top-down fuzzy decision making with partial preference information

Top-Down Fuzzy Decision Making with PartialPreference Information

HSIAO-FAN WANG1 [email protected]

Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu

(30043), Taiwan, ROC

ZHI-HAO HUANG1

Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu

(30043), Taiwan, ROC

Abstract. This paper proposes a multi-stage decision procedure to cope with a hierarchical multiple objective

decision environment in which the upper-level DM only provides partial preference information and the lower-

level DM is fuzzy about the tradeoff questions such that to achieve substantially more than or equal to some

values is delivered to maximize the objectives. Therefore, the procedure consists of two levels, a upper-level and a

lower-level. The main idea is that after the upper-level provides partial preference information to the lower-level

as a guideline of decision, the lower-level DM determines a satisfactory solution from the reduced non-dominated

set in the framework of multi-objective fuzzy programs.

Keywords: top-down decision procedure, partial information, satisfactory solution, fuzzy programming, MOLP

1. Introduction

When solving a decision problem that is in the form of an Multi-Objective Linear Program

(MOLP) below, one has to consider the decision environment in which the problem occurs

(Chankong and Haimes (1983)).

Model (P)

Maximize Z ¼ Cx

ð1Þs:t: X ¼ fx j Ax � b; x >�� 0g

where x is an n 1 vector of decision variable, C is a q n cost coefficient matrix, Z ¼Cx is an objective value vector, A is an m n constraint matrix and b is an m 1 vector of

the right hand sides (RHS).

One of the common situations is a multi-level decision structure in which multi-level

programs are described (Bracken and McGill (1973), Davis and Talavage (1977), Stanley

Lee and Hsu-Shih (1999), Wen and Hsu (1991)). In a multi-level decision environment, a

decision maker (DM) at one level of the hierarchy may have his own objectives and

decision space (variables), but the decision of the lower level may be influenced by that of

the upper level. This is often referred to the decision environment when the decision is

Fuzzy Optimization and Decision Making, 1, 161–176, 2002# 2002 Kluwer Academic Publishers. Printed in The Netherlands.

made between different but hierarchical departments of an organization and thus, the

decision makers at different levels will control different decision variables. With such

situation, Model (P) can be rewritten into a Bi-level Program (BLP) as below in which the

decision process is similar to the static two person Stackelberg game (Stanley Lee and

Hsu-Shih (1999)):

Model(BLP)

Maximizex1 Z1 ¼ cT11x1 þ cT12x2 ðupper levelÞ

where x2 solves

Maximizex2 Z2 ¼ cT21x1 þ cT22x2 ðlower levelÞ

s:t: X ¼ fx j A1x1 þ A2x2 � b; x ¼ ðx1; x2Þ >�� 0g ð2Þ

where A1 and A2 are mxn1 and mxn2 matrices, respectively; c11, c21 and x1 are

n1-dimensional vectors; c12, c221 and x2 are n2-dimensional vectors; x1 and x2 are decision

variables controlled by upper and lower decision makers respectively.

In the other hand, if the decision is made within one department, the hierarchical

structure in making decisions would still exist for different levels of DMs. In such case,

both upper and lower decision makers would face same objectives with same decision

space but with different weights of importance towards the objectives. Since the lower

level usually makes the final decision based on the upper level’s decision, therefore, a two-

stage decision procedure with top-down order will be performed with a supporting model

transformed from Model (P) as below (Wang and Huang (2001)) and this is what we are

interested in this study:

Model(TDP)

Finding N 0 ¼ fx j x 2 X ; Maximize Z ¼ ETCxg ðUpper levelÞ

where x solves

Maximize Z ¼ Cx ðLower levelÞ

s:t: x 2 N 0 ð3Þ

where E 2 Rqp represents the extreme points of the weight convex hull obtained from the

preference structure of the upper level (Carrizosa et al (1995), Davis and Talavage (1977),

Marmol et al (1998)).

