What is the optimal amount of each special ingredient for

Question

Q1 Linear Programming -- Product Output

A company produces 5 different products, A, B, C, D, and E. 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	Product D	Product E	Available
Melting	7 hrs	2 hrs	0 hrs	2 hrs	3 hrs	2250 hrs
Cooling	3 hrs	3 hrs	1 hrs	3 hrs	4 hrs	2100 hrs
Shaping	4 hrs	2 hrs	1 hrs	1 hrs	3 hrs	1750 hrs
Sanding	1 hrs	3 hrs	3 hrs	3 hrs	1 hrs	1700 hrs
Painting	2 hrs	2 hrs	2 hrs	2 hrs	1 hrs	1400 hrs

Suppose product A sells for $250/unit, product B sells for $200/unit, product C sells for $125/unit, product D sells for $165/unit, and each product E sells for $185/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 placed a number of restrictions on the number of products to be manufactured. No one product can be more than half of all units manufactured. Also, the number of As, Bs, and Cs combined must be at least equal to the number of Ds and Es combined. Finally, there must be at least 1 product D and 1 product E manufactured. 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?

b) How much revenue would be made if they followed your recommendation?

Q2 Linear Programming- Product Output

Suppose your company sells 4 different kinds of pies, Apple Pies, Banana Pies, Cherry Pies, and Danish Pies, where each Apple Pie sells for $12, each Banana Pie sells for $13, each Cherry Pie sells for $11, and each Danish Pie sells for $14. 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 2 hours on machine #1 and 1 hour on machine #2. To make one Banana Pie requires 3 hours on machine #1 and 2 hours on machine #2. To make one Cherry Pie requires 1 hour on machine #1 and 2 hours on machine #2. And to make one Danish Pie requires 1.5 hours on machine #1 and 0.75 hours on machine #2. Finally, suppose there can be no more than a total of 250 hours of processing time on machine number 1 and no more than a total of 250 hours of processing time on machine number 2. No one pie should account for more than half of all pies produced. 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?

b) How much revenue would be made if they followed your recommendation?

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
d) What are the business implications of your answer to question d?
e) Given the business implications, what concrete business recommendations do you have for the company?

#	Data
1	258
2	294
3	276
4	223.5
5	316.5
6	192
7	151.5
8	265.5
9	318
10	198
11	252
12	192
13	157.5
14	339
15	256.5
16	51
17	303
18	294
19	190.5
20	220.5
21	330
22	199.5
23	261
24	258
25	309
26	223.5
27	226.5
28	129
29	237
30	205.5
31	90
32	271.5

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? (4 points)

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 18 customers per hour (assume Poisson distribution) and the cashier seems to be able to serve an average of 24 customers per hour (assuming exponential distribution).

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

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

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

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

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

Q6 Queuing

Part 1 What is the service rate, µ, if the average time to serve a customer is 7 minutes?

Part 2 If, on average, it takes 90 seconds to serve a customer then what is the the hourly service rate, µ?

Part 3 An 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 40 customers per hour.

Part 3a What is the probability that the server is busy?
Part 3b What is the average number of customers waiting in line for service?
Part 3c What is the average time a customer spends waiting in line for service?
Part 3d If the arrival rate remains at 28 customers per hour and the stand's manager wants to have the average total time a customer spends in the system to be a maximum of 4 minutes on average, then the service rate must change to what value?

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 6 gallons of special ingredient A, and must contain at least 6 gallons of special ingredient B, and must contain at least 12 gallons of special ingredient C. Plus each keg must also contain exactly 50 gallons in total of across all 3 of these special ingredients.

2) Each keg of Yolandas requires at least 8 gallons of special ingredient A, and must contain at least 6 gallons of special ingredient B, and must contain at least 4 gallons of special ingredient C. Plus each keg must also contain exactly 36 gallons in total of across all 3 of these special ingredients.

3) Each keg of Zorgons requires at least 16 gallons of special ingredient A, and must contain at least 18 gallons of special ingredient B, and must contain at least 16 gallons of special ingredient C. Plus each keg must also contain exactly 60 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 2 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 $36.
1 gallon of special ingredient B costs the manufacturer $40.
1 gallon of special ingredient C costs the manufacturer $30.

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?