Aggregation of Pentagonal Fuzzy Numbers with Ordered Weighted Averaging Operator based VIKOR

Volume 119 No. 9 2018, 295-311 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Aggregation of Pentagonal Fuzzy Numbers with Ordered Weighted Averaging Operator based VIKOR S. Johnson Savarimuthu 1 and T. Pathinathan 2 1 Department of Mathematics, St. Joseph s College of Arts and Science, Cuddalore-1, Tamil Nadu, India 1 johnson22970@gmail.com 2 P.G. and Research Department of Mathematics, Loyola College, Chennai - 34, Tamil Nadu, India 2 pathinathan@gmail.com Abstract Conflicting opinions arise while making decisions in groups. VIKOR is a technique to determine a compromise solution among the group decision makers to resolve the contradictions. In this paper, we introduce two different methodologies to aggregate the group decision makers opinion characterized by Pentagonal Fuzzy Numbers (PFNs). First, we utilize the generalized induced ordered weighted averaging (GIOWA) operator to aggregate the group decision makers opinion represented by PFNs. Then, we aggregate the group decision makers opinion by combining the linguistic GIOWA operator into pentagonal VIKOR method. The newly introduced two methodologies are compared with the previously used method using a case study. We have employed these methodologies to choose a most suitable crop for cultivation in Villupuram District, Tamil Nadu, India. AMS Subject Classification:03E72 1 295

Keywords: Pentagonal fuzzy number, VIKOR, OWA operator, GIOWA operator, compromise solution. 1 Introduction The real life group decision making process encounters conflicting opinions, disagreements, and contradictions. Several decision making techniques [3] are proposed to resolve the disagreements among the decision makers. VIKOR (VIseKriterijumska Optimizacija I Kompromisno Resenje) is one such technique, which primarily focus on the conflicting situations among the decision makers and largely employed to resolve the disagreements. The principal idea behind the VIKOR technique is to determine the compromise solution [26] among the group multiple solutions. Ronald Yager [2, 19 24] extensively studied and extended the work of Richard Bellman and Lotfi Zadeh [1] in decision making environments. Bellman and Zadeh developed a decision making theory and used fuzzy intersection (minimum) operation to combine the importance of the criteria. Ronald Yager [19] [23] extended the theory by developing the concept of quantifier guided aggregation to combine the importance of the criteria. Also, he suggested that the criteria weight importance calculated through quantifier guided aggregation [2] [16 18] [19 24] provides an overall group assessment over each criterion. T. Pathinathan and S. Johnson Savarimuthu [5] made a historical review on VIKOR multi-criteria decision making technique. Also, they extended the VIKOR method [12], where the decision makers opinion is characterized by pentagonal fuzzy numbers. Earlier, they had developed several decision making techniques such as TOPSIS combined with dual hesitant fuzzy set [6], TOPSIS with pentagonal hesitant fuzzy sets [7] and weight based intuitionistic fuzzy set (WBIFS) [9] into analytic hierarchy process (AHP). In all the earlier methods [6 9] [11] [12] [14] [15], the Experts opinion are collected and processed to establish an ideal solution. In early 2017, T. Pathinathan and S. Johnson Savarimuthu introduced a new fuzzy set named: weight based intuitionistic fuzzy set (WBIFS) [9] to study the impact of the external factors over the selection of best alternative. Then they applied the newly defined 2 296

weight based intuitionistic fuzzy set concept into analytic hierarchy process, where the entries of the decision matrix are characterized by WBIFS. In some cases, the conflicts arise among the decision makers in choosing a best alternative among the multiple alternatives. In such conflicting situations, the methods which we had developed earlier are found to be inconsistent to obtain the best alternative. In this paper, we propose two different methodologies which primarily order the importance of the opinion based on the importance of the criteria. We have employed the generalized ordered weighted averaging operator to order the importance of the criteria. Then we have proposed the generalized ordered weighted averaging operator combined with VIKOR for the multiple group decision analysis which is characterized by pentagonal fuzzy numbers. This paper is organized in the following manner. Section Two provides the proposed algorithm (1) which uses generalized ordered weighted averaging operator for the quantification. Section Three discusses the algorithmic approach of the newly proposed GIOWA- VIKOR decision making technique. Section Four gives the experimental verification of the proposed algorithms followed by comparison of the results obtained from both the algorithms in section five. Finally the paper is concluded in section Six. 2 Proposed algorithm on aggregating the multiple group decisions using GIOWA operator We propose an algorithm for newly extended multiple group decision making technique which uses GIOWA operator, where the decision entries are characterized by pentagonal fuzzy number. The newly proposed multi-criteria decision making technique aggregate the group subjective opinions by fixing the positions. Step 1: The subjective opinions from the group of decision makers (experts) are gathered. Step 2: The collected subjective opinions are of vague statements 3 297

