Paul A. Roach

Container stowage planning: a methodology for generating computerised solutions

The container stowage problem concerns the suitable placement of containers in a container-ship on a multi-port journey; it requires consideration of the consequences each placement has on decisions at subsequent ports. A methodology for the automatic generation of computerised solutions to the container stowage problem is shown; objective functions that provide a basis for evaluating solutions are given in addition to the underlying structures and relationships that embody this problem. The methodology progressively refines the placement of containers within the cargo-space of a container ship until each container is specifically allocated to a stowage location. The methodology embodies a two stage process to computerised planning, that of a generalised placement strategy and a specialised placement procedure. Heuristic rules are built into objective functions for each stage that enable the combinatorial tree to be explored in an intelligent way, resulting in good, if not optimal, solutions for the problem in a reasonable processing time.

Principles of Combinatorial Optimization Applied to Container-Ship Stowage Planning

In this paper, a methodology for generating automated solutions to the container stowage problem is shown. The methodology was derived by applying principles of combinatorial optimization and, in particular, the Tabu Search metaheuristic. The methodology progressively refines the placement of containers, using the Tabu search concept of neighbourhoods, within the cargo-space of a container ship until each container is specifically allocated to a stowage location. Heuristic rules are built into objective functions for each stage that enable the combinatorial tree to be explored in an intelligent way, resulting in good, if not optimal, solutions for the problem in a reasonable processing time.

Container stowage pre-planning: using search to generate solutions, a case study

Container-ships are vessels possessing an internal structure that facilitates the handling of containerised cargo. At each port along the vessels journey, containers destined for those ports are unloaded and additional containers destined for subsequent ports are loaded. Determining a viable arrangement of containers that facilitates this process, in a cost-effective way, constitutes the deep-sea container-ship stowage problem. This paper outlines a computer system that generates good sub-optimal solutions to the stowage pre-planning problem. This is achieved through an intelligent analysis of the domain allowing the problem to be divided into sub-problems: a generalised placement strategy and a specialised placement procedure. This methodology progressively refines the arrangement of containers within the cargo-space of a container ship until each container is specifically allocated to a stowage location. Good, if not optimal, solutions for the problem are obtained in a reasonable processing time through the use of heuristics incorporated into objective functions for each stage.

Construction of Heuristics for a Search-Based Approach to Solving Sudoku

Sudoku is a logic puzzle, consisting of a 9×9 grid and further subdivided into ‘mini-grids’ of size 3×3. Each row, column, and 3×3 mini-grid contains the numbers 1 to 9 once, with a true Sudoku grid having a unique solution. Sudoku, along with similar combinatorial structures, has relationships with a range of real-world problems. Much published work on the solution of Sudoku puzzles has acknowledged the link between Sudoku and Latin Squares, thereby recognising the scale of any search space of possible solutions and that the generalization of the puzzle to larger grid sizes is NPcomplete. However, most published approaches to the solution of Sudoku puzzles have focussed on the use of constraint satisfaction algorithms that effectively mimic solution by hand, rather than directly exploiting features of the problem domain to reduce the size of the search space and constructing appropriate heuristics for the application of search techniques. This paper highlights important features of the search space to arrive at heuristics employed in a modified steepest ascent hill-climbing algorithm, and proposes a problem initialization and neighbourhood that greatly speed solution through a reduction of problem search space. Results shown demonstrate that this approach is sufficient to solve even the most complex rated puzzles, requiring relatively few moves. An analysis of the nature of the problem search space is offered.

Activist model of political party growth

The membership of British political parties has a direct influence on their political effectiveness. This paper applies the mathematics of epidemiology to the analysis of the growth and decline of such memberships. The party members are divided into activists and inactive members, where all activists influence the quality of party recruitment, but only a subset of activists recruit and thus govern numerical growth. The activists recruit for only a limited period, which acts as a restriction on further party growth. This Limited Activist model is applied to post-war and recent memberships of the Labour, Scottish National and Conservative parties. The model reproduces data trends, and relates realistically to historical narratives. It is concluded that the political parties analysed are not in danger of extinction but experience repeated periods of growth and decline in membership, albeit at lower numbers than in the past.

