The LTOPSIS: An alternative to TOPSIS decision-making approach for linguistic variables

Expert Systems with Applications 39 (2012) 2119–2126

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

The LTOPSIS: An alternative to TOPSIS decision-making approach for linguistic variables Elio Cables a, M. Socorro García-Cascales b,⇑, M. Teresa Lamata c a

Dpto. Informática, Universidad de Holguín, Oscar Lucero Moya, Holguín, Cuba Dpto. Ciencias de la Computación e Inteligencia Artificial, Universidad de Granada, 18071 Granada, Spain c Dpto. de Electrónica, Tecnología de Computadoras y Proyectos, Universidad Politécnica de Cartagena, Murcia, Spain b

a r t i c l e

i n f o

a b s t r a c t

Keywords: TOPSIS method Fuzzy numbers Linguistic variables

This paper develops an evaluation approach based on the Technique for Order Performance by Similarity to Ideal Solution (TOPSIS). When the input for a decision process is linguistic, it can be understood that the output should also be linguistic. For that reason, in this paper we propose a modification of the TOPSIS algorithm which develops the above idea and which can also be used as a linguistic classifier. In this new development, modifications to the classic algorithm have been considered which enable linguistic outputs and which can be checked through the inclusion of an applied example to demonstrate the goodness of the new model proposed. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) approach is a method for the arrangement of ratings to an ideal solution by similarity. The TOPSIS approach was developed by (Hwang & Yoon, 1981), and improved by the same authors in 1987 and 1992. Lai, Liu, and Hwang (1994) and Zeleny (1982) and many other researchers have also worked on this theme. Some examples using the fuzzy set theory can be seen in Braglia, Frosolini, and Ontanari (2003), Chu (2002a, 2002b), Jahanshsloo, Hosseinzadeh, and Izadikhah (2006), Kelemenis and Askonus (2010) and Garcia-Cascales and Lamata (2009a). Most of the time the decision-maker is not able to define the importance of the criteria or the goodness of the alternatives with respect to each criterion in a numeric way. In many situations, we use measures or quantities which are not exact but approximate. In these situations, a more realistic approach may be to use linguistic assessments instead of numerical values, that is, to suppose that the ratings and/or weights of the criteria are assessed by means of linguistic variables. It is well known that fuzzy sets have been employed in handling inexact and vague information, since they can employ natural languages in terms of linguistic variables. Aristoteles explained that a sign of a well-trained mind was to not seek to find greater accuracy than that which the nature of the problem allows. Taking this assertion into account, our discussion will be focused on developing a model, the TOPSIS model, in such a way that both the inputs and the outputs are linguistic terms. In the classical TOPSIS method, the performance ratings and the weights of the criteria are given as real values, with the outputs being an index, whose value belongs to the interval [0, 1]. In this article we seek to not only obtain a ranking but also the possibility

In multiple criteria decision analysis, a number of alternatives have to be evaluated and compared using several criteria. The aim of is to provide support to the decision-makers in the process of making the choice among different options. In this way, practical problems found in business, services or manufacturing are often characterized by several conflicting criteria, and there may be no solution which satisfies all the criteria simultaneously, that is to say, that there is no one decision which is the best for all the criteria. Thus, the solution is a compromise solution according to the decision-maker’s preferences. MCDA has been an area of very rapid growth in recent decades. These techniques can be used to identify a single preferred option; to rank options; or to list a limited number of alternatives for subsequent evaluation. These decision problems involve six components (Keeney & Raiffa, 1976): A goal or a set of goals the decision-maker seeks to achieve. A set of criteria. The set of decision alternatives. The set of weights associated with the criteria. The set of outcomes or consequences associated with each alternative/criteria pair.  The decision-maker or group of decision-makers involved in the decision making process with their preferences.     

