分类数很大时的成组排序问题

成组排序具有深刻的实际应用背景,是近年来国外研究得较多的一个热点.已有的某些动态规划算法的复杂性随分类数的增长呈指数型增长趋势,本文用“归并”和解不超过四个新的子问题的方法把分类数较大时的问题转化为分类数较小时的相应问题,简化了问题的求解.     

On-line scheduling on a batch processing machine with unbounded batch size to minimize the makespan

We present on-line algorithms to minimize the makespan on a single batch processing machine. We consider a parallel batching machine that can process up to b jobs simultaneously. Jobs in the same batch complete at the same time. Such a model of a batch processing machine has been motivated by burn-in ovens in final testing stage of semiconductor manufacturing. We deal with the on-line scheduling problem when jobs arrive over time. We consider a set of independent jobs. Their number is not known in advance. Each job is available at its release date and its processing requirement is not known in advance. This general problem with infinite machine capacity is noted 1∣p − batch, r_j, b = ∞∣C_max. Deterministic algorithms that do not insert idle-times in the schedule cannot be better than 2-competitive and a simple rule based on LPT achieved this bound [Z. Liu, W. Yu, Scheduling one batch processor subject to job release dates, Discrete Applied Mathematics 105 (2000) 129–136]. If we are allowed to postpone start of jobs, the performance guarantee can be improved to 1.618. We provide a simpler proof of this best known lower bound for bounded and unbounded batch sizes. We then present deterministic algorithms that are best possible for the problem with unbounded batch size (i.e., b = ∞) and agreeable processing times (i.e., there cannot exist an on-line algorithm with a better performance guarantee). We then propose another algorithm that leads to a best possible algorithm for the general problem with unbounded batch size. This algorithm improves the best known on-line algorithm (i.e. [G. Zhang, X. Cai, C.K. Wong, On-line algorithms for minimizing makespan on batch processing machines, Naval Research Logistics 48 (2001) 241–258]) in the sense that it produces a shortest makespan while ensuring the same worst-case performance guarantee.     

On scheduling an unbounded batch machine

A batch machine is a machine that can process up to c jobs simultaneously as a batch, and the processing time of the batch is equal to the longest processing time of the jobs assigned to it. In this paper, we deal with the complexity of scheduling an unbounded batch machine, i.e., c=+∞. We prove that minimizing total tardiness is binary NP-hard, which has been an open problem in the literature. Also, we establish the pseudopolynomial solvability of the unbounded batch machine scheduling problem with job release dates and any regular objective. This is distinct from the bounded batch machine and the classical single machine scheduling problems, most of which with different release dates are unary NP-hard. Combined with the existing results, this paper provides a nearly complete mapping of the complexity of scheduling an unbounded batch machine.     

A note on two-agent scheduling on an unbounded parallel-batching machine with makespan and maximum lateness objectives

This paper studies the two-agent scheduling on an unbounded parallel-batching machine. In the problem, there are two agents A and B with each having their own job sets. The jobs of a common agent can be processed in a common batch. Moreover, each agent has an objective function to be minimized. The objective function of agent A is the makespan of his jobs and the objective function of agent B is maximum lateness of his jobs. Yazdani Sabouni and Jolai [M.T. Yazdani Sabouni, F. Jolai, Optimal methods for batch processing problem with makespan and maximum lateness objectives, Appl. Math. Model. 34 (2010) 314–324] presented a polynomial-time algorithm for the problem to minimize a positive combination of the two agents’ objective functions. Unfortunately, their algorithm is incorrect. We then dwell on the problem and present a polynomial-time algorithm for finding all Pareto optimal solutions of this two-agent parallel-batching scheduling problem.     

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

In this paper we consider the single machine past-sequence-dependent (p-s-d) setup times scheduling problems with general position-dependent and time-dependent learning effects. By the general position-dependent and time-dependent learning effects, we mean that the actual processing time of a job is not only a function of the total normal processing times of the jobs already processed, but also a function of the job’s scheduled position. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the θth (θ ? 0) power of job completion times, the total lateness, the total weighted completion time, the maximum lateness, the maximum tardiness and the number of tardy jobs. We show that the problems of makespan, the total completion time, the sum of the θth (θ ? 0) power of job completion times and the total lateness can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem, the maximum lateness minimization problem, maximum tardiness minimization problem and the number of tardy jobs minimization problem can be solved in polynomial time under certain conditions.     

Batch scheduling with controllable setup and processing times to minimize total completion time

We study a problem of scheduling n jobs on a single machine in batches. A batch is a set of jobs processed contiguously and completed together when the processing of all jobs in the batch is finished. Processing of a batch requires a machine setup time dependent on the position of this batch in the batch sequence. Setup times and job processing times are continuously controllable, that is, they are real-valued variables within their lower and upper bounds. A deviation of a setup time or job processing time from its upper bound is called a compression. The problem is to find a job sequence, its partition into batches, and the values for setup times and job processing times such that (a) total job completion time is minimized, subject to an upper bound on total weighted setup time and job processing time compression, or (b) a linear combination of total job completion time, total setup time compression, and total job processing time compression is minimized. Properties of optimal solutions are established. If the lower and upper bounds on job processing times can be similarly ordered or the job sequence is fixed, then O(n³ log n) and O(n⁵) time algorithms are developed to solve cases (a) and (b), respectively. If all job processing times are fixed or all setup times are fixed, then more efficient algorithms can be devised to solve the problems.     

