Batch scheduling on a two‐machine jobshop with machine‐dependent setup times

The problem of minimum makespan on an m machine jobshop with unit execution time (UET) jobs (m ≥ 3) is known to be strongly NP‐hard even with no setup times. We focus in this article on the two‐machine case. We assume UET jobs and consider batching with batch availability and machine‐dependent setup times. We introduce an efficient \begin{align*}(O(\sqrt{n}))\end{align*} algorithm, where n is the number of jobs. We then introduce a heuristic for the multimachine case and demonstrate its efficiency for two interesting instances. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Single machine just‐in‐time scheduling problems with two competing agents

In scheduling problems with two competing agents, each one of the agents has his own set of jobs to be processed and his own objective function, and both share a common processor. In the single‐machine problem studied in this article, the goal is to find a joint schedule that minimizes the total deviation of the job completion times of the first agent from a common due‐date, subject to an upper bound on the maximum deviation of job completion times of the second agent. The problem is shown to be NP‐hard even for a nonrestrictive due‐date, and a pseudopolynomial dynamic program is introduced and tested numerically. For the case of a restrictive due‐date (a sufficiently small due‐date that may restrict the number of early jobs), a faster pseudopolynomial dynamic program is presented. We also study the multiagent case, which is proved to be strongly NP‐hard. A simple heuristic for this case is introduced, which is tested numerically against a lower bound, obtained by extending the dynamic programming algorithm. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 61: 1–16, 2014     

Batch scheduling with step‐deteriorating processing times to minimize flowtime

Both topics of batch scheduling and of scheduling deteriorating jobs have been very popular among researchers in the last two decades. In this article, we study a model combining these two topics. We consider a classical batch scheduling model with unit‐jobs and batch‐independent setup times, and a model of step‐deterioration of processing times. The objective function is minimum flowtime. The optimal solution of the relaxed version (allowing non‐integer batch sizes) is shown to have a unique structure consisting of two consecutive decreasing arithmetic sequences of batch sizes. We also introduce a simple and efficient rounding procedure that guarantees integer batch sizes. The entire solution procedure requires an effort of O(n) (where nis the number of jobs.) © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

Flow‐shop batch scheduling with identical processing‐time jobs

In many practical situations of production scheduling, it is either necessary or recommended to group a large number of jobs into a relatively small number of batches. A decision needs to be made regarding both the batching (i.e., determining the number and the size of the batches) and the sequencing (of batches and of jobs within batches). A setup cost is incurred whenever a batch begins processing on a given machine. This paper focuses on batch scheduling of identical processing‐time jobs, and machine‐ and sequence‐independent setup times on an m‐machine flow‐shop. The objective is to find an allocation to batches and their schedule in order to minimize flow‐time. We introduce a surprising and nonintuitive solution for the problem. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Heuristics for multimachine minmax scheduling problems with general earliness and tardiness costs

We consider the problem of scheduling N jobs on M parallel machines so as to minimize the maximum earliness or tardiness cost incurred for each of the jobs. Earliness and tardiness costs are given by general (but job-independent) functions of the amount of time a job is completed prior to or after a common due date. We show that in problems with a nonrestrictive due date, the problem decomposes into two parts. Each of the M longest jobs is assigned to a different machine, and all other jobs are assigned to the machines so as to minimize their makespan. With these assignments, the individual scheduling problems for each of the machines are simple to solve. We demonstrate that several simple heuristics of low complexity, based on this characterization, are asymptotically optimal under mild probabilistic conditions. We develop attractive worst-case bounds for them. We also develop a simple closed-form lower bound for the minimum cost value. The bound is asymptotically accurate under the same probabilistic conditions. In the case where the due date is restrictive, the problem is more complex only in the sense that the set of initial jobs on the machines is not easily characterized. However, we extend our heuristics and lower bounds to this general case as well. Numerical studies exhibit that these heuristics perform excellently even for small- or moderate-size problems both in the restrictive and nonrestrictive due-date case. © 1997 John Wiley & Sons, Inc.     

Determining optimal sizes of bounded batches with rejection via quadratic min‐cost flow

In this article, we consider a single machine scheduling problem, in which identical jobs are split into batches of bounded sizes. For each batch, it is allowed to produce less jobs than a given upper bound, that is, some jobs in a batch can be rejected, in which case a penalty is paid for each rejected job. The objective function is the sum of several components, including the sum of the completion times, total delivery cost, and total rejection cost. We reduce this problem to a min‐cost flow problem with a convex quadratic function and adapt Tamir's algorithm for its solution. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 217–224, 2017     

Due‐window assignment with unit processing‐time jobs

In due‐window assignment problems, jobs completed within a designated time interval are regarded as being on time, whereas early and tardy jobs are penalized. The objective is to determine the location and size of the due‐window, as well as the job schedule. We address a common due‐window assignment problem on parallel identical machines with unit processing time jobs. We show that the number of candidate values for the optimal due‐window starting time and for the optimal due‐window completion time are bounded by 2. We also prove that the starting time of the first job on each of the machines is either 0 or 1, thus introducing a fairly simple, constant‐time solution for the problem. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

Scheduling a maintenance activity on parallel identical machines

We study a problem of scheduling a maintenance activity on parallel identical machines, under the assumption that all the machines must be maintained simultaneously. One example for this setting is a situation where the entire system must be stopped for maintenance because of a required electricity shut‐down. The objective is minimum flow‐time. The problem is shown to be NP‐hard, and moreover impossible to approximate unless P = NP. We introduce a pseudo‐polynomial dynamic programming algorithm, and show how to convert it into a bicriteria FPTAS for this problem. We also present an efficient heuristic and a lower bound. Our numerical tests indicate that the heuristic provides in most cases very close‐to‐optimal schedules. © 2008 Wiley Periodicals, Inc. Naval Research Logistics 2009