• No results found

A New Proposed Fuzzy Programming Approach for Solving a Fully Fuzzy Linear Programming Problem

N/A
N/A
Protected

Academic year: 2022

Share "A New Proposed Fuzzy Programming Approach for Solving a Fully Fuzzy Linear Programming Problem"

Copied!
11
0
0

Loading.... (view fulltext now)

Full text

(1)

ISSN 1450-216X / 1450-202X Vol. 150 No 4 November, 2018, pp. 478-488 http://www. europeanjournalofscientificresearch.com

A New Proposed Fuzzy Programming Approach for Solving a Fully Fuzzy Linear Programming Problem

H. A. Khalifa

Department of Operations Research Institute of Statistical Studies and Research

Cairo University, Giza, Egypt.

Elshimaa A. Elgendi

Corresponding Author, Department of Operations Research and Decision Support Faculty of Computers and Information, Cairo University, Giza, Egypt

E-mail: e.elgendi@fci-cu.edu.eg Tel: +2 12 06851111 Abdul Hadi N. Ahmed

Department of Mathematical Statistics Institute of Statistical Studies and Research

Cairo University, Giza, Egypt

Abstract

In this paper, a new fully fuzzy linear programming (FFLP) problem with equality and/or inequality constraints is formulated. The FFLP problem is considered by incorporating triangular fuzzy numbers in all parameters and variables. A proposed method to find the optimal fuzzy solution for the FFLP problem is suggested. As the proposed method is applied to the FFLP problem, a classical multi-objective linear programming (MOLP) problem is obtained. Then, a fuzzy programming approach is applied to the MOLP problem by defining linear as well as non-linear membership functions to find the optimal compromise solution. The approach leads to an efficient solution as well as optimal compromise solution. The methodology is illustrated with the help of solved numerical examples using GAMS modeling language.

Keywords: Fully fuzzy linear programming; Triangular fuzzy numbers; Multi-objective linear programming; Fuzzy programming; Linear and non-linear membership function; Optimal fuzzy solution; GAMS.

1. Introduction

Linear programming (LP) is one of the most frequently applied operationsresearch techniques. LP is useful in developing new approaches in order tobe better fit the real world problems. LP models in reality involve a lot ofparameters whose values are assigned by experts. However, both expertsand decision makers frequently do not precisely know the values of thoseparameters. Therefore, it is useful to consider the knowledge of expertsabout the parameters as being a fuzzy data.

Fully fuzzy linear programming problems (FFLP) are fuzzy linear programming problems with either equality or inequality constraints, in whichall the parameters and the variables are represented by fuzzy numbers.

(2)

In this paper, a proposed method for solving FFLP problem is suggested.As a result of the proposed method, a MOLP problem is then obtained. Afuzzy programming approach is applied to the MOLP by introducing membership functions, and an optimal compromise solution is obtained.

Membership functions used are the linear membership function, nonlinear membership function, and hyperbolic membership function.

The paper is organized as follows. Related research work is shown inSection 2. In Section 3, some basic definitions and arithmetic operationsof two triangular fuzzy numbers are reviewed. A FFLP problem, specificdefinition and properties are presented in Section 4. In Section 5, a fewmembership functions are defined. In Section 6,the proposed solution procedurefor the FFLP problem is illustrated.

Numerical examples are given in Section 7. Finally,some concluding remarks are reported in Section 8.

2. Related Work

Many research works were devoted to solve fuzzy linear programmingproblems [1-6]. However, fuzzy linear programming have been studied inthese works not all parts of the problem were assumed to be fuzzy, e.g., onlythe right hand side or the objective function coefficients were fuzzy but thevariables were not fuzzy.

Formulation Fuzzy linear programming (FLP) problem with fuzzy coefficients is reported in [1]

and [2]. A study on the complete solution set forthe most generalized symmetric FLP problems is shown in [5], in which bothfuzzy equality and inequality constraints are included, and both linear andnonlinear membership functions are allowed. Linear Programming problemwith Interval number and fuzzy number coefficients are approached in fuzzydecisive set approach and interval number linear programming approach withseveral membership levels [6]. The complementary slackness theorem is usedto solve FLP with fuzzy parameters without the need of a simplex tableau [7].

