A study of cubical fuzzy possibility degree measure and its applications to multiple attribute decision-making problems

Abstract

In recent years, the extensions of fuzzy sets are much more familiar in almost all fields as they are reliable in defining the imprecise information of every decision-making situation. In this sequence of extensions, the cubical fuzzy sets are very efficient in dealing with imprecise information as it extends picture and spherical fuzzy sets. This article is interested in developing a new improved cubical fuzzy possibility degree measure. The desirable properties of the developed measure are also discussed. The advantage of the proposed measure is that it is capable of comparing the cubical fuzzy numbers in fuzzy nature itself and provides the degrees of preference relations between them. A comparison study is made with the existing ranking measures to exhibit the feasibility and validity of the proposed approach. Based on the improved measure, a method for ranking cubical fuzzy numbers is constructed. A solution approach to a cubical fuzzy multiple attribute decision-making problem is presented. To exhibit the potency and the practical applicability of the proposal, two real-life instances of selecting the best-cutting fluid for cutting gears have been illustrated. The results are compared with the literature.

Keywords

Cubical fuzzy set cubical fuzzy number (CFN)possibility degree measure (PDM)improved possibility degree measure (IPDM)ranking

1 Introduction

During a decision-making (DM) process, every decision-maker has to deal with uncertain/fuzzy information. Zadeh [1] introduced the fuzzy set theory to handle this type of fuzzy information by assuming a membership degree μ to each and every element of the set, provided the membership degree should lie between 0 and 1. In later years, Atanassov [2, 3] developed an intuitionistic fuzzy set [IFS] as an effective tool to handle the fuzzy information in a two-dimensional space, by extending the fuzzy set with an additional degree of non-membership ν satisfying the condition that the total of the two degrees should be bounded by 0 and 1.

Later, the picture fuzzy set (PFS) developed by B.C. Cuong et al. [5], spherical fuzzy set (SFS) defined by S. Ashraf et al. [6] and cubical fuzzy set (CFS) proposed by Asghar Khan et al. [7] are more powerful in dealing with the uncertainties as they consider three degrees like membership μ, neutral η and non-membership ν for each element of the sets considered. The origin of the CFS is shown in Fig. 1. The conditions of PFS and SFS limit the decision-maker in assigning the three degrees in the linear and spherical space whereas CFS gives freedom to the decision-maker to assign them in the broader cubical space of a fuzzy environment. CFS is clearly, the superset of PFS and SFS. Comparison of CFS, SFS, and PFS is explained using Table 2 and diagrammatically shown in Fig. 2. This motivates us to address the cubical fuzzy environment for the DM process.

Fig. 1

Origin of CFS.

Table 1

Comparison of PFS, SFS and CFS

Fuzzy set	Space covered	Deals with the values satisfying the condition
PFS [5]	Linear	0 ≤ μ + η + ν ≤ 1
SFS [6]	Spherical (also covers linear)	0 ≤ μ² + η² + ν² ≤ 1
CFS [7]	Cubical (also covers linear and spherical)	0 ≤ μ³ + η³ + ν³ ≤ 1

Table 2

Advantages of the proposed measures over the existing ranking measures

Ranking measure	Compares all types of CFNs	Compares CFNs in Fuzzy Nature itself	Provides relative closeness information	Applied with real life illustration
Score function [7]	No	No	No	Yes
Accuracy function [7]	No	No	No	Yes
Certainty function [7]	No	No	No	Yes
PDM [Proposed]	No	Yes	Yes	Yes
IPDM [Proposed]	Yes	Yes	Yes	Yes

Fig. 2

Diagrammatic representation of CFS.

During the cubical fuzzy DM process, the comparison of CFNs plays a major role in ordering them. Several ranking measures for comparing CFNs are available in the literature. Asghar Khan et al. [7] employed a variety of measures like score, accuracy, and certainty measures to rank different CFNs. They compare the CFNs by converting them into their equivalent crisp values. However, it is found that in some circumstances these measures fail to order CFNs. For example, if two different CFNs have the same score, accuracy, and certainty values, then it is difficult to distinguish them. To overcome the lack of existing ranking methods, it is necessary to develop some new measures for the effective comparison of the CFNs.

In this aspect, an alternative to probability theory called the possibility theory, helps to deal with some types of uncertainty as it compares two dissimilar items using the possibility degree measure, which measures the chance of one of them happening. Xu and Da [8] introduced “The possibility degree approach, in the context of the probabilistic method of ranking interval numbers”. Tao et al. [9] “Ranked the interval-valued (IV) intuitionistic fuzzy numbers (IFNs) with possibility degree and applied it to solve fuzzy multiple attribute decision-making (MADM) problem”. Wei and Tang [10] developed the PDM for ranking IFNs. Wan and Dong [11] constructed “A new ranking approach for IVIFNs based on the possibility degree and then applied it to solve a multi-attribute group decision-making problem”. Zhang et al. [12] proposed “A new way to rank distinct IVIFNs using the possibility degree method”. Gao [13] discussed the possibility degree and comprehensive priority of the interval numbers. “An overview of possible measurements of IVIFSs [4] and their applications to multiple criteria decision-making (MCDM)” was done by Dammak et al. [14]. Samir and Animesh [15] used “The PDMs for ranking IVIFNs in solving MCDM problems”. Jun Ye [16] developed “A MADM method based on the possibility degree ranking of interval neutrosophic numbers”. Harish Garg and Kamal Kumar [17] improved the existing possibility measure for IFNS. Dammak et al. [19] made the comparison between the existing intuitionistic fuzzy possibility measures by implementing the MCDM method. Xu and Da [20] prioritized the interval judgment matrices based on the possibility method. Lan et al. [21] introduced a new method for ranking interval numbers using two-dimensional priority degrees. Qiao et al. [22] used the possibility degree of Z-numbers to develop a multi-criteria PROMETHEE method under an uncertain linguistic environment. Garg and Kumar [[23, 24]] developed an IVIF possibility degree using the connection number of set pair analysis and defined a possibility degree for linguistic IVIFN for solving a group decision-making problem. Shekhovtsov et al. [25] discussed the usage of possibility degree in solving multi-criteria decision analysis problems.

As a result of the preceding investigations conducted in the possibility degrees of IFS and neutrosophic contexts, this paper aims at introducing new PDMs and ranking methods based on them to deal with the MADM problem under a cubical fuzzy (CF) environment. These measures provide the probability of the superiority of one CFN over another effectively by addressing the inadequacies of existing ranking measures and by utilizing the notion of the three-dimensional random vector from a probability standpoint. The advantages of the proposed measures are shown in Table 2. To achieve this, the pivotal contributions of the proposed study may be listed as follows.

For an effective comparison of CFNs, basically, a new PDM that overcomes the shortcomings of the existing ranking measures is proposed and some numerical examples to reveal its advantage.

However, the proposed PDM is insufficient to compare all types of CFNs. To conquer this limitation, an IPDM is developed and some numerical examples under the CF environment are discussed to show its potency.

Some desirable properties like normative, complementary, reflexivity, and transitivity of the IPDM are verified.

Ranking methods for ordering the CFNs are constructed based on the proposed PDMs.

A DM approach based on the proposed PDMs for solving cubical fuzzy MADM problems is developed.

Developed ranking methods are compared with existing ones in the literature to validate the proposal.

Numerical illustrations of two real-life instances of selecting the best fluid for cutting spur gears are considered to show the efficacy of the proposed CFMADM approach.

This shows that the proposed methods provide the probability of relative closeness relation between the CFNs which is an infirmity in the existing ranking measures.

Also, the proposed PDMs compare the CFNs in the fuzzy nature itself without converting them into crisp values which is a shortfall of other existing ranking measures.

The content of this article is outlined as follows: Section 2 exhibits the fundamentals of CFS. Section 3 defines a new PDM and a ranking method based on it. This section also provides the advantage of the proposed method over the existing ranking methods. Section 4 defines an IPDM and a ranking method based on it. This section also discusses the noteworthy properties of improved measure and its superiority over PDM and the available works in the literature. Section 5 presents a comparative study between the proposed PDMs and several existing measures. Section 6 deals with DM approaches to solve MADM problems based on the proposed PDM and IPDM ranking methods in the CFS environment. Section 7 uses real-life instances to demonstrate the effectiveness, feasibility, and superiority of the proposed cubical fuzzy DM approaches. Section 8 brings the article to a close.

2 Preliminaries

This section discusses some fundamental concepts of CFS whose membership degrees are defined in a unit interval I = [0, 1] on a universal set $\ddot{U}$ .

Definition 2.1 [1]

Fuzzy set (FS) F in $\ddot{U}$ can be defined as the collection of ordered pairs $F = {〈 α, μ_{F} (α) 〉 | α \in \ddot{U}}$ where μ_F (α) is the membership degree for all α in F and is defined by $μ_{F} : \ddot{U} \to I$ .

Definition 2.2 [2]

Intuitionistic fuzzy set (IFS) T in $\ddot{U}$ is defined as $T = {〈 α, μ_{T} (α), ν_{T} (α) 〉 | α \in \ddot{U}}$ where $μ_{T} : \ddot{U} \to I$ and $ν_{T} : \ddot{U} \to I$ are the membership and non-membership degrees respectively and satisfying the condition 0 ≤ μ_T (α) + ν_T (α) ≤1 for all $α \in \ddot{U}$ to the set T. For any T in $\ddot{U}$ , the degree of refusal $π_{T} : \ddot{U} \to I$ for all α in T is defined by π_T (α) =1 - μ_T (α) - ν_T (α).

Definition 2.3 [4]

Picture fuzzy set (PFS) E in $\ddot{U}$ is defined as $E = {〈 α, μ_{E} (α), η_{E} (α), ν_{E} (α) 〉 | α \in \ddot{U}}$ where $μ_{E} : \ddot{U} \to I$ , $η_{E} : \ddot{U} \to I$ and $ν_{E} : \ddot{U} \to I$ are the membership, neutral and non-membership degrees respectively and satisfying the condition 0 ≤ μ_E (α) + η_E (α) + ν_E (α) ≤1 for all $α \in \ddot{U}$ to the set E. For any E in $\ddot{U}$ , the degree of refusal $π_{E} : \ddot{U} \to I$ for all α in E is defined by π_E (α) =1 - μ_E (α) - η_E (α) - ν_E (α).

