T. R. Jefferson

Allocation of resources in project management

Resource allocation on a project planning network is modelled as a minimax convex program. A limited budget is to be distributed over the activities of a project so as to minimize the duration of the project or critical path. Conjugate duality is used to simplify the formulation for computational purposes

The analysis of entropy models with equality and inequality constraints

Entropy models are emerging as valuable tools in the study of various social problems of spatial interaction. With the development of the modelling has come diversity. Increased flexibility in the model can be obtained by allowing certain constraints to be relaxed from equality to inequality. To provide a better understanding of these entropy models they are analysed by geometric programming. Dual mathematical programs and algorithms are obtained.

Generalized geometric programming applied to problems of optimal control: Part I, theory

A modification to the formulation in Ref. 1 is given.

Duality in infinite-dimensional mathematical programming: Convex integral functionals

Abstract A completely symmetric duality theory is derived for convex integral functionals. As an example we derive an infinite-dimensional version of prototype geometric programming. In this case the dual problem may turn out to be finite-dimensional. Examples from statistics are included.

Quadratic geometric programming with application to machining economics

Geometric Programming is extended to include convex quadratic functions. Generalized Geometric Programming is applied to this class of programs to obtain a convex dual program. Machining economics problems fall into this class. Such problems are studied by applying this duality to a nested set of three problems. One problem is zero degree of difficulty and the solution is obtained by solving a simple system of equations. The inclusion of a constraint restricting the force on the tool to be less than or equal to the breaking force provides a more realistic solution. This model is solved as a program with one degree of difficulty. Finally the behavior of the machining cost per part is studied parametrically as a function of axial depth.

Scheduling sugar cane plant and ratoon crops and a fallow—a constrained Markov model

Abstract A simple model of sugar cane crop rotation is developed to optimize the return to a single sugar cane farm with respect to harvest scheduling. The value of the cane crop varies during the harvesting season due to changing sugar content and harvest timing alters the possible growth period of the next crop in the rotation. The model is formulated as a deterministic version of a Markov process. This structure retains the dynamic nature of the deterministic problem and will allow future inclusion of stochastic parameters (e.g. due to climate). The solution is complicated by a constraint on the decisions of the process due to external restrictions on allowable harvest area. The solution method is an extension of Howards Policy Iteration Algorithm.

The entropy approach to material balancing in mineral processing

Abstract The material balancing problem seeks to transform observed data taken from a mineral processing circuit into a data set satisfying mass conservation. Here entropy forms the basis for this analysis. Developed to model matter in physics and chemistry using the multinomial distribution, entropy has been valuable in determining the most probable state of an observed physical system. For this problem the entropy model combines disparate data such as flow rates, assays and particle size distributions to produce a consistent data set. Two actual data sets are analyzed using entropy and results are compared with a least-squares approach.

Duality for minmax programs

Abstract A duality theory is derived for minimizing the maximum of a finite set of convex functions subject to a convex constraint set generated by both linear and nonlinear inequalities. The development uses the theory of generalised geometric programming. Further, a particular class of minmax program which has some practical significance is considered and a particularly simple dual program is obtained.

A generalisation of geometric programming with an application to information theory

Abstract A duality theory for convex functionals is developed. As an illustration, we consider an entropy maximising model associated with information science.

Duality of a nonconvex sum of ratios

For mathematical programs with objective involving a sum of ratios of affine functions, there are few theoretical results due to the nonconvex nature of the program. In this paper, we derive a duality theory for these programs by establishing their connection with geometric programming. This connection allows one to bring to bear the powerful theory and computational algorithms associated with geometric programming.