The singing wineglass: an exercise in mathematical modelling

Lecturers in mathematical modelling courses are always on the lookout for new examples to illustrate the modelling process. A physical phenomenon, documented as early as the nineteenth century, was recalled: when a wineglass ‘sings’, waves are visible on the surface of the wine. These surface waves are used as an exercise in mathematical modelling. Based on assumptions about the wine in the glass and observations illustrated with photographs, a mathematical problem is set up. This problem includes a non-homogeneous Neumann boundary condition on the lateral side of the glass. The solution to the mathematical problem is animated using Mathematica?. The predictions of the model are tested by comparing them with the known facts. The predictions of the model agree with the actual observations.     

Probability Modelling and Optimal Location of a Travelling Salesman

The ‘random part’ of an operations research model may be less satisfactory than the ‘deterministic part’, and it may thus be desirable to design algorithms that require few probability assumptions and make few calls to a suitable ‘probability oracle’. We consider here the problem of locating a service facility on a tree network so as to minimize the expected length of a travelling salesman tour through a random set of demand nodes.     

The Feeder-bus Network-design Problem

The potential for improving the cost-effectiveness of public transport operations by designing better integrated feeder-bus/rail rapid transit systems has been widely recognized. This paper defines the feeder-bus network-design problem (FBNDP) as that of designing a feeder-bus network to access an existing rail system. The FBNDP is considered under two different demand patterns, many-to-one (M-to-1) and many-to-many (M-to-M). We present a mathematical programming model for the M-to-1 FBNDP, and show that it can be generalized to the M-to-M FBNDP. The FBNDP is a large and difficult vehicle-routeing-type problem with an additional decision variable—operating frequency. A heuristic model is presented, which generalizes the ‘savings approach’ to incorporate operating frequency. The computational analysis shows that the proposed heuristic provides reasonable feeder-bus networks and consistent responses to ‘what if’ questions. A comparison indicates that the proposed heuristic provides solutions that are superior to manually designed networks. The advantages of this heuristic are particularly significant under variable demand.     

Developing a Visual Interactive Model for Corporate Cash Management

The operational research/management science journals contain an extensive literature that addresses the corporate cash management problem; yet few, if any, companies make use of any of this published work in their daily cash-management decision making. A review of the literature suggests that the reason for this lack of applications may well be poor problem formulation—the problems that are solved in the literature as ‘cash management’ problems evolve from a ‘hard systems’ view of real-world cash management. However, the problem as perceived by cash managers involves both dynamic and loosely structured components which are difficult to model using classical (i.e. ‘hard systems’) approaches.We therefore decided to approach the cash management problem as an experiment in the use of a novel visual interactive problem solving (VIPS) methodology. The aim of the experiment was to develop an implementable, visual interactive model to support daily cash management decision making. Working closely with a corporate cash manager, we first developed a visual model of his daily decision problem and then agreed on the feasible options and the interactive requirements. At this stage, the problem was sufficiently well defined for a mathematical model to be built and the visual model made ‘smart’.This paper discusses the results of this experiment and suggests that VIPS may have distinct advantages as a problem-solving technique in loosely structured, ‘messy’ problem situations.     

Mathematic education in a cultural setting

The paper develops some proposals for curriculum development in mathematics based on an analysis of the intercultural transmission of mathematical knowledge. By introducing a concept of culture which calls for an analysis of individual and social behaviours, we are lead to recognize ‘ethnomathematics’ as a form of structured knowledge and to recognize ‘matheracy’ as a characteristic behaviour of human beings. Upon these two concepts, we introduce a theoretical framework for curriculum development in mathematics.

Curricular space is conceptualized as a three‐dimensional space with components, contents, methods and objectives considered solidarily. This relies upon an epistemology of action, based on an integration of episteme, techne and praxis. This allows for an approach in which theory and practice are in a dialectical relationship. Special reference is made to the problems of mathematical education for culturally differentiated groups, and in particular to the situation in third‐world countries.     

