J Appl Res Ind Eng Vol 4 No 1 (2017) 24ndash38

Journal of Applied Research on Industrial

Engineering wwwjournal-apriecom

A Generalized Model for Fuzzy Linear Programs with

Trapezoidal Fuzzy Numbers

Seyed Hadi Nasseri 1 Hadi Zavieh1 Seyedeh Maedeh Mirmohseni2

1Department of Mathematics University of Mazandaran Babolsar Iran

2School of Mathematics and Information Science Key Laboratory of Mathematics and Interdisciplinary

Sciences of Guangdong Higher Education Institutes Guangzhou University Guangzhou 510006 China

A B S T R A C T P A P E R I N F O

In this paper a linear programming problem with symmetric trapezoidal fuzzy

number which is introduced by Ganesan et al in [4] is generalized to a general kind

of trapezoidal fuzzy number In doing so we first establish a new arithmetic

operation for multiplication of two trapezoidal fuzzy numbers in order to prepare a

method for solving the fuzzy linear programming and the primal simplex algorithm a

general linear ranking function has been used as a convenient approach in the

literature In fact our main contribution in this work is based on 3 items 1)

Extending the current fuzzy linear program to a general kind which doesnrsquot

essentially include the symmetric trapezoidal fuzzy numbers 2) Defining a new

multiplication role of two trapezoidal fuzzy numbers 3) Establishing a fuzzy primal simplex algorithm for solving the generalized model We in particular emphasize that

this study can be used for establishing fuzzy dual simplex algorithm fuzzy primal-

dual simplex algorithm fuzzy multi objective linear programming and the other

similar methods which are appeared in the literature

Chronicle Received 06 June 2017

Revised 10 August 2017

Accepted 24 August 2017

Available 24 August 2017

Keywords Fuzzy linear programming

Fuzzy arithmetic

Fuzzy ordering

Fuzzy primal simplex

algorithm

1 Introduction

Fuzzy mathematical programming has been developed for treating uncertainty about the

setting optimization problems In recent years various attempts have been made to study the

solution of fuzzy linear programming problems either from theoretical or computational

point of view After the pioneering works on this area many authors have considered various

kinds of the FLP problems and have proposed several approaches to solve these problems

[23891013] Some authors have made a comparison between fuzzy numbers and in

particular linear ranking function to solve the fuzzy linear programming problems Of course

ranking functions have been proposed by researchers to meet their requirements regarding the

Corresponding author

E-mail address nhadi57gmailcom DOI 1022105jarie201749602

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 25

problem under consideration and conceivably there are no generally accepted criteria for

application of ranking functions Using the concept of comparison of fuzzy numbers Maleki

et al [10] proposed a new method to solve Fuzzy Number Linear Programming (FNLP)

problems Then Mahdavi-Amiri and Nasseri [9] used a certain linear ranking function to

define the dual of FNLP problems as a concept that gives an efficient method called the dual

simplex algorithm [11] for solving FNLP problems Also Mahdavi-Amiri and Nasseri [8]

proposed another approach to define dual of FNLP problems as Linear Programming with

Fuzzy Variables (FVLP) problems It introduced a dual simplex algorithm for solving FVLP

problems Nasseri and Ebrahimnejad [12] suggested a fuzzy primal simplex algorithm in

order to solve the flexible linear programming problem Next they recommended the fuzzy

primal simplex method to solve the flexible linear programming problems directly without

solving any auxiliary problem Hosseinzadeh Lofi et al [5] discussed full fuzzy linear

programming (FFLP) problems whose parameters and variables are triangular fuzzy

numbers They used the concept of the symmetric triangular fuzzy number and introduced an

approach to defuzzify a general fuzzy quantityIn order to deal with the problem first of all

the fuzzy triangular number is approximated to its nearest symmetric triangular number on

the assumption that all decision variables are symmetric triangular Kumar et al [7] proposed

a new method to find the fuzzy optimal solution of the same type of fuzzy linear

programming problems Ganesan and Veeramani [4] introduced a new type of fuzzy

arithmetic for symmetric trapezoidal fuzzy numbers and proposed a method for solving FLP

problems without converting them to the crisp linear programming problems After that

Sapan Kumar Das and et al [1] proposed a mathematical model for solving fully fuzzy linear

programming problem with trapezoidal fuzzy numbers However the proposed approach

could modify the mentioned model to a simpler one while it couldnrsquot be applied for the

general kind of the trapezoidal fuzzy numbers Hence in this paper we first extended the

recently quoted model by Ganesan and Veeramani to the general model and then based it on

a new multiplication role of two trapezoidal fuzzy numbers We then established a fuzzy

primal simplex algorithm to solve the mentioned generalized model We also emphasized that

the proposed approach could be used for re-establishing the similar format of solving

algorithm such as fuzzy two-phase method fuzzy dual simplex algorithm etc We finally

illustrated our work by dealing with some numerical examples which are convenient samples

of problems It will be observed that this is a more appropriate method for the decision

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 26

makers to adapt to the real situations and it in particular can solve these problems as simple

as possible

2 Preliminaries

21 Definitions and notations

In this section some notations concepts new definitions and some fundamental results

have been presented on fuzzy arithmetic in the lake of symmetric assumption and a new

generalized definition has been particularly proposed for multiplication of the general

trapezoidal fuzzy numbers

Definition 21 A fuzzy set on ℝ is said to be a trapezoidal fuzzy number if there exists real

numbers 1198861 1198862 where 1198861 le 1198862 and ℎ1 ℎ2 ge 0 such that

(119909) =

119909

ℎ1+ℎ1 minus 1198861ℎ1

119891119900119903 119909 isin (1198861 minus ℎ1 1198861)

1 119891119900119903 119909 isin (1198861 1198862) minus119909

ℎ2+1198862 + ℎ2ℎ2

119891119900119903 119909 isin (1198862 1198862 + ℎ2)

0 119900119905ℎ119890119903119908119894119904119890

where (119909) is the membership function of fuzzy number

We denoted it by = [1198861 1198862 ℎ1 ℎ2]

In the above definition when ℎ1 = ℎ2 we called it as symmetric trapezoidal fuzzy number

Moreover when ℎ1 = ℎ2 = 0 = [1198861 1198862]

We use ℱ(ℝ) to denote the set of all trapezoidal fuzzy numbers

Definition 22 Let = [1198861 1198862 ℎ1 ℎ2] and = [1198871 1198872 1198961 1198962] be two trapezoidal fuzzy

numbers If so the arithmetic operations on and will be defined by

(i) Addition + = [1198861 1198862 ℎ1 ℎ2] + [1198871 1198872 1198961 1198962] = [1198861 + 1198871 1198862 + 1198872 ℎ1 + 1198961 ℎ2 + 1198962]

(ii) Subtraction minus = [1198861 1198862 ℎ1 ℎ2] minus [1198871 1198872 1198961 1198962] = [1198861 minus 1198872 1198862 minus 1198871 ℎ1 + 1198962 ℎ2 + 1198961]

22 Ranking functions

One convenient approach for solving the FLP problems is based on the concept of

comparison of fuzzy numbers by use of ranking functions (see [14]) An effective approach

27 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

for ordering the elements of ℱ(ℝ) is to define a ranking function ℛℱ(ℝ) ⟶ ℝ which maps

each fuzzy number into the real line where a natural order exists

Definition 23 We defined orders on ℱ(ℝ) by

≽ if and only if ℛ() ge ℛ()

≻ if and only if ℛ() ≻ ℛ()

≃ if and only if ℛ() ≃ ℛ()

where and b are in ℱ(ℝ) Also we write ≼ if and only if ≽

Attention has been exclusively paid to linear ranking functions that is a ranking function

ℛ such that

ℛ(119896 + ) = 119896ℛ() + ℛ()

for any and belonging to ℱ(ℝ) and any k isin ℝ

Now using the above approach we may rank a big category of fuzzy numbers where the

symmetric property is extended to the non-symmetric form too

Remark 1 For any trapezoidal fuzzy number the relation ≽ 0 holds if there exists 120576 gt 0

and 120572 gt 0 such that ≽ (minus120576 120576 120572 120572) We realized that ℛ(minus120576 120576 120572 120572) = 0 We also

considered asymp 0 only if ℛ() = 0 Thus without loss of generality throughout the paper

we regard 0 = (0 0 0 0) as the zero trapezoidal fuzzy number

The following lemma is now immediately at hand

Lemma 21 Let ℛ be any linear ranking function Then

(i) ≽ if and only if minus ≽ 0 if and only if minus ≽ minus

(ii) If ≽ and 119888 ≽ then + 119888 ≽ +

The linear ranking function has been considered on ℱ(ℝ) as

ℛ() = 119888119897119886119897 + 119888119906119886

119906 + 119888120572120572 + 119888120573120573

where = (119886119897 119886119906 120572 120573) and 119888119897 119888119906 119888120572 119888120573 are constants at least one of them is nonzero A

special version of the above linear ranking function was first proposed by Yager [14] (see

also [8] and [9])

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 28

Proposition 21 For any trapezoidal fuzzy numbers and 119888 we have

(i) 119888 ( + ) asymp (119888 + 119888 )

(ii) 119888 ( minus ) asymp (119888 minus 119888 )

Theorem 21 (i) The relation ≼ is a partial order relation on the set of trapezoidal fuzzy

numbers

(ii) The relation ≼ is a linear order relation on the set of trapezoidal fuzzy numbers

(iii) For any two trapezoidal fuzzy numbers and if ≼ then ≼ (1 minus 120582) + 120582 ≼

for all λ 0 le λ le 1

3 A new role in fuzzy arithmetic and fuzzy ordering

A definition of the multiplication of two symmetric fuzzy numbers is given by Ganesan and

Veeramani in [7] based on the Extension Principle (see [12]) However fuzzy arithmetic and

a fuzzy ordering role based on the given definition is established by many researchers

Although many valuable works are appeared in the literature there are a big limitation in the

basic definition In fact the symmetric assumption is not reasonable in practice since there

are many real situations that need to be dealt with by decision makers especially when the

main parameters of the system is formulated in the more general case and frankly in the form

of non-symmetric fuzzy number Moreover by introducing a new definition of the length of

120596 we may keep the length of the resulted fuzzy number which is obtained based on the

given multiplication role

For defining the multiplication of the two (non-symmetric) trapezoidal fuzzy numbers we

first needed to define the following notations

Let

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 120574 = 119860119907119890119903119886119892119890 120579

Then define

120572 = | 120574 minus 119898119894119899120579 | and 120573 = |119898119886119909120579 minus 120574|

Let 120596 = |120573minus120572

2| and we now may define the multiplication of two non-symmetric trapezoidal

fuzzy numbers as an extended role of the given definition for the symmetric form of fuzzy

numbers which is defined by Ganesan and Veeramani in [4] as follows

29 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Definition 31 Let = [1198861 1198862 ℎ1 ℎ2] and = [1198871 1198872 1198961 1198962] be two trapezoidal fuzzy

numbers Then the arithmetic operations on and are given by

Multiplication = [1198861 1198862 ℎ1 ℎ2][1198871 1198872 1198961 1198962]

= [(1198861 + 11988622

)(1198871 + 11988722

)minus 120596 (1198861 + 11988622

) (1198871 + 11988722

) +120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

From the above definition it is clear that

120582 = [1205821198861 1205821198862 120582ℎ1 120582ℎ2] 119891119900119903 120582 ge 0 [1205821198862 1205821198861 minus120582ℎ2 minus120582ℎ1] 119891119900119903 120582 lt 0

Remark 2 Depending upon the need we overcame the limitation of the multiplication role

which is given just for the symmetric kind of trapezoidal fuzzy numbers See in [4]

Definition 32 The model

119898119886119909 = sum 119888119909119899119895=1

(2-1)

119878 119905 sum 119886119894119895119909119899

119895=1≼ 119887 119894 = 1hellip 119898

119909 ≽ 0 119895 = 1hellip 119899

if 119886119894119895 isin ℝ and 119888 119909 119887 isin ℱ(ℝ) 119894 = 1hellip 119898 119895 = 1hellip 119899 is called a Semi-Fuzzy Linear

Programming (SFLP) problem

Definition 33 Any = (1199091 1199092 hellip 119909 ) isin ℱ119899(ℝ) where each 119909 isin ℱ(ℝ) which satisfies

the constraints and non-negativity restrictions of (2-1) is said to be a fuzzy feasible solution to

(2-1)

Definition 34 Let Q be the set of all fuzzy feasible solutions of (1) A fuzzy feasible

solution 119883 isin 119876 is said to be a fuzzy optimum solution to (1) if 119883 ≽ for all isin 119876

where = (1198881 1198882 hellip 119888 ) and = 11988811199091 + 11988821199092 +⋯+ 119888119909

32 Fuzzy Basic feasible solution

The concept of fuzzy basic feasible solution is similar to the given definition in [8] We

give a new definition for fuzzy solution associated to the discussed model as below

Definition 35 Let = (1199091 1199092 hellip 119909 ) isin ℱ119899(ℝ) suppose that = (119861

119879 119873119879) where 119909 asymp

119861minus1 119909 asymp 0 to be a fuzzy basic feasible solution of the system 119860 asymp If 119909 =

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 30

[minus120572119895 120572119895 ℎ119895 ℎ119895] for some 120572119894 gt 0 ℎ119895 and ℎ119895 ge 0 that is 119909 asymp 0 and every basic variable of the

corresponding to every feasible basic 119861 is positive is said to be a degenerated fuzzy basic

feasible solution

The following theorem concerns the so-called nondegenerate FNLP problems

Theorem 31 Let the FNLP problem be nondegenerate A basic feasible solution 119909 asymp

119861minus1 119909 asymp 0 is optimal to (2-1) only if 119911 ≽ 119888 for all 119895 1 le 119895 le 119899

Proof Suppose that 119909lowast = (119861119879 119873

119879)119879 is a basic feasible solution to (2-1) where 119909 asymp

119861minus1 119909 asymp 0 Then = 119888119909 = 119888119861minus1 On the other hand for every feasible solution

we have asymp 119860 asymp 119861119909 +119873119909 Hence we obtain

= 119888 minussum (119911 minus 119888)119909119895ne119861119894

(3 minus 1)

Then

= 119911lowast = 119888119909 + 119888119909 = 119888119861minus1 minussum (119888119861

minus1119886119895 minus 119888)119909119895ne119861119894

The proof can now be completed using (3-1) and Theorem 32 given in Section 3 1048576

Now we are going to devise a fuzzy primal simplex algorithm for solving the Problem (2-1)

33 Primal simplex method in tableau format for the fuzzy linear Problems

Consider the fuzzy linear programing problem as in (2-1) We rewrite the fuzzy linear

programing problem as

119872119886119909 = 119888119909 + 119888119909

119904 119905 119861119909 + 119873119909 =

119909 ≽ 0 119909 ≽ 0

Hence we have 119909 + 119861minus1119873119909 asymp 119861

minus1 Therefore + (119888119861minus1119873 minus 119888)119909 asymp 119888119861

minus1 With

119909 asymp 0 we have 119909 = 119861minus1 asymp 119910 and asymp 119888119861minus1 Thus we rewrited the above problem as

in Table 1

Table 1 The primal simplex tableau

Basis 119909 119909 RHS

0 119911 minus 119888 = 119888119861minus1119873 minus 119888 119910119900 = 119888119861

minus1

119909 119868 119884 = 119861minus1119873 119910 asymp 119861minus1

31 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Remark 3 Table 1 gives all the information needed to proceed with the simplex method The

fuzzy cost row in Table 1 is 119900119879 = 119888119861

minus1119860 minus 119888 where 119900119895 = 119888119861minus1119886119895 minus 119888 = 119911 minus 119888 1 le 119895 le

119899 with 119900119895 asymp 0 for 119895 = 119861119894 1 le 119894 le 119898 According to the optimality conditions (Theorem 31)

we are at the optimal solution if 119900119895 ≽ 0 for all 119895 ne 119861119894 1 le 119894 le 119898 On the other hand if

119900119895 ≼ 0 for some 119896 ne 119861119894 1 le 119894 le 119898 the problem is either unbounded or an exchange of a

basic variable 119909119861 and the nonbasic variable 119909 can be made to increase the rank of the

objective value (under nondegeneracy assumption) The following results established in [9]

help us devise the fuzzy primal simplex algorithm

Theorem 32 If there is a column 119896 (not in basis) in a fuzzy primal simplex tableau as 119910119900 =

119911 minus 119888 ≺ 0 and 119910119894119896 le 0 1 le 119894 le 119898 the problem (2-1) is unbounded

Theorem 33 If a nonbasic index 119896 exists in a fuzzy primal simplex tableau like 119910119900 = 119911 minus

119888 ≺ 0 and there exists a basic index 119861119894 like 119910119894119896 ge 0 a pivoting row 119903 can be found so that

pivoting on 119910119903119896 can yield a feasible tableau with a corresponding nondecreasing (increasing

under nondegeneracy assumption) fuzzy objective value

Remark 4 (see [9]) If there exists k such that 119910119900 ≺ 0 and the problem is not unbounded r

can be chosen as

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

Where it is the minimum fuzzy value of the above ratio

in order to replace 119909119861 in the basis by 119909 resulting in a new basis 119861 =

(119886119861119894 1198861198612 hellip 119886119861119903minus1 119886119896 119886119861119903+1 hellip 119886119861119898) The new basis is primal feasible and the

corresponding fuzzy objective value is nondecreasing (increasing under nondegeneracy

assumption) It can be shown that the new simplex tableau is obtained by pivoting on 119910119903119896

ie doing Gaussian elimination on the 119896 th column by using the pivot row 119903 with the pivot

119910119903119896 to transform the 119896 th column to the unit vector 119890119903 It is easily seen that the new fuzzy

objective value is 119910119900 = 119910119900 minus 119910119900119910119903119900

119910119903119896≽ 119910119900 where 119900119896 ≼ 0 and

119910119903119900

119910119903119896 (if the problem is

nondegenerate and consequently 119910119903119900

119910119903119896gt 0 and hence119910119900 asymp 119910119900)

We now describe the pivoting strategy

34 Pivoting and change of basis

If 119909 enters the basis and 119909119861 leaves the basis pivoting on 119910119903119896 in the primal simplex tableau

is carried out as follows

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 32

1) Divide row r by 119910119903119896

2) For 119894 = 01 119898 and 119894 ne 119903 update the 119894 th row by adding to it minus119910119894119896 times the new 119903th

row

We now present the primal simplex algorithm for the FNLP problem

35 The main steps of fuzzy primal simplex algorithm

Algorithm 31 The fuzzy primal simplex method

Assumption A basic feasible solution with basis B and the corresponding simplex tableau is

at hand

1 The fuzzy basic feasible solution is given by 119909 = 119910= 119861minus1 and 119909 asymp 0 The fuzzy

objective value is = 119910119900 = 119888119861minus1

2 Calculate 119900119895 = 119911 minus 119888 1 le 119895 le 119899 with for 119895 ne 119861119894 1 le 119894 le 119898

Let 119910119900 = 1198981198941198991le119895le119899119910119900 If 119910119900 ≽ 0 then stop the current solution is optimal

3 If 119910119900 ≼ 0 then stop the problem is unbounded Otherwise it determines an index 119903

corresponding to a variable 119909119861 leaving the basis as follows

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

4 Pivot on and update the simplex tableau Go to step 2

Remark 5 (limitation of existing method) Investigation of the current models and methods

show that almost all of them are disable to formulate the real problems when the symmetric

assumption is not at hand Furthermore by defining a new product role we may to define a

practical method for solving the generalized model while the methods cannot work by the

pioneering approach

Now we are on the point of giving some illustrative examples In the next subsection we

give two examples The first problem is concerning to the symmetric form of fuzzy number

that discussed by Ganesan et al in [4] This example shows how we can solve the existing

shortcomings of the first model which is given in Example 31

36 Numerical examples

Here we are going to explore the proposed approach for solving a linear programming

problem with symmetric fuzzy numbers which is considered by Ganesan et al in [4]

33 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Example 31 We consider the fuzzy mathematical model which is given by Ganesan et al

in [4] The corresponding model is given in the below

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 1 asymp [475 5056 6]

141 + 133 + 2 asymp [460 4808 8]

121 + 15 2 + 3 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Using the multiplication role which is established by Ganesan et al the fuzzy optimal

solution and the fuzzy optimal value of objective function is as follows

1199091 = [0000] 1199092 = [415

1691045

169174

169174

169] 1199093 = [

460

13480

138

138

13]

1 = [0000] 2 = [0000] 3 = [62910

16977430

1693455

1693455

169]

119885119881 = [94235

169120265

16919819

16919819

169]

Using a ranking function such as Yager we obtain 119877(119881) =120783120782120789120784120787120782

120783120788120791

By substituting the fuzzy values of 2 and 3 in the objective function and using the new role

of fuzzy multiplication we have

119867 = 11988811 + 11988822 + 11988833

11199091 = [13152 2][00 0 0] = [00 0 0]

21199092 = [415

1691045

169174

169174

169] [121433] = [

9175

1699805

1695571

1695571

169]

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 34

31199093 = [151722] [460

13480

138

138

13] = [

97630

16997890

1691096

1691096

169]

119885119867 = [00 0 0]+ [9175

1699805

1695571

1695571

169] + [

97630

16997890

16914248

16914248

169]

= [106805

169107695

16919819

16919819

169]

We clearly obtain 119877(119867) =120783120782120789120784120787120782

120783120788120791

So according to the ordering role which is given in Definition 22 it is concluded that

119881 asymp 119867

In fact it has been shown in this example that the proposed method can solve the symmetric

version of the given fuzzy numbers as well as Ganesanrsquos method

Example 32 In this example we consider a general form of trapezoidal fuzzy number for

coefficients in the objective function if it is not necessary to be symmetric

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 4 asymp [475505 6 6]

141 + 133 + 5 asymp [460 4808 8]

121 + 15 2 + 6 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

The first tableau of the fuzzy primal simplex algorithm is given as below

Table 2 The first simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (-15-1343) (-14-1254) (-17-1543) 0 0 0 0

1199094

1199095

1199096

12 13 12 1 0 0

14 0 13 0 1 0

12 15 0 0 0 1

(47550566)

(46048088)

(46549555)

35 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

The first Since (01 02 03) = ((minus15minus1343) (minus14minus1254) (minus17minus1543)) and

(ℛ(01) ℛ(02)ℛ( 03)) = (minus1425minus1325minus1625) then 3 inters the basis and based

on the minimum ratio test the leaving variable is 5 Pivoting on 11991053 = 13

After calculating the amount of RHS column in Table 1it has been found that the

multiplication role which was defined by Ganesan and Veeramani in [4] is not satisfying and

we must hence use Definition 22 for obtaining the amount of multiplication as follows

119910119900119900119899119890119908 = 0 + (

460

13480

13 8

13 8

13)(151734)

Now assume that = (460

13480

13 8

13 8

13) and =(151734) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 6900

137820

137200

138160

13

120574 = 119860119907119890119903119886119892119890 120579 =7520

13

120572 = | 120574 minus 119898119894119899120579 | =620

13 120573 = |119898119886119909120579 minus 120574| =

640

13

So 120596 = |120573minus120572

2| =

10

13

= [(1198861 + 1198862

2) (

1198871 + 1198872

2) minus 120596 (

1198861 + 1198862

2) (

1198871 + 1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

= [7510

137530

131576

132056

13]

Updating the data is based on the about calculation concluded in Table 2

Table 3 The second simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (15

1369

13 94

13 95

13) (-14-1254) 0 0 (

15

1317

133

134

13) 0 (

7510

137530

13 1576

13 2056

13)

1199094

1199093

1199096

minus12

13 13 0

14

13 0 1

12 15 0

1 minus12

13 0

0 1

13 0

0 0 1

(415

131045

13 174

13 174

13)

(460

13480

13 8

13 8

13)

(46549555)

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 36

Since the problem is maximization the optimality condition is not valid and hence 2 enters

the basis and the leaving variable is 4 The next tableau based on the following calculations

is shown in Table 3

119910119900119900119899119890119908 = (

7510

137530

131576

132056

13) + (

415

1691045

169 174

169 174

169)(121445)

