Single machine scheduling with release dates and job delivery to minimize the makespan

AbstractIn single machine scheduling with release dates and job delivery, jobs are processed on a single machine and then delivered by a capacitated vehicle to a single customer. Only one vehicle is employed to deliver these jobs. The vehicle can deliver at most c jobs at a shipment. The delivery completion time of a job is defined as the time at which the delivery batch containing the job is delivered to the customer and the vehicle returns to the machine. The objective is to minimize the makespan, i.e., the maximum delivery completion time of the jobs. When preemption is allowed to all jobs, we give a polynomial-time algorithm for this problem. When preemption is not allowed, we show that this problem is strongly NP-hard for each fixed c≥1. We also provide a 53-approximation algorithm for this problem, and the bound is tight

Online scheduling on a single machine with one restart for all jobs to minimize the weighted makespan

In this paper, we consider the online scheduling problem on a single machine to minimize the weighted makespan. In this problem, all jobs arrive over time and they are allowed to be restarted only once. For the general case when the processing times of all jobs are arbitrary, we show that there is no online algorithm with a competitive ratio of less than 2, which matches the lower bound of the problem without restart. That is, only one restart for all jobs is invalid for improving the competitive ratio in the general case. For the special case when all jobs have the same processing time, we present the best possible online algorithm with a competitive ratio of 1.4656, which improves the competitive ratio of $\frac{1+\sqrt{5}}{2}\approx1.618$ for the problem without restart

A PTAS for single-machine scheduling with release dates and job delivery to minimize makespan

We consider the single-machine scheduling problem with release dates and job delivery to minimize makespan. Preemption is not allowed in the processing of the jobs. All jobs are first processed on a single machine and then delivered by a capacitated vehicle to a single customer. The vehicle can deliver at most c ≥ 1 jobs in each shipment. The round-trip transportation time between the machine and customer is a constant T > 0. The problem was proved to be strongly NP-hard and a 3/2-approximation algorithm was presented in the literature. In this paper we provide a polynomial-time approximation scheme (PTAS) for the problem

Optimal algorithms for single-machine scheduling with rejection to minimize the makespan

In this paper we consider the single-machine scheduling problem with rejection. Each job has an arrival time, a processing time and a rejection penalty. The objective is to minimize the sum of the makespan of the accepted jobs and the total rejection penalty of the rejected jobs. First, we present a polynomial-time algorithm for the off-line problem when split is allowed. Second, for the on-line problem with arbitrary arrival times, we provide a determined on-line algorithm with the best-possible competitive ratio 2. Finally, for the on-line problem with at most two distinct arrival times, we provide a determined on-line algorithm with the best-possible competitive ratio .Scheduling Rejection penalty On-line algorithm Competitive ratio

SINGLE MACHINE SCHEDULING WITH JOB DELIVERY TO MINIMIZE MAKESPAN

In the single machine scheduling problem with job delivery to minimize makespan, jobs are processed on a single machine and delivered by a capacitated vehicle to their respective customers. We first consider the special case with a single customer, that is, all jobs have the same transportation time. Chang and Lee (2004) proved that this case is strongly NP-hard. They also provided a heuristic with the worst-case performance ratio $\frac{5}{3}$, and pointed out that no heuristic can have a worst-case performance ratio less than $\frac{3}{2}$ unless P = NP. In this paper, we provide a new heuristic which has the best possible worst-case performance ratio $\frac{3}{2}$. We also consider an extended version in which the jobs have non-identical transportation times and the transportation time of a delivery batch is defined as the maximum transportation time of the jobs contained in it. We provide a heuristic with the worst-case performance ratio 2 for the extended version, and show that this bound is tight.Scheduling, job delivery, heuristic, worst-case performance ratio

New Results on Single-Machine Scheduling with Rejection to Minimize the Total Weighted Completion Time

In this paper, we study eight single-machine scheduling problems with rejection and position-dependent parameters. We consider two position-dependent parameters as follows: (1) position-dependent weights and (2) position-dependent processing times. In addition, we also introduce a weight-modifying activity or a rate-modifying activity into our problems. In the first six problems, the task is to minimize the sum of the total weighted completion time of accepted jobs and the total rejection cost of rejected jobs. We show that all six problems can be solved in polynomial time. In the last two problems, the task is to minimize the total weighted completion time of accepted jobs under the constraint that the total rejection cost of rejected jobs can not exceed a given upper bound. We show that these two problems are binary NP-hard and each problem admits an FPTAS

A note on the complexity of flow shop scheduling with transportation constraints

International audienceThis note investigates two-machine flow shop scheduling with transportation constraints to minimize makespan. Recently, Soukhal et al. [A. Soukhal, A. Oulamara, P. Martineau, Complexity of flow shop scheduling problems with transportation constraints, European Journal of Operational Research 161 (2005) 32-41] proved that this problem is strongly NP-hard when the capacity of the truck is limited to two or three parts. The considered problem with blocking constraints is also proved to be strongly NP-hard by Soukhal et al. Unfortunately, their proofs contain mistakes. We point out their proofs' invalidity and then show that, when the capacity of the truck is limited to two parts, the problem is binary NP-hard, and when the capacity of the truck is limited to three parts the problem is strongly NP-hard even if the jobs have a common processing time on machine one and all jobs have the same transportation time. We show also that the last result can be generalized to any fixed c (c ⩾ 3) parts