Pergamon Computers ind. Engng Vol. 27, Nos 1--4, pp. 457-460, 1994

Copyright © 1994 Elsevier Science Ltd 0360-8352(94)00148-0 Printed in Great Britain. All fights reserved

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F o r m u l a t i o n a n d A n a l y s i s o f F u z z y L i n e a r P r o g r a m m i n g P r o b l e m s

b y U s e r O r i e n t e d R a n k i n g C r i t e r i a

Yozo N a k a h a r a * a n d M i t s u o G e n t

*Sh izuoka Se ika Col lege , Ya izu 425, J a p a n

t A s h i k a g a I n s t i t u t e of Techno logy , A s h i k a g a 326, J a p a n

Abstract

We propose new ranking criteria of FNs(fuzzy numbers). We also propose a formulation of lineal- progranmfing prob- lems with FN coefficients, by using the proposed criteria, and show an algorithm for solving the formulated problems in some cases.

Key words: Ranking criteria, fuzzy linear programming

1 Introduction

When we model a decision making problem in a real world by the mathematical programming, crisp coefficients or crisp target values are set, conventionally. But, in some cases, such crisp values have some vagueness or ambiguity at the problem in fact. Then, mathematical programming which can take such vagueness or ambiguity into account with fuzzy theory has been tried to develop. And the mathe- matical programming problems of which coefficients or goals are fuzzy sets or FNs have been proposed and researched[l], [2].

As well known, Dubois and Prade have proposed four in- dices of inequality between FNs by the concept of possibility and necessity[3]. Using the ranking criteria produced ordi- nary from these ranking indices, the four type treatments of constraint with inequality between FNs have been proposed by Dubois[4]. These indices and the treatments of con- straints are very useful, but in some cases, a more detailed treatment of the constraints is required[5]~[10]. Moreover, application of the ranking criterion for the treatment of an objective function has been limited in special cases.

In this paper, we will propose new ranking criteria of FNs each of which is defined using two parameters, and show that the proposed criteria include the criteria pro- duced ordinary from the indices proposed by Dubois and Prade. Moreover, we will propose quantitative formulation of LP(linear programming) problems with FN coefficients, by using the proposed ranking criteria, and show an algo- rithm for solving the formulated problems in some cases. In this formulation, DM(decision make)'s request for the treat- ment of a constraint can be reflected more precisely than in

the treatments of a constraint proposed by Dubois. Also, the optimization of an objective function with FN coeffi- cients is formulated as the user oriented extension of the optimization of an objective function with real coefficients by the proposed ranking criteria.

2 Notations and Arithmetics

We denote {x 6 R ia < x < b} by [a,b] for any a, b E R such as a _< b and we call this a closed interval. We also denote the right-hand side of any closed interval 1 by I n and the left-hand side by I L.

A fuzzy set is denoted by the letter(s) with tilde on its top as 5 or ~ . We denote a-level set of ~ by ~(a) or [~]~ or a~ for any a E [0, 1]. For any FN ~, a-level set is a closed interval for a E [0, 1].

Clearly, the following relation holds for any FN (-(':

~ L ~ R

[O ]o= [ [ c ] o , [c ]o ] for any ,:, E [O, ~]. (~)

In this paper, we define the multiplication of a interval by a real number and the summation(addition) or subtraction of intervals usually, and the multiplication of a fnzzy number by a real number and the summation(addition) or subtrac- tion of fuzzy numbers are defined by u s in~the extension principle as usual. For arbitrary FNs C, C1 and ~.'~, and for arbitrary real numbers a E [0, 1] and k, the following relations hold:

[c, + c~]o = [5;,]o + [c-;]o L - - R [[O;,]o" + [ ~]~, , [5;,],, n + = [C~]o ], (2)

[c, - C~lo = [O;,]o - [~]o [[O-;,]o": [+ , [~ ] ,n [~]o,:], (a)

[kS]o = [~k]o = k[~]o ~nk_Ft [k[C L , ' [C]o ], f o , . k > O

= k - n k - L - (4) [ - [c] , , , .[c]o 1, for k < o.

For any point al, a2 in R and q 6 R, ala2(q)~ is defined as the number(i.e, point in R) which divides the vector cqa~

457

4 5 8 Selected papers from the 16th Annual Conference on Computers and Industrial Engineering

by q : 1 - q. Clearly, we have the following relation for arbitrary a~, a:, q ~ R:

a - ~ ( q ) = a-~(1 - q). (5)

For any closed interval lax, a~] and [b~, b~] and any real num- ber q, we define [a,, a~](-< q)[b,, b2l as follows[5],-~[9]:

[a,, a~](-< q)[bi, b~] qa~ + (1 - q)a, < qbx + (1 - q)b~. (6)

For arbitrary fuzzy set ~ and b, the possibility measure H and the necessity measure N were defined as follows:

IIz(b) = sup{tq(y) A #;(Y)} (7) Y

Nz(b) = i~f{(1 - #~(Y)) Vg~(y)} . (8)

In the above equations, a A b means min{a,b} and a V b means max{a, b} for any real number a and b.