⇑ Corresponding author. Address: Dpto. de Electrónica, Tecnología de Computadoras y Proyectos, Universidad Politécnica de Cartagena, Murcia, Spain. Tel.: +34 968 326 574; fax: +34 968 326 400. E-mail address: [email protected] (M.S. García-Cascales). 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.07.119

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Fig. 1. Structure of the system.

to obtain a linguistic output and thus treat the TOPSIS method as a classifier. To do so we shall employ the input of data to the system by means of linguistic variables which will be modeled as fuzzy numbers, which will then be treated using the TOPSIS method, obtaining as a result not only a numerical ranking but also a linguistic output, as can be appreciated in Fig. 1. The paper is organized as follows: In the following section a literature review is carried out of the different developments of the TOPSIS method. Section 3 introduces the linguistic variables and the fuzzy sets are described. In Section 4, the framework for TOPSIS evaluation and the modifications for linguistic variables are detailed. Section 5 examines an illustrative example. The final section outlines the most important conclusions. 2. Literature review In recent years, diverse papers have appeared in the literature in distinct applied fields utilizing the TOPSIS method as the multi-criteria decision making method, either on its own or in a hybrid form with other methodologies, and thus we present some of the most noteworthy communications published recently. A wide variety of publications exist in which fuzzy logic has been employed together with the TOPSIS method in order to manage uncertainty or lack or accuracy in different applications. Therefore, we have a number of examples such as in the case of Kahraman, Cevik, Ates, and Gulbay (2007a) who apply the fuzzy TOPSIS method for the evaluation of industrial robotic systems. Kahraman, Ates, Cevik, Gulbay, and Erdogan (2007b) also propose a fuzzy TOPSIS approach to resolve a problem in logistic information technology. The total quality management consultant selection under fuzzy environment is viewed in Saremi, Mousavi, and Sanayei (2009) and the applications in aggregate planning in Wang and Liang (2004); whereas in Wang and Chang (2007), the application is related with the Air Force Academy in Taiwan to evaluate the initial training aircraft. Sun and Lin (2009) develop a fuzzy TOPSIS method for evaluating the competitive advantages of shopping websites. Mobile telephone alternatives are studied in Isßıklar and Büyüközkan (2007). In Tansel and Yurdakul (2010) a quick credibility scoring decision support system is developed for banks to determine the credibility of manufacturing firms in Turkey, the proposed credit scoring model is based on financial ratios and the fuzzy TOPSIS approach. A new fuzzy TOPSIS for evaluating alternatives by integrating using subjective and objective weights is developed in Wang and Lee (2009). Similarly, there is a wide range of literature on group decisions in which the TOPSIS method is used in group in conjunction with fuzzy logic. Examples of this can be seen in Chen (2000) and Chu (2002a, 2002b), who give the extension for group decision, the former for solving supplier selection problems in a fuzzy environment and the latter for problems in location selection. Fan and Liu (2010) propose a method to solve the group decision-making problem with multi-granularity uncertain linguistic information with an appropriate extension of the classical TOPSIS to a group fuzzy TOPSIS.

Other authors have utilized an AHP method to determine the importance weights of the criteria, and TOPSIS to obtain the performance ratings of the alternatives. This hybrid approach is used by Tsaur, Chang, and Yen (2002) to evaluate airline service quality, by Garcia-Cascales, Lamata, and Verdegay (2007a) and Garcia-Cascales and Lamata (in press) for the best parts cleaning system in an engine factory, Yurdakul and Tansel (2005) developed a performance model for manufacturing companies, Lin, Wang, Chen, and Chang (2008) integrate AHP and TOPSIS approaches into the customer-driven product design process and Buyukozkan and Ruan (2007) combine both e-government and website quality assessment methodologies to improve the evaluation phase and include all aspects related to service quality through the website. Also, Gharehgozli, Rabbani, Zaerpour, and Razmi (2008) work with this methodology in the acceptance/rejection of incoming orders, Ertugrul and Karakasoglu (2009) used the methodology in the evaluation of Turkish cement firms and Amiri (2010) utilized the methodology in project selection for oil-fields. It is also possible to find other hybrid methodologies in the literature such as Celik, Kandakoglu, and Deha (2009) which combines SWOT (strengths, weaknesses, opportunities and threats) with fuzzy AHP and fuzzy TOPSIS for a systematic decision aid mechanism which could be adopted into the official recruitment procedures of academic administrations. Amiri, Zandieh, Soltani, and Vahdani (2009) present a hybrid multi-criteria decision-making model to evaluate the competence of the firms with an adaptative AHP approach with the use of interval data and TOPSIS method. Chen and Chen (2010) present a conjunctive multi-criteria decision-making approach based on decision-making trial and evaluation laboratory DEMATEL, fuzzy analytic network process FANP and TOPSIS as an innovations support system for Taiwanese higher education. Finally, it is possible to find papers in the literature which compare TOPSIS and VIKOR approaches. In this sense, we emphasize the works of Opricovic and Tzeng (2004, 2007) and Chu, Shyu, Tzeng, and Khosla (2007).

