author: Francisco Javier Cabrerizo, Department of Software Engineering and Computer Systems, Universidad Nacional de Educaci´on a Distancia, C/ Juan del Rosal 16, Madrid 28040, Spain; E-mail: [email protected].
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There exist several different representation formats in which decision makers can express their preferences. However, fuzzy preference relations [1,2] have been widely used because they have proved to be a very expressive representation format and also because they present good properties that allow to operate with them easily [1,2]. The effort to complete pairwise evaluations is far more manageable in comparison to any experimental overhead we need when assigning membership grades to all alternatives of the universe in a single step, which implies that the decision maker must be able to judge each alternative against all the others as a whole, which can be a difficult task. The pairwise comparison helps the decision maker focus only on two elements once at a time thus reducing uncertainty and hesitation while leading to the higher of consistency. It is obvious that consistent information, which does not imply any kind of contradiction, is more relevant or important than the information containing some contradictions. However, due to the complexity of most GDM problems, decision makers’ preferences can be inconsistent. Fortunately, the lack of consistency can be quantified and monitored [2,3,4]. Consistency requires that each decision maker has to allow a certain degree of flexibility and be ready to make an adjustment of his/her firts choices and, here, information granularity [5,6,7] may come into play. Information granularity is an important design asset and may offer to each decision maker a real level of flexibility using some initial opinions expressed by each decision maker that can be modified with the intent to reach a higher level of consistency. Assuming that each decision maker expresses his/her preferences using a fuzzy preference relation, this required flexibility is brought into the fuzzy preference relations by allowing them to be granular rather than numeric. That is, we consider the entries of the fuzzy preference relations are not plain numbers but information granules, say fuzzy sets, rough sets, intervals, probability density functions, etc. In summary, information granularity that is present here serves as an important modeling asset, offering an ability of the decision maker to exercise some flexibility to be used in adjusting his/her initial position. To do so, the fuzzy preference relation is abstracted to its granular format. In this contribution, we propose an allocation of information granularity as a key component to facilitate the improvement of consistency. To do so, in the realization of the granular representation of the fuzzy preference relations, we introduce a certain level of granularity supplying the required flexibility to increase the level of consistency in the preferences given by the decision maker. This proposed concept of granular fuzzy preference relation is used to optimize a performance index, which expresses the level of consistency of the decision maker. Given the nature of the required optimization, the ensuing optimization problem is solved by engaging a machinery of population-based optimization, namely Particle Swarm Optimization (PSO) [8]. This contribution is set out as follows. In Section 2, we present the method to obtain the consistency achieved by a decision maker when expressing his/her preferences using a fuzzy preference relation. Section 3 is concerned with the improvement of consistency through an allocation of information granularity. To illustrate it, an example is reported in Section 4. Finally, we point out some conclusions in Section 5.
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2. Preliminaries In a classical GDM situation, there is a problem to solve, a solution set of possible alternatives, X = {x1 , x2 , . . . , xn }, (n ≥ 2), and a group of two or more decision makers, E = {e1 , e2 , . . . , em }, (m ≥ 2), characterized by their own motivations, attitudes, ideas and knowledge, who express their opinions about this set of alternatives to achieve a common solution [9,10,11]. The objective is to classify the alternatives from best to worst, associating with them some degrees of preference. Among the different representation formats that decision makers may use to express their opinions, fuzzy preference relations [10,12,13] are one of the most used because of their effectiveness as a tool for modelling decision processes and their utility and easiness of use when we want to aggregate decision makers’ preferences into group ones [10,14]. Definition 1. A fuzzy preference relation PR on a set of alternatives X is a fuzzy set on the Cartesian product X × X, i.e., it is characterized by a membership function μPR : X × X → [0, 1]. A fuzzy preference relation PR may be represented by the n × n matrix PR = (pri j ), being pri j = μPR (xi , x j ) (∀i, j ∈ {1, . . . , n}) interpreted as the preference degree or intensity of the alternative xi over x j : pri j = 0.5 indicates indifference between xi and x j (xi ∼ x j ), pri j = 1 indicates that xi is absolutely preferred to x j , and pri j > 0.5 indicates that xi is preferred to x j (xi x j ). Based on this interpretation we have that prii = 0.5 ∀i ∈ {1, . . . , n} (xi ∼ xi ). Since prii ’s (as well as the corresponding elements on the main diagonal in some other matrices) do not matter, we will write them as ‘–’ instead of 0.5 [10,15]. When it is assumed that pri j + pr ji = 1 (∀i, j ∈ {1, . . . , n}) the preference relation is called reciprocal preference relation and it is more easily interpreted as a stochastic relation [16,17,18]. However, as it is always not the case [15,19], this assumption is not made in this study. Due to the complexity of most decision making problems, decision makers’ preferences may not satisfy formal properties that fuzzy preference relations are required to verify. Consistency is one of them, and it is associated with the transitivity property. Coherence in preference modeling has been introduced in standard decision making frameworks, taking many different formulations in each context, as a need in order to assure consistent decision making procedures [20]. In the classical numeric (Boolean) context, preferences use to be assumed to be transitive in order to assure consistent behavior. In the fuzzy framework, transitivity plays a crucial role in coherence modeling, since crisp behavior should appear as a particular case. Hence, crisp transitivity has been generalized into fuzzy preference modeling, existing a great variety of fuzzy transitivity properties, each one offering a different consistency assumption [2]. Alternatively, consistency has been understood by Cutello and Montero [3] as a rationality measure, therefore allowing degrees of performance. A key argument was that most standard fuzzy transitivity conditions in literature were crisp in nature, i.e., they either hold or not hold. But it is apparent that some situations are extremely intransitive while sometimes we only find small or unexpected transitivity violations that can be in some way bypassed in practice. Consistency in most cases allows different degrees, and it should be measured. The axiomatic approach of Cutello and Montero [3] was a first proposal in this direction, proposing a particular family of conditions any rationality measure should verify within preference modeling. Definition 1 dealing with a preference relation does not imply any kind of consistency property. In fact, preference values of a fuzzy preference relation can be contradic-
