Minimizing total weighted tardiness on heterogeneous batch processing machines with incompatible job families

This paper addresses a specific class of scheduling parallel batching problem, which is observed in steel casting industries. The focus of this research is to minimize the total weighted tardiness on heterogeneous batch processing machines under conditions of dynamic job arrivals, incompatible job families and non-identical job sizes. This type of parallel batching problem arises in a number of different settings, including diffusion in wafer fabrication, heat treatment operations in aircraft industries, and metal working. The problem is viewed as a three stage-decision-problem: the first stage involves selecting a machine from the heterogeneous batch processing machines for scheduling; the second stage involves the selection of a job family from the available incompatible job families; and the third stage involves the selection of a set of jobs to create a batch from the selected job family based on the capacity of the selected batch-processing machine. Since the problem is NP-hard, a few greedy heuristics are proposed. The computational experiments show that the proposed greedy heuristic algorithms are capable of consistently obtaining near-optimal solutions (statistically estimated) in very reasonable computational time on a Pentium III 650 Mz with 128 MB RAM.     

Heuristics for makespan minimization on parallel batch processing machines with unequal job ready times

This research aims at minimizing the makespan of a set of identical batch processing machines arranged in parallel. Each job is defined by its processing time, ready time, and size. Each machine can process several jobs simultaneously as long as the machine capacity is not exceeded. The batch processing and ready times depend upon the batch composition. The batch processing time is equal to the longest processing job in the batch, and the batch ready time is equal to the largest ready time among those jobs in the batch. The problem under study is NP-hard. Consequently, a constructive heuristic is proposed and its performance with respect to solution quality and computational cost is compared against other solution approaches found in the literature. The computational experiments on a set of randomly generated instances show that the performance of the proposed heuristic is competitive with respect to solution quality and requires little computational cost.     

ACO-based multi-objective scheduling of parallel batch processing machines with advanced process control constraints

This research was motivated by a scheduling problem in the dry strip operations of a semiconductor wafer fabrication facility. The machines were modeled as parallel batch processing machines with incompatible job families and dynamic job arrivals, and constraints on the sequence-dependent setup time and the qual-run requirements of advanced process control. The optimization had multiple objectives, the total weighted tardiness (TWT) and makespan, to consider simultaneously. Since the problem is NP-hard, we used an Ant Colony Optimization (ACO) algorithm to achieve a satisfactory solution in a reasonable computation time. A variety of simulation experiments were run to choose ACO parameter values and to demonstrate the performance of the proposed method. The simulation results showed that the proposed ACO algorithm is superior to the common Apparent Tardiness Cost-Batched Apparent Tardiness Cost rule for minimizing the TWT and makespan. The arrival time distribution and the number of jobs strongly affected the ACO algorithm’s performance.     

Minimizing total weighted tardiness on heterogeneous batch processors with incompatible job families

In this paper, we address a scheduling problem for minimizing total weighted tardiness. The background for the paper is derived from the automobile gear manufacturing process. We consider the bottleneck operation of heat treatment stage of gear manufacturing. Real-life scenarios like unequal release times, incompatible job families, non-identical job sizes, heterogeneous batch processors, and allowance for job splitting have been considered. We have developed a mathematical model which takes into account dynamic starting conditions. The problem considered in this study is NP-hard in nature, and hence heuristic algorithms have been proposed to address it. For real-life large-size problems, the performance of the proposed heuristic algorithms is evaluated using the method of estimated optimal solution available in literature. Extensive computational analyses reveal that the proposed heuristic algorithms are capable of consistently obtaining near-optimal statistically estimated solutions in very reasonable computational time.     

具有柔性加工路径的作业车间批量调度优化研究

古典作业车间调度问题已经被研究了几十年并证明为 NP- hard问题。柔性作业车间调度是古典作业车间调度问题的扩展 ,它允许工序可以由一个机床集合中的多台机床完成加工 ,调度的目的是将工序分配给各机床 ,并对各机床上的工序进行排序以使完成所有工序的时间最小化。本文采用遗传算法进行柔性作业车间调度研究 ,针对柔性作业车间问题提出了一种新颖直观的基因编码方法以适用于批量调度 ,并分析了几种批量调度方案 ,最后给出了这些调度的仿真结果 ,证明单件最佳调度不适合扩展成批量最佳调度     

Genetic algorithms for minimizing makespan in a flow shop with two capacitated batch processing machines

This paper considers a flow shop with two batch processing machines. The processing times of the job and their sizes are given. The batch processing machines can process multiple jobs simultaneously in a batch as long as the total size of all the jobs in a batch does not exceed its capacity. When the jobs are grouped into batches, the processing time of the batch is defined by the longest processing job in the batch. Batch processing machines are expensive and a bottleneck. Consequently, the objective is to minimize the makespan (or maximize the machine utilization). The scheduling problem under study is NP-hard, hence, a genetic algorithm (GA) is proposed. The effectiveness (in terms of solution quality and run time) of the GA approach is compared with a simulated annealing approach, a heuristic, and a commercial solver which was used to solve a mixed-integer formulation of the problem. Experimental study indicates that the GA approach outperforms the other approaches by reporting better solution.     

