A Primal-dual Simplex Algorithm for Solving Linear Programming Problems with Symmetric Trapezoidal Fuzzy Numbers

Two existing methods for solving a class of fuzzy linear programming (FLP) problems involving symmetric trapezoidal fuzzy numbers without converting them to crisp linear programming problems are the fuzzy pri-mal simplex method proposed by Ganesan and Veeramani [1] and the fuzzy dual simplex method proposed by Ebrahimnejad and Nasseri [2]. The former method is not applicable when a primal basic feasible solution is not easily at hand and the later method needs to an initial dual basic feasible solution. In this paper, we develop a novel approach namely the primal-dual simplex algorithm to overcome mentioned shortcomings. A numerical example is given to illustrate the proposed approach.


Introduction
In optimizing real world systems, one usually ends up with a linear or nonlinear programming problem.For many cases, the coefficients involved in the objective and constraint functions are imprecise in nature and have to be interpreted as fuzzy numbers to reflect the real world situation.The resulting mathematical problem is therefore referred to as a fuzzy mathematical programming problem.After the pioneering work on fuzzy linear programming by Tanaka et al. [3,4] and Zimmermann [5], several kinds of fuzzy linear programming problems have appeared in the literature and different methods have been proposed to solve such problems [6][7][8][9][10][11][12].One important class of these methods that has been highlighted by many researches is based on comparing of fuzzy numbers using ranking functions.Based on this idea, Maleki et al. [13] proposed a simple method for solving fuzzy number linear programming (FNLP) problems.They also applied an special kind of FNLP problems, involving fuzzy numbers only in objective function, as an auxiliary problem for solving fuzzy variable linear programming (FVLP) problems.Ebrahimnejad et al. [14] developed their method for solving bounded linear programming with fuzzy cost coefficients.Then Mahdavi-Amiri and Nasseri [15] used the certain linear ranking function to define the dual of FNLP problem as a similar problem that lead to an efficient algorithm called the dual simplex algorithm [16] for solving FNLP problems.Based on these algorithms, Ebrahimnejad [17] investigated the concept of sensitivity analysis in FNLP problems.Of course, Mahdavi-Amiri and Nasseri [18] and Mahdavi-Amiri et al. [19] proposed two efficient algorithms for solving FVLP problems directly without need of any auxiliary problem.Moreover, Nasseri and Ebrahimnejad [20] suggested the fuzzy primal simplex method to solve the flexible linear programming problems directly without solving any auxiliary problem.Then, Ebrahimnejad et al. [6] gave another efficient method namely primal-dual simplex method to obtain the fuzzy solution of FVLP problems.Ebrahimnejad and Nasseri [21] used the complementary slackness for solving both FNLP problem and FVLP problem.Hosseinzadeh Lotfi et al. [9] discussed full fuzzy linear programming (FFLP) problems of which all parameters and variable are triangular fuzzy numbers.They used the concept of the symmetric triangular fuzzy number and proposed an approach to defuzzify a general fuzzy quantity.After that Kumar et al. [11] proposed a new method to find the fuzzy optimal solution of same type of fuzzy linear programming problems.
Recently Ganesan and Veeramani [1] introduced a new method based on primal simplex algorithm for solving linear programming problem with symmetric trapezoidal fuzzy numbers without converting them to crisp linear programming problems.Ebrahimnejad et al. [7] extended their method for situations in which some or all variables are restricted to lie within fuzzy lower and fuzzy upper bounds.After that, Nasseri and Mahdavi-Amiri [22] and Nasseri et al. [23] developed the concept of duality of such problems that led to a new method based on dual simplex algorithm [2].However, dual simplex algorithm begins with a basic (not necessarily feasible) dual solution and proceeds by pivoting through a series of dual basic fuzzy solution until the associated complementary primal basic solution is feasible.In this paper, we describe a new method for solving linear programming problem with symmetric trapezoidal fuzzy numbers, called the primal-dual algorithm, similar to the dual simplex method, which begins with dual feasibility and proceeds to obtain primal feasibility while maintaining complementary slackness.An important difference between the dual simplex method and the dual simplex method is that the primal-dual simplex method does not require a dual feasible solution to be basic.This paper is organized as follows: In Section 2, we give some necessary concepts of fuzzy set theory.A review of linear programming problems with symmetric trapezoidal fuzzy numbers and two methods for solving such fuzzy problems are given in Section 3. We develop and present a fuzzy primal-dual algorithm to solve the fuzzy linear programming problems in Section 4 and explain it by an illustrative example.Finally, we conclude in Section 5.

