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# Quantitative Analysis for Management

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Problem 1
Jane wants to setup a photo shop. The cost to rent an office is \$150 per week. The variable cost of making one photo is \$20 and she can sell it for \$50.
1. Jane has to sell ------- photos per week to break even.
2. If Jane sells 10 units, her profits would be ------- dollars. (Please only enter an integer and include no units.)

Problem 2
Paul wants to choose one of the two investment opportunities over three possible scenarios. Investment 1 will yield a return of \$10,000 in Scenario 1, \$2,000 in Scenario 2, and a negative return of -\$5,000 in Scenario 3. Investment 2 will yield a return of \$6,000 in Scenario 1, \$4,000 in Scenario 2, and zero in Scenario 3. The probability for Scenario 1 is 0.2, for Scenario 2 is 0.3, and for Scenario 3 is 0.5.
1. If you were to choose the investment that maximizes Paul's Expected Money Value (EMV), then you should choose __________.
A. Investment 1
B. Investment 2
C. Indifferent
2. If Paul is uncertain about the return for Investment 1 in Scenario 1, then this return has to be ------- dollars in order to make Paul indifferent between these two investments (i.e. the two investments would have the same EMV.) (Please only enter an integer and include no units.)

Problem 3
Sam has a cleaning service. To better allocate his resources, he would like to forecast his weekly orders based on the order number he received in the past 13 weeks as shown in the following table.
Week Demand
Week 1 11
Week 2 14
Week 3 16
Week 4 10
Week 5 15
Week 6 17
Week 7 11
Week 8 14
Week 9 17
Week 10 12
Week 11 14
Week 12 16
Week 13 15
1. Using a three week moving average, Sam's forecast for his Week 14 order number is ------- . (Please round to two decimal points and include no units.)
2. Using a three week weighted moving average with weights 3, 2, and 1 given to the most recent, second most recent, and third most recent week, respectively, Sam's forecast for his Week 14 order number is ------ . (Please round to two decimal points and include no units.)
3. If the MAD for moving average is 4.17 and the MAD for weighted moving average is 2.38, then which forecast is more accurate?
A. Moving average
B. Weighted moving average
C. The same
D. Not enough information to evaluate.

Problem 4
A grocery store needs to sell 3,000 cartons of 2L 2% milk per month. The sales is relatively constant throughout the month. The owner of this grocery store purchases milk from a supplier 50 miles away for \$2 per carton, and it takes a day to restock. The holding cost per carton per month is \$1.5, and the ordering cost per order is about \$18.5 including labor, gas and depreciation. Consider a month of 30 days.
1. The optimal order quantity is about ------ cartons of milk, and the average inventory is about cartons. (Please round to the closest integer and include no units.)
2. Given the optimal order quantity calculated above, if the average inventory is 136 cartons, then the monthly holding cost is ------- dollars, and the total cost including the cost of supply, holding and ordering is ------- dollars. (Please round to two decimal points and include no units.)
3. The reorder point is ------ cartons. (Please only enter an integer and include no units.)

Problem 5
A cafeteria wants to introduce a new burger, with bread and beef together weighing at least 1 ounce. The cafeteria manage also wants the new burger to meet a new nutrition standard, i.e. contains at least 7 units of Vitamin A and 10 units of Vitamin B. Each ounce of beef contains 1 unit Vitamin A and 6 units of Vitamin B, while each ounce of bread contains 2 units Vitamin A and 1 units of Vitamin B. The price of beef is \$0.5 per ounce and the price of bread is \$0.1 per ounce.
1. To minimize the cost, the cafeteria should use ----- ounces of beef and ----- ounces of bread to make the new burger. (Please round to two decimal points and include no units.)
2. If the cafeteria uses 1.18 ounces of beef and 2.91 ounces of bread to make the new burger, the total cost of the new burger (excluding other ingredients) is ---- dollars, (Please round to two decimal points and include no units.) and the content of Vitamin A is ----- while that for Vitamin B is ------ . (Please round to the closest integer and include no units for the last two answers.)

