Non-LR Type Fuzzy Parameters in Mathematical ... - Semantic Scholar
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Non-L-R Type Fuzzy Parameters in Mathematical Programming Problems Cheng-Feng Hu, Murat Adivar, and Shu-Cherng Fang
Abstract—The triangular norm-based operations in fuzzy logic usually lead to non-L-R type fuzzy sets. This study considers mathematical programming problems with non-L-R type fuzzy parameters. It shows that the fuzzy solutions to such problems can be obtained by solving an optimization problem on a mixed domain. The necessary and sufficient conditions for solving the resulting optimization problems are investigated by employing the theory of convex optimization on mixed domains. This is the first attempt to solve the fuzzy optimization problem with non-L-R type membership functions in view of optimization problems on a mixed domain. Index Terms—Fuzzy decision making, fuzzy optimization, fuzzy set theory, mathematical programming, membership function, triangular norms.
A˜ ˜ B μA˜ μB˜ S f˜ g˜i T R R+ Z Z+ N N0 σ ρ Λn ej convΛ n cintΛ n
NOMENCLATURE Fuzzy set. Fuzzy set. ˜ Corresponding membership function of A. ˜ Corresponding membership function of B. Triangular conorm. Fuzzy mapping. Fuzzy mappings, i = 1, 2, . . . , m. Time scale. Real numbers. Positive real numbers Integer numbers. Positive integer numbers. Natural numbers. Positive natural numbers. Forward jump operator. Backward jump operator. Product of time scales Ti , i = 1, 2, . . . , n. jth unit vector Convex hull in Λn . Convex interior in Λn .
conv Rn
Interior of convR n in Rn .
◦
Manuscript received January 22, 2013; revised April 19, 2013 and June 29, 2013; accepted August 7, 2013. Date of publication August 26, 2013; date of current version October 2, 2014. This work was supported in part by the National Science Council of China, under Grant NSC 101-2221-E-214-045. C.-F. Hu is with the Department of Applied Mathematics, National Chiayi University, Chiayi, Taiwan (e-mail: [email protected]). M. Adivar is with the Department of Mathematics, Izmir University of Economics, Izmir 35330, Turkey (e-mail: [email protected]). S.-C. Fang is with the Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695-7906 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2013.2279691
fˆ ξ ∂ f (x) i x i ∂ f (x) i x i Ni± n
Bi
Restricted function. Subgradient of a function. Partial -derivative of f with respect to xi . Partial -derivative of f with respect to xi . Sets in the time scale Ti , i = 1, 2, . . . , n. Frame at the point in Λn .
i=1
r ri c˜ ˜bi c˜j ˜bi,k Ti X ˜ h F (W ) μY˜ a ˜ μa˜ μc˜j μ˜b i , k μc˜ μ˜b i ˜ Z ˜i B μB˜ i μZ˜ (˜ c)α (˜ cj )α (˜bi )α (˜bi,k )α ˜ α (Z) ˜i )α (B ˜α T i,k ˜α T m +1,j
Number of fuzzy parameters in the fuzzy objective of (2). Number of fuzzy parameters in the ith fuzzy constraint of (2). Fuzzy vector. Fuzzy vectors, i = 1, 2, . . . , m. Fuzzy parameters, j = 1, 2, . . . , r. Fuzzy parameters, k = 1, 2, . . . , ri , i = 1, 2, . . . , m. Time scales, i = 1, 2, . . . , n. Closed and bounded interval in R. Fuzzy mapping. Set of all fuzzy subsets on W . Membership function of Y˜ . Fuzzy vector. Membership function of a ˜. Membership functions of c˜j , j = 1, 2, . . . , r. Membership functions of ˜bi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m. Membership function of c˜. Membership functions of ˜bi , i = 1, 2, . . . , m. Fuzzy set. Fuzzy sets, i = 1, 2, . . . , m ˜i , i = 1, 2, . . . , m Membership functions of B Membership function of Z˜ α–level sets of c. ˜ α–level sets of c˜j , i = 1, 2, . . . , n α–level sets of ˜bi , i = 1, 2, . . . , m α–level sets of ˜bi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m ˜ α–level sets of Z. ˜i , i = 1, 2, . . . , m. α–level sets of B α–level sets of ˜bi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m. α–level sets of c˜j , j = 1, 2, . . . , r. I. INTRODUCTION
UZZY optimization has been one of the most well-studied topics in the area of soft computing. Its applications, as well as practical realizations, can be found in many real-life problems. Overviews on the main stream of fuzzy optimization can be found in [12] and [38]. Lai and Hwang [27], [28]
F
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HU et al.: NON-L-R TYPE FUZZY PARAMETERS IN MATHEMATICAL PROGRAMMING PROBLEMS
also provided an insightful survey on fuzzy mathematical programming and fuzzy multiple objective decision making. Comparisons between fuzzy optimization and stochastic optimization for multiobjective programming problems were considered in [39]. Corresponding to linear programming in conventional optimization, fuzzy linear programming methods have been a constantly studied subject in the fuzzy context following the seminal paper of Bellman and Zadeh [4]. Models and methods for solving fuzzy linear programming problems with imprecision in the constraints or the objectives can be easily found in the literature [10], [11], [16], [19], [40], [47]. As stated by Rubin and Narasimhan [34], the heart of the methodology for fuzzy linear programming lies in the construction of membership functions for the objection coefficients, technical coefficients, resource variables, and decision variables [43]. The various types of membership functions have been employed in fuzzy linear programming problems with applications, including the linear membership function [17], [45], [46], linear fuzzy constraints [15], tangent type membership function [31], interval linear membership function [18], exponential membership function [9], inverse tangent membership function [35], logistic type membership function [44], concave piecewise linear membership function [23], piecewise linear membership function [21], [22], dynamics membership function [5], and S-curve membership function [26], [32], [33], [43]. Since the tangent type membership function, exponential membership function, and hyperbolic membership function are nonlinear functions, a fuzzy mathematical programming problem (FMP) using such membership functions results in a nonlinear program [43]. In the literature, a linear membership function is commonly employed to avoid involving nonlinearity. Nevertheless, a linear membership function might not be a suitable representation in many practical situations [44]. Nowadays, fuzzy optimization still receives considerable attention. It is important to look into the foundations of fuzzy optimization in order to meet the current trends, and to develop future research lines [8]. When the concept and methods of fuzzy optimization are applied to real-world decision making, the fuzzy parameters are usually represented by means of fuzzy subsets on the real line, known as fuzzy numbers. Linear programming with fuzzy coefficients has been developed mainly under the assumption that fuzzy coefficients are L-R type fuzzy numbers introduced by Dubois [14]. The reason is that the operations on this type of fuzzy number can be easily performed algebraically, and the resulting fuzzy mathematical programming problems can be reduced to a traditional mathematical programming problem. However, in real applications, the L-R type fuzzy number assumption may not be the most appropriate for modeling the fuzzy parameters [24], [30]. For instance, consider a transportation distribution problem of soft drinks with a fuzzy forecast demand at each destination [29]. Then, the union of two fuzzy sets representing the demand for summer or winter seasons might be considered, which automatically leads to fuzzy forecast demands with non-L-R type fuzzy membership functions. Another good example is to consider assessing the demand of a new product, such as i-watch, by asking professional/consulting opinions from experts. It is very
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Fig. 1. Membership function of the fuzzy set that is obtained by the triangular ˜ conorm-based operation of fuzzy sets A˜ and B.
possible that an optimistic opinion will provide a high demand and a pessimistic opinion will project a low demand. Even both opinions of high and low demands are represented in the L-R type membership functions, the resulting demand membership can hardly be represented by an L-R type membership function using the logic “or” operator. It is well known that the conjunction in fuzzy logic is usually interpreted by triangular norms [25]. The triangular norm-based operations of fuzzy sets often result in membership functions with multiple peaks. This idea can be illustrated clearly by the following simple example. ˜ be two fuzzy sets with the corExample 1: Let A˜ and B responding L-R type membership functions μA˜ (·) and μB˜ (·) respectively, as shown in Fig. 1. Consider a conjunction whose interpretation is given by a triangular conorm (t-conorm for short) S: [0, 1]2 → [0, 1] that is defined as follows [48]: S(μA˜ (x), μB˜ (x)) = max{μA˜ (x), μB˜ (x)}.
(1)
Then, the triangular conorm-based operation of fuzzy sets A˜ and ˜ defined by (1) automatically leads to a non-L-R type fuzzy B set with multiple peaks. To facilitate a more general setting in the formulation of fuzzy decision-making problems, the non-L-R type fuzzy parameters are introduced in this paper. Consider an optimization problem with fuzzy parameters in the following form: maximize
f˜(x, c˜)
s.t. g˜i (x, ˜bi ) ≤ 0, i = 1, 2, . . . , m x ≥ 0,
(2)
where f˜, g˜i , i = 1, 2, . . . , m, are fuzzy mappings, x = (x1 , x2 · · · xn )T is an n-dimensional vector of decision variables, c˜ and ˜bi , i = 1, 2, . . . , m, are fuzzy vectors consisting of fuzzy parameters. Following [10], we show that the fuzzy solution of problem (2) with non-L-R type fuzzy parameters can be obtained by solving an optimization problem over the domain of the product of time scales, where a time scale is defined to be a nonempty closed subset of real numbers. An introduction to the theory of calculus on time scales can be found in the original work by Hilger [20] and the more recent works [1], [2], and [13]. The
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notion of convexity for functions of one variable on a time scale has been introduced in [13]. Moreover, the notions of convexity for functions of several variables and convexity for the subsets in a product space of arbitrary time scales, called a mixed domain, have been thoroughly investigated in [1]. Following the work of [1], the necessary and sufficient conditions for solving the resulting optimization problems on the mixed domain are discussed. This is the first attempt of employing the theory of convex optimization on time scales for solving fuzzy mathematical programming problems. Hopefully, the proposed work is not only a generalization of existing theory but also an initial step for the development of new models and algorithms for fuzzy optimization. The rest of this paper is organized as follows. Some basic concepts and analytic properties regarding the convex sets of a time scale and convex functions defined on the mixed domain are presented in Section II. In Section III, we show that the fuzzy solution of the problem (2) with non-L-R type fuzzy parameters can be obtained by solving an optimization problem on a mixed domain. The necessary and sufficient conditions for solving problem (2) with linear objective function and non-L-R type fuzzy parameters are provided in Section IV. In Section V, numerical examples are included to illustrate the associated results. Conclusions are provided in Section VI. II. OPTIMIZATION ON MIXED DOMAINS In this section, some basic concepts of convex optimization on mixed domains are introduced. All proofs are omitted to keep the paper succinct and readable. Readers may refer to [6] and [7] for a comprehensive review on time scales, and to [1] for the detailed analysis on the theory of convex optimization on mixed domains. It is important to mention that, throughout this study, we assume that a time scale, denoted T, has the topology inherited from the standard topology on the real numbers R. Moreover, we denote the set of integer numbers by Z, natural numbers by N, and positive natural numbers by N0 . Definition 1 [Time scale T]: An arbitrary nonempty closed subset T of real numbers is called a time scale. Definition 2 [Operations on time scales]: Let T be a time scale. The forward jump operator σ: T → T and backward jump operator ρ: T → T are defined by σ(t) = inf{s ∈ T: s > t}
TABLE I OPERATORS σ AND ρ ON SOME PARTICULAR TIME SCALES
TABLE II CLASSIFICATION OF THE POINTS OF A TIME SCALE
{M }. Otherwise Tκ = T. If T has a right-scattered minimum m, then Tκ = T − {m}. Otherwise Tκ = T. Hereafter, we use the notation Λn to denote the product T1 × T2 × · · · × Tn of the time scales Ti , i = 1, 2, . . . , n, and ej to represent the jth unit vector whose jth coordinate is 1 and other coordinates are 0. Definition 3 [Convex set] [1]: The set S ⊂ Λn is said to n i be convex in Λ if and only if m i=1 λi x ∈ S for all m ∈ + 1 2 m S and λ1 , λ2 , . . . , λm ∈ [0, 1] such that Z m, x , x , . . . , x ∈ m i n λ = 1 and i=1 i i=1 λi x ∈ Λ . This definition can alternatively be stated as as follows. Corollary 1 [1]: Let S be a subset of Λn . The set S is convex if and only if convR n (S) ∩ Λn = S, where convR n (S) denotes the convex hull of the set S in Rn . Definition 4 [Convex hull] [1]: The convex hull of a set S in Λn , denoted by convΛ n (S), is the collection of all convex combinations of elements of S in Λn . In other words, x ∈ convΛ n (S) if and only if x=
m
λi xi ∈ Λn
i=1
for some integer m > 0, λi ∈ [0, 1] ∈ R such that and x1 , x2 , . . . , xm ∈ S. Corollary 2 [1]: For any set S in Λn
m i=1
λi = 1,
convΛ n (S) = convR n (S) ∩ Λn .
and ρ(t) = sup{s ∈ T: s < t} respectively. In Table I, we illustrate the operators σ and ρ on some particular time scales. In Table I, q N := {q n : n ∈ N and q > 1} and N20 := {n2 : n ∈ N}. The points of a time scale t ∈ T, can be classified in the following manner. The sets Tκ and Tκ are derived from the time scale T as follows. If T has a left-scattered maximum M, then Tκ = T −
Corollary 3 [1]: Let S ⊂ Λn ; then S is convex in Λn if and only if S = convΛ n (S). We now define more details of a convex set in the product of time scales. Definition 5 [1]: Let S ⊂ Λn be a convex set in Λn . The convex-interior, denoted by cintΛ n (S), are defined by ◦
n cintΛ n (S) =conv R n (S) ∩Λ ◦
n where conv R n (S) indicates the interior of convR n (S) in R .
