a generalization of fuzzy goal programming with preemptive structure

Pergamon PII: S0305-.0548(96)00096--2

Computers Ops Res. Vol. 24, No. 9, pp. 819-828, 1997 © 1997 Elsevier Science Ltd

All rights reserved. Printed in Great Britain 0305-0548/97 $17.00+0.00

A GENERALIZATION OF FUZZY GOAL PROGRAMMING WITH

PREEMPTIVE STRUCTURE

Hsiao-Fan Wang'~ and Ching-Chun Fu§ Department of Industrial Engineering, National Tsing Hua University, Hsinchu, 30043, Taiwan

(Received July 1996; in revised form December 1996)

Scope and P u ~ F u z z y goal programming (FGP) with preemptive structure is an important skill in decision making. Based on the properties of the structure, the proposed model can describe the preference attitude of a decision maker in an uncertain environment, and the developed transformation procedure can increase the computation efficiency.

Abstract--This article proposes an alternative method that utilizes a penalty cost to solve a FGP problem with preemptive structure. By investigating the properties of this FGP, we have transformed an FGP into a simpler form which can be solved with 1 iteration. Besides, the proposed method allows the mix-use of the linguistic terms of "around", "at least", and "at most" with different types of membership functions. With designed operations of dilation and contraction of these membership functions, the preference attitude of a decision maker in uncertain environment can also be described. © 1997 Elsevier Science Ltd

1. I N T R O D U C T I O N

Goal programming (GP) [1] is a common tool used in decision making, but providing crisp goals can be a problem for a decision maker. Since Zadeh [2] proposed the concept of fuzzy sets, Bellman and Zadeh [3] have developed a basic framework for decision making in a fuzzy environment. Thereafter, research followed [4-6] in which Narasimhan [7] and Hannan [8] extended the fuzzy set theory to the field of goal programming. While Hannan's method [8] is very complicated in computing the membership functions with absolute forms, Narasimhan's method [7] cannot find the priority order when fuzzy priorities of fuzzy goals are considered, such as "very important", "moderately important" and "important". In general, we often classify the goals by their levels of importance in a GP problem. Then, the problem is solved by their orders of levels. This type of preemptive structure in fuzzy environment was first studied

K by Tiwari e t a l . [9] with symmetric triangular membership functions. Tiwari et al.'s FGP requires Y. 2"

i=1

iterations where K is the number of priority levels and mi is the number of goals in the i 'h level of priority. However, Chen [10] indicated that when the membership functions are bounded above by 1, Tiwari e t

a l . ' s algorithm can be sped up to the order of K. Therefore, based on Chen's results, we shall try to develop a general form of FGP problem with a preemptive structure and propose an alternative computation method in this study.This study first reconstructs a fuzzy goal programming model with a preemptive structure in Section 2 where different types of membership functions to reflect different decision behaviour are specified. Then, a solution procedure is developed in Section 3 with an illustrative example in Section 4. Finally, conclusions are presented in Section 5.

2. A M O D E L OF FUZZY GOAL P R O G R A M M I N G

Consider a generalized model of FGP with preemptive structure as below.

Model 1

k. K w k ~ k • G it'g i, (xl .... ,x~)----(Ax) i , -Bi, , t l E I t,k ~ 14,

k. K . B ~ : i 2 ~ l : , k e I 4 , Gi:gi2(Xl .... ,X.)-- (Ax) ~ <

(1)

(2)

1" To whom all correspondence should he addressed (email: [email protected]). ~: Hsiao-Fan Wang has been teaching at the Department of Industrial Engineering, National Tsing Hua University, since she

graduated from Cambridge University, UK. Her primary areas of research are multicriteria decision making and fuzzy mathematics.

§ Ching-Chun Fu is a Ph.D. candidate at the Department of Industrial Engineering, National Tsing Hua University. His research interest~ are project management, resource allocation, and fuzzy goal programming.

