Intuitionistic Hesitant Fuzzy VIKOR method for Multi-Criteria Group Decision Making

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1 Inter national Journal of Pure and Applied Mathematics Volume 113 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Intuitionistic Hesitant Fuzzy VIKOR method for Multi-Criteria Group Decision Making S.Narayanamoorthy 1 and S.Geetha 2 1 Department of Mathematics, Bharathiar University, Coimbatore, Tamil Nadu, India snm phd@yahoo.co.in 2 Department of Mathematics, Bharathiar University, Coimbatore, Tamil Nadu, India geethakarkuzhali@gmail.com February 10, 2017 Abstract Intuitionistic hesitant fuzzy set is a beneficial tool to indenture with vagueness and ambiguity in the multi-criteria decision making problems. The VIse kriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method of compromise ranking determines a compromise solution, provides a maximum group utility for the majority and a minimum of an individual regret for the opponent. The main purpose of this paper is to confine the VIKOR method for intuitionistic hesitant fuzzy set. The degree of importance of each criterion determined by entropy method. AMS Subject Classification:94D05, 90B50, 91B06. Key Words and Phrases:: Intuitionistic hesitant fuzzy, VIKOR ijpam.eu

2 1 Introduction Fuzzy set theory is one of the theories enabling us to group the phenomenon of vagueness in a formal way. Zadeh [15] suggested employing the fuzzy set theory as a modeling tool for complex systems in As a generalization of fuzzy set Krassimir Atanassov [1] introduced the concept of intuitionistic fuzzy set (IFS). Torra and Narukawa [9] and Torra [8] proposed the hesitant fuzzy set and discussed the relationship between a hesitant fuzzy set and an intuitionistic fuzzy set. Zhu [18] proposed dual/intuitionistic hesitant fuzzy sets [DHFS/ IHFS] which encompass fuzzy sets, intuitionistic fuzzy sets and hesitant fuzzy sets as special case. Entropy is uncertainty, it quantified the fuzziness of discourse system and connects naturally with conditioning. Zadeh [16] proposed several entropy formulas based on shannon s function. Zeng and Li [17] proposed new concept of entropy for interval valued fuzzy sets. Xia and Xu [13] proposed some cross-entropy and entropy fromulas for intuitionistic fuzzy set and applied them to group decision making. Xu and Xia [14] developed the concept of entropy and cross-entropy for hesitant fuzzy information and discussed their properties. Na Zhao and Zeshui Xu [desktop] investigate the entropy models on dual hesitant fuzzy information. Opricovic [5] developed the VIKOR method as a multi-criteria decision making method to solve a discrete decision problem with non-commensurable and conflicting criteria. VIKOR method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria. Opricovic and Tzeng [6] developed fuzzy VIKOR with incomplete infromation for analyze of land use stratergies. Vahdani Hadipour and Sadaghiani Amiri [10] proposed an interval-valued fuzzy VIKOR method for solving MCDM problems using interval-valued fuzzy concepts. Park Cho and Young [7] extended the VIKOR method for multiple criteria group decision making in interval-valued intuitionistic fuzzy environment. Wan Wang and Dong [11] extended the VIKOR method with triangular intuitionistic fuzzy numbers for solving the multi-criteria decision making problems. Huchang [2] extend the classical VIKOR method to hesitant fuzzy circumstances. In this paper, we propose an intuitionistic hesitant fuzzy VIKOR method using entropy weights. ijpam.eu

3 Definition 1. Let X be a fixed set. The hesitant fuzzy set on X is in terms of a function α that when applied to X returns a subset of [0, 1], which can be represented as the following mathematical symbol: A = ( x, α(x) /x X) Where α = (x) is a set of some values in[0, 1], denoting the possible membership degrees of the element x X to the set A. Definition 2. A DHFS/ IHFS D on X is defined in terms of two functions h(x), g(x), both return a subset of [0, 1] when they are applied to X. Mathematically, it is represented by following expression: D = x, {h(x), g(x)}, x X Where {h(x)} and {g(x)} represent the possible membership and non-membership degrees of the elements x Xto the set D, respectively and satisfy the following conditions: γ 0, η 1, 0 γ + + η + 1 γ h(x), η g(x) γ +, η + are defined as follows: γ + h + (x) = η + g + (x) = γ h(x) η g(x) max {γ} x X max {η} x X Definition 3. An Entropy on IHFE α = µ, η is a real-valued function E : H [0, 1], which is related to δ σ(i) = µσ(i) η σ(i) and φ σ(i) = 1 µσ(i) η σ(i) satisfies the following axiomatic requirements: (i) E(α) = 0, if and only if α = {0}, {1} or α = {1}, {0} ; (ii) E(α) = 1, if and only if α = {0}, {0} ; (iii) E(α) = E(α c ) (iv) E is monotonic decreasing with regard to δ σ(i) and monotonic increasing with respect to E is monotonic decreasing with regard to φ σ(i) for all i 1, 2,, l, where l = max(l µ, l η ) Definition 4. Let α 1 = {µ α1, η α1 } and α 2 = {µ α2, η α2 } be two IHFSs on X = {x 1, x 2,, x n } then the distance measure between ijpam.eu

