Solving fuzzy linear programming problem using ...

Int. J. Operational Research, Vol. 15, No. 3, 2012

Solving fuzzy linear programming problem using symmetric fuzzy number approximation C. Veeramani* Department of Mathematics and Computer Applications, PSG College of Technology, Coimbatore, Tamilnadu, India, E-mail: [email protected] *Corresponding author

C. Duraisamy Department of Mathematics, Kongu Engineering College, Perundurai, Tamilnadu, India E-mail: [email protected] Abstract: In this paper, we discussed the fully fuzzy linear programming (FFLP) problems in which all the parameters and variables are triangular fuzzy numbers. We proposed a new approach of solving FFLP problem using the concept of nearest symmetric triangular fuzzy number approximation with preserve expected interval. Using this technique, FFLP problems model turned into two crisp linear problems: first problem is designed in which the centre objective value will be calculated. Second is designed to obtain the margin from the objective of the principal problem. Finally, the fuzzy approximate solution of the FFLP problem is obtained from the solution of two linear programming problems. Keywords: fuzzy number; symmetric triangular approximation; linear programming; multi-objective linear programming. Reference to this paper should be made as follows: Veeramani, C. and Duraisamy, C. (2012) ‘Solving fuzzy linear programming problem using symmetric fuzzy number approximation’, Int. J. Operational Research, Vol. 15, No. 3, pp.321–336. Biographical notes: C. Veeramani received his BSc and MSc degrees in Mathematics from Bharathidasan University, India, in 2003 and 2005, respectively. Mphil degree in Mathematics from Periyar University, India, in 2007. Currently, he is pursuing his research in the area of fuzzy optimisation at Anna University, Coimbatore, India. Now, he is working as an Assistant Professor in the Department of Mathematics and Computer Applications in PSG College of Technology Coimbatore, India. He has presented papers in national and international conferences and also published four research papers in international journals.

Copyright © 2012 Inderscience Enterprises Ltd.

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322

C. Veeramani and C. Duraisamy C. Duraisamy received BSc degree in Mathematics from Madras University, India, in 1982. MSc, MPhil and PhD degrees in Mathematics from Bharathiyar University, India, in 1986, 1996 and 2004, respectively. Currently, he is working as a Professor and a Dean in the School of Science and Humanities in Kongu Engineering College, India. He has presented several papers in national and international conferences and conducted many workshops and conferences. He has published many papers in national and international journals. Currently, he is supervising six research scholars.

1

Introduction

Modelling and optimisation under a fuzzy environment are called fuzzy modelling and fuzzy optimisation. Fuzzy linear programming is one of the most frequently applied fuzzy decision-making techniques. Although, it has been investigated and expanded for more than decades by many researchers and from the various point of views, it is still useful to develop new approaches to better fit the real-world problems within the framework of fuzzy linear programming. Traditional optimisation techniques and methods have been successfully applied for years to solve problems with a well-defined structure. Such optimisation problems are usually well-formulated by crisply specific objective functions and specific system of constraints, and solved by precise mathematics. Unfortunately, real-world situations are often not deterministic. There exist various types of uncertainties in social, industrial and economic systems, such as randomness of occurrence of events, imprecision and ambiguity of system data and linguistic vagueness, etc., which come from many ways, including errors of measurement, deficiency in history and statistical data, insufficient theory, incomplete knowledge expression and the subjectivity and preference of human judgement, etc. As pointed out by Zimmerman (1978), various kinds of uncertainties can be categorised as stochastic uncertainty and fuzziness. Stochastic uncertainty relates to the uncertainty of occurrences, of phenomena or of events. Its characteristics lie in that descriptions of information are crisp and welldefined; however, they vary in their frequency of occurrence. Therefore, systems with this type of uncertainty are called stochastic systems, which can be solved by stochastic optimisation techniques using probability theory. In some other situations, the decision-maker does not think that the frequently used probability distribution is always appropriate, especially when the information is vague, relating to human language and behaviour, imprecise/ambiguous system data or when the information could not be described and defined well due to limited knowledge and deficiency in its understanding. Such types of uncertainty are called fuzziness. For example, in the real-life problems, the following situation may occur: if a company wants to launch new product in the market, then there may exist uncertainty about the profit (or cost) and availability (or demand) of the product. In such a situation, the profit (or cost) and availability (or demand) may be represented by the fuzzy numbers. It cannot be formulated and solved effectively by traditional mathematics-based optimisation

