Strict Valued Preference Relations and Choice Functions in Decision ...

Strict Valued Preference Relations and Choice Functions in Decision-Making Procedures 1

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Ildar Batyrshin , Natalja Shajdullina , and Leonid Sheremetov

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Mexican Petroleum Institute, Eje Central Lazaro, 152, Mexico, D.F., 07730 {batyr,sher}@imp.mx 2 Kazan State Technological University, K. Marx st., 68, Kazan, Russia [email protected]

Abstract. Fuzzy (valued) preference relations (FPR) give possibility to take into account the intensity of preference between alternatives. The refinement of crisp (non-valued) preference relations by replacing them with valued preference relations often transforms crisp preference relations with cycles into acyclic FPR. It gives possibility to make decisions in situations when crisp models do not work. Different models of rationality of strict FPR defined by the levels of transitivity or acyclicity of these relations are considered. The choice of the best alternatives based on given strict FPR is defined by a fuzzy choice function (FCF) ordering alternatives in given subset of alternatives. The relationships between rationality of strict FPR and rationality of FCF are studied. Several valued generalizations of crisp group decision-making procedures are proposed. As shown on examples of group decision-making in multiagent systems, taking into account the preference values gives possibility to avoid some problems typical for crisp procedures.

1 Introduction The problem of decision-making (DM) may be considered as the problem of ranking of elements of some set of alternatives X or looking for the “best” alternatives from this set [6, 9]. Different approaches to these problems are varying in the structure of the set X, in the initial information about these elements, in the criteria used for ranking and evaluation of the “best” alternatives, etc. Most of intelligent systems include as a part some DM procedures, e.g. crisp and valued preference relations are used for modeling decision making in multiagent systems [5, 11, 12]. Valued (fuzzy) preference relations (FPR) give possibility to take into account the intensity of preference between alternatives. Different models of DM based on FPR have been considered in literature [1-3, 5-8, 10-12]. These models are usually based on a weak fuzzy preference relation R:X×X→L defined on the set of alternatives X such that for all alternatives x,y the value R(x,y) is understood as a degree to which the proposition “a not worse than b” is true, or as intensity of preference of x over y etc. Usually it is supposed that the set of true values L coincides with interval [0,1]. In this case the operations on the set of FPR may be defined by means of fuzzy logic operations given on L. Generally L may denote some linearly ordered set of preference values [2, 7] For example, L may be a set of numerical values, the set of R. Monroy et al. (Eds.): MICAI 2004, LNAI 2972, pp. 332–341, 2004. © Springer-Verlag Berlin Heidelberg 2004

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scores {0, 1, 2, 3, 4, 5, 6} or the set of linguistic evaluations such as “very small preference”, “small preference”, “strong preference” etc. Usually a weak FPR and associated with it strict, indifference and incomparability fuzzy relations are considered [5, 6]. The properties of rationality of DM procedures are related with the properties of consistency of underlying FPR. These consistency properties are usually formulated in the form of transitivity or acyclicity of weak FPR and associated strict FPR. The types of consistency of strict FPR in the form of types of transitivity and acyclicity of these relations are considered in this work. The absence of desired requirement of consistency may be used for correction of given strict FPR by some formal procedure or for overestimation of preference values for some pair of alternatives. The more traditional approach considers the rationality of fuzzy choice function (FCF) with respect to the crisp set of non-dominated alternatives, which may be obtained as a result of the use of FCF [2, 3, 6, 8, 10]. The existence of such nondominated set of alternatives is related with acyclicity of underlying FPR [2, 3]. This approach really reduces the problem to a non-valued, crisp choice functions and crisp acyclic relations and makes little use of information about valued preferences. In our work, we consider FCF as a ranking function and rationality conditions of FCF are formulated as rationality of rankings on all possible subsets of alternatives. The paper is organized as follows. The properties of consistency of strict fuzzy preference relations in terms of possible types of transitivity and acyclicity are studied in Section 2. The rationality conditions for FCF are considered in Section 3. The relationships between the consistency properties of strict FPR and rationality conditions of FCF are studied in Section 4. Example of application of DM procedures in multiagent systems is discussed in Section 5. Finally the conclusions and further directions of extension of proposed models are discussed.

