The defence planning problem – force production under uncertain future purchasing power

The central purpose of defence planning is to ensure balance between the armed forces’ tasks, their economic resources and force structure. The defence leadership’s decision problem consists of acquiring and maintaining a force structure that maximises expected defence capability, given the tasks that the armed forces must solve, the purchasing power of future defence budgets, and the stock of input factors (personnel, materiel systems, supply, and physical and digital infrastructure) at the beginning of the planning period.

We analyse the relationship between resource use, force structure and defence capability as a mathematical optimisation problem with the aim of creating a framework for a comprehensive analysis of the defence planning problem and a basis for further research and model development. The target audience are mathematically trained researchers in and outside of the defence sector. We highlight the tension between 1) the need to prioritise acquisition of durable inputs in a time-limited period during which defence capability is to be strengthened and 2) how uncertain purchasing power of future defence budgets induces significant uncertainty around the optimal composition of the force structure. We model the defence planning problem at two levels: 1) by production and cost functions that map acquisition of input factors under a budget constraint to force production, and 2) a defence function that maps the force structure into expected goal attainment in a scenario portfolio. The analysis assumes that decision makers have limited possibilities to substitute between input factors in production, and that operations in the scenario portfolio require both depth and width in the force structure.

Expected operational success increases in each force structure element (FSE), first convexly and then concavely as a sigmoid curve. Through combined arms, the marginal contribution from any FSE to overall defence capability increases with the availability of complementary FSEs. When purchasing power in the defence budget is low, optimal resource use will generally entail that certain FSEs are not acquired. The intuition is that the decision maker, when forced to prioritise between different FSEs, achieves higher expected goal attainment by ensuring critical depth in combat-decisive FSEs rather than distributing the budget broadly, resulting in all FSEs having a low marginal contribution to expected goal attainment. The optimal composition of the force structure is therefore sensitive to changes in the purchasing power of the defence budget when purchasing power is low. Only at high purchasing power will the budget shares of the FSEs stabilise as an approximately linear function of the missions’ depth and width requirements. The clearest implication is for small countries with small defence budgets that face demanding missions, where small changes in future purchasing power can lead to substantial changes in the optimal composition of the planned force structure.

We have formulated the model with restrictive assumptions. This allows for analytical solutions and isolates the interaction between force structure requirements and purchasing power as a source of risk in long-term planning. We see potential in extending the model to analyse how the solution to the defence planning problem depends, among others, on uncertainty about future developments in purchasing power, the national security situation, allied and enemy courses of action, and the decision maker’s objective function.