The Hellinger distance in Multicriteria Decision Making: An illustration to the TOPSIS and TODIM methods

Expert Systems with Applications 41 (2014) 4414–4421

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

The Hellinger distance in Multicriteria Decision Making: An illustration to the TOPSIS and TODIM methods Rodolfo Lourenzutti a,⇑, Renato A. Krohling b,1 a

Graduate Program in Computer Science, PPGI, UFES Federal University of Espirito Santo, Av. Fernando Ferrari, 514, CEP 29075-910 Vitória, Espírito Santo, ES, Brazil Department of Production Engineering & Graduate Program in Computer Science, PPGI, UFES Federal University of Espirito Santo, Av. Fernando Ferrari, 514, CEP 29075-910 Vitória, Espírito Santo, ES, Brazil b

a r t i c l e

i n f o

Keywords: Hellinger distance Multicriteria Decision Making (MCDM) Probability distribution Stochastic dominance degree TOPSIS TODIM

a b s t r a c t Due to the difficulty in some situations of expressing the ratings of alternatives as exact real numbers, many well-known methods to support Multicriteria Decision Making (MCDM) have been extended to compute with many types of information. This paper focuses on the information represented as probability distribution. Many of the methods that deal with probability distribution use the concept of stochastic dominance, which imposes very strong restrictions to differentiate two probability distributions, or uses the probability distributions to obtain a quantity that will be used to rank the alternatives. This paper brings the Hellinger distance concept to the MCDM context to assist the models to deal with probability distributions in a direct way without any transformation. Transformations in the data or summary quantities may miss represent the original information. For direct comparisons among probability distributions we use the stochastic dominance degree (SDD). We illustrate how simple it can be to adapt the existing methods to deal with probability distributions through the Hellinger distance and SDD by adapting the TOPSIS and TODIM (an acronym in Portuguese of Interactive and Multicriteria Decision Making) methods. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Multicriteria Decision Making (MCDM) problems have becoming increasingly complicated over the years. It is necessary to consider plenty of alternatives and criteria which make the problems easily overwhelming to human reasoning. Methods to support decision making are now essential and a lot of efforts has been made in the past few decades to advance the field. The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), proposed by Hwang and Yoon (1981), is one of the most used methods to support MCDM. The main idea of TOPSIS is that the solution should be as far as possible from the worst possible solution and as close as possible from the best possible solution. This method is quite simple and intuitive, presenting a satisfactory performance in many applications. For a broad survey about the TOPSIS method we refer the interest reader to Behzadian, Otaghsara, Yazdani, and Ignatius (2012).

⇑ Corresponding author. Tel.: +55 27 8135 0081. E-mail addresses: [email protected] (R. Lourenzutti), krohling. [email protected] (R.A. Krohling). 1 Prof. Krohling had originally the idea of bringing the Hellinger Distance into Multicriteria Decision Making. http://dx.doi.org/10.1016/j.eswa.2014.01.015 0957-4174/Ó 2014 Elsevier Ltd. All rights reserved.

Despite the importance of the TOPSIS method, there are some other difficulties in the MCDM problems. As shown by Kahneman and Tversky (1979), human thinking is not completely rational presenting strong bias in some situations. For instance, people are more sensitive to losses than they are to gains. In this same work, the Prospect Theory, which describes how individuals make decisions in situations involving risks, was proposed. In order to consider the human bias in the MCDM methods, Gomes and Lima (1992) proposed the TODIM (an acronym in Portuguese of Interactive and Multicriteria Decision Making) method, one of the first MCDM methods based on the Prospect Theory. The TODIM method makes use of the prospect function to calculate the dominance of one alternative over another. This method has been successfully applied to MCDM problems, e.g., Gomes and Rangel (2009) and Gomes, Rangel, and Maranhão (2009). In the standard formulations, the TOPSIS and TODIM methods deal with crisp numbers only. This is a serious drawback. Crisp numbers are very precise information and sometimes this desired accuracy is not possible to achieve. Several types of information have been considered in the MCDM models. For example, there are many TOPSIS adaptations to deal with interval numbers (Dymova, Sevastjanov, & Tikhonenko, 2013; Jahanshahloo, Lotfi, & Davoodi, 2009; Yue, 2011), probability distribution information

