A novel group decision-making approach based on partitioned Hamy mean operators in q-rung orthopair fuzzy context

The motivation and contribution of the study can be explained in three phases. Inspiration for the first phase of the proposed work came from [13, 24, 34], where the partitioning concept was integrated with various AOs. Therefore, as the HM operator has a dual form [18, 45], the above references spiritualize us to introduce partitioning in dual HM (DHM). This new AO, partitioned DHM (PDHM), can intuitively and logically model the interrelationships among attributes. In the second phase, we were encouraged by the fact that work has yet to be presented on partitioned HM (PHM) for the orthopair family of fuzzy sets that include IFS, PFS, and q-ROFS. Therefore, the following papers [7, 34, 48, 51] motivated us to introduce the PHM and PDHM operators for generalized orthopair fuzzy sets. Based on these extensions, the proposed AOs are named q-ROFPHM, q-ROFWPHM, q-ROFPDHM, and q-ROFWPDHM. Finally, motivated by the role of multiple experts in real-world decision-making scenarios, as studied in the following papers [13, 15, 20, 22], this paper contributes an MAGDM approach for handling real-world decision-making problems by applying two new AOs, q-ROFWPHM and q-ROFWPDHM.

Firstly, PDHM’s introduction addresses the need to handle attribute partitioning in MAGDM. This operator can effectively integrate attribute groups while preserving the internal relationships within each group, enabling more accurate decision outcomes. Furthermore, the extension of PDHM and PHM operators to the generalized orthopair fuzzy environment, leading to the development of the q-ROFPHM, q-ROFWPHM, q-ROFPDHM, and q-ROFWPDHM, introduces novel methodologies for dealing with fuzzy and uncertain information. The proposed MAGDM approach, utilizing the innovative AOs, also presents a novel framework for solving decision problems involving multiple attributes and group preferences. This approach addresses a significant gap in the existing MAGDM research and provides decision-makers with a robust methodology for complex decision scenarios. Moreover, the practical case study conducted in this research, focusing on selecting a suitable industry for investment, demonstrates the practical applicability and effectiveness of the proposed approach.

Overall, the innovations presented in this research contribute to the advancement of MAGDM. These contributions enhance the accuracy, reliability, and feasibility of decision outcomes in complex scenarios. Therefore, it can be useful for solving urban logistics problems, opening up new possibilities for effective decision-making in supply chain management and various other domains.

The concise structure of this paper is as follows: First, a brief overview of related literature has been provided in the section “Literature review”. Some fundamental definitions of the q-ROFS, HM, DHM, and PHM operators are given in the section “Preliminaries”. Then, the section “Partitioned dual Hamy mean (PDHM)” introduces the PDHM operator, its desirable properties, and the refinement of partition mean inequality. The section “q-Rung orthopair fuzzy partitioned Hamy mean operators” presents PHM, while the section “q-Rung orthopair fuzzy partitioned dual Hamy mean operators” proposes PDHM AOs under the q-ROF environment and names these AOs as q-ROFPHM, q-ROFWPHM, q-ROFPHM, and q-ROFWPHM. These AOs are defined in decision-making, where all attributes may or may not be correlated. The essential properties of these AOs are also discussed in their respective sections. Furthermore, an MAGDM approach based on q-ROFWPHM and q-ROFWPDHM operators is present in the section “An MAGDM approach based on the q-ROFWPHM and q-ROFWPDHM operators”. The section Some practical applications of the proposed methodology” provides the application of the presented approach by analyzing two MAGDM problems adopted from [24] and [34] with detailed sensitivity and comparative analyses. Finally, the section “Conclusion” delivers the conclusion of the paper and some recommendations for future work. The structure of the proposed study is also presented in Fig. 1.