HU et al.: NON-L-R TYPE FUZZY PARAMETERS IN MATHEMATICAL PROGRAMMING PROBLEMS
Corollary 4 [1]: Let S ⊂ Λn be a convex set in Λn . Then the convex-interior cintΛ n (S) is a convex set in Λn . After defining convex sets in Λn , we turn our attention to the convex functions defined on Λn . Definition 6 [Convex function] [1]: Let S be a convex set in Λn . The function f : S → R is said to be convex if and only if m m i f λi x ≤ λi f (xi ) i=1
i=1
for all m ∈ Z+ , xi ∈ S, i = 1, 2, . . . , m, and λi ∈ [0, 1], i = m i n λ = 1 and 1, 2, . . . , m, such that m i=1 i i=1 λi x ∈ Λ . Observe that, if f : S → R is a convex function, then f (a + λ(b − a)) ≤ f (a) + λ(f (b) − f (a)) holds for all a, b ∈ S and λ ∈ [0, 1] such that a + λ(b − a) ∈ Λn . However, the converse of this statement may not be true for an arbitrary function f defined on mixed domains. Corollary 5 [1]: Let S be a convex set in Λn . If the function f : Rn → R is convex on convR n (S), then the restricted function fˆ := f |S is convex on S. Once convex functions on Λn are defined, we give the first order information by the subgradients. Definition 7 [Subgradient] [1]: Let S be a nonempty convex set in Λn , and f : S → R be a convex function on S. Then, ¯ ∈ S if ξ ∈ Rn is called a subgradient of f at x ¯) for all x ∈ S. f (x) ≥ f (¯ x) + ξ T (x − x Similarly, let f : S → R be a concave function on S (i.e., −f is convex on S). Then, ξ ∈ Rn is called a subgradient of f at x ¯ ∈ S if f (x) ≤ f (¯ x) + ξ T (x − x ¯) for all x ∈ S. Let S be a nonempty convex set in Λn . By Corollary 4, we know that cintΛ n (S) is convex. Then, we have the following two results. Theorem 1 [1]: Let S be a nonempty convex set in Λn and f : S → R be a convex function on S. Then, for x ¯ ∈ cintΛ n (S), there exists a vector ξ such that ¯) for all x ∈ S f (x) ≥ f (¯ x) + ξ T (x − x i.e., ξ is a subgradient of f at x ¯. Theorem 2 [1]: Let S be a nonempty convex set in Λn and f : S → R. If for every point x ¯ ∈ cintΛ n (S) there exists a subgradient vector ξ such that f (x) ≥ f (¯ x) + ξ T (x − x ¯) for all x ∈ S then f is convex on cintΛ n (S). To characterize convex functions in terms of subgradients on mixed domains, the following definitions can be found in [36] and [37] . Definition 8: Let Ti , i ∈ {1, x, . . . , n} be time scales and f : Λn → R be a function. The partial -derivative of f with
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respect to xi ∈ Tκi is defined by f (x1 , . . . , σi (xi ), xi+1 , . . . , xn ) ∂f (x) := slim i →x i i xi σi (xi ) − si s i = σ i ( x i ) f (x1 , x2 , . . . , si , xi+1 , . . . , xn ) − (3) σi (xi ) − si where σi : Ti → Ti is the forward jump operator on the ith time scale Ti . Similarly, the partial -derivative of f : Λn → R is defined by f (x1 , . . . , ρi (xi ), xi+1 , . . . , xn ) ∂f (x) := slim →x i i i xi ρi (xi ) − si s i = ρ i ( x i ) f (x1 , x2 , . . . , si , xi+1 , . . . , xn ) − (4) ρi (xi ) − si where ρi : Ti → Ti is the backward jump operator on the ith time scale Ti . In preparation for the next result, let us define the set n
Bi (x, h± i )
i=1
as a frame at the point x = (x1 , x2 , . . . , xn ) in Λn , where ⎧ ⎫ ⎪ ⎪ n ⎨ ⎬ ± i j ± ± Bi (x, hi ) = se + xj e : s ∈ Ni (x, hi ) ⎪ ⎪ ⎩ ⎭ j=1 j = i
i
e , i = 1, 2, . . . , n, are the unit vectors whose components are determined by 1, i = j eij = δij = 0, i = j the sets Ni± (x, h± i ), i = 1, 2, . . . , n, given by {xi , σi (xi )} if μi (xi ) > 0 ) := Ni+ (x, h+ i [xi , xi + h+ i ) if μi (xi ) = 0 where h+ i > 0, and Ni− (x, h− i )
:=
{ρi (xi ), xi } (xi −
h− i , xi ]
if νi (xi ) > 0 if νi (xi ) = 0
h− i
> 0, are the sets in the time scale Ti . where ∂ f (x) In order for the partial derivatives ∂fi(x) x i and i x i given by (3) and (4) to be well defined at a point x = (x1 , x2 , . . . , xn ) ∈ S, one has to assume that n Bi (x, h± (5) i )⊂S i=1
to guarantee that the vectors (x1 , x2 , . . . , σi (xi ), xi+1 , . . . , xn ) and (x1 , x2 , . . . , si , xi+1 , . . . , xn ), s ∈ Ni± (x, h± i ), i = 1, 2, . . . , n, are in S. Note that the condition (5) also implies x ∈ cintΛ n (S). Then, we are ready to state the next result. Theorem 3 [1]: Let S be a nonempty convex set in Λn . Let f : S → R be a function such that the partial derivatives ∂ f (x) ∂ f (x) ¯ and i x i |x= x ¯ , i = 1, 2, . . . , n, exist at any point i x i |x= x
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x ¯ = (¯ x1 , x ¯2 , . . . , x ¯n ) ∈ cintΛ n (S) satisfying (5). If f is convex x) ∈ [0, 1], i = 1, 2, . . . , n, such on S, then there exist scalars λi (¯ that the vector n ∂f (x) ∂f (x) x) + (1 − λi (¯ x)) ξ(¯ x) := λi (¯ ei i xi x= x¯ i xi x= x¯ i=1 (6) is a subgradient of f at any point x ¯ ∈ cintΛ n (S), i.e f (x) ≥ f (¯ x) + ξ(¯ x)T (x − x ¯) for all x ∈ S. For convex functions defined on Λn , we now study their minimum or maximum solutions. Definition 9 [1]: For a given function f : Λn → R and a given set S ⊂ Λn , consider the following problem: minimize f (x) subject to x ∈ S.
(7)
A point x ∈ S is called a feasible solution to the problem. If x ¯ ∈ S and f (x) ≥ f (¯ x) for all x ∈ S, then x ¯ is called an optimal solution or a global optimal solution [3]. The collection of optimal solutions are called alternative optimal solutions. Theorem 4 [1]: Let f : Λn → R be a convex function and S ¯ ∈ S is an optimal be a nonempty convex set in Λn . The point x solution to the problem (22) if and only if f has a subgradient ξ at x ¯ such that
corresponding to (8) is given as follows: s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m x ∈ T1 × T 2 × · · · × T n
h: T1 × T2 × · · · × Tn × F (Xn ) → F (R) , where F (W ) is the set of all fuzzy subsets on W . Let aj , j = 1, 2, . . . , n, be the fuzzy sets on X with corresponding membership functions μa j , j = 1, 2, . . . , n, respectively. The membership function of the fuzzy set Y := h(x, a) is defined as follows: sup{μa (a): y = h(x, a)}, if {a: y = h(x, a)} = ∅ μY (y) = 0, otherwise a2 , . . . , an )T is a fuzzy vector with the memwhere a = ( a1 , bership function given by μa (a) :=
Corollary 6 [1]: Under the assumptions of Theorem 5, if S ⊂ Λn is convex with ◦
n cintΛ n (S) =conv R n (S) ∩Λ = ∅
Let r be the number of fuzzy parameters in the fuzzy objective of (2), ri , i = 1, 2, . . . , m, be the number of fuzzy parameters in the ith fuzzy constraint of (2), and T1 , T2 , . . . , Tn be time scales. Replacing the nonnegativity condition x ≥ 0 in (2) with x ∈ T1 × T2 × · · · × Tn , the problem (2) can be described in the following form: f(x, c)
{μa j (aj ): aj ∈ aj } .
cj := {(cj , μc j (cj )): cj ∈ X}, j = 1, 2, . . . , r bi,k := {(bi,k , μ (bi,k )): bi,k ∈ X}, k = 1, 2, . . . , ri bi , k for each i ∈ {1, 2, . . . , m} with corresponding membership functions μc j , j = 1, 2, . . . , r, and μb i , k , k = 1, 2, . . . , ri , for each i ∈ {1, 2, . . . , m}. The fuzzy vectors c and bi , i = 1, 2, . . . , m, are described as fuzzy sets c := {(c, μc (c)): c ∈ Xr }
(10)
and bi := {(bi , μ (bi )): bi ∈ Xr i }, bi
i = 1, 2, . . . , m
with corresponding membership functions μc and μb i , i = 1, 2, . . . , m, defined by
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m x ∈ T1 × T 2 × · · · × T n
min
j =1,2,...,n
The fuzzy parameters cj , j = 1, 2, . . . , r, and bi,1 , bi,2 , . . . , bi,r , i = 1, 2, . . . , m, in (8) can be described as fuzzy sets on i X as follows:
then x ¯ ∈ cintΛ n (S) is an optimal solution to the problem (22) if and only if there exists a zero subgradient of f at x ¯. III. FUZZY MATHEMATICAL PROGRAMMING PROBLEM WITH NON-L-R TYPE FUZZY PARAMETERS
(9)
where c := (c1 , c2 , . . . , cr )T ∈ Rr , and bi := (bi,1 , bi,2 , . . . , bi,r i )T ∈ Rr i , i = 1, 2, . . . , m, are real vectors. Inspired by [41, Definition 1], we first define a fuzzification of the function h as follows. Definition 10: The fuzzification of the function h: T1 × T2 × · · · × Tn × Xn → R is a fuzzy mapping denoted by
¯) ≥ 0 for all x ∈ S. ξ T (x − x
max
f (x, c)
max
(8)
where c := ( c1 , c2 , . . . , cr )T and bi := (bi,1 , bi,2 , . . . , bi,r i )T , i = 1, 2, . . . , m, are fuzzy vectors with cj , j = 1, 2, . . . , r, and bi,1 , bi,2 , . . . , bi,r i , i = 1, 2, . . . , m, being fuzzy parameters, and T1 , T2 , . . . , Tn are time scales consisting of nonnegative reals. Suppose that X is a closed and bounded interval in R. Let f : T1 × T2 × · · · × Tn × Xr → R, gi : T1 × T2 × ·s × Tn × Xr i → R, i = 1, 2, . . . , m, be crisp functions. A conventional mathematical programming problem on time scales
μc (c) :=
min {μc j (cj ): c = (c1, c2 , . . . , cr )T and cj ∈ X}
j =1,2,...,r
(11) and μb i (bi ) :=
min
k =1,2,...,r i
{μb i , k (bi,k ): bi = (bi,1 , bi,2 , . . . , bi,r i )T and bi,k ∈ X}, i = 1, 2, . . . , m,
respectively. As in Definition 10, the fuzzy mappings f: T1 × T2 × · · · × Tn × F (Xr ) → F (E)
HU et al.: NON-L-R TYPE FUZZY PARAMETERS IN MATHEMATICAL PROGRAMMING PROBLEMS
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and gi : T1 × T2 × · · · × Tn × F (Xr i ) → F (R), i = 1, 2, . . . , m in (8) are defined by means of fuzzy sets Z := f˜(x, c) and Bi := g˜i (x, bi ), i = 1, 2, . . . , m, whose membership functions are given by sup{μc (c) : z = f (x, c)}, if {c: z = f (x, c)} = ∅ μZ (z) = 0, otherwise and μB i (βi ) sup{μb i (bi ): βi = gi (x, bi )}, if {bi : βi = gi (x, bi )} = ∅ = 0, otherwise
Fig. 2.
Membership function of the non-L-R type fuzzy parameter.
Since α = {z: z = f (x, c) and μ (z) ≥ α} (Z) Z
i = 1, 2, . . . , m, respectively. For each α ∈ [0, 1] and c = (c1 , c2 , . . . , cr ) ∈ Xr , we have
= {z: z = f (x, c) and μc (c) ≥ α}
μc (c) ≥ α ⇔ μc j (cj ) ≥ α, for all j = 1, 2, . . . , r.
