Lot-sizing with setup times and costs

A classical example involving fixed charges is the lot-sizing model, which is essentially a generalization of the basic EOQ model to take into account multiple items, limited production capacity, and time-varying demand. To see why such a model arises, note that in the multiperiod planning models (12.26) and (12.27) we did not consider at all the need for machine setup before starting production. Suppose that, in order to produce a lot of item i, we need to spend a setup time  for each resource m. This setup time does not depend on the lot size, and it gives us an incentive to stock an item, rather than producing it in each time bucket. By the same token, we may have a fixed cost f_i associated with each setup for item i; this may depend, e.g., on material which is scrapped at the beginning of a lot because of the need of adjusting machines. In purchasing, setup times play no role, but we may need to tackle similar issues, e.g., when there is a fixed component in the transportation cost. The decision of producing a lot of item i during time bucket t is a logical decision; either we do it or we do not. Hence, we introduce a binary decision variable

and link x_it and δ_it using the big-M constraint

In practice, one way to quantify the big-M is to consider that there is no economic reason to produce more than what we can sell in the remaining time to the end of the planning horizon; therefore, we may choose

The resulting model, in the case of cost minimization, is a fairly straightforward extension of (12.27):

This is a rather innocent-looking MILP problem, which can be solved by commercial branch and bound code. In practice, it is very hard to solve to optimality; in Section 12.6.3 we consider a suitable reformulation that improves model solvability considerably.