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.     

A new Lagrangian Relaxation Algorithm for scheduling dissimilar parallel machines with release dates

In this article we investigate the parallel machine scheduling problem with job release dates, focusing on the case that machines are dissimilar with each other. The goal of scheduling is to find an assignment and sequence for a set of jobs so that the total weighted completion time is minimised. This type of production environment is frequently encountered in process industry, such as chemical and steel industries, where the scheduling of jobs with different purposes is an important goal. This article formulates the problem as an integer linear programming model. Because of the dissimilarity of machines, the ordinary job-based decomposition method is no longer applicable, a novel machine-based Lagrangian relaxation algorithm is therefore proposed. Penalty terms associated with violations of coupling constraints are introduced to the objective function by Lagrangian multipliers, which are updated using subgradient optimisation method. For each machine-level subproblem after decomposition, a forward dynamic programming algorithm is designed together with the weighted shortest processing time rule to provide an optimal solution. A heuristics is developed to obtain a feasible schedule from the solution of subproblems to provide an upper bound. Numerical results show that the new approach is computationally effective to handle the addressed problem and provide high quality schedules.     

A hybrid scheduling approach for a two-stage flexible flow shop with batch processing machines

In this paper, we discuss a flexible flow shop scheduling problem with batch processing machines at each stage and with jobs that have unequal ready times. Scheduling problems of this type can be found in semiconductor wafer fabrication facilities (wafer fabs). We are interested in minimizing the total weighted tardiness of the jobs. We present a mixed integer programming formulation. The batch scheduling problem is NP-hard. Therefore, an iterative stage-based decomposition approach is proposed that is hybridized with neighborhood search techniques. The decomposition scheme provides internal due dates and ready times for the jobs on the first and second stage, respectively. Each of the resulting parallel machine batch scheduling problems is solved by variable neighborhood search in each iteration. Based on the schedules of the subproblems, the internal due dates and ready times are updated. We present the results of designed computational experiments that also consider the number of machines assigned to each stage as a design factor. It turns out that the proposed hybrid approach outperforms an iterative decomposition scheme where a fairly simple heuristic based on time window decomposition and the apparent tardiness cost dispatching rule is used to solve the subproblems. Recommendations for the design of the two stages with respect to the number of parallel machines on each stage are given.     

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.     

Generating subproblems in branch and bound algorithms for parallel machines scheduling problem

A branch and bound algorithm (B&B) has been widely used in various discrete and combinatorial optimization fields. To obtain optimal solutions as soon as possible for scheduling problems, three tools, which are branching, bounding and dominance rules, have been developed in the B&B algorithm. One of these tools, a branching is a method for generating subproblems and directly determines size of solution to be searched in the B&B algorithm. Therefore, it is very important to devise effective branching scheme for the problem.In this note, a survey of branching schemes is performed for parallel machines scheduling (PMS) problems with n independent jobs and m machines and new branching schemes that can be used for identical and unrelated PMS problems, respectively, are suggested. The suggested branching methods show that numbers of generated subproblems are much smaller than that of other methods developed earlier and therefore, it is expected that they help to reduce a lot of CPU time required to obtain optimal solutions in the B&B algorithm.     

Scheduling a hybrid flowshop with batch production at the last stage

In this paper, we address the problem of scheduling n$n$ jobs in an s$s$-stage hybrid flowshop with batch production at the last stage with the objective of minimizing a given criterion with respect to the completion time. The batch production at stage s$s$ is referred to as serial batches by Hopp and Spearman where the processing time of a batch is equal to the sum of the processing times of all jobs included in it. This paper establishes an integer programming model and proposes a batch decoupling based Lagrangian relaxation algorithm for this problem. In this algorithm, after capacity constraints are relaxed by Lagrangian multipliers, the relaxed problem is decomposed based on a batch, unlike the commonly used job decoupling, so that it can be decomposed into batch-level subproblems, each for a specific batch. An improved forward dynamic programming algorithm is then designed for solving these subproblems where all operations within a batch form an in-tree structure and the precedence relations exist not only between the operations of a job but between the jobs in this batch at the last stage. A computational comparison is provided for the developed algorithm and the commonly used Lagrangian relaxation algorithm which, after capacity constraints and precedence relations within a batch are relaxed, decomposes the relaxed problem into job-level subproblems and solves the subproblems by using dynamic programming. Numerical results show that the designed Lagrangian relaxation method provides much better schedules and converges faster for small to medium sized problems, especially for larger sized problems.     

Approaches to modeling train scheduling problems as job-shop problems with blocking constraints

The problem of scheduling a set of trains traveling through a given railway network consisting of single tracks, sidings and stations is considered. For every train a fixed route and travel times, an earliest departure time at the origin and a desired arrival time at the destination are given. A feasible schedule has to be determined which minimizes total tardiness of all trains at their destinations. This train scheduling problem is modeled as a job-shop scheduling problem with blocking constraints, where jobs represent trains and machines constitute tracks or track sections. Four MIP formulations without time-indexed variables are developed based on two different transformation approaches of parallel tracks and two different types of decision variables leading to job-shop scheduling problems with or without routing flexibility. A computational study is made on hard instances with up to 20 jobs and 11 machines to compare the MIP models in terms of total tardiness values, formulation size and computation time.     

