Zhi-Long Chen

Integrated Production and Outbound Distribution Scheduling: Review and Extensions

In many applications involving make-to-order or time-sensitive (e.g., perishable, seasonal) products, finished orders are often delivered to customers immediately or shortly after the production. Consequently, there is little or no finished product inventory in the supply chain such that production and outbound distribution are very intimately linked and must be scheduled jointly to achieve a desired on-time delivery performance at minimum total cost. Research on integrated scheduling models of production and outbound distribution is relatively recent but is growing very rapidly. In this paper, we provide a survey of such existing models. We present a unified model representation scheme, classify existing models into several different classes, and for each class of the models give an overview of the optimality properties, computational tractability, and solution algorithms for the various problems studied in the literature. We clarify the tractability of some open problems left in the literature and some new problems by providing intractability proofs or polynomial-time exact algorithms. We also identify several problem areas and issues for future research.

Integrated Scheduling of Production and Distribution Operations

Motivated by applications in the computer and food catering service industries, we study an integrated scheduling model of production and distribution operations. In this model, a set of jobs (i.e., customer orders) are first processed in a processing facility (e.g., manufacturing plant or service center) and then delivered to the customers directly without intermediate inventory. The problem is to find a joint schedule of production and distribution such that an objective function that takes into account both customer service level and total distribution cost is optimized. Customer service level is measured by a function of the times when the jobs are delivered to the customers. The distribution cost of a delivery shipment consists of a fixed charge and a variable cost proportional to the total distance of the route taken by the shipment. We study two classes of problems under this integrated scheduling model. In the first class of problems, customer service is measured by the average time when the jobs are delivered to the customers; in the second class, customer service is measured by the maximum time when the jobs are delivered to the customers. Two machine configurations in the processing facility--single machine and parallel machine--are considered. For each of the problems studied, we provide an efficient exact algorithm, or a proof of intractability accompanied by a heuristic algorithm with worst-case and asymptotic performance analysis. Computational experiments demonstrate that the heuristics developed are capable of generating near-optimal solutions. We also investigate the possible benefit of using the proposed integrated model relative to a sequential model where production and distribution operations are scheduled sequentially and separately. Computational tests show that in many cases a significant benefit can be achieved by integration.

Machine scheduling with transportation considerations

In most manufacturing and distribution systems, semi-finished jobs are transferred from one processing facility to another by transporters such as automated guided vehicles (AGVs) and conveyors, and finished jobs are delivered to customers or warehouses by vehicles such as trucks. Most machine scheduling models assume either that there are an infinite number of transporters for delivering jobs or that jobs are delivered instantaneously from one location to another without transportation time involved. In this paper, we study machine scheduling problems with explicit transportation considerations. Models are considered for two types of transportation situations. The first situation involves transporting a semi-finished job from one machine to another for further processing. The second appears in the environment of delivering a finished job to the customer or warehouse. Both transportation capacity and transportation times are explicitly taken into account in our models. We study this class of scheduling problems by analysing their complexity. We show that many problems are computationally difficult and propose polynomial or pseudo-polynomial algorithms for some problems. Copyright

Solving a Practical Pickup and Delivery Problem

We consider a pickup and delivery vehicle routing problem commonly encountered in real-world logistics operations. The problem involves a set of practical complications that have received little attention in the vehicle routing literature. In this problem, there are multiple carriers and multiple vehicle types available to cover a set of pickup and delivery orders, each of which has multiple pickup time windows and multiple delivery time windows. Orders and carrier/vehicle types must satisfy a set of compatibility constraints that specify which orders cannot be covered by which carrier/vehicle types and which orders cannot be shipped together. Order loading and unloading sequence must satisfy the nested precedence constraint that requires that an order cannot be unloaded until all the orders loaded into the truck later than this order are unloaded. Each vehicle trip must satisfy the drivers work rules prescribed by the Department of Transportation which specify legal working hours of a driver. The cost of a trip is determined by several factors including a fixed charge, total mileage, total waiting time, and total layover time of the driver. We propose column generation based solution approaches to this complex problem. The problem is formulated as a set partitioning type formulation containing an exponential number of columns. We apply the standard column generation procedure to solve the linear relaxation of this set partitioning type formulation in which the resulting master problem is a linear program and solved very efficiently by an LP solver, while the resulting subproblems are computationally intractable and solved by fast heuristics. An integer solution is obtained by using an IP solver to solve a restricted version of the original set partitioning type formulation that only contains the columns generated in solving the linear relaxation. The approaches are evaluated based on lower bounds obtained by solving the linear relaxation to optimality by using an exact dynamic programming algorithm to solve the subproblems exactly. It is shown that the approaches are capable of generating near-optimal solutions quickly for randomly generated instances with up to 200 orders. For larger randomly generated instances with up to 500 orders, it is shown that computational times required by these approaches are acceptable.

