Scheduling two-stage hybrid flow shops with parallel batch, release time, and machine eligibility constraints

This paper investigates a difficult scheduling problem on a specialized two-stage hybrid flow shop with multiple processors that appears in semiconductor manufacturing industry, where the first and second stages process serial jobs and parallel batches, respectively. The objective is to seek job-machine, job-batch, and batch-machine assignments such that makespan is minimized, while considering parallel batch, release time, and machine eligibility constraints. We first propose a mixed integer programming (MIP) formulation for this problem, then gives a heuristic approach for solving larger problems. In order to handle real world large-scale scheduling problems, we propose an efficient dispatching rule called BFIFO that assigns jobs or batches to machines based on first-in-first-out principle, and then give several reoptimization techniques using MIP and local search heuristics involving interchange, translocation and transposition among assigned jobs. Computational experiments indicate our proposed re-optimization techniques are efficient. In particular, our approaches can produce good solutions for scheduling up to 160 jobs on 40 machines at both stages within 10?min.     

Unrelated parallel machine scheduling with setup times and a total weighted tardiness objective

This paper presents several search heuristics and their performance in batch scheduling of parallel, unrelated machines. Identical or similar jobs are typically processed in batches in order to decrease setup times and/or processing times. The problem accounts for allotting batched work parts into unrelated parallel machines, where each batch consists of a fixed number of jobs. Some batches may contain different jobs but all jobs within each batch should have an identical processing time and a common due date. Processing time of each job of a batch is determined according to the machine group as well as the batch group to which the job belongs. Major or minor setup times are required between two subsequent batches depending on batch sequence but are independent of machines. The objective of our study is to minimize the total weighted tardiness for the unrelated parallel machine scheduling. Four search heuristics are proposed to address the problem, namely (1) the earliest weighted due date, (2) the shortest weighted processing time, (3) the two-level batch scheduling heuristic, and (4) the simulated annealing method. These proposed local search heuristics are tested through computational experiments with data from dicing operations of a compound semiconductor manufacturing facility.     

Scheduling unrelated parallel batch processing machines with non-identical job sizes and unequal ready times

This research analyzes the problem of scheduling a set of n jobs with arbitrary job sizes and non-zero ready times on a set of m unrelated parallel batch processing machines so as to minimize the makespan. Unrelated parallel machine is a generalization of the identical parallel processing machines and is closer to real-world production systems. Each machine can accommodate and process several jobs simultaneously as a batch as long as the machine capacity is not exceeded. The batch processing time and the batch ready time are respectively equal to the largest processing time and the largest ready time among all the jobs in the batch. Motivated by the computational complexity and the practical relevance of the problem, we present several heuristics based on first-fit and best-fit earliest job ready time rules. We also present a mixed integer programming model for the problem and a lower bound to evaluate the quality of the heuristics. The small computational effort of deterministic heuristics, which is valuable in some practical applications, is also one of the reasons that motivates this study. The results show that the heuristic proposed in this paper has a superior performance compared to the heuristics based on ideas proposed in the literature.     

The two-stage assembly scheduling problem to minimize total completion time with setup times

We address the two-stage assembly scheduling problem where there are m machines at the first stage and an assembly machine at the second stage. The objective is to schedule the available n jobs so that total completion time of all n jobs is minimized. Setup times are treated as separate from processing times. This problem is NP-hard, and therefore we present a dominance relation and propose three heuristics. The heuristics are evaluated based on randomly generated data. One of the proposed heuristics is known to be the best heuristic for the case of zero setup times while another heuristic is known to perform well for such problems. A new version of the latter heuristic, which utilizes the dominance relation, is proposed and shown to perform much better than the other two heuristics.     

Variable neighborhood search approaches for scheduling jobs on parallel machines with sequence-dependent setup times, precedence constraints, and ready times

In this paper, we discuss a scheduling problem for jobs on identical parallel machines. Ready times of the jobs, precedence constraints, and sequence-dependent setup times are considered. We are interested in minimizing the performance measure total weighted tardiness that is important for achieving good on-time delivery performance. Scheduling problems of this type appear as subproblems in decomposition approaches for large scale job shops with automated transport of the jobs as, for example, in semiconductor manufacturing. We suggest several variants of variable neighborhood search (VNS) schemes for this scheduling problem and compare their performance with the performance of a list based scheduling approach based on the Apparent Tardiness Cost with Setups and Ready Times (ATCSR) dispatching rule. Based on extensive computational experiments with randomly generated test instances we are able to show that the VNS approach clearly outperforms heuristics based on the ATCSR dispatching rule in many situations with respect to solution quality. When using the schedule obtained by ATCSR as an initial solution for VNS, then the entire scheme is also fast and can be used as a subproblem solution procedure for complex job shop decomposition approaches.     

