A new approach to avoid rank reversal cases in the TOPSIS method

Computers & Industrial Engineering 132 (2019) 84–97

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A new approach to avoid rank reversal cases in the TOPSIS method a

b,⁎

Renan Felinto de Farias Aires , Luciano Ferreira a b

T

Department of Applied Social Sciences, UFERSA, Mossoró, Brazil Management School, UFRGS, Porto Alegre, RS, Brazil

ARTICLE INFO

ABSTRACT

Keywords: Multi-criteria decision-making TOPSIS Rank reversal Normalization

During recent decades, different methods of Multicriteria Decision Support have been used to help decisionmakers select better alternatives for various decision problems. However, these methods have been criticized in the literature because they present a problem called rank reversal. In particular, analyzing this problem in relation to the TOPSIS method is still limited. Our review of the literature showed that papers are limited to analyzing cases of rank reversal by adding and removing alternatives and the solutions proposed are limited to case studies and can be improved in order to widen the scope of their application. Thus, we initially performed an analysis to determine the main cases of the rank reversal presented in the literature and to identify the main gaps in relation to the TOPSIS method. Next, we define a framework for evaluating both the TOPSIS method and the proposed model in relation to different cases of rank reversal. Finally, this paper puts forward a new method called R-TOPSIS, which proved to be robust in the experiments performed, since there were no cases of rank reversal for either the simulated cases nor for the real case used to validate it. The proposed method was also validated using statistics of dispersion and similarity to evaluate its adherence to the classic TOPSIS method.

1. Introduction The main focus of the multi-criteria decision making (MCDM) field is to introduce procedures, methods as well as tools for solving problems and consequently to support decision-makers (DM) to make better decisions. In MCDM problems, the overall performance of the alternatives is evaluated with respect to several and conflicting criteria, and the objectives are combined based on the DM’s preferences (Hwang & Yoon, 1981). Despite the great growth and evolution of the MCDM field, these methods have presented a problem called rank reversal (RRP), for which the literature has given different interpretations. Saaty, for example, states “Assume that an individual has expressed preference among a set of alternatives, and that as a result, he or she has developed a ranking for them. Can and should that individual’s preferences and the resulting rankings order of the alternatives be affected if alternatives are added to or deleted from the set?” Researchers such as Saaty and Vargas (1984b), Saaty (1987) and Millet and Saaty (2000) also argued for the legitimacy of this problem. For some authors such as Saaty and Vargas (1984b) and Millet and Saaty (2000) the reversals do not occur often and thus having rank reversals in certain occasions and of certain types may not be indicative of faulty decision making. On the other hand, Salem and Awasthi (2018) mentioned that this

⁎

problem could drive some DMs away from using methods known to have rank reversal (RR). In fact, Anbaroglu, Heydecker, and Cheng (2014), for example, used the Weighted Product Model instead of the classical AHP because it does not suffer from any kind of RR. In addition, problems that are usually solved with MCDM methods involve complex situations and risks, such as those related to industrial engineering, and can occur in dynamic environments, such as those presented in Campanella and Ribeiro (2011), where new alternatives can be inserted, removed or changed over time. In these situations the RRP is highly undesirable, since a non-optimal alternative can be selected and the feasibility of using the method can be questioned. Mufazzal and Muzakkir (2018) also claim that RRP is a serious limitation of the MCDM field, which could lead DMs to misunderstand the difference between the alternatives selected and the rationality used in the decision-making process. In the MCDM literature, the RRP was initially associated with the change in ranking of the alternatives as a consequence, for example, of adding or deleting an alternative. This means that a DM’s preference ordering between two alternatives changes when an alternative is added or removed and this clearly contradicts the Principle of Independence from irrelevant alternatives. Finan and Hurley (2002) and Lin, Chou, Chouhuang, and Hsu (2008), for example, associated RRP with a non-discriminating criterion. In this case, as the DM is

Corresponding author. E-mail address: [email protected] (L. Ferreira).

https://doi.org/10.1016/j.cie.2019.04.023 Received 28 August 2018; Received in revised form 12 February 2019; Accepted 15 April 2019 Available online 18 April 2019 0360-8352/ © 2019 Elsevier Ltd. All rights reserved.

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R.,.d.F. Aires and L. Ferreira

indifferent among the alternatives when they are compared against that criterion, it is presumably safe to eliminate them from further consideration. Finally, Wang and Triantaphyllou (2008a) emphasized that the ranking of alternatives by an effective MCDM method should follow the transitivity property. The TOPSIS method has been widely used in the literature. Behzadian, Khanmohammadi Otaghsara, Yazdani, and Ignatius (2012), for example, surveyed 269 papers with applications of this method, while Salih, Zaidan, Zaidan, and Ahmed (2019) analyzed 170 papers with applications of the fuzzy-TOPSIS. Other hybrid models and extensions have also been proposed, such as Fuzzy-TOPSIS-ELECTRE (Ferreira, Borenstein, & Santi, 2016), AHP-TOPSIS (Goyal & Kaushal, 2019; Kaur, Singh, & Kumar, 2018), AHP-TOPSIS-QFD (Pramanik, Haldar, Mondal, Naskar, & Ray, 2017), TOPSIS-VIKOR (Baccour, 2018; Ploskas & Papathanasiou, in press; Wu, Xu, Jiang, & Zhong, 2019) and DEA-TOPSIS (Subbaiah, Shekhar, & Kandukuri, 2014; Wang et al., 2019). However, the classical version of the method, as proposed by Hwang and Yoon (1981), was chosen as the focus of this paper based on the literature review conducted by de Farias Aires and Ferreira (2018) and also because, to our knowledge, only four papers were published on RRP in the TOPSIS method: (i) García-Cascales and Lamata (2012) proposed modifications to the normalization procedure and the introduction of fictitious alternatives; (ii) Senouci, Mushtaq, Hoceini, and Mellouk (2016) analyzed the effect of four normalization procedures on RRP; (iii) Mufazzal and Muzakkir (2018) proposed the incorporation of two new measures called the Weighted Proximity Index and the Overall Proximity Value to minimize the RRP; and (iv) Cables, Lamata, and Verdegay (2016) propose a new concept for an ideal solution called the Reference Ideal Method. Although these papers have presented interesting ideas and solutions, they have presented some limitations, such as: (i) they consider only the addition and removal of alternatives in order to evaluate cases of rank reversal. Therefore, they do not include an analysis on the important property of transitivity and other possible RR situations, as detailed in 2.2; (ii) they typically use case studies. This approach hinders the generalization of results; (iii) they have limited applications. For example, García-Cascales and Lamata (2012) consider that all criteria must have the same range of values; (iv) some of the proposals do not solve the problem, for example, those by Senouci et al. (2016) and Mufazzal and Muzakkir (2018); and (v) they include modifications to the method that may make them difficult for DMs to use, see, for example, Cables et al. (2016). Therefore, this paper seeks to answer the following research question: “How can a robust approach to solve the RRP in the classic TOPSIS method be developed?”. We assume that the changes to the TOPSIS method should be minimal and that the initial characteristics of the method proposed by Hwang and Yoon (1981) should be preserved, since these are factors that have contributed to this method having spread to and being used in widely different areas of knowledge. In addition, we set out to evaluate and demonstrate the robustness of the solution proposed in this paper by examining the five main types of RR found in the literature, see Section 2.2.1, and comparing these with the classic TOPSIS method, and also by using a real case in which the environment is dynamic and propitious in order to demonstrate the importance of RRP. In summary, the contributions of this paper are as follows: (i) we develop a new extension to solve the RRP in the classic TOPSIS method, since our review of the literature showed that the recent solutions have not completely resolved the RRP like the one presented in this paper; (ii) we define a framework to evaluate cases of RR in the TOPSIS method which involve the RR criteria available in the literature, simulated decision problems and a real case; (iii) we validate the proposed solution using the proposed framework, which can also be used to evaluate other MCDM methods. The remainder of this paper is organized as follows: Section 2 presents the basic concepts of the TOPSIS method and a review of the

