an approximation approach for representing s-shaped membership functions

412 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 18, NO. 2, APRIL 2010

An Approximation Approach for RepresentingS-Shaped Membership Functions

Ching-Ter Chang

AbstractIn general, to formulate a fuzzy-linear-programmingproblem with n S-shaped utility (membership) functions, tradi-tional methods require n or more extra binary variables becauseS-shaped curves are neither convex nor concave in all places.Adding binary variables does not improve the bound of the linear-programming relaxation. On the contrary, added binary vari-ables increase the computational burden in the solution process ifproblems get large. Therefore, a formulation without binary vari-ables should be more efficient. Accordingly, this study proposes apiecewise-linear approach to formulate an S-shaped membershipfunction (MF) without adding any extra binary variables, whichimproves the efficiency of fuzzy-linear programming in solving de-cision/management problems with S-shaped MFs. Finally, a com-putational experiment is provided to demonstrate the superiorityof the proposed models. An illustrative example is also provided toshow the usefulness of the proposed method.

Index TermsBinary variable, fuzzy-goal programming (FGP),fuzzy-linear programming (FLP), membership function (MF),piecewise-linear function (PLF).

I. INTRODUCTION

THE S-shaped membership function (MF) is the mostwidely used technique for prospect theory in the vari-ous fields of decision science. Several studies have attemptedto achieve an efficient representation of S-shaped MF, althoughno previous works have provided an efficient and compact wayto deal with such problems. This study develops a piecewise-linear approach to represent the S-shaped MF. In the proposedapproach, extra binary variables are no longer required. The pro-posed approach represents a linear form that can be efficientlysolved using common linear-programming packages.

Fuzzy-linear programming (FLP) was first proposed byZimmermann [36] to solve multiple objective problems in whichsome objectives are conflicting, imprecise, or fuzzy in na-ture. Goal programming (GP) was introduced by Charnes andCooper [7] and has since been applied successfully in manyreal-world problems. One advantage of applying fuzzy-set the-ory in GP is that imprecise aspiration levels can be added to thedecision model, thus leading to fuzzy GP (FGP). This can beexpressed as follows.

Manuscript received June 12, 2009; revised November 16, 2009; acceptedJanuary 21, 2010. First published February 17, 2010; current version publishedApril 2, 2010.

The author is with the Department of Information Management, Chang GungUniversity, Tao-Yuan 333, Taiwan (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TFUZZ.2010.2042961

A solution set x is found for FGP that contains n fuzzy goalzi(x) given by

zi(x) zi , for i = 1, 2, . . . , n,zi(x) z0i , for i = n+ 1, . . . ,ms.t. x F (whereF is a feasible set)

where zi(x) zi indicates that the ith fuzzy goal is approxi-mately greater than or equal to the aspiration level zi . zi(x) z0imeans that the ith fuzzy goal is smaller than or equal to the as-piration level z0i .

In the MaxMin approach, as proposed by Zimmermann [36],an FGP with n objectives can be expressed as the followingmathematical programming:

Max

s.t. ij (zi(x)) 0, for i = 1, 2, . . . , nx F (whereF is a feasible set)

where is a continuous variable, x is a n 1 vector of thedecision variable, and ij (zi(x)) is an MF of the ith objective.

The main advantage of FLP is that the vague aspirations ofa decision maker (DM) can be qualified using natural languageand vague phenomena, where the degree of natural language andvague phenomena are usually represented by MF. In addition,the preference structure of the DM is one of the important fac-tors utilized in many decisions about management and economicproblems to judge DMs level of satisfaction with the options of-fered to them [4]. To represent the various preference structuresof the DM, previous studies often employed various MFs or util-ity functions, such as pure linear, piecewise linear, tangent, trian-gular, convex, concave, quasi-concave, trapezoidal, exponential,dynamics, V-shaped, U-shaped, S-shaped, and reverse S-shaped[4], [6], [8], [9], [12], [13], [21], [24], [26], [28], [33], [35], [36].According to the survey by Dacey [9], pure concave or convexMF may not be realistic in some situations. In fact, S-shaped MFis quite often suggested, particularly when the MF is interpretedas a utility function [9], [37]. Furthermore, it is well known thatS-shaped utility functions have been proposed in prospect the-ory [17]. The S-shaped MF is a relatively general MF, which isbetter suited to meet practical needs because it is more flexiblein expressing the vagueness in fuzzy parameters for nonlinearproblems [33]. It has also been applied successfully in many ap-plications, such as psychology, economics, knapsack, financialinvestment, project management, production planning, and or-ganizational behavior [2], [8], [9], [28], [33]. However, since theS-shaped MF represents a nonlinear form, it is difficult to solvethis problem in a straightforward fashion. In order to deal with