The generality of q-ROFS over IFS and PFS makes it a popular research topic in decision sciences. Using an AO-based decision-making approach, q-ROFS has been successfully integrated with several mean-type operators to evaluate multi-attribute decision-making (MADM) or MAGDM problems. Specifically, Liu and Wang [27] constructed some basic arithmetic operations for q-rung orthopair fuzzy numbers (q-ROFNs), and in addition to that, the AM and GM for q-ROFNs are also introduced with a new MADM approach. Liu and Liu [25] developed algebraic norms-based BM operators under a q-ROF environment for MAGDM problems, while Liang et al. [21] extended the Choquet integral for q-ROFNs and determined fuzzy measures using entropy and cross-entropy. Wang et al. [43] constructed the HM and DHM operators for q-ROFNs and used them to solve MADM problems. Xing et al. [45] extended Wang et al.’s work [43] by employing the interactional operational laws of q-ROFNs. Garg and Chen [15] introduced some q-ROF neutral weighted averaging AOs by fusing the interaction between membership degrees and membership coefficient sum. Zhang et al. [50] studied the q-ROFSs within the multi-granularity three-way decisions approach and developed a novel MAGDM method. To model the conjunctive and disjunctive behavior of q-ROFNs, Rawat and Komal [36] utilized Hamacher T-norm (TN) and T-conorm (TC) and introduced a family of MM-based AOs for q-ROF-MADM problems. Deveci et al. [11] utilized the q-ROF-CODAS method to identify the best rehabilitation strategy after the closure of a mining site. Tang et al. [41] constructed the q-ROF Zhenyuan integral and used BWM and optimization-based Shapley values to develop a new decision-making model. Recently, Senapati et al. [40] constructed Aczel–Alsina (AA) TN and TC-based average AOs under a q-ROF environment and studied an MADM problem of appropriate global partner selection for companies. Further, Akram et al. [2] infuse the concept of prioritization with the AA TN and TC and develop two robust AOs for a GDM problem related to energy resource selection. Farid and Riaz [14] utilized the AA TN and TC and linear programming to develop a new MAGDM method. Some other recent work on q-ROFSs-based decision-making and discussed paper are summarized in Table 1.

In the above-studied papers, it is assumed that all the attributes depend on each other without classifying their characteristics. However, in real-life MADM problems, there are situations when all the attributes can be classified into several independent groups according to their characteristics. Also, attributes in a particular group may interact with each other in the form of interdependency, which needs to be addressed appropriately. It is worth noticing that the interdependency between attributes can be reflected through the correlation between corresponding arguments in the aggregation process. For such a scenario, Dutta and Guha [13] introduced the partitioned BM, which avoids the unwanted influence of irrelevant attributes in the aggregation process. This has inspired many researchers and has used the concept of attribute partitioning in decision-making through various AOs, such as BM, HM, MSM, MM, and Heronian mean [7, 23, 24, 30, 34, 48, 51]. A brief literature review on HM is also presented in the following section.

Due to the advantages of HM, many researchers have analyzed real-life MADM or MAGDM problems through this function. There are several extensions to HM. Specifically, Qin [33] investigated it for symmetric triangular interval type-2 fuzzy numbers. Rong et al. [38] explored HM for hesitant fuzzy linguistic numbers, Ali et al. [4] constructed HM in a complex interval-valued q-ROF environment, and recently, Akram et al. [1] introduced HM for two-tuple linguistic complex q-ROFNs. These HM operators are also utilized for AO-based decision-making approaches in their papers. Liu and Liu [26] adopted linguistic IFS to develop HM-based AOs for GDM problems. Under the generalized orthopair fuzzy environment, Wang et al. [43] introduced HM and DHM operators, while Xing et al. [45] proposed interactional HM and DHM operators. Liu et al. [29] presented a normal wiggly hesitant fuzzy linguistic term set, and under this environment, some power HM operators were also developed. Using the concept of partitioning, Liu et al. [30] constructed the PHM under an interval-valued intuitionistic uncertain linguistic environment and investigated a plant location selection problem.

A generalized orthopair fuzzy set or q-ROFS A is a collection of elements from a domain of discourse say X, with their membership grades and is written as follows:where \(q\ge 1\) and \(\mu _A(x),\nu _A(x):X \rightarrow [0,1]\) are membership grade functions whose values represent the levels of agreement and disagreement for membership of x in A, respectively. The term \(\pi _A(x) = \left( 1-\mu ^q_A(x)-\nu ^q_A(x)\right) ^{1/q}\) is known as the degree of hesitancy of x in A. The orthopair \((\mu _A(x),\nu _A(x))\) is a q-ROFN and conveniently written as \((\mu ,\nu )\).

