ContributionsTree-visibility orders

We introduce a new class of partially ordered sets, called tree-visibility orders, containing interval orders, duals of generalized interval orders and height one orders. We give a characterization of tree-visibility orders by an infinite family of minimal forbidden suborders. Furthermore, we present an efficient recognition algorithm for tree-visibility orders.     

Comparability Invariance Results for Tolerance Orders

We prove comparability invariance results for three classes of ordered sets: bounded tolerance orders (equivalent to parallelogram orders), unit bitolerance orders (equivalent to point-core bitolerance orders) and unit tolerance orders (equivalent to 50% tolerance orders). Each proof uses a different technique and relies on the alternate characterization.     

Integer programming approach to reactive scheduling in make-to-order manufacturing

New algorithms based on mixed integer programming formulations are proposed for reactive scheduling in a dynamic, make-to-order manufacturing environment. The problem objective is to update a long-term production schedule subject to service level and inventory constraints, whenever the customer orders are modified or new orders arrive. Different rescheduling policies are proposed, from a total reschedule of all remaining and unmodified customer orders to a non-reschedule of all such orders. In addition, a medium restrictive policy is considered for rescheduling only a subset of remaining customer orders awaiting material supplies. Numerical examples modeled after a real-world scheduling/rescheduling of customer orders in the electronics industry are presented and some results of computational experiments are reported.     

Order acceptance and due-date quotation in low machine rates

In real manufacturing environments, some customer orders include multiple jobs. However, a single due-date should be assigned to each order. Further, machine processing rate is not constant at all times. In effect, in many manufacturing operations, the machine processing rate decreases to a subnormal level during time and needs a special type of maintenance to bring the normal state back. Due to this reduction in capacity, production schedulers may decide to reject some orders. In this paper, the novel extensive problem of selecting a subset of orders, assigning due-dates to selected orders, scheduling the selected orders and jobs within each one, and scheduling the rate-modifying maintenance is studied. The objective function is minimizing total cost of lost-sales of rejected orders together with due-date length and maximum of earliness and tardiness of selected orders. The problem is proved polynomial and an optimal approach is developed.     

Some positive dependence stochastic orders

In this paper we study some stochastic orders of positive dependence that arise when the underlying random vectors are ordered with respect to some multivariate hazard rate stochastic orders, and have the same univariate marginal distributions. We show how the orders can be studied by restricting them to copulae, we give a number of examples, and we study some positive dependence concepts that arise from the new positive dependence orders. We also discuss the relationship of the new orders to other positive dependence orders that have appeared in the literature.     

Decomposition of partial orders

A decomposition theory for partial orders which arises from the split decomposition of submodular functions is introduced. As a consequence of this theory, any partial order has a unique decomposition consisting of indecomposable partial orders and certain highly decomposable partial orders. The highly decomposable partial orders are completely characterized. As a special case of partial orders, we consider lattices and distributive lattices. It occurs, that the highly decomposable distributive lattices are precisely the Boolean lattices.     

Chain Dominated Orders

We study finite partial orders which have a chain such that every element of the order either belongs to this chain or has all its covers in this chain. We show that such orders are exactly the orders being both interval orders and truncated lattices. We prove that their jump number is polynomially tractable and that their dimension is unbounded. We also show that every order admits a visibility model having such an order as host.     

Unit and Proper Tube Orders

In 2005, we defined the n-tube orders, which are the n-dimensional analogue of interval orders in 1 dimension, and trapezoid orders in 2 dimensions. In this paper we consider two variations of n-tube orders: unit n-tube orders and proper n-tube orders. It has been proven that the classes of unit and proper interval orders are equal, and the classes of unit and proper trapezoid orders are not. We prove that the classes of unit and proper n-tube orders are not equal for all n ≥ 3, so the general case follows the situation in 2 dimensions.     

Free Triangle Orders

We define a new class of ordered sets, called free triangle orders. These are ordered sets represented by a left-to-right ordering on geometric objects contained in a horizontal strip in the plane. The objects are called 'free triangles', and have one vertex on each of the two boundaries of the strip and one vertex in its interior. These ordered sets generalize the classes of trapezoid and triangle orders studied by Bogart, Möhring, and Ryan, represented by trapezoids and triangles respectively, contained within a strip in the plane, and are a special case of the orders of tube dimension 2 introduced by Habib and co-workers, which are represented by any set of convex bodies contained within a strip in the plane. Our main result is that the class of free triangle orders strictly contains the class of trapezoid orders.     

Proper and Unit Trapezoid Orders and Graphs

We show that the class of trapezoid orders in which no trapezoid strictly contains any other trapezoid strictly contains the class of trapezoid orders in which every trapezoid can be drawn with unit area. This is different from the case of interval orders, where the class of proper interval orders is exactly the same as the class of unit interval orders.     

