“I’m not very good at solving problems”: An exploration of students’ problem solving behaviours

This paper reports one aspect of a larger study which looked at the strategies used by a selection of grade 6 students to solve six non-routine mathematical problems. The data revealed that the students exhibited many of the behaviours identified in the literature as being associated with novice and expert problem solvers. However, the categories of ‘novice’ and ‘expert’ were not fully adequate to describe the range of behaviours observed and instead three categories that were characteristic of behaviours associated with ‘naïve’, ‘routine’ and ‘sophisticated’ approaches to solving problems were identified. Furthermore, examination of individual cases revealed that each student's problem solving performance was consistent across a range of problems, indicating a particular orientation towards naïve, routine or sophisticated problem solving behaviours. This paper describes common problem solving behaviours and details three individual cases involving naïve, routine and sophisticated problem solvers.     

When does data envelopment analysis outperform a naïve efficiency measurement model?

This paper compares the results from data envelopment analysis (DEA) to a naïve efficiency measurement model, which generates a scalar efficiency score by averaging all output–input ratios. Random data and real-life data are used to test the relative performance of the naïve model against various DEA models. The results suggest that the proposed the naïve model replicates DEA efficiency scores almost perfectly for constant return-to-scales and low heterogeneity in output–input data. It is therefore concluded that heterogeneity in output–input data is important to take advantage of the capability of DEA. It is also shown that heterogeneity is more relevant to efficiency measurement than the number of dimensions.     

Ship-to-order supplies: Contract breachability and the impact of a manufacturer-owned direct channel

We address the issue of contract breachability in a supply chain involving a retailer and a manufacturer operating under ship-to-order contract terms and stochastic demands. The manufacturer is required to fulfill the retailer’s demands on a continuous basis with little or no advance notice. The issue in such an environment is whether the retailer can “naively” assume that she will get a very high fill rate from the manufacturer and therefore has no need for contract penalties in case the manufacturer’s inventory falls short. We suggest a stochastic calculus framework to study the problem and derive a condition when the retailer’s naïve assumption is justified since the probability of stock-outs of the manufacturer is negligible. That is, the ship-to-order contract will not be breached and the fill rate will be more than a predetermined threshold. Furthermore we find that although the manufacturer-owned direct channel generates more revenue and may reduce the volatility of both inventory and production orders, the ratio between expected direct channel and retail sales affects the benefits.     

Optimizing a dynamic order-picking process

This research studies the problem of batching orders in a dynamic, finite-horizon environment to minimize order tardiness and overtime costs of the pickers. The problem introduces the following trade-off: at every period, the picker has to decide whether to go on a tour and pick the accumulated orders, or to wait for more orders to arrive. By waiting, the picker risks higher tardiness of existing orders on the account of lower tardiness of future orders. We use a Markov decision process (MDP) based approach to set an optimal decision making policy. In order to evaluate the potential improvement of the proposed approach in practice, we compare the optimal policy with two naïve heuristics: (1) “Go on tour immediately after an order arrives”, and, (2) “Wait as long as the current orders can be picked and supplied on time”. The optimal policy shows a considerable improvement over the naïve heuristics, in the range of 7–99%, where the specific values depend on the picking process parameters. We have found that one measure, the slack percentage of the picking process, associated with the difference between the promised lead time and the single item picking time, predicts quite accurately the cost reduction generated by the optimal policy. Since relatively small-scale problems could be solved by the optimal algorithm, a heuristic was developed, based on the structure and properties of the optimal solutions. Numerical results show that the proposed heuristic, MDP-H, outperforms the naïve heuristics in all experiments. As compared to the optimal solution, MDP-H provides close to optimal results for a slack of up to 40%.     

Measuring retail company performance using credit scoring techniques

This paper discusses models for evaluating credit risk in relation to the retailing industry. Hunt’s [Hunt, S.D., 2000. A General Theory of Competition. Sage Publications Inc., California] Resource–Advantage Theory of Competition is used as a basis for variable selection, given the theory’s relevancy to retail competition. The study focuses on the US retail market. Four standard credit scoring methodologies: Naïve Bayes, Logistic Regression, Recursive Partitioning and Artificial Neural Network, are compared with Sequential Minimal Optimization (SMO), using a sample of 195 healthy companies and 51 distressed firms over five time periods from 1994 to 2002.     

Bounding the Elliptic Mahler Measure II

The theory of heights for rational points on arithmetic ellipticcurves is becoming well known. An important fact in the basictheory is the relationship between the naïve and the canonicalheight of a rational point; in fact, they differ by a uniformlybounded amount. The paper provides a generalisation of thisfact from the point of view of heights of polynomials, ratherthan heights of points. This concept is extended to polynomialsin several variables.     

