Comparing Rankings from Using TODIM and a Fuzzy Expert System Procedia Computer Science 55 ( 2015 ) 126 – 138 1877-0509 © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ITQM 2015 doi: 10.1016/j.procs.2015.07.019 ScienceDirect Available online at www.sciencedirect.com Information Technology and Quantitative Management (ITQM 2015) Comparing rankings from using TODIM and a fuzzy expert system Valério Antonio Pamplona Salomonb*, Luís Alberto Duncan Rangela aUFF, Av. dos Trabalhadores 420, 27255-125, Volta Redonda, RJ, Brazil bUNESP, Av. Ariberto Pereira da Cunha, 333, 12.516-410, Guaratingueta, SP, Brazil Abstract TODIM is, in its original formulation, an MCDA method developed to solve ranking problems. As an MCDA method TODIM combines the use of a multi-attribute value function as well as elements of the Outranking Approach, being founded on Prospect Theory. Recent advances in TODIM incorporate concepts from Fuzzy Sets. Although modelling multi- criteria decision problems with Fuzzy Sets has been utilized when the available data are imprecise, their use in MCDA is slightly controversial, because the data fuzzification can invalidate the outcome. Following a mixed qualitative-quantitative research strategy, our aim is to prove that for the ranking problems, TODIM can provide better solutions than Fuzzy Sets. Ranks from TODIM are linear, or strong, in a sense that it has no ties between the alternative solutions. The rank obtained with a Fuzzy Expert System can be weaker, that is, it may be a number of ties. The research strategy extends this result to ranking problems with the occurrence of crisp criteria. © 2015 The Authors. Published by Elsevier B.V. Selection and/or peer-review under responsibility of the organizers of ITQM 2015 Keywords: Fuzzy Expert Systems; Ranking Problem; TODIM 1. Introduction “On any one day people face a plethora of different decisions” [1]. This way, Multi-Criteria Decision Analysis (MCDA) methods have been developed to support decision makers in their decision problems. One reason for different MCDA methods is that there are different decision problems. First classifications of decision problems are Discrete Problem versus Continuous Problem. A Discrete Problem involves a discrete set of alternative solutions. A Continuous Problem involves a case where the number of possible alternatives are infinite [2]. There are four types of Discrete Problem: selecting an alternative solution (Choice Problem), * Corresponding author. Tel.: +55-12-3123-2232; fax: +55-12-3123-2468 E-mail address: salomon@feg.unesp.br © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ITQM 2015 http://crossmark.crossref.org/dialog/?doi=10.1016/j.procs.2015.07.019&domain=pdf 127 Valério Antonio Pamplona Salomon and Luís Alberto Duncan Rangel / Procedia Computer Science 55 ( 2015 ) 126 – 138 grouping alternatives (Sorting Problem), ordering alternatives from best to worst (Ranking Problem), or better descripting alternatives (Description Problem) [3]. This work focuses the Ranking Problem. The large number of MCDA methods engendered classifications for the methods. American School and European School [4] are, perhaps, the most well-known. These classifications are criticized, not only for xenophobia, but also to difficult developments by international teams [5]. Aggregation Approach and Outranking Approach are better classifications. However, both sets of classification shave often the same result. For instance, Analytic Hierarchy Process (AHP) and Multi-Attribute Utility Theory (MAUT) are MCDA methods for the Aggregation Approach, and they are from American School [6-7]; Elimination et Choix Traduisant la Realité (ELECTRE) and Preference Ranking Organization Method for Enriched Evaluation (PROMETHEE) are MCDA methods for Outranking Approach, and they are from European School [8-9]. An exception is Measuring Attractiveness by a Categorical Based Evaluation Technique (MACBETH), which is a MCDA method for Aggregation Approach, and is from European School [10]. Matter of fact, the choice of an MCDA method shall be based on