Multiple-source learning precedence graph concept for the automotive industry

In modern production systems, customized mass production of complex products, such as automotive or white goods, is often realized at assembly lines with a high degree of manual labor. For firms that apply assembly systems, the assembly line balancing problem (ALBP) arises, which is to assign optimally tasks to stations or workers with respect to some constraints and objectives. Although the literature provides a number of relevant models and efficient solution methods for ALBP, firms, in most cases, do not use this knowledge to balance their lines. Instead, the planning is mostly performed manually by numerous planners responsible for small sub-problems. This is because of the lack of data, like the precedence relations between the tasks to be performed. Such data is hard to collect and to maintain updated.     

A Decomposition Method for Transfer Line Life Cycle Cost Optimisation

A new method to search best parameters of a transfer line so that the cost of each manufactured part will be minimised. The synchronised transfer lines with parallel machining are considered. Such lines are widely used in mass and large-scale mechanical production. The objective is to minimise the line life cycle cost per part under the given productivity and technological constraints. The design decisions to be optimised are: number of spindles and workstations. This will be accomplished by defining subsets of tasks which are performed by one spindle head and cutting conditions for each spindle. The paper focuses on a mathematical model of the problem and methods used to solve it. This model is formulated in terms of mixed (discrete and non-linear) programming and graph theory. A special decomposition scheme based on the parametric decomposition technique is proposed. For solving the sub-problems obtained after decomposition, a Branch-and-Bound algorithm as well as a shortest path technique are used.     

Balancing modular transfer lines with serial-parallel activation of spindle heads at stations

The paper deals with an as yet unexplored combinatorial optimization problem concerning balancing complex transfer lines in the machining/process environment. In contrast to similar problems for assembly lines, in transfer line balancing, tasks are grouped into blocks. All tasks of each block are executed simultaneously (in parallel) by one piece of equipment (spindle head). For the transfer lines considered in this paper, spindle heads at each station are activated in serial-parallel order. The set of all available spindle heads is known beforehand. Precedence, cycle time, compatibility, and parallelism constraints for the blocks and tasks are given. The line investment cost is estimated by the sum of block and station costs. The problem is to assign all tasks (using the available blocks) such that all constraints are respected and line investment cost is at a minimum. This paper focuses on solving the problem via a branch-and-bound algorithm. An approach for obtaining an efficient lower bound is offered, based on a reduction of the initial problem to a set partitioning problem. Computational experiments reveal that the proposed approach is efficient mathematically and can be used to solve practical transfer line design problems of a medium size.     

An Algorithm for a Letter-Based Representation of All-Pairwise Comparisons

All-pairwise comparisons among a set of t treatments or groups are one of the most frequent tasks in applied statistics. Users of statistical software are accustomed to the familiar lines display, in which treatments that do not differ significantly, are connected by a common line or letter. Availability of the lines display is restricted mainly to the balanced analysis of variance setup. This limited availability is at stark variance with the diversity of statistical methods and models, which call for multiple comparisons. This article describes a general method for graphically representing any set of t(t?1)/2 all-pairwise significance statements (p values) for t treatments by a familiar letter display, which is applicable regardless of the underlying data structure or the statistical method used for comparisons. The method reproduces the familiar lines display in case of the balanced analysis of variance. Its broad applicability is demonstrated using data from an international multienvironment wheat yield trial and from a fish catching survey.     

Computational Construction as a Means to Coordinate Representations of Infinity

In this paper, we describe a design experiment aimed at helping students to explore and develop concepts of infinite processes and objects. Our approach is based on the design and development of a computational microworld, which afforded students the means to construct a range of representational models (symbolic, visual and numeric) of infinity-related objects (infinite sequences, in particular). We present episodes based on four students’ activities, seeking to illustrate how the available tools mediated students’ understandings of the infinite in rich ways, allowing them to discriminate subtle process-oriented features of infinite processes. We claim that the microworld supported students in the coordination of hitherto unconnected or conflicting intuitions concerning infinity, based on a constructive articulation of different representational forms we name as ‘representational moderation’.

