The document proposes a new "bound and decomposition method" to find an optimal fuzzy solution for fully fuzzy linear programming problems. The method decomposes the given fully fuzzy linear programming problem into three crisp linear programming problems: a middle level problem, upper level problem, and lower level problem. The three crisp linear programming problems are then solved separately. Using the optimal solutions from these problems, an optimal fuzzy solution for the original fully fuzzy linear programming problem is obtained. The proposed method does not require fuzzy ranking functions or restrictions on the coefficients matrix elements.
A New Hendecagonal Fuzzy Number For Optimization Problemsijtsrd
A new fuzzy number called Hendecagonal fuzzy number and its membership function is introduced, which is used to represent the uncertainty with eleven points. The fuzzy numbers with ten ordinates exists in literature. The aim of this paper is to define Hendecagonal fuzzy number and its arithmetic operations. Also a direct approach is proposed to solve fuzzy assignment problem (FAP) and fuzzy travelling salesman (FTSP) in which the cost and distance are represented by Hendecagonal fuzzy numbers. Numerical example shows the effectiveness of the proposed method and the Hendecagonal fuzzy number M. Revathi | Dr. M. Valliathal | R. Saravanan | Dr. K. Rathi"A New Hendecagonal Fuzzy Number For Optimization Problems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-5 , August 2017, URL: http://www.ijtsrd.com/papers/ijtsrd2258.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/2258/a-new-hendecagonal-fuzzy-number-for-optimization-problems/m-revathi
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searchalgorithms: Deterministic and Randomized.It has been found that deterministic methods when run on hard CSP instances are frequently very slow in execution.A deterministic method always outputs a solution in the end, but it can take an enormous amount of time to do so.This has led to the development of randomized search algorithms like GSAT, which are typically based on local (i.e., neighbourhood) search. Such methodshave been applied very successfully to find good solutions to hard decision problems
Determination of Optimal Product Mix for Profit Maximization using Linear Pro...IJERA Editor
This paper demonstrates the use of liner programming methods in order to determine the optimal product mix for
profit maximization. There had been several papers written to demonstrate the use of linear programming in
finding the optimal product mix in various organization. This paper is aimed to show the generic approach to be
taken to find the optimal product mix.
A New Hendecagonal Fuzzy Number For Optimization Problemsijtsrd
A new fuzzy number called Hendecagonal fuzzy number and its membership function is introduced, which is used to represent the uncertainty with eleven points. The fuzzy numbers with ten ordinates exists in literature. The aim of this paper is to define Hendecagonal fuzzy number and its arithmetic operations. Also a direct approach is proposed to solve fuzzy assignment problem (FAP) and fuzzy travelling salesman (FTSP) in which the cost and distance are represented by Hendecagonal fuzzy numbers. Numerical example shows the effectiveness of the proposed method and the Hendecagonal fuzzy number M. Revathi | Dr. M. Valliathal | R. Saravanan | Dr. K. Rathi"A New Hendecagonal Fuzzy Number For Optimization Problems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-5 , August 2017, URL: http://www.ijtsrd.com/papers/ijtsrd2258.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/2258/a-new-hendecagonal-fuzzy-number-for-optimization-problems/m-revathi
SATISFIABILITY METHODS FOR COLOURING GRAPHScscpconf
The graph colouring problem can be solved using methods based on Satisfiability (SAT). An instance of the problem is defined by a Boolean expression written using Boolean variables and the logical connectives AND, OR and NOT. It has to be determined whether there is an assignment of TRUE and FALSE values to the variables that makes the entire expression true.A SAT problem is syntactically and semantically quite simple. Many Constraint Satisfaction Problems (CSPs)in AI and OR can be formulated in SAT. These make use of two kinds of
searchalgorithms: Deterministic and Randomized.It has been found that deterministic methods when run on hard CSP instances are frequently very slow in execution.A deterministic method always outputs a solution in the end, but it can take an enormous amount of time to do so.This has led to the development of randomized search algorithms like GSAT, which are typically based on local (i.e., neighbourhood) search. Such methodshave been applied very successfully to find good solutions to hard decision problems
