multi-objective fuzzy linear programming and its...

Tamsui Oxford Journal of Mathematical Sciences 21(2) (2005) 243-268 Aletheia University

Multi-Objective Fuzzy Linear Programming and Its Application in Transportation Model

Bablu Jana and Tapan Kumar Roy*

Department of Mathematics, Bengal Engineering college (Deemed University), Howrah, West-Bengal, Pin 711103, India.

Received July 27, 2004, Accepted December 31, 2004.

Abstract In this study, the solution procedure of Multi-objective Fuzzy Linear

Programming Problem (MOFLPP) with mixed constraints and its application in solid transportation problem, is going to be presented. There are two parts in this paper. In the first part, a Multi-objective Linear Programming Problem MOLPP) with fuzzy coefficients occurring in constraints and objective functions and fuzzy constraint goals, has been considered. Here fuzzy constraint goals and coefficients of objective and constraint functions are characterised by Triangular Fuzzy Numbers (TFNs). Using Bellman and Zadeh’s (1970) multicriteria fuzzy decision-making process, the very problem has been converted to a crisp non-linear programming problem. Then it has been solved using fuzzy decisive set method. In other part, a linear multi-objective solid transportation problem with mixed constraint as well as additional restriction in fuzzy environment is considered. In this transportation problem, cost coefficients of objective functions and additional restriction function, the supply, demand and conveyance capacity have been expressed as TFNs. This MOFLPP is solved by fuzzy decisive set method as before. Numerical examples have been provided for two parts to illustrate the solution procedure.

Keywords and Phrases: Fuzzy multi-objective linear programming, Solid transportation problem, Triangular fuzzy number.

* E-mail: [email protected]

Bablu Jana and Tapan Kumar Roy

244

1. Introduction Generally, in a Multi-Objective Linear Programming Problem (MOLPP), coefficients (of objective and constraint functions) as well as constraint goals are assumed to be fixed in value. But there are many practical situations where this assumtions are not valid. These coefficients as well as constraint goals may not be well defined due to lack of information of data and/or uncertain market situations. For this reasons, the different coefficients and constraint goals may be chacterised by fuzzy numbers.

The idea of fuzzy set was first proposed by Zadeh [11 ], as a mean of handling uncertainty that is due to imprecision rather than to randomness. After that Bellman and Zadeh [11] proposed that a fuzzy decision might be defined as the fuzzy set, defined by the intersection of fuzzy objective and constraint goals. From this view point , Tanakka and Asai [6], Zimmermann [7] introduced fuzzy linear programming problem in fuzzy environment. Tong [14], Gasimov and Yenilmez [12] among others, considered single objective mathematical programming with all fuzzy parameters. Tong considered the fuzzy linear programming problem with fuzzy constraints. After defuzzification he solved the so-obtained crisp problem by fuzzy decisive set method proposed by Sakawa and Yano [10]. Gasimov and Yenilmez considered fuzzy linear programming (FLP) problem with less than type constraints. In their paper Coefficients of constraints were taken as fuzzy numbers. They solved it by fuzzy decisive set method and modified sub-gradient method. Lai-Hawng [15] considered MOLPP with all parameters, having a triangular possibility distribution. They used an auxiliary model and it was solved by multi-objective linear programming methods. Chanas, [3] proposed a fuzzy programming in multi-objective linear programming and it was solved by parametric approach. Zimmermann [7] proposed a fuzzy multicriteria decision making set, defined as the intersection of all fuzzy goals and constraints.

There are so many Transportation models where FLP have been applied. Bit et al [1] considered fuzzy programming approach to multicriteria decision making transportation problem in which the constraints are of equality types. Latter Bit et al [2] also considered a fuzzy programming approach to multi-objective solid transportation problem in which the supply, demand and capacity constraints are of equality and inequality types. They solved it by fuzzy programming technique. Das et al [13] considered the multi-objective transportation problem with interval cost, source and destination parameters. They converted the interval cost, source and destination parameters into deterministic one and finally it was solved by fuzzy programming technique.

Multi-Objective Fuzzy Linear Programming and Its Application

245

In this paper, we have proposed a MOFLPP with mixed constraints in which right hand side of constraints are fuzzy numbers. Using Bellman and Zadeh’s fuzzy decision-making process, the MOFLPP is converted into an equivalent crisp LPP. Then it is solved by simplex method. Next we have also considered MOFLPP with coefficients of objective as well as constraint functions and right hand sides of constraints are TFNs. Converting it into an equivalent crisp non-linear programming problem, it is also solved by fuzzy decisive set method.

We have also considered an application of MOFLPP on a transportation model .We have considered a multi-objective solid transportation model with an additional restriction and mixed constraints in which coefficients of objective functions, additional restriction function, demand, supply and conveyance capacities are expressed as TFNs. It is then solved by fuzzy decisive set method as shown before. 2. Triangular Fuzzy Number (TFN) Let F(ℜ) be a set of all triangular fuzzy number in a real line ℜ. A triangular

fuzzy number~A (∈F(ℜ)) is a normal and convex fuzzy set with the following

membership function ~A

µ : ℜ→[1,0] (which satisfies both normality i.e ~A

µ (x) =1 for at

least one x ∈ R and convexity i.e ~A

µ (x/) ≥ min.( ~A

µ (x1) , ~A

µ (x2) ) where ~A

µ (x) ∈

[1,0] and ∀ x/ ∈ [x1,x2] ). µ

~A

µ (x) = 12

1

aaax−− for a1 ≤ x ≤ a2

= 23

3

aaxa

−−

for a2 ≤ x ≤ a3

= 0 elsewhere O a1 a2 a3 x

Figure-1.TFN~A =(a1 ,a2 ,a3 )

It is parameterized by a triplet (a1,a2,a3) where a1,a3 are the lower and upper limits

of support of ~A and a2 is the pick value of

~A (fig.-1).

Triangular fuzzy numbers are very often used in different application (e.g. fuzzy controls, managerial decision making, business and finance, social sciences etc.).


