MNG221 - Management Science

Forecasting

Lecture Outline

• Forecasting basics• Moving average• Exponential smoothing• Linear trend line• Forecast accuracy

Forecasting BasicsForecasting

Forecasting Basics

• A Forecast – is a prediction of something that is likely to occur in the future.

• A variety of forecasting methods exist, and their applicability is dependent on the:–time frame of the forecast (i.e., how

far in the future we are forecasting),

Forecasting Basics

• A variety of forecasting methods exist, and their applicability is dependent on the:–the existence of patterns in the

forecast (i.e., seasonal trends, peak periods), and

–the number of variables to which the forecast is related.

Forecasting ComponentsForecasting

Forecasting Components

• Time Frames of Forecast: –Short Range - encompass the

immediate future and are concerned with the daily operations rarely goes beyond a couple months into the future.

Forecasting Components

• Time Frames of Forecast: –Medium Range - encompasses

anywhere from 1 or 2 months to 1 year.

–More closely related to a yearly production plan and will reflect such items as peaks and valleys in demand

Forecasting Components

• Time Frames of Forecast: –Long Range - encompasses a period

longer than 1 or 2 years. –It is Related to management's attempt

to plan new products for changing markets, build new facilities, or secure long-term financing.

Forecasting Components

• Forecasts can exhibit patterns or trend: –A trend is a long-term movement of

the item being forecast–Random variations are movements

that are not predictable and follow no pattern (and thus are virtually unpredictable).

Forecasting Components

Forecasts can exhibit patterns or trend: A cycle is an undulating movement in

demand, up and down, that repeats itself over a lengthy time span (i.e., more than 1 year).

A seasonal pattern is an oscillating movement in demand that occurs periodically (in the short run) and is repetitive. Seasonality is often weather related.

Forecasting Components: Forecast Patterns

Forms of forecast movement: (a) trend, (b) cycle, (c) seasonal pattern, and (d) trend with seasonal pattern

Forecasting MethodsForecasting

Forecasting Methods

The forecasting component determines to a certain extent the type of forecasting method that can or should be used.

• Time Series - is a category of statistical techniques that uses historical data to predict future behavior.

Forecasting Methods

• Regression (or causal) methods - attempt to develop a mathematical relationship (in the form of a regression model) between the item being forecast and factors that cause it to behave the way it does.

Forecasting Methods

• Qualitative methods - use management judgment, expertise, and opinion to make forecasts.

• Often called "the jury of executive opinion,"

• They are the most common type of forecasting method for the long-term strategic planning process.

Time Series AnalysisForecasting Methods

Time Series Methods

• Time series methods tend to be most useful for short-range forecasting, (although they can be used for longer-range forecasting) and relate to only one factor time.

Time Series Methods

• Two types of time series methods: 1. The Moving Average

a)Simple Moving Averageb)Weighted Moving Average

2. Exponential Smoothing.

Time Series – Moving Average

Moving Averages• The moving average method uses

several values during the recent past to develop a forecast.

• The moving average method is good for stable demand with no pronounced behavioral patterns.

Time Series – Moving Average

Moving Averages• Moving averages are computed for

specific periods, such as 3 months or 5 months, depending on how much the forecaster desires to smooth the data.

Time Series – Moving Average

Simple Moving Averages• Moving average forecast may be computed for

specified time period as follows:

where

n = number of periods in the moving average

D = data in period i

n

D

MA

n

ii

ti

1

,

Time Series – Moving AverageSimple Moving Averages - Delivery Orders for 10-month period

Month Orders Delivered per Month

January 120

February 90

March 100

April 75

May 110

June 50

July 75

August 130

September 110

October 90

Time Series – Moving AverageSimple Moving Averages Example

• The moving average from the demand for orders for the last 3 months in the sequence:

Time Series – Moving AverageSimple Moving Averages Example

• The 5-month moving average is computed from the last 5 months of demand data, as follows:

Time Series – Moving AverageSimple 3- and 5- month Moving Average

Time Series – Moving AverageSimple 3- and 5- month Moving Average

Longer-period moving averages react more slowly to recent demand changes than do shorter-period moving averages.

