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IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making
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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
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Short Communication
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IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making
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a 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 Department of Informatics, UFES – Federal University of Espirito Santo, Av. Fernando Ferrari, 514, CEP 29075-910 Vitória, Espírito Santo, ES, Brazil
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a r t i c l e
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Renato A. Krohling a,⇑, André G.C. Pacheco b, André L.T. Siviero b
i n f o
Article history: Received 28 February 2013 Received in revised form 19 June 2013 Accepted 22 August 2013 Available online xxxx
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Keywords: Multi-criteria decision making (MCDM) Intuitionistic fuzzy numbers Prospect function value IF-TODIM
a b s t r a c t The recently developed fuzzy TODIM (an acronym in Portuguese for iterative multi-criteria decision making) method using fuzzy numbers has been applied to uncertain MCDM problems with promising results. In this paper, a more general approach to the fuzzy TODIM, which takes into account the membership and the non-membership of the fuzzy information is considered. So, the fuzzy TODIM method has been extended to handle intuitionistic fuzzy information. This way, it is possible to tackle more challenging MCDM problems. Two case studies are used to illustrate and show the suitability of the developed method. 2013 Elsevier B.V. All rights reserved.
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1. Introduction
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Multi-criteria decision making (MCDM) problems occur in different areas of science and engineering [12]. Several research efforts have been made in order to develop new methods or to improve existing ones. Typical challenges for MCDM methods are uncertainty, risk, etc. In this sense, the theory of fuzzy sets and fuzzy logic developed by Zadeh [28] has been used to model uncertainty or lack of knowledge and applied to a variety of MCDM problems. Bellman and Zadeh [3] introduced the theory of fuzzy sets in MCDM problems as an effective approach to treat vagueness, lack of knowledge and ambiguity inherent in the human decision making process which are known as fuzzy multi-criteria decision making (FMCDM). For real world-problems the decision matrix is affected by uncertainty, which can be modeled using fuzzy numbers [7]. Another important aspect of decision making is to consider the risk attitude/preferences of the decision maker in MCDM. Prospect theory developed by Kahneman and Tversky [13] is a descriptive model of individual decision making under condition of risk. In turn, Tversky and Kahneman [21] proposed the cumulative prospect theory, which capture psychological aspects of decision making under risk. In prospect theory, the outcomes are expressed by means of gains and losses from a reference alternative [18]. The
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⇑ Corresponding author. Tel.: +55 2799475139. E-mail addresses: [email protected] (R.A. Krohling), pacheco.comp@ gmail.com (A.G.C. Pacheco), [email protected] (A.L.T. Siviero).
value function in prospect theory assumes a S-shape concave above the reference alternative, which reflects the aversion of risk in face of gains; and the convex part below the reference alternative reflects the propensity to risk in case of losses. One of the first MCDM methods based on prospect theory was proposed by Gomes and Lima [9]. Despite its effectiveness and simplicity in concept, this method presents some shortcomings because of its inability to deal with uncertainty and imprecision inherent in the process of decision making. In the original formulation of TODIM (an acronym in Portuguese for Iterative Multi-criteria Decision Making), the rating of alternatives, which composes the decision matrix, is represented by crisp values. Since the TODIM method [10] is not able to handle uncertainty, Krohling and de Souza [15] proposed a fuzzy TODIM to tackle uncertain MCDM problems. A clear advantage of this method is its ability to treat uncertain information using fuzzy numbers. Recently, Fan et al. [8] have presented an extension of the TODIM method, whereas the attribute values (crisp numbers, interval numbers and fuzzy numbers) are expressed in the format of random variables with cumulative distribution functions and next the classical TODIM can be applied. Atanasov [1] proposed a more general theory for fuzzy numbers, known as intuitionistic fuzzy numbers, which are described by a membership function and a non-membership function. In the last few years, intuitionistic fuzzy numbers have been applied to solve MCDM problems [2,27,16,17,4,5,19,22–26]. In this paper, based on the fuzzy TODIM method [15] and intuitionistic fuzzy
0950-7051/$ - see front matter 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.knosys.2013.08.028
Q1 Please cite this article in press as: R.A. Krohling et al., IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.08.028
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numbers [1], we propose the intuitionistic fuzzy TODIM method, for short, IF-TODIM to handle uncertain MCDM problems. The remainder of this article is organized as follows. In Section 2, some preliminary background on intuitionistic fuzzy numbers is provided. In Section 3, an intuitionistic fuzzy TODIM method is developed, which contains uncertainty in the decision matrix modeled by intuitionistic trapezoidal fuzzy numbers. In Section 4, case studies are presented to illustrate the method and the results show the feasibility of the approach. In Section 5, some conclusions and directions for future work are given.
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2. Intuitionistic fuzzy multi-criteria decision making
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Trapezoidal intuitionistic fuzzy numbers are commonly used for solving decision-making problems, where the available information is imprecise. Next, some basic definitions of intuitionistic fuzzy sets and fuzzy numbers are provided [1,6,11,20].
