5742 |
P a g e M a r c h 2 3 , 2 0 1 5
On multi-objective linear programming problems with inexact
rough interval-fuzzy coefficients
E. E. Ammar
*, M. L. Hussein
**and A. M. Khalifa
*1
Dept. of Math., Faculty of Science, Tanta University, Tanta, Egypt.
Dept. of Math., Faculty of Science, Kafrelsheikh University, Egypt.
[email protected]
Abstract
This paper deals with a multi-objective linear programming problem with an inexact rough interval fuzzy coefficients IRFMOLP. This problem is considered by incorporating an inexact rough interval fuzzy number in both the objective function and constrains. The concept of "Rough interval" is introduced in the modeling framework to represent dual-uncertain parameters. A suggested solution procedure is given to obtain rough interval solution for IRFLP(w) problem. Finally,two numerical example is given to clarify the obtained results in this paper.
Keywords
Multi-objective linear programming; Rough-interval fuzzy coefficients; Weighting method; Rough interval solution.
1
Corresponding author [email protected]
Council for Innovative Research
Peer Review Research Publishing System
Journal:
INTERNATIONAL JOURNAL OF COMPUTERS & TECHNOLOGY
Vol.14, No.5
5743 |
P a g e M a r c h 2 3 , 2 0 1 5
1 Introduction
The theory of rough sets proposed by Pawlak (1982) [16], can be regarded as an effective mathematical vehicle for dealing with imprecise and ambiguous data analyses, which can be subsequently applied to pattern recognition, machine learning, and knowledge discovery [[1]-[7]]. The equivalence relation is a key notion in Pawlak’s rough set model. All equivalence classes form a partition of a universe of discourse. Using equivalence classes, an arbitrary subset can be approximated by two subsets called the lower approximation and the upper approximation. However, the equivalence relation is a stringent condition that may limit the applications of rough sets in practical problems. Hence, various extensions of Pawlak’s rough set were developed from an equivalence relation into a more general mathematical concept, e.g., binary relations by Slowinski (2000) [18]. The theory of rough set deals with approximation of an arbitrary subset of universe by two definable or observable subsets called lower and upper approximation. Dubois and Prade (1990) [5] pointedly that the rough fuzzy set is a special case of the fuzzy rough set. The notion of interval-valued fuzzy set was suggested for the first time by Turksan (1986) [19] and Gorse (1986) [9]. They are applied to the fields of the approximate inference gave Miyamoto (2004)[14] fuzzy sets and
-level sets .Shaocheng (1994)[17] introduced two kinds of linear programming with fuzzy numbers. They are called interval number and fuzzy number linear programming. Guijun and Xiapong (1998)[10] define interval-valued fuzzy number and interval distribution numbers and gave their extended operations. Zhong et al. (1994)[23] study fuzzy random linear programming having fuzzy random variable coefficients and the decision vector of fuzzy random variable. It is generally accepted that these two theories are related, but a listing and complementary, to each other ( Miyamoto (2004) [14], XU (2012) [20] and Yao (1996) [21]). establishes a rough multiple objective programming model for a solid transportation problem. Gong et al., (2008) [8]and Li et al., (2007) [12], introduce an valued fuzzy information system by means of iterating the classical Pawlak’s rough set theory with the interval-valued fuzzy set theory and oles cusses the basic rough set theory for the interval-interval-valued fuzzy information system. There are at least two approaches for the development of the fuzzy rough sets theory: The constructive and axiomatic approaches (Yao (1998) [22]). In the constructive approach, the relation to the universe is the primitive notion. The lower and upper approximation operator are contracted by means of this notion. On the other hand, the axiomatic approach takes the lower and upper approximation operator as primitive notion. In this approaches, a set of axioms is used to characterize approximation operators. Zhang (2012) [24] presents a general framework for the study of interval type-2 rough fuzzy sets by using both constructive and axiomatic approaches. In this paper, multi-objective linear programming problem with an inexact rough interval fuzzy coefficients IRFMOL is introduced.The problem is transformed into the corresponding IRFLP problem using the weighting method, A solution procedure is given to obtain the rough interval solution for the IRFLP(w) problem.The paper is organized as in the following sections: In section 2, some preliminaries are introduced. In section 3, problem formulation is introduced as specific definition and properties. In section 4, a solution procedure is given to obtain a rough interval solution for the IRFMOLP problem in section 3. In section 5, numerical example is given to clarify the obtained results. Finally, some concluding remarks are reported in section 6.
