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.
- A taxi company limits its drivers to one eight-hour shift per day, with one half-hour break for a meal. A taxi cannot carry more than three passengers, all in the back seat, and cannot combine stops; that is, all the passengers must be going to the same destination.
- Acme Unlimited sells custom machine tools. The sales department must send each order to Engineering for approval. There are no exceptions to this rule, which creates delays.
- An online university offers six-week terms, with three two-week modules per term. Each module is supposedly limited to one topic, or set of related topics. Faculty members find it difficult to create interesting courses, and students express frustration with their learning experiences.
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.
A = number of Model A cabinets purchased
B = number of Model B cabinets purchased
A*( 3 cubic feet) + B*(4.5 cubic feet) ≥ 17 cubic feet
A*( 3 cubic feet) + B*(4.5 cubic feet) GE 17 cubic feet
- Patty makes pottery in her home studio. She works a total of six hours (360 minutes) per day. It takes her 10 minutes to make a cup, 15 minutes to make a bowl, and 30 minutes to make a vase. Write a constraint governing the joint production of cups, bowls and vases in a day. (Trans: “joint” = determined together; each affected by the others.)
C = number of cups made
B = number of bowls made
V = number of vases made
- An aid agency is buying generators for storm survivors. The generators will be shipped on a convoy of flatbed trucks having a total usable cargo area of 1,350 square feet. Generator A has a footprint (floor space required) of 8 square feet, and Generator B has a footprint of 12 square feet. Write a constraint governing the number of generators of each type than can be shipped.
GenA = number of type A generators shipped
GenB= number of type B generators shipped
- Patty (see 2.1. above) earns the following profit on each product:
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 refinery produces gasoline and fuel oil under the following constraints.
gas = number of gallons of gasoline produced per day
fuel = number of gallons of fuel oil produced per day
Minimum daily demand for fuel oil = 3 million gallons (fuel ≥ 3)
Maximum daily demand for gasoline = 6.4 million gallons (gas ≤ 6.4)
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:|
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.
- There are no page limits. Write what you need to write, neither more nor less. Make each sentence count! (Having said that; it’s unlikely that one page would be enough, and very likely that eight pages would be too much.)
- Ensure that your answer reflects your detailed understanding of the theory and techniques taught in this module.