A direct interval extension of TOPSIS method Expert Systems with Applications 40 (2013) 4841–4847 Contents lists available at SciVerse ScienceDi rect Expert Systems with Applic ations j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e s w a A direct interval extension of TOPSIS method 0957-4174/$ - see front matter � 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.eswa.2013.02.022 ⇑ Corresponding author. Tel./fax: +48 34 3250 589. E-mail address: sevast@icis.pcz.pl (P. Sevastjanov). Ludmila Dymova, Pavel Sevastjanov ⇑, Anna Tikhonenko Institute of Comp. & Information Sci., Technical University of Czestochowa, Dabrowskiego 73, 42 201 Czestochowa, Poland a r t i c l e i n f o a b s t r a c t Keywords: TOPSIS Interval extension The TOPSIS method is a technique for order preference by similar ity to ideal solution. This technique cur- rently is one of the most popular methods for Multiple Criteria Decision Making (MCDM). The TOPSIS method was primary developed for dealing with only real-valued data. In many cases, it is hard to present precisely the exact ratings of alternatives with respect to local criteria and as a result these ratings are considered as intervals. There are some papers devoted to the interval extensions of TOPSIS method, but these extensions are based on different heuristic approaches to definition of positive and negative ideal solutions. These ideal solutions are presented by real values or intervals, which are not attainable in a decision matrix. Since this is in contradiction with basics of classical TOPSIS method, in this paper we propose a new direct approach to interval extension of TOPSIS method which is free of heuristic assumptions and limitations of known methods. Using numerical examples we show that ‘‘direct interval extension of TOPSIS method’’ may provide the final ranking of alternatives which is substantially different from the results obtained using known methods. � 2013 Elsevier Ltd. All rights reserved. 1. Introduction The technique for order performanc e by similarity to ideal solu- tion (TOPSIS) (Lai, Liu, & Hwang, 1994 ) is one of known classical MCDM method. It was first developed by Hwang and Yoon (1981) for solving MCDM problems . The basic principle of the TOPSIS method is that the chosen alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. There exist a large amount of literature involving TOPSIS theory and applications. Is was shown by Garca-Cascal es and Lamata (2012) and Wang and Luo (2009) that ‘‘One of the problems attributable to TOPSIS is that it can cause the phenomeno n known as rank reversal. In this phenomeno n the alternative’s order of preference changes when an alternative is added to or removed from the decision problem. In some cases this may lead to what is called total rank reversal, where the order of preferences is totally inverted, that is to say, that the alternative considered the best, with the inclusion or re- moval of an alternative from the process, then becomes the worst. Such a phenomeno n in many cases may not be acceptable’’. Wang and Luo (2009) showed that rank reversal phenomeno n occurs not only in the TOPSIS method, but in many other decision making ap- proaches such as Analytic Hierarchy Process (AHP), the Borda–Ken- dall (BK) method for aggregating ordinal preferences, the simple additive weighting (SAW) method, and the cross-efficiency evalua- tion method in data envelopment analysis (DEA). Therefore, we can say that this problem is typical for known method of MCDM. In Garca-Cas cales and Lamata (2012), the authors proposed a new method for the solution of this problem in the framework of TOPSIS method. Nevertheless it was pointed out in Garca-Cascal es and La- mata (2012) that ‘‘the two methods ‘‘the classical’’ and ‘‘the new’’ do not have to give the same order. This is especially so in the case of evaluating alternatives which are very close’’. In other words, ‘‘the classical’’ and ‘‘the new’’ methods may provide different re- sults based on the same decision matrix. Hence, it is not obvious that ‘‘the new’’ method performs better than classical one if there is no need to add or remove an alternative from the decision problem. Therefore, hereinafter we shall consider only classical TOPSIS method and its interval extension. In classical MCDM methods, the ratings and weights of criteria are known precisely. A survey of these methods is presented in Hwang and Yoon (1981). In the classical TOPSIS method, the ratings of alternatives and the weights of criteria are presented by real values. Neverthel ess, sometimes it is difficult to determine precisely the real values of ratings of alternatives with respect to local crite- ria, and as a result, these ratings are presented by intervals. Jahansha hlo, Hosseinzade, and Izadikhah (2006) and Jahan- shahloo, Hosseinzade