2 Preliminaries
In this section, we introduce some notions related to the IRFMOLP problem. The following concepts can be found in [11], [15] and [24]. Number of
-cuts, leading to a series of dual intervals being generated. Among these intervals, the internal has two limits (i.e.~
a
NL(
)
and
~
a
NL(
))
reflect the conservative feature of fuzzy information, while the external ones (i.e.a
~
NU(
)
and
a
~
NU(
)
) correspond to the optimistic characteristics. LetF
NR(R
)
denote the set of all compact rough fuzzy numbers on real line (R), that is,~
a
F
NR(
R
)
R
. It is
a
~
L lower bounded membership grades consist of a membership function located inside the other one formed by the upper bounded gradesa
~
U,a
~
R Satisfies:1.
a
~
R is normal( i.e
x
a
~
R=
{
x
:
a
~
L
,
a
~
U
,
~
a
L
~
a
U,
(0,1]}
) such thata
R(x
)
=
1.
2. For any
(0,1]
,
a
R=
[
a
(
)
U,
a
(
U)]
:
[
a
(
)
L,
a
(
)
L]
is a rough interval number onR
such that
[
a
L,
a
L]
[
a
U,
a
U].
Definition 1
Suppose that
[
(
),
(
)]
:
[
(
),
(
)]
,
=
Ra
Ua
Ua
La
La
[
(
),
(
)]
:
[
(
),
(
)]
.
=
Rb
Ub
Ub
Lb
Lb
we define
5744 |
P a g e M a r c h 2 3 , 2 0 1 5
1.
a
R
b
R=
[
a
U(
)
b
U(
),
a
U(
)
b
U(
)]
:
[
a
L(
)
b
L(
),
a
L(
)
b
L(
)]
,
2.
a
R
b
R=
[
a
U(
)
b
U(
),
a
U(
)
b
U(
)]
:
[
a
L(
)
b
L(
),
a
L(
)
b
L(
)]
,
3.
a
R
b
R=
[
a
U(
)
b
U(
),
a
U(
)
b
U(
)]
:
[
a
L(
)
b
L(
),
a
L(
)
b
L(
)]
,
4. The order relation
"
"
is defined by).
(
)
(
)
(
)
(
Rb
Riff
a
Lb
Land
a
Ub
Ua
5. Let{
[
a
(
),
a
(
)]
:
[
a
(
),
a
(
i)]
,
i
I
}
L i L i U i U
I
is the index set and
I(
i I)
U>
ia
, then
[
(
),
(
)]
:
[
(
),
(
i)]
=
L i L i U i U I i
a
a
a
a
[
(
),
(
)]
:
[
(
),
(
i)]
,
L I i i L I i i U I i i U I ia
a
a
a
6. let
{
[
a
U(
i),
a
U(
i)]
:
[
a
L(
i),
a
L(
i)]
,
i
I
}
I
is the index set and
I(
i I)
U<
ia
, then
[
U(
i),
U(
i)]
:
[
L(
i),
L(
i)]
=
I i
a
a
a
a
[
iIa
U(
i),
iIa
U(
i)]
:
[
iIa
L(
i),
iIa
L(
i)]
.
Definition 2
LetF
N(R
)
denote the set of all compact fuzzy numbers, that is, for anya
~
F
N(
R
)
, satisfies:1.
