Download Simplex Method for Fuzzy Variable Linear Programming Problems

World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:3, No:10, 2009
Simplex Method for Fuzzy Variable Linear
Programming Problems
S.H. Nasseri, and E. Ardil
International Science Index, Mathematical and Computational Sciences Vol:3, No:10, 2009 waset.org/Publication/4359
Abstract—Fuzzy linear programming is an application of fuzzy
set theory in linear decision making problems and most of these
problems are related to linear programming with fuzzy variables. A
convenient method for solving these problems is based on using of
auxiliary problem. In this paper a new method for solving fuzzy
variable linear programming problems directly using linear ranking
functions is proposed. This method uses simplex tableau which is
used for solving linear programming problems in crisp environment
before.
Keywords—Fuzzy variable linear programming, fuzzy number,
ranking function, simplex method.
Z
I. INTRODUCTION
[10] proposed the first formulation of fuzzy
linear programming. Fang and Hu [4] considered linear
programming with fuzzy constraint coefficients. Vasant and et
al [9] applied linear programming with fuzzy parameters for
decision making in industrial production planning. Maleki and
et al [6, 7] introduced a linear programming problem with
fuzzy variables and proposed a new method for solving these
problems using an auxiliary problem. Mahdavi-Amiri and
Nasseri [5] described duality theory for the fuzzy variable
linear programming (FVLP) problems. This study focuses on
FVLP problems. Hence, first some important concepts of
fuzzy theory are reviewed and concept of the comparison of
fuzzy numbers by introducing a linear ranking function is
described. Moreover, fuzzy basic feasible solution for the
FVLP problems and also optimality conditions along with
fuzzy simplex algorithm for solving the fuzzy variable linear
programming problems is proposed.
IMMERMANN
μa% ( x)
is called the membership function of x in a% which
maps R to a subset of the nonnegative real numbers whose
supremum is finite. If
sup x μa% ( x) = 1
the fuzzy set a% is
called normal.
Definition 2.2. The support of a fuzzy set a% on R is the
crisp set of all x ∈ R such that
μa% ( x) > 0 .
Definition 2.3. The set of elements that belong to the fuzzy set
a% on R at least to the degree α is called the α − cut set:
aα = {x ∈ R | μa% ( x) ≥ α } .
Definition 2.4. A fuzzy set a%
on
R is convex if
μa% (λ x + (1 − λ ) y ) ≥ min{μ a% ( x), μa% ( y )} ,
x, y ∈ R, and λ ∈ [0,1] .
Note that, a fuzzy set is convex if all α − cuts are convex.
Definition 2.5. 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% ( x0 ) = 1 .
2) μa% ( x) is piecewise continuous.
A fuzzy number a% is a trapezoidal fuzzy number if the
membership function of it be in the following form:
II. DEFINITIONS AND NOTATIONS
In this section, some of the fundamental definitions and
concepts of fuzzy sets theory initiated by Zadeh [2] (taken
from Bezdek [3]) are reviewed.
Definition 2.1. A fuzzy set a% in
aL −α
aL
aU
aU + β
Fig. 1 Trapezoidal Fuzzy Number
R is a set of ordered pairs:
a% = {( x, μa% ( x)) | x ∈ R}
We may show any trapezoidal fuzzy number by
a% = (a L , aU , α , β ) , where the support of
S.H. Nasseri, Department of Mathematical and Computer Sciences,
Sharif University of Technology, Tehran, Iran (corresponding author: e-mail:
[email protected]).
E. Ardil is with Department of Computer Engineering, Trakya University,
Edirne Turkey (e-mail: [email protected]).
International Scholarly and Scientific Research & Innovation 3(10) 2009
a%
is
(a − α , a + β ) , and the core of a% is [a , a ] . Let
F (R ) be the set of trapezoidal fuzzy numbers. Note that, we
consider F ( R ) throughout this paper.
