This year I was given the opportunity to present a poster of my work so far, allowing me to discuss my progress with academics and industry leaders, and to receive useful guidance on which future ideas seem promising and useful given current problems. In particular, I was able to showcase my new model, and some of my early results demonstrating its outperformance compared to existing benchmarks. My poster is shown below.

In the problem we are considering, we have a single-block warehouse like that shown above. The warehouse is divided into parallel aisles, each of which are split into bays. We have a front cross-aisle and a back cross-aisle, and pickers (staff responsible for picking products from shelves) are permitted to travel bidirectionally through the aisles and cross-aisles. This layout is the most widely-studied of warehouse layouts, and is reminiscent of many warehouses used in practice by retail companies.

The aim of the problem is to minimise the distance travelled by pickers, and it can naturally be broken down into three distinct stages.

• Storage Assignment: Products are assigned to slots in the warehouse, where each slot is represented by an (aisle, bay) pair
• Order Batching: Orders are batched together before they are picked
• Picker Routing: Routes are selected for pickers to take when collecting orders

These stages can be solved in a sequential or an integrated framework. In true operations there is a place for both – for full warehouse reshuffles, for example once per season, all stock can be moved and an integrated approach taken. For daily orders, only the batching and routing needs to be solved. Naturally, techniques developed for the former can easily be extended to the latter.

During my PhD, I will be using stochastic programming approaches to construct optimisation models aimed at solving the SLAP in an integrated way. A key priority will be in improving scalability. This can be done in three distinct ways: firstly, modelling techniques can be used to create tight formulations with small technology matrices and few variables, particularly auxiliary variables. This, used in conjunction with tailored solution algorithms, can be applied broadly to all new models. To complement this, the second approach is to reduce the size of the problem by leveraging problem-specific operational constraints, such as one-way systems and the clustering of products based on slot capacities. Thirdly, input reduction techniques can be used to reduce the size of problems through the careful selection of a small subset of orders which well-approximate the global set whilst reducing the requirement of the model to optimise over large inputs. Using these, we aim to scale stochastic programmes which accurately model the problem to warehouses of a size which is of genuine interest and use to industry.

Robustness in portfolio selection is not a novel concept, having been reviewed extensively in financial literature. It aims to address the downfalls of traditional portfolios by enforcing a lower bound on the losses that can be incurred under a range of market scenarios. The objective of the project was to assess whether classical frameworks for portfolio optimisation such as Markowitz and CVaR can be improved by adding a robust element.

Robust and non-robust versions of the CVaR and Markowitz models were created. They were each firstly trained on historical returns data for 55 stocks from between 1995 and 2025, and their split assessed using three criterion – their stock allocation, their division between industries, and their geographical spread. These are shown in the diagrams below:

Contrary to expectation, the robust models focus a larger proportion of capital into a smaller number of “safe-haven” assets. These tend to be well-established companies in traditionally safe sectors, with a focus on steady returns rather than outsized and volatile growth. Some of the companies to consistently receive a large proportion of capital include Johnson & Johnson (JNJ), The Commonwealth Bank of Australia (CBA.AX), and Nestle (NESN.SW).

The robust models show an expected exodus from risky sectors such as consumer discretionary, and an inflow into the sectors of healthcare and consumer staples which empirically weather financial shocks well.

The geographical split indicates that portfolios are dominated by American listed companies, representative of the large share of the global stock market which the US represents. The main change that the robust models exhibited is a movement from EU-based companies to Australian and Canadian entities. Although this could be indicative of factors such as a lower correlation between the latter and American stocks, it is likely this is random and based more on the specific chosen companies.

Having assessed the allocations, we next wanted to assess how well the portfolios performed. We did this firstly using an in- and out-of-sample method on the thirty years of financial data. This yielded the following graphs:

The best of the four models in the scatter plot is the robust CVaR, which has the greatest clustering in the top-left (high-return, low-variance). We are also able to pick up the 2008/2009 financial crash, shown by the four points in the bottom-right of the chart. The risk-return trade-off is similar for the four models as shown in the Sharpe ratio over time, although the CVaR maintains a slight edge over the other models in most time periods.

To further assess model performance, a copula was used to simulate returns data for testing on. A copula is a function that can model both marginal distributions as well as co-dependence structures. Forty years of data was sampled from the copula, and this was split into one-month, 21-day periods to assess short-term performance. The performances of each of the stocks are shown in the below histograms:

The robust models exhibited a greater concentration around the central values, resembling a t-distribution of the returns. There we also fewer values in the extremes, which is to be expected due to the extra precautions taken to ensure more security in the portfolio. Of greater interest though is the reduction in variances, with the robust models dominated by months of low variance, compared to the non-robust models having much longer tails and many months of high variance.

Th results of the report made it evident that robustness can have huge benefits when applied even to simple frameworks, and that the reduction in volatility does not need to come at a compromise to expected return.

The full report, including much more detail on data pre-processing, the models and general metrics achieved, can be found Here.

The data used in the report was related to the severity of Cardiac Allograft Vasculopathy (CAV) of patients who had received a heart transplant. Post-transplant, patients went between “Well”, “Mild/Moderate”, “Severe” and “Death”, and it was of interest to see which patient covariates had the largest effect on patient health decline. The covariates considered were the patient age, the age of the heart donor, the patient sex, whether the patients were diagnosed with Ischaemic Heart Disease (IHD) prior to transplant, and the cumulative rejection episodes experienced by the patient. Models considering all combinations of the covariates were considered, and their Akaike’s Information Criterion plotted in Figure 1.

Based on this, we used a model including the age of the heart donor, the IHD diagnoses and the cumulative rejection episodes. For each, we investigated the hazard ratios between each pair of states, shown below.

The hazard ratios, with some exceptions, generally seemed to indicate that increases in the covariate values increase the hazard. Note that the hazard ratio for the cumulative rejection episodes and for the donor age is the increase in hazard for each episode/year, whereas IHD is binary. To test our hypothesis, we took a patient in the 20th percentile of each covariate, and a patient in the 80th percentile, and evaluated their progression between states. The graphs corresponding to the patient at the 20th percentile are shown below:

The graphs show a lower-risk profile associated with the above individual – it takes a long time for the patient to transition out of the well state, and their overall survival probability is higher than that of the overall group at all time point at a 5% level of confidence. The same graphs were also plotted for the 80th percentile individual.

The survival in this group is comparably much lower. Particularly, there is a much shorter time spent in the well state, with greater transitions to both mild/moderate and straight to death. The survival probability is also much reduced across all time points, indicating that these covariates are indeed responsible for increased risk to patients.

Such results are useful to patients and doctors as they allow treatment to be tailored to the specific needs of the patient – patients identified as being high-risk can be afforded more appointments and stronger medication, and the lowest risk patients can be allowed fewer side-effects and a fuller life.

The full article can be found Here.

One talk of particular interest was given by Dries Goosens of Ghent University. His research is in Sports Scheduling, and he has been part of a team developing a categorisation of such problems with the aim of better choosing metaheuristics based on problem-specific features. I think that this will be an invaluable contribution to the field, providing structure in what can be a mess of problems and solutions, and would be useful in a range of other areas of OR such as vehicle routing.

I’m looking forward to attending the next conference and hopefully seeing some familiar faces!

The results found showed that DQL could indeed be applied to the TSP, but the results were inferior to that of the GA, both with regards to runtime and final cost.

My full report can be viewed .