中 華 大 學 碩 士 論 文

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中 華 大 學 碩 士 論 文

題目:應用轉換特性求解多目標線性分數規劃問 題

Applying Transformation Characteristics in Solving Multiple Objective Linear

Fractional Programming Problem

系 所 別:經 營 管 理 研 究 所 學號姓名:M09119019 吳 信 毅 指導教授:裴 文 博 士

中華民國九十三年六月

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應用轉換特性求解多目標線性分數規劃問題

學生:吳信毅 指導教授:裴文博士

摘 要

近幾年來,於模糊多目標規劃問題範疇,已有許多專家學者發 表多種解題技巧。但是,其大多數皆以考慮線性目標方程式為主。

時至今日,分數規劃型態之目標函數卻與之漸增,僅考慮單純線性 規劃已顯不足。在當前所提出之求解多目標線性分數規劃方法論 中,多數又需透過繁複且多餘的計算過程才能順利得到一個良好的 妥協解供決策者參酌。

有鑑於此,本研究提出運用線性分數規劃型態問題之轉換特性 簡化其繁雜之求解過程。並於文末以兩個採自 Chakraborty and Gupta 兩位學者發表過之例題演算,導證本研究提出之求解方法確 實可行。

關鍵字:多目標、線性分數規劃、模糊集、模糊線性規劃、轉換特 性

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Applying Transformation Characteristics in Solving Multiple Objective Linear Fractional Programming

Problem

Student: Hsin-Yi Wu Advisor: Dr. Wen Pei

ABSTRACT

In the past few years, various types of formulations or solution methods have been proposed with fuzzy multiobjective programming problems, but most of them exclusively take linear functions as objective functions. However some goals that can be expressed with a ratio equation are available. Therefore, models that can handle such a fractional objective are preferable. Unfortunately, most of the present methodologies in solving multiple objective linear fractional programming problems are computationally burdensome. Accordingly we propose such a useful method to solve this kind of problems by using transformation characteristics in order to reduce the complexity in solution procedure.

Furthermore, two numerical examples adopted from Chakraborty and Gupta’s research are used to prove the methodology mentioned in this paper does surely work in solving multiple objective linear fractional programming problems.

Keywords: Multiple objective, Linear fractional programming, Fuzzy sets, Fuzzy linear programming, Transformation characteristic

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ACKNOWLEDGEMENTS

本研究承恩師裴文博士不辭辛勞的指導,始得以順利完成,在 他嚴謹的治學理念以及宏觀的態度下,使我獲益良多,且在做人處 事方面學習到不少獨特的見解,謹致上最崇高的謝意!並衷心感謝 口試委員黃祥熙博士與葉鳴朗博士於百忙之中,仍不吝提供寶貴意 見,使本論文更臻完善。感謝中華大學應用數學系各位老師對我數 學觀念的栽培,使我得以善用並完成此篇論文。

此外,感激曾明郎老師在我求學過程中給予之鼓勵、肯定與細 心指導。謝謝雄及佩珊與我一同成長,還要感謝摯友靖華,於論文 製作過程中所給予的精神鼓勵和支持,讓我在心情低落和遇到挫折 時得以順利度過。

感謝鼓勵並支持我完成學業的家人,以及所有關心我的朋友。

最後僅以此論文獻給我的父親─吳火丁先生,以及母親─李宜芳女 士。

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CONTENTS

CHINESE ABSTRACT... i

ENGLISH ABSTRACT... ii

ACKNOWLEDGEMENTS...iii

CONTENTS ... iv

Chapter 1. INTRODUCTION... 1

Chapter 2. LITERATURE REVIEW ... 4

Chapter 3. THE CONCEPTS OF FLP AND THE TRANSFORMATION CHARACTERISTICS OF MOLFPP .... 8

3.1 Fuzzy Linear Programming ...8

3.2 Linear Fractional Programming...9

3.3 Multiple Objective Linear Fractional Programming Problem...12

Chapter 4. SOLUTION PROCEDURE AND NUMERICAL EXAMPLES ... 17

4.1 Solution Procedure...17

4.2 Numerical Examples ...18

Chapter 5. CONCLUSIONS... 25

REFERENCES ... 27

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Chapter 1 INTRODUCTION

Management programming problems are based upon many estimated values. Therefore lots of uncertain factors are included in.