Counting general and self-dual interval orders

In this paper, we present a new method to derive formulas for the generating functions of interval orders, counted with respect to their size, magnitude, and number of minimal and maximal elements. Our method allows us not only to generalize previous results on refined enumeration of general interval orders, but also to enumerate self-dual interval orders with respect to analogous statistics.Using the newly derived generating function formulas, we are able to prove a bijective relationship between self-dual interval orders and upper-triangular matrices with no zero rows. Previously, a similar bijective relationship has been established between general interval orders and upper-triangular matrices with no zero rows and columns.     

Line directionality of orders

Some orders can be represented by translating convex figures in the plane. It is proved thatN-free and interval orders admit such representations with an unbounded number of directions. Weak orders, tree-like orders and two-dimensional orders of height one are shown to be two- directional. In all cases line segments can be used as convex sets.     

Single-peaked orders on a tree

Inada (1969) and Sen and Pattanaik (1969) have characterized the sets of preference orders which ensure the transitivity of the strict majority rule, no matter how each voter selects his own order in the set. But a problem remains untouched: which domains of orders guarantee the existence of a majority winner without necessarily ensuring the transitivity of the strict majority rule. We provide in this paper domains, called sets of single-peaked linear orders on a tree, which enjoy such a property. They appear as a generalization of the well-known sets of single-peaked linear orders.     

Selectivity in division algebras

A commutative order in a central simple algebra over a number field is said to be selective if it embeds in some, but not all, maximal orders in the algebra. We completely characterize selective orders in central division algebras, of dimension 9 or greater, in terms of the characterization of selective orders given by Chinburg and Friedman in the quaternionic case.     

Continuous time trading of a small investor in a limit order market

We provide a mathematical framework to model continuous time trading of a small investor in limit order markets. We show how elementary strategies can be extended in a suitable way to general continuous time strategies containing orders with infinitely many different limit prices. The general limit buy order strategies are predictable processes with values in the set of nonincreasing demand functions. It turns out that our strategy set of limit and market orders is closed, but limit orders can turn into market orders when passing to the limit, and any element can be approximated by a sequence of elementary strategies.     

Global Dependence Stochastic Orders

Two basic ideas, that give rise to global dependence stochastic orders, are introduced and studied. The similarities and differences between the resulting global dependence orders, and the known common positive dependence orders, are discussed. Some desirable properties that global dependence orders may expected to satisfy are listed and checked. Three particular global dependence orders, that come up from the two general ideas, are studied in detail. It is shown, among other things, how these orders can be verified. Finally, some applications in auction theory, in reliability theory, and in economics, are described.     

The construction of numerically Calabi–Yau orders on projective surfaces

In this paper, we construct a vast collection of maximal numerically Calabi–Yau orders utilising a noncommutative analogue of the well-known commutative cyclic covering trick. Such orders play an integral role in the Mori program for orders on projective surfaces and although we know a substantial amount about them, there are relatively few known examples.     

指令驱动市场中的组合下单策略研究

已有研究一般直接假设指令驱动市场中的知情交易者总是选择市价单进行交易，本文则基于现实交易状况．建立了一个理论模型．分析拥有私有信息的知情交易者组合使用限价委托单和市价委托单的决策问题．具体给出了不同私有信息状态下知情交易者的组合下单选择。     

A hybrid of adaptive large neighborhood search and tabu search for the order-batching problem

Given a set of customer orders and a routing policy, the goal of the order-batching problem?(OBP) is to group customer orders to picking orders (batches) such that the total length of all tours through a rectangular warehouse is minimized. Because order picking is considered the most labor-intensive process in warehousing, effectively batching customer orders can result in considerable savings. The OBP is NP-hard if the number of orders per batch is greater than two, and the exact solution methods proposed in the literature are not able to consistently solve larger instances. To address larger instances, we develop a metaheuristic hybrid based on adaptive large neighborhood search and tabu search, called ALNS/TS. In numerical studies, we conduct an extensive comparison of ALNS/TS to all previously published OBP methods that have used standard benchmark sets to investigate their performance. ALNS/TS outperforms all comparison methods with respect to both average solution quality and run-time. Compared to the state-of-the-art, ALNS/TS shows the clearest advantages on the larger instances of the existing benchmark sets, which assume a higher number of customer orders and higher capacities of the picking device. Finally, ALNS/TS is able to solve newly generated large-scale instances with up to 600 customer orders and six articles per customer order with reasonable run-times and convincing scaling behavior and robustness.     

Order acceptance and scheduling with machine availability constraints

We consider an order acceptance and scheduling model with machine availability constraints. The manufacturer (machine) is assumed to be available to process orders only within a number of discontinuous time intervals. To capture the real-life behavior of a typical manufacturer who has restrictions of time availability to process orders, our model allows the manufacturer to reject or outsource some of the orders. When an order is rejected or outsourced, an order-dependent cost of penalty will occur. The objective is to minimize the makespan of all accepted orders plus the total penalty of all rejected/outsourced orders. We study the approximability of the model and some of its important special cases.