mcgraw-hill/irwin & modified by jim grayson for quan 6610 © the mcgraw-hill companies, inc.,...

© The McGraw-Hill Companies, Inc., 200313.1McGraw-Hill/Irwin & modified by Jim Grayson for Quan 6610

Table of ContentsChapter 13 (Forecasting)

A Case Study: The Computer Club Warehouse Problem (Section 13.2) 13.5–13.9Applying Time-Series Forecasting to the Case Study (Section 13.3) 13.10–13.26The Time-Series Forecasting Methods in Perspective (Section 13.5) 13.35–13.39Causal Forecasting with Linear Regression (Section 13.6) 13.40–13.44Forecasting in Practice (Section 13.8) 13.46–13.47


The Computer Club Warehouse (CCW)

• The Computer Club Warehouse (CCW) sells computer products at bargain prices by taking telephone orders (as well as website and fax orders) directly from customers.

• Products include computers, peripherals, supplies, software, and computer furniture.

• The CCW call center is never closed. It is staffed by dozens of agents to take and process customer orders.

• A large number of telephone trunks are provided for incoming calls. If an agent is not free when a call arrives, it is placed on hold. If all the trunks are in use (called saturation), the call receives a busy signal.

• An accurate forecast of the demand for agents is needed.

Question: How should the demand for agents be forecasted?


Average Daily Call Volume (3 Years of Data)

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CCW's Average Daily Call Volume

Year Quarter Call Volume1 1 6,8091 2 6,4651 3 6,5691 4 8,2662 1 7,2572 2 7,0642 3 7,7842 4 8,7243 1 6,9923 2 6,8223 3 7,9493 4 9,650


Measuring the Forecast Error

• The mean absolute deviation (called MAD) measures the average forecasting error.

MAD = (Sum of forecasting errors) / (Number of forecasts)

• The mean square error (often abbreviated MSE) measures the average of the square of the forecasting error.

MSE = (Sum of square of forecasting errors) / (Number of forecasts).

• The MSE increases the weight of large errors relative to the weight of small errors.


Considering Seasonal Effects

• When there are seasonal patterns in the data, they can be addressed by forecasting methods that use seasonal factors.

• The seasonal factor for any period of a year (a quarter, a month, etc.) measures how that period compares to the overall average for an entire year.

Seasonal factor = (Average for the period) / (Overall average)

• It is easier to analyze data and detect new trends if the data are first adjusted to remove the seasonal patterns.

Seasonally adjusted data = (Actual call volume) / (Seasonal factor)


Calculation of Seasonal Factors for CCW

QuarterThree-Year

AverageSeasonalFactor

1 7,019 7,019 / 7,529 = 0.93

2 6,784 6,784 / 7,529 = 0.90

3 7,434 7,434 / 7,529 = 0.99

4 8,880 8,880 / 7,529 = 1.18

Total = 30,117

Average = 30,117 / 4 = 7,529


Excel Template for Calculating Seasonal Factors

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Estimating Seasonal Factors for CCW

True Year Quarter Value Type of Seasonality

1 1 6,809 Quarterly1 2 6,4651 3 6,5691 4 8,266 Estimate for2 1 7,257 Quarter Seasonal Factor2 2 7,064 1 0.93232 3 7,784 2 0.90102 4 8,724 3 0.98733 1 6,992 4 1.17943 2 6,8223 3 7,9493 4 9,650


Seasonally Adjusted Time Series for CCW

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Seasonally Adjusted Time Series for CCW

Seasonal Actual Seasonally AdjustedYear Quarter Factor Call Volume Call Volume

1 1 0.93 6,809 7,3221 2 0.90 6,465 7,1831 3 0.99 6,569 6,6351 4 1.18 8,266 7,0052 1 0.93 7,257 7,8032 2 0.90 7,064 7,8492 3 0.99 7,784 7,8632 4 1.18 8,724 7,3933 1 0.93 6,992 7,5183 2 0.90 6,822 7,5803 3 0.99 7,949 8,0293 4 1.18 9,650 8,178


Outline for Forecasting Call Volume

1. Select a time-series forecasting method.

2. Apply this method to the seasonally adjusted time series to obtain a forecast of the seasonally adjusted call volume for the next time period.

3. Multiply this forecast by the corresponding seasonal factor to obtain a forecast of the actual call volume (without seasonal adjustment).


The Moving-Average Forecasting Method

• The moving-average forecasting method averages the data for only the most recent time periods.

n = Number of recent periods to consider as relevant for forecasting

Forecast = Average of last n values

• The moving-average forecasting method is a good one to use when conditions don’t change much over the number of time periods included in the average.

• However, the moving-average method is slow to respond to changing conditions. It places the same weight on each of the last n values even though the older values may be less representative of current conditions than the last value observed.


