A New Approach for Solving Fully Fuzzy Bilevel Linear Programming Problems

Int. J. IndustrialMathematics (ISSN 2008-5621)

Vol. 12, No. 1, 2020 Article ID IJIM-1318, 11 pages Research Article

A New Approach for Solving Fully Fuzzy Bilevel Linear Programming

Problems

S. F. Tayebnasab∗, F. Hamidi †‡, M. Allahdadi §

Received Date: 2018-10-04 Revised Date: 2019-07-04 Accepted Date: 2019-08-28

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Abstract

This paper addresses a type of fully fuzzy bilevel linear programming (FFBLP) wherein all the co-efficients and decision variables in both the objective functions and constraints are triangular fuzzy numbers. In order to obtain the fuzzy optimal solution, a new efficient method for FFBLP with unconstrained variables and parameters has been proposed, . This proposed approach is based on crisp bilevel programming. Some examples have been provided to illustrate these methods.

Keywords: Bilevel linear programming; Triangular fuzzy numbers; Ranking function; Optimal solu-tion; Unconstrain variables.

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1

Introduction

T

he bilevel programming is a hierarchical

optimization problem involving two levels wherein the upper level constraint region is im-plicitly specified by the lower level problem. This kind of problem is nonconvex and very hard to solve due to its structure. Over the past few decades, the bilevel programming problem has possessed a lot of consideration and has been widely applied to a variety of fields such as electricity markets [22], energy networks [8] , location-allocation problem [19]. The recent surveys on this topic were given by Dempe [5],

∗_{Department of Mathematics, University of Sistan and}

Baluchestan, Zahedan, Iran.

†_{Corresponding author. f [email protected], Tel:}

+98(915)1415886.

‡_{Department of Mathematics, University of Sistan and}

Baluchestan, Zahedan, Iran.

§_{Department of Mathematics, University of Sistan and}

Baluchestan, Zahedan, Iran.

price setting problem [11], principal-agent prob-lems [3, 20], and so forth. Detailed study on bilevel programming problem can be found in the most recent books (Dempe [4]; Dempe et al. [6]). Considering its structure, this type of problem is nonconvex and very difficult to solve. Consid-ering its structure, this type of problem is non-convex and very difficult to solve. In common bilevel programming problems, it is assumed that the parameters are precisely defined, but under real conditions, the decision making coefficients and variables may not be crisp. The fuzzy sets theory is a powerful tool for dealing with inac-curate or vague information and in recent years the fuzzy bilevel programming problem wherein the coefficients are fuzzy numbers in both the ob-jective function and constraints has been studied by some authors. Sakawa et al. [18] formulated the fuzzy bilevel linear programming problem and developed the fuzzy programming method for its solution. . Zhang et al. [23] used the fuzzy

Tucker, fuzzy Kth-best, and fuzzy branch and bound methods to solve fuzzy bilevel linear pro-gramming problems; however, the decision vari-ables are not fuzzy in any of the mentioned works. Ruziyeva [16] proposed a membership function approach for solving the linear fuzzy bilevel op-timization problem. Allahviranloo et al. [1] pro-posed a new method for solving fully fuzzy lin-ear programming problems by the use of rank-ing function. Hosseinzadeh Lotfi et al. [7] trans-formed the fully fuzzy linear programming prob-lem into two corresponding linear programming problems based on the concept of the symmet-ric triangular fuzzy number and developed a lex-icography method to solve such a problem. Ku-mar et al. [10] proposed a new method to find the optimal fuzzy solution of the fully fuzzy lin-ear programming problem with equal constraint. This paper aims at developing a new method for the solution of the fully fuzzy linear programming problems wherein all the coefficients and decision variables are expressed as triangular fuzzy num-bers. Safaei and Saraj [17] decomposed the FF-BLP problem into three deterministic linear pro-gramming problems and then obtained its fuzzy optimal solution. Ren [13] investigated the feasi-bility of fuzzy solutions of the generalized fuzzy bilevel linear programming problem and devel-oped a stepwise interactive algorithm to solve the problem. Ren [14] a new approach based on de-viation degree measures and a ranking function method is proposed. Ren et al. [15] converted the FFBLP problem to crisp form under different constraint feasibility degrees based on a fuzzy re-lation for the ranking of the fuzzy numbers. This paper has been so organized as to present the def-initions and operators between triangular fuzzy numbers in Section2, express the new approaches in the FFBLP using the ranking function in Sec-tion3, provide some numerical examples to illus-trate how methods are implemented in Section4, and, finally, present the conclusions in Section5.

2

Prerequisites

This section presents the definitions needed to ob-tain the results [10,12,9,2].

Definition 2.1 If X is a universal set, the fuzzy subsetAeofX can be expressed by its membership

function as follows:

µ_Ae:X→[0,1]

where gives a real numberµ_Ae(x)to every element

x∈Xin the interval[0,1]and the value ofµ_Ae(x)

at x shows the membership grade of x in Ae. A fuzzy subset Ae can be characterized as a set of ordered pairs of elementxand gradeµ_Ae(x)shown often as follows:

e

A={(x, µ_Ae(x));x∈X}.

Definition 2.2 A fuzzy numberAe= (a, b, c)will be triangular if its membership function is as fol-lows:

µ_Ae(x) =

    

(x−a)

(b−a), a≤x≤b (c−x)

(c−b), b≤x≤c

0 O.W.

Definition 2.3 A triangular fuzzy number Ae= (a, b, c) will be non-negative only if a ≥ 0; the related non-negative set can be shown by F(R+).

Definition 2.4 A triangular fuzzy number Ae= (a, b, c) will be unrestricted if a, b, c∈R. The set of which is shown by F(R).

Definition 2.5 if Ae= (a, b, c) and Be = (d, e, f)

are two triangular fuzzy numbers, then:

(1)Ae⊕Be = (a, b, c)⊕(d, e, f) = (a+d, b+e, c+ f),

(2) ⊖Ae=⊖(a, b, c) = (−c,−b,−a),

(3)Ae⊖Be= (a, b, c)⊖(d, e, f) = (a−f, b−e, c−d),

(4)

λAe=   

(λa, λb, λc), λ≥0 (λc, λb, λa), λ <0.

(5) Ae⊗ Be = (min(γ), be, max(γ)) where,γ =

{ad, af, cd, cf}

(6) Be = (d, e, f) be a non-negative triangular fuzzy number then:

e

A⊗Be=       

Definition 2.6 Two triangular fuzzy numbers e

A = (a, b, c) and Be = (d, e, f) are equal only if

a=d, b=e andc=f.

