Ijraset Journal For Research in Applied Science and Engineering Technology
Authors: S. Samuel Nirmalsingh, P. John Robinson
DOI Link: https://doi.org/10.22214/ijraset.2024.64250
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The paper discusses about a new defuzzification method called the Comprehensive LiSamuel ranking for Triangular Intuitionistic Fuzzy Numbers (TrIFNs) which is improved from the LiSamuel ranking method associated with membership function and LiSamuel ranking method associated with nonmembership function. The Comprehensive LiSamuel ranking for TrIFNs is utilised to derive the weights for the Multiple Attribute Decision Making (MADM) problems under TrIFN. The MADM problem is solved based on twoperson zerosum game which is converted into a pair of LPP by using Comprehensive LiSamuel ranking. The process of transforming decision matrix into a matrix game and deriving decision maker’s and attribute weights from the matrix game are discussed. Numerical illustration is given to justify the viability and effectiveness of the proposed method and comparisons are made with existing ranking methods. The comparison study reveals that the proposed method provides the best maximum optimal solution for the decision problem than the existing methods.
I. INTRODUCTION
Multiple Attribute Decision Making (MADM) problems are more prevalent in reallife situations. The problem of MADM is to find the desirable solution from a limited number of possible alternatives tested for multiple attributes, both standardized and limited. To choose a desirable solution, the decision maker often provides his or her preferred information in the form of numerical values. However, in most cases, numerical values are insufficient to model reallife decision making problems. Indeed, human judgments including the details of preference information can be stated in the form of intuitionistic fuzzy knowledge. Therefore, the problem of MADM under intuitionistic fuzzy nature are an interesting study area for recent researchers. The details about the weights of the attribute can sometimes be known, partially known or be completely unknown. MADM problems are assumed to have a predetermined, limited number of alternatives. MADM problem solving involves filtering and ranking and it can be viewed as another way to integrate information into decision matrix together with additional information from the decision maker to obtain final ranking or selecting from the set of alternatives. In addition to the information contained in the decision matrix, it requires additional information from the decision maker to reach at a final ranking / selection. In many cases decision makers have vague information about alternatives with respect to attributes. The Triangular Intuitionistic Fuzzy Set (TrIFS), which has its membership function and nonmembership function is applied by many researchers in decision making theory. Robinson [14] and Robinson & Amirtharaj [15],[16],[17],[18] proposed correlation coefficient for various higher order IFS and applied them in MAGDM problems. Robinson & Amirtharaj [19] have given a MAGDM analysis for triangular and trapezoidal intuitionistic fuzzy sets. Robinson et al. [20] has proposed a decision support systems miner algorithm to solve MAGDM problems. A biological application of MAGDM was proposed in Robinson et al. [13]. A method for solving Triangular Intuitionistic Fuzzy Linear Programming Problem was introduced in Nirmalsingh & Robinson [11]. Li [5],[7], Li & Yang [8], Li & Wan [3] and Li et al. [9] introduced some linear programming approaches to multi attribute decision making with IFS. Li [6] presented the decision making process and game theory under IFS. Nan & Li [10] and An & Li [1] introduced linear programming approach to solve matrix games under IFS. Li [4] has given some arithmetic operations on triangular intuitionistic fuzzy sets and proposed a ratio ranking method to rank the triangular intuitionistic fuzzy numbers. Wang & Kerre [21] have introduced some properties for ordering the fuzzy values. Bhaumik et al. [2] has introduced a linear programming approach to solve triangular intuitionistic fuzzy matrix games using robust ranking method. Nirmalsingh et al.
[12] have proposed LiSamuel ranking method associated with membership function and LiSamuel ranking method associated with nonmembership function for defuzzification of TrIFNs. Xu et al. [22] have converted the decision information of the MADM problem to twoperson zerosum game and used linear programming to solve the matrix game.
Definition 1 [12]: The LiSamuel ranking method associated with membership function, for is defined as:

(1) 
Definition 2 [12]: The LiSamuel ranking method associated with nonmembership function, for TrIFN is defined as:

(2) 
The LiSamuel ranking associated with membership function eq.(1) considers the membership, and not the nonmembership degree and LiSamuel ranking associated with nonmembership function eq.(2) considers the nonmembership, and not the membership degree. This urges the need to propose a new ranking method which considers all the parameters of the TrIFNs, given in the following section.
II. A NOVEL DEFUZZIFICATION METHOD FOR TRIFNS
The LiSamuel ranking which is defined in definition 1 and 2 are for membership and nonmembership functions separately. The new Comprehensive LiSamuel ranking takes the membership and nonmembership into account and it is defined as follows:

(3) 
Where represents the decision maker’s preference level and means that the decision maker’s preference level is pessimistic; means that the decision maker’s preference level is optimistic; means that the decision maker’s preference level maintains neutrality. Thus, the new Comprehensive LiSamuel ranking reflects the decision maker’s subjective preference level.
Definition 3: Let and be two TrIFNs. Based on the Comprehensive LiSamuel ranking, for , we have:
Theorem 1: is a continuous nonincreasing function of the parameter .
Proof. is a linear function of the variable . Hence is continuous on . The Partial derivative of with respect to can be calculated as follows: .It is noted that . Hence, . Therefore, is a nonincreasing function of the parameter .
Theorem 2: Let and be two TrIFNs with and . Then for any ,
is true.
Proof. We know that .
Since and then let the minimum of membership is and maximum of nonmembership is .
Now, the Comprehensive LiSamuel ranking of for is as follows:
III. SOLVING TRIFN TWOPERSON ZEROSUM GAME WITH LINEAR PROGRAMMING TECHNIQUES
Let and be the sets of all pure strategies for player I and player II in a matrix game respectively, where be the payoff matrix with TrIFN entries. The TrIFN matrix game is denoted by where is the mixed strategies of the Player I and is the mixed strategies of the Player II. Each element of where informs us about the knowledge player I and player II indicate on their own payoffs provided that the player I and player II chooses pure strategy. The TrIFNs in the payoff matrix are assumed to be positive for simplicity.
The optimal strategies can be found using a pair of intuitionistic fuzzy linear programming problems which is given by;

(4) 
and

(5) 
By using the Comprehensive LiSamuel ranking (3) the TrIFNs are converted into crisp numbers and the pair of intuitionistic fuzzy LPP are transformed into classic LPP by defuzzification method.
and
The above LPP can be expanded into the following form,
(6) 
and
(7) 
where , , and are decision variables; is a parameter. For a specific the optimal solutions of the pair of linear programming problem are the optimal strategies of the TrIFN twoperson zerosum games.
IV. ALGORITHM FOR MADM PROBLEM UNDER TRIFN WITH DEFUZZIFICATION METHOD
V. NUMERICAL ILLUSTRATION
A numerical illustration is given here to demonstrate the process of the proposed method. A selection of video monitoring system for a school campus is taken as a decision problem in [22]. The same decision problem is considered here in order to give a comparative study of the proposed method. The normalized decision matrix with alternatives , , and and with attributes , , , and is given below:
Table 1: Normalized Decision Matrix
(0.3,0.4,0.9; 0.6,0.1) 
(0.3,0.5,0.7; 0.5,0.4) 
(0.5,0.7,0.8; 0.5,0.2) 
(0.6,0.8,0.9; 0.7,0.3) 
(0.4,0.5,0.6; 0.5,0.3) 

(0.4,0.5,1; 0.5,0.2) 
(0.5,0.6,0.7; 0.4,0.2) 
(0.4,0.5,0.6; 0.5,0.1) 
(0.6,0.7,0.8; 0.6,0.2) 
(0.3,0.4,1; 0.6,0.4) 

(0.2,0.4,0.5; 0.5,0.3) 
(0.6,0.8,1; 0.7,0.1) 
(0.4,0.6,0.8; 0.5,0.1) 
(0.4,0.6,1; 0.4,0.4) 
(0.4,0.5,0.6; 0.6,0.2) 

(0.3,0.4,0.8; 0.6,0.1) 
(0.4,0.5,0.6; 0.5,0.2) 
(0.6,0.8,1; 0.4,0.3) 
(0.4,0.6,0.8; 0.6,0.2) 
(0.5,0.6,0.7; 0.6,0.3) 
The conditions for attribute weights given by the decision maker are:
.
Step 2: The above decision matrix is transformed into a twoperson zerosum game with alternatives and attributes (j=1,2,…,5) as pure strategies. and are mixed strategies of decision maker and attribute. The decision matrix represents the payoff matrix.
Step 3: The TrIFN matrix game is converted to a pair of TrIFN linear programming problem given below:
Step 4: Applying the Comprehensive LiSamuel ranking method for as a decision parameter to convert the pair TrIFN linear programming to classic linear programming problems are given below:
Step 5: By solving the above problems we can obtain the optimal strategy , where and .
Step 6: Calculate the expected score of the alternatives by using the decision maker weight and attribute weight which are given below:
Step 7: Ranking all the alternatives with Comprehensive LiSamuel ranking method for are given below:
Step 8: The order of the alternatives according to the proposed ranking method is . Hence the alternative with the greatest value is selected as the best.
Table 2: Comparison of proposed method with existing methods.
Model 
Optimal solution of the LPP 
Ranking 
Xu et al. [22] 
Max X=0.2879 Min Y=0.3037 

