Optimality conditions for linear programming problems with fuzzy coefficients

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Optimality conditions for linear programming problems with fuzzy coefficients

Hsien-Chung Wu

Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan Received 22 November 2006; accepted 28 September 2007

Abstract

The optimality conditions for linear programming problems with fuzzy coefficients are derived in this paper. Two solution concepts are proposed by considering the orderings on the set of all fuzzy numbers. The solution concepts proposed in this paper will follow from the similar solution concept, called the nondominated solution, in the multiobjective programming problem. Under these settings, the optimality conditions will be naturally elicited.

c

2007 Elsevier Ltd. All rights reserved.

Keywords:Fuzzy numbers; Nondominated solutions; (crisp) Fuzzy constraints

1. Introduction

The occurrence of fuzziness in the real world is inevitable owing to some unexpected situations. Therefore, imposing fuzziness upon conventional optimization problems becomes an interesting research topic. The collection of papers on fuzzy optimization edited by Słowi´nski [1] and Delgado et al. [2] gives the main stream of this topic. Lai and Hwang [3,4] also give an insightful survey. On the other hand, the book edited by Słowi´nski and Teghem [5] provides comparisons between fuzzy optimization and stochastic optimization for multiobjective programming problems.

Bellman and Zadeh [6] inspired the development of fuzzy optimization by providing the aggregation operators, which combined the fuzzy goals and fuzzy decision space. After this motivation and inspiration, there appeared a lot of articles dealing with fuzzy optimization problems. Some interesting articles are Buckley [7,8], Julien [9] and Luhandjula et al. [10] using possibility distribution, Herrera et al. [11] and Zimmermann [12,13] using fuzzified constraints and objective functions, Inuiguchi et al. [14,15] using modality measures, Tanaka and Asai [16] using fuzzy parameters, and Lee and Li [17–19] considering the de Novo programming problem.

The duality of the fuzzy linear programming problem was firstly studied by Rodder and Zimmermann [20]

considering the economic interpretation of the dual variables. After that, many interesting results regarding the duality of the fuzzy linear programming problem was investigated by Bector et al. [21–23], Liu et al. [24], Ram´ık [25], Verdegay [26] and Wu [27]. In this paper, we investigate the optimality conditions for linear programming problems with fuzzy coefficients.

E-mail address:hcwu@nknucc.nknu.edu.tw.

0898-1221/$ - see front matter c2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.camwa.2007.09.004

In Section2, we introduce some basic properties and arithmetics of fuzzy numbers. In Section3, we formulate two linear programming problems with fuzzy coefficients. One considers crisp (conventional) linear constraints, and the other considers fuzzy linear constraints. Two solution concepts are proposed for these two problems. In Section4, we derive the optimality conditions for these two problems by introducing the multipliers. Finally, in Section5, three examples are provided to illustrate the discussions in linear programming problems with fuzzy coefficients.

2. Arithmetics of fuzzy numbers

LetRbe the set of all real numbers. The fuzzy subseta˜ofRis defined by a functionξa˜ :R→ [0,1], which is called amembership function. Theα-level set ofa, denoted by˜ a˜_α, is defined bya˜_α = {x∈R:ξa˜(x)≥α}for allα∈(0,1]. The 0-level seta˜0is defined as the closure of the set{x∈R:ξa˜(x) >0}, i.e.,a˜0=cl({x∈R:ξa˜(x) >0}). Definition 2.1. We denote byF(R)the set of all fuzzy subsetsa˜ ofRwith membership functionξa˜ satisfying the following conditions:

(i) a˜ is normal, i.e., there exists anx∈Rsuch thatξa˜(x)=1;

(ii) ξa˜ is quasi-concave, i.e.,ξa˜(λx+(1−λ)y)≥min{ξa˜(x), ξa˜(y)}for allx,y∈Randλ∈ [0,1];

(iii) ξa˜ is upper semicontinuous, i.e.,{x∈R:ξa˜(x)≥α} = ˜a_αis a closed subset ofUfor eachα∈(0,1]; (iv) the 0-level seta˜0is a compact subset ofR.

The membera˜inF(R)is called afuzzy number.

Suppose now thata˜ ∈F(R). From Zadeh [28], theα-level seta˜_α ofa˜is a convex subset ofRfor eachα∈ [0,1] from condition (ii). Combining this fact with conditions (iii) and (iv), theα-level seta˜_α ofa˜is a compact and convex subset of Rfor eachα ∈ [0,1], i.e., a˜_α is a closed interval inR for eachα ∈ [0,1]. Therefore, we also write a˜_α =

a˜_α^L,a˜^U_α .

Definition 2.2. Leta˜ be a fuzzy number. We say thata˜ isnonnegativeifa˜_α^L ≥0 for allα∈ [0,1]. We say thata˜ is positiveifa˜_α^L >0 for allα∈ [0,1]. We say thata˜isnonpositiveifa˜^U_α ≤0 for allα∈ [0,1]. We say thata˜isnegative ifa˜^U_α <0 for allα∈ [0,1].

Remark 2.1. Leta˜be a fuzzy number. Thena˜_α^L ≤ ˜a^U_α for allα∈ [0,1]. Therefore ifa˜ is nonnegative thena˜_α^L ≥0 anda˜^U_α ≥0 for allα ∈ [0,1], and ifa˜ is positive thena˜_α^L >0 anda˜_α^U >0 for allα ∈ [0,1]. We can have similar consequences for nonpositive and negative fuzzy numbers.