Definition 2.7 [10]. A ranking function is in the form F(R)→R; it maps every fuzzy number on the real-line where there is natural order. If

e

A = (a, b, c) is a triangular fuzzy number, then

R(Ae) = 1₄(a+ 2b+c).

Definition 2.8 If Ae = (a, b, c) is non-negative triangular fuzzy number and Be = (e, f, g) is an unrestricted triangular fuzzy number, then:

e

A ⊗ Be = (min(ae, ce), bf, max(ag, cg))

where min(ae, ce) = ae+₂ce − |ae−₂ce| and

max(ag, cg) = ag+₂cg +|ag−₂cg|. since (a, b, c) is a triangular fuzzy number (i.e. a ≤ b ≤ c), so

min(ae, ce) andmax(ag, cg) can be written as:

min(ae, ce) = (a+₂c)e − (c−₂a)|e| and

max(ag, cg) = (a+₂c)g+ (c−₂a)|g|.

Definition 2.9 If Ae= (a, b, c) and Be = (x, y, z)

are two triangular fuzzy numbers, then :

e

A⊗Be=                       

(min(ax, cx), by, max(az, cz)), a≥0

(min(cx, az), by, max(cz, ax)), a <0, c≥o

(min(az, cz), by, max(ax, cx)), c <0.

or

e

A⊗Be=                                         

({0.5(a+c)x−0.5(c−a)|x|} , by,{0.5(a+c)z

,+0.5(c−a)|z|}), a≥0 ({0.5(az+cx)−0.5|cx−az|} , by,{0.5(ax+cz)

,+0.5|cz−ax|}), a <0, c≥o ({0.5(a+c)z−0.5(c−a)|z|} , by,{0.5(a+c)x

+0.5(c−a)|x|}), c <0.

3

Proposed algorithms

Next, some new algorithms are proposed to find the fuzzy optimal solution for the FF-BLP problem in three cases, unconstrained fuzzy coefficients corresponding to the variables and non-negative fuzzy variables, non-negative fuzzy coefficients corresponding to the variables and

unconstrained fuzzy variables and finally, un-constrained fuzzy coefficients corresponding to the variables and unconstrained fuzzy variables. Consider the following FFBLP problem and sup-pose the decision maker at the ith level each of which controls its own decision vector, ˜xi:

maximize

˜ x1

˜

F(˜x1,x˜2) = ˜c11⊗x˜1⊕c˜12⊗x˜2

s.t. (3.1)

maximize

˜ x2

˜

f(˜x1,x˜2) = ˜c21⊗x˜1⊕c˜22⊗x˜2

s.t. A˜⊗x˜1⊕B˜⊗x˜2 = ˜b

where ex1 , ex1 are triangular fuzzy numbers,

e

A = (eaij)m×n1 , Be = (ebij)m×n2 , eb = (ebi)m×1

and ec11 = (ec11j)1×n1 , ec12 = (ec12j)1×n2 ,

e

c21 = (ec21j)1×n1 , ec22 = (ec22j)1×n2 ,n1+n2 =n

, eaij , ebij, ebi ∈ F(R) , xe1 , xe1 ∈ F(R+) for all

1≤i≤m, 1≤j ≤n.

Let the parameters ec11 , ec12, ec21 , ec22 ,

e

aij , ebij, xe1 , ex2 and ebi be the triangular

fuzzy number (p1j, q1j, r1j), (p2j, q2j, r2j),

(p3j, q3j, r3j), (p4j, q4j, r4j), (a1ij, a2ij, a3ij),

(b1_ij, b2_ij, b3_ij), (x1j, y1j, t1j), (x2j, y2j, t2j) and

(mi, ni, pi) respectively. So the problem (3.1)

can, then, be written as follows:

maximize

(x1j,y1j,t1j)

˜

F(˜x1,x˜2) =

∑n1

j=1(p1j, q1j, r1j) ⊗

(x1j, y1j, t1j)⊕ ∑n2

j=1(p2j, q2j, r2j)⊗(x2j, y2j, t2j)

s.t. (3.2)

maximize

(x2j,y2j,t2j)

˜

f(˜x1,x˜2) =

∑n1

j=1(p3j, q3j, r3j) ⊗

(x1j, y1j, t1j) ⊕ ∑n2

j=1 = (p4j, q4j, r4j) ⊗

(x2j, y2j, t2j) s.t.

∑n1

j=1(a1ij, a2ij, a3ij) ⊗

(x1j, y1j, t1j)⊕ ∑n2

j=1(b1ij, b2ij, b3ij)⊗(x2j, y2j, t2j)

= (mi, ni, pi),∀i= 1,2, ..., m.

where (x1j, y1j, t1j),(x2j, y2j, t2j) are

trian-gular fuzzy numbers.

Now, using Definition 2.5 , we will rewrite this problem (3.2) as follows:

maximize

(x1j,y1j,t1j)

˜

F(˜x1,x˜2) =

∑n1

j=1 p_1j≥0

(p1j, q1j, r1j) ⊗

(x1j, y1j, t1j)⊕ ∑n1

j=1 p_1j<0 r_1j≥0

(p1j, q1j, r1j)⊗(x1j, y1j, t1j)

⊕ ∑n1

j=1 r_1j<0

(p1j, q1j, r1j) ⊗ (x1j, y1j, t1j) ⊕ ∑n2

j=1 p_2j≥0

⊕∑n2

j=1 p_2j<0 r2j≥0

(p2, q2j, r2j) ⊗ (x2j, y2j, t2j) ⊕

∑n2

j=1 r2j<0

(p2j, q2j, r2j)⊗(x2j, y2j, t2j)

s.t. (3.3)

maximize

(x2j,y2j,t2j)

˜

f(˜x1,x˜2) =

∑n1

j=1 p_3j≥0

(p3j, q3j, r3j) ⊗

(x1j, y1j, t1j)⊕ ∑n1

j=1 p_3j<0 r_3j≥0

(p3j, q3j, r3j)⊗(x1j, y1j, t1j)

⊕∑n1

j=1 r_3j<0

(p3j, q3j, r3j) ⊗ (x1j, y1j, t1j) ⊕ ∑n2

j=1 p_4j≥0

(p4j, q4j, r4j)⊗(x2j, y2j, t2j)

