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Concepts of generalized concavity based on aggregation functions
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Fuzzy Sets and Systems 198 (2012) 112 – 127 www.elsevier.com/locate/fss
Concepts of generalized concavity based on aggregation functions夡 Masamichi Kona,∗ , Hiroaki Kuwanob a Graduate School of Science and Technology, Hirosaki University, 3 Bunkyo, Hirosaki, Aomori 036-8561, Japan b Faculty of Business Administration and Information Science, Kanazawa Gakuin University, 10 Sue, Kanazawa, Ishikawa 920-1392, Japan
Received 1 February 2011; received in revised form 22 June 2011; accepted 1 October 2011 Available online 10 October 2011
Abstract In the present paper, the quasiconcavity of membership functions is generalized based on conjunctive aggregation functions, and the properties of the generalized quasiconcavity are investigated. Fuzzy multicriteria and scalar programming problems are then considered, and the properties of Pareto optimal solutions and compromise solutions as well as their relationships are derived. Finally, we discuss the application of the obtained results to facility location problems. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. Keywords: Aggregation function; Quasiconcavity; Fuzzy programming problem
1. Introduction Motivated by extensive applications of quasiconcave functions in economics and optimization, several generalizations of quasiconcavity of functions have been introduced and investigated (see [5] and the references therein). In the present paper, we consider primarily functions from the n-dimensional Euclidean space Rn into the unit interval [0, 1]. Such functions can be regarded as membership functions of fuzzy sets on Rn . Therefore, it is natural to use the terminology of fuzzy theory. Quasiconcavity is defined using the minimum operation (see Definition 8). In [5,7,8], for membership functions, quasiconcavity was generalized by allowing arbitrary triangular norms instead of the minimum operation. In the present paper, we generalize quasiconcavity by allowing arbitrary conjunctive aggregation functions instead of the minimum operation. The generalization of the present study is a further generalization of the generalization reported in [5,7,8]. Aggregation functions are very important for the generalization of operations of fuzzy sets and have been widely investigated (see [1,4,5] and the references therein). On the other hand, for fuzzy multicriteria programming problems and fuzzy scalar programming problems, the properties of their solutions related to generalized quasiconcavity based on triangular norms have been investigated 夡 The present study was supported by a Grant-in-Aid for Scientific Research (No. 22510133) from the Japan Society for the Promotion of Science. ∗ Corresponding author. Tel.: +81 172 39 3538; fax: +81 172 39 3526.
E-mail address: [email protected] (M. Kon). 0165-0114/$ - see front matter Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2011.10.001
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in [5–8]. Fuzzy scalar programming problems are scalarized problems of fuzzy multicriteria programming problems. In the present paper, for fuzzy multicriteria programming problems and fuzzy scalar programming problems, we investigate the properties of their solutions as related to generalized quasiconcavity based on conjunctive aggregation functions. Furthermore, we discuss the application of the obtained results to facility location problems. The remainder of the present paper is organized as follows. In Section 2, we present some definitions and properties of aggregation functions. In Section 3, we generalize the quasiconcavity of membership functions defined on Rn based on conjunctive aggregation functions and investigate the properties of the present generalized quasiconcavity. In Section 4, we consider applications of the present generalized quasiconcavity of membership functions to fuzzy multicriteria programming problems and fuzzy scalar programming problems. In Section 5, we discuss applications of a fuzzy multicriteria programming problem and a fuzzy scalar programming problem to facility location problems. Finally, conclusions are presented in Section 6.
2. Aggregation functions In this section, we investigate the properties of aggregation functions. Aggregation functions are used in order to generalize the quasiconcavity of membership functions. For details of aggregation functions, see [1,4,5,9,10]. Throughout the present paper, N is the set of all natural numbers, m ∈ N is fixed, I = {1, . . . , m}, and Ik = {1, . . . , k}, k ∈ N. For a, b ∈ R ∪ {−∞, ∞}, we set [a, b] = {x ∈ R : a ≤ x ≤ b}, [a, b[= {x ∈ R : a ≤ x < b}, ]a, b] = {x ∈ R : a < x ≤ b}, and ]a, b[= {x ∈ R : a < x < b}. First, aggregation functions are defined. Definition 1 (See, for example, Beliakov et al. [1]). Let k ∈ N and G k : [0, 1]k → [0, 1]. For k = 1, G 1 is called an aggregation function if G 1 (x) = x, x ∈ [0, 1]. For k ≥ 2, G k is called an aggregation function if the following two axioms are satisfied: (G1) if xi , yi ∈ [0, 1], xi ≤ yi , i ∈ Ik , then G k (x1 , . . . , xk ) ≤ G k (y1 , . . . , yk ) (monotonicity), and (G2) G k (0, . . . , 0) = 0 and G k (1, . . . , 1) = 1 (boundary condition). Next, the definitions of the properties of aggregation functions are given. Definition 2 (See, for example, Beliakov et al. [1] except for (iii)). Let k ∈ N, and let G k : [0, 1]k → [0, 1] be an aggregation function. (i) G k is said to be conjunctive if G k (x1 , . . . , xk ) ≤ min{x1 , . . . , xk } for any xi ∈ [0, 1], i ∈ Ik . (ii) G k is said to be strongly monotone increasing if xi , yi ∈ [0, 1], xi ≤ yi , i ∈ Ik and x j < y j for some j ∈ Ik imply that G k (x1 , . . . , xk ) < G k (y1 , . . . , yk ). (iii) G k is said to be strictly monotone increasing if xi , yi ∈ [0, 1], xi < yi , i ∈ Ik imply that G k (x1 , . . . , xk ) < G k (y1 , . . . , yk ). (iv) G k is said to be idempotent if G k (x, . . . , x) = x for any x ∈ [0, 1]. (v) When k = 2, G 2 is said to be commutative if G 2 (x1 , x2 ) = G 2 (x2 , x1 ) for any xi ∈ [0, 1], i ∈ I2 . (vi) When k = 2, G 2 is said to be associative if G 2 (x1 , G 2 (x2 , x3 )) = G 2 (G 2 (x1 , x2 ), x3 ) for any xi ∈ [0, 1], i ∈ I3 . Let k ∈ N, and let G k : [0, 1]k → [0, 1] be an aggregation function. If G k is strongly monotone increasing, then G k is strictly monotone increasing. When k = 1, G 1 is strongly monotone increasing if and only if G 1 is strictly monotone increasing. Note that there does not exist any strongly monotone increasing conjunctive aggregation function when k ≥ 2. One of the important classes of aggregation functions G 2 : [0, 1]2 → [0, 1] is the class of triangular norms defined as follows. Definition 3 (See, for example, Klement et al. [4]). A triangular norm (t-norm for short) is a binary operation T on [0, 1], that is, a function T : [0, 1]2 → [0, 1], such that for any xi ∈ [0, 1], i ∈ I4 the following four axioms are satisfied: (T1) T (x1 , x2 ) = T (x2 , x1 ) (commutativity), (T2) T (x1 , T (x2 , x3 )) = T (T (x1 , x2 ), x3 ) (associativity), (T3) if x1 ≤ x3 , x2 ≤ x4 , then T (x1 , x2 ) ≤ T (x3 , x4 ) (monotonicity), and (T4) T (x1 , 1) = x1 (boundary condition).
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Example 1. Important examples of t-norms are the minimum TM and the drastic product TD defined as TM (x1 , x2 ) = min{x1 , x2 }, TM (x1 , x2 ) if max{x1 , x2 } = 1, TD (x1 , x2 ) = 0 otherwise. Here, TM is the strongest t-norm, and TD is the weakest t-norm. In other words, TD ≤ T ≤ TM for any t-norm T. For details on these t-norms and other interesting t-norms, see, for example, [4]. Let T be a t-norm. If we define T1 : [0, 1] → [0, 1] as T1 (x) = x, x ∈ [0, 1], and T2 = T , and Tk : [0, 1]k → [0, 1] as Tk (x1 , . . . , xk−1 , xk ) = T (Tk−1 (x1 , . . . , xk−1 ), xk ) for k ≥ 3, then each Tk , k ∈ N is an aggregation function. When we consider a t-norm as an aggregation function, we are considering the aggregation function associated with the t-norm under consideration. Note that each t-norm is a conjunctive aggregation function. Definition 4 (See, for example, Saminger et al. [9]). Let k ∈ N, and let G k : [0, 1]k → [0, 1] and G 2 : [0, 1]2 → [0, 1] be aggregation functions. G k is said to dominate G 2 (G k ?G 2 ) if G k (G 2 (x1 , y1 ), . . . , G 2 (xk , yk )) ≥ G 2 (G k (x1 , . . . , xk ), G k (y1 , . . . , yk )) for any xi , yi ∈ [0, 1], i ∈ Ik . An aggregation function G 1 : [0, 1] → [0, 1] and a t-norm TM dominate any aggregation function G 2 : [0, 1]2 → [0, 1]. Complete characterization of the class of all aggregation functions that dominate TM is given in [9]. j For each k ∈ N and each aggregation function G 2 : [0, 1]2 → [0, 1], we define aggregation functions G k : [0, 1]k → j j [0, 1], j ∈ I2 as G 1 (x) = x, x ∈ [0, 1], j ∈ I2 when k = 1, and G 2 = G 2 , j ∈ I2 when k = 2, and G 1k (x1 , . . . , xk ) = G 2 (G 1k−1 (x1 , . . . , xk−1 ), xk ),
(1)
G 2k (x1 , . . . , xk ) = G 2 (x1 , G 2k−1 (x2 , . . . , xk ))
(2)
for xi ∈ [0, 1], i ∈ Ik when k ≥ 3. j
Lemma 1. Let k ∈ N. Let G 2 : [0, 1]2 → [0, 1] be an aggregation function, and let G k : [0, 1]k → [0, 1], j ∈ I2 be aggregation functions defined by (1) and (2). j
(i) If G 2 is commutative and associative or G 2 ?G 2 , then G k ?G 2 , j ∈ I2 . j (ii) If G 2 is strongly monotone increasing, then G k , j ∈ I2 are also strongly monotone increasing. j (iii) If G 2 is strictly monotone increasing, then G k , j ∈ I2 are also strictly monotone increasing. Proof. We show only the results (i) and (ii) for G 1k . The proof of (iii) for G 1k is similar to the proof of (ii) for G 1k . The other results for G 2k can be shown in a similar manner. (i) Assume that G 2 is commutative and associative, and let xi , yi ∈ [0, 1], i ∈ Ik . Then, G 1k (G 2 (x1 , y1 ), . . . , G 2 (xk , yk )) = G 2 (G 1k (x1 , . . . , xk ), G 1k (y1 , . . . , yk )). Therefore, G 1k ?G 2 . Next, assume that G 2 ?G 2 . When k = 1, 2, G 1k ?G 2 . When k ≥ 2, we show that G 1k+1 ?G 2 providing that G 1k ?G 2 . Let xi , yi ∈ [0, 1], i ∈ Ik+1 . Then, G 2 (G 1k+1 (x1 , . . . , xk+1 ), G 1k+1 (y1 , . . . , yk+1 )) = G 2 (G 2 (G 1k (x1 , . . . , xk ), xk+1 ), G 2 (G 1k (y1 , . . . , yk ), yk+1 )) ≤ G 2 (G 2 (G 1k (x1 , . . . , xk ), G 1k (y1 , . . . , yk )), G 2 (xk+1 , yk+1 )) ≤ G 2 (G 1k (G 2 (x1 , y1 ), . . . , G 2 (xk , yk )), G 2 (xk+1 , yk+1 )) = G 1k+1 (G 2 (x1 , y1 ), . . . , G 2 (xk+1 , yk+1 )). Therefore, G 1k+1 ?G 2 .
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(ii) Assume that G 2 is strongly monotone increasing. When k = 1, 2, G 1k is strongly monotone increasing. When k ⱖ 2, we show that G 1k+1 is strongly monotone increasing providing that G 1k is strongly monotone increasing. Let xi , yi ∈ [0, 1], i ∈ Ik+1 , and assume that xi ≤ yi , i ∈ Ik+1 and that x j < y j for some j ∈ Ik+1 . Then, G 1k+1 (x1 , . . . , xk+1 ) = G 2 (G 1k (x1 , . . . , xk ), xk+1 ) < G 2 (G 1k (y1 , . . . , yk ), yk+1 ) = G 1k+1 (y1 , . . . , yk+1 ). Therefore, G 1k+1 is strongly monotone increasing. 3. Generalization of quasiconcavity In this section, we generalize the quasiconcavity of membership functions defined on Rn based on conjunctive aggregation functions. 3.1. G-quasiconcavity Before we generalize the quasiconcavity of membership functions defined on Rn based on conjunctive aggregation functions, some definitions that are needed later herein are presented. Definition 5 (See, for example, Ramík and Vlach [5]). Let X ⊂ Rn , X ∅. Let f : Rn → R, and let ∈ R. Then, U X ( f, ) = {x ∈ X : f (x) ≥ } is called the level set of f on X. Definition 6 (See, for example, Ramík and Vlach [5]). Let X ⊂ Rn , X ∅, and let : Rn → [0, 1]. Then, Core X () = {x ∈ X : (x) = 1} is called the core of on X. Definition 7. Let X ⊂ Rn , X ∅, and let : Rn → [0, 1]. Then, is said to be level-closed, level-bounded, or level-compact on X if U X (, ) is closed, bounded, or compact for any ∈]0, 1], respectively. Now, recall the standard definitions of quasiconcavity of functions. Definition 8 (See, for example, Ramík and Vlach [5]). Let X ⊂ Rn , X ∅ be a convex set, and let f : Rn → R. (i) f is said to be quasiconcave on X if f (x + (1 − )y) ≥ min{ f (x), f (y)} for any x, y ∈ X and any ∈]0, 1[. (ii) f is said to be strictly quasiconcave on X if f is quasiconcave on X and f (x + (1 − )y) > min{ f (x), f (y)}
(3)
for any x, y ∈ X, f (x) f (y) and any ∈]0, 1[. (iii) f is said to be strongly quasiconcave on X if (3) holds for any x, y ∈ X, xy and any ∈]0, 1[. We will also use the following concept of quasiconcavity of a function from a point. Let X ⊂ Rn , X ∅, and let f : Rn → R. Let x, y ∈ X, xy, and let L X (x, y) = {t ∈ R : x + t(y − x) ∈ X }. We define f x,y : R → R as f (x + t(y − x)) if t ∈ L X (x, y), f x,y (t) = 0 otherwise. Definition 9 (See, for example, Ramík and Vlach [5]). Let X ⊂ Rn , X ∅ be a convex set, and let f : Rn → R. A function f is said to be quasiconcave, strictly quasiconcave, or strongly quasiconcave on X from y ∈ X if f x,y is quasiconcave, strictly quasiconcave, or strongly quasiconcave on L X (x, y), respectively, for any x ∈ X, xy.