Determination of Optimal Product Mix for Profit Maximization using Linear Pro...IJERA Editor
This paper demonstrates the use of liner programming methods in order to determine the optimal product mix for
profit maximization. There had been several papers written to demonstrate the use of linear programming in
finding the optimal product mix in various organization. This paper is aimed to show the generic approach to be
taken to find the optimal product mix.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
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IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
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Heptagonal Fuzzy Numbers by Max Min MethodYogeshIJTSRD
In this paper, we propose another methodology for the arrangement of fuzzy transportation problem under a fuzzy environment in which transportation costs are taken as fuzzy Heptagonal numbers. The fuzzy numbers and fuzzy values are predominantly used in various fields. Here, we are converting fuzzy Heptagonal numbers into crisp value by using range technique and then solved by the MAX MIN method for the transportation problem. M. Revathi | K. Nithya "Heptagonal Fuzzy Numbers by Max-Min Method" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-3 , April 2021, URL: https://www.ijtsrd.com/papers/ijtsrd38280.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathamatics/38280/heptagonal-fuzzy-numbers-by-maxmin-method/m-revathi
Solving Fuzzy Maximal Flow Problem Using Octagonal Fuzzy NumberIJERA Editor
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By ranking the octagonal fuzzy numbers it is possible to compare them and using this we convert the fuzzy
valued maximal flow algorithm to a crisp valued algorithm . It is proved that a better solution is obtained when
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this a numerical example is solved and the obtained result is compared with the existing results . If there is no
uncertainty about the flow between source and sink then the proposed algorithm gives the same result as in crisp
maximal flow problems.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
New Method for Finding an Optimal Solution of Generalized Fuzzy Transportatio...BRNSS Publication Hub
In this paper, a proposed method, namely, zero average method is used for solving fuzzy transportation problems by assuming that a decision-maker is uncertain about the precise values of the transportation costs, demand, and supply of the product. In the proposed method, transportation costs, demand, and supply are represented by generalized trapezoidal fuzzy numbers. To illustrate the proposed method, a numerical example is solved. The proposed method is easy to understand and apply to real-life transportation problems for the decision-makers.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Some Properties of Determinant of Trapezoidal Fuzzy Number MatricesIJMERJOURNAL
ABSTRACT: The fuzzy set theory has been applied in many fields such as management, engineering, matrices and so on. In this paper, some elementary operations on proposed trapezoidal fuzzy numbers (TrFNs) are defined. We also defined some operations on trapezoidal fuzzy matrices (TrFMs). The notion of Determinant of trapezoidal fuzzy matrices are introduced and discussed. Some of their relevant properties have also been verified.
Heptagonal Fuzzy Numbers by Max Min MethodYogeshIJTSRD
In this paper, we propose another methodology for the arrangement of fuzzy transportation problem under a fuzzy environment in which transportation costs are taken as fuzzy Heptagonal numbers. The fuzzy numbers and fuzzy values are predominantly used in various fields. Here, we are converting fuzzy Heptagonal numbers into crisp value by using range technique and then solved by the MAX MIN method for the transportation problem. M. Revathi | K. Nithya "Heptagonal Fuzzy Numbers by Max-Min Method" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-3 , April 2021, URL: https://www.ijtsrd.com/papers/ijtsrd38280.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathamatics/38280/heptagonal-fuzzy-numbers-by-maxmin-method/m-revathi
Solving Fuzzy Maximal Flow Problem Using Octagonal Fuzzy NumberIJERA Editor
In this paper a general fuzzy maximal flow problem is discussed . A crisp maximal flow problem can be solved
in two methods : linear programming modeling and maximal flow algorithm . Here I tried to fuzzify the
maximal flow algorithm using octagonal fuzzy numbers introduced by S.U Malini and Felbin .C. kennedy [26].
