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International Journal of Uncertainty,

Fuzziness and Knowledge-Based Systems

Vol. 19, No. 6 (2011) 9991012

World Scientific Publishing Company

DOI: 10.1142/S021848851100743X

999

DEGENERACY IN FUZZY LINEAR PROGRAMMING

AND ITS APPLICATION

LEILA ALIZADEH SIGARPICHDepartment of Mathematics, Islamic Azad University,

Shahr-e Rey Branch, P. O. Box: 18155-144, 1311963651 Tehran, Iran

[email protected]

TOFIGH ALLAHVIRANLOO, FARHAD HOSSEINZADEH LOTFI and NARSIS AFTAB KIANI

Islamic Azad University, Science and Research Branch,

P. O. Box: 14515-775, 1477893855 Tehran, Iran

Received 21 October 2008

Revised 19 September 2011

In this paper, by a definite linear function for ranking symmetric triangular fuzzy numbers, in a Fuzzy Linear

Programming problem (FLP) model, we introduced a Fuzzy Degenerate Solution (FDS). In the physical

meaning, occurrence of degeneracy in a Fuzzy Minimal Cost Flow Network is investigated. To prevent of

falling into Cycling phenomenon in optimization process, we defined two new techniques of Cycling prevention

proper to fuzzy environment.

Keywords: Symmetric triangular fuzzy number; fuzzy numbers ranking; fuzzy linear programming; fuzzy

degeneracy; cycling prevention.

1. Introduction

The concept of decision making in fuzzy environment was first proposed by Bellman and

Zadeh.5

Subsequently, Tanaka et al.22

made use of this concept in mathematical

programming. A fuzzy linear programming problem (FLP) has various types. One of

them is the full fuzzy linear programming problem. FLP has been studied by many

scholars.13, 6, 9, 1113,15,1921

We discuss on a fuzzy linear programming problem, which the right hand side of the

coefficient matrix in the constraints and variables are fuzzy numbers. The fuzzy linear

programming problem with fuzzy coefficients was proposed by Negoita.20

Maleki et al.

introduced a linear programming problem with fuzzy variables and proposed a method

for solving it.19

Maleki used a certain ranking function to solve fuzzy linear programming

problems.1

He also introduced a new method for solving linear programming problems

with vagueness in constraints using a linear ranking function.

In this paper we focus on the kind of linear programming problem in which the right

hand side of the constraints and variables are fuzzy numbers. In this field, some papers

have been published.18

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L. A. Sigarpich et al.1000

In the solving process of a linear programming problem, if some iterations are carried

out in simplex method without any improvement in the objective function, it means that

there is a tie for the minimum ratio to specify the existing variable, so we will have a

degenerate solution.14,17 We will extend this concept to fuzzy linear programmingproblems. An interesting example for the physical meaning of this phenomenon in an

FLP is a special kind of fuzzy minimal-cost network flow problem. The model is called

flow conservation or nodal balance.17

This kind of problem might arise in a logistics

network, communication systems, oil pipeline systems, distribution problems in the

power networks and various other areas.

In the solving process, when degeneracy occurs, if the same sequence of iterations

appear many times in the simplex method, we shall cycle forever among the bases

1 2 1, ,...., tB B B B= without reaching an optimal solution; consequently, we call this

phenomenon "falling into cycling".Some rules for preventing cycling have been discussed in Refs. 14 and 17. In the

present work, FLPs with fuzzy (R.H.S.) right hand side are considered. We define fuzzy

degeneracy (FD) in an FLP, then the rules that are applied for the prevention of cycling

prevention are shown. One of the rules is "fuzzy perturbation technique" which is used in

FLP; another one is the "lexicographic rule", also applied in a fuzzy environment.

The paper is organized as follows:

In Sec. 2, some definitions of a fuzzy number are discussed. In Sec. 3, several

theorems are explained for proving the equivalence of an FLP to two crisp LPs. In Sec. 4,

fuzzy degeneracy and its occurrence in a fuzzy network is defined and the algorithms for

preventing cycling in the problem are illustrated by solving some numerical examples.