3. Linguistic variable and fuzzy sets 3.1. Linguistic variable Natural language to express perception or judgement is always subjective, uncertain, or vague. Since words are less precise than numbers, the concept of a linguistic variable approximately characterizes phenomena which are poorly defined to be described with conventional quantitative terms (Delgado, Verdegay, & Vila, 1992, 1993; Herrera & Herrera-Viedma, 2002). The concept of a linguistic variable is very useful in dealing with situations which are too complex or not well defined to be reasonably described in conventional quantitative expressions (Zimmermann, 1996), where fuzzy numbers are introduced to appropriately express linguistic variables. To resolve the vagueness, ambiguity, and subjectivity of human judgement, fuzzy sets theory was introduced to express the linguistic terms in decision-making processes. Bellman and

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Zadeh (1970) were the first researchers to consider the decisionmaking problem using fuzzy sets. In general, for the decision-maker it is easier when he/she evaluates his/her judgements by means of linguistic terms. In those cases, the concept of fuzzy number is more adequate than that of real number. We are particularly interested in the role of linguistic variables as an ordinal scale and their associated terms, which in this case will be modeled by means of a triangular fuzzy number, and which will be used (treated) in the multi-criteria decision-making. Then we have identified the linguistic variable with a fuzzy set (Bellman & Zadeh, 1970; Kacprzyk & Yager, 2001; Kerre, 1982).

We observe that in this case, the vocabulary consists of seven linguistic terms. The relation ‘‘linguistic term/fuzzy number’’ is associated to the perception that the decision-maker has of the reality. This relation can be consulted in Garcia-Cascales and Lamata (2007b). An example of the semantic rule that such an association establishes is:

Definition 1. A linguistics variable is characterized by the following five elements: {X; T(X); U; G; M} where:

3.2. Fuzzy sets

    

X is the name of the variable, T(X) is the finite sets of terms of X (the set of linguistic values), U is the universe of discourse, (G) is the syntactic rule that generates T(X) elements, and M is the semantic rule which associates a fuzzy number with each of the linguistic terms of X.

On most occasions, the decision-maker is unable to define the importance of the criteria or the quality of the alternatives with regard to each criterion in a conclusive manner. When problems are presented in which it is necessary to evaluate phenomena related to perceptions and relations between human beings (esthetics, taste, goodness, etc.), words from natural language are usually used instead of numerical values to issue valuations. In the example developed in epigraph 5 and which will be employed to evaluate the method; we will utilize the variable ‘‘Goodness’’ as the linguistic labels for the evaluation of the alternatives, which is represented in Fig. 2.

Medium poor ! ½2:9; 4:1; 4:2 Good ! ½6:8; 7:5; 7:9 Below, we introduce the ideas of fuzzy sets related with the terms of the ‘‘vocabulary’’ associated to the linguistic variable.

