Regularized Cost Function Research Articles

Scheduling a batching machine

We address the problem of scheduling n jobs on a batching machine to minimize regular scheduling criteria that are non-decreasing in the job completion times. A batching machine is a machine that can handle up to b jobs simultaneously. The jobs that are processed together form a batch, and all jobs in a batch start and complete at the same time. The processing time of a batch is equal to the largest processing time of any job in the batch. We analyse two variants: the unbounded model, where b⩾n; and the bounded model, where b<n. For the unbounded model, we give a characterization of a class of optimal schedules, which leads to a generic dynamic programming algorithm that solves the problem of minimizing an arbitrary regular cost function in pseudopolynomial time. The characterization leads to more efficient dynamic programming algorithms for specific cost functions: a polynomial algorithm for minimizing the maximum cost, an O(n3) time algorithm for minimizing the number of tardy jobs, an O(n2) time algorithm for minimizing the maximum lateness, and an O(n log n) time algorithm for minimizing the total weighted completion time. Furthermore, we prove that minimizing the weighted number of tardy jobs and the total weighted tardiness are NP-hard problems. For the bounded model, we derive an O(nb(b−1)) time dynamic programming algorithm for minimizing total completion time when b>1; for the case with m different processing times, we give a dynamic programming algorithm that requires O(b2m22m) time. Moreover, we prove that due date based scheduling criteria give rise to NP-hard problems. Finally, we show that an arbitrary regular cost function can be minimized in polynomial time for a fixed number of batches. © 1998 John Wiley & Sons, Ltd.

High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system

Some imaging systems employ detector arrays which are not su‐ciently dense so as to meet the Nyquist criteria during image acquisition. This is particularly true for many staring infrared imagers. Thus, the full resolution afiorded by the optics is not being realized in such a system. This paper presents a technique for estimating a high resolution image, with reduced aliasing, from a sequence of undersampled rotated and translationally shifted frames. Such an image sequence can be obtained if an imager is mounted on a moving platform, such as an aircraft. Several approaches to this type of problem have been proposed in the literature. Here we extend some of this previous work. In particular, we deflne an observation model which incorporates knowledge of the optical system and detector array. The high resolution image estimate is formed by minimizing a new regularized cost function which is based on the observation model. We show that with the proper choice of a tuning parameter, our algorithm exhibits robustness in the presence of noise. We consider both gradient descent and conjugate gradient optimization procedures to minimize the cost function. Detailed experimental results are provided to illustrate the performance of the proposed algorithm using digital video from an infrared imager.

General stochastic single-machine scheduling with regular cost functions

We examine a stochastic scheduling model with n jobs and a single machine, where the processing times of the jobs are random variables with arbitrary distributions, and the machine is subject to stochastic breakdowns. Two types of regular performance measures, the total cost and the maximum cost, with general cost functions are investigated. We derive conditions under which a sequence in nondecreasing stochastic order of processing times minimizes the total expected cost, whereas a sequence in nondecreasing stochastic order of due dates minimizes the maximum expected cost. In addition, results on almost sure minimization of the total stochastic cost or the maximum stochastic cost are also provided. Applications of the general results to various special cases with commonly studied measures like mean weighted flowtime, mean weighted tardiness, weighted number of tardy jobs, maximum tardiness, etc., are discussed, which extend the previous results on these cases to more general situations.

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.

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.

List scheduling revisited

We consider the problem of scheduling n jobs on m identical parallel machines to minimize a regular cost function. The standard list scheduling algorithm converts a list into a feasible schedule by focusing on the job start times. We prove that list schedules are dominant for this type of problem. Furthermore, we prove that an alternative list scheduling algorithm, focusing on the completion times rather than the start times, yields also dominant list schedules for problems with sequence dependent setup times.

From Data Distributions to Regularization in Invariant Learning

Ideally pattern recognition machines provide constant output when the inputs are transformed under a group G of desired invariances. These invariances can be achieved by enhancing the training data to include examples of inputs transformed by elements of G, while leaving the corresponding targets unchanged. Alternatively the cost function for training can include a regularization term that penalizes changes in the output when the input is transformed under the group. This paper relates the two approaches, showing precisely the sense in which the regularized cost function approximates the result of adding transformed examples to the training data. We introduce the notion of a probability distribution over the group transformations, and use this to rewrite the cost function for the enhanced training data. Under certain conditions, the new cost function is equivalent to the sum of the original cost function plus a regularizer. For unbiased models, the regularizer reduces to the intuitively obvious choice—a term that penalizes changes in the output when the inputs are transformed under the group. For infinitesimal transformations, the coefficient of the regularization term reduces to the variance of the distortions introduced into the training data. This correspondence provides a simple bridge between the two approaches.

Schedule-induced posets

Schedules are a basic tool in the treatment of project networks and sequencing problems. For many questions, however, it is not the schedule itself but a partial order (poset) naturally induced on the set of activities that is really relied on. In fact, this correspondence of the usually uncountably many schedules to a finite system of posets was a key to many recent results in this field and constitutes the discrete character of the described problems, all the more so as additional properties of the respective schedules can very often be reflected in related properties of these induced posets. The present paper gives insights into the class of those posets that can be induced by schedules. An immediate observation is that schedule-induced posets are just the interval orders. More involved characterizations are concerned with the possible ‘local optimality’ of such structures. One such property is the existence of activity durations that will result in a project duration of the considered poset that is shorter (simultaneously) than the duration of all its extensions. A similar, related characterization for general regular cost functions (in particular tardiness cost) is also included, as are insights into ‘best’ activity durations with the mentioned properties. The paper closes with some remarks and numerical data on the considered class of posets as a whole.

Preselective strategies for the optimization of stochastic project networks under resource constraints

AbstractThis article deals with a stochastic version of the optimization problem for project networks under resource constraints. In this, activity durations are assumed to be realized according to some joint probability distribution and the aim of optimization is to minimize the expected overall project cost (monotonically increasing with project duration). Certain strategies are known that constitute feasible solutions to this problem, the best studied of which are the so‐called ES strategies (“earliest start” with regard to fixed project structures). In this paper, a considerably broader class of strategies is introduced, namely preselective strategies. It is shown that this generalization, for which an algorithmic approach remains possible, preserves almost all the desirable behavior known for ES strategies. In particular, the number of “essential” strategies remains finite and even minimal optimum‐determining sets of such strategies can, in general, be characterized. Also, the analytic behavior is still proper and there is considerable “stability” to weak convergence of the joint distribution of activity durations as well as to a. e. convergence of the cost function. Last but not least, possible generalization to arbitrary regular cost functions is again imminent.