**Methods for Multiple-Attribute Group Decision Making with** *q***-Rung Interval-Valued Orthopair Fuzzy Information and Their Applications to the Selection of Green Suppliers**

#### **Jie Wang 1, Hui Gao 1, Guiwu Wei 1,\* and Yu Wei 2,\***


Received: 15 November 2018; Accepted: 26 December 2018; Published: 6 January 2019

**Abstract:** In the practical world, there commonly exist different types of multiple-attribute group decision making (MAGDM) problems with uncertain information. Symmetry among some attributes' information that is already known and unknown, and symmetry between the pure attribute sets and fuzzy attribute membership sets, can be an effective way to solve this type of MAGDM problem. In this paper, we investigate four forms of information aggregation operators, including the Hamy mean (HM) operator, weighted HM (WHM) operator, dual HM (DHM) operator, and the dual-weighted HM (WDHM) operator with the *q*-rung interval-valued orthopair fuzzy numbers (*q*-RIVOFNs). Then, some extended aggregation operators, such as the *q*-rung interval-valued orthopair fuzzy Hamy mean (*q*-RIVOFHM) operator; *q*-rung interval-valued orthopairfuzzy weighted Hamy mean (*q*-RIVOFWHM) operator; *q*-rung interval-valued orthopair fuzzy dual Hamy mean (*q*-RIVOFDHM) operator; and *q*-rung interval-valued orthopair fuzzy weighted dual Hamy mean (*q*-RIVOFWDHM) operator are presented, and some of their precious properties are studied in detail. Finally, a real example for green supplier selection in green supply chain management is provided, to demonstrate the proposed approach and to verify its rationality and scientific nature.

**Keywords:** multiple attribute group decision making (MAGDM); Pythagorean fuzzy set (PFSs); *q*-rung orthopair fuzzy sets (*q*-RIVOFSs); *q*-RIVOFWHM operator; *q*-RIVOFWDHM operator; green suppliers selection

#### **1. Introduction**

For the indeterminacy of decision makers and decision-making issues, we cannot always give accurate evaluation values for alternatives to select the best project in real multiple-attribute decision making (MADM) problems. To overcome this disadvantage, fuzzy set theory, as defined by Zadeh [1] in 1965, originally used the membership function to describe the estimation results, rather than an exact real number. Atanassov [2,3] presents the intuitionistic fuzzy set (IFS) by considering another measurement index which names a non-membership function. Hereafter, the IFS and its extension has aroused the attention of a large number of scholars since its appearance [4–25]. More recently, the Pythagorean fuzzy set (PFS) [26,27] has emerged as a useful tool for describing the indeterminacy of the MADM problems. Zhang and Xu [28] proposed the detailed mathematical expression for PFS and presented the definition of Pythagorean fuzzy numbers (PFNs). Wei and Lu [29] proposed some Maclaurin Symmetric Mean Operators with PFNs. Peng and Yang [30] studied the division and subtraction operations of PFNs. Wei and Lu [31] defined some power aggregation operators with PFNs based on the traditional power aggregation operators [32–37]. Beliakov and James [38] presented

the average aggregation functions of PFNs. Reformat and Yager [39] studied the collaborative-based recommender system under the Pythagorean fuzzy environment. Gou et al. [40] proposed some desirable properties of the continuous Pythagorean fuzzy number. Wei and Wei [41] defined some similar measures of Pythagorean fuzzy sets, based on cosine functions with traditional similarity measures [42–45]. Ren et al. [46] applied the Pythagorean fuzzy TODIM model in MADM. Garg [47] combines the Einstein Operations and Pythagorean fuzzy information to propose a new aggregation operator. Zeng et al. [48] provided a Pythagorean fuzzy hybrid method to study MADM. Garg [49] presents a novel accuracy function based on interval-valued Pythagorean fuzzy information for solving MADM problems. Wei et al. [50] propose the Pythagorean hesitant fuzzy Hamacher operators in MADM. Wei and Lu [51] develop the dual hesitant Pythagorean fuzzy Hamacher operators in MADM. Lu et al. [52] develop the hesitant Pythagorean fuzzy Hamacher aggregation operators in MADM.

In addition to this, based on the fundamental theories of IFS and PFS, Yager [53] further defined the *q*-rung orthopair fuzzy sets (*q*-ROFSs), in which the sum of the *q*th power of the degrees of membership and the *<sup>q</sup>*th power of the degrees of non-membership is satisfied the condition *<sup>μ</sup><sup>q</sup>* + *<sup>ν</sup><sup>q</sup>* ≤ 1. It is clear that the *q*-ROFSs are more general for IFSs and PFSs, as they are all special cases. Therefore, we can express a wider range of fuzzy information by using *q*-ROFSs. Liu and Wang [54] develop the *q*-rung orthopair, fuzzy weighted averaging (*q*-ROFWA) operator and the *q*-rung orthopair, fuzzy weighted geometric (*q*-ROFWG) operator to fuse the evaluation information. Liu and Liu [55] proposes a *q*-rung orthopair, fuzzy Bonferroni mean (*q*-ROFBM) aggregation operator, by considering the *q*-rung orthopair fuzzy information and the Bonferroni mean (BM) operator. Wei et al. [56] combine the *q*-rung orthopair fuzzy numbers (*q*-ROFNs) with a generalized Heronian mean (GHM) operator to present some aggregation operators, and applied them into MADM problems. Wei et al. [57] define some *q*-rung orthopair, fuzzy Maclaurin symmetric mean operators for the potential evaluation of emerging technology commercialization.

Nevertheless, in many practical decision-making problems, for the uncertainty of the decision-making environment and the subjectivity of decision makers (DMs), it is always difficult for DMs to exactly describe their views with a precise number; however, they can be expressed by an interval number within [0, 1]. This denotes that it is necessary to introduce the definition of *q*-rung interval-valued orthopair fuzzy sets (*q*-RIVOFSs), of which the degrees of positive membership and negative membership are given by an interval value. This kind of situation is more or less like that encountered in interval-valued, intuitionistic fuzzy environments [58,59]. It should be noted that when the upper and lower limits of the interval values are same, *q*-RIVOFSs reduce to *q*-ROFSs, meaning that the latter is a special case of the former.

This research has four main purposes. The first is to develop a comprehensive MAGDM method for selecting the best green supplier with *q*-RIVOFNs. The second purpose lies in exploring several aggregation operators based on traditional Hamy mean (HM) operators with *q*-RIVOFNs. The third is to establish an integrated outranking decision-making method by the *q*-RIVOFWHM (*q*-RIVOFWDHM) operators. The final purpose is to demonstrate the application, practicality, and effectiveness of the proposed MADM method for selecting the best green supplier.

To further study the *q*-RIVOFSs, our paper combines the Hamy mean (HM) operator, which considers the relationship between the attribute's estimation values with *q*-rung interval-valued orthopair fuzzy numbers to investigate MAGDM problems. For the sake of clarity, the rest of this research is organized as follows. Firstly, we briefly introduce the fundamental theories, such as definition, score, and accuracy functions, and operational laws of the *q*-ROFSs and *q*-RIVOFSs in Section 2. Then, based on *q*-RIVOFSs and Hamy mean (HM) operators, we propose four aggregation operators, including the *q*-rung interval-valued orthopair, fuzzy Hamy mean (*q*-RIVOFHM) operator; the *q*-rung interval-valued orthopair, fuzzy weighted Hamy mean (*q*-RIVOFWHM) operator; the *q*-rung interval-valued orthopair, fuzzy dual Hamy mean (*q*-RIVOFDHM) operator; and the *q*-rung interval-valued orthopair, fuzzy weighted dual Hamy mean (*q*-RIVOFWDHM) operator in Section 3. Meanwhile, some important properties of these operators are also studied. Thereafter, the models

which apply the proposed aggregation operators to solve MAGDM problems are presented in Section 4, and an illustrative example to select the best green supplier is developed. Some comments are provided to summarize this article in Section 5.

#### **2. Preliminaries**

#### *2.1. q-Rung Interval-Valued Orthopair Fuzzy Sets (q-RIVOFSs)*

According to the *q*-rung orthopair fuzzy sets (*q*-ROFSs) [53] and interval-valued Pythagorean fuzzy sets (IVPFSs) [49], we develop the definition of the *q*-rung interval-valued orthopair fuzzy sets (*q*-RIVOFSs).

**Definition 1.** *Let X be a fixed set. A q-RIVOFS is an object having the form*

$$\tilde{Q} = \left\{ \left< \mathbf{x}, \left( \tilde{\mu}\_{\bar{Q}}(\mathbf{x}), \tilde{\nu}\_{\bar{Q}}(\mathbf{x}) \right) \right> | \mathbf{x} \in X \right\} \tag{1}$$

*where <sup>μ</sup>Q*(*x*) <sup>⊂</sup> [0, 1] *and <sup>ν</sup>Q*(*x*) <sup>⊂</sup> [0, 1] *are interval numbers, and <sup>μ</sup>Q*(*x*) <sup>=</sup> *μL <sup>Q</sup>*(*x*), *<sup>μ</sup><sup>R</sup> <sup>Q</sup>*(*x*) , *<sup>ν</sup>Q*(*x*) <sup>=</sup> *νL <sup>Q</sup>*(*x*), *<sup>ν</sup><sup>R</sup> <sup>Q</sup>*(*x*) *with the condition* 0 ≤ *μR <sup>Q</sup>*(*x*) *q* + *νR <sup>Q</sup>*(*x*) *q* ≤ 1*,* ∀*x* ∈ *X, q* ≥ 1*. The numbers <sup>μ</sup>Q*(*x*), *<sup>ν</sup>Q*(*x*) *represent, respectively, the function of positive membership degree (PMD) and negative membership degree (NMD) of the element <sup>x</sup> to <sup>Q</sup>. Then, for <sup>x</sup>* <sup>∈</sup> *X, <sup>π</sup>Q*(*x*) <sup>=</sup> *πL <sup>Q</sup>*(*x*), *<sup>π</sup><sup>R</sup> <sup>Q</sup>*(*x*) = 4 *q* 1 − *μ<sup>R</sup> <sup>Q</sup>*(*x*) *q* + *νR <sup>Q</sup>*(*x*) *q* , *q* 1 − *μ<sup>L</sup> <sup>Q</sup>*(*x*) *q* + *νL <sup>Q</sup>*(*x*) *q*5 *denotes the function of the refusal membership degree (RMD) of the element x to Q.*

As a matter of convenience, we called *<sup>q</sup>* <sup>=</sup> *u<sup>L</sup> <sup>q</sup>*, *<sup>u</sup><sup>R</sup> q* , *vL q*, *vR q* a *q*-rung interval-valued orthopair fuzzy number (*q*-RIVOFN). Let *<sup>q</sup>* <sup>=</sup> *u<sup>L</sup> <sup>q</sup>*, *<sup>u</sup><sup>R</sup> q* , *vL q*, *vR q* be a *<sup>q</sup>*-RIVOFN, then *<sup>S</sup>*(*q*) <sup>=</sup> 1 4 1 + *uL q q* − *vL q q* + 1 + *uR q q* − *vR q q* and *<sup>H</sup>*(*q*) <sup>=</sup> *uL q q* + *uR q q* + *vL q q* + *vR q q* <sup>2</sup> are the score and accuracy function of a *<sup>q</sup>*-RIVOFN *<sup>q</sup>*.

**Definition 2.** *Let <sup>q</sup>*<sup>1</sup> <sup>=</sup> *u<sup>L</sup> q*1 , *u<sup>R</sup> q*1 , *vL q*1 , *v<sup>R</sup> q*1 *and <sup>q</sup>*<sup>2</sup> <sup>=</sup> *u<sup>L</sup> q*2 , *u<sup>R</sup> q*2 , *vL q*2 , *v<sup>R</sup> q*2 *be two q-RIVOFNs; <sup>S</sup>*(*q*1) *and <sup>S</sup>*(*q*2) *be the scores of <sup>q</sup>*<sup>1</sup> *and <sup>q</sup>*2*, respectively; and let <sup>H</sup>*(*q*1) *and <sup>H</sup>*(*q*2) *be the accuracy degrees of <sup>q</sup>*<sup>1</sup> *and <sup>q</sup>*2*, respectively. Then, if <sup>S</sup>*(*q*1) <sup>&</sup>lt; *<sup>S</sup>*(*q*2)*, then <sup>q</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>q</sup>*2*; if <sup>S</sup>*(*q*1) <sup>=</sup> *<sup>S</sup>*(*q*2)*, then (1) if <sup>H</sup>*(*q*1) <sup>=</sup> *<sup>H</sup>*(*q*2)*, then <sup>q</sup>*<sup>1</sup> <sup>=</sup> *<sup>q</sup>*2*; (2) if H*(*q*1) <sup>&</sup>lt; *<sup>H</sup>*(*q*2)*, then <sup>q</sup>*<sup>1</sup> <sup>&</sup>lt; *<sup>q</sup>*2*.*

**Definition 3.** *Let <sup>q</sup>*<sup>1</sup> <sup>=</sup> *u<sup>L</sup> q*1 , *u<sup>R</sup> q*1 , *vL q*1 , *v<sup>R</sup> q*1 *, <sup>q</sup>*<sup>2</sup> <sup>=</sup> *u<sup>L</sup> q*2 , *u<sup>R</sup> q*2 , *vL q*2 , *v<sup>R</sup> q*2 *, and <sup>q</sup>* <sup>=</sup> *u<sup>L</sup> <sup>q</sup>*, *<sup>u</sup><sup>R</sup> q* , *vL q*, *vR q be three q-RIVOFNs, and some basic operation rules for them are shown as follows:* *Symmetry* **2019**, *11*, 56

(1) *<sup>q</sup>*<sup>1</sup> <sup>⊕</sup> *<sup>q</sup>*<sup>2</sup> <sup>=</sup> ⎛ ⎜⎜⎝ ⎡ ⎢ ⎢ ⎣ *q uL q*1 *q* + *uL q*2 *q* − *uL q*1 *q uL q*2 *q* , *q uR q*1 *q* + *uR q*2 *q* − *uR q*1 *q uR q*2 *q* ⎤ ⎥ ⎥ ⎦, *vL q*1 *vL q*2 , *v<sup>R</sup> q*1 *vR q*2 ⎞ ⎟⎟⎠ ; (2)*q*<sup>1</sup> <sup>⊗</sup> *<sup>q</sup>*<sup>2</sup> <sup>=</sup> ⎛ ⎜⎜⎝ *μL q*1 *vL q*2 , *μ<sup>R</sup> q*1 *vR q*2 , ⎡ ⎢ ⎢ ⎣ *q vL q*1 *q* + *vL q*2 *q* − *vL q*1 *q vL q*2 *q* , *q vR q*1 *q* + *vR q*2 *q* − *vR q*1 *q vR q*2 *q* ⎤ ⎥ ⎥ ⎦ ⎞ ⎟⎟⎠ ; (3) *<sup>λ</sup>q* <sup>=</sup> *<sup>q</sup>* 1 − 1 − *uL q q<sup>λ</sup>* , *q* 1 − 1 − *uR q q<sup>λ</sup>* , 4 *vL q λ* , *vR q λ* 5 , *λ* > 0; (4) (*q*) *<sup>λ</sup>* = 4 *μL q λ* , *μR q λ* 5 , *q* 1 − 1 − *vL q q<sup>λ</sup>* , *q* 1 − 1 − *vR q q<sup>λ</sup>* , *λ* > 0; (5) *<sup>q</sup><sup>c</sup>* <sup>=</sup> *v<sup>L</sup> q*, *vR q* , *μL <sup>q</sup>*, *<sup>μ</sup><sup>R</sup> q* .

#### *2.2. Hamy Mean Operator*

**Definition 4 [60].** *The HM operator is defined as follows:*

$$\text{HM}^{(\text{x})}(\tilde{q}\_1, \tilde{q}\_2, \dots, \tilde{q}\_n) = \frac{\sum\_{1 \le i\_1 < \dots < i\_n \le n} \left(\prod\_{j=1}^x \tilde{q}\_{i\_j}\right)^{\frac{1}{x}}}{\mathbb{C}\_n^x} \tag{2}$$

*where x is a parameter and x* = 1, 2, ... , *n, i*1, *i*2, ... , *ix are x integer values taken from the set* {1, 2, . . . , *n*} *of k integer values; C<sup>x</sup> <sup>n</sup> denotes the binomial coefficient and C<sup>x</sup> <sup>n</sup>* = *<sup>n</sup>*! *x*!(*n*−*x*)! *.*

#### **3. Some Hamy Mean Operators with** *q***-RIVOFNs**

#### *3.1. q-RIVOFHM Operator*

In this chapter, consider both HM operator and *q*-RIVOFNs, we propose the *q*-rung interval-valued orthopair fuzzy Hamy mean (*q*-RIVOFHM) operator.

**Definition 5.** *Let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*j* = 1, 2, ... , *n*) *be a set of q-RIVOFNs. The q-RIVOFHM operator is*

$$q\text{-RIVOFHM}^{(\text{x})}(\tilde{q}\_1, \tilde{q}\_2, \dots, \tilde{q}\_n) = \frac{\stackrel{\stackrel{\leftrightarrow}{\oplus}}{1 \le i\_1 < \dots < i\_n \le n} \binom{\stackrel{\text{x}}{\otimes} \tilde{q}\_{\tilde{i}}}{\stackrel{\text{j}}{\cdot} \text{l}^{\text{j}}}^{\frac{\text{j}}{\text{k}}}}{\text{C}\_n^{\text{x}}} \tag{3}$$

**Theorem 1.** *Let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*j* = 1, 2, ... , *n*) *be a set of q-RIVOFNs. The fused value by using q-RIVOFHM operator is also a q-RIVOFN, where*

*Symmetry* **2019**, *11*, 56

*q*-RIVOFHM(*x*) (*q*1, *<sup>q</sup>*2, ··· , *<sup>q</sup>n*) <sup>=</sup> <sup>⊕</sup> <sup>1</sup>≤*<sup>i</sup>* 1<...<*ix*≤*n <sup>x</sup>* ⊗ *j*=1 *qi j*  1 *x Cx n* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎢ ⎣ *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *uL qj q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* , *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *uR qj q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎣ ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup> q* 1 − *x* ∏ *j*=1 1 − *vL qj q* <sup>1</sup> *x* ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup> q* 1 − *x* ∏ *j*=1 1 − *vL qj q* <sup>1</sup> *x* ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎦ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎭ (4)

**Proof.**

$$\boldsymbol{\hat{f}}\_{\vec{i}\vec{j}}^{\vec{x}}\overset{\textstyle \vec{\mathcal{G}}}{\mathop{\otimes}}\boldsymbol{\tilde{q}}\_{l\_{\vec{j}}} = \left\{ \left[ \prod\_{j=1}^{\mathsf{x}} \boldsymbol{u}\_{\vec{q}\_{j}}^{\mathsf{L}} \prod\_{j=1}^{\mathsf{x}} \boldsymbol{u}\_{\vec{q}\_{j}}^{\mathsf{R}} \right] / \left[ \sqrt[q]{1 - \prod\_{j=1}^{\mathsf{x}} \left(1 - \left(\boldsymbol{v}\_{\vec{q}\_{j}}^{\mathsf{L}}\right)^{q}\right)} , \sqrt[q]{1 - \prod\_{j=1}^{\mathsf{x}} \left(1 - \left(\boldsymbol{v}\_{\vec{q}\_{j}}^{\mathsf{R}}\right)^{q}\right)} \right] \right\} \tag{5}$$

Thus,

$$\begin{aligned} \left( \mathop{\otimes}\_{j=1}^{\boldsymbol{x}} \widehat{q}\_{l\_{j}} \right)^{\frac{1}{\bar{\boldsymbol{x}}}} &= \left\{ \left[ \left( \prod\_{j=1}^{\boldsymbol{x}} \boldsymbol{u}\_{\bar{q}\_{j}}^{\boldsymbol{L}} \right)^{\frac{1}{\bar{\boldsymbol{x}}}} , \left( \prod\_{j=1}^{\boldsymbol{x}} \boldsymbol{u}\_{\bar{q}\_{j}}^{\boldsymbol{R}} \right)^{\frac{1}{\bar{\boldsymbol{x}}}} \right], \left[ \underbrace{\sqrt[q]{1 - \left( \prod\_{j=1}^{\boldsymbol{x}} \left( 1 - \left( \boldsymbol{v}\_{\bar{q}\_{j}}^{\boldsymbol{L}} \right)^{q} \right)}\_{\boldsymbol{q}\_{j}}^{\frac{1}{\bar{\boldsymbol{x}}}}} \right] \right] \right\} \end{aligned} \tag{6}$$

Thereafter,

<sup>⊕</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup> <sup>x</sup>* ⊗ *j*=1 *qij* 1 *x* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎣ *q* <sup>1</sup> <sup>−</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *uL qj q x* ⎞ <sup>⎠</sup>, *<sup>q</sup>* <sup>1</sup> <sup>−</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *uR qj q x* ⎞ ⎠ ⎤ ⎥ ⎦, ⎡ ⎢ <sup>⎣</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup> q* 1 − *x* ∏ *j*=1 1 − *vL qj q* <sup>1</sup> *x* , ∏ 1≤*i*1<...<*ix*≤*n q* 1 − *x* ∏ *j*=1 1 − *vR qj q* <sup>1</sup> *x* ⎤ ⎥ ⎦ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎭ (7)

Therefore,

$$= \left\{ \begin{array}{l} q\text{-RIVOFHM}^{(x)}(\tilde{q}\_{1},\tilde{q}\_{2},\ldots,\tilde{q}\_{\tilde{n}}) = \frac{\stackrel{\stackrel{\scriptstyle\rightarrow}{\mathbb{Q}}\_{\left(\tilde{x}\_{i}\right)}\left(\frac{\tilde{x}\_{i}}{\tilde{q}\_{1}}\right)^{\frac{1}{2}}}{\mathrm{C}\_{\tilde{n}}}\\ = \left\{ \left[ \sqrt{\frac{1}{1-\left(\prod\_{1\leq i\_{1}}^{\prime}\prod\_{i\_{1}}^{\prime}\left(1-\left(\prod\_{j=1}^{\prime}{\tilde{x}\_{i\_{j}}^{\prime}}\right)^{\frac{1}{2}}\right)}\right)^{\frac{1}{2}}}, \sqrt{\prod\_{1\leq i\_{1},\ldots,i\_{\ell}\leq n}\left(1-\left(\prod\_{j=1}^{\prime}{\tilde{x}\_{i\_{j}}^{\prime}}\right)^{\frac{1}{2}}\right)}\right],\\ \left\{ \left(\prod\_{1\leq i\_{1}<\cdots$$

Hence, Equation (4) is kept.

