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Introduction to Management Science

8th Edition

by

Bernard W. Taylor III

Chapter 5

Forecasting

Chapter 5 - Forecasting

- Forecasting Components
- Time Series Methods
- Forecast Accuracy
- Time Series Forecasting Using Excel
- Time Series Forecasting Using QM for Windows
- Regression Methods

Chapter 5 - Forecasting

- A variety of forecasting methods are available for use depending on the time frame of the forecast and the existence of patterns.
- Time Frames:
- Short-range (one to two months)
- Medium-range (two months to one or two years)
- Long-range (more than one or two years)

- Patterns:
- Trend
- Random variations
- Cycles
- Seasonal pattern

Chapter 5 - Forecasting

Patterns (1 of 2)

- Trend - A long-term movement of the item being forecast.
- Random variations - movements that are not predictable and follow no pattern.
- Cycle - A movement, up or down, that repeats itself over a lengthy time span.
- Seasonal pattern - Oscillating movement in demand that occurs periodically in the short run and is repetitive.

Chapter 5 - Forecasting

Forecasting Components

Patterns (2 of 2)

Figure 5.1

Forms of Forecast Movement: (a) Trend, (b) Cycle, (c) Seasonal Pattern, (d) Trend with Seasonal Pattern

Chapter 5 - Forecasting

Forecasting Methods

- Times Series - Statistical techniques that use historical data to predict future behavior.
- Regression Methods - Regression (or causal ) methods that attempt to develop a mathematical relationship between the item being forecast and factors that cause it to behave the way it does.
- Qualitative Methods - Methods using judgment, expertise and opinion to make forecasts.

Chapter 5 - Forecasting

Qualitative Methods

- Qualitative methods, the “jury of executive opinion,” is the most common type of forecasting method for long-term strategic planning.
- Performed by individuals or groups within an organization, sometimes assisted by consultants and other experts, whose judgments and opinion are considered valid for the forecasting issue.
- Usually includes specialty functions such as marketing, engineering, purchasing, etc. in which individuals have experience and knowledge of the forecasted item.
- Supporting techniques include the Delphi Method, market research, surveys, etc.

Chapter 5 - Forecasting

Overview

- Statistical techniques that make use of historical data collected over a long period of time.
- Methods assume that what has occurred in the past will continue to occur in the future.
- Forecasts based on only one factor - time.

Chapter 5 - Forecasting

Moving Average (1 of 5)

- Moving average uses values from the recent past to develop forecasts.
- This dampensor smoothesout random increases and decreases.
- Useful for forecasting relatively stable items that do not display any trend or seasonal pattern.
- Formula for:

Chapter 5 - Forecasting

Moving Average (2 of 5)

- Example: Instant Paper Clip Supply Company forecast of orders for the next month.
- Three-month moving average:
- Five-month moving average:

Chapter 5 - Forecasting

Moving Average (3 of 5)

Figure 5.2

Three- and Five-Month Moving Averages

Chapter 5 - Forecasting

Moving Average (4 of 5)

Figure 5.2

Three- and Five-Month Moving Averages

Chapter 5 - Forecasting

Moving Average (5 of 5)

- Longer-period moving averages react more slowly to changes in demand than do shorter-period moving averages.
- The appropriate number of periods to use often requires trial-and-error experimentation.
- Moving average does not react well to changes (trends, seasonal effects, etc.) but is easy to use and inexpensive.
- Good for short-term forecasting.

Chapter 5 - Forecasting

Weighted Moving Average (1 of 2)

- In a weighted moving average, weights are assigned to the most recent data.
- Formula:

Chapter 5 - Forecasting

Weighted Moving Average (2 of 2)

- Determining precise weights and number of periods requires trial-and-error experimentation.

Chapter 5 - Forecasting

Exponential Smoothing (1 of 11)

- Exponential smoothing weights recent past data more strongly than more distant data.
- Two forms: simpleexponential smoothing and adjusted exponential smoothing.
- Simple exponential smoothing:
Ft + 1 = Dt + (1 - )Ft

where: Ft + 1 = the forecast for the next period

Dt = actual demand in the present period

Ft = the previously determined forecast for the present period

= a weighting factor (smoothing constant)

use F1 = D1.

