The product line pricing problem is generally defined as a seller's task to determine the optimal prices for each product in a product line while accounting for both demand-side and supply-side restrictions. In this paper, we contribute to the existing literature by incorporating consumers’ budget considerations into the seller's optimisation problem. Despite playing an important role in many applications, such as the pricing of tickets for sporting or theatre seasons, budget constraints have not been considered in the academic literature in the context of product line pricing to date. By building on the assumptions and the model formulation recently proposed in a previous study by [Burkart, W. R. et al. (2012). Product line pricing for services with capacity constraints and dynamic substitution. European Journal of Operational Research, 219(2), 347--359] for standard product line pricing, we propose a number of new mixed-integer linear formulations assuming budget-constrained consumers. The formulations differ in terms of how consumers handle their individual budget limitations. To solve the underlying problems, we propose a customised branch-and-bound procedure that relies on various novel problem-specific bound arguments explicitly exploiting the consumers’ budget limitations. Experimental tests show that the branch-and-bound procedure clearly outperforms IBM ILOG CPLEX, which is unable to solve problems for even medium-sized instances. Furthermore, based on a number of scenarios, we derive managerial insights regarding, e.g., the overall impact of considering budget constraints on the seller's revenue as well as the impact of correctly anticipating the type of consumer purchase behaviour. 
Optimal product line pricing in the presence of budget-constrained consumers