characteristics from the decision problem, including necessary data and expected results. Nevertheless, the choice for an MCDA method have been a matter of opinion. This way, decision maker chooses to apply a familiar MCDA method to solve a new decision problem. Since decision maker has already applied this method, the new application gains in feasibility. On another way, the decision maker may choose a not familiar method. Or else, a method never applied before, just to expand decision maker’s knowledge in MCDA practice. Different MCDA methods may yield different results when applied to the same problem [11]. Still, a single method application can lead to different ranks. This can be a result from different individuals providing data, or it can be resulted from time-lapses in data collection. This work addresses the divergence between ranks from different MCDA applications with the concepts of rank correlation [12]. Multi-Criteria Interactive Decision-Making (shorted as TODIM, from Portuguese) is an MCDA method [13] developed to solve the Ranking Problem, TODIM combines elements from both Aggregation Approach and Outranking Approach. It first application was to rank projects with environmental impacts. Later, TODIM incorporated elements from Prospect Theory [14-15]. The previous case on environmental impacts was analyzed. The ranks with Prospect Theory diverge from the original rank. However, the ranks have some degree of correlation. The most well-know TODIM application was to rank residential properties [16]. It was recently revisited, considering criteria interactions [17]. The ranks with criteria interactions and with TODIM also diverge each other. However, the top two alternatives are the same from both applications. That is, the ranks are correlated each other. Recent advances in TODIM incorporate concepts from Fuzzy Sets [18-19]. Fuzzy Sets Theory (FST) was firstly proposed for the Classification Problem [20]. “A fuzzy set is a class of object with a continuum of grades of memberships. Such a set is characterized by a membership function which assigns to each object a grade of membership ranging between zero and one” [21]. In Classical Sets Theory (CST), sets are crisp. That is, an element belongs to a set or not. Then, when the available data are imprecise, FST is expected to better solve Classification Problem than CST. Therefore, new versions of classical MCDA methods were developed to incorporate FST [22-24]. FST have also been successfully applied to the Ranking Problem [25-26]. However, the use of Fuzzy Sets in MCDA is slightly controversial [27]. When these data are allowed to vary in choice over the values of a scale, as in AHP, these data are themselves already fuzzy [28]. Then, data fuzzification can invalidate the outcome. On the same issue, in real life crisp sets do exist [29]. Therefore, the use of Fuzzy Sets may result in loss of information, when transforming precise data in imprecise information. This work takes place on the side of questioning the indiscriminate use of FST in MCDA. Our aim is to prove that, for the Ranking Problem, TODIM can provide a better solution than FST. A mixed qualitative- quantitative research strategy [30] was adopted. That is, the goal is not to consider exhaustive cases. 128 Valério Antonio Pamplona Salomon and Luís Alberto Duncan Rangel / Procedia Computer Science 55 ( 2015 ) 126 – 138 Next section presents concepts on rank correlation, TODIM, and FST. Section 3 presents a case of Ranking Problem from a Brazilian real estate market. This problem involves fifteen alternatives and eight criteria, with a crisp criterion. The rank from TODIM is linear, or strong, in a sense that it has no ties between the alternative solutions. The rank obtained with a FST is weaker, that is, it has a number of ties. The research strategy extends this result to ranking problems with the occurrence of crisp criteria. Section 4 presents some conclusions and proposal for future researches. 