R. Lourenzutti, R.A. Krohling / Expert Systems with Applications 41 (2014) 4414–4421

(Wentao & Huan, 2010), fuzzy information (Chen, 2000, Krohling & Campanharo, 2011, Lee, Jun, & Chung, 2014), intuitionistic fuzzy information (Boran, Gen, Kurt, & Akay, 2009), interval-valued intuitionistic fuzzy information (Park, Park, Kwun, & Tan, 2011; Tan, 2011; Ye, 2010), and so on. On the other hand, the TODIM method was extended first to deal with fuzzy numbers (Krohling & de Souza, 2012), intuitionistic fuzzy information (Krohling, Pacheco, & Siviero, 2013) and intuitionistic fuzzy information in a random environment (Lourenzutti & Krohling, 2013). Also, a hybrid TODIM was proposed in Fan, Zhang, Chen, and Liu (2013) which deals with crisp numbers, interval-valued numbers and fuzzy numbers at the same time. A problem that arises with many methods that deal with more complex types of information or which consider heterogeneous information is that they do not use a direct approach to analyze the data in hands. Usually, or they use some heuristics, sometimes unjustified, or they perform a transformation in the dataset. For example, Wentao and Huan (2010) summarize the probability distribution into a confidence interval. Fan et al. (2013) perform transformations in the data types: the interval-valued number is transformed into a uniform distribution and the fuzzy information into a probability distribution. While this may be a helpful approach, it may cause some problems. Transformations in the data or summary quantities may miss represent the original information. For instance, the interval of confidence used in Wentao and Huan (2010) to summarize the probability distribution is centered in the expected value. But what if the expected value does not exist? This is the case for the Cauchy distribution. We could choose infinitely many other confidence intervals. What about for a very asymmetric distribution? In this case the median should be preferred to the mean? And for a multi modal distribution? We choose the mean to center the interval or we choose one of the peaks? In the latter case, which peak should we choose? Also, Dymova et al. (2013) pointed out some problems with many extensions of the TOPSIS method that deal with interval-valued numbers and proposed a direct approach. In some situations, the ratings of the alternatives with respect to the criteria are best described as probability distributions. This could occur because there are some random factors that could affect the performances or simply because the expert’s knowledge is provided in a probability distribution form. For example, if we are evaluating the time of arrival of an elevator in a building, it is natural to express this evaluation as a probability distribution since this time is not deterministic. Also, when analyzing investment projects we do not know exactly how the projects will perform in the future, so in this case probability distributions are very helpful to express the expert’s knowledge. Probability distributions are a complex type of information to work with in MCDM. This is because MCDM methods are always comparing alternatives’ ratings to determine which one is preferable, but comparisons between probability distributions are not a trivial task. Given two probability distributions it is hard to state which one is higher/preferable. One of the most used concepts to address this problem is the stochastic dominance (Nowak, 2004a; Nowak, 2004b; Zaras, 2004; Zhang, Fan, & Liu, 2010). However, the stochastic dominance is too restrictive, i.e., given two probability distributions, hardly one of them will dominate the other. Recently, Liu, Fan, and Zhang (2011) proposed a very interesting, intuitive and easy to use method, called stochastic dominance degree. In Liu et al. (2011), the stochastic dominance degree was combined with the PROMETHEE II to rank the alternatives. Their method first calculates the stochastic dominance degree matrices for each criterion; second, these matrices are aggregated into the overall dominance matrix; third, the outgoing flow, entering flow and net flow are calculated using this matrix; finally, the final ranks are determined based on the obtained net flows.

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In PROMETHEE II we first calculate the deviation based on pairwise comparisons (the difference of the ratings) and then apply this to a preference function (Behzadian, Kazemzadeh, Albadvi, & Aghdasi, 2010). While the method in Liu et al. (2011) is useful and interesting, they applied the preference function to the stochastic dominance degrees instead of using the expert’s information in the original form. Therefore, it is an indirect use of the original information. As mentioned above, many MCDM methods that deal with probability distributions are based on a too restrictive concept, which is the stochastic dominance, or use an indirect approach by transforming the original information. To avoid some drawbacks of using these approaches (as already mentioned) we want to adapt well-established methods to be capable of dealing with this type of information in a direct way. In this paper we introduce new versions of the TOPSIS and TODIM methods that are capable of dealing directly with the probability distributions as alternatives’ ratings. In order to do so, it is necessary to calculate distances between these probability distributions. The distances will play the role of the difference in the standard formulation. Here we will use the Hellinger distance (Hellinger, 1909). The Hellinger distance is a well established metric for calculating distance between probability distributions and has been broadly applied over the years, making it well tested and reliable. The fundamental difference, besides the underlying methods, between our approach and the Liu et al. (2011) method is that we will use the stochastic dominance only to determine which alternative is preferable under a certain criterion. The TOPSIS and TODIM method are applied directly to the probability distributions ratings. We emphasize that this paper is not about comparisons between methods. Instead it is about bringing a very useful metric, the Hellinger distance, to the context of MCDM. The remaining of the paper is organized as follows: In Section 2 we present some necessary concepts. The methods are introduced in Section 3. In Section 4 some applications are discussed. The final remarks are presented in Section 5.

2. Preliminary concepts Consider the problem of selecting one between m alternatives. Each alternative is evaluated with respect to n criteria. Let A ¼ fA1 ; A2 ; . . . ; Am g be a set with the m alternatives and C ¼ fC 1 ; C 2 ; . . . ; C n g be a set with the n criteria. We can summarize the Multicriteria Decision Making (MCDM) problem into the following matrix:

ð1Þ

where X ij represents the rating of the ith alternative evaluated with respect to the jth criterion. In this work X ij is considered to be a random variable with distribution fij , denoted by X ij  fij . Since the distribution of X ij provides a complete characterization of the random variable X ij , in order to simplify the formulation of the problem, the decision matrix will be written in terms of the distributions instead of random variables. In order to define a method that deals directly with information in the form of probability distributions without the need of any kind of transformation, we need to answer two questions: (i) given two probability distributions which one is higher/preferable? (ii)

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how far are they from each other? We try to answer these questions with the next two definitions. Definition 2.1 (Hellinger, 1909). Let f and g be two probability density functions (pdf). The Hellinger distance between f and g is given by

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z qffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi2 1 f ðxÞ  gðxÞ dx HDðf ; gÞ ¼ 2 R

ð2Þ

In discrete case it is given by

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n  u1 X pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi2 HDðf ; gÞ ¼ t pðxi Þ  qðxi Þ 2 i¼1