By ranking the octagonal fuzzy numbers it is possible to compare them and using this we convert the fuzzy
valued maximal flow algorithm to a crisp valued algorithm . It is proved that a better solution is obtained when
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this a numerical example is solved and the obtained result is compared with the existing results . If there is no
uncertainty about the flow between source and sink then the proposed algorithm gives the same result as in crisp
maximal flow problems.
A NEW ALGORITHM FOR SOLVING FULLY FUZZY BI-LEVEL QUADRATIC PROGRAMMING PROBLEMSorajjournal
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Solving Fuzzy Maximal Flow Problem Using Octagonal Fuzzy NumberIJERA Editor
In this paper a general fuzzy maximal flow problem is discussed . A crisp maximal flow problem can be solved
in two methods : linear programming modeling and maximal flow algorithm . Here I tried to fuzzify the
maximal flow algorithm using octagonal fuzzy numbers introduced by S.U Malini and Felbin .C. kennedy [26].
By ranking the octagonal fuzzy numbers it is possible to compare them and using this we convert the fuzzy
valued maximal flow algorithm to a crisp valued algorithm . It is proved that a better solution is obtained when
it is solved using fuzzy octagonal number than when it is solved using trapezoidal fuzzy number . To illustrate
this a numerical example is solved and the obtained result is compared with the existing results . If there is no
uncertainty about the flow between source and sink then the proposed algorithm gives the same result as in crisp
maximal flow problems.
Numerical Solutions of Second Order Boundary Value Problems by Galerkin Resid...iosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
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profit maximization. There had been several papers written to demonstrate the use of linear programming in
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taken to find the optimal product mix.
FUZZY ASSIGNMENT PROBLEM BASED ON TRAPEZOIDAL APPROXIMATIONIJESM JOURNAL
In the fuzzy optimization problem researchers used triangular, trapezoidal, LR fuzzy numbers etc. Most of the results depend upon the shape of the fuzzy numbers when operating with fuzzy numbers. Less regular membership functions leads to more complex calculations. Trapezoidal approximation of fuzzy number is adopted for any given fuzzy number. In this paper authors analysed which type of fuzzy number is best in zero’s and one’s method of assignment problem. Numerical examples show that the assignment problem handling LR fuzzy numbers with trapezoidal approximation is an effective one in zero’s assignment.
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The boundary value problem differential operator on the graph of a specific structure is discussed in this article. The graph has degree 1 vertices and edges that are linked at one common vertex. The differential operator expression with real valued potentials, the Dirichlet boundary conditions, and the conventional matching requirements define the boundary value issue. There are a finite number of eig nv lu s in this problem.The residues of the diagonal elements of the Weyl matrix in the eigenvalues are referred to as weight numbers. The ig nv lu s are monomorphic functions with simple poles.The weight numbers under consideration generalize the weight numbers of differential operators on a finite interval, which are equal to the reciprocals of the squared norms of eigenfunctions. These numbers, along with the eig nv lu s, serve as spectral data for unique operator reconstruction. The contour integration is used to obtain formulas for surfacethe weight numbers, as well as formulas for the sums in the case of superficial near ig nv lu s. On the graphs, the formulas can be utilized to analyze inverse spectral problems. Ghulam Hazrat Aimal Rasa "Formulas for Surface Weighted Numbers on Graph" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-6 | Issue-3 , April 2022, URL: https://www.ijtsrd.com/papers/ijtsrd49573.pdf Paper URL: https://www.ijtsrd.com/mathemetics/calculus/49573/formulas-for-surface-weighted-numbers-on-graph/ghulam-hazrat-aimal-rasa
Survival and hazard estimation of weibull distribution based on
Aj24247254
1. M.Jayalakshmi, P. Pandian / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 4, July-August 2012, pp.247-254
A New Method for Finding an Optimal Fuzzy Solution For
Fully Fuzzy Linear Programming Problems
M.Jayalakshmi and P. Pandian
Department of Mathematics, School of Advanced Sciences,
VIT University, Vellore-14, India.
