Scheduling resumable simple linear deteriorating jobs on a single machine with an availability constraint to minimize makespan

We consider a single-machine scheduling problem in which the processing time of each job is a simple linear deteriorating function of its waiting time. The machine is subject to an availability constraint. Jobs interrupted by machine unavailability can resume their processing. The objective is to minimize the makespan. We first show that the problem can be solved optimally by 0–1 integer programming. We then prove that the problem is NP-hard in the ordinary sense and there exists a fully polynomial time approximation scheme for it.     

Bounded parallel-batching scheduling with two competing agents

We consider a scheduling problem in which two agents, each with a set of non-preemptive jobs, compete to perform their jobs on a common bounded parallel-batching machine. Each of the agents wants to minimize an objective function that depends on the completion times of its own jobs. The goal is to schedule the jobs such that the overall schedule performs well with respect to the objective functions of both agents. We focus on minimizing the makespan or the total completion time of one agent, subject to an upper bound on the makespan of the other agent. We distinguish two categories of batch processing according to the compatibility of the agents. In the case where the agents are incompatible, their jobs cannot be processed in the same batch, whereas all the jobs can be processed in the same batch when the agents are compatible. We show that the makespan problem can be solved in polynomial time for the incompatible case and is NP-hard in the ordinary sense for the compatible case. Furthermore, we show that the latter admits a fully polynomial-time approximation scheme. We prove that the total completion time problem is NP-hard and is polynomially solvable for the incompatible case with a fixed number of job types.     

Parallel-machine scheduling of simple linear deteriorating jobs

We consider parallel-machine scheduling problems in which the processing time of a job is a simple linear increasing function of its starting time. The objectives are to minimize the makespan, total machine load, and total completion time. We show that all the problems are strongly NP-hard with an arbitrary number of machines and NP-hard in the ordinary sense with a fixed number of machines. For the former two problems, we prove that there exists no polynomial time approximation algorithm with a constant worst-case bound when the number of machines is arbitrary unless P=NP. When the number of machines is fixed, we propose two similar fully polynomial-time approximation schemes for the former two problems.     

Scheduling on parallel machines with preemption and transportation delays

This paper deals with an identical parallel machines scheduling problem, where independent jobs can be preempted and transported from one machine to another. The transportation of a preempted job requires a time called the transportation delay. The goal is to find a solution that minimizes the total completion time (makespan). We first study the case of equal-size jobs where new complexity results are given. Then, to solve the problem with two identical machines, we present a dynamic programming algorithm and a fully polynomial time approximation scheme (FPTAS). Experimental results show the efficiency of the FPTAS compared to a previously published heuristic.     

Two-machine flow-shop scheduling with rejection

We study a scheduling problem with rejection on a set of two machines in a flow-shop scheduling system. We evaluate the quality of a solution by two criteria: the first is the makespan and the second is the total rejection cost. We show that the problem of minimizing the makespan plus total rejection cost is NP-hard and for its solution we provide two different approximation algorithms, a pseudo-polynomial time optimization algorithm and a fully polynomial time approximation scheme (FPTAS). We also study the problem of finding the entire set of Pareto-optimal points (this problem is NP-hard due to the NP-hardness of the same problem variation on a single machine [20]). We show that this problem can be solved in pseudo-polynomial time. Moreover, we show how we can provide an FPTAS that, given that there exists a Pareto optimal schedule with a total rejection cost of at most R and a makespan of at most K, finds a solution with a total rejection cost of at most (1+?)R and a makespan value of at most (1+?)K. This is done by defining a set of auxiliary problems and providing an FPTAS algorithm to each one of them.     

A two-stage flow shop scheduling problem on a batching machine and a discrete machine with blocking and shared setup times

Motivated by applications in iron and steel industry, we consider a two-stage flow shop scheduling problem where the first machine is a batching machine subject to the blocking constraint and the second machine is a discrete machine with shared setup times. We show that the problem is strongly NP-hard when the objective is to minimize the makespan. When solved with a heuristic priority rule, the worst case ratio with the minimum makespan is 2. For a more general objective, the minimization of a linear combination of the makespan and the total blocking time, a quadratic mixed integer program is presented first. Then we pinpoint two cases with polynomial time algorithms: the case without blocking constraint and the case with a given job sequence. Also for the general objective, we analyze an approximation algorithm. Finally, we evaluate the algorithms, giving experimental results on randomly generated test problems.     

