An effective hybrid multi-objective genetic algorithm for bi-criteria scheduling on a single batch processing machine with non-identical job sizes

This paper addresses the problem of scheduling jobs with non-identical sizes on a single batch processing machine. A batch processing machine is one which can process multiple jobs simultaneously as a batch as long as the total size of jobs being processed does not exceed the machine capacity. The batch processing time is equal to the longest processing time among all jobs in the batch. For the simultaneous minimization of the bi-criteria of makespan and maximum tardiness, we propose two different multi-objective genetic algorithms based on different representation schemes. While the first algorithm do search via generating sequences of jobs using genetic operators and then batching jobs keeping their order in the sequence, the second algorithm uses the idea of generating batches of jobs directly using genetic operators and ensures feasibility through using heuristic procedures. The type of representation used in the second algorithm allows introducing heuristics with the ability of biasing the search towards each objective and also allows hybridization with a local search heuristic that gives the ability of finding Pareto-optimal or locally efficient Pareto-solutions. Computational results show that the non-dominated solutions obtained by the latter algorithm are very superior in closeness to the true Pareto-optimal solutions and to keep diversity in the obtained Pareto-set, as the problem size increases.     

Integrated batch sizing and scheduling on a single machine

In this paper, we address the integrated batch sizing and scheduling problem. We consider a single machine which can handle at most one customer order at a time and for which the nominal production rate is the same for all the customer orders. Demand is deterministic, and all the orders are ready to be processed at time zero and must be delivered at a given due date. Each order can be satisfied from different batches. Upper and lower bounds on the size of the batches are considered. We seek a feasible schedule that minimizes the sum of the tardiness costs and the setup costs incurred by creating a new batch. We present some structural properties of the optimal schedules for both single-order and multiple-order problems and then propose dynamic programming algorithms based on these properties. Computational results that show the efficiency of the method are reported.     

Bicriterion scheduling with equal processing times on a batch processing machine

In this paper we consider the problem of scheduling jobs with equal processing times on a single batch processing machine so as to minimize a primary and a secondary criteria. We provide optimal polynomial time algorithms for various combinations of the primary and secondary criteria.     

On scheduling unbounded batch processing machine(s)

The computational complexity of scheduling jobs with released dates on an unbounded batch processing machine to minimize total completion time and on parallel unbounded batch processing machines to minimize total weighted completion time remains open. In this note we show that the first problem is NP-hard with respect to id-encoding, and the second one is strongly NP-hard.     

基于自由搜索算法求解单机差异工件批调度问题*

采用自由搜索（free search,FS）算法对单机差异工件批调度问题的制作跨度进行优化。针对该问题的离散优化特征以及自由搜索算法的不足,将自由搜索算法与实数编码遗传算法相结合,在标准FS算法的基础上引入两种杂交算子和精英保留策略,提出混合自由搜索（hybrid free search,HFS）算法。仿真实验结果表明,该算法表现出良好的鲁棒性和收敛性,与标准FS、FFLPT以及BFLPT算法相比,HFS算法提高了寻优精度。     

Scheduling a single batch processing machine with secondary resource constraints

This paper studies the problems of minimizing total completion time (ΣC_i) and makespan (Cmax) on a single batch processing machine with job families and secondary resource constraints. The motivation for this problem is the burn-in operation in the final testing stage of semiconductor manufacturing, where both oven capacity and the number of boards available may constrain scheduling decisions. Because both problems are NP-hard, integer programming formulations are developed for special cases and are then used to develop heuristics. Extensive computational experiments show that the heuristics are capable of consistently obtaining good solutions in modest CPU times.     

An ACO-based intercell scheduling approach for job shop cells with multiple single processing machines and one batch processing machine

Intercell transfers are inevitable for the manufacturing of complicated products, which disrupts the philosophy of cellular manufacturing and leads new challenges to the field of production scheduling. The issue of intercell scheduling is analyzed in the context of a cellular manufacturing system consisting of multiple single-processing machines and one batch-processing machine, which is derived from the actual manufacturing of complicated assemblies in the equipment manufacturing industry. Since the two types of machines are different from, even contrary to, each other in some constraint conditions, a combinational ant colony optimization (CACO) approach is developed in this paper, which designs two structures for the single-processing machines and the batch-processing machine, respectively. By updating pheromone trails integratedly and scheduling the single-processing operations and the batch-processing operations simultaneously, cooperative optimization for the two types of machines is achieved in the CACO. Minimizing the maximum completion time is taken as the scheduling objectives. Computational results show that the CACO has significant advantages comparing with other approaches and the CPLEX, and is especially suitable for the large dimension problems.     

Lot scheduling on a single machine

In a practical situation, a manufacturer receives different orders from its customers. Different orders may contain different quantities of the product. Therefore, the manufacturer has to decide how to group these orders into different lots based on the capacity of the lot processing machine (such as integrated circuit tester, heated container, etc.) and then decides the sequence of these lots. In this paper, we study a lot scheduling problem with orders which can be split. The objective is to minimize the total completion time of all orders. We show that this problem can be solved in polynomial time.     

The complexity of single machine scheduling with two distinct deadlines and identical decreasing rates of processing times

The makespan problem on a single machine for a set of tasks with two distinct deadlines and identical decreasing rates of processing times is considered in this paper. Ho et al. [1] have proposed a model of a task system in which the processing time of a task decreases with its starting time. When the decreasing rate is identical, the computational complexity of the makespan problem with two distinct deadlines is posed as an open problem. In this paper we show that the problem is NP-complete. It follows that both the corresponding flow time problem and maximum lateness problem are also NP-complete.     

