Simplifying the Search for Effective Ranking of Fuzzy Numbers

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1 1.119/TFUZZ , IEEE Transactions on Fuzzy Systems IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL., NO., 1 Simplifying the Search for Effective Ranking of Fuzzy Numbers Adrian I. Ban, Lucian Coroianu Abstract We prove that a ranking index which generates an order on some subset of fuzzy numbers can be reduced to a ranking index with a simpler form and which generates an equivalent order. In this way, the finding of ranking indices generating orders with desirable properties can be essentially simplified such that we need to search only from additive or scale invariant ranking indices. We determine exactly, making abstraction of equivalent orders, the class of ranking indices on trapezoidal fuzzy numbers which generate orders satisfying all the basic requirements. Finally, ranking indices used to rank trapezoidal fuzzy numbers are extended to ranking indices used to rank arbitrary fuzzy numbers so that the desired properties are preserved. Index Terms Fuzzy number; Trapezoidal fuzzy number; Ranking; Ranking index I. INTRODUCTION In the last decades many papers have been devoted to studies on fuzzy number ranking procedures. It is meaningful that Web of Science and Scopus return over 13 results for the search fuzzy number ranking. Basically, we can distinguish two approaches: - Based on so called ranking indices, which are functions from fuzzy numbers to real values. Then, a ranking is generated by a procedure based on the standard ordering of reals (see, e.g., [1]-[4], [6]-[7], [15]-[16], [2]-[21], [26], [31], [34], [37], [41], [44], [46], [51], [54]). This method can be extended even for other classes of fuzzy sets which are not necessarily fuzzy numbers (see e.g. [17], [39]). - Based on fuzzy binary relations (see, e.g., [8], [24], [45]). This approach can be also extended to more general settings than fuzzy numbers (see [5]). Even from this enumeration we see that there are numerous ways to rank fuzzy numbers. Some comparative studies can be found in the papers [14], [54] or in the recent paper [12]. We have several reasons to justify our research. First of all, the main results of this paper could be very helpful for studying which reasonable properties are satisfied by a certain ranking procedure. Then, our results are very general since they describe precisely the shape of a ranking index which generates an order that satisfies a specific requirement. The impact of a good choice of ranking fuzzy numbers is decisive in many applications related to decision theory, optimization, artificial intelligence, approximate reasoning, socioeconomic systems, etc. As Matarazzo and Munda pointed out in [43], A.I. Ban is with the Department of Mathematics and Informatics, University of Oradea, 4187 Oradea, Romania ( aiban@uoradea.ro) L. Coroianu is with the Department of Mathematics and Informatics, University of Oradea, 4187 Oradea, Romania ( lcoroianu@uoradea.ro) Manuscript received ; revised a key issue in operationalizing fuzzy set theory is how to compare fuzzy numbers. In the present paper we discuss only ranking approaches obtained from ranking indices. Especially in the very recent papers the authors try to impose a certain ranking method by finding some examples in which their approach gives better results comparing to others (see, e.g., [2], [6], [7], [31], [39], [46], [47], [55]). But obviously there is a kind of subjectivity because from a few examples we cannot conclude which approach is better. For this reason, the goal of this paper is to characterize ranking approaches rather than classifying them. We will do that by taking as a rough guide the reasonable properties of Wang and Kerre ([54]) presented in Section III. Because we are mostly interested in sets which are closed under addition and/or scalar multiplication and not necessarily closed under fuzzy multiplication (such sets are the set of trapezoidal fuzzy numbers and the set of triangular fuzzy numbers) we slightly modified the requirement If A B then A C B C for every C comparing with [54]. Actually, apart from a particular case of Adamo s approach in [4], we are not aware of the existence of an ordering generated by a ranking index which would satisfy the above property. The main result in Section III proves that, making abstraction of equivalent orders, one needs to discuss only ranking indices with the property that they belong to the support of the fuzzy number. In this way, the study is drastically simplified, the best proof being the following result in Section IV, very useful in the searching of orders that satisfy all the basic requirements: if a ranking index is defined on a closed under addition and scalar multiplication set of fuzzy numbers and it generates an order satisfying all the reasonable properties, then there exists a linear ranking index which generates an equivalent order. In Section V we adapt all these results for the ranking of trapezoidal fuzzy numbers. In fact, we prove that, making abstraction of the equivalent orders, we can determine accurately the class of ranking indices which generate orders satisfying all the basic requirements. In addition, we characterize classes of ranking indices that generate orders which satisfy just a part of the reasonable properties. Since in most applications the researchers use only triangular or trapezoidal fuzzy numbers (see, e.g., [11], [18], [42], [53]), these results are useful when someone would be interested in ranking such fuzzy numbers. We apply the results from this section in order to study the reasonable properties of some ranking procedures between trapezoidal fuzzy numbers which can be found in the literature. It is worth noticing that the theoretical results of the paper are very easy to handle in concrete examples. We just need to verify whether the ranking index belongs to the classes (c) 213 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

