Machine scheduling with job class setup and delivery considerations

We study machine scheduling problems in which the jobs belong to different job classes and they need to be delivered to customers after processing. A setup time is required for a job if it is the first job to be processed on a machine or its processing on a machine follows a job that belongs to another class. Processed jobs are delivered in batches to their respective customers. The batch size is limited by the capacity of the delivery vehicles and each shipment incurs a transport cost and takes a fixed amount of time. The objective is to minimize the weighted sum of the last arrival time of jobs to customers and the delivery (transportation) cost. For the problem of processing jobs on a single machine and delivering them to multiple customers, we develop a dynamic programming algorithm to solve the problem optimally. For the problem of processing jobs on parallel machines and delivering them to a single customer, we propose a heuristic and analyze its performance bound.     

Scheduling jobs on identical machines with agreement graph

We consider the following problem of scheduling with agreements: a set of jobs must be scheduled non-preemptively on identical machines subject to constraints that only some specific jobs can be scheduled concurrently on different machines. These constraints are represented by an agreement graph and the aim is to minimize the makespan. This problem is NP-hard. We study the complexity of the problem for two machines and arbitrary bipartite agreement graphs, in particular we prove the NP-hardness of the open problem proposed in the literature which is the case of two machines with processing times at most 3. We propose list algorithms with empirical results for the problem in the general case.     

A branch-and-bound algorithm for single-machine scheduling with batch delivery and job release times

This paper addresses scheduling a set of jobs with specified release times on a single machine for delivery in batches to customers or to other machines for further processing. This problem is a natural extension of minimizing the sum of flow times in the presence of release time by considering the possibility of delivering jobs in batches and introducing batch delivery costs. The scheduling objective adopted is that of minimizing the sum of 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 scheme. Computational experiments show significant improvement over an existing dynamic programming algorithm.     

Scheduling of inventory releasing jobs to minimize a regular objective function of delivery times

In this note we provide new complexity and algorithmic results for scheduling inventory releasing jobs, a new class of single machine scheduling problems proposed recently by Boysen et al. We focus on tardiness related criteria, while known results are concerned with inventory levels between fixed delivery points. Our interest is motivated by the fact that deciding whether a feasible schedule exists is NP-hard in the strong sense, provided that all delivery deadlines are fixed, and there are no restrictions on the amount of products released by the jobs, nor on the job processing times. We will establish NP-hardness results, or provide polynomial or pseudo-polynomial time algorithms for various special cases, and describe a fully polynomial approximation scheme for one of the variants with the maximum tardiness criterion.     

Single machine scheduling with job delivery to multiple customers

We investigate a single machine scheduling problem with job delivery to multiple customers. In this problem, each job needs to be processed on the single machine, and then delivered by a single vehicle to its customer, where the job has a physical size representing the fraction of space it occupies on the vehicle. The vehicle delivers a shipment from the machine to a customer and has to return back to the machine for delivering the next shipment. It takes different constant time for the round trips between the machine and the different customers. The goal is to minimize the makespan, by that time all the jobs are processed and delivered to their respective customers, and the vehicle returns back to the machine. We propose a 2-approximation algorithm for the general case; when there are only two customers, we present an improved 5/3-approximation algorithm. The design and performance analysis of these two algorithms integrate several known results and techniques for the single machine scheduling problem, the bin-packing problem, and the knapsack problem.     

Single CNC machine scheduling with controllable processing times to minimize total weighted tardiness

Advanced manufacturing technologies, such as CNC machines, require significant investments, but also offer new capabilities to the manufacturers. One of the important capabilities of a CNC machine is the controllable processing times. By using this capability, the due date requirements of customers can be satisfied much more effectively. Processing times of the jobs on a CNC machine can be easily controlled via machining conditions such that they can be increased or decreased at the expense of tooling cost. Since scheduling decisions are very sensitive to the processing times, we solve the process planning and scheduling problems simultaneously. In this study, we consider the problem of scheduling a set of jobs on a single CNC machine to minimize the sum of total weighted tardiness, tooling and machining costs. We formulated the joint problem, which is NP-hard since the total weighted tardiness problem (with fixed processing times) is strongly NP-hard alone, as a nonlinear mixed integer program. We proposed a DP-based heuristic to solve the problem for a given sequence and designed a local search algorithm that uses it as a base heuristic.     