Given \(\alpha _1= (\mu _1,\nu _1)\) and \(\alpha _2= (\mu _2,\nu _2)\) are q-ROFNs. The probabilistic sum and product based operations on these two q-ROFNs are Some useful fundamental properties of these operations are as follows:

For a q-ROFN \(\alpha _i = (\mu _i, \nu _i)\), the score function (S) is given by Eq. (2) and its range is \(S(\alpha _i) \in [-1,1]\)If two q-ROFNs having the same score values, then find their accuracy values through a function called accuracy function \(H(\alpha )\in [0,1]\) and it is defined by Eq. (3)The following method is commonly used to provide the ordering between any two q-ROFNs, say \(\alpha _1 = (\mu _1, \nu _1)\) and \(\alpha _2 = (\mu _2, \nu _2)\):

Given any set \(A=\{a_1,a_2, \ldots ,a_n\} \subset \mathbb {R}_+\) and a natural number \(k \le n\), the HM operator on A is defined by Eq. (4)where, \(i_1, i_2, \ldots ,i_k\) are k natural numbers, such that \(1\le i_1< \cdots <i_k\le n\) and \(C_n^k=\frac{n!}{k!(n-k)!}\). The parameter k is also known as a granularity parameter. The \(\mathbb {R}_+\) is the set of non-negative real numbers.

Given any set \(A=\{a_1,a_2, \ldots ,a_n\} \subset \mathbb {R}_+\) and a natural number \(k \le n\), the DHM operator on A is defined by Eq. (5)where \(i_1, i_2, \ldots ,i_k\) are k natural numbers, such that \(1\le i_1< \cdots <i_k\le n\) and the value of \(C_n^k\) is \(\frac{n!}{k!(n-k)!}\).

Suppose, a set \(A=\{a_1,a_2, \ldots ,a_n\} \subset \mathbb {R}_+\) is partitioned into d-number of groups \(P_1, P_2, \ldots , P_d\), such that \(P_i\cap P_j=\emptyset \) and \(\cup _{t=1}^{d}P_t=A\), then the PHM operator on A is defined by Eq. (6)where \((i_1, i_2, \ldots , i_{k_t})\) is \(k_t\)-tuple full combination of \((1,2, \ldots , |p_t|)\), \(k_t= 1, 2, \ldots ,|P_t|\) is the granularity parameter for the partition \(P_t\), and \(|P_t|\) denotes the cardinality of \(P_t\).

Suppose, a set \(\{a_1,a_2, \ldots ,a_n\} \subset \mathbb {R}_+\) is partitioned into d-number of groups \(P_1, P_2, \ldots , P_d\), then the PDHM operator on a given set is defined as follows:where \(i_1, i_2, \ldots ,i_{k_t}\) are \(k_t\) natural numbers, such that \(1\le i_1< \cdots <i_{k_t}\le |P_t|\), \(k_t= 1, 2, \ldots ,|P_t|\) is the granularity parameter for the partition \(P_t\), \(|P_t|\) denotes the cardinality of \(P_t\) (tth partition), and \(C_{|P_t|}^k=\frac{|P_t|!}{k!(|P_t|-k)!}\). Figure 2 depicts the functioning and structure of the PDHM operators.

The calculation process for the proposed PDHM operator can be explain through a example: Consider \(A=\{0.4,0.7,0.1,0.9,0.3,0.5\}\) be the set of six input arguments which is partitioned into two groups \(P_1=\{a_1,a_2,a_5\}=\{0.4,0.7,0.3\}\) and \(P_2=\{a_3,a_4,a_6\}=\{0.1,0.9,0.5\}\). Therefore, \(|P_1|=3\) and \(|P_2|=3\). Taking the granularity values \(k_1=k_2=2\), the aggregated value of A by applying PDHM operator is computed as follows:Now, using the traditional DHM operator as shown in Eq. (5), we getIt is observed that the aggregated values obtained from the PDHM and DHM operators differ, because the traditional DHM operator does not consider partitioning of inputs, whereas the PDHM operator splits the given set and provides a more refined and appropriate result.