Parallel machine scheduling with earliness-tardiness penalties and space limits

In order to maximize an availability of machine and utilization of space, the parallel machines scheduling problem with space limit is frequently discussed in the industrial field. In this paper, we consider the parallel machine scheduling problem in which n jobs having different release times, due dates, and space limits are to be scheduled on m parallel machines. The objective function is to minimize the weighted sum of earliness and tardiness. To solve this problem, a heuristic is developed which is divided into three modules hierarchically: job selection, machine selection and job sequencing, and solution improvement. To illustrate its effectiveness, a proposed heuristic is compared with genetic algorithm (GA), hybrid genetic algorithm (HGA), and tabu search (TS), which are well-known meta-heuristics in a large number of randomly generated test problems based on the field situation. Also, we determine the job selection rule that is suitable to the problem situation considered in this paper and show the effectiveness of our heuristic method.     

A new hybrid parallel algorithm for consistent-sized batch splitting job shop scheduling on alternative machines with forbidden intervals

Considering alternative machines for operations, forbidden intervals during which machines cannot be available and a job’s batch size greater than one in the real manufacturing environment, this paper studies the batch splitting scheduling problem on alternative machines with forbidden intervals, based on the objective to minimize the makespan. A scheduling model is established, taking before-arrival set-up, processing, and transfer time into account. And a new hybrid parallel algorithm, based on differential evolution and genetic algorithm, is brought forward to solve both the batch splitting problem and the batch scheduling problem by assuming a common number of sub-batches in advance. A solution consists of the actual optimum number of sub-batches for each job, the optimum batch size for each sub-batch, and the optimum sequence of operations for these sub-batches. Experiments on the performance of the proposed algorithm under different common numbers of sub-batches are carried out. The results of simulations indicate that the algorithm is feasible and efficient.     

NEW NONSTANDARD JOB SHOP SCHEDULING ALGORITHM

Considering the complex constraint between operations in nonstandard job shop scheduling problem （NJSSP）, critical path of job manufacturing tree is determined according to priority scheduling function constructed. Operations are divided into dependent operations and independent operations with the idea of subsection, and corresponding scheduling strategy is put forward according to operation characteristic in the segment and the complementarities of identical function machines. Forward greedy rule is adopted mainly for dependent operations to make operations arranged in the right position of machine selected, then each operation can be processed as early as possible, and the total processing time of job can be shortened as much as possible. For independent operations optimum scheduling rule is adopted mainly, the inserting position of operations will be determined according to the gap that the processing time of operations is subtracted from idle time of machine, and the operation will be inserted in the position with minimal gap. Experiments show, under the same conditions, the result that operations are scheduled according to the object function constructed, and the scheduling strategy adopted is better than the result that operations are scheduled according to efficiency scheduling algorithm.     

Minimizing total weighted tardiness on a batch-processing machine with non-agreeable release times and due dates

This study considers the scheduling problem observed in the burn-in operation of semiconductor final testing, where jobs are associated with release times, due dates, processing times, sizes, and non-agreeable release times and due dates. The burn-in oven is modeled as a batch-processing machine which can process a batch of several jobs as long as the total sizes of the jobs do not exceed the machine capacity and the processing time of a batch is equal to the longest time among all the jobs in the batch. Due to the importance of on-time delivery in semiconductor manufacturing, the objective measure of this problem is to minimize total weighted tardiness. We have formulated the scheduling problem into an integer linear programming model and empirically show its computational intractability. Due to the computational intractability, we propose a few simple greedy heuristic algorithms and meta-heuristic algorithm, simulated annealing (SA). A series of computational experiments are conducted to evaluate the performance of the proposed heuristic algorithms in comparison with exact solution on various small-size problem instances and in comparison with estimated optimal solution on various real-life large size problem instances. The computational results show that the SA algorithm, with initial solution obtained using our own proposed greedy heuristic algorithm, consistently finds a robust solution in a reasonable amount of computation time.     