Kumar et al. proposed a new method to solve FFLP problems wherethe constraints are all inequality [8] and equality [9]. Nasseri et al. [10]introduced a novel approach to solve FFLP based on the definition of membership function and used a convenient technique for solving the classicalmulti- objective programming. Rajarajeswari and Sudha [11] proposed a newmethod for solving FFLP problem based on a ranking method. Sudha [12]studied FFLP problem using ranking function with the help of linear system of equation through the trapezoidal fuzzy numbers. Shamooshaki etal. [13]

proposed a new method to solve the FFLP problem with L-R fuzzynumber and the Lexicography method. Hosseinzadeh and Edalatpanah [14]designed a new model for solving the FFLP problem considering the L-Rfuzzy numbers and the Lexicography method in conjunction with crisp linear programming.

Many variations of FLP and FFLP are shown in literature such as multi-objective linear programming problem (MOLP), where coefficients of theobjective functions and/ or of the constraints are known exactly but imprecisely [4]. Large-scale FLP solved by a revised interior point-method presented in [15]. MOLP with fuzzy numbered cost coefficients is introducedby Wang and Wang [16].

Bitran [3] and Steuer [17] developed different algorithms to solve MOLP of which the cost coefficients are also interval valued.They applied the vector-maximum theory [18] to find the coefficients extreme points. A fully fuzzy MOLP also introduced where the parametersand variables are triangular fuzzy numbers, the solution procedure based onthe comparison of fuzzy numbers by a linear ranking function [19].

3. Preliminaries

In this section, relevant background and notation of fuzzy numbers are reviewed. For a detailed exposition on fuzzy numbers, see, e.g., [10,20,21], among others.

(3)

3.1. Definitions and Notation

Definition1. Let Rbe the set of real numbers, the fuzzy numbers p~is a mappingµ~p:R→[0,1], with the following properties:

(i) µ~p(x)is upper semi-continuous membership function.

(ii) p~is a convex fuzzy set, i.e., µ~px+(1−λ)y)≥ min

{

µ~p(x),µ~p(y)

}

, for all x,yR;0≤λ≤1. (iii)p~is normal, i.e., there exists x0Rfor which µ~p(x0)=1.

(iv) Supp (~p)=

{

x:µ~p(x)>0

}

is the support of a fuzzy set ~p.

Definition 2. A triangular fuzzy numberp~ is defined by a triplet (p1,p2,p3)with a membership function µ~p(x) is defined by





>

− ≤

− ≤

<

=

. ,

0

, ,

, ,

, 0

) (

3 3 2

2 3

3

2 1

1 2

1

1

~

p x

p x p p

p x p

p x p p

p p x

p x

p x µ

Definition3. A triangular fuzzy number ~ ( , , )

3 2

1 p p

p

p= is said to be nonnegative if and only if .

1≥0 p

3.2. Arithmetic Operations of Triangular Fuzzy Numbers

In this subsection, some arithmetic operations on triangular fuzzy numbers are introduced.

Let~ ( , , )

3 2

1 p p

p

p= and~ ( , , )

3 2 1 q q q

q= be two triangular fuzzy numbers.

(a) Addition:

).

, ,

(

) , , )(

( ) , ,

~ ( )

~(

3 3 2 2 1 1

3 2 1 3

2 1

q p q p q p

q q q p

p p q p

+ +

+

=

+

= +

(b) Subtraction:

).

, ,

(

) , , ( ) ( ) , ,

~ ( )

~(

1 3 2 2 3 1

3 2 1 3

2 1

q p q p q p

q q q p

p p q p

=

=

(c) The symmetric (image):

).

, , (

) , , (

~) (

1 2 3

3 2 1

p p p

p p p p

=

=

(d) Multiplication:





<

<

=

=

. 0 ),

, , (

, 0 , 0 ),

, , (

0 ),

, , (

) , , ( ) )(

, ,

~ ( )

~(

3 1

3 2 2 3 1

3 1

3 3 2 2 3 1

1 3

3 2 2 1 1

3 2 1 3 2 1

p q

p q p q p

p p

q p q p q p

p q

p q p q p

q q q p p p q p

(4)

(e) The order relation (≤) inF(R), the set of all triangular fuzzy numbers on R is defined by:

) , , ( ) ( ) , ,

(p1 p2 p3q1 q2 q3 if and only if p1q1,p2q2,p3q3.

4. Problem Formulation

In this section we reformulate linear programming problem in a fully fuzzy framework with equality and inequality constraints.