characterized into a linguistic variables. Step 3: Construct a decision matrix (DM) where the decision entries of the matrix are characterized by a pentagonal fuzzy number and it is given by: DM = [f ij ] n m (1) Step 4: Construct an aggregated decision matrix from the group decision opinions (12). Step 5: The importance of the criteria and its respective weights has been calculated. Step 6: Construct aggregated subjective weights of each criterion (12). Step 7: The entries in the decision matrix are ordered from largest value to the smallest value. Then the respective criteria weights are ordered and the importance of the ordered criteria weights (i.e., the ordered weighted averaging weights) is calculated by the formula: w i (x) = Q ( ik=1 ) ( c i 1 k k=1 nk=1 Q c ) k c nk=1 k c k where i=1,2,3,...n represents n criteria s and T represents the total sum of the importance of criteria s and it is given by, and linguistic quantifier Q is defined as: (2) n T = c k (3) k=1 Q(x) = x 2 (4) 4 298

Step 8: Then the group overall assessment is calculated by the function, generalized induced ordered weighted averaging operator (GIOWA) is defined as follows: n GIOW A w ( A 1, C 1, x 11, A 1, C 2, x 12..., A 1, C n, x 1n ) = w i x 1i i=1 (5) Step 9: Ranking of a best alternative has been calculated as minimum gets first and maximum gets last. 3 Proposed algorithm on aggregating the multiple group decisions using VIKOR- GIOWA operator method We propose an algorithm which combines GIOWA operator into VIKOR decision making technique, where the decision entries are characterized by pentagonal fuzzy number. Step 1: The subjective opinions from the group of decision makers (Experts) are gathered. Step 2: The collected subjective opinions are of vague statements characterized into a linguistic variables. Step 3: Construct a decision matrix (DM) where the decision entries of the matrix are characterized by a pentagonal fuzzy number and it is given by: DM = [f ij ] n m (6) 5 299

Step 4: Construct an aggregated decision matrix from the group decision opinions (12). Step 5: The importance of the criteria and its respective weights has been calculated. Step 6: Construct aggregated subjective weights of each criterion using equations (12). Step 7: Construct a normalized decision matrix using the following equations x + ij5 = max {x ij5 }, C j B (7) i f ij = f ij = x ij1 = min {x ij1 }, C j C (8) i, x ij2 x +, x ij3 ij5 x +, x ij4 ij5 x +, x ) ij5 ij5 x +, C j B (9) ij5 ( xij1 x + ij5 ( xij1 x ij5, x ij2 x ij5, x ij3 x ij5, x ij4 x, x ij5 ij5 x ij5 ), C j C (10) Step 8: Obtain a best value and worst value by using the following equations: f j + = max f ij (11) i f j = min i f ij (12) where f + j and f j are the best and worst values of all criterion function. Step 9: Calculate the values of S i and R i as follows: n ( f + ) S i = w f j ij j f i + fi j=1 (13) where, w j is calculated by using the ordered weighted averaging (OWA) operator function, which has been adopted from step 7 of the previous algorithm. { ( f + )} R i = max w j f ij j j f i + fi (14) 6 300

where w j are the ordered weighted averaging weights of the criteria. Step 10: Calculate the values of Q i as follows: ( Si S ) ( Ri R ) Q j = v + (1 v) S S R R (15) where v is the weight introduced for the strategy of maximum group utility, and 1 v is the weight of the individual regret. S = min S i i S = max i R = min i R = max i Step 11: Rank the alternatives sorting by values S, R and Q in an ascending order. In VIKOR, ascending order is used for ranking. The minimum value gets the maximum rank. The minimum value maintains the cooperative group utility in choosing a compromise solution. Step 12: Alternative which is the best ranked by the measure Q should satisfy the following two conditions: C 1. Acceptable advantage C 2. Acceptable stability in decision making S i R i R i 4 Case Study Study area includes all the 22 blocks of Villupuram district, Tamil Nadu, South India. Through interviews the opinions have been collected from 142 respondents and based on the farming experience we have chosen the following 8 decision makers. The following table (1) represents the farming experience of the farmers who are cultivating maximum number of crops in the Villupuram district. 7 301