We suppose that = (415

1691045

169 174

169 174

169) and =(121445) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 4980

16912540

1695810

16914630

169

120574 = 119860119907119890119903119886119892119890 120579 =9490

169

120572 = | 120574 minus 119898119894119899120579 | =4510

169 120573 = |119898119886119909120579 minus 120574| =

5140

169

Hence

120596 = |120573minus120572

2| =

315

169

Then

= [(1198861+1198862

2) (

1198871+1198872

2) minus 120596 (

1198861+1198862

2) (

1198871+1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ] = [

18600

16919230

1696616

1697661

169]

Then the objective value in this basis is as follows

+(7510

137530

131576

132056

13) = [

116230

169117120

16927104

16934389

169]

Table 4 The optimal simplex tableau Basic 1 2 3 4 5 6 RHS

Z (27

169753

169 1282

169 1283

169) 0 0 (

12

1314

134

135

13) (

27

16977

169 99

169 100

169) 0 (

116230

169117120

169 27104

169 34389

169)

1199092

1199093

1199096

minus12

169 1 0

14

13 0 1

2208

169 0 0

1

13

minus12

169 0

0 1

13 0

minus15 180

169 1

(415

1691045

169 174

169 174

169)

(460

13480

13 8

13 8

13)

(62910

16977430

169 1019

169 1019

169)

This table is optimal and the optimal value of the variables and the objective function are as

follows

37 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

2lowast = (

415

1691045

169 174

169 174

169) 3

lowast = (460

13480

13 8

13 8

13) 6

lowast = (62910

16977430

169 1019

169 1019

169)

and 1lowast = 4

lowast = 5lowast = 0 and 119911lowast = (

116230

169117120

169 27104

169 34389

169)

Where

119885119867 = [13 15 3 4]1lowast + [12 14 4 5]2

lowast + [1517 3 4]3lowast

= [1315 3 4]0+ (121445)(415

1691045

169174

169174

169) + (151734)(

460

13480

138

138

13)

= (116230

169117120

16927104

16934389

169)

Thus it has been seen that the proposed approach gave the same results as the mentioned

problem in the method proposed by Ganesan et al In particular the proposed arithmetic

allows the decision makers to model as a general type where the promoters can be

formulated as a general from of trapezoidal fuzzy numbers which is more appropriate for

real situations than just in the symmetric form

4 Conclusion

In the paper a new role for the multiplication of two general forms of trapezoidal fuzzy

numbers has been defined In particular we saw that the new model sounds to be more

appropriate for the real situation while in the pioneering model which was suggested by

Ganessan et al [4] and subsequently Das et all [1] the trapezoidal fuzzy numbers are

assumed to be essential in the symmetric form Therefore these tools can be useful for

preparing some solving algorithms fuzzy primal dual simplex algorithms and so on

5 Acknowledgment

The authors would like to thanks the anonymous referees for their valuable comments to lead

us for improving the earlier version of the mentioned manuscript

References

[1] Das S K Mandal T amp Edalatpanah S A (2017) A mathematical model for solving fully fuzzy linear

programming problem with trapezoidal fuzzy numbers Applied Intelligence 46(3) 509-519 [2] Ebrahimnejad A amp Nasseri S H (2009) Using complementary slackness property to solve linear

programming with fuzzy parameters Fuzzy Information and Engineering 1(3) 233-245

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 38 [3] Ebrahimnejad A Nasseri S H Lotfi F H amp Soltanifar M (2010) A primal-dual method for linear

programming problems with fuzzy variables European Journal of Industrial Engineering 4(2) 189-209 [4] Ganesan K amp Veeramani P (2006) Fuzzy linear programs with trapezoidal fuzzy numbers Annals of

Operations Research 143(1) 305-315 [5] Lotfi F H Allahviranloo T Jondabeh M A amp Alizadeh L (2009) Solving a full fuzzy linear

programming using lexicography method and fuzzy approximate solution Applied Mathematical

Modelling 33(7) 3151-3156

[6] Klir G amp Yuan B (1995) Fuzzy sets and fuzzy logic Theory and Applications Prentice-Hall PTR

New Jersey

[7] Kumar A Kaur J amp Singh P (2011) A new method for solving fully fuzzy linear programming

problems Applied Mathematical Modelling 35(2) 817-823 [8] Mahdavi-Amiri N amp Nasseri S H (2007) Duality results and a dual simplex method for linear

programming problems with trapezoidal fuzzy variables Fuzzy sets and systems 158(17) 1961-1978 [9] Mahdavi-Amiri N amp Nasseri S H (2006) Duality in fuzzy number linear programming by use of a

certain linear ranking function Applied Mathematics and Computation 180(1) 206-216 [10] Maleki H R Tata M amp Mashinchi M (2000) Linear programming with fuzzy variables Fuzzy sets

and systems 109(1) 21-33 [11] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear

programming problem Advances in Fuzzy Sets and Systems 5(2) 81-95 [12] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for

solving flexible linear programming problems European Journal of Industrial Engineering 4(3) 372-

389 [13] Nasseri S H Ebrahimnejad A amp Mizuno S (2010) Duality in fuzzy linear programming with

symmetric trapezoidal numbers Applications and Applied Mathematics 5(10) 1467-1482 [14] Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval Information sciences

24(2) 143-161

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 26

makers to adapt to the real situations and it in particular can solve these problems as simple

as possible

2 Preliminaries

21 Definitions and notations

In this section some notations concepts new definitions and some fundamental results

have been presented on fuzzy arithmetic in the lake of symmetric assumption and a new

generalized definition has been particularly proposed for multiplication of the general

trapezoidal fuzzy numbers

Definition 21 A fuzzy set on ℝ is said to be a trapezoidal fuzzy number if there exists real

numbers 1198861 1198862 where 1198861 le 1198862 and ℎ1 ℎ2 ge 0 such that

(119909) =

119909

ℎ1+ℎ1 minus 1198861ℎ1

119891119900119903 119909 isin (1198861 minus ℎ1 1198861)

1 119891119900119903 119909 isin (1198861 1198862) minus119909

ℎ2+1198862 + ℎ2ℎ2

119891119900119903 119909 isin (1198862 1198862 + ℎ2)

0 119900119905ℎ119890119903119908119894119904119890

where (119909) is the membership function of fuzzy number

We denoted it by = [1198861 1198862 ℎ1 ℎ2]

In the above definition when ℎ1 = ℎ2 we called it as symmetric trapezoidal fuzzy number

Moreover when ℎ1 = ℎ2 = 0 = [1198861 1198862]

We use ℱ(ℝ) to denote the set of all trapezoidal fuzzy numbers

Definition 22 Let = [1198861 1198862 ℎ1 ℎ2] and = [1198871 1198872 1198961 1198962] be two trapezoidal fuzzy

numbers If so the arithmetic operations on and will be defined by

(i) Addition + = [1198861 1198862 ℎ1 ℎ2] + [1198871 1198872 1198961 1198962] = [1198861 + 1198871 1198862 + 1198872 ℎ1 + 1198961 ℎ2 + 1198962]

(ii) Subtraction minus = [1198861 1198862 ℎ1 ℎ2] minus [1198871 1198872 1198961 1198962] = [1198861 minus 1198872 1198862 minus 1198871 ℎ1 + 1198962 ℎ2 + 1198961]

22 Ranking functions

One convenient approach for solving the FLP problems is based on the concept of

comparison of fuzzy numbers by use of ranking functions (see [14]) An effective approach

27 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

for ordering the elements of ℱ(ℝ) is to define a ranking function ℛℱ(ℝ) ⟶ ℝ which maps

each fuzzy number into the real line where a natural order exists

Definition 23 We defined orders on ℱ(ℝ) by

≽ if and only if ℛ() ge ℛ()

≻ if and only if ℛ() ≻ ℛ()

≃ if and only if ℛ() ≃ ℛ()

where and b are in ℱ(ℝ) Also we write ≼ if and only if ≽

Attention has been exclusively paid to linear ranking functions that is a ranking function

ℛ such that

ℛ(119896 + ) = 119896ℛ() + ℛ()

for any and belonging to ℱ(ℝ) and any k isin ℝ

Now using the above approach we may rank a big category of fuzzy numbers where the

symmetric property is extended to the non-symmetric form too

Remark 1 For any trapezoidal fuzzy number the relation ≽ 0 holds if there exists 120576 gt 0

and 120572 gt 0 such that ≽ (minus120576 120576 120572 120572) We realized that ℛ(minus120576 120576 120572 120572) = 0 We also

considered asymp 0 only if ℛ() = 0 Thus without loss of generality throughout the paper

we regard 0 = (0 0 0 0) as the zero trapezoidal fuzzy number

The following lemma is now immediately at hand

Lemma 21 Let ℛ be any linear ranking function Then

(i) ≽ if and only if minus ≽ 0 if and only if minus ≽ minus

(ii) If ≽ and 119888 ≽ then + 119888 ≽ +

The linear ranking function has been considered on ℱ(ℝ) as

ℛ() = 119888119897119886119897 + 119888119906119886

119906 + 119888120572120572 + 119888120573120573

where = (119886119897 119886119906 120572 120573) and 119888119897 119888119906 119888120572 119888120573 are constants at least one of them is nonzero A

special version of the above linear ranking function was first proposed by Yager [14] (see

also [8] and [9])

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 28

Proposition 21 For any trapezoidal fuzzy numbers and 119888 we have

(i) 119888 ( + ) asymp (119888 + 119888 )

(ii) 119888 ( minus ) asymp (119888 minus 119888 )

Theorem 21 (i) The relation ≼ is a partial order relation on the set of trapezoidal fuzzy

numbers

(ii) The relation ≼ is a linear order relation on the set of trapezoidal fuzzy numbers

(iii) For any two trapezoidal fuzzy numbers and if ≼ then ≼ (1 minus 120582) + 120582 ≼

for all λ 0 le λ le 1

3 A new role in fuzzy arithmetic and fuzzy ordering

A definition of the multiplication of two symmetric fuzzy numbers is given by Ganesan and

Veeramani in [7] based on the Extension Principle (see [12]) However fuzzy arithmetic and

a fuzzy ordering role based on the given definition is established by many researchers

Although many valuable works are appeared in the literature there are a big limitation in the

basic definition In fact the symmetric assumption is not reasonable in practice since there

are many real situations that need to be dealt with by decision makers especially when the

main parameters of the system is formulated in the more general case and frankly in the form

of non-symmetric fuzzy number Moreover by introducing a new definition of the length of

120596 we may keep the length of the resulted fuzzy number which is obtained based on the

given multiplication role

For defining the multiplication of the two (non-symmetric) trapezoidal fuzzy numbers we

first needed to define the following notations

Let

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 120574 = 119860119907119890119903119886119892119890 120579

Then define

120572 = | 120574 minus 119898119894119899120579 | and 120573 = |119898119886119909120579 minus 120574|

Let 120596 = |120573minus120572

2| and we now may define the multiplication of two non-symmetric trapezoidal

fuzzy numbers as an extended role of the given definition for the symmetric form of fuzzy

numbers which is defined by Ganesan and Veeramani in [4] as follows

29 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Definition 31 Let = [1198861 1198862 ℎ1 ℎ2] and = [1198871 1198872 1198961 1198962] be two trapezoidal fuzzy

numbers Then the arithmetic operations on and are given by

Multiplication = [1198861 1198862 ℎ1 ℎ2][1198871 1198872 1198961 1198962]

= [(1198861 + 11988622

)(1198871 + 11988722

)minus 120596 (1198861 + 11988622

) (1198871 + 11988722

) +120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

From the above definition it is clear that

120582 = [1205821198861 1205821198862 120582ℎ1 120582ℎ2] 119891119900119903 120582 ge 0 [1205821198862 1205821198861 minus120582ℎ2 minus120582ℎ1] 119891119900119903 120582 lt 0

Remark 2 Depending upon the need we overcame the limitation of the multiplication role

which is given just for the symmetric kind of trapezoidal fuzzy numbers See in [4]

Definition 32 The model

119898119886119909 = sum 119888119909119899119895=1

(2-1)

119878 119905 sum 119886119894119895119909119899

119895=1≼ 119887 119894 = 1hellip 119898

119909 ≽ 0 119895 = 1hellip 119899

if 119886119894119895 isin ℝ and 119888 119909 119887 isin ℱ(ℝ) 119894 = 1hellip 119898 119895 = 1hellip 119899 is called a Semi-Fuzzy Linear

Programming (SFLP) problem

Definition 33 Any = (1199091 1199092 hellip 119909 ) isin ℱ119899(ℝ) where each 119909 isin ℱ(ℝ) which satisfies

the constraints and non-negativity restrictions of (2-1) is said to be a fuzzy feasible solution to

(2-1)

Definition 34 Let Q be the set of all fuzzy feasible solutions of (1) A fuzzy feasible

solution 119883 isin 119876 is said to be a fuzzy optimum solution to (1) if 119883 ≽ for all isin 119876

where = (1198881 1198882 hellip 119888 ) and = 11988811199091 + 11988821199092 +⋯+ 119888119909

32 Fuzzy Basic feasible solution

The concept of fuzzy basic feasible solution is similar to the given definition in [8] We

give a new definition for fuzzy solution associated to the discussed model as below

Definition 35 Let = (1199091 1199092 hellip 119909 ) isin ℱ119899(ℝ) suppose that = (119861

119879 119873119879) where 119909 asymp

119861minus1 119909 asymp 0 to be a fuzzy basic feasible solution of the system 119860 asymp If 119909 =

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 30

[minus120572119895 120572119895 ℎ119895 ℎ119895] for some 120572119894 gt 0 ℎ119895 and ℎ119895 ge 0 that is 119909 asymp 0 and every basic variable of the

corresponding to every feasible basic 119861 is positive is said to be a degenerated fuzzy basic

feasible solution

The following theorem concerns the so-called nondegenerate FNLP problems

Theorem 31 Let the FNLP problem be nondegenerate A basic feasible solution 119909 asymp

119861minus1 119909 asymp 0 is optimal to (2-1) only if 119911 ≽ 119888 for all 119895 1 le 119895 le 119899

Proof Suppose that 119909lowast = (119861119879 119873

119879)119879 is a basic feasible solution to (2-1) where 119909 asymp

119861minus1 119909 asymp 0 Then = 119888119909 = 119888119861minus1 On the other hand for every feasible solution

we have asymp 119860 asymp 119861119909 +119873119909 Hence we obtain

= 119888 minussum (119911 minus 119888)119909119895ne119861119894

(3 minus 1)

Then

= 119911lowast = 119888119909 + 119888119909 = 119888119861minus1 minussum (119888119861

minus1119886119895 minus 119888)119909119895ne119861119894

The proof can now be completed using (3-1) and Theorem 32 given in Section 3 1048576

Now we are going to devise a fuzzy primal simplex algorithm for solving the Problem (2-1)

33 Primal simplex method in tableau format for the fuzzy linear Problems

Consider the fuzzy linear programing problem as in (2-1) We rewrite the fuzzy linear

programing problem as

119872119886119909 = 119888119909 + 119888119909

119904 119905 119861119909 + 119873119909 =

119909 ≽ 0 119909 ≽ 0

Hence we have 119909 + 119861minus1119873119909 asymp 119861

minus1 Therefore + (119888119861minus1119873 minus 119888)119909 asymp 119888119861

minus1 With

119909 asymp 0 we have 119909 = 119861minus1 asymp 119910 and asymp 119888119861minus1 Thus we rewrited the above problem as

in Table 1

Table 1 The primal simplex tableau

Basis 119909 119909 RHS

0 119911 minus 119888 = 119888119861minus1119873 minus 119888 119910119900 = 119888119861

minus1

119909 119868 119884 = 119861minus1119873 119910 asymp 119861minus1

31 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Remark 3 Table 1 gives all the information needed to proceed with the simplex method The

fuzzy cost row in Table 1 is 119900119879 = 119888119861

minus1119860 minus 119888 where 119900119895 = 119888119861minus1119886119895 minus 119888 = 119911 minus 119888 1 le 119895 le

119899 with 119900119895 asymp 0 for 119895 = 119861119894 1 le 119894 le 119898 According to the optimality conditions (Theorem 31)

we are at the optimal solution if 119900119895 ≽ 0 for all 119895 ne 119861119894 1 le 119894 le 119898 On the other hand if

119900119895 ≼ 0 for some 119896 ne 119861119894 1 le 119894 le 119898 the problem is either unbounded or an exchange of a

basic variable 119909119861 and the nonbasic variable 119909 can be made to increase the rank of the

objective value (under nondegeneracy assumption) The following results established in [9]

help us devise the fuzzy primal simplex algorithm

Theorem 32 If there is a column 119896 (not in basis) in a fuzzy primal simplex tableau as 119910119900 =

119911 minus 119888 ≺ 0 and 119910119894119896 le 0 1 le 119894 le 119898 the problem (2-1) is unbounded

Theorem 33 If a nonbasic index 119896 exists in a fuzzy primal simplex tableau like 119910119900 = 119911 minus

119888 ≺ 0 and there exists a basic index 119861119894 like 119910119894119896 ge 0 a pivoting row 119903 can be found so that

pivoting on 119910119903119896 can yield a feasible tableau with a corresponding nondecreasing (increasing

under nondegeneracy assumption) fuzzy objective value

Remark 4 (see [9]) If there exists k such that 119910119900 ≺ 0 and the problem is not unbounded r

can be chosen as

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

Where it is the minimum fuzzy value of the above ratio

in order to replace 119909119861 in the basis by 119909 resulting in a new basis 119861 =

(119886119861119894 1198861198612 hellip 119886119861119903minus1 119886119896 119886119861119903+1 hellip 119886119861119898) The new basis is primal feasible and the

corresponding fuzzy objective value is nondecreasing (increasing under nondegeneracy

assumption) It can be shown that the new simplex tableau is obtained by pivoting on 119910119903119896

ie doing Gaussian elimination on the 119896 th column by using the pivot row 119903 with the pivot

119910119903119896 to transform the 119896 th column to the unit vector 119890119903 It is easily seen that the new fuzzy

objective value is 119910119900 = 119910119900 minus 119910119900119910119903119900

119910119903119896≽ 119910119900 where 119900119896 ≼ 0 and

119910119903119900

119910119903119896 (if the problem is

nondegenerate and consequently 119910119903119900

119910119903119896gt 0 and hence119910119900 asymp 119910119900)

We now describe the pivoting strategy

34 Pivoting and change of basis

If 119909 enters the basis and 119909119861 leaves the basis pivoting on 119910119903119896 in the primal simplex tableau

is carried out as follows

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 32

1) Divide row r by 119910119903119896

2) For 119894 = 01 119898 and 119894 ne 119903 update the 119894 th row by adding to it minus119910119894119896 times the new 119903th

row

We now present the primal simplex algorithm for the FNLP problem

35 The main steps of fuzzy primal simplex algorithm

Algorithm 31 The fuzzy primal simplex method

Assumption A basic feasible solution with basis B and the corresponding simplex tableau is

at hand

1 The fuzzy basic feasible solution is given by 119909 = 119910= 119861minus1 and 119909 asymp 0 The fuzzy

objective value is = 119910119900 = 119888119861minus1

2 Calculate 119900119895 = 119911 minus 119888 1 le 119895 le 119899 with for 119895 ne 119861119894 1 le 119894 le 119898

Let 119910119900 = 1198981198941198991le119895le119899119910119900 If 119910119900 ≽ 0 then stop the current solution is optimal

3 If 119910119900 ≼ 0 then stop the problem is unbounded Otherwise it determines an index 119903

corresponding to a variable 119909119861 leaving the basis as follows

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

4 Pivot on and update the simplex tableau Go to step 2

Remark 5 (limitation of existing method) Investigation of the current models and methods

show that almost all of them are disable to formulate the real problems when the symmetric

assumption is not at hand Furthermore by defining a new product role we may to define a

practical method for solving the generalized model while the methods cannot work by the

pioneering approach

Now we are on the point of giving some illustrative examples In the next subsection we

give two examples The first problem is concerning to the symmetric form of fuzzy number

that discussed by Ganesan et al in [4] This example shows how we can solve the existing

shortcomings of the first model which is given in Example 31

36 Numerical examples

Here we are going to explore the proposed approach for solving a linear programming

problem with symmetric fuzzy numbers which is considered by Ganesan et al in [4]

33 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Example 31 We consider the fuzzy mathematical model which is given by Ganesan et al

in [4] The corresponding model is given in the below

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 1 asymp [475 5056 6]

141 + 133 + 2 asymp [460 4808 8]

121 + 15 2 + 3 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Using the multiplication role which is established by Ganesan et al the fuzzy optimal

solution and the fuzzy optimal value of objective function is as follows

1199091 = [0000] 1199092 = [415

1691045

169174

169174

169] 1199093 = [

460

13480

138

138

13]

1 = [0000] 2 = [0000] 3 = [62910

16977430

1693455

1693455

169]

119885119881 = [94235

169120265

16919819

16919819

169]

Using a ranking function such as Yager we obtain 119877(119881) =120783120782120789120784120787120782

120783120788120791

By substituting the fuzzy values of 2 and 3 in the objective function and using the new role

of fuzzy multiplication we have

119867 = 11988811 + 11988822 + 11988833

11199091 = [13152 2][00 0 0] = [00 0 0]

21199092 = [415

1691045

169174

169174

169] [121433] = [

9175

1699805

1695571

1695571

169]

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 34

31199093 = [151722] [460

13480

138

138

13] = [

97630

16997890

1691096

1691096

169]

119885119867 = [00 0 0]+ [9175

1699805

1695571

1695571

169] + [

97630

16997890

16914248

16914248

169]

= [106805

169107695

16919819

16919819

169]

We clearly obtain 119877(119867) =120783120782120789120784120787120782

120783120788120791

So according to the ordering role which is given in Definition 22 it is concluded that

119881 asymp 119867

In fact it has been shown in this example that the proposed method can solve the symmetric

version of the given fuzzy numbers as well as Ganesanrsquos method

Example 32 In this example we consider a general form of trapezoidal fuzzy number for

coefficients in the objective function if it is not necessary to be symmetric

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 4 asymp [475505 6 6]

141 + 133 + 5 asymp [460 4808 8]

121 + 15 2 + 6 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

The first tableau of the fuzzy primal simplex algorithm is given as below

Table 2 The first simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (-15-1343) (-14-1254) (-17-1543) 0 0 0 0

1199094

1199095

1199096

12 13 12 1 0 0

14 0 13 0 1 0

12 15 0 0 0 1

(47550566)

(46048088)

(46549555)

35 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

The first Since (01 02 03) = ((minus15minus1343) (minus14minus1254) (minus17minus1543)) and

(ℛ(01) ℛ(02)ℛ( 03)) = (minus1425minus1325minus1625) then 3 inters the basis and based

on the minimum ratio test the leaving variable is 5 Pivoting on 11991053 = 13

After calculating the amount of RHS column in Table 1it has been found that the

multiplication role which was defined by Ganesan and Veeramani in [4] is not satisfying and

we must hence use Definition 22 for obtaining the amount of multiplication as follows

119910119900119900119899119890119908 = 0 + (

460

13480

13 8

13 8

13)(151734)

Now assume that = (460

13480

13 8

13 8

13) and =(151734) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 6900

137820

137200

138160

13

120574 = 119860119907119890119903119886119892119890 120579 =7520

13

120572 = | 120574 minus 119898119894119899120579 | =620

13 120573 = |119898119886119909120579 minus 120574| =

640

13

So 120596 = |120573minus120572

2| =

10

13

= [(1198861 + 1198862

2) (

1198871 + 1198872

2) minus 120596 (

1198861 + 1198862

2) (

1198871 + 1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

= [7510

137530

131576

132056

13]

Updating the data is based on the about calculation concluded in Table 2

Table 3 The second simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (15

1369

13 94

13 95

13) (-14-1254) 0 0 (

15

1317

133

134

13) 0 (

7510

137530

13 1576

13 2056

13)

1199094

1199093

1199096

minus12

13 13 0

14

13 0 1

12 15 0

1 minus12

13 0

0 1

13 0

0 0 1

(415

131045

13 174

13 174

13)

(460

13480

13 8

13 8

13)

(46549555)