When making decisions, it is sometimes difficult for a decision maker to completely

specify his or her preference structure either because of the complexity of the problem or

because he/she is unclear about the hypothetical tradeoff questions that are used to assess

WANG AND HUANG162

the evaluation functions. In hierarchical structure, this especially happens to the upper-

level decision makers because of the limit of time and also they often take a broad

viewpoint towards the decision problems and thus detailed and precise tradeoffs among

objectives are not what they concern. Therefore, articulation of complete information

regarding the weighting constants is almost impossible in practice, especially, for the

upper-level DM (Marmol et al (1998)). Therefore, in this study, partial information of the

upper-level DM’s preference structure is considered for analysis.

Furthermore, in a top-down decision procedure, once the upper level provides partial

preference information, the non-dominated solution set will be reduced which facilitates

the lower level to make final decision. However, if the lower-level DM is fuzzy about

the tradeoff questions such that only vague statements such as ‘‘to achieve some

objectives substantially more ( less) than or equal to some values of objectives’’ can

be addressed, then this is a fuzzy programming structure (Sakawa et al (2000),

Zimmermann (1978)) and an interactive procedure will be developed in this study to

support lower-level to make decisions.

In a word, the main idea of the proposed procedure is that when facing a multi-objective

decision problem which can be described by a multi-objective linear problem (MOLP),

while the upper-level only provides partial preference information, the analyst can make

use of this preference information to reduce the non-dominated solution set. Then based on

the fuzzy programming analysis (Sakawa (1993), Zimmermann (1978)), the analyst

supports the final decision made by the lower-level based on the procedure of Bellman

and Zadeh (1970) of which the membership functions of fuzzy goals for all of the

objective functions and their minimal satisfactory levels are articulated for finding a final

satisfactory solution. This procedure is shown in Figure 1.

This paper is organized as follows: In Section 2, we shall address the issues of top-down,

two stage decision procedure when the partial and fuzzy information are provided

respective to upper and lower level DMs. In Section 3, a decision procedure for

hierarchical decision makers is proposed. Section 4 gives an illustrative example for the

proposed decision procedure. Then, it follows a discussion and evaluation of the proposed

method in Section 5. Finally, summary and conclusions are drawn in Section 6.

Upper-level

Lower-level

-satisfactory solution

Partial preference information

Linguistic preference information

Figure 1. The Proposed Procedure for Hierarchical Decision Structure.

TOP-DOWN FUZZY DECISION MAKING 163

2. Issues of Top-Down Decision Making with Partial and Fuzzy Preference

Information

In this section, we shall discuss some critical issues related to a multiobjective decision

procedure for hierarchical decision makers with respect to their different perception

towards the same decision criteria.

It has been noted that articulation of a decision maker’s preference structure is one of the

most difficult issues in multi-criteria decision problems. In top-down decision structure,

the upper level DM considers the problems from macro viewpoint, whereas the lower level

DM who has to practically solve the problems and thus he/she should compromise among

different objectives under resource constraints to achieve the maximal overall profit of the

department. Because two levels take different viewpoints towards same objectives, when

the DMs of both levels can not present their weights of importance precisely for those

objectives, their ways of presenting their preferences are different and thus in Sections 2.1

and 2.2, we shall first discuss this issues respectively.

2.1. Partial Preference Information of the Upper-Level DM

In this section we shall discuss the functions and impacts of partial preference presenta-

tions of the upper-level DM.

2.1.1. Preference Presentation with Partial Information

The responsibility of the upper-level is to provide a guideline for a decision problem. Such

guideline towards the relative weights of importance among the objectives can be stated in

either ordinal or cardinal scales. If ordinal scale is taken to present preference, then

homogeneous linear relation can be described. However, if the DM can specify those in

cardinal scale, then non-homogeneous linear relation can be adopted (Carrizosa et al

(1995), Marmol et al (1998), Wang and Huang (2001)). Because ordinal preferences can

be obtained by marginal substitution between the objectives or even the ranking order of

the objectives, but the cardinal preferences require more information, therefore ordinal

scale is comparatively easier for the decision makers to address. However, the non-

homogeneous linear relation with cardinal information is more precise and thus is more

useful in analysis and applications.