2 Strict Valued Preference Relations A valued relation on a universal set of alternatives Ω is a function P:Ω×Ω→LP, where LP is a linearly ordered set of preference values with minimum and maximum elements denoted as 0 and I respectively. We will consider here the set of preference values LP= [0,1] used in fuzzy logic with ordering relation < defined by the linear ordering of real numbers, and with 0 = 0, I = 1. Generally, many results related with strict FPR and FCF may be extended on the case of finite scale LP= {a0, a1, …, an} with linearly ordered grades a0 < a1< …< an. Such a scale may contain numerical grades like LP= {0,1,2,3,4,5,6} or linguistic grades LP = {absence of preference, very small preference, small preference, average preference, strong preference, very strong preference, absolute preference}. For this reason we will consider here the terms valued preference relation and fuzzy preference relation as synonyms [2, 7]. The linear ordering relation < on L defines the operations ∧ and ∨ on L: a∧b=a and a∨b=b iff a ≤ b (i.e. a < b or a=b) for all a,b from L. The negation operation ′ may be introduced on LP as follows: a′ = 1 - a for LP= [0,1] and ak′ = an-k for finite scale with n+1 grades. The operations on LP satisfy De Morgan laws: (a∧b)′=a′∨b′ and (a∨b)′=a′∧b′ and the involution law: ak′′= ak.

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P will be called a FPR if it satisfies on Ω the asymmetry condition: P(x,y)∧P(y,x)= 0. We will write P(x,y) ≥ 0 if P(y,x) = 0. P(x,y) will be understood as a preference degree or intensity of preference of x over y. The following types of transitivity reflect the different types of consistency of P: − Weak transitivity: WT. From P(x,y) > 0 and P(y,z) > 0 it follows P(x,z) > 0. − Negative transitivity: NT. From P(x,y) ≥ 0 and P(y,z) ≥ 0 it follows P(x,z) ≥ 0. − Transitivity: T. From P(x,y) > 0 and P(y,z)> 0 it follows P(x,z)≥ P(x,y)∧P(y,z). − Strong transitivity: ST. From P(x,y) ≥ 0 and P(y,z) ≥ 0 it follows P(x,z)≥ P(x,y)∨P(y,z). − Quasi-series: QS. From P(x,y) ≥ 0 and P(y,z) ≥ 0 it follows P(x,z)=P(x,y)∨P(y,z). − Super-strong transitivity: strong transitivity together with the property SST. From P(x,y) > 0 and P(y,z)> 0 it follows P(x,z)> P(x,y)∨P(y,z). Suppose P is a strict FPR P on Ω and x0,x1, …, xn (n ≥ 2) are some elements of Ω. Consider the following types of cycles, which may by induced by P in Ω: − 0-cycle: P(x0,x1) > 0, P(x1,x2)> 0, …, P(xn-1,xn) > 0, P(xn,x0) > 0; − a-cycle: P(x0,x1)≥ a, P(x1,x2)≥ a, … P(xn-1,xn)≥ a, P(xn,x0)≥ a, where a∈L, a > 0; − max-cycle: P(x0,x1)= P(x1,x2)=…= P(xn-1,xn) = P(xn,x0) = a ≥ P(xi,xk) for some a∈L, a ≥ 0, and all i,k ∈ {0, 1, …, n}. As a special case of a-cycle with a= I, I-cycle will be considered. It is clear that any I-cycle is a max-cycle. We will say that a strict FPR P satisfies one of the properties acyclicity (0-AC), a-acyclicity (a-AC), I-acyclicity (I-AC) and maxacyclicity (MAC) if it does not contain correspondingly 0-cycles, a-cycles, I-cycles and max-cycles. Proposition 1. The transitive and acyclic classes of strict FPR are partially ordered by inclusion as follows: QS ⊆ ST⊆ NT ⊆ WT, SST ⊆ ST ⊆ T ⊆ WT⊆ 0-AC⊆ ai-AC⊆ aj-AC⊆ I-AC, MAC⊆ I-AC, where ai and aj are elements of L such that 0 ≤ ai ≤ aj ≤ I. The fuzzy quasi-series is a direct fuzzification of crisp quasi-series considered in [2, 10]. A fuzzy quasi-series is related with a fuzzy quasi-ordering relation [13] and defines some hierarchical partition of the set of alternatives on the ordered classes of alternatives. The class of these relations due to their special structure is the narrowest class of strict FPR, whereas the class of I-AC is the widest class of strict FPR.

