Modelling the dynamics of genetic algorithms using statistical mechanics

A formalism for modelling the dynamics of Genetic Algorithms (GAs) using methods from statistical mechanics, originally due to Prugel-Bennett and Shapiro, is reviewed, generalized and improved upon. This formalism can be used to predict the averaged trajectory of macroscopic statistics describing the GA's population. These macroscopics are chosen to average well between runs, so that fluctuations from mean behaviour can often be neglected. Where necessary, non-trivial terms are determined by assuming maximum entropy with constraints on known macroscopics. Problems of realistic size are described in compact form and finite population effects are included, often proving to be of fundamental importance. The macroscopics used here are cumulants of an appropriate quantity within the population and the mean correlation (Hamming distance) within the population. Including the correlation as an explicit macroscopic provides a significant improvement over the original formulation. The formalism is applied to a number of simple optimization problems in order to determine its predictive power and to gain insight into GA dynamics. Problems which are most amenable to analysis come from the class where alleles within the genotype contribute additively to the phenotype. This class can be treated with some generality, including problems with inhomogeneous contributions from each site, non-linear or noisy fitness measures, simple diploid representations and temporally varying fitness. The results can also be applied to a simple learning problem, generalization in a binary perceptron, and a limit is identified for which the optimal training batch size can be determined for this problem. The theory is compared to averaged results from a real GA in each case, showing excellent agreement if the maximum entropy principle holds. Some situations where this approximation brakes down are identified. In order to fully test the formalism, an attempt is made on the strong sc np-hard problem of storing random patterns in a binary perceptron. Here, the relationship between the genotype and phenotype (training error) is strongly non-linear. Mutation is modelled under the assumption that perceptron configurations are typical of perceptrons with a given training error. Unfortunately, this assumption does not provide a good approximation in general. It is conjectured that perceptron configurations would have to be constrained by other statistics in order to accurately model mutation for this problem. Issues arising from this study are discussed in conclusion and some possible areas of further research are outlined.

Noisy fitness evaluation in genetic algorithms and the dynamics of learning

A theoretical model is presented which describes selection in a genetic algorithm (GA) under a stochastic fitness measure and correctly accounts for finite population effects. Although this model describes a number of selection schemes, we only consider Boltzmann selection in detail here as results for this form of selection are particularly transparent when fitness is corrupted by additive Gaussian noise. Finite population effects are shown to be of fundamental importance in this case, as the noise has no effect in the infinite population limit. In the limit of weak selection we show how the effects of any Gaussian noise can be removed by increasing the population size appropriately. The theory is tested on two closely related problems: the one-max problem corrupted by Gaussian noise and generalization in a perceptron with binary weights. The averaged dynamics can be accurately modelled for both problems using a formalism which describes the dynamics of the GA using methods from statistical mechanics. The second problem is a simple example of a learning problem and by considering this problem we show how the accurate characterization of noise in the fitness evaluation may be relevant in machine learning. The training error (negative fitness) is the number of misclassified training examples in a batch and can be considered as a noisy version of the generalization error if an independent batch is used for each evaluation. The noise is due to the finite batch size and in the limit of large problem size and weak selection we show how the effect of this noise can be removed by increasing the population size. This allows the optimal batch size to be determined, which minimizes computation time as well as the total number of training examples required.

Representation and parameterisation issues in genetic algorithms

A multi-chromosome GA (Multi-GA) was developed, based upon concepts from the natural world, allowing improved flexibility in a number of areas including representation, genetic operators, their parameter rates and real world multi-dimensional applications. A series of experiments were conducted, comparing the performance of the Multi-GA to a traditional GA on a number of recognised and increasingly complex test optimisation surfaces, with promising results. Further experiments demonstrated the Multi-GA's flexibility through the use of non-binary chromosome representations and its applicability to dynamic parameterisation. A number of alternative and new methods of dynamic parameterisation were investigated, in addition to a new non-binary 'Quotient crossover' mechanism. Finally, the Multi-GA was applied to two real world problems, demonstrating its ability to handle mixed type chromosomes within an individual, the limited use of a chromosome level fitness function, the introduction of new genetic operators for structural self-adaptation and its viability as a serious real world analysis tool. The first problem involved optimum placement of computers within a building, allowing the Multi-GA to use multiple chromosomes with different type representations and different operators in a single individual. The second problem, commonly associated with Geographical Information Systems (GIS), required a spatial analysis location of the optimum number and distribution of retail sites over two different population grids. In applying the Multi-GA, two new genetic operators (addition and deletion) were developed and explored, resulting in the definition of a mechanism for self-modification of genetic material within the Multi-GA structure and a study of this behaviour.

