An Improved Multi-objective Water Cycle Algorithm to Modify Inconsistent Matrix in Analytic Hierarchy Process

Abstract

Analytic Hierarchy Process (AHP) is a prevalent decision-making system for multi-criteria decision-making problems (MCDM). An essential step in AHP is to evaluate the pair-wise comparison matrix (PCM) for its consistency, which is affected by decision-makers’ judgments. A PCM that fails to satisfy the consistency represents the inconsistent judgments and requires modification in it. This paper proposes an enhanced population-based multi-objective water cycle algorithm, also called (MOWCA), to modify inconsistent PCM and obtain an optimally consistent PCM. The proposed multi-objective algorithm (MOA) utilizes the Cosine Consistency index (CCI) for consistency evaluation and the Cosine Maximization method (CM) for modifying the entries in inconsistent PCM until the CCI values are optimized. The proposed MOA aims to find an optimally consistent PCM that not only satisfies the desired CCI level but also has a lower distance from original PCM to preserve decision-maker initial decisions. In the proposed MOA, the concept of salt concentration and infiltration is introduced into the evaporation rate (ER), which extends MOWCA to improved MOWCA. The proposed MOA is tested on several inconsistent PCMs. Subsequently, its performance is compared with existing meta-heuristics. The derived optimum CCI values are validated using a statistical significance test. The experimental findings indicate that compared with existing approaches, the proposed MOA generates optimum CCI values.