The multilevel principle applied to sorting

Sorting algorithms are developed in the setting of iterative multilevel methods. These algorithms borrow aggregation techniques from algorithms used for the numerical solution of elliptic partial differential equations which are of optimal order in running time and storage space for structured problems. A computationally inexpensive preconditioner drives random data chosen from known distributions towards a special case for which the new sorting algorithms are of optimal order.     

Piecewise lexicographic programming: A new model for practical decision problems

A new model for practical decision problems is presented. It allows one to consider lexicographic preference structures by introducing the new class of piecewise lexicographic functions which impose a total order in the objective-and-constraint space. In this way, the concepts of objective and constraints are merged into a new unified notion of co-objective. Moreover, the lexicographic preference structure may be applied not only among different coobjectives, but also among different ranges of the same decision variable. The main merits of this model appear to be its versatility (it is able to deal with different types of multiobjective optimization situations without requiring user interaction) and its compactness (it does not require one to increase the original number of decision variables and constraints). A linear version of the model is investigated in more detail.     

Hash table reorganization

We consider the operation of permuting in place the records of an open address hash table in order to correspond to a different hashing function. Our emphasis is primarily on minimizing the amount of work space used. Lower and upper bounds are derived on the unrestricted problem, that is, without making any assumptions about the probing discipline used. For the special case of linear probing, we give an algorithm which, in practice, requires no work space outside the table and which runs in linear time with respect to the table size.     

基于遗传算法的多通道密集式移动货架订单拣选研究

密集式移动货架越来越多地应用到仓储实践中,提高了仓储空间利用率,但增加了订单拣选的时间成本。本文根据密集式移动货架的仓储布局特点,针对多条通道可同时打开的情况,将货架移动时间转换成通道移动距离进行计算,提出了多条通道依次移动的优化规则,以整批订单拣选所耗费的总时间最少为目标,建立了订单拣选顺序优化的数学模型。针对该模型的特点,设计了实数编码且全局寻优的遗传算法,并进行了不同规模的算例模拟。计算结果表明,该算法具有较强的适用性,针对不同规模的问题,均有显著的优化效果;货架数量、订单数量以及移动通道数量的小幅度增减,将会导致总拣选时间较大幅度的波动;多条移动通道初始位置居于中部或均匀分散,总拣选时间略优于其集中于仓储系统一端。     

Rough set based rule induction from two decision tables

We study rule induction from two decision tables as a basis of rough set analysis of more than one decision tables. We regard the rule induction process as enumerating minimal conditions satisfied with positive examples but unsatisfied with negative examples and/or with negative decision rules. From this point of view, we show that seven kinds of rule induction are conceivable for a single decision table. We point out that the set of all decision rules from two decision tables can be split in two levels: a first level decision rule is positively supported by a decision table and does not have any conflict with the other decision table and a second level decision rule is positively supported by both decision tables. To each level, we propose rule induction methods based on decision matrices. Through the discussions, we demonstrate that many kinds of rule induction are conceivable.     

Large Scale Simulation with Scaled Boundary Finite Element Method

Nowadays scientific and engineering applications often require wave propagation in infinite or unbounded domains. In order to model such applications we separate our model into near-field and far-field. The near-field is represented by the well-known finite element method (FEM), whereas the far-field is mapped by a scaled boundary finite element (SBFE) approach. This latter approach allows wave propagation in infinite domains and suppresses the reflection of waves at the boundary, thus being a suitable method to model wave propagation to infinity. It is non-local in time and space. From a computational point of view, those characteristics are a drawback because they lead to storage consuming calculations with high computational time-effort. The non-locality in space causes fully populated unit-impulse acceleration influence matrices for each time step, leading to immense storage consumption for problems with a large number of degrees of freedom. Additionally, a different influence matrix has to be assembled for each time step which yields unacceptable storage requirements for long simulation times. For long slender domains, where many nodes are rather far from each other and where the influence of the degrees of freedom of those distant nodes is neglectable, substructuring represents an efficient method to reduce storage requirements and computational effort. The presented simulation with substructuring still yields satisfactory results. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decision making under uncertainty comprising complete ignorance and probability