Abstract
A new method namely, bound and decomposition method is proposed to find an optimal fuzzy
solution for fully fuzzy linear programming (FFLP) problems. In the proposed method, the given FFLP
problem is decomposed into three crisp linear programming (CLP) problems with bounded variables
constraints, the three CLP problems are solved separately and by using its optimal solutions, the fuzzy
optimal solution to the given FFLP problem is obtained. Fuzzy ranking functions and addition of nonnegative
variables were not used and there is no restriction on the elements of coefficient matrix in the proposed
method. The bound and decomposition method is illustrated by numerical examples.
Keywords: Linear programming, Fuzzy variables, Fuzzy linear programming, Bound and Decomposition method,
Optimal fuzzy solution
1. Introduction
Linear programming (LP) is one of the most frequently applied operations research techniques. Although, it is
investigated and expanded for more than six decades by many researchers from the various point of views, it is still
useful to develop new approaches in order to better fit the real world problems within the framework of linear
programming. In the real world situations, a linear programming model involves a lot of parameters whose values
are assigned by experts. However, both experts and decision makers frequently do not precisely know the value of
those parameters. Therefore, it is useful to consider the knowledge of experts about the parameters as fuzzy data
[25]. Using the concept of decision making in fuzzy environment given by Bellman and Zadeh [5], Tanaka et al.
[24] proposed a method for solving fuzzy mathematical programming problems. Zimmerman [26] developed a
method for solving fuzzy LP problems using multiobjective LP technique. Campos and Verdegay [8] proposed a
method to solve LP problems with fuzzy coefficients in both matrix and right hand of the constraint. Inuiguchi et al.
[15] used the concept of continuous piecewise linear membership function for FLP problems. Cadenas and
Verdegay [7] solved a LP problem in which all its elements are defined as fuzzy sets. Fang et al. [12] developed a
method for solving LP problems with fuzzy coefficients in constraints. Buckley and Feuring [6] proposed a method
to find the solution for a fully fuzzified linear programming problem by changing the objective function into a
multiobjective LP problem. Maleki et al. [19] solved the LP problems by the comparison of fuzzy numbers in which
all decision parameters are fuzzy numbers. Liu [17] introduced a method for solving FLP problems based on the
satisfaction degree of the constraints.
Maleki [20] proposed a method for solving LP problems with vagueness in constraints by using ranking
function. Zhang et al. [27] introduced a method for solving FLP problems in which coefficients of objective function
are fuzzy numbers. Nehi et al. [22] developed the concept of optimality for LP problems with fuzzy parameters by
transforming FLP problems into multiobjective LP problems. Ramik [23] proposed the FLP problems based on
fuzzy relations. Ganesan and Veeramani [13] proposed an approach for solving FLP problem involving symmetric
trapezoidal fuzzy numbers without converting it into crisp LP problems. Hashemi et al. [14] introduced a two phase
approach for solving fully fuzzified linear programming. Jimenez et al. [16] developed a method using fuzzy ranking
method for solving LP problems where all the coefficients are fuzzy numbers .
Allahviranloo et al. [1] solved fuzzy integer LP problem by reducing it into two crisp integer LP problems.
Allahviranloo et al. [2] proposed a method based on ranking function for solving FFLP problems. Nasseri [21]
solved FLP problems by using classical linear programming. Ebrahimnejad and Nasseri [10] solved FLP problem
with fuzzy parameters by using the complementary slackness theorem. Ebrahimnejad et al. [11] proposed a new
primal-dual algorithm for solving LP problems with fuzzy variables by using duality theorems. Dehghan et al. [9]
proposed a FLP approach for finding the exact solution of fully fuzzy linear system of equations which is applicable
only if all the elements of the coefficient matrix are non-negative fuzzy numbers. Lotfi et al. [18] proposed a new
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2. M.Jayalakshmi, P. Pandian / International Journal of Engineering Research and Applications
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Vol. 2, Issue 4, July-August 2012, pp.247-254
method to find the fuzzy optimal solution of FFLP problems with equality constraints which can be applied only if
the elements of the coefficient matrix are symmetric fuzzy numbers and the obtained solutions are approximate but
not exact. Amit Kumar et al. [3, 4] proposed a method for solving the FFLP problems by using fuzzy ranking
function in the fuzzy objective function.
