2. Forecasting The Process of Predicting the Future presented by Your Local Engineering Management Office (LEMO) My concern is the future since I plan to spend the rest of my life there. "Those who have knowledge, don't predict. Those who predict, don't have knowledge. " --Lao Tzu, 6th Century BC Chinese Poet Wise words from a long time ago.

3. Applications sales strategic planning financial investments inventory levels production levels work force sizing energy requirements economic planning –unemployment –housing starts –inflation rates

4. Typical Forecasts Product Sales Replacement part demands Lead-times Machine break rates Expenditures Market share Unit costs Labor rates

5. Forecasts are used in manufacturing engineering in: –Inventory models –Machine loading –Production planning models –MRP systems –Manufacturing simulations A good forecast model is –accurate –computationally efficient –robust (to changes in patterns)

6. Laws of Forecasting First Law: Forecasts are always wrong! Second Law: Forecasts always change! Third Law: The further into the future, the less reliable the forecast! Fourth Law: A good forecast includes a measure of error! Fifth Law: Aggregate forecasts are more accurate! Sixth Law: Forecasts should not replace known values!

7. Accuracy of forecasts depends on Accuracy of data sample size stability of the random process –variability –stationary vs non-stationary process length of forecasting period method used model selected

8. Forecasting Methods Qualitative (subjective) –historical analogy –market research customer surveys –Expert opinion –Delphi technique –sales force composites Quantitative Models –regression analysis (causal models) –time-series models moving averages exponential smoothing Box-Jenkins auto-regressive

9. Time Horizon Short Term –Sales, shift schedules, material and part requirements, equipment failures –Days and weeks Intermediate –Product sales, labor requirements, resources –Weeks and months Long-term –Capacity requirements, long-term sales patterns, growth trends, resource and labor costs –Months and years

10. Time Series Time Series: random variable indexed on time e.g. D t = demands during month t D 1, D 2, …, D t,… form a time series Basic Premise: Can predict the future from the past - the underlying process will continue as it has in the (recent) past. Forecast: where a i is the weight placed on the i th observation

11. Elements of Time Series Data Trend (G t ) –Constant (stationary) (G t = b) –linear (constant) (G t = bt) –quadratic (accelerated) (G t = bt 2 ) –exponential (growth) (G t = b t ) Seasonal (S t ) Cyclical (C t ) Randomness (Noise) (e t ) –no recognizable pattern

12. Trends time population

13. Seasonal Snow blower sales spring summer fall winter trend present

14. Cycles Unemployment Rate 1990199119921993199419951996

15. Cycles long swings away from trend due to factors other than seasonality –generally occurs over a number of years difficult to model –not as stable –rarely repeats at fixed intervals –amplitude varies –need several years of data (complete cycles) to distinguish from trends causes of cycles include –psychological forces (fashions, music, food) –population demographics (college enrollments) –institutional (public policy, business practices, tax policies) –replacement cycles (technology changes, obsolescence) –education

16. The Road Ahead Stationary (constant) process –moving averages –exponential smoothing Trend only process –linear regression –Holt’s method (double exponential smoothing) Seasonal process –seasonal factors (stationary process) –Winter’s method (trend process) but first a detour…

17. Evaluating Forecasts Forecast Error e t = F t - D t errors Bad forecast Wrong again (measures bias)

18. Forecast Error Example e = F - D

19. Forecast Model Model historical dataforecast error Observed value Multiplicative Model F t = Trend x seasonal x cyclical x irregular = aG t S t C t e t Additive Model F t = Trend + seasonal + cyclical + irregular = a + G t + S t + C t + e t

20. Notation Given D 1, D 2, …, D t demands have been observed and assuming an additive linear model: F t,t+  = forecast made at time t for period t+  let F t = F t-1,t D 1, D 2, …, D t are observed values of demand during periods 1, 2, …, t. At time t, we have observed D t, D t-1, …

21. 2. Average forecast: 3. Moving averages: F t = (D t-1 + D t-2 + … + D t-N ) / N and F t-1,t+  = F t 1. Last data point (LDP) Forecast:F t = D t-1 Potential Forecasts

22. Time Series - Moving Averages no trends/no cycles/no seasonal effects Model: D t =  +  t Underlying constant of the process E[  t ] = 0 and Var[  t ] =  2 Let’s hear it for the moving average model!

