A mean-variance optimisation approach to collectively pricing warranty policies

Warranty policy can influence the profit and cost of a product. In practice, a manufacturer commonly produces more than one product, or a portfolio of products, and provides warranty servicing for them. Many authors have attempted to optimise warranty policy to maximise the profit or minimise the cost of each individual product. Warranty claims of the products produced by the same manufacturer, however, may be due to common causes, since the products may be designed by the same engineer team or using the same type of components. This implies that the numbers of warranty claims of different products may be related, and optimisation of warranty policies for each individual product may therefore cause biased decisions. To overcome this disadvantage, this paper aims to collectively optimise a manufacturer's total profit for a portfolio of different products by using a mean-variance optimisation approach. A tool from the probability theory, {\it copulas}, is used to depict the dependence among the warranty claims of different products. Numerical examples are provided to illustrate the application of the proposed methods.