L
884
U
scholar.waset.org/1999.7/4359
L
U
World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:3, No:10, 2009
A. Arithmetic on Fuzzy Numbers
Since in this paper we only consider the trapezoidal fuzzy
numbers therefore we define arithmetic on the elements of
~
F (R ) . Let a~ = (a L , a U , α , β ) and b = (b L , bU , γ , θ )
be two trapezoidal fuzzy numbers and x ∈ R . Then, we
IV. FUZZY LINEAR PROGRAMMING
In this section, we introduce fuzzy linear programming
(FLP) problems. In order to the definition of fuzzy linear
programming problems it is necessary to introduce linear
programming problems.
define
A. Linear Programming
A linear programming (LP) problem is defined as:
Max z = cx
s.t. Ax = b
International Science Index, Mathematical and Computational Sciences Vol:3, No:10, 2009 waset.org/Publication/4359
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 )
~
a~ + b = (a L + b L , a U + bU , α + γ , β + θ )
~
a~ − b = (a L − bU , a U − b L , α + θ , β + γ )
III. RANKING FUNCTIONS
A convenient method for comparing of the fuzzy numbers
is by use of ranking functions. We define a ranking
function ℜ : F ( R ) → R , which maps each fuzzy number
into the real line. Now, suppose that a% and b% be two
trapezoidal fuzzy numbers. Therefore, we define orders on
F (R ) as following:
~
(1)
a~ ≥ b if and only if ℜ(a% ) ≥ ℜ(b% )
ℜ
~
(2)
a~ > b if and only if ℜ(a% ) > ℜ(b% )
ℜ
~
a~ = b if and only if ℜ(a% ) = ℜ(b% )
(3)
ℜ
~ and b~ are in F ( R) . Also we write a~ ≤ b~ if and
where a
ℜ
~ ~
only if b ≥ a .
ℜ
In the above problem the all of parameters are crisp [1].
Now, if the some of parameters be fuzzy numbers we obtain a
fuzzy linear programming which is defined in the next
subsection.
B. Fuzzy Linear Programming
Suppose that in the linear programming problem some
parameters be fuzzy number. Then, we have a fuzzy linear
programming problem. Hence, it is possible the some
coefficients of the problem in the objective function, technical
coefficients, the right-hand side coefficients or decision
making variables be fuzzy number [5], [6], [7], [8]. Here, we
focus on the linear programming problems with fuzzy
variables which is defined in the next section
V. FUZZY VARIABLE LINEAR PROGRAMMING
A fuzzy variable linear programming (FVLP) problem is
defined as follows:
Max ~
z = c~
x
ℜ
Lemma 3.1. Let ℜ be any linear ranking function. Then
~
x =b
s.t. A~
ℜ
~
x ≥0
~ ≥ 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
(7)
ℜ
ℜ
where b% ∈ (F (R)) , x% ∈ (F (R)) , A∈ R
is a linear ranking function.
m
These are many numbers ranking function for comparing
fuzzy numbers. Here, we use from linear ranking functions,
that is, a ranking function ℜ such that
ℜ(ka% + b% ) = k ℜ(a% ) + ℜ(b% ) .
(6)
x≥0
T
where c = (c1,..., cn ), b = (b1,..., bm ) , and A = [ aij ]m×n .
(4)
One suggestion for a linear ranking function as following:
1
ℜ(a~ ) = a L + a U + ( β − α ).
(5)
2
L
U
where a% = (a , a , α , β ) ∈ F ( R ) , and adopted by
Maleki and et al [6, 7].
~ = (a L , a U , α , β )
Then, for trapezoidal fuzzy numbers a
~
b = (b L , bU , γ , θ ) , we have
~
1
1
a~ ≥ b if and only if aL + aU + (β −α) ≥ bL + bU + (θ −γ ).
ℜ
2
2
and
International Scholarly and Scientific Research & Innovation 3(10) 2009
n
m×n
, cT ∈ Rn , and ℜ
Definition5.1. We say that fuzzy vector x% ∈ ( F ( R )) is a
n
fuzzy feasible solution to (7) if and only if
constraints of the problem.
Definition 5.2. A fuzzy feasible solution
~
x satisfies the
~
x* is a fuzzy
optimal solution for (7), if for all fuzzy feasible solution ~
x for
~
~
(7), we have cx* ≥ cx .
ℜ
A. Fuzzy Basic Feasible Solution
Here, we describe fuzzy basic feasible solution (FBFS) for
the FVLP problems which established by Mahdavi-Amiri and
Nasseri [5].
