a fuzzy multiobjective linear programming

E L S E V I E R Fuzzy Sets and Systems 86 (1997) 61-72

sets and systems

A fuzzy multiobjective linear programming

H s i a o - F a n W a n g * , M i a o - L i n g W a n g

Institute of Industrial Engineering, National Tsing Hua University, Hsinchu, Taiwan, ROC

Received May 1995; revised November 1995

Abstract

In this study, we focus on an MOLP problem with fuzzy-numbered cost coefficients. Based on the membership functions, the problem is transformed into a multiobjective problem with parametrically interval-valued MOLP problem. Then by the proposed tolerance analysis of a nondominated set, a fuzzy set of the efficient extreme bases can be obtained. Therefore, the fuzzy solutions can be focused to provide complete information for the final decision. Theoretical developments are provided with numerical illustrations.

Keywords: Fuzzy number; Parametrically interval values; MOLP

I. Introduction

When modeling a multiobjective linear programming (MOLP), how to estimate the exact values of the coefficients is a problematic task. Normally, the coefficients are either given by a decision maker (DM) subjectively or by statistical inference from historical data. Therefore, to reflect this uncertainty we need to construct a model with inexact coefficients. Many authors [3, 13, 14, 16] considered this problem as a fuzzy linear program (FLP) with fuzzy coefficients of which a membership function was defined for each fuzzy coefficient. Thus, a fuzzy or crisp solution can be obtained.

Alternatively, from the viewpoint of operations research, this problem was regarded to be an interval- valued mathematical program. Oettli and Prager [7] are the pioneers. Then, several researchers [2, 5, 8, 10] followed and mainly searched for the approximate bounds of a solution set. Among them, Rohn [10] has proposed a solution procedure with theoretical support for finding the exact bounds. However, although the exact bounds of a solution set have been obtained, the exact solution mixes still remain unknown. Thus, no decision can be made because a solution that is selected arbitrarily within the bounds is most likely infeasible.

As regard to a multiple-objective optimization problem, Bitran [1] and Steuer [11] developed different algorithms to solve an M O L P of which the cost coefficients are also interval-valued. They applied the vector-maximum theory [19] to find the efficient extreme points. However, the algorithms are not easily

*Corresponding author.

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62 H.-F. Wang, M.-L. Wang / Fuzzy Sets and Systems 86 (1997) 61-72

implemented and the nondominated set cannot be obtained. By introducing the extreme inequalities constructed by the maximum and minimum values of the intervals, Tong [,16] proposed a solution procedure to cope with interval-valued linear program (LP) problems. Although an interval-valued optimal solution can be obtained, the exact solution-mixes for the final decision is still unknown. For this, the authors have proposed both theories and solution procedures for an M O L P with interval-valued coefficients [,181. The nondominated set can be obtained and the exact solution-mixes are derived to provide complete information. In particular, based on a lexicographical ordering, a final solution can be made by the proposed procedures.

In this study, we shall extend the developed method for an M O L P with interval-valued coefficients to an M O L P with general fuzzy cost coefficients. That is, we focus on a fuzzy M O L P in the following form of

Max Z(cO = (31 (~), fz(cO . . . . . fK(~)) t = ( U x , ~ 2 x . . . . , ~ r x ) ~

s.t. Ax ~ b, (1)

x~>0,

where t means "transpose", A = [ a i j ] , for i = 1, . . . , m , j = 1 . . . . , n is an m × n constraint matrix; b = [b~], for i = 1, ... ,m is the column vector of the right-hand sides (RHSs); t? k = [~1, J = 1 . . . . ,n; k = 1 . . . . . K is a column vector with fuzzy numbers. By introducing the concepts of the membership functions and the cone polyhedral theory, the properties of Model (1) will be investigated in Section 2 where a solution procedure is proposed to find the fuzzy nondominated extreme solutions. Furthermore, from the analysis, the nondominated solution sets of a parametrically interval-valued M O L P can also be obtained. Numerical examples are provided for illustration. Discussion and conclusions are made in Section 3.

