Firms often utilize promotion (such as coupons, advertisements, recruitment of excellent salespeople, and leafleting etc.) and dynamic adjustment of price to manage customers as well as proper production/inventory plan to satisfy the customers to get maximal profit in a firm. The decision on promotion and pricing and the decision on production/inventory must support each other. This paper addresses coordinated decision on pricing, promotion(non-price promotion) and inventory management. Specifically, we study a single item, periodic review model. The demand function is a linear demand function in which the market scale can be affected by the promotion conducted in that period. Unsatisfied demands are fully backlogged. We characterize the structure of the optimal policy that simultaneously determines the price, the promotion and the ordering quantity to maximize the total discounted profit with finite and infinite period problems. We show that the optimal replenishment policy is the quasi base stock list price target promotion policy, i.e., there exist a critical inventory level, a list price and a target promotion such that it is optimal to order up to the critical level, charge the list price and conduct the target promotion when the initial inventory is below the critical level and order nothing, conduct a higher promotion and charge a proper price to increase the demand otherwise. We also prove that the expected demand and the optimal promotion are increasing in the inventory, and price and promotion are complementary. Then we extend the problem to the case with capacity constraint. We show that the modified quasi base stock list price target promotion policy is optimal. For the joint decision problem on pricing, promotion and inventory control with positive fixed setup cost, we show that the optimal policy is (s, S, p, e) policy. That is, there exist two critical inventory levels s(t) and S-t (s(t) <= S-t), a list price p(t) and a target promotion e(t) in period t such that order up to S-t, charge price p(t) and conduct promotion e(t) when the initial inventory is less than s(t) and order nothing, charge a proper price and conduct a proper promotion otherwise.