⊕∑n2

j=1 p_4j<0 r_4j≥0

(p4, q4j, r4j) ⊗ (x2j, y2j, t2j) ⊕

∑n2

j=1 r_4j<0

(p4j, q4j, r4j) ⊗ (x2j, y2j, t2j) s.t. ∑n1

j=1 a1

ij≥0

(a1_ij, a2_ij, a3_ij) ⊗ (x1j, y1j, t1j) ⊕ ∑n1

j=1 a1

ij<0

a3_ij≥0

(a1_ij, a2_ij, a3_ij)⊗(x1j, y1j, t1j)

⊕∑n1

j=1 a3

ij<0

(a1_ij, a2_ij, a3_ij) ⊗ (x1j, y1j, t1j) ⊕ ∑n2

j=1 b1

ij≥0

(b1_ij, b2_ij, b3_ij)⊗(x2j, y2j, t2j)

⊕ ∑n2

j=1 b1

ij<0

b3_ij≥0

(b1_ij, b2_ij, b3_ij) ⊗ (x2j, y2j, t2j) ⊕

∑n2

j=1 b1

ij<0

(b1_ij, b2_ij, b3_ij)⊗(x2j, y2j, t2j)

= (mi, ni, pi)∀, i= 1,2, ..., m,

where (x1j, y1j, t1j),(x2j, y2j, t2j) are triangular

fuzzy numbers.

First, we describe an algorithm for solving a FFBLP problem with unrestricted fuzzy coeffi-cients corresponding to the variables and non-negative fuzzy variables.

3.1 Algorithm 1

Step 1. Using Definition 2.5, the FFBLP problem (3.3) can be written as follows:

maximize

(x1j,y1j,t1j)

˜

F(˜x1,x˜2) =

∑n1

j=1 p_1j≥0

(p1jx1j, q1jy1j, r1jt1j) ⊕ ∑n1

j=1 p1j<0

r_1j≥0

(p1jt1j, q1jy1j, r1jt1j)

⊕∑n1

j=1 r_1j<0

(p1jt1j, q1jy1j, r1jx1j) ⊕ ∑n2

j=1 p_2j≥0

(p2jx2j, q2jy2j, r2jt2j)

⊕∑n2

j=1 p_2j<0 r_2j≥0

(p2t2j, q2jy2j, r2jt2j) ⊕

∑n2

j=1 r_2j<0

(p2jt2j, q2jy2j, r2jx2j)

s.t.

maximize

(x2j,y2j,t2j)

˜

f(˜x1,x˜2) =

∑n1

j=1 p_3j≥0

(p3jx1j, q3jy1j, r3jt1j) ⊕ ∑n1

j=1 p_3j<0 r_3j≥0

(p3jt1j, q3jy1j, r3jt1j)

⊕∑n1

j=1 r_3j<

(p3jt1j, q3jy1j, r3jx1j) ⊕ ∑n2

j=1 p_4j≥0

(p4jx2j, q4jy2j, r4jt2j)

⊕∑n2

j=1 p_4j<0 r4j≥0

(p4t2j, q4jy2j, r4jt2j) ⊕

∑n2

j=1 r4j<0

(p4jt2j, q4jy2j, r4jx2j)

s.t. ∑n1

j=1 a1_ij≥0

(a1_ijx1j, a2ijy1j, a3ijt1j) ⊕ ∑n1

j=1 a1_ij<0 a3

ij≥0

(a1_ijt1j, a2ijy1j, a3ijt1j)

⊕∑n1

j=1 a3

ij<0

(a1_ijt1j, a2ijy1j, a3ijx1j) ⊕

∑n2

j=1 b1

ij≥0

(b1_ijx2j, b2ijy2j, b3ijt2j)

⊕∑n2

j=1 b1

ij<0

b3_ij≥0

(b1_ijt2j, b2ijy2j, b3ijt2j) ⊕

∑n2

j=1 b1

ij<0

(b1_ijt2j, b2ijy2j, b3ijx2j)

= (mi, ni, pi),∀i= 1,2, ..., m.

where (x1j, y1j, t1j),(x2j, y2j, t2j) are

non-negative triangular fuzzy numbers.

Step 2. Using Definition 2.7, the fuzzy objective function at each level of the FFBLP problem found in Step 1, can be transformed into the crisp objective function at each level and the FFBLP problem can be written as follows:

maximize

x1j,y1j,t1j

R[∑n1

j=1 p_1j≥0

(p1jx1j, q1jy1j, r1jt1j)] ⊕

R[∑n1

j=1 p_1j<0 r_1j≥0

(p1jt1j, q1jy1j, r1jt1j)]

⊕ R[∑n1

j=1 r_1j<0

(p1jt1j, q1jy1j, r1jx1j)] ⊕

R[∑n2

j=1 p_2j≥0

(p2jx2j, q2jy2j, r2jt2j)]

⊕R[∑n2

j=1 p_2j<0 r_2j≥0

(p2t2j, q2jy2j, r2jt2j)] ⊕

R[∑n2

j=1 r_2j<0

(p2jt2j, q2jy2j, r2jx2j)]

s.t.

maximize

x2j,y2j,t2j

f =R[∑n1

j=1 p_3j≥0

(p3jx1j, q3jy1j, r3jt1j)]⊕

R[∑n1

j=1 p3j<0

r_3j≥0

(p3jt1j, q3jy1j, r3jt1j)]

⊕R[∑n1

j=1 r_3j<

(p3jt1j, q3jy1j, r3jx1j)] ⊕

R[∑n2

j=1 p_4j≥0

(p4jx2j, q4jy2j, r4jt2j)]

⊕R[∑n2

j=1 p_4j<0 r_4j≥0

(p4t2j, q4jy2j, r4jt2j)] ⊕

R[∑n2

j=1 r_4j<0

(p4jt2j, q4jy2j, r4jx2j)]

s.t. ∑n1

j=1 a1

ij≥0

(a1_ijx1j, a2ijy1j, a3ijt1j) ⊕

∑n1

j=1 a1

ij<0

a3 ij≥0

(a1_ijt1j, a2ijy1j, a3ijt1j)

⊕∑n1

j=1 a3

ij<0

(a1_ijt1j, a2ijy1j, a3ijx1j) ⊕

∑n2

j=1 b1

ij≥0

(b1_ijx2j, b2ijy2j, b3ijt2j)

⊕∑n2

j=1 b1

ij<0

b3 ij≥0

(b1_ijt2j, b2ijy2j, b3ijt2j) ⊕

∑n2

j=1 b1

ij<0

(b1_ijt2j, b2ijy2j, b3ijx2j)

= (mi, ni, pi)∀i= 1,2, ..., m,

where (x1j, y1j, t1j),(x2j, y2j, t2j) are

non-negative triangular fuzzy numbers.