4 α 1 and α 2 is defined as d(α 1, α 2 ), which satisfies the following properties: (1)0 d(α 1, α 2 ) 1; (2)d(α 1, α 2 ) = 0 if and only if d(α 1 = α 2 ) (3)d(α 1, α 2 ) = d(α 2, α 1 ) (4) Let α 3 be any IHFS, if α 1 α 2 α 3, then d(α 1, α 2 ) d(α 1, α 2 ) and d(α 2, α 3 ) d(α 2, α 3 ) The intuitionistic hesitant normalized Hamming distance: d = 1 2n n i=1 [ 1 l l µ σ(j) α 1 j=1 µ σ(j) α m m ηα σ(j) 1 j=1 ] ηα σ(j) 2 and a intuitionistic hesitant normalized Euclidean distance: d = 1 2n n i=1 [ 1 l l µ σ(j) α 1 j=1 µ σ(j) α m m η σ(j) α 1 j=1 η σ(j) α 2 2 ] 1/2 where µ σ(j) α 1, µ α σ(j) 2 and ηα σ(j) 1, ηα σ(j) 2 are the j th largest values of memberships degree and non-membership degree of α 1 and α 2, respectively. Definition 5. Let d 1 = {h 1, g 1 } and d 2 = {h 2, g 2 } be any two IHFEs; then the score function of d i (i = 1, 2) is, S(d i ) = 1 µ 1 η(i = 1, 2) n(h i ) n(g i ) µ h i η g i and the accuracy function of d i (i = 1, 2) is P (d i ) = 1 µ + 1 η(i = 1, 2) l(h i ) l(g i ) µ h i η g i where l(h i ) and l(g i ) are the numbers of the elements in h 1 and g 1, respectively, then, (i) if S(d 1 ) > S(d 2 ), then d 1 is superior to d 2, denoted by d 1 > d 2. (ii) if S(d 1 ) = S(d 2 ), then (1)P (d 1 ) = P (d 2 ), then d 1 is equivalent to d 2, denoted by d 1 d 2. (2) P (d 1 ) > P (d 2 ), then d 1 is superior to d 2, denoted by d 1 > d 2. ijpam.eu

5 2 Algorithm for the Proposed Method In this section, we propose intuitionistic hesitant fuzzy VIKOR method for MCGDM problems with Entropy using Normalized Euclidean distance for IHFEs. Let A = A 1, A 2,, A n and C = C 1, C 2,, C m be set of n alternatives and of m criteria respectively. Assume that the performance of the alternative A i (i = 1, 2,, n)with respect to criteria C j (j = 1, 2,, m) is measured by an intuitionistic hesitant fuzzy element, h ij = { γ ij h } ij = { } ( µ γij ), ( η γij )/ µ ij, η ij γ ij. If two (or) more decision makers provide the same value, then the values comes only once in the h ij. Step 1: The information about the weight W j of the criteria C j is unknown completely, then we use an exact model of entropy weights for criteria weights. The intuitionistic hesitant fuzzy entropy measure is, E(A i ) = 1 l µ σ(i) η σ(i) λ + [ µ σ(i) + η σ(i)] λ 1 (1) l 2 i=1 Step 2: The intuitionistic hesitant fuzzy positive ideal solution (IHF-PIS) denoted by f + and intuitionistic hesitant fuzzy negative ideal solution (IHF-NIS) denoted by f, are given by following expressions: For benefit criteria, { h + = {(max µ γij ), (min η γij)} / γ ij h } ij (2) { h = {(min µ γij ), (max η γij)} / γ ij h } ij (3) For cost criteria, { h + = {(min µ γij ), (max η γij)} / γ ij h } ij { h = {(max µ γij ), (min η γij)} / γ ij h } ij Step 6: Compute the group utility and individual regret values of the alternatives as, l W j d(h + j U l = L h ij) 1,l d(h + j h j ) (6) j=1 (4) (5) ijpam.eu

