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The Unreliable Job Selection and Sequencing Problem

Alessandro Agnetis, Roel Leus, Emmeline Perneel, Ilaria Salvadori

TL;DR

This paper defines the Unreliable Job Selection and Sequencing Problem (UJSSP), a stochastic single-machine scheduling model where the subset of jobs and their order are chosen to maximize the expected net profit $z(S,\sigma)$. It establishes NP-hardness in the general case via a PPP reduction and identifies polynomially solvable special cases with identical costs or identical success probabilities, solvable by greedy methods. The authors contribute a compact MILP formulation, a dynamic programming approach (pseudopolynomial for integer costs), and two novel stepwise exact algorithms that exploit affine dependency and upper-envelope techniques to prune the search space. Computational experiments show the methods solving up to thousands of jobs (and PPP-derived instances) efficiently, highlighting practical performance and potential for broader application in submodular selection contexts and reliability-aware scheduling.

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.

The Unreliable Job Selection and Sequencing Problem

TL;DR

This paper defines the Unreliable Job Selection and Sequencing Problem (UJSSP), a stochastic single-machine scheduling model where the subset of jobs and their order are chosen to maximize the expected net profit . It establishes NP-hardness in the general case via a PPP reduction and identifies polynomially solvable special cases with identical costs or identical success probabilities, solvable by greedy methods. The authors contribute a compact MILP formulation, a dynamic programming approach (pseudopolynomial for integer costs), and two novel stepwise exact algorithms that exploit affine dependency and upper-envelope techniques to prune the search space. Computational experiments show the methods solving up to thousands of jobs (and PPP-derived instances) efficiently, highlighting practical performance and potential for broader application in submodular selection contexts and reliability-aware scheduling.

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.

Paper Structure

This paper contains 36 sections, 4 theorems, 61 equations, 11 figures, 4 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Complexity
  4. Identical costs
  5. Identical probabilities
  6. The general case
  7. A MILP formulation
  8. An upper bound
  9. Dynamic programming
  10. Stepwise exact algorithms
  11. Foundations of the stepwise exact algorithms
  12. Optimality conditions
  13. Parameterizing the unknowns
  14. Upper envelope computation
  15. Bounding the unknowns
  16. ...and 21 more sections

Key Result

Theorem 1

olszewski2016 Given an instance of SSP, if: then SSP is optimally solved by the greedy algorithm.

Figures (11)

  • Figure 1: Comparison of sets to get $\mathcal{S}^{c^*}_{\leq 1}$.
  • Figure 2: Comparison of sets to get $\mathcal{S}^{c^*}_{\leq 2}$.
  • Figure 3: Comparison of sets to get $\mathcal{S}^{c^*}_{\leq 3}$.
  • Figure 4: Comparison of sets to get $S^{c^*}_{\geq 4}$.
  • Figure 5: Comparison of sets to get $S^{c^*}_{\geq 3}$.
  • ...and 6 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Example 1
  • Lemma 1
  • proof
  • Example 2