Solution of Fuzzy Multi-Objective Linear Programming Problems using Fuzzy Programming Techniques Based on Hyperbolic Membership Functions

Solution of Fuzzy Multi-Objective Linear Programming Problems using Fuzzy Programming Techniques Based

on Hyperbolic Membership Functions

Priyadarsini Rath and Rajani B. Dash Department of Mathematics,

Ravenshaw University, Cuttack, Odisha, INDIA.

email: [email protected], [email protected] (Received on: December 4, 2016)

ABSTRACT

In this paper a Fuzzy Multi Objective LPP is first reduced to crisp Multi Objective LPP using ranking function. The crisp Multi Objective LPP is then solved by Zimmerman Technique using hyperbolic membership functions. The comparison of the results obtained using this method with those using trapezoidal membership function method is presented.

AMS 2000 Subject Classification: 90C29,90C32.

Keywords: Multi Objective Linear Programming Problem, Fuzzy Multiobjective Linear Programming Problem, Hyperbolic Membership Function.

1. INTRODUCTION

Most of the real world problems are inherently characterized by multiple, conflicting and incommensurate aspect of evaluation. These area of evolution are generally operationalized by objective functions to be optimized in the framework of multiple objective linear programming models. Furthermore, when addressing real world problems, frequently the parameters are imprecise numerical quantities. Fuzzy quantities are very adequate for modeling these situations. Bellmann and Zadeh

introduced the concept of fuzzy quantities and also the concept of fuzzy decision making. The most common approach to solve fuzzy linear programming problem is to change them into corresponding deterministic linear program. Some methods based on comparison of fuzzy numbers have been suggested by H.R.

Maleki

, A. Ebrahimnejad, S.H.Nasser

, F. Roubens

. A. Munoz. Zimmermann

has

introduced fuzzy programming approach to solve crisp multi objective linear programming

problem. H.M. Nehi et al.,

used ranking function suggested by Delgodo et al.,

to solve fuzzy

MOLPP. Leberling

used a special type non-linear (hyperbolic) membership function for the vector maximum linear programming problem. Dhingra and Moskowitz

defined other type of non-linear (exponential, quadratic and logarithmic ) membership functions and applied them to an optimal design problem. Verma, Bishwal and Biswas

used the fuzzy programming technique with some non-linear (hyperbolic and exponential) membership function to solve a multi objective transportation problem. R.B. Dash and P.D.P Dash

introduced a method in which a fuzzy MOLLP is first reduced to crisp MOLLP using ranking function suggested by F. Roubens

. Then he solved crisp MOLPP using Zimmerman technique based on trapezoidal membership function.

In this paper, following R.B. Dash

we reduce Fuzzy MOLPP to crisp MOLPP using Roben’s Ranking function.Then we solve the crisp problem applying hyperbolic membership function and the results obtained using this method are compared with those obtained using Trapezoidal membership function method through a numerical study.

2. MULTI OBJECTIVE LINEAR PROGRAMMING

The problem to optimize multiple conflicting objective functions simultaneously under given constraints is called multi objective linear programming problem and can be formulated by following optimization problem.

1 2

Max f (x)  ( f (x), f (x)...f (x) )

s.t. x  X={x  R | g (x)

 0 , j 1, 2...m}...(2.1) 

1 2

where f (x), f (x)...f (x) are k-distincts

objective functions of the decision variable and X is the feasible set of constrained decisions.

Definition 2.1:

x* is said to be a complete optimize solution for (2.1) if there exist x* ϵ X s. t. f

(x*) ≥ f

(x), I = 1, 2, 3….k

For all x ϵ X

3. HYPERBOLIC MEMBERSHIP FUNCTION FOR FUZZY NUMBERS

A hyperbolic membership function is defined by

1 if Z

 L

H

(x) Z

 =

{((U L )/2) Z (x) } {((U L )/2) Z (x) } {((U L )/2) Z (x) } {((U L )/2) Z (x) }

1 1

2 2

p p p p p p p p

p p p p p p p p

e e

e e

 

 

    

    

 

 if L

 Z

 U

0 if Z

 U

Where 6

U L

p

p p

 

 P = 1, 2, 3…..p

If the membership function 

Z

(x) is piecewise linear, then it is referred as trapezoidal fuzzy number and is usually denoted by A  (a , a , a , a )

. If a

 a

then trapezoidal fuzzy number is turned into a triangular fuzzy number A  (a , a , a )

A fuzzy number A = (a, b, c) is said to be a triangular fuzzy number if its membership function is given by

x a b a



 a   x b



Z

(x) = 1 x  b x b

b c



 b   x c 0 Otherwise

Assume that R: F(R) → R. R is linear ordered function that maps each fuzzy number into the real number, in which F(R) denotes the whole fuzzy numbers. Accordingly for any two fuzzy numbers a and b we have.

