Fast divide-and-conquer algorithms for preemptive scheduling problems with controllable processing times – A polymatroid optimization approach

We consider a variety of preemptive scheduling problems with controllable processing times on a single machine and on identical/uniform parallel machines, where the objective is to minimize the total compression cost. In this paper, we propose fast divide-and-conquer algorithms for these scheduling problems. Our approach is based on the observation that each scheduling problem we discuss can be formulated as a polymatroid optimization problem. We develop a novel divide-and-conquer technique for the polymatroid optimization problem and then apply it to each scheduling problem. We show that each scheduling problem can be solved in O(Tfeas(n) log n) time by using our divide-and-conquer technique, where n is the number of jobs and Tfeas(n) denotes the time complexity of the corresponding feasible scheduling problem with n jobs. This approach yields faster algorithms for most of the scheduling problems discussed in this paper

Some topics on deterministic scheduling problems

Sequencing and scheduling problems are motivated by allocation of limited resources over time. The goal is to find an optimal allocation where optimality is defined by some problem specific objectives. This dissertation considers the scheduling of a set of ri tasks, with precedence constraints, on m \u3e= 1 identical and parallel processors so as to minimize the makespan. Specifically, it considers the situation where tasks, along with their precedence constraints, are released at different times, and the scheduler has to make scheduling decisions without knowledge of future releases. Both preemptive and nonpreemptive schedules are considered. This dissertation shows that optimal online algorithms exist for some cases, while for others it is impossible to have one. The results give a sharp boundary delineating the possible and the impossible cases. Then an O(n log n)-time implementation is given for the algorithm which solves P|pj = 1, rj, outtree| ΣCj and P|pmtn, pj=1,rj,outtree|ΣCj. A fundamental problem in scheduling theory is that of scheduling a set of n unit-execution-time (UET) tasks, with precedence constraints, on m \u3e 1 parallel and identical processors so as to minimize the mean flow time. For arbitrary precedence constraints, this dissertation gives a 2-approximation algorithm. For intrees, a 1.5-approximation algorithm is given. Six dual criteria problems are also considered in this dissertation. Two open problems are first solved. Both problems are single machine scheduling problems with the number of tardy jobs as the primary criterion and with the total completion time and the total tardiness as the secondary criterion, respectively. Both problems are shown to be NP-hard. Then it focuses on bi-criteria scheduling problems involving the number of tardy jobs, the maximum weighted tardiness and the maximum tardiness. NP-hardness proofs are given for the scheduling problems when the number of tardy jobs is the primary criterion and the maximum weighted tardiness is the secondary criterion, or vice versa. It then considers complexity relationships between the various problems, gives polynomial-time algorithms for some special cases, and proposes fast heuristics for the general case

Agent-based Three Layer Framework of Assembly-Oriented Planning and Scheduling for Discrete Manufacturing Enterprises

To solve the cost burden caused by delivery tardiness for small and medium sized packaging machinery enterprises, the assembly-oriented planning and scheduling is studied based on the multi-agent technology. Taking into account the due date, the planning and scheduling are optimized iteratively with the rule-based algorithms complying with the machining and assembling process constraints. To make the realistic large-scale production planning scheme tailored to fit the optimization algorithms, a multi-agent system is utilized to conceptually construct a three-layer framework corresponding to three planning horizons of different task level. These planning horizons are quarter/month of product/subassembly level, week of part level, and day of operation level. With the strategy of combining top-down task decomposition and bottom-up plan update (where the quarterly orders are decomposed into the monthly, weekly, and daily tasks according to the product processing structure while the resulting plans of every layer are updated iteratively based on the bottom layer optimization), the proposed framework not only addresses the planning but also the periodic and event-driven scheduling of the processes. In this paper, a gravure printing machine is considered as a test case for evaluating the proposed framework. The simulation results with the historical data of the orders for this machine demonstrate the effectiveness of the proposed scheme on the control of the distribution of idle-time. It can also provide a resolution to tardiness penalty for small and medium sized enterprises

Dominance inequalities for scheduling around an unrestrictive common due date

The problem considered in this work consists in scheduling a set of tasks on a single machine, around an unrestrictive common due date to minimize the weighted sum of earliness and tardiness. This problem can be formulated as a compact mixed integer program (MIP). In this article, we focus on neighborhood-based dominance properties, where the neighborhood is associated to insert and swap operations. We derive from these properties a local search procedure providing a very good heuristic solution. The main contribution of this work stands in an exact solving context: we derive constraints eliminating the non locally optimal solutions with respect to the insert and swap operations. We propose linear inequalities translating these constraints to strengthen the MIP compact formulation. These inequalities, called dominance inequalities, are different from standard reinforcement inequalities. We provide a numerical analysis which shows that adding these inequalities significantly reduces the computation time required for solving the scheduling problem using a standard solver.Comment: 30 pages, 7 figures and 4 table

