To express the opinions of decision-makers, q-rung orthopair hesitant fuzzy sets (q-ROHFSs) have been employed extensively. Therefore, it is necessary to construct q-rung orthopair hesitant fuzzy preference relations (q-ROHFPRs) as a crucial decision-making tool for decision-makers. The goal of this paper aims to define a new consistency and consensus approach for solving q-ROHFPR group decision-making (GDM) problems. To do this, we first state the definitions of q-ROHFPRs and additive consistent q-ROHFPRs based on q-ROHFSs, an additive consistency index and acceptable additive consistent q-ROHFPRs. Second, based on minimizing the deviation, we establish an acceptable goal programming model for unacceptable additive consistent q-ROHFPRs. Third, an iterative algorithm is created for achieving acceptable consistency and reaching a rational consensus. The degree of rational consensus among individual q-ROHFPRs is quantified by a distance-based consensus index. Afterward, a non-linear programming model is formulated to derive the priority vector of alternatives, which are q-rung orthopair hesitant fuzzy numbers (q-ROHFNs). Based on this model, a GDM model for q-ROHFPRs is then developed. To demonstrate the validity and utility of the proposed GDM model, a case study on the risk assessment of hypertension is provided. The finding of sensitivity and comparison analyses supports the feasibility and efficacy of the suggested approach.
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Introduction
Decision-makers (DMs) find it extremely difficult and rigid to quantify preferences in terms of real numbers in real-life challenges due to the increasing complicacy of ambiguous information [1, 2]. In addition, the fuzzy sets (FSs) [3] have some restrictions that must be considered when DMs are reluctant to give specific numbers for their preference opinions. Torra [4] suggested the concept of the hesitant fuzzy sets (HFSs) that the hesitant membership degree is conveyed via some distinct values rather than a singular one, and it has been widely used to address the Multi-criteria decision-making (MCDM) and Multi-criteria group decision-making (MCGDM) problems [5‐8]. Li et al. [9] established a novel similarity degree for HFSs and discussed their properties. Xu and Xia [10, 11] investigated several novel aggregation operators of HFSs, and HFSs have been widely used in industrial production and daily life. Later on, researchers extended HFSs to some new fuzzy set types, such as hesitant Pythagorean fuzzy sets (HPFSs) [12], dual hesitant fuzzy sets (DHFSs) [13], hesitant fuzzy N-soft sets [14], interval-valued hesitant fuzzy sets (IVHFSs) [15], and interval-valued hesitant fuzzy N-Soft sets [16]. Especially, based on the idea of q-rung orthopair fuzzy sets (q-ROFSs) and HFSs [4, 55], researchers have further explored new fuzzy sets [17‐23], such as q-rung orthopair hesitant fuzzy sets (q-ROHFSs) [17] which can make DMs express ambiguous information freely and accurately, q-rung orthopair hesitant fuzzy rough sets (q-ROHFRSs) [18], q-rung orthopair probabilistic hesitant fuzzy rough sets (q-ROPHFRSs) [19], etc. In recent years, researchers have focused on combining q-ROFSs and HFSs because of the freedom and accuracy of expressing fuzzy information.
Researchers have focused a lot of attention on preference relations as a crucial decision-making tool. Since Satty [24] proposed the multiplicative preference relations (MPRs) which can be utilized to rid of ranking multiple alternatives, researchers paid a lot of attention to MCDM based on such preference relations [25, 26]. In literature, researchers are extending MPRs to FSs and HFSs in order to increase the decision-making process’s freedom and accuracy. For example, Liu et al. [27] defined the hesitant fuzzy preference relations (HFPRs) while Zhao et al. [28] stated the dual hesitant fuzzy preference relations (DHFPRs). Tang and Meng [29] developed the interval-valued hesitant fuzzy preference relations (IVHFPRs), while Gou [30] described double hierarchy hesitant fuzzy linguistic preference relations (DHHFLPRs) which are used to tackle MCDM problems in more complicated circumstances [31]. However, to date, there has been no research on the preference relations in which the preference judgments of DMs are represented by q‐ROHFSs. To make up this kind of gap, we propose the q‐ROHFPRs which can be used to solve MCDM or MCGDM problems in more complicated circumstances, and which makes DMs more freedom express judgment values.
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The main aspects of MCGDM problem solving are preference relationship, consistency, and consensus. However, when DMs use preference relations to deliver their evaluation data, it becomes exceedingly challenging to maintain consistency and consensus. As a result, numerous consistent optimization models for HFPRs, DHFPRs, IVHFPRs, and DHHFLPRs have been developed by researchers. For example, Zhang et al. [32] constructed several mixed-linear programming models for adjusting those HFPRs that did not satisfy additive consistency. Based on α-normalization methods, Zhu et al. [33] developed a programming algorithm to extract priorities from HFPRs and presented some consistency indexes to measure acceptable consistent degrees based on β-normalization methods. Li et al. [34] introduced an interval consistency index to evaluate the consistency degrees of hesitant fuzzy linguistic preference relations. Tang et al. [35] developed a new method for deriving priority from DHFPRs. In order to automatically adjust the consistency, Zhang et al. [36] constructed an automatic algorithm to adjust the unacceptable consistent HFPRs and applied the logarithmic least squares approach for deriving priorities. Meng and An [37] exploited an interactive algorithm based on multiplicative consistency and consensus HFPRs for dealing with GDM problems. In addition, Gou et al. [38] introduced an interval consistency index and an average consistency index of DHHFLPRs. For the MCGDM problem, some scholars have established optimization models based on consistency and consensus. Gou et al. [39] devised a double hierarchy linguistic distance-based clustering method for large-scale group decision-making. Wu et al. [40] put forward two heuristic approaches to achieve consistency and consensus level in an analytical hierarchy process. Zhang et al. [41] applied iterative algorithms for improving the consistency and consensus degree in GDM. He and Xu [42] established a non-linear programming model for MCGDM of hesitant MPR. Gou et al. [43] designed a feedback mechanism based to make whole individual DHHFLPRs with acceptable multiplicative consistency and a new consensus method based on distance measures to get the consensus degrees. Zhang et al. [44] developed two optimization models to generate adjustment advice for DMs in the consensus-reaching process. Li et al. [45] constructed a minimum adjustment optimization model to promote consensus. Gao and Zhang [46] developed a consensus-reaching algorithm with non-cooperative behavior management to deal with PIS-based social network GDM problems. Li et al. [47] studied a consensus approach to reaching consistency control for solving incomplete hesitant fuzzy linguistic preference relations GDM problems. Through investigation, the optimization methods of consistency and consensus of q-ROHFPRs need to be explored, which will be researched in this paper.
In light of the aforementioned studies, the q-ROHFSs have been considered a useful tool to express the judgments of DMs. At the same time, the preference relations between objects are considered an important decision-making tool to solve the priority of objects. However, for q-ROHHPRs, there are the following problems to be solved:
(1)
The q-ROHFNs give DMs more flexibility and freedom when expressing their membership degree and non-membership degrees with a variety of real numbers [17], and they have been widely used in practical production and life. However, the q-ROFHPRs are not given, which extends the definitions of HFPRs [27] and q-ROFPRs [48].
(2)
To the best of our knowledge, the additive consistent q-ROHFPRs, which can be useful for providing evaluation values between objects for DMs have not been explored. Also, the additive consistent preference relation has been widely used in production and real life because the priority of objectives can be solved by the optimization method.
(3)
The inconsistent q-ROHFPRs and low degree of consensus level of q-ROHFPRs cannot be resolved by the existing GDM methods in other fuzzy environments [29, 30] and therefore, the rationality of ranking outcomes and the consensus of individual viewpoints cannot be guaranteed.
From the above investigation and analysis, this study focuses on studying q-ROHFPRs, additive consistent q-ROHFPRs, acceptable additive consistent q-ROHFPRs, acceptable consensus, and constructing a group decision-making algorithm of q-ROHFPRs. The following is the list of the contributions to the proposed study:
(1)
A concept of q-ROHFPRs is introduced based on q-ROHFSs, and some of its properties are examined.
(2)
Inspired by Zhang [48], the additive consistent q-ROHFPRs are introduced and the acceptable additive consistent q-ROHFPRs are designed based on an additive consistent index. To get the acceptable additive consistent q-ROHFPRs, we establish an acceptable goal programming model based on minimizing the deviation for unacceptable additive consistent q-ROHFPRs.
(3)
We introduce the consensus index of q-ROHFPRs, and provide a consensus index for q-ROHFPRs. In order to obtain q-ROHFPRs with acceptable consensus, we develop an iterative algorithm.
