Polynomial time algorithms for two special classes of the proportionate multiprocessor open shop

Approximation algorithms for the multiprocessor open shop scheduling problem

On the complexity and some properties of multi-stage scheduling problems with earliness and tardiness penalties

We consider the routing open shop problem being a generalization of the open shop and the metric travelling salesman problems. The jobs are located at nodes of some transportation network, and the machines travel on the network to execute the jobs in the open shop environment. The machines are initially located at the same node (depot) and must return to the depot after completing all the jobs. It is required to find a non-preemptive schedule that minimizes the makespan. The problem is NP-hard even on a two-node network with two machines. We present new polynomial-time approximation algorithms with worst-case performance guarantees.KeywordsRouting open shopapproximation algorithmworst-case analysis

Scheduling ordered open shops

Scheduling two-machine preemptive open shops to minimize total completion time

We consider the flow shop and the open shop problems with m machines and n jobs: M is the total processing time of the most loaded machine, t* is the maximum operation length. Two functions are defined, μ(m) is the minimum number such that the optimum of any flow shop is at most M + μ(m)t*; η(m) is the minimum number for which the inequality M ≥ η(m)t* ensures that the optimum of the open shop is equal to M Upper and lower bounds for functions μ(m) and η(m) are derived and two polynomial algorithms are suggested: the first one delivers an optimal schedule of an open shop provided M ≥ η′(m)t* (for some function η′(m) ≥ η(m)); the second one computes a near-optimal approximation schedule of a flow shop. Both algorithms are based on a vector summation algorithm in m-dimensional banach space.

An efficient tabu search approach for the two-machine preemptive open shop scheduling problem

The open shop problem with routing and allowed preemption is a generalization of the two classical discrete optimization problems: the NP-hard metrical traveling salesman problem and the polynomially solvable scheduling problem, i.e., the open shop with allowed preemption. In the paper, a partial case of this problem is considered when the transportation network consists of two nodes. It is proved that the problem with two machines is polynomially solvable, while the problem is NP-hard in the strong sense in the case of not fixed number of machines.

The Open Shop Problem is a Production Scheduling problem where jobs do not have predestined processing routes, which increases the solution space and, therefore, the complexity of the problem. This work proposes a novel mathematical model that addresses the Open Shop theme with worker flexibility and no-wait. We consider the resource constraint of flexible and multiskilled labor. That is, the workers can execute their activities in more than one workstation. Simultaneously, the latter the continued production is contemplated, which hinders the reachability of feasible solutions and therefore increases the problem complexity. To validate the model, we generate several benchmarks, and the results obtained highlight the importance of worker flexibility in achieving greater problem feasibility.

This is a summary of the PhD thesis defended by the author in September 2011 at the Ecole des Mines de Nantes (EMNantes). The thesis was advised by Narendra Jussien (EMNantes) and co-advised by Louis-Martin Rousseau (Ecole Polytechnique de Montreal—Poly), Christelle Gueret (EMNantes) and Andre Langevin (Poly). The manuscript is written in French and can be obtained from http://tel.archives-ouvertes. fr/tel-00630122/fr/ or by email from arnaud.malapert@unice.fr. The focus of the dissertation is to study the resolution of shop and batch scheduling problems derived from real-world applications using constraint programming. The remainder of this abstract summarizes the results of the dissertation. In shop problems, n jobs, consisting each of m tasks, must be processed on m machines. The processing orders of tasks which belong to a job can vary: global order (flow shop); order per job (job shop); no order (open shop). We consider the construction of non-preemptive schedules of minimal makespan, which is NP-Hard for m ≥ 3. The study and classification of models and search algorithms show that one of the major challenge of solving optimally these problems is to provide good solutions as quick as the metaheuristics. Therefore, we introduce a new algorithm characterized by its: simplicity, most of its components are available in constraint solvers; flexibility, two constraint models offer different trade-offs among propagation strength, speed and simplicity; genericity, it is applicable to any disjunctive scheduling problem. The algorithm relies on recent constraint filtering algorithm and (randomized) branching techniques with new upper bound heuristics. The algorithm starts with a randomized constructive heuristic (without propagation) which initializes the upper bound so that the selection and propagation of initial choices are improved. During the search, the filtering is based on the disjunctive graph model and disjunctive global constraints, whereas the branching is conducted by adding precedences to the disjunctive graph with the profile strategy. Besides, randomized restart policies combined with nogood recording allow to search diversification and learning from restarts. Indeed, nogoods