Then, we need to prove that Equation (4) is a *q*-RIVOFN. We need to prove 0 ≤ *μR <sup>Q</sup>*(*x*) *q* + *νR <sup>Q</sup>*(*x*) *q* ≤ 1.

Let

$$\begin{aligned} \mu\_{\boldsymbol{Q}}^{\mathbb{R}}(\boldsymbol{x}) &= \sqrt[q]{1 - \left(\prod\_{1 \le i\_1 < \ldots < i\_x \le n} \left(1 - \left(\prod\_{j=1}^x u\_{\widehat{q}\_j}^{\mathbb{R}}\right)^{\frac{q}{1-\widehat{q}\_j}}\right)\right)^{\frac{1}{C\_n^{\alpha}}}} \\ \nu\_{\boldsymbol{Q}}^{\mathbb{R}}(\boldsymbol{x}) &= \left(\prod\_{1 \le i\_1 < \ldots < i\_x \le n} \sqrt[q]{1 - \left(\prod\_{j=1}^x \left(1 - \left(\nu\_{\widehat{q}\_j}^{\mathbb{R}}\right)^{\boldsymbol{q}}\right)\right)^{\frac{1}{C\_n^{\alpha}}}}\right)^{\frac{1}{C\_n^{\alpha}}} \end{aligned}$$

**Proof.**

$$\begin{split} 0 \leq & \left(\mu\_{\widehat{Q}}^{R}(\mathbf{x})\right)^{q} + \left(\nu\_{Q}^{R}(\mathbf{x})\right)^{q} \\ = & 1 - \left(\prod\_{1 \leq i\_{1} < \dots < i\_{n} \leq n} \left(1 - \left(\prod\_{j=1}^{x} u\_{\widehat{q}\_{j}}^{q}\right)^{\frac{q}{x}}\right)\right)^{\frac{1}{1-q}} + \left(\prod\_{1 \leq i\_{1} < \dots < i\_{n} \leq n} \left(1 - \left(\prod\_{j=1}^{x} \left(1 - \left(\mathbb{P}\_{\widehat{q}\_{j}}^{R}\right)^{q}\right)\right)^{\frac{1}{n}}\right)\right)^{\frac{1}{1-q}} \\ \leq & 1 - \left(\prod\_{1 \leq i\_{1} < \dots < i\_{n} \leq n} \left(1 - \left(\prod\_{j=1}^{x} \left(1 - \left(\mathbb{P}\_{\widehat{q}\_{j}}^{R}\right)^{q}\right)\right)^{\frac{1}{1-q}}\right)\right)^{\frac{1}{1-q}} + \left(\prod\_{1 \leq i\_{1} < \dots < i\_{n} \leq n} \left(1 - \left(\prod\_{j=1}^{x} \left(1 - \left(\mathbb{P}\_{\widehat{q}\_{j}}^{R}\right)^{q}\right)\right)^{\frac{1}{1-q}}\right)\right)^{\frac{1}{1-q}} \\ = & 1 - 1 \end{split}$$

So 0 ≤ *μR <sup>Q</sup>*(*x*) *q* + *νR <sup>Q</sup>*(*x*) *q* ≤ 1 is maintained.

**Example 1.** *Let* ([0.5, 0.8], [0.4, 0.5]),([0.3, 0.5], [0.6, 0.7]),([0.5, 0.7], [0.2, 0.3]) *and* ([0.4, 0.8], [0.1, 0.2]) *be four q-RIVOFNs, and suppose x* = 2, *q* = 3*—then, according to Equation (4), we have*

*q*-RIVOFHM(*x*) (*q*1, *<sup>q</sup>*2, ··· , *<sup>q</sup>n*) <sup>=</sup> <sup>⊕</sup> <sup>1</sup>≤*<sup>i</sup>* 1<...<*ix*≤*n <sup>x</sup>* ⊗ *j*=1 *qi j*  1 *x Cx n* = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 3 1 − ⎛ ⎝ 1 − (0.5 × 0.3) 3 2 × 1 − (0.5 × 0.5) 3 2 × 1 − (0.5 × 0.4) 3 2 × 1 − (0.3 × 0.4) 3 2 × 1 − (0.3 × 04) 3 2 × 1 − (0.5 × 0.4) 3 2 ⎞ ⎠ 1 *C*2 4 , 3 1 − ⎛ ⎝ 1 − (0.8 × 0.5) 3 2 × 1 − (0.8 × 0.7) 3 2 × 1 − (0.8 × 0.8) 3 2 × 1 − (0.5 × 0.7) 3 2 × 1 − (0.5 × 0.8) 3 2 × 1 − (0.7 × 0.8) 3 2 ⎞ ⎠ 1 *C*2 4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ 3 ⎛ ⎜⎜⎜⎜⎜⎜⎝ 1 − --<sup>1</sup> − 0.43 × - <sup>1</sup> − 0.63 1 2 × 1 − --<sup>1</sup> − 0.43 × - <sup>1</sup> − 0.2<sup>3</sup> 1 2 × 1 − --<sup>1</sup> − 0.43 × - <sup>1</sup> − 0.13 1 2 × 1 − --<sup>1</sup> − 0.63 × - <sup>1</sup> − 0.23 1 2 × 1 − --<sup>1</sup> − 0.6<sup>3</sup> × - <sup>1</sup> − 0.13 1 2 × 1 − --<sup>1</sup> − 0.23 × - <sup>1</sup> − 0.13 1 2 ⎞ ⎟⎟⎟⎟⎟⎟⎠ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ 1 *C*2 4 , ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ 3 ⎛ ⎜⎜⎜⎜⎜⎜⎝ 1 − --<sup>1</sup> − 0.53 × - <sup>1</sup> − 0.7<sup>3</sup> 1 2 × 1 − --<sup>1</sup> − 0.5<sup>3</sup> × - <sup>1</sup> − 0.3<sup>3</sup> 1 2 × 1 − --<sup>1</sup> − 0.5<sup>3</sup> × - <sup>1</sup> − 0.23 1 2 × 1 − --<sup>1</sup> − 0.73 × - <sup>1</sup> − 0.33 1 2 × 1 − --<sup>1</sup> − 0.7<sup>3</sup> × - <sup>1</sup> − 0.23 1 2 × 1 − --<sup>1</sup> − 0.33 × - <sup>1</sup> − 0.23 1 2 ⎞ ⎟⎟⎟⎟⎟⎟⎠ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ 1 *C*2 4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ([ 0.4261, 0.7072], [ 0.3604, 0.4605])

*The q-RIVOFHM satisfies the following three properties.*

**Property 1.** *Idempotency: if <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*j* = 1, 2, . . . , *n*) *are equal, then*

$$q\text{-RIVOFHM}^{(x)}(\tilde{q}\_1, \tilde{q}\_2, \dots, \tilde{q}\_n) = \tilde{q} \tag{9}$$

**Proof.** Since *<sup>q</sup><sup>j</sup>* <sup>=</sup> *<sup>q</sup>* <sup>=</sup> *u<sup>L</sup> <sup>q</sup>*, *<sup>u</sup><sup>R</sup> q* , *vL q*, *vR q* , then

*q*-RIVOFHM(*x*) (*q*, *<sup>q</sup>*, ··· , *<sup>q</sup>*) <sup>=</sup> <sup>⊕</sup> <sup>1</sup>≤*<sup>i</sup>* 1<...<*ix*≤*n <sup>x</sup>* ⊗ *j*=1 *q*  1 *x Cx n* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎢ ⎣ *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *uL q q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* , *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *uR q q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎣ ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup> q* 1 − *x* ∏ *j*=1 1 − *vL q q* <sup>1</sup> *x* ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup> q* 1 − *x* ∏ *j*=1 1 − *vR q q* <sup>1</sup> *x* ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎦ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎣ *q* 1 − ⎛ ⎝ 1 − *u<sup>L</sup> q x <sup>q</sup> x <sup>C</sup><sup>x</sup> n* ⎞ ⎠ 1 *Cx n* , *q* 1 − ⎛ ⎝ 1 − *u<sup>R</sup> q x <sup>q</sup> x <sup>C</sup><sup>x</sup> n* ⎞ ⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎦, ⎡ ⎢ ⎢ ⎣ ⎛ ⎜⎝ ⎛ ⎝ *q* : 1 − 1 − *vL q q<sup>x</sup>* 1 *x* ⎞ ⎠ *Cx n* ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ ⎛ ⎝ *q* : 1 − 1 − *vR q q<sup>x</sup>* 1 *x* ⎞ ⎠ *Cx n* ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎦ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ = *u<sup>L</sup> <sup>q</sup>*, *<sup>u</sup><sup>R</sup> q* , *vL q*, *vR q* <sup>=</sup> *<sup>q</sup>* 

**Property 2.** *Monotonicity: let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*<sup>j</sup>* <sup>=</sup> 1, 2, ... , *<sup>n</sup>*) *and <sup>q</sup> <sup>j</sup>* = 4 *uL qj* , *uR qj* 5 , 4 *vL qj* , *vR qj* 5(*<sup>j</sup>* <sup>=</sup> 1, 2, ... , *<sup>n</sup>*) *be two sets of q-RIVOFNs. If <sup>u</sup><sup>L</sup> <sup>q</sup><sup>j</sup>* <sup>≤</sup> *uL qj* , *u<sup>R</sup> <sup>q</sup><sup>j</sup>* <sup>≤</sup> *uR qj* , *v<sup>L</sup> <sup>q</sup><sup>j</sup>* <sup>≥</sup> *vL qj and v<sup>L</sup> <sup>q</sup><sup>j</sup>* <sup>≥</sup> *vR qj hold for all j, then q*-RIVOFHM(*x*) (*q*1, *<sup>q</sup>*2, ··· , *<sup>q</sup>n*) <sup>≤</sup> *<sup>q</sup>*-RIVOFHM(*x*) - *q* 1, *q* 2, ··· , *<sup>q</sup> n* (10)

**Proof.** Given that *u<sup>L</sup> <sup>q</sup><sup>j</sup>* <sup>≤</sup> *uL qj* , we can obtain

$$\left(\prod\_{j=1}^{x} u\_{\overline{q}}^{L}\right)^{\frac{q}{x}} \le \left(\prod\_{j=1}^{x} \left(u\_{\overline{q}}^{L}\right)'\right)^{\frac{q}{x}} \tag{11}$$

$$\left(\prod\_{1 \le i\_1 < \ldots < i\_x \le n} \left(1 - \left(\prod\_{j=1}^x u\_{\overline{q}}^L\right)^{\frac{q}{x}}\right)\right)^{\frac{1}{C\_n^2}} \ge \left(\prod\_{1 \le i\_1 < \ldots < i\_x \le n} \left(1 - \left(\prod\_{j=1}^x \left(u\_{\overline{q}}^L\right)'\right)^{\frac{q}{x}}\right)\right)^{\frac{1}{C\_n^2}}\tag{12}$$
  $\text{proafter}$ 

Thereafter,

$$\sqrt[n]{1 - \left(\prod\_{1 \le i\_1 < \dots < i\_t \le n} \left(1 - \left(\prod\_{j=1}^x u\_{\tilde{q}}^L\right)^{\frac{2}{\tilde{q}}}\right)\right)^{\frac{1}{L\_n^{\tilde{q}}}}} \le \sqrt[n]{1 - \left(\prod\_{1 \le i\_1 < \dots < i\_t \le n} \left(1 - \left(\prod\_{j=1}^x \left(u\_{\tilde{q}}^L\right)^{\frac{2}{\tilde{q}}}\right)\right)^{\frac{1}{L\_n^{\tilde{q}}}}\right)^{\frac{1}{L\_n^{\tilde{q}}}}} \tag{13}$$

That means *u<sup>L</sup> <sup>q</sup>* <sup>≤</sup> *uL q* . Similarly, we can obtain *u<sup>R</sup> <sup>q</sup>* <sup>≤</sup> *uR q* , *v<sup>L</sup> <sup>q</sup>* <sup>≥</sup> *vL q* and *v<sup>L</sup> <sup>q</sup>* <sup>≥</sup> *vR q* . Thus, the proof is complete.

$$\begin{array}{rclclcl}\textbf{Property 3.} & \quad \textbf{Boundedness:} & \quad \textbf{let } \overrightarrow{q}\_{\langle\rangle} &=& \left( \left[ u\_{\overrightarrow{q}\_{\langle}}^{\textit{L}}, u\_{\overrightarrow{q}\_{\langle\rangle}}^{\textit{R}} \right], \left[ v\_{\overrightarrow{q}\_{\langle}}^{\textit{L}}, v\_{\overrightarrow{q}\_{\langle}}^{\textit{R}} \right] \right) (j & = & 1, 2, \dots, n) & \textbf{be a set } \textbf{of} \\ \text{of } \textit{q-RIVOFNs.} & \quad \textit{If } \widehat{q}^{+} &=& \left( \left[ \max\_{i} \left( u\_{\overrightarrow{q}\_{\langle}}^{\textit{L}} \right), \max\_{i} \left( u\_{\overrightarrow{q}\_{\langle}}^{\underline{R}} \right) \right], \left[ \min\_{i} \left( v\_{\overrightarrow{q}\_{\langle}}^{\underline{L}} \right), \min\_{i} \left( v\_{\overrightarrow{q}\_{\langle}}^{\underline{R}} \right) \right] \right) \right) & \text{and} & \widehat{q}^{-} &=& \\ \left( \left[ \min\_{i} \left( u\_{\overrightarrow{q}\_{\langle}}^{\underline{L}} \right), \min\_{i} \left( u\_{\overrightarrow{q}\_{\langle}}^{\underline{R}} \right) \right], \left[ \max\_{i} \left( v\_{\overrightarrow{q}\_{\langle}}^{\underline{L}} \right), \max\_{i} \left( v\_{\overrightarrow{q}\_{\langle}}^{\underline{R}} \right) \right] \right) \right) & \text{then} & \\ & & & \\ & & \dots & \end{array}$$

$$
\hat{q}^- \le q \text{-RIVOFHM}^{(x)}(\tilde{q}\_1, \tilde{q}\_2, \dots, \tilde{q}\_n) \le \tilde{q}^+ \tag{14}
$$

From Property 1,

$$\begin{array}{c} q\text{-RIVOFHM}^{(x)}\left(\hat{q}\_1^-,\hat{q}\_2^-,\cdots,\hat{q}\_n^-\right) = \hat{q}^-\\ q\text{-RIVOFHM}^{(x)}\left(\hat{q}\_1^+,\hat{q}\_2^+,\cdots,\hat{q}\_n^+\right) = \hat{q}^+ \end{array}$$

From Property 2,

$$\widetilde{q}^{-} \le q \text{-RIVOFHM}^{(\ge)}(\widetilde{q}\_1, \widetilde{q}\_2, \dots, \widetilde{q}\_n) \le \widetilde{q}^{+} $$

#### *3.2. The q-RIVOFWHM Operator*

In practical MADM problems, it is important to take the attribute weights into account. This section will develop the *q*-rung interval-valued orthopair, fuzzy weighted Hamy mean (*q*-RIVOFWHM) operator.

