Forecasting Problem 1 (10 points) | |||||

The following equation summarizes the trend portion of quarterly sales of automatic dishwashers over a long cycle. Sales also exhibit seasonal variations. | |||||

Ft = 40 – 6.5t + 2t2 | |||||

where | |||||

Ft = Unit sales (in 000 units) | |||||

t = 1 at the first quarter of 2010 | |||||

Quarter | Relative | ||||

1 | 110 | ||||

2 | 100 | ||||

3 | 60 | ||||

4 | 130 | ||||

a) Using the information given, prepare a seasonalized forecast of sales for each quarter of 2014. | |||||

Year | Qtr | t | Seasonally Unadjusted Forecast | Quarterly Index | Seasonally Adjusted Forecast |

2014 | 1 | 17 | 507.50 | 110% | 558.25 |

2 | 18 | 571.00 | 100% | 571.00 | |

3 | 19 | 638.50 | 60% | 383.10 | |

4 | 20 | 710.00 | 130% | 923.00 | |

b) Prepare a quarter-by-quarter time series plot showing the forecast trend (seasonally unadjusted forecast) from the first quarter of 2010 to the fourth quarter of 2014. Superimpose on this graph the seasonally adjusted forecast for the same time period. | |||||

Year | Qtr | t | Seasonally Unadjusted Forecast | Quarterly Index | Seasonally Adjusted Forecast |

2010 | 1 | 1 | 35.50 | 110% | 39.05 |

2 | 2 | 35.00 | 100% | 35.00 | |

3 | 3 | 38.50 | 60% | 23.10 | |

4 | 4 | 46.00 | 130% | 59.80 | |

2011 | 1 | 5 | 57.50 | 110% | 63.25 |

2 | 6 | 73.00 | 100% | 73.00 | |

3 | 7 | 92.50 | 60% | 55.50 | |

4 | 8 | 116.00 | 130% | 150.80 | |

2012 | 1 | 9 | 143.50 | 110% | 157.85 |

2 | 10 | 175.00 | 100% | 175.00 | |

3 | 11 | 210.50 | 60% | 126.30 | |

4 | 12 | 250.00 | 130% | 325.00 | |

2013 | 1 | 13 | 293.50 | 110% | 322.85 |

2 | 14 | 341.00 | 100% | 341.00 | |

3 | 15 | 392.50 | 60% | 235.50 | |

4 | 16 | 448.00 | 130% | 582.40 | |

2014 | 1 | 17 | 507.50 | 110% | 558.25 |

2 | 18 | 571.00 | 100% | 571.00 | |

3 | 19 | 638.50 | 60% | 383.10 | |

4 | 20 | 710.00 | 130% | 923.00 |

Number of Dishwashers Sold

2010-2014

Seasonally Unadjusted Forecast 2010.0 2011.0 2012.0 2013.0 2014.0 35.5 35.0 38.5 46.0 57.5 73.0 92.5 116.0 143.5 175.0 210.5 250.0 293.5 341.0 392.5 448.0 507.5 571.0 638.5 710.0 Seasonally Adjusted Forecast 2010.0 2011.0 2012.0 2013.0 2014.0 39.05 35.0 23.1 59.8 63.25000000000001 73.0 55.5 150.8 157.85 175.0 126.3 325.0 322.85 341.0 235.5 582.4 558.25 571.0 383.1 923.0

## Fore Prob 2

Forecasting Problem 2 (10 points) | |||||

The data below represent the relative shares (by quarter) of call volumes over 16 quarters from a call center at a major financial institution. | |||||

2010 | 2011 | 2012 | 2013 | Average Qtr Percent Share | |

Q1 | 23.2% | 23.0% | 23.3% | 21.9% | 22.8% |

Q2 | 25.1% | 24.6% | 26.2% | 25.3% | 25.3% |

Q3 | 28.5% | 28.8% | 28.6% | 29.8% | 28.9% |

Q4 | 23.2% | 23.6% | 21.9% | 23.1% | 22.9% |

Total | 100% | 100% | 100% | 100% | 100% |

(a) Using the average quarter percent share column (Column G), generate a pie chart and a bar chart to show the quarterly percent shares of call volumes to the call center. | |||||

(b) Using the percentage table above, what are the indices for each of the four quarters? | |||||

Index | |||||

Q1 | 91.3% | ||||

Q2 | 101.2% | ||||

Q3 | 115.7% | ||||

Q4 | 91.8% | ||||

400.0% | |||||

(c) Assume that the projected number of calls for the year 2014 is 50,000,000, what are the seasonally adjusted forecasts for the number of calls for Q1, Q2, Q3, and Q4? | |||||

Forecast Number of Calls | |||||

2014 | 50,000,000 | ||||

Forecast, By Quarter | |||||

Q1 | 11,413,027 | ||||

Q2 | 12,651,833 | ||||

Q3 | 14,465,680 | ||||

Q4 | 11,469,460 | ||||

50,000,000 |

Typical Volume of Call Center, Percent Distribution by Quarter

Average Qtr Percent Share

0.228260546938608 0.253036651885006 0.289313598547655 0.229389202628731

Typical Volume of Call Center, Percent Distribution by Quarter

Average Qtr Percent Share

0.228260546938608 0.253036651885006 0.289313598547655 0.229389202628731

Typical Volume of Call Center, Percent Distribution by Quarter

Average Qtr Percent Share

0.228260546938608 0.253036651885006 0.289313598547655 0.229389202628731

Typical Volume of Call Center, Percent Distribution by Quarter

Average Qtr Percent Share

0.228260546938608 0.253036651885006 0.289313598547655 0.229389202628731

## Fore Prob 3

Forecasting Problem 3 (15 points) | |||||||||

The data below represent the call volumes over 16 quarters from a call center at a major financial institution. Develop a forecasting model for the volume of calls (in 000 units). | |||||||||

2010 | 2011 | 2012 | 2013 | ||||||

Q1 | 473 | 544 | 628 | 709 | |||||

Q2 | 513 | 582 | 707 | 725 | |||||

Q3 | 582 | 681 | 773 | 854 | |||||

Q4 | 474 | 557 | 592 | 661 | |||||

(a) Create a time series graph showing the: (1) actual data, (2) trend line for the data, and (3) deseasonalized actual data. Label the graph appropriately. | |||||||||

Year | Qtr | t | Actual Volume of Calls | Deseasonalized Actual Volume of Calls | |||||

2010 | 1 | 1 | 473 | 483.18 | |||||

2 | 2 | 513 | 503.68 | ||||||

3 | 3 | 582 | 515.76 | ||||||

4 | 4 | 474 | 542.25 | ||||||

2011 | 1 | 5 | 544 | 555.71 | |||||

2 | 6 | 582 | 571.43 | ||||||

3 | 7 | 681 | 603.49 | ||||||

4 | 8 | 557 | 637.20 | ||||||

2012 | 1 | 9 | 628 | 641.52 | |||||

2 | 10 | 707 | 694.16 | ||||||

3 | 11 | 773 | 685.02 | ||||||

4 | 12 | 592 | 677.23 | ||||||

2013 | 1 | 13 | 709 | 724.27 | |||||

2 | 14 | 725 | 711.83 | ||||||

3 | 15 | 854 | 756.80 | ||||||

4 | 16 | 661 | 756.17 | ||||||

(b) Develop the quarterly (Q1, Q2, Q3, Q4) indexes for the volume of calls | |||||||||

See page117-118 of text for similar problem | |||||||||

Year | Qtr | t | Volume of Calls | MA4 | MA2 | Y/MA2 | |||

2010 | 1 | 1 | 473 | ||||||

2 | 2 | 513 | 510.5 | ||||||

3 | 3 | 582 | 528.25 | 519.375 | 1.120578 | Unadjusted Index | Adjusted Index | ||

4 | 4 | 474 | 545.5 | 536.875 | 0.882887 | Qtr | |||

2011 | 1 | 5 | 544 | 570.25 | 557.875 | 0.975129 | 1 | 97.8% | 97.9% |

2 | 6 | 582 | 591 | 580.625 | 1.002368 | 2 | 101.7% | 101.9% | |

3 | 7 | 681 | 612 | 601.5 | 1.132170 | 3 | 112.7% | 112.8% | |

4 | 8 | 557 | 643.25 | 627.625 | 0.887473 | 4 | 87.3% | 87.4% | |

2012 | 1 | 9 | 628 | 666.25 | 654.75 | 0.959145 | 399.5% | 400.0% | |

2 | 10 | 707 | 675 | 670.625 | 1.054240 | ||||

3 | 11 | 773 | 695.25 | 685.125 | 1.128261 | ||||

4 | 12 | 592 | 699.75 | 697.5 | 0.848746 | ||||

2013 | 1 | 13 | 709 | 720 | 709.875 | 0.998767 | |||

2 | 14 | 725 | 737.25 | 728.625 | 0.995025 | ||||

3 | 15 | 854 | |||||||

4 | 16 | 661 | |||||||

a | 473.68 | ||||||||

(c ) Using the trend line you developed in Part (a), what are your seasonally unadjusted and seasonally adjusted forecasts for the four quarters of 2014? | b | 18.207 | |||||||

Year | Qtr | t | Seasonally Unadjusted Forecast | Quarterly Index | Seasonally Adjusted Forecast | ||||

