Batch scheduling of identical jobs with controllable processing times

Abstract

In scheduling models with controllable processing times, the job processing times can be controlled (i.e. compressed) by allocating additional resources. In batch scheduling a large number of jobs may be grouped and processed as separate batches, where a batch processing time is identical to the total processing times of the jobs contained in the batch, and a setup time is incurred when starting a new batch.

A model combining these two very popular and practical phenomena is studied. We focus on identical jobs and linear compression cost function. Two versions of the problem are considered: in the first we minimize the sum of the total flowtime and the compression cost, and in the second we minimize the total flowtime subject to an upper bound on the maximum compression. We study both problems on a single machine and on parallel identical machines. In all cases we introduce closed form solutions for the relaxed version (allowing non-integer batch sizes). Then, a simple rounding procedure is introduced, tested numerically, and shown to generate extremely close-to-optimal integer solutions. For a given number of machines, the total computational effort required by our proposed solution procedure is $O(\sqrt{n})$, where $n$ is the number of jobs.

Introduction

Vickson [1], [2] introduced a new class of scheduling models with controllable processing times. In this model the job processing times are not given constants as in classical scheduling, but can be controlled (i.e. compressed) by allocating additional resources. Various versions of this very practical scheduling setting have been studied by many researchers, as reflected in the recent survey of Shabtai and Steiner [3]. Some of the early papers are: Van Wassenhove and Baker [4], Nowicki and Zdrzalka [5], Janiak and Kovalyov [6], Wan et al. [7], Hoogeveen and Woeginger [8], Janiak et al. [9], Shakhlevich and Strusevich [10], Wang [11] Akturk et al. [12] and Wang and Xia [13]. More recently, Tseng et al. [14] studied a single machine setting with controllable processing times and an objective of minimum total tardiness; Turkcan et al. [15] studied a setting of parallel machines and objective functions of minimum manufacturing cost and total weighted earliness and tardiness; Shakhlevich et al. [16] focused on the trade-off between the maximum cost (which is a function of the completion times) and the total compression cost; Shabtay et al. [17] addressed due date assignment problems in a group technology environment; Gurel et al. [18] considered failures of the machine and repair time, and focused on an anticipative scheduling approach; Wan et al. [19] studied the problem of scheduling jobs of two-agents on a single machine with controllable processing times; Choi et al. [20] focused on minimizing weighted completion time subject to an upper bound on the maximum compression cost; Leyvand et al. [21] considered just-in-time scheduling on parallel machines; Yin and Wang [22] studied a model combining controllable processing times and learning; Wang and Wang [23] addressed the single machine problem of minimizing the total resource consumption subject to an upper bound on the total weighted flowtime; Wei et al. [24] focused on a model in which the job processing times are a function of both resource consumption and the job starting times; Uruk et al. [25] studied a two-machine flowshop problem with flexible operations and controllable processing times; and Wang and Wang [26] focused on convex resource dependent processing times and job deterioration.

In batch scheduling a large number of jobs may be grouped and processed as separate batches. Such batching is generally based on the existence of some similarity between jobs belonging to the same class. A batch processing time is identical to the total processing times of the jobs contained in the batch. A setup time is incurred when starting a new batch. In their classical paper, Santos and Magazine [27] solved a single machine batch scheduling problem to minimize total flowtime. They assumed a constant (i.e. batch independent) setup time, and batch availability, i.e., jobs are completed only when their entire batch is completed. Dobson et al. [28], Naddef and Santos [29] and Coffman et al. [30] extended the basic results obtained by Santos and Magazine for the “relaxed version” of the problem (when integer batch sizes are not required). Later, Shallcross [31] and Mosheiov et al. [32] solved the integer version. We refer the reader also to the extensive survey of Allahverdi et al. [33] on batch scheduling under different machine settings and objective functions.

In this paper we combine (to our knowledge for the first time) the two phenomena of batch scheduling and controllable processing times. Specifically, with respect to batching, we consider the setting of Santos and Magazine [27], i.e. a single machine, batch independent processing times, batch availability and the objective of minimum flowtime. With respect to the option of controlling job processing times, we assume linear compression cost. Two problems are solved: in the first we minimize the total flowtime plus the compression cost, and in the second we minimize flowtime, subject to an upper bound on the maximum batch compression. We first focus on a single machine setting, and show that in both cases, the solution for the relaxed version consists of a decreasing arithmetic sequence of batch sizes, for which closed form solutions are obtained. The total running time of the procedure is $O(\sqrt{n})$, where $n$ is the number of jobs. An integer solution is obtained by a simple rounding procedure, requiring additional $O(\sqrt{n})$ time. Our numerical tests indicate that the integer solutions are extremely close to those of the relaxed versions (which are lower bounds on the optimal solutions). We then consider the setting of parallel identical machines. [We refer the reader to Mor and Mosheiov [34], who studied batch scheduling on parallel identical machines without the option of compression.] In this case, we show that the solution of both problems consists of identical decreasing arithmetic sequences of the batch sizes on all the machines. The total computational effort of this more complicated solution procedure becomes $O(m\sqrt{n})$, where $m$ is the number of machines. A rounding procedure and its evaluation are provided for this setting as well.

In Section 2 we present the notation and the formulation. In 3 Problem P1:, 4 Problem P2:, we solve the relaxed versions of the two different problems on a single machine. In Section 5 we propose the rounding procedure to obtain an integer solution for both problems. 6 Numerical examples, 7 Parallel identical machines are devoted to numerical examples and numerical tests, respectively. Section 8 contains the extension to parallel identical machines.