3 Fuzzy Choice Functions Suppose LC is a linearly ordered set of evaluations of the respective quality of alternatives in the sets of alternatives X⊆Ω. The minimum 0 and the maximum I elements of LC will be considered as evaluations of the “worst” and the “best” alternatives in X. Generally we will suppose that such the “worst” and the “best” alternatives may be absent in X which may contain for example only “good” or “not bad” alternatives. The set LC will be considered as a set of possible values of FCF related with the fuzzy preference relation P defined on Ω. For this reason LC will be tied with the set of values LP of correspondent FPR. In fuzzy context, it will be

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supposed that LC = [0,1], but generally it may be a set of scores LC= {0, 1, 2, 3, 4, 5, 6} or a set of suitable linguistic values. A fuzzy choice function C is a correspondence which defines for each finite set of alternatives X⊆Ω the function CX:X→LC. In such definition, the FCF is really a score function measuring alternatives in the scale LC and defining some linear ordering of alternatives from given set X. The possible properties of rationality of these orderings on different sets of alternatives X⊆Ω are discussed in [2]. We consider here several new conditions of rationality of FCF. In the following, for any FCF the fulfillment of the trivial choice property will be required: TC. (∀x⊆Ω) C{x}(x) = I. This condition says that any element x is “the best” in the set containing only this element. Another, more strong condition requires that in any set of alternatives “the best” alternative exists: BC. (∀X⊆Ω)(∃x∈X) CX(x) = I. This axiom is a very strong requirement because the set of “the best” alternatives in general may be empty. As shown in DM theory, the choice functions generated by preference relations fulfill the similar condition if the correspondent preference relation is acyclic [2, 3]. This problem will be discussed also below. The following two conditions are some weakening of the previous one. b-UAC. (∀X⊆Ω)(∃x∈X) CX(x) > b, where b∈L, (b ≥ 0) is some level of unacceptability of the quality of alternatives chosen from a given set of alternatives. The “good” alternative should have the quality, which is greater than this level. The next special case of the previous condition requires the existence of “not the worst” alternatives: NWC. (∀X⊆Ω)(∃x∈X) CX(x) > 0. This rationality condition requires that in any set of alternatives the rational choice function can select “not the worst” alternatives. Another possible requirement on FCF requires the existence of nontrivial ordering of alternatives: NTO. (∀X⊆Ω) ((X ≥ 2)→(∃x,y∈X) (x≠y)&(CX(x) > CX(y)). The stronger condition on FCF requires that the “best” alternatives should be “standard” element: ES. (∀X⊆Ω) (∀x∈X)((CX(x)=I)→(∀y∈X)(CX(y)= C{x,y}(y))). This condition is very strong. For choice functions satisfying this condition it follows, for example, that if we have “the best” element in some set X then for evaluating the quality of another alternatives in X it is sufficient to compare these alternatives only with this standard (or “ideal”) element and other alternatives in X may not be considered in this evaluation. The condition of dependence of strict orderings: DSO. (∀X,Y⊆Ω)(∀x,y∈X∩Y)((CX(x) > CX(y)) → (CY(x) > CY(y))), requires that if x has a higher level of choice function than y in some set X then such situation takes place also in any set Y containing both alternatives.

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Proposition 2. BC ⊆ bj-UAC ⊆ bi-UAC ⊆ NWC, NTO⊆ NWC, ES∩BC⊆DSO, where bi and bj are elements of L such that 0 ≤ bi ≤ bj ≤ I. As shown in the following section, if the FCF is generated by strict FPR then all these conditions are characterized by some requirements on transitivity or acyclicity of this relation.

4 Choice Functions Generated by Strict FPR Fuzzy choice functions may be generated by some FPR [10, 2] as follows:

C X ( x) = ( max P ( y, x)) ′ = min ( P ( y, x))′. y∈ X y∈ X It is clear that the properties of choice function and strict preference relation generating this choice function are interrelated. It is clear also from asymmetry of fuzzy strict preference relations P and from definition of choice function that any choice function satisfies the property TC. We will need also in the following condition of weak completeness of linguistic strict preference relation: WC. From x≠y it follows P(x,y)∨P(y,x)>0. Theorem 3. The diagram on Fig. 1 characterizes the FCFs and fuzzy strict preference relations generating these choice functions.

On this diagram, A ↔ B denotes that the choice function satisfies the property A if and only if the strict preference relation generating this choice function satisfies the property B. A→ B denotes that from A follows B. For example, choice function satisfies the condition of b-UAC iff the strict preference relation P generating this choice function is a-acyclic with a= b′.

Fig.1. Relationships between the classes of choice functions and linguistic strict preference relations generating these choice functions.