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2.1. Preliminaries on intuitionistic fuzzy sets and intuitionistic fuzzy numbers Definition 1. Let X be the universe of discourse. An intuitionistic e is characterized by a subset of X defined by fuzzy set A A = {hx, lA(x), mA(x)ijx 2 X}, where lA:X ? [0; 1] and mA:X ? [0; 1] with the condition 0 6 lA(x) + mA(x) 6 1j"x 2 X. The numeric values lA(x) and mA(x) stands for the degree of membership and the degree of non-membership of x in A, respectively. ~ is Definition 2. An intuitionistic trapezoidal fuzzy number a ~ ¼ ða1 ; a2 ; a3 ; a4 ; la~ ; ma~ Þ with membership function defined by a given by [22,24]:
la~ ðxÞ ¼ 118
8 xa 1 ~ a~ ; a1 6 x < a2 l > > a2 a1 > > > > ~ a~ ; > :l > a4 a3 > > > : 0;
a3 < x 6 a4 otherwise;
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while the non-membership function is given by:
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8 ða xÞþm~ ðxa Þ 2 1 a~ > ; a1 6 x < a2 > a2 a1 > > > > 3 a~ 4 > ; a3 < x 6 a4 > a4 a3 > > > : 0; otherwise:
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ð1Þ
ð2Þ
~a~ represent the maximum value of member~ a~ and m The values l ~, respectively. For ship degree and non-membership degree of a instance, consider the intuitionistic trapezoidal fuzzy number (ITFN) hVG; 0.6, 0.3i = h0.5, 0.75, 0.95, 1; 0.6, 0.3i. In this case, a decision maker not only assess the rating of an alternative by using the linguistically defined trapezoidal fuzzy number (TFN) VG (Very Good) but also provides the degree of membership and non-membership, 0.6 and 0.3 respectively. ~ ¼ ða1 ; a2 ; a3 ; Definition 3. Let a trapezoidal fuzzy number a ~a~ Þ, then its expected value is calculated as ~ a~ ; m a4 ; l ~Þ ¼ ½ða1 þ a2 þ a3 þ a4 Þ ð1 þ l ~ a~ m ~a~ Þ=8. In addition, definitions Iða ~Þ ¼ Iða ~ Þ ðl ~a~ Þ and Hða ~Þ ¼ Iða ~Þ ðl ~a~ Þ are presented, ~ a~ m ~ a~ þ m for Sða which are known as score function and accuracy function, respectively [22]. Definition 4. Let two intuitionistic trapezoidal fuzzy numbers ~ ¼ ðb1 ; b2 ; b3 ; b4 ; l ~ ¼ ða1 ; a2 ; a3 ; a4 ; l ~a~ Þ and b ~b~ Þ, then [22]: ~ a~ ; m ~ b~ ; m a
~ then a ~ ~Þ > SðbÞ ~ > b. If Sða ~ ~ then a ~ ~ ~ ~ ¼ b. If SðaÞ ¼ SðbÞ and If HðaÞ ¼ HðbÞ ~ and Hða ~ then a ~ ~Þ ¼ SðbÞ ~Þ > HðbÞ ~ > b. If Sða
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Definition 5. Let two intuitionistic trapezoidal fuzzy numbers ~ ¼ ðb1 ; b2 ; b3 ; b4 ; l ~ ¼ ða1 ; a2 ; a3 ; a4 ; l ~a~ Þ and b ~b~ Þ, then the dis~ a~ ; m ~ b~ ; m a tance between them is calculated as [22]:
~ ¼ ~; bÞ dða
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1 ~a~ Þ a1 ð1 þ l ~b~ Þ b1 j þ jð1 þ l ~ a~ m ~ b~ m ~ a~ jð1 þ l 8 ~a~ Þ a2 ð1 þ l ~b~ Þ b2 j þ jð1 þ l ~a~ Þ a3 ~ b~ m ~ a~ m m ~b~ Þ ~ b~ m ð1 þ l
~ a~ m ~a~ Þ a4 ð1 þ l ~ b~ m ~b~ Þ b4 j : b3 jþjð1 þ l
ð3Þ
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~ ¼ ðVL; 0:8; 0:1Þ ¼ ð0:1; 0:2; 0:3; a For instance, consider ~ ¼ ðEH; 0:7; 0:2Þ ¼ ð0:7; 0:8; 0:9; 0:95; 0:7; 0:2Þ, 0:4; 0:8; 0:1Þ and b where VL and EH are linguistic definitions of the trapezoidal fuzzy numbers, Very Low and Extremely High, respectively. The distance ~ ¼ 0:4156. ~; bÞ between them is dða
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2.2. Decision making problem with uncertain decision matrix
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Let us consider the fuzzy decision matrix A, which consists of alternatives and criteria, described by:
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xij where A1, A2, . . . , Am are alternatives, C1, C2, . . . , Cn are criteria, ~ intuitionistic trapezoidal fuzzy numbers that indicates the rating of the alternative Ai with respect to criterion Cj. The weight vector W = (w1, w2, . . . , wn) composed of the individual weights wj P (j = 1, . . . , n) for each criterion Cj satisfying nj¼1 wj ¼ 1. In the following section, the method is presented.
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2.3. IF-TODIM – An intuitionistic fuzzy TODIM method
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For information on the TODIM method the reader is referred to Gomes and Rangel [10]. The intuitionistic fuzzy TODIM method, for short, IF-TODIM, which is an extension of the fuzzy TODIM method [15], is described in the following steps:
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Step 1: The criteria are normally classified into two types: benefit and cost. The intuitionistic trapezoidal fuzzy-decision e ¼ ½~ matrix A xij mxn with i = 1, . . . , m, and j = 1, . . . , n is normalized which results the correspondent fuzzy decision matrix e ¼ ½~rij . The fuzzy normalized value ~r ij is calculated as: R mxn
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max a4ij akij r kij ¼ with k ¼ 1; 2; 3; 4 for cost criteria maxi a4ij mini a1ij akij min a1ij with k ¼ 1; 2; 3; 4 r kij ¼ maxi a4ij mini a1ij for benefit criteria
ð4Þ
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We denote ak of aij = {(a1, a2, a3, a4)} as akij , e.g., a3ij ¼ a3 . e i over each Step 2: Calculate the dominance of each alternative R e alternative R j using the following expression: m X ei; R ejÞ ¼ ei; R e j Þ 8ði; jÞ dð R /c ð R
ð5Þ
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Q1 Please cite this article in press as: R.A. Krohling et al., IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.08.028
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Fig. 1. Prospect value function.
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where
ffiffiffiffiffiffiffiffiffiffiffiffi 8v u wrc dð~r ic ; ~xjc Þ if ð~r ic > ~r jc Þ > m u > > tX > > w rc > > > c¼1 > > < if ð~r ic ¼ ~r jc Þ e e /c ð R i ; R j Þ ¼ 0; vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ! > u > m > u X > > u > wrc > t > > c¼1 : 1 dð~ric ; ~xjc Þ if ð~r ic < ~r jc Þ h wrc
ð6Þ
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3. Experimental results
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3.1. Case study 1 – supply chain selection
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In the first case study, the supply chain selection problem investigated by Wei and Wang [26] is used as the benchmark, who also developed a decision making method using intuitionistic trapezoidal fuzzy numbers. In this problem, the alternatives are five suppliers, evaluated according to four criteria: product quality (C1), service (C2), delivery (C3) and price (C4). It is known that C1, C2 and C3 are benefit criteria and C4 is a cost criterion. The weight vector associated to each criterion is W = (w1, w2, w3, w4) = (0.35, 0.2333, 0.3, 0.1167). The intuitionistic trapezoidal fuzzy decision matrix is described in Table 1. The factor of attenuation of losses h, was set to h = 1 [10] but the value h = 2.5 has also been used [15]. The IF-TODIM method was applied to the decision matrix given in Table 1. The plot of the prospect function value for each one of the partial dominance using h = 1 and h = 2.5 is depicted in Fig. 2. The order of the alternatives obtained is A2 A5 A4 A3 A1, which is in agreement with those obtained in [26]. As we can notice in Fig. 2 the prospect function value in the negative quadrant for h = 1 presents a deeper slope than for h = 2.5.
