Single-Machine Scheduling with Release Times and Tails

We study the problem of scheduling jobs with release times and tails on a single machine with the objective to minimize the makespan. This problem is strongly NP-hard, however it is known to be polynomially solvable if all jobs have equal processing time P. We generalize this result and suggest an O(n ² log nlog P) algorithm for the case when the processing times of some jobs are restricted to either P or 2P.     

The two-machine no-wait general and proportionate open shop makespan problem

We consider the two-machine no-wait open shop minimum makespan problem in which the determination of an optimal solution requires an optimal pairing of the jobs followed by the optimal sequencing of the job pairs. We show that the required enumeration can be curtailed by reducing the pair sequencing problem for a given pair set to a traveling salesman problem which is equivalent to a two-machine no-wait flow shop problem solvable in O(n log n) time. We then propose an optimal O(n log n) algorithm for the proportionate problem with equal machine speeds in which each job has the same processing time on both machines. We show that our O(n log n) algorithm also applies to the more general proportionate problem with equal machine speeds and machine-specific setup times. We also analyze the proportionate problem with unequal machine speeds and conclude that the required enumeration can be further curtailed (compared to the problem with arbitrary job processing times) by eliminating certain job pairs from consideration.     

Two machine preemptive scheduling problem with release dates,equal processing times and precedence constraints

We consider a scheduling problem with two identical parallel machines and n jobs. For each job we are given its release date when job becomes available for processing. All jobs have equal processing times. Preemptions are allowed. There are precedence constraints between jobs which are given by a (di)graph consisting of a set of outtrees and a number of isolated vertices. The objective is to find a schedule minimizing mean flow time. We suggest an O(n²) algorithm to solve this problem.The suggested algorithm also can be used to solve the related two-machine open shop problem with integer release dates, unit processing times and analogous precedence constraints.     

Scheduling with controllable release dates and processing times: Makespan minimization

The paper deals with the single-machine scheduling problem in which job processing times as well as release dates are controllable parameters and they may vary within given intervals. While all release dates have the same boundary values, the processing time intervals are arbitrary. It is assumed that the cost of compressing processing times and release dates from their initial values is a linear function of the compression amount. The objective is to minimize the makespan together with the total compression cost. We construct a reduction to the assignment problem for the case of equal release date compression costs and develop an O(n²) algorithm for the case of equal release date compression costs and equal processing time compression costs. For the bicriteria version of the latter problem with agreeable processing times, we suggest an O(n²) algorithm that constructs the breakpoints of the efficient frontier.     

Complexity of minimizing the total flow time with interval data and minmax regret criterion

We consider the minmax regret (robust) version of the problem of scheduling n jobs on a machine to minimize the total flow time, where the processing times of the jobs are uncertain and can take on any values from the corresponding intervals of uncertainty. We prove that the problem in NP-hard. For the case where all intervals of uncertainty have the same center, we show that the problem can be solved in O(nlogn) time if the number of jobs is even, and is NP-hard if the number of jobs is odd. We study structural properties of the problem and discuss some polynomially solvable cases.     

Batch scheduling with controllable setup and processing times to minimize total completion time

We study a problem of scheduling n jobs on a single machine in batches. A batch is a set of jobs processed contiguously and completed together when the processing of all jobs in the batch is finished. Processing of a batch requires a machine setup time dependent on the position of this batch in the batch sequence. Setup times and job processing times are continuously controllable, that is, they are real-valued variables within their lower and upper bounds. A deviation of a setup time or job processing time from its upper bound is called a compression. The problem is to find a job sequence, its partition into batches, and the values for setup times and job processing times such that (a) total job completion time is minimized, subject to an upper bound on total weighted setup time and job processing time compression, or (b) a linear combination of total job completion time, total setup time compression, and total job processing time compression is minimized. Properties of optimal solutions are established. If the lower and upper bounds on job processing times can be similarly ordered or the job sequence is fixed, then O(n³ log n) and O(n⁵) time algorithms are developed to solve cases (a) and (b), respectively. If all job processing times are fixed or all setup times are fixed, then more efficient algorithms can be devised to solve the problems.     

Bicriteria scheduling on a series-batching machine to minimize maximum cost and makespan

This paper studies the bicriteria problem of scheduling n jobs on a serial-batching machine to minimize maximum cost and makespan simultaneously. A serial-batching machine is a machine that can handle up to b jobs in a batch and jobs in a batch start and complete respectively at the same time and the processing time of a batch is equal to the sum of the processing times of jobs in the batch. When a new batch starts, a constant setup time s occurs. We confine ourselves to the unbounded model, where b ≥ n. We present a polynomial-time algorithm for finding all Pareto optimal solutions of this bicriteria scheduling problem.     

A matheuristic approach for the two-machine total completion time flow shop problem

This paper deals with the two-machine total completion time flow shop problem. We present a so-called matheuristic post processing procedure that improves the objective function value with respect to the solutions provided by state of the art procedures. The proposed procedure is based on the positional completion times integer programming formulation of the problem with O(n ²) variables and O(n) constraints.     

