Formulation of the second part

If the available capacity is not allocated completely in the first part of the rule we continue with the second part. In this second part we shall consider the possibility of producing some extra orders to avoid capacity problems in the future and we determine the effect of this production on the costs on the aggregated level. Calculating this effect exactly is usually impossible, because of the complex state and action spaces. Therefore the first step is to simplify the state and action space by means of aggregation. The order state on the aggregated level takes the form  

Let Cai be the expected number of orders on the aggregated level that arrive during the last i periods before their due date period t  

Then we can describe the expected number of orders that should be produced before the end of the i-th period, denoted by F,-, in the following way  

We do not consider any variability, but we just consider the penalty costs we would have if the number of orders would equal F,- and if, from the second period on, we would use all available capacity, by taking the maximum action C . The penalty cost function in this model is given by  

In the summation we can stop after H periods, where H can be chosen freely, or as soon as we have a non-positive term for i N. 

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