Students’ reflections on their learning experiences: lessons from a longitudinal study on the development of mathematical ideas and reasoning

This paper characterizes the views on mathematical learning of five high school students based on the students’ reflections on their mathematical experiences in a longitudinal study that focused on the development of mathematical ideas and reasoning in particular research conditions. The students’ views are presented according to five themes about learning which describe the students’ views on the nature of knowledge and what it means to know, source of knowledge, motivation to engage in learning, certainty in knowing, and how the students’ views vary with particular areas of mathematical activity. The study addresses the need for more research on epistemological beliefs of students below college age. In particular, the results provide evidence that challenge the existing assumption that, prior to college, students exhibit naïve epistemological beliefs.     

On Waring’s problem for several algebraic forms

We reconsider the classical problem of representing a finite number of forms of degree d in the polynomial ring over n + 1 variables as scalar combinations of powers of linear forms. We define a geometric construct called a grove, which, in a number of cases, allows us to determine the dimension of the space of forms which can be so represented for a fixed number of summands. We also present two new examples, where this dimension turns out to be less than what a naïve parameter count would predict.     

Application of machine learning techniques for supply chain demand forecasting

Full collaboration in supply chains is an ideal that the participant firms should try to achieve. However, a number of factors hamper real progress in this direction. Therefore, there is a need for forecasting demand by the participants in the absence of full information about other participants’ demand. In this paper we investigate the applicability of advanced machine learning techniques, including neural networks, recurrent neural networks, and support vector machines, to forecasting distorted demand at the end of a supply chain (bullwhip effect). We compare these methods with other, more traditional ones, including naïve forecasting, trend, moving average, and linear regression. We use two data sets for our experiments: one obtained from the simulated supply chain, and another one from actual Canadian Foundries orders. Our findings suggest that while recurrent neural networks and support vector machines show the best performance, their forecasting accuracy was not statistically significantly better than that of the regression model.     

Tabu search and GRASP for the maximum diversity problem

In this paper, we develop new heuristic procedures for the maximum diversity problem (MDP). This NP-hard problem has a significant number of practical applications such as environmental balance, telecommunication services or genetic engineering. The proposed algorithm is based on the tabu search methodology and incorporates memory structures for both construction and improvement. Although proposed in seminal tabu search papers, memory-based constructions have often been implemented in naïve ways that disregard important elements of the fundamental tabu search proposals. We will compare our tabu search construction with a memory-less design and with previous algorithms recently developed for this problem. The constructive method can be coupled with a local search procedure or a short-term tabu search for improved outcomes. Extensive computational experiments with medium and large instances show that the proposed procedure outperforms the best heuristics reported in the literature within short computational times.     

The benefits of differential variance-based constraints in portfolio optimization

The main problem of portfolio optimization is parameter estimation error. Various methods have been suggested to mitigate this problem, among which are shrinkage, resampling, Bayesian updating, naïve diversification, and imposing constraints on the portfolio weights. This study suggests two substantial extensions of the constrained optimization approach: the Variance-Based Constraints (VBC), and the Global Variance-Based Constraints (GVBC) methods. By the VBC method the constraint imposed on the weight of a given stock is inversely proportional to its standard deviation: the higher a stock’s sample standard deviation, the higher the potential estimation error of its parameters, and therefore the tighter the constraint imposed on its weight. GVBC employs a similar idea, but instead of imposing a sharp boundary constraint on each stock, a quadratic “cost” is assigned to deviations from the naive 1/N weight, and a single global constraint is imposed on the total cost of all deviations. Comparing ten optimization methods we find that the two new suggested methods typically yield the best performance, as measured by the Sharpe ratio. GVBC ranks first. These results are obtained for two different datasets, and are also robust to the number of assets under consideration.     

Smoothed Monte Carlo estimators for the time-in-the-red in risk processes

We consider a modified version of the de Finetti model in insurance risk theory in which, when surpluses become negative the company has the possibility of borrowing, and thus continue its operation. For this model we examine the problem of estimating the time-in-the red over a finite horizon via simulation. We propose a smoothed estimator based on a conditioning argument which is very simple to implement as well as particularly efficient, especially when the claim distribution is heavy tailed. We establish unbiasedness for this estimator and show that its variance is lower than the naïve estimator based on counts. Finally we present a number of simulation results showing that the smoothed estimator has variance which is often significantly lower than that of the naïve Monte-Carlo estimator.     

Runge-Kutta stability on a Floquet problem

This work examines the stability of explicit Runge-Kutta methods applied to a certain linear ordinary differential equation with periodic coefficients. On this problem naïve use of the eigenvalues of the Jacobian results in misleading conclusions about stable behaviour. It is shown, however, that a valid analogue of the classical absolute stability theory can be developed. Further, using a suitable generalisation of the equilibrium theory of Hall [ACM Trans. on Math. Soft. 11 (1985), pp. 289–301], accurate predictions are made about the performance of modern, adaptive algorithms.Supported by the University of Dundee Research Initiatives Fund.     