In this paper, a new method namely, bound and decomposition method is proposed for finding an optimal fuzzy
solution to FFLP problems. In this method, the FFLP problem is decomposed into three CLP problems and then
using its solutions, we obtain an optimal fuzzy solution to the given FFLP problem. With the help of numerical
examples, the bound and decomposition method is illustrated. The advantage of the bound and decomposition
method is that there is no restriction on the elements of the coefficient matrix, fuzzy ranking functions and
nonnegative variables are not used, the obtained results exactly satisfy all the constraints and the computation in the
proposed method is more easy and also, simple because of the LP technique and the level wise computation. The
bound and decomposition method can serve managers by providing an appropriate best solution to a variety of
linear programming models having fuzzy numbers and variables in a simple and effective manner.
2. Preliminaries
We need the following definitions of the basic arithmetic operators and partial ordering relations on fuzzy
triangular numbers based on the function principle which can be found in [2, 4, 25,26 ].
Definition 2.1 A fuzzy number ~
a is a triangular fuzzy number denoted by (a1, a2 , a3 ) where a1, a2 and a3
are real numbers and its member ship function a ( x) is given below:
~
( x a1) /( a2 a1) for a1 x a2
a ( x) (a3 x) /( a3 a2 ) for a2 x a3
~
0 otherwise
Definition 2.2 Let (a1, a2 , a3 ) and (b1, b2 , b3 ) be two triangular fuzzy numbers. Then
(i) (a1, a2 , a3 ) (b1, b2 , b3 ) = (a1 b1, a2 b2 , a3 b3 ) .
(ii) (a1, a2 , a3 ) (b1, b2 , b3 ) = (a1 b3 , a2 b2 , a3 b1) .
(iii) k (a1, a2 , a3 ) = (ka1, ka2 , ka3 ) , for k 0 .
(iv) k (a1, a2 , a3 ) = (ka3 , ka2 , ka1 ) , for k 0 .
(a1b1 , a2 b2 , a3b3 ), a1 0,
(v) (a1, a2 , a3 ) (b1 , b2 , b3 ) (a1b3 , a2 b2 , a3b3 ), a1 0, a3 0,
(a b , a b , a b ), a3 0.
1 3 2 2 3 1
Let F (R) be the set of all real triangular fuzzy numbers.
~ ~
A (a1 , a2 , a3 ) and B (b1 , b2 , b3 ) be in F (R) , then
Definition 2.3 Let
~ ~ ~ ~
(i) A B iff ai bi , i 1,2, 3 ; (ii) A B iff ai bi , i 1,2, 3
~ ~ ~ ~
(iii) A B iff ai bi , i 1,2, 3 and A 0 iff ai 0 , i 1,2, 3 .
3. Fully Fuzzy Linear Programming Problem
Consider the following fully fuzzy linear programming problems with m fuzzy inequality/equality constraints
and n fuzzy variables may be formulated as follows:
(P) Maximize (or Minimize) ~ c ~ z ~T x
248 | P a g e
3. M.Jayalakshmi, P. Pandian / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 4, July-August 2012, pp.247-254
~ ~
subject to A ~ {, , } b ,
x (1)
~ ~
x 0 (2)
where c c j , A aij , ~ ( ~j )nx1 and b (bi )mx1 and
~ ~ ~ ~ ~ ~ x ~
~T ~
1n mn
x x aij , c j , ~j , bi F ( R) , for all
1 j n and 1 i m .