Chapter 5 - Forecasting

Exponential Smoothing (2 of 11)

- The most commonly used values of are between.10 and .50.
- Determination of is usually judgmental and subjective and often based on trial-and -error experimentation.

Chapter 5 - Forecasting

Exponential Smoothing (3 of 11)

Example: PM Computer Services (see Table 5.4).

- Exponential smoothing forecasts using smoothing constant of .30.
- Forecast for period 2 (February):
F2 = D1 + (1- )F1 = (.30)(37) + (.70)(37) = 37 units

- Forecast for period 3 (March):
F3 = D2 + (1- )F2 = (.30)(40) + (.70)(37) = 37.9 units

Chapter 5 - Forecasting

Exponential Smoothing (4 of 11)

Using

F1 = D1

Table 5.4

Exponential Smoothing Forecasts, = .30 and = .50

Chapter 5 - Forecasting

Exponential Smoothing (5 of 11)

- The forecast that uses the higher smoothing constant (.50) reacts more strongly to changes in demand than does the forecast with the lower constant (.30).
- Both forecasts lag behind actual demand.
- Both forecasts tend to be consistently lower than actual demand.
- Low smoothing constants are appropriate for stable data without trend; higher constants appropriate for data with trends.

Chapter 5 - Forecasting

Exponential Smoothing (6 of 11)

Figure 5.3

Exponential Smoothing Forecasts

Chapter 5 - Forecasting

Exponential Smoothing (7 of 11)

- Adjusted exponential smoothing: exponential smoothing with a trend adjustment factor added.
- Formula:
AFt + 1 = Ft + 1 + Tt+1

where: Tt = an exponentially smoothed trend factor:

Tt + 1 = (Ft + 1 - Ft) + (1 - )Tt

Tt = the last (previous) period’s trend factor

= smoothing constant for trend ( a value between zero and one).

- Reflects the weight given to the most recent trend data.
- Determined subjectively.

Chapter 5 - Forecasting

Exponential Smoothing (8 of 11)

Example: PM Computer Services exponential smoothed

forecasts with = .50 and = .30 (see Table 5.5).

- Start with T2 = 0.00
- Adjusted forecast for period 3:
T3 = (F3 - F2) + (1 - )T2

= (.30)(38.5 - 37.0) + (.70)(0) = 0.45

AF3 = F3 + T3 = 38.5 + 0.45 = 38.95

Chapter 5 - Forecasting

Exponential Smoothing (9 of 11)

Tt + 1 = (Ft + 1 - Ft) + (1 - )Tt

Table 5.5

Adjusted Exponentially Smoothed Forecast Values

Chapter 5 - Forecasting

Exponential Smoothing (10 of 11)

- Adjusted forecast is consistently higher than the simple exponentially smoothed forecast.
- It is more reflective of the generally increasing trend of the data.

Chapter 5 - Forecasting

Exponential Smoothing (11 of 11)

Figure 5.4

Adjusted Exponentially Smoothed Forecast

Chapter 5 - Forecasting

Linear Trend Line (1 of 5)

- When demand displays an obvious trend over time, a least squares regression line , orlinear trend line, can be used to forecast.
- Formula:

Chapter 5 - Forecasting

Linear Trend Line (2 of 5)

Example: PM Computer Services (see Table 5.6)

Chapter 5 - Forecasting

Linear Trend Line (3 of 5)

Table 5.6

Least Squares Calculations

Chapter 5 - Forecasting

Linear Trend Line (4 of 5)

- A trend line does not adjust to a change in the trend as does the exponential smoothing method.
- This limits its use to shorter time frames in which trend will not change.

Chapter 5 - Forecasting

Seasonal Adjustments (1 of 4)

- A seasonal pattern is a repetitive up-and-down movement in demand.
- Seasonal patterns can occur on a monthly, weekly, or daily basis.
- A seasonally adjusted forecast can be developed by multiplying the normal forecast by a seasonal factor.
- A seasonal factor can be determined by dividing the actual demand for each seasonal period by total annual demand:
Si =Di/D

Chapter 5 - Forecasting

Seasonal Adjustments (2 of 4)

- Seasonal factors lie between zero and one and represent the portion of total annual demand assigned to each season.
- Seasonal factors are multiplied by annual demand to provide adjusted forecasts for each period.

