Interactive fuzzy programming for random fuzzy two-level programming problems through possibility-based fractile model

Expert Systems with Applications 39 (2012) 12599–12604

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Interactive fuzzy programming for random fuzzy two-level programming problems through possibility-based fractile model Masatoshi Sakawa ⇑, Takeshi Matsui Faculty of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan

a r t i c l e

i n f o

Keywords: Two-level programming Random fuzzy programming Possibility Fractile criteria optimization Interactive programming

a b s t r a c t This paper focuses on interactive decision making methods for random fuzzy two-level linear programming problems. Considering the probabilities that the decision makers’ objective function values are smaller than or equal to target variables, fuzzy goals of the decision makers are introduced. Using the fractile model to optimize the target variables under the condition that the degrees of possibility with respect to the attained probabilities are greater than or equal to certain permissible levels, the original random fuzzy two-level programming problems are reduced to deterministic ones. Interactive fuzzy nonlinear programming to obtain a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented. An illustrative numerical example demonstrates the feasibility and efficiency of the proposed method. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Decision making problems in hierarchical managerial or public organizations are often formulated as two-level mathematical programming problems (Sakawa & Nishizaki, 2009; Shimizu, Ishizuka, & Bard, 1997). In the context of two-level programming, the decision maker at the upper level first specifies a strategy, and then the decision maker at the lower level specifies a strategy so as to optimize the objective with full knowledge of the action of the decision maker at the upper level. In conventional multi-level mathematical programming models employing the solution concept of Stackelberg equilibrium, it is assumed that there is no communication among decision makers, or they do not make any binding agreement even if there exists such communication (Bialas & Karwan, 1984; Nishizaki & Sakawa, 2000; Shimizu et al., 1997; Simaan & Cruz, 1973). Compared with this, for decision making problems in such as decentralized large firms with divisional independence, it is quite natural to suppose that there exists communication and some cooperative relationship among the decision makers (Sakawa & Nishizaki, 2009). For two-level linear programming problems or multi-level ones such that decisions of decision makers in all levels are sequential and all of the decision makers essentially cooperate with each other, Lai (1996) and Shih, Lai, and Lee (1996) proposed fuzzy interactive approaches. In their methods, the decision makers identify membership functions of the fuzzy goals for their objective functions, and in particular, the decision maker at the upper level ⇑ Corresponding author. E-mail address: [email protected] (M. Sakawa). 0957-4174/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.eswa.2012.05.024

also specifies those of the fuzzy goals for the decision variables. The decision maker at the lower level solves a fuzzy programming problem with a constraint with respect to a satisfactory degree of the decision maker at the upper level. Unfortunately, there is a possibility that their method leads a final solution to an undesirable one because of inconsistency between the fuzzy goals of the objective function and those of the decision variables. In order to overcome the problem in their methods, by eliminating the fuzzy goals for the decision variables, Sakawa et al. have proposed interactive fuzzy programming for two-level or multi-level linear programming problems to obtain a satisfactory solution for decision makers (Sakawa, Nishizaki, & Uemura, 1998, 2000a). Extension to two-level linear fractional programming problems (Sakawa, Nishizaki, & Uemura, 2001), decentralized two-level linear programming problems (Sakawa & Nishizaki, 2002a; Sakawa, Nishizaki, & Uemura, 2002), two-level linear fractional programming problems with fuzzy parameters (Sakawa, Nishizaki, & Uemura, 2000b) twolevel nonconvex programming problems with fuzzy parameters (Sakawa & Nishizaki, 2002b), two-level integer programming problems (Sakawa, Katagiri, & Matsui, 2010) and fuzzy random twolevel programming problems (Sakawa & Kato, 2009; Sakawa, Nishizaki, & Katagiri, 2011) were also provided. The subsequent works on two-level or multi-level programming have been appearing (Abo-Sinna & Baky, 2007; Lee, 2001; Pramanik & Roy, 2007; Roghanian, Sadjadi, & Aryanezhad, 2007; Sinha, 2003). Recently, from a viewpoint of ambiguity and randomness different from fuzzy random variables (Kwakernaak, 1978; Puri & Ralescu, 1986; Wang & Qiao, 1993), by considering the experts’ ambiguous understanding of means and variances of random variables, a concept of random fuzzy variables was proposed, and