Definition 1. A real fuzzy number A is described as any fuzzy subset of the real line R with membership function fA which processes the following properties:

(a) fA(x) is a continuous mapping fromR to the closed interval [0, 1]; (b) fA(x) = 0, for all x 2 (1, a]; (c) fA(x) is strictly increasing on [a, b]; (d) fA(x) = w, for all x 2 [b, c]; (e) fA(x) is strictly decreasing on [c, d]; (f) fA(x) = 0, for all x 2 (d, 1], where a, b, c, d are real numbers (Zadeh, 1965). Definition 2. The fuzzy number A will be triangular if its membership function is given in Fig. 3, where a, b and c are real numbers. Then we designate A = (a, b, c) as the triangular fuzzy number. In this formula, c and a are the upper and lower values of the support of A, respectively, and b is the median value of A.

Fig. 2. Example of the linguistic variable ‘‘Goodness’’.

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Fig. 3. Triangular fuzzy number.

The basic theory of the triangular fuzzy number (TFN) is described in Dubois and Prade (1980), Kaufmann and Gupta (1991), Klir and Folger (1988) and Klir and Yuan (1995), where a fuzzy number is considered as a convex and normalized fuzzy set. Starting from Fig. 1 and taking into account the aggregation processes which will be undertaken for each alternative we will have to apply the steps outlined in Fig. 4. Given a set of words of a natural language which define the performance of an alternative for each of the criteria and which we note by a vector T(X)n, we need to also obtain a linguistic output T(X), which is the output of the aggregation process of the n values. For this reason, and according to Zadeh (1975), we associate the fuzzy representation to each word, Fig. 2 given by the semantic rule; in this case obtained directly from the decision-maker, as can be seen in Garcia-Cascales and Lamata (2007b). In this sense and with regard to the fuzzy numbers, we will show only the mathematical operations that will be used throughout the development of the paper. Definition 3. If T1 and T2 are two TFN defined by the triplets (a1, b1, c1) and (a2, b2, c2), respectively. For this case, the necessary arithmetic operations with positive fuzzy numbers are:

(a) Addition

T 1  T 2 ¼ ½a1 þ a2 ; b1 þ b2 ; c1 þ c2 

ð2Þ

(b) Scalar multiplication

k  T 1 ¼ ðk  a1 ; k  a2 ; k  a3 Þ

ð3Þ

(c) Maximum

MaxðT 1 ; T 2 Þ ¼ ½Maxða1 ; a2 Þ; Maxðb1 ; b2 Þ; Maxðc1 ; c2 Þ

Fig. 4. Scheme showing the transition in the aggregation process.

(d) Minimum

MinðT 1 ; T 2 Þ ¼ ½Minða1 ; a2 Þ; Minðb1 ; b2 Þ; Minðc1 ; c2 Þ

ð5Þ

(e) Distance

dðT 1 ; T 2 Þ ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ½ða1  a2 Þ2 þ ðb1  b2 Þ2 þ ðc1  c2 Þ2  3

ð6Þ

4. Existing framework for TOPSIS evaluation The TOPSIS method is particularly useful for those problems in which the valuations of the alternatives on the basis of the criteria are not represented in the same units. For example, if in the purchase of a car we include as decision criteria the price and the power of the engine in horsepower, we must take into amount the fact that the units to measure these are not the same, thus we could consider that for a medium range car the price would be between approximately 12,000 and 30,000 Euros, whilst the horsepower may be between 90 and 130. Therefore, although both are numerical we cannot operate with them as they are, since they are not comparable. Thus the obligatory step is to normalize them. TOPSIS is a well-known classical method for MCDA which was developed in Hwang et al. (1981). The principle steps of this method are: Step 1. Decision matrix construction. Step 2. Normalized decision matrix construction. Step 3. Weighted normalized decision matrix construction. Step 4. Determining the positive and negative ideal solutions. Step 5. Calculating the distance of each alternative to the positive and negative ideal solutions.  Step 6. Calculating the relative proximity.  Step 7. Ranking the preference order.     

