Analysis of the multi x,T -rule

The analysis of this production rale will be done in much the same way as the analysis of the uncapacitated (x,T)-rule. For every type of product we consider a Markov chain in which the state is given by the number of penalty points. The interaction between the different states is found in the transition probabilities. We will consider the situation in which there are orders with different priorities. Every order has a fixed 

1) the demand po period per type and priority is integer valued, finite, independent of the demand for other periods or types and stochastically stationary. 

2) the penalty points of an order depend on the type and priority only and they are integer valued, preferably with one penalty-point for an order of the lowest priority class, in order to keep the state space as small as possible. 

3) the penalty points of a product type is defined as the sum of the penalty points of all orders for that type and a second element. This second element is the product of a type dependent constant C,- and the number of periods that has elapsed since the first arrival of an unfinished order for this type. The number of penalty points of the product type is the states in the Markov chain for that type. 

4) the types are ordered according to an increasing average demand, in order to simplify the choice of the type in situations in which two types have the same number of penalty points. 

---