Chapter 10, Problem 6P

Using the data in problem 2, prepare exponentially smoothed forecasts for the following cases:

1. a. α = .1 and F₁ = 90
2. b. α = .3 and F₁ = 90

Summary Introduction

To develop: The exponential smoothed forecast

Introduction:

Exponential smoothing:

In the exponential smoothing forecast method, older data are given lesser importance, and newer data are given more importance. It is efficient in making short-term forecasts.

Explanation of Solution

Given information:

Forecast for period 1 (F1) = 90

Smoothing constant (α) = 0.1

Period	Dt
1	92
2	127
3	106
4	165
5	125
6	111
7	178
8	97

Formula for exponential smoothing:

$\begin{array}{c}{\text{F}}_{\text{Period}}=F_t+\left[\alpha \left(D_t-F_t\right)\right]\\ {\text{F}}_{\text{Period}}=\text{Forecast for the period calculated}\text{.}\\ {\text{F}}_{\text{t}}=\text{Forecasted}\text{sales}\\ {\text{D}}_{\text{t}}=\text{Previous}\text{demand}\\ \text{α}=\text{Smoothing}\text{constant}\end{array}$

Calculation of forecast:

The forecast for period 1 is 90.

Period 2:

$\begin{array}{c}{\text{F}}_{\text{2}}=F_1+\left[\alpha \left(D_1-F_1\right)\right]\\ =90+\left[0.1\left(92-90\right)\right]\\ =90+\left(0.1\times 2\right)\\ =90+0.2\\ =90.2\end{array}$

The forecast for period 2 is 90.2.

Period 3:

$\begin{array}{c}{\text{F}}_3=F_2+\left[\alpha \left(D_2-F_2\right)\right]\\ =90.2+\left[0.1\left(127-90.2\right)\right]\\ =90.2+\left(0.1\times 36.8\right)\\ =90.2+3.68\\ =93.9\end{array}$

The forecast for period 3 is 93.9.

Period 4:

$\begin{array}{c}{\text{F}}_4=F_3+\left[\alpha \left(D_3-F_3\right)\right]\\ =93.9+\left[0.1\left(106-93.9\right)\right]\\ =93.9+\left(0.1\times 12.1\right)\\ =93.9+1.21\\ =95.1\end{array}$

The forecast for period 4 is 95.1.

Period 5:

$\begin{array}{c}{\text{F}}_{\text{5}}=F_4+\left[\alpha \left(D_4-F_4\right)\right]\\ =95.1+\left[0.1\left(165-95.1\right)\right]\\ =95.1+\left(0.1\times 70.1\right)\\ =95.1+7.01\\ =102.1\end{array}$

The forecast for period 5 is 102.1.

Period 6:

$\begin{array}{c}{\text{F}}_6=F_5+\left[\alpha \left(D_5-F_5\right)\right]\\ =102.1+\left[0.1\left(125-102.1\right)\right]\\ =102.1+\left(0.1\times 22.9\right)\\ =102.1+2.29\\ =104.4\end{array}$

The forecast for period 6 is 104.4.

Period 7:

$\begin{array}{c}{\text{F}}_7=F_6+\left[\alpha \left(D_6-F_6\right)\right]\\ =104.4+\left[0.1\left(111-104.4\right)\right]\\ =104.4+\left(0.1\times 6.6\right)\\ =104.4+0.66\\ =105\end{array}$

The forecast for period 7 is 105.

Period 8:

$\begin{array}{c}{\text{F}}_8=F_7+\left[\alpha \left(D_7-F_7\right)\right]\\ =105+\left[0.1\left(178-105\right)\right]\\ =105+\left(0.1\times 73\right)\\ =105+7.3\\ =112.3\end{array}$

The forecast for period 8 is 112.3.

Summary Introduction

To develop: The exponential smoothed forecast

Introduction:

Exponential smoothing:

In the exponential smoothing forecast method, older data are given lesser importance, and newer data are given more importance. It is efficient in making short-term forecasts.

Explanation of Solution

Given information:

Forecast for period 1 (F1) = 90

Smoothing constant (α) = 0.3

Period	Dt
1	92
2	127
3	106
4	165
5	125
6	111
7	178
8	97

Formula for exponential smoothing:

$\begin{array}{c}{\text{F}}_{\text{Period}}=F_t+\left[\alpha \left(D_t-F_t\right)\right]\\ {\text{F}}_{\text{Period}}=\text{Forecast for the period calculated}\text{.}\\ {\text{F}}_{\text{t}}=\text{Forecasted}\text{sales}\\ {\text{D}}_{\text{t}}=\text{Previous}\text{demand}\\ \text{α}=\text{Smoothing}\text{constant}\end{array}$

Calculation of forecast:

The forecast for period 1 is 90.

Period 2:

$\begin{array}{c}{\text{F}}_{\text{2}}=F_1+\left[\alpha \left(D_1-F_1\right)\right]\\ =90+\left[0.3\left(92-90\right)\right]\\ =90+\left(0.3\times 2\right)\\ =90+0.6\\ =90.6\end{array}$

The forecast for period 2 is 90.6.

Period 3:

$\begin{array}{c}{\text{F}}_3=F_2+\left[\alpha \left(D_2-F_2\right)\right]\\ =90.6+\left[0.3\left(127-90.6\right)\right]\\ =90.6+\left(0.3\times 36.8\right)\\ =90.6+10.92\\ =101.5\end{array}$

The forecast for period 3 is 101.5.

Period 4:

$\begin{array}{c}{\text{F}}_4=F_3+\left[\alpha \left(D_3-F_3\right)\right]\\ =101.5+\left[0.3\left(106-101.5\right)\right]\\ =101.5+\left(0.3\times 4.5\right)\\ =101.5+1.35\\ =102.9\end{array}$

The forecast for period 4 is 102.9.

Period 5:

$\begin{array}{c}{\text{F}}_{\text{5}}=F_4+\left[\alpha \left(D_4-F_4\right)\right]\\ =102.9+\left[0.3\left(165-102.9\right)\right]\\ =102.9+\left(0.3\times 62.1\right)\\ =102.9+18.63\\ =121.5\end{array}$

The forecast for period 5 is 121.5.

Period 6:

$\begin{array}{c}{\text{F}}_6=F_5+\left[\alpha \left(D_5-F_5\right)\right]\\ =121.5+\left[0.3\left(125-121.5\right)\right]\\ =121.5+\left(0.3\times 3.5\right)\\ =121.5+1.041\\ =122.6\end{array}$

The forecast for period 6 is 122.6.

Period 7:

$\begin{array}{c}{F}_7=F_6+\left[\alpha \left(D_6-F_6\right)\right]\\ =122.6+\left[0.3\left(111-122.6\right)\right]\\ =122.6+\left(0.3\times \left(-11.6\right)\right)\\ =122.6-3.48\\ =119.1\end{array}$

The forecast for period 7 is 119.1.

Period 8:

$\begin{array}{c}{\text{F}}_8=F_7+\left[\alpha \left(D_7-F_7\right)\right]\\ =119.1+\left[0.3\left(178-119.1\right)\right]\\ =119.1+\left(0.3\times 58.9\right)\\ =119.1+17.67\\ =136.8\end{array}$

The forecast for period 8 is 136.8.