59
A Novel Method to Solve Fuzzy Integer Linear
Programming Problems
M. Evangeline Jebaseeli
1, Jijendra Kumar
2, D. Paul Dhayabaran
3PG and Research Department of Mathematics1,2,3, Bishop Heber College, Tiruchirappalli
[email protected], [email protected], [email protected]
Abstract - This paper deals with Fuzzy Integer Linear Programming Problems in which all the parameters and variables are pentagonal fuzzy numbers. In this study, a method based on the aggregated structure of m-LPPs for solving Fuzzy Integer Linear Programming Problems is introduced. The proposed method wipes out the complexity of solving Fuzzy Integer Linear Programming Problems.
Keywords: Fuzzy Integer Linear Programming Problem – Pentagonal Fuzzy Number – Aggregated structure of m-LPPs.
1. INTRODUCTION
Optimization is an extremely important are in Science and Technology which provides useful tools and techniques for the formulation and solution of a multitude of problems. Unfortunately, most of the available mathematical tools are crisp, deterministic or precise in nature, that is, parameters of a process are never precisely fixed to a definite value.
In real situations, crisp or deterministic tools cannot be applied because the complete description of a real system require more detailed data than what a human being can recognize, process or understand. The available data cannot be put in a precise form due to involved uncertainties. There exist various types of uncertainties in social, industrial and economic systems, such as, randomness of occurrence of events, imprecision and ambiguity of system data and linguistic vagueness. Fuzzy optimization provides a useful and efficient tool for modeling and optimizing such system. In 1965, Zadeh introduced Fuzzy set theory as a means of representing and manipulating data that are not precise, but rather fuzzy. It was specifically designed to represent mathematically the uncertainty and vagueness and to provide formalized tools for dealing with imprecision, intrinsic to many problems. Fuzzy Optimization aims at solving the fuzzy model optimally by means of optimization techniques.
Linear Programming is a technique for the optimization of a linear objective function subject to linear equality and linear inequality constraints. Linear Programming has important application in many areas of engineering and management. Since the real- world problems are very complex, the
parameters of Linear Programming are usually represented by fuzzy numbers.
The concept of Fuzzy Linear Programming was introduced by Tanaka (1974). It plays a vital role in Fuzzy optimization, which can formulate the uncertainty. Nasseri (2008) has proposed a new method for solving the Fuzzy Linear Programming Problems in which he has used the fuzzy ranking method to convert the fuzzy objective function into crisp objective function.
Linear Programming Problems with integer restriction on decision variables are called Integer Programming Problems. It forms a special class of Linear Programming Problem. Integer Programming 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 interms 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.
The concept of fuzzy decision making was proposed by Bell Man & Zadeh (1970). An application fuzzy optimization technique to Linear Programming Problem with multi-objective has been presented by Zimmerman.
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variables by using classical integer linear Programming. Stephen Dinagar (2018) introduced a new method Fuzzy Integer Linear Programming Problems with Pentagonal Fuzzy Number.
2. PRELIMINARIES
In this section, some basic definitions of fuzzy set, fuzzy number, triangular and trapezoidal fuzzy number, pentagonal fuzzy number and arithmetic operations of triangular fuzzy numbers, which are the basis for the development of fuzzy set theory, are reviewed. (Dubois, Parade 1980 and Kaufmann and Gupta, 1985)
Definition 2.1: Fuzzy set
Let Xbe a classical set of objects, called the universe. The characteristic function fA of crisp set
A
X
assigns a value either 0 or 1 to each member in X. This function can be generalized to a functionf
A~such that the value assigned to the element of the universal set X falls within a specified range (that is
]
f
A ). The assigned value indicates the membership grade of the element in set A. The Definition 2.2: Fuzzy NumberA fuzzy
A
~
defined on the universal set of real numbers
, is said to be fuzzy number if its membership function has the following properties:i.
A
~
is convex, that isii.