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5 Group Decisions In group DM, the intensity of preferences often plays important role. Consider an example. Five friends want to make decision where to go in the evening. Three of them slightly prefer bar to restaurant but other two strongly prefer restaurant to bar. If the intensity of preferences is not taken into account then applying for example simple majority rule, bar should be chosen. But usually, the intensity of preferences influences on the group decision and in this case, the restaurant may be chosen if the intensities of preference of restaurant by two friends are very strong. Different methods of aggregation of preference intensities have been proposed [6, 7]. Several methods of aggregation of FPR, which generalize the classical crisp methods, are considered further in this section. As shown by Arrow, a group decision procedure satisfying several axioms or rationality does not exist. Any such proposed procedure may be criticized from one or another point of view. The generalizations of some of these procedures on the case of valued preferences are not free from critique as well but they give the possibility to take into account the intensity of preferences and as a result, to diminish the drawbacks of crisp procedures. 5.1 Valued Simple Majority Rule

The draft formulation of the simple majority rule is the following: an alternative is a winner if it is placed on the first position by majority of agents. It may happen that several alternatives receive equal number of votes. In this case, some additional procedure of resolving such situations may be used [4]. The possible generalization of this method on fuzzy preference relations for linguistic evaluations of intensity was considered in [7]. We propose here a new method, which uses the fuzzy evaluations of intensity in strict FPR. The crisp simple majority rule takes into account only information about the best alternatives and may be considered as a procedure operating with the individual choice functions. The alternative x receives the vote Vi(x) =1 if it belongs to the choice function of i-th agent. The sum of these votes defines the winner. The vote that alternative received from some agent may be considered as the value of characteristic function of choice function defined by linear ordering of all alternatives by this agent. Table 1 shows example of preference profile for 5 agents a1 ,…, a5 on the set of three alternatives X={x,y,z}, where, for example the first column denotes the ordering x > y > z of alternatives correspondent to the agent a1. Here x and y have the maximum scores equal 2 correspondent to the number of their location on the first place in the preference profile. Table 2 contains the correspondent values of votes of alternatives in this preference profile. Table 1. Profile of 5 preferences

a1

a2

a3

a4

a5

x y z

y z x

z x y

x y z

y z x

Table 2. Choice functions and sum of votes for profile from Table 1.

x y z

a1

a2

a3

a4

a5

SUMi(Vi)

1 0 0

0 1 0

0 0 1

1 0 0

0 1 0

2 2 1

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Consider possible fuzzy generalization of simple majority rule based on the profile of strict FPR. Each strict FPR is replaced by a correspondent FCF, linearly ordering alternatives with respect to the value of choice function. The averaged sum of membership values in the FCFs for each alternative obtained for all agents is calculated. The alternative with the maximum value is considered as the solution of the group decision problem. For example of group decision problem for 5 friends considered above, the crisp and fuzzy simple majority rules give the following results. Suppose the fuzzy strict preferences between bar and restaurant for these 5 friends have the following values: P1(bar, restaurant) = 0.2, P2(bar, restaurant) = 0.3, P3(bar, restaurant) = 0.2, P4(restaurant, bar) = 0.8, P5(restaurant, bar) = 0.7. The corresponding matrixes of fuzzy strict preference relations are presented on Table 3. Table 4 contains corresponding crisp ranking when the intensity of preferences is not taken into account. Table 5 contains the scores obtained by all alternatives by the crisp simple majority rule. Table 6 contains FCFs defined by FPRs and the resulting scores of alternatives. As it can be seen, the crisp and fuzzy simple majority rules give different results because the fuzzy approach gives possibility to take into account the intensity of preferences which are not considered by the crisp approach. Table 3. Example of 5 fuzzy strict preference relations

P1

bar

rest.

P2

bar

rest.

P3

bar

rest.

bar rest. P4

0 0 bar

0.2 0 rest.

bar rest. P5

0 0 bar

0.3 0 rest.

bar rest.

0 0

0.2 0

bar rest.

0 0.8

0 0

bar rest.