246

More generally, the left~A and right

~A are branches of the TFN. They can be denoted

by left~A =(a1 , a2 , a2 ) and right

~A =(a2 ,a2 ,a3 ).

i) The left TFN ~A =(a1 ,a2 ,a2 ) (fig.-2) is suitable to represent positive large

or words with similar meaning (e.g. old age, big profit , high risk, etc.) provided that a2 > a1 . It is represented by the following membership functions:

µ

~A

µ (x) = 0 for x ≤ a1 ,

= 12

1

aaax−− for a1 ≤ x ≤ a2 1

= 1 for x ≥ a2 , O a1 a2 x

Figure-2. Left TFN~A = (a1 ,a2 ,a2 )

ii) The right TFN ~A = (a2 ,a2 ,a3 ) (fig.-3)

~A is suitable to represent positive small

or words with similar meaning (e.g. young age, small cost, small risk etc.) provided that a3 > a2 . It is represented by the following membership functions:

~A

µ (x) = 1 for x ≤ a2 , µ

= 23

3

aaxa

−−

for a2 ≤ x ≤ a3 , 1

= 0 for x ≥ a3 , O a2 a3 x

Figure-3. right TFN~A = (a2 ,a2 ,a3 )

Note: A TFN = (a1,a2,a3) is positive (negative) if a1 ≥ 0 (a3 < 0 ).


247

3. Multi-Objective Linear Programming Problem (MOLPP) with Fuzzy Resources The General Multi-Objective Linear Programming Problem (GMOLPP) with mixed constraints may be written as follows: Minimize Z =[ Z1,Z2, Z3, LL , ZK ] (3.1)

subject to ∑=

n

jjij xa

1

≥ bi for i = 1,2,3, ,LL m1

∑=

n

jjij xa

1

≤ bi for i = m1+1, m1+2, ,LL m2

∑=

n

jjij xa

1 = bi for i = m2+1 , m2+2, ,LL m

xj ≥ 0 , j = 1, 2, 3, ,LL n where

Zk = ∑=

n

jj

kj xc

1

, k = 1, 2, 3, ,LL K

3.1 MOLPP with fuzzy resources When constraint goals are TFNs, (3.1) becomes Minimize Z =[ Z1,Z2, Z3, ,LL ZK ] (3.2)

subject to ∑=

n

jjij xa

1

≥ ib~

for i = 1,2,3, ,LL m1

∑=

n

jjij xa

1 ≤ ib

~ for i = m1+1, m1+2, ,LL m2

∑=

n

jjij xa

1 = ib

~ for i = m2+1 , m2+2, ,LL m

xj ≥ 0 , j = 1, 2, 3, ,LL n where

Zk = ∑=

n

jj

kj xc

1

, k = 1, 2, 3, ,LL K

We will accept some assumptions.


248

Assumption1: ib~

are considered as the following positive TFNs:

Left TFN ib~

=( bi - 0ib , bi , bi )with tolerance 0

ib (<bi) for∑=

n

jjij xa

1 ≥ ib

~; i = 1,2,3, ,LL m1 ;

RightTFN ib~

=(bi, i, bi+ 0ib ) with tolerance 0

ib (>0) for∑=

n

jjij xa

1≤ ib

~;i=m1+1,m1+2, ,LL m2;

and TFN ib~

= ( bi- lib ,bi ,bi+ r

ib ) with tolerances lib (< bi), r

ib ( > 0) for ∑=

n

jjij xa

1 = ib

~ ,

i = m2+1 , m2+2, ,LL m. The problem (3.2) becomes with extreme tolerances as Minimize Z =[ Z1,Z2, Z3, ,LL ZK ] (3.3)

subject to ∑=

n

jjij xa

1 ≥ bi- 0

ib i = 1,2,3, ,LL m1

∑=

n

jjij xa

1 ≤ bi + 0

ib i = m1+1, m1+2, ,LL m2

∑=

n

jjij xa

1 ≥ bi- l

ib i = m2+1 , m2+2, ,LL m

∑=

n

jjij xa

1 ≤ bi+ r

ib i = m2+1 , m2+2, ,LL m

xj ≥ 0 , j = 1, 2, 3, ,LL n. where

Zk = ∑=

n

jj

kj xc

1

, k = 1, 2, 3, ,LL K

3.2 Fuzzy Programming Technique for the Solution of MOLPP with Fuzzy Resources The MOFLPP can be considered as a Vector Minimum Problem (VMP). Let Lk and Uk be the lower and upper bound for the k-th objective, where Lk = aspired level of achivement for the k-th objective function, and Uk = highest acceptable level of achivement for the k-th objective function. When the aspiration levels for each objective have been specified, we formed a fuzzy model. Our next step is to transform the fuzzy model into a crisp model (i.e a coventional LPP). The details of the foregoing steps may be presented as follows:


249

Algorithm Step-1. Solve the MOLPPs (3.1) and (3.3) as a single objective LPP using each time

only one objective and ignore all others. Step-2. From the results of step-1, determine the corresponding value for every

objective functions at each solutions. Step-3. Find upper and lower bounds (i.e Uk and Lk) for kth objective from the 2k

objective values derived in step-2. Step-4. The initial fuzzy model is equivalent to following: Find { xj ; j = 1, 2, 3, ,LL n } (3.4) so as to satisfy Zk ≤~ Lk for k = 1, 2, 3, ,LL K.

∑=

n

jjij xa

1≥~ bi i = 1,2,3, ,LL m1

∑=

n

jjij xa

1≤~ bi i = m1+1, m1+2, ,LL m2

∑=

n

jjij xa

1≅ bi i = m2+1 , m2+2, ,LL m

xj ≥ 0, j=1,2,3, ,LL n Here membership functions for fuzzy constraints of (3.4) are defined as: (for kth constraints kG~ (k = 1, 2, 3, ,LL K)

)(1

~ ∑=

n

jj

kjG xc

kµ = 1 for ∑

=

n

jj

kj xc

1 ≤ Lk , k = 1, 2, 3, ,LL K.