A nearest neighbor sweep circle algorithm for computing discrete Voronoi tessellations

An algorithm for computing discrete, 2-dimensional, Euclidean Voronoi tessellations is presented. The algorithm combines a limiting sweep circle approach with a nearest neighbor cellular approach. It reduces the computational cost of the naïve approach while at the same time giving the Euclidean Voronoi tessellations that simple nearest neighbor algorithms are unable to produce. The algorithm is shown, through analytical methods, to produce good approximations to corresponding continuous Voronoi tessellations depending on the definition of neighbor used in the nearest neighbor step and the mesh size. The quality of different types of neighbor definitions are discussed as well as the computational cost. The algorithm is general enough to be easily extended to higher dimensions and nonuniform meshes. The analysis lays the groundwork for the computation of discrete centroidal Voronoi tessellations where some kind of numerical integration is required.     

A mixed integer programming model for long term capacity expansion planning: A case study from The Hunter Valley Coal Chain

The Hunter Valley Coal Chain is the largest coal export operation in the world with a throughput in excess of 100 million tonnes per annum (Mtpa). Coal is delivered to the shipping terminal from 40 mines using 27 coal load points spread across the Hunter Valley region. This paper describes an MILP model for determining the capacity requirements, and the most cost effective capacity improvement initiatives, to meet demand while minimising the total cost of infrastructure and demurrage. We present results from computational experiments on the model’s performance along with a comparison of the model’s output with detailed analyses by the coal chain analysts and planners.     

A simple analytic method for computing the natural frequencies and mode shapes of tall buildings

A simple, yet accurate analytical approach based on energy principles is developed for quick computation of natural frequencies and mode shapes of multistory buildings constructed using framed tube, shear core and double belt trusses systems. The approach here is based on development of a continuum model that would be equivalent, in major motions, to the actual multistory building. Models studied here are cantilever beams with concentrated moments placed at belt truss locations. Governing equation and boundary conditions of the equivalent beam and moment system were derived using the energy method and Hamilton’s principle. Separation of variables technique is then applied to model’s partial differential equation to obtain the required eigensystem. Robustness and correctness of the proposed method are demonstrated through several numerical examples. Here, 40, 55 and 70-storey tall buildings with combined system of framed tube, shear core and double belt trusses, in which the results obtained from the proposed method, are compared with those obtained from three-dimensional analyses using SAP2000 software. Comparative analyses reveal that the proposed method is simple and efficient; and it provides reasonably accurate results quickly, a feature that is vital during the early stages of building design.     

Predicting the event and time horizon of bankruptcy using financial ratios and the maturity schedule of long-term debt

We study the problem of simultaneous and coherent assessment the probability of a firm’s bankruptcy at various time horizons in future. In contrast with usual (one-period) formulations of the problem, such multi-period formulation better matches the nature of bankruptcy process (bankruptcy occurs in time) and allows an easier and more natural incorporation of bankruptcy (default) prognoses in valuation of risky debt and equity, optimization of corporate capital structure etc. The study uses a new mathematical apparatus—multi-alternative decision rules of statistical decision theory. We investigate a new type of predictive variables that can be extracted from the maturity schedule of a firm’s long-term debt. The study develops Bayesian-type forecasting rules that use both maturity schedule factors and traditional financial ratios. These rules noticeably enhance bankruptcy prediction (compared with the familiar one-period Z-score rules of Altman) for bankruptcy within the first 1, 2 or 3 years. Predictive factors derived from schedule information enhance bankruptcy prediction at distant time horizons.     

Application of artificial neural networks for the prediction of roll force and roll torque in hot strip rolling process

This paper introduces an artificial neural network (ANN) application to a hot strip mill to improve the model’s prediction ability for rolling force and rolling torque, as a function of various process parameters. To obtain a data basis for training and validation of the neural network, numerous three dimensional finite element simulations were carried out for different sets of process variables. Experimental data were compared with the finite element predictions to verify the model accuracy. The input variables are selected to be rolling speed, percentage of thickness reduction, initial temperature of the strip and friction coefficient in the contact area. A comprehensive analysis of the prediction errors of roll force and roll torque made by the ANN is presented. Model responses analysis is also conducted to enhance the understanding of the behavior of the NN model. The resulted ANN model is feasible for on-line control and rolling schedule optimization, and can be easily extended to cover different aluminum grades and strip sizes in a straight-forward way by generating the corresponding training data from a FE model.     

Order-enriched categorical models of the classical sequent calculus

It is well-known that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the well-known categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cut-reduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish non-deterministic choices of cut-elimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.     

On testing equality of pairwise rank correlations in a multivariate random vector

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical tests for the hypothesis that all pairwise Spearman’s rank correlation coefficients in a multivariate random vector are equal. The tests are nonparametric and their asymptotic distributions are derived based on the asymptotic behavior of the empirical copula process. Only weak assumptions on the distribution function, such as continuity of the marginal distributions and continuous partial differentiability of the copula, are required for obtaining the results. A nonparametric bootstrap method is suggested for either estimating unknown parameters of the test statistics or for determining the associated critical values. We present a simulation study in order to investigate the power of the proposed tests. The results are compared to a classical parametric test for equal pairwise Pearson’s correlation coefficients in a multivariate random vector. The general setting also allows the derivation of a test for stochastic independence based on Spearman’s rho.