2. Theory Background 2.1. Correlation between ranks As observed in Section 1, different MCDA methods may yield different ranks to the same problem [11]. The rank correlation coefficient is a measure of ranks agreement [12]. For two ranks of n elements, A and B, Kendall coefficient, b, is obtained by Equation 1, when aij and bij are score matrices obtained by Equation 2. 1 1 1 nn ba n i n j ijij b (1) tiedare and objects0 object of behind ranked is object 1 object of ahead ranked is object 1 ji ji ji aij (2) Two identical ranks will have agreements in all positions, aij = bij, then b = 1. For opposite ranks, that is two ranks with no agreement on any position, b = –1. For instance, if A = (1, 2, 3, 4), B = (1, 3, 2, 4), and C = (4, 3, 2, 1), we have b (A, B) ≈ 0.67, b (A, C) = –1, and b (B, C) ≈ – 0.67. Ranging from –1 to 1, Kendall coefficient measures the closeness of correspondence between two given ranks. In other words, it measures the compatibility between two ranks [31]. Kendall coefficient performs satisfactory to linear ranks. However, it presents some difficulties with partial and weak ranks. Linear ranks has no ties; weak ranks permits ties; two or more ranks are partial when at least one of them is incomplete, in the sense that each ranker may not necessarily rank all of the objects [32]. Emond-Mason coefficient, x, differs from Kendall coefficient by using a value of one for aij and bij to represent ties instead of the value of zero used by Kendall’s. By extending this interchange to acco mmodate ties, x is not flawed as b for weak ranks or for partial ranks [12]. An MCDA application may result weak ranks; a Fuzzy System may result weak ranks, too. Then, x will be adopted in this work, instead of b. 2.2. TODIM method TODIM has similarities with other MCDA methods as ELECTRE [33] and PROMETHEE [34]. However, while practically all other MCDA methods start from the premise that the decision maker always looks for some maximum overall value, TODIM method makes use of a measurement of overall value calculable according to Prospect Theory. TODIM application requires numerical values for the evaluation of the alternatives regarding the criteria. For qualitative criteria, alternatives can be evaluated in a verbal scale, but it must be then transformed into a cardinal scale. The numerical evaluation for the alternatives regarding to all the criteria composes the matrix of 129 Valério Antonio Pamplona Salomon and Luís Alberto Duncan Rangel / Procedia Computer Science 55 ( 2015 ) 126 – 138 evaluation. This matrix must be normalized, for each criterion: the value for one alternative must be divided by the sum of values for all the alternatives. This way, a stochastic matrix is obtained, that is, a matrix where all the components are in-between zero to one, and every column sums equal to one. This is the matrix of normalized alternatives’ scores against criteria, P = pnm, with n indicating the number of alternatives and m the number of criteria. The next step is the attribution of weights for the criteria. Usually, weights are attributed by DM using a linear 1 to 5 scale, similar to the Likert scale (Likert, 1932) [35] The decision makers must indicate a criterion r as the reference criterion. The criterion with the highest weight is usually chosen. The vector of weights, wr = wrc, is composed by the weight of the criterion c divided by the weight of the reference criterion r. The measurement of dominance (Ai, Aj) of each alternative Ai over each alternative Aj, incorporate concepts of Prospect Theory, according to Equation 3. m c jicji jiAAAA 1 ),(),,(),( (3) Where: 0)(if )()( 1 0)(if0 0)(if )( ),( 1 1 jcic rc icjc m c rc jcic jcicm c rc jcicrc jic PP w PPw PP PP w PPw AA The expression c(Ai, Aj) is the contribution of criterion c to the dominance of alternative Ai over alternative Aj. If pic was greater than pjc, it will represent a gain for (Ai, Aj); if pic and pjc were equal, then a zero will assigned to (Ai, Aj); if pic was less than pjc, then c(Ai, Aj) will be a loss to (Ai, Aj). The function c(Ai, Aj) allows the adjustment of problem data to the Prospect Theory, that is, considering the aversion to risk and the propensity to risk. This function has the shape of an ‘‘S’’, as presented in Figure 1. 