Bruce van Brunt

Optimal location of transshipment depots

A number of agricultural commodities are either transported once harvested to a destination outside the production region or processed (or consumed) at a centrally located facility within the production region. One or more depots (or assembly points) can be located optimally within the harvesting region to reduce the overall shipment costs when transshipment at a lower freight rate can be achieved from the depot. Processing at the depot to concentrate the commodity, or to select desired fractions from the raw material, can achieve additional reductions in freight costs and increase the economic benefit. A model based on a circular harvesting region with uniform production per hectare and a linear transport function was used to define the optimal location of the depots. In the case of transshipment out of the supply region, a single depot was considered. A transshipment depot can be placed within the supply region for a net reduction in transport costs as long as the transshipment freight rate is less than approximately 85% of the collection freight rate. For transshipment to a central processing facility, the supply region was divided into uniform sectors and the optimal depot placement in a sector was located. Given a non-negative transshipment freight rate and that all production passes through the depot, the optimal depot placement cannot lie beyond 70.7% of the radius. A minimum reduction in the ratio of the freight rates is also required for the shipment point to move from the centre of the region to within a sector. Division of the production region into an infinite number of sectors was examined and it was found that convergence to the maximum benefit (at infinite number) was rapid so that, with just eight or nine sectors, 80% of the limiting benefit was obtained. The model was developed with the harvesting of milk in mind, but appears to relate to a wide range of harvestable low value bulky agricultural commodities. For a circular harvesting region, it was found that the economic benefit varied as the cube of the radius of the production region and linearly with the production intensity, regardless of whether transshipment was internal or external. The economic benefit was quantified for both variants of the model using selected parameters pertinent to milk harvesting.

A model for asymmetrical cell division

We present a model that describes the growth, division and death of a cell population structured by size. The model is an extension of that studied by Hall and Wake (1989) and incorporates the asymmetric division of cells. We consider the case of binary asymmetrical splitting in which a cell of size ε divides into two daughter cells of different sizes and find the steady size distribution (SSD) solution to the non-local differential equation. We then discuss the shape of the SSD solution. The existence of higher eigenfunctions is also discussed.

EIGENFUNCTIONS ARISING FROM A FIRST-ORDER FUNCTIONAL DIFFERENTIAL EQUATION IN A CELL GROWTH MODEL

A boundary-value problem for cell growth leads to an eigenvalue problem. In this paper some properties of the eigenfunctions are studied. The first eigenfunction is a probability density function and is of importance in the cell growth model. We sharpen an earlier uniqueness result and show that the distribution is unimodal. We then show that the higher eigenfunctions have nested zeros. We show that the eigenfunctions are not mutually orthogonal, but that there are certain orthogonality relations that effectively partition the set of eigenfunctions into two sets. doi:10.1017/S1446181111000575

A mechanistic model of hydrogen-methanogen dynamics in the rumen.

Existing mathematical models to estimate methane production in the rumen are based on calculation of hydrogen balances without considering the presence of methanogens. In this study, a mechanistic model of methane production is proposed that depicts the interaction between hydrogen concentration and methanogens in the rumen. Analytical results show that it meets biological expectations, namely increased fractional passage rate leads to a greater growth rate of methanogens, and a greater steady state hydrogen concentration. This model provides a basis on which to develop a more comprehensive model of methane production in the rumen that includes thermodynamics and feed fermentation pathways.

State Feedback Minimax and Guaranteed Cost Controllers for Linear Systems with Integral Quadratic Constraint Uncertainty Descriptions

Abstract This paper considers the design of minimax and optimal guaranteed cost controllers for uncertain linear systems. The uncertainty is permitted to have a structured description and is bounded by an integral quadratic constraint. For known initial conditions on the system state, it is shown that minimax control can be realised by static full state feedback controllers. A new proof of minimax optimality is given using standard methods for linear systems. The design of such controllers requires the optimal solution of a single Riccati equation dependent upon a set of parameters whose dimension is equal to the number of uncertainty blocks. It is shown that this multivariable optimisation is convex and thus numerically tractable. No explicit criteria to determine the existence of a solution are known; however, if solutions exist, results pertaining to their location are given to assist numerical optimisation. The extension of the approach to arbitrary unknown initial conditions is considered but is shown, in general, not to be possible. A tractable approach, considering the initial condition to be a random variable whose occurrence satisfies a certain probability density function, is presented. The resulting optimisation problem is shown to be convex and thus numerically tractable.

The Fixed-Point Problem

The goal of this chapter is to devise a method for approximating solutions of equations. This method is called fixed point iteration and is a process whereby a sequence of more and more accurate approximations is found. The convergence of this sequence to the desired solution is discussed. The procedure is then refined to give Newton’s method.

Limits of Functions

The concept of a limit of a function is introduced and theorems proved which assist in the calculation of such limits where they exist. The concept is then modified to give one-sided limits and infinite limits.

Taylor Polynomials and Taylor Series

Taylor polynomials are used to approximate values of functions at specified points. The error incurred is investigated by means of Taylor’s theorem. A method for ensuring that the approximation is accurate to within a specified error tolerance is illustrated. Taylor polynomials are then used to define Taylor series. Several techniques for finding these series are also discussed.

Sequences of Functions

Here sequences of functions are studied. The question of interchanging the order of the limit and integration processes, or the limit and differentiation processes, is raised. The concept of uniform convergence of a sequence of functions is introduced as a justification of such interchanges. The idea of uniform convergence is extended to series of functions, and the book concludes with a number of tests for uniform convergence of series.

Evolution equations having conservation laws with flux characteristics

A class of evolution equations in divergence form is studied in this paper. Specifically, we develop conditions under which the spatial divergence term, the flux, corresponds to the characteristic of a conservation law. The KdV equation is a prominent example of an equation having a flux term that is also a characteristic for a conservation law. We show that the flux term must be self-adjoint. General equations for the corresponding conservation laws and Hamiltonian densities are derived and supplemented with examples.