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tory. However, the study of consistency is crucial for avoiding misleading solutions in GDM [15]. To make a rational choice, properties to be satisfied by such fuzzy preference relations have been suggested [2]. In this contribution, we make use of the additive transitivity property which facilitates the verification of consistency in the case of fuzzy preference relations. As it is shown in [2], additive transitivity for fuzzy preference relations can be seen as the parallel concept of Saaty’s consistency property for multiplicative preference relations [21]. The mathematical formulation of the additive transitivity was given by [14]: (pri j − 0.5) + (pr jk − 0.5) = (prik − 0.5), ∀i, j, k ∈ {1, . . . , n}
(1)
Additive transitivity implies additive reciprocity. Indeed, because prii = 0.5 ∀i, if we make k = i in Eq. (1) then we have: pri j + pr ji = 1, ∀i, j ∈ {1, . . . , n}. Eq. (1) can be rewritten as follows: prik = pri j + pr jk − 0.5, ∀i, j, k ∈ {1, . . . , n}
(2)
A fuzzy preference relation is considered to be “additively consistent” when for every three options encountered in the problem, say xi , x j , xk ∈ X , their associated preference degrees, pri j , pr jk , prik , fulfil Eq. (2). Given a fuzzy preference relation, Eq. (2) can be used to calculate an estimated value of a preference degree using other preference degrees. Indeed, using an intermediate alternative x j , the following estimated value of prik (i = k) can be obtained in three different ways [15]: • From prik = pri j + pr jk − 0.5 we obtain the estimate (epik ) j1 = pri j + pr jk − 0.5
(3)
• From pr jk = pr ji + prik − 0.5 we obtain the estimate (epik ) j2 = pr jk − pr ji + 0.5
(4)
• From pri j = prik + prk j − 0.5 we obtain the estimate (epik ) j3 = pri j − prk j + 0.5
(5)
Then, we can estimate the value of a preference pik according to the following expression: n
∑
epik =
(epik ) j1 + (epik ) j2 + (epik ) j3
j=1 j=i,k
3(n − 2)
(6)
When information provided is completely consistent then (epik ) jl = prik ∀ j, l. However, because decision makers are not always fully consistent, the assessment made by
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an decision maker may not verify Eq. (2) and some of the estimated preference degree values (epik ) jl may not belong to the unit interval [0, 1]. We note, from (3)–(5), that the maximum value of any of the preference degrees (epik ) jl (l ∈ {1, 2, 3}) is 1.5 while the minimum one is −0.5. Taking this into account, the error between a preference value and its estimated one in [0, 1] is computed as follows [15]:
ε pik =
2 · |epik − prik | 3
(7)
Thus, it can be used to define the consistency degree cdik associated to the preference degree prik as follows: cdik = 1 − ε pik
(8)
When cdik = 1, then ε pik = 0 and there is no inconsistency at all. The lower the value of cdik , the higher the value of ε pik and the more inconsistent is prik with respect to the rest of information. In the following, we define the consistency degrees associated with individual alternatives and the overall fuzzy preference relation: • The consistency degree, cdi ∈ [0, 1], associated to a particular alternative xi of a fuzzy preference relation is defined as: cdi =
∑nk=1;i=k (cdik + cdki ) 2(n − 1)
(9)
• The consistency degree, cd ∈ [0, 1], of fuzzy preference relation is defined as follows: cd =
∑ni=1 cdi n
(10)
When cd = 1, the fuzzy preference relation is fully consistent. Otherwise, the lower cd the more inconsistent the fuzzy preference relation is.
3. Improving Consistency Through an Allocation of Information Granularity The improvement of consistency when the decision makers express their opinions by means of fuzzy preference relations becomes a very important aspect in order to avoid misleading solutions. It is needless to say that it calls for some flexibility exhibited by the decision maker. The changes of opinions, when the decision makers express their opinions using fuzzy preference relations, are articulated through alterations of the entries of the fuzzy preference relations. If the pairwise comparisons of the fuzzy preference relations are not managed as single numeric values, which are inflexible, but rather as information granules, it will bring the essential factor of flexibility. This means that the fuzzy preference relation is elevated to its granular format. The notation G(PR) is used to accentuate the fact that we are interested in granular fuzzy
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preference relations. In such a case, G(.) represents a specific granular formalism being used here (for instance, intervals, fuzzy sets, rough sets, probability density functions, and alike). We present the concept of granular fuzzy preference relation and emphasize a role of information granularity being regarded here as an important conceptual and computational resource which can be exploited as a means to increase the level of consistency achieved in the preferences given by the decision maker. The level of granularity is treated as synonymous of the level of flexibility, which makes easy the improvement of the consistency. The higher level of granularity is allowed to the decision maker, the higher the feasibility of arriving at higer level of consistency. In this contribution, the granularity of information is articulated through intervals and, therefore, the length of such intervals (entries of the fuzzy preference relations) can be sought as a level of granularity α . As we are using interval-valued fuzzy preference relations, G(PR) = P(PR), where P(.) denotes a family of intervals. The flexibility given by the level of granularity can be effectively used to optimize a certain optimization criterion to capture the essence of the consistency. In the following subsection, the optimization criterion which has to be optimized is described and its optimization using the PSO framework is presented. 3.1. The Optimization Criterion We suppose each decision maker feels equally comfortable when choosing any fuzzy preference relation whose values are placed within the bounds established by the fixed level of granularity α . This level of granularity is used to improve the level of consistency within the fuzzy preference relation. This improvement is effectuate at the level of individual decision maker using the following performance criterion: Q=
1 m ∑ cdl m l=1
(11)
Therefore, the optimization problem reads as follows: MaxPR1 ,PR2 ,...,PRm ∈P(PR) Q
(12)
The aforementioned maximization problem is carried out for all interval-valued fuzzy preference relations admissible because of the introduced level of information granularity α . This fact is underlined by including a granular form of the fuzzy preference relations allowed in the problem, i.e., PR1 , PR2 , . . ., PRm , are elements of the family of interval-valued fuzzy preference relations, namely, P(PR). This optimization task is not an easy one. In light of the form of the optimization criterion, we can consider alternatives such as genetic algorithms, evolutionary optimization, PSO, simulated annealing, ant colonies, and the like, to optimize it. Here, the optimization of the fuzzy preference relations, coming from the space of interval-valued fuzzy preference relations, is realized by means of the PSO, which is especially attractive given its less significant computing overhead in comparison with other techniques of global optimization [22] (say, genetic algorithms). However, one could think of the usage of some other optimization mechanisms as well. In what follows, we recall the basis of the method and associate the generic representation scheme of the PSO with the format of the problem at hand.