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CHANG: APPROXIMATION APPROACH FOR REPRESENTING S-SHAPED MEMBERSHIP FUNCTIONS 413

nonlinear MF, Inuiguchi et al. [15] proposed a technique us-ing a standard linear programming to solve a problem in whichMFs are strictly quasi-concave. Romero [29] derived a gen-eral structure of the achievement function for piecewise-penaltyfunctions. Keha et al. [19] proposed a branch-and-cut algorithmfor nonconvex piecewise-linear optimization problems. The ma-jor advantage of the piecewise-linear function (PLF) is that anarbitrary function of one variable can be approximated by PLF,with the quality of the approximation controlled by a numberof linear segments. Jimenez et al. [16] showed that any typeof nonlinear MF can be approximated by a PLF. It has alsobeen demonstrated that, due to their simplicity and efficiencywith respect to computability, PLFs are preferred to representthe nonlinear S-shaped MF in practical applications, such asportfolio selection, electronic design, and the optimization of agas networks [10]. However, to formulate a nonconvex PLF, ex-tra binary variables are required in traditional approaches. Suchvariables lead to a complex mixed integer program [18]. Yanget al. [35] created an efficient piecewise approach to formulatethe S-shaped MF, and this was further developed by Li andYu [22], Lin and Chen [25], and Chang [4]. In order to formu-late an S-shaped MF with both convex and concave segments,the above studies required extra binary variables. However, asshown by Keha et al. [18], adding binary variables does notimprove the bound of the linear-programming relaxation. Onthe contrary, the binary variables increase the computationalburden in the solution process if the problem size becomeslarge.

Our central interest here is to formulate a novel approach foran S-shaped MF in which an extra binary variable is not re-quired. Based on the observation of Keha et al. [18], this couldsignificantly improve the efficiency of FLP with S-shaped MFsfor decision/management problems. Furthermore, the proposedapproach can be applied to any kind of MF in addition to theS-shaped MF. The numerical study performed here indicatesthat the suggested model to formulate the S-shaped MF per-forms quite well, and the computational results also indicatethe superiority of the proposed method over the mixed-integermodels of Yang et al. [35] and Li and Yu [24]. The improvedefficiency of the proposed model is especially apparent as thenumber of binary variables increases.

The rest of this paper is organized as follows. The problemdescription and related methods are described in Section II.The proposed models are established in Section III. Computa-tional experience and discussions are provided in Section IV. Inorder to demonstrate the usefulness of the proposed methods,an illustrative example is provided in Section V. Finally, someconclusions are given in Section VI.

II. PROBLEM DESCRIPTION

In the theory of expected utility, the utility functionij (zi(x))describes the risk attitude of DMs [20]. The risk attitude reflectswhether DMs are risk-averse (i.e., a concave utility function) orrisk-seeking (i.e., a convex utility function). In fact, the utilityfunction is equivalent to the fuzzy-set MF [11]. In the MaxMin

Fig. 1. S-shaped MF.

model, the utility functions are selected to have the same shapeas the MFs [34]. In this study, we are interested in an FLP prob-lem with n MFs, ij (zi(x))(i = 1, . . . , n), where ij (zi(x))is a convex nondecreasing function for 0 zi(x) i anda concave function for zi(x) i . In general, the member-ship value of ij (zi(x)) lies in the interval [0,1] mapping with0 zi(x) . Similarly, the value used to measure the rangeof possible outcomes in a utility function is usually assigned autility of 1 for the best outcome and 0 for the worst outcome.This is a typical S-shaped MF, as depicted in Fig. 1.

Let ij (zi(x)) denote the set of subgradients of ij (zi(x))at zi(x). Because ij (zi(x)) is neither convex nor concaveeverywhere, thus, the following two subgradients are con-ducted: For zi(x) [0, i ], we have ij (zi(x)) ij (zi(x)) +(zi(x) zi(x)), where ij (zi(x)). For zi(x) [i,],we have ij (zi(x)) ij (zi(x)) + (zi(x) zi(x)), where ij (zi(x)).

Hannan [12] developed a novel approach to formulate inter-polated MF, ij (zi(x)), i = 1, 2, . . . , n, as follows.

Hannans method [12]:

ij (zi(x)) =n

k=1

i,k |zi(x) zi,k |+ izi(x) + i

where i,k = (1/2)(ti,k+1 ti,k ), i = (1/2)(ti,n+1 + ti,1),and i = (1/2)(si,n+1 + si,1). It is also assumed thatij (zi(x)) = ti,k zi(x) + si,k for each segment zi,k1 zi(x) zi,k , k = 1, 2, . . . , n, ti,k is the slope, and si,k is the in-tercept for the section of curve initiated at zi,k1 and terminatedat zi,k .

Tversky and Kahneman [32] employed a nonlinear approachto fit the parameters of constant relative risk aversion/preferenceutility function as follows.

Tversky and Kahnemans method [32]:ij (zi(x)) = zi (x), for zi(x) 0= (zi (x)), for zi(x) < 0.Tversky and Kahneman found that the estimators are =

= 0.88, and = 2.25.


Fig. 2. Approximated S-shaped MF, where ij (zi (x)) = i1 (i2 i3 )in which means the union, and means the intersection.

Following Hannan [12], Hu and Fang [14] proposed an algo-rithm to solve fuzzy inequalities with piecewise-linear MFs asfollows.

Hu and Fangs method [14]:

MaxxRn

Mini=1,...,m

{ n

k=1

i,k (d+i,k + di,k ) + izi(x) + i

}

s.t. zi(x) d+i,k + di,k = zi,k , d+i,k d

i,k = 0

d+i,k 0, di,k 0x F (whereF is a feasible set)

where d+i,k = max(0, zi(x) zi,k ) and di,k = max(0, zi,k zi(x)) are deviational variables for |zi(x) zi,k |, and othervariables are defined as in Hannans method.