In fact, a variety of these problems that have been solved by linear programming (LP) techniques are realized to be more complicated.

Frequently these problems have multiple objectives to be optimized rather than a single objective. To face such a situation, fuzzy multi-objective programming has been developed.

In multi-objective analysis, some objectives can be expressed with a ratio equation such as return on investment, ratio of operating profit to net-sales, etc. Thus multiple objective fractional programming (MOFP) models that can handle such a fractional objective are preferable. If both the numerator and the denominator of these ratios are linear functions and some technical linear restrictions are to be satisfied, then it is called a multiple objective linear fractional programming problem (MOLFPP).

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After Charnes and Copper [7] have shown that a linear fractional programming problem can be optimized by reducing it to two linear programs, there were several methodologies developed to solve MOLFPP. Though the notions in solving MOLFPP seemed to expand rapidly, most of these methodologies are computationally burdensome [10]. The application of fuzzy set theory is to overcome the difficulty.

Chakraborty and Gupta [5] have proposed a novel methodology for reducing the complexity in solving the MOLFPP. But it looks like to make some mistake in practicing numerical examples. We may mention here that the example taken from [5] could be solved by using the transformation model evolved from its own characteristics.

We now give a concise description of the forthright concept in solving the MOLFPP. If the readers have found the original MOLFPP that can not be solved by using the reducing procedure proposed by Cooper [7] directly.

The paper is organized as follows. We review linear fractional programming (LFP) literature in Section 2. In Section 3, we concisely describe relevant LFP characteristic and the complete method proposed

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by us. The numerical example is represented to support the model in section 4. Finally, conclusions are given in section 5.

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Chapter 2

LITERATURE REVIEW

During the mid-1960s and early 1970s of the last century, fractional programming (FP) was studied extensively. In contrast to the single objective FP, multi-objective fractional programming (MOFP) has not been discussed that extensively and only a few approaches have appeared in the literature. The multiplicity of objectives added to their fractional structures exacerbates the difficulties obstructing the successful application of mathematical programming techniques. It has pointed out that in most of MOFP approaches, the problems are converted into single objective FP problems and then solved employing the method of Charnes and Cooper or Bitran and Novaes [24].

Kornbluth and Steuer [18] have presented an algorithm for solving the MOLFPP by combining aspects of multiple objective linear fractional programming, single objective fractional programming and goal programming. The method enhances the possible application of multiple objective programming to a wider variety of problems.

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Nykowski et al. [22] have proposed a compromise procedure for the MOLFPP. Both two methodologies mentioned are computationally burdensome.

After Zadeh introduced the concept of fuzzy set theory, Zimmermann [28、29] first applied fuzzy set theory concept with suitable choices of membership functions and derived a fuzzy linear program which is identical to the maximum program, and he showed that solutions obtained by fuzzy linear programming (FLP) which are always efficient solutions and also gives an optimal compromise solution. Luhandjula [19] solved MOLFPP by applying fuzzy approach to overcome the computational difficulties of using conventional fractional programming (FP) approaches to solve multiple objective fractional programming problem (MOFPP). Dutta et al. [10]

reconsidered the problem of Luhandjula [19] and pointed out some fallacies. One drawback of this approach in [19] is that the aggregation of membership functions is done with a compensatory operator which does not guarantee the efficiency of the optimal solution. Dutta et al [10] modified the linguistic approach of Luhandjula [19] by constructing

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the desirable membership functions which combines the linguistic aspirations and also taking the view that the objective ratios should be close to the maximum value ( maximum value of the numerator/minimum value of denominator). They proposed a “Simple Additive Weighting” (SAW) model of MOLFPP. The decision makers can use the approach to put relative importance among the proximities by giving weights to the membership functions.