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3.2. Case study 2 – oil spill in the sea
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This decision making problem is composed of ten alternatives and two criteria. According to our notation, the criteria C1 oil at the coast (OC) is a cost criterion and the criterion C2 oil intercepted (OI) is a benefit criterion. Firstly, the decision matrix has been normalized. In this study, data are affected by uncertainty because the simulation of oil spots depends on several factors such as quantity and type of oil spilled, location of spill, weather and ocean conditions, among others. For a detailed description of how the data have been obtained the reader is referred to Krohling and Campanharo [14]. The first experiment was to examine the new method considering that all information is 100% certain, i.e., the degree of membership is 1 and the degree of non-membership is 0. For validation purposes our results are compared with those obtained by fuzzy TODIM as reference [15]. Similar results are expected for this particular case. The IF-TODIM method has been applied to the decision matrix with intuitionistic trapezoidal fuzzy numbers (see Table 2). Q4 Similar to the previous case, experiments with IF-TODIM using h = 1 and h = 2.5 have been carried out. The ranking of the alternatives using h = 1 is listed in Table 3 and depicted in Fig. 3. The ranking results of the experiments using h = 2.5 are not shown since there is no change in the ranking of the alternatives. The plot of the prospect function value for each one of the partial dominance values using h = 1 and h = 2.5 is depicted in Fig. 4.
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ei; R e j Þ, denoted by partial dominance, represents the The term /c ð R ei; R e j Þ when comcontribution of the criterion c to the function dð R paring the alternative i with alternative j. The values ~r ic and ~r jc are the rating of the alternatives i and j, respectively with respect to criterion c. The value wrc represents the weight of criterion c divided c by the weight of the reference r, i.e., wrc ¼ w , whereas the latter is wr the criterion that has the greater weight. The term dð~ric ; ~rjc Þ stands for the distance between the two intuitionistic fuzzy numbers ~r ic and ~r jc , calculated by Eq. (3). Three cases can occur in Eq. (6): (i) if ð~ric > ~r jc Þ, it represents a gain; (ii) if ð~r ic ¼ ~r jc Þ, it is nil; and (iii) if ð~ric < ~r jc Þ, it represent a loss. Definitions (3) and (4) are used in each case. The parameter h represents the attenuation factor of the losses. Different values of this parameter lead to different shapes of the prospect value function in the negative quadrant as illustrated in Fig. 1. The final matrix of dominance is obtained by summing up the partial matrices of dominance for each criterion. Step 3: Calculate the global value of the alternative i by normalizing the final matrix of dominance according to the following expression:
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P
ni ¼
P
dði; jÞ min dði; jÞ P P max dði; jÞ min dði; jÞ
ð5Þ
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Sorting the values ni provides the rank of each alternative. The best alternatives are those that have higher value ni. Next, the approach is illustrated on two case studies.
Table 1 Decision matrix for the supply chain selection [26].
Q1 Please cite this article in press as: R.A. Krohling et al., IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.08.028
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Fig. 2. Prospect function value using h = 1 and h = 2.5.
~ ¼ 1, Fig. 3. Ranking of the alternatives obtained using F-TODIM and IF-TODIM for l m~ ¼ 0.
Table 2 Decision matrix for the oil spill in the sea [14]. Alternatives
C1 (103)
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10
[(7.76, [(8.85, [(9.34, [(7.38, [(5.27, [(7.30, [(6.16, [(5.16, [(5.27, [(5.64,
8.20, 9.35, 9.86, 7.79, 5.56, 7.70, 6.50, 5.45, 5.57, 5.96,
C2 (103) 9.06, 9.49)] 10.33, 10.82)] 10.89, 11.41)] 8.61, 9.02)] 6.15, 6.44)] 8.51, 8.92)] 7.19, 7.53)] 6.02, 6.31)] 6.15, 6.44)] 6.58, 6.90)]
[(4.70, [(3.62, [(3.15, [(5.09, [(7.19, [(5.21, [(6.37, [(7.41, [(7.37, [(7.03,
4.96, 3.82, 3.32, 5.38, 7.59, 5.50, 6.73, 7.83, 7.78, 7.42,
5.48, 4.22, 3.67, 5.94, 8.39, 6.08, 7.44, 8.65, 8.60, 8.20,
5.75)] 4.43)] 3.84)] 6.22)] 8.79)] 6.37)] 7.79)] 9.06)] 9.01)] 8.59)]
Table 3 ~ ¼ 1, m ~ ¼ 0. Ranking of alternative using F-TODIM and IF-TODIM for l
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Alternative
F-TODIM
IF-TODIM
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10
0.3528 0.1123 0 0.4617 0.9583 0.4951 0.7264 1.0000 0.9213 0.8889
0.3475 0.1136 0 0.4568 0.9577 0.4944 0.7250 1.0000 0.9290 0.8882
It is noted from Table 3 and Fig. 3 that the ranking of the alternatives provided by IF-TODIM is in agreement with the results obtained by F-TODIM as expected. The slightly small difference in the ni values in Table 3 are due to the formulae used to calculate the distance between the trapezoidal fuzzy numbers [15] and between intuitionistic trapezoidal fuzzy numbers presented in this work according to Eq. (3). Next, the degree of membership is changed to 80% and its degree of non-membership to 20%. By changing the degree of ~ A8 ¼ 1; membership and non-membership for alternative 8 from l ~A8 ¼ 0:2, the alternative 8 is no longer the ~ A8 ¼ 0:8; m m~A8 ¼ 0 to l best as shown in Fig. 5. In this case, the alternative 5 became the best one. Next, a sensitivity study was carried out in order to investigate the influence of the degree of membership and non-membership, m and l respectively, by varying them for the second best alternative A5. It was altered gradually the degree of membership and
Fig. 4. Prospect function value for the case of oil spill in the sea using h = 1 and h = 2.5.