However, in order to solve the above nonlinear problem, theirmodel adopted a complicated algorithm, which is called en-tropic regularization procedure. Dacey [9] claimed that theS-shaped utility function provides an explanation for the typesof observed behavior that is superior to the explanations sup-ported by a concave utility function/a convex utility functionthat represents a single attitude toward risk. A two-piece utilityfunction is often adopted in order to represent either a con-vex segment or concave segment in the S-shaped utility func-tion [28]. Therefore, an extra binary variable is needed to selectthe correct segment for the convex or concave area, as shownin Fig. 1 [27]. PLFs are still widely used to approximate theS-shaped MF because it is inherently efficient in terms of thelinear model. Among these functions, the following one can behighlighted. The S-shaped MF is constructed by using conjunc-tion (i.e., intersection) and disjunction (i.e., union) operatorswithin the following PLF approach [35].

Yang et al.s method [35]: This method is shown in Fig. 2.Notably, the accuracy of the approximated S-shaped MF

depends heavily upon the selected number of break points(e.g., ai4 and ai5 in Fig. 2) for the presentation of theS-shaped MF in Fig. 1. As the number of break pointsincreases, the error between the S-shaped MF and the ap-proximated S-shaped MF decreases, i.e., the quality of theapproximation is controlled by the size of the linear seg-ments in the approximated S-shaped MF according to theneeds of DMs. It should be noted that the accuracy of the

Fig. 3 (a) Convex function. (b) Convex function with break point.

approximated S-shaped MF, as discussed above, depends heav-ily on the number of break points assigned. For a strictly con-vex function z(y) if z(tb0 + (1 t)b) < tz(b0) + (1 t)z(b),where b0 y = tb0 + (1 t)b b, as depicted in Fig. 3(a), theerror of piecewise linearizing z(y) by z(y) is computed as

Error = eb0 + (eb eb0 )y b0b b0

ey .

By taking partial Error/y = 0, the maximal error occursat

y = lneb eb0b b0

.

By doing so, we obtain the first break point at y, as shown inFig. 3(b). Based on this choice, the minimal error of functionz(y) is obtained, as depicted in Fig. 3(b).

Similarly, for a strictly concave function z(x) ifz(ta0 + (1 t)a) > tz(a0) + (1 t)z(a), 0 < a0 x =ta0 + (1 t)a a, as depicted in Fig. 4, the error of piecewiselinearizing z(x) by z(x) is computed as

Error = z(x)[z(a0) + (z(a) z(a0))

(x a0a a0

)].

By taking Error/x = 0, the maximal error occurs at x =(a a0)/(z(a) z(a0)), which can be chosen as the locationof the first break point.


Fig. 4. Concave function.

All of the above procedures can be iterated until the error iswithin the predefined fault tolerance (for more details, see [23]).Using the model proposed by Yang et al. [35], an S-shaped MFcan be approximated by the intersection and union of three ramp-type linear functions, as shown in Fig. 2, where i1 , i2 , andi3 (i.e., i1(zi(x)), i2(zi(x)), and i3(zi(x))) by piecewiseapproximation, which is expressed as follows:

i1(zi(x)) =

1, if zi(x) ai81 ai8 zi(x)

ai8 ai2, if ai2 zi(x) < ai8

0, otherwise

i2(zi(x)) =



0, otherwise

and

i3(zi(x)) =



0, otherwise.

(1)

To implement the above S-shaped MF [i.e., (1)], one morebinary variable is required. This because the union of such ramp-type linear functions must be interpreted as an eitheror rela-tionship. Therefore, Yang et al. [35] formulated the associatedfuzzy-programming problem with an S-shaped MF as follows:

Max

s.t.

1 ai8 zi(x)ai8 ai2

+M(1 bi),

1 ai6 zi(x)ai6 ai3

+Mbi,

1 ai7 zi(x)ai7 ai1

+Mbi,

i = 1, 2, . . . ,n

(2)

where M represents a big positive value, and bi is a binaryvariable.

Yang et al. [35] stated that in order to achieve reasonableaccuracy, only the binary variables necessary to approximate

Fig. 5. Piecewise-linear function.

the S-shaped MF should be used. However, as can be seen in themodel of Yang et al. [35],

ni=1 mi binary variables are required

to solve the S-shaped MF problem, where mi represents thenumber of unions between the concave and convex functions inij (zi(x)). Following Yang et al. [35], two extension modelscan be seen in the works of Li and Yu [22] and Lin and Chen [25].

Li and Yus method [24]: Li and Yu [24] proposed a PLF inorder to solve fuzzy-multiobjective problem with quasi-concaveMFs as follows.

Let z(x) be the PLF of x, as depicted in Fig. 5, where ai , i =1, 2, . . . , n are the break points of z(x), and si , i = 1, 2, . . . , nare the slopes of line segments between ai and ai+1 . z(x) canbe expressed as the sum of absolute terms as follows:

z(x) = a1 + s1(x a1) +n1

i=2

si si12

(|x ai |+ x ai).

(3)

Equation (3) can be examined as follows.1) If x = a1 , then z(x) = z(a1).2) If x a2 , then z(x) = z(a1) + (z(a2) z(a1))/(a2

a1)(x a1) = z(a1) + s1(x a1).3) If x a3 , then z(x) = z(a1) + s1(x a1) + s2(x

a2) = z(a1) + s1(x a1) + ((s2 s1)/2)(|x a2 |+x a2).