2. An MOLP with fuzzy-numbered cost coefficients

Let ~3 be a set of bases. If/~ is the efficient extreme bases of Problem (1), then,/~ is a fuzzy subset of defined by

/~ = {(B, ~g(B))IVB e ~ } . (2)

where #g(B) is the degree of the membership of B in/~. Furthermore, l e t /~ be the membership function of c~ E R for each j , k . Then, an a-cut of the fuzzy set ~ is a crisp interval and can be found in the form [MincCa{c[~t~(c)/> ~}, Maxc~a{cl#~(c) i> ~}1. Furthermore, by the convexity of a fuzzy number, the interval can be obtained as [inf {(/l~)-~ (e)}, sup {(#~)-1 (e)}]- That is, the bounds of intervals are functions of e. Therefore, Model (1) can be transformed into a crisp M O L P with parametrically interval-valued cost coefficients as defined in the following:

Max Z(~) = (z 1(~), Z2(~) . . . . . zK(~) ) t = (C l ( 0 0 x , C2(00X, . . . ,cK(o~)X) t

s.t. Ax ~< b, (3)

x~>0,

where c k(~) = I-c~ (~)1, c~ (a) e [-inf {(#~) -1 (~)}, sup {(~t~) -1 (a)} ], for k = 1 . . . . , K, j = 1 . . . . , n with inf {(#~) -1 (~)} and sup {(~t~)-1(~)} being lower and upper bounds of the cost coefficients, respectively. When ~ increases from 0, the compatibility of cost coefficients decreases. Therefore, the intervals are getting narrower. That is, the larger the ~, the smaller is the cost interval. Equivalently, giving ~1 and ~2 in interval [,0, 11 with ~1 < ~2, we have [inf { (~) - 1(~ 2)}, sup {(g~)- ~ (~e)}] -~ [inf{(#~)- ~(~1 )}, sup {(/~)-1 (~1)} ] for each k, j. That is, the intervals of a cost coefficient are nested with respective to ~.

H.-F. Wang, M.-L. Wang / Fuzzy Sets and Systems 86 (1997) 61-72 6 3

According to our previous s tudy [18], when a cost coefficient, say ~(c~), varies in an interval [inf{(p~)- ~ (c0}, sup {(p~)- l(e)}], the gradient of the objective function varies, respectively. Therefore, there are infinite gradients, and any single objective function will become a mult iple or infinite objective functions. Because the gradients can be expressed as a convex combina t ion of the K x 2" extreme gradients [10], say {e (n- ~)×2"+'(c0, I = 1, ... ,2"} = {e(c~)e N"[e(~) = [-ej(~x)], ej(~) 6 {inf{(p~)-~(c0}, sup{(#~)-~(c0}}, j = 1 . . . . ,n, k = 1, K}, there exists some 2~, i = 1, , K x ~" x "~ × ~" ~ = 1, such that any gradient 9(~) can be writ ten " ' " ' . . . . ' g - . i = l "~i

V~×2",i~ei(ct). In consequence, P rob l em (3) can be t rans formed into a pa ramet r ic v e c t o r - m a x i m u m a s ~-.i = 1

prob lem with K x 2" objectives as stated in the following p rob lem (4):

M a x Z = E(~)tx = (e~(~)x, e2(c0x . . . . . eK×2"(~)x)t

s.t. Ax ~< b, (4)

x ~ O ,

where E(~) = [ei(c0], i = 1, ... , K x 2", is a (K x 2") x (K x 2") matr ix of the cri terion cone of P rob lem (4). O n the other hand, it can be seen tha t once a value of e is given, P rob lem (3) becomes an interval-valued

p rob lem with cons tant bounds as denoted by P(c 0. In addit ion, due to the nested cost intervals, for any pair Of~l , ~2 ~: [-0, 1] with ~1 < ez, any gradient of P(~z) will be an element of P(e l ) . Therefore, the cri terion cone of P(e2) is embedded in that of P(ea). Fur the rmore , the relative interior of the cri terion cone of P(e2) is a subset of that of P (e l ). Thus, f rom the cone polyhedral theory [19], we define the semi-posit ive polar cones of P(c~i) by E ~ (~i) = {Y E ~ " l E ( ~ ) y / > 0, E(cq)y q: 0} w {0 e ~"} and the domina t ion set of any feasible point x in the feasible set S by Dx(cq ) = {x} ~ E ~ (~), i = 1, 2, respectively. Then, we have E ~ (~1) ~ E ~ (~2) for ~x < c~2.