This paper investigates a model of decision making under uncertainty comprising opposite epistemic states of complete ignorance and probability. In the first part, a new utility theory under complete ignorance is developed that combines Hurwicz–Arrow's theory of decision under ignorance with Anscombe–Aumann's idea of reversibility and monotonicity used to characterize subjective probability. The main result is a representation theorem for preference under ignorance by a particular one-parameter function – the τ-anchor utility function. In the second part, we study decision making under uncertainty comprising an ignorant variable and a probabilistic variable. We show that even if the variables are independent, they are not reversible in Anscombe–Aumann's sense. This insight leads to the development of a new proposal for decision under uncertainty represented by a preference relation that satisfies the weak order and monotonicity assumptions but rejects the reversibility assumption. A distinctive feature of the new proposal is that the certainty equivalent of a mapping from the state space of uncertain variables to the prize space depends on the order in which the variables are revealed. Explicit modeling of the order of variables explains some of the puzzles in multiple-prior model and the models for decision making with Dempster–Shafer belief function.     

An orthogonal spline collocation alternating direction implicit method for second-order hyperbolic problems

On a rectangular region, we consider a linear second-order hyperbolicinitial-boundary value problem involving a mixed derivativeterm, continuous variable coefficients and non-homogeneous Dirichletboundary conditions. In comparison to the alternating directionimplicit Laplace-modified method of Fernandes (1997), we formulateand analyse a new parameter-free alternating direction implicitscheme in which the standard central difference formula is usedfor the time approximation and orthogonal spline collocationis used for the spatial discretization. We establish unconditionalstability of the scheme, and its optimal order in the discretemaximum norm in time and the H¹ norm in space. Numerical experimentsindicate that the new scheme, which has the same order as themethod of Fernandes (1997, Numer. Math., 77, 223–241),is more accurate. We also show that the new scheme is easilygeneralized to the second-order hyperbolic problems on rectangularpolygons. Extensions of the scheme to problems with discontinuouscoefficients, nonlinear problems, and problems with other boundaryconditions are also discussed.     

Scheduling batches in flowshop with limited buffers in the shampoo industry

In this paper we address the problem of planning a temporary storage area in a real production system. This temporary storage area is composed of parallel temporary storage units with distinct capacities. The storage operation of a job, also called a batch, has to answer time restrictions such as release dates, due dates, restricted family dependent setup times and time lags, and also a space constraint which is the capacity of the temporary storage unit. The goal is to schedule the batches on the storage units in order to minimize the total setup times and the maximum lateness. First, we model the problem on a single storage unit as a two-machine flowshop problem with a limited buffer capacity and we show that it is NP-hard. We also show that the particular case in which no lateness is allowed is solvable in polynomial time under special conditions on the buffer capacity, both for single or parallel temporary storage units. Next we provide three heuristics: a greedy algorithm, a hybrid heuristic based on Ant Colony Optimization and Simulated Annealing and finally a dedicated heuristic. The latter strongly exploits the structural properties shown in this paper. We provide experimental results which highlight the efficiency of the dedicated heuristic in comparison with the two other heuristics.     

A high order HODIE finite difference scheme for 1D parabolic singularly perturbed reaction-diffusion problems

This paper deals with the numerical approximation of the solution of 1D parabolic singularly perturbed problems of reaction-diffusion type. The numerical method combines the standard implicit Euler method on a uniform mesh to discretize in time and a HODIE compact fourth order finite difference scheme to discretize in space, which is defined on a priori special meshes condensing the grid points in the boundary layer regions. The method is uniformly convergent having first order in time and almost fourth order in space. The analysis of the uniform convergence is made in two steps, splitting the contribution to the error from the time and the space discretization. Although this idea has been previously used to prove the uniform convergence for parabolic singularly perturbed problems, here the proof is based on a new study of the asymptotic behavior of the exact solution of the semidiscrete problems obtained after the time discretization by using the Euler method. Some numerical results are given corroborating in practice the theoretical results.