Conclusion is drawn in Sec. 5.

2. Preliminaries

We present an arbitrary fuzzy number by an ordered pair of functions ( ( ), ( )), 0 1,u r u r r which satisfy the following requirements:

( )u ris a bounded left continuous nondecreasing function over [0 , 1].

( )u ris a bounded left continuous nonincreasing function over [0 , 1].

( )u rand ( )u r are right continuous in 0 .

( )u r ( )u r , 0 1r .

Therefore, the ordered pair ( ( ) , ( ))u r u r

is the conventional parametric form of a

triangular fuzzy number.

A crisp number is simply represented by ( )u r = ( )u r = , 0 1r .

2.1.Definitions

2.1.1.Triangular fuzzy number

( ) (1) (1)uC Core u u u= = =% % is the most promising value of u~ (may be in the

economics literature), (0)u is the smallest possible value of u~

and (0)u is the largestpossible value of u~ .

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Degeneracy in Fuzzy Linear Programming and its Application 1001

( )u r = (0)u + ( (0)) .uC u r% ( )u r = )0(u + ( (0))uC u r% , 0 1r (ris a real number).

(0) 0Lu uw C u= % % and (0) 0

Ru uw u C= % % are the left and right margins of u% ,

respectively.

2.1.2. Symmetric triangular fuzzy number( STFN)

IfL Ru uw w=% % , it means that the right and left margins of the fuzzy number are equal, then it is

called the symmetric triangular fuzzy number.

2.1.3.A new representation of ( STFN)

We write u~ , as the following new representation:

( (1 ) , (1 )) ( , ),u u u u u uu C w r C w r C w= + =% % % % % %% 0 1 ,r then ( , )u uC w% % will be a

STFN. From Definition 2.1.1, we will have (1 ) ( )u uC w r u r =% % and

(1 ) ( )u uC w r u r + =% % where , .u uC w R% %

2.1.4. Fuzzy arithmetics

Let T.S be the set of all STFNs, 1 1 2 2( , ), ( , )t C w u C w= =% and ,Rk by using the

extension principle in fuzzy arithmetics, we can define:

1. u~t = if and only if21

CC = and21

ww = .

2. t u%+ =1 2 1 2

( , )C C w w+ + .

3. 1 1( , )kt kC k w= .1

2.1.5. Fuzzy number ranking

To define an ordering on T.S , let =t~

(11

w,C ) and u~ = (22

w,C ) be in T.S . We say

u~t~ < if and only if:

1.21

CC <

or

2.1 2

( )C C= 2 1

( )w w< .

Supposem n

A R

(in a special case we could have nm = ), =X~

( 1 2, ,...., nT T T )

Tand Y

~= ( 1 2, ,...., nU U U )

T, which means that X

~, Y

~ T.S . Now we

have Core ( X~

+ Y~

) = Core ( X~

) + Core ( Y~

).

1. Core ( X~

A ) = ACore ( X~

).

2. A ( X~

+ Y~

) = X~

A + Y~

A .13

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Lemma 1. Let 1 2, 0 ,w w then 1 2w w if and only if for any 0 1r ,

1 1 2 2w w r w w r .

Proof. Proof is clear.

Lemma 2. Consider two sSTFN =T ( 1 1 1 1,w w r w w r + ) and U = ( 2 2 ,w w r + 2 2w w r ) whose cores are zero, then

T U if and only if 2 2 1 1w w r w w r , for any 0 1r .

Proof. Proof is clear.

3. Fuzzy Linear Programming Problems

Consider the following FLP

==

=

).,(~

),(~

.~

,~

0~

~~.

