An Extension of TODIM with VIKOR approach based on Gini Simpson Index of Diversity under Picture fuzzy framework to Evaluate Opinion Polls

Cuong and Kreinovich was the first who gives the idea of Picture fuzzy set (PFS), which is an extension of intuitionistic fuzzy set (IFS) by cosidering positive, negative and neutral membership of element. In this paper, we have been worked on new entropy measure of PFS from the probabilistic view point and it‟s properties are examined from mathematical point of view. A hybrid aproach is presented with the assistance of TODIM (Portuguese abbreviation for Interactive MultiCriteria Decision Making) and VIKOR (Vlsekriterijumska Optimizacija I Kompromisno Resenje) methods. Further, we applied it to MCDM (multi criterion decision making) problems with picture fuzzy numbers (PFNs), where the information about criteria synthetic weights is partially known and completely unknown and show its existence with the help of some practical cases. After getting the output, we are able to infer that the proposed hybrid approach is comparatively better so as to handle the uncertainty and vulnerabilities for the decision making problems. Based upon these two approaches we can determine the opinion poll of voting outcomes and then, we compare its result with other MCDM approaches that exists in the literature.

In the first section, we discuss the work done by many researchers in this field also the motivational source. Second section identi application to voting model by using the the proposed hybrid MCDM technique. In the last segment, the paper is presented with Conclusions .

Preliminaries
Some basic definitions and concepts related to FS, IFS and PFS over * = 1 , 2 , . . . , has been discussed in this section. For an IFS, the pair ( ( ), ( )) is described as an IFN and denotes the membership and non membership degree of set E.
There is a drawback in the IFS of Atanassov. In IFS, he has not defined the concept of "degree of refusal" which restricts it"s extent of application .This drawback was eliminated by Cuong and Kreinovich they added the "degree of refusal membership" in Picture fuzzy set (PFS)". Obviously, when ( ) = 0 , then the PFSs reduce into IFS, while if ( ), ( ) = 0 then the PFSs become FSs. In the voting, those who are abstain can be interpreted as: on one hand, they vote for; on the other hand, they vote against. Meanwhile, those who are refusal of the voting can be explained as they are not care about this voting.
(b). If ( 1 ) = ( 2 ), implies that 1 is equivalent to 2 , denoted by 1 ≡ 2 ; Here ( ) = − represents goal difference and ( ) = + + repesents an effective degree of voting. As score increases, then the number of peoples are more who vote for" and who vote against" and people who refuse of voting become less. So, ( ) demonstrates the effective degree of voting.

Entropy Concept for PFSs
The fuzzy entropy measures the uncertainty of a FS and denote it"s degree of fuzziness and to measure the fuzziness, the four axioms are proposed by De Luca and Termini [29] :

Novel Parametric Measure for PFSs
In this section, we proposed a new PF information measure based on Gini Simpson entropy.

Proposed PF Information Measure:
For any ∈ , we define    , then (4.8) and (4.9) hold. Szmidt and Kacpryzk [30] proposed the distance between two IFSs as the distance between their parametres, that is ( , , ). To determine the distance between two IFSs we uses Euclidean or Hamming distance measure.We may concluded from proposition (4.1) PFS is closer to maximum value ( . By using Hessian matrix we can prove that ( ) is a concave function at the stationary points . (4.11) is said to be strictly concave if If ( ) is negative definite at a point in its domain and The Hessian of ( ) is given by . Therefore, by using proposition (4.1), we see that 2 ( ) satisfies the condition 4.
Corollary 4.1. For any Picture fuzzy set and (complement of ), we have (4.20)

An extension of TODIM based on VIKOR for Picture Fuzzy MCDM Problem
In this section we will use hybrid TODIM-VIKOR method to MCDM problems for opinion poll based on the proposed entropy measure for PFSs. To show validity and practical reasonability, we apply proposed measure in a MCDM problem, involving partially known and unknown information about criteria weights for alternatives in PF information.
Let us consider a case of nation where elections will be held in near future. Let Þ 1 , Þ 2 , Þ 3 , Þ 4 , Þ 5 are are different political parties are contesting and they are contesting on different issues say: (1) National security (2) economy (3) employment (4) stability (5) corruption. A survey on 1000 people has been conducted by news channel for opinion poll to determine the possible outcomes of elections. To get the best possible outcome we applied the PF TODIM-VIKOR approaches to this kind of problems with PFNs, and the procedure for hybrid TODIM-VIKOR method is as follows :

VIKOR Method
The idea of VIKOR was given Opricovic et al. [26] to determine a compromise solution which is very near to the consistent solution. This solution is helpful to find the best solution by taking the majority and minimize the ("opponent") with conflicting criteria.
By considering alternative Þ corresponding to each critera is given as where + = max and − = min givs the best and worst solutions. shows the weight criteria and , gives the distance of alternative Þ to the best solution.In the VIKOR method 1, (as ) and ∞, (as ), = 1,2, . . . are used to formulate as "boundary measures". The main steps for the new PF TODIM-VIKOR method based on the proposed entropy measure as follows .

2)
Step 2:Convert the decision matrix = ( ) × into a normalized PF decision matrix which is denoted as : = ( ) , for cost criteria , for benefit criteria (5.3) where = ( , , ) is complement of . After that we will obtain a new PF decision matrix = ( ) × Step 3:

Methodology 1: If the criteria weights are partially known
We will solve the MCDM problem if the attribute weight are partially known and also completely known by collecting all PF information under distinct conditions and then we will compare the final PF values . Here we used the proposed measure to find the attribute weights by using the below said formula: Methodology 2: When the Criteria Weights are completely unknown The attribute weight can be calculated by using the below said equation if attribute weights are completely unknown. where is the weight of the criterion , = max 1 , 2 , . . . , and 0 ≤ ≤ 1. In this step, each criteria carries equal importance, any criterion may be choosen.