Production of blends and fairings by Fourier methods

Design and manipulation of curves and surfaces are required in many different areas of technology, for example in the definition of products such as cars and the casings for electrical devices. Fourier methods have long been used in image processing but have seen much less use in computer aided design (CAD). This paper suggests the use of Fourier methods for the generation of blends, which provide smooth transitions between model surfaces, and fairings, which smooth large regions of the surface of an object. A description is given of the use of Fourier methods to allow predictable and simple control over curved shape in a two- dimensional system, and results of the system are presented and analyzed. This allows informed assertions to be made regarding the extension to a surface smoothing system, which, at the time of writing, has not been developed. In particular we note that under certain circumstances the Fourier smoothing methods reduce to spline production, and the connection with spline theory is made clear where applicable.© (1992) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Automation of the Solution of Kakuro Puzzles

Kakuro puzzles, also called cross sum puzzles, are grids containing clues to the completion of numerical ‘words’. Being structured in a similar way to crossword puzzles, Kakuro grids contain overlapping continuous runs that are exclusively either horizontal or vertical. The ‘clues’ take the form of specified run totals, and a puzzle is solved by placing a value in each cell such that every run sums to its specified total, and no run contains duplicate values. While most puzzles have only a single solution, longer runs may be satisfied using many arrangements of values, leading to the puzzle having a deceptively large search space. The associated, popular Sudoku puzzle has been linked with important real-world applications including timetabling and conflict free wavelength routing, and more recently, coding theory due to its potential usefulness in the construction of erasure correction codes. It is possible that Kakuro puzzles will have similar applications, particularly in the construction of codes, where run totals may form a generalised type of parity check. A project has begun to investigate the properties of the class of Kakuro puzzles, and thereby establish its potential usefulness to real-world applications including coding theory. This paper reports some early findings from that project, specifically concerning puzzle complexity and the appropriateness of heuristic approaches for its automated solution. It highlights the use of heuristics to guide search by a backtracking solver, in preference to local search optimisation, and reports on the effectiveness of two heuristics and a pruning technique for reducing solution time. The authors believe this to be the first published work in the use of heuristics, in combination with pruning, for the automated solution of Kakuro puzzles.

A Mathematical Model of a Novel 3D Fractal-Inspired Piezoelectric Ultrasonic Transducer

Piezoelectric ultrasonic transducers have the potential to operate as both a sensor and as an actuator of ultrasonic waves. Currently, manufactured transducers operate effectively over narrow bandwidths as a result of their regular structures which incorporate a single length scale. To increase the operational bandwidth of these devices, consideration has been given in the literature to the implementation of designs which contain a range of length scales. In this paper, a mathematical model of a novel Sierpinski tetrix fractal-inspired transducer for sensor applications is presented. To accompany the growing body of research based on fractal-inspired transducers, this paper offers the first sensor design based on a three-dimensional fractal. The three-dimensional model reduces to an effective one-dimensional model by allowing for a number of assumptions of the propagating wave in the fractal lattice. The reception sensitivity of the sensor is investigated. Comparisons of reception force response (RFR) are performed between this novel design along with a previously investigated Sierpinski gasket-inspired device and standard Euclidean design. The results indicate that the proposed device surpasses traditional design sensors.

THE EFFECTIVENESS OF A SIERPINSKI CARPET-INSPIRED TRANSDUCER

Piezoelectric ultrasonic transducers have the ability to act both as a receiver and a transmitter of ultrasound. Standard designs have a regular structure and therefore operate effectively over narrow bandwidths due to their single length scale. Naturally occurring transducers benefit from a wide range of length scales giving rise to increased bandwidths. It is therefore of interest to investigate structures which incorporate a range of length scales, such as fractals. This paper applies an adaptation of the Green function renormalization method to analyze the propagation of an ultrasonic wave in a series of pre-fractal structures. The structure being investigated here is the Sierpinski carpet. Novel expressions for the non-dimensionalized electrical impedance and the transmission and reception sensitivities as a function of the operating frequency are presented. Comparisons of metrics between three new designs alongside the standard design (Euclidean structure) and the previously investigated Sierpinski gasket device are performed. The results indicate a significant improvement in the reception sensitivity of the device, and improved bandwidth in both the receiving and transmitting responses.

The Structure of Reduced Sudoku Grids and the Sudoku Symmetry Group

A Sudoku grid is a constrained Latin square. In this paper a reduced Sudoku grid is described, the properties of which differ, through necessity, from that of a reduced Latin square. The Sudoku symmetry group is presented and applied to determine a mathematical relationship between the number of reduced Sudoku grids and the total number of Sudoku grids for any size. This relationship simplifies the enumeration of Sudoku grids and an example of the use of this method is given.