Solving FLP problem using symmetric fuzzy number approximation

323

techniques and probability-based stochastic optimisation approaches. There are number of researchers who have exhibited their interest to solve the fuzzy linear programming problems (Allahviranloo et al., 2008; Ebrahimnejad and Nasseri, 2009, 2010; Marbini and Tavana, 2011, etc.).

2

Literature review

Fuzzy set theory was developed by Zadeh (1965). The concept of fuzzy decision and the decision model under fuzzy environments were proposed by Bellman and Zadeh (1970). Tanaka et al. (1973) adopted this concept for solving mathematical programming problems. Zimmerman (1978) proposed the first formulation of fuzzy linear programming. Chanas (1983) proposed the possibility of the identification of a complete fuzzy decision in fuzzy linear programming by the use of parametric programming technique. Werners (1987) introduced an interactive system which supports a decisionmaker in solving programming models with crisp or fuzzy constraints and crisp or fuzzy goals. Campos and Verdegay (1989) considered linear programming problems in which both matrix and resources are fuzzy numbers. Inuiguchi et al. (1990) dealt with the fuzzy linear programming problems with continuous piecewise linear membership function. Cadenas and Verdegay (1997) studied a linear programming problem in which all of its elements are defined as fuzzy numbers. Fang et al. (1999) presented a method for solving linear programming problems with fuzzy coefficients in constraints. Buckley and Feuring (2000) introduced a method to find the solution for fully fuzzified linear programming problems with all the parameters and variables as fuzzy numbers by changing the objective function into a multi-objective fuzzy linear programming problem. Liu (2001) proposed a method for solving fuzzy linear programming problems based on the satisfaction degree of the constraints. Maleki (2002) solved the linear programming problems in which all decision parameters are fuzzy numbers by the comparison of fuzzy numbers. Zhang et al. (2003) proposed a method for solving fuzzy linear programming problems which involve fuzzy numbers in coefficients of objective functions. Nehi et al. (2004) defined the concept of optimality for linear programming problems with fuzzy parameters by transforming fuzzy linear programming problems into multi-objective linear programming problems. Ramik (2005) proposed the fuzzy linear programming problems based on fuzzy relations. Ganesan and Veeramani (2006) proposed an approach to solve a fuzzy linear programming problem involving symmetric trapezoidal fuzzy numbers without converting it into crisp linear programming problem. Hashemi et al. (2006) proposed a two-phase approach to find the optimal solutions of class of fuzzy linear programming problems, where all decision parameters and variables are fuzzy numbers. Jebaraj and Iniyan (2007) developed a fuzzy-based linear programming optimal energy model that minimises the cost and determines the optimum allocation of different energy sources for the centralised and decentralised power generation in India. Rommelfanger (2007) proposed a new method for solving stochastic linear programming problems with fuzzy parameters. Jimenez et al. (2007) proposed a method for solving

