multiple objective decision making in past, present, and future

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大葉學報第十二卷第二期民國十二年

Jour nal of Da-Yeh University, Vol. 12, No. 2, pp. 1-8 (2003)

1

Multiple-Objective Decision-Making in the Past,

Present and Future

GWO-HSHIUNG TZENG

Institute of Management Technology and Environmental Research Group, National Chiao Tung University

1001 Ta-Hsueh Rd., Hsinchu, Taiwan

ABSTRACT

Since Kuhn and Tucker [14] originally proposed the concept of a proper noninferior solution for

solving nonlinear programming problems, which was later modified by Geoffrion [8], Yu [35] further

introduced a compromise solution method to cope with multicriteria decision-making problems. In

addition, Charnes [4] presented a goal-programming method, and Bellman and Zadeh [2] proposedthe concepts of decision-making in a fuzzy environment. Thus, many distinguished works have

guided research in this field. This study reviews some methods concerning basic mathematical

concepts of models applied to multiple-objective decision-making problems, including fuzzy

multiobjective linear programming (FMOLP), fuzzy goal programming (FGP), a two-phase method,

an achievement function, data envelopment analysis (DEA), and De Novo Programming. According

to the past and current developments in multiobjective programming, researchers are able to

determine what they are interested in for future multicriteria decision-making problems.

Key Words: fuzzy, multi-objective, data envelopment analysis (DEA), De-Novo programming

曾國雄

1001

摘要

自從Kuhn

及Tucker

在1951

年提出向量最適化及非劣解的觀念以來，歷經Geoffrion

及游

伯龍等學者的補強，使得多目標決策理論如今得以發揚光大。此外， Charnes 在 1955 年提出的

目標規劃，及 Bellman 與 Zadeh 在 1970 年提出的模糊決策概念，亦使得多目標決策領域吸引許

多學者投入，研究成果也更為活潑及多樣化。本研究回顧前人心血，包括一些具指標性意義的

研究，如模糊多目標規劃，模糊目標規劃，資料包絡分析及 De-Novo 規劃等。冀由回顧多目標

決策的過去及了解多目標決策的現在發展，以利有心研究者發覺更多的新課題及未來研究方

向。

關鍵詞模糊，多目標，資料包絡分析， De-Novo 規劃



Journal of Da-Yeh University, Vol. 12, No. 2, 2003

2

Fuzzy Sets

(Zadeh,1965)

DM in fuzzy environment

(Bellman & Zadeh, 1970)

Grey Theory

(Deng,1982)

Fuzzy Combinatorial MODM with GA

(Sakawa, 1994)

Evolutionary Computation

GA in search, Opt. and Machine Learning

(Goldberg, D.E. 1989)

GA + Data Structure = Evolutionary Programming

(Michalewicz, Z. & Schoenauer, M. 1996)

Multi-Objective Optimization using Evolutionary

Algorithms (Deb, K., 2001)

Vector Optimization

(Kuhn-Tucker, 1951) (Koopmans,1951)

.ε -constraints

.weighting (parameter)

0

..

)](),...,([minmax/ 1

≥

≤

x

b Axt s

x f x f k

Compromise solution

(Yu, 1971,1973) (Yu & Zeleny, 1972)

Habitual Domain (HD)

Multistage Multiobjective

(Yu, 1980)

Fuzzy + HD

Multiobjective Game

(Sakawa & Nishizaki, 1990s)

De Novo

Programming

(Zeleny,1986)

Goal Programming

(Charnes & Cooper, 1955)

Data Envelopment Analysis, DEA

(Charnes, Cooper & Rhodes, 1978)

Multiple Criteria

Multiple Constraints

Level (MC2)

(Yu, et al., 1979)

Combine Together in 1990s

Fuzzy De Novo

(Lee etc.)

