Momentum Ranking Function of Z-Numbers and its Application to Game Theory

: After Zadeh introduced the concept of z-number scientists in various fields have shown keen interest in applying this concept in various applications. In applications of z-numbers, to compare two z-numbers, a ranking procedure is essential. While a few ranking functions have been already proposed in the literature there is a need to evolve some more good ranking functions. In this paper, a novel ranking function for z-numbers is proposed-"the Momentum Ranking Function"(MRF). Also, game theoretic problems where the payoff matrix elements are z-numbers are considered and the application of the momentum ranking function in such problems is demonstrated.


Introduction:
Operations research techniques are widely applied in many real-life problems. However, while collecting data for the construction of the appropriate mathematical model, it is frequently seen that the data available is imprecise and not fully reliable. In 2011, Zadeh 1 developed the concept of fuzzy numbers and introduced znumbers. z-numbers contain both information and an estimate of the reliability of the information. Hence z numbers have the potential for application in various types of operations research problems. Zainb Hassan Radhy 2 and et.al. dealt fuzzy assignment model by using a linguistic variable. Computations involving z-numbers were studied by Shahila Bhanu and Velammal 3 . In dealing with applications of z-numbers there are two challenges 1. How to perform arithmetic operations on z numbers? 2. How to compare z numbers?
The first challenge can be overcome by using the novel R-type arithmetic operations introduced by Stephen 4 . The second problem-the problem of ranking or ordering z-numbers is a topic of interest to those who wish to study applications of znumbers. So far, a few ranking methods have been proposed. Rasha Jalal Mitlif 5 describes an efficient algorithm for Fuzzy Linear Fractional Programming Problems via Ranking Function. Siddhartha Sankar Biswas 6 defines Z 1 as strongly greater than Z 2 as componentwise comparisons. Wen Jiang, ChunheXie, Yu Luo, and YongchuanTang 7 proposed a new method for ranking Z-numbers by evaluating generalized fuzzy numbers. Iden Hassan Hussein and Rasha Jalal Mitilif 8 introduced a ranking function to solve fuzzy multiple objective functions. Amir HoseinMahmoodiI D et. al. 9 gave the comparison of linguistic z-numbers with the max-score rule. Bingyi Kanga, Gyan Chhipi-Shrestha, Yong Denga, Kasun Hewage, RehanSadiq 10 gave the evolutionary games with the z-number. Iden Hassan Hussein, Zainab Saad Abood 11 solved fuzzy game problems by using three different ranking functions. Mujahid Abdullahi 12 and others gave a new ranking method for Z-by converting z-number into a fuzzy number, and then the centroid, point method, and decision rules are utilized to rank the obtained fuzzy numbers. Parameswari 13 proposed a Lexicographic order-based ranking on z numbers. However, there is scope for further research in this area.
In this paper, a novel ranking function for znumbers is proposed-the momentum ranking function. Also, game theoretic problems where the payoff matrix elements are z numbers, that is, zpayoff matrix are considered and the application of

Preliminary Definitions: Definition 1: Formal definition of z-number
Consider an ordered pair of (C, D) where C is a fuzzy set defined on the real line and D is a fuzzy number whose support is contained in [0,1]. Then (C, D) is called a z-number.

Definition 2: Formal definition of z-valuation
Let X be an uncertain variable. The z-valuation 'X is z (C, D)' is equivalent to an assignment statement "X is (C, D )". It means that FEP (X is C) is D.

Definition 3: Ranking Function
A ranking function r k on a set of fuzzy numbers F, a real-valued function, A 1 ≤ A 2 if and only if r k (A 1 ) ≤ r k (A 2 ), where A 1 , A 2 ∈ F.

Definition 4: MIN R Operation
Let * be any one of the basic arithmetic operations addition, subtraction, multiplication, or division.

Momentum Ranking Function of z-number Definition 7: Momentum Ranking Function [MRF]
Let r 1 and r 2 be any two ranking functions for fuzzy numbers. Then for the z-number (A, B), define the Momentum Ranking Function[MRF] by ( ) = M(r 1 , r 2 )(z) = r 1 (A)r 2 (B) The MRF function can then be used to rank or order a list of z-numbers.

Definition 9: z-Games
Two persons zero sum z-game is a two-person zerosum game with the elements of the payoff matrix znumbers. Player 1 has m strategies and player 2 has n strategies.
The entry z ij gives information regarding the z-payoff to player 1 when strategy i is used by the first player and strategy j is used by the second player. If z ij = (A ij , B ij ), then A ij is the fuzzy estimate of player 1's gain and B ij is the reliability of this estimate.
Step 1: Choose any two ranking functions r 1 and r 2 , the MRF function M(r 1 , r 2 ) can be used to order the entries in any row or column. Hence the maximum of each column and the minimum of every row can be found. If a row minimum coincides with a column maximum, then it can be considered as a z-saddle point, then continue, otherwise, if a z-saddle point does not exist, then go to Step 4 Step 2: If (l, m) is a z-saddle point then the optimal strategy for players 1 and 2 is ( , ), and the value of the game is ( , ), where is calculated by using step 3.
Step 3: However, what is the reliability of this estimate.? Since all the entries of the z-payoff matrix play a role in the computation, the reliability of the expected gain is = B MIN = min{ B ij |1 ≤ i ≤ m, 1 ≤ j ≤ n}, where the minimum of fuzzy numbers is calculated by ordering the list of fuzzy numbers by the r 2 ranking function. Hence the optimum strategy for player 1 at the z-saddle point (l, m) is(A lm , B MIN ).
Step 4: If the z-saddle point of the given z-game does not exist and the z-payoff is 2×2, then go to Step 6, or if the given z-payoff matrix is of any m×n, where m ≥ 3 and n ≥ 3, then continue the following steps.
Step 5: Reduce the given z-payoff matrix into 2 × 2 by using the z-dominance property: (a) Find the value of M(r 1 , r 2 )(z ij ), i = 1 to m, and j = 1 to n by using any two-ranking functions r 1 and r 2 . (b) All the ranks of the k th row or any convex linear combination of two or more strategies (rows) are less than or equal to the corresponding ranks of any other r th row, then the k th row is dominated by the r th row.
(c) All the ranks of the k th column or any convex linear combination of two or more strategies (columns) are greater than or equal to the corresponding ranks of any other r th column, then the k th column is dominated by the r th column. (d) To reduce the size of the z-payoff matrix, delete the dominated rows or columns. (e) Do the above steps (b) to (d) repeatedly until get the z-game's z-payoff as 2 ×2.
Step 6: For any 2×2 two-person zero-sum z-game without any z-saddle point having the payoff matrix as:  .

Numerical Computation
Example 3: Consider the following z-payoff, present in Table 1: for the trapezoidal fuzzy number C(a,b,c,d) and find MRF(Z ij ), i = 1 to 3, j = 1 to 3, then put it in Table 2.  . .

Conclusion:
A novel ranking procedure for z-numbers has been presented here. Its application to twoperson zero-sum z-games has been highlighted. MRF is easy to implement and will prove to be a very useful tool in z versions of various optimization problems.