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mathematical programming problems with random fuzzy variables were formulated together with the development of a simulation-based approximate solution method (Liu, 2002). Under these circumstances, in this paper, assuming cooperative behavior of the decision makers, we consider solution methods for decision making problems in hierarchical organizations under random fuzzy environments. Considering the probabilities that the decision makers’ objective function values are smaller than or equal to target variables, we introduce fuzzy goals of the decision makers for the probabilities. Then we adopt the fractile model to optimize the target variables under the condition that the degrees of possibility with respect to the attained probabilities are greater than or equal to certain permissible levels. Interactive fuzzy programming to obtain a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented. It is shown that all of the problems to be solved in the proposed interactive fuzzy nonlinear programming become nonlinear programming problems and approximate optimal solutions can be obtained through the use of particle swarm optimization for nonlinear programming (PSONLP) (Matsui, Sakawa, Kato, Uno, & Tamada, 2008). An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method.

In the framework of stochastic programming, it is implicitly assumed that the uncertain parameter which well represents the stochastic factor of real systems can be definitely expressed as a single random variable. However, from the expert’s experimental point of view, the experts may think of a collection of random variables to be appropriate to express stochastic factors rather than only a single random variables. In this case, reflecting the expert’s conviction degree that each of random variables properly represents the stochastic factor, it would be quite reasonable to assign the different degrees of possibility to each of random variables. For handling such an uncertain parameter, a random fuzzy variable was defined by Liu (2002) as a function from a possibility space to a collection of random variables, which is considered to be an extended concept of fuzzy variable (Nahmias, 1978). It should be noted here that the fuzzy variables can be viewed as another way of dealing with the imprecision which was originally represented by fuzzy sets. Although we can employ Liu’s definition, for consistently discussing various concepts in relation to the fuzzy sets, we define the random fuzzy variables by extending not the fuzzy variables but the fuzzy sets. Definition 1 (Random fuzzy variable). Let C be a collection of e is defined by its random variables. Then, a random fuzzy variable C membership function

  Nð90; 102 Þ if c   Nð100; 102 Þ if c

  Nð110; 102 Þ if c otherwise;

e is a random fuzzy variable. More generally, when the mean then C values are expressed as fuzzy sets or fuzzy numbers, the corresponding random variable with the fuzzy mean is represented by a random fuzzy variable. 3. Random fuzzy two-level programming

9 e 11 x1 þ C e 12 x2 > z1 ðx1 ; x2 Þ ¼ C > > > > > = e e minimize z2 ðx1 ; x2 Þ ¼ C 21 x1 þ C 22 x2 for DM2 > > > > subject to A1 x1 þ A2 x2 6 b > > ; x1 P 0; x2 P 0; minimize for DM1

ð2Þ

where the two objective functions z1 and z2 are those of DM1 and DM2, respectively, and ‘‘minimize’’ and ‘‘minimize’’ mean that for DM1

for DM2

DM1 and DM2 are minimizers for their objective functions. Moreover, x1 is an n1 dimensional decision variable column vector for the decision maker at the upper level (DM1), x2 is an n2 dimensional decision variable column vector for the decision maker at the lower level (DM2), Aj, j = 1, 2 are m  nj coefficient matrices, and b is an m dimensional column vector. Observing that the real data with uncertainty are often distributed normally, from the practical point of view, we assume that e ljk , k = 1, 2, . . . , nj of C e lj ; l ¼ 1; 2; j ¼ 1; 2 is the Gaussian each of C e ljk which is represented random variable with fuzzy mean value M by an L-R fuzzy number characterized by the membership function

8 m s < L aljk if mljk P s  ljk  l Me ðsÞ ¼ s m ljk :R ljk if mljk < s; b

ð3Þ

ljk

where the shape functions L and R are nonincreasing continuous functions from [0, 1) to [0, 1], mljk is the mean value, and aljk and bljk