Fuzzy MC2

(Shi, etc)

Fuzzy DEA

Fuzzy Multiobjective for DEA

(Chiang & Tzeng, 2000)

Fuzzy + HD + Dynamic + Multistage Multiobjective Decision Making

TOPSIS for MODM

(Hwang et al., 1994)

Fuzzy Multiobjective

Programming

(Sakawa etc. 1980s)

In the Future

Two-level Multiobjective

Multi-level Multiobjective

Coalition

Multiobjective Optim. With Linguistic Logic Model

(Carlson & Fuller, 2002)

Fig. 1. Development of multiple objective decision making

I. INTRODUCTION

Since Kuhn and Tucker [14] published multiple

objectives using vector optimization concept, and Yu [35]

proposed compromise solution method to cope with

multicriteria decision-making problems, there have abundant

work for applications such as in transportation investment and

planning, econometric and development planning, financial

planning, business conducting and investment portfolio

selecting, land-use planning, water resource management,

public policy and environmental issues, and so on. After

Bellman and Zadeh [2] proposed the concepts of

decision-making in fuzzy environment, many distinguished

work guide person study in this field such as Hwang and Yoon

[10], Zimmermann [40], Sakawa [22, 23], Lee and Li [17], and

so on. This aforementioned evolution of multi-criteria

decision making (MCDM) is shown in Fig. 1.

Fuzzy multiobjective linear programming (FMOLP)

formulates the objectives and the constraints as fuzzy sets,

characterized by their individual linear membership functions.

The decision set is defined as the intersection of all fuzzy sets

and the relevant hard constraints. A crisp solution generated

by selecting the optimal solution, such that it has the highest

degree of membership in the decision set. For further

discussions refer to Zimmermann [40], Werners [34],

Martinson [18].

This paper organized as follows, the FMOLP model

highlighted in Section II. The fuzzy goal programming (FGP)

model presented in Section III. The fuzzy goal and fuzzy



GWO-HSHIUNG TZENG: Multiple-Objective Decision-Making in the Past, Present and Future

3

constraint programming model presented in Section IV. Two

phase approach for solving FMOLP problems illustrated in

Section V. Three models of goal programming with

achievement function introduced in Section VI. We propose a

new multiple objectives programming approach to data

envelopment analysis (DEA) in Section VII. De Novo

programming method in multi-criteria optimal system design

presents in Section VIII. Finally we summarize most of the

methods for multiple objective decision making problems and

point out the future direction of our research.

II. FUZZY MULTIPLE OBJECTIVES

LINEAR PROGRAMMING

FMOLP problems usually has the following format:

max ∑=

==n

j jkjk qk xc z

11,...,2,1,~~

min ∑=

+==n

j jkjk qqk xcw

11 ,...,1,~~

s.t. ∑=

=≤n

ji jij mib xa

11;,...,2,1,

~~ ∑=

+=≥n

ji jij mmib xa

121 ,...,1,

~~

∑=

+==n

ji jij mmib xa

12 ;,...,1,

~~ n j x j ,...,2,1,0 =≥ (1)

where kjc% is the j-th fuzzy coefficient of the k -th objective, ãij

is the j-th fuzzy coefficient of the i-th constraint and ib~

is the

fuzzy right hand side of the i-th constraint. Problem (1) can

solve by transferring it into a crisp model shown as (2).

max ∑=

==n

j j

U kjk qk xc z

11,...,2,1,)()( α α

min ∑=

+==n

j j

Lkjk qqk xcw

11 ,...,1,)()( α α

s.t. ∑= ≤

n

j

U

i j

L

ij b xa1

,)()( 　α α i=1,2,…,m1, m2+1,…, m

∑=

≥n

j

Li j

U ij b xa

1

,)()( α α i=m1+1,…,m2; x j≥ 0, j=1,2,..,n

(2)

where U kjc α )( and L

kjc α )( , U ija α )( and L

ija α )( and U ib α )(

and Lib α )( are upper and lower bound of fuzzy number kjc~ ,

ãij and ib~

, respectively, by taking α -level cut. Problem (2)

can be solved by fuzzy algorithm interactively. For details,

see Zimmermann [40], Lee and Li [17], Sakawa [24, 25],

Shibano et al. [29], Shih et al. [30], Ida and Gen [11], Shih and

Lee [31] etc.