2个加工时间不同工序组批的嵌套批综合调度算法

针对已有的批加工工序数为2的批综合调度算法,没有考虑组批工序不同和后续工序中存在组批工序的情况,不能适用于更复杂的批综合调度问题,提出求解2个加工时间不同工序组批的嵌套批综合调度算法。该算法根据批处理工序加工时间不同的特点,定义串行衔接时间和并行衔接时间;提出判断组批的余差比较策略;当可与等待工序一同批处理的工序不唯一时,采用组批前移最大化策略确定组批处理工序;当准备组批处理工序的后续工序中存在组批工序时,在余差比较策略中加入嵌套优化策略确定该工序是否组批;由于组批工序的后续工序较多且对调度结果影响较大,采用前续工序优先策略使组批工序尽早加工。理论分析和实例表明,提出的算法可解决2个加工时间不同工序组批的嵌套批综合调度问题。     

Combined column generation and constructive heuristic for a proportionate flexible flow shop scheduling

In a proportionate flow shop problem, jobs have to be processed through a fixed sequence of machines, and processing time for each job is equal on all machines. Such a problem has seldom been tackled. Proportionate flexible flow shop (PFFS) scheduling problems combine the properties of proportionate flow shop scheduling problems and parallel machine scheduling problems. This study presents a combined approach based on column generation (CG) for a PFFS problem with the criterion to minimize the objective of the total weighted completion time (TWCT). Minimizing TWCT in a PFFS problem significantly differs from the parallel-identical-machine scheduling problem, an optimal schedule in which jobs on each machine are in the weighted shortest processing time (WSPT) order. This combined approach adopts a CG approach to effectively handle job assignments to machines, and a constructive heuristic to obtain an optimal sequence for a single machine. Experimental results show the effectiveness of the combined approach in obtaining excellent quality solutions in a reasonable time, especially for large-scale problems.     

Minimization of maximum tardiness on unrelated parallel machines with process restrictions and setups

We consider the problem of scheduling N jobs on M unrelated parallel machines to minimize maximum tardiness. Each job has a due date and requires a single stage of processing. A setup for dies is incurred if the type of the job scheduled is different from the previous one on that machine. For each die type, the number of dies is restricted. Because of the mechanical structure of the machines and the fitness of dies to each machine, the processing time depends on both the job and the machine. In this paper, an efficient heuristic based on guided search, record-to-record travel, and tabu lists is presented to minimize maximum tardiness. Computational characteristics of the proposed heuristic are evaluated through extensive experiments, which show that the proposed heuristic outperforms a simulated annealing method tested and is able to prescribe the optimal solutions for problems in small scales.     

Optimal parallel machines scheduling with machine availability and eligibility constraints

In reality, the machine might become unavailable due to machine breakdowns or various inevitable reasons, and machine might have different capability to processing job. Motivated by this, we consider the problem of scheduling n non-preemptive and independent jobs on m identical machines incorporating machine availability and eligibility constraints while minimizing the maximum lateness. Each machine is capable of processing at specific availability intervals. We develop a branch and bound algorithm applying several immediate selection rules for solving this scheduling problem.     

Makespan minimization in a job shop with a BPM using simulated annealing

A scheduling problem commonly observed in the metal working industry has been studied in this research effort. A job shop equipped with one batch processing machine (BPM) and several unit-capacity machines has been considered. Given a set of jobs, their process routes, processing requirements, and size, the objective is to schedule the jobs such that the makespan is minimized. The BPM can process a batch of jobs as long as its capacity is not exceeded. The batch processing time is equal to the longest processing job in the batch. If no batches were to be formed, the scheduling problem under study reduces to the classical job shop problem with makespan objective, which is known to be nondeterministic polynomial time-hard. A network representation of the problem using disjunctive and conjunctive arcs, and a simulated annealing (SA) algorithm are proposed to solve the problem. The solution quality and run time of SA are compared with CPLEX, a commercial solver used to solve the mathematical formulation and with four dispatching rules. Experimental study clearly highlights the advantages, in terms of solution quality and run time, of using SA to solve large-scale problems.     