A fully fuzzy linear programming (FFLP) problem with equality and inequality constraints is formulated as follows:

Max

)~

~ (

~

1

j n

j

j x

c Z

= (1)

Subject to

~ ~

, , , 2 , 1

~, ) , ,

~ ( )

~ (

1

m i

b x

a j i

n

j

ij ⋅ ≤ ≈ ≥ = ⋅⋅⋅

= (2)

, , 2 , 1 ,

~0

~xjj = ⋅⋅⋅ n

(3)

Where ~) , ~ (~ ) ,

~ ( ,

~ )

~ ( ,

~ )

~ (

1 1

1× = × = × = ×

= cj n A aij m n b bi m x xj n

c and ~,~ ( ),

~ ,

~ ,c b x F R aij j i j

; ,..., 2 ,

1 m

i= .

,..., 2 ,

1 n

j=

Let

; ,..., 2 , 1

; ,..., 2 , 1 ), , ,

~ (

; ,..., 2 , 1 ), , ,

~ (c d e j n a p q r i m j n

cj = j j j = ij = ij ij ij = = ~ ( , , ),

i i i

i u v w

b = i=1,2,...,m, and ~xj =(xj,yj,tj),j=1,2,...,n.

Then, the FFLP problem (1- 3) can be rewritten in the following form

Max

) , , ( ) ( ) , , (

1

j j j j j n

j

j d e x y t

c

Z =

= (4)

Subject to

, ,..., 2 , 1 ), , , ( ) , , ( ) , ( ) ( ) , , (

1

m i

w v u t

y x r q

p ij ij j j j i i i

n

j

ij ⋅ ≤ = ≥ =

= (5)

≤ ≤ and (xj,yj,tj)≥0, j=1,2,...,n.

()6

5. Membership Functions

In this section, a few membership functions are introduced to solve multi-objective linear and nonlinear programming problems.

(i) Linear membership function: The linear membership function [22] for a vector maximum problem is given by:



− ≤

=

k k

k k

k k

k

k k

k k

k

L x Z

U x Z L L

U

L x Z

L x Z x

) ( ,

1

, ) ( ) ,

(

, ) ( ,

0 ) (

) ( ) ( )

(

) (

µ (7)

Where µk(x)is the membership function of the kth objective function and, LkandUk are the lower and upper bounds ofz(k)(x), LkUk and they are calculated as:

(5)

Uk

= maxZ(k)(x),xX and Lk

=minZ(k)(x), k=1,2,...,K.

(ii) Nonlinear membership function: using product-operator [22],the nonlinear membership function is given as:

, ) ( )

(

1

=

=

K

k

k x

x µ

µ

(8) where, µk(x)is the membership function of the k th objective function and is given by:

= , < < (9)

(iii) Hyperbolic membership function: The hyperbolic membership function (Leberling, 1981) is given as:

{ }

[ ]

2 2 1

/ ) (

) ( 2tanh

) 1

( = (k)k + k k +

k x Z x U L α

µ

(10) where, Lk

and Uk

are, respectively the lower and upper bounds of Z(k)(x) andαk is a constant and ).

( /

6 k k

k = U +L

α

6. Solution Procedure

The suggested procedure of solution is described in the following steps.

Step 1: Decompose the problem (4-6) using the arithmetic operations and the partial order relation of triangular fuzzy numbers as follows:

Max ( ( , , )( )( , , ) )

1

1 j j

n

j

j j

j

j d e x y t

c of

value Lower

Z

=

=

Max ( ( , , )( )( , , ))

1

2 j j

n

j

j j

j

j d e x y t

c of value Middle

Z

=

=

Max ( ( , , )( )( , , ))

1

3 j j j j j

n

j

j d e x y t

c of

value Upper

Z =

=

Subject to

. ,..., 2 , 1 , 0 ) , , (

, ,..., 2 , 1 , ) , , ( ) ) , , ( ) ( ) , , ( (

, ,..., 2 , 1 , ) , , ( ) ) , , ( ) ( ) , , ( (

, ,..., 2 , 1 , ) , , ( ) ) , , ( ) ( ) , , ( (

1 1 1

n j

t y x

m i

w t

y x r

q p of

value Upper

m i

v t

y x r

q p of

value Middle

m i

u t

y x r

q p of

value Lower

j j j

i j

j j n

j

ij ij ij

i j

j j ij

n

j

ij ij

i j

j j ij

ij n

j ij

=

=

=

=

=

=

=

=

=

=

(11) Step 2: Solve the problem (11), using each time, only one objective at a time and ignoring the others.