~

~~~~bbXX

WCbWCXTSXX

bXAts

XCMax

(1)

Where 0 (0. ), 0, ( ), ( ), , 0X Xb b

C Core X C Core b W W z z= = = % % % %%% % are the

margins of X~

and b

~, respectively and STX

~ . m nA R , nC R

and b% is an

arbitrary fuzzy vector. In this model, one of the parameters that affects the feasibility or

infeasibility of the FLP is parameter ""; for feasibility, this parameter must be largeenough (with the definite rank, defining a margin for the fuzzy zero is essential).

In this section, we are going to reduce FLP (1) to two crisp LPs.

Theorem 1.If problem (1) has a feasible solution then b~ T.S .

Proof. Note that every combination of STFNs is an STFN. We have1

n

j jja x b

== %% .

b~ T.S , and the proof is completed. Problem (1) is equivalent to the following

problem:

(2)

=

TSXWC

WCWCAts

WCCMax

XX

bbXX

XX

.~

,0,0

),(),(.

),(

~~

~~~~

~~

where , ,X X b

C W C% % % and

bW R% . By (1), this problem is equivalent to the following

problem:

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=

.0,0

),(),(.

),(

~~

~~~~

~~

XX

bbXX

XX

WC

WCWAACts

WCCCMax

(3)

Remark 1. Problem. (1) is reduced to problem (3). By Definition 2.1.5, the following

two problems (4) and (5) are derived from (3):

=

.0,0

.

~~

~~

~

XX

bX

X

WC

CACts

CCMax

(4)

which is a crisp linear programming problem and the constraint " 0X

W z % " isredundant; and the i -th entry of

bC% is the core of the i -th entry of b

~.

=

.0,0

.

~~

~~

~

XX

bX

X

WC

WWAts

WCMax

(5)

and the constraint " 0X

C % " is redundant.

Proposition 1. We notice that problems (4) and (5) are solved independently from each

other. Then the solution of the original problem is derived by combining those solutions,

in the parametric form of a fuzzy number. The equivalence of problem (2) to (4) and (5)

could be explained by the following simple remark:

Remark 2.fX% is a feasible solution of (3) if and only if there exist two feasible

solutions cf

X and wf

X of problems (4) and (5), respectively, such thatf

X% =

( cf

X , wf

X ).

Proof. Let fX% be a feasible solution of problem (2). Set:cfX : = fX

C%

.

wfX : = fXW

%.

It is clear thatf

X% = ( cf

X , wf

X ) = ( fXC

%, fXW

%). Now 0 cfX and

cf

f

XAX AC=

%= Core (

fAX% ) = Core( b

~) =

bC% .

Moreover 0 wfX = fXW% andw

f

f

bXA X A W W= = %

%

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( A is the absolute value of the components of matrix A ). So we have two feasible

solutions ,c wf f

X X of problems (4) and (5), respectively, where ( , )c wf ff

X X X=% .

Conversely, if we have two feasible solutions ,c wf f

X X of problems (4) and (5),

respectively, then , 0c ff

XbXAX AC C C= = % %% and , 0w f

f

b bXA X AW W W z= = % %% ,

thus ( . )X X

A C W% % = ( . ).b bC W% % This means that ,b~

X~

A = then =X~

( . )X X

C W% % is the

feasible solution of (2), and the proof is completed.

Corollary 1. Problem (2) is infeasible if and only if either problem (4) or problem (5) is

infeasible.

Theorem 2. If ,c wf f

X X

are the optimal solutions of problems (4) and (5),

respectively, then

( , )c wf ff

X X X =% is the optimal solution of (2).

Proof. Suppose ,c wf f

X X

are the optimal solutions of problems (4) and (5),

respectively. Letf

X =% ( , )c wf fX X and fX% be a feasible solution of problem (2). By

Remark 2, cf

X = Core( fX~

) and ( )wf f

X W X= % are feasible solutions of problems (4)

and (5) , respectively, so

c cf fCX CX

w wf f

C X C X

( C is the absolute value of the components of vector C). Thus, by Definition 2.1.5 we

have

( , ) ( , )c w c wf f f fCX C X CX C X

So3 4 *

( , ) ( , ) .c wf ff f

CX CX C X CX C X CX = =% % % %

Hence,f

X% is the optimal solution of problem (2), and the proof is completed.

Theorem 3.Iff

X% is the optimal solution of problem (2), then there exist two optimal

solutions of problems (4) and (5), say, ,c wf f

X X

, such that

( , ).c wf ff

X X X =%

Proof. Supposef

X% is the optimal solution of problem (2). Let cf