Time Series – Moving Average

Weighted Moving Average• The major disadvantage of the Simple

Moving Average method is that it does not react well to variations that occur for a reason, such as trends and seasonal effects (although this method does reflect trends to a moderate extent).

Time Series – Moving Average

Weighted Moving Average• The Simple Moving Average method can

be adjusted to reflect more closely more recent fluctuations in the data and seasonal effects.

• This adjusted method is referred to as a Weighted Moving Average method.

Time Series – Moving Average

• Weighted Moving Average - is a time series forecasting method in which the most recent data are weighted.

• It may be computed for specified time period using the following:

Time Series – Moving Average• Weighted Moving Average -

Where: Wi = the weight for period i, is

between 0% and 100% ∑Wi =1.00Di = data in period i

Time Series – Moving AverageWeighted Moving Average

For example, if the Instant Paper Clip Supply Company wants to compute a 3-month weighted moving average with a weight of 50% for the October data, a weight of 33% for the September data, and a weight of 17% for August, it is computed as.

Time Series – Moving AverageWeighted Moving Average - Table

Time Series – Exponential Smoothing

• The Exponential Smoothing forecast method is an averaging method that weights the most recent past data more strongly than more distant past data.

• There are two forms of exponential smoothing: 1. Simple Exponential Smoothing 2. Adjusted Exponential Smoothing

(adjusted for trends, seasonal patterns, etc.)

Time Series – Exponential Smoothing

Simple Exponential Smoothing • The simple exponential smoothing

forecast is computed by using the formula:

ttt FDF )1(1

Time Series – Exponential Smoothing

Simple Exponential Smoothing

where

Ft+1 = the forecast for the next period

Dt = the actual demand for the present period

Ft = the previously determined forecast for the present periods

α = a weighting factor referred to as the smoothing constant

ttt FDF )1(1

Time Series – Exponential Smoothing

Simple Exponential Smoothing • The smoothing constant, α, is betw. 0 & 1. • It reflects the weight given to the most

recent demand data.»For example, if α = .20,

»Ft+1 = .20Dt + .80Ft

• This means that our forecast for the next period is based on 20% of recent demand (Dt) and 80% of past demand.

Time Series – Exponential Smoothing

Simple Exponential Smoothing • The higher α is (the closer α is to one),

the more sensitive the forecast will be to changes in recent demand.

• Alternatively, the closer α is to zero, the greater will be the dampening or smoothing effect.

Time Series – Exponential Smoothing

Simple Exponential Smoothing • The most commonly used values of α

are in the range from .01 to .50.

• However, the determination of α is usually judgmental and subjective and will often be based on trial-and-error experimentation.

Time Series – Exponential SmoothingSimple Exponential Smoothing Example

Period Month Demand

1 January 37

2 February 40

3 March 41

4 April 37

5 May 45

6 June 50

7 July 43

8 August 47

9 September 56

10 October 52

11 November 55

12 December 54

•A company - PM Computer Services has accumulated demand data in table for its computers for the past 12 months.

•It wants to compute exponential smoothing forecasts, using smoothing constants (α) equal to 0.30 and 0.50.

Time Series – Exponential Smoothing

Simple Exponential Smoothing Example• To develop the series of forecasts for the data

i, start with period 1 (January) and compute the forecast for period 2 (February) by using α = 0.30.

• The formula for exponential smoothing also requires a forecast for period 1, which we do not have, so we will use the demand for period 1 as both demand and the forecast for period 1.

