Ang, Keng Cheng

Many types of pricing models incorporating different forms of demand functions have emerged in the past years. In an earlier work, a piecewise- defined Complementarity-Constrained Demand Function (CCDF) was discussed to correct certain weaknesses in commonly used demand functions. The authors introduced a Complementarity-Constrained (CC) pricing model incorporating the CCDF in that same work. However, there was a lack of numerical implementations therein. Hence in a separate work, we developed an algorithm using MATLAB to compare a generic pricing model and a CC pricing model. Experiments were performed to compare the revenues from the two models for certain ranges of parameters defining the demand function. In this work, we conduct further numerical testing by simulating the bidding behaviours of different types of customers and using simulated demand data to compare the models. We find that the use of the CC model leads to higher revenues for certain simulated scenarios.

Queuing theory is usually introduced to students from second year onwards in a university undergraduate programme, as the mathematical principles governing queues can be fairly demanding, making it challenging to introduce any earlier. However, we often see queues and experience queuing in real life. It would therefore be appropriate, relevant and useful to introduce the concept of queuing theory to preuniversity students or first- year undergraduates. The approach suggested is through simulation models supported by suitable technology. In doing so, students can understand some basic probability theory and statistical concepts, such as the Poisson process and exponential distribution, and learn how queues may be modelled through simulation, without the need to know all about classical queuing theory. In this paper, we will discuss the role that simulation can play in a classroom to create real world learning experiences for students. To provide a concrete illustration, a set of real data collected in a simple ATM queue will be used to explain how students can systematically be engaged in a modelling activity involving queues. Following that, queues at cinema ticketing counters are studied to discuss the modelling of a more complex queue system.

Mathematical modelling has been gaining attention and becoming a part of classroom practice in many countries. In Singapore, despite recognizing its importance and relevance, curriculum planners and teachers face various challenges in including and incorporating mathematical modelling in their teaching curriculum. Nonetheless, the recommended practice is to expose students to learning experiences in mathematical modelling whenever and wherever possible. In this paper, a framework which serves a practical guide for teachers in planning instruction in mathematical modelling will be introduced. Examples illustrating the application of this framework by teachers in crafting classroom learning experience in mathematical modelling will be presented. In addition, a learning experience implemented for a group of teachers in an in-service course will also be discussed. It is no coincidence that technology had featured quite prominently in these examples as mathematical modelling in practice would often involve the use of some specific technological tools.

We study the SIR model of transmission of dengue fever with two pathogen strains. A model is constructed to study the effects of different factors on the course of the epidemic. The difference between the two strains is not discussed. Instead, we focus on the trends of primary infection and secondary infection. Our analysis shows that factors related to the host (such as host population) do not change the pattern the spread significantly. In contrast, factors related to the vector (such as vector population, vector life span and biting rate) have a more significant effect on the outbreak of secondary infection.

This paper posits that teachers’ pedagogical content knowledge in mathematical modelling instruction can be demonstrated in the crafting of action plans and expected teaching and learning moves via their lesson images (Schoenfeld, 1998). It can also be developed when teachers shape appropriate teaching moves in response to students’ learning actions. Such adaptive development of teachers’ pedagogical content knowledge may in turn be supported by their knowledge of the mathematical modelling process and Ang’s (to appear) proposed framework for planning mathematical modelling instruction.

In the last few decades, there have been abundant discussions among mathematicians and mathematics educators on promoting mathematical modelling (a process of using mathematics to tackle real world problems) as a classroom practice. Mathematics educators and curriculum planners have been advocating the teaching of mathematical modelling in schools for some time now. Despite the consensus on its importance and relevance, mathematical modelling remains a difficult activity for both teachers and learners to fully engage in. In this paper, we examine some of these difficulties and discuss how technology can play a pivotal role in providing the essential support to make mathematical modelling a more accessible mathematical activity amongst students. Through a series of examples drawn from different fields and topics, we illustrate how a range of technological tools may be successfully and efficiently utilized in modelling tasks. In addition, we discuss the need for an optimal use of technology to balance between achieving the objectives of the tasks and attaining the goals of learning mathematics.

Mathematical modelling can play a very important role in cancer research. In particular, modelling the growth of tumour has the potential of shedding light on the mechanisms of tumour cell growth and proliferation. In this paper, we examine and analyze one such model with the aid of MATLAB. The model, first proposed by Sherratt and Chaplain in 2001, is based on a set of partial differential equations. The equations describe the growth, movement and death of tumour cells, accompanied by a supply of nutrients. This spatial-temporal model depends on a number of parameter values as well as rate functions. The model is solved numerically using finite difference method implemented on MATLAB. Effects and influence of the parameter values and rate functions are analyzed. Results are validated against a set of known experimental data, and good agreement is observed.

In this paper, we examine the use of a simple stochastic di erential equation in the modelling of an epidemic. Real data for the Singapore SARS outbreak are used for a detailed study. The model is solved numerically and implemented on matlab, with further analysis and re nement. This article is built around several matlab programs and serves to provide a practical and accessible introduction to numerical methods for a stochastic model for epidemics.