Consistency and Consensus of Distributed Preference Relations Based on Stochastic Optimal Allocation in GDM Problems

Abstract

Consistency and consensus are two key challenges under uncertain circumstances in pairwise comparison-based group decision-making (GDM), especially in a distributed preference relation (DPR) environment. In this paper, a comprehensive framework designed to tackle GDM problems evaluated by DPR is developed. First, after discussing two types of inconsistency in a complete DPR matrix: uncertainty-caused inconsistency and preference-caused inconsistency, two targeted optimization models to generate a consistent certain DPR matrix are proposed based on the definitions of stochastic additive strong/weak consistency and the similarity measure between DPRs. These models can effectively derive a DPR matrix closest to the original certain judgment by stochastic optimal allocation of uncertainties. Second, a new group consensus degree is introduced to measure the consensus in the group. Then a consensus improving model is given to reach an acceptable consensus by adjusting the DPR matrix of the decision maker (DM) with the least consensus degree. Third, the DMs’ weights are determined based on the expected consistency of the original DPR matrix by stochastic simulation instead of subjective judgment, and then the aggregated DPR matrix is obtained to derive a final solution using the weighted averaging operator. Finally, an automobile selection example is given to verify the validity and rationality of the proposed models.

Appendix

1.1 Proofs

1.1.1 1. Proof of Theorem 2

Proof:

If D is a stochastic additive weak consistent DPR matrix, it means that there exists $d_{ij}^{Sto} \in d_{ij}$, $d_{jk}^{Sto} \in d_{jk}$ and $d_{ik}^{Sto} \in d_{ik}$, $i,j,k = 1,2, \cdots ,M$, satisfying $CI($$S^{Sto} ) = 0$, where $S^{Sto}$ is the score matrix of $D^{Sto}$. Then.

$$ s_{ij}^{ - } \le s_{ij}^{Sto} \le s_{ij}^{ + } , $$

$$s_{jk}^{ - } \le s_{jk}^{Sto} \le s_{jk}^{ + } ,$$

and the sum of above equations leads to the following inequalities.

$$ s_{ij}^{ - } + s_{jk}^{ - } \le s_{ik}^{Sto} \le s_{ij}^{ + } + s_{jk}^{ + } . $$

Since Eq. (26) holds for any $j = 1,2, \cdots ,M$, $\mathop {\max }\limits_{j} (s_{ij}^{ - } + s_{jk}^{ - } ) \le \mathop {\min }\limits_{j} (s_{ij}^{ + } + s_{jk}^{ + } )$ for any $i,j,k = 1,2, \cdots ,M$.

Conversely, if $\mathop {\max }\limits_{j} (s_{ij}^{ - } + s_{jk}^{ - } ) \le \mathop {\min }\limits_{j} (s_{ij}^{ + } + s_{jk}^{ + } )$ for any $i,j,k = 1,2, \cdots ,M$, then $s_{ik}^{ - }$$\le s_{ik}^{Sto} \le s_{ik}^{ + }$ for any $i,k = 1,2, \cdots ,M$; thus, there exists $S^{Sto}$ satisfying $CI(S^{Sto} ) = 0$ which means that D is a stochastic additive weak consistent DPR matrix.

1.1.2 Proof of Theorem 5

Proof:

$$ GCD^{update} - GCD $$

$$ = \frac{1}{T}\sum\limits_{t = 1}^{T} {CD(E_{t} )_{{}}^{update} - } \frac{1}{T}\sum\limits_{t = 1}^{T} {CD(E_{t} )} $$

$$ = \frac{1}{T}\sum\limits_{t = 1}^{T} {(CD(E_{t} )_{{}}^{update} - CD(E_{t} ))} $$

$$ = \frac{1}{T} \cdot \frac{1}{T - 1}\left[ {\sum\limits_{t = 1}^{T} {\left( {\sum\limits_{k = 1,k \ne t}^{T} {SD^{update} (D_{k} ,D_{t} )} - \sum\limits_{k = 1,k \ne t}^{T} {SD(D_{k} ,D_{t} )} } \right)} } \right] $$

$$ = \frac{1}{T} \cdot \frac{1}{T - 1} \cdot 2\left( {\sum\limits_{k = 1,k \ne l}^{T} {\left[ {SD(D_{l}^{update} ,D_{k} ) - SD(D_{l}^{{}} ,D_{k} )} \right]} } \right) $$

$$ = \frac{2}{T}\left( {CD(E_{l} )_{{}}^{update} - CD(E_{l} )} \right) $$

1.1.3 3. Proof of Theorem 6

Proof:

Based on Eqs. (25), (38) and (39), we obtain.

$$ CI(S^{C} ) = \frac{2}{M \cdot (M - 1) \cdot (M - 2)}\sum\limits_{i = 1}^{M - 2} {\sum\limits_{j = i + 1}^{M - 1} {\sum\limits_{k = j + 1}^{M} {\left| {s_{ij}^{C} + s_{jk}^{C} - s_{ik}^{C} } \right|} } } $$

$$ = \frac{2}{M \cdot (M - 1) \cdot (M - 2)}\sum\limits_{i = 1}^{M - 2} {\sum\limits_{j = i + 1}^{M - 1} {\sum\limits_{k = j + 1}^{M} {\left| {\sum\limits_{n = - N}^{N} {m_{ij}^{C} (H_{n} ) \cdot s(H_{n}^{{}} )} + \sum\limits_{n = - N}^{N} {m_{jk}^{C} (H_{n} ) \cdot s(H_{n}^{{}} )} - \sum\limits_{n = - N}^{N} {m_{ik}^{C} (H_{n} ) \cdot s(H_{n}^{{}} )} } \right|} } } $$

$$ = \frac{2}{M \cdot (M - 1) \cdot (M - 2)}\sum\limits_{i = 1}^{M - 2} {\sum\limits_{j = i + 1}^{M - 1} {\sum\limits_{k = j + 1}^{M} {\left| {\sum\limits_{n = - N}^{N} {\sum\limits_{l = 1}^{T} {w_{l} m_{ij}^{l} (H_{n} )} \cdot s(H_{n}^{{}} )} + \sum\limits_{n = - N}^{N} {\sum\limits_{l = 1}^{T} {w_{l} m_{jk}^{l} (H_{n} )} \cdot s(H_{n}^{{}} )} - \sum\limits_{n = - N}^{N} {\sum\limits_{l = 1}^{T} {w_{l} m_{ik}^{l} (H_{n} )} \cdot s(H_{n}^{{}} )} } \right|} } } $$

$$ = \frac{2}{M \cdot (M - 1) \cdot (M - 2)}\sum\limits_{i = 1}^{M - 2} {\sum\limits_{j = i + 1}^{M - 1} {\sum\limits_{k = j + 1}^{M} {\left| {\sum\limits_{l = 1}^{T} {\sum\limits_{n = - N}^{N} {w_{l} m_{ij}^{l} (H_{n} )} \cdot s(H_{n}^{{}} )} + \sum\limits_{l = 1}^{T} {\sum\limits_{n = - N}^{N} {w_{l} m_{jk}^{l} (H_{n} )} \cdot s(H_{n}^{{}} )} - \sum\limits_{l = 1}^{T} {\sum\limits_{n = - N}^{N} {w_{l} m_{ik}^{l} (H_{n} )} \cdot s(H_{n}^{{}} )} } \right|} } } $$

$$ = \frac{2}{M \cdot (M - 1) \cdot (M - 2)}\sum\limits_{i = 1}^{M - 2} {\sum\limits_{j = i + 1}^{M - 1} {\sum\limits_{k = j + 1}^{M} {\left| {\sum\limits_{l = 1}^{T} {w_{l} s_{ij}^{l} } + \sum\limits_{l = 1}^{T} {w_{l} s_{jk}^{l} } - \sum\limits_{l = 1}^{T} {w_{l} s_{ik}^{l} } } \right|} } } $$

$$ = \frac{2}{M \cdot (M - 1) \cdot (M - 2)}\sum\limits_{i = 1}^{M - 2} {\sum\limits_{j = i + 1}^{M - 1} {\sum\limits_{k = j + 1}^{M} {\left| {\sum\limits_{l = 1}^{T} {w_{l} (s_{ij}^{l} + s_{jk}^{l} - s_{ik}^{l} )} } \right|} } } $$

$$ = \sum\limits_{l = 1}^{T} {w_{l} \cdot CI(S^{l} )} \le \sum\limits_{l = 1}^{T} {w_{l} \cdot \mathop {\max }\limits_{l} \{ CI(S^{l} )\} } = \mathop {\max }\limits_{l} \{ CI(S^{l} )\} $$