A GRA approach to a MAGDM problem with interval-valued q-rung orthopair fuzzy information

Abstract

Interval-valued q-rung orthopair fuzzy numbers (IVq-ROFNs), as a generalization of the q-rung orthopair fuzzy numbers, are a more robust and reliable tool when dealing with uncertain information during decision-making processes, and can therefore be applied to a broader range of situations. This paper presents an approach to a multi-attribute group decision-making (MAGDM) problem in an IVq-ROF environment. In decision-making, the most sensitive part is information fusion (information aggregation); for this purpose, we extend the Einstein geometric aggregation operator for IVq-ROFNs. Einstein operators are valuable in information fusion, as they consider the interrelationship between arguments. Thus, while dealing with the information fusion process, the interrelationship between arguments ensures that aggregated values do not lose information. We use the traditional grey relational analysis (GRA) approach to rank the alternatives based on the attributes. In the GRA approach, we use positive and negative ideal solutions to obtain the grey relational coefficient (GRC). The GRCs of alternatives are calculated based on a new distance measure, which utilizes the hesitancy or indeterminacy degree of IVq-ROFNs. Utilizing the hesitancy or indeterminacy degree in distance measures reduces information loss significantly. The proposed approach considers three cases of attributes’ weights: partially known, completely unknown, and known. Consideration of three cases of attributes’ weights allows the approach to be applied to any appropriate MAGDM problem. We establish an optimization model to compute partially known attributes’ weights; we use the entropy weight determination method to compute unknown attributes’ weights. Finally, we discuss a real-world case study to validate the proposed approach.

A Appendix

Definition 6

(Atanassov 1999a) An IVIFS $\Re$ in $\Omega$ is defined as

$$\begin{aligned} \Re= & {} \{\langle \ell , \mu _{\Re }(\ell ), \vartheta _{\Re }(\ell ) \rangle |\ell \in \Omega \}, \end{aligned}$$

(24)

where the intervals $\mu _{\Re }(\ell ) = [\mu ^{L}_{\Re }(\ell ), \mu ^{U}_{\Re }(\ell )]$ and $\vartheta _{\Re }(\ell ) = [\vartheta ^{L}_{\Re }(\ell ), \vartheta ^{U}_{\Re }(\ell )]$ $\subseteq [0,1]$ represents the membership and non-membership degrees, respectively, such that $0 \le \sup \mu _{\Re }(\ell ) + \sup \vartheta _{\Re }(\ell ) \le 1$ for all elements $\ell \in \Omega$. For convenience, an IVIFN is represented as $\Re = \langle [\mu ^{L}_{\Re }, \mu ^{U}_{\Re }], [\vartheta ^{L}_{\Re }, \vartheta ^{U}_{\Re }] \rangle$, with the condition that $0 \le \mu ^{U}_{\Re } + \vartheta ^{U}_{\Re } \le 1$.

Definition 7

(Khan and Abdullah 2018; Peng and Yang 2016) An IVPFS $\mathfrak {P}$ in $\Omega$ is defined as

$$\begin{aligned} \mathfrak {P}= & {} \{\langle \ell , \mu _{\mathfrak {P}}(\ell ), \vartheta _{\mathfrak {P}}(\ell ) \rangle |\ell \in \Omega \}, \end{aligned}$$

(25)

where intervals $\mu _{\mathfrak {P}}(\ell ) = [\mu ^{L}_{\mathfrak {P}}(\ell ), \mu ^{U}_{\mathfrak {P}}(\ell )]$ and $\vartheta _{\mathfrak {P}}(\ell ) = [\vartheta ^{L}_{\mathfrak {P}}(\ell ), \vartheta ^{U}_{\mathfrak {P}}(\ell )]$ $\subseteq [0,1]$ represents the membership and non-membership degrees, respectively, such that $0 \le (\sup \mu _{\mathfrak {P}}(\ell ))^2 + (\sup \vartheta _{\mathfrak {P}}(\ell ))^2 \le 1$ for all elements $\ell \in \Omega$. For the convenience, the IVPFN is represented as $\mathfrak {P} = \langle [\mu ^{L}_{\mathfrak {P}}, \mu ^{U}_{\mathfrak {P}}], [\vartheta ^{L}_{\mathfrak {P}}, \vartheta ^{U}_{\mathfrak {P}}] \rangle$, with the condition $0 \le (\mu ^{U}_{\mathfrak {P}})^2 + (\vartheta ^{U}_{\mathfrak {P}})^2 \le 1$.