Definition 2.4 [5]

Spherical fuzzy set (SFS) H on $\ddot{U}$ is defined as $H = {〈 α, μ_{H} (α), η_{H} (α), ν_{H} (α) 〉 | α \in \ddot{U}}$ where $μ_{H} : \ddot{U} \to I$ , $η_{H} : \ddot{U} \to I$ and $ν_{H} : \ddot{U} \to I$ are the membership, neutral and non-membership degrees respectively and satisfying the condition 0 ≤ (μ_H (α)) ² + (η_H (α)) ² + (ν_H (α)) ² ≤ 1 for all $α \in \ddot{U}$ to the set H. For any H in $\ddot{U}$ , the degree of refusal $π_{H} : \ddot{U} \to I$ for all α in H is defined by $π_{H} (α) = \sqrt{1 - (μ_{H} (α))^{2} - (η_{H} (α))^{2} - (ν_{H} (α))^{2}}$ .

Definition 2.5 [7]

Cubical fuzzy set (CFS) L on $\ddot{U}$ is defined as $L = {〈 α, μ_{L} (α), η_{L} (α), ν_{L} (α) 〉 | α \in \ddot{U}}$ where $μ_{L} : \ddot{U} \to I$ , $η_{L} : \ddot{U} \to I$ and $ν_{L} : \ddot{U} \to I$ are the membership, neutral and non-membership degrees respectively and satisfying the condition 0 ≤ (μ_L (α)) ³ + (η_L (α)) ³ + (ν_L (α)) ³ ≤ 1 for all $α \in \ddot{U}$ to the L. For any L in $\ddot{U}$ , the degree of refusal $π_{L} : \ddot{U} \to I$ for all α in L is defined by $π_{L} (α) = \sqrt[3]{1 - (μ_{L} (α))^{3} - (η_{L} (α))^{3} - (ν_{L} (α))^{3}}$ .

For any CFS $L = {〈 α, μ_{L} (α), η_{L} (α), ν_{L} (α) 〉 | α \in \ddot{U}}$ the cubical pair 〈μ_L (α) , η_L (α) , ν_L (α) 〉 is called a CFN. For simplicity we denote a CFN by L = 〈μ_L, η_L, ν_L〉, where μ_L, η_L, ν_L ∈ I, 0 ≤ (μ_L) ³ + (η_L) ³ + (ν_L) ³ ≤ 1 for all $α \in \ddot{U}$ . Let $L$ be the set of all the CFNs.

Definition 2.6 [7]

Let $L_{i} = 〈 μ_{L_{i}}, η_{L_{i}}, ν_{L_{i}} 〉 \in L$ (i = 1, 2) then

L₁ = L₂ iff μ_{L
₁} = μ_{L
₂}, η_{L
₁} = η_{L
₂}, ν_{L
₁} = ν_{L
₂};

L₁ ⪯ L₂ iff μ_{L
₁} ≤ μ_{L
₂}, η_{L
₁} ≤ η_{L
₂}, ν_{L
₁} ≤ ν_{L
₂};

L_i^c = 〈ν_{L
_i}, η_{L
_i}, μ_{L
_i}〉 for all i;

$L_{1} + L_{2} = 〈 \sqrt[3]{(μ_{L_{1}})^{3} + (μ_{L_{2}})^{3} - (μ_{L_{1}})^{3} (μ_{L_{2}})^{3}}, η_{L_{1}} η_{L_{2}}, ν_{L_{1}} ν_{L_{2}} 〉$ ;

$L_{1} L_{2} = 〈 μ_{L_{1}} μ_{L_{2}}, η_{L_{1}} η_{L_{2}}, \sqrt[3]{(ν_{L_{1}})^{3} + (ν_{L_{2}})^{3} - (ν_{L_{1}})^{3} (ν_{L_{2}})^{3}} 〉$ ;

For some $λ > 0, λ L_{i} = 〈 \sqrt[3]{1 - (1 - (μ_{L_{i}})^{3})^{λ}}, (η_{L_{i}})^{λ}, (ν_{L_{i}})^{λ} 〉$ for all i;

For some $λ > 0, L_{i}^{λ} = 〈 (μ_{L_{i}})^{λ}, (η_{L_{i}})^{λ}, \sqrt[3]{1 - (1 - (ν_{L_{i}})^{3})^{λ}} 〉$ for all i;

Definition 2.7 [7]

The score function Ş of $L = 〈 μ_{L}, η_{L}, ν_{L} 〉 \in L$ is defined as $Ş (L) = μ_{L}^{3} - ν_{L}^{3} \in [- 1, 1]$ (1)

Definition 2.8 [7]

The accuracy function Ą of $L = 〈 μ_{L}, η_{L}, ν_{L} 〉 \in L$ is defined as $Ą (L) = μ_{L}^{3} + ν_{L}^{3} \in I$ (2)

Definition 2.9 [7]

The certainty function Ç of $L = 〈 μ_{L}, η_{L}, ν_{L} 〉 \in L$ is defined as $Ç (L) = μ_{L}^{3} \in I$ (3)

Definition 2.10 [7]

Let L₁ = 〈μ_{L
₁}, η_{L
₁}, ν_{L
₁}〉 and L₂ = 〈μ_{L
₂}, η_{L
₂}, ν_{L
₂}〉 be any two CFNs and Ş(L_i), Ą(L_i), Ç(L_i) (i = 1, 2) be their respective score, accuracy and certainty values, then we arrive at the following results:

If Ş(L₁) > Ş(L₂), then L₁ ≻ L₂;

Ş(L₁) < Ş(L₂), then L₁ ≺ L₂;

Ş(L₁) = Ş(L₂) and Ą(L₁) > Ą(L₂) then L₁ ≻ L₂;

Ş(L₁) = Ş(L₂) and Ą(L₁) < Ą(L₂) then L₁ ≺ L₂;

Ş(L₁) = Ş(L₂), Ą(L₁) = Ą(L₂) and Ç(L₁) > Ç(L₂) then L₁ ≻ L₂;

Ş(L₁) = Ş(L₂), Ą(L₁) = Ą(L₂) and Ç(L₁) < Ç(L₂) then L₁ ≺ L₂;

Ş(L₁) = Ş(L₂), Ą(L₁) = Ą(L₂) and Ç(L₁) = Ç(L₂) then L₁ ∼ L₂.

Example 2.1

Let L₁ = (0.80, 0.50, 0.63) and L₂ = (0.80, 0.60, 0.63) be any two CFNs whose score and accuracy values are equal. Then by Definition 2.9, we conclude that L₁ ∼ L₂ as we get Ç(L₁) = Ç(L₂) = 0.5120 by (11).

This example shows that the available ranking measures fails to compare some CFNs.

Definition 2.11 [7]

Cubical fuzzy weighted average (CFWA) operator with respect to weight $w = (w_{1}, w_{2}, \dots, w_{n}); w_{j} \in [0, 1]; \sum_{j = 1}^{n} w_{j} = 1$ is defined as CFWA_w (L₁, L₂, …, L_n) = $〈 \sqrt[3]{1 - \prod_{j = 1}^{n} (1 - (μ_{L_{j}})^{3})^{w_{j}}}, \prod_{j = 1}^{n} (η_{L_{j}})^{w_{j}}, \prod_{j = 1}^{n} (ν_{L_{j}})^{w_{j}} 〉$ (4)

Definition 2.12 [7]

Cubical fuzzy weighted geometric (CFWG) operator with respect to weight $w = (w_{1}, w_{2}, \dots, w_{n}); w_{j} \in [0, 1]; \sum_{j = 1}^{n} w_{j} = 1$ is defined as CFWG_w (L₁, L₂, …, L_n) = $〈 \prod_{j = 1}^{n} (μ_{L_{j}})^{w_{j}}, \prod_{j = 1}^{n} (η_{L_{j}})^{w_{j}}, \sqrt[3]{1 - \prod_{j = 1}^{n} (1 - (ν_{L_{j}})^{3})^{w_{j}}} 〉$ (5)

3 PDM for comparing CFNs

In this section, we have developed a new PDM for the efficient comparison of CFNs with examples to overcome the drawback of the existing measures discussed in the previous section.

Definition 3.1

Let L₁ = 〈μ_{L
₁}, η_{L
₁}, ν_{L
₁}〉 and L₂ = 〈μ_{L
₂}, η_{L
₂}, ν_{L
₂}〉 be any two CFNs whose refusal degrees are given by (π_{L
₁}) ³ = 1 - (μ_{L
₁}) ³ - (η_{L
₁}) ³ - (ν_{L
₁}) ³ and (π_{L
₂}) ³ = 1 - (μ_{L
₂}) ³ - (η_{L
₂}) ³ - (ν_{L
₂}) ³ then the PDM, denoted by p_o (L₁ ≥ L₂) of L₁ ≥ L₂ is proposed as $\begin{matrix} p_{o} (L_{1} \geq L_{2}) \\ = \frac{1}{3} {\begin{matrix} p_{o} (μ_{L_{1}} \geq μ_{L_{2}}) + p_{o} (η_{L_{1}} \geq η_{L_{2}}) \\ + p_{o} (ν_{L_{1}} \geq ν_{L_{2}}) \end{matrix}} \end{matrix}$ (6) where $p_{o} (μ_{L_{1}} \geq μ_{L_{2}}) = \frac{〈 \begin{matrix} max {0, ((μ_{L_{1}})^{3} + (π_{L_{1}})^{3}) - (μ_{L_{2}})^{3}} \\ - max {0, (μ_{L_{1}})^{3} - ((μ_{L_{2}})^{3} + (π_{L_{2}})^{3})} \end{matrix} 〉}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}}$ $p_{o} (η_{L_{1}} \geq η_{L_{2}}) = \frac{〈 \begin{matrix} max {0, ((η_{L_{1}})^{3} + (π_{L_{1}})^{3}) - (η_{L_{2}})^{3}} \\ - max {0, (η_{L_{1}})^{3} - ((η_{L_{2}})^{3} + (π_{L_{2}})^{3})} \end{matrix} 〉}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}}$

$p_{o} (ν_{L_{1}} \geq ν_{L_{2}}) = \frac{〈 \begin{matrix} max {0, ((ν_{L_{1}})^{3} + (π_{L_{1}})^{3}) - (ν_{L_{2}})^{3}} \\ - max {0, (ν_{L_{1}})^{3} - ((ν_{L_{2}})^{3} + (π_{L_{2}})^{3})} \end{matrix} 〉}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}}$

provided either (π_{L
₁}) ³ ≠ 0 or (π_{L
₂}) ³ ≠ 0.