Considering z(x) in (3), if si > si1 , then z(x) is convexwithin the interval ai1 x ai+1 ; if si < si1 , then z(x)is concave within the interval ai1 x ai+1 . According toa height-balanced tree, as proposed by Adelson-Velskii andLandis [1] (which is called the AVL tree), Chang [4] recentlyextended the model of Yang et al. [35] to prove that for an FLPproblem with n S-shaped MFs, only lnn/ ln 2 extra binaryvariables are needed. This seems to be the most efficient tech-nique for S-shaped MF. However, as shown by Keha et al. [18],adding binary variables does not improve the bound of the linearprogramming relaxation. It is clear that the major difficulty inimplementing the above methods [4], [22], [23], [25], [35] is thatthey require the addition of many binary variables if the problemsize becomes large or better approximation quality is required(i.e., the number of break points of the approximated S-shapedMF needs to be increased). This increases the computationalburden in a solution processes using the branch-and-bound


Fig. 6. Example of the RSMF.

approach. In other words, current methods are not computa-tionally efficient to solve the FLP with S-shaped MFs. In fact,the formulation of an S-shaped MF with binary variables is aclassical nondeterministic polynomial-time (NP) hard combina-torial optimization problem [2], [19]. Therefore, the formulationof an S-shaped MF without binary variables should be more ef-ficient. This study proposes a novel method to represent theS-shaped MF without binary variables, which seems to be moreefficient, i.e., the new S-shaped MF represents a linear formof GP. In fact, not only can this S-shaped MF be determined inpolynomial time, but it can also be solved using common-linear-programming packages. This improves the utility of S-shapedMF to solve decision/management problems.

III. PROPOSED MODELS

In order to improve the efficiency of FLP with S-shaped MFsto solve decision/management problems, this section investi-gates how to model an FLP problem with n S-shaped MFs,ij (zi(x))(i = 1, . . . , n) in which binary variables are not re-quired. In general, the S-shaped MF can be classified into twoforms: 1) right S-shaped MF (RSMF) and 2) left S-shaped MF(LSMF), as shown in Figs. 6 and 7, respectively. For the sakeof simplicity, but without the loss of generality, let us look ata simple example of RSMF to briefly demonstrate how to con-struct the achievement model without adding any binary vari-ables, where pij (j = 1, 2, and 3) are the positive deviationsdistances from the break points bij (j = 1, 2, and 3), respec-tively. If zi(x) = (i.e., ij (8) = 0.85), as shown in Fig. 6,how exactly can we obtain the membership value of RSMF?Based on the principle of the triangle and GP technique, anymembership value of RSMF ij () can be exactly representedby summing the following three simple linear terms:

[ij (bi2) ij (bi1)]pi1 pi2bi2 bi1

[ij (bi3) ij (bi2)]pi2 pi3bi3 bi2

, and

[ij (bi4) ij (bi3)]pi3

bi4 bi3.

Fig. 7. Example of the LSMF.

In other words, it is possible to formulate an FLP problem withn S-shaped MFs without adding binary variables, if appropriateconstraints are added to the achievement model. Accordingly,the following two achievement models (i.e., Models II and III)are derived to exactly represent the behavior of RSMF.

In Fig. 6, sij (j = 1, 2, and 3) are slopes of line segmentbetween two break points bij and bij+1 and (j = 1, 2, and 3).Let us consider the positive and negative deviational variablesnij and pij (j = 1, 2, and 3) for ith RSMF, such that

pij ={zi(x) bij , if zi(x) bij 00, otherwise

nij ={bij zi(x), if zi(x) bij < 00, otherwise.

According to the principle of GP, to avoid the loss of generality,it is customary to assume that pijnij = 0 and zi(x) bij =pij nij in almost all GP problems. However, it is still possibleto eliminate half of the deviation variables because RSMF is amonotonically increasing step function, and the new model canbe expressed as follows.

Model I:

Max i

Min pi1 + pi2 + pi3

s.t.

i = [ij (bi2) ij (bi1)]pi1 pi2bi2 bi1

+ [ij (bi3) ij (bi2)]

pi2 pi3bi3 bi2

+ [ij (bi4) ij (bi3)]pi3

bi4 bi3(4)

zi(x) pi1 + ni1 = bi1 (5)zi(x) pi2 + ni2 = bi2 (6)zi(x) pi3 + ni3 = bi3 (7)x F (whereF is a feasible set)


where bij (j = 1, . . . , 4) are break points of RSMF, pij andnij (j = 1, 2, and 3) are positive and negative deviations fromthe target values (break points) bij (j = 1, 2, and 3), andi (0 i 1) is the additional continuous variable that repre-sents the membership value (i.e., the utility value) of RSMF. Itis clear that Model I is a typical two-objective MaxMin programthat initially minimizes deviation pij between fuzzy goal zi(x)and break point bij , while also maximizing the membershipvalue i . According to the novel approach of multichoice GP,as proposed by Chang [5], the achievement function of Model Ican be formulated as in the following model.

