On-Demand Meal Delivery: A Markov Model for Circulating Couriers

On-demand meal delivery has become a feature of most cities around the world as a result of platforms and apps that facilitate it as well as the pandemic, which for a period, closed restaurants. Meals are delivered by couriers, typically on bikes, e-bikes, or scooters, who circulate collecting meals from kitchens and delivering them to customers, who usually order online. A Markov model for circulating couriers with n + 1 parameters, where n is the number of kitchens plus customers, is derived by entropy maximization. There is one parameter for each kitchen and customer representing the demand for a courier, and there is one parameter representing the urgency of delivery. It is shown how the mean and variance of delivery time can be calculated once the parameters are known. The Markov model is irreducible. Two procedures are presented for calibrating model parameters on a data set of orders. Both procedures match known order frequencies with fitted visit probabilities; the first inputs an urgency parameter value and outputs mean delivery time, whereas the second inputs mean delivery time and outputs the corresponding urgency parameter value. Model calibration is demonstrated on a publicly available data set of meal orders from Grubhub. Grubhub data are also used to validate the calibrated model using a likelihood ratio. By changing the location of one kitchen, it is shown how the calibrated model can estimate the resulting change in demand for its meals and the corresponding mean delivery time. The Markov model could also be used for the assignment of courier trips to a street network.