Part I: Identifying Constraints
Each short scenario describes a business situation. For each scenario, please write a short paragraph explaining which constraint or constraints is/are present, and why.
Part II: Describing Constraints
For each situation, write down the appropriate constraint. Please use the standard symbols “+” and “-” for plus and minus, and “ * ” for multiplication; however, if you find the symbols “≤” and “≥” difficult to keep straight, you may write “LE” for “Less than or equal to,” and “GE” for “Greater than or equal to.”
Example: An office is buying filing cabinets. The Model A cabinet holds a maximum of 3 cubic feet of files. Model B holds a maximum of 4.5 cubic feet of files. At any time, the office will have a maximum of 17 cubic feet of files that need to be stored. (A bit more cabinet space wouldn’t be a problem, but not enough would be. The files can’t be stacked on the floor.) Write a constraint on the number of cabinets the office should acquire.
Variables:
A = number of Model A cabinets purchased
B = number of Model B cabinets purchased
Answer:
A*( 3 cubic feet) + B*(4.5 cubic feet) ≥ 17 cubic feet
Or
A*( 3 cubic feet) + B*(4.5 cubic feet) GE 17 cubic feet
Variables:
C = number of cups made
B = number of bowls made
V = number of vases made
Variables:
GenA = number of type A generators shipped
GenB= number of type B generators shipped
3. Patty (see 2.1. above) earns the following profit on each product:
Vase: $1.35
Cup: $0.75
Bowl: $2.40
Using the variable labels given in 2.1, write the profit equation for one of Patty’s days.
Part III: Solving an Allocation Problem
A small Excel application, LP Estimation.xlsx is available to help you with this part of the Case.
A refinery produces gasoline and fuel oil under the following constraints.
Let
gas = number of gallons of gasoline produced per day
fuel = number of gallons of fuel oil produced per day
Demand constraints:
Minimum daily demand for fuel oil = 3 million gallons (fuel ≥ 3)
Maximum daily demand for gasoline = 6.4 million gallons (gas ≤ 6.4)
Production constraints:
Refining one gallon of fuel oil produces at least 0.5 gallons of gasoline.
(fuel ≤ 0.5*gas; gas ≥ 2 fuel
Wholesale prices (earned by the refinery):
Gas: $1.90 per gallon
Fuel oil: $1.5 per gallon.
Your job is to maximize the refinery’s daily profit by determining the optimum mix of fuel oil and gasoline that should be produced. The correct answer consists of a number for fuel oil, and a number for gasoline, that maximizes the following profit equation:
P = (1.90)*gas + (1.5)*fuel (Answer will be in millions of dollars.)
Run at least 10 trials, and enter the data into a table that should look something like this:
| Trial production values: | Profit: | |
| Fuel oil | Gasoline | |
| 3.1 | 6.3 | 16.62 |
| Etc. | ||
Here’s how the “LP Estimation” worksheet is set up:
The tentative production goals are between the allowable minimum and maximum for each. Note that the maximum fuel oil that can be produced depends upon the gas production. Conversely, the minimum gas that can be produced depends upon the fuel oil production. If you don’t use the worksheet, the challenge will be to find test values for both oil and gas production that jointly satisfy the constraints.
Note: This is NOT how such a problem is usually solved. Rather, it is solved using LP. The purpose of this exercise is to acquaint you with the type of problem that’s usually solved using LP, and give you an appreciation of how difficult such a problem would be without LP.
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