Time Series – Exponential Smoothing

Simple Exponential Smoothing Example• Thus the forecast for February is:

–F2 = αD1 + (1 - α)F1 –= (.30)(37) + (.70)(37) = 37 units

Time Series – Exponential Smoothing

Simple Exponential Smoothing Example• The forecast for period 3 is computed

similarly: F3 = α D2 + (1 - α)F2

= (.30)(40) + (.70)(37) = 37.9 units• The final forecast is for period 13, January,

and is the forecast of interest to PM Computer Services: F13 = α D12 + (1 - α)F12

= (.30)(54) + (.70)(50.84) = 51.79 units

Time Series – Exponential SmoothingSimple Exponential Smoothing Example

Period  Month  DemandForecast, Ft + 1

a = 0.30 a = 0.50

1 January 37

2 February 40 37.00 37.00

3 March 41 37.90 38.50

4 April 37 38.83 39.75

5 May 45 38.28 38.37

6 June 50 40.29 41.68

7 July 43 43.20 45.84

8 August 47 43.14 44.42

9 September 56 44.30 45.71

10 October 52 47.81 50.85

11 November 55 49.06 51.42

12 December 54 50.84 53.21

13 January 51.79 53.61

Time Series – Exponential SmoothingSimple Exponential Smoothing Example

• In general, when demand is relatively stable, without any trend, using a small value for α is more appropriate to simply smooth out the forecast.

• Alternatively, when actual demand displays an increasing (or decreasing) trend, as is the case, a larger value of α is generally better.

Time Series – Exponential Smoothing

Adjusted Exponential Smoothing • The adjusted exponential smoothing

forecast consists of the exponential smoothing forecast with a trend adjustment factor added to it.

• The formula for the adjusted forecast is:AFt+1 = Ft+1 + Tt+1

where

T = an exponentially smoothed trend factor

Time Series – Exponential Smoothing

Adjusted Exponential Smoothing • The trend factor is computed much the same

as the exponentially smoothed forecast. • It is, in effect, a forecast model for trend:

Tt+1 = β(Ft+1 - Ft) + (1 - β)Tt

where

Tt = the last period trend factor

β = a smoothing constant for trend

Time Series – Exponential Smoothing

Adjusted Exponential Smoothing • Like α, β is a value between 0 and 1. • It reflects the weight given to the most

recent trend data.• Also like α, β is often determined

subjectively, based on the judgment of the forecaster.

Time Series – Exponential Smoothing

Adjusted Exponential Smoothing • A high β reflects trend changes more

than a low β.• It is not uncommon for β to equal α in

this method.• The closer β is to one, the stronger a

trend is reflected.

Time Series – Exponential Smoothing

Adjusted Exponential Smoothing Example • PM Computer Services now wants to develop

an adjusted exponentially smoothed forecast, using the same 12 months of demand.

• The adjusted forecast for February, AF2, is the same as the exponentially smoothed forecast because the trend computing factor will be zero (i.e., F1 and F2 are the same and T2 = 0).

Time Series – Exponential Smoothing

Adjusted Exponential Smoothing Example • Thus, we will compute the adjusted forecast

for March, AF3, as follows, starting with the determination of the trend factor, T3:

–T3 = β (F3 - F2) + (1 β)T2 = (.30)(38.5 - 37.0) + (.70)(0) = 0.45, and

–AF3 = F3 + T3 = 38.5 + 0.45 = 38.95

Time Series – Exponential Smoothing

Adjusted Exponential Smoothing • Period 13 is computed as follows:

–T13 = β(F13 - F12) + (1 β)T12 –= (.30)(53.61 - 53.21) + (.70)(1.77) =

1.36and• AF13 = F13 + T13 = 53.61 + 1.36 = 54.96

units

Time Series – Exponential Smoothing

Period Month DemandForecast

(Ft +1)Trend (Tt +1)

Adjusted Forecast (AFt +1)

1 January 37 37.00

2 February 40 37.00 0.00 37.00

3 March 41 38.50 0.45 38.95

4 April 37 39.75 0.69 40.44

5 May 45 38.37 0.07 38.44

6 June 50 41.68 1.04 42.73

7 July 43 45.84 1.97 47.82

8 August 47 44.42 0.95 45.37

9 September 56 45.71 1.05 46.76

10 October 52 50.85 2.28 53.13

11 November 55 51.42 1.76 53.19

12 December 54 53.21 1.77 54.98

13 January 53.61 1.36 54.96

Time Series – Exponential SmoothingSimple Exponential Smoothing Example

Time Series – Linear Trend Line

Linear Trend Line• Linear regression is most often thought

of as a causal method of forecasting in which a mathematical relationship is developed between demand and some other factor that causes demand behavior.