Fig. 5. Ranking of the alternatives for the case degree of membership and non~A8 ¼ 0 as compared to l ~A8 ¼ 0:2. ~ A8 ¼ 1, m ~ A8 ¼ 0:8, m membership l
Q1 Please cite this article in press as: R.A. Krohling et al., IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.08.028
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l~ A5
m~A5
ni
Alternative order
1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.50 0.15
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.50 0.80
0.9471 0.9029 0.8568 0.8393 0.6952 0.6866 0.6252 0.5321 0.4800
A8 A5 A9 A10 A7 A6 A4 A1 A2 A3 A8 A9 A5 A10 A7 A6 A4 A1 A2 A3 A8 A9 A10 A5 A7 A6 A4 A1 A2 A3 A8 A9 A10 A5 A7 A6 A4 A1 A2 A3 A8 A9 A10 A7 A5 A6 A4 A1 A2 A3 A8 A9 A10 A7 A5 A6 A4 A1 A2 A3 A8 A9 A10 A7 A5 A6 A4 A1 A2 A3 A8 A9 A10 A7 A5 A6 A4 A1 A2 A3 A8 A9 A10 A7 A6 A5 A4 A1 A2 A3
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non-membership for the first criterion, Oil at the cost (OC), and was kept the same settings for the second criterion, Oil intercepted (OI). So, it is possible to observe in Table 4 that when the degree of nonmembership increases, the alternative 5 begins to fall in the ranking.
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4. Conclusions
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In this article, an intuitionistic fuzzy TODIM method, named IF-TODIM has been presented, which is able to tackle MCDM problems affected by uncertainty represented by intuitionistic trapezoidal fuzzy numbers using its membership and non-membership degree. The IF-TODIM method has been investigated on two case studies: Firstly, for a supply chain selection, and next for determination of the best combat responses in case of oil spill in the sea. It was noticed that the rank of alternatives obtained by IF-TODIM depends not only on the membership degree but also on the non-membership degree of the TIFN. The fuzzy TODIM, in its formulation, is only applicable to a fuzzy decision matrix. On the other hand, the presented IF-TODIM method can be applied to more challenging MCDM problems. The method is currently being expanded to group decision making problems.
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Acknowledgements
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The authors thank Talles T.M. de Souza for the plots of the prospect functions and Rodolfo Lourenzutti for discussions on intuitionistic fuzzy numbers. We are very grateful to reviewers for the valuable suggestions, which improve the quality of the manuscript. R.A. Krohling would like to thank the financial support of the Brazilian Agency CNPq under Grant No. 303577/2012-6.
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References
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[1] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87– 96. [2] K.T. Atanassov, G. Pasi, R. Yager, Intuitionistic fuzzy interpretations of multicriteria multi-person and multi-measurement tool decision making, International Journal of Systems Science 36 (2005) 859–868.
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[3] R.E. Bellman, L.A. Zadeh, Decision-making in a fuzzy environment, Management Science 17 (1970) 141–164. [4] F.E. Boran, M. Kurt, D. Akay, A multi-criteria intuitionistic fuzzy group decision making for selection of supplier with TOPSIS method, Expert Systems with Applications 36 (2009) 11363–11368. [5] Y. Chen, B. Li, Dynamic multi-attribute decision making model based on triangular intuitionistic fuzzy numbers, Scientia Iranica 18 (2011) 268–274. [6] D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk, H. Prade, Terminological difficulties in fuzzy set theory-the case of intuitionistic fuzzy sets, Fuzzy Sets and Systems 156 (2005) 485–491. [7] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980. [8] Z.-P. Fan, X. Zhang, F.-D. Chen, Y. Liu, Extended TODIM method for hybrid multiple attribute decision making problems, Knowledge-Based systems 42 (2013) 40–48. [9] L.F.A.M. Gomes, M.M.P.P. Lima, TODIM: basics and application to multicriteria ranking of projects with environmental impacts, Foundations of Computing and Decision Sciences 16 (1992) 113–127. [10] L.F.A.M. Gomes, L.A.D. Rangel, An application of the TODIM method to the multicriteria rental evaluation of residential properties, European Journal of Operational Research 193 (2009) 204–211. [11] P. Grzegorzewski, Distances between intuitionistic fuzzy sets and/or intervalvalued fuzzy sets based on the Hausdorff metric, Fuzzy Sets and Systems 148 (2004) 319–328. [12] C.L. Hwang, K. Yoon, Multiple Attribute Decision Making, Springer-Verlag, Berlin, 1981. [13] D. Kahneman, A. Tversky, Prospect theory: an analysis of decision under risk, Econometrica 47 (1979) 263–292. [14] R.A. Krohling, V.C. Campanharo, Fuzzy TOPSIS for group decision making: a case study for accidents with oil spill in the sea, Expert Systems with Applications 38 (2011) 4190–4197. [15] R.A. Krohling, T.T.M. de Souza, Combining prospect theory and fuzzy numbers to multi-criteria decision making, Expert Systems with Applications 39 (2012) 11487–11493. [16] L. Lin, X.H. Yuan, Z.Q. Xia, Multicriteria fuzzy decision-making methods based on intuitionistic fuzzy sets, Journal of Computer and System 73 (2007) 84–88. [17] H.W. Liu, G.J. Wang, Multi-criteria decision-making methods based on intuitionistic fuzzy sets, European Journal of Operational Research 179 (2007) 220–233. [18] P. Salminen, Solving the discrete multiple criteria problem using linear prospect theory, European Journal of Operational Research 72 (1994) 146–154. [19] L. Shen, H. Wang, X. Feng, Some arithmetic aggregation operators within intuitionistic trapezoidal fuzzy setting and their application to group decision making, MSIE (2011) 1053–1059. [20] E. Szmidt, J. Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems 114 (2000) 505–518. [21] A. Tversky, D. Kahneman, Advances in prospect theory: cumulative representation of uncertainty, Journal of Risk and Uncertainty 5 (1992) 297– 323. [22] J.Q. Wang, Z. Zhang, Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision making problems, Journal of Systems Engineering and Electronics 20 (2009) 321–326. [23] J.Q. Wang, Z. Zhang, Multi-criteria decision-making method with incomplete certain information based on intuitionistic fuzzy number, Control and Decision 24 (2009) 226–230. [24] G.W. Wei, Some arithmetic aggregation operators with intuitionistic trapezoidal fuzzy numbers and their application to group decision making, Journal of Computers 5 (2010) 345–351. [25] G.-W. Wei, Maximizing deviation method for multiple attribute decision making in intuitionistic fuzzy setting, Knowledge-Based systems 21 (2008) 833–836. [26] G. Wei, H. Wang, ITFWG operator and its application to decision making, Chinese Control and Decision Conference (2009) 5904–5909. [27] Z. Xu, Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making, Fuzzy Optimization and Decision Making 6 (2007) 109–121. [28] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353.