Let “” be any binary operations⊕or⊗between two fuzzy numbersa˜ andb. The membership function of˜ a˜ ˜b is defined by

ξ_{a ˜˜ b}(z)= sup

x◦y=z

min{ξa˜(x), ξb˜(y)}

using the extension principle in Zadeh [29], where the operations = ⊕and⊗correspond to the operations◦ = + and×, respectively. Then we have the following results.

Proposition 2.1. Leta˜,b˜∈F(R). Then we have (i) a˜⊕ ˜b∈F(R)and

(a˜⊕ ˜b)α =h

a˜_α^L + ˜b_α^L,a˜^U_α + ˜b^U_αi

; (ii) a˜⊗ ˜b∈F(R)and

(a˜⊗ ˜b)α =h minn

a˜_α^Lb˜_α^L,a˜_α^Lb˜^U_α,a˜^U_αb˜_α^L,a˜^U_αb˜^U_αo ,maxn

a˜_α^Lb˜_α^L,a˜_α^Lb˜^U_α,a˜^U_αb˜_α^L,a˜^U_αb˜^U_αoi . The following proposition is very useful for discussing the optimality conditions.

Proposition 2.2. Let a be a nonnegative fuzzy number and˜ b be a nonpositive fuzzy number. If˜ a˜ ⊗ ˜b = ˜0, then a˜_α^Lb˜^L_α = ˜a_α^Lb˜^U_α = ˜a_α^Ub˜^L_α = ˜a^U_αb˜^U_α =0for allα∈ [0,1].

Proof. FromProposition 2.1(ii), we immediately have thata˜⊗ ˜b= ˜0 implies

a˜_α^Ub˜_α^L =0= ˜a_α^Lb˜^U_α (1)

for allα∈ [0,1]. Therefore, we need to show thata˜_α^Lb˜_α^L = ˜a_α^Ub˜^U_α =0 for allα∈ [0,1]. We consider the following cases.

(i) Suppose thatb˜^U_α 6=0. Thenb˜_α^L 6=0 sinceb˜_α^L ≤ ˜b^U_α. Therefore,a˜_α^L = ˜a^U_α =0 by(1), i.e.,a˜_α^Lb˜_α^L = ˜a_α^Ub˜_α^U =0.

(ii) Suppose thatb˜^U_α =0 andb˜_α^L =0. Then it is easy to see thata˜_α^Lb˜_α^L = ˜a^U_αb˜^U_α =0.

(iii) Suppose thatb˜^U_α =0 andb˜_α^L 6=0. Then it remains to showa˜_α^Lb˜_α^L =0. Sinceb˜_α^L 6=0 anda˜_α^Ub˜_α^L =0 by(1), we havea˜^U_α =0. Sincea˜is nonnegative, 0≤ ˜a_α^L ≤ ˜a^U_α =0 showsa˜_α^Lb˜_α^L =0. We complete the proof.

We say thata˜ is acrisp numberwith valuemif its membership function is given by ξa˜(r)=

1 ifr =m 0 otherwise.

We also use the notation1˜_{m}to represent the crisp number with valuem. It is easy to see that(1˜_{m})_α^L =(1˜_{m})^U_α =m for allα∈ [0,1]. Let us remark that a real numbermcan be regarded as a crisp number1˜_{m}.

3. Solution concept

LetA = [a^L,a^U]andB = [b^L,b^U]be two closed intervals inR. We writeB ≤ Aif and only ifb^L ≤a^L and b^U ≤a^U, andB< Aif and only if the following conditions are satisfied:

(b^L <a^L b^U ≤a^U or

(b^L ≤a^L b^U <a^U or

(b^L <a^L b^U <a^U .

Leta˜andb˜be two fuzzy numbers. Thena˜_α = [ ˜a_α^L,a˜_α^U]andb˜_α = [ ˜b_α^L,b˜^U_α]are two closed intervals inRfor all α∈ [0,1]. We writeb˜ ˜aif and only ifb˜_α ≤ ˜a_α for allα∈ [0,1], or equivalently,b˜_α^L ≤ ˜a_α^L andb˜^U_α ≤ ˜a_α^U for all α∈ [0,1]. It is easy to see that “” is a partial ordering onF(R).

For notational convenience, we denote by0 the crisp number˜ 1˜_{0}with value 0. Now we consider the following two linear programming problems with fuzzy coefficients:

(FLP1) min (c˜1⊗ ˜1_x1)⊕ · · · ⊕(c˜n⊗ ˜1_xn)

subject to a_j1x₁+ · · · +a_{j n}x_n+b_j ≤0 for j =1, . . . ,m x₁, . . . ,x_n≥0

and

(FLP2) min (c˜1⊗ ˜1x₁)⊕ · · · ⊕(c˜n⊗ ˜1x_n)

subject to (a˜_j1⊗ ˜1_x1)⊕ · · · ⊕(a˜_{j n}⊗ ˜1_xn)⊕ ˜b_j ˜0 for j =1, . . . ,m x₁, . . . ,x_n≥0

where1˜_{xi} is a crisp number with value x_i for i = 1, . . . ,n. Problem (FLP1) considers the crisp (conventional) constraints, and problem (FLP2) considers fuzzy constraints. For convenient presentation, we also write

f˜(x)=(c˜1⊗ ˜1_x1)⊕ · · · ⊕(c˜n⊗ ˜1_xn) g_j(x)=a_j1x₁+ · · · +a_{j n}x_n+b_j

g˜j(x)=(a˜j1⊗ ˜1_x1)⊕ · · · ⊕(a˜j n⊗ ˜1_xn)⊕ ˜bj.

