A NOTE ON “COMMON DUE WINDOW SCHEDULING”

Kramer and Lee recently addressed a common due window scheduling problem with earliness and tardiness penalties, where earliness and tardiness penalty factors are constant and the common window size is given. They showed that the problem is polynomial when the location of the due window is a decision variable. For the case where the location of the due window is given, the problem is also polynomial when the latest due date is greater than or equal to the makespan, and they proposed a pseudopolynomial dynamic programming algorithm to find an optimal schedule when the latest due date is less than the makespan. In this note we address the problem for the case where the location of the due window is given. Specifically, we show that the problem is polynomial if the window location is unrestricted, and present a more efficient dynamic program algorithm to optimally solve the problem if the window location is restricted. The concepts of unrestricted and restricted window locations are defined in this note.     

A FRAMEWORK FOR MACHINE SCHEDULING PROBLEMS WITH CONTROLLABLE PROCESSING TIMES

This is a study of single and parallel machine scheduling problems with controllable processing time for each job. The processing time for job j depends on the position of the job in the schedule and is a function of the number of resource units allocated to its processing. Processing time functions and processing cost functions are allowed to be nonlinear. The scheduling problems considered here have important applications in industry and include many of the existing scheduling models as special cases. For the single machine problem, the objective is minimization of total compression costs plus a scheduling measure. The scheduling measures include makespan, total flow time, total differences in completion times, total differences in waiting times, and total earliness and tardiness with a common due date for all jobs. Except when the total earliness and tardiness measure is involved, each case the problem is solved efficiently. Under an assumption typically satisfied in just-in-time systems, the problem with total earliness and tardiness measure is also solved efficiently. Finally, for a large class of processing time functions; parallel machine problems with total flow time and total earliness and tardiness measures are solved efficiently. In each case we reduce the problem to a transportation problem.     

Single-machine batch delivery scheduling with an assignable common due window

This paper addresses a batch delivery single-machine scheduling problem in which jobs have an assignable common due window. Each job will incur an early (tardy) penalty if it is early (tardy) with respect to the common due window under a given schedule. 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 the optimal size and location of the window, the optimal dispatch date for each job, as well as an optimal job sequence to minimize a cost function based on earliness, tardiness, holding time, window location, window size, and batch delivery. We show that the problem can be optimally solved in O(n⁸)$O(n^8)$ time by a dynamic programming algorithm under a reasonable assumption on the relationships among the cost parameters. A computational experiment is also conducted to evaluate the performance of the proposed algorithm. We also show that some special cases of the problem can be optimally solved by lower order algorithms.     

Single-machine scheduling with periodic due dates to minimize the total earliness and tardy penalty

We consider a single-machine scheduling problem such that the due dates are assigned to each job depending on its order, and the lengths of the intervals between consecutive due dates are identical. The objective is to minimize the total penalty for the earliness and tardiness of each job. The early penalty proportionally increases according to the earliness amount, while the tardy penalty increases according to the step function. We show that the problem is strongly NP-hard, and furthermore, polynomially solvable if the two types of processing times exist.

     

Experimental earliness/tardiness production planning with the due window system

In order to use the philosophy of JIT to improve the production planning method of MRP-II, we propose the experimental software system of the earliness/tardiness produc tion planning problem with due window. By means of the approaches and model reported in this paper, the optimal production planning can be achieved. The recommended model extends the problem of due window from the shop scheduling level into the aggregated planning level of mass manufacturing systems. Simulation results have demonstrated that the experimental software is a useful tool for the production management of repetitive manufacturing enterprises.     