The Moving-Average Method Applied to CCW

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Moving Average Forecasting Method with Seasonality for CCW

Seasonally Seasonally True Adjusted Adjusted Actual Forecasting Number of previous

Year Quarter Value Value Forecast Forecast Error periods to consider1 1 6,809 7,322 n = 41 2 6,465 7,1831 3 6,569 6,635 Type of Seasonality1 4 8,266 7,005 Quarterly2 1 7,257 7,803 7,036 6,544 7132 2 7,064 7,849 7,157 6,441 623 Quarter Seasonal Factor2 3 7,784 7,863 7,323 7,250 534 1 0.932 4 8,724 7,393 7,630 9,003 279 2 0.903 1 6,992 7,518 7,727 7,186 194 3 0.993 2 6,822 7,580 7,656 6,890 68 4 1.183 3 7,949 8,029 7,589 7,513 4363 4 9,650 8,178 7,630 9,004 6464 1 7,826 7,2794 24 34 45 15 25 35 4 Mean Absolute Deviation6 1 MAD = 4376 26 3 Mean Square Error6 4 MSE = 238,816


The Exponential Smoothing Forecasting Method

• The exponential smoothing forecasting method places the greatest weight on the last value in the time series and then progressively smaller weights on the older values.

Forecast = (Last value) + (1 – ) (Last forecast)

is the smoothing constant between 0 and 1.

• This method places a weight of a on the last value, (1–) on the next-to-last value, (1–)2 on the next prior value, etc.

– For example, when = 0.5, the method places a weight of 0.5 on the last value, 0.25 on the next-to-last, 0.125 on the next prior, etc.

– A larger value of places more emphasis on the more recent values, a smaller value places more emphasis on the older values.

• The choice of the value of the smoothing constant a has a substantial effect on the forecast.

– A small value (say, = 0.1) is appropriate if conditions are relatively stable.

– A larger value (say, = 0.5) is appropriate if significant changes occur frequently.


The Exponential Smoothing Method Applied to CCW

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Exponential-Smoothing Forecasting Method with Seasonality for CCW

Seasonally Seasonally True Adjusted Adjusted Actual Forecasting Smoothing Constant

Year Quarter Value Value Forecast Forecast Error 0.51 1 6,809 7,322 7,500 6,975 1661 2 6,465 7,183 7,411 6,670 205 Initial Estimate1 3 6,569 6,635 7,297 7,224 655 Average = 7,5001 4 8,266 7,005 6,966 8,220 462 1 7,257 7,803 6,986 6,497 760 Type of Seasonality2 2 7,064 7,849 7,394 6,655 409 Quarterly2 3 7,784 7,863 7,622 7,545 2392 4 8,724 7,393 7,742 9,136 412 Quarter Seasonal Factor3 1 6,992 7,518 7,568 7,038 46 1 0.933 2 6,822 7,580 7,543 6,789 33 2 0.903 3 7,949 8,029 7,561 7,486 463 3 0.993 4 9,650 8,178 7,795 9,199 451 4 1.184 1 7,987 7,4284 24 34 45 15 25 35 46 16 2 Mean Absolute Deviation6 3 MAD = 3246 47 1 Mean Square Error

MSE = 157,836


A Time Series with Trend(Population of a State over Time)

1995 2000 2005 Year

Population(Millions)

4.8

5.0

5.2

5.4

Trendline


Exponential Smoothing with Trend Forecasting Method

• The exponential smoothing with trend forecasting method uses the recent values in the time series to estimate any current upward or downward trend in these values.

Trend = Average change from one time-series value to the next

• The formula for forecasting the next value in the time series adds the estimated trend.

Forecast = (Last value) + (1 – ) (Last forecast) + Estimated trend

is the smoothing constant between 0 and 1.

• Exponential smoothing also is used to obtain and update the estimated trend.

Estimated trend = (Latest trend) + (1 – ) (Last estimate of trend)

is the trend smoothing constant.

• The formula for forecasting n periods from now is

Forecast = (Last value) + (1 – ) (Last forecast) + n (Estimated trend)


Exponential Smoothing with Trend Applied to CCW

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Exponential-Smoothing with Trend Forecasting Method with Seasonality for CCW

Seasonally Seasonally True Adjusted Latest Estimated Adjusted Actual Forecasting Smoothing Constant

Year Quarter Value Value Trend Trend Forecast Forecast Error 0.31 1 6,809 7,322 0 7,500 6,975 166 0.31 2 6,465 7,183 -54 -16 7,430 6,687 2221 3 6,569 6,635 -90 -38 7,318 7,245 676 Initial Estimate1 4 8,266 7,005 -243 -100 7,013 8,276 10 Average = 7,5002 1 7,257 7,803 -102 -100 6,910 6,427 830 Trend = 02 2 7,064 7,849 167 -20 7,158 6,442 6222 3 7,784 7,863 187 42 7,407 7,333 451 Type of Seasonality2 4 8,724 7,393 179 83 7,627 9,000 276 Quarterly3 1 6,992 7,518 13 62 7,619 7,085 933 2 6,822 7,580 32 53 7,642 6,877 55 Quarter Seasonal Factor3 3 7,949 8,029 34 47 7,670 7,594 355 1 0.933 4 9,650 8,178 155 80 7,858 9,272 378 2 0.904 1 176 108 8,062 7,498 3 0.994 2 4 1.184 34 45 15 25 35 46 16 26 36 4 Mean Absolute Deviation7 1 MAD = 345

Mean Square ErrorMSE = 180,796


MAD and MSE for the Various Forecasting Method

Forecasting Method MAD MSE

CCW’s 25 percent rule 424 317,815

Last-value method 295 145,909

Averaging method 400 242,876

Moving-average method 437 238,816

Exponential smoothing 324 157,836

Exponential smoothing with trend 345 180,796


Comparison of the Forecasting Methods

• Last value method: Suitable for a time series that is so unstable that even the next-to-last value is not considered relevant for forecasting the next value.