A
~
is normal, that is there exists exactly onex
0
withf
A~(
x
0)
1
iii.
f
~(
x
)
A is piecewise continuous
Definition 2.3: Pentagonal Fuzzy Number
A pentagonal fuzzy number
and
a
5 are real numbers and its membership is given below:Definition 2.4: Arithmetic Operations on Pentagonal fuzzy numbers
Let
A
~
p
(
a
1,
a
2,
a
3,
a
4,
a
5)
and(iii) Scalar Multiplication:
)
Definition 2.5:
Let
A
~
p
(
a
1,
a
2,
a
3,
a
4,
a
5)
andDefinition 2.6:
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Definition 3.7:
A real fuzzy vector
b
~
(
b
~
)
m1is called non-negative and denoted byb
~
0
if each element ofb
~
is a non-negative real fuzzy number.3. FUZZY INTEGER LINEAR
PROGRAMMING PROBLEM
Consider the Integer Programming Problem with fuzzy variables,
Max
~
z
c
x
~
subject toA
x
~
b
~
,~
x
0
andMariappan and Antony (2016) introduced the concept of aggregated structure of m-Linear Programming Problems.
Assumptions:
m – number of LPPs are considered.
All the objective functions should be of the Maximization (If anyone/more objective is of minimization, convert the same).
In each LPP all the variables should be non-negative. (if not, do the modification in the corresponding LPP)
In each LPP all the right hand side values of the constraints should be non-negative. (if not modify)
General LPP structure of ith problem (i=1,2,…m) can be given as:
Max =
Subject to the conditions :
1
Aggregated structure of m – LPPs is, Max Z = ∑ ∑ Subject to the conditions:
INTEGER LINEAR PROGRAMMING PROBLEMS
Pandian (2010) introduced the decomposition method for Fuzzy Integer Linear Programming Problem. Based on this, the decomposed form of Fuzzy Integer Linear
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Max
z
3
cx
3 , subject toAx
3
b
3 ,0
3
x
and are integers (P3) Maxz
2
cx
2 , subject toAx
2
b
2 ,0
,
20 3
2
x
x
x
and are integers (P2)where
x
30is a optimal solution of (P3) Maxz
1
cx
1 , subject toAx
1
b
1 ,0
,
10 3
1
x
x
x
and are integers (P1)Max
z
4
cx
4 , subject to ,Ax
4
b
40
,
40 3
4
x
x
x
and are integers (P4)Max
z
5
cx
5 , subject toAx
5
b
5 ,0
,
50 3
5
x
x
x
and are integers (P5)Aggregated structure of (P1), (P2), (P3), (P4) and (P5) is,
Max
z
cx
1
cx
2
cx
3
cx
4
cx
5Subject to
1 1
b
Ax
,Ax
2
b
2 ,Ax
3
b
3 ,4 4
b
Ax
,Ax
5
b
5,
0 3 2
x
x
x
1
x
30,
x
4
x
30,x
5
x
30,5
,
4
,
3
,
2
,
1
0
i
x
i andx
i are integers (AP)
i
1
,
2
,
3
,
4
,
5
Theorem 5.1
A fuzzy vector
x
~
0
(
x
10,
x
20,
x
30,
x
40,
x
50)
is an optimal solution of the problem (P) if and only ifx
30 andx
0are optimal solution of the crisp integer linear programming problems (P3) and (AP) respectively, where,Max
z
3
cx
3 , subject toAx
3
b
3 ,0
3
x
and are integers (P3)Max
z
cx
1
cx
2
cx
3
cx
4
cx
5Subject to :
Ax
1
b
1 ,Ax
2
b
2 ,3 3
b
Ax
,Ax
4
b
4,Ax
5
b
5,
0 3 2
x
x
x
1
x
30,
x
4
x
30,x
5
x
30,5
,
4
,
3
,
2
,
1
0
i
x
i andx
i are integers(AP)
i
1
,
2
,
3
,
4
,
5
6. SOLVING PROCEDURE
Convert the given fuzzy integer linear programming problem (P) into crisp integer linear programming problems (P3), (P2), (P1), (P4) and (P5) as given in section ().
Reduced the above decomposed crisp integer linear programming problems into pair of crisp integer linear programming problems (P3) and (AP) using the aggregated structure of m-LPPs given in section ().
Find the optimal solutions of the reduced pair of crisp linear programming problems (P3) and (AP).
Find the optimal solution of the given Fuzzy Integer Linear Programming Problem (P) by substituting the obtained values from the above step.
