Scheduling in dynamic assembly job-shops to minimize the sum of weighted earliness,weighted tardiness and weighted flowtime of jobs

In many manufacturing systems, jobs that are completed early are held as finished-goods inventory until their due-dates, and hence we incur earliness costs. Similarly, jobs that are completed after their due-dates incur penalty. The objective in such situations would, therefore, be to meet the due-dates of the respective jobs as closely as possible, and consequently minimize the sum of earliness and tardiness of jobs because earliness and tardiness of jobs greatly influence the performance of a schedule with respect to cost. In addition, a job incurs holding cost from the time of its arrival until its completion. Most studies on scheduling in such manufacturing systems assume unit earliness cost, unit tardiness cost and unit holding cost of a job. However, in reality such an assumption need not always hold and it is quite possible that there exist different costs of earliness, tardiness and holding for different jobs. In addition, most studies on job-shop scheduling assume that jobs are independent and that no assembly operations exist. The current study addresses the problem of scheduling in dynamic assembly job-shops (i.e. shops that manufacture multi-level jobs) with the consideration of jobs having different earliness, tardiness and holding costs. An attempt is made in this paper to present dispatching rules by incorporating the relative costs of earliness, tardiness and holding of jobs in the form of scalar weights. In the first phase of the study, relative costs (or weights for) earliness and tardiness of jobs are considered, and the dispatching rules are presented in order to minimize the sum of weighted earliness and weighted tardiness of jobs. In the second phase of the study, the objective considered is the minimization of the sum of weighted earliness, weighted tardiness and weighted flowtime of jobs, and the dispatching rules are presented by incorporating the relative costs of earliness, tardiness and flowtime of jobs. Simulation studies have been conducted separately for both phases of the current study, the performance of the scheduling rules have been observed independently, and the results of the simulation study have been reported. The proposed rules are found to be effective in minimizing the mean and maximum values of the measures of performance.     

Scheduling in dynamic assembly job-shops to minimize the sum of weighted earliness,weighted tardiness and weighted flowtime of jobs

In many manufacturing systems, jobs that are completed early are held as finished-goods inventory until their due-dates, and hence we incur earliness costs. Similarly, jobs that are completed after their due-dates incur penalty. The objective in such situations would, therefore, be to meet the due-dates of the respective jobs as closely as possible, and consequently minimize the sum of earliness and tardiness of jobs because earliness and tardiness of jobs greatly influence the performance of a schedule with respect to cost. In addition, a job incurs holding cost from the time of its arrival until its completion. Most studies on scheduling in such manufacturing systems assume unit earliness cost, unit tardiness cost and unit holding cost of a job. However, in reality such an assumption need not always hold and it is quite possible that there exist different costs of earliness, tardiness and holding for different jobs. In addition, most studies on job-shop scheduling assume that jobs are independent and that no assembly operations exist. The current study addresses the problem of scheduling in dynamic assembly job-shops (i.e. shops that manufacture multi-level jobs) with the consideration of jobs having different earliness, tardiness and holding costs. An attempt is made in this paper to present dispatching rules by incorporating the relative costs of earliness, tardiness and holding of jobs in the form of scalar weights. In the first phase of the study, relative costs (or weights for) earliness and tardiness of jobs are considered, and the dispatching rules are presented in order to minimize the sum of weighted earliness and weighted tardiness of jobs. In the second phase of the study, the objective considered is the minimization of the sum of weighted earliness, weighted tardiness and weighted flowtime of jobs, and the dispatching rules are presented by incorporating the relative costs of earliness, tardiness and flowtime of jobs. Simulation studies have been conducted separately for both phases of the current study, the performance of the scheduling rules have been observed independently, and the results of the simulation study have been reported. The proposed rules are found to be effective in minimizing the mean and maximum values of the measures of performance.     

Parallel machine earliness and tardiness scheduling with proportional weights

In this paper we study the problem of scheduling n jobs with a common due date and proportional early and tardy penalties on m identical parallel machines. We show that the problem is NP-hard and propose a dynamic programming algorithm to solve it. We also propose two heuristics to tackle the problem and analyze their worst-case error bounds.Scope and purposeScheduling problems to minimize the total weighted earliness and tardiness (WET) arise in Just-in-time manufacturing systems, where one of the objectives is to complete each job as close to its due date as possible. The earliness and tardiness weights of a job in WET tend to increase with the value of the job. Because processing time is often a good surrogate for the value of a job, it is reasonable to consider weights that are proportional to job processing times. In this paper we study the parallel identical machine WET problem with proportional weights. We propose both exact and approximation algorithms to tackle the problem.     