literature on the RRP; Section 3 presents a framework for evaluating cases of RR in MCDM and the results obtained by applying it in the TOPSIS method; Section 4 presents the proposed method R-TOPSIS, as well as the validation steps conducted; and finally Section 5 draws some conclusions and makes suggestions for future research. 2. Background 2.1. TOPSIS method The TOPSIS method is one of the most widely used multi-criteria decision analysis methods, see for example Behzadian et al. (2012) and Ferreira, Borenstein, Righi, and de Almeida Filho (2018). It was proposed by Hwang and Yoon (1981) and extended by Yoon (1987). In this method, the best alternative is the one nearest to the positive ideal solution (PIS) and farthest from the negative ideal solution (NIS). PIS is a hypothetical alternative that maximizes the benefit criteria (B) and simultaneously minimizes the cost criteria (C). On the contrary, NIS maximizes the cost criteria and simultaneously minimizes the benefit criteria. The alternative which has the least Euclidean distance from PIS while being farthest from NIS is the best one of all (Mufazzal & Muzakkir, 2018). In the last step, a closeness coefficient (CCi ) is calculated for each alternative, and the alternatives are ranked in descending order by using the CCi obtained. Formally, let A = [ai |i = 1, …, m] a set of alternatives, C = [cj |j = 1, …, n] a set of criteria and W = [wj |j = 1, …, n], wj > 0 and n w =1 the importance level of the criteria, j=1 j X = [xij |i = 1, …, m;j = 1, …, n] the decision matrix, where x ij is the performance rating of the alternative ai with respect to the criterion cj . Thus, Algorithm 1 describes the steps of the TOPSIS method as proposed by Hwang and Yoon (1981). Algorithm 1. TOPSIS method Step 1: Calculate the normalized decision matrix (nij ) as:

nij =

xij m 2 i = 1 x ij

, i = 1, …, m;j = 1, …, n

Step 2: Calculate the weighted normalized fuzzy decision matrix (rij ) as:

rij = wj × nij , i = 1, …, m;j = 1, …, n Step 3: Obtain the positive (PIS) and negative (NIS) ideal solutions as: + + PIS = {r1+, …, r + j , …, rn }, where vj =

NIS = {r1 , …, r j , …, rn }, where v j =

max(rij | i = 1, …, m), if j

B

min(rij | i = 1, …, m), if j

C

min(rij | i = 1, …, m), if j

B

max(rij | i = 1, …, m), if j

C

Step 4: Calculate the distances of each alternative i in relation to the ideal solutions as:

Si+ =

Si =

n j=1

(rij

2 r+ j ) , i = 1, …, m

(rij

r j ) 2 , i = 1, …, m

n j =1

Step 5: Calculate the closeness coefficient of the alternatives (CCi ) as:

CCi =

Si Si+ + Si

Step 6: Sort the alternatives in descending order. The highest CCi value indicates the best performance in relation to the evaluation criteria.

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2.2. Literature review

According to de Farias Aires and Ferreira (2018), Type #1 is the most widely used in the literature and many methods have not been tested for all the types, including the TOPSIS method with respect to Type #3, Type #4, and Type #5. These authors also classify the studies into five clusters according to their research goal: Survey, Application, Problem solution, Simulation and Problem identification. The majority of the them were related to identifying the problem (61.54%), where the authors studied the nature of the problem, and outlined its causes and conditions. 14.62% of the papers involved how to find a solution for the problem, for which the authors designed a new procedure to solve the RRP. Simulations in which the authors investigated the performance of different MCDM methods or procedures represented 13.85% of the papers. Applications that involved some multicriteria decision-making method and used ranking reversal assumptions to improve the reliability of the results produced were addressed by 9.23% of the studies. Finally, 0.8% of the papers were systematic literature reviews. From this initial literature review, we decided to focus the development of this paper on the cluster “Problem solution” and the TOPSIS method, given the small number of papers found in the literature that address the RRP in this method as well as because its use is extensive in the MCDM literature.

The first discussions on rank reversal in MCDM methods are described in the seminal papers authored by Belton and Gear (1983) and Saaty and Vargas (1984a, 1984b) with respect to the AHP method. Belton and Gear (1983) proposed a new form of normalization called BG (Belton & Gear) in which the priorities of the alternatives should be measured by the maximum value instead of their sum. This paper modified the classical AHP, thus preventing the occurrence of reversals and demonstrated RRP for the first time in the literature. Saaty and Vargas (1984a) responded to Belton & Gear by pointing out that the method that the latter proposed also suffered from rank reversal, and discussed the main inconsistencies in Saaty and Vargas (1984b). Since this initial discussion, the number of papers on RRP has grown considerably, and this expansion is related both to other aspects that generate rank reversal in AHP, e.g., issues regarding the property of transitivity (Wang & Triantaphyllou, 2008a), and a non-discriminating criterion (Finan & Hurley, 2002; Jan, Tung, & Deng, 2011), just as in other MCDM methods, such as TOPSIS (García-Cascales & Lamata, 2012), ELECTRE (Figueira, Greco, Roy, & Słowiński, 2013), and PROMETHEE (De Keyser & Peeters, 1996). Recently, de Farias Aires and Ferreira (2018) extended the literature review conducted by Maleki and Zahir (2013) by analyzing 130 papers on different MCDM methods from 37 journals. They were examined from five aspects: the multicriteria technique; the year and journal in which the papers were published; the co-authorship network; the rank reversal situations; and the research goal. The approach which tackled RRP most was AHP, since this method was associated with 76.1% of the papers analyzed, 16.2% of which used two or more MCDM methods in the analysis and experiments, 3.1% dealt with the RRP in PROMETHEE, 2.3% specifically discussed the RRP in the ELECTRE family, and 2.3% dealt with the RRP in TOPSIS. On the other hand, the addition and/or removal of irrelevant alternatives has been the most used criterion to evaluate RR irregularities in the literature, see Maleki and Saadat (2013) and de Farias Aires and Ferreira (2018). Moreover, four other criteria have been also used, as follows: alteration of the indication of the best alternative; decomposition of the decision problem; the non-discriminating criterion; and the transitivity property. Thus, it is possible to characterize different typologies of rank reversals from the literature, as described in the following:

2.2.1. Rank reversal in the TOPSIS method This section presents an analysis of the studies related to RRP in the TOPSIS method. It focuses on the crisp version of this method, as proposed by Hwang and Yoon (1981). However, other studies on RRP regarding the fuzzy version of this method (Chen, 2000; Chen, Lin, & Huang, 2006) can be found in the literature, see for example Salem and Awasthi (2018) and Roszkowska and Kacprzak (2016), and so too can hybrid approaches, see for example Iç (2014). Initially, García-Cascales and Lamata (2012) pointed out that the RRP in the TOPSIS method is due to the normalization procedure and also the modifications in the positive and negative ideal solutions (PIS and NIS). For this, the authors propose a procedure for absolute normalization, not a relative one, as is done originally, and the use of fictitious alternatives with extreme values to represent PIS and NIS. Based on this solution, the authors show that their method avoids RR. Next, Mufazzal and Muzakkir (2018) proposed a new method to minimize the RRP in the TOPSIS method. They modified Algorithm 1 from step 3. After calculating the weighted and normalized decision matrix, the new algorithm computes the weighted proximity index (WPI) in order to calculate the proximity of each alternative from the best available in the range of a given decision problem. WPI is a measure of the deviation of each alternative from the best values, determined by picking the maximum value for the positive attribute and the minimum for the negative attribute, corresponding to each criterion. Next, the overall proximity value (OPV) is computed for each alternative by summing the WPI corresponding to each criterion. The best alternative is the one with the least value of OPV. Unfortunately, according to the results presented, these modifications were not sufficient to solve the RRP, although they did reduce the effects of the RRP by adding alternatives to or removing alternatives from the problem. Senouci et al. (2016) analyzed the effect of four normalization procedures to avoid or reduce cases of RR in the TOPSIS method. The results presented by the authors are important, but they too do not completely eliminate the problem, since it is necessary to better define how to connect the normalization procedures proposed with the definition of PIS and NIS, as explained in our paper. The authors are clear when they explain the problem of the proposed solution: if new alternatives are added, the RR problems will depend on the values of the new alternative. If one or more attributes are greater or less than the original maximum or minimum values, RR may occur. Cables et al. (2016) propose a new ideal solution concept called RIM – Reference Ideal Mode, where values can vary between the minimum and maximum value of each criterion. Unlike the classic TOPSIS, ideal

1. Type #1: the final rank order of the alternatives changes if an irrelevant alternative is added to (or removed from) the problem. See, for example, Buede and Maxwell (1995), Zanakis, Solomon, Wishart, and Dublish (1998), Wang and Luo (2009), García-Cascales and Lamata (2012), Verly and De Smet (2013) and Cinelli, Coles, and Kirwan (2014). 2. Type #2: the indication of the best alternative changes if a nonoptimal alternative is replaced by another worse one. See, for example, Triantaphyllou and Shu (2001) and Wang and Triantaphyllou (2008a). 3. Type #3: the transitivity property is violated if an irrelevant alternative is added to (or removed from) the problem. See, for example, Triantaphyllou and Shu (2001) and Wang and Triantaphyllou (2008a). 4. Type #4: the transitivity property is violated if the initial decisionproblem is decomposed into sub-problems, i. e., for the same decision problem and when the same MCDM method is used, the rankings of the smaller problems are in conflict with the overall ranking of the alternatives. See, for example, Triantaphyllou and Shu (2001) and Wang and Triantaphyllou (2008a). 5. Type #5: the final rank order of the alternatives changes if a nondiscriminating criterion is removed from the problem. See, for example, Finan and Hurley (2002), Lin et al. (2008), Jung, Wou, Li, and Julian (2009), Jan et al. (2011) and Verly and De Smet (2013). 86

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solutions do not necessarily have to be at the extreme values and the input data for the decision problem can be different types. The authors explain that the proposed method does not suffer from RR when irrelevant alternatives are added or removed. Mousavi-Nasab and Sotoudeh-Anvari (2018) conducted a comparative analysis between the TOPSIS, COPRAS, SAW and VIKOR methods in relation to the RRP for which they used decision problems obtained from the literature on selecting materials. Among their results what stands out is that the TOPSIS method presented the worst performance of all the methods analyzed. In summary, among the papers analyzed for the classic TOPSIS method, only García-Cascales and Lamata (2012) and Cables et al. (2016) demonstrate solutions to solve the problem. However, there are still gaps that can be better explored. First, the main limitation of García-Cascales and Lamata (2012) is the use of the maximum value of the entire decision matrix as a denominator in the normalization procedure. This procedure is valid in the context presented by the authors, where all the criteria have the same range of values. However, for situations where the criteria have different ranges of values, the normalization strategy that the authors used may affect the preference relationships and the trade-off established by the DMs involved in solving the problem. Cables et al. (2016) proposes a new way of defining the ideal solutions and a new form of normalization, which is different from the classic TOPSIS method. Our goal is to maintain the basic principles and characteristics of the classical method. Secondly, all authors evaluate cases of RRs classified as Type #1, see Section 2.2, and thus fail to consider the other typologies of RR presented in the RR literature, which characterize important requirements to guarantee the robustness of a MCDM method. Therefore, new propositions and solutions for the RRP in the classic TOPSIS method need to be better explored and validated by using different decision problems and comparative statistical analysis to draw attention to the adherence of the new method to the concepts of the classic TOPSIS method, as performed by Ferreira et al. (2016), Chamodrakas, Leftheriotis, and Martakos (2011) and Zanakis et al. (1998). In addition, consideration should be given to the different cases of evaluating RR, including to the typologies of RR presented in Section 2.2.

2. Number of alternatives: 5, 7, 9, 11; 3. Performance of the alternatives: randomly generated by using a uniform distribution in the interval [0–200]; 4. Weights of the criteria: three types of weights were defined for the experiments. In the first case, weights were equal for all criteria; in the second case, randomly generated weights were used in the interval [0–1] for all criteria through a uniform distribution; and finally, in the third case, a U-shaped beta distribution was used to generate the criteria weights; 5. Number of replications: 100 cases for each combination, thereby generating 4800 different decision problems; 6. Type of normalization: four types of normalization were used in the experiments: Vectorial, Max-Min, Max and Sum, as Appendix A. Based on these aspects, the TOPSIS method was evaluated for each of the 4,800 decision problems generated randomly using seven RR cases found in the literature, as highlighted in Section 2.2: 1. Type #11: evaluation of changing the indication of the best alternative by adding an irrelevant alternative. For this, we used two decision matrices as input; the first was the original decision matrix, generated randomly, and the second was constructed by adding an irrelevant alternative to the initial decision matrix. Then, the TOPSIS method was applied to each of the decision matrices. If the result of choosing the best alternative was different, it was considered that the method presented RR for this instance of analysis; 2. Type #12: the procedure is similar to the above. However, we evaluate the alteration of the indication of the best alternative by excluding an irrelevant alternative; 3. Type #2: evaluation of the change in the indication of the best alternative by replacing an irrelevant alternative with one that performs worse. In this case, two decision matrices were considered as input, where the first was the original decision matrix, generated randomly, while the second one was constructed by replacing a nonoptimal alternative of the initial decision matrix with another one that performed between 10% and 40% lower in each criterion. Next, the TOPSIS method was applied to each of the decision matrices. If the result of choosing the best alternative was different, it was considered that the method presented RR for this instance of analysis; 4. Type #31: evaluation of the transitivity rule by adding an irrelevant alternative. Just as in the previous cases, two decision matrices were considered as input: the first one was also the randomly generated decision matrix, while the second was a decision matrix generated from the first to which a non-optimal alternative was added. Then, the TOPSIS method was applied to each of the decision matrices. If the rankings produced in the two cases were not compatible with the property of transitivity, the method was considered to present RR for this instance of analysis; 5. Type #32: the procedure is similar to the above. However, we assess the propriety of transitivity by excluding an irrelevant alternative; 6. Type #4: Three decision matrices were considered as input. Just as in previous situations, the first was also the decision matrix randomly generated, with m alternatives, while the second was a matrix with the m/2 alternatives of the first matrix and the third was a matrix with the remaining m m/2 alternatives. Afterwards, the TOPSIS method was applied to the three decision matrices. If the rankings produced are not compatible with the property of transitivity, it was considered that the method presented RR for this instance of analysis; 7. Type #5: evaluation of changing the indication of the best alternative by excluding a non-discriminating criterion. For this, we use two decision matrices as input and two weight vectors. The first decision matrix was generated randomly with the mandatory inclusion of a non-discriminating criterion. The second decision matrix was constructed from the first and excluded the non-discriminating

