Approximation for Scheduling on Parallel Machines with Fixed Jobs or Unavailability Periods

We survey results that address the problem of non-preemptive scheduling on parallel machines with fixed jobs or unavailability periods with the purpose of minimizing the maximum completion time. We consider both identical and uniform processors, and also address the special case of scheduling on nonsimultaneous parallel machines, which may start processing at different times. The discussed results include polynomial-time approximation algorithms that achieve the best possible worst-case approximation bound of 1.5 in the class of polynomial algorithms unless P = NP for scheduling on identical processors with at most one fixed job on each machine and on uniform machines with at most one fixed job on each machine. The presented heuristics have similarities with the LPT algorithm or the MULTIFIT algorithm and they are fast and easy to implement. For scheduling on nonsimultaneous machines, experiments suggest that they would perform well in practice. We also include references to the relevant work in this area that contains more complex algorithms. We then discuss the main methods of argument used in the approximation bound proofs for the simple heuristics, and comment upon current challenges in this area by describing aspects of related practical problems from the automotive industry

Scheduling Problems

Scheduling is defined as the process of assigning operations to resources over time to optimize a criterion. Problems with scheduling comprise both a set of resources and a set of a consumers. As such, managing scheduling problems involves managing the use of resources by several consumers. This book presents some new applications and trends related to task and data scheduling. In particular, chapters focus on data science, big data, high-performance computing, and Cloud computing environments. In addition, this book presents novel algorithms and literature reviews that will guide current and new researchers who work with load balancing, scheduling, and allocation problems

Scheduling on uniform nonsimultaneous parallel machines

Abstract We consider the problem of scheduling on uniform processors which may not start processing at the same time with the purpose of minimizing the maximum completion time. We give a variant of the Multifit algorithm which generates schedules which end within 1.382 times the optimal maximum completion time for the general problem, and within √ 6/2 times the optimal maximum completion time for problem instances with at most two processors. This results from properties of a variant of the Multifit algorithm for scheduling on uniform processors with simultaneous start times. We also show that if a better approximation bound of the Multifit variant for scheduling on uniform processors will be found in the future, this bound will also apply to our Multifit variant for scheduling on nonsimultaneous uniform processors

A hierarchical heuristic approach for machine loading problems in a partially grouped environment

The loading problem in a Flexible Manufacturing System (FMS) lies in the allocation of operations and associated cutting tools to machines for a given set of parts subject to capacity constraints. This dissertation proposes a hierarchical approach to the machine loading problem when the workload and tool magazine capacity of each machine are restrained. This hierarchical approach reduces the maximum workload of the machines by partially grouping them. This research deals with situations where different groups of machines performing the same operation require different processing times and this problem is formulated as an integer linear problem. This work proposes a solution that is comprised of two phases. In the first phase (Phase I), demand is divided into batches and then operations are allocated to groups of machines by using a heuristic constrained by the workload and tool magazine capacity of each group. The processing time of the operation is different for each machine group, which is composed of the same identical machines; however, these machines can perform different sets of operations if tooled differently. Each machine and each group of machines has a limited time for completing an operation. Operations are allocated to groups based on their respective workload limits. In the second phase (Phase II), demand is divided into batches again and operations are assigned to machines based on their workload and tool magazine capacity defined by Longest Processing Time (LPT) and Multifit algorithms. In Phase II, like Phase I, partial grouping is more effective in balancing the workload than total grouping. In partial grouping, each machine is tooled differently, but they can assist one another in processing each individual operation. Phase I demonstrates the efficiency of allocating operations to each group. Phase II demonstrates the efficiency of allocating operations to each machine within each group. This two-phase solution enhances routing flexibility with the same or a smaller number of machines through partial grouping rather than through total grouping. This partial grouping provides a balanced solution for problems involving a large number of machines. Performance of the suggested loading heuristics is tested by means of randomly generated tests

An EPTAS for scheduling jobs on uniform processors using an MILP relaxation with a constant number of integral variables

We present an efficient polynomial time approximation scheme (EPTAS) for scheduling on uniform processors, i.e. finding a minimum length schedule for a set of n independent jobs on m processors with different speeds (a fundamental NP-hard scheduling problem). The previous best polynomial time approximation scheme (PTAS) by Hochbaum and Shmoys has a running time of (n/\epsilon)^{O(1/\epsilon^2)}. Our algorithm, based on a new mixed integer linear programming (MILP) formulation with a constant number of integral variables and an interesting rounding method, finds a schedule whose length is within a relative error $\epsilon$ of the optimum, and has running time 2^{O(1/\epsilon^2 \log(1/\epsilon)^3)} + poly(n)