**Definition 6.** *Let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*j* = 1, 2, ... , *n*) *be a set of q-RIVOFNs, with their weight vector as wi* = (*w*1, *w*2,..., *wn*) *T, thereby satisfying wi* <sup>∈</sup> [0, 1] *and* <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *wi* = 1*. Then we can define the q-RIVOFWHM operator as follows:*

$$q\text{-RIVOFWH}\_{w}^{(x)}(\tilde{q}\_{1},\tilde{q}\_{2},\dots,\tilde{q}\_{n}) = \frac{\stackrel{\bigoplus}{1\le i\_{1}\le\ldots\le i\_{x}\le n} \left(\stackrel{\text{x}}{\otimes}(\tilde{q}\_{i\_{j}})^{w\_{i\_{j}}}\right)^{\frac{1}{\mathfrak{X}}}}{\text{C}\_{n}^{x}}\tag{15}$$

**Theorem 2.** *Let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*j* = 1, 2, ... , *n*) *be a set of q-RIVOFNs. The fused value obtained by using q-RIVOFWHM operator is also a q-RIVOFN, where*

*<sup>q</sup>*-RIVOFWHM(*x*) *<sup>w</sup>* (*q*1, *<sup>q</sup>*2,..., *<sup>q</sup>n*) = <sup>⊕</sup> <sup>1</sup>≤*<sup>i</sup>* 1<...<*ix*≤*n <sup>x</sup>* ⊗ *j*=1 *qi j wi j*  1 *x Cx n* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎢ ⎣ *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *uL qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* , *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *uR qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎣ ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − *vL qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − *vR qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎦ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (16)

**Proof.** From Definition 3, we can obtain

$$\left( \left( \check{q}\_{i\_{\vec{l}}} \right)^{w\_{\vec{i}\_{\vec{j}}}} = \left\{ \left[ \left( u\_{\vec{q}\_{\vec{l}}}^{L} \right)^{w\_{\vec{i}\_{\vec{j}}}}, \left( u\_{\vec{q}\_{\vec{j}}}^{R} \right)^{w\_{\vec{i}\_{\vec{j}}}} \right], \left[ \sqrt[q]{1 - \left( 1 - \left( v\_{\vec{q}\_{\vec{j}}}^{L} \right)^{q} \right)^{w\_{\vec{i}\_{\vec{j}}}}}, \sqrt[q]{1 - \left( 1 - \left( v\_{\vec{q}\_{\vec{j}}}^{R} \right)^{q} \right)^{w\_{\vec{i}\_{\vec{j}}}}} \right] \right\} \tag{17}$$

Thus,

$$\begin{aligned} \left(\begin{array}{l} \underset{j=1}{\overset{\text{x}}{\otimes}} \left(\tilde{q}\_{i\_{j}}\right)^{\text{w}\_{i\_{j}}} = \left\{ \begin{array}{l} \quad & \left[\prod\limits\_{j=1}^{\text{x}} \left(\boldsymbol{\mu}\_{\tilde{q}\_{j}}^{\text{L}}\right)^{\text{w}\_{i\_{j}}}, \prod\limits\_{j=1}^{\text{x}} \left(\boldsymbol{\mu}\_{\tilde{q}\_{j}}^{\text{R}}\right)^{\text{w}\_{i\_{j}}} \right], \\\ \left[\sqrt{1-\prod\limits\_{j=1}^{\text{x}} \left(1-\left(\boldsymbol{\upsilon}\_{\tilde{q}\_{j}}^{\text{L}}\right)^{\text{q}}\right)^{\text{w}\_{i\_{j}}}, \sqrt[q]{1-\prod\limits\_{j=1}^{\text{x}} \left(1-\left(\boldsymbol{\upsilon}\_{\tilde{q}\_{j}}^{\text{R}}\right)^{\text{q}}\right)^{\text{w}\_{i\_{j}}}} \right] \end{array} \right\} \end{aligned} \tag{18}$$

Therefore,

$$\left( \begin{matrix} \frac{\mathbf{x}}{\bigotimes} \left( \tilde{q}\_{\hat{i}\_{j}} \right)^{w\_{\hat{j}}} \\ j=1 \end{matrix} \right)^{\frac{1}{\tilde{\mathbf{x}}}} = \left\{ \begin{matrix} \left( \prod\_{j=1}^{\mathbf{x}} \left( u\_{\tilde{q}\_{\hat{j}}}^{L} \right)^{w\_{\hat{j}}} \right)^{\frac{1}{\tilde{\mathbf{x}}}} \left( \prod\_{j=1}^{\mathbf{x}} \left( u\_{\tilde{q}\_{\hat{j}}}^{R} \right)^{w\_{\hat{j}}} \right)^{\frac{1}{\tilde{\mathbf{x}}}} \\ \end{matrix} \right\}, \\\left\{ \begin{matrix} \left( \prod\_{j=1}^{\mathbf{x}} \left( 1 - \left( v\_{\tilde{q}\_{\hat{j}}}^{L} \right)^{q} \right)^{w\_{\hat{j}}} \end{matrix} \right\}^{\frac{1}{\tilde{\mathbf{x}}}} \begin{matrix} \left( u\_{\tilde{q}\_{\hat{j}}}^{R} \right)^{w\_{\hat{j}}} \\ 1 - \left( \prod\_{j=1}^{\mathbf{x}} \left( 1 - \left( v\_{\tilde{q}\_{\hat{j}}}^{R} \right)^{q} \right)^{\frac{1}{\tilde{\mathbf{x}}}} \end{matrix} \end{matrix} \right\} \right\} \tag{19}$$

Thereafter,

$$= \left\{ \begin{array}{l} \stackrel{\bigoplus}{1}\_{1 \leq i\_{1} < \cdots \leq i\_{t} \leq n} \left( \stackrel{\sum}{j\_{t}} \left( \boldsymbol{\mu}\_{i\_{t}} \right)^{w\_{i\_{j}}} \right)^{\frac{1}{2}} \\ = & \left\{ \sqrt[n]{1 - \prod\_{1 \leq i\_{1} < \cdots \leq i\_{t} \leq n} \left( 1 - \left( \prod\_{j=1}^{x} \left( \boldsymbol{\mu}\_{\boldsymbol{\widehat{q}}\_{i}}^{1} \right)^{w\_{i\_{j}}} \right)^{\frac{1}{2}} \right)}, \sqrt[n]{1 - \prod\_{1 \leq i\_{1} < \cdots \leq i\_{t} \leq n} \left( 1 - \left( \prod\_{j=1}^{x} \left( \boldsymbol{\mu}\_{\boldsymbol{\widehat{q}}\_{j}}^{R} \right)^{w\_{j\_{j}}} \right)^{\frac{1}{2}} \right)} \right] \\ = & \left\{ \prod\_{1 \leq i\_{1} < \cdots \leq i\_{t} \leq n} \left( \sqrt[n]{1 - \left( \prod\_{j=1}^{x} \left( 1 - \left( \boldsymbol{\mu}\_{\boldsymbol{\widehat{q}}\_{j}}^{1} \right)^{w\_{j\_{t}}} \right)^{\frac{1}{2}}} \right)^{\frac{1}{2}} \right)\_{1 \leq i\_{1} < \cdots \leq i\_{t} \leq n} \left( \sqrt[n]{1 - \left( \prod\_{j=1}^{x} \left( 1 - \left( \boldsymbol{\mu}\_{\boldsymbol{\widehat{q}}\_{j}}^{R} \right)^{w\_{j\_{t}}} \right)^{\frac{1}{2}}} \right) \right)} \end{array} \tag{20}$$

Furthermore,

*<sup>q</sup>*-RIVOFWHM(*x*) *<sup>w</sup>* (*q*1, *<sup>q</sup>*2,..., *<sup>q</sup>n*) = <sup>⊕</sup> <sup>1</sup>≤*<sup>i</sup>* 1<...<*ix*≤*n <sup>x</sup>* ⊗ *j*=1 *qi j wi j*  1 *x Cx n* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎢ ⎣ *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *uL qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* , *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *uR qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎣ ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − *vL qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − *vR qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎦ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (21)

Hence, Equation (16) is kept.

Then we need to prove that Equation (16) is a *q*-RIVOFN. We need to prove that 0 ≤ *μR <sup>Q</sup>*(*x*) *q* + *νR <sup>Q</sup>*(*x*) *q* ≤ 1. Let

$$\begin{aligned} \mu\_{\overline{Q}}^{\underline{R}}(\mathbf{x}) &= \sqrt[q]{1 - \left(\prod\_{1 \le i\_1 < \ldots < i\_k \le n} \left(1 - \left(\prod\_{j=1}^x \left(\mu\_{\overline{q}\_j}^{\mathbb{R}}\right)^{w\_{i\_j}}\right)^{\frac{x}{\overline{\pi}}}\right)\right)^{\frac{1}{\overline{C}\_n}}} \\ \nu\_{\overline{Q}}^{\underline{R}}(\mathbf{x}) &= \left(\prod\_{1 \le i\_1 < \ldots < i\_k \le n} \left(\sqrt[q]{1 - \left(\prod\_{j=1}^x \left(1 - \left(\nu\_{\overline{q}\_j}^{\mathbb{R}}\right)^q\right)^{w\_{i\_j}}\right)^{\frac{1}{\overline{\pi}}}}\right)\right)^{\frac{1}{\overline{C}\_n}} \end{aligned}$$

**Proof.**

$$\begin{split} 0 &\leq \left(\mu^{R}\_{Q}(\mathbf{x})\right)^{q} + \left(\nu^{R}\_{Q}(\mathbf{x})\right)^{q} \\ &= 1 - \left(\prod\_{1 \leq i\_{1} < \ldots < i\_{k} \leq n} \left(1 - \left(\prod\_{j=1}^{x} \left(\mu^{R}\_{\overline{q}\_{j}}\right)^{w\_{j}}\right)^{2}\right)\right)^{\frac{1}{1-q}} + \left(\prod\_{1 \leq i\_{1} < \ldots < i\_{k} \leq n} \left(1 - \left(\prod\_{j=1}^{x} \left(1 - \left(\nu^{R}\_{\overline{q}\_{j}}\right)^{q}\right)^{w\_{j}}\right)\right)^{\frac{1}{1-q}}\right) \\ &\leq 1 - \left(\prod\_{1 \leq i\_{1} < \ldots < i\_{k} \leq n} \left(1 - \left(\prod\_{j=1}^{x} \left(1 - \left(\nu^{R}\_{\overline{q}\_{j}}\right)^{q}\right)^{w\_{j}}\right)^{\frac{1}{1-q}}\right)\right)^{\frac{1}{1-q}} + \left(\prod\_{1 \leq i\_{1} < \ldots < i\_{k} \leq n} \left(1 - \left(\prod\_{j=1}^{x} \left(1 - \left(\nu^{R}\_{\overline{q}\_{j}}\right)^{q}\right)^{w\_{j}}\right)^{\frac{1}{1-q}}\right)^{\frac{1}{1-q}}\right)^{\frac{1}{1-q}} \\ &= 1 \end{split}$$

Therefore, 0 ≤ *μR <sup>Q</sup>*(*x*) *q* + *νR <sup>Q</sup>*(*x*) *q* ≤ 1 is maintained.

**Example 2.** *Let* ([0.5, 0.8], [0.4, 0.5]),([0.3, 0.5], [0.6, 0.7]),([0.5, 0.7], [0.2, 0.3]) *and* ([0.4, 0.8], [0.1, 0.2]) *be four q-RIVOFNs, and w* = (0.2, 0.1, 0.3, 0.4)*; in addition, suppose x* = 2, *q* = 3*. Then, according to Equation (16), we have*

$$=\begin{pmatrix}q\text{-IVON}\text{NH}\_{2}^{0,1}(\hat{q}\_{1},\hat{q}\_{2},\ldots,\hat{q}\_{n})=\frac{\psi\_{\text{out}}\left(\hat{s}\_{0}\left(\hat{s}\_{y}\right)^{\top}\right)^{\dagger}}{\mathbf{1}}\\\\\quad\times\left(\begin{bmatrix}\displaystyle\displaystyle\displaystyle\displaystyle\mathbbm{1}-\left(\left(1-\left(0.9^{\mathsf{L}}\otimes\boldsymbol{0}^{\mathsf{-2}\mathsf{d}}\otimes\boldsymbol{0}^{\mathsf{-2}\mathsf{d}}\right)^{\dagger}\right)\times\left(1-\left(0.9^{\mathsf{L}}\otimes\boldsymbol{0}^{\mathsf{-2}\mathsf{d}}\right)^{\dagger}\right)\times\left(1-\left(0.9^{\mathsf{z}\_{0}}\otimes\boldsymbol{0}^{\mathsf{-2}\mathsf{d}}\right)^{\dagger}\right)\\\quad\times\left(\left(1-\left(0.9^{\mathsf{L}}\otimes\boldsymbol{0}^{\mathsf{-2}\mathsf{d}}\right)^{\dagger}\right)\times\left(1-\left(0.9^{\mathsf{z}\_{0}}\otimes\boldsymbol{0}^{\mathsf{-2}\mathsf{d}}\right)^{\dagger}\right)\times\left(1-\left(0.9^{\mathsf{z}\_{0}}\otimes\boldsymbol{0}^{\mathsf{-2}\mathsf{d}}\right)^{\dagger}\right)^{\dagger}\\\quad\times\left(\begin{pmatrix}1-\left(1-\left(0.8^{\mathsf{L}}\otimes\boldsymbol{0}^{\mathsf{-2}\mathsf{d}}\right)^{\dagger}\right)\times\left(1-\left(0.8^{\mathsf{L}}\otimes\boldsymbol{0}^{\mathsf{-2}\mathsf{d}}\right)^{\dagger}\right)\times\left(1-\left(0.9^{\mathsf{z}\_{0}}\otimes\boldsymbol{0}^{\mathsf{-2}\mathsf{d}}\right)^{\dagger}\right)\\\quad\times\left(1-\left(1-\left(0.9^{\mathsf{L}}\otimes\boldsymbol{0}^{\mathsf{-2}\mathsf{d}}\right)^{\dagger}\right)^{\$$

The *q*-RIVOFWHM operator satisfies the following properties.

**Property 4.** *Monotonicity: let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*<sup>j</sup>* <sup>=</sup> 1, 2, ... , *<sup>n</sup>*) *and <sup>q</sup> <sup>j</sup>* = 4 *uL qj* , *uR qj* 5 , 4 *vL qj* , *vR qj* 5(*<sup>j</sup>* <sup>=</sup> 1, 2, ... , *<sup>n</sup>*) *be two sets of q-RIVOFNs. If <sup>u</sup><sup>L</sup> <sup>q</sup><sup>j</sup>* <sup>≤</sup> *uL qj* , *u<sup>R</sup> <sup>q</sup><sup>j</sup>* <sup>≤</sup> *uR qj* , *v<sup>L</sup> <sup>q</sup><sup>j</sup>* <sup>≥</sup> *vL qj and v<sup>L</sup> <sup>q</sup><sup>j</sup>* <sup>≥</sup> *vR qj hold for all j, then q*-RIVOFWHM(*x*) (*q*1, *<sup>q</sup>*2, ··· , *<sup>q</sup>n*) <sup>≤</sup> *<sup>q</sup>*-RIVOFWHM(*x*) - *q* 1, *q* 2, ··· , *<sup>q</sup> n* (22)

The proof is similar to *q*-RIVOFHM, so it is omitted here.

**Property 5.** *Boundedness: let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*j* = 1, 2, ... , *n*) *be a set of q-RIVOFNs. If <sup>q</sup>*<sup>+</sup> <sup>=</sup> *maxi uL qj* , *maxi uR qj* , *mini vL qj* , *mini vR qj and <sup>q</sup>*<sup>−</sup> <sup>=</sup> *mini uL qj* , *mini uR qj* , *maxi vL qj* , *maxi vR qj then <sup>q</sup>*<sup>−</sup> <sup>≤</sup> *<sup>q</sup>*-RIVOFWHM(*x*) (*q*1, *<sup>q</sup>*2, ··· , *<sup>q</sup>n*) <sup>≤</sup> *<sup>q</sup>*<sup>+</sup> (23)

From Theorem 2, we get

*q*-RIVOFWHM(*x*) - *q*− <sup>1</sup> , *<sup>q</sup>*<sup>−</sup> <sup>2</sup> , ··· , *<sup>q</sup>*<sup>−</sup> *n* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 min *uL qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* , *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 min *uR qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − max *vL qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − max *vR qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (24) *q*-RIVOFWHM(*x*) - *q*+ <sup>1</sup> , *<sup>q</sup>*<sup>+</sup> <sup>2</sup> , ··· , *<sup>q</sup>*<sup>+</sup> *n* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 max *uL qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* , *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 max *uR qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − min *vL qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − min *vR qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (25)

From Property 4, we get

$$
\hat{q}^{\cdots} \le q \text{-RIVOFWH}^{(x)}(\hat{q}\_1, \hat{q}\_2, \cdots, \hat{q}\_n) \le \hat{q}^+ \tag{26}
$$

It is obvious that the *q*-RIVOFWHM operator lacks the property of idempotency.

#### *3.3. The q-RIVOFDHM Operator*

Wu et al. [61] define the dual Hamy mean (DHM) operator.

*Symmetry* **2019**, *11*, 56

**Definition 7 [61].** *The DHM operator can be defined as:*

$$\text{DHM}^{(\text{x})}(\tilde{q}\_1, \tilde{q}\_2, \dots, \tilde{q}\_n) = \left( \prod\_{1 \le i\_1 < \dots < i\_r \le n} \left( \frac{\sum\_{j=1}^{\infty} \tilde{q}\_{i\_j}}{\infty} \right) \right)^{\frac{1}{C\_n^{\text{th}}}} \tag{27}$$

*where x is a parameter, and x* = 1, 2, ... , *n, i*1, *i*2, ... , *ix are x integer values taken from the set* {1, 2, . . . , *n*} *of k integer values; C<sup>x</sup> <sup>n</sup> denotes the binomial coefficient and C<sup>x</sup> <sup>n</sup>* = *<sup>n</sup>*! *x*!(*n*−*x*)! *.*

In this section, we will propose the *q*-rung interval-valued orthopair, fuzzy DHM (*q*-RIVOFDHM) operator.

**Definition 8.** *Let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*j* = 1, 2, ... , *n*) *be a set of q-RIVOFNs. The q-RIVOFDHM operator is*

$$q\text{-RIVOFDHM}^{(x)}(\tilde{q}\_1, \tilde{q}\_2, \dots, \tilde{q}\_n) = \left( \underset{1 \le i\_1 < \dots < i\_x \le n}{\odot} \left( \frac{\stackrel{\stackrel{\mathcal{X}}{\odot}}{\underset{j=1}^x} \tilde{q}\_{i\_j}}{\text{x}} \right) \right)^{\frac{1}{C\_n^x}} \tag{28}$$

**Theorem 3.** *Let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*j* = 1, 2, ... , *n*) *be a set of q-RIVOFNs. The fused value by using q-RIVOFDHM operators is also a q-RIVOFN, where*

*q*-RIVOFDHM(*x*) (*q*1, *<sup>q</sup>*2, ··· , *<sup>q</sup>n*) <sup>=</sup> ⎛ <sup>⎝</sup> <sup>⊗</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎝ *x* ⊕ *j*=1 *qi j x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎣ ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup> q* 1 − *x* ∏ *j*=1 1 − *uL qj q* <sup>1</sup> *x* ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup> q* 1 − *x* ∏ *j*=1 1 − *uR qj q* <sup>1</sup> *x* ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎦, ⎡ ⎢ ⎢ ⎢ ⎣ *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *vL qj q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* , *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *vR qj q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎦ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (29)

**Proof.**

$$\boldsymbol{\hat{\upleftrightarrow}}\_{j=1}^{\boldsymbol{x}} \boldsymbol{\tilde{q}}\_{l\_{j}} = \left\{ \left[ \sqrt[q]{1 - \prod\_{j=1}^{\boldsymbol{x}} \left( 1 - \left( \boldsymbol{u}\_{\overline{q}\_{j}}^{L} \right)^{q} \right)}, \sqrt[q]{1 - \prod\_{j=1}^{\boldsymbol{x}} \left( 1 - \left( \boldsymbol{u}\_{\overline{q}\_{j}}^{R} \right)^{q} \right)} \right], \left[ \prod\_{j=1}^{\boldsymbol{x}} \boldsymbol{v}\_{\overline{q}\_{j'}}^{L} \prod\_{j=1}^{\boldsymbol{x}} \boldsymbol{v}\_{\overline{q}\_{j}}^{R} \right] \right\} \tag{30}$$

Thus,

$$\frac{\stackrel{\circ}{q}\_{j=1}^{\frac{\pi}{4}}\tilde{q}\_{j}}{\stackrel{\circ}{\mathbf{x}}}=\left\{ \left[ \begin{array}{c} \sqrt[q]{1-\left(\prod\_{j=1}^{\pi}\left(1-\left(\boldsymbol{\mu}\_{\overline{q}\_{j}}^{\mathrm{L}}\right)^{q}\right)}^{\frac{1}{\pi}}\\ \hline \sqrt[q]{1-\left(\prod\_{j=1}^{\pi}\left(1-\left(\boldsymbol{\mu}\_{\overline{q}\_{j}}^{\mathrm{R}}\right)^{q}\right)\right)^{\frac{1}{\pi}}}^{\frac{1}{\pi}} \end{array} \right], \left[\left(\prod\_{j=1}^{\pi}\boldsymbol{v}\_{\overline{q}\_{j}}^{\mathrm{L}}\right)^{\frac{1}{\pi}}, \left(\prod\_{j=1}^{\pi}\boldsymbol{v}\_{\overline{q}\_{j}}^{\mathrm{R}}\right)^{\frac{1}{\pi}}\right] \right\}\tag{31}$$

Thereafter,

$$= \left\{ \begin{array}{l} \stackrel{\scriptstyle\otimes}{\mathbb{Q}}\_{1\leq i\_{1}<\cdots$$

Therefore,

*q*-RIVOFDHM(*x*) (*q*1, *<sup>q</sup>*2, ··· , *<sup>q</sup>n*) <sup>=</sup> ⎛ <sup>⎝</sup> <sup>⊗</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎝ *x* ⊕ *j*=1 *qi j x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎣ ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup> q* 1 − *x* ∏ *j*=1 1 − *uL qj q* <sup>1</sup> *x* ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup> q* 1 − *x* ∏ *j*=1 1 − *uR qj q* <sup>1</sup> *x* ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎦, ⎡ ⎢ ⎢ ⎢ ⎣ *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *vL qj q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* , *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *vR qj q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎦ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (33)

Hence, Equation (29) is kept.