Model II:

Min pi1 + pi2 + pi3 + i(e+i + ei )

s.t.

i e+i + ei = 1

i = [ij (bi2) ij (bi1)]pi1 pi2bi2 bi1

+ [ij (bi3) ij (bi2)]

pi2 pi3bi3 bi2


bi4 bi3zi(x) pi1 + ni1 = bi1zi(x) pi2 + ni2 = bi2zi(x) pi3 + ni3 = bi3 (8)x F (whereF is a feasible set)

where e+i and ei are positive and negative derivations from the

highest membership value of RSMF, i is the weight attachedto the sum of the deviations of |i 1|, and other variables aredefined as in Model I.

Notably, Model II also adds no extra binary variables, sinceit represents a linear form of GP. Therefore, it can be efficientlysolved using common-linear-programming packages [18]. Infact, it correctly represents the behavior of RSMF. For a briefexplanation of the correctness of Model II, Fig. 5 can be formu-lated as follows.

Program P1:

Min pi1 + pi2 + pi3 + i(e+i + ei )

s.t.

i =[0.3

pi1 pi24 1 + 0.4

pi2 pi36 4 + 0.3

pi310 6

](9)

i e+i + ei = 1 (10)zi(x) pi1 + ni1 = 1 (11)zi(x) pi2 + ni2 = 4 (12)zi(x) pi3 + ni3 = 6 (13)x F (whereF is a feasible set)

Proposition 1: Program P1 and Fig. 6 are equivalent in thesense that they have the same optimal solutions.

Proof:1) If zi(x) = 8 (lies on line segment [bi3 , bi4] in Fig. 6),

then 8 7 = 1 [from (11)], 8 4 = 4 [from (12)],and 8 2 = 6 [from (13)]. It can be revealed that

approaching highest value of RSMF as close as possiblecan be ensured by (10), if appropriate weight value ofi is given in the objective function. This forces i =[0.3[(7 4)/(4 1)] + 0.4[(4 2)/(6 4)] + 0.3[(2)/(10 6)]] = 0.85 [from (9)]. It is the exact membershipvalue of P1. The value is exactly the same as in Fig. 6.

2) Similarly, other values are arbitrarily selected as zi(x) = 2and 5 (lies on other line segments [bi1 , bi2] and [bi2 , bi3]in Fig. 6), which can easily be checked in P1 to obtain allexact membership values.

This is obviously the same as P1 in Fig. 6. This completes theproof of Proposition 1.

Similarly, this idea can be easily reused to construct other pro-posed models (Models IIIVII) corresponding to specific needs.Referring to the work of Chang [3], the auxiliary constraints in(5) and (6) of Model I can be reduced without affecting thecorrectness. This leads to the following concise model.

Model III:

Min wi1pi1 + wi2pi2 + wi3pi3 + i(e+i + ei )

s.t.

i = [ij (bi2) ij (bi1)]pi1

bi2 bi1+ [ij (bi3) ij (bi2)]

pi2bi3 bi2


bi4 bi3(14)

i e+i + ei = 1 (15)zi(x) pi1 pi2 pi3 bi1 (16)wi1 < wi2 < wi3 (17)

0 pi1 bi2 bi1 , 0 pi2 bi3 bi20 pi3 bi4 bi3 (18)

x F (whereF is a feasible set)

where wij (j = 1, 2, and 3) are weights attached to posi-tive deviations, pij (j = 1, 2, and 3) 0 pij bij+1 bij arebounded positive deviations from the target value bi1 , and othervariables are defined as in Model I.

Similarly, Fig. 6, for instance, can be expressed usingModel III as follows.

Program P2:

Min wi1pi1 + wi2pi2 + wi3pi3 + i(e+i + ei )

s.t.

i =[0.3

pi13

+ 0.4pi22

+ 0.3pi34

](19)

i e+i + ei = 1 (20)zi(x) pi1 pi2 pi3 1 (21)wi1 < wi2 < wi3 (22)

0 pi1 3, 0 pi2 2, 0 pi3 4 (23)x F (whereF is a feasible set)

Proposition 2: Program P2 and Fig. 6 are equivalent in thesense that they have the same optimal solutions.


TABLE ISIZE OF VARIOUS METHODS

Proof: If zi(x) = 8 (lies on line segment [bi3 , bi4] in Fig. 6),then 8 3 2 2 1 [from (21)] because wi1 , wi2 , andwi3 are ranked by (22) as wi1 < wi2 < wi3 , and they areattached to deviations pi1 , pi2 , and pi3 in the objective func-tion, respectively. As a result, the deviation priority rank or-der of the deviations becomes pi1 > pi2 > pi3 and all devia-tions pij (j = 1, 2, and 3) are bounded [from (23)]. Therefore,the value of i is obtained as i = [0.3(3/3) + 0.4(2/2) +0.3(2/4)] = 0.85 [from (19)]. This is the exact membershipvalue of P2. This result is exactly the same as that in Fig. 6.Similarly, other values are arbitrarily selected as zi(x) = 2 and5 (lies on other line segments [bi1 ,bi2] and [bi2 ,bi3] in Fig. 6),which can easily be checked in P2 to obtain all exact member-ship values. This completes the proof of Proposition 2. Let usconsider the following LSMF shown in Fig. 7.

Following the same idea, as in Models II and III, two achieve-ment models for the LSMF are derived as follows.