Let B(~) denote the set of efficient bases of P(~) . We have the following result.

Theorem 1. I f O <<. ~1 <~ ~2 ~ "'" <<- an <<. 1, then B(1) _ B(~n) _ -.- ~_ B(eE) ~- B(0~I) c::: B(0).

Proof. Suppose x is an efficient extreme solut ion of P(~2) with the efficient basis B e B(ez). Tha t is, D x ( ~ 2 ) ~ S = {x}.

F r o m the p roper ty of direct sum of set theory, since E~(cq ) c E~(e2) , we have {x} @ E ~ ( ~ I ) c { x } ® E ~ (ez). Thus, f rom the definition of the domina t ion set, we have Dx(ex) _ Dx(c~2).

Therefore, it can be obta ined that DAcq)c~S ~ D~(c~z)nS. Since x is efficient, that is to say that {x} = DAz2)c~S, we have Dx(e~)nS __ {x}.

Conversely, since x is feasible, i.e. x e S and f rom the definition of D~, x e D x ( e ~ ) too. Tha t is, { X } ~ Dx(~I)~S .

Thus, we have D~(~ I )nS = {x}. Tha t is, x is also an efficient solution of P(a~). So, B E B(~I) and B(a2) -- B(al) , for any ~1 ~ CX2'

Finally, let us take different levels of ai as defined, we have the stated sequence. [ ]

T h e o r e m 1 says that any efficient set of different levels of c~ is a subset of the efficient set B(0) and they are also nested. Thus, once the set of B(0) is found, all possible efficient extreme solutions of P rob lem (3) can be found. When ~ is given to be 0, it is equivalent to our considered p rob lem [18] of which a solution procedure has been developed to find the n o n d o m i n a t e d set and the exact solut ion mixes. When c~ varies, an al ternat ive procedure to find all of efficient sets is p roposed below.

First, let us take a look at P(0). P(0) can be t ransformed into a general M O L P with K x 2" objectives as we have ment ioned above. The concept of the irreducible genera tor of the criterion cone p roposed by Telgen [15] can be used to reduce the n u m b e r of objectives [18]. Then, an efficient set of P(0) can be searched to obta in B(0), a simplified M O L P p rob lem [-4, 9].


Fig. 1. The distribution of the efficient extreme bases.

So far, we have all possible efficient bases. The remaining question is how these bases distributed with respect to different ~. To cope with this issue, we shall turn to Problem (3). Problem (3) can also be transformed into a K x 2" objective LP but with parametric cost coefficients. To obtain all possible efficient bases with respective to ~, Wang and Ou-Yang [17] have provided both theoretical evidence and algorithms to deal with this problem. When ~ varies, the nonbasic reduced cost matrix F(~) corresponding to each efficient basis will be the function of ~. A basis remains nondominant if and only if there exists a vector of weights W= (wl, . . . , W K x 2 , ) t with all wi > 0 such that g t F ( ~ ) ~> 0. Thus, for a vector W, we can find an interval of a to satisfy the relation. The set of a corresponding to all possible W's is the largest allowable interval of parameter ~. By solving this nonlinear problem, the largest allowable region (critical region) of each basis can be obtained. Conversely, for different selection of ~, we can find the corresponding bases if

falls in their critical regions. Thus, the distribution of the efficient bases is obtained. Furthermore, since all bases are derived from P(0), 0 is a common element in all allowable intervals. Without loss of generality, let the allowable interval of basis B~ be [0, ~i] with 0 ~< al ~< "'" ~< ~q ~< 1, the efficient sets can be obtained from B(1) to B(0) by decreasing a, which constructs a nested structure as described in Fig. 1.

