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Applied Mathematical Modelling 35 (2011) 4526–4540
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Applied Mathematical Modelling
journal homepage: www.elsevier .com/locate /apm
Some new results in linear programs with trapezoidal fuzzy numbers:Finite convergence of the Ganesan and Veeramani’s methodand a fuzzy revised simplex method
A. EbrahimnejadIslamic Azad University, Qaemshahr Branch, Department of Mathematics, Qaemshahr, Iran
a r t i c l e i n f o a b s t r a c t
Article history:Received 20 September 2010Received in revised form 26 February 2011Accepted 8 March 2011Available online 17 March 2011
Keywords:Fuzzy linear programmingFuzzy primal simplex algorithmRankingTrapezoidal fuzzy number
0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.03.021
E-mail addresses: a.ebrahimnejad@srbiau.ac.ir, a
In a recent paper, Ganesan and Veermani [K. Ganesan, P. Veeramani, Fuzzy linear programswith trapezoidal fuzzy numbers, Ann. Oper. Res. 143 (2006) 305–315] considered a kind oflinear programming involving symmetric trapezoidal fuzzy numbers without convertingthem to the crisp linear programming problems and then proved fuzzy analogues of someimportant theorems of linear programming that lead to a new method for solving fuzzy lin-ear programming (FLP) problems. In this paper, we obtain some another new results for FLPproblems. In fact, we show that if an FLP problem has a fuzzy feasible solution, it also has afuzzy basic feasible solution and if an FLP problem has an optimal fuzzy solution, it has anoptimal fuzzy basic solution too. We also prove that in the absence of degeneracy, themethod proposed by Ganesan and Veermani stops in a finite number of iterations. Then,we propose a revised kind of their method that is more efficient and robust in practice.Finally, we give a new method to obtain an initial fuzzy basic feasible solution for solvingFLP problems.
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1. Introduction
Fuzzy linear programming (FLP) has been developed for treating uncertainty in the setting of optimization problems. Inrecent years, various attempts have been made to study the solution of FLP problems, either from theoretical or computa-tional point of view. The concept of fuzzy linear programming on general level was first proposed by Tanaka et al. [1] in theframework of the fuzzy decision of Bellman and Zadeh [2]. Afterwards many authors have considered various kinds of theFLP problems and have proposed several approaches for solving these problems [3–7]. Some authors have used the conceptof comparison of fuzzy numbers and linear ranking function to solve the fuzzy linear programming problems. Of course,ranking functions have been proposed by researchers to suit their requirements of the problem under consideration and con-ceivably there are no generally accepted criteria for application of ranking functions. Nevertheless, usually in such methodsauthors define a crisp model which is equivalent to the FLP problem and then use optimal solution of the model as the opti-mal solution of the FLP problem. Maleki et al. [3] using the concept of comparison of fuzzy numbers, proposed a new methodfor solving fuzzy number linear programming (FNLP) problems. Then Mahdavi-Amiri and Nasseri [4] used the certain linearranking function to define the dual of FNLP problems as FNLP problems again that lead to an efficient method called the dualsimplex algorithm [8] for solving FNLP problems. Also, Mahdavi-Amiri and Nasseri [5] proposed another approach to definedual of FNLP problems as fuzzy variable linear programming (FVLP) problems leading to a dual simplex algorithm for solvingFVLP problems. Then, Ebrahimnejad et al. [9] introduced another efficient method namely primal–dual simplex algorithm to
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A. Ebrahimnejad / Applied Mathematical Modelling 35 (2011) 4526–4540 4527
obtain a fuzzy solution of FVLP problems. Also, Ebrahimnejad and Nasseri [10] used the complementary slackness to solveFNLP and FVLP problems without the need of a simplex tableau. Then Nasseri and Ebrahimnejad [11] proposed a fuzzy pri-mal simplex algorithm for solving the flexible linear programming problem and then suggested the fuzzy primal simplexmethod to solve the flexible linear programming problems directly without solving any auxiliary problem. Hosseinzadeh Lofiet al. [12] discussed full fuzzy linear programming (FFLP) problems of which all parameters and variable are triangular fuzzynumbers. They used the concept of the symmetric triangular fuzzy number and introduced an approach to defuzzify a gen-eral fuzzy quantity. For such a problem, first, the fuzzy triangular number is approximated to its nearest symmetric trian-gular number, with the assumption that all decision variables are symmetric triangular. Kumar et al. [13] proposed a newmethod to find the fuzzy optimal solution of same type of fuzzy linear programming problems. Ebrahimnejad [14] basedon fuzzy simplex algorithms for solving fuzzy number linear programming and using the general linear ranking functionson fuzzy numbers generalized a concept of sensitivity analysis in FNLP problems.
Recently, Ganesan and Veeramani [15] introduced a new type of fuzzy arithmetic for symmetric trapezoidal fuzzynumbers and proposed a method for solving FLP problems without converting them to the crisp linear programmingproblems. Ebrahimnejad et al. [16] generalized their method for solving bounded linear programming problems withsymmetric trapezoidal fuzzy numbers. Nasseri and Mahdavi-Amiri [17] and Nasseri et al. [18] used their results to definethe dual of fuzzy linear programming. In this paper based on this new arithmetic, we obtain some other new results forFLP problems. We also prove that in the absence of degeneracy, the method proposed by Gaesan and Veermani [15]stops in a finite number of iterations. Then we propose a revised kind of their method that is more efficient and robustin practice.
This paper is organized as follows: in Section 2, we give some necessary concepts and backgrounds of fuzzy arithmetic. InSection 3, we first review the method proposed by Gaesan and Veermani [15] for solving FLP problems and then prove somenew results about these problems and give a tableau format of the fuzzy primal method. The fuzzy revised simplex algorithmto solve FLP problems is given in Section 4. We propose a method for solving FLP problems with the assumption that an ini-tial basic feasible solution is not readily available in Section 5. Finally, we conclude in Section 6.
2. Preliminaries
In this section we introduce some of the basic terminologies of fuzzy set theory and the main concepts needed in the restof the paper.
Definition 2.1. Let R be the universal set. ~a is called a fuzzy set in R if ~a is a set of ordered pairs ~a ¼ fðx;l~aðxÞÞjx 2 Rg, wherel~að:Þ is membership function of ~a and assigns to each element x 2 R, a real number l~aðxÞ in the interval [0,1].
Definition 2.2. The a-cut or a-level of a fuzzy set ~a is defined as an ordinary set ½~a�a for which the degree of its membershipfunction exceeds the level a, that is, ½~a�a ¼ fx 2 Rjl~aðxÞP ag.
Definition 2.3. The support of a fuzzy set ~a is a set of elements in R for which l~aðxÞ is positive, that is,supp ~a ¼ fx 2 Rjl~aðxÞ > 0g.
Definition 2.4. A fuzzy set ~a is called convex if for each x; y 2 R and each k 2 ½0;1�; l~aðkxþ ð1� kÞyÞP minfl~aðxÞ;l~aðyÞg.
Definition 2.5. A fuzzy set ~a is called a normal fuzzy set if supfl~aðxÞjx 2 Rg ¼ 1:
Definition 2.6. A fuzzy number is a convex normalized fuzzy set of the real line R; whose membership function is piecewisecontinuous.
Definition 2.7. An LR type flat fuzzy number [19], is denoted as ~a ¼ ðaL; aU ;a; bÞLR, if
l~aðxÞ ¼
L aL�xa
� �for aL � a 6 x 6 aL;
1 for aL6 x 6 aU ;
R x�aU
b
� �for aU
6 x 6 aU þ b;
0 else;
8>>>>><>>>>>:ð1Þ
where the symmetric non-increasing function L : ½0;1Þ ! ½0;1� is the left shape function, that L (0) = 1. Also, a right shapefunction R(.) is similarly defined as L(.).