A study of the flexible job shop scheduling problem with parallel machines and reentrant process

This paper develops a scheduling algorithm for the job shop scheduling problem with parallel machines and reentrant process. This algorithm includes two major modules: the machine selection module (MSM) and the operation scheduling module (OSM). An order has several jobs and each job has several operations in a hierarchical structure. The MSM helps an operation to select one of the parallel machines to process it. The OSM is then used to schedule the sequences and the timing of all operations assigned to each machine. A real-life weapons production factory is used as a case study to evaluate the performance of the proposed algorithm. Due to the high penalty of delays in military orders, the on-time delivery rate is the most important performance measure and then makespan is the next most important measure. Well-known performance measures in the scheduling literature, such as maximum lateness and average tardiness, are also evaluated. The simulation results demonstrate that the MSM and OSM using the combination of earliest due date (EDD), the operations’ lowest level code (LLC) of the bill of materials (BOM), and the longest processing time (LPT) outperforms the other scheduling methods.     

Algorithms for flexible flow shop problems with unrelated parallel machines, setup times, and dual criteria

In textile industries, production facilities are established as multi-stage production flow shop facilities, where a production stage may be made up of parallel machines. This known as a flexible or hybrid flow shop environment. This paper considers the problem of scheduling n independent jobs in such an environment. In addition, we also consider the general case in which parallel machines at each stage may be unrelated. Each job is processed in ordered operations on a machine at each stage. Its release date and due date are given. The preemption of jobs is not permitted. We consider both sequence- and machine-dependent setup times. The problem is to determine a schedule that minimizes a convex combination of makespan and the number of tardy jobs. A 0–1 mixed integer program of the problem is formulated. Since this problem is NP-hard in the strong sense, we develop heuristic algorithms to solve it approximately. Firstly, several basic dispatching rules and well-known constructive heuristics for flow shop makespan scheduling problems are generalized to the problem under consideration. We sketch how, from a job sequence, a complete schedule for the flexible flow shop problem with unrelated parallel machines can be constructed. To improve the solutions, polynomial heuristic improvement methods based on shift moves of jobs are applied. Then, genetic algorithms are suggested. We discuss the components of these algorithms and test their parameters. The performance of the heuristics is compared relative to each other on a set of test problems with up to 50 jobs and 20 stages.     

A mixed integer programming formulation for scheduling of virtual manufacturing cells (VMCs)

We present a multi-objective mixed integer programming formulation for job scheduling in virtual manufacturing cells (VMCs). In a VMC, machines are dedicated to a part family as in a regular cell, but machines are not physically relocated in a contiguous area. Cell configurations are therefore temporary, and assignments are made to optimize the scheduling objective under changing demand conditions. We consider the case where there are multiple jobs with different processing routes. There are multiple machine types with several identical machines in each type and are located in different locations in the shop floor. The two scheduling objectives are makespan minimization and minimizing total traveling distance. Since batch splitting is permitted in the system, scheduling decisions must tell us the (a) assignment of jobs to the machines, (b) the job starting time at each machine, and (c) the part quantity processed on different machines due to batch splitting. Under these decision variables, the objective function is to minimize the sum of the makespan and total traveling distance/cost. Illustrative examples are given to demonstrate the implementation of the model.     

Unrelated parallel machine scheduling with secondary resource constraints

This paper considers unrelated parallel machine scheduling with secondary resource constraints. There are n jobs, each needing to be processed on one of the fitted machines. A setup that includes detaching one die and attaching another from the fitted die type is incurred if the type of job scheduled is different from the last job on that machine. For each kind of die type, the number of dies available is limited. Due to the mechanical structure of the machines, the processing time of a job depends on the machine on which the job is processed, and some jobs are restricted to be processed only on certain machines. In this paper, a heuristic with a capability relative to a runtime and solution quality is developed to minimise the makespan. The performance of the presented heuristic is evaluated through extensive computational experiments. Computational results show that the presented heuristic outperforms the search method tested. It is expected that this research can be applied in industry where unrelated parallel machines are used to process different components and setups for auxiliary equipments are required.     

A hybrid genetic algorithm to minimize makespan for the single batch machine dynamic scheduling problem

This paper considers a single batch machine dynamic scheduling problem, which is readily found in the burn-in operation of semiconductor manufacturing. The batch machine can process several jobs as a batch simultaneously, within the capacity limit of the machine, and the processing time is represented by the longest processing time among all jobs in a batch. For a single batch machine problem with arbitrary job release time, we proposed an improved algorithm (merge-split procedure) to refine the solution obtained by the LPT-BFF heuristic, and two versions of a hybrid genetic algorithm (GA) are introduced in this paper. Each version of the hybrid GA diversifies job sequences using the GA operators in stage 1, forms batches in stage 2, and finally sequence the batches in stage 3. The difference is that merge-split procedures are involved in the second version of the hybrid GA. Computational experiments showed that the hybrid GA would obtain satisfactory average solution quality and the merge-split procedures would be good at reinforcing the solution consistency of the hybrid GA.