A Dynamic Multiple Stage, Multiple Objective Optimization Model With An Application To A Wastewater Treatment System

Abstract

Decision-making for complex dynamic systems involves multiple objectives. Various methods balance the tradeoffs of multiple objectives, the most popular being weighted-sum and constraint-based methods. Under convexity assumptions an optimal solution to the constraint-based problem can also be obtained by solving the weighted-sum problems, and all Pareto optimal solutions can be obtained by systematically varying the weights or constraint limits. The challenge is to generate meaningful weights or constraint limits that yield practical solutions. In this dissertation, we utilize the Analytic Hierarchy Process (AHP) and develop a methodology to generate weight vectors successively for a dynamic multiple stage, multiple objective (MSMO) problem. Our methodology has three phases: (1) the input phase obtains judgments on pairs of objectives for the first stage and on dependencies from one stage to the next, (2) the matrix generation phase uses the input phase information to compute pairwise comparison matrices for subsequent stages, and (3) the weighting phase applies AHP concepts, with the necessary weight vectors obtained from expert opinions. We develop two new geometric-mean based methods for computing pairwise comparison matrices in the matrix generation phase. The weight ratios in the pairwise comparison matrices conform to the subjective ratio scale of AHP, and the geometric mean maintains this scale at each stage. Finally, for these two methods, we discuss the consistency of computed pairwise comparison matrices, note the convergence behavior, and apply our three-phase methodology to a problem of evaluating technological processes/units at each stage of an MSMO Wastewater Treatment System (WTS). The WTS is a 20-dimensional, continuous-state, 17-stage, 6-objective, stochastic problem.