Balancing stochastic two-sided assembly lines: A chance-constrained,piecewise-linear,mixed integer program and a simulated annealing algorithm

Two-sided assembly lines are often designed to produce large-sized products, such as automobiles, trucks and buses. In this type of a production line, both left-side and right-side of the line are used in parallel. In all studies on two-sided assembly lines, the task times are assumed to be deterministic. However, in real life applications, especially in manual assembly lines, the tasks may have varying execution times defined as a probability distribution. The task time variation may result from machine breakdowns, loss of motivation, lack of training, non-qualified operators, complex tasks, environment, etc. In this paper, the problem of balancing two-sided assembly lines with stochastic task times (STALBP) is considered. A chance-constrained, piecewise-linear, mixed integer program (CPMIP) is proposed to model and solve the problem. As a solution approach a simulated annealing (SA) algorithm is proposed. To assess the effectiveness of CPMIP and SA algorithm, a set of test problems are solved. Finally, computational results indicating the effectiveness of CPMIP and SA algorithm are reported.     

Simple heuristics for the assembly line worker assignment and balancing problem

We propose simple heuristics for the assembly line worker assignment and balancing problem. This problem typically occurs in assembly lines in sheltered work centers for the disabled. Different from the well-known simple assembly line balancing problem, the task execution times vary according to the assigned worker. We develop a constructive heuristic framework based on task and worker priority rules defining the order in which the tasks and workers should be assigned to the workstations. We present a number of such rules and compare their performance across three possible uses: as a stand-alone method, as an initial solution generator for meta-heuristics, and as a decoder for a hybrid genetic algorithm. Our results show that the heuristics are fast, they obtain good results as a stand-alone method and are efficient when used as a initial solution generator or as a solution decoder within more elaborate approaches.     

Balancing U-lines in a multiple U-line facility

U-shaped production lines and facilities consisting of many such lines are important parts of modem manufacturing systems. The problem of balancing and rebalancing U-line facilities is studied in this paper. Like the traditional line balancing problem this problem is NP-hard. The objective is to assign tasks to a minimum number of regular, crossover, and multiline stations while satisfying cycle time, precedence, location, and station-type constraints. A secondary objective is to concentrate the idle time in one station so that improvement efforts can be focused there in accordance with modern just-in-time principles. A reaching dynamic programming algorithm is presented for determining optimal balances. It is effective for balancing and rebalancing facilities with any number of U-lines, provided that individual U-lines do not have more than 22 tasks and do not have wide, sparse precedence graphs.     

Fifth graders’ understanding of fractions on the number line

The goal of this research was to examine fifth graders’ understanding of fractions on the number line. This case‐study design focused on the various ways that students represented fractions on number lines. Students responded to task‐based interview questions by identifying fractions as a number on the number line as well as equivalency and problem solving. The tasks were administered individually to 26 fifth‐grade students over a 15‐minute time frame in their respective schools. The two groups of 10‐year‐old students answered most questions in written form with pencil and paper and were often asked to explain how they arrived at an answer. Student performance was highest when instructed to plot ½ on a number line of 0 to 1 as well as naming a fraction less than ½. The students performed lowest when they attempted to plot ½, ¼, and 1 on a number line with a predetermined unit 0 to 1/3. Other low performing concepts consisted of plotting ¼ on a number line from 0‐3, identifying ¼ on a non‐routine number line, and plotting a unit fraction with an equivalent fraction as well as an improper fraction on a common number line.     

Third‐grade teachers’ self‐reported use of multiplication and division models

Visual representations and manipulatives are a highly advocated mathematical tool for the teaching and learning of multiplication and division. Although there is some prior research on elementary teachers’ general use of manipulatives and visual representations, there is little to no specific focus on use of such representations on a specific mathematical concept. The present study examined third grade teachers’ reported use of visual representations for teaching multiplication and division. Findings indicate prevalent use of discrete models and infrequent use of continuous models. Length models and number lines are rarely used across all Common Core standards focusing on multiplication/division, with numeric‐only representations being reported frequently across all standards. Groups‐of and array models were the most prevalent visual model reported by third grade teachers. Although teachers report higher degrees of access to certain materials than previous reports on manipulative use, interview data suggests this may have more to do with purchase agreements between school districts and textbook companies than pedagogical preferences of classroom teachers. Supporting findings in prior decades, teachers in the present study report prevalent use of flashcards, charts and grid paper, and variations of counters.     

