• Forecasting: Estimating the customer demand as a function of the price chosen (especially hard for products with no sales history or infrequent sales).

• Objective function: What exactly should Tesco aim to optimise? Sales volume? Profit? Profit margin? Conversion rates?

• Optimisation: How to choose prices for many related products to optimise the chosen objective function.

• Evalution: How to demonstrate that the chosen prices are optimal, especially to people without a mathematical background.

Aggregate sales data was provided for about 400 products over about 2 years so that quantitive approaches could be tested. For some products competitors’ prices were also provided.

The group recognised that capillary suction was the dominant process by which the contaminant spreads in the porous substrate. Therefore, in the first instance the absorption of the contaminant was modelled using Darcy’s law. At the next level of complication a diffuse interface model based on Richards’ equation was employed. The results of the two models were found to agree at early times, while at later times we found that the diffuse interface model predicted the more realistic scenario in which the contaminant has seeped deeper into the substrate even in the absence of further contaminant being supplied at the surface.

The decontamination process was modelled in two cases; first, where the product of the decontamination reaction was water soluble, and the second where the reaction product formed soluble in the contaminant phase and of similar density. These simple models helped explain some of the key physics involved in the process, and how the decontamination process might be optimised. We found that decontamination was most effective in the first of these two cases.

The group then sought to incorporate hydrodynamic effects into the reaction model. In the long wavelength limit, the governing equations reduced to a one-dimensional Stefan model similar to the one considered earlier. More detailed approximations and numerical simulations of this model were beyond the scope of this study group, but provide an entry point for future research in this area.

The team has developed three complementary models, each with different strengths and weaknesses so that, depending on the information desired, one model may be more useful than another. The three models are:

1. A continuum model giving a macroscopic description of the filter. The governing equations are derived from first-principle consider- ations of conservation of mass and momentum. Constitutive relations for this model are derived by considering the processes going on in the filter at a microscopic level.

2. A stochastic model based on a Markov Decision Process. Each droplet is modelled as a single entity that can merge or move stochastically. This leads to a Markov simulation of the filter and the computation of average quantities.

3. A Lattice-Boltzmann model. The droplets are modelled to interact with each other and with the filter, using a Boltzmann distribution for their speed. This simulates the hydrodynamic behaviour of the droplet inside the filter.