It is noted that in practice, to incorporate the upper-level’s preference is to reduce the

non-dominated solution set. Therefore, let us first consider the initial set of admissible

weights by the simplex of � ¼ {w 2 Rq | w >�� 0, etw ¼ 1}, where et ¼ (1, . . . ,1), qrepresents the number of objectives and w represents the relative weight of importance of

each objective perceived by a DM. Then for ordinal preference, we have

Definition 2.1 (Carrizosa et al (1995)) Let the information presented by a homogeneous

linear relation be Mw >�� 0, M 2 Rqq, M�1 >�� 0, then the convex hull of weights is

PðMÞ ¼ fw 2 � j Mw >�� 0g with M�1 >�� 0: ð4Þ

WANG AND HUANG164

For cardinal preference, we have

Definition 2.2 (Marmol et al (1998)) Let the information presented by a non-

homogeneous linear relation be � <�� Mw <�� , M 2 Rqq, M�1 >�� 0 and �, � 2 Rq, then

the convex hull of weights is

P�;�ðMÞ ¼ fw 2 � j � <�� Mw <�� g with M�1 >�� 0: ð5Þ

To reduce the non-dominated solution set of a multi-objective linear programming

problem, we must elicit the extreme points of the weight convex hull of the DM. While

Carrizosa et al (1995) have shown that such extreme points of P(M) are just the columns of

M�1 normalized to have unit sum, Marmol et al (1998)’s algorithm can be adopted to

obtain the extreme points of P�,� (M).

When the matrix E 2 Rqp in Model (TDP) represents the extreme points of the weight

convex hull, then, the columns of matrix E represent the extreme points and p is the

number of extreme points.

2.1.2. Properties Derived from the Partial Preference Information

At first, let us assume that in all discussion that follows, Model (P) has a bounded, feasible

solution set X ¼ {x | Ax <�� b , x >�� 0}.

When partial preference information of the upper level is provided, within the

framework of an MOLP, the following Corollary suggests the existence of a non-

dominated set which satisfies upper level’s preference structure:

COROLLARY 2.1 (Wang and Huang (2001)) Let N 0 represent the non-dominated solution

set with respect to the partial preference information of a DM, then we can find N 0 by the

following model:

Model (P1)

Maximize Z ¼ ETCx

s:t: x 2 X

ð6Þ

The relation between Model (P) and Model (P1) has the following property:

COROLLARY 2.2 (Wang and Huang (2001)) The inefficient extreme points in model ( P)

retain inefficient in model ( P1).

Definition 2.3 (Steuer (1986)) Let C>�� be the semi-positive polar cone generated by the

gradients of the q objective functions, then

C>�� ¼ fy 2 Rn j Cy >�� 0; Cy 6¼ 0g [ f0 2 Rng: ð7Þ


THEOREM 2.1 (Steuer (1986)) Let Dx be the domination set at x 2 X, then x is efficient if

and only if , Dx, \ X ¼ {x}.

Theorem 2.1 provides a tool for detecting efficient points that can be geometrically

visualized. If the intersection of the domination set and the feasible region only contains x,

then x is efficient. If there are other points in the intersection, then x is inefficient.

COROLLARY 2.3 (Steuer (1986)) The larger the domination set Dx is, the greater the

possibility that x 2 X is inefficient. Similarly, the smaller Dx, the greater the possibility

that x 2 X is efficient.

Definition 2.4 (Wang and Huang (2001)) Let C�1 be the semi-positive polar cone

generated by the gradients of eiT C, i ¼ 1, 2 . . . p, then

C�1 ¼ fy 2 Rn j ETCy >�� 0; ETCy 6¼ 0g [ f0 2 Rng: ð8Þ

Let D1x be the domination set when the semi-positive polar cone is C�1 at x, then we have

COROLLARY 2.4 Dx � D1x

Proof: Since for any ei, i ¼ 1, 2 . . . p is strictly positive andPq

j¼1 eij ¼ 1, so C� � C�i .

Hence, x þ C� � x þ C�1 , it means that the number dominate x in C�

1 is larger than that in

C�, i.e. Dx � D1x .