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 36

Since the problem is maximization the optimality condition is not valid and hence 2 enters

the basis and the leaving variable is 4 The next tableau based on the following calculations

is shown in Table 3

119910119900119900119899119890119908 = (

7510

137530

131576

132056

13) + (

415

1691045

169 174

169 174

169)(121445)

We suppose that = (415

1691045

169 174

169 174

169) and =(121445) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 4980

16912540

1695810

16914630

169

120574 = 119860119907119890119903119886119892119890 120579 =9490

169

120572 = | 120574 minus 119898119894119899120579 | =4510

169 120573 = |119898119886119909120579 minus 120574| =

5140

169

Hence

120596 = |120573minus120572

2| =

315

169

Then

= [(1198861+1198862

2) (

1198871+1198872

2) minus 120596 (

1198861+1198862

2) (

1198871+1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ] = [

18600

16919230

1696616

1697661

169]

Then the objective value in this basis is as follows

+(7510

137530

131576

132056

13) = [

116230

169117120

16927104

16934389

169]

Table 4 The optimal simplex tableau Basic 1 2 3 4 5 6 RHS

Z (27

169753

169 1282

169 1283

169) 0 0 (

12

1314

134

135

13) (

27

16977

169 99

169 100

169) 0 (

116230

169117120

169 27104

169 34389

169)

1199092

1199093

1199096

minus12

169 1 0

14

13 0 1

2208

169 0 0

1

13

minus12

169 0

0 1

13 0

minus15 180

169 1

(415

1691045

169 174

169 174

169)

(460

13480

13 8

13 8

13)

(62910

16977430

169 1019

169 1019

169)

This table is optimal and the optimal value of the variables and the objective function are as

follows

37 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

2lowast = (

415

1691045

169 174

169 174

169) 3

lowast = (460

13480

13 8

13 8

13) 6

lowast = (62910

16977430

169 1019

169 1019

169)

and 1lowast = 4

lowast = 5lowast = 0 and 119911lowast = (

116230

169117120

169 27104

169 34389

169)

Where

119885119867 = [13 15 3 4]1lowast + [12 14 4 5]2

lowast + [1517 3 4]3lowast

= [1315 3 4]0+ (121445)(415

1691045

169174

169174

169) + (151734)(

460

13480

138

138

13)

= (116230

169117120

16927104

16934389

169)

Thus it has been seen that the proposed approach gave the same results as the mentioned

problem in the method proposed by Ganesan et al In particular the proposed arithmetic

allows the decision makers to model as a general type where the promoters can be

formulated as a general from of trapezoidal fuzzy numbers which is more appropriate for

real situations than just in the symmetric form

4 Conclusion

In the paper a new role for the multiplication of two general forms of trapezoidal fuzzy

numbers has been defined In particular we saw that the new model sounds to be more

appropriate for the real situation while in the pioneering model which was suggested by

Ganessan et al [4] and subsequently Das et all [1] the trapezoidal fuzzy numbers are

assumed to be essential in the symmetric form Therefore these tools can be useful for

preparing some solving algorithms fuzzy primal dual simplex algorithms and so on

5 Acknowledgment

The authors would like to thanks the anonymous referees for their valuable comments to lead

us for improving the earlier version of the mentioned manuscript

References

[1] Das S K Mandal T amp Edalatpanah S A (2017) A mathematical model for solving fully fuzzy linear

programming problem with trapezoidal fuzzy numbers Applied Intelligence 46(3) 509-519 [2] Ebrahimnejad A amp Nasseri S H (2009) Using complementary slackness property to solve linear

programming with fuzzy parameters Fuzzy Information and Engineering 1(3) 233-245

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 38 [3] Ebrahimnejad A Nasseri S H Lotfi F H amp Soltanifar M (2010) A primal-dual method for linear

programming problems with fuzzy variables European Journal of Industrial Engineering 4(2) 189-209 [4] Ganesan K amp Veeramani P (2006) Fuzzy linear programs with trapezoidal fuzzy numbers Annals of

Operations Research 143(1) 305-315 [5] Lotfi F H Allahviranloo T Jondabeh M A amp Alizadeh L (2009) Solving a full fuzzy linear

programming using lexicography method and fuzzy approximate solution Applied Mathematical

Modelling 33(7) 3151-3156

[6] Klir G amp Yuan B (1995) Fuzzy sets and fuzzy logic Theory and Applications Prentice-Hall PTR

New Jersey

[7] Kumar A Kaur J amp Singh P (2011) A new method for solving fully fuzzy linear programming

problems Applied Mathematical Modelling 35(2) 817-823 [8] Mahdavi-Amiri N amp Nasseri S H (2007) Duality results and a dual simplex method for linear

programming problems with trapezoidal fuzzy variables Fuzzy sets and systems 158(17) 1961-1978 [9] Mahdavi-Amiri N amp Nasseri S H (2006) Duality in fuzzy number linear programming by use of a

certain linear ranking function Applied Mathematics and Computation 180(1) 206-216 [10] Maleki H R Tata M amp Mashinchi M (2000) Linear programming with fuzzy variables Fuzzy sets

and systems 109(1) 21-33 [11] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear

programming problem Advances in Fuzzy Sets and Systems 5(2) 81-95 [12] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for

solving flexible linear programming problems European Journal of Industrial Engineering 4(3) 372-

389 [13] Nasseri S H Ebrahimnejad A amp Mizuno S (2010) Duality in fuzzy linear programming with

symmetric trapezoidal numbers Applications and Applied Mathematics 5(10) 1467-1482 [14] Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval Information sciences

24(2) 143-161

27 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

for ordering the elements of ℱ(ℝ) is to define a ranking function ℛℱ(ℝ) ⟶ ℝ which maps

each fuzzy number into the real line where a natural order exists

Definition 23 We defined orders on ℱ(ℝ) by

≽ if and only if ℛ() ge ℛ()

≻ if and only if ℛ() ≻ ℛ()

≃ if and only if ℛ() ≃ ℛ()

where and b are in ℱ(ℝ) Also we write ≼ if and only if ≽

Attention has been exclusively paid to linear ranking functions that is a ranking function

ℛ such that

ℛ(119896 + ) = 119896ℛ() + ℛ()

for any and belonging to ℱ(ℝ) and any k isin ℝ

Now using the above approach we may rank a big category of fuzzy numbers where the

symmetric property is extended to the non-symmetric form too

Remark 1 For any trapezoidal fuzzy number the relation ≽ 0 holds if there exists 120576 gt 0

and 120572 gt 0 such that ≽ (minus120576 120576 120572 120572) We realized that ℛ(minus120576 120576 120572 120572) = 0 We also

considered asymp 0 only if ℛ() = 0 Thus without loss of generality throughout the paper

we regard 0 = (0 0 0 0) as the zero trapezoidal fuzzy number

The following lemma is now immediately at hand

Lemma 21 Let ℛ be any linear ranking function Then

(i) ≽ if and only if minus ≽ 0 if and only if minus ≽ minus

(ii) If ≽ and 119888 ≽ then + 119888 ≽ +

The linear ranking function has been considered on ℱ(ℝ) as

ℛ() = 119888119897119886119897 + 119888119906119886

119906 + 119888120572120572 + 119888120573120573

where = (119886119897 119886119906 120572 120573) and 119888119897 119888119906 119888120572 119888120573 are constants at least one of them is nonzero A

special version of the above linear ranking function was first proposed by Yager [14] (see

also [8] and [9])

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 28

Proposition 21 For any trapezoidal fuzzy numbers and 119888 we have

(i) 119888 ( + ) asymp (119888 + 119888 )

(ii) 119888 ( minus ) asymp (119888 minus 119888 )

Theorem 21 (i) The relation ≼ is a partial order relation on the set of trapezoidal fuzzy

numbers

(ii) The relation ≼ is a linear order relation on the set of trapezoidal fuzzy numbers

(iii) For any two trapezoidal fuzzy numbers and if ≼ then ≼ (1 minus 120582) + 120582 ≼

for all λ 0 le λ le 1

3 A new role in fuzzy arithmetic and fuzzy ordering

A definition of the multiplication of two symmetric fuzzy numbers is given by Ganesan and

Veeramani in [7] based on the Extension Principle (see [12]) However fuzzy arithmetic and

a fuzzy ordering role based on the given definition is established by many researchers

Although many valuable works are appeared in the literature there are a big limitation in the

basic definition In fact the symmetric assumption is not reasonable in practice since there

are many real situations that need to be dealt with by decision makers especially when the

main parameters of the system is formulated in the more general case and frankly in the form

of non-symmetric fuzzy number Moreover by introducing a new definition of the length of

120596 we may keep the length of the resulted fuzzy number which is obtained based on the

given multiplication role

For defining the multiplication of the two (non-symmetric) trapezoidal fuzzy numbers we

first needed to define the following notations

Let

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 120574 = 119860119907119890119903119886119892119890 120579

Then define

120572 = | 120574 minus 119898119894119899120579 | and 120573 = |119898119886119909120579 minus 120574|

Let 120596 = |120573minus120572

2| and we now may define the multiplication of two non-symmetric trapezoidal

fuzzy numbers as an extended role of the given definition for the symmetric form of fuzzy

numbers which is defined by Ganesan and Veeramani in [4] as follows

29 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Definition 31 Let = [1198861 1198862 ℎ1 ℎ2] and = [1198871 1198872 1198961 1198962] be two trapezoidal fuzzy

numbers Then the arithmetic operations on and are given by

Multiplication = [1198861 1198862 ℎ1 ℎ2][1198871 1198872 1198961 1198962]

= [(1198861 + 11988622

)(1198871 + 11988722

)minus 120596 (1198861 + 11988622

) (1198871 + 11988722

) +120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

From the above definition it is clear that

120582 = [1205821198861 1205821198862 120582ℎ1 120582ℎ2] 119891119900119903 120582 ge 0 [1205821198862 1205821198861 minus120582ℎ2 minus120582ℎ1] 119891119900119903 120582 lt 0

Remark 2 Depending upon the need we overcame the limitation of the multiplication role

which is given just for the symmetric kind of trapezoidal fuzzy numbers See in [4]

Definition 32 The model

119898119886119909 = sum 119888119909119899119895=1

(2-1)

119878 119905 sum 119886119894119895119909119899

119895=1≼ 119887 119894 = 1hellip 119898

119909 ≽ 0 119895 = 1hellip 119899

if 119886119894119895 isin ℝ and 119888 119909 119887 isin ℱ(ℝ) 119894 = 1hellip 119898 119895 = 1hellip 119899 is called a Semi-Fuzzy Linear

Programming (SFLP) problem

Definition 33 Any = (1199091 1199092 hellip 119909 ) isin ℱ119899(ℝ) where each 119909 isin ℱ(ℝ) which satisfies

the constraints and non-negativity restrictions of (2-1) is said to be a fuzzy feasible solution to

(2-1)

Definition 34 Let Q be the set of all fuzzy feasible solutions of (1) A fuzzy feasible

solution 119883 isin 119876 is said to be a fuzzy optimum solution to (1) if 119883 ≽ for all isin 119876

where = (1198881 1198882 hellip 119888 ) and = 11988811199091 + 11988821199092 +⋯+ 119888119909

32 Fuzzy Basic feasible solution

The concept of fuzzy basic feasible solution is similar to the given definition in [8] We

give a new definition for fuzzy solution associated to the discussed model as below

Definition 35 Let = (1199091 1199092 hellip 119909 ) isin ℱ119899(ℝ) suppose that = (119861

119879 119873119879) where 119909 asymp

119861minus1 119909 asymp 0 to be a fuzzy basic feasible solution of the system 119860 asymp If 119909 =

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 30

[minus120572119895 120572119895 ℎ119895 ℎ119895] for some 120572119894 gt 0 ℎ119895 and ℎ119895 ge 0 that is 119909 asymp 0 and every basic variable of the

corresponding to every feasible basic 119861 is positive is said to be a degenerated fuzzy basic

feasible solution

The following theorem concerns the so-called nondegenerate FNLP problems

Theorem 31 Let the FNLP problem be nondegenerate A basic feasible solution 119909 asymp

119861minus1 119909 asymp 0 is optimal to (2-1) only if 119911 ≽ 119888 for all 119895 1 le 119895 le 119899

Proof Suppose that 119909lowast = (119861119879 119873

119879)119879 is a basic feasible solution to (2-1) where 119909 asymp

119861minus1 119909 asymp 0 Then = 119888119909 = 119888119861minus1 On the other hand for every feasible solution

we have asymp 119860 asymp 119861119909 +119873119909 Hence we obtain

= 119888 minussum (119911 minus 119888)119909119895ne119861119894

(3 minus 1)

Then

= 119911lowast = 119888119909 + 119888119909 = 119888119861minus1 minussum (119888119861

minus1119886119895 minus 119888)119909119895ne119861119894

The proof can now be completed using (3-1) and Theorem 32 given in Section 3 1048576

Now we are going to devise a fuzzy primal simplex algorithm for solving the Problem (2-1)

33 Primal simplex method in tableau format for the fuzzy linear Problems

Consider the fuzzy linear programing problem as in (2-1) We rewrite the fuzzy linear

programing problem as

119872119886119909 = 119888119909 + 119888119909

119904 119905 119861119909 + 119873119909 =

119909 ≽ 0 119909 ≽ 0

Hence we have 119909 + 119861minus1119873119909 asymp 119861

minus1 Therefore + (119888119861minus1119873 minus 119888)119909 asymp 119888119861

minus1 With

119909 asymp 0 we have 119909 = 119861minus1 asymp 119910 and asymp 119888119861minus1 Thus we rewrited the above problem as

in Table 1

Table 1 The primal simplex tableau

Basis 119909 119909 RHS

0 119911 minus 119888 = 119888119861minus1119873 minus 119888 119910119900 = 119888119861

minus1

119909 119868 119884 = 119861minus1119873 119910 asymp 119861minus1

31 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Remark 3 Table 1 gives all the information needed to proceed with the simplex method The

fuzzy cost row in Table 1 is 119900119879 = 119888119861

minus1119860 minus 119888 where 119900119895 = 119888119861minus1119886119895 minus 119888 = 119911 minus 119888 1 le 119895 le

119899 with 119900119895 asymp 0 for 119895 = 119861119894 1 le 119894 le 119898 According to the optimality conditions (Theorem 31)

we are at the optimal solution if 119900119895 ≽ 0 for all 119895 ne 119861119894 1 le 119894 le 119898 On the other hand if

119900119895 ≼ 0 for some 119896 ne 119861119894 1 le 119894 le 119898 the problem is either unbounded or an exchange of a

basic variable 119909119861 and the nonbasic variable 119909 can be made to increase the rank of the

objective value (under nondegeneracy assumption) The following results established in [9]

help us devise the fuzzy primal simplex algorithm

Theorem 32 If there is a column 119896 (not in basis) in a fuzzy primal simplex tableau as 119910119900 =

119911 minus 119888 ≺ 0 and 119910119894119896 le 0 1 le 119894 le 119898 the problem (2-1) is unbounded

Theorem 33 If a nonbasic index 119896 exists in a fuzzy primal simplex tableau like 119910119900 = 119911 minus

119888 ≺ 0 and there exists a basic index 119861119894 like 119910119894119896 ge 0 a pivoting row 119903 can be found so that

pivoting on 119910119903119896 can yield a feasible tableau with a corresponding nondecreasing (increasing

under nondegeneracy assumption) fuzzy objective value

Remark 4 (see [9]) If there exists k such that 119910119900 ≺ 0 and the problem is not unbounded r

can be chosen as

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

Where it is the minimum fuzzy value of the above ratio

in order to replace 119909119861 in the basis by 119909 resulting in a new basis 119861 =

(119886119861119894 1198861198612 hellip 119886119861119903minus1 119886119896 119886119861119903+1 hellip 119886119861119898) The new basis is primal feasible and the

corresponding fuzzy objective value is nondecreasing (increasing under nondegeneracy

assumption) It can be shown that the new simplex tableau is obtained by pivoting on 119910119903119896

ie doing Gaussian elimination on the 119896 th column by using the pivot row 119903 with the pivot

119910119903119896 to transform the 119896 th column to the unit vector 119890119903 It is easily seen that the new fuzzy

objective value is 119910119900 = 119910119900 minus 119910119900119910119903119900

119910119903119896≽ 119910119900 where 119900119896 ≼ 0 and

119910119903119900

119910119903119896 (if the problem is

nondegenerate and consequently 119910119903119900

119910119903119896gt 0 and hence119910119900 asymp 119910119900)

We now describe the pivoting strategy

34 Pivoting and change of basis

If 119909 enters the basis and 119909119861 leaves the basis pivoting on 119910119903119896 in the primal simplex tableau

is carried out as follows

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 32

1) Divide row r by 119910119903119896

2) For 119894 = 01 119898 and 119894 ne 119903 update the 119894 th row by adding to it minus119910119894119896 times the new 119903th

row

We now present the primal simplex algorithm for the FNLP problem

35 The main steps of fuzzy primal simplex algorithm

Algorithm 31 The fuzzy primal simplex method

Assumption A basic feasible solution with basis B and the corresponding simplex tableau is

at hand

1 The fuzzy basic feasible solution is given by 119909 = 119910= 119861minus1 and 119909 asymp 0 The fuzzy

objective value is = 119910119900 = 119888119861minus1

2 Calculate 119900119895 = 119911 minus 119888 1 le 119895 le 119899 with for 119895 ne 119861119894 1 le 119894 le 119898

Let 119910119900 = 1198981198941198991le119895le119899119910119900 If 119910119900 ≽ 0 then stop the current solution is optimal

3 If 119910119900 ≼ 0 then stop the problem is unbounded Otherwise it determines an index 119903

corresponding to a variable 119909119861 leaving the basis as follows

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

4 Pivot on and update the simplex tableau Go to step 2

Remark 5 (limitation of existing method) Investigation of the current models and methods

show that almost all of them are disable to formulate the real problems when the symmetric

assumption is not at hand Furthermore by defining a new product role we may to define a

practical method for solving the generalized model while the methods cannot work by the

pioneering approach

Now we are on the point of giving some illustrative examples In the next subsection we

give two examples The first problem is concerning to the symmetric form of fuzzy number

that discussed by Ganesan et al in [4] This example shows how we can solve the existing

shortcomings of the first model which is given in Example 31

36 Numerical examples

Here we are going to explore the proposed approach for solving a linear programming

problem with symmetric fuzzy numbers which is considered by Ganesan et al in [4]

33 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Example 31 We consider the fuzzy mathematical model which is given by Ganesan et al

in [4] The corresponding model is given in the below

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 1 asymp [475 5056 6]

141 + 133 + 2 asymp [460 4808 8]

121 + 15 2 + 3 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Using the multiplication role which is established by Ganesan et al the fuzzy optimal

solution and the fuzzy optimal value of objective function is as follows

1199091 = [0000] 1199092 = [415

1691045

169174

169174

169] 1199093 = [

460

13480

138

138

13]

1 = [0000] 2 = [0000] 3 = [62910

16977430

1693455

1693455

169]

119885119881 = [94235

169120265

16919819

16919819

169]

Using a ranking function such as Yager we obtain 119877(119881) =120783120782120789120784120787120782

120783120788120791

By substituting the fuzzy values of 2 and 3 in the objective function and using the new role

of fuzzy multiplication we have

119867 = 11988811 + 11988822 + 11988833

11199091 = [13152 2][00 0 0] = [00 0 0]

21199092 = [415

1691045

169174

169174

169] [121433] = [

9175

1699805

1695571

1695571

169]

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 34

31199093 = [151722] [460

13480

138

138

13] = [

97630

16997890

1691096

1691096

169]

119885119867 = [00 0 0]+ [9175

1699805

1695571

1695571

169] + [

97630

16997890

16914248

16914248

169]

= [106805

169107695

16919819

16919819

169]

We clearly obtain 119877(119867) =120783120782120789120784120787120782

120783120788120791

So according to the ordering role which is given in Definition 22 it is concluded that

119881 asymp 119867

In fact it has been shown in this example that the proposed method can solve the symmetric

version of the given fuzzy numbers as well as Ganesanrsquos method

Example 32 In this example we consider a general form of trapezoidal fuzzy number for

coefficients in the objective function if it is not necessary to be symmetric

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 4 asymp [475505 6 6]

141 + 133 + 5 asymp [460 4808 8]

121 + 15 2 + 6 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

The first tableau of the fuzzy primal simplex algorithm is given as below

Table 2 The first simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (-15-1343) (-14-1254) (-17-1543) 0 0 0 0

1199094

1199095

1199096

12 13 12 1 0 0

14 0 13 0 1 0

12 15 0 0 0 1

(47550566)

(46048088)

(46549555)

35 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

The first Since (01 02 03) = ((minus15minus1343) (minus14minus1254) (minus17minus1543)) and

(ℛ(01) ℛ(02)ℛ( 03)) = (minus1425minus1325minus1625) then 3 inters the basis and based

on the minimum ratio test the leaving variable is 5 Pivoting on 11991053 = 13

After calculating the amount of RHS column in Table 1it has been found that the

multiplication role which was defined by Ganesan and Veeramani in [4] is not satisfying and

we must hence use Definition 22 for obtaining the amount of multiplication as follows

119910119900119900119899119890119908 = 0 + (

460

13480

13 8

13 8

13)(151734)

Now assume that = (460

13480

13 8

13 8

13) and =(151734) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 6900

137820

137200

138160

13

120574 = 119860119907119890119903119886119892119890 120579 =7520

13

120572 = | 120574 minus 119898119894119899120579 | =620

13 120573 = |119898119886119909120579 minus 120574| =

640

13

So 120596 = |120573minus120572

2| =

10

13

= [(1198861 + 1198862

2) (

1198871 + 1198872

2) minus 120596 (

1198861 + 1198862

2) (

1198871 + 1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

= [7510

137530

131576

132056

13]

Updating the data is based on the about calculation concluded in Table 2

Table 3 The second simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (15

1369

13 94

13 95

13) (-14-1254) 0 0 (

15

1317

133

134

13) 0 (

7510

137530

13 1576

13 2056

13)

1199094

1199093

1199096

minus12

13 13 0

14

13 0 1

12 15 0

1 minus12

13 0

0 1

13 0

0 0 1

(415

131045

13 174

13 174

13)

(460

13480

13 8

13 8

13)

(46549555)

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 36

Since the problem is maximization the optimality condition is not valid and hence 2 enters

the basis and the leaving variable is 4 The next tableau based on the following calculations

is shown in Table 3

119910119900119900119899119890119908 = (

7510

137530

131576

132056

13) + (

415

1691045

169 174

169 174

169)(121445)

We suppose that = (415

1691045

169 174

169 174

169) and =(121445) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 4980

16912540

1695810

16914630

169

120574 = 119860119907119890119903119886119892119890 120579 =9490

169

120572 = | 120574 minus 119898119894119899120579 | =4510

169 120573 = |119898119886119909120579 minus 120574| =

5140

169

Hence

120596 = |120573minus120572

2| =

315

169

Then

= [(1198861+1198862

2) (

1198871+1198872

2) minus 120596 (

1198861+1198862

2) (

1198871+1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ] = [

18600

16919230

1696616

1697661

169]

Then the objective value in this basis is as follows

+(7510

137530

131576

132056

13) = [

116230

169117120

16927104

16934389

169]

Table 4 The optimal simplex tableau Basic 1 2 3 4 5 6 RHS

Z (27

169753

169 1282

169 1283

169) 0 0 (

12

1314

134

135

13) (

27

16977

169 99

169 100

169) 0 (

116230

169117120

169 27104

169 34389

169)

1199092

1199093

1199096

minus12

169 1 0

14

13 0 1

2208

169 0 0

1

13

minus12

169 0

0 1

13 0

minus15 180

169 1

(415

1691045

169 174

169 174

169)

(460

13480

13 8

13 8

13)

(62910

16977430

169 1019

169 1019

169)

This table is optimal and the optimal value of the variables and the objective function are as

follows

37 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

2lowast = (

415

1691045

169 174

169 174

169) 3

lowast = (460

13480

13 8

13 8

13) 6

lowast = (62910

16977430

169 1019

169 1019

169)

and 1lowast = 4

lowast = 5lowast = 0 and 119911lowast = (

116230

169117120

169 27104

169 34389

169)