4528 A. Ebrahimnejad / Applied Mathematical Modelling 35 (2011) 4526–4540
Trapezoidal fuzzy numbers (TRFN) are special cases of LR fuzzy numbers with L(x) = R(x) = 1 � x and the following mem-bership function:
l~aðxÞ ¼
x�ðaL�aÞa for aL � a 6 x 6 aL;
1 for aL6 x 6 aU ;
ðaUþbÞ�xb for aU
6 x 6 aU þ b;
0 else:
8>>>><>>>>: ð2Þ
Remark 2.1. If a ¼ b in trapezoidal fuzzy number ~a ¼ ðaL; aU ;a; bÞ, we obtain a symmetric trapezoidal fuzzy number, and wedenote it as ~a ¼ ðaL; aU ;a;aÞ. Also, we denote the set of symmetric trapezoidal fuzzy numbers on R by FðRÞ.
While modeling certain problems in the physical sciences and engineering, it is often observed that the parameters of theproblem are not known precisely but rather are as fuzzy numbers. Now, we define arithmetic on symmetric trapezoidal fuz-zy numbers (taken from [15]). Let ~a1 ¼ aL
1; aU1 ;a1;a1
� �and ~a2 ¼ aL
2; aU2 ;a2;a2
� �be two trapezoidal fuzzy numbers. Then the
arithmetic operations on ~a1 and ~a2 are given by:
Addition: ~a1 þ ~a2 ¼ aL1 þ aL
2; aU1 þ aU
2 ;a1 þ a2;a1 þ a2� �
.
Subtraction: ~a1 � ~a2 ¼ aL1 � aU
2 ; aU1 � aL
2;a1 þ a2;a1 þ a2� �
.
Multiplication: ~a1~a2 �aL
1þaU1
2
� �aL
2þaU2
2
� �� t;
aL1þaU
12
� �aL
2þaU2
2
� �þ t; aU
1 a2 þ aU2 a2
�� ��; aU1 a2 þ aU
2 a2
�� ��� �, where
t ¼ t2 � t1
2; t1 ¼ min aL
1aL2; a
L1aU
2 ; aU1 aL
2; aU1 aU
2
� �; t2 ¼ max aL
1aL2; a
L1aU
2 ; aU1 aL
2; aU1 aU
2
� �:
If in the definition of multiplication two symmetric trapezoidal fuzzy numbers, we let ~a ¼ ~a1 ¼ ðaL; aU ;a;aÞ and~a2 ¼ ðk; k;0;0Þ, then it can be seen that t ¼ ðaU�aLÞjkj
2 . Hence, we have:
k P 0; k 2 R; k~a ¼ ðkaL; kaU ; ka; kaÞ;k < 0; k 2 R; k~a ¼ ðkaU ; kaL;�ka;�kaÞ:
Note that depending upon the need, one can also use a smaller t in the definition of multiplication involving symmetric trap-ezoidal fuzzy numbers.
Ranking of fuzzy numbers is an important issue in the study of fuzzy set theory. Ranking procedures are also useful invarious applications and one of them will be in the study of fuzzy linear programming problems. Chu and Tsao [20] em-ployed an area between the centroid and original points to rank fuzzy numbers. However there were some problems withthe ranking method. Wang and Lee [21] indicated these problems of Chu and Tsao’s method, and then proposed a revisedmethod which can avoid these problems for ranking fuzzy numbers. Wu and Mendel [22] evaluated several ranking meth-ods, similarity measures and uncertainty measures for interval type-2 fuzzy sets using real survey data. Wang et al. [23] pro-posed a novel approach to ranking fuzzy numbers based on the left and right deviation degree (LR deviation degree). In thispaper, we focus on ranking procedure proposed by Gaesan and Veermani [15] as follows.
Definition 2.8. Let ~a1 ¼ aL1; a
U1 ;a1;a1
� �and ~a2 ¼ aL
2; aU2 ;a2;a2
� �be two symmetric trapezoidal fuzzy numbers. We say ~a1 � ~a2
if we have:
aL1 � a1
� �þ aU
1 þ a1� �
2<
aL2 � a2
� �þ aU
2 þ a2� �
2; that is
aL1 þ aU
1
2<
aL2 þ aU
2
2:
Definition 2.9. Let ~a1 ¼ aL1; a
U1 ;a1;a1
� �and ~a2 ¼ aL
2; aU2 ;a2;a2
� �be two symmetric trapezoidal fuzzy numbers. We say ~a1 � ~a2
if at least one of the following conditions be hold:
(i) aL1þaU
12 ¼ aL
2þaU2
2 ; aL2 < aL
1 and aU1 < aU
2 ,
(ii) aL1þaU
12 ¼ aL
2þaU2
2 ; aL2 ¼ aL
1; aU1 ¼ aU
2 and a1 6 a2.
Remark 2.2. Two symmetric trapezoidal fuzzy numbers ~a1 ¼ aL1; a
U1 ;a1;a1
� �and ~a2 ¼ aL
2; aU2 ;a2;a2
� �are equivalent if and
only if aL1þaU
12 ¼ aL
2þaU2
2 .
Definition 2.10. For any trapezoidal fuzzy number ~a, we define ~a � ~0, if there exist e P 0 and a P 0 such that~a � ð�e; e;a;aÞ. We also denote ð�e; e;a;aÞ by ~0. Note that ~0 is equivalent to ð0;0;0; 0Þ ¼ 0. Naturally, one may consider~0 ¼ ð0;0;0;0Þ as the zero symmetric trapezoidal fuzzy number.
A. Ebrahimnejad / Applied Mathematical Modelling 35 (2011) 4526–4540 4529
Remark 2.3. If ~x � ~0, then ~x is said to be a zero symmetric trapezoidal fuzzy number. It is to be noted that if ~x ¼ ~0, then ~x � ~0,but the converse need not be true. If ~x 6� ~0 (that is ~x is not equivalent to ~0), then it is said to be a non-zero symmetric trap-ezoidal fuzzy number. It is to be noted that if ~x 6� ~0, then ~x – ~0, but the converse need not be true. If ~x � ~0ð~x � ~0Þ and ~x 6� ~0,then is said to be a positive (negative) symmetric trapezoidal fuzzy number and is denoted by ~x � ~0ð~x � ~0Þ. Now if~a1; ~a2 2 FðRÞ, it is easy to show that if ~a1 � ~a2, then ~a1 � ~a2 � ~0.
The following lemma immediately follows form the definition of the mentioned arithmetic operations on symmetrictrapezoidal fuzzy numbers.
Lemma 2.1. If ~a1; ~a2 2 FðRÞ, and c 2 R such that c – 0, then
(i) ~a1~a2 � ~a2~a1.(ii) cð~a1~a2Þ � ðc~a1Þ~a2 � ~a1ðc~a2Þ.
The two following results are taken from Gaesan and Veermani [15] and we omit the proofs.
Lemma 2.2. For any symmetric trapezoidal fuzzy number ~a1; ~a2 and ~a3, we have:
(i) ~a3ð~a1 þ ~a2Þ � ð~a3~a1 þ ~a3~a2Þ.(ii) ~a3ð~a1 � ~a2Þ � ð~a3~a1 � ~a3~a2Þ.
Lemma 2.3. If ~a1; ~a2 2 FðRÞ, then
(i) The relation � is a partial order relation on the set of symmetric trapezoidal fuzzy numbers.(ii) The relation � is a linear order relation on the set of symmetric trapezoidal fuzzy numbers.