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linear programming problems where all the coefficients are fuzzy numbers and used a fuzzy ranking method to rank the fuzzy objective values and to deal with the inequality relation on constraints. Allahviranloo et al. (2007) solved the fuzzy integer linear programming problem by reducing it into a crisp integer linear programming problem. Allahviranloo et al. (2008) proposed a new method for solving fully fuzzy linear programming (FFLP) problems by the use of ranking function. Youssef and Rebai (2008) proposed a new approach, which consists in solving the classification problems via fuzzy linear programming models. Ebrahimnejad and Nasseri (2009) used the complementary slackness theorem to solve fuzzy linear programming problem with fuzzy parameters without the need of a simplex tableau. Lotfi et al. (2009) proposed a method for solving FFLP problem using lexicography method and provided fuzzy approximate solution. Nagoorgani et al. (2009) to solve FLPP using C–R representation. Ebrahimnejad et al. (2010) proposed a new primal-dual algorithm for solving linear programming problems with fuzzy variables by using duality results. Ebrahimnejad and Nasseri (2010) introduced a new method called the bounded dual simplex method for bounded fuzzy number linear programming problems. Recently, Marbini and Tavana (2011) proposed an extension of the linear programming method with fuzzy parameters. In this paper, consider FFLP problems of which all the parameters and variables are triangular fuzzy numbers. We use the concept of the symmetric triangular fuzzy number approximation with preserved expected interval to defuzzify a general fuzzy quantity. Every FFLP problems model turned into two crisp complex linear problems: first problem is designed in which the centre objective value will be calculated. Second problem is designed to obtain the margin from the objective of the principal problem. The rest of this paper is organised as follows: Section 3 is devoted to mathematical preliminaries, which will be used further in this paper. A brief discussion of nearest symmetric triangular fuzzy number approximation (NSTFNA) which preserves expected interval is given in Section 4. In Section 5, a detailed treatment of solving FFLP problem is presented. In Section 6, the example illustrated the fact that the developed method can be successfully applied. Weakness of the existing method, and advantages and limitation of proposed method are discussed in Section 7 and conclusion is drawn in Section 8. Throughout this paper R stands for set of all real numbers, F(R) stands for the set of all fuzzy numbers on R, F T ( R) stands for the set of all symmetric triangular fuzzy number. Let A is a fuzzy number and P is its membership function and T ( A ) is its A

NSTFNA preserving the expected interval.

3

Preliminaries

This section is devoted to mathematical preliminaries, which will be used further in this paper. The basic definitions involving fuzzy sets, fuzzy numbers and operations on fuzzy numbers are outlined. Definition 1: A fuzzy subset A of the real line R with membership function P A : R o [0,1] is called fuzzy number if 1

A is normal, i.e. there exists an element x0 such that P A ( x0 ) 1.

Solving FLP problem using symmetric fuzzy number approximation 2

325

A is fuzzy convex, i.e. P A (O x1 (1 O ) x2 ) t P A ( x1 ) P A ( x2 ), for all x1 , x2 R for

all O [0,1]. 3

P A is upper semi-continuous.

4

Support of A is bounded, i.e. supp ( A )

cl({x R : P A ( x ) ! 0}) and cl is the closer

operator. The core of a fuzzy set A is defined by Core( A ) {x R : P A ( x) 1}. A required tool for dealing fuzzy numbers is their r-cuts The r-cut of a fuzzy number A is defined as Ar

^ x R : P A ( x) t r`

(1)

A family { Ar : r (0,1]} is a set representation of the fuzzy number A . According to the definition of fuzzy number, it is seen at once that every r-cut of a fuzzy number is a closed interval. Hence, we have Ar

[ A L (r ),

AU (r )]

(2)

where A L (r )

^

`

inf x R : P A ( x) t r ,

AU (r )

^

`

sup x R : P A ( x) t r .

3.1 Triangular fuzzy number A triangular fuzzy number A is denoted by A

(a (1) , a (2) , a (3) ) where a (1) is Core( A ) ,

a (2) is the left fuzziness and a (3) is the right fuzziness, then the membership function can be defined as,

P A ( x)

x a (1) a (2) , a (1) a (2) d x d a (1) ° (2) a ° °° a (1) a (3) x , a (1) d x d a (1) a (3) ® (3) a ° °0 otherwise ° °¯

The parametric form of a triangular fuzzy number is represented by A

ª a (1) a (2) (1 r ), a (1) a (3) (1 r )º ¬ ¼

If a (2)

a (3) , then the triangular fuzzy number A is called symmetric triangular fuzzy number. It is denoted as A ( x0 , V ) , where x0 is Core( A ) and V is the left and right width of A . The addition and scalar multiplication of fuzzy numbers are defined by the

extension principle as follows:

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Definition 2: If A [ A L (r ), AU (r )] and B [ B L (r ), BU (r )] are arbitrary fuzzy triangular fuzzy numbers, then the addition and scalar multiplication are defined by 1 A B (r ) A (r ) B (r ), A B (r ) A (r ) B (r )

2

L KA L (r )

L

L

KA U (r )

KA L (r ),

U

U

U

KAU (r ), where K ! 0

Definition 3: If A [ A L (r ), AU (r )] and B [ B L (r ), BU (r )] are arbitrary fuzzy triangular fuzzy numbers, then the multiplication is defined by

hL (r ),

h

AB

hU (r )

where

^ max ^ A L (r ) B L (r ),

` `

hL (r )

min A L (r ) B L (r ), AU (r ) BU (r ), AU (r ) B L (r ), A L (r ) BU (r )

hU (r )

AU (r ) BU (r ), AU (r ) B L (r ), A L (r ) BU (r )

Suppose A (a (1) , a (2) , a (3) ) and B fuzzy numbers, then

(b(1) , b(2) , b(3) ) are the two positive triangular

ª a (1) b(1) a (1) b(2) (1 r ) a (2) b(1) (r 1) a (2) b(2) (r 1)2 , ¬ a (1) b(1) a (1) b(3) (1 r ) a (3) b(1) (1 r ) a (3)b(3) (r 1) 2 º¼

AB

Definition 4: For arbitrary fuzzy numbers A and B , their parametric form is [ A L (r ), AU (r )], [ B L (r ), BU (r )] , respectively, then the distance between A and B is defined as,

d A , B

1

2

1

³ A (r ) B (r ) dr ³ A 0

L

L

0

U

2

(r ) BU (r ) dr

(3)

Simply, D A , B

d 2 A , B

1

1

³0 AL (r ) B L (r ) 2dr ³0 AU (r ) BU (r ) 2dr

(4)

Another essential notion connected with fuzzy numbers is an expected interval EI(A) of a fuzzy number A. It is given by EI( A)

ª « ¬

³

1

0

A L (r ) dr ,

1

º AU (r ) dr » 0 ¼

³

Theorem 1: Let X , Y F n u n ( R ), then

1

Core X Y

2

Core AX

3

A X Y

Core( X ) Core(Y )

Core A Core X AY AX

Proof: The proof is obvious.

(5)


327

Theorem 2: Let A ( x0 , V1 ) and B ( y0 , V 2 ) are the symmetric triangular fuzzy number. We say A B if and only if x0 y0 or x0 y0 and V1 ! V 2 .

4

Nearest symmetric triangular fuzzy number approximation

In this section, an approximation operator T : F ( R) o F T ( R) which produces a symmetric triangular fuzzy numbers that is closest to given original fuzzy number is investigated. The symmetric triangular fuzzy number has identical expected interval as the original one. Therefore, this operator will be called as the NSTFNA operator preserving the expected interval. Suppose A is a fuzzy number and [ A L (r ), AU (r )] is its r-cut. To find a symmetric triangular fuzzy number T ( A ) , this is nearest to A with respect to metric ‘d’ defined in Equation (5). Let [(T ( A ) L (r )), (T ( A )U (r ))] denote the r-cut of T ( A ) . Thus, we have to minimise d ( A ), T ( A )

1

³ A (r) T ( A ) (r ) L

0

L

2

dr

1

³ A

U

0

(r ) T ( A )U (r )

2

dr

(6)

It is easily seen that the r-cut of T ( A ) is equal to [ x0 V V r , x0 V V r ]. Therefore, Equation (6) reduces to

d A , T ( A )

1

³ A (r ) x L

0

0

2

V V r dr

1

³ A 0

U

2

(r ) x0 V V r dr

(7)

and we try to minimise Equation (7) with respect to x0 and V. However, we want to find an NSTFNA which is not only closest to given fuzzy number, but also which preserves the expected interval of that fuzzy number. Thus our problem is to find such real number x0 and V t 0 that minimise Equation (7) with respect to the condition