Single machine scheduling problems with controllable processing times and total absolute differences penalties

In this paper, we consider single machine scheduling problem in which job processing times are controllable variables with linear costs. We concentrate on two goals separately, namely, minimizing a cost function containing total completion time, total absolute differences in completion times and total compression cost; minimizing a cost function containing total waiting time, total absolute differences in waiting times and total compression cost. The problem is modelled as an assignment problem, and thus can be solved with the well-known algorithms. For the case where all the jobs have a common difference between normal and crash processing time and an equal unit compression penalty, we present an O(n log n) algorithm to obtain the optimal solution.     

On the single machine serial batching scheduling problem to minimize total completion time with precedence constraints, release dates and identical processing times

We consider the single machine, serial batching, total completion time scheduling problem with precedence constraints, release dates and identical processing times in this paper. The complexity of this problem is reported as open in the literature. We provide an O(n⁵) time algorithm to solve this problem.     

Scheduling jobs with release dates,equal processing times,and inclusive processing set restrictions

We consider the problem of scheduling a given set of n jobs with equal processing times on m parallel machines so as to minimize the makespan. Each job has a given release date and is compatible to only a subset of the machines. The machines are ordered and indexed in such a way that a higher-indexed machine can process all the jobs that a lower-indexed machine can process. We present a solution procedure to solve this problem in O(n²+mnlogn) time. We also extend our results to the tree-hierarchical processing sets case and the uniform machine case.     

Optimal preemptive scheduling on uniform machines with discounted flowtime objectives

We consider the problem of scheduling n jobs on m parallel machines. Each job has a deterministic processing time and a weight associated with it. For uniform machines we show that discounted flowtime is minimized by serving jobs preemptively in increasing order of their remaining processing times, assigning the job with the shortest remaining processing time to the fastest available machine.     

Single machine common flow allowance scheduling with controllable processing times

In this paper, we consider single machine SLK due date assignment scheduling problem in which job processing times are controllable variables with linear costs. The objective is to determine the optimal sequence, the optimal common flow allowance and the optimal processing time compressions to minimize a total penalty function based on earliness, tardiness, common flow allowance and compressions. We solve the problem by formulating it as an assignment problem which is polynomially solvable. For some special cases, we present an O(n logn) algorithm to obtain the optimal solution respectively.     

Minimizing number of tardy jobs on a batch processing machine with incompatible job families

In this paper we consider the problem of minimizing number of tardy jobs on a single batch processing machine. The batch processing machine is capable of processing up to B jobs simultaneously as a batch. We are given a set of n jobs which can be partitioned into m incompatible families such that the processing times of all jobs belonging to the same family are equal and jobs of different families cannot be processed together. We show that this problem is NP-hard and present a dynamic programming algorithm which has polynomial time complexity when the number of job families and the batch machine capacity are fixed. We also show that when the jobs of a family have a common due date the problem can be solved by a pseudo-polynomial time procedure.     

Scheduling linearly shortening jobs under precedence constraints

We consider the problem of scheduling a set of dependent jobs on a single machine with the maximum completion time criterion. The processing time of each job is variable and decreases linearly with respect to the starting time of the job. Applying a uniform approach based on the calculation of ratios of expressions that describe total processing times of chains of jobs, we show basic properties of the problem. On the basis of these properties, we prove that if precedence constraints among jobs are in the form of a set of chains, a tree, a forest or a series–parallel digraph, the problem can be solved in O(n log n) time, where n denotes the number of the jobs.     

A faster algorithm for a due date assignment problem with tardy jobs

The single-machine due date assignment problem with the weighted number of tardy jobs objective, (the TWNTD problem), and its generalization with resource allocation decisions and controllable job processing times have been solved in O(n⁴) time by formulating and solving a series of assignment problems. In this note, a faster O(n²) dynamic programming algorithm is proposed for the TWNTD problem and for its controllable processing times generalization in the case of a convex resource consumption function.     

Due-date assignment and single machine scheduling with deteriorating jobs

We study a scheduling problem with deteriorating jobs, that is, jobs whose processing times are an increasing function of their start times. We consider the case of a single machine and linear job-independent deterioration. The problem is to determine an optimal combination of the due-date and schedule so as to minimize the sum of due-date, earliness and tardiness penalties. We give an O(n log n) time algorithm to solve this problem.     

Two-machine open shop scheduling with special transportation times

The paper considers a problem of scheduling n jobs in a two-machine open shop to minimise the makespan, provided that preemption is not allowed and the interstage transportation times are involved. In general, this problem is known to be NP-hard. We present a linear time algorithm that finds an optimal schedule if no transportation time exceeds the smallest of the processing times. We also describe an algorithm that creates a heuristic solution to the problem with job-independent transportation times. Our algorithm provides a worst-case performance ratio of 8/5 if the transportation time of a job depends on the assigned processing route. The ratio reduces to 3/2 if all transportation times are equal.