(4)
A priority vector of alternatives expressed by acceptable additive consistent q-ROHFPRs is derived using a non-linear programming model.
(5)
A new MCGDM model for q-ROHFPRs is stated using a consensus iteration algorithm, an acceptable consistent goal programming model, and a priority vector optimization model.
(6)
A blood pressure risk assessment case is provided to verify the viability and efficacy of the presented GDM model. Results from sensitivity and comparison analyses demonstrate the viability and effectiveness of the proposed method.
The remainder of this paper is organized as follows: Sect. "Preliminaries" reviews the basic preliminaries, Sect. "Q-ROHFPRs and additive consistent q-ROHFPRs" presents the additive consistency and acceptable additive consistent goal programming models of q-ROHFPRs. Sect. "MCGDM model of q-ROHFPRs" constructs a consensus model and MCGDM model of q-ROHFPRs. A case is presented in Sect. "Numerical Example". Sect. "Conclusion" concludes our work.
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Preliminaries
Definition 1
[17]. Let \({\text{X}} = \left\{ {x_{1} ,x_{2} , \ldots ,x_{n} } \right\}\) be a q-ROHFS \(Q\) on \(X\) is shown in Eq. (1), \(q \ge 1\) and \(i = 1,2, \ldots ,n\).
In Eq. (1), \(g\left( {x_{i} } \right)\) and \(h\left( {x_{i} } \right)\) are two non-empty subsets on \(\left [ {0,1} \right]\), denoting non-membership and membership functions of \(Q\), respectively. For \(\forall x_{i} \in X\), \(h\left( {x_{i} } \right)\) and \(g\left( {x_{i} } \right)\) satisfy \(0 \le \left( {\mu^{ + } } \right)^{q} + \left( {v^{ + } } \right)^{q} \le 1\), \(0 \le \mu^{ + } \le 1\), \(0 \le v^{ + } \le 1\), where \(\mu \in h\left( {x_{i} } \right)\), \(v \in g\left( {x_{i} } \right)\),\({ }\mu^{ + } = {\rm max}_{{\mu \in h\left( {x_{i} } \right)}} \left\{ \mu \right\}\), \(v^{ + } = {\rm max}_{{v \in g\left( {x_{i} } \right)}} \left\{ v \right\}\). For \(\forall x_{i} \in X\), the hesitation function is described as \(\pi_{Q} \left( {x_{i} } \right) = \sqrt [q]{{1 - \mu^{q} - v^{q} }}\).
We denote such pair as a \(Q = \left( {h\left( x \right),g\left( x \right)} \right) \) which is named as a q-ROHFN.
Definition 2
[17]. The \(Q = \left( {h,g} \right)\),\({ }Q_{1} = \left( {h_{1} ,g_{1} } \right)\) and \(Q_{2} = \left( {h_{2} ,g_{2} } \right)\) are three q-ROHFNs, the basic operations of q-ROHFNs are defined as follows, where \(q \ge 1\) and \({\uplambda } > 0\).
[17]. The \(Q = \left( {h,g} \right) = \left( {\left\{ {\mu_{1} ,\mu_{2} , \ldots ,\mu_{N\left( h \right)} } \right\},\left\{ {v_{1} ,v_{2} , \ldots ,v_{N\left( g \right)} } \right\}} \right)\) is a q-ROHFN, then the score function of \(Q\) is given in Eq. (2) and the accuracy function of \(Q\) is given in Eq. (3).
In Eqs. (2) and (3), \(N\left( h \right){ }\) and \(N\left( g \right)\) mean the element numbers included in \(h\) and \(g\), separately.
Definition 4
[17]. The \(Q_{1} = \left( {h_{1} ,g_{1} } \right)\) and \(Q_{2} = \left( {h_{2} ,g_{2} } \right)\) are two q-ROHFNs, the comparison regulations are presented as follows:
(1)
If \({\text{S}}_{{Q_{1} }} > {\text{S}}_{{Q_{2} }}\), then \(Q_{1} > Q_{2}\);
(2)
If \({\text{S}}_{{Q_{1} }} < {\text{S}}_{{Q_{2} }}\), then \(Q_{1} < Q_{2}\);
(3)
If \({\text{S}}_{{Q_{1} }} = {\text{S}}_{{Q_{2} }}\), (a) if \({\text{D}}_{{Q_{1} }} > {\text{D}}_{{Q_{2} }}\), then \(Q_{1} > Q_{2}\); (b) if \({\text{D}}_{{Q_{1} }} < {\text{D}}_{{Q_{2} }}\), then \(Q_{1} < Q_{2}\); (c) if \({\text{D}}_{{Q_{1} }} = {\text{D}}_{{Q_{2} }}\), then \(Q_{1} = Q_{2}\).
Definition 5
[17]. The \(Q_{1} = \left( {h_{1} ,g_{1} } \right) = \Big( \Big\{ \mu_{1}^{s} \Big( x \Big) \Big\},\Big\{ v_{1}^{{\text{s}}} \Big( x \Big) \Big\} \Big)\) and \(Q_{2} = \left( {h_{2} ,g_{2} } \right) = \left( {\left\{ {\mu_{2}^{s} \left( x \right)} \right\},\left\{ {v_{2}^{{\text{s}}} \left( x \right)} \right\}} \right)\) are two q-ROHFNs, the \({\rm Hamming}\) distance of \(Q_{1}\) and \(Q_{2}\) is presented as Eq. (4),where \(s = 1,2, \ldots ,N\)
This section introduces the idea of q-ROHFPRs and describes its properties. Also, an additively consistent and acceptable consistent goal programming model of q-ROHFPRs is provided.
q-ROHFPRs
When DMs go with decisions on a group of alternatives \( X = \left\{ {x_{1} ,x_{2} , \ldots ,x_{{\text{n}}} } \right\}\), all of them make use of q-ROHFNs to express the preference relation between two alternatives. Inspired by q-ROHFSs [17] and q-ROFPRs [48], the concept of q-ROHFPRs is introduced in Definition 6.
Definition 6.
The q-ROHFPR \(A\) on \(X\) is characterized by a matrix \(A = \left( {a_{ij} } \right)_{n \times n} \subseteq X \times X\), where \(a_{ij} = \left( {h_{ij} ,g_{ij} } \right)\) is a q-ROHFN for all \(i,j = 1,2, \ldots ,n\). \(h_{ij} = \left\{ {\mu_{ij}^{s} } \right\}\) indicates a set of membership degrees, \(g_{ij} = \left\{ {v_{ij}^{s} } \right\}\) represents a set of non-membership degrees, \(x_{i} \in X\), \(s = 1, \ldots ,N\), \(q \ge 1 { }\) with conditions:
In Eq. (5), \( \mu_{ij}^{s} \in h_{ij} ,{ }v_{ij}^{s} \in g_{ij} ,\) and \(\mu_{ij}^{s}\) and \(v_{ij}^{s}\) represent the s-th element in \(h_{ij}\) and \(g_{ij}\), separately. The membership degree \(\mu_{ij}^{s}\) means \(x_{i}\) is preferred to \(x_{j}\), and the non-membership degree \(v_{ij}^{s}\) means \(x_{j}\) is preferred to \(x_{i}\). Inspired by q-ROFPRs [48], if \(\left( {\mu_{ij}^{s} } \right)^{q}\) = \(\left( {v_{ij}^{s} } \right)^{q}\) = \(0.5\) (\(s = 1, \ldots ,N\)), that means that the alternatives \(x_{i}\) and \(x_{j}\) are indistinguishable. And for any alternative itself that is indistinguishable, we can get \(\mu_{ii}^{s} = v_{ii}^{s} = \sqrt [q]{0.5}\), so the diagonal elements of q-ROHFPR are designed as \(\left( {\left\{ {\sqrt [q]{0.5}} \right\},\left\{ {\sqrt [q]{0.5}} \right\}} \right)\).
For each element given by DMs in q-ROHFPR, \({\mu }_{ij}^{s}\) and \({v}_{ij}^{s} \text{must satisfy}\)\(0\le {\mu }_{ij}^{s}\le 1, 0\le {v}_{ij}^{s}\le 1,\) and \({\left({\mu }_{ij}^{s}\right)}^{q}+{\left({v}_{ij}^{s}\right)}^{q}\le 1\). And the minimum integer \(q\) value can be determined by the traversal method in practical applications, for example, when the judgment value is \(\left(\left\{0.8\right\},\left\{0.7\right\}\right)\), the minimum integer value of q is 3( \({0.8}^{3}+{0.7}^{3}=0.855<1)\). The elements are ascended in \({h}_{ij}\) and \({g}_{ij}\), and \(N\) represents the element numbers of \({h}_{ij}\) and \({g}_{ij}\). The preference relation of n alternatives can be expressed as a \(n\times n\) matrix of q-ROHFPR, as shown in Eq. (6).
