# FORECASTING

### Learning Objectives:

• To recognize that different forecasting methods are appropriate in different situations
• To become familiar with the various methods of forecasting
• To learn measures for analyzing the performance of forecast methods
• To learn to use Excel for different types of forecasting problems

## Contents

Introduction to Forecasting

Forecasting Methods:

• Time Series

FORECASTING - a method for translating past experience into estimates of the future

Read: The University Bookstore Student Computer Purchase Program page 497 in the text.

Key questions which must be answered:

• what is the purpose of the forecast?
• what specifically do we wish to forecast?
• how important is the past in predicting the future?
• what system will be used to make the forecast?

forecasting horizons:

• long-term: more than 2 years
• medium-term: 3 months to 2 years
• short-term: 0 to 3 months

forecasting methods:

qualitative methods

quantitative methods

- causal methods

- time series methods

### QUALITATIVE FORECASTING METHODS

qualitative forecasting methods are based on educated opinions of appropriate persons

1. delphi method: forecast is developed by a panel of experts who anonymously answer a series of questions; responses are fed back to panel members who then may change their original responses

- very time consuming and expensive

- new groupware makes this process much more feasible

2. market research: panels, questionnaires, test markets, surveys, etc.

3. product life-cycle analogy: forecasts based on life-cycles of similar products, services, or processes

4. expert judgement by management, sales force, or other knowledgeable persons

### QUANTITATIVE FORECASTING METHODS

#### TIME SERIES FORECASTING METHODS

time series forecasting methods are based on analysis of historical data (time series: a set of observations measured at successive times or over successive periods). They make the assumption that past patterns in data can be used to forecast future data points.

1. moving averages (simple moving average, weighted moving average): forecast is based on arithmetic average of a given number of past data points

2. exponential smoothing (single exponential smoothing, double exponential smoothing): a type of weighted moving average that allows inclusion of trends, etc.

3. mathematical models (trend lines, log-linear models, Fourier series, etc.): linear or non-linear models fitted to time-series data, usually by regression methods

4. Box-Jenkins methods: autocorrelation methods used to identify underlying time series and to fit the "best" model

COMPONENTS OF TIME SERIES DEMAND

1. average: the mean of the observations over time

2. trend: a gradual increase or decrease in the average over time

3. seasonal influence: predictable short-term cycling behaviour due to time of day, week, month, season, year, etc.

4. cyclical movement: unpredictable long-term cycling behaviour due to business cycle or product/service life cycle

5. random error: remaining variation that cannot be explained by the other four components

#### SIMPLE MOVING AVERAGE

moving average techniques forecast demand by calculating an average of actual demands from a specified number of prior periods

each new forecast drops the demand in the oldest period and replaces it with the demand in the most recent period; thus, the data in the calculation "moves" over time

simple moving average: At = Dt + Dt-1 + Dt-2 + ... + Dt-N+1

N

where N = total number of periods in the average

forecast for period t+1: Ft+1 = At

Key Decision: N - How many periods should be considered in the forecast

Tradeoff: Higher value of N - greater smoothing, lower responsiveness

Lower value of N - less smoothing, more responsiveness

- the more periods (N) over which the moving average is calculated, the less susceptible the forecast is to random variations, but the less responsive it is to changes

- a large value of N is appropriate if the underlying pattern of demand is stable

- a smaller value of N is appropriate if the underlying pattern is changing or if it is important to identify short-term fluctuations

a weighted moving average is a moving average where each historical demand may be weighted differently

average: At = W1 Dt + W2 Dt-1 + W3 Dt-2 + ... + WN Dt-N+1

where:

N = total number of periods in the average

Wt = weight applied to period t's demand

Sum of all the weights = 1

forecast: Ft+1 = At = forecast for period t+1

exponential smoothing gives greater weight to demand in more recent periods, and less weight to demand in earlier periods

average: At = a Dt + (1 - a) At-1 = a Dt + (1 - a) Ft

forecast for period t+1: Ft+1 = At

where:

At-1 = "series average" calculated by the exponential smoothing model to period t-1

a = smoothing parameter between 0 and 1

the larger the smoothing parameter , the greater the weight given to the most recent demand

#### DOUBLE EXPONENTIAL SMOOTHING

when a trend exists, the forecasting technique must consider the trend as well as the series average ignoring the trend will cause the forecast to always be below (with an increasing trend) or above (with a decreasing trend) actual demand

double exponential smoothing smooths (averages) both the series average and the trend

forecast for period t+1: Ft+1 = At + Tt

average: At = aDt + (1 - a) (At-1 + Tt-1) = aDt + (1 - a) Ft

average trend: Tt = B CTt + (1 - B) Tt-1

current trend: CTt = At - At-1

forecast for p periods into the future: Ft+p = At + p Tt

where:

At = exponentially smoothed average of the series in period t

Tt = exponentially smoothed average of the trend in period t

CTt = current estimate of the trend in period t

a = smoothing parameter between 0 and 1 for smoothing the averages

B = smoothing parameter between 0 and 1 for smoothing the trend

#### MULTIPLICATIVE SEASONAL METHOD

What happens when the patterns you are trying to predict display seasonal effects?

What is seasonality? - It can range from true variation between seasons, to variation between months, weeks, days in the week and even variation during a single day or hour.