2 1.119/TFUZZ , IEEE Transactions on Fuzzy Systems 2 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL., NO., obtained in Section V. Basically this is very easy to verify since we need only to observe whether the ranking index is additive or scale invariant. In Section VI we extend orders between trapezoidal fuzzy numbers to orders between arbitrary fuzzy numbers so that some (or all) of the basic desirable properties are preserved. We can state that the main idea is the same as in the topic of the approximation of fuzzy numbers by fuzzy numbers with simpler form: to use simpler representations so that some parameters are preserved (see, e.g., [9], [1], [35]). In our case it can be used an order on the set of trapezoidal fuzzy numbers to compare arbitrary fuzzy numbers, with the property that the extended order satisfies all the properties which hold on the set of trapezoidal fuzzy numbers. The paper ends with conclusions where, besides a resume of the main results, we point out some possible further research in this topic. II. PRELIMINARIES We begin by recalling some basic definitions used in this paper. Definition 1: (see [28]) A fuzzy number A is a fuzzy subset of the real line, A : R [, 1], where A (x) denotes the value of the membership function of A in x, satisfying the following properties: (i) A is normal (i. e. there exists x R such that A (x ) = 1); (ii) A is fuzzy convex (i. e. A (λx 1 + (1 λ) x 2 ) min (A (x 1 ), A (x 2 )), for every x 1, x 2 R and λ [, 1]); (iii) A is upper semicontinuous in every x R (i. e. ε >, δ > such that A (x) A (x ) < ε, whenever x x < δ); (iv) supp (A) is bounded, where supp (A) = cl {x R : A (x) > } and cl (M) denotes the closure of the set M. The α cut, α (, 1], of a fuzzy number A is a crisp set defined as A α = {x R : A(x) α}. Every α cut, α [, 1], of a fuzzy number A is a closed interval where A α = [A L (α), A U (α)], A L (α) = inf{x R : A(x) α}, (1) A U (α) = sup{x R : A(x) α} (2) for any α (, 1], with the convention A = [A L (), A U ()] := supp A. Fuzzy numbers with simple membership functions are preferred in practice. The most often used are so-called trapezoidal fuzzy numbers, that is fuzzy numbers with α cuts given by T α = [t 1 + (t 2 t 1 )α, t 4 (t 4 t 3 )α], α [, 1], where t 1, t 2, t 3, t 4 R, t 1 t 2 t 3 t 4. We denote by T = (t 1, t 2, t 3, t 4 ) such a fuzzy number. When t 2 = t 3 we obtain a triangular fuzzy number. When t 1 = t 2 = t 3 = t 4 = r we obtain the real number r. Throughout this paper we denote by F (R) the set of all fuzzy numbers, by F T (R) the set of all trapezoidal fuzzy numbers and by F (R) the set of all triangular fuzzy numbers. In what follows, we recall some important characteristics of a fuzzy number. The expected value EV (A) of a fuzzy number A is given by (see [29], [36]) EV (A) = 1 2 (A L (α) + A U (α)) dα. (3) The ambiguity Amb(A) and the value V al(a) of a fuzzy number A are given by ([25]) Amb(A) = V al(a) = α(a U (α) A L (α))dα, (4) α(a U (α) + A L (α))dα. (5) Let A, B F (R) and λ R. We consider (see e.g. [27], p. 4) the addition A + B by (A + B) α = A α + B α = [A L (α) + B L (α), A U (α) + B U (α)] and the scalar multiplication λ A by { [λal (α), λa (λ A) α = λa α = U (α)], for λ, [λa U (α), λa L (α)], for λ <. If T = (t 1, t 2, t 3, t 4 ), S = (s 1, s 2, s 3, s 4 ) F T (R) then S + T = (t 1 + s 1, t 2 + s 2, t 3 + s 3, t 4 + s 4 ), λ T = (λt 1, λt 2, λt 3, λt 4 ) for λ and λ T = (λt 4, λt 3, λt 2, λt 1 ) for λ <, therefore F T (R) is closed under addition and scalar multiplication. III. REASONABLE PROPERTIES FOR RANKING FUZZY NUMBERS In their famous paper ([54]) Wang and Kerre proposed a list of reasonable properties for the ordering approaches over a set S F (R). We will adapt these requirements for the case when the ordering approach is induced by a binary relation over S. In this setting, comparing A and B in terms of does not depend on any subset (finite or not) of S which contains A and B. This is an important observation because Wang and Kerre considered a more general setting in which the ordering approach also depends on the finite subset of S on which is applied. However, motivated by the numerous papers dealing with the ordering of fuzzy numbers by using a given binary relation over S, in this paper we will not deal with the second case. This is why the requirements of Wang and Kerre will be rewritten with respect to on S. But before that let us make a more rigorous approach to. First of all we assume that is a total binary relation over S. This means that for any (A, B) S 2, we have A B or B A. Then, it is well known from Order Theory that we can construct other binary relations induced by on S. At first, we construct on S the relation which is the negation of, that is A B (A, B) / S, where S = {(A, B) S 2 : A B}. Next, we construct the relation on S, where A B B A. Obviously this implies that from A B it results that A B. If not, then it easily results that A B and B A which contradicts the fact that is a total binary relation on S. Note that if would not be a total binary relation then the correct definition of would be A B A B (c) 213 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

3 1.119/TFUZZ , IEEE Transactions on Fuzzy Systems BAN AND COROIANU: SIMPLIFYING THE SEARCH FOR EFFECTIVE RANKING OF FUZZY NUMBERS 3 and B A. Furthermore, we construct the relation on S, where A B A B and B A. It is immediate that is an equivalence relation over S. Also, A B A B or A B for all A, B S. Similarly, we can define on S the relations,, and respectively. We do not go into details since their construction is obvious. Having in mind the above considerations, we consider the following basic requirements for a total binary relation on S and for the orders and respectively, generated by on S. A 1 ) A A for any A S. A 2 ) For any (A, B) S 2, from A B and B A results A B. A 3 ) For any (A, B, C) S 3, from A B and B C results A C. A 4 ) For any (A, B) S 2, from inf supp(a) sup supp(b) results A B. A 4) For any (A, B) S 2, from inf supp(a) >sup supp(b) results A B. A 5 ) Let A, B, A+C and B+C be elements of S. If A B, then A + C B + C. A 5) Let A, B, A+C and B+C be elements of S. If A B, then A + C B + C. A 6 ) For any (A, B) S 2 and λ R such that λ A, λ B S, from A B results λ A λ B if λ and λ A λ B if λ. Remark 2: The preliminary assumption together reflexivity A 1 ) and transitivity A 3 ) assure that the binary relation is a total preorder on S. In Order Theory these are the minimal conditions for comparing the objects in S. On the other hand we observe that A 2 is nothing else but the definition of. Therefore, it is questionable whether property A 2 should be listed since it holds trivially. One argument would be if and respectively, were not introduced according to Order Theory but rather according to the necessity of a specific application. However in all the ranking approaches that we know from the literature, the relations and respectively, always agree with their standard definition according to Order Theory. Remark 3: All the above reasonable properties are derived from the list of Wang and Kerre. Some of them are modified according to the new trends concerning the ranking of