Genetic algorithms for discovery of matrix multiplication methods

We present a parallel genetic algorithm for nding matrix multiplication algo-rithms. For 3 x 3 matrices our genetic algorithm successfully discovered algo-rithms requiring 23 multiplications, which are equivalent to the currently best known human-developed algorithms. We also studied the cases with less mul-tiplications and evaluated the suitability of the methods discovered. Although our evolutionary method did not reach the theoretical lower bound it led to an approximate solution for 22 multiplications.

Using genetic algorithms for improved discrete sequence prediction

A statistics-based method using genetic algorithms for predicting discrete sequences is presented. The prediction of the next value is based upon a fixed number of previous values and the statistics offered by the training data. According to the statistics, in similar past cases different values occurred next. If these values are considered with the appropriate weights, the forecast is successful. Weights are generated by genetic algorithms.

The dynamics of a genetic algorithm for a simple learning problem

A formalism for describing the dynamics of Genetic Algorithms (GAs) using method s from statistical mechanics is applied to the problem of generalization in a perceptron with binary weights. The dynamics are solved for the case where a new batch of training patterns is presented to each population member each generation, which considerably simplifies the calculation. The theory is shown to agree closely to simulations of a real GA averaged over many runs, accurately predicting the mean best solution found. For weak selection and large problem size the difference equations describing the dynamics can be expressed analytically and we find that the effects of noise due to the finite size of each training batch can be removed by increasing the population size appropriately. If this population resizing is used, one can deduce the most computationally efficient size of training batch each generation. For independent patterns this choice also gives the minimum total number of training patterns used. Although using independent patterns is a very inefficient use of training patterns in general, this work may also prove useful for determining the optimum batch size in the case where patterns are recycled.

The dynamics of a genetic algorithm under stabilizing selection

A formalism recently introduced by Prugel-Bennett and Shapiro uses the methods of statistical mechanics to model the dynamics of genetic algorithms. To be of more general interest than the test cases they consider. In this paper, the technique is applied to the subset sum problem, which is a combinatorial optimization problem with a strongly non-linear energy (fitness) function and many local minima under single spin flip dynamics. It is a problem which exhibits an interesting dynamics, reminiscent of stabilizing selection in population biology. The dynamics are solved under certain simplifying assumptions and are reduced to a set of difference equations for a small number of relevant quantities. The quantities used are the population's cumulants, which describe its shape, and the mean correlation within the population, which measures the microscopic similarity of population members. Including the mean correlation allows a better description of the population than the cumulants alone would provide and represents a new and important extension of the technique. The formalism includes finite population effects and describes problems of realistic size. The theory is shown to agree closely to simulations of a real genetic algorithm and the mean best energy is accurately predicted.

A hybrid genetic algorithm for the multi-depot vehicle routing problem

The distribution of finished products from depots to customers is a practical and challenging problem in logistics management. Better routing and scheduling decisions can result in higher level of customer satisfaction because more customers can be served in a shorter time. The distribution problem is generally formulated as the vehicle routing problem (VRP). Nevertheless, there is a rigid assumption that there is only one depot. In cases, for instance, where a logistics company has more than one depot, the VRP is not suitable. To resolve this limitation, this paper focuses on the VRP with multiple depots, or multi-depot VRP (MDVRP). The MDVRP is NP-hard, which means that an efficient algorithm for solving the problem to optimality is unavailable. To deal with the problem efficiently, two hybrid genetic algorithms (HGAs) are developed in this paper. The major difference between the HGAs is that the initial solutions are generated randomly in HGA1. The Clarke and Wright saving method and the nearest neighbor heuristic are incorporated into HGA2 for the initialization procedure. A computational study is carried out to compare the algorithms with different problem sizes. It is proved that the performance of HGA2 is superior to that of HGA1 in terms of the total delivery time.

Using non-parametric search algorithms to forecast daily excess stock returns

Are the learning procedures of genetic algorithms (GAs) able to generate optimal architectures for artificial neural networks (ANNs) in high frequency data? In this experimental study,GAs are used to identify the best architecture for ANNs. Additional learning is undertaken by the ANNs to forecast daily excess stock returns. No ANN architectures were able to outperform a random walk,despite the finding of non-linearity in the excess returns. This failure is attributed to the absence of suitable ANN structures and further implies that researchers need to be cautious when making inferences from ANN results that use high frequency data.