A note: Common due date assignment for a single machine scheduling with the rate-modifying activity

In this paper we study the single machine common due date assignment and scheduling problem with the possibility to perform a rate-modifying activity (RMA) for changing the processing times of the jobs following this activity. The objective is to minimize the total weighted sum of earliness, tardiness and due date costs. Placing the RMA to some position in the schedule can decrease the objective function value. Several properties of the problem are considered which in some cases can reduce the complexity of the solution algorithm.     

Soft Due Window Assignment and Scheduling on Parallel Machines

We study problems of scheduling jobs on identical parallel machines, in which a due window has to be assigned to each job. If a job is completed within its due window, then it incurs no scheduling cost. Otherwise, it incurs earliness or tardiness cost. Two due window models are considered. In both models, the due window size is a decision variable common for all jobs. In the first model, called a constant due window, the due window starting time is a decision variable common for all jobs, and in the second, called a slack due window, the due window starting time is equal to the job processing time plus a decision variable common for all jobs. The objective is to find a job schedule as well as the size and location(s) of the due window(s) such that a weighted maximum or sum of costs associated with job earliness, job tardiness, and due window size is minimized. We establish the properties of optimal solutions of these minmax and minsum problems. For a constant due window model, we prove that the minmax problem with arbitrary weights and the minsum problem with equal weights are polynomially equivalent to the classical parallel machine scheduling problem to minimize the makespan. We further show that the problems for a constant due window model and slack due window model with the same objective function are reversible in the sense that their optimal solutions are mirror images of each other. These results imply O(n) and O(n log n) time algorithms for the considered problems when m=1.     

交货期窗口下带有附加惩罚的单机提前/拖期调度问题

交货期窗口下的交货期确定和排序问题是调度领域研究的一个方面,本文对交货期口下的单机作业问题进行了研究,目标函数不仅考虑提前／拖期惩罚,还考虑附加惩罚,假设如果任务在交货期窗口内完工,则不受提前／拖期片罚;如果在交货期窗口外完工,将导致提前／拖期惩罚,本文确定了最优公共交货期,给出了相庆的最优排序,并提出了一个多项式时间算法确定了使目标函数为最小的最优调度,最后的数值例子说明了算法的有效性。     

Minimizing total earliness and tardiness through unrelated parallel machine scheduling using distributed release time control

Using unrelated parallel machine scheduling to minimize the total earliness and tardiness of jobs with distinct due dates is a nondeterministic polynomial-hard problem. Delayed customer orders may result in penalties and reduce customer satisfaction. On the other hand, early completion creates inventory storage costs, which increase the total cost. Although parallel machines can increase productivity, machine assignments also increase the complexity of production. Therefore, the challenge in parallel machine scheduling is to dynamically adjust the machine assignment to complete the job within the shortest possible time. In this paper, we address an unrelated parallel machine scheduling problem for jobs with distinct due dates and dedicated machines. The objective is to dynamically allocate jobs to unrelated parallel machines in order to minimize the total earliness and tardiness time. We formulate the problem as a mixed integer linear programming (MILP) model and develop a modified genetic algorithm (GA) with a distributed release time control (GARTC) mechanism to obtain the near-optimal solution. A preliminary computational study indicates that the developed GARTC not only provides good quality solutions within a reasonable amount of time, but also outperforms the MILP model, a classic GA and heuristic approaches described in the literature.     

Single machine common flow allowance scheduling with a rate-modifying activity

In this paper we consider single machine SLK due date assignment scheduling problem with a rate-modifying activity. In this model, the machine has a rate-modifying activity that can change the processing rate of machine under consideration. Hence the actual processing times of jobs vary depending on whether the job is scheduled before or after the rate-modifying activity. We need to make a decision on when to schedule the rate-modifying activity, the optimal common flow allowance and the sequence of jobs to minimize total earliness, tardiness and common flow allowance cost. We introduce an efficient (polynomial time) solution for this problem.     

