Chapter 4, Problem 20P

Resolve Problem 4.19 with α = .1 and β =.8. Using MSE, determine which smoothing constants provide a better forecast.

Summary Introduction

To determine: Compute MSE using the given smoothing constants and find the better forecasting smoothing constants using trend-adjusted exponential smoothing method.

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 uniform time interval. Moving average method and trend projections are two of the time series methods which use weights to prioritize past data.

Answer to Problem 20P

On comparing MSE from two smoothing constants (refer to equations (1) and (2)), it can be inferred that smoothing constant with α=0.1 and β=0.8 provides better forecast because it minimizes the error.

Explanation of Solution

Given information:

Time period	Month	Income ($ in thousands)
1	February	70
2	March	68.5
3	April	64.8
4	May	71.7
5	June	71.3
6	July	72.8

Formula to calculate the MSE &forecasted demand

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

$\begin{array}{c}\text{Forecast}\text{including}\text{trend}{\text{(FIT}}_{\text{t}}\text{)}=\left(\begin{array}{l}\text{Exponentially}\text{smoothed}\text{forecast}\text{average}\left({\text{F}}_{\text{t}}\right)\\ +\text{Exponetially}\text{smoothed}\text{Trend}\left({\text{T}}_{\text{t}}\right)\end{array}\right)\\ {\text{FIT}}_{\text{t}}={\text{F}}_{\text{t}}+{\text{T}}_{\text{t}}\end{array}$

$\begin{array}{l}{\text{F}}_{\text{t}}=\text{α}\left({\text{A}}_{\text{t-1}}\right)+1-\text{α}\left({\text{F}}_{\text{t-1}}+{\text{T}}_{\text{t-1}}\right)\\ {\text{T}}_{\text{t}}=\text{β}\left(\text{F}-{\text{F}}_{\text{t-1}}\right)+\left(1-\text{β}\right){\text{T}}_{\text{t-1}}\end{array}$

Where,

$\begin{array}{c}{\text{F}}_{\text{t}}=\text{Exponentially}\text{Smoothed}\text{forecast}\text{average}\text{of}\text{the}\text{data}\text{series}\text{in}\text{period}\text{t}\\ {\text{T}}_{\text{t}}=\text{Exponentially}\text{Smoothed}\text{forecast}\text{average}\text{of}\text{the}\text{data}\text{series}\text{in}\text{period}\text{t}\\ {\text{A}}_{\text{t}}=\text{Actual}\text{demand}\text{in}\text{period}\text{t}\end{array}$

$\begin{array}{c}\text{α}=\text{smoothing}\text{constant}\\ \text{β}=\text{smoothing}\text{constant}\text{for}\text{trend}\end{array}$

Calculation of FIT and MSE usingα=0.1 andβ=0.2:

Time period	Month	Income ($ in thousands)	Ft ($ in thousands)	Tt	FIT	Error	Sq. Error
1	February	70	65	0	65	5	25
2	March	68.5	65.500	0.100	65.600	2.9	8.41
3	April	64.8	65.890	0.158	66.048	-1.25	1.56
4	May	71.7	65.923	0.133	66.056	5.64	31.85
5	June	71.3	66.621	0.246	66.867	4.43	19.66
6	July	72.8	67.310	0.335	67.644	5.16	26.58
Total							113.05
MSE							18.84

Table 1

Excel worksheet:

Calculation of FIT for February:

$\begin{array}{c}\text{FIT}={\text{F}}_1+{\text{T}}_1\\ =\text{65}+\text{0}\\ =\$\text{65}\end{array}$

To calculate FIT for February, compute F₁ and T₁. The forecast F₁ is $ 65 and trend T₁ is 0. The sum of both values gives FIT, which is equal to $ 65.

Calculation of FIT for March:

$\begin{array}{c}{\text{F}}_{\text{2}}=\text{α}\left({\text{A}}_{\text{1}}\right)+\text{1}-\text{α}\left({\text{F}}_{\text{1}}+{\text{T}}_{\text{1}}\right)\\ =\text{0}\text{.1}\times \text{70}+\text{1}-\text{0}\text{.1}\left(\text{65}+\text{0}\right)\\ =\$\text{65}\text{.5}\end{array}$

$\begin{array}{c}{\text{T}}_{\text{2}}=\text{β}\left({\text{F}}_{\text{2}}-{\text{F}}_{\text{1}}\right)+\left(\text{1}-\text{β}\right){\text{T}}_{\text{1}}\\ =\text{0}\text{.2}\left(\text{65}\text{.5}-\text{65}\right)+\left(\text{1}-\text{0}\text{.2}\right)\text{0}\\ =\text{0}\text{.1}\end{array}$

$\begin{array}{c}\text{FIT}={\text{F}}_{\text{2}}+{\text{T}}_{\text{2}}\\ =\text{65}\text{.5}+\text{0}\text{.10}\\ =\$\text{65}\text{.60}\end{array}$

To calculate FIT for February, compute F₂ and T₂. The forecast F₂ is $ 65.5 and trend T₂ is 0.1. The sum of both values gives FIT, which is equal to $ 65.60.

