Maple sugaring requires three main operations. First, in the late fall or early winter, seasoned wood is cut and split for running the boiler ("evaporator") that boils the sap down into syrup. Then, in mid-winter, maple trees are tapped using a drill ("auger"), and a tap is inserted in the tree. The taps are connected via tubing to a sugarhouse. Finally, when the sap starts to run (usually in late February or early March), it is collected at the sugarhouse, where it is boiled down into maple syrup. Farmer Moore has 1200 worker (labor) hours available during the entire sugaring season, and her workers cut the wood, tap the trees, and run the boiler. Cutting/splitting wood and tapping trees can be done individually, but running the boiler requires two workers (i.e., 2 worker hours = 1 boiling hour).
The table below summarizes the labor requirements, maximum availability of hours for specific steps, and corresponding production rates. Assume all labor costs $18 per hour.
|Step/Process||Input Labor||Output Rate||Max Available|
|Cutting/Splitting Wood||1 Worker Hour||¼ Cord of Wood||----|
|Tapping Trees||1 Worker Hour||25 Taps||480 Worker Hours|
|Boiling Sap (2-person operation)||1 Worker Hour||1/2 Hour of Boiling||240 Boiling Hours|
Additionally, there are several important input/output relationships or conversion rates.
|Input||Needed to Produce (Output)|
|1 Tap||18 Gallons of Sap|
|1 Gallon Sap||1/40 Gallon of Maple Syrup|
|1 Cord of Wood (in the boiler)||70 Gallons of Maple Syrup|
|1 Hour of Boiling||35 gallons of Maple Syrup|
Farmer Moore sells maple syrup for $50 per gallon.
1A. Write out an LP model formulation that determines how much time is needed for tapping the trees, running the boiler, and cutting and splitting wood so that Farmer Moore can maximize her profit (or net contribution). Solve your model in Excel and report your optimal solution and optimal profit.
Hint: I used 5 variables: time tapping (TT), time boiling (TB), time cutting/splitting wood (TC), sap produced (S), and syrup produced (SY). You will need constraints that limit the production of sap and syrup; each row in the last table led to a simple constraint in my model. This set up is helpful for part b. (There may be other simple ways to model this.)
1B. There is now a cottage industry for people that tap maple trees and sell the sap (but do not own a sugarhouse for converting it into syrup). Suppose maple sap can be purchased in the local market for $.60 per gallon (in unlimited quantities). Write out a new LP model formulation that allows Farmer Moore to augment her sap production with additional sap purchases to maximize profits. Solve your model in Excel and report your new optimal solution and new profit.
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