𝑎̃ 

𝑏̃ 𝑖𝑓𝑓 𝑅(𝑎̃) ≥ 𝑅(𝑏̃) 𝑎̃ 

𝑏̃ 𝑖𝑓𝑓 𝑅(𝑎̃) ≥ 𝑅(𝑏̃) 𝑎̃

R

 𝑏̃ 𝑖𝑓𝑓 𝑅(𝑎̃) ≥ 𝑅(𝑏̃)

We restrict our attention to linear ranking function, that is a ranking function R such that (ka b) kR(a) R(b)

R    for any a and b in F (R) and any k  R . Ruben’s ranking function:

The ranking function suggested by F. Rubens is defined by

1

0

(a) 1 (inf a sup a )

R  2 



d 

This reduces to

1 2

(a) 1 (a a ( )) 2

R 

 

   For a trapezoidal number

(a

a, a , a , a

) a  

 

Solving Fuzzy multi objective Linear programming (FMOLP)

A fuzzy multi objective linear programming problem is defined as followed

Max Z

j

c x

  p=1, 2…….q

s.t.

1, 2...

j

a x  b i  m

 (3.2)

where x

 0

𝑎̃

and 𝑐̃

are in the above relation are in trapezoidal form as 𝑎̃

= (𝑎

, 𝑎

, 𝑎

, 𝑎

)

𝑐̃

= (𝑐

, 𝑐

, 𝑐

, 𝑐

) Definition 3.2:

x  X is said to be feasible solution to the FMOLP problem (3.2) if it satisfies constraints of (3.2).

Definition 3.3:

x

 X is said to be an optimal solution to this FMOLP problem (3.2) if there does not exist another x ϵ X such that 𝑧̃

(𝑥) ≥ 𝑧̃

(𝑥

) for all i =1, 2…q. Now the FMOLP can be transformed to a classic form of a MOLP by considering µ

as an exponential function. By implementing µ

on the above model, we can get the classical form of MOLP problem

𝑀𝑎𝑥 𝜇

(𝑧̃

) = ∑ 𝜇

(𝑐̃

)𝑥

p = 1, 2…q s.t. ∑ 𝜇

(𝑎̃

)𝑥

≤ 𝜇

(𝑏̃

) I = 1, 2…m

x

≥ 0 So we have

𝑀𝑎𝑥 𝑧

= ∑ 𝑐

𝑥

p=1, 2…q

s.t. ∑

𝑎

𝑥

≤ 𝑏

i=1, 2…m (3.3)

x

≥ 0

Where 𝑎

, 𝑏

, 𝑐

are real numbers corresponding to the fuzzy numbers 𝑎̃

, 𝑏̃

, 𝑐̃

with respect to the exponential function, respectively.

Lemma 3.4:

The optimum solution of (3.2) and (3.3) are equivalent.

Proof:

Let M

, M

be set of all feasible solutions of (3.2) and (3.3) respectively.

Then x ϵ M

if ∑

𝜇

(𝑎̃

)𝑥

≤ 𝜇

(𝑏̃

) i=1, 2…m By considering µ

as an exponential function we have

∑ 𝜇

(𝑎̃

)𝑥

≤ 𝜇

(𝑏̃

) i=1, 2…m

 ∑ 𝑎

𝑥

≤ 𝑏

Hence x ϵ M

Thus M

=M

Let’s x* ϵ X be the complete optimal solution of (3.2) The 𝑧̃

(𝑥

) ≥ 𝑧̃

(𝑥) for all x ϵ X

Where ‘X’ is a feasible set of solutions

 𝜇

(𝑧̃

(𝑥

)) ≥ 𝜇

(𝑧̃

(𝑥))

 𝜇

(∑ 𝑐̃

𝑥

) ≥ 𝜇

(∑ 𝑐̃

𝑥

)

 (∑ 𝜇

𝑐̃

𝑥

) ≥ (∑ 𝜇

𝑐̃

𝑥

) ∀ j=1, 2..q

 ∑ 𝑐

𝑥

≥ ∑ 𝑐

𝑥

∀ j=1, 2..q

 𝑧

(𝑥

) ≥ 𝑧

(𝑥) ∀ x

4. FUZZY PROGRAMMING TECHNIQUE To solve MOLLP

𝑀𝑎𝑥 𝑧

= ∑ 𝑐

𝑥

p=1, 2,…l

s.t. ∑

𝑎

𝑥

≤ 𝑏

i=1, 2,…n

We use fuzzy programming technique suggested by Zimmermann. The method is presented briefly in the following steps.

