Chapter 6. Intuitionistic Fuzzy PROMETHEE Technique. AIDS stands for acquired immunodeficiency syndrome. AIDS is the final stage. 6.

Transcription

1 Chapter 6 Intuitionistic Fuzzy PROMETHEE Technique 6.1 Introduction AIDS stands for acquired immunodeficiency syndrome. AIDS is the final stage of HIV infection, and not everyone who has HIV advances to this stage. AIDS is the stage of infection that occurs when your immune system is badly damaged and one become vulnerable to opportunistic infections. When the number of CD4 cells falls below 200 cells per cubic millimeter of blood (200 cells/mm3), the person is considered to have progressed to AIDS (The CD4 count of an uninfected adult/adolescent who is generally in good health ranges from 500 cells/mm3 to 1,600 cells/mm3). One can also be diagnosed with AIDS if it develops one or more opportunistic infections, regardless of CD4 count. Without treatment, people who are diagnosed with AIDS typically survive about 3 years. Once someone has a dangerous opportunistic illness, life expectancy without treatment falls to about 1 year. People with AIDS need medical treatment to prevent death. 98

2 Due to advances in epidemiological modelling, the literature on uncertainty of infectious diseases has significantly grown. Recently, several authors have developed the concept of fuzzy set theory in the modelling of diseases [72, 108, 121, 130, 136]. In our day-to-day life, multi-criteria decision making (MCDM) problems are commonly occurred in every decision, such as in selecting a car to purchase, choosing an electronic product from paytm or snapdeal, selecting a person in an interview for a senior executive position, to rank the ARV drugs etc. MCDM is the process of selecting the best option from a set of alternatives for the purpose of covering personal needs or social requirements with respect to set of independent decisive factors or criteria. Brans [22] developed the concept of PROMETHEE, to obtain the partial or complete ranking of the alternatives on the basis of positive outranking flow, negative outranking flow and net outranking flow. In MCDM, PROMETHEE is used to recognize whether the alternatives are preferable, incomparable or indifferent to the others with respect to the set of criteria. There are numerous types of PROMETHEE techniques that have been developed for different situations with crisp input data [1, 2, 15, 23, 24, 25, 50, 206], but it cannot be applied to handle the fuzziness arises in the day to day life. To avoid the limitation arises in these techniques with crisp values, Le Teno & Mareschal [89] generalized the PROMETHEE in which the assessment values of alternatives over criteria are characterized into interval numbers. Consequently, Goumas & Lygerou [45] illustrated the fuzzy PROMETHEE into the situation where assessment values are represented in fuzzy numbers. In a while, various researchers extended the works on PROMETHEE technique based on different types of fuzzy illustration such as interval numbers, fuzzy numbers, fuzzy interval numbers and generalized fuzzy numbers [21, 43, 50, 93, 206]. These extensional forms of PROMETHEE technique cannot be applied to represent the support and 99

3 opposition information of decision maker. Therefore, to shirk the disadvantages of this technique with fuzzy values, Liao & Xu [96] developed the PROMETHEE technique for IFS, which is more valid in managing uncertain information. In this chapter, the intuitionistic fuzzy PROMETHEE (IF-PROMETHEE) technique is discussed to rank the various ARV drugs for HIV/AIDS treatment. In this chapter, section 6.2 presents the uncertain problem of HIV/AIDS infection, a method for transforming FSs in IFSs and the basic idea of PROMETHEE technique. Intuitionistic fuzzy entropy measure is developed in section 6.3 and compared with various existing measures, which reveals the effectiveness of the proposed entropy measure. In section 6.4, intuitionistic fuzzy PROMETHEE (IF-PROMETHEE) technique for multi-criteria decision making problems is discussed with the help of proposed entropy measure and the intuitionistic fuzzy preferences. To find the optimal ARV drug to reduce the infection of HIV, the proposed technique is used in section 6.5. Finally, conclusions are given in section Basic Concepts In this section, a decision making problem to rank the ARV drugs and the basic idea of PROMETHEE technique is presented Identified Problem of HIV/AIDS To find the optimal drug for the treatment of HIV/AIDS, a set of decision makers want to rank the performance of ARV drugs on the basis of different criteria. The set of ARV drugs are Didanosine (K 1 ), Nevirapine (K 2 ), Zidovudine (K 3 ) and Lamivudine (K 4 ). These drugs {K 1, K 2, K 3, K 4 } have to be ranked on their performance with respect to the (i) market price (H 1 ); (ii) effect on patients 100