Approximation algorithms for minimizing the total weighted number of late jobs with late deliveries in two-level supply chains

We study a supply chain scheduling problem in which n jobs have to be scheduled on a single machine and delivered to m customers in batches. Each job has a due date, a processing time and a lateness penalty (weight). To save batch-delivery costs, several jobs for the same customer can be delivered together in a batch, including late jobs. The completion time of each job in the same batch coincides with the batch completion time. A batch setup time has to be added before processing the first job in each batch. The objective is to find a schedule which minimizes the sum of the weighted number of late jobs and the delivery costs. We present a pseudo-polynomial algorithm for a restricted case, where late jobs are delivered separately, and show that it becomes polynomial for the special cases when jobs have equal weights and equal delivery costs or equal processing times and equal setup times. We convert the algorithm into an FPTAS and prove that the solution produced by it is near-optimal for the original general problem by performing a parametric analysis of its performance ratio.     

An improved approximation algorithm for single machine scheduling with job delivery

In single machine scheduling with release times and job delivery, jobs are processed on a single machine and then delivered by a capacitated vehicle to a single customer. Only one vehicle is employed to deliver these jobs. The vehicle can deliver at most c jobs in a shipment. The delivery completion time of a job is defined as the time in which the delivery batch containing the job is delivered to the customer and the vehicle returns to the machine. The objective is to minimize the makespan, i.e., the maximum delivery completion time of the jobs. We provide an approximation algorithm for this problem which is better than that given in the literature, improving the performance ratio from 5/3 to 3/2.     

Non-approximability of just-in-time scheduling

We consider the following single machine just-in-time scheduling problem with earliness and tardiness costs: Given n jobs with processing times, due dates and job weights, the task is to schedule these jobs without preemption on a single machine such that the total weighted discrepancy from the given due dates is minimum. NP-hardness of this problem is well established, but no approximation results are known. Using the gap-technique, we show in this paper that the weighted earliness–tardiness scheduling problem and several variants are extremely hard to approximate: If n denotes the number of jobs and b∈ℕ is any given constant, then no polynomial-time algorithm can achieve an approximation which is guaranteed to be at most a factor of O(b ⁿ ) worse than the optimal solution unless P = NP.     

Scheduling jobs with simultaneous considerations of controllable processing times and learning effect

This paper considers a scheduling problem with general job-dependent learning curves and controllable processing times on a single machine. The objective is to determine the optimal compressions of the processing times and the optimal sequence of jobs so as to minimize some total cost functions, which consist of regular and non-regular functions and the processing time compressions. It shows that the problem can be solved by an assignment problem and thus can be solved in polynomial time. Some extensions of the problem are also given.

     

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.     

Single machine due date assignment scheduling problem with customer service level in fuzzy environment

Due date assignment scheduling problems with deterministic and stochastic parameters have been studied extensively in recent years. In this paper, we consider a single machine due date assignment scheduling problem with uncertain processing times and general precedence constraint among the jobs. The processing times of the jobs are assumed to be fuzzy numbers. We first propose an optimal polynomial time algorithm for the problem without precedence constraints among jobs. Then, we show that if general precedence constraint is involved, the problem is NP-hard. Finally, we show that if the precedence constraint is a tree or a collection of trees, the problem is still polynomially solvable.     

Semi-online two-level supply chain scheduling problems

We consider two-level supply chain scheduling problems where customers release jobs to a manufacturer that has to process the jobs and deliver them to the customers. Processed jobs are grouped into batches, which are delivered to the customers as single shipments. The objective is to minimize the total cost which is the sum of the total flow time and the total delivery cost. Such problems have been considered in the off-line environment where future jobs are known, and in the online environment where at any time there is no information about future jobs. It is known that the best possible competitive ratio for an online algorithm is 2. We consider the problem in the semi-online environment, assuming that a lower bound P for all processing times is available a priori, and present a semi-online algorithm with competitive ratio \(\frac{2D}{D+P}\) where D is the cost of a delivery. Also, for the special case where all processing times are equal, we prove that the algorithm is \(1.045\sqrt{\frac{2-u}{u}}\)-competitive, where u is the density of the instance.     

