Convergence of Stochastic Optimization and Decision Analysis in the Engineering Design of Aquifer Remediation
Authors: Freeze R. Allan , Gorelick Steven M. 
Published: August 2005

This paper compares and contrasts stochastic optimization and decision analysis as frameworks for the design of remedial pump‐and‐treat systems in contaminated aquifers. Both decision‐making frameworks (1) seek a least‐cost, low‐risk remedial design; (2) consider uncertainty due to partial knowledge of field environments, which causes imperfect predictive capability of simulation; (3) target predictive uncertainty due to spatially variable hydraulic conductivities and handle it by invoking geostatistical uncertainty theory, and (4) deal with the design and economic impacts of uncertainty by employing the concept of reliability or its complement the probability of failure. The fundamental difference between the two approaches lies in the fact that decision analysis considers a broad suite of technological strategies from which one of many predetermined design alternatives is selected as the best, while stochastic optimization determines the optimal pump‐and‐treat design but considers only one technological strategy at a time. The early stochastic optimization formulations sought to quantify the cost of overdesign needed to achieve greater performance reliability. The procedure involved a cost minimization that led to the development of a trade‐off curve of cost versus reliability. For each point on the trade‐off curve a single‐valued optimum was achieved by defining a preset level of desired reliability. Decision analysis has always involved a cost‐risk minimization, in which a single‐valued optimum is obtained by simultaneously accounting for all costs, including the risk costs associated with the probability of failure. Risk costs are assigned a dollar value based on the level of expected reliability; a trade‐off curve is not needed. More‐recent formulations using stochastic optimization follow the philosophy of the decision‐analysis framework by accounting for risk costs through a penalty cost. Using the latter approach, we show that the objective functions in both frameworks are virtually identical.

A decision maker should adopt a decision‐analysis framework if he or she (1) wants to minimize total system cost by selecting the best design alternatives from among a specified set, (2) has a known risk‐cost preference (utility function), (3) wants to consider a broad suite of technological alternatives, and (4) is willing to accept the numbers, locations, and pumping rates for wells that are the best of those under consideration but are not necessarily optimal. The advantages of decision analysis lie in the ease with which capital costs can be incorporated, and the ability to examine alternative designs that span multiple technologies. The disadvantages revolve around the difficulty in determining a decision maker's utility function, selecting a single‐valued design as the best from the predefined set of design alternatives, and the inefficiencies introduced by the need for a full enumeration of the design alternatives. A decision maker should adopt optimization if he or she (1) is interested in a truly optimal selection of well locations and pumping rates, (2) has an unknown or uncertain risk‐cost preference, and (3) is comfortable considering a single remedial technology at a time. The advantages of optimization lie in its clever and efficient methodologies for identifying a global optimum. The main disadvantages lie in the difficulties associated with a rigorous consideration of capital costs for nonlinear problems, and the fact that solutions do not typically span multiple technologies. The choice of whether to employ optimization or decision analysis as a design tool is not necessarily an either/or proposition, and we suggest possible avenues for their combined use.