= kk

n

jj

kjk

LU

xcU

−

−∑=1 for Lk ≤ ∑

=

n

jj

kj xc

1 ≤ Uk ,

= 0 for ∑=

n

jj

kj xc

1 > Uk ,

(for the ith constraints ~

iC (i = 1,2,3, ,LL m1))

)(~ iC

bi

µ = 1 for bi ≤ ∑=

n

jjij xa

1- 0

ib


250

= 01

i

n

jijij

b

bxa∑=

− for ∑

=

n

jjij xa

1- 0

ib ≤ bi ≤ ∑=

n

jjij xa

1

= 0 for bi > ∑=

n

jjij xa

1


iC (i = m1+1, m1+2, ,LL m2 ))

)(~ iC

bi

µ = 0 for bi < ∑=

n

jjij xa

1

= 01

i

n

jjiji

b

xab ∑=

− for ∑

=

n

jjij xa

1 ≤ bi ≤ ∑

=

n

jjij xa

1+ 0

ib

= 1 for bi ≥ ∑=

n

jjij xa

1+ 0

ib


iC (i = m2+1 , m2+2, ,LL m ))

)(~ iC

bi

µ = 0 , for bi > ∑=

n

jjij xa

1+ l

ib

= )(~ iC

bR

i

µ = li

ili

n

jjij

b

bbxa −+∑=1 for ∑

=

n

jjij xa

1 < bi ≤ ∑

=

n

jjij xa

1+ l

ib

= )(~ iC

bL

i

µ = ri

ri

n

jjiji

b

bxab +−∑=1 for ∑

=

n

jjij xa

1- r

ib ≤ bi < ∑=

n

jjij xa

1

= 0 for bi ≤ ∑=

n

jjij xa

1- r

ib

Step-5. Using the max-min operator (as Zimmermann [7]) crisp LPP for (3.2) is

formulated as follows: Max λ (3.5)

subject to, ∑=

n

jj

kj xc

1+ λ( Uk - Lk) ≤ Uk for k = 1, 2, 3, ,LL K.


251

∑=

n

jjij xa

1-λ 0

ib ≥ bi , for i = 1,2,3, ,LL m1

∑=

n

jjij xa

1+ λ 0

ib ≤ bi , for i = m1+1, m1+2, ,LL m2

∑=

n

jjij xa

1- λ l

ib ≥ bi - lib , for i = m2+1 , m2+2, ,LL m

∑=

n

jjij xa

1+λ r

ib ≤ bi + rib , for i = m2+1 , m2+2, ,LL m

0 ≤ λ ≤ 1 , xj ≥ 0 . j=1,2,3, ,LL n It can be solved by any simplex method. Numerical Example1: MinimizeZ1= 1

1c x1+ 12c x2 (1)

Minimize Z2 = 21c x1+ 2

2c x2 subject to a11x1 + a12x2 ≥ 1b~ , a21x1+a22x2 ≥ 2b~ , x1 , x2 ≥ 0

where kjc =

7235

, ija =

1142

, i ,j, k =1, 2 and 1b~ =~

20 = (18, 20, 20) ;

2b~ =~

10= (9, 10, 10) respectively constraint goals. To solve this problem, we first solve the following four Sub-Problems (SPs): Minimize Z11 = 5x1+3x2 … (1.1) Minimize Z12 = 5x1+3x2 .…………(1.2) subject to 2x1+4x2 ≥ 20 , subject to 2x1+4x2 ≥ 18 , x1+x2 ≥ 10 , x1+x2 ≥ 9 , x1 , x2 ≥ 0 ; x1 , x2 ≥ 0 ; Minimize Z21 = 2x1+7x2 ……(1.3) Minimize Z22 = 2x1+7x2 ……………(1.4) subject to 2x1+4x2 ≥ 20 , subject to 2x1+4x2 ≥ 18 , x1+x2 ≥ 10 , x1+x2 ≥ 9 , x1 , x2 ≥ 0 ; x1 , x2 ≥ 0 ;


252

So the optimal solutions of (1.1), (1.2), (1.3) and (1.4) are x11* = (x *11

1 , *112x ) = ( 0, 10 ) , Z11*( x11*) = 30,

x12* = (x *121 , *12

2x ) = ( 0, 9 ) , Z12*( x12*) = 27, x21* = (x *21

1 , *212x ) = ( 10, 0 ) , Z21*( x21*) = 20,

x22* = (x *221 , *22

2x ) = ( 9, 0 ) , Z22*( x22*) = 18 respectively. So L1 = min { Z1*( x11*), Z1*( x12*), Z1( x21*), Z1( x22*) }= min{30, 27, 50, 45} = 27 and U1 = max {Z1*( x11*), Z1*( x12*), Z1( x21*), Z1( x22*) } = max{30, 27, 50, 45 } = 50 L2 = min {Z2( x11*), Z2( x12*), Z2*( x21*), Z2*( x22*) } = min{70, 63, 20, 18} = 18 and U2 = max { Z2( x11*), Z2( x12*), Z2*( x21*), Z2*( x22*) } = max{70, 63, 20, 18} = 70 Following the step-4, the problem (1) is equivalent to Find {xj , j = 1, 2.} (1.5) so as to satisfy 5x1+3x2 ≤~ 27 , 2x1+7x2 ≤~ 18 , 2x1+4x2 ≥~ 20 , x1+x2 ≥~ 9 , x1 , x2 ≥ 0. Here membership functions for fuzzy constraints of (1.5) are defined as: )35( 21~

1xxG +µ = 1 for 5x1+3x2 ≤ 27 ,

= 23

3550 21 )xx( +− for 27 ≤ 5x1+3x2 ≤ 50 ,

= 0 for 5x1+3x2 > 50 , )72( 21~

2xxG +µ = 1 for 2x1+7x2 ≤ 18 ,

= 52

7270 21 )xx( +− for 18 ≤ 2x1+7x2 ≤ 70 ,

= 0 for 2x1+7x2 > 70 ,

)20(1

~C

µ = 1 for 2x1+4x2 > 22 ,

= 2

2042 22 −+ xx for 20 ≤ 2x1+4x2 ≤ 22 ,

= 0 for 2x1+4x2 < 20 ,


253

)10(2

~C

µ = 1 for x1+x2 >11 ,

= x2+x2 -10 for 10 ≤ x1+x2 ≤ 11 , = 0 for x1+x2 < 10 , Using the max-min operator (as Zimmermann [7]) crisp LPP for (1) is formulated as follows: Max λ (1.6) 5x1+3x2+23λ ≤ 50, 2x1+7x2 +52λ ≤ 70, 2x1+4x2 -2λ ≥ 20, x1+x2 -λ ≥ 10 , 0 ≤ λ ≤ 1, x1,x2≥ 0 . So, optimal solution of MOFLPP (1) are x *