Optimization of PCB component placements for the collect-and-place machines

This paper presents two hybrid genetic algorithms (HGAs) to optimize the component placement operation for the collect-and-place machines in printed circuit board (PCB) assembly. The component placement problem is to optimize (i) the assignment of components to a movable revolver head or assembly tour, (ii) the sequence of component placements on a stationary PCB in each tour, and (iii) the arrangement of component types to stationary feeders simultaneously. The objective of the problem is to minimize the total traveling time spent by the revolver head for assembling all components on the PCB. The major difference between the HGAs is that the initial solutions are generated randomly in HGA1. The Clarke and Wright saving method, the nearest neighbor heuristic, and the neighborhood frequency heuristic are incorporated into HGA2 for the initialization procedure. A computational study is carried out to compare the algorithms with different population sizes. It is proved that the performance of HGA2 is superior to HGA1 in terms of the total assembly time.

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.

Integrated component scheduling models for chip shooter machines

This paper focuses on minimizing printed circuit board (PCB) assembly time for a chipshootermachine, which has a movable feeder carrier holding components, a movable X–Y table carrying a PCB, and a rotary turret with multiple assembly heads. The assembly time of the machine depends on two inter-related optimization problems: the component sequencing problem and the feeder arrangement problem. Nevertheless, they were often regarded as two individual problems and solved separately. This paper proposes two complete mathematical models for the integrated problem of the machine. The models are verified by two commercial packages. Finally, a hybrid genetic algorithm previously developed by the authors is presented to solve the model. 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 assembly time.

Evolutionary neural networks : models and applications

The scaling problems which afflict attempts to optimise neural networks (NNs) with genetic algorithms (GAs) are disclosed. A novel GA-NN hybrid is introduced, based on the bumptree, a little-used connectionist model. As well as being computationally efficient, the bumptree is shown to be more amenable to genetic coding lthan other NN models. A hierarchical genetic coding scheme is developed for the bumptree and shown to have low redundancy, as well as being complete and closed with respect to the search space. When applied to optimising bumptree architectures for classification problems the GA discovers bumptrees which significantly out-perform those constructed using a standard algorithm. The fields of artificial life, control and robotics are identified as likely application areas for the evolutionary optimisation of NNs. An artificial life case-study is presented and discussed. Experiments are reported which show that the GA-bumptree is able to learn simulated pole balancing and car parking tasks using only limited environmental feedback. A simple modification of the fitness function allows the GA-bumptree to learn mappings which are multi-modal, such as robot arm inverse kinematics. The dynamics of the 'geographic speciation' selection model used by the GA-bumptree are investigated empirically and the convergence profile is introduced as an analytical tool. The relationships between the rate of genetic convergence and the phenomena of speciation, genetic drift and punctuated equilibrium arc discussed. The importance of genetic linkage to GA design is discussed and two new recombination operators arc introduced. The first, linkage mapped crossover (LMX) is shown to be a generalisation of existing crossover operators. LMX provides a new framework for incorporating prior knowledge into GAs.Its adaptive form, ALMX, is shown to be able to infer linkage relationships automatically during genetic search.

Unsupervised KDD to creatively support managers' decision making with fuzzy association rules:a distribution channel application

To be competitive in contemporary turbulent environments, firms must be capable of processing huge amounts of information, and effectively convert it into actionable knowledge. This is particularly the case in the marketing context, where problems are also usually highly complex, unstructured and ill-defined. In recent years, the development of marketing management support systems has paralleled this evolution in informational problems faced by managers, leading to a growth in the study (and use) of artificial intelligence and soft computing methodologies. Here, we present and implement a novel intelligent system that incorporates fuzzy logic and genetic algorithms to operate in an unsupervised manner. This approach allows the discovery of interesting association rules, which can be linguistically interpreted, in large scale databases (KDD or Knowledge Discovery in Databases.) We then demonstrate its application to a distribution channel problem. It is shown how the proposed system is able to return a number of novel and potentially-interesting associations among variables. Thus, it is argued that our method has significant potential to improve the analysis of marketing and business databases in practice, especially in non-programmed decisional scenarios, as well as to assist scholarly researchers in their exploratory analysis. © 2013 Elsevier Inc.

A genetic algorithm for the generalised transportation problem

The generalised transportation problem (GTP) is an extension of the linear Hitchcock transportation problem. However, it does not have the unimodularity property, which means the linear programming solution (like the simplex method) cannot guarantee to be integer. This is a major difference between the GTP and the Hitchcock transportation problem. Although some special algorithms, such as the generalised stepping-stone method, have been developed, but they are based on the linear programming model and the integer solution requirement of the GTP is relaxed. This paper proposes a genetic algorithm (GA) to solve the GTP and a numerical example is presented to show the algorithm and its efficiency.