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3.2. PSO Environment in Optimization of Fuzzy Preference Relations PSO algorithm is a population based stochastic optimization technique inspired by bird flocking and fish schooling originally designed and introduced by Kennedy and Eberhart [8]. It is based on communication and interaction between the members of the swarm, what means that each member of the PSO algorithm, named as particle, determines its position by combining the history of its own best location with those of others members of the swarm. The algorithm flow of the PSO starts with a population of particles whose positions are the potential solutions of the problem, and the velocities are randomly initialized in the problem search space. In each iteration/generation, the search for optimal position (solution) is performed by updating the particles velocities and positions based on a predefined fitness function. The velocity of each particle is updated using two best positions, namely personal best position and neighborhood best position. The personal best position is the best position the particle has visited and neighborhood best position is the best position the particle and its neighbors have visited [23,24,25]. The optimization of the fuzzy preference relations coming from the space of interval-valued fuzzy preference relations is realized by means of the PSO. In the following, we elaborate on the fitness function, its realization, and the PSO optimization along with the corresponding formation of the components of the swarm. 3.2.1. Particle An important point in a PSO algorithm is finding an appropriate mapping between problem solution and the particle’s representation. In our framework, each particle represents a vector whose entries are located in the interval [0, 1]. Basically, if there is a group of m decision makers and a set of n alternatives, the number of entries of the particle is m · n(n − 1). Starting with the initial fuzzy preference relation provided by the decision maker and assuming a given level of granularity α (located in the unit interval), let us consider an entry pri j . The interval of admissible values of this entry of P(PR) implied by the level of granularity is equal to: [a, b] = [max(0, pri j − α /2), min(1, pri j + α /2)]
(13)
Let assume that the entry of interest of the particle is x. It is transformed linearly according to the expression z = a + (b − a)x. For example, consider that pri j is equal to 0.8, the admissible level of granularity α = 0.1, and the corresponding entry of the particle is x = 0.6. Then, the corresponding interval of the granular fuzzy preference relation computed as given by Eq. (13) becomes equal to [a, b] = [0.75, 0.85]. Subsequently, z = 0.81, and, therefore, the modified value of pri j becomes equal to 0.81. The overall particle is composed of the individual segments, where each of them is concerned with the optimization of the parameters of the fuzzy preference relations. 3.2.2. Fitness Function In the PSO, the performance of each particle during its movement is assessed by means of some performance index (fitness function). Here, the aim of the PSO is the maximization
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of the individual consistency achieved by each decision maker. Therefore, the fitness function, f , associated with the particle is defined as: f =Q
(14)
being Q the optimization criterion presented in Section 3.1. The higher the value of f , the better the particle is. 3.2.3. Algorithm We use the generic form of the PSO algorithm. In such a way, the updates of the velocity of a particle are realized in the form v(t + 1) = w × v(t) + c1 a · (z p − z) + c2 b · (zg − z) where “t” is an index of the generation and · denotes a vector multiplication realized coordinatewise. z p denotes the best position reported so far for the particle under discussion while zg is the best position overall and developed so far across the entire population. The current velocity v(t) is scaled by the inertia weight (w) which emphasizes some effect of resistance to change the current velocity. The value of the inertia weight is kept constant through the entire optimization process and equal to 0.2 (this value is commonly encountered in the existing literature [22]). By using the inertia component, we form the memory effect of the particle. The two other parameters of the PSO, that is a and b, are vectors of random numbers drawn from the uniform distribution over the [0, 1] interval. These two update components help form a proper mix of the components of the velocity. The second expression governing the change in the velocity of the particle is particularly interesting as it nicely captures the relationships between the particle and its history as well as the history of overall population in terms of their performance reported so far. The next position (in iteration step “t+1”) of the particle is computed in a straightforward manner: z(t + 1) = z(t) + v(t + 1). When it comes to the representation of the solutions, the particle z is composed of “m · n(n − 1)” entries positioned in the [0,1] interval that corresponds to the search space. In addition, one should note that while PSO optimizes the fitness function, there is no guarantee that the result is optimal, rather than that we can refer to the solution as the best one being formed by the PSO.