h Lotfi, and Davoodi (2009) extended the con- cept of TOPSIS method to develop a methodology for solving MCDM problem with interval data. The main limitatio n of this approach is that the ideal solutions are presente d by real values, not by http://dx.doi.org/10.1016/j.eswa.2013.02.022 mailto:sevast@icis.pcz.pl http://dx.doi.org/10.1016/j.eswa.2013.02.022 http://www.sciencedirect.com/science/journal/09574174 http://www.elsevier.com/locate/eswa 4842 L. Dymova et al. / Expert Systems with Applications 40 (2013) 4841–4847 intervals. The similar approach to determining ideal solutions is used in Yue (2011) and Sayadi, Heydari, and Shahanaghi (2009) this approach is used in context of the so-called VIKOR method which is based on the measure of ‘‘closenes s’’ to the ideal solutions, too. In this paper, we show that these extensions may lead to the wrong results especially in the case of intersect ion of some inter- vals representing the ratings of alternatives . In Chen (2011), Jahanshahloo et al. (2009), Jahanshahlo o, Khodabakhshi , Hosseinz adeh Lotfi, and Moazam i Goudarzi (2011), Tsaur (2011), Ye and Li (2009) and Yue (2011), the different definitions of interval positive and negative ideal solutions are pro- posed. They will be analysed in the next section, but their common limitation is that they are based on the heuristic assumptions (usu- ally without any analysis and clear justification) and provide inter- val ideal solutions that are not always attainable in the interval- valued decision matrix. Since this is in contradiction with basics of classical TOPSIS method, in this paper, we propose a new direct approach to inter- val extension of TOPSIS method which is free heuristic assump- tions and limitations of known methods. This approach makes it possible to obtain the positive and negative ideal solutions in the interval form such that (opposite to the known methods) these interval-valued solutions are always attainable on the initial interval-valued decision matrix. Since this approach is based on the interval comparison, a new simple, but well-justified method for interval comparison is develope d and presented in the special section. It is worth noting that the most general approach to the solu- tion of MCDM problems in the fuzzy setting is the presenta tion of all fuzzy values by correspond ing sets of a-cuts. There are no restrictions on the shape of membership functions of fuzzy values in this approach and the fuzzy TOPSIS method is reduced to the solution of MCDM problems using interval extended TOPSIS meth- od on the corresponding a-cuts (Wang & Elhag, 2006 ). Therefore, the development of a reliable interval extension of TOPSIS method may be considered as a first step in the solution of MCDM problems using the fuzzy TOPSIS method. The rest of the paper is set out as follows. In Section 2, we pres- ent the basics of TOPSIS method and analyse its known interval extensions. Section 3 presents the direct interval extension of TOP- SIS method and the method for interval comparison which is needed to develop this extension. The results obtained using the method proposed in Jahanshahlo et al. (2006) and Jahanshahlo o et al. (2009) are compare d with those obtained by the develope d new method. Section 4 concludes with some remarks. 2. The basics of TOPSIS method and known approaches to its interval extension The classical TOPSIS method is based on the idea that the best alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. It is assumed that if each local criterion is monotonically increasing or decreasing, then it is easy to define an ideal solution. The positive ideal solution is compose d of all the best achiev- able values of local criteria, while the negative ideal solution is composed of all the worst achievable values of local criteria. Suppose a MCDM problem is based on m alternatives A1,A2, . . . ,Am and n criteria C1, C2,. . .,Cn. Each alternative is evaluated with respect to the n criteria. All the ratings are assigned to alternatives and presented in the decision matrix D[xij]m�n, where xij is the rating of alternative Ai with respect to the criterion Cj. Let W = (w1,w2, . . . ,wn) be the vector of local criteria weights satisfying Pn j¼1wj ¼ 1. The TOPSIS method consists of the following steps: 1. Normaliz e the decision matrix: rij ¼ xijffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPm k¼1x 2 kj q ; i ¼ 1; . . . ; m; j ¼ 1; . . . ; n: ð1Þ Multipl y the columns of normalized decision matrix by the associ- ated weights: v ij ¼ wj � rij; i ¼ 1; . . . ; m; j ¼ 1; . . . ; n: ð2Þ 2. Determine the positive ideal and negative ideal solutions, respectivel y, as follows: Aþ ¼ vþ1 ; v þ 2 ; . . . ; v þ n � � ¼fðmax i v ijjj 2 K bÞðmin i v ijjj 2 K cÞg; ð3Þ A� ¼ v�1 ; v � 2 ; . . . ; v � n � � ¼fðmin i v ijjj 2 K bÞðmax i v ijjj 2 K cÞg; ð4Þ where Kb is the set of benefit criteria and Kc is the set of cost criteria. 