23. Moving Averages Lag behind Trends

24. (Simple) Exponential Smoothing Why F t is just the weighted sum of the current observation and the previous estimate. previous forecast error from previous forecast

25. More (Simple) Exponential Smoothing continuing: Note that the weights sum to one:

26. Moving Averages versus Exponential Smoothing Average age of data: moving average = (1/N) (1 + 2 + 3 + … + N) = (1/N) N (N+1)/2 = (N + 1)/2 exp smooth = equating ages: Example if N = 10, then  =.18182 if  =.1, then N = 19

27. A Little Math Trick

28. More Moving Averages versus Exponential Smoothing age of data 123456123456 exponential smoothing with  =.3 moving average with n = 6 weight placed on i th value

29. Considerations in the selection of the smoothing constant If  is small – response to change will be slow If  is large – response to change will be fast Normally.1 <  <.3  Average age of the data:  Set:  = 2/(n+1) to correspond to n-period moving average  Minimize forecast error (MAD, MSE, RMSE, etc.) Remember to go to Excel!

30. Moving Averages versus Exponential Smoothing – A Comparison Similarities Assume stationary process (with adjustment of shifts in the mean) Single parameter model (N and α) Lags behind trend data Similar levels of accuracy Differences Smoothing uses all past data, MA uses only the last N values Need to save N data points for MA MA weighs each observation by N while smoothing weights the N th observation by  (1-  ) N-1

31. Trend Based Methods The Journey Continues… "Wall Street indices predicted nine out of the last five recessions ! " --Paul A. Samuelson in Newsweek, Science and Stocks, 19 Sep. 1966.

32. Trend Data x x x x x x Model: D t =  + Bt +  t E[  t ] = 0 and Var[  t ] =  2 time demands

33. Regression Analysis y t = A + Bt + e t with t = 1, 2, …, n F t+k = a +b (t + k) where a & b are Least Square estimates

34. Least-Squares Estimates and D i is the demand for time period i, i = 1,2, …, n

35. The Necessary Example QuarterIndexEngine Demands (failures) 1/20071 20 2/20072 25 3/20073 22 4/20074 28 1/20085 30 2/20086 32 3/20087 33 4/20088 31 GE's F110 engine family provides the most reliable power for the F-16C/D fighter aircraft. With a reputation for stall-free operation, the F110 continues to be the choice for F-16 operators and has been selected for twin-engine F-15 application. Go to Excel…

36. A Second Example e.g. F 6 = 88.4 + 87.8 (6) = 615.2

37. Trend Data - continued Double Exponential Smoothing (Holt’s method) Model: D t =  + Bt +  t Value of the series (intercept) Value of the trend (slope)

38. A Holt’s Method Example Set α =  =.1; S 0 = 200; G 0 = 10 S 1 =(.1)(200) + (.9)(200+10)=209.0 G 1 = (.1)(209 - 200) + (.9)(10) = 9.9 S 2 =(.1)(250) + (.9)(209+9.9)= 222.0 G 2 = (.1)(222 - 209) + (.9)(9.9) = 10.2 S 3 =(.1)(175) + (.9)(222+10.2)=226.5 G 3 = (.1)(226.5 - 222) + (.9)(10.2) = 9.6 F 3,4 = 226.5 + 9.6 = 236.1 and F 3,5 = 226.5 +(2) 9.6 = 245.7 D t 200 250 175 186 225 285 305 190

39. Forecasting with Seasonal Effects Experience the ups and downs of the four seasons "I always avoid prophesying beforehand because it is much better to prophesy after the event has already taken place. " --Winston Churchill

40. Seasonal Data with no trend Model:D t = c k  +  t, 1  k  N where  = average (annual) demand c k = seasonal factor for period k  t = random component N = number of periods in a season The road ahead is no longer straight

41. Seasonal Factors for a Series with no Trend 1.Compute the sample mean of the data 2.Divide each observation by the sample mean 3.Average the factors for like periods within each season. 4.Result are N seasonal factors I can do that.