fuzzy numbers. They will be mentioned in the following remarks. The only property from the list of Wang and Kerre which has no corresponding property in the above list is the property denoted with A 5 in the paper of Wang and Kerre. This property is needed in the case when the ordering approach is variable depending on the finite subsets of S and basically it states that considering two finite subsets of S the ordering approach gives the same result on their intersection. We can adapt it to our case by considering on any subset A (finite or not) of S the restriction of to A which practically does not change anything when it comes to compare two elements of S and hence property A 5 from the original list of Wang and Kerre holds trivially. Remark 4: Property A 6 replaces in some sense the stronger corresponding property from the same paper of Wang and Kerre ([54]), which in our setting is equivalent with Let A, B, A C and B C be elements of S and C. If A B then A C B C, where C means that the support of the fuzzy number C is included in [, ). However, we prefer A 6 in the present form because in this case from A B it results that A B, a property which is considered very important in many papers (see e.g. [3], [6], [31]) and its lack produces shortcomings. Another reason why we consider A 6 in this form is that if S coincides with the set of trapezoidal (or triangular) fuzzy numbers and A, B are trapezoidal (or triangular) fuzzy numbers then A B may fail to be a trapezoidal (triangular) fuzzy number. In most of the cases, ranking fuzzy numbers assumes to associate for each fuzzy number (or, more generally, fuzzy quantity) a real number and so fuzzy numbers are ranked through these real values. In this way, if S is a subset of F (R) then a ranking index P : S R generates on S a ranking, denoted by P, where A P B if and only if P (A) P (B). Obviously, P is a total preorder on S, therefore it satisfies A 1, A 2 and A 3. Now, constructing the rankings P, P, P and P respectively, by the standard procedure presented earlier, we have: A P B if and only if P (A) > P (B), A P B if and only if P (A) < P (B), A P B if and only if P (A) = P (B), A P B if and only if P (A) P (B). The ranking indices were often introduced without a clear justification and without satisfying a minimal set of conditions. Immediate shortcomings were found for most of them and it is obvious that the consequences of using unsuitable rankings are not only theoretical. We give here only the following example, other discussions will be included later. Example 5: The parametrized ranking index M r, r >, introduced in [3] becomes M r (T ) = ( 2(t r+2 1 t r+2 2 )(t 3 t 4) 2(t r+2 3 t r+2 (r+2)(r+1)(t 1 t 2)(t 3 t 4)(t 1+t 2 t 3 t 4) for T = (t 1, t 2, t 3, t 4 ). We have M 2 ( 1, 2, 1, ) = ( 1247 > ( 2 3 ) = M 2 (1, 2, 3, 4), 4 )(t 1 t 2) ) 1 2 ) 1 r therefore ( 1, 2, 1, ) M2 (1, 2, 3, 4), an obvious contradiction with our intuition. Below we propose some natural requirements for the case when the order is generated by a ranking index. Remark 6: In the present paper we assume that R S and P R : R R is continuous, for any ranking index P : S R. The first assumption is absolutely natural. The second one is not at all restrictive since one can easily observe that it is satisfied for most ranking indices (e.g., those introduced in [1]- [3], [7], [19]-[21], [25], [26], [41], [46], [47], [5]). Actually, we could not find an approach that uses an index having discontinuities on R. This would also look quite unnatural. Remark 7: (i) The requirement A 4 seems stronger than its (c) 213 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

4 1.119/TFUZZ , IEEE Transactions on Fuzzy Systems 4 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL., NO., correspondent in [54], which adapted to our setting becomes For any (A, B) S 2, from inf supp(a) > sup supp(b) (6) results A B. If the ranking is generated by a ranking index then they are equivalent. Indeed, let us consider (A, B) S 2 such that inf supp(a) = sup supp(b). We denote a = inf supp(a). Then a 1 n < inf supp(a), for every n N, and from (6) we have A P a 1 n, for every n N. This implies P (A) P ( a n) 1, for every n N, therefore (from the continuity of P on R, see Remark 6) we obtain P (A) P (a). On the other hand, we have a + 1 n > sup supp(b) for every n N and, in the same way, we obtain P (a) P (B). As a conclusion, P (A) P (B), that is A P B. (ii) As just above or taking into account Proposition 3.1 in [54] we obtain that A 4 is stronger than A 4, for any ordering generated by a ranking index. Definition 8: Two orderings 1 and 2 on the set S are said to be equivalent if, for any A, B S, from A 1 B it results A 2 B and from A 1 B it results A 2 B. If 1 and 2 are total binary relations (for example if they are generated by ranking indices) then the second requirement in the above definition is equivalent with the requirement that from A 1 B it results A 2 B. It is immediate that if 1 has a property A {A 4,A 4,A 5,A 5,A 6 } and 2 is equivalent with 1 then 2 has the property A too. In some recent papers (see, e.g., [3], [7], [46] ) the following requirement is considered as an important reasonable property for a ranking index P : S R. A 4) For any A S, P (A) supp(a). Such a ranking index satisfies in fact a basic requirement of a defuzzifier (see [38], [48] ). It is not a surprise, some wellknown defuzzifiers are considered as ranking indices too, the best example being the expected value (see (3)). For any order over S which is generated by a ranking index P : S R, if A 4 holds then A 4 (and implicitly A 4 ) holds. We notice that the converse implication does not hold in general. Indeed, by taking S = F (R) and P : S R, P (A) = (A L ()) 3, then we observe that A 4 holds while A 4 does not hold in general. On the other hand, in some sense we can say that it suffices to study only ranking indices for which A 4 holds. This is certified by the following theorem. Theorem 9: If P : S R is a ranking index on S such that P satisfies A 4 then there exists a ranking index R : S R which satisfies A 4 and R is equivalent with P. Proof. Let us choose arbitrarily A S and suppose that supp(a) = [a, b]. Since P satisfies A 4 (and implicitly A 4 ) it results that P (a) P (A) P (b). The continuity of P R (see Remark 6) implies that there exists x A [a, b] such that P (x A ) = P (A). In addition, x A is unique with this property. Indeed, for any x [a, b], x x A, either P (x A ) < P (x) or P (x A ) > P (x) because otherwise requirement A 4 is violated. Therefore, we can define the ranking index R : S R, R(A) = x