On the other hand, if (π_{L
₁}) ³ = (π_{L
₂}) ³ = 0, then we define by (12) $p_{o} (L_{1} \geq L_{2}) = {\begin{matrix} 1; & (μ_{L_{1}})^{3} > (μ_{L_{2}})^{3} \\ 0; & (μ_{L_{1}})^{3} < (μ_{L_{2}})^{3} \\ 0.5; & (μ_{L_{1}})^{3} = (μ_{L_{2}})^{3} \end{matrix}$ (7)

Definition 3.2

Let L₁ and L₂ be two CFNs.

If p_o (L₁ ≥ L₂) > p_o (L₂ ≥ L₁) then L₁ is called superior to L₂ to the extent of degree p_o (L₁ ≥ L₂), denoted by $L_{1} \overset{p_{o} (L_{1} \geq L_{2})}{≻} L_{2}$ .

If p_o (L₁ ≥ L₂) < p_o (L₂ ≥ L₁) then L₂ is called superior to L₁ to the extent of degree p_o (L₂ ≥ L₁), denoted by $L_{2} \overset{p_{o} (L_{2} \geq L_{1})}{≻} L_{1}$ .

If p_o (L₁ ≥ L₂) = p_o (L₂ ≥ L₁) =0.5 then L₁ is said to be indifferent to L₂, denoted by L₁ ∼ L₂.

The following is a numerical example under the CF environment that shows the applicability of the proposed PDM.

Example 3.1

Consider two CFNs L₁ = (0.70, 0.50, 0.54) and L₂ = (0.61, 0.75, 0.40). Their refusal degrees are (π_{L
₁}) ³ = 0.3745 and (π_{L
₂}) ³ = 0.2871 respectively. Apply (13) on the above two CFNs to get $\begin{matrix} p_{O} (L_{1} \geq L_{2}) \\ = \frac{1}{3} {\frac{\begin{matrix} 〈 \begin{matrix} max {0, {0.70}^{3} + 0.3745 - {0.61}^{3}} \\ - max {0, {0.70}^{3} - {0.61}^{3} - 0.2871} \end{matrix} 〉 \\ + 〈 \begin{matrix} max {0, {0.50}^{3} + 0.3745 - {0.75}^{3}} \\ - max {0, {0.50}^{3} - {0.75}^{3} - 0.2871} \end{matrix} 〉 \\ + 〈 \begin{matrix} max {0, {0.54}^{3} + 0.3745 - {0.40}^{3}} \\ - max {0, {0.54}^{3} - {0.40}^{3} - 0.2871} \end{matrix} 〉 \end{matrix}}{0.3745 + 0.2871}} \end{matrix}$ = 0.5220.

Similarly, obtain p_o (L₂ ≥ L₁) =0.4780.

Hence we observe that L₁ is greater than L₂ by 52.2% and L₂ is greater than L₁ by 47.8%. However, a ranking methodology is to be developed to rank the CFNs. The following section develops a ranking method for ordering the CFNs based on PDM.

3.1 Method of ranking CFNs based on the proposed PDM

In this section, we have developed new ranking methods for ordering CFNs based on the proposed PDM.

A new cubical fuzzy ranking methodology based on the proposed PDM is structured as follows: For ranking CFNs, the PDM matrix denoted by $\hat{P_{o}} = (p_{o_{ij}})_{n \times n}$ corresponding to the possibility measures p_{o
_ij} = p_o (L_i ≥ L_j) ; (i, j = 1, 2, …, n) is given by

$\hat{P_{o}} = [\begin{matrix} p_{o_{11}} & p_{o_{12}} & \dots & p_{o_{1 n}} \\ p_{o_{21}} & p_{o_{22}} & \dots & p_{o_{2 n}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ p_{o_{m 1}} & p_{o_{m 2}} & \dots & p_{o_{mn}} \end{matrix}]$ (8) Now, the ranking values [17] for the CFNs L_i (i = 1, 2, ⋯ , n) is given by $ω_{i} = \frac{1}{n (n - 1)} [\sum_{j = 1}^{n} p_{o_{ij}} + \frac{n - 2}{2}]$ (9) Then, the order of all CFNs L_i (i = q, r, s) is made in accordance with the descending order of ω_i’s and is given as $L_{s} \overset{p_{o_{sr}}}{>} L_{r} \overset{p_{o_{rq}}}{>} L_{q} for ω_{s} > ω_{r} > ω_{q} .$ (10) which is the most likely ranking order showing the relative information of the possibility degree that L_s greater than L_r is p_{o
_sr} and the possibility degree that L_r greater than L_q is p_{o
_rq} and hence we choose L_s as the best one. If ω_q = ω_r then L_q ∼ L_r.

For Example 3.1, the PDM matrix using (12) is $\begin{matrix} \begin{matrix} L_{1} & L_{2} \end{matrix} \\ {\hat{P}}_{o} = \begin{matrix} L_{1} \\ L_{2} \end{matrix} [\begin{matrix} 0.5 & 0.5220 \\ 0.4780 & 0.5 \end{matrix}] \end{matrix}$ . From the matrix, using (19) the ranking values of the CFNs L₁ and L₂ are obtained as ω₁ = 0.5110 and ω₂ = 0.4890. Using ( PDMrank order) we conclude the ranking order as $L_{1} \overset{0.5220}{>} L_{2}$ since ω₁ > ω₂ with the relative information of L₁ is preferred over L₂ by 52.2% and so L₁ is the greatest CFN.

3.2 Advantage of the proposed PDM

In this section, we present the advantage of the proposed PDM over the existing ranking methods [7] using the numerical example discussed in Section 2.

Example 3.2

If we apply (12) on the data considered in the Example 2.1 for ordering L₁ and L₂,

we have p_o (L₁ ≥ L₂) = and p_o (L₂ ≥ L₁) =. Therefore, using (7) the PDM matrix is constructed as $\begin{matrix} \begin{matrix} L_{1} & L_{2} \end{matrix} \\ ∥ {\hat{P}}_{o} = \begin{matrix} L_{1} \\ L_{2} \end{matrix} [\begin{matrix} 0.5 & 0.6079 \\ 0.3921 & 0.5 \end{matrix}] \end{matrix}$ .Thus, the ranking values for the CFNs L₁ and L₂ computed using the equation (10) are ω₁ = 0.5540 and ω₂ = 0.4461. Here ω₁ > ω₂. Hence, using (10) we conclude that $L_{1} \overset{0.6079}{>} L_{2}$ which indicates the possibility that L₁ superior to L₂ is 0.6079 (60.8%) and the greatest CFN is L₁.

Example 3.3

Let L₁ = (0.80, 0.30, 0.60) and L₂ = (0.60, 0.30, 0.80) be two CFNs whose refusal degrees are (π_{L
₁}) ³ = 0.2450 and (π_{L
₂}) ³ = 0.2450 respectively. Apply (8) on these two CFNs to get p_o (L₁ ≥ L₂) =0.5. Similarly, obtain p_o (L₂ ≥ L₁) =0.5.

We observe that L₁ is superior to L₂ by 50% and L₂ is superior to L₁ by 50% and by definition 3.2 we conclude that L₁ ∼ L₂. It is clear that at certain conditions it is unable to conclude the comparison of CFNs using (6).

Even then, for some CFNs, the proposed PDM also fails to order. As a result, an improved measure must be developed that can work well in all circumstances. Motivated by this, in the next section we proposed an improved PDM to meet the requirements.

4 IPDM for comparing CFNs

This section presents an improved PDM called IPDM to compare the CFNs with examples. Some basic properties of the IPDM are discussed.

Definition 4.1

Let L₁ = 〈μ_{L
₁}, η_{L
₁}, ν_{L
₁}〉 and L₂ = 〈μ_{L
₂}, η_{L
₂}, ν_{L
₂}〉 be any two CFNs whose refusal degrees are given by (π_{L
₁}) ³ = 1 - (μ_{L
₁}) ³ - (η_{L
₁}) ³ - (ν_{L
₁}) ³ and (π_{L
₂}) ³ = 1 - (μ_{L
₂}) ³ - (η_{L
₂}) ³ - (ν_{L
₂}) ³, then the improved possibility degree measure (IPDM), denoted by p_I (L₁ ≥ L₂), of L₁ ≥ L₂ is proposed as $\begin{matrix} p_{I} (L_{1} \geq L_{2}) = \\ min {max (\frac{1 - (μ_{L_{1}})^{3} - 2 (μ_{L_{2}})^{3} - (η_{L_{1}})^{3} - (ν_{L_{1}})^{3}}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}}, 0), 1} \end{matrix}$ (11) Or $\begin{matrix} p_{I} (L_{1} \geq L_{2}) = \\ min {max (\frac{(μ_{L_{1}})^{3} - (μ_{L_{2}})^{3} + (π_{L_{2}})^{3}}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}}, 0), 1} \end{matrix}$ (12) provided either (π_{L
₁}) ³ ≠ 0 or (π_{L
₂}) ³ ≠ 0.