By Corollaries 2.2, 2.3, 2.4, a non-dominated solution set can be reduced by the

preference information of a DM which facilitates making a decision.

2.2. Vague Preference Information of Lower-Level DM

2.2.1. Preference Presentation with Vague Information

Facing the same decision problem as upper-level, the lower-level DM should take from

more operational viewpoint to achieves substantially more ( less) than or equal to some

levels of objectives under the upper level’s guideline. Then such vague goals of objectives

can be presented by linear membership functions in fuzzy programming (Sakawa (1993),

Zimmermann (1978)).

To elicit a membership function ui (zi (x)) for a maximization objective functions zi of

model (P), a strictly monotonic increasing form is considered by first calculating the mini-

mum and maximum values of an objective function under the given constraints as follows:

zmini ¼ min

x2XziðxÞ; i ¼ 1; 2 . . . q ð9Þ

zmaxi ¼ max

x2XziðxÞ; i ¼ 1; 2 . . . q ð10Þ

WANG AND HUANG166

zimin and zi

max can be regarded as the worst and the best achievement of the i-th objective.

Taking the closed interval [zimin, zi

max] as reference, the lower level DM is asked to specify

the pessimistic value zi0 of the objective function with the zero degree of satisfaction and

the optimistic value zi1 of the objective function with the degree of satisfaction as one such

that [zi0, zi

1] � [zimin, zi

max]. Therefore, the values which are less than zi0 are undesirable with

membership values ui (zi(x)) ¼ 0; and for those larger than zi1 are the desired values with

the membership value ui (zi(x)) ¼ 1.

Therefore, a linear membership function ui (zi(x)) of the i-th objective function is

formulated as below and shown in Figure 2 for which the satisfactory level of each

objective function for a decision x can be presented by �i2 [0, 1], and thus ui (zi (x)) >�� i,for i ¼ 1, . . . ,q is desired.

uiðziÞ ¼

0 if zi � z0i

zi � z0i

z1i� z0

i

if z0i < zi < if z1i

1 if z1i � zi

8>>>><>>>>:

ð11Þ

2.2.2. Properties Derived from Vague Preference Information

Once the membership function is defined, finding a satisfactory solution of model (P1) by

lower level DM is equivalent to find a solution with the largest degree of satisfaction with

all objective values from the non-dominated solution set, N 0. That is,

Model (P2)

Maximize uiðziðxÞÞ i ¼ 1; 2 . . . q

s:t: x 2 N 0ð12Þ

1

0

)( ii Zu

iZ1iZ0

iZ

Figure 2. Membership Function of a Maximization Objective Function zi.


To derive a satisfactory solution of model (P2), we first find the maximal solution of

fuzzy decision problem proposed by Bellman and Zadeh (1970). That is, the following

problem should be solved to obtain a solution which maximizes the smallest satisfactory

degree among all of the fuzzy goals:

Max�min fu1ðz1ðxÞÞ; . . . . . . ; uqðzqðxÞÞ

s:t: x 2 N 0:ð13Þ

Then, let � ¼ mini¼1;...q

uiðziðxÞÞ ¼ mini¼1;...q

�i, the problem above can be transformed into the

following model:

Model (P2 0 )

Maximize �

s:t: x 2 N 0

uiðziðxÞÞ >�� ; i ¼ 1; . . . ; q

� 2 ½0; 1�:

ð14Þ

With definition (11), this model is a simple linear program which can be solved by

Simplex method. If the DM satisfies the optimal solution x* of model (P20 ) with objective

value �*, then a satisfactory solution is found; otherwise, the DM is asked to specify the

desired satisfactory degrees �j >�� * with �j2 [0, 1] for some objectives zj, j 2 {1, . . . ,q}.Then the following problem is solved

Model (P200)

Maximize �

s:t: x 2 N 0

ujðzjðxÞÞ >�� j; for some j 2 f1; . . . ; qg

uiðziðxÞÞ >�� ; for i 2 f1; . . . ; qg=j

� j 2 ½0; 1�; j 2 f1; . . . ; qg; � 2 ½0; 1�:

