World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:1, No:10, 2007
Simplex Method for Solving Linear
Programming Problems with Fuzzy Numbers
S. H. Nasseri, E. Ardil, A. Yazdani, and R. Zaefarian
International Science Index, Mathematical and Computational Sciences Vol:1, No:10, 2007 waset.org/Publication/2936
Abstract—The fuzzy set theory has been applied in many fields,
such as operations research, control theory, and management
sciences, etc. In particular, an application of this theory in decision
making problems is linear programming problems with fuzzy
numbers. In this study, we present a new method for solving fuzzy
number linear programming problems, by use of linear ranking
function. In fact, our method is similar to simplex method that was
used for solving linear programming problems in crisp environment
before.
Keywords—Fuzzy number
function, simplex method.
F
linear
programming,
ranking
I. INTRODUCTION
UZZY
linear programming first formulated by
Zimmermann [10]. Recently, these problems are
considered in several kinds, that is, it is possible that some
coefficients of the problem in the objective function, technical
coefficients, the right-hand side coefficients or decision
making variables be fuzzy number [3], [4], [5], [6], [7], [8],
[9]. In this work, we focus on the linear programming
problems with fuzzy numbers in the objective function.
Verdegay and et al [4], [9] proposed the equivalent parametric
linear programming problems for these problems by use of a
certain membership function and proposed a dual method for
fuzzy number linear programming problems. Here, we first
explain the concept of the comparison of fuzzy numbers by
introducing a linear ranking function. Moreover, we describe
basic feasible solution for the FNLP problems and state
optimality conditions for these problems. Finally, we provide
some important results for FNLP problems and we propose
simplex algorithm for solving these problems.
II. DEFINITIONS AND NOTATIONS
We first review necessary backgrounds of fuzzy sets theory.
S.H. Nasseri, Department of Mathematical Sciences, Sharif University of
Technology, Tehran, Iran (corresponding author: [email protected] ).
E. Ardil is with Department of Computer Engineering, Trakya University,
Edirne, Turkey ([email protected]).
A. Yazdani, Faculty of Basic Sciences, Mazandaran University, Babolsar,
Iran ( [email protected] ).
R. Zaefarian, Industrial Engineering Department, Sharif University of
Technology, Tehran, Iran ([email protected]).
International Scholarly and Scientific Research & Innovation 1(10) 2007
A. Fuzzy Sets
Let X be a classical set of objects, called the universe,
whose generic elements are denoted by x . The membership
in a crisp subset of X is often viewed as characteristic
function μ A ( x ) from X to {0,1} such that:
μA (x ) = 1
, if
x ∈A
= 0 , otherwise
where {0,1} is called a valuation set.
[0,1] ,
A is called a fuzzy set proposed by Zadeh [2]. μA (x ) is the
degree of membership of x in A . The closer the value of
μ A (x ) is to 1 , the more x belong to A . Therefore, A is
If the valuation set is allowed to be the real interval
completely characterized by the set of ordered pairs:
A = {(x , μ A (x )) | x ∈ X } .
The support of a fuzzy set A is the crisp subset of X and
is presented as:
SuppA = {x ∈ X | μA (x ) > 0} .
The α − level ( α − cut ) set of a fuzzy set A is a crisp
subset of X and is denoted by
Aα = {x ∈ X | μA (x ) ≥ α } .
A
fuzzy
set
A
in
X
is
convex
μA (λ x + (1 − λ ) y ) ≥ min{μ A (x ), μA ( y )}
if
x , y ∈ X , and λ ∈ [0,1] . Alternatively, a fuzzy set is
convex if all α − level sets are convex. Note that in this
paper we suppose that X = R .
A fuzzy number A is a convex normalized fuzzy set on
the real line R such that
1) It exists at least one x0 ∈ R with μ A (x 0 ) = 1 .
2)
μA (x ) is piecewise continuous.
Among the various types of fuzzy numbers, triangular and
trapezoidal fuzzy numbers are of the most important. Note
that, in this study we only consider trapezoidal fuzzy numbers.
A fuzzy number is a trapezoidal fuzzy number if the
membership function of it be in the following form:
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World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:1, No:10, 2007
1
ℜ(a~ ) = a L + a U + ( β − α )
2
L
U
%
where a = ( a , a , α , β ) ∈ F ( R ) .