3. Obtain the distances of the existing alternativ es from the posi- tive ideal and negative ideal solutions: two Euclidean distances for each alternatives are, respectivel y, calculated as follows: Sþi ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn j¼1 ðv ij � vþj Þ 2 vuut ; i ¼ 1; . . . ; m; S�i ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn j¼1 ðv ij � v�j Þ 2 vuut ; i ¼ 1; . . . ; m: ð5Þ 4. Calculate the relative closeness to the ideal alternativ es: RCi ¼ S�i Sþi þ S � i ; i ¼ 1; 2; . . . ; m; 0 6 RCi 6 1: ð6Þ 5. Rank the alternativ es accordin g to the relative closeness to the ideal alternatives: the bigger is the RCi, the better is the alterna- tive Ai. In Jahansha hlo et al. (2006) and Jahanshahlo o et al. (2009), an interval extension of classical TOPSIS method was proposed. This approach may be described as follows. Let ½xij� ¼ ½xLij; x U ij � be an interval value of jth criterion for ith alter- native (xLij and x U ij are the lower and upper bounds of interval, respectivel y), W = (w1,w2, . . . ,wn) be the weight vector satisfying Pn j¼1wj ¼ 1. Then D½½xLij; x U ij ��m�n is the interval-valued decision ma- trix. The method proposed in Jahanshahlo et al. (2006) and Jahan- shahloo et al. (2009) consists of the following steps: 1. Normaliz ing the decision matrix using the following expressions : rLij ¼ xLij Pm k¼1 x L kj � �2 þ xUkj � �2� �� �12 ; i¼1;. . .;m; j¼1;. . .;n; ð7Þ rUij ¼ xUij Pm k¼1 x L kj � �2 þ xUkj � �2� �� �12 ; i¼1;. . .;m; j¼1;. . .;n: ð8Þ 2. Taking into account the importance of criteria, the weighted normalized interval-valued decision matrix is obtained using the following expressions: v Lij ¼ wj � r L ij; v U ij ¼ wj � r U ij ; i ¼ 1; . . . ; m; j ¼ 1; . . . ; n: L. Dymova et al. / Expert Systems with Applications 40 (2013) 4841–4847 4843 3. The positive and negative ideal solutions are obtained as follows: Aþ ¼ vþ1 ; v þ 2 ; . . . ; v þ n � � ¼ max i v Uij jj 2 K b � � ; min i v Lijjj 2 K c � � ; ð9Þ A� ¼ v�1 ; v � 2 ; . . . ; v � n � � ¼ min i v Lijjj 2 K b � � ; max i v Uij jj 2 K c � � : ð10Þ 4. The separation of each alternative from the positive ideal solu- tion is calculated using the n-dimensional Euclidean distance: Sþi ¼ X j2Kb v Lij � v þ j � �2 þ X j2K c v Uij � v þ j � �2( )12 ; i ¼ 1; . . . ; m: ð11Þ Similarly, the separation from the negative ideal solution is calcu- lated as follows: S�i ¼ X j2Kb v Uij � v � j � �2 þ X j2K c v Lij � v � j � �2( )12 ; i ¼ 1; . . . ; m: ð12Þ 5. Calculate the relative closeness to the ideal alternatives : RCi ¼ S�i Sþi þ S � i ; i ¼ 1; 2; . . . ; m; 0 6 RCi 6 1: ð13Þ Table 1 Decision matrix. C1 C2 A1 [5, 7] [22, 32] A2 [0, 10] [25, 27] 6. Rank the alternatives according to the relative closeness to the ideal alternativ es: the bigger is the RCi, the better is the alterna- tive Ai. In Jahanshahloo et al. (2009) and Jahansha hloo et al. (2011), the interval extension of TOPSIS method based on interval-val ued ideal solution was proposed . This method is based on the assumption that the ideal and negative-ideal solutions are changed for each alternative. Nevertheles s, analysing this approach we can see that obtained interval ideal solutions are not always attainable in the interval decision matrix. In Ye and Li (2009), an interval TOPSIS method is used in the framework of group multi-attribute decision model. For the deci- sion maker k, the positive and negative interval-valued solutions are presented in Ye and Li (2009) (in our notation) as follows: Aþk ¼ mþkU1 ; m þkU 2 ; . . . ; m þkU n � � � ¼ max i mkLij ; max i mkUij � � ; A�k ¼ mþkU1 ; m þkU 2 ; . . . ; m þkU n � � � ¼ min i mkLij ; min i mkUij � � : There is no any justification of this approac h in Ye and Li (2009) and only what we can say about it is that in the case of only benefit cri- teria the upper bound of A+k may be obtained from (9) and the low- er bound of A�k may be obtained from (10). It is also important that interval-v alued ideal solution s obtained using the method proposed by Ye and Li (2009) are not always attainabl e in the interval -valued decision matrix. The similar problem may be found in Chen (2011). A more complicated approach to the definition of interval-val- ued ideal solutions was proposed by Tsaur (2011), where the author wrote ‘‘Theoretica lly, a for pivot value m _þ j ¼ ½m _þL j ; m _þU j � for criterion j in the positive ideal solution, we know that both of m _þL j and m _þU j might be obtained from different alternatives.’’