混合离散差分进化算法在单机批处理调度中的应用

本文研究单机批处理调度问题,批处理机有批次容量限制,批处理时间由每个批次所含作业中的最长作业处理时间决定。每个作业具有不同的大小、处理时间、提前拖期惩罚权重,所有作业具有公共交货期,且交货期无限晚。目标函数为最小化所有作业的加权提前拖期惩罚之和。该问题已被证明为NP难题,本研究找到了其最优解具有的一些性质,在此基础上利用它们提出了一种动态规划(DP)与差分进化(DE)算法相结合的混合离散差分进化(HDDE)算法来求解该问题,通过与传统的遗传算法、模拟退火算法和迭代贪婪算法进行对比,HDDE算法显示了更加强大的全局搜索能力。     

Optimal Policy for a Stochastic Scheduling Problem with Applications to Surgical Scheduling

We consider the stochastic, single‐machine earliness/tardiness problem (SET), with the sequence of processing of the jobs and their due‐dates as decisions and the objective of minimizing the sum of the expected earliness and tardiness costs over all the jobs. In a recent paper, Baker ( 2014 ) shows the optimality of the Shortest‐Variance‐First (SVF) rule under the following two assumptions: (a) The processing duration of each job follows a normal distribution. (b) The earliness and tardiness cost parameters are the same for all the jobs. In this study, we consider problem SET under assumption (b). We generalize Baker's result by establishing the optimality of the SVF rule for more general distributions of the processing durations and a more general objective function. Specifically, we show that the SVF rule is optimal under the assumption of dilation ordering of the processing durations. Since convex ordering implies dilation ordering (under finite means), the SVF sequence is also optimal under convex ordering of the processing durations. We also study the effect of variability of the processing durations of the jobs on the optimal cost. An application of problem SET in surgical scheduling is discussed.     

TECHNICAL NOTE: A SIMPLE MODEL FOR OPTIMIZING THE SINGLE MACHINE EARLY/TARDY PROBLEM WITH SEQUENCE-DEPENDENT SETUPS

A simple mixed integer programming model for the N job/single machine scheduling problem with possibly sequence-dependent setup times, differing earliness/tardiness cost penalties, and variable due dates is proposed and evaluated for computational efficiency. Results indicated that the computational effort required to reach optimality rose with the number of jobs to be scheduled and with decreased variance in due dates. Though computational effort was significant for the largest problems solved, the model remained viable for optimizing research scale problems.     

Minmax common flow-allowance problems with convex resource allocation and position-dependent workloads

We study minmax due-date based on common flow-allowance assignment and scheduling problems on a single machine, and extend known results in scheduling theory by considering convex resource allocation. The total cost function of a given job consists of its earliness, tardiness and flow-allowance cost components. Thus, the common flow-allowance and the actual jobs’ processing times are decision variables, implying that the due-dates and actual processing times can be controlled by allocating additional resource to the job operations. Consequently, our goal is to optimize a cost function by seeking the optimal job sequence, the optimal job-dependent due-dates along with the actual processing times. In all addressed problems we aim to minimize the maximal cost among all the jobs subject to a constraint on the resource consumption. We start by analyzing and solving the problem with position-independent workloads and then proceed to position-dependent workloads. Finally, the results are generalized to the method of common due-window. For all studied problems closed form solutions are provided, leading to polynomial time solutions.

     

A bicriteria approach to scheduling a single machine with job rejection and positional penalties

Single machine scheduling problems have been extensively studied in the literature under the assumption that all jobs have to be processed. However, in many practical cases, one may wish to reject the processing of some jobs in the shop, which results in a rejection cost. A solution for a scheduling problem with rejection is given by partitioning the jobs into a set of accepted and a set of rejected jobs, and by scheduling the set of accepted jobs among the machines. The quality of a solution is measured by two criteria: a scheduling criterion, F1, which is dependent on the completion times of the accepted jobs, and the total rejection cost, F2. Problems of scheduling with rejection have been previously studied, but usually within a narrow framework—focusing on one scheduling criterion at a time. This paper provides a robust unified bicriteria analysis of a large set of single machine problems sharing a common property, namely, all problems can be represented by or reduced to a scheduling problem with a scheduling criterion which includes positional penalties. Among these problems are the minimization of the makespan, the sum of completion times, the sum and variation of completion times, and the total earliness plus tardiness costs where the due dates are assignable. Four different problem variations for dealing with the two criteria are studied. The variation of minimizing F1+F2 is shown to be solvable in polynomial time, while all other three variations are shown to be $\mathcal{NP}$ -hard. For those hard problems we provide a pseudo polynomial time algorithm. An FPTAS for obtaining an approximate efficient schedule is provided as well. In addition, we present some interesting special cases which are solvable in polynomial time.     