Now, let us turn to our original problem (1). From the above discussion, since the basis Bi is an efficient basis with the largest degree ~. That is, B~ is a member of /~ with degree a~. We have /~= {(B1, ~1),--. ,(Bq, ~q)} with a-level subset B(~) = {BI#/~(B) ~> ~}. Then, the Decomposition Theorem [6] holds as shown in the following.

Theorem 2 (Decomposition theorem).

B = V ~ ' B ( a ) , 0 <. ~ <~ 1, at

where

f l /f #n(B) ~> ~, /~B(~)(B) = /f #n(B) < ce.

Proof . See [6] . [ ]

Furthermore, when the membership degree is given, since the cost coefficients are interval-valued, so do the objective values. Let us denote the objective value as [Z_(a), Z(~)] = ([_zl(~), ~1(~)] . . . . . [_ZK(~), ~K(~)]t, then, the following theorem states its property.

H.-F. Wang, M.-L. Wang / Fuzzy Sets and Systems 86 (1997) 61-72 65

Theorem 3. / f0 <~ 0{1 ~< "'" ~< % ~< 1, then [Z_(I), 2(1)] ~ [Z_(%), 2(%)] ~ ... _ [Z_(0{x), 2(0{,)] _ [-Z(0), 2(0)] for each nondominated solution.

Proof. First, from Theorem 1, the efficient solution x of B(0{2) is also efficient in B(0{1). Second, by its nonnegativity, x t> 0. And, finally, [_c k (0{2), ~k(0{2)] - [-ck(0{1), ~k(0{1)] is according to its nested structure. We have [c_k(0{2),ck(0{2)']X CO_ [c_k(0{1),ck(0{1)]X, i.e., [zk(0{),zk(0{) "] ~. [-zk(0{),zk(0{) "] for all k = 1, ... ,K. Thus, [-Z(0{2), 2(0{2) ] C2 [-Z(0{1), Z(0{1)].

Similarly, let us take different levels of 0{ and the result is obtained. []

Let us summarize the above solution procedure in the following: (1) Transform an fuzzy M O L P problem into a crisp interval-valued problem P(0{) defined in (1). (2) Set 0{ = 0 and find all of efficient bases of the interval-valued problem P(0) [18]. (3) Find the critical region of the derived basis by Wang and Ou-Yang's algorithm. (4) Distribute the efficient extreme bases with respective to 0{ and find their corresponding solutions. Let us use an example to explain the proposed procedure.

Example 1.

Max ?~xl + ?~x2 + ?~x3

Max ?~Xl + 72x2 + ?~x3

Max galXl + CIX3

s.t. 3xl - - X2 "At- 3X3 ~ 6

- - X 1 + 2 X 2 - x 3 ~ < 9

- - X l + 5 X 2 + 4 X 3 ~ < 8

5xl --x3 ~< 10

- - X 1 - - X 2 +2X3 ~ 8

Xi,X2,X3 >~ O.

(5)

.l(cl) = (6)

~(c2 i) = { (7)

U~(c~) = t (8)

c~ - 5, cl e [5, 6];

1, c~ ~ [6, 8];

(10 - c~)/2, c~ ~ [-8, 103;

0 otherwise;

c2 I + 1, c ~ e [ - - 1 , 0 ] ;

0 otherwise;

(c~ + 2)/0.5, c3 x ~ [ - 2 , -1.53;

( - 1 - c ~ ) / 0 . 5 , c l e [ - 1 . 5 , - 1 ] ;

0 otherwise;

The functions of these fuzzy coefficients are defined as follows and are shown in Fig. 2 where ~2 ~ , ~3 ~ , 5 2, ~2, ~ and ~3 a are triangular fuzzy numbers; ~ is a trapezoidal and ~23 is a nonlinear fuzzy number:


l 1 1 .., (c,)

5 6 10 P-C[

1 1 ,u2 (%)

/ 1

-1 (a)

I C 2

| I

-2 -1.5 -1 (c)

2 2 ~*~ (¢~)

2 t

% 1 3 5

(d)

2 2

3 4 6

(e)

2 2 C 2 ~ ~-~ C 3

0 1.5 3

(0

-1 1 5

3 3 p3(c3)

3

0 1 c3

(g) (h)