~
a
is normal ( i.e
x
a
=
{
x
:
a
~
(
x
)
,
(0,1]}
) such that~ x
a
(
)
=
1,
2. For any
(0,1]
,
a
=
a
L(
),
a
U(
)
is closed interval number onR
,
a
L
a
U.
Suppose that,a
=
a
L(
),
a
U(
)
,
b
=
[
b
L(
),
b
U(
)
, we define1.
a
b
=
[
a
L(
),
a
U(
)]
[
b
L(
),
b
U(
)]
=
[
a
L(
)
b
L(
),
a
U(
)
b
U(
)],
(1) 2.a
b
=
[
a
L(
),
a
U(
)]
[
b
L(
),
b
U(
)]
=
[
a
L(
)
b
U(
),
a
U(
)
b
L(
)],
(2) 3.a
b
=
[
a
L(
),
a
U(
)]
[
b
L(
),
b
U(
)]
=
[
a
L(
)
b
L(
)
a
U(
)
b
L(
)
a
L(
)
b
U(
)
)],
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
),
(
)
(
U
L
L
L
U
U
L
U
U
Ub
a
b
a
b
a
b
a
b
a
(3)5745 |
P a g e M a r c h 2 3 , 2 0 1 5
4. The order relation
"
"
is defined by),
(
)
(
),
(
)
(
)]
(
),
(
[
)]
(
),
(
[
a
L
a
U
b
L
b
U
iff
a
L
b
L
a
U
b
U
5. Leta
a
ii
I
R
I
U i L,
}
)],
(
),
(
{[
is the index set and
iIa
(
iI)
L>
, then)],
(
),
(
[
=
)]
(
),
(
[
i U i i L i i U i L ia
a
a
a
(4) 6. Leta
a
ii
I
R
I
U i L,
}
)],
(
),
(
{[
is the index set and
iIa
(
iI)
U<
, then)].
(
),
(
[
=
)]
(
),
(
[
i U i i L i i U i L ia
a
a
a
(5)Lemma 1
Leta
~
,
b
~
F
N(
R
)
, then for any
(0,1]
, we have
a
b
b
a
)
=
(
(6)
is any operation defined in (1)-(6).3 Problem formulation
Consider the rough fuzzy multiobjective linear programming problem (IRFMOLP), as in the following form: • Model 1
R R R R R
R M R xX
D
X
D
f
IRFMOLP
min
~
(
~
,
~
)
=
~
~
~ (7) Subject to:0}
~
,
~
~
~
:
~
{
=
)
~
,
~
(
~
R R R R
R R
x
B
X
A
x
B
A
M
whereD
~
R=
[
d
~
liR]
kn,
A
~
R=
[
a
~
jiR]
mn,
B
~
R=
(
b
~
1R,
b
~
2R,...,
b
~
mR),
X
R=
(
x
1R,
x
2R,...,
x
nR)
T , and~
;
~
;
~
R R RB
A
D
are compact rough fuzzy numbers. The above problem can be reformulated by use the weighting method as in the following form:• Model 2
R R R R R
R M R xX
C
X
C
f
w
IRFLP
(
)
min
~
(
~
,
~
)
=
~
~
~ ~ (8) Subject to:0}
,
~
~
:
{
=
~
R R R R
R R
x
B
X
A
x
M
where[
C
~
]
nR=
[
W
]
1k[
D
~
]
kRn,=
{
0
,
=
1,2,...,
,
=
1}
1 = i k i ii
k
w
w
w
,)
,...,
,
(
=
),
~
,...,
~
,
~
(
=
~
,
]
~
[
=
~
,
]
~
[
=
~
2 1 2 1 T R n R R R R m R R R n m R ji R n k R li Rx
x
x
X
b
b
b
B
a
A
d
D
, andC
~
R;
A
~
R;
B
~
Rare compact rough fuzzy numbers.