X

= Core (f

X% )

and wf

X

= W ( )fX% . By theorem (2) these are feasible solutions of problems (4) and

(5), respectively.

Now, if cf

X and wf

X are feasible solutions of problems (4) and (5) , respectively,

we know that ( , )c wf ff

X X X=% is a feasible solution of problem (2). Since fX% is the

optimal solution of problem (2), so by definition (5) ( )cf f

CX Core CX = %

**( ) cff

Core CX CX =% , hence *c cf fCX CX and thus * cfX is the optimal solution

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of problem (4). On the other hand, if* w wf fC X C X > then 2Y =%

*( , ) ( , )c w c wf f f f fCX C X CX C X CX

< = % and ( * ,c wf fX X ) is a feasible solution

of problem (2). But this is a contradiction, so*w wf fC X C X and hence * wfX is the

optimal solution of problem (5), and the proof is completed.

Remark 3. Problem (2) is reduced to problems (4) and (5).

Corollary 2.fX% is the unique optimal solution of problem (2) if and only if each of the

problems (4) and (5) have unique optimal solutions cf

X

and wf

X

, respectively, such

that* *

( , ).c wf ff

X X X =% Also * wfX

ois an alternative solution of problem (2) if and

only if at least one of the problems (4) or (5) has an alternative solution.

Proof. This is a summary of the previous results.

Remark 4. Problem (2) is unbounded if and only if either (4) or (5) have an unbounded

solution.

Remark 5. The above mentioned proofs are analogue for the minimization case.

4. Fuzzy Degeneracy

Let X be an optimal solution of problem (4) and Y be an optimal solution of problem

(5) then

( ; )X X Y

=%

This solution could be written in the form* * *

1 2( , ..., )nX x x x =% , which is an optimal

solution of problem (2). By this point and concluding from the previous theorems, we are

going to define Fuzzy degeneracy:

4.1.Definitions

4.1.1. Strong degeneracy

A fuzzy linear programming problem has a strongly degenerate basic feasible solution ifthe number of the components of the solution in the simplex method of the form ( 0 , 0 )

or (0,) are more than " n m " (n is the number of the variables and m is the rank of the

coefficient matrix in the system of simultaneous fuzzy equations "A". ("A" is the

coefficient matrix in the (1) till (5)) in Sec.3, therefore we have a strongly degenerate

basic feasible solution for fuzzy linear programming problems.

4.1.2. Weak degeneracy

A fuzzy linear programming problem has a weakly degenerate basic feasible solution if

the number of the fuzzy zeroes ( w,0

), of any basic feasible solution of the problem aremore than " n m ", therefore we have a weakly degenerate basic feasible solution for

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fuzzy linear programming problems. We know that " w " always satisfies the condition

0 w z .

Notation: In our methodology, related to crisp problems, the original problem has a stronglydegenerate basic feasible solution if both of the crisp problems have degenerate basic feasible

solutions and the (zero, zero) or (zero,

z), the respective components of the degenerate

solutions of the two crisp problems are corresponding. But if only the problem related to the

core has a degenerate basic feasible solution or the corresponding components of the solutions

of the two crisp problems are not simultaneously (zero, zero) or (zero, z) , then the original

problem will have a fuzzy solution that is a weakly degenerate basic feasible solution.

4.2.Degeneracy in capacity ascertainment of the electricity transmission network lines

with a limited energy loss

We know that the structure of a network is analogues with a graph. There are some nodes

and arcs in a network. With each node i in G we associate a fuzzy numberi

b~

, that is, the

available (approximate) fuzzy supply of an item (if 0ib >% % ) or the required (approximate)

fuzzy demand for the item (if 0ib

% % ) are

called sources, and nodes with ( 0ib

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