3 1 (C) 1 (d) 2 (e) 2 (f) c3 2, (g) c 3, (h) c 3 Fig. 2. The graphs of the membership functions of (a) c~, (b) c 2, c 3, c 1 , c 2,

(c~ - 1 ) / 2 , c~ e [1 , 3];

/~2(c2) = ( 5 - c2) /2 , c~ ~ [3 ,5 ] ;

0 otherwise;

c~ 2 -- 3, c l e [3, 4];

/~2(c22) = ( 6 - c2) /2 , c 2 ~ [4, 6];

0 otherwise;

# 2 ( c ~ ) = { ~ 0 1 - - [ ( c 2 - - 1 " 5 ) / 1 " 5 1 2 ' c~ ~ [0, 3];

o therwise;

(9)

(lO)

(11)


[ ( c ~ + 1 ) / 2 , c ~ s [ - 1 , 1 ] ;

p3(c3) = (5 - c~)/4, c~ ~ [1 ,5] ;

0 otherwise;

{c~ - - 1, c 3 e [ O , 1 3 ;

~ ( c 3 ) = 0 otherwise.

(12)

(13)

2.1. a-level transformation

For each e-level, we can obtain the corresponding cost intervals. Take gl as an example, we have

f c l - 5 >i e, c l ~ [5, 6], (14) / ~ { ( c l ) / > e i f f ~ ( 1 0 _ c l ) / 2 ) e, c I ~ [ 8 , 1 0 ] .

Then, we have cl E [5 + e; 10 - 2e]. Similarly, we have c I e [ - 1 + e, 0]; c~ e _ [ - 2 + 0.5e, - 1 - 0.5c~];

c 2 e [1 + 2c~, 5 - 2e];c 2 e [3 + c~, 6 - 2e];c~ e [1.5 - 1.5x/1 - e 2, 1.5 + 1.5x/1 - e2];c~ e [ - 1 + 2e, 5 - 4el and c3 3 E [1 + 5~, 6]. Thus, Problem (5) is t ransformed into the following:

Max [5 + c~, 10 - 2e]Xl -+- [ - 1 + e, 0]x2 + [ - 2 + 0 . 5 e , - 1 - 0 .Se]x3

20~]X 1 + [3 + a, 6 -- 2e]x2 + [1.5 -- 1.5x/1 -- e 5 , 1.5 + 1.5x/i- -- c~2]x3

4 e ] x l + [1 + 5e, 6]X 3

Max [1 + 2c~, 5 -

Max [ - 1 + 2c~, 5 -

s.t. 3xl - - X 2 -~- 3X 3 ~ 6

- - X1 -~ 2X2 - - x3 ~ 9

- - X t -k 5X2 A-4X3 ~ 8

5X 1 - - X 3 ~ 10

- - X 1 - - X 2 -{- 2X3 ~< 8

X 1 , X 2 , X 3 ~ O.

2.2. Finding all possible efficient bases

For e = 0, we have P(0) as the following problem (16):

[5, 10]Xl + [ - 1 , 0 ] x 2 + [ - 2 , - - 1 ] X 3

P(O):

[1, 5 ]x l + [3, 6]x2

[ - - 1, 5 ] X 1

3xi - x2 + 3x3 ~< 6

- - X 1 + 2 X 2 - - X 3 ~ < 9

- - X 1 "nt-5X2 +4x3~<8

5xl -- x3 ~< 10

- - X 1 - - X 2 + 2 X 3 ~<8

X 1 , X 2 , X 3 ~ O.

+ [0, 3] x3

+ [1, 6]x3

M a x

Max

Max

S.t.

