Coordinated Logistics Scheduling for In-House Production and Outsourcing

In this paper, we address a new scheduling model for a firm with an option of outsourcing. A job can be processed by either in-house production or outsourcing. All outsourced jobs have to be transported back to the firm in batches, and the transportation costs have to be taken into account. We model the situation as a scheduling problem with transportation considerations. We discuss four commonly used objective functions, and solve them by dynamic programming algorithms. Note to Practitioners-An efficient supply chain management needs the coordination of production and transportation. Such problems exist in many different scenarios. This research considers a particular problem for a firm that has an option of using a subcontractor to fulfill part of its orders. The production schedule has to be coordinated with logistics issues for the transportation from the subcontractor to the firm. The purposes of this paper are twofold. First, we build models and provide optimal solutions for the specific cases discussed in this paper. Second, we hope to raise the issue of coordinated logistics scheduling, and motivate future research on more complicated models.     

On the Complexity of Non-preemptive Shop Scheduling with Two Jobs

In this note, we investigate the time complexity of non-preemptive shop scheduling problems with two jobs. First we study mixed shop scheduling where one job has a fixed order of operations and the operations of the other job may be executed in arbitrary order. This problem is shown to be binary NP-complete with respect to all traditional optimality criteria even if distinct operations of the same job require different machines. Moreover, we devise a pseudo-polynomial time algorithm which solves the problem with respect to all non-decreasing objective functions. Finally, when the job with fixed order of operations may visit a machine more than once, the problem becomes unary NP-complete. Then we discuss shop scheduling with two jobs having chain-like routings. It is shown that the problem is unary NP-complete with respect to all traditional optimality criteria even if one of the jobs has fixed order of operations and the jobs cannot visit a machine twice. Received July 28, 2001; revised May 15, 2002 Published online: July 26, 2002     

带尺寸约束的二机流水车间生产运输协调博弈调度问题

研究二机流水车间生产运输协调调度问题,当工件在第1台机器加工完成后,由1台带有容量限制的运输车分批次运输到第2台机器加工,运输过程考虑工件尺寸约束,目标函数为最小化最大完工时间.考虑到源于不同客户的工件对机器及运输设备的竞争,以工件为博弈方,工件在生产运输过程中等待时间为策略,各工件完工时间为收益,建立非合作博弈模型.通过将问题转化为马尔可夫决策过程,设计线性逼近值函数的Q-learning算法求解纳什均衡调度.实验结果表明Q-learning算法求得的纳什均衡调度具有较好的全局最优性,从而能够在满足客户的利益下,提高企业的生产效率,实现客户与企业的双赢.     

A non-delayed relax-and-cut algorithm for scheduling problems with parallel machines,due dates and sequence-dependent setup times

Consider the problem of scheduling a set of jobs to be processed exactly once, on any machine of a set of unrelated parallel machines, without preemption. Each job has a due date, weight, and, for each machine, an associated processing time and sequence-dependent setup time. The objective function considered is to minimize the total weighted tardiness of the jobs.This work proposes a non-delayed relax-and-cut algorithm, based on a Lagrangean relaxation of a time indexed formulation of the problem. A Lagrangean heuristic is also developed to obtain approximate solutions.Using the proposed methods, it is possible to obtain optimal solutions within reasonable time for some instances with up to 180 jobs and six machines. For the solutions for which it is not possible to prove optimality, interesting gaps are obtained.     

