Obtainable sizes of topologies on finite sets

We study the smallest possible number of points in a topological space having k open sets. Equivalently, this is the smallest possible number of elements in a poset having k order ideals. Using efficient algorithms for constructing a topology with a prescribed size, we show that this number has a logarithmic upper bound. We deduce that there exists a topology on n points having k open sets, for all k in an interval which is exponentially large in n. The construction algorithms can be modified to produce topologies where the smallest neighborhood of each point has a minimal size, and we give a range of obtainable sizes for such topologies.     

Travel time estimation and order batching in a 2-block warehouse

Order batching problem (OBP) is the problem of determining the number of orders to be picked together in one picking tour. Although various objectives may arise in practice, minimizing the average throughput time of a random order is a common concern. In this paper, we consider the OBP for a 2-block rectangular warehouse with the assumptions that orders arrive according to a Poisson process and the method used for routing the order-pickers is the well-known S-shape heuristic. We first elaborate on the first and second moment of the order-picker’s travel time. Then we use these moments to estimate the average throughput time of a random order. This enables us to estimate the optimal picking batch size. Results from simulation show that the method provides a high accuracy level. Furthermore, the method is rather simple and can be easily applied in practice.     

The Wijsman and Attouch-Wets topologies on hyperspaces revisited

In this paper we will prove that, for an arbitrary metric space X and a fairly arbitrary collection Σ of subsets of X, it is possible to endow the hyperspace CL(X) of all nonempty closed subsets of X (to be identified with their distance functionals) with a canonical distance function having the topology of uniform convergence on members of Σ as topological coreflection and the Hausdorff metric as metric coreflection. For particular choices of Σ, we obtain canonical distance functions overlying the Wijsman and Attouch-Wets topologies. Consequently we apply the general theory of spaces endowed with a distance function and compare the results with those obtained for the classical hyperspace topologies. In all cases we are able to prove results which are both stronger and more general than the classical ones.     

On the order of a group of even order

We present an upper bound for |G| of a group G of even order possessing a unique conjugacy class of involutions.     

Banach lattices which are order uniformly noncreasy

The class of order uniformly noncreasy Banach lattices is introduced and studied. A fixed point theorem for those lattices is proved.     

The determinants of fourth order dissipative operators with transmission conditions

In this paper, discontinuous non-self-adjoint differential operators in Weyl?s limit circle are studied. We give the determinant of perturbation connected with the dissipative operator L   generated by fourth order differential expression in L²(I)$L^2(I)$, where discontinuity of operator is dealt with transmission conditions. We obtain the Green?s function, then, using characteristic determinant, we prove the completeness of the system of eigenfunctions and associated functions of this dissipative operator.     

Characterization of the order weak compactness of semi-compact operators

We study the relation between the class of semi-compact operators and the class of order weakly compact operators and we give some interesting consequences.     

Tabu search heuristics for the order batching problem in manual order picking systems

In a manual order picking system, order pickers walk or ride through a distribution warehouse in order to collect items requested by (internal or external) customers. In order to perform these operations efficiently, it is usually required that customer orders be combined into (more substantial) picking orders that are limited in size. The order batching problem considered in this paper deals with the question of how a given set of customer orders should be combined into picking orders such that the total length of all picker tours necessary for all of the requested items to be collected is minimized. For the solution of this problem the authors suggest two approaches based on the tabu search principle. The first is a (classic) tabu search (TS), and the second is the attribute-based hill climber (ABHC). In a series of extensive numerical experiments, these approaches are benchmarked against other solution methods put forward in the current literature. It is demonstrated that the proposed methods are superior to the existing methods and provide solutions which may allow distribution warehouses to operate more efficiently.     

Profitability of online order fulfillment in multi-channel retailing

Store-based retailers have been concerned about the high fulfillment costs and low gross margins of their online sales. This paper incorporates such issues in a non-linear mixed-integer profit maximization model of the online order fulfillment process for multi-channel retailers. A solution approach based on an iterative outer-approximation technique is developed. Using empirical data from a large U.S. retailer, the profitability of different options used by multi-channel retailers to fill online orders is evaluated. Managerial insights are developed through a factorial research design that highlights the effect of relevant logistics processes and fulfillment methods on the profitability of online sales for retailers.     

Pawlak代数及其性质

本文用公理化方法给出了Ｐａｗｌａｋ粗集代数的格形式,即Ｐａｗｌａｋ代数并研究了其构造,从而初步展示了这种代数结构与序结构、拓扑结构之间的关系     

Using a TSP heuristic for routing order pickers in warehouses

In this paper, we deal with the sequencing and routing problem of order pickers in conventional multi-parallel-aisle warehouse systems. For this NP-hard Steiner travelling salesman problem (TSP), exact algorithms only exist for warehouses with at most three cross aisles, while for other warehouse types literature provides a selection of dedicated construction heuristics. We evaluate to what extent reformulating and solving the problem as a classical TSP leads to performance improvements compared to existing dedicated heuristics. We report average savings in route distance of up to 47% when using the LKH (Lin–Kernighan–Helsgaun) TSP heuristic. Additionally, we examine if combining problem-specific solution concepts from dedicated heuristics with high-quality local search features could be useful. Lastly, we verify whether the sophistication of ‘state-of-the-art’ local search heuristics is necessary for routing order pickers in warehouses, or whether a subset of features suffices to generate high-quality solutions.     