819

820 Hsiao-Fan Wang and Ching-Chun Fu

G~:g.r,,(Xl .... ,x,)--(Ax)~ > B~,,i 3 ~13,k ~ I4, (3)

P~= {G~ilij~Ijj= 1,2,3,k e 14}, (4)

xi>-O,i = 1,2 ... . . n, (5)

I~ = { 1,2 ..... l, }, 12 = { l, + 1 ..... 12], 13 = { 12 + 1 ..... m }, 14 = { 1,2 ..... K}.

where G~ denotes the i 'h goal of a type j problem, j = 1, 2, 3, with a linear or nonlinear function g~x that K

has the k 'h priority to reach a target value B ~ , j e { 1,2,3,}. If each o f k levels has m k goals, then E mk=m. k = l

The symbols < , > , and ---- denote the fuzzified aspiration levels with respect to the linguistic terms

of "at most", "at least", and "around" and we refer them to be type 1, 2, 3 problems, respectively. Under "at most" situation, we allow the goal i to be spread to the right-hand-side of Bi2 with a certain range ri2. Similarly, with "at least", I~3 is the allowed left spread of B~3 and with "around", r~ and l;~ are the respective spreads of right and left to Bi~, where B~j can be a single value or an interval value denoted by [Bin, B j . Figures 1, 2 and 3 show these membership functions in linear case. Pk, k= 1, 2 .... Kdenotes

c-i

I -

Bi I Bill Bilr 0 D

(AX)il

Fig. 1. The membership function shapes of type 1.

0 D

Bi 2

(Ax)i2

Fig. 2. The piecewise linear membership function shape of type 2.

A generalization of fuzzy goal programming 821

5

I -

0 t, / ~ ' ~ l i 3 " ~

Bi 3

(Ax)i3

Fig. 3. The piecewise linear membership function shape of type 3.

the set of goals with k 'h level of priority. The smaller the value of k, the higher is P / s priority, i.e. P~>P2>...>Pg. The rule of preemption is that the lower levels of priority cannot be considered unless

the higher levels of priority are satisfied.Now, let us define three types of fuzzified goals as below: (T1: "around"):

O, if (Ax)i <-Bid- li:

(1 [B6~-(Ax)6] )ci,, if Bid-lit~(Ax)6<Bid, li~

I, Zi~(Ax)= 1, if Bi,l<-(Ax)i <-Bi,,., (6)

(1 [(Ax)i~-Bi:] )die if B6,.<.(Ax)i<_Bi:+ri,,

0, if (Ax)~>-Bi:+r~:

where cij, dil ~- {0.5,1,2 }, i I all. pj~(A x) denotes the degree of membership of type-1 goal. If c;~=2, d~=2, we have a contraction. If

C~l =0.5, d~ =0.5, we have a dilation. We allow the membership functions of type-1 goals to have different shapes of deviation simultaneously.

(T2: "at most"):

/z~,(Ax) = { ( 1

where di:~ {0.5,1,2}, i2E12.

1, if ( ix)i--<Bi: ,

[(Ax)i2 - Bi2] ~di: if Bj~<-(Ax)i~<--Bi~ + ri: ri 2 /

O, if (Ax)i>-Bi2+ri:.

(7)

If di2=l, it is a piecewise linear membership function [11] with a right spread. Otherwise it is a nonlinear membership function with di2 = 2 of contraction or d~2 =0.5 of dilation.

(T3: "at least"):

where c~3e {0.5,1,2}, i3ei3.

0, if (Ax)i3<Bi3 - lie

. -16 Ci, ' ifBi3-16<(Ax)i3<Bi: (8)

1, if (Ax)i~>-Bi,,,


u21

u22

u23 0

Membership function

I

. . . . . . . Risk-averse . . . . . .

Risk-neutral ~

Risk-seeking D

I I

x I x 2 x 3 x 4 x 5 x 6

(Ax ) i

u51 -~

u52

u53

Fig. 4. The spread shapes o f a membership function.

If ci3 ~. 1, it is a piecewise linear membership function with a left spread. Otherwise it is a nonlinear membership function with ci3=2 of contraction or ci3 =0.5 of dilation.