We need to interpret the meaning of minimization in problems (FLP1) and (FLP2). Since “” is a partial ordering, not a total ordering, on F(R), we may follow the similar solution concept (the nondominated solution) used in multiobjective programming problem to interpret the meaning of minimization in problems (FLP1) and (FLP2).

Two types of nondominated solution will be considered. We writea˜≺_Ib˜if and only ifa˜_α ≤ ˜b_α for allα∈ [0,1] and there exists anα^∗∈ [0,1]such thata˜_α^∗<b˜_α^∗, i.e.,

(a˜_α^L∗ <b˜_α^L∗

a˜_α^U∗ ≤ ˜b^U_α∗

or

(a˜_α^L∗ ≤ ˜b_α^L∗

a˜^U_α∗ <b˜^U_α∗

or

(a˜_α^L∗<b˜_α^L∗

a˜^U_α∗<b˜^U_α∗

. (2)

Therefore, we see thata˜≺_Ib˜meansa˜ ˜banda˜6= ˜b. On the other hand, we writea˜≺_IIb˜if and only ifa˜_α <b˜_αfor allα∈ [0,1], i.e.,(2)is satisfied for eachα∈ [0,1].

Definition 3.1. Letx^∗ be a feasible solution of problem (FLP1). We say thatx^∗ is anondominated type-I solution (resp.nondominated type-II solution) of problem (FLP1) if there exists no feasible solutionx¯such that f˜(x¯)≺_I f˜(x^∗) (resp. f˜(x¯)≺_II f˜(x^∗)). The nondominated type-I and type-II solutions can be similarly considered for problem (FLP2).

In what follows, we are going to provide the optimality conditions for nondominated solution of problems (FLP1) and (FLP2).

4. The optimality conditions

In order to derive the optimality conditions of problems (FLP1) and (FLP2), we need to recall the Karush–Kuhn–Tucker conditions for nonlinear programming problem. Let f andg_j, j = 1, . . . ,m, be real-valued functions defined onRⁿ. Then we consider the following (conventional) nonlinear programming problem:

(NLP) min f(x)= f(x₁, . . . ,x_n) subject to g_j(x)≤0,j =1, . . . ,m.

Suppose that the constraint functions gj are convex on Rⁿ for each j = 1, . . . ,m. Then the well-known Karush–Kuhn–Tucker optimality conditions for problem (NLP) (e.g., see Horst et al. [30] or Bazarra et al. [31]) is stated below.

Theorem 4.1. Assume that the constraint functions g_j : Rⁿ → R are convex on Rⁿ for j = 1, . . . ,m. Let X = {x ∈ Rⁿ : g_j(x) ≤ 0,i = 1, . . . ,m} be a feasible set and a point x^∗ ∈ X . Suppose that the objective function f :Rⁿ → Ris convex atx^∗, and f , g_j, j =1, . . . ,m, are continuously differentiable atx^∗. If there exist (Lagrange) multipliers0≤µj ∈R, j=1, . . . ,m, such that

(i) ∇f(x^∗)+Pm

i=1µj∇g_j(x^∗)=0;

(ii) µjgj(x^∗)=0for all j =1, . . . ,m,

thenx^∗is an optimal solution of problem(NLP).

Now, we are in a position to derive the optimality conditions for problems (FLP1) and (FLP2).

4.1. Crisp (conventional) constraints

Now we consider the problem (FLP1) with crisp constraints. We adopt the following notations:

c^L_α =





 c˜₁^L_α

...

c˜_n^L_α





, c^U_α =





 c˜^U_1α

...

c˜^U_nα





, and a_j =





 a_j1

...

aj n





,

wherec˜_i^L_α =(c˜_i)_α^Landc˜^U_iα =(c˜_i)^U_α fori=1, . . . ,n. We also see thatc˜_i^L_α ≤ ˜c^U_iαfor allα∈ [0,1]and alli=1, . . . ,m.

We also denote bye_kthe unit vector inRⁿfork=1, . . . ,n, i.e., thekth component ofe_kis 1 and the other components ofe_kare zero.

Theorem 4.2. Let x^∗ = (x^∗₁, . . . ,x_n^∗)be a feasible solution of problem(FLP1). If there exist positive real-valued functions λ^L and λ^U defined on[0,1], and nonnegative real-valued functions µj and λk for j = 1, . . . ,m and k=1, . . . ,n defined on[0,1]such that the following conditions are satisfied:

(i) λ^L(α)·c_α^L+λ^U(α)·c^U_α +Pm

j=1µj(α)·a_j−Pn

k=1λk(α)·e_k =0for allα∈ [0,1];

(ii) µj(α) · gj(x^∗) = 0 = λk(α) · x_k^∗ for all α ∈ [0,1], all j = 1, . . . ,m and all k = 1, . . . ,n, where gj(x^∗)=aj1x₁^∗+ · · · +aj nx_n^∗+bj,

thenx^∗is a nondominated type-I solution of problem(FLP1).

