A joint GA+DP approach for single burn-in oven scheduling problems with makespan criterion

This research is motivated by the scheduling problem found in the burn-in operation of semiconductor final testing, where jobs are associated with release times, processing times, and sizes. The burn-in ovens are modeled as batch-processing machines 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. Moreover, this paper attempts to schedule jobs on a single batch-processing machine to minimize makespan. A joint GA+DP algorithm is proposed involving two stages: (1) the formation of job sequence by genetic algorithm operators, and (2) the formation of batches by a dynamic programming algorithm. Computational experiments are given to examine the performance of the proposed algorithm. The experimental results indicate that the joint GA+DP approach has well improved on all instances with respect to solution quality and runtime.     

Minimizing makespan in a two-machine no-wait flow shop with batch processing machines

Given a set of jobs and two batch processing machines (BPMs) arranged in a flow shop environment, the objective is to batch the jobs and sequence the batches such that the makespan is minimized. The job sizes, ready times, and processing times on the two BPMs are known. The batch processing machines can process a batch of jobs as long as the total size of all the jobs assigned to a batch does not exceed its capacity. Once the jobs are batched, the processing time of the batch on the first machine is equal to the longest processing job in the batch; processing time of the batch on the second machine is equal to the sum of processing times of all the jobs in the batch. The batches cannot wait between two machines (i.e., no-wait). The problem under study is NP-hard. We propose a mathematical formulation and present a particle swarm optimization (PSO) algorithm. The solution quality and run time of PSO is compared with a commercial solver used to solve the mathematical formulation. Experimental study clearly highlights the advantages, in terms of solution quality and run time, of using PSO to solve large-scale problems.     

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.     

Particle swarm optimization for scheduling batch processing machines in a permutation flowshop

Batch processing machines are capable of processing several jobs in a batch simultaneously. These machines are used in many real-life applications. This paper presents solution approaches to schedule batch processing machines arranged in a permutation flowshop in order to minimize its makespan (or completion time of the last batch). The processing time of each job on all the machines and their sizes are given. Each machine can process a batch of jobs as long as its capacity is not violated. The batch processing time is equal to the longest processing job in the batch. Since the problem under study is NP-hard, commercial mixed-integer solvers may require prohibitively long run time to solve even modest sized problems. Consequently, a particle swarm optimization (PSO) algorithm is proposed. Three heuristics to update the particle’s positions are also proposed. The effectiveness of the proposed PSO algorithm is compared with a commercial solver (which was used to solve a mathematical model) and several heuristics from the literature. The experimental study conducted indicates that the proposed PSO algorithm outperforms both the commercial solver and the heuristics in terms of solution quality. The commercial solver requires longer run times compared to PSO.     

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.     

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.     

GRASP to minimize makespan for a capacitated batch-processing machine

This paper presents a Greedy Randomized Adaptive Search Procedure (GRASP) to minimize the makespan of a capacitated batch-processing machine. Given a set of jobs and their processing times and sizes, the objective is to group these jobs into batches and schedule the batches on a single batch-processing machine such that the time taken to complete the last batch of jobs (or makespan) is minimized. The batch-processing machine can process a batch of jobs simultaneously as long as the total size of all the jobs in that batch does not exceed the machine capacity. The batch-processing time is equal to the longest processing time for a job in the batch. It has been shown that the problem under study is non-deterministic polynomial-time hard. Consequently, a GRASP approach was developed. The solution quality of GRASP was compared to other solution approaches such as simulated annealing, genetic algorithm, and a commercial solver through an experimental study. The study helps to conclude that GRASP outperforms other solution approaches, especially on larger problem instances.     

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.     

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.     

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.     

Flow shop scheduling with two batch processing machines and nonidentical job sizes

A batch processing machine can process several jobs simultaneously. In this research, we consider the problem of a two-stage flow shop with two batch processing machines to minimize the makespan. We assume that the processing time of a batch is the longest processing time among all the jobs in that batch and the sizes of the jobs are nonidentical. There is a limitation on batch sizes and the sum of job sizes in a batch must be less than or equal to the machine capacity. Since this problem is strongly nondeterministic polynomial time hard, we propose two heuristic algorithms. The first one is knowledge-based and the other is based on the batch first fit heuristic proposed previously. To further enhance the solution quality, two different simulated annealing (SA) algorithms based on the two constructive heuristics is also developed. Since heuristic methods for this problem has not been proposed previously, a lower bound is developed for evaluating the performance of the proposed methods. Several test problems have been solved by SAs and lower bound method and the results are compared. Computational studies show that both algorithms provide good results but the first SA (ARSA) algorithm considerably outperforms the second one (FLSA). In addition, the results of ARSA algorithm, optimal solutions, and lower bounds are compared for several small problems. The comparisons show that except for one instance, the ARSA could find the optimal solutions and the proposed lower bound provides small gaps comparing with the optimal solutions.     