Time Series – Linear Trend Line

Linear Trend Line• However, when demand displays an

obvious trend over time, a least squares regression line, or linear trend line, can be used to forecast demand.

• A linear trend line is a linear regression model that relates demand to time.

Time Series – Linear Trend Line

Linear Trend Line• The linear regression takes form of a

linear equation as follows:

where

a = intercept

b = slope of the line

x = the time period

y = forecast for demand for period x

bxay

Time Series – Linear Trend Line

Linear Trend Line• The parameters of the trend line may be

calculated as follows:

and

whereand

22 xnx

yxnxyb

n

xx

xbya

n

yy

Time Series – Linear Trend LineLinear Trend Line Example

x (period) y (demand) xy x2

1 37 37 12 40 80 43 41 123 94 37 148 165 45 225 256 50 300 367 43 301 498 47 376 649 56 504 81

10 52 520 10011 55 605 12112 54 648 14478 557 3,867 650

Time Series – Linear Trend Line

Linear Trend Line Example• Using these values for ẋ and ӯ the values, the

parameters for the linear trend line are computed as follows:

Time Series – Linear Trend Line

Linear Trend Line ExampleTherefore, the linear trend line is

y = 35.2 + 1.72x•To calculate a forecast for period 13, x = 13 would be substituted in the linear trend line:

y = 35.2 + 1.72(13) = 57.56

• A linear trend line will not adjust to a change in trend as will exponential smoothing.

Time Series – Linear Trend Line

Linear Trend Line Example

Time Series – Seasonal Adjustments

Seasonal Adjustments• Many demand items exhibit seasonal

behavior or pattern, that is, a repetitive up-and-down movement in demand.

• It is possible to adjust the seasonality of a normal forecast by multiplying it by a seasonal factor.

Time Series – Seasonal Adjustments

Seasonal Adjustments• A seasonal factor, which is a numerical

value is multiplied by the normal forecast to get a seasonally adjusted forecast.

Time Series – Seasonal Adjustments

Seasonal Adjustments• One method for developing a demand for

seasonal factors is dividing the actual demand for each seasonal period by the total annual demand, according to the following formula:

Time Series – Seasonal Adjustments

Seasonal Adjustments• The resulting seasonal factors are

between 0 and 1. • These seasonal factors are thus

multiplied by the annual forecasted demand to yield seasonally adjusted forecasts for each period.

Time Series – Seasonal Adjustments

Seasonal Adjustments ExampleDemand (1,000s)

Year QUARTER 1 QUARTER 2 QUARTER 3 QUARTER 4 TOTAL2003 12.6 8.6 6.3 17.5 45.02004 14.1 10.3 7.5 18.2 50.12005 15.3 10.6 8.1 19.6 53.6Total 42.0 29.5 21.9 55.3 148.7

Next, multiply the forecasted demand for the next year, 2006, by each of the seasonal factors to get the forecasted demand for each quarter.

Time Series – Seasonal AdjustmentsSeasonal Adjustments Example

Demand (1,000s)

Year QUARTER 1 QUARTER 2 QUARTER 3 QUARTER 4 TOTAL

2003 12.6 8.6 6.3 17.5 45.0

2004 14.1 10.3 7.5 18.2 50.1

2005 15.3 10.6 8.1 19.6 53.6

Total 42.0 29.5 21.9 55.3 148.7

• However, to accomplish this, we need a demand forecast for 2006.

