An Approach to Improve Practical Application ofFuzzy Measures in Multicriteria Decision Making

Amol S. WagholikarSchool ofCommunication and Information Technology

Griffith UniversityGold Coast, Queensland, Australia

a.wagholikar(griffith.edu.au

Abstract - There has been continuous development in research contributes to the theory of fuzzy measures,approaches that can model human reasoning process in decision multicriteria decision-making and similarity-based reasoning.problems. Application of fuzzy measures and fuzzy integrals is a Such combination of important theories illustrates thesignificant development in this pursuit. Different decision modelsare used in building modern decision support systems that signif e ofaths reserch lTi or ffeasneprovide intelligent assistance to the decision maker. Most of the perspectioworeal-life decision problems involve interactive criteria. In these determination.problems, unlike additive measures non-additive measures suchas fuzzy measures are applied to determine the overall worth of II. ABOUT NON-ADDITIVE MEASURESan alternative due to their inherent ability to model theinteraction between the criteria. However, determination of fuzzy We are addressing the issue of fuzzy measure determination inmeasures in practical applications is problematic. This paper the context of multicriteria decision-making (MCDM).discusses this issue and proposes an approach to resolve the issue. MCDM is a systematic and formal decision model. It consists

of evaluation of given set of alternatives against a given set ofI. INTRODUCTION criteria. The chosen criteria can be dependent or independent.

Multicriteria decision-making is a systematic and formal They can be positively or negatively related with each otherdecision model [1]. It is one of the common models used in [7]. The interaction between the criteria should be consideredsolving decision problems involving interactive criteria. Most while aggregating the criteria evaluations. Normally, additiveof the real-life decision problems are characterized by the measures such as weighted average are used for aggregation.presence of interactive criteria. The interaction between the Due to the additive nature of these measures, these measurescriteria must be considered to provide intelligent decision are inadequate to model the phenomenon of interaction.support to the decision maker. Traditional additive measuressuch as weighted average are inadequate to model the A. Non-additive measuresinteraction between the criteria [2]. This calls for novel Non-additive measures such as fuzzy measures and fuzzyapproaches that can address this issue. Non-additive measures integrals have the ability to model the issue of dependencesuch as fuzzy measures are capable of modeling the betweenthe criteria.interaction between the criteria. However, there are certainissues in transferring the theoretical ability in practical B. Fuzzy Measuressituations. This is mainly due the complexity in determining Fuzzy measures model the combined importance of a sub-setfuzzy measure coefficients in practical applications. It is of criteria. These are monotone set functions defined over aalways important to determine practical ways of applying universal set C and a non-empty family P of sub-sets (i.e.fuzzy measures in decision problems [3]. In this pursuit, the * 11main research objective of this work is to investigate the issue power set) of C. It is defined as

=IP L[ ,where

of fuzzy measure acquisition and develop a methodology to iIiP) = 0 & 8(X) = 1 The fuzzy measures possess theresolve the issue in practical problems. This work proposes a property of super additivity and sub additivity, which modelsnovel approach for determining fuzzy measures in the positive and negative type of interaction respectively.multicriteria decision-making.

The proposed approach rests on the conceptual foundationof similarity-based reasoning [4]. There are various C. Fuzzy Integralsapproaches suggested for resolving the issue of fuzzy measureacquisition [5] [6]. These approaches mainly suggest the use Fuzzy integrals are the incremental summation of the productof past precedents for determination of fuzzy measures. This of the criteria evaluat1on and Its fuzzy measures. There are twowork has extended this notion by applying similarity-based types of commonly used fuzzy integrals viz Choquet integralreasoning for resolving this issue. This work illustrates a [8] and Sugeno integral [9]. Choquet integral is used for asignificant application of similarity-based reasoning especially quantitative setting whereas Sugeno integral is proposed for acase-based reasoning for solving MCDM problems. This

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qualitative setting [10]. Since we have assumed a quantitative n

setting, we would use Choquet integral for this work. E WiLet C = {cl, c2, ....,c1,} be the set of elements. Fi be the fuzzy decision maker such that i=1 For this above decisionmeasure on C. The elements of C can be criteria in MCDM, problem, we have case-base in the following form.players in cooperative games or set of beliefs in Demester- i. A set of attributes same as the new decision problem orshafer theory. The Choquet integral of the function f: C - target problem mentioned above.[0,1] with respect to Fi is given by. ii. Absolute values of each attribute.

