Combining two-stage stochastic programming and recoverable robustness to minimize the number of late jobs in the case of uncertain processing times

Abstract

Minimizing the number of late jobs on a single machine is a classic scheduling problem, which can be used to model the situation that from a set of potential customers, we have to select as many as possible whom we want to serve, while selling no to the other ones. This problem can be solved by Moore-Hodgson's algorithm, provided that all data are deterministic. We consider a stochastic variant of this problem, where we assume that there is a small probability that the processing times differ from their standard values as a result of some kind of disturbance. When such a disturbance occurs, then we must apply some recovery action to make the solution feasible again. This leads us to the area of recoverable robustness, which handles this uncertainty by modeling each possible disturbance as a scenario; in each scenario, the initial solution must then be made feasible by applying a given, simple recovery algorithm to it. Since we cannot accept previously rejected customers, our only option is to reject customers that would have been served in the undisturbed case. Our problem therefore becomes to find a solution for the undisturbed case together with a feasible recovery to every possible disturbance. Our goal hereby is to maximize the expected number of served customers; we assume here that we know the probability that a given scenario occurs. In this respect, our problem falls outside the area of the 'standard' recoverable robustness, which contains the worst-case recovery cost as a component of the objective. Therefore, we consider our approach as a combination of two-stage stochastic programming and recoverable robustness. We show that this problem is N P-hard in the ordinary sense even if there is only one scenario, and we present some sufficient conditions that allow us to find a part of the optimal solution in polynomial time. We further evaluate several solution methods to find an optimal solution, among which are dynamic programming, branch-and-bound, and branch-and-price.

Algorithms for rescheduling jobs with a LIFO buffer to minimize the weighted number of late jobs

“…In the literature, various other situations of rescheduling were considered. One research branch going back to the seminal work of Hall and Potts (2004) considers the arrival of new jobs to be integrated into the sequence of old jobs (see also Hall et al 2007 andRener et al 2022). Another direction moving into the area of robust optimization deals with changes in the original data of the jobs.…”

Robust Predictive-Reactive Scheduling: An Information-Based Decision Tree Model

“…In order to provide solutions that are well balanced between robustness and quality, many hybrid methods have been suggested. The recoverable robustness framework [1] considers two stage decisions where robust decisions are taken at the first stage and where recovery algorithms are used to restore feasibility for a realised scenario on the second stage. This framework is linked to adjustable robust optimisation [12] that roughly transposes the concept of two-stage stochastic programming to robust optimisation: some recourse variables can be adjusted to the realised scenario.…”

Adjustable robust optimization with nonlinear recourses

“…recourse actions by adding, to the possibility of keeping or rejecting a job at the second stage, the option of repairing it at the expense of an extra processing time. Third, the uncertainty is modeled in van denAkker et al [2018] by a finite set of discrete scenarios that affects the processing times only, while we consider a polyhedral uncertainty set defining the objective function. Both papers aim at optimizing the worst-case cost, added to average cost in van den Akker et al[2018].…”

“…In terms of methodology, although the two works use branch-and-price algorithms, the reformulations used are totally different. The work in van denAkker et al [2018] is based on a classical deterministic equivalent formulation, where the recourse decisions for each scenario are modeled using one set of variables and constraints, whereas the current work is based on a robust two-stage programming formulation, which is rewritten as a static robust program of very large size.…”