Step 3. Using Definitions2.6,2.7, the FFBLP problem, obtained in Step 2, is transformed into the following crisp bilevel linear programming problem:

maximize

x1j,y1j,t1j

F = 1₄[∑n1

j=1 p_1j≥0

(p1jx1j, q1jy1j, r1jt1j)]⊕

1 4[

∑n1

j=1 p_1j<0 r_1j≥0

(p1jt1j, q1jy1j, r1jt1j)]

⊕ 1

4[

∑n1

j=1 r1j<0

(p1jt1j, q1jy1j, r1jx1j)] ⊕

1 4[

∑n2

j=1 p_2j≥0

(p2jx2j, q2jy2j, r2jt2j)]

⊕1 4[

∑n2

j=1 p_2j<0 r_2j≥0

(p2t2j, q2jy2j, r2jt2j)] ⊕

1 4[

∑n2

j=1 r_2j<0

(p2jt2j, q2jy2j, r2jx2j)]

s.t.

maximize

x2j,y2j,t2j

f = 1₄[∑n1

j=1 p_3j≥0

(p3jx1j, q3jy1j, r3jt1j)]⊕

1 4[

∑n1

j=1 p_3j<0 r3j≥0

(p3jt1j, q3jy1j, r3jt1j)]

⊕1 4[

∑n1

j=1 r_3j<

(p3jt1j, q3jy1j, r3jx1j)] ⊕

1 4[

∑n2

j=1 p_4j≥0

(p4jx2j, q4jy2j, r4jt2j)]

⊕1 4[

∑n2

j=1 p_4j<0 r_4j≥0

(p4t2j, q4jy2j, r4jt2j)] ⊕

1 4[

∑n2

j=1 r4j<0

(p4jt2j, q4jy2j, r4jx2j)]

s.t. ∑n1

j=1 a1

ij≥0

(a1_ijx1j) + ∑n1

j=1 a1

ij<0

a3_ij≥0

(a1_ijt1j) +

∑n1

j=1 a3

ij<0

(a1_ijt1j) + ∑n2

j=1 b1

ij≥0

(b1_ijx2j)

+ ∑n2

j=1 b1

ij<0

b3_ij≥0

(b1_ijt2j) + ∑n2

j=1 b1

ij<0

(b1_ijt2j) = mi,∀i =

1,2, ..., m.

∑n1

j=1 a1

ij≥0

(a2_ijy1j)+ ∑n1

j=1 a1

ij<0

a3 ij≥0

(a2_ijy1j)+ ∑n1

j=1 a3

ij<0

(a2_ijy1j)+

∑n2

j=1 b1

ij≥0

(b2_ijy2j)

+ ∑n2

j=1 b1

ij<0

b3 ij≥0

(b2_ijy2j) + ∑n2

j=1 b1

ij<0

(b2_ijy2j) = ni,∀i =

1,2, ..., m.

∑n1

j=1 a1

ij≥0

(a3_ijt1j) + ∑n1

j=1 a1

ij<0

a3 ij≥0

(a3_ijt1j) + ∑n1

j=1 a3

ij<0

(a3_ijx1j) +

∑n2

j=1 b1

ij≥0

+ ∑n2

j=1 b1_ij<0 b3

ij≥0

(b3_ijt2j) + ∑n2

j=1 b1_ij<0

(b3_ijx2j) = pi,∀i =

1,2, ..., m. y1j−x1j ≥0, t1j−y1j ≥0, y2j−x2j ≥

0, t2j−y2j ≥0, x1j ≥0, x1j ≥0,∀j= 1, ..., n.

Step 4. Find the optimal solution x1j, y1j, t1j, x2j, y2j, t2j, by solving the bilevel

linear prpgramming problem obtained in Step 3.

Step 5. Find the fuzzy optimal solution by putting the values of x1j, y1j, t1j, x2j, y2j, t2j

inxe1 = (x1j, y1j, t1j), ex2 = (x2j, y2j, t2j).

Step 6. Find the fuzzy optimal value by putting

e

x1, ex2 in: F˜(˜x1,x˜2) =

∑n1

j=1(p1j, q1j, r1j) ⊗

(x1j, y1j, t1j)⊕ ∑n2

j=1(p2j, q2j, r2j)⊗(x2j, y2j, t2j)

˜

f(˜x1,x˜2) =

∑n1

j=1(p3j, q3j, r3j)⊗(x1j, y1j, t1j)⊕

∑n2

j=1(p4j, q4j, r4j)⊗(x2j, y2j, t2j).

Now, we will present an algorithm for the FFBLP problem with non-negative fuzzy coefficients cor-responding to the variables and unrestricted fuzzy variables .

3.2 Algorithm 2

Step 1. Using Definition 2.8, the FFBLP problem (3) can be written as follows:

maximize

(x1j,y1j,t1j)

˜

F(˜x1,x˜2) =

∑n1

j=1(min(p1jx1j, r1jx1j), q1jy1j,

max(p1jt1j, r1jt1j))

⊕∑n2

j=1(min(p2jx2j, r2jx2j), q2jy2j,

max(p2jt2j, r2jt2j))

s.t.

maximize

(x2j,y2j,t2j)

˜

f(˜x1,x˜2) =

∑n1

j=1(min(p3jx1j, r3jx1j), q3jy1j,

max(p3jt1j, r3jt1j))

⊕∑n2

j=1(min(p4jx2j, r4jx2j), q4jy2j,

max(p4jt2j, r4jt2j))

s.t.∑n1

j=1(min(a1ijx1j, a3ijx1j), a2ijy1j,

max(a1_ijt1j, a3ijt1j))

⊕∑n2

j=1(min(b1ijx2j, b3ijx2j), b2ijy1j,

max(b1_ijt2j, b3ijt2j))

= (mi, ni, pi)∀i= 1,2, ..., m.

(x1j, y1j, t1j),(x2j, y2j, t2j)∈F(R), j = 1, ..., n.

