Chapter 4, Problem 11P

Use exponential smoothing with a smoothing constant of 0.3 to forecast the registrations at the seminar given in Problem 4.10. To begin the procedure, assume that the forecast for year 1 was 5,000 people signing up.

a) What is the MAD?

b) What is the MSE?

Summary Introduction

To determine: Forecast the registrations at the seminar using exponential smoothing and hence compute MAD.

Introduction: Forecasting is used to predict future changes or demand patterns. It involves different approaches and varies with different time periods. Exponential Smoothing and Naïve forecasting methods are two of the time series methods of forecasting which use past data to forecast the future.

Answer to Problem 11P

Using exponential smoothing, the registrations at the seminar are forecasted and the computed MAD is 2.44.

Explanation of Solution

Given information:

Year	Registrations (000)
1	4
2	6
3	4
4	5
5	10
6	8
7	7
8	9
9	12
10	14
11	15

$\begin{array}{c}\text{Initial forecast for year 1}=\text{5000}\\ \text{Smoothing}\text{constant}=\text{0}\text{.3}\end{array}$

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:

Year	Registrations (000)	Ft (000)
1	4	5
2	6	4.700
3	4	5.090
4	5	4.763
5	10	4.834
6	8	6.384
7	7	6.869
8	9	6.908
9	12	7.536
10	14	8.875
11	15	10.412

Table 1

Excel calculation:

Calculation of forecast for year 2:

$\begin{array}{c}{\text{F}}_2={\text{F}}_1+\text{α}\left({\text{A}}_1{\text{-F}}_1\right)\\ =5+0.3\left(4-5\right)\\ =4.70\end{array}$

To calculate the forecast for year 2, substitute the value of the forecast of year 1, smoothing constant, and difference of actual and forecasted demand in the above formula. The result of forecast for year 2 is 4.70

Calculation of forecast for year 3:

$\begin{array}{c}{\text{F}}_3={\text{F}}_2+\text{α}\left({\text{A}}_2{\text{-F}}_2\right)\\ =4.70+0.3\left(6-4.7\right)\\ =5.09\end{array}$

To calculate the forecast for year 3, substitute the value of the forecast of year 1, smoothing constant, and difference of actual and forecasted demand in the above formula. The result of forecast for year 3 is 5.09

Calculation of forecast for year 4:

$\begin{array}{c}{\text{F}}_4={\text{F}}_3+\text{α}\left({\text{A}}_3{\text{-F}}_3\right)\\ =5.09+0.3\left(4-5.09\right)\\ =4.763\end{array}$

To calculate the forecast for year 4, substitute the value of the forecast of year 3, smoothing constant, and difference of actual and forecasted demand in the above formula. The result of forecast for year 3 is 4.763

Calculation of forecast for year 5:

$\begin{array}{c}{\text{F}}_5={\text{F}}_4+\text{α}\left({\text{A}}_4{\text{-F}}_4\right)\\ =4.763+0.3\left(5-4.763\right)\\ =4.834\end{array}$

To calculate the forecast for year 5, substitute the value of the forecast of year 4, smoothing constant, and difference of actual and forecasted demand in the above formula. The result of forecast for year 5 is 4.834

Calculation of forecast for year 6:

$\begin{array}{c}{\text{F}}_6={\text{F}}_5+\text{α}\left({\text{A}}_5{\text{-F}}_5\right)\\ =4.834+0.3\left(10-4.834\right)\\ =6.384\end{array}$

To calculate the forecast for year 6, substitute the value of the forecast of year 5, smoothing constant, and difference of actual and forecasted demand in the above formula. The result of forecast for year 5 is 6.384

Calculation of forecast for year 7:

$\begin{array}{c}{\text{F}}_7={\text{F}}_6+\text{α}\left({\text{A}}_6{\text{-F}}_6\right)\\ =6.384+0.3\left(8-6.384\right)\\ =6.869\end{array}$

To calculate the forecast for year 7, substitute the value of the forecast of year 6, smoothing constant, and difference of actual and forecasted demand in the above formula. The result of forecast for year 7 is 6.869

Calculation of forecast for year 8:

$\begin{array}{c}{\text{F}}_8={\text{F}}_7+\text{α}\left({\text{A}}_7{\text{-F}}_7\right)\\ =6.869+0.3\left(7-6.869\right)\\ =6.908\end{array}$

