Single-machine batch scheduling minimizing weighted flow times and delivery costs

This paper addresses scheduling a set of jobs on a single machine for delivery in batches to one customer or to another machine for further processing. The problem is a natural extension of that of minimising the sum of weighted flow times, considering the possibility of delivering jobs in batches and introducing batch delivery costs. The scheduling objective adopted is that of minimising the sum of weighted flow times and delivery costs. The extended problem arises in the context of coordination between machine scheduling and a distribution system in a supply chain network. Structural properties of the problem are investigated and used to devise a branch-and-bound solution method. For the special case, when the maximum number of batches is fixed, the branch-and-bound scheme provided shows significant improvements over an existing dynamic-programming algorithm.     

Coordinating multi-location production and customer delivery

We study two parallel machine scheduling problems with equal processing time jobs and delivery times and costs. The jobs are processed on machines which are located at different sites, and delivered to a customer by a single vehicle. The first objective considered is minimizing the sum of total weighted completion time and total vehicle delivery costs. The second objective considered is minimizing the sum of total tardiness and total vehicle delivery costs. We develop several interesting properties of an optimal scheduling and delivery policy, and show that both problems can be solved by reduction to the Shortest-Path problem in a corresponding network. The overall computational effort of both algorithms is O(n ^m2+m+1) (where n and m are the number of jobs and the number of machines, respectively) by the application of the Directed Acyclic Graph (DAG) method. We also discuss several special cases for which the overall computational effort can be significantly reduced.     

具有可变配送费用和固定配送时刻的单机排序问题

研究了单机环境下生产与配送的协同排序问题.有多个工件需要在一台机器上进行加工,加工完的工件需要分批配送到一个客户.每批工件只能在固定的几个配送时刻出发,不同的配送时刻对应着不同的配送费用.我们的目标是找到生产与配送的协同排序,极小化排序的时间费用与配送费用的加权和.研究了排序理论中主要的四个目标函数,构建了单机情况下的具体模型,分析了问题的复杂性,对于配送费用单调非增的情况给出了它们的最优算法.     

A Branch and Bound Algorithm to Minimize the Total Weighed Number of Tardy Jobs and Delivery Costs

This paper addresses the production and delivery scheduling integration problem; a manufacturer receives ₋ orders from one customer while the orders need to be processed on one or two machines and be sent to the customer in batches. Sending several jobs in batches will reduce the transportation cost but it may increase the number of tardy jobs. The objective is to minimize the sum of the total weighted number of tardy jobs and the delivery costs. The structural properties of the problem for a single machine and special cases of the two-machine flow shop problem are investigated and used to set up a new branch and bound algorithm. A heuristic algorithm for upper bound calculation and two approaches for lower bound calculation are also introduced. Results of computational tests show significant improvement over an existing dynamic programming method.     

Minimizing the weighted number of tardy jobs with due date assignment and capacity-constrained deliveries for multiple customers in supply chains

In this paper, an integrated due date assignment and production and batch delivery scheduling problem for make-to-order production system and multiple customers is addressed. Consider a supply chain scheduling problem in which n orders (jobs) have to be scheduled on a single machine and delivered to K customers or to other machines for further processing in batches. A common due date is assigned to all the jobs of each customer and the number of jobs in delivery batches is constrained by the batch size. The objective is to minimize the sum of the total weighted number of tardy jobs, the total due date assignment costs and the total batch delivery costs. The problem is NP-hard. We formulate the problem as an Integer Programming (IP) model. Also, in this paper, a Heuristic Algorithm (HA) and a Branch and Bound (B&B) method for solving this problem are presented. Computational tests are used to demonstrate the efficiency of the developed methods.     

Batch delivery scheduling with batch delivery cost on a single machine

We consider a scheduling problem in which n independent and simultaneously available jobs are to be processed on a single machine. The jobs are delivered in batches and the delivery date of a batch equals the completion time of the last job in the batch. The delivery cost depends on the number of deliveries. The objective is to minimize the sum of the total weighted flow time and delivery cost. We first show that the problem is strongly NP-hard. Then we show that, if the number of batches is B, the problem remains strongly NP-hard when B ? U for a variable U ? 2 or B ? U for any constant U ? 2. For the case of B ? U, we present a dynamic programming algorithm that runs in pseudo-polynomial time for any constant U ? 2. Furthermore, optimal algorithms are provided for two special cases: (i) jobs have a linear precedence constraint, and (ii) jobs satisfy the agreeable ratio assumption, which is valid, for example, when all the weights or all the processing times are equal.     