Step 2. Using Definition 2.8, the FFBLP

problem obtained in Step 1 can be written as follows:

maximize

(x1j,y1j,t1j)

˜

F(˜x1,x˜2) =

∑n1

j=1( 1

2(p1j + r1j)x1j −

1

2(r1j−p1j)|x1j|, q1jy1j, 1

2(p1j +r1j)t1j

+ 1₂(r1j − p1j)|t1j|) ⊕ ∑n2

j=1(12(p2j + r2j)x2j − 1

2(r2j−p2j)|x2j|, q2jy2j, 1

2(p2j+r2j)t2j+ 1

2(r2j−p2j)|t2j|)

s.t. maximize

(x2j,y2j,t2j)

˜

f(˜x1,x˜2) =

∑n1

j=1( 1

2(p3j +r3j)x1j −

1

2(r3j−p3j)|x1j|, q3jy1j, 1

2(p3j +r3j)t1j

+1₂(r3j − p3j)|t1j|) ⊕ ∑n2

j=1( 1

2(p4j + r4j)x2j − 1

2(r4j−p4j)|x2j|, q4jy2j, 1

2(p4j + r4j)t2j + 1

2(r4j − p4j)|t2j|)

s.t. ∑n1

j=1(12(a 1

ij + a3ij)x1j − 1₂(a3ij −

a1_ij)|x1j|, a2ijy1j,1₂(a1ij +a3ij)t1j +1₂(aij3 −a1ij)|t1j|)

⊕∑n2

j=1(12(b1ij + bij3)x2j − 1₂(b3ij −

b1_ij)|x2j|, b2ijy2j,1₂(b1ij+b3ij)t2j+1₂(bij3 −b1ij)|t2j|)

= (mi, ni, pi)∀i= 1,2, ..., m.

(x1j, y1j, t1j),(x2j, y2j, t2j)∈F(R), j = 1, ..., n.

Step 3. Using Definitions 2.6, 2.7, the FF-BLP problem, obtained in Step 2, is transformed into the following crisp bilevel nonlinear pro-gramming problem:

maximize

x1j,y1j,t1j

F = 1₄[∑n1

j=1(12(p1j + r1j)x1j − 1

2(r1j−p1j)|x1j|+2(q1jy1j) + 1

2(p1j +r1j)t1j

+ 1₂(r1j −p1j)|t1j|)] + 1₄[ ∑n2

j=1(12(p2j +r2j)x2j − 1

2(r2j−p2j)|x2j|

+ 2(q2jy2j) +1₂(p2j+r2j)t2j+1₂(r2j−p2j)|t2j|)]

s.t.

maximize

x2j,y2j,t2j

f = 1₄[∑n1

j=1(12(p3j +r3j)x1j − 1 2(r3j−

p3j)|x1j|2(q3jy1j) +1₂(p3j+r3j)t1j

+ 1₂(r3j −p3j)|t1j|)] + 1₄[ ∑n2

j=1(12(p4j +r4j)x2j − 1

2(r4j−p4j)|x2j|

+ 2(q4jy2j) + 2(1₂(p4j+r4j)t2j) +1₂(r4j−p4j)|t2j|)]

s.t. ∑n1

j=1( 1 2(a

1

ij + a3ij)x1j − 1₂(a3ij −

a1_ij)|x1j|+ ∑n2

j=1( 1

2(b1ij+b3ij)x2j

−1

2(b3ij−b1ij)|x2j|=mi,∀i= 1, ..., m. ∑n1

j=1a2ijy1j+ ∑n2

j=1b2ijy2j =ni∀i= 1, ..., m. ∑n1

j=1(12(a 1

ij + t3ij)x1j + 1₂(a3ij −

a1_ij)|t1j|+ ∑n2

j=1(12(b 1

ij +b3ij)x2j

y1j − x1j ≥ 0, y2j − x2j ≥ 0, t1j − y1j ≥

0, t2j−y2j ≥0

y1j, x1j, t1j, y2j, x2j, t2j ∈ R,∀ : 1 ≤ i ≤ m,1 ≤

j≤n,∀j= 1, ..., n.

Step 4. Find the optimal solution x1j, y1j, t1j, x2j, y2j, t2j, by solving the bilevel

nonlinear programming problem obtained in Step 3.

Step 5. Find the fuzzy optimal solution by putting the values of x1j, y1j, t1j, x2j, y2j, t2j in e

x1 = (x1j, y1j, t1j), ex2 = (x2j, y2j, t2j).

Step 6. Find the fuzzy optimal value by putting xe1, xe2 in: F˜(˜x1,x˜2) =

∑n1

j=1(p1j, q1j, r1j) ⊗ (x1j, y1j, t1j) ⊕

∑n2

j=1(p2j, q2j, r2j)⊗(x2j, y2j, t2j)

˜

f(˜x1,x˜2) =

∑n1

j=1(p3j, q3j, r3j)⊗(x1j, y1j, t1j)⊕ ∑n2

j=1(p4j, q4j, r4j)⊗(x2j, y2j, t2j).

And finally, we will present an algorithm which is used for all FFBLP problems with uncon-strained fuzzy coefficients corresponding to the variables and unconstrained fuzzy variables.

3.3 Algorithm 3

Step 1. Using Definition 2.9, the FFBLP problem (3.1) can be written as follows:

maximize

(x1j,y1j,t1j)

˜

F(˜x1,x˜2) =

∑n1

j=1 p_1j≥0

(min(p1jx1j, r1jx1j)), q1jy1j

, max(p1jt1j, r1jt1j))

⊕∑n1

j=1 p_1j<0 r_1j≥0

(min(r1jx1j, p1jt1j), q1jy1j,

max(r1jt1j, p1jx1j))

⊕∑n1

j=1 r_1j<0

(minp1jt1j, r1jt1j), q1jy1j,

max(p1jx1j, r1jx1j))

⊕∑n2

j=1 p_2j≥0

(min(p2jx2j, r2jx2j), q2jy2j,

max(p2jt2j, r2jt2j))

⊕∑n2

j=1 p2j<0

r_2j≥0

(min(r2jx2j, p2jt2j), q2jy2j,

max(r2jt2j, p2jx2j))

⊕∑n2

j=1 r_2j<0

(min(p2jt2j, r2jt2j), q2jy2j,

max(p2jx2j, r2jx2j))

s.t.

maximize

(x2j,y2j,t2j)

˜

f(˜x1,x˜2) =

∑n1

j=1 p_3j≥0

(min(p3jx1j, r3jx1j)), q3jy1j,

max(p3jt1j, r3jt1j))

⊕∑n1

j=1 p_3j<0 r_3j≥0

(min(r3jx1j, p3jt1j)), q3jy1j,

max(r3jt1j, p3jx1j))

⊕∑n1

j=1 r3j<0

(min(p3jt1j, r3jt1j)), q3jy1j,

max(p3jx1j, r3jx1j))