(iii) For any two symmetric trapezoidal fuzzy numbers ~a1 and ~a2, if ~a1 � ~a2 then ~a1 � ð1� kÞ~a1 þ k~a2 � ~a2, for all k; 0 6 k 6 1.
Here we give some new results (see also Ebrahimnejad et al. [16]).
Lemma 2.4. If ~a1; ~a2 2 FðRÞ, then
(i) ~a1~a2 � ~0, if and only if ~a1 � ~0 and ~a2 � ~0 or ~a2 � ~0 and ~a1 � ~0.(ii) ~a1~a2 � ~0, if and only if ~a1 � ~0 and ~a2 � ~0, or ~a1 � ~0 and ~a2 � ~0.
(iii) ~a1 � ~a2, if and only if k~a1 � k~a2, for any k < 0; k 2 R.
Proof. Straightforward. h
Lemma 2.5. Suppose ~a1; ~a2; ~a3 2 FðRÞ such that ~a1 � ~a2. We have:
(i) If ~a3 � ~0, then ~a3~a1 � ~a3~a2.(ii) If ~a3 � ~0, then ~a3~a1 � ~a3~a2.
Proof. From ~a1 � ~a2, we have ~a2 � ~a1 � ~0. Hence from Lemma 2.4, we have ~a3ð~a2 � ~a1Þ � ~0, if ~a3 � ~0, and also ~a3ð~a2 � ~a1Þ � ~0,if ~a3 � ~0. Now the results follow from Lemma 2.2. h
Lemma 2.6. Suppose ~a1; ~a2; ~a3 2 FðRÞ such that ~a1 � ~a2; ~a3 6� ~0. Then, ~a3~a1 � ~a3~a2.
Proof. Let ~a1 ¼ aL1; a
U1 ;a1;a1
� �; ~a2 ¼ aL
2; aU2 ;a2;a2
� �and ~a3 ¼ aL
3; aU3 ;a3;a3
� �. Since ~a1 � ~a2 and ~a3 6� ~0, then aL
1þaU1
2 ¼ aL2þaU
22 and
aL3þaU
32 – 0. It follows that aL
3þaU3
2
� �aL
1þaU1
2
� �¼ aL
3þaU3
2
� �aL
2þaU2
2
� �: Therefore, from Remark 2.2, we have ~a3~a1 � ~a3~a2. h
3. Fuzzy linear programming and some new results
For the linear programming problems in the crisp environment [24,25], the aim is to maximize or minimize a linear objec-tive function under linear constraints. But in many practical situations, the decision maker may not be in a position to specifythe objective and/or constraint functions precisely but rather can specify them in a fuzzy sense. In such situations, it is desir-able to use some fuzzy linear programming type of modeling so as to provide more flexibility to the decision maker. Mishraand Ghosh [26] have proposed an interactive fuzzy programming method for obtaining a satisfactory solution to bi-level
4530 A. Ebrahimnejad / Applied Mathematical Modelling 35 (2011) 4526–4540
quadratic fractional programming problems with essentially cooperative decision makers, interacting with their optimalsolutions. In that interactive method, after determining the fuzzy goals of the decision makers at both levels, a satisfactorysolution was efficiently derived by updating the minimal satisfactory levels of decision makers at upper levels with consid-erations of overall satisfactory balance between both levels. Gaesan and Veermani [15] considered a kind of linear program-ming problems where the cost coefficients in objective function, the right hand side vector and the variables are symmetrictrapezoidal fuzzy numbers, simultaneity. They named this kind of problems as fuzzy linear programming (FLP) problems andproved fuzzy analogues of some important theorems of linear programming. In this section, we review these results and thenprove some other new results.
Gaesan and Veermani [15] defined the standard form of FLP problem as follows:
max ~z ’ ~c~x;
s:t: A~x � ~b;
~x � ~0;
ð3Þ
where ~b 2 ðFðRÞÞm; ~cT 2 ðFðRÞÞn; A 2 Rmn are given and ~x 2 ðFðRÞÞn is to be determined.
Definition 3.1. Any fuzzy vector ~x 2 ðFðRÞÞn which satisfies the constraints and nonnegative restrictions of (3) is said to be afuzzy feasible solution.
Definition 3.2. Let S be the set of all fuzzy feasible solutions of (3). Any fuzzy vector ~x 2 S is said to be a fuzzy optimumsolution to (3) if ~c~x � ~c~x for all ~x 2 S, where ~c ¼ ð~c1; ~c2; . . . ; ~cnÞ and ~c~x ¼ ~c1~x1 þ ~c2~x2 þ . . .þ ~cn~xn.
Definition 3.3 (Fuzzy basic solution; [15]). Suppose ~�x ¼ ð~�x1; ~�x2; . . . ; ~�xnÞ solves A~x � ~b. If all ~�xj � ð��xj; �xj;aj;ajÞ for some �xj P 0and aj P 0, then ~�x is said to be a fuzzy basic solution. If ~�xj 6� ð��xj; �xj;aj;ajÞ for some �xj P 0 and aj P 0, then ~�x has some non-zero components, say ~�x1; ~�x2; � � � ; ~�xk; 1 6 k 6 m. Then A~�x � ~b can be written as:
a1~�x1 þ a2
~�x2 þ � � � þ ak~�xk þ akþ1ð��xkþ1; �xkþ1;akþ1;akþ1Þ þ � � � þ anð��xn; �xn;an;anÞ � ~b:
If the columns a1; a2; . . . ; ak corresponding to these non-zero components ~�x1; ~�x2; . . . ; ~�xk are linear independent, then ~�x is said tobe fuzzy basic solution.
Theorem 3.1. If the fuzzy system of linear equality constraints in nonnegative variables of the FLP (3) has a fuzzy feasible solution,it also has a fuzzy basic feasible solution.