= {f (x, c) and c ∈ ( c)α }
This along with (11) yields c1 )α × ( c2 )α × · · · × ( cr )α ( c)α = (
(12)
cj )α , j = 1, 2, . . . , r, are the α-level sets of c where ( c)α and ( and cj , j = 1, 2, . . . , r, respectively, and defined by
and i )α = {βi : βi = gi (x, bi ) and μ (βi ) ≥ α} (B Bi = {βi : βi = gi (x, bi ) and μb i (bi ) ≥ α}
( c)α := {c ∈ X : μc (c) ≥ α} r
and
= {gi (x, bi ) and bi ∈ (bi )α }
( cj )α := cj ∈ X: μc j (cj ) ≥ α , j = 1, 2, . . . , r.
by using the tolerance approach in [8], the problem (8) can be restated as follows:
Since α-level sets are the crisp sets whose elements have degrees of membership greater than or equal to α, the larger the value of α the higher the degree of satisfaction. Similarly, for each i ∈ {1, 2, . . . , m}, we have (bi )α = (bi,1 )α × (bi,2 )α × · · · × (bi,r i )α
(14)
and (bi,k )α := {bi,k ∈ X: μb i , k (bi,k ) ≥ α}, k = 1, 2, . . . , ri . (15) Suppose that the fuzzy parameters in the problem (8) are represented as fuzzy sets whose highest membership values are clustered around a couple of real numbers as shown in Fig. 2 and the membership functions μc j , j = 1, 2, . . . , r, and μb i , k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m, are lower semicontinuous. This along with closeness of X implies that the sets ( cj )α , j = 1, 2, . . . , r, and (bi,k )α , k = 1, 2, . . . , ri , for each i ∈ {1, 2, . . . , m}, are closed, i.e., ( cj )α , j = 1, 2, . . . , r, and (bi,k )α , k = 1, 2, . . . , ri , for each i ∈ {1, 2, . . . , m}, are time scales. Hence, by (13) we get (bi )α := Tαi,1 × · · · × Tαi,r i where Tαi,1 :=(bi,k )α for each i ∈ {1, 2, . . . , m}.
f (x, c)
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m bi,k ∈ (bi,k )α , k = 1, 2, . . . , ri , i = 1, 2, . . . , m
(13)
where (bi )α and (bi,k )α , k = 1, 2, . . . , ri , are the α-level sets of bi and bi,k , k = 1, 2, . . . , ri , respectively, and defined by (bi )α := {bi ∈ Xr i : μb i (bi ) ≥ α}
max
cj ∈ ( cj )α , j = 1, 2, . . . , r x ∈ T1 × T 2 × . . . × T n .
(16)
Let Tαi,k := (˜bi,k )α , k = 1, 2, . . . , ri , for each i ∈ {1, 2, . . . , cj )α , j = 1, 2, l, r. The problem (16) is equivm}, Tαm +1,j := (˜ alent to max
f (x, c)
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m bi,k ∈ Tαi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m cj ∈ Tαm +1,j , j = 1, 2, . . . , r x ∈ T1 × T2 ×, · · · , ×Tn .
(17)
Let K := r1 + r2 + · · · + rm . For each α ∈ [0, 1], define the mixed domain Λnα +K := T1 × · · · × Tn × (b1 )α × · · · × (bm )α
(18)
where (bi )α := Tαi,1 × · · · × Tαi,r i , i = 1, 2, . . . , m. When the nonlinear objective function f (x, c) is nondecreasing with respect to its second argument, we can eliminate the parameter c
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from the objective function f (x, c) by defining the function z(x, b) = max{f (x, c): c ∈
Tαm +1,1
×
Tαm +1,2
many fuzzy mathematical programming problems which have not been proposed in the existing literature. For instance, the FMP with exponential decision parameters
× · · · × Tαm +1,r } which actually does not include the parameter b := (b1 , b2 , . . . , bm )T where bi = (bi,1 , . . . , bi,r i )T , i = 1, 2, . . . , m. Hence, the problem (17) turns into the following problem: max
max
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m bi,k ∈ Tαi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m
z (x, b)
cj ∈ Tαm +1,j , j = 1, 2, . . . , r n x ∈ 2N
gi (x, bi ) ≤ 0, i = 1, 2, . . . , m bi,k ∈ Tαi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m (x, b) ∈ Λnα +K . This along with Theorem 4 yields the following result: Theorem 5: Suppose that f is nondecreasing with respect to its second argument, and, for each grade α ∈ [0, 1], the set Sα := (x, b) ∈ Λnα +K : gi (x, bi ) ≤ 0, i = 1, 2, . . . , m
where 2N = {2t : t ∈ N} or the FMP with binary decision parameters max
n
(xi − xαi , bi − bαi )ei ≥ 0 for all (x, b) ∈ Sα
i=1
where α
α
ξ (x , b ) =
n
∂f λi (x , β ) Δi xi (x α ,β α ) i=1 α
α
∂f ei . + (1 − λi (x , β )) ∇i xi (x α ,β α ) α
bi,k ∈ Tαi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m cj ∈ Tαm +1,j , j = 1, 2, . . . , r x ∈ {0, 1}n . IV. FUZZY MATHEMATICAL PPROGRAMMING PROBLEM WITH LINEAR OBJECTIVE AND NON-L-R TYPE FUZZY COEFFICIENTS In this section, we suppose that all the assumptions we made in the previous section are valid. If we let f (x, c) = nj=1 cj xj , then (2) turns into the following FMP:
α
max
cj xj
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m x ∈ T1 × T 2 × · · · × T n
(20)
f (x, c)
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m bi,k ∈ Tαi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m cj ∈ Tαm +1,j , j = 1, 2, . . . , r n x ∈ R+
(19)
with continuous decision parameters, but also the problem max
n j =1
Remark 1: Note that the decision variables x in (17) are supposed to be in T1 × T2 × · · · × Tn . Hence, the problem (17) corresponding to (2) covers not only the problem max
f (x, c)
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m
is a convex set in Λnα +K satisfying cintΛ nα + K (Sα ) = ∅. Then, for each grade α ∈ [0, 1] , (xα , bα ) = (xα1 , . . . , xαn , bα1 , . . . , bαm ) is an optimal solution to problem (8) if and only if there exist scalars λi (xα , bα ) , i = 1, 2, . . . , n + ri , such that ξ T (xα , bα )
f (x, c)
where cj , j = 1, 2, . . . , n are fuzzy parameters, gi , i = 1, 2, . . . , m, are fuzzy mappings associated with the crisp functions gi , i = 1, 2, . . . , m, bi := (bi,1 , bi,2 · · · bi,r i )T are fuzzy vectors consisting of fuzzy parameters bi,1 , bi,2 , . . . , bi,r i , i = 1, 2, . . . , m. According to the similar discussion in Section III, the fuzzy solution of (20) can be obtained by solving the following problem:
f (x, c)
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m bi,k ∈ Tαi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m cj ∈ Tαm +1,j , j = 1, 2, . . . , r x ∈ Nn with integer decision parameters. Since there are time scales other than N an R+ , the problem (16) has a potential to cover
max
n
cj xj
j =1
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m bi,k ∈ bi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m α
cj )α , j = 1, 2, . . . , n, cj ∈ ( x ∈ T1 × T 2 × · · · × T n .
(21)
HU et al.: NON-L-R TYPE FUZZY PARAMETERS IN MATHEMATICAL PROGRAMMING PROBLEMS
Since x ≥ 0 and X is a closed and bounded interval, (21) can be expressed as follows: ⎛ ⎞ n −min z(x, b) = ⎝− ( cj )U xj ⎠ α
j =1
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m (x, b) ∈ Λnα +K where
(22)
cj )α , j = 1, 2, . . . , n ( cj )Uα := max cj : cj ∈ (
and Λnα +K is defined in (18). Observe that for a linear function h(x) = nj=1 γj xj , where x = (x1 , x2 , . . . , xn )T , we have
Fig. 3.
Membership function of fuzzy parameter c˜1 .
∂h(x) ∂h(x) = = γj . j xj j xj Hence, combining Theorem 3 and Theorem 4, we obtain the following result. Theorem 6: Suppose that, for each α ∈ [0, 1] , the set Sα := (x, b) ∈ Λnα +K : gi (x, bi ) ≤ 0, i = 1, 2, . . . , m is a convex set in Λnα +K satisfying cintΛ nα + K (Sα ) = ∅. Then, for each grade α ∈ [0, 1] , (xα , bα ) ∈ Sα is an optimal solution to the problem (20) if and only if Fig. 4. Membership function of fuzzy parameter c˜2 . ⎛ ⎞T n +K ⎝ ( cj )Uα ej⎠ ((x, b) − (xα − bα )) ≥ 0 for all (x, b) ∈ Sα j =1
where ( cj )Uα := max cj : cj ∈ ( cj )α for j = 1, 2, . . . , n, and ( cj )Uα := 0 for j = n + 1, n + 2, . . . , n + K. In this study, we are developing the fuzzy-time scale analogue of conventional geometric method in convex analysis. The related issues of computation time analysis or convergence analysis can be found in [3]. V. NUMERICAL EXAMPLES To illustrate the associated results derived in the previous section, numerical examples are considered in this section. The numerical experiments are performed on the HP Compaq dx2810 MT under the Windows 7 operating system. Example 2: Consider the following mathematical programming problem with fuzzy parameters: max
Fig. 5.
Membership function of fuzzy parameter c˜3 .
Fig. 6.
Membership function of fuzzy parameter ˜b1 .
c˜1 x1 + c˜2 x2 + c˜3 x3
s.t. 6x1 + 4x2 − x3 ≤ ˜b1 2x1 + 4x2 + x3 ≤ ˜b2 x1 − x2 ≤ ˜b3 x1 , x2 , x3 ≥ 0
(23)
where c˜1 , c˜2 , c˜3 , ˜b1 , ˜b2 , and ˜b3 are fuzzy parameters with corresponding membership functions μc˜1 , μc˜2 , μc˜3 , μ˜b 1 , μ˜b 2 , and μ˜b 3 , which can be graphically described in Figs. 3–8.
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b1 ∈ [6, 9] b2 ∈ [4, 9] b3 ∈ [4, 5] x1 , x2 , x3 ≥ 0. max s.t.
6x1 + 2x2 − x3 6x1 + 4x2 − x3 − b1 ≤ 0 2x1 + 4x2 + x3 − b2 ≤ 0 x1 − x2 − b3 ≤ 0
Fig. 7.
b1 ∈ [12, 18]
Membership function of fuzzy parameter ˜b2 .
b2 ∈ [4, 9] b3 ∈ [4, 5] x1 , x2 , x3 ≥ 0. max s.t.
6x1 + 2x2 − x3 6x1 + 4x2 − x3 − b1 ≤ 0 2x1 + 4x2 + x3 − b2 ≤ 0 x1 − x2 − b3 ≤ 0 b1 ∈ [6, 9] b2 ∈ [4, 9]
Fig. 8.
b3 ∈ [8, 10]
Membership function of fuzzy parameter ˜b3 .
x1 , x2 , x3 ≥ 0.
For α = 0.8, problem (23) can be considered as follows: max s.t.
s.t.
z(x1 , x2 , x3 , b1 , b2 , b3 ) = 6x1 + 2x2 − x3
b1 ∈ [12, 18]
x1 − x2 − b3 ≤ 0 b1 ∈ [6, 9] [12, 18]
b2 ∈ [4, 9] b3 ∈ [8, 10] x1 , x2 , x3 ≥ 0.
[8, 10]
x1 , x2 , x3 ≥ 0.
(24)
Since the constraints b1 ∈ [6, 9] ∪ [12, 18] and b3 ∈ [4, 5] ∪ [8, 10] in (24) can be interpreted as b1 ∈ [6, 9] or b1 ∈ [12, 18] and b3 ∈ [4, 5] or b3 ∈ [8, 10], respectively, the solution of problem (24) can be obtained by choosing the solution with the largest objective values of the following linear programming problems max s.t.