A tardiness-augmented approximation scheme for rejection-allowed multiprocessor rescheduling

In the multiprocessor scheduling problem to minimize the total job completion time, an optimal schedule can be obtained by the shortest processing time rule and the completion time of each job in the schedule can be used as a guarantee for scheduling revenue. However, in practice, some jobs will not arrive at the beginning of the schedule but are delayed and their delayed arrival times are given to the decision-maker for possible rescheduling. The decision-maker can choose to reject some jobs in order to minimize the total operational cost that includes three cost components: the total rejection cost of the rejected jobs, the total completion time of the accepted jobs, and the penalty on the maximum tardiness for the accepted jobs, for which their completion times in the planned schedule are their virtual due dates. This novel rescheduling problem generalizes several classic NP-hard scheduling problems. We first design a pseudo-polynomial time dynamic programming exact algorithm and then, when the tardiness can be unbounded, we develop it into a fully polynomial time approximation scheme. The dynamic programming exact algorithm has a space complexity too high for truthful implementation; we propose an alternative to integrate the enumeration and the dynamic programming recurrences, followed by a depth-first-search walk in the reschedule space. We implemented the alternative exact algorithm in C and conducted numerical experiments to demonstrate its promising performance.

     

MINIMIZING ABSOLUTE AND SQUARED DEVIATION OF COMPLETION TIMES FROM DUE DATES

This is a study of a single-machine scheduling problem with the objective of minimizing the sum of a function of earliness and tardiness called the earliness and tardiness (ET) problem. I will show that if priority weights of jobs are proportional to their processing times, and if earliness and tardiness cost functions are linear, the problem will be equivalent to the total weighted tardiness problem. This proves that the et problem is np -hard. In addition, I present a heuristic algorithm with worst case bound for the et problem based on the equivalence relation between the two. When earliness and tardiness cost functions are quadratic, I consider the problem for a common due date for all jobs and for different job due dates. In general, the et problem with quadratic earliness and tardiness cost functions and all job weights equal to one is np -hard. I show that in many cases, when weights of jobs are proportional to their processing times, the problem can be solved efficiently. In the published results on the et problem with quadratic earliness and tardiness cost functions other researchers have assumed a zero starting time for the schedule. I discuss the advantages of a nonzero starting time for the schedule.     

Single machine scheduling: A comparison of two solution procedures

Recent research on the single machine scheduling problem has focused on the treatment of multiple scheduling objectives. Most works have used some combination of mean flowtime, maximum tardiness, or total tardiness as scheduling criteria. Previous research has largely ignored earliness as a scheduling criterion. This paper presents a model that employs the criteria of flowtime as a measure of work-in-process (WIP) inventory and total job earliness to represent finished goods inventory. Total tardiness is used to represent customer satisfaction. The three criteria are used to form a single, weighted-sum objective function for guiding the choice of the best processing sequence. Two procedures are presented that might be used to solve this problem. The first is an enumeration scheme using bounding and dominance criteria that have been developed to aid efficient solution, and the second is a mixed integer linear programming (LP) formulation. Computational experience with the two models is also presented.     

Approximation algorithm for the parallel-machine scheduling problem with release dates and submodular rejection penalties

In this paper, we consider the parallel-machine scheduling problem with release dates and submodular rejection penalties. In this problem, we are given m identical parallel machines and n jobs. Each job has a processing time and a release date. A job is either rejected, in which case a rejection penalty has to be paid, or accepted and processed on one of the m identical parallel machines. The objective is to minimize the sum of the makespan of the accepted jobs and the rejection penalty of the rejected jobs which is determined by a submodular function. Our main work is to design a 2-approximation algorithm based on the primal-dual framework.