On the other hand, if (π_{L
₁}) ³ = (π_{L
₂}) ³ = 0, then we define by (10) $p_{I} (L_{1} \geq L_{2}) = {\begin{matrix} 1; & (μ_{L_{1}})^{3} > (μ_{L_{2}})^{3} \\ 0; & (μ_{L_{1}})^{3} < (μ_{L_{2}})^{3} \\ 0.5; & (μ_{L_{1}})^{3} = (μ_{L_{2}})^{3} \end{matrix}$ (13)

Definition 4.2

Let L₁ and L₂ be two CFNs.

If p_I (L₁ ≥ L₂) > p_I (L₂ ≥ L₁) then L₁ is called superior to L₂ to the extent of degree p_I (L₁ ≥ L₂), denoted by $L_{1} \overset{p_{I} (L_{1} \geq L_{2})}{≻} L_{2}$ .

If p_I (L₁ ≥ L₂) < p_I (L₂ ≥ L₁) then L₂ is called superior to L₁ to the extent of degree p_I (L₂ ≥ L₁), denoted by $L_{2} \overset{p_{I} (L_{2} \geq L_{1})}{≻} L_{1}$ .

If p_I (L₁ ≥ L₂) = p_I (L₂ ≥ L₁) =0.5 then L₁ is said to be indifferent to L₂, denoted by L₁ ∼ L₂.

Theorem 4.1

Let L₁ = 〈μ_{L
₁}, η_{L
₁}, ν_{L
₁}〉 and L₂ = 〈μ_{L
₂}, η_{L
₂}, ν_{L
₂}〉 be two CFNs, then

0 ≤ p_I (L₁ ≥ L₂) ≤1 (Boundedness or Normative property);

p_I (L₁ ≥ L₂) =0.5 if L₁ = L₂ (Reflexivity property);

p_I (L₁ ≥ L₂) + p_I (L₂ ≥ L₁) =1 (Complementary property).

Proof

let $s = \frac{(μ_{L_{1}})^{3} - (μ_{L_{2}})^{3} + (π_{L_{2}})^{3}}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}}$ Then we discuss the following cases:

If s ≥ 1, then p_I (L₁ ≥ L₂) = min{max(s,0),1} = 1.

If 0 < s < 1, then p_I (L₁ ≥ L₂) = min{max(s,0),1} = s.

If s ≤ 0, then p_I (L₁ ≥ L₂) = min{max(s,0),1} = 0.

Therefore, in all the cases, we have 0 ≤ p_I (L₁ ≥ L₂) ≤1.

Let L₁ = L₂, this implies that μ_{L
₁} = μ_{L
₂}, η_{L
₁} = η_{L
₂}, ν_{L
₁} = ν_{L
₂}. Also, (π_{L
₁}) ³ = (π_{L
₂}) ³. Then (12) becomes $\begin{matrix} p_{I} (L_{1} \geq L_{2}) = \\ min {max (\frac{(μ_{L_{1}})^{3} - (μ_{L_{2}})^{3} + (π_{L_{2}})^{3}}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}}, 0), 1} \end{matrix}$ $\begin{matrix} = min {max (\frac{(μ_{L_{2}})^{3} - (μ_{L_{2}})^{3} + (π_{L_{2}})^{3}}{(π_{L_{2}})^{3} + (π_{L_{2}})^{3}}, 0), 1} \end{matrix}$

$\begin{matrix} = min {max (\frac{(π_{L_{2}})^{3}}{2 (π_{L_{2}})^{3}}, 0), 1} \end{matrix}$ = 0.5

For two CFNs L₁ and L₂, let us assume that $s = \frac{(μ_{L_{1}})^{3} - (μ_{L_{2}})^{3} + (π_{L_{2}})^{3}}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}}$ , $t = \frac{(μ_{L_{2}})^{3} - (μ_{L_{1}})^{3} + (π_{L_{1}})^{3}}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}}$ ,such that $s + t = \frac{(μ_{L_{1}})^{3} - (μ_{L_{2}})^{3} + (π_{L_{2}})^{3} + (μ_{L_{2}})^{3} - (μ_{L_{1}})^{3} + (π_{L_{1}})^{3}}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}}$ = 1.Then we discuss the following cases

If s ≤ 0 and t ≥ 1 then

p_I (L₁ ≥ L₂) + p_I (L₂ ≥ L₁)

= min{max(s,0),1} + min{max(t,0),1} = 1.

If 0 < s, t < 1 then

p_I (L₁ ≥ L₂) + p_I (L₂ ≥ L₁)

= min{max(s,0),1} + min{max(t,0),1} = 1.

If s ≥ 1 and t ≤ 0 then

p_I (L₁ ≥ L₂) + p_I (L₂ ≥ L₁)

= min{max(s,0),1} + min{max(t,0),1} = 1.

Therefore, in all the cases we have

p_I (L₁ ≥ L₂) + p_I (L₂ ≥ L₁) =1.

Theorem 4.2

For two CFNs L₁ = 〈μ_{L
₁}, η_{L
₁}, ν_{L
₁}〉 and L₂ = 〈μ_{L
₂}, η_{L
₂}, ν_{L
₂}〉, the proposed IPDM p_I (L₁ ≥ L₂) for L₁ ≥ L₂ satisfies the following properties:

p_I (L₁ ≥ L₂) =1 if (μ_{L
₁}) ³ - (π_{L
₁}) ³ ≥ (μ_{L
₂}) ³;

p_I (L₁ ≥ L₂) =0 if (μ_{L
₂}) ³ - (π_{L
₂}) ³ ≥ (μ_{L
₁}) ³.

proof. For two CFNs L₁ = 〈μ_{L
₁}, η_{L
₁}, ν_{L
₁}〉 and L₂ = 〈μ_{L
₂}, η_{L
₂}, ν_{L
₂}〉,

Let (μ_{L
₁}) ³ - (π_{L
₁}) ³ ≥ (μ_{L
₂}) ³

Now, $\frac{(μ_{L_{1}})^{3} - (μ_{L_{2}})^{3} + (π_{L_{2}})^{3}}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}} \geq \frac{(μ_{L_{1}})^{3} - (μ_{L_{1}})^{3} + (π_{L_{1}})^{3} + (π_{L_{2}})^{3}}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}} = 1$ .

Therefore, $\min {\max (\frac{(μ_{L_{1}})^{3} - (μ_{L_{2}})^{3} + (π_{L_{2}})^{3}}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}}, 0), 1} = 1$ . Hence,p_I (L₁ ≥ L₂) =1.

Let (μ_{L
₂}) ³ - (π_{L
₂}) ³ ≥ (μ_{L
₁}) ³

Now, $\frac{(μ_{L_{1}})^{3} - (μ_{L_{2}})^{3} + (π_{L_{2}})^{3}}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}} \leq \frac{(μ_{L_{2}})^{3} - (π_{L_{2}})^{3} - (μ_{L_{2}})^{3} + (π_{L_{2}})^{3}}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}} = 0$ .

Therefore, $\min {\max (\frac{(μ_{L_{1}})^{3} - (μ_{L_{2}})^{3} + (π_{L_{2}})^{3}}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}}, 0), 1} = 0$ . Hence,p_I (L₁ ≥ L₂) =0.

Theorem 4.3

Let L_i = 〈μ_{L
_i}, η_{L
_i}, ν_{L
_i}〉 (i=1,2,3) be three CFNs. Then, the proposed IPDM p_I (L₁ ≥ L₂) for L₁ ≥ L₂ satisfies the following properties:

p_I (L₁ ≥ L₂) ≥0.5 iff (μ_{L
₁}) ³ + (η_{L
₂}) ³ + (η_{L
₂}) ³ ≥ (μ_{L
₂}) ³ + (η_{L
₁}) ³ + (η_{L
₁}) ³;

p_I (L₁ ≥ L₂) ≥0.5 and p_I (L₂ ≥ L₃) ≥0.5 then p_I (L₁ ≥ L₃) ≥0.5(weak transitivity property).

proof For three CFNs L_i = 〈μ_{L
_i}, η_{L
_i}, ν_{L
_i}〉 (i=1,2,3), it is evident that

(π_{L_{i}})^{3} = 1 - (μ_{L_{i}})^{3} - (η_{L_{i}})^{3} - (ν_{L_{i}})^{3}

(14)

Let p_I (L₁ ≥ L₂) ≥0.5

$\Leftrightarrow \frac{(μ_{L_{1}})^{3} - (μ_{L_{2}})^{3} + (π_{L_{2}})^{3}}{(π_{L_{1}})^{3} + (π_{L_{2}})^{3}} \geq 0.5$ (By Eq. (12))

⇔2 ((μ_{L
₁}) ³ - (μ_{L
₂}) ³ + (π_{L
₂}) ³) ≥ (π_{L
₁}) ³ + (π_{L
₂}) ³.

⇔ (μ_{L
₁}) ³ + (η_{L
₂}) ³ + (ν_{L
₂}) ³ ≥ (μ_{L
₂}) ³ + (η_{L
₁}) ³ + (ν_{L
₁}) ³ (By Eq. (14))

Let p_I (L₁ ≥ L₂) ≥0.5 and p_I (L₂ ≥ L₃) ≥0.5

By property (i),

p_I (L₁≥ L₂) ≥0.5implies (μ_{L
₁}) ³ + $(η_{L_{2}})^{3} + (η_{L_{2}})^{3} \geq (μ_{L_{2}})^{3} + (η_{L_{1}})^{3} + (η_{L_{1}})^{3}$ (15) and

p_I (L₂≥ L₃) ≥0.5implies (μ_{L
₃}) ³ + $(η_{L_{3}})^{3} + (η_{L_{3}})^{3} \geq (μ_{L_{3}})^{3} + (η_{L_{2}})^{3} + (η_{L_{2}})^{3}$ (16)

From inequalities (15) and (6),

(μ_{L
₁}) ³ + (η_{L
₃}) ³ + (η_{L
₃}) ³ ≥ (μ_{L
₃}) ³ + (η_{L
₁}) ³ + (η_{L
₁}) ³

impliesp_I (L₁ ≥ L₃) ≥0.5 (By property (i)).

The following numerical example under the CF environment discusses the applicability of the proposed IPDM.