Preliminaries
In this section, we review the fundamental notions of fuzzy set theory (see [1,7,24]).
Definition 2.1.A fuzzy number on (real line) is said to be a symmetric trapezoidal fuzzy number if there exist real numbers , , t ab ab a b a b . 2 = ma t From the above definition it can be seen tha he relations  and  as given below: n cases (2) and (3), we also write Note that i a b    num-an d say that a  and b  are equivalent.
Remark 2.1.Two symmetric trapezoidal fuzzy bers , if there exist we define 0 a    0 . We also Note that 0  is equivalent to   0, 0, 0 = 0 .Natural , one may c r The following lemma immediately follows De- results ar

Lem
 sym uzzy 2) The relation r order r symmetric trape idal fuzzy numbers.
3) For any two symmetric trapezoidal fuzzy numbers Here, we first review these new no b useful in our further consideration.After that we review the concept of duality for such problems proposed by Nasseri and Mahdavi-Amiri [22] and Nasseri et al. [23].

A Fuzzy Primal Simplex Algorithm efinition A linear programming problem with tra-
are given and is a feasible solution to (1) if and only if is to be deter nition 3.2.We say that a f where the parameters of the problem are as defined in (1).x  , where , then stop solution is optim , go to (2). 2) Let here T is the index set of the current nonbasic variab If ; the current 0 and determine the index r as follows: 3) Update the tableau by pivoting at fuzzy basic and nonbasic variables where x  leaves the basis, and go t

A Fuzzy Dual Simplex Algorithm
) is (5) is including the fuzzy variab nstraints of problem (1).In fact, the roblem as the DFLP problem.
shall discuss here the re is defined f oblem (1).We name this p or the ith constraint of pr We lationships between the FLP problem and its corresponding dual and omit the proof asseri and Mahdavi-Amiri [22] and Nasseri et al. [23]).Theore to FLP and DFLP problems, respectivel x  is fuzzy optimal solution of the primal problem then the fuzzy vector * = w c B 1 , where B is the optimal basis, is a fuzzy optimal solution of the dual problem).
Theorem 3.6.(Complementary slackness).Suppose * x  and * w  are feasible solutions of the FLP problem and its corresponding dual, the DFLP problem, respectively.Then * x  and w  are respectively optimal if and only if Ebrahimnejad and Nasseri [2] using the above results, introduced a new fuzz dual algorithm for solving problem (2).Algorit y hm 3.2.A fuzzy dual simplex algorithm Initialization step Suppose that basis be dual feasible for the problem (2).Form the Tableau 3.1 as an initial dual feasible simplex tableau.Suppose   for all j .

Main step 1) Suppose
 , then Stop; the current fuzzy solution is optimal.
Else suppose for all Stop pivot column by t ng test: e note that the method which is propo by Ganesan and Veeramani in [1], starts with a fuzzy basic feasible solution for FLP and moves to an optimal walking through a sequence of fuzzy feasib FLP.All the bases with the possible excep ptimal basis obtained in fuzzy primal simplex algorithm o don't satisfy the optimality criteria for FLP or feasibility condition for DFLP.But their method has no efficient when a primal basic feasible solution is not at hand.So, Ebrahimnejad and Nasseri [2] developed a new dual simplex algorithm to overcome this shortcoming by using the duality results which have been proposed by Nasseri and Mahdavi-Amiri [22] and Nasseri et al. [23].This algorithm starts with a dual basic feasible solution, but primal basic infeasible solution and walks to an optimal solution by moving among adjacent dual basic feasible solutions.However, the dual simplex method for solving FLP problem needs to an initial dual basic feasible solution.Here, we develop the fuzzified version of conventional primal-dual method of linear programming problems that any dual feasible solution, whether basic or not, is adequate to initiate this method.
Corollary 4.1.[22,23] The optimality criteria 0  , for all j , for the FLP problem is equivalent to the feasibility condition for the DFLP problem.
To explain the main strategy employed by this method, we consider the following standard FLP: be the row of dual vector variables.The Dual of FLP problem (6)   0 = 1, , ŵ  ose dual feasible solution.Now consider the fol- . ng problem known as the restricted fuzzy oblem corresponding to ŵ  .min 0 1 x    , x  and ŵ  are optimal to (6) and (7), respectively.If 0 0 x    , let v  be the optimal solution to the dual of the restricted zzy primal problem (9).ow, we construct the new fuzzy dual solution for ( 6) such that all th basic imal variables in the restricted fuzzy primal problem remain in the new restricted fuzzy primal problem and in addition, at least one riables that did not belong to the set  would get to res icted fuzzy primal problem.Furthermore, this variable would reduced 0 In order to construct such a dual fuzzy vector, consider the following fuzzy dual w  , where > 0 Now, if ˆj x  with j   is a basic variable in the restricted fuzzy primal pro m, then omp ble from c l acknes ementary sl s va 0 j   a ce nd hen 0 , per primal problem.If mitting j j in the new restricted fuzzy    and ˆj va   0  , then 0 with at least one compone equal t y zero.First, we show in this case for each j there is an . That is, i e choose j  as above, then 0 j j wa c      .Now, we define  as follows: In ad e show dition, w 0  odifying the dual fuzzy vector leads to a new feasible dual fuzzy Also, all the variables that belonged to the ed fuzzy primal problem basis are passed to the new basis.In addition, a new fuzzy variable m solution.restrict k x  t can e basis, is passed to th fu lem.Thus, we continu present restricted fuzzy primal prob hat is a didate to enter th e restricted zzy primal prob e from the lem basis by entering k x  , which leads to a potential reduction in 0 x  .This process is continued until 0 0 x    in which case we have an optimal solution or else 0 0 x    and ˆ0 j va    all j for    .We explain this case as Theorem 4.1 as below.