ð3Þ

2.2. The standard TODIM Next we describe the TODIM method, proposed in Gomes and Lima (1992). In this proposal all the X ij are real numbers. Let w ¼ ðw1 ; w2 ; . . . ; wn Þ be the weight vector of the criteria P C 1 ; C 2 . . . ; C n , where 0 6 wi 6 1 and ni¼1 wi ¼ 1. It is necessary that the decision maker defines a reference criterion, usually the criterion with the highest weight. Let the C r ; 1 6 r 6 n be such criterion. Define wrj ¼ wj =wr . The TODIM (h), h > 0, method consists in: 1. Define and normalize the decision matrix. 2. Calculate the final dominance of Ai over each alternative Aj P by dðAi ; Aj Þ ¼ nc¼1 /c ðAi ; Aj Þ; 8ði; jÞ where,

8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pwrc ðX ic  X jc Þ > if X ic P X jc > wrc < c r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi /c ðAi ; Aj Þ ¼ P > wrc > c :  1h ðX jc  X ic Þ otherwise wrc

The Hellinger distance satisfies: 1. 0 6 HDðf ; gÞ 6 1 2. HDðf ; gÞ ¼ HDðg; f Þ

3. The global valor of alternative i is obtained by

P Definition 2.2 (Liu et al., 2011). Let X 1  f1 and X 2  f2 be two random variables, where f1 and f2 are the respective distributions. Then the (stochastic) dominance degree of f1 over f2 is given by Df1 f2 ¼ P ðX 1 P X 2 Þ  0:5P ðX 1 ¼ X 2 Þ. 2.1. The standard TOPSIS There are many methods proposed to deal with Multicriteria Decision Making problems. The TOPSIS method proposed by Hwang and Yoon (1981) suggests that the best alternative should be as far as possible of the worst possible solution and as close as possible to the best possible solution. This method has been widely used in MCDM problems. For a recent survey about TOPSIS applications the interested reader is referred to Behzadian et al. (2012). The TOPSIS is described in the following steps: 1. Define and normalize the decision matrix. The normalization is usually made through

X ij rij ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pm 2 k¼1 X kj

ð4Þ

2. Aggregate the weights to the decision matrix by making v ij ¼ wj rij . 3. Define the positive ideal solution (PIS), v þ j , and the negative ideal solution (NIS), v  j , for each criterion. Usually v þj ¼ maxfv 1j ; . . . ; v mj g and v j ¼ minfv 1j ; . . . ; v mj g for benefit criteria and v þj ¼ minfv 1j ; . . . ; v mj g and v j ¼ maxfv 1j ; . . . ; v mj g for cost criteria. 4. Calculate the separation measures for each alternative

Sþi

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 2 u n  þ ¼t v j  v ij ;

ð8Þ

i ¼ 1; . . . ; m

ð5Þ

ei ¼

P  mini j d ðAi ; Aj Þ P j d ðAi ; Aj Þ  mini j d ðAi ; Aj Þ

j d ðAi ; Aj Þ

maxi

P

4. Sort the alternatives by their value

ð9Þ

ei .

The parameter h in TODIM controls the impact caused in case of losses. We have that, if h < 1 the losses are amplified and if h > 1 the losses are attenuated. The prospect theory states that the individuals are more sensitive to losses than to gains, suggesting h < 1. However, in most application of TODIM, it is used h P 1. This parameter can considerably affect the ranking order of the alternative. If we choose a small h we are looking for an alternative that provides small losses in all criteria, on the other hand, if we choose big values for h we are looking for an alternative that provides more gains, even if we have losses in some criteria. We illustrate this characteristic in Example 1. Recently, Lourenzutti and Krohling (2013) pointed out an unexpected behavior of the TODIM method and suggested the following modification in the /c function,

/c ðAi ; Aj Þ ¼

( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wc ðX ic  X jc Þ if X ic P X jc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1h wc ðX jc  X ic Þ otherwise

ð10Þ

Also, Gomes, Machado, and Rangel (2013), based on the Cumulative Prospect Theory, proposed the following modification in /c ,

8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w > if X ic P X jc < P rcwrc ðX ic  X jc Þ c /c ðAi ; Aj Þ ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi > :  1h Pwrcw ðX jc  X ic Þ otherwise c

ð11Þ

rc

It is not difficult to see that both modifications will lead to the same values for ei ; i ¼ 1; 2; . . . ; m; therefore they will provide equivalent rank order.

j¼1

3. The Hellinger distance in Multicriteria Decision Making

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX 2 u n    Si ¼ t v j  v ij ;

i ¼ 1; . . . ; m

ð6Þ

j¼1

5. Calculate the closeness coefficients to the ideal solution for each alternative

CC i ¼

Si

Si þ Sþi

Suppose that the ratings of the alternatives provided by the decision makers are in probability distribution form. In this section we adapt the TOPSIS and the TODIM methods to be capable to deal with this type of information. 3.1. Hellinger-TOPSIS (H-TOPSIS)

ð7Þ

6. Rank the alternatives according to CC i . The bigger CC i is, the better alternative Ai will be.

Since our information is in probability distribution form we cannot use the TOPSIS directly. First, we need an objective way to determine the PIS and NIS, i.e., to determine the ’’biggest’’