Then, we need to prove that Equation (29) is a *q*-RIVOFN. We need to prove that 0 ≤ *μR <sup>Q</sup>*(*x*) *q* + *νR <sup>Q</sup>*(*x*) *q* ≤ 1. Let 1

*n*

$$\nu\_{\overline{Q}}^{R}(\mathbf{x}) = \underbrace{\left(\prod\_{1 \le i\_1 < \ldots < i\_k \le n} \sqrt[q]{1 - \left(\prod\_{j=1}^{\infty} \left(1 - \left(\mu\_{\overline{q}\_j}^R\right)^{\eta}\right)\right)^{\frac{1}{\eta}}}\right)^{\frac{1}{\binom{\eta}{\eta}}}}\_{\nu\_{\overline{Q}}(\mathbf{x}) = \sqrt[q]{1 - \left(\prod\_{1 \le i\_1 < \ldots < i\_k \le n} \left(1 - \left(\prod\_{j=1}^{\infty} \nu\_{\overline{q}\_j}^R\right)^{\frac{\eta}{\tau}}\right)\right)^{\frac{1}{\binom{\eta}{\tau}}}}$$

**Proof.**

$$\begin{split} 0 \leq & \left(\mu\_{\widehat{Q}}^{R}(\mathbf{x})\right)^{q} + \left(\nu\_{\widehat{Q}}^{R}(\mathbf{x})\right)^{q} \\ = & 1 - \left(\prod\_{1 \leq i\_{1} < \ldots < i\_{n} \leq n} \left(1 - \left(\prod\_{j=1}^{\mathbf{x}} w\_{\widehat{q}\_{j}}^{q}\right)^{\frac{q}{n}}\right)\right)^{\frac{1}{q\_{n}}} + \left(\prod\_{1 \leq i\_{1} < \ldots < i\_{n} \leq n} \left(1 - \left(\prod\_{j=1}^{\mathbf{x}} \left(1 - \left(u\_{\widehat{q}\_{j}}^{R}\right)^{q}\right)\right)^{\frac{1}{q\_{n}}}\right)\right)^{\frac{1}{q\_{n}}} \\ \leq & 1 - \left(\prod\_{1 \leq i\_{1} < \ldots < i\_{n} \leq n} \left(1 - \left(\prod\_{j=1}^{\mathbf{x}} \left(1 - \left(u\_{\widehat{q}\_{j}}^{R}\right)^{q}\right)\right)^{\frac{1}{q\_{n}}}\right)\right)^{\frac{1}{q\_{n}}} + \left(\prod\_{1 \leq i\_{1} < \ldots < i\_{n} \leq n} \left(1 - \left(\prod\_{j=1}^{\mathbf{x}} \left(1 - \left(u\_{\widehat{q}\_{j}}^{R}\right)^{q}\right)\right)^{\frac{1}{q\_{n}}}\right)\right)^{\frac{1}{q\_{n}}} \\ = & 1 - 1 \end{split}$$

Therefore, 0 ≤ *μR <sup>Q</sup>*(*x*) *q* + *νR <sup>Q</sup>*(*x*) *q* ≤ 1 is maintained. **Example 3.** *Let* ([0.5, 0.8], [0.4, 0.5]),([0.3, 0.5], [0.6, 0.7]),([0.5, 0.7], [0.2, 0.3]) *and* ([0.4, 0.8], [0.1, 0.2]) *be four q-RIVOFNs, and suppose x* = 2, *q* = 3*; then according to Equation (29), we have*

*q*-RIVOFDHM(*x*) (*q*1, *<sup>q</sup>*2, ··· , *<sup>q</sup>n*) <sup>=</sup> ⎛ <sup>⎝</sup> <sup>⊗</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎝ *x* ⊕ *j*=1 *qi j x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ 3 ⎛ ⎜⎜⎜⎜⎜⎜⎝ <sup>1</sup> − --<sup>1</sup> <sup>−</sup> 0.53 × - <sup>1</sup> <sup>−</sup> 0.3<sup>3</sup> 1 2 × <sup>1</sup> − --<sup>1</sup> <sup>−</sup> 0.5<sup>3</sup> × - <sup>1</sup> <sup>−</sup> 0.5<sup>3</sup> 1 2 × <sup>1</sup> − --<sup>1</sup> <sup>−</sup> 0.53 × - <sup>1</sup> <sup>−</sup> 0.43 1 2 × <sup>1</sup> − --<sup>1</sup> <sup>−</sup> 0.33 × - <sup>1</sup> <sup>−</sup> 0.53 1 2 × <sup>1</sup> − --<sup>1</sup> <sup>−</sup> 0.33 × - <sup>1</sup> <sup>−</sup> 0.43 1 2 × <sup>1</sup> − --<sup>1</sup> <sup>−</sup> 0.53 × - <sup>1</sup> <sup>−</sup> 0.43 1 2 ⎞ ⎟⎟⎟⎟⎟⎟⎠ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ 1 *C*2 4 , ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ 3 ⎛ ⎜⎜⎜⎜⎜⎜⎝ <sup>1</sup> − --<sup>1</sup> <sup>−</sup> 0.83 × - <sup>1</sup> <sup>−</sup> 0.5<sup>3</sup> 1 2 × <sup>1</sup> − --<sup>1</sup> <sup>−</sup> 0.8<sup>3</sup> × - <sup>1</sup> <sup>−</sup> 0.7<sup>3</sup> 1 2 × <sup>1</sup> − --<sup>1</sup> <sup>−</sup> 0.83 × - <sup>1</sup> <sup>−</sup> 0.83 1 2 × <sup>1</sup> − --<sup>1</sup> <sup>−</sup> 0.53 × - <sup>1</sup> <sup>−</sup> 0.73 1 2 × <sup>1</sup> − --<sup>1</sup> <sup>−</sup> 0.53 × - <sup>1</sup> <sup>−</sup> 0.83 1 2 × <sup>1</sup> − --<sup>1</sup> <sup>−</sup> 0.73 × - <sup>1</sup> <sup>−</sup> 0.83 1 2 ⎞ ⎟⎟⎟⎟⎟⎟⎠ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ 1 *C*2 4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 3 1 − ⎛ ⎝ 1 − (0.4 × 0.6) 3 2 × 1 − (0.4 × 0.2) 3 2 × 1 − (0.4 × 0.1) 3 2 × 1 − (0.6 × 0.2) 3 2 × 1 − (0.6 × 0.1) 3 2 × 1 − (0.2 × 0.1) 3 2 ⎞ ⎠ 1 *C*2 4 , 3 1 − ⎛ ⎝ 1 − (0.5 × 0.7) 3 2 × 1 − (0.5 × 0.3) 3 2 × 1 − (0.5 × 0.2) 3 2 × 1 − (0.7 × 0.3) 3 2 × 1 − (0.7 × 0.2) 3 2 × 1 − (0.3 × 0.2) 3 2 ⎞ ⎠ 1 *C*2 4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ( [0.4348, 0.7214 ], [0.3283, 0.4291 ])

The *q*-RIVOFDHM has the following three operators.

$$\mathbf{Property 6. Idempotency: if } \widetilde{q}\_{\widetilde{l}} = \left( \left[ u\_{\widetilde{l}\widetilde{q}}^{\mathcal{L}}, u\_{\widetilde{q}\widetilde{l}}^{\mathcal{R}} \right], \left[ v\_{\widetilde{q}\widetilde{l}}^{\mathcal{L}}, v\_{\widetilde{q}\widetilde{l}}^{\mathcal{R}} \right] \right) (j = 1, 2, \dots, n) \text{ are equal, then} $$

$$q\text{-RIVOFDHM}^{(x)}(\widetilde{q}\_{1}, \widetilde{q}\_{2}, \dots, \widetilde{q}\_{n}) = \widetilde{q} \tag{34}$$

**Proof.** Since *<sup>q</sup><sup>j</sup>* <sup>=</sup> *<sup>q</sup>* <sup>=</sup> *u<sup>L</sup> <sup>q</sup>*, *<sup>u</sup><sup>R</sup> q* , *vL q*, *vR q* , then *Symmetry* **2019**, *11*, 56

*q*-RIVOFDHM(*x*) (*q*1, *<sup>q</sup>*2, ··· , *<sup>q</sup>n*) <sup>=</sup> ⎛ <sup>⎝</sup> <sup>⊗</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎝ *x* ⊕ *j*=1 *qi j x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎣ ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup> q* 1 − *x* ∏ *j*=1 1 − *uL qj q* <sup>1</sup> *x* ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup> q* 1 − *x* ∏ *j*=1 1 − *uR qj q* <sup>1</sup> *x* ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎦, ⎡ ⎢ ⎢ ⎢ ⎣ *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *vL qj q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* , *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *vR qj q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎦ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎣ ⎛ ⎜⎝ ⎛ ⎝ *q* : 1 − 1 − *uL q q<sup>x</sup>* 1 *x* ⎞ ⎠ *Cx n* ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ ⎛ ⎝ *q* : 1 − 1 − *uR q q<sup>x</sup>* 1 *x* ⎞ ⎠ *Cx n* ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎦, ⎡ ⎢ ⎢ ⎣ *q* 1 − ⎛ ⎝ 1 − *v<sup>L</sup> q x <sup>q</sup> x <sup>C</sup><sup>x</sup> n* ⎞ ⎠ 1 *Cx n* , *q* 1 − ⎛ ⎝ 1 − *v<sup>R</sup> q x <sup>q</sup> x <sup>C</sup><sup>x</sup> n* ⎞ ⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎦, ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ = *u<sup>L</sup> <sup>q</sup>*, *<sup>u</sup><sup>R</sup> q* , *vL q*, *vR q* <sup>=</sup> *<sup>q</sup>*

**Property 7.** *Monotonicity: let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*<sup>j</sup>* <sup>=</sup> 1, 2, ... , *<sup>n</sup>*) *and <sup>q</sup> <sup>j</sup>* = 4 *uL qj* , *uR qj* 5 , 4 *vL qj* , *vR qj* 5(*<sup>j</sup>* <sup>=</sup> 1, 2, ... , *<sup>n</sup>*) *be two sets of q-RIVOFNs. If <sup>u</sup><sup>L</sup> <sup>q</sup><sup>j</sup>* <sup>≤</sup> *uL qj* , *u<sup>R</sup> <sup>q</sup><sup>j</sup>* <sup>≤</sup> *uR qj* , *v<sup>L</sup> <sup>q</sup><sup>j</sup>* <sup>≥</sup> *vL qj and v<sup>L</sup> <sup>q</sup><sup>j</sup>* <sup>≥</sup> *vR qj hold for all j, then*

$$q\text{-RIVOFDHM}^{(\text{x})}(\tilde{q}\_1, \tilde{q}\_2, \dots, \tilde{q}\_n) \le q\text{-RIVOFDHM}^{(\text{x})}(\tilde{q}\_1', \tilde{q}\_2', \dots, \tilde{q}\_n')\tag{35}$$

**Proof.** Given that *u<sup>L</sup> <sup>q</sup><sup>j</sup>* <sup>≤</sup> *uL qj* , we can obtain

$$\prod\_{j=1}^{x} \left( 1 - \left( u\_{\overline{q}\_j}^L \right)^q \right) \ge \prod\_{j=1}^{x} \left( 1 - \left( \left( u\_{\overline{q}\_j}^L \right)' \right)^q \right) \tag{36}$$

$$1 - \left(\prod\_{j=1}^{x} \left(1 - \left(u\_{\vec{q}\_j}^L\right)^q\right)\right)^{\frac{1}{x}} \le 1 - \left(\prod\_{j=1}^{x} \left(1 - \left(\left(u\_{\vec{q}\_j}^L\right)'\right)^q\right)\right)^{\frac{1}{x}} \tag{37}$$

Thereafter,

$$\left(\prod\_{1 \le l < \dots < l\_1 \le n} \sqrt[q]{1 - \left(\prod\_{j=1}^{\chi} \left(1 - \left(u\_{\tilde{q}\_l}^L\right)^q\right)\right)^{\frac{1}{q}}}\right)^{\frac{1}{q}} \le \left(\prod\_{1 \le l\_1 < \dots < l\_2 \le n} \sqrt[q]{1 - \left(\prod\_{j=1}^{\chi} \left(1 - \left(\left(u\_{\tilde{q}\_l}^L\right)^q\right)^q\right)\right)^{\frac{1}{q}}}\right)^{\frac{1}{q-1}}\tag{38}$$

That means that *u<sup>L</sup> <sup>q</sup>* <sup>≤</sup> *uL q* . Similarly, we can obtain *u<sup>R</sup> <sup>q</sup>* <sup>≤</sup> *uR q* , *v<sup>L</sup> <sup>q</sup>* <sup>≥</sup> *vL q* and *v<sup>L</sup> <sup>q</sup>* <sup>≥</sup> *vR q* . Thus, the proof is complete.

**Property 8.** *Boundedness: let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*j* = 1, 2, ... , *n*) *be a set of q-RIVOFNs. If <sup>q</sup>*<sup>+</sup> <sup>=</sup> *maxi uL qj* , *maxi uR qj* , *mini vL qj* , *mini vR qj and <sup>q</sup>*<sup>−</sup> <sup>=</sup> *mini uL qj* , *mini uR qj* , *maxi vL qj* , *maxi vR qj then*

$$
\hat{q}^- \le q \text{-RIVOFDHM}^{(\mathbf{x})}(\tilde{q}\_1, \tilde{q}\_2, \dots, \tilde{q}\_n) \le \tilde{q}^+ \tag{39}
$$

From Property 6,

$$\begin{array}{c} q\text{-RIVOFDHHM}^{(x)}\left(\hat{q}\_1^-,\hat{q}\_2^-,\cdots,\hat{q}\_n^-\right) = \hat{q}^-\\ q\text{-RIVOFDHHM}^{(x)}\left(\hat{q}\_1^+,\hat{q}\_2^+,\cdots,\hat{q}\_n^+\right) = \hat{q}^+ \end{array}$$

From Property 7,

$$\widetilde{q}^{-} \le q \text{-RIVOFDHM}^{(x)}(\widetilde{q}\_1, \widetilde{q}\_{2^\*} \cdots \cdot, \widetilde{q}\_n) \le \widetilde{q}^{+-}$$

#### *3.4. The q-RIVOFWDHM Operator*

In real MADM problems, it's of necessity to take attribute weights into account; we will propose the *q*-rung interval-valued orthopair fuzzy weighted DHM (*q*-RIVOFWDHM) operator in this chapter.

**Definition 9.** *Let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*j* = 1, 2, ... , *n*) *be a set of q-RIVOFNs, with their weight vector as wi* = (*w*1, *w*2,..., *wn*) *T, thereby satisfying wi* <sup>∈</sup> [0, 1] *and* <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *wi* = 1*. If*

$$q\text{-RIVOFWDHM}^{(\mathbf{x})}(\tilde{q}\_1, \tilde{q}\_2, \dots, \tilde{q}\_n) = \left( \underset{1 \le i\_1 < \dots < i\_t \le n}{\odot} \left( \frac{\stackrel{\text{\(\)}}{j-1} w\_{i\_j} \tilde{q}\_{i\_j}}{\text{x}} \right) \right)^{\frac{1}{\tilde{C}\_n}} \tag{40}$$

**Theorem 4.** *Let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*j* = 1, 2, ... , *n*) *be a set of q-RIVOFNs. The fused value by using q-RIVOFWDHM operators is also a q-RIVOFN, where*

*q*-RIVOFWDHM(*x*) (*q*1, *<sup>q</sup>*2, ··· , *<sup>q</sup>n*) <sup>=</sup> ⎛ <sup>⎝</sup> <sup>⊗</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎝ *x* ⊕ *j*=1 *wi j qi j x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎣ ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − *uL qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − *uR qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *vL qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* , *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *vR qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎦ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (41)

**Proof.**

$$w\_{\vec{i}\_{j}}\widetilde{q}\_{\vec{i}\_{j}} = \left\{ \left[ \sqrt[q]{1 - \left( 1 - \left( u\_{\frac{\vec{i}}{q\_{\vec{i}}}} \right)^{q\_{\vec{i}}} \right)^{w\_{\vec{i}\_{j}}}}, \sqrt[q]{1 - \left( 1 - \left( u\_{\frac{\vec{a}}{q\_{\vec{i}}}} \right)^{q} \right)^{w\_{\vec{i}\_{j}}}} \right], \left[ \left( v\_{\frac{\vec{a}}{q\_{\vec{j}}}}^{L} \right)^{w\_{\vec{i}\_{j}}}, \left( v\_{\frac{\vec{a}}{q\_{\vec{j}}}}^{R} \right)^{w\_{\vec{i}\_{j}}} \right] \right\} \tag{42}$$

Thus,

$$\begin{aligned} \left( \mathop{\rm tr}\_{j=1}^{\frac{\mathbf{x}}{\mathbf{x}\_{j}}} \left( w\_{\bar{i}\_{j}} \widetilde{q}\_{\bar{i}\_{j}} \right) = \left\{ \left[ \mathop{\rm tr}\_{j=1}^{\frac{\mathbf{x}}{\mathbf{x}}} \left( 1 - \left( \boldsymbol{\mu}\_{\bar{i}\_{j}}^{\mathbf{L}} \right)^{q} \right)^{w\_{\bar{i}\_{j}}}, \sqrt[q]{1 - \prod\_{j=1}^{\frac{\mathbf{x}}{\mathbf{x}}} \left( 1 - \left( \boldsymbol{\mu}\_{\bar{i}\_{j}}^{\mathbf{R}} \right)^{q} \right)^{w\_{\bar{i}\_{j}}}} \right], \right. \right. \end{aligned} \tag{43}$$

Therefore,

$$\frac{\begin{array}{c} \mathop{\rm t}^{\boldsymbol{x}}\_{\boldsymbol{j}} \left( \boldsymbol{w}\_{i} \widetilde{\boldsymbol{\eta}}\_{i} \right) \\ \hline \boldsymbol{\chi} \\ \end{array}}{\mathop{\rm t}} = \left\{ \left[ \begin{array}{c} \left[ \sqrt{1 - \left( \prod\_{j=1}^{\boldsymbol{x}} \left( 1 - \left( \boldsymbol{u}\_{\widetilde{\boldsymbol{q}}\_{j}}^{\boldsymbol{\bot}} \right)^{\boldsymbol{w}\_{i}} \right)^{\boldsymbol{w}\_{i}} \right)^{\frac{1}{\boldsymbol{\varpi}}}}, \sqrt{1 - \left( \prod\_{j=1}^{\boldsymbol{x}} \left( 1 - \left( \boldsymbol{u}\_{\widetilde{\boldsymbol{q}}\_{j}}^{\boldsymbol{R}} \right)^{\boldsymbol{w}\_{i}} \right)^{\frac{1}{\boldsymbol{\varpi}}} \right)^{\frac{1}{\boldsymbol{\varpi}}}} \right] \; \right. \\ \left[ \left( \prod\_{j=1}^{\boldsymbol{x}} \left( \boldsymbol{v}\_{\widetilde{\boldsymbol{q}}\_{j}}^{\boldsymbol{L}} \right)^{\boldsymbol{w}\_{j}} \right)^{\frac{1}{\boldsymbol{\varpi}}}, \left( \prod\_{j=1}^{\boldsymbol{x}} \left( \boldsymbol{v}\_{\widetilde{\boldsymbol{q}}\_{j}}^{\boldsymbol{R}} \right)^{\boldsymbol{w}\_{i}} \right)^{\frac{1}{\boldsymbol{\varpi}}} \right] \; \end{array} \right\} \; \tag{44}$$

Thereafter,

$$= \left\{ \begin{array}{l} \text{\$\n\n\n\$} \\ \text{\$\n\n\n\$} \text{\$\n\n\n\$} \\ = \left\{ \begin{array}{l} \text{\$\n\n\n\$} \\ 1 \le i\_{1} < \ldots < i\_{L} \le n \end{array} \left( \sqrt[4]{1 - \left( \prod\_{j=1}^{\infty} \left( 1 - \left( {}\_{\text{i}\_{\text{i}}}^{L} \right)^{q} \right)^{w\_{j}}} \right)^{\frac{1}{2}}} \right) \\ \left\{ \begin{array}{l} \text{\$\n\n\n\$} \\ 1 - \prod\_{1 \le i\_{1} < \ldots < i\_{L} \le n} \left( 1 - \left( \prod\_{j=1}^{\infty} \left( {}\_{\text{i}\_{\text{i}}}^{L} \right)^{w\_{j}} \right)^{\frac{1}{2}} \right), \end{array} \left( \sqrt[4]{1 - \left( {}\_{\text{i}\_{\text{i}}}^{L} \left( 1 - \left( {}\_{\text{i}\_{\text{i}}}^{L} \right)^{q} \right)^{w\_{j}} \right)^{\frac{1}{2}}} \right) \\ \left\{ \begin{array}{l} \text{\$\n\n\n} \\ 1 - \prod\_{1 \le i\_{1} < \ldots < i\_{L} \le n} \left( 1 - \left( {}\_{\text{i}\_{\text{i}}}^{L} \left( {}\_{\text{i}\_{\text{i}}}^{L} \right)^{w\_{j}} \right)^{\frac{1}{2}} \right) \\ \end{array} \right\} \end{array} \right\} \tag{45}$$

Furthermore,

*q*-RIVOFWDHM(*x*) (*q*1, *<sup>q</sup>*2, ··· , *<sup>q</sup>n*) <sup>=</sup> ⎛ <sup>⎝</sup> <sup>⊗</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎝ *x* ⊕ *j*=1 *wi j qi j x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎣ ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − *uL qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − *uR qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *vL qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* , *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 *vR qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎦ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (46)

Hence, Equation (41) is kept.