3. Framework for evaluating rank reversal in the TOPSIS method This section presents an analysis of the TOPSIS method regarding the rank reversal problem. Unlike the previous studies, we have defined a framework for evaluating RR cases, which comprises seven (7) possible cases of RR collected from the MCDM literature, decision problems simulated by using different combinations of normalization procedures, different numbers of criteria, alternatives and weights, and similarity and dispersion statistics. This set of experiments using the TOPSIS method and later the proposed method were based on the following papers: Wang and Triantaphyllou (2008a), Zanakis et al. (1998), Chamodrakas et al. (2011) and Ferreira et al. (2016). According to these authors, simulation is a very flexible and adaptable method that allows different samples and controlled and replicable experiments to be used which provide a large set of results and from which the pattern of solutions obtained by different methods can be studied (Chamodrakas et al., 2011). By using simulations, MCDM methods can be evaluated, which consider different sample sizes and parameters, in terms of the number of criteria and alternatives, and different ways of rating alternatives and distributing weight criteria (Ferreira et al., 2016). Therefore, we implemented in Java Programming Language® the methods TOPSIS and R-TOPSIS, as well as procedures to evaluate the types of RR and the statistics used for the analysis. The main input parameters used during the simulations were the following: 1. Number of criteria: 5, 10, 15, 20; 87

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criterion. Then, the weights of the criteria for the first decision problem were generated, as explained previously, and weights for the second decision problem were adjusted using the procedure proposed by Saaty and Vargas (2006). Thereafter, the TOPSIS method was applied to each of the decision problems. If the result of choosing the best alternative was different, it was considered that the method presented RR for this instance of analysis.

In the second round of experiments, we replaced the Vet normalization procedure with the Max Min . The first difference is that we did not obtain RR for all the simulated cases from Type #11 to Type #4, since cases of RR were not found for Type #11 and Type #31. However, despite presenting a smaller number of RR cases for Type #12 and Type #32, the use of Max Min normalization produced the worst RR analysis for Type #2 cases. In the third round of experiments, we used the Max normalization procedure. This type of normalization did not produce RR for Type #11, Type #31 and Type #5, as the Max Min normalization did. In addition, it presented the lowest RR average of all the cases analyzed and the lowest RR index for Type #4 (27.83%), which was considered the most problematic one during the experiments. In the fourth round of experiments, we used the Sum normalization procedure. In this case, as in the Vet normalization, there are cases of RR for all cases analyzed from Type #11 to Type #4. It can also be verified that this method presents the highest RR average for the simulated cases. It should be noted that this procedure is used by the classical AHP method and was considered by many authors as the main cause of RR in this method, see for example Ramanathan (2013) and Zahir (2009). In the last round, we repeated the previous four experiments using the “TOPSIS #2” version, where PIS and NIS are fixed. In general terms, the results obtained are similar and the “TOPSIS #2” also did not present any RR cases for Type #5. Finan and Hurley (2002) also demonstrated that if there is only one level of criteria, the rank order of alternatives is unaffected if the non-discriminating criterion is left out. Lastly, we discuss the causes of RR presented at the beginning of this section. First, previous studies on RR in the TOPSIS method have highlighted that the vectorial normalization method affects the independence between alternatives since the addition or exclusion of alternatives distorts the original values of the problem and violates the Principle of Independence from irrelevant alternatives. This is not difficult to verify. Note that the denominator of Vet involves the sum of the score of the “m” alternatives of the problem for each criterion, as well as the Sum denominator. On the other hand, the Max Min denominator involves the score of two alternatives for each criterion, the best and the worst performance, while the Sum denominator involves the score of only one alternative for each criterion. Thus, it can be said that the greater the number of alternatives involved in the normalization process, the greater is the dependency created between the scores of the alternatives for each criterion in the resulting normalized and weighted decision matrix, and the greater the chance of RR cases being associated with this, as shown in Table 1. Second, for “TOPSIS #1”, as the PIS and NIS are obtained from the normalized and weighted decision matrix, any modification in the score of the alternatives and any addition or exclusion of alternatives, can result in a new PIS and/or NIS, which can generate cases of RR. On the other hand, for “TOPSIS #2”, the results obtained with the previous experiments showed that it is not enough to fix the ideal solutions to solve the RR problem, since the values of the new decision matrix that

In addition, dispersion and similarity statistics were used to compare the classic TOPSIS method with the proposed method. The following statistics were used in our experiments (Chamodrakas et al., 2011; Ferreira et al., 2016; Zanakis et al., 1998): 1. Similarity: Spearman’s rank correlation (SRC) and Mean absolute error of ranks (MAER); 2. Dispersion: difference between the CCi of the best and worst alternative (DBW), difference between the CCi of the first and second alternative (DFS) and the standard deviation of the CCi of each ranking (DSP). 3.1. Results Initially, this section aims to demonstrate the causes of RR in the TOPSIS method by using the framework defined in the previous section. In the MCDM literature there is consensus that the RR in the TOPSIS method is caused by the normalization procedure and also by the procedure used to obtain the PIS and NIS, see for example GarcíaCascales and Lamata (2012), Senouci et al. (2016) and Cables et al. (2016). Thus, in this section we evaluate the impact of the normalization procedures on the independence of the alternatives and RR on the TOPSIS method using different simulated cases by varying the number of criteria, alternatives, the distribution of the weights and the range of the alternatives scores, as previously explained. In addition, we evaluated two versions of the TOPSIS method. The first one, called “TOPSIS #1”, corresponds to the classic idea of the method, where PIS and NIS are obtained in a relative way from the scores of the alternatives, as Algorithm 1. The second version, called “TOPSIS #2”, was inspired by the idea proposed by Chen (2000) for the FuzzyTOPSIS method, where PIS and NIS have fixed values. Each of these versions was combined with the four types of normalization presented in Appendix A. Table 1 presents the results obtained for each case. In the first four round of experiments, we used the “TOPSIS #1” and the normalization procedures described in Appendix A. First of all, we used the Vet normalization procedure. In this case, we obtained RR for all cases from Type #11 to Type #4. Type #4 presented the worst performance (39.0%) followed by the other cases of RR that also involve the property of transitivity, which account for 20% (Type #31) and 18% (Type #32) of cases. Finally, the RR cases related to changing the indication of the best alternative presented the lowest RR indices, these accounting for 4% (Type #11), 7.5% (Type #12) and 2.0% (Type #2) of cases. Table 1 TOPSIS: Rank reversal analysis. TOPSIS #1

TOPSIS #2 (PIS = 1 and NIS = 0)