4 (H 2 ); (iii) viral load (H 3 ); (iv) side effects (H 4 ); (v) growth of the virus (H 5 ) Basic Idea of PROMETHEE Technique Brans [22] proposed the idea of PROMETHEE technique, to handle the circumstances in which incomparability happens in pair wise comparisons [25]. This technique provides the information between and within the criteria, which are obvious and comprehensible for the forecaster and decision maker [96]. The procedural steps involved in classical PROMETHEE technique are as follows: Step I: Estimate the alternatives K = {K 1, K 2,..., K p } with respect to the criteria H = {H 1, H 2,..., H q }, which is given as in Table 6.1: Table 6.1: Decision matrix H 1 (.) H 2 (.)... H j (.)... H q (.) K 1 H 1 (K 1 ) H 2 (K 1 )... H j (K 1 )... H q (K 1 ) K 2 H 1 (K 2 ) H 2 (K 2 )... H j (K 2 )... H q (K 2 ) K i H 1 (K i ) H 2 (K i )... H j (K i )... H q (K k ) K p H 1 (K p ) H 2 (K p )... H j (K p )... H q (K p ) The deviations based on pair wise comparisons between all the alternatives with respect to each criterion are determined as follows: D k (K i, K j ) = H k (K i ) H k (K j ), (6.2.1) where D k (K i, K j ) represents the difference between the evaluations of the alternatives K i and K j on the criterion H k. Step II: Compute the degree of preference of the alternative K i to the alterna- 101

5 tive K j using the following function: P k (K i, K j ) = f k (D k (K i, K j )), K i, K j Q, (6.2.2) where the preference function f k transforms the difference between the evaluations of the alternatives K i and K j on the criterion H k into a preference degree ranging from 0 to 1. These functions are preferred by decision maker according to their quandary. Brans & Mareschal [23, 24, 25] has developed various types of preference function or generalized criteria, which are (i) usual criterion (ii) quasi criterion (iii) criterion with linear preference (iv) level criterion (v) criterion with indifference area and (vi) Gaussian criterion. Step III: Estimate the overall degree of preference or global preference index Π(K i, K j ) = q w k P k (K i, K j ), (6.2.3) k=1 where the weight w k (0 w k 1) represents the relative importance of the criterion H k and q k=1 w k = 1. In particular, when the number of criteria is not too large, then the decision makers can provide the weights. Step IV: To obtain the ranking of the alternatives, compute the positive and the negative outranking flows by using the following functions: and φ + (K i ) = φ (K i ) = 1 (p 1) 1 (p 1) Π(K i, δ) (6.2.4) δ P Π(δ, K i ), (6.2.5) where the positive outranking flow φ + (K i ) signifies the preference of the alternative K i over the other alternatives and the negative outranking flow φ (K i ) signifies that preference of other alternatives over the alternative K i. The greater value of φ + (K i ) and the smaller value of φ (K i ) denote the optimality of alternative K i. According to the value of the positive and negative outranking flows, the partial outranking can be derived as 102 δ P

6 (i) K i outranks K j if φ + (K i ) φ + (K j ) and φ (K i ) φ (K j ). (ii) Indifference between two alternatives happens when there is an equality in φ + and φ. (iii) φ + (K i ) > φ + (K j ) and φ (K i ) > φ (K j ) or φ + (K i ) < φ + (K j ) and φ (K i ) < φ (K j ) implies the incomparability between the alternatives. Step V: Obtain the complete ranking according to the value of the net outranking flow φ(k i ) = φ + (K i ) φ (K i ) for each alternative. 6.3 Entropy Measure for IFSs In this section, entropy measure for IFSs based on sine function is introduced and compared with existing entropy measures Intuitionistic Fuzzy Entropy Measure For every A IF S( Q), intuitionistic fuzzy entropy E(A) is defined as follows [62]: E(A) = 1 n n sin i=1 {( 1 µa (u i ) ν A (u i ) + π A (u i ) 1 + µ A (u i ) ν A (u i ) + π A (u i ) ) } π 2 (6.3.1) Theorem 6.1. The function E(A) given by (6.3.1) is an entropy measure for IFSs. Proof. It can easily be shown that E(A) satisfies the conditions (P1)-(P3) [Definition 1.8]. Now, we have to only prove the condition (P4). Let µ A (u i ) µ B (u i ), ν B (u i ) ν A (u i ) for µ B (u i ) ν B (u i ). Since µ A (u i ) µ B (u i ) ν B (u i ) ν A (u i ), it follows that µ A (u i ) ν A (u i ) µ B (u i ) ν B (u i ) and µ A (u i ) ν A (u i ) µ B (u i ) ν B (u i ). It implies 1 µ A (u i ) ν A (u i ) 1 µ B (u i ) ν B (u i ), 103