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distribution and the ‘‘smallest’’ distribution. We use the degree of stochastic dominance to deal with this problem. Second, observe that the separation measures in Eqs. (5) and (6) are n-dimensional Euclidean distances. Also, since v ij ¼ wj rij , we have that

1. Define the decision matrix. 2. Calculate the final dominance of Ai over each alternative Aj P by d ðAi ; Aj Þ ¼ nc¼1 /c ðAi ; Aj Þ; 8ði; jÞ where,

8 pffiffiffiffiffiffi if Dfic fjc > Dfjc fic wc HD ðfic ; fjc Þ > < pffiffiffiffiffiffi 1 /c ðAi ; Aj Þ ¼  h wc HD ðfic ; fjc Þ if Dfic fjc < Dfjc fic > : 0 if Dfic fjc ¼ Dfjc fic

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX n  n 2 u 2 u  2 u n  þ uX uX Sþi ¼ t v j  v ij ¼ t wj r þj  wj r ij ¼ t w2j rþj  r ij j¼1

j¼1

j¼1

ð12Þ This means that, instead of computing the weights directly into the decision matrix, one could consider the weights into the separation measures. This fact is very useful since the direct application of the weights into the decision matrix could lead to tiresome calculations of the probability distribution of the new rating wj X ij . The H-TOPSIS uses the Hellinger distance to calculate the distance in each component of the separation measure. Also, there is no need for normalization since 0 6 HDð:; :Þ 6 1. The H-TOPSIS is described as follows: 1. Define the decision matrix. 2. Calculate the PIS, fjþ , and the NIS, fj , for each criterion. The PIS, fjþ 2 ff1j ; . . . ; fmj g is such that Df þ fij P Dfij f þ , j

j

8i ¼ 1; . . . ; m, for benefit criteria and Dfjþ fij 6 Dfij fjþ for cost criteria. Analogously, the NIS, fj 2 ff1j ; . . . ; fmj g is such that Dfj fij P Dfij fj , D

fj fij

6D

fij fj

8i ¼ 1; . . . ; m,

for

cost

criteria

and

for benefit criteria.

3. Calculate the separation measures for each alternative

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX u n 2 2 þ  Sþi ¼ t wj HD fj ; fij ;

i ¼ 1; . . . ; m

ð13Þ

i ¼ 1; . . . ; m

ð14Þ

j¼1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX u n 2 2   Si ¼ t wj HD fj ; fij ; j¼1

4. Calculate the closeness coefficients to the ideal solution for each alternative

CC i ¼

Si

3. Calculate the separation measures for each alternative



 wj HD fjþ ; fij ;

i ¼ 1; . . . ; m

ð16Þ

  wj HD fj ; fij ;

i ¼ 1; . . . ; m

ð17Þ

j¼1

Si ¼

n X

8 pffiffiffiffiffiffi if Dfic fjc < Dfjc fic wc HD ðfic ; fjc Þ > < pffiffiffiffiffiffi 1 /c ðAi ; Aj Þ ¼  h wc HD ðfic ; fjc Þ if Dfic fjc > Dfjc fic > : 0 if Dfic fjc ¼ Dfjc fic

ð19Þ

if c is a cost criterion. 3. The global valor of alternative i is obtained by

P

ei ¼

P  mini j d ðAi ; Aj Þ P j d ðAi ; Aj Þ  mini j d ðAi ; Aj Þ

j d ðAi ; Aj Þ

maxi

P

4. Sort the alternatives by their value

ð20Þ

ei .

Again, no normalization is necessary since the Hellinger distance value is always between 0 and 1. The Hellinger-TODIM deals with the stochastic nature of the information, considering the prospect theory, in a direct, easy and natural way. Observe that the Hellinger distance is outside the squared root function in Eqs. (18) and (19). Despite the fact that the original TODIM uses the difference inside the squared root function we can find both versions in its generalizations. For example, in Krohling and de Souza (2012) and Krohling et al. (2013) the distance measure is used outside the squared root function and in Lourenzutti and Krohling (2013) it is used inside the squared root function. Now, by looking the Equations (2) and (3), we can see that the internal part of the Hellinger distance (i.e. the HD2 ) is responsible for analyzing the divergence between the distributions. Then, the Hellinger distance includes a new component, which is the squared root function, to this divergence. Due to this reason we used the Hellinger distance outside the squared root function.

ð15Þ

Since the Hellinger distance value is always between 0 and 1, there is no necessity to apply the squared and the squared root functions. Therefore we adapt the Step 3 as follows.

Sþi ¼

if c is a benefit criterion and

4. Experimental results

Si þ Sþi

5. Rank the alternatives according the CC i . The bigger CC i is, the better is the alternative Ai .

n X

ð18Þ

j¼1

3.2. Hellinger-TODIM (H-TODIM) In order to compare two alternatives, the Hellinger-TODIM method first uses the concept of (stochastic) dominance degree to determine if we are facing gain or losses. Next, it evaluates the difference between the alternatives through the Hellinger distance. Only then this difference is applied to the prospect function. The H-TODIM (h) is described in the following steps:

In this section we illustrate the H-TOPSIS and H-TODIM methods in two examples. The first one is a simpler one where all criteria are normally distributed and there are no cost criteria. In the second example, there is a cost criterion and different distribution families for the criteria. 4.1. Example 1 This problem is discussed in Lahdelma, Makkonen, and Salminen (2006) and Liu et al. (2011). Consider the problem of choosing the most appropriated strategy for an electricity retailer. The alternatives are evaluated according to 4 criteria: C 1 long-term profit, Table 1 Mean and standard deviation of each evaluation ðlij ; rij Þ. All evaluations are assumed to be normally distributed. Data drawn from Liu et al. (2011). Alternatives