According to the properties; dilation, contraction and motionless, of the membership functions, we can use them to present different decision behaviour. Based on the decision theory, the characters of decision makers can be divided into three categories of risk-averse, risk-seeking, and risk-neutral [12]. When a goal should be achieved at least (at most) to a certain level, a unit less than (more than) that level will cause a lower degree of satisfaction for a risk-seeker than that for a risk-averser. Figure 4 shows the respective membership functions.

Therefore, the proposed FGP model generalizes the existing models by allowing different combinations of membership functions and types of goals. As the form of this model is not easy to compute, we shall first transform it into a simpler form by introducing the concept of penalty cost. Then, we shall show that with this transformed model, the existing software package can be adopted to solve it in one iteration.

3. A M E T H O D T O S O L V E T H E M O D E L O F F G P

The rule of preemption states that the lower levels of priority can not be considered unless the higher levels of priority are satisfied. This is equivalent to saying that if P~>P2>. . .>Px and if 1-->A~->A2-->...-->AK->A-->0, then A must satisfy the two following conditions.

(i) If the first ( K - 1) levels are satisfied, then A=AK and A~= 1, VkE { 1, 2 ..... K - 1 }. (ii) If the first ( l - 1) levels ( l - K - 2 ) are satisfied, once At<l, then there are no feasible A and Ak, Vk~ {/+i ..... K}.

This means that these goals with the same k rh level of priority have the same value of A k and if that level of the goals is achieved, then Ak= 1. A denotes the minimal degree among Ak, k~14. Once a level appears to be unable to reach its goal, then the second condition will force the model to provide an infeasible solution.

Now, let M k, k~ { 1, 2 ..... K} be the penalty cost of the k th level of goals with O<M]<M2<. . .<Mr, then, there exists a A'(->0) that satisfies the following constraints.

M1A'_>I - A t

M2A'-> 1 - A2

: (9)

MrA'>_I - A x

MrA' +A-<l


Theorem 1. The set of inequalities (9) satisfies two conditions of the preemption rule.

P r o o f Because;

l->aj>-,~:->-..->Ax->a->0~ max A= max(rain i a3, (10)

therefore,

0-<1 - Al--<...--< 1 - Ar-<l - A-<l~ min(l - A)= min(max ~ (1 - A3). (11)

Now, let us consider the two conditions below.

(i) Since the first ( K - 1) levels are satisfied, we have Ak= 1, Vk= 1, 2 ..... K - I. Then, inequalities (9) become

M~,t'---O,k= 1 ..... K - 1

MrA'>-I - Ar This leads to A<-Ar. Therefore, max A=ar. MrA'+A_<I

(ii) If the first (l - 1) levels are satisfied with l<-K - 2, then ,t 1 =A2= . . . . A~_ ~= 1, but ,it< 1. Then

Then

MkA'>-O,k= 1,2 ..... I - 1 1 - A /

M t A ' > I - A ~ , ~ A ' > M,

{ MI+,A,>I _ AI+I:=~At~ 1 - A I + 1

Mt+l

1 - Ar M r A ' > I - Ar,~A'__ - -

Mr MrA'+A_<l

} l- s implies A'= max; ~ = ~ ,Vi,s ~ [l.k]

MrA' +A <- 1

EO,

~max A{ =0,

Mr ~- ~(1- ,L) ,

~ M r ( 1 - A ~ ) +f i~l

Mr' (1 - A~)+A-- < 1

M~ M,~

-<or<O, i f ( l -A~)> M~ Mr

(

~A{-<O, if(1 -As)= M, M~

k .

~(0,1), i f ( l - A , ) < Ms Mr

Since A--O,

i f ( l -A,)> Ms Mr

if (1 - A~) = M s Mr

i f ( l -A~)< Ms Mr

(12)


Since Mx.-.*~ implies MJMx--*O, therefore we can always give a big Mx to prohibit the occurrence of the third case of (12). Thus, max A=0 or a e O .

