This case competition was put on by Amazon Logistics’ Research team. The challenge was to determine what wage to offer three different types of truck-driver for last-mile delivery (delivery from distribution center to home).
Each type of driver had a different level of risk associated with their wage:
guaranteed drivers are paid whether or not there are packages available
option drivers are paid a reservation price but not a full wage if there are no packages for them to deliver
spot drivers are paid only if there are packages for them to deliver.
Given a normal distribution of packages, we needed to model profits as a function of the wages offered to different drivers. We used game theory to formulate a nonlinear optimization problem. Our two-person team won second place.
Background research on wages for truck drivers
Understand that implicit in the challenge is deciding how many of each type of driver to hire
Propose that drivers will accept or reject a contract based on their perceived risk of not getting hired (ie, they will demand a risk premium if their wage is not guaranteed)
Define variables, such as the day’s demand and wage paid to each driver type
Define parameters such as the market wage and the probability of getting hired by another company
Define relating equations based on classical risk aversion functions and the normal distribution
Model Total Expected Cost as a function of the wage variables, using relationships defined previously to eliminate other variables
Perform a grid search to find the wages that minimize Total Expected Cost
Much of the work was formulating the model, which was done analytically (math by hand). Our final optimization was a fairly complicated nonlinear model. In order to do any optimization, we needed to build a model in R so that we could perform a grid search.
The six-page white paper, with mathematical and code appendix, is available here.