A nonlinear multidimensional knapsack problem in the optimal design of mixture experiments - Goos, P.; Syafitri, U.; Sartono, B.; Vazquez, A.R. (European journal of operational research, 2020)
•We define a novel type of nonseparable multidimensional knapsack problem.•The new knapsack problem arises in the statistical design of mixture experiments.•We propose a variable descent algorithm to compute D- and I-optimal designs.•We also propose a mixed integer nonlinear programming formulation.•We study designs for first-order and second-order Scheffé regression models. Mixture experiments usually involve various constraints on the proportions of the ingredients of the mixture under study. In this paper, inspired by the fact that the available stock of certain ingredients is often limited, we focus on a new type of constraint, which we refer to as an ingredient availability constraint. This type of constraint substantially complicates the search for optimal designs for mixture experiments. One difficulty, for instance, is that the optimal number of experimental runs is not known a priori. The resulting optimal experimental design problem belongs to the class of nonlinear nonseparable multidimensional knapsack problems. We present a variable neighborhood search algorithm as well as a mixed integer nonlinear programming approach to tackle the problem to identify D- and I-optimal designs for mixture experiments when there is a limited stock of certain ingredients, and we show that the variable neighborhood search algorithm is highly competitive in terms of solution quality and computing time.