TODIM Method
The TODIM method is a discrete multicriteria method used for qualitative as well as quantitative criteria based on probability theory. The dominance degree of Þ over each alternative Þ w.r.t. each criterion is given by: where ( , 1 )find the distance between the two PFNs and 1 and the parameter represents the attenuation factor of losses. By definition if > 1 , then (Þ , Þ 1 ) signifies a gain ; if < 1 , then (Þ , Þ 1 ) signifies loss.
Step 11:Then find the compromise solution, if the solution satisfies the following two conditions will be the most desirable solution. If the conditions X1 and X2 are not simultaneously satisfied, at that point we look for the compromise solution as follows: (a) (Acceptable advantage):, The alternatives Þ (1) and Þ (2) gives the compromised solutions If only condition X2 is not satisfied. for maximum If the condition X1 is not satisfied.

Method 1:For Partially Known Criteria Weights
The PF decision matrix depicted in table 2 shows the overall views of the public. To find the PFN"s ( , , ) we proceed as follows: Since 1000 people have been considered for survey , consider 300 people support the party Þ 1 corresponding to criteria 1 that is "yes", 200 people are neutral and 100 people do not support the party Þ 1 or say no. Then, the PFN ( 11 ,11 ,11  In a similar way, we can obtain other entries of PF decision matrix.
Results study with proposed method in PF framework: Step 1.  The following optimal model is used to find the attribute weights :
Step 4. We can form the dominance matrices 1 − 5 by assuming the value of = 2.5 from the  Step5. By using (5.9), We can determine the overall dominance of each alternative Þ w.r.t the alternatives Þ 1 and shows the overall dominance matrix : Step 6. With the help of (5.11) and (5.12), we calculate the positive and the negative solution denoted by p respectively : Steps 7-8. In this step, we calculate and as below: 1 = 0.5990, 2 = 0.6040, 3 = 0.3628, 4 = 0.5489, 5 = 0.5087 1 = 0.3449, 2 = 0.3900, 3 = 0.1500, 4 = 0.2756, 5 = 0.2196 Step 9. Determine ( = 1,2,3,4,5) by using the value of = 0.5 1 = 0.8956, 2 = 1.000, 3 = 0.000, 4 = 0.6475, 5 = 0.4474 Step 10.Ranking and compromised solution of the alternatives by taking the values , and is shown in the    In general, we discuss the influence of to the value of . The results are appeared in Table 4. From Table 4, we can say that those distribution graphics have the same distribution when the weight ≤ 0.5 or ≥ 0.5 and the values of values of five possible projects have the same change rate as the weight increases. So, the best project is different as the weight increases. With the results in table 4 and visualized results in Fig. 1 and Fig. 2, the schemes provided the best choice Þ 3 or ( Þ 3 , Þ 5 ).
Step 11. Results in Table 4 demonstrate that alternatives Þ 3 and Þ 5 places at the first two positions in the ranking. However, by using the condition X1, Þ 3 − Þ 5 = 0.000 − 0.4474 = −0.4474 < 1 5−1 = 0.25. Which shows that the condition X1 is not satified. Therefore , we look for the compromise solution given below : (Þ (5) ) − (Þ (3) ) = 0.4474 − 0.000 = 0.4474 < In this subsection, when the criteria weights are completely unkown then we solve the same example : 1. By using (5.5), we will determine the values of criteria weights as follows:  Therefore, the compromised solutions remains the same by both methods.

Comparative analysis
To verify the effectiveness of our proposed entropy we compare it with the method proposed by (  The ranking of alternatives so obtained is given by : Þ 3 ≻ Þ 5 ≻ Þ 4 ≻ Þ 1 ≻ Þ 2 , thus Þ 3 as the most suitable alternative. In our proposed method, Þ 3 is best choice, but ranking order does not matter for other alternatives. In the former methods, the weights criteria are assumed by experts or determined by aggregation operators, which can be unreasonable to be attained practically. Compared with the existing methods, the latter (proposed approach) has some valuable advantages as follows: In a complex decision making context, using PFNs that involve various types of evaluating results to represent experts view is a good choice.
(b) The entropy approach is used for the calculation of the criterion weight and this approach is more reasonable and flexible.
(c) The advantages of entropy information, experts behaviours , group utility and minimum individual regret are fully used

Conclusions
In this paper, we conclude that a new fuzzy information has been successfully introduced and validated it in light of newly proposed framework for PFSs which is an extension of IFS. Realizing the vital role of criteria weights in MCDM problems, the proposed MCDM problem has dicussed by applying two different approaches that is partially known and completely unkown criteria weight. PFSs are appropriate in describing and addressing the uncertainty and vagueness information measure occurring in MCDM problems. Additionally, the operating of proposed MADM method is throughly explained with the help of numerical example based on the concept of PF VIKOR-TODIM supported opinion polls for predicting the output of elections. To check the viability and applicability of the proposed MCDM method , we compare the resulting output with the existing MCDM in literatures. With better practical decision making value, the eminent achivements of the present research can forward an effective and reliable scientific approach for solving the multi criteria picture fuzzy decision making problem. The proposed MCDM method is applied to various complicated problems like site choice, venture establishment, health department , insurance sector where it helps to determine the risk factor, to establish new venture and so on.