Step 3: Using the solutions obtained in Step 2, the corresponding value of all objective functions of each of the solutions are found.

Step 4: The lower and upper bounds (LkandUk,k=1,2,3) for each of the objective function are obtained from Step 3.

Step 5: Using a membership function (as discussed in Section 4) the corresponding fuzzy linear (nonlinear) programming problem is found. The details are presented below.

Let a vector- maximum problem be of the type

(6)

MaxZ(k)(x)=(C(k))Tx, k =1,2,...,K Subject to

. 0

,

x

b Ax

(12) (a) Using a linear membership function (7), problem (12) becomes

Max Z=λ, Subject to

. 1 0

, 0

,

..., , 2 , 1 ) ,

)(

(

− =

≤ −

λ λ

x b Ax

K L k

U

L x Z

k k

k k

(13) (b) Using a nonlinear membership function with product- operator (8)-(9) becomes

Max

, ) (

1

=

=

K

k

k x

Z µ

Subject to . 0

,

x

b Ax

(14) (c) Using hyperbolic membership function (10) becomes

MaxZ=λ , Subject to

[ ]

. 1 0

, 0

,

..., , 2 , 1 2, } 1

2 / ) (

) ( { 2tanh

1 ( )

= + +

λ

α λ

x b Ax

K k

L U x

Z k k k k

(15)

Or equivalently, Max β

Subject to

. 0 , 0

,

..., , 2 , 1 , 2 / )

) (

(

=

∗ +

β

α β

α x

b Ax

K k

L U

Z k k k k k

(16)

Step 6: Finally, solve the linear programming / nonlinear programming problem to find the optimal compromise solution and hence the optimal fuzzy solution (xj,yj,tj), j =1,2,...,n with the corresponding optimum fuzzy valueZ=(Z1,Z2,Z3).

7. Numerical Examples

In this section, we present numerical examples to illustrate the solution procedure proposed in this paper.

Example 7.1. Consider the following FFLP problem Max Z =(1,2,3)(⋅)~x1(+)(2,3,4)(⋅)~x2

Subject to

(7)

), 28 , 11 , 2

~ ( ) )(

2 , 1 , 0 ( )

~ ( ) ( ) 3 , 2 , 1 (

), 27 , 10 , 1

~ ( ) ( ) 3 , 2 , 1 ( )

~( ) ( ) 2 , 1 , 0 (

2 1

2 1

⋅ +

⋅ +

x x

x x

where, ~x1 and ~x2 are non-negative triangular fuzzy numbers.

Let ~ ( , , )

1 1 1

1 x y t

x = and ~ ( , , )

2 2 2

2 x y t

x = . Then the given FFLP problem may be written as follows:

MaxZ =(1,2,3)(⋅)(x1,y1,t2)(+)(2,3,4)(⋅)(x2,y2,t2) Subject to

. 0 ) , , ( , 0 ) , , (

), 28 , 11 , 2 ( ) , , ( ) ( ) 2 , 1 , 0 ( ) ( ) , , )(

( ) 3 , 2 , 1 (

), 27 , 10 , 1 ( ) , , ( ) ( ) 3 , 2 , 1 ( ) ( ) , , ( ) ( ) 2 , 1 , 0 (

2 2 2 1

1 1

2 2 2 1

1 1

2 2 2 1

1 1

⋅ +

⋅ +

t y x t

y x

t y x t

y x

t y x t

y x

Step 1: The above problem can be written as follows Max Z1=x1+2x2 ,

maxZ2 =2y1+3y2 maxZ3 =3t1+4t2 Subject to

. 0 , , , , ,

, 28 2 3 , 11 1

2 , 2 0 1

, 27 3 2 , 10 2

1 , 1 1 0

2 1 2 1 2 1

2 1 2

1 2

1

2 1 2

1 2

1

≤ +

≤ +

≤ +

≤ +

≤ +

≤ +

t t y y x x

t t y

y x

x

t t y

y x

x

Step 2: The solution of each single objective linear programming problem is:

).