Where

119885119867 = [13 15 3 4]1lowast + [12 14 4 5]2

lowast + [1517 3 4]3lowast

= [1315 3 4]0+ (121445)(415

1691045

169174

169174

169) + (151734)(

460

13480

138

138

13)

= (116230

169117120

16927104

16934389

169)

Thus it has been seen that the proposed approach gave the same results as the mentioned

problem in the method proposed by Ganesan et al In particular the proposed arithmetic

allows the decision makers to model as a general type where the promoters can be

formulated as a general from of trapezoidal fuzzy numbers which is more appropriate for

real situations than just in the symmetric form

4 Conclusion

In the paper a new role for the multiplication of two general forms of trapezoidal fuzzy

numbers has been defined In particular we saw that the new model sounds to be more

appropriate for the real situation while in the pioneering model which was suggested by

Ganessan et al [4] and subsequently Das et all [1] the trapezoidal fuzzy numbers are

assumed to be essential in the symmetric form Therefore these tools can be useful for

preparing some solving algorithms fuzzy primal dual simplex algorithms and so on

5 Acknowledgment

The authors would like to thanks the anonymous referees for their valuable comments to lead

us for improving the earlier version of the mentioned manuscript

References

[1] Das S K Mandal T amp Edalatpanah S A (2017) A mathematical model for solving fully fuzzy linear

programming problem with trapezoidal fuzzy numbers Applied Intelligence 46(3) 509-519 [2] Ebrahimnejad A amp Nasseri S H (2009) Using complementary slackness property to solve linear

programming with fuzzy parameters Fuzzy Information and Engineering 1(3) 233-245

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 38 [3] Ebrahimnejad A Nasseri S H Lotfi F H amp Soltanifar M (2010) A primal-dual method for linear

programming problems with fuzzy variables European Journal of Industrial Engineering 4(2) 189-209 [4] Ganesan K amp Veeramani P (2006) Fuzzy linear programs with trapezoidal fuzzy numbers Annals of

Operations Research 143(1) 305-315 [5] Lotfi F H Allahviranloo T Jondabeh M A amp Alizadeh L (2009) Solving a full fuzzy linear

programming using lexicography method and fuzzy approximate solution Applied Mathematical

Modelling 33(7) 3151-3156

[6] Klir G amp Yuan B (1995) Fuzzy sets and fuzzy logic Theory and Applications Prentice-Hall PTR

New Jersey

[7] Kumar A Kaur J amp Singh P (2011) A new method for solving fully fuzzy linear programming

problems Applied Mathematical Modelling 35(2) 817-823 [8] Mahdavi-Amiri N amp Nasseri S H (2007) Duality results and a dual simplex method for linear

programming problems with trapezoidal fuzzy variables Fuzzy sets and systems 158(17) 1961-1978 [9] Mahdavi-Amiri N amp Nasseri S H (2006) Duality in fuzzy number linear programming by use of a

certain linear ranking function Applied Mathematics and Computation 180(1) 206-216 [10] Maleki H R Tata M amp Mashinchi M (2000) Linear programming with fuzzy variables Fuzzy sets

and systems 109(1) 21-33 [11] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear

programming problem Advances in Fuzzy Sets and Systems 5(2) 81-95 [12] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for

solving flexible linear programming problems European Journal of Industrial Engineering 4(3) 372-

389 [13] Nasseri S H Ebrahimnejad A amp Mizuno S (2010) Duality in fuzzy linear programming with

symmetric trapezoidal numbers Applications and Applied Mathematics 5(10) 1467-1482 [14] Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval Information sciences

24(2) 143-161

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 28

Proposition 21 For any trapezoidal fuzzy numbers and 119888 we have

(i) 119888 ( + ) asymp (119888 + 119888 )

(ii) 119888 ( minus ) asymp (119888 minus 119888 )

Theorem 21 (i) The relation ≼ is a partial order relation on the set of trapezoidal fuzzy

numbers

(ii) The relation ≼ is a linear order relation on the set of trapezoidal fuzzy numbers

(iii) For any two trapezoidal fuzzy numbers and if ≼ then ≼ (1 minus 120582) + 120582 ≼

for all λ 0 le λ le 1

3 A new role in fuzzy arithmetic and fuzzy ordering

A definition of the multiplication of two symmetric fuzzy numbers is given by Ganesan and

Veeramani in [7] based on the Extension Principle (see [12]) However fuzzy arithmetic and

a fuzzy ordering role based on the given definition is established by many researchers

Although many valuable works are appeared in the literature there are a big limitation in the

basic definition In fact the symmetric assumption is not reasonable in practice since there

are many real situations that need to be dealt with by decision makers especially when the

main parameters of the system is formulated in the more general case and frankly in the form

of non-symmetric fuzzy number Moreover by introducing a new definition of the length of

120596 we may keep the length of the resulted fuzzy number which is obtained based on the

given multiplication role

For defining the multiplication of the two (non-symmetric) trapezoidal fuzzy numbers we

first needed to define the following notations

Let

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 120574 = 119860119907119890119903119886119892119890 120579

Then define

120572 = | 120574 minus 119898119894119899120579 | and 120573 = |119898119886119909120579 minus 120574|

Let 120596 = |120573minus120572

2| and we now may define the multiplication of two non-symmetric trapezoidal

fuzzy numbers as an extended role of the given definition for the symmetric form of fuzzy

numbers which is defined by Ganesan and Veeramani in [4] as follows

29 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Definition 31 Let = [1198861 1198862 ℎ1 ℎ2] and = [1198871 1198872 1198961 1198962] be two trapezoidal fuzzy

numbers Then the arithmetic operations on and are given by

Multiplication = [1198861 1198862 ℎ1 ℎ2][1198871 1198872 1198961 1198962]

= [(1198861 + 11988622

)(1198871 + 11988722

)minus 120596 (1198861 + 11988622

) (1198871 + 11988722

) +120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

From the above definition it is clear that

120582 = [1205821198861 1205821198862 120582ℎ1 120582ℎ2] 119891119900119903 120582 ge 0 [1205821198862 1205821198861 minus120582ℎ2 minus120582ℎ1] 119891119900119903 120582 lt 0

Remark 2 Depending upon the need we overcame the limitation of the multiplication role

which is given just for the symmetric kind of trapezoidal fuzzy numbers See in [4]

Definition 32 The model

119898119886119909 = sum 119888119909119899119895=1

(2-1)

119878 119905 sum 119886119894119895119909119899

119895=1≼ 119887 119894 = 1hellip 119898

119909 ≽ 0 119895 = 1hellip 119899

if 119886119894119895 isin ℝ and 119888 119909 119887 isin ℱ(ℝ) 119894 = 1hellip 119898 119895 = 1hellip 119899 is called a Semi-Fuzzy Linear

Programming (SFLP) problem

Definition 33 Any = (1199091 1199092 hellip 119909 ) isin ℱ119899(ℝ) where each 119909 isin ℱ(ℝ) which satisfies

the constraints and non-negativity restrictions of (2-1) is said to be a fuzzy feasible solution to

(2-1)

Definition 34 Let Q be the set of all fuzzy feasible solutions of (1) A fuzzy feasible

solution 119883 isin 119876 is said to be a fuzzy optimum solution to (1) if 119883 ≽ for all isin 119876

where = (1198881 1198882 hellip 119888 ) and = 11988811199091 + 11988821199092 +⋯+ 119888119909

32 Fuzzy Basic feasible solution

The concept of fuzzy basic feasible solution is similar to the given definition in [8] We

give a new definition for fuzzy solution associated to the discussed model as below

Definition 35 Let = (1199091 1199092 hellip 119909 ) isin ℱ119899(ℝ) suppose that = (119861

119879 119873119879) where 119909 asymp

119861minus1 119909 asymp 0 to be a fuzzy basic feasible solution of the system 119860 asymp If 119909 =

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 30

[minus120572119895 120572119895 ℎ119895 ℎ119895] for some 120572119894 gt 0 ℎ119895 and ℎ119895 ge 0 that is 119909 asymp 0 and every basic variable of the

corresponding to every feasible basic 119861 is positive is said to be a degenerated fuzzy basic

feasible solution

The following theorem concerns the so-called nondegenerate FNLP problems

Theorem 31 Let the FNLP problem be nondegenerate A basic feasible solution 119909 asymp

119861minus1 119909 asymp 0 is optimal to (2-1) only if 119911 ≽ 119888 for all 119895 1 le 119895 le 119899

Proof Suppose that 119909lowast = (119861119879 119873

119879)119879 is a basic feasible solution to (2-1) where 119909 asymp

119861minus1 119909 asymp 0 Then = 119888119909 = 119888119861minus1 On the other hand for every feasible solution

we have asymp 119860 asymp 119861119909 +119873119909 Hence we obtain

= 119888 minussum (119911 minus 119888)119909119895ne119861119894

(3 minus 1)

Then

= 119911lowast = 119888119909 + 119888119909 = 119888119861minus1 minussum (119888119861

minus1119886119895 minus 119888)119909119895ne119861119894

The proof can now be completed using (3-1) and Theorem 32 given in Section 3 1048576

Now we are going to devise a fuzzy primal simplex algorithm for solving the Problem (2-1)

33 Primal simplex method in tableau format for the fuzzy linear Problems

Consider the fuzzy linear programing problem as in (2-1) We rewrite the fuzzy linear

programing problem as

119872119886119909 = 119888119909 + 119888119909

119904 119905 119861119909 + 119873119909 =

119909 ≽ 0 119909 ≽ 0

Hence we have 119909 + 119861minus1119873119909 asymp 119861

minus1 Therefore + (119888119861minus1119873 minus 119888)119909 asymp 119888119861

minus1 With

119909 asymp 0 we have 119909 = 119861minus1 asymp 119910 and asymp 119888119861minus1 Thus we rewrited the above problem as

in Table 1

Table 1 The primal simplex tableau

Basis 119909 119909 RHS

0 119911 minus 119888 = 119888119861minus1119873 minus 119888 119910119900 = 119888119861

minus1

119909 119868 119884 = 119861minus1119873 119910 asymp 119861minus1

31 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Remark 3 Table 1 gives all the information needed to proceed with the simplex method The

fuzzy cost row in Table 1 is 119900119879 = 119888119861

minus1119860 minus 119888 where 119900119895 = 119888119861minus1119886119895 minus 119888 = 119911 minus 119888 1 le 119895 le

119899 with 119900119895 asymp 0 for 119895 = 119861119894 1 le 119894 le 119898 According to the optimality conditions (Theorem 31)

we are at the optimal solution if 119900119895 ≽ 0 for all 119895 ne 119861119894 1 le 119894 le 119898 On the other hand if

119900119895 ≼ 0 for some 119896 ne 119861119894 1 le 119894 le 119898 the problem is either unbounded or an exchange of a

basic variable 119909119861 and the nonbasic variable 119909 can be made to increase the rank of the

objective value (under nondegeneracy assumption) The following results established in [9]

help us devise the fuzzy primal simplex algorithm

Theorem 32 If there is a column 119896 (not in basis) in a fuzzy primal simplex tableau as 119910119900 =

119911 minus 119888 ≺ 0 and 119910119894119896 le 0 1 le 119894 le 119898 the problem (2-1) is unbounded

Theorem 33 If a nonbasic index 119896 exists in a fuzzy primal simplex tableau like 119910119900 = 119911 minus

119888 ≺ 0 and there exists a basic index 119861119894 like 119910119894119896 ge 0 a pivoting row 119903 can be found so that

pivoting on 119910119903119896 can yield a feasible tableau with a corresponding nondecreasing (increasing

under nondegeneracy assumption) fuzzy objective value

Remark 4 (see [9]) If there exists k such that 119910119900 ≺ 0 and the problem is not unbounded r

can be chosen as

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

Where it is the minimum fuzzy value of the above ratio

in order to replace 119909119861 in the basis by 119909 resulting in a new basis 119861 =

(119886119861119894 1198861198612 hellip 119886119861119903minus1 119886119896 119886119861119903+1 hellip 119886119861119898) The new basis is primal feasible and the

corresponding fuzzy objective value is nondecreasing (increasing under nondegeneracy

assumption) It can be shown that the new simplex tableau is obtained by pivoting on 119910119903119896

ie doing Gaussian elimination on the 119896 th column by using the pivot row 119903 with the pivot

119910119903119896 to transform the 119896 th column to the unit vector 119890119903 It is easily seen that the new fuzzy

objective value is 119910119900 = 119910119900 minus 119910119900119910119903119900

119910119903119896≽ 119910119900 where 119900119896 ≼ 0 and

119910119903119900

119910119903119896 (if the problem is

nondegenerate and consequently 119910119903119900

119910119903119896gt 0 and hence119910119900 asymp 119910119900)

We now describe the pivoting strategy

34 Pivoting and change of basis

If 119909 enters the basis and 119909119861 leaves the basis pivoting on 119910119903119896 in the primal simplex tableau

is carried out as follows

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 32

1) Divide row r by 119910119903119896

2) For 119894 = 01 119898 and 119894 ne 119903 update the 119894 th row by adding to it minus119910119894119896 times the new 119903th

row

We now present the primal simplex algorithm for the FNLP problem

35 The main steps of fuzzy primal simplex algorithm

Algorithm 31 The fuzzy primal simplex method

Assumption A basic feasible solution with basis B and the corresponding simplex tableau is

at hand

1 The fuzzy basic feasible solution is given by 119909 = 119910= 119861minus1 and 119909 asymp 0 The fuzzy

objective value is = 119910119900 = 119888119861minus1

2 Calculate 119900119895 = 119911 minus 119888 1 le 119895 le 119899 with for 119895 ne 119861119894 1 le 119894 le 119898

Let 119910119900 = 1198981198941198991le119895le119899119910119900 If 119910119900 ≽ 0 then stop the current solution is optimal

3 If 119910119900 ≼ 0 then stop the problem is unbounded Otherwise it determines an index 119903

corresponding to a variable 119909119861 leaving the basis as follows

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

4 Pivot on and update the simplex tableau Go to step 2

Remark 5 (limitation of existing method) Investigation of the current models and methods

show that almost all of them are disable to formulate the real problems when the symmetric

assumption is not at hand Furthermore by defining a new product role we may to define a

practical method for solving the generalized model while the methods cannot work by the

pioneering approach

Now we are on the point of giving some illustrative examples In the next subsection we

give two examples The first problem is concerning to the symmetric form of fuzzy number

that discussed by Ganesan et al in [4] This example shows how we can solve the existing

shortcomings of the first model which is given in Example 31

36 Numerical examples

Here we are going to explore the proposed approach for solving a linear programming

problem with symmetric fuzzy numbers which is considered by Ganesan et al in [4]

33 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Example 31 We consider the fuzzy mathematical model which is given by Ganesan et al

in [4] The corresponding model is given in the below

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 1 asymp [475 5056 6]

141 + 133 + 2 asymp [460 4808 8]

121 + 15 2 + 3 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Using the multiplication role which is established by Ganesan et al the fuzzy optimal

solution and the fuzzy optimal value of objective function is as follows

1199091 = [0000] 1199092 = [415

1691045

169174

169174

169] 1199093 = [

460

13480

138

138

13]

1 = [0000] 2 = [0000] 3 = [62910

16977430

1693455

1693455

169]

119885119881 = [94235

169120265

16919819

16919819

169]

Using a ranking function such as Yager we obtain 119877(119881) =120783120782120789120784120787120782

120783120788120791

By substituting the fuzzy values of 2 and 3 in the objective function and using the new role

of fuzzy multiplication we have

119867 = 11988811 + 11988822 + 11988833

11199091 = [13152 2][00 0 0] = [00 0 0]

21199092 = [415

1691045

169174

169174

169] [121433] = [

9175

1699805

1695571

1695571

169]

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 34

31199093 = [151722] [460

13480

138

138

13] = [

97630

16997890

1691096

1691096

169]

119885119867 = [00 0 0]+ [9175

1699805

1695571

1695571

169] + [

97630

16997890

16914248

16914248

169]

= [106805

169107695

16919819

16919819

169]

We clearly obtain 119877(119867) =120783120782120789120784120787120782

120783120788120791

So according to the ordering role which is given in Definition 22 it is concluded that

119881 asymp 119867

In fact it has been shown in this example that the proposed method can solve the symmetric

version of the given fuzzy numbers as well as Ganesanrsquos method

Example 32 In this example we consider a general form of trapezoidal fuzzy number for

coefficients in the objective function if it is not necessary to be symmetric

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 4 asymp [475505 6 6]

141 + 133 + 5 asymp [460 4808 8]

121 + 15 2 + 6 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

The first tableau of the fuzzy primal simplex algorithm is given as below

Table 2 The first simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (-15-1343) (-14-1254) (-17-1543) 0 0 0 0

1199094

1199095

1199096

12 13 12 1 0 0

14 0 13 0 1 0

12 15 0 0 0 1

(47550566)

(46048088)

(46549555)

35 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

The first Since (01 02 03) = ((minus15minus1343) (minus14minus1254) (minus17minus1543)) and

(ℛ(01) ℛ(02)ℛ( 03)) = (minus1425minus1325minus1625) then 3 inters the basis and based

on the minimum ratio test the leaving variable is 5 Pivoting on 11991053 = 13

After calculating the amount of RHS column in Table 1it has been found that the

multiplication role which was defined by Ganesan and Veeramani in [4] is not satisfying and

we must hence use Definition 22 for obtaining the amount of multiplication as follows

119910119900119900119899119890119908 = 0 + (

460

13480

13 8

13 8

13)(151734)

Now assume that = (460

13480

13 8

13 8

13) and =(151734) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 6900

137820

137200

138160

13

120574 = 119860119907119890119903119886119892119890 120579 =7520

13

120572 = | 120574 minus 119898119894119899120579 | =620

13 120573 = |119898119886119909120579 minus 120574| =

640

13

So 120596 = |120573minus120572

2| =

10

13

= [(1198861 + 1198862

2) (

1198871 + 1198872

2) minus 120596 (

1198861 + 1198862

2) (

1198871 + 1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

= [7510

137530

131576

132056

13]

Updating the data is based on the about calculation concluded in Table 2

Table 3 The second simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (15

1369

13 94

13 95

13) (-14-1254) 0 0 (

15

1317

133

134

13) 0 (

7510

137530

13 1576

13 2056

13)

1199094

1199093

1199096

minus12

13 13 0

14

13 0 1

12 15 0

1 minus12

13 0

0 1

13 0

0 0 1

(415

131045

13 174

13 174

13)

(460

13480

13 8

13 8

13)

(46549555)

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 36

Since the problem is maximization the optimality condition is not valid and hence 2 enters

the basis and the leaving variable is 4 The next tableau based on the following calculations

is shown in Table 3

119910119900119900119899119890119908 = (

7510

137530

131576

132056

13) + (

415

1691045

169 174

169 174

169)(121445)

We suppose that = (415

1691045

169 174

169 174

169) and =(121445) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 4980

16912540

1695810

16914630

169

120574 = 119860119907119890119903119886119892119890 120579 =9490

169

120572 = | 120574 minus 119898119894119899120579 | =4510

169 120573 = |119898119886119909120579 minus 120574| =

5140

169

Hence

120596 = |120573minus120572

2| =

315

169

Then

= [(1198861+1198862

2) (

1198871+1198872

2) minus 120596 (

1198861+1198862

2) (

1198871+1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ] = [

18600

16919230

1696616

1697661

169]

Then the objective value in this basis is as follows

+(7510

137530

131576

132056

13) = [

116230

169117120

16927104

16934389

169]

Table 4 The optimal simplex tableau Basic 1 2 3 4 5 6 RHS

Z (27

169753

169 1282

169 1283

169) 0 0 (

12

1314

134

135

13) (

27

16977

169 99

169 100

169) 0 (

116230

169117120

169 27104

169 34389

169)

1199092

1199093

1199096

minus12

169 1 0

14

13 0 1

2208

169 0 0

1

13

minus12

169 0

0 1

13 0

minus15 180

169 1

(415

1691045

169 174

169 174

169)

(460

13480

13 8

13 8

13)

(62910

16977430

169 1019

169 1019

169)

This table is optimal and the optimal value of the variables and the objective function are as

follows

37 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

2lowast = (

415

1691045

169 174

169 174

169) 3

lowast = (460

13480

13 8

13 8

13) 6

lowast = (62910

16977430

169 1019

169 1019

169)

and 1lowast = 4

lowast = 5lowast = 0 and 119911lowast = (

116230

169117120

169 27104

169 34389

169)

Where

119885119867 = [13 15 3 4]1lowast + [12 14 4 5]2

lowast + [1517 3 4]3lowast

= [1315 3 4]0+ (121445)(415

1691045

169174

169174

169) + (151734)(

460

13480

138

138

13)

= (116230

169117120

16927104

16934389

169)

Thus it has been seen that the proposed approach gave the same results as the mentioned

problem in the method proposed by Ganesan et al In particular the proposed arithmetic

allows the decision makers to model as a general type where the promoters can be

formulated as a general from of trapezoidal fuzzy numbers which is more appropriate for

real situations than just in the symmetric form

4 Conclusion

In the paper a new role for the multiplication of two general forms of trapezoidal fuzzy

numbers has been defined In particular we saw that the new model sounds to be more

appropriate for the real situation while in the pioneering model which was suggested by

Ganessan et al [4] and subsequently Das et all [1] the trapezoidal fuzzy numbers are

assumed to be essential in the symmetric form Therefore these tools can be useful for

preparing some solving algorithms fuzzy primal dual simplex algorithms and so on

5 Acknowledgment

The authors would like to thanks the anonymous referees for their valuable comments to lead

us for improving the earlier version of the mentioned manuscript

References

[1] Das S K Mandal T amp Edalatpanah S A (2017) A mathematical model for solving fully fuzzy linear

programming problem with trapezoidal fuzzy numbers Applied Intelligence 46(3) 509-519 [2] Ebrahimnejad A amp Nasseri S H (2009) Using complementary slackness property to solve linear

programming with fuzzy parameters Fuzzy Information and Engineering 1(3) 233-245

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 38 [3] Ebrahimnejad A Nasseri S H Lotfi F H amp Soltanifar M (2010) A primal-dual method for linear

programming problems with fuzzy variables European Journal of Industrial Engineering 4(2) 189-209 [4] Ganesan K amp Veeramani P (2006) Fuzzy linear programs with trapezoidal fuzzy numbers Annals of

Operations Research 143(1) 305-315 [5] Lotfi F H Allahviranloo T Jondabeh M A amp Alizadeh L (2009) Solving a full fuzzy linear

programming using lexicography method and fuzzy approximate solution Applied Mathematical

Modelling 33(7) 3151-3156

[6] Klir G amp Yuan B (1995) Fuzzy sets and fuzzy logic Theory and Applications Prentice-Hall PTR

New Jersey

[7] Kumar A Kaur J amp Singh P (2011) A new method for solving fully fuzzy linear programming

problems Applied Mathematical Modelling 35(2) 817-823 [8] Mahdavi-Amiri N amp Nasseri S H (2007) Duality results and a dual simplex method for linear

programming problems with trapezoidal fuzzy variables Fuzzy sets and systems 158(17) 1961-1978 [9] Mahdavi-Amiri N amp Nasseri S H (2006) Duality in fuzzy number linear programming by use of a

certain linear ranking function Applied Mathematics and Computation 180(1) 206-216 [10] Maleki H R Tata M amp Mashinchi M (2000) Linear programming with fuzzy variables Fuzzy sets

and systems 109(1) 21-33 [11] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear

programming problem Advances in Fuzzy Sets and Systems 5(2) 81-95 [12] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for

solving flexible linear programming problems European Journal of Industrial Engineering 4(3) 372-

389 [13] Nasseri S H Ebrahimnejad A amp Mizuno S (2010) Duality in fuzzy linear programming with

symmetric trapezoidal numbers Applications and Applied Mathematics 5(10) 1467-1482 [14] Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval Information sciences

24(2) 143-161

29 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Definition 31 Let = [1198861 1198862 ℎ1 ℎ2] and = [1198871 1198872 1198961 1198962] be two trapezoidal fuzzy

numbers Then the arithmetic operations on and are given by

Multiplication = [1198861 1198862 ℎ1 ℎ2][1198871 1198872 1198961 1198962]