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.     

Lot streaming in a two-machine mixed shop

Most classical scheduling models overlook the fact that products are often produced in job lots and assume that job lots are indivisible single entities, although an entire job lot consists of many identical items. However, splitting an entire lot (process batch) into sublots (transfer batches) to be moved to downstream machines allows the overlapping of different operations on the same product while work needs to be completed on the upstream machine. This approach is known as lot streaming in scheduling theory. In this study, the lot streaming problem of multiple jobs in a two-machine mixed shop where there are two different job types as flow shop and open shop is addressed so as to minimize the makespan. The optimal solution method is developed for the mixed shop scheduling problem in which lot streaming can improve the makespan.     

A game-theory approach for job scheduling in networked manufacturing

This paper presents a new kind of scheduling solution for jobs in networked manufacturing environments. The main contributions of this study can be focused on three points: The first is to distinguish the concepts and requirements of job scheduling in the networked manufacturing environment form those in the traditional manufacturing environment. The second is to construct a game-theory mathematical model to deal with this new job scheduling problem. In this presented mathematical model, this new job scheduling problem is formulated as an N-person non-cooperative game with complete information. The players correspond to the jobs submitted, respectively, by related customers and the payoff of each job is defined as its makespan. Each player has a set of strategies which correspond to the feasible geographical distributive machines. Therefore, obtaining the optimal scheduling results is determined by the Nash equilibrium (NE) point of this game. In order to find the NE point, the last point is to design and develop a genetic algorithm (GA)-based solution algorithm to effectively solve this mathematical model. Finally, a numerical example is presented to demonstrate the feasibility of the approach.     

Scheduling optimization of flexible manufacturing system using cuckoo search-based approach

In the present work, a cuckoo search (CS)-based approach has been developed for scheduling optimization of a flexible manufacturing system by minimizing the penalty cost due to delay in manufacturing and maximizing the machine utilization time. To demonstrate the application of cuckoo search (CS)-based scheme to find the optimum job, the proposed scheme has been applied with slight modification in its Levy flight operator because of the discrete nature of the solution on a standard FMS scheduling problem containing 43 jobs and 16 machines taken from literature. The CS scheme has been implemented using Matlab, and results have been compared with other soft computing-based optimization approaches like genetic algorithm (GA) and particle swarm optimization found in the literature. The results shown by CS-based approach have been found to outperform the results of existing heuristic algorithms such as GA for the given problem.     

Heuristics to minimize makespan of parallel batch processing machines

Batch-processing machines can process several jobs simultaneously. These machines are commonly used to test Printed Circuit Boards (PCBs). The processing time and the dimensions of the PCB are given. Each batch is formed such that the total size of all the PCBs in the batch does not exceed the machine capacity. The batch processing time is equal to the longest processing time of all the PCBs in the batch. These batch processing machines are expensive and a bottleneck. Scheduling PCBs on these parallel batch processing machines to minimize their makespan is NP-hard. Consequently, we propose several heuristics. The performance of the proposed heuristics is compared to a simulated annealing approach and a commercial solver.     

An improved mixed integer linear formulation and lower bounds for minimizing makespan on a flow shop with batch processing machines

This paper considers a flow shop scheduling problem with batch processing machines. Each batch processing machine has a limited capacity and can process a group of jobs, each of them having a different known capacity requirement, simultaneously. Job processing time on each machine is known and arbitrary. The processing time of a batch on each machine is the longest processing time of all jobs in the batch. We improve the only existing mixed integer linear formulation (MILF) of the problem through significant reduction in size complexity of the model. Results justify that the improved MILF is clearly more efficient in reducing the required time for obtaining optimal makespan of small-size problems, in comparison with the existing MILF. Motivated by relaxing variety of the problem assumptions, several valid lower bounds on the optimal makespan are also proposed that can furtheraccelerate obtaining optimal solution through proposed MILF. Robustness evaluation of each bound under the different problem settings is reported through computations.     

一种基于时间窗的多阶段混合流水车间调度方法

考虑同时包含单处理机和批处理机的多阶段混合流水车间调度问题,设计一种基于时间窗的蚁群算法,以最小化最大完工时间为优化目标。在解决整体调度问题的过程中,通过蚁群算法实现工序分派、排序、组批三个阶段的协同优化调度;另外在工件组批阶段加入时间窗策略,利用前瞻性算法动态构建组批方案,通过分析批处理机的时空关系设计合理的组批时间窗,以获得较高的机器利用率。试验结果表明,与无时间窗的蚁群算法相比,时间窗策略在确保最小化最大完工时间的同时,提高了批处理机的利用率;与CPLEX相比,基于时间窗的蚁群算法在最大完工时间和计算效率方面均有较大优势。蚁群算法与时间窗的结合兼顾了多个阶段多种机型的生产特点,适用于解决多阶段混合流水车间的调度问题。     

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.     

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.