J. Appl. Res. Ind. Eng. Vol. 4, No. 1 (2017) 24–38 Journal of Applied Research on Industrial Engineering www.journal-aprie.com A Generalized Model for Fuzzy Linear Programs with Trapezoidal Fuzzy Numbers Seyed Hadi Nasseri *, 1 , Hadi Zavieh 1 , Seyedeh Maedeh Mirmohseni 2 1 Department of Mathematics, University of Mazandaran, Babolsar, Iran. 2 School 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 doesn’t 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 [2,3,8,9,10,13]. 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: [email protected]DOI: 10.22105/jarie.2017.49602
15
Embed
A Generalized Model for Fuzzy Linear Programs …...fuzzy numbers as an extended role of the given definition for the symmetric form of fuzzy numbers which is defined by Ganesan and
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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)
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
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
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
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 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)
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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