Informatica

Fuzzy set (Zadeh, 1965) was first introduced for assigning each criterion to a vague or an uncertain information with the help of membership degree which ranges between 0 and 1. Then, intuitionistic fuzzy set (IFS) (Atanassov, 1986) was proposed to take into account both membership and non-membership degrees such that their sum is less than or equal to 1. This property is used to describe an object which satisfies a criterion more definitively and further precisely compared with that in fuzzy environment (Cong, 2014; Si et al., 2019).

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 $\frac{\sqrt{3}}{2}$, and the degree of dissatisfying is $\frac{1}{2}$. In this case, we easily see that $\frac{\sqrt{3}}{2}+\frac{1}{2}\geqslant 1$, that is, the current case is not able to be properly described by the use of IFS concept. Instead, the satisfaction of ${\big(\frac{\sqrt{3}}{2}\big)^{2}}+{\big(\frac{1}{2}\big)^{2}}\leqslant 1$ shows the need of employing another concept for describing the latter-mentioned case, and that concept is nothing else except the concept of Pythagorean fuzzy set (PFS) (Yager, 2013).

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 $\frac{\sqrt[3]{7}}{2}$, and the degree of dissatisfying is $\frac{1}{2}$. Then, it is easily seen that $\frac{\sqrt[3]{7}}{2}+\frac{1}{2}\geqslant 1$ and also ${\big(\frac{\sqrt[3]{7}}{2}\big)^{2}}+{\big(\frac{1}{2}\big)^{2}}\geqslant 1$ which indicate that the current case cannot be described with the help of concepts of IFS and PFS. This is while, ${\big(\frac{\sqrt[3]{7}}{2}\big)^{3}}+{\big(\frac{1}{2}\big)^{3}}\leqslant 1$. Thus, such an implication makes clear the need of defining a more general concept than the IFS and the PFS concepts. Indeed, this fact made a study of introducing a more general concept than that of IFS and PFS which is called p-rung orthopair fuzzy set (p-ROFS) (Yager, 2017).

The p-ROFS concept describes the degree of an object satisfying a criterion together with the degree of dissatisfying such that the sum of p-power of both satisfaction and dissatisfaction degrees is less than or equal to 1. It can be easily deduced that the space of p-ROFS membership grades is greater than those of Pythagorean and intuitionistic membership grades. That is, any Pythagorean or intuitionistic membership grade is also a p-ROFS membership grade, but in general, all p-ROFS membership grades are not in the forms of Pythagorean or intuitionistic membership grades. This obviously implies that we are able to implement the concept of p-ROFS in the cases where we do not allow to implement the concepts of PFS or IFS.

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 (2018) introduced and investigated the p-rung orthopair fuzzy weighted averaging operator and the p-rung orthpair fuzzy weighted geometric operator. Moreover, the p-rung orthopair fuzzy Bonferroni mean and the p-rung orthopair fuzzy Heronian mean were proposed respectively by Liu and Liu (2018) and Wei et al. (2018). Subsequently, the p-rung orthopair fuzzy Archimedean Bonferroni mean operators were developed by Liu and Wang (2019). Besides the latter-mentioned application of p-ROFSs, there is an increasing demand and a growing number of researches for other application fields, for instance, Du (2018) proposed Minkowski-type distance measures for p-ROFS by emphasizing on their application in decision making, and Zhang C. et al. (2019, in press) dealt with additive and multiplicative consistency analysis for p-ROF preference relation.

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 (2017) and Wei et al. (2018) in which non-algorithmic ranking techniques for p-ROFSs are described. As it will be demonstrated later, Yager’s (2017) and Wei et al.’s (2018) score functions cannot differentiate many p-ROFNs in some situations. The other is that given by Liu and Wang (2018) as an algorithmic ranking technique which uses both score and accuracy functions of p-ROFSs. By taking the impact of membership and non-membership degrees together with the hesitation information into account, Peng et al. (2018) introduced another algorithmic ranking technique for p-ROFSs. Anyway, both Liu and Wang’s (2018) and Peng et al.’s (2018) techniques are in algorithmic form and they need definitely more computations than the non-algorithmic techniques. Moreover, since Liu and Wang’s (2018) technique does not consider the influence of abstention, the information of p-ROFS may be lost. Furthermore, the curve function $f(x)=\frac{\exp (x)}{\exp (x)+1}$ which appears in Peng et al.’s (2018) technique leads to more complexity in the evaluation of the score function.