Scheduling parallel machines with single vehicle delivery

We investigate the integrated production and distribution scheduling problem in a supply chain. The manufacturer’s production environment is modeled as a parallel machine system. A single capacitated vehicle is employed to deliver products in batches to multiple customers. The scheduling problem can also be viewed as either parallel machines with delivery considerations or a flexible flowshop. Different inventory holding costs, job sizes (volume or storage space required in the transportation unit), and job priorities are taken into account. Efficient mathematical modeling and near-optimal heuristic approaches are presented for minimizing total weighted completion time.     

The Coordination of Scheduling and Batch Deliveries

This paper considers several scheduling problems where deliveries are made in batches with each batch delivered to the customer in a single shipment. Various scheduling costs, which are based on the delivery times of the jobs, are considered. The objective is to minimize the scheduling cost plus the delivery cost, and both single and parallel machine environments are considered. For many combinations of these, we either provide efficient algorithms that minimize total cost or show that the problem is intractable. Our work has implications for the coordination of scheduling with batch delivery decisions to improve customer service.     

On-line supply chain scheduling problems with preemption

We consider supply chain scheduling problems where customers release jobs to a manufacturer that has to process the jobs and deliver them to the customers. The jobs are released on-line, that is, at any time there is no information on the number, release and processing times of future jobs; the processing time of a job becomes known when the job is released. Preemption is allowed. To reduce the total costs, processed jobs are grouped into batches, which are delivered to customers as single shipments; we assume that the cost of delivering a batch does not depend on the number of jobs in the batch. The objective is to minimize the total cost, which is the sum of the total flow time and the total delivery cost. For the single-customer problem, we present an on-line two-competitive algorithm, and show that no other on-line algorithm can have a better competitive ratio. We also consider an extension of the algorithm for the case of m customers, and show that its competitive ratio is not greater than 2m if the delivery costs to different customers are equal.     

具有多个不可用时间段的单机供应链排序问题

在单机供应链排序问题中, 机器会有多个长度确定的不可用时间段,它仅可以在可用时间段内加工工件,且每个可用时间段的长度不大于给定的常数.多个完工工件可组成一批由一个容量无限制的运输工具发送给客户.问题的目标是如何 安排工件的加工、发送以及不可用时间段,以使总发送时间与总发送费用之和达到最小. 对于工件加工可恢复的情况,可在多项式时间 O(n^2) 内得到最优序. 对于工件加工不可恢复的情况,证明了问题是强NP-难的, 并提出了~2-近似算法.     

Scheduling with job delivery coordination on single machine

This paper investigates a single machine scheduling problem with job delivery coordination, in which each job demands different amount of storage space during transportation. In this problem, a set of independent jobs from a customer must first be processed on a machine without preemption and then delivered by two homogeneous vehicles to the customer in batches. To minimize the makespan, we develop a best possible polynomial-time heuristic with a worst-case ratio of 2.     

带有拒绝的单机和同型机排序问题

研究了带有拒绝的单机和同型机排序问题. 对于单机情形, 工件的惩罚费用是对应加工时间的\alpha倍.如果工件有到达时间, 目标为最小化时间表长与惩罚费用之和, 证明了这个问题是可解的.如果所有工件在零时刻到达, 目标为最小化总完工时间与惩罚费用之和, 也证明了该问题是可解的.对于同型机排序问题, 研究了工件分两批在线实时到达的情形, 目标为最小化时间表长与惩罚费用之和.针对机器台数2和m, 分别给出了竞争比为2和4-2/m的在线算法.     

Minimizing the weighted number of tardy jobs with due date assignment and capacity-constrained deliveries

We study a supply chain scheduling problem, where a common due date is assigned to all jobs and the number of jobs in delivery batches is constrained by the batch size. Our goal is to minimize the sum of the weighted number of tardy jobs, the due-date-assignment costs and the batch-delivery costs. We show that some well-known NP\mathcal{NP}-hard problems reduce to our problem. Then we propose a pseudo-polynomial algorithm for the problem, establishing that it is NP\mathcal{NP}-hard only in the ordinary sense. Finally, we convert the algorithm into an efficient fully polynomial time approximation scheme.     