Model IV:

Min ni1 + ni2 + ni3 + i(e+i + ei )

s.t.

i = [ij (bi2) ij (bi1)]ni1 ni2bi1 bi2

+ [ij (bi3) ij (bi2)]

ni2 ni3bi2 bi3

+ [ij (bi4) ij (bi3)]ni3

bi3 bi4(24)

i e+i + ei = 1 (25)zi(x) + ni1 = bi1 (26)

zi(x) + ni2 = bi2 (27)

zi(x) + ni3 = bi3 (28)


where all variables are defined as in Model II.Model V:

Min wi1ni1 + wi2ni2 + wi3ni3 + i(e+i + ei )

s.t.

i = [ij (bi2) ij (bi1)]ni1


ni2bi2 bi3

+ [ij (bi4) ij (bi3)]ni3

bi3 bi4(29)

i e+i + ei = 1 (30)zi(x) + ni1 + ni2 + ni3 bi1 (31)wi1 < wi2 < wi3 (32)

0 ni1 bi1 bi2 , 0 ni2 bi2 bi3 ,0 ni3 bi3 bi4 (33)


where all variables are defined as in Model III.In order to save space, the proofs of Models IV and V have

been omitted. For the sake of simplicity and without loss ofgenerality, we compare all models with the model proposed byYang et al. [35] below. For the FLP problem with n RSMFs,the extra binary variables, additional continuous variables, andauxiliary constraints in various methods are compared in Table I.

As seen in Table I, Models III and V are better than the othermodels (i.e., Models I, II, and IV) because they have fewerauxiliary constraints and additional continuous variables, whileModels III and V are significantly better than the Yang et al.model (YM) [35] because they do not require extra binary vari-ables. This reduces the complexity of the problem from NPhard to polynomial time. In contrast, all of the previous mod-els, which are discussed above, can only solve S-shaped MFswith one union and one intersection. In order to formulate anS-shaped MF with multiunion and multiintersection, a generalachievement model should be developed. For the sake of sim-plicity, but without the loss of generality, a case of RSMF withtwo unions and two intersections is considered, as shown inFig. 8.

The achievement model of Fig. 8 can be expressed as follows.Model VI:

Minwi1pi1 + wi2pi2 + wi3pi3 + wi4pi4

+ wi5pi5 + i(e+i + ei )

s.t.

i = [ij (bi2) ij (bi1)]pi1


pi2bi3 bi2


bi4 bi3


Fig. 8. RSMF with two unions and two intersections.



pi5bi6 bi5

(34)

i e+i + ei = 1 (35)zi(x) pi1 pi2 pi3 pi4 pi5 bi1 (36)wi1 < wi2 < wi3 < wi4 < wi5 (37)

0 pi1 bi2 bi1 , 0 pi2 bi3 bi2 , 0 pi3 bi4 bi30 pi4 bi5 bi4 , 0 pi5 bi6 bi5 (38)

x F (whereF is a feasible set).

Following the idea of Model VI, a general RSMF can beformulated as follows.

Model VII:

Minn

j=1

wij pij + i(e+i + ei )

s.t.

i =n

j=2

[ij (bij ) ij (bij1)

pij1bij bij1

](39)

i e+i + ei = 1 (40)

zi(x)n1

j=1

pij bi1 (41)

wi1 < wi2 < < win1 (42)0 pij1 bij bij1 , j = 2, . . . , n (43)x F (whereF is a feasible set).

As seen in Model VII, a general RSMF with n unions orintersections, two auxiliary constraints, and n additional con-tinuous variables are required. None of the binary variables is

needed, no matter how the number of unions and intersectionsare increased.

IV. COMPUTATIONAL EXPERIENCE AND DISCUSSION

A. Computational Experience

The superiority of the proposed method can be demonstratedthrough some test examples. These test examples follow thesame pattern, as do FLP with S-shaped MF. Five groups of thetest examples are characterized by the number of S-shaped MFs(n = 50, 100, and 200) for each FLP. For each of the five groups,ten FLP with S-shaped MFs are randomly generated, therebyforming a set of 50 test problems. According to the simplexmethod, Models III and V are more efficient and compact thanthe other models given in Table I. Thus, each of the test examplesare formulated as FLPs with S-shaped MF by all the models andthen solved by LINGO [31] on a PC with Pentium 4 Duo CPU(2.2 GHz). The performances of all the models, as measured byCPU time, are compared in Table II.

As can be seen in Table II, all models can be solved within ashort computational time, even with a large number of S-shapedMFs. Based on Table I, Models III and V are better than theother models because they have fewer auxiliary constraints andadditional continuous variables. In order to demonstrate the su-periority of the proposed methods, another computational test isconducted. For sake of simplicity, but without the loss of gener-ality, we compare YM, Li and Yus model (LM), and Model III,as measured by CPU time and the number of iterations. Eachof the test examples is formulated as an FLP with a number ofS-shaped MF (n = 20, 30, 40, and 50) by the YM, LM, andModel III and then solved by LINGO [31] on a PC with a Pen-tium 4 Duo CPU (2.2 GHz). The CPU time and number ofiterations of the models are compared in Tables III and IV.

As can be seen in Tables III and IV, Model III can be solvedwith shorter computational time and fewer iterations, even with alarge number of S-shaped MFs. This is not true for YM and LM.The superior performance of Model III becomes more evidentwhen the number of S-shaped MFs is increased. This outcomeis expected because Model III adds no extra binary variables.In contrast, n extra binary variables are required to represent nS-shaped MFs in YM and LM. In other words, the superiority ofModel III becomes more pronounced as the number of S-shapedMFs increases.