ð15Þ

If an optimal solution to model (P200 ) exists, then �* (P20 ) >�� * (P200 ) for objectives

i 2 {1, . . . ,q}/j and �j >�� * (F 0 ) for objectives j ¼ 1, 2 . . . ,q. Therefore, the DM

obtains a satisfactory solution with satisfactory degrees of some objectives j larger than

or equal to the minimal satisfactory levels �j which are specified by DM. Otherwise, we

WANG AND HUANG168

ask DM to specify the minimal satisfactory level by neglecting those objective functions

he/she considers the less important. If an optimal solution exists, then stop; otherwise we

continue to ask DM to specify the minimal satisfactory level by neglecting those he/she

considers the less important objective functions among the remaining objective functions

till an optimal solution is found.

3. The Two-Stage Decision Procedure

Based on the properties derived from upper and lower levels’ preference information as

discussed in Section 2, in this section we propose a two-stage decision procedure for

hierarchical decision makers such that a satisfactory solution can be found.

Before proceed our presentation, let us first summarize the discussion above by

redefining model (TDP) into a Top-Down Fuzzy Program model (TDFP) as follows so

that the properties of the preference structures of both levels can be presented:

Model(TDFP)

Finding N 0 ¼ fx j x 2 X; Maximize Z ¼ ETCxg ðUpper levelÞ

where x solves

Maximize uiðziðxÞÞ i ¼ 1; 2 . . . q ðLower levelÞ

s:t: x 2 N 0 ð16Þ

Then, the procedure is outlined as follows:

Step1: The upper level is asked to provide his/her preference information. Then, based on

Definitions 2.1 and 2.2, the extreme points of the respective weight convex hulls

P(M) and P�,� (M) are generated.

Step2: Use those extreme points to derive objective vectors ETC and then the irreducible

objective vectors is formed (Telgen (1982)) to construct model (P1).

Step3: Use model (P1) to reduce the non-dominated extreme point set N to obtain the

reduced set N 0.

Step4: The minimum and maximum values of each objective function under the reduced

set N 0 are presented to lower-level and the DM is asked to specify the values of zi0

and zi1 for each objective function. Then a membership function ui (zi(x)) can be

determined when a strictly monotonic increasing function with respect to zi (x) is

assumed.

Step5: Find the non-dominated extreme point set of the following model (P3) which has

the satisfactory degree � of all points in set N 0.


Model (P3)

Maximize Z ¼ ETCx

Maximize �

s:t: uiðziðxÞÞ >�� ; i ¼ 1; . . . ; q

x 2 X : ð17Þ

Step6: Choose a solution x* with the maximal satisfactory degree �* from the points in set

N0. If DM satisfies this solution, then stop; otherwise ask DM to specify the

satisfactory degrees �j for some objective functions j 2 {1, . . . ,q} with �j >�� * andgo to Step7.

Step7: With these satisfactory levels �j, we formulate the following model (P4):

Model (P4)

Maximize Z ¼ ETCx

Maximize �

s:t: x 2 X

ujðzjðxÞÞ >�� j; for some j 2 f1; . . . ; qg

uiðziðxÞÞ � �; for i 2 f1; . . . ; qg=j ð18Þ

Step8: Find the non-dominated extreme point set of model (P4), if there are solutions

which belong to set N 0, then choose a solution which maximizes the satisfactory

degree � and then stop; otherwise we ask DM to specify the minimal satisfactory

level by neglecting those with the less important objective functions among the

remaining objective functions, then go to Step7.

This procedure is illustrated by the following example :

4. An Numerical Example

Now let us consider a three objective linear problem, Max {Cx| x 2 X }, where the

objective matrix C is given by C ¼

"3 1 2 1

1 �1 2 4

�1 5 1 2

#and X is defined by the

constrains: 2x1þ x2þ 4x3þ 3x4 <�� 60, 3x1þ 4x2þ x3þ 2x4 <�� 60, x1, x2, x3, x4 >�� 0.