Especially, when \(\mu_{ij}^{1} = \mu_{ij}^{2} = \ldots = \mu_{ij}^{N} = \mu_{ij}\) and \(v_{ij}^{1} = v_{ij}^{2} = \ldots = v_{ij}^{N} = v_{ij}\), then the q-ROHFPR \(A\) retrogrades to a q-ROFPRs. q-ROHFPRs not only reflect the preferred direction of alternatives, but also express the strength of preference. To ensure the consistency of the DMs’ preference relations, Definition 6 should be satisfied \({ }N\left( {h_{ij} } \right) = N\left( {g_{ij} } \right)\). For three alternatives \(x_{i} ,x_{j} ,x_{k}\) which satisfy \( \mu_{ij}^{s} + \mu_{jk}^{s} + \mu_{ki}^{s} \ne \mu_{ik}^{s} + \mu_{kj}^{s} + \mu_{ji}^{s}\), the experts can easily determine the preference relation between two alternatives, but when there are more than three alternatives, it becomes more challenging. If each q-ROHFN includes the same number of preference membership degrees and preference non-membership degrees in the preference relation matrix, inspired by Zhang [48], the following six properties of q-ROFHPRs can be employed to determine the relationship among multiple alternatives which are presented by \(A\), where \(k,q \ge 1,i,j = 1,2, \ldots ,n\).
(1)
Triangle inequality property. If \(a_{ik} \oplus a_{kj} \ge a_{ij}\) is held, for \(\forall a_{ik} ,a_{ij} ,a_{kj} \in A,\) the triangle inequality of q-ROFHPRs is satisfied. Let alternatives \(x_{i} ,x_{j} ,x_{k}\) be three vertices of a triangle, and their length of sides are \(a_{ij} ,a_{ik}\), and \(a_{kj}\), then \(a_{ij}\) should be smaller than the sum of \(a_{ik}\) and \(a_{kj}\).
(2)
Weak transitivity property. Inspired by q-ROFPRs [48], in order to make q-ROHFPRs satisfy additive consistent, \(a_{ik} ,a_{ij} ,a_{kj}\) needs to compare with \(\left( {\left\{ {\sqrt [q]{0.5}} \right\},\left\{ {\sqrt [q]{0.5}} \right\}} \right) {\text{using}} \) Definitions 3 and 4. If \(a_{ik} \ge \left( {\left\{ {\sqrt [q]{0.5}} \right\},\left\{ {\sqrt [q]{0.5}} \right\}} \right)\) and \(a_{kj} \ge \left( {\left\{ {\sqrt [q]{0.5}} \right\},\left\{ {\sqrt [q]{0.5}} \right\}} \right) {\text{are}}\)\({\text{satisfied}},\) and then the \(a_{ij} \ge \left( {\left\{ {\sqrt [q]{0.5}} \right\},\left\{ {\sqrt [q]{0.5}} \right\}} \right)\) can be gotten, for \(\forall a_{ik} ,a_{ij} ,a_{kj} \in A\), the weak transitivity of q-ROFHPRs is satisfied. The property means that if alternative \(x_{i}\) is better than \(x_{k}\), and \(x_{k}\) is better than \(x_{j}\), then \(x_{i}\) should take precedence over \(x_{j}\). It is worth noting that weak transitivity is the minimum requirement to judge whether the q-ROHFPRs satisfy the consistency or not.
(3)
Max–min transitivity property. If \(a_{ij} \ge {\text{min}}\left\{ {a_{ik} ,a_{kj} } \right\}\) is held for \(\forall a_{ij} ,a_{ik} ,a_{kj} \in A\), the max–min transitivity of q-ROHFPRs is satisfied.
(4)
Max–max transitivity property. If \(a_{ij} \ge {\text{max}}\left\{ {a_{ik} ,a_{kj} } \right\}\) is held for \(\forall a_{ij} ,a_{ik} ,a_{kj} \in A\), the max–max transitivity of q-ROHFPRs is satisfied.
(5)
Restricted max–min transitivity property. If \(a_{ik} \ge \left( {\left\{ {\sqrt [q]{0.5}} \right\},\left\{ {\sqrt [q]{0.5}} \right\}} \right)\) and \(a_{kj} \ge \left( {\left\{ {\sqrt [q]{0.5}} \right\},\left\{ {\sqrt [q]{0.5}} \right\}} \right)\)\({\text{are held}}\), \({\text{then}} {\text{the}} a_{ij} \ge {\text{min}}\left\{ {a_{ik} ,a_{kj} } \right\}\) is also held for \(\forall a_{ik} ,a_{ij} ,a_{kj} \in A\), the restricted max–min transitivity of q-ROHFPRs is satisfied.
(6)
Restricted max–max transitivity property. If \(a_{ik} \ge \left( {\left\{ {\sqrt [q]{0.5}} \right\},\left\{ {\sqrt [q]{0.5}} \right\}} \right)\) and \(a_{kj} \ge \left( {\left\{ {\sqrt [q]{0.5}} \right\},\left\{ {\sqrt [q]{0.5}} \right\}} \right){ }\)\({\text{are held}}\), \({\text{and then the }}a_{ij} \ge {\text{max}}\left\{ {a_{ik} ,a_{kj} } \right\}\) is also held, for \(\forall a_{ik} ,a_{ij} ,a_{kj} \in A\), the restricted max-max transitivity of q-ROHFPRs is satisfied.
The triangular inequality of q-ROHFPRs is the vital relationship among the three alternatives in terms of the aforementioned characteristics of q-ROHFPRs, and weak transitivity is a prerequisite condition for q-ROHFPRs consistency.
Additive consistency of q-ROHFPRs
In the decision-making process, the lack of consistency may lead to fallacious conclusions, and the necessary condition of consistency should satisfy the transitivity, which states three alternatives \(x_{i} ,x_{j} ,x_{k}\). If alternative \(x_{i}\) is better than alternative \(x_{j}\), and alternative \(x_{j}\) is better than alternative \(x_{k}\), then we can get that the alternative \(x_{i}\) is better than alternative \(x_{k}\). There is often either additive transitivity or multiplicative transitivity based on the preference values provided by DMs. A definition of additive consistency of the q-ROHFPRs is put forth based on the additive transitivity definition of Zhang [48].
Definition 7.
Let \({\text{A}} = \left( {a_{ij} } \right)_{n \times n}\) be a q-ROHFPR matrix. If all elements meet Eq. (7), the q-ROHFPR A is additively consistent.
In Eq. (7), \(s = 1, \ldots ,N;{ }j,k,i = 1,2, \ldots ,n\). The \(\mu_{ij}^{s}\) and \(v_{ij}^{s}\) are the s-th membership degree of \(a_{ij}\) and the s-th non-membership degree of \(a_{ij}\), respectively. The Eq. (7) is derived from the fuzzy preference relation (FPR) \({\text{B}} = \left( {b_{ij} } \right)_{n \times n}\) additive transitivity definition \(b_{ij} + b_{jk} + b_{ki} = b_{ik} + b_{kj} + b_{ji}\) presented by Tanino [34]. What’s more, if there is an element in \(a_{ij}\), that is, when \(\left\{ {\mu_{ij}^{s} } \right\}\) and \(\left\{ {v_{ij}^{s} } \right\}\) in \(\left( {\left\{ {\mu_{ij}^{s} } \right\},\left\{ {v_{ij}^{s} } \right\}} \right)\) reduce to a real number \(\mu_{ij}\) and \(v_{ij}\), separately. And Eq. (7) can be turned into \(\left\{ {\begin{array}{*{20}c} {\mu_{ij} + \mu_{jk} + \mu_{ki} = \mu_{ik} + \mu_{kj} + \mu_{ji} } \\ {v_{ij} + v_{jk} + v_{ki} = v_{ik} + v_{kj} + v_{ji} } \\ \end{array} } \right.\), which is the same as the additive consistency equation of q-ROFPR in Zhang [48].
According to Definition 7 of the additively consistent q-ROHFPRs, another additive consistency of q-ROHFPRs is put forward.
Definition 8.