C1

C2

C3

C4

A1 A2 A3 A4 A5 A6 A7 A8 A9

ð439; 143Þ ð426; 125Þ ð264; 135Þ ð444; 125Þ ð605; 115Þ ð449; 126Þ ð449; 107Þ ð457; 126Þ ð453; 107Þ

ð163; 36Þ ð159; 31Þ ð104; 32Þ ð163; 31Þ ð220; 31Þ ð166; 32Þ ð164; 27Þ ð165; 32Þ ð163; 27Þ

ð12:1; 0:5Þ ð12:1; 0:5Þ ð13:1; 0:5Þ ð12:1; 0:5Þ ð11:0; 0:5Þ ð12:1; 0:5Þ ð12:1; 0:5Þ ð12:1; 0:5Þ ð12:1; 0:5Þ

ð9:3; 6:5Þ ð14:8; 6:5Þ ð9:3; 6:5Þ ð9:3; 6:5Þ ð9:3; 6:5Þ ð4:3; 6:5Þ ð9:3; 6:5Þ ð9:3; 6:5Þ ð14:8; 6:5Þ

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C 2 short-term profit; C 3 market share and C 4 green market share. Admit that we have 9 alternatives. The weight vector used in Liu et al. (2011) is w ¼ ð0:33; 0:35; 0:15; 0:17Þ. In Table 1 is presented the mean and standard deviation of each evaluation. All evaluations are assumed to be normally distributed. Let f be the pdf of a N ðl1 ; r1 Þ and let g be the pdf of a N ðl2 ; r2 Þ. The Hellinger distance between f and g is given by

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u 2r1 r2 ðl1  l2 Þ2 t Hðf ; gÞ ¼ 1  exp 0:25 2 r21 þ r22 r1 þ r22

ð21Þ

For the Normal distribution it is quite easy to determine which alternative is preferable according to Definition 2.2, it is the one with the biggest mean. 4.1.1. H-TOPSIS For the H-TOPSIS, we first need to define the PIS and NIS for each criterion. In this case, it is easily obtained just by looking the data matrix. Table 2 presents the PIS and NIS for the criteria. Once we have defined the PIS and NIS we must calculate the Hellinger distance between the alternatives’ ratings and the corresponding PIS and NIS, as presented in Tables 3 and 4.  Finally, we calculate the Sþ i and Si , both are presented in Table 5, and rank the alternatives by their closeness coefficient. The final results are presented in Table 6. The results obtained using H-TOPSIS fully agrees with the results obtained by Liu et al. (2011). 4.1.2. H-TODIM In the H-TODIM method, for each criterion it is necessary to calculate the Hellinger distance between all pairs of ratings. The Hellinger distances between the alternatives evaluations for the Criterion 1, are shown in Table 7. For the remaining criteria Eq. (21) should be used. Once the Hellinger distances are calculated, the partial and final dominance matrices are easily obtained. We present the final dominance matrix in Table 8. The rank order obtained by Liu et al. (2011) and by the Hellinger-TODIM are presented in Table 9. The results provided by H-TODIM are compatible to those achieved by Liu et al. (2011). For h P 1:2 the rank order are the same for both methods. We recall that for h > 1, the losses are attenuated and, consequently, the H-TODIM(h) searches for an alternative that provides more gains, even if it provides more losses. Next, we should analyze the impact of the h parameter in the rank order. We compare the change in the first two positions when we go from H-TODIM (2.5) (A5  A9 ) to H-TODIM (0.3) (A9  A5 ). Tables 10–13 show the lines 5 and 9 of the matrices /1 to /4 for h ¼ 2:5 and h ¼ 0:3. Since the parameter h controls the impact of losses, the gains of one alternative over another is unaffected by the variation of h, as we can see in Tables 10–13. By Tables 10 and 11 it is clear how the alternative A5 presents the most gains for the more important criteria under consideration, C 1 and C 2 . However, by Tables 12 and 13 the alternative A5 faces many cases of losses. When we use h ¼ 2:5 these losses are attenuated, making the gains of A5 much more important than the losses. However for h ¼ 0:3 the losses are amplified and, even the criteria C 3 and C 4 having such small weights values, the amplification of the losses of A5 are very expensive, making A9 , an alternative that provides less gains but at the same time less losses, preferred.

Table 2 The PIS and NIS for each criteria for the problem of electricity retailer. Criterion

C1

C2

C3

C4

PIS NIS

A5 A3

A5 A3

A3 A5

A2 (A9 ) A6

Table 3 Hellinger distance between the alternatives and the positive ideal solutions for the problem of electricity retailer.

A1 A2 A3 A4 A5 A6 A7 A8 A9

PIS1

PIS2

PIS3

PIS4

0.441 0.494 0.778 0.450 0.000 0.436 0.469 0.416 0.458

0.553 0.619 0.904 0.587 0.000 0.555 0.612 0.563 0.620

0.627 0.627 0.000 0.627 0.943 0.627 0.627 0.627 0.627

0.293 0.000 0.293 0.293 0.293 0.528 0.293 0.293 0.000

Table 4 Hellinger distance between the alternatives and the negative ideal solution for the problem of electricity retailer.

