3.8: End of Chapter Problems
- Page ID
- 117741
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Problem #1
Below are monthly sales of light bulbs from the lighting store.
| Month | Sales |
|---|---|
| Jan | 255 |
| Feb | 298 |
| Mar | 357 |
| Apr | 319 |
| May | 360 |
| June |
.
Forecast sales for June using the following
- Naïve method
- Three- month simple moving average
- Three-month weighted moving average using weights of .5, .3 and .2
- Exponential smoothing using an alpha of .2 and a May forecast of 350.
Solution
- 360
- (357 + 319 + 360) / 3 = 345.3
- 360 x .5 + 319 x .3 + 357 x .2 = 347.1
- 350 + .2(360 – 350) = 352
Problem #2
Demand for aqua fit classes at a large Community Center are as follows for the first six weeks of this year.
| Week | Demand |
|---|---|
| 1 | 162 |
| 2 | 158 |
| 3 | 138 |
| 4 | 190 |
| 5 | 182 |
| 6 | 177 |
| 7 |
.
You have been asked to experiment with several forecasting methods. Calculate the following values:
- a) Forecast for weeks 3 through week 7 using a two-period simple moving average
- b) Forecast for weeks 4 through week 7 using a three-period weighted moving average with weights of .6, .3 and .1
- c) Forecast for weeks 4 through week 7 using exponential smoothing. Begin with a week 3 forecast of 130 and use an alpha of .3
Solution
|
Week |
Demand |
a) |
b) |
c) |
|---|---|---|---|---|
|
1 |
162 |
|||
|
2 |
158 |
|||
|
3 |
138 |
(162 + 158) / 2 = 160 |
130 |
|
|
4 |
190 |
(158 + 138) / 2 = 148 |
138 x .6 + 158 x .3 + 162 x .1 = 146.4 |
130 + .3 x (138 – 130) = 132.4 |
|
5 |
182 |
(138 + 190) / 2 = 164 |
190 x .6 + 138 x .3 + 158 x .1 = 171.2 |
132.4 + .3 x (190 – 132.4) = 149.7 |
|
6 |
177 |
(190 + 182) / 2 = 186 |
182 x .6 + 190 x .3 + 138 x .1 = 180 |
149.7 + .3 x (182 – 149.7) = 159.4 |
|
7 |
(182 + 177) / 2 = 179.5 |
177 x .6 + 182 x .3 + 190 x .1 = 179.8 |
159.4 + .3 x (177 – 159.4) = 164.7 |
Problem #3
Sales of a new shed has grown steadily from the large farm supply store. Below are the sales from the past five years. Forecast the sales for 2018 and 2019 using exponential smoothing with an alpha of .4. In 2015, the forecast was 360. Calculate a forecast for 2016 through to 2020.
|
Year |
Sales |
Forecast |
|---|---|---|
|
2015 |
348 |
360 |
|
2016 |
372 |
|
|
2017 |
311 |
|
|
2018 |
371 |
|
|
2019 |
365 |
|
|
2020 |
.
Solution
|
Year |
Sales |
Forecast |
|---|---|---|
|
2015 |
348 |
360 |
|
2016 |
372 |
360 + .4 x (348 – 360) = 355.2 |
|
2017 |
311 |
355.2 + .4 x (372 – 355.2) = 361.9 |
|
2018 |
371 |
361.9 + .4 x (311 – 361.9) = 341.6 |
|
2019 |
365 |
341.6 + .4 x (371 – 341.6) = 353.3 |
|
2020 |
353.3 + .4 x (365-353.3) = 358.0 |
Problem #4
Below is the actual demand for X-rays at a medical clinic. Two methods of forecasting were used. Calculate a mean absolute deviation for each forecast method. Which one is more accurate?
| Week | Actual Demand | Forecast #1 | Forecast #2 |
|---|---|---|---|
| 1 | 48 | 50 | 50 |
| 2 | 65 | 55 | 56 |
| 3 | 58 | 60 | 55 |
| 4 | 79 | 70 | 85 |
Solution
|
Week |
Actual Demand |
Forecast #1 |
IerrorI |
Forecast #2 |
IerrorI |
|---|---|---|---|---|---|
|
1 |
48 |
50 |
2 |
50 |
2 |
|
2 |
65 |
55 |
10 |
56 |
9 |
|
3 |
58 |
60 |
2 |
55 |
3 |
|
4 |
79 |
70 |
9 |
85 |
6 |
|
Mean Abs Deviation: |
5.75 |
Mean Abs Deviation: |
5 |