7407 FACTA UNIVERSITATIS Series:Mechanical Engineering Vol. 20, No 2, 2022, pp. 381 - 398 https://doi.org/10.22190/FUME210315056S © 2022 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper TOWARDS RELIABLE DECISION-MAKING IN THE GREEN URBAN TRANSPORT DOMAIN Andrii Shekhovtsov, Jakub Więckowski, Bartłomiej Kizielewicz, Wojciech Sałabun Research Team on Intelligent Decision Support Systems, Department of Artificial Intelligence Methods and Applied Mathematics, Faculty of Computer Science and Information Technology, West Pomeranian University of Technology, Szczecin, Poland Abstract. Operational research is a scientific discipline related to the decision theory that allows determining solutions for specific problems related to, for example, widely understood transport. Increasingly popular in this field are issues related to the domain of the green urban transport. In order to support the decision-making process in this area, methods of multi-criteria decision analysis (MCDA) are used more and more often. However, if we solve a specific problem using different MCDA methods, we get different rankings, as each method has a different methodological basis. Therefore, the challenge is how to make a reliable decision. This paper presents a numerical example from the green urban transport domain, which is solved by six different MCDA methods that return a complete ranking. We measure the similarity of these rankings using coefficients rw and WS, and then we propose a simple way of determining a compromise solution. The obtained compromise ranking is guaranteed to be the best match to the selected MCDA methods' rankings, which is proved in the paper. Finally, possible directions for further development work are identified. Key Words: MCDA, Transport Selection, Green Urban Transport, Operational Research 1. INTRODUCTION Decision-making problems appear in every aspect of our lives. Many factors are involved in making a choice, which makes it much more difficult [1]. Sometimes decision-making is influenced by emotions, making the choices made biased. However, making a choice should always maximize benefits and minimize possible losses. To support decision-making and to remain objective in the process, many methods have been Received March 15, 2021 / Accepted July 30, 2021 Corresponding author: Wojciech Sałabun West Pomeranian University of Technology, Szczecin; ul. Żołnierska 49, 71-210 Szczecin, Poland E-mail: wojciech.salabun@zut.edu.pl 382 A. SHEKHOVTSOV, J. WIĘCKOWSKI, B. KIZIELEWICZ, W. SAŁABUN developed to support the decision-maker [2, 3]. The models based on these methods are designed to indicate which of the considered alternatives are better than the others. Nevertheless, a more critical aspect is whether getting even different rankings can contribute to a more reliable solution. There are often used methods belonging to the multi-criteria decision analysis (MCDA) methods in support systems, which work on the basis of evaluating a set of alternatives and giving them preference values thus assessing their quality concerning other options [4]. The group of MCDA methods is constantly extended by newly created methods, making it difficult to choose a method to solve the selected task [5, 6]. Moreover, another difficulty may be determining whether choosing a different method to solve the same problem may give different results [1]. It makes it worthwhile to lean into answering the question if the choice of MCDA method matters. MCDA methods can be divided into three main groups, the American school of methods, the European school of methods and rule-based methods [7, 8]. Each group is based on slightly different assumptions about how the problem should be solved, but in general, each attempts to obtain a ranking that evaluates the alternatives under consideration. The effectiveness of MCDA methods has been tested many times, and they have been used to solve problems of selecting industrial locations [9, 10], material suppliers [11, 12], in sports [13, 14] or medicine [15]. These methods are willingly used by experts, making them increasingly applicable to a wide range of problems. Moreover, a particular decision problem can often be solved using more than one method but then we get different rankings. It is related to assumptions used in the algorithms, which can be exemplary based on the distance from the best or the best and the worst element. Therefore, these solutions are not incorrect from the methodological point of view. However, it should be considered how to decide on the basis of such rankings