Optimizing Selection of Technologies in a Multiple Stage, Multiple Objective Wastewater Treatment System

Abstract

In this paper, a multiple stage wastewater treatment system (WTS) is solved for the selection of technological options at each stage to minimize (economic cost, size, odour emissions) and to maximize (nutrient recovery, robustness, global desirability). Stages in the wastewater treatment system are the levels of treatment. There are 17 levels of treatment, where the first 11 levels are for the liquid treatment and the last 6 levels are for the solid treatment. This results in a 20-dimensional, continuous-state, 17-stage, 6-objective, stochastic optimization problem. The resulting multiple stage, multiple objective (MSMO) WTS is solved using the three-phase methodology in conjunction with the multiple objective version of highdimensional, continuous-state, stochastic dynamic programming (SDP). The three-phase methodology comprises the input phase, the matrix generation phase and the weighting phase. The primary goal of three-phase methodology is to obtain weight vectors at each stage of the WTS utilizing expert's opinions in the input phase, computing pairwise comparison matrices at each stage using the geometric mean-based methods in the matrix generation phase, and then calculating weight vectors at each stage using the eigenvector method in the weighting phase. The weight vectors are then used to scalarize the vector optimization problem, which is solved using the high-dimensional, continuous-state SDP augmented for handling multiple objectives at each stage.The results obtained are practical as evidenced by the selection of new technologies in levels 1 and 5 thereby validating expert's decision to include them in the evaluation process. In addition to encouraging reviews from WTS experts, the implementation results satisfy a set of external constraints in the form of interstage dependencies between technological options in the WTS. Furthermore, the solution technique presented here utilizes expert's opinions in the solution development process, and is quite general in its application to a variety of large-scale MSMO problems.

Divergence Behaviour of the Successive Geometric Mean Method of Pairwise Comparison Matrix Generation for a Multiple Stage, Multiple Objective Optimization Problem

“…(), which has been augmented to handle multiple objectives using the three‐phase methodology explained in Tarun () and Tarun et al . ().…”

“…All the matrices generated in the input phase are k × k , where k denotes the number of objectives. The matrix generation phase uses the first‐stage pairwise comparison matrix and the interstage diagonal transformation matrices to compute the pairwise comparison matrices from stages 2 through T (Tarun et al ., ; Tarun, ; Tarun et al ., ). The weighting phase computes the weight vectors, W τ , across the stages using the principal eigenvector approximation method (Saaty, , ).…”

“…() addresses this issue by introducing a three‐phase methodology and extending Tsai's high‐dimensional, continuous‐state stochastic dynamic programming method to handle multiple objectives at each stage of an MSMO optimization problem. The three‐phase methodology is integrated with the augmented high‐dimensional, continuous‐state stochastic dynamic programming method to solve a large‐scale, real‐world MSMO optimization problem (Tarun, ; Tarun et al ., ). This MSMO solution methodology is elaborated in Section 1.2.…”

“…(, ), Tarun () and Tarun et al . () provides a practical way to transform an MSMO optimization problem into an MSSO optimization problem using the weighted sum of objective functions scalarization method. The MSSO optimization problem formulation (shown schematically in Figure ) can be expressed as follows.…”

“…Tarun et al . () presents the working of three‐phase methodology in conjunction with the augmented (for MSMO optimization problem) high‐dimensional, continuous‐state stochastic dynamic programming method by solving a 17‐stage, 20‐dimensional, 6‐objective wastewater treatment system. This is the largest MSMO optimization problem that has been solved in the literature.…”