Chakraborty and Gupta [5] proposed a different methodology for solving MOLFPP. The approach mentioned that suitable transformation should have been applied to formulate an equivalent multi objective linear programming problem (MOLPP) and the resulting MOLPP should has been solved based on fuzzy set theoretic approach.

The equivalency has also been established and hence, the feasibility region is not affected.

The notional convention of two sets, I and Ic, and the concept of

Zi were utilized to develop the focal model in [5]. Though the nature of objectives might be not known explicitly, the way to classify its set is also defined in the proposed methodology. The steps were taken as

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follows:

1. Maximize each objective function Zi(x) subject to the given set of constraints using the method proposed in [7]. Let Zi* be the maximum value of Zi(x) for i=1,2,...,k .

2. Examine the nature of Zi* for all values of i=1,2,...,k. If Zi* 0, then iI, and if Zi* <0, then iIc.

Now the index set I and Ic are known completely.

3. If iI , then we may assume the maximum aspiration level is

* i Zi

Z = for iI and if iIc, then we may assume the maximum aspiration level Zi =1/Zi*.

By employ of the proposed methodology in solving MOLFPP, it is said to be always yielding an efficient solution, reducing the complexity, and much easier in computing procedure. But, there seems to be some absurdity occurred in the illustrated numerical examples in [5].

As a result of the situation, this paper offers another way to consider the examples adopted from [5] by using the transformation characteristics of MOLFPP.

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Chapter 3

THE CONCEPTS OF FLP AND

THE TRANSFORMATION CHARACTERISTICS OF MOLFPP

3.1 Fuzzy Linear Programming

Fuzzy linear programming is fuzzy set theory applied to linear multicriteria decision making problems. The MOLPP can be considered as a vector optimizing problem (for convenience, take minimizing problem for example). The first step is to assign two values Uk and Lk as upper and lower bounds for each objective function Zk:

Uk = Highest acceptable level of achievement for objective k Lk = Aspired level of achievement for objective k

Let

dk = Uk Lk = the degradation allowance for objective k.

Once the aspiration levels and degradation for each objective have been specified, the fuzzy model formed. The next step takes An element X that has a degree of membership in the k-th objective, denoted by a membership function μk( X), to transform the fuzzy model

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into a crisp single objective linear programming model of λ. The range of the membership function is [0,1].

<

<

=

. 0

, 1

, 1

) (

k k

k k k k

k k k

k k

k

U Z if

U Z L L if

U L Z

L Z if μ X

In the case of the multiobjective linear problems, the fuzzy linear programming technique always gives an optimal compromise solution.

This approach is similar, in many respects, to the weighted linear goal programming method. Weighted linear goal programming depends on the development of weights, whereas fuzzy programming uses fuzzy membership functions.

3.2 Linear Fractional Programming

The general format of a classical linear fractional programming problem (LFPP) [7] can be stated as

Max

β α + + x d

x c

T T

subject to

=

=

X x Rn Ax b x b Rm

x , 0, , (1)

where c,dRn; α,βR, X is nonempty and bounded.

The general format of LFPP seems not that easy to be solved.

Therefore, some coordinator has to be utilized to translate.

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Theorem 1 (Equivalent of LFP and LP) If dTx*+β =γ >0 for x*, an optimal solution of LFP (1), and

(

y*, t*

)

is an optimal solution of the following linear programming (LP):

Max cTy+αt, s.t. dTy+ tβ 1

0

− bt

Ay ,

y,t 0, yRn, tR. (2) Then *

*

t

y is also an optimal solution of LFP (1).

Proof. Suppose the theorem was false, i.e. assume that there exists an optimal xX , such that

β α β

α

+

> + + +

) (

) (

T

T

T

T

t d y

t c y

x d

x c

Let dTx +β =θγ , for some θ >0 . Consider yˆ=θ1x , tˆ=θ1 t

x yˆ= *ˆ

.