in order to make the decision as reliable as possible. However, in the cases where MCDA methods guarantee different results within one problem, it is worthwhile to determine the extent to which the obtained results are similar [16]. For this purpose, correlation coefficients can be used to assess the similarity of the analyzed rankings through numerical values. These coefficients include Spearman’s weighted correlation coefficient and WS similarity coefficient [8, 17]. Firstly, it is an opportunity to determine how strongly different results were returned by the methods used. Secondly, it can be used to establish a compromise ranking. For this purpose, measures need to be defined to determine whether a given ranking is a better compromise. In this paper, we propose a new approach to determine the compromise ranking based on the similarity rankings. The proposed approach is extremely simple and is intended to compromise ranking based on the rankings obtained from various MCDA methods. We present our methodology by using a theoretical multi-criteria problem in the form of the selection of the electric bus. For this purpose, we have selected six MCDA methods that give a full ranking as a result. The obtained results from TOPSIS, VIKOR, PROMETHEE II, COPRAS, COMET, and SPOTIS methods are then compared using the similarity coefficients to check how the rankings correspond to each other. On this basis, we define two measures indicating which of the resulting rankings the best compromise ranking is. Then, we determine the compromise ranking whose result is the closest to all the rankings considered. This results in a final compromise ranking that indicates the most reliable solutions in the absence of knowledge of the reference solution. Towards Reliable Decision-Making in the Green Urban Transport Domain 383 The rest of the paper is organized as follows. In Sections 2.1-2.6, the preliminaries of selected MCDA methods are presented. The ranking similarity coefficients are presented in Section 2.7. Section 3 includes an empirical study case, in which the comparison of six given methods application, namely TOPSIS, VIKOR, PROMETHEE II, COPRAS, COMET, and SPOTIS, is made. Section 4 describes the proposed approach to obtain compromise ranking and a short discussion, and finally, the conclusions from the research are drawn in Section 5. 2. PRELIMINARIES Multi-Criteria Decision Analysis (MCDA) methods solve a multi-criteria problem in various areas [13, 14, 18]. With an extended trend in using these methods to solve problems, more extensions and new techniques are being developed [19, 20]. In this section, we recall the algorithms of the MCDA methods (Sections 2.1-2.6) and the used similarity coefficients (Sections 2.7 and 2.8). It is necessary because there are many versions of these algorithms in the literature [21, 22, and 23]. 2.1. The TOPSIS method The Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) method was developed in 1992 [24]. Chen and Hwang proposed an approach to examining the set of alternatives based on the calculation of the distance to the ideal solution. To evaluate the alternatives’ preferences, the Positive Ideal Solution (PIS) and Negative Ideal Solution (NIS) are used [25, 26]. The method application requires an expert to define a weights vector, which describes the relevance of criteria [27]. The first step is to normalize the decision matrix in order to obtain the correct final result. Next, a weighted normalized decision matrix should be calculated using the following Eq. (1). , 1, , ; 1, ,=  =  =  ij i ij v w r j J i n (1) where wi is the value of the i-th weighting, rij is the normalized attribute for the j-th alternative against the i-th criterion, vij is the weighted normalized attribute for the j-th alternative against the i-th criterion, n is the number of criteria, and J is the number of alternatives. Positive (A*) and negative (A-) ideal solutions for a defined decision-making problem should also be identified as Eq. (2): 1 1 { , , } {(max | ), (min | )} { , , } {(min | ), (max | )} P C n j ij j ij P C n j ij j ij A v v v i I v i I A v v v i I v i I    − − − =  =   =  =   (2) where IC stands for cost type criteria and IP for profit type. Negative and positive distance from an ideal solution should be calculated using the n-dimensional Euclidean distance. To apply such calculations, the formula presented below should be used Eq. (3): 384 A. SHEKHOVTSOV, J. WIĘCKOWSKI, B. KIZIELEWICZ, W. SAŁABUN 2 1 2 1 ( ) , 1, , ( ) , 1, , n j ij i i n j ij i i D v v j J D v v j J   = − − = = − =  = − =    (3) The last step is to calculate the relative closeness to the ideal solution by Eq. (4): , 1, , ( ) j j j j D C j J D D −   − = =  + (4) 2.2. The VIKOR method The VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje) method was developed by Opricovic in 1998 [28]. The final preferences for alternatives are calculated using expert knowledge and the closeness of the solution to the ideal solution [29, 30]. The input data does not need to be normalized as in the TOPSIS method [31]. Each of the criteria is initially defined as a cost or profit type of criteria by Eq. (5). The cost type shows that we want it to achieve the lowest possible values, while the profit type should achieve the highest possible values. , ; , i j ij i j ij i j ij i j ij f max f f min f if the i th criteria is a profi type f min f f max f if the i th criteria is a cost type  −  − = = − = = − (5) j ij j j i f f w f f   − −  − (6) For each of the criteria, the best fi * and the worst fi - values are defined. Then, using the formula presented in Eq. (6), preference values for each criterion are calculated, taking into account the weights for criteria defined at the beginning. On this basis, the closeness to the ideal solution is calculated, considered in three different rankings calculated from Eq. (7), Eq. (8) and Eq. (9) for S, R, and Q, respectively. 1 N j ij i j j j i f f S w f f   − = − =  −  (7) j ij i j j j i f f R max w f f   −  − =   −   (8) (1 )i i i S S R R Q v v S S R R   −  −  − − =  + −  − − (9) Towards Reliable Decision-Making in the Green Urban Transport Domain 385 2.3. The PROMETHEE II method The Preference Ranking Organization METHod for Enrichment of Evaluations (PROMETHEE) is a family of MCDA methods developed by Brans [32, 33]. It uses input data similar to other methods, but it optionally requires choosing preference function and some other variables. In this paper, we use the PROMETHEE II method because the output of this method is a full ranking of the alternatives. It is the approach where a complete ranking of the actions is based on the multicriteria net flow. It includes preferences and indifferences (preorder) [34]. According to [32, 35], PROMETHEE II is designed to solve the following multicriteria problems: 1 2 { ( ), ( ), ( ) | } n max g a g a g a a A  (10) where A is a finite set of alternatives and gi(·) is a set of evaluation criteria either to be maximized or minimized. In other words, gi(aj) is a value of criteria i for alternative aj. With this values and weights we can define evaluation table. Table 1 Evaluation table a g1(·) g2(·) … gn(·) w1 w2 … wn a1 g1(a1) g2(a1) … gn(a1) a1 g1(a2) g2(a2) … gn(a2) … … … … … am g1(am) g2(am) … gn(am) Step 1. After defining the problem as described above, calculate preference function values. It defined as Eq. (11) for profit criteria. ( , ) [ ( , )], ,P a b F d a b a b A=   (11) where d(a,b) is the difference between two actions (pairwise comparison): ( , ) ( ) ( )d a b g a g b= − (12) and the value of preference function P is always between 0 and 1 and it is calculated for each criterion. Step 2. Calculate aggregated preference indices by Eq. (13). 1 1 ( ) ( ) ( , ) , ( , ) , n j j j n j j j a b P a b w b a P b a w   = =  =    =     (13) where a and b are alternatives and π(a,b) shows how much alternative a is preferred to b over all of the criteria. There are some properties Eq. (14) which must be true for all alternatives set A. 386 A. SHEKHOVTSOV, J. WIĘCKOWSKI, B. KIZIELEWICZ, W. SAŁABUN ( , ) 0 0 ( , ) 1 0 ( , ) 1 0 ( , ) ( , ) 1 a a a b b a a b b a      =         +  (14) Step 3. Next, calculate positive Eq. (15) and negative Eq. (16) outranking flows. 1 ( ) 1 ( , ) x A a m a x  +  = −  (15) 1 ( ) 1 ( , ) x A x m a a  −  = −  (16) Step 4. In this paper, we use only PROMETHEE II, which results in a complete ranking of alternatives. Ranking is based on the net flow Φ Eq. (17). ( ) ( ) ( )a a a + −  =  − (17) A larger value of Φ(a) means a higher position in the ranking. 2.4. The COPRAS method The COmplex PRoportional ASsessment (COPRAS), introduced by Zavadskas [36, 37] assumes a direct and proportional relationship of the importance of the investigated variants on a system of criteria adequately describing the decision variants as well as on values and weights of the criteria [38]. This method ranks alternatives based on their relative importance (weight). Final ranking is creating using the positive and negative ideal solutions [37, 39]. Assuming that we have decision matrix with m alternatives and n criteria is represented as X = fij(Ai)m×n, COPRAS method is defined in the following five steps.: Step 1. Calculate normalized decision matrix using Eq. (18). 