1 =4.758065; x *2 = 5.645161;

Z1* = 40.72581, Z2* = 49.03226 with aspiration level *λ = 0.403226. 4. MOLPP with Fuzzy Coefficients and Fuzzy Resources

When the objective function’s coefficients, technological coefficients and also right hand side of constraints are fuzzy numbers then (3.1) becomes Minimize Z =[ Z1,Z2, Z3, ,LL ZK ] (4.1)

subject to ∑=

n

jjij xa

1

~ ≥ ib

~ for i = 1,2,3, ,LL m1

∑=

n

jjij xa

1

~ ≤ ib

~ for i = m1+1, m1+2, ,LL m2

∑=

n

jjij xa

1

~ = ib

~ for i = m2+1, m2+2, ,LL m

xj ≥ 0 . j=1, 2, 3, ,LL n where

Zk = ∑=

n

jj

kj xc~

1

, k = 1, 2, 3, ,LL K

Assumption1: Fuzzy objective and constraints coefficients are considered as the following positive TFN’s:


254

Right TFN kjc = ( k

jc , kjc , k

jc + kjp ) with tolerance k

jp ( > 0) for the objective

function j

n

j

kj xc∑

=1

~ for k = 1, 2, 3, ,LL K.

Left TFNs ija~

= (aij - 0ijd , aij , aij ) with tolerance 0

ijd (< aij) and ib~

= ( bi- 0ib , bi , bi )

with tolerance 0ib (< bi) for ∑

=

n

jjij xa

1

~≥ ib

~; i = 1,2,3, ,LL m1.

Right TFNs ija~

= (aij , aij , aij + 0ijd ) with tolerance 0

ijd ( > 0) and ib~

= (bi , bi , bi

+ 0ib ) with tolerance 0

ib ( > 0) for ∑=

n

jjij xa

1

~≤ ib

~; i =m1+ 1, m1+2 , ,LL m2. and

TFNs ija~

= ( aij - lijd , aij , aij + r

ijd ) with tolerances lijd ( < aij ) , r

ijd ( > 0) and ib~

=

( bi - lib , bi , bi + r

ib ) with tolerances lijb ( < bi) , r

ijb (> 0) for ∑=

n

jjij xa

1

~= ib

~; i = m2+1,

m2+2, ,LL m . For the calculation of upper (Uk) and lower (Lk) bounds of the k-th( k = 1, 2, 3,

,LL K) objective function, we first construct the following eight sub-problems(4.2 – 4.9) :

Minimize Zk1 = j

n

j

kj xc∑

=1

(4.2)

subject to ∑=

n

jjij xa

1 ≥ bi , for i = 1,2,3, ,LL m1

∑=

n

jjij xa

1 ≤ bi , for i = m1+1, m1+2, ,LL m2

∑=

n

jjij xa

1 = bi, for i = m2+1, m2+2, ,LL m

xj ≥ 0 . j=1,2,3, ,LL n .

Minimize Zk2 = ∑=

+n

jj

kj

kj x)pc(

1

(4.3)

subject to same constraints of (4.2)

MinimizeZk3 = j

n

j

kj xc∑

=1

(4.4)


255

subject to ∑=

n

jjij xa

1 ≥ bi- 0

ib , for i = 1,2,3, ,LL m1

∑=

n

jjij xa

1 ≤ bi + 0

ib , for i = m1+1, m1+2, ,LL m2

∑=

n

jjij xa

1 ≥ bi- l

ib , for i = m2+1 , m2+2, ,LL m

∑=

n

jjij xa

1 ≤ bi+ r

ib , for i = m2+1 , m2+2, ,LL m

xj ≥ 0 . j=1,2,3, ,LL n .

Minimize Zk4 = ∑=

+n

jj

kj

kj x)pc(

1

(4.5)

subject to same constraints of (4.4)

Minimize Zk5= j

n

j

kj xc∑

=1

(4.6)

subject to ∑=

−n

jjijij xda

1

0 )( ≥ bi , for i = 1,2,3, ,LL m1

∑=

+n

jjijij xda

1

0 )( ≤ bi , for i = m1+1, m1+2, ,LL m2

∑=

−n

jj

lijij xda

1)( ≥ bi , for i = m2+1 , m2+2, ,LL m

∑=

+n

jj

rijij xda

1)( ≤ bi , for i = m2+1 , m2+2, ,LL m

xj ≥ 0 . j=1,2,3, ,LL n .

Minimize Zk6 = ∑=

+n

jj

kj

kj x)pc(

1

(4.7)

subject to same constraints of (4.6).

Minimize Zk7 = j

n

j

kj xc∑

=1

(4.8)

subject to ∑=

−n

jjijij xda

1

0 )( ≥ bi- 0ib , for i = 1,2,3, ,LL m1


256

∑=

+n

jjijij xda

1

0 )( ≤ bi + 0ib , for i = m1+1, m1+2, ,LL m2

∑=

−n

jj

lijij xda

1)( ≥ bi - l

ib , for i = m2+1 , m2+2, ,LL m

∑=

+n

jj

rijij xda

1)( ≤ bi+ r

ib , for i = m2+1 , m2+2, ,LL m

xj ≥ 0 . j =1,2,3, ,LL n .

