Group Decision Making with Heterogeneous Preference Structures: An Automatic Mechanism to Support Consensus Reaching

Abstract

In real-world decision problems, decision makers usually express their opinions with different preference structures. In order to deal with the heterogeneous preference information in group decision making, this paper presents an optimization-based consensus model for group decision making with heterogeneous preference structures (utility values, preference orderings, multiplicative preference relations and additive preference relations). This proposal seeks to minimize the information loss between decision makers’ heterogeneous preference information and individual preference vectors and also seeks the collective solution with a consensus. Meanwhile, in order to justify the consensus model, we discuss its internal aggregation operator between the obtained individual and group preference vectors, demonstrate that the proposed model satisfies the Pareto principle of social choice theory, and prove the uniqueness of the solution to the optimization model. Furthermore, based on the proposed optimization-based consensus model, we present an automatic mechanism to support consensus reaching in the group decision making with heterogeneous preference structures. In the consensus reaching process, the obtained individual and group preference vectors are considered as a decision aid which decision makers can use as a reference to adjust their preference opinions. Finally, detailed simulation experiments and comparison analysis are conducted to demonstrate the feasibility and effectiveness of our proposed model.

Appendices

Appendix A: Proofs

Proof of Theorem 2

Without loss of generality, let $w_{i}^{k*}$ and $\overline{{w_{i}^{c*} }}$ be the optimal solution to the MILCM. Then, we have that $\hbox{min} \; \sum\nolimits_{k = 1}^{m} {\alpha_{k} \left( {w^{k} } \right)^{T} G^{k} w^{k} } = \sum\nolimits_{k = 1}^{m} {\alpha_{k} \left( {w^{k*} } \right)^{T} G^{k} w^{k*} }$ and $\sqrt {\frac{1}{2m}\sum\nolimits_{k = 1}^{m} {\sum\nolimits_{i = 1}^{n} {\left( {w_{i}^{k*} - \overline{{w_{i}^{c*} }} } \right)^{2} } } } \le 1 - \varepsilon .$

Let $F\left( {w_{1}^{c} ,w_{2}^{c} , \ldots ,w_{n}^{c} } \right) = \sqrt {\frac{1}{2m}\sum\nolimits_{k = 1}^{m} {\sum\nolimits_{i = 1}^{n} {\left( {w_{i}^{k*} - w_{i}^{c} } \right)^{2} } } }$. We have that the minimum value of F is

$$ \hbox{min} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} F = F\left( {\frac{1}{m}\sum\limits_{k = 1}^{m} {w_{1}^{k*} } ,\frac{1}{m}\sum\limits_{k = 1}^{m} {w_{2}^{k*} } , \ldots ,\frac{1}{m}\sum\limits_{k = 1}^{m} {w_{n}^{k*} } } \right). $$

Then, the following can be obtained

$$ \sqrt {\frac{1}{2m}\sum\limits_{k = 1}^{m} {\sum\limits_{i = 1}^{n} {\left( {w_{i}^{k*} - \frac{1}{m}\sum\limits_{k = 1}^{m} {w_{i}^{k*} } } \right)^{2} } } } \le \sqrt {\frac{1}{2m}\sum\limits_{k = 1}^{m} {\sum\limits_{i = 1}^{n} {\left( {w_{i}^{k*} - \overline{{w_{i}^{c*s} }} } \right)^{2} } } } \le 1 - \varepsilon $$

Therefore, $w_{i}^{k*}$ and $w_{i}^{c*} = \frac{1}{m}\sum\nolimits_{k = 1}^{m} {w_{i}^{k*} }$ is the optimal solution to the MILCM. This completes the proof of Theorem 2.

Proof of Theorem 3

Without loss of generality, according to Theorem 2, let $w_{i}^{c*} { = }\frac{1}{m}\sum\nolimits_{k = 1}^{m} {w_{i}^{k*} } \left( {i = 1,2, \ldots ,n} \right)$. Since $w_{i}^{k*} \ge w_{j}^{k*}$ for $k = 1,2, \ldots ,m$, we easily obtain that $w_{i}^{c*} \ge w_{j}^{c*}$. This completes the proof of Theorem 3.