1 ij ij m ij i x r x = =  (18) Step 2. Calculate difficult normalized decision matrix, which represents multiplication of the normalized decision matrix elements with the appropriate weight coefficients using Eq. (19). ij ij j v r w=  (19) Step 3. Determine the sums of difficult normalized values which were calculated previously. Eq. (20) should be used for profit criteria and Eq. (21) for cost criteria. 1 k i ij j S v + = =  (20) 1 n i ij j k S v − = + =  (21) Towards Reliable Decision-Making in the Green Urban Transport Domain 387 Step 4. Calculate the relative significance of alternatives using Eq. (22). 1 1 m min i i i i m min i i i S S Q S S S S − − = + − − = −  = +          (22) Step 5 Final ranking is performed according Ui values Eq. (23). 100% i i max i Q U Q =  (23) where Qi max stands for maximum value of the utility function. Better alternatives have higher Ui value, e.g. the best alternative has Ui = 100. 2.5. The COMET method The Characteristic Objects Method (COMET) belongs to the rule-based MCDA methods [41]. The final preferences for the alternatives are calculated on the rule base, which is obtained by defining the Characteristic Objects and pairwise comparison made by expert [42]. Moreover, it is worth noticing that COMET is the first method to be completely free of the rank reversal phenomenon [42, 43]. The formal notation of this method can be shortly recalled in the following five steps [44, 45, 46, 47]. Step 1. Define the Space of the Problem – the expert determines the dimensionality of the problem by selecting number r of criteria, C1, C2, ..., Cr. Then, the set of fuzzy numbers for each criterion Ci is selected Eq. (24): 1 2 ,..., }{ , r rn r r rn C C C C= (24) where nr is a number of the fuzzy numbers for criterion r. Step 2. Generate Characteristic Objects – The characteristic objects (CO) are obtained by using the Cartesian Product of fuzzy numbers cores for each criterion as follows Eq. (25): 1 2 ( ) ( ) ( ) r CO C C C C C C=   (25) Step 3. Rank the Characteristic Objects – the expert determines the Matrix of Expert Judgment (MEJ). It is a result of pairwise comparison of the COs by the problem expert. The MEJ matrix contains results of comparing characteristic objects by the expert, where αij is the result of comparing COi and COj by the expert. Function fexp denotes the mental function of the expert. It depends solely on the knowledge of the expert and can be presented as Eq. (26). Afterwards, the vertical vector of the Summed Judgments (SJ) is obtained as follows Eq. (27). exp exp exp exp exp exp 0.0, ( ) ( ) 0.5, ( ) ( ) 1.0, ( ) ( ) i j ij i j i j f CO f CO f CO f CO f CO f CO     = =   (26) 1 t i ij j SJ  = =  (27) 388 A. SHEKHOVTSOV, J. WIĘCKOWSKI, B. KIZIELEWICZ, W. SAŁABUN Finally, the values of preference are approximated for each characteristic object. As a result, vertical vector P is obtained, where i-th row contains the approximate value of preference for COi. Step 4. The Rule Base – each characteristic object and value of preference is converted to a fuzzy rule as follows Eq. (28): 1 2 ( ) ( ) i i i C CIF C AND C AND THEN P (28) In this way, the complete fuzzy rule base is obtained. Step 5 Inference and Final Ranking – each alternative is presented as a set of crisp numbers (e.g., Ai = {a1i, a2i, ..., ari}). This set corresponds to criteria C1, C2, ..., Cr. Mamdani’s fuzzy inference method is used to compute preference of i-th alternative. The rule base guarantees that the obtained results are unequivocal. 2.6. The SPOTIS method The Stable Preference Ordering Towards Ideal Solution (SPOTIS) method is a recently developed method [48]. Similarly to the COMET method, it is declared fully resistant to the rank reversal phenomenon. This method’s main assumption is to define the data boundaries, which are used to determine Ideal Solution Point (ISP). Based on it, further calculations to obtain the final preferences for alternatives are being made [49]. The definition of the data boundaries requires to select maximum Sj max and minimum Sj min bound for each criterion Cj. Ideal Solution Point Sj * is defined as Sj * = Sj max for profit and as Sj * = Sj min for cost type of criterion. More necessary transformations during the method application are presented below. Step 1. Calculation of the normalized distances to Ideal Solution Point Eq. (29). ),( ij j ij i j max min j j S S d A S S S   − = − (29) Step 2. Calculation of weighted normalized distances d(Ai, S *) ∈ [0,1], according to Eq. (30). 