In this section, some fundamental properties, such as boundedness, monotonicity, and idempotency for PDHM operator, are verified. The section also proved that \(PDHM ^{k_t}\) is non-decreasing and \(PHM ^{k_t}\) is non-increasing in nature with respect to \(k_t\).

1. Idempotency: Given a set, \(\{a_1,a_2, \ldots ,a_n\}\), such that \(a_i=a\) \(\forall \) \(i \in N=\{1,2, \ldots ,n\}\). The \(PDHM ^{k_t}(a_1,a_2, \ldots ,a_n)=a\).

2. Monotonicity: Given two sets, \(\{a_1,a_2, \ldots ,a_n\}\) and \(\{b_1,b_2, \ldots ,b_n\}\), such that \(a_i\le b_i\) \(\forall \) \(i \in N\). The \(PDHM ^{k_t}(a_1,a_2, \ldots ,a_n) \le PDHM ^{k_t}(b_1,b_2, \ldots ,b_n)\).

3. Boundedness: Given a set \(\{a_1,a_2, \ldots ,a_n\}\), whose \(\min _i a_i\) \(= a^-\) and \(\max _i a_i=a^+\), where \(i \in N\). We have \(a^- \le PDHM ^{k_t}(a_1,a_2, \ldots ,a_n) \le a^+.\)

Remark For \(k_t=1~\forall ~t\), the PHM reduces to the partitioned AM (PAM) and the PDHM to the partitioned GM (PGM). Further, if \(k_t=1~\forall ~t\) and the cardinality of all partition (\(|P_t|\)) are same, then the PHM and PDHM will reduce to the AM and GM, respectively.

Given a set \(\{\alpha _1,\alpha _2, \ldots ,\alpha _n\}\) of q-ROFNs corresponding to the attributes set \(\{\gamma _1, \gamma _2, \ldots , \gamma _n\}\) which can be partitioned into d-number of groups \(P_1, P_2, \ldots , P_d\). Then, the q-ROFPHM operator is a map from \(\left( \alpha _1,\alpha _2, \ldots ,\alpha _n\right) \) to \(\left( [0,1],[0,1]\right) \) and defined aswhere \(|P_t|\) denotes the cardinality of tth partition \(P_t=\{\gamma _{t_1}, \gamma _{t_2}, \ldots , \gamma _{t_{|P_t|}}\}\); \(k_t= 1, 2, \ldots ,|P_t|\) is the granularity parameter of partition \(P_t\); \(i_1, i_2, \ldots ,i_{k_t}\) are \(k_t\) natural numbers taken from the set \(\{1, 2, \ldots , |P_t|\}\) and \(C_{|P_t|}^{k_t}=\frac{|P_t|!}{k_t!(|P_t|-k_t)!}\) is the binomial coefficient.

1. Idempotency: For a set \(\{\alpha _1,\alpha _2, \ldots ,\alpha _n\}\) of q-ROFNs with \(\alpha _i= \alpha =(\mu ,\nu ),~\forall ~i \in N\). The \(q-ROFPHM ^{k_t}(\alpha _1,\alpha _2, \ldots ,\alpha _n)=\alpha .\)

2. Monotonicity: Given two sets \(\{\alpha _1,\alpha _2, \ldots ,\alpha _n\}\) and \(\{\alpha _1',\alpha _2', \ldots ,\alpha _n'\}\) of q-ROFNs, such that \(\alpha _i \le \alpha _i'\) i.e., \((\mu _i,\nu _i) \le (\mu _i',\nu _i')\) or \(\mu _i\le \mu _i'\) and \(\nu _i\ge \nu _i'\), \(\forall ~i \in N\). The \(q-ROFPHM ^{k_t}(\alpha _1,\alpha _2, \ldots ,\alpha _n)\) \(\le \) \(q-ROFPHM ^{k_t}(\alpha _1',\alpha _2', \ldots ,\alpha _n')\).

