# applied statistics

Q1 Linear Programming — Product Output

A company produces 3 different products, A, B, and C.  All products require time in up to 5 different processing departments.  The amount of time each product requires with each department is listed in the table below:

 Product A Product B Product C Available Melting 6 hrs 2 hr 0 hrs 1200 hrs Cooling 3 hrs 2 hrs 1 hr 1240 hrs Shaping 3 hrs 2 hrs 1 hr 2000 hrs Sanding 2 hrs 3 hr 3 hrs 1875 hrs Painting 1 hr 2 hrs 2 hrs 1550 hrs

Suppose product A sells for \$200/unit, product B sells for \$225/unit, and product C sells for \$125/unit.  And suppose that all products have a guaranteed buyer (i.e., no matter how many are produced it will be sold for the asking price).  Management has asked for your recommendations about how many of each product type to produce that maximize revenue.  So question…

Question:
a)How much of each product do you recommend be produced that you believe maximizes revenue?

Question 2: Suppose your company sells 4 different kinds of pies, Apple Pies, Banana Pies, Cherry Pies, and Danish Pies, where each Apple Pie sells for \$8, each Banana Pie sells for \$9, each Cherry Pie sells for \$7, and each Danish Pie sells for \$10.  Further, suppose to make these pies, each must be processed on 2 different machines in your factory and each takes a different amount of time on the two machines.  Specifically, to make one Apple Pie requires 1 hour on machine #1 and 2 hours on machine #2.  To make one Banana Pie requires 2 hours on machine #1 and 3 hours on machine #2.  To make one Cherry Pie requires 2 hours on machine #1 and 1 hour on machine #2.  And to make one Danish Pie requires 1 hour on machine #1 and 0.5 hours on machine #2.  Finally, suppose there can be no more than a total of 150 hours of processing time on machine number 1 and no more than a total of 175 hours of processing time on machine number 2.  Management has asked for your recommendation about how many of each product type to produce that maximizes revenue.

Question:
a)How much of each product do you recommend be produced that you believe maximizes revenue?

Question 3: Q3  Control Charts

Your local cell phone company records how mani milliseconds for each cell phone to connect
to the tower when a call is made.  You randomly selected a sample of cell phone connection times
times in order to determine if the connection times are in control are not.

Below are the connection times of different phone calls that were randomly selected.

Questions:  Both tell you answer and explain/justify it.
a)Are the connection times normally distributed?
b)How many values are outliers?
c)Is the connection time in control or out of control?  Explain
e)Given the business implications, what concrete business recommendations do you have for the company?

#Data
160
2184
3127
4206
5133
6174
786
834
9151
10147
11149
12220
13132
14168
15202
16158
17101
18177
19172
20196
21137
22211
23171
24128
25226
26149
27172
28212
29105
30128
31196
32181

Q4  Statistcal Process Control: A set of data has been collected and evaluated for process failures using Statistical Process Controls. After the quantitiate analysis had been completed, several observations were made.

Questions:  Both tell you answer and explain/justify it.
a)If no values are outside of the control limits, what are the implications for the business and what would you recommend they do?
b)If 10% of the values are outside of the +/-1 standard deviation control limits, what are the implications for the business and what would you recommend they do?
c)If 10% of the values are outside of the +/-3 standard deviation control limits, what are the implications for the business and what would you recommend they do?
d)If 10% of the values are higher than the +3 standard deviation control limit and no values are below the -3 standard deviation control limit, what are the implications for the business and what would you recommend they do?

Q5  Queuing : The BUS670 store is a small facility that sells all sorts of supplies, snacks, etc.  It has one checkout counter where one employee operates the cash register.  Once customers select their items for purchase, they line up at the counter in a first-come-first-served basis to pay for their selections.

You monitored the store for a few days and found that customers arrive on the waiting line to pay for the selections at an average of 24 customers per hour (assume Poisson distribution) and the cashier seems to be able to serve an average of 30 customers per hour (assuming exponential distribution).

The owner of the store has hired you to answer the following questions:

Part 1In a typical 8 hour work day, how much time in total will the cashier be idle?

Part 2How many customers on average are waiting in line to pay?

Part 3What is the average amount of time a customer is waiting in line?

Part 4What is the probability a customer will have to wait in the line before being able to pay?
Q6  Queuing

Part 1If the average time to serve a customer is 3 minutes, then the service rate, µ, would be
2 points

a. 3 per hour
b. 12 per hour
c. 16 per hour
d. 20 per hour

Part 2If, on average, it takes 90 seconds to serve a customer then the hourly service rate, µ, would be
2 points

a. 90 per hour
b. 40 per hour
c. 30 per hour
d. 1.5 per hour

Part 3An espresso stand has a single server. Customers arrive to the stand at the rate of 28 per hour according to a Poisson distribution.  Service times are exponentially distributed with a service rate of 35 customers per hour.

Part 3aThe probability that the server is busy is
a. 0.20
b. 0.60
c. 0.80
d. 1.00

Part 3bThe average number of customers waiting in line for service is
a. 4.0
b. 3.8
c. 3.5
d. 3.2

Part 3cThe average time a customer spends waiting in line for service is
a. 0.114 minute
b. 0.143 minute
c. 6.857 minutes
d. 8.58 minutes

Part 3dIf the arrival rate remains at 28 customers per hour and the stand’s manager wants to have the average time a customer spends in the system (i.e., wait time line and service time) to be a maximum of 6 minutes on average, then the service rate must

a. decrease by 2 to 33 customers per hour
b. decrease by 3 to 32 customers per hour
c. increase by 3 to 38 customers per hour
d. increase by 2 to 37 customers per hour

Q7  Others

a)Why are Control Charts important?
b)What are the key steps from implementing Statistical Process Control and what are the benefits?

Extra Credit — Linear Programming — Inventory Management

An alcohol manufacturer specializes in 3 different drinks, called Xena, Yolanda, and Zorgon.  Each of these drinks has an ingredient unique to it, but these also share 3 ingredients in common.

1)Each keg of Xenas requires at least 3 gallons of special ingredient A, and must contain at least 3 gallons of special ingredient B, and must contain at least 6 gallons of special ingredient C.  Plus each keg must also contain exactly 25 gallons in total of across all 3 of these special ingredients.

2)Each keg of Yolandas requires at least 4 gallons of special ingredient A, and must contain at least 3 gallons of special ingredient B, and must contain at least 2 gallons of special ingredient C.  Plus each keg must also contain exactly 18 gallons in total of across all 3 of these special ingredients.

3)Each keg of Zorgons requires at least 8 gallons of special ingredient A, and must contain at least 9 gallons of special ingredient B, and must contain at least 8 gallons of special ingredient C.  Plus each keg must also contain exactly 30 gallons in total of across all 3 of these special ingredients.

The mixture of these special ingredients cannot be dominated by any one particular ingredient.  Specifically, in each keg, no one individual special ingredient can have more than double the amount of any other special ingredient.

The manufacturer must make 1 keg of each of the three drinks, but must also attempt to reduce the total cost of the special ingredients.

·         1 gallon of special ingredient A costs the manufacturer \$18.
·         1 gallon of special ingredient B costs the manufacturer \$20.
·         1 gallon of special ingredient C costs the manufacturer \$15.

HINT:  Note to solve this problem, you will need to have 9 variables (3 for Xena, 3 for Yolanda, and 3 for Zorgon; each of these 3 corresponding to the number of gallons of ingredient A, B, and C they each require).

Questions:
a)What is the optimal amount of each special ingredient for each drink?
b)What is the optimal cost of the special ingredients in total?