Single-machine batch delivery scheduling with job release dates,due windows and earliness,tardiness, holding and delivery costs

This paper deals with a single-machine scheduling problem in which jobs are released in different points in time but delivered to customers in batches. A due window is associated with each job. The objective is to schedule the jobs, to form them into batches and to decide the delivery date of each batch so as to minimize the sum of earliness, tardiness, holding, and delivery costs. A mathematical model of the problem is presented, and a set of dominance properties is established. To solve this NP-hard problem efficiently, a solution method is then proposed by incorporating the dominance properties with an imperialist competitive algorithm. Unforced idleness and forming discontinuous batches are allowed in the proposed algorithm. Moreover, the delivery date of a batch may be decided to be later than the completion time of the last job in the batch. Finally, computational experiments are conducted to evaluate the proposed model and solution procedure, and results are discussed.     

Common due date assignment and scheduling with a rate-modifying activity to minimize the due date,earliness, tardiness,holding, and batch delivery cost

e consider a single-machine batch delivery scheduling and common due date assignment problem. In addition to making decisions on sequencing the jobs, determining the common due date, and scheduling job delivery, we consider the option of performing a rate-modifying activity on the machine. The processing time of a job scheduled after the rate-modifying activity decreases depending on a job-dependent factor. Finished jobs are delivered in batches. There is no capacity limit on each delivery batch, and the cost per batch delivery is fixed and independent of the number of jobs in the batch. The objective is to find a common due date for all the jobs, a location of the rate-modifying activity, and a delivery date for each job to minimize the sum of earliness, tardiness, holding, due date, and delivery cost. We provide some properties of the optimal schedule for the problem and present polynomial algorithms for some special cases.     

Common due date assignment and scheduling with ready times

We consider the problem of scheduling a set of nonsimultaneously available jobs on one machine. Each job has a ready time only at or after which the job can be processed. All the jobs have a common due date, which needs to be determined. The problem is to determine a due date and a schedule so as to minimize a total penalty depending on the earliness, tardiness and due date. We show that this problem is strongly NP-hard and give an efficient algorithm that finds an optimal due date and schedule when either the job sequence is predetermined or all jobs have the same processing time. We also propose three approximation algorithms for the general and special cases together with their experimental analysis.

Scope and purpose

We consider the single machine due date assignment problem for scheduling jobs which are ready for processing at different times. The problem under consideration arises in production planning and scheduling concerning the setting of appropriate due dates for a number of customer orders arriving over time. Most of the earlier publications on this subject assumed that the jobs are ready for processing simultaneously. This assumption is too restrictive for real-life production systems where jobs arrive at different times. We show that the problem with unequal ready times is NP-hard and develop fast heuristic algorithms for it, and exact algorithms for two special cases.     

Single‐machine common flow allowance scheduling with aging effect,resource allocation,and a rate‐modifying activity

In this paper, we consider scheduling of deteriorating jobs on a single machine with slack (SLK) due date assignment, resource allocation, and a rate‐modifying activity. The rate‐modifying activity can change jobs’ processing rates such that the actual processing time of a job depends on whether the job is processed before or after the rate‐modifying activity. In addition, the actual processing time of a job also depends on its position in a processing sequence (i.e., the aging effect) and the amount of resource allocated to it. The objective is to determine the optimal sequence, optimal common flow allowance, optimal resource allocation, and optimal location of the rate‐modifying activity to minimize a total penalty function comprising the earliness, tardiness, common flow allowance, and resource allocation costs. We consider two variants of the problem associated with two different processing time functions and provide a polynomial‐time algorithm to solve each variant.     

Group scheduling and job-dependent due window assignment based on a common flow allowance

We study a single-machine group scheduling and job-dependent due window assignment problem in which each job is assigned an individual due window based on a common flow allowance. In the group technology environment, the jobs are divided into groups in advance according to their processing similarities and all the jobs of the same group are processed consecutively in order to improve production efficiency. A sequence-independent machine setup time precedes the processing of the first job of each group. A job completed earlier (later) than its due window will incur an earliness (tardiness) penalty. Our goal is to find the optimal sequence for both the groups and jobs, together with the optimal due window assignment, to minimize the total cost that comprises the earliness and tardiness penalties, and the due window starting time and due window size costs. We give an O(n log n)time algorithm to solve this problem.     

