A p-rung orthopair fuzzy set (p-ROFS) describes a generalization of intuitionistic fuzzy set and Pythagorean fuzzy set in the case where we face a larger representation space of acceptable membership grades, and moreover, it gives a decision maker more flexibility in expressing his/her real preferences. Under the p-rung orthopair fuzzy environment, we are going to propose a novel and parametrized score function of p-ROFSs by incorporating the idea of weighted average of the degree of membership and non-membership functions. In view of this fact, this study is further undertaken to investigate and present different properties of the proposed score function for p-ROFSs. Moreover, we indicate that this ranking technique reduces some of the drawbacks of the existing ones. Eventually, we develop an approach based on the above-mentioned ranking technique to deal with multiple criteria decision making problems with p-rung orthopair fuzzy information.

Fuzzy set (Zadeh,

Since the sum of both satisfaction and dissatisfaction degrees of an IFS is not always less than or equal to 1, it would be interesting to consider the sum of their square to be less than or equal to 1. In this regard, the concept of IFS clearly fails to deal with such a situation. In order to clarify this fact, we suppose that an expert would like to express his/her preference in a decision making situation where the degree of an object satisfying a criterion is

Besides that, we may consider a more general situation in which an expert would like to express his/her preference by the degree of an object satisfying a criterion as

The p-ROFS concept describes the degree of an object satisfying a criterion together with the degree of dissatisfying such that the sum of

Apart from the above-mentioned advantages, an immediate and very interesting benefit of p-ROFS definition is that it provides the experts with greater freedom in modelling the forms of imprecise information.

There exist a large number of researches that deal mainly with the concept of p-ROFS which has been applied in many different contexts. Liu and Wang (

One of the leading topics of the recent development of p-ROFSs has been the focus on the ranking function which plays an essential role in the decision making problems. The pioneer works in this regard are those proposed by Yager (

Regarding the above-mentioned deficiencies of the existing p-ROFS ranking techniques, we are motivated here to investigate an effective score function for p-ROFSs in the form of non-algorithmic ranking technique which is constructed by considering the impact of membership and non-membership degrees together with the hesitation information.

The present contribution is organized as follows: Introduction of p-ROFS concept and a brief review of some preliminaries are given in Section

Throughout this section, we are willing to firstly review the concepts of IFS and PFS, and then we will focus mainly on the concept of p-rung orthopair fuzzy set (p-ROFS) and its essential set and algebraical operations.

Now, if we are interested in describing a situation in which an expert would like to express his/her preference by the degree of an object satisfying a criterion as

Moreover, for notational convenience, we name

Comparison of spaces of p-ROFSs for the parameters

As can be seen from Fig.

In what follows, we are interested in reviewing a number of set and algebraical operations on p-ROFNs.

In an analogous manner similar to IFNs and PFNs, the subset relation of p-ROFNs is defined as the following:

An immediate consequence from the above definition is that

Throughout the present section, we first review the existing ranking techniques of p-ROFN, and in the second stage, we propose a new parametrical score function for p-ROFNs by taking both the membership and the hesitation degree of a p-ROFN into account.

Assume that

With the help of this setting, we are able to present the comparison rule between the two p-ROFNs

if

if

There exists another p-ROFN score function introduced by Wei

With the help of this setting, we are able to present another comparison rule between the two p-ROFNs

if

if

Following Yager’s (

Using this setting, the comparison rule between the two p-ROFNs

if

if

if

if

In continuation of Liu and Wang’s (

Based on the above score function

For any two p-ROFNs

if

if

if

if

In view of this setting, Peng

Furthermore, Peng

We are now in a position to provide a brief summary of advantages and disadvantages of the ranking techniques described above:

Following from the non-algorithmic techniques of Yager (

Both Liu and Wang’s (

On the basis of the above-mentioned deficiencies of the existing techniques, we still believe that an effective score function in the form of non-algorithmic ranking technique should be constructed by considering the impact of membership and non-membership degrees together with the hesitation information.

