# Case Assignment

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 â€ś+â€ť and â€ś-â€ť

for plus and minus, and â€ś * â€ť for multiplication; however, if you find the

symbols â€śâ‰¤â€ť and â€śâ‰Ąâ€ť difficult to keep straight, you may write â€śLEâ€ť for â€śLess

than or equal to,â€ť and â€śGEâ€ť for â€śGreater than or equal to.â€ť

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 wouldnâ€™t be a problem, but not enough would be. The

files canâ€™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) â‰Ą 17 cubic feet

Or

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: â€śjointâ€ť = determined together; each

affected by the others.)

Variables:

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.

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 Pattyâ€™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 â‰Ą 3)

Maximum daily demand for

gasoline = 6.4 million gallons (gas â‰¤ 6.4)

Production constraints:

Refining one gallon of

fuel oil produces at least 0.5 gallons of gasoline.

(fuel â‰¤ 0.5*gas; gas â‰Ą 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 refineryâ€™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.

Hereâ€™s how the â€śLP Estimationâ€ť 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 donâ€™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 thatâ€™s usually solved using LP, and give you an appreciation of

how difficult such a problem would be without LP.

Assignment Expectations

There are

no page limits. Write what you need to write, neither more nor less. Make each

sentence count! (Having said that; itâ€™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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