John A. George

Packing different-sized circles into a rectangular container

Abstract This paper is motivated by the problem of fitting pipes of different diameters into a shipping container. Here we study the subproblem of fitting circles of different sizes into a rectangle since that problem is a central part of the larger problem. We formulate this situation as a nonlinear mixed integer programming problem and develop a number of heuristic procedures for (approximately) solving this problem. The heuristics are based on a variety of solution building rules that emulate the process of packing a container. Some of these methods, including a genetic algorithm, were based on a more structured design intended to provide solutions which are ‘stable’ from a stowage viewpoint. These heuristics are described in detail and their relative performances are compared for a sample set of 66 randomly generated problems. Based on this sample, the best performing heuristic methods were a quasi-random technique and a genetic algorithm of the ‘stable’ solution structure.

Integrated container loading software for pulp and paper industry

Abstract In this paper we describe a container loading software package for a pulp and paper manufacturer. The program takes, as input, orders for paper products in sheets or reels. As output it produces precise order quantities, and stowage plans for shipping containers. The program includes a heuristic algorithm for two-dimensional stowing of reels of a common diameter, a bin packing algorithm for combining reels into stacks and a two-dimensional algorithm for stowing pallets of various sizes. Since the order sizes are usually greater than one container load all algorithms are designed for the multi-container case. The program is currently in use and has been effective in several decision situations including (a) good sizes for reel diameters (b) efficient stowage patterns, and (c) production quantities that are compatible with cost efficient shipping. It also reduces the time taken to produce stowage and shipping documents.

Political district determination using large-scale network optimization

Abstract Political redistricting involves the amalgamation of a large number of geographic or statistical units into a smaller set of electoral districts such that the districts satisfy various criteria prescribed by law, judicial mandate, or historical precedent. This paper describes a network-based optimization model that is currently being used by governmental decision-makers in New Zealand to assist in the preliminary stages of determining Parliamentary district boundaries. The model is capable of solving large-scale problems. In the New Zealand case, the procedure has been applied to problems involving over 35,000 geographic units (called “meshblocks”) and 95 Parliamentary districts. The model generates electoral districts that have roughly equal population and are generally compact in shape. One of the key features of the model is its flexibility. On the one hand, it can take into account currently existing features such as topographic terrain or previously existing electoral boundaries. On the other hand, it can also be used to generate an entirely new set of political boundaries as was mandated by the voters in a recent referendum in New Zealand.

Controlled rounding for tables with subtotals

The problem of rounding in statistical tables to protect confidentiality is an important problem in the area of data publication, especially for official statistics. Controlled rounding involves rounding the table data to a prespecified base while ensuring additivity to totals. Previous research provided a formulation of the controlled rounding problem of a simple two-way table as a transportation problem. This paper extends that work to tables with subtotals by using a capacitated transshipment formulation. It is shown that some forms of tables with subtotals always have a controlled rounding solution. Other table structures cannot be guaranteed such a solution under “zero-restrictedness”. Initial computational experience suggests that the method is viable for use in practical situations.

Confidentiality control of tabulated data: Some practical network models

The problem of confidentiality control is a significant issue for statistics gathering organisations. Most existing methods suffer from a variety of difficulties for practical implementation. This paper presents techniques for confidentiality control of tabulated data using network formulations to overcome many of the weaknesses of the current methods. These techniques are based on the established methodology of controlled rounding, which preserves additivity in the resulting solution, while adding a number of pieces of new material. Firstly, a controlled rounding formulation is introduced that is designed to provide an unbiased solution. Then we construct three variations of a controlled perturbation technique using different forms of bounds on the cells. Finally, a methodology is explained for disturbing only the small-valued cells of the table using each of the four basic methods. A simple example of the methods and results from experimentation on the use of the methods show that they satisfy the needs of confidentiality and are computationally viable.

Pricing strategies for a non-replenishable item under variable demand and inflation

Abstract In this paper a model is developed for the pricing of non-replenishable inventory. Pricing strategies are examined that determine the minimum special price for immediate disposal of the entire stock. These are assessed using the return from inventory, net of holding costs, available for financing overheads and profits. Previous studies [2] and [3] have presented models for pricing the immediate disposal case. These have assessed the strategy on the basis of the lump sum generated at the end of a certain period. Their results gave, in many instances, very low special prices. This papers result do not support their contentions in most instances. Indeed for many practical situations a special price of at least 80% of normal price is required. Substantially lower special prices are only justified when declining demand causes units of inventory to be sold at scrap value.

A fast algorithm for large-scale controlled rounding of 3-dimensional census tables

Abstract Confidentiality control is a major concern of official statistics authorities. Various measures are taken at different stages of the collection and publication of the data. In this regard, controlled rounding has been developed as a technique for confidentiality control of tabular data when it is published. The literature to date presents algorithms for both 2-dimensional and 3-dimensional tables. While fast algorithms are available for the 2-dimensional problem, current methods for 3-dimensional controlled rounding are regarded as too slow for practical implementation. The emphasis of this paper is thus a method for large-scale tables which is sufficiently fast and robust for use in a commercial implementation. It is a two phase method, wherein the first phase solves a network to create a partially feasible solution, while a heuristic employing tabu search methods completes the rounding in the second phase. Minor infeasibility is permissible for the sake of computational speed. Aspects of the methods and the table structure that affect speed of solution are discussed. Results are presented for large tables with up to 200,000 primary entries. Results indicate that the method produces good solutions in acceptable time.