A which satisfies requirement A 4. The equivalence between P and R is immediate by the construction of R. Remark 1: Keeping the hypothesis and notations in the above theorem, let us introduce P R : R P (R), P R (x) = P R (x) = P (x) (basically P R and P R represent the same function if we make abstraction of their domains of values). Since P satisfies A 4, it is immediate that P R is strictly increasing and hence invertible. Moreover for any A S we have R(A) = x A = P 1 R (P R(x A )) = P 1 (P (x A)) = P 1 (P (A)), R R that is R = P 1 1 R P. In addition, PR is strictly increasing and continuous. Remark 11: Requirements A 4 and A 4 are natural for an ordering between fuzzy numbers, their absence leads to shortcomings and any subsequent discussion could be stopped. Therefore (taking into account Remark 6 and Theorem 9) we can focus on ranking indices which satisfy A 4. Example 12: Applying Theorem 9 to the above index P : F (R) R, P (A) = (A L ()) 3 we obtain R : F (R) R, R(A) = A L (), where R satisfies A 4 and R is equivalent with P. Below we apply Theorem 9 for two known ranking indices. Example 13: A ranking index based on centroids of fuzzy numbers was introduced in [21]. It reduces to P Chu T sao (A) = (a+b+c)(a+4b+c) 9(a+2b+c), for every A = (a, b, c) F (R), satisfies A 4 on S = F (R) and P Chu T sao R : R R is continuous. From P Chu T sao (x A ) = P Chu T sao (A) we obtain x A = 2P Chu T sao (A). According with Theorem 9 the ranking PChu T sao is equivalent with RChu T sao, where R Chu T sao : F (R) R given by R Chu T sao (A) = 2P Chu T sao (A) satisfies A 4. Example 14: The ranking index introduced in [2] was intensely cited (e.g. [3], [7], [31], [32], [54], etc.). It is introduced in a more general framework, but becomes P Cho Li (A) ( 1 = 2(M m) A L (α)dα + ) A U (α)dα 2m, for every A F (R), where m, M R, M m, reflect the attitude (of the decision maker) towards risk. It is immediate that PCho Li satisfies A 4 and P Cho Li R is continuous. According with Theorem 9 and its proof, R Cho Li (A) = 1 2 A L (α)dα A U (α)dα satisfies A 4 and RCho Li is equivalent with PCho Li. But R Cho Li = EV, therefore (at least for fuzzy numbers in the sense of the present paper) we obtain the same ranking as the one generated by the expected value, often used in applications (see, e.g., [11], [22]). Table 1 in [54] offers a list of other ranking indices for which Theorem 9 is applicable (c) 213 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

5 1.119/TFUZZ , IEEE Transactions on Fuzzy Systems BAN AND COROIANU: SIMPLIFYING THE SEARCH FOR EFFECTIVE RANKING OF FUZZY NUMBERS 5 IV. CHARACTERIZATION OF EFFECTIVE RANKING INDICES In what follows we provide important properties of a ranking index starting from some reasonable properties fulfilled by the generated order. At the end of this section we characterize the valuable ranking indices (on a set S F (R) satisfying suitable properties) with respect to A 1, A 2, A 3, A 4, A 4, A 5, A 5, A 6 and A 4. Theorem 15: Let S F (R) such that S + S S and R : S R a ranking index which satisfies A 4. If R satisfies A 5 on S then R is additive on S and in addition R satisfies A 5 on S. Proof. First of all, let us observe again that R satisfies A 2 and A 3 on S since R is generated by a ranking index. Since R satisfies A 4 on S it easily results that R(R(A)) = R(A), for all A S, which implies that A R R(A), for all A S. Let us now choose arbitrarily A, B S. We have A R R(A) which by A 2 and A 5 implies that A R(A) R and then again by A 2 and A 5 we get that A + B R(A) R(B) R B R(B) and since B R(B) R, by A 2 and A 3 we get that A + B R(A) R(B) R and applying again A 2 and A 5 we obtain A + B R R(A) + R(B). This implies R(A + B) = R(R(A) + R(B)) and since by A 4 clearly we have R(R(A) + R(B)) = R(A) + R(B) it results that R(A + B) = R(A) + R(B). Now, since R is additive on S it is immediate that A 5 is satisfied by R on S and this ends the proof. Sometimes it is difficult to prove that A 5 is satisfied or not for a ranking index. The above result helps us to obtain negative results from the non-additivity of the restrictions of the ranking indices to suitable families of fuzzy numbers. We exemplify this by the following. Example 16: The restriction of the ranking index introduced in [4] (see also [49]) to trapezoidal fuzzy numbers becomes R Lee Li : F T (R) R given by R Lee Li (A) = a2 b 2 +c 2 +d 2 ab+cd 3( a b+c+d), for every A = (a, b, c, d) F T (R). The property A 4 is verified. Because R Lee Li (1, 2, 3, 4) + R Lee Li (2, 3, 4, 6) = = R Lee Li (3, 5, 7, 1) = R Lee Li ((1, 2, 3, 4) + (2, 3, 4, 6)) from Theorem 15 we obtain that RLee Li does not satisfy A 5 on F T (R) and implicitly on F (R). Analyzing carefully the proof of the previous theorem, we obtain the following equivalence. Corollary 17: Let S F (R) such that S + S S and R : S R a ranking index which satisfies A 4. The order R on S satisfies A 5 on S if and only if it satisfies A 5 on S. Proof. In view of the previous theorem we need to prove only the converse implication. But this is immediate since we observe that the reasoning in the proof of Theorem 15 is not influenced at all if instead of A 5 we use everywhere A 5. For example, from A R R(A) it is said in the proof of Theorem 15 that by A 2 and A 5 we get that A R(A) R. But this holds too if instead of A 5 we use A 5. Suppose by way of contradiction that A R(A) R does not hold. Then either A R(A) R or A R(A) R. In the first case, by A 5 we obtain A R R(A) and this obviously is a contradiction. Similarly, we obtain a contradiction in the second case too. Thus, we must have A R(A) R. Similarly, everywhere in the proof of Theorem 15 we can replace the hypothesis that R satisfies A 5 on S with the hypothesis that R satisfies A 5 on S. Corollary 18: Let S F (R) such that S + S S. If R : S R is a ranking index so that A 4 and A 5 are satisfied by R on S, then A 5 is satisfied too by R on S. In addition, there exists an additive ranking index R : S R which satisfies A 4 on S and R is equivalent with R. Proof. Since R satisfies A 4, by Theorem 9 there exists R : S R which satisfies A 4 and such that the order R is equivalent with R. Therefore, noting the hypothesis we get that A 5 is satisfied by R on S. As R satisfies all the hypothesis from Theorem 15 (by taking R := R there) it results that R is additive and that R satisfies A 5 on S. Since R is equivalent with R, it easily results now that R satisfies A 5 on S and hence the proof is complete. Example 19: With the notations in Corollary 18, if R = P Cho Li in Example 14 then R = EV. Theorem 2: Let S F (R) such that λ S S for all λ R and suppose that R : S R is a ranking index which satisfies A 4. If R satisfies A 6 on S then R is scale invariant, that is R(λ A) = λr(a), for every λ R and A S. Proof. Again, we begin the proof by noticing that A 1, A 2 and A 3 are satisfied by R on S. Now, let us choose arbitrarily A S and λ R. We have A R R(A) which by A 2 and A 6 implies that λ A R λr(a). Therefore, we obtain R(λ A) = R(λR(A)) and since obviously R(λR(A)) = λr(a) we get R(λ A) = λr(a) and the proof is complete. Theorem 2 furnish us a method to prove that A 6 is not satisfied. It is useful especially for sophisticated ranking indices. We choose a suitable family of fuzzy numbers on which the ranking index is not scale invariant. We immediately obtain that the generated order does not satisfy A 6. We exemplify the method below. Example 21: The restriction of the ranking index introduced in [41] to triangular fuzzy numbers becomes R θ Liou W ang : F (R) R given by R θ Liou W ang (A) = 1 θ 2 a b + θ 2 c, for every A = (a, b, c) F (R), where θ reflects the decision maker s optimistic or pessimistic attitude. Because R θ Liou W ang (, 1, 2) = θ 1 2 θ 3 2 = R θ Liou W ang ( 2, 1, ) = R θ Liou W ang ( (, 1, 2)) for every θ 1 2, from Theorem 2 we obtain that R θ Liou W ang does not satisfy A 6 on F (R) and implicitly on F (R). Corollary 22: Let S F (R) such that λ S S for all λ R and suppose that R : S R is a ranking index. If R satisfies A 4 and A 6 on S, then there exists a scale invariant ranking index R : S R which satisfies A 4 on S and which generates on S an equivalent order with R. Proof. The proof easily follows by combining Theorems 9 and (c) 213 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

6 1.119/TFUZZ , IEEE Transactions on Fuzzy Systems 6 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL., NO., Remark 23: The idea in Remark 1 is applicable for the ranking indices in Corollaries 18 and 22. Obviously, an effective ranking between fuzzy numbers from a set S should satisfy the requirements from the beginning of Section III. For this reason, let us denote M(S) = {P : S R P satisfies A 1, A 2, A 3, A 4, A 4, A 5, A 5, A 6 }. Then, inspired by Theorem 9, we consider M (S) = {P : S R P satisfies A 4 and P satisfies A 1, A 2, A 3, A 5, A 5, A 6 }. From the above considerations it results that in general M (S) M(S). But from Theorem 9 it also results that if P M(S) then there exists P M (S) such that P is equivalent with P on S. Therefore, in order to find effective orders over S it suffices to study the elements from M (S). This might just simplify the procedure since A 4 should simplify the calculations part. This clearly is the case when S = F T (R) as it will be seen in the next section. But before that, we conclude this section with some useful results in which we can characterize the elements from the classes M(S) and M (S), respectively. Theorem 24: Let S F (R) such that S +S S and λ S S for all λ R and P : S R a ranking index. Then we have: (i) P M (S) if and only if P satisfies A 4 on S and P is linear on S; (ii) P M(S) if and only if there exists P M (S) such that P and P are equivalent on S. Proof. (i) We prove only the direct implication since the converse one results by simple verifications. Since P M (S) it is immediate that A 4 holds. This, together with the fact that the hypothesis implies that obviously A 5 and A 6 hold, all these imply that the hypothesis of Theorems 15 and 2, respectively, are fulfilled and applying the conclusions of these theorems we get the linearity of P. (ii) At first, we prove the direct implication. The existence of P is guaranteed by Theorem 9 as this theorem clearly is applicable in our case. Since P and P are equivalent on S and since P M(S), it easily results that P M (S). To prove the converse implication it suffices to notice that obviously by P M (S) it results P M(S) and the desired conclusion easily results taking into account that P and P are equivalent on S. V. CHARACTERIZATION OF VALUABLE RANKING INDICES ON TRAPEZOIDAL FUZZY NUMBERS As stated in the title, in this section we determine M (F T (R)) and hence, making abstraction of equivalent orders over F T (R), we find all ranking indices P : F T (R) R with the property that P satisfies the reasonable properties A 1, A 2, A 3, A 4, A 4, A 5, A 5 and A 6 on F T (R). There are several reasons why one should investigate on orders over the set F T (R). First of all, since trapezoidal fuzzy numbers have a simple representation and they are so often used in applications, we should investigate as much as possible this family of fuzzy numbers. This is why if someone chooses randomly a paper that studies the ranking of fuzzy numbers, almost all the numerical examples are performed on trapezoidal or triangular fuzzy numbers. There are papers (see e.g. [3], [32]) which study only the ranking of trapezoidal or triangular fuzzy numbers. Then, as it will be seen in Section VI, we can extend orders defined on F T (R) to orders defined on F (R) so that the basic requirements are preserved. The idea (although the approach is different comparing with the construction from the final section) to rank fuzzy numbers through trapezoidal fuzzy numbers can be found in the paper [9], where fuzzy numbers are ranked using their trapezoidal approximations preserving the ambiguity and value. Actually, the topic of the approximation of fuzzy numbers by fuzzy numbers with simpler form (triangular, trapezoidal, semi-trapezoidal, etc.) and the topic of the ranking of fuzzy numbers with simpler form are quite related since the purpose is to simplify as much as possible the handling of the information which appears as a fuzzy number. In this section, and in the followings too, we use a different notation for trapezoidal fuzzy numbers which seems a more suitable representation in the obtaining of the main results. Namely, we consider the α-cut of a trapezoidal fuzzy number T in the form (see e.g. [3]) T α = [x σ + σα, y + β βα], α [, 1], (7) where x, y, σ, β R, σ, β and x y and we denote T = [x, y, σ, β]. Comparing to the classical representation of T, that is T = (t 1, t 2, t 3, t 4 ), it is immediate that t 1 = x σ, (8) t 2 = x, (9) t 3 = y, (1) t 4 = y + β. (11) According with (3)-(5), after some simple calculations, we get the expected value, ambiguity and value respectively, of a trapezoidal fuzzy number T = [x, y, σ, β] as EV (T ) = 1 2 x y 1 4 σ + 1 4β, (12) Amb(T ) = 1 2 x y σ + 1 6β, (13) V al(t ) = 1 2 x y 1 6 σ + 1 6β. (14) For a trapezoidal fuzzy number T = [x, y, σ, β] let us consider the quantity R(T ) = ax + by + cσ + dβ (15) where a, b, c, d R are fixed. It is immediate that the function R : F T (R) R is additive and positively homogenous. Let us introduce the set Ω = { R : F T (R) R a, b, c, d R such that R (x, y, σ, β) = ax + by + cσ + dβ}. Later we will prove that M (F T (R)) Ω which, by the relation between M (F T (R)) and M(F T (R)), justify a detailed study of the set Ω. We do that by finding necessary (c) 213 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