Step-1:

Solve the multi objective linear programming problem by considering one objective

at a time and ignoring all others. Repeat the process ‘q’ times for ‘q’ different objective functions.

Let X

, X

,….,X

be the ideal situation for respective functions.

Step-2:

Using all the above ideal q solutions in the step-1 construct a pay-off matrix of size q by q. Then from pay-off matrix find the lower bound(L

) and upper bound(U

) for the objective function

𝑧

as :𝐿

≤ 𝑧

≤ 𝑈

p=1, 2,…q

Step-3:

If we use the hyperbolic membership function as defined (3.1) then an equivalent crisp model for the fuzzy model can be formulated as follows:

Min λ

{((U L )/2) Z (x) } {((U L )/2) Z (x) } {((U L )/2) Z (x) } {((U L )/2) Z (x) }

1 1

2 2

p p p p p p p p

p p p p p p p p

e e

e e

 

 

 

^





p = 1, 2...q

∑ 𝑎

𝑥

≤ 𝑏

i=1, 2...m

λ ≥ 0 , x

≥0, j=1, 2…n

The above problem can be further simplified as:

Min x

Subject to

𝛼

𝑧

(𝑥) + 𝑥

≥ 𝛼

(𝑈

+ 𝐿

) 2 ⁄ p=1, 2…q

∑ 𝑎

𝑥

≤ 𝑏

i=1, 2...m

x

≥0, j=1, 2…n

𝑥

≥ 0

Where 𝑥

= tan ℎ

(2𝜆 − 1)

Step-4:

Solve crisp model to find the optimal compromise solutions. Evaluate the values of objective functions at the obtained compromise solution.

5. NUMERICAL EXAMPLE

𝑀𝑎𝑥 ∶ 𝑧̃

(𝑥) = 10 ̃𝑥

+ 11 ̃𝑥

+ 15 ̃𝑥

𝑀𝑎𝑥 ∶ 𝑧̃

(𝑥) = 5̃𝑥

+ 4̃𝑥

+ 9̃𝑥

S.t 1̃𝑥

+ 1̃𝑥

+ 1̃𝑥

≤ 15 ̃ 7̃𝑥

+ 5̃𝑥

+ 3̃𝑥

≤ 30 ̃ 3̃𝑥

+ 4̃𝑥

+ 10 ̃𝑥

≤ 100 ̃ 𝑥

, 𝑥

, 𝑥

≥ 0

Where

1̃ = (0.94, 1, 1.1) 1̃ = (0.98, 1, 1.06) 1̃ = (0.96, 1, 1.12) 7̃ = (6.2, 6.4, 7.4, 7.6) 5̃ = (4.3, 4.4, 5.2, 5.7) 3̃ = (2.2, 2.3, 3.3, 3.8) 3̃ = (2.3, 2.5, 3.3, 3.5) 4̃ = (3.2, 3.4, 4.2, 4.8) 10 ̃ = (9.3, 10, 10.3)

15 ̃ = (14.4, 14.6, 15.2, 15.6) 80 ̃ = (79.2, 79.3, 80.3, 80.8) 100 ̃ = (99.3, 99.4, 100.2, 100.7) 10 ̃ = (9.2, 9.4, 10.2, 10.4) 11 ̃ = (10.3, 10.6, 11.2, 11.5) 15 ̃ = (14.4, 14.5, 15.1, 15.6) 5̃ = (4.9, 5, 5.5)

4̃ = (3.2, 4, 4.4) 9̃ = (8.6, 9, 9.6)