Minimize Zk8 = ∑=

+n

jj

kj

kj x)pc(

1

(4.9)

subject to same constraints of (4.8). 4.1 Fuzzy Programming Technique for the Solution of MOLPP with Fuzzy Coefficients and Fuzzy Resources The MOFLPP can be considered as a vector minimum problem. Let Lk and Uk be the lower and upper bound for the k-th objective, where Lk = aspired level of achivement for the k-th objective function, and Uk = highest acceptable level of achivement for the k-th objective function. When the aspiration levels for each objectives have been specified, we formed a fuzzy model. Our next step is to transform the fuzzy model into a crisp model The foregoing steps may be presented as follows: Step-1. Solve the MOLPPs (4.2), (4.3), (4.4), (4.5), (4.6), (4.7), (4.8) and (4.9) for

each kth objectives ( k = 1, 2, 3, …., K). Step-2. From the results of step-1, determine the corresponding value for every

objective function at each solution. Step-3. Find upper and lower bounds (i.e Uk and Lk ) for kth objective from the 8k

objective values derived in step-2, as follows: Lk = min {Zk(xrs*) } k= 1, 2, 3, ,LL K 1≤ r ≤ K 1≤ s ≤ 8 Uk = max {Zk(xrs*) } k= 1, 2, 3, ,LL K 1≤ r ≤ K 1≤ s ≤ 8 Step-4. The initial fuzzy model becomes (in terms of aspiration levels with each

objectives)


257

Find { xj ; j = 1, 2, 3, ,LL n } (4.10) so as to satisfy Zk ≤~ Lk for k = 1, 2, 3, ,LL K.

∑=

n

jjij xa

1

≥~ bi i = 1,2,3, ,LL m1

∑=

n

jjij xa

1≤~ bi i = m1+1, m1+2, ,LL m2

∑=

n

jjij xa

1≅ bi i = m2+1 , m2+2, ,LL m

Here the membership functions for the fuzzy constraints of (4.10) are defined as: (for kth constraints kG~ (k = 1, 2, 3, ,LL K)

)U( kG~kµ = 0 , for Uk < ∑

=

n

jj

kj xc

1

= Mxp

xcU

n

jj

kj

n

jj

kjk

+

−

∑

∑

=

=

1

1 for ∑=

n

jj

kj xc

1 ≤ Uk ≤ ∑

=

+n

jj

kj

kj xpc

1)( + M

= 1 for Uk ≥ ∑=

+n

jj

kj

kj xpc

1)( + M

where M = Uk – Lk, k = 1, 2, 3, ,LL K.

(for the ith constraints i~C (i = 1,2,3, ,LL m1))

)b( iC~

i

µ = 1, for bi ≤ ∑=

−n

jjijij xda

1

0 )( - 0ib

= 0

1

0

1

i

n

jjij

n

jijij

bxd

bxa

+

−

∑

∑

=

= , for ∑=

−n

jjijij xda

1

0 )( - 0ib ≤ bi ≤ ∑

=

n

jjij xa

1


258

= 0, for bi > ∑=

n

jjij xa

1

(for the ith constraints i~C (i = m1+1, m1+2, ,LL m2 ))

)b( iC~

i

µ = 0, for bi < ∑=

n

jjij xa

1

= 0

1

0

1

i

n

jjij

n

jjiji

bxd

xab

+

−

∑

∑

=

= , for ∑=

n

jjij xa

1 ≤ bi ≤ ∑

=

+n

jjijij xda

1

0 )( + 0ib

= 1, for bi ≥ ∑=

+n

jjijij xda

1

0 )( + 0ib

(for the ith constraints i~C (i = m2+1 , m2+2, ,LL m ))

)b( iC~

i

µ = 0, for bi < ∑=

−n

jj

lijij xda

1)( - r

ib

= )b( iLCi

µ = ri

n

jj

lij

n

j

riij

lijiji

bxd

bxdab

+

+−−

∑

∑

=

=

1

1)(

, for ∑=

−n

jj

lijij xda

1)( - r

ib ≤ bi ≤ ∑=

n

jjij xa

1

= )b( iRCi

µ = ∑

∑

=

=

+

−++

n

j

lij

ri

n

ji

lij

rijij

bxd

bbxda

1

1)(

, for ∑=

n

jjij xa

1≤ bi ≤ ∑

=

+n

jj

rijij xda

1)( + l

ib

= 0, for bi > ∑=

+n

jjijij xda

1

0 )( + lib

Step-5. Using the max-min operator (as Zimmermann [ 15]) crisp LPP for (4.1) is

formulated as follows: Max λ (4.11)

subject to, ∑=

+n

jj

kj

kj xpc

1

0 )( λ +λ ( kUZ – k

LZ ) ≤ kUZ for k = 1, 2, 3, ,L K.

∑=

−n

jjijij xda

1

0 )( λ -λ 0ib ≥ bi , for i = 1,2,3, ,LL m1


259

∑=

+n

jjijij xda

1

0 )( λ +λ 0ib ≤ bi , for i = m1+1, m1+2, ,LL m2

∑=

−−n

jj

lijij xda

1))1(( λ + λ r

ib ≤ bi + rib , for i = m2+1, m2+2, ,LL m

∑=

−+n

jj

rijij xda

1))1(( λ - λ l

ib ≥ bi - lib , for i = m2+1, m2+2, ,LL m

0 ≤ λ ≤ 1, xj ≥ 0 j =1,2,3, ,LL n Note: The constraints in problem (4.11) containing cross product terms λ xj( j = 1, 2, 3, ,L n) which are not convex. Therefore the solution of this problem requires the special approach adopted for solving general non-convex application problems. It may be solved by fuzzy decisive set method [12]. 4.2 The algorithm of the fuzzy decisive set method ( Sakawa and Yano [ 12] ):

This method is based on the idea that for a fixed value of λ, the problem (4.11) are linear programming problem. Obtains the optimal solution λ* to the problem (4.11) is equivalent to determining the maximum value of λ so that the feasible set is non-empty. The algorithm to this method for the problem (4.11) is presented below. Algorithm:

Step1- Set λ =1 and test whether a feasible set satisfying the constraints of the problem (31) exist or not, using phase one of the Simplex method. If a feasible set exist, set λ =1. Otherwise, set λL=0 and λR =1 and go to the next step.

Step2- For the value of λ=2

RL λλ + up date the value of λL and λR using the bisection

method as follows: λL = λ if feasible set is non-empty for λ , λR = λ if feasible set is empty for λ .

Consequently for each λ , test whether a feasible set of the problem (4.11) exists or not exists , using phase one of the simplex method and determine the maximum value of λ* satisfying the constraints of the problem (4.11). Numerical Example 2: Minimize Z1 = 1

1~c x1+ 1

2~c x2 (2)


260

Minimize Z2 = 21

~c x1+ 22

~c x2 subject to 11

~a x1+ 12~a x2 ≥ 1

~b 21

~a x1+ 22~a x2 ≥ 2

~b x1 , x2 ≥ 0

where 11

~c =~5 = (5, 5, 6) ; 1

2~c =

~3 = (3, 3, 4.5) and 2

1~c =

~2 = (2,2, 4) ; 2

2~c =

~7 = (7, 7, 7.5)

respectively for objective coefficients.