1 ,( )( , ) N i j ij i j j d A S w d A S   = =  (30) Step 3. Final ranking should be determined based on d(Ai, S *) values. Smaller values d(Ai, S *) which are preferences of alternatives result in better position in general ranking. 2.7. The ranking similarity coefficients The idea of using the rankings similarity coefficients is not new and has been the subject of many works [50, 51]. Particularly interesting are works related to the weighted rank measure of correlation rw [52, 53]. Recently, a new coefficient is proposed by Sałabun [8]. It is an asymmetric measure, where the weight of a given comparison is determined based on the significance of the position in the reference ranking. These coefficients can be presented as Eq. (31) and Eq. (32) respectively: Towards Reliable Decision-Making in the Green Urban Transport Domain 389 2 1 3 2 6 ( ) ( 1) ( 1) 1 ( 1) ( ) n xi yi xi yi i w R n R n R r R n n n n =  − − + + − + = −  + − −  (31) 1 | | 1 2 max{| |,| |}1 xi n xi yiR i xixi R WS N R R R − = − −  = −   −   (32) where rw is a value of the weighted rank measure of correlation, WS is a value of similarity coefficient, n is a length of ranking, Rxi and Ryi mean the place in the ranking for i-th element in, respectively, ranking x and ranking y. 3. NUMERICAL EXAMPLE IN GREEN URBAN TRANSPORT DOMAIN The problem of electric bus selection is presented as an exemplary problem from the green urban transport domain. This problem considers seven crit eria and nine alternatives, where the choice of criteria was taken by the most critical elements of importance in assessing electric buses’ quality [54, 55, 56]. We assume that a reference ranking is not available, and our task is to find the best compromi se solution. The defined criteria are presented in Table 2, where the criterion name, its type and the unit in which the values will be given are shown. On the other hand, the decision matrix for the selected nine alternatives, which is used to calculate the final preference values in each of the MCDA methods used, is included in Table 3. Each alternative contains the name of the electric bus model and the manufacturer’s values for the criteria considered. Six vehicle manufacturers were selected, and one of the selected models was presented in four different alternatives. Table 2 Considered criteria Ci to electric bus selection Ci Name Type Units C1 Battery capacity Profit kWh C2 Engine power Profit kW C3 Range Profit Km C4 Price Cost Thousands PLN C5 All places Profit Unit C6 Seating places Profit Unit C7 Places for disabled Profit Unit Six MCDA methods were selected to solve the problem of selecting the optimal electric bus to compare the results and check whether the obtained results would be significantly different from each other. TOPSIS method with minmax normalization, VIKOR method without using normalization, PROMETHEE II method with usual type preference function, COPRAS method in standard configuration, COMET method with object evaluation by TOPSIS method and SPOTIS method in the standard configuration were used in the research. 390 A. SHEKHOVTSOV, J. WIĘCKOWSKI, B. KIZIELEWICZ, W. SAŁABUN Table 3 Decision matrix with set of alternatives Ai Ai Name C1 C2 C3 C4 C5 C6 C7 A1 Ursus City Smile 18M 230 120 150 2051.74 83 24 1 A2 Solaris Urbino 12 Electric 210 160 150 2212.86 85 32 1 A3 Byd K9 324 180 250 2101.78 80 25 1 A4 Volvo 7900E 250 200 200 2550.00 83 30 1 A5 Proterra Catalyst FC 94 190 109 2020.00 80 40 0 A6 Proterra Catalyst XR 220 190 264 2465.00 80 40 0 A7 Proterra Catalyst E2 440 190 491 2885.00 80 40 0 A8 Proterra Catalyst E2 Max 660 190 685 3370.00 80 40 0 A9 New Flyer Xcelsior CHARGE 40’s 300 159 210 3293.75 83 40 2 The obtained preference values using the mentioned methods are shown in Table 4. When TOPSIS, PROMETHEE II, COPRAS and COMET methods are used, a higher preference value means a better-evaluated alternative. On the other hand, when preferences obtained from applying the VIKOR and SPOTIS methods are analyzed, a smaller value indicates a better choice. To facilitate the interpretation of the results, the obtained rankings in the form of positional rankings are included in Table 5. A histogram shows the positional rankings’ visualization for the different methods and alternatives in Fig. 1. Table 4 Obtained preferences of alternatives for selected MCDA methods Ai TOPSIS VIKOR PROM. II COPRAS COMET SPOTIS A1 0.3939 1.0000 -0.1786 0.7446 0.2908 0.6589 A2 0.5139 0.5813 -0.0714 0.7831 0.5722 0.4810 A3 0.4399 0.9144 0.0179 0.8580 