Forecasting: On-Line Lesson

� Lesson 1- Introduction and Examples �

Forecasting involves predicting future events. Forecasting, as the name implies, is not certain. Forecasts are based on historical data and/or personal judgment. This lesson will focus on creating forecasts from historical data. So, if a forecast is essentially a "guess", why are they necessary? Forecasts are required in the daily activities of an organization in order to guide them into the future.

On-Line Lesson

Forecasting- Introduction and Examples

Lesson 1-Introduction

Forecasting involves predicting future events. Forecasting, as the name implies, is not certain. Forecasts are based on historical data and/or personal judgment. This lesson will focus on creating forecasts from historical data. So, if a forecast is essentially a �guess�, why are they necessary? Forecasts are required in the daily activities of an organization in order to guide them into the future.

� Time Horizon �

� Types of Forecast �

� Time Series Components �

� Last Period Demand Forecast �

� Example - Last Period Demand Forecasting �

� Moving Average Forecast �

� Example � Two Month and Three Month Moving Average Forecast �

� Forecast Error �

� Example � Calculate MAD for the 2-Month Moving Average Problem �

� Example � Calculate MAD for the 3-Month Moving Average Problem �

� Weighted Moving Average Forecast �

� Example � Weighted 3-Month Moving Average �

� Exponential Smoothing �

� Example � Exponential Smoothing Forecast �

� Seasonally Adjusted Forecast �

� Causal Forecasting Model: Linear Regression �

� Example � Linear Regression �

Time Horizons

A forecast is typically categorized by the amount of time into the future it is predicting. Short-Range forecasts predict up to three months into the future. Medium-Range forecasts predict from three months to three years into the future. Long-Range forecasts predict in excess of three years into the future. Remember, as the Time Horizon of the forecast increases, the certainty of the forecast decreases.

Types of Forecasts

Generally speaking, there are two types of forecasts. Quantitative Forecasts rely on mathematical models based on historical data in order to predict the future. Qualitative Forecasts rely on personal or professional opinion in order to predict the future. Quantitative Forecasts are comprised of Time Series Models and Causal Models. Time Series Models predict on the assumption that the future is a function of the past. In other words, they look at what has happened over a period of time and use a series of past data to make a forecast. Causal Models incorporate the variables or factors that might influence the quantity being forecast (Heizer and Render, 1999).

In this lesson, the following Time Series Models will be presented:

� Last Period Demand

� Moving Average

� Weighted Moving Average

� Exponential Smoothing

� Seasonally Adjusted

Additionally, the following Causal Model will be presented: Linear Regression.

Time Series Components

Time Series have the following components:

1. Trend is a gradual upward or downward data pattern over time. For example, the sale of personal computers had an upward trend during the 1980�s and 1990�s.

2. Seasonality is a data pattern that repeats itself after a repeatable amount of time. For example, the sale of coats will increase annually during winter.

3. Random variation is caused by chance and can not be predicted.

Last Period Demand Forecast

This model assumes that the demand in the last period will repeat itself in the next period. The Last Period Demand forecast requires only the last period of historical data. For example, let�s say we are trying to forecast the sales of notebooks. If the actual number of notebooks sold in March, 1999, were 1,234 notebooks, and we are interested in creating a forecast for April, 1999, the forecast would be 1,234 notebooks.

So, the formula for calculating the Last Period Demand Forecast would be as follows:

F_t = A_t-1

where,

F_t = Forecasted demand for period t.

A_t-1= Actual demand for period (t � 1).

Example - Last Period Demand

The following table depicts actual demand data for bowling ball sales (in thousands):

Table FCST-1: Last Period Demand Example

Period (t)	Month	Actual Demand
1	Jan	10
2	Feb	15
3	Mar	12
4	Apr	14
5	May	16

What would the forecast for June (Period 6) be?

F₆ = A_(6-1)

F₆ = A₅ = 16

So, the forecast for June (Period 6) would be 16.

Moving Average Forecast

This model averages x periods of historical actual demand as the forecast. This method is useful when the historical demand is steady. The more periods included in the moving average, the less responsive the forecast will be. The Moving Average Forecast formula is as follows:

where,

F_t = Forecasted demand for period t.

A_i= Actual Demand for period i.

X = Number of periods in the moving average.

Example � Two Month and Three Month Moving Average Forecast

The following table depicts actual demand for mechanical pencil sales (in millions):

Table FCST-2: Moving Average Example

Period (t)	Month	Actual Demand
1	Jan	5
2	Feb	6
3	Mar	8
4	Apr	6
5	May	7

What would a 2-Month and 3-Month Moving Average Forecast be for June (Period 6)?

Calculate the 2-Month Moving Average Forecast for Period 6. Remember, in this case, x = 2 since this is a 2-month moving average, and t = 6 since the forecast for Period 6 is desired.