Q1 Please cite this article in press as: R.A. Krohling et al., IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.08.028
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Contents lists available at ScienceDirect
Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
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Short Communication
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IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making
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a 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 Department of Informatics, UFES – Federal University of Espirito Santo, Av. Fernando Ferrari, 514, CEP 29075-910 Vitória, Espírito Santo, ES, Brazil
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a r t i c l e
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Renato A. Krohling a,⇑, André G.C. Pacheco b, André L.T. Siviero b
i n f o
Article history: Received 28 February 2013 Received in revised form 19 June 2013 Accepted 22 August 2013 Available online xxxx
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Keywords: Multi-criteria decision making (MCDM) Intuitionistic fuzzy numbers Prospect function value IF-TODIM
a b s t r a c t The recently developed fuzzy TODIM (an acronym in Portuguese for iterative multi-criteria decision making) method using fuzzy numbers has been applied to uncertain MCDM problems with promising results. In this paper, a more general approach to the fuzzy TODIM, which takes into account the membership and the non-membership of the fuzzy information is considered. So, the fuzzy TODIM method has been extended to handle intuitionistic fuzzy information. This way, it is possible to tackle more challenging MCDM problems. Two case studies are used to illustrate and show the suitability of the developed method. 2013 Elsevier B.V. All rights reserved.
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1. Introduction
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Multi-criteria decision making (MCDM) problems occur in different areas of science and engineering [12]. Several research efforts have been made in order to develop new methods or to improve existing ones. Typical challenges for MCDM methods are uncertainty, risk, etc. In this sense, the theory of fuzzy sets and fuzzy logic developed by Zadeh [28] has been used to model uncertainty or lack of knowledge and applied to a variety of MCDM problems. Bellman and Zadeh [3] introduced the theory of fuzzy sets in MCDM problems as an effective approach to treat vagueness, lack of knowledge and ambiguity inherent in the human decision making process which are known as fuzzy multi-criteria decision making (FMCDM). For real world-problems the decision matrix is affected by uncertainty, which can be modeled using fuzzy numbers [7]. Another important aspect of decision making is to consider the risk attitude/preferences of the decision maker in MCDM. Prospect theory developed by Kahneman and Tversky [13] is a descriptive model of individual decision making under condition of risk. In turn, Tversky and Kahneman [21] proposed the cumulative prospect theory, which capture psychological aspects of decision making under risk. In prospect theory, the outcomes are expressed by means of gains and losses from a reference alternative [18]. The
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⇑ Corresponding author. Tel.: +55 2799475139. E-mail addresses: [email protected] (R.A. Krohling), pacheco.comp@ gmail.com (A.G.C. Pacheco), [email protected] (A.L.T. Siviero).
value function in prospect theory assumes a S-shape concave above the reference alternative, which reflects the aversion of risk in face of gains; and the convex part below the reference alternative reflects the propensity to risk in case of losses. One of the first MCDM methods based on prospect theory was proposed by Gomes and Lima [9]. Despite its effectiveness and simplicity in concept, this method presents some shortcomings because of its inability to deal with uncertainty and imprecision inherent in the process of decision making. In the original formulation of TODIM (an acronym in Portuguese for Iterative Multi-criteria Decision Making), the rating of alternatives, which composes the decision matrix, is represented by crisp values. Since the TODIM method [10] is not able to handle uncertainty, Krohling and de Souza [15] proposed a fuzzy TODIM to tackle uncertain MCDM problems. A clear advantage of this method is its ability to treat uncertain information using fuzzy numbers. Recently, Fan et al. [8] have presented an extension of the TODIM method, whereas the attribute values (crisp numbers, interval numbers and fuzzy numbers) are expressed in the format of random variables with cumulative distribution functions and next the classical TODIM can be applied. Atanasov [1] proposed a more general theory for fuzzy numbers, known as intuitionistic fuzzy numbers, which are described by a membership function and a non-membership function. In the last few years, intuitionistic fuzzy numbers have been applied to solve MCDM problems [2,27,16,17,4,5,19,22–26]. In this paper, based on the fuzzy TODIM method [15] and intuitionistic fuzzy
0950-7051/$ - see front matter 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.knosys.2013.08.028
Q1 Please cite this article in press as: R.A. Krohling et al., IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.08.028
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numbers [1], we propose the intuitionistic fuzzy TODIM method, for short, IF-TODIM to handle uncertain MCDM problems. The remainder of this article is organized as follows. In Section 2, some preliminary background on intuitionistic fuzzy numbers is provided. In Section 3, an intuitionistic fuzzy TODIM method is developed, which contains uncertainty in the decision matrix modeled by intuitionistic trapezoidal fuzzy numbers. In Section 4, case studies are presented to illustrate the method and the results show the feasibility of the approach. In Section 5, some conclusions and directions for future work are given.
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2. Intuitionistic fuzzy multi-criteria decision making
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Trapezoidal intuitionistic fuzzy numbers are commonly used for solving decision-making problems, where the available information is imprecise. Next, some basic definitions of intuitionistic fuzzy sets and fuzzy numbers are provided [1,6,11,20].
