John Glen

Feature Article-Mathematical Models in Farm Planning: A Survey

Farm planning has increased in complexity and importance as agriculture in the developed world has become concentrated in larger, more specialized farm units. These changes have stimulated the development of formal planning techniques based on mathematical models. Although this approach is characteristic of operations research, the professions direct involvement in agricultural planning has been limited: much of the published work is associated with agricultural economics. In this paper, we provide an OR-oriented introduction to the problems involved in agricultural planning, particularly at the farm level. We describe the planning problems of both the crop and livestock sectors and outline the models that have been proposed for solving these problems. Researchers, and agricultural extension and advisory services, have been the main users of these models, but the widespread availability of microcomputers gives considerable scope for developing models for use by farmers.

Classification accuracy in discriminant analysis: a mixed integer programming approach

Classification models can be developed by statistical or mathematical programming discriminant analysis techniques. Variable selection extensions of these techniques allow the development of classification models with a limited number of variables. Although stepwise statistical variable selection methods are widely used, the performance of the resultant classification models may not be optimal because of the stepwise selection protocol and the nature of the group separation criterion. A mixed integer programming approach for selecting variables for maximum classification accuracy is developed in this paper and the performance of this approach, measured by the leave-one-out hit rate, is compared with the published results from a statistical approach in which all possible variable subsets were considered. Although this mixed integer programming approach can only be applied to problems with a relatively small number of observations, it may be of great value where classification decisions must be based on a limited number of observations.

An iterative mixed integer programming method for classification accuracy maximizing discriminant analysis

Linear discriminant functions which maximize the number of correctly classified observations in a training sample can be generated by a mixed integer programming (MIP) discriminant analysis model in which a binary variable is associated with each observation, but because of the computational requirements this model can only be applied to relatively small problems. In this paper, an iterative MIP method is developed to allow classification accuracy maximizing discriminant functions to be generated for problems with many more observations than can be considered by the standard MIP formulation. Using minimization of the sum of deviations as the objective, a mathematical programming discriminant analysis model is first used to generate a discriminant function for the complete set of observations. A neighborhood of observations about this function is then defined and a MIP model is used to generate a discriminant function that maximizes classification accuracy within this neighborhood. The process of defining a neighborhood about the most recently generated discriminant function and solving a neighborhood MIP model is repeated until there is no improvement in the total number of observations classified correctly. This new iterative MIP method is applied to a two-group problem involving 690 observations.

A mathematical programming model for improvement planning in a semi-subsistence farm

Abstract In some semi-subsistence agriculture systems, long fallows have traditionally been used to maintain soil fertility, but fallow periods are often shortened because of increased pressure on land, resulting in reduced crop yields. In such cases crop yields can often be increased by adopting agricultural methods based on the use of new crop varieties, fertilisers and herbicides. These improved cultivation techniques must be introduced over a number of years, but the transition process has received little attention in evaluating improvement of semi-subsistence cultivation systems. In this paper a mathematical programming approach is developed for planning the introduction of improved cultivation systems in a semi-subsistence farm in northern Chiapas, Mexico. This new approach first uses a linear programming model to determine capital dependent steady state cultivation policies. Results from this steady model are then incorporated into a multiperiod mixed integer programming model for determining steady state policy and the associated improvement plan.

A comparison of standard and two-stage mathematical programming discriminant analysis methods

Abstract Classification models can be developed using standard or two-stage mathematical programming (MP) discriminant analysis methods. In standard MP discriminant analysis methods, discriminant functions are generated by solving a single MP model. In two-stage MP methods, observations that are difficult to classify are identified in the first stage, with greater emphasis being given to these observations in the second stage MP model for discriminant function generation. In this paper, two two-stage methods are described and compared with two standard MP models, the model for minimisation of the sum of deviations and the model for maximisation of classification accuracy. The performance of these MP discriminant analysis methods and Fisher’s linear discriminant analysis, a parametric statistical technique, is then evaluated on a published data set and on a number of simulated data sets.

Mean-variance portfolio rebalancing with transaction costs and funding changes

Investment portfolios should be rebalanced to take account of changing market conditions and changes in funding. Standard mean-variance (MV) portfolio selection methods are not appropriate for portfolio rebalancing, as the initial portfolio, change in funding and transaction costs are not considered. A quadratic mixed integer programming portfolio rebalancing model, which takes account of these factors is developed in this paper. The transaction costs in this portfolio rebalancing model are composed of fixed charges and variable costs, including the market impact costs associated with large market trades of individual securities, where these variable transaction costs are assumed to be non-linear functions of traded value. The use of this model is demonstrated and it is shown that when initial portfolio, funding changes and transaction costs are taken into account in portfolio construction and rebalancing, MV efficient portfolios that include risk-free lending do not have the structure expected from portfolio theory.

Mathematical programming models for piecewise-linear discriminant analysis

Mathematical programming (MP) discriminant analysis models are widely used to generate linear discriminant functions that can be adopted as classification models. Nonlinear classification models may have better classification performance than linear classifiers, but although MP methods can be used to generate nonlinear discriminant functions, functions of specified form must be evaluated separately. Piecewise-linear functions can approximate nonlinear functions, and two new MP methods for generating piecewise-linear discriminant functions are developed in this paper. The first method uses maximization of classification accuracy (MCA) as the objective, while the second uses an approach based on minimization of the sum of deviations (MSD). The use of these new MP models is illustrated in an application to a test problem and the results are compared with those from standard MCA and MSD models.

A mixed integer programming model for fertiliser policy evaluation

Abstract Commercial fertiliser and trace element mixtures are widely used in crop production to supply part of the nutrient requirements of the crop. In evaluating fertiliser policy, the crop producer must consider the mixtures to be used and the blending and application policy. In this paper a mixed integer programming model is developed to determine the policy for sourcing, blending and application of commercial fertiliser and trace mixtures to supply specified crop nutrient requirements at minimum cost. Results from the model are presented and the advantages and disadvantages of the approach are discussed.

Heuristics for feature selection in mathematical programming discriminant analysis models

In developing a classification model for assigning observations of unknown class to one of a number of specified classes using the values of a set of features associated with each observation, it is often desirable to base the classifier on a limited number of features. Mathematical programming discriminant analysis methods for developing classification models can be extended for feature selection. Classification accuracy can be used as the feature selection criterion by using a mixed integer programming (MIP) model in which a binary variable is associated with each training sample observation, but the binary variable requirements limit the size of problems to which this approach can be applied. Heuristic feature selection methods for problems with large numbers of observations are developed in this paper. These heuristic procedures, which are based on the MIP model for maximizing classification accuracy, are then applied to three credit scoring data sets.

An additive utility mixed integer programming model for nonlinear discriminant analysis

Mathematical programming (MP) discriminant analysis models can be used to develop classification models for assigning observations of unknown class membership to one of a number of specified classes using values of a set of features associated with each observation. Since most MP discriminant analysis models generate linear discriminant functions, these MP models are generally used to develop linear classification models. Nonlinear classifiers may, however, have better classification performance than linear classifiers. In this paper, a mixed integer programming model is developed to generate nonlinear discriminant functions composed of monotone piecewise-linear marginal utility functions for each feature and the cut-off value for class membership. It is also shown that this model can be extended for feature selection. The performance of this new MP model for two-group discriminant analysis is compared with statistical discriminant analysis and other MP discriminant analysis models using a real problem and a number of simulated problem sets.