Comprehensive LiSamuel ranking 
Max X=0.4152 Min Y=0.4345 

Comprehensive LiSamuel ranking coupled with Xu et al. [22] 
Max X=0.3484 Min Y=0.3578 
In Table 2, the result of MADM proplem using Xu et al. [22] method, the Comprehensive LiSamuel ranking method and Comprehensive LiSamuel ranking coupled with Xu et al. [22] method are observed to be consistent. The defuzzification in the Comprehensive LiSamuel ranking method is done by converting each TrIFN coefficient in LPP into crisp values, whereas in Comprehensive LiSamuel ranking coupled with Xu et al. [22] method, the defuzzification is done by taking the minimum of membership degree and maximum of nonmembership in each constraint of the LPP.
Table 3: Computations with Comprehensive LiSamuel Ranking
Sl.no. 

Ranking 
1 
0 

2 
0.1 

3 
0.2 

4 
0.3 

5 
0.4 

6 
0.5 

7 
0.6 

8 
0.7 

9 
0.8 

10 
0.9 

11 
1 
VI. DISCUSSION
In order to overcome the limitation of the LiSamuel ranking method, it is improved to consider both membership and nonmembership functions which is called Comprehensive LiSamuel ranking. Some properties and theorems are proved for the Comprehensive LiSamuel ranking method are given. An algorithm for MADM problem under TrIFN with the Comprehensive LiSamuel ranking method is proposed in which the decision matrix is considered as a TrIF twoperson zerosum game and solved using linear programming techniques. The TrIF twoperson zerosum game is solved by constructing a pair of TrIF linear programming problems then the TrIFNs are transformed into crisp values using the proposed Comprehensive LiSamuel ranking method with neutral preference level then the pair of LPPs are solved and the optimal solutions are taken as attribute weights with which the collective overall TrIFN of alternatives are found using expected score. Finally the alternatives are ranked using the proposed Comprehensive LiSamuel ranking method and the best alternative is selected. In the numerical illustration the same decision problem from [22] is considered here to give a comparative study of the proposed method. The result comparison between the current method and the Xu et al. [22] is given in Table 2. In that comparison all the result shows that is the best alternative even the ranking orders are the same but the optimal solution of the LPP constructed by the proposed Comprehensive LiSamuel ranking method gives the maximum value of X which is greater than the maximum X value given by the LPP by Xu et al. [22]. Further the Xu et al. [22] method uses only the minimum membership value and maximum membership value of the matrix game while constructing the LPP whereas in the proposed method Comprehensive LiSamuel ranking the membership and nonmembership value of each TrIFN element in the matrix game is considered while constructing the LPP. It is shown in Table 3 that for different value of same order of ranking occurs. In Xu et al. [22] for the alternative obtains the highest value, for the alternative obtains the highest value.
Even though the order of ranking in all the above discussed methods are same, the maximum value for the optimal solution is obtained from the newly proposed method.
In this article, the LiSamuel ranking method is improved and defined Comprehensive LiSamuel ranking method. Then the process of transforming the decision matrix of the MADM problem into a twoperson zerosum matrix game is defined. Then the matrix game is solved by converting it into a pair of TrIFN linear programming problems and the problems are changed into crisp liner programming problem using the proposed Comprehensive LiSamuel ranking method. The optimal solution of the linear programming problems are the optimal mixed strategies of the matrix game. Then the optimal mixed strategies are used as decision maker’s weights to find the expected score of the alternatives. The alternatives are ranked using the proposed Comprehensive LiSamuel ranking method. The alternative with the highest ranking is selected as the best alternative. A numerical illustration of a video monitoring system selection problem is solved using the proposed MADM algorithm and the results are compared with the existing methods presented in the literature. The comparative study reveals the effectiveness and applicability of the proposed methods and algorithms.
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Copyright © 2024 S. Samuel Nirmalsingh, P. John Robinson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Paper Id : IJRASET64250
Publish Date : 20240916
ISSN : 23219653
Publisher Name : IJRASET
DOI Link : Click Here