## 1 Introduction  
Multi-criteria group decision-making (MCGDM) is a familiar decision activity, such as investment decision-making, medical diagnosis, personnel examination and military system efficiency evaluation (see [1]). In MCGDM, different experts usually give different judgments on some alternatives over a set of evaluation criteria, which are used to rank the alternatives. Denote \(A_{i}\) and \(C_{j}\), \(i=1,2,\cdots,n\), \(j=1,2,\cdots,m\), the alternatives \(i\) and the criterion \(j\), respectively. An MCGDM problem with \(n\) alternatives, \(p\) experts and \(m\) criteria can be mathematically described as follows. Let \(x_{ij}^{k}\) be the \(k\)-th expert's evaluation on the \(i\)-th alternative by the \(j\)-th criterion, where \(k=1,2,\cdots,p\). We call \(X_{k}=(x_{ij}^{k})_{n\times m}\), \(k=1,2,\cdots,p\), the score matrices. \(u^{k}=(u_{1}^{k},u_{2}^{k},\cdots,u_{m}^{k})\) is a given vector, where \(u_{j}^{k}\) is referred to as the weight of the \(j\)-th criterion by the \(k\)-th expert. Let \(w=(w_{1},w_{2},\cdots,w_{p})\) be a given weight of the experts, whose component reflects the importance of the evaluation of the \(k\) expert. Then, an overall evaluation of alternative \(i\) is obtained by:  
\[S_{i}=\sum_{k=1}^{p}a_{ik}w_{k}, \tag{1}\]  
where:  
\[a_{ik}=\sum_{j=1}^{m}x_{ij}^{k}u_{j}^{k}. \tag{2}\]  
It is clear that the weight vectors \(u^{k}\) and \(w\) play a fundamental role in ranking the preference of the alternatives on the basis of the given matrices \(X_{k}\), \(k=1,2,\cdots,p\). However, the determination of the weight vectors has been regarded as one of the main difficulties in solving the MCGDM problem. In the existing results, the methods to specify the weight vectors can be classified as the subjective and objective ones. For example, [PERSON] and [PERSON] in [2] presented a simple method, which uses the decision-makers' own subjective opinions to calculate the expert's weights. In [3], [PERSON] _et al._ proposed a flexible consensus scheme to establish the order. [PERSON] in [4] established another decision-making term to give weights to the initial decision-making members and worked out the weights by measuring the additional preference value deviations. [PERSON] in [5] improved [PERSON]'s method and proposed a more direct method to calculate the weights. Since the above methods are associated with the experience of the decision-makers, they basically belong to the subjective method. However, we should reduce decision bias and improve the objectiveness of decision-making although, experience is an important reference. In contrast, as an objective method, the obvious feature is to choose the suitable weights on the basis of computational models, which can mine useful information from the score matrices. For example, in [6], the variation coefficient method was presented to choose the weights of criteria by calculating the standard deviation. The entropy-based method (EBM) was proposed in [7] to calculate the weights of criteria based on the concept of entropy. In [8], a distance-based group decision-making methodology was proposed to solve unconventional multi-person multi-criteria emergency decision-making problems. The above three methods are designed to obtain the weights, such that the degree of discrimination and deviation from each alternative is improved. Since the weights in these methods are determined directly based on the data themselves, they are relatively objective and against the possible bias of the decision-maker. However, the existing objective methods seem to neglect the fact that each expert holds a uniform viewpoint of the importance (weights) of each criterion when he/she needs to rank all the alternatives. For an MCGDM problem with incomplete information on attribute weights, linear programming models were established to find the compromise weights in [9,10]. By virtue of the notion of the significance degree, a zero-one mixed integer linear programming model was constructed in [11] to identify the weights.