An effective methodology is reported for determining the optimal capacity (lot size) of batch processing and storage networks which include material recycle or reprocessing streams.It is assumed that any given storage unit can store one mterial type which can be purchased from suppliers,internally produced,and internally consumed and/or sold to customers.It is further assumed that a storage unit is connected to all processing stages that use or produce the material to which that storage unit is dedicated.Each processing stage transforms a set of feedstock materials or intermediates into a set of products with constant conversion factors.The objective for optimization is to minimize the total cost composed of raw material procurement,setup,and inventory holding costs,as well asthe capital costs of processing stages and storage units.A novel production and inventory analysis formulation,the PSW model,provides useful expressions for the upper/lower bounds and average level of the storage inventory holdup.The expressions for the Kuhn-Tucker conditions of the optimization problem can be reduced to two subproblems.The first yields analytical solutions for determining batch sizes,while the second is a separable concave minimization network flow subproblem whose solution yields the average material flow rates through the networks.For the special case in which the number of storage units is equal to the number of process stages and raw materials storage units,a complete analytical solution for average flow rates can be derived.
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