7 1.119/TFUZZ , IEEE Transactions on Fuzzy Systems BAN AND COROIANU: SIMPLIFYING THE SEARCH FOR EFFECTIVE RANKING OF FUZZY NUMBERS 7 and sufficient conditions for a, b, c, d such that R Ω can be used to rank effectively trapezoidal fuzzy numbers. We have already mentioned in the previous section that requirements A 1, A 2 and A 3 hold for R on F T (R) since R is generated by a ranking index. Then, since one can easily prove that R(T + S) = R(T ) + R(S) for all R Ω and T, S F T (R), it results that properties A 5 and A 5 hold too. Therefore, it remains to find necessary and sufficient conditions such that properties A 4, A 4, A 4 and A 6 would hold. Theorem 25: Let R Ω, R(x, y, σ, β) = ax +by +cσ+ dβ. The order R satisfies A 4 if and only if and a (16) b (17) c (18) d (19) a + b + c (2) a + b d. (21) Proof. ( ) We suppose that A 4 holds. We consider particular cases for T = [x, y, σ, β] and T = [x, y, σ, β ] such that inf supp(t ) sup supp(t ) is satisfied, until we obtain that conditions (16)-(21) are satisfied. Note that since we have supposed that A 4 holds it results that T R T and therefore R(T ) R(T ). Firstly, let us consider the particular case when x = y = y > x > and σ = β = σ = β =. Since R(T ) R(T ) implies ax ax and since x > x >, it is immediate that a. Now, let us consider the particular case when x = σ = β = x = y = σ = β = and y >. Since R(T ) R(T ) implies by and since y >, it is immediate that b. Consider now the case when x = y = σ = x = y = σ = β = and β >. Since R(T ) R(T ) implies dβ and since β >, we obtain d. Consider now the case when x = y = σ = β = x = y = β = and σ >. R(T ) R(T ) implies cσ and since σ >, we obtain c. Now, we consider the case when x = y = σ = 1 and β = x = y = σ = β =. It is immediate that from R(T ) R(T ) we get a + b + c. Finally, we consider the case when x = y = β = 1, σ = β = σ =, x = y =. It is immediate that from R(T ) R(T ) we get a + b d. Collecting the inequalities obtained in the particular cases from above, we obtain that (16)-(21) hold. ( ) Let a, b, c, d be real numbers satisfying (16)-(21) and let T = [x, y, σ, β], T = [x, y, σ, β ] denote two arbitrary trapezoidal fuzzy numbers such that inf supp(t ) sup supp(t ). This immediately implies that x x σ+β and y y σ + β. Noting that the hypothesis imply cσ and dβ, by direct calculations we get R(T ) R(T ) = a(x x ) + b(y y ) + c(σ σ ) + d(β β ) a(σ + β ) + b(σ + β ) + cσ dβ = σ(a + b + c) + β (a + b d). This implies T R T and the theorem is proved. Theorem 26: Let R Ω, R(x, y, σ, β) = ax +by +cσ+ dβ. The order R satisfies A 4 if and only if and a (22) b (23) a + b > (24) c (25) d (26) a + b c (27) a + b d. (28) Proof. ( ) Since A 4 holds, by the previous theorem, it suffices to prove that a+b >. For this purpose let us consider T = [x, y, σ, β] and T = [x, y, σ, β ] in the particular case when x = y = 1 and σ = β = x = y = σ = β =. Clearly we have inf supp(t ) >sup supp(t ), therefore R(T ) > R(T ), that is a + b >. ( ) Let a, b, c, d be real numbers such that (22)-(28) are satisfied and let T = [x, y, σ, β], T = [x, y, σ, β ] denote two arbitrary trapezoidal fuzzy numbers such that inf supp(t ) >sup supp(t ). Then it is easy to check that x x > σ + β and y y > σ + β. Then, from (22)-(24) combined with the previous two inequalities we obtain a(x x ) + b(y y ) > a(σ + β ) + b(σ + β ) and this implies R(T ) R(T ) = a(x x ) + b(y y ) + c(σ σ ) + d(β β ) > a(σ + β ) + b(σ + β ) + c(σ σ ) + d(β β ) a(σ + β ) + b(σ + β ) + cσ dβ = σ(a + b + c) + β (a + b d). Therefore, we obtain R(T ) > R(T ), that is T R T and the proof is complete. Example 27: Delgado, Vila and Voxman [25] considered that any comparison procedure ought to take into account the magnitude assessment as well as the imprecision involved in any fuzzy number, therefore value and ambiguity appear to be good parameters to be used together for these purposes. They proposed the following approach, by introducing ri(λ, δ) (A) = λv al (A) + δamb (A), for every fuzzy number A, where λ [, 1] and δ [ 1, 1] are given. When V al (A) = V al (B) and Amb (A) = Amb (B) two fuzzy numbers A and B should be considered to be (c) 213 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

8 1.119/TFUZZ , IEEE Transactions on Fuzzy Systems 8 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL., NO., equal in their opinion. The comparison procedure should be mainly determined by the value, therefore δ λ, where means much higher, must be imposed and δ represents the decision-maker s attitude against the uncertainty. Now, if T = [x, y, σ, β] is a trapezoidal fuzzy number, by simple calculations (see (13), (14)) we get ri(λ, δ) (T ) = λ δ 2 x + λ+δ 2 y + δ λ 6 σ + λ+δ 6 β and it is immediate that ri Ω for all choices of λ and δ. From Theorem 26 we obtain ri(λ,δ) satisfies A 4, for every λ and δ. Theorem 28: The ranking index R Ω, R(x, y, σ, β) = ax + by + cσ + dβ satisfies A 4 if and only if and a (29) b (3) a + b = 1 (31) c [ 1, ] (32) d [, 1]. (33) Proof. ( ) Firstly, let us notice that if A 4 holds then it is immediate that A 4 holds too. Therefore, comparing (22)-(28) and (29)-(33), it suffices to prove (31)-(33). Since we have supposed that A 4 holds, it results that for any trapezoidal fuzzy number T = [x, y, σ, β], we have x σ ax + by + cσ + dβ y + β. (34) Take x = y = 1 and σ = β =. Replacing in (34) we get 1 a + b 1 and thus we obtain (31). Now, if we take x = y = β =, σ = 1, then replacing in (34) we get 1 c which implies that (32) holds. Finally, take x = y = σ =, β = 1. Then d 1 and it follows that (33) holds. ( ) Let a, b, c, d be real numbers such that (29)-(33) are satisfied and let T = [x, y, σ, β] denote a trapezoidal fuzzy number. Conditions (29)-(31) and (33) imply R(T ) = ax + by + cσ + dβ ax + by + dβ (a + b)y + β = y + β. On the other hand, (29)-(32) imply R(T ) = ax + by + cσ + dβ (a + b)x + cσ (a + b)x σ = x σ, and the theorem is proved. Example 29: The ranking index ri (λ, δ) in Example 27 satisfies A 4 if and only if λ = 1. Theorem 3: Let R Ω, R(x, y, σ, β) = ax +by +cσ+ dβ. The order R satisfies A 6 if and only if and a = b (35) c + d =. (36) Proof. ( ) Let