Example 4.1

Let L₁ = 〈0.7, 0.5, 0.6〉 and L₂ = 〈0.4, 0.8, 0.5〉 be two CFNs. Then by Equation (12) $p_{I} (L_{1} \geq L_{2}) = \min {\max (\frac{0 . 7^{3} - 0 . 4^{3} + 0.30}{0.32 + 0.30}, 0), 1} = 0.93$ .

Simlarly, p_I (L₂ ≥ L₁) =0.07.

From this example, by definition 4.2 we observe that L₁ is superior to L₂ by 93% and L₂ is superior to L₁ by 7%. However, a ranking methodology is to be developed to rank the CFNs. The following section develops a ranking method for ordering the CFNs based on IPDM.

4.1 Method of ranking CFNs based on the proposed IPDM

In this section, we have developed new ranking methods for ordering CFNs based on the proposed PDM.

A new cubical fuzzy ranking methodology based on the proposed IPDM is structured as follows: For ranking CFNs, the IPDM matrix denoted by $\hat{P_{I}} = (p_{I_{ij}})_{n \times n}$ corresponding to the improved possibility degree measures p_{I
_ij} = p_I (L_i ≥ L_j) ; (i, j = 1, 2, …, n) is given by $\hat{P_{I}} = [\begin{matrix} p_{I_{11}} & p_{I_{12}} & \dots & p_{I_{1 n}} \\ p_{I_{21}} & p_{I_{22}} & \dots & p_{I_{2 n}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ p_{I_{m 1}} & p_{I_{m 2}} & \dots & p_{I_{mn}} \end{matrix}]$ (17) Now, the ranking values [17] for the CFNs L_i (i = 1, 2, ⋯ , n) is given by $ω_{i} = \frac{1}{n (n - 1)} [\sum_{j = 1}^{n} p_{I_{ij}} + \frac{n - 2}{2}]$ (18) Then, the order of all CFNs L_i (i = q, r, s) is made in accordance with the descending order of ω_i’s and is given as $L_{r} \overset{p_{I_{rs}}}{>} L_{s} \overset{p_{I_{sq}}}{>} L_{q} for ω_{r} > ω_{s} > ω_{q} .$ (19) which gives the relative information of the possibility degree that L_r superior than L_s is p_{I
_rs} and the possibility degree that L_s superior than L_q is p_{I
_sq} and hence we choose L_r as the best one. If ω_i’s are equal then we say that the CFNs are equivalent (i.e., if ω_r ∼ ω_s then L_r ∼ L_s).

The following are some numerical examples that show the ranking of CFNs using the proposed IPDM ranking method.

Example 4.1

Let L₁ = 〈0.83, 0.53, 0.63〉, L₂ = 〈0.80, 0.50, 0.30〉 and L₃ = 〈0.91, 0.21, 0.11〉 be three CFNs. To obtain the ranking order, we construct the IPDM matrix using (11) and (13) as $\begin{matrix} \begin{matrix} L_{1} & L_{2} & L_{3} \end{matrix} \\ ∥ {\hat{P}}_{I} = \begin{matrix} L_{1} \\ L_{2} \\ L_{3} \end{matrix} [\begin{matrix} 0.5 & 1 & 0.21 \\ 0 & 0.5 & 0 \\ 0.80 & 1 & 0.5 \end{matrix}] \end{matrix}$ . Based on this matrix, the ranking values of the CFNs (12) is ω₁ = 0.37, ω₂ = 0.17 and ω₃ = 0.47 and we obtain ω₃ > ω₁ > ω₂. Using (19), the ranking order of the given CFNs is obatined as $L_{3} \overset{0.80}{>} L_{1} \overset{1.0}{>} L_{2}$ . From this result, we observe the possibility that L₃ superior to L₁ is 0.80 (80%) and the possibility of L₁ to be superior to L₂ is 1.0 (100%) and hence we conclude that the greatest number is L₃. In other words, we can say that L₃ has 80% possibility of being superior while comparing with L₁ and L₁ has 100% possibility of being superior in comparing with L₂ and L₃ is the greatest one since it is superior among all the three.

4.2 Advantage of the proposed IPDM

In this section, we present the advantage of the proposed IPDM over the existing ranking methods [7] and the proposed PDM, using the numerical examples discussed in sections 2 and 3.

Example 4.2

If we apply (12) on the data considered in the Example 3.2 for comparing L₁ and L₂, then we have p_I (L₁ ≥ L₂) =1 and p_I (L₂ ≥ L₁) =0. Using (17) we obtain the IPDM matrix as $\begin{matrix} \begin{matrix} L_{1} & L_{2} \end{matrix} \\ ∥ {\hat{P}}_{I} = \begin{matrix} L_{1} \\ L_{2} \end{matrix} [\begin{matrix} 0.5 & 1 \\ 1 & 0.5 \end{matrix}] \end{matrix}$ Thus, the ranking values for the CFNs L_i (i = 1, 2) using (17) are ω₁ = 0.75 and ω₂ = 0.25. Here ω₁ > ω₂. Hence, we conclude that $L_{1} \overset{1}{>} L_{2}$ by (19). This shows that the possibility of L₁ to be superior than L₂ is 100% and the greatest CFN to be chosen is L₁.

Example 4.3

If we apply (12) on the data considered in the Example 2.1 for comparing L₁ and L₂, then we have p_I (L₁ ≥ L₂) =0.1630 and p_I (L₂ ≥ L₁) =0.8370. Therefore, using (17) the IPDM matrix is constructed as $\begin{matrix} \begin{matrix} L_{1} & L_{2} \end{matrix} \\ ∥ {\hat{P}}_{I} = \begin{matrix} L_{1} \\ L_{2} \end{matrix} [\begin{matrix} 0.5 & 0.1630 \\ 0.8370 & 0.5 \end{matrix}] \end{matrix}$ . Thus, the ranking values for the CFNs L_i (i = 1, 2) computed using the equation (18) are ω₁ = 0.3315 and ω₂ = 0.6685. Here ω₂ > ω₁. Hence, using (19) we conclude that $L_{2} \overset{0.8370}{>} L_{1}$ which indicates the possibility that L₂ superior to L₁ is 0.8370 (83.7%) and the greatest CFN is L₂.

5 Comparison study

In this section, the feasibility of the proposed PDMs is validated with the ranking measures existing in [7] with numerical examples of different CFNs and accumulated in Table 3 to reveal the comparison.

Table 3
Comparison of proposed ranking methods with the existing methods

Ranking method Case 1 Case 2 Case 3 Case 4

L₁ = 〈0.7, 0.5, 0.7〉 L₁ = 〈0.4, 0.6, 0.8〉 L₁ = 〈0.8, 0.5, 0.6〉 L₁ = 〈0.91, 0.21, 0.11〉

L₂ = 〈0.6, 0.5, 0.6〉 L₂ = 〈0.8, 0.5, 0.4〉 L₂ = 〈0.8, 0.4, 0.6〉 L₂ = 〈0.84, 0.50, 0.45〉

Score function [7] L₁ ∼ L₂ L₁ ∼ L₂ L₁ ∼ L₂ L₁ > L₂

Accuracy function [7] L₁ > L₂ L₁ ∼ L₂ L₁ ∼ L₂ L₁ > L₂

Certainty function [7] L₁ > L₂ L₂ > L₁ L₁ ∼ L₂ L₁ > L₂

PDM [Proposed] $L_{2} \overset{0.5670}{>} L_{1}$ $L_{1} \overset{0.5299}{>} L_{2}$ $L_{2} \overset{0.5286}{>} L_{1}$ $L_{1} \overset{0.5174}{>} L_{2}$

(L₂ is preferred (L₁ is preferred (L₂ is preferred (L₁ is preferred

over L₁ by 56.70%) over L₂ by 52.99%) over L₁ by 52.86%) over L₂ by 51.74%)

IPDM [Proposed] $L_{1} \overset{0.856}{>} L_{2}$ $L_{2} \overset{1.000}{>} L_{1}$ $L_{1} \overset{0.586}{>} L_{2}$ $L_{1} \overset{0.824}{>} L_{2}$

(L₁ is preferred (L₂ is preferred (L₁ is preferred (L₁ is preferred

over L₂ by 85.6%) over L₁ by 100%) over L₂ by 58.6%) over L₂ by 82.4%)

Ranking method	Case 1	Case 2	Case 3	Case 4
	L₁ = 〈0.7, 0.5, 0.7〉	L₁ = 〈0.4, 0.6, 0.8〉	L₁ = 〈0.8, 0.5, 0.6〉	L₁ = 〈0.91, 0.21, 0.11〉
	L₂ = 〈0.6, 0.5, 0.6〉	L₂ = 〈0.8, 0.5, 0.4〉	L₂ = 〈0.8, 0.4, 0.6〉	L₂ = 〈0.84, 0.50, 0.45〉
Score function [7]	L₁ ∼ L₂	L₁ ∼ L₂	L₁ ∼ L₂	L₁ > L₂
Accuracy function [7]	L₁ > L₂	L₁ ∼ L₂	L₁ ∼ L₂	L₁ > L₂
Certainty function [7]	L₁ > L₂	L₂ > L₁	L₁ ∼ L₂	L₁ > L₂
PDM [Proposed]	$L_{2} \overset{0.5670}{>} L_{1}$	$L_{1} \overset{0.5299}{>} L_{2}$	$L_{2} \overset{0.5286}{>} L_{1}$	$L_{1} \overset{0.5174}{>} L_{2}$
	(L₂ is preferred	(L₁ is preferred	(L₂ is preferred	(L₁ is preferred
	over L₁ by 56.70%)	over L₂ by 52.99%)	over L₁ by 52.86%)	over L₂ by 51.74%)
IPDM [Proposed]	$L_{1} \overset{0.856}{>} L_{2}$	$L_{2} \overset{1.000}{>} L_{1}$	$L_{1} \overset{0.586}{>} L_{2}$	$L_{1} \overset{0.824}{>} L_{2}$
	(L₁ is preferred	(L₂ is preferred	(L₁ is preferred	(L₁ is preferred
	over L₂ by 85.6%)	over L₁ by 100%)	over L₂ by 58.6%)	over L₂ by 82.4%)

6 DM approaches based on the proposed PDMs for solving cubical fuzzy MADM problems

This section presents a DM method based on the proposed PDMs for solving MADM problems in the CFS environment. Consider a cubical fuzzy MADM (CFMADM) problem consists of m alternatives A₁, A₂, …, A_m under n different criteria C₁, C₂, …, C_n in which the decision maker’s rating values L_ij’s are arranged in a CF decision matrix as $D = [L_{ij}]_{m \times n} = \begin{matrix} \begin{matrix} C_{1} & C_{2} & \dots & C_{n} \end{matrix} \\ \begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{m} \end{matrix} [\begin{matrix} L_{11} & L_{12} & \dots & L_{1 n} \\ L_{21} & L_{22} & \dots & L_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ L_{m 1} & L_{m 2} & \dots & L_{mn} \end{matrix}] \end{matrix}$

where [L_ij] = 〈μ_{L
_ij}, η_{L
_ij}, ν_{L
_ij}〉.