III. FUZZY GOAL PROGRAMMING

In most FGP problems can mathematically be

represented as:

max )](~

),...,(~

),(~

[ 21 x f x f x f k

s.t. Ax≤b; x≥0 (3)

where x, b are vector of variables and right hand side [15, 35]

defined the membership function of fuzzy goal as follows:

≤

<

≤

>

−

−−=

−

−−

)(

)()(

)()(

)()(

,)()(

)()(

,0

1

,1

)( *

*

*

*

x f

x f x f

x f x f

x f x f

x f x f

x f x f x i

ii

ii

ii

ii

ii g i

　µ

(4)

where )(* x f i and )( x f i− represent the positive ideal

solution and negative ideal solution, respectively. We can

transfer (3) to λ expression method as follows:

λ 　 x

max

s.t. ,)()(

)()(*

x f x f

x f x f

ii

ii−

−

−

−≤λ i=1,…,k

Ax≤b; x≥ 0 (5)

We also can employ max-min method to transfer (3) as follows:

i xminmax λ

s.t. Ax≤b; x≥0 (6)

IV. FUZZY GOAL AND FUZZY

CONSTRAINT PROGRAMMING

The fuzzy goal and fuzzy constraint programming

problems can be represented as:

max )](~

,),(~

),(~

[ 21 x f x f x f k ⋅⋅⋅

s.t. b x A~~

≤ ; x≥0 (7)

where x is the vector of variables and b~

is vector of fuzzy

right hand side. First, we define the membership function of

fuzzy goal as follows:

*

**

*

1, ( ) ( )

( ) ( )( ) 1 , ( ) ( ) ( )

( ) ( )0, ( ) ( )

i

i i

i i g i i i

i i

i i

f x f x

f x f x x f x f x f x

f x f x f x f x

µ −

−

−

>

−= − ≤ ≤

− <

(8)




4

1, ( )

( )( ) 1 , ( )

0, ( )

j

j j

j j

C j j j j

j

j j j

Ax b

Ax b x b Ax b p

p

Ax b p

µ

<

−= − ≤ ≤ +

> +

(9)

In this case, we can transfer (7) to λ expression method as

follows:

λ 　 x

max

s.t.)()(

)()(1

* x f x f

x f x f

ii

ii−

−

−

−−≤λ , i=1,2,…,k

j

j j

p

b Ax −−≤

)(1λ , j=1,2,…,m; x≥0 (10)

We also can employ max-min method to transfer (4) as follows:

λ 　 ji x ,

minmax

s.t. x≥ 0 (11)

V. TWO PHASE APPROACH FOR SOLVING

FMOLP PROBLEM

Usually there are two or more goals in FMOLP

problems, here we illustrate two phase approach for solving the

following mathematical programming:

)],~(~

,),,~(~

),,~(~

[max112211 xc f xc f xc f k k

x⋅⋅⋅　

)],~(~

,),,~(~

),,~(~

[min 2211 1111 xc f xc f xc f k k k k k k

x⋅⋅⋅++++　

s.t. ;~~b x A ⇑ x≥ 0 (12)

where “ ⇑ ” represents binary relation and defined as follows:

}{}{}{}{}{}{ =∨<∨≤∨≥∨>=⇑ , “ ∨ ” means “or”.

First, we consider crisp multi-objective linear programming

problems as following programming:

)],~

(~

,),,~

(~

),,~

(~

[max112211 xC f xC f xC f

U k k

U U α α α ⋅⋅⋅　

)],~

(~

,),,~

(~

),,~

(~

[min,22,11

1111 xC f xC f xC f

Lk k

Lk k

Lk k α α α ⋅⋅⋅

++++　

s.t. U Lb x A α α )()( ≤ ; LU

b x A α α )()( ≥ ; x≥0; x∈ X α (13)

Zimmermann [40] indicated that two important relation

between α and β :

(1) Optimal level of α and β , that is α= β ;

(2) Having trade-off relation between α and β .