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In [5,8], the quasiconcavity of functions defined on Rn is generalized using the concept of star-shaped sets. Definition 10 (See, for example, Ramík and Vlach [5]). Let X ⊂ Rn . (i) X is said to be star-shaped from y ∈ X if I(x, y) = {z ∈ Rn : z = x + (y − x), ∈ [0, 1]} ⊂ X for any x ∈ X . (ii) The set of all points in X from which X is star-shaped is called the kernel of X and is denoted by Ker(X ). (iii) X is said to be star-shaped if Ker(X )∅ or X = ∅. Definition 11 (See, for example, Ramík and Vlach [5]). Let X ⊂ Rn , X ∅ be a star-shaped set, and let f : Rn → R. A function f is said to be star-shaped on X if U X ( f, ) is star-shaped for any ∈ R. Let X ⊂ Rn , X ∅ be a convex set, and let f : Rn → R. Since any convex set of Rn is star-shaped, if f is quasiconcave on X, then f is star-shaped on X. When n = 1, X ⊂ R is an interval, and the class of all quasiconcave functions on X coincides with the class of all star-shaped functions on X. In [5,7,8], the quasiconcavity of membership functions is generalized by allowing arbitrary t-norms instead of the minimum operation in Definition 8. Therefore, we generalize the quasiconcavity of membership functions by allowing arbitrary conjunctive aggregation functions instead of the minimum operation in Definition 8. This is a further generalization of the generalization of quasiconcavity presented in [5,7,8]. This further generalization, which is the primary definition of the present study, is defined as follows: Definition 12. Let X ⊂ Rn , X ∅ be a convex set, and let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function. In addition, let : Rn → [0, 1]. (i) is said to be G 2 -quasiconcave (or G-quasiconcave) on X if (x + (1 − )y) ≥ G 2 ((x), (y)) for any x, y ∈ X and any ∈]0, 1[. (ii) is said to be strictly G 2 -quasiconcave (or strictly G-quasiconcave) on X if is G-quasiconcave on X and (x + (1 − )y) > G 2 ((x), (y))
(4)
for any x, y ∈ X, (x)(y) and any ∈]0, 1[. (iii) is said to be strongly G 2 -quasiconcave (or strongly G-quasiconcave) on X if (4) holds for any x, y ∈ X, xy and any ∈]0, 1[. There are implicational relationships among the well-known quasiconcavities in Definition 8 and the G-quasiconcavities presented in Definition 12. For a convex set X ⊂ Rn , X ∅ and a conjunctive aggregation function G 2 : [0, 1]2 → [0, 1] and : Rn → [0, 1], is strongly quasiconcave ⇒ strictly quasiconcave ⇒ quasiconcave ⇓ ⇓ ⇓ strongly G-quasiconcave ⇒ strictly G-quasiconcave ⇒ G-quasiconcave on X from Definition 12. 3.2. Examples of G-quasiconcavity In the following, we present some examples of G-quasiconcavity.
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3.2.1. First examples Let ∈]0, 1[. Let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function, and let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function defined as follows: 0 if x ∈ [0, 1 − ] or y ∈ [0, 1 − ], G 2 (x, y) = G 2 (x, y) otherwise for x, y ∈ [0, 1]. Set X = R. (i) A membership function : R → [0, 1] defined as ⎧ 0 if x ∈] − ∞, 0] ∪ [5, ∞[, ⎪ ⎪ ⎪ ⎨ x − 2 + 1 if x ∈ [1, 2], (x) = ⎪ 1 if x ∈ [2, 3], ⎪ ⎪ ⎩ −x + 3 + 1 if x ∈ [3, 4], where (x) ∈ [0, 1 − ] if x ∈]0, 1[∪]4, 5[ is G-quasiconcave on X and is not strictly G-quasiconcave on X (Fig. 1), as shown in the following. Let x, y ∈ R, and let ∈]0, 1[. If x ∈] − ∞, 1[∪]4, ∞[ or y ∈] − ∞, 1[∪]4, ∞[, then we have (x + (1 − )y) ≥ 0 = G 2 ((x), (y)). If x, y ∈ [1, 4], then we have (x + (1 − )y) ≥ min{(x), (y)} ≥ G 2 ((x), (y)) because is quasiconcave on [1, 4]. Therefore, is G-quasiconcave on X. For x0 = −1, y0 = 1, we have ( 21 x0 + 21 y0 ) = (0) = 0G 2 ((x0 ), (y0 )) = G 2 (0, 1 − ) = 0. Therefore, is not strictly G-quasiconcave on X. (ii) A membership function : R → [0, 1] defined as ⎧ if x ∈ [0, 1], ⎪ ⎨ x + 1 − if x ∈ [1, 2], (x) = 1 ⎪ ⎩ −x + 2 + 1 if x ∈ [2, 3], where (x) ∈]0, 1 − ] if x ∈] − ∞, 0[∪]3, ∞[ is strictly G-quasiconcave on X and is not strongly G-quasiconcave on X (Fig. 2), as shown in the following. Let x, y ∈ R, and let ∈]0, 1[. First, assume that (x)(y). If x ∈] − ∞, 0[∪]3, ∞[ or y ∈] − ∞, 0[∪]3, ∞[, then we have (x + (1 − )y) > 0 = G 2 ((x), (y)). If x, y ∈ [0, 3], then we have (x + (1 − )y) > min{(x),
Fig. 1. (i) (x) ( = 0.75).
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Fig. 2. (ii) (x) ( = 0.75).
Fig. 3. (iii) (x) ( = 0.75).
(y)} ≥ G 2 ((x), (y)) because is strictly quasiconcave on [0,3]. Next, assume that (x) = (y). If x ∈] − ∞, 0[∪]3, ∞[ or y ∈] − ∞, 0[∪]3, ∞[, we have (x + (1 − )y) > 0 = G 2 ((x), (y)). If x, y ∈ [0, 3], then we have (x + (1 − )y) ≥ min{(x), (y)} ≥ G 2 ((x), (y)) because is quasiconcave on [0, 3]. Therefore, is strictly G-quasiconcave on X. For x0 = 1, y0 = 2, we have ( 21 x0 + 21 y0 ) = ( 23 ) = 1G 2 ((x0 ), (y0 )) = G 2 (1, 1) = 1. Therefore, is not strongly G-quasiconcave on X. (iii) A membership function : R → [0, 1] defined as x + 1 − if x ∈ [0, 1], (x) = −x + 1 + if x ∈ [1, 2], where (x) ∈]0, 1 − ] if x ∈] − ∞, 0[∪]2, ∞[ is strongly G-quasiconcave on X (Fig. 3), as shown in the following. Let x, y ∈ R, xy, and let ∈]0, 1[. If x ∈]−∞, 0[∪]2, ∞[ or y ∈]−∞, 0[∪]2, ∞[, then we have (x +(1−)y) > 0 = G 2 ((x), (y)). If x, y ∈ [0, 2], then we have (x + (1 − )y) > min{(x), (y)} ≥ G 2 ((x), (y)) because is strongly quasiconcave on [0,2]. Therefore, is strongly G-quasiconcave on X.
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Examples (i)–(iii) above show that G-quasiconcavity allows some differences from quasiconcavity and that such differences can be controlled by conjunctive aggregation functions. Fuzzy numbers are often used to model real-world problems. A fuzzy number is a fuzzy set on R with some restrictions and is interpreted as the fuzzy set of real numbers around a for some real number a ∈ R. Let a : R → [0, 1] be the membership function of this fuzzy set. Quasiconcavity of a is usually assumed. For each x ∈ R, the value a (x) indicates the grade of x in the fuzzy set. For a conjunctive aggregation function G 2 : [0, 1]2 → [0, 1], such as that considered in this section, if we assume G-quasiconcavity of a rather than quasiconcavity of a , then we can also consider a generalization of fuzzy numbers. Such generalization allows some differences from quasiconcavity. If membership functions of fuzzy sets on Rn are determined from, for example, observations or experiments and if the obtained membership functions exhibit differences from quasiconcavity and their differences are not small enough to neglect, then G-quasiconcavity can be expected to be useful and suitable for representations of the real world. 3.2.2. Second examples ( p) For each p ∈ [1, ∞[, let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function defined as ( p)
G 2 (x, y) = [min{x, y}] p for x, y ∈ [0, 1]. Set X = R. For each ∈ [0, 21 [, let : R → [0, 1] be a membership function defined as (Fig. 4) ⎧ 0 if x ∈] − ∞, 0] ∪ [6, ∞[, ⎪ ⎪ ⎪ ⎪ 1 ⎪ if x ∈ [0, 1], ⎪ ⎪ 2x ⎪ ⎪ ⎨ sin 4x + 1 if x ∈ [1, 2] ∪ [4, 5], 2 (x) = 1 1 ⎪ x − if x ∈ [2, 3], ⎪ 2 ⎪2 ⎪ ⎪ 1 5 ⎪ −2x + 2 if x ∈ [3, 4], ⎪ ⎪ ⎪ ⎩ 1 −2x + 3 if x ∈ [5, 6]. Then, 0 is quasiconcave on X, and not every , ∈]0, 21 [ is quasiconcave on X. For ∈]0, 21 [, we show that is G ( p) -quasiconcave on X for p ∈ [ p , ∞[, and that is not G ( p) -quasiconcave on X for p ∈ [1, p [, where p = log( 21 − )/log( 21 + ). Fix any ∈]0, 21 [, and set a = 1 − 2, b = 2 + 2, c = 4 − 2, and d = 5 + 2 (Fig. 4). Solving ( 21 + ) p ≤ 21 − in p, we have p ≥ log( 21 − )/log( 21 + ) = p . First, we assume that p ∈ [ p , ∞[. Let x, y ∈ R, and let ∈ [0, 1]. If x ∈ [a , b ]∪[c , d ] or y ∈ [a , b ]∪[c , d ], ( p) then we have (x + (1 − )y) ≥ 21 − ≥ ( 21 + ) p ≥ [min{ (x), (y)}] p = G 2 ( (x), (y)) because (x) ≤ ( p) 1 1 2 + or (y) ≤ 2 + . If x, y ∈]b , c [, then we have (x + (1 − )y) ≥ min{ (x), (y)} ≥ G 2 ( (x), (y)).
Fig. 4. (x) ( = 0.25).
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If x ∈] − ∞, a [∪]d , ∞[ or y ∈] − ∞, a [∪]d , ∞[, then we have (x + (1 − )y) ≥ min{ (x), (y)} ≥ ( p) G 2 ( (x), (y)). Therefore, is G ( p) -quasiconcave on X. 13 1 Next, we assume that p ∈ [1, p [. Set x0 = 98 , y0 = 37 8 and 0 = 14 . Then, since (x 0 ) = (y0 ) = + 2 and ( p) ( p) (0 x0 + (1 − 0 )y0 ) = − + 21 , we have G 2 ( (x0 ), (y0 )) = G 2 ( + 21 , + 21 ) = [min{ + 21 , + 21 }] p = ( + 21 ) p > 21 − = (0 x0 + (1 − 0 )y0 ). Therefore, is not G ( p) -quasiconcave on X. 3.3. Properties of G-quasiconcavity In the following, we investigate the properties of G-quasiconcavity. The following theorem gives conditions such that the concept of quasiconcavity of the membership functions defined on R coincides with the concept of the G-quasiconcavity of the membership functions. Theorem 1. Let X ⊂ R, X ∅ be a convex set, and let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function such that G 2 ≥ TD . Let : R → [0, 1] such that Core X ()∅. If is G-quasiconcave, strictly G-quasiconcave, or strongly G-quasiconcave on X, then is quasiconcave, strictly quasiconcave, or strongly quasiconcave on X, respectively. Proof. We show only that if is G-quasiconcave on X then is quasiconcave on X. The proofs for strict G-quasiconcavity and strong G-quasiconcavity are similar and so are omitted. Since Core X ()∅, there exists x ∈ X such that (x) = 1. Since G 2 is conjunctive and G 2 ≥ TD , we have G 2 (, 1) = G 2 (1, ) = , ∈ [0, 1].