5 , 6 , 0 , 0 , 0 , 0 (

), 0 , 0 , 3 , 4 , 0 , 0 (

), 0 , 0 , 0 , 0 , 1 , 2 (

3 2

1

=

=

=

X X

X

Step 3: The objective function values are

. 38 ) ( , 0 ) ( , 0 ) (

, 0 ) ( , 17 ) ( , 0 ) (

, 0 ) ( , 0 ) ( , 4 ) (

3 3 3

2 3

1

2 3 2

2 2

1

1 3 1

2 1

1

=

=

=

=

=

=

=

=

=

X Z X

Z X

Z

X Z X

Z X

Z

X Z X

Z X

Z

Step 4: The lower and upper bounds of the objective functions are

, 0 ,

38

, 0 ,

17

, 0 ,

4

3 3

2 2

1 1

=

=

=

=

=

=

L U

L U

L U

Step 5: (a) Using the linear membership function and applying the fuzzy programming approach, one formulates the following problem

Max λ, subject to

. 1 0

, 0 , , , , ,

, 28 2 3 , 11 1

2 , 2 0 1

, 27 3 2 , 10 2

1 , 1 1 0

, 0 38 4 3

, 0 17 3

2

, 0 4 2

2 1 2 1 2 1

2 1 2

1 2

1

2 1 2

1 2

1 2 1

2 1

2 1

≤ +

≤ +

≤ +

≤ +

≤ +

≤ +

− +

− +

− +

λ λ

λ λ

t t y y x x

t t y

y x

x

t t y

y x

x t t

y y

x x

The obtained compromise solution is

(8)

. 38 ,

17 ,

4

, 1 , 5 , 6 , 3 , 4 , 1 , 2

3 2

1

2 1

2 1

2 1

=

=

=

=

=

=

=

=

=

=

Z Z

Z

t t

y y

x

x λ

(b) Using the product type of membership function, a fuzzy nonlinear programming problem is formulated as follows

Max λ =λ1λ2λ3, subject to

. 0 , , , 0 , , , , ,

, 28 2 3 , 11 1

2 , 2 0 1

, 27 3 2 , 10 2

1 , 1 1 0

, 0 38 4 3

, 0 17 3

2

, 0 4 2

3 2 1 2

1 2 1 2 1

2 1 2

1 2

1

2 1 2

1 2

1

3 2

1

2 2

1

1 2 1

≤ +

≤ +

≤ +

≤ +

≤ +

≤ +

=

− +

=

− +

=

− +

λ λ λ λ

λ λ

t t y y x x

t t y

y x

x

t t y

y x

x t t

y y

x x

The obtained compromise solution is:

. 38 ,

17 ,

4

1 , 1 , 1 , ,

5 ,

6 , 3 , 4 ,

1 , 2

3 2

1

3 2

1 2

1 2

1 2

1

=

=

=

=

=

=

=

=

=

=

=

=

=

Z Z

Z

t t

y y

x

x λ λ λ λ

(c) Hyperbolic membership function: Using hyperbolic membership function, we formulate fuzzy nonlinear programming problem as

Max β , subject to

. 0 , 0 , , , , ,

, 28 2 3 , 11 1

2 , 2 0 1

, 27 3 2 , 10 2

1 , 1 1 0

, 57 19 4 3

, 51 17 3

2

, 6 2 2

2 1 2 1 2 1

2 1 2

1 2

1

2 1 2

1 2

1 2 1

2 1

2 1

≤ +

≤ +

≤ +

≤ +

≤ +

≤ +

− +

− +

− +

β β

β β

t t y y x x

t t y

y x

x

t t y

y x

x t t

y y

x x

The obtained compromise solution is:

. 38 ,

17 ,

4

, 5 , 6 , 3 , 4 ,

1 , 2

3 2

1

2 1

2 1

2 1

=

=

=

=

=

=

=

=

=

Z Z

Z

t t

y y

x x

Example7.2. Here an FFLP problem presented in Kumar et al [9],is used and its solution by the proposed method is presented below

MaxZ =(1,6,9)(⋅)~x1(+)(2,3,8)(⋅)~x2 , subject to

), 30 , 17 , 1

~ ( ) ( ) 4 , 3 , 1 ( )

~ ( ) ( ) 2 , 1 , 1 (

), 30 , 16 , 6

~ ( ) ( ) 3 , 2 , 1 ( )

~ ( ) ( ) 4 , 3 , 2 (

2 1

2 1

=

⋅ +

=

⋅ +

x x

x x

where, ~x and 1 ~x are non-negative triangular fuzzy numbers. 2 Let ~ ( , , )

1 1 1

1 x y t

x = and ~ ( , , )

