Emergent models in a reinvention activity for learning the slope of a curve

Introducing the slope of a curve as the limit of the slope of secant lines is a well-known challenge in mathematics education. As an alternative, three other approaches can be recognized, based on linear approximation, based on multiplicities, or based on transition points. In this study we investigated which of these approaches fits students most by analyzing students’ inventions during a lesson scenario revolving around a design problem. The problem is set in a context that is meaningful to students and invites them to invent methods to construct a tangent line to a curve: an implementation of the guided reinvention principle from Realistic Mathematics Education (RME). The teaching scenario is based on the phased lesson structure of the Theory of Didactical Situations (TDS). The scenario was tested with 44 groups of three students in six grade 9 or 10 classrooms. We classified the strategies used by students and, using the emergent models-principle from RME, investigated to which of the four approaches the student strategies connect best. The results show that the groups produced a variety of strategies in each classroom and these strategies contributed to a meaningful institutionalization of the notion of slope of a curve.     

Optimizing strategic behaviour in a dynamic setting in professional team sports

This article develops a dynamic game-theoretic model of optimizing strategic behaviour by football (soccer) teams. Teams choose between defensive and attacking formations and between a non-violent and a violent playing style, and can vary these choices continuously throughout each match. Starting from the end of the match and working backwards, the teams’ optimal strategies conditional on the current state of the match are determined by solving a series of two-player non-cooperative subgames. Numerical simulations are used to explore the sensitivity of strategic behaviour to variations in the structural parameters. The analysis demonstrates that the strategic behaviour of football teams can be rationalized in accordance with game-theoretic principles of optimizing strategic behaviour by agents when payoffs are uncertain and interdependent.     

A simulated annealing heuristic for the team orienteering problem with time windows

This paper presents a simulated annealing based heuristic approach for the team orienteering problem with time windows (TOPTW). Given a set of known locations, each with a score, a service time, and a time window, the TOPTW finds a set of vehicle tours that maximizes the total collected scores. Each tour is limited in length and a visit to a location must start within the location’s service time window. The proposed heuristic is applied to benchmark instances. Computational results indicate that the proposed heuristic is competitive with other solution approaches in the literature.     

Flexible software reliability growth model with testing effort dependent learning process

A lot of importance has been attached to the testing phase of the Software Development Life Cycle (SDLC). It is during this phase it is checked whether the software product meets user requirements or not. Any discrepancies that are identified are removed. But testing needs to be monitored to increase its effectiveness. Software Reliability Growth Models (SRGMs) that specify mathematical relationships between the failure phenomenon and time have proved useful. SRGMs that include factors that affect failure process are more realistic and useful. Software fault detection and removal during the testing phase of SDLC depend on how testing resources (test cases, manpower and time) are used and also on previously identified faults. With this motivation a Non-Homogeneous Poisson Process (NHPP) based SRGM is proposed in this paper which is flexible enough to describe various software failure/reliability curves. Both testing efforts and time dependent fault detection rate (FDR) are considered for software reliability modeling. The time lag between fault identification and removal has also been depicted. The applicability of our model is shown by validating it on software failure data sets obtained from different real software development projects. The comparisons with established models in terms of goodness of fit, the Akaike Information Criterion (AIC), Mean of Squared Errors (MSE), etc. have been presented.     

A consumer purchasing model with learning and departure behaviour

The purpose of this paper is to extend the model of negative binominal distribution used in consumer purchasing models so as to incorporate the consumer's learning and departure behaviours. The regularity of interpurchase time and its unobserved heterogeneity are also included. Due to these extensions, this model can be used to determine during a given period how many purchases are made by an experienced or an inexperienced customer. This model also allows the determination of the probability that a customer with a given pattern of purchasing behaviour still remains, or has departed, at any time after k≥1 purchases are made. An illustration of the approach is conducted using consumer purchase data for tea. As assessed by comparing results with Theil's U, the integrated model developed gives the best results and shows that learning and departure are important factors which influence consumer's purchase behaviour, especially, when evaluating the behaviour of inexperienced customers.     

A learning approach to the two person decentralized team problem with incomplete information

A learning approach to the two person decentralized team problem with incomplete information and with a 2X2 payoff matrix is considered. It is shown that if the payoff matrix is unimodal, there exists a proper choice of parameters of the learning algorithm that will ensure asymptotically an expected payoff as close to maximum payoff as desired. It is shown that a team problem with multimodal payoff matrix gives rise to interesting class of open problems.     