⊕∑n2

j=1 p_4j≥0

(min(p4jx2j, r4jx2j), q4jy2j,

max(p4jt2j, r4jt2j))

⊕∑n2

j=1 p_4j<0 r4j≥0

(min(r4jx2j, p4jt2j), q4jy2j,

max(r4jt2j, p4jx2j))

⊕∑n2

j=1 r_4j<0

(min(p4jt2j, r4jt2j), q4jy2j,

max(p4jx2j, r4jx2j)) s.t. ∑n1

j=1 a1_ij≥0

(min(a1_ijx1j, aij3x1j), a2ijy1j,

max(a1_ijt1j, a3ijt1j))

⊕∑n1

j=1 a1_ij<0 a3

ij≥0

(min(a3_ijx1j, aij1t1j), a2ijy1j,

max(a3_ijt1j, a1ijx1j))

⊕∑n1

j=1 a3_ij<0

(min(a1_ijt1j, aij3t1j), a2ijy1j,

max(a1_ijx1j, a3ijx1j))

⊕∑n2

j=1 b1_ij≥0

(min(b1_ijx2j, bij3x2j), b2ijy2j,

max(b1_ijt2j, b3ijt2j))

⊕∑n2

j=1 b1_ij<0 b3_ij≥0

((min(b3_ijx2j, bij1t2j), b2ijy2j,

max(b3_ijt2j, b1ijx2j))

⊕∑n2

j=1 b1_ij<0

(min(b1_ijt2j, bij3t2j), b2ijy2j,

max(b1_ijx2j, b3ijx2j))

= (mi, ni, pi),∀i =

1,2, ..., m,(x1j, y1j, t1j),(x2j, y2j, t2j) ∈

F(R), j = 1, ..., n.

Step 2. Using Definition 2.9, the problem in Step 1, can be written as follows:

maximize

(x1j,y1j,t1j)

˜

F(˜x1,x˜2) =

∑n1

j=1 p_1j≥0

(0.5(p1j+r1j)x1j−

Table 1: Results of solving three FFBLP problems by the proposed algorithms.

Example fuzzy optimal solution lower level fuzzy optimal value upper level fuzzy optimal value

4.1 ex1= (1,2,3) x2e = (4,5,6) fe(x1,e ex2) = (13,32,39) Fe(x1,e x2e ) = (9,27,75) 4.2 x1e = (−3,−2,7₅) x2e = (1,2,121₂₅) fe(ex1,x2e ) = (−14,2,156₅ ) Fe(x1,e ex2) = (−7,4,831₂₅) 4.3 ex1= (1,3₂,2) ex2= (−3,−2,1₂) fe(ex1,x2e ) = (−13,−5₂,7₂) Fe(x1,e x2e ) = (−14,−3,5)

+0.5(r1j−p1j)|t1j|⊕ ∑n1

j=1 p_1j<0 r_1j≥0

(0.5(r1jx1j+p1jt1j)−

0.5|r1jx1j−p1jt1j|, q1jy1j,

0.5(r1jt1j + p1jx1j) + 0.5|r1jt1j − p1jx1j|) ⊕ ∑n1

j=1 r1j<0

(0.5(p1j+r1j)t1j

− 0.5(r1j − p1j)|t1j|, q1jy1j,0.5(p1j + r1j)x1j +

0.5(r1j−p1j)|x1j|)

⊕ ∑n2

j=1 p_2j≥0

(0.5(p2j + r2j)x2j − 0.5(r2j −

p2j)|x2j|, q2jy2j,0.5(p2j +r2j)t2j

+ 0.5(r2j − p2j)|t2j|) ⊕ ∑n2

j=1 p_2j<0 r_2j≥0

(0.5(r2jx2j +

p2jt2j)−0.5|r2jx2j−p2jt2j|,

q2jy2j,0.5(r2jt2j+p2jx2j) + 0.5|r2jt2j−p2jx2j|)⊕ ∑n2

j=1 r_2j<0

(0.5(p2j+r2j)t2j

− 0.5(r2j − p2j)|t2j|, q2jy2j,0.5(p2j + r2j)x2j) +

0.5(r2j−p2j)|x2j|)

s.t.

maximize

(x2j,y2j,t2j)

˜

f(˜x1,x˜2) =

∑n1

j=1 p_3j≥0

(0.5(p3j+r3j)x1j−

0.5(r3j−p3j)|x1j|, q3jy1j,0.5(p3j+r3j)t1j

+0.5(r3j−p3j)|t1j|⊕ ∑n1

j=1 p_3j<0 r_3j≥0

(0.5(r3jx1j+p3jt1j)−

0.5|r3jx1j−p3jt1j|, q3jy1j,

0.5(r3jt1j + p3jx1j) + 0.5|r3jt1j − p3jx1j|) ⊕ ∑n1

j=1 r_3j<0

(0.5(p3j+r3j)t1j

− 0.5(r3j − p3j)|t1j|, q3jy1j,0.5(p3j + r3j)x1j +

0.5(r3j−p3j)|x1j|)

⊕ ∑n2

j=1 p_4j≥0

(0.5(p4j + r4j)x2j − 0.5(r4j −

p4j)|x2j|, q4jy2j,0.5(p4j +r4j)t2j

+ 0.5(r4j − p4j)|t2j|) ⊕ ∑n2

j=1 p_4j<0 r_4j≥0

(0.5(r4jx2j +

p4jt2j)−0.5|r4jx2j−p4jt2j|,

q4jy2j,0.5(r4jt2j+p4jx2j) + 0.5|r4jt2j−p4jx2j|)⊕ ∑n2

j=1 r4j<0

(0.5(p4j+r4j)t2j

− 0.5(r4j − p4j)|t2j|, q4jy2j,0.5(p4j + r4j)x2j) +

0.5(r4j−p4j)|x2j|)

s.t. ∑n1

j=1 a1_ij≥0

(0.5(a1_ij + a3_ij)x1j − 0.5(a3ij −

a1_ij)|x1j|, a2ijy1j,0.5(a1ij + a3ij)t1j + 0.5(a3ij −

a1 ij)|t1j|)

⊕ ∑n1

j=1 a1_ij<0 a3

ij≥0

(0.5(a3

ijx1j + a1ijt1j) − 0.5|a3ijx1j −

a1

ijt1j|, a2ijy1j,0.5(a1ijx1j + a3ijt1j) + 0.5|a3ijt1j −

a1 ijx1j)

