(635a) A Scalable Stochastic Programming Approach for Designing Flexible Systems

Flexibility is the ability of a system to maintain feasible operation in the face of externalities. A number of approaches have been proposed to quantify and analyze system flexibility and to design flexible systems [1]. In the context of optimal design, these approaches have given rise to a class of design problems where design variables are selected to minimize cost and either satisfy a fixed uncertainty set (the flexibility test problem) or maximize the size of the uncertainty set (the flexibility index problem) [2,3,4,5]. However, studies reported in the literature find the latter problem to be largely intractable due to its multi-objective complexity [1,6]. Thus, the optimal design problem has traditionally been formulated such that it enforces feasibility over a fixed uncertainty set. Such an approach is similar to robust programming techniques and similarly can be overly conservative [7].

The stochastic flexibility (SF) index is a flexibility measure that exactly quantifies the probability of finding feasible operation and thus avoids the conservative behavior associated with uncertainty sets [8]. Specifically, it models uncertain parameters as random variables with an associated probability density function that can be integrated over the feasible region to determine the probability of satisfying system constraints (i.e., having feasible operation). It can be computed rigorously by evaluating the feasibility of Monte Carlo samples [9]. The SF index can be used as a metric to guide the design of flexible systems. Such a design problem would seek a design that minimizes a design cost while ensuring that the system remains feasible with a given SF index. This approach gives rise to a joint chance constraint problem that is computationally challenging to solve. In particular, the joint chance constraint often needs to be reformulated by using binary variables [10].

In this presentation, we demonstrate the utility of this design problem and provide a more scalable approach to reformulate the joint chance constraint using continuous variables. The approach relies on the observation that the joint chance constraint problem provides a Pareto solution for the conflict resolution (multi-objective) problem that seeks to minimize cost and maximize the SF index. We will demonstrate that we can recover the Pareto set for this problem to high accuracy by solving a continuous formulation. This thus provides a scalable approach to solve large-scale design problems that would otherwise be intractable. Furthermore, this approach has implications for general joint chance constraint problems that have been traditionally bottle-necked by binary reformulations.

References:

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[9] A. Shapiro. Sample Average Approximation. In Encyclopedia of Operations Research and Management Science, pages 1350–1355. Springer, 2013.

[10] J. Luedtke and S. Ahmed. A Sample Approximation Approach for Optimization with Probabilistic Constraints. SIAM Journal on Optimization, 19(2):674–699, 2008.