Two-machine flowshop scheduling with flexible operations and controllable processing times

We consider a two-machine flowshop scheduling problem with identical jobs. Each of these jobs has three operations, where the first operation must be performed on the first machine, the second operation must be performed on the second machine, and the third operation (named as flexible operation) can be performed on either machine but cannot be preempted. Highly flexible CNC machines are capable of performing different operations. Furthermore, the processing times on these machines can be changed easily in albeit of higher manufacturing cost by adjusting the machining parameters like the speed and/or feed rate of the machine. The overall problem is to determine the assignment of the flexible operations to the machines and processing times for each operation to minimize the total manufacturing cost and makespan simultaneously. For such a bicriteria problem, there is no unique optimum but a set of nondominated solutions. Using ?-constraint$?\mathrm{-}\mathrm{constraint}$ approach, the problem could be transformed to be minimizing total manufacturing cost for a given upper limit on the makespan. The resulting single criterion problem can be reformulated as a mixed integer nonlinear problem with a set of linear constraints. We use this formulation to optimally solve small instances of the problem while a heuristic procedure is constructed to solve larger instances in a reasonable time.     

Two‐stage hybrid flow shop with recirculation

In this paper, we present two scheduling hybrid flow shop problems to minimize the makespan. In each problem, we have two stages. In the first problem, one machine at each stage is considered with recirculation of jobs in the second stage (machine). We prove that this first problem is polynomial and we present an algorithm for its resolution. The second problem consists of one machine in the first stage and two identical parallel machines in the second. Jobs can be recirculated a fixed number of times in the second stage. We show that the problem is NP‐hard and a polynomial subproblem is proposed. Linear program and heuristics are also presented with numerical experimentations.     

Two-stage hybrid flowshop scheduling with simultaneous processing machines

Simultaneous processing machines, common in processing industries such as steel and food production, can process several jobs simultaneously in the first-in, first-out manner. However, they are often highly energy-consuming. In this paper, we study a new two-stage hybrid flowshop scheduling problem, with simultaneous processing machines at the first stage and a single no-idle machine with predetermined job sequence at the second stage. A mixed integer programming model is proposed with the objective of minimizing the total processing time to reduce energy consumption and improve production efficiency. We give a sufficient and necessary condition to construct feasible sequencing solutions and present an effective approach to calculate the time variables for a feasible sequencing solution. Based on these results, we design a list scheduling heuristic algorithm and its improvement. Both heuristics can find an optimal solution under certain conditions with complexity O(nlogn), where n is the number of jobs. Our experiments verify the efficiency of these heuristics compared with classical heuristics in the literature and investigate the impacts of problem size and processing times.     

A new Lagrangian relaxation algorithm for hybrid flowshop scheduling to minimize total weighted completion time

We investigate the problem of scheduling n jobs in s-stage hybrid flowshops with parallel identical machines at each stage. The objective is to find a schedule that minimizes the sum of weighted completion times of the jobs. This problem has been proven to be NP-hard. In this paper, an integer programming formulation is constructed for the problem. A new Lagrangian relaxation algorithm is presented in which precedence constraints are relaxed to the objective function by introducing Lagrangian multipliers, unlike the commonly used method of relaxing capacity constraints. In this way the relaxed problem can be decomposed into machine type subproblems, each of which corresponds to a specific stage. A dynamic programming algorithm is designed for solving parallel identical machine subproblems where jobs may have negative weights. The multipliers are then iteratively updated along a subgradient direction. The new algorithm is computationally compared with the commonly used Lagrangian relaxation algorithms which, after capacity constraints are relaxed, decompose the relaxed problem into job level subproblems and solve the subproblems by using the regular and speed-up dynamic programming algorithms, respectively. Numerical results show that the new Lagrangian relaxation method produces better schedules in much shorter computation time, especially for large-scale problems.     

时间约束混合流水车间调度问题综述

车间调度是智能制造领域中的核心问题之一, 在经典流水车间调度中, 所有工件按照相同的加工顺序在指 定机床上加工. 混合流水车间调度(HFS)作为流水车间调度的特例, 相比前者增加了机床选择的灵活性, 可以显著 优化系统目标, 但同时也增加了问题求解的难度. 由于时间约束HFS相比基本HFS问题更贴近实际生产过程, 近年 来, 综合考虑各类时间相关约束的HFS问题得到了深入研究. 因此, 本文围绕基本HFS、有限等待时间HFS、带准备 时间HFS、模糊/随机加工时间HFS、多时间约束HFS、时间约束相关多目标HFS等问题开展研究. 针对每一类时间 约束HFS问题, 按照问题规模对当前研究成果进行分类描述, 按照确定性算法、启发式方法、元启发式方法、算法混 合对相关成果进行算法分类, 按照实际工业应用对文献进行归类分析. 另一方面, 围绕交货期、能耗、成本等3类性 能指标, 分析了在各类时间约束HFS问题中的多目标优化相关成果. 最后详细分析了带时间约束HFS问题在问题层 面、算法层面和应用层面存在的挑战性问题和未来研究的方向.     

Hybrid flexible flowshops with sequence-dependent setup times and machine availability constraints

One of the common assumptions in the field of scheduling is that machines are always available in the planning horizon. This may not be true in realistic problems since machines might be busy processing some jobs left from previous production horizon, breakdowns or preventive maintenance activities. Another common assumption is the consideration of setup times as a part of processing times, while in some industries, such as printed circuit board and automobile manufacturing, not only setups are an important factor but also setup magnitude of a job depends on its immediately preceding job on the same machine, known as sequence-dependent setup times. In this paper, we consider hybrid flexible flowshops with sequence-dependent setup times and machine availability constraints caused by preventive maintenance. The optimization criterion is the minimization of makespan. Since this problem is NP-hard in the strong sense, we propose three heuristics, based on SPT, LPT and Johnson rule and two metaheuristics based on genetic algorithm and simulated annealing. Computational experiments are performed to evaluate the efficiencies of the algorithms.     

Integrating the lot-sizing and sequencing decisions for scheduling a capacitated flow line

In this paper we consider a general problem of scheduling a single flow line consisting of multiple machines and producing a given set of jobs. The manufacturing environment is characterized by sequence dependent set-up times, limited intermediate buffer space, and capacity constraints. In addition, jobs are assigned with due dates that have to be met. The objectives of the scheduling are: (1) to meet the due dates without violating the capacity constraints, (2) to minimize the makespan, and (3) to minimize the inventory holding costs. While most of the approaches in the literature treat the problem of scheduling in flow lines as two independent sub-problems of lot-sizing and sequencing, our approach integrates the lot-sizing and sequencing heuristics. The integrated approach uses the Silver-Meal heuristic (modified to include lot-splitting) for lot-sizing and an improvement procedure applied to Palmer's heuristic for sequencing, which takes into account the actual sequence dependent set-up times and the limited intermedite buffer capacity. We evaluate the performance of the integrated approach and demonstrate its efficacy for scheduling a real world SMT manufacturing environment.     

Heuristic methods for the identical parallel machine flowtime problem with set-up times

We consider the scheduling of N jobs divided into G families for processing on M identical parallel machines. No set-up is necessary between jobs belonging to the same family. A set-up must be scheduled when switching from the processing of family i jobs to those of another family j, i≠j, the duration of this set-up being the sequence-independent set-up time s_j for family j. We propose heuristics for this problem and computationally evaluate the performance of the heuristics relative to lower bounds and solutions obtained using an exact algorithm.Scope and purposeWe study a machine-scheduling problem within which we have identical parallel machines, jobs arranged into families, and sequence-independent set-up times between jobs of different families on these machines. Our purpose is to develop simple, effective and efficient heuristics for this problem, and we seek to maximise the use of ideas and algorithms that have appeared previously in the literature for related problems. In our computational experiments, we seek to study the behaviour of these heuristics and uncover relevant properties of the scheduling problem. Within this experiment, we compare the observed performance of the heuristics relative to lower bounds and optimal solutions.     

An artificial immune system heuristic for two-stage multi-machine assembly scheduling problem to minimize total completion time

We address the two-stage multi-machine assembly scheduling problem. The first stage consists of m independently working machines where each machine produces its own component. The second stage consists of two independent and identical assembly machines. The objective is to come up with a schedule that minimizes total or mean completion time for all jobs. The problem has been addressed in the scheduling literature and several heuristics have been proposed. In this paper, we propose a new heuristic called artificial immune system (AIS). We conduct experimental analysis for comparing the newly proposed heuristic AIS with the best known heuristic in the literature. Experimental results show that our proposed heuristic AIS performs better than the best known existing heuristic. More specifically, our new heuristic AIS reduces the error of the best known heuristic by 60% while the computational times of both AIS and the best known heuristic are almost the same.     