7 which can be written as Therefore, 1 µ A (u i ) ν A (u i ) + 1 µ A (u i ) ν A (u i ) 1 µ B (u i ) ν B (u i ) + 1 µ B (u i ) ν B (u i ), u i Q. 1 µ A (u i ) ν A (u i ) + 1 µ A (u i ) ν A (u i ) 1 + µ A (u i ) ν A (u i ) + 1 µ A (u i ) ν A (u i ) 1 µ B(u i ) ν B (u i ) + 1 µ B (u i ) ν B (u i ) 1 + µ B (u i ) ν B (u i ) + 1 µ B (u i ) ν B (u i ), u i Q. It implies {( ) } 1 µa (u i ) ν A (u i ) + 1 µ A (u i ) ν A (u i ) π sin 1 + µ A (u i ) ν A (u i ) + 1 µ A (u i ) ν A (u i ) 2 {( ) } 1 µb (u i ) ν B (u i ) + 1 µ B (u i ) ν B (u i ) π sin, u i 1 + µ B (u i ) ν B (u i ) + 1 µ B (u i ) ν B (u i ) 2 Q. Therefore, E(A) E(B). It follows that E(A) satisfies the conditions (P1)-(P4) [Definition 1.8]. Hence, E(A) is an entropy measure for IF S( Q). The effectiveness of proposed intuitionistic fuzzy entropy is expressed by comparing it with some existing intuitionistic fuzzy entropy measures [62, 119, 180, 205], as given below: Suppose that A be an IFS in the universe of discourse Q. Now, let us remind some existing entropy measures which are as listed below: Huang & Liu [62]: E HL (A) = 1 n n i =1 1 µ A (u i ) ν A (u i ) + π A (u i ) 1 + µ A (u i ) ν A (u i ) + π A (u i ). (6.3.2) Mishra et al. [119]: E MJH (A) = 1 n Wei & Zhang [180]: E W Z (A) = 1 n n sin i=1 i=1 {( µa (u i ) ν A (u i ) µ A (u i ) ν A (u i ) ) } π. (6.3.3) 2 n { } (µa (u i ) ν A (u i )) π cos. (6.3.4) 2(1 + π A (u i )) 104

8 Zhang & Jiang [205]: E ZJ (A) = 1 n n i=1 µ A (u i ) ν A (u i ) µ A (u i ) ν A (u i ). (6.3.5) Example 6.1. Let A 1 = { u i, 0.1, 0.9 : u i Q}, A 2 = { u i, 0.2, 0.7 : u i Q}, A 3 = { u i, 0.2, 0.5 : u i Q}, A 4 = { u i, 0.2, 0.4 : u i Q} and A 5 = { u i, 0.4, 0.5 : u i Q}. These existing entropy measures (6.3.2)-(6.3.5) satisfy the set of postulates for entropy measure [Definition 1.8]. From Table 6.2, it can be observed that the nearer the membership to the non-membership, the greater the intuitionistic fuzzy entropy measure. Table 6.2: Comparison with existing entropy measures A 1 A 2 A 3 A 4 A 5 E HL E MJH E W Z E ZJ E Comparing the result of the proposed measure with the existing intuitionistic fuzzy entropy measures, it can be concluded that the proposed measure is in synchronization with the existing intuitionistic fuzzy entropy measures. The comparison results of the intuitionistic fuzzy entropy measures are as follows: E HL (A 1 ) < E HL (A 2 ) < E HL (A 3 ) < E HL (A 4 ) < E HL (A 5 ), E MJH (A 1 ) < E MJH (A 2 ) < E MJH (A 3 ) < E MJH (A 4 ) < E MJH (A 5 ), E W Z (A 1 ) < E W Z (A 2 ) < E W Z (A 3 ) < E W Z (A 4 ) < E W Z (A 5 ), 105

