Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Solution of a bilevel optimistic scheduling problem on parallel machines

Quentin Schau, Olivier Ploton, Vincent T'kindt, Han Hoogeveen, Federico Della Croce, Jippe Hoogeveen

TL;DR

The uniform parallel machines scheduling problem is considered in the context of optimistic bilevel optimization, and it is shown that this problem is NP-hard in the strong sense by providing a reduction from the Numerical 3-Dimensional Matching problem and a moderately exponential-time dynamic programming algorithm.

Abstract

We consider the uniform parallel machines scheduling problem in the context of optimistic bilevel optimization, where two speed options are considered. In this scenario, the leader aims to minimize the weighted number of tardy jobs, while the follower seeks to minimize the total completion time on a set of uniform machines. This problem has practical applications in Industry 4.0. We show that this problem is NP-hard in the strong sense by providing a reduction from the Numerical 3-Dimensional Matching problem and we provide a moderately exponential-time dynamic programming algorithm. The problem is solved by means of a concise MIP formulation and a branch-and-bound algorithm that embeds a column generation approach for the lower bound computation. Computational experiments are presented for instances with up to 80 jobs and 4 machines while larger problems are out of reach for the proposed approaches.

Solution of a bilevel optimistic scheduling problem on parallel machines

TL;DR

The uniform parallel machines scheduling problem is considered in the context of optimistic bilevel optimization, and it is shown that this problem is NP-hard in the strong sense by providing a reduction from the Numerical 3-Dimensional Matching problem and a moderately exponential-time dynamic programming algorithm.

Abstract

We consider the uniform parallel machines scheduling problem in the context of optimistic bilevel optimization, where two speed options are considered. In this scenario, the leader aims to minimize the weighted number of tardy jobs, while the follower seeks to minimize the total completion time on a set of uniform machines. This problem has practical applications in Industry 4.0. We show that this problem is NP-hard in the strong sense by providing a reduction from the Numerical 3-Dimensional Matching problem and we provide a moderately exponential-time dynamic programming algorithm. The problem is solved by means of a concise MIP formulation and a branch-and-bound algorithm that embeds a column generation approach for the lower bound computation. Computational experiments are presented for instances with up to 80 jobs and 4 machines while larger problems are out of reach for the proposed approaches.
Paper Structure (19 sections, 9 theorems, 18 equations, 3 figures, 4 tables)

This paper contains 19 sections, 9 theorems, 18 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Properties, modeling and complexity issues
  3. Block structure
  4. Mixed Integer Programming
  5. Complexity results
  6. The follower's problem
  7. The bilevel problem
  8. A related polynomially solvable bilevel scheduling problem
  9. Properties of the solutions
  10. A branch-and-bound algorithm
  11. Branching scheme
  12. Memorization
  13. Initial upper bound computation
  14. Lower bound computation by Column Generation
  15. Generating Columns
  16. ...and 4 more sections

Key Result

Theorem 2.1

The $Q|V_i \in \{V_{0},V_{1}\}|\sum_j C_j$ problem is solvable in $\mathcal{O}(n \log n)$ time.

Figures (3)

  • Figure 1: Scheduling pattern with block structure for the solution of $Q|V_i \in \{V_{1},V_{0}\}|\sum_j C_j$
  • Figure 2: Machine schedules for the problem $P2||Lex \left( \sum_j C_j,\sum_j U_j \right)$
  • Figure 3: Machine schedules for the $P||Lex \left( \sum_j C_j,\sum_j U_j \right)$

Theorems & Definitions (18)

  • Theorem 2.1
  • Remark
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Corollary 2.4.1
  • proof
  • ...and 8 more