Since if (1 -A~)>MJMK, there is no feasible solution of A, so are the values of A t, Vke {/+ 1 ..... K}. Hence, we discuss the case of max a = 0 where (1 -A~)=MJMK.

M~ Since (1 - as)= Mxx and M X - > I - ai,Vie {1+ 1 ..... K - 1 }

/

~M,( 1 -as _ )->1 ai %

Mi >--1 -hi ~(l - a~)

Ms Mi

Mi ~ ai > -- 1 - -- . MK

~ a ~ I, because M~<M K implies M~ 4 0 .

But ai~a~< 1, V ie {/+ 1 ..... K - 1},

.'. a i e 0 , V i e { / + l ..... K - 1}.

But, how large should MK be given? Since MK>-MJl - a,, from experiences, 1/1 - as} -- 10 K is easy to calculate. Hence, in general, M~=(10K) ~- ~ for k= 1,2 ..... K is suggested.

Now, we need to use the skill of a-cut set [13] to transform the Model 1 from fuzzy type into crisp type. After combining the crisp type of Model 1 and the equation (9), we wish that the larger the value of A, the higher the satisfactory level of a decision maker. This implies that max a is our goal in a new model as below.

Model 2

s.t.

max A, (13)

Ak< [ { (1-- [B~- (Ax)~] ) c'J} ]k'je { l'3 }'ijel~ kJ j (14)

Ak<--I { (1-- [(Ax)~-B~] ) d~J} ]k, je{1,2},ijellt, Jlz,kel 4 (15)

MkA' + A~>-l,keI4, (16)

A+MKa'- - I . (17)

k= 1,2 ..... K.

Therefore, the FGP was transformed into an LP or an NLP to maximize the total degree of membership (a) which satisfies all priority levels of goals. Formulae (14) and (15) are used to classify the same k 'h level of priority with the same degree of a~. for different types. Formulae (16) and (17) satisfy the rule of preemption. When any of k levels violates the rule of preemption, the larger the Mk, the sooner does the a approach zero. This model can be solved by commonly used software packages such as LINDO for


linear cases or GINO for nonlinear cases. In either case, only one iteration is required to obtain its solution. We have applied the method to solve the example used by Tiwari et al. [9] and Chen [10], and obtained the same solution [14]. As they only consider triangular membership functions in their example, we hereby use another example to discuss different cases in the FGP problems.

4. A NUMERICAL EXAMPLE

A manufacture produces three products. Two machines are functioned in sequence. Product 1 needs 2 h per unit on machine I and 3 h on machine II. Similarly, product 2 needs 6 h and 3 h, and product 3 needs 5 h and 8 h on respective machines. In addition, to produce products 1, 2, and 3 per unit need 4 tons, 3 tons, and 6 tons of material, and can make $ 30, $ 40, and $ 50 of unit profits respectively. The plant manager plans to make a total profit of "around $ 20000" by selling "at least 100 units" of each product in which "around 2500 h" and "between 2000 and 2050 h" of the total working hours are operated on machine I and II respectively when "at most 1500 tons" of the amount of material is used. How much of each product should be produced in order to achieve the manager's goals?

Let X~, )(2, and X3 denote the respective amounts of products 1, 2, and 3 that should be produced. We can describe the above goals without priority levels as below.

G:30X, +40X2 + 50X3=20000,

G2:2X, +6X2 +5X3=2500,

G3:3X l + 3X 2 + 8X 3- [2000,2050],

G4:4XI+3X2+6X 3 < 1500,

Gs:X ~ > 100,

G6:X 2 > 100,

GT:X 3 > 100,

X,~2~3-->0.

This means that I, = { 1, 2, 3}, 12= {4}, and 13= {5, 6, 7} in Model 1. According to the plant manager's experience, the parameters of the membership functions with respect to each goal are provided to be lj =5000, r I = 10000, 1z=13=50, r2=r3= 100, r4=500, 15=16=17= I0. Therefore, their membership functions can be defined below.