Definition 3
The rough fuzzy vectorx
R(
A
~
R,
B
~
R)
which satisfies the conditions in model (2), is said to be a rough fuzzy optimal solution of mode (2) IRFLP(w) if)
~
,
~
(
~
)
~
,
~
(
~
R R R R R Rx
C
f
x
C
f
for each
~
x
R
M
~
R(
A
~
R,
B
~
R)
, furthermore ifx
R is a rough fuzzy vector. then it is said to be a rough fuzzy optimal solution of model 2: for any
(0,1]
the
level cuts ofc
iR,
a
jiR,
b
jR,
x
R are rough intervals, furthermore, if5746 |
P a g e M a r c h 2 3 , 2 0 1 5
R
x
is a rough fuzzy vector, then it is said to be a rough fuzzy optimal solution of model 2: for any
(0,1]
the
level cuts ofc
iR,
a
jiR,
b
jR,
x
R are rough intervals denote them by:
[
(
),
(
)]
:
[
(
),
(
)]
,
=
)
(
L
i L i U i U i R ic
c
c
C
c
[
(
),
(
)]
:
[
(
),
(
)]
,
=
)
(
a
jiR a
jiU
a
jiU
a
jiL
a
jiL
[
(
),
(
)]
:
[
(
),
(
)]
,
=
)
(
b
jR b
jU
b
jU
b
jL
b
jL
[
(
),
(
)]
:
[
(
),
(
)]
,
=
)
(
i L
L i U i U i R ix
x
x
x
x
1,2,...,
=
1,2,...,
=
n
j
m
i
Definition 4
(Rough fuzzy efficient solution) The rough fuzzy vectorx
R(
A
~
R,
B
~
R)
which satisfies the condition in model 1 IRFMOLP, is called a rough fuzzy efficient solution of mode 1 IRFMOLP if and only if there does not exist anotherx
R(
A
~
R,
B
~
R)
M
~
R such that)
~
,
~
(
~
)
~
,
~
(
~
R R i R R iC
x
f
C
x
f
for all
i
and(
)
i(
R)
Ri
x
f
x
f
for at least onei
=
1,2,...,
k
Remark 1
For anyw
*
, any rough fuzzy optimal solution of modle 2 is an rough fuzzy efficient solution of problem modle 1Also, for any
(0,1]
we transfer model 2 to two boundary models, an upper approximation fuzzy interval programming problemP
~
UU and lower approximation fuzzy interval programming problemP
~
LL as follows:
U U U
U M x U Uf
C
X
P
~
:
min
~
=
~
~
~ (9) Subject to:0}
,
~
~
:
{
=
)
~
,
~
(
~
x
B
X
A
x
B
A
M
U U U U U
L L L
L M x L Lf
C
X
P
~
:
min
~
=
~
~
~ (10) Subject to:0}
,
~
~
:
{
=
)
~
,
~
(
~
x
B
X
A
x
B
A
M
L L L L L4 Solution procedure
(a) Convert model (2) to two boundary corresponding to submodels
P
UU,P
LL, respectively, if the objective is to be minimized, thenP
UU is desired.(b) Convert each boundary model to two corresponding to submodels the upper and lower approximation intervals, respectively.