(5)
Assume that is G-quasiconcave on X. Let x, y ∈ X , and let ∈]0, 1[. Then, without loss of generality, assume that x ≤ y. Assume that x + (1 − )y ≤ x. Then, since x ≤ x + (1 − )y ≤ x, there exists ∈ [0, 1] such that x + (1 − )y = x + (1 − )x. We have (x + (1 − )y) = (x + (1 − )x) = (x) = 1 ≥ min{(x), (y)} when = 0, and (x + (1 − )y) = (x + (1 − )x) = (x) ≥ min{(x), (y)} when = 1. Thus, we assume that ∈]0, 1[. Then, from G-quasiconcavity of on X and (5), we have (x + (1 − )y) = (x + (1 − )x) ≥ G 2 ((x), (x)) = (x) ≥ min{(x), (y)}. Similarly, when x ≤ x + (1 − )y, it can be shown that (x + (1 − )y) ≥ min{(x), (y)}. Therefore, is quasiconcave on X. The following examples reveal that neither the condition G 2 ≥ TD nor the condition Core X ()∅ in Theorem 1 can be eliminated. Example 2. (i) Set X = R, and define a conjunctive aggregation function G 2 : [0, 1]2 → [0, 1] as 1 if x = y = 1, G 2 (x, y) = 0 otherwise for x, y ∈ [0, 1], and define : R → [0, 1] as 1 if x = 0, (x) = 1 1 3 sin x + 2 otherwise for x ∈ R. In this case, G 2 ⱖ TD and Core X () = {0}∅. Since is strongly G-quasiconcave on X, is strictly G-quasiconcave on X and is G-quasiconcave on X. However, is not quasiconcave on X. Thus, is neither strictly quasiconcave on X nor strongly quasiconcave on X. (ii) Set X = R and G 2 = TD and define : R → [0, 1] as (x) = 13 sin x + 21 for x ∈ R. In this case, G 2 ≥ TD and Core X () = ∅. Since is strongly G-quasiconcave on X, is strictly G-quasiconcave on X and is quasiconcave on X. However, is not quasiconcave on X. Thus, is neither strictly quasiconcave on X nor strongly quasiconcave on X. The following theorem provides a relationship among the star-shaped character of membership functions, their quasiconcavity from a point, and their TD -quasiconcavity.
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Theorem 2 ( Ramík and Vlach [5, Theorem 4.34]). Let X ⊂ Rn , X ∅ be a convex set, and let : Rn → [0, 1]. Then, the following statements are equivalent. (i) Core X () ⊂ ∈[0,1] Ker(U X (, )). (ii) is quasiconcave on X from x for any x ∈ Core X (). (iii) is TD -quasiconcave on X. The following theorem provides a relationship between the strict quasiconcavity of membership functions from a point and their strict TD -quasiconcavity. Theorem 3. Let X ⊂ Rn , X ∅ be a convex set, and let : Rn → [0, 1]. Then, the following statements hold. (i) If is strictly TD -quasiconcave on X, then is strictly quasiconcave on X from x for any x ∈ Core X (). (ii) Assume that (x) > 0, x ∈ X . If is strictly quasiconcave on X from x for any x ∈ Core X (), then is strictly TD -quasiconcave on X. Proof. (i) Let x ∈ Core X (). Since is strictly TD -quasiconcave on X, is TD -quasiconcave on X. From Theorem 2, is quasiconcave on X from x. Thus, in order to show that is strictly quasiconcave on X from x, it is sufficient to show that, for each x ∈ X, xx, x,x (t) = 1, t ∈ [0, 1] if (x) = 1 and x,x is strictly monotone increasing on [0, sx ] if (x) < 1, where sx = inf{t ∈ L X (x, x) : x,x (t) = 1} ≤ 1. When x ∈ X , xx, (x) < 1, based on the quasiconcavity of x,x on L X (x, x), we have sx ≥ 0. Let x ∈ X, xx. First, assume that (x) = 1. Then, x,x (0) = (x) = 1, x,x (1) = (x) = 1, and, based on the TD -quasiconcavity of on X, we have 1 ≥ x,x (t) = (x + t(x − x)) ≥ TD ((x), (x)) = TD (1, 1) = 1 for any t ∈]0, 1[. Next, assume that (x) < 1. This is trivial when sx = 0. Thus, assume that sx > 0. Choose any u, v ∈ R such that 0 ≤ u < v ≤ sx , and set x1 = x + u(x − x), x2 = x + v(x − x). Since u < sx , x,x (u) = (x1 ) < 1. When v = sx , if x,x (sx ) = 1, then x,x (u) < 1 = x,x (sx ) = x,x (v), and if x,x (sx ) < 1, then sx < 1 and there exists ∈]0, 1[ such that x2 = x1 + (1 − )x, then, based on the strict TD -quasiconcavity of on X, we have x,x (v) = (x2 ) = (x1 + (1 − )x) > TD ((x1 ), (x)) = (x1 ) = x,x (u). When v < sx , there exists ∈]0, 1[ such that x2 = x1 + (1 − )x, and, based on the strict TD -quasiconcavity of on X, we have x,x (v) = (x2 ) = (x1 + (1 − )x) > TD ((x1 ), (x)) = (x1 ) = x,x (u). (ii) Based on the assumption, x,x is strictly quasiconcave on L X (x, x) for any x ∈ Core X () and any x ∈ X, xx. Thus, x,x is quasiconcave on L X (x, x) for any x ∈ Core X () and any x ∈ X, xx. In other words, is quasiconcave on X from x for any x ∈ Core X (). From Theorem 2, is TD -quasiconcave on X. Let x , x ∈ X, (x )(x), and let ∈]0, 1[. First, assume that (x ) = 1. Then, since x ∈ Core X (), x,x is strictly quasiconcave on L X (x, x ). Thus, we have (x + (1 − )x) > TM ((x ), (x)) ≥ TD ((x ), (x)). A similar result can be obtained when (x) = 1. Next, assume that (x ) < 1, (x) < 1. Then, we have (x + (1 − )x) > 0 = TD ((x ), (x)). The following theorem presents a relationship between the strong quasiconcavity of membership functions from a point and their strong TD -quasiconcavity. Theorem 4. Let X ⊂ Rn , X ∅ be a convex set, and let : Rn → [0, 1]. Then, the following statements hold. (i) If is strongly TD -quasiconcave on X, then is strongly quasiconcave on X from x for any x ∈ Core X (). (ii) Assume that (x) > 0, x ∈ X . If is strongly quasiconcave on X from x for any x ∈ Core X (), then is strongly TD -quasiconcave on X. Proof. (i) Let x ∈ Core X (). Then, in order to show that is strongly quasiconcave on X from x, it is sufficient to show that x,x is strictly monotone increasing on [0, 1] for each x ∈ X, xx. In order to show this, it is sufficient to show that x,x (0) < x,x () for each x ∈ X, xx and any ∈]0, 1]. Let x ∈ X, xx, and let ∈]0, 1]. If (x) = 1, then from the strong TD -quasiconcavity of on X, 1 ≥ (x+(1−)x) > TD ((x), (x)) = TD (1, 1) = 1 for any ∈]0, 1[, which is a contradiction. Thus, (x) < 1. If = 1, then we have x,x (0) = (x) < 1 = (x) = x,x (1) = x,x (). Thus, assume that ∈]0, 1[. Then, we have x,x () = (x + (x − x)) > TD ((x), (x)) = TD (1, (x)) = (x) = x,x (0) from the strong TD -quasiconcavity of on X.
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(ii) Based on the assumption, x,x is strongly quasiconcave on L X (x, x) for any x ∈ Core X () and any x ∈ X, xx. Let x , x ∈ X, x x, and let ∈]0, 1[. First, assume that (x ) = 1. Then, since x ∈ Core X (), x,x is strongly quasiconcave on L X (x, x ). Thus, we have (x + (1 − )x) > TM ((x ), (x)) ≥ TD ((x ), (x)). A similar result can be obtained when (x) = 1. Next, assume that (x ) < 1, (x) < 1. Then, we have (x + (1 − )x) > 0 = TD ((x ), (x)). The following example reveals that the condition (x) > 0, x ∈ X in Theorems 3(ii) and 4(ii) cannot be eliminated. Example 3. Let x = (x, y) ∈ R2 , and set X = {x ∈ R2 : |x| + |y| ≤ 1}. Define : R2 → [0, 1] as (x) = 2 max{1 − |x| − |y|, 0} and define : R → [0, 1] as 1 − 21 (|x| + |y|) if x ∈ [−1, 1] × {0}, (x) = (x) otherwise. In this case, Core X () = {0} and not (x) > 0, x ∈ X . Since is strongly quasiconcave on X from 0, is also strictly quasiconcave on X from 0. On the other hand, if we set y = (1, 0), z = (0, 1), then (y) = 21 (z) = 0 and ( 21 y + 21 z) = 0 = TD ((y), (z)). Thus, is not strictly TD -quasiconcave on X. In addition, is not strongly TD -quasiconcave on X. Theorem 5. Let X ⊂ Rn , X ∅ be a convex set. In addition, let : Rn → [0, 1] such that Core X ()∅, and let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function such that G 2 ≥ TD . Assume that is G-quasiconcave on X. Then, is star-shaped on X. Proof. Since is G-quasiconcave on X and G 2 ≥ TD , (x + (1 − )y) ≥ G 2 ((x), (y)) ≥ TD ((x), (y)) for any x, y ∈ X and any ∈]0, 1[. Thus, is TD -quasiconcave on X. Since ∅Core X () ⊂ ∈[0,1] Ker(U X (, )) from Theorem 2, is star-shaped on X. Example 2 above indicates that neither the condition Core X ()∅ nor the condition G 2 ≥ TD in Theorem 5 can be eliminated. 3.4. Aggregation of G-quasiconcave membership functions When several G-quasiconcave membership functions are aggregated using an aggregation function, a new aggregated membership function can be obtained. We investigate the properties of such aggregated membership functions. Let i : Rn → [0, 1], i ∈ I , and let G m : [0, 1]m → [0, 1] be an aggregation function. Then, we define a function : Rn → [0, 1] as (x) = G m (1 (x), . . . , m (x)), x ∈ Rn ,
(6)
and we investigate its properties related to G-quasiconcavity. Theorem 6. Let X ⊂ Rn , X ∅ be a convex set. Let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function such n m that G 2 ≥ T D , and let G m : [0, 1] → [0, 1] be an aggregation function. Then, let i : R → [0, 1], i ∈ I . Moreover, assume that i∈I Core X (i )∅ and that i , i ∈ I are G -quasiconcave on X. Then, a function : Rn → [0, 1] defined by (6) is star-shaped on X. Proof. Let ∈ R. If > 1, then U X (, ) = ∅ is star-shaped. Thus, assume that ≤ 1. In order to show that U X (, ) is star-shaped, we fix any x ∈ i∈I Core X (i ) and show that x ∈ Ker(U X (, )). Since 1 (x) = · · · = m (x) = 1, we have (x) = G m (1 (x), . . . , m (x)) = G m (1, . . . , 1) = 1 ≥ . Thus, x ∈ U X (, ). Let x ∈ U X (, ), and let ∈]0, 1[. Then, we set z = x + (1 − )x. If x = x, then z = x + (1 − )x ∈ U X (, ). Thus, we assume that xx. Since i , i ∈ I are G -quasiconcave on X and G 2 ≥ TD , i , i ∈ I are also TD -quasiconcave on X. From Theorem 2, (i )x,x , i ∈ I are quasiconcave on L X (x, x). Thus, i (x) ≤ i (z), i ∈ I . Based on the
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monotonicity of G m , G m (1 (x), . . . , m (x)) ≤ G m (1 (z), . . . , m (z)). Since x ∈ U X (, ), we have ≤ (x) ≤ (z). Thus, z = x + (1 − )x ∈ U X (, ). Consequently, x ∈ Ker(U X (, )) and U X (, ) is star-shaped. Theorem 7. Let X ⊂ Rn , X ∅ be a convex set. Let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function, and let G m : [0, 1]m → [0, 1] be an aggregation function such that G m ?G 2 . Let i : Rn → [0, 1], i ∈ I , and let : Rn → [0, 1] be a function defined by (6). If i , i ∈ I are G -quasiconcave, strictly G -quasiconcave, or strongly G -quasiconcave on X, then is G -quasiconcave, strictly G -quasiconcave, or strongly G -quasiconcave on X, respectively. Proof. We show only that if i , i ∈ I are G -quasiconcave on X, then is G -quasiconcave on X. The other results can be proven in a similar manner. Let x, y ∈ X and ∈]0, 1[. Since i , i ∈ I are G -quasiconcave on X, i (x+(1−)y) ≥ G 2 (i (x), i (y)), i ∈ I . Since G m ?G 2 , we have (x + (1 − )y) = G m (1 (x + (1 − )y), . . . , m (x + (1 − )y)) ≥ G m (G 2 (1 (x), 1 (y)), . . . , G 2 (m (x), m (y))) ≥ G 2 (G m (1 (x), . . . , m (x)), G m (1 (y), . . . , m (y))) = G 2 ((x), (y)). We have the following corollary from Lemma 1 and Theorem 7. Corollary 1. Let X ⊂ Rn , X ∅ be a convex set. Let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function, and let i : Rn → [0, 1], i ∈ I . Let 1 : Rn → [0, 1] be a function defined as 1 (x) = TM (1 (x), . . . , m (x)), x ∈ Rn , j j j and let 2 : Rn → [0, 1], j ∈ I2 be functions defined as 2 (x) = G m (1 (x), . . . , m (x)), x ∈ Rn , j ∈ I2 , where j G m : [0, 1]m → [0, 1], j ∈ I2 are aggregation functions defined by (1) and (2). (i) If i , i ∈ I are G-quasiconcave on X, then 1 is G-quasiconcave on X. Moreover, if G 2 is commutative and j associative or G 2 ?G 2 , then 2 , j ∈ I2 are G-quasiconcave on X. (ii) If i , i ∈ I are strongly G-quasiconcave on X, then 1 is strongly G-quasiconcave on X. Moreover, if G 2 is j strictly monotone increasing, and if G 2 is commutative and associative or G 2 ?G 2 , then 2 , j ∈ I2 are strongly G-quasiconcave on X. 4. Fuzzy programming problems In this section, we consider the application of G-quasiconcavity of membership functions to fuzzy programming problems. 4.1. Problem formulation We consider a fuzzy multicriteria programming problem formulated as max (x) = ( (x), . . . , (x)) 1 m F (P (X )) s.t. x ∈ X and a fuzzy scalar programming problem formulated as max (x) = G ( (x), . . . , (x)) m 1 m F (PG (X )) s.t. x ∈ X where X ⊂ Rn , X ∅ and i : Rn → [0, 1], i ∈ I and F = {i : i ∈ I }, and G m : [0, 1]m → [0, 1] is an aggregation function. The fuzzy scalar programming problem (PGF (X )) is a problem that scalarizes the fuzzy multicriteria programming problem by aggregating membership functions in the objective function of the fuzzy multicriteria programming problem using an aggregation function. Consider the following situation. I is a set of m decision makers, and X ⊂ Rn , X ∅ is a set of feasible actions (or alternatives). For each feasible action x ∈ X and each i ∈ I , i (x) represents the degree of satisfaction for the feasible action x with respect to the decision maker i. Then, (P F (X )) can be interpreted as the problem of finding
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a feasible action that maximizes the degree of satisfaction with respect to each decision maker, and (PGF (X )) can be interpreted as the problem of finding a feasible action that maximizes a mean of the degree of satisfaction with respect to each decision maker. Furthermore, the G-quasiconcavity of i , i ∈ I allows some differences from quasiconcavity, as mentioned in Section 3.2.1. Definition 13 (See, for example, Ramík and Vlach [5]). (i) A point x0 ∈ X is called a weak Pareto optimal solution of (P F (X )) if there is no x ∈ X such that (x0 ) < (x). (ii) A point x0 ∈ X is called a Pareto optimal solution of (P F (X )) if there is no x ∈ X such that (x0 ) ≤ (x) and (x0 )(x). (iii) A point x0 ∈ X is called a strictly Pareto optimal solution of (P F (X )) if there is no x ∈ X such that xx0 and (x0 ) ≤ (x). F , X F , and X F be sets of all weak Pareto, Pareto, and strictly Pareto optimal solutions of (P F (X )), respecLet X W P P SP F . tively. From Definition 13, X SFP ⊂ X PF ⊂ X W P F Let X G be the set of all optimal solutions of (PGF (X )). An optimal solution of (PGF (X )) is called a compromise solution. In [8], the properties of compromise solutions related to T-quasiconcavity for a t-norm T are investigated. In [5–8], the properties of compromise solutions are investigated when an aggregation function G m in (PGF (X )) is an arbitrary t-norm or TM . All of these approaches are generalizations of the max–min approach in [2].
