Chapter 4, Problem 17P

Summary Introduction

To determine: Find the forecast of sales using exponential smoothing with smoothing constant 0.6 and 0.9 and infer the effect of exponential smoothing on forecast. Using MAD, determine the accurate forecast of exponential smoothing with given smoothing constant 0.3, 0.6 and 0.9.

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

Answer to Problem 17P

On Comparing MAD from exponential smoothing with smoothing constant 0.3, 0.6 and 0.9 (refer to equations (1), (2) and (3)), it can be inferred that the MAD of exponential smoothing with smoothing constant is most accurate because of least value of MAD.

Explanation of Solution

Forecast of sales using exponential smoothing with smoothing constant 0.6:

Given information:

Year	Sales
1	450
2	495
3	518
4	563
5	584

$\begin{array}{c}\text{Initial forecast for year 1}=410\\ \text{Smoothing}\text{constant}=\text{0}\text{.6}\end{array}$

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,

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

Smoothing constant=0.6
Year	Sales	Forecast	Absolute error
1	450	410	40
2	495	434	61
3	518	470.6	47.4
4	563	499.04	63.96
5	584	537.416	46.584
6		565.3664
		Total	258.944
		MAD	51.7888

Excel worksheet:

Calculation of the forecast for year 2:

$\begin{array}{c}{\text{F}}_2={\text{F}}_1+\text{α}\left({\text{A}}_1-{\text{F}}_1\right)\\ =410+0.6\left(450-410\right)\\ =434\end{array}$

To calculate the forecast for year 2, substitute the value of forecast of year 1, smoothing constant and the difference of actual and forecasted demand in the above formula. The result of the forecast for year 2 is 434.

Calculation of the forecast for year 3:

$\begin{array}{c}{\text{F}}_3={\text{F}}_2+\text{α}\left({\text{A}}_2-{\text{F}}_2\right)\\ =434+0.6\left(495-434\right)\\ =470.6\end{array}$

To calculate the forecast for year 3, substitute the value of forecast of year 2, smoothing constant and the difference of actual and forecasted demand in the above formula. The result of the forecast for year 3 is 470.6.

Calculation of the forecast for year 4:

$\begin{array}{c}{\text{F}}_4={\text{F}}_3+\text{α}\left({\text{A}}_3-{\text{F}}_3\right)\\ =470.60+0.6\left(518-470.60\right)\\ =499.04\end{array}$

To calculate the forecast for year 4, substitute the value of forecast of year 3, smoothing constant and the difference of actual and forecasted demand in the above formula. The result of the forecast for year 4 is 499.04.

Calculation of the forecast for year 5:

$\begin{array}{c}{\text{F}}_5={\text{F}}_4+\text{α}\left({\text{A}}_4-{\text{F}}_4\right)\\ =499.04+0.6\left(563-499.04\right)\\ =537.416\end{array}$

To calculate the forecast for year 5, substitute the value of forecast of year 4, smoothing constant and the difference of actual and forecasted demand in the above formula. The result of the forecast for year 5 is 537.416.

Calculation of the forecast for year 6:

$\begin{array}{c}{\text{F}}_6={\text{F}}_5+\text{α}\left({\text{A}}_5-{\text{F}}_5\right)\\ =537.416+0.6\left(584-537.416\right)\\ =565.36\end{array}$

To calculate the forecast for year 5, substitute the value of forecast of year 5, smoothing constant and the difference of actual and forecasted demand in the above formula. The result of the forecast for year 6 is 565.36.

Calculation of MAD using exponential smoothing with smoothing constant α=0.6:

Formula to calculate the Mean Absolute Deviation:

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

Calculation of the absolute error for year 1:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|450-410\right|\\ =\left|40\right|\\ =40\end{array}$

The absolute error for year 1 is the modulus of the difference between 450 and 410, which corresponds to 40. Therefore, the absolute error for year 1 is 40.

Calculation of the absolute error for year 2:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|495-434\right|\\ =\left|61\right|\\ =61\end{array}$

The absolute error for year 2 is the modulus of the difference between 495 and 434, which is correspond to 61. Therefore, the absolute error for year 2 is 61.

Calculation of the absolute error for year 3:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|518-470.6\right|\\ =\left|47.4\right|\\ =47.4\end{array}$

The absolute error for year 3 is the modulus of the difference between 518and 470.6, which is correspond to 47.4. Therefore, the absolute error for year 3 is 47.4.

Calculation of the absolute error for year 4:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|563-499.04\right|\\ =\left|63.96\right|\\ =63.96\end{array}$

The absolute error for year 4 is the modulus of the difference between 563and499.04, which is correspond to 63.96. Therefore, the absolute error for year 4 is 63.96.

Calculation of the absolute error for year 5:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|584-537.416\right|\\ =\left|46.584\right|\\ =46.584\end{array}$

The absolute error for year 5 is the modulus of the difference between 584and537.416, which is correspond to 46.584. Therefore, the absolute error for year 5 is 46.584.