Proof. We are going to prove this result by contradiction. Suppose that conditions (i) and (ii) are satisfied andx^∗is not a nondominated type-I solution. Then there exists a feasible solutionx¯such that f˜(x¯)≺_I f˜(x^∗), i.e., from(2),

(f˜_α^L∗(x¯) < f˜_α^L∗(x^∗) f˜_α^U∗(x¯)≤ ˜f_α^U∗(x^∗) or

(f˜_α^L∗(x¯)≤ ˜f_α^L∗(x^∗) f˜_α^U∗(x¯) < f˜_α^U∗(x^∗) or

(f˜_α^L∗(x¯) < f˜_α^L∗(x^∗)

f˜_α^U∗(x¯) < f˜_α^U∗(x^∗) (3) for someα^∗ ∈ [0,1]. Sincexi ≥ 0 fori = 1, . . . ,n andc˜_i^L_α ≤ ˜c^U_iα for allα ∈ [0,1]and alli = 1, . . . ,n, using Proposition 2.1, we have

f˜_α^L(x)= ˜c₁^L_α·x₁+ · · · + ˜c_n^L_α·x_n and f˜_α^U(x)= ˜c^U_1α·x₁+ · · · + ˜c^U_nα·x_n. (4) We also have that

∇ ˜f_α^L(x)=c_α^L and ∇ ˜f_α^U(x)=c^U_α. (5) We now define a real-valued function

f(x)=

λ^L(α^∗)· ˜c_1α^L ∗+λ^U(α^∗)· ˜c^U_1α∗

·x1+ · · · +

λ^L(α^∗)· ˜c_n^L_α∗+λ^U(α^∗)· ˜c^U_nα∗

·x_n. (6)

Then we see that

f(x)=λ^L(α^∗)· ˜f_α^L∗(x)+λ^U(α^∗)· ˜f_α^U∗(x). (7) Combining(3)and(7), we see that

f(x¯) < f(x^∗) (8)

sinceλ^L(α^∗) >0 andλ^U(α^∗) >0. Furthermore, from(5)and(7), we have

∇f(x)=λ^L(α^∗)· ∇ ˜f_α^L∗(x)+λ^U(α^∗)· ∇ ˜f_α^U∗(x)=λ^L(α^∗)·c^L_α∗+λ^U(α^∗)·c^U_α∗. (9) We consider the following constrained optimization problem:

(P1) min f(x)

subject to gj(x)=aj1x1+ · · · +aj nxn+bj ≤0,j=1, . . . ,m, gm+k(x)= −xk≤0,k=1, . . . ,n,

where f is defined in(7). Then problems (P1) and (FLP1) have identical feasible regions. Since conditions (i) and (ii) of this theorem are satisfied for allα ∈ [0,1], anda_j = ∇gj(x)for all j =1, . . . ,m, according to(9)and for any fixedα^∗∈ [0,1], we can obtain the following two new conditions by lettingµjα^∗ =µj(α^∗)≥0 for j =1, . . . ,m, g_m+k(x)= −x_k andλkα^∗=λk(α^∗)fork=1, . . . ,n:

(i⁰) ∇f(x^∗)+Pm

j=1µjα^∗· ∇g_j(x^∗)+Pn

k=1λkα^∗· ∇g_m+k(x^∗)=λ^L(α^∗)·c_α^L∗+λ^U(α^∗)·c^U_α∗+Pm

j=1µj(α^∗)· a_j −Pn

k=1λk(α^∗)·e_k=0;

(ii⁰) µjα^∗·gj(x^∗)=0=λkα^∗·gm+k(x^∗)for all j =1, . . . ,mand allk=1, . . . ,n.

UsingTheorem 4.1, we see that the above conditions (i’) and (ii’) are the KKT conditions for problem (P1). Therefore, we conclude thatx^∗is an optimal solution of problem (P1), i.e., f(x^∗)≤ f(x¯), which contradicts(8). This completes the proof.

Remark 4.1. The positive real-valued functionsλ^L andλ^U, and nonnegative real-valued functions µj andλk for j =1, . . . ,mandk =1, . . . ,n can be constructed as follows. For any fixedα ∈ [0,1], if there exist positive real numbersλ_α^L andλ^U_α, and nonnegative real numbersλkα andµjα for j =1, . . . ,m andk = 1, . . . ,n such that the following conditions are satisfied:

(a) λ_α^L·c_α^L+λ^U_α ·c^U_α +Pm

j=1µjα·a_j−Pn

k=1λkα·e_k =0;

(b) µjα·gj(x^∗)=0=λkα·x_k^∗for all j =1, . . . ,mandk=1, . . . ,n,

then we can define the positive real-valued functions λ^L(α) = λ_α^L andλ^U(α) = λ^U_α for all α ∈ [0,1], and the nonnegative real-valued functions µj(α) = µjα andλk(α) = λkα for allα ∈ [0,1], all j = 1, . . . ,m and all k=1, . . . ,n. Therefore, if the above conditions (a) and (b) are satisfied for allα∈ [0,1], thenx^∗is a nondominated type-I solution of problem (FLP1) byTheorem 4.2.

Theorem 4.3. Letx^∗ =(x₁^∗, . . . ,x_n^∗)be a feasible solution of problem(FLP1). If there exist positive real numbers λ^L andλ^U, nonnegative real numbersµj andλk for j =1, . . . ,m and k =1, . . . ,n, andα^∗ ∈ [0,1]such that the following conditions are satisfied:

(i) λ^L ·c^L_α∗+λ^U ·c^U_α∗+Pm

j=1µj·a_j−Pn

k=1λk·e_k =0;

(ii) µj·g_j(x^∗)=0=λk·x_k^∗for all j =1, . . . ,m and all k=1, . . . ,n, thenx^∗is a nondominated type-II solution of problem(FLP1).