us suppose that A 6 holds. Let us choose T = [ 1, 1,, ] and O = [,,, ]. If R(T ) R(O) then the hypothesis imply R( T ) R(O). Since R(O) = and T = T, it immediately follows that R(T ) = that is a + b = and we get a = b. If R(T ) R(O) then the reasoning is similar therefore we omit the details. Now, let us choose T = [,, 1, 1]. Again, it is easy to check that T = T and reasoning as above we get R(T ) = that is c + d =. ( ) Let us consider the reals a, b, c, d such that (35)-(36) hold. So, if T = [x, y, σ, β] then R(T ) = ax + ay dσ + dβ. To prove that A 6 holds it suffices to prove that the operator R is scale invariant. Now, if λ then one can easily prove that R(λ T ) = λr(t ). On the other hand, if λ < then λ T = [λy, λx, λβ, λσ]. Then, by direct calculations we obtain R(λ T ) = λay + λax + λdβ λdσ = λ(ax + ay dσ + dβ) = λr(t ). We thus obtained that R is scale invariant which proves the converse implication and the theorem. Example 31: The ordering ri(λ,δ) in Example 27 satisfies A 6 if and only if δ =. Example 32: A method of ranking fuzzy numbers with integral value was proposed in [41]. The ranking index reduces in the case of trapezoidal fuzzy numbers to I k Liou W ang ([x, y, σ, β]) = (1 k) x + ky + k 1 2 σ + k 2 β, where k [, 1] represents the degree of optimism of a decision maker. According with Theorem 3 the order generated by ILiou W k ang satisfies A 6 if and only if k = 1 2. If k 1 2 then the validity and superiority of this method in comparison with other methods, as it was proved by examples in [41] is at least debatable. In fact, it is easy to find a shortcoming of the method by choosing A = [1, 1, 1, 1], B = [ 6 5, 6 5, 2, 2] and the degree of optimism k = 3 4. Indeed, I 3 4 Liou W ang (A) = 5 4 < 17 1 = I 3 4 Liou W ang (B), therefore B 3 I 4Liou W A. On the other hand, because ang A = [ 1, 1, 1, 1] and B = [ 6 5, 6 5, 2, 2] our expectation is that A 3 I 4Liou W B. Nevertheless, ang I 3 4 Liou W ang ( A) = 3 4 < 7 1 = I 3 4 Liou W ang ( B), that is B 3 I 4Liou W A. ang The following important theorem, an immediate consequence of the theoretical results obtained so far in this section, characterizes the element of Ω which are in M (F T (R)). Theorem 33: Let R Ω, R(T ) = ax + by + cσ + dβ for T = [x, y, σ, β]. Then R M (F T (R)) if and only if a = b = 1 2 (37) c [ 1, ] (38) (c) 213 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

9 1.119/TFUZZ , IEEE Transactions on Fuzzy Systems BAN AND COROIANU: SIMPLIFYING THE SEARCH FOR EFFECTIVE RANKING OF FUZZY NUMBERS 9 and c + d =. (39) Proof. The proof is immediate taking into account that (16)- (21), (29)-(33) and (35)-(36) have to be satisfied simultaneously. Example 34: ri(λ, δ) M(F T (R)) and hence ri(λ,δ) (see Example 27) satisfies A 1, A 2, A 3, A 4, A 4, A 5, A 5 and A 6 if and only if δ =. If λ < 1 then ri(λ, δ) M(F T (R)) \ M (F T (R)). Moreover, if λ = 1 then ri(λ, δ) = ri(1, ) = V al M (F T (R)), a ranking index proposed for the first time in [33]. By Theorem 33 it is easy to deduce that some already introduced ranking indices are elements of M (F T (R)) and implicitly they satisfy A 1, A 2, A 3, A 4, A 4, A 5, A 5 and A 6. Example 35: Let us consider EV : F T (R) R, which for any trapezoidal fuzzy number T = [x, y, σ, β] associates its expected value that is EV (T ) = 1 2 x y 1 4 σ + 1 4β. It is immediate that EV M (F T (R)). It seems that Yager ([5]) has been the first who proposed to rank fuzzy numbers through their expected values. This procedure was considered also more recently in the paper [7]. Example 36: In the paper [3], the authors considered the magnitude of a trapezoidal fuzzy number, namely the function Mag f : F T (R) R, ( ) Mag f (T ) = 1 2 (T L (α) + T U (α) + x + y )f(α)dα where T = [x, y, σ, β] is an arbitrary trapezoidal fuzzy number and f is a nonnegative and nondecreasing function 1 on [, 1] with f() =, f(1) = 1 and f(α)dα = 1/2. Since by simple calculations we get Mag f (T ) = 1 2 x y + σ 2 + β 2 f(α)(α 1)dα (4) f(α)(1 α)dα, (41) one can easily obtain (see Theorem 33) that Mag f M (F T (R)). In the paper [3], the authors dealt with the particular case f(α) = α when Mag(T ) = 1 2 x y 1 12 σ β. Example 37: The restriction to F T (R) of the ranking index introduced in [46] becomes M f,f ([x, y, σ, β]) = 1 2 x y + σ 2 + β 2 f(α)(1 α)dα, f(α)(α 1)dα where the weighting functions f, f : [, 1] R are non-negative, monotone increasing such that f(α)dα = 1 f(α)dα = 1. It is immediate M f,f M (F T (R)). Example 38: In [26] a ranking procedure is proposed via so called valuation functions. The authors consider a strictly monotonous function (valuation) f : [, 1] [, ) and the ranking index R D Y : F T (R) R by R D Y ([x, y, σ, β]) = 2 ω 2 x + 2 ω 2 y 1 ω 2 σ + 1 ω 2 β, where ω = αf(α)dα 1. Since < ω < 1 for any valuation f(α)dα f, by Theorems 26 and 3 respectively, we easily observe that R D Y generates an order which satisfies A 4 and A 6 and hence R D Y M(F T (R)), for any valuation f. On the other hand, by Theorem 28, we observe that R D Y / M (F T (R)). It is easily seen that (R D Y ) M (F T (R)), where (R D Y ) = 1 2 ω R D Y and (RD Y ) is equivalent with RD Y (see Theorem 9). Having in mind the result from above, it would be important to know whether there exists any other ranking index R M (F T (R)) which does not belong to Ω. The answer to this question is negative and hence we can prove now the main result of this section. Theorem 39: Let us consider a ranking index R : F T (R) R. Then R M (F T (R)) if and only if there exists c [ 1, ] such that for some T F T (R), T = [x, y, σ, β], we have R(T ) = 1 2 x y + cσ cβ. (42) Proof. Taking into account Theorem 33, it is easily seen that we can obtain the desired conclusion by proving that R M (F T (R)) implies R Ω. Firstly, let us observe that from Theorem 24, (i) it results that R is linear on F T (R). We continue with a constructive proof although we could use the Riesz representation theorem for linear functionals. Let T F T (R), T = (t 1, t 2, t 3, t 4 ), in the usual representation proposed in Section II, because it is more suitable for this proof. Now, let us consider the trapezoidal fuzzy numbers v 1 = (,,, 1), v 2 = (,, 1, 1), v 3 = (, 1, 1, 1) and v 4 = (1, 1, 1, 1). Having in mind the addition and the scalar multiplication of fuzzy numbers we get that T = t 1 v 4 + (t 2 t 1 )v 3 + (t 3 t 2 )v 2 + (t 4 t 3 )v 1. The linearity of R implies R(T ) = t 1 R(v 4 ) + (t 2 t 1 )R(v 3 ) + (t 3 t 2 )R(v 2 ) + (t 4 t 3 )R(v 1 ). Returning now to the other parametric representation of T, that is T = [x, y, σ, β] and taking into account (8)-(11), we obtain R(T ) = x (R(v 4 ) R(v 2 ))+y R(v 2 )+σ(r(v 3 ) R(v 4 ))+βr(v 1 ). Clearly, this last relation implies that R Ω and the proof is complete. In what follows, we characterize classes of ranking indices over F T (R) which generate orders satisfying entirely or just a part of the basic requirements on F T (R). Moreover, by Theorem 9 too, we can simplify the searching of such orders using equivalent orders that satisfy requirement A 4 on F T (R). Corollary 4: (i) If R M(F T (R)) then there exists R M (F T (R)) such that R and R are equivalent. In addition (by Theorem 39), there exists c [ 1, ] such that for some T = [x, y, σ, β] F T (R) we have R (T ) = 1 2 x y + cσ cβ. (ii) If R : F T (R) R is a ranking index such that R satisfies A 4 and A 5 then R satisfies A 5 too and moreover there exists an additive ranking index R + : F T (R) R, which satisfies A 4 on F T (R) and which generates on F T (R) an equivalent order with R (c) 213 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