(c) Solve the first desired upper bounded submodel and obtain solution of
x
U andP
UU.(d) Add constraint
x
L
x
U to the second upper bounded submodel, solve it and obtain solution ofx
L andP
LL .(e) Add constraint
x
L
x
L to the first first lower bounded submodel, solve it and obtain solution ofx
L andP
LL5747 |
P a g e M a r c h 2 3 , 2 0 1 5
(f) Add constraint
x
U
x
L to the first second lower bounded submodel, solve it and obtain solution ofx
U and UU
P
.(g) Incorporate the solution of all submodel to the obtain optimal solution as following :
[
,
]
:
[
,
]
=
[
,
]
:
[
,
]
=
U U L L R U U L L Rf
f
f
f
f
x
x
x
x
x
After deduce the following programming problems for a upper approximation interval and lower approximation interval • Model 3: (a)
min
:
X
C
P
U U M x U U (b)min
:
X
C
P
L L M x L L • Model 4: (a)min
:
X
C
P
U U M x U U (b)min
:
X
C
P
L L M x L L • Model 5: (a)min
:
X
C
P
U U M x U U (b)min
:
X
C
P
L L M x L L • Model 6: (a)min
:
X
C
P
U U M x U U (b)min
:
X
C
P
L L M x L L • Model7: (a)P
UU and U UP
X
C
and
X
C
U U U M x U U U M x min
min
) , ( ) , ( (b)P
LL andP
LL5748 |
P a g e M a r c h 2 3 , 2 0 1 5
X
C
and
X
C
L L L M x L L L M xmin
min
) , ( ) , ( • Model 8: (a)min
:
) , ( X
C
P
U U U M x U U (b)min
:
) , ( X
C
P
L L L M x L L It is clear that for any
(0,1]
Models (3-8) are linear programming problem with rough interval numbers solved them by simplex method .Lemma 2
Suppose that(
A
~
)
L
0
(i.ea
jiL
0
and
a
jiL
0
) for any
(0,1]
if~
x
M
~
, thenL L
M
x
andx
L
M
L . Similarly, whenA
~
U
0
and~
x
M
~
U thenx
U
M
U andx
U
M
U .Lemma 3
Suppose that(
A
~
)
L
0
(i.ea
jiL
0
and
a
jiL
0
) for any
(0,1]
if~
x
L
M
~
L, thenL L
M
x
andx
L
M
L . Similarly, whenA
~
U
0
and~
x
U
M
~
U thenx
U
M
U andx
U
M
U.Lemma 4
Suppose thatA
L
0
andA
L
0
(i.ea
jiL
0
and
a
jiL
0
) for any
in
(0,1]
if) , (
~
~
L LM
x
, thenx
L
M
(L,L) . Similarly, whenA
U
0
andA
U
0;
~
x
M
~
(U,U), then) , ( U U U
M
x
5 Basic results
In the following we shall prove some theorems which point out that a rough fuzzy optimal solution of models
P
~
UUand
P
~
LL may by resolved into a class of optimal solution of modules (model 3-8)Theorem 1 Suppose that
A
~
U
0
andC
~
U
0
ifx
~
U is upper approximation (rough) fuzzy optimal solution of modelsP
~
UU, then for any
(0,1]
, we have1. U U U M U x U U U U M U x U
X
C
X
C
=
min
,
=
min
2.)
,
(
=
~
,
~
~
min
=
~
~ ~ U U U U U U M x UX
C
(11)3.
(
x
)
U,
(
x
)
U are upper approximation (rough) optimal solution of models ( 5) (problems,
LL)
U UP
P
and (6)(problems,
LL)
U UP
P
, respectively.Proof Suppose that
~
x
U is upper approximation fuzzy solution of modelP
(U), that is, U T U U T U U Ux
C
Min
x
C
and
M
x
~
(
~
)
~
=
(
~
)
~
(12) SinceA
~
U
0
, by lemma(1), we have(
x
)
U
M
U,
(
x
)
U
M
U,5749 |
P a g e M a r c h 2 3 , 2 0 1 5
}
~
~
:
{
=
}
~
~
:
{
=
x
x
M
and
M
x
x
M
M
U U
U U