9 E ZJ (A 1 ) < E ZJ (A 2 ) < E ZJ (A 3 ) < E ZJ (A 4 ) < E ZJ (A 5 ) and E(A 1 ) < E(A 2 ) < E(A 3 ) < E(A 4 ) < E(A 5 ). Thus, Table 6.2 reveals the effectiveness of the proposed intuitionistic fuzzy entropy. 6.4 Intuitionistic Fuzzy PROMETHEE Technique In this section, the IF-PROMETHEE technique is developed and applied to solve MCDM problems in intuitionistic fuzzy environment Intuitionistic Fuzzy Preference Relation In the process of decision making, the preference over the alternatives is usually expressed by the decision makers. At a time, the decision maker is not confident about their findings in decision making, thus, it is appropriate to state the decision maker s preferred values in terms of intuitionistic fuzzy values rather than the real values [188]. Since the IFS may not only afford the degree of preferred but also the degree of non-preferred and imprecise, and therefore, the demonstration of hesitant preferences with IFS is an important feature for decision makers [93, 159, 188]. An intuitionistic fuzzy preference relation is defined as follows: Definition 6.1 [188]. An intuitionistic fuzzy preference relation I R on the set Q = {u 1, u 2,..., u p } is expressed by a matrix I R = (r ij ) p p with r ij = < (u i, u j ), µ(u i, u j ), ν(u i, u j ) >, where i, j = 1, 2,..., p. For convenience, let r ij = (µ ij, υ ij ) is an intuitionistic fuzzy value consisting of the degree µ ij and ν ij, where µ ij indicates the degree to which u i is preferred to the object u j, ν ij denotes the degree to which the object u i is not preferred to the object u j and π(u i, u j ) = 1 µ(u i, u j ) υ(u i, u j ) is inferred as the hesitancy degree with the 106

10 condition: µ ij, ν ij [0, 1], µ ij + ν ij 1, µ ij = ν ji, µ ji = ν ij, µ ii = ν ii = 0.5, π ij = 1 µ ij ν ij, i, j = 1, 2,...p Procedure for IF-PROMETHEE Technique Suppose the evaluation of alternatives with respect to each criterion is given by the decision makers in terms of crisp values. Now, the preferences µ ij between the alternatives K i and K j over the criteria H k can be computed by determining the deviations based on pair wise comparisons and the V-shape with indifference criterion, which is shown as 0, D k (K i, K j ) η µ (k) D ij = k (K i, K j ) η, ρ D η ρ k (K i, K j ) η 1, D k (K i, K j ) > η. µ (k) p1. (6.4.1) Then, the preference matrix over the criterion H k is defined as follows: µ (k) µ (k) 1p µ (k) µ (k) 2p U (k) = (µ (k) ij ) p p = (6.4.2) µ (k) p2... The non-membership degree of an intuitionistic fuzzy value can be computed by the equations ν ji = µ ij and ν ij = µ ji, therefore, the intuitionistic fuzzy preference relation I (k) R I (k) R = (r(k) ij ) p p = over the criteria β k is calculated as follows: (µ (k) 12, ν (k) 12 )... (µ (k) 1p, ν (k) 1p ) (µ (k) 21, ν (k) 21 )... (µ (k) 2p, ν (k) 2p ) (6.4.3) (µ (k) p1, ν (k) p1 ) (µ (k) p2, ν (k) p2 )