环上的拓扑和偏序

在环R上引入了拓扑O[R]和偏序≤R,证明了(R,O[R])是可分的,第一可数的局部紧空间,并得出了如下结论:(1)(R*,O*[R])是T1的当且仅当O*[R]是离散的当且仅当R中的任一元r满足r=r2=-r;(2)若(R,O[R])是T0的,则U∈O[R]当且仅当U=↓U;(3)若R是伪有限的且对任意r都有〈r〉>2,则(R,≤R)是代数Domain;(4)若环R的特征数chR为2,则R是伪有限的当且仅当Rop是代数Domain。     

The Cartesian Closedness of the Category CDL and Function Spaces on Topological CDL''''s

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An order expediting policy for continuous review systems with manufacturing lead-time

Expedited shipments are often seen in practice. When the inventory level of an item gets dangerously low after an order has been placed, material managers are often willing to expedite the order at extra fixed and/or variable costs. This paper proposes a single-item continuous-review order expediting inventory policy, which can be considered as an extension of ordinary (s,Q)$(s\text{,}Q)$ models. Besides the two usual operational parameters: reorder point s and order quantity Q, it consists of a third parameter called the expedite-up-to level R. If inventory falls below R at the end of the manufacturing lead-time, the buyer can request the upstream supplier to deliver part of an outstanding order via a fast transportation mode. The amount expedited will raise inventory to R, while the remaining order is delivered via a slow (regular) supply mode. Simple procedures are developed to obtain optimal operational parameters. Computational results show that the proposed policy can save large costs for a firm if service level is high, demand variability is large, the extra cost for expediting is small, or the manufacturing lead-time is long.     

Lower and upper topologies in the Hausdorff partial order on a fixed set

In the partial order of Hausdorff topologies on a fixed infinite set there may exist topologies τ?σ in which there is no Hausdorff topology μ satisfying σ?μ?τ. τ and σ are lower and upper topologies in this partial order, respectively. Alas and Wilson showed that a compact Hausdorff space cannot contain a maximal point and therefore its topology is not lower. We generalize this result by showing that a maximal point in an H-closed space is not a regular point. Furthermore, we construct in ZFC an example of a countably compact, countably tight lower topology, answering a question of Alas and Wilson. Finally, we characterize topologies that are upper in this partial order as simple extension topologies.     

Exact order of convergence of the secant method

We study the exact order of convergence of the secant method when applied to the problem of finding a zero of a nonlinear function defined from into . Under the standard assumptions for which Newton's method has the exact Q-order of convergencep, wherep is some positive integer, we establish that the secant method has the Q-order and the exact R-order of convergence . We prove also that, forp=2 andp=3, the secant method has the exact Q-order of convergenceS(p). Moreover, we present a counterexample to show that, forp4, it may not have an exact Q-order of convergence.The author wishes to thank Florian Potra, Richard Tapia, and the referees for helpful comments and suggestions.This paper was prepared while the author was Visiting Professor, Department of Mathematics, University of Kentucky, Lexington, Kentucky.     

Mixed-width aisle configurations for order picking in distribution centers

Space required for the order picking area and labor required to perform the picking activity are two significant costs for a distribution center (DC). Traditionally, DCs employ either entirely wide or entirely narrow aisles in their picking systems. Wide aisles allow pickers to pass each other, which reduces blocking, and requires fewer pickers than their narrow-aisle counterpart for the same throughput. However, the amount of space required for wide-aisle configurations is high. Narrow aisles utilize less space than wide aisles, but are less efficient because of the increased likelihood of congestion experienced by pickers. We propose a variation to the traditional orthogonal aisle designs where both wide and narrow aisles are mixed within the configuration, with a view that mixed-width aisles may provide a compromise between space and labor. To analyze these new mixed-width aisle configurations, we develop analytical models for space and travel time considering randomized storage and traversal routing policies. Through a cost-based optimization model, we identify system parameters for which mixed-width aisle configurations are optimal. Experimental results indicate that annual cost savings of up to $48,000 can be realized over systems with pure wide or narrow aisle configurations.     

A parametric total order on fuzzy numbers and a fuzzy shortest route problem

A parametric total order relation is introduced in the form of a certain modification of the "fuzzy max" order on the class of fuzzy numbers generated by a shape function. By the parametric relation. fuzzy numbers unordered with respect of the fuzzy max order can be ordered according either to their value of center or the size of ambiguity. A fuzzy shortest route problem in which are distances are given by fuzzy numbers is discussed under the criterion of the parametric total order, and solved by the dynamic programming approach. A method is proposed to find all of fuzzy routes mimmal in the sense of the fuzzy max order.     

Design and control of warehouse order picking: A literature review

Order picking has long been identified as the most labour-intensive and costly activity for almost every warehouse; the cost of order picking is estimated to be as much as 55% of the total warehouse operating expense. Any underperformance in order picking can lead to unsatisfactory service and high operational cost for the warehouse, and consequently for the whole supply chain. In order to operate efficiently, the order-picking process needs to be robustly designed and optimally controlled. This paper gives a literature overview on typical decision problems in design and control of manual order-picking processes. We focus on optimal (internal) layout design, storage assignment methods, routing methods, order batching and zoning. The research in this area has grown rapidly recently. Still, combinations of the above areas have hardly been explored. Order-picking system developments in practice lead to promising new research directions.     

Discrete system order reduction using multipoint step response matching

The proposed method of linear time invariant discrete system order reduction is based on multipoint step response matching for both pole and zero evaluation of the low order model. Depending on the number of zeros and poles of the low order model, the number of points are selected on the time axis of the unit step response such that the unknown poles and zeros can be determined by solving a set of nonlinear equations using Newton's method.