. There- fore, the ideal solutions obtained using the method proposed by Tsaur (2011) are not always attainable in the interval-valued deci- sion matrix. In Yue (2011), the group decision making problem was solved using the modified interval extension of TOPSIS method. In this ap- proach, the positive and negative ideal solutions were presented by interval-valued matrices. For example, the negative ideal solution was presented as follows A� ¼ ð½m�Lij ; m �U ij �Þm�n , where m�Lij ¼ minkm kL ij ; m �U ij ¼ maxkm kU ij (k is a number of decision maker). It is easy to see that obtained A� is not always attainable in the interval decision matrices provided by decision makers. Summaris ing, we can say that the common limitatio n of known approach es to interval extension of TOPSIS method is that they (based on the different assumptions) provide interval-valued ideal solutions which are not always attainable in correspondi ng inter- val-value d decision matrices. This is in contradiction with basics of classical TOPSIS method and is a consequence of heuristic assumpti ons which are not usually justified enough. In the most of analysed approaches, the upper bound of positive interval-valued solution is calculated as in the expression (9) and the lower bound of negative interval solution is calculated as in (10). Hence, we can say that the approach developed in Jahanshahlo et al. (2006) and Jahansha hloo et al. (2009) providing real-valued ideal solutions attainable in the interval-valued decision matrix seems to be more justified than the other analysed here ap- proaches providing interval-val ued ideal solutions, which are not always attainable in the interval-valued decision matrix. Therefore, to compare a direct interval extension of TOPSIS method we pro- pose in this paper with other known approach es, it seems to be en- ough to compare the results obtained by our method with those obtained using the method develope d in Jahanshahlo et al. (2006) and Jahansha hloo et al. (2009) (see expressions (7)–(13)). 3. A new approach to the interval extension of TOPSIS method 3.1. The problem formulation Therefore, a more correct and straightforw ard approach to cal- culation of ideal solutions is representing them in the interval form using the expressions : Aþ ¼ vþL1 ; v þU 1 � � ; vþL2 ; v þU 2 � � ; . . . ; vþLn ; v þU n � �� � ¼ max i v Lij; v U ij h i jj 2 K b � � ; min i v Lij; v U ij h i jj 2 K c � � ; ð14Þ A� ¼ v�L1 ; v �U 1 � � ; v�L2 ; v �U 2 � � ; . . . ; v�Ln ; v �U n � �� � ¼ min i v Lij; v U ij h i jj 2 K b � � ; max i v Lij; v U ij h i jj 2 K c � � : ð15Þ As there are no any type reduction s (representation of interv als by real values) and additi onal assumpt ions concerned with expres- sions (14) and (15), we call our approach Direct Interval Extension of TOPSIS method. It is easy to see that expressions (14) and (15) provide the posi- tive and negative interval-valued ideal solutions which are always attainable in the corresponding interval-valued decision matrix. To perform the difference of proposed approach from known ones it is enough to compare it with the method developed in Jah- anshahlo et al. (2006) and Jahansha hloo et al. (2009) (see explana- tion at the end of previous section). 4844 L. Dymova et al. / Expert Systems with Applications 40 (2013) 4841–4847 Suppose we deal with the interval-valued decision matrix pre- sented in Table 1, where [x11] = [5, 7], [x12] = [22, 32], [x21] = [0,10] and [x22] = [25, 27] represent the ratings of alternatives A1 and A2 with respect to the benefit criteria C1 and C2. Since we deal with the only benefit criteria C1 and C2, then expression (9) in our case is reduced to Aþ ¼fmþ1 ; m þ 2 g¼ maxiðx U ijÞ. Using this expression, from the first column of Table 1 we get mþ1 ¼ 10 and from the second column m þ 1 ¼ 32. Therefore Aþ ¼fmþ1 ; m þ 2 g¼f10; 32g. Similarly , from (10) we get A ¼fm1; m2g¼ miniðx L ijÞ¼ f0; 22g. On the other hand, using any method for interval comparison (see Sevastjanov (2007), Wang & Kerre (2001) and subSection 3.2) we obtain that [x21] < [x11], [x22] < [x12] and for the positive and negative interval ideal solutions we get: ½Aþ�¼ f½m�þ1 ; ½m� þ 2 g¼ f½5; 7�; ½22; 32�g and ½A��¼ f½m��1 ; ½m� � 2 g¼f½0; 10�; ½25; 27�g. It is easy to see that fmþ1 ; m þ 2 g¼f10; 32g is not included in f½m�þ1 ; ½m� þ 2 g¼f½5; 7�; ½22; 32�g and fm � 1 ; m � 2 g¼f0; 22g is not included in f½m��1 ; ½m� � 2 g¼f½0; 10�; ½25; 27�g. Thus, we can say that in the cases when some intervals in the decision matrix intersect, the approach proposed in Jahanshahlo et al. (2006) and Jahansha hloo et al. (2009) may lead to wrong results. It is seen that our method provides the interval-valued ideal solutions which are strongly attainable in the considered decision matrix (see Table 1.). Since the other known methods analysed in previous section may produce interval-valued ideal solutions which are not attainable in the considered decision matrix (see Ta- ble 1.), they may produce the wrong results too. We use here the words ‘‘wrong results’’ to emphasize that only our approach based on the expressions (14), (15) guarantees that obtained interval-valued ideal solutions will be always attainable in the considered interval-valued decision matrix and this is in compliance with basics of classical TOPSIS method. As in (14) and (15) the minimal and maximal intervals must be chosen, the main difficulty in the implementati on of the above method is the problem of interval comparison. 