Proof of Lemma 1

Let $\overline{{w_{i}^{k} }}$ and $\overline{{w_{i}^{c} }}$ be the optimal solution to the MILCM. Based on Theorem 2, $\overline{{w_{i}^{k} }}$ and $w_{i}^{c} = \frac{1}{m}\sum\nolimits_{k = 1}^{m} {\overline{{w_{i}^{k} }} }$ are also the optimal solution to the MILCM.

Because $w_{i}^{k*} \left( {k = 1,2, \ldots ,m;i = 1,2, \ldots ,n} \right)$ is the optimal solution to P₂, we have that

$$ \begin{aligned} \sqrt {\frac{1}{2m}\sum\limits_{k = 1}^{m} {\sum\limits_{i = 1}^{n} {\left( {w_{i}^{k*} - \frac{1}{m}\sum\limits_{k = 1}^{m} {w_{i}^{k*} } } \right)^{2} } } } \le 1 - \varepsilon ; \hfill \\ \frac{1}{m}\sum\limits_{k = 1}^{m} {w_{1}^{k*} } + \frac{1}{m}\sum\limits_{k = 1}^{m} {w_{2}^{k*} } + , \ldots , + \frac{1}{m}\sum\limits_{k = 1}^{m} {w_{n}^{k*} } = \frac{1}{m}\sum\limits_{k = 1}^{m} {\sum\limits_{i = 1}^{n} {w_{i}^{k*} } } = 1. \hfill \\ \end{aligned} $$

Thus, $w_{i}^{k*} \left( {k = 1,2, \ldots ,m;i = 1,2, \ldots ,n} \right)$ and $w_{i}^{c*} = \frac{1}{m}\sum\nolimits_{k = 1}^{m} {w_{i}^{k*} } \left( {i = 1,2, \ldots ,n} \right)$ is a feasible solution to the MILCM. Then, the following can be obtained $\sum\nolimits_{k = 1}^{m} {\alpha_{k} \, \left( {\overline{{w^{k} }} } \right)^{T} G^{k} \overline{{w^{k} }} } \le \sum\nolimits_{k = 1}^{m} {\alpha_{k} \, \left( {w^{k*} } \right)^{T} G^{k} w^{k*} }$. Based on the proof of Theorem 2, we can obtain that $\sqrt {\frac{1}{2m}\sum\nolimits_{k = 1}^{m} {\sum\nolimits_{i = 1}^{n} {\left( {\overline{{w_{i}^{k} }} - \frac{1}{m}\sum\limits_{k = 1}^{m} {\overline{{w_{i}^{k} }} } } \right)^{2} } } } \le \sqrt {\frac{1}{2m}\sum\limits_{k = 1}^{m} {\sum\limits_{i = 1}^{n} {\left( {\overline{{w_{i}^{k} }} - \overline{{w_{i}^{c} }} } \right)^{2} } } } \le 1 - \varepsilon .$

Thus, $\overline{{w_{i}^{k} }} \left( {k = 1,2, \ldots ,m,i = 1,2, \ldots ,n} \right)$ is also a feasible solution to P₂. Consequently, we can obtain that $\sum\nolimits_{k = 1}^{m} {\alpha_{k} \, \left( {\overline{{w^{k} }} } \right)^{T} G^{k} \overline{{w^{k} }} } \ge \sum\nolimits_{k = 1}^{m} {\alpha_{k} \, \left( {w^{k*} } \right)^{T} G^{k} w^{k*} }$. Thus, we have that $\sum\nolimits_{k = 1}^{m} {\alpha_{k} \, \left( {w^{k*} } \right)^{T} G^{k} w^{k*} } = \sum\nolimits_{k = 1}^{m} {\alpha_{k} \, \left( {\overline{{w^{k} }} } \right)^{T} G^{k} \overline{{w^{k} }} }$. Therefore, $w_{i}^{k*} \left( {k = 1,2, \ldots ,m;i = 1,2, \ldots ,n} \right)$ and $w_{i}^{c*} = \frac{1}{m}\sum\nolimits_{k = 1}^{m} {w_{i}^{k*} } \left( {i = 1,2, \ldots ,n} \right)$ is the optimal solution to the MILCM, and $w_{i}^{k*} \left( {k = 1,2, \ldots ,m;i = 1,2, \ldots ,n} \right)$ is also the optimal solution to $w_{i}^{k} \left( {k = 1,2, \ldots ,m;i = 1,2, \ldots ,n} \right)$ in the MILCM.