Proof. If ~b � ~0, then ~x � ~0 is a fuzzy basic solution of (3) by definition. If ~b 6� ~0, then ~x � ~0 is not fuzzy feasible to (3). In thiscase, suppose ~�x ¼ ð~�x1; ~�x2; . . . ; ~�xnÞ is a fuzzy feasible solution, where ~�xj ¼ ~�xL
j ;~�xU
j ;aj;aj
� �. Then A~�x �
Pnj¼1aj
~�xj � ~b. Also, withoutloss of generality suppose faj : ~�xj � ~0g ¼ fa1; a2; � � � ; akg. Hence, ~�xj � ~0 for all j R f1;2; � � � ; kg. So, we have
a1~�x1 þ a2
~�x2 þ � � � þ ak~�xk � ~b: ð4Þ
Now, if fa1; a2; � � � ; akg is linear independent, ~�x is a fuzzy basic feasible solution of (3) and we have proved the theorem. If not,there must be exist real numbers y1; y2; � � � ; yk, not all zero, such that
a1y1 þ a2y2 þ � � � þ akyk ¼ 0: ð5Þ
Let ~yj ¼ ðyj; yj;0;0Þ for j ¼ 1;2; . . . ; k. Thus, from (5) we have
a1~y1 þ a2~y2 þ � � � þ ak~yk � ~0: ð6Þ
Using Eqs. (4) and (6), we get
a1ð~�x1 þ h~y1Þ þ a2ð~�x2 þ h~y2Þ � � � þ akð~�xk þ h~ykÞ � ~b: ð7Þ
where h is a real number.Define the fuzzy vector ~�xðhÞ ¼ ð~�x1ðhÞ; ~�x2ðhÞ; . . . ; :~�xnðhÞÞ, where
~�xjðhÞ ¼~�xk þ h~yk for j ¼ 1;2; . . . ; k;~0 for j ¼ kþ 1; kþ 2; . . . ;n:
(ð8Þ
A. Ebrahimnejad / Applied Mathematical Modelling 35 (2011) 4526–4540 4531
Clearly, ~xðhÞ satisfies A~x � b. Define
h1 ¼max16j6k
~�xLj þ ~�xU
j
yjjyj > 0
( );
h2 ¼ min16j6k
~�xLj þ ~�xU
j
yjjyj < 0
( ):
Then for all values of h satisfying h1 6 h 6 h2; ~�xðhÞ is a fuzzy feasible of (3). Since ðy1; y2; . . . ; ykÞ – ð0;0; . . . ;0Þ, at least one ofthe h1 or h2 must be finite. Let, d be either h1 or h2, which is finite. Clearly ~�xðdÞ is a fuzzy feasible solution in which at mostðk� 1Þ of the fuzzy variables are fuzzy positive. Hence, starting with a feasible solution ~�x in which k variables are fuzzy po-sitive, we have constructed another fuzzy feasible solution ~�xðdÞ in which at most ðk� 1Þ variables are fuzzy positive. Either~�xðdÞ is fuzzy basic feasible solution, in which case we are done, or we can apply the same procedure on it and construct an-other fuzzy feasible solution of (3) in which the number of fuzzy positive variables is at least on less than the correspondingnumber of ~�xðdÞ. When this procedure is repeated, we are guaranteed to find a fuzzy basic feasible solution of (3) after at mostðk� 1Þ applications of the procedure. h
Remark 3.1. Consider the fuzzy system of constraints (3) where A is a matrix of order ðm nÞ and rankðAÞ ¼ m. Any ðmmÞmatrix B formed by m linearly independent columns of A is known as a basis for this fuzzy system. The column vectors of Aand the fuzzy variables in the problem, can be partitioned into the basic and the nonbasic part with respect to this basis B.Each column vector of A, which is in the basis B, is known as a basic column vector. All the remaining column vectors of A arecalled the nonbasic column vectors.
Remark 3.2. Let ~xB be the vector of the variables associated with the basic column vectors. The variables in ~xB are known asthe fuzzy basic variables with respect to basis B, and ~xB is the fuzzy basic vector. Also, let ~xN and N be the vector and thematrix of the remaining variables and columns, which are called the fuzzy nonbasic variables and nonbasic matrix, respec-tively. In this case, ~x ¼ ð~xB; ~xNÞ ¼ ðB�1~b; ~0Þ is also a fuzzy basic solution.
Definition 3.4. Suppose ~�x is a fuzzy basic feasible solution of (3). If the number of fuzzy positive variables ~�x are exactly m,then ~�x is called a nondegenerate fuzzy basic feasible solution, i.e. ~�xB ¼ ð~�x1; . . . ; ~�xmÞ � ð~0; ~0; . . . ; ~0Þ. If the number of fuzzy posi-tive ~�xj is less than of m, then ~�x is called a degenerate fuzzy basic feasible solution.
Example 3.1. Consider the following systems of equalities:
~x1 þ ~x2 þ ~x3 � ð3;9;3;3Þ;~x2 þ ~x4 � ð1;3;1;1Þ;~x1 þ 2~x2 þ ~x5 � ð4;12;4;4Þ:
Note that A ¼ ½a1; a2; a3; a4; a5� ¼1 1 1 0 00 1 0 1 01 2 0 0 1
24 35. In this case the fuzzy basic feasible solution ~xB ¼~x1~x2~x3
24 35 ¼ ð2;6;2;2Þð1;3;1;1Þð0;0;0; 0Þ
24 35corresponding to basis B ¼ ½a1; a2; a3� is degenerate since the fuzzy basic variable ~x3 � ~0.
Suppose ~�x is a fuzzy basic feasible solution of (3). Let yk and ~w be the solutions to Byk ¼ ak and ~wB ¼ ~cB, respectively. De-fine ~zj ¼ ~waj, where ~cB ¼ ð~cB1 ; . . . ; ~cBm Þ. Now, we are in a position to state some important theorems of fully fuzzy linear pro-gramming problems concerning to improving a fuzzy feasible solution, unbounded criteria and the optimality conditions(taken from [15]).
Theorem 3.2. If we have a fuzzy basic feasible solution with fuzzy objective value ~z such that ~zk � ~ck for some nonbasic variablexk, and yki0, then it is possible to obtain a new basic feasible solution with new fuzzy objective value ~�z, that satisfies ~�z � ~z.
Theorem 3.3. If we have a fuzzy basic feasible solution with ~zk � ~ck for some nonbasic variable xk, and yk 6 0, then the problem(3) has an unbounded optimal solution.
Theorem 3.4 (Optimality conditions). If a fuzzy basic solution ~xB ¼ B�1~b; ~xN � ~0 is feasible to (3) and ~zj � ~cj for all j, 1 6 j 6 n,then the fuzzy basic solution is a fuzzy optimal solution to (3).
Gaesan and Veermani [15] based on these theorems proposed a new algorithm for solving FLP problems. Here, we give asummary of their method.
4532 A. Ebrahimnejad / Applied Mathematical Modelling 35 (2011) 4526–4540
Algorithm 3.1. A fuzzy primal method for FLPInitialization StepChoose a starting basic feasible solution with basic B. (In Section 5, we explain a procedure for finding an initial fuzzy
feasible basis)
Main step
(1) The basic feasible solution is given by ~xB � B�1~b and ~xN � ~0. The fuzzy objective value is: ~z � ~cB~xB.(2) Let ~w ¼ ~cBB�1 and calculate ~zj � ~cj for all nonbasic variables. Suppose ~zj � ~cj ¼ hL
j ;hUj ;hj;hj
� �. Let
hUk þ hL
k ¼minj2R hUj þ hL
j
n owhere R is the index set of the current nonbasic variables. If hU
k þ hLk P 0, then stop; the cur-
rent solution is optimal. Otherwise go to step 3 with ~xk as entering variable.(3) Let yk ¼ B�1ak. If yk 6 0, then stop; the problem is unbounded. Otherwise, suppose ~�bi ¼ �bU
i ;�bL
i ;ai;ai� �
and determinethe index of the variable ~xBr leaving the basis as follows:
�bUr þ �bL
r
yrk¼ min
16i6m
�bUi þ �bL
i
yikjyik > 0
( ):
(4) Update the basic B where ak replaces aBr , and go to (1).
Remark 3.3. In the step 4 of the above mentioned method, new basis Bnew, new fuzzy basic solution, and new fuzzy objectiveare obtained as follows, respectively:
Bnew ¼ faB1 ; . . . ;aBr�1 ;ak;aBrþ1 ; . . . ;aBmg; ð9Þ
~̂xBi¼ ~�xBi
�~�xBr
yrkyrj
!; i – r and ~̂xk ¼ ~̂xBr ¼
~�xBr
yrk; ð10Þ
~̂z ¼ ~�z�~�xBr
yrkð~zk � ~ckÞ: ð11Þ
We note that the fuzzy primal method for FLP problems, starting a fuzzy basic feasible solution moves to another fuzzy basicsolution with a better (at least not worse) fuzzy objective value until it finds an optimal fuzzy basic feasible solution after a finitenumber of steps. Here, we first prove that in the absence of degeneracy, the fuzzy primal method stops in a finite number ofiterations. Then, we show that if an FLP problem has an optimal fuzzy solution, it has an optimal fuzzy basic solution too.
