Scheduling on uniform parallel machines to minimize maximum lateness

Abstract

We consider the uniform parallel machine scheduling problem with the objective of minimizing maximum lateness. We show that an extension of the EDD rule to a uniform parallel machine setting yields a maximum lateness value which does not exceed the optimal value by more than p_max, where p_max is the maximum job processing time.

Introduction

One of the earliest results in scheduling theory is that the earliest due date (EDD) rule minimizes maximum lateness on a single machine [6]. The objective of this paper is to extend the EDD rule to a uniform parallel machine setting where the maximum lateness problem is known to be NP-hard. We show that the implementation of the EDD rule to the uniform parallel machine problem yields a maximum lateness value that does not exceed the optimal value by more than p_max, where p_max is the maximum job processing time.

We formally define the uniform parallel machines scheduling problem with the maximum lateness objective, Qm//L_max, as follows. There are n jobs to be processed without preemption on m continuously available uniform parallel machines. Each machine can process only one job at a time, and each job can be processed on only one machine. Job $\text{J}{}_{\mathrm{j}}\text{,}\text{j=1,…,n}$, becomes available at time zero, requires p_j units of processing and has a due date d_j. Machine $\text{M}{}_{\mathrm{i}}\text{,}\text{i=1,…,m}$, has a speed $\text{s}{}_{\mathrm{i}}\text{,}\text{s}{}_{\mathrm{i}}\text{⩾1}$. The impact of speed s_i is that machine M_i can carry out s_i units of processing in one time unit. Without loss of generality, we may assume that s₁⩾s₂⩾⋯⩾s_m=1. If job J_j is assigned to machine M_i then it requires p_j/s_i time units to be completed. The objective is to determine a schedule so that the maximum lateness $\text{L}{}_{\mathrm{<mtext>max</mtext>}}\text{=}\text{max}{}_{\mathrm{1⩽j⩽n}}\text{L}{}_{\mathrm{j}}$ is minimized, where L_j=C_j−d_j is the lateness and C_j is the completion time of job J_j.

The Qm//L_max problem is known to be NP-hard even for m=2 [3]. Let S_H be a schedule generated by a heuristic H and let $\text{S}{}^{\mathrm{\ast }}$ be an optimal schedule for Qm//L_max. Heuristic H is said to provide the worst-case ratio bound ρ if for any problem instance $\text{L}{}_{\mathrm{<mtext>max</mtext>}}\text{(S}{}_{\mathrm{H}}\text{)/L}{}_{\mathrm{<mtext>max</mtext>}}\text{(S}{}^{\mathrm{\ast }}\text{)⩽ρ}$. Since it is possible that $\text{L}{}_{\mathrm{<mtext>max</mtext>}}\text{(S}{}^{\mathrm{\ast }}\text{)=0}$, Lenstra [8] suggested an equivalent formulation of the Pm//L_max problem (with identical parallel machines) which avoids this difficulty. Let $\text{q}{}_{\mathrm{j}}\text{=d}{}_{\mathrm{<mtext>max</mtext>}}\text{−d}{}_{\mathrm{j}}\text{,}\text{j=1,…,n}$, be the delivery time (or tail) of job J_j, where $\text{d}{}_{\mathrm{<mtext>max</mtext>}}\text{=}\text{max}{}_{\mathrm{1⩽j⩽n}}\text{d}{}_{\mathrm{j}}$ is the maximum due date. Consider the related problem Pm/q_j/C_max, where C_max=max_1⩽j⩽n(C_j+q_j). Observe that C_max=max_1⩽j⩽n(C_j+d_max−d_j)=max_1⩽j⩽n(C_j−d_j)+d_max=L_max+d_max. Thus Pm//L_max and Pm/q_j/C_max are equivalent. The above relationships are applicable to the Qm//L_max case as well. Until now, no worst-case ratio bounds have been obtained in the literature for either Qm//L_max or Qm/q_j/C_max. Gusfield [4] implemented the EDD heuristic for the Pm/r_j/L_max problem (with unequal job release times r_j) and obtained the bound $\text{L}{}_{\mathrm{<mtext>max</mtext>}}\text{(S}{}_{\mathrm{<mtext>EDD</mtext>}}\text{)−L}{}_{\mathrm{<mtext>max</mtext>}}\text{(S}{}^{\mathrm{\ast }}\text{)⩽[(2m−1)/m]p}{}_{\mathrm{<mtext>max</mtext>}}$. Using the same heuristic as Gusfield [4], Masuda et al. [9] obtained a modified worst-case ratio bound for the Pm//L_max problem of the form $\text{(L}{}_{\mathrm{<mtext>max</mtext>}}\text{(S}{}_{\mathrm{<mtext>EDD</mtext>}}\text{)−L}{}_{\mathrm{<mtext>max</mtext>}}\text{(S}{}^{\mathrm{\ast }}\text{))/(L}{}_{\mathrm{<mtext>max</mtext>}}\text{(S}{}^{\mathrm{\ast }}\text{)+d}{}_{\mathrm{<mtext>max</mtext>}}\text{)⩽1−1/m}$. Carlier [1] also considered the same heuristic as Gusfield [4] applied to the Pm/r_j,q_j/C_max problem and obtained the bound $\text{C}{}_{\mathrm{<mtext>max</mtext>}}\text{(S}{}_{\mathrm{<mtext>EDD</mtext>}}\text{)−C}{}_{\mathrm{<mtext>max</mtext>}}\text{(S}{}^{\mathrm{\ast }}\text{)⩽2(p}{}_{\mathrm{<mtext>max</mtext>}}\text{−1)}$. Finally, Hall and Shmoys [5] considered the problem Pm/r_j,q_j,prec/C_max (with precedence constraints) and proved that for general list scheduling heuristic (LS) $\text{C}{}_{\mathrm{<mtext>max</mtext>}}\text{(S}{}_{\mathrm{<mtext>LS</mtext>}}\text{)/C}{}_{\mathrm{<mtext>max</mtext>}}\text{(S}{}^{\mathrm{\ast }}\text{)<2}$. They also developed a polynomial approximation scheme for this problem. For more on the related literature, see Lawler et al. [7] and Tanaev et al. [10].

In this paper, we obtain ratio bounds for the Qm//L_max and Qm/q_j/C_max problems. Our bounds yield the result of Masuda et al. [9] when $\text{s}{}_{\mathrm{i}}\text{=1,}\text{i=1,…,m}$, that is when Qm//L_max reduces to the Pm//L_max problem.