~
Let the parameters aij , c j , ~ j and bi be the triangular fuzzy number ( p j , q j , r j ) , ( x j , y j , t j ) ,
~ ~ x
( aij , bij , cij ) and ( bi , g i , hi ) respectively. Then, the problem (P) can be written as follows:
(P) Maximize (or Minimize) z1 , z 2 , z3 n
p j , q j , rj x j , y j ,t j
j 1
subject to
n
aij , bij , cij x j , y j , t j { , , } bi , gi , hi , for all i 1,2,..., m
j 1
x j , y j,t j ~
0.
Now, since x j , y j , t j is a triangular fuzzy number, then
xj yj tj , j =1,2,…,m . (3)
The relation (3) is called bounded constraints.
Now, using the arithmetic operations and partial ordering relations, we decompose the given FLPP as follows:
Maximize z1 = n
lower value of ( p j , q j , r j ) ( x j , y j , t j )
j 1
z 2 = middle value of ( p j , q j , r j ) ( x j , y j , t j )
n
Maximize
j 1
upper value of ( p j , q j , r j ) ( x j , y j , t j )
n
Maximize z3 =
j 1
subject to
n
lower value of (aij , bij , cij ) ( x j , y j , t j ) {, , }bi , for all i 1,2,..., m ;
j 1
n
middle value of (aij , bij , cij ) ( x j , y j , t j ) {, , }gi , for all i 1,2,...,m ;
j 1
upper value of (aij , bij , cij ) ( x j , y j , t j ) {, , }hi , for all i 1,2,...,m
n
j 1
and all decision variables are non-negative.
From the above decomposition problem, we construct the following CLP problems namely, middle level
problem (MLP), upper level problem (ULP) and lower level problem (LLP) as follows:
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4. M.Jayalakshmi, P. Pandian / International Journal of Engineering Research and Applications
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Vol. 2, Issue 4, July-August 2012, pp.247-254
z 2 = middle value of ( p j , q j , r j ) ( x j , y j , t j )
n
(MLP) Maximize
j 1
subject to
Constraints in the decomposition problem in which atleast one decision
variable of the (MLP) occurs and all decision variables are non-negative.
upper value of ( p j , q j , r j ) ( x j , y j , t j )
n
(ULP) Maximize z3 = j 1
subject to
upper value of ( p j , q j , rj ) ( x j , y j , t j ) z2 ;
n
j 1
Constraints in the decomposition problem in which atleast one decision
variable of the (ULP) occurs and are not used in (MLP) ;
all variables in the constraints and objective function in (ULP) must
satisfy the bounded constraints ;
replacing all values of the decision variables which are obtained in (MLP) and all
decision variables are non-negative.
and (LLP) Maximize z1 = n
lower value of ( p j , q j , r j ) ( x j , y j , t j )
j 1
subject to
lower value of ( p j , q j , rj ) ( x j , y j , t j ) z 2 ;
n
j 1
Constraints in the decomposition constraints in which atleast one decision
variable of the (LLP) occurs which are not used in (MLP) and (ULP);
all variables in the constraints and objective function in (LLP) must
satisfy the bounded constraints;
replacing all values of the decision variables which are
obtained in the (MLP) and (ULP) and all decision variables are non-negative,
where z2 is the optimal objective value of (MLP).
Remark 3.1: In the case of LP problem involving trapezoidal fuzzy numbers and variables, we decompose it into
four CLP problems and then, we solve the middle level problems ( second and third problems) first. Then, we solve
the upper level and lower level problems and then, we obtain the fuzzy optimal solution to the given FLP problem
involving trapezoidal fuzzy numbers and variables.
4. The Bound and Decomposition Method
Now, we prove the following theorem which is used in the proposed method namely. Bound and
decomposition method to solve the FFLP problem.