Multiprocessor Scheduling with Machine Allotment and Parallelism Constraints

Abstract. Modern computer systems distribute computation among several machines to speed up the execution of programs. Yet, setup and communication costs, as well as parallelism constraints, bound the number of machines that can share the execution of a given application, and the number of machines by which it can be processed simultaneously . We study the resulting scheduling problem, stated as follows. Given a set of n jobs and m uniform machines, assign the jobs to the machines subject to parallelism and machine allotment constraints, such that the overall completion time of the schedule (or makespan ) is minimized. Indeed, the multiprocessor scheduling problem (where each job can be processed by a single machine) is a special case of our problem; thus, our problem is strongly NP-hard. We present a (1+ α) -approximation algorithm for this problem, where α ∈ (0,1] depends on the minimal number of machine allotments and the minimal parallelism allowed for any job. Also, we show that when the maximal number of machines that can share the execution of a job is some fixed constant, our problem has a polynomial time approximation scheme ; for other special cases we give optimal polynomial time algorithms. Finally, through the relation of our problem to the classic preemptive scheduling problem on multiple machines, we shed some fresh light on what is known in scheduling folklore as the power of preemption.     

Scheduling Parallel Machines for the Customer Order Problem

This paper considers scheduling problems where jobs are dispatched in batches. The objective is to minimize the sum of the completion times of the batches. While a machine can process only one job at a time, multiple machines can simultaneously process jobs in a batch. This simple environment has a variety of real world applications such as part kitting and customer order scheduling.A heuristic is presented for the parallel machine version of the problem. Also, a tight worst case bound on the relative error is found. For the case of two parallel machines, we examine two heuristics, which are based on simple scheduling rules. We find tight worst case bounds of 6/5 and 9/7 on the relative error and show that neither procedure is superior for all instances. Finally, we empirically evaluate these two heuristics. For large problems, the methods find solutions that are close to optimal.     

A GRASP approach for makespan minimization on parallel batch processing machines

In this paper we consider the problem of scheduling a set of identical batch processing machines arranged in parallel. A Greedy Randomized Adaptive Search Procedure (GRASP) approach is proposed to minimize the makespan under the assumption of non-zero job ready times, arbitrary job sizes and arbitrary processing times. Each machine can process simultaneously several jobs as a batch as long as the machine capacity is not violated. The batch processing time is equal to the largest processing time among those jobs in the batch. Similarly, the batch ready time is equal to the largest ready time among those jobs in the batch. The performance of the proposed GRASP approach was evaluated by comparing its results to a lower bound and heuristics published in the literature. Experimental study suggests that the solution obtained from the GRASP approach is superior compared to other heuristics.     

Scheduling job classes on uniform machines

We study a scheduling problem with job classes on parallel uniform machines. All the jobs of a given class share a common due-date. General, non-decreasing and class-dependent earliness and tardiness cost functions are assumed. Two objectives are considered: (i) minmax, where the scheduler is required to minimize the maximum earliness/tardiness cost among all the jobs and (ii) minmax-minsum, where the scheduler minimizes the sum of the maximum earliness/tardiness cost in all job classes. The problem is easily shown to be NP-hard, and we focus here on the introduction of simple heuristics. We introduce LPT (Largest Processing Time first)-based heuristics for the allocation of jobs to machines within each class, followed by a solution of an appropriate non-linear program, which produces for this job allocation an optimal schedule of the classes. We also propose a lower bound, based on balancing the load on the machines. Our numerical tests indicate that the heuristics result in very small optimality gaps.     

A two-stage hybrid flowshop scheduling problem with a function constraint and unrelated alternative machines

This article addresses a two-stage hybrid flowshop scheduling problem with unrelated alternative machines. The problem to be studied has m unrelated alternative machines at the first machine center followed by a second machine center with a common processing machine in the system. The objective is to minimize the makespan of the system. For the processing of any job, it is assumed that the operation can be partially substituted by other machines in the first center, depending on its machining constraints. Such scheduling problems occur in certain practical applications such as semiconductors, electronics manufacturing, airplane engine production, and petrochemical production. We demonstrate that the addressed problem is NP-hard and then provide some heuristic algorithms to solve the problem efficiently. The experimental results show that the combination of the modified Johnson's rule and the First-Fit rule provides the best solutions within all proposed heuristics.Scope and purpose     

Scheduling of manufacturing systems using the Lagrangian relaxationtechnique

The practical solutions for three manufacturing scheduling problems are examined. As each problem is formulated, constraints are added or modified to reflect increasing real world complexity. The first problem considers scheduling single-operation jobs on identical machines. The second problem is concerned with scheduling multiple-operation jobs with simple fork/join precedence constraints on identical machines. The third problem is the job shop problem in which multiple-operation jobs with general precedence constraints are scheduled on multiple machine types Langrangian relaxation is used to decompose each of the scheduling problems into job- or operation-level subproblems. The subproblems are easier to solve than the original problem and have intuitive appeal. This technique results in algorithms which generate near-optimal schedules efficiently, while giving a lower bound on the optimal cost. In resolving the scheduling problem from one time instant to the next, the Lagrange multipliers from the last schedule can be used to initialize the multipliers, further reducing the computation time