11 By using (6.4.3), all individual intuitionistic fuzzy preferences of the alternatives over each criterion are established and then the global intuitionistic fuzzy preference index for each alternative can be derived. The global intuitionistic fuzzy preference index of the alternative K i to K j on all criteria can be determined as follows: r ij = r(k i, K j ) = (µ ij, ν ij ) = ( q k=1 w k µ (k) ij, q k=1 w k ν (k) ij where w k is the calculated weight of the criterion H k (k = 1, 2,..., q). ), (6.4.4) Thus, the global or overall intuitionistic fuzzy preference relation can be established as (µ 12, ν 12 )... (µ 1p, ν 1p ) (µ 21, ν 21 )... (µ 2p, ν 2p ) I R = (r ij ) p p = (6.4.5) (µ p1, ν p1 ) (µ p2, ν p2 ) (µ p3, ν p3 ) An algorithm for proposed intuitionistic fuzzy PROMETHEE technique consists of nine steps and the flowchart, which are as follows: Step I: For multi-criteria decision making problem, generate a set of alternatives K = {K 1, K 2,..., K p } and a set of criteria H = {H 1, H 2,..., H q }. In this step, the evaluation values of the alternatives over the criteria are in crisp numerical values. Step II: In this step, proper fuzzy sets are constructed for each criterion from crisp numerical values. Then, create a decision matrix of intuitionistic fuzzy values from fuzzy values by using construction theorem [Definition 5.1]. Step III: Using entropy (6.3.1), calculate the information of each intuitionistic fuzzy value in the intuitionistic fuzzy assessment matrix and the information matrix of this assessment matrix is M = (E ik ) p q, where E ik = E 1 ( r ik ). Normalize the information values in the above decision matrix by using Ē ik = E ik max E ik, i = 1, 2,..., p; k = 1, 2,..., q. (6.4.6) 108

12 The normalized information matrix is expressed as M = (E ik ) p q. Step IV: Calculate the weight vector w = (w 1, w 2,..., w q ) T, where w i 0 and q k =1 w k = 1, using the formula w k = 1 p i=1 Ēik q q p ; i = 1, 2,..., p; k = 1, 2,..., q. (6.4.7) k=1 i=1 Ēik Step V: Compute the deviations of each pair of alternatives over different criteria H k (k = 1, 2,..., q) via (6.2.1). Identify the decision maker s preference function, i. e., establish the parameter η as a strict preference threshold and ρ as an indifference threshold. Then, calculate the preferences µ (k) ij for the alternative K i against alternative K j with respect to the criterion H k by using V-shape indifference criterion (6.4.1). Thereafter, the preference matrix U (k) (k = 1, 2,..., q) can be constructed with the use of (6.4.2). Step VI: Calculate the intuitionistic fuzzy preference relation I (k) R over the criteria H k (k = 1, 2,..., q). = (r(k) ij ) p p Step VII: Construct the collective intuitionistic fuzzy preference relation I R = (r ij ) p p by using formulae (6.4.4) and (6.4.5). Step VIII: Compute the intuitionistic fuzzy positive outranking flow φ + (K i ) and the intuitionistic negative ranking flow φ (K i ) for the alternative K i by using the formulae (6.2.4) and (6.2.5), respectively. Step IX: To compare the intuitionistic fuzzy values φ + (K i ) and φ (K i ), Szmidt & Kacprzyk s [157] method is easy to use and is given as ρ(φ(k i )) = 0.5 (1 + π φ(ki )) (1 µ φ(ki )). (6.4.8) The intuitionistic fuzzy net outranking flow cannot be directly calculated because the intuitionistic fuzzy set doesn t have subtraction operation. Therefore, the deviation between the Szmidt & Kacprzyk s [157] function of the intuitionistic fuzzy positive outranking flow and the intuitionistic fuzzy negative outranking flow is computed by using ρ(φ(k i )) = ρ(φ + (K i )) ρ(φ (K i )). (6.4.9) 109

13 MCDM problem Construction of intuitionistic fuzzy decision matrix Information matrix of IFVs Normalized information matrix Calculate weight vector Compute the deviations of each pair of alternatives and create the preference matrix Calculate the intuitionistic fuzzy preference relation Construct the collective intuitionistic fuzzy preference relation Compute the intuitionistic fuzzy positive outranking flow and the intuitionistic negative outranking flow flow Net outranking flow Figure 6.1: Flow chart of IF-PROMETHEE technique Thus, a partial ranking will be created by comparing φ + (K i ) and φ (K i ) of the alternatives through (6.4.9). Otherwise, a complete ranking is obtained according to the deviation between the score values of the intuitionistic fuzzy positive outranking flow and that of the intuitionistic fuzzy negative outranking flow. 6.5 Identified Problem with HIV/AIDS In this section, to rank the set of ARV drugs discussed in section 6.2, IF- PROMETHEE technique is applied, with respect to the set of criteria. The 110

