## 1 Introduction
In [12], a linear programming model was also constructed to obtain the weights for the MCGDM problem with imprecise information. In some existing results available in the literature, it is also often that the subjective judgment and the objective model are combined to choose the weight vectors. For example, [PERSON] _et al._ in [13] proposed a linguistic ordered weighted averaging operator to calculate the weights. In [14], there were four different operators presented. [PERSON] in [15] presented the so-called REMBRANDT method (ratio estimation in magnitudes of decibels to rate alternatives which are non-dominated) to determine the weights. Actually, the REMBRANDT method is a combination of AHP (Analytic Hierarchy Process) and the sample multi-attribute rating technique (SMART) to quantify the decision-makers' experiences. The linguistic probabilistic weighted average (LPWA) was presented in [16]. In addition, the above methods have also been extended to solve uncertain group decision-making problems. For example, in [17], a model was constructed by maximizing the comprehensive membership coefficient to determine the weights of decision-makers, as the expert's score matrices are involved with intuitionistic interval fuzzy information. The basic idea is to determine the weights by different definitions on the degree of discrimination and deviation, as done in a certain environment (see [18,19]). In summary, compared to the objective method, subjective judgment is often deemed a possible threat to the fairness of ranking in practice. However, as for an objective method, it is still a challenging task to construct a more reasonable computational model to mine the information from the score matrices. Different from all the methods mentioned above, it is noted that in [20,21], a so-called robust portfolio modeling (RPM) method was presented to solve the multi-criterion project portfolio problems without the determination of weights in advance. However, compared to the first type of methods, RPM needs an efficient algorithm to find all the non-dominated portfolios to compute the core index of each portfolio. As pointed out in [20], the search for all the non-dominated portfolios is far more difficult than the solution of a knapsack problem. Thus, no polynomial-time algorithm exists in general to find the optimal project portfolio. Owing to the advantage of the methods with predetermined weights in MCGDM, instead of the RPM approach, this paper intends to present two Min-Max optimization models to determine the weights of criteria and experts, respectively. In addition, based on the existing smooth optimization techniques, efficient and convergent algorithms will also be developed to solve the Min-Max problems in this paper. It is clear that with the optimal point estimates of weights, the complexity in solving the problem of ranking the alternatives is greatly reduced. Specifically, to choose the weights of criteria, we will take into consideration the uniform viewpoint possessed by the same expert on the importance (weighted value) of each criterion when he/she needs to rank all the alternatives. Examples will be constructed to show the difference between our method and the ones available in the literature, which are based on the degree of discrimination and deviation from each alternative. On the other hand, the weights of experts will be optimized such that the overall score of each alternative, calculated by the obtained weights, can collect the perspectives of the experts as many as possible. In this case, the objective function is to minimize the maximal variation between the actual and ideal score vectors of the \(p\) experts, such that the optimal weights can reduce the difference among the experts in evaluating the same alternative. Finally, applications of the models will be employed to show the significance of the models and algorithm. The rest of paper is organized as follows. In next section, two Min-Max models are constructed to optimize the weight vectors.