Theorem 3.5. In the absence of degeneracy, the fuzzy primal method stops in a finite number of iterations, either with an optimalfuzzy basic feasible solution or with the conclusion that the optimal value is unbounded.
Proof. In the absence of degeneracy, every fuzzy basic feasible solution has exactly m fuzzy positive components and has aunique associated basis. Also, at each iteration of the method, one of the following three actions is executed. It may be stop
with an optimal fuzzy basic solution if hUk þ hL
k P 0, i.e. ~zk � ~ck � ~0; it may stop with an unbounded optimal solution if
hUk þ hL
k < 0 and yk 6 0; or else it gives a new fuzzy basic feasible solution if hUk þ hL
k < 0 and yki0. In the absence of degen-
eracy, �bUr þ �bL
r > 0, i.e. ~�xB ¼ ~�br � ~0 and hence~�xBryrk� ~0. By (11), the difference between the fuzzy objective values at the previous
iteration and the current iteration is~�xBryrkð~zk � ~ckÞ � ~0. Thus, the fuzzy objective value increases strictly in each iteration. Hence
a basis that appears once in the course of method can never reappear. Also the total number of bases for (3) is less than or
equal to nm
: Hence, the method would stop in a finite number of steps with a finite optimal fuzzy basic solution or with an
unbounded optimal solution. h
Theorem 3.6. If FLP (3) has an optimum fuzzy feasible solution, then it has a fuzzy basic feasible solution that is optimal.
Proof. Suppose ~�x be an optimum fuzzy feasible solution. Let faj : ~�xj � ~0g ¼ fa1; a2; . . . ; akg If fa1; a2; . . . ; akg is linear indepen-dent, ~�x is a fuzzy basic feasible solution of (3) and we are done. Suppose this set is linearly dependent. So, there existsy ¼ ðy1; y2; . . . ; ykÞ– ð0;0; . . . ;0Þ such that
a1y1 þ a2y2 þ � � � þ akyk ¼ 0: ð12Þ
Let ~yj ¼ ðyj; yj;0;0Þ for j ¼ 1;2; . . . ; k. Thus, from (12) we have
a1~y1 þ a2~y2 þ � � � þ ak~yk � ~0: ð13Þ
A. Ebrahimnejad / Applied Mathematical Modelling 35 (2011) 4526–4540 4533
We now show that the assumption ~�x is optimal implies that any ~y ¼ ð~y1; ~y2; . . . ; ~ykÞ satisfying (13) must also satisfy
Table 1The cur
Basis
~z
zL þ~x1
..
.
~xr
..
.
~xm
c1~y1 þ c2~y2 þ � � � þ ck~yk � ~0: ð14Þ
Suppose not. Define ~�xðhÞ, h1 and h2 as the proof of Theorem 3.1. Then, ~zð~�xðhÞÞ ¼ ~zð~�xÞ þ hðc1~y1 þ c2~y2 þ � � � þ ck~ykÞ. It is clear thath1 > 0; h2 < 0 and at least one of them is finite. Let 0 < e < minfh1; jh2jg. Therefore, ~�xðhÞ is a fuzzy feasible solution for all hsatisfying �e < h < e. If c1~y1 þ c2~y2 þ � � � þ ck~yk � ~0, let p ¼ e and if c1~y1 þ c2~y2 þ � � � þ ck~yk � ~0, let p ¼ �e. Then ~�xðpÞ is a fuz-zy feasible solution of (3) and ~zð~�xðpÞÞ � ~zð~�xÞ, which contradicts the assumption of ~zð~�xÞ is an optimum fuzzy solution of (3).Hence, (14) must hold. Similar to the proof of Theorem 3.1, using (13) we can obtain another feasible solution ~̂x in which thenumber of fuzzy positive variables is at least on less than the corresponding number of ~�x. By (14), any such fuzzy feasible ~̂xthat we obtain must also satisfy ~zð~�xÞ � ~zð~̂xÞ, and also ~̂x is also an optimum fuzzy feasible solution. Hence, when this procedureis applied repeatedly, an optimal fuzzy basic feasible solution of (3) will be obtained after at most ðk� 1Þ applications of theprocedure. h
Now, we describe the fuzzy primal method in tableau method. Suppose that we have a starting fuzzy basic solution ~x withbasis B. The FLP problem (3) can be represented as follows.
max ~z;
s:t: ~z� ~cB~xB � ~cN~xN � ~0;
B~xB þ N~xN � ~b;
~xB; ~xN � ~0:
ð15Þ
From the second constraint of (15) we have
~xB þ B�1N~xN � B�1~b: ð16Þ
Multiplying (16) by ~cB and adding to ~z� ~cB~xB � ~cN~xN � ~0, we get
~zþ ~0~xB þ ð~cBB�1N � ~cNÞ~xN � ~cBB�1~b: ð17Þ
Currently ~xN � 0, and from (16) and (17) we get ~xB � B�1~b and ~z � ~cBB�1~b. Also. from (16) and (17), we can represent the cur-rent fuzzy basic solution with basis B in Table 1.
We note that in Table 1, there exist two objective rows: the first gives the ~zj � ~cj and the second is used to choose thepivoting column. Also, we have two right-hand- side columns: the first gives the current fuzzy basic variables values andthe second gives the real numbers corresponding to the fuzzy numbers given in the first column to choose pivoting row.In fact, similar to the simplex method for solving linear programming problems in crisp environment, we can design a fuzzysimplex method in tableau format for solving fuzzy linear programming (similar to [27]). We now describe the fuzzy primalmethod, proposed by Ganesan and Veermani, in tableau format as follows:
Algorithm 3.2. A fuzzy primal simplex method for FFLPInitialization StepFind an initial fuzzy basic feasible solution with basis B. Form the initial tableau similar to Table 1.Main step
(1) Calculate ~zj � ~cj for all nonbasic variables. Suppose ~zj � ~cj ¼ hLj ;h
Uj ;hj;hj
� �. Let
hUk þ hL
k ¼minj2R
hUj þ hL
j
n o;
where R is the index set of the current nonbasic variables. If hUk þ hL
k P 0, then stop; the current solution is optimal.Otherwise go to step 2.
rent fuzzy basic feasible solution.
~x1 . . . ~xr . . . ~xm . . . ~xj . . . ~xk . . . RHS R
~0 . . . ~0 . . . ~0 . . . ~zj � ~cj . . . ~zk � ~ck . . . ~�z � ~cB~�b –
zU 0 . . . 0 . . . 0 . . . hLj þ hU
j. . . hL
k þ hUk
. . . – �zL þ �zU
1 . . . 0 . . . 0 . . . y1j . . . y1k . . . ~�b1�bL
1 þ �bU1
..
. . . . ... . . . ..
. . . . ... . . . ..
. . . . ... ..
.
0 . . . 1 . . . 0 . . . yrj . . . yrk . . . ~�br�bL
r þ �bUr
..
. . . . ... . . . ..
. . . . ... . . . ..
. . . . ... ..
.
0 . . . 0 . . . 1 . . . ymj . . . ymk . . . ~�bm�bL
m þ �bUm
Table 2The data of Example 3.1.
Machines Time per unit (min) Machine capacity (min/day)
P1 P2 P3
M1 12 13 12 490M2 14 – 13 470M3 12 15 – 480
4534 A. Ebrahimnejad / Applied Mathematical Modelling 35 (2011) 4526–4540
(2) Let yk ¼ B�1ak. If yk 6 0, then stop; the problem is unbounded. Otherwise, suppose ~�bi ¼ �bUi ;
�bLi ;ai;ai
� �and determine
the index r as follows:
�bUr þ �bL
r
yrk¼ min
16i6m
�bUi þ �bL
i
yikjyik > 0
( ):
(3) Update the tableau by pivoting at yrk. Update the fuzzy basic and nonbasic variables where ~xk enters the basis and ~xBr
leaves the basis, and go to (1).
