The Wagner-Whitin approach

The two extra problems, the partly unknown demand and the moving horizon, will be approached in the following way. In our algorithm we do not consider the demand distribution, but we will replace the unknown future demands by their expected value, still assuming that we can not produced expected orders before their arrival date. The second problem is known as the effect of terminal conditions in the rolling schedule. Baker (1981) studies this problem in a special quadratic production-inventory model. In his solution the terminal conditions are based on the profile of states that occurs in a deterministic finite horizon model. Implemented in a situation with uncertain demand, this solution achieves a near-optimal performance. Although we do not have this quadratic production-inventory model, we will also consider terminal conditions. Therefore we assume some simple production mle to be used after the horizon to measure the effect of an action sequence on later periods. 

Li our Wagner-Whitin approach we want to find the action sequence (a, a2.aH) with minimal costs. The costs of an action sequence consist of three parts the direct costs in the first period, the expected costs in periods 2,3. and the indirect costs the effect of the action sequence on the costs in later periods. The effect of the production of ord s with a due date after the horizon will be measured by determining the marginal costs for the expected order state at the end of the H-Ha. period. Therefore we introduce a salvage function LQ, which can be compared with the function v (r) in the optimal policy (cf. (4.2.4)). 

Let Ah be the set of all the possible action sequences with H elements. Let z,- be the projected order state vector during the i-th period for an action sequence AeA//. Hence  

Now the production rule takes the following form given a state reR determine the action sequence Ae Ah for which  

In this paragraph we will consider the indirect costs. Let AeAh be a given action sequence. Then we denote by /(A)=max(f I a, 0) the last period in which we produced and by j(A)=max( i+a the last period for which some orders have been 

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Due to the increased state space, the use of the Wagner-Whitin approach and the use of the optimal policy will become much more complex, because in these production rules we have to consider the order state during a number of periods. In the Silver-Meal approach and in the (x,7>-nile we only consider the action for the first period. In order to determine the penalty costs, if we do not produce during this first period, we have to know the sum of the ri values for the different priorities weighted with the... 

The new state vector r does not provide the necessary information for a good use of the optimal policy and for the Wagner-Whitin approach. For these rules it is indeed necessary to know the order state vector for each of the different priorities separately. The (x,T)-rule and the Silver-Meal approach can be used straightforwardly in this situation. Of course the fixed cycle production rule n (cf. (4.2.10)) can also be used without any complications. The use of these rules is now exactly the same as described in 4.2.4.2. and 4.2.4.4. 

In the model we assume that M types of products can be produced on the same machine. The length of a period is C time units and we will express the delivery times and the lead times in an integer number of periods. The maximum lead time will be N periods. The service time, that is the time to produce one order, is supposed to be one time unit for all types. Every set-up takes S time units. Producing a type of product implies a set-up, disregarding the possibility that the same type may have been produced in the previous period. This assumption is made because this element is quite common in periodic review models, for instance in the uncapacitated Wagner-Whitin approach. The assumption has also a large influence upon the production rules, but the rules that will be presented can easily be adapted to a situation in which no set-up would be needed if production continues in the next period. By these assumptions, the maximum number of orders that can be produced in one period, without working overtime, is equal to C-S. 

Gilbert [64] focuses on a similar problem but assumes that demand is a multiplicative function of seasonality, i.e., dt p) = PtD p), where D p) represents the intensity of demand, thus demand the ratio of demands in any two periods does not depend on price. Gilbert also assumes that holding costs and production set-up costs are invariant over time, and the total revenue is concave. He develops a solution approach that guarantees optimality for the problem, employing a Wagner-Whitin time approach for determining production periods. 

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