On-line supply chain scheduling problems with preemption

We consider supply chain scheduling problems where customers release jobs to a manufacturer that has to process the jobs and deliver them to the customers. The jobs are released on-line, that is, at any time there is no information on the number, release and processing times of future jobs; the processing time of a job becomes known when the job is released. Preemption is allowed. To reduce the total costs, processed jobs are grouped into batches, which are delivered to customers as single shipments; we assume that the cost of delivering a batch does not depend on the number of jobs in the batch. The objective is to minimize the total cost, which is the sum of the total flow time and the total delivery cost. For the single-customer problem, we present an on-line two-competitive algorithm, and show that no other on-line algorithm can have a better competitive ratio. We also consider an extension of the algorithm for the case of m customers, and show that its competitive ratio is not greater than 2m if the delivery costs to different customers are equal.     

Single machine batch scheduling to minimize the weighted number of late jobs

The single machine batch scheduling problem to minimize the weighted number of late jobs is studied. In this problem,n jobs have to be processed on a single machine. Each job has a processing time, a due date and a weight. Jobs may be combined to form batches containing contiguously scheduled jobs. For each batch, a constant set-up time is needed before the first job of this batch is processed. The completion time of each job in the batch coincides with the completion time of the last job in this batch. A job is late if it is completed after its due date. A schedule specifies the sequence of jobs and the size of each batch, i.e. the number of jobs it contains. The objective is to find a schedule which minimizes the weighted number of late jobs. This problem isNP-hard even if all due dates are equal. For the general case, we present a dynamic programming algorithm which solves the problem with equal weights inO(n ³) time. We formulate a certain scaled problem and show that our dynamic programming algorithm applied to this scaled problem provides a fully polynomial approximation scheme for the original problem. Each algorithm of this scheme has a time requirement ofO(n ³/ +n ³ logn). A side result is anO(n logn) algorithm for the problem of minimizing the maximum weight of late jobs.Supported by INTAS Project 93-257.     

Direct algorithm for computation of derivatives of the Daubechies basis functions

We extend the direct algorithm for computing the derivatives of the compactly supported Daubechies N-vanishing-moment basis functions. The method yields exact values at dyadic rationals for the nth derivative (0  n  N − 1) of the basis functions, when it exists. Example results are shown for the first derivatives of the basis functions from the Daubechies N-vanishing-moment extremal phase orthonormal family (for N = 3, 4, and 5), and the CDF(2, N) spline-based biorthogonal family (for N = 6, 8, and 10).     

Single machine batch scheduling with resource dependent setup and processing times

Jobs are processed by a single machine in batches. A batch is a set of jobs processed contiguously and completed together when the processing of all jobs in the batch is finished. Processing of a batch requires a machine setup time common for all batches. Both the job processing times and the setup time can be compressed through allocation of a continuously divisible resource. Each job uses the same amount of the resource. Each setup also uses the same amount of the resource, which may be different from that for the jobs. Polynomial time algorithms are presented to find an optimal batch sequence and resource values such that either the total weighted resource consumption is minimized, subject to meeting job deadlines, or the maximum job lateness is minimized, subject to an upper bound on the total weighted resource consumption. The algorithms are based on linear programming formulations of the corresponding problems.     

Hierarchical optimization with double due dates on an unbounded parallel-batching machine to minimize maximum lateness

This paper studies a hierarchical optimization problem on an unbounded parallel-batching machine, in which two objective functions are maximum lateness induced by two sets of due dates, representing different purposes of two decision-makers. By a hierarchical optimization problem, we mean the problem of optimizing the secondary criterion under the constraint that the primary criterion is optimized. A parallel-batching machine is a machine that can handle several jobs in a batch in which all jobs start and complete respectively at the same time. We present an \(O(n\log P)\)-time algorithm and an \(O(n^3)\)-time algorithm for this hierarchical scheduling problem, where P is the total processing time of all jobs.     

On the Positivity of the Fundamental Polynomials for Generalized Hermite–Fejér Interpolation on the Chebyshev Nodes

It is shown that the fundamental polynomials for (0, 1, …, 2m+1) Hermite–Fejér interpolation on the zeros of the Chebyshev polynomials of the first kind are non-negative for −1x1, thereby generalising a well-known property of the original Hermite–Fejér interpolation method. As an application of the result, Korovkin's 10theorem on monotone operators is used to present a new proof that the (0, 1, …, 2m+1) Hermite–Fejér interpolation polynomials offC[−1, 1], based onnChebyshev nodes, converge uniformly tofasn→∞.     