• Averaging method: Suitable for a very stable time series where even its first few values are considered relevant for forecasting the next value.

• Moving-average method: Suitable for a moderately stable time series where the last few values are considered relevant for forecasting the next value.

• Exponential smoothing method: Suitable for a time series in the range from somewhat unstable to rather stable, where the value of the smoothing constant needs to be adjusted to fit the anticipated degree of stability.

• Exponential smoothing with trend: Suitable for a time series where the mean of the distribution tends to follow a trend either up or down, provided that changes in the trend occur only occasionally and gradually.


Causal Forecasting

Causal forecasting obtains a forecast of the quantity of interest (the dependent variable) by relating it directly to one or more other quantities (the independent variables) that drive the quantity of interest.

Type of ForecastingPossible Dependent

VariablePossible Independent

Variable

Sales Sales of a product Amount of advertising

Spare parts Demand for spare parts Usage of equipment

Economic trends Gross domestic product Various economic factors

CCW call volume Call volume Total sales

Any quantity This same quantity Time


Adding a Trendline to the Graph

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CCW's Average Daily Sales and Call Volume

Sales CallYear Quarter ($thousands) Volume

1 1 4,894 6,8091 2 4,703 6,4651 3 4,748 6,5691 4 5,844 8,2662 1 5,192 7,2572 2 5,086 7,0642 3 5,511 7,7842 4 6,107 8,7243 1 5,052 6,9923 2 4,985 6,8223 3 5,576 7,9493 4 6,647 9,650


Linear Regression

• When doing causal forecasting with a single independent variable, linear regression involves approximating the relationship between the dependent variable (call volume for CCW) and the independent variable (sales for CCW) by a straight line.

• This linear regression line is drawn on a graph with the independent variable on the horizontal axis and the dependent variable on the vertical axis. The line is constructed after plotting a number of points showing each observed value of the independent variable and the corresponding value for the dependent variable.

• The linear regression line has the formy = a + bx

wherey = Estimated value of the dependent variablea = Intercept of the linear regression line with the y-axisb = Slope of the linear regression linex = Value of the independent variable


Excel Template for Linear Regression

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Linear Regression of Call Volume vs. Sales Volume for CCW

Time Independent Dependent Estimation Square Linear Regression LinePeriod Variable Variable Estimate Error of Error y = a + bx

1 4,894 6,809 6,765 43.85 1,923 a = -1,223.862 4,703 6,465 6,453 11.64 136 b = 1.633 4,748 6,569 6,527 42.18 1,7804 5,844 8,266 8,316 49.93 2,4935 5,192 7,257 7,252 5.40 29 Estimator6 5,086 7,064 7,079 14.57 212 If x = 5,0007 5,511 7,784 7,772 11.66 1368 6,107 8,724 8,745 21.26 452 then y= 6,938.189 5,052 6,992 7,023 31.07 96510 4,985 6,822 6,914 91.70 8,40811 5,576 7,949 7,878 70.55 4,97712 6,647 9,650 9,627 23.24 540


Forecasting in Practice

• A survey of forecasting practices at 500 U.S. corporations indicates that judgmental forecasting methods are somewhat more widely used than statistical methods.

• Among judgmental methods, the most popular is a jury of executive opinion. When forecasting sales, manager’s opinion is a close second.

• Statistical forecasting methods also are fairly widely used, especially in companies with high sales.

• Among statistical methods, the moving-average method and linear regression are the most widely used. Both exponential smoothing and the last-value method also receive considerable use.


The Forecasting Method Used in Actual Applications

Organization Quantity Being Forecasted Forecasting Method

Merit Brass Co. Sales of finished goods Exponential smoothing

Hidroelétrica Español Energy demand ARIMA (Box-Jenkins), etc.

American AirlinesDemand for different fare classes

Exponential smoothing

American AirlinesNeed for spare parts to repair airplanes

Causal forecasting with linear regression

Albuquerque Microelectronics

Production yield in wafer fabrication

Exponential smoothing with trend

U.S. Department of LaborUnemployment insurance payments

Causal forecasting with linear regression

United AirlinesDemand at reservations offices and airports

ARIMA (Box-Jenkins)

Taco Bell Number of customer arrivals Moving average

L.L. Bean Staffing needs at call center ARIMA (Box-Jenkins)

All references available for download at www.mhhe.com/hillier2e/articles

mcgraw-hill/irwin & modified by jim grayson for quan 6610 © the mcgraw-hill companies, inc.,...

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