High-multiplicity cyclic job shop scheduling

Abstract

We consider the High-Multiplicity Cyclic Job Shop Scheduling Problem. There are two objectives of interest: the cycle time and the flow time. We give several approximation algorithms after showing that a very restricted case is APX-hard.

Introduction

Job shop scheduling is a widely studied and difficult combinatorial optimization problem [10]. We are given a collection of jobs and a set of machines. Each job consists of a sequence of operations, which must be performed in order. Each operation has a particular processing time and must be performed on a specific machine. Here we consider what we term the High-Multiplicity Cyclic Job Shop Scheduling Problem, in which the jobs are all the same and the schedule must process them in a cyclic fashion; that is, the schedule must repeat the same pattern of operations every $\tau$ time steps for some cycle time $\tau$ where $\tau$ measures the throughput of a schedule. We will also be interested in the flow time $F$, that is, the time it takes to complete one job. This measure of latency is of interest by itself, and it also indicates the amount of work-in-progress at any time during steady state.

Practical sources of this and similar problems are in integrated circuit (IC) fabrication [6], [17], machine shops [6], and printed circuit board manufacturing [18]. In IC fabrication, for instance, large numbers of identical silicon wafers are produced. Each wafer has multiple layers of circuitry and thus must pass through the stages of the photolithographic process multiple times. The problem is important in other environments as well. Shop floors are considered easier to manage if production follows a repeating pattern [11]. Compilers in high-performance computing environments schedule loops of code to be repeated many times [9]. It is desirable to maximize throughput and at the same time minimize work-in-progress in many manufacturing environments [9].

We now formally define the High-Multiplicity Cyclic Job Shop Scheduling Problem. We are given a set of identical jobs that must be processed on a given set $M=\{M_1,\dots ,M_m\}$ of $m$ machines. The number of jobs is not important, as will be seen; we may assume the set is infinite. Each job $j$ consists of a sequence of $\mu$ operations $O_1,\dots ,O_{\mu }$ that must be processed in this order, that is called the precedence relation. Operation $O_k$ must be processed on machine $M_{{\pi }_k}$, during $p_k$ time units. A machine can process at most one operation at a time, and each job may be processed by at most one machine at any time. It will be convenient to number the jobs starting at 0 and also to index time starting at 0; we assume job 0 starts at time 0. We require that there exists some $\tau$, the cycle time, such that at every time $t\ge 0$ and for every machine $i$, if $i$ processes operation $O_k$ of job $j$ at time $t$, then $i$ processes operation $O_k$ of job $j+1$ at time $t+\tau$. Let $S_j$ and $C_j$ be the starting and completion times of job $j$. Thus a new job starts every $\tau$ time steps; i.e., $S_j=\tau j$. There are two objectives of interest in this scheduling problem: the cycle time $\tau$ and the flow time $F=C_j-S_j=C_0$.

For each machine $i$, let ${\ell }_i$ be the total amount of work in a single job processed by machine $i$. Let $\ell ={\sum }_{i\in M}{\ell }_i={\sum }_{k=1}^{\mu }{p}_k$ denote the total length of each job and let $\bar{\tau }={max}_{i=1,\dots ,m}{\ell }_i$ denote the per-job maximum load on any machine. Obviously we must have $F\ge \ell$ and $\tau \ge \bar{\tau }$. We will make use of the following folklore result.

Proposition 1

For any instance, there exists a schedule with cycle time $\tau =\bar{\tau }$ .

Proof

For a given operation $k$, let $i$ be the machine that processes $k$ and let $p$ be the sum of all $p_{k^{\prime }}$ for operations processed by $i$ with $k^{\prime }<k$. We schedule operation $k$ in job $j$ to start at time $\bar{\tau }(j+k)+p$ and finish at time $\bar{\tau }(j+k)+p+p_k$. It should be clear that all processing times and precedence constraints are satisfied and that our schedule is cyclic with $\tau =\bar{\tau }$.

Fix a time $t\ge 0$ and a machine $i$. If there is no operation in any job scheduled on machine $i$ at time $t$, we do not need to prove anything. Otherwise we need to prove that only one operation is scheduled on $i$ at time $t$. Let $r=tmod\bar{\tau }$. Suppose $t=\bar{\tau }(j_1+k_1)+r=\bar{\tau }(j_2+k_2)+r$. By our schedule’s definition $r$ determines a unique $k$ such that only operations $k$ (in any jobs) might be scheduled on machine $i$ at time $t$. Thus $k_1=k_2$ and then $j_1=j_2$, and we are done. □

Note that the flow time of the schedule constructed in the proof of Proposition 1 might be much greater than the minimal flow time over all schedules with cycle time $\tau =\bar{\tau }$.

Graves et al. [6] initiated the study of the High-Multiplicity Cyclic Job Shop Scheduling Problem and gave a heuristic for minimizing flow time subject to a given cycle time. Roundy [15] showed NP-hardness of the problem of minimizing flow time subject to the minimum cycle time. McCormick and Rao [11] further showed that even if each machine has only two operations in each job, but with different processing times, and the “cyclic machine structure” (see [15] or [11] for a definition) is fixed, it is still NP-hard to minimize flow time. Mittendorf and Timkovsky [13] give a simple greedy algorithm with an absolute error bound of ${\sum }_{i=1}^m(l_i^2-l_i)/2$ on the flow time and a more complicated algorithm with a slightly better bound, improving somewhat on the trivial flow time of $\bar{\tau }\ell$ which follows from the proof of Proposition 1. They also show the NP-hardness of the special case in which all machines but one have a single unit time operation in each job. Hall, Lee, and Posner [7] study the problem in which there are many copies of each of a small number of different job types and give various NP-hardness results and polynomial time algorithms. Timkovsky [17] surveys a wide range of environments for periodic scheduling including the problem studied in this paper. Hanen and Munier [9] also survey periodic variants of various scheduling problems. We refer the reader to these surveys for details.