3.2. The methods for interval comparison The problem of interval comparison is of perennial interest, be- cause of its direct relevance in practical modeling and optimization of real-world processes. To compare intervals, usually the quantitative indices are used (see reviews in Sevastjanov (2007) and Wang & Kerre (2001)). Wang, Yang, and Xu (2005) proposed a simple heuristic method which provides the degree of possibility that an interval is great- er/lesser than another one. For intervals B = [bL, bU], A = [aL, aU], the possibilities of B P A and A P B are defined in Wang et al. (2005) Wang, Yang, and Xu (2005) as follows: PðB P AÞ¼ maxf0; bU � aLg� maxf0; bL � aUg aU � aL þ bU � bL ; ð16Þ PðA P BÞ¼ maxf0; aU � bLg� maxf0; aL � bUg aU � aL þ bU � bL : ð17Þ The similar expression s were proposed earlier by Facchinett i, Ricci, and Muzzioli (1998) and by Xu and Da (2002). Xu and Chen (2008) showed that the expressions proposed in Facchine tti et al. (1998), Wang et al. (2005)
and Xu and Da (2002) are equivalent ones. A separate group of methods is based on the so-called probabi- listic approach to the interval comparis on (see review in Sevastja- nov (2007)). The idea to use the probability interpretation of interval is not a novel one. Nevertheles s, only in Sevastjanov (2007) the complete consistent set of interval and fuzzy interval relations involving separated equality and inequalit y relations develope d in the framewor k of probability approach is presented. Neverthel ess, the results of interval comparison obtained using expressions (16) and (17) generally are similar to those obtained with the use of probabilistic approach to the interval comparison. The main limitations of described above methods is that they provide an extent to which an interval is greater/l esser than an- other one if they have a common area (the intersection and inclu- sion cases should be considered separately (Sevastjanov (2007))). If there are no intersections of compared intervals, the extent to which an interval is greater/l esser than another one is equal to 0 or 1 regardles s of the distance between intervals. For example, Let A = [1, 2], B = [3, 4] and C = [100, 200]. Then using described above approach es we obtain: P(C > A) = P(B > A) = 1, P(A > B) = 0. Thus, we can say that in the case of overlapping intervals the above methods provide the possibility (or probabili ty) that an interval is greater/lesser than another one and this possibility (or probabili ty) can be treated as the strength of inequality or (in some sense) as the distance between compared intervals. On the other hand, the above methods can not provide the mea- sure of intervals inequality (distance) when they have no a com- mon area. Of course, the Hamming distance dH ¼ 1 2 ðjaL � bLjþ jaU � bUjÞ: ð18Þ or Euclidean distance dE ¼ 1 2 ððaL � bLÞ2 þðaU � bUÞ2Þ 1 2 ð19Þ can be used as the distance between intervals, but these distances give no information about which interval is greater /lesser. It can be seen that they can not be used directly for interval comparis on especially when an interval is included into another one. Therefore, here we propose to use directly the operation of interval subtracti on (Moore, 1966 ) instead of Hamming and Euclidean distances. This method makes it possible to calculate the possibilit y (or probabili ty) that an interval is greater/l esser that another one when they have a common area and when they do not intersect . So for intervals A = [aL, aU] and B = [bL, bU], the result of subtrac- tion is the interval C = A � B = [cL, cU]; cL = aL � bU, cU = aU � bL. It is easy to see that in the case of overlappi ng intervals A and B, we al- ways obtain a negative left bound of interval C and a positive right bound. Therefore, to get a measure of distance between intervals which additional ly indicate which interval is greater/l esser, we propose here to use the following value: DA�B ¼ 1 2 ððaL � bUÞþðaU � bLÞÞ: ð20Þ It is easy to prove that for intervals with common center, DA�B is al- ways equal to 0. Really, expression (20) may be rewritten as follows: DA�B ¼ 1 2 ðaL þ aUÞ� 1 2 ðbU þ bLÞ � � : ð21Þ We can see that expression (21) represen ts the distance between the centers of compared intervals A and B. This is not a surprising result as Wang et al. (2005) noted that most of the proposed meth- ods for interval comparison are ‘‘totally based on the midpoint