Rescheduling for machine disruption to minimize makespan and maximum lateness

We consider a scheduling problem where a set of jobs has already been scheduled to minimize some cost objective on a single machine when the machine becomes unavailable for a period of time. The decision-maker needs to reschedule the jobs without excessively disrupting the original schedule. The disruption is measured as the maximum time deviation, for any given job, between the original and new schedules. We examine a general model where the maximum time disruption appears both as a constraint and as part of the cost objective. For a scheduling cost modeled as the makespan or maximum lateness, we provide a pseudopolynomial time optimal algorithm, a constant factor approximation algorithm, and a fully polynomial time approximation scheme. The approximation algorithm has an asymptotically achievable worst-case performance ratio of 2 and has average performance close to optimal. Managerial insights are given on how scheduling costs are affected by machine disruption and the approximation algorithm.     

Scheduling a batching machine with convex resource consumption functions

In various industries jobs undergo a batching, or burn in, process where different tasks are grouped into batches and processed simultaneously. The processing time of each batch is equal to the longest processing time among all jobs contained in the batch. All to date studies dealing with batching machines have considered fixed job processing times. However, in many real life applications job processing times are controllable through the allocation of a limited resource. The most common and realistic model assumes that there exists a non-linear and convex relationship between the amount of resource allocated to a job and its processing time. The scheduler?s task when dealing with controllable processing times is twofold. In addition to solving the sequencing problem, one must establish an optimal resource allocation policy. We combine these two widespread models on a single machine setting, showing that both the makespan and total completion time criteria can be solved in polynomial time. We then show that our proposed approach can be applied to general bi-criteria objective comprising of the makespan and the total completion time.     

Preemptive scheduling on uniformly related machines: minimizing the sum of the largest pair of job completion times

We revisit the classic problem of preemptive scheduling on m uniformly related machines. In this problem, jobs can be arbitrarily split into parts, under the constraint that every job is processed completely, and that the parts of a job are not assigned to run in parallel on different machines. We study a new objective which is motivated by fairness, where the goal is to minimize the sum of the two maximal job completion times. We design a polynomial time algorithm for computing an optimal solution. The algorithm can act on any set of machine speeds and any set of input jobs. The algorithm has several cases, many of which are very different from algorithms for makespan minimization (algorithms that minimize the maximum completion time of any job), and from algorithms that minimize the total completion time of all jobs.     

Minimizing Mean Completion Time in a Batch Processing System

We consider batch processing jobs to minimize the mean completion time. A batch processing machine can handle up to $B$ jobs simultaneously. Each job is represented by an arrival time and a processing time. Jobs processed in a batch have the same completion time, i.e., their common starting time plus the processing time of their longest job. For batch processing, non-preemptive scheduling is usually required and we discuss this case. The batch processing problem reduces to the ordinary uniprocessor system scheduling problem if $B=1$. We focus on the other extreme case $B=+\infty$. Even for this seemingly simple extreme case, we are able to show that the problem is NP-hard for the weighted version. In addition, we establish a polynomial time algorithm for a special case when there are only a constant number of job processing times. Finally, we give a polynomial time approximation scheme for the general case.     

Fully Polynomial Approximation Schemes for a Symmetric Quadratic Knapsack Problem and its Scheduling Applications

We design a fully polynomial-time approximation scheme (FPTAS) for a knapsack problem to minimize a symmetric quadratic function. We demonstrate how the designed FPTAS can be adopted for several single machine scheduling problems to minimize the sum of the weighted completion times. The applications presented in this paper include problems with a single machine non-availability interval (for both the non-resumable and the resumable scenarios) and a problem of planning a single machine maintenance period; the latter problem is closely related to a single machine scheduling problem with two competing agents. The running time of each presented FPTAS is strongly polynomial.     

On the complexity of bi-criteria scheduling on a single batch processing machine

This paper considers hierarchical bi-criteria scheduling on a single batch processing machine where the primary criterion is the makespan. We show that the problem where the secondary criterion is the total completion time can be solved in polynomial time for a given machine capacity and the problem where the secondary criterion is the (weighted) number of late jobs is (strongly) NP-hard.     

Scheduling on parallel processors with varying processing times

In this paper, we construct the pseudopolynomial dynamic programming algorithm that optimally solves the parallel identical processor scheduling problem to minimize the maximum job completion times (makespan) under varying processing times. They can be described by an arbitrary monotonic function dependent on the number of previously processed jobs, which can model learning or aging effects. Beside the canonical dynamic programming algorithm, we provide its efficient parallel fast version, which solves moderate problem instances of the problem within reasonable time and memory usage. Additionally, on the basis of the constructed algorithm, a fully polynomial time approximation scheme for the considered problem is provided.     