2 2 2

2 x y t

x = . Then the given FFLP problem may be written as follows

MaxZ =(1,6,9)(⋅)(x1,y1,t2)(+)(2,3,8)(⋅)(x2,y2,t2), subject to

(9)

. 0 ) , , ( , 0 ) , , (

), 30 , 17 , 1 ( ) , , ( ) ( ) 4 , 3 , 1 ( ) ( ) , , )(

( ) 2 , 1 , 1 (

), 30 , 16 , 6 ( ) , , ( ) ( ) 3 , 2 , 1 ( ) ( ) , , ( ) ( ) 4 , 3 , 2 (

2 2 2 1

1 1

2 2 2 1

1 1

2 2 2 1

1 1

=

⋅ +

=

⋅ +

t y x t

y x

t y x t

y x

t y x t

y x

Step 1: The above problem can be re-written as follows Max Z1 =1x1+2x2 ,

maxZ2 =6y1+3y2 maxZ3 =9t1+8t2 subject to

. 0 , , , , ,

, 30 4 2 , 17 3

1 , 1 1 1

, 30 3 4 , 16 2

3 , 6 1 2

2 1 2 1 2 1

2 1 2

1 2

1

2 1 2

1 2

1

= +

= +

= +

= +

= +

= +

t t y y x x

t t y

y x

x

t t y

y x

x

Step 2: The solution of each single objective linear programming problem is:

).

6 , 3 , 0 , 0 , 0 , 0 (

), 0 , 0 , 5 , 2 , 0 , 0 (

), 0 , 0 , 0 , 0 , 667 . 2 , 667 . 1 (

3 2

1

=

=

=

X X

X

Step3: The objective function values are

. ) (X Z , ) (X Z , ) (X Z

, ) (X Z , ) (X Z , ) (X Z

, (X Z , ) (X Z , ) (X Z

75 0

0

0 27

0

0 ) 0

7

3 3 3

2 3

1

2 3 2

2 2

1

1 3 1

2 1

1

=

=

=

=

=

=

=

=

=

Step4: lower and upper bounds of the objective functions are

, 0 ,

75

, 0 ,

27

, 0 ,

7

3 3

2 2

1 1

=

=

=

=

=

=

L U

L U

L U

Step5: Linear and nonlinear membership functions cannot be applied in the FFLP problems with equality constraints

(a) Using a hyperbolic membership function, we formulate a fuzzy nonlinear programming problem as follows:

Max β , subject to

. 0 , 0 , , , , ,

, 30 4 2 , 17 3

1 , 1 1 1

, 30 3 4 , 16 2

3 , 6 1 2

, 75 25 8 9

, 27 9 3 6

, 21 7 6 3

2 1 2 1 2 1

2 1 2

1 2

1

2 1 2

1 2

1 2 1

2 1

2 1

= +

= +

= +

= +

= +

= +

− +

− +

− +

β β

β β

t t y y x x

t t y

y x

x

t t y

y x

x t t

y y

x x

The obtained compromise solution is:

. 75 ,

27 ,

7

, 6 , 3 , 5 , 2 , 667 . 2 , 667 . 1

3 2

1

2 1

2 1

2 1

=

=

=

=

=

=

=

=

=

Z Z

Z

t t

y y

x x

(10)

8. Concluding Remarks

In this paper, we have introduced a fully fuzzy linear programming (FFLP) problem with equality and inequality constraints and triangular fuzzy numbers in all the parameters and variables. The FFLP problem has been converted into the classical multi-objective linear programming (MOLP) problem and then a fuzzy programming is applied using different membership functions. An optimal fuzzy solution for the FFLP problem is then obtained using model instance of the generic model that is designed using GAMS modeling language.

References

[1] C. Negoita, Fuzziness in management, OPSA/TIMS, Miami (1970).

[2] H. Tanaka, T. Okuda, K. Asai, on fuzzy-mathematical programming,Journal of Cybernetics 3 (4) (1973) 37-46.

[3] G. R. Bitran, Linear multiple objective problems with interval coefficients, Management Science 26 (7) (1980) 694_706.

[4] H. Rommelfanger, R. Hanuscheck, J. Wolf, Linear programming withfuzzy objectives, Fuzzy Sets and Systems 29 (1) (1989) 31-48.

[5] R. Zhao, R. Govind, G. Fan, The complete decision set of the generalizedsymmetrical fuzzy linear programming problem, Fuzzy Sets and Systems51 (1) (1992) 53-65.