= [(1198861 + 11988622

)(1198871 + 11988722

)minus 120596 (1198861 + 11988622

) (1198871 + 11988722

) +120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

From the above definition it is clear that

120582 = [1205821198861 1205821198862 120582ℎ1 120582ℎ2] 119891119900119903 120582 ge 0 [1205821198862 1205821198861 minus120582ℎ2 minus120582ℎ1] 119891119900119903 120582 lt 0

Remark 2 Depending upon the need we overcame the limitation of the multiplication role

which is given just for the symmetric kind of trapezoidal fuzzy numbers See in [4]

Definition 32 The model

119898119886119909 = sum 119888119909119899119895=1

(2-1)

119878 119905 sum 119886119894119895119909119899

119895=1≼ 119887 119894 = 1hellip 119898

119909 ≽ 0 119895 = 1hellip 119899

if 119886119894119895 isin ℝ and 119888 119909 119887 isin ℱ(ℝ) 119894 = 1hellip 119898 119895 = 1hellip 119899 is called a Semi-Fuzzy Linear

Programming (SFLP) problem

Definition 33 Any = (1199091 1199092 hellip 119909 ) isin ℱ119899(ℝ) where each 119909 isin ℱ(ℝ) which satisfies

the constraints and non-negativity restrictions of (2-1) is said to be a fuzzy feasible solution to

(2-1)

Definition 34 Let Q be the set of all fuzzy feasible solutions of (1) A fuzzy feasible

solution 119883 isin 119876 is said to be a fuzzy optimum solution to (1) if 119883 ≽ for all isin 119876

where = (1198881 1198882 hellip 119888 ) and = 11988811199091 + 11988821199092 +⋯+ 119888119909

32 Fuzzy Basic feasible solution

The concept of fuzzy basic feasible solution is similar to the given definition in [8] We

give a new definition for fuzzy solution associated to the discussed model as below

Definition 35 Let = (1199091 1199092 hellip 119909 ) isin ℱ119899(ℝ) suppose that = (119861

119879 119873119879) where 119909 asymp

119861minus1 119909 asymp 0 to be a fuzzy basic feasible solution of the system 119860 asymp If 119909 =

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 30

[minus120572119895 120572119895 ℎ119895 ℎ119895] for some 120572119894 gt 0 ℎ119895 and ℎ119895 ge 0 that is 119909 asymp 0 and every basic variable of the

corresponding to every feasible basic 119861 is positive is said to be a degenerated fuzzy basic

feasible solution

The following theorem concerns the so-called nondegenerate FNLP problems

Theorem 31 Let the FNLP problem be nondegenerate A basic feasible solution 119909 asymp

119861minus1 119909 asymp 0 is optimal to (2-1) only if 119911 ≽ 119888 for all 119895 1 le 119895 le 119899

Proof Suppose that 119909lowast = (119861119879 119873

119879)119879 is a basic feasible solution to (2-1) where 119909 asymp

119861minus1 119909 asymp 0 Then = 119888119909 = 119888119861minus1 On the other hand for every feasible solution

we have asymp 119860 asymp 119861119909 +119873119909 Hence we obtain

= 119888 minussum (119911 minus 119888)119909119895ne119861119894

(3 minus 1)

Then

= 119911lowast = 119888119909 + 119888119909 = 119888119861minus1 minussum (119888119861

minus1119886119895 minus 119888)119909119895ne119861119894

The proof can now be completed using (3-1) and Theorem 32 given in Section 3 1048576

Now we are going to devise a fuzzy primal simplex algorithm for solving the Problem (2-1)

33 Primal simplex method in tableau format for the fuzzy linear Problems

Consider the fuzzy linear programing problem as in (2-1) We rewrite the fuzzy linear

programing problem as

119872119886119909 = 119888119909 + 119888119909

119904 119905 119861119909 + 119873119909 =

119909 ≽ 0 119909 ≽ 0

Hence we have 119909 + 119861minus1119873119909 asymp 119861

minus1 Therefore + (119888119861minus1119873 minus 119888)119909 asymp 119888119861

minus1 With

119909 asymp 0 we have 119909 = 119861minus1 asymp 119910 and asymp 119888119861minus1 Thus we rewrited the above problem as

in Table 1

Table 1 The primal simplex tableau

Basis 119909 119909 RHS

0 119911 minus 119888 = 119888119861minus1119873 minus 119888 119910119900 = 119888119861

minus1

119909 119868 119884 = 119861minus1119873 119910 asymp 119861minus1

31 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Remark 3 Table 1 gives all the information needed to proceed with the simplex method The

fuzzy cost row in Table 1 is 119900119879 = 119888119861

minus1119860 minus 119888 where 119900119895 = 119888119861minus1119886119895 minus 119888 = 119911 minus 119888 1 le 119895 le

119899 with 119900119895 asymp 0 for 119895 = 119861119894 1 le 119894 le 119898 According to the optimality conditions (Theorem 31)

we are at the optimal solution if 119900119895 ≽ 0 for all 119895 ne 119861119894 1 le 119894 le 119898 On the other hand if

119900119895 ≼ 0 for some 119896 ne 119861119894 1 le 119894 le 119898 the problem is either unbounded or an exchange of a

basic variable 119909119861 and the nonbasic variable 119909 can be made to increase the rank of the

objective value (under nondegeneracy assumption) The following results established in [9]

help us devise the fuzzy primal simplex algorithm

Theorem 32 If there is a column 119896 (not in basis) in a fuzzy primal simplex tableau as 119910119900 =

119911 minus 119888 ≺ 0 and 119910119894119896 le 0 1 le 119894 le 119898 the problem (2-1) is unbounded

Theorem 33 If a nonbasic index 119896 exists in a fuzzy primal simplex tableau like 119910119900 = 119911 minus

119888 ≺ 0 and there exists a basic index 119861119894 like 119910119894119896 ge 0 a pivoting row 119903 can be found so that

pivoting on 119910119903119896 can yield a feasible tableau with a corresponding nondecreasing (increasing

under nondegeneracy assumption) fuzzy objective value

Remark 4 (see [9]) If there exists k such that 119910119900 ≺ 0 and the problem is not unbounded r

can be chosen as

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

Where it is the minimum fuzzy value of the above ratio

in order to replace 119909119861 in the basis by 119909 resulting in a new basis 119861 =

(119886119861119894 1198861198612 hellip 119886119861119903minus1 119886119896 119886119861119903+1 hellip 119886119861119898) The new basis is primal feasible and the

corresponding fuzzy objective value is nondecreasing (increasing under nondegeneracy

assumption) It can be shown that the new simplex tableau is obtained by pivoting on 119910119903119896

ie doing Gaussian elimination on the 119896 th column by using the pivot row 119903 with the pivot

119910119903119896 to transform the 119896 th column to the unit vector 119890119903 It is easily seen that the new fuzzy

objective value is 119910119900 = 119910119900 minus 119910119900119910119903119900

119910119903119896≽ 119910119900 where 119900119896 ≼ 0 and

119910119903119900

119910119903119896 (if the problem is

nondegenerate and consequently 119910119903119900

119910119903119896gt 0 and hence119910119900 asymp 119910119900)

We now describe the pivoting strategy

34 Pivoting and change of basis

If 119909 enters the basis and 119909119861 leaves the basis pivoting on 119910119903119896 in the primal simplex tableau

is carried out as follows

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 32

1) Divide row r by 119910119903119896

2) For 119894 = 01 119898 and 119894 ne 119903 update the 119894 th row by adding to it minus119910119894119896 times the new 119903th

row

We now present the primal simplex algorithm for the FNLP problem

35 The main steps of fuzzy primal simplex algorithm

Algorithm 31 The fuzzy primal simplex method

Assumption A basic feasible solution with basis B and the corresponding simplex tableau is

at hand

1 The fuzzy basic feasible solution is given by 119909 = 119910= 119861minus1 and 119909 asymp 0 The fuzzy

objective value is = 119910119900 = 119888119861minus1

2 Calculate 119900119895 = 119911 minus 119888 1 le 119895 le 119899 with for 119895 ne 119861119894 1 le 119894 le 119898

Let 119910119900 = 1198981198941198991le119895le119899119910119900 If 119910119900 ≽ 0 then stop the current solution is optimal

3 If 119910119900 ≼ 0 then stop the problem is unbounded Otherwise it determines an index 119903

corresponding to a variable 119909119861 leaving the basis as follows

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

4 Pivot on and update the simplex tableau Go to step 2

Remark 5 (limitation of existing method) Investigation of the current models and methods

show that almost all of them are disable to formulate the real problems when the symmetric

assumption is not at hand Furthermore by defining a new product role we may to define a

practical method for solving the generalized model while the methods cannot work by the

pioneering approach

Now we are on the point of giving some illustrative examples In the next subsection we

give two examples The first problem is concerning to the symmetric form of fuzzy number

that discussed by Ganesan et al in [4] This example shows how we can solve the existing

shortcomings of the first model which is given in Example 31

36 Numerical examples

Here we are going to explore the proposed approach for solving a linear programming

problem with symmetric fuzzy numbers which is considered by Ganesan et al in [4]

33 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Example 31 We consider the fuzzy mathematical model which is given by Ganesan et al

in [4] The corresponding model is given in the below

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 1 asymp [475 5056 6]

141 + 133 + 2 asymp [460 4808 8]

121 + 15 2 + 3 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Using the multiplication role which is established by Ganesan et al the fuzzy optimal

solution and the fuzzy optimal value of objective function is as follows

1199091 = [0000] 1199092 = [415

1691045

169174

169174

169] 1199093 = [

460

13480

138

138

13]

1 = [0000] 2 = [0000] 3 = [62910

16977430

1693455

1693455

169]

119885119881 = [94235

169120265

16919819

16919819

169]

Using a ranking function such as Yager we obtain 119877(119881) =120783120782120789120784120787120782

120783120788120791

By substituting the fuzzy values of 2 and 3 in the objective function and using the new role

of fuzzy multiplication we have

119867 = 11988811 + 11988822 + 11988833

11199091 = [13152 2][00 0 0] = [00 0 0]

21199092 = [415

1691045

169174

169174

169] [121433] = [

9175

1699805

1695571

1695571

169]

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 34

31199093 = [151722] [460

13480

138

138

13] = [

97630

16997890

1691096

1691096

169]

119885119867 = [00 0 0]+ [9175

1699805

1695571

1695571

169] + [

97630

16997890

16914248

16914248

169]

= [106805

169107695

16919819

16919819

169]

We clearly obtain 119877(119867) =120783120782120789120784120787120782

120783120788120791

So according to the ordering role which is given in Definition 22 it is concluded that

119881 asymp 119867

In fact it has been shown in this example that the proposed method can solve the symmetric

version of the given fuzzy numbers as well as Ganesanrsquos method

Example 32 In this example we consider a general form of trapezoidal fuzzy number for

coefficients in the objective function if it is not necessary to be symmetric

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 4 asymp [475505 6 6]

141 + 133 + 5 asymp [460 4808 8]

121 + 15 2 + 6 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

The first tableau of the fuzzy primal simplex algorithm is given as below

Table 2 The first simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (-15-1343) (-14-1254) (-17-1543) 0 0 0 0

1199094

1199095

1199096

12 13 12 1 0 0

14 0 13 0 1 0

12 15 0 0 0 1

(47550566)

(46048088)

(46549555)

35 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

The first Since (01 02 03) = ((minus15minus1343) (minus14minus1254) (minus17minus1543)) and

(ℛ(01) ℛ(02)ℛ( 03)) = (minus1425minus1325minus1625) then 3 inters the basis and based

on the minimum ratio test the leaving variable is 5 Pivoting on 11991053 = 13

After calculating the amount of RHS column in Table 1it has been found that the

multiplication role which was defined by Ganesan and Veeramani in [4] is not satisfying and

we must hence use Definition 22 for obtaining the amount of multiplication as follows

119910119900119900119899119890119908 = 0 + (

460

13480

13 8

13 8

13)(151734)

Now assume that = (460

13480

13 8

13 8

13) and =(151734) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 6900

137820

137200

138160

13

120574 = 119860119907119890119903119886119892119890 120579 =7520

13

120572 = | 120574 minus 119898119894119899120579 | =620

13 120573 = |119898119886119909120579 minus 120574| =

640

13

So 120596 = |120573minus120572

2| =

10

13

= [(1198861 + 1198862

2) (

1198871 + 1198872

2) minus 120596 (

1198861 + 1198862

2) (

1198871 + 1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

= [7510

137530

131576

132056

13]

Updating the data is based on the about calculation concluded in Table 2

Table 3 The second simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (15

1369

13 94

13 95

13) (-14-1254) 0 0 (

15

1317

133

134

13) 0 (

7510

137530

13 1576

13 2056

13)

1199094

1199093

1199096

minus12

13 13 0

14

13 0 1

12 15 0

1 minus12

13 0

0 1

13 0

0 0 1

(415

131045

13 174

13 174

13)

(460

13480

13 8

13 8

13)

(46549555)

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 36

Since the problem is maximization the optimality condition is not valid and hence 2 enters

the basis and the leaving variable is 4 The next tableau based on the following calculations

is shown in Table 3

119910119900119900119899119890119908 = (

7510

137530

131576

132056

13) + (

415

1691045

169 174

169 174

169)(121445)

We suppose that = (415

1691045

169 174

169 174

169) and =(121445) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 4980

16912540

1695810

16914630

169

120574 = 119860119907119890119903119886119892119890 120579 =9490

169

120572 = | 120574 minus 119898119894119899120579 | =4510

169 120573 = |119898119886119909120579 minus 120574| =

5140

169

Hence

120596 = |120573minus120572

2| =

315

169

Then

= [(1198861+1198862

2) (

1198871+1198872

2) minus 120596 (

1198861+1198862

2) (

1198871+1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ] = [

18600

16919230

1696616

1697661

169]

Then the objective value in this basis is as follows

+(7510

137530

131576

132056

13) = [

116230

169117120

16927104

16934389

169]

Table 4 The optimal simplex tableau Basic 1 2 3 4 5 6 RHS

Z (27

169753

169 1282

169 1283

169) 0 0 (

12

1314

134

135

13) (

27

16977

169 99

169 100

169) 0 (

116230

169117120

169 27104

169 34389

169)

1199092

1199093

1199096

minus12

169 1 0

14

13 0 1

2208

169 0 0

1

13

minus12

169 0

0 1

13 0

minus15 180

169 1

(415

1691045

169 174

169 174

169)

(460

13480

13 8

13 8

13)

(62910

16977430

169 1019

169 1019

169)

This table is optimal and the optimal value of the variables and the objective function are as

follows

37 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

2lowast = (

415

1691045

169 174

169 174

169) 3

lowast = (460

13480

13 8

13 8

13) 6

lowast = (62910

16977430

169 1019

169 1019

169)

and 1lowast = 4

lowast = 5lowast = 0 and 119911lowast = (

116230

169117120

169 27104

169 34389

169)

Where

119885119867 = [13 15 3 4]1lowast + [12 14 4 5]2

lowast + [1517 3 4]3lowast

= [1315 3 4]0+ (121445)(415

1691045

169174

169174

169) + (151734)(

460

13480

138

138

13)

= (116230

169117120

16927104

16934389

169)

Thus it has been seen that the proposed approach gave the same results as the mentioned

problem in the method proposed by Ganesan et al In particular the proposed arithmetic

allows the decision makers to model as a general type where the promoters can be

formulated as a general from of trapezoidal fuzzy numbers which is more appropriate for

real situations than just in the symmetric form

4 Conclusion

In the paper a new role for the multiplication of two general forms of trapezoidal fuzzy

numbers has been defined In particular we saw that the new model sounds to be more

appropriate for the real situation while in the pioneering model which was suggested by

Ganessan et al [4] and subsequently Das et all [1] the trapezoidal fuzzy numbers are

assumed to be essential in the symmetric form Therefore these tools can be useful for

preparing some solving algorithms fuzzy primal dual simplex algorithms and so on

5 Acknowledgment

The authors would like to thanks the anonymous referees for their valuable comments to lead

us for improving the earlier version of the mentioned manuscript

References

[1] Das S K Mandal T amp Edalatpanah S A (2017) A mathematical model for solving fully fuzzy linear

programming problem with trapezoidal fuzzy numbers Applied Intelligence 46(3) 509-519 [2] Ebrahimnejad A amp Nasseri S H (2009) Using complementary slackness property to solve linear

programming with fuzzy parameters Fuzzy Information and Engineering 1(3) 233-245

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 38 [3] Ebrahimnejad A Nasseri S H Lotfi F H amp Soltanifar M (2010) A primal-dual method for linear

programming problems with fuzzy variables European Journal of Industrial Engineering 4(2) 189-209 [4] Ganesan K amp Veeramani P (2006) Fuzzy linear programs with trapezoidal fuzzy numbers Annals of

Operations Research 143(1) 305-315 [5] Lotfi F H Allahviranloo T Jondabeh M A amp Alizadeh L (2009) Solving a full fuzzy linear

programming using lexicography method and fuzzy approximate solution Applied Mathematical

Modelling 33(7) 3151-3156

[6] Klir G amp Yuan B (1995) Fuzzy sets and fuzzy logic Theory and Applications Prentice-Hall PTR

New Jersey

[7] Kumar A Kaur J amp Singh P (2011) A new method for solving fully fuzzy linear programming

problems Applied Mathematical Modelling 35(2) 817-823 [8] Mahdavi-Amiri N amp Nasseri S H (2007) Duality results and a dual simplex method for linear

programming problems with trapezoidal fuzzy variables Fuzzy sets and systems 158(17) 1961-1978 [9] Mahdavi-Amiri N amp Nasseri S H (2006) Duality in fuzzy number linear programming by use of a

certain linear ranking function Applied Mathematics and Computation 180(1) 206-216 [10] Maleki H R Tata M amp Mashinchi M (2000) Linear programming with fuzzy variables Fuzzy sets

and systems 109(1) 21-33 [11] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear

programming problem Advances in Fuzzy Sets and Systems 5(2) 81-95 [12] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for

solving flexible linear programming problems European Journal of Industrial Engineering 4(3) 372-

389 [13] Nasseri S H Ebrahimnejad A amp Mizuno S (2010) Duality in fuzzy linear programming with

symmetric trapezoidal numbers Applications and Applied Mathematics 5(10) 1467-1482 [14] Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval Information sciences

24(2) 143-161

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 30

[minus120572119895 120572119895 ℎ119895 ℎ119895] for some 120572119894 gt 0 ℎ119895 and ℎ119895 ge 0 that is 119909 asymp 0 and every basic variable of the

corresponding to every feasible basic 119861 is positive is said to be a degenerated fuzzy basic

feasible solution

The following theorem concerns the so-called nondegenerate FNLP problems

Theorem 31 Let the FNLP problem be nondegenerate A basic feasible solution 119909 asymp

119861minus1 119909 asymp 0 is optimal to (2-1) only if 119911 ≽ 119888 for all 119895 1 le 119895 le 119899

Proof Suppose that 119909lowast = (119861119879 119873

119879)119879 is a basic feasible solution to (2-1) where 119909 asymp

119861minus1 119909 asymp 0 Then = 119888119909 = 119888119861minus1 On the other hand for every feasible solution

we have asymp 119860 asymp 119861119909 +119873119909 Hence we obtain

= 119888 minussum (119911 minus 119888)119909119895ne119861119894

(3 minus 1)

Then

= 119911lowast = 119888119909 + 119888119909 = 119888119861minus1 minussum (119888119861

minus1119886119895 minus 119888)119909119895ne119861119894

The proof can now be completed using (3-1) and Theorem 32 given in Section 3 1048576

Now we are going to devise a fuzzy primal simplex algorithm for solving the Problem (2-1)

33 Primal simplex method in tableau format for the fuzzy linear Problems

Consider the fuzzy linear programing problem as in (2-1) We rewrite the fuzzy linear

programing problem as

119872119886119909 = 119888119909 + 119888119909

119904 119905 119861119909 + 119873119909 =

119909 ≽ 0 119909 ≽ 0

Hence we have 119909 + 119861minus1119873119909 asymp 119861

minus1 Therefore + (119888119861minus1119873 minus 119888)119909 asymp 119888119861

minus1 With

119909 asymp 0 we have 119909 = 119861minus1 asymp 119910 and asymp 119888119861minus1 Thus we rewrited the above problem as

in Table 1

Table 1 The primal simplex tableau

Basis 119909 119909 RHS

0 119911 minus 119888 = 119888119861minus1119873 minus 119888 119910119900 = 119888119861

minus1

119909 119868 119884 = 119861minus1119873 119910 asymp 119861minus1

31 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Remark 3 Table 1 gives all the information needed to proceed with the simplex method The

fuzzy cost row in Table 1 is 119900119879 = 119888119861

minus1119860 minus 119888 where 119900119895 = 119888119861minus1119886119895 minus 119888 = 119911 minus 119888 1 le 119895 le

119899 with 119900119895 asymp 0 for 119895 = 119861119894 1 le 119894 le 119898 According to the optimality conditions (Theorem 31)

we are at the optimal solution if 119900119895 ≽ 0 for all 119895 ne 119861119894 1 le 119894 le 119898 On the other hand if

119900119895 ≼ 0 for some 119896 ne 119861119894 1 le 119894 le 119898 the problem is either unbounded or an exchange of a

basic variable 119909119861 and the nonbasic variable 119909 can be made to increase the rank of the

objective value (under nondegeneracy assumption) The following results established in [9]

help us devise the fuzzy primal simplex algorithm

Theorem 32 If there is a column 119896 (not in basis) in a fuzzy primal simplex tableau as 119910119900 =

119911 minus 119888 ≺ 0 and 119910119894119896 le 0 1 le 119894 le 119898 the problem (2-1) is unbounded

Theorem 33 If a nonbasic index 119896 exists in a fuzzy primal simplex tableau like 119910119900 = 119911 minus

119888 ≺ 0 and there exists a basic index 119861119894 like 119910119894119896 ge 0 a pivoting row 119903 can be found so that

pivoting on 119910119903119896 can yield a feasible tableau with a corresponding nondecreasing (increasing

under nondegeneracy assumption) fuzzy objective value

Remark 4 (see [9]) If there exists k such that 119910119900 ≺ 0 and the problem is not unbounded r

can be chosen as

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

Where it is the minimum fuzzy value of the above ratio

in order to replace 119909119861 in the basis by 119909 resulting in a new basis 119861 =

(119886119861119894 1198861198612 hellip 119886119861119903minus1 119886119896 119886119861119903+1 hellip 119886119861119898) The new basis is primal feasible and the

corresponding fuzzy objective value is nondecreasing (increasing under nondegeneracy

assumption) It can be shown that the new simplex tableau is obtained by pivoting on 119910119903119896

ie doing Gaussian elimination on the 119896 th column by using the pivot row 119903 with the pivot

119910119903119896 to transform the 119896 th column to the unit vector 119890119903 It is easily seen that the new fuzzy

objective value is 119910119900 = 119910119900 minus 119910119900119910119903119900

119910119903119896≽ 119910119900 where 119900119896 ≼ 0 and

119910119903119900

119910119903119896 (if the problem is

nondegenerate and consequently 119910119903119900

119910119903119896gt 0 and hence119910119900 asymp 119910119900)

We now describe the pivoting strategy

34 Pivoting and change of basis

If 119909 enters the basis and 119909119861 leaves the basis pivoting on 119910119903119896 in the primal simplex tableau

is carried out as follows

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 32

1) Divide row r by 119910119903119896

2) For 119894 = 01 119898 and 119894 ne 119903 update the 119894 th row by adding to it minus119910119894119896 times the new 119903th

row

We now present the primal simplex algorithm for the FNLP problem

35 The main steps of fuzzy primal simplex algorithm

Algorithm 31 The fuzzy primal simplex method

Assumption A basic feasible solution with basis B and the corresponding simplex tableau is

at hand

1 The fuzzy basic feasible solution is given by 119909 = 119910= 119861minus1 and 119909 asymp 0 The fuzzy

objective value is = 119910119900 = 119888119861minus1

2 Calculate 119900119895 = 119911 minus 119888 1 le 119895 le 119899 with for 119895 ne 119861119894 1 le 119894 le 119898