Calculation of FIT for April:

$\begin{array}{c}{\text{F}}_3=\text{α}\left({\text{A}}_2\right)+\text{1}-\text{α}\left({\text{F}}_2+{\text{T}}_2\right)\\ =\text{0}\text{.1}\times 68.5+\text{1}-\text{0}\text{.1}\left(\text{65}\text{.5}+\text{0}\text{.1}\right)\\ =\$\text{65}\text{.890}\end{array}$

$\begin{array}{c}{\text{T}}_3=\text{β}\left({\text{F}}_3-{\text{F}}_2\right)+\left(\text{1}-\text{β}\right){\text{T}}_2\\ =\text{0}\text{.2}\left(\text{65}\text{.71}-\text{65}\text{.5}\right)+\left(\text{1}-\text{0}\text{.2}\right)0.1\\ =\text{0}\text{.158}\end{array}$

$\begin{array}{c}\text{FIT}={\text{F}}_3+{\text{T}}_3\\ =\text{65}\text{.890}+\text{0}\text{.158}\\ =\$\text{66}\text{.048}\end{array}$

To calculate FIT for March, compute F₃ and T₃. The forecast F₃ is $ 65.890 and trend T₃ is 0.158. The sum of both values gives FIT, which is equal to $ 66.048.

Calculation of FIT for May:

$\begin{array}{c}{\text{F}}_4=\text{α}\left({\text{A}}_3\right)+\text{1}-\text{α}\left({\text{F}}_3+{\text{T}}_3\right)\\ =\text{0}\text{.1}\times 64.8+\text{1}-\text{0}\text{.1}\left(\text{65}\text{.710}+\text{0}\text{.122}\right)\\ =\$\text{65}\text{.923}\end{array}$

$\begin{array}{c}{\text{T}}_4=\text{β}\left({\text{F}}_4-{\text{F}}_3\right)+\left(\text{1}-\text{β}\right){\text{T}}_3\\ =\text{0}\text{.2}\left(\text{65}\text{.509}-\text{65}\text{.710}\right)+\left(\text{1}-\text{0}\text{.2}\right)0.122\\ =\text{0}\text{.133}\end{array}$

$\begin{array}{c}\text{FIT}={\text{F}}_4+{\text{T}}_4\\ =\text{65}\text{.923}+\text{0}\text{.133}\\ =\$\text{66}\text{.056}\end{array}$

To calculate FIT for May, compute F₄ and T₄. The forecast F₄ is $ 65.923 and trend T₄ is 0.133. The sum of both values gives FIT, which is equal to $ 66.056.

Calculation of FIT for June:

$\begin{array}{c}{\text{F}}_5=\text{α}\left({\text{A}}_4\right)+\text{1}-\text{α}\left({\text{F}}_4+{\text{T}}_4\right)\\ =\text{0}\text{.1}\times \text{71}\text{.7}+\text{1}-\text{0}\text{.1}\left(\text{65}\text{.509}+\text{0}\text{.057}\right)\\ =\$\text{66}\text{.621}\end{array}$

$\begin{array}{c}{\text{T}}_5=\text{β}\left({\text{F}}_5-{\text{F}}_4\right)+\left(\text{1}-\text{β}\right){\text{T}}_4\\ =\text{0}\text{.2}\left(\text{66}\text{.077}-\text{65}\text{.509}\right)+\left(\text{1}-\text{0}\text{.2}\right)0.057\\ =\text{0}\text{.246}\end{array}$

$\begin{array}{c}\text{FIT}={\text{F}}_5+{\text{T}}_5\\ =\text{66}\text{.621}+\text{0}\text{.246}\\ =\$\text{66}\text{.867}\end{array}$

To calculate FIT for June, compute F₅ and T₅. The forecast F₅ is $ 66.621 and trend T₅ is 0.246. The sum of both values gives FIT, which is equal to $ 66.867.

