A mathematical modelling technique for the analysis of the dynamics of a simple continuous EDA

This paper presents some initial attempts to mathematically model the dynamics of a continuous estimation of distribution algorithm (EDA) based on a Gaussian distribution and truncation selection. Case studies are conducted on both unimodal and multimodal problems to highlight the effectiveness of the proposed technique and explore some important properties of the EDA. With some general assumptions, we show that, for ID unimodal problems and with the (mu, lambda) scheme: (1). The behaviour of the EDA is dependent only on the general shape of the test function, rather than its specific form; (2). When initialized far from the global optimum, the EDA has a tendency to converge prematurely; (3). Given a certain selection pressure, there is a unique value for the proposed amplification parameter that could help the EDA achieve desirable performance; for ID multimodal problems: (1). The EDA could get stuck with the (mu, lambda) scheme; (2). The EDA will never get stuck with the (mu, lambda) scheme.

DEA-R:ratio-based comparative efficiency model, its mathematical relation to DEA and its use in applications

We propose a new mathematical model for efficiency analysis, which combines DEA methodology with an old idea-Ratio Analysis. Our model, called DEA-R, treats all possible ratios "output/input" as outputs within the standard DEA model. Although DEA and DEA-R generate different summary measures for efficiency, the two measures are comparable. Our mathematical and empirical comparisons establish the validity of DEA-R model in its own right. The key advantage of DEA-R over DEA is that it allows effective integration of the model with experts' opinions via flexible restrictive conditions on individual "output/input" pairs. © 2007 Springer Science+Business Media, LLC.

A mathematical model for assembly line balancing model to consider disordering sequence of workstations

Purpose – A binary integer programming model for the simple assembly line balancing problem (SALBP), which is well known as SALBP-1, was formulated more than 30 years ago. Since then, a number of researchers have extended the model for the variants of assembly line balancing problem.The model is still prevalent nowadays mainly because of the lower and upper bounds on task assignment. These properties avoid significant increase of decision variables. The purpose of this paper is to use an example to show that the model may lead to a confusing solution. Design/methodology/approach – The paper provides a remedial constraint set for the model to rectify the disordered sequence problem. Findings – The paper presents proof that the assembly line balancing model formulated by Patterson and Albracht may lead to a confusing solution. Originality/value – No one previously has found that the commonly used model is incorrect.

An integrated scheduling problem of PCB components on sequential pick-and-place machines:mathematical models and heuristic solutions

This paper formulates several mathematical models for determining the optimal sequence of component placements and assignment of component types to feeders simultaneously or the integrated scheduling problem for a type of surface mount technology placement machines, called the sequential pick-andplace (PAP) machine. A PAP machine has multiple stationary feeders storing components, a stationary working table holding a printed circuit board (PCB), and a movable placement head to pick up components from feeders and place them to a board. The objective of integrated problem is to minimize the total distance traveled by the placement head. Two integer nonlinear programming models are formulated first. Then, each of them is equivalently converted into an integer linear type. The models for the integrated problem are verified by two commercial packages. In addition, a hybrid genetic algorithm previously developed by the authors is adopted to solve the models. The algorithm not only generates the optimal solutions quickly for small-sized problems, but also outperforms the genetic algorithms developed by other researchers in terms of total traveling distance.