     

Single facility scheduling with dual criteria: minimizing maximum tardiness subject to minimum number of tardy jobs

This paper considers the static single machine scheduling problem with the objective of minimizing the maximum tardiness of any job subject to the constraint that the total number of lardy jobs is minimum. Based on simple dominance conditions an o(n²) heuristic algorithm is proposed to find an approximate solution to this problem. The effectiveness of the proposed heuristic algorithm is empirically evaluated by solving a large number of problems and comparing them to the optimal solutions obtained through the branch and bound algorithm.     

Discrete parallel machine makespan ScheLoc problem

Scheduling–Location (ScheLoc) problems integrate the separate fields of scheduling and location problems. In ScheLoc problems the objective is to find locations for the machines and a schedule for each machine subject to some production and location constraints such that some scheduling objective is minimized. In this paper we consider the discrete parallel machine makespan ScheLoc problem where the set of possible machine locations is discrete and a set of n jobs has to be taken to the machines and processed such that the makespan is minimized. Since the separate location and scheduling problem are both \(\mathcal {NP}\)-hard, so is the corresponding ScheLoc problem. Therefore, we propose an integer programming formulation and different versions of clustering heuristics, where jobs are split into clusters and each cluster is assigned to one of the possible machine locations. Since the IP formulation can only be solved for small scale instances we propose several lower bounds to measure the quality of the clustering heuristics. Extensive computational tests show the efficiency of the heuristics.     

带工期指派的产品服务系统订单随机调度问题研究

针对由一个制造工厂和多个区域服务中心组成的服务型制造企业,研究了考虑生产时间和服务时间均具有随机性且工期可指派的产品服务系统（PSS）订单调度问题。首先以最小化订单提前、误工和工期指派费用的期望总额为目标构建问题的优化模型,然后分析目标函数近似值的最优性条件,据此提出加权最短平均生产时间排序规则,并结合该规则与插入邻域局部搜索设计了启发式算法对问题进行求解,最后通过数值仿真验证算法的可行性和有效性。研究表明,提前费用偏差对PSS订单调度与工期指派决策的影响很小,因此企业管理者无需准确估计库存费用也能制定出比较有效的PSS订单调度策略;而工期指派费用偏差对决策结果的影响非常大,因此企业管理者在决策时必须谨慎估计该项费用。     

Particle swarm optimization based-algorithms to solve the two-machine cross-docking flow shop problem: just in time scheduling

Cross-docking is an innovative logistical strategy which provides less inventory holding costs, less transportation costs and fast customer deliveries without storage in between or less than 24 hours. In this paper, we address the two-machine cross-docking scheduling problem within a Just-In-Time (JIT) context. This latter requires the punctuality and exactness of product deliveries. To satisfy this target, we aim to minimize the total earliness and tardiness, then early or tardy deliveries are discouraged. This study presents a great contribution in solving such NP-hard problem while applying different versions of the PSO (Particle Swarm Optimization) algorithm. One of them is hybridized with the Genetic Algorithm (GA). This latter is then shown to be the best one over computational experiments using different sized instances and by determining a percentage deviation from a developed lower bound.

     

The web proxy location problem in general tree of rings networks

The Web proxy location problem in general networks is an NP-hard problem. In this paper, we study the problem in networks showing a general tree of rings topology. We improve the results of the tree case in literature and get an exact algorithm with time complexity O(nhk), where n is the number of nodes in the tree, h is the height of the tree (the server is in the root of the tree), and k is the number of web proxies to be placed in the net. For the case of networks with a general tree of rings topology we present an exact algorithm with O(kn ²) time complexity.This research has been supported by NSF of China (No. 10371028) and the Educational Department grant of Zhejiang Province (No. 20030622).     

Scheduling with release times and rejection on two parallel machines

In this paper we study scheduling with release times and job rejection on two parallel machines. In our scheduling model each job is either accepted and then processed by one of the two machines at or after its release time, or it is rejected and then a rejection penalty is paid. The objective is to minimize the makespan of the accepted job plus the total penalty of all rejected jobs. The scheduling problem is NP-hard in the ordinary sense. In this paper, we develop a \(1.5+\epsilon \)-approximation algorithm for the problem, where \(\epsilon \) is any given small positive constant.