14 procedural steps for IF-PROMETHEE technique are as follows: Step I: The alternatives {K 1, K 2, K 3, K 4 } are to be appraised under the above five criteria and are listed in Table 6.3. Table 6.3: Fuzzy decision matrix H 1 H 2 H 3 H 4 H 5 K K K K Intuitionistic fuzzy decision matrix is created in Table 6.4 from fuzzy decision matrix given by Table 6.3 with the help of Definition 5.1. Table 6.4: Intuitionistic fuzzy decision matrix H 1 H 2 H 3 H 4 H 5 K 1 (0.23, 0.587) (0.61, 0.2) (0.192, 0.63) (0.22, 0.75) (0.196, 0.62) K 2 (0.26, 0.554) (0.2, 0.61) (0.63, 0.192) (0.094, 0.875) (0.62, 0.196) K 3 (0.62, 0.197) (0.61, 0.2) (0.259, 0.56) (0.31, 0.66) (0.227, 0.59) K 4 (0.197, 0.62) (0.36, 0.454) (0.337, 0.484) (0.15, 0.82) (0.322, 0.50) Step III: By using the formula (6.3.1), the information measure of each intuitionistic fuzzy value of the above decision matrix is computed and obtained the 111

15 following information matrix: M = (E ij ) 4 5 = The above information matrix is altered into the normalized information matrix by using formula (6.4.6), is given as follows: M = (E ij ) 4 5 = Step IV: Calculate the criterion weight vectors by using (6.4.7) in normalized information matrix. Then, the weight vector of all the decision attributes are obtained as W = (0.2195, , , , ) T. Step V: The deviations of each pair of alternatives over different criteria can be computed with the help of (6.2.1). The preferences µ (k) ij for the alternative K i against the alternative K j are obtained by using V-shape indifference criterion (6.4.1), where the indifference threshold ρ is taken as zero for all criteria and the strict preference threshold η are considered as η = 53 for first criterion H 1, η = 9.5 for second criterion H 2, η = 23 for third criterion H 3, η = 2.5 for fourth criterion H 4 and η = 25 for fifth criterion H 5. Then, the preference matrices U (k) (k = 1, 2, 3, 4, 5) are evaluated as follows: U (1) =, U (2) =,

16 U (3) = , U (4) = U (5) = , Step VI: The intuitionistic fuzzy preference relations I (k) R = (r (k) ij ) p p obtained by using the equations ν ji = µ ij and ν ij = µ ji over the criteria H k (k = 1, 2, 3, 4, 5), which are given as (0, ) (0, ) (0.0687, 0) I (1) R = (0.0530, 0) (0, ) (0.1217, 0), (0.7464, 0) (0.6934, 0) (0.8151, 0) (0, ) (0, ) (0, ) (0.8421, 0) (0, 0) (0.5263, 0) I (2) R = (0, ) (0, ) (0, ), (0, 0) (0.8421, 0) (0.5263, 0) (0, ) (0.3158, 0) (0, ) (0, ) (0, ) (0, ) I (3) R = (0.9778, 0) (0.8283, 0) (0.6522, 0), (0.1496, 0) (0, ) (0, ) (0.3257, 0) (0, ) (0.1761, 0) (0.5320, 0) (0, ) (0.3000, 0) I (4) R = (0, ) (0, ) (0, ), (0.3680, 0) (0.9000, 0) (0.6680, 0) (0, ) (0.2320, 0) (0, ) 113 are