Minimizing weighted earliness–tardiness on parallel machines using hybrid metaheuristics

We consider the problem of scheduling a set of jobs on a set of identical parallel machines where the objective is to minimize the total weighted earliness and tardiness penalties with respect to a common due date. We propose a hybrid heuristic algorithm for constructing good solutions, combining priority rules for assigning jobs to machines and a local search with exact procedures for solving the one-machine subproblems. These solutions are then used in two metaheuristic frameworks, Path Relinking and Scatter Search, to obtain high quality solutions for the problem.The algorithms are tested on a large number of test instances to assess the efficiency of the proposed strategies.The results show that our algorithms consistently outperform the best reported results for this problem.     

Scheduling jobs on identical machines with agreement graph

We consider the following problem of scheduling with agreements: a set of jobs must be scheduled non-preemptively on identical machines subject to constraints that only some specific jobs can be scheduled concurrently on different machines. These constraints are represented by an agreement graph and the aim is to minimize the makespan. This problem is NP-hard. We study the complexity of the problem for two machines and arbitrary bipartite agreement graphs, in particular we prove the NP-hardness of the open problem proposed in the literature which is the case of two machines with processing times at most 3. We propose list algorithms with empirical results for the problem in the general case.     

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.     

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.     

A new ILS algorithm for parallel machine scheduling problems

This paper addresses the non-preemptive scheduling problem of scheduling jobs on identical parallel machines to minimize the maximum completion time or makespan. The problem has been proved to be NP-hard in the strong sense. The NP-hardness of the problem motivates us to develop a new methodology to obtain near-optimal solutions. We formulate the problem as an integer programming and then propose a new iterated local search (ILS) algorithm based on a variable number of cyclic exchanges to solve it. The properties of the solutions are derived and the results are used to improve the computational efficiency of our algorithm. Computational experiments show that the cyclic exchange neighborhood embedded in an iterated local search framework is effective for solving the scheduling problems with up to 1000 jobs and 40 machines within a reasonable amount of computation time. Received: April 2005 / Accepted: January 2006     

The total completion time open shop scheduling problem with a given sequence of jobs on one machine

This paper addresses the open shop scheduling problem to minimize the total completion time, provided that one of the machines has to process the jobs in a given sequence. The problem is NP-hard in the strong sense even for the two-machine case. A lower bound is derived based on the optimal solution of a relaxed problem in which the operations on every machine may overlap except for the machine with a given sequence of jobs. This relaxed problem is NP-hard in the ordinary sense, however it can be quickly solved via a decomposition into subset-sum problems. Both heuristic and branch-and-bound algorithm are proposed. Experimental results show that the heuristic is efficient for solving large-scaled problems, and the branch-and-bound algorithm performs well on small-scaled problems.Scope and purposeShop scheduling problems, widely used in the modeling of industrial production processes, are receiving an increasing amount of attention from researchers. To model practical production processes more closely, additional processing restrictions can be introduced, e.g., the resource constraints, the no-wait in process requirement, the precedence constraints, etc. This paper considers the total completion time open shop scheduling problem with a given sequence of jobs on one machine. This model belongs to a new class of shop scheduling problems under machine-dependent precedence constraints. This problem is NP-hard in the strong sense. A heuristic is proposed to efficiently solve large-scaled problems and a branch-and-bound algorithm is presented to optimally solve small-scaled problems. Computational experience is also reported.     

带释放时间的并行机调度问题的ILS & SS算法

研究工件带释放时间的两类并行机最小化总完成时间的调度问题.针对问题提出了一种新的基于变深度环交换邻域结构的Iterated local search(ILS)算法.1)提出了变深度环交换邻域结构.2)基于变深度环交换和传统Swap的混合邻域,提出了带有两种kick策略的ILS算法.3)为了加强ILS逃出局部最优的能力,将Scatter search (SS)搜索方法引入了ILS算法中;算法将当前最好解和次好解进行分散处理,再从处理后的解开始继续迭代.为了验证算法的有效性,对两类并行机问题分别随机产生100组数据进行试验.实验结果表明:对于同构并行机问题,引入SS的ILS算法的计算结果与下界的平均偏差为0.99%,而没有引入SS的ILS算法的为1.06%;对于无关并行机问题,引入SS搜索方法后,ILS算法的计算结果 改进了6.06%,并明显优于多点下降算法.     