0~

(1 -

1,

( 1 -

0,

[20000 - 30X~ - 40X2 - 50X 3] Cl, 5000

[30Xj +40X2 + 50X 3 - 20000] )d,,

10000

0r

l, " /~(G2)= 1,

O,

if 30X~ +40X2 + 50X3 -< 15000,

if 15000--<30X1 +40X2+50X3_<20000

if 30Xl +40X2 + 50X3 = 20000,

if 20000-30X I +40X2 + 50X3-<30000

if 30Xl +40X 2 + 50X3>30000.

if 2X t + 6X2 +5X3<2450,

if 2450<2X~ + 6X 2 + 5X3_<2500,

if 2X~ + 6X 2 + 5X3--- 2500,

if 2500<2X~ +6X2 + 5X3-<2600,

if 2X~ + 6X 2 + 5X3-->2600.


0, if 3Xx + 3X 2 + 8X3_< 1950,

(1 - [2000 - 3X 1 - 3X 2 - 8X3] 50 )c~, if 1950<-3X~+3X2+8X3<_2000,

/z3(G3) = l , if 2000--<3X~ +3X2+ 8X3<_2050,

(1 - [3XI + 3X2 + 8X3 - 2050] 100 )a3, if 2050<-3Xj +3X2+8X3_2150 ,

0, if 3X~ +3X2 + 8Xs-2150.

1, if 4Xl+3X2+6X3<_1500,

bt4(G4)= (1 - [4X~-3X2+6X 3 - 1500] )e~ if 1500~-4XI+3X2+6X3<-2000, 500

0, if 4X~ + 3X2 + 6X 3 ~ 2000.

0, if X~-< 90,

/zs(Gs) = (1 [ 1 0 0 - Xl] 10 )c~, if 90<_X~_<100,

1, if XI>-100.

0, if X2-<90,

[100-X21 ~ 6 ( G 6 ) - - ' (1 10 )c6, i f90_X2<100,

1, if X2->100.

I 0, if X3-~90,

~ (G7)= (1 [100-X3] )c7, if90_~X3__.100, 10

1, if X3->100.

where cip dij~ { .5, 1, 2}, ij~ { 1, 2 ..... 7 } , j s { 1, 2, 3}. Since the plant manager intends to evaluate different operation strategies, four situations on the priority levels of goals are considered with different operations of membership functions. Specifically, they are described below.

Four considerations on goals.(C1) All goals are at the same level of priority.(C2) G~ is at the first priority level, the remainders are at the same level of the second priority. These are Pt = {Gt~} and

tG2 G21 P 2 = / 2 ..... 7j .(C3) G1 is at the first priority level. Gs, G6, and G 7 are at the second level. G2, G3, and G4 are at the third level. These are Pt = { G ~ }, - 2 2 2 P2- {G5,G6,G7}, and P3 = {G~,G3,G34}.(C4) Gj is at the first priority level. G5, G6, and G 7 are at the second level. G2 and G 3 are at the third level. G4 is at the last level. These are PI=GI , _ 2 2 2 _ 3 3 P2-G s,G 6,G T, P3-G 2,G 3, and P4= {G4}.

We summarize four considerations in Table 1. Four operations on membership functions:

(F1) c~= 'd,, = 1, i~ ~ { 1,2,3}, d4 = l, and c~= 1, i3~ {5,6,7}. This considers piecewise linear membership functions with a risk-neutral decision behaviour. (F2) c~ =d~ =2, i~ ~ { 1,2,3}, d4=2, and c i=2, i3~ {5,6,7}. These nonlinear forms attribute to contraction membership functions which reflect a risk-seeking behaviour. (F3) c~, =d~ = 0.5, ij e { 1,2,3}, d4 = 0.5, and c~3 =0.5 e {5,6,7}. By contrast, these nonlinear forms represent dilation membership functions. This, in turn, reflects a risk-aversion behaviour. (F4) c~ =dl =2, c~. =di~ =0.5, i~ e {2,3 }, d4=0.5, and ci~= l, i 3 ~ {5,6,7}. This is a combined case to represent different attitudes of a decision maker toward different issues in a problem.