Chapter 5 - Forecasting

Seasonal Adjustments (3 of 4)

- Example: Wishbone Farms

Table 5.7

Demand for Turkeys at Wishbone Farms

S1 = D1/ D = 42.0/148.7 = 0.28

S2 = D2/ D = 29.5/148.7 = 0.20

S3 = D3/ D = 21.9/148.7 = 0.15

S4 = D4/ D = 55.3/148.7 = 0.37

Chapter 5 - Forecasting

Seasonal Adjustments (4 of 4)

- Multiply forecasted demand for entire year by seasonal factors to determine quarterly demand.
- Forecast for entire year (trend line for data in Table 5.7):
y = 40.97 + 4.30x = 40.97 + 4.30(4) = 58.17

- Seasonally adjusted forecasts:
SF1 = (S1)(F5) = (.28)(58.17) = 16.28

SF2 = (S2)(F5) = (.20)(58.17) = 11.63

SF3 = (S3)(F5) = (.15)(58.17) = 8.73

SF4 = (S4)(F5) = (.37)(58.17) = 21.53

Note the potential for confusion: the Si are “seasonal factors” (fractions),

whereas the SFi are “seasonally adjusted forecasts” (commodity values)!

Chapter 5 - Forecasting

Overview

- Forecasts will always deviate from actual values.
- Difference between forecasts and actual values referred to as forecast error.
- Would like forecast error to be as small as possible.
- If error is large, either technique being used is the wrong one, or parameters need adjusting.
- Measures of forecast errors:
- Mean Absolute deviation (MAD)
- Mean absolute percentage deviation (MAPD)
- Cumulative error (E-bar)
- Average error, or bias (E)

Chapter 5 - Forecasting

Mean Absolute Deviation (1 of 7)

- MAD is the average absolute difference between the forecast and actual demand.
- Most popular and simplest-to-use measures of forecast error.
- Formula:

Chapter 5 - Forecasting

Mean Absolute Deviation (2 of 7)

Example: PM Computer Services (see Table 5.8).

- Compare accuracies of different forecasts using MAD:

Chapter 5 - Forecasting

Mean Absolute Deviation (3 of 7)

Table 5.8

Computational Values for MAD

Chapter 5 - Forecasting

Mean Absolute Deviation (4 of 7)

- The lower the value of MAD relative to the magnitude of the data, the more accurate the forecast.
- When viewed alone, MAD is difficult to assess.
- Must be considered in light of magnitude of the data.

Chapter 5 - Forecasting

Mean Absolute Deviation (5 of 7)

- Can be used to compare accuracy of different forecasting techniques working on the same set of demand data (PM Computer Services):
- Exponential smoothing ( = .50): MAD = 4.04
- Adjusted exponential smoothing ( = .50, = .30): MAD = 3.81
- Linear trend line: MAD = 2.29

- Linear trend line has lowest MAD; increasing from .30 to .50 improved smoothed forecast.

Chapter 5 - Forecasting

Mean Absolute Deviation (6 of 7)

- A variation on MAD is the mean absolute percent deviation(MAPD).
- Measures absolute error as a percentage of demand rather than per period.
- Eliminates problem of interpreting the measure of accuracy relative to the magnitude of the demand and forecast values.
- Formula:

Chapter 5 - Forecasting

Mean Absolute Deviation (7 of 7)

MAPD for other three forecasts:

Exponential smoothing ( = .50): MAPD = 8.5%

Adjusted exponential smoothing ( = .50, = .30):

MAPD = 8.1%

Linear trend: MAPD = 4.9%

Chapter 5 - Forecasting

Cumulative Error (1 of 2)

- Cumulative error is the sum of the forecast errors (E =et).
- A relatively large positive value indicates forecast is biased low, a large negative value indicates forecast is biased high.
- If preponderance of errors are positive, forecast is consistently low; and vice versa.
- Cumulative error for trend line is always almost zero, and is therefore not a good measure for this method.
- Cumulative error for PM Computer Services can be read directly from Table 5.8.
- E = et = 49.31 indicating forecasts are frequently below actual demand.