130 Valério Antonio Pamplona Salomon and Luís Alberto Duncan Rangel / Procedia Computer Science 55 ( 2015 ) 126 – 138 Figure 1. Value function of the TODIM method [15] Above the horizontal axis, that is for value equal to zero, there is a concave curve representing the gains; below the horizontal axis, there is a convex curve representing the losses. The concave part reflects the aversion to risk in the face of gains and the convex part, in turn, symbolizes the propensity to risk when dealing with losses. is the attenuation factor of the losses. Different choices of lead to different shapes of the prospect theoretical value function in the negative quadrant. The overall value for alternative Ai, i, is obtained with Equation 4. n j ji n j ji n j ji n j ji i AAAA AAAA 11 11 ),(min),(max ),(min),( (4) 2.3. Fuzzy expert systems Expert system is an information system that emulates the decision-making ability of a human expert [36]. An expert system is typically made of three parts: a Knowledge Base, an Inference Engine and a Working Memory [37]. The Working Memory is the stored information gained by the user of the system. The Inference Engine uses the Knowledge Base together with information from the problem to provide an expert solution. If–Then rules are popular schemes for knowledge representation as “If premise then conclusion”. In a Fuzzy Expert System, premises and conclusion are fuzzy propositions, as in “If X is small then Y is large with a certainty factor equal to 0.8” [28], for instance, because Small and Large are fuzzy sets. Several membership functions can be used in the definition of a fuzzy set. One of the most used is the triangular function [38].As presented in Figure 2, a triangular fuzzy set has a triangular membership function. A triangular fuzzy set is usually represented as a vector, (x1, x2, x3), where A (x1) = A (x1) = 0, and A (x1) = 1. Figure 2. Triangular fuzzy set A = (x1, x2, x3) One of the most popular Fuzzy Expert System is the Mamdani Model [39]. In a Mamdani Model, If–Then rules may have several clauses as If A and B and C… Then Z, where all A, B, C… and Z are fuzzy propositions. For every clause in the rule, the matching degree between the current value for the variable and a linguistic label must be computed. The clauses are aggregated, using the minimum fuzzy operator. If more than one rule implies in the same result, the rules must be aggregated, using the maximum fuzzy operator. The overall matching degree can be obtained, also using the minimum fuzzy operator. This degree is referred as alpha-cut, 131 Valério Antonio Pamplona Salomon and Luís Alberto Duncan Rangel / Procedia Computer Science 55 ( 2015 ) 126 – 138 or -cut [40]. The -cut level will generate a new fuzzy set, with a trapezoidal membership function, as presented in ! . A real number may be obtained from the centroid of gravity (COG) of the resulting fuzzy set, within a process referred as defuzzification [41]. Figure 3. Defuzzification by the centroid of gravity 3. Illustrative Case 3.1. Data collection Volta Redonda is a Brazilian city situated in the south of the State of Rio de Janeiro. The city has approximately 260,000 inhabitants [42]. There is a large number of properties, residential and commercial, rented or available for rent. The major steel plant installed in the city in the 1940’s is a landmark of Brazilian industrialization. Because of this industrial vocation, Volta Redonda was nicknamed Steel City. However, its economy is quite diverse on services as education and transportation, to name a few. Local real estate agents mentioned eight criteria for the selection of a residential property: Location (C1), Constructed Area (C2), Construction Quality (C3), State of Conservation (C4), Garage Spaces (C5), Rooms (C6), Attractions (C7), and Security (C8). Criteria C2, C5, C6, and C8 are quantitative; C2 is measured in m 2; C5 and C6 are measured in unities of rooms or space garages; and C8is a crisp criterion, since a residential property has security