Scheduling jobs and maintenance activities on parallel machines

Most machine scheduling models assume that the machines are available all of the time. However, in most realistic situations, machines need to be maintained and hence may become unavailable during certain periods. In this paper, we study the problem of processing a set of n jobs on m parallel machines where each machine must be maintained once during the planning horizon. Our objective is to schedule jobs and maintenance activities so that the total weighted completion time of jobs is minimized. Two cases are studied in this paper. In the first case, there are sufficient resources so that different machines can be maintained simultaneously if necessary. In the second case, only one machine can be maintained at any given time. In this paper, we first show that, even when all jobs have the same weight, both cases of the problem are NP-hard. We then propose branch and bound algorithms based on the column generation approach for solving both cases of the problem. Our algorithms are capable of optimally solving medium sized problems within a reasonable computational time. We note that the general problem where at most j machines, 1 ≤ j ≤ m, can be maintained simultaneously, can be solved similarly by the column generation approach proposed in this paper.

Solving Parallel Machine Scheduling Problems by Column Generation

We consider a class of problems of scheduling n jobs on m identical, uniform, or unrelated parallel machines with an objective of minimizing an additive criterion. We propose a decomposition approach for solving these problems exactly. The decomposition approach first formulates these problems as an integer program, and then reformulates the integer program, using Dantzig-Wolfe decomposition, as a set partitioning problem. Based on this set partitioning formulation, branch-and-bound exact solution algorithms can be designed for these problems. In such a branch-and-bound tree, each node is the linear relaxation problem of a set partitioning problem. This linear relaxation problem is solved by a column generation approach where each column represents a schedule on one machine and is generated by solving a single machine subproblem. Branching is conducted on variables in the original integer programming formulation instead of variables in the set partitioning formulation such that single machine subproblems are more tractable. We apply this decomposition approach to two particular problems: the total weighted completion time problem and the weighted number of tardy jobs problem. The computational results indicate that the decomposition approach is promising and capable of solving large problems.

Order Assignment and Scheduling in a Supply Chain

We consider the supply chain of a manufacturer who produces time-sensitive products that have a large variety, a short life cycle, and are sold in a very short selling season. The supply chain consists of multiple overseas plants and a domestic distribution center (DC). Retail orders are first processed at the plants and then shipped from the plants to the DC for distribution to domestic retailers. Due to variations in productivity and labor costs at different plants, the processing time and cost of an order are dependent on the plant to which it is assigned. We study the following static and deterministic order assignment and scheduling problem faced by the manufacturer before every selling season: Given a set of orders, determine which orders are to be assigned to each plant, find a schedule for processing the assigned orders at each plant, and find a schedule for shipping the completed orders from each plant to the DC, such that a certain performance measure is optimized. We consider four different performance measures, all of which take into account both delivery lead time and the total production and distribution cost. A problem corresponding to each performance measure is studied separately. We analyze the computational complexity of various cases of the problems by either proving that a problem is intractable or providing an efficient exact algorithm for the problem. We propose several fast heuristics for the intractable problems. We analyze the worst-case and asymptotic performance of the heuristics and also computationally evaluate their performance using randomly generated test instances. Our results show that the heuristics are capable of generating near-optimal solutions quickly.

Scheduling and common due date assignment with earliness-tardiness penalties and batch delivery costs

Abstract We consider a single machine scheduling problem involving both the scheduling of job processing and the scheduling of job delivery. A common due date for all the jobs and a delivery date for each job need to be determined in order to minimize the sum of earliness penalties, tardiness penalties, due date penalty, and delivery costs. Finished jobs are delivered in batches. There is no capacity limitation on a batch delivery and the cost per batch delivery is fixed and independent of the number of jobs in the batch. All the jobs completed before or at the due date are delivered in one batch at the due date. We present in this paper a polynomial dynamic programming algorithm for solving this problem.

Parallel machine scheduling with time dependent processing times

Abstract We consider a parallel machine scheduling problem in which the processing time of a job is a simple linear function of its starting time. The objective is to minimize total completion times. We show that the problem is NP-hard in the strong sense even with a fixed number of machines. When the number of machines is arbitrary, we show that there is no polynomial time heuristic with a constant worst-case bound unless P = NP. Under the two-machine case, we derive a data dependent worst-case bound for a simple polynomial time heuristic whose performance can be arbitrarily bad. However, for the case of a fixed number of machines, the question whether there exists a polynomial time heuristic with a constant worst-case bound remains open.

Dynamic Column Generation for Dynamic Vehicle Routing with Time Windows

We consider a dynamic vehicle routing problem with hard time windows, in which a set of customer orders arrives randomly over time to be picked up within their time windows. The dispatcher does not have any deterministic or probabilistic information on the location and size of a customer order until it arrives. The objective is to minimize the sum of the total distance of the routes used to cover all the orders. We propose a column-generation-based dynamic approach for the problem. The approach generates single-vehicle trips (i.e., columns) over time in a real-time fashion by utilizing existing columns, and solves at each decision epoch a set-partitioning-type formulation of the static problem consisting of the columns generated up to this time point. We evaluate the performance of our approach by comparing it to an insertion-based heuristic and an approach similar to ours, but without computational time limit for handling the static problem at each decision epoch. Computational results on various test problems generalized from a set of static benchmark problems in the literature show that our approach outperforms the insertion-based heuristic on most test problems.