Proof. We are going to prove this result by contradiction. Suppose that conditions (i) and (ii) are satisfied andx^∗is not a nondominated type-II solution. Then there exists a feasible solutionx¯such that f˜(x¯)≺_II f˜(x^∗), i.e.,(3)is satisfied for allα∈ [0,1]. Forα^∗in conditions (i) and (ii), we can define a real-valued function

f(x)=(λ^L· ˜c₁^L_α∗+λ^U · ˜c^U_1α∗)·x₁+ · · · +(λ^L · ˜c_n^L_α∗+λ^U · ˜c^U_nα∗)·x_n. (10) Then we see that

f(x)=λ^L· ˜f_α^L∗(x)+λ^U· ˜f_α^U∗(x) (11)

and

∇f(x)=λ^L· ∇ ˜f_α^L∗(x)+λ^U· ∇ ˜f_α^U∗(x)=λ^L·c_α^L∗+λ^U·c^U_α∗. (12) Now we consider the constrained optimization problem (P1) as in the proof ofTheorem 4.2. Then we can obtain the following two new conditions:

(i⁰) ∇f(x^∗)+Pm

j=1µj· ∇gj(x^∗)+Pn

k=1λk· ∇gm+k(x^∗)=λ^L·c_α^L∗+λ^U·c^U_α∗+Pm

j=1µj·a_j−Pn

k=1λk·e_k=0;

(ii⁰) µj·g_j(x^∗)=0=λk·g_m+k(x^∗)for all j=1, . . . ,mand allk=1, . . . ,n. The remaining proof follows from the similar arguments ofTheorem 4.2.

We are going to present another type of optimality conditions for a nondominated type-II solution without consideringα^∗.

Definition 4.1. Leta˜ be a fuzzy number. We say thata˜ is acanonical fuzzy numberif the functionsη1(α)= ˜a_α^L and η2(α)= ˜a_α^U are continuous on [0,1].

Remark 4.2. Leta˜ be a fuzzy number and its membership function be strictly increasing on the interval a˜₀^L,a˜₁^L and strictly decreasing on the interval

a˜^U₁,a˜^U₀

. Then, from the fact of strict monotonicity,a˜_α^Landa˜^U_α are continuous functions with respect to the variableαon [0,1]. This shows thata˜ is a canonical fuzzy number.

Let f : [a,b] → Rⁿ be a vector-valued function defined on the closed interval such that each component f_i, i =1, . . . ,n, is continuous on[a,b]. Then the Riemann integral offon[a,b]is defined to be the Riemann integral of each component f_ion[a,b]. More precisely, we have

Z b a

f(x)dx = Z b

a

f1(x)dx, . . . ,Z b a

f_n(x)dx

.

Now we are in a position to present another type of optimality conditions by considering the Riemann integral.

Theorem 4.4. Suppose that the fuzzy coefficientsc˜_i for i =1, . . . ,n in the fuzzy-valued objective function f are now˜ assumed to be canonical fuzzy numbers. Letx^∗=(x₁^∗, . . . ,x_n^∗)be a feasible solution of problem(FLP1). If there exist positive real numbersλ^L andλ^U, and nonnegative real numbersµj andλkfor j =1, . . . ,m and k=1, . . . ,n such that the following conditions are satisfied:

(i) λ^L·R1

0 c^L_αdα+λ^U ·R1

0 c^U_αdα+Pm

j=1µj ·a_j −Pn

k=1λk·e_k=0;

(ii) µj·g_j(x^∗)=0=λk·x_k^∗for all j =1, . . . ,m and all k=1, . . . ,n, thenx^∗is a nondominated type-II solution of problem(FLP1).

Proof. For any fixedx, since f˜_α^L(x)and f˜_α^U(x)are continuous on [0,1] with respect to the variableαby definition, they will be Riemann integrable on [0,1] with respect toα. Therefore, we can define a real-valued function as follows:

f(x)=λ^L· Z 1

0

f˜_α^L(x)dα+λ^U· Z 1

0

f˜_α^U(x)dα. (13)

For any fixedα, since f˜_α^L(x)and f˜_α^U(x)are linear functions, that is, they are continuously differentiable, by Rudin [32, Theorem 9.42], we have

∇f(x)=λ^L· Z 1

0

∇ ˜f_α^L(x)dα+λ^U· Z 1

0

∇ ˜f_α^U(x)dα=λ^L · Z 1

0

c_α^Ldα+λ^U· Z 1

0

c^U_αdα. (14)

It also says that condition (i) of this theorem is well defined. We are going to prove this result by contradiction. Suppose thatx^∗is not a nondominated type-II solution. Then there exists a feasible solutionx¯such that f˜(x¯)≺_II f˜(x^∗), i.e.,(3) is satisfied for allα∈ [0,1]. Therefore, we have

λ^L · ˜f_α^L(x¯)+λ^U · ˜f_α^U(x¯) < λ^L· ˜f_α^L(x^∗)+λ^U· ˜f_α^U(x^∗)

for allα∈ [0,1]sinceλ^L >0 andλ^U >0. By taking integration with respect toαon [0,1] and using(13), we obtain f(x¯) < f(x^∗). Now we consider the constrained optimization problem (P1) as in the proof ofTheorem 4.2. Applying (14)to conditions (i) and (ii) of this theorem, we obtain the following two new conditions:

(i⁰) ∇f(x^∗)+Pm

j=1µj · ∇gj(x^∗)+Pn

k=1λk· ∇gm+k(x^∗)=0;

(ii⁰) µj ·g_j(x^∗)=0=λk·g_m+k(x^∗)for all j =1, . . . ,mand allk=1, . . . ,n.