A survey on problems and methods in generalized assembly line balancing

Assembly lines are traditional and still attractive means of mass and large-scale series production. Since the early times of Henry Ford several developments took place which changed assembly lines from strictly paced and straight single-model lines to more flexible systems including, among others, lines with parallel work stations or tasks, customer-oriented mixed-model and multi-model lines, U-shaped lines as well as unpaced lines with intermediate buffers.In any case, an important decision problem, called assembly line balancing problem, arises and has to be solved when (re-) configuring an assembly line. It consists of distributing the total workload for manufacturing any unit of the product to be assembled among the work stations along the line.Assembly line balancing research has traditionally focused on the simple assembly line balancing problem (SALBP) which has some restricting assumptions. Recently, a lot of research work has been done in order to describe and solve more realistic generalized problems (GALBP). In this paper, we survey the developments in GALBP research.     

Pattern Generalization with Graphs and Words: A Cross-Sectional and Longitudinal Analysis of Middle School Students' Representational Fluency

Cross-sectional and longitudinal data from students as they advance through the middle school years (grades 6-8) reveal insights into the development of students' pattern generalization abilities. As expected, students show a preference for lower-level tasks such as reading the data, over more distant predictions and generation of abstractions. Performance data also indicate a verbal advantage that shows greater success when working with words than graphs, a replication of earlier findings comparing words to symbolic equations. Surprisingly, students show a marked advantage with patterns presented in a continuous format (line graphs and verbal rules) as compared to those presented as collections of discrete instances (point-wise graphs and lists of exemplars). Student pattern-generalization performance also was higher when words and graphs were combined. Analyses of student performance patterns and strategy use contribute to an emerging developmental model of representational fluency. The model contributes to research on the development of representational fluency and can inform instructional practices and curriculum design in the area of algebraic development. Results also underscore the impact that perceptual aspects of representations have on students' reasoning, as suggested by an Embodied Cognition view.     

Solving a robotic assembly line balancing problem using efficient hybrid methods

In this paper we are studying a robotic assembly line balancing problem. The goal is to maximize the efficiency of the line and to balance the different tasks between the robots by defining the suitable tasks and components to assign to each robot. We are interested in a robotic line which consists of seizing the products on a moving conveyor and placing them on different location points. The performances evaluations of the system are done using a discret event simulation model. This latter has been developed with C++ language. As in our industrial application we are bounded by the execution time, we propose some resolution methods which define the suitable component and point positions in order to define the strategy of pick and place for each robot. These methods are based on the ant colony optimization, particle swarm optimization and genetic algorithms. To enhance the quality of the developed algorithms and to avoid local optima, we have coupled these algorithms with guided local search. After that, an exact method based on full enumeration is also developed to assess the quality of the developed methods. Then, we try to select the best algorithm which is able to get the best solutions with a small execution time. This is the main advantage of our methods compared to exact methods. This fact represents a great interest taking in consideration that the selected methods are used to manage the functioning of real industrial robotic assembly lines. Numerical results show that the selected algorithm performs optimally for the tested instances in a reasonable computation time and satisfies the industrial constraint.     

Self-Organized Task Allocation for Service Tasks in Computing Systems with Reconfigurable Components

A self-organized scheme for the allocation service tasks in adaptive or organic computing systems is proposed. Such computing systems are highly self-organized and the components ideally adapt to the needs of users or the environment. Typically, the components of such systems need some service from time to time in order perform their work efficiently. Since the type of service tasks will often change in this systems it is attractive to use reconfigurable hardware to perform the service tasks. The studied system consists of normal worker components and the helper components which have reconfigurable hardware and can perform different service tasks. The speed with which a service task is executed by a helper depends on its actual configuration. Different strategies for the helpers to decide about service task acceptance and reconfiguration are proposed. These task acceptance strategies are inspired by stimulus-threshold models that are used to explain task allocation in social insects. Analytical results for a system with two reconfigurable helpers are presented together with simulation results for larger systems. This work was supported by the German Research Foundation (DFG) through the project Organisation and Control of Self-Organising Systems in Technical Compounds within SPP 1183.     