A1 A2 A3 A4 A5 A6 A7 A8 A9

NIS1

NIS2

NIS3

NIS4

0.425 0.421 0.000 0.463 0.778 0.472 0.510 0.490 0.519

0.561 0.563 0.000 0.596 0.904 0.612 0.637 0.604 0.629

0.674 0.674 0.943 0.674 0.000 0.674 0.674 0.674 0.674

0.267 0.528 0.267 0.267 0.267 0.000 0.267 0.267 0.528

Table 5 The Sþ and S of each alternative for the problem of electricity retailer.

Sþ S

A1

A2

A3

A4

A5

A6

A7

A8

A9

0.483 0.483

0.474 0.527

0.623 0.187

0.498 0.508

0.191 0.618

0.522 0.471

0.513 0.538

0.478 0.520

0.462 0.582

Table 6 The ranking of the alternatives obtained by Liu et al. (2011) and by the H-TOPSIS. Method

Rank

Liu et al. (2011) H-TOPSIS

1

2

3

4

5

6

7

8

9

A5 A5

A9 A9

A2 A2

A8 A8

A7 A7

A4 A4

A1 A1

A6 A6

A3 A3

Table 7 Hellinger distances between the alternatives evaluations under criterion C 1 . We remind that the Hellinger distance is symmetric. HD (.,.)

A1

A2

A3

A4

A5

A6

A7

A8

A9

A1 A2 A3 A4 A5 A6 A7 A8 A9

0

0.075 0

0.425 0.421 0

0.068 0.051 0.463 0

0.441 0.494 0.778 0.450 0

0.068 0.065 0.472 0.015 0.436 0

0.146 0.104 0.510 0.079 0.469 0.081 0

0.079 0.087 0.490 0.037 0.416 0.022 0.085 0

0.148 0.113 0.519 0.082 0.458 0.082 0.013 0.082 0

4.2. Example 2 This section illustrates the use of the methods discussed above with different probability distribution family for the criteria and also with the presence of two cost criteria. This problem is discussed in Wentao and Huan (2010). A company wants to choose an investment project. There are six options, each one was evaluated according to three criteria, C 1 initial investment, C 2 risk

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R. Lourenzutti, R.A. Krohling / Expert Systems with Applications 41 (2014) 4414–4421 Table 8 Final dominance matrix, with h ¼ 2:5, for the problem of electricity retailer.

A1 A2 A3 A4 A5 A6 A7 A8 A9

A1

A2

A3

A4

A5

A6

A7

A8

A9

0 0.091 0.02 0.15 0.694 0.218 0.443 0.309 0.444

0.241 0 0.109 0.167 0.653 0.194 0.275 0.236 0.364

0.695 0.917 0 0.725 0.919 0.65 0.76 0.739 0.984

0.06 0.121 0.032 0 0.712 0.098 0.318 0.208 0.388

0.011 0.193 0.052 0.018 0 0.095 0.024 0.008 0.199

0.092 0.174 0.177 0.14 0.906 0 0.143 0.274 0.393

0.177 0.077 0.046 0.127 0.729 0.09 0 0.34 0.262

0.124 0.093 0.038 0.083 0.687 0.057 0.136 0 0.087

0.178 0.146 0.136 0.155 0.638 0.006 0.048 0.251 0

Table 9 The ranking of the alternatives provided by Liu et al. (2011) and obtained by HTODIM, using different values for h for the electricity retailer problem. Method

Rank

Liu et al. (2011) H-TODIM (2.5) H-TODIM (2) H-TODIM (1.8) H-TODIM (1.6) H-TODIM (1.4) H-TODIM (1.2) H-TODIM (1) H-TODIM (0.8) H-TODIM (0.6) H-TODIM (0.4) H-TODIM (0.3)

1

2

3

4

5

6

7

8

9

A5 A5 A5 A5 A5 A5 A5 A5 A5 A5 A9 A9

A9 A9 A9 A9 A9 A9 A9 A9 A9 A9 A5 A8

A2 A2 A2 A2 A2 A2 A2 A8 A8 A8 A8 A5

A8 A8 A8 A8 A8 A8 A8 A2 A2 A2 A7 A7

A7 A7 A7 A7 A7 A7 A7 A7 A7 A7 A2 A2

A4 A4 A4 A4 A4 A4 A4 A4 A4 A4 A4 A4

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

A6 A6 A6 A6 A6 A6 A6 A6 A6 A6 A6 A6

A3 A3 A3 A3 A3 A3 A3 A3 A3 A3 A3 A3

of the investment and C 3 the benefit of the investment. The criteria C 1 and C 2 are cost criteria and C 3 is a benefit criterion. The evaluations of the alternatives are provided by Table 14. Wentao and Huan (2010) considered incomplete weight information. Thus, the weight information were provided as restrictions inequalities instead a constant vector. This is not our case here. We will use the following weight vector value, w ¼ ð0:3; 0:2; 0:5Þ, which respects the restrictions imposed in Wentao and Huan (2010). We shall remind the reader that it is not our objective to compare the methods as much as to illustrate the methods introduced in this work. However, we present the results provided by Wentao and Huan (2010) for illustrative purposes. To calculate the Hellinger distance and the degree of stochastic dominance for the uniform distribution family is a simple task. However, in some situations it can be very difficult and/or tiresome to calculate analytically all the dominance degree of the alternatives.