Two Machine Scheduling under Disruptions with Transportation Considerations

Effective logistics scheduling requires synchronization of manufacturing and delivery to optimize customer service at minimum total cost. In this paper, we study a new scheduling problem that arises in a disruption environment. Such a problem occurs when a disruption unexpectedly happens, and consequently, some machines become unavailable for certain periods. Jobs that are assigned to the disrupted machines and have not yet been processed can either be moved to other available machines for processing, which may involve additional transportation time and cost, or can be processed by the same machine after the disruption. Our goal is to reschedule jobs so that an objective function, including the original cost function, and possibly transportation costs and disruption cost caused by deviating from the originally planned completion times, is minimized. In this paper, we focus on the two-machine case to demonstrate some major properties, and hope that these properties can provide insights for solving other general problems, such as multiple (more than two) machine scheduling and machine scheduling in other configurations (job shop or flow shop) under disruption. We study problems with different related costs. In each problem, we either provide a polynomial algorithm to solve the problem optimally, or show its NP-hardness. If the problem is NP-hard in the ordinary sense, we also present a pseudo-polynomial algorithm to solve the problem optimally. This research is supported in part by Hong Kong RGC grant HKUST 6145/03E and in part by NSF Grant DMI-0300156.     

Integrating the lot-sizing and sequencing decisions for scheduling a capacitated flow line

In this paper we consider a general problem of scheduling a single flow line consisting of multiple machines and producing a given set of jobs. The manufacturing environment is characterized by sequence dependent set-up times, limited intermediate buffer space, and capacity constraints. In addition, jobs are assigned with due dates that have to be met. The objectives of the scheduling are: (1) to meet the due dates without violating the capacity constraints, (2) to minimize the makespan, and (3) to minimize the inventory holding costs. While most of the approaches in the literature treat the problem of scheduling in flow lines as two independent sub-problems of lot-sizing and sequencing, our approach integrates the lot-sizing and sequencing heuristics. The integrated approach uses the Silver-Meal heuristic (modified to include lot-splitting) for lot-sizing and an improvement procedure applied to Palmer's heuristic for sequencing, which takes into account the actual sequence dependent set-up times and the limited intermedite buffer capacity. We evaluate the performance of the integrated approach and demonstrate its efficacy for scheduling a real world SMT manufacturing environment.     

Learning-based algorithms in scheduling

The aim of the paper is to present a conception of intelligent learning-based algorithms for scheduling. A general knowledge based model of a vast class of discrete deterministic processes is given. The model is a basis for the method of the synthesis of intelligent, learning-based algorithms, that is described in the paper. The designing simulation experiments that use learning is also described. To illustrate the presented ideas, the scheduling algorithm for a special NP-hard problem is given. The significant feature of the problem is that the retooling time depends not only on a pair of jobs to be processed directly one after the other, but also on the subset of jobs already performed. The proof of the NP-hardness of the problem is also given in the paper.     

Scheduling of inventory releasing jobs to satisfy time-varying demand: an analysis of complexity

This paper studies a new class of single-machine scheduling problems, which are faced by Just-in-Time-suppliers satisfying a given demand. In these models the processing of jobs leads to a release of a predefined number of product units into inventory. Consumption is triggered by predetermined time-varying, and product-specific demand requests. While all demands have to be fulfilled, the objective is to minimize the resulting product inventory. We investigate different subproblems of this general setting with regard to their computational complexity. For more restricted problem versions strongly polynomial time algorithms are presented. In contrast to this, NP-hardness in the strong sense is proven for more general problem versions. Moreover, for the most general version, even finding a feasible solution is shown to be strongly NP-hard.     

Setting due dates to minimize the total weighted possibilistic mean value of the weighted earliness-tardiness costs on a single machine

In this paper, it is investigated how to sequence jobs with fuzzy processing times and predict their due dates on a single machine such that the total weighted possibilistic mean value of the weighted earliness-tardiness costs is minimized. First, an optimal polynomial time algorithm is put forward for the scheduling problem when there are no precedence constraints among jobs. Moreover, it is shown that if general precedence constraints are involved, the problem is NP-hard. Then, four reduction rules are proposed to simplify the constraints without changing the optimal schedule. Based on these rules, an optimal polynomial time algorithm is proposed when the precedence constraint is a tree or a collection of trees. Finally, a numerical experiment is given.     

A Multiple-Criterion Model for Machine Scheduling

We consider a scheduling problem involving a single processor being utilized by two or more customers. Traditionally, such scenarios are modeled by assuming that each customer has the same criterion. In practice, this assumption may not hold. Instead of using a single criterion, we examine the implications of minimizing an aggregate scheduling objective function in which jobs belonging to different customers are evaluated based on their individual criteria. We examine three basic scheduling criteria: minimizing makespan, minimizing maximum lateness, and minimizing total weighted completion time. Although determining a minimum-cost schedule according to any one of these criteria is polynomially solvable, we demonstrate that when minimizing a mix of these criteria, the problem becomes NP-hard.     

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.