Now, calculate the 3-Month Moving Average Forecast for Period 6. Remember, in this case, x = 3, and t = 6.

The 2-Month Moving Average for Period was 6.5, and the 3-Month Moving Average for Period 6 was 7.0. So, which forecast is better? This can be answered by calculating the Forecast Error for each forecast.

Forecast Error

Forecast Error is used to determine the overall accuracy of the forecast by measuring the overall average error. This is accomplished by comparing the forecast to actual demand for several historical periods. One such measure is the Mean Absolute Deviation (MAD). MAD is calculated as follows:

where,

|A_t - F_t| = Forecast Error

F_t = Forecasted demand for period t.

A_t = Actual demand for period t.

n = Number of periods that the forecast error is calculated.

Therefore, larger MAD values indicate that the forecast is performing poorly.

Example � Calculate MAD for the 2-Month Moving Average Problem

Table FCST-3: 2-Month Moving Average MAD

Period (t)	Month	Actual Demand A_t	2-Month Moving Average F_t	\|A_t - F_t\|
1	Jan	5	---	---
2	Feb	6	---	---
3	Mar	8	5.5	2.5
4	Apr	6	7.0	1
5	May	7	7.0	0
6	Jun	?	6.5	---

For this example, n = 3 because the forecast error can only be calculated for three periods (3, 4, and 5). Since this is a 2-Month Moving Average, a forecast can not be calculated until period 3. Now, the MAD can be calculated:

Example � Calculate MAD for the 3-Month Moving Average Problem

Table FCST-4: 3-Month Moving Average MAD

Period (t)	Month	Actual Demand A_t	3-Month Moving Average F_t	\|A_t - F_t\|
1	Jan	5	---	---
2	Feb	6	---	---
3	Mar	8	---	---
4	Apr	6	6.3	0.3
5	May	7	6.7	0.3
6	Jun	?	7.0	---

For this example, n = 2 since only two forecast errors could be calculated. The MAD calculation follows:

For this problem the 3-Month Moving Average had a smaller MAD than the 2-Month Moving Average. Therefore, the 3-Month Moving Average model would be selected as the preferred forecasting method due to the smaller MAD value.

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Weighted Moving Average Forecast

This model applies weights to periods in order to give more emphasis on recent periods as opposed to the simple Moving Average model. The formula for the Weighted Moving Average model is a follows:

F_t = Forecasted demand for period t.

W_i = Weight for period i (the sum of all weights must equal to one)

A_i = Actual demand for period i.

X = Number of periods in the moving average.

Example � Weighted 3-Month Moving Average

The following table depicts actual demand for mechanical pencil:

Table FCST-5: Weighted Moving Average Example

Period (t)	Month	Actual Demand
1	Jan	5
2	Feb	6
3	Mar	8
4	Apr	6
5	May	7

Calculate a Weighted 3-Month Moving Average Forecast for June (Period 6), May (Period 5), and April (Period 4) using the following weights:

W_t-1 = Weight to be applied to the actual demand for the last period.

W_t-2 = Weight to be applied to the actual demand two periods ago.

W_t-3 = Weight to be applied to the actual demand three periods ago.

Remember, the sum of all weights must equal one. In this case, all three weights do add up to one. For this example, X = 3 (three periods in the moving average), and t = 6. Now, calculate the Period 6 forecast:

Now, calculate the forecast for periods 5, and 4:

F₅ = 0.1x6 + 0.2x8 + 0.7x6 = 6.4

F₄ = 0.1x5 + 0.2x6 + 0.7x8 = 7.3

So, how does the Weighted 3-Month Moving Average forecast compare to the simple 3-Month Moving Average? This can be evaluated using MAD as indicated below:

Table FCST-6: Weighted Moving Average Forecast Errors

Period (t)	Month	Actual Demand A_t	3-Month Weighted Moving Average Forecast F_t	\|A_t - F_t\|
1	Jan	5	---	---
2	Feb	6	---	---
3	Mar	8	---	---
4	Apr	6	7.3	1.3
5	May	7	6.4	0.6
6	June	?	6.9	---

For this example, n = 2 since only two forecast errors could be calculated. The MAD calculation follows:

The 3-Month Moving Average MAD was 0.3, the 3-Month Weighted Moving Average MAD is 1.0. So, in this case, the simple 3-Month Moving Average outperformed the 3-Month Weighted Moving Average.

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Exponential Smoothing

Exponential Smoothing is another form of the weighted moving average model. The Exponential Smoothing formula follows:

F_t = aA_t-1 + (1-a)F_t-1

F_t = Forecasted demand for period t.

a = Smoothing Constant (0< a <1).

A_t-1 = Actual demand for period (t �1).

F_t-1= Forecasted demand for period (t �1).