4.2. Auxiliary results In the following, we present some auxiliary results. Lemma 2. Let X ⊂ Rn , X ∅, and let : Rn → [0, 1]. (i) If all U X (, ), ∈]0, 1] are closed, then is upper semicontinuous on X. (ii) If X is closed and is upper semicontinuous on X, then all U X (, ), ∈ [0, 1] are closed. Proof. (i) Let x0 ∈ X , and let > 0. Then, in order to show that there exists > 0 such that (x) < (x0 ) + for any x ∈ B(x0 , ) = {x ∈ X : x − x0 < }, where · is Euclidean norm, assume that for any > 0 there exists x ∈ B(x0 , ) such that (x) ≥ (x0 ) + . For each k ∈ N, there exists xk ∈ B(x0 , 1/k) such that (xk ) ≥ (x0 ) + . Set = (x0 )+ > 0. Based on the assumption, U X (, ) is closed. Thus, since {xk } ⊂ U X (, ) and limk→∞ xk = x0 , we have x0 ∈ U X (, ), (x0 ) ≥ = (x0 ) + , and ≤ 0, which contradicts the statement that > 0. (ii) Let ∈ [0, 1]. Let {xk } ⊂ U X (, ) be any convergent sequence, and let x0 ∈ Rn be its limit. Since {xk } ⊂ X and X is closed, x0 ∈ X . Then, for each k ∈ N, since xk ∈ U X (, ), (xk ) ≥ . From the upper semicontinuity of on X, we have ≤ lim supk→∞ (xk ) ≤ (x0 ). Therefore, x0 ∈ U X (, ), and U X (, ) is closed. Based on fundamental calculations in real analysis, we obtain the following theorem. Theorem 8. Let X ⊂ Rn , X ∅, and let i : Rn → [0, 1], i ∈ I . Assume that i , i ∈ I are upper semicontinuous on X. Let G m : [0, 1]m → [0, 1] be an upper semicontinuous aggregation function. Then, a function : Rn → [0, 1] defined by (6) is upper semicontinuous on X. 4.3. Properties of compromise solutions We present some valuable results on compromise solutions. The following theorem gives the sufficient conditions for the existence of compromise solutions. Theorem 9. Let X ⊂ Rn , X ∅ be a closed set. Assume that i : Rn → [0, 1], i ∈ I are level-compact on X. Let F = {i : i ∈ I }, and let G m : [0, 1]m → [0, 1] be an upper semicontinuous idempotent aggregation function. Then, X GF ∅.
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Proof. Let ∈]0, 1]. Then, we show that U X (, ) is compact. First, in order to show that U X (, ) is bounded, assume that U X (, ) is unbounded. Then, there exists {xk } ⊂ U X (, ) such that limk→∞ xk = ∞. Fix any such that 0 < < . Since U X (i , ), i ∈ I are bounded, there / U X (i , ) for any i ∈ I and any k ≥ k0 . Based on the monotonicity and idempotency exists k0 ∈ N such that xk ∈ of G m , we have (xk ) = G m (1 (xk ), . . . , m (xk )) ≤ G m ( , . . . , ) = < for any k ≥ k0 , which contradicts the statement that {xk } ⊂ U X (, ). Therefore, U X (, ) is bounded. Since i , i ∈ I are level-closed on X, i , i ∈ I are upper semicontinuous on X from Lemma 2(i). From Theorem 8, is upper semicontinuous on X. Thus, U X (, ) is closed from Lemma 2(ii). Consequently, U X (, ) is compact, and is level-compact on X. If sup{(x) : x ∈ X } = 0, then X GF = X ∅. Assume that sup{(x) : x ∈ X } > 0, and fix any such that 0 < < sup{(x) : x ∈ X }. Since U X (, )∅ is compact and is upper semicontinuous on U X (, ), there exists x∗ ∈ U X (, ) ⊂ X such that sup{(x) : x ∈ X } = sup{(x) : x ∈ U X (, )} = (x∗ ). Therefore, X GF ∅. The following theorem gives a sufficient condition for a compromise solution to be a weak Pareto optimal solution of (P F (X )). Theorem 10. Let X ⊂ Rn , X ∅, and let i : Rn → [0, 1], i ∈ I . Let F = {i : i ∈ I }, and let G m : [0, 1]m → [0, 1] F . be a strictly monotone increasing aggregation function. Then, X GF ⊂ X W P F , assume that x∗ ∈ F . Then, there exists x ∈ X such that Proof. Let x∗ ∈ X GF . In order to show that x∗ ∈ X W / XW P P ∗ i (x ) < i (x ), i ∈ I . On the other hand, from the strict monotonicity of G m , we have G m (1 (x∗ ), . . . , m (x∗ )) < G m (1 (x ), . . . , m (x )), which contradicts the statement that x∗ ∈ X GF .
For Pareto and strictly Pareto optimal solutions of (P F (X )), results similar to Theorem 10 can be obtained, and the proofs of these solutions are similar to the proof of Theorem 10. Theorem 11. Let X ⊂ Rn , X ∅, and let i : Rn → [0, 1], i ∈ I . Let F = {i : i ∈ I }, and let G m : [0, 1]m → [0, 1] be a strongly monotone increasing aggregation function. Then, X GF ⊂ X PF . Theorem 12. Let X ⊂ Rn , X ∅, and let i : Rn → [0, 1], i ∈ I . Let F = {i : i ∈ I }, and let G m : [0, 1]m → [0, 1] be an aggregation function. Assume that X GF is a singleton. Then, X GF ⊂ X SFP . The following theorem gives sufficient conditions for a compromise solution to be unique when the compromise solution exists. Theorem 13. Let X ⊂ Rn , X ∅ be a convex set. Assume that i : Rn → [0, 1], i ∈ I are strongly quasiconcave on X. Let F = {i : i ∈ I }, and let G m : [0, 1]m → [0, 1] be a strictly monotone increasing aggregation function. Assume also that G m ?TM . If x∗ ∈ X GF , then X GF = {x∗ }. Proof. First, we show that is strongly quasiconcave on X. Let x, y ∈ X, xy, and let ∈]0, 1[. Since i , i ∈ I are strongly quasiconcave on X, we have i (x + (1 − )y) > TM (i (x), i (y)), i ∈ I . Since G m is strictly monotone increasing and G m ?TM , we have (x + (1 − )y) = G m (1 (x + (1 − )y), . . . , m (x + (1 − )y)) > G m (TM (1 (x), 1 (y)), . . . , TM (m (x), m (y))) ≥ TM (G m (1 (x), . . . , m (x)), G m (1 (y), . . . , m (y))) = TM ((x), (y)). Therefore, is strongly quasiconcave on X. Next, assume that there exists x ∈ X GF such that x x∗ . Then, (x ) = (x∗ ). Based on the strong quasiconcavity of on X, we have (x + (1 − )x∗ ) > TM ((x ), (x∗ )) = (x∗ ) for any ∈]0, 1[, which contradicts the statement that x∗ ∈ X GF . The following theorem gives a relationship between local and global optimal solutions of (PGF (X )). Theorem 14. Let X ⊂ Rn , X ∅ be a convex set, and let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function such that G 2 ≥ TD . Let i : Rn → [0, 1], i ∈ I , and let F = {i : i∈ I }. Let G m : [0, 1]m → [0, 1] be an aggregation function. Assume that i , i ∈ I are G -quasiconcave on X and that i∈I Core X (i )∅. If x ∈ X is a strict local optimal solution of (PGF (X )), then x is a unique global optimal solution of (PGF (X )).
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Proof. From Theorem 6, is star-shaped on X. Set = (x). Then, U X (, ) is star-shaped. Since x is a strict local optimal solution of (PGF (X )), there exists an open ball B ⊂ Rn with the center at x such that (x) < (x) for any x ∈ B ∩ X, xx. Now, assume that x is not a global optimal solution of (PGF (X )). Then, there exists v ∈ X such that (x) < (v). Since x, v ∈ U X (, ) and U X (, ) is star-shaped, for any fixed y ∈ Ker(U X (, )), P(x, v) = I(x, y) ∪ I(v, y) ⊂ U X (, ), and there exists z ∈ P(x, v) ∩ B, zx. Since z ∈ U X (, ), (z) ≥ . On the other hand, since z ∈ B ∩ X, zx, we have (z) < (x) = , which is a contradiction. Therefore, x is a global optimal solution of (PGF (X )). Similarly, based on the assumption that x is a strict local optimal solution of (PGF (X )), it can be shown that x is a unique global optimal solution of (PGF (X )). 5. Application in facility location problems In this section, we discuss the application of a fuzzy multicriteria programming problem (P F (X )) and a fuzzy scalar programming problem (PGF (X )) to facility location problems, and the necessity and importance of G-quasiconcavity is revealed. Let I = {1, . . . , m} be the set of customers of a certain facility to be located, and let pi = (ai , bi ) ∈ R2 , i ∈ I be locations associated with customers. Here, pi , i ∈ I are called demand points. Then, the problem of locating the facility is called a single facility location problem. Let x = (x, y) ∈ R2 be the variable location of the facility, and let X ⊂ R2 be a convex feasible region in which the facility can be located. For simplicity, we assume that X = R2 . We also assume that the travel distance from each demand point pi , i ∈ I to the facility x is measured by d(pi , x) = (|ai − x|1/2 + |bi − y|1/2 )2 . In [3], such d is called hyper-rectilinear distance and it is used in facility location problems. We define each membership function i : R2 → [0, 1], i ∈ I as i (x) = 1/(1 + d(pi , x)), x ∈ R2 . The value i (x) represents the degree of satisfaction for location x of the facility with respect to customer i, and customer i prefers shorter travel distances. In this case, the fuzzy multicriteria programming problem (P F (X )) is formulated as
1 1 max , . . . , 1 + d(p1 , x) 1 + d(pm , x) s.t. x ∈ X, and the fuzzy scalar programming problem (PGF (X )) is formulated as
1 1 max G m , ... , 1 + d(p1 , x) 1 + d(pm , x) s.t. x ∈ X. Each i , i ∈ I is a translated function of (x, y) = 1/(1 + (|x|1/2 + |y|1/2 )2 ), x, y ∈ R by pi . Fig. 5 shows the function : R2 → [0, 1]. From simple calculations, since is strongly quasiconcave on X from 0, is strongly TD -quasiconcave
Fig. 5. (x, y) = 1/(1 + (|x|1/2 + |y|1/2 )2 ).
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on X from Theorem 4. This means that there exists a conjunctive aggregation function G 2 : [0, 1]2 → [0, 1] such that is G -quasiconcave on X. Again, from simple calculations, since the level sets of are not convex, is not quasiconcave on X. The method by which to determine a proper aggregation function is not investigated herein, and we can only state that if quasiconcavity (i.e. TM -quasiconcavity) is not appropriate, we can attempt to apply some other aggregation function. 6. Conclusions In the present paper, the quasiconcavity of membership functions was generalized by allowing arbitrary conjunctive aggregation functions rather than the minimum operation. The generalized quasiconcavity of the present study was called G-quasiconcavity. Then, the properties of the G-quasiconcavity of membership functions were investigated. Furthermore, a fuzzy multicriteria programming problem and a fuzzy scalar programming problem were considered. The properties of weak Pareto, Pareto, strictly Pareto optimal solutions, and compromise solutions, and their relationships were then derived. Furthermore, we discussed the application of the obtained results to facility location problems. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
G. Beliakov, A. Pradera, T. Calvo, Aggregation Functions: A Guide for Practitioners, Springer-Verlag, 2007. R.E. Bellman, L.A. Zadeh, Decision-making in a fuzzy environment, Manage. Sci. 17 (1970) 141–164. H. Juel, R. Love, The facility location problem for hyper-rectilinear distances, IIE Trans. 17 (1985) 94–98. E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, 2000. J. Ramík, M. Vlach, Generalized Concavity in Fuzzy Optimization and Decision Analysis, Kluwer Academic Publishers, 2002. J. Ramík, M. Vlach, Pareto-optimality of compromise decisions, Fuzzy Sets Syst. 129 (2002) 119–127. J. Ramík, M. Vlach, Concepts of generalized concavity based on triangular norms, J. Stat. Manage. Syst. 5 (2002) 87–106. J. Ramík, M. Vlach, Aggregation of generalized quasiconcave functions in optimization, CEJOR Cent. Eur. J. Oper. Res. 12 (2004) 35–45. S. Saminger, R. Mesiar, U. Bodenhofer, Domination of aggregation operators and preservation of transitivity, Int. J. Uncertainty Fuzziness Knowledge-Based Syst. 10 (2002) 11–35. [10] S. Saminger-Platz, R. Mesiar, D. Dubois, Aggregation operators and commuting, IEEE Trans. Fuzzy Syst. 15 (2007) 1032–1045.