Proof of Lemma 3

Let $w = \left( {w^{1} ,w^{2} , \ldots ,w^{m} } \right)^{T}$. According to Theorem 1 and Eq. (7), we can obtain that

$$ \begin{aligned} & w^{T} H\left( w \right)w = w^{T} \nabla^{2} f{\kern 1pt} w = 2w^{T} G{\kern 1pt} w = 2f \\ & \quad = 2\sum\limits_{k = 1}^{{m_{1} }} {\alpha_{k} \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {\left( {w_{i}^{k} u_{j}^{k} - w_{j}^{k} u_{i}^{k} } \right)^{2} } } } + 2\sum\limits_{{k = m_{1} + 1}}^{{m_{2} }} {\alpha_{k} \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {\left( {w_{i}^{k} \left( {n - o_{j}^{k} } \right) - w_{j}^{k} \left( {n - o_{i}^{k} } \right)} \right)^{2} } } } \\ & \quad \quad + 2\sum\limits_{{k = m_{2} + 1}}^{{m_{3} }} {\alpha_{k} \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {\left( {w_{i}^{k} - w_{j}^{k} a_{ij}^{k} } \right)^{2} } } } + 2\sum\limits_{{k = m_{3} + 1}}^{m} {\alpha_{k} \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {\left( {w_{i}^{k} - \left( {w_{i}^{k} + w_{j}^{k} } \right)p_{ij}^{k} } \right)^{2} } } } \\ \end{aligned} $$

where H is the Hessian matrix of the objective function.

Because for any i, j and k, there exists at least one of the inequalities $w_{i}^{k} u_{j}^{k} \ne w_{j}^{k} u_{i}^{k} ,k \in \left[ {1,m_{1} } \right]$, $w_{i}^{k} \left( {n - o_{j}^{k} } \right) \ne w_{i}^{k} ,k \in \left[ {m_{1} + 1,m_{2} } \right]$, $w_{i}^{k} \ne w_{j}^{k} a_{ij}^{k} ,k \in \left[ {m_{2} + 1,m_{3} } \right]$ and $w_{i}^{k} \ne \left( {w_{i}^{k} + w_{j}^{k} } \right)p_{ij}^{k} ,k \in \left[ {m_{3} + 1,m} \right]$, we can obtain that $w^{T} H\left( w \right)w > 0$. Thus, the objective function of P₂ is a strictly convex function in the feasible region. Thus, this completes the proof of Lemma 3.

Proof of Theorem 4

Since $\varepsilon \in \left[ {0,1} \right]$, let $w_{i}^{1} = w_{i}^{2} = \cdots = w_{i}^{m} \left( {i = 1,2, \ldots ,n} \right)$, $\sum\nolimits_{i = 1}^{n} {w_{i}^{k} } = 1$ and $w_{i}^{k} \ge 0$. Then, $\sqrt {\frac{1}{2m}\sum\limits_{k = 1}^{m} {\sum\limits_{i = 1}^{n} {\left( {w_{i}^{k} - \frac{1}{m}\sum\limits_{k = 1}^{m} {w_{i}^{k} } } \right)^{2} } } } = 0 \le 1 - \varepsilon$. Thus, $w_{i}^{k} \left( {k = 1,2, \ldots ,m,i = 1,2, \ldots ,n} \right)$ is a feasible solution to P₂.

Let $\varOmega$ be the feasible region of P₂. Based on Lemmas 2 and 3, the feasible region of P₂ is a closed convex set and the objective function of P₂ is a strictly convex function in the feasible region. Thus, there exists an optimal solution to P₂ in the feasible region. Without loss of generality, let $\overline{{w_{i}^{k} }} \left( {k = 1,2, \ldots ,m;i = 1,2, \ldots ,n} \right)$ be the optimal solution to P₂, and let $\sum\nolimits_{k = 1}^{m} {\alpha_{k} \, \left( {\overline{{w^{k} }} } \right)^{T} G^{k} \overline{{w^{k} }} } = f^{*}$. Then, we can obtain that $\sum\nolimits_{k = 1}^{m} {\alpha_{k} \, \left( {w^{k} } \right)^{T} G^{k} w^{k} } \ge f^{*}$. Suppose $\overline{\overline{{w_{i}^{k} }}} \left( {k = 1,2, \ldots ,m;i = 1,2, \ldots ,n} \right)$ is also the optimal solution to P₂. We can obtain that $\sum\limits_{k = 1}^{m} {\alpha_{k} {\kern 1pt} \left( {\overline{\overline{{w^{k} }}} } \right)^{T} G^{k} \overline{\overline{{w^{k} }}} } = f^{*}$. Let $\underline{{w_{i}^{k} }} = \alpha \overline{{w_{i}^{k} }} + \left( {1 - \alpha } \right)\overline{\overline{{w_{i}^{k} }}} \left( {k = 1,2, \ldots ,m;i = 1,2, \ldots ,n} \right),\alpha \in \left( {0,1} \right)$. Because the feasible region of P₂ is a closed convex set, we have that $\underline{{w_{i}^{k} }} \left( {k = 1,2, \ldots ,m;i = 1,2, \ldots ,n} \right) \in\Omega$.