Problem 6
At a car wash station, on average, there are 4 cars coming in for the service every 10 minutes. The average wash time is 2 minutes. The Poisson distribution is appropriate for the arrival rate and service times are exponentially distributed. Please convert all rates into cars per hour and answer the following questions.
1. The average time a car spent in the waiting line is ----- hours, and the total time a car spent in this car wash station is ---- hour. (Please round to two decimal points and include no units.)
2. The average number of cars in this car wash station is ----- . (Please round to the closest integer and include no units.)
3. The probability that there are no cars in this station is ------ . (Please round to one decimal points and include no units.)
4. The probability that there are exactly two cars in this station is ------- . (Please round to three decimal points and include no units.)

Problem 7
To study the weight accuracy of a 50lb fertilizer bag, 12 samples of 12 bags of fertilizer in each sample were taken and the results are as follows.
Mean Range
Sample 1 47 1.1
Sample 2 46 1.31
Sample 3 46 0.91
Sample 4 47 1.1
Sample 5 48 1.21
Sample 6 50 0.82
Sample 7 49 0.86
Sample 8 49 1.11
Sample 9 51 1.12
Sample 10 52 0.99
Sample 11 50 0.86
Sample 12 51 1.2
1. The overall average weight of a bag of fertilizer is ---- pound, and the average range is ----- pound. (Please round to two decimal points and include no units.)
2.The upper control limit for a 99.7% control chart for the mean is ----- pound, and the lower control limit is ----- pound. (Please round to two decimal points and include no units. Please enter the upper limit first.)
3. The upper control limit for a 99.7% control chart for the range is ----- pound, and the lower control limit is ----- pound. (Please round to two decimal points and include no units. Please enter the upper limit first.)

#### Solution Preview

Please refer attached files for complete solutions.

Problem 1
Jane wants to setup a photo shop. The cost to rent an office is \$150 per week. The variable cost of making one photo is \$20 and she can sell it for \$50.

1. Fixed Cost =F=\$150
Variable Cost per unit=V=\$20
Price per unit=P=\$50
BEP=F/(P-V)=150/(50-20)=5
Jane has to sell 5 photos per week to break even.

2. Total Revenue=P*Q=50*10=\$500
Total Cost=F+V*Q=150+20*10=\$350
Total Profit=TR-TC=500-350=\$150

If Jane sells 10 units, her profits would be 150 dollars.

Problem 2
Paul wants to choose one of the two investment opportunities over three possible scenarios. Investment 1 will yield a return of \$10,000 in Scenario 1, \$2,000 in Scenario 2, and a negative return of -\$5,000 in Scenario 3. Investment 2 will yield a return of \$6,000 in Scenario 1, \$4,000 in Scenario 2, and zero in Scenario 3. The probability for Scenario 1 is 0.2, for Scenario 2 is 0.3, and for Scenario 3 is 0.5.

Expected Return For investment 1=0.2*10000+0.3*2000+0.5*(-5000)=\$100
Expected Return For investment 2=0.2*6000+0.3*4000+0.5*0=\$2400
Expected Return (EMV) is higher in case of investment 2. It should be selected.

1. If you were to choose the investment that maximizes Paul's Expected Money Value (EMV), then you should choose Investment 2.

2. Expected Return For investment 1=0.2*X+0.3*2000+0.5*(-5000)=0.2X-1900
Expected Return For investment 2=0.2*6000+0.3*4000+0.5*0=\$2400
0.2X-1900=2400
0.2X=4300
X=4300/0.2=\$21500
If Paul is uncertain about the return for Investment 1 in Scenario 1, then this return has to be 21500 dollars in order to make Paul indifferent between these two investments (i.e. the two investments would have the same EMV.) ...

#### Solution Summary

There are 7 problems related to quantitative reasoning. Solution to first problem uses CVP approach to find the BEP and profit. Solution to second problem uses EMV approach to choose the best alternative. Solution to third problem uses moving average and weighted moving average methods to forecast the desired value. Solution to 4th problem depicts the steps to find EOQ and inventory costs. Solution to 5th problem uses LP model to find the optimal combination of ingredients. Solution to 6th problem uses waiting line model concepts to find desired probabilities. Solution to 7th problem depicts the steps to find out the upper and lower control limits of mean and range.

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