s of interv al numbers’’. It easy to see that the result of subtrac tion of interv als with common centers is an interval centered around 0. In the framewo rk of interval analysis, such interval is treated as the interv al 0. Table 2 Results of interval comparison. Method 1 2 3 4 5 6 7 P(A P B) 0 0.06 0.22 0.5 0.78 1 1 P(A 6 B) 1 0.94 0.78 0.5 0.22 0 0 dE 10.82 10 7.81 6 7.81 10 10.82 dH 9 8 6 6 6 8 9 DA�B �9 �8 �5 0 5 8 9 L. Dymova et al. / Expert Systems with Applications 40 (2013) 4841–4847 4845 More strictly, if a is a real value, then 0 can be defined as a � a. Similarly, if A is an interval, then interval zero may be defined as an interval A � A = [aL � aU, aU � aL] which is centered around 0. Therefore, the value of DA�B equal to 0 for A and B having a com- mon center may be treated as a real-valued representat ion of inter- val zero. The similar situation we have in statistics. Let A and B be sam- ples of measure ments with corresponding uniform probability dis- tributions such that they have a common mean (meanA = meanB), but different variances (rA > rB). Then using statistical methods it is impossibl e to prove that the sample B is greater than the sample A or that the sample A is greater than the sample B. Taking into account the above considerati on we can say that the interval comparison based on the assumption that intervals having a common center are equal ones seems to be justified and reasonable. Obviously, the comparison of intervals based on comparison of their centers seems to be too simple. Nevertheles s, as it is shown above, this approach is based directly on conventional operation of interval subtraction. Therefore it is not a heuristic one. Moreover this method coincides better with common sense than more com- plicated known approach es. Let us consider two intervals A = [3, 5] and B = [1, 4]. Since aU > bU and aL > bL, then according to the Moore (1966) and common sense we have A > B. Since A and B are not identical and have no a common center there is no chance for A and B to be equal ones. Finally, according to common sense in this case the possibility of A < B should be equal to 0. There is no chance for B to be greater than A as a whole, although some point belong- ing to B in the common area of A and B may be greater than the points of A in this area. Nevertheles s, in our case from (16) and (17) we get PðB P AÞ¼ 15 and PðA P BÞ¼ 45. Thus, we can see that the known approaches may provide counterintu itive results. In Table 1, we present the values of P(A P B), P(B P A) (see expressions (16), (17)), the Hammin g dH and Euclidean dH dis- tances (see expressions (18), (19)) between Ai and B, and DAi�B for intervals A1 = [4, 7], A2 = [5, 8], A3 = [8, 11], A4 = [13, 16], A5 = [18, 21], A6 = [21, 24], A7 = [22, 25] and B = [7, 22] placed as it is shown in Fig. 1. The numbers in the first row in Table 1 corre- spond to the numbers of intervals Ai, i = 1 to 7. (see Table 2). We can see that the values of DAi�B are negative when Ai 6 B and become positive for Ai P B. These estimates coincide (at least qual- itatively) with P(Ai P B) and P(B P Ai). So we can say that the sign of DAi�B indicates which interval is greater/lesser and the values of absðDAi�BÞ may be treated as the distances between intervals since these values are close the to the values of dE and dH in both cases: Fig. 1. Compared intervals. when intervals have a common area and when the there is no such an area. 3.3. The comparison of the direct interval extension of TOPSIS method with the known method Using DA�B, it is easy to obtain from (14), (15) the ideal interval solutions Aþ ¼f½vþL1 ; v þU 1 �; ½v þL 2 ; v þU 2 �; . . . ; ½v þL n ; v þU n �g; A� ¼f½v�L1 ; v �U 1 �; ½v �L 2 ; v �U 2 �; . . . ; ½v �L n ; v �U n �g: As in the framewo rk of our approac h the distance between interval s A and B is presented by the value of DA�B, there is no need to use Hamming or Euclidean distances for calculatio n of Sþi and S � i . Since DA�B is the subtraction of the midpoints of A and B, the values of Sþi and S � i may be calculated as follows: Sþi ¼ 1 2 X j2K B ððvþLj þ v þU j Þ�ðv L ij þ v U ijÞÞþ 1 2 X j2KC ððv Lij þ v U ijÞ �ðvþLj þ v þU j ÞÞ: ð22Þ S�i ¼ 1 2 X j2K B ððv Lij þ v U ijÞ�ðv �L j þ v �U j ÞÞþ 1 2 X j2KC ððv�Lj þ v �U j Þ �ðv Lij þ v U ijÞÞ: ð23Þ Finally , using expression (13) we obtain the relativ e closeness RCi to the ideal alternativ e. Let us consider some illustrative examples. Example 1. Suppose we deal with three alternatives Ai, i = 1 to 3 and four local criteria Cj, j = 1 to 4 presented by intervals in Table 3, where C1 and C2 are benefit criteria, C3 and C4 are cost criteria. Suppose W = (0.25, 0.25, 0.25, 0.25). To stress the advantag es of our method, in this example many intervals representi ng the val- ues of ratings intersect . Then using