Robust Sampling Budget Allocation Under Deep Uncertainty

A novel methodology is introduced for optimally allocating a sampling budget. Sampling budget allocation problems arise frequently in various settings. For example, in the design of complex engineering systems, given both the complexity of these systems and the imperfect information on new technologies, designers often face deep uncertainty as to system performance. Consequently, designers need to sample multiple alternative designs under a limited budget. This article proposes a minimax regret approach to allocate the sampling budget in the presence of deep uncertainty pertaining to system performance. The objective is to maximize the probability of selecting the design with the minimum-maximum regret under a limited sampling budget and imperfect information. To effectively solve the minimax regret problem, an approximation methodology that provides good solutions with quantifiable uncertainty is developed. The essence of the methodology, which has the added benefit of being generally applicable to any multilevel optimization, is that all but the first level of multilevel optimization can be eliminated via a response surface. By sampling many values of a higher level decision's variables, solving the next lower level optimization given those samples values, and calibrating a response surface to the objective function value eliminate one required optimization. Doing this repeatedly reduces the complexity of the multilevel optimization to a standard optimization. Regardless of the number of levels in the optimization, repeating this process ultimately leaves one with a single optimization whose objective function can be directly computed, given the highest level variables. Numerical experiments with two sampling allocation examples demonstrate both the benefit of the robust sampling budget allocation versus nonrobust formulations and the effectiveness of the proposed solution approach.