(4)
IV. FUZZY LINEAR PROGRAMMING
aL − α
aL
aU
In this section, we introduce fuzzy linear programming
(FLP) problems. So, we first define linear programming
problems.
aU + β
Fig. 1 Trapezoidal Fuzzy Number
We
show
any
trapezoidal
fuzzy
number
a% = (a L , aU , α , β ) , where the support of
U
L
A. Linear Programming
A linear programming (LP) problem is defined as:
Max z = cx
s.t. Ax = b
is
(a − α , a + β ) , and the modal set of a% is [a , a ] .
Let F ( R ) be the set of trapezoidal fuzzy numbers. In the
next subsection we describe arithmetic on F ( R ) .
L
International Science Index, Mathematical and Computational Sciences Vol:1, No:10, 2007 waset.org/Publication/2936
by
a%
U
B. Arithmetic on Fuzzy Numbers
~ = (a L , a U , α , β ) and b~ = (b L , bU , γ , θ ) be two
Let a
trapezoidal fuzzy numbers and x ∈ R . Then, the results of
applying fuzzy arithmetic on the trapezoidal fuzzy numbers as
shown in the following:
x≥0
T
where c = (c1,..., cn ), b = (b1,..., bm ) , and A = [ aij ]m×n .
(5)
In the above problem, all of the parameters are crisp [1].
Now, if some of the parameters be fuzzy numbers we obtain a
fuzzy linear programming which is defined in the next
subsection.
x > 0, x a% = ( x a L , x a U , x α , x β )
x < 0, x a% = ( x a U , x a L , − x β , − x α )
B. Fuzzy Linear Programming
Suppose that in the linear programming problem some
parameters be fuzzy numbers. Hence, it is possible that some
coefficients of the problem in the objective function, technical
coefficients, the right-hand side coefficients or decision
making variables be fuzzy number [3], [4], [5], [6], [7], [8],
[9]. Here, we consider the linear programming problems with
fuzzy numbers in the objective function.
III. RANKING FUNCTIONS
V. FUZZY NUMBER LINEAR PROGRAMMING
Image of a% : −a% = ( −a , −a , β , α )
U
~
~ + b = (a
Addition: a
Scalar Multiplication:
L
L
+ b L , a U + bU , α + γ , β + θ )
A convenient method for comparing of the fuzzy numbers
is by use of ranking functions. A ranking function is a map
from F ( R ) into the real line. Now, we define orders on
A fuzzy number linear programming (FNLP) problem is
defined as follows:
Max ~
z = c~x
F (R ) as following:
~
(1)
a~ ≥ b if and only if ℜ(a% ) ≥ ℜ(b% )
ℜ
~
a~ > b if and only if ℜ(a% ) > ℜ(b% )
(2)
ℜ
~
a~ = b if and only if ℜ(a% ) = ℜ(b% )
(3)
ℜ
~ and b~ are in F ( R) . It is obvious that we may
where a
~ ≤ b~ if and only if b~ ≥ a~ . Since there are many
write a
s.t. Ax = b
ℜ
ℜ
ranking function for comparing fuzzy numbers we only apply
linear ranking functions. So, it is obvious that if we suppose
that ℜ be any linear ranking function, then
~ ≥ b~ if and only if a~ − b~ ≥ 0 if and only if − b~ ≥− a~
i) a
ℜ
ℜ
~ ≥ b~ and c~ ≥ d~ , then a~ + c~ ≥ b~ + d~ .
ii) If a
ℜ
ℜ
ℜ
ℜ
One suggestion for a linear ranking function as following:
International Scholarly and Scientific Research & Innovation 1(10) 2007
ℜ
(6)
x≥0
m
n
m×n T
n
where b ∈ R , x ∈ R , A∈ R , c% ∈ (F (R)) , and ℜ is a
linear ranking function.
Definition 5.1. We say that vector x ∈ R is a feasible
solution to (6) if and only if x satisfies the constraints of the
problem.