from eqs (1) (3), (6) and since
C
~
U
0
, then the the first side of eqn12]
)
)(
)
(
,
)
(
)
(
[
=
]
)
)(
(
,
)
(
)
[(
1 = 1 = U j U j n j U j U j n j U U U Ux
c
x
c
x
C
x
C
]
)
(
)
(
,
)
(
)
[(
=
1 = U j U j U j U j n jx
c
x
c
]
)
(
,
)
)][(
)
(
,
)
[(
=
1 = U j U j U j U j n jx
x
c
c
U j U j n jx
c
)
(
~
)
~
(
=
1 =
]
~
~
[
=
C
UX
U (13)From eqs (1),(3),(4),(5) and (6), then the second side of eqn(12)
U T U U M U x U T U U M U xx
C
x
C
~
)
~
=
min
(
~
)
~
(
min
~ ~ ~ ~
U j U j n j U M U xx
c
)
~
(
)
~
(
min
=
1 = ~ ~
]
)
(
,
)
)][(
)
(
,
)
[(
min
=
1 = ~ ~ U j U j U j U j n j U M U xx
x
c
c
]
)
(
)
)
(
,
)
(
)
[(
min
=
1 = ~ ~ U j U j U j U j n j U M U xx
c
x
c
]
)
(
)
)
(
,
)
(
)
[(
min
=
~ ~ U U U U U M U xx
c
x
c
]
)
(
)
)
(
min
,
)
(
)
(
min
[
=
~ ~ ~ ~ U U U M U X U U U M U XX
C
X
C
(14)From eqs (13) and (14) there followes (11), we also obtain, for any
(0,1]
X
C
X
C
U U M X U U)
(
min
=
)
(
)
(
(15) X
C
X
C
U U M X U U)
(
min
=
)
(
)
(
(16)eqs (15) and (16) show that the part of the theorem is correct, too the proof is compleat.
Theorem 2 Suppose that
A
~
U
0
andC
~
U
0
if}
~
~
:
{
~
}
~
~
:
{
~
U U U U U U U UM
x
x
M
and
M
x
x
M
Ux
~
is upper approximation (rough) fuzzy optimal5750 |
P a g e M a r c h 2 3 , 2 0 1 5
1.
C
X
C
UX
U M x U U U M x Umin
=
min
=
2.)
,
(
=
~
~
~
min
=
~
~ ~ U U U U U U M U x UX
C
(17)3.
x
U,
x
U are upper approximation (rough) optimal solution of models ( 5) (problemsP
UU(
)
P
LL(
)
) and (6) (problems
P
UU(
),
P
LL(
)
), respectively;Proof
Suppose that~
x
U is upper approximation fuzzy optimal solution of modelP
~
UU (problem ?? ), that is U T U U T U U Ux
C
x
C
and
M
x
~
(
~
)
=
min
(
~
)
~
(18) SinceA
~
U
0
by lemma 1, we have(
x
)
U
M
U,
(
x
)
U
M
U,
M
U=
{
x
U:
~
x
U
M
~
U}
and}
~
~
:
{
=
U U U UM
x
x
M
From eqs (1) (3), (6) and since
C
~
U
0
, then the the first side of eqn18]
)
)(
)
(
,
)
(
)
(
[
=
]
)
)(
(
,
)
(
)
[(
1 = 1 = U j U j n j U j U j n j U U U Ux
c
x
c
x
C
x
C
]
)
)(
)
(
,
)
(
)
[(
=
1 = U j U j U j U j n jx
c
x
c
]
)
(
,
)
)][(
)
(
,
)
[(
=
1 = U j U j U j U j n jx
x
c
c
U j U j n jx
c
)
(
)
~
(
=
1 =
]
~
~
[
=
C
UX
U (19)From eqs (1),(3),(4),(5) and (6), then the second side of eqn(18)
U T U U M U x U T U U M U xx
C
x
C
~
)
=
min
(
~
)
(
min
~ ~ ~ ~
U j U j n j U M U xc
x
min
)
(
)
~
(
=
1 = ~ ~
]
)
(
,
)
)][(
)
(
,
)
[(
min
=
1 = ~ ~ U j U j U j U j n j U M U xx
x
c
c
]
)
(
)
)
(
,
)
(
)
[(
min
=
1 = ~ ~ U j U j U j U j n j U M U xx
c
x
c
[(
)
(
)
,
(
)
)
(
)
]
min
=
~ ~ U U U U U M U xx
C
x
C
5751 |
P a g e M a r c h 2 3 , 2 0 1 5
]
)
(
)
)
(
min
,
)
(
)
(
min
[
=
~ ~ ~ ~ U U U M U x U U U M U xx
C
x
C
(20)From eqs (19) and (20) there followes (17), also obtain, for any
(0,1]
X
Min
C
X
C
U U M X U U)
(
=
)
(
)
(
(21) U U U M X U UX
C
X
C
(
)
=
min
(
)
(
)
)
(
(22)eqs (21) and (22) show that the part of the theorem is correct, too the proof is compleat.
