Response Surface Methodology

This chapter first summarizes Response Surface Methodology (RSM), which started with Box and Wilson’s 1951 article on RSM for real, non-simulated systems. RSM is a stepwise heuristic that uses first-order polynomials to approximate the response surface locally. An estimated polynomial metamodel gives an estimated local gradient, which RSM uses in steepest ascent (or descent) to decide on the next local experiment. When RSM approaches the optimum, the latest first-order polynomial is replaced by a second-order polynomial. The fitted second-order polynomial enables the estimation of the optimum. This chapter then focuses on simulated systems, which may violate the assumptions of constant variance and independence. A variant of RSM that provably converges to the true optimum under specific conditions is summarized, and an adapted steepest ascent that is scale-independent is presented. Next, the chapter generalizes RSM to multiple random responses, selecting one response as the goal variable and the other responses as the constrained variables. This generalized RSM is combined with mathematical programming to estimate a better search direction than the steepest ascent direction. To test whether the estimated solution is indeed optimal, bootstrapping may be used. Finally, the chapter discusses robust optimization of the decision variables, while accounting for uncertainties in the environmental variables.