Theorem 3

Let $I_i = \langle [\mu ^{L}_i, \mu ^{U}_i], [\vartheta ^{L}_i, \vartheta ^{U}_i] \rangle$, $(i=1,2,\ldots , n)$ be n IVq-ROFNs, then the IVq-ROFEWG operator satisfies the following basic properties:

• I Commutativity: If $I'_1, I'_2, \ldots , I'_n$ is any permutation of $I_1, I_2, \ldots , I_n$, then

  $$\begin{aligned} \text {IV}q{} & {} -\text {ROFEWG}_{\omega }\left( I_1, I_2, \ldots , I_n \right) = \text {IV}q\nonumber \\{} & {} -\text {ROFEWG}_{\omega }\left( I'_1, I'_2, \ldots , I'_n \right) . \end{aligned}$$

  (26)

• II Idempotency: If $I_i = I$, $\forall i$, then

  $$\begin{aligned} \text {IV}q-\text {ROFEWG}_{\omega }\left( I_1, I_2, \ldots , I_n \right) = I. \end{aligned}$$

  (27)

• III Boundedness: If $I_{\max } =\max \limits _{1 \le i \le n} \left( I_i \right)$ and $I_{\min } =\min \limits _{1 \le i \le n} \left( I_i \right)$, then

  $$\begin{aligned} I_{\min } \le \text {IV{ q}-ROFEWG}_{\omega }\left( I_1, I_2, \ldots , I_n \right) \le I_{\max }. \end{aligned}$$

  (28)

• IV Monotonicity: If $I_i \le I'_i$, $\forall i$, then

  $$\begin{aligned} \text {IV}q{} & {} -\text {ROFEWG}_{\omega }\left( I_1, I_2, \ldots , I_n \right) \nonumber \\{} & {} \le \text {IV{ q}-ROFEWG}_{\omega }\left( I'_1, I'_2, \ldots , I'_n \right) . \end{aligned}$$

  (29)

Proof

Here, we will show only the proof of properties [I.] and [II.]; the rest of the two can be proved easily.

Commutativity: Here, we have

$$\begin{aligned} \text {IV}q{} & {} -\text {ROFEWG}_{\omega }\left( I_1, I_2, \ldots , I_n \right) \nonumber \\{} & {} = \left( I_1 \right) ^{\omega _1} \otimes _e \left( I_2 \right) ^{\omega _2} \otimes _e \ldots \otimes _e \left( I_n \right) ^{\omega _n}, \end{aligned}$$

(30)

and

$$\begin{aligned} \text {IV}q{} & {} -\text {ROFEWG}_{\omega }\left( I'_1, I'_2, \ldots , I'_n \right) = \left( I'_1 \right) ^{\omega _1} \otimes _e\nonumber \\{} & {} \left( I'_2 \right) ^{\omega _2} \otimes _e \ldots \otimes _e \left( I'_n \right) ^{\omega _n}. \end{aligned}$$

(31)

Since $\left( I'_1, I'_2, \ldots , I'_n \right)$ is any permutation of $\left( I_1, I_2, \ldots , I_n \right)$. Thus from Eqs. (30) and (31), Eq. (26) always holds.

Idempotency: We have

$$\begin{aligned}{} & {} {\text {IV}q-\text {ROFEWG}}_{\omega }\left( I_1, I_2, \ldots , I_n \right) \\{} & {} = \left\langle \left[ \frac{\root q \of { 2 \prod _{i=1}^{n} \left( \left( \mu ^{L}_i \right) ^q \right) ^{\omega _i}}}{\root q \of {\prod _{i=1}^{n} \left( 2 - \left( \mu ^{L}_i \right) ^q \right) ^{\omega _i} + \prod _{i=1}^{n} \left( \left( \mu ^{L}_i \right) ^q \right) ^{\omega _i} }},\right. \right. \\{} & {} \left. \left. \frac{\root q \of { 2 \prod _{i=1}^{n} \left( \left( \mu ^{U}_i \right) ^q \right) ^{\omega _i}}}{\root q \of {\prod _{i=1}^{n} \left( 2 - \left( \mu ^{U}_i \right) ^q \right) ^{\omega _i} + \prod _{i=1}^{n} \left( \left( \mu ^{U}_i \right) ^q \right) ^{\omega _i} }} \right] , \right. \\{} & {} \left. \left[ \frac{\root q \of {\prod _{i=1}^{n} \left( 1 + \left( \vartheta ^{L}_i \right) ^q \right) ^{\omega _i} - \prod _{i=1}^{n} \left( 1 - \left( \vartheta ^{L}_i \right) ^q \right) ^{\omega _i} }}{\root q \of {\prod _{i=1}^{n} \left( 1 + \left( \vartheta ^{L}_i \right) ^q \right) ^{\omega _i} + \prod _{i=1}^{n} \left( 1 - \left( \vartheta ^{L}_i \right) ^q \right) ^{\omega _i} }},\right. \right. \\{} & {} \left. \left. \frac{\root q \of {\prod _{i=1}^{n} \left( 1 + \left( \vartheta ^{U}_i \right) ^q \right) ^{\omega _i} - \prod _{i=1}^{n} \left( 1 - \left( \vartheta ^{U}_i \right) ^q \right) ^{\omega _i} }}{\root q \of {\prod _{i=1}^{n} \left( 1 + \left( \vartheta ^{U}_i \right) ^q \right) ^{\omega _i} + \prod _{i=1}^{n} \left( 1 - \left( \vartheta ^{U}_i\right) ^q \right) ^{\omega _i} }} \right] \right\rangle . \end{aligned}$$