考虑多个RMAs的单机调度问题

在处理时间不断恶化的情况下,针对插入多个机器维护阶段(RMAs)和考虑交货期安排的单机调度问题展开研究,目标是最小化提前和拖期惩罚。产品加工过程中,在处理工件之前插入多个RMAs可以降低恶化现象从而恢复机器的生产效率,目的是同时找到最优序列、最优松弛时间和RMAs的最优位置以使提前和拖期惩罚最小。根据问题的特点,提出了相关的性质和定理,通过证明得出了最优的松弛时间。最后,证明了该问题在多项式时间内是可解的。     

An Agent-Based Approach for Scheduling Multiple Machines

We present a new agent-based solution approach for the problem of scheduling multiple non-identical machines in the face of sequence dependent setups, job machine restrictions, batch size preferences, fixed costs of assigning jobs to machines and downstream considerations. We consider multiple objectives such as minimizing (weighted) earliness and tardiness, and minimizing job-machine assignment costs. We use an agent-based architecture called Asynchronous Team (A-Team), in which each agent encapsulates a different problem solving strategy and agents cooperate by exchanging results. Computational experiments on large instances of real-world scheduling problems show that the results obtained by this approach are significantly better than any single algorithm or the scheduler alone. This approach has been successfully implemented in an industrial scheduling system.     

Single-machine scheduling with a common due window

We study several single-machine non-preemptive scheduling problems to minimize the sum of weighted earliness–tardiness, weighted number of early and tardy jobs, common due window location, and flowtime penalties. We allow the due window location to be either a decision variable or a given parameter. We assume that the due window location has a tolerance and the window size is a given parameter. We further make the assumption that the ratios of the job processing times to the earliness–tardiness weights are agreeable for the first problem. We propose pseudo-polynomial dynamic programming algorithms to optimally solve the problems. We also provide polynomial time algorithms for several special cases.Scope and purpose The widespread use of Just-In-Time philosophy in manufacturing to eliminate inventories leads to a new class of scheduling problems in which the earliness and/or number of early jobs are penalized as well as the tardiness and/or tardy jobs. In this type of environments, the jobs are sometimes associated with a period of time within which they incur no penalty since the customers will generally allow a time interval for the delivery of the products. This time period is called a due window. There are a variety of applications with due windows in factory automation, production maintenance, and so on. In this paper, we consider the common due window problems to minimize the weighted earliness–tardiness, weighted number of early–tardy jobs and weighted flowtime on a single machine. The main contributions of this paper are identifying the computational complexity of the problems, developing dynamic programming algorithms to optimally solve them, and providing efficient and exact polynomial algorithms for the special cases.     

Fast neighborhood search for the single machine earliness–tardiness scheduling problem

This paper addresses the one machine scheduling problem in which n jobs have distinct due dates with earliness and tardiness costs. Fast neighborhoods are proposed for the problem. They are based on a block representation of the schedule. A timing operator is presented as well as swap and extract-and-reinsert neighborhoods. They are used in an iterated local search framework. Two types of perturbations are developed based, respectively, on random swaps and earliness and tardiness costs. Computational results show that very good solutions for instances with significantly more than 100 jobs can be derived in a few seconds.     

具有多个维修区间的单机调度问题

针对工件具有位置退化效应,机器具有多个维修区间的单机调度问题。工件的加工时间为位置相关的函数。每次机器维修后回到初始的水平。目标函数为总的提前费用,误工费用,共同交货期的窗时费用和开始时间费用。对于共同交货期分为包括维修区间和不包括维修区间两种情形进行讨论,采用线性规划建立指派问题的数学模型,并分别提出最优序列的一些最优性质和相应的多项式时间算法。     

A note to due-window assignment and single machine scheduling with deteriorating jobs and a rate-modifying activity

This note considers a single machine scheduling and due-window assignment problem, in which the processing time of a job is a linear function of its starting time and the job-independent deterioration rates are identical for all jobs. We allow an option for performing a rate-modifying activity for changing the normal processing times of the jobs following this activity. The objective is to schedule the jobs, the due-window and the rate-modifying activity so as to minimize the sum of earliness, tardiness and due-window starting time and due-window size costs. We introduce a polynomial solution for the problem.     

Single-machine batch delivery scheduling with an assignable common due date and controllable processing times

We consider single-machine batch delivery scheduling with an assignable common due date and controllable processing times, which vary as a convex function of the amounts of a continuously divisible common resource allocated to individual jobs. Finished jobs are delivered in batches and there is no capacity limit on each delivery batch. We first provide an O(n⁵) dynamic programming algorithm to find the optimal job sequence, the partition of the job sequence into batches, the assigned common due date, and the resource allocation that minimize a cost function based on earliness, tardiness, job holding, due date assignment, batch delivery, and resource consumption. We show that a special case of the problem can be solved by a lower-order polynomial algorithm. We then study the problem of finding the optimal solution to minimize the total cost of earliness, tardiness, job holding, and due date assignment, subject to limited resource availability, and develop an O(nlog n) algorithm to solve it.