ð4Þ The main idea of the TOPSIS method is represented in Fig. 5, where we would have to evaluate the seven alternatives (A, B, C, D, E, F, G), and where, for questions related to the graphical representation, we only consider two criteria. Thus, the solution is based upon the concept that the chosen alternative should have the shortest distance from the positive ideal solution, ( PIS) = {MaxC1, MaxC2}, in our case (C, D, E, F) and the farthest from the negative ideal solution, (NIS) = {MinC1, MinxC2} in our case (A, B, C, G). In this way, between all those alternatives which are at the same distance from the NIS, we will choose the alternative that moves farthest away from the straight line that joins PIS with NIS, in our case, C > B > G > A. On the contrary, amongst all the

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F2 = (P, . . . , P), . . . , F7 = (VG, . . . , VG) when VP = [0, 1, 1.9], . . . , VG=[7.9, 9, 10]. Each fictitious variable will be a vector of dimension m with all the values the same and the same as the linguistic term they represent. Step 4: Identify the Fuzzy Positive Ideal Solution FPIS (Aþ i ; i = 1, 2, . . . , m) which is made of all the best performance scores

 þ ¼ fv~ þ ; v~ þ ; :::; v~ þ g ¼ fMaxðv~ ij Þg; A 1 2 m

i ¼ 1; 2; . . . ; m

ð8Þ

Each one of the components in the m-dimensional vector is 1 2 3 ~þ determined as v i ¼ ðv i ; v i ; v i Þ. Fig. 5. PIS and NIS representation.

Proposition 1. When considering the alternatives from the previous step it is verified by the construction itself that

alternatives which have the same best PIS, in the TOPSIS sense the best will be the one which is nearest to the straight line PIS_NIS, that is to say C > E > D > F. 4.1. The proposed LTOPSIS algorithm When considering linguistic rather than numerical valuations, we need to operate a function with them in which each linguistic term is associated a fuzzy number. To do so it will be necessary to consider modifying the algorithm, taking this premise into account. Therefore, the new fuzzy TOPSIS algorithm for linguistic variables (LTOPSIS) consists of the following steps: Step 1: Identify the evaluation criteria and the appropriate linguistic variables for the importance weight of the criteria and determine the set of feasible alternatives with the linguistic score for alternatives in terms of each criterion.

Aþ ¼ fv~ þ1 ; v~ þ2 ; . . . ; v~ þm g ¼ fTðXÞmax ; TðXÞmax ; . . . ; TðXÞmax g

ð9Þ

where T(X)max represents the greatest of the linguistic terms in the vocabulary. And the Fuzzy Negative Ideal Solution (FNIS) (A i ; j = 1, 2, . . . , n), is made of all the worst performance scores

  ¼ fv~  ; v~  ; :::; v~  g ¼ fMinðv~ ij Þg; A 1 2 m

i ¼ 1; 2; . . . ; m

ð10Þ

Each one of the components in the m-dimensional vector is deter1 2 3 mined as v~  i ¼ ðv i ; v i ; v i Þ. Proposition 2. When considering the alternatives from the previous step it is verified by the construction itself that

A ¼ fv~ 1 ; v~ 2 ; . . . ; v~ m g ¼ fTðXÞmin ; TðXÞmin ; . . . ; TðXÞmin g

ð11Þ

where T(X)min represents the lowest of the linguistic terms in the vocabulary. Step 5: It is necessary to obtain the distance of each alternative to the FPIS, that is to say, to (T(X)max) and to the FNIS (T(X)min) We shall use the two-multidimensional Euclidean distance adapted to the case of fuzzy numbers, according to (6). þ

So, the distance of an alternative to the ideal solution di , þ

di ¼ where each representing the ratings are fuzzy numbers associated with the linguistic terms; Ai, i = 1, 2, . . . , m, represent the possible alternatives; and Cj, j = 1, 2, . . . , n represent the criteria. Remark: If we consider the TOPSIS algorithm, the following step would be the normalization of the matrix, but when utilizing linguistic terms for the evaluation of the alternatives with respect to all the criteria this step is unnecessary. ~ ij  is calculated Step 2: The weighted decision matrix V ¼ ½v ~ kij ¼ ðv 1ij ; v 2ij ; v 3ij Þ. The using Eq. (7), having as representation v weight of the criterion j is represented by wj in Eq. (7):

v~ ij ¼ wj  n~kij ;