Using ranking function suggested by Ruben

the problem reduces to 𝑚𝑎𝑥 ∶ 𝑧

(𝑥) = 𝑅(10 ̃)𝑥

+ 𝑅(11 ̃)𝑥

+ 𝑅(15 ̃)𝑥

𝑚𝑎𝑥 ∶ 𝑧

(𝑥) = 𝑅(5̃)𝑥

+ 𝑅(4̃)𝑥

+ 𝑅(9̃)𝑥

s.t

𝑅(1̃)𝑥

+ 𝑅(1̃)𝑥

+ 𝑅(1̃)𝑥

≤ 𝑅(15 ̃) 𝑅(7̃)𝑥

+ 𝑅(5)𝑥

+ 𝑅(3̃)𝑥

≤ 𝑅(80 ̃) 𝑅(3̃)𝑥

+ 𝑅(4̃)𝑥

+ 𝑅(10 ̃)𝑥

≤ 𝑅(100 ̃ ) 𝑥

, 𝑥

, 𝑥

≥ 0

⇒ 𝑚𝑎𝑥 ∶ 𝑧

(𝑥) = 9.8𝑥

+ 10.9𝑥

+ 14.9𝑥

(5.1) 𝑚𝑎𝑥 ∶ 𝑧

(𝑥) = 5.1𝑥

+ 3.9𝑥

+ 9.1𝑥

(5.2) s.t

1.01𝑥

+ 1.01𝑥

+ 1.02𝑥

≤ 14.95

6.9𝑥

+ 4.9𝑥

+ 2.9𝑥

≤ 79.9 (5.3)

2.9𝑥

+ 3.9𝑥

+ 9.9𝑥

≤ 99.9

𝑥

, 𝑥

, 𝑥

≥ 0

Solving (5.1) and (5.3) we get

𝑧

=

, 𝑧

=

Solving (5.2) and (5.3) we get

𝑧

=

2347

, 𝑧

=

7041

Function LB UB

𝑧

185.94 188.87

𝑧

94.24 107.77

If we use hyperbolic membership function then the equation crisp model can be formulated as

Min x

Subject to

𝛼

𝑧

(𝑥) + 𝑥

≥ 𝛼

(𝑈

+ 𝐿

) 2 ⁄ 𝛼

𝑧

(𝑥) + 𝑥

≥ 𝛼

(𝑈

+ 𝐿

) 2 ⁄

∑ 𝑎

𝑥

≤ 𝑏

1, 2... m

i 

1, 2... n j 

As per Step-4, let us solve the problem which is used by hyperbolic membership function and the problem reduced to

Min X

1836.569𝑥

+ 2042.7145𝑥

+ 2792.3345𝑥

− 𝑥

≥ 3 515.1255𝑥

+ 393.9195𝑥

+ 919.1455𝑥

− 𝑥

≥ 3

1.01𝑥

+ 1.01𝑥

+ 1.02𝑥

≤ 14.95 (5.4)

6.9𝑥

+ 4.9𝑥

+ 2.9𝑥

≤ 79.9 2.9𝑥

+ 3.9𝑥

+ 9.9𝑥

≤ 99.9 𝑥

, 𝑥

, 𝑥

≥ 0

Solving we get

X

= 0.0000

X

= 7.6576980568012 X

= 7.074240159442 X

= 184.38146668077

Now the optimal value of the objective functions of FMOLPP (5.4) becomes 𝑧

= 10 ̃𝑥

+ 11 ̃𝑥

+ 15 ̃𝑥

= ( 9.2, 9.4, 10.2, 10.4 )𝑥

+ (10.3, 10.6, 11.2, 11.5 )𝑥

+ ( 14.4, 14.5, 15.1, 15.6 )𝑥

= ( 180.7433483, 183.7480817, 192.5872446, 198.4216741)

𝑧

= 5̃𝑥

+ 4̃𝑥

+ 9̃𝑥

= (4.9, 5, 5.5 )𝑥

+ ( 3.2, 4, 4.4 )𝑥

+ ( 8.6, 9, 9.6 )𝑥

= ( 85.34309915, 94.29895366, 103.021425)

The membership functions corresponding to the fuzzy objective function are as follows.

0 x  180.7433483

180.7433483

3.0047334 x 

180.7433483   x 183.7480817

1

(x)

H



= 1 183.7480817   x 192.5872446

198.4216741 5.8344295

 x

192.5872446   x 198.4216741

0 x  198.4216741

0 x  85.34309915 _,

021425 .

 103 x

85.34309915 8.95585451 x 

85.34309915   x 94.29895366

2

(x)

H





103.021425 8.72247134

 x

94.29895366   x 103.021425

CONCLUSION

On comparison we see that the result obtained in this paper is very close to that obtained in

using trapezoidal membership function in Zimmerman’s algorithm. Thus one can use hyperbolic in place of trapezoidal membership function in the Zimmerman’s algorithm.

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