Then θ1(dTx +β)=dTyˆ+tˆβ =γ and (yˆ,tˆ) also satisfies Ayˆ− tbˆ0 ,

, tˆ0. But

( )

(

β

)

γ α

θ

α θ

β

α c y t

x d

x c x

d x LHS c

T T

T T

T ˆ ˆ

* 1

* 1

*

* +

+ =

= + +

= +

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γ α β

α β

α T

T

T

T

T

t y c t y d

t y c

t d y

t c y

RHS +

+ =

= + +

= +

) (

) (

Now

β α β

α

+

> + + +

) (

) (

T

T

T

T

t d y

t c y

x d

x

c +α > ( )+α

T

T

t c y x

c .

Contradicted to (y,t) is the optimal solution for (1). Thus, the equivalence is proved. Q.E.D.

Since the maximizing LFPP that had been shown above to be equivalent to linear programming problem is practicable, it is clearly to define minimizing LFPP as the following equivalent LP problem:

Min cTy+αt, s.t. dTy+ tβ 1

0

− bt

Ay ,

y,t 0, yRn, tR.

Though the transformed skill is useful to help most decision makers (DMs) in solving LFPP, but when solving the about LFPP, if

=0

t there is no way to find xi (

t

xi = yi ). Thus the original LFPP can

not be solved. In this paper, we proposed to use the basic transformed characteristic of the original objective to solve the problem. As we

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know about the property of LP, with the same constraints, it can be denoted that

Δ

x

Max F(x)

Δ

x

Min ( ) 1

x F .

Following this postulate, Chakraborty and Gupta perform the transformation of the LFPP as follows:

) (

) ( )

( ) ( )

( ) (

x N

x Max D x

D x Min N x

D x MaxN

x x

x

Δ

Δ

Δ

(3)

We propose the following identical transformation derived from (3) in this paper,

) (

) ( )

( ) ( )

( ) (

x N

x Min D x

N x MinD x

D x MaxN

x x

x

Δ Δ

Δ

.

With these identical equations, we can transform the hardly solved LFPP into equivalent problem to overcome the situation that t =0.

3.3 Multiple Objective Linear Fractional Programming Problem

The general format of maximizing MOLFPP can be written as

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Max

+

= + +

= + +

= +

=

i T k

i T k k

i T

i T

i T

i T

x d

x x c

Z

x d

x x c

Z

x d

x x c

Z

x Z

β α β α β α

) (

) (

) (

) (

2 2 2

1 1 1

subject to

=

=

X x Rn Ax b x b Rm

x , 0, , (4)

where ci,diRn ; αi,βiR , i=1,2,...k , k 2 , X is nonempty and bounded.

Similarly, minimum problem can also be defined as

Min Z(x)=[Z1(x),Z2(x),...,Zk(x)]

s.t.

=

=

X x Rn Ax b x b Rm

x , 0, , (5)

where ci,diRn ; αi,βiR , i=1,2,...k , k 2 , X is nonempty and bounded,

with

) (

) ) (

( D x

x N x

d x x c

Z

i i i T i

i T i

i =

+

= +

β

α .

By using the general idea of theorem 1 in handling the minimum MOLFPP, we can reduce the general format as the following equivalent

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MOLPP:

Min cTi y+αit, s.t. diTy+ tβi =γ

0

− tb y

Ai i ,

y,t 0, i=1,2,...k, k 2.

Chakraborty and Gupta [5] utilize the above result to develop a MOLFPP model. The membership functions for Ni(x) and Di(x) are as followed:

If iI, then )) ( (tNi y t μi

<

<

=

i i i i i

i

i

Z t y tN f i

Z t y tN Z if

t y tN

t y tN if

) ( 1

) ( 0 0

0 ) (

0 ) ( 0

If iIc, then )) ( (tDi y t μi

<

<

=

i i i i i

i

i

Z t y tD f i

Z t y tD Z if

t y tD

t y tD if

) ( 1

) ( 0 0

0 ) (

0 ) ( 0

and then using Zimmermann’s min operator to transform the

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equivalent MOLPP into the crisp model as

Max λ

s.t. μi(tNi(y t))λ for iI, μi(tDi(y t))λ for iIc, tDi(y t)1 for iI, tNi(y t)1 for iIc, A(y t)− b0,

t>0, y0, i=1,2,...k, k 2.