Let \({\text{A}} = \left( {a_{ij} } \right)_{n \times n}\) be a q-ROHFPR matrix, if it satisfies Eq. (8),
According to Definition 6, and \(\mu_{ij}^{s} = v_{ji}^{s}\), the Eq. (8) can be reduced to \(\left( {\mu_{ij}^{s} } \right)^{q} + \left( {\mu_{jk}^{s} } \right)^{q} + \left( {\mu_{ki}^{s} } \right)^{q} = \left( {v_{ij}^{s} } \right)^{q} + \left( {v_{jk}^{s} } \right)^{q} + \left( {v_{ki}^{s} } \right)^{q}\) for all \(i < j < k\), obviously \(\mu_{ij}^{s}\),\({ }\mu_{jk}^{s}\) are placed in the upper triangle of \({\text{A}}\), and \(\mu_{ki}^{s}\),\( \mu_{ji}^{s}\) are positioned in the lower triangle of \({\text{A}}\). Consequently, for the sake of computation, \(v_{ik}^{s}\) can be substituted with \(\mu_{ki}^{s}\), and similarly, \(\mu_{ik}^{s}\) can be substituted with \(v_{ki}^{s}\) to obtain the Eq. (8).
After that, considering that the probability of completely consistent q-ROHFPRs is rare in the decision-making process, we introduce the definition of acceptable additive consistency. Based on Eq. (8), the additive consistency index (CI) is presented as Definition 9.
Definition 9.
Let \({\text{A}} = \left( {a_{ij} } \right)_{n \times n} {\text{be}} \) a q-ROHFPR matrix, the \({\text{CI}}\left( {\text{A}} \right)\) is presented as Eq. (9).
where \(q \ge 1\), \(0 \le \left( {\mu_{ij}^{s} } \right)^{q} + \left( {v_{ij}^{s} } \right)^{q} \le 1\), \(s = 1, \ldots ,N\), \(j,k,i = 1,2, \ldots ,n.\) In Eq. (9), it can be easily inferred that \(0 \le {\text{CI}}\left( A \right) \le 1\)\(\left( {0 \le \left( {\mu_{ij}^{s} } \right)^{q} + \left( {v_{ij}^{s} } \right)^{q} \le 1} \right)\). And we can see that the smaller of the \({\text{CI}}\left( A \right)\), the higher consistency level of \(A\). The consistency of q-ROHFPR \(A\) is entirely additive if \({\text{CI}}\left( A \right) = 0\). Nevertheless, there are few completely additive consistent q-ROHFPRs in real GDM problems. Therefore, the CI is usually applied to judge the acceptable level of additive consistency, specifically, if \({\text{CI}}\left( {\text{A}} \right)\) is less than or equal to \(\overline{{{\text{CI}}}}\)(given in advance, it is known as the threshold), then q-ROHFPR \({\text{A}}\) is an acceptable additive consistency. According to the \(\overline{{{\text{CI}}}}\), the acceptable additive consistency of q-ROHFPR A is defined in Definition 10.
Definition 10.
Let \({\text{A}}\) be a q-ROHFPR matrix. If \({\text{CI}}\left( {\text{A}} \right) \le \overline{{{\text{CI}}}}\), then q-ROHFPR \({\text{A}}\) satisfies acceptable additive consistency. Otherwise, the q-ROHFPR \({\text{A}}\) does not satisfy additive consistency. If \({\text{CI}}\left( {\text{A}} \right) = 0\), the q-ROHFPR \({\text{A}}\) is completely additive consistent, where \(\overline{{{\text{CI}}}}\)(\(0 \le \overline{{{\text{CI}}}} \le 1\)) is given in advance.
In real life, there are very few situations where \({\text{CI}}\left( {\text{A}} \right)\) is equal to 0. Despite the decision cost, the value of \({\text{CI}}\left( {\text{A}} \right)\) is expected to be as low as possible. Yang [56] suggested \(\overline{{{\text{CI}}}} = 0.1 \) as the appropriate \(\overline{{{\text{CI}}}}\) that needs to be provided in front of the decision-making process.
Goal programming model solving acceptable consistent q-ROHFPRs
The consistent level of q-ROHFPRs shows their rationality during decision-making. However, it is difficult to meet acceptable consistency, and even more difficult to meet complete consistency in the experts’ judgment matrix. In order to obtain more reasonable results, a non-linear goal programming model is constructed to adjust the q-ROHFPR matrix \(A\) which is an unacceptable additive consistency, and to get the acceptable additive consistent q-ROHFPR \(\tilde{A} = \left( {\tilde{a}_{ij} } \right)_{n \times n}\) which should be closer to \(A\). According to Definition 5, the Hamming distance between q-ROHFPR \(A\) and q-ROHFPR \(\tilde{A}\) is described as Definition 11.
Definition 11.
Let \(A\) and \(\tilde{A}\) be two q-ROHFPR matrices. The distance between q-ROHFPR \(A\) and \(\tilde{A}\) is defined in Eq. (10).
In real-world decision-making, expressing more consistent preferences leads to more reasonable outcomes. Thus, for an unacceptable consistency matrix \(A\), we propose a new q-ROHFPR that satisfies the acceptable additive consistency and retains the original preference information as much as possible. According to Definition 6, we can get \(\tilde{\mu }_{ij}^{s} = \tilde{v}_{ji}^{s}\) and \(\tilde{\mu }_{ji}^{s} = \tilde{v}_{ij}^{s}\), and then it can be determined that the elements of the upper and lower triangle are symmetrical; thus, the consistency level of the upper triangle element of matrix \(A\) should be improved. From the perspective of goal programming, when the q-ROHFPR matrix \(A\) and the threshold \(\overline{CI}\) are given, the goal programming model is established as shown in model \(\left( {M - 1} \right)\), which minimizes the deviation between the adjusted q-ROHFPR \(\tilde{A} \) and the original q-ROHFPR \(A\).
where model \(\left( {M - 1} \right)\), the q-ROHFPR matrix \(A\) is offered by a DM, and the threshold \(\overline{CI}\) is predetermined. And the model \(\left( {M - 1} \right)\) is a non-linear goal programming model. \(\tilde{A}\) is an acceptable additive consistent q-ROHFPR matrix, and \(dev\left( {A,\tilde{A}} \right)\) is the objective function. And the \(\tilde{A}\) can be calculated by the model \(\left( {M - 1} \right)\) which satisfies acceptable additive consistency. Under the assumption that the constraints are satisfied, the model \(\left( {M - 1} \right)\) seeks to solve \(\tilde{A}\) with the smallest distance from \(A\). Based on the analysis presented above, the absolute deviation is minimized by the model \(\left( {M - 1} \right)\), in addition, the \(CI\left( {\tilde{A}} \right)\) should satisfy \(CI\left( {\tilde{A}} \right) \le \overline{CI}\). Substituting Eqs. (9) and (11) into model \(\left( {M - 1} \right)\), and the model \(\left( {M - 1} \right)\) can be represented as model \(\left( {M - 2} \right)\).
By computing model \(\left( {M - 2} \right)\), we can derive all top triangular matrix elements of \(\tilde{A}\), which is an acceptable additive consistent q-ROHFPRs. Subsequently, the complete matrix \(\tilde{A}\) can be obtained through Definition 6, which is the goal we are seeking.