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2.1. Preliminaries on intuitionistic fuzzy sets and intuitionistic fuzzy numbers Definition 1. Let X be the universe of discourse. An intuitionistic e is characterized by a subset of X defined by fuzzy set A A = {hx, lA(x), mA(x)ijx 2 X}, where lA:X ? [0; 1] and mA:X ? [0; 1] with the condition 0 6 lA(x) + mA(x) 6 1j"x 2 X. The numeric values lA(x) and mA(x) stands for the degree of membership and the degree of non-membership of x in A, respectively. ~ is Definition 2. An intuitionistic trapezoidal fuzzy number a ~ ¼ ða1 ; a2 ; a3 ; a4 ; la~ ; ma~ Þ with membership function defined by a given by [22,24]:
la~ ðxÞ ¼ 118
8 xa 1 ~ a~ ; a1 6 x < a2 l > > a2 a1 > > > >
a3 < x 6 a4 otherwise;
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while the non-membership function is given by:
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8 ða xÞþm~ ðxa Þ 2 1 a~ > ; a1 6 x < a2 > a2 a1 > > > >
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137 138
ð1Þ
ð2Þ
~a~ represent the maximum value of member~ a~ and m The values l ~, respectively. For ship degree and non-membership degree of a instance, consider the intuitionistic trapezoidal fuzzy number (ITFN) hVG; 0.6, 0.3i = h0.5, 0.75, 0.95, 1; 0.6, 0.3i. In this case, a decision maker not only assess the rating of an alternative by using the linguistically defined trapezoidal fuzzy number (TFN) VG (Very Good) but also provides the degree of membership and non-membership, 0.6 and 0.3 respectively. ~ ¼ ða1 ; a2 ; a3 ; Definition 3. Let a trapezoidal fuzzy number a ~a~ Þ, then its expected value is calculated as ~ a~ ; m a4 ; l ~Þ ¼ ½ða1 þ a2 þ a3 þ a4 Þ ð1 þ l ~ a~ m ~a~ Þ=8. In addition, definitions Iða ~Þ ¼ Iða ~ Þ ðl ~a~ Þ and Hða ~Þ ¼ Iða ~Þ ðl ~a~ Þ are presented, ~ a~ m ~ a~ þ m for Sða which are known as score function and accuracy function, respectively [22]. Definition 4. Let two intuitionistic trapezoidal fuzzy numbers ~ ¼ ðb1 ; b2 ; b3 ; b4 ; l ~ ¼ ða1 ; a2 ; a3 ; a4 ; l ~a~ Þ and b ~b~ Þ, then [22]: ~ a~ ; m ~ b~ ; m a
~ then a ~ ~Þ > SðbÞ ~ > b. If Sða ~ ~ then a ~ ~ ~ ~ ¼ b. If SðaÞ ¼ SðbÞ and If HðaÞ ¼ HðbÞ ~ and Hða ~ then a ~ ~Þ ¼ SðbÞ ~Þ > HðbÞ ~ > b. If Sða
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Definition 5. Let two intuitionistic trapezoidal fuzzy numbers ~ ¼ ðb1 ; b2 ; b3 ; b4 ; l ~ ¼ ða1 ; a2 ; a3 ; a4 ; l ~a~ Þ and b ~b~ Þ, then the dis~ a~ ; m ~ b~ ; m a tance between them is calculated as [22]:
~ ¼ ~; bÞ dða
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1 ~a~ Þ a1 ð1 þ l ~b~ Þ b1 j þ jð1 þ l ~ a~ m ~ b~ m ~ a~ jð1 þ l 8 ~a~ Þ a2 ð1 þ l ~b~ Þ b2 j þ jð1 þ l ~a~ Þ a3 ~ b~ m ~ a~ m m ~b~ Þ ~ b~ m ð1 þ l
~ a~ m ~a~ Þ a4 ð1 þ l ~ b~ m ~b~ Þ b4 j : b3 jþjð1 þ l
ð3Þ
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~ ¼ ðVL; 0:8; 0:1Þ ¼ ð0:1; 0:2; 0:3; a For instance, consider ~ ¼ ðEH; 0:7; 0:2Þ ¼ ð0:7; 0:8; 0:9; 0:95; 0:7; 0:2Þ, 0:4; 0:8; 0:1Þ and b where VL and EH are linguistic definitions of the trapezoidal fuzzy numbers, Very Low and Extremely High, respectively. The distance ~ ¼ 0:4156. ~; bÞ between them is dða
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2.2. Decision making problem with uncertain decision matrix
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Let us consider the fuzzy decision matrix A, which consists of alternatives and criteria, described by:
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xij where A1, A2, . . . , Am are alternatives, C1, C2, . . . , Cn are criteria, ~ intuitionistic trapezoidal fuzzy numbers that indicates the rating of the alternative Ai with respect to criterion Cj. The weight vector W = (w1, w2, . . . , wn) composed of the individual weights wj P (j = 1, . . . , n) for each criterion Cj satisfying nj¼1 wj ¼ 1. In the following section, the method is presented.
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2.3. IF-TODIM – An intuitionistic fuzzy TODIM method
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For information on the TODIM method the reader is referred to Gomes and Rangel [10]. The intuitionistic fuzzy TODIM method, for short, IF-TODIM, which is an extension of the fuzzy TODIM method [15], is described in the following steps:
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Step 1: The criteria are normally classified into two types: benefit and cost. The intuitionistic trapezoidal fuzzy-decision e ¼ ½~ matrix A xij mxn with i = 1, . . . , m, and j = 1, . . . , n is normalized which results the correspondent fuzzy decision matrix e ¼ ½~rij . The fuzzy normalized value ~r ij is calculated as: R mxn
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max a4ij akij r kij ¼ with k ¼ 1; 2; 3; 4 for cost criteria maxi a4ij mini a1ij akij min a1ij with k ¼ 1; 2; 3; 4 r kij ¼ maxi a4ij mini a1ij for benefit criteria
ð4Þ
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We denote ak of aij = {(a1, a2, a3, a4)} as akij , e.g., a3ij ¼ a3 . e i over each Step 2: Calculate the dominance of each alternative R e alternative R j using the following expression: m X ei; R ejÞ ¼ ei; R e j Þ 8ði; jÞ dð R /c ð R
ð5Þ
c¼1
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Fig. 1. Prospect value function.
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where
ffiffiffiffiffiffiffiffiffiffiffiffi 8v u wrc dð~r ic ; ~xjc Þ if ð~r ic > ~r jc Þ > m u > > tX > > w rc > > > c¼1 > > < if ð~r ic ¼ ~r jc Þ e e /c ð R i ; R j Þ ¼ 0; vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ! > u > m > u X > > u > wrc > t > > c¼1 : 1 dð~ric ; ~xjc Þ if ð~r ic < ~r jc Þ h wrc
ð6Þ
3
3. Experimental results
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3.1. Case study 1 – supply chain selection
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In the first case study, the supply chain selection problem investigated by Wei and Wang [26] is used as the benchmark, who also developed a decision making method using intuitionistic trapezoidal fuzzy numbers. In this problem, the alternatives are five suppliers, evaluated according to four criteria: product quality (C1), service (C2), delivery (C3) and price (C4). It is known that C1, C2 and C3 are benefit criteria and C4 is a cost criterion. The weight vector associated to each criterion is W = (w1, w2, w3, w4) = (0.35, 0.2333, 0.3, 0.1167). The intuitionistic trapezoidal fuzzy decision matrix is described in Table 1. The factor of attenuation of losses h, was set to h = 1 [10] but the value h = 2.5 has also been used [15]. The IF-TODIM method was applied to the decision matrix given in Table 1. The plot of the prospect function value for each one of the partial dominance using h = 1 and h = 2.5 is depicted in Fig. 2. The order of the alternatives obtained is A2 A5 A4 A3 A1, which is in agreement with those obtained in [26]. As we can notice in Fig. 2 the prospect function value in the negative quadrant for h = 1 presents a deeper slope than for h = 2.5.