To calculate the forecast for year 8, substitute the value of the forecast of year 7, smoothing constant, and difference of actual and forecasted demand in the above formula. The result of forecast for year 8 is 6.908

Calculation of forecast for year 9:

$\begin{array}{c}{\text{F}}_9={\text{F}}_8+\text{α}\left({\text{A}}_8{\text{-F}}_8\right)\\ =6.908+0.3\left(7-6.908\right)\\ =7.536\end{array}$

To calculate the forecast for year 9, substitute the value of the forecast of year 8, smoothing constant, and difference of actual and forecasted demand in the above formula. The result of forecast for year 8 is 7.536

Calculation of forecast for year 10:

$\begin{array}{c}{\text{F}}_{10}={\text{F}}_9+\text{α}\left({\text{A}}_9{\text{-F}}_9\right)\\ =7.536+0.3\left(12-7.536\right)\\ =8.875\end{array}$

To calculate the forecast for year 10, substitute the value of the forecast of year 9, smoothing constant, and difference of actual and forecasted demand in the above formula. The result of forecast for year 9 is 8.875

Calculation of forecast for year 11:

$\begin{array}{c}{\text{F}}_{11}={\text{F}}_{10}+\text{α}\left({\text{A}}_{10}{\text{-F}}_{10}\right)\\ =8.875+0.3\left(12-8.875\right)\\ =10.412\end{array}$

To calculate the forecast for year 11, substitute the value of the forecast of year 10, smoothing constant, and difference of actual and forecasted demand in the above formula. The result of forecast for year 10 is 10.412

The forecasted value of registrations using exponential smoothing is provided in table 1

Calculation of MAD using exponential smoothing

Formula to calculate MAD

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

Table 1 provides the adequate forecasted data to compute MAD.

Year	Registrations (000)	Ft (000)	Absolute error
1	4	5.000	1
2	6	4.700	1.30
3	4	5.090	1.09
4	5	4.763	0.24
5	10	4.834	5.17
6	8	6.384	1.62
7	7	6.869	0.13
8	9	6.908	2.09
9	12	7.536	4.46
10	14	8.875	5.13
11	15	10.412	4.59
		Total	26.81
		MAD	2.44

Table 2

Excel calculation:

Mean Absolute Deviation:

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

Mean Absolute Deviation is obtained by dividing the summation of absolute values by the number of years. Absolute error is obtained by taking modulus for the difference between actual and forecasted values.

Calculation of absolute error for year 1

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual-Forecast}\right|\\ =\left|4-5\right|\\ =\left|-1\right|\\ =1\end{array}$

The absolute error for year 1 is the modulus of the difference between 4 and 5, which corresponds to 1. Therefore absolute error for year 1 is 1.

Calculation of absolute error for year 2

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual-Forecast}\right|\\ =\left|6-4.7\right|\\ =\left|1.30\right|\\ =1.30\end{array}$

The absolute error for year 2 is the modulus of the difference between 6 and 4.7, which corresponds to 1.30. Therefore absolute error for year 2 is 1.30

Calculation of absolute error for year 3

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual-Forecast}\right|\\ =\left|4-5.09\right|\\ =\left|-1.09\right|\\ =1.09\end{array}$

The absolute error for year 3 is the modulus of the difference between 4 and 5.09, which corresponds to 1.09. Therefore absolute error for year 1.09

Calculation of absolute error for year 4

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual-Forecast}\right|\\ =\left|5-4.763\right|\\ =\left|0.24\right|\\ =0.24\end{array}$

The absolute error for year 4 is the modulus of the difference between 5 and 4.763, which corresponds to 0.24. Therefore absolute error for year 4 is 0.24

Calculation of absolute error for year 5

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual-Forecast}\right|\\ =\left|10-4.834\right|\\ =\left|5.17\right|\\ =5.17\end{array}$

The absolute error for year 5 is the modulus of the difference between 10 and 4.834, which corresponds to 5.17. Therefore absolute error for year 5 is 5.17

Calculation of absolute error for year 6

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual-Forecast}\right|\\ =\left|8-6.384\right|\\ =\left|1.62\right|\\ =1.62\end{array}$