For the FVLP problem is defined in (7), consider the
system Ax% = b% and
ℜ
885
x% ≥ 0 . Let A = [aij ]m ×n . Assume
ℜ
scholar.waset.org/1999.7/4359
World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:3, No:10, 2009
rank (A ) = m . Partition A as [ B
N ] where B ,
m × m , is nonsingular. It is obvious that rank ( B ) = m .
Let y j be the solution to By =a j . It is apparent that the
x%B + B −1 Nx% N = B −1b% , and also for objective function
ℜ
z% + (cB B N − cN ) x% N = cB B −1b%.
−1
ℜ
−1
−1
Currently x% N = 0 , and then x% B = B b% , and z% = cB B b% .
basic solution
~
~
xB = (~
x B1 ,..., ~
x Bm ) T = B −1b , ~
xN = 0
(8)
ℜ
ℜ
~
is a solution of A~
x = b . We call x% , accordingly partitioned
ℜ
ℜ
Then, we may rewrite the above FVLP problem in the
following tableau format:
ℜ
TABLE I
FUZZY SIMPLEX TABLEAU
~
x NT ) T , a fuzzy basic solution corresponding to the
as ( x%
x B ≥ 0 , then the fuzzy basic solution is feasible
basis B . If ~
T
B
and the corresponding fuzzy objective value is ~
z = cB ~
xB ,
z%
where c B = (c B1 ,..., c Bm ) .
x%B
International Science Index, Mathematical and Computational Sciences Vol:3, No:10, 2009 waset.org/Publication/4359
ℜ
nonbasic
Now, corresponding to every
~
x j , 1 ≤ j ≤ n, j ≠ Bi , and
variable
i = 1,..., m, define
z j = c B y j = c B B −1 a j .
(9)
If x% B > 0 , then is x% called a nondegenerate fuzzy basic
ℜ
feasible solution, and if at least one component of x% B is zero,
then x% is called a degenerate fuzzy 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 FVLP problem is nondegenerate. A
~
fuzzy basic feasible solution ~
x B = B −1b , ~
x N = 0 is optimal
ℜ
ℜ
to (7) if and only if z j ≥ c j for all 1 ≤
j ≤ n.
Maleki and et al [6, 7] was proposed a method for solving
FVLP problems by use of solving an auxiliary problem. They
discuss on the some relations between the FVLP problem and
the auxiliary problem. Then, they used from these results for
solving the FVLP problems. Here, we propose simplex
method for solving FVLP problems.
VI. SIMPLEX METHOD FOR THE FVLP PROBLEMS
A. Fuzzy Simplex Method in Tableau Format
Consider the FVLP problem as is defined in (7).
Max
z% = cB x% B + c N x% N
ℜ
s.t. Bx% B + Nx% N = b%
ℜ
(10)
x%B , x% N ≥ 0
ℜ
−1
−1
Then, it is possible we write x% B = B b% − B Nx% N and
ℜ
z% = cB ( B b% + B −1 Nx% N ) + cN x% N . Also, we may rewrite
−1
ℜ
International Scholarly and Scientific Research & Innovation 3(10) 2009
x%B
z%
ℜ
fuzzy
ℜ
1
0
0
I
x% N
R.H.S.
cB B −1 N − cN
cB B −1b%
B −1b%
B −1 N
The above tableau gives us all the information we need to
proceed with the simplex method. The cost row in the above
tableau is
(γ j ) j ≠ Bi = (cB B −1a j − c j ) j ≠ Bi = ( z j − c j ) j ≠ Bi .
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
γl < 0 ,
exchange x% Br
with
yl = B −1al .
If
for
a l ≠ Bi
then we may
x%l . Then we compute the vector
yl ≤ 0 , then
x%l can be increase
indefinitely, and then the optimal objective is unbounded. On
the other hand, if yl has at least one positive component, then
the increase in will be blocked by one of the current basic
variables, which drops to zero.
B. Pivoting
If x%l enters the basis and x% Br leaves the basis, then
pivoting on yrl can be stated as follows:
r by yrl .
2) For i = 1,..., m and i ≠ r , update the i th row by adding
to it − yil times the new r th row.
3) Update row zero by adding to it γ l times the new r th row.
1) Divide row
Theorem 6.1. If in a fuzzy simplex tableau, an l exists such
that zl − cl < 0 and there exists a basic index i such that
yil > 0 , then a pivoting row r can be found so that pivoting
on yrl will yield a fuzzy feasible tableau with a corresponding
nondecreasing objective value.