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 2. Section 3 deals with reviewing three kinds of p-ROFS ranking orders, and then this section continues with introducing an innovative score function for p-ROFSs which possesses different properties. Section 4 is devoted to the application of p-ROFS score function in multiple criteria decision making (MCDM) problems by emphasizing the superiority of the proposed score function over the other existing ones. Finally, this article concludes in Section 5.

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 $\frac{\sqrt[3]{7}}{2}$, and the degree of dissatisfying as $\frac{1}{2}$, then we immediately find that none of the existing concepts of IFS and PFS can explain such a situation because of $\frac{\sqrt[3]{7}}{2}+\frac{1}{2}\leqslant ̸1$ and ${\big(\frac{\sqrt[3]{7}}{2}\big)^{2}}+{\big(\frac{1}{2}\big)^{2}}\leqslant ̸1$. This impulses Yager (2017) to define a more general concept than the IFS and the PFS concepts which is not only able to improve such a shortcoming, but it is also well-obeyed in this case.

Moreover, for notational convenience, we name $({\mu _{{A_{\text{p-ROFS}\hspace{2.5pt}}}}}(x),{\nu _{{A_{\text{p-ROFS}\hspace{2.5pt}}}}}(x))$ a p-rung orthopair fuzzy number (p-ROFN), and it is simply indicated hereafter by $({\mu _{{A_{\text{p-ROFN}\hspace{2.5pt}}}}},{\nu _{{A_{\text{p-ROFN}\hspace{2.5pt}}}}})$.

As can be seen from Fig. 1, the intuitionistic membership degrees are those points being located in the area of the graph $x+y\leqslant 1$, and the Pythagorean membership degrees are those points being located in the area of the graph ${x^{2}}+{y^{2}}\leqslant 1$. This is while, the p-rung orthopair membership degrees are those points being located in the area of the graph ${x^{p}}+{y^{p}}\leqslant 1$ where $p\in [1,\infty )$. This implies that the p-rung orthopair membership degrees provide us with a wider representation of non-standard membership degrees than the intuitionistic and the Pythagorean membership degrees. Indeed, any intuitionistic fuzzy number (IFN) and Pythagorean fuzzy number (PFN) can be considered as a p-ROFN, but the reverse is not held, that is, all p-ROFNs are not always IFNs or PFNs.

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 for any $k>0$.

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.

Multiple criteria decision making (MCDM) is an active research area, and there exist a large number of researches ( Farhadinia, 2014, 2016a, 2016b; Farhadinia and Herrera-Viedma, 2018; Farhadinia and Xu, 2017) in which the decision maker is going to provide a list of alternatives ranking in accordance with a given set of criteria.

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 $X=\{{x_{1}},{x_{2}},\dots ,{x_{m}}\}$ describes a set of alternatives, and the set of criteria is in the form of $C=\{{c_{1}},{c_{2}},\dots ,{c_{n}}\}$. Furthermore, we denote the associated weight vector of criteria by $w=({w_{1}},{w_{2}},\dots ,{w_{n}})$ such that $0\leqslant {w_{j}}\leqslant 1$ $(j=1,2,\dots ,n)$ with the property ${\textstyle\sum _{j=1}^{n}}{w_{j}}=1$. Assume that a decision maker group is organized to evaluate the characteristics of each alternative with respect to each criterion with the help of p-ROFN concept. In this regard, the p-rung orthopair fuzzy decision matrix will be where $0\leqslant {\mu _{{D_{\text{p-ROFN}\hspace{2.5pt}}^{(ij)}}}^{p}}+{\nu _{{D_{\text{p-ROFN}\hspace{2.5pt}}^{(ij)}}}^{p}}\leqslant 1$ for any $p\in [1,\infty )$.

