The dynamic programming rule

At the end of every period, we have to take a decision about the lead times for the new orders that have been arrived during the period. Then the clients can accept these lead times or withdraw the order. If the process of acceptance has taken place, still at the end of the period, a decision about the production has to be made. The production rule for this situation can be rather simple. If there are no required deliveries for the first period, then it makes no sense to produce already and therefore action a = 0 is the only reasonable option. Because of the assumption that all accepted ordos will be produced in time, we have to produce if th are required deliveries for the first period. Usually most of the orders wiU have about the same due date. A reasonable action in this situation is to produce all orders, in other words action a = N, where N is the maximal lead time of an ordo. However, we can also think of a situation in which there are clearly two groups of ord one group with a residual lead time of one period and another group of orders, which is not too small, with a residual lead time of three or more periods. In this situation the most reasonable action can be the production of only the orders that have to be delivered by the end of the first period, or in other words take action a = 1. Of course, we can only come to a situation like this if the rules for the lead times make an occurrence of such a situation possible. 

If we use this production rule, the two most important elements for the rules for generating lead times are the total number of orders and the minimal value of the residual lead times of the accepted orders. If the lead time for the new orders is the same as the minimal value of the residual lead times, the lead time decision does not imply extra holding costs. If the lead time for the new orders is shorter than the minimal value, then we have to pay holding cost for the difference in periods. In the situation in which we decide only to produce the new orders, as far as they are accepted, in the first period, then we do not have holding costs. From this discussion it follows that the original state space, with the state space vector r=(ri,r2.rsh where r,- denotes the number of other with a residual lead time of i periods, can be limited to the two most important elements the total number of accepted orders and the minimal values of the residual lead times of the accepted orders. We will denote a state by (y,t), where y is the total number of orders, yeN, and t the number of periods, 0. The state in which there are no orders will be denoted by (0,N). In the states with r = 0 we have produced all orders during the period. 

Now we come to the description of the DPR. In this rule we make the decision about the lead times and the action for the production simultaneously. In a situation where we have decided to produce and where all orders for the first period have been withdrawn, the decision to produce will be cancelled. If no new orders have arrived during a period, the possible actions for the production will be a =N if there are required deliveries for the first period, or u = 0 otherwise. In the DPR we have to make a choice for every state (y,t) in combination with j new orders. This choice will be denoted by Ch (y,t,j). If there are no new orders during a period, no decision has to be made. The value of Ch(y,t,j) will be the offered lead time to the J new orders, if this lead time is 2 periods or more. If the offered lead time to the J new orders is equal to 1, then we will write Ch(y,t,j) = i if the lead time decision is combined with the decision to produce all orders, or Ch(y,t,j) = 0 if the lead time decision is combined with the decision to produce only the new orders, as far as they are accepted. In the states (y,t) with t=0,l,2, the production rule does not allow the choice Ch(y,t,j) = 0. If the choice Ch(y,t,h) = 0 is made, the cost effect of the new orders will be calculated immediately. Therefore, the effect of this choice upon the future periods is the same as the effect of having no new orders during a period. 

In order to make the correct choices Ch(y,t,j), we will use a value function v(y,t) which denotes the marginal profit until the next production period. In the DPR we choose this value function similar to the relative cost function v(r) in (4.2.6)  

Due to the cost structure where the holding costs are paid immediately, where the revenues are received in the production period and where the set-up costs are paid in the production period, we have the following direct costs. If in the state (y,t) J new orders have arrived and the choice Ch is made, the direct costs of this choice, analogue to the direct costs in (4.2.6), will be denoted by dc(y,t,j,Ch). The value of dc(y,t,J,Ch) is given by  

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