⊕ ∑n1

j=1 a3_ij<0

(0.5(a1_ij + a3_ij)t1j − 0.5(a3ij −

a1

ij)|t1j|, a2ijy1j,0.5(a1ij + a3ij)x1j + 0.5(a3ij −

a1_ij)|x1j|)

⊕ ∑n2

j=1 b1_ij≥0

((0.5(b1_ij + b3_ij)x2j − 0.5(b3ij −

b1_ij)|x2j|, bij2y2j,0.5(b1ij+b3ij)t2j+ 0.5(bij3 −b1ij)|t2j|)

⊕ ∑n2

j=1 b1_ij<0 b3

ij≥0

(0.5(b3_ijx2j + b1ijt2j) − 0.5|b3ijx2j −

b1_ijt2j|, b2ijy2j,0.5(b1ijx2j + b3ijt2j) + 0.5|b3ijt2j −

b1_ijx2j)

⊕ ∑n2

j=1 b1_ij<0

(0.5(b1_ij + b3_ij)t2j − 0.5(b3ij −

b1_ij)|t2j|, bij2y2j,0.5(b1ij+b3ij)x2j+0.5(bij3 −b1ij)|x2j|)

= (mi, ni, pi),∀i =

1,2, ..., m,(x1j, y1j, t1j),(x2j, y2j, t2j) ∈

F(R), j = 1, ..., n.

Step 3. Using Definitions 2.6,2.7, the FF-BLP problem, obtained in Step 2, is transformed into the following crisp bilevel linear program-ming problem:

maximize

x1j,y1j,t1j

F = 1₄[∑n1

j=1 p_1j≥0

(0.5(p1j + r1j)x1j −

0.5(r1j−p1j)|x1j|+2q1jy1j+ 0.5(p1j+r1j)t1j

+ 0.5(r1j − p1j)|t1j|] + 1₄[ ∑n1

j=1 p_1j<0 r_1j≥0

(0.5(r1jx1j +

p1jt1j)−0.5|r1jx1j−p1jt1j|+2q1jy1j

1 4[

∑n1

j=1 r_1j<0

(0.5(p1j +r1j)t1j

−0.5(r1j−p1j)|t1j|+2q1jy1j+ 0.5(p1j+r1j)x1j+

0.5(r1j−p1j)|x1j|)]

+ 1₄[∑n2

j=1 p_2j≥0

(0.5(p2j + r2j)x2j − 0.5(r2j −

p2j)|x2j|+2q2jy2j + 0.5(p2j+r2j)t2j

+ 0.5(r2j − p2j)|t2j|)] + 1₄[ ∑n2

j=1 p_2j<0 r2j≥0

(0.5(r2jx2j +

p2jt2j)−0.5|r2jx2j−p2jt2j|

+ 2q2jy2j + 0.5(r2jt2j + p2jx2j) + 0.5|r2jt2j −

p2jx2j|)] + 1₄[ ∑n2

j=1 r_2j<0

(0.5(p2j +r2j)t2j

−0.5(r2j−p2j)|t2j|+2q2jy2j+ 0.5(p2j+r2j)x2j) +

0.5(r2j−p2j)|x2j|)]

s.t.

maximize

x2j,y2j,t2j

f = 1₄[∑n1

j=1 p_3j≥0

(0.5(p3j + r3j)x1j −

0.5(r3j−p3j)|x1j|+2q3jy1j+ 0.5(p3j +r3j)t1j

+ 0.5(r3j − p3j)|t1j|] + 1₄[ ∑n1

j=1 p_3j<0 r_3j≥0

(0.5(r3jx1j +

p3jt1j)−0.5|r3jx1j−p3jt1j|+2q3jy1j2

+ 0.5(r3jt1j +p3jx1j) + 0.5|r3jt1j −p3jx1j|)] + 1

4[

∑n1

j=1 r3j<0

(0.5(p3j +r3j)t1j

−0.5(r3j−p3j)|t1j|+2q3jy1j+ 0.5(p3j+r3j)x1j+

0.5(r3j−p3j)|x1j|)]

+ 1₄[∑n2

j=1 p_4j≥0

(0.5(p4j + r4j)x2j − 0.5(r4j −

p4j)|x2j|+2q4jy2j + 0.5(p4j+r4j)t2j

+ 0.5(r4j − p4j)|t2j|)] + 1₄[ ∑n2

j=1 p4j<0

r_4j≥0

(0.5(r4jx2j +

p4jt2j)−0.5|r4jx2j−p4jt2j|

+ 2q4jy2j + 0.5(r4jt2j + p4jx2j) + 0.5|r4jt2j −

p4jx2j|)] + 1₄[ ∑n2

j=1 r_4j<0

(0.5(p4j +r4j)t2j

−0.5(r4j−p4j)|t2j|+2q4jy2j+ 0.5(p4j+r4j)x2j) +

0.5(r4j−p4j)|x2j|)]

s.t. ∑n1

j=1 a1

ij≥0

(0.5(a1_ij + a3_ij)x1j − 0.5(a3ij −

a1_ij)|x1j|) + ∑n1

j=1 a1

ij<0

a3 ij≥0

(0.5(a3_ijx1j + a1ijt1j) −

0.5|a3_ijx1j−a1ijt1j|)

+∑n1

j=1 a3

ij<0

(0.5(a1_ij +a3_ij)t1j−0.5(a3ij −a1ij)|t1j|) +

∑n2

j=1 b1

ij≥0

(0.5(b1_ij+b3_ij)x2j −0.5(b3ij −b1ij)|x2j|)

+∑n2

j=1 b1

ij<0

b3 ij≥0

(0.5(b3_ijx2j+b1ijt2j)−0.5|b3ijx2j−b1ijt2j)|)+

∑n2

j=1 b1_ij<0

(0.5(b1_ij+b3_ij)t2j−0.5(b3ij−b1ij)|t2j|) =mi,

∀i= 1, ...m.

