Optimal capacity assignment

An important step in the optimization of production processes is to assign the production capacity in a certain period to the several products that can be produced on a production line. For simplicity, we ignore changeover time here. In this situation, the optimal capacity assignment is defined as follows. 

Assume that there are n products with demand densities 8, ...8n, already received orders ri,... rn, and available stock si,... sn. Assume that ti, - h time units are needed to produce one unit of product on a given production train and that pi,... pn are the marginal profits achieved per additional unit of each product. The optimal capacity assignment is the vector x which maximizes the function X ,"=i M ( S+.r , +% ) Pi subject to the constraints Xi > 0 and XiU < t, t being the length of the period. 

Figure 6.10 shows the data flow of the software tool BayAPS PP for optimal capacity assignment for given stochastic demands. Transaction data about demand and inventories is typically imported from SAP R/3 as indicated, production capacity master data and side conditions are stored in the software tool. Forecasts can be taken from a forecast tool or from SAP R/3. The output ofthe tool is a list ofpriorities of products and their lot sizes, which are optimal based on the presently available information. Only the next production orders are realized before the computation is repeated, and the subsequently scheduled production is only a prediction. 

This section deals with production lines for more than one product. In the process industries it is often a problem to assign the capacity of one production line to several products, all or some of which have uncertain demand. We want to optimize the overall service level for such a production line. 

The core algorithm assigns the production capacity to the competing products of the same production line, such that the overall expected sales are maximal. If the capacity is critical this basically means that production capacity is designed to the more likely parts of the uncertain demand. If there is plenty of production capacity, safety stock is allocated reflecting the product-specific uncertainty of demand. When looking at all products, usually the situation is between these extremes and the algorithm provides an optimal compromise. 

The algorithm assigns more capacity to product B01 than to B02. Why The capacity is critical, because only 20 + 168 units of product are available, but the total demand is 400. Under these circumstances the optimal portion of production capacity is diverted from the other products to B01, the demand for which is more certain because of the orders that were already received. For A01 and A02 the same number of available units of product after production is still the same. 

By definition, all interpretive methods of optimization require knowledge of the capacity factors of all individual solutes. This is the fundamental difference between the simultaneous and sequential methods of optimization (sections 5.2 and 5.3, respectively) and the interpretive methods of section 5.5. Moreover, in the specific cases in which only a limited number of components is of interest or in which weighting factors are assigned to the individual solutes (see section 4.6.1) it is also necessary to recognize the individual peaks (at least the relevant ones) in each chromatogram. In section 5.5 we have tacitly assumed that it would be possible to obtain the retention data (capacity factors) of all the individual solutes at each experimental location. 

There is another personnel assignment question for which a network model may be useful. There is a set of jobs and a set of people where for each person there is a list of the jobs that the person is qualified for. The objective is to Maximize the total number of jobs that can be filled with qualified people. This question can be modeled as a maximmn flow problem, as shown in Figure 8. All arcs in the network have a capacity of one. The objective function is to maximize the flow between nodes s and t. An arc connecting nodes P and Jj inchoates that person i is qualified for job 7. In the optimal flow pattern, person i is assigned to job j if the flows on arcs (B , Jj), (s, P ) and (/, f) are all one. Since every arc (Jj, t) has a flow capacity of one, the total flow into node Jj can be at most one and so job j can be assigned to at most one person. For this model, there are no direct considerations of cost but instead the objective is to maximize the number of jobs flUed by qualified people. 

Generalized assignment problems involve optimal arranging of objects i = 1,. . . , m of known size Sj into locations j = 1,. . . , n of known capacity Kj. An (ILF) formulation is... 

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