• In this case, because the demand data in the table seem to exhibit a generally increasing trend, we compute a linear trend line for the 3 years of data in the table to use as a rough forecast estimate:

y = 40.97 + 4.30x = 40.97 + 4.30(4) = 58.17 or 58,170 turkeys.

Time Series – Seasonal AdjustmentsSeasonal Adjustments Example

Demand (1,000s)Year QUARTER 1 QUARTER 2 QUARTER 3 QUARTER 4 TOTAL2003 12.6 8.6 6.3 17.5 45.02004 14.1 10.3 7.5 18.2 50.12005 15.3 10.6 8.1 19.6 53.6Total 42.0 29.5 21.9 55.3 148.7

Using this annual forecast of demand, the seasonally adjusted forecasts, SFi, for 2006 are as follows:

Forecast AccuracyForecasting

Forecast Accuracy

• It is not probable that a forecast will be completely accurate.

• Forecasts will always deviate from the actual demand resulting in a Forecast error

• A Forecast Error is the difference between the forecast and actual demand.

Forecast Accuracy

• There are different measures of forecast error:–Mean Absolute Deviation (MAD), –Mean Absolute Percent Deviation

(MAPD), –Cumulative Error (E), –Average Error or Bias (Ē), –Mean Squared Error (MSE).

Forecast Accuracy

Mean Absolute Deviation (MAD) is the average, absolute difference between the forecast and the demand and is computed by the following formula:

Forecast Accuracy

Mean Absolute Percent Deviation (MAPD)

–is a absolute error as a percentage of demand.

Forecast Accuracy

Cumulative error – sum of the forecast error.

Average error – is the per-period average of cumulative error.

teE

Forecast Accuracy

Mean Squared Error (MSE) • Each individual error value is squared,

and then these values are summed and averaged.

• As with other measures of forecast accuracy, the smaller the MSE, the better

Forecast accuracy – Worked Example

Regression MethodsForecasting

Regression Methods

• In contrast to times series techniques, regression is a forecasting technique that measures the relationship of one variable to one or more other variables.

• The simplest form of regression is linear regression.

Regression Methods

• Simple Linear Regression relates one dependent variable to one independent variable in the form of a linear equation:

Regression Methods

Simple Linear Regression• To develop the linear equation, the slope, b,

and the intercept, a, must first be computed by using the following least squares formulas:

Regression Methods

Simple Linear Regression• Where

Regression MethodsSimple Linear Regression

x (wins) y (attendance, 1,000s)

xy x2

4 36.3 145.2 16

6 40.1 240.6 36

6 41.2 247.2 36

8 53.0 424.0 64

6 44.0 264.0 36

7 45.6 319.2 49

5 39.0 195.0 25

7 47.5 332.5 49

49 346.7 2,167.7 311

Regression Methods

Simple Linear Regression

Regression Methods

Simple Linear Regression• Substituting these values for a and b into

the linear equation line, we have• y = 18.46 + 4.06x• Thus, for x = 7 (wins), the forecast for

attendance is• y = 18.46 + 4.06(7) = 46.88 or 46,880

Regression Methods

Correlation• Correlation in a linear regression equation is a

measure of the strength of the relationship between the independent and dependent variables. The formula for the correlation coefficient is:

Regression Methods

Correlation• The value of r varies between -1.00 and

+1.00, with a value of ±1.00 indicating a strong linear relationship between the variables.

Regression Methods

Correlation ExampleWe can determine the correlation coefficient for the linear regression equation determined in our State University example by substituting most of the terms calculated for the least squares formula (except for Sy2) into the formula for r:

Regression Methods

Coefficient of Determination• Another measure of the strength of the

relationship between the variables in a linear regression equation is the coefficient of determination.

• The coefficient of determination is the percentage of the variation in the dependent variable that results from the independent variable.

Regression Methods

Coefficient of Determination• It is computed by simply squaring the

value of r. • For our example, r = .948; thus, the

coefficient of determination is: