Lot sizing problems¶
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Given a set \(T=\{1,2,\dots,m\}\) of periods with a demand \(d_t\) in every period \(t\in T\), a solution fixes the periods with a setup and the quantity produced in each of them, so that every demand is satisfied.
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The objective is to minimize the total cost of setups, production and inventory holding; the members differ in the capacities, in the number of products and in the treatment of initial stock, overtime and unsatisfied demand.
| Problem | Acronym | Difference |
|---|---|---|
| Capacitated lot sizing | CLS | The central problem of the family. |
| Uncapacitated lot sizing problem | ULS | Special case of the CLS in which the capacities are not binding, for example \(cap_t \ge \sum_u d_u\). |
| Capacitated lot sizing with constant capacity | CLS-C | Special case of the CLS; the zero-inventory ordering property is replaced by a weaker one: between two periods with empty inventory at most one period produces a quantity strictly between \(0\) and \(C\). |
| Lot sizing with several products and setup times | MLS | Generalizes the CLS, which is the case of one product with \(vt_t=1\) and no setup time. |
| Lot sizing with initial stock | LS-SI | Removes the assumption of an empty initial stock of the CLS. |
| Lot sizing with overtime | LS-OT | Replaces the fixed capacity \(cap_t\) of the CLS by two levels of capacity with different costs. |
| Lot sizing with unsatisfied demand | LS-UD | Relaxes the demand constraints of the CLS, with a penalty in the objective. |