Table 1: Sample respondents and their farming experience Name Age Farming Experience D 1 R. Ezhumalai 46 Owns 4.5 acres of agricultural land, with 25 years of farming experience, Sadakatti village. D 2 N. Sivasakthi 47 Owns 4.5 acres, with 15 years of experience, Kandamangalam village. D 3 V. Vedagiri 56 Owns 8 acres, with 20 years of farming experience, Marakkanam. D 4 M. Gopal 71 Owns 12 acres, with 50 years of farming experience, Sennagonam village. D 5 P. Kuppusamy 62 Owns 6 acres, with 50 years of farming experience, Olakkoor village. D 6 P. Pakkiri 50 Owns 7.5 acres, with 26 years of farming experience, Kannaarampattu village. D 7 G. Narasingam 49 Owns 6.75 acres, with 25 years of farming experience, Thirumoondicharam village. D 8 S.Kudiyarasumani 60 Owns 10 acres, with 40 years of farming experience, Mettatthur village. 4.1 Adaptation of the problem The opinions are collected for the following alternatives based on the criterion which is stated as follows: 4.1.1 Alternatives A 1 Paddy A 2 Sugarcane A 3 Urad A 4 Groundnut A 5 Tapioca 8 302

4.1.2 Criteria C 1 Profit and loss in the yield C 2 Seed quality C 3 Soil quality C 4 Climatic (Sunlight) condition C 5 Water availability C 6 Assistance from government agencies C 7 Assistance from private agencies C 8 Level of underground water C 9 Fixation price of grains C 10 Agriculture loan discount The aggregated pentagonal decision matrix [12] from our previous paper has been taken again to make a comparative study. The criteria are classified with the help of following linguistic variable and its fuzzy linguistic scale values [12] Then the decision matrix characterized by pentagonal fuzzy number is aggregated (12) and the criteria weights are aggregated (12). The following table (2) shows the fuzzy centre value entries of the aggregated pentagonal decision matrix. Table 2: Fuzzy center value of the decision matrix C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 10 w j 0.5729 0.4727 0.4805 0.4688 0.3281 0.4063 0.3594 0.3333 0.5521 0.4375 A 1 0.5833 0.6094 0.6094 0.5938 0.4688 0.4063 0.3594 0.3333 0.5625 0.4375 A 2 0.6041 0.4531 0.4805 0.4688 0.3281 0.4063 0.3594 0.3333 0.5417 0.4375 A 3 0.6389 0.4844 0.4531 0.4375 0.3333 0.4167 0.3750 0.3333 0.6667 0.4444 A 4 0.5833 0.5078 0.4805 0.4688 0.3281 0.4063 0.3594 0.3333 0.5417 0.4375 A 5 0.5833 0.6250 0.6250 0.6250 0.2500 0.3750 0.2500 0.3333 0.3333 0.3333 Then, by using step 7 of a proposed algorithm 1, the ordered weighted averaging weights has been calculated as follows: 9 303

Table 3: Ordered weighted averaging (OWA) weights of each alternative w 1j w 2j w 3j w 4j w 5j w 6j w 7j w 8j w 9j w 10j A 1 0.0115 0.0352 0.0572 0.1006 0.1288 0.0914 0.1391 0.1468 0.1440 0.1454 A 2 0.0169 0.0482 0.0674 0.0886 0.1122 0.1243 0.1331 0.1319 0.1342 0.1432 A 3 0.0157 0.0494 0.0661 0.0908 0.1033 0.1325 0.1331 0.1319 0.1320 0.1454 A 4 0.0169 0.0482 0.0661 0.0908 0.1114 0.1243 0.1331 0.1319 0.1342 0.1432 A 5 0.0115 0.0352 0.0572 0.1006 0.0918 0.0880 0.1708 0.1576 0.1311 0.1563 Then by using the step 8 of the proposed algorithm 1, the generalized induced ordered weighted averaging weights have been obtained as follows: Table 4: Overall evaluation using generalized induced ordered weighted aeraging (GIOWA) of each alternative A 1 * w 1 (x) A 2 * w 2 (x) A 3 * w 3 (x) A 4 * w 4 (x) A 5 * w 5 (x) C 2 0.0070 0.0102 0.0104 0.0098 0.0072 C 3 0.0215 0.0261 0.0315 0.0261 0.0220 C 4 0.0340 0.0324 0.0320 0.0336 0.0358 C 1 0.0587 0.0415 0.0411 0.0436 0.0587 C 9 0.0725 0.0509 0.0459 0.0522 0.0344 C 5 0.0429 0.0544 0.0580 0.0544 0.0293 C 10 0.0609 0.0541 0.0555 0.0541 0.0569 C 6 0.0596 0.0474 0.0494 0.0474 0.0525 C 7 0.0517 0.0447 0.0440 0.0447 0.0328 C 8 0.0485 0.0470 0.0485 0.0470 0.0391 Total 0.4571 0.4086 0.4164 0.4129 0.3686 4.2 Experimental results based on algorithm 1 Then the rank has been obtained as follows: 10 304