6x1 + 4x2 − x3 − b1 ≤ 0 x1 − x2 − b3 ≤ 0
2x1 + 4x2 + x3 − b2 ≤ 0
b3 ∈ [4, 5]
6x1 + 2x2 − x3 2x1 + 4x2 + x3 − b2 ≤ 0
6x1 + 4x2 − x3 − b1 ≤ 0
b2 ∈ [4, 9]
max
6x1 + 2x2 − x3 6x1 + 4x2 − x3 − b1 ≤ 0 2x1 + 4x2 + x3 − b2 ≤ 0 x1 − x2 − b3 ≤ 0
Therefore, the optimal solution to the problem (24) is (xα1 , xα2 , xα3 , bα1 , bα2 , bα3 ) = (3.1583, 0, 0.9499, 18, 8.3467, 4.3962) with the subgradient ξ(xα1 , xα2 , xα3 , bα1 , bα2 , bα3 ) = (−6, −2, 1, 0, 0, 0)T . It can be observed that for any (x1 , x2 , x3 , b1 , b2 , b3 ) in the feasible domain of the problem (24), we have ξ(xα1 , xα2 , xα3 , bα1 , bα2 , bα3 )T ((x1 , x2 , x3 , b1 , b2 , b3 ) − (xα1 , xα2 , xα3 , bα1 , bα2 , bα3 )) ≥ 0. According to Figs. 3–8, we assume that for the case of c2 )Uα = 2.3, (˜ c3 )Uα = −0.8, (˜b1 )α ∈ [5.5, α = 0.5, (˜ c1 )Uα = 7, (˜ ˜ 9.5] ∪ [11.5, 18.5], (b2 )α ∈ [4, 11], (˜b3 )α ∈ [3, 5.5] ∪ [7.5, 10.5], c2 )Uα = 2.7, (˜ c3 )Uα = and for the case of α = 0.2, (˜ c1 )Uα = 8, (˜ ˜ ˜ ˜ −0.5, (b1 )α ∈ [5, 10] ∪ [11, 19], (b2 )α ∈[2, 12], (b3 )α ∈ [2, 6] ∪ [7, 11]. Table III shows the computational results for different values of α.
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TABLE III COMPUTATIONAL RESULTS FOR DIFFERENT VALUES OF α
TABLE V VALUE OF (˜ cj )U α
TABLE IV COMPUTATIONAL RESULTS BY GA FOR DIFFERENT VALUES OF α
0.025x4 + 0.015x5 + 0.021x8 ≤ ˜b15 0.0017x4 + 0.0035x9 ≤ ˜b16 0.0015x4 + 0.0015x5 ≤ ˜b17 To compare the results obtained in Table III with other metaheuristics method, the computational results of solving (24) by GA are shown in Table IV. Example 3: An industrial management problem of fuzzy linear programming with fuzzy resources and fuzzy coefficients for the objective function has been studied in Vasant et al. [43] using real-life data from a vegetable oil company. The company produces a variety of products such as vegetable oils, solid oils, toilet soap, solid cleaner, liquid cleaner, car soap, toothpaste, and washing powder. There are n products to be manufactured by mixing m raw materials with different proportion and by using k varieties of processing methods. There are also some constraints imposed by the marketing department such as product mix requirements, main product line requirements, and the lower and upper limit of demand for each product. The aforementioned requirements and conditions are fuzzy, and the objective is to obtain the optimal units of products with a certain degree of satisfaction. The fuzzy linear programming model can be described as follows: max
c˜1 x1 + c˜2 x2 + c˜3 x3 + c˜4 x4 + c˜5 x5 + c˜6 x6 + c˜7 x7 + c˜8 x8 + c˜9 x9 + c˜10 x10
s.t. 1.03x1 ≤ ˜b1 0.025x1 + 0.021x2 ≤ ˜b2 0.003x1 + 0.00035x2 ≤ ˜b3 1.053x2 + 0.78x3 + 0.66x6 ≤ ˜b4 0.0035x2 + 0.0045x4 ≤ ˜b5 0.28x3 + 0.25x6 ≤ ˜b6 0.00124x3 + 0.0065x7 + 0.0017x10 ≤ ˜b7 0.5x3 + 0.5x6 + 0.035x10 ≤ ˜b8 0.5x3 + 0.25x6 + 0.6x10 ≤ ˜b9 0.0003x3 + 0.0035x10 ≤ ˜b10 0.018x3 ≤ ˜b11 0.00775x3 + 0.0057x6 + 0.00225x10 ≤ ˜b12 0.0745x4 + 0.04x7 + 0.22x9 ≤ ˜b13 0.016x4 ≤ ˜b14
0.002x5 + 0.00225x8 ≤ ˜b18 0.0205x4 + 0.02785x5 + 0.012x8 ≤ ˜b19 0.34x5 + 0.285x8 ≤ ˜b20 0.265x5 + 0.225x8 ≤ ˜b21 0.022x5 + 0.022x8 ≤ ˜b22 0.0175x5 + 0.0125x7 + 0.035x8 ≤ ˜b23 0.275x6 ≤ ˜b24 0.275x7 ≤ ˜b25 0.3x8 ≤ ˜b26 0.0055x9 ≤ ˜b27 150x1 ≤ ˜b28 15.25x2 ≤ ˜b29 28.65x3 ≤ ˜b30 32.2x4 ≤ ˜b31 204x5 ≤ ˜b32 45.9x6 ≤ ˜b33 238.5x7 ≤ ˜b34 64.46x8 ≤ ˜b35 32.23x9 ≤ ˜b36 212.5x10 ≤ ˜b37 x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 , x10 ≥ 0.
(25)
Assume that the fuzzy resources and fuzzy coefficients for the objective function are non-L-R type fuzzy parameters. For a given α = 0.8, the values of (˜ cj )Uα , j = 1, 2, . . . , 10 and the sets (˜bi )α , i = 1, 2, . . . , 37, are listed in Tables V and VI. Table VII shows the result of solving (25) by using MATLAB and its optimization toolbox for the case of α = 0.8. The results obtained in [43] for the case of the level of satisfaction μ = 0.7994 are also provided in Table VII. The different assumptions of fuzzy parameters for the fuzzy resources and fuzzy coefficients for the objective function lead to different optimal units of products for the same proposed problem.
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TABLE VI VALUE OF (˜bi )α
VI. CONCLUSION This paper has been studying mathematical programming problems with non-L-R type fuzzy parameters. Such a problem can be converted into an optimization problem on a mixed domain. The necessary and sufficient conditions for solving the resulting optimization problem with the domain of the product of time scales are investigated. The main contribution of the paper is the employment of the theory of convex optimization on mixed domains to the resolution of the fuzzy optimization problem with non-L-R type membership functions. To the best of our knowledge, the fuzzy optimization problems with nonL-R type parameters considered in this study are completely new to the literature and have not been studied elsewhere before. Probably, an alternative model of the proposed approach may be considered in which more objectives and constraints are included allowing the resolution of the problem through the use of L-R fuzzy parameters. However, solving this alternative model with L-R fuzzy parameters could be much more complex from a computational point of view. This study essentially provides a new concept and broadens the scope for solving fuzzy optimization problems. ACKNOWLEDGMENT The authors would like to thank referees for their very constructive comments. REFERENCES
TABLE VII COMPUTATIONAL RESULT OF SOLVING PROBLEM (25)
[1] M. Adivar and S.-C. Fang, “Convex optimization on mixed domains,” J. Ind. Manag. Optim., vol. 8, no. 1, pp. 189–227, 2012. [2] R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,” Results Math., vol. 35, pp. 3–22, 1999. [3] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming Theory and Algorithms. New York, NY, USA: Wiley, 2004. [4] R. E. Bellman and L. A. Zadeh, “Decision making in a fuzzy environment,” Manag. Sci., vol. 17(B), no. 4, pp. 141–164, 1970. [5] A. Buller, “Fuzzy sets with dynamic memberships,” in Proc. 1st Int. Conf. Fuzzy Syst. Knowl. Discov., 2002, pp. 564–565. [6] M. Bohner and A. C. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications. Boston, MA, USA: Birkh¨auser, 2001. [7] M. Bohner and A. C. Peterson, Advances in Dynamic Equations on Time Scales. Boston, MA, USA: Birkh¨auser, 2003. [8] J. M. Cadenas and J. L. Verdegay, “Towards a new strategy for solving fuzzy optimization problems,” Fuzzy Optim. Decision Making, vol. 8, pp. 231–244, 2009. [9] C. Carlsson and P. Korhonen, “A parametric approach to fuzzy linear programming,” Fuzzy Sets Syst., vol. 20, pp. 17–30, 1986. [10] M. Delgado, J. L. Verdegay, and M. A. Vila, “Imprecise costs in mathematical programming problems,” Control Cybern., vol. 16, no. 2, pp. 113–121, 1987. [11] M. Delgado, J. L. Verdegay, and M. A. Vila, “A general model for fuzzy linear programming,” Fuzzy Sets Syst., vol. 29, pp. 21–29, 1989. [12] M. Delgado, J. Kacprzyk, J.-L. Verdegay, and M. A. Vila, Eds., Fuzzy Optimization: Recent Advances. New York, NY, USA: Physica-Verlag, 1994. [13] C. Dinu, “Convex functions on time scales,” Ann. Univ. Craioa, Math. Comp. Sci. Ser., vol. 35, pp. 87–96, 2008. [14] D. Dubois and H. Prade, “Operations on fuzzy numbers,” Int. J. Syst. Sci., vol. 9, no. 6, pp. 613–626, 1978. [15] D. Dubois and H. Prade, “Systems of linear fuzzy constraints,” Fuzzy Sets Syst., vol. 3, pp. 37–48, 1980. [16] D. Dubois and H. Prade, “Ranking of fuzzy numbers in the setting of possibility theory,” Inf. Sci., vol. 30, pp. 183–244, 1983. [17] R. N. Gasimov and K. Yenilmez, “Solving fuzzy linear programming problems with linear membership functions,” Turkish J. Math., vol. 26, pp. 375–396, 2002.
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[18] E. L. Hannan, “Linear programming with multiple fuzzy goals,” Fuzzy Sets Syst., vol. 6, pp. 235–248, 1981. [19] F. Herrera and J. L. Verdegay, “Three models of fuzzy integer linear programming,” Eur. J. Oper. Res., vol. 83, no. 3, pp. 581–593, 1995. [20] S. Hilger, “Analysis on measure chains; a unified approach to continuous and discrete calculus,” Results Math., vol. 18, pp. 18–56, 1990. [21] C.-F. Hu and S.-C. Fang, “Solving fuzzy inequalities with concave membership functions,” Fuzzy Sets Syst., vol. 99, pp. 233–240, 1998. [22] C.-F. Hu and S.-C. Fang, “Solving fuzzy inequalities with piecewise linear membership functions,” IEEE Trans. Fuzzy Syst., vol. 7, no. 2, pp. 230– 235, Apr. 1999. [23] M. Inuiguchi, H. Ichihashi, and Y. Kume, “A solution algorithm for fuzzy linear programming with piecewise linear membership functions,” Fuzzy Sets Syst., vol. 34, pp. 15–31, 1990. [24] M. Inuiguchi and T. Tanino, “Possibilistic linear programming with fuzzy if-then rule coefficients,” Fuzzy Optim. Decision Making, vol. 1, pp. 65– 91, 2002. [25] E. P. Klement and M. Navara, “A survey on different triangular normbased fuzzy logics,” Fuzzy Sets Syst., vol. 101, pp. 241–251, 1999. [26] V. B. Kuz’min, “A parametric approach to description of linguistic values of variables and hedges,” Fuzzy Sets Syst., vol. 6, pp. 27–41, 1981. [27] Y.-J. Lai and C.-L. Hwang, “Lecture notes in economics and mathematical systems,” in Fuzzy Mathematical Programming: Methods Applications, New York, NY, USA: Springer-Verlag, 1992. [28] Y.-J. Lai and C.-L. Hwang, “Lecture notes in economics and mathematical systems,” in Fuzzy Multiple Objective Decision Making: Methods Applications, New York, NY, USA: Springer-Verlag, 1994. [29] T.-F. Liang, “Distribution planning decisions using interactive fuzzy multiobjective linear programming,” Fuzzy Sets Syst., vol. 157, pp. 1303–1316, 2006. [30] B. Liu, “Toward fuzzy optimization without mathematical ambiguity,” Fuzzy Optim. Decision Making, vol. 1, pp. 43–63, 2002. [31] H. Leberling, “On finding compromise solutions in multicriteria problems using the fuzzy min-operator,” Fuzzy Sets Syst., vol. 6, pp. 105–118, 2002. [32] M. D. Madronero, D. Peidro, and P. Vasant, “Vendor selection problem by using an interactive fuzzy multi-objective approach with modified s-curve membership functions,” Comput. Math. Appl., vol. 60, pp. 1038–1048, 2010. [33] D. Peidro and P. Vasant, “Transportation planning with modified scurve membership functions using an interactive fuzzy multi-objective approach,” Appl. Soft Comput., vol. 11, pp. 2656–2663, 2011. [34] P. A. Rubin and R. Narasimhan, “Fuzzy goal programming with nested priorities,” Fuzzy Sets Syst., vol. 14, pp. 115–129, 1984. [35] M. Sakawa, “Interactive computer program for fuzzy linear programming with multiple objectives,” J. Man-Mach. Stud., vol. 18, pp. 489–503, 1983. [36] M. Z. Sarikaya, N. Aktan, H. Yildirim, and K. Ilarslan, “Partial diferentiation for multivariable functions on n-dimensional time scales,” J. Math. Inequalities, vol. 3, pp. 277–291, 2009. [37] M. Z. Sarikaya, N. Aktan, H. Yildirim, and K. Ilarslan, “Directional derivative and curves on n-dimensional time scales,” Acta. Appl. Math, vol. 105, pp. 45–63, 2009. [38] R. Slowi´nski, Ed., Fuzzy Sets in Decision Analysis, Operations Research and Statistics. Boston, MA, USA: Kluwer, 1998. [39] R. Slowi´nski, Ed., Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty. Boston, MA, USA: Kluwer, 1990. [40] H. Tanaka, H. Ichihashi, and F. Asai, “A formulation of fuzzy linear programming problems based a comparison of fuzzy numbers,” Control Cybern., vol. 13, pp. 185–194, 1984. [41] H. Tanaka and F. Asai, “Fuzzy linear programming based on fuzzy mappings,” Bull. Osaka Prefect., vol. 29, no. 2, pp. 113–125, 1981. [42] P. Vasant, “Fuzzy decision making of profit function in production planning using S-curve membership function,” Comput. Ind. Eng., vol. 51, pp. 715–725, 2006. [43] P. Vasant, I. Elamvazuthi, and J. F. Webb, “Fuzzy technique for optimization of objective function with uncertain resource variables and technological coefficients,” Int. J. Model., Simulat., Sci. Comput., vol. 1, no. 3, pp. 349–367, 2010. [44] J. Watada, “Fuzzy portfolio selection and its applications to decision making,” Tatra Mount. Math. Publication, vol. 13, pp. 219–248, 1997.