ð1Þ

C

In Definition 1, the membership function

8 0:5 > > > < le ðcÞ ¼ 0:7 > C 0:3 > > : 0

Consider the random fuzzy two-level linear programming problems formulated as

2. Random fuzzy variables

le : C ! ½0; 1:

emphasized here that if mean and/or variance of random variables are specified by the expert as a set of real values or fuzzy sets, such uncertain parameters can be represented by not fuzzy random variables but random fuzzy variables. As a simple example of random fuzzy variables, we consider a Gaussian random variable whose mean value is not definitely specified as a constant. For example, when some random parameter c is represented by the Gaussian random variable N(si, 102) where the expert identifies a set {s1, s2, s3} of possible mean values as (s1, s2, s3) = (90, 100,110), if the membership function l is defined by eC

le assigns each ranC

Þ. It should be noted here  2 C to a real number l ðc dom variable c eC that if C is defined as R, then (1) becomes equivalent to the membership function of an ordinary fuzzy set. In this sense, a random fuzzy variable can be regarded as an extended concept of fuzzy g sets. On the other hand, if C is defined as a singleton C ¼ fc e Þ ¼ 1, then the corresponding random fuzzy variable C and l ðc eC can be viewed as an ordinary random variable. When taking account of the imprecise nature of the realized values of random variables, it would be appropriate to employ the concept of fuzzy random variables. However, it should be

μ M~ ljk(τ) 1 L

0

R

αljk

mljk

τ

β ljk

Fig. 1. An example of the membership function

l Me ðsÞ. ljk

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are positive numbers which represent left and right spreads. Fig. 1 illustrates an example of the membership function l e ðsÞ.

Let C be a collection of all possible Gaussian random variables

e ljk is expressed N(s, r2) where s 2 (1, 1) and r2 2 (0, 1). Then, C as a random fuzzy variable with the membership function



le ðcljk Þ ¼ l Me ðsljk Þjcljk ljk C ljk

   N sljk ; r2ljk ;

8cljk 2 C:

μP~l(p)

ð4Þ

le ðul Þ ¼ sup C lx

min

16k6nj ;j¼1;2

cl

(

le

C ljk

)  nj 2 X X   ljk Þu l ¼ ðc cljk xjk j¼1 k¼1

!)  nj 2 X X  l  N min l e ðsljk Þu sljk xjk ; V l ðxÞ ; ¼ sup 16k6n ;j¼1;2 M ljk s j

l

j¼1 k¼1

Pe ðG1 Þ

l11 ; . . . ; c l1n1 ; c l21 ; . . . ; c l2n2 Þ; sl ¼ ðsl11 ; . . . ; sl1n1 ; s121 ; . . . ; sl2n2 Þ,  l ¼ ðc where c and

maximize

Pe ðG2 Þ

r2ljk x2jk :

Assuming that the decision makers (DMs) concerns about the probe l x are smaller abilities that their own objective function values C than or equal to certain target values fl, l = 1, 2, we introduce the e l ðxÞx 6 fl Þ which are expressed as fuzzy sets P el probabilities Pðxj C with the membership functions



leP ðpl Þ ¼ sup le l u

C lx

  l Þjpl ¼ Pðxjul ðxÞ 6 fl Þ ; ðu

ð6Þ

where fl, l = 1, 2 are target values specified by the DMs as constants. Considering the imprecise nature of the DMs’ judgments for the e l with respect to the random fuzzy objective funcprobabilities P e l ; l ¼ 1; 2 e l x; l ¼ 1; 2, we introduce the fuzzy goals G tion values C e such as ‘‘ P l should be greater than or equal to a certain value’’. Such e l ; l ¼ 1; 2 can be quantified by eliciting corresponding fuzzy goals G membership functions

8 if p 6 p0l > <0 leG ðpÞ ¼ g l ðpÞ if p0l 6 p 6 p1l ; > l : 1 if p1l 6 p;

l ¼ 1; 2

ð7Þ

P2

for DM2

ð9Þ

> subject to A1 x1 þ A2 x2 6 b > > > > ; x1 P 0; x2 P 0





e l Þ ¼ sup min l ðp Þ; l ðp Þ ; Pe ð G e l e l Pl

pl

Gl

maximize

l1 ðf1 Þ

maximize

l2 ðf2 Þ

for DM1 for DM2

e 1 Þ P h1 subject to Pe ð G P1 f2 Þ P h2 Pe ð G P2

l ¼ 1; 2:

where h1 and h2 are permissible possibility levels specified by the DMs, and l1 and l2 are the membership functions of the fuzzy goals for the target variables f1 and f2, respectively. e l Þ P hl ; l ¼ 1; 2 in (10) is equivaFrom (8), the constraints Pe ð G Pl lently replaced by the condition that there exists a p such that leP ðpl Þ P hl and leG ðpl Þ P hl , namely, l l ( !) nj 2 X X sup min l Me ðsljk Þjpl ¼ Pðxjul ðxÞ 6 fl Þ; ul  N sljk xjk ; V l ðxÞ ljk s 16k6nj ;j¼1;2 j¼1 k¼1

l

ð11Þ

and pl P l ðhl Þ; l ¼ 1; 2, where l ðhl Þ are pseudo inverse funceG l eG l tions defined as lH ðhl Þ ¼ inffpl jle ðpl Þ P hl g; l ¼ 1; 2. This implies eG l Gl  l Þ; l ¼ 1; 2 such that that there exists a vector ðpl ; sl ; u H

ð8Þ min

H

l Me ðsljk Þ P hl ; ul  N ljk

 l ðxÞ 6 fl Þ; pl ¼ Pðxju

μG~l (p)

0

pl1

Fig. 2. An example of a membership function

nj 2 X X

!

sljk xjk ; V l ðxÞ ;

j¼1 k¼1

H pl P le ; Gl

which can be equivalently transformed into the condition that there  l Þ such that exists a vector ðsl ; u

l Me ðsljk Þ P hl ; ul  N ljk

pl0

ð10Þ

> > > > > > > > > A1 x1 þ A2 x2 6 b > > > ; x1 P 0; x2 P 0;

16k6nj ;j¼1;2

1

9 > > > > > > > > > > > > =

P hl

where gl(p), l = 1, 2 are nondecreasing functions. Fig. 2 illustrates a e l. possible shape of the membership function for the fuzzy goal G Recalling that the membership function is regarded as a possiel bility distribution, the degree of possibility that the probability P e attains the fuzzy goal G l is expressed as Pl

9 > > > > > =

P1

for DM1

or equivalently

j¼1 k¼1

l

p

e l Þ. Fig. 3 illustrates the degree of possibility Pe ð G Pl Now, assuming that the DMs are willing to maximize the degrees of possibility with respect to the attained probability, we consider the possibility-based probability model for random fuzzy two-level programming problems formulated as

maximize

nj 2 X X

pl1

e l Þ. Fig. 3. The degree of possibility Pe ð G Pl

ð5Þ

V l ðxÞ ¼

pl0

0

Through the Zadeh’s extension principle, in view of (4), the membership function of a random fuzzy variable corresponding to each of objective functions zl(x1, x2), l = 1, 2 is given as

(

μG~l (p)

~1 ΠP~l (G)

M ljk

p el. leG ðpÞ of a fuzzy goal G l

H ; 6 fl Þ P le Gl

nj 2 X X

!

sljk xjk ; V l ðxÞ ;

 l ð xÞ Pðxju

j¼1 k¼1

l ¼ 1; 2; j ¼ 1; 2; k ¼ 1; . . . ; nj :

In view of (3), it follows that

l Me ðsljk Þ P hl () sljk 2 ½mljk  LH ðhl Þaljk ; mljk þ RH ðhl Þbljk ; ljk

ð12Þ

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M. Sakawa, T. Matsui / Expert Systems with Applications 39 (2012) 12599–12604

where Lw(hl) and Rw(hl) are pseudo inverse functions defined as Lw(hl) = sup{tjL(t) P hl} and Rw(hl) = sup{tjL(t) P hl}. Hence, (12) l is rewritten as the equivalent condition that there exists a u such that

l ðxÞ 6 fl Þ P lH ; Pðxju e Gl

ð13Þ

 l ðxÞ 6 fl Þ is transformed into Since Pðxju

P2 Pnj

 LH ðhl Þaljk gxjk pffiffiffiffiffiffiffiffiffiffiffi V l ðxÞ ! P2 Pnj fl  j¼1 k¼1 fmljk  LH ðhl Þaljk gxjk pffiffiffiffiffiffiffiffiffiffiffi 6 ; V l ðxÞ k¼1 fmljk