Vehicle capacity planning system: a case study on vehicle routing problem with time windows

In this paper, we consider a local logistic company that provides transportation service for moving empty and laden containers within Singapore. Due to the limited capacity of its own fleet of vehicles, the company cannot handle all the job orders and have to outsource some orders to other smaller local transportation companies. The current operation of assigning jobs for outsourcing goes through two steps. In the first step, a certain percentage of jobs will be preselected for outsourcing according to some simple rules. Then at the second step, the rest of the jobs will be put into an in-house computer system which assigns jobs to its internal fleet of vehicles according to some greedy rules and the remaining jobs that cannot be served by the internal fleet of vehicles will be outsourced. This paper presents a vehicle capacity planning system (VCPS), which models the problem as a vehicle routing problem with time window constraints (VRPTW) and tabu search (TS) is applied to find a solution for the problem. From the simulation results, some new rules on how to assign jobs for outsourcing are derived, which are shown to be about 8% better than existing rules currently adopted by the company.     

复杂并行机调度问题基于分解的优化算法

针对纺织生产过程中广泛存在的带特殊工艺约束的大规模并行机调度问题，提出了一种基于分解的优化算法。首先将原调度问题分解为机台选择和工件排序两个子问题，然后针对机台选择子问题提出一种进化规划算法，并采用一种具有多项式时间复杂度的最优算法求解工件排序子问题，以得到问题特征信息（即每台机器对应拖期工件数的最小值），该问题特征信息用以指导进化规划算法的迭代过程。不同规模并行机调度问题的数值计算结果及实际制造企业应用效果表明，本文提出的算法是有效的。     

Unrelated parallel machine scheduling with setup times using simulated annealing

This paper presents a scheduling problem for unrelated parallel machines with sequence-dependent setup times, using simulated annealing (SA). The problem accounts for allotting work parts of L jobs into M parallel unrelated machines, where a job refers to a lot composed of N items. Some jobs may have different items while every item within each job has an identical processing time with a common due date. Each machine has its own processing times according to the characteristics of the machine as well as job types. Setup times are machine independent but job sequence dependent. SA, a meta-heuristic, is employed in this study to determine a scheduling policy so as to minimize total tardiness. The suggested SA method utilizes six job or item rearranging techniques to generate neighborhood solutions. The experimental analysis shows that the proposed SA method significantly outperforms a neighborhood search method in terms of total tardiness.     

Scheduling in a manufacturing shop with sequence-dependent setups

In this paper, the problem of scheduling multiple jobs in a flexible manufacturing cell with multiple machine stations is addressed. Due to the large capital investments that usually characterize flexible manufacturing systems (FMS), an area of control of great interest to system users is that of maximizing the system performance through the minimization of machine idle and setup times. The magnitude of total time spent on machine setups and idle times is influenced by the availability of jobs, job mix, similarities of jobs and job scheduling procedure used. Similar jobs on the same machine require less setup times. Similarly, the use of an adequate scheduling method also reduces total idle and setup times. Such reduction improves the flow times of jobs. In this paper, a heuristic algoritm for scheduling jobs with sequence dependent setup times in a FMS is presented. The measure of performance for evaluating schedule adequacy is the production makespan.     

A dominant class of schedules for malleable jobs in the problem to minimize the total weighted completion time

This paper is about scheduling parallel jobs, i.e. which can be executed on more than one machine at the same time. Malleable jobs is a special class of parallel jobs. The number of machines a malleable job is executed on may change during its execution.In this work, we consider the NP-hard problem of scheduling malleable jobs to minimize the total weighted completion time (or mean weighted flow time). For this problem, we introduce the class of “ascending” schedules in which, for each job, the number of machines assigned to it cannot decrease over time while this job is being processed.We prove that, under a natural assumption on the processing time functions of jobs, the set of ascending schedules is dominant for the problem. This result can be used to reduce the search space while looking for an optimal solution.     

Heuristics for scheduling problems with an unavailability constraint and position-dependent processing times

We study a single machine scheduling problem, where the machine is unavailable for processing for a pre-specified time period. We assume that job processing times are position-dependent. The objective functions considered are minimum makespan, minimum total completion time and minimum number of tardy jobs. All these problems are known to be NP-hard even without position-dependent processing times. For all three cases we introduce simple heuristics which are based on solving the classical assignment problem. Lower bounds, worst case analysis and asymptotic optimality are discussed. All heuristics are shown numerically to perform extremely well.     