Gabor dual spline windows

A method is presented for constructing dual Gabor window functions that are polynomial splines. The spline windows are supported in [−1,1], with a knot at x=0, and can be taken C^m smooth and symmetric. The translation and modulation parameters satisfy a=1 and 0<b1/2. The full range 0<ab<1 is handled by introducing an additional knot. Many explicit examples are found.     

On a problem of H. Shapiro

Let μ be a real measure on the line such that its Poisson integral M(z) converges and satisfies|M(x+iy)|Ae^−cyα, y→+∞,for some constants A,c>0 and 0<α1. We show that for 1/2<α1 the measure μ must have many sign changes on both positive and negative rays. For 0<α1/2 this is true for at least one of the rays, and not always true for both rays. Asymptotical bounds for the number of sign changes are given which are sharp in some sense.     

Scheduling jobs with release dates on parallel batch processing machines

In this paper we consider the problem of scheduling jobs with release dates on parallel unbounded batch processing machines to minimize the maximum lateness. We show that the case where the jobs have deadlines is strongly NP-hard. We develop a polynomial-time approximation scheme for the problem to minimize the maximum delivery completion time, which is equivalent to minimizing the maximum lateness from the optimization viewpoint.     

Estimates for n-widths of the Hardy-type operators (Addendum to “Improved estimates for the approximation numbers of the Hardy-type operators”)

Consider the Hardy-type operator T : L^p(a,b)→L^p(a,b),-∞a<b∞, which is defined by

It is shown that

where ρ_n(T) stands for any of the following: the Kolmogorov n-width, the Gel’fand n-width, the Bernstein n-width or the nth approximation number of T.     

Approximation algorithms for two-machine flow shop scheduling with batch setup times

In many practical situations, batching of similar jobs to avoid setups is performed while constructing a schedule. This paper addresses the problem of non-preemptively scheduling independent jobs in a two-machine flow shop with the objective of minimizing the makespan. Jobs are grouped into batches. A sequence independent batch setup time on each machine is required before the first job is processed, and when a machine switches from processing a job in some batch to a job of another batch. Besides its practical interest, this problem is a direct generalization of the classical two-machine flow shop problem with no grouping of jobs, which can be solved optimally by Johnson's well-known algorithm. The problem under investigation is known to be NP-hard. We propose two O(n logn) time heuristic algorithms. The first heuristic, which creates a schedule with minimum total setup time by forcing all jobs in the same batch to be sequenced in adjacent positions, has a worst-case performance ratio of 3/2. By allowing each batch to be split into at most two sub-batches, a second heuristic is developed which has an improved worst-case performance ratio of 4/3. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

The de Schipper formula and squares of Riesz spaces

In this paper we introduce and study the square mean and the geometric mean in Riesz spaces. We prove that every geometric mean closed Riesz space is square mean closed and give a counterexample to the converse. We define for positive a, b in a square mean closed Riesz space E an addition via the formulaab=sup {(cos x)a + (sin x)b: 0 x 2π},which goes back to a formula by de Schipper. In case that E is geometric mean closed this turns the mldeflying set of the positive cone of E into a lattice ordered semigroup, which in turn is the positive cone ofa Riesz space E^□. We prove, under the additional condition that E is geometric mean closed, that E^□ is Riesz isomorphic to the square of E as introduced earlier by Buskes and van Rooij.     

Bicriteria scheduling on a series-batching machine to minimize maximum cost and makespan

This paper studies the bicriteria problem of scheduling n jobs on a serial-batching machine to minimize maximum cost and makespan simultaneously. A serial-batching machine is a machine that can handle up to b jobs in a batch and jobs in a batch start and complete respectively at the same time and the processing time of a batch is equal to the sum of the processing times of jobs in the batch. When a new batch starts, a constant setup time s occurs. We confine ourselves to the unbounded model, where b ≥ n. We present a polynomial-time algorithm for finding all Pareto optimal solutions of this bicriteria scheduling problem.     

Minimizing the weighted number of tardy jobs with due date assignment and capacity-constrained deliveries for multiple customers in supply chains

In this paper, an integrated due date assignment and production and batch delivery scheduling problem for make-to-order production system and multiple customers is addressed. Consider a supply chain scheduling problem in which n orders (jobs) have to be scheduled on a single machine and delivered to K customers or to other machines for further processing in batches. A common due date is assigned to all the jobs of each customer and the number of jobs in delivery batches is constrained by the batch size. The objective is to minimize the sum of the total weighted number of tardy jobs, the total due date assignment costs and the total batch delivery costs. The problem is NP-hard. We formulate the problem as an Integer Programming (IP) model. Also, in this paper, a Heuristic Algorithm (HA) and a Branch and Bound (B&B) method for solving this problem are presented. Computational tests are used to demonstrate the efficiency of the developed methods.