Furthermore, using a weight vector w = (w₁, w₂, …, w_n) where w_j ∈ [0, 1] and sum of $\sum_{j = 1}^{n} w_{j} = 1$ to consider the benefits of the provided attributes.

Then, using the proposed PDMs, we design the following methods for solving CFMADM.

6.1 DM approach for solving CFMADM using PDM ranking method

Convert the CF decision matrix to a normalized form, if required using (20) $L_{ij} = {\begin{matrix} 〈 μ_{L_{ij}}, η_{L_{ij}}, ν_{L_{ij}} 〉; & for benefit type attribute \\ 〈 ν_{L_{ij}}, η_{L_{ij}}, μ_{L_{ij}} 〉; & for cost type attribute \end{matrix}$ (20)

Use the CFWA operator (21) to aggregate the different rating values L_ij (j = 1, 2, …, n) of the alternatives A_i (i = 1, 2, …, m) into L_i (i = 1, 2, …, m). (i.e., CFWA(L_i1, L_i2, …, L_in) = L_i = 〈μ_Li, η_{L
_i}, ν_{L
_i}〉)

Compute the possibility degree measures p_{o
_ij} = p_o (L_iL_j) (i, j = 1, 2, …, m) using (14) and obtain the PDM matrix using (12) as $\hat{P_{o}} = (p_{o_{ij}})_{m \times m}$ .

Evaluate the ranking values ω_i (i = 1, 2, …, m) using (19).

Finally rank the alternatives A_i’s (i = 1, 2, …, m) according to their respective ranking values ω_i (i = 1, 2, …, m) using (17).

6.2 DM approach for solving CFMADM using IPDM ranking method

Convert the CF decision matrix to a normalized form, if required using (11) $L_{ij} = {\begin{matrix} 〈 μ_{L_{ij}}, η_{L_{ij}}, ν_{L_{ij}} 〉; & for benefit type attribute \\ 〈 ν_{L_{ij}}, η_{L_{ij}}, μ_{L_{ij}} 〉; & for cost type attribute \end{matrix}$ (21)

Use the CFWA operator (4) to aggregate the different rating values L_ij (j = 1, 2, …, n) of the alternatives A_i (i = 1, 2, …, m) into L_i (i = 1, 2, …, m). (i.e., CFWA(L_i1, L_i2, …, L_in) = L_i = 〈μ_Li, η_{L
_i}, ν_{L
_i}〉)

Compute the improved possibility degree measures p_{I
_ij} = p_I (L_iL_j) (i, j = 1, 2, …, m) using (6) and obtain the IPDM matrix using (8) as $\hat{P_{I}} = (p_{I_{ij}})_{m \times m}$ .

Evaluate the ranking values ω_i (i = 1, 2, …, m) using (14).

Finally rank the alternatives A_i’s (i = 1, 2, …, m) according to their respective ranking values ω_i (i = 1, 2, …, m) using (9).

7 Numerical illustrations

In this section, the constructed cubical fuzzy DM approaches are applied to the existing MADM problem in [7] to validate the efficacy of the ranking methods based on the proposed PDMs. Table 4 shows the comparison followed by a conclusion. MADM problems of two real-time instances of the cutting fluid selection for cutting gears [18] under the CFS environment are considered and then solved to demonstrate the computational efficiency and applicability of the developed ranking methods.

Table 4
Efficacy of the proposed ranking methods

Ranking method Aggregation operator Ranking Best choice Least considerable

Score function [7] CFWA A₁ ≻ A₅ ≻ A₄ ≻ A₃ ≻ A₂ A ₁ A ₂

CFWG A₁ ≻ A₅ ≻ A₃ ≻ A₄ ≻ A₂ A ₁ A ₂

PDM [Proposed] CFWA $A_{3} \overset{0.5087}{≻} A_{2} \overset{0.5124}{≻} A_{4} \overset{0.5021}{≻} A_{5} \overset{0.5371}{≻} A_{1}$ A ₃ A ₁

CFWG $A_{3} \overset{0.5311}{≻} A_{2} \overset{0.5504}{≻} A_{1} \overset{0.5886}{≻} A_{5} \overset{0.5604}{≻} A_{4}$ A ₃ A ₄

IPDM [Proposed] CFWA $A_{1} \overset{0.8548}{≻} A_{5} \overset{0.5276}{≻} A_{4} \overset{0.5721}{≻} A_{3} \overset{0.5108}{≻} A_{2}$ A ₁ A ₂

CFWG $A_{1} \overset{1.0000}{≻} A_{4} \overset{0.6320}{≻} A_{2} \overset{0.5455}{≻} A_{5} \overset{0.2692}{≻} A_{3}$ A ₁ A ₃

Ranking method	Aggregation operator	Ranking	Best choice	Least considerable
Score function [7]	CFWA	A₁ ≻ A₅ ≻ A₄ ≻ A₃ ≻ A₂	A ₁	A ₂
	CFWG	A₁ ≻ A₅ ≻ A₃ ≻ A₄ ≻ A₂	A ₁	A ₂
PDM [Proposed]	CFWA	$A_{3} \overset{0.5087}{≻} A_{2} \overset{0.5124}{≻} A_{4} \overset{0.5021}{≻} A_{5} \overset{0.5371}{≻} A_{1}$	A ₃	A ₁
	CFWG	$A_{3} \overset{0.5311}{≻} A_{2} \overset{0.5504}{≻} A_{1} \overset{0.5886}{≻} A_{5} \overset{0.5604}{≻} A_{4}$	A ₃	A ₄
IPDM [Proposed]	CFWA	$A_{1} \overset{0.8548}{≻} A_{5} \overset{0.5276}{≻} A_{4} \overset{0.5721}{≻} A_{3} \overset{0.5108}{≻} A_{2}$	A ₁	A ₂
	CFWG	$A_{1} \overset{1.0000}{≻} A_{4} \overset{0.6320}{≻} A_{2} \overset{0.5455}{≻} A_{5} \overset{0.2692}{≻} A_{3}$	A ₁	A ₃

From Table 4, we observe that the IPDM ranking method is suitable for comparing all CFNs. The results obtained for the proposed methods are consistent with the existing results, implying that the PDMs ranking methods can successfully address the DM problem. Clearly, the eventual ranking of the proposed ranking methods differs from the existing methods due to the mobility of the cubical fuzzy DM environment.

Two real-life instances for the selection of cutting fluid for cutting gears are discussed below as an application of the cubical fuzzy DM approaches based on the proposed PDMs ranking methods.

Example 7.1

A DM situation involving the best-cutting fluid selection for cutting alloy steel spur gears is considered in this example. Gears are cut using a machining method called hobbing, which is a slow-speed machining process with a heavy and continuous cut in which gear teeth are gradually created through a succession of cuts with a cutting tool called a hob. Carbide is one of the tool materials used to cut gears. The dimensional accuracy and desirable polish of the spur gears are negatively affected by a rise in temperature at the tool and workpiece interface. Wetting compounds (usually fatty esters) improve the capacity of the cutting fluids to coat the metal fines, cutting tools and workpieces, improving lubrication and protecting from micro soldering in such harsh applications. Clean oils containing intense pressure EP additives, like sulphur and chlorine, have long been known to perform well in machining operations on alloy steels with a high degree of difficulty. Cutting fluids containing chlorine additives extend tool life when cutting at slower rates, whereas high alloy steels require sulphur additives. The viscosity of a cutting fluid is very important in terms of performance and maintenance.The viscosity of the suspension allowed micro-chips and debris to settle out. Micro-chips and debris settle out of suspension in lower viscosity fluids. The disposal of cutting fluid has become more difficult and expensive as environmental regulations have become more stringent. Minimizing the environmental imprint of cutting fluids has become a major goal for both manufacturers and end-users in recent years. It is widely acknowledged that neat cutting oils are best suited to low-speed and heavy-duty cuts, as they require a lot of effort. As a result, seven neat cutting oils were considered in this example based on the following cutting fluid properties: viscosity at 40°C in cst, chlorine by weight (%), sulphur by weight (%), fatty ester by weight (%) and disposability on a relative (R) scale, all of them can have a major impact on the provided machining needs. The R scale is used to compare the physical and chemical qualities of cutting fluid that have an impact on the environment. The R scale values of higher and lower represent the cutting fluid options, highest and lowest disposability, respectively. The greater the value for disposability, the less destructive the effect of cutting fluid disposal on the environment. Table 5 displays the CF decision matrix, which contains all the values connected to the considered criterion for the various alternatives. Moreover, the prioritized weights of each criterion assigned for consideration are given in the last row of Table 5 to eliminate any subjectivity in the DM process. Following the methods outlined in the previous part of the ranking CFNs based on PDM and IPDM, the most potent neat cutting oil for cutting alloy steel gears is chosen from the identified neat cutting oil alternatives.