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[45] H. J. Zimmermann, “Description and optimization of fuzzy systems,” Int. J. General Syst., vol. 2, pp. 209–215, 1976. [46] H. J. Zimmermann, “Fuzzy programming and linear programming with several objective functions,” Fuzzy Sets Syst., vol. 1, pp. 45–55, 1978. [47] H. J. Zimmermann, Fuzzy Sets, Decision Making, and Expert Systems. Boston, MA, USA: Kluwer, 1987. [48] H. J. Zimmermann, Fuzzy Set Theory and Its Applications. Boston, MA, USA: Kluwer, 1991.
Cheng-Feng Hu received the B.S. degree from National Tsing Hua University, Hsinchu City, Taiwan and the Ph.D. degree from North Carolina State University Raleigh, NC, USA, in 1993 and 1997 respectively. Her research interests include fuzzy optimization and decision making and financial engineering. She is currently with the Department of Applied Mathematics, National Chiayi University, Chiayi, Taiwan.
Murat Adivar received the Bachelor’s, Master’s, and Ph.D. degrees, in 1996, 1998, and 2003 respectively, from Department of Mathematics, Ankara University, Ankara, Turkey. He is currently with the Department of Mathematics, Izmir University of Economics (IUE), Izmir, Turkey, and also serves as a Vice Rector in charge of Academic Affairs with IUE. He is also working with E. P. Fitts in the Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, USA, on a joint research project funded by TUBITAK. His Ph.D. thesis, which was in the field of spectral analysis, was funded by The Scientific and Technological Research Council of Turkey, International Ph.D. Scholarship Programme for Raising Young Scientists. His research interests include spectral analysis, functional analysis, analysis of qualitative and quantitative properties of dynamic equations on time scales, and global optimization Dr. Adivar received a NATO Fellowship for post doctoral studies from the Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA. He has been an invited speaker to several international meetings and conferences and has works published in many internationally recognized journals.
Shu-Cherng Fang received the B.S. degree from National Tsing Hua University, Hsinchu City, Taiwan, and the Ph.D. degree from Northwestern University in Evanston, Chicago, IL, USA. He holds the Walter Clark Chair Professorship and Alumni Distinguished Graduate Professorship with North Carolina State University, Raleigh, NC, USA. He was the Senior Member of the Research Staff at Western Electric Engineering Research Center, a Supervisor at AT&T Bell Labs, and Department Manager at the Corporate Headquarters of AT&T Technologies before joining the NC State. He has published more than 180 refereed journal articles. He authored the books Linear Optimization and Extensions: Theory and Algorithms (Englewood Cliffs, NJ, USA: Prentice Hall, 1993) with S. C. Puthenpura, Entropy Optimization and Mathematical Programming (Norwell, MA, USA: Kluwer Academic, 1997) with J.R. Rajasekera and H.-S. Tsao, and Linear Conic Optimization (San Francisco, CA, USA: Scientific Press, 2013) with W. Xing. His research interests include linear and nonlinear programming, fuzzy optimization and decision making, soft computing, logistics, and supply chain management. Dr. Fang is the Founding Editor-in-Chief of Fuzzy Optimization and Decision Making, and currently serves on the editorial boards of 25 scientific journals.
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 5, OCTOBER 2014
Non-L-R Type Fuzzy Parameters in Mathematical Programming Problems Cheng-Feng Hu, Murat Adivar, and Shu-Cherng Fang
Abstract—The triangular norm-based operations in fuzzy logic usually lead to non-L-R type fuzzy sets. This study considers mathematical programming problems with non-L-R type fuzzy parameters. It shows that the fuzzy solutions to such problems can be obtained by solving an optimization problem on a mixed domain. The necessary and sufficient conditions for solving the resulting optimization problems are investigated by employing the theory of convex optimization on mixed domains. This is the first attempt to solve the fuzzy optimization problem with non-L-R type membership functions in view of optimization problems on a mixed domain. Index Terms—Fuzzy decision making, fuzzy optimization, fuzzy set theory, mathematical programming, membership function, triangular norms.
A˜ ˜ B μA˜ μB˜ S f˜ g˜i T R R+ Z Z+ N N0 σ ρ Λn ej convΛ n cintΛ n
NOMENCLATURE Fuzzy set. Fuzzy set. ˜ Corresponding membership function of A. ˜ Corresponding membership function of B. Triangular conorm. Fuzzy mapping. Fuzzy mappings, i = 1, 2, . . . , m. Time scale. Real numbers. Positive real numbers Integer numbers. Positive integer numbers. Natural numbers. Positive natural numbers. Forward jump operator. Backward jump operator. Product of time scales Ti , i = 1, 2, . . . , n. jth unit vector Convex hull in Λn . Convex interior in Λn .
conv Rn
Interior of convR n in Rn .
◦
Manuscript received January 22, 2013; revised April 19, 2013 and June 29, 2013; accepted August 7, 2013. Date of publication August 26, 2013; date of current version October 2, 2014. This work was supported in part by the National Science Council of China, under Grant NSC 101-2221-E-214-045. C.-F. Hu is with the Department of Applied Mathematics, National Chiayi University, Chiayi, Taiwan (e-mail: [email protected]). M. Adivar is with the Department of Mathematics, Izmir University of Economics, Izmir 35330, Turkey (e-mail: [email protected]). S.-C. Fang is with the Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695-7906 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TFUZZ.2013.2279691
fˆ ξ ∂ f (x) i x i ∂ f (x) i x i Ni± n
Bi
Restricted function. Subgradient of a function. Partial -derivative of f with respect to xi . Partial -derivative of f with respect to xi . Sets in the time scale Ti , i = 1, 2, . . . , n. Frame at the point in Λn .
i=1
r ri c˜ ˜bi c˜j ˜bi,k Ti X ˜ h F (W ) μY˜ a ˜ μa˜ μc˜j μ˜b i , k μc˜ μ˜b i ˜ Z ˜i B μB˜ i μZ˜ (˜ c)α (˜ cj )α (˜bi )α (˜bi,k )α ˜ α (Z) ˜i )α (B ˜α T i,k ˜α T m +1,j
Number of fuzzy parameters in the fuzzy objective of (2). Number of fuzzy parameters in the ith fuzzy constraint of (2). Fuzzy vector. Fuzzy vectors, i = 1, 2, . . . , m. Fuzzy parameters, j = 1, 2, . . . , r. Fuzzy parameters, k = 1, 2, . . . , ri , i = 1, 2, . . . , m. Time scales, i = 1, 2, . . . , n. Closed and bounded interval in R. Fuzzy mapping. Set of all fuzzy subsets on W . Membership function of Y˜ . Fuzzy vector. Membership function of a ˜. Membership functions of c˜j , j = 1, 2, . . . , r. Membership functions of ˜bi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m. Membership function of c˜. Membership functions of ˜bi , i = 1, 2, . . . , m. Fuzzy set. Fuzzy sets, i = 1, 2, . . . , m ˜i , i = 1, 2, . . . , m Membership functions of B Membership function of Z˜ α–level sets of c. ˜ α–level sets of c˜j , i = 1, 2, . . . , n α–level sets of ˜bi , i = 1, 2, . . . , m α–level sets of ˜bi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m ˜ α–level sets of Z. ˜i , i = 1, 2, . . . , m. α–level sets of B α–level sets of ˜bi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m. α–level sets of c˜j , j = 1, 2, . . . , r. I. INTRODUCTION
UZZY optimization has been one of the most well-studied topics in the area of soft computing. Its applications, as well as practical realizations, can be found in many real-life problems. Overviews on the main stream of fuzzy optimization can be found in [12] and [38]. Lai and Hwang [27], [28]
F
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HU et al.: NON-L-R TYPE FUZZY PARAMETERS IN MATHEMATICAL PROGRAMMING PROBLEMS
also provided an insightful survey on fuzzy mathematical programming and fuzzy multiple objective decision making. Comparisons between fuzzy optimization and stochastic optimization for multiobjective programming problems were considered in [39]. Corresponding to linear programming in conventional optimization, fuzzy linear programming methods have been a constantly studied subject in the fuzzy context following the seminal paper of Bellman and Zadeh [4]. Models and methods for solving fuzzy linear programming problems with imprecision in the constraints or the objectives can be easily found in the literature [10], [11], [16], [19], [40], [47]. As stated by Rubin and Narasimhan [34], the heart of the methodology for fuzzy linear programming lies in the construction of membership functions for the objection coefficients, technical coefficients, resource variables, and decision variables [43]. The various types of membership functions have been employed in fuzzy linear programming problems with applications, including the linear membership function [17], [45], [46], linear fuzzy constraints [15], tangent type membership function [31], interval linear membership function [18], exponential membership function [9], inverse tangent membership function [35], logistic type membership function [44], concave piecewise linear membership function [23], piecewise linear membership function [21], [22], dynamics membership function [5], and S-curve membership function [26], [32], [33], [43]. Since the tangent type membership function, exponential membership function, and hyperbolic membership function are nonlinear functions, a fuzzy mathematical programming problem (FMP) using such membership functions results in a nonlinear program [43]. In the literature, a linear membership function is commonly employed to avoid involving nonlinearity. Nevertheless, a linear membership function might not be a suitable representation in many practical situations [44]. Nowadays, fuzzy optimization still receives considerable attention. It is important to look into the foundations of fuzzy optimization in order to meet the current trends, and to develop future research lines [8]. When the concept and methods of fuzzy optimization are applied to real-world decision making, the fuzzy parameters are usually represented by means of fuzzy subsets on the real line, known as fuzzy numbers. Linear programming with fuzzy coefficients has been developed mainly under the assumption that fuzzy coefficients are L-R type fuzzy numbers introduced by Dubois [14]. The reason is that the operations on this type of fuzzy number can be easily performed algebraically, and the resulting fuzzy mathematical programming problems can be reduced to a traditional mathematical programming problem. However, in real applications, the L-R type fuzzy number assumption may not be the most appropriate for modeling the fuzzy parameters [24], [30]. For instance, consider a transportation distribution problem of soft drinks with a fuzzy forecast demand at each destination [29]. Then, the union of two fuzzy sets representing the demand for summer or winter seasons might be considered, which automatically leads to fuzzy forecast demands with non-L-R type fuzzy membership functions. Another good example is to consider assessing the demand of a new product, such as i-watch, by asking professional/consulting opinions from experts. It is very
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Fig. 1. Membership function of the fuzzy set that is obtained by the triangular ˜ conorm-based operation of fuzzy sets A˜ and B.
possible that an optimistic opinion will provide a high demand and a pessimistic opinion will project a low demand. Even both opinions of high and low demands are represented in the L-R type membership functions, the resulting demand membership can hardly be represented by an L-R type membership function using the logic “or” operator. It is well known that the conjunction in fuzzy logic is usually interpreted by triangular norms [25]. The triangular norm-based operations of fuzzy sets often result in membership functions with multiple peaks. This idea can be illustrated clearly by the following simple example. ˜ be two fuzzy sets with the corExample 1: Let A˜ and B responding L-R type membership functions μA˜ (·) and μB˜ (·) respectively, as shown in Fig. 1. Consider a conjunction whose interpretation is given by a triangular conorm (t-conorm for short) S: [0, 1]2 → [0, 1] that is defined as follows [48]: S(μA˜ (x), μB˜ (x)) = max{μA˜ (x), μB˜ (x)}.