10 1.119/TFUZZ , IEEE Transactions on Fuzzy Systems 1 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL., NO., (iii) If R : F T (R) R is a ranking index such that R satisfies A 4 and A 6 then there exists a scale invariant ranking index R : F T (R) R, which satisfies A 4 on F T (R) and which generates on F T (R) an equivalent order with R. Proof. (i) From the proof of Theorem 9 it results the existence of R : F T (R) R which satisfies A 4 and such that the order R on F T (R) is equivalent with R. All these together easily imply that R M (F T (R)). The rest of the proof follows immediately from Theorem 39. (ii) and (iii) The proofs are immediate by Corollaries 18 and 22 respectively, by taking there S = F T (R). The following example confirms the result in Corollary 4, (i). Example 41: In the very recent paper [47] a new ranking index was proposed. It becomes Q µ S D ([x, y, σ, β]) = (1 µ) x + (1 µ) y (1 µ)2 2 σ + (1 µ)2 2 β, for every trapezoidal fuzzy number T = [x, y, σ, β], where µ [, 1] is a decision level. According with Theorem 33, Q µ S D M (F T (R)) if and only if µ = 1 2. It is interesting to note that Q µ S D M(F T (R)) for every µ < 1, therefore (see Theorem 9) there exists a ranking index ( Q µ ) S D defined by ( Q µ S D ) ([x, y, σ, β]) = 1 2 x y 1 µ 4 σ + 1 µ 4 β, such that ( Q µ ) S D M (F T (R)) and the orders generated by ( Q µ ) S D and Qµ S D are equivalent. Remark 42: In [3] the authors suggest two other desirable properties for a ranking index R on trapezoidal fuzzy numbers: (i) If T is a symmetric trapezoidal fuzzy number with respect to the origin, that is T = [ x, x, σ, σ], then R(T ) =. (ii) If T 1 is symmetric with respect to T 2 (or, equivalently, T 2 is symmetric with respect to T 1 ), that is T 1 = [x, y, σ, σ] and T 2 = [x, y, β, β], then R(T 1 ) = R(T 2 ). It is immediate that both properties hold for any R M (F T (R)). Remark 43: As we can see from Theorem 39, the elements of M (F T (R)) depend on only one parameter. It would be interesting if, depending on a specific application, we could add a new basic requirement so that the interpretation would be meaningful. VI. RANKING FUZZY NUMBERS THROUGH TRAPEZOIDAL FUZZY NUMBERS So far, we found the class M (F T (R)) consisting of all ranking indices that generate orders over the space of trapezoidal fuzzy numbers satisfying requirements A 1, A 2, A 3, A 4, A 5, A 5 and A 6. In what follows, we prove that for any R M (F T (R)) there exists R M (F (R)) such that R(T ) = R(T ) for any trapezoidal fuzzy number T. This means that R is an extension of R on F (R) so that all the desirable properties A 1, A 2, A 3, A 4, A 5, A 5 and A 6 hold on F (R). Theorem 44: Let us consider a trapezoidal valued operator T : F (R) F T (R) and R M (F T (R)). If T is linear and supp (T (A)) supp (A), for every A F (R), then R : F (R) R given by R(A) = R(T (A)) is linear and R M (F (R)). Proof. By Theorem 24, (i) it results that R is linear on F T (R). Thus, R is linear on F (R) since R is the composition of the linear operators R and T. It remains to prove that requirements A 1, A 2, A 3, A 4, A 5, A 5 and A 6 hold on F (R) for R and R. Clearly, A 1, A 2 and A 3 are satisfied. Moreover, since R is linear, we easily obtain that A 5, A 5 and A 6 are satisfied too. Now, ( considering an arbitrary fuzzy number A, since R M F T (R) ), we get that R(T (A)) supp (T (A)). Since supp (T (A)) supp (A) and R(T (A)) = R(A) we get that R(A) supp (A) which proves that A 4 holds for R on F (R). The theorem is proved. Example 45: Let us consider EV : F (R) R the operator which associates to a fuzzy number its expected value (see (3)) and T : F (R) F T (R) given by [ T (A) = A L (α)dα, ] A U (α)dα. Since A L () A L(α)dα A U (α)dα A U (), for every A F (R), it results that supp (T (A)) supp (A). Moreover, considering R = EV in Theorem 44 and R the restriction of EV to F T (R) it is immediate that R(A) = R(T (A)). With the present notations, it is equivalent with EV (A) = EV (T (A)). Since T is linear and EV M (F T (R)) (see Example 35), it results that R, R and T satisfy all the hypothesis of Theorem 44, therefore EV M (F (R)). Even if the expected value EV generates an order over F (R) which satisfies all the basic requirements, there are situations when it is not practical. We refer here to the case when we are dealing with a fuzzy number A for which 1 A L(α)dα + A U (α)dα is difficult to compute. So, it is natural to ask ourselves if it is possible to provide orders over F (R) which are easy to use from computational point of view and which satisfy the basic requirements considered in this paper. Before giving a general result in this sense, in the following example we propose another method to extend an order from F T (R) to F (R). Example 46: If T = (t 1, t 2, t 3, t 4 ) is a trapezoidal fuzzy number then it is known 1 that EV (T ) = 4 (t 1 + t 2 + t 3 + t 4 ) = 1 4 (T L() + T L (1) + T U (1) + T U ()). Let us introduce the operator T : F (R) F T (R) given by T (A) = (A L (), A L (1), A U (1), A U ()). In other words, T (A) is the unique trapezoidal fuzzy number which preserves the support and the core of A F (R). We take R : F (R) R given by R(A) = 1 4 (A L() + A L (1) + A U () + A U (1)) and R the restriction of R on the subset F T (R) that is R(T ) = R(T ) = EV (T ) for all T F T (R). It is easy to check that all the hypothesis in Theorem 44 are satisfied for R, R and T, therefore R M (F (R)). The above proposed operator R has the great advantage that it is very convenient from computational point of view. It can (c) 213 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See