j¼1

l  u

U

j¼1

 LH ðhl Þaljk gxjk pffiffiffiffiffiffiffiffiffiffiffi V l ðxÞ

! H P le ðhl Þ; Gl

ð14Þ

where U is a probability distribution function of the standard Gaussian random variable N(0, 1). From the monotone increasingness of U, (14) is rewritten as nj 2 X X pffiffiffiffiffiffiffiffiffiffiffi H fmljk  LH ðhl Þaljk gxjk þ U1 ðle ðhl ÞÞ V l ðxÞ 6 fl ; Gl

j¼1 k¼1

min

þ U1

nj 2 X X

ð15Þ

fmljk  L ðhl Þaljk gxjk

pffiffiffiffiffiffiffiffiffiffiffi leH ðhl Þ V l ðxÞ Gl

ð16Þ

l1 ðf1 Þ

ð17Þ or equivalently

for DM1

maximize for DM2

  l1 Z P1 ;F ðx1 ; x2 Þ   l2 Z P2 ;F ðx1 ; x2 Þ

9 > > > > > > > =

> > > subject to A1 x1 þ A2 x2 6 b > > > > ; x1 P 0; x2 P 0; where







l1 Z P1 ;F ðx1 ; x2 Þ ; l2 Z P2 ;F ðx1 ; x2 Þ

o 9 > > = > > ;

x2 P 0

ð20Þ

ð21Þ

Although the membership function does not always need to be linear, for the sake of simplicity, we adopt a linear membership function which characterizes the fuzzy goal of each decision maker. The linear membership functions ll, l = 1, 2 are defined as

ll



 ;F ZP ðx1 ; x2 Þ ¼ l

8 > 1 > > < Z

> > > :

P;F l

;F if Z P ðx1 ; x2 Þ 6 z1l l ðx1 ;x2 Þz0l z1l z0l

;F 0 if z1l < Z P l ðx1 ; x2 Þ < zl

ð22Þ

;F if Z P ðx1 ; x2 Þ P z0l : l

0

9 > > > > > H > > subject to fm1jk  L ðh1 Þa1jk gxjk > > > j¼1 k¼1 > >  > > p ffiffiffiffiffiffiffiffiffiffiffiffi

 > 1 > þU leH ðh1 Þ V 1 ðxÞ 6 z11  z01 v þ z01 > > > > G1 = nj 2 XX > fm2jk  LH ðh2 Þa2jk gxjk > > > > j¼1 k¼1 > >  > > p ffiffiffiffiffiffiffiffiffiffiffiffi

 1 H 1 0 0> > þU le ðh2 Þ V 2 ðxÞ 6 z2  z2 v þ z2 > > > G2 > > > > > A1 x1 þ A2 x2 6 b > ; x1 P 0; x2 P 0:

v

nj 2 X X

9 > > > > > > > maximize l2 ðf2 Þ > > for DM2 > > >  n > j 2 X > X p ffiffiffiffiffiffiffiffiffiffiffiffi > H 1 H > fm1jk  L ðh1 Þa1jk gx1jk þ U le ðh1 Þ V 1 ðxÞ 6 f1 > subject to = G1 j¼1 k¼1 >  nj > 2 X > X pffiffiffiffiffiffiffiffiffiffiffiffi > > fm2jk  LH ðh2 Þa2jk gx2jk þ U1 lH ðh2 Þ V 2 ðxÞ 6 f2 > > > eG 2 > > j¼1 k¼1 > > > > > A1 x1 þ A2 x2 6 b > > ; x1 P 0; x2 P 0

maximize

n

9 >   > > P;F > subject to l1 Z 1 ðx1 ; x2 Þ P v > > > =   P;F l2 Z 2 ðx1 ; x2 Þ P v > > > > > A1 x1 þ A2 x2 6 b > > ; x1 P 0; x2 P 0:

maximize

j¼1 k¼1



Consequently, (9) is equivalently transformed into for DM1

ð19Þ

Then, (21) is equivalently transformed into the convex programming problem

H

6 fl :

maximize

l

v

where U1 is the inverse function of U. From (11)–(15), it holds that

Pl

pffiffiffiffiffiffiffiffiffiffiffi V l ðxÞ;

are convex functions of (x1, x2). In order to obtain an initial candidate for an overall satisfactory solution to (9) or (17), it would be useful for DM1 to find a solution which maximize the smaller degree of satisfaction between the two DMs by solving the maximin problem

maximize H

k¼1 fmljk

e l Þ P hl () Pe ð G

Gl

x1 P 0;

