1. 4 - 1© 2011 Pearson Education, Inc. publishing as Prentice Hall 4 4 Forecasting PowerPoint presentation to accompany Heizer and Render Operations Management, 10e Principles of Operations Management, 8e PowerPoint slides by Jeff Heyl

2. 4 - 2© 2011 Pearson Education, Inc. publishing as Prentice Hall What is Forecasting?  Process of predicting a future event  Underlying basis of all business decisions  Production  Inventory  Personnel  Facilities ??

3. 4 - 3© 2011 Pearson Education, Inc. publishing as Prentice Hall  Short-range forecast  Up to 1 year, generally less than 3 months  Purchasing, job scheduling, workforce levels, job assignments, production levels  Medium-range forecast  3 months to 3 years  Sales and production planning, budgeting  Long-range forecast  3 + years  New product planning, facility location, research and development Forecasting Time Horizons

4. 4 - 4© 2011 Pearson Education, Inc. publishing as Prentice Hall Distinguishing Differences  Medium/long range  Medium/long range forecasts deal with more comprehensive issues and support management decisions regarding planning and products, plants and processes  Short-term  Short-term forecasting usually employs different methodologies than longer-term forecasting  Short-term  Short-term forecasts tend to be more accurate than longer-term forecasts

5. 4 - 5© 2011 Pearson Education, Inc. publishing as Prentice Hall Influence of Product Life Cycle  Introduction and growth require longer forecasts than maturity and decline  As product passes through life cycle, forecasts are useful in projecting  Staffing levels  Inventory levels  Factory capacity Introduction – Growth – Maturity – Decline

6. 4 - 6© 2011 Pearson Education, Inc. publishing as Prentice Hall Product Life Cycle Best period to increase market share R&D engineering is critical Practical to change price or quality image Strengthen niche Poor time to change image, price, or quality Competitive costs become critical Defend market position Cost control critical IntroductionGrowthMaturityDecline Company Strategy/Issues Figure 2.5 Internet search engines Sales Drive-through restaurants CD-ROMs Analog TVs iPods Boeing 787 LCD & plasma TVs Twitter Avatars Xbox 360

7. 4 - 7© 2011 Pearson Education, Inc. publishing as Prentice Hall Product Life Cycle Product design and development critical Frequent product and process design changes Short production runs High production costs Limited models Attention to quality IntroductionGrowthMaturityDecline OM Strategy/Issues Forecasting critical Product and process reliability Competitive product improvements and options Increase capacity Shift toward product focus Enhance distribution Standardization Fewer product changes, more minor changes Optimum capacity Increasing stability of process Long production runs Product improvement and cost cutting Little product differentiation Cost minimization Overcapacity in the industry Prune line to eliminate items not returning good margin Reduce capacity Figure 2.5

8. 4 - 8© 2011 Pearson Education, Inc. publishing as Prentice Hall Types of Forecasts  Economic forecasts  Address business cycle – inflation rate, money supply, housing starts, etc.  Technological forecasts  Predict rate of technological progress  Impacts development of new products  Demand forecasts  Predict sales of existing products and services

9. 4 - 9© 2011 Pearson Education, Inc. publishing as Prentice Hall Strategic Importance of Forecasting  Human Resources – Hiring, training, laying off workers  Capacity – Capacity shortages can result in undependable delivery, loss of customers, loss of market share  Supply Chain Management – Good supplier relations and price advantages

10. 4 - 10© 2011 Pearson Education, Inc. publishing as Prentice Hall Forecasting Approaches  Used when situation is vague and little data exist  New products  New technology  Involves intuition, experience  e.g., forecasting sales on Internet Qualitative Methods

11. 4 - 11© 2011 Pearson Education, Inc. publishing as Prentice Hall Forecasting Approaches  Used when situation is ‘stable’ and historical data exist  Existing products  Current technology  Involves mathematical techniques  e.g., forecasting sales of color televisions Quantitative Methods

12. 4 - 12© 2011 Pearson Education, Inc. publishing as Prentice Hall Overview of Quantitative Approaches 1.Naive approach 2.Moving averages 3.Exponential smoothing 4.Trend projection 5.Linear regression time-series models associative model

13. 4 - 13© 2011 Pearson Education, Inc. publishing as Prentice Hall  Set of evenly spaced numerical data  Obtained by observing response variable at regular time periods  Forecast based only on past values, no other variables important  Assumes that factors influencing past and present will continue influence in future Time Series Forecasting

14. 4 - 14© 2011 Pearson Education, Inc. publishing as Prentice Hall Trend Seasonal Cyclical Random Time Series Components