WANG AND HUANG170

Then, the efficient extreme point solutions are listed in Table 1. Since solutions G, H and

I are dominated ( G by A, H by 0.5B and 0.5C, and I by F ), therefore the non-dominated

extreme point set N = {A, B, C, D, E, F}.

Step1: Assume that the upper-level gives the preference structure as

0 <�� w1<�� w2; 0 <�� w1

<�� w3 1with w1 þ w2 þ w3 ¼ 1:

Because the information is homogeneous linear relations, so we can use the method

of Carrizosa et al (1995) to find the extreme points of P(M) ¼ {w2 �| Mw >�� 0}

where

M ¼

1 0 0

�1 1 0

�1 0 1

266664

377775 and M�1 ¼

1 0 0

1 1 0

1 0 1

266664

377775:

From the non-negativity of M�1, the extreme points of P(M) are the columns of

matrix

E ¼

1=3 0 0

1=3 1 0

1=3 0 1

266664

377775:

Step2: The irreducible objective vectors are (1, 5/3, 5/3, 7/3), (1, �1, 2, 4), (�1, 5, 1, 2)

respectively.

Table 1. Efficient extreme points with objective function values.

Decision variables Slack variable Objective function values

Point x1 x2 x3 x4 x5 x6 Z1 Z2 Z3

A 18 0 6 0 0 0 66 30 -12

B 12 0 0 12 0 0 48 60 12

C 0 12 12 0 0 0 36 12 72

D 0 6 0 18 0 0 24 66 66

E 0 15 0 0 45 0 15 -15 75

F 0 0 0 20 0 20 20 80 40

G 20 0 0 0 20 0 60 20 -20

H 0 0 15 0 0 45 30 30 15

I 0 0 0 0 60 60 0 0 0


Step3: Reducing non-dominated extreme point set N by model (1):

Model (1)

Max

x1 þ 5=3x2 þ 5=3x3 þ 7=3x4

x1 � x2 þ 2x3 þ 4x4

�x1 þ 5x2 þ x3 þ 2x4

8>>>><>>>>:

s:t:

2x1 þ x2 þ 4x3 þ 3x4 � 60

3x1 þ 4x2 þ x3 þ 2x4 � 60

x1; x2; x3; x4 � 0

8>>>><>>>>:

The non-dominated extreme point set of model (1) is N 0 ¼ {C, D, E, F}.

Step4: The minimum and maximum values from objective 1 to objective 3 are [15, 36],

[�15, 80], [40, 75] respectively. By taking account of the minimum and maximum

of each objective function, lower level DM specifies the acceptable range of each

objective function as [z10, z1

1] ¼ [18, 30], [z20, z2

1] ¼ [50, 70], [z30, z3

1] ¼ [55, 70].

Then the membership functions ui(zi(x)) for i ¼ 1,2,3 can be defined by a linear form as

follows:

u1ðz1ðxÞÞ ¼

0 if z1 � 18

z1�1830�18

if 18 < z1 < 30

1 if 30 � z1

8>>>><>>>>:

u2ðz2ðxÞÞ ¼

0 if z2 � 50

z2�5070�50

if 50 < z2 < 70

1 if 70 � z2

8>>>><>>>>:

u3ðz3ðxÞÞ ¼

0 if z3 � 55

z3�5570�55

if 55 < z3 < 70

1 if 70 � z3

8>>>><>>>>:

WANG AND HUANG172

Step5: We construct the following model to find the non-dominated extreme points,

objective function value and the satisfactory degree � and the result are shown in

Table 2:

Model (2):

Max

x1 þ 5=3x2 þ 5=3x3 þ 7=3x4

x1 � x2 þ 2x3 þ 4x4

�x1 þ 5x2 þ x3 þ 2x4

o

8>>>>>>>><>>>>>>>>:

s:t:

2x1 þ x2 þ 4x3 þ 3x4 � 60

3x1 þ 4x2 þ x3 þ 2x4 � 60

uiðziðxÞÞ � �; i ¼ 1; 2; 3

x1; x2; x3; x4 � 0

8>>>>>>>><>>>>>>>>:

Step6: Because the solution D ¼ (0, 6, 0, 18) has the maximal satisfactory degree � ¼ 0.5

in set N 0 with the corresponding objective values of (24, 66, 66), and the DM

satisfies this solution, so we stop.