Let 119910119900 = 1198981198941198991le119895le119899119910119900 If 119910119900 ≽ 0 then stop the current solution is optimal

3 If 119910119900 ≼ 0 then stop the problem is unbounded Otherwise it determines an index 119903

corresponding to a variable 119909119861 leaving the basis as follows

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

4 Pivot on and update the simplex tableau Go to step 2

Remark 5 (limitation of existing method) Investigation of the current models and methods

show that almost all of them are disable to formulate the real problems when the symmetric

assumption is not at hand Furthermore by defining a new product role we may to define a

practical method for solving the generalized model while the methods cannot work by the

pioneering approach

Now we are on the point of giving some illustrative examples In the next subsection we

give two examples The first problem is concerning to the symmetric form of fuzzy number

that discussed by Ganesan et al in [4] This example shows how we can solve the existing

shortcomings of the first model which is given in Example 31

36 Numerical examples

Here we are going to explore the proposed approach for solving a linear programming

problem with symmetric fuzzy numbers which is considered by Ganesan et al in [4]

33 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Example 31 We consider the fuzzy mathematical model which is given by Ganesan et al

in [4] The corresponding model is given in the below

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 1 asymp [475 5056 6]

141 + 133 + 2 asymp [460 4808 8]

121 + 15 2 + 3 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Using the multiplication role which is established by Ganesan et al the fuzzy optimal

solution and the fuzzy optimal value of objective function is as follows

1199091 = [0000] 1199092 = [415

1691045

169174

169174

169] 1199093 = [

460

13480

138

138

13]

1 = [0000] 2 = [0000] 3 = [62910

16977430

1693455

1693455

169]

119885119881 = [94235

169120265

16919819

16919819

169]

Using a ranking function such as Yager we obtain 119877(119881) =120783120782120789120784120787120782

120783120788120791

By substituting the fuzzy values of 2 and 3 in the objective function and using the new role

of fuzzy multiplication we have

119867 = 11988811 + 11988822 + 11988833

11199091 = [13152 2][00 0 0] = [00 0 0]

21199092 = [415

1691045

169174

169174

169] [121433] = [

9175

1699805

1695571

1695571

169]

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 34

31199093 = [151722] [460

13480

138

138

13] = [

97630

16997890

1691096

1691096

169]

119885119867 = [00 0 0]+ [9175

1699805

1695571

1695571

169] + [

97630

16997890

16914248

16914248

169]

= [106805

169107695

16919819

16919819

169]

We clearly obtain 119877(119867) =120783120782120789120784120787120782

120783120788120791

So according to the ordering role which is given in Definition 22 it is concluded that

119881 asymp 119867

In fact it has been shown in this example that the proposed method can solve the symmetric

version of the given fuzzy numbers as well as Ganesanrsquos method

Example 32 In this example we consider a general form of trapezoidal fuzzy number for

coefficients in the objective function if it is not necessary to be symmetric

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 4 asymp [475505 6 6]

141 + 133 + 5 asymp [460 4808 8]

121 + 15 2 + 6 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

The first tableau of the fuzzy primal simplex algorithm is given as below

Table 2 The first simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (-15-1343) (-14-1254) (-17-1543) 0 0 0 0

1199094

1199095

1199096

12 13 12 1 0 0

14 0 13 0 1 0

12 15 0 0 0 1

(47550566)

(46048088)

(46549555)

35 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

The first Since (01 02 03) = ((minus15minus1343) (minus14minus1254) (minus17minus1543)) and

(ℛ(01) ℛ(02)ℛ( 03)) = (minus1425minus1325minus1625) then 3 inters the basis and based

on the minimum ratio test the leaving variable is 5 Pivoting on 11991053 = 13

After calculating the amount of RHS column in Table 1it has been found that the

multiplication role which was defined by Ganesan and Veeramani in [4] is not satisfying and

we must hence use Definition 22 for obtaining the amount of multiplication as follows

119910119900119900119899119890119908 = 0 + (

460

13480

13 8

13 8

13)(151734)

Now assume that = (460

13480

13 8

13 8

13) and =(151734) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 6900

137820

137200

138160

13

120574 = 119860119907119890119903119886119892119890 120579 =7520

13

120572 = | 120574 minus 119898119894119899120579 | =620

13 120573 = |119898119886119909120579 minus 120574| =

640

13

So 120596 = |120573minus120572

2| =

10

13

= [(1198861 + 1198862

2) (

1198871 + 1198872

2) minus 120596 (

1198861 + 1198862

2) (

1198871 + 1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

= [7510

137530

131576

132056

13]

Updating the data is based on the about calculation concluded in Table 2

Table 3 The second simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (15

1369

13 94

13 95

13) (-14-1254) 0 0 (

15

1317

133

134

13) 0 (

7510

137530

13 1576

13 2056

13)

1199094

1199093

1199096

minus12

13 13 0

14

13 0 1

12 15 0

1 minus12

13 0

0 1

13 0

0 0 1

(415

131045

13 174

13 174

13)

(460

13480

13 8

13 8

13)

(46549555)

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 36

Since the problem is maximization the optimality condition is not valid and hence 2 enters

the basis and the leaving variable is 4 The next tableau based on the following calculations

is shown in Table 3

119910119900119900119899119890119908 = (

7510

137530

131576

132056

13) + (

415

1691045

169 174

169 174

169)(121445)

We suppose that = (415

1691045

169 174

169 174

169) and =(121445) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 4980

16912540

1695810

16914630

169

120574 = 119860119907119890119903119886119892119890 120579 =9490

169

120572 = | 120574 minus 119898119894119899120579 | =4510

169 120573 = |119898119886119909120579 minus 120574| =

5140

169

Hence

120596 = |120573minus120572

2| =

315

169

Then

= [(1198861+1198862

2) (

1198871+1198872

2) minus 120596 (

1198861+1198862

2) (

1198871+1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ] = [

18600

16919230

1696616

1697661

169]

Then the objective value in this basis is as follows

+(7510

137530

131576

132056

13) = [

116230

169117120

16927104

16934389

169]

Table 4 The optimal simplex tableau Basic 1 2 3 4 5 6 RHS

Z (27

169753

169 1282

169 1283

169) 0 0 (

12

1314

134

135

13) (

27

16977

169 99

169 100

169) 0 (

116230

169117120

169 27104

169 34389

169)

1199092

1199093

1199096

minus12

169 1 0

14

13 0 1

2208

169 0 0

1

13

minus12

169 0

0 1

13 0

minus15 180

169 1

(415

1691045

169 174

169 174

169)

(460

13480

13 8

13 8

13)

(62910

16977430

169 1019

169 1019

169)

This table is optimal and the optimal value of the variables and the objective function are as

follows

37 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

2lowast = (

415

1691045

169 174

169 174

169) 3

lowast = (460

13480

13 8

13 8

13) 6

lowast = (62910

16977430

169 1019

169 1019

169)

and 1lowast = 4

lowast = 5lowast = 0 and 119911lowast = (

116230

169117120

169 27104

169 34389

169)

Where

119885119867 = [13 15 3 4]1lowast + [12 14 4 5]2

lowast + [1517 3 4]3lowast

= [1315 3 4]0+ (121445)(415

1691045

169174

169174

169) + (151734)(

460

13480

138

138

13)

= (116230

169117120

16927104

16934389

169)

Thus it has been seen that the proposed approach gave the same results as the mentioned

problem in the method proposed by Ganesan et al In particular the proposed arithmetic

allows the decision makers to model as a general type where the promoters can be

formulated as a general from of trapezoidal fuzzy numbers which is more appropriate for

real situations than just in the symmetric form

4 Conclusion

In the paper a new role for the multiplication of two general forms of trapezoidal fuzzy

numbers has been defined In particular we saw that the new model sounds to be more

appropriate for the real situation while in the pioneering model which was suggested by

Ganessan et al [4] and subsequently Das et all [1] the trapezoidal fuzzy numbers are

assumed to be essential in the symmetric form Therefore these tools can be useful for

preparing some solving algorithms fuzzy primal dual simplex algorithms and so on

5 Acknowledgment

The authors would like to thanks the anonymous referees for their valuable comments to lead

us for improving the earlier version of the mentioned manuscript

References

[1] Das S K Mandal T amp Edalatpanah S A (2017) A mathematical model for solving fully fuzzy linear

programming problem with trapezoidal fuzzy numbers Applied Intelligence 46(3) 509-519 [2] Ebrahimnejad A amp Nasseri S H (2009) Using complementary slackness property to solve linear

programming with fuzzy parameters Fuzzy Information and Engineering 1(3) 233-245

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 38 [3] Ebrahimnejad A Nasseri S H Lotfi F H amp Soltanifar M (2010) A primal-dual method for linear

programming problems with fuzzy variables European Journal of Industrial Engineering 4(2) 189-209 [4] Ganesan K amp Veeramani P (2006) Fuzzy linear programs with trapezoidal fuzzy numbers Annals of

Operations Research 143(1) 305-315 [5] Lotfi F H Allahviranloo T Jondabeh M A amp Alizadeh L (2009) Solving a full fuzzy linear

programming using lexicography method and fuzzy approximate solution Applied Mathematical

Modelling 33(7) 3151-3156

[6] Klir G amp Yuan B (1995) Fuzzy sets and fuzzy logic Theory and Applications Prentice-Hall PTR

New Jersey

[7] Kumar A Kaur J amp Singh P (2011) A new method for solving fully fuzzy linear programming

problems Applied Mathematical Modelling 35(2) 817-823 [8] Mahdavi-Amiri N amp Nasseri S H (2007) Duality results and a dual simplex method for linear

programming problems with trapezoidal fuzzy variables Fuzzy sets and systems 158(17) 1961-1978 [9] Mahdavi-Amiri N amp Nasseri S H (2006) Duality in fuzzy number linear programming by use of a

certain linear ranking function Applied Mathematics and Computation 180(1) 206-216 [10] Maleki H R Tata M amp Mashinchi M (2000) Linear programming with fuzzy variables Fuzzy sets

and systems 109(1) 21-33 [11] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear

programming problem Advances in Fuzzy Sets and Systems 5(2) 81-95 [12] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for

solving flexible linear programming problems European Journal of Industrial Engineering 4(3) 372-

389 [13] Nasseri S H Ebrahimnejad A amp Mizuno S (2010) Duality in fuzzy linear programming with

symmetric trapezoidal numbers Applications and Applied Mathematics 5(10) 1467-1482 [14] Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval Information sciences

24(2) 143-161

31 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Remark 3 Table 1 gives all the information needed to proceed with the simplex method The

fuzzy cost row in Table 1 is 119900119879 = 119888119861

minus1119860 minus 119888 where 119900119895 = 119888119861minus1119886119895 minus 119888 = 119911 minus 119888 1 le 119895 le

119899 with 119900119895 asymp 0 for 119895 = 119861119894 1 le 119894 le 119898 According to the optimality conditions (Theorem 31)

we are at the optimal solution if 119900119895 ≽ 0 for all 119895 ne 119861119894 1 le 119894 le 119898 On the other hand if

119900119895 ≼ 0 for some 119896 ne 119861119894 1 le 119894 le 119898 the problem is either unbounded or an exchange of a

basic variable 119909119861 and the nonbasic variable 119909 can be made to increase the rank of the

objective value (under nondegeneracy assumption) The following results established in [9]

help us devise the fuzzy primal simplex algorithm

Theorem 32 If there is a column 119896 (not in basis) in a fuzzy primal simplex tableau as 119910119900 =

119911 minus 119888 ≺ 0 and 119910119894119896 le 0 1 le 119894 le 119898 the problem (2-1) is unbounded

Theorem 33 If a nonbasic index 119896 exists in a fuzzy primal simplex tableau like 119910119900 = 119911 minus

119888 ≺ 0 and there exists a basic index 119861119894 like 119910119894119896 ge 0 a pivoting row 119903 can be found so that

pivoting on 119910119903119896 can yield a feasible tableau with a corresponding nondecreasing (increasing

under nondegeneracy assumption) fuzzy objective value

Remark 4 (see [9]) If there exists k such that 119910119900 ≺ 0 and the problem is not unbounded r

can be chosen as

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

Where it is the minimum fuzzy value of the above ratio

in order to replace 119909119861 in the basis by 119909 resulting in a new basis 119861 =

(119886119861119894 1198861198612 hellip 119886119861119903minus1 119886119896 119886119861119903+1 hellip 119886119861119898) The new basis is primal feasible and the

corresponding fuzzy objective value is nondecreasing (increasing under nondegeneracy

assumption) It can be shown that the new simplex tableau is obtained by pivoting on 119910119903119896

ie doing Gaussian elimination on the 119896 th column by using the pivot row 119903 with the pivot

119910119903119896 to transform the 119896 th column to the unit vector 119890119903 It is easily seen that the new fuzzy

objective value is 119910119900 = 119910119900 minus 119910119900119910119903119900

119910119903119896≽ 119910119900 where 119900119896 ≼ 0 and

119910119903119900

119910119903119896 (if the problem is

nondegenerate and consequently 119910119903119900

119910119903119896gt 0 and hence119910119900 asymp 119910119900)

We now describe the pivoting strategy

34 Pivoting and change of basis

If 119909 enters the basis and 119909119861 leaves the basis pivoting on 119910119903119896 in the primal simplex tableau

is carried out as follows

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 32

1) Divide row r by 119910119903119896

2) For 119894 = 01 119898 and 119894 ne 119903 update the 119894 th row by adding to it minus119910119894119896 times the new 119903th

row

We now present the primal simplex algorithm for the FNLP problem

35 The main steps of fuzzy primal simplex algorithm

Algorithm 31 The fuzzy primal simplex method

Assumption A basic feasible solution with basis B and the corresponding simplex tableau is

at hand

1 The fuzzy basic feasible solution is given by 119909 = 119910= 119861minus1 and 119909 asymp 0 The fuzzy

objective value is = 119910119900 = 119888119861minus1

2 Calculate 119900119895 = 119911 minus 119888 1 le 119895 le 119899 with for 119895 ne 119861119894 1 le 119894 le 119898

Let 119910119900 = 1198981198941198991le119895le119899119910119900 If 119910119900 ≽ 0 then stop the current solution is optimal

3 If 119910119900 ≼ 0 then stop the problem is unbounded Otherwise it determines an index 119903

corresponding to a variable 119909119861 leaving the basis as follows

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

4 Pivot on and update the simplex tableau Go to step 2

Remark 5 (limitation of existing method) Investigation of the current models and methods

show that almost all of them are disable to formulate the real problems when the symmetric

assumption is not at hand Furthermore by defining a new product role we may to define a

practical method for solving the generalized model while the methods cannot work by the

pioneering approach

Now we are on the point of giving some illustrative examples In the next subsection we

give two examples The first problem is concerning to the symmetric form of fuzzy number

that discussed by Ganesan et al in [4] This example shows how we can solve the existing

shortcomings of the first model which is given in Example 31

36 Numerical examples

Here we are going to explore the proposed approach for solving a linear programming

problem with symmetric fuzzy numbers which is considered by Ganesan et al in [4]

33 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Example 31 We consider the fuzzy mathematical model which is given by Ganesan et al

in [4] The corresponding model is given in the below

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 1 asymp [475 5056 6]

141 + 133 + 2 asymp [460 4808 8]

121 + 15 2 + 3 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Using the multiplication role which is established by Ganesan et al the fuzzy optimal

solution and the fuzzy optimal value of objective function is as follows

1199091 = [0000] 1199092 = [415

1691045

169174

169174

169] 1199093 = [

460

13480

138

138

13]

1 = [0000] 2 = [0000] 3 = [62910

16977430

1693455

1693455

169]

119885119881 = [94235

169120265

16919819

16919819

169]

Using a ranking function such as Yager we obtain 119877(119881) =120783120782120789120784120787120782

120783120788120791

By substituting the fuzzy values of 2 and 3 in the objective function and using the new role

of fuzzy multiplication we have

119867 = 11988811 + 11988822 + 11988833

11199091 = [13152 2][00 0 0] = [00 0 0]

21199092 = [415

1691045

169174

169174

169] [121433] = [

9175

1699805

1695571

1695571

169]

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 34

31199093 = [151722] [460

13480

138

138

13] = [

97630

16997890

1691096

1691096

169]

119885119867 = [00 0 0]+ [9175

1699805

1695571

1695571

169] + [

97630

16997890

16914248

16914248

169]

= [106805

169107695

16919819

16919819

169]

We clearly obtain 119877(119867) =120783120782120789120784120787120782

120783120788120791

So according to the ordering role which is given in Definition 22 it is concluded that

119881 asymp 119867

In fact it has been shown in this example that the proposed method can solve the symmetric

version of the given fuzzy numbers as well as Ganesanrsquos method

Example 32 In this example we consider a general form of trapezoidal fuzzy number for

coefficients in the objective function if it is not necessary to be symmetric

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 4 asymp [475505 6 6]

141 + 133 + 5 asymp [460 4808 8]

121 + 15 2 + 6 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

The first tableau of the fuzzy primal simplex algorithm is given as below

Table 2 The first simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (-15-1343) (-14-1254) (-17-1543) 0 0 0 0

1199094

1199095

1199096

12 13 12 1 0 0

14 0 13 0 1 0

12 15 0 0 0 1

(47550566)

(46048088)

(46549555)

35 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

The first Since (01 02 03) = ((minus15minus1343) (minus14minus1254) (minus17minus1543)) and

(ℛ(01) ℛ(02)ℛ( 03)) = (minus1425minus1325minus1625) then 3 inters the basis and based

on the minimum ratio test the leaving variable is 5 Pivoting on 11991053 = 13

After calculating the amount of RHS column in Table 1it has been found that the

multiplication role which was defined by Ganesan and Veeramani in [4] is not satisfying and

we must hence use Definition 22 for obtaining the amount of multiplication as follows

119910119900119900119899119890119908 = 0 + (

460

13480

13 8

13 8

13)(151734)

Now assume that = (460

13480

13 8

13 8

13) and =(151734) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 6900

137820

137200

138160

13

120574 = 119860119907119890119903119886119892119890 120579 =7520

13

120572 = | 120574 minus 119898119894119899120579 | =620

13 120573 = |119898119886119909120579 minus 120574| =

640

13

So 120596 = |120573minus120572

2| =

10

13

= [(1198861 + 1198862

2) (

1198871 + 1198872

2) minus 120596 (

1198861 + 1198862

2) (

1198871 + 1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

= [7510

137530

131576

132056

13]

Updating the data is based on the about calculation concluded in Table 2

Table 3 The second simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (15

1369

13 94

13 95

13) (-14-1254) 0 0 (

15

1317

133

134

13) 0 (

7510

137530

13 1576

13 2056

13)

1199094

1199093

1199096

minus12

13 13 0

14

13 0 1

12 15 0

1 minus12

13 0

0 1

13 0

0 0 1

(415

131045

13 174

13 174

13)

(460

13480

13 8

13 8

13)

(46549555)

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 36

Since the problem is maximization the optimality condition is not valid and hence 2 enters

the basis and the leaving variable is 4 The next tableau based on the following calculations

is shown in Table 3

119910119900119900119899119890119908 = (

7510

137530

131576

132056

13) + (

415

1691045

169 174

169 174

169)(121445)

We suppose that = (415

1691045

169 174

169 174

169) and =(121445) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 4980

16912540

1695810

16914630

169

120574 = 119860119907119890119903119886119892119890 120579 =9490

169

120572 = | 120574 minus 119898119894119899120579 | =4510

169 120573 = |119898119886119909120579 minus 120574| =

5140

169

Hence

120596 = |120573minus120572

2| =

315

169

Then

= [(1198861+1198862

2) (

1198871+1198872

2) minus 120596 (

1198861+1198862

2) (

1198871+1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ] = [

18600

16919230

1696616

1697661

169]

Then the objective value in this basis is as follows

+(7510

137530

131576

132056

13) = [

116230

169117120

16927104

16934389

169]

Table 4 The optimal simplex tableau Basic 1 2 3 4 5 6 RHS

Z (27

169753

169 1282

169 1283

169) 0 0 (

12

1314

134

135

13) (

27

16977

169 99

169 100

169) 0 (

116230

169117120

169 27104

169 34389

169)

1199092

1199093

1199096

minus12

169 1 0

14

13 0 1

2208

169 0 0

1

13

minus12

169 0

0 1

13 0

minus15 180

169 1

(415

1691045

169 174

169 174

169)

(460

13480

13 8

13 8

13)

(62910

16977430

169 1019

169 1019

169)

This table is optimal and the optimal value of the variables and the objective function are as

follows

37 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

2lowast = (

415

1691045

169 174

169 174

169) 3

lowast = (460

13480

13 8

13 8

13) 6

lowast = (62910

16977430

169 1019

169 1019

169)

and 1lowast = 4

lowast = 5lowast = 0 and 119911lowast = (

116230

169117120

169 27104

169 34389

169)

Where

119885119867 = [13 15 3 4]1lowast + [12 14 4 5]2

lowast + [1517 3 4]3lowast

= [1315 3 4]0+ (121445)(415

1691045

169174

169174

169) + (151734)(

460

13480

138

138

13)

= (116230

169117120

16927104

16934389

169)

Thus it has been seen that the proposed approach gave the same results as the mentioned

problem in the method proposed by Ganesan et al In particular the proposed arithmetic

allows the decision makers to model as a general type where the promoters can be

formulated as a general from of trapezoidal fuzzy numbers which is more appropriate for

real situations than just in the symmetric form

4 Conclusion

In the paper a new role for the multiplication of two general forms of trapezoidal fuzzy

numbers has been defined In particular we saw that the new model sounds to be more

appropriate for the real situation while in the pioneering model which was suggested by

Ganessan et al [4] and subsequently Das et all [1] the trapezoidal fuzzy numbers are

assumed to be essential in the symmetric form Therefore these tools can be useful for

preparing some solving algorithms fuzzy primal dual simplex algorithms and so on

5 Acknowledgment

The authors would like to thanks the anonymous referees for their valuable comments to lead

us for improving the earlier version of the mentioned manuscript

References

[1] Das S K Mandal T amp Edalatpanah S A (2017) A mathematical model for solving fully fuzzy linear

programming problem with trapezoidal fuzzy numbers Applied Intelligence 46(3) 509-519 [2] Ebrahimnejad A amp Nasseri S H (2009) Using complementary slackness property to solve linear

programming with fuzzy parameters Fuzzy Information and Engineering 1(3) 233-245

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 38 [3] Ebrahimnejad A Nasseri S H Lotfi F H amp Soltanifar M (2010) A primal-dual method for linear

programming problems with fuzzy variables European Journal of Industrial Engineering 4(2) 189-209 [4] Ganesan K amp Veeramani P (2006) Fuzzy linear programs with trapezoidal fuzzy numbers Annals of

Operations Research 143(1) 305-315 [5] Lotfi F H Allahviranloo T Jondabeh M A amp Alizadeh L (2009) Solving a full fuzzy linear

programming using lexicography method and fuzzy approximate solution Applied Mathematical

Modelling 33(7) 3151-3156

[6] Klir G amp Yuan B (1995) Fuzzy sets and fuzzy logic Theory and Applications Prentice-Hall PTR

New Jersey

[7] Kumar A Kaur J amp Singh P (2011) A new method for solving fully fuzzy linear programming

problems Applied Mathematical Modelling 35(2) 817-823 [8] Mahdavi-Amiri N amp Nasseri S H (2007) Duality results and a dual simplex method for linear

programming problems with trapezoidal fuzzy variables Fuzzy sets and systems 158(17) 1961-1978 [9] Mahdavi-Amiri N amp Nasseri S H (2006) Duality in fuzzy number linear programming by use of a

certain linear ranking function Applied Mathematics and Computation 180(1) 206-216 [10] Maleki H R Tata M amp Mashinchi M (2000) Linear programming with fuzzy variables Fuzzy sets

and systems 109(1) 21-33 [11] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear

programming problem Advances in Fuzzy Sets and Systems 5(2) 81-95 [12] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for

solving flexible linear programming problems European Journal of Industrial Engineering 4(3) 372-

