Single-machine scheduling simultaneous with position-based and sum-of-processing-times-based learning considerations under group technology assumption

This paper considers the problems of scheduling with the effect of learning on a single-machine under group technology assumption. We propose a new learning model where the job actual processing time is linear combinations of the scheduled position of the job and the sum of the normal processing time of jobs already processed. We show that the makespan minimization problem is polynomially solvable. We also prove that the total completion time minimization problem with the group availability assumption remains polynomially solvable under agreeable conditions.     

Comments on “A bicriteria parallel machine scheduling with a learning effect of setup and removal times”

The purpose of this paper is to point out some errors in Eren [Applied Mathematical Modelling, 33 (2009) 1141–1150]. We also suggest some methods to fix them.     

Simple and U-type assembly line balancing problems with a learning effect

In this paper, we introduced learning effect into assembly line balancing problems. In many realistic settings, the produced worker(s) (or machine(s)) develops continuously by repeated the same or similar activities. Therefore, the production time of product shortens if it is processed later. We show that polynomial solutions can be obtained for both simple assembly line balancing problem (SALBP) and U-type line balancing problem (ULBP) with learning effect.     

Derivative reproducing properties for kernel methods in learning theory

The regularity of functions from reproducing kernel Hilbert spaces (RKHSs) is studied in the setting of learning theory. We provide a reproducing property for partial derivatives up to order s when the Mercer kernel is C^2s. For such a kernel on a general domain we show that the RKHS can be embedded into the function space C^s. These observations yield a representer theorem for regularized learning algorithms involving data for function values and gradients. Examples of Hermite learning and semi-supervised learning penalized by gradients on data are considered.     

Single-machine scheduling problems with a learning effect

In many situations, the skills of workers continuously improve when repeating the same or similar tasks. This phenomenon is known as the “learning effect” in the literature. In most studies, the learning phenomenon is implemented by assuming the actual job processing time is a function of its scheduled position [D. Biskup, Single-machine scheduling with learning considerations, Eur. J. Oper. Res. 115 (1999) 173–178]. Recently, a new model is proposed where the actual job processing time depends on the sum of the processing times of jobs already processed [C. Koulamas, G.J. Kyparisis, Single-machine and two-machine flowshop scheduling with general learning functions, Eur. J. Oper. Res. 178 (2007) 402–407]. In this paper, we extend their models in which the actual job processing time not only depends on its scheduled position, but also depends on the sum of the processing times of jobs already processed. We then show that the single-machine makespan and the total completion time problems remain polynomially solvable under the proposed model. In addition, we show that the total weighted completion time has a polynomial optimal solution under certain agreeable solutions.     

Machine-part cell formation in group technology using a modified ART1 method

Group Technology (GT) is a useful way of increasing the productivity for manufacturing high quality products and improving the flexibility of manufacturing systems. Cell formation (CF) is a key step in GT. It is used in designing good cellular manufacturing systems using the similarities between parts in relation to the machines in their manufacture. It can identify part families and machine groups. Recently, neural networks (NNs) have been widely applied in GT due to their robust and adaptive nature. NNs are very suitable in CF with a wide variety of real applications. Although Dagli and Huggahalli adopted the ART1 network with an application in machine-part CF, there are still several drawbacks to this approach. To address these concerns, we propose a modified ART1 neural learning algorithm. In our modified ART1, the vigilance parameter can be simply estimated by the data so that it is more efficient and reliable than Dagli and Huggahalli’s method for selecting a vigilance value. We then apply the proposed algorithm to machine-part CF in GT. Several examples are presented to illustrate its efficiency. In comparison with Dagli and Huggahalli’s method based on the performance measure proposed by Chandrasekaran and Rajagopalan, our modified ART1 neural learning algorithm provides better results. Overall, the proposed algorithm is vigilance parameter-free and very efficient to use in CF with a wide variety of machine/part matrices.     

Erratum to “Single machine past-sequence-dependent setup times scheduling with general position-dependent and time-dependent learning effects” [Appl. Math. Modell. 35 (2011) 1388–1395]

The aim of this paper is to show by a counterexample that Theorem 7 and Corollary 8 in Wang and Li [Appl. Math. Modell. 35 (2011) 1388–1395.] are incorrect.     

About the concept of angle in elementary school: Misconceptions and teaching sequences

This paper reports classroom research dealing with the difficulties encountered by schoolchildren in the acquisition of angle concept. Two obstacles were pointed out in previous studies: the side-length obstacle and the salience of the prototypical right angle. The first aim of the present study is to determine the extent to which a teaching sequence based on a concrete situation in the meso-space can enable pupils to progress in their conceptualization of angles. This problem situation is based on the notion of visual field. The angle appears in real space between two infinite directions that correspond to two lines of sight. The specificity of this situation is to confront pupils with an angle between two infinite directions in space. The second goal of this research is to study the links between the two obstacles. To answer these research questions, we compared two versions of the teaching sequence, one dynamic (the angle varies) and one static (the angle does not vary) in 3rd and 4th grade classes. The unfolding of the sequence was analyzed and pupils were tested individually before and after the sequence. They were requested to draw angles and angle variations. The results showed that (1) the sequence helped the pupils progress (2) the obstacle of side-length is not the only difficulty faced by pupils; the salience of the prototypical right angle constitutes a real learning obstacle and (3) the type of angle produced and the ability to change its size are linked. In conclusion, the implications for teaching are presented.     

Learning trajectories in teacher education: Supporting teachers’ understandings of students’ mathematical thinking

Recent work by researchers has focused on synthesizing and elaborating knowledge of students’ thinking on particular concepts as core progressions called learning trajectories. Although useful at the level of curriculum development, assessment design, and the articulation of standards, evidence is only beginning to emerge to suggest how learning trajectories can be utilized in teacher education. Our paper reports on two studies investigating practicing and prospective elementary teachers’ uses of a learning trajectory to make sense of students’ thinking about a foundational idea of rational number reasoning. Findings suggest that a mathematics learning trajectory supports teachers in creating models of students’ thinking and in restructuring teachers’ own understandings of mathematics and students’ reasoning.     

The optimal solution of multi-kernel regularization learning

In regularized kernel methods, the solution of a learning problem is found by minimizing a functional consisting of a empirical risk and a regularization term. In this paper, we study the existence of optimal solution of multi-kernel regularization learning. First, we ameliorate a previous conclusion about this problem given by Micchelli and Pontil, and prove that the optimal solution exists whenever the kernel set is a compact set. Second, we consider this problem for Gaussian kernels with variance σ∈(0,∞), and give some conditions under which the optimal solution exists.