带有分批费用的容量有界的单机平行分批排序问题

考虑的问题是在添加工资费用或包装费用等附加的分批费用下,如何使单机平行分批中总完工时间和分批费用之和达到最小.首先我们假定工件和批处理机都在零时刻到达,工件被成批地进行加工,一旦开始加工就不允许中断,每批的加工时间等于该批中最大的加工时间,而且假设每分一批都产生一个分批费用.然后对具有m个不同的加工时间,批容量有界且为固定值b的情形下目标函数为∑C_j与分批费用之和这一排序问题,利用动态规划的方法给出了多项式时间算法,时间界为O(b²m2m²222^m).     

Batch scheduling and common due-date assignment on a single machine

We consider the problem of scheduling n groups of jobs on a single machine where three types of decisions are combined: scheduling, batching and due-date assignment. Each group includes identical jobs and may be split into batches; jobs within each batch are processed jointly. A sequence independent machine set-up time is needed between each two consecutively scheduled batches of different groups. A due-date common to all jobs has to be assigned. A schedule specifies the size of each batch, i.e. the number of jobs it contains, and a processing order for the batches. The objective is to determine a value for the common due-date and a schedule so as to minimize the sum of the due date assignment penalty and the weighted number of tardy jobs. Several special cases of this problem are shown to be ordinary NP-hard. Some cases are solved in O(n log n) time. Two pseudopolynomial dynamic programming algorithms are presented for the general problem, as well as a fully polynomial approximation scheme.     

Single Machine Scheduling with Major and Minor Setup Times: A Tabu Search Approach

The paper deals with a single machine scheduling problem in which jobs are grouped into families that require major setup times, and where jobs within families require for processing minor setup times. The former are sequence independent and the latter have special triangular structure. The problem is to find a partition of families into batches, sequences of jobs in particular batches and a sequence of batches which minimize a general regular cost function. The tabu search algorithm for finding near-optimal schedules is proposed and results of computational experiments are presented.     

Flow shop scheduling algorithms for minimizing the completion time variance and the sum of squares of completion time deviations from a common due date

We consider two problems of m-machine flow shop scheduling in this paper: one, with the objective of minimizing the variance of completion times of jobs, and the other with the objective of minimizing the sum of squares of deviations of job completion times from a common due date. Lower bounds on the sum of squares of deviations of job completion times from the mean completion time of jobs for a given partial sequence are first presented. Using these lower bounds, a branch and bound algorithm based on breadth-first search procedure for scheduling n jobs on m-machines with the objective of minimizing completion time variance (CTV) is developed to obtain the best permutation sequence. We also present two lower bounds and thereafter, a branch and bound algorithm with the objective of minimizing the sum of squares of deviations of job completion times from a given common due date (called the MSD problem). The computational experience with the working of the two proposed branch and bound algorithms is also reported. Two heuristics, one for each of the two problems, are developed. The computational experience on the evaluation of the heuristics is discussed.     

Lot Streaming in a Two-stage Flow Shop with Set-up,Processing and Removal Times Separated

This paper considers the job (lot) scheduling problem for two-stage flow shops in which the movement of transfer batches (sublots) from the first stage to the next are allowed. Set-up, processing and removal times are considered as separable and independent of the order in which jobs are processed at any of two stages. An optimal transfer batch sizing and scheduling algorithm which has an objective of minimizing the maximum flow time (makespan) is developed and demonstrated by a numerical example.     

Batch Delivery Scheduling on a Single Machine

The problem of partitioning a set of independent and simultaneously available jobs into batches and sequencing them for processing on a single machine is presented. Jobs in the same batch are to be delivered together, upon completion of the last job in the batch. Jobs finished before this time have to wait until delivery. There are a delivery cost depending on the number of batches formed and an earliness cost for jobs finished before delivery. The dynamic programming approach to minimizing the total cost is considered, yielding two pseudopolynomial algorithms when the number of batches has a fixed upper bound. A polynomial algorithm for a special case of the problem is also presented.     

An iterative approach for the serial batching problem with parallel machines and job families

In this paper, we consider a parallel machine scheduling problem to minimize the total completion time. Each job belongs to a certain family. All jobs of one family have identical processing times. Major setups occur between jobs of different families, and we include sequence dependencies. Batches of jobs belonging to the same family can be formed to avoid these setups. Furthermore, we assume serial batching and batch availability. Therefore, the processing time of a batch is the sum of the processing times of all jobs grouped into the corresponding batch. An iterative method is developed for solving this specific problem. This approach alternates between varying batch sizes using an efficient heuristic and sequencing batches based on variable neighborhood search (VNS). Computational results demonstrate that the iterative heuristic outperforms heuristics based on a fixed batch size and list scheduling.