MCGDM model of q-ROHFPRs
In this section, we stated an MCGDM problem using q-ROHFPRs. The rating of each expert is provided in terms of the q-ROHFPRs, to guarantee the decision is sensible, that must satisfy acceptable additive consistency. Aggregated matrices can be used to solve the optimal solution. To select the best alternative from multiple alternatives, we exploit an automatic method to measure and reach a consensus among the DMs’ opinions. For combining the individual acceptable additive consistent q-ROHFPRs into a collective matrix, an aggregation operator is first introduced. Then, a consensus index is proposed. Subsequently, a convergent iterative approach is proposed to obtain a collection of consensus matrices. The priority-based method is exploited to calculate the alternatives’ priority vectors. The MCGDM model is constructed to generate the final ranking of alternatives. To describe this MCGDM model more conveniently, the GDM method is given as follows:
Let \(X = \left\{ {x_{1} ,x_{2} , \ldots ,x_{n} } \right\}\) be alternatives set, \(D = \left\{ {d_{1} ,d_{2} , \ldots ,d_{m} } \right\}\) be DMs set, and the DM’s weight vectors satisfy \(\lambda_{t} \ge 0\),\(\mathop \sum \nolimits_{t = 1}^{m} \lambda_{t} = 1\left( {t = 1,2, \ldots ,m} \right)\). DM \(d_{t}\) offers an individual q-ROHFPR matrix \(A_{t} = \left( {a_{ijt} } \right)_{n \times n}\), where \(a_{ijt} = \left( {\mu_{ijt} ,v_{ijt} } \right)\). \( \mu_{ijt}\) indicates the membership degree of \(a_{ijt}\), and \(\mu_{ijt} = \left\{ {\mu_{ijt}^{1} ,\mu_{ijt}^{2} , \ldots ,\mu_{ijt}^{N} } \right\}\). \(v_{ijt}\) represents the non-membership degree of \(a_{ijt}\), and \(v_{ijt} = \left\{ {v_{ijt}^{1} ,v_{ijt}^{2} , \ldots ,v_{ijt}^{N} } \right\}\). Let \({ }\tilde{A}_{t} = \left( {\tilde{a}_{ijt} } \right)_{n \times n}\) be the acceptable additive consistent q-ROHFPR of \(A_{t}\) which is derived from model \(\left( {M - 2} \right)\), where \(\tilde{a}_{ijt} = \left( {\tilde{\mu }_{ijt} ,\tilde{v}_{ijt} } \right)\),\( \tilde{\mu }_{ijt} = \left\{ {\tilde{\mu }_{ijt}^{1} ,\tilde{\mu }_{ijt}^{2} , \ldots ,\tilde{\mu }_{ijt}^{N} } \right\}\) and \(\tilde{v}_{ijt} = \left\{ {\tilde{v}_{ijt}^{1} ,\tilde{v}_{ijt}^{2} , \ldots ,\tilde{v}_{ijt}^{N} } \right\}\). \(A = \left( {a_{ij} } \right)_{n \times n}\) be the collective matrix of \(\tilde{A}_{t}\), where \(a_{ij} = \left( {\mu_{ij} ,v_{ij} } \right)\), \(\mu_{ij} = \left\{ {\mu_{ij}^{1} ,\mu_{ij}^{2} , \ldots ,\mu_{ij}^{N} } \right\}\) and \(v_{ij} = \left\{ {v_{ij}^{1} ,v_{ij}^{2} , \ldots ,v_{ij}^{N} } \right\}\).\( A^{\prime} = \left( {a_{ij}^{{\prime}} } \right)_{n \times n}\) be the acceptable consensus q-ROHFPR, where \(a_{ij}^{{\prime}} = \left( {\mu_{ij}^{{\prime}} ,v_{ij}^{{\prime}} } \right)\), \(\mu_{ij}^{{\prime}} = \left\{ {\mu_{ij}^{1^{\prime}} ,\mu_{ij}^{2^{\prime}} , \ldots ,\mu_{ij}^{N^{\prime}} } \right\}\) and \(v_{ij}^{{\prime}} = \left\{ {v_{ij}^{1^{\prime}} ,v_{ij}^{2^{\prime}} , \ldots ,v_{ij}^{N^{\prime}} } \right\}\). Let \(\omega = \left( {\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} } \right)^{T} \) be the alternatives priority vector corresponding to \(A^{\prime}\), and their values are also q-ROHFNs, where \(\omega_{i} = \left( {\omega_{i\mu } ,\omega_{iv} } \right) = \left( {\left\{ {\omega_{i\mu }^{s} } \right\},\left\{ {\omega_{iv}^{s} } \right\}} \right)\),which satisfies \(\omega_{i\mu }^{s} \ge 0\), \(\mathop \sum \nolimits_{i = 1}^{n} \omega_{i\mu }^{s} = 1\), \(\omega_{iv}^{s} \ge 0\), \(\mathop \sum \nolimits_{i = 1}^{n} \omega_{iv}^{s} = 1\) and \(\left( {\omega_{i\mu }^{s} } \right)^{q} + \left( {\omega_{iv}^{s} } \right)^{q} \le 1\). where \(s = 1, \ldots ,N\), \(q \ge 1, j,i = 1,2, \ldots ,n\).
Consensus matrix of q-ROHFPRs
Consensus is the term for when individual ideas are unanimous and represent the group’s position. To obtain the representative ranking, it is necessary to give a commonly recognized analysis of group opinions. When individual q-ROHFPRs in the MCGDM problem are acceptable for additive consistency, it is assumed that aggregated q-ROHFPRs will similarly be acceptable for additive consistency. Therefore, an aggregation operator is put forward, as shown in Definition 13.
Definition 13.
Let q-ROHFPR matrix \(\tilde{A}_{t}\) be of acceptable additive consistency. \(A\) is the collective q-ROHFPR matrix of \(\tilde{A}_{t}\), which can be aggregated by the following Eq. (12).
According to Eqs. (12) and (13), we can get aggregated matrix \(A\), and prove that \(A\) satisfies acceptable additive consistency.
Theorem 1.
Suppose each q-ROHFPR matrix\(\tilde{A}_{t}\)meets acceptable additive consistency, the q-ROHFPR matrix A, computed by Eqs. (12) and (13), should meet the acceptable additive consistency.
To measure the consensus degree between individual opinions and group opinions, and obtain a scientific and effective ranking, based on Definition 11, the distance between \(\tilde{A}_{t}\) and \(A\) can be calculated as in Eq. (14).
In Eq. (14), \(s = 1, \ldots ,N\), \(j, i = 1, \ldots ,n, t = 1, \ldots ,m\). According to Eq. (14), the consensus index based on distance is introduced as Definition 14.
Definition 14.
Let \(\tilde{A}_{t} \) be the q-ROHFPRs matrix, and \(A \) be the aggregated q-ROHFPRs, then the consensus index \(\left( {GCI} \right)\) of \(\tilde{A}_{t}\) can be defined as
In Eq. (16), \(D\left( {\tilde{A}_{t} ,{\text{A}}} \right)\) is the distance between \(\tilde{A}_{t}\) and \({\text{A}}\). The less \(GCI\left( {\tilde{A}_{t} } \right)\) is, the more consensual the DMs’ views are. If \(GCI\left( {\tilde{A}_{t} } \right) = 0\), then the DMs’ viewpoints reach completely consensual.
However, it is a challenge for DMs to provide q-ROHFPRs meeting complete consensus during decision-making in reality, then the value of \(\overline{GCI}\)(\(0 \le \overline{GCI} \le 1\)) is given in advance which is called the consensus threshold, as shown in Definition 15. The \(\overline{GCI}\) will be utilized to determine whether q-ROHFPRs meet acceptable consensus levels.
Definition 15.
For a \(\overline{GCI} \left( {0 \le \overline{GCI} \le 1} \right)\) given in advance, if \(GCI\left( {\tilde{A}_{t} } \right) \le \overline{GCI}\), the \(\tilde{A}_{t}\) satisfies q-ROHFPR acceptable consensus. If not, the \(\tilde{A}_{t}\) is a q-ROHFPR unacceptable consensus. Especially, if \(GCI\left( {\tilde{A}_{t} } \right) = 0\), \(\tilde{A}_{t}\) is said to be a q-ROHFPR complete consensus.
In real life, it is rare to reach a complete consensus. In order to control the cost of decision-making, the \(\overline{GCI}\) needs to be predetermined before decision-making. In this paper, the smaller \(\overline{GCI}\) is, the higher the degree of reaching consensus is.
Consensus iterating algorithm
The most crucial step in the MCGDM decision-making process is reaching consensus; this requires satisfying the consistency requirements for each preference matrix as well as a consensus between the individual preference matrix and the aggregated matrix. If the evaluation opinions given by DMs differ significantly, the calculated results will not be convincing.
In an MCGDM environment, if \(GCI\left({\widetilde{A}}_{t}\right)>\overline{GCI}\), then \({\widetilde{A}}_{t}\) is a q-ROHFPR with unacceptable consensus. There may be an approach to handle this situation, that is, \({\widetilde{A}}_{t}\) must be returned to the DM \({d}_{t}\) to reconstruct a new q-ROHFPR, and follow this procedure until \(GCI\left({\widetilde{A}}_{t}\right)\le \overline{GCI}\). This way is reliable and accurate but impracticable because of too large an amount and too long a period of work needed. Inspired by Xu [49], we design a convergent iterative algorithm to achieve acceptable consensus among all the experts’ opinions.
In our approach, let \(\theta \left( {\theta \ge 1} \right)\) be iterative times, \(\theta_{max}\)\(\left( {\theta_{max} \ge 1} \right)\) be the maximum iterative times, \(\zeta \left( {0 < \zeta < 1} \right)\) be the adjusted parameter inspired by Xu [49], \(\tilde{A}_{t}^{\left( \theta \right)} = \left( {\tilde{a}_{ijt}^{\left( \theta \right)} } \right)_{n \times n}\) be the modified after \(\theta\) iterations that make \(\tilde{A}_{t}^{\left( \theta \right)}\) is an acceptable consensus q-ROHFPR, and \(\tilde{A}_{t}^{\left( 0 \right)} = \tilde{A}_{t}\), \(\tilde{A}_{t}^{\left( 0 \right)}\) be the initially acceptable additive consistent q-ROHFPR.