(1)
Then, the triangular conorm-based operation of fuzzy sets A˜ and ˜ defined by (1) automatically leads to a non-L-R type fuzzy B set with multiple peaks. To facilitate a more general setting in the formulation of fuzzy decision-making problems, the non-L-R type fuzzy parameters are introduced in this paper. Consider an optimization problem with fuzzy parameters in the following form: maximize
f˜(x, c˜)
s.t. g˜i (x, ˜bi ) ≤ 0, i = 1, 2, . . . , m x ≥ 0,
(2)
where f˜, g˜i , i = 1, 2, . . . , m, are fuzzy mappings, x = (x1 , x2 · · · xn )T is an n-dimensional vector of decision variables, c˜ and ˜bi , i = 1, 2, . . . , m, are fuzzy vectors consisting of fuzzy parameters. Following [10], we show that the fuzzy solution of problem (2) with non-L-R type fuzzy parameters can be obtained by solving an optimization problem over the domain of the product of time scales, where a time scale is defined to be a nonempty closed subset of real numbers. An introduction to the theory of calculus on time scales can be found in the original work by Hilger [20] and the more recent works [1], [2], and [13]. The
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 5, OCTOBER 2014
notion of convexity for functions of one variable on a time scale has been introduced in [13]. Moreover, the notions of convexity for functions of several variables and convexity for the subsets in a product space of arbitrary time scales, called a mixed domain, have been thoroughly investigated in [1]. Following the work of [1], the necessary and sufficient conditions for solving the resulting optimization problems on the mixed domain are discussed. This is the first attempt of employing the theory of convex optimization on time scales for solving fuzzy mathematical programming problems. Hopefully, the proposed work is not only a generalization of existing theory but also an initial step for the development of new models and algorithms for fuzzy optimization. The rest of this paper is organized as follows. Some basic concepts and analytic properties regarding the convex sets of a time scale and convex functions defined on the mixed domain are presented in Section II. In Section III, we show that the fuzzy solution of the problem (2) with non-L-R type fuzzy parameters can be obtained by solving an optimization problem on a mixed domain. The necessary and sufficient conditions for solving problem (2) with linear objective function and non-L-R type fuzzy parameters are provided in Section IV. In Section V, numerical examples are included to illustrate the associated results. Conclusions are provided in Section VI. II. OPTIMIZATION ON MIXED DOMAINS In this section, some basic concepts of convex optimization on mixed domains are introduced. All proofs are omitted to keep the paper succinct and readable. Readers may refer to [6] and [7] for a comprehensive review on time scales, and to [1] for the detailed analysis on the theory of convex optimization on mixed domains. It is important to mention that, throughout this study, we assume that a time scale, denoted T, has the topology inherited from the standard topology on the real numbers R. Moreover, we denote the set of integer numbers by Z, natural numbers by N, and positive natural numbers by N0 . Definition 1 [Time scale T]: An arbitrary nonempty closed subset T of real numbers is called a time scale. Definition 2 [Operations on time scales]: Let T be a time scale. The forward jump operator σ: T → T and backward jump operator ρ: T → T are defined by σ(t) = inf{s ∈ T: s > t}
TABLE I OPERATORS σ AND ρ ON SOME PARTICULAR TIME SCALES
TABLE II CLASSIFICATION OF THE POINTS OF A TIME SCALE
{M }. Otherwise Tκ = T. If T has a right-scattered minimum m, then Tκ = T − {m}. Otherwise Tκ = T. Hereafter, we use the notation Λn to denote the product T1 × T2 × · · · × Tn of the time scales Ti , i = 1, 2, . . . , n, and ej to represent the jth unit vector whose jth coordinate is 1 and other coordinates are 0. Definition 3 [Convex set] [1]: The set S ⊂ Λn is said to n i be convex in Λ if and only if m i=1 λi x ∈ S for all m ∈ + 1 2 m S and λ1 , λ2 , . . . , λm ∈ [0, 1] such that Z m, x , x , . . . , x ∈ m i n λ = 1 and i=1 i i=1 λi x ∈ Λ . This definition can alternatively be stated as as follows. Corollary 1 [1]: Let S be a subset of Λn . The set S is convex if and only if convR n (S) ∩ Λn = S, where convR n (S) denotes the convex hull of the set S in Rn . Definition 4 [Convex hull] [1]: The convex hull of a set S in Λn , denoted by convΛ n (S), is the collection of all convex combinations of elements of S in Λn . In other words, x ∈ convΛ n (S) if and only if x=
m
λi xi ∈ Λn
i=1
for some integer m > 0, λi ∈ [0, 1] ∈ R such that and x1 , x2 , . . . , xm ∈ S. Corollary 2 [1]: For any set S in Λn
m i=1
λi = 1,
convΛ n (S) = convR n (S) ∩ Λn .
and ρ(t) = sup{s ∈ T: s < t} respectively. In Table I, we illustrate the operators σ and ρ on some particular time scales. In Table I, q N := {q n : n ∈ N and q > 1} and N20 := {n2 : n ∈ N}. The points of a time scale t ∈ T, can be classified in the following manner. The sets Tκ and Tκ are derived from the time scale T as follows. If T has a left-scattered maximum M, then Tκ = T −
Corollary 3 [1]: Let S ⊂ Λn ; then S is convex in Λn if and only if S = convΛ n (S). We now define more details of a convex set in the product of time scales. Definition 5 [1]: Let S ⊂ Λn be a convex set in Λn . The convex-interior, denoted by cintΛ n (S), are defined by ◦
n cintΛ n (S) =conv R n (S) ∩Λ ◦
n where conv R n (S) indicates the interior of convR n (S) in R .
HU et al.: NON-L-R TYPE FUZZY PARAMETERS IN MATHEMATICAL PROGRAMMING PROBLEMS
Corollary 4 [1]: Let S ⊂ Λn be a convex set in Λn . Then the convex-interior cintΛ n (S) is a convex set in Λn . After defining convex sets in Λn , we turn our attention to the convex functions defined on Λn . Definition 6 [Convex function] [1]: Let S be a convex set in Λn . The function f : S → R is said to be convex if and only if m m i f λi x ≤ λi f (xi ) i=1
i=1
for all m ∈ Z+ , xi ∈ S, i = 1, 2, . . . , m, and λi ∈ [0, 1], i = m i n λ = 1 and 1, 2, . . . , m, such that m i=1 i i=1 λi x ∈ Λ . Observe that, if f : S → R is a convex function, then f (a + λ(b − a)) ≤ f (a) + λ(f (b) − f (a)) holds for all a, b ∈ S and λ ∈ [0, 1] such that a + λ(b − a) ∈ Λn . However, the converse of this statement may not be true for an arbitrary function f defined on mixed domains. Corollary 5 [1]: Let S be a convex set in Λn . If the function f : Rn → R is convex on convR n (S), then the restricted function fˆ := f |S is convex on S. Once convex functions on Λn are defined, we give the first order information by the subgradients. Definition 7 [Subgradient] [1]: Let S be a nonempty convex set in Λn , and f : S → R be a convex function on S. Then, ¯ ∈ S if ξ ∈ Rn is called a subgradient of f at x ¯) for all x ∈ S. f (x) ≥ f (¯ x) + ξ T (x − x Similarly, let f : S → R be a concave function on S (i.e., −f is convex on S). Then, ξ ∈ Rn is called a subgradient of f at x ¯ ∈ S if f (x) ≤ f (¯ x) + ξ T (x − x ¯) for all x ∈ S. Let S be a nonempty convex set in Λn . By Corollary 4, we know that cintΛ n (S) is convex. Then, we have the following two results. Theorem 1 [1]: Let S be a nonempty convex set in Λn and f : S → R be a convex function on S. Then, for x ¯ ∈ cintΛ n (S), there exists a vector ξ such that ¯) for all x ∈ S f (x) ≥ f (¯ x) + ξ T (x − x i.e., ξ is a subgradient of f at x ¯. Theorem 2 [1]: Let S be a nonempty convex set in Λn and f : S → R. If for every point x ¯ ∈ cintΛ n (S) there exists a subgradient vector ξ such that f (x) ≥ f (¯ x) + ξ T (x − x ¯) for all x ∈ S then f is convex on cintΛ n (S). To characterize convex functions in terms of subgradients on mixed domains, the following definitions can be found in [36] and [37] . Definition 8: Let Ti , i ∈ {1, x, . . . , n} be time scales and f : Λn → R be a function. The partial -derivative of f with
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respect to xi ∈ Tκi is defined by f (x1 , . . . , σi (xi ), xi+1 , . . . , xn ) ∂f (x) := slim i →x i i xi σi (xi ) − si s i = σ i ( x i ) f (x1 , x2 , . . . , si , xi+1 , . . . , xn ) − (3) σi (xi ) − si where σi : Ti → Ti is the forward jump operator on the ith time scale Ti . Similarly, the partial -derivative of f : Λn → R is defined by f (x1 , . . . , ρi (xi ), xi+1 , . . . , xn ) ∂f (x) := slim →x i i i xi ρi (xi ) − si s i = ρ i ( x i ) f (x1 , x2 , . . . , si , xi+1 , . . . , xn ) − (4) ρi (xi ) − si where ρi : Ti → Ti is the backward jump operator on the ith time scale Ti . In preparation for the next result, let us define the set n
Bi (x, h± i )
i=1
as a frame at the point x = (x1 , x2 , . . . , xn ) in Λn , where ⎧ ⎫ ⎪ ⎪ n ⎨ ⎬ ± i j ± ± Bi (x, hi ) = se + xj e : s ∈ Ni (x, hi ) ⎪ ⎪ ⎩ ⎭ j=1 j = i
i
e , i = 1, 2, . . . , n, are the unit vectors whose components are determined by 1, i = j eij = δij = 0, i = j the sets Ni± (x, h± i ), i = 1, 2, . . . , n, given by {xi , σi (xi )} if μi (xi ) > 0 ) := Ni+ (x, h+ i [xi , xi + h+ i ) if μi (xi ) = 0 where h+ i > 0, and Ni− (x, h− i )
:=
{ρi (xi ), xi } (xi −
h− i , xi ]
if νi (xi ) > 0 if νi (xi ) = 0
h− i
> 0, are the sets in the time scale Ti . where ∂ f (x) In order for the partial derivatives ∂fi(x) x i and i x i given by (3) and (4) to be well defined at a point x = (x1 , x2 , . . . , xn ) ∈ S, one has to assume that n Bi (x, h± (5) i )⊂S i=1
to guarantee that the vectors (x1 , x2 , . . . , σi (xi ), xi+1 , . . . , xn ) and (x1 , x2 , . . . , si , xi+1 , . . . , xn ), s ∈ Ni± (x, h± i ), i = 1, 2, . . . , n, are in S. Note that the condition (5) also implies x ∈ cintΛ n (S). Then, we are ready to state the next result. Theorem 3 [1]: Let S be a nonempty convex set in Λn . Let f : S → R be a function such that the partial derivatives ∂ f (x) ∂ f (x) ¯ and i x i |x= x ¯ , i = 1, 2, . . . , n, exist at any point i x i |x= x
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x ¯ = (¯ x1 , x ¯2 , . . . , x ¯n ) ∈ cintΛ n (S) satisfying (5). If f is convex x) ∈ [0, 1], i = 1, 2, . . . , n, such on S, then there exist scalars λi (¯ that the vector n ∂f (x) ∂f (x) x) + (1 − λi (¯ x)) ξ(¯ x) := λi (¯ ei i xi x= x¯ i xi x= x¯ i=1 (6) is a subgradient of f at any point x ¯ ∈ cintΛ n (S), i.e f (x) ≥ f (¯ x) + ξ(¯ x)T (x − x ¯) for all x ∈ S. For convex functions defined on Λn , we now study their minimum or maximum solutions. Definition 9 [1]: For a given function f : Λn → R and a given set S ⊂ Λn , consider the following problem: minimize f (x) subject to x ∈ S.