Type of RR

Vet

Max-Min

Max

Sum

Vet

Max-Min

Max

Vet

Type Type Type Type Type Type Type

4.00% 7.50% 2.00% 20.00% 18.33% 39.00% 0.00%

0.00% 4.25% 2.75% 0.00% 10.58% 40.83% 0.00%

0.00% 2.08% 0.75% 0.00% 7.25% 28.50% 0.00%

6.83% 8.92% 1.92% 28.25% 18.83% 43.08% 0.00%

3.75% 5.67% 2.58% 19.83% 16.08% 38.25% 0.00%

0.00% 4.33% 3.67% 0.00% 10.33% 42.08% 0.00%

0.00% 2.92% 3.00% 0.00% 8.17% 32.83% 0.00%

6.00% 8.42% 1.75% 26.33% 21.00% 46.25% 0.00%

15.14%

9.74%

6.43%

17.97%

14.36%

10.07%

7.82%

18.29%

#11 #12 #2 #31 #32 #4 #5

Average

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has been produced after applying the RR evaluation cases (Type #1 …Type #5) can be different from the initial decision matrix, and can alter the value of the sum of distances of each alternative in relation to the ideal solutions, which in turn can generate changes in the CC, and may have an impact on the indication of the best alternative and/or on the transitivity relationships. Therefore, it can be concluded that a robust solution for the RRP in the TOPSIS method should consider the following aspects simultaneously: (i) choosing a normalization procedure that minimizes the effects of dependence between the alternatives; (ii) defining parameters for normalization procedures that are fixed for the decision problem and its derived problems; and, (iii) using the same NIS and PIS even if the set of alternatives is modified, be it to do with the addition, exclusion or modifications of its scores.

Secondly, the PIS and NIS ideas should be fixed. Unlike the solution proposed by Cables et al. (2016), we will consider that the ideal solutions should be in the extreme values in order to maintain compatibility with the original ideas of the method. In addition, the values of the ideal solutions should be used as a parameter in the normalization procedures, since it must be ensured that there will be no change in the values of the normalized and weighted decision matrix after modifications have been introduced in the initial decision problem, such as the addition and removal of alternatives or a change in the score of the alternatives. Based on these considerations, this paper proposes the use of an additional input parameter to the TOPSIS method called “domain”, i.e., a numeric value (integer or real) representing the range of possible values that each criterion can take. The domain concept that we use in our proposal is based on the ideas of García-Cascales and Lamata (2012) and Cables et al. (2016). However, we consider that it is only a question of being a numerical value and that each criterion can be a proper range of values defined by the DM, by experts or by the interviewees. Arising from these considerations, Algorithm 2 presents the main steps proposed for the R-TOPSIS. In relation to García-Cascales and Lamata (2012), our model offers the analyst the flexibility to define a domain for each criterion (Dj ) and the possibility of choosing between the Max-Min or Max normalization procedure, in accordance with step 5 of Algorithm 2. Moreover, our model does not introduce fictitious alternatives into the problem. In relation to Cables et al. (2016), our model is simpler to apply, because it is similar to the classic model and does not require the learning of additional operations. In addition, we consider that the decision matrix of the problem consists only of numerical values.

4. Proposed method This section presents a new procedure to solve the RRP in the TOPSIS method arising from the experiments presented in the previous section. First of all, a normalization procedure should be chosen to minimize the impact on the dependence on alternatives. Thus, in view of the results presented in the previous section, both the Max-Min normalization and the Max normalization can be chosen, provided that their parameters are adjusted as explained in the course of this section. Algorithm 2. R-TOPSIS method Step 1: Define a set of alternatives ( A = [ai]m ); Step 2: Define a set of criteria (C = [cj]n ), as well as a subdomain of real numbers D = [dj ]2 × n , where dj , to evaluate the rating of the alternatives, where d1j is the minimum value of Dj and d2j is the maximum value of Dj . Step 3: Estimate the performance rating of the alternatives as X = [xij ]m × n ; n Step 4: Elicit the criteria weights as W = [wj]n , where wj > 0 and j = 1 wj = 1; Step 5: Calculate the normalized decision matrix (nij ) by using Max or Max-Min as: Step 5.1: Max

nij =

xij d2j

4.1. Validation of the R-TOPSIS method This section presents the results obtained with the following procedures used to validate the proposed method: (i) evaluation of the RTOPSIS by using different simulated decision problems and cases of RR; (ii) comparing the proposed model with the classic version of the TOPSIS method by using similarity and dispersion statistics; and (iii) evaluating the two methods by using the cases of RR described in Section 2.2 and the case study presented in de Farias Aires, Ferreira, de Araujo, and Borenstein (2018). In this latter case, we did not use Type #5 because there was no record of RR occurring during the experiments. Table 2 presents the results obtained with the proposed model for the different cases of RR evaluation presented in Section 3. It can be seen that the R-TOPSIS did not present any case of RR for all the simulated cases, thereby proving that the model is robust. Table 3 presents the comparative results between the TOPSIS and RTOPSIS methods considering the dispersion (DSP, BW and DFS) and similarity (SRC and MAER) statistics, for which a mean of three rounds of simulation for 4,800 simulated cases was used. In relation to the dispersion statistics for the CC resulting from each method, it can be seen that the R-TOPSIS presents lower values for the DSP, DBW and DFS when compared to the TOPSIS method for the same normalization

, i = 1, 2, …, m; j = 1, 2, …, n

Step 5.2: Max-Min

nij =

xij

d1j

d2j

d1j

, i = 1, 2, …, m ; j = 1, 2, …, n

Step 6: Calculate the weighted normalized decision matrix (rij ) as:

rij = wj × nij , i = 1, 2, …, m ; j = 1, 2, …, n. Step 7: Set the negative (NIS) and positive (PIS) ideal solutions as:

NIS = {r1 , …, rn }, where r j =

d1j d2j

wj if j

PIS = {r1+, …, rn+}, where r + j = wj if j

B and r j = wj if j

B and r + j =

d1j d2j

wj if j

C

C

Step 8: Calculate the distances of each alternative i in relation to the ideal solutions as:

Si+ =

n j=1

(rij

2 r+ j ) , i = 1, 2, …, m; Si =

n j=1

(rij

r j )2 , i = 1, 2, …, m.

Table 2 R-TOPSIS: Rank reversal analysis.

Step 9: Calculate the closeness coefficient of the alternatives (CCi ) as:

S CCi = + i Si + Si Step 10: Sort the alternatives in descending order. The highest CCi value indicates the best performance in relation to the evaluation criteria.

89

Type of RR

Max

Max-Min

Type Type Type Type Type Type Type

0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

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performed throughout this section, it is possible to deduce the validity of the proposed model for the different cases of RR evaluation proposed in Section 3, as well as its adherence to the philosophy of the classic TOPSIS method.