To find the acceptable consensus q-ROHFPR of an MCGDM problem, first set \(0 < \zeta < 1\) and \(\tilde{A}_{t}^{\left( 0 \right)} = \tilde{A}_{t}\), when \(\theta = 0\). And a new q-ROHFPR \(\tilde{A}_{t}^{{\left( {\theta + 1} \right)}}\) can be obtained from the \(\tilde{A}_{t}^{\left( \theta \right)}\) according to an iterative Eq. (17).
which is closer to the required acceptable consensus q-ROHFPR than the initial q-ROHFPR \(\tilde{A}_{t}^{\left( 0 \right)}\).
Let \(A^{\left( \theta \right)} = \left( {a_{ij}^{\left( \theta \right)} } \right)_{n \times n}\) be the modified collective q-ROHFPR after \(\theta\) iterations, and\(A^{\left( 0 \right)} = A\). \(A^{\prime} = \left( {a_{ij}^{{\prime}} } \right)_{n \times n}\) be the acceptable consensus collective q-ROHFPR. Then let \(\theta = \theta + 1\) and take the new q-ROHFPR as the initial q-ROHFPR, i.e., \(\tilde{A}_{t}^{{\left( {\theta + 1} \right)}} \to \tilde{A}_{t}^{\left( \theta \right)}\). Later, find \(\tilde{A}_{t}^{{\left( {\theta + 1} \right)}} { }\) again in the same way, repeating this process until \(GCI\left( {\tilde{A}_{t}^{\left( \theta \right)} { }} \right) \le \overline{GCI}\). Calculate the final collective q-ROHFPR \(A^{\left( \theta \right)}\), let \(A^{\prime} = A^{\left( \theta \right)}\), output \(\tilde{A}_{t}^{\left( \theta \right)} \) and \(A^{\prime}\). At this point, \(\tilde{A}_{t}^{\left( \theta \right)}\) can be taken as the acceptable consensus q-ROHFPR. The algorithm (Algorithm 1) for checking and improving the consensus is described as follows.
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The iterative algorithm described above converges. In addition, Theorem 2 can be demonstrated.
Theorem 2.
A q-ROHFPR\(\tilde{A}_{t}^{\left( \theta \right)}\)is an acceptable additive consistent and acceptable consensus if\(\tilde{A}_{t}^{\left( \theta \right)}\)is generated by Algorithm1.
The outstanding feature of Algorithm 1 is that it can automatically modify individual opinions to reach a consensus in the group opinions, and avoid forcing DMs to alter their decision, providing great convenience for its practical application.
Deriving priority vector of alternatives
To determine the ultimate rank of the alternatives, the priority vector should be determined. Once the collective q-ROHFPR has been obtained with acceptable additive consistency and acceptable consensus, the priority vectors can be derived from it. Assume that \(A^{\prime} = \left( {a_{ij}^{{\prime}} } \right)_{n \times n}\) is the collective q-ROHFPR matrix which satisfies acceptable additive consistency and consensus, and the elements of \(A^{\prime}\) are q-ROHFNs.
Based on Definition 7 and Ref. [48], we can consider the extracted matrices to derive priority vectors of q-ROHFPRs, therefore the extracted \({R}^{s}\) matrix can be defined as Definition 16.
Definition 16
For a collective q-ROHFPR \({A}^{{\prime}}\), let \({R}^{s}={({r}_{ij}^{s})}_{n\times n}\) be an extracted matrix from \({A}^{{\prime}}\), which are arranged at
In Eq. (18), \(s=1,\dots ,N\), and \(j,i={\rm 1,2},\dots ,n\). From Ref. [48], we can see that the sth matrix \({R}^{s}\) is a q-ROFPRs, and the diagonal elements of q-ROFPRs are \(\left(\sqrt [q]{0.5},\sqrt [q]{0.5}\right)\) which are extracted from \({A}^{{\prime}}\). The \({\mu }_{ij}^{s}\) is extracted from the s-th element in \({h}_{ij}\) of \({A}^{{\prime}}\), the \({v}_{ij}^{s}\) is extracted from the s-th element in \({g}_{ij}\) of \({A}^{{\prime}}\). Because \({A}^{{\prime}}\) is an acceptable additive consistency, we can future investigate that \({R}^{s}\) is an acceptable additive consistency.
Theorem 3
If\({A}^{{\prime}}\)be an acceptable additive consistent q-ROHFPR, then\({R}^{s}\)be a q-ROFPR with acceptable additive consistency.
It is easy to give proof of Theorem 3 based on Definition 9, so it is omitted here.
Because \({R}^{s}\) is an acceptable additive consistent matrix, we can derive the priority vector according to an additive consistent method. Motivated by Zhang et al. [48], the acceptable additive consistent q-ROFPR \(R^{s}\) can be obtained as
In Eq. (19), \(q \ge 1\), \(j,i = 1,\)\(\dots ,n\) and \(s=1,\dots , N\). \({w}_{iL}^{s}\) and \({w}_{iR}^{s}\) be the bottom and top margins of \({w}_{i}^{s}\), which stand for the bottom and top limitations of the extent of importance for an alternative \({x}_{i}\), separately. The interval weight vector of \({R}^{s}\) can be presented \({w}^{s}={\left({w}_{1}^{s},{w}_{2}^{s},\dots ,{w}_{n}^{s}\right)}^{T}={\left(\left [{w}_{1L}^{s},{w}_{1R}^{s}\right],\left [{w}_{2L}^{s},{w}_{2R}^{s}\right],\dots ,\left [{w}_{nL}^{s},{w}_{nR}^{s}\right]\right)}^{T}\). The subjective assessments of each DM, however, are usually not 100% accurate because of the existence of complexity, indeterminacy and subjectivity. In other words, Eq. (19) may not hold exactly to a certain degree. According to Eq. (19), we can generate the below deviation variables to measure the discrepancy between the interval weight vector and the q-ROHFPRs with acceptable additive consistency and acceptable consensus relations.
In Eq. (20), \(q \ge 1,{\text{and }} s = 1,2, \ldots , N;\)\(j,i = 1,2, \ldots ,n\). To obtain the interval priority vectors, based on Eq. (20), the absolute values of \(\sum\nolimits_{i=1}^{n-1}\sum\nolimits_{j=i+1}^{n}\sum\nolimits_{s=1}^{N}\left|{\alpha }_{ij}^{s}\right|+\left|{\beta }_{ij}^{s}\right|\) should be kept as small as possible. The smaller the absolute values, the higher the level of acceptable additive consistency. Consequently, the following non-linear programming model can be constructed for calculating the interval weight vector \({w}^{s}\), as shown in model (M-3).
The interval weight vector \(w^{s}\) should be acquired via computing the above model (M-3). Based on the relationship between intervals and q-ROFNs, \(w_{i}^{s} = \left [ {w_{iL}^{s} ,w_{iR}^{s} } \right]\) can be transformed into a q-ROFN \(\omega_{i}^{s} = \left( {\omega_{i\mu }^{s} ,\omega_{iv}^{s} } \right)\) by Eq. (21). Subsequently, the q-rung orthopair fuzzy (q-ROF) priority vectors \(\omega^{s} = \left( {\omega_{1}^{s} ,\omega_{2}^{s} , \ldots ,\omega_{n}^{s} } \right)^{T}\) of \(R^{s}\) can be derived after all \(\omega_{i}^{s}\) are calculated, where \(i = 1,2, \ldots ,n.\)
By model (M-3) and Eq. (21), the \({\omega }^{s}\) of \({R}^{s}\) can be solved. According to Definition 16 and Theorem 3, we combine \({\omega }^{s}\) for all \(s={\rm 1,2},\dots ,N\) to acquire the vector \(\omega \) of \({A}^{{\prime}}\), the \(\omega \) can be gotten by Eq. (22).
We make use of the approach for comparison and sorting of q-ROHFNs to get the final ranking result of alternatives.
MCGDM model of q-ROHFPRs
The proposed MCGDM is an integrated model designed for q-ROHFPRs fuzzy environment. In the decision environment of q-ROHFPRs, judgment matrices and DMs’ weights are given by DMs. Then Eq. (9) is used to check each judgment matrix whether satisfies acceptable additive consistency. If there is a judgment matrix that does not satisfy acceptable additive consistency, the model \(\left(M-2\right)\) is used to adjust the judgment matrix into acceptable additive consistent matrices. If all judgment matrices satisfy acceptable additive consistency, the aggregation operator is used to get a collective matrix by Eq. (13). If an acceptable additive consistent matrix does not reach consensus with the collective matrix, Algorithm 1 is used to adjust the judgment matrix until acceptable additive consistency and acceptable consensus are satisfied. Then, the priority vector of the alternatives is derived by formulating a non-linear programming model \(\left(M-3\right)\). According to scores of alternatives priority vector, the ranking result of alternatives is got. By the above discussion, the procedures of the presented approach for the MCGDM problem with q-ROHFPRs are described as follows, and Fig. 1 further illustrates the decision-making process of MCGDM.