EI T ( A )

EI( A )

(8)

By Equation (5), we can write Equation (8) as follows:

V Vº ª «¬ x0 2 , x0 2 »¼

ª « ¬

³

1

0

A L (r ) dr ,

³

1

0

º AU (r ) dr » ¼

(9)

It is easily seen that to minimise d ( A, T ( A)), it is enough to minimise the function f ( x , V ) d 2 ( A , T ( A )) with respect to the following conditions: 0

x0

V 2

³

1

0

A L (r ) dr

0 and

x0

V 2

³

1

0

AU (r ) dr

0

328


Using Lagrangian multipliers method, the problem is reduced to minimise: 1

³ A (r ) x

g (t )

L

0

V § O1 ¨ x0 2 ©

2

V V r dr

0

1

³ A

U

0

1

V · § A L (r ) dr ¸ O2 ¨ x0 2 2 ¹ ©

³

2

(r ) x0 V V r dr 1

· AU (r )dr ¸ 2 ¹

³

where O1 and O 2 are the real numbers, called the Lagrangian multipliers and t To minimise Equation (11), we consider wg wx0

2

wg wV

1

³ A (r ) x L

0

V V r dr 2

0

1

³ A 0

U (r )

x0 V V r dr O1 O2

(10)

( x0 , V ).

0 (11)

1

³ A (r ) x V V r (1 r ) dr ³ A (r ) x V V r (1 r ) dr L

0

0

1

U

0

wg wO1

x0

wg wO 2

x0

V 2

V 2

0

³

1

0

³

1

0

(12) 0

A L (r ) dr

0

(13)

AU (r ) dr

0

(14)

By solving the above Equations (11)–(14), the solution is x0

V

1 2

³

1

0

1 A L (r )dr 0 2

³

1

AU (r ) dr

³

1

0

³

1

0

AU (r ) dr

A L (r ) dr

(15) (16)

Moreover, we can get the Hessian matrix ª§ 2 w g x0 , V · det « ¨ « © wxi wx j ¸¹ xi , x j «¬

º » » x0 ,V » ¼

16 !0 3

(17)

Equation (17) shows that x0 and V given by Equations (15) and (16) actually minimise d 2 ( A , T ( A )) and simultaneously minimise d ( A , T ( A )). Moreover, this minimum is unique, because of the shape of the function d ( A , T ( A )) is also global. Therefore, the symmetric triangular fuzzy number T ( A ) with x0 and V given by Equations (15) and (16) is indeed the NSTFNA of fuzzy number A with respect to metric ‘d’ preserving the expected interval.


5

329

Solving FFLP problem using symmetric triangular fuzzy number approximation

This section studies a linear programming problem with fuzzy coefficients and fuzzy parameters. To describe this problem, we consider the following linear programming problem in the conventional form: Max Z C T X ° AX b ®s.t. °X t 0 ¯

(18)

where C and X are the n-dimensional column vectors, A is an m u n(m d n) matrix and b is an m-dimensional column vector. In this model, all the coefficients of A, b and C are crisp numbers, and each constraint must be satisfied strictly. However, in the real-world decision problems, a decision-maker does not always know the exact values of the coefficients taking part in the problem and that vagueness in the coefficients may not be a probabilistic type. In this situation, the decision-maker can model inexactness by means of fuzzy parameter. C T X b

Max Z °° ®s.t. AX ° °¯ X t 0

(19)

where X ( x(1) , x (2) , x (3) ), b C (c (1) , c (2) , c (3) ), x (1) Core( X ), b(1) x (2) , b(2) , A(2) , c (2) and x (3) , b(3) , A(3) , c (3) respectively, also A(1) A(2) t 0, Equation (19) can be written as:

(b(1) , b(2) , b(3) ), Core(b ), A(1)

A ( A(1) , A(2) , A(3) ), Core( A ), c (1) Core(C ), are the left and right margins of X , b , A , C ,

c (1) c (2) t 0

and

Therefore,

Max Z c (1) , c (2) , c (3) x (1) , x(2) , x(3) ° ° (1) (2) (3) x (1) , x (2) , x (3) b(1) , b(2) , b(3) ®s.t. A , A , A ° (1) (2) °x x t 0 ¯

b(1) b(2) t 0.