(7)
A point x ∈ S is called a feasible solution to the problem. If x ¯ ∈ S and f (x) ≥ f (¯ x) for all x ∈ S, then x ¯ is called an optimal solution or a global optimal solution [3]. The collection of optimal solutions are called alternative optimal solutions. Theorem 4 [1]: Let f : Λn → R be a convex function and S ¯ ∈ S is an optimal be a nonempty convex set in Λn . The point x solution to the problem (22) if and only if f has a subgradient ξ at x ¯ such that
corresponding to (8) is given as follows: s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m x ∈ T1 × T 2 × · · · × T n
h: T1 × T2 × · · · × Tn × F (Xn ) → F (R) , where F (W ) is the set of all fuzzy subsets on W . Let aj , j = 1, 2, . . . , n, be the fuzzy sets on X with corresponding membership functions μa j , j = 1, 2, . . . , n, respectively. The membership function of the fuzzy set Y := h(x, a) is defined as follows: sup{μa (a): y = h(x, a)}, if {a: y = h(x, a)} = ∅ μY (y) = 0, otherwise a2 , . . . , an )T is a fuzzy vector with the memwhere a = ( a1 , bership function given by μa (a) :=
Corollary 6 [1]: Under the assumptions of Theorem 5, if S ⊂ Λn is convex with ◦
n cintΛ n (S) =conv R n (S) ∩Λ = ∅
Let r be the number of fuzzy parameters in the fuzzy objective of (2), ri , i = 1, 2, . . . , m, be the number of fuzzy parameters in the ith fuzzy constraint of (2), and T1 , T2 , . . . , Tn be time scales. Replacing the nonnegativity condition x ≥ 0 in (2) with x ∈ T1 × T2 × · · · × Tn , the problem (2) can be described in the following form: f(x, c)
{μa j (aj ): aj ∈ aj } .
cj := {(cj , μc j (cj )): cj ∈ X}, j = 1, 2, . . . , r bi,k := {(bi,k , μ (bi,k )): bi,k ∈ X}, k = 1, 2, . . . , ri bi , k for each i ∈ {1, 2, . . . , m} with corresponding membership functions μc j , j = 1, 2, . . . , r, and μb i , k , k = 1, 2, . . . , ri , for each i ∈ {1, 2, . . . , m}. The fuzzy vectors c and bi , i = 1, 2, . . . , m, are described as fuzzy sets c := {(c, μc (c)): c ∈ Xr }
(10)
and bi := {(bi , μ (bi )): bi ∈ Xr i }, bi
i = 1, 2, . . . , m
with corresponding membership functions μc and μb i , i = 1, 2, . . . , m, defined by
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m x ∈ T1 × T 2 × · · · × T n
min
j =1,2,...,n
The fuzzy parameters cj , j = 1, 2, . . . , r, and bi,1 , bi,2 , . . . , bi,r , i = 1, 2, . . . , m, in (8) can be described as fuzzy sets on i X as follows:
then x ¯ ∈ cintΛ n (S) is an optimal solution to the problem (22) if and only if there exists a zero subgradient of f at x ¯. III. FUZZY MATHEMATICAL PROGRAMMING PROBLEM WITH NON-L-R TYPE FUZZY PARAMETERS
(9)
where c := (c1 , c2 , . . . , cr )T ∈ Rr , and bi := (bi,1 , bi,2 , . . . , bi,r i )T ∈ Rr i , i = 1, 2, . . . , m, are real vectors. Inspired by [41, Definition 1], we first define a fuzzification of the function h as follows. Definition 10: The fuzzification of the function h: T1 × T2 × · · · × Tn × Xn → R is a fuzzy mapping denoted by
¯) ≥ 0 for all x ∈ S. ξ T (x − x
max
f (x, c)
max
(8)
where c := ( c1 , c2 , . . . , cr )T and bi := (bi,1 , bi,2 , . . . , bi,r i )T , i = 1, 2, . . . , m, are fuzzy vectors with cj , j = 1, 2, . . . , r, and bi,1 , bi,2 , . . . , bi,r i , i = 1, 2, . . . , m, being fuzzy parameters, and T1 , T2 , . . . , Tn are time scales consisting of nonnegative reals. Suppose that X is a closed and bounded interval in R. Let f : T1 × T2 × · · · × Tn × Xr → R, gi : T1 × T2 × ·s × Tn × Xr i → R, i = 1, 2, . . . , m, be crisp functions. A conventional mathematical programming problem on time scales
μc (c) :=
min {μc j (cj ): c = (c1, c2 , . . . , cr )T and cj ∈ X}
j =1,2,...,r
(11) and μb i (bi ) :=
min
k =1,2,...,r i
{μb i , k (bi,k ): bi = (bi,1 , bi,2 , . . . , bi,r i )T and bi,k ∈ X}, i = 1, 2, . . . , m,
respectively. As in Definition 10, the fuzzy mappings f: T1 × T2 × · · · × Tn × F (Xr ) → F (E)
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and gi : T1 × T2 × · · · × Tn × F (Xr i ) → F (R), i = 1, 2, . . . , m in (8) are defined by means of fuzzy sets Z := f˜(x, c) and Bi := g˜i (x, bi ), i = 1, 2, . . . , m, whose membership functions are given by sup{μc (c) : z = f (x, c)}, if {c: z = f (x, c)} = ∅ μZ (z) = 0, otherwise and μB i (βi ) sup{μb i (bi ): βi = gi (x, bi )}, if {bi : βi = gi (x, bi )} = ∅ = 0, otherwise
Fig. 2.
Membership function of the non-L-R type fuzzy parameter.
Since α = {z: z = f (x, c) and μ (z) ≥ α} (Z) Z
i = 1, 2, . . . , m, respectively. For each α ∈ [0, 1] and c = (c1 , c2 , . . . , cr ) ∈ Xr , we have
= {z: z = f (x, c) and μc (c) ≥ α}
μc (c) ≥ α ⇔ μc j (cj ) ≥ α, for all j = 1, 2, . . . , r.
= {f (x, c) and c ∈ ( c)α }
This along with (11) yields c1 )α × ( c2 )α × · · · × ( cr )α ( c)α = (
(12)
cj )α , j = 1, 2, . . . , r, are the α-level sets of c where ( c)α and ( and cj , j = 1, 2, . . . , r, respectively, and defined by
and i )α = {βi : βi = gi (x, bi ) and μ (βi ) ≥ α} (B Bi = {βi : βi = gi (x, bi ) and μb i (bi ) ≥ α}
( c)α := {c ∈ X : μc (c) ≥ α} r
and
= {gi (x, bi ) and bi ∈ (bi )α }
( cj )α := cj ∈ X: μc j (cj ) ≥ α , j = 1, 2, . . . , r.
by using the tolerance approach in [8], the problem (8) can be restated as follows:
Since α-level sets are the crisp sets whose elements have degrees of membership greater than or equal to α, the larger the value of α the higher the degree of satisfaction. Similarly, for each i ∈ {1, 2, . . . , m}, we have (bi )α = (bi,1 )α × (bi,2 )α × · · · × (bi,r i )α
(14)
and (bi,k )α := {bi,k ∈ X: μb i , k (bi,k ) ≥ α}, k = 1, 2, . . . , ri . (15) Suppose that the fuzzy parameters in the problem (8) are represented as fuzzy sets whose highest membership values are clustered around a couple of real numbers as shown in Fig. 2 and the membership functions μc j , j = 1, 2, . . . , r, and μb i , k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m, are lower semicontinuous. This along with closeness of X implies that the sets ( cj )α , j = 1, 2, . . . , r, and (bi,k )α , k = 1, 2, . . . , ri , for each i ∈ {1, 2, . . . , m}, are closed, i.e., ( cj )α , j = 1, 2, . . . , r, and (bi,k )α , k = 1, 2, . . . , ri , for each i ∈ {1, 2, . . . , m}, are time scales. Hence, by (13) we get (bi )α := Tαi,1 × · · · × Tαi,r i where Tαi,1 :=(bi,k )α for each i ∈ {1, 2, . . . , m}.
f (x, c)
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m bi,k ∈ (bi,k )α , k = 1, 2, . . . , ri , i = 1, 2, . . . , m
(13)
where (bi )α and (bi,k )α , k = 1, 2, . . . , ri , are the α-level sets of bi and bi,k , k = 1, 2, . . . , ri , respectively, and defined by (bi )α := {bi ∈ Xr i : μb i (bi ) ≥ α}
max
cj ∈ ( cj )α , j = 1, 2, . . . , r x ∈ T1 × T 2 × . . . × T n .
(16)
Let Tαi,k := (˜bi,k )α , k = 1, 2, . . . , ri , for each i ∈ {1, 2, . . . , cj )α , j = 1, 2, l, r. The problem (16) is equivm}, Tαm +1,j := (˜ alent to max
f (x, c)
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m bi,k ∈ Tαi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m cj ∈ Tαm +1,j , j = 1, 2, . . . , r x ∈ T1 × T2 ×, · · · , ×Tn .
(17)
Let K := r1 + r2 + · · · + rm . For each α ∈ [0, 1], define the mixed domain Λnα +K := T1 × · · · × Tn × (b1 )α × · · · × (bm )α
(18)
where (bi )α := Tαi,1 × · · · × Tαi,r i , i = 1, 2, . . . , m. When the nonlinear objective function f (x, c) is nondecreasing with respect to its second argument, we can eliminate the parameter c
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from the objective function f (x, c) by defining the function z(x, b) = max{f (x, c): c ∈
Tαm +1,1
×
Tαm +1,2
many fuzzy mathematical programming problems which have not been proposed in the existing literature. For instance, the FMP with exponential decision parameters
× · · · × Tαm +1,r } which actually does not include the parameter b := (b1 , b2 , . . . , bm )T where bi = (bi,1 , . . . , bi,r i )T , i = 1, 2, . . . , m. Hence, the problem (17) turns into the following problem: max
max
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m bi,k ∈ Tαi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m
z (x, b)
cj ∈ Tαm +1,j , j = 1, 2, . . . , r n x ∈ 2N
gi (x, bi ) ≤ 0, i = 1, 2, . . . , m bi,k ∈ Tαi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m (x, b) ∈ Λnα +K . This along with Theorem 4 yields the following result: Theorem 5: Suppose that f is nondecreasing with respect to its second argument, and, for each grade α ∈ [0, 1], the set Sα := (x, b) ∈ Λnα +K : gi (x, bi ) ≤ 0, i = 1, 2, . . . , m
where 2N = {2t : t ∈ N} or the FMP with binary decision parameters max
n
(xi − xαi , bi − bαi )ei ≥ 0 for all (x, b) ∈ Sα
i=1
where α
α
ξ (x , b ) =
n
∂f λi (x , β ) Δi xi (x α ,β α ) i=1 α
α
∂f ei . + (1 − λi (x , β )) ∇i xi (x α ,β α ) α
bi,k ∈ Tαi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m cj ∈ Tαm +1,j , j = 1, 2, . . . , r x ∈ {0, 1}n . IV. FUZZY MATHEMATICAL PPROGRAMMING PROBLEM WITH LINEAR OBJECTIVE AND NON-L-R TYPE FUZZY COEFFICIENTS In this section, we suppose that all the assumptions we made in the previous section are valid. If we let f (x, c) = nj=1 cj xj , then (2) turns into the following FMP:
α
max
cj xj
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m x ∈ T1 × T 2 × · · · × T n
(20)
f (x, c)
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m bi,k ∈ Tαi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m cj ∈ Tαm +1,j , j = 1, 2, . . . , r n x ∈ R+
(19)
with continuous decision parameters, but also the problem max
n j =1
Remark 1: Note that the decision variables x in (17) are supposed to be in T1 × T2 × · · · × Tn . Hence, the problem (17) corresponding to (2) covers not only the problem max
f (x, c)
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m
is a convex set in Λnα +K satisfying cintΛ nα + K (Sα ) = ∅. Then, for each grade α ∈ [0, 1] , (xα , bα ) = (xα1 , . . . , xαn , bα1 , . . . , bαm ) is an optimal solution to problem (8) if and only if there exist scalars λi (xα , bα ) , i = 1, 2, . . . , n + ri , such that ξ T (xα , bα )
f (x, c)
where cj , j = 1, 2, . . . , n are fuzzy parameters, gi , i = 1, 2, . . . , m, are fuzzy mappings associated with the crisp functions gi , i = 1, 2, . . . , m, bi := (bi,1 , bi,2 · · · bi,r i )T are fuzzy vectors consisting of fuzzy parameters bi,1 , bi,2 , . . . , bi,r i , i = 1, 2, . . . , m. According to the similar discussion in Section III, the fuzzy solution of (20) can be obtained by solving the following problem:
f (x, c)
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m bi,k ∈ Tαi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m cj ∈ Tαm +1,j , j = 1, 2, . . . , r x ∈ Nn with integer decision parameters. Since there are time scales other than N an R+ , the problem (16) has a potential to cover
max
n
cj xj
j =1
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m bi,k ∈ bi,k , k = 1, 2, . . . , ri , i = 1, 2, . . . , m α
cj )α , j = 1, 2, . . . , n, cj ∈ ( x ∈ T1 × T 2 × · · · × T n .
(21)
HU et al.: NON-L-R TYPE FUZZY PARAMETERS IN MATHEMATICAL PROGRAMMING PROBLEMS
Since x ≥ 0 and X is a closed and bounded interval, (21) can be expressed as follows: ⎛ ⎞ n −min z(x, b) = ⎝− ( cj )U xj ⎠ α
j =1
s.t. gi (x, bi ) ≤ 0, i = 1, 2, . . . , m (x, b) ∈ Λnα +K where
(22)
cj )α , j = 1, 2, . . . , n ( cj )Uα := max cj : cj ∈ (
and Λnα +K is defined in (18). Observe that for a linear function h(x) = nj=1 γj xj , where x = (x1 , x2 , . . . , xn )T , we have
Fig. 3.
Membership function of fuzzy parameter c˜1 .
∂h(x) ∂h(x) = = γj . j xj j xj Hence, combining Theorem 3 and Theorem 4, we obtain the following result. Theorem 6: Suppose that, for each α ∈ [0, 1] , the set Sα := (x, b) ∈ Λnα +K : gi (x, bi ) ≤ 0, i = 1, 2, . . . , m is a convex set in Λnα +K satisfying cintΛ nα + K (Sα ) = ∅. Then, for each grade α ∈ [0, 1] , (xα , bα ) ∈ Sα is an optimal solution to the problem (20) if and only if Fig. 4. Membership function of fuzzy parameter c˜2 . ⎛ ⎞T n +K ⎝ ( cj )Uα ej⎠ ((x, b) − (xα − bα )) ≥ 0 for all (x, b) ∈ Sα j =1
where ( cj )Uα := max cj : cj ∈ ( cj )α for j = 1, 2, . . . , n, and ( cj )Uα := 0 for j = n + 1, n + 2, . . . , n + K. In this study, we are developing the fuzzy-time scale analogue of conventional geometric method in convex analysis. The related issues of computation time analysis or convergence analysis can be found in [3]. V. NUMERICAL EXAMPLES To illustrate the associated results derived in the previous section, numerical examples are considered in this section. The numerical experiments are performed on the HP Compaq dx2810 MT under the Windows 7 operating system. Example 2: Consider the following mathematical programming problem with fuzzy parameters: max
Fig. 5.