Because the objective function of P₂ is a strictly convex function in the feasible region, the following can be obtained $\sum\nolimits_{k = 1}^{m} {\alpha_{k} \, \left( {\underline{{w^{k} }} } \right)^{T} G^{k} \underline{{w^{k} }} } < \alpha \sum\nolimits_{k = 1}^{m} {\alpha_{k} \, \left( {\overline{{w^{k} }} } \right)^{T} G^{k} \overline{{w^{k} }} } + \left( {1 - \alpha } \right)\sum\nolimits_{k = 1}^{m} {\alpha_{k} \, \left( {\overline{\overline{{w^{k} }}} } \right)^{T} G^{k} \overline{\overline{{w^{k} }}} } = f^{*}$. This contradicts the fact that $f^{*}$ is the minimum value in Ω. Thus, $\overline{{w_{i}^{k} }} \left( {k = 1,2, \ldots ,m;i = 1,2, \ldots ,n} \right)$ is the unique optimal solution to P₂ in $\varOmega$. This completes the proof of Theorem 4.

Appendix B: OBM Presented by Fan et al. (2004) and Ma et al. (2006)

The OBM has been proposed in Fan et al. (2004) and Ma et al. (2006). The basic notations in the OBM are as follows. Let $G^{k} { = }\left( {g_{ij}^{k} } \right)_{n \times n} ,k = 1,2, \ldots ,m$, where $g_{ij}^{k}$ is calculated according to Eqs. (11–18). Let $G = \sum\nolimits_{k = 1}^{m} {\alpha_{k} G^{k} }$, where $\alpha_{k}$ is calculated according to Eq. (19). Let $e = \left( {1,1, \ldots ,1} \right)^{T}$. Let $w^{k} = \left( {w_{1}^{k} ,w_{2}^{k} , \ldots ,w_{n}^{k} } \right)^{T}$ be the individual preference vector of $e_{k}$. Let $w^{c} = \left( {w_{1}^{c} ,w_{2}^{c} , \ldots ,w_{n}^{c} } \right)^{T}$ be the group preference vector.

The OBM seeks to find the group preference vector based on Model (A1):

$$ \left\{ \begin{array}{l} \hbox{min} \, \left( {w^{c} } \right)^{T} Gw^{c} \hfill \\ s.t.\left\{ \begin{array}{l} e^{T} w^{c} = 1 \hfill \\ w^{c} \ge 0 \hfill \\ \end{array} \right. \hfill \\ \end{array} \right. $$

(A1)

where the group preference vector $w^{c}$ is the decision variables in Mode (A1).

According to Eqs. (11–19), the objective function of Model (A1) is to minimize the information loss between decision makers’ heterogeneous preference information and group preference vectors. Ma et al. (2006) showed that the optimal group preference vector $w^{c}$ can be obtained by Eq. (A2):

$$ w^{c} = \frac{{G^{ - 1} e}}{{e^{T} G^{ - 1} e}} $$

(A2)

Similar to Model (A1), the OBM can be used to find the optimal individual preference vector $w^{k}$, which can be described by Model (A3):

$$ \left\{ \begin{array}{l} \hbox{min} \, \left( {w^{k} } \right)^{T} G^{k} w^{k} \hfill \\ s.t.\left\{ \begin{array}{l} e^{T} w^{k} = 1 \hfill \\ w^{k} \ge 0 \hfill \\ \end{array} \right. \hfill \\ \end{array} \right. $$

(A3)

where the individual preference vector $w^{k}$ is the decision variable in Mode (A3).