j ¼ 1; 2; . . . ; m; k ¼ 1; 2; 3

ð7Þ

where wj is the weight of the jth attribute or criterion, and such P that 1 ¼ nj¼1 wj . Step 3: Introduction of fictitious alternatives. The same number of alternatives as there are in the set of linguistic terms will be added to the set of valuations of the alternatives for each of the criteria. These variables are considered as ‘‘fictitious variables’’ and there will be as many as there are linguistic terms defined as the vocabulary, thus for the example which is developed in epigraph 5 it will have F1 = (VP, . . . , VP),

n X

dðv~ ij ; v~ þj Þ;

i ¼ 1; . . . ; m

ð12Þ

j¼1 

and from the fuzzy negative ideal solution di , 

di ¼

n X

dðv~ ij ; v~ j Þ;

i ¼ 1; . . . ; m

ð13Þ

j¼1

with the results being real numbers. Step 6: The ranking score Ri is calculated using Eq. (14). The ranking scores obtained represent the alternatives’ performance achievement within their status. A higher score corresponds to a better performance 

Ri ¼

 di

di þ; þ di

i ¼ 1; . . . ; m

ð14Þ

If Ri = 1 ? Ai ¼ Aþ . If Ri = 0 ? Ai ¼ A . where the Ri value lies between 0 and 1. The closer the Ri = 1 value implies a higher priority of the ith alternative. Step 7: Rank the preference order and the linguistic output. Once Ri has been obtained, we should obtain the linguistic output.

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To do this, the numerical output of each alternative is associated to the linguistic variable to which it is closest, that is to say, which is the shortest distance from it. So, we can choose any distance, for example the Manhattan distance for its simplicity

dðAi ; Lj Þ ¼ jAi  Lj j

ð15Þ

5. Illustrative Example To illustrate the procedure we will use the problem presented in Garcia-Cascales and Lamata (in press). In this, we study a decision-making maintenance problem in an engine factory that specializes in the production, sale, and maintenance of four-stroke, high and medium-speed diesel engines for naval applications, tank propulsion, electricity generating plants, turnkey cogeneration plants, and rail traction. One of the most important steps that should be carried out in a maintenance process and in the engine reparation is the cleaning Table 1 Definition of the linguistic labels. Linguistic variable

Legend

Fuzzy numbers

VP P MP F MG G VG

Very poor Poor Medium poor Fair Medium good Good Very good

[0, 1, 1.9] [1.9, 2.5, 2.9] [2.9, 4.1, 4.2] [4.2, 4.8, 5] [5, 5.8, 6.8] [6.8, 7.5, 7.9] [7.9, 9, 10]

Fig. 6. Decision matrix.

of each and every component. The testing process and reconditioning of every component requires that every piece has a high quality cleaning; if not, the reparation process will not be appropriate. The problems concerned with this are that there are pieces with diverse degrees of dirt and with very different geometry, and a work process that demands speed and flexibility. The global objective of the problem is to decide which the best system for cleaning the pieces is. We are dealing with a problem characterized by the following components: (1) Objective: Choose the best cleaning system (2) Alternatives:  A1: Conventional cleaning  A2: Chemical cleaning  A3: Ultrasonic cleaning (3) Criteria:  C1: Total annual operation cost  C2: System productivity  C3: System load capacity  C4: Cleaning efficiency  C5: Harmful effects In this way, we suppose a problem with one decision-maker (the assembly workshop manager), five criteria Cj, and three alternatives Ai, In our case, let us assume that the decision-maker uses the linguistic score set S for the rating on each qualitative criterion respectively. S = {VP, P, MP, F, MG, G, VG}, where T(X)max = VG > G > MG > F > MP > P > VP = T(X)min. As has been seen above in Section 3, fuzzy sets have been employed in handling inexact and vague information because of their capability to use natural languages in terms of linguistic variables. To obtain the fuzzy values associated with the linguistic terms the procedure developed in Garcia-Cascales and Lamata (2007b) will be used, as mentioned above, with a scale which varies in the interval [0, 10] and the result of which can be seen in Table 1. The final result to evaluate the suitability of each alternative under each of the criteria can be seen in Fig. 6. The weights for each of the criteria utilized were found by means of Analytic Hierarchy Process (Garcia-Cascales & Lamata, 2009a).

