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Chapter 11

Solved Problems

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Exhibit 11.2 Example Linear and Nonlinear Trend Patterns

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Exhibit 11.3 Seasonal Pattern of Home Natural Gas Usage

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Exhibit Extra Trend and Business Cycle Characteristics (each data point is 1 year apart)

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Exhibit 11.4

Call Center Volume

Example of a time series with trend and seasonal components:

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Exhibit 11.5 Chart of Call Volume

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Basic Concepts in Forecasting

• Forecast error is the difference between the observed value of the time series and the forecast, or At – Ft .

Mean Square Error (MSE)

Mean Absolute Deviation Error (MAD)

Mean Absolute Percentage Error (MAPE)

Σ(At – Ft )2

MSE = [11.1]T

MAD = [11.2]׀ At – Ft׀T

Σ׀(At – Ft )/At ׀ X 100MAPE = [11.3]

T

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Exhibit 11.6 Forecast Error of Example Time Series Data

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Solved Problem

Develop three-period and four-period moving-average forecasts and single exponential smoothing forecasts with a = 0.5. Compute the MAD, MAPE, and MSE for each. Which method provides a better forecast?

Period Demand Period Demand

1 86 7 91

2 93 8 93

3 88 9 96

4 89 10 97

5 92 11 93

6 94 12 95

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Based on these error metrics (MAD, MSE, MAPE), the 3-month moving average is the best method among the three.

80

82

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90

92

94

96

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1 2 3 4 5 6 7 8 9 10 11 12

Period

Moving Average Forecasts

Solved Problem

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Exhibit 11.7 Summary of 3-Month Moving-Average Forecasts

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Exhibit 11.8 Milk-Sales Forecast Error Analysis

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Single Exponential Smoothing

• Single Exponential Smoothing (SES) is a forecasting technique that uses a weighted average of past time-series values to forecast the value of the time series in the next period.

Ft+1 = At + (1 – )Ft = Ft + (At – Ft)

[11.5]

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Exhibit 11.9 Summary of Single Exponential Smoothing Milk-Sales Forecasts with α = 0.2

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Exhibit 11.10 Graph of Single Exponential Smoothing Milk-Sales Forecasts with α = 0.2

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Regression as a Forecasting Approach

• Regression analysis is a method for building a statistical model that defines a relationship between a single dependent variable and one or more independent variables, all of which are numerical.

Yt = a + bt (11.7)- Simple linear regression finds the best values of a and b using

the method of least squares.- Excel provides a very simple tool to find the best-fitting

regression model for a time series by selecting the Add Trendline option from the Chart menu.

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Exhibit 11.11 Factory Energy Costs

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Exhibit 11.12

Format Trendline Dialog Box

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Exhibit 11.13 Least-Squares Regression Model for Energy Cost Forecasting

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Exhibit 11.14 Gasoline Sales Data

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Exhibit 11.15 Chart of Sales versus Time

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Exhibit 11.16 Multiple Regression Results