In a similar way, the individual preference vectors $w^{k} \left( {k = 1,2, \ldots ,m} \right)$ can be obtained by Eq. (A4):

$$ w^{k} = \frac{{\left( {G^{k} } \right)^{ - 1} e}}{{e^{T} \left( {G^{k} } \right)^{ - 1} e}} $$

(A4)

Thus, based on Eqs. (A2) and (A4), the OBM can obtain individual preference vectors $w^{k} \left( {k = 1,2, \ldots ,m} \right)$ and the group preference vector $w^{c} .$

Appendix C: The IR-DR CM: Algorithm 2

Herrera-Viedma et al. (2002) proposed a consensus model in GDM with heterogeneous preference structures, in which the IR and DR are employed. Its basic notations are as follows. Let $P_{{}}^{k}$ and $P_{{}}^{c}$ be the individual and collective additive preference relations. Let wd and wa be the associated weight vector of decision makers and alternatives. Let $\alpha$ be the adjustment parameter. Let $w^{k} = \left( {w_{1}^{k} ,w_{2}^{k} , \ldots ,w_{n}^{k} } \right)^{T}$ be the individual preference vector of $e_{k}$.

In the IR-DR CM, the transformation functions, presented by Chiclana et al. (1998, 2001), are used to transform heterogeneous preference information into additive preference relations. The individual additive preference relations $P_{{}}^{k} = \left( {p_{ij}^{k} } \right)_{n \times n} \left( {k = 1,2, \ldots ,m} \right)$ can be obtained by Eq. (A5):

$$ p_{ij}^{k} = \left\{ {\begin{array}{*{20}l} {{{\left( {u_{i}^{k} } \right)^{2} } \mathord{\left/ {\vphantom {{\left( {u_{i}^{k} } \right)^{2} } {\left( {\left( {u_{i}^{k} } \right)^{2} + \left( {u_{j}^{k} } \right)^{2} } \right)}}} \right. \kern-0pt} {\left( {\left( {u_{i}^{k} } \right)^{2} + \left( {u_{j}^{k} } \right)^{2} } \right)}}} \hfill & {e_{k} \in E^{U} } \hfill \\ {p_{ij}^{k} = 0.5\left( {1 + {{\left( {o_{j}^{k} - o_{i}^{k} } \right)} \mathord{\left/ {\vphantom {{\left( {o_{j}^{k} - o_{i}^{k} } \right)} {\left( {n - 1} \right)}}} \right. \kern-0pt} {\left( {n - 1} \right)}}} \right)} \hfill & {e_{k} \in E^{O} } \hfill \\ {p_{ij}^{k} = 0.5\left( {1 + \log_{9} a_{ij}^{k} } \right)} \hfill & {e_{k} \in E^{A} } \hfill \\ \end{array} } \right. $$

(A5)

Then, the IR-DR CM includes two processes:

• (1) Selection process

In the selection process, we use ordered weighting operators (OWA) to aggregate decision makers’ opinions. An OWA operator of dimension n is a function as $OWA_{w} \left( {x_{1} ,x_{2} , \ldots ,x_{n} } \right) = \sum\nolimits_{i = 1}^{n} {d_{i} y_{\left( i \right)} }$, where $y_{\left( i \right)}$ is the ith largest element of $\left\{ {y_{1} ,y_{2} , \ldots ,y_{n} } \right\}$, and $d = \left( {d_{1} ,d_{2} , \ldots ,d_{n} } \right)^{T}$ is the associated weight vector, such that $d_{i} \ge 0\left( {i = 1,2, \ldots ,n} \right)$ and $\sum\nolimits_{i = 1}^{n} {d_{i} } = 1$. Linguistic quantifiers are used to calculate the weight vector of decision makers $wd$ and the weight vector of alternatives $wa$ in the IR-DR CM.