Calculation of the Mean Absolute Deviation using exponential smoothing with smoothing constant 0.6:

$\begin{array}{c}\text{MAD}=\frac{{\displaystyle \sum \left|\text{Actual}-\text{Forecast}\right|}}{n}\\ =\frac{40+61+47.4+63.96+46.584}{5}\\ =\frac{258.944}{5}\\ =51.7888\end{array}$ (1)

Upon the substitution of summation value of the absolute error for 5 years, that is, 258.944 is divided by the number of years. That is, 5 yields MAD of 51.7888.

The forecast of sales using exponential smoothing with 0.6 as smoothing constant is 565.36 and MAD is 51.7888.

Forecast of sales using exponential smoothing with smoothing constant 0.9:

Given information:

Year	Sales
1	450
2	495
3	518
4	563
5	584

$\begin{array}{c}\text{Initial forecast for year 1}=410\\ \text{Smoothing}\text{constant}=\text{0}\text{.9}\end{array}$

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,

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

Smoothing constant=0.9
Year	Sales	Forecast	Absolute error
1	450	410	40
2	495	446	49
3	518	490.1	27.9
4	563	515.21	47.79
5	584	558.221	25.779
6		581.4221
		Total	190.469
		MAD	38.0938

Excel worksheet:

Calculation of the forecast for year 2:

$\begin{array}{c}{\text{F}}_2={\text{F}}_1+\text{α}\left({\text{A}}_1-{\text{F}}_1\right)\\ =410+0.9\left(450-410\right)\\ =446\end{array}$

To calculate the forecast for year 2, substitute the value of forecast of year 1, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 2 is 446.

Calculation of the forecast for year 3:

$\begin{array}{c}{\text{F}}_3={\text{F}}_2+\text{α}\left({\text{A}}_2-{\text{F}}_2\right)\\ =446+0.9\left(495-446\right)\\ =490.1\end{array}$

To calculate the forecast for year 3, substitute the value of forecast of year 2, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 3 is 490.1.

Calculation of the forecast for year 4:

$\begin{array}{c}{\text{F}}_4={\text{F}}_3+\text{α}\left({\text{A}}_3-{\text{F}}_3\right)\\ =490.1+0.9\left(518-490.1\right)\\ =515.21\end{array}$

To calculate the forecast for year 4, substitute the value of forecast of year 3, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 4 is 515.21.

Calculation of the forecast for year 5:

$\begin{array}{c}{\text{F}}_5={\text{F}}_4+\text{α}\left({\text{A}}_4-{\text{F}}_4\right)\\ =515.21+0.9\left(563-515.21\right)\\ =558.221\end{array}$

To calculate the forecast for year 5, substitute the value of 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 558.221.

Calculation of the forecast for year 6:

$\begin{array}{c}{\text{F}}_6={\text{F}}_5+\text{α}\left({\text{A}}_5-{\text{F}}_5\right)\\ =558.221+0.9\left(584-558.221\right)\\ =581.42\end{array}$

To calculate the forecast for year 6, substitute the value of forecast of year 5, smoothing constant and difference of actual and forecasted demand in the above formula. The result of forecast for year 6 is 581.42.

Calculation of MAD using exponential smoothing with smoothing constant α=0.9:

Formula to calculate the Mean Absolute Deviation:

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

Calculation of the absolute error for year 1:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|450-410\right|\\ =\left|40\right|\\ =40\end{array}$

The absolute error for year 1 is the modulus of the difference between 450 and 410, which corresponds to 40. Therefore, the absolute error for year 1 is 40.

Calculation of the absolute error for year 2:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|495-446\right|\\ =\left|49\right|\\ =49\end{array}$

The absolute error for year 2 is the modulus of the difference between 495 and 446, which corresponds to 49. Therefore, the absolute error for year 2 is 49.

Calculation of the absolute error for year 3:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|518-490.1\right|\\ =\left|27.9\right|\\ =27.9\end{array}$

The absolute error for year 3 is the modulus of the difference between 518and490.1, which corresponds to 27.9. Therefore, the absolute error for year 3 is 27.9.

Calculation of the absolute error for year 4:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|563-515.21\right|\\ =\left|47.79\right|\\ =47.79\end{array}$

The absolute error for year 4 is the modulus of the difference between 563and515.21, which corresponds to 4.254. Therefore, the absolute error for year 4 is 47.79.

Calculation of the absolute error for year 5:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|584-558.221\right|\\ =\left|25.779\right|\\ =25.779\end{array}$

The absolute error for year 5 is the modulus of the difference between 584and558.221, which corresponds to 25.779. Therefore, the absolute error for year 5 is 25.779.

Calculation of the Mean Absolute Deviation using exponential smoothing:

$\begin{array}{c}\text{MAD}=\frac{{\displaystyle \sum \left|\text{Actual}-\text{Forecast}\right|}}{n}\\ =\frac{40+49+27.9+47.79+25.779}{5}\\ =\frac{190.469}{5}\\ =38.093\end{array}$ (2)

Upon the substitution of summation value of absolute error for 5 years, that is, 190.469 is divided by the number of years. That is, 5 yields MAD of 38.093.