Now, with the help of Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) technique (Liou et al., 2011) and Technique for Order of Preference by Similarity to the Ideal Solution (TOPSIS) (Lai et al., 1994) together with implementing the innovative score function of p-ROFNs, we will be able to describe a MCDM algorithm as the following:

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:

We assume that the decision values are described by p-ROFNs in the form of the decision matrix: Needless to say that the above decision matrix is not in the form of an intuitionistic fuzzy matrix as considered in Chen et al. (2016) because the entries $({\mu _{{D_{\text{p-ROFN}\hspace{2.5pt}}^{(21)}}}},{\nu _{{D_{\text{p-ROFN}\hspace{2.5pt}}^{(21)}}}})$ and $({\mu _{{D_{\text{p-ROFN}\hspace{2.5pt}}^{(41)}}}},{\nu _{{D_{\text{p-ROFN}\hspace{2.5pt}}^{(41)}}}})$ are not IFNs.

Step 1. We determine the best and the worst values in correspondence with all benefit criteria as the following:

Step 2. By keeping the score functions $S{c_{YLW}}$, $S{c_{\mathit{PDG}}}$ and $S{c_{F}}$ given respectively by (10), (12) and (14) into account, we are able to construct the score matrices for $p=2$; for $p=3$; for $p=2$; for $p=3$; for $p=2$ $(\lambda =0.5)$; for $p=3$ $(\lambda =0.5)$.

In order to save more space for convenient storage, we do not state here the calculation of ${S_{D-F}}={[S{c_{F}}({D_{\text{p-ROFN}\hspace{2.5pt}}^{(ij)}})]_{5\times 4}}$ for $\lambda =0,1$ (as samples of values $\lambda \in [0,1]$), and only the corresponding results will be reported in Step 5.

By the way, we obtain here:

(For $p=2$) (For $p=3$) (For $p=2$) (For $p=3$) (For $p=2$ $(\lambda =0.5)$) (For $p=3$ $(\lambda =0.5)$)

Hereafter, we do not give the details of computation of p-ROFNs for $p=3$, and only the corresponding results for $p=2$ are returned.

Step 3. We construct the normalized nearest-best and farthest-worst solution matrices as follows: for $p=2$, for $p=2$, for $p=2$ $(\lambda =0.5)$, and for $p=2$, for $p=2$, for $p=2$ $(\lambda =0.5)$.

Step 4. If we keep the above normalized nearest-best and farthest-worst solution matrices together with the weighting vector $w=({w_{1}}=0.25,{w_{2}}=0.40,{w_{3}}=20,{w_{4}}=0.15)$ in mind, then we will be able to calculate the group utility and the individual regret for each alternative ${x_{i}}$ in both best and worst cases:

(Best case: for $p=2$) (Worst case: for $p=2$)

In this case, we omit the calculations of ${S_{i}^{\mathit{worst}}}$ and ${R_{i}^{\mathit{worst}}}$ because they are similar to those given above.

Step 5. We construct the normalized nearest-best group utility and nearest-best individual regret values, respectively, of alternative ${x_{i}}$ as the following: (for $p=2$) The calculations of the normalized farthest-worst group utility values ${\underline{\underline{S}}_{\hspace{0.1667em}i}}$ and farthest-worst individual regret values ${\underline{\underline{R}}_{\hspace{0.1667em}i}}$ are omitted due to lack of space.

On the basis of Step 6 and Step 7, we can determine the relative closeness degree of each alternative ${x_{i}}$ for both cases $p=2,3$:

(For $p=2$) (For $p=3$) where the smaller value of the relative closeness degree indicates the better preference order of alternative ${x_{i}}$. In this regard, the preference orders of the alternatives are given in Table 7.

From Table 7, we can observe that the results of ranking orders for suppliers based on the existing score functions of Yager (2017), Wei et al. (2018), Liu and Wang (2018), and Peng et al. (2018) compared to the proposed score function $S{c_{F}}$ remain unchanged for the values $\lambda =0,0.5$, and more or less different for $\lambda =1$. However, a decision maker may select the diverse values of parameter λ in accordance with his/her diverse preferences and attitudes in real and actual decision making cases. Therefore, the application range of proposed non-algorithmic ranking technique of p-ROFNs is wider than the existing ones, and it can flexibly handle more general decision information compared to the algorithmic ranking techniques. Moreover, by taking the new transformation function MIN into account, we can prevent violence of division by zero which may occur in the traditional version of VIKOR techniques. These superiorities of the proposed score-based MCDM algorithm compared to the existing approaches indicate that it is more suitable for the actual situations.

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 (2018) does not work logically for the present problem.

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