By taking a closer look at the Exponential Smoothing formula, it is apparent that each subsequent forecast is comprised of the exponentially smoothed forecasts from the previous period(s) in the term F_t-1. Actually, the Exponential Smoothing model places more emphasis on recent demand and less emphasis on older data. In terms of a, the larger the value (closer to one), the more reactive to recent data the model will behave. With smaller values of a (closer to zero), the less reactive, or smoother the forecast will be.

Example � Exponential Smoothing Forecast

The following table depicts actual historical demand. Calculate two forecasts using Exponential Smoothing for a = 0.3 and a = 0.5 and determine which value of a performed best.

Table FCST-7: Exponential Smoothing Data

Period (t)	Actual Demand A_t
1	34
2	44
3	42
4	30
5	?

Since Exponential Smoothing builds each subsequent period forecast from the previous, the model must be initialized by assuming that the 1^st period forecast is equal to the actual demand:

F₁ = A₁ = 34

Now, calculate the forecasts for periods 2 through 5:

As can be seen above, each subsequent forecast carries forward the previous forecasted values. Now, calculate the corresponding MAD values for each a used:

Table FCST-8: Exponential Smoothing Error Calculations

Period (t)	Actual Demand A_t	Forecast a =0.3 F_t	Forecast a =0.5 F_t	a =0.3 \|A_t - F_t\|	a =0.5 \|A_t - F_t\|
1	34	34	34	0	0
2	44	34	34	10	10
3	42	37	39	5	3
4	30	38.5	40.5	8.5	10.5
5	?	36	35.3	---	---

MAD for a = 0.5:

So, for this example, both a = 0.3 and a = 0.5 performed identically when compared using MAD.

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Seasonally Adjusted Forecast

The Seasonally Adjusted forecast takes into account seasonality by calculating indices or factors that are then applied to the forecast. For example, consider the following table which depicts quarterly sales of boots for years 1996 through 1998:

Table FCST-9: Seasonal Demand Example

Qtr	Season	�96	�97	�98
1	Fall	100	110	105
2	Winter	130	125	140
3	Spring	150	160	155
4	Summer	85	95	100

In order to generate a Seasonally Adjusted forecast, perform the following steps:

1. Calculate the average seasonal demand.

2. Calculate the total average demand.

3. Calculate the seasonal indices for each season:

4. Calculate the Seasonally Adjusted forecast.

For this example, each quarter represents a season as depicted below:

As can be seen above, each quarter represents a season with Winters having the smallest demand and Falls having the greatest demand. If the total annual forecast for 1999 is 550 units, calculate a Seasonally Adjusted forecast for Spring, Summer, Fall, and Winter 1999.

Step 1: Calculate the Average Seasonal Demand:

Season	Average Seasonal Demand
Spring	(100+110+105)/3 = 105
Summer	(130+125+140)/3 = 132
Fall	(150+160+155)/3 = 155
Winter	(85+95+100)/3 = 93

Step 2: Calculate the Total Average Demand:

Total Average Demand = (105+132+155+93)/4 = 121

Step 3: Calculate the seasonal indices:

Spring Index = (Average Spring Demand) / (Total Average Demand)

Summer Index = (Average Summer Demand) / (Total Average Demand)

Fall Index = (Average Fall Demand) / (Total Average Demand)

Winter Index = (Average Winter Demand) / (Total Average Demand)

Step 4: Calculate the Seasonally Adjusted forecast for 1999. In order to accomplish this, the given annual forecast for 1999 of 550 units must be equally distributed across all four quarters in 1999 then factored by the corresponding seasonal index:

Spring 1999 Seasonally Adjusted forecast: (550/4) x (0.87) = 119.63

Summer 1999 Seasonally Adjusted forecast: (550/4) x (1.09) = 149.88

Fall 1999 Seasonally Adjusted forecast: (550/4) x (1.28) = 176.00

Winter 1999 Seasonally Adjusted forecast: (550/4) x (0.77) = 105.88

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Causal Forecasting Model: Linear Regression

In all of the previous forecasting models, time was always the independent variable with some measure of demand as the dependent variable. For example, using the Time Series models, some measure of demand was forecasted over time. However, using Causal models, Linear Regression can be used to forecast non time dependent events. For example, Linear Regression could be used to forecast the number of fishing licenses sold given a number of bass boats sold during a year.

Simple Linear Regression has the following formula:

= value of the dependent variable (what is to be forecasted).

a = y-axis intercept.

b = slope of the regression line.

x = independent variable.

Additionally, b and a can be calculated as follows:

n = number of data points.

= the mean of the x data values

= the mean of the y data values

For a complete overview of Linear Regression refer to the Statistics module.

Example � Linear Regression

The operations manager of a musical instrument distributor feels that demand for bass drums may be related to the number of television appearances by the popular rock group Green Shades during the previous month. The manager has collected the data shown in the following table:

Table FCST-10: Linear Regression Example