Math540 probability function

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MATH540 Week 3 Assignment, Chapter 14, Jet Copies, Set up

Provided by Professor Aungst, Supplemental Instruction

Information from Jet Copies Case Study:

– Students bought an $18,000 copier to start their own copy business.

– Wanted to purchase a smaller copier for $8,000 as back-up

– Created a simulation to estimate the amount of revenue that would be lost if they did not have a

backup

– Time between breakdowns is 0 weeks to 6 weeks (see probability function on page 679, and

provided later in this set up

– Developed following probability distribution of repair times:

Repair Time (days)

Probability

1

0.20

2

0.45

3

0.25

4

0.10

– Estimated they would sell between 2,000 and 8,000 copies per day at 10 cents (0.10) per copy

– Used a uniform probability distribution between 2,000 and 8,000 to estimate how many copies

they would sell per day

– If loss of revenue due to machine downtime during 1 year is greater than or equal to $12,000,

then they should purchase the back-up copier

– Decided to conduct a manual simulation of this process for 1 year to see if the model was

working correctly

– Our assignment is to perform this manual simulation for JET copies and determine the

loss of revenue for 1 year.

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Here’s some preliminary Set Up information:

The probability function for time between repairs, f(x), is,

f(x) = x/18, 0 <= x <= 6

and, r = x^2/36

x2 = 36r

x = 6*sqrt of r (use this formula in the column you designate as time between repairs)

You could develop the cumulative distribution and random number ranges for the distribution of

repair times for reference if you would like that for reference.

Repair Time

Repair Time

y (days)

1

2

3

P(y)

0.2

0.45

0.25

Cumulative

Probability

RN Ranges

4

0.10

The probability function for daily demand is developed by determining the linear function

for the uniform distribution, which is,

f(z) = 1 / b – a which equals 1/6

Letting F(z) = r in the Integrated Function, and solving for z we get: z = 6r + 2 (this is the

formula for copies lost)

There are various ways to set up the Monte Carlo simulation in Excel using the formulas we

learned in Chapter 14 … namely Random Number Generation (which is =RAND) and

VLOOKUP which allows us to “point back” to a probability table and insert a probability based

on that Random Number and the Probability associated with it in the table.

Most students start with developing the probability table for Repair time to later be used as the

VLOOKUP Table for Repair Time probability.

P(x)

Cumulative

Repair

Time

The Simulation itself would be for 52 weeks (which would be when the cumulative “time

between breakdowns” reached 52 weeks). You could begin with a Random Number (r1) which

would be multiplied by column 2, the Time Between Breakdown (in weeks) formula of 6*square

root of r1

You could then sum those variables in a cumulative list in column 3 (so you could tell when the

simulation reached 52 weeks).

In column 4 you could generate another random number (say, r2) to calculate the column 5

Repair time in y days.

That r2 could be used in a column 5 for Repair Time in y days which could be calculated by

using the =VLOOKUP function which would relate that r2 to probabilities in the Repair Time

probability table originally set up.

You might then set up some random number columns and result columns for repairs taking 1 day,

2 days, 3 days and 4 days.

At some point, you would need to figure out how to calculate copies lost in a day in thousands

and that would probably include the formula z = 6r + 2

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