Definition 5.2. A feasible solution x* is an optimal solution
for (6), if for all feasible solution x for (6), we have
c~x* ≥ c~x .
n
ℜ
A. Fuzzy Basic Feasible Solution
Here, we introduce basic feasible solutions for FNLP
problems. Consider the system Ax = b and x ≥ 0, where
A is an m × n matrix and b is an m vector. Now, suppose
rank ( A, b) = rank ( A) = m . Partition after possibly
rearranging the columns of A as [ B , N ] where B ,
that
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scholar.waset.org/1999.7/2936
World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:1, No:10, 2007
International Science Index, Mathematical and Computational Sciences Vol:1, No:10, 2007 waset.org/Publication/2936
m × m , is nonsingular. It is obvious that rank (B ) = m .
T
T T
−1
The point x = ( xB , xN ) where xB = B b , xN = 0 is
called a basic solution of the system. If xB ≥ 0 , then x is
called a basic feasible solution (BFS) of the system. Here B
is called the basic matrix and N is called the nonbasic
matrix. The components of xB are called a basic variables,
and the components of xN are called nonbasic variables.
If xB > 0 , then is x called a nondegenerate basic feasible
solution, and if at least one component of xB is zero, then x
is called a degenerate basic feasible solution.
The following theorem characterizes optimal solutions. The
result corresponds to the so-called nondegenerate problems,
where all fuzzy basic variables corresponding to every basis B
are nonzero (and hence positive) [5].
Theorem 5.1. Assume the FNLP problem is nondegenerate. A
basic feasible solution
x B = B −1b, x N = 0 is optimal to (6)
if and only if z% j ≥ c% j for all 1 ≤
j ≤ n.
ℜ
Proof. Suppose that x * = ( x
solution
to
(6),
x )
T
N
T
B
T
is a basic feasible
x B = B −1b, x N = 0 .
where
Then
z%* = c%B xB = c%B B −1b. On the other hand, for all feasible
ℜ
ℜ
solution x , we have b = Ax = BxB + NxN . Hence, we may
obtain
% = c%B xB + c%N xN = c%B B b − ∑ (c%B B a j − c% j ) x j
z% = cx
−1
ℜ
ℜ
−1
ℜ
j ≠ Bi
z% = z%* − ∑ ( z% j − c% j ) x j
Then,
ℜ
(7)
j ≠ Bi
z%
xB
z%
1
0%
xB
0
I
R.H.S.
xN
c%B B −1 N − c%N
c%B B −1b
B −1 N
B −1b
The above tableau gives us all the information we need to
proceed with the simplex method. The fuzzy cost row in the
above tableau is γ% =(c%B B a j − c% j ) j ≠ Bi , which consists of
−1
ℜ
the
γ% j = z% j − c% j ’s
ℜ
for the nonbasic variables. According to
the optimality condition for these problems we are at the
optimal solution if γ% j ≥ 0 for all j ≠ Bi . On the other hand, if
ℜ
γ%k < 0 , for
ℜ
a k ≠ Bi then we may exchange xBr with xk .
Then we compute the vector yk
= B −1ak . If yk ≤ 0 , then
xk can be increase indefinitely, and then the optimal objective
is unbounded. On the other hand, if yk has at least one
positive component, then the increase in will be blocked by
one of the current basic variables, which drops to zero.
Theorem 6.1. If in a simplex tableau, an l exists such that
z%l − c%l < 0 and there exists a basic index i such that yil > 0 ,
ℜ
then a pivoting row r can be found so that pivoting on
yrl will yield a feasible tableau with a corresponding
nondecreasing fuzzy objective value.
Proof. We need a criterion for choosing a basic variable to
leave the basis so that the new simplex tableau will remain
feasible and the new objective value is nondecreasing. Assume
column l is the pivot column. Also, suppose that
is x = ( xB , xN )
T
T T
a basic feasible solution to the FNLP
Therefore, the results follow immediately from (7) and the
assumptions of theorem.
xB = B −1b , and xN = 0 . Then, the
corresponding fuzzy objective value is z% = c%B B−1b = c%B y0 .
In the next section, we propose simplex method for solving
FNLP problems.
On the other hand, for any basic feasible solution to the
FNLP problem, we have
problem, where
ℜ
xB + ∑ y j x j = y0
VI. SIMPLEX METHOD FOR THE FNLP PROBLEMS
Consider the FNLP problem as is defined in (6).
max z% =ℜ c%B xB + c%N x N
s.t.