Table 3 R-TOPSIS: Statistics of dispersion and similarity. Method

Normalization

DSP

DBW

DFS

SRC

MAER

R-TOPSIS TOPSIS Diff

Max Max

0.0903 0.1071 −18.60%

0.2615 0.3104 −18.70%

0.0536 0.0677 −26.31%

0.9932

0.2669

R-TOPSIS TOPSIS Diff

Max-Min Max-Min

0.0821 0.0986 −20.10%

0.2357 0.2832 −20.15%

0.0483 0.0581 −20,29%

0.9722

5. Conclusions and final remarks 0.8193

This paper has presented a detailed analysis of the TOPSIS method in relation to the RRP. From a series of computational experiments which use simulated cases, we explain that the causes of RR in this method are brought about by the normalization procedures and by choosing PIS and NIS. Then, we presented a new extension for the method by defining a domain for each criterion and by using the Max and Max Min normalization procedures. We also validated this method by using procedures for internal and external validation. After analyzing the results of the computational experiments, it can be said that the R-TOPSIS method proved to be robust and an excellent alternative to the classic TOPSIS method, since it did not generate any case of reversion of ranking for either the simulated cases or for the case study used for its validation. Another advantage is that R-TOPSIS does not require additional learning related to the classic method, and it also maintains the correspondence of rankings with regard to TOPSIS. As a disadvantage, an additional input parameter, the domain of each criterion, must be defined by the DMs. Thus, the main contributions of our paper are the following: (i) the rigorous analysis of the TOPSIS method in relation to the RRP, considering the different cases of RR presented in the literature; (ii) the demonstration of the causes of the RRP in the method; (iii) the demonstration of the effect of normalization procedures in the RRP; (iv) proposing a new extension for the classic TOPSIS method, which is immune to the RRP; (v) defining a framework to evaluate cases of RR and to validate the proposed solution which consists of simulated cases, dispersion and similarity statistics and the use of a real case to demonstrate the importance of the RRP for cases of dynamic multi-criteria decision-making. Finally, as future research possibilities, the framework defined in this paper can be used to analyze the RRP in other MCDM methods, such as ELECTRE and PROMETHEE. In addition, the computational experiments performed can be enhanced by using other combinations of cases of RR presented in Section 2.2, for example, to evaluate the effect of the nondiscriminating criterion on transitivity; new ways of modeling DMs’ preferences can also be used, see for example de Almeida Filho, Clemente, Morais, and de Almeida (2018); based on García-Cascales and Lamata (2012), a mathematical proof for R-TOPSIS can be developed. In relation to the proposed model, it is expected that it can be used in different studies and areas, in order to disseminate and further validate its use.

procedures. On the other hand, in relation to the statistics used to evaluate the similarity between the rankings produced by the two methods, it can be verified that there is a high correspondence among the rankings, thus demonstrating that the R-TOPSIS maintains correspondence with the principles of the classical method. In the last step of validating the method, we used the case study presented in de Farias Aires et al. (2018) as reference to compare the RTOPSIS with the classic version of the TOPSIS method in relation to the types of RR presented in Section 3, in which a model for selection of engineering students was developed. This problem is especially relevant for the analysis of the RRP, since it can be characterized as a decision-making problem in a dynamic context (Campanella & Ribeiro, 2011), where new students can join, give up or change the selection process depending on the competition or new opportunities that can arise at any point in their academic life. In this context, RR problems are extremely undesirable. The sample used in de Farias Aires et al. (2018) consisted of 91 students. However, we used only the data from the top 20 participants of the selection process to facilitate the analysis, interpretation and presentation of the results, and the different rounds of simulation with the models. The criteria used in the model were: C1 – Academic performance index (IRA); C2 – Efficiency rating in workload hours (IECH); e C3 – Efficiency rating in academic periods (IEPL). The weights of the criteria were as follows: IRA = 0.3633; IECH = 0.3298; e IEPL = 0.3069. From this input data, different operations were performed to verify the occurrence of RR in the TOPSIS method, see Appendix B. Then, these operations were performed with the R-TOPSIS method, Appendix C. In relation to the TOPSIS method, once again, we demonstrated the causes and different situations of RR. For example, see Tables B.5 and B.6, the addition of an irrelevant alternative, represented by the alternative AA was fundamental for increasing the general sum of each column and the change in the normalized matrix. This factor also directly affected the PIS values, since the values changed from {0.0892, 0.0746, 0.0686} to {0.2439, 0.0724, 0.0670}; and the NIS, because the values were changed from {0.0693, 0.0707, 0.0686} to {0.1894, 0.0686, 0.0670}. On the other hand, taking as an example the same situation for the proposed model, see Tables C.11 and C.12, the use of the minimum and maximum values of the domain for each of the criteria did not result in altering the normalized values of the alternatives as well as the PIS values (0.3633, 0.3298 and 0.3069) and NIS (0.00, 0.00 and 0.00). Finally, from the results obtained with the validation procedures

Acknowledgements This research has been supported by CAPES and National Council for Scientific and Technological Development (Grant 301453/2013-6) of Ministry of Science and Technology, Brazil.

Appendix A. Normalization procedures The four normalization procedures used in our experiments are briefly described below, as Chakraborty and Yeh (2007). 1. The Vectorial normalization method divides the performance ratings of the decision matrix by its norm.

x ij

nij =

n j=1

x ij2

, for i = 1, …, m ,

2. The Max-Min method considers both the maximum and minimum performance ratings of the criteria when the calculation is being made.

nij =

x ij xijmax

x ijmin x ijmin

, for i = 1, …, m , j

Benefit

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nij =

x ijmax xijmax

x ij x ijmin

, for i = 1, …, m , j

Cost

where x ijmax and x ijmin are the maximum and minimum performance rating among alternatives for the criterion j, respectively. 3. The Max method divides the performance ratings by the maximum performance rating of each criterion j.

nij =

x ij xijmax

nij = 1

, for i = 1, …, m , j

x ij x ijmax

Benefit

, for i = 1, …, m , j

Cost

4. The Sum method divides the performance ratings by the sum of the performance ratings of each criterion j.

nij =

x ij n j=1

x ij

, for i = 1, …, m

Appendix B. Case study: rank reversal in the TOPSIS method This appendix demonstrates the occurrences of RR in the TOPSIS method by using the case study presented in de Farias Aires et al. (2018). To simplify viewing the results, a sample of twenty students was used, as presented in Table B.4. The results obtained by applying the TOPSIS method are presented in Table B.5. Table B.4 Decison Matrix. Alternative

IRA

IECH

IEPL

A1 A2 A5 A8 A12 A13 A54 A55 A56 A59 A61 A67 A71 A72 A76 A83 A87 A88 A90 A91 wj

8.5218 7.0455 7.9625 8.4073 8.5000 7.0291 8.3702 6.7018 7.8455 8.2291 7.7618 7.5909 7.6564 7.5273 8.6286 7.7673 8.3273 7.0983 7.4224 8.3304 0.3633

1.0000 1.0000 0.9821 0.9649 1.0000 1.0000 0.9649 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9821 1.0000 1.0000 0.9483 0.9483 0.9821 0.3298

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.3069

The first operation undertaken was to add an irrelevant alternative called AA to demonstrate the occurrence of RR type #11 and #31. Table B.6 presents the results obtained by applying the TOPSIS method, where the violation of the choice of the best alternative can be verified. A76 became the best one; and from the rule of transitivity, since there were alterations in the positions of the alternatives A8 , A87 , A54 , A91 and A88 . The second operation was to exclude alternative A12 to demonstrate the occurrence of RR type #12 and #32. Table B.7 shows the results obtained by applying the TOPSIS method. In this case, there was a violation of the indication of the best alternative, because A76 became the best alternative instead of the alternative A1; and there were changes in the positions of alternatives A8 , A54 , A87 and A88 . The third operation was to alter the scores of alternative A12 by values of 40% lower for all criteria. It was thus possible to demonstrate the occurrence of RR type #2. Table B.8 presents the results obtained by applying the TOPSIS method. In this case, there was a change in the indication of the best alternative, since A76 became the best alternative. In addition, there were changes in the positions of alternatives A8 , A54 , A87 and A8 , thus also violating the property of transitivity. The fourth operation was to split the initial decision problem (Table B.4) into two subproblems and to apply the TOPSIS method to both of them, as shown in Tables B.9 and B.10. To verify the case of RR type #4, it is enough to verify that there is inversion in the rank of alternatives A8 and A12 , when comparing the initial decision problem with Subproblem #1. In Subproblem #2, the initial order remains the same.