4. Example of Application In this section, we show an example which helps quantifying the performance of the proposed method. Proceeding with the details of the optimization environment, PSO was used with the following values of the parameters which were selected as a result of intensive experimentation: • The size of the swarm consisted of 100 particles. This size of the population was found to produce “stable” results meaning that very similar or identical results were reported in successive runs of the PSO. Due to the search space, this particular size of the population was suitable to realize a search process. • The number of generations (or iterations) was set to 100. It was observed that after 100 generations there were no further changes of the values of the fitness function.
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• The parameters in the update equation for the velocity of the particle were set as c1 = c2 = 2. These values are commonly encountered in the existing literature. Let us suppose the following four fuzzy preference relations coming from four decision makers E = {e1 , e2 , e3 , e4 }: ⎞ − 0.4 0.3 0.3 ⎜ 0.4 − 0.7 0.7 ⎟ ⎟ PR1 = ⎜ ⎝ 0.4 0.1 − 0.2 ⎠ 0.2 0.3 0.7 −
⎞ − 0.2 0.6 0.3 ⎜ 0.6 − 0.5 0.3 ⎟ ⎟ PR2 = ⎜ ⎝ 0.2 0.3 − 0.5 ⎠ 0.1 0.3 0.9 −
⎞ − 0.2 0.5 0.1 ⎜ 0.4 − 0.2 0.8 ⎟ ⎟ PR3 = ⎜ ⎝ 0.5 0.4 − 0.9 ⎠ 0.9 0.1 0.4 −
⎞ − 0.6 0.2 0.6 ⎜ 0.4 − 0.6 0.2 ⎟ ⎟ PR4 = ⎜ ⎝ 0.8 0.6 − 0.5 ⎠ 0.4 0.6 0.6 −
⎛
⎛
⎛
⎛
1.00
Considering a given level of granularity α , Figure 1 illustrates the performance of the PSO quantified in terms of the fitness function (optimization criterion) obtained in successive generations. The most notable improvement is noted as the very beginning of the optimization, and afterwards, there is a clearly visible stabilization, where the values of the fitness function remain constant. To put the obtained optimization results in a certain context, we report the performance obtained when considering the entries of the fuzzy preference relations are sin-
f
0.80
0.85
0.90
0.95
α = 0.2 α = 0.4 α = 0.6 α = 0.8 α = 1.0 α = 2.0
0
20
40
60
80
100
generation
Figure 1. Fitness function f in successive PSO generations for selected values of α .
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gle numeric values, that is, when no granularity is allowed (α = 0). In such a case, the corresponding consistency degrees of the four fuzzy preference relations are cd1 = 0.85, cd2 = 0.85, cd3 = 0.76, and cd4 = 0.83. Therefore, the value of the fitness function f is is 0.82. Comparing with the values obtained by the PSO, the fitness function f takes on now lower values. As we can see in Figure 1, the higher the admitted level of granularity α , the higher the values obtained by the fitness function f . It is due to the fact that the higher the level of granularity α , the higher the level of flexibility introduced in the fuzzy preference relations and, therefore, the possibility of achieving higher level of consistency. In particular, when each entry of the granular preference relation is treated as the whole [0, 1] interval (it occurs when α = 2.0), the value of the fitness function is the maximum one, which is 1. However, when the level of granularity is very high, the values of the entries of the fuzzy preference relation could be very different in comparison with the original values provided by the decision maker and, therefore, he/she could reject them. Once the consistency of the fuzzy preference relations has been increased, a selection process could be applied in order to find the solution set of alternatives according to the preferences provided by the decision makers. This selection process is composed of two steps [26]: aggregation of preferences provided by the decision makers and exploitation of the aggregated preference obtained previously.
5. Conclusions In this contribution, we have proposed a method based on an allocation of information granularity as an important asset to increase the consistency. Here, information granularity is an important and useful asset that support to increase consistency in the fuzzy preference relations provide by the decision makers. It offers a badly needed flexibility so that the granular fuzzy preference relations can produce numeric realizations so that they are of higher consistency. To do so, the PSO environment has been shown to serve a suitable optimization framework. The granular representation of fuzzy preference relations discussed in this contribution was the one using intervals. However, any other formalism of granular computing, especially fuzzy sets, could be equally applicable here. Furthermore, in the scenario analyzed in this contribution, a uniform allocation of granularity has been discussed, where the same level of granularity α has been allocated across all the fuzzy preference relations. However, a nonuniform distribution of granularity could be considered, where these levels are also optimized so that each decision maker might have an individual value of α becoming available to his/her disposal.
Acknowledgments The authors would like to acknowledge FEDER financial support from the Project FUZZYLING-II Project TIN2010-17876, and also the financial support from the Andalusian Excellence Projects TIC-05299 and TIC-5991.
