Comparison of Methods to Derive Relative Weights.

Identifying critical factors, objectives, criteria, or alternatives in operating research methods is a vital decision-making problem in academic society. Besides, fuzzy theorems, inventory models, and optimization approaches, the analytic hierarchy process is an important tool to decide relative weights for several factors, objectives, criteria, or alternatives. This paper will examine the relationship among several methods to derive the relative weight. Saaty and Vargas provided an example to prove that the relative weight is the same by the Eigenvalue Method (EM) and by the Logarithmic Least Square Method (LLSM). Saaty and Vargas did not know the relative weight by the Least Square Method (LSM). It is an open question proposed by Saaty and Vargas. Saaty is the founding father of the Analytic Hierarchy Process (AHP) and Vargas is one of the main contributors to AHP, respectively. Hence, in this paper, we present a detailed examination of the open question. We obtain that there are two (non-consistent) comparison matrices under the Saaty-Vargas construction which satisfied the property of the relative weights derived by three methods: EM, LLSM, and LSM are the same. We not only solve the open question proposed by Saaty and Vargas but also extend it to a more general setting. We consider all possible conditions for a ∈ {2,3,...,9} such that for a = 2, we provided an analytical proof; for a = 3, we show a hybrid method such that a discrete sensitivity analysis for the open question proposed by Saaty and Vargas is completely solved by us; for a ∈ {4,5,...,9}, we present numerical approaches. Our findings help researchers to decide which evaluation method will be adopted to derive the relative weights for their future research. Our results will help practitioners to decide the relative weights for factors, objectives, criteria, or alternatives in research methods. [ABSTRACT FROM AUTHOR]