Jackson's semi-preemptive scheduling on a single machine

We propose an effective improvement of the well-known Jackson's preemptive schedule lower bound for the single machine scheduling problem with heads and tails. The semi-preemptive scheduling concept roughly consists in reducing the preemption impact by constraining some particular job parts to be processed in reduced time intervals. The impact of semi-preemptive scheduling is twofold: it yields a lower bound which dominates the preemptive one, and enables more effective adjustments of the heads and tails. Our experimental study revealed that suitably embedding our procedure within Carlier's algorithm makes feasible to solve all of the hard instances which could not be solved by its original variant.     

Single machine batch scheduling problem with fuzzy batch size

In a batch scheduling problem, jobs are grouped (group is called batch) and scheduled in batches, and a setup time is incurred when starting a new batch. Processing times are assumed to be identical for all jobs. Setup times are assumed to be identical for all batches. Though all batch sizes cannot exceed a common upper bound, the upper bound is flexible and satisfaction degree with respect to the upper limit to be maximized is given. Also the other two objectives, i.e., the maximum completion time and the flow-time are to be minimized. Usually there exists no solution optimizing three objectives at a time. Therefore we define non-dominated solutions consisting of batch size, batch number and allocation of jobs to batches. First we propose an efficient algorithm for a sub-problem with fixed upper limit of batch size, fixed batch number based on a Lagrange relaxation procedure. Based on the properties of non-dominated solutions clarified in this paper, we propose an efficient algorithm to find some non-dominated solutions. Finally we summarize the results in this paper and discuss further research problems.     

On scheduling a single machine with resource dependent release times

We consider a single machine scheduling problem with resource dependent release times that can be controlled by a non-increasing convex resource consumption function. The objective is to minimize the weighted total resource consumption and sum of job completion times with an initial release time greater than the total processing times. It is known that the problem is polynomially solvable in O(n⁴) with n the number of jobs.     

Meta-heuristics for stable scheduling on a single machine

This paper presents a model for single-machine scheduling with stability objective and a common deadline. Job durations are uncertain, and our goal is to ensure that there is little deviation between planned and actual job starting times. We propose two meta-heuristics for solving an approximate formulation of the model that assumes that exactly one job is disrupted during schedule execution, and we also present a meta-heuristic for the global problem with independent job durations.     

Knowledge-based single machine scheduling

Production scheduling remains as one of the most important functions in manufacturing systems. Knowledge-based systems provide new insight to production scheduling. The increasing complexity of selecting the right procedure, understanding the procedure and solving an instant can be simplified with the help of knoweldge-based expert systems. This paper explains a prototype system developed using a rule-based expert system shell, M.1, for an application limited to single machine scheduling only.     

Coupled-task scheduling on a single machine subject to a fixed-job-sequence

This paper investigates single-machine coupled-task scheduling where each job has two tasks separated by an exact delay. The objective of this study is to schedule the tasks to minimize the makespan subject to a given job sequence. We introduce several intriguing properties of the fixed-job-sequence problem under study. While the complexity status of the studied problem remains open, an O(n²) algorithm is proposed to construct a feasible schedule attaining the minimum makespan for a given permutation of 2n tasks abiding by the fixed-job-sequence constraint. We investigate several polynomially solvable cases of the fixed-job-sequence problem and present a complexity graph of the problem.     

Dynamic non-preemptive single machine scheduling

Considering a dynamic single machine problem in which operations cannot be split, we first develop a decision theory based heuristic called DT-TD (Decision Theory-Tactically Delayed) of computational complexity O(n²). Using a simple look-ahead procedure, it produces, actically delayed (TD) schedules. We then develop a branch-and-bound (BB) algorithm (which uses DT-TD to obtain the initial upper bound) to obtain the optimum schedule. The optimum schedules are examined to identify conditions where TD schedules are necessary. Results based on 540 test problems suggest that TD schedules are important, for job shop scheduling under the range of conditions examined, when due dates are arbitrary and utilization is low. Additional test results indicate that the difference between the optimum schedule and the optimum non-delay schedule could be substantial. Finally, the performance of the DT-TD heuristic is analyzed by comparing its solution to the optimum solution obtained using the BB algorithm. The results indicate that the DT-TD heuristic is effective.     

Scheduling a batch processing machine with incompatible job families

The problem of scheduling batch processors is important in some industries and, at a more fundamental level, captures an element of complexity common to many practical scheduling problems. We describe a branch and bound procedure applicable to a batch processor model with incompatible job families. Jobs in a given family have identical job processing times, arbitrary job weights, and arbitrary job sizes. Batches are limited to jobs from the same family. The scheduling objective is to minimize total weighted completion time. We find that the procedure returns optimal solutions to problems of up to about 25 jobs in reasonable CPU time, and can be adapted for use as a heuristic for larger problems.     

Single machine flow-time scheduling with a single breakdown

Summary We consider the problem of scheduling tasks on a single machine to minimize the flowtime. The machine is subject to breakdowns during the processing of the tasks. The breakdowns occur at a random times and the machine is unavailable until it is repaired. The times for repair are random and independent of each other and of the breakdown process. A task that is preempted due to a breakdown must be restarted and otherwise preemptions are not allowed. We show in the case of a single breakdown that if the distribution function of the time to breakdown is concave then Shortest Processing Time (SPT) first scheduling stochastically minimizes the flowtime. For the case of multiple breakdowns we show that SPT minimizes the expected flowtime when the times to breakdown are exponentially distributed. If the time for a single breakdown is known before scheduling begins, and the processing times of the tasks are also known, then we show that the problem of deciding whether there is a schedule with flowtime less than or equal to a given value is NP-complete. Finally, we bound the performance of SPT scheduling in the deterministic case when there is a single breakdown.The work of this author was partially supported by The Lady Davis Fellowship Trust through a Lady Davis Visiting Professorship at the Technion, Haifa, Israel