Proof. We need a criterion for choosing a fuzzy 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
886
scholar.waset.org/1999.7/4359
World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:3, No:10, 2009
Now, if we enter x%l into the basis, then we have x%l
FVLP problem, where x% B = B b% , and
x% j = 0 , for all j ≠ Bi ∪ l . Since yil ≤ 0 , i = 1,..., m , hence
ℜ
−1
ℜ
x% N = 0 . Then, the
ℜ
y%i 0 − yil x%l ≥ 0
corresponding fuzzy objective value is z% = cB B b% = cB y%0 .
ℜ
On the other hand, for any fuzzy basic feasible solution to the
FVLP problem, we have
ℜ
j ≠ Bi
(11)
Therefore, the current fuzzy basic solution will remain
feasible. Now, the value of ẑ for the above fuzzy feasible
solution as following:
m
zˆ = cB x% B + cN x% N = ∑ cBi ( y%i 0 − yil x%l ) + cl x%l
ℜ
−1
where y j = B a j .
ℜ
(12)
ℜ
ℜ
m
m
i =1
i =1
= cB y% 0 − (cB yl − cl ) x%l = z% − ( zl − cl ) x%l
ℜ
Since, we want x% B be feasible, hence
ℜ
So,
y%i 0 − yil x%l ≥ 0 , for all i = 1,..., m.
zˆ = z% − ( zl − cl ) x%l .
ℜ
If yil ≤ 0 , then it is obvious that the above condition is hold.
Hence, for all yil > 0 , we need to have
x%l ≤
ℜ
(21)
ℜ
Hence, we can enter x%l into the basis with arbitrarily large
fuzzy value. Then, from (21) we have unbounded solution.
y%i 0
yil
(13)
y% r 0
y%
= min{ i 0 | yil > 0}
yrl ℜ
yil
(14)
Also, for any fuzzy basic feasible solution to the FVLP
problem, we have
z% = cB y% 0 − ∑ ( z j − c j )x% j
(15)
So, if we enter x%l into the basis we have
ℜ
−1
1.The basic feasible solution is given by x% B = B b% = y% 0 and
ℜ
ℜ
x% N = 0 . The fuzzy objective z% = cB B b% = cB y% 0 .
−1
ℜ
ℜ
2.Calculate w = cB B
−1
, and
ℜ
y 0 = ℜ( y% 0 ) . For each
γ j = z j − c j = cB B −1a j − c j
γ l = min j {γ j } . If γ l ≥ 0 , then stop; the
nonbasic variable, calculate
j ≠ Bi
z% = cB y% 0 − ( zl − cl ) x%l
VII. A FUZZY SIMPLEX METHOD
Suppose that we are given a basic feasible solution with
basis B . Then:
To satisfy (5) it is sufficient to let
ℜ
i =1
= ∑ cBi y%i 0 − (∑ cBi yil − cl ) x%l
So, if x%l enters into the basis we may write
x%B = y% 0 − yl x%l
(20)
ℜ
ℜ
x%B + ∑ y j x% j = y% 0
ℜ
ℜ
−1
International Science Index, Mathematical and Computational Sciences Vol:3, No:10, 2009 waset.org/Publication/4359
> 0 , and
x% =( x%BT , x% N T )T is a fuzzy basic feasible solution to the
(16)
= wa j − c j . Let
current solution is optimal. Otherwise go to step 3.
−1
If yl ≤ 0 , then stop; the optimal
We note that the new objective value is nondecreasing, since
z% = cB y% 0 − ( zl − cl ) x%l ≥ cB y% 0
(17)
3.Calculate yl = B al .
Using the fact that ( zl − cl ) x%l ≤ 0 .
yr 0
y
= min{ i 0 | yil > 0}
yrl 1≤i ≤ m yil
y%
Update y% i 0 by replacing y% i 0 − r 0 yil for i ≠ r and y% r 0 by
yrl
y% r 0
. Also, update z%
by replacing
replacing
yrl
y%
z% − r 0 ( zl − cl ) . Then, update B by replacing aBr with al
yrl
ℜ
ℜ
ℜ
Theorem 6.2. If for any fuzzy basic feasible solution to the
FVLP problem there is some column not in basis for which
zl − cl < 0 and yil ≤ 0 , i = 1,..., m , then the FVLP problem
has an unbounded solution.