Theorem 4.1: Let [ xM ] { x j , x j M } be an optimal solution of (MLP) , [ xU ] { x j , x j U } be an
optimal solution of (ULP) and [ xL ] { xj , xj L } be an optimal solution of (LLP) where L, M and U
are sets of decision variables in the ( LLP), (MLP) and (ULP) respectively. Then
~ ( x1 , x 2 , x3 ), j 1,2,..., n
xj j j j is an optimal fuzzy solution to the given problem (P) where each one of
x1, x2 and x3 , j 1,2,..., n is an element of L, M and U .
j j j
250 | P a g e
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(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 4, July-August 2012, pp.247-254
Proof: Let [ ~ j ] {~ j , j 1,2,..., n} be a feasible solution of (P). Clearly, [ yM ],[ yU ] and [ yL ] are feasible
y y
solutions of (MLP), (ULP) and ( LLP) respectively.
Now, since [ xM ],[ xU ] and [ xL ] are optimal solutions of (MLP), (ULP) and ( LLP) respectively, we have
Z1 ([ xL ]) Z1 ([ yL ]); Z 2 ([ xM ]) Z 2 ([ yM ]) and Z3 ( [ xU ]) Z3 ([ yU ])
x
This implies that Z ([ ~ ]) Z ([ ~ ]) , for all feasible solution of the problem (P).
j y j
Therefore, {~ ( x1, x2 , x3 ), j 1,2,..., n} is an optimal fuzzy solution to the given problem (P) where
xj j j j
each one of x1, x2 and x3 , j 1,2,..., n is an element of L, M and U .
j j j
Hence the theorem.
Now, we propose a new method namely, bound and decomposition method for solving a FFLP problem.
The bound and decomposition method proceeds as follows.
Step 1: Construct (MLP), (ULP) and (LLP) problems from the given the FFLP problems.
Step 2: Using existing linear programming technique, solve the (MLP) problem, then the (ULP) problem and then,
the (LLP) problem in the order only and obtain the values of all real decision variables x j , y j and t j and
values of all objectives z1 , z2 and z3 . Let the decision variables values be x j , y j and t j , j = 1,2,…,m and
objective values be z1 , z2 and z3 .
Step 3: An optimal fuzzy solution to the given FFLP problems is ~ ( x , y , t ) ,
xj j=1,2,…,m and the
j j j
maximum fuzzy objective is ( z1 , z 2 , z3 ) ( by the Theorem 4.1.).
Now, we illustrate the proposed method using the following numerical examples
Example 4.1: Consider the following fully fuzzy linear programming problem (Objective function contains negative
term)
Maximize z 1, 2, 3 ~ 2, 3, 4 ~2
x1 x
subject to
0, 1, 2 ~1 1, 2, 3 ~2 ( 2, 10, 24 );
x x
1, 2, 3 ~1 0, 1, 2 ~2 ( 1, 8, 21 );
x x
~ and ~ ~ .
x1 x2 0
Let ~1 ( x1 , y1 , t1 ) , ~2 ( x2 , y2 , t 2 ) and ~ ( z1 , z 2 , z3 ) .
x x z
Now, the decomposition problem of the given FLPP is given below:
Maximize z1 t1 2x2
Maximize z2 2 y1 3 y2
Maximize z3 3t1 4t2
subject to
0 x1 x2 2 ; x1 0 x2 1 ;
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Vol. 2, Issue 4, July-August 2012, pp.247-254
y1 2 y2 10 ; 2 y1 y2 8 ; .
2t1 3t 2 24 ; 3t1 2t 2 21 ;
x1 , x2 0 , y1 , y2 0 , t1 , t 2 0 .
Now, the Middle Level problem is given below:
( P2 ) : Maximize z2 2 y1 3 y2
subject to
y1 2 y2 10 ; 2 y1 y2 8 ; y1 , y2 0 .
Now, solving the problem ( P2 ) using simplex method, we obtain the optimal solution y1 2 ; y2 4 and
z 2 16 .