3. Boundedness: Given a set of q-ROFNs \(\{\alpha _1,\alpha _2, \ldots ,\alpha _n\}\), whose \(\left( \min _i \mu _i,~\max _i \nu _i \right) =\alpha ^-\) and \(\left( \max _i\mu _i,~\min _i \nu _i\right) =\alpha ^+\), where \(i \in N\). We have \(\alpha ^-\le q-ROFPHM ^{k_t}(\alpha _1,\alpha _2, \ldots ,\) \(\alpha _n)\le \alpha ^+\).

Given a set \(\{\alpha _1,\alpha _2, \ldots ,\alpha _n\}\) of q-ROFNs corresponding to the attributes set \(\{\gamma _1, \gamma _2, \ldots , \gamma _n\}\) which can be partitioned into d-number of groups \(P_1, P_2, \ldots , P_d\). Let \(w_i\in [0,1]\) represents the importance of the ith attribute (\(\gamma _i\)), such that \(\sum _{i=1}^nw_i=1\). The q-ROFWPHM operator is a map from \((\alpha _1,\alpha _2, \ldots ,\alpha _n)^n\) to ([0, 1], [0, 1]) and defined aswhere \(|P_t|\) denotes the cardinality of tth partition \(P_t=\{\gamma _{t_1}, \gamma _{t_2}, \ldots , \gamma _{t_{|P_t|}}\}\); \(k_t= 1, 2, \ldots ,|P_t|\) is the granularity parameter of the partition \(P_t\); \(i_1, i_2, \ldots ,i_{k_t}\) are \(k_t\) natural numbers taken from the set \(\{1, 2, \ldots , |P_t|\}\) and \(C_{|P_t|}^{k_t}=\frac{|P_t|!}{k_t!(|P_t|-k_t)!}\) is the binomial coefficient.

All three essential properties of an AO are satisfied by the q-ROFWPHM operator and can be easily proved.

Given a set \(\{\alpha _1,\alpha _2, \ldots ,\alpha _n\}\) of q-ROFNs corresponding to the attributes set \(\{\gamma _1, \gamma _2, \ldots , \gamma _n\}\) which can be partitioned into d-number of groups \(P_1, P_2, \ldots , P_d\). Then, the q-ROFPDHM operator is a map from \(\alpha _1,\alpha _2, \ldots ,\alpha _n\) to \(\left( [0,1],[0,1]\right) \) and defined aswhere \(|P_t|\) denotes the cardinality of tth partition \(P_t=\{\gamma _{t_1}, \gamma _{t_2}, \ldots , \gamma _{t_{|P_t|}}\}\); \(k_t= 1, 2, \ldots ,|P_t|\) is the granularity parameter of the partition \(P_t\); \(i_1, i_2, \ldots ,i_{k_t}\) are \(k_t\) natural numbers taken from the set \(\{1, 2, \ldots , |P_t|\}\) and \(C_{|P_t|}^{k_t}=\frac{|P_t|!}{k_t!(|P_t|-k_t)!}\) is the binomial coefficient.

Some elemental properties of the q-ROFPDHM operator are discussed hereafter:

1. Idempotency: For a set \(\{\alpha _1,\alpha _2, \ldots ,\alpha _n\}\) of q-ROFNs with \(\alpha _i=\alpha =(\mu ,\nu )\), \(\forall ~i \in N\). The \(q-ROFPDHM ^{k_t}(\alpha _1,\alpha _2, \ldots ,\) \(\alpha _n)=\alpha \).

2. Monotonicity: Given two sets, \(\{\alpha _1,\alpha _2, \ldots ,\alpha _n\}\) and \(\{\alpha _1',\alpha _2', \ldots ,\alpha _n'\}\) of q-ROFNs, such that \(\alpha _i \le \alpha _i'\) i.e., \((\mu _i,\nu _i) \le (\mu _i',\nu _i')\) or \(\mu _i\le \mu _i'\) and \(\nu _i\ge \nu _i'\), \(\forall ~i \in N\). The \(q-ROFPDHM ^{k_t}(\alpha _1,\alpha _2, \ldots ,\alpha _n)\) \(\le \) \(q-ROFPDHM ^{k_t}(\alpha _1',\alpha _2', \) \(\ldots ,\alpha _n')\).