Els
l dual fuzzy solution to the restricted fuzzy primal pro 3)  , , , , , 0  , , , , , 0 The dual problem is given by the following: Now we solve the FLP problem ( 13) by the fuzzy primal-dual simplex algorithm.
Initialization step: The initial fuzzy dual solution is given by  min 1,1,0,0  where 8 x  and 9 x  are the artificial fuzzy variables.Solving this problem by Ganesan's met optimal fuzzy solution and the optima value as follows: hod [7] gives the l objective fuzzy x  So, we have , and .Thus, Complementary slackness gives the dual solution as  is determined as follows: and we replace 3 1,5,1,1 , , , 0 The optimal fuzzy solution to this problem is given by x and the fuzzy artificial fuzzy variables 8 x and 9 x , which minimize the sum of the artificial fu va les to obtain a initial fuzzy basic feasible solution: zzy riab     , , , , , , , , 0 x x x x x x x x x ,  n t ly at hand.

Conclusions
Ganesan and Veeramani in [1] proposed a new approach based on primal simplex algorithm to obtain the fuzzy solution of fuzzy linear programming problem sy ased on the interesting been established by Ganesan and ch can be expected to be After finding a initial fuzzy basic feasible solution by solving the problem (16), we must minimize the original objective fu ction of he problem (11).So, this process is time consuming and has no efficiency computationally for solving such problems in which an initial fuzzy basic solution is not easi with mmetric trapezoidal fuzzy numbers without converting them to crisp linear programming problems.In this paper, we reviewed the dual of a linear programming problem with symmetric trapezoidal fuzzy numbers.Then, we introduced a fuzzy primal-dual algorithm for solving the FLP problems directly without converting them to crisp near programming problems, b li results which have eeramani [1].This approa V efficient if an initial dual fuzzy solution can be computed readily.This algorithm is also useful specially for solving minimum cost flow problem with fuzzy parameters in which finding an initial dual feasible solution turns out to be a trivial task.However, development of network primal-dual simplex algorithm for solving such problem in fuzzy environment may also produce intersecting results.

Acknowledgments
The author would like to thank Prof. Tian Huang, Editorial Assistant in AM Editorial Board, and anonymous referees for the various suggestions which have led to an improvement in both the quality and clarity of the paper.Finally, the author greatly appreciates to the office of vice chancellor for research of Islamic Azad University-Qaemshahr Branch.
We denote a symmetric trapezoidal fuzzy number a  by a  and b are given by (taken from[1]): depending upon the need, one can also use a smaller  in the definition of multiplication involving symmetric trapezoidal fuzzy numbers.

x
 and 0 w  are fuzzy feasible so-

5 .
any one of the FLP or DFLP problem is unbounded, then the other problem has no 0 (Strong duality.)If any one of the FLP or DFLP problem has an fuzzy optimal solution, then both problems have fuzzy optimal solutions and the two optimal objective fuzzy values are equal.(In fact, if * ) is now solved a x  .In this process, once an artificial variable ja x  dr ut of the basic variables, discard it from the problem, respectively.If 0 0 and P. Veeramani, "Fuzzy Linear Program uzzy Numbers," Annals of Op-43, No. 1, 2006, pp.305-315.

Table 1 . The initial primal simplex tableau.
dual constraint, we find that the last two dual constraints hold at equality.Thus, (11)e want to solve the problem(11)directly by use of Algorithm 3.1 proposed by and Veeramani [1], we must first solve the linear programming problem with introducing the slack variables