Maintenance grouping for multi-component systems with availability constraints and limited maintenance teams

International audienceThe paper deals with a maintenance grouping approach for multi-component systems whose components are connected in series. The considered systems are required to serve a sequence of missions with limited breaks/stoppage durations while maintenance teams (repairmen) are limited and may vary over time. The optimization of the maintenance grouping decision for such multi-component systems leads to a NP-complete problem. The aim of the paper is to propose and to optimize a dynamic maintenance decision rule on a rolling horizon. The heuristic optimization scheme for the maintenance decision is developed by implementing two optimization algorithms (genetic algorithm and MULTIFIT) to find an optimal maintenance planning under both availability and limited repairmen constraints. Thanks to the proposed maintenance approach, impacts of availability constraints or/and limited maintenance teams on the maintenance planning and grouping are highlighted. In addition, the proposed grouping approach allows also updating online the maintenance planning in dynamic contexts such as the change of required availability level and/or the change of repairmen over time. A numerical example of a 20-component system is introduced to illustrate the use and the advantages of the proposed approach in the maintenance optimization framework

Multiprocessor Scheduling with Availability Constraints

We consider the problem of scheduling a given set of tasks on multiple pro- cessors with predefined periods of unavailability, with the aim of minimizing the maximum completion time. Since this problem is strongly NP-hard, polynomial ap- proximation algorithms are being studied for its solution. Among these, the best known are LPT (largest processing time first) and Multifit with their variants. We give a Multifit-based algorithm, FFDL Multifit, which has an optimal worst- case performance in the class of polynomial algorithms for same-speed processors with at most two downtimes on each machine, and for uniform processors with at most one downtime on each machine, assuming that P 6= NP. Our algorithm finishes within 3/2 the maximum between the end of the last downtime and the end of the optimal schedule. This bound is asymptotically tight in the class of polynomial algorithms assuming that P 6= NP. For same-speed processors with at most k downtimes on each machine our algorithm finishes within ( 3 2 + 1 2k ) the end of the last downtime or the end of the optimal schedule. For problems where the optimal schedule ends after the last downtime, and when the downtimes represent fixed jobs, the maximum completion time of FFDL Multifit is within 3 2 or ( 3 2+ 1 2k ) of the optimal maximum completion time. We also give an LPT-based algorithm, LPTX, which matches the performance of FFDL Multifit for same-speed processors with at most one downtime on each machine, and is thus optimal in the class of polynomial algorithms for this case. LPTX differs from LPT in that it uses a specific order of processors to assign tasks if two processors become available at the same time. For a similar problem, when there is at most one downtime on each machine and no more than half of the machines are shut down at the same time, we show that a bound of 2 obtained in a previous work for LPT is asymptotically tight in the class of polynomial algorithms assuming that P 6= NP

Algorithms for Scheduling Problems and Integer Programming

The first part of this thesis gives approximation results to scheduling problems. The classical makespan minimization problem on identical parallel machines asks for a distribution of a set of jobs to a set of machines such that the latest job completion time is minimized. For this strongly NP-complete problem we give a new EPTAS algorithm. In fact, it admits a practical implementation which beats the currently best approximation ratio of the MULTIFIT algorithm. A well-studied extension of the problem is the partition of the jobs into classes which impose a class-specific setup time on a machine whenever the processing switches to a job of a different class. For these so-called scheduling problems with batch setup times we present a 1.5-approximation algorithm for each of the three major settings. We achieve similar results for the likewise natural variant of many shared resources scheduling (MSRS) where instead of imposing a setup time each class is identified by a resource which can be occupied by at most one of its jobs at a time. For MSRS we present a 1.5-approximation and two EPTAS results. The second part provides results for fixed-priority uniprocessor real-time scheduling and variants of block-structured integer programming. We give a new approach to compute worst-case response times which admits a polynomial-time algorithm for harmonic periods even in the presence of task release jitters. In more detail, we prove a duality between Response Time Computation (RTC) and the Mixing Set problem. Furthermore, both problems can be expressed as block-structured integer programs which are closely related to simultaneous congruences. However, the setting of the famous Chinese Remainder Theorem is that each congruence has to have a certain remainder. We relax this setting such that the remainder of each congruence may lie in a given interval. We show that the smallest solution to these congruences can be computed in polynomial time if the set of divisors is harmonic