Here, for an illustration of the above method we consider the following example that has been solved by Gaesan andVeermani [15].
Example 3.2 [15]. A company produces three products P1, P2 and P3. These products on three different machines M1, M2
and M3. The time required to manufacture one unit of each products and the daily capacity of the machines are given inTable 2:
Note that the time availability can vary from day to day due to break down of machines, overtime work etc. Finally theprofit for each product can also vary due to variations in price. At the same time the company wants to keep the profit some-what close to Rs:14 for P1, Rs:13 for P2 and Rs:16 for P3. The company wants to determine the range of each product to beproduced per day to maximize its profit. It is assumed that all the amounts produced are consumed in the market.
Since the profit from each product and the time availability on each machine are uncertain, the number of units to beproduced on each product will also be uncertain. So we will model the problem as a fuzzy linear programming problem.We use symmetric tarpezoidal fuzzy numbers for each uncertain value.
Profit for P1 which is close to 14 is modelled as (13,15,2,2). Similarly the other parameters also modelled as symmetrictrapezoidal fuzzy numbers taking into account the nature of the problem and the other requirements. So the problem is for-mulated as follows:
max ~z � ð13;15;2;2Þ~x1 þ ð12;14;3;3Þ~x2 þ ð15;17;2;2Þ~x3;
s:t: 12~x1 þ 13~x2 þ 12~x3 � ð475;505;6;6Þ;14~x1 þ 13~x3 � ð460;480;8;8Þ;12~x1 þ 15~x2 � ð465;495;5;5Þ;~x1; ~x2; ~x3 � ~0:
Now the standard form of the fuzzy linear programming problem becomes
max ~z � ð13;15;2;2Þ~x1 þ ð12;14;3;3Þ~x2 þ ð15;17;2;2Þ~x3;
s:t: 12~x1 þ 13~x2 þ 12~x3 þ ~x4 � ð475;505;6;6Þ;14~x1 þ 13~x3 þ ~x5 � ð460;480;8;8Þ;12~x1 þ 15~x2 þ ~x6 � ð465;495;5;5Þ;~x1; ~x2; ~x3; ~x4; ~x5; ~x6 � ~0;
where ~x4; ~x5 and ~x6 are the slack fuzzy variables.We may write the first fuzzy primal simplex tableau as Table 3.Now ~x3 is an entering variable and ~x5 is a leaving variable. Then by pivoting on y23 ¼ 13, we obtain the next tableau as
Table 4.In this case ~x2 is an entering variable and ~x4 is a leaving variable. The new tableau is given as Table 5 by pivoting on
y12 ¼ 13.This is the optimal tableau since ~zj � ~cj � ~0 for all nonbasic fuzzy variables. The optimal solution matched with the solu-
tion obtained by Ganesan and Veermani, is given as follows:
~x1 � ~0; ~x2 ¼415169
;1045169
;174169
;174169
; ~x3 ¼
46013
;48013
;8
13;
813
~z � 94235
169;120265
169;19819
169;19819
169
:
Table 3The initial fuzzy simplex tableau.
Basis ~x1 ~x2 ~x3 ~x4 ~x5 ~x6 RHS R
z �28 �26 �32 0 0 0 – 0~z (�15,�13,2,2) (�14,�12,3,3) (�17,�15,2,2) ~0 ~0 ~0 (0,0,0) –~x4 12 13 12 1 0 0 (475,505,6,6) 980~x5 14 0 13 0 1 0 (460,480,8,8) 940~x6 12 15 0 0 0 1 (465,495,5,5) 960
Table 4The first iteration.
Basis ~x1 ~x2 ~x3 ~x4 ~x5 ~x6 RHS R
z 8413
�26 0 0 3213
0 – 1504013
~z 1513 ;
6913 ;
5413 ;
5413
� �(�14,�12,3,3) ~0 ~0 15
13 ;1713 ;
213 ;
213
� �~0 6890
13 ; 815013 ; 1096
13 ; 109613
� �–
~x4 � 1213
13 0 1 � 1213
0 41513 ;
104513 ; 174
13 ;17413
� �1460
13~x3 14
130 1 0 1
130 460
13 ;48013 ;
813 ;
813
� �94013
~x6 12 15 0 0 0 1 (465,495,5,5) 960
Table 5The optimal solution.
Basis ~x1 ~x2 ~x3 ~x4 ~x5 ~x6 RHS R
z 789169
�26 0 2613
104169
0 – 214500169
~z 51169 ;
729169 ;
738169 ;
738169
� �~0 ~0 12
13 ;1413 ;
313 ;
313
� �27
169 ;77
169 ;62
169 ;62
169
� �~0 94235
169 ; 120265169 ; 19819
169 ; 19819169
� �–
~x2�12169
1 0 113
�12169
0 415169 ;
1045169 ;
174169 ;
174169
� �1460169
~x3 1413
0 1 0 113
0 46013 ;
48013 ;
813 ;
813
� �94013
~x6 1848169
0 0 �1513
�180169
1 62910169 ; 77430
169 ; 3455169 ;
3455169
� �140340
169
A. Ebrahimnejad / Applied Mathematical Modelling 35 (2011) 4526–4540 4535
4. The fuzzy revised simplex method
In this section we describe some special implementations of the fuzzy simplex method discussed in previous section orslight modification of it. We know that in the fuzzy simplex method using canonical tableau, many computation are per-formed at every pivoting step. Every time a pivot is performed, it is carried out on every column of the tableau. This canbe very time-consuming. In the fuzzy revised simplex method, all necessary computations are carried out restricting the piv-oting operations on inverse matrix. In fact, the fuzzy revised simplex method is a systematic procedure for implementing thesteps of the fuzzy simplex method in a smaller array, thus saving storage space [24].
Suppose that we have a fuzzy basic feasible solution with a known B�1. Table 6, called the fuzzy revised simplex tableau,is constructed where ~w ¼ ~cBB�1 and ~�b ¼ B�1~b.
Note that similar to canonical tableau, there are two right-hand-side columns: the first gives the current fuzzy basic vari-ables values and the second gives the real numbers corresponding to fuzzy numbers to do minimum ratio test. Also, the
Table 6The fuzzy revised simplex tableau.
Basic inverse RHS R
w – cB�b
~w ~cB~�b –
B�1 ~�b �b
Table 7Adding the updated column of ~xk .
Basic inverse RHS R ~xk
w – cB�b hL
k þ hUk
~w ~cB~�b – �bL þ �bU
B�1 ~�b �b yk
Table 8The initial fuzzy revised simplex tableau.
Basic inverse RHS R
z 0 0 0 – 0~z ~0 ~0 ~0 ~0 –~x4 1 0 0 (475,505,6,6) 980~x5 0 1 0 (460,480,8,8) 940~x6 0 0 1 (465,495,5,5) 960
4536 A. Ebrahimnejad / Applied Mathematical Modelling 35 (2011) 4526–4540
objective rows are used to check optimality conditions. In fact, the first piece of information that we need to see is the ~zj � ~cj
values. Since ~w is known, these values calculated as in step 1 in Algorithm 3.2 to check for optimality. Suppose that~zj � ~cj � ~0. Then we wish to examine the updated column of ~xk. Using B�1, we may compute yk ¼ B�1ak. If yk 6 0, then stopwith the indication that the optimal solution value is unbounded. Otherwise, the updated column of ~xk can be appended tothe fuzzy revised simplex tableau as shown, while the rest of the tableau is still kept hidden (Table 7).