Table 5: Ranking of the alternatives by GIOWA GIOWA Rank weights Paddy 0.4571 1 st (A 1 ) Sugarcane 0.4086 4 th (A 2 ) Urad (A 3 ) 0.4164 2 nd Groundnut 0.4129 3 rd (A 4 ) Tapioca (A 5 ) 0.3686 5 th Also, the numerical decision opinions from our previous paper has been processed by the newly proposed algorithm 2 which combines VIKOR characterized by pentagonal decision entries into ordered weighted averaging (OWA) weights. The value of S i and R i is calculated by using step 8 of the proposed algorithm 2, as follows: S 1 = 0.12376 S 2 = 0.370461 S 3 = 0.27743 S 4 = 0.3438 S 5 = 0.5775 Then the value of R i is calculated using the equation 14 as follows: R 1 = 0.03959 R 2 = 0.07589 R 3 = 0.10214 R 4 = 0.0753 R 5 = 0.1708 11 305

The value of each Q i by equation 15 is calculated as follows: Q 1 = (0.5) (0.12376 0.12376) (1 0.5) (0.03959 0.03959) + = 0 (0.5755 0.12376) (0.1708 0.03959) Q 2 = 0.41018067 Q 3 = 0.407695 Q 4 = 0.3836001 Q 5 = 1.0000 4.3 Experimental results based on algorithm 2 Table 6: The value of S i and R i S i R i Q i A 1 0.1237 0.0396 0.0000 A 2 0.3705 0.0759 0.4102 A 3 0.2774 0.1021 0.4077 A 4 0.3484 0.0753 0.3836 A 5 0.5775 0.1078 1.0000 Table 7: method The ranking of the alternatives by GIOWA-VIKOR Values induced by GIOWA Rank VIKOR method Paddy 0.0000 1 st (A 1 ) Sugarcane 0.4102 4 th (A 2 ) Urad (A 3 ) 0.4077 3 rd Groundnut 0.3836 2 nd (A 4 ) Tapioca (A 5 ) 1.0000 5 th 12 306

5 Comparison of the results obtained from both the algorithms The results obtained from the proposed algorithm 1 (Table 5) and 2 (Table 7) are compared with the result obtained through the extended VIKOR method [12]. From the table 8, we make the following observations. The alternative (A 1 ) Paddy and (A 5 ) Tapioca ranks first and fifth respectively in both the proposed algorithms and whereas Paddy ranks 2 nd in the extended VIKOR technique. The algorithm based on generalized induced ordered weighted averaging operator considers the importance of the criteria weights and evaluates the criteria based on the ordered (decreasing order) decision opinions. By the proposed algorithm GIOWA VIKOR technique, the ordered criteria weight evaluation through VIKOR technique yield a remarkable variation in the values and that leads to the ranking. The alternatives Sugarcane (A 2 ), Urad (A 3 ) and Groundnut (A 4 ) ranks fourth, second and third position by the algorithm 1 and whereas fourth, third and second position form GIOWA VIKOR technique. 13 307

Table 8: Comparative results of extended VIKOR, GIOWA, and GIOWA-VIKOR method Alternatives Extended VIKOR Rank GIOWA weights Rank GIOWA VIKOR method Rank Paddy 0.4999 2 nd 0.4571 1 st 0.0000 1 st (A 1 ) Sugarcane 0.5888 4 th 0.4086 4 th 0.4102 4 th (A 2 ) Urad (A 3 ) 0.5237 3 rd 0.4164 2 nd 0.4077 3 rd Groundnut 0.2121 1 st 0.4129 3 rd 0.3836 2 nd (A 4 ) Tapioca (A 5 ) 1.0000 5 th 0.3686 5 th 1.0000 5 th 6 Conclusion The newly proposed VIKOR techniques shows A 1 (Paddy) is the best compromise crop by satisfying all such criterias involved in evaluating it. Whereas, Sugarcane (A 2 ), Urad (A 3 ) and Groundnut (A 4 ) are the three crops which shows sizeable differences in reaching the best optimum. References 1. R. E. Bellman and L. A. Zadeh, Decision making in a fuzzy environment, Management Sciences, 17, 4, (1970), 141-164. 2. D. Filev and R. R. Yager, On the issue of obtaining OWA operator weights, Fuzzy Sets and Systems, Elsevier Science Publishers, 94, (1998), 157-169. 3. T. Pathinathan and S. Johnson Savarimuthu, A Historical Overview of VIKOR Model (VIseKriterijumska Optimizacija I Kompromisno Resenje), International Journal of Multidisciplinary Research and Modern Education, 3, 1, (2017), 1-16. 14 308

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