In the proposed methodology, I is a set such that I ={i:Ni(x)0 for some xΔ} and Ic ={i:Ni(x)<0 for each xΔ} where

} ,..., 2 , 1

{ k

I

IΥ c = . And the knack of computing Zi , the maximum aspiration levels, is proceeded as “if iI, then it may assume the maximum aspiration level is Zi =Zi*, and if iIc, then Zi =1 Zi*.”

Though the methodology has been yielding an efficient solution all the times, reducing the complexity, and much easier in computing procedure.

It is not that easy and correct in applying to solve MOLFPP.

The method proposed in this paper suggests that if the follow-up researchers meet the analogous problems with the examples in [5],

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which get the same result, t =0, that cannot easily be solved by directly reducing into MOLPP, they can try to use the transformed characteristics mentioned above to simplify the MOLFPP into MOLPP.

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Chapter 4

SOLUTION PROCEDURE AND NUMERICAL EXAMPLES

In this section we will introduce the whole procedure of using the transformed skill. The MOLFPP examples adopted from Chakraborty and Gupta [5] will be used to show if the solution procedure we had introduced before does surely work if DMs get met the problem obtains

=0

t as its solution.

4.1 Solution Procedure

Since t =0 shows up the correct solution could not be found by directly reducing MOLFPP into MOLPP sometimes, we should utilize the transformation characteristics well to form the easily solved MOLFPP from the original problems. Inasmuch as the situation, we have developed the following procedure to check the feasibility of reducing the original MOLFPP and what should we do if me meet t =0

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Step 1. Check if the original MOLFPP can be solved by directly

reducing into MOLPP [7].

Step 2. If it gets the result of t=0 , try to use the helpful transformation form in translating the original MOLFPP into the opposite optimizing problem.

Step 3. After reducing the transformed MOLFPP into MOLPP, use

Zimmermann’s min operator to transform the equivalent MOLPP into the crisp model.

4.2 Numerical Examples

Example 1. Let’s consider a MOLFPP with two objectives as follows:

Max

⎟⎟

⎜⎜

+ +

= +

+ +

+

=

=

1 2 5 ) 7 (

3 , 2 ) 3

( )

(

2 1

2 1 2

2 1

2 1 1

x x

x x x

Z

x x

x x x

Z x Z

s.t. x1− x2 1,

15 3

2x1+ x2 ,

1 3 x ,

0

xi , i=1,2.

First, we convert the above MOLFPP to the equivalent MOLPP as

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Max ⎟⎟

⎜⎜

+

=

+

=

) 7

) , (

) 2 3 ) , (

2 1 2

2 1 1

y y t y f

y y t

y f

s.t. y1 +y2 +3t1,

1 2

5y1+ y2 +t ,

2 0

1y t

y ,

0 15 3

2y1 y2 t ,

0

1− t3

y ,

yi, t0, i=1,2.

After solving the maximum problem by considering one objective each time, we’ll get y1 =0, y2 =0, t =0 for f1(y,t), and y1 =0.194805,

2 =0

y , t =0.025974 for f2(y,t). Thus

) 584415 .

0 , 0 ( ) ,

(U1 L1 = ,

) 0 , 3636 . 1 ( ) , (U2 L2 =

With these values and Zimmerman’s approach, the above MOLPP could be more easily solved after being transformed as a linear programming problem. And the solution of the LP problem is obtained as λ =0.7741981, y1 =0.131962, y2 =0.131962, and t =0. Since we get t =0, it results in xi cannot be calculated by

t xi = yi .