Comparing German and Taiwanese secondary school students’ knowledge in solving mathematical modelling tasks requiring their assumptions

Mathematization is critical in providing students with challenges for solving modelling tasks. Inadequate assumptions in a modelling task lead to an inadequate situational model, and to an inadequate mathematical model for the problem situation. However, the role of assumptions in solving modelling problems has been investigated only rarely. In this study, we intentionally designed two types of assumptions in two modelling tasks, namely, one task that requires non-numerical assumptions only and another that requires both non-numerical and numerical assumptions. Moreover, conceptual knowledge and procedural knowledge are also two factors influencing students’ modelling performance. However, current studies comparing modelling performance between Western and non-Western students do not consider the differences in students’ knowledge. This gap in research intrigued us and prompted us to investigate whether Taiwanese students can still perform better than German students if students’ mathematical knowledge in solving modelling tasks is differentiated. The results of our study showed that the Taiwanese students had significantly higher mathematical knowledge than did the German students with regard to either conceptual knowledge or procedural knowledge. However, if students of both countries were on the same level of mathematical knowledge, the German students were found to have higher modelling performance compared to the Taiwanese students in solving the same modelling tasks, whether such tasks required non-numerical assumptions only, or both non-numerical and numerical assumptions. This study provides evidence that making assumptions is a strength of German students compared to Taiwanese students. Our findings imply that Western mathematics education may be more effective in improving students’ ability to solve holistic modelling problems.     

On the asymptotic behaviour of linear learning processes with partial forgetting

For linear learning processes with partial forgetting a recursion of the form $$X_{n + 1} = \frac{1}{n}\sum\limits_{k = j_n }^n {(B_k X_k + b_k ) with \frac{{j_n }}{n}} \to \alpha \in [0,1)$$ in a real separable Banach spaceB is investigated. Results on a.s. convergence and rate of convergence of (X _n ) (loglog and distributional invariance principle) are obtained under corresponding assumptions on the sequences of arithmetic means of the random elementsb _n andB _n inB andL(B), resp., together with suitable spectral conditions on the (deterministic) Cesàro limit of (B _n ). By the special casej _n ≡1 results on stochastic approximation are comprehended. The limit process in the invariance principles satisfies a stochastic differential equation with delay.     

Exploring a structure for mathematics lessons that initiate learning by activating cognition on challenging tasks

While there is widespread agreement on the importance of incorporating problem solving and reasoning into mathematics classrooms, there is limited specific advice on how this can best happen. This is a report of an aspect of a project that is examining the opportunities and constraints in initiating learning by posing challenging mathematics tasks intended to prompt problem solving and reasoning to students, not only to activate their thinking but also to develop an orientation to persistence. Data were sought from teachers and students in middle primary classes (students aged 8–10 years) via online surveys. One lesson focusing on the concept of equivalence is described in detail although mention is made of other lessons. The research questions focused on the teachers’ reactions to the lesson structure and the specifics of the implementation in a particular school. The results indicate that student learning is facilitated by the particular lesson structure. This article reports on the implementation of this lesson structure and also on the finding that students’ responses to the lessons can be used to inform subsequent learning experiences.     

Emergent behaviors of a thermodynamic Cucker-Smale flock with a time-delay on a general digraph

We present emergent flocking dynamics of a thermodynamic Cucker-Smale (TCS) flock on a general digraph with spanning trees under the effect of communication time-delays. The TCS model describes a temporal evolution of mechanical and thermodynamic observables such as position, velocity and temperature of CS particles. In this paper, we study how variations in mechanical and thermodynamic variables can decay to zero along a time-independent network with position dependent weights from initial state configuration. For this, we provide a sufficient framework for a mechanical and thermodynamical flocking in terms of initial configuration, network topology, and system parameters. We also present several numerical examples and compare them with analytical results.     

Distributed computing viewed as a team problem with constraints

The optimal distribution of the workload in a system of interconnected computer units is considered. Formulated as a team decision problem with a singular cost criterion and with equality and inequality constraints, it is shown that the problem admits always a unique piecewise linear strategy which is globally optimal. Some interesting particular cases are studied.The research reported in this paper was made possible through support from the Office of Naval Research under the Joint Services Electronics Program by Contract No. N00014-75-C-0648 and Contract No. N00014-77-C-0531 and by the National Science Foundation, Grant No. ENG-76-11824.     

