Branch-and-bound method for minimizing the weighted completion time scheduling problem on a single machine with release dates
Highlights
► In this paper, we consider a single machine scheduling problem with release dates. ► We propose two new lower bounds that can be, respectively, computed in and in . ► We present a new efficient heuristic which is complementary with the existence results. ► We propose some dominance properties and a branch-and-bound algorithm capable of solving problems with up to 120 jobs.
Introduction
In this paper, we consider the scheduling of a set N of n jobs with release dates on a single machine, with the aim of minimizing the total weighted completion time. The machine can process at most one job at a time. Every job Ji has a positive processing time pi, a positive release date ri and a positive weight wi. Given a schedule S of jobs, we denote the completion time of job Ji as Ci(S). Without loss of generality, we assume that the jobs are sorted in nondecreasing order of their release dates. The aim is to find a schedule to minimize the total weighted completion time . According to the standard machine scheduling classification, this problem is denoted as [1]. Given the aim of this paper, we recall the following related works.
Lenstra et al. [2] showed that this problem is strongly NP-hard. Labetoulle et al. [3] proved that this problem remains strongly NP-hard even if preemption is allowed. When all release dates are equal, the problem can be solved by the shortest weighted processing time (SWPT) rule [4] in which jobs are sorted in nondecreasing order of . For the problem with arbitrary weights, Rinaldi and Sassano [5] and Bianco and Ricciardelli [6] proposed several dominance properties. The exploitation of various mixed integer programming formulations to generate lower bounds was investigated by Dyer and Wolsey [7]. A branch-and-bound algorithm was proposed by Hariri and Potts [8] in which the lower bound is obtained using a Lagrangian relaxation. Belouadah et al. [9] proposed a new lower bound based on the job splitting principle and new dominance rules. They developed a branch-and-bound algorithm able to solve problems with up to 50 jobs. Jouglet et al. [10] elaborated other dominance properties and they proposed a branch-and-bound method able to solve problems with up to 100 jobs. Pan [11] proposed an improving branch-and-bound method for the problem based on the works of Potts and Van Wassenhove [12] and Posner [13].
The development of polynomial time approximation algorithms received much attention. For , Goemans et al. [14] proposed 1.47-approximation algorithm. Such a result improves the 2-approximation algorithm of Hall et al. [15]. Skutella [16] improved this result by a 4/3-approximation algorithm and he constructed a 1.6853-approximation algorithm for . Afrati et al. [17] proposed an efficient polynomial time approximation scheme for the single-machine case and extended it to the parallel-machine problem.
This paper is organized as follows. Section 2 describes the two proposed lower bounds. In Section 3, the dominance rules and a heuristic algorithm are proposed. Section 4 presents the branch-and-bound algorithm that we design. The different numerical experiments are summarized in Section 5. Finally, some concluding remarks are discussed in Section 6.
Section snippets
Lower bounds
In this section, we present two lower bounding schemes which can be useful to evaluate the quality of a feasible schedule. The first bound is based on exchanging weights and job splitting. The second proposed bound is based on a new splitting scheme, which allows us to exploit the good property of the Shortest Remaining Processing Time (SRPT) rule that can optimally solve [18].
Dominance properties
We define a dominance relation between partial schedules, which can help us to prune nodes. We denote the total weighted completion time for jobs in the partial schedule , the completion time of the last job in and is the schedule composed of and completed by the optimal partial schedule of the remaining jobs.
Let and denote, respectively, the earliest starting time, earliest completion time of job i at time and by the completion
Branch-and-bound algorithm
The proposed branch-and-bound algorithm uses a basic scheme. During the computation, we keep the list of nonexplored nodes arranged in increasing order of the corresponding lower bounds (ties are broken according to the nonincreasing order of the number of scheduled jobs). Each node represents a partial schedule which is also a partial list. The algorithm always tries to develop the head of the list. The branching from a node consists in creating new nodes by adding an unscheduled job to the
Computational results
This section describes the computational results obtained for the proposed branch-and-bound algorithm. The branch-and-bound algorithm was coded in the C Language on 2.80 GHz Pentium D processor with 2 Giga RAM. The branching strategy is the breadth-first strategy to take advantage of Theorem 3.1.
For each job j, an integer processing times pj from the uniform distribution and an integer wj from the uniform distribution was generated. Release dates were generated from the uniform
Conclusion
In this paper, we investigated the single-machine scheduling problem with the aim of minimizing the total weighted completion time, under release dates assumption. The main contribution of this paper lies in the two proposed new lower bounds. We showed that these lower bounds are complementary to an existing one. For this reason, our choice was to compute all of them to get the tightest possible bound. Our current work consists in analyzing the cases where our new bounds are smaller than the
Acknowledgments
The authors are grateful to the anonymous referees for their constructive remarks and suggestions.
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