Table 2 The values of the alternatives and the fictitious alternatives are represented the corresponding weighted values and the FIS and PIS. C1 0.3461

C2 0.2975

C3 0.0686

C4 0.1812

C5 0.1066

A1 A2 A3

(5.0, 5.8, 6.8) (5.0, 5.8, 6.8) (4.2 , 4.8, 5.0)

(5.0, 5.8, 6.8) (4.2, 4.8, 5.0) (7.9, 9.0, 10.0)

(4.2, 4.8, 5.0) (7.9, 9.0, 10.0) (5.0, 5.8, 6.8)

(4.2, 4.8, 5.0) (5.0, 5.8, 6.8) (5.0, 5.8, 6.8)

(5.0, 5.8, 6.8) (0.0, 1.0, 1.9) (5.0, 5.8, 6.8)

VP p MP F MG G VG

(0.0, 1.0, 1.9) (1.9, 2.5, 2.9) (2.9, 4.1, 4.2) (4.2, 4.8, 5.0) (5, 5.8, 6.8.0) (6.8, 7.5, 7.9) (7.9, 9.0, 10.0)

(0.0, 1.0, 1.9) (1.9, 2.5, 2.9) (2.9, 4.1, 4.2) (4.2, 4.8, 5.0) (5, 5.8, 6.8.0) (6.8, 7.5, 7.9) (7.9, 9.0, 10.0)

(0.0, 1.0, 1.9) (1.9, 2.5, 2.9) (2.9, 4.1, 4.2) (4.2, 4.8, 5.0) (5, 5.8, 6.8.0) (6.8, 7.5, 7.9) (7.9, 9.0, 10.0)

(0.0, 1.0, 1.9) (1.9, 2.5, 2.9) (2.9, 4.1, 4.2) (4.2, 4.8, 5.0) (5, 5.8, 6.8) (6.8, 7.5, 7.9) (7.9, 9.0, 10.0)

(0.0, 1.0, 1.9) (1.9, 2.5, 2.9) (2.9, 4.1, 4.2) (4.2, 4.8, 5.0) (5, 5.8, 6.8) (6.8, 7.5, 7.9) (7.9, 9.0, 10.0)

v A1 v A2 v A3

(1.73, 2.00, 2.35) (1.73, 2.00, 2.35) (1.45, 1.66, 1.73)

(1.48, 1.72, 2.02) (1.25, 1.43, 1.48) (2.35, 2.67, 2.97)

(0.28, 0.33, 0.34) (0.54, 0.62, 0.68) (0.34, 0.39, 0.46)

(0.76, 0.87, 0.90) (0.90, 1.05, 1.23) (0.90, 1.05, 1.23)

(0.53, 0.62, 0.72) (0.00, 0.10, 0.20) (0.53, 0.62, 0.72)

vVP vP vMP vF vMG vG vVG

(0.00, 0.34, 0.66) (0.66, 0.86, 1.01) (1.01, 1.42, 1.45) (1.45, 1.66, 1.73) (1.73, 2.01, 2.35) (2.35, 2.59, 2.73) (2.73, 3.11, 3.46)

(0.00, 0.29, 0.56) (0.56, 0.74, 0.86) (0.86, 1.22, 1.25) (1.25, 1.43, 1.49) (1.49, 1.72, 2.02) (2.02, 2.23, 2.35) (2.35, 2.68, 2.97)

(0.00, 0.07,0.13) (0.13, 0.17, 0.20) (0.20, 0.28, 0.29) (0.29, 0.33, 0.34) (0.34, 0.40, 0.46) (0.46, 0.51, 0.54) (0.54, 0.62, 0.68)

(0.00, 0.18, 0.34) (0.34, 0.45, 0.52) (0.52, 0.74, 0.76) (0.76, 0.87, 0.91) (0.91, 1.05, 1.23) (1.23, 1.36, 1.43) (1.43, 1.63, 1.81)