We summarized four operations of membership functions in Table 2. It can be seen that we allow

Table 1. The lists of f our situations

Items G~ G2 G3 G4 G~ G6 G7

(el) Pi PI PI Pi Pt Pi Pt (C2) P~ P: P~ P2 P2 P2 P~ (C3) P~ Ps P3 P3 P2 P: P2 (C4) Pt Ps Ps P~ P~ P: P2

A generalization of fuzzy goal programming

Table 2. The lists of four operations about membership functions

Forms Items G, G., G 3 G~ Gs G6 G7

FI c,. 1 I 1 0 1 1 1 d~ 1 1 1 l 0 0 0 1~ 5000 50 50 0 10 10 10

$

r~j 10000 100 100 500 0 0 0 Type T1 TI TI T2 T3 T3 T3 Pref. n n n n n n n I.IN L L L L L L L

F2 2 2 2 0 2 2 2 c~ d~ 2 2 2 2 0 0 0

J

l~ 5000 100 100 0 10 10 10 Ty r'lae 10000 100 100 500 0 0 0

TI TI TI T2 T3 T3 T3 Pref. s s s s s s s L/N N N N N N N N

F3 c~j 0.5 0.5 0.5 0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0 0 0

/d:i 5000 100 100 0 10 10 10 J

r~ 10000 100 100 500 0 0 0 Type TI TI TI T2 T3 T3 T3 Pref. a a a a a a a L/N N N N N N N

F4 c~ 2 0.5 0.5 0 1 1 1 dj 2 0.5 0.5 0.5 0 0 0 l / 5000 100 100 0 10 10 10

1

Ty r'lae 10000 100 100 500 0 0 0 T1 TI TI T2 T3 T3 T3

Pref. s a a a n n n L/N N N N N L L L

827

different linguistic terms (T1, T2, and T3), risk preference (risk-neutral (n), risk-seeking (s), and risk- averse (a)), membership function (linear (L) and nonlinear (N)), symmetric shape (/i=ri), and asymmetric shape (l~j#ri) to appear in a model simultaneously.

We use the software package of LINDO to solve F1 and GINO to solve F2, F3, and F4. The results are shown in Table 3.

From Table 3, when F1 is used at situation (C1), the amount of three products are Xl =93, X2=297, and X 3 = 100 and all goals are achieved at the same level of A =0.27. At situation (C2) with two priority levels, the amount of three products are XI = 93, X: = 314, and X3 = 93. Goal (G I) of the first priority has achieved (,t] = 1), whereas the satisfactory level of goals (G 22 ..... G72) at the second priority is A2 =0.26. At situations (C3) and (C4), we can observe that both of them produce the same amounts of three products with the same value of A.

We can observe that, in general, the value of A decreases as the priority level increases at any of four operations. This is because the greater the number of higher levels, the less resources may be used at the lower levels. This leads to an overall lower satisfaction level.

Alternatively, the behaviour of a decision maker at F1, F2, F3 and F4 operations are with respect to risk-neutral, risk-seeking, risk-aversion, and the combinations. With four operations F1, F2, F3 and F4 of situation (C1), we can find that the amount of three products among F1, F2, and F3 situations are the same, but satisfactory levels to achieve all goals are 0.27, 0.08, and 0.52 respectively. This means that

Table 3. The results of the numerical example

T C h X~ X~ X~ ,t~ ,,/2 a3 .L

FI (CI) 0.27 93 297 100 (C2) 0.26 93 314 93 1 0.26 (C3) 0.2 100 300 100 1 1 0.2 (C4) 0.2 100 300 100 1 l l 0.2

F2 (CI) 0.08 93 297 100 (C2) 0.08 93 297 100 1 0.08 (C3) 0.04 100 299 100 1 1 0.4 (C4) 0.04 100 300 100 1 1 1 0.04

F3 (CI) 0.52 93 297 100 (C2) 0.52 93 297 100 1 0.52 (C3) 0.49 100 294 100 1 1 0.49 (C4) 0.45 100 300 100 1 1 1 0.45

F4 (CI) 0.51 96 297 99 (C2) 0.51 96 297 99 1 0.51 (C3) 0.49 100 294 100 1 l 0.49 (C4) 0.45 100 300 100 1 1 1 0.45


according to the subjective preference attitude of a decision maker, a risk-averser has a higher satisfactory level to achieve goals than that of a risk-seeker when the same amount of resources can be utilized in a fuzzy environment. Although the decision maker has combined risk attitude among goals at operation F4, the amounts of three products and satisfactory levels to achieve goals in different situations all are similar to or the same as those of F3. It means that the overall preferred attitude tends to be risk-aversion.