Then, we need to prove that Equation (41) is a *q*-RIVOFN. We need to prove that 0 ≤ *μR <sup>Q</sup>*(*x*) *q* + *νR <sup>Q</sup>*(*x*) *q* ≤ 1. Let

$$\begin{aligned} \mu\_{\boldsymbol{Q}}^{\boldsymbol{R}}(\boldsymbol{x}) &= \underbrace{\prod\_{1 \le i\_1 < \ldots < i\_x \le n} \left( \sqrt[q]{1 - \left( \prod\_{j=1}^x \left( 1 - \left( \boldsymbol{u}\_{\overline{q}\_j}^{\boldsymbol{R}} \right)^{q} \right)^{\boldsymbol{w}\_j} \right)^{\frac{1}{\boldsymbol{z}}}}\_{\boldsymbol{\nu}\_{\boldsymbol{Q}}^{\boldsymbol{R}}(\boldsymbol{x}) = \sqrt[q]{1 - \left( \prod\_{j=1}^x \left( 1 - \left( \prod\_{j=1}^x \left( \boldsymbol{v}\_{\overline{q}\_j}^{\boldsymbol{R}} \right)^{\boldsymbol{w}\_j} \right)^{\frac{1}{\boldsymbol{z}}} \right) \right)^{\frac{1}{\boldsymbol{c}\_n^{\boldsymbol{R}}}}} \end{aligned}$$

**Proof.**

$$\begin{split} 0 \le & \left(\mu^{R}\_{Q}(\mathbf{x})\right)^{d} + \left(\nu^{R}\_{Q}(\mathbf{x})\right)^{d} \\ = & \left(\prod\_{1 \le i\_{1} < \ldots < i\_{L} \le n} \left(1 - \left(\prod\_{j=1}^{x} \left(1 - \left(\mu^{R}\_{\vec{q}\_{j}}\right)^{q}\right)^{w\_{j}}\right)\right)^{\frac{1}{1-\xi}}\right) + 1 - \left(\prod\_{1 \le i\_{1} < \ldots < i\_{L} \le n} \left(1 - \left(\prod\_{j=1}^{x} \left(\nu^{R}\_{\vec{q}\_{j}}\right)^{w\_{j}}\right)^{2}\right)\right)^{\frac{1}{1-\xi}} \\ \le & \left(\prod\_{1 \le i\_{1} < \ldots < i\_{L} \le n} \left(1 - \left(\prod\_{j=1}^{x} \left(1 - \left(\mu^{R}\_{\vec{q}\_{j}}\right)^{q}\right)^{w\_{j}}\right)^{\frac{1}{2}}\right)\right)^{\frac{1}{1-\xi}} + 1 - \left(\prod\_{1 \le i\_{1} < \ldots < i\_{L} \le n} \left(1 - \left(\prod\_{j=1}^{x} \left(1 - \left(\mu^{R}\_{\vec{q}\_{j}}\right)^{q}\right)^{w\_{j}}\right)^{\frac{1}{1-\xi}}\right)\right)^{\frac{1}{1-\xi}} \\ = & 1 - 1 \end{split}$$

Therefore, 0 ≤ *μR <sup>Q</sup>*(*x*) *q* + *νR <sup>Q</sup>*(*x*) *q* ≤ 1 is maintained.

**Example 4.** *Let* ([0.5, 0.8], [0.4, 0.5]),([0.3, 0.5], [0.6, 0.7]),([0.5, 0.7], [0.2, 0.3]) *and* ([0.4, 0.8], [0.1, 0.2]) *be four q-RIVOFNs; suppose x* = 2, *q* = 3*, and ω* = (0.2, 0.1, 0.3, 0.4)*. Then, based on Equation (41), we can get*

*q*-RIVOFWDHM(*x*) (*q*1, *<sup>q</sup>*2, ··· , *<sup>q</sup>n*) <sup>=</sup> ⎛ <sup>⎝</sup> <sup>⊗</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎝ *x* ⊕ *j*=1 *wi j qi j x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* = ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ 3 ⎛ ⎜⎜⎜⎜⎜⎜⎝ 1 − - <sup>1</sup> − 0.5<sup>3</sup> 0.2 <sup>×</sup> - <sup>1</sup> − 0.3<sup>3</sup> 0.1<sup>1</sup> 2 × 1 − - <sup>1</sup> − 0.53 0.2 <sup>×</sup> - <sup>1</sup> − 0.5<sup>3</sup> 0.3<sup>1</sup> 2 × 1 − - <sup>1</sup> − 0.53 0.2 <sup>×</sup> - <sup>1</sup> − 0.43 0.4<sup>1</sup> 2 × 1 − - <sup>1</sup> − 0.3<sup>3</sup> 0.1 <sup>×</sup> - <sup>1</sup> − 0.5<sup>3</sup> 0.3<sup>1</sup> 2 × 1 − - <sup>1</sup> − 0.33 0.1 <sup>×</sup> - <sup>1</sup> − 0.43 0.4<sup>1</sup> 2 × 1 − - <sup>1</sup> − 0.5<sup>3</sup> 0.3 <sup>×</sup> - <sup>1</sup> − 0.4<sup>3</sup> 0.4<sup>1</sup> 2 ⎞ ⎟⎟⎟⎟⎟⎟⎠ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ 1 *C*2 4 , ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ 3 ⎛ ⎜⎜⎜⎜⎜⎜⎝ 1 − - <sup>1</sup> − 0.8<sup>3</sup> 0.2 <sup>×</sup> - <sup>1</sup> − 0.5<sup>3</sup> 0.1<sup>1</sup> 2 × 1 − - <sup>1</sup> − 0.83 0.2 <sup>×</sup> - <sup>1</sup> − 0.7<sup>3</sup> 0.3<sup>1</sup> 2 × 1 − - <sup>1</sup> − 0.83 0.2 <sup>×</sup> - <sup>1</sup> − 0.83 0.4<sup>1</sup> 2 × 1 − - <sup>1</sup> − 0.5<sup>3</sup> 0.1 <sup>×</sup> - <sup>1</sup> − 0.7<sup>3</sup> 0.3<sup>1</sup> 2 × 1 − - <sup>1</sup> − 0.53 0.1 <sup>×</sup> - <sup>1</sup> − 0.83 0.4<sup>1</sup> 2 × 1 − - <sup>1</sup> − 0.7<sup>3</sup> 0.3 <sup>×</sup> - <sup>1</sup> − 0.8<sup>3</sup> 0.4<sup>1</sup> 2 ⎞ ⎟⎟⎟⎟⎟⎟⎠ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ 1 *C*2 4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 3 1 − ⎛ ⎜⎜⎝ 1 − - 0.40.2 <sup>×</sup> 0.60.1 <sup>3</sup> 2 × 1 − - 0.40.2 <sup>×</sup> 0.20.3 <sup>3</sup> 2 × 1 − - 0.40.2 <sup>×</sup> 0.10.4 <sup>3</sup> 2 × 1 − - 0.60.1 <sup>×</sup> 0.20.3 <sup>3</sup> 2 × 1 − - 0.60.1 <sup>×</sup> 0.10.4 <sup>3</sup> 2 × 1 − - 0.20.3 <sup>×</sup> 0.10.4 <sup>3</sup> 2 ⎞ ⎟⎟⎠ 1 *C*2 4 , 3 1 − ⎛ ⎜⎜⎝ 1 − - 0.50.2 <sup>×</sup> 0.70.1 <sup>3</sup> 2 × 1 − - 0.50.2 <sup>×</sup> 0.30.3 <sup>3</sup> 2 × 1 − - 0.50.2 <sup>×</sup> 0.20.4 <sup>3</sup> 2 × 1 − - 0.70.1 <sup>×</sup> 0.30.3 <sup>3</sup> 2 × 1 − - 0.70.1 <sup>×</sup> 0.20.4 <sup>3</sup> 2 × 1 − - 0.30.3 <sup>×</sup> 0.20.4 <sup>3</sup> 2 ⎞ ⎟⎟⎠ 1 *C*2 4 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ = ([0.2819, 0.4954], [0.7249, 0.7855])

We will then study some precious properties of *q*-RIVOFWDHM operator.

**Property 9.** *Monotonicity: let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*<sup>j</sup>* <sup>=</sup> 1, 2, ... , *<sup>n</sup>*) *and <sup>q</sup> <sup>j</sup>* = 4 *uL qj* , *uR qj* 5 , 4 *vL qj* , *vR qj* 5(*<sup>j</sup>* <sup>=</sup> 1, 2, ... , *<sup>n</sup>*) *be two sets of q-RIVOFNs. If <sup>u</sup><sup>L</sup> <sup>q</sup><sup>j</sup>* <sup>≤</sup> *uL qj* , *u<sup>R</sup> <sup>q</sup><sup>j</sup>* <sup>≤</sup> *uR qj* , *v<sup>L</sup> <sup>q</sup><sup>j</sup>* <sup>≥</sup> *vL qj and v<sup>L</sup> <sup>q</sup><sup>j</sup>* <sup>≥</sup> *vR qj hold for all j, then q*-RIVOFWDHM(*x*) (*q*1, *<sup>q</sup>*2, ··· , *<sup>q</sup>n*) <sup>≤</sup> *<sup>q</sup>*-RIVOFWDHM(*x*) - *q* 1, *q* 2, ··· , *<sup>q</sup> n* (47)

This proof is similar to *q*-RIVOFDHM, so it is omitted here.

**Property 10.** *(Boundedness) Let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*j* = 1, 2, ... , *n*) *be a set of q-RIVOFNs. If <sup>q</sup>*<sup>+</sup> <sup>=</sup> *maxi uL qj* , *maxi uR qj* , *mini vL qj* , *mini vR qj and <sup>q</sup>*<sup>−</sup> <sup>=</sup> *mini uL qj* , *mini uR qj* , *maxi vL qj* , *maxi vR qj then <sup>q</sup>*<sup>−</sup> <sup>≤</sup> *<sup>q</sup>*-RIVOFWDHM(*x*) (*q*1, *<sup>q</sup>*2, ··· , *<sup>q</sup>n*) <sup>≤</sup> *<sup>q</sup>*<sup>+</sup> (48)

From Theorem 4, we get

*q*-RIVOFWDHM(*x*) - *q*− <sup>1</sup> , *<sup>q</sup>*<sup>−</sup> <sup>2</sup> , ··· , *<sup>q</sup>*<sup>−</sup> *n* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − min *uL qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − min *uR qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 max *vL qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* , *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 max *vR qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (49) *q*-RIVOFWDHM(*x*) - *q*+ <sup>1</sup> , *<sup>q</sup>*<sup>+</sup> <sup>2</sup> , ··· , *<sup>q</sup>*<sup>+</sup> *n* = ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − max *uL qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* , ⎛ ⎜⎝ <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ ⎜⎝ *q* 1 − *x* ∏ *j*=1 1 − max *uR qj qwi j* 1 *x* ⎞ ⎟⎠ ⎞ ⎟⎠ 1 *Cx n* ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ *q* 1 − ⎛ <sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>* ⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup> *x* ∏ *j*=1 min *vL qj wi j q x* ⎞ ⎠ ⎞ ⎠ 1 *Cx n* , ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (50)

From Property 9, we get

⎢ ⎢ ⎢ ⎢ ⎣

*q*

1 − ⎛

<sup>⎝</sup> <sup>∏</sup> <sup>1</sup>≤*i*1<...<*ix*≤*<sup>n</sup>*

⎛ <sup>⎝</sup><sup>1</sup> <sup>−</sup>

$$
\hat{q}^- \le q \text{-RIVOFWDHM}^{(x)}(\hat{q}\_1, \hat{q}\_2, \dots, \hat{q}\_n) \le \hat{q}^+ \tag{51}
$$

1 *Cx n*

⎥ ⎥ ⎥ ⎥ ⎦

It is obvious that the *q*-RIVOFWDHM operator is short of the property of idempotency.

 *x* ∏ *j*=1 min *vR qj wi j q x* ⎞ ⎠ ⎞ ⎠

#### **4. Application of Green Supplier Selection**

#### *4.1. Numerical Example*

With the rapid development of economic globalization, and the growing enterprise competition environment, the competition between modern enterprises has become the competition between supply chains. The diversity of the people consuming is increasing, and the new product life cycles are getting shorter. The volatility of the demand market and from external factors drives enterprises for effective supply chain integration and management, as well as strategic alliances with other enterprises to enhance core competitiveness and resist external risk. The key measure to achieving this goal is supplier selection. Therefore, the supplier selection problem has gained a lot of attention, whether in regard to supply chain management theory or in actual production management problems [62–70]. In order to illustrate our proposed method in this article, we provide a numerical example for selecting green suppliers in green supply chain management using *q*-RIVOFNs. There is a panel with five possible green suppliers in green supply chain management to select: *<sup>Q</sup>i*(*<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4, 5). The experts select four attributes to evaluate the five possible green suppliers: (1) C1 is the product quality factor; (2) C2 is the environmental factors; (3) C3 is the delivery factor; and (4) C4 is the price factor. The five possible green suppliers *<sup>Q</sup>i*(*<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4, 5) are to be evaluated by the decision maker using the *q*-RIVOFNs, under the above four attributes (whose weighting vector *ω* = (0.3, 0.2, 0.3, 0.2), and expert weighting vector *ω* = (0.2, 0.2, 0.6)) which are listed in Tables 1–3.

**Table 1.** The *q*-RIVOFN decision matrix 1 (*R*1) by expert one.


**Table 2.** The *q*-RIVOFN decision matrix 1 (*R*2) by expert two.


**Table 3.** The *q*-RIVOFN decision matrix 1 (*R*3) by expert three.


In the following, we utilize the approach developed to select green suppliers in green supply chain management.

**Step 1.** According to *<sup>q</sup>*-RIVOFNs *<sup>q</sup>ij*(*<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4, 5, *<sup>j</sup>* <sup>=</sup> 1, 2, 3, 4), we can aggregate all *<sup>q</sup>*-RIVOFNs *<sup>q</sup>ij* by using the *<sup>q</sup>*-RIVOFWA (*q*-RIVOFWG) operator, to get the overall *<sup>q</sup>*-RIVOFNs *<sup>Q</sup>i*(*<sup>i</sup>* <sup>=</sup> 1, 2, 3, 4, 5) of the green suppliers *<sup>Q</sup>i*. Then, the fused values are given in Table 4. (Let *<sup>q</sup>* <sup>=</sup> 3).

**Definition 10.** *Let <sup>q</sup><sup>j</sup>* <sup>=</sup> *u<sup>L</sup> qj* , *u<sup>R</sup> qj* , *vL qj* , *v<sup>R</sup> qj* (*j* = 1, 2, ... , *n*) *be a set of q-RIVOFNs, with their weight vector as wi* = (*w*1, *w*2,..., *wn*) *T, thereby satisfying wi* <sup>∈</sup> [0, 1] *and* <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *wi* = 1*. Then we can obtain*

$$\begin{split} & q \text{-RVOFWA} \left( \widetilde{q}\_{1}, \widetilde{q}\_{2}, \ldots, \widetilde{q}\_{n} \right) = \sum\_{j=1}^{n} w\_{j} \widetilde{q}\_{j} \\ &= \left\langle \left[ \sqrt{1 - \prod\_{j=1}^{n} \left( 1 - u\_{\widetilde{q}\_{j}}^{L} \right)^{w\_{j}}}, \sqrt[3]{1 - \prod\_{j=1}^{n} \left( 1 - u\_{\widetilde{q}\_{j}}^{R} \right)^{w\_{j}}} \right], \left[ \prod\_{j=1}^{n} \left( v\_{\widetilde{q}\_{j}}^{L} \right)^{w\_{j}}, \prod\_{j=1}^{n} \left( v\_{\widetilde{q}\_{j}}^{R} \right)^{w\_{j}} \right] \right\rangle \end{split} \tag{52}$$

$$\begin{split} &q\text{-RIVOFWG}(\tilde{q}\_{1},\tilde{q}\_{2},\ldots,\tilde{q}\_{l}) = \prod\_{j=1}^{n} \left(\tilde{q}\_{j}\right)^{w\_{j}} \\ &= \left\langle \left[\prod\_{j=1}^{n} \left(u\_{\tilde{q}\_{j}}^{\perp}\right)^{w\_{j}}, \prod\_{j=1}^{n} \left(u\_{\tilde{q}\_{j}}^{\tilde{R}}\right)^{w\_{j}}\right], \left[\sqrt[q]{1 - \prod\_{j=1}^{n} \left(1 - v\_{\tilde{q}\_{j}}^{\perp}\right)^{w\_{j}}}, \sqrt[q]{1 - \prod\_{j=1}^{n} \left(1 - v\_{\tilde{q}\_{j}}^{\mathbb{R}}\right)^{w\_{j}}}\right] \right\rangle \end{split} \tag{53}$$

**Table 4.** The fused results from the *q*-RIVOFWA operator.


**Step 2.** Based on Table 4, we can fuse all *<sup>q</sup>*-RIVOFNs *<sup>q</sup>ij* by the *<sup>q</sup>*-RIVOFWHM (*q*-RIVOFWDHM) operator to get the results of *q*-RIVOFNs. Let *x* = 2, then the fused values are given in Table 5.

**Table 5.** The fused values of the *q*-rung interval-valued orthopair, fuzzy weighted Hamy mean (*q*-RIVOFWHM) and the *q*-rung interval-valued orthopair, fuzzy weighted dual Hamy mean (*q*-RIVOFWDHM)) operators.


**Step 3.** Based on the fused values given in Table 5, and the score functions of *q*-RIVOFNs, the green suppliers' scores are shown in Table 6.


**Table 6.** The score values *s Qi* of the green suppliers.

**Step 4.** Rank all the alternatives by the values of Table 6, and the ordering results are shown in Table 7. Obviously, the best selection is *<sup>Q</sup>*3.


**Table 7.** Ordering of the green suppliers.

#### *4.2. Influence of the Parameter x*

In order to show the effects on the ranking results, by changing parameters of *x* in the *q*-RIVOFWHM (*q*-RIVOFWDHM) operators, all of the results are shown in Tables 8 and 9. (Let *q* = 3).