42. Seasonal Indices – a real nice example Divide by the Mean = 16.425 Example 2.6 Cars on a toll bridge Data is in 1,000 forecasts 16.05 12.15 13.78 17.1 23.05 x 16.425

43. Another example Qtr200120022003avgindex 1124.5157.4144.1142.00.691 2181.0192.3178.4183.90.896 3287.1281.8251.5273.41.332 4240.1217.1208.6221.91.081 205.31 Data are quarterly sales in 1,000 of gallons and are normally distributed with a mean of 200 and a std. dev. of 20

44. De-seasonalizing Our Example QtrRawDe-seasonnal 20011124.5180.0 2181.0202.1 3287.1215.5 4240.1222.1 20021157.4227.6 2192.3214.7 3281.8211.6 4217.1200.8 20031144.1208.4 2178.4199.1 3251.5188.8 4208.6193.0 Qtrindex 10.691 20.896 31.332 41.081 for example: 124.5/.691 = 180.0 181.0/.896 = 202.1

45. Applying a MA Forecasting Model Qtrsales 1-qtr 2-qtr 3-qtr 4-qtr 5-qtr 6-qtr 7-qtr

46. The Forecast for the next year 1-qtr 2-qtr 3-qtr 4-qtr 5-qtr 6-qtr 7-qtr

47. Seasonal Data with Trend Model:D t = (  + Gt)c t +  t where  = the base constant at t = 0 G = slope of trend component c t = seasonal factor for period t  t = random component The road ahead not only curves but also climbs.

48. Another better way when trend is present… Could you review with us the 8 easy steps to applying the moving average method?

49. The 8 easy steps… 1. Obtain moving averages where N = length of season 2. Average and center adjacent values 3. Divide results of #2. into D t 4. For each season, compute the “average” (i.e. the mean or median) 5. Adjust so sum is N by multiply each average by N / Total 6. Deseasonalize series by dividing each D t by corresponding seasonal index 7. Forecast deseasonalized series using appropriate model 8. Apply corresponding seasonal indices to “reseasonalize” the series He is right. There are 8 steps.

50. Example of the first 6 easy steps…

51. Steps 1and 2 of the first 6 easy steps…

52. More Steps 1 and 2 of the first 6 easy steps…

53. Step 3 of the first 6 easy steps… 26 / 18.50 = 1.405 17 / 19.13 =.889

54. Step 4 of the first 6 easy steps… (.600 +.516 +.478)/3 =.531

55. Another Step 4 of the first 6 easy steps… (1.082+ 1.143 + 1.143)/3 = 1.123

56. Step 5 of the first 6 easy steps… (4/ 3.995).531 =.532

57. More Step 5 of the first 6 easy steps… (4/ 3.995) 1.123 = 1.124

58. Step 6 of the first 6 easy steps… 12 /.532 = 22.6

59. More Step 6 of the first 6 easy steps… 23 / 1.124 = 20.5

60. Forecastperiodindices #7 - Deseason Forecast Y5 Qtr 1170.53228.0 Y5 Qtr 2181.12428.7 Y5 Qtr 3191.37429.3 Y5 Qtr 4200.97029.9 Now easy step 7 … y = 0.6402x + 17.137 R 2 = 0.8444 7. Forecast using deseasonalized values.6402 (18) + 17.137 = 28.7

61. Forecastperiodindices #7 - Deseason Forecast #8 - season adjusted Y5 Qtr 1170.53228.014.9 Y5 Qtr 2181.12428.732.2 Y5 Qtr 3191.37429.340.3 Y5 Qtr 4200.97029.929.0 Finally easy step 8… y = 0.6402x + 17.137 8. Seasonalize forecast.532 x 28.0 = 14.9 1.124 x 28.7 = 32.2

62. Now Begins Winter’s Model LetD t = demand in period t N = the number of periods (length of season) S t = estimate of deseasonalized series in period t G t = estimate of trend term in period t c t = estimate of seasonal component for period t Yikes, this model has it all!