Then the mathematical programming (13) become as

follows:

β 　 x

max

s.t. )((max)

xi g µ β ≤ ; )(

(min) x

i g µ β ≥ ; x∈ X α (14)

where

−

−

−

−=

α α

α α µ (max)

*(max)

(max)(max) ),()(

(max)

ii

iU ii

g f f

f xC f x

i, i=1,2,…,k 1

*(min)(min)

(min)(min) ),()(

(min)

α α

α α µ ii

Liii

g f f

xC f f x

i −

−=

−

−

, i=k 1+1,k 1+2,…,k

Furthermore, using iteration procedure to find the

optimal solution, when α ≅ β , then stop. That is, only to

findλ in second phase, such that: λ=min{α, β }Lee and Li [17] proposed algorithm for this problems as

follows:

Step 1. Setting tolerable error τ , step width ε and initial α

-cut (α =1.0), iterative frequency t=1;

Step 2. Putting α =α -t ε, solve c-LP problem, then obtained β

and x;

Step 3. If |α- β | ≤ τ , let λ =min{α, β }, go to step 4; otherwise, go

back step 2. If width ε is too large, let ε = ε /2 and

t=1, go back step 2;

Step 4. Obtained λ , α , β and x; end.

Therefore, we can solve c-LP2 problems as above two

phases algorithm. Moreover, Ida and Gen [11] proposed

following programming to solve this problems:

∑=

=k

ii

k 1

1max β β 　

s.t.−

−

−

−≤≤

α α

α α β β (max)

*(max)

(max)(max) ),(

ii

iU ii

i f f

f xC f , i=1,2,…,k 1

*(min)(min)

(min)(min) ),(

α α

α α β β ii

Liii

i f f

xC f f

−

−≤≤

−

−

, i=k 1+1, k 1+2,…,k

x∈ X α , β , β i ∈[0,1] (15)

VI. GOAL PROGRAMMING WITH

ACHIEVEMENT FUNCTIONS

Goal programming (GP) is an analytical approach

devised to address decision-making problems where targets

have been assigned to all the attributes and where the

decision-maker is interested in minimizing the

non-achievement of the corresponding goals [21].

Initially conceived as an application of single objective

linear programming by Charnes and Cooper [3, 4], goal programming gained popularity in the 1960s and 70s from the

works of Ijiri [13], Lee [16], and Ignizio [12]. A key element




5

of a GP model is the achievement function that represents a

mathematical expression of the unwanted deviation variables.

Each type of achievement function leads to a different GP

variant. Tamiz and others [32] show that around 65% of GP

applications reported use lexicographic achievement functions,

21% weighted achievement functions and the rest other types

of achievement functions, such as a min-max structure in which

the maximum deviation is minimized.

The weighted achievement model lists the unwanted

deviation variables, each weighted according to importance, the

programming shown as [12]:

Min )(∑ +− +i

iiii d d β α

s.t. iii g d d x f =−+ +−)( ; 0=⋅ +− ii d d ;

0,0 ≥≥ +−ii d d 　 (16)

where

α i=wi k i if−id is unwanted, otherwise α i=0;

β i=wi k i if+id is unwanted, otherwise β i=0.

The parameters wi and k i are the weights reflecting

preferential and normalizing purposes attached to achievement

of the i-th goal.

The second model, lexicography achievement model, is

made up of an ordered vector whose dimension coincides with

the Q number of priority levels established in the model.

Each component in this vector represents the unwanted

deviation variables of the goals placed in the corresponding

priority level [12].