B. Discussion

As can be seen in Tables IIIV, according to the efficiencyof the various methods, all proposed models are better thanthe YM and LM models because all of the proposed modelsrepresent linear forms. The YM and LM models represent mixedinteger forms. The convergence and termination criteria of YMand LM follow the branch-and-bound approach. In contrast,the convergence and termination criteria of all of the proposedmodels follow the principle of the simplex method. The maximalerror of the approximated piecewise-linear approach can be seen


TABLE IIPERFORMANCE OF ALL MODELS (CPU TIME: SS)

TABLE IIIPERFORMANCE OF ALL MODELS (CPU TIME: MM:SS)

TABLE IVPERFORMANCE OF ALL MODELS (NUMBER OF ITERATIONS)

Fig. 9. Convex function with break points.

in Section II. The efficient algorithm to select break points isdescribed as follows.

Step 1 (Select initial break points)1) For each convex function z(y), during the interval [b0 , b],

as shown in Fig. 9, the first point is selected at point y bythe method described in Section II.

2) For each convex function z(x), during the interval [a0 , a],as shown in Fig. 10, the first point is selected at point xby the method described in Section II.

Step 2 (Formulate the piecewise function)The proposed models can be used to approximately linearize

S-shaped function z(x).

Fig. 10. Concave function with break points.

Step 3 (Solve the program and assess acceptable error):Solve the formulated model to obtain the solution x =(

x1 , x2 , . . . , x

n

). If , for all i, where zi(xi ) is the ap-

proximate linear function, then terminate the solution process;otherwise, go to Step 4.

Step 4 (Add finer break points):If bk xi bk , for convex function z(y), or ak xi

ak , for concave function z(x), add a new break point within theinterval, and repeat Step 2.

V. ILLUSTRATIVE EXAMPLE

In this section, we consider the following slightly modifiedFLP example used by Chang [4]. A university plans to promotetheir programs using E-learning approaches via the Internet toincrease the number of students enrolled. In order to achievethis objective, a new E-learning web server (ELWS) should beset up in the university. It is important to note that according toa marketing survey, many students are enrolled in the univer-sity due to the higher user satisfaction of ELWS. In addition,the survey indicates that with a higher number of ELWS, thestudents feel more satisfied. Thus, a major issue of the plan ishow to develop the maximum number of ELWSs with limitedresources to offer the greatest user satisfaction. The survey alsoshows that the relationship between the number of ELWS, andthe students satisfaction can be represented by an S-shaped MF,as depicted in Fig. 11. The parameters of the problem are givenin Table V. Corresponding to the resource limitations of the uni-versity, some constraints are also given as follows: 1) The totalavailable investment budget must not exceed $1.5 million; 2) at


Fig. 11. Number of ELWS versus satisfaction of students (which is called therate of student satisfaction).

least seven ELWSs should be implemented; and 3) the systemmust serve at least 10 000 students.

The problem can be formulated using the approach proposedby Yang et al. [35] as follows.

P1:

Max

s.t.

(1 12 0(x)

10

)+M(1 1) (44)

(1 8 1(x)

5

)+M1 (45)

(1 9 2(x)

7.5

)+M1 (46)

0(x) 12, 1(x) 8, 2(x) 9 (47)0(x) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9

+ x10 + x11 + x12 (Number of ELWSs) (48)

1(x) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9


2(x) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9


0.2x1 + 0.1x2 + 0.2x3 + 0.2x4 + 0.25x5 + 0.2x6 + 0.2x7

+ 0.2x8 + 0.3x9 + 0.2x10 + 0.2x11

+ 0.2x12 1.5 (Budget) (51)x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11

+ x12 7 (At least seven EWLSs) (52)3x1 + 1.5x2 + 2x3 + 2.5x4 + 1.2x5 + 0.8x6 + 0.7x7 + 0.9x8

+ 0.4x9 + 1.5x10 + 1.1x11 + 1.3x12

10 (Serve at least 10 thousand students) (53)

where xi (i = 1, 2, . . . , 12), and 1 is a binary variable,i(x)(i = 0, 1, and 2) are three ramp-type MFs, and (0 1) is the additional continuous variable that represents themembership value of the S-shaped MF. This problem is solvedusing LINGO [31] on a PC with Pentium 4 Duo CPU (2.2 GHz)to obtain the solution as (x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 , x10 ,

x11 , x12 , 1) = (1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0), the number ofELWSs is 8, the E-learning system serves 12 500 students, andthe membership value = 0.866 (i.e., the rate of satisfaction is86.66%).

We assume that the value of i is 30, which is large enoughto force the membership value to approach the highest levelof MF (i.e., value of one) as closely as possible. The problemcan be expressed using Model II as follows.

P2:

Min p1 + p2 + p3 + 30(e+i + ei )

s.t.