The index r of step 2 in Algorithm 3.2 can be calculated by the usual minimum test. Also, pivoting on yrk gives as usual thenew ~w; B�1;
~�b and ~cB~�B and the process is repeated. The following is a summary of the fuzzy revised simplex method.
Algorithm 4.1. The fuzzy revised simplex methodInitialization StepFind an initial fuzzy basic feasible solution with basis B. Calculate ~w ¼ ~cBB�1;
~�b and form the fuzzy revised simple tableau(Table 6).
Main step
(1) Calculate ~zj � ~cj for all nonbasic variables. Suppose ~zj � ~cj ¼ hLj ;h
Uj ;hj;hj
� �. Let hU
k þ hLk ¼minj2R hU
j þ hLj
n owhere R is the
index set of the current nonbasic variables. If hUk þ hL
k P 0, then stop; the current solution is optimal. Otherwise go tostep 2.
(2) Let yk ¼ B�1ak. If yk 6 0, then stop; the problem is unbounded. Otherwise, insert the columnhLþhK~zk�~ck
yk
" #to the right of the
fuzzy revised simplex tableau as Table 7. Determine the index r as follows:
Table 9Adding
z~z~x4
~x5
~x6
Table 1First ite
z~z~x4
~x3
~x6
�bUr þ �bL
r
yrk¼ min
16i6m
�bUi þ �bL
i
yikjyik > 0
( ):
(3) Pivot at yrk. This update the fuzzy tableau. Now the column corresponding to ~xk is completely eliminated from the tab-leau and the main step is repeated.
Example 3.3. For an illustration of the fuzzy revised method, we solve the FLP given in Example 3.2 by Algorithm 4.1.The initial basic is eB ¼ ½a4; a5; a6�. Also, ~w ¼ ~cBB�1 ¼ ð~0; ~0; ~0Þ and ~�b ¼ ½~�b1;
~�b2;~�b3� ¼ ½ð475;505;6;6Þ; ð460;480;8;8Þ;
ð465;495;5;5Þ�.
the updated column of ~x3.
Basic inverse RHS R ~x3
0 0 0 – 0 �32~0 ~0 ~0 ~0 – (�17,�15,2,2)
1 0 0 (475,505,6,6) 980 120 1 0 (460,480,8,8) 940 130 0 1 (465,495,5,5) 960 0
0ration of the fuzzy revised algorithm.
Basic inverse RHS R
0 3213
0 – 1504013
~0 1513 ;
1713 ;
213 ;
213
� �~0 6890
13 ; 815013 ; 1096
13 ; 109613
� �–
1 �1213
0 41513 ;
104513 ; 174
13 ;17413
� �1460
13
0 113
0 46013 ;
48013 ;
813 ;
813
� �94013
0 0 1 (465,495,5,5) 960
A. Ebrahimnejad / Applied Mathematical Modelling 35 (2011) 4526–4540 4537
First Iteration:Here ~w ¼ ð~0; ~0; ~0Þ. Noting that ~zj � ~cj ¼ ~waj � ~cj, we get
Table 1Adding
z~z~x4
~x3
~x6
~z1 � ~c1 ¼ ð�15;�13;2;2Þ; ~z2 � ~c2 ¼ ð�14;�12;3;3Þ; ~z3 � ~c3 ¼ ð�17;�15;2;2Þ
hL1 þ hU
1 ¼ �28; hL2 þ hU
2 ¼ �26; hL3 þ hU
3 ¼ �32;
Thus k ¼ 3 and ~x3 enters the basis:
y3 ¼ B�1a3 ¼1 0 00 1 00 0 1
264375 12
130
264375 ¼ 12
130
264375:
Insert the vector
hL3þhU
3~z3�~c3
y3
24 35 ¼�32
ð�17;�15;2;2Þ
12130
26643775
to the right of Table 8 leading to Table 9.Now pivoting at y23 ¼ 13 gives the new solution as reported in Table 10.Second Iteration:Here ~w ¼ ~0; 15
13 ;1713 ;
213 ;
213
� �; ~0
� �. Noting that ~zj � ~cj ¼ ~waj � ~cj, we get
~z1 � ~c1 ¼1513
;6913
;5413
;5413
; ~z2 � ~c2 ¼ ð�14;�12;3;3Þ; ~z5 � ~c5 ¼
1513
;1713
;2
13;
213
hL1 þ hU
1 ¼8413
; hL2 þ hU
2 ¼ �26; hL3 þ hU
3 ¼3213
;
Thus k ¼ 2 and ~x2 enters the basis:
y2 ¼ B�1a3 ¼1 0 0�1213
113 0
0 0 1
264375 13
015
264375 ¼ 13
015
264375;
Insert the vector
hL2þhK
2~z2�~c2
y3
24 35 ¼�26
ð�14;�12;3;3Þ
130
15
26643775
to the right of Table 10 and obtain Table 11.Now pivoting at y12 = 13 gives the new solution as reported in Table 12.Third Iteration:Here ~w ¼ 12
13 ;1413 ;
313 ;
313
� �; 27
169 ;77
169 ;62
169 ;62
169
� �; ~0
� �. Noting that ~zj � ~cj ¼ ~waj � ~cj, we get
~z1 � ~c1 ¼51
169;729169
;738169
;738169
; ~z4 � ~c4 ¼
1213
;1413
;3
13;
313
; ~z5 � ~c5 ¼
27169
;77
169;
62169
;62
169
hL
1 þ hU1 ¼
789169
; hL4 þ hU
4 ¼ 13; hL5 þ hU
5 ¼104169
:
Since hLj þ hU
j P 0 for all fuzzy nonbasic variables, the fuzzy basic solution presented in Table 12 is optimal.
1the updated column of ~x2.
Basic inverse RHS R x̂2
0 3213
0 – 1504013
�26~0 15
13 ;1713 ;
213 ;
213
� �~0 6890
13 ; 815013 ; 1096
13 ; 109613
� �– (�14,�12,3,3)
1 �1213
0 41513 ;
104513 ; 174
13 ;17413
� �1460
1313
0 113
0 46013 ;
48013 ;
813 ;
813
� �94013
0
0 0 1 (465,495,5,5) 960 15
Table 12Second iteration of the fuzzy revised algorithm.
Basic inverse RHS R
z 2613
104169
0 – 214500169
~z 1213 ;
1413 ;
313 ;
313
� �27
169 ;77
169 ;62
169 ;62
169
� �~0 94235
169 ; 120265169 ; 19819
169 ; 19819169
� �–
~x21
13�12169
0 415169 ;
1045169 ;
174169 ;
174169
� �1460169
~x3 0 113
0 46013 ;
48013 ;
813 ;
813
� �94013
~x6 �1513
�180169
1 62910169 ; 77430
169 ; 3455169 ;
3455169
� �140340
169
Table 13The initial tableau.
Basic ~x1 ~x2 ~x3 ~x4 ~x5 ~x6 ~x7 RHS R
z 0 0 0 0 0 –2 –2 – 0~z ~0 ~0 ~0 ~0 ~0 (�1,�1,0,0) (�1,�1,0,0) ~0 –~x6 1 1 –1 0 0 1 0 (1,3,2,2) 4~x7 �1 1 0 –1 0 0 1 (0,2,1,1) 2~x5 0 1 0 0 1 0 0 (2,4,1,1) 6
Table 14The revised initial tableau.