Calculation of FIT for July:

$\begin{array}{c}{\text{F}}_6=\text{α}\left({\text{A}}_5\right)+\text{1}-\text{α}\left({\text{F}}_5+{\text{T}}_5\right)\\ =\text{0}\text{.1}\times \text{71}\text{.3}+\text{1}-\text{0}\text{.1}\left(\text{66}\text{.077}+\text{0}\text{.159}\right)\\ =\$\text{67}\text{.310}\end{array}$

$\begin{array}{c}{\text{T}}_6=\text{β}\left({\text{F}}_6-{\text{F}}_5\right)+\left(\text{1}-\text{β}\right){\text{T}}_5\\ =\text{0}\text{.2}\left(\text{66}\text{.455}-\text{66}\text{.077}\right)+\left(\text{1}-\text{0}\text{.2}\right)0.159\\ =\text{0}\text{.335}\end{array}$

$\begin{array}{c}\text{FIT}={\text{F}}_6+{\text{T}}_6\\ =\text{67}\text{.310}+\text{0}\text{.335}\\ =\$\text{67}\text{.644}\end{array}$

To calculate FIT for July, compute F₆ and T₆. The forecast F₆ is $ 67.310 and trend T₆ is 0.335. The sum of both values gives FIT, which is equal to $ 67.644.

Calculation of MSE:

MSE is obtained by dividing the summation value of the square of the difference between actual and forecasted sales with the number of years n; n=6.

Table 1provides the values for square of the difference between actual and forecasted sales.

$\begin{array}{c}\text{MSE}=\frac{{{\displaystyle \sum \left(\text{Actual}-\text{Forecast}\right)}}^2}{n}\\ =\frac{25+8.41+1.56+31.85+19.66+26.58}{6}\\ =\frac{113.05}{6}\\ =18.84\end{array}$ (1)

MSE using α=0.1 and β=0.2 is 18.84.

Calculation of FIT and MSE using α=0.1 and β=0.8:

Time period	Month	Income ($ in thousands)	Ft ($ in thousands)	Tt	FIT	Error	Sq. Error
1	February	70	65	0	65	5	25
2	March	68.5	65.500	0.400	65.900	2.6	6.76
3	April	64.8	66.160	0.608	66.768	-1.968	3.87
4	May	71.7	66.571	0.451	67.022	4.678	21.89
5	June	71.3	67.490	0.825	68.314	2.986	8.91
6	July	72.8	68.613	1.064	69.677	3.123	9.76
Total							76.19
MSE							12.70

Table 2

Excel worksheet:

Calculation of FIT for February:

$\begin{array}{c}\text{FIT}={\text{F}}_1+{\text{T}}_1\\ =\text{65}+\text{0}\\ =\$\text{65}\end{array}$

To calculate FIT for February, compute F₁ and T₁. The forecast F₁ is $ 65 and trend T₁ is 0. The sum of both values gives FIT, which is equal to $ 65.

Calculation of FIT for March:

$\begin{array}{c}{\text{F}}_{\text{2}}=\text{α}\left({\text{A}}_{\text{1}}\right)+\text{1}-\text{α}\left({\text{F}}_{\text{1}}+{\text{T}}_{\text{1}}\right)\\ =\text{0}\text{.1}\times \text{70}+\text{1}-\text{0}\text{.1}\left(\text{65}+\text{0}\right)\\ =\$\text{65}\text{.5}\end{array}$

$\begin{array}{c}{\text{T}}_{\text{2}}=\text{β}\left({\text{F}}_{\text{2}}-{\text{F}}_{\text{1}}\right)+\left(\text{1}-\text{β}\right){\text{T}}_{\text{1}}\\ =\text{0}\text{.8}\left(\text{65}\text{.5}-\text{65}\right)+\left(\text{1}-\text{0}\text{.8}\right)\text{0}\\ =\text{0}\text{.4}\end{array}$

$\begin{array}{c}\text{FIT}={\text{F}}_{\text{2}}+{\text{T}}_{\text{2}}\\ =\text{65}\text{.5}+\text{0}\text{.4}\\ =\$\text{65}\text{.90}\end{array}$

To calculate FIT for February, compute F₂ and T₂. The forecast F₂ is $ 65.5 and trend T₂ is 0.4. The sum of both values gives FIT, which is equal to $ 65.90.

Calculation of FIT for April:

$\begin{array}{c}{\text{F}}_3=\text{α}\left({\text{A}}_2\right)+\text{1}-\text{α}\left({\text{F}}_2+{\text{T}}_2\right)\\ =\text{0}\text{.1}\times 68.5+\text{1}-\text{0}\text{.1}\left(\text{65}\text{.5}+\text{0}\text{.4}\right)\\ =\$\text{66}\text{.160}\end{array}$

$\begin{array}{c}{\text{T}}_3=\text{β}\left({\text{F}}_3-{\text{F}}_2\right)+\left(\text{1}-\text{β}\right){\text{T}}_2\\ =\text{0}\text{.8}\left(\text{65}\text{.440}-\text{65}\text{.5}\right)+\left(\text{1}-\text{0}\text{.8}\right)0.4\\ =\text{0}\text{.608}\end{array}$

$\begin{array}{c}\text{FIT}={\text{F}}_3+{\text{T}}_3\\ =\text{66}\text{.160}+\text{0}\text{.608}\\ =\$\text{66}\text{.786}\end{array}$

To calculate FIT for March, compute F₃ and T₃. The forecast F₃ is $ 66.160 and trend T₃ is 0.608. The sum of both values gives FIT, which is equal to $ 66.786.