∑n1

j=1(a3ijy1j) + ∑n2

j=1(b3ijy2j) =ni,∀i= 1, ..., m. ∑n1

j=1 a1

ij≥0

(0.5(a1_ij + a3_ij)t1j + 0.5(a3_ij − a1_ij)|t1j|) +

∑n1

j=1 a1

ij<0

a3_ij≥0

(0.5(a1_ijx1j+a3_ijt1j) + 0.5|a3_ijt1j−a1_ijx1j)

+∑n1

j=1 a3

ij<0

(0.5(a1_ij+a3_ij)x1j+ 0.5(a3_ij−a1_ij)|x1j|) + ∑n2

j=1 b1

ij≥0

(0.5(b1_ij+b3_ij)t2j+ 0.5(b3_ij −b1_ij)|t2j|)

+∑n2

j=1 b1

ij<0

b3_ij≥0

(0.5(b1_ijx2j+b3_ijt2j)+0.5(b3_ijt2j−b1_ijx2j)+

∑n2

j=1 b1

ij<0

(0.5(b1_ij+b3_ij)x2j+ 0.5(b3_ij−b1_ij)|x2j|) =pi,

∀i = 1, ...m., y1j − x1j ≥ 0, y2j − x2j ≥

0, t1j−y1j ≥0, t2j−y2j ≥0

, y1j, x1j, t1j, y2j, x2j, t2j ∈ R,∀ : 1 ≤ i≤ m,1 ≤

j≤n

Step 4. Find the optimal solution x1j, y1j, t1j, x2j, y2j, t2j, by solving the bilevel

nonlinear programming problem obtained in Step 3.

Step 5. Find the fuzzy optimal solution by putting the values of x1j, y1j, t1j, x2j, y2j, t2j in e

x1= (x1j, y1j, t1j), ex2 = (x2j, y2j, t2j).

Step 6. Find the fuzzy optimal value by putting

e

x1, xe2 in: F˜(˜x1,x˜2) =

∑n1

j=1(p1j, q1j, r1j) ⊗

(x1j, y1j, t1j)⊕ ∑n2

j=1(p2j, q2j, r2j)⊗(x2j, y2j, t2j)

˜

f(˜x1,x˜2) =

∑n1

j=1(p3j, q3j, r3j)⊗(x1j, y1j, t1j)⊕ ∑n2

j=1(p4j, q4j, r4j)⊗(x2j, y2j, t2j).

4

Numerical examples

Here, we give some numerical examples and solve them by proposed algorithms in previous section. Examples4.1,4.2and 4.3have been solved by the algorithms 1,2 and 3 respectively. Table 1, shows the results of FFBLP problems solved by the proposed algorithms.

Example 4.1 Solve the following

FF-BLP problem using the first algorithm:

maximize

˜ x1

˜

s.t. maximize

˜ x2

˜

f(˜x1,x˜2) = (1,6,3)⊗x˜1⊕(3,4,5)⊗

˜ x2

s.t. (2,3,4)⊗x˜1⊕(1,2,3)⊗x˜2= (6,16,30)

(−1,1,2) ⊗ x˜1 ⊕ (1,3,4) ⊗ x˜2 = (1,17,30), where ex1 , ex1 are non-negative triangular fuzzy numbers.

Example 4.2 Solve the following

FF-BLP problem using the second algorithm:

maximize

˜ x1

˜

F(˜x1,x˜2) = (1,2,3)⊗x˜1⊕(2,4,6)⊗x˜2

s.t. maximize

˜ x2

˜

f(˜x1,x˜2) = (1,2,5)⊗x˜1⊕(1,3,4)⊗

˜ x2

s.t. (0,1,2)⊗x˜1⊕(1,3,5)⊗x˜2= (−5,4,27)

(2,4,7)⊗x˜1⊕(2,3,5)⊗x˜2 = (−19,−2,34), where ex1 , ex1 are unrestricted triangular fuzzy numbers.

Example 4.3 Solve the following FFBLP prob-lem using the third algorithm:

maximize

˜ x1

˜

F(˜x1,x˜2) = (−1,2,3)⊗x˜1⊕(2,3,4)⊗

˜ x2

s.t. maximize

˜ x2

˜

f(˜x1,x˜2) = (−2,1,2) ⊗ x˜1 ⊕

(1,2,3)⊗x˜2

s.t. (−3,−2,0) ⊗ x˜1 ⊕ (6,7,8) ⊗ x˜2 =

(−30,−17,−3)

(2,4,6)⊗x˜1⊕(−2,−1,2)⊗x˜2= (−4,8,18),where

e

x1 , ex1 are unrestricted triangular fuzzy numbers.

5

Conclusion

This paper presents new algorithms to find the fuzzy optimal solution of the FFBLP problem with equality constraints and non-negative fuzzy variables or unconstrained fuzzy variables. In these algorithms, the fuzzy bilevel linear pro-gramming problem is transformed to a determin-istic bilevel programming problem using the rank-ing function method, and the solution of the de-terministic problem is achieved using the com-mon Kth-best method. The proposed methods are quite useful in solving the real-world prlems where the information is inexact. To ob-tain the fuzzy optimal solution, new efficient algo-rithms have been proposed for FFBLP problems and some numerical examples have been solved to illustrate the proposed methods.

References

[1] T. Allahviranloo, F. H. Lotfi, M. K. Kiasary, N. A. Kiani, L. Alizadeh, Solving fully fuzzy linear programming problem by the ranking function, Applied Mathematical Sciences 2 (2008) 19-32.

[2] R. E. Bellman, L. A. Zadeh, Decision Mak-ing in A Fuzzy Environment, Management Science 17 (1970) 141-164.

[3] M. Cecchini, J. Ecker, M. Kupferschmid, R. Leitch, Solving nonlinear principal-agent problems using bilevel programming,Eur. J. Oper. Res.230 (2013) 364-373.

[4] S. Dempe, Foundations of bilevel program-ming,Kluwer, Dordrecht, 2002.

[5] S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization

52 (2003) 333-359.

[6] S. Dempe, V. Kalashnikov, G. A. Prez-Valds, N. Kalashnykova, Bilevel program-ming problems,Springer, Berlin, 2015.

[7] F. Hosseinzadeh Lotfi, T. Allahviranloo, M. Alimardani Jondabeh, L. Alizadeh, Solving a full fuzzy linear programming using lexi-cography method and fuzzy approximate so-lution, Applied Mathematical Modelling 33 (2009) 3151-3156.

[8] V. V. Kalashnikov, S. Dempe, G. A. Prez-Valds, N. I. Kalashnykova, J. F. Camacho-Vallejo, Bilevel programming and applica-tions, Mathematical Problems in Engineer-ing (2015) http://dx.doi.org/10.1155/ 2015/310301/.

[9] J. Kaur, A. Kumar, Exact fuzzy opti-mal solution of fully fuzzy linear program-ming problems with unrestricted fuzzy vari-ables,Appl. Intell.37 (2012) 145-154.

[10] A. Kumar, J. Kaur, P. Singh, A new method for solving fully fuzzy linear programming problems, Applied Mathematical Modelling