Chapter 4, Problem 59P

Sales of tablet computers at Ted Glickman’s electronics store in Washington, D.C., over the past 10 weeks are shown in the table below:

a) Forecast demand for each week, including week 10, using exponential smoothing with α = .5 (initial forecast = 20).

b) Compute the MAD.

c) Compute the tracking signal.

Summary Introduction

To determine: To forecast the demand for 10 weeks using exponential smoothing.

Introduction: A sequence of data points in successive order is known as time series. Time series forecasting is the prediction based on past events, which are at a uniform time interval. Exponential smoothing is one of the time series methods which use a smoothing constant to emphasis past data.

Answer to Problem 59P

Using exponential smoothing, the forecast for week 10 is done.

Explanation of Solution

Given information:

Week	Demand
1	20
2	21
3	28
4	37
5	25
6	29
7	36
8	22
9	25
10	28

$\begin{array}{c}\text{Smoothing}\text{constant}\text{α}=\text{0}\text{.5}\\ \text{Initial}\text{forecast}=\text{20}\end{array}$

Formula to calculate the forecasted demand:

${\text{F}}_{\text{t}}={\text{F}}_{\text{t-1}}+\text{α}\left({\text{A}}_{\text{t-1}}-{\text{F}}_{\text{t-1}}\right)$

Where,

$\begin{array}{c}{\text{F}}_{\text{t}}=\text{new}\text{forecast}\\ {\text{F}}_{\text{t-1}}=\text{Previous}\text{period's}\text{forecast}\\ \text{α}=\text{smoothing}\text{constant}\\ {\text{A}}_{\text{t-1}}=\text{Previous}\text{period}\text{actual}\text{Demand}\end{array}$

Calculation to forecast demand using exponential smoothing:

Week	Demand	Ft
1	20	20
2	21	20
3	28	20.50
4	37	24.25
5	25	30.63
6	29	27.81
7	36	28.41
8	22	32.20
9	25	27.10
10	28	26.05

Excel worksheet:

Calculation of the forecast for week 2:

$\begin{array}{c}{\text{F}}_2={\text{F}}_1+\text{α}\left({\text{A}}_1-{\text{F}}_1\right)\\ =20+0.5\left(20-20\right)\\ =20\end{array}$

To calculate the forecast for week 2, substitute the value of forecast of week 1, smoothing constant and difference between actual and forecasted demand in the above formula. The result of forecast for week 2 is 20.

Calculation of the forecast for week 3:

$\begin{array}{c}{\text{F}}_3={\text{F}}_2+\text{α}\left({\text{A}}_2-{\text{F}}_2\right)\\ =20+0.5\left(21-20\right)\\ =20.50\end{array}$

To calculate the forecast for week 3, substitute the value of forecast of week 1, smoothing constant and difference between actual and forecasted demand in the above formula. The result of forecast for week 3 is 20.50.

Calculation of the forecast for week 4:

$\begin{array}{c}{\text{F}}_4={\text{F}}_3+\text{α}\left({\text{A}}_3-{\text{F}}_3\right)\\ =20.50+0.5\left(28-20.50\right)\\ =24.25\end{array}$

To calculate the forecast for week 4, substitute the value of forecast of week 3, smoothing constant and difference between actual and forecasted demand in the above formula. The result of forecast for week 4 is 24.25.

Calculation of the forecast for week 5:

$\begin{array}{c}{\text{F}}_5={\text{F}}_4+\text{α}\left({\text{A}}_4-{\text{F}}_4\right)\\ =24.25+0.5\left(27-24.25\right)\\ =30.63\end{array}$

To calculate the forecast for week 5, substitute the value of forecast of week 4, smoothing constant and difference between actual and forecasted demand in the above formula. The result of forecast for week 5 is 30.63.

Calculation of the forecast for week 6:

$\begin{array}{c}{\text{F}}_6={\text{F}}_5+\text{α}\left({\text{A}}_5-{\text{F}}_5\right)\\ =30.63+0.5\left(25-30.63\right)\\ =27.81\end{array}$

To calculate the forecast for week 6, substitute the value of forecast of week 5, smoothing constant and difference between actual and forecasted demand in the above formula. The result of forecast for week 6 is 27.81.