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3.2. Case study 2 – oil spill in the sea
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This decision making problem is composed of ten alternatives and two criteria. According to our notation, the criteria C1 oil at the coast (OC) is a cost criterion and the criterion C2 oil intercepted (OI) is a benefit criterion. Firstly, the decision matrix has been normalized. In this study, data are affected by uncertainty because the simulation of oil spots depends on several factors such as quantity and type of oil spilled, location of spill, weather and ocean conditions, among others. For a detailed description of how the data have been obtained the reader is referred to Krohling and Campanharo [14]. The first experiment was to examine the new method considering that all information is 100% certain, i.e., the degree of membership is 1 and the degree of non-membership is 0. For validation purposes our results are compared with those obtained by fuzzy TODIM as reference [15]. Similar results are expected for this particular case. The IF-TODIM method has been applied to the decision matrix with intuitionistic trapezoidal fuzzy numbers (see Table 2). Q4 Similar to the previous case, experiments with IF-TODIM using h = 1 and h = 2.5 have been carried out. The ranking of the alternatives using h = 1 is listed in Table 3 and depicted in Fig. 3. The ranking results of the experiments using h = 2.5 are not shown since there is no change in the ranking of the alternatives. The plot of the prospect function value for each one of the partial dominance values using h = 1 and h = 2.5 is depicted in Fig. 4.
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ei; R e j Þ, denoted by partial dominance, represents the The term /c ð R ei; R e j Þ when comcontribution of the criterion c to the function dð R paring the alternative i with alternative j. The values ~r ic and ~r jc are the rating of the alternatives i and j, respectively with respect to criterion c. The value wrc represents the weight of criterion c divided c by the weight of the reference r, i.e., wrc ¼ w , whereas the latter is wr the criterion that has the greater weight. The term dð~ric ; ~rjc Þ stands for the distance between the two intuitionistic fuzzy numbers ~r ic and ~r jc , calculated by Eq. (3). Three cases can occur in Eq. (6): (i) if ð~ric > ~r jc Þ, it represents a gain; (ii) if ð~r ic ¼ ~r jc Þ, it is nil; and (iii) if ð~ric < ~r jc Þ, it represent a loss. Definitions (3) and (4) are used in each case. The parameter h represents the attenuation factor of the losses. Different values of this parameter lead to different shapes of the prospect value function in the negative quadrant as illustrated in Fig. 1. The final matrix of dominance is obtained by summing up the partial matrices of dominance for each criterion. Step 3: Calculate the global value of the alternative i by normalizing the final matrix of dominance according to the following expression:
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P
ni ¼
P
dði; jÞ min dði; jÞ P P max dði; jÞ min dði; jÞ
ð5Þ
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Sorting the values ni provides the rank of each alternative. The best alternatives are those that have higher value ni. Next, the approach is illustrated on two case studies.
Table 1 Decision matrix for the supply chain selection [26].
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Fig. 2. Prospect function value using h = 1 and h = 2.5.
~ ¼ 1, Fig. 3. Ranking of the alternatives obtained using F-TODIM and IF-TODIM for l m~ ¼ 0.
Table 2 Decision matrix for the oil spill in the sea [14]. Alternatives
C1 (103)
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10
[(7.76, [(8.85, [(9.34, [(7.38, [(5.27, [(7.30, [(6.16, [(5.16, [(5.27, [(5.64,
8.20, 9.35, 9.86, 7.79, 5.56, 7.70, 6.50, 5.45, 5.57, 5.96,
C2 (103) 9.06, 9.49)] 10.33, 10.82)] 10.89, 11.41)] 8.61, 9.02)] 6.15, 6.44)] 8.51, 8.92)] 7.19, 7.53)] 6.02, 6.31)] 6.15, 6.44)] 6.58, 6.90)]
[(4.70, [(3.62, [(3.15, [(5.09, [(7.19, [(5.21, [(6.37, [(7.41, [(7.37, [(7.03,
4.96, 3.82, 3.32, 5.38, 7.59, 5.50, 6.73, 7.83, 7.78, 7.42,
5.48, 4.22, 3.67, 5.94, 8.39, 6.08, 7.44, 8.65, 8.60, 8.20,
5.75)] 4.43)] 3.84)] 6.22)] 8.79)] 6.37)] 7.79)] 9.06)] 9.01)] 8.59)]
Table 3 ~ ¼ 1, m ~ ¼ 0. Ranking of alternative using F-TODIM and IF-TODIM for l
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Alternative
F-TODIM
IF-TODIM
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10
0.3528 0.1123 0 0.4617 0.9583 0.4951 0.7264 1.0000 0.9213 0.8889
0.3475 0.1136 0 0.4568 0.9577 0.4944 0.7250 1.0000 0.9290 0.8882
It is noted from Table 3 and Fig. 3 that the ranking of the alternatives provided by IF-TODIM is in agreement with the results obtained by F-TODIM as expected. The slightly small difference in the ni values in Table 3 are due to the formulae used to calculate the distance between the trapezoidal fuzzy numbers [15] and between intuitionistic trapezoidal fuzzy numbers presented in this work according to Eq. (3). Next, the degree of membership is changed to 80% and its degree of non-membership to 20%. By changing the degree of ~ A8 ¼ 1; membership and non-membership for alternative 8 from l ~A8 ¼ 0:2, the alternative 8 is no longer the ~ A8 ¼ 0:8; m m~A8 ¼ 0 to l best as shown in Fig. 5. In this case, the alternative 5 became the best one. Next, a sensitivity study was carried out in order to investigate the influence of the degree of membership and non-membership, m and l respectively, by varying them for the second best alternative A5. It was altered gradually the degree of membership and
Fig. 4. Prospect function value for the case of oil spill in the sea using h = 1 and h = 2.5.