For any fuzzy number ~, ]~, oo) and [~, ~ ) are defined as the fuzzy numbers with following membership functions respectively:

~1~,~)(~) = }n~(1 - ~ ( ~ ) ) (9)

#[~,,,o)(r) -- sup#~(u) (10) r>_u

3 R a n k i n g C r i t e r i a

Dubois and Prade defined the following four indices of an inequality for arbitrary fuzzy numbers ~ and b[3]:

Pos(~ < b) = H~([5, oo)) (11) Pos(~<b) = H~(]a,~)) (12) Nes(5<_b) = N~([H, oo)) (13)

N e s ( a < b ) = N~(]~,co)) (14)

The following Theorem 1 is well known:

T h e o r e m 1 For any a E [0,1], the following ,,elations hold:

Pos(~ -< "b) >_ c~ ¢=:v 7~(a) L -< b(a) R (15)

Pos(~ < b) >_ a ¢:=v a(1 - a ) " -< b(a) a (16)

Nes(a < ~) >_ ~ ~=~ ~(a)L _< ~(1 - a) L (17)

Nes(a < 5) >_ a ~=* a(1 - a)R -< i(1 - a)L (iS)

Hence, by anyone of indices and cutting level a E [0,1] set by DM, one ranking criterion can be made. But DM wants to use another ranking criterion in some cases. Then, we define the following four type of new ranking criteria for FNs set by DM with two parameters:

T y p e l c r i t e r i on : B is larger than A if and only if fi(< (Typel(a, q)))B

Type2 criterion: B is larger than A if and only if 7t(<_ (Type2(a, q)))[~

T y p e 3 c r i t e r i on : /~ is larger than A if and only if A(< (Type3(a, q)))B

T y p e 4 c r i t e r ion : B is larger than A, if and only if 71(-< (Type4(a, q)))B

where for arbitrary FNs .4 and B, any q C R. and ally a E [0, 11, we define .4(< Tyl~el(a,q))B, A(<_Type2(a,q))B, A(-< Type3(a,q))B and A(< Type4(~,q))B as tollows:

A(-< Typel(a, q))B

A~L~A~ (q) -< B~R~B~ (q) (19)

fl(-< Type2(a, q) )[~

¢==~ AI_~,RAI_.L(q) -< B~,nB.L(q) (20)

A( <_ Type3( a, q) )B

¢::::> A~LA~R(q) <_ Bx_~LBx_~R(q) (21)

~( <<_ Type4(., q) )~ ¢=:> A,_~RAx_~}~(q) -< Bx_~LBx_~(q) (22)

These ranking criteria include conventional ranking criteria as the case q = 0, clearly. For aa'bitrary FNs A and /~, we have following relations fi'om (6) and interval arithmetics, clearly:

A~L--~A~ (q) -< ~ ( q ) ¢=~ qA~ R + (1 - q)A~ L

-< qB. L + (1 - q)B. R

¢=~ A.( < q)B~, ¢==> ( -As)( -< (1 - q))(-B~), (23)

AI_~nA,_.L(q) <_ B.RB.L(q) qAl_~ L + (i - q)A,_~ n

< qB~ L + (1 - q)B~. n

¢=:v (-Ax_~)( < q)B~ AI-~(_< (1 - q))(-B~), (24)

AoLA.R(q) <_ Bl_~LBx_.n(q) ~:==> qA~ R + (1 - q)Ao L

-< qBl_~ R + (1 -- q)Bl_~ L

¢==> A~(-< q ) ( - B , - . ) ¢:=* (-A.)(_< (1 - q))Bx-~, (25)

AI-.RAI-ff3(q) -< BI-~,LBI-.~(q)

¢==> qAl_c, L + (1 - q)Ax_~ n < qBx_~, R + (1 - q)Bl-o L

( - A I - ~ ) ( _ < q ) ( - B l - o ) ¢:::* A,_.(_< (1 - q))Bx-.. (26)

4 M a x O p e r a t o r s

W e denote the following set (27) by (28) [10]:

{S e FI 9($(<_ Typek(a,q))'}' and -~(T(< Typek(~,q)),_q))

for all : F E F }

max(Typek(a, q) ) F

(27)

(2s)