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New Trends in Software Methodologies, Tools and Techniques H. Fujita et al. (Eds.) IOS Press, 2014 © 2014 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-61499-434-3-194
Improving Consistency in Fuzzy Preference Relations with an Allocation of Information Granularity Francisco Javier CABRERIZO a,1 , Witold PEDRYCZ b,c,d , Francisco CHICLANA e , and Enrique HERRERA-VIEDMA c,f a Dept. of Software Engineering and Computer Systems, Universidad Nacional de Educaci´on a Distancia, Spain b Dept. of Electrical & Computer Engineering, University of Alberta, Canada c Dept. of Electrical & Computer Engineering, King Abdulaziz University, Saudi Arabia d Systems Research Institute, Polish Academy of Sciences, Poland e School of Computer Science and Informatics, De Montfort University, UK f Dept. of Computer Science and Artificial Intelligence, University of Granada, Spain Abstract. An important issue to bear in mind in Group Decision Making situations is that of consistency. However, the expression of consistent preferences is often a very difficult task for the decision makers, specially in decision problems with a high number of alternatives and when decision makers use fuzzy preference relations to provide their opinions. It leads to situations where a decision maker may not be able to express all his/her preferences properly and without contradiction. To overcome this problem, we propose the concept of the information granularity being regarded as an important and useful asset supporting the goal to reach consistent fuzzy preference relations. To do so, we develop a concept of granular fuzzy preference relation where each pairwise comparison is formed as a certain information granule instead of a single numeric value. As being more abstract, the granular format of the preference model offers the required flexibility to increase the level of consistency. Keywords. Consistency, granularity of information, particle swarm optimization, fuzzy preference relation
1. Introduction A Group Decision Making (GDM) problem is usually understood as a decision problem which consists in finding the best alternative(s) from a set of feasible alternatives, X = {x1 , . . . , xn }, according to the preferences provided by a group of decision makers, E = {e1 , . . . , em }, characterized by their experience and knowledge. To do this, decision makers have to convey their preferences or opinions by means of a set of evaluations over a set of possible alternatives. 1 Corresponding Author: Francisco Javier Cabrerizo, Department of Software Engineering and Computer Systems, Universidad Nacional de Educaci´on a Distancia, C/ Juan del Rosal 16, Madrid 28040, Spain; E-mail: [email protected].
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There exist several different representation formats in which decision makers can express their preferences. However, fuzzy preference relations [1,2] have been widely used because they have proved to be a very expressive representation format and also because they present good properties that allow to operate with them easily [1,2]. The effort to complete pairwise evaluations is far more manageable in comparison to any experimental overhead we need when assigning membership grades to all alternatives of the universe in a single step, which implies that the decision maker must be able to judge each alternative against all the others as a whole, which can be a difficult task. The pairwise comparison helps the decision maker focus only on two elements once at a time thus reducing uncertainty and hesitation while leading to the higher of consistency. It is obvious that consistent information, which does not imply any kind of contradiction, is more relevant or important than the information containing some contradictions. However, due to the complexity of most GDM problems, decision makers’ preferences can be inconsistent. Fortunately, the lack of consistency can be quantified and monitored [2,3,4]. Consistency requires that each decision maker has to allow a certain degree of flexibility and be ready to make an adjustment of his/her firts choices and, here, information granularity [5,6,7] may come into play. Information granularity is an important design asset and may offer to each decision maker a real level of flexibility using some initial opinions expressed by each decision maker that can be modified with the intent to reach a higher level of consistency. Assuming that each decision maker expresses his/her preferences using a fuzzy preference relation, this required flexibility is brought into the fuzzy preference relations by allowing them to be granular rather than numeric. That is, we consider the entries of the fuzzy preference relations are not plain numbers but information granules, say fuzzy sets, rough sets, intervals, probability density functions, etc. In summary, information granularity that is present here serves as an important modeling asset, offering an ability of the decision maker to exercise some flexibility to be used in adjusting his/her initial position. To do so, the fuzzy preference relation is abstracted to its granular format. In this contribution, we propose an allocation of information granularity as a key component to facilitate the improvement of consistency. To do so, in the realization of the granular representation of the fuzzy preference relations, we introduce a certain level of granularity supplying the required flexibility to increase the level of consistency in the preferences given by the decision maker. This proposed concept of granular fuzzy preference relation is used to optimize a performance index, which expresses the level of consistency of the decision maker. Given the nature of the required optimization, the ensuing optimization problem is solved by engaging a machinery of population-based optimization, namely Particle Swarm Optimization (PSO) [8]. This contribution is set out as follows. In Section 2, we present the method to obtain the consistency achieved by a decision maker when expressing his/her preferences using a fuzzy preference relation. Section 3 is concerned with the improvement of consistency through an allocation of information granularity. To illustrate it, an example is reported in Section 4. Finally, we point out some conclusions in Section 5.