n iii. A set of individual weights defined by the decision makerCh, {f (c (i))- f(c (i-1))} !I (A (i)) (1) at the previous decision making instance.

i =1 iv. A set S of fuzzy measure coefficients defined previously.Where .(i) indicates that the indices have been permuted such We assume that these coefficients are constructed usingthat - expert's feedback.0 c f(c(1))f(c (2)) c........cf(c (n)) 1 and A(i) { c v. Global evaluation of the object obtained by using the(i), , c (n)}. We can observe that for aggregating "n" Choquet integral and the fuzzy measure coefficients obtainedelements, we need to determine 2'-1 fuzzy measure from the experts.coefficients. Hence their practical determination is the main 2. After constructing the above data set, the similarity of theissue especially for higher values of "n". cases w.r.t to the new decision problem is determined using

absolute value-based City-block distance metric or Manhattandistance metric [16]. The Manhattan distance metric is defined

III. DETERMINATION OF FUZZY MEASURES as the absolute difference between the values of pair ofattributes or criteria I.e.

As mentioned earlier, fuzzy measure is the coefficient of the n

power set of sub-sets of criteria. The issue of determining the d = I Ixk - xIkfuzzy measures is mainly marked by the complexity due to k=1 (2)high number of fuzzy measure coefficients. For e.g. when n Where xi & xj are the values of the attribute "k". For=3, we can have 7 coefficients including individual as well as computing the similarity, the distance values should besub-sets of criteria. The number of fuzzy measure coefficients normalized in the range [0,1] by the maximum distance dmaxcan be very large for higher values of n, say when n is equal to between the corresponding attribute values in the new problem7. To resolve this issue certain data-driven methods [11], and the case base. The global similarity of a given case will beinteractive methods [12] have been suggested. Also some calculated by simple weighted average of the local similaritiesapproaches suggest the use of pre-defined fuzzy measures to of the given attributes. Thus the distance between the casesdetermine the new fuzzy measures or use of global evaluations will befrom the training data [13]. We are extending this notion of nusing training data about the past decisions. We are proposing E wkdia new method where the training data is in the form of a case- d(Caset, Caseu) = k=1base. The case-base will consist of data about the past n

decisions made under MCDM setting. k=1

A. Suggested Approach. Based on this distance, the similarity is calculated as

We are extending the notion of using learning data about the Sim(Caset,Caseu) = 1- d(Caset,Caseu) (4)past decisions. Reasoning by analogy is the main theoretical Where Caset = Target case i.e. the new decision problem &foundation of our approach [14][15]. New problems can be Caseu= Case from the case library i.e. past decision problems.solved using solutions of the past similar problems. We areproposing a new method where the learning data is in the form 3. After such similarity computation the case with maximumof a case-base. Case-based reasoning (CBR) allows solving similarity would be retrieved and the values of its fuzzynew problems based on the solutions from the past similar measures would be considered for computing the globalproblems. Hence CBR can be a good tool for this purpose. evaluation in the new decision problem. The worth of

alternatives will incorporate the interaction between theB. Proposed CBR-based Interactive model criteria.We are proposing a 2-phase model. These phases are asfollows- D. Phase II of the Suggested Approach

The decision maker is likely to have different importanceC. Phase I of the Suggested Approach values. Hence using a direct rating method [17]. We propose1. Let C = {C1 C2.Cfl} be the set of criteria against which an Importance scale designed to elicit the new fuzzy measure

asetAof lterntive {A1 . Am}ito eeautd h coefficients from the decision maker. The importance scale is

individual weights, wi on the criteria is elicited from the dsge oaktedcso ae bu h motneo

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sub-set of criteria. The decision maker can directly rate the Good Goodsub-set of criteria in a range between 0 and 1. The intuitively Case 4 $20,000 1500 12 Good Gooddesigned scale is shown below.

