关键词: TOPSIS, 排序方法, 弱等效方案, 接近度

Abstract: In decision-making processes, the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) is a widely recognized multi-criteria decision analysis (MCDA) method. However, a significant challenge arises when multiple alternatives have identical ranking indices, particularly when these alternatives are located on the same multi-dimensional sphere. This leads to ambiguities in the ranking process, obscuring decision-making and reducing the clarity and utility of the TOPSIS method. Given the increasing complexity of decision-making environments in both theoretical and practical scenarios, there is a pressing need to refine the TOPSIS method to address this limitation.
   This paper aims to enhance the TOPSIS methodology by introducing a modified approach that effectively differentiates between strongly and weakly equivalent alternatives, thereby providing a more robust and clear ranking system. The proposed modification to the TOPSIS method involves several key steps to address the issue of equivalent alternatives. Firstly, this paper categorizes equivalent alternatives into strongly equivalent alternatives and weakly equivalent alternatives. Strongly equivalent alternatives are those where different alternatives have equal distances to the ideal solution, while weakly equivalent alternatives are those where the ratios of distances to the ideal solution are equal but the distances themselves are different. Using a two-dimensional target coordinate system to describe the TOPSIS system, the following characteristics are identified: (1)All alternatives are located in a specific region. (2)Strongly equivalent alternatives are at the same point, while weakly equivalent alternatives lie on a specific line. Based on the relative positions of equivalent alternatives and the positive and negative ideal solutions in the two-dimensional target space, a modified TOPSIS ranking method is proposed for evaluation systems with weakly equivalent alternatives. This method involves grouping and intergroup ranking of alternatives, and then redefining the ranking index to sort the equivalent alternatives within each group. The integration of intragroup and intergroup ranking forms a ranking sequence that achieves full ranking of systems with weakly equivalent alternatives, consistent with the classic TOPSIS method.
   The improved TOPSIS method retains the original ranking characteristics while enabling full ranking of weakly equivalent alternatives. The numerical analysis verifies the reasonableness and effectiveness of this method. The results indicate that the new method effectively addresses the ranking issue caused by weakly equivalent alternatives, enhancing decision-making accuracy and practicality. By resolving these ambiguities, the modified TOPSIS method increases the robustness and reliability of the decision-making process.
   Future research could further explore the effectiveness of this method in different application scenarios and larger datasets. Additionally, integrating this method into other MCDA techniques could lead to even more precise decision-making tools. Expanding this approach could also involve investigating its applicability in real-world decision-making situations, offering deeper insights and further validation.

Key words: TOPSIS, ranking method, weakly equivalent alternatives, ranking index