A new robust optimization approach for scheduling under uncertainty: II. Uncertainty with known probability distribution

In this work, we consider the problem of scheduling under uncertainty where the uncertain problem parameters can be described by a known probability distribution function. A novel robust optimization methodology, originally proposed by Lin, Janak, and Floudas [Lin, X., Janak, S. L., & Floudas, C. A. (2004). A new robust optimization approach for scheduling under uncertainty: I. Bounded uncertainty. Computers and Chemical Engineering, 28, 1069–1085], is extended in order to consider uncertainty described by a known probability distribution. This robust optimization formulation is based on a min–max framework and when applied to mixed-integer linear programming (MILP) problems, produces “robust” solutions that are immune against data uncertainty. Uncertainty is considered in the coefficients of the objective function, as well as the coefficients and right-hand-side parameters of the inequality constraints in MILP problems. Robust optimization techniques are developed for uncertain data described by several known distributions including a uniform distribution, a normal distribution, the difference of two normal distributions, a general discrete distribution, a binomial distribution, and a poisson distribution. The robust optimization formulation introduces a small number of auxiliary variables and additional constraints into the original MILP problem, generating a deterministic robust counterpart problem which provides the optimal/feasible solution given the (relative) magnitude of the uncertain data, a feasibility tolerance, and a reliability level. The robust optimization approach is then applied to the problem of short-term scheduling under uncertainty. Using the continuous-time model for short-term scheduling developed by Floudas and co-workers [Ierapetritou, M. G. & Floudas, C. A. (1998a). Effective continuous-time formulation for short-term scheduling: 1. Multipurpose batch processes. Ind. Eng. Chem. Res., 37, 4341–4359; Lin, X. & Floudas, C. A. (2001). Design, synthesis and scheduling of multipurpose batch plants via an effective continuous-time formulation. Comp. Chem. Engng., 25, 665–674], three of the most common sources of uncertainty in scheduling problems are explored including processing times of tasks, market demands for products, and prices of products and raw materials. Computational results on several examples and an industrial case study are presented to demonstrate the effectiveness of the proposed approach.