Selected papers from the 16th Annual Conference on Computers and Industrial Engineering 4 5 9

where F is a set of FNs, k • {1,2,3,4}, a E [0,1] and q • R. Let us consider a problem of finding the following

sets:

max(Typek(a,q)) {Z(x) [ x • G}, (29)

{x • G IZ(~x) E max(Typek(a,q)) {Z(y) [ y • G}} (30)

where G is any set and 2 is any mapping from G into set of all FNs. We denote this problem by the following (31) and (32).

max(Typek(a, q)) Z(x) (31)

s.t. x • G. (32)

We denote an element of the set (29) by a max(Typek(a, q)) objective function value of the problem (31) and (32). We also denote an element of the set (30) by a solution of the problem (31) and (32). Note that for arbitrary c~ • [0,1] and q • R max(Typek(a q)) can be regarded as the operator from W s into W s where

W s = {SWI S W C the set of all FNs}. (33)

We have the following Theorem and it's Corollary:

T h e o r e m 2 For arbitrary FN J,, a • [0,1] and q E ( -oo, 1/2], the following relation holds:

fi E max(Typel (a ,q)){Z(x) I x E G} (34)

2 ~ {Z(x) I x • a } , (35)

A LA~a(1 -- q) >_ fl (36)

where G is any set, 2 is any mapping fi'om G into set of all FNs and [J is given by the following relation:

R ~ L 3=max{[Z(x) ]~ [Z(x)], ( l - q ) i x • G } (37)

Corol lary 1 For arbitrary a' • [0, 1] and q • (-oa, 1/2], the following relations hold:

max(Typel(a,q)) {Z(x) I x e G} (38)

: {J, • {Z(-x) I x • G} I (36)} (39) )

- - - - L - - R

= {Z(x) I x • G, [Z(x)]° [Z(x)l . (1 - q) >_ ,8}, (40)

{x • a I Z(x) • max(Typel(a,q)) {Z(x) I x • 6'}}

L ~ /~ = {x • c; I [Z(x)] . [Z(x)]~ (1 - q) > ~}. (41)

where G is any set, Z is any mapping fi'om G into set of all FNs and ~ is given by (37).

5 L P P r o b l e m s w i t h F N s

ALP problem with FN coefficients is formulated as follows: n

max Z(x) = ~ C3x j (42) j = l

s.t. E Aijxj _< Bi, i = 1,- . . , m (43) j = l

xj>_0, j = l , 2 , . . - , n (44)

where

C.AIE ZT:I/4-EE

: the objective FN coefficient of the j - th decision vari- able,

Ai-"j : the FN coefficient of the j - th decision variable in the i-th constraint,

: the right-hand side FN in the i-th constraint,

n : the number of decision variables,

xj : the j-th decision variable.

We propose the following method for quantitatively formu- lating the above LP problem with FN coefficients by using the ranking criteria:

step 1: Select k • {1,2,3,4}, a • [0,1] and q • R.

step 2: Select k~ • {1, . . . ,4}, a, • [0,1] and q, • R for any i • {1,.. . ,m}.

s t e p 3: Formulate the following problem:

max(Typek(a, q)) Z(--X) = ~_. ~x , (45) J = l

s . t . X • G 1 N " ' ' I~l ( J ' m (46)

where for any i • {1,.. . ,m}

Gi = {y • R ~ I L AI--~JYJ( <- Typeki(ai, qi))B~i, j = l

yj _> 0 for Vj • {1, . . . ,n}} (47)

Note that for any i E {1 , . . . ,m},

n

~ ~Ojyj(<_ Typeki(a~,qi))B;, (,18) j = l

yj >_ 0 tbr Vj • {1, . . - ,n} (49)

is a linear system of y • R n. Ill fact, for arbitr~try x, >_ 0(j = 1 , . . - ,n ) and i • {1, - . . ,m}, we have the following relations:

n

~ y / <_ rypel(~, q~) )~ j = l

n - - R - - L

¢ = . y~(q[Au] . + (1 - q)[A0]~ )xj j = l

- - L ~ R

<_ q[Bi]o +(1 - q)[Ui]o , (50)

3 = 1

n ~ L ~ R

¢:=* ~,(q[Aij]l-~ + ( 1 - q ) [ A , j l l - o ),rj j = l

L - - R

< q[B& + (1 - q ) [B& , (51)

L Ai'-~jyj(<_ Type3(al, q , ) )~ j = l

n - - R ~ L

*=* ~_,(q[Ao],~ + (1 - q)[Ao]o ):% j = l

- R , (52) <_ q[B,h-o + (1 - q ) [ ~ h _ o ~

460 Selected papers from the 16th Annual Conference on Computers and Industrial Engineering

n

~, ~ijyi( <_ rype4(~i, qi) )'~ j = l

n - - L ~ R

~_.(q[Aij]a-. + (1 - q)[Ais]l-. )xj j = l

R ~ L < q[Ui]x-~ + (1 a)[BI (53) _ - _ , _ ~ _ ~ .