For the above data set, we conducted several experiments forIMPORTANCE SCALE determining similarity of the alternatives in table 1.2 with

respect to the case-base in table 1.3 using SPSS 14.0. WeSub-set Extremely 0

conducted experiments with different similarity measures.[ Sub-set|ExtremelIy 0° l Following table shows the results using City-block distance

is Less Highly 0.1 measures chosen for this approach.important Very 0.2

Strongly 0.3 TABLE IVPROXIMITY MATRIX - SPSS OUTPUT FOR CASE SIMILARITYModerately 0.4 USING CITY-BLOCK DISTANCE MEASURE

_ _ | Equally 1 0.5 _Sub-set Moderately 0.6 ] Absolute City Block Distanceis More Strongly | 0.7 Case 1 Case 2 Case 3 Case 4important Very 0.8 ] Car A .008 .009 .000 .008________ | Highly 0.9 Car B .035 .017 .027 .035

L________ | Extremely 1 Car C .005 .018 .008 .000Based on this scale, the new fuzzy measures can be elicited Car D .060 .068 .058 .050from the decision maker. Fuzzy measure values computed inthe second phase indicate the agreement between the systemvlsnthesc dpsecisdinmaker the firseeb twn an systeo This is a dissimilarity matrix. From this dissimilarity matrix,values and the decision maker's values in the first and second

wecnosreta a s iia oCs ^CrBiphasesrepciey we can observe that Car A is similar to Case 3, Car B issimilar to Case 2, Car C is similar to Case 4, Car D is similarto Case 4.Based on the above similarities and correspondingfuzzy measures of the similar cases, the global evaluations of

IV. ILLUSTRATION all the cars is given by the proposed case-based decisionsupport system. We used RDBMS technology for the system

Let us consider a classic car selection problem where the development. The agreement of the decision maker with thefollowing data is assumed. There is no standard data for the above set of results is determined by the direct rating methodapplication of the proposed algorithm. Hence we have built a using the proposed importance scale.customized data set for a car selection problem. The data set From the above dissimilarity matrix, we can observe that -consists of a set of 4 alternatives. These alternatives are Car A is similar to Case 3evaluated against a set of 5 criteria. The absolute values of thecriteria for a given alternative are expressed in numeric as well Car B is similar to Case 2as linguistic terms.

Car C is similar to Case 4TABLE II SAMPLE

DATA SET FOR CAR SELECTION PROBLEM Car Dissimilar to Case 4Price Power Fuel Space Comfort

Economy Based on the above similarities and corresponding fuzzyWeights 0.3 0.2 0.2 0.1 0.2 measures of the similar cases, the global evaluations of all theCarA $30,000 2000 -10 -Good GoodCarB $30,000 2000 1 O OK cars is given by the proposed case-based decision supportCar C $25,000 1000 12 OK OKCar C $20,000 1500 12 OK OK system. We propose to use RDBMS technology for the systemCar D $35,000 2500 8 Very Very development. The sample output from the system is shown

Good Good below. The output shows the global evaluation of the cars. Itcan be observed that the global evaluation of the cars is

Suppose for the above data set, we have the following case- intuitive. Data from Table II supports the results given by thebase. system.

TABLE III CASEBASE OF PAST DECISIONS

Price Power Fuel Space ComfortEconomy

Case 1 $40,000 3000 8 Very VeryGood Good

Case 2 $35,000 2000 10 Good GoodCase 3 $45,000 3000 7 Very Very

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Since this is a case-based decision support system, itsGlobal Evaluations for the Cars performance will depend upon the number of cases in the case

library. If the number of cases is large, then the resultsC||A 0 expected from the system should be more accurate. However,CarB 07 the system performance may degrade due to higher retrieval

time. Hence there may need to be a trade off between theC| C| 68 number of cases and the system performance. The overallCIit 0.6751 ranking of the alternatives is also an important indicator ofCarlll es Zsystem performance. The results given by the system

prototype are intuitive as they are model the interactionOK between the criteria correctly. The evaluation of the proposed

systems illustrates the usefulness of this research for solvingfuzzy measure acquisition problem in practical applications.

Fig. 1.1. Sample Output from the System Prototype

The agreement of the decision maker with the above set of VI. CONCLUSIONresults is determined by the direct rating method using theproposed importance scale. The sample output for the direct We discussed an important issue of acquisition of fuzzyrating method is shown in the following interface. measures in practical decision-making problems. This issue

was investigated in detail. In this paper, we presented aninteractive model for determining non-additive measures inthe context of multi-criteria decision-making to resolve the

/'price,power,con~~~forL undertaken issue. We have proposed a model using similarity-F:i based reasoning. We have illustrated the prototype of theF

xmproposed system with a sample data set and through an

-mtart E j& |experimental implementation. This model demonstrated ther 0. original, extensive and significant contribution of our research.F _|MH This research has made an important contribution inF simplifying the complex problem of fuzzy measure acquisition

mmis_" p MM-in practical applications.lmig rlrF

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