BxB + NxN = b
xB , x N ≥ 0
Hence, we may write
−1
xB + B NxN = B b . Therefore,
−1
−1
z% + (c%B B N − c%N ) xN = c%B B b . Currently xN = 0 , and
ℜ
then xB
−1
−1
= B b , and z% = c%B B b . Then we rewrite the
ℜ
−1
where y j = B a j .
So, if xl enters into the basis we may write
Since, we want x B be feasible, hence
(9)
x Bi ≥ 0 , or
yi 0 − yil xl ≥ 0 , for all i = 1,..., m.
If yil ≤ 0 , then it is obvious that the above condition is hold.
Hence, for all yil > 0 , we need to have
above FNLP problem in the following tableau format:
International Scholarly and Scientific Research & Innovation 1(10) 2007
(8)
j ≠ Bi
xB = y0 − yl xl
−1
ℜ
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scholar.waset.org/1999.7/2936
World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:1, No:10, 2007
yi 0
yil
xl ≤
(10)
To satisfy (10) it is sufficient to let
yr 0
y
= min{ i 0 | yil > 0}
yrl
yil
(11)
VII. A NUMERICAL EXAMPLE
For an illustration of the above method we solve a FNLP
problem by use of simplex method.
Example 8.1.
Max z% =(5,8, 2,5) x 1 + (6,10, 2, 6)x 2
ℜ
Also, for any basic feasible solution to the FNLP problem, we
have
z% = c%B y0 − ∑ ( z% j − c% j )x j
ℜ
5 x1 + 4 x2 ≤ 10
(12)
j ≠ Bi
x1 , x2 ≥ 0
So, if we enter xl into the basis we have
z% = c%B y0 − ( z%l − c%l ) xl .
ℜ
International Science Index, Mathematical and Computational Sciences Vol:1, No:10, 2007 waset.org/Publication/2936
s.t. 2 x1 + 3 x2 ≤ 6
(13)
We may rewrite
We note that the new objective value is nondecreasing, since
(14)
z% = c%B y0 − ( z%l − c%l ) xl ≥ c%B y0 ,
ℜ
Using the fact that
x1 , x2 , x3 , x4 ≥ 0
ℜ
( z%l − c%l ) xl ≤ 0 .
2 x1 + 3x2 + x3 = 6
5 x1 + 4 x2 + x4 = 10
We may write the first feasible simplex tableau as follows:
ℜ
Theorem 6.2. If for any basic feasible solution to the FNLP
problem there is some column not in basis for which
z%l − c%l < 0 and yil ≤ 0 , i = 1,..., m , then the FNLP problem
x1
basis
z%
ℜ
x2
(−8, −5,5,2) (−10, −6,6,2)
x3
x4
0%
0%
0%
R.H .S .
has an unbounded solution.
Proof. Suppose that xB is a basic solution to the FNLP
x3
2
3
1
0
6
problem, so
x4
5
4
0
1
10
xBi + ∑ yij x j = yi 0 , i = 1,..., m, j = 1,..., n,
(15)
xBi = yi 0 − ∑ yij x j , i = 1,..., m, j = 1,..., n.
(16)
j ≠ Bi
or
j ≠ Bi
Now, if we enter xl into the basis, then we have xl > 0 , and
x j = 0 , for all j ≠ Bi ∪ l . Since yil ≤ 0 , i = 1,..., m , hence
yi 0 − yil xl ≥ 0
Since ( z%1 − c%1 , z% 2 − c%2 ) =(( −8, −5,5, 2), (−10, −6, 6, 2)) ,
ℜ
and (γ 1 , γ 2 ) = (ℜ(γ%1 ), ℜ(γ%2 )) = (−14.5, −18) ,
enters the basis and the leaving variable is x3 .The new
tableau is:
(17)
Therefore, the current basic solution will remain feasible.
Now, the value of ẑ for the above feasible solution as
following:
basis
z%
m
zˆ = c%B xB + c%N xN = ∑ c%Bi ( yi 0 − yil xl ) + c%l xl
ℜ
ℜ
m
m
i =1
i =1
i =1
= ∑ c%Bi yi 0 − (∑ c%Bi yil − c%l ) xl
ℜ
= c%B y0 − (c%B yl − c%l ) xl = z% − ( z%l − c%l ) xl
ℜ
So,
zˆ = z% − ( z%l − c%l ) xl .