(20)

We apply the fuzzy production of two positive fuzzy asymmetric parameter numbers, and with the Equations (15) and (16), ( X 0CX , V CX ), ( X 0 AX , V AX ), ( X 0G , V b ) will be , AX and b, symmetric triangular fuzzy number with preserve expected interval of CX respectively, and that are derived from the following relations:

V CX

1

³ c x c x (1 r ) c x (1 r ) c x (1 r ) dr ³ c x c x (1 r ) c x (1 r ) c x (1 r ) dr (1) (1)

(1) (3)

(3) (1)

(3) (3)

2

0

1

(1) (1)

0 (2) (1)

c

x 2

(1) (2)

(2) (1)

(2) (2)

c (3) x(1) c (1) x (2) c (2) x (2) c (1) x (3) c(3) x (3) 2 2 3 2 3

2

330

C. Veeramani and C. Duraisamy X0

CX

1 2

1

³ c x c x (1 r ) c x (1 r) c x (1 r ) dr 1 ³ c x c x (1 r ) c x (1 r ) c x (1 r ) dr 2 (1) (1)

(1) (2)

(2) (1)

(2) (2)

2

0

1

(1) (1)

(1) (3)

(3) (1)

(3) (3)

2

0

c (1) x (1)

c (3) x(1) c (2) x (1) c(2) x(2) c (1) x (2) c(1) x(3) c (3) x(3) 4 4 6 4 4 6

And for the constrains we have

Vk AX

1

³ A x ³ A

(1) (1)

0

1

(1) (1)

x

0 (2) (1)

A(1) x (3) (1 r ) A(3) x (1) (1 r ) A(3) x (3) (1 r )2 dr

A(1) x (2) (1 r ) A(2) x (1) (1 r ) A(2) x(2) (1 r ) 2 dr

A x A(3) x (1) A(1) x(2) A(2) x (2) A(1) x (3) A(3) x (3) 2 2 2 3 2 3 1 1 X0 A(1) x (1) A(1) x(2) (1 r ) A(2) x (1) (1 r ) A(2) x (2) (1 r ) 2 dr AX 2 0 1 1 (1) (1) A x A(1) x (3) (1 r ) A(3) x (1) (1 r ) A(3) x (3) (1 r )2 dr 2 0 A(3) x (1) A(2) x(1) A(2) x (2) A(1) x (2) A(1) x (3) A(3) x (3) A(1) x (1) 4 4 6 4 4 6 For the right-hand side of the constrains, we have

³

³

V b X0 b

1

³ b

(1)

0

1 2

1

³ b 0

b(3) (1 r ) dr (1)

1

³ b

(1)

0

b(2) (1 r ) dr

1 2

1

³ b 0

b(2) b(3) 2

b(2) (1 r ) dr (1)

b(3) (1 r ) dr

b(2) b(3) 4 4 We could view the problem (20) as multi-objective linear programming problem. The problem (21) is designed in which the centre objective value will be calculated. The problem (22) is designed to obtain the margin from the objective of the principal problem. For maximising the objective function of the problem (19), we must solve the problems (21) and (22). d b, x(1) x (2) t 0, X F n ( R) X / AX Suppose that S b(1)

^

`

We know the preference of the core of solution respect to margin is ordinal and then by applying the lexicography rule, we will have the following formal representation for the centre and fuzziness problems: ° Max ® °¯ s.t.

