The Study On The Invalidity Of Fuzzy AHP And A Weighted Geometric Method Of AHP

In economic management, the Analytic Hierarchy Process (AHP) is a widely used decision analysis tool, especially in complex decision environment with subjective motives, which need to measure subjective judgment with different dimensions. The AHP with a relative judgment mode can tackle the multi-dimension issue into dimensionless and derive the final results by the hierarchical decomposition structure.However, the AHP has met too much criticism since its birth; among the criticism the fuzzification of the fundamental scale of the AHP is an important issue. This paper pays much attention to this issue and calls into question the validity of applying fuzzy logic on AHP. We not only analyze the invalidity of the basic mathematical logic of fuzzy AHP and but also analyze the invalid outcome of two types of fuzzy AHP methods which are widely used. Based on the above analyses, a weighted geometric AHP approach is proposed to tackle the current problems in the AHP.The main contents of this paper are as followed:1. We give general analysis on the basic mathematical logic of fuzzy AHP and find it has many problems:1) fuzzy AHP violates the basic logic of fuzzy set theory, which are the circular definition paradox of fuzzy numbers, membership grade never employ in fuzzy AHP and the results of the a-cut method is unbelievable;2) fuzzy AHP violates the basic principals of the AHP which include the axiom of reciprocal, the operational rule of consistency and the continuity axiom.2. In dealing with the outcomes, fuzzy AHP does not give an acceptable method to rank fuzzy numbers and a way to check the validity of the results.3. We discuss the validity of the Analytic Hierarchy/Network Process (AHP/ANP) in complex and uncertain environments and find that fuzzy ANP is a false proposition because there is no fuzzy priority in the super matrix which provides the basis for the ANP.4. We analysis the triangular fuzzy AHP which was proposed by Laarhoven and Pedrycz (1983) and find it has two significant problems. To address the mentioned problems, we propose that multiple fuzzy judgments for a single comparison need to be combined into one unit, and then further derive an analytic solution for the improved model. We successfully prove that the improved triangular fuzzy AHP approach is more accurate than existing approach. Through theoretical and numerical analysis, we demonstrate that even with the improved approach the result of triangular fuzzy AHP is invalid due to its potential violation to the basic assumption of the triangular fuzzy number that1) the values of lower, modal and upper numbers should be in non-decreasing order and2) the lower/upper values of the final weight should only relate to the lower/upper value of the triangular fuzzy judgments.5. We focus on an extent analysis method on fuzzy AHP which was proposed by Chang (1996) and pointed out the proposed method contains two mistakes and five issues. In this research we call attention to the errors and reapply the example and find the core equation of the extent analysis method could cause zero-weight, poor robustness, unreasonable priorities, information loss and the violation of Saaty’s AHP is a special case of the extent analysis method.6. We analysis the general critiques on traditional AHP and point out that the fuzzification of fundamental scale and rank reversal are two main challenges on traditional AHP. Meanwhile, the theoretical limitation of rank preservation methods are domestratred.7. A weighted-geometric AHP approach was proposed to tackle the challenges in the AHP. The proposed approach obeys the four basic axioms of the AHP and can derive expected outcome.