Scheduling Production in a Heavily Constrained Plant – Anode Manufacture in an Aluminium Smelter

In this paper a mathematical model of an anode manufacturing plant is developed with the objective of assisting in the planning and scheduling of production for up to a month ahead. A model of the overall smelter already exists and is based on a monthly time frame. The anode plant is a part of this model but is a very heavily constrained area and it is necessary to have a daily planning model that will help to achieve the optimal level of operations as dictated by the overall model of the smelter. The anode area is responsible for the production of carbon blocks (called anodes) which are an integral part of the aluminium smelting process. The plant is also responsible for the delivery of all raw materials within the smelter. The model developed is a daily one replicated for up to a month ahead, interconnected by opening and closing stocks.     

Spatial fuzzy consumer's decision making: A multicriteria analysis

This paper is devoted to a multicriteria analysis of the consumer's behavior when the decision maker is acting in a fuzzy space and manifesting an imprecise attitude. At first, the process of decision making is described with the help of three relationships between the set of goods which are supplied in several locations, the set of their characteristics and the set of the consumer's a priori possible behaviors. All these relations are fuzzy. The model applies the theory of fuzzy relations equations. Then, the stages of the decision process are analyzed. Often fuzzy behavior relations are like ‘black boxes’. The mathematical solution of the model indicates in which conditions their valuations are possible. The main interest of this method is not to use additive operations on subjective items and to use operators which are coherent with the fuzzy nature of the variables.     

Towards a multi-objective performance assessment and optimization model of a two-echelon supply chain using SCOR metrics

This paper aims at multi-objective performance assessment and optimization of a multi-period two-echelon supply chain consisting of a supplier and a manufacturer. On the basis of the assessment system of the supply-chain operations reference model, the supply chain’s performance is investigated with respect to costs, assets, agility, reliability and responsiveness. First, methods to quantify these five performance attributes are put forward. Then a multi-objective mathematical programming model is developed for production decision making of components and products so that the supply chain’s performance frontier formed with Pareto efficient performance values can be achieved. Thereafter a simple augmented \(\epsilon \) -constraint method is proposed for searching for all Pareto efficient solutions of the multi-objective mathematical programming problem. Finally, efficiency of the method is demonstrated with a numerical example and a sensitivity analysis is implemented to reveal effects of capacity expansion on supply chains’ performance.     

With a focus on ‘Grundvorstellungen’ Part 1: a theoretical integration into current concepts

Current comparative studies such as PISA assess individual achievement in an attempt to grasp the concept of competence. Working with mathematics is then put into concrete terms in the area of application. Thereby, mathematical work is understood as a process of modelling: At first, mathematical models are taken from a real problem; then the mathematical model is solved; finally the mathematical solution is interpreted with a view to reality and the original problem is validated by the solution. During this cycle the main focus is on the transition between reality and the mathematical level. Mental objects are necessary for this transition. These mental objects are described in the German didactic with the concept of Grundvorstellungen'. In the delimitation to related educational constructs, ‘Grundvorstellungen’ can be described as mental models of a mathematical concept.     

Model validation in operations research

Numerous articles have appeared in the literature expressing different degrees of concern with the methodology of OR in general and with the validation of OR models in particular. Suggestions have been formulated to remove some of the shortcomings of the methodology as currently practised and to introduce modifications in the approach because of the changing nature of the problems tackled. Advances in modeling capabilities and solution techniques have also had considerable impact on the way validation is perceived. Large scale computer-based mathematical models and especially simulation models have brought new dimensions to the notion of validation. Terms like ‘confidence’, ‘credibility and reliability’, ‘model assessment and evaluation’, ‘usefulness and usability of the model’ have become rather common. This paper is an attempt, through an interpretation of the literature, to put model validation and related issues in a framework that may be of use both to model-builders and to decision-makers.     

Changed views on mathematical knowledge in the course of didactical theory development—independent corpus of scientific knowledge or result of social constructions?