The collective additive preference relation $P_{{}}^{c} = \left( {p_{ij}^{c} } \right)_{n \times n}$ can be obtained by Eq. (A6):

$$ p_{ij}^{c} = OWA_{wd} \left( {p_{ij}^{1} ,p_{ij}^{2} , \ldots ,p_{ij}^{m} } \right) $$

(A6)

The individual and group preference vectors $w^{k}$ and $w^{c}$ can be obtained by Eqs. (A7) and (A8):

$$ w_{i}^{k} = OWA_{wa} \left( {p_{i1}^{k} ,p_{i2}^{k} , \ldots ,p_{in}^{k} } \right) $$

(A7)

$$ w_{i}^{c} = OWA_{wa} \left( {p_{i1}^{c} ,p_{i2}^{c} , \ldots ,p_{in}^{c} } \right) $$

(A8)

The individual and group preference vectors are transformed into the standardized individual and group preference vectors, $w^{k*}$ and $w^{c*}$, where

$$ w_{i}^{k*} = {{w_{i}^{k} } \mathord{\left/ {\vphantom {{w_{i}^{k} } {\sum\limits_{i = 1}^{n} {w_{i}^{k} } }}} \right. \kern-0pt} {\sum\limits_{i = 1}^{n} {w_{i}^{k} } }} $$

(A9)

$$ w_{i}^{c*} = {{w_{i}^{c} } \mathord{\left/ {\vphantom {{w_{i}^{c} } {\sum\limits_{i = 1}^{n} {w_{i}^{c} } }}} \right. \kern-0pt} {\sum\limits_{i = 1}^{n} {w_{i}^{c} } }} $$

(A10)

• (2) Consensus process

Let

$$ DM\left( {e_{k} } \right) = \sqrt {\frac{1}{2}\sum\limits_{i = 1}^{n} {\left( {w_{i}^{k*} - w_{i}^{c*} } \right)^{2} } } $$

(A11)

If $DM\left( {e_{q} } \right) = \mathop {\hbox{max} }\limits_{k} \left\{ {DM\left( {e_{k} } \right)|k = 1,2, \ldots ,m} \right\}$, based on the IR, we identify the decision maker $e_{q}$ who has the worst consensus level. Based on the DR, we guide the decision maker $e_{q}$ to move his/her opinion to the collective opinion.

In the IR-DR CM, we repeat the selection process and consensus process until all the decision makers’ opinions reach the consensus threshold.

The detailed IR-DR CM can be described as Algorithm 2.

Algorithm 2.

• Input: The original preference information $U^{k} \left( {k = 1,2, \ldots ,m_{1} } \right)$, $O^{k} \left( {k = m_{1} + 1,m_{1} + 2, \ldots ,m_{2} } \right)$, $A^{k} \left( {k = m_{2} + 1,m_{2} + 2, \ldots ,m_{3} } \right)$, $P^{k} \left( {k = m_{3} + 1,m_{3} + 2, \ldots ,m} \right)$, the established consensus threshold $\varepsilon$, the weight vector of decision makers $wd$, the weight vector of alternatives $wa$ and the adjustment parameter $\alpha$

• Output: The individual preference vectors $w^{k}$, the group preference vectors $w^{c}$ and the number of iterations $h$

• Step 1: Let $h = 0$ and $U_{h}^{k} = U^{k} \left( {k = 1,2, \ldots ,m_{1} } \right)$, $O_{h}^{k} = O^{k} \left( {k = m_{1} + 1,m_{1} + 2, \ldots ,m_{2} } \right)$, $A_{h}^{k} = A^{k} \left( {k = m_{2} + 1,m_{2} + 2, \ldots ,m_{3} } \right)$, $P_{h}^{k} = P^{k} \left( {k = m_{3} + 1,m_{3} + 2, \ldots ,m} \right)$.

• Step 2: Based on Eqs. (A5–A10), transform $U_{h}^{k}$, $O_{h}^{k}$, $A_{h}^{k}$ and $P_{h}^{k}$ into the individual and group preference vectors $w_{h}^{k}$ and $w_{h}^{c}$. Using Eq. (6) calculate the consensus level, i.e. $CM^{h} = 1 - \sqrt {\frac{1}{2m}\sum\limits_{k = 1}^{m} {\sum\limits_{i = 1}^{n} {\left( {w_{h,i}^{k} - w_{h,i}^{c} } \right)^{2} } } }$. If $CM^{h} \ge \varepsilon$, go to Step 4, otherwise, go to the next step.

