Commitment under uncertainty: production-inventory policies associated with environmental considerations

We consider a manufacturer that produces in response to a stochastic demand and emits pollution during the production process. Industrial pollutants released into the air are characterised by spatial variability and heterogeneity, and the precision of instruments for measuring stochastic pollution stocks varies widely across pollutants. Consequently, commitment (open-loop) strategies to control production and associated emissions are considered more practical than contingent (feedback) approaches when manufacturers are concerned about environmental consequences. In this article, we derive an optimal, open-loop control policy over an infinite time horizon and show a condition that ensures that such a policy induces asymptotic convergence of expected inventory stock and pollution stock trajectories to unique steady states. Moreover, we find that the variance in the pollution stock also converges asymptotically to a unique steady state, which is critical for prevention of irreversible environmental consequences. We show that environmental uncertainty affects inventories, leading, in the long run, to lower expected steady-state inventory stocks. We compare feedback and open-loop production policies and show that though there are losses associated with the inability to track inventory and pollution stocks accurately and thus applying the open-loop control instead of feedback, the expected long-term inventory and pollution stocks are not affected.