Coordinating a supply chain with effort and price dependent stochastic demand

This paper investigates the issue of channel coordination for a supply chain facing stochastic demand that is sensitive to both sales effort and retail price. In the standard newsvendor setting, the returns policy and the revenue sharing contract have been shown to be able to align incentives of the supply chain’s members so that the decentralized supply chain behaves as well as the integrated one. When the demand is influenced by both retail price and retailer sales effort, none of the above traditional contracts can coordinate the supply chain. To resolve this issue, we explore a variety of other contract types including joint return policy with revenue sharing contract, return policy with sales rebate and penalty (SRP) contract, and revenue sharing contract with SRP. We find that only the properly designed returns policy with SRP contract is able to achieve channel coordination and lead to a Pareto improving win–win situation for supply chain members. We then provide analytical method to determine the contract parameters and finally we use a numerical example to illustrate the findings and gain more insights.     

Route problem with constraints depending on a list of tasks

An additive route problem with preceding conditions is considered in which the cost function and the move constraints both depend on a list of tasks that have not been performed by the current time. The problem is solved by applying a dynamic programming method that takes into account both these factors and is implemented in the construction of a (generally) incomplete array of Bellman function values.     

Complexity of sales effort and carbon emission reduction effort in a two-parallel household appliance supply chain model

This paper studies the complexity of sales effort and carbon emission reduction effort in a Bertrand household appliance supply chain system. A two-parallel model consisting of a traditional supply chain without any carbon emission reduction effort and a low-carbon supply chain with carbon emission reduction effort is established. The chain to chain competition is analyzed in three scenarios, one of which is a horizontal Nash game, and the others are the traditional supply chain Stackelberg game and the low-carbon supply chain Stackelberg game. The optimal solutions of the horizontal Nash game and the Stackelberg game are obtained and three models’ dynamic evolutions based on the bounded rationality are investigated. A dynamic Stackelberg game model with respect to five variables is proposed and investigated via the stable region, the bifurcation, and the maximum Lyapunov exponent. The profits of the two-parallel household appliance supply chains are compared in three dynamic game structures. An interesting phenomenon that the system will enter a six-cycle state after the two-cycle state and fall into chaos directly is discovered in the traditional supply chain Stackelberg game. Our results suggest that the adjustments of price would affect the stability and profits much more than the sales effort and carbon emission reduction effort in all three game structures. Every supply chain should take the suitable adjustment speeds for the price and sales effort to keep the system in the stable state. Each adjustment should not exceed the domain of attraction.     

Strategic behaviour in a tandem queue with alternating server

Queueing Systems - This paper considers an unobservable two-site tandem queueing system attended by an alternating server. We study the strategic customer behaviour under two threshold-based...     

Working with tasks for the learning of problem solving in maths teaching as an issue of the first teacher training phase

This article describes learning goals of teacher training for the working with tasks in maths lessons. Selected common and different features of tasks intended for the learning and performing are especially referred to     

Optimal timing of a sequence of tasks with general completion costs

Scheduling a sequence of tasks––in the acceptation of finding the execution times––is not a trivial problem when the optimization criterion is irregular as for instance in earliness–tardiness problems. This paper presents an efficient dynamic programming algorithm to solve the problem with general cost functions depending on the end time of the tasks, idle time costs and variable durations also depending on the execution time of the tasks. The algorithm is also valid when the precedence graph is a tree and it can be adapted to determine the possible execution windows for each task not exceeding a maximum fixed cost.     

Scheduling with a position-weighted learning effect

In general, human learning takes time to build up, which results from a worker gaining experience from repeating similar operations over time. In the early stage of processing a given set of similar jobs, a worker is not familiar with the operations, so his learning effect on the jobs scheduled early is not apparent. On the other hand, when the worker has gained experience in processing the jobs his learning improves. So a worker’s learning effect on a job depends not only on the total processing time of the jobs that he has processed but also on the job position. In this paper we introduce a position-weighted learning effect model for scheduling problems. We provide optimal solutions for the single-machine problems to minimize the makespan and the total completion time, and an optimal solution for the single-machine problem to minimize the total tardiness under an agreeable situation. We also consider two special cases of the flowshop problem.     

Flow-shop scheduling with a learning effect

The paper is devoted to some flow-shop scheduling problems with a learning effect. The objective is to minimize one of the two regular performance criteria, namely, makespan and total flowtime. A heuristic algorithm with worst-case bound m for each criteria is given, where m is the number of machines. Furthermore, a polynomial algorithm is proposed for both of the special cases: identical processing time on each machine and an increasing series of dominating machines. An example is also constructed to show that the classical Johnson's rule is not the optimal solution for the two-machine flow-shop scheduling to minimize makespan with a learning effect. Some extensions of the problem are also shown.