3. Boundedness: Given a set of q-ROFNs, \(\{\alpha _1,\alpha _2, \ldots ,\alpha _n\}\) whose \(\left( \min _{i} \mu _i,~\max _{i}\nu _i \right) =\alpha ^-\) and \(\left( \max _{i}\mu _i,~\min _{i}\nu _i\right) =\alpha ^+\), where \(i \in N\). We have \(\alpha ^-\le q-ROFPDHM ^P(\alpha _1,\alpha _2, \ldots ,\) \(\alpha _n)\le \alpha ^+\).

In addition, the q-ROF dual Hamy mean (q-ROFDHM), the q-ROF geometric average (q-ROFGA), and the q-ROF averaging (q-ROFA) operators are some specific cases of the developed q-ROFPHM operator.

Case 1: For \(d=1\), that is \(k_t=k\) and \(|P_t|=n\), the q-ROFPDHM operator converts to the q-ROFDHM operatorCase 2: For \(d=1\) and \(k_t=k=1\), the q-ROFPHM operator converts to the q-ROFA operatorCase 3: For \(d=1\) and \(k_t=k=n\), the q-ROFPHM operator converts to the q-ROFGA operator

Given a set \(\{\alpha _1,\alpha _2, \ldots ,\alpha _n\}\) of q-ROFNs corresponding to the attributes set \(\{\gamma _1, \gamma _2, \ldots , \gamma _n\}\) which can be partitioned into d-number of groups \(P_1, P_2, \ldots , P_d\). Let \(w_i\in [0,1]\) represents the importance of the ith attribute (\(\gamma _i\)), such that \(\sum _{i=1}^nw_i=1\). The q-ROFWPDHM operator is a map from \((\alpha _1,\alpha _2, \ldots ,\alpha _n)\) to ([0, 1], [0, 1]) and defined aswhere \(|P_t|\) denotes the cardinality of tth partition \(P_t=\{\gamma _{t_1}, \gamma _{t_2}, \ldots , \gamma _{t_{|P_t|}}\}\); \(k_t= 1, 2, \ldots , |P_t|\) is the granularity parameter of the partition \(P_t\); \(i_1, i_2, \ldots ,i_{k_t}\) are \(k_t\) natural numbers taken from the set \(\{1, 2, \ldots , |P_t|\}\) and \(C_{|P_t|}^{k_t}=\frac{|P_t|!}{k_t!(|P_t|-k_t)!}\) is the binomial coefficient.

All three essential properties of an AO are satisfied by the q-ROFWPDHM operator and can be proved easily.

To illustrate the suggested MAGDM approach, an MAGDM problem of selecting the best personnel from Liu et al. [24] has been considered for the discussion. In this problem, four feasible alternatives, \(X_1, X_2, X_3,\) and \(X_4\), are there, which will be evaluated based on seven different attributes that capture the competency framework of teachers. All these benefit type attributes are learning ability \((\gamma _1)\), management innovation ability \((\gamma _2)\), communication and coordination ability \((\gamma _3)\), professional quality \((\gamma _4)\), moral character \((\gamma _5)\), professional knowledge level \((\gamma _6)\), and legal awareness \((\gamma _7)\). The weights \(w_j,~j=1,2,..,7\) assign to these attributes are 0.1, 0.1, 0.1, 0.2, 0.2, 0.2, and 0.1, respectively. Now, considering their interrelationship pattern, attributes’ set can be partitioned into two groups \(P_1\) and \(P_2\), where \(P_1=\{\gamma _1,\gamma _2,\gamma _3,\gamma _6\}\) and \(P_2=\{\gamma _4,\gamma _5,\gamma _7\}\) representing the individual ability level and the personal accomplishment, respectively. There are three DMs, \(D_1,D_2\), and \(D_3\), with their preference weights \(\omega _1=0.4, \omega _2=0.3\), and \(\omega _3=0.3\), respectively. The DM (\(D_k\)) evaluates each alternative \(X_i\) in accordance to every attribute \(\gamma _j\) and construct the decision matrices of q-ROFNs denoted by \(M^k=[\alpha _{ij}^k]=[(\mu _{ij}^k,\nu _{ij}^k)]_{4\times 7},~i=1,2,3,4;~j=1,2,..,7\) and \(k=1,2,3\). The decision matrices associated with three DMs are shown in Tables 3, 4, and 5, respectively. Now, the discussed MAGDM method has been applied to analyze this problem. The procedural steps are listed hereafter.