To Develop a Knowledge-Based System for Parallel Machine Operations Scheduling

As the industrialized world develops, more and more resources are becoming critical. Machines, manpower and facilities are now commonly thought of as critical resources in production and service activities. Scheduling these resources leads to increased efficiency, utilization, and ultimately, profitability. Current operation scheduling methods require complex mathematically modeling techniques that demand the substantial and extensive knowledge. Otherwise, the simple methods may not provide good results. The project is intended to engage in the issue of parallel machine operation scheduling from the knowledge based system perspective. The deal between both fields will emerge a new development of formulation and integration in artificial intelligence at area of industrial scheduling. Eventhough there are diversity techniques in manufacturing industry, the scope of the study is only limited to identical parallel operation scheduling due to time constraint towards the completion of the project. The goal of this project is to produce a working model of knowledge system for parallel machine operation scheduling. The execution of the project will be conducted in two semesters. For the First Semester, the data gathering and carrying out the associated case studies were explicitly nurtured as to aid better understanding of the project. The case studies performed were: (1) Parallel Processing - Jobs of Equal Weight, (2) Parallel Processing - Weighted Jobs, and (3) Parallel Processing - Jobs with Due Dates. In the Second Semester, the knowledge system was effectively developed. The development of the knowledge system was done in the expert interface which consists of six parts. The first step is theory familiarization, followed by user interface development, the inference engine development, dry run or testing and verification of the system. When all the steps are taken, the knowledge system can be considered as complete

Scheduling based on earliness and tardiness criteria in assembly job shops

In this research, the following four scheduling problems have been studied: (1) single machine problem with earliness cost minimization, (2) single machine problem with the sum of the weighted earliness and weighted tardiness cost minimization, (3) assembly job shop problem with earliness cost minimization, and (4) assembly job shop problem with the sum of weighted earliness and weighted tardiness cost minimization. Four mathematical models based on these four scheduling problems were developed in an effort to obtain optimal solutions. Six heuristic algorithms have been developed to solve the problems. The performances of the heuristic algorithms were demonstrated on some sample test problems. Quality of solutions and CPU time of solutions were the factors of interest. Several properties of optimal solutions for the single machine scheduling problem with the objective of minimizing the weighted earliness penalty have been identified in the research. Algorithms I, III, V, and VI were developed based on these identified properties while the algorithms II and IV were developed based on the tabu search concept;Algorithms I and II were developed to solve the first case (1) problem. The results indicate that these two algorithms are able to produce solutions close to optimal in small size problems. The results also show that algorithm I is relatively better than algorithm II in large size problem;Algorithms III and IV were developed to solve the second case (2) problem. Both algorithms obtained a small average deviation solutions (i.e., less than 2%) from optimal in small size test problems. For all problems tested, the algorithm IV is the best algorithm for solving the earliness/tardiness problems compared to algorithm III and the Ow & Morton algorithm;Algorithm V was developed to solve the third case (3) problem. It obtained an average deviation solutions less than 1% from the optimal. Algorithm VI was developed to solve the fourth case (4) problem. Algorithm VI obtained an average deviation solutions of 2.53% from the optimal;In testing all developed heuristics the computational requirements for solving the problems are less than 2 second in all test problems

Least space-time first scheduling algorithm : scheduling complex tasks with hard deadline on parallel machines

Both time constraints and logical correctness are essential to real-time systems and failure to specify and observe a time constraint may result in disaster. Two orthogonal issues arise in the design and analysis of real-time systems: one is the specification of the system, and the semantic model describing the properties of real-time programs; the other is the scheduling and allocation of resources that may be shared by real-time program modules. The problem of scheduling tasks with precedence and timing constraints onto a set of processors in a way that minimizes maximum tardiness is here considered. A new scheduling heuristic, Least Space Time First (LSTF), is proposed for this NP-Complete problem. Basic properties of LSTF are explored; for example, it is shown that (1) LSTF dominates Earliest-Deadline-First (EDF) for scheduling a set of tasks on a single processor (i.e., if a set of tasks are schedulable under EDF, they are also schedulable under LSTF); and (2) LSTF is more effective than EDF for scheduling a set of independent simple tasks on multiple processors. Within an idealized framework, theoretical bounds on maximum tardiness for scheduling algorithms in general, and tighter bounds for LSTF in particular, are proven for worst case behavior. Furthermore, simulation benchmarks are developed, comparing the performance of LSTF with other scheduling disciplines for average case behavior. Several techniques are introduced to integrate overhead (for example, scheduler and context switch) and more realistic assumptions (such as inter-processor communication cost) in various execution models. A workload generator and symbolic simulator have been implemented for comparing the performance of LSTF (and a variant -- LSTF+) with that of several standard scheduling algorithms. LSTF\u27s execution model, basic theories, and overhead considerations have been defined and developed. Based upon the evidence, it is proposed that LSTF is a good and practical scheduling algorithm for building predictable, analyzable, and reliable complex real-time systems. There remain some open issues to be explored, such as relaxing some current restrictions, discovering more properties and theorems of LSTF under different models, etc. We strongly believe that LSTF can be a practical scheduling algorithm in the near future