2014 | 1 | 17 | 783.199 | 97.9% | 766.69 | ||||

2 | 18 | 801.406 | 101.9% | 816.23 | |||||

3 | 19 | 819.613 | 112.8% | 924.88 | |||||

4 | 20 | 837.820 | 87.4% | 732.37 |

Volume of Calls to Call Center 2010 to 2013

Actual Volume of Calls

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 2010 2011 2012 2013 473.0 513.0 582.0 474.0 544.0 582.0 681.0 557.0 628.0 707.0 773.0 592.0 709.0 725.0 854.0 661.0 Deseasonalized Actual Volume of Calls 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 2010 2011 2012 2013 483.1849331666253 503.6807630447924 515.7593318992456 542.2452597374343 555.713749773032 571.4272984250861 603.4915893872617 637.195379058546 641.522490546809 694.1565291864878 685.0205559417816 677.2345860011836 724.2666334358083 711.8295384161296 756.8014938865219 756.1690225452405 0.0 Linear 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 2010 2011 2012 2013 1.0

## Fore Prob 4

Forecasting Problem 4 (15 points) | ||||||||||

Many supply managers use a monthly reported survey result known as the purchasing managers’ index (PMI) as a leading indicator to forecast future sales for their businesses. Suppose that the PMI and your business sales data for the last 10 months are the following: | ||||||||||

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

Month | ||||||||||

PM | 43 | 43.1 | 41.5 | 38.5 | 40.5 | 45.2 | 46.2 | 48.1 | 49 | 53 |

Sales (in $000) | 122 | 124 | 125 | 123 | 119 | 120 | 125 | 127 | 135 | 136 |

A. Develop a regression model that can be used by supply managers in forecasting future sales for businesses. Explain what forecasting model approach you used and why you chose it. Show complete work (cut and paste from Excel if used in the analysis). (10 points) | ||||||||||

B. Develop a sales forecast for the 11th and 12th months using the model you developed in part A when PMIs are 52 and 50, respectively. (10 points) | ||||||||||

Y= 1.0529x + 78.421 | = | 50 | = | 131.066 | ||||||

52 | = | 133.1718 |

PMI Vs. Sales

Sales (in $000)

43.0 43.1 41.5 38.5 40.5 45.2 46.2 48.1 49.0 53.0 122.0 124.0 125.0 123.0 119.0 120.0 125.0 127.0 135.0 136.0

PMI

Sales (in 000)

## Fore Prob5

Forecasting Problem 5 (15 pts) | |||||||||

An electrical contractor’s records during the last 5 weeks indicate the number of job requests: | |||||||||

Week | Actual Requests | ||||||||

1 | 20 | ||||||||

2 | 22 | ||||||||

3 | 18 | ||||||||

4 | 21 | ||||||||

5 | 22 | ||||||||

a) | Graph the actual request data using appropriate labels, and provide insights about the time series (describe what you observe re the behavior of the sales over the period under review). | ||||||||

b) | What is the forecast for Week 6 using a 2-period moving average? | ||||||||

21.5 | |||||||||

c) | What is the forecast for Week 6 using the Naïve method? | ||||||||

22 | |||||||||

Compute the MAD, MAPE, and MSE for the two-period moving average and Naïve models and compare your results. Explain which of the two forecasting models you prefer and why. | |||||||||

d) | |||||||||

Forecasts | MAD | MSE | MAPE | ||||||

Week | Actual Requests | MA2 | Naïve | MA2 | Naïve | MA2 | Naïve | MA2 | Naïve |

1 | 20 | ||||||||

2 | 22 | 20 | 2 | 4.000 | 9.09% | ||||

3 | 18 | 21 | 22 | 3 | 4 | 9.000 | 16.000 | 16.67% | 22.22% |

4 | 21 | 20 | 18 | 1 | 3 | 1.000 | 9.000 | 4.76% | 14.29% |

5 | 22 | 19.5 | 21 | 2.5 | 1 | 6.250 | 1.000 | 11.36% | 4.55% |

MAD | 2.2 | 2.5 | MSE | 8.1 | 10.0 | MAPE | 10.93% | 12.54% | |

e) | Graph the actual number of requests, the 2-period and Naïve forecasts. Use appropriate labels for your graphs |

Actual Requests

Actual 20.0 22.0 18.0 21.0 22.0

Actual Requests Actual vs. MA2 and Naive Forecasrs

Actual Requests 20.0 22.0 18.0 21.0 22.0 MA2 21.0 20.0 19.5 Naïve 20.0 22.0 18.0 21.0

Actual Requests

Actual 20.0 22.0 18.0 21.0 22.0

Actual Requests Actual vs. MA2 and Naive Forecasrs

Actual Requests 20.0 22.0 18.0 21.0 22.0 MA2 21.0 20.0 19.5 Naïve 20.0 22.0 18.0 21.0