 L ðhl Þaljk gxjk pffiffiffiffiffiffiffiffiffiffiffi  Nð0; 1Þ; V l ðxÞ

P2 Pnj

leH ðhl Þ

subject to A1 x1 þ A2 x2 6 b

(13) is equivalently transformed as

fl 



or equivalently

k¼1 fmljk

j¼1

þ U1

maximize

in consideration of

P2 Pnj

fmljk  LH ðhl Þaljk gxjk

¼ 1; 2

j¼1 k¼1

l  u

nj 2 X X j¼1 k¼1

ul

! nj 2 X X N fmljk  LH ðhl Þaljk gxjk ; V l ðxÞ :

P xj

;F ZP l ðx1 ; x2 Þ ¼

ð23Þ If DM1 is satisfied with the membership function values   ll Z Pl ;F ðx1 ; x2 Þ ; l ¼ 1; 2, the corresponding optimal solution x⁄ to (21) is regarded as the satisfactory solution. Otherwise, by introduc  ;F ing the constraint that l1 Z P is larger than or equal to the 1 ðxÞ minimal satisfactory level d 2 (0, 1) specified by DM1, we consider the problem of maximizing the membership function   l2 Z P2 ;F ðx1 ; x2 Þ formulated as

ð18Þ









maximize

l2 Z P2 ;F ðx1 ; x2 Þ

subject to

l1 Z P1 ;F ðx1 ; x2 Þ P d A1 x1 þ A2 x2 6 b x1 P 0; x2 P 0

9 > > > > > = > > > > > ;

ð24Þ

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M. Sakawa, T. Matsui / Expert Systems with Applications 39 (2012) 12599–12604

or equivalently, we obtain the convex programming problem

4. Numerical example

9  nj 2 X X pffiffiffiffiffiffiffiffiffiffiffiffi > H 1 H > > minimize fm2jk  L ðh2 Þa2jk gxjk þ U le ðh2 Þ V 1 ðxÞ > > > G2 > j¼1 k¼1 > > > >  nj > 2 X X > p ffiffiffiffiffiffiffiffiffiffiffiffi = H 1 fm1jk  L ðh1 Þa1jk gxjk þ U leH ðh1 Þ V 2 ðxÞ subject to G1 j¼1 k¼1 > > >

 > > 6 z11  z01 d þ z01 > > > > > > A1 x1 þ A2 x2 6 b > > ; x1 P 0; x2 P 0: ð25Þ In general, when the objective functions of DM1 and DM2 conflict with each other, it should be noted here that the larger the minimal satisfactory level d for l1 is specified by DM1, the smaller the satisfactory degree for l2 becomes, which may lead to the improper satisfactory balance between DM1 and DM2 due to the large difference between the membership function values of both DMs. In order to derive the satisfactory solution which has wellbalanced membership function values between both DMs, by introducing the ratio D expressed as

  l2 Z P2 ;F ðx1 ; x2 Þ ; D¼  l1 Z P1 ;F ðx1 ; x2 Þ

To demonstrate the feasibility and efficiency of the proposed method, consider the following two-level linear programming problem involving random fuzzy variable coefficients:

9 e 11 x1 þ C e 12 x2 > z1 ðx1 ; x2 Þ ¼ C > > for DM1 > > > > > e e minimize z2 ðx1 ; x2 Þ ¼ C 21 x1 þ C 22 x2 > > > > for DM2 > > > > subject to a11 x1 þ a12 x2 6 b1 > > > = a x þa x 6b minimize

21 1

22 2

2

a31 x1 þ a32 x2 6 b3 a41 x1 þ a42 x2 6 b4 a51 x1 þ a52 x2 6 b5 x1 ¼ ðx11 ; x12 ; x13 ÞT P 0 x2 ¼ ðx21 ; x22 ; x23 ÞT P 0:

ð27Þ

> > > > > > > > > > > > > > > > > > > ;

Table 1 shows values of coefficients of constraints ai, i = 1, 2, 3, 4, 5 and bi, i = 1, 2, 3, 4, 5 and Table 2 shows values of parameters of random fuzzy variables mljk, aljk and r2ljk ; l ¼ 1; 2; j ¼ 1; 2; k ¼ 1; . . . ; 6, where triangular fuzzy numbers are assumed for

l Me ðsÞ. ljk

ð26Þ

the lower bound Dmin and the upper bound Dmax of D, specified by DM1, are introduced to determine whether or not the ratio D is appropriate. To be more explicit, if it holds that

D 2 ½Dmin ; Dmax ; then DM1 regards the corresponding solution as a preferable candidate for the satisfactory solution with well-balanced membership function values. Now we summarize a procedure of interactive fuzzy programming for the possibility-based fractile model in order to derive a satisfactory solution. Interactive fuzzy programming in the possibility-based fractile model Step 1: Ask DMs to specify the membership functions ll, l = 1, 2. Step 2: Ask DM1 to specify the permissible possibility levels hl, l = 1, 2. Step 3: For the current hl, l = 1, 2, solve the maximin problem (20). Step 4: DM1 is supplied with the current values of the membership functions l1 and l2 for the optimal solution obtained in step 3. If DM1 is satisfied with the current membership function values, then stop. If DM1 is not satisfied and prefers to update hl, l = 1, 2, ask DM1 to update hl, and return to step 3. Otherwise, ask the minimal sat DM1 to specify  isfactory level d for l1 Z 1P;F ðx1 ; x2 Þ and the permissible range [Dmin, Dmax] of the ratio D. Step 5: For the current minimal satisfactory level d, solve the convex programming problem (25). Step 6: DM1 is supplied with the current values of the membership function l1, l2 and the ratio D. If D 2 [Dmin, Dmax] and DM1 is satisfied with the current membership function values, then stop. Otherwise, ask DM1 to update the minimal satisfactory level d, and return to step 5. In this section, from the viewpoint of taking the possibility of coordination or bargaining between the DMs into account, it is assumed that there exist communication and a cooperative relationship between the DMs.

Through the use of this numerical example, it is now appropriate to illustrate the proposed interactive fuzzy programming. The parameter values of particle swarm optimization for nonlinear programming (PSONLP) are set as swarm size N = 50, maximal search generation number Tmax = 3000, c1 = 2.0, c2 = 2.0, w0 = 1.2 and wT max ¼ 0:1. For illustrative purposes, assume that the DMs subjectively e1 determine the membership functions (7) for the probabilities P 0 0 1 1 e and P 2 as linear ones by assessing p1 ¼ p2 ¼ 0:2; p1 ¼ p2 ¼ 0:9 and

g 1 ðpÞ ¼ g 2 ðpÞ ¼

p  0:2 : 0:7

Also assume that the DMs specify the permissible possibility levels as h1 = 0.7 and h2 = 0.7. Furthermore, assume that the fuzzy goals for the target variables f1 and f2 are determined by the linear membership functions

ll ðfl Þ ¼

8 1; > > <

fl fl0

f 1 f 0 > > :l l 0;

if f l < fl1 ; if fl1 6 fl 6 fl0 if f l > fl0 :

Here the parameter values fl1 and fl0 are respectively determined as  



lo lo  jo lo fl1 ¼ flmin ¼ fl xlo and fl0 ¼ flm ¼ fl xjo 1 ; x2 1 ; x2 ; l – j, where x1 ; x2 is a feasible solution minimizing fl(x1, x2). Through PSONLP, the parameter values characterizing the linear membership functions are determined as f11 ¼ 172:22; f10 ¼ 90:61, f21 ¼ 120:87, and f20 ¼ 21:81. The obtained result is shown at the column labeled ‘‘1st’’ in Table 3. DM1 is not satisfied with this solution, but he does not desire to update fl, l = 1, 2. Thus, DM1 determines the minimal satisfactory level ^ d ¼ 0:90 to improve l1(f1) at the expense of l2(f2). Furthermore, DM1 specifies the upper bound Dmax = 0.90 Table 1 Values of coefficients in constraints.