Chapter 5 - Forecasting

Cumulative Error (2 of 2)

- Cumulative error for other forecasts:
Exponential smoothing ( = .50): E = 33.21

Adjusted exponential smoothing ( = .50, =.30):

E = 21.14

- Average error (bias) is the per period average of cumulative error.
- Average error for exponential smoothing forecast:
- A large positive value of average error indicates a forecast is biased low; a large negative error indicates it is biased high.

Chapter 5 - Forecasting

Example Forecasts by Different Measures

Table 5.9

Comparison of Forecasts for PM Computer Services

- Results consistent for all forecasts:
- Larger value of alpha is preferable.
- Adjusted forecast is more accurate than exponential smoothing forecasts.
- Linear trend is more accurate than all the others.

Chapter 5 - Forecasting

Overview

- Time series techniques relate a single variable being forecast to time.
- Regression is a forecasting technique that measures the relationship of one variable to one or more other variables.
- Simplest form of regression is linear regression.

Chapter 5 - Forecasting

Linear Regression

- Linear regression relates demand (dependent variable ) to an independent variable.

Linear function

Chapter 5 - Forecasting

Linear Regression Example (1 of 3)

- State University athletic department.

Chapter 5 - Forecasting

Linear Regression Example (3 of 3)

Figure 5.6

Linear Regression Line

Chapter 5 - Forecasting

Correlation (1 of 2)

- Correlationis a measure of the strength of the relationship between independent and dependent variables.
- Formula:
- Value lies between +1 and -1.
- Value of zero indicates little or no relationship between variables.
- Values near 1.00 and -1.00 indicate strong linear relationship: correlation and anti-correlation.

Chapter 5 - Forecasting

Coefficient of Determination

- The Coefficient of determination is the percentage of the variation in the dependent variable that results from the independent variable.
- Computed by squaring the correlation coefficient, r.
- For State University example:
r = .948, r2 = .899

- This value indicates that 89.9% of the amount of variation in attendance can be attributed to the number of wins by the team, with the remaining 10.1% due to other, unexplained, factors.

Chapter 5 - Forecasting

Multiple Regression with Excel (1 of 4)

- Multiple regression relates demand to two or more independent variables.
General form:

y = 0 + 1x1 + 2x2 + . . . + kxk

where 0 = the intercept

1 . . . k = parameters representing contributions of the independent variables

x1 . . . xk = independent variables

Chapter 5 - Forecasting

Computer Software Firm (1 of 4)

Problem Statement:

- For data below, develop an exponential smoothing forecast using = .40, and an adjusted exponential smoothing forecast using = .40 and = .20.
- Compare the accuracy of the forecasts using MAD and cumulative error.

Chapter 5 - Forecasting

Computer Software Firm (2 of 4)

Step 1: Compute the Exponential Smoothing Forecast.

Ft+1 = Dt + (1 - )Ft

Step 2: Compute the Adjusted Exponential Smoothing

Forecast

AFt+1 = Ft +1 + Tt+1

Tt+1 = (Ft +1 - Ft) + (1 - )Tt

Chapter 5 - Forecasting

Computer Software Firm (4 of 4)

Step 3: Compute the MAD Values

Step 4: Compute the Cumulative Error.

E(Ft) = 35.97

E(AFt) = 30.60

Chapter 5 - Forecasting

Building Products Store (1 of 5)

Problem Statement:

For the following data,

- Develop a linear regression model
- Determine the strength of the linear relationship using correlation.
- Determine a forecast for lumber given 10 building permits in the next quarter.

Chapter 5 - Forecasting

Building Products Store (3 of 5)

Step 1: Compute the Components of the Linear Regression

Equation.

Chapter 5 - Forecasting

Building Products Store (4 of 5)

Step 2: Develop the Linear regression equation.

y = a + bx, y = 1.36 + 1.25x

Step 3: Compute the Correlation Coefficient.

Chapter 5 - Forecasting

Building Products Store (5 of 5)

Step 4: Calculate the forecast for x = 10 permits.

Y = a + bx = 1.36 + 1.25(10) = 13.86 or 1,386 board ft

Chapter 5 - Forecasting

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