Dynasearch for the earliness–tardiness scheduling problem with release dates and setup constraints

doi:10.1016/j.orl.2005.06.005

Operations Research Letters

Volume 34, Issue 5, September 2006, Pages 591-598

https://doi.org/10.1016/j.orl.2005.06.005 Get rights and content

Abstract

A large dynasearch neighborhood is introduced for the one-machine scheduling problem with sequence-dependent setup times and costs and earliness–tardiness penalties. Finding the best schedule in this neighborhood is NP-complete in the ordinary sense but can be done in pseudo-polynomial time. We also present experimental results.

Introduction

Local search algorithms are widely used as a practical approach for solving combinatorial optimization problems. Starting with a feasible solution, these algorithms iteratively try to improve the current solution by searching a better solution in the neighborhood of the current solution until a local minimum is found. The efficiency of these algorithms critically depends on the definition of the neighborhood: with a larger neighborhood, the quality of the local minimum is generally better but the computation time required to explore the neighborhood is longer. So, in practice, large neighborhood are not useful unless it can efficiently be explored. Such an approach is referred to as a very large-scale neighborhood search technique, this term is popularized in the survey by Ahuja et al. [1].

In the scheduling literature, Congram et al. [2] present a large neighborhood—called dynasearch—for the single-machine total weighted tardiness scheduling problem. This neighborhood can be seen as the composition of an arbitrary number of independent swap operations. Its size is exponential but a dynamic programming algorithm is proposed to compute the optimal schedule in the neighborhood in polynomial time. Based on this work, Grosso et al. [3] propose an enhanced dynasearch neighborhood that combines the swap operator used in [2] with a second “extract and reinsert” operator.

In this paper, we extend this approach to a more general and a more practical problem that appears in particular in manufacturing scheduling problems, and that is called L2-STC-NCOS in MASCLIB, the library of manufacturing scheduling problems from industry proposed by ILOG [4]. More precisely, our problem deals with setup times and setup costs, release dates, deadlines and a general end-time-dependent costs that can for example model the commonly used earliness–tardiness penalties. The reader is referred to [4], [5] for a presentation of the relevance of this problem and for related references in the just-in-time and setup scheduling literature. Mathematical programming formulations of the problem, a branch-and-bound algorithm and an efficient heuristic procedure are also presented in [5].

Section 2 introduces the scheduling problem. The associated dynasearch problem, which consists in finding the optimal solution in the so-called dynasearch neighborhood, is shown to be NP-complete in the ordinary sense. Section 3 is devoted to the presentation of a pseudo-polynomial algorithm to search the neighborhood and experimental results are eventually presented in Section 4.

Section snippets

Problem definition and complexity

A single machine has to process n tasks $J_{1}, \dots, J_{n}$ such that at most one task is performed at any time. Preemption of tasks is not allowed but idle time can be inserted between two tasks. Each task $J_{i}$ has a processing time $p_{i}$ and belongs to a group (or family) $g_{i} \in {1, \dots, q}$ (with $q ⩽ n$ ). Setup or changeover times and costs, which are given as two $q \times q$ matrices, are associated to these groups. This means that in a schedule where $J_{j}$ is processed immediately after $J_{i}$ , there must be a setup time of at least

Dynasearch algorithm

Based on the approach introduced by [6], let us define $Σ_{k}^{g} (t)$ as the cost of optimally scheduling the tasks $J_{1}, \dots, J_{k}$ subject to the constraints:

$•$
the sequence is in the dynasearch neighborhood of $(J_{1}, \dots, J_{k})$ ,
$•$
the group of the last task (i.e. the task in position k) is g,
$•$
the last task completes before time t.

Clearly,

Σ_{k}^{g}

is a real function. The value

Σ_{k}^{g} (t)

is set to

\infty

when there is no feasible schedule ending before t (for instance, if

t < \sum_{i = 1}^{k} p_{i}

). We also define

Σ_{0}^{g} (t)

as the cost of setting up the

Experimental results

In order to evaluate the behavior of the dynasearch neighborhood, we compared it to a simple descent algorithm based on the union of the three neighborhoods SWAP, EBSR and EFSR—by contrast, the dynasearch neighborhood is seen as the composition of these neighborhoods. This algorithm is described in [5] and is used to find an initial feasible solution in the branch-and-bound procedure. Even if a single descent only finds a local optimum, the iteration of several descent procedure from randomly

Acknowledgements

The author is grateful to an anonymous referee for helpful comments that improved the presentation of the theoretical and experimental results.

References (6)

A. Grosso et al.
An enhanced dynasearch neighborhood for the single-machine total weighted tardiness scheduling problem
Oper. Res. Lett.
(2004)
F. Sourd
Earliness–tardiness scheduling with setup considerations
Comput. Oper. Res.
(2005)
F. Sourd
Optimal timing of a sequence of tasks with general completion costs
European J. Oper. Res.
(2005)

There are more references available in the full text version of this article.

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