389 [13] Nasseri S H Ebrahimnejad A amp Mizuno S (2010) Duality in fuzzy linear programming with

symmetric trapezoidal numbers Applications and Applied Mathematics 5(10) 1467-1482 [14] Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval Information sciences

24(2) 143-161

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 32

1) Divide row r by 119910119903119896

2) For 119894 = 01 119898 and 119894 ne 119903 update the 119894 th row by adding to it minus119910119894119896 times the new 119903th

row

We now present the primal simplex algorithm for the FNLP problem

35 The main steps of fuzzy primal simplex algorithm

Algorithm 31 The fuzzy primal simplex method

Assumption A basic feasible solution with basis B and the corresponding simplex tableau is

at hand

1 The fuzzy basic feasible solution is given by 119909 = 119910= 119861minus1 and 119909 asymp 0 The fuzzy

objective value is = 119910119900 = 119888119861minus1

2 Calculate 119900119895 = 119911 minus 119888 1 le 119895 le 119899 with for 119895 ne 119861119894 1 le 119894 le 119898

Let 119910119900 = 1198981198941198991le119895le119899119910119900 If 119910119900 ≽ 0 then stop the current solution is optimal

3 If 119910119900 ≼ 0 then stop the problem is unbounded Otherwise it determines an index 119903

corresponding to a variable 119909119861 leaving the basis as follows

119910119903119910119903119896

= 1198981198941198991le119894le119898 119910119894119910119894119896 119910119894119896 gt 0

4 Pivot on and update the simplex tableau Go to step 2

Remark 5 (limitation of existing method) Investigation of the current models and methods

show that almost all of them are disable to formulate the real problems when the symmetric

assumption is not at hand Furthermore by defining a new product role we may to define a

practical method for solving the generalized model while the methods cannot work by the

pioneering approach

Now we are on the point of giving some illustrative examples In the next subsection we

give two examples The first problem is concerning to the symmetric form of fuzzy number

that discussed by Ganesan et al in [4] This example shows how we can solve the existing

shortcomings of the first model which is given in Example 31

36 Numerical examples

Here we are going to explore the proposed approach for solving a linear programming

problem with symmetric fuzzy numbers which is considered by Ganesan et al in [4]

33 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Example 31 We consider the fuzzy mathematical model which is given by Ganesan et al

in [4] The corresponding model is given in the below

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 1 asymp [475 5056 6]

141 + 133 + 2 asymp [460 4808 8]

121 + 15 2 + 3 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Using the multiplication role which is established by Ganesan et al the fuzzy optimal

solution and the fuzzy optimal value of objective function is as follows

1199091 = [0000] 1199092 = [415

1691045

169174

169174

169] 1199093 = [

460

13480

138

138

13]

1 = [0000] 2 = [0000] 3 = [62910

16977430

1693455

1693455

169]

119885119881 = [94235

169120265

16919819

16919819

169]

Using a ranking function such as Yager we obtain 119877(119881) =120783120782120789120784120787120782

120783120788120791

By substituting the fuzzy values of 2 and 3 in the objective function and using the new role

of fuzzy multiplication we have

119867 = 11988811 + 11988822 + 11988833

11199091 = [13152 2][00 0 0] = [00 0 0]

21199092 = [415

1691045

169174

169174

169] [121433] = [

9175

1699805

1695571

1695571

169]

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 34

31199093 = [151722] [460

13480

138

138

13] = [

97630

16997890

1691096

1691096

169]

119885119867 = [00 0 0]+ [9175

1699805

1695571

1695571

169] + [

97630

16997890

16914248

16914248

169]

= [106805

169107695

16919819

16919819

169]

We clearly obtain 119877(119867) =120783120782120789120784120787120782

120783120788120791

So according to the ordering role which is given in Definition 22 it is concluded that

119881 asymp 119867

In fact it has been shown in this example that the proposed method can solve the symmetric

version of the given fuzzy numbers as well as Ganesanrsquos method

Example 32 In this example we consider a general form of trapezoidal fuzzy number for

coefficients in the objective function if it is not necessary to be symmetric

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 4 asymp [475505 6 6]

141 + 133 + 5 asymp [460 4808 8]

121 + 15 2 + 6 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

The first tableau of the fuzzy primal simplex algorithm is given as below

Table 2 The first simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (-15-1343) (-14-1254) (-17-1543) 0 0 0 0

1199094

1199095

1199096

12 13 12 1 0 0

14 0 13 0 1 0

12 15 0 0 0 1

(47550566)

(46048088)

(46549555)

35 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

The first Since (01 02 03) = ((minus15minus1343) (minus14minus1254) (minus17minus1543)) and

(ℛ(01) ℛ(02)ℛ( 03)) = (minus1425minus1325minus1625) then 3 inters the basis and based

on the minimum ratio test the leaving variable is 5 Pivoting on 11991053 = 13

After calculating the amount of RHS column in Table 1it has been found that the

multiplication role which was defined by Ganesan and Veeramani in [4] is not satisfying and

we must hence use Definition 22 for obtaining the amount of multiplication as follows

119910119900119900119899119890119908 = 0 + (

460

13480

13 8

13 8

13)(151734)

Now assume that = (460

13480

13 8

13 8

13) and =(151734) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 6900

137820

137200

138160

13

120574 = 119860119907119890119903119886119892119890 120579 =7520

13

120572 = | 120574 minus 119898119894119899120579 | =620

13 120573 = |119898119886119909120579 minus 120574| =

640

13

So 120596 = |120573minus120572

2| =

10

13

= [(1198861 + 1198862

2) (

1198871 + 1198872

2) minus 120596 (

1198861 + 1198862

2) (

1198871 + 1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

= [7510

137530

131576

132056

13]

Updating the data is based on the about calculation concluded in Table 2

Table 3 The second simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (15

1369

13 94

13 95

13) (-14-1254) 0 0 (

15

1317

133

134

13) 0 (

7510

137530

13 1576

13 2056

13)

1199094

1199093

1199096

minus12

13 13 0

14

13 0 1

12 15 0

1 minus12

13 0

0 1

13 0

0 0 1

(415

131045

13 174

13 174

13)

(460

13480

13 8

13 8

13)

(46549555)

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 36

Since the problem is maximization the optimality condition is not valid and hence 2 enters

the basis and the leaving variable is 4 The next tableau based on the following calculations

is shown in Table 3

119910119900119900119899119890119908 = (

7510

137530

131576

132056

13) + (

415

1691045

169 174

169 174

169)(121445)

We suppose that = (415

1691045

169 174

169 174

169) and =(121445) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 4980

16912540

1695810

16914630

169

120574 = 119860119907119890119903119886119892119890 120579 =9490

169

120572 = | 120574 minus 119898119894119899120579 | =4510

169 120573 = |119898119886119909120579 minus 120574| =

5140

169

Hence

120596 = |120573minus120572

2| =

315

169

Then

= [(1198861+1198862

2) (

1198871+1198872

2) minus 120596 (

1198861+1198862

2) (

1198871+1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ] = [

18600

16919230

1696616

1697661

169]

Then the objective value in this basis is as follows

+(7510

137530

131576

132056

13) = [

116230

169117120

16927104

16934389

169]

Table 4 The optimal simplex tableau Basic 1 2 3 4 5 6 RHS

Z (27

169753

169 1282

169 1283

169) 0 0 (

12

1314

134

135

13) (

27

16977

169 99

169 100

169) 0 (

116230

169117120

169 27104

169 34389

169)

1199092

1199093

1199096

minus12

169 1 0

14

13 0 1

2208

169 0 0

1

13

minus12

169 0

0 1

13 0

minus15 180

169 1

(415

1691045

169 174

169 174

169)

(460

13480

13 8

13 8

13)

(62910

16977430

169 1019

169 1019

169)

This table is optimal and the optimal value of the variables and the objective function are as

follows

37 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

2lowast = (

415

1691045

169 174

169 174

169) 3

lowast = (460

13480

13 8

13 8

13) 6

lowast = (62910

16977430

169 1019

169 1019

169)

and 1lowast = 4

lowast = 5lowast = 0 and 119911lowast = (

116230

169117120

169 27104

169 34389

169)

Where

119885119867 = [13 15 3 4]1lowast + [12 14 4 5]2

lowast + [1517 3 4]3lowast

= [1315 3 4]0+ (121445)(415

1691045

169174

169174

169) + (151734)(

460

13480

138

138

13)

= (116230

169117120

16927104

16934389

169)

Thus it has been seen that the proposed approach gave the same results as the mentioned

problem in the method proposed by Ganesan et al In particular the proposed arithmetic

allows the decision makers to model as a general type where the promoters can be

formulated as a general from of trapezoidal fuzzy numbers which is more appropriate for

real situations than just in the symmetric form

4 Conclusion

In the paper a new role for the multiplication of two general forms of trapezoidal fuzzy

numbers has been defined In particular we saw that the new model sounds to be more

appropriate for the real situation while in the pioneering model which was suggested by

Ganessan et al [4] and subsequently Das et all [1] the trapezoidal fuzzy numbers are

assumed to be essential in the symmetric form Therefore these tools can be useful for

preparing some solving algorithms fuzzy primal dual simplex algorithms and so on

5 Acknowledgment

The authors would like to thanks the anonymous referees for their valuable comments to lead

us for improving the earlier version of the mentioned manuscript

References

[1] Das S K Mandal T amp Edalatpanah S A (2017) A mathematical model for solving fully fuzzy linear

programming problem with trapezoidal fuzzy numbers Applied Intelligence 46(3) 509-519 [2] Ebrahimnejad A amp Nasseri S H (2009) Using complementary slackness property to solve linear

programming with fuzzy parameters Fuzzy Information and Engineering 1(3) 233-245

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 38 [3] Ebrahimnejad A Nasseri S H Lotfi F H amp Soltanifar M (2010) A primal-dual method for linear

programming problems with fuzzy variables European Journal of Industrial Engineering 4(2) 189-209 [4] Ganesan K amp Veeramani P (2006) Fuzzy linear programs with trapezoidal fuzzy numbers Annals of

Operations Research 143(1) 305-315 [5] Lotfi F H Allahviranloo T Jondabeh M A amp Alizadeh L (2009) Solving a full fuzzy linear

programming using lexicography method and fuzzy approximate solution Applied Mathematical

Modelling 33(7) 3151-3156

[6] Klir G amp Yuan B (1995) Fuzzy sets and fuzzy logic Theory and Applications Prentice-Hall PTR

New Jersey

[7] Kumar A Kaur J amp Singh P (2011) A new method for solving fully fuzzy linear programming

problems Applied Mathematical Modelling 35(2) 817-823 [8] Mahdavi-Amiri N amp Nasseri S H (2007) Duality results and a dual simplex method for linear

programming problems with trapezoidal fuzzy variables Fuzzy sets and systems 158(17) 1961-1978 [9] Mahdavi-Amiri N amp Nasseri S H (2006) Duality in fuzzy number linear programming by use of a

certain linear ranking function Applied Mathematics and Computation 180(1) 206-216 [10] Maleki H R Tata M amp Mashinchi M (2000) Linear programming with fuzzy variables Fuzzy sets

and systems 109(1) 21-33 [11] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear

programming problem Advances in Fuzzy Sets and Systems 5(2) 81-95 [12] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for

solving flexible linear programming problems European Journal of Industrial Engineering 4(3) 372-

389 [13] Nasseri S H Ebrahimnejad A amp Mizuno S (2010) Duality in fuzzy linear programming with

symmetric trapezoidal numbers Applications and Applied Mathematics 5(10) 1467-1482 [14] Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval Information sciences

24(2) 143-161

33 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

Example 31 We consider the fuzzy mathematical model which is given by Ganesan et al

in [4] The corresponding model is given in the below

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 2 2] 1 + [12 14 3 3] 2 + [15 17 2 2] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 1 asymp [475 5056 6]

141 + 133 + 2 asymp [460 4808 8]

121 + 15 2 + 3 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Using the multiplication role which is established by Ganesan et al the fuzzy optimal

solution and the fuzzy optimal value of objective function is as follows

1199091 = [0000] 1199092 = [415

1691045

169174

169174

169] 1199093 = [

460

13480

138

138

13]

1 = [0000] 2 = [0000] 3 = [62910

16977430

1693455

1693455

169]

119885119881 = [94235

169120265

16919819

16919819

169]

Using a ranking function such as Yager we obtain 119877(119881) =120783120782120789120784120787120782

120783120788120791

By substituting the fuzzy values of 2 and 3 in the objective function and using the new role

of fuzzy multiplication we have

119867 = 11988811 + 11988822 + 11988833

11199091 = [13152 2][00 0 0] = [00 0 0]

21199092 = [415

1691045

169174

169174

169] [121433] = [

9175

1699805

1695571

1695571

169]

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 34

31199093 = [151722] [460

13480

138

138

13] = [

97630

16997890

1691096

1691096

169]

119885119867 = [00 0 0]+ [9175

1699805

1695571

1695571

169] + [

97630

16997890

16914248

16914248

169]

= [106805

169107695

16919819

16919819

169]

We clearly obtain 119877(119867) =120783120782120789120784120787120782

120783120788120791

So according to the ordering role which is given in Definition 22 it is concluded that

119881 asymp 119867

In fact it has been shown in this example that the proposed method can solve the symmetric

version of the given fuzzy numbers as well as Ganesanrsquos method

Example 32 In this example we consider a general form of trapezoidal fuzzy number for

coefficients in the objective function if it is not necessary to be symmetric

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 4 asymp [475505 6 6]

141 + 133 + 5 asymp [460 4808 8]

121 + 15 2 + 6 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

The first tableau of the fuzzy primal simplex algorithm is given as below

Table 2 The first simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (-15-1343) (-14-1254) (-17-1543) 0 0 0 0

1199094

1199095

1199096

12 13 12 1 0 0

14 0 13 0 1 0

12 15 0 0 0 1

(47550566)

(46048088)

(46549555)

35 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

The first Since (01 02 03) = ((minus15minus1343) (minus14minus1254) (minus17minus1543)) and

(ℛ(01) ℛ(02)ℛ( 03)) = (minus1425minus1325minus1625) then 3 inters the basis and based

on the minimum ratio test the leaving variable is 5 Pivoting on 11991053 = 13

After calculating the amount of RHS column in Table 1it has been found that the

multiplication role which was defined by Ganesan and Veeramani in [4] is not satisfying and

we must hence use Definition 22 for obtaining the amount of multiplication as follows

119910119900119900119899119890119908 = 0 + (

460

13480

13 8

13 8

13)(151734)

Now assume that = (460

13480

13 8

13 8

13) and =(151734) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 6900

137820

137200

138160

13

120574 = 119860119907119890119903119886119892119890 120579 =7520

13

120572 = | 120574 minus 119898119894119899120579 | =620

13 120573 = |119898119886119909120579 minus 120574| =

640

13

So 120596 = |120573minus120572

2| =

10

13

= [(1198861 + 1198862

2) (

1198871 + 1198872

2) minus 120596 (

1198861 + 1198862

2) (

1198871 + 1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

= [7510

137530

131576

132056

13]

Updating the data is based on the about calculation concluded in Table 2

Table 3 The second simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (15

1369

13 94

13 95

13) (-14-1254) 0 0 (

15

1317

133

134

13) 0 (

7510

137530

13 1576

13 2056

13)

1199094

1199093

1199096

minus12

13 13 0

14

13 0 1

12 15 0

1 minus12

13 0

0 1

13 0

0 0 1

(415

131045

13 174

13 174

13)

(460

13480

13 8

13 8

13)

(46549555)

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 36

Since the problem is maximization the optimality condition is not valid and hence 2 enters

the basis and the leaving variable is 4 The next tableau based on the following calculations

is shown in Table 3

119910119900119900119899119890119908 = (

7510

137530

131576

132056

13) + (

415

1691045

169 174

169 174

169)(121445)

We suppose that = (415

1691045

169 174

169 174

169) and =(121445) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 4980

16912540

1695810

16914630

169

120574 = 119860119907119890119903119886119892119890 120579 =9490

169

120572 = | 120574 minus 119898119894119899120579 | =4510

169 120573 = |119898119886119909120579 minus 120574| =

5140

169

Hence

120596 = |120573minus120572

2| =

315

169

Then

= [(1198861+1198862

2) (

1198871+1198872

2) minus 120596 (

1198861+1198862

2) (

1198871+1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ] = [

18600

16919230

1696616

1697661

169]

Then the objective value in this basis is as follows

+(7510

137530

131576

132056

13) = [

116230

169117120

16927104

16934389

169]

Table 4 The optimal simplex tableau Basic 1 2 3 4 5 6 RHS

Z (27

169753

169 1282

169 1283

169) 0 0 (

12

1314

134

135

13) (

27

16977

169 99

169 100

169) 0 (

116230

169117120

169 27104

169 34389

169)

1199092

1199093

1199096

minus12

169 1 0

14

13 0 1

2208

169 0 0

1

13

minus12

169 0

0 1

13 0

minus15 180

169 1

(415

1691045

169 174

169 174

169)

(460

13480

13 8

13 8

13)

(62910

16977430

169 1019

169 1019

169)

This table is optimal and the optimal value of the variables and the objective function are as

follows

37 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

2lowast = (

415

1691045

169 174

169 174

169) 3

lowast = (460

13480

13 8

13 8

13) 6

lowast = (62910

16977430

169 1019

169 1019

169)

and 1lowast = 4

lowast = 5lowast = 0 and 119911lowast = (

116230

169117120

169 27104

169 34389

169)

Where

119885119867 = [13 15 3 4]1lowast + [12 14 4 5]2

lowast + [1517 3 4]3lowast

= [1315 3 4]0+ (121445)(415

1691045

169174

169174

169) + (151734)(

460

13480

138

138

13)

= (116230

169117120

16927104

16934389

169)

Thus it has been seen that the proposed approach gave the same results as the mentioned

problem in the method proposed by Ganesan et al In particular the proposed arithmetic

allows the decision makers to model as a general type where the promoters can be

formulated as a general from of trapezoidal fuzzy numbers which is more appropriate for

real situations than just in the symmetric form

4 Conclusion

In the paper a new role for the multiplication of two general forms of trapezoidal fuzzy

numbers has been defined In particular we saw that the new model sounds to be more

appropriate for the real situation while in the pioneering model which was suggested by

Ganessan et al [4] and subsequently Das et all [1] the trapezoidal fuzzy numbers are

assumed to be essential in the symmetric form Therefore these tools can be useful for

preparing some solving algorithms fuzzy primal dual simplex algorithms and so on

5 Acknowledgment

The authors would like to thanks the anonymous referees for their valuable comments to lead

us for improving the earlier version of the mentioned manuscript

References

[1] Das S K Mandal T amp Edalatpanah S A (2017) A mathematical model for solving fully fuzzy linear

programming problem with trapezoidal fuzzy numbers Applied Intelligence 46(3) 509-519 [2] Ebrahimnejad A amp Nasseri S H (2009) Using complementary slackness property to solve linear

programming with fuzzy parameters Fuzzy Information and Engineering 1(3) 233-245

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 38 [3] Ebrahimnejad A Nasseri S H Lotfi F H amp Soltanifar M (2010) A primal-dual method for linear

programming problems with fuzzy variables European Journal of Industrial Engineering 4(2) 189-209 [4] Ganesan K amp Veeramani P (2006) Fuzzy linear programs with trapezoidal fuzzy numbers Annals of

Operations Research 143(1) 305-315 [5] Lotfi F H Allahviranloo T Jondabeh M A amp Alizadeh L (2009) Solving a full fuzzy linear

programming using lexicography method and fuzzy approximate solution Applied Mathematical

Modelling 33(7) 3151-3156

[6] Klir G amp Yuan B (1995) Fuzzy sets and fuzzy logic Theory and Applications Prentice-Hall PTR

New Jersey

[7] Kumar A Kaur J amp Singh P (2011) A new method for solving fully fuzzy linear programming

problems Applied Mathematical Modelling 35(2) 817-823 [8] Mahdavi-Amiri N amp Nasseri S H (2007) Duality results and a dual simplex method for linear

programming problems with trapezoidal fuzzy variables Fuzzy sets and systems 158(17) 1961-1978 [9] Mahdavi-Amiri N amp Nasseri S H (2006) Duality in fuzzy number linear programming by use of a

certain linear ranking function Applied Mathematics and Computation 180(1) 206-216 [10] Maleki H R Tata M amp Mashinchi M (2000) Linear programming with fuzzy variables Fuzzy sets

and systems 109(1) 21-33 [11] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear

programming problem Advances in Fuzzy Sets and Systems 5(2) 81-95 [12] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for

solving flexible linear programming problems European Journal of Industrial Engineering 4(3) 372-

389 [13] Nasseri S H Ebrahimnejad A amp Mizuno S (2010) Duality in fuzzy linear programming with

symmetric trapezoidal numbers Applications and Applied Mathematics 5(10) 1467-1482 [14] Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval Information sciences

24(2) 143-161

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 34

31199093 = [151722] [460

13480

138

138

13] = [

97630

16997890

1691096

1691096

169]

119885119867 = [00 0 0]+ [9175

1699805

1695571

1695571

169] + [

97630

16997890

16914248

16914248

169]

= [106805

169107695

16919819

16919819

169]

We clearly obtain 119877(119867) =120783120782120789120784120787120782

120783120788120791

So according to the ordering role which is given in Definition 22 it is concluded that

119881 asymp 119867

In fact it has been shown in this example that the proposed method can solve the symmetric

version of the given fuzzy numbers as well as Ganesanrsquos method

Example 32 In this example we consider a general form of trapezoidal fuzzy number for

coefficients in the objective function if it is not necessary to be symmetric

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 ≼ [475 5056 6]

141 + 133 ≼ [460480 8 8]

121 + 15 2 ≼ [465495 5 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

Now the standard form of the fuzzy linear programming problem becomes

119898119886119909 asymp [13 15 3 4] 1 + [12 14 4 5] 2 + [15 17 3 4] 3

119904119906119887119895119890119888119905 119905119900 121 + 132 + 123 + 4 asymp [475505 6 6]

141 + 133 + 5 asymp [460 4808 8]

121 + 15 2 + 6 asymp [465 4955 5]

1 ≽ 0 2 ≽ 0 3 ≽ 0

The first tableau of the fuzzy primal simplex algorithm is given as below

Table 2 The first simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (-15-1343) (-14-1254) (-17-1543) 0 0 0 0

1199094

1199095

1199096

12 13 12 1 0 0

14 0 13 0 1 0

12 15 0 0 0 1

(47550566)

(46048088)

(46549555)

35 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

The first Since (01 02 03) = ((minus15minus1343) (minus14minus1254) (minus17minus1543)) and

(ℛ(01) ℛ(02)ℛ( 03)) = (minus1425minus1325minus1625) then 3 inters the basis and based

on the minimum ratio test the leaving variable is 5 Pivoting on 11991053 = 13

After calculating the amount of RHS column in Table 1it has been found that the

multiplication role which was defined by Ganesan and Veeramani in [4] is not satisfying and

we must hence use Definition 22 for obtaining the amount of multiplication as follows

119910119900119900119899119890119908 = 0 + (

460

13480

13 8

13 8

13)(151734)

Now assume that = (460

13480

13 8

13 8

13) and =(151734) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 6900

137820

137200

138160

13

120574 = 119860119907119890119903119886119892119890 120579 =7520

13

120572 = | 120574 minus 119898119894119899120579 | =620

13 120573 = |119898119886119909120579 minus 120574| =

640

13

So 120596 = |120573minus120572

2| =

10

13

= [(1198861 + 1198862

2) (

1198871 + 1198872

2) minus 120596 (

1198861 + 1198862

2) (

1198871 + 1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

= [7510

137530

131576

132056

13]

Updating the data is based on the about calculation concluded in Table 2

Table 3 The second simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (15

1369

13 94

13 95

13) (-14-1254) 0 0 (

15

1317

133

134

13) 0 (

7510

137530

13 1576

13 2056

13)

1199094

1199093

1199096

minus12

13 13 0

14

13 0 1

12 15 0

1 minus12

13 0

0 1

13 0

0 0 1

(415

131045

13 174

13 174

13)

(460

13480

13 8

13 8

13)

(46549555)

