Single-machine scheduling with past-sequence-dependent setup times and learning effects: a parametric analysis

In this article, we consider the single-machine scheduling problem with past-sequence-dependent (p-s-d) setup times and a learning effect. The setup times are proportional to the length of jobs that are already scheduled; i.e. p-s-d setup times. The learning effect reduces the actual processing time of a job because the workers are involved in doing the same job or activity repeatedly. Hence, the processing time of a job depends on its position in the sequence. In this study, we consider the total absolute difference in completion times (TADC) as the objective function. This problem is denoted as 1/LE, s _psd /TADC in Kuo and Yang (2007 Kuo, WH and Yang, DL. 2007. Single Machine Scheduling with Past-sequence-dependent Setup Times and Learning Effects. Information Processing Letters, 102: 22–26. [Crossref], [Web of Science ®] , [Google Scholar]) (‘Single Machine Scheduling with Past-sequence-dependent Setup Times and Learning Effects’, Information Processing Letters, 102, 22–26). There are two parameters a and b denoting constant learning index and normalising index, respectively. A parametric analysis of b on the 1/LE, s _psd /TADC problem for a given value of a is applied in this study. In addition, a computational algorithm is also developed to obtain the number of optimal sequences and the range of b in which each of the sequences is optimal, for a given value of a. We derive two bounds b* for the normalising constant b and a* for the learning index a. We also show that, when a?a* or b?>?b*, the optimal sequence is obtained by arranging the longest job in the first position and the rest of the jobs in short processing time order.     

Single-machine scheduling with truncated sum-of-processing-times-based learning effect including proportional delivery times

The single-machine scheduling problem with truncated sum-of-processing-times-based learning effect and past-sequence-dependent job delivery times is considered. Each job’s delivery time depends on its waiting time of processing. For some regular objective functions, it is proved that the problems can be solved by the smallest processing time first rule. For some special cases of the total weighted completion time and the maximum lateness objective functions, the thesis shows that the problems can be solved in polynomial time.     

Single machine scheduling with past-sequence-dependent setup times and learning effects

This paper studies a single machine scheduling problem with setup times and learning considerations. The setup times are proportional to the length of the already scheduled jobs. That is, the setup times are past-sequence-dependent. It is assumed that the learning process reflects a decrease in the process time as a function of the number of repetitions, i.e., as a function of the job position in the sequence. The following objectives are considered: the makespan, the total completion time, the total absolute differences in completion times and the sum of earliness, tardiness and common due-date penalty. Polynomial time algorithms are proposed to optimally solve the above objective functions.     

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.     

Single machine scheduling with past-sequence-dependent setup times and deteriorating jobs

This paper considers single machine scheduling problems with setup times and deteriorating jobs. The setup times are proportional to the length of the already processed jobs, that is, the setup times are past-sequence-dependent (p-s-d). It is assumed that the job processing times are defined by functions dependent on their starting times. The following objectives are considered: the makespan, the total completion time, and the sum of earliness, tardiness, and due-window starting time and size penalties. We propose polynomial time algorithms to solve these problems.     

Single machine scheduling with past-sequence-dependent setup times and deteriorating jobs

This paper considers single machine scheduling problems with setup times and deteriorating jobs. The setup times are proportional to the length of the already processed jobs, that is, the setup times are past-sequence-dependent (p-s-d). It is assumed that the job processing times are defined by functions dependent on their starting times. The following objectives are considered: the makespan, the total completion time, and the sum of earliness, tardiness, and due-window starting time and size penalties. We propose polynomial time algorithms to solve these problems.     

Single-machine scheduling with learning effect and resource-dependent processing times

We consider resource allocation scheduling with learning effect in which the processing time of a job is a function of its position in a sequence and its resource allocation. The objective is to find the optimal sequence of jobs and the optimal resource allocation separately. We concentrate on two goals separately, namely, minimizing a cost function containing makespan, total completion time, total absolute differences in completion times and total resource cost; minimizing a cost function containing makespan, total waiting time, total absolute differences in waiting times and total resource cost. We analyse the problem with two different processing time functions. For each combination of these, we provide a polynomial time algorithm to find the optimal job sequence and resource allocation.     

Single machine batch scheduling with release times and delivery costs

We study single machine batch scheduling with release times. Our goal is to minimize the sum of weighted flow times (or completion times) and delivery costs. Since the problem is strongly $\mathcal{NP}$ -hard even with no delivery cost and identical weights for all jobs, an approximation algorithm is presented for the problem with identical weights. This uses the polynomial time solution we give for the preemptive version of the problem. We also present an evolutionary metaheuristic algorithm for the general case. Computational results show very small gaps between the results of the metaheuristic and the lower bound.     