0.3897 0.5853 A4 0.5018 0.2783 0.1250 0.8191 0.5320 0.4977 A5 0.4535 0.9360 -0.1786 0.5988 0.4141 0.5893 A6 0.4584 0.8167 -0.0357 0.6863 0.4278 0.5661 A7 0.5008 0.6577 0.1071 0.8487 0.5304 0.4987 A8 0.5277 0.5000 0.1071 1.0000 0.5988 0.4464 A9 0.5178 0.0443 0.1071 0.9983 0.5848 0.4738 Table 5 Positional rankings of obtained results from application of MCDA methods Ai TOPSIS VIKOR PROM. II COPRAS COMET SPOTIS A1 9 9 9 7 9 9 A2 3 4 7 6 3 3 A3 8 7 5 3 8 7 A4 4 2 1 5 4 4 A5 7 8 8 9 7 8 A6 6 6 6 8 6 6 A7 5 5 4 4 5 5 A8 1 3 3 1 1 1 A9 2 1 2 2 2 2 Towards Reliable Decision-Making in the Green Urban Transport Domain 391 Fig. 1 Visualization of the positional rankings comparison of used MCDA methods It is worth noting that for the results obtained for the PROMETHEE II method, the preference values for the alternatives A7, A8 and A9 were the same, as they were for the pair of alternatives A1 and A5. Differences in preference values were very small, which may be caused by errors in the numerical representation of floating-point numbers. The results obtained with the PROMETHEE method with ties is presented in Table 6, where the preference values and the positional ranking for the set of alternatives from the decision matrix are listed. Table 6 Obtained preferences for PROMETHEE I and II application Ai ɸ - ɸ+ ɸ PROM. II A1 0.5357 0.3571 -0.1786 8.5 A2 0.5000 0.4286 -0.0714 7 A3 0.4286 0.4464 0.0179 5 A4 0.3929 0.5179 0.1250 1 A5 0.4643 0.2857 -0.1786 8.5 A6 0.3929 0.3571 -0.0357 6 A7 0.3214 0.4286 0.1071 3 A8 0.3214 0.4286 0.1071 3 A9 0.3929 0.5000 0.1071 3 When the preference values are rounded to four significant decimal places, the obtained ranking correlation values may change. The rankings’ obtained similarity values when taking into account the rounding of the preference values are presented in Table 7. A visualization of the newly obtained correlation relationships between the obtained rankings is shown in Fig. 2, where the correlation values were represented by histograms for the analyzed alternatives and methods. Table 7 Correlation between PROMETHEE with ties and other methods Method rw WS TOPSIS 0.5869 0.6991 VIKOR 0.7971 0.7832 PROM. II 1.0000 1.0000 COPRAS 0.6619 0.7476 COMET 0.5869 0.6991 392 A. SHEKHOVTSOV, J. WIĘCKOWSKI, B. KIZIELEWICZ, W. SAŁABUN Fig. 2 Alternate comparison of obtained rankings from MCDA methods Analyzing the obtained values for the preference values derived from the application of the selected six MCDA methods, it can be observed that the PROMETHEE II method guaranteed equal preference values for the alternatives A7 - A9 and for the alternatives A1 and A5. It is worth noting that the raw preference values obtained by the MCDA methods included in Table 4 are challenging to compare, due to the variety of values obtained. On the other hand, when analyzing the obtained positional rankings shown in Table 5, which are more representative, it can be observed that some of the alternatives have different orders depending on the method used. Alternative A8 can be considered the optimal choice, having been best ranked by four of the six methods. On the other hand, the second place in the rankings analyzed was most often obtained by alternative A9, which was ranked in this position five times. When the preference values obtained for the PROMETHEE method were considered and rounded to four decimal places, it is worth noting that the rankings’ similarity values differed from those when no rounding was considered. The correlation values presented in Table 7 showed that only when comparing the rounded preference values with those obtained from the VIKOR method, the correlation increased slightly. In contrast, the correlation decreased for the other rankings. It shows that, in addition to the methods used to solve a given multi-criteria problem, the results are also affected by inaccuracies and numerical errors arising from the execution of operations and machine representation of floating-point numbers. 