Fuzzy Sets and Systems 198 (2012) 112 – 127 www.elsevier.com/locate/fss
Concepts of generalized concavity based on aggregation functions夡 Masamichi Kona,∗ , Hiroaki Kuwanob a Graduate School of Science and Technology, Hirosaki University, 3 Bunkyo, Hirosaki, Aomori 036-8561, Japan b Faculty of Business Administration and Information Science, Kanazawa Gakuin University, 10 Sue, Kanazawa, Ishikawa 920-1392, Japan
Received 1 February 2011; received in revised form 22 June 2011; accepted 1 October 2011 Available online 10 October 2011
Abstract In the present paper, the quasiconcavity of membership functions is generalized based on conjunctive aggregation functions, and the properties of the generalized quasiconcavity are investigated. Fuzzy multicriteria and scalar programming problems are then considered, and the properties of Pareto optimal solutions and compromise solutions as well as their relationships are derived. Finally, we discuss the application of the obtained results to facility location problems. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. Keywords: Aggregation function; Quasiconcavity; Fuzzy programming problem
1. Introduction Motivated by extensive applications of quasiconcave functions in economics and optimization, several generalizations of quasiconcavity of functions have been introduced and investigated (see [5] and the references therein). In the present paper, we consider primarily functions from the n-dimensional Euclidean space Rn into the unit interval [0, 1]. Such functions can be regarded as membership functions of fuzzy sets on Rn . Therefore, it is natural to use the terminology of fuzzy theory. Quasiconcavity is defined using the minimum operation (see Definition 8). In [5,7,8], for membership functions, quasiconcavity was generalized by allowing arbitrary triangular norms instead of the minimum operation. In the present paper, we generalize quasiconcavity by allowing arbitrary conjunctive aggregation functions instead of the minimum operation. The generalization of the present study is a further generalization of the generalization reported in [5,7,8]. Aggregation functions are very important for the generalization of operations of fuzzy sets and have been widely investigated (see [1,4,5] and the references therein). On the other hand, for fuzzy multicriteria programming problems and fuzzy scalar programming problems, the properties of their solutions related to generalized quasiconcavity based on triangular norms have been investigated 夡 The present study was supported by a Grant-in-Aid for Scientific Research (No. 22510133) from the Japan Society for the Promotion of Science. ∗ Corresponding author. Tel.: +81 172 39 3538; fax: +81 172 39 3526.
E-mail address: [email protected] (M. Kon). 0165-0114/$ - see front matter Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2011.10.001
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in [5–8]. Fuzzy scalar programming problems are scalarized problems of fuzzy multicriteria programming problems. In the present paper, for fuzzy multicriteria programming problems and fuzzy scalar programming problems, we investigate the properties of their solutions as related to generalized quasiconcavity based on conjunctive aggregation functions. Furthermore, we discuss the application of the obtained results to facility location problems. The remainder of the present paper is organized as follows. In Section 2, we present some definitions and properties of aggregation functions. In Section 3, we generalize the quasiconcavity of membership functions defined on Rn based on conjunctive aggregation functions and investigate the properties of the present generalized quasiconcavity. In Section 4, we consider applications of the present generalized quasiconcavity of membership functions to fuzzy multicriteria programming problems and fuzzy scalar programming problems. In Section 5, we discuss applications of a fuzzy multicriteria programming problem and a fuzzy scalar programming problem to facility location problems. Finally, conclusions are presented in Section 6.
2. Aggregation functions In this section, we investigate the properties of aggregation functions. Aggregation functions are used in order to generalize the quasiconcavity of membership functions. For details of aggregation functions, see [1,4,5,9,10]. Throughout the present paper, N is the set of all natural numbers, m ∈ N is fixed, I = {1, . . . , m}, and Ik = {1, . . . , k}, k ∈ N. For a, b ∈ R ∪ {−∞, ∞}, we set [a, b] = {x ∈ R : a ≤ x ≤ b}, [a, b[= {x ∈ R : a ≤ x < b}, ]a, b] = {x ∈ R : a < x ≤ b}, and ]a, b[= {x ∈ R : a < x < b}. First, aggregation functions are defined. Definition 1 (See, for example, Beliakov et al. [1]). Let k ∈ N and G k : [0, 1]k → [0, 1]. For k = 1, G 1 is called an aggregation function if G 1 (x) = x, x ∈ [0, 1]. For k ≥ 2, G k is called an aggregation function if the following two axioms are satisfied: (G1) if xi , yi ∈ [0, 1], xi ≤ yi , i ∈ Ik , then G k (x1 , . . . , xk ) ≤ G k (y1 , . . . , yk ) (monotonicity), and (G2) G k (0, . . . , 0) = 0 and G k (1, . . . , 1) = 1 (boundary condition). Next, the definitions of the properties of aggregation functions are given. Definition 2 (See, for example, Beliakov et al. [1] except for (iii)). Let k ∈ N, and let G k : [0, 1]k → [0, 1] be an aggregation function. (i) G k is said to be conjunctive if G k (x1 , . . . , xk ) ≤ min{x1 , . . . , xk } for any xi ∈ [0, 1], i ∈ Ik . (ii) G k is said to be strongly monotone increasing if xi , yi ∈ [0, 1], xi ≤ yi , i ∈ Ik and x j < y j for some j ∈ Ik imply that G k (x1 , . . . , xk ) < G k (y1 , . . . , yk ). (iii) G k is said to be strictly monotone increasing if xi , yi ∈ [0, 1], xi < yi , i ∈ Ik imply that G k (x1 , . . . , xk ) < G k (y1 , . . . , yk ). (iv) G k is said to be idempotent if G k (x, . . . , x) = x for any x ∈ [0, 1]. (v) When k = 2, G 2 is said to be commutative if G 2 (x1 , x2 ) = G 2 (x2 , x1 ) for any xi ∈ [0, 1], i ∈ I2 . (vi) When k = 2, G 2 is said to be associative if G 2 (x1 , G 2 (x2 , x3 )) = G 2 (G 2 (x1 , x2 ), x3 ) for any xi ∈ [0, 1], i ∈ I3 . Let k ∈ N, and let G k : [0, 1]k → [0, 1] be an aggregation function. If G k is strongly monotone increasing, then G k is strictly monotone increasing. When k = 1, G 1 is strongly monotone increasing if and only if G 1 is strictly monotone increasing. Note that there does not exist any strongly monotone increasing conjunctive aggregation function when k ≥ 2. One of the important classes of aggregation functions G 2 : [0, 1]2 → [0, 1] is the class of triangular norms defined as follows. Definition 3 (See, for example, Klement et al. [4]). A triangular norm (t-norm for short) is a binary operation T on [0, 1], that is, a function T : [0, 1]2 → [0, 1], such that for any xi ∈ [0, 1], i ∈ I4 the following four axioms are satisfied: (T1) T (x1 , x2 ) = T (x2 , x1 ) (commutativity), (T2) T (x1 , T (x2 , x3 )) = T (T (x1 , x2 ), x3 ) (associativity), (T3) if x1 ≤ x3 , x2 ≤ x4 , then T (x1 , x2 ) ≤ T (x3 , x4 ) (monotonicity), and (T4) T (x1 , 1) = x1 (boundary condition).
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Example 1. Important examples of t-norms are the minimum TM and the drastic product TD defined as TM (x1 , x2 ) = min{x1 , x2 }, TM (x1 , x2 ) if max{x1 , x2 } = 1, TD (x1 , x2 ) = 0 otherwise. Here, TM is the strongest t-norm, and TD is the weakest t-norm. In other words, TD ≤ T ≤ TM for any t-norm T. For details on these t-norms and other interesting t-norms, see, for example, [4]. Let T be a t-norm. If we define T1 : [0, 1] → [0, 1] as T1 (x) = x, x ∈ [0, 1], and T2 = T , and Tk : [0, 1]k → [0, 1] as Tk (x1 , . . . , xk−1 , xk ) = T (Tk−1 (x1 , . . . , xk−1 ), xk ) for k ≥ 3, then each Tk , k ∈ N is an aggregation function. When we consider a t-norm as an aggregation function, we are considering the aggregation function associated with the t-norm under consideration. Note that each t-norm is a conjunctive aggregation function. Definition 4 (See, for example, Saminger et al. [9]). Let k ∈ N, and let G k : [0, 1]k → [0, 1] and G 2 : [0, 1]2 → [0, 1] be aggregation functions. G k is said to dominate G 2 (G k ?G 2 ) if G k (G 2 (x1 , y1 ), . . . , G 2 (xk , yk )) ≥ G 2 (G k (x1 , . . . , xk ), G k (y1 , . . . , yk )) for any xi , yi ∈ [0, 1], i ∈ Ik . An aggregation function G 1 : [0, 1] → [0, 1] and a t-norm TM dominate any aggregation function G 2 : [0, 1]2 → [0, 1]. Complete characterization of the class of all aggregation functions that dominate TM is given in [9]. j For each k ∈ N and each aggregation function G 2 : [0, 1]2 → [0, 1], we define aggregation functions G k : [0, 1]k → j j [0, 1], j ∈ I2 as G 1 (x) = x, x ∈ [0, 1], j ∈ I2 when k = 1, and G 2 = G 2 , j ∈ I2 when k = 2, and G 1k (x1 , . . . , xk ) = G 2 (G 1k−1 (x1 , . . . , xk−1 ), xk ),
(1)
G 2k (x1 , . . . , xk ) = G 2 (x1 , G 2k−1 (x2 , . . . , xk ))
(2)
for xi ∈ [0, 1], i ∈ Ik when k ≥ 3. j
Lemma 1. Let k ∈ N. Let G 2 : [0, 1]2 → [0, 1] be an aggregation function, and let G k : [0, 1]k → [0, 1], j ∈ I2 be aggregation functions defined by (1) and (2). j
(i) If G 2 is commutative and associative or G 2 ?G 2 , then G k ?G 2 , j ∈ I2 . j (ii) If G 2 is strongly monotone increasing, then G k , j ∈ I2 are also strongly monotone increasing. j (iii) If G 2 is strictly monotone increasing, then G k , j ∈ I2 are also strictly monotone increasing. Proof. We show only the results (i) and (ii) for G 1k . The proof of (iii) for G 1k is similar to the proof of (ii) for G 1k . The other results for G 2k can be shown in a similar manner. (i) Assume that G 2 is commutative and associative, and let xi , yi ∈ [0, 1], i ∈ Ik . Then, G 1k (G 2 (x1 , y1 ), . . . , G 2 (xk , yk )) = G 2 (G 1k (x1 , . . . , xk ), G 1k (y1 , . . . , yk )). Therefore, G 1k ?G 2 . Next, assume that G 2 ?G 2 . When k = 1, 2, G 1k ?G 2 . When k ≥ 2, we show that G 1k+1 ?G 2 providing that G 1k ?G 2 . Let xi , yi ∈ [0, 1], i ∈ Ik+1 . Then, G 2 (G 1k+1 (x1 , . . . , xk+1 ), G 1k+1 (y1 , . . . , yk+1 )) = G 2 (G 2 (G 1k (x1 , . . . , xk ), xk+1 ), G 2 (G 1k (y1 , . . . , yk ), yk+1 )) ≤ G 2 (G 2 (G 1k (x1 , . . . , xk ), G 1k (y1 , . . . , yk )), G 2 (xk+1 , yk+1 )) ≤ G 2 (G 1k (G 2 (x1 , y1 ), . . . , G 2 (xk , yk )), G 2 (xk+1 , yk+1 )) = G 1k+1 (G 2 (x1 , y1 ), . . . , G 2 (xk+1 , yk+1 )). Therefore, G 1k+1 ?G 2 .
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(ii) Assume that G 2 is strongly monotone increasing. When k = 1, 2, G 1k is strongly monotone increasing. When k ⱖ 2, we show that G 1k+1 is strongly monotone increasing providing that G 1k is strongly monotone increasing. Let xi , yi ∈ [0, 1], i ∈ Ik+1 , and assume that xi ≤ yi , i ∈ Ik+1 and that x j < y j for some j ∈ Ik+1 . Then, G 1k+1 (x1 , . . . , xk+1 ) = G 2 (G 1k (x1 , . . . , xk ), xk+1 ) < G 2 (G 1k (y1 , . . . , yk ), yk+1 ) = G 1k+1 (y1 , . . . , yk+1 ). Therefore, G 1k+1 is strongly monotone increasing. 3. Generalization of quasiconcavity In this section, we generalize the quasiconcavity of membership functions defined on Rn based on conjunctive aggregation functions. 3.1. G-quasiconcavity Before we generalize the quasiconcavity of membership functions defined on Rn based on conjunctive aggregation functions, some definitions that are needed later herein are presented. Definition 5 (See, for example, Ramík and Vlach [5]). Let X ⊂ Rn , X ∅. Let f : Rn → R, and let ∈ R. Then, U X ( f, ) = {x ∈ X : f (x) ≥ } is called the level set of f on X. Definition 6 (See, for example, Ramík and Vlach [5]). Let X ⊂ Rn , X ∅, and let : Rn → [0, 1]. Then, Core X () = {x ∈ X : (x) = 1} is called the core of on X. Definition 7. Let X ⊂ Rn , X ∅, and let : Rn → [0, 1]. Then, is said to be level-closed, level-bounded, or level-compact on X if U X (, ) is closed, bounded, or compact for any ∈]0, 1], respectively. Now, recall the standard definitions of quasiconcavity of functions. Definition 8 (See, for example, Ramík and Vlach [5]). Let X ⊂ Rn , X ∅ be a convex set, and let f : Rn → R. (i) f is said to be quasiconcave on X if f (x + (1 − )y) ≥ min{ f (x), f (y)} for any x, y ∈ X and any ∈]0, 1[. (ii) f is said to be strictly quasiconcave on X if f is quasiconcave on X and f (x + (1 − )y) > min{ f (x), f (y)}
(3)
for any x, y ∈ X, f (x) f (y) and any ∈]0, 1[. (iii) f is said to be strongly quasiconcave on X if (3) holds for any x, y ∈ X, xy and any ∈]0, 1[. We will also use the following concept of quasiconcavity of a function from a point. Let X ⊂ Rn , X ∅, and let f : Rn → R. Let x, y ∈ X, xy, and let L X (x, y) = {t ∈ R : x + t(y − x) ∈ X }. We define f x,y : R → R as f (x + t(y − x)) if t ∈ L X (x, y), f x,y (t) = 0 otherwise. Definition 9 (See, for example, Ramík and Vlach [5]). Let X ⊂ Rn , X ∅ be a convex set, and let f : Rn → R. A function f is said to be quasiconcave, strictly quasiconcave, or strongly quasiconcave on X from y ∈ X if f x,y is quasiconcave, strictly quasiconcave, or strongly quasiconcave on L X (x, y), respectively, for any x ∈ X, xy.
