Exponential Smoothing and Week Auto Sales

Practice Problems: Chapter 4, Forecasting

Problem 1:

Auto sales at Carmen’s Chevrolet are shown below. Develop a 3-week moving average.

|Week |Auto Sales |
|1 |8 |
|2 |10 |
|3 |9 |
|4 |11 |
|5 |10 |
|6 |13 |
|7 |- |

Problem 2:

Carmen’s decides to forecast auto sales by weighting the three weeks as follows:

|Weights Applied |Period |
|3 |Last week |
|2 |Twoweeks ago |
|1 |Three weeks ago |
|6 |Total |

Problem 3:

A firm uses simple exponential smoothing with [pic] to forecast demand. The forecast for the week of January 1 was 500 units whereas the actual demand turned out to be 450 units. Calculate the demand forecast for the week of January 8.

Problem 4:

Exponential smoothing is used to forecast automobile battery sales. Two value of [pic] are examined, [pic] and [pic] Evaluate the accuracy of each smoothing constant. Which is preferable? (Assume the forecast for January was 22 batteries.) Actual sales are given below:

|Month |Actual |Forecast |
| |Battery Sales| |
|January |20 |22 |
|February |21 | |
|March |15 | |
|April |14 | |
|May |13 | |
|June |16 | |

Problem 5:

Use the sales data given below to determine: (a) the least squares trend line, and (b) the predicted value for 2003 sales.

|Year |Sales (Units)|
|1996 |100 |
|1997 |110 |
|1998 |122 |
|1999 |130 |
|2000 |139 |
|2001 |152 |
|2002 |164 |

To minimize computations, transform the value of x (time) to simpler numbers. In this case, designate year 1996 as year 1, 1997 as year 2, etc.

Problem 6:

Given the forecast demand and actual demand for 10-foot fishing boats, compute the tracking signal and MAD.

|Year |Forecast Demand|Actual Demand |
|1 |78 |71 |
|2 |75 |80 |
|3 |83 |101 |
|4 |84 |84 |
|5 |88 |60 |
|6 |85 |73 |

Problem: 7

Over the past year Meredith and Smunt Manufacturing had annual sales of 10,000 portable water pumps. The average quarterly sales for the past 5 years have averaged: spring 4,000, summer 3,000, fall 2,000 and winter 1,000. Compute the quarterly index.

Problem: 8

Using the data in Problem, Meredith and Smunt Manufacturing expects sales of pumps to grow by 10% next year. Compute next year’s sales and the sales for each quarter.

ANSWERS:

Problem 1:

[pic]

|Week |Auto Sales |Three-Week Moving Average |
|1 |8 | |
|2 |10 | |
|3 |9 | |
|4 |11 |(8 + 9 + 10) / 3 = 9 |
|5 |10 |(10 + 9 + 11) / 3 = 10 |
|6 |13 |(9 + 11 + 10) / 3 = 10 |
|7 |- |(11 + 10 + 13) / 3 = 11 1/3 |

Problem 2:

[pic]

|Week |Auto Sales |Three-Week Moving Average |
|1 |8 | |
|2 |10 | |
|3 |9 | |
|4 |11 |[(3*9) + (2*10) + (1*8)] / 6 = 9 1/6 |
|5 |10 |[(3*11) + (2*9) + (1*10)] / 6 = 10 1/6 |
|6 |13 |[(3*10) + (2*11) + (1*9)] / 6 = 10 1/6 |
|7 |- |[(3*13) + (2*10) + (1*11)] / 6 = 11 2/3 |

Problem 3:

[pic]

Problem 4:

|Month |Actual Battery Sales |Rounded Forecast with|Absolute Deviation |Rounded Forecast |Absolute Deviation |
| | |a =0.8 |with a =0.8 |with a =0.5 |with a =0.5 |
|January |20 |22 |2 |22 |2 |
|February |21 |20 |1 |21 |0 |
|March |15 |21 |6 |21 |6 |
|April |14 |16 |2 |18 |4 |
|May |13 |14 |1 |16 |3 |
|June |16 |13 |3 |14.5 |1.5 |
| | | | | | |
| | | |S = 15 | |S = 16 |
| |2.56 | |2.95 |
|[pic] | | | |
|SE |3.5 | |3.9 |

On the basis of this analysis, a smoothing constant of a = 0.8 is preferred to that of a = 0.5 because it has a smaller MAD.

Problem 5:

|Year |Time Period |Sales |X2 |XY |
| |(X) |(Units) (Y) | | |
|1996 |1 |100 |1 |100 |
|1997 |2 |110 |4 |220 |
|1998 |3 |122 |9 |366 |
|1999 |4 |130 |16 |520 |
|2000 |5 |139 |25 |695 |
|2001 |6 |152 |36 |912 |
|2002 |7 |164 |49 |1148 |
| |S X = 28 |S Y =917 |S X2=140 |S XY = 3961|

[pic]

[pic]

[pic]

[pic]

Therefore, the least squares trend equation is:

[pic]

To project demand in 2003, we denote the year 2003 as [pic] and:

Sales in [pic]

Problem 6:

|Year |Forecast Demand|Actual Demand |Error |RSFE |
|1 |78 |71 |-7 |-7 |
|2 |75 |80 |5 |-2 |
|3 |83 |101 |18 |16 |
|4 |84 |84 |0 |16 |
|5 |88 |60 |-28 |-12 |
|6 |85 |73 |-12 |-24 |

[pic]

|Year |Forecast Demand|Actual Demand ||Forecast Error||Cumulative Error |MAD |Tracking Signal |
|1 |78 |71 |7 |7 |7.0 |-1.0 |
|2 |75 |80 |5 |12 |6.0 |-0.3 |
|3 |83 |101 |18 |30 |10.0 |+1.6 |
|4 |84 |84 |0 |30 |7.5 |+2.1 |
|5 |88 |60 |28 |58 |11.6 |-1.0 |
|6 |85 |73 |12 |70 |11.7 |-2.1 |

[pic]

Problem 7:

Sales of 10,000 units annually divided equally over the 4 seasons is [pic] and the seasonal index for each quarter is: spring [pic] summer [pic] fall [pic] winter [pic]

Problem 8:

Next years sales should be 11,000 pumps [pic] Sales for each quarter should be 1/4 of the annual sales [pic] the quarterly index.

[pic]

[pic]

[pic]

[pic]