National Paper runs a paper plant in Maine that buys pulp and turns it into three types of paper products: food contact paper, high-speed inkjet paper, and envelope paper. They need to plan their production schedule for the next quarter. The plant has two "machines" (each the size of a football field), and each machine works independently to produce the product it has been assigned. The plants run 24 hours per day, seven days a week, and so an entire quarter equals 2000 hours of useable production time (some hours during the quarter are set aside for cleaning and maintenance and are not included in this 2000 hours). Despite the plant's nonstop production, recent demand for their three products has skyrocketed, and the plant cannot satisfy demand for all of their products.

 

Quarterly forecasted demands for the three types of paper products are currently: 480000 tons of food contact paper, 280000 tons of inkjet paper, and 120000 tons of envelope paper. The two machines that make these three products are of different types and produce paper products at different rates. These rates (tons per hour) are given in the table below:

Machine	Food Contact	Inkjet	Envelope
Machine 1	150 tons/hour	200 tons/hour	180 tons/hour
Machine 2	135 tons/hour	200 tons/hour	170 tons/hour

Margins (profit margins), measured in dollars per ton, are given in the table below.

Food Contact	Inkjet	Envelope
$1250/ton	$1000/ton	$1100/ton

 

• a. Assuming machines can switch to different products without incurring any downtime, write out an LP formulation that helps National Paper decide which products to make as well as the appropriate production targets (quantities) of each type of paper product.
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• b. Solve your formulation in Excel and report the optimal production schedule and the optimal value.
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• Now suppose each machine must first be set up to make each type of paper product, and this requires 48 hours of preparation, which comes out of that machine's useable production time of 2000 hours. At the start of the quarter, assume neither machine is set up to make anything, and so a setup is required.
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• c. Write out a new formulation in its entirety that incorporates this set up requirement. Clearly label all decision variables and their types.
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• d. Solve your model in Excel and report your new optimal production schedule and your new optimal value.