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In [5,8], the quasiconcavity of functions defined on Rn is generalized using the concept of star-shaped sets. Definition 10 (See, for example, Ramík and Vlach [5]). Let X ⊂ Rn . (i) X is said to be star-shaped from y ∈ X if I(x, y) = {z ∈ Rn : z = x + (y − x), ∈ [0, 1]} ⊂ X for any x ∈ X . (ii) The set of all points in X from which X is star-shaped is called the kernel of X and is denoted by Ker(X ). (iii) X is said to be star-shaped if Ker(X )∅ or X = ∅. Definition 11 (See, for example, Ramík and Vlach [5]). Let X ⊂ Rn , X ∅ be a star-shaped set, and let f : Rn → R. A function f is said to be star-shaped on X if U X ( f, ) is star-shaped for any ∈ R. Let X ⊂ Rn , X ∅ be a convex set, and let f : Rn → R. Since any convex set of Rn is star-shaped, if f is quasiconcave on X, then f is star-shaped on X. When n = 1, X ⊂ R is an interval, and the class of all quasiconcave functions on X coincides with the class of all star-shaped functions on X. In [5,7,8], the quasiconcavity of membership functions is generalized by allowing arbitrary t-norms instead of the minimum operation in Definition 8. Therefore, we generalize the quasiconcavity of membership functions by allowing arbitrary conjunctive aggregation functions instead of the minimum operation in Definition 8. This is a further generalization of the generalization of quasiconcavity presented in [5,7,8]. This further generalization, which is the primary definition of the present study, is defined as follows: Definition 12. Let X ⊂ Rn , X ∅ be a convex set, and let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function. In addition, let : Rn → [0, 1]. (i) is said to be G 2 -quasiconcave (or G-quasiconcave) on X if (x + (1 − )y) ≥ G 2 ((x), (y)) for any x, y ∈ X and any ∈]0, 1[. (ii) is said to be strictly G 2 -quasiconcave (or strictly G-quasiconcave) on X if is G-quasiconcave on X and (x + (1 − )y) > G 2 ((x), (y))
(4)
for any x, y ∈ X, (x)(y) and any ∈]0, 1[. (iii) is said to be strongly G 2 -quasiconcave (or strongly G-quasiconcave) on X if (4) holds for any x, y ∈ X, xy and any ∈]0, 1[. There are implicational relationships among the well-known quasiconcavities in Definition 8 and the G-quasiconcavities presented in Definition 12. For a convex set X ⊂ Rn , X ∅ and a conjunctive aggregation function G 2 : [0, 1]2 → [0, 1] and : Rn → [0, 1], is strongly quasiconcave ⇒ strictly quasiconcave ⇒ quasiconcave ⇓ ⇓ ⇓ strongly G-quasiconcave ⇒ strictly G-quasiconcave ⇒ G-quasiconcave on X from Definition 12. 3.2. Examples of G-quasiconcavity In the following, we present some examples of G-quasiconcavity.
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3.2.1. First examples Let ∈]0, 1[. Let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function, and let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function defined as follows: 0 if x ∈ [0, 1 − ] or y ∈ [0, 1 − ], G 2 (x, y) = G 2 (x, y) otherwise for x, y ∈ [0, 1]. Set X = R. (i) A membership function : R → [0, 1] defined as ⎧ 0 if x ∈] − ∞, 0] ∪ [5, ∞[, ⎪ ⎪ ⎪ ⎨ x − 2 + 1 if x ∈ [1, 2], (x) = ⎪ 1 if x ∈ [2, 3], ⎪ ⎪ ⎩ −x + 3 + 1 if x ∈ [3, 4], where (x) ∈ [0, 1 − ] if x ∈]0, 1[∪]4, 5[ is G-quasiconcave on X and is not strictly G-quasiconcave on X (Fig. 1), as shown in the following. Let x, y ∈ R, and let ∈]0, 1[. If x ∈] − ∞, 1[∪]4, ∞[ or y ∈] − ∞, 1[∪]4, ∞[, then we have (x + (1 − )y) ≥ 0 = G 2 ((x), (y)). If x, y ∈ [1, 4], then we have (x + (1 − )y) ≥ min{(x), (y)} ≥ G 2 ((x), (y)) because is quasiconcave on [1, 4]. Therefore, is G-quasiconcave on X. For x0 = −1, y0 = 1, we have ( 21 x0 + 21 y0 ) = (0) = 0G 2 ((x0 ), (y0 )) = G 2 (0, 1 − ) = 0. Therefore, is not strictly G-quasiconcave on X. (ii) A membership function : R → [0, 1] defined as ⎧ if x ∈ [0, 1], ⎪ ⎨ x + 1 − if x ∈ [1, 2], (x) = 1 ⎪ ⎩ −x + 2 + 1 if x ∈ [2, 3], where (x) ∈]0, 1 − ] if x ∈] − ∞, 0[∪]3, ∞[ is strictly G-quasiconcave on X and is not strongly G-quasiconcave on X (Fig. 2), as shown in the following. Let x, y ∈ R, and let ∈]0, 1[. First, assume that (x)(y). If x ∈] − ∞, 0[∪]3, ∞[ or y ∈] − ∞, 0[∪]3, ∞[, then we have (x + (1 − )y) > 0 = G 2 ((x), (y)). If x, y ∈ [0, 3], then we have (x + (1 − )y) > min{(x),
Fig. 1. (i) (x) ( = 0.75).
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Fig. 2. (ii) (x) ( = 0.75).
Fig. 3. (iii) (x) ( = 0.75).
(y)} ≥ G 2 ((x), (y)) because is strictly quasiconcave on [0,3]. Next, assume that (x) = (y). If x ∈] − ∞, 0[∪]3, ∞[ or y ∈] − ∞, 0[∪]3, ∞[, we have (x + (1 − )y) > 0 = G 2 ((x), (y)). If x, y ∈ [0, 3], then we have (x + (1 − )y) ≥ min{(x), (y)} ≥ G 2 ((x), (y)) because is quasiconcave on [0, 3]. Therefore, is strictly G-quasiconcave on X. For x0 = 1, y0 = 2, we have ( 21 x0 + 21 y0 ) = ( 23 ) = 1G 2 ((x0 ), (y0 )) = G 2 (1, 1) = 1. Therefore, is not strongly G-quasiconcave on X. (iii) A membership function : R → [0, 1] defined as x + 1 − if x ∈ [0, 1], (x) = −x + 1 + if x ∈ [1, 2], where (x) ∈]0, 1 − ] if x ∈] − ∞, 0[∪]2, ∞[ is strongly G-quasiconcave on X (Fig. 3), as shown in the following. Let x, y ∈ R, xy, and let ∈]0, 1[. If x ∈]−∞, 0[∪]2, ∞[ or y ∈]−∞, 0[∪]2, ∞[, then we have (x +(1−)y) > 0 = G 2 ((x), (y)). If x, y ∈ [0, 2], then we have (x + (1 − )y) > min{(x), (y)} ≥ G 2 ((x), (y)) because is strongly quasiconcave on [0,2]. Therefore, is strongly G-quasiconcave on X.
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Examples (i)–(iii) above show that G-quasiconcavity allows some differences from quasiconcavity and that such differences can be controlled by conjunctive aggregation functions. Fuzzy numbers are often used to model real-world problems. A fuzzy number is a fuzzy set on R with some restrictions and is interpreted as the fuzzy set of real numbers around a for some real number a ∈ R. Let a : R → [0, 1] be the membership function of this fuzzy set. Quasiconcavity of a is usually assumed. For each x ∈ R, the value a (x) indicates the grade of x in the fuzzy set. For a conjunctive aggregation function G 2 : [0, 1]2 → [0, 1], such as that considered in this section, if we assume G-quasiconcavity of a rather than quasiconcavity of a , then we can also consider a generalization of fuzzy numbers. Such generalization allows some differences from quasiconcavity. If membership functions of fuzzy sets on Rn are determined from, for example, observations or experiments and if the obtained membership functions exhibit differences from quasiconcavity and their differences are not small enough to neglect, then G-quasiconcavity can be expected to be useful and suitable for representations of the real world. 3.2.2. Second examples ( p) For each p ∈ [1, ∞[, let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function defined as ( p)
G 2 (x, y) = [min{x, y}] p for x, y ∈ [0, 1]. Set X = R. For each ∈ [0, 21 [, let : R → [0, 1] be a membership function defined as (Fig. 4) ⎧ 0 if x ∈] − ∞, 0] ∪ [6, ∞[, ⎪ ⎪ ⎪ ⎪ 1 ⎪ if x ∈ [0, 1], ⎪ ⎪ 2x ⎪ ⎪ ⎨ sin 4x + 1 if x ∈ [1, 2] ∪ [4, 5], 2 (x) = 1 1 ⎪ x − if x ∈ [2, 3], ⎪ 2 ⎪2 ⎪ ⎪ 1 5 ⎪ −2x + 2 if x ∈ [3, 4], ⎪ ⎪ ⎪ ⎩ 1 −2x + 3 if x ∈ [5, 6]. Then, 0 is quasiconcave on X, and not every , ∈]0, 21 [ is quasiconcave on X. For ∈]0, 21 [, we show that is G ( p) -quasiconcave on X for p ∈ [ p , ∞[, and that is not G ( p) -quasiconcave on X for p ∈ [1, p [, where p = log( 21 − )/log( 21 + ). Fix any ∈]0, 21 [, and set a = 1 − 2, b = 2 + 2, c = 4 − 2, and d = 5 + 2 (Fig. 4). Solving ( 21 + ) p ≤ 21 − in p, we have p ≥ log( 21 − )/log( 21 + ) = p . First, we assume that p ∈ [ p , ∞[. Let x, y ∈ R, and let ∈ [0, 1]. If x ∈ [a , b ]∪[c , d ] or y ∈ [a , b ]∪[c , d ], ( p) then we have (x + (1 − )y) ≥ 21 − ≥ ( 21 + ) p ≥ [min{ (x), (y)}] p = G 2 ( (x), (y)) because (x) ≤ ( p) 1 1 2 + or (y) ≤ 2 + . If x, y ∈]b , c [, then we have (x + (1 − )y) ≥ min{ (x), (y)} ≥ G 2 ( (x), (y)).
Fig. 4. (x) ( = 0.25).
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If x ∈] − ∞, a [∪]d , ∞[ or y ∈] − ∞, a [∪]d , ∞[, then we have (x + (1 − )y) ≥ min{ (x), (y)} ≥ ( p) G 2 ( (x), (y)). Therefore, is G ( p) -quasiconcave on X. 13 1 Next, we assume that p ∈ [1, p [. Set x0 = 98 , y0 = 37 8 and 0 = 14 . Then, since (x 0 ) = (y0 ) = + 2 and ( p) ( p) (0 x0 + (1 − 0 )y0 ) = − + 21 , we have G 2 ( (x0 ), (y0 )) = G 2 ( + 21 , + 21 ) = [min{ + 21 , + 21 }] p = ( + 21 ) p > 21 − = (0 x0 + (1 − 0 )y0 ). Therefore, is not G ( p) -quasiconcave on X. 3.3. Properties of G-quasiconcavity In the following, we investigate the properties of G-quasiconcavity. The following theorem gives conditions such that the concept of quasiconcavity of the membership functions defined on R coincides with the concept of the G-quasiconcavity of the membership functions. Theorem 1. Let X ⊂ R, X ∅ be a convex set, and let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function such that G 2 ≥ TD . Let : R → [0, 1] such that Core X ()∅. If is G-quasiconcave, strictly G-quasiconcave, or strongly G-quasiconcave on X, then is quasiconcave, strictly quasiconcave, or strongly quasiconcave on X, respectively. Proof. We show only that if is G-quasiconcave on X then is quasiconcave on X. The proofs for strict G-quasiconcavity and strong G-quasiconcavity are similar and so are omitted. Since Core X ()∅, there exists x ∈ X such that (x) = 1. Since G 2 is conjunctive and G 2 ≥ TD , we have G 2 (, 1) = G 2 (1, ) = , ∈ [0, 1].
