Constructive procedures to solve 2-dimensional bin packing problems with irregular pieces and guillotine cuts

This paper presents an approach for solving a new real problem in cutting and packing. At its core is an innovative mixed integer programme model that places irregular pieces and defines guillotine cuts. The two-dimensional irregular shape bin packing problem with guillotine constraints arises in the glass cutting industry, for example, the cutting of glass for conservatories. Almost all cutting and packing problems that include guillotine cuts deal with rectangles only, where all cuts are orthogonal to the edges of the stock sheet and a maximum of two angles of rotation are permitted. The literature tackling packing problems with irregular shapes largely focuses on strip packing i.e. minimizing the length of a single fixed width stock sheet, and does not consider guillotine cuts. Hence, this problem combines the challenges of tackling the complexity of packing irregular pieces with free rotation, guaranteeing guillotine cuts that are not always orthogonal to the edges of the stock sheet, and allocating pieces to bins. To our knowledge only one other recent paper tackles this problem. We present a hybrid algorithm that is a constructive heuristic that determines the relative position of pieces in the bin and guillotine constraints via a mixed integer programme model. We investigate two approaches for allocating guillotine cuts at the same time as determining the placement of the piece, and a two phase approach that delays the allocation of cuts to provide flexibility in space usage. Finally we describe an improvement procedure that is applied to each bin before it is closed. This approach improves on the results of the only other publication on this problem, and gives competitive results for the classic rectangle bin packing problem with guillotine constraints.     

Optimization models for degrouping population data

In certain countries population data are available in grouped form only, usually as quinquennial age groups plus a large open-ended range for the elderly. However, official statistics call for data by individual age since many statistical operations, such as the calculation of demographic indicators, require the use of ungrouped population data. In this paper a number of mathematical models are proposed which, starting from population data given in age groups, enable these ranges to be degrouped into age-specific population values without leaving a fractional part. Unlike other existing procedures for disaggregating demographic data, ours makes it possible to process several years' data simultaneously in a coherent way, and provides accurate results longitudinally as well as transversally. This procedure is also shown to be helpful in dealing with degrouped population data affected by noise, such as those affected by the age-heaping phenomenon.     

Selection of supply portfolio under disruption risks

This paper deals with the optimal selection of supply portfolio in a make-to-order environment in the presence of supply chain disruption risks. Given a set of customer orders for products, the decision maker needs to decide from which supplier to purchase custom parts required for each customer order to minimize total cost and mitigate the impact of disruption risks. The selection of suppliers and allocation of orders is based on price and quality of purchased parts and reliability of delivery. The two types of disruption scenarios are considered: scenarios with independent local disruptions of each supplier and scenarios with local and global disruptions that may result in all suppliers disruption simultaneously. The problem is formulated as a single- or bi-objective mixed integer program and a value-at-risk and conditional value-at-risk approach is applied to control the risk of supply disruptions. The proposed portfolio approach is capable of optimizing the supply portfolio by calculating value-at-risk of cost per part and minimizing expected worst-case cost per part simultaneously. Numerical examples are presented and some computational results are reported.     

关于Diophantine方程x3-8=py2

在p是奇素数的假设下,证明了如果p=12r2 1,其中r是偶数,则方程x3-8=py2没有适合gcd(x,y)=1的正整数解(x,y).     

A Novel Design Method for SAW Bandpass Filter

提出了一种新的声表面波（ＳＡＷ）滤波器设计方法。先对设计目标函数采样减少变量数，用Ｚ变换零点分开并通过内插重构得到两组加权初值。假定其中一个固定而对另一加权系数利用标准的雷米兹交换算法进行优化。对Ｂｌａｃｋｍａｎ谱成形成滤波器的模拟设计表明，该方法计算精确方便，简明实用。     

关于联立Pell方程组x2-2y2=1,y2-Dz2=4

设D是无平方因子正整数,证明了当D是偶数时,如果D没有适合p≡1(mod8)的素因数p,则方程组x2-2y2=1和y2-Dz2=4仅有整数解(x,y,z)=(±3,±2,0).     

整周模糊度动态快速求解

提出了一种整周模糊度动态快速解算方法。该方法结合了最小二乘分组搜索方法和整数高斯变换方法,减少了备选整周模糊度的组合数。通过采用多种检验方法剔除不正确的模糊度组合,使整周模糊度的解算具有较高的可靠性和较短的时间。且无论在动态还是静态的情况下,整周模糊度解算时间均比较小。     

WDM网络中的光迹保护研究

介绍了光迹的基本概念和一种在网络节点没有波长转换功能时的连接保护光迹设计方案,并将该方案描述成整数线性规划模型,用线性规划软件进行仿真。仿真结果表明:采用光迹设计方案大大提高了波长通道的利用率,且波长通道上的每一个连接请求都得到了100％的保护．     

On the fair optimization of cost and customer service level in a supply chain under disruption risks

This paper presents a new decision-making problem of a fair optimization with respect to the two equally important conflicting objective functions: cost and customer service level, in the presence of supply chain disruption risks. Given a set of customer orders for products, the decision maker needs to select suppliers of parts required to complete the orders, allocate the demand for parts among the selected suppliers, and schedule the orders over the planning horizon, to equitably optimize expected cost and expected customer service level. The supplies of parts are subject to independent random local and regional disruptions. The fair decision-making aims at achieving the normalized expected cost and customer service level values as much close to each other as possible. The obtained combinatorial stochastic optimization problem is formulated as a stochastic mixed integer program with the ordered weighted averaging aggregation of the two conflicting objective functions. Numerical examples and computational results, in particular comparison with the weighted-sum aggregation of the two objective functions are presented and some managerial insights are reported. The findings indicate that for the minimum cost objective the cheapest supplier is usually selected, and for the maximum service level objective a subset of most reliable and most expensive suppliers is usually chosen, whereas the equitably efficient supply portfolio usually combines the most reliable and the cheapest suppliers. While the minimum cost objective function leads to the largest expected unfulfilled demand and the expected production schedule for the maximum service level follows the customer demand with the smallest expected unfulfilled demand, the equitably efficient solution ensures a reasonable value of expected unfulfilled demand.     

关于P≤k(n)=Pk(n+k)的另一种证法

本文给出了整数分拆中的一个重要计数公式的一种新证法,并得到了一个新的计数公式,从而而为整数分拆的计算提供了方便。