7. NUMERICAL EXAMPLES Example 7.1
Consider the following integer linear programming problem with fuzzy variables,
Max
Z
~
5
~
x
1
15
~
x
2Subject to:
4
x
~
1
8
~
x
2
(
28
,
44
,
65
,
68
,
73
)
,)
46
,
38
,
33
,
22
,
18
(
~
7
~
2 1
x
x
,0
~
,
0
~
2 1
x
x
and~
x
1,
~
x
2are integers.Let
Z
(
Z
1,
Z
2,
Z
3,
Z
4,
Z
5)
,)
,
,
,
,
(
~
1 1 1 1 1
1
u
v
x
y
t
x
,~
x
2
(
u
2,
v
2,
x
2,
y
2,
t
2)
Decomposed crisp integer linear programming problems of the above fuzzy integer linear programming problem are,Max
Z
3
5
x
1
15
x
2 , Subject to:65
8
4
x
1
x
2
,x
1
7
x
2
33
,x
1
0
,
x
2
0
, andx
1,
x
2are integers.63
Max
Z
2
5
v
1
15
v
2 , Subject to:44
8
4
v
1
v
2
,v
1
7
v
2
22
,v
1
0
,
v
2
0
,and
v
1,
v
2 are integers. (P2)Max
Z
1
5
u
1
15
u
2 , Subject to:28
8
4
u
1
u
2
,u
1
7
u
2
18
,u
1
0
,
u
2
0
,and
u
1,
u
2are integers. (P1)Max
Z
4
5
y
1
15
y
2 , Subject to:68
8
4
y
1
y
2
,y
1
7
y
2
38
,0
,
0
21
y
y
, andy
1,
y
2 are integers. (P4)Max
Z
5
5
t
1
15
t
2, Subject to:4
t
1
8
t
2
73
,46
7
21
t
t
,t
1,
t
2
0
, andt
1,
t
2 are integers. (P5)From the above, the reduced pair of crisp integer linear programming problems are:
Max
Z
3
5
x
1
15
x
2 , Subject to:65
8
4
x
1
x
2
,x
1
7
x
2
33
,x
1
0
,
x
2
0
, andx
1,
x
2are integers.(P3) Solution of the crisp integer linear programming problem (P3) is,
10
0 1
x
,x
20
3
The Aggregated Problem (AP) of (P3), (P2), (P1), (P4) and (P5) is,
Max 1,2,3,4,5 1 2 1 2 1
2 1 2 1 2
5 15 5 15 5 15 5 15 5 15
Z u u v v x
x y y t t
,
Subject to:
4
u
1
8
u
2
28
,u
1
7
u
2
18
,3
,
10
21
u
u
44
8
4
v
1
v
2
,v
1
7
v
2
22
,3
,
10
21
v
v
65
8
4
x
1
x
2
,x
1
7
x
2
33
,68
8
4
y
1
y
2
,y
1
7
y
2
38
,3
,
10
2 1
y
y
73
8
4
t
1
t
2
,t
1
7
t
2
46
,t
1
10
,
t
2
3
0
,
0
21
u
u
,v
1
0
,
v
2
0
,y
1
0
,
y
2
0
0
,
0
21
x
x
andt
1
0
,
t
2
0
2 1
,
u
u
,v
1,
v
2,x
1,
x
2y
1,
y
2andt
1,
t
2are integers. The solution of the aggregated problem(AP) is,1 2 1 2 1 2
1 2 1 2
3,
2;
7,
2;
10,
3;
11,
3;
12,
4
u
u
v
v
x
x
y
y
t
t
.
By theorem 5.1, the solution of the given Fuzzy Integer Linear Programming Problem (P) is
)
12
,
11
,
10
,
7
,
3
(
)
,
,
,
,
(
~
1 1 1 1 1
1
u
v
x
y
t
x
,)
4
,
3
,
3
,
2
,
2
(
)
,
,
,
,
(
~
2 2 2 2 2
2
u
v
x
y
t
x
The objective value is
Z
~
(
45
,
65
,
95
,
100
,
120
)
8. CONCLUSION
In this paper, a new method for solving Fuzzy Integer Linear Programming Problem with fuzzy RHS and fuzzy decision variables is discussed. The underlying idea of the proposed method is the aggregated structure of m-LPPs. Based on aggregated structure of the problem; the given problem can be converted into pair of crisp integer linear programming problems. If both the crisp integer linear programming problems have optimal solution then the Fuzzy Integer Linear Programming Problem under consideration will have an optimal solution. This proposed method provides an optimal solution to fuzzy integer linear programming problems without the ranking functions and applying classical integer linear programming.
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