## Rel Prob 1

Reliability Problem 1 (15 points) | |||

One of the industrial robots designed by a leading producer of servomechanisms has three major components. Components’ reliabilities are 80, 85, and 95%. All of the components must function in order for the robot to operate effectively. | |||

a. Compute the reliability of the robot. | |||

0.8 | 0.85 | 0.95 | |

Reliability = | 0.646 | ||

b. Designers want to improve the reliability by adding a backup component. Due to space limitations, only one backup can be added. The backup for any component will have the same reliability as the unit for which it is the backup. Which component should get the backup in order to achieve the highest reliability? Show proof of your answer by computing the overall reliabilities of the three options (assume 100% reliable backup switch) | |||

0.96 | |||

Tring all components with the formula above, we got component with 80% reliability | |||

c. If one backup with a reliability of 99% can be added to any of the main components, which component should get it to obtain the highest overall reliability? Show proof of your choice by computing the overall reliabilities of the three options (assume a backup switch with 100% reliability). | |||

Switch with 1 | |||

0.8 | 0.99 | 0.99800 | 0.998 |

0.85 | 0.99 | 0.99850 | 0.9985 |

0.95 | 0.99 | 0.99950 | 0.9995 |

So, Component with 80% reliability |

## Rel Prob 2

Reliability Problem 2 (10 points) | ||||

Lucky Lumen light bulbs have an expected life that is exponentially distributed with a mean of 20,000 hours. Determine the probability that one of these light bulbs will last: | ||||

a. At least 24,000 hours | ||||

t | 24000 | |||

MTBF | 20000 | |||

P(T>=24000) | 0.3011942119 | |||

b. No longer than 4,000 hous | ||||

t | 4000 | |||

MTBF | 20000 | |||

P(T<4000) | 0.8187307531 | |||

Faliure Rate | 1-P(T<4000) | 0.1812692469 | ||

c. Between 4,000 and 24,000 hours | ||||

t | 24000 | t | 4000 | |

MTBF | 20000 | MTBF | 20000 | |

P(T>=24000) | 0.3011942119 | P(T<4000) | 0.8187307531 | 0.517537 |

## Rel Prob 3

Reliability Problem 3 (10 points) | |||

An office manager has received a report from a consultant that includes a section on equipment replacement. The report indicates that scanners have a service life that is normally distributed with a mean of 41 months and a standard deviation of 4 months. | |||

On the basis of this information, determine the percentage of scanners that can be expected to fail in the following time periods. | |||

a. Before 38 months of service. | |||

t | 38 | ||

Mean | 41 | ||

SD | 4 | ||

Z | -0.75 | ||

P(T<38) | 22.66274% | ||

b. Between 40 and 45 months of service. | |||

t | 40 | 45 | |

Mean | 41 | 41 | |

SD | 4 | 4 | |

Z | -0.25 | 1 | |

P(40 <t<45) | 0.4012936743 | 0.8413447461 | 44.00511% |

c. Within +/- 2 months of the average service life. | |||

t | 39 | 43 | |

Mean | 41 | 41 | |

SD | 4 | 4 | |

Z | -0.5 | 0.5 | |

P(39<t<43) | 0.3085375387 | 0.6914624613 | 38.29% |

d. If the manufacturer of the scanner offers a service contracts of 3 years on these scanners, what percentage of scanners can be expected to fail from wear-out during the service period? | |||

t | 36 | ||

Mean | 41 | ||

SD | 4 | ||

Z | -1.25 | ||

p(t<36m) | 10.56498% | ||

e. If the cost of replacement each scanner is $250, and if 1,000 units of this scanner are sold, what is the expected warranty replacement cost to the manufacturer. | |||

$250 | |||

1000 | |||

$ 26,412.44 |

## Rel Prob 4

Reliability Problem 4 (10 points) | ||||||

How high must reliability be? Prime business customers expect public carrier-class communications data links to be available 99.999 percent of the time. The so-called five nines rule implies only 5 minutes of downtime per year. Such high reliability is needed not only in telecommunications but also for mission-critical systems such as airline reservation systems or banking fund transfers. | ||||||

Suppose a certain network web server is up only 90 percent of the time (i.e. its probability of being down is 0.10). How many independent servers are needed to ensure that the system is up at least 99.999 percent of the time? | ||||||

Show your work and explain your answer. | ||||||

Server Rel. | 90% | |||||

Machine Failure Rate | 10% | |||||

Number of Servers | 1 | 2 | 3 | 4 | 5 | 6 |

P(failure) | 0.100000 | 0.010000 | 0.001000 | 0.000100 | 0.000010 | 0.000001 |

P(Reliable) | 0.900000 | 0.990000 | 0.999000 | 0.999900 | 0.999990 | 0.999999 |

% | 90 | 99 | 99.9 | 99.99 | 99.999 | 99.9999 |