Calculation of the forecast for week 7:

$\begin{array}{c}{\text{F}}_7={\text{F}}_6+\text{α}\left({\text{A}}_6-{\text{F}}_6\right)\\ =27.81+0.5\left(29-27.81\right)\\ =28.41\end{array}$

To calculate the forecast for week 7, substitute the value of forecast of week 6, smoothing constant and difference between actual and forecasted demand in the above formula. The result of forecast for week 7 is 28.41.

Calculation of the forecast for week 8:

$\begin{array}{c}{\text{F}}_8={\text{F}}_7+\text{α}\left({\text{A}}_7-{\text{F}}_7\right)\\ =28.41+0.5\left(36-28.41\right)\\ =32.20\end{array}$

To calculate the forecast for week 8, substitute the value of forecast of week 7, smoothing constant and difference between actual and forecasted demand in the above formula. The result of forecast for week 8 is 32.20.

Calculation of the forecast for week 9:

$\begin{array}{c}{\text{F}}_9={\text{F}}_8+\text{α}\left({\text{A}}_8-{\text{F}}_8\right)\\ =32.20+0.5\left(22-32.20\right)\\ =27.10\end{array}$

To calculate the forecast for week 9, substitute the value of forecast of year 8, smoothing constant and difference between actual and forecasted demand in the above formula. The result of forecast for week 9 is 27.10.

Calculation of the forecast for week 10:

$\begin{array}{c}{\text{F}}_{10}={\text{F}}_9+\text{α}\left({\text{A}}_9-{\text{F}}_9\right)\\ =27.10+0.5\left(25-27.10\right)\\ =26.05\end{array}$

To calculate the forecast for week 10, substitute the value of forecast of year 9, smoothing constant and difference between actual and forecasted demand in the above formula. The result of forecast for week 10 is 26.05.

Thus, using exponential smoothing, the forecast for week 10 is done.

Summary Introduction

To determine: To compute MAD.

Answer to Problem 59P

MAD is 4.99.

Explanation of Solution

Given information:

Week	Demand
1	20
2	21
3	28
4	37
5	25
6	29
7	36
8	22
9	25
10	28

$\begin{array}{c}\text{Smoothing}\text{constant}\text{α}=\text{0}\text{.5}\\ \text{Initial}\text{forecast}=\text{20}\end{array}$

Formula to calculate MAD:

$\text{MAD}=\frac{{\displaystyle \sum \left|\text{Actual}-\text{Forecast}\right|}}{n}$

Calculation of MAD:

Week	Demand	Ft	Absolute error
1	20	20	0
2	21	20	1
3	28	20.50	7.50
4	37	24.25	12.75
5	25	30.63	5.63
6	29	27.81	1.19
7	36	28.41	7.59
8	22	32.20	10.20
9	25	27.10	2.10
10	28	26.05	1.95
Total			49.91
MAD			4.99

Table 1

Excel worksheet:

Calculation of the absolute error for week 2:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|21-20\right|\\ =\left|-1\right|\\ =1\end{array}$

The absolute error for week 1 is the modulus of the difference between 21 and 20, which corresponds to 1. Therefore, the absolute error for week 2 is 1.

Calculation of the absolute error for week 3:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|28-20.50\right|\\ =\left|7.50\right|\\ =7.50\end{array}$

The absolute error for week 3 is the modulus of the difference between 28 and 20.50, which corresponds to 7.50. Therefore, the absolute error for week 3 is 7.50.

Calculation of the absolute error for week 4:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|37-24.25\right|\\ =\left|12.75\right|\\ =12.75\end{array}$

The absolute error for week 4 is the modulus of the difference between 37 and 24.25, which corresponds to 12.75. Therefore, the absolute error for week 4 is 12.75.

Calculation of the absolute error for week 5:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|25-30.63\right|\\ =\left|-5.63\right|\\ =5.63\end{array}$

The absolute error for week 5 is the modulus of the difference between 25 and 30.63, which corresponds to 5.63. Therefore, the absolute error for week 5 is 5.63.