(5)
Assume that is G-quasiconcave on X. Let x, y ∈ X , and let ∈]0, 1[. Then, without loss of generality, assume that x ≤ y. Assume that x + (1 − )y ≤ x. Then, since x ≤ x + (1 − )y ≤ x, there exists ∈ [0, 1] such that x + (1 − )y = x + (1 − )x. We have (x + (1 − )y) = (x + (1 − )x) = (x) = 1 ≥ min{(x), (y)} when = 0, and (x + (1 − )y) = (x + (1 − )x) = (x) ≥ min{(x), (y)} when = 1. Thus, we assume that ∈]0, 1[. Then, from G-quasiconcavity of on X and (5), we have (x + (1 − )y) = (x + (1 − )x) ≥ G 2 ((x), (x)) = (x) ≥ min{(x), (y)}. Similarly, when x ≤ x + (1 − )y, it can be shown that (x + (1 − )y) ≥ min{(x), (y)}. Therefore, is quasiconcave on X. The following examples reveal that neither the condition G 2 ≥ TD nor the condition Core X ()∅ in Theorem 1 can be eliminated. Example 2. (i) Set X = R, and define a conjunctive aggregation function G 2 : [0, 1]2 → [0, 1] as 1 if x = y = 1, G 2 (x, y) = 0 otherwise for x, y ∈ [0, 1], and define : R → [0, 1] as 1 if x = 0, (x) = 1 1 3 sin x + 2 otherwise for x ∈ R. In this case, G 2 ⱖ TD and Core X () = {0}∅. Since is strongly G-quasiconcave on X, is strictly G-quasiconcave on X and is G-quasiconcave on X. However, is not quasiconcave on X. Thus, is neither strictly quasiconcave on X nor strongly quasiconcave on X. (ii) Set X = R and G 2 = TD and define : R → [0, 1] as (x) = 13 sin x + 21 for x ∈ R. In this case, G 2 ≥ TD and Core X () = ∅. Since is strongly G-quasiconcave on X, is strictly G-quasiconcave on X and is quasiconcave on X. However, is not quasiconcave on X. Thus, is neither strictly quasiconcave on X nor strongly quasiconcave on X. The following theorem provides a relationship among the star-shaped character of membership functions, their quasiconcavity from a point, and their TD -quasiconcavity.
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Theorem 2 ( Ramík and Vlach [5, Theorem 4.34]). Let X ⊂ Rn , X ∅ be a convex set, and let : Rn → [0, 1]. Then, the following statements are equivalent. (i) Core X () ⊂ ∈[0,1] Ker(U X (, )). (ii) is quasiconcave on X from x for any x ∈ Core X (). (iii) is TD -quasiconcave on X. The following theorem provides a relationship between the strict quasiconcavity of membership functions from a point and their strict TD -quasiconcavity. Theorem 3. Let X ⊂ Rn , X ∅ be a convex set, and let : Rn → [0, 1]. Then, the following statements hold. (i) If is strictly TD -quasiconcave on X, then is strictly quasiconcave on X from x for any x ∈ Core X (). (ii) Assume that (x) > 0, x ∈ X . If is strictly quasiconcave on X from x for any x ∈ Core X (), then is strictly TD -quasiconcave on X. Proof. (i) Let x ∈ Core X (). Since is strictly TD -quasiconcave on X, is TD -quasiconcave on X. From Theorem 2, is quasiconcave on X from x. Thus, in order to show that is strictly quasiconcave on X from x, it is sufficient to show that, for each x ∈ X, xx, x,x (t) = 1, t ∈ [0, 1] if (x) = 1 and x,x is strictly monotone increasing on [0, sx ] if (x) < 1, where sx = inf{t ∈ L X (x, x) : x,x (t) = 1} ≤ 1. When x ∈ X , xx, (x) < 1, based on the quasiconcavity of x,x on L X (x, x), we have sx ≥ 0. Let x ∈ X, xx. First, assume that (x) = 1. Then, x,x (0) = (x) = 1, x,x (1) = (x) = 1, and, based on the TD -quasiconcavity of on X, we have 1 ≥ x,x (t) = (x + t(x − x)) ≥ TD ((x), (x)) = TD (1, 1) = 1 for any t ∈]0, 1[. Next, assume that (x) < 1. This is trivial when sx = 0. Thus, assume that sx > 0. Choose any u, v ∈ R such that 0 ≤ u < v ≤ sx , and set x1 = x + u(x − x), x2 = x + v(x − x). Since u < sx , x,x (u) = (x1 ) < 1. When v = sx , if x,x (sx ) = 1, then x,x (u) < 1 = x,x (sx ) = x,x (v), and if x,x (sx ) < 1, then sx < 1 and there exists ∈]0, 1[ such that x2 = x1 + (1 − )x, then, based on the strict TD -quasiconcavity of on X, we have x,x (v) = (x2 ) = (x1 + (1 − )x) > TD ((x1 ), (x)) = (x1 ) = x,x (u). When v < sx , there exists ∈]0, 1[ such that x2 = x1 + (1 − )x, and, based on the strict TD -quasiconcavity of on X, we have x,x (v) = (x2 ) = (x1 + (1 − )x) > TD ((x1 ), (x)) = (x1 ) = x,x (u). (ii) Based on the assumption, x,x is strictly quasiconcave on L X (x, x) for any x ∈ Core X () and any x ∈ X, xx. Thus, x,x is quasiconcave on L X (x, x) for any x ∈ Core X () and any x ∈ X, xx. In other words, is quasiconcave on X from x for any x ∈ Core X (). From Theorem 2, is TD -quasiconcave on X. Let x , x ∈ X, (x )(x), and let ∈]0, 1[. First, assume that (x ) = 1. Then, since x ∈ Core X (), x,x is strictly quasiconcave on L X (x, x ). Thus, we have (x + (1 − )x) > TM ((x ), (x)) ≥ TD ((x ), (x)). A similar result can be obtained when (x) = 1. Next, assume that (x ) < 1, (x) < 1. Then, we have (x + (1 − )x) > 0 = TD ((x ), (x)). The following theorem presents a relationship between the strong quasiconcavity of membership functions from a point and their strong TD -quasiconcavity. Theorem 4. Let X ⊂ Rn , X ∅ be a convex set, and let : Rn → [0, 1]. Then, the following statements hold. (i) If is strongly TD -quasiconcave on X, then is strongly quasiconcave on X from x for any x ∈ Core X (). (ii) Assume that (x) > 0, x ∈ X . If is strongly quasiconcave on X from x for any x ∈ Core X (), then is strongly TD -quasiconcave on X. Proof. (i) Let x ∈ Core X (). Then, in order to show that is strongly quasiconcave on X from x, it is sufficient to show that x,x is strictly monotone increasing on [0, 1] for each x ∈ X, xx. In order to show this, it is sufficient to show that x,x (0) < x,x () for each x ∈ X, xx and any ∈]0, 1]. Let x ∈ X, xx, and let ∈]0, 1]. If (x) = 1, then from the strong TD -quasiconcavity of on X, 1 ≥ (x+(1−)x) > TD ((x), (x)) = TD (1, 1) = 1 for any ∈]0, 1[, which is a contradiction. Thus, (x) < 1. If = 1, then we have x,x (0) = (x) < 1 = (x) = x,x (1) = x,x (). Thus, assume that ∈]0, 1[. Then, we have x,x () = (x + (x − x)) > TD ((x), (x)) = TD (1, (x)) = (x) = x,x (0) from the strong TD -quasiconcavity of on X.
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(ii) Based on the assumption, x,x is strongly quasiconcave on L X (x, x) for any x ∈ Core X () and any x ∈ X, xx. Let x , x ∈ X, x x, and let ∈]0, 1[. First, assume that (x ) = 1. Then, since x ∈ Core X (), x,x is strongly quasiconcave on L X (x, x ). Thus, we have (x + (1 − )x) > TM ((x ), (x)) ≥ TD ((x ), (x)). A similar result can be obtained when (x) = 1. Next, assume that (x ) < 1, (x) < 1. Then, we have (x + (1 − )x) > 0 = TD ((x ), (x)). The following example reveals that the condition (x) > 0, x ∈ X in Theorems 3(ii) and 4(ii) cannot be eliminated. Example 3. Let x = (x, y) ∈ R2 , and set X = {x ∈ R2 : |x| + |y| ≤ 1}. Define : R2 → [0, 1] as (x) = 2 max{1 − |x| − |y|, 0} and define : R → [0, 1] as 1 − 21 (|x| + |y|) if x ∈ [−1, 1] × {0}, (x) = (x) otherwise. In this case, Core X () = {0} and not (x) > 0, x ∈ X . Since is strongly quasiconcave on X from 0, is also strictly quasiconcave on X from 0. On the other hand, if we set y = (1, 0), z = (0, 1), then (y) = 21 (z) = 0 and ( 21 y + 21 z) = 0 = TD ((y), (z)). Thus, is not strictly TD -quasiconcave on X. In addition, is not strongly TD -quasiconcave on X. Theorem 5. Let X ⊂ Rn , X ∅ be a convex set. In addition, let : Rn → [0, 1] such that Core X ()∅, and let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function such that G 2 ≥ TD . Assume that is G-quasiconcave on X. Then, is star-shaped on X. Proof. Since is G-quasiconcave on X and G 2 ≥ TD , (x + (1 − )y) ≥ G 2 ((x), (y)) ≥ TD ((x), (y)) for any x, y ∈ X and any ∈]0, 1[. Thus, is TD -quasiconcave on X. Since ∅Core X () ⊂ ∈[0,1] Ker(U X (, )) from Theorem 2, is star-shaped on X. Example 2 above indicates that neither the condition Core X ()∅ nor the condition G 2 ≥ TD in Theorem 5 can be eliminated. 3.4. Aggregation of G-quasiconcave membership functions When several G-quasiconcave membership functions are aggregated using an aggregation function, a new aggregated membership function can be obtained. We investigate the properties of such aggregated membership functions. Let i : Rn → [0, 1], i ∈ I , and let G m : [0, 1]m → [0, 1] be an aggregation function. Then, we define a function : Rn → [0, 1] as (x) = G m (1 (x), . . . , m (x)), x ∈ Rn ,
(6)
and we investigate its properties related to G-quasiconcavity. Theorem 6. Let X ⊂ Rn , X ∅ be a convex set. Let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function such n m that G 2 ≥ T D , and let G m : [0, 1] → [0, 1] be an aggregation function. Then, let i : R → [0, 1], i ∈ I . Moreover, assume that i∈I Core X (i )∅ and that i , i ∈ I are G -quasiconcave on X. Then, a function : Rn → [0, 1] defined by (6) is star-shaped on X. Proof. Let ∈ R. If > 1, then U X (, ) = ∅ is star-shaped. Thus, assume that ≤ 1. In order to show that U X (, ) is star-shaped, we fix any x ∈ i∈I Core X (i ) and show that x ∈ Ker(U X (, )). Since 1 (x) = · · · = m (x) = 1, we have (x) = G m (1 (x), . . . , m (x)) = G m (1, . . . , 1) = 1 ≥ . Thus, x ∈ U X (, ). Let x ∈ U X (, ), and let ∈]0, 1[. Then, we set z = x + (1 − )x. If x = x, then z = x + (1 − )x ∈ U X (, ). Thus, we assume that xx. Since i , i ∈ I are G -quasiconcave on X and G 2 ≥ TD , i , i ∈ I are also TD -quasiconcave on X. From Theorem 2, (i )x,x , i ∈ I are quasiconcave on L X (x, x). Thus, i (x) ≤ i (z), i ∈ I . Based on the
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monotonicity of G m , G m (1 (x), . . . , m (x)) ≤ G m (1 (z), . . . , m (z)). Since x ∈ U X (, ), we have ≤ (x) ≤ (z). Thus, z = x + (1 − )x ∈ U X (, ). Consequently, x ∈ Ker(U X (, )) and U X (, ) is star-shaped. Theorem 7. Let X ⊂ Rn , X ∅ be a convex set. Let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function, and let G m : [0, 1]m → [0, 1] be an aggregation function such that G m ?G 2 . Let i : Rn → [0, 1], i ∈ I , and let : Rn → [0, 1] be a function defined by (6). If i , i ∈ I are G -quasiconcave, strictly G -quasiconcave, or strongly G -quasiconcave on X, then is G -quasiconcave, strictly G -quasiconcave, or strongly G -quasiconcave on X, respectively. Proof. We show only that if i , i ∈ I are G -quasiconcave on X, then is G -quasiconcave on X. The other results can be proven in a similar manner. Let x, y ∈ X and ∈]0, 1[. Since i , i ∈ I are G -quasiconcave on X, i (x+(1−)y) ≥ G 2 (i (x), i (y)), i ∈ I . Since G m ?G 2 , we have (x + (1 − )y) = G m (1 (x + (1 − )y), . . . , m (x + (1 − )y)) ≥ G m (G 2 (1 (x), 1 (y)), . . . , G 2 (m (x), m (y))) ≥ G 2 (G m (1 (x), . . . , m (x)), G m (1 (y), . . . , m (y))) = G 2 ((x), (y)). We have the following corollary from Lemma 1 and Theorem 7. Corollary 1. Let X ⊂ Rn , X ∅ be a convex set. Let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function, and let i : Rn → [0, 1], i ∈ I . Let 1 : Rn → [0, 1] be a function defined as 1 (x) = TM (1 (x), . . . , m (x)), x ∈ Rn , j j j and let 2 : Rn → [0, 1], j ∈ I2 be functions defined as 2 (x) = G m (1 (x), . . . , m (x)), x ∈ Rn , j ∈ I2 , where j G m : [0, 1]m → [0, 1], j ∈ I2 are aggregation functions defined by (1) and (2). (i) If i , i ∈ I are G-quasiconcave on X, then 1 is G-quasiconcave on X. Moreover, if G 2 is commutative and j associative or G 2 ?G 2 , then 2 , j ∈ I2 are G-quasiconcave on X. (ii) If i , i ∈ I are strongly G-quasiconcave on X, then 1 is strongly G-quasiconcave on X. Moreover, if G 2 is j strictly monotone increasing, and if G 2 is commutative and associative or G 2 ?G 2 , then 2 , j ∈ I2 are strongly G-quasiconcave on X. 4. Fuzzy programming problems In this section, we consider the application of G-quasiconcavity of membership functions to fuzzy programming problems. 4.1. Problem formulation We consider a fuzzy multicriteria programming problem formulated as max (x) = ( (x), . . . , (x)) 1 m F (P (X )) s.t. x ∈ X and a fuzzy scalar programming problem formulated as max (x) = G ( (x), . . . , (x)) m 1 m F (PG (X )) s.t. x ∈ X where X ⊂ Rn , X ∅ and i : Rn → [0, 1], i ∈ I and F = {i : i ∈ I }, and G m : [0, 1]m → [0, 1] is an aggregation function. The fuzzy scalar programming problem (PGF (X )) is a problem that scalarizes the fuzzy multicriteria programming problem by aggregating membership functions in the objective function of the fuzzy multicriteria programming problem using an aggregation function. Consider the following situation. I is a set of m decision makers, and X ⊂ Rn , X ∅ is a set of feasible actions (or alternatives). For each feasible action x ∈ X and each i ∈ I , i (x) represents the degree of satisfaction for the feasible action x with respect to the decision maker i. Then, (P F (X )) can be interpreted as the problem of finding
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a feasible action that maximizes the degree of satisfaction with respect to each decision maker, and (PGF (X )) can be interpreted as the problem of finding a feasible action that maximizes a mean of the degree of satisfaction with respect to each decision maker. Furthermore, the G-quasiconcavity of i , i ∈ I allows some differences from quasiconcavity, as mentioned in Section 3.2.1. Definition 13 (See, for example, Ramík and Vlach [5]). (i) A point x0 ∈ X is called a weak Pareto optimal solution of (P F (X )) if there is no x ∈ X such that (x0 ) < (x). (ii) A point x0 ∈ X is called a Pareto optimal solution of (P F (X )) if there is no x ∈ X such that (x0 ) ≤ (x) and (x0 )(x). (iii) A point x0 ∈ X is called a strictly Pareto optimal solution of (P F (X )) if there is no x ∈ X such that xx0 and (x0 ) ≤ (x). F , X F , and X F be sets of all weak Pareto, Pareto, and strictly Pareto optimal solutions of (P F (X )), respecLet X W P P SP F . tively. From Definition 13, X SFP ⊂ X PF ⊂ X W P F Let X G be the set of all optimal solutions of (PGF (X )). An optimal solution of (PGF (X )) is called a compromise solution. In [8], the properties of compromise solutions related to T-quasiconcavity for a t-norm T are investigated. In [5–8], the properties of compromise solutions are investigated when an aggregation function G m in (PGF (X )) is an arbitrary t-norm or TM . All of these approaches are generalizations of the max–min approach in [2].