Single-machine group scheduling problems under the effects of deterioration and learning

This paper investigates single-machine group scheduling problems with simultaneous considerations of deteriorating and learning effects to minimize the makespan and the total completion time of all jobs. The group setup time is assumed to follow a simple linear time-dependent deteriorating model. Two models of learning for the job processing time are examined in this study. We provided polynomial time solutions for the makespan minimization problems. We also showed that the total completion time minimization problems remain polynomially solvable under agreeable conditions.     

Single-machine scheduling with deteriorating jobs

In this article, we study a single-machine scheduling problem in which the processing time of a job is a nonlinear function of its basic processing time and starting time. The objectives are to minimise the makespan, the sum of weighted completion times and the sum of the kth powers of completion times. We show that the makespan minimisation problem can be solved in polynomial time. However, the total completion time and the sum of the kth powers of completion times minimisation problems can be solved in polynomial time in some cases. Besides, some useful properties are also provided for the sum of weighted completion times problem under certain conditions.     

Scheduling jobs with deterioration effect and controllable processing time

This article considered the single machine scheduling with controllable processing time (resource allocation) and deterioration effect concurrently. It discussed the minimization of three objectives, which involve the weighted sum of the makespan and the total resource consumption cost, the total resource consumption cost under the condition that the makespan (total flow time) is restricted to a fixed constant and the optimal resource allocation and the optimal job sequence is what we need to make decision. Considering the makespan constraint, it proved that these problems can be solved in polynomial time. A special case of the last problem can be solved in polynomial time with respect to the total flow time constraint.

     

Fully polynomial time approximation scheme for the weighted flow-time minimization on a single machine with a fixed non-availability interval

In a recent paper [Theoretical Computer Science 363, 257–265], He, Zhong and Gu considered the non-resumable case of the scheduling problem with a fixed non-availability interval under the non-resumable scenario. They proposed a polynomial time approximation scheme (PTAS) to minimize the total completion time.In this paper, we propose a fully polynomial-time approximation scheme to minimize the total weighted completion time. The FPTAS has O(n²/ε²) time complexity, where n is the number of jobs and ε is the required error bound. The proposed FPTAS outperforms all the previous approximation algorithms designed for this problem and its running time is strongly polynomial.     

Computational performances of a simple interchange heuristic for a scheduling problem with an availability constraint

This paper deals with a scheduling problem on a single machine with an availability constraint. The problem is known to be NP-complete and admits several approximation algorithms. In this paper we study the approximation scheme described in He et al. [Y. He, W. Zhong, H. Gu, Improved algorithms for two single machine scheduling problems, Theoretical Computer Science 363 (2006) 257–265]. We provide the computation of an improved relative error of this heuristic, as well as a proof that this new bound is tight. We also present some computational experiments to test this heuristic on random instances. These experiments include an implementation of the fully-polynomial time approximation scheme given in Kacem and Ridha Mahjoub [I. Kacem, A. Ridha Mahjoub, Fully polynomial time approximation scheme for the weighted flow-time minimization on a single machine with a fixed non-availability interval, Computers and Industrial Engineering 56 (2009) 1708–1712].     

Single-machine scheduling with deteriorating jobs under a series–parallel graph constraint

This paper considers single-machine scheduling problems with deteriorating jobs, i.e., jobs whose processing times are an increasing function of their starting times. In addition, the jobs are related by a series–parallel graph. It is shown that for the general linear problem to minimize the makespan, polynomial algorithms exist. It is also shown that for the proportional linear problem of minimization of the total weighted completion time, polynomial algorithms exist, too.     

Algorithms for scheduling incompatible job families on single batching machine with limited capacity

Motivated by applications in food processing and semiconductor manufacturing industries, we consider the scheduling problem of a batching machine with jobs of multiple families. The machine has a limited capacity to accommodate jobs. The jobs are in arbitrary sizes and multiple families. Jobs from different families cannot be processed in a batch. We show the problems of minimizing makespan and total batch completion time are both NP-hard in the strong sense. We present a mixed integer programming model for the problems. Then we propose two polynomial time heuristics based on longest processing time first rule and first fit rule. For the special case where a larger job also has a longer processing time, the heuristic for minimizing makespan is optimal. For the general case, we show the performance guarantee of the methods for the two objectives respectively.