Calculation of FIT for May:

$\begin{array}{c}{\text{F}}_4=\text{α}\left({\text{A}}_3\right)+\text{1}-\text{α}\left({\text{F}}_3+{\text{T}}_3\right)\\ =\text{0}\text{.1}\times 64.8+\text{1}-\text{0}\text{.1}\left(\text{65}\text{.440}+\text{0}\text{.608}\right)\\ =\$\text{66}\text{.571}\end{array}$

$\begin{array}{c}{\text{T}}_4=\text{β}\left({\text{F}}_4-{\text{F}}_3\right)+\left(\text{1}-\text{β}\right){\text{T}}_3\\ =\text{0}\text{.8}\left(\text{65}\text{.347}-\text{65}\text{.440}\right)+\left(\text{1}-\text{0}\text{.8}\right)0.608\\ =0.451\end{array}$

$\begin{array}{c}\text{FIT}={\text{F}}_4+{\text{T}}_4\\ =\text{65}\text{.347}+0.451\\ =\$\text{67}\text{.022}\end{array}$

To calculate FIT for May, compute F₄ and T₄. The forecast F₄ is $66.571 and trend T₄ is 0.451. The sum of both values gives FIT, which is equal to $ 67.022.

Calculation of FIT for June:

$\begin{array}{c}{\text{F}}_5=\text{α}\left({\text{A}}_4\right)+\text{1}-\text{α}\left({\text{F}}_4+{\text{T}}_4\right)\\ =\text{0}\text{.1}\times \text{71}\text{.7}+\text{1}-\text{0}\text{.1}\left(\text{65}\text{.347}+0.451\right)\\ =\$\text{67}\text{.490}\end{array}$

$\begin{array}{c}{\text{T}}_5=\text{β}\left({\text{F}}_5-{\text{F}}_4\right)+\left(\text{1}-\text{β}\right){\text{T}}_4\\ =\text{0}\text{.8}\left(\text{67}\text{.490}-\text{66}\text{.571}\right)+\left(\text{1}-\text{0}\text{.8}\right)0.451\\ =\text{0}\text{.825}\end{array}$

$\begin{array}{c}\text{FIT}={\text{F}}_5+{\text{T}}_5\\ =\text{67}\text{.490}+\text{0}\text{.825}\\ =\$\text{68}\text{.314}\end{array}$

To calculate FIT for June, compute F₅ and T₅. The forecast F₅ is $ 67.490 and trend T₅ is 0.825. The sum of both values gives FIT, which is equal to $ 68.314.

Calculation of FIT for July:

$\begin{array}{c}{\text{F}}_6=\text{α}\left({\text{A}}_5\right)+\text{1}-\text{α}\left({\text{F}}_5+{\text{T}}_5\right)\\ =\text{0}\text{.1}\times \text{71}\text{.3}+\text{1}-\text{0}\text{.1}\left(\text{67}\text{.490}+\text{0}\text{.825}\right)\\ =\$68.613\end{array}$

$\begin{array}{c}{\text{T}}_6=\text{β}\left({\text{F}}_6-{\text{F}}_5\right)+\left(\text{1}-\text{β}\right){\text{T}}_5\\ =\text{0}\text{.8}\left(\text{68}\text{.613}-\text{67}\text{.490}\right)+\left(\text{1}-\text{0}\text{.8}\right)0.825\\ =\text{1}\text{.064}\end{array}$

$\begin{array}{c}\text{FIT}={\text{F}}_6+{\text{T}}_6\\ =\text{68}\text{.613}+\text{1}\text{.064}\\ =\$\text{69}\text{.677}\end{array}$

To calculate FIT for July, compute F₆ and T₆. The forecast F₆ is $ 68.613 and trend T₆ is 1.064. The sum of both values gives FIT, which is equal to $ 69.677.

Calculation of MSE:

MSE is obtained by dividing the summation value of the square of the difference between actual and forecasted sales with the number of years n=6.

Table 2provides the values for square of the difference between actual and forecasted sales.

$\begin{array}{c}\text{MSE}=\frac{{{\displaystyle \sum \left(\text{Actual}-\text{Forecast}\right)}}^2}{n}\\ =\frac{25+6.76+3.87+21.89+8.91+9.76}{6}\\ =\frac{76.19}{6}\\ =12.70\end{array}$ (2)

MSE using α=0.1 and β=0.8 is 12.70.

On comparing MSE from two smoothing constants (refer to equations(1)&(2)), it can be inferred that smoothing constant with α=0.1 and β=0.8 provides better forecast because it minimizes the error.