The Mixturegram: A Visualization Tool for Assessing the Number of Components in Finite Mixture Models

Certain practical and theoretical challenges surround the estimation of finite mixture models. One such challenge is how to determine the number of components when this is not assumed a priori. Available methods in the literature are primarily numerical and lack any substantial visualization component. Traditional numerical methods include the calculation of information criteria and bootstrapping approaches; however, such methods have known technical issues regarding the necessary regularity conditions for testing the number of components. The ability to visualize an appropriate number of components for a finite mixture model could serve to supplement the results from traditional methods or provide visual evidence when results from such methods are inconclusive. Our research fills this gap through development of a visualization tool, which we call a mixturegram. This tool is easy to implement and provides a quick way for researchers to assess the number of components for their hypothesized mixture model. Mixtures of univariate or multivariate data can be assessed. We validate our visualization assessments by comparing with results from information criteria and an ad hoc selection criterion based on calculations used for the mixturegram. We also construct the mixturegram for two datasets.     

Demystifying DEA — A Visual Interactive Approach Based on Multiple Criteria Analysis

Both Data Envelopment Analysis (DEA) and Multiple Criteria Analysis (MCA) can be used to assess the efficiency with which units perform similar tasks. This paper describes an approach derived from the integration of data envelopment analysis and a multi-attribute value function. This approach is implemented as a visual interactive decision support system, the use of which is illustrated by a practical application. The authors feel that this approach overcomes some of the limitations of the original DEA approach and, in particular, increases users' understanding of DEA. The approach is particularly well suited to the analysis of the efficiency of a small number of units.     

Binary fuzzy goal programming approach to single model straight and U-shaped assembly line balancing

Assembly line balancing generally requires a set of acceptable solutions to the several conflicting objectives. In this study, a binary fuzzy goal programming approach is applied to assembly line balancing. Models for balancing straight and U-shaped assembly lines with fuzzy goals (the number of workstations and cycle time goals) are proposed. The binary fuzzy goal programming models are solved using the methodology introduced by Chang [Chang, C.T., 2007. Binary fuzzy goal programming. European Journal of Operational Research 180 (1), 29–37]. An illustrative example is presented to demonstrate the validity of the proposed models and to compare the performance of straight and U-shaped line configurations.     

Bees Algorithm for constrained fuzzy multi-objective two-sided assembly line balancing problem

Bees Algorithm is one of the swarm intelligence based heuristics which tries to model natural behaviour of honey bees in food foraging and used to solve optimization problems. On the other hand, Two-sided Assembly Line Balancing Problem is a generalization of simple Assembly Line Balancing Problem where different assembly tasks are carried out on the same product in parallel at both left and right sides of the line. Two-sided assembly lines are generally employed for the assembly of large-sized products such as buses and trucks. Furthermore, many real life problems contain imprecise objectives and Fuzzy Multi-objective Programming gives an opportunity to handle such situations. In this study, Two-sided Assembly Line Balancing Problem is considered more realistically by employing positional, zoning and synchronous task constraints and by utilizing fuzzy approaches so as to maximize work slackness index and line efficiency, and minimize total balance delay. For solving this problem Bees Algorithm is used as a search mechanism for obtaining good solutions and extensive computational results are presented.     

Orchards in elliptic curves over finite fields

Consider a set of n points on a plane. A line containing exactly 3 out of the n points is called a 3-rich line. The classical orchard problem asks for a configuration of the n points on the plane that maximizes the number of 3-rich lines. In this note, using the group law in elliptic curves over finite fields, we exhibit several (infinitely many) group models for orchards wherein the number of 3-rich lines agrees with the expected number given by Green-Tao (or, Burr, Grünbaum and Sloane) formula for the maximum number of lines. We also show, using elliptic curves over finite fields, that there exist infinitely many point-line configurations with the number of 3-rich lines exceeding the expected number given by Green-Tao formula by two, and this is the only other optimal possibility besides the case when the number of 3-rich lines agrees with the Green-Tao formula.