63. The smoothing equations: Series Trend Seasonal Forecast

64. Step 1: Calculate the average of each of the seasons: V 1 = Davg 1 ; V 2 = Davg 2 …, V m = Davg m Step 2: Set G 0 = (V m – V 1 ) / [(m-1)N] (initial slope estimate) Step 3: Calculate S 0 = Davg m + G 0 (N-1)/2 (value of series at t = 0) a. Step 4: Calculate the seasonal factors: V i = average of season i, j = period of season Could you review with us the 6 easy steps to applying Winter’s Model?

65. Step 4 Explained a. Step 4: Calculate the seasonal factors: V i = average of season i, j = period of season mean of the i th year initial slope estimate for N = 4 (quarters) j=1,2,3,4 series + trend

66. Step 4 b. Average the seasonal factors Step 4 c. Normalize the seasonal factors

67. Step 5 : Forecast for period t+  F t+  = (S t +  G t ) c t+  Step 6: Next period, update model parameters with new data point.

68. Example 2.8 G 0 = (V m – V 1 ) / [(m-1)N] S 0 = Davg 2 + G 0 (N-1)/2

69. Sales in 100's alpha=.2 beta=.1 gamma=.1 before yr3 observed DtFtStGtCt forecast yr 3 Qtr 11614.1924.580.9400.595 14.190 yr 3 Qtr 23328.2926.411.0291.131 27.504 yr 3 Qtr 33437.9026.910.9761.350 35.476 yr 3 Qtr 42625.5928.010.98830.915 24.376 More of Example 2.8 F t+ 1 = (S t + G t ) c  ; t = 0 F t+ 1 = (S t +  G t ) c t+  ; t = 0,  = 1,2,3,4

70. Year 4 forecast Sales in 100's alpha=.2 beta=.1gamma=.1Normalize DtFtStGtCt yr 3 yr 3 Qtr 11614.1924.580.9400.5950.5964 yr 3 Qtr 23328.2926.411.0291.1311.1334 yr 3 Qtr 33437.9026.910.9761.3501.3533 yr 3 Qtr 42625.5928.0150.98830.9150.9170 Yr 4 Qtr 1117.303.9904.000 Yr 4 Qtr 2233.99 Yr 4 Qtr 3 3 41.92 Yr 4 Qtr 4 4 29.31 F t+ 1 = [28.015 +.9883].5964 = 17.30 F t+ 2 = [28.015 + 2(.9883)] 1.1334 = 33.99 F t+ 2 = [28.015 + 3(.9883)] 1.3533 = 41.92 F t+ 2 = [28.015 + 4(.9883)].9170 = 29.31

71. Bonus Topic Casual Regression

72. Regression Model (Causal) Model:D t = a + bx t + e t where D t = (demand) forecast for period t x t = independent variable for period t e t = random noise a,b slope, intercept to be estimated using least-squares Two requirements are necessary to use this model: 1. A causal relationship exists between x and D 2. The value for x can be determined prior to period t (a time lag exists)

73. Least-Squares Formulae Why not show the class how it works in Excel?

74. Example – casual model

75. The Regression Model

76. A Multiple Regression Example Y t = drywall sales in month t x t-1 = housing starts in month t-1 x t-2 = housing starts in month t-2 x t-3 = housing starts in month t-3 linear trend

77. Demand  Sales Demands for items that are not in stock will result in backorders or lost sales Lost sales data is usually not available Assume true monthly demand is normal with a mean of 100 and a standard deviation of 30 –If 110 items are stock each month –Then Pr{Demands > 110} .37 What about lost Sales?

78. Can we have some homework problems? Chapter 2: 12, 13,16-22, 24, 28- 30, 33 – 36. "The future isn't what it used to be !" -- anonymous