The study tries to show one line of how the German didactical tradition has evolved in response to new theoretical ideas and new—empirical—research approaches in mathematics education. First, the classical mathematical didactics, notably ‘stoffdidaktik’ as one (besides other) specific German tradition are described. The critiques raised against ‘stoffdidaktik’ concepts [for example, forms of ‘progressive mathematisation’, ‘actively discovering learning processes’ and ‘guided reinvention’ (cf. Freudenthal, Wittmann)] changed the basic views on the roles that ‘mathematical knowledge’, ‘teacher’ and ‘student’ have to play in teaching–learning processes; this conceptual change was supported by empirical studies on the professional knowledge and activities of mathematics teachers [for example, empirical studies of teacher thinking (cf. Bromme)] and of students’ conceptions and misconceptions (for example, psychological research on students’ mathematical thinking). With the interpretative empirical research on everyday mathematical teaching–learning situations (for example, the work of the research group around Bauersfeld) a new research paradigm for mathematics education was constituted: the cultural system of mathematical interaction (for instance, in the classroom) between teacher and students.     

An analytical study of a swapping operation in the money market

Investors in money markets have found a class of transaction in which return from investment can be increased without increasing the risk taken. We formally analyse one of these operations (usually referred to as a ‘swap’) and obtain the condition under which it is possible to obtain a swap profit.From this condition, an indicator is obtained to aid the decision of when to engage a swap. The indicator evidences the role played by the spread between yield rates, time to maturity and timing, in a swap.Finally the indicator is used to derive some mathematical properties of the swapping operation. We show that a trend toward widening spreads is neither a necessary nor a sufficient condition for a swap-profit to be obtained. We give examples using hard data to support our theoretical results.     

Division in a binary representation for complex numbers

Computer operations involving complex numbers, essential in such applications as Fourier transforms or image processing, are normally performed in a ‘divide-and-conquer’ approach dealing separately with real and imaginary parts. A number of proposals have treated complex numbers as a single unit but all have foundered on the problem of the division process without which it is impossible to carry out all but the most basic arithmetic. This paper resurrects an early proposal to express complex numbers in a single ‘binary’ representation, reviews basic complex arithmetic and is able to provide a fail-safe procedure for obtaining the quotient of two complex numbers expressed in the representation. Thus, while an outstanding problem is solved, recourse is made only to readily accessible methods. A variety of extensions to the work requiring similar basic techniques are also identified. An interesting side-line is the occurrence of fractal structures, and the power of the ‘binary’ representation in analysing the structure is briefly discussed.     

An application of probability to combinatorics: a proof of Vandermonde identity

In this paper, we give possible suggestions for a classroom lesson about an application of probability using basic mathematical notions. We will approach to some combinatoric results without using ‘induction’, ‘polynomial identities’ nor ‘generating functions’, and will give a proof of the ‘Vandermonde Identity’ using elementary notions of probability.     

The concept ‘model’ and its potential role in mathematics education

Under the general term ‘mathematical models’ is now subsumed a sufficiently diversified collection of distinct types of models to merit categorization and an attempt at a unified definition. In this study a broad ‘working definition’ for the concept ‘model’ is proposed; basic types of models are identified and illustrated, and the impact of conscious instruction of models on mathematics education is analysed.     

A Symbolic Dance: The Interplay Between Movement,Notation, and Mathematics on a Journey Toward Solving Equations

This article analyzes the use of the software Grid Algebra with a mixed ability class of 21 nine-to-ten-year-old students who worked with complex formal notation involving all four arithmetic operations. Unlike many other models to support learning, Grid Algebra has formal notation ever present and allows students to “look through” that notation and interpret it either in terms of physical journeys on a grid or in terms of mathematical operations. A dynamic fluidity was found between the formal notation, imagery of movements on a grid, and the process of mathematical operations. This fluidity is interpreted as a “dance” between these three. The significant way in which this dynamic took place reflects the scaffolding and fading offered by the software, which was crucial to the students’ fluency with formal notation well beyond what has been reported from students of that age.     

Delivering the shape and space programme of study in the UK national mathematics curriculum using Zeno

The UK national curriculum in mathematics, through the shape and space programme of study, requires that all pupils engage in activities which lead to an understanding of transformation of the plane. This paper introduces Zeno, a package which offers vector operations and transformation primitives in addition to Logo turtle graphics‐like features. It runs on any IBM‐compatible (with CGA or better graphics) or Apple Macintosh personal computer. The paper demonstrates how the Zeno environment can enable pupils to ‘discover’ the central mathematical principles which underpin transformation of the plane.