Convergence of stochastic optimization and decision analysis in the engineering design of aquifer remediation

This paper compares and contrasts stochastic optimization and decision anal ysis as frameworks for the design of remedial pump-and-treat systems in con taminated 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 sim ulation; (3) target predictive uncertainty due to spatially variable hydrau lic conductivities and handle it by invoking geostatistical uncertainty the ory, and (4) deal with the design and economic impacts of uncertainty by em ploying the concept of reliability or its complement the probability of fai lure. The fundamental difference between the two approaches lies in the fac t that decision analysis considers a broad suite of technological strategie s from which one of many predetermined design alternatives is selected as t he best, while stochastic optimization determines the optimal pump-and-trea t design but considers only one technological strategy at a time. The early stochastic optimization formulations sought to quantify the cost of overde sign needed to achieve greater performance reliability. The procedure invol ved 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-va lued 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 co sts, including the risk costs associated with the probability of failure. R isk costs are assigned a dollar value based on the level of expected reliab ility; a trade-off curve is not needed. More-recent formulations using stoc hastic optimization follow the philosophy of the decision-analysis framewor k by accounting for risk costs through a penalty cost. Using the latter app roach, we show that the objective functions in both frameworks are virtuall y 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 alternat ives from among a specified set, (2) has a known risk-cost preference (util ity function), (3) wants to consider a broad suite of technological alterna tives, and (4) is willing to accept the numbers, locations, and pumping rat es for wells that are the best of those under consideration but are not nec essarily optimal. The advantages of decision analysis lie in the ease with which capital costs can be incorporated, and the ability to examine alterna tive designs that span multiple technologies. The disadvantages revolve aro und the difficulty in determining a decision maker's utility function, sele cting a single-valued design as the best from the predefined set of design alternatives, and the inefficiencies introduced by the need for a full enum eration of the design alternatives. A decision maker should adopt optimizat ion if he or she (1) is interested in a truly optimal selection of well loc ations and pumping rates, (2) has an unknown or uncertain risk-cost prefere nce, and (3) is comfortable considering a single remedial technology at a t ime. The advantages of optimization lie in its clever and efficient methodo logies for identifying a global optimum. The main disadvantages he in the d ifficulties associated with a rigorous consideration of capital costs for n onlinear problems, and the fact that solutions do not typically span multip le technologies. The choice of whether to employ optimization or decision a nalysis as a design tool is not necessarily an either/or proposition, and w e suggest possible avenues for their combined use.