Optimal assignment of due-dates for a single processor scheduling problem

This article examines proportionate flow-shop scheduling with general earliness/tardiness costs under two due-date assignment models: the common and slack models. The objective is to determine the optimal job sequence and due dates to minimize a total penalty function that includes earliness and tardiness penalties, the number of early and delayed jobs, and the due-date assignment cost. Several optimal structural properties are derived, and two polynomial-time dynamic programming (DP) algorithms are proposed to solve the problems, with a time complexity of O(u 4 ), and u is the number of jobs.

Single machine scheduling with due date assignment to minimize the total weighted lead time penalty and late work

This paper considers single machine scheduling and due date assignment problems in which the jobs need to be delivered to customers after processing. It is assumed that the delivery times are proportional to the length of the already processed jobs, and a job's processing time depends on its position in a sequence. The objective functions include total earliness, the weighted number of tardy jobs and the cost of due date assignment. We analyze the problems with two different due date assignment methods and conclude that the problems are polynomial time solvable. We provide a dynamic programming algorithm with O(n3) times for the problems.

We studied single machine scheduling problems in which the jobs need to be delivered to customers after processing. It is assumed that the delivery times are proportional to the length of the already processed jobs, and a job's processing time depended on its position in a sequence. The objective functions include total earliness, the weighted number of tardy jobs and the cost of due date assignment. We analyzed these problems with two different due date assignment methods and conclude that the problems are polynomial time solvable.

We consider a stochastic counterpart of the well-known earliness-tardiness scheduling problem with a common due date, in which n stochastic jobs are to be processed on a single machine. The processing times of the jobs are independent and normally distributed random variables with known means and known variances that are proportional to the means. The due dates of the jobs are random variables following a common probability distribution. The objective is to minimize the expectation of a weighted combination of the earliness penalty, the tardiness penalty, and the flow-time penalty. One of our main results is that an optimal sequence for the problem must be V-shaped with respect to the mean processing times. Other characterizations of the optimal solution are also established. Two algorithms are proposed, which can generate optimal or near-optimal solutions in pseudopolynomial time. The proposed algorithms are also extended to problems where processing times do not satisfy the assumption in the model above, and are evaluated when processing times follow different probability distributions, including general normal (without the proportional relation between variances and means), uniform, Laplace, and exponential. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44, 531–557, 1997.

A problem of scheduling n jobs with exponential processing times on a single machine is discussed, the objective is to find an optimal schedule to minimize the expectation of total weighted absolute deviations of completion times about a deterministic common due date. This problem is a typical scheduling model in just-in-time manufacturing system where both earliness penalties and tardiness penalties are considered. The deterministic equivalent of the objective function is derived, and /spl Lambda/-shaped property, with respect to the products of jobs' weights and their processing time rates, of the optimal schedules of this problem is established. The /spl Lambda/-shaped property of the optimal schedule can reduce the candidates of optimal schedule from n! to 2/sup n/.

This paper considers two types of stochastic group scheduling problems, one involves a single machine, and the other involves an m -machine flow shop. In both problems, jobs are classified into groups. The jobs of the same group are to be processed one after another, and a sequence independent setup time is incurred between each two consecutively scheduled groups. The job processing times are independent random variables and group setup times are functions of the job processing times. The objective is to find both a group sequence and a job sequence that minimizes the expected total flow time. An optimal group sequence and an optimal job sequence are obtained by processing jobs in increasing stochastic order of their mean processing times and group setup times for a single machine problem. A precedence relationship is derived for the m -machine flow shop problem.

Benchmarks for scheduling on a single machine against restrictive and unrestrictive common due dates

Contents volume 51

Earliness and tardiness penalties are designed for such scheduling problems where the popular Just‐In‐Time (JIT) concept is considered to be of significant importance. In this paper, a self evolution (SE) algorithm is proposed to solve the problem of single‐machine total earliness and tardiness penalties with a common due date. Up to now, no specific attention has been paid to straddling V‐shaped schedules of such problems, which may be better than pure V‐shaped schedules for the early due date cases; and no specific discussions have been made on the start time setting of the first job in a schedule. Therefore, in this research, efforts have been made on digging out the straddling V‐shaped schedules, improving the efficiency of setting the start time of a schedule, and reducing the execution time. In addition, a new RHRM approach is proposed to create the initial solution for evolution, which helps in achieving the fast contingency of the algorithm. The performance of the proposed algorithm has been tested on 280 benchmark instances ranging from 10 to 1000 jobs from the OR Library, and the results show that the proposed SE algorithm delivers much higher efficiency in finding optimal or near‐optimal solutions with both better results in total penalties and significant execution time reduction. Copyright © 2008 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

We study the problem of two-machine no-wait flowshop scheduling with learning effect and convex resource-dependent processing times. Under the condition of the due-date assignment with common flow allowance (i.e. slack (SLK) due-date assignment), we provide a bi-criteria analysis where the first criterion is to minimise the scheduling criteria (i.e. the weighted sum of earliness, tardiness and flow allowance costs), and the second criterion is to minimise the resource consumption cost (i.e. the weighted sum of resource consumption cost). The objective is to determine the optimal job sequence, resource allocations and common (flow allowance) slack time that minimise the three different versions of the two criteria. We prove that these problems can be solved in polynomial time.

We consider a single machine static and deterministic scheduling problem in which jobs have a common due window. Jobs completed within the window incur no penalties, other jobs incur either earliness or tardiness penalties. The objective is to find the optimal size and location of the window as well as an optimal sequence to minimise a cost function based on earliness, tardiness, window size, and window location. We propose an O(n log n) algorithm to solve the problem.

We consider single-machine group scheduling with group-dependent multiple due windows assignment. In the group technology environment, the jobs are divided into groups in advance according to their processing similarities, and all the jobs of the same group are processed consecutively in order to improve production efficiency. A sequence-independent machine set-up time precedes the processing of the first job of each group. Each group has group-dependent multiple due windows. The objective is to find the optimal job sequence, the set of jobs assigned to each due window sequence, the optimal group sequence, and the optimal due window assignment to minimise a total cost that comprises the earliness and tardiness penalties and the due window starting time and due window size costs. For the case where the number of jobs assigned to each due window in each group is given in advance, we show that the problem is solvable in time, where n is the total number of jobs. For the case where the number of jobs assigned to each due window in each group is unknown, we give an time algorithm to solve the problem, where and hi is the number of due window of the ith group.

This brief studies the linear quadratic regulation problem of discrete-time switched linear systems via dynamic programming. Discrete decision variables (referred as switch inputs) are introduced to represent the subsystem selection. This formulation leads to a mixed integer quadratic programming problem, which is known to be computationally hard in general. Instead of deriving suboptimal policies with analytical bounds on its optimality, the optimal switching sequence is computed in a computationally efficient manner when the input matrix is restricted as a vector. The unique contribution of this brief is an analytical expression of both the optimal switching condition (determining the subsystem selection) and the optimal control law. Proof of optimality is presented by fractional optimization and the practical merits of our approach is validated through simulations on a second-order system in comparison with recent pruning methods.

This paper considers the problem of scheduling n jobs on a single machine to minimize the number of tardy (or late) jobs. Each job has a release date, a processing time and a due date. The general case with non-equal release dates and different due dates is considered. Using new and efficient lower bounds and several dominance rules, a branch and bound scheme is proposed based on the definition of a master sequence, i.e. a sequence containing at least one optimal sequence. With this procedure, 95% of 140-job instances are optimally solved in a maximum of one-hour CPU time.