• Step 3: Using Eq. (A11) obtains $DM^{h} \left( {e_{k} } \right) = \sqrt {\frac{1}{2}\sum\limits_{i = 1}^{n} {\left( {w_{h,i}^{k} - w_{h,i}^{c} } \right)^{2} } }$. Without loss of generality, suppose that $DM^{h} \left( {e_{q} } \right) = \mathop {\hbox{max} }\limits_{k} \left\{ {DM^{h} \left( {e_{k} } \right)|k = 1,2, \ldots ,m} \right\}$. Based on the GPV-based rules, we obtain the adjusted suggestions $SU_{h,k}^{GPV}$, $SO_{h,k}^{GPV}$, $SA_{h,k}^{GPV}$ and $SP_{h,k}^{GPV}$. For decision maker $e_{q}$, if $e_{q} \in E^{U}$, let $U_{h + 1}^{k} = \left( {u_{h + 1,1}^{k} ,u_{h + 1,2}^{k} , \ldots ,u_{h + 1,n}^{k} } \right)^{T}$, where $u_{h + 1,i}^{k} { = }\alpha {\text{I}}nf\left\{ {SU_{h,k,i}^{GPV} } \right\} + \left( {1{ - }\alpha } \right){\text{Sup}}\left\{ {SU_{h,k,i}^{GPV} } \right\}$; if $e_{q} \in E^{O}$, let $O_{h + 1}^{k} = \left( {o_{h + 1,1}^{k} ,o_{h + 1,2}^{k} , \ldots ,o_{h + 1,n}^{k} } \right)^{T}$ and $ot_{h,i}^{k} { = }\alpha {\text{I}}nf\left\{ {SO_{h,k,i}^{GPV} } \right\} + \left( {1{ - }\alpha } \right){\text{Sup}}\left\{ {SO_{h,k,i}^{GPV} } \right\}$, where $o_{h + 1,i}^{k} = t$ when $ot_{h,i}^{k}$ is the t th smallest value in $\left\{ {ot_{h,1}^{k} ,ot_{h,2}^{k} , \ldots ,ot_{h,n}^{k} } \right\}$; If $e_{q} \in E^{A}$, let $A_{h + 1}^{k} = \left( {a_{h + 1,ij}^{k} } \right)_{n \times n}$, where

  $$ a_{h + 1,ij}^{k} { = }\left\{ {\begin{array}{*{20}l} {\alpha {\text{I}}nf\left\{ {SA_{h,k,ij}^{GPV} } \right\} + \left( {1 - \alpha } \right){\text{Sup}}\left\{ {SA_{h,k,ij}^{GPV} } \right\}} \hfill & {i \ge j} \hfill \\ {1/a_{h + 1,ji}^{k} } \hfill & {i < j} \hfill \\ \end{array} } \right. $$

  if $e_{q} \in E^{P}$, let $P_{h + 1}^{k} = \left( {p_{h + 1,ij}^{k} } \right)_{n \times n}$, where

  $$ p_{h + 1,ij}^{k} { = }\left\{ {\begin{array}{*{20}l} {\alpha {\text{I}}nf\left\{ {SP_{h,k,ij}^{GPV} } \right\} + \left( {1{ - }\alpha } \right){\text{Sup}}\left\{ {SP_{h,k,ij}^{GPV} } \right\}} \hfill & {i \ge j} \hfill \\ {1 - p_{h + 1,ji}^{k} } \hfill & {{\kern 1pt} i < j} \hfill \\ \end{array} } \right.. $$

  For decision makers $e_{k} \left( {k \ne q} \right)$, $U_{h + 1}^{k} = U_{h}^{k}$, $O_{h + 1}^{k} = O_{h}^{k}$, $A_{h + 1}^{k} = A_{h}^{k}$ and $P_{h + 1}^{k} = P_{h}^{k}$. Then, let $l = l + 1$, and go to Step 2.

• Step 4: Let $w^{k} = w_{h}^{k}$ and $w^{c} = w_{h}^{c}$. Output $w^{k}$, $w^{c}$ and $h.$

Then, Algorithm 2 is called the IR-DR CM.

Appendix D: The OBM CM: Algorithm 3

Because the OBM in Fan et al. (2004) and Ma et al. (2006) cannot guarantee a high consensus level among the decision makers, motivated by the IR-DR CM, we propose the OBM CM. In Step 2 of Algorithm 2, we delete two input parameters wd and wa in the input and replace Step 2 with Step 2*. Then, we can obtain the OBM CM, Algorithm 3.