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2. Preliminaries In a classical GDM situation, there is a problem to solve, a solution set of possible alternatives, X = {x1 , x2 , . . . , xn }, (n ≥ 2), and a group of two or more decision makers, E = {e1 , e2 , . . . , em }, (m ≥ 2), characterized by their own motivations, attitudes, ideas and knowledge, who express their opinions about this set of alternatives to achieve a common solution [9,10,11]. The objective is to classify the alternatives from best to worst, associating with them some degrees of preference. Among the different representation formats that decision makers may use to express their opinions, fuzzy preference relations [10,12,13] are one of the most used because of their effectiveness as a tool for modelling decision processes and their utility and easiness of use when we want to aggregate decision makers’ preferences into group ones [10,14]. Definition 1. A fuzzy preference relation PR on a set of alternatives X is a fuzzy set on the Cartesian product X × X, i.e., it is characterized by a membership function μPR : X × X → [0, 1]. A fuzzy preference relation PR may be represented by the n × n matrix PR = (pri j ), being pri j = μPR (xi , x j ) (∀i, j ∈ {1, . . . , n}) interpreted as the preference degree or intensity of the alternative xi over x j : pri j = 0.5 indicates indifference between xi and x j (xi ∼ x j ), pri j = 1 indicates that xi is absolutely preferred to x j , and pri j > 0.5 indicates that xi is preferred to x j (xi x j ). Based on this interpretation we have that prii = 0.5 ∀i ∈ {1, . . . , n} (xi ∼ xi ). Since prii ’s (as well as the corresponding elements on the main diagonal in some other matrices) do not matter, we will write them as ‘–’ instead of 0.5 [10,15]. When it is assumed that pri j + pr ji = 1 (∀i, j ∈ {1, . . . , n}) the preference relation is called reciprocal preference relation and it is more easily interpreted as a stochastic relation [16,17,18]. However, as it is always not the case [15,19], this assumption is not made in this study. Due to the complexity of most decision making problems, decision makers’ preferences may not satisfy formal properties that fuzzy preference relations are required to verify. Consistency is one of them, and it is associated with the transitivity property. Coherence in preference modeling has been introduced in standard decision making frameworks, taking many different formulations in each context, as a need in order to assure consistent decision making procedures [20]. In the classical numeric (Boolean) context, preferences use to be assumed to be transitive in order to assure consistent behavior. In the fuzzy framework, transitivity plays a crucial role in coherence modeling, since crisp behavior should appear as a particular case. Hence, crisp transitivity has been generalized into fuzzy preference modeling, existing a great variety of fuzzy transitivity properties, each one offering a different consistency assumption [2]. Alternatively, consistency has been understood by Cutello and Montero [3] as a rationality measure, therefore allowing degrees of performance. A key argument was that most standard fuzzy transitivity conditions in literature were crisp in nature, i.e., they either hold or not hold. But it is apparent that some situations are extremely intransitive while sometimes we only find small or unexpected transitivity violations that can be in some way bypassed in practice. Consistency in most cases allows different degrees, and it should be measured. The axiomatic approach of Cutello and Montero [3] was a first proposal in this direction, proposing a particular family of conditions any rationality measure should verify within preference modeling. Definition 1 dealing with a preference relation does not imply any kind of consistency property. In fact, preference values of a fuzzy preference relation can be contradic-
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tory. However, the study of consistency is crucial for avoiding misleading solutions in GDM [15]. To make a rational choice, properties to be satisfied by such fuzzy preference relations have been suggested [2]. In this contribution, we make use of the additive transitivity property which facilitates the verification of consistency in the case of fuzzy preference relations. As it is shown in [2], additive transitivity for fuzzy preference relations can be seen as the parallel concept of Saaty’s consistency property for multiplicative preference relations [21]. The mathematical formulation of the additive transitivity was given by [14]: (pri j − 0.5) + (pr jk − 0.5) = (prik − 0.5), ∀i, j, k ∈ {1, . . . , n}
(1)
Additive transitivity implies additive reciprocity. Indeed, because prii = 0.5 ∀i, if we make k = i in Eq. (1) then we have: pri j + pr ji = 1, ∀i, j ∈ {1, . . . , n}. Eq. (1) can be rewritten as follows: prik = pri j + pr jk − 0.5, ∀i, j, k ∈ {1, . . . , n}
(2)
A fuzzy preference relation is considered to be “additively consistent” when for every three options encountered in the problem, say xi , x j , xk ∈ X , their associated preference degrees, pri j , pr jk , prik , fulfil Eq. (2). Given a fuzzy preference relation, Eq. (2) can be used to calculate an estimated value of a preference degree using other preference degrees. Indeed, using an intermediate alternative x j , the following estimated value of prik (i = k) can be obtained in three different ways [15]: • From prik = pri j + pr jk − 0.5 we obtain the estimate (epik ) j1 = pri j + pr jk − 0.5
(3)
• From pr jk = pr ji + prik − 0.5 we obtain the estimate (epik ) j2 = pr jk − pr ji + 0.5
(4)
• From pri j = prik + prk j − 0.5 we obtain the estimate (epik ) j3 = pri j − prk j + 0.5
(5)
Then, we can estimate the value of a preference pik according to the following expression: n
∑
epik =
(epik ) j1 + (epik ) j2 + (epik ) j3
j=1 j=i,k
3(n − 2)
(6)
When information provided is completely consistent then (epik ) jl = prik ∀ j, l. However, because decision makers are not always fully consistent, the assessment made by
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an decision maker may not verify Eq. (2) and some of the estimated preference degree values (epik ) jl may not belong to the unit interval [0, 1]. We note, from (3)–(5), that the maximum value of any of the preference degrees (epik ) jl (l ∈ {1, 2, 3}) is 1.5 while the minimum one is −0.5. Taking this into account, the error between a preference value and its estimated one in [0, 1] is computed as follows [15]:
ε pik =
2 · |epik − prik | 3
(7)
Thus, it can be used to define the consistency degree cdik associated to the preference degree prik as follows: cdik = 1 − ε pik
(8)
When cdik = 1, then ε pik = 0 and there is no inconsistency at all. The lower the value of cdik , the higher the value of ε pik and the more inconsistent is prik with respect to the rest of information. In the following, we define the consistency degrees associated with individual alternatives and the overall fuzzy preference relation: • The consistency degree, cdi ∈ [0, 1], associated to a particular alternative xi of a fuzzy preference relation is defined as: cdi =
∑nk=1;i=k (cdik + cdki ) 2(n − 1)
(9)
• The consistency degree, cd ∈ [0, 1], of fuzzy preference relation is defined as follows: cd =
∑ni=1 cdi n
(10)
When cd = 1, the fuzzy preference relation is fully consistent. Otherwise, the lower cd the more inconsistent the fuzzy preference relation is.