Table 5

Cubical fuzzy decision matrix for the best neat cutting oil selection problem

Criteria	Viscosity at 40°C	Sulphur wgt. (%)	Chlorine wgt. (%)	Fatty ester wgt. (%)	Sulphur wgt. (%)
Alternative	(C₁)	(C₂)	(C₃)	(C₄)	(C₅)
A ₁	〈0.88, 0.43, 0.39〉	〈0.85, 0.35, 0.45〉	〈0.43, 0.63, 0.70〉	〈0.41, 0.21, 0.91〉	〈0.82, 0.23, 0.63〉
A ₂	〈0.85, 0.35, 0.45〉	〈0.89, 0.39, 0.49〉	〈0.30, 0.89, 0.39〉	〈0.21, 0.51, 0.81〉	〈0.89, 0.53, 0.50〉
A ₃	〈0.73, 0.59, 0.35〉	〈0.57, 0.36, 0.76〉	〈0.24, 0.85, 0.65〉	〈0.60, 0.20, 0.80〉	〈0.89, 0.53, 0.50〉
A ₄	〈0.83, 0.53, 0.63〉	〈0.68, 0.58, 0.63〉	〈0.66, 0.74, 0.50〉	〈0.60, 0.20, 0.80〉	〈0.82, 0.23, 0.63〉
A ₅	〈0.91, 0.41, 0.31〉	〈0.63, 0.50, 0.70〉	〈0.60, 0.60, 0.59〉	〈0.60, 0.20, 0.80〉	〈0.71, 0.45, 0.63〉
A ₆	〈0.90, 0.25, 0.42〉	〈0.55, 0.78, 0.45〉	〈0.45, 0.87, 0.45〉	〈0.35, 0.43, 0.85〉	〈0.71, 0.45, 0.63〉
A ₇	〈0.80, 0.50, 0.43〉	〈0.48, 0.37, 0.80〉	〈0.82, 0.40, 0.42〉	〈0.82, 0.62, 0.44〉	〈0.82, 0.23, 0.61〉
Weight	0.092	0.162	0.274	0.275	0.197

Now we rank the alternatives A₁, A₂, …, A₇ based on the criteria C₁, C₂, …, C₅ by applying the CFMADM approaches using the proposed PDM and IPDM ranking methods as discussed in Section 6.

Using the CFWA operator (17), we obtain the aggregated CF values for the seven neat oils as

L₁ = 〈0.72, 0.34, 0.65〉L₂ = 〈0.75, 0.55, 0.53〉L₃ = 〈0.69, 0.44, 0.63〉L₄ = 〈0.72, 0.38, 0.63〉L₅ = 〈0.69, 0.39, 0.63〉L₆ = 〈0.62, 0.55, 0.53〉 and

L₇ = 〈0.79, 0.41, 0.51〉

By step 3 of section 6.1 and 6.2 respectively, we obtain the possibility degree matrices $\hat{P_{o}} = (p_{o_{ij}})_{m \times m}$ and $\hat{P_{I}} = (p_{I_{ij}})_{m \times m}$ as follows: $\hat{P_{o}} = [\begin{matrix} 0.5 & 0.514 & 0.495 & 0.498 & 0.488 & 0.471 & 0.502 \\ 0.486 & 0.5 & 0.480 & 0.484 & 0.474 & 0.457 & 0.488 \\ 0.505 & 0.520 & 0.5 & 0.503 & 0.494 & 0.477 & 0.508 \\ 0.502 & 0.516 & 0.497 & 0.5 & 0.491 & 0.473 & 0.505 \\ 0.512 & 0.526 & 0.506 & 0.509 & 0.5 & 0.483 & 0.514 \\ 0.529 & 0.543 & 0.523 & 0.527 & 0.517 & 0.5 & 0.531 \\ 0.498 & 0.512 & 0.492 & 0.495 & 0.486 & 0.469 & 0.5 \end{matrix}]$ $\hat{P_{I}} = [\begin{matrix} 0.5 & 0.372 & 0.587 & 0.507 & 0.603 & 0.765 & 0.300 \\ 0.628 & 0.5 & 0.717 & 0.634 & 0.729 & 0.888 & 0.41 2 \\ 0.413 & 0.283 & 0.5 & 0.421 & 0.519 & 0.686 & 0.219 \\ 0.493 & 0.366 & 0.579 & 0.5 & 0.595 & 0.756 & 0.295 \\ 0.397 & 0.271 & 0.481 & 0.405 & 0.5 & 0.664 & 0.211 \\ 0.235 & 0.112 & 0.314 & 0.244 & 0.336 & 0.5 & 0.067 \\ 0.700 & 0.588 & 0.781 & 0.705 & 0.789 & 0.933 & 0.5 \end{matrix}]$

6.1 and 6.2 respectively, we obtain the ranking values as

ω₁ = 0.142, ω₂ = 0.140, ω₃ = 0.143, ω₄ = 0.142, ω₅ = 0.144, ω₆ = 0.147 and ω₇ = 0.142.

and ω₁ = 0.146, ω₂ = 0.167, ω₃ = 0.132, ω₄ = 0.145, ω₅ = 0.129, ω₆ = 0.103 and ω₇ = 0.178. The corresponding ranking order of the considered alternatives in accordance with the ranking values obtained above, using step 5 of section 6.1 and 6.2 are: The ranking order based on the PDM ranking method is obtained as

A_{6} \overset{0.517}{≻} A_{5} \overset{0.506}{≻} A_{3} \overset{0.505}{≻} A_{1} \overset{0.498}{\sim} A_{4} \overset{0.505}{\sim} A_{7} \overset{0.512}{≻} A_{2} .

This method ranks some of the alternatives with equivalency.

The ranking order based on the IPDM ranking method $A_{7} \overset{0.588}{≻} A_{2} \overset{0.628}{≻} A_{1} \overset{0.507}{≻} A_{4} \overset{0.579}{≻} A_{3} \overset{0.519}{≻} A_{5} \overset{0.664}{≻} A_{6} .$ This method ranks all the alternatives effectively.

From Table 6, we observe that the ranking method based on the proposed IPDM by including neutral and refusal degrees in its formula is efficient in ranking the CF alternatives than the other ranking approaches as they fail to compare some alternatives and suggests that A₇ is the best one and A₆ is the least considerable. Hence, we conclude that the proposed ranking method based on IPDM is good in comparing all types of CF alternatives with a better ranking result.

Table 6

Comparison of the results with the existing ranking approaches

Ranking method	Ranking
Score function [7]	A₇ ≻ A₂ ≻ A₄ ≻ A₁ ≻ A₆ ≻ A₅ ∼ A₃.
Accuracy function [7]	A₁ ≻ A₇ ≻ A₄ ≻ A₃ ∼ A₅ ≻ A₂ ≻ A₆.
Certainty function [7]	A₇ ≻ A₂ ≻ A₁ ∼ A₄ ≻ A₃ ∼ A₅ ≻ A₆.
PDM [Proposed]	$A_{6} \overset{0.517}{≻} A_{5} \overset{0.506}{≻} A_{3} \overset{0.505}{≻} A_{1} \overset{0.498}{\sim} A_{4} \overset{0.505}{\sim} A_{7} \overset{0.512}{≻} A_{2}$ .
IPDM [Proposed]	$A_{7} \overset{0.588}{≻} A_{2} \overset{0.628}{≻} A_{1} \overset{0.507}{≻} A_{4} \overset{0.579}{≻} A_{3} \overset{0.519}{≻} A_{5} \overset{0.664}{≻} A_{6}$ .

From this example, we conclude that the neat oil A₇ is the best fluid for cutting gears as it is probably 58.8% better than A₂, 70% better than A₁, 70.5% better than A₄, 78.1% better than A₃, 78.9% better than A₅ and 93.3% better than A₆. Likewise, we can judge the worthiness of other alternatives through this method in a manner that A₂ is 62.8% superior to A₁, A₁ is 50.7% superior to A₄, A₄ is 57.9% superior to A₃, A₃ is 51.9% superior to A₅ and A₅ is 66.4% superior to A₆.

Example 7.2

This example shows how to choose a cutting fluid for turning operations based on copper materials with steel tools (ST) of high-speed. Dry machining is only possible with a few copper-based materials. In other circumstances, however, dry cutting fails to deliver the desired tool life and surface finish. In addition, lubrication of various elements of the machine tool requires the use of cutting fluid. Furthermore, it is common knowledge that ST of high-speed can only maintain their hardness at temperatures of roughly 550 - 600°C. As a result, the cutting fluid must be able to cool when even being cut with high speed ST tools. Since the cutting fluids containing sulphur have a tendency to respond with copper, a cutting fluid free from sulphur should be chosen. Soluble cutting oils with reduced viscosity are chosen as the light cut and high-speed operation necessitates efficient cooling. They must contain corrosion inhibitors and biocide, which aid in the prevention of rancidity. As a result, nine soluble oil alternatives are evaluated in this cutting fluid selection problem. Four cutting fluid parameters, such as flash point in degrees Celsius, viscocity in cst at 40°C, density in g/cmş at 29°C, and corrosion resistance in R scale, were chosen as key factors for this selection problem. Table 7 provides the CF decision matrix, which contains all important data for the alternatives analyzed in relation to the evaluation criteria used for this topic. Table 7 also includes the priority weights for the four criteria.

