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International Journal of Fuzzy Mathematics and Systems.
ISSN 2248-9940 Volume 2, Number 3 (2012), pp. 307-314
© Research India Publications
http://www.ripublication.com
Solving Fuzzy Integer Programming Problem as
Multiobjective Integer Programming Problem
Rajani B. Dash and P.D.P. Dash
S.C.S (A) College, Puri Odisha-752001, India
E-mail: [email protected]
Shishu Ananta Mahavidyalaya, Balipatna, Khurda, Odisha, India
E-mail: [email protected]
Abstract
This paper proposes the method to the solution of fuzzy integer programming
problem with the help of Multi objective integer programming problem when
both constraint matrix and the cost coefficients are fuzzy in nature. The
method is explained with the help of an illustrative example.
Keywords: Fuzzy linear programming problem, Fuzzy integer programming
problem, Multi Objective fuzzy linear programming problem, Multi Objective
Fuzzy Integer programming problem, Fuzzy set.
Mathematics Subject classification: 49M37
Introduction
Linear programming problems with integer restriction on decision variable are called
Integer programming problems. It forms a special class of linear programming
problem. This type of problem is of particular importance in business and industry
where quite often discrete nature of variables is involved in many decision making
situations. For example in manufacturing, the problem is frequently scheduled in
terms of batches, lots, in distribution, a shipment must involve a discrete number of
trucks, aircrafts, or freight cars. Integer Programming Problem has been applied to
solve many real world problems. But it fails to deal with imprecise data. Many
researchers have succeeded in capturing imprecise information by Fuzzy Linear
Programming Problem.[1-2], [5].
This Concept of fuzzy decision making was first propoposed by Bell Man and
Zadeh. Recently much attention has been focused on Fuzzy Linear Programming
Problem.[2-4], [7]. An application of fuzzy optimization technique to LPP with Multi
308
Rajani B. Dash and P.D.P. Dash
Objective [7] has been presented by Zimmerman. Tanaka et.al presented a fuzzy
approach to Multi Objective Linear Programming Problem. Negoita has formulated
FLPP with fuzzy coefficient matrix. Zhang G.et.al [5] formulated a FLPP as four
objective constraind optimization problems where the cost coefficients are fuzzy and
also presented its solution. Thakre P.A. et.al [8] provided a method to solve FLPP
where both the coefficient matrix of the constraint and the cost coefficient are fuzzy in
nature.
In this paper we introduced fuzzy decision making to Integer programming
problem. Here also both the coefficient matrix of the constraint and cost coefficients
are taken to be Fuzzy in nature.
Each problem, first converted in to equivalent crisp integer programming problem
which are then solved by standard method such as fractional cut ( Gomery’s) method.
Preliminaries
Definition 1: A sub set A of a set X is said to be fuzzy set if µ : X → [0, 1], where
µ denote the degree of belongingness of A in X.
Definition 2 : A fuzzy set A of a set X is said to be normal if µ (X) = 1, For all x ε X
.
Definition 3: The height of A is defined and denoted as h (A) =
sup  A  X 
xX
Definition 4: The α – cut and strong α – cut is defined and denoted respectively as

A

 x /  ( x)  
A

αA+ = { x / µA (x) > α}
Definition 5: If a fuzzy number a is fuzzy set A on R, it must possess at least three
properties

a
( x)  1
{ x∈ R /
{ x∈R/


a
a
( x) >α} is a closed interval for every α∈(0, 1].
( x) > 0} is bounded and it is denoted by [aλl, aλR]
Theorem 1: A fuzzy set A on R
 ( x1  (1   ) x 2)  min[   x1 ,   x 2 ]
A
A
is
convex
if
and
only
A
For all x1, x2∈ X and for all λ∈ [0, 1], where min denotes the minimum operator.
if
Solving Fuzzy Integer Programming Problem
309
Proof : Obvious.
Theorem 2: Let a be a fuzzy set on R then a ∈ f(R) if and only if
 1,

 a (x) =  L( x),
 R( x ),

for
for
x  [m, n]
xm
for
xn

a
satisfies
Where L(x) is the right continuous monotonic increasing function, 0≤ L(x)≤1 and
lim L(x) =0, R(x) is a left continuous monotonic decreasing function, 0≤ R(x)≤1
and lim R(x) =0
x
x
Proof : Obvious
Theorem 3: For any two triangular fuzzy numbers A =
s ,l , r
2
2
2
s ,l , r
1
1
1
and B =
, A ≤B if and only if s1≤s2 and s1- l1 ≤ s2 – l2 and s1 +r1 ≤ s2 +r2
Proof : Obvious
Fuzzy Integer Programming Problem
We consider the fuzzy integer programming problem (FIPP) in which the cost of the
decision variables as well as the coefficient matrix of constraints are fuzzy in nature.
n
i.e c, x  f  x   max z   c j x j
( 1)
j 1
subject to
n
 a x  b
ij
j 1
j
ij
,1  i  m, x j  0
1


 a  d  x
a
 ij  x    ij ij
d ij


0

and are integers
x  a ij
for
for
for
a
ij
 x  a ij  d ij
x  a ij  d ij
310
Rajani B. Dash and P.D.P. Dash

