A fluctuating demand rate

In the previous section, we assumed that orders that are delivered too late had no influence upon futiue demand. The effect of unsatisfied clients was expressed in the penalty costs p, but not in the demand. This way of modelling unsatisfied clients is very common, and it can be found in many production-inventory models. Nevertheless, in a practical situation orders that are delivered too late may have an 

The situation in which the population of clients is subject to different kinds of changes, as described above, is modelled in the following way  

Let N(f) be the population size at the beginning of period t, D(t) the decrease in the population size during period t, which is independent of the lateness of the orders, L(t) the lateness-dependent decrease in the population size and I(t) the number of new clients during period t. The stochastic processes are independent of each other, except that in every period the total decrease cannot exceed the number of clients in the population. Therefore the population size in period r+1 cannot be less than the number of new clients during period t. Hence the number of clients in the population can be described by the following formula 

For a firm, the population size is not that important. Much more important is the resulting demand by these clients. In our model, all clients have the same properties with respect to leaving, arriving as a new client and with respect to the demand they order. This implies that the distribution of the demand for a certain period, having n clients is the n-fold convolution of the distribution of the demand with 1 client. We will assume that every client can order products in every period for each of the N possible lead times. 

For a given distribution of the demand with 1 client and given distributions for the stochastic processes D(r),L(0 and /(/), we have to determine the action as a function of the population size N t). The object should be the maximisation of the profit. The profit is given by the revenues of the orders minus the costs for set-ups and the holding costs. We do not have penalty costs in this situation, since the costs of lateness are now expressed in the demand and the number of clients. 

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