**Table 8.** Ordering results for different *x* values by the *q*-RIVOFWHM operator.


**Table 9.** Ordering results for different *x* values by the *q*-RIVOFWDHM operator.


#### *4.3. Influence of the Parameter q*

In order to show the effects on the ranking results by changing the parameters of *q* in the *q*-RIVOFWHM (*q*-RIVOFWDHM) operators, all of the results are shown in Tables 10 and 11. (Let *x* = 2).


**Table 10.** Ordering results for different *q* by the *q*-RIVOFWHM operator.

**Table 11.** Ordering results for different *q* by the *q*-RIVOFWDHM operator.


#### *4.4. Comparative Analysis*

In this chapter, we compare the *q*-RIVOFWHM and *q*-RIVOFWDHM operators with the *q*-RIVOFWA and *q*-RIVOFWG operators. The comparative results are shown in Table 12.



From above, we can see that we get the same optimal green suppliers, which shows the practicality and effectiveness of the proposed approaches. However, the *q*-RIVOFWA operator and *q*-RIVOFWG operator do not consider the information about the relationship between arguments being aggregated, and thus cannot eliminate the influence of unfair arguments on decision results. Our proposed *q*-RIVOFWHM and *q*-RIVOFWDHM operators consider the information about the relationship among arguments being aggregated.

At the same time, Liu and Wang [54] develop the *q*-rung orthopair, fuzzy weighted averaging (*q*-ROFWA) operator, as well as the *q*-rung orthopair, fuzzy weighted geometric (*q*-ROFWG) operator. Liu and Liu [55] propose some *q*-rung orthopair, fuzzy Bonferroni mean (*q*-ROFBM) aggregation operators. Wei et al. [56] define the generalized Heronian mean (GHM) operator to present some aggregation operators, and apply them into MADM problems. Wei et al. [57] define some *q*-rung orthopair, fuzzy Maclaurin symmetric mean operators. However, all of these operators can only deal with *q*-rung orthopair fuzzy sets (*q*-ROFSs), and cannot deal with *q*-rung interval-valued orthopair fuzzy sets (*q*-RIVOFSs). The main contribution of this paper is to study the MAGDM problems based on the *q*-rung interval-valued orthopair fuzzy sets (*q*-RIVOFSs), and to utilize the Hamy mean

(HM) operator, weighted Hamy mean (WHM) operator, dual Hamy mean (DHM) operator, and weighted dual Hamy mean (WDHM) operator, to develop some Hamy mean aggregation operators with *q*-RIVOFNs.

#### **5. Conclusions**

In this paper, we study the MAGDM problems with *q*-RIVOFNs. Then, we utilize the Hamy mean (HM) operator, weighted Hamy mean (WHM) operator, dual Hamy mean (DHM) operator, and weighted dual Hamy mean (WDHM) operator, in order to develop some Hamy mean aggregation operators with *q*-RIVOFNs. The prominent characteristic of each of these proposed operators is studied. Then, we have utilized these operators to develop some approaches to solve the MAGDM problems with *q*-RIVOFNs. Finally, a practical example for green supplier selection is given to show the developed approach. Using the illustrated example, we have roughly shown the effects on the ranking results by changing parameters in the *q*-RIVOFWHM (*q*-RIVOFWDHM) operators. In the future, the application of the proposed fused operators of *q*-RIVOFNs needs to be explored in decision making [71–74], risk analysis [75,76], and many other fields under uncertain environments [77–81].

**Author Contributions:** J.W., H.G., G.W. and Y.W. conceived and worked together to achieve this work, J.W. compiled the computing program by Excel and analyzed the data, J.W. and G.W. wrote the paper. Finally, all the authors have read and approved the final manuscript.

**Funding:** The work was supported by the National Natural Science Foundation of China under Grant No. 71571128 and the Humanities and Social Sciences Foundation of Ministry of Education of the People's Republic of China (17XJA630003) and the Construction Plan of Scientific Research Innovation Team for Colleges and Universities in Sichuan Province (15TD0004).

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **New Analytic Solutions of Queueing System for Shared–Short Lanes at Unsignalized Intersections**

**Ilija Tanackov 1,\*, Darko Dragi´c 1, Siniša Sremac 1, Vuk Bogdanovi´c 1, Bojan Mati´c <sup>1</sup> and Milica Milojevi´c <sup>2</sup>**


Received: 24 November 2018; Accepted: 25 December 2018; Published: 6 January 2019

**Abstract:** Designing the crossroads capacity is a prerequisite for achieving a high level of service with the same sustainability in stochastic traffic flow. Also, modeling of crossroad capacity can influence on balancing (symmetry) of traffic flow. Loss of priority in a left turn and optimal dimensioning of shared-short line is one of the permanent problems at intersections. A shared–short lane for taking a left turn from a priority direction at unsignalized intersections with a homogenous traffic flow and heterogeneous demands is a two-phase queueing system requiring a first in–first out (FIFO) service discipline and single-server service facility. The first phase (short lane) of the system is the queueing system M(*pλ*)/M(*μ*)/1/∞, whereas the second phase (shared lane) is a system with a binomial distribution service. In this research, we explicitly derive the probability of the state of a queueing system with a short lane of a finite capacity for taking a left turn and shared lane of infinite capacity. The presented formulas are under the presumption that the system is Markovian, i.e., the vehicle arrivals in both the minor and major streams are distributed according to the Poisson law, and that the service of the vehicles is exponentially distributed. Complex recursive operations in the two-phase queueing system are explained and solved in manuscript.

**Keywords:** sustainability; left turn; intersections; lane capacity

#### **1. Introduction**

The initial considerations of the queuing systems for shared–short lanes at unsignalized intersections were based on the proven procedure by Harders [1], where the lengths of the short lanes are considered either as infinite or zero. In his paper, Harders had presented a limiting analytic frame of the queuing system.

Wu [2] used a pure queuing system, considering in detail the Markovian and non-Markovian systems depending on the distribution of the service, in both the steady and unsteady states of the working regimes of an unsignalized intersection. Wu noticed that the relation between the vehicles in the minor stream of a shared–short lane introduces very complex recursive operations in the queuing system, especially for left turns.

A basic problem while dimensioning shared–short Lanes is the occurrence of the "short lane domino effect" phenomena, queue overflow/short lane saturation [2] and inevitable consequences, time delay. This phenomenon is inevitable when demand exceeds its capacity. Increasing the capacity of road engineering according to Li et al. [3] has become an important way of solving traffic problems. Obligatory consequence of saturation is [4] increases in fuel consumption and [5] increased air pollution at intersections [6].

For now research on signalized intersectionsare dominant. Applying queuing theory in solutions for signalized intersections has a classic theme status and tradition longer than 60 years [7,8], with developed analytical models car-following models, macro traffic flow models, complex networks approaches, cellular automata models, traffic sensing technologies-based approaches, etc. [9]. Apart from Markovian queuing systems non-Markovian queuing systems can be found on signalized intersections [10]. Key spot in these solutions is presented in the form of shared–short Lanes for the left turn [11,12].

The solution of Shared–Short Lanes optimization for unsignalized intersections [13] is more difficult than it is for signalized intersections. Detailed explanation for different analytical approach has been given by Nielsen, Frederiksen and Simonsen [14]. Due to sequential distribution of priority, on signalized intersections deterministic solutions can be applied as well [15]. A probabilistic solution in deterministic time sequences of signalized intersection [16] and right turn solutions [17] cannot be applied for left turn. Unlike unsignalized intersections, on signalized intersections solutions can be found even under conditions of great variations of capacity [18]. One of the basic reasons is the relation between priorities [19].

A homogenous vehicle flow entering an unsignalized intersection system is characterized by simultaneous heterogeneous demands (left turn, though, and right turn). Owing to the traffic rules, the flow is separated into vehicles that are prioritized (through and right turn) and those that lose their priority (left turn). This differentiation of the flow is associated with the significant role of the binomial distribution of the arrival process. A queuing system with such specificity has been observed and explained by Yajima and Phung-Duc [20]. The previously mentioned complex recursive operations noticed by Wu [2] are based on such a binomial distribution, which determines the values and approaches to the transition between the different states of the system. Binomial distribution laws are determined on intersections as well [21] but are signalized. However binomial distribution analytically "favours" Markovian process. Poisson expression of binomial probabilities directly introduces Poisson flows into analytical tools of the queuing system, and with it exponential distribution according to the Palm theory.

The capacity of a short lane performs the spatial selection of vehicles based on the demands alters their prioritization. The basic objective is to ensure that the flow of vehicles that lose priority (left turn) according to the traffic rules do not slow down or entirely block the flow of the vehicles that retain their priority (through and right turn) owing to the first in-first out (FIFO) discipline of the service. Therefore, the proper dimensioning of a short lane has a significant effect on the capacity of the intersection and losses in time.

New analytic solutions of a Queuing system for shared–short lanes at unsignalized intersections have been presented through the following chapters after introduction:

2. Queueing system phenomenon


5.1. Binomial distribution of the vehicle service in a system with only a finite-capacity shared lane

5.2. Solving queuing systems with only an infinite-capacity shared lane

6. Solving a two-phase queuing system with a finite-capacity short lane *i*=*constant* and an infinite-capacity shared lane *j*∈[1, ∞)

6.1. Probabilities of the states of the two-phase queuing system


9. Conclusions

#### **2. Queueing System Phenomenon**

Queuing theory is generally considered a branch of operations research as sub-field of applied mathematics. This was founded just over 100 years ago, by the publication of works and by successful practical application by the Danish mathematician, statistician and engineer Agner Krarup Erlang (1878–1929). However, after initial success in its application, this avant-garde probabilistic methods for making decisions about the resources needed to provide a service, has provided numerous analytical limitations.

Queuing theory is based on elementary system theory, on entity structure and relations. A dominant part in queuing systems is occupied by relations—randomly distributed continuous-time. Entities are system states. They are always whole numbers and represent number of clients in the system. Based on relations between system's intersections and systems entities, the probability of each system state is calculated.

Primary classification of queuing system depends on probabilistic distribution of time. If distribution density is exponential f(t) = *λ*(t)e−*<sup>λ</sup>*(t), queuing system is Markovian. In case of any other time distribution, the system is non-Markovian. Markovian systems are by rule analytically available. Otherwise, if distribution density is not exponential, analytical calculation is extremely difficult and in some cases even today unsolvable. This classification has been established as an honor to Andrei Andreyevich Markov (1856–1922). David George Kendall (1918–2007) adjusted basic systematization and notation of queuing systems to primary classification.

Secondary classification has also been based on a system's relations. If the average value of probabilistic distribution of time is constant, a queuing system is stationary. Stationary Markovian queuing system has exponential distribution density f(t) = *λ*e−*λ*<sup>t</sup> , *λ*(t) = *λ* = const. The method for the analytical solution of unstationary Markovian queuing system was presented in 1931 by Nikolaevich Kolmogorov (1903–1987) [22]. Solution determined by Kolmogorov for Markovian queuing systems is principally same as for non-Markovian systems. It is based on a system of differential equations. The number of the equation is always equal to number of states, which can be infinite as well! Application of Laplace transformation for solving system of differential equation is much easier in case of Markovian queuing systems. Also, it is understood that queuing system is ergodic.

Tertiary classification is based on the use of system entity, for service and waiting. During this the queuing system can have different service disciplines: FIFO, LIFO (last in–first out), stochastic choice of service, group service, priorities in service etc. For waiting as a rule the FQFS (first in queue–first on service) discipline is used. The basic structure of the queuing system is dominantly based on tertiary classification.

Quartic classification is based on client flow. This structure can be homogenous or inhomogeneous or in other words heterogeneous. Classification has a dual nature. The simplest queuing system concept is when homogenous clients demand homogenous service. In any case of inhomogeneousness of clients and service, queuing system structure becomes delicate to solve.

Many great mathematicians and engineers had contributed to development of queuing theory: Félix Pollaczek (1892–1981) [23],Aleksandr YakovlevichKhinchin (1894–1959), our contemporary, Sir John Frank Charles Kingman (born 1939) [24], David George Kendall (1918–2007), and our other contemporary Jonh Dutton Conant Little (born 1928), etc. Their research has been dominantly pointed towards solving non-Markovian queuing systems. However, an approach towards solving nonstationary non-Markovian queuing systems has been lacking. The development of personal computers of the 1980s and 1990s made the prognosis that each queuing system could be solved by the use of simulations. This attitude has somewhat discouraged further efforts in the analytical approach of the queuing theory and was consistently described by Koenigsberg in the set and reasoned antithesis [25]. His absolutely correct assessment of the necessity of analytical approach and positive development prognosis, confirmed Schwartz, Selinka and Stoletz, especially for non-stationary time-dependent non-Markovian queuing systems [26]. The analytical approach to solving the queuing

system remains an imperative. This imperative does not exist in itself, it is encouraged by the practical application of the queuing system and the lifeblood of queuing theory lies in its applications [27].

#### **3. Unsignalised Intersections and Queueing**

During the first research in the 1930s, probabilistic nature of traffic had been determined. Determined Poisson distribution in research of road infrastructure capacity in papers by Kinzer [28] and Adams [29] had for the first time proven Markovian structure of traffic flow through Conrad Palma's (1907–1951) theorem. Whole number clients (vehicles in traffic) and exponential distribution of time between consecutive cars in free traffic flow, presented an ideal basis for the application of queuing theory. Traffic flow intensity is by rule time-dependent, or unstationary. However, dimensioning traffic infrastructure capacity of most frequent intensity or maximal intensity can be chosen and declared as stationary. This depends on the solving strategy of queuing system. This effectively expands the first two classifications.

An intersection is a queuing system. However, circumstances on intersections get extremely complicated in the parts of the third and the fourth classification. Thw parallel approach of numerous exponential flows, priority distributions, client heterogeneousness (pedestrians, cyclists, different vehicles: cars, busses, trucks, etc.), different demands (driving straight, left turn, right turn) results in a large number of interactions and complex probabilistic conditioning. Apart from this permanent imperative traffic safety, always presents additional conditions into complex probabilistic conditionality.

This conditionality is greater on unsignalized intersections. On signalized intersections in calculated time sequences, priorities are strictly distributed, which in great measure reduces probabilistic conditionality of antagonistic flows.

This paper treats intersection as Markovian stationary queuing system with FIFO service discipline, one service channel, the final capacity of short lane, endless number of places in queue/shared lane, homogenous vehicle flow with heterogeneous demands: driving straight and left turn. Demand distribution is a stationary discrete random variable of binominal distribution. Even though only one intersection segment had been considered, very complex recursive operations assumed by Wu [2], had been solved within this manuscript after 25 years.

#### **4. Limiting Analytic Framework of a Queueing System**

The probability "*p*" with which vehicles from the priority direction decide to make a left turn can be statistically determined based on the classic Laplacian definition of probability. It is equal to the quotient of number of vehicles turning left and total number of vehicles arriving at the intersection.

If the flow of vehicles is independent, then the Poisson flow with arbitrarily assumed average arrival rate *λ* can be described as two independent Poisson flows (1):

$$
\lambda = p\lambda + (1 - p)\lambda \tag{1}
$$

Distribution of the service time for taking a left turn has been the subject of various analyses [30,31] starting from the first concrete application of the queueing systems to the latest research results. The approaches for the utilization of these intervals are in the domain of the time differences from minimally accepted to maximally rejected intervals of priority Poisson flow. The approximation for the service rate of a left turn as an exponential distribution of intensity *μ* is not very effective for a queueing system at unsignalized intersections owing to the dispersed data points. However, for the first complete analytical solution of the complex recursive operations, it is necessary to remain in the Markovian domain [12,32–34]. Thus, the service rate exponential distribution has been adopted here, and its derived solutions have a complete theoretical and practical relevance.

According to Harders [1], depending on the number of locations on an individual lane for taking a left turn, there are two limiting cases: minimal and maximal average number of vehicles in the system.

A minimal average number of vehicles in the system is achieved if the intersection is designed with a separate lane of unlimited capacity for taking a left turn. The vehicles that plan to move forward in the intersection have separate reserved server, and based on the priority achieve the maximal level of service. A queue is formed only by those vehicles planning to take a left turn at the intersection. In this system, average arrival rate is "*pλ*"and average service rate is "*μ*". The discipline of the service is FIFO. There is a single server and an unlimited number of positions in the queue. This system is a classic Erlang system in which no vehicles are rejected with Kendal markings M(*pλ*)/M(*μ*)/1/∞ (Figure 1), and it has already been considered by Wu [2]. The states are determined by the number of vehicles in a separate lane for making a left turn. Accordingly, there are two indices in X*i,j*, where index "*i*" denotes the number of vehicles in the lane for making a left turn (short lane), whereas index "*j*" denotes the number of locations in the lane for all the vehicles in the traffic (shared lane). In the queueing system (Figure 1), the indices of the states have values *i*∈[0,∞) and *j* = 0.

**Figure 1.** Queueing system with a separate lane for left–turn maneuvering.

According to the formula from for queueing system with Kendal denotation M(*pλ*)/M(*μ*)/1/∞, the minimal average number of vehicles in the system is defined in (2).

$$k\_{\min} = \left(\frac{\mu - p\lambda}{\mu}\right) \sum\_{i=0}^{\infty} i \left(\frac{p\lambda}{\mu}\right)^i = \frac{p\lambda}{\mu - p\lambda} \tag{2}$$

The maximal average number of vehicles in the system is achieved for all the vehicles that are present only in a single-shared lane. In the queueing system (Figure 2), the indices of the states have values *i* = 0 and *j*∈[0,∞). In this system, the arrival rate is not the same for all the states. The system crosses from state X0,0 into state X0,1 only on arrival of a vehicle that plans to make a left turn at the intersection with probability "*p*." In this case, the arrival rate equals "*pλ*". When waiting for a service owing to the FIFO discipline of the service, the server is occupied for all the other vehicles arriving at the intersection with *λ* intensity, and a queue is formed in the system by all the vehicles. The intensity of the service is not equal for all the states. Only the intensity of service *μ* from state X0,1 is known for the vehicles performing the left-turn maneuver.

If a consecutive vehicle in state X0,2 at the intersection plans a left−turn maneuver with probability "*p*", then after servicing the vehicles from state X0,1 in the queuing system, it switches from state X0,2 into state X0,1 with intensity "*pμ*."

However, if a consecutive vehicle in state X0,2 at the intersection plans to go forward with probability (1 − *p*), then after servicing the vehicles from state X0,1 of the queuing system, it directly transfers from state X0,2 to X0,0 state with "(1− *p*)*μ*" intensity.

Depending on the binomial distribution of the vehicles, the system from state X0,3 can transite to states X0,2, X0,1, or X0,0 with different service levels. In general, the system can transite from state X0,*<sup>j</sup>* into any of the previous states with different intensities, with a final summation of "*μ*" (Figure 2). It should be noticed that each state can be achieved from any of the following states.

**Figure 2.** Queueing system with a shared lane.

The maximal average number of vehicles in this system is obtained by (3), which will be explained later in Section 5.2.

$$k\_{\text{max}} = \frac{\mu - p\lambda}{p((1-p)\lambda + \mu)} \sum\_{j=0}^{\infty} j \left(\frac{\lambda}{(1-p)\lambda + \mu}\right)^j = \frac{\lambda}{\mu - p\lambda} \tag{3}$$

For an intersection that has a short lane designed for taking left turns with final capacity "*i*" and has a shared lane with unlimited number of shared places *j*∈[1,∞) for cars, the queueing system is presented in Figure 3.

**Figure 3.** Queueing system with a shared–short lane.

The average number of vehicles in such a queuing system is within the limits of the minimal (2) and maximal (3). The relation between the limiting values can be obtained from expression (4).

$$k\_{\rm min} = \frac{p\lambda}{\mu - p\lambda} < k\_i < \frac{\lambda}{\mu - p\lambda} = \frac{k\_{\rm min}}{p} = k\_{\rm max} \tag{4}$$

The procedure for the calculation of expression (4) will be presented in detail in Section 6.3.