the known method for interval extension of TOPSIS method Jahanshahlo et al. (2006) and Jahanshahlo o et al. (2009) (expressions (7)–(13)) we obtain R1 = 0.5311, R2 = 0.6378, R3 = 0.3290 and therefore R2 > R1 > R3, whereas with the use of our method (expressions 7, 8, 22, 23 and 13) we get R1 = 0.7688, R2 = 0.7528, R3 = 0.0717 and therefore R1 > R2 > R3. We can see that there is a considerable difference between the final ranking obtained by the known method and using our method based on the direct extension of TOPSIS method. This can be ex- plained by the fact that the method proposed in Jahansha hlo et al. (2006) and Jahanshahlo o et al. (2009) has some limitatio ns Table 3 Decision matrix. C1 C2 C3 C4 A1 [6, 22] [10, 15] [16, 21] [18, 20] A2 [15, 18] [8, 11] [20, 30] [19, 28] A3 [9, 13] 12, 17] [42, 48] [40, 49] Table 4 Decision matrix. C1 C2 C3 C4 A1 [6, 22] [10, 15] [13, 19] [40, 48] A2 [3, 4] [17, 21] [20, 30] [22, 28] A3 [25, 28] [8, 10] [42, 48] [18, 20] 4846 L. Dymova et al. / Expert Systems with Applications 40 (2013) 4841–4847 concerned with the presentation of intervals by real values in the calculation of ideal solutions and using the Euclidean distance when intervals intersect. Example 2. In this example, there are no intersecting intervals in the columns of decision matrix (see Table 4). As in the previous example, C1 and C2 are benefit criteria, C3 and C4 are cost criteria, W = (0.25, 0.25, 0.25, 0.25). Then using the known method proposed in Jahanshahlo et al. (2006) and Jahanshahloo et al. (2009), in this example we get R1 = 0.4825, R2 = 0.4984, R3 = 0.5413 and therefore R3 > R2 > R1. With the use of our method we obtain R1 = 0.5671, R2 = 0.4675, R3 = 0.4488 and therefore R1 > R2 > R3. Hence, we can conclude that even in the case when there are no intersecting intervals in the col- umns of interval valued decision table, the known method pro- posed in Jahanshahlo et al. (2006) and Jahanshahloo et al. (2009) and our methods may provide very different final rankings of alter- natives. That may be explained by the fact that in the method pro- posed in Jahanshahlo et al. (2006) and Jahanshahlo o et al. (2009), the real-valued ideal solutions are used, whereas in our method they are presented by intervals attainable in the interval-valued decision table. Nevertheles s, when there are no intersecting intervals in the columns of interval-valued decision table, the final ratings ob- tained by the method proposed in Jahanshahlo et al. (2006) and Jahanshahlo o et al. (2009) may coincide with those obtained using our method. Consider the illustrative examples. Example 3. In this example we will use the decision table presented in Table 4, where C1 and C2 are benefit criteria, C3 and C4 are cost criteria, but W = (0.5, 0.1, 0.25, 0.15). Then using the method from Jahanshahlo et al. (2006) and Jahanshahlo o et al. (2009) we obtain R1 = 0.4812, R2 = 0.2232, R3 = 0.5798 and therefore R3 > R1 > R2. Using our method we get R1 = 0.6653, R2 = 0.2758, R3 = 0.6363 and therefore R3 > R1 > R2. Thus, in this case two considered methods provide coincided ratings. Example 4. Let us consider the decision matrix presente d in Table 5, which differs from Table 4 by only one element [x31] so that [x31] intersects with [x11]. There are no other intersection s in the columns of Table 5. Using, as in previous case W = (0.5, 0.1, 0.25, 0.15), and the method from Jahanshahlo et al. (2006) and Jahansha hloo et al. (2009) we get R1 = 0.4812, R2 = 0.25322, R3 = 0.5798 and therefore R3 > R1 > R2 as in the previous example, whereas with the use of our method we obtain R1 = 0.6653, R2 = 0.2758, R3 = 0.6363 and therefore R1 > R3 > R2. Table 5 Decision matrix. C1 C2 C3 C4 A1 [6, 22] [10, 15] [13, 19] [40, 48] A2 [3, 4] [17, 21] [20, 30] [22, 28] A3 [12, 28] [8, 10] [42, 48] [18, 20] Thus, we can see that even the change of only one element in a decision matrix which leads to the appearance of intersecting intervals in the correspondi ng column, may lead to the significant changes in the results obtained by our method, whereas the known method (Jahanshahlo et al. (2006) and Jahansha hloo et al. (2009)) does not provide different final ratings of compared alternatives. 