Table 7

Cubical fuzzy decision matrix for the best soluble cutting oil selection problem

Criteria	Corrosion resistance	Flash point	Density at 29°C	Viscosity at40°C
Alternative	(C₁)	(C₂)	(C₃)	(C₄)
A ₁	〈0.9, 0.3, 0.4〉	〈0.9, 0.3, 0.4〉	〈0.7, 0.8, 0.4〉	〈0.4, 0.6, 0.8〉
A ₂	〈0.8, 0.4, 0.5〉	〈0.7, 0.6, 0.5〉	〈0.7, 0.5, 0.6〉	〈0.5, 0.6, 0.7〉
A ₃	〈0.9, 0.4, 0.2〉	〈0.8, 0.5, 0.4〉	〈0.8, 0.6, 0.3〉	〈0.6, 0.7, 0.5〉
A ₄	〈0.9, 0.4, 0.2〉	〈0.9, 0.4, 0.2〉	〈0.8, 0.5, 0.4〉	〈0.8, 0.5, 0.4〉
A ₅	〈0.9, 0.4, 0.2〉	〈0.8, 0.5, 0.4〉	〈0.8, 0.7, 0.5〉	〈0.6, 0.5, 0.7〉
A ₆	〈0.9, 0.4, 0.2〉	〈0.9, 0.5, 0.4〉	〈0.6, 0.7, 0.5〉	〈0.7, 0.6, 0.5〉
A ₇	〈0.9, 0.4, 0.2〉	〈0.7, 0.6, 0.5〉	〈0.5, 0.6, 0.7〉	〈0.5, 0.8, 0.7〉
A ₈	〈0.8, 0.4, 0.5〉	〈0.8, 0.6, 0.4〉	〈0.7, 0.6, 0.5〉	〈0.5, 0.7, 0.6〉
A ₉	〈0.9, 0.3, 0.4〉	〈0.7, 0.6, 0.5〉	〈0.9, 0.5, 0.4〉	〈0.3, 0.7, 0.8〉
Weight	0.332	0.141	0.274	0.253

Now we rank the alternatives A₁, A₂, …, A₉ based on the criteria C₁, C₂, C₃, C₄ by applying the CFMADM approaches using the proposed PDM and IPDM ranking methods as discussed in section 6.

Using the CFWA operator (17), we obtain the aggregated CF values for the nine soluble oils as

L₁ = 〈0.77, 0.52, 0.52〉L₂ = 〈0.71, 0.50, 0.57〉L₃ = 〈0.82, 0.53, 0.31〉L₄ = 〈0.86, 0.45, 0.29〉L₅ = 〈0.82, 0.51, 0.39〉L₆ = 〈0.82, 0.53, 0.36〉L₇ = 〈0.76, 0.56, 0.44〉L₈ = 〈0.73, 0.55, 0.51〉 and

L₉ = 〈0.83, 0.47, 0.49〉

By step 3 of Section 6.1 and 6.2 respectively, we obtain the possibility degree matrices $\hat{P_{o}} = (p_{o_{ij}})_{m \times m}$ and $\hat{P_{I}} = (p_{I_{ij}})_{m \times m}$ as follows:

$\hat{P_{o}} = [\begin{matrix} 0.5 & 0.480 & 0.497 & 0.504 & 0.501 & 0.503 & 0.489 & 0.485 & 0.520 \\ 0.520 & 0.5 & 0.517 & 0.525 & 0.521 & 0.523 & 0.508 & 0.506 & 0.539 \\ 0.503 & 0.483 & 0.5 & 0.507 & 0.503 & 0.505 & 0.491 & 0.488 & 0.522 \\ 0.496 & 0.475 & 0.493 & 0.5 & 0.496 & 0.498 & 0.484 & 0.481 & 0.514 \\ 0.499 & 0.479 & 0.497 & 0.504 & 0.5 & 0.502 & 0.487 & 0.484 & 0.519 \\ 0.497 & 0.477 & 0.495 & 0.502 & 0.498 & 0.5 & 0.485 & 0.483 & 0.517 \\ 0.511 & 0.492 & 0.509 & 0.516 & 0.513 & 0.515 & 0.5 & 0.497 & 0.531 \\ 0.515 & 0.494 & 0.512 & 0.519 & 0.516 & 0.517 & 0.503 & 0.5 & 0.534 \\ 0.480 & 0.461 & 0.478 & 0.486 & 0.481 & 0.483 & 0.469 & 0.466 & 0.5 \end{matrix}]$

$\hat{P_{I}} = [\begin{matrix} 0.5 & 0.725 & 0.329 & 0.134 & 0.312 & 0.307 & 0.565 & 0.661 & 0.196 \\ 0.275 & 0.5 & 0.127 & 0 & 0.108 & 0.102 & 0.346 & 0.436 & 0 \\ 0.671 & 0.873 & 0.5 & 0.315 & 0.488 & 0.484 & 0.723 & 0.815 & 0.391 \\ 0.866 & 1 & 0.685 & 0.5 & 0.677 & 0.674 & 0.907 & 0.998 & 0.596 \\ 0.688 & 0.892 & 0.512 & 0.323 & 0.5 & 0.496 & 0.740 & 0.834 & 0.402 \\ 0.693 & 0.898 & 0.516 & 0.326 & 0.504 & 0.5 & 0.746 & 0.840 & 0.406 \\ 0.435 & 0.654 & 0.277 & 0.093 & 0.260 & 0.254 & 0.5 & 0.591 & 0.146 \\ 0.339 & 0.564 & 0.185 & 0.002 & 0.166 & 0.160 & 0.409 & 0.5 & 0.047 \\ 0.804 & 1 & 0.609 & 0.404 & 0.598 & 0.594 & 0.854 & 0.953 & 0.5 \end{matrix}]$

By step 4 of sections 6.1 and 6.2 respectively, obtain the ranking values as

ω₁ = 0.111, ω₂ = 0.113, ω₃ = 0.111, ω₄ = 0.110, ω₅ = 0.111, ω₆ = 0.110, ω₇ = 0.112, ω₈ = 0.113 and ω₉ = 0.108.

and ω₁ = 0.100, ω₂ = 0.075, ω₃ = 0.122, ω₄ = 0.144, ω₅ = 0.123, ω₆ = 0.124, ω₇ = 0.093, ω₈ = 0.082 and ω₉ = 0.136.

The corresponding ranking order of the considered alternatives in accordance with the ranking values obtained above, using step 5 of sections 6.1 and6.2 are:

The ranking order based on the PDM ranking method $A_{2} \overset{0.506}{\sim} A_{8} \overset{0.503}{≻} A_{7} \overset{0.511}{≻} A_{1} \overset{0.497}{\sim} A_{3} \overset{0.503}{\sim} A_{5} \overset{0.504}{≻} A_{4}$

$\overset{0.498}{\sim} A_{6} \overset{0.517}{≻} A_{9}$ .

This method ranks some of the alternatives with equivalency.

The ranking order based on the IPDM ranking method

$A_{4} \overset{0.596}{≻} A_{9} \overset{0.594}{≻} A_{6} \overset{0.504}{≻} A_{5} \overset{0.512}{≻} A_{3} \overset{0.671}{≻} A_{1} \overset{0.565}{≻} A_{7}$

$\overset{0.591}{≻} A_{8} \overset{0.564}{≻} A_{2}$ .

This method compares all the alternatives effectively.

From Table 8, we observe that the ranking method based on the proposed IPDM by including neutral and refusal degrees in its formula is efficient in ranking the CF alternatives than the other ranking approaches as they fail to compare some alternatives and suggests that A₄ is the best one and A₂ is the least considerable. Hence, we conclude that the proposed ranking method based on IPDM is good in comparing all types of CF alternatives with a better ranking result.

Table 8

Comparison of the results with the existing ranking approaches

Ranking method	Ranking
Score function [7]	A₄ ≻ A₃ ≻ A₆ ≻ A₅ ≻ A₉ ≻ A₇ ≻ A₁ ≻ A₈ ≻ A₂
Accuracy function [7]	A₉ ≻ A₄ ≻ A₅ ≻ A₆ ≻ A₁ ≻ A₃ ≻ A₇ ≻ A₈ ≻ A₂
Certainty function [7]	A₄ ≻ A₉ ≻ A₃ ∼ A₅ ∼ A₆ ≻ A₁ ≻ A₇ ≻ A₈ ≻ A₂
PDM [Proposed]	$A_{2} \overset{0.506}{\sim} A_{8} \overset{0.503}{≻} A_{7} \overset{0.511}{≻} A_{1} \overset{0.497}{\sim} A_{3} \overset{0.503}{\sim} A_{5} \overset{0.504}{≻} A_{4} \overset{0.498}{\sim} A_{6} \overset{0.517}{≻} A_{9}$ .
IPDM [Proposed]	$A_{4} \overset{0.596}{≻} A_{9} \overset{0.594}{≻} A_{6} \overset{0.504}{≻} A_{5} \overset{0.512}{≻} A_{3} \overset{0.671}{≻} A_{1} \overset{0.565}{≻} A_{7} \overset{0.591}{≻} A_{8} \overset{0.564}{≻} A_{2}$ .

From this example, we conclude that the soluble oil A₄ is the best fluid for cutting gears as it is probably 59.6% better than A₉, 67.4% better than A₆, 67.7% better than A₅, 68.5% better than A₃, 86.6% better than A₁, 90.7% better than A₇, 99.8% better than A₈ and 100% better than A₂. Likewise, we can judge the worthiness of other alternatives through this method in a manner that A₉ is 59.4% superior to A₆, A₆ is 50.4% superior to A₅, A₅ is 51.2% superior to A₃, A₃ is 67.1% superior to A₁, A₁ is 56.5% superior to A₇, A₇ is 59.1% superior to A₈ and A₈ is 56.4% more than A₂.

8 Conclusion

This article proposes new possibility degree measures for ranking CFNs. Initially, we offer an overview of some of the existing ways of ranking CFNs and their shortcomings. The current research introduces a new PDM and an IPDM that successfully overcome the shortcomings of available measures in the literature. The most noteworthy properties of the IPDM are investigated. Using these measures, new ranking methods for ordering the CFNs have been developed. A comparative study was conducted with the existing ranking methods to show the efficacy of the proposal. Two real-life numerical examples for selecting the best cutting fluids for cutting gears are used to demonstrate the effectiveness, feasibility, applicability, and superiority of the developed CFMADM approaches, as well as a comparison is made with an existing result for a MADM problem in the literature. The proposed methods are straightforward in solving MADM problems. They provide the degree of possibility as supplementary information to the ranking of alternatives and compare the alternatives in their fuzzy nature without converting them into crisp form. A most generic fuzzy framework for MADM problems to be adopted for complex real-life MADM applications When the space of CFN increases. This can be tackled by extending our present work to other fuzzy and uncertain environments to solve complex fuzzy DM problems. The results of this paper can be used to solve other cubical fuzzy multi-objective, transportation, assignment and group-DM problems.

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