Now, let’s translate the original problem into the following

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minimizing MOLFPP, and see if the approach mentioned in this paper before does work:

Min

⎟⎟

⎜⎜

+ +

= +

+

+

= +

=

2 1

2 1 2

2 1

2 1 1

7

1 2 ) 5

(

2 , 3 ) 3 ( )

(

x x

x x x

Z

x x

x x x

Z x Z

⎟⎟

⎜⎜

+ +

= +

=

=

2 1

2 1 2

2 1

2 1 1

7

1 2 ) 5

(

2 , 3 ) 3 (

x x

x x x

Z

x x

x x x

Z

s.t. x1− x2 1,

15 3

2x1+ x2 ,

1 3 x ,

0

xi , i=1,2.

The above MOLFPP problem will be equivalent to the following MOLPP:

Min ⎟⎟

⎜⎜

+ +

=

=

t y y t y f

t y y t y f

2 1 2

2 1 1

2 5 ) , (

3 )

, (

s.t. 3y1 − y2 2 1,

1 7y1+ y2 ,

2 0

1y t

y ,

0 15 3

2y1+ y2 t ,

0

1− t3

y ,

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yi, i=1,2, t 0.

Solve the minimum problem only one objective each time to find the values Uk and Lk for every objective. The result value will be

) 3478 . 0 , 0 ( ) ,

(U1 L1 = and (U2,L2)=(0.8696,0). Applying Zimmerman’s approach to reduce the above MOLP problem as a crisp model (LP), then the solution obtained by using standard LP packages: λ =0.49998,

065215 .

1 =0

y , y2 =0.043477, and t=0.021738. So, the solution of the original problem is: x1 3, x2 2,

8 5

1

=

Z ,

20 23

2 =

Z .

Example 2. Let’s consider an another MOLFPP with three objectives as follows:

Max

⎟⎟

⎜⎜

+ +

= +

+ +

= +

+ +

+

=

=

2 3 2 ) 4 (

1 2 5 ) 7 (

3 , 2 ) 3

( )

(

2 1

2 1 3

2 1

2 1 2

2 1

2 1 1

x x

x x x

Z

x x

x x x

Z

x x

x x x

Z

x Z

s.t. x1− x2 1,

15 3

2x1+ x2 ,

9 9 2

1 + x

x ,

1 3 x ,

0

xi , i=1,2.

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After checking the result of directly reducing the original MOLFPP, we find that it meets the same situation as example 1, t=0. The procedure of using the methodology proposed by this paper is showed as follows.

1. Translate the original maximizing MOLFPP into the equivalent minimizing MOLFPP.

Min

⎟⎟

⎜⎜

+ +

= +

+ +

= +

=

=

2 1

2 1 3

2 1

2 1 2

2 1

2 1 1

4 2 3 ) 2

(

7

1 2 ) 5

(

2 , 3 ) 3 ( )

(

x x

x x x

Z

x x

x x x

Z

x x

x x x

Z

x Z

s.t. x1− x2 1,

15 3

2x1+ x2 ,

9 9 2

1 + x

x ,

1 3 x ,

0

xi , i=1,2.

2. Reduce the transformed minimizing MOLFPP into the following MOLPP by using the skill of

t x= . y

Min

+ +

=

+ +

=

=

t y y t y f

t y y t y f

t y y t y f

2 3 2 ) , (

2 5 ) , (

3 )

, (

2 1 3

2 1 2

2 1 1

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s.t. 3y1 − y2 2 1,

1 7y1+ y2 ,

1 4 2

1+ y

y ,

2 0

1y t

y ,

0 15 3

2y1+ y2 t ,

0 9 9 2

1+ y t

y ,

0

1− t3 y

0

1− t3

y ,

yi, i=1,2, t 0.

3. Similarly, to solve the minimum problem only one objective each time to find the values Uk and Lk for every objective. And the result value will be (U1,L1)=(0,0.32) , (U2,L2)=(0.8,0) , and

) 0 , 56 . 0 ( ) ,

(U3 L3 = .

4. Applying Zimmerman’s approach to reduce the above MOLP problem as a crisp model (LP)

Min λ

s.t. y1y2 3t+0.320.32λ, λ

8 . 0 2

5y1+ y2 +t ,

Figure

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