This note gives a simple proof that in a (r, q) system the average outstanding backorders and the average stockouts per unit time are jointly convex in the two control variables q and r.

We consider a supply chain in which two suppliers compete for supply to a customer. Pricing and delivery-frequency decisions in the system are analyzed by two three-stage noncooperative games with different decision rights designated to the parties involved. The customer first sets the price (or...