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 36

Since the problem is maximization the optimality condition is not valid and hence 2 enters

the basis and the leaving variable is 4 The next tableau based on the following calculations

is shown in Table 3

119910119900119900119899119890119908 = (

7510

137530

131576

132056

13) + (

415

1691045

169 174

169 174

169)(121445)

We suppose that = (415

1691045

169 174

169 174

169) and =(121445) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 4980

16912540

1695810

16914630

169

120574 = 119860119907119890119903119886119892119890 120579 =9490

169

120572 = | 120574 minus 119898119894119899120579 | =4510

169 120573 = |119898119886119909120579 minus 120574| =

5140

169

Hence

120596 = |120573minus120572

2| =

315

169

Then

= [(1198861+1198862

2) (

1198871+1198872

2) minus 120596 (

1198861+1198862

2) (

1198871+1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ] = [

18600

16919230

1696616

1697661

169]

Then the objective value in this basis is as follows

+(7510

137530

131576

132056

13) = [

116230

169117120

16927104

16934389

169]

Table 4 The optimal simplex tableau Basic 1 2 3 4 5 6 RHS

Z (27

169753

169 1282

169 1283

169) 0 0 (

12

1314

134

135

13) (

27

16977

169 99

169 100

169) 0 (

116230

169117120

169 27104

169 34389

169)

1199092

1199093

1199096

minus12

169 1 0

14

13 0 1

2208

169 0 0

1

13

minus12

169 0

0 1

13 0

minus15 180

169 1

(415

1691045

169 174

169 174

169)

(460

13480

13 8

13 8

13)

(62910

16977430

169 1019

169 1019

169)

This table is optimal and the optimal value of the variables and the objective function are as

follows

37 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

2lowast = (

415

1691045

169 174

169 174

169) 3

lowast = (460

13480

13 8

13 8

13) 6

lowast = (62910

16977430

169 1019

169 1019

169)

and 1lowast = 4

lowast = 5lowast = 0 and 119911lowast = (

116230

169117120

169 27104

169 34389

169)

Where

119885119867 = [13 15 3 4]1lowast + [12 14 4 5]2

lowast + [1517 3 4]3lowast

= [1315 3 4]0+ (121445)(415

1691045

169174

169174

169) + (151734)(

460

13480

138

138

13)

= (116230

169117120

16927104

16934389

169)

Thus it has been seen that the proposed approach gave the same results as the mentioned

problem in the method proposed by Ganesan et al In particular the proposed arithmetic

allows the decision makers to model as a general type where the promoters can be

formulated as a general from of trapezoidal fuzzy numbers which is more appropriate for

real situations than just in the symmetric form

4 Conclusion

In the paper a new role for the multiplication of two general forms of trapezoidal fuzzy

numbers has been defined In particular we saw that the new model sounds to be more

appropriate for the real situation while in the pioneering model which was suggested by

Ganessan et al [4] and subsequently Das et all [1] the trapezoidal fuzzy numbers are

assumed to be essential in the symmetric form Therefore these tools can be useful for

preparing some solving algorithms fuzzy primal dual simplex algorithms and so on

5 Acknowledgment

The authors would like to thanks the anonymous referees for their valuable comments to lead

us for improving the earlier version of the mentioned manuscript

References

[1] Das S K Mandal T amp Edalatpanah S A (2017) A mathematical model for solving fully fuzzy linear

programming problem with trapezoidal fuzzy numbers Applied Intelligence 46(3) 509-519 [2] Ebrahimnejad A amp Nasseri S H (2009) Using complementary slackness property to solve linear

programming with fuzzy parameters Fuzzy Information and Engineering 1(3) 233-245

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 38 [3] Ebrahimnejad A Nasseri S H Lotfi F H amp Soltanifar M (2010) A primal-dual method for linear

programming problems with fuzzy variables European Journal of Industrial Engineering 4(2) 189-209 [4] Ganesan K amp Veeramani P (2006) Fuzzy linear programs with trapezoidal fuzzy numbers Annals of

Operations Research 143(1) 305-315 [5] Lotfi F H Allahviranloo T Jondabeh M A amp Alizadeh L (2009) Solving a full fuzzy linear

programming using lexicography method and fuzzy approximate solution Applied Mathematical

Modelling 33(7) 3151-3156

[6] Klir G amp Yuan B (1995) Fuzzy sets and fuzzy logic Theory and Applications Prentice-Hall PTR

New Jersey

[7] Kumar A Kaur J amp Singh P (2011) A new method for solving fully fuzzy linear programming

problems Applied Mathematical Modelling 35(2) 817-823 [8] Mahdavi-Amiri N amp Nasseri S H (2007) Duality results and a dual simplex method for linear

programming problems with trapezoidal fuzzy variables Fuzzy sets and systems 158(17) 1961-1978 [9] Mahdavi-Amiri N amp Nasseri S H (2006) Duality in fuzzy number linear programming by use of a

certain linear ranking function Applied Mathematics and Computation 180(1) 206-216 [10] Maleki H R Tata M amp Mashinchi M (2000) Linear programming with fuzzy variables Fuzzy sets

and systems 109(1) 21-33 [11] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear

programming problem Advances in Fuzzy Sets and Systems 5(2) 81-95 [12] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for

solving flexible linear programming problems European Journal of Industrial Engineering 4(3) 372-

389 [13] Nasseri S H Ebrahimnejad A amp Mizuno S (2010) Duality in fuzzy linear programming with

symmetric trapezoidal numbers Applications and Applied Mathematics 5(10) 1467-1482 [14] Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval Information sciences

24(2) 143-161

35 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

The first Since (01 02 03) = ((minus15minus1343) (minus14minus1254) (minus17minus1543)) and

(ℛ(01) ℛ(02)ℛ( 03)) = (minus1425minus1325minus1625) then 3 inters the basis and based

on the minimum ratio test the leaving variable is 5 Pivoting on 11991053 = 13

After calculating the amount of RHS column in Table 1it has been found that the

multiplication role which was defined by Ganesan and Veeramani in [4] is not satisfying and

we must hence use Definition 22 for obtaining the amount of multiplication as follows

119910119900119900119899119890119908 = 0 + (

460

13480

13 8

13 8

13)(151734)

Now assume that = (460

13480

13 8

13 8

13) and =(151734) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 6900

137820

137200

138160

13

120574 = 119860119907119890119903119886119892119890 120579 =7520

13

120572 = | 120574 minus 119898119894119899120579 | =620

13 120573 = |119898119886119909120579 minus 120574| =

640

13

So 120596 = |120573minus120572

2| =

10

13

= [(1198861 + 1198862

2) (

1198871 + 1198872

2) minus 120596 (

1198861 + 1198862

2) (

1198871 + 1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ]

= [7510

137530

131576

132056

13]

Updating the data is based on the about calculation concluded in Table 2

Table 3 The second simplex tableau Basic 1199091 1199092 1199093 1199094 1199095 1199096 RHS

Z (15

1369

13 94

13 95

13) (-14-1254) 0 0 (

15

1317

133

134

13) 0 (

7510

137530

13 1576

13 2056

13)

1199094

1199093

1199096

minus12

13 13 0

14

13 0 1

12 15 0

1 minus12

13 0

0 1

13 0

0 0 1

(415

131045

13 174

13 174

13)

(460

13480

13 8

13 8

13)

(46549555)

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 36

Since the problem is maximization the optimality condition is not valid and hence 2 enters

the basis and the leaving variable is 4 The next tableau based on the following calculations

is shown in Table 3

119910119900119900119899119890119908 = (

7510

137530

131576

132056

13) + (

415

1691045

169 174

169 174

169)(121445)

We suppose that = (415

1691045

169 174

169 174

169) and =(121445) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 4980

16912540

1695810

16914630

169

120574 = 119860119907119890119903119886119892119890 120579 =9490

169

120572 = | 120574 minus 119898119894119899120579 | =4510

169 120573 = |119898119886119909120579 minus 120574| =

5140

169

Hence

120596 = |120573minus120572

2| =

315

169

Then

= [(1198861+1198862

2) (

1198871+1198872

2) minus 120596 (

1198861+1198862

2) (

1198871+1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ] = [

18600

16919230

1696616

1697661

169]

Then the objective value in this basis is as follows

+(7510

137530

131576

132056

13) = [

116230

169117120

16927104

16934389

169]

Table 4 The optimal simplex tableau Basic 1 2 3 4 5 6 RHS

Z (27

169753

169 1282

169 1283

169) 0 0 (

12

1314

134

135

13) (

27

16977

169 99

169 100

169) 0 (

116230

169117120

169 27104

169 34389

169)

1199092

1199093

1199096

minus12

169 1 0

14

13 0 1

2208

169 0 0

1

13

minus12

169 0

0 1

13 0

minus15 180

169 1

(415

1691045

169 174

169 174

169)

(460

13480

13 8

13 8

13)

(62910

16977430

169 1019

169 1019

169)

This table is optimal and the optimal value of the variables and the objective function are as

follows

37 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

2lowast = (

415

1691045

169 174

169 174

169) 3

lowast = (460

13480

13 8

13 8

13) 6

lowast = (62910

16977430

169 1019

169 1019

169)

and 1lowast = 4

lowast = 5lowast = 0 and 119911lowast = (

116230

169117120

169 27104

169 34389

169)

Where

119885119867 = [13 15 3 4]1lowast + [12 14 4 5]2

lowast + [1517 3 4]3lowast

= [1315 3 4]0+ (121445)(415

1691045

169174

169174

169) + (151734)(

460

13480

138

138

13)

= (116230

169117120

16927104

16934389

169)

Thus it has been seen that the proposed approach gave the same results as the mentioned

problem in the method proposed by Ganesan et al In particular the proposed arithmetic

allows the decision makers to model as a general type where the promoters can be

formulated as a general from of trapezoidal fuzzy numbers which is more appropriate for

real situations than just in the symmetric form

4 Conclusion

In the paper a new role for the multiplication of two general forms of trapezoidal fuzzy

numbers has been defined In particular we saw that the new model sounds to be more

appropriate for the real situation while in the pioneering model which was suggested by

Ganessan et al [4] and subsequently Das et all [1] the trapezoidal fuzzy numbers are

assumed to be essential in the symmetric form Therefore these tools can be useful for

preparing some solving algorithms fuzzy primal dual simplex algorithms and so on

5 Acknowledgment

The authors would like to thanks the anonymous referees for their valuable comments to lead

us for improving the earlier version of the mentioned manuscript

References

[1] Das S K Mandal T amp Edalatpanah S A (2017) A mathematical model for solving fully fuzzy linear

programming problem with trapezoidal fuzzy numbers Applied Intelligence 46(3) 509-519 [2] Ebrahimnejad A amp Nasseri S H (2009) Using complementary slackness property to solve linear

programming with fuzzy parameters Fuzzy Information and Engineering 1(3) 233-245

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 38 [3] Ebrahimnejad A Nasseri S H Lotfi F H amp Soltanifar M (2010) A primal-dual method for linear

programming problems with fuzzy variables European Journal of Industrial Engineering 4(2) 189-209 [4] Ganesan K amp Veeramani P (2006) Fuzzy linear programs with trapezoidal fuzzy numbers Annals of

Operations Research 143(1) 305-315 [5] Lotfi F H Allahviranloo T Jondabeh M A amp Alizadeh L (2009) Solving a full fuzzy linear

programming using lexicography method and fuzzy approximate solution Applied Mathematical

Modelling 33(7) 3151-3156

[6] Klir G amp Yuan B (1995) Fuzzy sets and fuzzy logic Theory and Applications Prentice-Hall PTR

New Jersey

[7] Kumar A Kaur J amp Singh P (2011) A new method for solving fully fuzzy linear programming

problems Applied Mathematical Modelling 35(2) 817-823 [8] Mahdavi-Amiri N amp Nasseri S H (2007) Duality results and a dual simplex method for linear

programming problems with trapezoidal fuzzy variables Fuzzy sets and systems 158(17) 1961-1978 [9] Mahdavi-Amiri N amp Nasseri S H (2006) Duality in fuzzy number linear programming by use of a

certain linear ranking function Applied Mathematics and Computation 180(1) 206-216 [10] Maleki H R Tata M amp Mashinchi M (2000) Linear programming with fuzzy variables Fuzzy sets

and systems 109(1) 21-33 [11] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear

programming problem Advances in Fuzzy Sets and Systems 5(2) 81-95 [12] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for

solving flexible linear programming problems European Journal of Industrial Engineering 4(3) 372-

389 [13] Nasseri S H Ebrahimnejad A amp Mizuno S (2010) Duality in fuzzy linear programming with

symmetric trapezoidal numbers Applications and Applied Mathematics 5(10) 1467-1482 [14] Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval Information sciences

24(2) 143-161

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 36

Since the problem is maximization the optimality condition is not valid and hence 2 enters

the basis and the leaving variable is 4 The next tableau based on the following calculations

is shown in Table 3

119910119900119900119899119890119908 = (

7510

137530

131576

132056

13) + (

415

1691045

169 174

169 174

169)(121445)

We suppose that = (415

1691045

169 174

169 174

169) and =(121445) then

120579 = 11988611198871 11988611198872 11988621198871 11988621198872 = 4980

16912540

1695810

16914630

169

120574 = 119860119907119890119903119886119892119890 120579 =9490

169

120572 = | 120574 minus 119898119894119899120579 | =4510

169 120573 = |119898119886119909120579 minus 120574| =

5140

169

Hence

120596 = |120573minus120572

2| =

315

169

Then

= [(1198861+1198862

2) (

1198871+1198872

2) minus 120596 (

1198861+1198862

2) (

1198871+1198872

2) + 120596 |11988621198961 + 1198872ℎ1| |11988621198962 + 1198872ℎ2| ] = [

18600

16919230

1696616

1697661

169]

Then the objective value in this basis is as follows

+(7510

137530

131576

132056

13) = [

116230

169117120

16927104

16934389

169]

Table 4 The optimal simplex tableau Basic 1 2 3 4 5 6 RHS

Z (27

169753

169 1282

169 1283

169) 0 0 (

12

1314

134

135

13) (

27

16977

169 99

169 100

169) 0 (

116230

169117120

169 27104

169 34389

169)

1199092

1199093

1199096

minus12

169 1 0

14

13 0 1

2208

169 0 0

1

13

minus12

169 0

0 1

13 0

minus15 180

169 1

(415

1691045

169 174

169 174

169)

(460

13480

13 8

13 8

13)

(62910

16977430

169 1019

169 1019

169)

This table is optimal and the optimal value of the variables and the objective function are as

follows

37 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

2lowast = (

415

1691045

169 174

169 174

169) 3

lowast = (460

13480

13 8

13 8

13) 6

lowast = (62910

16977430

169 1019

169 1019

169)

and 1lowast = 4

lowast = 5lowast = 0 and 119911lowast = (

116230

169117120

169 27104

169 34389

169)

Where

119885119867 = [13 15 3 4]1lowast + [12 14 4 5]2

lowast + [1517 3 4]3lowast

= [1315 3 4]0+ (121445)(415

1691045

169174

169174

169) + (151734)(

460

13480

138

138

13)

= (116230

169117120

16927104

16934389

169)

Thus it has been seen that the proposed approach gave the same results as the mentioned

problem in the method proposed by Ganesan et al In particular the proposed arithmetic

allows the decision makers to model as a general type where the promoters can be

formulated as a general from of trapezoidal fuzzy numbers which is more appropriate for

real situations than just in the symmetric form

4 Conclusion

In the paper a new role for the multiplication of two general forms of trapezoidal fuzzy

numbers has been defined In particular we saw that the new model sounds to be more

appropriate for the real situation while in the pioneering model which was suggested by

Ganessan et al [4] and subsequently Das et all [1] the trapezoidal fuzzy numbers are

assumed to be essential in the symmetric form Therefore these tools can be useful for

preparing some solving algorithms fuzzy primal dual simplex algorithms and so on

5 Acknowledgment

The authors would like to thanks the anonymous referees for their valuable comments to lead

us for improving the earlier version of the mentioned manuscript

References

[1] Das S K Mandal T amp Edalatpanah S A (2017) A mathematical model for solving fully fuzzy linear

programming problem with trapezoidal fuzzy numbers Applied Intelligence 46(3) 509-519 [2] Ebrahimnejad A amp Nasseri S H (2009) Using complementary slackness property to solve linear

programming with fuzzy parameters Fuzzy Information and Engineering 1(3) 233-245

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 38 [3] Ebrahimnejad A Nasseri S H Lotfi F H amp Soltanifar M (2010) A primal-dual method for linear

programming problems with fuzzy variables European Journal of Industrial Engineering 4(2) 189-209 [4] Ganesan K amp Veeramani P (2006) Fuzzy linear programs with trapezoidal fuzzy numbers Annals of

Operations Research 143(1) 305-315 [5] Lotfi F H Allahviranloo T Jondabeh M A amp Alizadeh L (2009) Solving a full fuzzy linear

programming using lexicography method and fuzzy approximate solution Applied Mathematical

Modelling 33(7) 3151-3156

[6] Klir G amp Yuan B (1995) Fuzzy sets and fuzzy logic Theory and Applications Prentice-Hall PTR

New Jersey

[7] Kumar A Kaur J amp Singh P (2011) A new method for solving fully fuzzy linear programming

problems Applied Mathematical Modelling 35(2) 817-823 [8] Mahdavi-Amiri N amp Nasseri S H (2007) Duality results and a dual simplex method for linear

programming problems with trapezoidal fuzzy variables Fuzzy sets and systems 158(17) 1961-1978 [9] Mahdavi-Amiri N amp Nasseri S H (2006) Duality in fuzzy number linear programming by use of a

certain linear ranking function Applied Mathematics and Computation 180(1) 206-216 [10] Maleki H R Tata M amp Mashinchi M (2000) Linear programming with fuzzy variables Fuzzy sets

and systems 109(1) 21-33 [11] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear

programming problem Advances in Fuzzy Sets and Systems 5(2) 81-95 [12] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for

solving flexible linear programming problems European Journal of Industrial Engineering 4(3) 372-

389 [13] Nasseri S H Ebrahimnejad A amp Mizuno S (2010) Duality in fuzzy linear programming with

symmetric trapezoidal numbers Applications and Applied Mathematics 5(10) 1467-1482 [14] Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval Information sciences

24(2) 143-161

37 A generalized model for fuzzy linear programs with trapezoidal fuzzy numbers

2lowast = (

415

1691045

169 174

169 174

169) 3

lowast = (460

13480

13 8

13 8

13) 6

lowast = (62910

16977430

169 1019

169 1019

169)

and 1lowast = 4

lowast = 5lowast = 0 and 119911lowast = (

116230

169117120

169 27104

169 34389

169)

Where

119885119867 = [13 15 3 4]1lowast + [12 14 4 5]2

lowast + [1517 3 4]3lowast

= [1315 3 4]0+ (121445)(415

1691045

169174

169174

169) + (151734)(

460

13480

138

138

13)

= (116230

169117120

16927104

16934389

169)

Thus it has been seen that the proposed approach gave the same results as the mentioned

problem in the method proposed by Ganesan et al In particular the proposed arithmetic

allows the decision makers to model as a general type where the promoters can be

formulated as a general from of trapezoidal fuzzy numbers which is more appropriate for

real situations than just in the symmetric form

4 Conclusion

In the paper a new role for the multiplication of two general forms of trapezoidal fuzzy

numbers has been defined In particular we saw that the new model sounds to be more

appropriate for the real situation while in the pioneering model which was suggested by

Ganessan et al [4] and subsequently Das et all [1] the trapezoidal fuzzy numbers are

assumed to be essential in the symmetric form Therefore these tools can be useful for

preparing some solving algorithms fuzzy primal dual simplex algorithms and so on

5 Acknowledgment

The authors would like to thanks the anonymous referees for their valuable comments to lead

us for improving the earlier version of the mentioned manuscript

References

[1] Das S K Mandal T amp Edalatpanah S A (2017) A mathematical model for solving fully fuzzy linear

programming problem with trapezoidal fuzzy numbers Applied Intelligence 46(3) 509-519 [2] Ebrahimnejad A amp Nasseri S H (2009) Using complementary slackness property to solve linear

programming with fuzzy parameters Fuzzy Information and Engineering 1(3) 233-245

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 38 [3] Ebrahimnejad A Nasseri S H Lotfi F H amp Soltanifar M (2010) A primal-dual method for linear

programming problems with fuzzy variables European Journal of Industrial Engineering 4(2) 189-209 [4] Ganesan K amp Veeramani P (2006) Fuzzy linear programs with trapezoidal fuzzy numbers Annals of

Operations Research 143(1) 305-315 [5] Lotfi F H Allahviranloo T Jondabeh M A amp Alizadeh L (2009) Solving a full fuzzy linear

programming using lexicography method and fuzzy approximate solution Applied Mathematical

Modelling 33(7) 3151-3156

[6] Klir G amp Yuan B (1995) Fuzzy sets and fuzzy logic Theory and Applications Prentice-Hall PTR

New Jersey

[7] Kumar A Kaur J amp Singh P (2011) A new method for solving fully fuzzy linear programming

problems Applied Mathematical Modelling 35(2) 817-823 [8] Mahdavi-Amiri N amp Nasseri S H (2007) Duality results and a dual simplex method for linear

programming problems with trapezoidal fuzzy variables Fuzzy sets and systems 158(17) 1961-1978 [9] Mahdavi-Amiri N amp Nasseri S H (2006) Duality in fuzzy number linear programming by use of a

certain linear ranking function Applied Mathematics and Computation 180(1) 206-216 [10] Maleki H R Tata M amp Mashinchi M (2000) Linear programming with fuzzy variables Fuzzy sets

and systems 109(1) 21-33 [11] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear

programming problem Advances in Fuzzy Sets and Systems 5(2) 81-95 [12] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for

solving flexible linear programming problems European Journal of Industrial Engineering 4(3) 372-

389 [13] Nasseri S H Ebrahimnejad A amp Mizuno S (2010) Duality in fuzzy linear programming with

symmetric trapezoidal numbers Applications and Applied Mathematics 5(10) 1467-1482 [14] Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval Information sciences

24(2) 143-161

S H Nasseri et al J Appl Res Ind Eng 4(1) (2017) 24-38 38 [3] Ebrahimnejad A Nasseri S H Lotfi F H amp Soltanifar M (2010) A primal-dual method for linear

programming problems with fuzzy variables European Journal of Industrial Engineering 4(2) 189-209 [4] Ganesan K amp Veeramani P (2006) Fuzzy linear programs with trapezoidal fuzzy numbers Annals of

Operations Research 143(1) 305-315 [5] Lotfi F H Allahviranloo T Jondabeh M A amp Alizadeh L (2009) Solving a full fuzzy linear

programming using lexicography method and fuzzy approximate solution Applied Mathematical

Modelling 33(7) 3151-3156

[6] Klir G amp Yuan B (1995) Fuzzy sets and fuzzy logic Theory and Applications Prentice-Hall PTR

New Jersey

[7] Kumar A Kaur J amp Singh P (2011) A new method for solving fully fuzzy linear programming

problems Applied Mathematical Modelling 35(2) 817-823 [8] Mahdavi-Amiri N amp Nasseri S H (2007) Duality results and a dual simplex method for linear

programming problems with trapezoidal fuzzy variables Fuzzy sets and systems 158(17) 1961-1978 [9] Mahdavi-Amiri N amp Nasseri S H (2006) Duality in fuzzy number linear programming by use of a

certain linear ranking function Applied Mathematics and Computation 180(1) 206-216 [10] Maleki H R Tata M amp Mashinchi M (2000) Linear programming with fuzzy variables Fuzzy sets

and systems 109(1) 21-33 [11] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy dual simplex method for fuzzy number linear

programming problem Advances in Fuzzy Sets and Systems 5(2) 81-95 [12] Nasseri S H amp Ebrahimnejad A (2010) A fuzzy primal simplex algorithm and its application for

solving flexible linear programming problems European Journal of Industrial Engineering 4(3) 372-

389 [13] Nasseri S H Ebrahimnejad A amp Mizuno S (2010) Duality in fuzzy linear programming with

symmetric trapezoidal numbers Applications and Applied Mathematics 5(10) 1467-1482 [14] Yager R R (1981) A procedure for ordering fuzzy subsets of the unit interval Information sciences

24(2) 143-161