**3. Results**

In this section, the presented BPP problem formulation will be demonstrated in several case studies of material requirements planning with data sets acquired from an industrial environment. All experimental data are available at [26]. The problem concerns raw materials which must be cut to satisfy the needs of work orders for input material. More specifically, raw materials must be available in desired quantities by a specific time, while the input (intermediate) material for one product has to be acquired from the same raw material batch. Purchase orders and work orders are not linked in the current configuration, and material requirements planning is done manually.

Raw materials are provided as defined by purchase orders. In the presented BPP formulation, purchase orders are described with bins, where *cj* determine the package size of raw material and *aj* the time when it will be available. *aj* is not necessarily the same for all raw materials of a particular purchase order. Every purchase order also has a label *FBl* , which designates a group of raw materials with the same properties. Work orders determine which materials (intermediates) are needed to produce one final product. Every intermediate is presented with an item in our problem formulation, where *si* specifies how much of a material is needed and *di* the work order's due date. As raw materials can slightly deviate in some parameters, we must deal with additional constraints. We must ensure that a group of intermediates used to make one final product are produced from the same raw material. Label *FIk*is used to determine items that belong to one work order, i.e., one final product.

In this way, the problem of material requirements planning can be translated into a generalized bin-packing problem. Depending on the situation encountered by the operator, different criterion with different constraints can be applied. We have analyzed two general situations that can occur in practical implementations. If the time constraints are feasible (Equation (4)), then we should apply the optimization problem presented with Equation (3), where the leftover is being minimizing. If this is not possible, softer constraints have to be chosen (Equation (5)), in which we optimize the problem from Equation (6). In this case, we are solving a multi-criterion problem, where leftover and tardiness are being minimized.