The remaining proof follows from the similar arguments ofTheorem 4.2.

Remark 4.3. From the proof ofTheorem 4.4, we see that if the function f in(13)is defined as f(x)=

Z 1 0

f˜_α^L(x)dα+ Z 1

0

f˜_α^U(x)dα,

thenTheorem 4.4still holds true if condition (i) is replaced by the following new condition:

Z 1 0

c^L_αdα+ Z 1

0

c^U_αdα+

m

X

j=1

µj ·a_j −

n

X

k=1

λk·e_k=0.

Now we are going to present the optimality conditions of problem (FLP1) in the fuzzy-valued form. We write 0˜ =(0˜, . . . ,0˜)^T. Letxbe ann-vector inRⁿ. Then the crisp vector1˜_{x}is defined as1˜_{x}=

1˜_{x1},1˜_{x2}, . . . ,1˜_{xn}

. Leta=(a₁, . . . ,a_n)be ann-vector inRⁿ. We say thatahas the same sign if and only ifa_i ≥0 for alli =1, . . . ,n simultaneously, ora_i <0 for alli =1, . . . ,nsimultaneously (i.e., the components of vectorahave the same sign).

Or, equivalently,ahas the same sign if and only ifa≥0ora<0.

Theorem 4.5. Letx^∗be a feasible solution of problem(FLP1). Letc˜ =(c˜1, . . . ,c˜n)^T. We assume that each vector a_j = ∇gj(x)has the same sign for j = 1, . . . ,m. If there exist nonnegative fuzzy numbersµ˜j,λ˜k ∈ F(R)for

j =1, . . . ,m and k=1, . . . ,n such that the following conditions are satisfied:

(i) c˜⊕h Lm

j=1

µ˜j⊗ ˜1{a_j}

i

⊕h Ln

k=1

λ˜k⊗ ˜1{−e_k}

i

= ˜0;

(ii) µ˜j⊗ ˜1_{gj(x^∗)}= ˜0= ˜λk⊗ ˜1_{x^∗

k}for all j =1, . . . ,m and k=1, . . . ,n, thenx^∗is a nondominated type-I solution of problem(FLP1).

Proof. We assume that conditions (i) and (ii) are satisfied. LetI+ ⊆ {1, . . . ,m}andI− ⊆ {1, . . . ,m}be the index sets defined by

I+= {j :a_j ≥0} and I−= {j :a_j <0}. Since

1˜_{aj}=

1˜_{aj1},1˜_{aj2}, . . . ,1˜_{{aj n}}

T

,

theith component of the formula in condition (i) is given by c˜_i⊕

" _m M

j=1

µ˜j⊗ ˜1_{{aj i}}

#

⊕

" _n M

k=1

λ˜k⊗ ˜1{−δki}

#

= ˜0, (15)

whereδki =1 ifi =kandδki =0 ifi 6=k. Sinceµ˜j andλ˜kare nonnegative fuzzy numbers for all j =1, . . . ,m and allk=1, . . . ,n, we have that(µ˜j)_α^L = ˜µ^L_jα,(µ˜j)^U_α = ˜µ^U_jα,(λ˜j)^L_α = ˜λ^L_jα and(λ˜j)^U_α = ˜λ^U_jα are nonnegative real numbers byRemark 2.1for allα ∈ [0,1], all j =1, . . . ,mand allk =1, . . . ,n. Taking theα-level set of(15)by usingProposition 2.1, we have

c˜_i^L_α+X

j∈I+

µ˜^L_jα·a_{j i}+X

j∈I−

µ˜^U_jα·a_{j i}−

n

X

k=1

λ˜^U_kα·δki =0= ˜c^U_iα+X

j∈I+

µ˜^U_jα·a_{j i}+X

j∈I−

µ˜^L_jα·a_{j i}−

n

X

k=1

λ˜_k^L_α·δki

for allα∈ [0,1]and alli=1, . . . ,n. Equivalently, in vector form, we have c^L_α +X

j∈I+

µ˜^L_jα·a_j+ X

j∈I−

µ˜^U_jα·a_j−

n

X

k=1

λ˜^U_kα·e_k=0=c^U_α +X

j∈I+

µ˜^U_jα·a_j+ X

j∈I−

µ˜^L_jα·a_j−

n

X

k=1

λ˜_k^L_α·e_k

for allα∈ [0,1], which also implies, by adding them together, c^L_α +c^U_α +

m

X

j=1

µjα·a_j−

n

X

k=1

λkα·e_k =0 (16)

for allα∈ [0,1], whereµjα = ˜µ^L_jα + ˜µ^U_jα andλkα = ˜λ_k^L_α+ ˜λ^U_kα are nonnegative real numbers for allα ∈ [0,1], all j =1, . . . ,mand allk=1, . . . ,n. We are going to prove this theorem by contradiction. Suppose thatx^∗is not a nondominated type-I solution. Then there exists a feasible solutionx¯such that f˜(x¯)≺_I f˜(x^∗), i.e.,(3)is satisfied for someα^∗ ∈ [0,1]. We now define a real-valued function f as in(6)or in(7). Sinceg_j(x^∗)≤0 for all j =1, . . . ,m andx_k^∗≥0 for allk=1, . . . ,n, taking theα-level set of condition (ii) by usingProposition 2.1, we obtain that