Step 1: Invite m DMs to evaluate n alternatives and get \({A}_{t}\)\((t={\rm 1,2},\dots ,m)\).
Step 2: Compute \({\rm CI}\left({A}_{t}\right)\) by Eq. (9). Check whether or not \({A}_{t}\) is an acceptable additive consistency. If they are all acceptably additive consistent, set \({\widetilde{A}}_{t}={A}_{t}\), transfer to Step 4. If not, transfer to Step 3.
Step 3: Adjust \({A}_{t}\) via model \(\left(M-2\right)\), and get acceptable consistent \({\widetilde{A}}_{t}\)\((t={\rm 1,2},\dots ,m)\).
Step 4: Aggregate \({\widetilde{A}}_{t}\), and obtain the collective \(A\) based on Eq. (13).
Step 5: Compute \({\rm GCI}\left({\widetilde{A}}_{t}\right)\) utilizing Eq. (16). And check whether or not \({\widetilde{A}}_{t}\) is an acceptable consensus, let \({\widetilde{A}}_{t}={\widetilde{A}}_{t}^{(0)}\), and move to Step 7. If not, move to the next.
Step 6: Utilize Algorithm 1 to get the consensus index and obtain an adjusted \({\widetilde{A}}_{t}^{(\theta )}\).
Step 7: As per Eq. (13), the final aggregated matrix \({A}^{{\prime}}\) of \({\widetilde{A}}_{t}^{(\theta )}\) is determined.
Step 8: Compute the extracted q-ROFPR \({R}^{s}\) from \({A}^{{\prime}}\) by Eq. (18).
Step 9: The \({w}^{s}\) (\(s={\rm 1,2},\dots ,N\)) can be acquired by model \(\left(M-3\right)\).
Step 10: The \({\omega }^{s}\)(\(s={\rm 1,2},\dots ,N\)) can be gotten via Eq. (21,
Step 11: The \(\omega \) can be obtained by Eq. (22).
Step 12: Utilize Eqs. (2) and (3) to calculate the score \({{\rm S}}_{Q}\left({\omega }_{i}\right)\) and accuracy value \({{\rm D}}_{Q}\left({\omega }_{i}\right)\) of \({\omega }_{i}\), (\(i={\rm 1,2},\dots ,n\)).
Step 13: Rank \({x}_{i}\) according to \({{\rm S}}_{Q}\left({\omega }_{i}\right)\) and \({{\rm D}}_{Q}\left({\omega }_{i}\right)\), and get the best alternative.
Numerical example
A case is taken for confirming the efficacy of the proposed MCGDM method of q-ROHFPRs. Moreover, the comparative analysis highlights the proposed approach’s advantages.
Numerical example
The research results confirm that the blood pressure risk assessment of residents can well promote the management health of residents [50]. When assessing the blood pressure risk of a resident, it is difficult to determine the priority of the four indices: lack of exercise (\(x_{1}\)), the habit of drinking (\(x_{2}\)), the habit of smoking (\(x_{3}\)), and insufficient sleep (\(x_{4}\)). These four indices are regarded as four alternatives \(X = \left\{ {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right\}\). Three experts \(D_{t}\,\, \left( {t = 1,2,3} \right)\), whose weights are \(\lambda = \left( {0.3,0.2,0.5} \right)\), are invited to estimate all of the alternatives and then offer their preference information in the form of q-ROHFPRs by pairwise comparisons denoted as \(A_{t} \,\,\left( {t = 1,2,3} \right)\). To sort alternatives and get an optimal alternative, the proposed method will be applied. The steps of our algorithm are realized as follows.
Step 1: Invite 3 DMs to evaluate 4 alternatives and get \({A}_{1},{A}_{2},{A}_{3}\), as shown in follows.
As introduced in Definition 11, \({A}_{1}\) and \({A}_{2}\) are acceptable additive consistency, set \({\widetilde{A}}_{1}={A}_{1}\) and \({\widetilde{A}}_{2}={A}_{2}\), \({A}_{3}\) need to be adjusted.
Step 3: Obtain the derived acceptable consistent matrix \({\widetilde{A}}_{3}\) by model \(\left(M-2\right)\).
Step 4: Utilize Eq. (13) to get the collective matrix \(A\) of \({\widetilde{A}}_{1}\), \({\widetilde{A}}_{2}\) and\({\widetilde{A}}_{3}\). \(A\) is obtained as follows.
Step 5: Substitute \({\widetilde{A}}_{1}\), \({\widetilde{A}}_{2}\) and \({\widetilde{A}}_{3}\), and \(A\) into Eq. (16), the \(GCI\) of \({\widetilde{A}}_{1}\), \({\widetilde{A}}_{2}\) and \({\widetilde{A}}_{3}\) can be got.
Identify the threshold of consensus \(\overline{GCI} = 0.19\). According to Definition 16, \(\tilde{A}_{1}\) need to be improved consensus degree, set \(\tilde{A}_{2}^{\left( 0 \right)} = \tilde{A}_{2}\) and \(\tilde{A}_{3}^{\left( 0 \right)} = \tilde{A}_{3}\).
Step 6: Using Algorithm 1 and setting \(\zeta = 0.5\), the acceptable consensus q-ROHFPR of \(\tilde{A}_{1}^{\left( 1 \right)}\) can be composed as.
Step 9: By solving model\(\left( {M - 3} \right)\), the interval weight vector \(w^{1}\), \(w^{2}\), \(w^{3}\) of \(R^{1}\),\(R^{2}\),\(R^{3}\) (\(s = 1,2,3\)) can be obtained as follows.
Step 13: Rank alternatives, the ranking result is \(x_{2} > x_{3} > x_{1} > x_{4}\), and the \(x_{2}\) is optimal.
As it can be seen from the ranking order \(x_{2} > x_{3} > x_{1} > x_{4}\) of the given alternatives, \(x_{2}\) is the most significant index that is consistent with the actual evaluations of experts followed by \(x_{3}\) and \(x_{1}\). The final ranking order further demonstrates the viability of the proposed approach. First, the acceptable additive consistency improving model \(\left( {M - 2} \right)\) based on the minimum deviation can reserve the decision data to the greatest extent possible; in the preceding numerical example, the values \(\left\{ {0.2100,0.2160,0.2280} \right\}\) are reserved. For acceptable additive consistency q-ROHPRs, the value of \(\overline{CI}\) is set to 0.1 which is suggested by Yang [56]. Then, the \(\zeta\) is set to \(0.5\), and the modified q-ROHFPRs of acceptable consensus can be quickly obtained based on Algorithm 1. Finally, after checking and improving the acceptable additive consistent degree and the acceptable consensus level, and the priority vector is derived from a collective q-ROHFPR, then the final ranking result is determined: \(x_{2} > x_{3} > x_{1} > x_{4}\).
Sensitivity analysis
To verify the impact of different values of q, \(\overline{GCI}\) and \(\zeta\) on the ranking results, we conducted sensitivity analysis about q, \(\overline{GCI}\) and \(\zeta\) using the MCGDM approach presented in this paper.
With diverse values of q, the ranking results are derived with \(\overline{GCI} = 0.19\) and \(\zeta = 0.5\), as shown in Fig. 2.
Fig. 1
q-ROHFPRs Decision-making flow of MCGDM
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It can be seen from Fig. 2, the ranking result of alternatives is \(x_{2} > x_{3} > x_{1} > x_{4}\) for distinct q values, which means that no matter how q changes, the optimal alternative is \(x_{2}\). The ranking orders demonstrate that the method is steady, because the variations in q can hardly affect the GDM problems.
With diverse values of \(\overline{GCI}\), the scores of alternatives are generated with q = 1 and \(\zeta = 0.5\), as shown in Fig. 3. From Fig. 3, it can be observed that the ranking results of these alternatives are identical for various values of \(\overline{GCI}\), and the scores of alternatives are gradually stable.
Fig. 2
Score of alternatives when the q changes
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With diverse values of \(\zeta\), the scores of the alternatives are calculated when the \(\overline{GCI} = 0.19\) and \(q = 1\), and the ranking results are shown in Fig. 4.
Fig. 3
Score of alternatives when the \(\overline{GCI }\) changes
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It can be seen from Fig. 4, the scores of the alternatives gradually converge as \(\zeta\) increases, and the ranking results of alternatives are consistently \(x_{2} > x_{3} > x_{1} > x_{4}\) with different values of \(\zeta\).