(0.00, 0.11, 0.20) (0.20, 0.26, 0.31) (0.31, 0.44, 0.45) (0.45, 0.51, 0.53) (0.53, 0.62, 0.72) (0.72, 0.80, 0.84) (0.84, 0.96, 1.06)

Aþ

(2.73, 3.11, 3.46)

(2.35, 2.68, 2.97)

(0.54, 0.62, 0.68)

(1.43, 1.63, 1.81)

(0.84, 0.96, 1.06)

A

(0.00, 0.34, 0.66)

(0.00, 0.29, 0.56)

(0.00, 0.07, 0.13)

(0.00, 0.18, 0.34)

(0.00, 0.11, 0.20)

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E. Cables et al. / Expert Systems with Applications 39 (2012) 2119–2126 Table 3 Numerical and linguistic results. Positive distance +

d A1 d+A2 d+A3 d+VP d+P d+MP d+F d+MG d+G d+MG

Negative distance 1.6761 1.9596 1.6395 4.0571 3.3133 2.6541 2.1807 1.5721 0.7945 0.0000



d A1 dA2 dA3 dVP d P dMP d F dMG d G dMG

Relative proximity 2.4071 2.2783 2.9157 0.0000 0.7438 1.4031 1.8764 2.4850 3.2626 4.0571

RA1 RA2 RA3 RVP RP RMP RF RMG RG RMG

With this information, the ratings in the decision matrix and the weights of the criteria are as follows: To this matrix, we apply the L-TOPSIS-algorithm. As already commented, in this paper we consider that the corresponding step of normalization is not necessary because all the ratings have the same reference set. For the determination of the Aþ and A , we take the  kij 8k ¼ 1; 2; 3; i = 1, 2, . . . , n for v þ  kij 8 maxi v and the mini v j  k ¼ 1; 2; 3; i = 1, 2, . . . ,n for v j . According to Proposition 1 we see how Aþ ¼ v VG; that is to say, the best of the labels, whilst taking into account proposition 2A = vVP, that is to say it would correspond to the worst of the linguistic terms. All these operations are reflected in Table 2. In Table 3 we can see the results for d+, d and R, as well as the linguistic outputs. From these it can be deduced that the third alternative would be the best. The order would be given by A3 > A2 > A1. In the linguistic valuations, although they are Medium good for the three, it is observed that the third would be in the interval [Medium good–Good], while the first two would be behind, in the [Fair–Medium good], and thus the best option will be A3. 6. Conclusions In a decision process we can find that the information to evaluate may be linguistic and/or numerical. To resolve this problem, one possible option is to work with the TOPSIS method. The classic TOPSIS method, as has been commented, provides output results by means of an index with values in a rank of [0, 1], when working with problems in which only information of a linguistic nature is available, it is interesting that both the input and the output of the data are in the same terms. In this way we propose a modification of the TOPSIS method which we have denominated LTOPSIS. This new method offers an important advantage over the classic method. This advantage is the possibility for the output of the system to be the same as the input, that is to say, linguistic. The algorithm will also provide us with the option of treating it as a linguistic classifier. To this end, two changes have been introduced into the algorithm proposed by Hwang et al. (1981). First, the normalization has been eliminated since it is not necessary, as all the values considered are within the same rank; and secondly the fictitious alternatives corresponding to the vocabulary terms have been introduced. Moreover, we have presented a practical example in order to show the goodness of the new algorithm proposed. Acknowledgements This work is partially supported by the DGICYT and Junta de Andalucía under Projects TIN2008-06872-C04-04, TIN201127696-Co2-01 and (P07-TIC02970), respectively.

0.5895 0.5376 0.6401 0.0000 0.1833 0.3458 0.4625 0.6125 0.8042 1.0000

Ranking score

Output

2 3 1

Fair–Medium good Fair–Medium good Medium good–Good Very poor Poor Medium poor Fair Medium good Good Very good

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