Corollary 1
Suppose thatA
U
0,
A
U
,
C
U
0,
and
C
U
0
let}
~
~
:
{
) , ( U U U U UM
x
x
M
, where
(0,1]
ifx
~
U is upper approximation (rough) fuzzy optimal solution of modelP
~
UU, then for any
(0,1],
x
U is upper approximation (rough) fuzzy optimal solution of model 8.Example 1
: Consider the following rough fuzzy linear programming problem:1
)
(IRFMOLP
R R R R R Rx
x
~
~
2
~
3
~
2
~
1
~
max
Subject to :0
~
~
,
5
1
~
4
5
~
7
2
~
~
~
1
~
3
~
6
~
8
~
6
~
2
~
2 1 2 1
R R R R R R R R R R R R Rx
x
x
x
Suppose for
=
0.5
the
Cut of the rough fuzzy numbers in the above can be write as:
R Rx
x
2 1[1.5,2.5]
:
[1,3]
[2.5,3.5]
:
[2,4]
[1.5,2.5]
:
[1,3]
[0.5,1.5]
:
[0,2]
max
subject to :
]
[14.5,15.5
:
[14,16]
]
[44.5,45.5
:
[44,46]
]
[26.5,27.5
:
[26,28]
[0.5,1.5]
:
[0,2]
[2.5,3.5]
:
[2,4]
[5.5,6.5]
:
[5,7]
[7.5,8.5]
:
[7,9]
[5.5,6.5]
:
[5,7]
[1.5,2.5]
:
[1,3]
2 1 R Rx
x
where
]
,
[
:
]
,
[
]
,
[
:
]
,
[
=
2 2 2 2 1 1 1 1 2 1 U U L L U U L L R Rx
x
x
x
x
x
x
x
x
x
5752 |
P a g e M a r c h 2 3 , 2 0 1 5
Convert the
(IRFMOLP
)
1 linear programming problem to single objective by weighting method0.5
=
=
,
=
)
(
x
w
1f
1w
2f
2at
w
1w
2f
, follows:)
(
1w
IRFLP
R Rx
x
2 1[1.5,2.5]
:
[1,3]
[1.5,2.5]
:
[1,3]
max
Subject to :
]
[14.5,15.5
:
[14,16]
]
[44.5,45.5
:
[44,46]
]
[26.5,27.5
:
[26,28]
[0.5,1.5]
:
[0,2]
[2.5,3.5]
:
[2,4]
[5.5,6.5]
:
[5,7]
[7.5,8.5]
:
[7,9]
[5.5,6.5]
:
[5,7]
[1.5,2.5]
:
[1,3]
2 1 R Rx
x
Now, we divided a problem above to a problems upper approximation problem
P
UU(
)
and lower approximation problem)
(
L LP
follows:
)
(
L LP
L L L Lx
x
x
x
Max
2 2 1 1,
,
[1.5,2.5]
[1.5,2.5]
Subject to :
14.5,15.5
44.5,45.5
26.5,27.5
,
,
0.5,1.5
2.5,3.5
5.5,6.5
7.5,8.5
5.5,6.5
1.5,2.5
2 2 1 1 L L L L
x
x
x
x
:
)
(
U UP
U U U Ux
x
x
x
2 2 1 1,
,
[1,3]
[1,3]
max
Subject to :
14,16
44,46
26,28
,
,
0,2
2,4
5,7
7,9
5,7
1,3
2 2 1 1 U U U U
x
x
x
x
Since