4. THE PROPOSED APPROACH The next phase of our research is to determine the similarity of the rankings obtained. For this purpose, it was decided to use the weighted Spearman correlation coefficient and the WS similarity coefficient. Both determinants are based on the obtained ranking values from the multi-criteria decision-making methods used. Table 8 shows the calculated correlations between the obtained rankings using the weighted Spearman correlation coefficient. Using this determinant of similarity guarantees obtaining values of the interval [-1.0, 1.0], where a value of -1.0 means a complete lack of similarity, while 1.0 means their equality. When examining the correlation of the ranking with itself, it will give a value of 1.0 in each case. In turn, the similarity results obtained using the similarity coefficient WS are included in Table 9. The values obtained using this determinant guarantee the similarity defined on the interval (0.0, 1.0], where the value 0.0 means no similarity of rankings, while Towards Reliable Decision-Making in the Green Urban Transport Domain 393 the value 1.0 means the identical rankings. The similarity obtained is highly influenced by changes noted on the first positions among the analyzed orders. The most correlated rankings were obtained by the TOPSIS, COMET and SPOTIS methods. The reason may be that these methods use the concept of reference objects. Despite using the same concept by the VIKOR method, the obtained rankings correlations differed significantly from those mentioned above. The divergence between TOPSIS, COPRAS, PROMETHEE II and VIKOR rankings is a frequent phenomenon appearing in the performance studies of MCDA methods. Meanwhile, using the WS similarity coefficient for the correlation study, it was noted that the most correlated rankings were obtained using TOPSIS, COMET and SPOTIS methods, which shows that both coefficients equally indicate the most similar rankings. The lowest similarity of rankings was noted for the PROMETHEE II method concerning the other rankings. Table 8 Rankings correlation for Spearman weighted correlation coefficient rw rw TOPSIS VIKOR PROM. II COPRAS COMET SPOTIS TOPSIS 1.0000 0.8667 0.6350 0.6650 1.0000 0.9917 VIKOR 0.8667 1.0000 0.8683 0.6233 0.8667 0.8750 PROM. II 0.6350 0.8683 1.0000 0.6783 0.6350 0.6650 COPRAS 0.6650 0.6233 0.6783 1.0000 0.6650 0.7300 COMET 1.0000 0.8667 0.6350 0.6650 1.0000 0.9917 SPOTIS 0.9917 0.8750 0.6650 0.7300 0.9917 1.0000 Table 9 Rankings correlation for WS similarity coefficient WS TOPSIS VIKOR PROM. II COPRAS COMET SPOTIS TOPSIS 1.0000 0.7916 0.7434 0.9051 1.0000 0.9981 VIKOR 0.8100 1.0000 0.8539 0.7434 0.8100 0.8119 PROM. II 0.7291 0.8698 1.0000 0.6841 0.7291 0.7375 COPRAS 0.8619 0.7098 0.7950 1.0000 0.8619 0.8830 COMET 1.0000 0.7916 0.7434 0.9051 1.0000 0.9981 SPOTIS 0.9981 0.7935 0.7438 0.9047 0.9981 1.0000 The proposed approach is to combine the usual voting approach and the similarity coefficients of the rankings. Each of the methods used takes a vote, where each alternative is given a number of points corresponding to a place in the ranking. A new ranking is then established where the highest-ranked alternative is the one that has received the least number of points. Of course, the compromise solution may vary due to the number of methods and the methods chosen. Table 10 shows the seven rankings that have been created based on the proposed approach, i.e., Rank 1 using all six methods, Rank 2 using five methods without TOPSIS, Rank 3 using five methods without VIKOR, Rank 4 using five methods without PROMETHEE II, Rank 5 using five methods without COPRAS, Rank 6 using five methods without COMET, and Rank 7 using five methods without SPOTIS. 394 A. SHEKHOVTSOV, J. WIĘCKOWSKI, B. KIZIELEWICZ, W. SAŁABUN Table 10 Position values for candidate compromise rankings Ai Rank 1 Rank 2 Rank 3 Rank 4 Rank 5 Rank 6 Rank 7 A1 9 9 9 9 9 9 9 A2 4 4.5 4 3.5 4 4.5 4.5 A3 6.5 6 6 7 7 6 6 A4 3 3 3 3.5 3 3 3 A5 8 8 8 8 8 8 8 A6 6.5 7 7 6 6 7 7 A7 5 4.5 5 5 5 4.5 4.5 A8 1 1.5 1 1 1.5 1.5 1.5 A9 2 1.5 2 2 1.5 1.5 1.5 In order to obtain a compromise ranking, the measure of its fit must first be defined. When the reference value is not known, one should select a ranking that best fits the selected MCDA methods' rankings. Thus, we use two matching measures, i.e., the minimum similarity value and the average similarity value. In Table 8, the best compromise ranking from among the rankings determined by the MCDA methods is the ranking obtained with the SPOTIS method, which has the highest minimum and average similarity value of the rankings, which amount to 0.665 and 0.876, respectively. On the other hand, the worst compromise ranking would be the ranking obtained using the COPRAS method. It has the lowest minimum and average value of ranking similarity, which are 0.623 and 0.727, respectively. An interesting situation is when comparing using the WS coefficient, where three times we obtain the same average value for the TOPSIS, COMET and SPOTIS methods. However, taking into account the highest minimum value, the ranking determined by the SPOTIS method is again the best (0.0004 higher than TOPSIS