Membership function of fuzzy parameter c˜3 .
Fig. 6.
Membership function of fuzzy parameter ˜b1 .
c˜1 x1 + c˜2 x2 + c˜3 x3
s.t. 6x1 + 4x2 − x3 ≤ ˜b1 2x1 + 4x2 + x3 ≤ ˜b2 x1 − x2 ≤ ˜b3 x1 , x2 , x3 ≥ 0
(23)
where c˜1 , c˜2 , c˜3 , ˜b1 , ˜b2 , and ˜b3 are fuzzy parameters with corresponding membership functions μc˜1 , μc˜2 , μc˜3 , μ˜b 1 , μ˜b 2 , and μ˜b 3 , which can be graphically described in Figs. 3–8.
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b1 ∈ [6, 9] b2 ∈ [4, 9] b3 ∈ [4, 5] x1 , x2 , x3 ≥ 0. max s.t.
6x1 + 2x2 − x3 6x1 + 4x2 − x3 − b1 ≤ 0 2x1 + 4x2 + x3 − b2 ≤ 0 x1 − x2 − b3 ≤ 0
Fig. 7.
b1 ∈ [12, 18]
Membership function of fuzzy parameter ˜b2 .
b2 ∈ [4, 9] b3 ∈ [4, 5] x1 , x2 , x3 ≥ 0. max s.t.
6x1 + 2x2 − x3 6x1 + 4x2 − x3 − b1 ≤ 0 2x1 + 4x2 + x3 − b2 ≤ 0 x1 − x2 − b3 ≤ 0 b1 ∈ [6, 9] b2 ∈ [4, 9]
Fig. 8.
b3 ∈ [8, 10]
Membership function of fuzzy parameter ˜b3 .
x1 , x2 , x3 ≥ 0.
For α = 0.8, problem (23) can be considered as follows: max s.t.
s.t.
z(x1 , x2 , x3 , b1 , b2 , b3 ) = 6x1 + 2x2 − x3
b1 ∈ [12, 18]
x1 − x2 − b3 ≤ 0 b1 ∈ [6, 9] [12, 18]
b2 ∈ [4, 9] b3 ∈ [8, 10] x1 , x2 , x3 ≥ 0.
[8, 10]
x1 , x2 , x3 ≥ 0.
(24)
Since the constraints b1 ∈ [6, 9] ∪ [12, 18] and b3 ∈ [4, 5] ∪ [8, 10] in (24) can be interpreted as b1 ∈ [6, 9] or b1 ∈ [12, 18] and b3 ∈ [4, 5] or b3 ∈ [8, 10], respectively, the solution of problem (24) can be obtained by choosing the solution with the largest objective values of the following linear programming problems max s.t.
6x1 + 4x2 − x3 − b1 ≤ 0 x1 − x2 − b3 ≤ 0
2x1 + 4x2 + x3 − b2 ≤ 0
b3 ∈ [4, 5]
6x1 + 2x2 − x3 2x1 + 4x2 + x3 − b2 ≤ 0
6x1 + 4x2 − x3 − b1 ≤ 0
b2 ∈ [4, 9]
max
6x1 + 2x2 − x3 6x1 + 4x2 − x3 − b1 ≤ 0 2x1 + 4x2 + x3 − b2 ≤ 0 x1 − x2 − b3 ≤ 0
Therefore, the optimal solution to the problem (24) is (xα1 , xα2 , xα3 , bα1 , bα2 , bα3 ) = (3.1583, 0, 0.9499, 18, 8.3467, 4.3962) with the subgradient ξ(xα1 , xα2 , xα3 , bα1 , bα2 , bα3 ) = (−6, −2, 1, 0, 0, 0)T . It can be observed that for any (x1 , x2 , x3 , b1 , b2 , b3 ) in the feasible domain of the problem (24), we have ξ(xα1 , xα2 , xα3 , bα1 , bα2 , bα3 )T ((x1 , x2 , x3 , b1 , b2 , b3 ) − (xα1 , xα2 , xα3 , bα1 , bα2 , bα3 )) ≥ 0. According to Figs. 3–8, we assume that for the case of c2 )Uα = 2.3, (˜ c3 )Uα = −0.8, (˜b1 )α ∈ [5.5, α = 0.5, (˜ c1 )Uα = 7, (˜ ˜ 9.5] ∪ [11.5, 18.5], (b2 )α ∈ [4, 11], (˜b3 )α ∈ [3, 5.5] ∪ [7.5, 10.5], c2 )Uα = 2.7, (˜ c3 )Uα = and for the case of α = 0.2, (˜ c1 )Uα = 8, (˜ ˜ ˜ ˜ −0.5, (b1 )α ∈ [5, 10] ∪ [11, 19], (b2 )α ∈[2, 12], (b3 )α ∈ [2, 6] ∪ [7, 11]. Table III shows the computational results for different values of α.
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TABLE III COMPUTATIONAL RESULTS FOR DIFFERENT VALUES OF α
TABLE V VALUE OF (˜ cj )U α
TABLE IV COMPUTATIONAL RESULTS BY GA FOR DIFFERENT VALUES OF α
0.025x4 + 0.015x5 + 0.021x8 ≤ ˜b15 0.0017x4 + 0.0035x9 ≤ ˜b16 0.0015x4 + 0.0015x5 ≤ ˜b17 To compare the results obtained in Table III with other metaheuristics method, the computational results of solving (24) by GA are shown in Table IV. Example 3: An industrial management problem of fuzzy linear programming with fuzzy resources and fuzzy coefficients for the objective function has been studied in Vasant et al. [43] using real-life data from a vegetable oil company. The company produces a variety of products such as vegetable oils, solid oils, toilet soap, solid cleaner, liquid cleaner, car soap, toothpaste, and washing powder. There are n products to be manufactured by mixing m raw materials with different proportion and by using k varieties of processing methods. There are also some constraints imposed by the marketing department such as product mix requirements, main product line requirements, and the lower and upper limit of demand for each product. The aforementioned requirements and conditions are fuzzy, and the objective is to obtain the optimal units of products with a certain degree of satisfaction. The fuzzy linear programming model can be described as follows: max
c˜1 x1 + c˜2 x2 + c˜3 x3 + c˜4 x4 + c˜5 x5 + c˜6 x6 + c˜7 x7 + c˜8 x8 + c˜9 x9 + c˜10 x10
s.t. 1.03x1 ≤ ˜b1 0.025x1 + 0.021x2 ≤ ˜b2 0.003x1 + 0.00035x2 ≤ ˜b3 1.053x2 + 0.78x3 + 0.66x6 ≤ ˜b4 0.0035x2 + 0.0045x4 ≤ ˜b5 0.28x3 + 0.25x6 ≤ ˜b6 0.00124x3 + 0.0065x7 + 0.0017x10 ≤ ˜b7 0.5x3 + 0.5x6 + 0.035x10 ≤ ˜b8 0.5x3 + 0.25x6 + 0.6x10 ≤ ˜b9 0.0003x3 + 0.0035x10 ≤ ˜b10 0.018x3 ≤ ˜b11 0.00775x3 + 0.0057x6 + 0.00225x10 ≤ ˜b12 0.0745x4 + 0.04x7 + 0.22x9 ≤ ˜b13 0.016x4 ≤ ˜b14
0.002x5 + 0.00225x8 ≤ ˜b18 0.0205x4 + 0.02785x5 + 0.012x8 ≤ ˜b19 0.34x5 + 0.285x8 ≤ ˜b20 0.265x5 + 0.225x8 ≤ ˜b21 0.022x5 + 0.022x8 ≤ ˜b22 0.0175x5 + 0.0125x7 + 0.035x8 ≤ ˜b23 0.275x6 ≤ ˜b24 0.275x7 ≤ ˜b25 0.3x8 ≤ ˜b26 0.0055x9 ≤ ˜b27 150x1 ≤ ˜b28 15.25x2 ≤ ˜b29 28.65x3 ≤ ˜b30 32.2x4 ≤ ˜b31 204x5 ≤ ˜b32 45.9x6 ≤ ˜b33 238.5x7 ≤ ˜b34 64.46x8 ≤ ˜b35 32.23x9 ≤ ˜b36 212.5x10 ≤ ˜b37 x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 , x10 ≥ 0.
(25)
Assume that the fuzzy resources and fuzzy coefficients for the objective function are non-L-R type fuzzy parameters. For a given α = 0.8, the values of (˜ cj )Uα , j = 1, 2, . . . , 10 and the sets (˜bi )α , i = 1, 2, . . . , 37, are listed in Tables V and VI. Table VII shows the result of solving (25) by using MATLAB and its optimization toolbox for the case of α = 0.8. The results obtained in [43] for the case of the level of satisfaction μ = 0.7994 are also provided in Table VII. The different assumptions of fuzzy parameters for the fuzzy resources and fuzzy coefficients for the objective function lead to different optimal units of products for the same proposed problem.
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TABLE VI VALUE OF (˜bi )α
VI. CONCLUSION This paper has been studying mathematical programming problems with non-L-R type fuzzy parameters. Such a problem can be converted into an optimization problem on a mixed domain. The necessary and sufficient conditions for solving the resulting optimization problem with the domain of the product of time scales are investigated. The main contribution of the paper is the employment of the theory of convex optimization on mixed domains to the resolution of the fuzzy optimization problem with non-L-R type membership functions. To the best of our knowledge, the fuzzy optimization problems with nonL-R type parameters considered in this study are completely new to the literature and have not been studied elsewhere before. Probably, an alternative model of the proposed approach may be considered in which more objectives and constraints are included allowing the resolution of the problem through the use of L-R fuzzy parameters. However, solving this alternative model with L-R fuzzy parameters could be much more complex from a computational point of view. This study essentially provides a new concept and broadens the scope for solving fuzzy optimization problems. ACKNOWLEDGMENT The authors would like to thank referees for their very constructive comments. REFERENCES
TABLE VII COMPUTATIONAL RESULT OF SOLVING PROBLEM (25)
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Cheng-Feng Hu received the B.S. degree from National Tsing Hua University, Hsinchu City, Taiwan and the Ph.D. degree from North Carolina State University Raleigh, NC, USA, in 1993 and 1997 respectively. Her research interests include fuzzy optimization and decision making and financial engineering. She is currently with the Department of Applied Mathematics, National Chiayi University, Chiayi, Taiwan.
Murat Adivar received the Bachelor’s, Master’s, and Ph.D. degrees, in 1996, 1998, and 2003 respectively, from Department of Mathematics, Ankara University, Ankara, Turkey. He is currently with the Department of Mathematics, Izmir University of Economics (IUE), Izmir, Turkey, and also serves as a Vice Rector in charge of Academic Affairs with IUE. He is also working with E. P. Fitts in the Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, USA, on a joint research project funded by TUBITAK. His Ph.D. thesis, which was in the field of spectral analysis, was funded by The Scientific and Technological Research Council of Turkey, International Ph.D. Scholarship Programme for Raising Young Scientists. His research interests include spectral analysis, functional analysis, analysis of qualitative and quantitative properties of dynamic equations on time scales, and global optimization Dr. Adivar received a NATO Fellowship for post doctoral studies from the Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA. He has been an invited speaker to several international meetings and conferences and has works published in many internationally recognized journals.
Shu-Cherng Fang received the B.S. degree from National Tsing Hua University, Hsinchu City, Taiwan, and the Ph.D. degree from Northwestern University in Evanston, Chicago, IL, USA. He holds the Walter Clark Chair Professorship and Alumni Distinguished Graduate Professorship with North Carolina State University, Raleigh, NC, USA. He was the Senior Member of the Research Staff at Western Electric Engineering Research Center, a Supervisor at AT&T Bell Labs, and Department Manager at the Corporate Headquarters of AT&T Technologies before joining the NC State. He has published more than 180 refereed journal articles. He authored the books Linear Optimization and Extensions: Theory and Algorithms (Englewood Cliffs, NJ, USA: Prentice Hall, 1993) with S. C. Puthenpura, Entropy Optimization and Mathematical Programming (Norwell, MA, USA: Kluwer Academic, 1997) with J.R. Rajasekera and H.-S. Tsao, and Linear Conic Optimization (San Francisco, CA, USA: Scientific Press, 2013) with W. Xing. His research interests include linear and nonlinear programming, fuzzy optimization and decision making, soft computing, logistics, and supply chain management. Dr. Fang is the Founding Editor-in-Chief of Fuzzy Optimization and Decision Making, and currently serves on the editorial boards of 25 scientific journals.