Scheduling jobs with ready times and precedence constraints on parallel batch machines using metaheuristics

In this paper, we discuss a scheduling problem for parallel batch machines where the jobs have ready times. Problems of this type can be found in semiconductor wafer fabrication facilities (wafer fabs). In addition, we consider precedence constraints among the jobs. Such constraints arise, for example, in scheduling subproblems of the shifting bottleneck heuristic when complex job shop scheduling problems are tackled. We use the total weighted tardiness as the performance measure to be optimized. Hence, the problem is NP-hard and we have to rely on heuristic solution approaches. We consider a variable neighborhood search (VNS) scheme and a greedy randomized adaptive search procedure (GRASP) to compute efficient solutions. We assess the performance of the two metaheuristics based on a large set of randomly generated problem instances and based on instances from the literature. The obtained computational results demonstrate that VNS is a very fast heuristic that quickly leads to high-quality solutions, whereas the GRASP is slightly outperformed by the VNS approach. However, the GRASP approach has the advantage that it can be parallelized in a more natural manner compared to VNS.     

An exact approach to early/tardy scheduling with release dates

In this paper, we consider the single machine earliness/tardiness scheduling problem with different release dates and no unforced idle time. The problem is decomposed into weighted earliness and weighted tardiness subproblems. Lower bounding procedures are proposed for each of these subproblems, and the lower bound for the original problem is the sum of the lower bounds for the two subproblems. The lower bounds and several versions of a branch-and-bound algorithm are then tested on a set of randomly generated problems, and instances with up to 30 jobs are solved to optimality. To the best of our knowledge, this is the first exact approach for the early/tardy scheduling problem with release dates and no unforced idle time.     

Minimizing makespan subject to minimum flowtime on two identical parallel machines

We consider the problem of scheduling jobs on two parallel identical machines where an optimal schedule is defined as one that gives the smallest makespan (the completion time of the last job) among the set of schedules with optimal total flowtime (the sum of the completion times of all jobs). We propose an algorithm to determine optimal schedules for the problem, and describe a modified multifit algorithm to find an approximate solution to the problem in polynomial computational time. Results of a computational study to compare the performance of the proposed algorithms with a known heuristic shows that the proposed heuristic and optimization algorithms are quite effective and efficient in solving the problem.Scope and purposeMultiple objective optimization problems are quite common in practice. However, while solving scheduling problems, optimization algorithms often consider only a single objective function. Consideration of multiple objectives makes even the simplest multi-machine scheduling problems NP-hard. Therefore, enumerative optimization techniques and heuristic solution procedures are required to solve multi-objective scheduling problems. This paper illustrates the development of an optimization algorithm and polynomially bounded heuristic solution procedures for the scheduling jobs on two identical parallel machines to hierarchically minimize the makespan subject to the optimality of the total flowtime.     

Scheduling algorithms to minimize the number of tardy jobs in two-stage hybrid flow shops

This paper considers a two-stage hybrid flow shop scheduling problem for the objective of minimizing the number of tardy jobs. Each job is processed through the two production stages in series, each of which has multiple identical parallel machines. The problem is to determine the allocation of jobs to the parallel machines as well as the sequence of the jobs assigned to each machine. To solve the problem, a branch and bound algorithm, which incorporates the methods to obtain the lower and upper bounds as well as the dominance properties to reduce the search space, is suggested that gives the optimal solutions. In addition, two-phase heuristic algorithms are suggested to obtain good solutions for large-size problems within a reasonable amount of computation time. To show the performances of the optimal and heuristic algorithms suggested in this paper, computational experiments are done on a number of randomly generated test problems, and the test results are reported.     

针对经典排序问题的一种新算法的近似比分析

给定m台平行机(同型机),n个工件,寻找一种分配方案,使得把这n个工件分配到m台机器后,整体完工时间尽可能短,这个NP-难问题被称为经典排序问题。如果每个工件的加工时间满足一定的条件,则有望能在多项式时间内有效地得到最优的分配方案。Yue等对加工时间满足整除性质的经典排序问题考虑了一种新的算法,该算法总是能得到这种特殊情况的最优分配。该算法在多项式时间内能够得到最优分配,是对于一般的经典排序问题的近似算法。文章在此基础上,考虑该新算法在一般问题上的近似比。文中考虑了这个新算法的两种版本,分别得到了3/2和2-1/2 ^q(q∈Z⁺)的近似比。紧例子表明,文中对算法的两个版本的分析都是最优的。     

Efficient symmetry breaking formulations for the job grouping problem

The job grouping problem consists of assigning a set of jobs, each with a specific set of tool requirements, to machines with a limited tool capacity in order to minimize the number of machines needed. Traditionally, a formulation has been used that assigns jobs to machines. However, such a formulation contains a lot of symmetry since the machines are identical and they can be permuted in any feasible solution. We propose a new formulation for this problem, based on the asymmetric representatives formulation (ARF) idea. This formulation eliminates the symmetry between the identical machines. We further propose various symmetry breaking constraints, including variable reduction and lexicographic ordering constraints, which can be added to the traditional formulation. These formulations are tested on a data set from the literature and newly generated data sets using a state-of-the-art commercial solver, which includes symmetry breaking features.