The paper surveys the complexity results for job shop, flow shop, open shop and mixed shop scheduling problems when the number n of jobs is fixed while the number r of operations per job is not restricted. In such cases, the asymptotical complexity of scheduling algorithms depends on the number m of machines for a flow shop and an open shop problem, and on the numbers m and r for a job shop problem. It is shown that almost all shop-scheduling problems with two jobs can be solved in polynomial time for any regular criterion, while those with three jobs are NP-hard. The only exceptions are the two-job, m-machine mixed shop problem without operation preemptions (which is NP-hard for any non-trivial regular criterion) and the n-job, m-machine open shop problem with allowed operation preemptions (which is polynomially solvable for minimizing makespan).

This chapter concentrates on compact and cyclic compact open shop scheduling. Compact scheduling is motivated by timetabling and cyclic compact scheduling by just-in-time systems. A compact schedule requires each job to be processed in a no-wait fashion and each machine to process operations without idle time. Compact schedules do not always exist. The chapter presents smallest instances for which compact schedules do not exist and tests to determine whether compact schedules exist for a given degree Δ. Such tests run in polynomial time for open shop instances with Δ ≤ 4 but become NP-complete for Δ = 5. The minimization of makespan over all compact schedules has polynomial-time solution only for Δ ≤ 3, and it is open even for (3, 4)-biregular open shops where every job has length equal 3 and each machine workload equal 4. Deficiency has been proposed as a measure of deviation from compactness. However, the problem of minimizing deficiency is not only NP-hard in the strong sense but also the existing integer programming (IP) and constraint programming (CP) models can hardly solve problems of size n + m = 10 in reasonable time.

In this thesis, we study the theoretical approach on energy-efficient scheduling problems arising in demand response management in the modern electrical smart grid. Consumers send in power requests with flexible feasible timeslots during which their requests can be served. The grid controller, upon receiving power requests, schedules each request within the specified interval. The electricity cost is measured by a convex function of the load in each timeslot. The objective is to schedule all requests with the minimum total electricity cost. We study the smart grid scheduling problem in different models. For the offline model, we prove the problem is NP-hard for the general case. We propose a polynomial time algorithm for special input where jobs have unit power request and unit time duration. By adapting the polynomial time algorithm for unit-size jobs, we propose an approximation algorithm for more general input. On the other hand, we also present an exact algorithm to find the optimal schedule for the problem with general input. For the online model, we propose an online algorithm for jobs with jobs with arbitrary power request, arbitrary time duration, and arbitrary contiguous feasible intervals. We also show a lower bound of the competitive ratio for the smart grid scheduling problem with unit height and arbitrary width. For special cases, we design different online algorithms with better competitive ratios. Finally, we look at other optimization problems and show how to solve them by adapting our techniques. We prove that our online algorithm can solve the machine minimization problem with an asymptotically optimal competitive ratio. We also show that our exact algorithm can be adapted to solve other demand response management problems.

Two-station single-track railway scheduling problem with trains of equal speed

We consider open shop problems with unit processing times,n jobs have to be processed onm machines. The order in which a given job is processed on the machines is not fixed. For each job a release time or a due date may be given. Additional, we consider the restriction that every machine must perform all corresponding operations without any delay time. Unit time open shop problems with release times to minimize total completion time were unsolved up to now for both allowed and forbidden delay times. We will solve these problems in the case of two and three machines. Furthermore we will give polynomial algorithms for several no-delay-problems with due dates.