Comprehensive Sensitivity Analysis in NLP Models in PSE Applications Using Space-Filling DOE Strategy
Chemical Engineering Transactions
Sensitivity analysis is an integral step in the interpretation of the solutions of optimization models, particularly when there are uncertainties in the numerical values of model parameters. Conventional approaches to sensitivity analysis rely on the use of shadow prices in linear models and Lagrange multipliers in non-linear models. Modern commercial optimization software packages are able to automatically generate such sensitivity coefficients to allow rapid post-optimality analysis. However,
... in the case of non-linear models, Lagrange multipliers have two distinct limitations. First, they represent only changes in the optimal value of an objective function with respect to small changes in parameter values, and thus remain valid only near the immediate vicinity of the nominal design point. Secondly, each Lagrange multiplier gives only the effect of the change of one parameter, assuming that all other parameters remain at their nominal values. Hence, they provide no information about joint effects or interactions caused by simultaneous changes in parameter values. In this paper, we present a strategy based on design of experiments (DOE) to generate a sensitivity surface, which we define as the mapping of the optimal model solution against a range of values of the optimization model parameters. Space-filling designs are used as a basis to generate proxy regression models with quadratic and interaction terms, in order to capture curvature of the sensitivity surface. The resulting proxy model contains more information than is available in conventional sensitivity analysis. In particular, this approach shows curvature and interaction effects that are not reflected when Lagrange multipliers are used. We present case studies based on problems drawn from process systems engineering (PSE) literature to illustrate this comprehensive sensitivity analysis strategy.