0 0.7

0 0

Table 4. Profile of crisp preferences corresponding to Table 3

a1

a2

a3

a4

a5

bar restaurant

bar restaurant

bar restaurant

restaurant bar

restaurant bar

Table 5. Crisp choice functions and sum of votes for profile from Table 4

a1

a2

a3

a4

a5

SUMi(Vi)

1 0

1 0

1 0

0 1

0 1

3 2

bar restaurant

Table 6. Fuzzy choice functions and sum of votes for profile from Table 3

bar restaurant

a1

a2

a3

a4

a5

SUMi(Vi)

1 0.8

1 0.7

1 0.8

0.2 1

0.3 1

3.5 4.3

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Table 7. Example on fuzzy Condorcet winner rule

P

x

y

z

x y z C

0 0 1/5 4/5

1/5 0 0 4/5

0 3/5 0 2/5

5.2 Fuzzy Condorcet Winner

The crisp Condorcet winner rule directly uses the information about pair-wise preferences. For given preference profile, an alternative x is a Condorcet winner if in more than half preference relations from the profile it is more preferable than each other alternative. Unfortunately the Condorcet winner does not always exist. This situation happens for the example presented in Table 1. The alternative that is better than all other alternatives for more than 2 agents does not exist: the alternative x is better than y for 3 agents, y is better than z for 4 agents and z is better that x for 3 agents. We obtain the circle: x > y, y > z, z > x which does not give us the possibility to select or reject one of three alternatives. But if we consider valued preference relations then one alternative may be rejected. Let us define strict valued preference relation on the set of alternatives in the following way. Denote V(x,y) the number of agents which say that x is better than y. Define P(x,y) = max{0,V(x,y)-V(y,x)}/N, where N is a total number of agents. Then for our example, we receive the strict valued preference relation shown in Table 7. The obtained strict FPR satisfies max-acyclicity MAX and weak completeness WC conditions and according to the Theorem 3 the choice function of this strict FPR satisfies the non-trivial ordering NTO condition and contains alternatives with different values of FCF. The FCF of this FPR is shown in the last string of Table 7. In comparison with x and y, the alternative z obtains the lower value of FCF and may be rejected. We should note that for the considered fuzzy Condorcet winner rule it is also possible to receive a strict FPR, which does not satisfy MAC condition such that all alternatives compose circle of preferences with equal values. But the possibility of such situation is much less than in the case of crisp Condorcet winner rule. It may be shown that for some classes of preference profiles the fuzzy Condorcet winner rule always will give FPR satisfying MAC condition and hence the correspondent FCF will satisfy the non-trivial ordering condition.

6 Conclusions In the paper, we have presented the models of valued strict preferences, which can be used by agents addressing the problems of ranking and aggregation of their fuzzy opinions. We considered the consistency properties of strict FPR separately from the properties of some weak FPR. First, it gives us the possibility to analyze more fine structures of FPR related with the rationality properties of DM procedures. Second, many situations exist when initially given information about preferences may be

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presented directly in the form of strict FPR, i.e. as asymmetric FRP. Such FPR may be received, for example, as a result of pair-wise comparison of all alternatives and replying on two questions: 1) What alternative from considered pair is more preferable? 2) If one of alternatives is more preferable, then what is the intensity of this preference? We think that for expert it is easier to evaluate his pair-wise preferences in a form given by strict FPR than to evaluate two intensities of preference: alternative a over b and alternative b over a for obtaining weak FPR. Another distinctive feature of considered approach is the way in which we study the rationality of the FCF. More traditional approach to FCF considers only crisp set of non-dominated alternatives, which lead to acyclicity of some underlying crisp preference relation and does not deal much with the intensity of preferences. Our approach really takes into account the information about intensity of pair-wise preferences in underlying FPR and gives a possibility to consider as the “good” alternatives, the alternatives dominated with a “low” value of intensity. This set of alternatives may be considered as a solution of a DM problem when the set of the “best”, non-dominated alternatives is empty. Our approach can give solution to the DM problem when more traditional approach does not work. The existence of the set of “good” alternatives is related with the properties of “weak” acyclicity of underlying strict FPR, which essentially use the intensity of pair-wise preferences. Such “weak” acyclicity properties admit some types of fuzzy circles in the strict FPR such that the choice of “good” alternatives may be done in the presence of such circles. The circles in crisp preference relations arise usually in multi-criteria evaluation of alternatives or like in Condorcet paradox when these relations are obtained as a result of aggregation of individual preference relations. In this case, the use of DM procedures based on strict FPR will decrease the possibility of arising circles in aggregated preference relation when the rational decision cannot be done. Central to this model is the incomparability relation that occurs when agents have conflicting information preventing them to come to a consensus. Here we have shown how the valued strict preferences can decrease the number of cycles and this way the number of conflicts for multi-agent DM. This model was shown to be applicable to both a single and multi-agent multi-criteria DM problem setting. Several generalizations of crisp group DM procedures have been proposed that give possibility to avoid some problems typical for crisp models. Acknowledgements. Partial support for this research work has been provided by the IMP, project D.00006 “Distributed Intelligent Computing”, and by RFBR grant 0301- 96245.

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