Step 1. Normalization of decision matrices:

The attributes of the selected problem are all benefit types. As a result, decision matrix normalization is not required.

Step 2. Calculation of comprehensive values \(({\tilde{\alpha }}_i^k)\):

Using Eqs. (18) and (19), calculate the comprehensive value \(({\tilde{\alpha }}_i^k)\) for each alternative \(X_i\) corresponding to DM-\(D_k\) using matrix \(M^k=[\alpha _{ij}^k]\), where j varies from \(1,2, \ldots ,n\). Here, we are given \(d=2\), \(|P_1|=4\) and \(|P_2|=3\). Since all the \(\alpha _{ij}^k\) are orthopair fuzzy numbers for \(1\le q<\infty \), so without loss of generality, choose \(q=1\). Also, the granularity parameter \(k_1=2\) and \(k_2=2\) are selected for computation. The computed values of \(({\tilde{\alpha }}_i^k)\) are provided in the Table 6 for q-ROFWPHM and q-ROFWPDHM operators, respectively.

Step 3. Computation of collective decision information \(({\tilde{\alpha }}_i)\):

As per the considered problem, there is no partitioning in the group of DMs, so \(d'=1\). For the computation, we have taken \(q=1\) and \(k_t'=2\). Then, apply the q-ROFWPHM and q-ROFWPDHM operators using Eqs. (20) and (21) to get a collective decision information \(({\tilde{\alpha }}_i)\) for the alternatives \(X_i\). The collective information is provided in Table 7.

Step 4. Calculation of score \(S({\tilde{\alpha }}_{i})\) and accuracy \(H({\tilde{\alpha }}_{i})\) values:

Use Eq. (2) to compute the score value \(S({\tilde{\alpha }}_{i})\) for each \({\tilde{\alpha }}_i\). The computed values are shown in Table 8.

Here, every score value is distinct, so we can proceed to the next step without calculating their accuracy values.

Step 5. Ranking of alternatives:

Utilizing the methodology described in the section “Score and accuracy functions”, rank the alternatives. The final ordering of alternatives based on the score values is summarized in Table 8.

The calculated results demonstrate that the suggested MAGDM strategy using the q-ROFWPHM and q-ROFW PDHM operators yields the same optimum choice (\(X_2\)) for the problem under consideration. Also, it is worth noticing that the score value of an alternative obtained by using the q-ROFWPDHM operator is higher than the score value from the q-ROFWPHM operator, which concludes that for optimistic types of decision preference, the q-ROFWPDHM operator can be applied. The q-ROFWPHM operator, the other operator, can aggregate all the information of a pessimistic nature. It signifies the role of the AOs in the proposed MAGDM method in the sense of flexibility based on DM preferences. The expert’s decision-making nature clearly impacts the final ordering of alternatives, which can be seen in Table 8.

The proposed MAGDM method is illustrated through a numerical example, adapted from [34], which involves selecting the most suitable industry for investment among five available options. This example serves to showcase the application and effectiveness of the proposed method.

