Dynamic Management of Self-Renewable Resources in Production-Inventory Systems

Abstract

We consider a production-inventory system in which production relies on a self-renewable natural resource. The resource regenerates at a rate that depends on its current stock as well as random environmental factors. In each period, the firm decides on the production quantity, facing stochastic demand and limited resource. This type of system is common in industries such as fishing and logging. We model the resource’s self-renewal using a generating function and formulate the production optimization problem as a dynamic program. We find that because of the renewable nature of the resource, the optimal stock level does not necessarily increase with the available resource quantity, and could become higher than the optimal stock level for a benchmark system where resource is unlimited. In a special case with deterministic demand and environment, we show that it could be optimal to halt production even with zero inventory left, in order to preserve the growth of resource. On the other hand, it could also be optimal to raise the inventory level to be higher than demand, when overpopulation becomes an issue. Moreover, a higher market price, which stems from adverse environmental conditions and signals lower future yields, can incentivize the firm to reduce production and conserve resources for future growth. In a deterministic environment where the inventory is fully perishable, we analytically characterize the long-run dynamics of the resource quantity. In particular, sustainable production is more achievable for low-margin products, and a myopic production firm does not necessarily deplete the resource. We conduct numerical experiments using real-world salmon population data to test the robustness of our findings in general systems with stochastic environment and non-perishable products. The results show that ignoring either the resource constraint or the self-renewal dynamics, could lead to high operational costs and undermine resource sustainability.