In contrast, if we change the priority sequence of goals, for example, let G 2, G3, and G4 be the first priority and the others be the second priority, then we cannot obtain the feasible solutions in any form of F1, F2, F3, and F4. As even the goals at the first priority level cannot be achieved (ex. A~ =0.29 in F1), it is needless to consider the goals at the lower levels and thus, the proposed model will show the infeasible values of A2 and A.

These explain that the proposed model can offer a decision maker an optimal solution according to his risk preferences and priority strategies with linguistic terms. When an infeasible solution appears, it tells that some levels of goals cannot be achieved under these sequence of priority. Hence, the decision maker may adjust the priority sequence of goals or supply more resources in order to achieve the goals.

5. CONCLUSIONS

This article develops an alternative method to solve a general FGP problem with preemptive structure. By using the concept of penalty cost, the defined FGP with different types of membership functions can be transformed into a simpler form so that the existing computer software can be applied easily. Although it needs additional K+ 1 constraints and variables, we can obtain final solutions with only one run. With the proposed model, a decision maker not only can present the preemptive structure of different goals, but also can use linguistic terms to express different preferences towards different considerations of goals and constraints with risk-seeking, risk-averse, and risk-neutral attitudes.

Acknowledgements--The authors gratefully acknowledge the financial support from the National Science Council, Taiwan, Taiwan, with project number #NSC84-2213-E007-009.

REFERENCES

I. Charnes, A. and Cooper, W.W., Management models and industrial application of linear programming. Wiley, New York, 1961.

2. Zadeh, L. A., Fuzzy sets. Information and Control, 1965, 8, 338-353. 3. Bellman, R. E. and Zadeh, L. A., Decision-making in a fuzzy environment. Management Sciences, 1970, 17, 141-164. 4. Luhandjula, M. K., Multiple objective programming problems with possibilistic coefficients. Fuzzy Sets and Systems, 1987, 21,

135-145. 5. Marfinson, E K., Fuzzy vs. minmax weighted multiobjective linear programming illustrative comparisons. Decision Sciences,

1994, 24, 809-824. 6. Tong, R. M. and Bonissone, P. P., A linguistic approach to decision-making with fuzzy sets. IEEE Transportation Systems,

Man, and Cybernetics, 1980, 10, 660-716. 7. Narasimhan, R., Goal programming in a fuzzy environment. Decision Sciences, 1980, 17, 325-336. 8. Hannan, E. L., Linear programming with multiple goals. Fuzy Sets and Systems, 1981, 6, 235-248. 9. Tiwari, R. N., Dharmar, S. and Rao, J. R., Priority structure in fuzzy goal programming. Fuzzy Sets and Systems, 1986, 19,

251-259. 10. Chert, H. K., A note on a fuzzy goal programming algorithm by Tiwari, Daharmar, and Rao. Fuzzy Sets and Systems, 1994,

62, 287-290. 11. Lai, Y. J. and Hwang, C. L., Fuzzy multiple objective decision making. Springer-Verlag, Berlin, 1994. 12. Holloway, C. A., Decision making under uncertainty models and choices. Hwatai, Taipei, 1982. 13. Zimmermann, H.-J., Fuzzy set theory and its applications, 2nd., edn. Kluwer, 1991. 14. Fu, C. C., A new approach to solving fuzzy goal programming problems with a preemptive structure. Journal of National

Defense Management College, 1996, 17, 52-55.

a generalization of fuzzy goal programming with preemptive structure

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