Table 10 The partial dominance of alternatives A5 and A9 over every other alternative with respect to C 1 , for h ¼ 2:5 (top) and h ¼ 0:3 (bottom). /1

A1

A2

A3

A4

A5

A6

A7

A8

A9

A5 A9

0.253 0.085

0.284 0.065

0.447 0.298

0.259 0.047

0.000 0.105

0.251 0.047

0.269 0.008

0.239 0.019

0.263 0.000

A5 A9

0.253 0.085

0.284 0.065

0.447 0.298

0.259 0.047

0.000 0.877

0.251 0.047

0.269 0.008

0.239 0.158

0.263 0.000

Table 11 The partial dominance of alternatives A5 and A9 over every other alternative with respect to C 2 , for h ¼ 2:5 (top) and h ¼ 0:3 (bottom). /2

A1

A2

A3

A4

A5

A6

A7

A8

A9

A5 A9

0.327 0.000

0.366 0.050

0.535 0.372

0.347 0.000

0.000 0.147

0.328 0.022

0.362 0.003

0.333 - 0.021

0.367 0.000

A5 A9

0.327 0.000

0.366 0.050

0.535 0.372

0.347 0.000

0.000 1.223

0.328 0.181

0.362 0.026

0.333 0.173

0.367 0.000

Table 12 The partial dominance of alternatives A5 and A9 over every other alternative with respect to C 3 , for h ¼ 2:5 (top) and h ¼ 0:3 (bottom). /3

A1

A2

A3

A4

A5

A6

A7

A8

A9

A5 A9

0.104 0.000

0.104 0.000

0.146 0.097

0.104 0.000

0.000 0.261

0.104 0.000

0.104 0.000

0.104 0.000

0.104 0.000

A5 A9

0.870 0.000

0.870 0.000

1.218 0.810

0.870 0.000

0.000 0.261

0.870 0.000

0.870 0.000

0.870 0.000

0.870 0.000

Table 13 The partial dominance of alternatives A5 and A9 over every other alternative with respect to C 4 , for h ¼ 2:5 (top) and h ¼ 0:3 (bottom). /4

A1

A2

A3

A4

A5

A6

A7

A8

A9

A5 A9

0.000 0.121

0.048 0.000

0.000 0.121

0.000 0.121

0.000 0.121

0.110 0.218

0.000 0.121

0.000 0.121

0.048 0.000

A5 A9

0.000 0.121

0.402 0.000

0.000 0.121

0.000 0.121

0.000 0.121

0.110 0.218

0.000 0.121

0.000 0.121

0.402 0.000

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R. Lourenzutti, R.A. Krohling / Expert Systems with Applications 41 (2014) 4414–4421

In such cases we can resort to computational technique to easily estimate these quantities with a reasonable precision. Table 15 shows the degree of stochastic dominance of the alternatives’ ratings with respect to criterion C 1 . 4.2.1. H-TOPSIS First, we need to calculate the PIS and NIS for each criterion. Based on the results presented in Table 15, we can see that A3 is Table 14 Data drawn from Wentao and Huan (2010) for the investment problem.

A1 A2 A3 A4 A5 A6

C1

C2

C3

U ½345; 405 U ½360; 430 U ½320; 390 U ½430; 505 U ½410; 490 U ½430; 540

N ð251; 63Þ N ð264; 82Þ N ð250; 75Þ N ð314; 100Þ N ð300; 113Þ N ð340; 126Þ

N ð130; 35Þ N ð138; 38Þ N ð131; 40Þ N ð165; 73Þ N ð157; 85Þ N ð183; 125Þ

Table 15 Stochastic dominance degree with respect to criterion 1 for the investment problem. Dfi fj A1 A2 A3 A4 A5 A6

A1 0.500 0.759 0.241 1.000 1.000 1.000

A2

A3

0.241 0.500 0.092 1.000 0.964 1.000

0.759 0.908 0.500 1.000 1.000 1.000

A4

A5

0.000 0.000 0.000 0.500 0.300 0.659

0.000 0.036 0.000 0.700 0.500 0.795

A6 0.000 0.000 0.000 0.341 0.205 0.500

Criterion

C1

C2

C3

PIS NIS

A3 A6

A3 A6

A6 A1

Table 17 Hellinger Distance between the alternatives and the positive ideal solutions for the investment problem. PIS1

PIS2

PIS3

0.553 0.756 0.000 1.000 1.000 1.000

0.087 0.077 0.000 0.287 0.269 0.383

0.556 0.525 0.517 0.265 0.207 0.000

Table 18 Hellinger Distance between the alternatives and the negative ideal solution for the investment problem.

A1 A2 A3 A4 A5 A6

NIS1

NIS2

NIS3

1.000 1.000 1.000 0.417 0.600 0.000

0.436 0.321 0.383 0.140 0.130 0.000

0.000 0.087 0.067 0.397 0.423 0.556

Table 19 The Sþ and S of each alternative for the problem of investment.

Sþ S

4.2.2. H-TODIM In Table 21 is presented the Hellinger distances between the probability distributions under criterion C 1 , which are described Table 20 The ranking of the alternatives obtained by the respective methods. Method

Table 16 The PIS and NIS for each criteria for the problem of investment.

