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Linear Programming

Product Mix Problem

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	Product Mix Problem

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The figure represents a manufacturing system producing two products labeled P and Q. The rounded rectangles at the top of the figure indicate the revenue per unit and the maximum sales per week. For instance we can sell as many as 100 units of P for $90 per unit. The circles show the raw materials used, and the rectangles indicate the operations that the products must pass through in the manufacturing process. Each rectangle designates a machine used for the operation and the time required.

For example product P consists of two subassemblies. To manufacture the first subassembly, one unit of RM1 passes through machine A for 15 minutes. The output of machine A is moved to machine C where it is processed for 10 minutes. The second subassembly starts with RM2 processed in machine B for 15 minutes. The output is taken to machine C for 5 minutes of processing. The two subassemblies are joined with a purchased part in machine D. The result is a finished unit of P. Product Q is manufactured by a similar process as indicated in the figure.

The rectangle at the upper left indicates that one machine of each type is available. Each machine operates for 2400 minutes per week. OE stands for operating expenses. For this case the operating expenses, not including the raw material cost is $6000. This amount is expended regardless of amounts of P and Q produced.

Our problems include the following: Find the product mix that maximizes profit. Identify the bottlenecks. For each product, find the range over which the unit profit can change without affecting the product mix. For each machine, identify the marginal benefit of adding one more minute of machine time. For each machine, find the range over which the time availability can change without affecting the identity of the bottleneck.