a1 a2 a3 a4 a5

al11

al12

al13

al21

al22

al23

b

2.00 0.00 5.00 3.00 3.50

3.00 0.00 2.50 8.00 4.00

8.00 0.00 1.00 4.00 4.00

0.00 2.50 5.00 7.00 5.50

0.00 9.00 5.00 8.00 4.00

0.00 4.00 8.00 8.00 1.00

500 425 400 600 300

12604

M. Sakawa, T. Matsui / Expert Systems with Applications 39 (2012) 12599–12604

Table 2 Values of mljk, aljk and

m1jk m2jk

a1jk a2jk

r21jk r22jk

r2ljk .

 ~cl11

 ~cl12

 ~ cl13

 ~cl21

 ~cl22

 ~cl23

8.00 5.00 0.70 0.90 1.00

5.00 4.00 0.50 0.80 0.90

6.00 5.00 0.80 0.70 1.10

4.00 3.00 0.40 0.40 1.20

6.00 5.00 0.70 0.60 1.00

3.00 3.00 0.60 0.50 0.90

0.80

0.90

1.00

1.20

0.70

1.00

in the proposed interactive fuzzy programming can be solved through particle swarm optimization for nonlinear programming (PSONLP). An illustrative numerical example demonstrated the feasibility and efficiency of the proposed method. Extensions to other stochastic programming models will be considered elsewhere. Also extensions to fuzzy random two-level linear programming problems with two decision makers under noncooperative environments will be reported in the near future. References

Table 3 Interaction process. Interaction

1st

2nd

3rd

4th

^ d x11 x12 x13 x21 x22 x23 l1(f1) l2(f2) D

–

0.90

0.70

0.80

11.37 18.13 11.42 21.42 0.53 22.08 0.67 0.67 1.0

18.20 35.21 5.40 6.49 4.59 20.06 0.90 0.48 0.54

15.02 22.09 0.61 1.12 34.82 11.17 0.72 0.69 0.97

13.04 27.49 0.38 10.17 17.80 15.73 0.80 0.63 0.78

and the lower bound Dmin = 0.60 for the ratio of objective functions D = l2(f2)/l1(f1). For the updated value of ^ d, (27) is solved by PSONLP. The obtained result is shown at the column labeled ‘‘2nd’’ in Table 3. Since the ratio of satisfactory degrees D is less than Dmin = 0.60, the second condition of termination of the interactive process is not fulfilled. Hence, DM1 updates the minimal satisfactory level ^ d from 0.90 to 0.70 for improving l2(f2) at the sacrifice of l1(f1). (27) is solved for the updated value of ^ d, and the obtained result is shown at the column labeled ‘‘3rd’’ in Table 3. DM1 considers that l1(f1) is improved but D is greater than Dmax. Hence, DM1 is not satisfied with this solution and updates the minimal satisfactory level ^ d from 0.70 to 0.80. For the updated value of ^ d, (27) is solved, and the obtained result is shown at the column labeled ‘‘4th’’ in Table 3. Since D exists in the interval [Dmin, Dmax] and DM1 is satisfied with the balance between l1(f1) and l2(f2), the interactive algorithm is terminated. In the proposed interactive fuzzy nonlinear programming, through a series of update procedures of the minimal satisfactory level ^ d and the target values fl, l = 1, 2, it can be possible to obtain a satisfactory solution where the satisfactory degree of DM1 is guaranteed to be greater than or equal to the minimal satisfactory level ^ d and is well balanced with that of DM2.

5. Conclusions In this paper, interactive decision making methods for random fuzzy two-level linear programming problems have been presented. Considering the probabilities that the decision makers’ objective function values are smaller than or equal to target variables, fuzzy goals of the decision makers for the probabilities were introduced. Through the use of fractile criterion optimization in stochastic programming, the original random fuzzy two-level programming problems were reduced to deterministic two-level programming ones. In order to obtain a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers, interactive fuzzy programming for random fuzzy two-level linear programming problems was proposed. It was shown that all of the problems to be solved

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