1

  p x
bi
i
 b  x   
i
pi


0

x  bi
for
for
b
i
 x  bi 
bp
for
i
i
p
i
x
Let us consider triangular fuzzy numbers which can be represented by three crisp
numbers s, l, r
n
Then (1) → max  c j x j
j 1
Such that
  s , l , r  x  t , u , v 
ij
ij
ij
ij
i
i
i
x0
0≤i≤m, 0≤j≤n
Where ( sij, lij, rij ) and (ti, ui, vi) are triangular fuzzy numbers.
Using theorem (3) the above problem can be written as
n
c, x  f  x   max  c j x j
j 1
Such that
n
s x t
ij
j
i
j 1
n
  s  l x  t  u
ij
ij
j
j
j
j 1
n
s  r x  t  v , x
ij
j 1
ij
j
i
i
i
0
and are integers.
Where the membership function
c
j
(x) is
(2)
Solving Fuzzy Integer Programming Problem
 0

 x  j

 1

 c j  x   
 i  x

 i   j

 0

x 
for
for


for

for
for
j
j
 x
j
j
 x 
j
j
 x 
j

j
311
x
Definition: A point x* ∈ X is said to be an optimial solution to theFIPP if
c , x *  c , x for all x∈X.
Multi Objective Integer Programming Problems With Fuzzy
Coefficients (MOIPP).
The above FIPP with fuzzy coefficients shall ultimately reduced to MOIPP as follows
Max  f  x  , f
1
xX
2

( x),............ f ( x )
k
Subject to (2)
Where fi : Rn → Ri and Rn is n dimensional Euclidean space
By considering weighting factor the MOIPP is defined as
max w f
1
i.e
xX
( x), w 2 f
1
( x) ,.................w f
2
k
k

( x)
k
max  w f
m
xX
m 1
m
( x)
Subject to (2)
Numerical Example
We illustrate the method by a numerical example
Solve the following FIPP
Max f (x1, x2) =
c x
1
1
 c 2 X2, xi ≥ 0 and are integers,
Subject to the constraints
(5, 4, 2)x1 + ( 4, 3, 1) x2≤ ( 11, 5, 3)
(4, 1, 3) x1 + ( 5, 4, 1)x2 ≤ ( 9, 5, 4)
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Rajani B. Dash and P.D.P. Dash
Where the membership functions of
 0
 x6

 1
 c 1  x   
 27  x
 15

 0
 0
 x  15

 1
 c 2( x)  
 30  x
 5

 0
c
for
x6
for
for
6x9
9  x  12
1
and
c
2
are
for 12  x  27
for
x  27
for
x  15
for 15  x  20
for 20  x  25
for
25  x  30
for
x  30
Let us write the above FIPP as
Max f(x1, x2) =
Where
c
2
c
1
c x  c x
1
1
2
2
= ( 6, 9, 12, 27)
= (15, 20, 25, 30)
Subject to
5x1+4x2≤11
4x1+5x2≤ 9
x1+x2≤6
3x1+x2≤4
7x1+5x2≤14
7x1+6x2≤13
x1, x2 ≥0 and are integers. ----------MOIPP
Max ( 6x1+15x2, 9x1+20x2, 12x1+25x2, 27x1+30x2 )
Subject to (3)
With weights, MOIPP becomes
Max (w) = {w1(6x1+15x2 )+ w2(9x1+20x2 )+w3(12x1+25x2 )+w4(27x1+30x2 )}
(3)
Solving Fuzzy Integer Programming Problem
313
Subject to (3)
Standard optimization techniques ( Fractional cut) is used to solve the problem
and solution is obtained for different weights.
For example, w1=0=w4, w2=1=w3
MOIPP max (w)= f(x1, x2)= 21x1+45x2
Subject to (3)
The optimal solution : (x1*, x2*) = ( 1, 1)
Max (w)= f(x1*, x2*)=f( 1, 1) =
c + c
 0
 x  21

 8
x

  f 1,1    1
 57  x

 20
 0
for
x  21
for
21  x  29
1
2
for 29  x  37
for 37  x  57
x  57
for
Following table lists the solution for above MOIPP for various weights and it also
shows that the solutions are independent of weights (wi, i=1, 2, 3, 4)
Sl.No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
W1
0
0
0.2
0.1
0
0.2
0.5
0
0
0.3
0.5
0
0.2
0.1
0
W2
1
1
.04
.02
.03
.04
0
.1
0
0.1
0.5
0
0.5
0.2
0.2
W3
1
.05
.05
.03
0
0.6
0.5
1
0
1
0.5
0.5
0.5
0.3
0
W4 (x1*, x2*)
0
(1, 1)
0
(1, 1)
0.2
(1, 1)
0.4
(1, 1)
0.4
(1, 1)
0.8
(1, 1)
0
(1, 1)
0
(1, 1)
0.5
(1, 1)
1
(1, 1)
0.5
(1, 1)
0.5
(1, 1)
0.5
(1, 1)
0.4
(1, 1)
0.2
(1, 1)
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Rajani B. Dash and P.D.P. Dash
Conclusion
We successfully discussed the solution of fuzzy integer programming problem with
the help of multi objective constrained integer programming problem where constraint
matrix and the cost coefficients are fuzzy quantities and also proved that the solutions
are independent of weights.
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