To sum up, whatever the values of the q, \(\overline{GCI}\) and \(\zeta\) are taken, the optimal alternative is always \(x_{2}\). It’s evident that the final ranking of a group of alternatives remains constant as q, \(\overline{GCI}\) and \(\zeta\) change. In other words, the ranking obtained using the acceptable additive consistent q-ROHFPRs and the acceptable consensus q-ROHFPRs are effective and reasonable. Thus, the proposed GDM method is adaptable and reliable.
Comparative analysis with other GDM methods
To further validate the feasibility and effectiveness of the MCGDM method, comparison analyses are investigated to demonstrate the advantages of the proposed methods. The q-ROHFPRs is a special case of dual hesitant fuzzy preference relations. Considering that within the fuzzy context, there have been no investigations proposed in “Group decision-making with q-ROHFPRs”, we compare our GDM methods with those developed by Tang et al. [35] and Zhao et al. [28] whose fuzzy environments are similar to ours.
Tang et al. [35] proposed a priority derivation method that can address inconsistent and incomplete DHFPRs. The DHFPR matrix needs to be transformed into several IFPR matrices, which are then aggregated into an additive consistent IFPR matrix. According to the consistent and consensus adjustment method, the priority vector is solved. And the consensus index is.
Considering that the method proposed by Tang et al. [35] is distinct from our method, here the DMs’ weight vector is \(\lambda = \left( {0.3,0.2,0.5} \right),\overline{CI} = 0.1,{ }\overline{GCI} = 0.81\), the score function is Eq. (2). the scores of priority vector are \({\text{S}}_{Q} \left( {\omega_{1} } \right) = 0.0118,{\text{ S}}_{Q} \left( {\omega_{2} } \right) = 0.1425,{\text{ S}}_{Q} \left( {\omega_{3} } \right) = 0.0163,{\text{ S}}_{Q} \left( {\omega_{4} } \right) = - 0.1706.\) then the ranking of alternatives is \(x_{2} > x_{3} > x_{1} > x_{4}\), and the \(x_{2}\) is optimal. The ranking result is the same as our proposed method, which verifies the feasibility and effectiveness of our method.
Since Zhao et al. [28] intended to handle the scenarios in which the values of DMs’ weights are unknown, here the DMs’ weight vector is \(\lambda = \left( {0.3,0.2,0.5} \right)\). Considering the principles of these two methods are distinct, the method in Ref. [28] adopts the compatibility degree and aggregation operator. Some main steps in Ref. [28] differ from ours as follows:
(1) Different aggregation methods. The \(A_{t} \,\,\left( {t = 1,2,3} \right)\) are collected into \(A_{c}\) by Eq. (24).
And for facilitating the comparisons of distinct approaches, Fig. 5 shows the exact sorting orders of the same alternatives obtained from distinct methods when q = 1.
Fig. 4
Score of alternatives when the \(\zeta\) changes
Fig. 5
Scores of alternatives obtained from different methods
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From Fig. 5, it can be easily seen that the ranking orders \(\left({x}_{2}>{x}_{1}>{x}_{3}>{x}_{4}\right)\) of the alternatives obtained by Zhao et al.’s [28] method are identical to those obtained by the proposed method in this paper, which demonstrates the practicability and effectiveness of the proposed methods in dealing with the GDM problems with q-ROHFPRs.
However, the method proposed by Zhao et al. [28] failed to consider individual consistency, whereas the proposed method is being given considers both individual consistency and group consensus. If the q-ROHFPRs have unsatisfactory additive consistency, a non-linear goal programming model (M-2) is used to compute the q-ROHFPRs with acceptable additive consistency and consensus. Additionally, the proposed method assigns priority weights to alternatives to rank them. As a result, the proposed method is more comprehensive and reasonable.
Conclusion
In this research, a novel MCGDM technique for q-ROHFPRs is presented. It combines a consensus iteration algorithm, a non-linear priority vector deriving programming model, and an acceptable additive consistent goal programming model. We propose the notion of q-ROHFPRs based on q-ROHFSs and describe some of its fundamental properties. The definition of additive consistent q-ROHFPRs and the acceptable additive consistent q-ROHFPRs are suggested, and then non-linear goal programming models are developed to adjust the consistency degree of the unacceptable additive consistent q-ROHFPRs. It is also suggested to use an aggregation operator to integrate multiple q-ROHFPR matrices into a collective q-ROHFPR matrix, which ensures the consistency of the collective q-ROHFPRs. In order to increase the consensus level between an individual q-ROHFPR and a group of them, the acceptable consensus of q-ROHFPRs is also presented, and a consensus-reaching approach is developed. Afterward, a blood pressure risk assessment case is offered to illustrate the feasibility and efficiency of the presented GDM model. Sensitivity analysis results show that no matter what the values of the q, \(\overline{GCI }\) and \(\zeta \) are, the optimal alternative is always the same, and the ranking result of alternatives is the same, it verifies that the proposed GDM model is flexible and stable. The comparative analysis results demonstrate that the new method is effective compared with methods proposed by Tang et al. [35] and Zhao et al. [28].
Although the proposed additive consistency q-ROHFPR can be effective in solving the priority of the alternatives, when the preference relation matrix is incomplete [35], the additive consistency optimization model proposed in this paper cannot be solved, so solving the incomplete q-ROHFPR will be our focus. Additionally, the multiplicative consistent preference relation is employed to resolve the priority of alternatives [37]. The multiplicative consistent q-ROHFPR is also the subject of our attention in order to improve the application scenarios. In future study, we shall extend our work to diverse applications, such as wind power plant sites selection [51], industrial design [52], computer-aided instruction [53], drug selection [54], building developers [58, 59], shipping industry 4.0 [60], etc.
Acknowledgements
The ideas of this paper have been printed in advance in arXiv: “Group Decision-Making with q-rung orthopair hesitant fuzzy preference relations” [57] by the same authors who finish this paper.
Declarations
Conflicts of interest
These authors state that there have been no competing interests among them.
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This article does not contain any studies with human participants or animals performed by any of the authors.
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This manuscript is the authors' original work and has not been published nor has it been submitted simultaneously elsewhere.
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Suppose that all q-ROHFPRs \(\tilde{A}_{t}\) are acceptable additive consistency, based on Definition 10, \({\text{CI}}\left( {\tilde{A}_{t} { }} \right) \le \overline{{{\text{CI}}}}\) can be obtained. To demonstrate that \(A\) satisfies acceptable additive consistency, and then we should testify \({\text{CI}}\left( A \right) \le \overline{{{\text{CI}}}}\).
According to the definition of CI, the below is obtained
Thus, according to Definition 10, \(A\) satisfies acceptable additive consistency, Theorem 1 is hold.
Appendix 2
Proof of Theorem 2
Proof. It can be proved in two cases when \(q=1\) and \(q>1\) are integers. For \(GCI\left({\widetilde{A}}_{t}^{(\theta )}\right)>\overline{GCI }\), according to Definition 14, we can get
(1) When \(q=1\), let \({\varvec{G}}{\varvec{C}}{\varvec{I}}{\varvec{q}}1\left({\widetilde{A}}_{t}^{\left(\theta +1\right)}\right)= GCI\left({\widetilde{A}}_{t}^{\left(\theta +1\right)}\right)\), and \({t}^{{\prime}}=t ( {t}^{{\prime}}=1,\dots ,m)\), we can get
It is all known \({|x}^{q}-{y}^{q}|=|\left(x-y\right) \left({x}^{q-1}+{x}^{q-2}{\rm y}+{x}^{q-3}{y}^{2}+\cdots +x{y}^{q-2}+{y}^{q-1}\right)|\le q|\left(x-y\right)|\), when \(x\in \left [{\rm 0,1}\right]\), \(y\in \left [{\rm 0,1}\right]\). Because \(0<\upzeta <1\), then we can get
Then we can get \(GCI\left( {\tilde{A}_{t}^{{\left( {\theta + 1} \right)}} } \right) \le q\zeta GCIq1\left( {\tilde{A}_{t}^{\left( \theta \right)} } \right).\) Because \(GCIq1\left( {\tilde{A}_{t}^{\left( \theta \right)} } \right)\) is convergent, \( q\,\, {\text{is constant}}\), \({\text{ and }}0 < \zeta < 1\), \(GCI\left( {\tilde{A}_{t}^{{\left( {\theta + 1} \right)}} } \right)\) is convergent. We can get that Algorithm 1 converges, which can be completely proved by the proof of Theorem 2.
