A polynomial time algorithm for makespan minimization on one machine with forbidden start and completion times

We consider the problem of scheduling independent jobs on a single resource under a special unavailability constraint: a set of forbidden instants is given, where no job is allowed to start or complete. We show that a schedule without idle time always exists if the number of forbidden instants is less than the number of distinct processing times appearing in the instance. We derive quite a fast algorithm to find such a schedule, based on an hybridization between a list algorithm and local exchange. As a corollary minimizing the makespan for a fixed number of forbidden instants is polynomial.     

Parallel machine scheduling with nested processing set restrictions

We consider the problem of scheduling a set of n independent jobs on m parallel machines, where each job can only be scheduled on a subset of machines called its processing set. The machines are linearly ordered, and the processing set of job j   is given by two machine indexes _aj$a_j$ and _bj$b_j$; i.e., job j   can only be scheduled on machines _aj,_aj+1,…,_bj$a_j\text{,}{a}_j+1\text{,}\dots \text{,}{b}_j$. Two distinct processing sets are either nested or disjoint. Preemption is not allowed. Our goal is to minimize the makespan. It is known that the problem is strongly NP-hard and that there is a list-type algorithm with a worst-case bound of 2-1/m$2-1/m$. In this paper we give an improved algorithm with a worst-case bound of 7/4. For two and three machines, the algorithm gives a better worst-case bound of 5/4 and 3/2, respectively.     

An Iterated Greedy heuristic for the sequence dependent setup times flowshop problem with makespan and weighted tardiness objectives

Iterated Greedy (IG) algorithms are based on a very simple principle, are easy to implement and can show excellent performance. In this paper, we propose two new IG algorithms for a complex flowshop problem that results from the consideration of sequence dependent setup times on machines, a characteristic that is often found in industrial settings. The first IG algorithm is a straightforward adaption of the IG principle, while the second incorporates a simple descent local search. Furthermore, we consider two different optimization objectives, the minimization of the maximum completion time or makespan and the minimization of the total weighted tardiness. Extensive experiments and statistical analyses demonstrate that, despite their simplicity, the IG algorithms are new state-of-the-art methods for both objectives.     

A bicriteria flowshop scheduling with a learning effect

In many situations, a worker’s ability improves as a result of repeating the same or similar tasks; this phenomenon is known as the learning effect. In this paper the learning effect is considered in a two-machine flowshop. The objective is to find a sequence that minimizes a weighted sum of total completion time and makespan. Total completion time and makespan are widely used performance measures in scheduling literature. To solve this scheduling problem, an integer programming model with n² + 6n variables and 7n constraints where n is the number of jobs is formulated. Because of the lengthy computing time and high computing complexity of the integer programming model, the problem with up to 30 jobs can be solved. A heuristic algorithm and a tabu search based heuristic algorithm are presented to solve large size problems. Experimental results show that the proposed heuristic methods can solve this problem with up to 300 jobs rapidly. According to the best of our knowledge, no work exists on the bicriteria flowshop with a learning effect.     

Scheduling problems in master-slave model

We consider scheduling problems in the master slave model, which was introduced by Sahni in 1996. The goal is to minimize the makespan and the total completion time. It has been shown that the problem of minimizing makespan is NP-hard. Sahni and Vairaktarakis developed some approximation algorithms to generate schedules whose makespan is at most constant times the optimal. In this paper, we show that the problem of minimizing total completion time is NP-hard in the strong sense. Then we develop algorithms to generate schedules whose total completion time and makespan are both bounded by some constants times their optimal values. Research supported in part by the National Science Foundation through grant DMI-0300156.     

Single-machine scheduling with deteriorating jobs and learning effects to minimize the makespan

This paper studies the single-machine scheduling problem with deteriorating jobs and learning considerations. The objective is to minimize the makespan. We first show that the schedule produced by the largest growth rate rule is unbounded for our model, although it is an optimal solution for the scheduling problem with deteriorating jobs and no learning. We then consider three special cases of the problem, each corresponding to a specific practical scheduling scenario. Based on the derived optimal properties, we develop an optimal algorithm for each of these cases. Finally, we consider a relaxed model of the second special case, and present a heuristic and analyze its worst-case performance bound.     

Job-shop local-search move evaluation without direct consideration of the criterion’s value

This article focuses on the evaluation of moves for the local search of the job-shop problem with the makespan criterion. We reason that the omnipresent ranking of moves according to their resulting value of a criterion function makes the local search unnecessarily myopic. Consequently, we introduce an alternative evaluation that relies on a surrogate quantity of the move’s potential, which is related to, but not strongly coupled with, the bare criterion. The approach is confirmed by empirical tests, where the proposed evaluator delivers a new upper bound on the well-known benchmark test yn2. The line of the argumentation also shows that by sacrificing accuracy the established makespan estimators unintentionally improve on the move evaluation in comparison to the exact makespan calculation, in contrast to the belief that the reliance on estimation degrades the optimization results.     

Approximations for the Two-Machine Cross-Docking Flow Shop Problem

We consider in this article the Two-Machine Cross-Docking Flow Shop Problem, which is a special case of scheduling with typed tasks, where we have two types of tasks and one machine per type. Precedence constraints exist between tasks, but only from a task of the first type to a task of the second type. The precedence relation is thus a directed bipartite graph. Minimizing the makespan is strongly NP-hard even with unit processing times, but any greedy method yields a 2-approximation solution. In this paper, we are interested in establishing new approximability results for this problem. More specifically, we investigate three directions: list scheduling algorithms based on the relaxation of the resources, the decomposition of the problem according to the connected components of the precedence graph, and finally the search of the induced balanced subgraph with a bounded degree.     

Graph models for scheduling systems with machine saturation property

Let be a set of n independent tasks and a set of m processors. During each time instant, each processor can be used by a single task at most. A schedule is for each task an allocation of one or more time intervals to one or more processors. A schedule is said to be optimal if it minimizes the maximum completion time. We say a schedule S has the machine saturation property (MS property) if, at any time instant of task execution, all the machines are simultaneously busy. In this paper, we analyze the conditions under which a parallel scheduling system allows a schedule with the MS property. While for some simple models the analytical conditions can be easily stated, a graph model approach is required when conflicts of processor usage are present. For this reason, we define the class of saturated graphs that correspond to scheduling systems with the MS property. We present efficient graph recognition algorithms to verify the MS property directly on some classes of saturated graphs