Proof. Suppose that x% B is a fuzzy basic solution to the FVLP
problem, so
x%Bi + ∑ yij x% j = y%i 0 , i = 1,..., m, j = 1,..., n, (18)
j ≠ Bi
or
ℜ
x%Bi = y%i 0 − ∑ yij x% j , i = 1,..., m, j = 1,..., n.
ℜ
(19)
solution is unbounded. Otherwise determine the index of the
variable x% Br leaving the basis as follows:
and go to step 2.
VIII. A NUMERICAL EXAMPLE
j ≠ Bi
For an illustration of the above method we solve a FVLP
problem by use of fuzzy simplex method.
International Scholarly and Scientific Research & Innovation 3(10) 2009
887
scholar.waset.org/1999.7/4359
World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:3, No:10, 2009
Example 8.1.
and introduced the fuzzy basic feasible solution for these
problems. Finally, we proposed a new algorithm for solving
these problems directly, by use of linear ranking function.
max z% = 3 x%1 + 4 x%2
ℜ
s.t.
3 x%1 + x%2 ≤(2, 4,1,3)
ℜ
REFERENCES
2 x%1 − 3 x%2 ≤(3,5, 2,1)
[1]
ℜ
x%1 , x%2 ≥ 0
ℜ
Now, we may rewrite the above problem in form (10):
3 x%1 + x%2 + x%3 =(2, 4,1,3)
ℜ
2 x%1 − 3 x%2 + x%4 =(3,5, 2,1)
International Science Index, Mathematical and Computational Sciences Vol:3, No:10, 2009 waset.org/Publication/4359
ℜ
x%1 , x%2 , x%3 , x%4 ≥ 0
ℜ
Therefore, using fuzzy simplex tableau (Table I), we obtain
first tableau as follow:
ℜ( R.H .S .)
basis
x%1
x%2 x%3
x%4
R.H.S.
z%
-3
-4
0
0
0%
0
x%3
x%4
3
1
1
0
(2,4,1,3)
7
2
-3
0
1
(3,5,2,1)
7.5
From
the
above
γ1 = z 1 − c1 = −3 < 0
Then,
tableau,
we
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] J.C. Bezdek, “Fuzzy models - What are they, and Why?”, IEEE
Transactions on Fuzzy Systems 1 (1993) 1--9.
[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] P. Vasant, R. Nagarajan, and S. Yaacab, ‘‘Decision making in industrial
production planning using fuzzy linear programming”, IMA, Journal of
Management Mathematics 15 (2004) 53--65.
[10] H. J. Zimmermann, “Fuzzy programming and linear programming with
several objective functions”, Fuzzy Sets and Systems 1 (1978) 45--55.
obtain
γ 2 = z 2 − c 2 = −4 < 0 .
γ 2 < γ 1 . Hence, related fuzzy nonbasic variable to γ 2 ,
and
that is x%2 is an entering variable. Therefore, according to the
minimum ration test is given in the step 3 of the fuzzy simplex
algorithm, x%3 is a leaving variable. Now, after pivoting (as
given in the part B of section 6) the new tableau is:
R.H.S.
ℜ( R.H .S .)
0
(8,16, 4,12)
28
1 1
0
(2,4,1,3)
7
0 3
1
(9,17,5,10)
28.5
basis
x%1
x%2 x%3
x%4
z%
x%2
9
0 4
3
x%4
11
According to above tableau, for fuzzy nonbasic variables
x%1 , x% 3 we have γ 1 = 9 > 0, γ 3 = 4 > 0 . Hence, using the
optimality condition for the FVLP problems is given in
Theorem 5.1, the optimal fuzzy solution is obtained
x% 1* = (0, 0, 0, 0) , x% *2 = (2, 4,1, 3) , x% *3 = (0, 0, 0, 0) ,
ℜ
ℜ
ℜ
x% *4 =(9,17,5,10) and z% =(8,16, 4,12) with ℜ( z% ) = 28 .
ℜ
ℜ
IX. CONCLUSION
We considered fuzzy variable linear programming problems
International Scholarly and Scientific Research & Innovation 3(10) 2009
888
scholar.waset.org/1999.7/4359