Branch-and-bound method for minimizing the weighted completion time scheduling problem on a single machine with release dates

Abstract

In this paper, we consider a single-machine scheduling problem with release dates. The aim is to minimize the total weighted completion time. This problem is known to be strongly NP-hard. We propose two new lower bounds that can be, respectively, computed in O(n²) and in $O(n\log n)$ time where n is the number of jobs. We prove a sufficient and necessary condition for local optimality, which can also be considered as a priority rule. We present an efficient heuristic using such a condition. We also propose some dominance properties. A branch-and-bound algorithm incorporating the heuristic, the lower bounds and the dominance properties is proposed and tested on a large set of instances.

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 J_i has a positive processing time p_i, a positive release date r_i and a positive weight w_i. Given a schedule S of jobs, we denote the completion time of job J_i as C_i(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 $\sum w_i{C}_i$. According to the standard machine scheduling classification, this problem is denoted as $1\left|r_i\right|\sum w_i\mathit{Ci}$ [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 $p_i/w_i$. 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 $1\left|d_i\right|\sum w_i{C}_i$ based on the works of Potts and Van Wassenhove [12] and Posner [13].

The development of polynomial time approximation algorithms received much attention. For $1|r_i,\mathit{pmtn}|\sum w_i{C}_i$, 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 $1\left|r_i\right|\sum w_i{C}_i$. 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.