AHP權重向量解法之理論性質與模擬實驗比較

Theoretical and Experimental Comparisons on AHP Judgment Matrix Analysis Methods

層級分析法，因其實施步驟與人們常用的分析步驟相近，目前已被廣泛地應用於工程與管理的各種不同領域，尤其是必需混合屬性與屬量分析的問題。林高正等人(2009) 從判斷矩陣的意涵出發，指出它是一個偏好結構之受到擾動的量測資料，進而指出 Saaty 的固有向量法 EM 之理論漏洞與實務應用上的缺點。相對地，以統計迴歸分析為基礎的方法，除有不錯的理論性質外，還有許多實務應用上的優點。因此，以迴歸分析為基礎應是進行判斷矩陣分析較佳的方向。他們並定義了判斷矩陣的圖形一致性與求解權重向量時的圖形一致性限制，進而探討了加入圖形一致性限制之對數最小平方法 LLSM-GC 與目標規劃法 GPM-GC 的性質。他們發現 LLSMGC與 GPM-GC 除有良好的理論性質外，還具有許多實務應用的優點。延續林高正等人 (2009) 的成果，本研究透過模擬實驗比較相關權重向量解法之估計能力與估計行為。實驗結果顯示，在實驗誤差為對數常態分配時，平均而言，LLSM-GC 與 GPMGC所得之因素排序顯著地比舊有的 EM、LLSM 與 GPM 更接近真正的因素排序。此外，本研究並以數值範例說明 Saaty 與 Vargas 所指的保序性並非指保持真實權重的順序，以及採用 CR 值評估判斷矩陣一致性並不是個嚴謹的方法。

The analytic hierarchy process (AHP) is one of the most commonly applied multiattributes decision analysis techniques for the reason that its implementation steps are very common to the people. It is even more popular in cases that quantitative and qualitative attributes combine. Lin et al. (2009) set out from the meaning of a judgment matrix, pointed out that its coefficients are perturbed measurement data of a preference structure, and then showed that Saaty’s eigenvector method has some theoretical weakness and practical disadvantages. On contrast, the methods based on statistical regression not only have nice properties in the decision theory, but also have several practical advantages. Therefore, analyzing the judgment matrix by regression is more proper. In additions, they defined the graphical consistency of a judgment matrix and the graphical consistency constraints for solving the priority vector, then considered the theoretical properties of the logarithm least squares method and the goal programming method with the graphical consistency constraints. It is shown that these two methods not only have nice properties but also several practical advantages. Since finding the priority vector is a problem of statistical estimation, the behavior and the ability of the estimators are as important as their theoretical properties. In this study, we perform a factorial experiment to compare the related methods, for the case that the logarithms of error terms are normally distributed. From the experimental results, it is found that LLSM-GC and GPM-GC perform better than the traditional methods in finding the priority order. In additions, in this experiment we have also found examples that the priority order found by EM is not the true order, and testing the consistency of a judgment matrix by using CR-value is not a serious method.