+++= ∑ ∑ ∑

∈ ∈ ∈

+−+−+−

1

)(),...,(),...,(hi hi hi

iiiiiiiiiiii

r Q

d d d d d d a Min Lex β α β α β α 　　

s.t. iiii g d d x f =−+ +−)( ; i∈{1,…,q}; i∈hr ; r ∈{1,…,Q}

x∈ F ; 0≥−id ; 0≥+

id (17)

where hr represents the index set of goals placed in the r -th

priority level. Lexicographic achievement functions imply a

non-compensatory structure of preferences. In other words,

there are no finite trade-offs among goals placed in different

priority levels [20].

The third model, minmax achievement model, seeks for

the minimization of the maximum deviation from any single

goal. If we represent by D this maximum deviation, the

mathematical programming of a linear goal programming (LGP)

model is the following [7]:

D Min x

　

s.t. Dd d iiii ≤+ +− β α

iiii g d d x f =−+ +−)( , i∈{1,…,q}

x∈ F , 0,0 ≥≥ +−

ii

d d 　 (18)

This model implies the optimization of a utility function

where the maximum deviation is minimized. It provides the

most balanced solution among the achievement of different

goals. Thus is, it is the solution of maximum equity among

the achievement of the different goals [33].

VII. MULTIPLE OBJECTIVE

PROGRAMMING WITH DEA

Data Envelopment Analysis (DEA) was developed by

Charnes, et al. [5] and extended by Banker et al. [1], is a

non-parametric programming method for estimating production

frontiers and evaluating the relative efficiency of decision

making units (DMUs), with multiple outputs and multiple

inputs. In CCR model, solving the relative efficiency of

DMUk as follows:

∑=

= s

j jk jk yuh Max

1

　

s.t. ∑=

=r

i

ik i xv

1

1, for k=1,…,n

011

≤− ∑∑==

r

iik i

s

j jk j xv yu , for k=1,…,n

vi ≥ ε >0, i=1,…,r ; u j ≥ ε >0, j=1,…, s (19)

The objective here is to find the largest sum of weighted

outputs of DMUk while keeping the sum of its weighted inputs

at unit value and forcing the ratio of the sum of weighted

outputs to the sum of weighted inputs for any DMU to be less

than one. Transferring the problem to dual program can then

find a minimal value for an intensity factor θ k that indicates the

potential of a proportional reduction in all the inputs of DMUk .

In BCC model adds another restriction to the

envelopment requirements. It requires that the reference point

on the production function for DMUk will be a convex

combination of the observed efficient DMUs. The primal

formulation for DMUk is written as:

∑=

−= s

jk jk jk u yuh Max

1

　

s.t. 1

1

0 =∑=

r

i

ii xv

011

≤−− ∑∑==

r

ik ik i

s

j jk j u xv yu , for k=1,…,n




6

vi ≥ ε >0, i=1,…,r ; u j ≥ ε >0, j=1,…, s (20)

The corresponding primal has a slightly different

objective from (19).

Furthermore, considering in CCR model, the efficiency

ratio of each DMU is calculated by its own best multipliers, not

by the common multipliers for all DMUs. Thus, this model

often results in too many DMUs may be identified as efficient.

We applied the concept of multiple objectives programming to

CCR model to find the common multipliers that could cause

the efficiency ratio for all DMU as large as possible. We

consider the efficiency ratio of all DMUs rather than k-th

DMUk in CCR model and then establish the following model:

⋅

⋅

=

⋅

⋅

=

⋅

⋅

=

∑

∑

∑

∑

∑

∑

=

=

=

=

=

=m

iini

s

r rnr

nm

iii

s

r r r

m

iii

s

r r r

xV

yU

z

xV

yU

z

xV

yU

z Max

1

1

12

12

2

11

11

1 ,...,,　　

s.t. n j

xV

yU

m

iiji

s

r rjr

,...,1,1

1

1 =≤

⋅

⋅

∑

∑

=

=

U r ≥ ε >0, r=1,…, s; V i ≥ ε >0, I=1,…,m (21)

We further transfer (19) to one objective programmingusing membership function with fuzzy multiple objectives

linear programming approach [19, 26, 27], we then conduct the

common multipliers to calculate the efficiency achievement for

all DMUs, the detail procedure can refer to Chiang and Tzeng

[6].

