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In the evidential reasoning (ER) approach, in addition to the weight of a criterion, reliability is another important concept in connection with the criterion although it attracts little attention. Additionally, a decision maker's risk attitude is of particular importance in decision-making. To simultaneously consider these two important factors, this paper proposes a new ER approach. In the approach, with the consideration of a decision maker's risk attitude, the combinational reliability of each criterion for each alternative is constructed from the original reliability of the criterion. By following the regression idea to learn the original reliability of each criterion, a unified optimization model is constructed, in which the maximum difference between combinational reliabilities and their estimations is minimized. Within the post-optimal solution space of criterion reliabilities found by solving the unified model, another optimization model is constructed, from which the minimum and maximum expected utilities of each alternative are determined. Solutions to multi-criteria decision-making problems are then generated from the expected utilities. The proposed approach is used to analyze a material supplier selection problem for a company located in Tongling, Anhui, China. The analysis of the problem demonstrates the applicability of the proposed approach. [ABSTRACT FROM AUTHOR]