ℜ
(18)
Hence, we can enter xl into the basis with arbitrarily large
value. Then, from (18) we have unbounded solution.
International Scholarly and Scientific Research & Innovation 1(10) 2007
x1
x2
x3
(−4, 53 , 193 ,6) 0%
x4
(2, 103 , 32 , 2) 0%
R.H .S .
(12,20,4,12)
x2
2
3
1
1
3
0
2
x4
7
3
0
−4
3
1
2
Now,
ℜ
then x2
from (γ%1 , γ%3 )
=((−4, 53 , 193 , 6), (2, 103 , 23 , 2))
ℜ
and
(ℜ(γ%1 ), ℜ(γ%3 )) = (γ 1 , γ 3 ) = ( −25 , 6) , it follows that x1 is an
entering variable and x4 is a leaving variable.
The last tableau is shown in the below.
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World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:1, No:10, 2007
basis
x3
x1 x2
z%
0% 0%
x2
0
1
x1
1
0
x4
( −72 , 307 , 307 , 387 ) ( −75 , 127 , 187 , 197 )
R.H .S .
(907 ,1487 , 327 , 907 )
5
7
−2
7
10
7
−4
7
3
7
6
7
w% = c%B B −1 =(c%2 , c%1 ) B −1 =(( −72 , 307 , 307 , 387 ), ( −75 , 127 , 187 , 197 ))
International Science Index, Mathematical and Computational Sciences Vol:1, No:10, 2007 waset.org/Publication/2936
ℜ
ℜ
ℜ
267
7
% − c%N ) =(( −72 , 307 , 307 , 387 ), ( −75 , 127 , 187 , 197 ))
(γ%3 , γ%4 ) =( wN
32 90
% −1
% = c%B B −1b =( 907 , 148
wb
7 , 7 , 7 ) , ℜ(c B B b ) =
ℜ
ℜ
ℜ
ℜ
15
(γ 3 , γ 4 ) = (ℜ(γ%3 ), ℜ(γ%4 )) = ( 327 , 14
) > 0 , γ%2 = γ%1 = 0% .
ℜ
ℜ
Now, using optimality condition there is not any variable that
enters the basis. Therefore, this basis is optimal.
VIII. CONCLUSION
We considered fuzzy number linear programming problems
and introduced the basic feasible solution for these problems.
Finally, we obtained some important results and we proposed
a new algorithm for solving these problems directly, by use of
linear ranking function.
REFERENCES
M.S. Bazaraa, J.J. Jarvis and H.D. Sherali, Linear Programming and
Network Flows, John Wiley, New York, Second Edition, 1990.
[2] R.E. Bellman and L.A. Zadeh, “Decision making in a fuzzy environment”,
Management Sci. 17 (1970) 141--164.
[3] M. Delgado, J.L. Verdegay, and M.A. Vila, “A general model for fuzzy
linear programming”, Fuzzy Sets and Systems 29 (1989) 21--29.
[4] S.C. Fang and C.F. Hu, “Linear programming with fuzzy coefficients in
constraint”, Comput. Math. Appl. 37 (1999) 63--76.
[5] N. Mahdavi-Amiri and S.H. Nasseri, “Duality in fuzzy variable linear
programming”, 4th World Enformatika Conference, WEC'05, June 2426, 2005, Istanbul, Turkey.
[6] H.R. Maleki, “Ranking functions and their applications to fuzzy linear
programming”, Far East J. Math. Sci. 4 (2002) 283--301.
[7] H.R. Maleki, M. Tata and M. Mashinchi, “Linear programming with
fuzzy variables”, Fuzzy Sets and Systems 109 (2000) 21--33.
[8] H. Rommelfanger, R. Hanuscheck and J. Wolf, “Linear programming
with fuzzy objective”, Fuzzy Sets and Systems 29 (1989) 31--48.
[9] J.L. Verdegay, “A dual approach to solve the fuzzy linear programming
problem”, Fuzzy Sets and Systems 14 (1984) 131--141.
[10] H. J. Zimmermann, “Fuzzy programming and linear programming with
several objective functions”, Fuzzy Sets and Systems 1 (1978) 45--55.
[1]
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