17 I (5) R = (0, ) (0, ) (0, ) (0.8880, 0) (0.8200, 0) (0.6240, 0) (0.0680, 0) (0, ) (0, ) (0.2640, 0) (0, ) (0.1960, 0). Step VII: With the use of formula (6.4.4) in above intuitionistic fuzzy preference relations, the overall intuitionistic fuzzy preference relation can be established as below: (0.2441, ) (0, ) (0.1645, ) (0.4364, ) (0.3751, ) (0.3172, ) I R =. (0.2491, 0) (0.4318, ) (0.3639, ) (0.1343, ) (0.0947, ) (0.0846, ) Step VIII: According to the formula (6.2.4) and (6.2.5), the intuitionistic fuzzy positive outranking flows are obtained as φ + (K 1 ) = (0.1420, ), φ + (K 2 ) = (0.3781, ), φ + (K 3 ) = (0.3526, 0), φ + (K 4 ) = (0.1048, ) and the intuitionistic fuzzy negative outranking flows are given as follows: φ (K 1 ) = (0.2844, 0), φ (K 2 ) = (0.2701, ), φ (K 3 ) = (0.1699, ), φ (K 4 ) = (0.2867, ). Step IX: With the use of (6.4.9), we get ρ + (φ(k 1 )) = , ρ + (φ(k 2 )) = , ρ + (φ(k 3 )) = , ρ + (φ(k 4 )) = , ρ (φ(k 1 )) = , ρ (φ(k 2 )) = , ρ (φ(k 3 )) = , ρ (φ(k 4 )) =

18 Now, comparing these intuitionistic fuzzy values, we have ρ(φ + (K 2 )) < ρ(φ + (K 3 )) < ρ(φ + (K 1 )) < ρ(φ + (K 4 )) and Therefore, and ρ(φ (K 3 )) > ρ(φ (K 1 )) > ρ(φ (K 4 )) > ρ(φ (K 2 )). φ + (K 2 ) > φ + (K 3 ) > φ + (K 1 ) > φ + (K 4 ) φ (K 3 ) < φ (K 1 ) < φ (K 4 ) < φ (K 2 ). Thus, by using (6.4.9), we have ρ(φ(k 1 )) = , ρ(φ(k 2 )) = , ρ(φ(k 3 )) = , ρ(φ(k 4 )) = Since ρ(φ(k 3 )) < ρ(φ(k 2 )) < ρ(φ(k 1 )) < ρ(φ(k 4 )), therefore, we obtain φ(k 3 ) > φ(k 2 ) > φ(k 1 ) > φ(k 4 ). Hence, the ranking of the four ARV drugs by IF-PROMETHEE is K 3 K 2 K 1 K 4. The ranking of these four alternatives is also compared with existing TOPSIS method and the comparison result is shown in Table 6.5. From Table 6.5, it can be concluded that K 3 is the optimal choice and the preference order of the remaining three options shows some diversities. 115

19 Table 6.5: Comparison with existing techniques Technique Ranking Optimal alternative Hung & Chen [67] technique Joshi & Kumar [75] technique Wei & Zhang [180] technique Proposed IF- PROMETHEE technique K 3 K 1 K 4 K 2 K 3 K 3 K 1 K 4 K 2 K 3 K 3 K 1 K 4 K 2 K 3 K 3 K 2 K 1 K 4 K H u n g & C h e n [6 7 ] J o s h i & K u m a r [7 5 ] W e i & Z h a n g [1 8 0 ] P ro p o s e d M e th o d R a n k in g 3 4 K 1 K 2 K 3 K 4 A R V D ru g s Figure 6.2: Ranking of ARV drugs selection The ranking order of four alternatives acquired from PROMETHEE and TOPSIS techniques, is graphically depicted in Figure 6.2 and thus, the optimal alternative 116

20 in this problem is K Conclusions In this chapter, a ranking problem of ARV drugs for HIV/AIDS under uncertain environment is presented. First of all, a new entropy measure based on sine function is developed in the intuitionistic fuzzy environment. Thereafter, numerical results are discussed to show the effectiveness of the proposed entropy. The PROMETHEE technique for MCDM problems is discussed in intuitionistic fuzzy environment to rank the ARV drugs for HIV/AIDS disease on the basis of their performance. In this technique, the proposed entropy measure is used to construct the intuitionistic fuzzy decision matrix and derive the weight of each criterion. After that, the preference matrices are established using V-shape indifference criterion. Later on, the intuitionistic fuzzy preference relations and overall intuitionistic fuzzy relation are calculated to acquire the positive and negative outranking flows for partial outranking of all alternatives. To obtain the complete ranking of the alternatives, a net outranking flow is computed for each alternative. Further, the ranking attained by proposed technique with the existing technique is discussed and the comparison exposes the validity of the proposed technique over others. Calculating the weight vector improves the efficiency of the proposed technique. The present technique is better than the other techniques as along with the ranking of the alternatives, it also compares the alternatives on the basis of their performance. 117