The average time that a vehicle spends in this system can be calculated based on the Little formula. From Figure 3, it is obvious that this is a two-phase queueing system. The first phase corresponds to the filling of the short lane for taking left turns, from state X0,0 to state X*i*,0. State X*i*,0 is the state connecting Phases I and II. The first phase finishes and the Phase II starts in the same state (X*i*,0).

#### **5. Calculation of the Maximal Number of Vehicles in a System with Only a Shared Lane**

*5.1. Binomial Distribution of the Vehicle Service in a System with Only a Finite-Capacity Shared Lane*

Until now we have explained the transition from state X0,2 into X0,0. Therefore, we next discuss the intensity of the vehicle service with capacity *j* = 3.

A queue in a joint lane is formed by vehicles that plan to take a left turn at the intersection with probability "*p*" and those vehicles that plan to drive with priority with probability "(1 − *p*)."

Therefore, the arrival rate is *λ* = *pλ* + (1 − *p*)*λ* (1). If the system is in state X0,0, on arrival of a vehicle with arrival rate priority "(1 − *p*)*λ*" it will not change its state (Figure 4).

**Figure 4.** Probabilities and intensities of the transitions from xk+3 state to the remaining states.

The system can only transit to state X0,1 with the arrival of a vehicle that plans to make a left turn with arrival rate "*pλ*" and service intensity "*μ*." This vehicle shuts down the server. Therefore, all the vehicles form a queue with arrival rate "*λ*." If the system is in state X0,1, i.e., *j* = 1, it can only transition into state X0,0 with intensity of left turn *μ*.

If the system is in state X0,2, it can undergo two transitions:


It should be noted that the total intensity of the transitions into state X0,2 is *<sup>p</sup>*2*<sup>μ</sup>* + (1 − *<sup>p</sup>*)*p<sup>μ</sup>* <sup>=</sup> *<sup>p</sup>μ*. It can be generalized that from each state X0,*j*, there are 2*j*−<sup>1</sup> possible transitions to each of the previous states. The balance (differential) equations of the steady-states are expressed in (5).

$$\begin{cases} P\_{0,0}'(t) = 0 = -\lambda p P\_{0,0} + \mu P\_{0,1} + (1-p)\mu P\_{0,2} + (1-p)^2 \mu P\_{0,3} \\ P\_{0,1}'(t) = 0 = +\lambda p P\_{0,0} - \lambda P\_{0,1} - \mu P\_{0,1} + p\mu P\_{0,2} + p(1-p)\mu P\_{0,3} \\ P\_{0,2}'(t) = 0 = +\lambda P\_{0,1} - \lambda P\_{0,2} - p\mu P\_{0,2} - (1-p)\mu P\_{0,2} + p\mu P\_{0,3} \\ P\_{0,3}'(t) = 0 = +\lambda P\_{0,2} - p\mu P\_{0,3} - (1-p)\mu p P\_{0,3} - (1-p)^2 \mu P\_{0,3} + \mu P\_{full} \\ P\_{full}'(t) = 0 = +\lambda P\_{0,3} - \mu P\_{full} \end{cases} \tag{5}$$

To observe the binomial laws, a system with capacity *j* = 4 is also considered (Figure 5).

**Figure 5.** Probabilities and intensities of the transitions from xk+4 state to the remaining states.

The balance (differential) equations of the steady-state of the queueing system with four places in the queue for vehicles in a joint traffic lane are given in (6).

$$\begin{aligned} P'\_{0,0}(t) &= 0 = -\lambda p P\_{0,0} + \mu P\_{0,1} + (1-p)^1 \mu P\_{0,2} + (1-p)^2 \mu P\_{0,3} + (1-p)^3 \mu P\_{0,4} \\ P'\_{0,1}(t) &= 0 = +\lambda p P\_{0,0} - \lambda P\_{0,1} - \mu P\_{0,1} + p\mu P\_{0,2} + p(1-p)\mu P\_{0,3} + p(1-p)^2 \mu P\_{0,4} \\ P'\_{0,2}(t) &= 0 = +\lambda P\_{0,1} - \lambda P\_{0,2} - p\mu P\_{0,2} + (1-p)\mu P\_{0,2} + p\mu P\_{0,3} + p(1-p)\mu P\_{0,4} \\ P'\_{0,3}(t) &= 0 = +\lambda P\_{0,2} - P\_{0,3}(\lambda + p\mu + p\mu(1-p) + \mu(1-p)^2) + p\mu P\_{0,4} \\ P'\_{0,4}(t) &= 0 = +\lambda P\_{0,3} - P\_{0,4}(\lambda + p\mu + p\mu(1-p) + p\mu(1-p)^2 + \mu(1-p)^3) + \mu P\_{full} \\ P'\_{full}(t) &= 0 = +\lambda P\_{0,4} - \mu P\_{full} \end{aligned} \tag{6}$$

The system can switch from state X0,4 into state X0,3 in four ways, which are included in the binomial expression in (7).

$$\mu \left( p^3 (1 - p)^0 + 2p^2 (1 - p)^1 + p^1 (1 - p)^2 \right) = p \mu \sum\_{k=0}^2 \binom{2}{k} p^k (1 - p)^{2 - k} \tag{7}$$

From each X0,*<sup>j</sup>* state there are (*j* − 2) ways to switch to X0,*<sup>j</sup>*−<sup>1</sup> state, which are included in the binomial expression for complementary probabilities, as expressed in (8):

$$p\mu \sum\_{k=0}^{j-2} \binom{j-2}{k} p^k (1-p)^{j-2-k} = p\mu (p + (1-p))^{j-2} = p\mu (1)^{j-2} = p\mu \tag{8}$$

and there are (*n* − 1) ways to switch from each states X0,*<sup>j</sup>*∈[2, <sup>∞</sup>] to state X0,*<sup>n</sup>*∈[1, *<sup>j</sup>*−1], which are included in the binomial expression of complementary probabilities given in (9), for *k*∈*N*.

$$p(1-p)^{j-n-1} \mu \sum\_{k=0}^{n-1} \binom{n-1}{k} p^k (1-p)^{n-1-k} = p \mu (p + (1-p))^{n-2} = p(1-p)^{j-n-1} \mu \tag{9}$$

*Symmetry* **2019**, *11*, 55

The exception is state X0,0 whose intensities are expressed as the product of the probabilities of each state in a geometric series (10).

$$\mu \sum\_{j=0}^{\infty} (1-p)^j P\_{0,j} \tag{10}$$

From each state *j*∈[1, *k*], *k*∈*N*, intensity of the vehicle service "*μ*" "spills" according to the partial geometric dependence defined in (11).

$$
\mu (1 - p)^{j - 1} + p\mu \sum\_{k=0}^{j - 2} (1 - p)^k + \\
= \mu (1 - p)^{j - 1} + p\mu \frac{1 - (1 - p)^{j - 1}}{1 - (1 - p)} \\
= \mu \tag{11}
$$

#### *5.2. Solving Queueing Systems with Only an Infinite-Capacity Shared Lane*

A queueing system in which no vehicles are rejected, because of the absence of a short lane (*i* = 0), in a heterogeneous vehicle flow with an infinite-capacity shared lane has states as presented in Figure 6.

The system of balance equations of the corresponding steady-state is given in (12), *k*∈*N*.

$$\begin{aligned} P'\_{0,0}(t) &= 0 = -p\lambda P\_{0,0} + \frac{1}{p} \sum\_{j=1}^{\infty} p(1-p)^{j-1} \mu P\_{0,j} \\ P'\_{0,1}(t) &= 0 = p\lambda P\_{0,0} - (\lambda + \mu)P\_{0,1} + \sum\_{j=2}^{\infty} p(1-p)^{j-2} \mu P\_{0,j} \\ P'\_{0,2}(t) &= 0 = \lambda P\_{0,1} - (\lambda + \mu)P\_{0,2} + \sum\_{j=3}^{\infty} p(1-p)^{j-3} \mu P\_{0,j} \\ &\dots \\ P'\_{0,k+1}(t) &= 0 = \lambda P\_{0,k} - (\lambda + \mu)P\_{0,k+1} + \sum\_{j=k+2}^{\infty} p(1-p)^{j-(k+2)} \mu P\_{0,j} \\ &\dots \end{aligned} \tag{12}$$

*Symmetry* **2019**, *11*, 55

From the balance equations of the steady-state of the system in (12), the relations between the probabilities and sums of the series of the geometric products of the probabilities are obtained, as given in (13).

$$\begin{aligned} p\lambda P\_{0,0} &= \frac{1}{p} \sum\_{j=1}^{\infty} p(1-p)^{j-1} \mu P\_{0,j} \\ (\lambda + \mu)P\_{0,1} - \lambda P\_{0,0} &= \sum\_{j=2}^{\infty} p(1-p)^{j-2} \mu P\_{0,j} \\ (\lambda + \mu)P\_{0,2} - \lambda P\_{0,1} &= \sum\_{j=3}^{\infty} p(1-p)^{j-3} \mu P\_{0,j} \\ \vdots \\ (\lambda + \mu)P\_{0,k} - \lambda P\_{0,k-1} &= \sum\_{j=k+1}^{\infty} p(1-p)^{j-k-1} \mu P\_{0,j} \\ \vdots \dots \end{aligned} \tag{13}$$

The relations between the sums should be noticed (14).

$$\begin{aligned} \sum\_{\substack{j=1\\j\ge 1}}^{\infty} p(1-p)^{j-1} \mu P\_{j} &= p\mu P\_{0,1} + (1-p) \sum\_{j=2}^{\infty} p(1-p)^{j-2} \mu P\_{0,j} \\ \sum\_{j=2}^{\infty} p(1-p)^{n-2} \mu P\_{0,j} &= p\mu P\_{0,2} + (1-p) \sum\_{j=3}^{\infty} p(1-p)^{j-3} \mu P\_{0,j} \\ \sum\_{j=3}^{\infty} p(1-p)^{n-3} \mu P\_{0,j} &= p\mu P\_{0,3} + (1-p) \sum\_{j=4}^{\infty} p(1-p)^{j-4} \mu P\_{0,j} \\ \vdots & \vdots \\ \sum\_{j=k}^{\infty} p(1-p)^{j-k} \mu P\_{0,j} &= p\mu P\_{0,k} + (1-p) \sum\_{j=k+1}^{\infty} p(1-p)^{j-(k+1)} \mu P\_{0,j} \\ \dots \end{aligned} \tag{14}$$

From the first balance equation of the steady-state in (12), expression (15) can be obtained.

$$p\lambda P\_{0,0} = \frac{1}{p} \sum\_{j=1}^{\infty} p(1-p)^{j-1} \mu P\_{0,j} \Leftrightarrow p^2 \lambda P\_{0,0} = p\mu P\_{0,1} + (1-p) \sum\_{j=2}^{\infty} p(1-p)^{j-2} \mu P\_{0,j} \tag{15}$$

From the second relation of the sums in (14), we can obtain expression (16).

$$\begin{aligned} p^2 \lambda P\_{0,0} &= p\mu P\_{0,1} + (1-p)[(\lambda+\mu)P\_{0,1} - p\lambda P\_{0,0}]\\ p\lambda P\_{0,0} &= \lambda P\_{0,1} + \mu P\_{0,1} - p\lambda P\_{0,1} \Leftrightarrow P\_{0,1} = \frac{p\lambda}{(1-p)\lambda + \mu} P\_{0,0} \end{aligned} \tag{16}$$

From the second balance equation in (12) of the steady-state, expression (17) can be obtained.

$$p(\lambda + \mu)P\_{0,1} - p\lambda P\_{0,0} = p\mu P\_{0,2} + (1 - p)\sum\_{j=3}^{\infty} p(1 - p)^{j - 3}\mu P\_{0,j} \tag{17}$$

From the third relation of the sums in (14), expression (18) can be derived.

$$(\lambda + \mu)P\_{0,1} - p\lambda P\_{0,0} = p\mu P\_{0,2} + (1 - p)[(\lambda + \mu)P\_{0,2} - \lambda P\_{0,1}]$$

$$[(1 - p)\lambda + \mu]P\_{0,2} = \frac{[2\lambda - p\lambda + \mu] - [\lambda - p\lambda + \mu]}{(1 - p)\lambda + \mu} p\lambda P\_{0,0} \Leftrightarrow P\_{0,2} = \frac{p\lambda^2 P\_{0,0}}{[(1 - p)\lambda + \mu]^2} \tag{18}$$

Furthermore, from the third balance equation of the steady-state of the system, expression (19) can be obtained.

$$p(\lambda + \mu)P\_{0,2} - \lambda P\_1 = \sum\_{j=3}^{\infty} p(1-p)^{j-3} \mu P\_{0,j} = p\mu P\_{0,3} + (1-p) \sum\_{j=4}^{\infty} p(1-p)^{j-4} \mu P\_{0,j} \tag{19}$$

From the fourth relation of the sums, expression (20) can be obtained.

$$(\lambda + \mu)P\_{0,2} - \lambda P\_1 = p\mu P\_{0,3} + (1 - p)[(\lambda + \mu)P\_{0,3} - \lambda P\_{0,2}]$$

$$[(1 - p)\lambda + \mu]P\_{0,3} = [(2 - p)\lambda + \mu]P\_{0,2} - \lambda P\_{0,1} \Leftrightarrow P\_{0,3} = \frac{p\lambda^3 P\_{0,0}}{[(1 - p)\lambda + \mu]^3} \tag{20}$$

From (16), (18) and (20) the recurrent relation for the probabilities of the states, *P*0,*<sup>j</sup>* is obtained analogously (21), and can be proved by mathematical induction.

$$P\_{0,j} = \frac{p\lambda^j P\_{0,0}}{[(1-p)\lambda + \mu]^j} \tag{21}$$

From the condition in (22):

$$P\_{0,0} + P\_{0,1} + P\_{0,2} + \dots = \sum\_{j=0}^{\infty} P\_{0,j} \tag{22}$$

The probability of the initial state in (23) is obtained. Under the stability condition *μ* ≥ *pλ* and 0 ≤ *p* ≤ 1, probability of the state without a vehicle is always 0 ≤ *P*0,0 ≤ 1.

$$P\_{0,0} = \frac{1}{p \sum\_{j=0}^{\infty} \left(\frac{\lambda}{(1-p)\lambda + \mu}\right)^j} = \frac{1}{p} \left(1 - \frac{\lambda}{(1-p)\lambda + \mu}\right) = \frac{\mu - p\lambda}{p((1-p)\lambda + \mu)}\tag{23}$$

From the recurrent relation of the probabilities of the states for *i* = 0 and *j*∈[0, ∞), (24) is obtained.

$$P\_{0,j} = \frac{\mu - p\lambda}{p((1-p)\lambda + \mu)} \frac{p\lambda^j}{[(1-p)\lambda + \mu]^j} = \frac{\mu - p\lambda}{((1-p)\lambda + \mu)} \frac{\lambda^j}{[(1-p)\lambda + \mu]^j} \tag{24}$$

The average number of vehicles in the system is defined in (25)

$$k\_{\text{max}} = \sum\_{j=0}^{\infty} kP\_{0,j} = \frac{\frac{\mu - p\lambda}{(1 - p)\lambda + \mu}}{1 - \frac{\mu - p\lambda}{(1 - p)\lambda + \mu}} = \frac{\lambda}{\mu - p\lambda} \tag{25}$$

This average number of vehicles is the maximal average number of vehicles *k*max in (2) achieved in the queueing system for the given values of *λ*, *μ*, and *p*. It is obvious that when *pλ = μ*, the average number of vehicles diverges, and under the stability condition *μ* ≥ *pλ*, the system does not fulfill its objective (26).

$$\lim\_{p\lambda \to \mu} k\_{\text{max}} = \lim\_{p\lambda \to \mu} \frac{\lambda}{\mu - p\lambda} = \infty \tag{26}$$

#### **6. Solving a Two-Phase Queueing System with a Finite-Capacity Short Lane** *i* **= Const and an Infinite-Capacity Shared Lane** *j*∈**[1,** ∞**)**

This queueing system is the usual state in practical conditions. The system has one server for taking a left turn. A separate lane for the left turn is designed, and it has finite capacity "*i*". A separate lane fills with a homogenous vehicle flow with a homogenous demand that becomes a heterogeneous demandon at unsignalized intersections when vehicles plan a left-turn maneuver. If product "*pλ*"converges to service rate "*μ*" or if there is high participation of "*p*" in the incoming flow, vehicles fill all the places "*i*" in the short lane, and then form a queue of heterogeneous vehicles in the shared lane with intensity "*λ*." The graph of the states of the queueing system is presented in Figure 7.

**Figure 7.** Two-phase queueing system of shared–short lane.

#### *6.1. Probabilities of the States of the Two-Phase Queueing System*

The connecting phases I and II is state *Pi*,0. The balance equations of the steady-state of the two-phase system are given in (27).

$$\begin{aligned} P'\_{0,0}(t) &= 0 = -p\lambda P\_{0,0} + \mu P\_{1,0} \\ P'\_{1,0}(t) &= 0 = +p\lambda P\_{0,0} - p\lambda P\_{1,0} - \mu P\_{1,0} + \mu P\_{2,0} \\ \vdots \\ P'\_{i-1,0}(t) &= 0 = +p\lambda P\_{i-2,0} - p\lambda P\_{i-1,0} - \mu P\_{i-1,0} + \mu P\_{i,0} \\ P'\_{i,0}(t) &= 0 = +p\lambda P\_{i-1,0} - p\lambda P\_{i,0} - \mu P\_{i,0} + \frac{1}{p} \sum\_{j=1}^{\infty} p(1-p)^{j-1}\mu P\_{i,j} \\ P'\_{i,1}(t) &= 0 = \lambda P\_{i,0} - (\lambda + \mu)P\_{i,1} + \sum\_{j=2}^{\infty} p(1-p)^{j-2}\mu P\_{i,j} \\ P'\_{i,2}(t) &= 0 = \lambda P\_{i,1} - (\lambda + \mu)P\_{i,2} + \sum\_{j=3}^{\infty} p(1-p)^{j-3}\mu P\_{i,j} \\ P'\_{i,3}(t) &= 0 = \lambda P\_{i,2} - (\lambda + \mu)P\_{i,3} + \sum\_{j=4}^{\infty} p(1-p)^{j-4}\mu P\_{i,j} \\ \vdots \end{aligned} \tag{27}$$

From state X0,0 to state X*i*,0, there are known relations based on the system M(*pλ*)/M(*μ*)/*i*/∞ (28).

$$\begin{aligned} p\lambda P\_{0,0} &= \mu P\_{1,0} \Leftrightarrow P\_{1,0} = \frac{p\lambda}{\mu} P\_{0,0} \\ \lambda \frac{p\lambda}{\mu} P\_{0,0} + \mu \frac{p\lambda}{\mu} P\_{0,0} - p\lambda P\_{0,0} &= \mu P\_{2,0} \Leftrightarrow P\_{2,0} = \left(\frac{p\lambda}{\mu}\right)^2 P\_{0,0} \\ P\_{3,0} &= \left(\frac{p\lambda}{\mu}\right)^3 P\_{0,0} \\ \dots \\ P\_{i-1,0} &= \left(\frac{p\lambda}{\mu}\right)^{i-1} P\_{0,0} \\ P\_{i,0} &= \frac{p\lambda}{\mu} P\_{i-1,0} = \left(\frac{p\lambda}{\mu}\right)^i P\_{0,0} \end{aligned} \tag{28}$$

For further solving the new relations between the sums, the expressions in (29) need to be noted (*k*∈*N*). <sup>∞</sup>

$$\begin{cases} \sum\_{j=1}^{\infty} p(1-p)^{j-1} \mu P\_{i,j} = p\mu P\_{i,1} + (1-p) \sum\_{j=2}^{\infty} p(1-p)^{j-2} \mu P\_{i,j} \\ \sum\_{j=2}^{\infty} p(1-p)^{j-2} \mu P\_{i,j} = p\mu P\_{i,2} + (1-p) \sum\_{j=3}^{\infty} p(1-p)^{j-3} \mu P\_{i,j} \\ \sum\_{j=3}^{\infty} p(1-p)^{j-3} \mu P\_{i,j} = p\mu P\_{i,3} + (1-p) \sum\_{j=4}^{\infty} p(1-p)^{j-4} \mu P\_{i,j} \\ \vdots \\ \sum\_{j=k}^{\infty} p(1-p)^{j-k} \mu P\_{i,k} = p\mu P\_{i,k} + (1-p) \sum\_{j=k+1}^{\infty} p(1-p)^{j-(k+1)} \mu P\_{i,j} \\ \vdots \end{cases} \tag{29}$$