It is interesting to note that the score function

From another point of view, the score function

However, such a consideration in defining a parametrized score function is quite common and reasonable, and it can be found in Zhang

The formula (

It is also of some interest to note that the above-introduced innovative score function

It is easily seen from Definition

Let us now investigate a number of other properties of innovative score function

For any p-ROFN

On the other hand, from the fact that

The proof comes from the fact that

(Necessity) The relation

From the relation (

Taking all the above relations into account, we get

From definition of the innovative score function

As a result, it is interesting to remark that in the case where

The proof is immediately apparent from calculating the first partial derivatives of

The result now follows from the fact that

In this part of the contribution, we re-consider once again the comparison results given in Peng

Needless to say that what we expect from Remark

The ranking results of the existing score functions.

p-ROFN | Ranking | Ranking | |||

Ranking | Ranking | ||

(Using |
|||

The ranking results of the proposed score function

p-ROFN | Ranking | |||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

The ranking results of the existing score functions.

p-ROFN | Ranking | Ranking | |||

Ranking | Ranking | ||

(Using |
|||

The ranking results of the proposed score function

p-ROFN | Ranking | |||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

The ranking results of the existing score functions.

Ranking | Ranking | ||||

Ranking | Ranking | ||

(Using |
|||

(Using |
|||

The ranking results of the proposed score function

p-ROFN | Ranking | |||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

∗ | ||||

The findings from Tables

The first collection of p-ROFNs demonstrates that

In spite of the existing score-based comparison techniques

Multiple criteria decision making (MCDM) is an active research area, and there exist a large number of researches ( Farhadinia,

In this part of the manuscript, we are facing a MCDM problem in which decision making is made by the use of a ranking procedure of p-ROFNs.

Suppose that

Now, with the help of Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) technique (Liou

Determine the best and the worst values with respect to all criteria which are denoted respectively by the p-ROFNs

(For the benefit criteria)

(For the cost criteria)

Construct the score matrix

Construct the following normalized nearest-best and farthest-worst solution matrices:

Keeping the above normalized nearest-best and farthest-worst solution matrices together with the weighting vector

(Best case)

Construct the normalized nearest-best group utility and nearest-best individual regret values, respectively, of alternative

Construct the nearest-best and farthest-worst score values of alternative

Compute the relative closeness degree of each alternative

Before going more into detail, we summarize again the superiorities of the above-mentioned MCDM algorithm compared to the existing approaches being based independently on VIKOR or TOPSIS techniques:

The proposed MCDM algorithm implements the innovative score function

By employing the new transformation function MIN, we are able to prevent violence of

The proposed MCDM algorithm ranks the alternatives based on the combination of VIKOR and TOPSIS outputs.

We assume that the decision values are described by p-ROFNs in the form of the decision matrix:

In order to save more space for convenient storage, we do not state here the calculation of

By the way, we obtain here:

(For

Hereafter, we do not give the details of computation of p-ROFNs for

(Best case: for

In this case, we omit the calculations of

On the basis of

(For

Rankings of alternatives for different score-based MCDM techniques under p-ROFN environment.

Score function | The final ranking | |

Proposed |
||

Proposed |
||

Proposed |
||

From Table 7, we can observe that the results of ranking orders for suppliers based on the existing score functions of Yager (

The purpose of this paper was to present an innovative and non-algorithmic ranking score function for p-ROFSs. The comparison of innovative score function for p-ROFSs with the existing non-algorithmic ranking ones showed some inherent advantages of the former one over the latter ones. Eventually, the performance of innovative score function for p-ROFSs compared to that of other score functions was demonstrated in a MCDM problem. Although, it usually seems that an algorithmic ranking technique should be more reliable than the proposed non-algorithmic ranking technique, but the existing algorithmic technique of Liu and Wang (

By the way, there exist a lot of fruitful research perspectives that can be productively pursued through the application of p-ROFS concept in conjunction with decision making situations. Indeed, the future works can be further extended by applying the proposed MCDM technique to the other fields which may be classified as

The scholars which may be considered for defining a class of reasonable comparative techniques, not only based on the score and the accuracy functions, but also based on more comparable rules of p-ROFSs;

The contributions which are based on the integration theory of p-ROFSs, specifically, those that are focusing on the aggregation operators of p-ROFSs;

The studies which deal with the information measures for p-ROFSs such as distance, similarity and entropy measures, and those studies that suggest a variety of systematic transformations of information measures;

Those working on the preference relations of p-ROFSs, and subsequently on the group consensus measures which are mainly divided into iterative and interactive categories;

The scholars which propose fruitful classes of decision making techniques under p-rung orthopair fuzzy environment.