Part I: Identifying
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.
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
B = number of Model B
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
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
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
3. Patty (see 2.1. above) earns the following profit on each
Using the variable labels given in 2.1, write the profit
equation for one of Patty’s days.
Part III: Solving an
A small Excel application,
Estimation.xlsx is available to help you with this part of the Case.
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
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
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:
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.
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.
References and citations are required. This requirement can be
satisfied by citing the module Home page.
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