P*

F0 X X S

(21)


331

and the problem of fuzziness is as follows: Max ° ®s.t. ° ¯

Z F1 ( X ) X S X 0k CX

(22)

P*

where P * is the fuzzy optimal value of the objective function (21). The last condition X 0CX P * is guarantee for satisfying the optimal solution of Equation (22) in Equation (21). The problems (21) and (22) can be written as follows:

c (3) x (1) c (2) x (1) c(2) x (2) c (1) x (2) c(1) x(3) c (3) x (3) 4 4 6 4 4 6 (3) (1) (2) (1) (2) (2) (1) (2) (1) (3) (3) (3) A x A x A x A x A x A x s.t. A(1) x (1) 4 4 6 4 4 6 (2) (3) b b d b(1) 4 4 (2) (1) (3) (1) A x A x A(1) x (2) A(2) x (2) A(1) x (3) A(3) x(3) b(2) b(3) d 2 2 2 3 2 3 2 x (1) x (2) t 0, x (2) t 0, x (3) t 0 Max P*

c (1) x (1)

(23)

and c (2) x (1) c(3) x (1) c (1) x (2) c (2) x (2) c (1) x (3) c (3) x (3) 2 2 2 3 2 3 (3) (1) (2) (1) (2) (2) (1) (2) (1) (3) A x A x A x A x A x A(3) x (3) s.t. A(1) x (1) 4 4 6 4 4 6 (2) (3) b b d b(1) 4 4 (2) (1) (3) (1) A x A x A(1) x(2) A(2) x (2) A(1) x (3) A(3) x (3) b(2) b(3) d 2 2 2 3 2 3 2 (1) (2) x x t 0, Max

(24)

x (2) t 0, x(3) t 0 X 0k P * CX

where P* is the optimal value of Equation (23). If problem (23) has a unique optimal solution, then we have obtained the optimal solution of problem (20), otherwise that problem (23) has alternative optimal solutions; we solve Equation (24) on the optimal solutions set of the problem (23). If the problem (23) has alternative solution such that, there are more solutions with unique objective function value it means that we derive more fuzzy solutions which their cores are the same; therefore, for ranking the solutions in the basis of Theorem 2, we must solve the problem related to margin.

332


Remark 1: Problem (20) is reduced to problems (23) and (24). Theorem 3: If X ( x(1) , x(2) , x(3) ) is an optimal solution of problems (23) and (24), then X ( x(1) , x(2) , x(3) ) is a optimal solution of problem (20). Proof: Suppose, if X * ( x (1) , x (2) , x (3) ) is an optimal solution of problems (23) and (24), but it is not optimal solution of problem (20). By Theorem 2, there exists a feasible ! X 0 k ) (V k0 d V CX solution of problem (20), say X 0 , such that ( X 0 k* ) and we CX

CX *

k CX 0

j know that X 0 is a feasible solution of Equations (23) and (24) as well, that is more X * and this is contradiction or ( X 0 k t X 0 k ) (V k0 V CX desirable than k k* ) that CX 0

CX *

CX

with the above analysis, we confronting with a contradiction. Remark 2: The problem (20) is unbounded if and only if the problem related to centre is unbounded.

6

Numerical example

In this section, the proposed method is illustrated by solving a numerical example. In the real-life problems, the following situation may occur: if a company wants to launch new product in the market, then there may exist uncertainty about the profit (or cost) and availability (or demand) of the product. In such a situation, the profit (or cost) availability (or demand) and products may be represented by triangular fuzzy numbers. Consider the following FFLP problem Max Z (2,1,1) X (3,1,1) X 1

2

s.t. (1,1,1) X 1 (2,1,1) X 2 d (10,9,17) (2,1,1) X (1,1,1) X d (11,9,17) 1

(25)

2

X 1 , X 2 t 0

Using the method described in Section 5, the above FLPP is reduced to two crisp problems. The first problem is related to core which is as follows: 1 2 7 11 2 x1(1) x1(2) x1(3) 3 x2(1) x2(2) x2(3) 3 3 12 12 1 5 1 2 s.t. x1(1) x1(2) x1(3) 2 x2(1) x2(2) x2(3) d 12 12 12 3 3 1 2 1 5 2 x1(1) x1(2) x1(3) x2(1) x2(2) x2(3) d 13 3 3 12 12 1 (2) 5 (3) 2 (2) 4 (3) (1) (1) x1 x1 x1 x2 x2 x2 d 13 6 6 3 3 2 4 1 5 x1(1) x1(2) x1(3) x2(1) x2(2) x2(3) d 13 3 3 6 6 Max