11~a =

~2 = (1.5,2,2) ; 12

~a = ~4 = (3,4,4) 21

~a =~1 = (.5,1,1) ; 22

~a =~1 = (1,1,1)

respectively for technological coefficients.

and 1~b =

~20 = (18, 20, 20) ; 2

~b =~

10 = (9, 10, 10) respectively for constraint goals.

To solve the problem (2), we first solve the following sixteen sub-problems: Minimize Z11 = 5x1+3x2 , -------- (2.1) Minimize Z12 = 6x1+4.5x2 , -------- (2.2) subject to subject to same constraints of (2.1). 2x1+4x2 ≥ 20 , x1+x2 ≥ 10 , x1 , x2 ≥ 0 ; Minimize Z13 = 5x1+3x2 , -------- (2.3) Minimize Z14 = 6x1+4.5x2 -------- (2.4) subject to subject to same constraints of (2.3).

2x1+4x2 ≥ 18 , x1+x2 ≥ 9 ,

x1 , x2 ≥ 0 ;

Minimize Z15 = 5x1+3x2 , -------- (2.5) Minimize Z16 = 6x1+4.5x2 , -------- (2.6) subject to subject to same constraints of (2.5).

1.5x1+3x2 ≥ 20 , 0.5x1+1x2 ≥ 10 , x1 , x2 ≥ 0 ; Minimize Z17 = 5x1+3x2 , -------- (2.7) Minimize Z18 = 6x1+4.5x2 , -------- (2.8) subject to subject to same constraints of (2.7). 1.5x1+3x2 ≥ 18 , 0.5x1+1x2 ≥ 9 ,

x1 , x2 ≥ 0 ;


261

Minimize Z21 = 2x1+7x2 , -------- (2.9) Minimize Z22 = 4x1+7.5x2 , ------(2.10) subject to subject to same constraints of (2.9). 2x1+4x2 ≥ 20 , x1+x2 ≥ 10 , x1 , x2 ≥ 0 ; Minimize Z23 = 2x1+7x2 , -------- (2.11) Minimize Z24 = 4x1+7.5x2 -------- (2.12)

subject to subject to same constraints of (2.11). 2x1+4x2 ≥ 18 ,

x1+x2 ≥ 9 , x1 , x2 ≥ 0 ;

Minimize Z25 = 2x1+7x2 , -------- (2.13) Minimize Z26 = 4x1+7.5x2 , --------(2.14) subject to subject to same constraints of (2.13).

1.5x1+3x2 ≥ 20 , 0.5x1+1x2 ≥ 10 , x1 , x2 ≥ 0 ; and Minimize Z27 = 2x1+7x2 , -------- (2.15) Minimize Z28 = 4x1+7.5x2 , ------(2.16) subject to subject to same constraints of (2.15). 1.5x1+3x2 ≥ 18 , 0.5x1+1x2 ≥ 9 ,

x1 , x2 ≥ 0 ; so the optimal solutions of the sub-problems ((2.1) – (2.16)) are x11* = (x *11

1 , *112x ) = ( 0, 10 ) , Z11*( x11*) = 30,

x12* = (x *121 , *12

2x ) = ( 0, 10 ) , Z12*( x12*) = 45, x13* = (x *13

1 , *132x ) = ( 0, 9 ) , Z13*( x13*) = 27,

x14* = (x *141 , *14

2x ) = ( 0, 9 ) , Z14*( x14*) = 40.5 , x15* = (x *15

1 , *152x ) = ( 0, 10 ) , Z15*( x15*) = 30 ,

x16* = (x *161 , *16

2x ) = ( 0, 10 ) , Z16*( x16*) = 45 , x17* = (x *17

1 , *172x ) = ( 0, 9 ) , Z17*( x17*) = 27,

x18* = (x *181 , *18

2x ) = ( 0, 9 ) , Z18*( x18*) = 40.5, and x21* = (x *21

1 , *212x ) = ( 10, 0 ) , Z21*( x21*) = 20,

x22* = (x *221 , *22

2x ) = ( 10.0 ) , Z22*( x22*) = 40,


262

x23* = (x *231 , *23

2x ) = ( 9, 0 ) , Z23*( x23*) = 18, x24* = (x *24

1 , *242x ) = ( 9, 0 ) , Z24*( x24*) = 36,

x25* = (x *251 , *25

2x ) = ( 20, 0 ) , Z25*( x25*) = 40, x26* = (x *26

1 , *262x ) = ( 0, 10 ) , Z26*( x26*) = 75,

x27* = (x *271 , *27

2x ) = ( 18,0 ) , Z27*( x27*) = 36, x28* = (x *28

1 , *282x ) = ( 0, 9 ) , Z28*( x28*) = 67.5,

So, L1 = min {Z1(xrs*) }=27 , U1 = max {Z1(xrs*) } = 100 1≤ r ≤ 2 1≤ r ≤ 2 1≤ s ≤ 8 1≤ s ≤ 8 L2 = min {Z2(xrs*) } = 18 , U2 = max {Z2(xrs*) } = 70 1≤ r ≤ 2 1≤ r ≤ 2 1≤ s ≤ 8 1≤ s ≤ 8 According to step-4 formulating membership functions and following step-5, crisp LPP of (2) is formulated as follows:

Max λ , (2.17) (5+1λ)x1+(3+1.5λ)x2+73 λ ≤ 100 , (2+2λ)x1+(7+.5λ)x2+52 λ ≤ 70 , (2-.5λ)x1+(4-1λ)x2 -2λ ≥ 20 , (1-.5λ)x1+1x2 -1λ ≥ 10 ,

0 ≤ λ ≤ 1 , x1 ,x2 ≥ 0 ; Using gradient based non- linear programming package the optimal solutions be x *

1 =10.88221, x *2 =2.041447, Z1* =60.53539, Z2* =36.05455 and aspiration level

*λ =0.4539063 . The problem (2.17) may also be solved by the fuzzy decisive set method.