3. Improving Consistency Through an Allocation of Information Granularity The improvement of consistency when the decision makers express their opinions by means of fuzzy preference relations becomes a very important aspect in order to avoid misleading solutions. It is needless to say that it calls for some flexibility exhibited by the decision maker. The changes of opinions, when the decision makers express their opinions using fuzzy preference relations, are articulated through alterations of the entries of the fuzzy preference relations. If the pairwise comparisons of the fuzzy preference relations are not managed as single numeric values, which are inflexible, but rather as information granules, it will bring the essential factor of flexibility. This means that the fuzzy preference relation is elevated to its granular format. The notation G(PR) is used to accentuate the fact that we are interested in granular fuzzy
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preference relations. In such a case, G(.) represents a specific granular formalism being used here (for instance, intervals, fuzzy sets, rough sets, probability density functions, and alike). We present the concept of granular fuzzy preference relation and emphasize a role of information granularity being regarded here as an important conceptual and computational resource which can be exploited as a means to increase the level of consistency achieved in the preferences given by the decision maker. The level of granularity is treated as synonymous of the level of flexibility, which makes easy the improvement of the consistency. The higher level of granularity is allowed to the decision maker, the higher the feasibility of arriving at higer level of consistency. In this contribution, the granularity of information is articulated through intervals and, therefore, the length of such intervals (entries of the fuzzy preference relations) can be sought as a level of granularity α . As we are using interval-valued fuzzy preference relations, G(PR) = P(PR), where P(.) denotes a family of intervals. The flexibility given by the level of granularity can be effectively used to optimize a certain optimization criterion to capture the essence of the consistency. In the following subsection, the optimization criterion which has to be optimized is described and its optimization using the PSO framework is presented. 3.1. The Optimization Criterion We suppose each decision maker feels equally comfortable when choosing any fuzzy preference relation whose values are placed within the bounds established by the fixed level of granularity α . This level of granularity is used to improve the level of consistency within the fuzzy preference relation. This improvement is effectuate at the level of individual decision maker using the following performance criterion: Q=
1 m ∑ cdl m l=1
(11)
Therefore, the optimization problem reads as follows: MaxPR1 ,PR2 ,...,PRm ∈P(PR) Q
(12)
The aforementioned maximization problem is carried out for all interval-valued fuzzy preference relations admissible because of the introduced level of information granularity α . This fact is underlined by including a granular form of the fuzzy preference relations allowed in the problem, i.e., PR1 , PR2 , . . ., PRm , are elements of the family of interval-valued fuzzy preference relations, namely, P(PR). This optimization task is not an easy one. In light of the form of the optimization criterion, we can consider alternatives such as genetic algorithms, evolutionary optimization, PSO, simulated annealing, ant colonies, and the like, to optimize it. Here, the optimization of the fuzzy preference relations, coming from the space of interval-valued fuzzy preference relations, is realized by means of the PSO, which is especially attractive given its less significant computing overhead in comparison with other techniques of global optimization [22] (say, genetic algorithms). However, one could think of the usage of some other optimization mechanisms as well. In what follows, we recall the basis of the method and associate the generic representation scheme of the PSO with the format of the problem at hand.
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3.2. PSO Environment in Optimization of Fuzzy Preference Relations PSO algorithm is a population based stochastic optimization technique inspired by bird flocking and fish schooling originally designed and introduced by Kennedy and Eberhart [8]. It is based on communication and interaction between the members of the swarm, what means that each member of the PSO algorithm, named as particle, determines its position by combining the history of its own best location with those of others members of the swarm. The algorithm flow of the PSO starts with a population of particles whose positions are the potential solutions of the problem, and the velocities are randomly initialized in the problem search space. In each iteration/generation, the search for optimal position (solution) is performed by updating the particles velocities and positions based on a predefined fitness function. The velocity of each particle is updated using two best positions, namely personal best position and neighborhood best position. The personal best position is the best position the particle has visited and neighborhood best position is the best position the particle and its neighbors have visited [23,24,25]. The optimization of the fuzzy preference relations coming from the space of interval-valued fuzzy preference relations is realized by means of the PSO. In the following, we elaborate on the fitness function, its realization, and the PSO optimization along with the corresponding formation of the components of the swarm. 3.2.1. Particle An important point in a PSO algorithm is finding an appropriate mapping between problem solution and the particle’s representation. In our framework, each particle represents a vector whose entries are located in the interval [0, 1]. Basically, if there is a group of m decision makers and a set of n alternatives, the number of entries of the particle is m · n(n − 1). Starting with the initial fuzzy preference relation provided by the decision maker and assuming a given level of granularity α (located in the unit interval), let us consider an entry pri j . The interval of admissible values of this entry of P(PR) implied by the level of granularity is equal to: [a, b] = [max(0, pri j − α /2), min(1, pri j + α /2)]
(13)
Let assume that the entry of interest of the particle is x. It is transformed linearly according to the expression z = a + (b − a)x. For example, consider that pri j is equal to 0.8, the admissible level of granularity α = 0.1, and the corresponding entry of the particle is x = 0.6. Then, the corresponding interval of the granular fuzzy preference relation computed as given by Eq. (13) becomes equal to [a, b] = [0.75, 0.85]. Subsequently, z = 0.81, and, therefore, the modified value of pri j becomes equal to 0.81. The overall particle is composed of the individual segments, where each of them is concerned with the optimization of the parameters of the fuzzy preference relations. 3.2.2. Fitness Function In the PSO, the performance of each particle during its movement is assessed by means of some performance index (fitness function). Here, the aim of the PSO is the maximization
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of the individual consistency achieved by each decision maker. Therefore, the fitness function, f , associated with the particle is defined as: f =Q
(14)
being Q the optimization criterion presented in Section 3.1. The higher the value of f , the better the particle is. 3.2.3. Algorithm We use the generic form of the PSO algorithm. In such a way, the updates of the velocity of a particle are realized in the form v(t + 1) = w × v(t) + c1 a · (z p − z) + c2 b · (zg − z) where “t” is an index of the generation and · denotes a vector multiplication realized coordinatewise. z p denotes the best position reported so far for the particle under discussion while zg is the best position overall and developed so far across the entire population. The current velocity v(t) is scaled by the inertia weight (w) which emphasizes some effect of resistance to change the current velocity. The value of the inertia weight is kept constant through the entire optimization process and equal to 0.2 (this value is commonly encountered in the existing literature [22]). By using the inertia component, we form the memory effect of the particle. The two other parameters of the PSO, that is a and b, are vectors of random numbers drawn from the uniform distribution over the [0, 1] interval. These two update components help form a proper mix of the components of the velocity. The second expression governing the change in the velocity of the particle is particularly interesting as it nicely captures the relationships between the particle and its history as well as the history of overall population in terms of their performance reported so far. The next position (in iteration step “t+1”) of the particle is computed in a straightforward manner: z(t + 1) = z(t) + v(t + 1). When it comes to the representation of the solutions, the particle z is composed of “m · n(n − 1)” entries positioned in the [0,1] interval that corresponds to the search space. In addition, one should note that while PSO optimizes the fitness function, there is no guarantee that the result is optimal, rather than that we can refer to the solution as the best one being formed by the PSO.
