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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
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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L. A. Sigarpich et al.1002
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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Degeneracy in Fuzzy Linear Programming and its Application 1003
=
.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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L. A. Sigarpich et al.1004
( 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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Degeneracy in Fuzzy Linear Programming and its Application 1005
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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L. A. Sigarpich et al.1006
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