(15)

(16)


Then, we have three variables, five functional constraints and three objective functions. Since the third objective has a constant cost, there are 20 objective functions to be considered as discussed in problem (3) and the criterion cone is stated in the following:

t 5 5 5 5 10 10 10 10 1 1 1 1 5 5 5 5 - 1 - 1 5 5

E = - 1 - 1 0 0 - 1 - 1 0 0 3 3 6 6 3 3 6 6 0 0 0 0

- 2 - 1 - 2 - 1 - 2 - 1 - 2 - 1 0 3 0 3 0 3 0 3 1 6 1 6

(17)

First, using Telgen's method [15] the number of objectives are reduced and an irreducible generator of the criterion cone is found. It is obtained that objectives 2, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 18, 19 and 20 are redundant and can be deleted from the criterion cone and the remaining ones form a minimal generator. Thus, we have a reduced problem with five objective functions as stated in Problem (18):

Max z a = 5 x a - x 2 - 2 x 3

Max z 2 = 5x1 - 2x3

Max z 3 = x l + 6 x 2

Max z 4 = X x + 6 X 2 + 3 x 3

Max z 5 = - x ~ +x3

s.t. 3xa - - X 2 "t- 3X3 ~< 6

--Xa +2X2 --X3~<9

-- Xl + 5X2 + 4X3 ~< 8 (18)

5x~ - x 3 ~< 10

- X x - x 2 + 2 x 3 ~ < 8

X1,Xz ,X 3 ~ O.

Problem (18) has 5 efficient extreme solutions:

Basis Solution

(x2, x4, Xs, xT, xs):(Xl, x2, x3) = (0, 1.6, O)

(Xl , X2, XS, X6, X 8 ) : ( X l , X2, X3) = (2, 0, 0)

(xl , x2, x4, xs , xs) : (x l , x2, x3) = (2, 2, 0) (19)

(x2, x3, Xs, x7, xs) : (x l , x2, x3) = (0, 0, 2)

(xl, x2, x3, xs, Xs):(xl, x2, x3) = (2.092, 1.651, 0.459).

That is, all possible efficient extreme bases of different a of Problem (15) are obtained by the combination of the above 5 bases.


2.2. Finding the critical regions of the bases

Now, we have the n o n d o m i n a t e d so lu t ions of P r o b l e m (15). Let us turn to P r o b l e m (15). P r o b l e m (15) can be t r ans fo rmed into a twenty object ive p r o b l e m as s ta ted in the following:

M a x z 1 = ( 5 + c 0 x l + ( - 1 + ~ ) x 2 + ( - 2 + 0 . 5 c t ) x 3

M a x z 2 = (5 -'~ ~)x 1 + ( - - 1 -~- 0~)x 2 + ( - 1 - 0.5ct)x3

M a x z 3 = (5 + ~)xl + ( - 2 + 0.5~)x3

M a x z 4 = (5 + c¢)xl + ( - 1 - 0.5e)x3

M a x z s = ( 1 0 - 2 e ) x l + ( - l + e ) x 2 + ( - 2 + 0 . 5 ~ ) x 3

M a x z 6 = (10 - 2~)x l + ( - 1 +e)xz + ( - 1 - 0.5c0x3

M a x z 7 = ( 1 0 - 2e)x l + ( - 2 + 0.5c0x3

M a x z 8 = ( 1 0 - 2e )x l + ( - 1 - 0.5e)x3

M a x z 9 = (1 + 2c<)xl + (3 + e)x2 + (1.5 - 1.5~/1 - e2)x3

M a x z 1° = (1 + 2e )x l + (3 + a)x2 + (1.5 + 1.5~/1 - c~2)x3

M a x z 11 = (1 + 2a )x l + ( 6 - 2e)x2 + (1.5 - 1.5~/1 - eZ)x3

M a x z 12 = (1 + 2c<)xl + (6 - 2~)x2 + (1.5 + 1 . 5 / 1 - aZ)x3

M a x z 12 -- (1 -]- 2~)X 1 ÷ (6 -- 2ct)x 2 + (3 -- ct)x3

M a x z 13 = (5 - 2~)xl + (3 + e)x2 + 2ex3

M a x z 14 = (5 - 2c~)xl + (3 + e)x2 + (3 - e)x3

M a x z 15 = (5 - 2e)Xl + (6 - 2e)x2 + 2~x3

M a x z a6 = (5 - 2~)xl + (6 - 2~)x2 + (3 - ~)x 3

M a x z 17 = ( - 1 + 2~)xa + ( 1 + 5a)x3

M a x z xs = ( - 1 + 2~)Xl + 6x3

M a x z 19 = (5 - 4C0Xl + (1 + 5~)x3

M a x z 2° = (5 - 4o0x I + 6x 3.