[6] T. Shaocheng, Interval number and fuzzy number linear programming,Fuzzy sets and systems 66 (3) (1994) 301-306.

[7] A. Ebrahimnejad, S. Nasseri, Using complementary slackness propertyto solve linear programming with fuzzy parameters, Fuzzy Informationand Engineering 1 (3) (2009) 233-245.

[8] A. Kumar, J. Kaur, P. Singh, Fuzzy optimal solution of fully fuzzylinear programming problems with inequality constraints, InternationalJournal of Mathematics and Computer Science (2010) 37-41.

[9] A. Kumar, J. Kaur, P. Singh, A new method for solving fully fuzzylinear programming problems, Applied Mathematical Modelling 35 (2)(2011) 817-823.

[10] S. H. Nasseri, F. Khalili, N. A. Taghi-Nezhad, S. M. Mortezania, Anovel approach for solving fully fuzzy linear programming problemsusing membership function concepts, Ann. Fuzzy Math. Inform 7 (3)(2014) 355-368.

[11] P. Rajarajeswari, A. S. Sudha, Solving a fully fuzzy linear programmingproblem by ranking, International Journal of Mathematics Trends andTechnology 9 (2) (2014) 159-164.

[12] A. S. Sudha, Solving fully fuzzy linear programming problem usingtrapezoidal ranking function, Journal of Global Research in Mathematical Archives (JGRMA) ISSN 2320-5822 2 (6) (2015) 21-26.

[13] M. M. Shamooshaki, A. Hosseinzadeh, S. A. Edalatpanah, A newmethod for solving fully fuzzy linear programming problems by usingthe lexicography method, Applied and Computational Mathematics 1(2015) 53-55.

[14] A. Hosseinzadeh, S. Edalatpanah, A new approach for solving fullyfuzzy linear programming by using the lexicography method, Advancesin Fuzzy Systems 2016.

[15] Y.-h. Zhong, Y.-l. Jia, D. Chen, Y. Yang, Interior point method forsolving fuzzy number linear programming problems using linear rankingfunction, Journal of Applied Mathematics 2013.

[16] H.-F. Wang, M.-L. Wang, A fuzzy multiobjective linear programming,Fuzzy Sets and Systems 86 (1) (1997) 61-72.

[17] R. E. Steuer, Multiple criteria optimization: theory, computation, andapplications, Wiley, 1986.

[18] P. L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, Journal of OptimizationTheory and Applications 14 (3) (1974) 319-377.

(11)

[19] H. M. Nehi, H. Hajmohamadi, A ranking function method for solving fuzzy multi-objective linear programming problem, Annals of fuzzyMathematics and Informatics 3 (1) (2012) 31-38.

[20] A. Kaufmann, M. M. Gupta, Fuzzy Mathematical Models in Engineeringand Management Science, Elsevier Science Inc., 1988.

[21] M. Sakawa, Fuzzy Sets and Interactive Multiobjective Optimization,Springer Science &

Business Media, 2013.

[22] H.-J. Zimmermann, Fuzzy programming and linear programming withseveral objective functions, Fuzzy sets and systems 1 (1) (1978) 45-55.

[23] H. Leberling, Onfinding compromise solutions in multicriteria problemsusing the fuzzy min- operator, Fuzzy sets and systems 6 (2) (1981) 105-118.

References

Related documents

3 &#34; In the South African constitutional context, the problem is two-fold: the operation of customary law practices in a society where equality

It is known that adult drivers who are most likely to be involved in motor vehicle accidents are those who are least likely to wear seat belts.’3 We do not know ifthe

The Foundation will continue to play its part in the ongoing debate including acting as an official observer in the consultations on the revision of the European Convention for

Experimental results show the Advanced Clustering Algorithm can improve the execution time, quality of SOM algorithm and works well on High Dimensional data

Ravi Kumar and Dwarakadasa (2000), while investi- gating the effect of matrix strength on the tensile proper- ties of SiC-reinforced Al-Zn-Mg alloy matrix composites, observed that

fected the bilirubin binding by serum albu- min in these infants or that while lowering the serum indirect bilirubin levels it did not effectively lower the total level of toxic

Al- though this is currently controversial, respiration is suggested to be critical for the physiology of life span because (i) the effect of CR on the RLS is mimicked by

Generally accepted principles for patient management that reflect a high degree of clinical cer- tainty (ie, based on class I evidence or, when circum-