Basic ~x1 ~x2 ~x3 ~x4 ~x5 ~x6 ~x7 RHS R
z 0 4 –2 –2 0 0 0 – 6~z ~0 (2,2,0,0) (�1,�1,0,0) (�1,�1,0,0) ~0 ~0 ~0 (1,5,0,0) –~x6 1 1 –1 0 0 1 0 (1,3,2,2) 4~x7 �1 1 0 –1 0 0 1 (0,2,1,1) 2~x5 0 1 0 0 1 0 0 (2,4,1,1) 6
4538 A. Ebrahimnejad / Applied Mathematical Modelling 35 (2011) 4526–4540
5. The initial fuzzy basic solution
Note that the fuzzy primal method for solving FLP problems starts with a fuzzy basic feasible solution and moves to animproved fuzzy basic solution, until the optimal solution is reached or else unboundedness of the fuzzy objective function isverified. However, in order to initialize this method, a basic B with B�1~b � ~0 must be available. In this section, we describe aprocedure to obtain an initial fuzzy basic solution with a very simple basis namely the identity.
In general any FLP problem can be transformed into a fuzzy problem of the following form:
max ~z ’ ~c~x;
s:t: A~x � ~b;
~x � ~0;
ð18Þ
where ~b � ~0.Suppose that A has no identity submatrix. To illustrate our procedure, suppose that we changed the restrictions by adding
an fuzzy artificial vector ~xa leading to the fuzzy system A~xþ ~xa � ~b; ~x � ~0; ~xa � ~0. Hence, we forced an identity matrix cor-responding to the fuzzy artificial vector. This gives an immediate fuzzy basic solution of the new fuzzy system, namely~xa � ~b; ~x � ~0. Thus, the fuzzy primal method can be applied. But, we must force these artificial vectors to fuzzy zero, becauseA~x � ~b if and only if A~xþ ~xa � ~b with ~xa � ~0. To eliminate the fuzzy artificial vectors, it is sufficient to mimimize the sum ofthe artificial vectors. In other words, we must solve the following fuzzy linear programming problems starting with the fuzzybasic feasible solution ~x � ~0 and ~xa � ~b.
min ~z ’ ~1~xa;
s:t: A~xþ ~xa � ~b;
~x � ~0; ~xa � ~0;
ð19Þ
where ~1 ¼ ðð1;1;0; 0Þ; . . . ; ð1;1;0;0ÞÞ.If the original FLP (18) has a fuzzy feasible solution, the optimal value of the FLP (19) is fuzzy zero, where all the fuzzy
artificial variables drop to fuzzy zero. In this case, we get a fuzzy basic solution of the (18) and the fuzzy primal method can
Table 15First iteration.
Basic ~x1 ~x2 ~x3 ~x4 ~x5 ~x6 ~x7 RHS R
z 4 0 –2 2 0 0 –4 – 2~z (2,2,0,0) ~0 (�1,�1,0,0) (1,1,0,0) ~0 ~0 (�2,�2,0,0) (�3,5,0,0) –~x6 2 0 �1 1 0 1 �1 (�1,3,3,3) 2~x2 �1 1 0 �1 0 0 1 (0,2,1,1) 2~x5 0 1 0 0 1 0 0 (0,4,2,2) 4
Table 16Second iteration.
Basic ~x1 ~x2 ~x3 ~x4 ~x5 ~x6 ~x7 RHS R
z 0 0 0 2 0 �2 �2 – 0~z ~0 ~0 ~0 ~0 ~0 (�1,�1,0,0) (–1,–1,0,0) (�6,6,0,0) � ~0 –~x1 1 0 �1
212
0 12
�12 ð�1
2 ;32 ;
32 ;
32Þ 1
2~x2 0 1 �1
2�12
0 12
12
�12 ;
72 ;
52 ;
52
� �3
~x5 0 0 12
12
1 �12
�12
�32 ;
92 ;
72 ;
72
� �3
A. Ebrahimnejad / Applied Mathematical Modelling 35 (2011) 4526–4540 4539
be started with the original objective ~c~x. Otherwise, at optimality ~xa 6� ~0 and the (18) has no fuzzy solution, because if there
is an ~x � ~0 with A~x � ~b, then~x~0
� �is a fuzzy feasible solution of the (19) and ~0ð~xÞ þ ~1ð~0Þ � ~0 � ~1~xa, violating optimality ~xa.
Example 5.1. Consider the following FLP problems:
max ~z � ð�2;0;1;1Þ~x1 þ ð1;3;2;2Þ~x2;
s:t: ~x1 þ ~x2 � ð1;3;2;2Þ;� ~x1 þ ~x2 � ð0;2;1;1Þ;~x2 � ð2;4;1;1Þ;~x1; ~x2 � ~0:
After introducing the slack variables ~x3; ~x4; ~x5 and the fuzzy artificial variables ~x6 and ~x7, the following FLP to eliminate fuzzyartificial variables is obtained.
min ~z � ð1;1;0;0Þ~x6 þ ð1;1;0; 0Þ~x7;
s:t: ~x1 þ ~x2 � ~x3 þ ~x6 � ð1;3;2;2Þ;� ~x1 þ ~x2 � ~x4 þ ~x7 � ð0;2;1;1Þ;~x2 þ ~x5 � ð2;4;1;1Þ;~x1; ~x2; ~x3; ~x4; ~x5; ~x6; ~x7 � ~0:
Now we may rewrite the above FLP problem in the primal simplex tableau format as Table 13.
Multiply rows 1 and 2 by (1,1,0,0) and add to row corresponding to ~z. Thus the new tableau is given as Table 14.Now ~x2 is an entering variable and ~x7 is a leaving variable. Then by pivoting on y22 ¼ 1, we obtain the next tableau as
Table 15.In this case ~x1 is an entering variable and ~x6 is a leaving variable. The new tableau is as Table 16.Since all the fuzzy artificial variables are at level fuzzy zero, we have an initial fuzzy basic solution with B ¼ ½a1; a2; a5� for
the original FLP problems. Now we can form the initial Table 1 with basis B and then use the Algorithm 3.2 for solving ori-ginal FLP problem.
6. Conclusions
The filed of fuzzy linear programming have recently attracted some interests. In this paper, we reviewed the fuzzy linearprogramming proposed by Gaesan and Veermani [15]. The key of the fuzzy primal method is that the optimal solution obtainat a fuzzy basic solution. Thus we proved that if an FLP problem has a fuzzy optimal solution, then it also has a fuzzy basicoptimal solution. We also showed that in the absence of degeneracy, the fuzzy primal method stops in a finite number ofiterations. In addition, the tableau format of the fuzzy primal method designed. Since in the fuzzy simplex method proposedin this paper, many computation are performed at ever pivoting step, we gave the fuzzy revised simplex method that ispurely an efficient computational scheme for applying the main ideas of the fuzzy simplex method. In other hand, in many
4540 A. Ebrahimnejad / Applied Mathematical Modelling 35 (2011) 4526–4540
cases for solving fuzzy linear programming problems, an initial fuzzy basic solution is not at hand. So we developed the fuzzysimplex method to obtain an initial fuzzy basic solution. We emphasize that the proposed method can not solve the FLPproblem when the variables are non symmetric trapezoidal fuzzy numbers. This will be an interesting research work inthe future.
Acknowledgments
The author thanks the anonymous referees for their constructive comments which contributed to the improvement of thepresent paper. We are especially grateful to Edition-in-chief, Prof. M. Cross, for his valuable comments. Finally, the authorgreatly appreciates to the office of vice chancellor for research of Islamic Azad University-Qamshahr Branch, for financiallysupport.
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