A Robust Optimization Approach for Scheduling with Uncertain Start-Time Dependent Costs

TL;DR

This paper develops a two-stage robust optimization model for single-machine scheduling with start-time dependent costs under uncertain, budgeted scenarios, addressing complexity and solution methods.

Contribution

It introduces a novel two-stage robust optimization framework for scheduling with uncertain costs and analyzes its computational complexity.

Findings

The problem is NP-hard and not approximable.

Evaluating a first-stage solution is NP-hard with discrete uncertainty.

Including uncertainty in the planning stage improves scheduling outcomes.

Abstract

In this work, we study a single-machine scheduling problem that aims at minimizing the total cost of a schedule subject to start-time dependent costs. This framework naturally captures scenarios where costs fluctuate throughout the day, such as time-varying energy or labor prices. To model more realistic scenarios, we assume that these costs lie within a budgeted uncertainty set and propose a two-stage robust optimization approach. In a first stage, the order in which activities should be executed is decided. After a cost scenario has been revealed, the starting times for each activity are established, subject to the ordering from the first stage. We demonstrate that the proposed problem is NP-hard and not approximable, implying the complexity of its robust counterpart. Furthermore, we show that already evaluating a first-stage solution is NP-hard when the uncertainty set is discrete. We develop models and solution methods for both continuous and discrete budgeted uncertainty. In computational experiments, we compare these approaches and demonstrate the advantages of including uncertainty beforehand.