Simplified dynamic programming rule

This simplified version of the dynamic progranuning rule will be called SDPR. The profit of the SDPR may be a little smaller than the profit of the DPR, but the choices for each state and number of new orders can be found much easier. Again we can start for instance with go= it and determine the choices Ch(y,t,J) in the same way as in the DPR, although now the values of v(y,t) are not directly influenced by this choice. From this set of choices we can determine the average costs of the rule in exactly the same way as in the DPR by using (4.4.7)-(4.4.15) and then use this value go in (4.4.17) and repeat this procedure again. 

As we already suggested, it is not necessary to consider all the possible choices in a particular state, with a particular demand. Therefore we can use the following properties. 

Property 4.3 for a given demand j the proposed lead time in state (y i,r), Ch(y, t,j) will be no longer than the proposed lead time in the state (y2,t),Ch(y2,t,j), with y i 2. 

Property 4.4 for a given state (y,t) the proposed lead time for demand j, Ch(y,t,Ji) will be no longer than the proposed lead time for demand j2, Ch (y,tj2), with j j2- 

Proof property 4.3 For given values of the state y,t and the demand , the lead time Ch(y,t,J) is determined in formula (4.4.4) by finding the k for which the function 

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First, we shall consider a cyclic production rule in combination with a cyclic rule for the lead times and determine the average profit of this approach. Then we will give a description of a dynamic programming rule (DPR) for the lead times. This DPR is based te reduction and the decision rule itself is determined by means of dynamic programming. We will conclude this section with the description of a simplified version of the DPR, which will be denoted as the SDPR. Then some examples will be given to compare the performance of the different decision rules. 

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