A two-phase grey fuzzy optimization approach for water quality management of a river system

An extension of the Grey Fuzzy Waste Load Allocation Model (GFWLAM) developed in an earlier work is presented here to address the problem of multiple solutions. Formulation of GFWLAM is based on the approach for solving fuzzy multiple objective optimization problems with max–min as the operator, which usually may not result in a unique solution. The multiple solutions of fuzzy multiobjective optimization model should be obtained as parametric equations or equations that represent a subspace. A two-phase optimization technique, two-phase GFWLAM, is developed to capture all alternative or multiple solutions of GFWLAM. The optimization model in Phase 1 is exactly same as the optimization model described in GFWLAM. The optimization model in Phase 2 maximizes the upper bounds of fractional removal levels of pollutants and minimizes the lower bounds of fractional removal levels of pollutants keeping the value of goal fulfillment level same as obtained from Phase 1. The widths of the interval-valued fractional removal levels play an important role in decision-making as these can be adjusted within their intervals by the decision-maker considering technical and economic feasibility in the final decision scheme. Two-phase GFWLAM widens the widths of interval-valued removal levels of pollutants, thus enhancing the flexibility in decision-making. The methodology is demonstrated with a case study of the Tunga-Bhadra river system in India