n,m-Rung Orthopair Fuzzy Sets With Applications to Multicriteria Decision Making

A q-rung orthopair fuzzy set is one of the effective generalizations of fuzzy set for dealing with uncertainties in information. Under this environment, in this study, we define a new type of extensions of fuzzy sets called n,m-rung orthopair fuzzy sets and investigate their relationship with Fermatean fuzzy sets, Pythagorean fuzzy sets and intuitionistic fuzzy sets. The n,m-rung orthopair fuzzy sets can supply with more doubtful circumstances than Fermatean fuzzy sets, Pythagorean fuzzy sets and intuitionistic fuzzy sets because of their larger range of depicting the membership grades. There is a symmetry between the values of this membership function and non-membership function. Here, any power function scales are utilized to widen the scope of the decision-making problems. In addition, the novel notion of an n,m-rung orthopair fuzzy set through double universes is more flexible when debating the symmetry between two or more objects that are better than the diffusing concept of an n-rung orthopair fuzzy set, as well as m-rung orthopair fuzzy set. The main advantage of n,m-rung orthopair fuzzy sets is that it can describe more uncertainties than Fermatean fuzzy sets, which can be applie in many decision-making problems. Then, we discover the essential set of operations for the n,m-rung orthopair fuzzy sets along with their several properties. Finally, we introduce a new operator, namely, n,m-rung orthopair fuzzy weighted power average (n,m-ROFWPA) over n,m-rung orthopair fuzzy sets and apply this operator to the MADM problems for evaluation of alternatives with n,m-rung orthopair fuzzy information.

dealing with uncertain objects. The merging between fuzzy 30 sets and some doubtfulness approaches like rough sets and 31 soft sets have been discussed in [1], [3], and [5]. Intuitionistic 32 fuzzy sets defined by Atanassov [2] and its one of the interest-33 ing generalizations of fuzzy sets with excellent applicability. 34 In various fields the applications of intuitionistic fuzzy sets 35 appear, including optimization problems decision-making, 36 medical diagnosis [8], [9], [10]. But, there are many cases 37 where the decision maker may give the degree of membership 38 and non-membership of a specific characteristic in such a 39 way that their sum is greater than one. Hence, Yager [24] 40 presented the concept of Pythagorean fuzzy set which is the 41 generalization of intuitionistic fuzzy sets and it is a more 42 effective tool to solve uncertain problems. The construct of 43 Pythagorean fuzzy sets can be used to describe doubtful 44 101 In this section, we study the notion of n,m-rung orthopair 102 fuzzy sets in details. For computations, we use only four 103 decimal places in whole paper. 104 A. n,m-RUNG ORTHOPAIR FUZZY SETS 105 Definition 1: Assume that N is a set of all natural numbers 106 and Z is a universal set. Then, the n,m-rung orthopair fuzzy 107 set (n,m-ROFS) , which is a set of ordered pairs over Z , 108 is defined by the following:  2) 1,2-rung orthopair fuzzy membership grades is smaller 139 than the set of Pythagorean membership grades. 3) Fermatean membership grades is larger than the set of 141 1,3-rung orthopair fuzzy and 2,3-rung orthopair fuzzy 142 membership grades.

143
Remark 4: From Figure 2, the set of 144 1) 2,m-rung orthopair fuzzy membership grades is larger 145 than the set of Pythagorean membership grades for 146 m > 2.

227
Similarly, we can get

229
It is obvious that These indicate that both of 1 ⊕ 2 and 1 ⊗ 2 are 240 n,m-ROFSs.

249
It is obvious that 0 ≤ ψ η ≤ 1, then we can get Similarly, we can also get

273
Proof: From Definition 8, we can obtain: 279 2) We can proof in a similar fashion to (1).
339 4) We can proof in a similar fashion to (3).
416 4) We can proof in a similar fashion to (3).

648
Step 1: Represent a MCDM problem under study using 649 n,m-rung orthopair fuzzy decision matrix.

653
Step 3: Using proposed n,m-ROFWPA operator to com-654 pute the alternative preference values with associated 655 weights.

656
Step 4: Find the score value for each alternative.

657
Step 5: Select the maximum score value to choose the best 658 alternative.  Table 1, where ϑ d i is the positive membership 668 degree for which alternative complies the given attribute and 669 ψ d i is the membership degree for which alternative does not 670 complies the given attribute such that 0  Table 2. 677 Now, we find the score value of each alternative and their 678 ranking as demonstrated in Table 3.

679
To explain the effect of the parameters n and m on MADM 680 end results, we have utilized different values of n and m 681 to rank the alternatives. The results of ranking order of the 682 alternatives based on n,m-ROFWPA operator are presented 683 in Table 3. When n, m = 2, 3, n, m = 3, 2, n, m = 4, 5 and 684 n, m = 10, 100, we obtained a rank of alternatives as J 4 685 J 2 J 1 J 3 J 5 , here, J 4 is the best choice and 686 J 2 is the best second choice, but, when n, m = 5, 4 and 687 n, m-ROFWPA( 1 , 2 , . . . , k ) ⊗ n, m-ROFWPA(L 1 , L 2 , . . . , L k )    n, m = 100, 10, we obtained a rank of alternatives as J 4 688 J 1 J 2 J 3 J 5 , here, J 4 is the best choice, and J 1 is 689 the best second choice. Thus, the overall best rank is J 4 .

690
To these MADM problems based on n,m-ROFWPA opera-  Table 4.

708
For express fuzzy information, n,m-rung orthopair fuzzy is 709 a good tool. It has a parameter n and m, so it holds a wider 710 range of fuzzy information than IFS, PFS, FFS and q-rung 711 orthopair fuzzy sets for q > 3. In this study, we have proposed 712 a new generalized intuitionistic fuzzy set is called n,m-rung 713 orthopair fuzzy sets and compared their relationship with 714 other kinds of the generalize of fuzzy sets. Moreover, some 715 operators on n,m-rung orthopair fuzzy sets are introduced and 716 their relationship have been proved. Also, we have presented 717 new weighted aggregated operator over n,m-rung orthopair 718 fuzzy sets and discussed their properties in details. Finally, 719 we have showed this procedure with one practical fully 720 developed example.

721
In future works, the weighted average operator, weighted 722 geometric operator and weighted power geometric opera-723 tor over n,m-rung orthopair fuzzy sets may be investigated 724 and study a MCDM methods depending on these operators. 725 In addition, we will try to define the topology on the collec-726 tion of n,m-rung orthopair fuzzy sets and introduce the ideas 727 of connectedness in n,m-rung orthopair fuzzy topology.