Fig. 5. Ranking of the alternatives for the case degree of membership and non~A8 ¼ 0 as compared to l ~A8 ¼ 0:2. ~ A8 ¼ 1, m ~ A8 ¼ 0:8, m membership l
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l~ A5
m~A5
ni
Alternative order
1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.50 0.15
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.50 0.80
0.9471 0.9029 0.8568 0.8393 0.6952 0.6866 0.6252 0.5321 0.4800
A8 A5 A9 A10 A7 A6 A4 A1 A2 A3 A8 A9 A5 A10 A7 A6 A4 A1 A2 A3 A8 A9 A10 A5 A7 A6 A4 A1 A2 A3 A8 A9 A10 A5 A7 A6 A4 A1 A2 A3 A8 A9 A10 A7 A5 A6 A4 A1 A2 A3 A8 A9 A10 A7 A5 A6 A4 A1 A2 A3 A8 A9 A10 A7 A5 A6 A4 A1 A2 A3 A8 A9 A10 A7 A5 A6 A4 A1 A2 A3 A8 A9 A10 A7 A6 A5 A4 A1 A2 A3
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non-membership for the first criterion, Oil at the cost (OC), and was kept the same settings for the second criterion, Oil intercepted (OI). So, it is possible to observe in Table 4 that when the degree of nonmembership increases, the alternative 5 begins to fall in the ranking.
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4. Conclusions
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In this article, an intuitionistic fuzzy TODIM method, named IF-TODIM has been presented, which is able to tackle MCDM problems affected by uncertainty represented by intuitionistic trapezoidal fuzzy numbers using its membership and non-membership degree. The IF-TODIM method has been investigated on two case studies: Firstly, for a supply chain selection, and next for determination of the best combat responses in case of oil spill in the sea. It was noticed that the rank of alternatives obtained by IF-TODIM depends not only on the membership degree but also on the non-membership degree of the TIFN. The fuzzy TODIM, in its formulation, is only applicable to a fuzzy decision matrix. On the other hand, the presented IF-TODIM method can be applied to more challenging MCDM problems. The method is currently being expanded to group decision making problems.
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Acknowledgements
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The authors thank Talles T.M. de Souza for the plots of the prospect functions and Rodolfo Lourenzutti for discussions on intuitionistic fuzzy numbers. We are very grateful to reviewers for the valuable suggestions, which improve the quality of the manuscript. R.A. Krohling would like to thank the financial support of the Brazilian Agency CNPq under Grant No. 303577/2012-6.
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References
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292 293 294 295 296 297 298 299 300 301 302 303
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313 314 315 316 317
[1] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87– 96. [2] K.T. Atanassov, G. Pasi, R. Yager, Intuitionistic fuzzy interpretations of multicriteria multi-person and multi-measurement tool decision making, International Journal of Systems Science 36 (2005) 859–868.
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[3] R.E. Bellman, L.A. Zadeh, Decision-making in a fuzzy environment, Management Science 17 (1970) 141–164. [4] F.E. Boran, M. Kurt, D. Akay, A multi-criteria intuitionistic fuzzy group decision making for selection of supplier with TOPSIS method, Expert Systems with Applications 36 (2009) 11363–11368. [5] Y. Chen, B. Li, Dynamic multi-attribute decision making model based on triangular intuitionistic fuzzy numbers, Scientia Iranica 18 (2011) 268–274. [6] D. Dubois, S. Gottwald, P. Hajek, J. Kacprzyk, H. Prade, Terminological difficulties in fuzzy set theory-the case of intuitionistic fuzzy sets, Fuzzy Sets and Systems 156 (2005) 485–491. [7] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980. [8] Z.-P. Fan, X. Zhang, F.-D. Chen, Y. Liu, Extended TODIM method for hybrid multiple attribute decision making problems, Knowledge-Based systems 42 (2013) 40–48. [9] L.F.A.M. Gomes, M.M.P.P. Lima, TODIM: basics and application to multicriteria ranking of projects with environmental impacts, Foundations of Computing and Decision Sciences 16 (1992) 113–127. [10] L.F.A.M. Gomes, L.A.D. Rangel, An application of the TODIM method to the multicriteria rental evaluation of residential properties, European Journal of Operational Research 193 (2009) 204–211. [11] P. Grzegorzewski, Distances between intuitionistic fuzzy sets and/or intervalvalued fuzzy sets based on the Hausdorff metric, Fuzzy Sets and Systems 148 (2004) 319–328. [12] C.L. Hwang, K. Yoon, Multiple Attribute Decision Making, Springer-Verlag, Berlin, 1981. [13] D. Kahneman, A. Tversky, Prospect theory: an analysis of decision under risk, Econometrica 47 (1979) 263–292. [14] R.A. Krohling, V.C. Campanharo, Fuzzy TOPSIS for group decision making: a case study for accidents with oil spill in the sea, Expert Systems with Applications 38 (2011) 4190–4197. [15] R.A. Krohling, T.T.M. de Souza, Combining prospect theory and fuzzy numbers to multi-criteria decision making, Expert Systems with Applications 39 (2012) 11487–11493. [16] L. Lin, X.H. Yuan, Z.Q. Xia, Multicriteria fuzzy decision-making methods based on intuitionistic fuzzy sets, Journal of Computer and System 73 (2007) 84–88. [17] H.W. Liu, G.J. Wang, Multi-criteria decision-making methods based on intuitionistic fuzzy sets, European Journal of Operational Research 179 (2007) 220–233. [18] P. Salminen, Solving the discrete multiple criteria problem using linear prospect theory, European Journal of Operational Research 72 (1994) 146–154. [19] L. Shen, H. Wang, X. Feng, Some arithmetic aggregation operators within intuitionistic trapezoidal fuzzy setting and their application to group decision making, MSIE (2011) 1053–1059. [20] E. Szmidt, J. Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems 114 (2000) 505–518. [21] A. Tversky, D. Kahneman, Advances in prospect theory: cumulative representation of uncertainty, Journal of Risk and Uncertainty 5 (1992) 297– 323. [22] J.Q. Wang, Z. Zhang, Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision making problems, Journal of Systems Engineering and Electronics 20 (2009) 321–326. [23] J.Q. Wang, Z. Zhang, Multi-criteria decision-making method with incomplete certain information based on intuitionistic fuzzy number, Control and Decision 24 (2009) 226–230. [24] G.W. Wei, Some arithmetic aggregation operators with intuitionistic trapezoidal fuzzy numbers and their application to group decision making, Journal of Computers 5 (2010) 345–351. [25] G.-W. Wei, Maximizing deviation method for multiple attribute decision making in intuitionistic fuzzy setting, Knowledge-Based systems 21 (2008) 833–836. [26] G. Wei, H. Wang, ITFWG operator and its application to decision making, Chinese Control and Decision Conference (2009) 5904–5909. [27] Z. Xu, Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making, Fuzzy Optimization and Decision Making 6 (2007) 109–121. [28] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353.
Q1 Please cite this article in press as: R.A. Krohling et al., IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria decision making, Knowl. Based Syst. (2013), http://dx.doi.org/10.1016/j.knosys.2013.08.028
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