In the minimum cost lot-sizing problem, we assumed that demand must be satisfied immediately; by a similar token, in the maximum profit lotsizing model, we assumed that any demand which is not satisfied immediately is lost. In other words, in both cases we assumed that customers are impatient.

• Write a model for cost minimization, assuming that customers are willing to wait, but there is a penalty. More precisely, backlog is allowed, which can be represented as "negative inventory holding." Clearly, the backlog cost 6¿ must be larger than the holding cost hi. Build a model to minimize cost.
• Now assume that customers are indeed patient, but they are willing to wait only for two time buckets; after two time buckets, any unsatisfied demand is lost. Build a model to maximize profit.
• In the classical lot-sizing model, we implicitly assume that each customer order may be satisfied by items that were produced in different batches.
In some cases, this is not acceptable; one possible reason is due to lot tracing; another possible reason is that there are little differences among batches (e.g., in color), that customers are not willing to accept. Then, we should explicitly account for individual order sizes and due dates.
Build a model to maximize profit.
• As a final generalization, assume that customers are impatient and that they order different items together (each order consists of several lines, specifying item type and quantity). If you cannot satisfy the whole order immediately, it is lost. Build a model to maximize profit.