Procurement decisions over multiple periods under piecewise-linear shortage costs and fixed capacity commitments

We study optimal procurement in a case where the buyer must match supply against uncertain demand using a combination of low-cost order-in-advance procurement contracts and a high-cost real-time balancing mechanism. The procurement contracts have a two-level structure in which the commitment to procure a fixed quantity for multiple periods has a lower unit-price than period-specific commitments. Moreover, the balancing mechanism implies a salvage value for unused supply and piecewise-linear shortage costs: small shortages (relative to the total quantity procured) are balanced with a lower unit cost than larger shortages. Minimizing procurement costs results in a stochastic non-linear multivariate optimization problem, which can be interpreted as a generalization of the classic newsvendor model. We derive the optimality conditions for this problem and show how they can be utilized to obtain a cost minimizing procurement strategy by solving a series of single variable equations. The model is then applied to support procurement decision making of a pulp & paper company that procures natural gas worth tens of millions of euros annually. This allows us to highlight some key practical implications of our work.