The forecast of sales using exponential smoothing with 0.9 as smoothing constant is 581.4221 and MAD is 38.093.

Forecast of sales using exponential smoothing with smoothing constant 0.3:

Given information:

Year	Sales
1	450
2	495
3	518
4	563
5	584

$\begin{array}{c}\text{Initial forecast for year 1}=410\\ \text{Smoothing}\text{constant}=\text{0}\text{.3}\end{array}$

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,

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

Smoothing constant=0.3
Year	Sales	Forecast	Absolute error
1	450	410	40
2	495	422	73
3	518	443.9	74.1
4	563	466.13	96.87
5	584	495.191	88.809
6		521.8337
		Total	372.779
		MAD	74.5558

Excel worksheet:

Calculation of the forecast for year 2:

$\begin{array}{c}{\text{F}}_2={\text{F}}_1+\text{α}\left({\text{A}}_1-{\text{F}}_1\right)\\ =410+0.3\left(450-410\right)\\ =422\end{array}$

To calculate the forecast for year 2, substitute the value of forecast of year 1, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 2 is 422.

Calculation of the forecast for year 3:

$\begin{array}{c}{\text{F}}_3={\text{F}}_2+\text{α}\left({\text{A}}_2-{\text{F}}_2\right)\\ =422+0.3\left(495-422\right)\\ =443.9\end{array}$

To calculate the forecast for year 3, substitute the value of forecast of year 2, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 3 is 443.9.

Calculation of the forecast for year 4:

$\begin{array}{c}{\text{F}}_4={\text{F}}_3+\text{α}\left({\text{A}}_3-{\text{F}}_3\right)\\ =443.9+0.3\left(518-443.9\right)\\ =466.13\end{array}$

To calculate the forecast for year 4, substitute the value of forecast of year 3, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 4 is 466.13.

Calculation of the forecast for year 5:

$\begin{array}{c}{\text{F}}_5={\text{F}}_4+\text{α}\left({\text{A}}_4-{\text{F}}_4\right)\\ =466.13+0.3\left(563-466.13\right)\\ =495.191\end{array}$

To calculate the forecast for year 5, substitute the value of forecast of year 4, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 5 is 495.191.

Calculation of the forecast for year 6:

$\begin{array}{c}{\text{F}}_6={\text{F}}_5+\text{α}\left({\text{A}}_5-{\text{F}}_5\right)\\ =495.191+0.9\left(584-495.191\right)\\ =521.833\end{array}$

To calculate the forecast for year 6, substitute the value of forecast of year 5, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 6 is 521.833.

Calculation of MAD using exponential smoothing with smoothing constant α=0.3:

Formula to calculate the Mean Absolute Deviation:

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

Calculation of the absolute error for year 1:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|450-410\right|\\ =\left|40\right|\\ =40\end{array}$

The absolute error for year 1 is the modulus of the difference between 450 and 410, which corresponds to 40. Therefore, the absolute error for year 1 is 40.

Calculation of the absolute error for year 2:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|495-422\right|\\ =\left|73\right|\\ =73\end{array}$

The absolute error for year 2 is the modulus of the difference between 495 and 422, which corresponds to 73. Therefore, the absolute error for year 2 is 73.

Calculation of the absolute error for year 3:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|518-443.9\right|\\ =\left|74.1\right|\\ =74.1\end{array}$

The absolute error for year 3 is the modulus of the difference between 518 and 443.9, which corresponds to 74.1. Therefore, the absolute error for year 3 is 74.1.

Calculation of the absolute error for year 4:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|563-466.13\right|\\ =\left|96.87\right|\\ =96.87\end{array}$

The absolute error for year 4 is the modulus of the difference between 563 and 466.13, which corresponds to 96.87. Therefore, the absolute error for year 4 is 96.87.

Calculation of the absolute error for year 5:

$\begin{array}{c}\text{Absolute}\text{error}=\left|\text{Actual}-\text{Forecast}\right|\\ =\left|584-495.191\right|\\ =\left|88.809\right|\\ =88.809\end{array}$

The absolute error for year 5 is the modulus of the difference between 584 and 495.191, which corresponds to 88.809. Therefore, the absolute error for year 5 is 88.809.

Calculation of the Mean Absolute Deviation using exponential smoothing with 0.3 as smoothing constant:

$\begin{array}{c}\text{MAD}=\frac{{\displaystyle \sum \left|\text{Actual}-\text{Forecast}\right|}}{n}\\ =\frac{40+73+74.1+96.87+88.809}{5}\\ =\frac{372.779}{5}\\ =74.5558\end{array}$ (3)

Upon the substitution of summation value of absolute error for 5 years, that is, 372.779 are divided by the number of years. That is, 5 yields MAD of 74.5558.

The forecast of sales using exponential smoothing with 0.3 as smoothing constant is 521.833 and MAD is 74.5558.

Hence, on comparing MAD from exponential smoothing with smoothing constant 0.3, 0.6 and 0.9 (refer to equations (1), (2) and (3)), it can be inferred that MAD of exponential smoothing with smoothing constant is most accurate because of least MAD.