4. Example of Application In this section, we show an example which helps quantifying the performance of the proposed method. Proceeding with the details of the optimization environment, PSO was used with the following values of the parameters which were selected as a result of intensive experimentation: • The size of the swarm consisted of 100 particles. This size of the population was found to produce “stable” results meaning that very similar or identical results were reported in successive runs of the PSO. Due to the search space, this particular size of the population was suitable to realize a search process. • The number of generations (or iterations) was set to 100. It was observed that after 100 generations there were no further changes of the values of the fitness function.
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• The parameters in the update equation for the velocity of the particle were set as c1 = c2 = 2. These values are commonly encountered in the existing literature. Let us suppose the following four fuzzy preference relations coming from four decision makers E = {e1 , e2 , e3 , e4 }: ⎞ − 0.4 0.3 0.3 ⎜ 0.4 − 0.7 0.7 ⎟ ⎟ PR1 = ⎜ ⎝ 0.4 0.1 − 0.2 ⎠ 0.2 0.3 0.7 −
⎞ − 0.2 0.6 0.3 ⎜ 0.6 − 0.5 0.3 ⎟ ⎟ PR2 = ⎜ ⎝ 0.2 0.3 − 0.5 ⎠ 0.1 0.3 0.9 −
⎞ − 0.2 0.5 0.1 ⎜ 0.4 − 0.2 0.8 ⎟ ⎟ PR3 = ⎜ ⎝ 0.5 0.4 − 0.9 ⎠ 0.9 0.1 0.4 −
⎞ − 0.6 0.2 0.6 ⎜ 0.4 − 0.6 0.2 ⎟ ⎟ PR4 = ⎜ ⎝ 0.8 0.6 − 0.5 ⎠ 0.4 0.6 0.6 −
⎛
⎛
⎛
⎛
1.00
Considering a given level of granularity α , Figure 1 illustrates the performance of the PSO quantified in terms of the fitness function (optimization criterion) obtained in successive generations. The most notable improvement is noted as the very beginning of the optimization, and afterwards, there is a clearly visible stabilization, where the values of the fitness function remain constant. To put the obtained optimization results in a certain context, we report the performance obtained when considering the entries of the fuzzy preference relations are sin-
f
0.80
0.85
0.90
0.95
α = 0.2 α = 0.4 α = 0.6 α = 0.8 α = 1.0 α = 2.0
0
20
40
60
80
100
generation
Figure 1. Fitness function f in successive PSO generations for selected values of α .
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gle numeric values, that is, when no granularity is allowed (α = 0). In such a case, the corresponding consistency degrees of the four fuzzy preference relations are cd1 = 0.85, cd2 = 0.85, cd3 = 0.76, and cd4 = 0.83. Therefore, the value of the fitness function f is is 0.82. Comparing with the values obtained by the PSO, the fitness function f takes on now lower values. As we can see in Figure 1, the higher the admitted level of granularity α , the higher the values obtained by the fitness function f . It is due to the fact that the higher the level of granularity α , the higher the level of flexibility introduced in the fuzzy preference relations and, therefore, the possibility of achieving higher level of consistency. In particular, when each entry of the granular preference relation is treated as the whole [0, 1] interval (it occurs when α = 2.0), the value of the fitness function is the maximum one, which is 1. However, when the level of granularity is very high, the values of the entries of the fuzzy preference relation could be very different in comparison with the original values provided by the decision maker and, therefore, he/she could reject them. Once the consistency of the fuzzy preference relations has been increased, a selection process could be applied in order to find the solution set of alternatives according to the preferences provided by the decision makers. This selection process is composed of two steps [26]: aggregation of preferences provided by the decision makers and exploitation of the aggregated preference obtained previously.
5. Conclusions In this contribution, we have proposed a method based on an allocation of information granularity as an important asset to increase the consistency. Here, information granularity is an important and useful asset that support to increase consistency in the fuzzy preference relations provide by the decision makers. It offers a badly needed flexibility so that the granular fuzzy preference relations can produce numeric realizations so that they are of higher consistency. To do so, the PSO environment has been shown to serve a suitable optimization framework. The granular representation of fuzzy preference relations discussed in this contribution was the one using intervals. However, any other formalism of granular computing, especially fuzzy sets, could be equally applicable here. Furthermore, in the scenario analyzed in this contribution, a uniform allocation of granularity has been discussed, where the same level of granularity α has been allocated across all the fuzzy preference relations. However, a nonuniform distribution of granularity could be considered, where these levels are also optimized so that each decision maker might have an individual value of α becoming available to his/her disposal.
Acknowledgments The authors would like to acknowledge FEDER financial support from the Project FUZZYLING-II Project TIN2010-17876, and also the financial support from the Andalusian Excellence Projects TIC-05299 and TIC-5991.
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