A $\text{3}\text{2}$-approximation algorithm for parallel machine scheduling with controllable processing times

Abstract

We derive a $\text{3}\text{2}$-approximation algorithm for the NP-hard parallel machine total weighted completion time problem with controllable processing times by the technique of convex quadratic programming relaxation.

Introduction

Scheduling models with job processing times being controllable through allocation of additional resources have received considerable attention in the last decade (see, e.g. [10], [2]). These models differ from the traditional scheduling models, where job processing times are assumed to be fixed, i.e. non-controllable. Many application examples of such models are described in the literature; for instances, the scheduling of tooling machines [14] and chemical processes [11] may be characterized as scheduling models with controllable job processing times.

In this paper, we consider the parallel machine total weighted completion time scheduling problem with controllable processing times. In this problem, we are given a set of n jobs J={1,2,…,n} to be processed on a set of m unrelated parallel machines M={1,2,…,m} such that each job only needs to be processed by one of the machines. Each job j∈J has a weight w_j∈Z⁺ and a normal processing time p_ij∈Z⁺∪{0} if it is processed on machine i∈M. The processing times of jobs are controllable in the following manner. The normal processing time of job j can be reduced by up to u_ij units (u_ij∈Z⁺∪{0} and u_ij⩽p_ij) if its processing on machine i is speeded up. Each unit reduction of processing time of job j on machine i requires a cost of c_ij due to the fact that additional resources are necessary for the speedup. In a given schedule, let t_ij denote the reduction of processing time and P_ij=p_ij−t_ij the actual processing time of job j if job j is processed on machine i. Let C_j denote the completion time of job j. The problem is to find a schedule of the jobs and a processing time reduction t_ij (t_ij⩽u_ij) for each job j if it is processed on machine i such that the total cost including the total weighted completion time of jobs and the total cost of speedup, i.e. $\text{∑}{}_{\mathrm{j\in J}}\text{w}{}_{\mathrm{j}}\text{C}{}_{\mathrm{j}}\text{+∑}{}_{\mathrm{i\in M}}\text{∑}{}_{\mathrm{j\in J}}\text{c}{}_{\mathrm{ij}}\text{t}{}_{\mathrm{ij}}$, is minimum. Following the three-field notation proposed by Lawler et al. [8], we denote this problem by R|cpt|Σw_jC_j+ΣΣc_ijt_ij, where the notation “cpt” stands for “controllable processing times”.

The problem R|cpt|Σw_jC_j+ΣΣc_ijt_ij is NP-hard even when there is only one machine [6]. No results on this problem have been reported in the literature. However, there are a handful of existing results on some special cases of the problem. Chen [1] gives a branch-and-bound exact solution algorithm for the problem P|cpt|Σw_jC_j+Σc_jt_j, where all the machines are identical, c_ij≡c_j, u_ij=u_j, and t_j is the processing time reduction of job j. Cheng et al. [3] show that the problem P|cpt|ΣC_j+Σc_jt_j is solvable by a polynomial-time algorithm. Vickson [15] shows that for the problem with one machine 1|cpt|Σw_jC_j+Σc_jt_j, there is an optimal schedule that satisfies the following all-or-none property: the processing time of each job j∈J is either fully reduced or not reduced at all, i.e. t_j∈{0,u_j}, and its actual processing time P_j∈{p_j,p_j−u_j}. Huang and Zhang [7] give polynomial-time algorithms for the problem 1|cpt|Σw_jC_j+Σc_jt_j with u_j≡u and c_j≡c.

In this paper, we propose a $\text{3}\text{2}$-approximation algorithm for the problem R|cpt|Σw_jC_j+ΣΣc_ijt_ij. The algorithm is based on a convex quadratic programming relaxation for an integer quadratic programming formulation of the problem. We are inspired by the recent successful application of convex quadratic and semidefinite programming relaxations by Skutella [12], [13] to a class of parallel-machine scheduling problems where processing times of jobs are not controllable. Skutella gives a $\text{3}\text{2}$-, 2-, and 2-approximation algorithm, respectively, for the problems R∥Σw_jC_j, R|r_ij|Σw_jC_j, and R|pmtn|Σw_jC_j. The design of these algorithms is based on the following idea. A given problem is first formulated as an integer quadratic program (IQP). Then a convex quadratic program (CQP) is derived by relaxing the IQP. An integer solution rounded from the fractional solution of CQP is feasible for IQP and used as an approximation solution of IQP. We extend Skutella's analysis on the problem R∥Σw_jC_j to the more general problem R|cpt|Σw_jC_j+ΣΣc_ijt_ij.

This paper is organized as follows. In Section 2 we formulate the problem R|cpt|Σw_jC_j+ΣΣc_ijt_ij as an IQP, and perform some preliminary analysis. In Section 3 we form a CQP relaxation and a strengthened relaxation for IQP and give a 2- and $\text{3}\text{2}$-approximation algorithm, respectively. Finally, we conclude the paper in Section 4.