VIII. DE NOVO PROGRAMMING METHOD

IN MODM

Dealing with a multi-objective decision making

(MODM) problem, we usually confront a situation that is

almost impossible to optimize all criteria in a given system.

This property is so-called trade-offs, which means that one

cannot increase the levels of satisfaction for a criterion without

decreasing that for another criterion. Zeleny [37, 38]

developed a De Novo programming for designing optimal

system by reshaping the feasible set. He suggested that

trade-offs are properties of inadequately designed system and

thus can be eliminated through designing better, preferably

optimal system. Zeleny [39] proposed the concept of optimal

portfolio of resources which is design of system resources in

the sense of integration, so that there are no trade-offs in a newdesigned system.

For example, when the budget of designing a new

optimal system is higher than total avail budget, Zeleny [39]

suggested an optimum-path ratio to contract the budget to

available budget along the optimal path. Along this line, Shi

[28] discussed different budgets from different point of views

and define six type optimum-path ratios to find alternatives for

optimal system design.

However, since the ideal point used in the De Novo

programming is not the ideal point in the ordinary system, the

budget for the redesigned system is always larger than the total

available budget. Consequently, no matter what

optimum-path ratio is used, it only can provide a certain path to

locate a solution in the decision space of the new system.

Assuming a MODM problem can be described as follows

[36]

Max Cx

s.t. Ax≤ b; x≥ 0 (22)

where C=C q× n and A=Am× n, b=(b1,…,bm)T ∈ Rm, and

x=( x1,…, x j,…, xn)T ∈ Rn. Let the k th row of C be denoted by

nk n

k j

k k RcccC ∈= ),...,...,( 1 , so that C k x,k=1,…,q, is the k th

criteria or objective function.

Assume that X={ x∈ Rn|Ax≤b, b≥0}, the ideal point of (22)

is T q f f f ),...,(**

1* = , where *

k f =sup{C k x|x∈ X } for k=1,…,q.

If there exists a nT n R x x x ∈= ),...,( **

1* , such that Cx*=

T q xC xC ),...,( **1 T q f f ),...,( **

1= , then the * x called the ideal

solution.

Because the components of b in (22) are determined in

advance, an ideal point usually is not attainable for the

properties of trade-offs among multiple criteria. When the

purpose is to design an optimal system rather than optimize a

given system, it is of interest to consider following problem:

Max Cx

s.t. Vx ≤ B; x ≥ 0 (23)

Then, we find the Min Vx for achieving ideal point, i.e.,

Min Vx

s.t. *k

k f xC ≥ , k=1,…, q

where V=pA=(V 1,…,V n)∈ Rn, p=( p1,…, pm)∈ Rm and B∈ R

present the unit prices of resources and total available budget

respectively. Formulation (23) implies that given the unit

prices of resources and total available budget allocate the budget, so that the resulting portfolio of resources maximizes

the values of the objective functions. There are three methods




7

of De Novo programming for locating a solution while dealing

with multi-criteria optimal system design problem: A

synthetic-optimal budget, meta-optimal budget, and

flexible-constraint meta-optimal budget. For further

discussion can refer to Shi [28].

IX. SUMMARY

We have briefly sketched seven important topics of

MODM problems, being space limit, it is difficult to list and

discuss many other methods adopted on MODM programming

such as fuzzy regression analysis, multiobjective

possibilistic/necessity programming, interactive programming

methods, two-level/multi-level/multi-stage multiobjective

programming, Habitual Domain, Genetic Algorithms and

Evolutionary Computing on MODM. We would like to

introduce these methods and its applications in near future.

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Received: Sep. 24, 2003 Revised: Oct. 15, 2003

Accepted: Oct. 31, 2003

multiple objective decision making in past, present, and future

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