Now, the Upper Level problem is given below:
( P3 ) : Maximize z3 3t1 4t 2
subject to
3t1 4t2 16 ; 2t1 3t 2 24 ; 3t1 2t 2 21 ; t1 y1 ; t2 y2 ; t1 , t 2 0 .
Now, solving the problem ( P3 ) with y1 2 ; y2 4 using simplex method , we obtain the optimal solution
and t1 3 ; t 2 6 and z3 33 .
Now, the Lower Level problem:
( P ) : Maximize z1 t1 2x2
1
subject to
t1 2 x2 16 ; 0 x1 x2 2 ; x1 0 x2 1 ; x1 y1 , x2 y2 ; x1 , x2 0 .
Now, solving the problem ( P ) with t1 3 ; y1 2 ; y 2 4 by simplex method, we obtain the optimal
1
solution x1 1 ; x2 2 and z1 1 .
Therefore, an optimal fuzzy solution to the given fully fuzzy linear programming problem is
~ ( 1, 2, 3 ) , ~ ( 2, 4, 6 ) and ~ ( 1, 16, 33 ) .
x1 x2 z
Remark 4.1: The solution of the Example 4.1 is obtained at the 8th iteration with 10 non-negative variables by Amit
Kumar et al.[4] method.
Example 4.2: Consider the following fully fuzzy linear programming problem:
Maximize z 1, 6, 9 ~ 2, 3, 8 ~2
x1 x
subject to
2, 3, 4 ~1 1, 2, 3 ~2 ( 6, 16, 30 );
x x
1, 1, 2 ~1 1, 3, 4 ~2 ( 1, 17, 30 );
x x
~ and ~ ~ .
x1 x2 0
Now, by solving the bound and decomposition method, an optimal fuzzy solution to the given fully fuzzy linear
programming problem is ~1 ( 1, 2, 3 ) , ~2 ( 4, 5, 6 ) and ~ ( 9, 27, 75 ) .
x x z
Remark 4.2: The solution of the Example 4.2, obtained by the bound and decomposition method is same as in Amit
Kumar et al. [4].
Example 4.3: Consider the following fully fuzzy linear programming problem:
Maximize z 1, 6, 9 ~1 2, 2, 8 ~2
x x
subject to
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Vol. 2, Issue 4, July-August 2012, pp.247-254
0, 1, 1 ~1 2, 2, 3 ~2 ( 4, 7, 14 );
x x
2, 2, 3 x1
~ 1, 4, 4 ~ ( 4, 14, 22 );
x2
2, 3, 4 x1
~ 1, 2, 3 ~ ( 12, 3, 6 );
x2
~ and ~ ~ .
x1 x2 0
Now, by solving the bound and decomposition method, an optimal fuzzy solution to the given fully fuzzy linear
programming problem is ~ ( 0, 1, 2 ) , ~2 ( 2, 3, 4 ) and ~ ( 4, 12, 50 ) .
x1 x z
Remark 4.3: The solution of the Example 4.3, obtained by the Bound and Decomposition method is same as in
Amit Kumar et al. [3].
Example 4.4: Consider the following fully fuzzy linear programming problem:
Maximize z 1, 2, 3 ~1 2, 3, 4 ~2
x x
subject to
0, 1, 2 ~1 1, 2, 3 ~2 ( 1, 10, 27 );
x x
1, 2, 3 x1
~ 0, 1, 2 ~ ( 2, 11, 28 );
x2
~ and ~ ~ .
x1 x2 0
Now, by solving the bound and decomposition method, an optimal fuzzy solution to the given fully fuzzy linear
programming problem is ~ ( 2, 4, 6 ) , ~2 ( 1, 3, 5 ) and ~ ( 4, 17, 38 ) .
x1 x z
Remark 4.4: The solution of the Example 4.4, obtained by the bound and decomposition method is same as in Amit
Kumar et al. [3].
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