multiple objective decision making in past, present, and future
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8/12/2019 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
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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
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GWO-HSHIUNG TZENG: Multiple-Objective Decision-Making in the Past, Present and Future
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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)
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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
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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
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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
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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
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