From the equation for *Pi*,1, by changing the sum for *j* = 3, (30) is obtained.

$$\begin{aligned} (\lambda + \mu)P\_{l,1} - \lambda P\_{l,0} &= \sum\_{j=2}^{\infty} p(1-p)^{j-2} \mu P\_{l,j} \\ (\lambda + \mu)P\_{l,1} - \lambda P\_{l,0} &= p\mu P\_{l,2} + (1-p)\sum\_{j=3}^{\infty} p(1-p)^{j-3} \mu P\_{l,j} \\ (\lambda + \mu)P\_{l,1} - \lambda P\_{l,0} &= p\mu P\_{l,2} - (1-p)\left(\lambda P\_{l,1} - (\lambda + \mu)P\_{l,2}\right) \\ \left[ (2-p)\lambda + \mu \right] \mu P\_{l,1} - \lambda P\_{l,0} &= \left[ (1-p)\lambda + \mu \right] P\_{l,2} \Leftrightarrow \lambda P\_{l,0} = \left[ (2-p)\lambda + \mu \right] P\_{l,1} - \left[ (1-p)\lambda + \mu \right] P\_{l,2} \\ P\_{l,0} &= \left[ (2-p) + \frac{\mu}{\lambda} \right] P\_{l,1} - \left[ (1-p) + \frac{\mu}{\lambda} \right] P\_{l,2} \end{aligned} \tag{30}$$

From the equation for *Pi*,2, by changing the sum for *j* = 4, (31) is obtained.

$$\begin{aligned} (\lambda + \mu)P\_{i,2} - \lambda P\_{i,1} &= \sum\_{j=3}^{\infty} p(1-p)^{j-3} \mu P\_{i,j} \\ (\lambda + \mu)P\_{i,2} - \lambda P\_{i,1} &= p\mu P\_{i,3} + (1-p) \sum\_{j=4}^{\infty} p(1-p)^{j-4} \mu P\_{i,j} \\ (\lambda + \mu)P\_{i,2} - \lambda P\_{i,1} &= p\mu P\_{i,3} - (1-p)(\lambda P\_{i,2} - (\lambda + \mu)P\_{i,3}) \\ &\quad \left[ (2-p)\lambda + \mu \right]P\_{i,2} - \lambda P\_{i,1} = \left[ (1-p)\lambda + \mu \right]P\_{i,3} \\ P\_{i,1} &= \left[ (2-p) + \frac{\mu}{\lambda} \right]P\_{i,2} - \left[ (1-p) + \frac{\mu}{\lambda} \right]P\_{i,3} \end{aligned} \tag{31}$$

Furthermore, a recurrent relation given in (32) is obtained.

$$P\_{i,k} = \left[ (2-p) + \frac{\mu}{\lambda} \right] P\_{i,k+1} - \left[ (1-p) + \frac{\mu}{\lambda} \right] P\_{i,k+2} \tag{32}$$

From the normative condition in (33):

$$\sum\_{k=0}^{i-1} P\_{k,0} + \sum\_{j=0}^{\infty} P\_{i,j} = \left(P\_{0,0} + P\_{1,0} + \dots + P\_{i-1,0}\right) + \left(P\_{i,0} + P\_{i,1} + P\_{i,2} + P\_{i,3} \dots\right) = 1\tag{33}$$

and applying the recurrent equation in (32), (34) can be derived.

$$\underbrace{\sum\_{k=0}^{i-1} P\_{k,0} + \left[ (2-p) + \frac{\mu}{\lambda} \right] P\_{i,1}}\_{\mathbf{P}\_{i,3}} - \underbrace{\left[ (1-p) + \frac{\mu}{\lambda} \right] P\_{i,2} + \left[ (2-p) + \frac{\mu}{\lambda} \right] P\_{i,2}}\_{\mathbf{P}\_{i,2}} \tag{34}$$

$$\underbrace{-\left[ (1-p) + \frac{\mu}{\lambda} \right] P\_{i,3} + \left[ (2-p) + \frac{\mu}{\lambda} \right] P\_{i,3} - \left[ (1-p) + \frac{\mu}{\lambda} \right] P\_{i,4} + \dots}\_{\mathbf{P}\_{i,4}} = 1$$

Then the normative condition becomes (35).

$$\sum\_{k=0}^{i-1} P\_{k,0} + \left[ (2-p) + \frac{\mu}{\lambda} \right] P\_{i,1} + P\_{i,2} + P\_{i,3} + \dots = 1 \tag{35}$$

Since,

$$\left[\left(2-p\right)+\frac{\mu}{\lambda}\right]P\_{i,1} = 2P\_{i,1} - pP\_{i,1} + \frac{\mu}{\lambda}P\_{i,1} = \left[\left(1-p\right)+\frac{\mu}{\lambda}\right]P\_{i,1} + P\_{i,1} \tag{36}$$

The normative condition can be given as (37).

$$\sum\_{k=0}^{i-1} P\_{k,0} + \left[ (1 - p) + \frac{\mu}{\lambda} \right] P\_{i,1} + P\_{i,1} + P\_{i,2} + \dots = 1 \tag{37}$$

By expanding (37) with (±*Pi*,0), the direct relation between probabilities *Pi*,0 and *Pi*,1 is obtained from the normative condition, and it is given in (38).

$$\sum\_{k=0}^{j-1} P\_{k,0} + \left[ (1-p) + \frac{\mu}{\lambda} \right] P\_{i,1} + \left( -P\_{i,0} \underbrace{+P\_{i,0}}\_{\right) + P\_{i,1} + P\_{i,2} + P\_{i,3} + \dots}\_{\sum\_{j=0}^{\infty} P\_{i,j}} = 1 \tag{38}$$

As a part of the normative condition in (33) is contained in (38), we can now obtain (39).

$$\underbrace{\sum\_{k=0}^{i-1} P\_{k,0} + \sum\_{j=0}^{\infty} P\_{i,j}}\_{from\ (33)-1} + \left[ (1-p) + \frac{\mu}{\lambda} \right] P\_{i,1} - P\_{i,0} = 1 \tag{39}$$

*Symmetry* **2019**, *11*, 55

By accepting the last balance equation given in (28), the value of first probability of the Phase II, *Pi*,1 is obtained through *P*0,0, as expressed in (40).

$$\begin{aligned} \left[ (1-p)\lambda + \mu \right] P\_{i,1} = \lambda P\_{i,0} \Leftrightarrow P\_{i,1} = \frac{\lambda P\_{i,0}}{\left[ (1-p)\lambda + \mu \right]} = \left( \frac{p\lambda}{\mu} \right)^{i} \frac{\lambda}{(1-p)\lambda + \mu} P\_{0,0} \\ P\_{i,j} = \frac{\lambda P\_{i,0}}{\left[ (1-p)\lambda + \mu \right]} = \left( \frac{p\lambda}{\mu} \right)^{j} \left( \frac{\lambda}{(1-p)\lambda + \mu} \right)^{j} P\_{0,0} \end{aligned} \tag{40}$$

The relation between the consecutive probabilities is maintained in the Phase II of the system, and so, it is the same as in (24). Therefore, the final expression for the normative condition through initial probability *P*0,0 becomes:

$$P\_{0,0} \sum\_{k=0}^{i-1} \left(\frac{p\lambda}{\mu}\right)^k + P\_{0,0} \left(\frac{p\lambda}{\mu}\right)^i \sum\_{j=0}^{\infty} \left(\frac{\lambda}{(1-p)\lambda + \mu}\right)^j = 1\tag{41}$$

The sums of the finite geometric series of the first phase and infinite geometric series of the Phase II are given in (42):

$$\begin{split} &P\_{0,0} \frac{1 - \left(\frac{p\lambda}{\mu}\right)^i}{1 - \left(\frac{p\lambda}{\mu}\right)}^i + P\_{0,0} \left(\frac{p\lambda}{\mu}\right)^i \frac{(1-p)\lambda + \mu}{\mu - p\lambda} = 1 \\ &P\_{0,0} \frac{\mu - \mu \left(\frac{p\lambda}{\mu}\right)^i}{\mu - p\lambda} + P\_{0,0} \frac{\lambda \left(\frac{p\lambda}{\mu}\right)^i - p\lambda \left(\frac{p\lambda}{\mu}\right)^i + \mu \left(\frac{p\lambda}{\mu}\right)^i}{\mu - p\lambda} = 1 \end{split} \tag{42}$$

and they yield final probability of the initial state *P*0,0 (43):

$$P\_{0,0} = \frac{\mu - p\lambda}{(1 - p)\lambda \left(\frac{p\lambda}{\mu}\right)^{\bar{l}} + \mu} \tag{43}$$

#### *6.2. Validation of Probability P0,0 of a State of the Two-Phase Queueing System*

The validation of probability *P*0,0 of a state of a two-phase queuing system can be achieved within the limiting conditions. Apart from the stability condition *μ* ≥ *pλ*, the first limiting condition is that when the number of places in the separate lane tends toward infinity, the well-known value of *P*0,0 is obtained for queueing system with Kendal denotation M(*pλ*)/M(*μ*)/1/∞, i.e., for a system with a distinct separate lane for left turns with infinite capacity (44).

$$\lim\_{i \to \infty} P\_{0,0} = \lim\_{i \to \infty} \frac{\mu - p\lambda}{(1 - p)\lambda \left(\frac{p\lambda}{\mu}\right)^i + \mu} = \frac{\mu - p\lambda}{\mu} = 1 - \frac{p\lambda}{\mu} \tag{44}$$

The second limiting case is when the number of places in the separate lane converges towards 0. The relation obtained in this case has already been defined in (24), with a difference in the value of "*p*" and in the denominator, derived from the differences in the two-phase queueing system with input intensity "*pλ*" for state *Pi*,0 when "i" converge to 0 (*i*→0) (45).

$$\lim\_{i \to 0} P\_{0,0} = \lim\_{i \to 0} \frac{\mu - p\lambda}{(1 - p)\lambda \left(\frac{p\lambda}{\mu}\right)^i + \mu} = \frac{\mu - p\lambda}{(1 - p)\lambda + \mu} \tag{45}$$

#### *6.3. Average Number of Vehicle in Short and Share Lane*

The general expression for the average number of vehicle switch capacity of short lane "*i*" is equal to the sum of the average number of vehicles per phase (46). It should be noticed that in the second phase, the short lane of capacity "*i*" fills up.

*Symmetry* **2019**, *11*, 55

$$k\_i = k\_{i(I)} + k\_{i(II)} = \underbrace{\sum\_{k=0}^{i-1} k P\_{k,0}}\_{\text{phase II}} + \underbrace{\sum\_{j=0}^{\infty} (i+j) P\_{i,j}}\_{\text{phase II}} \tag{46}$$

The first phase has a known value of the average number of vehicles based on the M(*pλ*)/M(*μ*)/1/*i* queueing system (47):

$$k\_{i(I)} = \sum\_{k=0}^{i-1} kP\_{k,0} = 1 - \frac{1 - \frac{p\lambda}{\mu}}{1 - \left(\frac{p\lambda}{\mu}\right)^{i+1}} + \left(\frac{p\lambda}{\mu}\right)^2 \frac{1 - \left(\frac{p\lambda}{\mu}\right)^{i-1} \left((i-1)(1 - \frac{p\lambda}{\mu}) + 1\right)}{\left(1 - \frac{p\lambda}{\mu}\right) \cdot \left[1 - \left(\frac{p\lambda}{\mu}\right)^{i+1}\right]} \tag{47}$$

Phase II can be separated into two sums (first phase is filled with "*i*" vehicles) (48):

$$k\_{i(II)} = \sum\_{j=0}^{\infty} (i+j)P\_{i,j} = \underbrace{i\sum\_{j=0}^{\infty} P\_{i,j}}\_{\text{Full short lane}} + \sum\_{j=0}^{\infty} jP\_{i,j} \tag{48}$$

The first sum is a pure geometric series, whereas the second is a known series for a system with infinite number of states, as given in (49).

$$k\_{i(II)} = \frac{i}{1 - \frac{\lambda}{(1 - p)\lambda + \mu}} + \frac{\frac{\lambda}{(1 - p)\lambda + \mu}}{1 - \frac{\lambda}{(1 - p)\lambda + \mu}} = \frac{i((1 - p)\lambda + \mu) + \lambda}{\mu - p\lambda} \tag{49}$$

#### **7. Results for Maximal Lane Capacity of Unsignalized Intersection**

Figure 8 presents the average number of vehicles in the system for *λ* = 500 vehicle/h, *μ* = 300 vehicle/h (*λ*+*μ* = 800 vehicle/h, maximal lane capacity of unsignalized intersection established by Lakkundi [35], *i*∈[5, 20], and *p*∈[0.10, 0.50]. The maximal number of vehicles in the system is marked in the figure at *i* = 0 and calculated according to (25) for different probabilities "*p*" (emphasized by red dots). The average number of vehicles has been calculated according to expressions (47)–(49). From Figure 8, it can be seen that probability "*p*" (with which vehicles from the priority direction decide to make a left turn) has more effect than number of places in the short lane "*i*".

For the given parameters, namely, *λ* = 500 vehicle/h and *μ* = 300 vehicle/h, two nomographic distributions for the queue lengths can be expressed by using the cumulative probabilities - analogous to the usage of (*P*0,0 + ... + *Pi*,0 + *Pi*,1 + *Pi*,2 + ... ). A 3D function of the independent variable of the short lane "*i*" and probability of the left-turn maneuver "*p*" is presented in Figure 9.

In the first case, for *i* = 5, it can be noted that 99% of the vehicles will be satisfied up to *p* = 0.20. This implies that for a short lane with a capacity of five places for vehicles, 99% of the vehicles will not use the capacity of the shared lane for forming a queue if *p* = 0.20, without queue overflow/short lane saturation. This is apriori consideration of capacity. Table 1 presents the calculation of cumulative probability. It is obvious that for *p* > 0.2 at given flow shared lane saturation begins.

In the second nomograph in Figure 9, an a posteriori problem is considered. Assumed, or in concrete case, statistical determined distribution of left turn is *p* = 0.27. From *p* = 0.27 ordinate on the surface of nomograph we reach intersection with line for percentile 99 - point "a". By following the 99 percentile line we reach point "b". From it, using the surface of nomograph we descend using ordinate reaching Queue lengths value which asses the value of capacity of short lane *i* ≈ 10! From the Table 1 concrete value can be calculated by interpolation. It is obvious that for *i* = 10 and *p* = 0.25 cumulative probability is 0.994 > 0.99, but for *i* = 10 and *p* = 0.30 cumulative probability is 0.986 > 0.99.

**Figure 8.** Average number of vehicles in the system for the chosen parameters.

**Figure 9.** Nomographs for parameters *λ* = 500 vehicle/h, *μ* = 300 vehicle/h, and short lane capacity *i* = 5 and *i* = 10.

The chosen examples have a simple message: to increase probability from *p* = 0.2 to *p* = 0.27 or for 35%, short lane capacity doubled respectively from *i* = 5 to *i* = 10! Further growth of probability "*p*" disproportionally increases needed capacity of short lane. For example, for *p* = 0.35, *λ* = 500 vehicle/h and *μ* = 300 vehicle/h, short lane will satisfy 99% of traffic flow if the designed short lane capacity *i* ≥ 17 (see values in Table 1, values on gray background are less than 99 percentile). For values of limiting *pmax* = 0.375 (50) queue overflow/short lane saturation is permanent and separate lane for left-turn maneuvering (Figure 1) needs to be designed.

$$p\_{\text{max}} = \frac{\mu}{\lambda + \mu} = \frac{300}{800} = 0.375 \tag{50}$$

To calculate the average time that a vehicle spends in a two-phase queueing system by using the Little formula, it is necessary to note that the intensity "*pλ*" changes to "*λ*" at state X*i*,1 (Figure 7). Thus, the average time that a vehicle spends in the system is defined in (51), and *ki*(I) is calculated according to (47) from states X*i*,0 to X*i*−1,0 (see Figure 7). Intensity "*pλ*" is the arrival rate for states X*i*,0 and X*i*,1. Consequently, Little's formula has the form:

$$\overline{t} = \frac{k\_{i(I)} + iP\_{i,0} + (i+1)P\_{i,1}}{p\lambda} + \frac{\sum\_{j=2}^{\infty} (j+i)P\_{i,j}}{\lambda} \tag{51}$$

The average time that a vehicle spends in the system is largely, but not entirely, proportional to the average number of vehicles in the system.


**Table 1.** Cumulative probability calculation for 3D nomograph *i*=5, short lane capacity *i*∈[1, 20], left turn probability *p*∈[0.05, 0.40].

#### **8. Discussion**

Existing methods for the calculation of shared–short lanes capacity are based on simulation software such as Highway Capacity Manual (HCM), Sidra Intersection, VISIT, etc. These methods are predominantly designed for signalized intersections.

For unsignalized intersections there is a previous method based on solving approximately queuing system from Wu [36] which is also verified by simulation. Development of this method lead to the presented analytical solution of complex recursive operations noted by Wu [2], too.

Transportation planners and policy makers have at their disposal a new analytical method for calculation of shared/short lanes for left turn capacity. According to Koenigsberg [25] it should be more precise in calculation and more sensitive to parameter variations when making decisions on design and reconstruction of capacity of shared/short lanes for the left turn.

For this reason, we emphasize that the key role in capacity calculation for shared/short lanes has the intensity of left turn "*μ*". In theory, if this intensity is large or *μ* >> *pλ*, needed capacity "*i*" of shared/short lanes for left turn converges to zero and all the needs are satisfied by the shared lane. However in the opposite case, capacity of shared/short lanes for left turn diverges for minimal values of "*p*" and "*λ*".This poses a question of justifiability of short lane for left turn design. Capacity with

intensity "*μ*" is conditioned by density of priority (opposite) flow which can be calculated on standard unsignalized intersections [34] and non-standard unsignalized intersections [37]. Ultimately, intensity of left turn "*μ*" is an indirect product of traffic safety!

#### **9. Conclusions**

The significance of signalized intersection has proven to be more important than unsignalized intersections. Signalized intersections service a higher part of the traffic flow. This is the reason why the participation of signalized intersections in traffic safety and service quality is greater. The prevalent direction of researchers is justifiably in the direction of signalized intersections. For these reasons there are significant difference in solutions for signalized and unsignalized intersection occurs. If the fact that there are significantly more unsignalized intersections than signalized ones is taken into consideration, created difference can be declared as theoreticaldeficit.

Optimal dimensioning of short-shared line for left turn from priority direction is the predominant problem of unsignalized intersections. The proposed model is solved only for homogenous car flows, but can easily be adapted for heterogeneous flows with cyclists and pedestrians. For the generalization and calculation of heterogeneous system it is not necessary to change the structure of thequeuing system. Analytic background for adaption is in the well-known characteristic of adding up Poisson flows of different intensities which is proven in Raikov's theorem from 1937. Based on a stated theoretical base, left turns from non−priority directions of unsignalized intersections can be easily solved.

With the possibility of practical application in intersection planning, this paper has brought a new solution in queuing theory. So far, binomial distribution through its relation with Poisson distribution has been used in monophasic homogenous Markovian queuing systems. The presented solution brings a new vision for solving multiphase heterogeneous queuing systems in which a client selectively decides on the service. In a small corps of existing solutions, once again it has been confirmed that the queuing system can be solved if client selection has binomial distribution. Future research can integrate different approaches like multi-criteria decision making methods [38,39].

**Author Contributions:** Each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content.

**Funding:** This research was funded by the Ministry of Science and Technological Development of Serbia, grant number TR 36012.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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