Imagine that you are the store manager and need to decide the promotion plan for two particular products in the next 12 weeks, Product A (full price is given to be pA for Product A) and Product B (full price is given to be pB for Product B). You are told that you can run promotions on this product for at most 4 out of the 12 weeks. Furthermore, the price for product A should always be at most the price of product B. An additional requirement is to space out ALL promotions (for either product A or product B) by at least 3 weeks. In addition, whenever you run a promotion, your supplier requires that you cannot discount more than 30 % of the full price of product A and not more than 50 % of the full price of product B. However, the prices at for product A and product B in every week take all values between the full price and the minimum discounted price.

You have estimated demand for week t to be dt. Assuming that there are no stock-outs, i.e., demand is equal to sales, formulate the promotion optimization problem for maximizing revenue over 12 weeks. Explain the objective function, decision variables, and constraints.