Scheduling unit length jobs with parallel nested machine processing set restrictions

This article explores the impact of restricting the machines upon which individual jobs may be scheduled. Even the simple case of a single stage of identical parallel machines cannot be solved to optimality in a reasonable time. We therefore focus on the case when job processing times are identical. In some applications the machine processing sets of jobs are structured in a nested fashion and do not partially overlap. We present efficient algorithms for solving this nested problem to optimality for each of the standard scheduling objective functions. In particular, an algorithm with constant running time minimises makespan on a fixed number of machines regardless of the number of jobs. Improvements in efficiency have been gained by attention to implementation issues, thus challenging the conventional approach to evaluating complexity.     

Scheduling with multi-attribute setup times

In this paper we address a scheduling problem with multi-attribute setup times originated from the manufacturing plant of a company producing PVC sheets. In the considered scheduling problem, each job has a number of attributes and each attribute has one or more levels. Because there is at least one different level of attribute between two adjacent jobs, it is necessary to make a setup adjustment whenever there is a switch to a different job. The objective of the problem is to determine a processing sequence so as to minimize the total setup time on a single machine.     

Robust scheduling on a single machine to minimize total flow time

In a real-world manufacturing environment featuring a variety of uncertainties, production schedules for manufacturing systems often cannot be executed exactly as they are developed. In these environments, schedule robustness that guarantees the best worst-case performance is a more appropriate criterion in developing schedules, although most existing studies have developed optimal schedules with respect to a deterministic or stochastic scheduling model. This study concerns robust single machine scheduling with uncertain job processing times and sequence-dependent family setup times explicitly represented by interval data. The objective is to obtain robust sequences of job families and jobs within each family that minimize the absolute deviation of total flow time from the optimal solution under the worst-case scenario. We prove that the robust single machine scheduling problem of interest is NP-hard. This problem is reformulated as a robust constrained shortest path problem and solved by a simulated annealing-based algorithmic framework that embeds a generalized label correcting method. The results of numerical experiments demonstrate that the proposed heuristic is effective and efficient for determining robust schedules. In addition, we explore the impact of degree of uncertainty on the performance measures and examine the tradeoff between robustness and optimality.     

A heuristic procedure for makespan minimization in job shops with multiple identical processors

Scheduling has been and continues to be a major issue in production planning. Job shop scheduling is one area where a considerable amount of research has been and continues to be pursued. Usual emphasis is on one machine per work center job shop scheduling. There appears to be very limited literature available on scheduling a job shop problem which requires scheduling of n jobs in m machine centers where each machine center may have k number of identical processors (though the number of identical processors may vary from one machine center to next). We discuss here, the problem of minimize of the makespan for such a job shop arrangement The problem can be represented by the symbol m x n x k.     

Single machine scheduling with interfering job sets

We consider two single machine bicriteria scheduling problems in which jobs belong to either of two different disjoint sets, each set having its own performance measure. The problem has been referred to as interfering job sets in the scheduling literature and also been called multi-agent scheduling where each agent's objective function is to be minimized. In the first problem (P1) we look at minimizing total completion time and number of tardy jobs for the two sets of jobs and present a forward SPT-EDD heuristic that attempts to generate the set of non-dominated solutions. The complexity of this specific problem is NP-hard; however some pseudo-polynomial algorithms have been suggested by earlier researchers and they have been used to compare the results from the proposed heuristic. In the second problem (P2) we look at minimizing total weighted completion time and maximum lateness. This is an established NP-hard problem for which we propose a forward WSPT-EDD heuristic that attempts to generate the set of supported points and compare our solution quality with MIP formulations. For both of these problems, we assume that all jobs are available at time zero and the jobs are not allowed to be preempted.     