Calculation of the absolute error for week 6:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|29-27.81\right|\\ =\left|1.19\right|\\ =1.19\end{array}$

The absolute error for week 6 is the modulus of the difference between 29 and 27.81, which corresponds to 1.19. Therefore, the absolute error for week 6 is 1.19.

Calculation of the absolute error for week 7:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|36-28.41\right|\\ =\left|7.59\right|\\ =7.59\end{array}$

The absolute error for week 7 is the modulus of the difference between 36 and 28.41, which corresponds to 7.59. Therefore, the absolute error for week 7 is 7.59.

Calculation of the absolute error for week 8:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|22-32.20\right|\\ =\left|-10.20\right|\\ =10.20\end{array}$

The absolute error for week 8 is the modulus of the difference between 22 and 32.20, which corresponds to 10.20. Therefore, the absolute error for week 8 is 10.20.

Calculation of the absolute error for week 9:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|25-27.10\right|\\ =\left|-2.10\right|\\ =2.10\end{array}$

The absolute error for week 9 is the modulus of the difference between 25 and 27.10, which corresponds to 2.10. Therefore, the absolute error for week 9 is 2.10.

Calculation of the absolute error for week 10:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|28-26.05\right|\\ =\left|2.10\right|\\ =2.10\end{array}$

The absolute error for week 10 is the modulus of the difference between 28 and 26.05, which corresponds to 2.10. Therefore, the absolute error for week 10 is 2.10.

Calculation of the Mean Absolute Deviation:

$\begin{array}{c}\text{MAD}=\frac{{\displaystyle \sum \left|\text{Actual}-\text{Forecast}\right|}}{n}\\ =\frac{0+1+7.50+12.75+5.63+1.19+7.59+10.20+2.10+1.95}{10}\\ =\frac{49.91}{10}\\ =4.99\end{array}$

The substitution of the summation value of absolute error for 10 weeks, divided by the number of weeks, which is 10 yields a MAD of 4.99.

The computed MAD is 4.99.

Summary Introduction

To determine: To compute the tracking signal.

Answer to Problem 59P

The tracking signal is 2.82.

Explanation of Solution

Given information:

Week	Demand
1	20
2	21
3	28
4	37
5	25
6	29
7	36
8	22
9	25
10	28

$\begin{array}{c}\text{Smoothing}\text{constant}\text{α}=\text{0}\text{.5}\\ \text{Initial}\text{forecast}=\text{20}\end{array}$

Formula to calculate tracking signal:

$\begin{array}{c}\text{Tracking}\text{Signal}=\frac{\text{Cumulative}\text{error}}{\text{MAD}}\\ =\frac{{\displaystyle \sum \left(\text{Actual}\text{demand}\text{in}\text{period}\text{i}-\text{Forecasted}\text{demand}\text{in}\text{period}\text{i}\right)}}{\text{MAD}}\\ \text{MAD}=\frac{{\displaystyle \sum \left|\text{Actual}-\text{Forecast}\right|}}{n}\end{array}$

Calculation of tracking signal:

Table 1 shows the calculation of MAD

Week	Demand	Ft	Absolute error	Error	Cumulative error
1	20	20	0	0	0
2	21	20	1	1	1
3	28	20.50	7.50	7.5	8.50
4	37	24.25	12.75	12.75	21.25
5	25	30.63	5.63	-5.63	15.63
6	29	27.81	1.19	1.19	16.81
7	36	28.41	7.59	7.59	24.41
8	22	32.20	10.20	-10.20	14.20
9	25	27.10	2.10	-2.10	12.10
10	28	26.05	1.95	1.95	14.05
		Total	49.91
		MAD	4.99	Tracking signal	2.82

Excel worksheet:

Calculation of tracking signal:

$\begin{array}{c}\text{Tracking}\text{Signal}=\frac{\text{Cumulative}\text{error}}{\text{MAD}}\\ =\frac{14.05}{4.99}\\ =2.82\end{array}$

The ratio of cumulative error and MAD is known as the tracking signal. The tracking signal of 2.82 is obtained by dividing 14.05 by 4.99.

Hence, the computed tracking signal is 2.82.