4.2. Auxiliary results In the following, we present some auxiliary results. Lemma 2. Let X ⊂ Rn , X ∅, and let : Rn → [0, 1]. (i) If all U X (, ), ∈]0, 1] are closed, then is upper semicontinuous on X. (ii) If X is closed and is upper semicontinuous on X, then all U X (, ), ∈ [0, 1] are closed. Proof. (i) Let x0 ∈ X , and let > 0. Then, in order to show that there exists > 0 such that (x) < (x0 ) + for any x ∈ B(x0 , ) = {x ∈ X : x − x0 < }, where · is Euclidean norm, assume that for any > 0 there exists x ∈ B(x0 , ) such that (x) ≥ (x0 ) + . For each k ∈ N, there exists xk ∈ B(x0 , 1/k) such that (xk ) ≥ (x0 ) + . Set = (x0 )+ > 0. Based on the assumption, U X (, ) is closed. Thus, since {xk } ⊂ U X (, ) and limk→∞ xk = x0 , we have x0 ∈ U X (, ), (x0 ) ≥ = (x0 ) + , and ≤ 0, which contradicts the statement that > 0. (ii) Let ∈ [0, 1]. Let {xk } ⊂ U X (, ) be any convergent sequence, and let x0 ∈ Rn be its limit. Since {xk } ⊂ X and X is closed, x0 ∈ X . Then, for each k ∈ N, since xk ∈ U X (, ), (xk ) ≥ . From the upper semicontinuity of on X, we have ≤ lim supk→∞ (xk ) ≤ (x0 ). Therefore, x0 ∈ U X (, ), and U X (, ) is closed. Based on fundamental calculations in real analysis, we obtain the following theorem. Theorem 8. Let X ⊂ Rn , X ∅, and let i : Rn → [0, 1], i ∈ I . Assume that i , i ∈ I are upper semicontinuous on X. Let G m : [0, 1]m → [0, 1] be an upper semicontinuous aggregation function. Then, a function : Rn → [0, 1] defined by (6) is upper semicontinuous on X. 4.3. Properties of compromise solutions We present some valuable results on compromise solutions. The following theorem gives the sufficient conditions for the existence of compromise solutions. Theorem 9. Let X ⊂ Rn , X ∅ be a closed set. Assume that i : Rn → [0, 1], i ∈ I are level-compact on X. Let F = {i : i ∈ I }, and let G m : [0, 1]m → [0, 1] be an upper semicontinuous idempotent aggregation function. Then, X GF ∅.
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Proof. Let ∈]0, 1]. Then, we show that U X (, ) is compact. First, in order to show that U X (, ) is bounded, assume that U X (, ) is unbounded. Then, there exists {xk } ⊂ U X (, ) such that limk→∞ xk = ∞. Fix any such that 0 < < . Since U X (i , ), i ∈ I are bounded, there / U X (i , ) for any i ∈ I and any k ≥ k0 . Based on the monotonicity and idempotency exists k0 ∈ N such that xk ∈ of G m , we have (xk ) = G m (1 (xk ), . . . , m (xk )) ≤ G m ( , . . . , ) = < for any k ≥ k0 , which contradicts the statement that {xk } ⊂ U X (, ). Therefore, U X (, ) is bounded. Since i , i ∈ I are level-closed on X, i , i ∈ I are upper semicontinuous on X from Lemma 2(i). From Theorem 8, is upper semicontinuous on X. Thus, U X (, ) is closed from Lemma 2(ii). Consequently, U X (, ) is compact, and is level-compact on X. If sup{(x) : x ∈ X } = 0, then X GF = X ∅. Assume that sup{(x) : x ∈ X } > 0, and fix any such that 0 < < sup{(x) : x ∈ X }. Since U X (, )∅ is compact and is upper semicontinuous on U X (, ), there exists x∗ ∈ U X (, ) ⊂ X such that sup{(x) : x ∈ X } = sup{(x) : x ∈ U X (, )} = (x∗ ). Therefore, X GF ∅. The following theorem gives a sufficient condition for a compromise solution to be a weak Pareto optimal solution of (P F (X )). Theorem 10. Let X ⊂ Rn , X ∅, and let i : Rn → [0, 1], i ∈ I . Let F = {i : i ∈ I }, and let G m : [0, 1]m → [0, 1] F . be a strictly monotone increasing aggregation function. Then, X GF ⊂ X W P F , assume that x∗ ∈ F . Then, there exists x ∈ X such that Proof. Let x∗ ∈ X GF . In order to show that x∗ ∈ X W / XW P P ∗ i (x ) < i (x ), i ∈ I . On the other hand, from the strict monotonicity of G m , we have G m (1 (x∗ ), . . . , m (x∗ )) < G m (1 (x ), . . . , m (x )), which contradicts the statement that x∗ ∈ X GF .
For Pareto and strictly Pareto optimal solutions of (P F (X )), results similar to Theorem 10 can be obtained, and the proofs of these solutions are similar to the proof of Theorem 10. Theorem 11. Let X ⊂ Rn , X ∅, and let i : Rn → [0, 1], i ∈ I . Let F = {i : i ∈ I }, and let G m : [0, 1]m → [0, 1] be a strongly monotone increasing aggregation function. Then, X GF ⊂ X PF . Theorem 12. Let X ⊂ Rn , X ∅, and let i : Rn → [0, 1], i ∈ I . Let F = {i : i ∈ I }, and let G m : [0, 1]m → [0, 1] be an aggregation function. Assume that X GF is a singleton. Then, X GF ⊂ X SFP . The following theorem gives sufficient conditions for a compromise solution to be unique when the compromise solution exists. Theorem 13. Let X ⊂ Rn , X ∅ be a convex set. Assume that i : Rn → [0, 1], i ∈ I are strongly quasiconcave on X. Let F = {i : i ∈ I }, and let G m : [0, 1]m → [0, 1] be a strictly monotone increasing aggregation function. Assume also that G m ?TM . If x∗ ∈ X GF , then X GF = {x∗ }. Proof. First, we show that is strongly quasiconcave on X. Let x, y ∈ X, xy, and let ∈]0, 1[. Since i , i ∈ I are strongly quasiconcave on X, we have i (x + (1 − )y) > TM (i (x), i (y)), i ∈ I . Since G m is strictly monotone increasing and G m ?TM , we have (x + (1 − )y) = G m (1 (x + (1 − )y), . . . , m (x + (1 − )y)) > G m (TM (1 (x), 1 (y)), . . . , TM (m (x), m (y))) ≥ TM (G m (1 (x), . . . , m (x)), G m (1 (y), . . . , m (y))) = TM ((x), (y)). Therefore, is strongly quasiconcave on X. Next, assume that there exists x ∈ X GF such that x x∗ . Then, (x ) = (x∗ ). Based on the strong quasiconcavity of on X, we have (x + (1 − )x∗ ) > TM ((x ), (x∗ )) = (x∗ ) for any ∈]0, 1[, which contradicts the statement that x∗ ∈ X GF . The following theorem gives a relationship between local and global optimal solutions of (PGF (X )). Theorem 14. Let X ⊂ Rn , X ∅ be a convex set, and let G 2 : [0, 1]2 → [0, 1] be a conjunctive aggregation function such that G 2 ≥ TD . Let i : Rn → [0, 1], i ∈ I , and let F = {i : i∈ I }. Let G m : [0, 1]m → [0, 1] be an aggregation function. Assume that i , i ∈ I are G -quasiconcave on X and that i∈I Core X (i )∅. If x ∈ X is a strict local optimal solution of (PGF (X )), then x is a unique global optimal solution of (PGF (X )).
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Proof. From Theorem 6, is star-shaped on X. Set = (x). Then, U X (, ) is star-shaped. Since x is a strict local optimal solution of (PGF (X )), there exists an open ball B ⊂ Rn with the center at x such that (x) < (x) for any x ∈ B ∩ X, xx. Now, assume that x is not a global optimal solution of (PGF (X )). Then, there exists v ∈ X such that (x) < (v). Since x, v ∈ U X (, ) and U X (, ) is star-shaped, for any fixed y ∈ Ker(U X (, )), P(x, v) = I(x, y) ∪ I(v, y) ⊂ U X (, ), and there exists z ∈ P(x, v) ∩ B, zx. Since z ∈ U X (, ), (z) ≥ . On the other hand, since z ∈ B ∩ X, zx, we have (z) < (x) = , which is a contradiction. Therefore, x is a global optimal solution of (PGF (X )). Similarly, based on the assumption that x is a strict local optimal solution of (PGF (X )), it can be shown that x is a unique global optimal solution of (PGF (X )). 5. Application in facility location problems In this section, we discuss the application of a fuzzy multicriteria programming problem (P F (X )) and a fuzzy scalar programming problem (PGF (X )) to facility location problems, and the necessity and importance of G-quasiconcavity is revealed. Let I = {1, . . . , m} be the set of customers of a certain facility to be located, and let pi = (ai , bi ) ∈ R2 , i ∈ I be locations associated with customers. Here, pi , i ∈ I are called demand points. Then, the problem of locating the facility is called a single facility location problem. Let x = (x, y) ∈ R2 be the variable location of the facility, and let X ⊂ R2 be a convex feasible region in which the facility can be located. For simplicity, we assume that X = R2 . We also assume that the travel distance from each demand point pi , i ∈ I to the facility x is measured by d(pi , x) = (|ai − x|1/2 + |bi − y|1/2 )2 . In [3], such d is called hyper-rectilinear distance and it is used in facility location problems. We define each membership function i : R2 → [0, 1], i ∈ I as i (x) = 1/(1 + d(pi , x)), x ∈ R2 . The value i (x) represents the degree of satisfaction for location x of the facility with respect to customer i, and customer i prefers shorter travel distances. In this case, the fuzzy multicriteria programming problem (P F (X )) is formulated as
1 1 max , . . . , 1 + d(p1 , x) 1 + d(pm , x) s.t. x ∈ X, and the fuzzy scalar programming problem (PGF (X )) is formulated as
1 1 max G m , ... , 1 + d(p1 , x) 1 + d(pm , x) s.t. x ∈ X. Each i , i ∈ I is a translated function of (x, y) = 1/(1 + (|x|1/2 + |y|1/2 )2 ), x, y ∈ R by pi . Fig. 5 shows the function : R2 → [0, 1]. From simple calculations, since is strongly quasiconcave on X from 0, is strongly TD -quasiconcave
Fig. 5. (x, y) = 1/(1 + (|x|1/2 + |y|1/2 )2 ).
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on X from Theorem 4. This means that there exists a conjunctive aggregation function G 2 : [0, 1]2 → [0, 1] such that is G -quasiconcave on X. Again, from simple calculations, since the level sets of are not convex, is not quasiconcave on X. The method by which to determine a proper aggregation function is not investigated herein, and we can only state that if quasiconcavity (i.e. TM -quasiconcavity) is not appropriate, we can attempt to apply some other aggregation function. 6. Conclusions In the present paper, the quasiconcavity of membership functions was generalized by allowing arbitrary conjunctive aggregation functions rather than the minimum operation. The generalized quasiconcavity of the present study was called G-quasiconcavity. Then, the properties of the G-quasiconcavity of membership functions were investigated. Furthermore, a fuzzy multicriteria programming problem and a fuzzy scalar programming problem were considered. The properties of weak Pareto, Pareto, strictly Pareto optimal solutions, and compromise solutions, and their relationships were then derived. Furthermore, we discussed the application of the obtained results to facility location problems. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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