Reference no: EM133154835

1. In the late winter every year, many New England farmers make maple syrup. They tap maple trees for their sap, which is then boiled down to make maple syrup. Farmer Moore is one such New Englander.

Maple sugaring requires three main steps. First, seasoned wood is cut and split for running the boiler ("evaporator") that boils the sap down into syrup. Then, during a critical two-week period in mid-February, the maple trees are tapped using a drill ("auger"), and a tap is inserted in the tree. Finally, the sap from the trees is collected via tubes and piped to a boiler in the sugarhouse, where it is boiled down into maple syrup. Farmer Moore has 1800 worker hours available during the 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.

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 Max Labor Hours Available
Cutting/Splitting Wood 1 Hour Labor ¼ Cord of Wood ----
Tapping Trees 1 Hour Labor 25 Taps 360
Boiling Sap (2-person operation) 1 Hour Labor 1/2 Hour of Boiling 480

Additionally, there are several important input/output relationships or conversion rates.

Input Will Produce (Output)

1 Tap 12 Gallons of Sap

1 Gallon Sap 1/40 Gallon of Maple Syrup

1 Cord of Wood (in the boiler) 20 Gallons of Maple Syrup

1 Hour of Boiling 14 gallons of Maple Syrup

Farmer Moore sells maple syrup for $50 per gallon.

1A. Write out an LP model 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. (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 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.