Environment pollution is a burning issue in recent era. Continuous emissions of carbon into the atmosphere have raised the threat to environment and existence of human civilization. Governments and regulatory bodies across the globe are responding sharply to the threat by introducing different carbon policies. In this study, we consider a strict carbon cap policy in which rregulatory bodies allow organizations to emit carbon within a specified level and the penalty for exceeding the level is infinitely large. Regulatory bodies and governments are considering seriously adopting this policy for some sectors. We develop a production-inventory model in a vendor-buyer supply chain under stochastic demand at the buyer with partial backorders. The vendor produces a product at a finite production rate and delivers the same to the buyer in several equal-sized shipments. A mathematical model has been formulated to determine the optimal order quantity, reorder point and number of shipments in a production cycle by minimizing the total expected cost of the supply chain. Further, sensitivity analyses gives insight about how with operational adjustment cost and emission could be controlled. The proposed model can be useful for the firms who are operating under strict carbon cap policy.