For λ =1 , the problem can be written as 6x1+4.5x2 ≤ 27 4x1+7.5x2 ≤ 18 1.5x1+3x2 ≥ 22 .5x1+x2 ≥ 11 x1, x2 ≥ 0 .


263

Since the feasible set is empty, by taking λL =0 & λR = 1, the new value of λ =2

10 +

=21 is tired.

For λ=21 =.5 , the problem (2.17) can be written as

5.5x1+3.75x2 ≤ 63.5 3x1+3.25x2 ≤ 44 1.75x1+3.5x2 ≥ 21 0.75x1+x2 ≥ 10.5 x1, x2 ≥ 0 .

Since the feasible set is empty, by taking λL =1 & λR =21 , the new value of λ

=2

210 +

=41 is tired. And so on.

The following values of λ are obtained in the next 25th iterations : λ = 0.25 ; λ = 0.375 ; λ = 0.4375 ; λ = 0.46875 ; λ = 0.453125 ; λ = 0.4609375 ; λ = 0.4570312 ; λ = 0.4550781 ; λ = 0.4541016 ; λ = 0.4536133 ; λ = 0.4538574 ; λ = 0.4539795 ; λ = 0.4539185 ; λ = 0.4538879 ; λ = 0.4539032 ; λ = 0.4539109 ; λ = 0.4539070 ; λ = 0.4539051 ; λ = 0.4539060 ; λ = 0.4539065 ; λ = 0.4539063 ; λ = 0.4539064 ; λ = 0.4539063 . Consequently, we obtain the optimal value λ* = 0.4539063 at the 25th iteration by using the fuzzy decisive set method and solutions of the problem (2) are x *

1 =10.88221 , x *2 = 2.041448 , Z1* =60.53539 , Z2* = 36.05455 and aspiration

level *λ =0.4539063 . 5. Application in Transportation Models The following model adopted from Bit, at al [3] is used to show that the above MOFLPP can be employed to solve the multi-objective transportation problems. Consider m origins ( or sources) Oi (i = 1, 2, ,LL m) and n destinations Dj (j = 1, 2, ,L n). At each origin Oi, let ai be the amount of homogeneous product which we want to transport to n destinations Dj to satify the demands for bj units of the product there. Let ek (k =1, 2, …., K) be the units of product which can be carried by K


264

different modes of transport called conveyance, such as trucks, air freight, freight train, ship, etc. A penalty p

ijkc associated with transportation of a unit of the product from source i to destination j by means of k-th conveyance for the p-th criterion. The penalty could represent transportation cost, deterioration amount, quantity goods delivered, under used capacity, etc. A variable xijk represents the unknown quantity to be transported from origin Oi to destination Dj by means of the k-th conveyance. A general Multi-Objective solid transportation model with mixed constraints, written as follows: Minimize Z =[ Z1,Z2, Z3, ,LL ZP ] (5.1) Subject to:

∑∑∈ ∈Jj

ijkKk

x ≥ , = , ≤ ai i∈I1, i∈I2 , i∈I3

∑∑∈ ∈Ii

ijkKk

x ≥ , = , ≤ bj j∈ J1 , j∈ J2 , j∈ J3

∑∑∈ ∈Ii

ijkJj

x ≥ , = , ≤ ek k∈ K1 , k∈ K2 , k∈ K3

xijk ≥ 0 , i ∈ I ,j∈ J , k∈ K . where Zp = ∑∑∑

∈ ∈ ∈Ii Jjijk

Kk

pijk xc , p = 1, 2, 3, ,LL P .

It is noted that the restriction on total delivery time is necessary for transportation of perishable goods, delivery of emergency supplies, etc. We are now adding an additional restriction to the above model that the total delivery time (∑∑∑

i jijkijk

k

xt )

is not more than T units. Here ijkt represents delivery time of unit item of transportation from i-th zone to j-th zone by means of k-th conveyance for the p-th criterion. In the above model, penalties, supply and demand amount, etc. are assumed to be fixed in value. In general, transportation penalties, delivery time, demand and supply amount are somewhat uncertain (non-stochastic) imprecise and vague in nature. So in real life situation, to depict this nature, all the parameters in the above model may be taken as fuzzy numbers. Then the above model (5.1) in fuzzy environment may be rewritten as Minimize Z =[ Z1,Z2, Z3, ,LL ZP ] (5.2) Subject to:

∑∑∑i j

ijkijkk

xt~ ≤ T~ i ∈ I , j ∈ J , k∈ K

∑∑∈ ∈Jj

ijkKk

x ≥ , = , ≤ ia~ i∈I1, i∈I2 , i∈I3


265

∑∑∈ ∈Ii

ijkKk

x ≥ , = , ≤ jb~ j∈ J1 , j∈ J2 , j∈ J3

∑∑∈ ∈Ii

ijkJj

x ≥ , = , ≤ ke~ k∈ K1 , k∈ K2 , k∈ K3

xijk ≥ 0 , i ∈ I ,j∈ J , k∈ K. where Zp(x) = ∑∑∑

∈ ∈ ∈Ii Jjijk

Kk

pijk xc~ .

It is a MOFLPP. It can be solved as before. Numerical Example3: 5.1 Fuzzy Multi-Objective Solid Transportation Model with Restriction on Total delivery Time Minimize Z1 = ∑∑∑

∈ ∈ ∈Ii Jjijk

Kkijk xc 1~ , (3)

Minimize Z2 = ∑∑∑∈ ∈ ∈Ii Jj

ijkKk

ijk xc 2~ ,

Minimize Z3 = ∑∑∑∈ ∈ ∈Ii Jj

ijkKk

ijk xc 3~ ,

Subject to:

∑∑∑i j

ijkijkk

xt~ ≤ T~ i ∈ I , j ∈ J , k∈ K

∑∑= =

3

1

3

11

j kjkx = 1

~a , ∑∑= =

3

1

3

12

j kjkx ≥ 1

~a , ∑∑= =

3

1

3

13

j kjkx ≤ 1

~a ,

∑∑= =

3

1

3

12

k ikix = 1

~b , ∑∑= =

3

1

3

12

k ikix ≥ 1

~b , ∑∑= =

3

1

3

12

k ikix ≤ 1

~b ,

∑∑= =

3

1

3

13

i jijx = 1

~e , ∑∑= =

3

1

3

13

i jijx ≥ 1

~e ,∑∑= =

3

1

3

13

i jijx ≤ 1

~e

xijk ≥ 0 , for i ∈ 1, 2, 3; j∈ 1, 2, 3; k∈ 1, 2, 3 where three penalties and delivery time are given as follows:

1C~ = (9,9,10) (12,12,12.5) (9,9,10) (6,6,7) (9,9,9) (7,7,7.5) (3,3,5) (7,7,7) (7,7,8) (5,5,6) (6,6,6) (5,5,7) (9,9,9.5) (11,11,11) (3,3,5) (6,6,7) (8,8,8) (6,6,7) (2,2,3) (2,2,4) (1,1,2) (2,2,2) (7,7,8) (7,7,7.5) (1,1,2) (9,9,9.5) (3,3,4)


266

2C~ =

(2,2,4) (9,9,9) (8,8,10) (1,1,2) (4,4,5) (1,1,2) (9,9,9) (9,9,9.5) (5,5,6) (2,2,3) (8,8,9) (1,1,3) (4,4,4.5) (5,5,6) (2,2,3) (8,8,8) (6,6,7) (9,9,9) (5,5,7) (2,2,3) (7,7,8) (8,8,10) (9,9,9) (7,7,8) (5,5,6) (2,2,4) (5,5,6)

3C~ = (2,2,3) (4,4,4) (6,6,7) (3,3,4) (6,6,7) (4,4,4) (8,8,8) (4,4,5) (9,9,10) (2,2,3) (5,5,6) (3,3,5) (5,5,5.5) (6,6,7) (6,6,6) (9,9,10) (6,6,7) (3,3,5) (1,1,3) (9,9,9) (1,1,2) (8,8,8) (3,3,5) (9,9,10) (5,5,5) (7,7,8) (11,11,11) t~ = (4,4,4) (5,5,6) (7,7,7) (5,5,5) (7,7,8) (5,5,5) (8,8,9) (5,5,5) (9,9,10) (3,3,4) (6,6,6) (4,4,4.5) (5.5,5.5,6) (7,7,8) (6,6,6) (10,10,10.5) (8,8,8) (5,5,6) (2,2,3) (9,9,9) (2,2,2.5) (9,9,9) (5,5,6) (9,9,9) (5,5,6) (9.5,9.6,10) (11,11,11) [Here =1

111c (9,9,10), =1112c (12,12,12.5), =1

113c (9,9,10) and similar representation for other elements.] where

T~ = 28~ =(82,82,85), 1~a = 8~ =(7,8,10), 2

~a = 9~ =(7.5,9,9), 3~a = 5~ =(5,5,8),

1~b = 7~ =(5,7,8), 2

~b = 6~ =(4,6,6), 3~b = 5~ =(5,5,6), 1

~e = 0~1 =(9.5,10,12), 2~e = 5~ =(4,5,5),

3~e = 6~ =(6,6,7), respectively for constraint goals.

The optimal solution of the above fuzzy transportation problem is *x121 = 4.805142, *x122 = 0.6301453, *x132 =1.872076, *x211= 5.556556, *x212 = 2.136081,

*x222 = 0.08009255, *x223 =1.099346, *x232 = 0.5889684,


267

Z1*(x*) = 97.09676, Z2*(x*) = 58.50913, Z3*(x*) = 58.08856 with aspiration level *λ =0.3073629.

6. Conclusion In this paper, we have proposed two types of FLPP. One is FLPP with fuzzy number resources and another is a FLPP with fuzzy number coefficients and resources. A procedure is developed for solving said FLPPs. It is also applied to a fuzzy solid transportation problem. This procedure may be very helpful for any fuzzy multi criteria decision making problem.

References [1] A. K. Bit, M. P. Biswal and S. S. Alam, Fuzzy programming approach to

multiobjective solid transportation problem, Fuzzy Sets and Systems 57 (1993) 183-194.

[2] A. K. Bit, M. P. Biswal and S. S. Alam, Fuzzy programming approach to multicriteria decision making transportation problem, Fuzzy sets and systems 50 (1992), 135 – 141.

[3] D. Chanas, Fuzzy programming in multiobjective linear programming- a parametric approach, Fuzzy Set and system 29 (1989) 303-313

[4] D. Klingman, and R. Russell, The transportation problems with mixed constraints. , Operational Research Quarterly 25 (1974) 447-455.

[5] G. M. Appa, The transportation problems with its variants, Operational Research Quarterly 24 (1973), 79-99.

[6] H. Tanaka, K. Asai, Fuzzy linear programming problems with fuzzy numbers. , Fuzzy Sets and Systems 13 (1984), 1-10.

[7] H. J. Zimmermann, Fuzzy programming and linear programming with several

objective functions., Fuzzy sets and System 1 (1978), 45- 55.


268

[8] Ishibuchi;Tanaka, Multiobjective programming in optimization of the interval objective function., European Journal of Operational Research 48 (1990), 219-225.

[9] M. Oheigeartaigh, A fuzzy transportation Algorithm, Fuzzy Sets and Systems

8 (1982), 235-243.

[10] M. Sakawa, and H. Yano, Interactive decision making for multi-objective linear fractional programming problems with fuzzy parameters. , Cybernetics Systems 16 (1985) 377-394.

[11] R. E. Bellman and L. A. Zadeh, Decision making in a fuzzy environment. Management Science 17 (1970), B141-B164.

[12] R. N. Gasimov and K. Yenilmez, Soving fuzzy linear programming with linear

membership funtions., Turk J Math 26 (2002), 375-396.

[13] S. K. Das, A. Goswami, and S. S. Alam, Multiobjective transportation problem with interval; cost, source and destination parameters., European Journal of Operational Research 117 (1999), 100-112.

[14] S. Tong, Interval number and fuzzy number linear programming. , Fuzzy Sets and Systems 66 (1994), 301-306.

[15] Y. J. Lai and C. L. Hawng, Fuzzy Mathematical Programming, Lecture notes in Economics and Mathematical systems., Springer-Verlag, (1992).

multi-objective fuzzy linear programming and its...

Documents