The absolute error for year 6 is the modulus of the difference between 8 and 6.384, which corresponds to 1.62. Therefore absolute error for year is 1.62

Calculation of absolute error for year 7

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual-Forecast}\right|\\ =\left|7-6.869\right|\\ =\left|0.13\right|\\ =0.13\end{array}$

The absolute error for year 7 is the modulus of the difference between 7 and 6.869, which corresponds to 0.13. Therefore absolute error for year is 0.13

Calculation of absolute error for year 8

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual-Forecast}\right|\\ =\left|9-6.908\right|\\ =\left|2.09\right|\\ =2.09\end{array}$

The absolute error for year 8 is the modulus of the difference between 9 and 6.908, which corresponds to 2.09. Therefore absolute error for year is 2.09

Calculation of absolute error for year 9

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual-Forecast}\right|\\ =\left|12-7.536\right|\\ =\left|4.46\right|\\ =4.46\end{array}$

The absolute error for year 9 is the modulus of the difference between 12 and 7.536, which corresponds to 4.46. Therefore absolute error for year is 4.46

Calculation of absolute error for year 10

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual-Forecast}\right|\\ =\left|14-8.875\right|\\ =\left|5.13\right|\\ =5.13\end{array}$

The absolute error for year 10 is the modulus of the difference between14 and 8.875, which corresponds to 5.13. Therefore absolute error for year is 5.13

Calculation of absolute error for year 11

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual-Forecast}\right|\\ =\left|15-10.412\right|\\ =\left|4.59\right|\\ =4.59\end{array}$

The absolute error for year 11 is the modulus of the difference between 15 and 10.412, which corresponds to 4.59. Therefore absolute error for year is 4.59

Calculation of MAD using exponential smoothing

$\begin{array}{c}\text{MAD}=\frac{{\displaystyle \sum \left|\text{Actual-Forecast}\right|}}{n}\\ =\frac{1+1.30+1.09+0.24+5.17+1.62+0.13+2.09+4.46+5.13+4.59}{11}\\ =\frac{26.81}{11}\\ =2.44\end{array}$

Upon substitution of summation, the value of absolute error for 11 years, that is, 26.81 is divided by number of years, that is, 11 yields MAD of 2.44

Hence, using exponential smoothing, the registrations at the seminar are forecasted and the computed MAD is 2.44

Summary Introduction

To determine: Forecast the registrations at the seminar using exponential smoothing and hence compute MSE.

Answer to Problem 11P

Using exponential smoothing, the registrations at the seminar are forecasted and the computed MSE is 9.53

Explanation of Solution

Given information:

Year	Registrations (000)
1	4
2	6
3	4
4	5
5	10
6	8
7	7
8	9
9	12
10	14
11	15

$\begin{array}{c}\text{Initial forecast for year 1}=\text{5000}\\ \text{Smoothing}\text{constant}=\text{0}\text{.3}\end{array}$

Formula to calculate MSE

$\text{MSE}=\frac{{{\displaystyle \sum \left(\text{Actual-Forecast}\right)}}^2}{n}$

Table 1 provides the required forecasted data which in turn is used to compute MSE.

Year	Registrations (000)	Ft (000)	Error2
1	4	5	1
2	6	4.700	1.69
3	4	5.090	1.19
4	5	4.763	0.06
5	10	4.834	26.69
6	8	6.384	2.61
7	7	6.869	0.02
8	9	6.908	4.38
9	12	7.536	19.93
10	14	8.875	26.27
11	15	10.412	21.05
		Total	104.87
		MSE	9.53

Table 3

Excel calculation:

Error is the difference between actual and forecasted values. Table 2 provides the value of Error for the forecasted and given values.

Calculation of MSE

$\begin{array}{c}\text{MSE}=\frac{{{\displaystyle \sum \left(\text{Actual-Forecast}\right)}}^2}{n}\\ =\frac{1+1.69+1.19+0.06+26.69+2.61+0.02+4.38+19.93+26.27+21.05}{11}\\ =\frac{104.87}{11}\\ =9.53\end{array}$

MSE is obtained by dividing the summation of the square of error (refer to Table (3)) with the n number of periods, that is, 11.

Hence, using exponential smoothing, the registrations at the seminar are forecasted and the computed MSE is 9.53.