Abstract: When the AHP method is applied to emergency situations such as warfare and natural disaster, there often exist missing data due to decision time constraint, incomplete information, and limited experience of the decision maker. In this paper, a complete solution based on optimization and analysis of the incomplete judgment matrix (IJM) is proposed. Specifically, some definitions and properties of the IJM are presented firstly, and the basic principles on judging the effectiveness of IJM are studied, based on which a connected graph method is proposed to identify the ineffective IJM. For the ineffective IJM, a method for making the IJM effective is explored by replenishing minimum unknown entries to realize a fully connected graph. When the IJM is effective, the missing data are first filled with unknown variables, and then an optimization model is built to solve the unknown variables based on minimizing the consistent ratio (CR). If the optimization model fails to find the optimum solution, there must exist great inconsistency in the IJM, which means the known entries may have constructed basic loops with great inconsistency. Therefore, the inconsistency adjustment method is proposed by analyzing the basic loops, where the element with greatest inconsistency is identified by comparing the sum total of CR values for all basic loops including the element, and modified by choosing the optimum value in the region[1/9,9]to make the sum total of CR values minimum. Based on the proposed method, a decision-making tool software is developed by Matlab programming. Finally, several examples are used to illustrate and validate the proposed method, and the results obtained by the self-developed software show that it is timely efficient, feasible and effective for IJM decision making problems under emergency situations. Also, comparison studies show that it is more effective than existing methods.

Key words: incomplete judgment matrix(IJM), incomplete information, effectiveness, optimization model, consistent ratio(CR)

摘要： 针对在紧急情况下,比如战争或灾难中利用AHP进行决策时,由于决策时间紧迫、信息掌握不完全、决策者经验限制等因素,通常会导致决策信息的不完全,提出一套完整的基于残缺判断矩阵的分析和优化的解决方案。首先,给出了残缺判断矩阵的相关定义和性质,研究了残缺判断矩阵的有效性判断的基本原理,并给出了连通图判定方法;对于无效残缺判断矩阵,通过增补最少的元素实现所有方案的互连通,从而使其成为有效残缺判断矩阵;对于有效残缺判断矩阵,提出以未知数填充残缺矩阵,构建以一致性比率最小为目标的优化决策模型;对于优化模型仍不能达到满意一致性的情形,从基本回路的不一致性分析入手,找出具有最大CR和的元素作为最不一致元素,在[1/9,9]区间上选出使得CR和最小的值作为该元素修正值,然后再构建优化模型实现最优化增补;根据以上原理,利用Matlab编程,开发了残缺判断矩阵的AHP相关决策工具软件。最后,通过算例分析验证了方法的可行性和有效性,证明了开发的软件能够满足紧急状态下决策的时效性要求,同时,通过与已有方法的对比证明了该方法更为有效。

关键词: 残缺判断矩阵, 不完全信息, 有效性, 优化模型, 一致性比率