These relations are obtained fi'om the definitions of the ranking criteria and the next relation for any i E {1, . . . , m}:

n

= Zt lox, j = l j = l

n ~ L n ~ R

= [Y~ [Aij], x j , ~ [Ai~], xj] (54) j = l j = l

where xj _> 0 for any j C {1, . . . ,n}.

6 A l g o r i t h m

We propose the following algorithm for solving the formu- lated problems (45) ~ (47) with k = 1 and q E (-c~, 1/2]:

step 1: Solve the following LP problem in order to obtain /3 given by (37):

max [Z(x)]o [Z(X)]o (1 - q) (55) s.t. x ~ G~ n . . . n Gm (56)

step 2: Solve the following LP problem in order to obtain 5' which is the set of a solution of this problem:

max ~ 0. xj (57) j = l

) L ~ B

s.t. [Z(x) L [Z(x)]° ( l - q ) > Z (ss) x e G~ n . . . n Gm (59)

where /3 is given by (37) in whicb G is replaced by C,~ n . . . N am.

step 3: Find the following set V:

v = { Z ( x ) I x ~ s } (60)

The Corollary 1 guarantees that S and V given by the above algorithm are the set of any solution of the prob- lem (45) ~ (47) with k=l and q _< (-o¢, 1/2] and the set of any max(Typel(a, q)) objective function value of the same problem, respectively.

7 C o n c l u s i o n

In this paper, we have proposed the new ranking criteria of FNs each of which is defined using two parameters, and have shown that the proposed criteria include the criteria produced ordinary from the indices proposed by Dubois. Moreover, we have proposed quantitative formulation of LP problems with FN coefficients, by using the proposed rank- ing criteria, and have shown an algorithm for solving the

formulated problems in some cases. In this formulation, DM's request for the treatment of a constraint can be re- flected more precisely than in the treatments of a constraint with an inequality between FNs proposed by Dubois. Also, the optimization of an objective function with FN coeffi- cients is formulated as the user oriented extension of the optimization of an objective function with real coeificients by the proposed ranking criteria.

R e f e r e n c e s

[1] C. V. Negoita, The Current Interest ill Fuzzy Opti- mization, Fuzzy Sets Syst., 6, 3, pp.261-269(1980).

[2] M. hmiguchi, H. Ichihashi, and H. Tanaka, Fuzzy programming: a survey of recent developments, in Stochastic versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uneertai~ty, pp.45- 68(1990), R. Slowinski and J.Teghem(eds.), Kluwer Academic publishers, Printed in the Netherlands.

[3] D. Dubois and H. Prade, Ranking fuzzy numbers in the setting of possibility theory, Information Science, 30, pp.183-224(1983).

[4] D. Dubois, Linear programming with fuzzy data in J.C.Bezek(ed.), Analysis of Fuzzy bzformation, Vol- ume 3: Applications in Engineerin 9 and ,5'cie~ce, CRC Press, Boca Raton, FL, pp.241-263(1987).

[5] H. Ishibuchi and H. Tanaka, Formulations and Analysis of Linear Programming Problem with Interval Coeffi- cients, Journal of Japan Industrial Manage me~*t Asso- ciation, 40, 5, pp.320-329(1988), in Japanese.

[6] Y. Nakahara, M. Sasaki, K. Ida and M. Gen, A Method for Solving 0-1 Linear Programming Problem with Interval Coefficients, Journal of Japan Industrial Management Association, 42, .5, pp.345-351(1991), in Japanese.

[7] Y. Nakahara, M. Sasaki and M. Gen, On the Lin- ear Programming Problems with Interval Coefficients, Computers and h~dustrial Engineering, 23. Nos 1-4, pp.301-304(1992).

[8] Y. Nakahara and M. Gen, A Method for Solving Lin- ear Programming Problems with Triangular Fuzzy Co- efficients Using New Ranking Index, Computers and Industrial Engineering, 24, pp. 1-4( 1993 ).

[9] Y. Nakahara and M. Gem New Ranking of Trapezoidal Fuzzy Numbers and Its Application to Linear Pro- gramming Problems, Proceedings in the Conference of the 9th Fuzzy System Symposiunz, pp.193-196(1993), in Japanese.

[10] Y. Nakahara and M. Gem New Ranking Criteria of Tri- angular Fuzzy Numbers and Application to Fuzzy Lin- ear Programming Problems, Proceedings in the Con- ference of Japan bzdustrial Manageme~t Association, pp.88-89(1993.5), in Japanese.