Minimizing the makespan on a batch machine with non-identical job sizes: an exact procedure

A batch processing machine can simultaneously process several jobs forming a batch. This paper considers the problem of scheduling jobs with non-identical capacity requirements, on a single-batch processing machine of a given capacity, to minimize the makespan. The processing time of a batch is equal to the largest processing time of any job in the batch. We present some dominance properties for a general enumeration scheme and for the makespan criterion, and provide a branch and bound method. For large-scale problems, we use this enumeration scheme as a heuristic method.Scope and purposeUsually in classical scheduling problems, a machine can perform only one job at a time. Although, one can find machines that can process several jobs simultaneously as a batch. All jobs of a same batch have common starting and ending times. Batch processing machines are encountered in many different environments, such as burn-in operations in semiconductor industries or heat treatment operations in metalworking industries. In the first case, the capacity of the machine is defined by the number of jobs it can hold. In the second case, each job has a certain capacity requirement and the total size of a batch cannot exceed the capacity of the machine. Hence, the number of jobs contained in each batch may be different. In this paper, we consider this second case (which is more difficult) and we provide an exact method for the makespan criterion (minimizing the last ending time).     

A Stronger Complexity Result for the Single Machine Multi-Operation Jobs Scheduling Problem to Minimize the Number of Tardy Jobs

We consider the single machine multi-operation jobs scheduling problem to minimize the number of tardy jobs. Each job consists of several operations that belong to different families. In a schedule, each family of job operations may be processed in batches with each batch incurring a setup time. A job completes when all of its operations have been processed. The objective is to minimize the number of tardy jobs. In the literature, this problem has been proved to be strongly NP-hard for arbitrary due-dates. We show in this paper that the problem remains strongly NP-hard even when the due-dates are common and all jobs have the same processing time.     

A hybrid metaheuristic for the prize-collecting single machine scheduling problem with sequence-dependent setup times

This paper investigates a single machine scheduling problem with strong industrial background, named the prize-collecting single machine scheduling problem with sequence-dependent setup times. In this problem, there are n candidate jobs for processing in a single machine, each job has a weight (or profit) and a processing time, and during processing a symmetric sequence-dependent setup time exists between two consecutive jobs. Since there is a maximum available time limitation of the machine, it is generally impossible to complete the processing of all the candidate jobs within this time limitation. The objective is to find a job processing sequence of maximal job weights (or profits) over a subset of all candidate jobs whose makespan does not exceed the given time limitation. This problem can be considered as an application of the orienteering problem (OP) in the field of discrete manufacturing. We formulate this problem as a mixed integer linear programming (MILP) model and propose a hybrid metaheuristic combining the structures of scatter search and variable neighborhood search. Computational results on a large number of randomly generated instances with different structures show that the proposed hybrid metaheuristic outperforms CPLEX and two metaheuristics proposed for the OP.     

Hybrid genetic algorithms with dispatching rules for unrelated parallel machine scheduling with setup time and production availability

This article considers the unrelated parallel machine scheduling problem with sequence- and machine-dependent setup times and machine-dependent processing times. Furthermore, the machine has a production availability constraint to each job. The objective of this problem is to determine the allocation policy of jobs and the scheduling policy of machines to minimize the total completion time. To solve the problem, a mathematical model for the optimal solution is derived, and hybrid genetic algorithms with three dispatching rules are proposed for large-sized problems. To assess the performance of the algorithms, computational experiments are conducted and evaluated using several randomly generated examples.     

Some unrelated parallel machine scheduling problems with past-sequence-dependent setup time and learning effects

In this paper, we study an unrelated parallel machine scheduling problem with setup time and learning effects simultaneously. The setup time is proportional to the length of the already processed jobs. That is, the setup time of each job is past-sequence-dependent. The objectives are to minimize the total absolute deviation of job completion times and the total load on all machines, respectively. We show that the proposed problem is polynomially solvable. We also discuss two special cases of the problem and show that they can be optimally solved by lower order algorithms.