4. Conclusion The critical analysis of known approach es to the interval exten- sion of TOPSIS method is presente d. It is shown that these exten- sions are based on different heuristic approaches to definition of positive and negative ideal solutions. These ideal solutions are pre- sented by real values or intervals, which are not attainable in a decision matrix. Since this is in contradictio n with basics of classical TOPSIS method, a new approach to the solution of MCDM problems with the use of TOPSIS method in the interval setting is proposed. This method called ‘‘direct interval extension of TOPSIS method’’ is free of heuristic limitations of known methods concerned with the def- inition of positive and negative ideal solutions and using the Euclidean distance when intervals in a decision matrix intersect. The main advantage of the proposed method is that (opposite to the known methods) it provides interval-valued positive and neg- ative ideal solutions which in compliance with the basics of classi- cal TOPSIS method are always attainable in the interval-valued decision matrix. It is shown that the use of known methods may lead to the wrong results as well as the use of the Euclidean distance when intervals representi ng the values of local criteria intersect. Using numerical examples, it is shown that the proposed ‘‘direct interval extension of TOPSIS method’’ may provide the final ranking of alternativ es which is substantially different from the results ob- tained using the known methods especially when some interval- valued ratings in the columns of decision table intersect . References Chen, T.-Y. (2011). Interval-valued fuzzy TOPSIS method with leniency reduction and a experimental analysis. Applied Soft Computing, 11, 4591–4606. Facchinetti, G., Ricci, R. G., & Muzzioli, S. (1998). Note on ranking fuzzy triangular numbers. International Journal of Intelligent Systems, 13, 613–622. Garca-Cascales, M. S., & Lamata, M. T. (2012). On rank reversal and TOPSIS method. Mathematical and Computer Modelling, 56, 123–132. Hwang, CL., & Yoon, K. (1981). Multiple Attribute Decision Making–Methods and Applications. Berlin Heidelberg: Springer. Jahanshahlo, G. R., Hosseinzade, L. F., & Izadikhah, M. (2006). An algorithmic method to extend TOPSIS for decision making problems with interval data. Applied Mathematics and Computation, 175 , 1375–1384. Jahanshahloo, G. R., Hosseinzadeh Lotfi, F., & Davoodi, A. R. (2009). Extension of TOPSIS for decision-making problems with interval data:Interval efficiency. Mathematical and Computer Modelling, 49, 1137–1142. Jahanshahloo, G. R., Khodabakhshi, M., Hosseinzadeh Lotfi, F., & Moazami Goudarzi, M. R. (2011). A cross-efficiency model based on super-efficiency for ranking units through the TOPSIS approach and its extension to the interval case. Mathematical and Computer Modelling, 53, 1946–1955. Lai, Y. J, Liu, T. Y., & Hwang, C. L. (1994). TOPSIS for MODM. European Journal of Operational Research, 76, 486–500. Moore, R. E. (1966). Interval analysis . Englewood Cliffs., N.J.: Prentice-Hall. Sayadi, M. K., Heydari, M., & Shahanaghi, K. (2009). Extension o VIKOR method for decision making problem with interval numbers. Applied Mathematical Modelling, 33, 2257–2262. Sevastjanov, P. (2007). Numerical methods for interval and fuzzy number comparison based on the probabilistic approach and Dempster–Shafer theory. Information Sciences, 177 , 4645–4661. Tsaur, R.-C. (2011). Decision risk analysis for an interval TOPSIS method. Applied Mathematics and Computation, 218 , 4295–4304. Wang, Y. M., & Elhag, T. M. S. (2006). Fuzzy TOPSIS method based on alpha level sets with an application to bridge risk assessment. Expert Systems with Applications, 31, 309–319. Wang, X., & Kerre, E. E. (2001). Reasonable properties for the ordering of fuzzy quantities (I) (II). Fuzzy Sets and Systems, 112 , 387–405. Wang, Y. M., & Luo, Y. (2009). On rank reversal in decision analysis. Mathematical and Computer Modelling, 49, 1221–1229. L. Dymova et al. / Expert Systems with Applications 40 (2013) 4841–4847 4847 Wang, Y. M., Yang, J. B., & Xu, D. L. (2005). A preference aggregation method through the estimation of utility intervals. Computers and Operations Research, 32, 2027–2049. Wang, Y. M., Yang, J. B., & Xu, D. L. (2005). A two-stage logarithmic goal programming method for generating weights from interval comparison matrices. Fuzzy Sets and Systems, 152 , 475–498. Xu, Z., & Da, Q. (2002). The uncertain OWA operator. International Journal of Intelligent Systems, 17, 569–575. Xu, Z., & Chen, J. (2008). Some models for deriving the priority weights from interval fuzzy preference relations. European Journal of Operational Research, 184 , 266–280. Ye, F., & Li, Y. N. (2009). Group multi-attribute decision model to partner selection in the formation of virtual enterprise under incomplete information. Expert Systems with Applications, 36, 9350–9357. Yue, Z. (2011). An extended TOPSIS for determining weights of decision makers with interval numbers. Knowledge-Based Systems, 24, 146–153. A direct interval extension of TOPSIS method 1 Introduction 2 The basics of TOPSIS method and known approaches to its interval extension 3 A new approach to the interval extension of TOPSIS method 3.1 The problem formulation 3.2 The methods for interval comparison 3.3 The comparison of the direct interval extension of TOPSIS method with the known method 4 Conclusion References