The Unreliable Job Selection and Sequencing Problem

2025-11-21

AI Summary

This paper studies the Unreliable Job Selection and Scheduling Problem (UJSSP) on a single unreliable machine: selecting a subset of jobs from a given set and sequencing them to maximize the expected net profit (reward minus processing cost). Job execution succeeds only if the machine remains operational throughout its processing; success probabilities are job-dependent and heterogeneous. We first prove UJSSP is NP-hard. Then, we formulate it as a mixed-integer linear program and a dynamic programming model. Two novel iterative exact algorithms are proposed, and their intrinsic connection to submodular optimization is revealed—showing that special cases admit polynomial-time solutions. Extensive experiments demonstrate that our methods are highly efficient and scalable, solving large-scale instances effectively. The approach is successfully applied to a real-world product segmentation scenario.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study a stochastic single-machine scheduling problem, denoted the Unreliable Job Selection and Sequencing Problem (UJSSP). Given a set of jobs, a subset must be selected for processing on a single machine that is subject to failure. Each job incurs a cost if selected and yields a reward upon successful completion. A job is completed successfully only if the machine does not fail before or during its execution, with job-specific probabilities of success. The objective is to determine an optimal subset and sequence of jobs to maximize the expected net profit. We analyze the computational complexity of UJSSP and prove that it is NP-hard in the general case. The relationship of UJSSP with other submodular selection problems is discussed, showing that the special cases in which all jobs have the same cost or the same failure probability can be solved in polynomial time. To compute optimal solutions, we propose a compact mixed-integer linear programming formulation, a dynamic programming algorithm, and two novel stepwise exact algorithms. We demonstrate that our methods are capable of efficiently solving large instances by means of extensive computational experiments. We further show the broader applicability of our stepwise algorithms by solving instances derived from the Product Partition Problem.

Problem

• Optimizing job selection and sequencing under machine failure uncertainty
• Maximizing expected net profit with stochastic job success probabilities
• Solving NP-hard scheduling problem with exact algorithms and formulations

Innovation

• Mixed-integer linear programming for job scheduling
• Dynamic programming algorithm for optimal sequencing
• Stepwise exact algorithms for large instances