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Table B.5 Results: TOPSIS. Alternative

IRA

IECH

IEPL

CCi

Ordering

A1 A76 A12 A87 A8 A91 A54 A59 A5 A56 A83 A61 A71 A67 A72 A90 A2 A13 A88 A55 wj PIS NIS

0.0881 0.0892 0.0879 0.0861 0.0869 0.0861 0.0866 0.0851 0.0823 0.0811 0.0803 0.0803 0.0792 0.0785 0.0778 0.0768 0.0729 0.0727 0.0734 0.0693 0.3633

0.0746 0.0732 0.0746 0.0746 0.0720 0.0732 0.0720 0.0746 0.0732 0.0746 0.0746 0.0746 0.0746 0.0746 0.0746 0.0707 0.0746 0.0746 0.0707 0.0746 0.3298

0.0746 0.0707

0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.0686 0.3069

0.0686 0.0686

0.9457 0.9378 0.9346 0.8470 0.8357 0.8352 0.8222 0.7974 0.6543 0.6057 0.5672 0.5646 0.5132 0.4816 0.4513 0.3634 0.2426 0.2368 0.2011 0.1622 –

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 –

0.0892 0.0693

– –

– –

Table B.6 Results TOPSIS: Addition of the alternative A12 . Alternative

IRA

IECH

IEPL

Rj

Ordering

A76 A1 A12 AA A8 A54 A87 A91 A59 A5 A56 A83 A61 A71 A67 A72 A90 A88 A2 A13 A55 PIS NIS

0.2439 0.2409 0.2402 0.2402 0.2376 0.2366 0.2354 0.2354 0.2326 0.2251 0.2217 0.2195 0.2194 0.2164 0.2145 0.2127 0.2098 0.2006 0.1991 0.1987 0.1894 0.2439 0.1894

0.0711 0.0724 0.0724 0.0724 0.0698 0.0698 0.0724 0.0711 0.0724 0.0711 0.0724 0.0724 0.0724 0.0724 0.0724 0.0724 0.0686 0.0686 0.0724 0.0724 0.0724 0.0724 0.0686

0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670 0.0670

0.9768 0.9447 0.9334 0.9334 0.8772 0.8592 0.8441 0.8439 0.7933 0.6543 0.5952 0.5549 0.5521 0.4978 0.4642 0.4315 0.3726 0.2052 0.1887 0.1808 0.0643 – –

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 – –

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Table B.7 Results TOPSIS: Exclusion of alternative A12 . Alternative

IRA

IECH

IEPL

Rj

Ordering

A76 A1 A8 A54 A87 A91 A59 A5 A56 A83 A61 A71 A67 A72 A90 A88 A2 A13 A55 PIS NIS

0.2575 0.2543 0.2509 0.2498 0.2485 0.2486 0.2456 0.2376 0.2341 0.2318 0.2316 0.2285 0.2265 0.2246 0.2215 0.2118 0.2102 0.2098 0.2000 0.2575 0.2000

0.0748 0.0761 0.0734 0.0734 0.0761 0.0748 0.0761 0.0748 0.0761 0.0761 0.0761 0.0761 0.0761 0.0761 0.0722 0.0722 0.0761 0.0761 0.0761 0.0761 0.0722

0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704 0.0704

0.9769 0.9447 0.8772 0.8592 0.8440 0.8439 0.7933 0.6543 0.5951 0.5548 0.5520 0.4978 0.4641 0.4315 0.3726 0.2052 0.1886 0.1807 0.0641 – –

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 – –

Table B.8 Results TOPSIS: Substitution of values in alternative A12 . Alternative

IRA

IECH

IEPL

Rj

Ordering

A76 A1 A8 A54 A87 A91 A59 A5 A56 A83 A61 A71 A67 A72 A90 A88 A2 A13 A55 A12 PIS NIS

0.2532 0.2500 0.2467 0.2456 0.2443 0.2444 0.2414 0.2336 0.2302 0.2279 0.2277 0.2246 0.2227 0.2209 0.2178 0.2083 0.2067 0.2062 0.1966 0.1496 0.2532 0.1496

0.0736 0.0749 0.0723 0.0723 0.0749 0.0736 0.0749 0.0736 0.0749 0.0749 0.0749 0.0749 0.0749 0.0749 0.0711 0.0711 0.0749 0.0749 0.0749 0.0450 0.0749 0.0450

0.0693 0.0693 0.0693 0.0693 0.0693 0.0693 0.0693 0.0693 0.0693 0.0693 0.0693 0.0693 0.0693 0.0693 0.0693 0.0693 0.0693 0.0693 0.0693 0.0416 0.0693 0.0416

0.9881 0.9719 0.9372 0.9281 0.9210 0.9207 0.8955 0.8260 0.7972 0.7774 0.7761 0.7496 0.7333 0.7176 0.6868 0.6080 0.6017 0.5979 0.5241 0.0000 – –

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 – –

Alternative

IRA

IECH

IEPL

Rj

Ordering

A76 A1 A12 A8 A54 A87 A91 A59 A5 A56 PIS NIS

0.3438 0.3396 0.3387 0.3350 0.3335 0.3318 0.3319 0.3279 0.3173 0.3126 0.3438 0.3126

0.1031 0.1049 0.1049 0.1013 0.1013 0.1049 0.1031 0.1049 0.1031 0.1049 0.1049 0.1013

0.0971 0.0971 0.0971 0.0971 0.0971 0.0971 0.0971 0.0971 0.0971 0.0971 0.0971 0.0971

0.9434 0.8648 0.8372 0.7008 0.6566 0.6195 0.6173 0.4969 0.1583 0.1055 – –

1 2 3 4 5 6 7 8 9 10 – –

Table B.9 Results TOPSIS: SubProblem 1.

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Table B.10 Results TOPSIS: SubProblem 2. Alternative

IRA

IECH

IEPL

Rj

Ordering

A83 A61 A71 A67 A72 A90 A2 A13 A88 A55 PIS NIS

0.3289 0.3287 0.3242 0.3215 0.3188 0.3143 0.2984 0.2977 0.3006 0.2838 0.3289 0.2838

0.1048 0.1048 0.1048 0.1048 0.1048 0.0994 0.1048 0.1048 0.0994 0.1048 0.1048 0.0994

0.0971 0.0971 0.0971 0.0971 0.0971 0.0971 0.0971 0.0971 0.0971 0.0971 0.0971 0.0971

1.0000 0.9949 0.8967 0.8359 0.7768 0.6620 0.3369 0.3225 0.3679 0.1073 – –

1 2 3 4 5 6 7 8 9 10 – –

Appendix C. Case study: rank reversal in the proposed method The objective of this section is to demonstrate the results obtained with the proposed method for the same operations that were carried out with the TOPSIS method in Appendix B. Therefore, initially in Table C.11, we present the results obtained by using the R-TOPSIS method, as well as the domain for each criterion used in the decision problem. The Max normalization procedure was used in the experiments presented in this section. Table C.11 Results R-TOPSIS. Alternative