Since $I_i = I$, $\forall i$, then we have

$$\begin{aligned}{} & {} {\text {IV}q-\text {ROFEWG}}_{\omega }\left( I_1, I_2, \ldots , I_n \right) \\{} & {} = \left\langle \left[ \frac{\root q \of { 2 \left( \left( \mu ^{L} \right) ^q \right) ^{\sum _{i=1}^{n}\omega _i}}}{\root q \of {\left( 2 - \left( \mu ^{L} \right) ^q \right) ^{\sum _{i=1}^{n}\omega _i} + \left( \left( \mu ^{L} \right) ^q \right) ^{\sum _{i=1}^{n}\omega _i} }},\right. \right. \\{} & {} \left. \left. \frac{\root q \of { 2 \left( \left( \mu ^{U} \right) ^q \right) ^{\sum _{i=1}^{n}\omega _i}}}{\root q \of { \left( 2 - \left( \mu ^{U} \right) ^q \right) ^{\sum _{i=1}^{n}\omega _i} + \left( \left( \mu ^{U} \right) ^q \right) ^{\sum _{i=1}^{n}\omega _i} }} \right] , \right. \nonumber \\{} & {} \left. \left[ \frac{\root q \of { \left( 1 + \left( \vartheta ^{L} \right) ^q \right) ^{\sum _{i=1}^{n}\omega _i} - \left( 1 - \left( \vartheta ^{L} \right) ^q \right) ^{\sum _{i=1}^{n}\omega _i} }}{\root q \of { \left( 1 + \left( \vartheta ^{L} \right) ^q \right) ^{\sum _{i=1}^{n}\omega _i} + \left( 1 - \left( \vartheta ^{L} \right) ^q \right) ^{\sum _{i=1}^{n}\omega _i} }},\right. \right. \\{} & {} \left. \left. \frac{\root q \of { \left( 1 + \left( \vartheta ^{U} \right) ^q \right) ^{\sum _{i=1}^{n}\omega _i} - \left( 1 - \left( \vartheta ^{U} \right) ^q \right) ^{\sum _{i=1}^{n}\omega _i} }}{\root q \of { \left( 1 + \left( \vartheta ^{U} \right) ^q \right) ^{\sum _{i=1}^{n}\omega _i} + \left( 1 - \left( \vartheta ^{U} \right) ^q \right) ^{\sum _{i=1}^{n}\omega _i} }} \right] \right\rangle . \end{aligned}$$

Since $\sum _{i=1}^{n}\omega _i =1$, thus ${\text {IV}q-\text {ROFEWG}}_{\omega }\left( I_1, I_2, \ldots , I_n \right) = I$.

1.1 A.1 Grey relational analysis

The GRA is a decision-making and data analysis method that assesses the relationship between a target and reference sequences. Its development can be attributed to Julong et al. (1989). GRA proves particularly valuable when confronted with incomplete, uncertain, or insufficient data scenarios. GRA finds widespread application across various disciplines, such as engineering, economics, management, and social sciences, where it evaluates the interdependence among multiple variables or factors. Its utility extends to addressing forecasting, optimization, decision-making, and performance evaluation challenges. At its core, GRA measures the similarity or proximity between two sequences by comparing their respective trends and patterns. Let $X_{0} = \left( x_{0}(1), x_{0}(2), \ldots , x_{0}(n) \right)$ is an ideal data set and $X_{k} = \left( x_{k}(1), x_{k}(2), \ldots , x_{k}(n) \right)$ are the alternative data sets of the same length. Then, the evaluating coefficient GRC $\digamma _{0k}^{j}$ of alternatives from the ideal data sets for the GRA approach is defined as follows:

$$\begin{aligned} \digamma _{0k}^{j} = \frac{ \min \limits _{k} \min \limits _{j} \left| x_{0}(j) - x_{k}(j) \right| + \rho \max \limits _{k} \max \limits _{j} \left| x_{0}(j) - x_{k}(j) \right| }{ \left| x_{0}(j) - x_{k}(j) \right| + \rho \max \limits _{k} \max \limits _{j} \left| x_{0}(j) - x_{k}(j) \right| },\nonumber \\ \end{aligned}$$

(32)

where $\rho \in (0,1]$ is the dynamic distinguishing coefficient. GRA offers a flexible and intuitive approach to examining variable relationships without precise mathematical models or complete data. By enabling DMs to consider multiple factors simultaneously, this methodology empowers them to make better-informed decisions based on complex systems analysis.