A1 A2 A3 A4 A5 A6

the PIS of criterion C 1 , since C 1 is a cost criterion. Table 16 presents the PIS and NIS for each criterion. Then, we obtain the Hellinger distances between the alternatives ratings and the PIS and NIS ratings, as presented in Tables 17 and 18. The Sþ and S for each alternative is presented in Table 19. Table 20 shows the final rank order of the TOPSIS for the investment problem. The biggest differences between the rank order of the two methods are the ranks of the alternatives A5 and A6 . The alternative A5 is smaller (in the sense of the stochastic dominance degree) in criteria C 1 and C 2 , therefore better, since both criteria are cost criteria. On the other hand, the alternative A6 is higher with respect to criterion C 3 , therefore better. Also, w3 ¼ w1 þ w2 . Thus, it is not intuitively clear which alternative is better. Besides, we remind that those methods are not directly comparable as explained in the beginning of the section.

A1

A2

A3

A4

A5

A6

0.461 0.387

0.505 0.408

0.259 0.410

0.490 0.351

0.457 0.417

0.377 0.278

Rank

Wentao and Huan (2010) H-TOPSIS

1

2

3

4

5

6

A3 A3

A1 A5

A6 A1

A2 A2

A4 A6

A5 A4

Table 21 Hellinger distances between the probability distributions under criterion C 1 of the investment problem. HD (.,.)

A1

A2

A3

A4

A5

A6

A1 A2 A3 A4 A5 A6

0

0.553 0

0.553 0.756 0

1 1 1 0

1 0.856 1 0.475 0

1 1 1 0.417 0.6 0

Table 22 Final dominance matrix, with h ¼ 2:5, for the investment problem.

A1 A2 A3 A4 A5 A6

A1

A2

A3

A4

A5

A6

0 0.085 0.389 0.001 0.021 0.096

0.343 0 0.429 0.014 0.046 0.095

0.156 0.131 0 0.026 0.004 0.078

0.587 0.546 0.578 0 0.271 0.071

0.577 0.451 0.562 0.058 0 0.008

0.586 0.543 0.573 0.216 0.328 0

Table 23 The ranking of the alternatives provided by Wentao and Huan (2010) and obtained by H-TODIM, using different values for h for the investment problem. Method

Wentao and Huan (2010) H-TODIM (2.5) H-TODIM (2) H-TODIM (1.8) H-TODIM (1.6) H-TODIM (1.4) H-TODIM (1.2) H-TODIM (1) H-TODIM (0.8) H-TODIM (0.6) H-TODIM (0.4) H-TODIM (0.3)

Rank 1

2

3

4

5

6

A3 A3 A3 A3 A3 A3 A3 A3 A3 A3 A3 A3

A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1 A1

A6 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2 A2

A2 A5 A5 A5 A5 A5 A5 A5 A5 A5 A5 A5

A4 A6 A6 A6 A6 A6 A6 A6 A4 A4 A4 A4

A5 A4 A4 A4 A4 A4 A4 A4 A6 A6 A6 A6

R. Lourenzutti, R.A. Krohling / Expert Systems with Applications 41 (2014) 4414–4421

by the uniform distributions family. The final dominance matrix is presented in Table 22. Table 23 shows the final rank order of the H-TODIM considering several values for h. Again, the rank order between the methods are compatible. The biggest distinction between the H-TODIM and the method of Wentao and Huan (2010) are the rank of A5 and A6 , the same that occurred with the H-TOPSIS.

5. Concluding remarks When facing information described as probability distributions, many methods use very restrictive concepts or perform transformations that can cause some undesired effects. To avoid these shortcomings we adapted two well-known methods to be capable of processing probability distributions in a direct and natural way. The TOPSIS and TODIM methods were modified to use the Hellinger distance, a new metric in the MCDM context. Hellinger distance is a well-known concept and it is much used in many other fields. In the modified methods, H-TOPSIS and H-TODIM, it is not necessary to transform the probability distributions in any other type of information. The methods are capable of computing directly from the probability distributions. Also, the methods do not use the usual restrictive concepts to deal with probability distributions, such as the stochastic dominance. Instead, the methods use a more relaxed concept that can be applied for any two alternatives’ ratings. Thus, potential problems caused by these factors are avoided. Both methods have a clear logic and they are simple to be computed. To analyze the behavior of the methods, they were applied in two examples already discussed in other works. Both methods behaved as expected and found the same best alternative as the other methods. Also, a careful analyze about the impact of h in H-TODIM (h) was performed. The h parameter brings flexibility by controlling the impact of alternatives’ losses to the rank order, a clear influence of the prospect theory. Since the majority of the MCDM methods are based on comparisons between the alternatives ratings, and these comparisons are made through difference/distance quantities, many other methods may be adapted. As shown in this work, this should not be hard and it is an interesting topic for future work. Another interesting topic is the adaptation of these methods to group decision making. Acknowledgements R. Lourenzutti would like to thank the Brazilian agency CAPES for the financial support and R. A. Krohling would like to thank the financial support of the Brazilian agency CNPq under Grant No. 303577/2012-6. Both authors would like to thank the reviewers for the insightful suggestions. References Behzadian, M., Kazemzadeh, R., Albadvi, A., & Aghdasi, M. (2010). PROMETHEE: A comprehensive literature review on methodologies and applications. European Journal of Operational Research, 200(1), 198–215. Behzadian, M., Otaghsara, S. K., Yazdani, M., & Ignatius, J. (2012). A state-of the-art survey of TOPSIS applications. Expert Systems with Applications, 39(17), 13051–13069.

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