To fully utilize dormant capital, the company’s board of directors made a strategic decision to invest in a new industry. Following preliminary research, five potential industries were identified as viable investment options. These alternatives encompass the medical industry (\(X_1\)), real-estate development industry (\(X_2\)), Internet industry (\(X_3\)), education and training industry (\(X_4\)), and manufacturing industry (\(X_5\)). To determine the optimal industry for investment, the board of directors assembled a panel of experts consisting of four individuals: \(D_1\), \(D_2\), \(D_3\), and \(D_4\). The relative significance of these experts was quantified using a weight vector \(\omega =(0.30, 0.22, 0.28, 0.20)\). Each of the four experts was tasked with evaluating the five alternative industries based on five attributes: capital profit potential (\(\gamma _1\)), market potential (\(\gamma _2\)), risk of capital loss (\(\gamma _3\)), growth potential (\(\gamma _4\)), and policy stability (\(\gamma _5\)). The weight vector \(w=(0.20, 0.20, 0.15, 0.25, 0.20)\) was utilized to assess the relative importance of these attributes. Based on the interrelationship structure, the five criteria were categorized into two partitions: \(P_1 = \{\gamma _1, \gamma _3, \gamma _5\}\) and \(P_2 = \{\gamma _2, \gamma _4\}\). Interrelationships were found among the three attributes within \(P_1\) and the two attributes within \(P_2\), while \(P_1\) and \(P_2\) remained independent. To allow for sufficient flexibility in evaluating the values of the attributes for each alternative industry, the experts were permitted to employ q-ROFNs. The evaluation outcomes provided by the four experts are individually documented in the following four matrices, as shown in Tables 18, 19, 20, and 21.

Now, the suggested MAGDM approach has been applied to analyze this problem. The procedural steps are listed hereafter.

Step 1. Normalization of decision matrices:

In this problem, four attributes \(\gamma _1\), \(\gamma _2\), \(\gamma _4\), and \(\gamma _5\) are of benefit types and one attribute \(\gamma _3\) is of cost type. Hence, the normalization of given data is essential and completed by utilizing Eq. 17.

Step 2. Calculation of comprehensive values \(({\tilde{\alpha }}_i^k)\):

Compute the comprehensive value \(({\tilde{\alpha }}_i^k)\) for each alternative \(X_i\) corresponding to DM \(D_k\) by utilizing the matrix \(M^k=[\alpha _{ij}^k]_{5\times 5}\) and Eqs. (18) and (19). Here, we are given \(d=2\), \(|P_1|=3\) and \(|P_2|=2\). Since the given information satisfy the orthopair fuzzy numbers condition for \(q\ge 3\), so without loss of generality, select \(q=3\). Also, select the granularity parameter \(k_1=2\) and \(k_2=2\). The computed values of \(({\tilde{\alpha }}_i^k)\) are provided in Tables 22 and 23 for q-ROFWPHM and q-ROFWPDHM operators, respectively.

Step 3. Computation of collective decision information \(({\tilde{\alpha }}_i)\):

As there is no partitioning of DMs group suggested in the problem, so implies \(d'=1\). For the further calculations, the values of parameters are \(q=3\) and \(k_t'=2\). Employ the q-ROFWPHM and q-ROFWPDHM operators using Eqs. (20) and (21) to get a collective decision information \(({\tilde{\alpha }}_i)\) for the alternatives \(X_i\). The collective information is provided in Table 24.

Step 4. Calculation of score \(S({\tilde{\alpha }}_{i})\) and accuracy \(H({\tilde{\alpha }}_{i})\) values:

Compute the score value \(S({\tilde{\alpha }}_{i})\) for each \({\tilde{\alpha }}_i\) using the score function mention in Eq. (2). The resultant values are shown in Table 25.

Step 5. Ranking of alternatives:

Using the approach discussed in the section“Score and accuracy functions”, rank the alternatives. The final ordering of alternatives is based on the score values and is summarized in Table 25.

Based on the computed results of problem 2, it is observed that the proposed MAGDM approach with either the q-ROFWPHM operator or the q-ROFWPDHM operator identifies the same best alternative, which is the Internet industry (\(X_2\)) following the medical industry (\(X_1\)). Internet industry investment offers significant growth potential, competitive advantages, favorable financial performance, expanding user engagement, and effective risk management strategies. However, it is important to note that investment decisions should also be tailored to individual risk tolerance, financial goals, and the specific investment opportunities available. Furthermore, comparing results between the q-ROFWPDHM operator and the q-ROFWPHM operator reveals a distinction between optimistic and pessimistic decision preferences. Specifically, the results obtained from the q-ROFWPDHM operator are higher than those obtained from the q-ROFWPHM operator, highlighting the differing optimism and pessimism inherent in the two operators.