Single-machine makespan minimization scheduling with nonlinear shortening processing times

In this paper, we consider the single-machine makespan minimization scheduling problem with nonlinear shortening processing times. By the nonlinear shortening processing times, we mean that the processing times of jobs are non-increasing nonlinear functions of their starting times. The computational complexity of the general problem remains an open problem, but we show that even with the introduction of nonlinear shortening processing times to job processing times, some special cases remain polynomially solvable. We also show that an optimal schedule of the general makespan minimization problem is V-shaped with respect to job normal processing times. A heuristic algorithm which utilize the V-shaped property is proposed, and computational experiments show that it is effective and efficient in obtaining near-optimal solutions.     

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.     

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.     

Preemptive scheduling with release times and deadlines

We consider the problem of deciding if there is a feasible preemptive schedule for a set of n independent tasks with release times and deadlines on m identical processors. The general problem is known to be solvable in O(n ³) time. In this paper, we study special cases for which faster algorithms exist. We introduce the notion of monotone schedules and study their properties. These properties are then exploited to devise fast algorithms for two special cases—the nested task systems and the non-overlapping task systems. We give an O(n log mn) time algorithm and an O(n log n+mn) time algorithm for nested task systems and non-overlapping task systems, respectively. Our algorithms generate at most O(n) and O(mn) preemptions for nested task systems and nonoverlapping task systems, respectively.Research supported in part by the ONR grant N00014-87-K-0833.     

Single-machine scheduling with deteriorating jobs and setup times to minimize the maximum tardiness

In many realistic production situations, a job processed later consumes more time than the same job when it is processed earlier. Production scheduling in such an environment is known as scheduling with deteriorating jobs. However, research on scheduling problems with deteriorating jobs has rarely considered explicit (separable) setup time (cost). In this paper, we consider a single-machine scheduling problem with deteriorating jobs and setup times to minimize the maximum tardiness. We provide a branch-and-bound algorithm to solve this problem. Computational experiments show that the algorithm can solve instances up to 1000 jobs in reasonable time.     

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。     

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。     

Single-machine scheduling with deteriorating jobs

In this article, we study a single-machine scheduling problem in which the processing time of a job is a nonlinear function of its basic processing time and starting time. The objectives are to minimise the makespan, the sum of weighted completion times and the sum of the kth powers of completion times. We show that the makespan minimisation problem can be solved in polynomial time. However, the total completion time and the sum of the kth powers of completion times minimisation problems can be solved in polynomial time in some cases. Besides, some useful properties are also provided for the sum of weighted completion times problem under certain conditions.     

heuristics for parallel machine scheduling with delivery times

A parallel machine scheduling problem is considered in which each job has a processing time and a delivery time. The objective is to find a schedule which minimizes the time by which all jobs are delivered. For a single machine this problem is easily solved in polynomial time, form2 machines it becomes NP-hard. Several heuristics using list scheduling as a subroutine are proposed and a tight worst-case analysis is given. The best one of our heuristics has a worst-case performance guarantee of 2–2/(m+1). For the on-line case we give a heuristic with the (best possible) worst-case performance of two.This research was supported by the Christian Doppler Laboratorium für Diskrete Optimierung.     

Single-machine scheduling with sum-of-logarithm-processing-times-based learning considerations

Scheduling with learning effects has attracted growing attention of the scheduling research community. A recent survey classifies the learning models in scheduling into two types, namely position-based learning and sum-of-processing-times-based learning. However, the actual processing time of a given job drops to zero precipitously as the number of jobs increases in the first model and when the normal job processing times are large in the second model. Motivated by this observation, we propose a new learning model where the actual job processing time is a function of the sum of the logarithm of the processing times of the jobs already processed. The use of the logarithm function is to model the phenomenon that learning as a human activity is subject to the law of diminishing return. Under the proposed learning model, we show that the scheduling problems to minimize the makespan and total completion time can be solved in polynomial time. We further show that the problems to minimize the maximum lateness, maximum tardiness, weighted sum of completion times and total tardiness have polynomial-time solutions under some agreeable conditions on the problem parameters.     

Online batch scheduling on parallel machines with delivery times

We study the online batch scheduling problem on parallel machines with delivery times. Online algorithms are designed on m parallel batch machines to minimize the time by which all jobs have been delivered. When all jobs have identical processing times, we provide the optimal online algorithms for both bounded and unbounded versions of this problem. For the general case of processing time on unbounded batch machines, an online algorithm with a competitive ratio of 2 is given when the number of machines m=2 or m=3, respectively. When m≥4, we present an online algorithm with a competitive ratio of 1.5+o(1).     

Single-machine scheduling with an availability constraint to minimize the weighted sum of the completion times

In this article, we consider a single-machine scheduling problem with one unavailability period, with the aim of minimizing the weighted sum of the completion times. We propose three exact methods for solving such a problem: a branch-and-bound method based on new properties and lower bounds, a mixed integer programming model, and a dynamic programming method. These methods were coded and tested on randomly generated instances, and their performances were analyzed. Our numerical experiments show that the branch-and-bound method and the dynamic programming method are complementary. Using these approaches, we are able to solve problems with up to 3000 jobs within a reasonable computation time.