P*

x1(1) x1(2) t 0, x2(1) x2(2) t 0

(26)

Solving FLP problem using symmetric fuzzy number approximation Solving the problem (26), the solution is P*

X 1

(4.67, 0, 0), X 2

333

(3.67, 0, 0) and

20.33. The second problem is related to margin which is as follows: 2 (2) 7 4 x x1(3) x2(1) x2(2) x2(3) 3 1 6 3 1 5 1 2 s.t. x1(1) x1(2) x1(3) 2 x2(1) x2(2) x2(3) d 12 12 12 3 3 1 2 1 5 2 x1(1) x1(2) x1(3) x2(1) x2(2) x2(3) d 13 3 3 12 12 1 5 2 4 x1(1) x1(2) x1(3) x2(1) x2(2) x2(3) d 13 6 6 3 3 2 (2) 4 (3) 1 (2) 5 (3) (1) (1) x1 x1 x1 x2 x2 x2 d 13 3 3 6 6 1 2 7 11 2 x1(1) x1(2) x1(3) 3 x2(1) x2(2) x2(3) 20.33 3 3 12 12 Max

z

x1(1)

x1(1) x1(2) t 0,

(27)

x2(1) x2(2) t 0

Solving the problem (27), the solution is X 1 (4.67, 0, 0), X 2 (3.67, 0, 0) and z 8.37. If we substitute the derived solution from the problem (26) related to core and the problem (27) related to margin into the objective function of the principal problem. Hence, the approximate optimal fuzzy solution of the principal problem is (20.33, 8.37, 8.37). Figure 1 indicates the membership function of fuzzy optimal solution of FFLP problem. Figure 1

Fuzzy approximate solution (see online version for colours)

334

7


Advantages and limitations of the proposed method

In this section, we discussed the advantages of the proposed method over the existing methods for solving FFLP problems, and further limitations of the proposed method are also discussed. The method discussed by Dehghan et al. (2006) for finding the exact solution of full fuzzy linear system is applicable only for non-negative fuzzy numbers. But this method is not applicable to negative fuzzy numbers which are often occur while modelling science and engineering problems. Lotfi et al. (2009) gave a new approach for solving the FFLP problem with equality constrain by converting the non-symmetric fuzzy numbers to a symmetric fuzzy numbers. In this method, while converting non-symmetric into a symmetric fuzzy number, the expected interval is not preserved. Kumar et al. (2011) presented a novel approach of converting FFLP problem into a LP problem by using ranking function R( A ) (a 2b c) / 4 where A (a, b, c), a d b d c. This method is not successful over FFLP problem with symmetric fuzzy number coefficients. The proposed method overcome all the difficulties discussed earlier. It can be used for FFLP problem with negative fuzzy numbers and inequality constrains. This method preserves expected interval while converting non-symmetric into a symmetric fuzzy numbers. By preserving the expected interval, we get more accurate results and the solutions are interpreted very easily. Basically, when we transform non-symmetric fuzzy number into symmetric fuzzy number, there may be a loss of information. But this loss of information will not highly affect the results. Moreover, the proposed method gives only an approximate solution.

8

Conclusion

In this paper, we have proposed a new method for solving the FFLP problems. Here, all the parameters and variables are triangular fuzzy numbers. The concept of NSTFNA is used and the FFLP problem is reduced to two linear programming problems. The fuzzy approximate solution has been obtained of the FFLP problem by solving the two linear programming problems. We gave an illustration for this method through an example. In future, the proposed work can be extended for solving problems, whose parameters and variables are trapezoidal fuzzy numbers, fully fuzzy multi-objective linear programming problem, fully linear fractional programming problems and fully fuzzy multi-objective linear fractional programming problems. They are under investigation.

Acknowledgements The authors would like to thank the Editor-in-Chief and anonymous referees for the various suggestions which have led to an improvement in both the quality and the clarity of this paper.


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