IRA

IECH

IEPL

Rj

Ordering

A76 A1 A12 A8 A87 A54 A91 A59 A5 A56 A83 A61 A71 A67 A72 A90 A2 A13 A88 A55 wj Dj

0.3135 0.3096 0.3088 0.3054 0.3025 0.3041 0.3026 0.2990 0.2893 0.2850 0.2822 0.2820 0.2782 0.2758 0.2735 0.2697 0.2560 0.2554 0.2579 0.2435 0.3633

0.3239 0.3298 0.3298 0.3182 0.3298 0.3182 0.3239 0.3298 0.3239 0.3298 0.3298 0.3298 0.3298 0.3298 0.3298 0.3127 0.3298 0.3298 0.3127 0.3298 0.3298

0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069

0.9157 0.9105 0.9093 0.9011 0.8993 0.8989 0.8984 0.8937 0.8775 0.8720 0.8676 0.8673 0.8615 0.8579 0.8544 0.8439 0.8284 0.8275 0.8264 0.8104 –

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 –

0.3633 0.0000

0.3298 0.0000

0.3069 0.0000

– –

– –

PIS NIS

0–10

0–1

0–1

0–1

–

Table C.12 demonstrates that there was no change in the indication of the best alternative and the ordering continues to be strictly the same as that of the original problem, after adding the irrelevant alternative AA . Therefore, RR types #11 and #31 were not characterized. Table C.12 Results R-TOPSIS: Addition of the alternative A12 . Alternative

IRA

IECH

IEPL

Rj

Ordering

A76 A1 A12 AA A8 A87 A54 A91 A59 A5

0.3135 0.3096 0.3088 0.3088 0.3054 0.3025 0.3041 0.3026 0.2990 0.2893

0.3239 0.3298 0.3298 0.3298 0.3182 0.3298 0.3182 0.3239 0.3298 0.3239

0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069

0.9157 0.9105 0.9093 0.9093 0.9011 0.8993 0.8989 0.8984 0.8937 0.8775

1 2 3 – 4 5 6 7 8 9

(continued on next page) 94

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Table C.12 (continued) Alternative

IRA

IECH

IEPL

Rj

Ordering

A56 A83 A61 A71 A67 A72 A90 A2 A13 A88 A55 PIS NIS

0.2850 0.2822 0.2820 0.2782 0.2758 0.2735 0.2697 0.2560 0.2554 0.2579 0.2435 0.3633 0.0000

0.3298 0.3298 0.3298 0.3298 0.3298 0.3298 0.3127 0.3298 0.3298 0.3127 0.3298 0.3298 0.0000

0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.0000

0.8720 0.8676 0.8673 0.8615 0.8579 0.8544 0.8439 0.8284 0.8275 0.8264 0.8104 – –

10 11 12 13 14 15 16 17 18 19 20 – –

Table C.13 shows that the indication of the best alternative and the ranking of the alternatives remained unchanged, without characterizing RR types #12 and #32. Table C.13 Results R-TOPSIS: Exclusion of alternative A12 . Alternative

IRA

IECH

IEPL

Rj

Ordering

A76 A1 A8 A87 A54 A91 A59 A5 A56 A83 A61 A71 A67 A72 A90 A2 A13 A88 A55 PIS NIS

0.3135 0.3096 0.3054 0.3025 0.3041 0.3026 0.2990 0.2893 0.2850 0.2822 0.2820 0.2782 0.2758 0.2735 0.2697 0.2560 0.2554 0.2579 0.2435 0.3633 0.0000

0.3239 0.3298 0.3182 0.3298 0.3182 0.3239 0.3298 0.3239 0.3298 0.3298 0.3298 0.3298 0.3298 0.3298 0.3127 0.3298 0.3298 0.3127 0.3298 0.3298 0.0000

0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.0000

0.9157 0.9105 0.9011 0.8993 0.8989 0.8984 0.8937 0.8775 0.8720 0.8676 0.8673 0.8615 0.8579 0.8544 0.8439 0.8284 0.8275 0.8264 0.8104 – –

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 – –

Table C.14 shows that the best alternative was maintained and so was the transitivity of the ordering after altering the A12 scores to values 40% lower for all criteria. Thus, RR type #2 was not characterized. Table C.14 Results R-TOPSIS: Substitution of values in alternative A12 . Alternative

IRA

IECH

IEPL

Rj

Ordering

A76 A1 A8 A87 A54 A91 A59 A5 A56 A83 A61 A71 A67 A72

0.3135 0.3096 0.3054 0.3025 0.3041 0.3026 0.2990 0.2893 0.2850 0.2822 0.2820 0.2782 0.2758 0.2735

0.3239 0.3298 0.3182 0.3298 0.3182 0.3239 0.3298 0.3239 0.3298 0.3298 0.3298 0.3298 0.3298 0.3298

0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069

0.9157 0.9105 0.9011 0.8993 0.8989 0.8984 0.8937 0.8775 0.8720 0.8676 0.8673 0.8615 0.8579 0.8544

1 2 3 4 5 6 7 8 9 10 11 12 13 14

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R.,.d.F. Aires and L. Ferreira

Table C.14 (continued) Alternative

IRA

IECH

IEPL

Rj

Ordering

A90 A2 A13 A88 A55 A12 PIS NIS

0.2697 0.2560 0.2554 0.2579 0.2435 0.5100 0.3633 0.0000

0.3127 0.3298 0.3298 0.3127 0.3298 0.6000 0.3298 0.0000

0.3069 0.3069 0.3069 0.3069 0.3069 0.6000 0.3069 0.0000

0.8439 0.8284 0.8275 0.8264 0.8104 0.5640 – –

15 16 17 18 19 20 – –

Finally, Tables C.15 and C.16 show that rankings obtained from using the R-TOPSIS method are compatible with the rankings of the original problem. Thus, RR type #4 also was not characterized. Table C.15 Results R-TOPSIS: SubProblem 1. Alternativ3

IRA

IECH

IEPL

Rj

Ordering

A1 A12 A8 A54 A59 A5 A56 A2 A13 A55 PIS NIS

0.3096 0.3088 0.3054 0.3041 0.2990 0.2893 0.2850 0.2560 0.2554 0.2435 0.3633 0.0000

0.3298 0.3298 0.3182 0.3182 0.3298 0.3239 0.3298 0.3298 0.3298 0.3298 0.3298 0.0000

0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.0000

0.9105 0.9093 0.9011 0.8989 0.8937 0.8775 0.8720 0.8284 0.8275 0.8104 – –

1 2 3 4 5 6 7 8 9 10 – –

Alternative

IRA

IECH

IEPL

Rj

Ordering

A76 A87 A91 A83 A61 A71 A67 A72 A90 A88 PIS NIS

0.3135 0.3025 0.3026 0.2822 0.2820 0.2782 0.2758 0.2735 0.2697 0.2579 0.3633 0.0000

0.3239 0.3298 0.3239 0.3298 0.3298 0.3298 0.3298 0.3298 0.3127 0.3127 0.3298 0.0000

0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.3069 0.0000

0.9157 0.8993 0.8984 0.8676 0.8673 0.8615 0.8579 0.8544 0.8439 0.8264 – –

1 2 3 4 5 6 7 8 9 10 – –

Table C.16 Results R-TOPSIS: SubProblem 2.

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