= 0.2p1 p2

2+ 0.4

p2 p32

+ 0.4p33

e+i + ei = 10(x) p1 = 20(x) p2 = 40(x) p3 = 60(x) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9

+ x10 + x11 + x12 (Number of ELWSs)

1(x) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9


2(x) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9


0.2x1 + 0.1x2 + 0.2x3 + 0.2x4 + 0.25x5 + 0.2x6 + 0.2x7

+ 0.2x8 + 0.3x9 + 0.2x10 + 0.2x11

+ 0.2x12 1.5 (Budget)x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11

+ x12 7 (At least seven EWLSs)3x1 + 1.5x2 + 2x3 + 2.5x4 + 1.2x5 + 0.8x6 + 0.7x7

+ 0.9x8 + 0.4x9 + 1.5x10 + 1.1x11

+ 1.3x12 10 (Serve at least 10 thousand students)

where all variables are defined as in Model II.This model is solved again using LINGO [31] on the same

PC to obtain the solution as (x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 ,x9 , x10 , x11 , x12 , p1 , p2 , p3) = (1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0,1, 6, 4, 2). The number of ELWSs is 8, the E-learning systemserves 12 700 students, and the membership value = 0.866(i.e., the rate of satisfaction is 86.66%). The number of ELWSis equal to P1. Thus, the satisfaction level is also equal to P1.It is noteworthy that P2 provides another alternative solution,which serves 12 700 students for DM reference. Obviously, noextra binary variables i are needed in P2. In addition, it is muchmore compact than P1. According to Proposition 2, we assumethat the weights given as 1, 2, and 3 are attached to deviationsp1 , p2 , and p3 , respectively, in order to assume the deviation


TABLE VELWS PARAMETERS

TABLE VIRESULTS OF ALL RIGHT S-SHAPED MF PROGRAMS

priority rank order of the deviation to be p1 > p2 > p3 . Theproblem can be formulated using Model III as follows.

P3:

Min p1 + 2p2 + 3p3 + 30(e+i + ei )

s.t.

= 0.2p12

+ 0.4p22

+ 0.4p33

e+i + ei = 10(x) p1 p2 p3 20 p1 2, 0 p2 2, 0 p3 30(x) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9


1(x) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9


2(x) = x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9


0.2x1 + 0.1x2 + 0.2x3 + 0.2x4 + 0.25x5 + 0.2x6 + 0.2x7

+ 0.2x8 + 0.3x9 + 0.2x10 + 0.2x11

+ 0.2x12 1.5 (Budget)x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11

+ x12 7 (At least seven EWLSs)3x1 + 1.5x2 + 2x3 + 2.5x4 + 1.2x5 + 0.8x6 + 0.7x7 + 0.9x8

+ 0.4x9 + 1.5x10 + 1.1x11 + 1.3x12

10 (Serve at least 10 thousand students).

This model is solved again using LINGO [31] to obtain thesolution as (x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 , x10 , x11 , x12 , p1 ,p2 , p3) = (1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 2, 2, 2). The number ofELWSs is 8, the E-learning system serves 12 500 students, andthe membership value = 0.866 (i.e., the rate of satisfactionis 86.66%). It is noteworthy that the optimal membership value = 0.866 is found in all RSMF programs (proposed ModelsIIII). For more details, the results of all of the RSMF programsare listed in Table VI. For the example with an RSMF or LSMF,the extra binary variables, additional continuous variables, and


TABLE VIISIZE OF VARIOUS METHODS

Fig. 12. Behavior of parameters affecting the achievement of goal.

auxiliary constraints of various methods are compared in Ta-ble VII. This table shows that no extra binary variables are usedin Models IV. In addition, Models III and V require fewer aux-iliary constraints, thereby making them computationally moreefficient than the model of Yang et al. [35] when the problemsize becomes large. In order to find out how by the changingof the weight values from 16 to 36 in the objective functionof Model III affects the target value (i.e., the rate of satisfac-tion) achieved, sensitivity analysis is performed. This problemis solved again using LINGO [31] sensitivity analysis to obtainthe optimal solution, as shown in Fig. 12, and the rate of satis-faction increases from 0.73 to 0.86 when the weight increasesfrom 22 to 23. No matter how much the weight increases, themaximum satisfaction level is 0.86.

VI. CONCLUSION

FLPs with S-shaped MF (or utility function) are often appliedin a wide variety of fields, such as psychology, organizationalbehavior, economics, and investment. However, in most pre-vious studies, extra binary variables are needed to formulateproblems, and these extra binary variables cause tedious com-putational burden when problems become large. To solve thisproblem, this study proposes several piecewise-approximationapproaches to formulate FLPs with an S-shaped MF in whichextra binary variables are not required. In addition, no extra bi-nary variables are needed for a precise approximation S-shapedMF, no matter how the number of unions and intersections areincreased. This reduces the complexity of the problem from anNP-hard problem to a polynomial-time problem. As a result,the superiority of the proposed method can be observed throughcomputational experience, which expresses the CPU time and

number of iterations. In addition, an illustrative example is alsoprovided to demonstrate the usefulness of the proposed meth-ods. Further research can examine nonlinear penalty functionsin the field of multiple objective decision making.

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Ching-Ter Chang received the M.S. and Ph.D. de-grees from the Department of Information Man-agement, National Chiao Tung University, Hsinchu,Taiwan, in 1992 and 1997, respectively.

He is currently a Full Professor with the ChangGung University, Tao-Yuan, Taiwan. His research in-terests include decision making, supply-chain man-agement, mathematical programming, and digitalapplications.

an approximation approach for representing s-shaped membership functions

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