1. Chapter 11 Solved Problems 1

2. Exhibit 11.2 Example Linear and Nonlinear Trend Patterns 2

3. Exhibit 11.3 Seasonal Pattern of Home Natural Gas Usage 3

4. Exhibit Extra Trend and Business Cycle Characteristics (each data point is 1 year apart) 4

5. Exhibit 11.4 Call Center Volume Example of a time series with trend and seasonal components: 5

6. Exhibit 11.5 Chart of Call Volume 6

7. Basic Concepts in Forecasting Forecast error is the difference between the observed value of the time series and the forecast, or A t – F t. Mean Square Error (MSE) Mean Absolute Deviation Error (MAD) Mean Absolute Percentage Error (MAPE) Σ ( A t – F t ) 2 MSE = [11.1] T ׀A t – F t ׀ MAD = [11.2] T Σ׀ ( A t – F t )/A t ׀ X 100 MAPE = [11.3] T 7

8. Exhibit 11.6 Forecast Error of Example Time Series Data 8

9. 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? PeriodDemandPeriodDemand 186791 2938 388996 4891097 5921193 6941295 9

10. Based on these error metrics (MAD, MSE, MAPE), the 3-month moving average is the best method among the three. Solved Problem 10

11. Exhibit 11.7 Summary of 3-Month Moving-Average Forecasts 11

12. Exhibit 11.8 Milk-Sales Forecast Error Analysis 12

13. 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. F t+1 =  A t + (1 –  )F t = F t +  (A t – F t ) [11.5] 13

14. Exhibit 11.9 Summary of Single Exponential Smoothing Milk-Sales Forecasts with α = 0.2 14

15. Exhibit 11.10 Graph of Single Exponential Smoothing Milk-Sales Forecasts with α = 0.2 15

16. 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. Y t = 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. 16

17. Exhibit 11.11 Factory Energy Costs 17

18. Exhibit 11.12 Format Trendline Dialog Box 18

19. Exhibit 11.13 Least-Squares Regression Model for Energy Cost Forecasting 19

20. Exhibit 11.14 Gasoline Sales Data 20

21. Exhibit 11.15 Chart of Sales versus Time 21

22. Exhibit 11.16 Multiple Regression Results 22