To deal with seasonal effects in forecasting two tasks must be completed:

1. a forecast for the entire period (ie year) must be made using whatever forecasting technique is appropriate. This forecast will be developed using whatever
2. the forecast must be adjust to reflect the seasonal effects in each period (ie month or quarter)

the multiplicative seasonal method adjusts a given forecast by multiplying the forecast by a seasonal factor

Step 1: calculate the average demand y per period for each year (y) of past data by dividing total demand for the year by the number of periods in the year

Step 2: divide the actual demand Dy,t for each period (t) by the average demand y per period (calculated in Step 1) to get a seasonal factor fy,t for each period; repeat for each year of data

Step 3: calculate the average seasonal factor t for each period by summing all the seasonal factors fy,t for that period and dividing by the number of seasonal factors

Step 4: determine the forecast for a given period in a future year by multiplying the average seasonal factor t by the forecasted demand in that future year

Seasonal Forecasting (multiplicative method)

Actual Demand

 Year Q1 Q2 Q3 Q4 Total Avg 1 100 70 60 90 320 80 2 120 80 70 110 380 95 3 134 80 70 100 381 96

Seasonal Factor

 Year Q1 Q2 Q3 Q4 1 1.25 .875 .75 1.125 2 1.26 .84 .74 1.16 3 1.4 .83 .73 1.04 Avg. Seasonal Factor 1.30 .85 .74 1.083

Seasonal Factor - the percentage of average quarterly demand that occurs in each quarter.

Annual Forecast for year 4 is predicted to be 400 units.

Average forecast per quarter is 400/4 = 100 units.

Quarterly Forecast = avg. forecast × seasonal factor.

• Q1: 1.303(100) = 130
• Q2: .85(100) = 85
• Q3: .74(100) = 74
• Q4: 1.083(100) = 108

#### CAUSAL FORECASTING METHODS

causal forecasting methods are based on a known or perceived relationship between the factor to be forecast and other external or internal factors

1. regression: mathematical equation relates a dependent variable to one or more independent variables that are believed to influence the dependent variable

2. econometric models: system of interdependent regression equations that describe some sector of economic activity

3. input-output models: describes the flows from one sector of the economy to another, and so predicts the inputs required to produce outputs in another sector

4. simulation modelling

MEASURING FORECAST ERRORS

There are two aspects of forecasting errors to be concerned about - Bias and Accuracy

Bias - A forecast is biased if it errs more in one direction than in the other

- The method tends to under-forecasts or over-forecasts.

Accuracy - Forecast accuracy refers to the distance of the forecasts from actual demand ignore the direction of that error.

Example: For six periods forecasts and actual demand have been tracked The following table gives actual demand Dt and forecast demand Ft for six periods:

 t Dt Ft Et (Et)2 |Et| | Et|/Dt 1 170 200 -30 900 30 17.6% 2 230 195 35 1225 35 15.2% 3 250 210 40 1600 40 16.0% 4 200 220 -20 400 20 10.0% 5 185 210 -25 625 25 13.5% 6 180 200 -20 400 20 11.1% Total -20 5150 170 83.5%

Forecast Measure

1. cumulative sum of forecast errors (CFE) = -20
2. mean absolute deviation (MAD) = 170 / 6 = 28.33
3. mean squared error (MSE) = 5150 / 6 = 858.33
4. standard deviation of forecast errors = 5150 / 6 = 29.30
5. mean absolute percent error (MAPE) = 83.4% / 6 = 13.9%

What information does each give?

1.
2.
3.
4.
5.

conclusions:

forecast has a tendency to over-estimate demand

average error per forecast was 28.33 units, or 13.9% of actual demand

sampling distribution of forecast errors has standard deviation of 29.3 units.

CRITERIA FOR SELECTING A FORECASTING METHOD

Objectives: 1. Maximize Accuracy and 2. Minimize Bias

Potential Rules for selecting a time series forecasting method. Select the method that

1. gives the smallest bias, as measured by cumulative forecast error (CFE); or
2. gives the smallest mean absolute deviation (MAD); or
3. gives the smallest tracking signal; or
4. supports management's beliefs about the underlying pattern of demand

or others. It appears obvious that some measure of both accuracy and bias should be used together. How?

What about the number of periods to be sampled?

• if demand is inherently stable, low values of and and higher values of N are suggested
• if demand is inherently unstable, high values of and and lower values of N are suggested

### FOCUS FORECASTING

"focus forecasting" refers to an approach to forecasting that develops forecasts by various techniques, then picks the forecast that was produced by the "best" of these techniques, where "best" is determined by some measure of forecast error.

FOCUS FORECASTING: EXAMPLE

For the first six months of the year, the demand for a retail item has been 15, 14, 15, 17, 19, and 18 units.

A retailer uses a focus forecasting system based on two forecasting techniques: a two-period moving average, and a trend-adjusted exponential smoothing model with = 0.1 and = 0.1. With the exponential model, the forecast for January was 15 and the trend average at the end of December was 1.

The retailer uses the mean absolute deviation (MAD) for the last three months as the criterion for choosing which model will be used to forecast for the next month.

a. What will be the forecast for July and which model will be used?

b. Would you answer to Part a. be different if the demand for May had been 14 instead of 19?