or has not. Tables 1 to 4 present the possible scores to evaluate alternatives according to qualitative criteria. Table 1. Possible scores to C1 Location Score Periphery 1 Between periphery and an average location 2 Average location 3 Good location 4 Excellent location 5 Table 2. Possible scores to C2 Construction Quality Score Low standard 1 Average standard 2 132 Valério Antonio Pamplona Salomon and Luís Alberto Duncan Rangel / Procedia Computer Science 55 ( 2015 ) 126 – 138 High standard 3 Table 3. Possible scores to C3 State of Conservation Score Bad 1 Average 2 Good 3 Very good 4 Table 4. Possible scores to C4 Attractions Score Without attractions 0 Backyard or terrace 1 Barbecue 2 Swimming pool 3 Swimming pool, barbecue and others 4 Weights from 1 to 5 must be assigned to the criteria, where 1 goes to the lowest important criterion and 5 to the highest important criterion. Location (C1) was indicated as the reference criterion. Table 5 presents the assigned weighted and the normalized weights, that is, summing equal to one. Table 5. Weights of criteria Criterion Assigned weight Normalized weight Localization (C1) 5 0.25 Constructed Area (C2) 3 0.15 Construction Quality(C3) 2 0.10 State of Conservation (C4) 4 0.20 Garage Spaces (C5) 1 0.05 Rooms (C6) 2 0.10 Attractions (C7) 1 0.05 Security (C8) 2 0.10 Fifteen residential properties in different neighborhoods of Volta Redonda were evaluated. These alternatives were simply named as A1 to A15. Table 6 presents the scores assigned to the alternatives according to the qualitative criteria (C1, C3, C4, and C7) and real data for the quantitative criteria (C2, C5, C6, and C8). Table 6. Data and assigned scores of residential properties Residential properties C1 C2 C3 C4 C5 C6 C7 C8 A1 3 290 3 3 1 6 4 0 A2 4 180 2 2 1 4 2 0 A3 3 347 1 2 2 5 1 0 A4 3 124 2 3 2 5 4 0 A5 5 360 3 4 4 9 1 1 A6 2 89 2 3 1 5 1 0 A7 1 85 1 1 1 4 0 1 A8 5 80 2 3 1 6 0 1 A9 2 121 2 3 0 6 0 0 A10 2 120 1 3 1 5 1 0 A11 4 280 2 2 2 7 3 1 A12 1 90 1 1 1 5 2 0 133 Valério Antonio Pamplona Salomon and Luís Alberto Duncan Rangel / Procedia Computer Science 55 ( 2015 ) 126 – 138 A13 2 160 3 3 2 6 1 1 A14 3 320 3 3 2 8 2 1 A15 4 180 2 4 1 6 1 1 Table 7 presents the normalized score for the residential properties against the criteria. Table 7. Normalized scores for the residential properties against the criteria Residential properties C1 C2 C3 C4 C5 C6 C7 C8 A1 0.068 0.103 0.100 0.075 0.045 0.069 0.174 0 A2 0.091 0.064 0.067 0.050 0.045 0.046 0.087 0 A3 0.068 0.123 0.033 0.050 0.091 0.057 0.043 0 A4 0.068 0.044 0.067 0.075 0.091 0.057 0.174 0 A5 0.114 0.127 0.100 0.100 0.182 0.103 0.043 0.143 A6 0.045 0.031 0.067 0.075 0.045 0.057 0.043 0 A7 0.023 0.030 0.033 0.025 0.045 0.046 0 0.143 A8 0.114 0.028 0.067 0.075 0.045 0.069 0 0.143 A9 0.045 0.043 0.067 0.075 0 0.069 0 0 A10 0.045 0.042 0.033 0.075 0.045 0.057 0.043 0 A11 0.091 0.099 0.067 0.050 0.091 0.080 0.130 0.143 A12 0.023 0.032 0.033 0.025 0.045 0.057 0.087 0 A13 0.045 0.057 0.100 0.075 0.091 0.069 0.043 0.143 A14 0.068 0.113 0.100 0.075 0.091 0.092 0.087 0.143 A15 0.091 0.064 0.067 0.100 0.045 0.069 0.043 0.143 The overall values presented in Table 8 were obtained simply by aggregating the normalized scores for the residential properties (Table 7), weighted by the normalized vector (Table 5). The bolded A5, A11, and A14 have the highest overall values; the stricken throughA7, A9, A10, and A12 have the lowest values. Table 8. Overall values for the residential properties Residential properties Overall value Rank A1 0.301 6 A2 0.241 10 A3 0.245 9 A4 0.257 8 A5 0.454 1 A6 0.192 11 A7 0.159 14 A8 0.311 5 A9 0.185 12 A10 0.185 12 A11 0.351 3 A12 0.125 15 A13 0.291 7 A14 0.366 2 A15 0.338 4 3.2. TODIM application As in many other TODIM applications [43], = 1 is adopted. To illustrate TODIM computation, from Table 2, let us consider the pair A2 and A4: For C1, p21>p41, then 0.075 For C2, p22>p42, then = 0.054 134 Valério Antonio Pamplona Salomon and Luís Alberto Duncan Rangel / Procedia Computer Science 55 ( 2015 ) 126 – 138 For C3, p23 =p43, then = 0 For C4, p24