and COMET). Tables 11 and 12 show the similarity between the candidate compromise rankings and those obtained using the MCDA methods. Thus Rank 1 has an average rw of 0.900 and a minimum similarity value of 0.7842. This means a much higher match than was obtained from each ranking obtained using the MCDA methods tested. The best fit was obtained for five methods when TOPSI, COMET or SPOTIS were eliminated (Rank 2, 6 and 7, respectively). This is logical because in these three cases, precisely the same ranking was obtained. The matching looks different for the WS measure. Different compromise ranks are determined using the average value of similarity of rankings and others in the minimum level criterion. In the first case, the best result is obtained for Rank 4 because here the average value of matching is 0.9065, but the minimum value is only 0.7682. Table 11 Values of similarity coefficient rw for candidate compromise rankings rw TOPSIS VIKOR PROM. II COPRAS COMET SPOTIS Rank 1 0.9623 0.9171 0.7992 0.7842 0.9623 0.9754 Rank 2 0.9265 0.9385 0.8542 0.8302 0.9265 0.9448 Rank 3 0.9483 0.9083 0.8033 0.8150 0.9483 0.9667 Rank 4 0.9862 0.9027 0.7331 0.7454 0.9862 0.9946 Rank 5 0.9629 0.9548 0.8108 0.7412 0.9629 0.9712 Rank 6 0.9265 0.9385 0.8542 0.8302 0.9265 0.9448 Rank 7 0.9265 0.9385 0.8542 0.8302 0.9265 0.9448 Towards Reliable Decision-Making in the Green Urban Transport Domain 395 Table 12 Values of similarity coefficient WS for candidate compromise rankings WS TOPSIS VIKOR PROM. II COPRAS COMET SPOTIS Rank 1 0.9621 0.8164 0.7840 0.9144 0.9621 0.9647 Rank 2 0.9043 0.8706 0.8302 0.8798 0.9043 0.9080 Rank 3 0.9586 0.8140 0.7836 0.9138 0.9586 0.9622 Rank 4 0.9821 0.8071 0.7682 0.9154 0.9821 0.9839 Rank 5 0.9177 0.8849 0.8161 0.8659 0.9177 0.9195 Rank 6 0.9043 0.8706 0.8302 0.8798 0.9043 0.9080 Rank 7 0.9043 0.8706 0.8302 0.8798 0.9043 0.9080 The compromise solution, in this case, should be selected as the order indicated by Rank 2, 6 or 7 (these rankings are identical), as this guarantees maximum matching for all the rankings examined. Finally, we also show how strongly similar solutions have been obtained using all methods or combinations of the five elements. Tables 13 and 14 show the similarity values of the rankings among themselves. As we can see, the rankings are very similar to each other, so the choice of methods is not a big problem in determining the compromise ranking, but the research presented in this paper should be extended and continued to generalize the observations of this paper. Table 13 Correlation matrix for rw values for candidate compromise rankings rw Rank 1 Rank 2 Rank 3 Rank 4 Rank 5 Rank 6 Rank 7 Rank 1 1.0000 0.9854 0.9971 0.9917 0.9900 0.9854 0.9854 Rank 2 0.9854 1.0000 0.9883 0.9662 0.9838 1.0000 1.0000 Rank 3 0.9971 0.9883 1.0000 0.9829 0.9812 0.9883 0.9883 Rank 4 0.9917 0.9662 0.9829 1.0000 0.9875 0.9662 0.9662 Rank 5 0.9900 0.9838 0.9812 0.9875 1.0000 0.9838 0.9838 Rank 6 0.9854 1.0000 0.9883 0.9662 0.9838 1.0000 1.0000 Rank 7 0.9854 1.0000 0.9883 0.9662 0.9838 1.0000 1.0000 Table 14 Correlation matrix for WS values for candidate compromise rankings WS Rank 1 Rank 2 Rank 3 Rank 4 Rank 5 Rank 6 Rank 7 Rank 1 1.0000 0.9387 0.9980 0.9813 0.9489 0.9387 0.9387 Rank 2 0.9408 1.0000 0.9430 0.9233 0.9858 1.0000 1.0000 Rank 3 0.9978 0.9407 1.0000 0.9789 0.9465 0.9407 0.9407 Rank 4 0.9817 0.9185 0.9795 1.0000 0.9348 0.9185 0.9185 Rank 5 0.9506 0.9854 0.9484 0.9362 1.0000 0.9854 0.9854 Rank 6 0.9408 1.0000 0.9430 0.9233 0.9858 1.0000 1.0000 Rank 7 0.9408 1.0000 0.9430 0.9233 0.9858 1.0000 1.0000 5. CONCLUSIONS The presented study shows a simple way of building a compromise ranking, which is quite helpful when the obtained ranking results are not unambiguous. Additionally, two criteria were given for the selection of the best compromise ranking. It allows obtaining a solution, which is maximally compatible with all methods involved in determining this 396 A. SHEKHOVTSOV, J. WIĘCKOWSKI, B. KIZIELEWICZ, W. SAŁABUN solution. The presented example for the selection of electric buses has shown that this solution is effective, and the proposed compromise solutions are better than any of the obtained solutions. Due to the fact that each method has a different methodological background, no solution can be discriminated against. Thanks to this approach, we obtain a solution that is a true compromise based on the available data. For further research directions, it is worth considering more MCDA methods to receive more benchmarkable results. 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