µ˜^U_jα·g_j(x^∗)=

µ˜j ⊗ ˜1_{gj(x^∗₎}

L

α =0=

µ˜j⊗ ˜1_{gj(x^∗₎}

U

α = ˜µ^L_jα·g_j(x^∗) for allα∈ [0,1]and all j =1, . . . ,mand

λ˜_k^L_α·x_k^∗=

λ˜k⊗ ˜1_{x^∗

k}

L α =0=

λ˜k⊗ ˜1_{x^∗

k}

U

α = ˜λ^U_jα·x_k^∗

for allα∈ [0,1]and allk=1, . . . ,n, which imply, by adding them together,

0= ˜µ^L_jα·g_j(x^∗)+ ˜µ^U_jα·g_j(x^∗)=µjα·g_j(x^∗) (17) for allα∈ [0,1]and all j =1, . . . ,mand

0= ˜λ^Lkα ·x_k^∗+ ˜λ^U_kα·x_k^∗=λkα·x^∗_k (18)

for allα∈ [0,1]and allk=1, . . . ,n. Now we consider the constrained optimization problem (P1) as in the proof of Theorem 4.2. According to Eqs.(16)–(18)and(9), we obtain the following two new conditions:

(i⁰) ∇f(x^∗)+Pm

j=1µjα^∗· ∇g_j(x^∗)+Pn

k=1λkα^∗· ∇g_m+k(x^∗)=0(note that(16)is satisfied for allα∈ [0,1]);

(ii⁰) µjα^∗ ·g_j(x^∗) = 0 = λkα^∗·g_m+k(x^∗)for all j = 1, . . . ,m and allk =1, . . . ,n (note that(17)and(18)are satisfied for allα∈ [0,1]).

UsingTheorem 4.1, we see thatx^∗is an optimal solution of problem (P1) by regarding conditions (i’) and (ii’) as the KKT conditions, i.e., f(x^∗)≤ f(x¯), which contradicts(8). This completes the proof.

4.2. Fuzzy constraints

Now we are going to derive the optimality conditions for problem (FLP2) with fuzzy constraints. We recall that the fuzzy constraints are presented as

g˜j(x)=(a˜j1⊗ ˜1x₁)⊕ · · · ⊕(a˜j n⊗ ˜1x_n)⊕ ˜bj

for j =1, . . . ,m. UsingProposition 2.1, we obtain

g˜^L_jα(x)≡(g˜_j)^L_α(x)= ˜a^L_j1α·x₁+ · · · + ˜a^L_{j nα}·x_n+ ˜b^L_jα g˜^U_jα(x)≡(g˜j)^U_α(x)= ˜a^U_j1α·x1+ · · · + ˜a^U_{j nα}·xn+ ˜b^U_jα forα∈ [0,1], where

b˜^L_jα =(b˜_j)_α^L,b˜^U_jα =(b˜_j)^U_α,a˜^L_{j iα} =(a˜_{j i})^L_α and a˜^U_{j iα}=(a˜_{j i})^U_α. We also write

a^L_jα =





 a˜^L_j1α

...

a˜^L_{j nα}





 and a^U_jα=





 a˜^U_j1α

...

a˜^U_{j nα}





.

Thena^L_jα = ∇ ˜g^L_jα(x)anda^U_jα = ∇ ˜g^U_jα(x). Now we are in a position to derive the optimality conditions of problem (FLP2).

Theorem 4.6. Letx^∗be a feasible solution of problem(FLP2).

(A)If there exist positive real-valued functionsλ^L and λ^U defined on [0,1], and nonnegative real-valued functions µj andλk for j=1, . . . ,m and k=1, . . . ,n defined on [0,1] such that the following conditions are satisfied:

(i) λ^L(α)·c_α^L+λ^U(α)·c^U_α +Pm

j=1µj(α)·a^L_jα−Pn

k=1λk(α)·ek =0for allα∈ [0,1]; (ii) µj(α)· ˜g^L_jα(x^∗)=0=λk(α)·x^∗_k for allα∈ [0,1], all j=1, . . . ,m and all k=1, . . . ,n, thenx^∗is a nondominated type-I solution of problem(FLP2).

(B)If there exist positive real-valued functionsλ^Landλ^Udefined on [0,1], and nonnegative real-valued functionsµj

andλk for j=1, . . . ,m and k=1, . . . ,n defined on [0,1] such that the following conditions are satisfied:

(iii) λ^L(α)·c^L_α+λ^U(α)·c^U_α +Pm

j=1µj(α)·a^U_jα−Pn

k=1λk(α)·e_k=0for allα∈ [0,1]; (iv) µj(α)· ˜g^U_jα(x^∗)=0=λk(α)·x_k^∗for allα∈ [0,1], all j=1, . . . ,m and all k=1, . . . ,n, thenx^∗is a nondominated type-I solution of problem(FLP2).

Proof. (A) We are going to prove this result by contradiction. Suppose that conditions (i) and (ii) are satisfied and x^∗is not a nondominated type-I solution. Then there exists a feasible solutionx¯such that f˜(x¯)≺_I f˜(x^∗), i.e.,(3)is satisfied for someα^∗ ∈ [0,1]. We now define a real-valued function f as in(6)or in(7)and consider the following constrained optimization problem:

(P2) min f(x)

subject to g˜^L_jα^∗(x)≤0,j =1, . . . ,m gk(x)= −xk≤0,k=1, . . . ,n.