6 W j d(h + j R l = L,l = max h ij) l d(h + j h j ) (7) Step 7: Calculate the value of Q i as follows, Q i = v (U l U ) (U + U ) + (l v) (R l R ) (R + R ) (8) Where, U + = min l U l, U = max l U l (9) R + = min l R l, R = max l R l (10) and v is the weight of the decision-making strategy the majority of criteria (or the maximum group utility). The compromise can be selected with voting by majority (v > 0.5), with consensus (v = 0.5), or with veto (v < 0.5). Step 8: Rank the alternatives according by the accuracy function of each U, R, and Q values in an increasing order. The result is a set of three ranking lists denoted as U [i], R [i], and Q [i]. Step 9: Propose the alternative i 1 corresponding to Q [1] (the smallest among Q i values) as a compromise solution if C1. The alternative i 1 has an acceptable advantage; in other words, Q [2] Q [1] DQ wheredq = 1/(n 1) and n is the number of alternatives. C2. The alternative i 1 is stable within the decision-making process; in other words, it is also the best ranked in U [i] orr [i]. If one of the above conditions is not satisfied, then a set of compromise solutions is proposed, which consists of: Alternatives i 1 and i 2 where i 1 = i 1 if only the condition C2 is not satisfied. Alternatives i 1, i 2,, i k if the condition C1 is not satisfied; and i k is determined by the relation Q [k] Q [1] < DQ for the maximum k where Q jk = Q [k] (the positions of these alternatives are in closeness) 3 Numerical Example Application of the proposed decision making method in a realistic scenario is demonstrate with a multiple criteria decision making ijpam.eu

7 problem adapted from Herrera and Herrera-Viedma [10] and Ye [11]. There is an investment company, which wants to invest a sum of money in the best option. There is a panel with four possible alternatives to invest the money: (1) A 1 is a car company; (2) A 2 is a food company; (3) A 3 is a computer company; (4) A 4 is a arms company. The investment company must take a decision according to the following three criteria: (1) C 1 is the risk analysis, (2) C 2 is the growth analysis; (3) C 3 is the environmental impact analysis. The four possible alternatives A i (i = 1, 2, 3, 4) are to be evaluated using the dual hesitant fuzzy information by three decision makers under the three criteria C j (j = 1, 2, 3) as listed in the following intuitionistic hesitant fuzzy decision matrix: {{0.5, 0.4, 0.3}, {0.4, 0.3}} {{0.6, 0.4}, {0.4, 02}} {{0.3, 0.2, 0.1}, {0.6, 0.5}} H = {{0.7, 0.6, 0.4}, {0.3, 0.2}} {{0.7, 0.6}, {0.3, 0.2}} {{0.7, 0.6, 0.4}, {0.2, 0.1}} {{0.6, 0.4, 0.3}, {0.3}} {{0.8, 0.7, 0.6}, {0.2, 0.1}} {{0.6, 0.5}, {0.3}} {{0.7, 0.6}, {0.2}} {{0.6, 0.5}, {0.3, 0.1}} {{0.4, 0.3}, {0.2, 0.3}} Solution: Step 1: The entropy values are calculated as follows by using the equation, w 1 = , w 2 = , w 3 = Step2: The intuitionistic hesitant fuzzy positive ideal solution (IHF- PIS) h + 1 = {{0.5, 0.4, 0.3}, {0.4, 0.3, 0.3}} h + 2 = {{0.7, 0.6, 0.6}, {0.2, 0.2, 0.2}} h + 3 = {{0.3, 0.2, 0.1}, {0.6, 0.5, 0.5}} The intuitionistic hesitant fuzzy negative ideal solution (IHF-NIS) h 1 = {{0.8, 0.7, 0.6}, {0.2, 0.1, 0.1}} h 2 = {{0.6, 0.4, 0.4}, {0.4, 0.3, 0.3}} h 3 = {{0.7, 0.6, 0.5}, {0.2, 0.1, 0.1}} Step 3: The group utility values and individual regrets values of the alternatives are calculated as follows, U + = {{0.7663, , }, {0.6174, , }} U = {{0.3756, , }, {0.5626, , }} ijpam.eu

8 The individual regrets values of the alternatives are calculated as follows, R = {{0.3917, , }, {0.3917, 0., 0.}} R + = {{0.0935, , }, {0.2805, , }} Step 4: The values of Q i are calculated by using equation (2.5), Q 1 = 2.6 Q 2 = Q 3 = 1.49 Q 4 = 0 Step 5: Rank the alternatives by sorting each Q values in an increasing order as follows, Q 4, Q 3, Q 2, Q 1. 4 Conclusion In the sophisticated world decision making is most prominent feature in human activities, in which the goals, constraints and the consequence of possible actions are not known precisely. VIKOR is a helpful tool for MCDM problems, particularly in a situation where the decision maker is not able or does not know to express his preferences at the beginning of system design. In this paper, we developed extension of the classical VIKOR method under intuitionistic hesitant fuzzy circumstances with hesitant fuzzy entropy. This method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria. References [1] Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Systems, 20(1) (1986), [2] Huchang Liao, Zeshui Xu, A VIKOR-based method for hesitant fuzzy multi-criteria decision making, Fuzzy Optimization and Decision Making, 12 (2013), [3] Hwang, C.L., Yoon, K. Multiple attribute decision making: Methods and applications, Berlin, Heidelberg: Verlag (1981). ijpam.eu

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