5. Evaluation and Discussion

The process of generating a compromise decision which satisfies different levels of needs

is a challenging task. Because of its efficiency and effectiveness in operation, top-down

decision procedure is one of the most commonly adopted styles in an organization. While

Table 2. Non-dominated solutions with satisfactory degrees.

Decision variables Objective function values Satisfactory Degree

Point x1 x2 x3 x4 Z1 Z2 Z3 �

C 0 12 12 0 36 12 72 0

D 0 6 0 18 24 66 66 0.5

E 0 15 0 0 15 �15 75 0

F 0 0 0 20 20 80 40 0


the upper level takes a broad and strategic views towards the problem; the lower level has

the responsibility to make final decision from an operational viewpoint under the

instruction of the upper level. However, providing preference information in terms of

the precise weighting constants has been recognized as a difficult task. Therefore, based

on the roles of the DMs at different levels, in this study we designed a procedure so that

with the support of analytic information by a computer, the requirements of DMs’

preference information can be minimized; and in the meantime, the effectiveness of DMs’

subjective judgments can be improved.

These results can be realized from the following:

(1) While the upper-level is only required to provide partial preference information; the

lower-level simply gives the preferred ranges of the objective functions with the

reference of the feasible ranges provided by the computer.

(2) The efficiency in reducing the complexity of the problems has also been achieved by

technically integrating such preference information into problem solving procedure so

that the non-dominated extreme solutions can be reduced. This idea can also been

referred to as Weighting Vector Space Reduction approach (Chankong and Haimes

(1983)).

(3) The interactive device provides a learning process for which the lower level of the final

decision maker can learn to recognize good solutions from adjusting the satisfactory

levels of the objectives.

Therefore, the proposed method provides integration-oriented, adaptation and dynamic

learning features by considering all possibilities of a specific domain of MOLP problems.

However, it also can be noted that the most time consuming step is to find the efficient

extreme point solutions of a vector-maximization problem at the 3rd step in which

ADBASE developed by Steuer (1983) can be used as a tool for this purpose.

6. Summary and Conclusion

In this study, we consider a decision environment of which when the decision makers in an

organization face a multi-objective decision problem, a top-down, multi-stage decision

procedure is performed. Within such environment the DMs at different levels of the

hierarchy have the same multiple objectives to deal with. However, because the upper

level plays a leading role, a decision made by the lower level DM should take account of

the upper level’s preference. Such decision problem when it can be described by a multi-

objective linear program has been described in this study by a Top-Down Decision model

as Model (TDP).

With such top-down structure, however, if the decision environment is uncertain that

both levels of DMs cannot precisely present their preference structure. This study

provides Model (TDFP) with a solution procedure to allow the upper level to present

his/her preference information that can be partially described in the form of ordinal or

cardinal relations. Since incorporating such information can reduce the non-dominated

WANG AND HUANG174

solution set, it facilitates the lower level to make final decision. However, if the lower

level is fuzzy about the weights of importance of the objectives, then based on the

membership function of fuzzy goals for all of the objective functions and by the procedure

of Bellman and Zadeh (1970) the lower-level can be supported to specify and adjust the

minimal satisfactory degrees for some or all of the fuzzy goals in order to find a final

satisfactory solution.

It has been realized that in reality, to obtain a satisfactory solution in a rapidly

changed society is a complicate and challenging problem. However, only the partial

preference information is required in the proposed procedure not only reduces such

complexity, but also facilitate the lower level to make a final decision within a small set

of efficient solutions. In the other hand, through fuzzy interactive technique, the lower

level will be more effective in dealing with the tradeoff problems among multiple

objectives.

Acknowledgments

The authors gratefully acknowledge the financial support from the National Science

Council, Taiwan, ROC with project number #89-2213-E007-038.

Note

1. Tel.: þ886-3-5742654; Fax: þ886-3-5722685.

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