We see that g˜_j(x^∗) ˜0 implies g˜^L_jα(x^∗) ≤ 0 and g˜^U_jα(x^∗) ≤ 0 for all α ∈ [0,1]. This shows that if x^∗ is a feasible solution of problem (FLP2), then x^∗ is also a feasible solution of problem (P2). From (9) and letting µjα^∗ = µj(α^∗) ≥ 0 for j = 1, . . . ,m andλkα^∗ = λk(α^∗)for k = 1, . . . ,n, we obtain the following two new conditions from conditions (i) and (ii):

(i⁰) ∇f(x^∗)+Pm

j=1µjα^∗· ∇ ˜g^L_jα∗(x^∗)+Pn

k=1λkα^∗· ∇g_k(x^∗)=λ^L(α^∗)·c_α^L∗+λ^U(α^∗)·c^U_α∗+Pm

j=1µj(α^∗)·a^L_jα∗− Pn

k=1λkα^∗e_k =0;

(ii⁰) µjα^∗· ˜g^L_jα∗(x^∗)=0=λkα^∗·g_k(x^∗)for all j =1, . . . ,mand allk=1, . . . ,n.

UsingTheorem 4.1, we see thatx^∗is an optimal solution of problem (P2) by regarding the above conditions (i’) and (ii’) as the KKT conditions, i.e., f(x^∗)≤ f(x¯), which contradicts(8).

(B) We now consider the following constrained optimization problem:

(P2⁰) min f(x)

subject to g˜^U_jα∗(x)≤0,j =1, . . . ,m g_k(x)= −x_k≤0,k=1, . . . ,n.

Then we see that ifx^∗is a feasible solution of problem (FLP2), thenx^∗is also a feasible solution of problem (P2’).

The above similar arguments can also be used. This completes the proof.

Theorem 4.7. Letx^∗be a feasible solution of problem(FLP2).

(A)If there exist positive real numbersλ^L and λ^U, nonnegative real numbers µj and λk for j = 1, . . . ,m and k=1, . . . ,n, andα^∗∈ [0,1]such that the following conditions are satisfied:

(i) λ^L ·c^L_α∗+λ^U ·c^U_α∗+Pm

j=1µj·a^L_jα∗−Pn

k=1λk·e_k=0;

(ii) µj· ˜g^L_jα∗(x^∗)=0=λk·x_k^∗for all j =1, . . . ,m and all k=1, . . . ,n, thenx^∗is a nondominated type-II solution of problem(FLP2).

(B)If there exist positive real numbersλ^L and λ^U, nonnegative real numbers µj and λk for j = 1, . . . ,m and k=1, . . . ,n, andα^∗∈ [0,1]such that the following conditions are satisfied:

(iii) λ^L·c_α^L∗+λ^U ·c^U_α∗+Pm

j=1µj·a^U_jα∗−Pn

k=1λk·e_k =0;

(iv) µj· ˜g^U_jα∗(x^∗)=0=λk·x_k^∗for all j=1, . . . ,m and all k=1, . . . ,n, thenx^∗is a nondominated type-II solution of problem(FLP2).

Proof. (A) We are going to prove this result by contradiction. Suppose that conditions (i) and (ii) are satisfied and x^∗is not a nondominated type-II solution. Then there exists a feasible solutionx¯such that f˜(x¯)≺_II f˜(x^∗), i.e.,(3)is satisfied for allα ∈ [0,1]. Forα^∗in conditions (i) and (ii), we now define a real-valued function f as in(10)or in (11). Then we have∇f(x)=λ^L·c_α^L∗+λ^U ·c^U_α∗from(12). Now we consider the constrained optimization problem (P2) as in the proof ofTheorem 4.6. Then we can obtain the following two new conditions:

(i⁰) ∇f(x^∗)+Pm

j=1µj· ∇ ˜g^L_jα∗(x^∗)+Pn

k=1λk· ∇g_k(x^∗)=λ^L·c^L_α∗+λ^U·c^U_α∗+Pm

j=1µj·a^L_jα∗−Pn

k=1λk·e_k=0;

(ii⁰) µj· ˜g^L_jα∗(x^∗)=0=λk·gk(x^∗)for all j =1, . . . ,mand allk=1, . . . ,n.

The remaining proof follows from the similar arguments ofTheorem 4.6.

(B) The above similar arguments can be used by considering problem(P2⁰).

Theorem 4.8. Suppose that the fuzzy coefficientsc˜_i in the fuzzy-valued objective function f and˜ a˜_{j i} in the fuzzy- valued constraint functionsg˜_j for i =1, . . . ,n and j =1, . . . ,m are now assumed to be canonical fuzzy numbers.

Letx^∗be a feasible solution of problem(FLP2).

(A)If there exist positive real numbersλ^L andλ^U, and nonnegative real numbersµj andλkfor j =1, . . . ,m and k=1, . . . ,n such that the following conditions are satisfied:

(i) λ^L ·R1

0 c_α^Ldα+λ^U·R1

0 c^U_αdα+Pm

j=1µj·R1

0 a^L_jαdα−Pn

k=1λk·e_k=0;

(ii) µj·R1

0 g˜^L_jα(x^∗)dα=0=λk·x_k^∗for all j =1, . . . ,m and all k=1, . . . ,n, thenx^∗is a nondominated type-II solution of problem(FLP2).