(20)

Thus, by W a n g and Ou-Yang ' s a lgo r i thm we have the cri t ical regions of the efficient ex t reme bases deno ted by B1 . . . . . B5, respectively, and are shown as follows:

Basis largest a l lowable in terval

B1 = (x2, x4, xs , xT, xs ) : [0 ,0 ]

B2 = (x l , x2, xs , x6, Xs): [0 ,0.96]

B 3 = (x1, X2, x4, x5, x8): [0, 1] (21)

B4 = (x2, x3, x5, xT, x s ) : [0, 1]

B5 = ( x l , x 2 , x 3 , x s , X s ) : [ 0 , 1 ] .


.1 /t

Fig. 3. The distribution of the efficient extreme bases of Example 1.

That is, the interval [0, 1] of ~ can be divided into 2 subregions by e = 0.96 and the bases can be partitioned into 3 groups as shown in Fig. 3.

2. 4. Finding the optimal solutions with respective to

Thus, the sets of efficient extreme bases of parametrically interval-valued problem (15) and the a-cut of the original problem can be derived as

I {B1, B2, B3, B4, Bs} for ~ = 0;

B(c0 = J{B2, B3, B4, Bs} for 0 < ~ ~< 0.96; (22)

[{B3, B4, Bs} for 0.96 < ~ ~< 1.

And, we have/~ = {(B1,0), (B2, 0.96), (B3, l), (B4, 1), (B5, 1)). Furthermore, B(0) = {B1, B2, B3, B4, Bs}, B(0.96) = {B2, B3, B4, Bs } and B(1) = {B3, B4, B5 }, thus B(1) c B(0.96) = B(0), this provide an evidence for Theorem 1.

Thus, if we choose the satisfactory level ~ to be 0.6, (2, 0, 0)t, (2, 2, 0) t, (0, 0, 2) t and (2.092, 1.651, 0.459) t are the efficient extreme solutions, and the objective values are

[11, 18]

[4.4, 7.6]

[0.4, 5.2]

for solution (2, 0, 0)t;

[10.4, 17.6]

[11.6, 17.2]

[0.4, 5.2]

for solution (2, 2, 0) t;

[ - 3 . 4 , - 2 .6 ] [ [0.3, 2.7] for solution (0,0,2) t and [10.2745, 17.8127] [8, 12] [10.6837, 17.1137]

[2.2544, 8.1932]

for solution (2.092, 1.651,0.459) t .

These values provide an evidence for the nondominance of the solutions. For different choices of ~, the proposed method gives different solutions for a D M to make proper evaluations.


On the other hand, let us take solution (2, 2, 0) t as an example. When e is chosen as 0.8, the objective value is

[11.2, 16.8]

[12.8, 15.6].

[1.2, 3.63

Since 1-11.2, 16.8] ~ [0.4, 17.6], [12.8, 15.6] c [11.6, 17.2] and [1.2, 3.6] c [0.4, 5.2], this shows the nested structure of the objective value as stated in Theorem 3.

3. Conclusion and discussion

In this paper, we focus on an M O L P problem whose cost coefficients can be any types of fuzzy numbers. When the membership degree is given as a parameter ~, the problem can be t ransformed into an M O L P with parametrical ly interval-valued cost coefficients and the intervals are functions of ~. We are shown that when

is set to zero, all efficient extreme solutions can be found. Then, by using W a n g and Ou-Yang ' s algorithm, the critical region of each solution can be found, and, the distr ibution of the solutions can be obtained. It has been proved that the larger the parameter , the less (or the same) is the number of the solutions. Thus, the solution sets are nested with respective to ~, so do the objective values. Then, the opt imal solutions can be obtained corresponding to different levels. This will provide complete information for the flexible decision makings and applications.

Acknowledgements

The authors would like to thank the Nat iona l Science Council, Republic of China for financial support of this manuscr ipt under the contrac t No. N S C 81-0415-E007-02.

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a fuzzy multiobjective linear programming

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