• Step 2*: Using Eqs. (A1) and (A2) obtains the group preference vector $w_{h}^{c}$, and using Eqs. (A3) and (A4) obtains the individual preference vectors $w_{h}^{k} \left( {k = 1,2, \ldots ,m} \right)$. Using Eq. (6) calculate the consensus level, i.e., $CM^{h} = 1 - \sqrt {\frac{1}{2m}\sum\limits_{k = 1}^{m} {\sum\limits_{i = 1}^{n} {\left( {w_{h,i}^{k} - w_{h,i}^{c} } \right)^{2} } } }$. If $CM^{h} \ge \varepsilon$, go to Step 4, otherwise, go to the next step.

Then, Algorithm 3 is called the OBM CM.

Appendix E: The MILCM CM: Algorithm 1′

In Algorithm 1, we replace Step 3 with Step 3′ to obtain Algorithm 1′.

• Step 3′: Based on IPV-based and GPV-based rules, we obtain the adjusted suggestions $SU_{h,k}^{IPV}$, $SO_{h,k}^{IPV}$, $SA_{h,k}^{IPV}$, $SP_{h,k}^{IPV}$, $SU_{h,k}^{GPV}$, $SO_{h,k}^{GPV}$, $SA_{h,k}^{GPV}$ and $SP_{h,k}^{GPV}$. Let $U_{h + 1}^{k} = \left( {u_{h + 1,1}^{k} ,u_{h + 1,2}^{k} , \ldots ,u_{h + 1,n}^{k} } \right)^{T}$, $A_{h + 1}^{k} = \left( {a_{h + 1,ij}^{k} } \right)_{n \times n}$ and $P_{h + 1}^{k} = \left( {p_{h + 1,ij}^{k} } \right)_{n \times n}$, where $u_{h + 1,i}^{k} { = }\alpha {\text{I}}nf\left\{ {SU_{h,k,i}^{IPV} \cap SU_{h,k,i}^{GPV} } \right\} + \left( {1{ - }\alpha } \right){\text{Sup}}\left\{ {SU_{h,k,i}^{IPV} \cap SU_{h,k,i}^{GPV} } \right\};$

  $$ a_{h + 1,ij}^{k} { = }\left\{ {\begin{array}{*{20}l} {\alpha {\text{I}}nf\left\{ {SA_{h,k,ij}^{IPV} \cap SA_{h,k,ij}^{GPV} } \right\} + \left( {1{ - }\alpha } \right){\text{Sup}}\left\{ {SA_{h,k,ij}^{IPV} \cap SA_{h,k,ij}^{GPV} } \right\}} \hfill & {i \ge j} \hfill \\ {1/a_{h + 1,ji}^{k} } \hfill & {i < j} \hfill \\ \end{array} } \right.; $$
  $$ p_{h + 1,ij}^{k} { = }\left\{ {\begin{array}{*{20}l} {\alpha {\text{I}}nf\left\{ {SP_{h,k,ij}^{IPV} \cap SP_{h,k,ij}^{GPV} } \right\} + \left( {1{ - }\alpha } \right){\text{Sup}}\left\{ {SP_{h,k,ij}^{IPV} \cap SP_{h,k,ij}^{GPV} } \right\}} \hfill & {i \ge j} \hfill \\ {1 - p_{h + 1,ji}^{k} } \hfill & {i < j} \hfill \\ \end{array} } \right.. $$

Let $O_{h + 1}^{k} = \left( {o_{h + 1,1}^{k} ,o_{h + 1,2}^{k} , \ldots ,o_{h + 1,n}^{k} } \right)^{T}$ and $ot_{h,i}^{k} { = }\alpha {\text{I}}nf\left\{ {SO_{h,k,i}^{IPV} \cap SO_{h,k,i}^{GPV} } \right\} + \left( {1{ - }\alpha } \right){\text{Sup}}\left\{ {SO_{h,k,i}^{IPV} \cap SO_{h,k,i}^{GPV} } \right\}$, where $o_{h + 1,i}^{k} = t$ when $ot_{h,i}^{k}$ is the t th smallest value in $\left\{ {ot_{h,1}^{k} ,ot_{h,2}^{k} , \ldots ,ot_{h,n}^{k} } \right\}$. Let $h = h + 1$. Then, go to Step 2.

Then, Algorithm 1′ is called the MILCM CM.