15. 4 - 15© 2011 Pearson Education, Inc. publishing as Prentice Hall Components of Demand Demand for product or service |||| 1234 Time (years) Average demand over 4 years Trend component Actual demand line Random variation Figure 4.1 Seasonal peaks

16. 4 - 16© 2011 Pearson Education, Inc. publishing as Prentice Hall  Persistent, overall upward or downward pattern  Changes due to population, technology, age, culture, etc.  Typically several years duration Trend Component

17. 4 - 17© 2011 Pearson Education, Inc. publishing as Prentice Hall  Regular pattern of up and down fluctuations  Due to weather, customs, etc.  Occurs within a single year Seasonal Component Number of PeriodLengthSeasons WeekDay7 MonthWeek4-4.5 MonthDay28-31 YearQuarter4 YearMonth12 YearWeek52

18. 4 - 18© 2011 Pearson Education, Inc. publishing as Prentice Hall  Repeating up and down movements  Affected by business cycle, political, and economic factors  Multiple years duration  Often causal or associative relationships Cyclical Component 05101520

19. 4 - 19© 2011 Pearson Education, Inc. publishing as Prentice Hall  Erratic, unsystematic, ‘residual’ fluctuations  Due to random variation or unforeseen events  Short duration and nonrepeating Random Component MTWTFMTWTF

20. 4 - 20© 2011 Pearson Education, Inc. publishing as Prentice Hall Naive Approach  Assumes demand in next period is the same as demand in most recent period  e.g., If January sales were 68, then February sales will be 68  Sometimes cost effective and efficient  Can be good starting point

21. 4 - 21© 2011 Pearson Education, Inc. publishing as Prentice Hall  MA is a series of arithmetic means  Used if little or no trend  Used often for smoothing  Provides overall impression of data over time Moving Average Method Moving average = ∑ demand in previous n periods n

22. 4 - 22© 2011 Pearson Education, Inc. publishing as Prentice Hall January10 February12 March13 April16 May19 June23 July26 Actual3-Month MonthShed SalesMoving Average (12 + 13 + 16)/3 = 13 2 / 3 (13 + 16 + 19)/3 = 16 (16 + 19 + 23)/3 = 19 1 / 3 Moving Average Example 101213 101213 (10 + 12 + 13)/3 = 11 2 / 3

23. 4 - 23© 2011 Pearson Education, Inc. publishing as Prentice Hall Graph of Moving Average ||||||||||||JFMAMJJASOND||||||||||||JFMAMJJASOND Shed Sales 30 – 28 – 26 – 24 – 22 – 20 – 18 – 16 – 14 – 12 – 10 – Actual Sales Moving Average Forecast

24. 4 - 24© 2011 Pearson Education, Inc. publishing as Prentice Hall  Used when some trend might be present  Older data usually less important  Weights based on experience and intuition Weighted Moving Average Weighted moving average = ∑ (weight for period n) x (demand in period n) ∑ weights

25. 4 - 25© 2011 Pearson Education, Inc. publishing as Prentice Hall January10 February12 March13 April16 May19 June23 July26 Actual3-Month Weighted MonthShed SalesMoving Average [(3 x 16) + (2 x 13) + (12)]/6 = 14 1 / 3 [(3 x 19) + (2 x 16) + (13)]/6 = 17 [(3 x 23) + (2 x 19) + (16)]/6 = 20 1 / 2 Weighted Moving Average 101213 131210 [(3 x 13) + (2 x 12) + (10)]/6 = 12 1 / 6 Weights AppliedPeriod 3 3Last month 2 2Two months ago 1 1Three months ago 6Sum of weights

26. 4 - 26© 2011 Pearson Education, Inc. publishing as Prentice Hall  Increasing n smooths the forecast but makes it less sensitive to changes  Do not forecast trends well  Require extensive historical data Potential Problems With Moving Average

27. 4 - 27© 2011 Pearson Education, Inc. publishing as Prentice Hall Moving Average And Weighted Moving Average 30 – 25 – 20 – 15 – 10 – 5 – Sales demand ||||||||||||JFMAMJJASOND||||||||||||JFMAMJJASOND Actual sales Moving average Weighted moving average Figure 4.2

28. 4 - 28© 2011 Pearson Education, Inc. publishing as Prentice Hall  Form of weighted moving average  Weights decline exponentially  Most recent data weighted most  Requires smoothing constant (  )  Ranges from 0 to 1  Subjectively chosen  Involves little record keeping of past data Exponential Smoothing

29. 4 - 29© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing New forecast =Last period’s forecast +  (Last period’s actual demand – Last period’s forecast) F t = F t – 1 +  (A t – 1 - F t – 1 ) whereF t =new forecast F t – 1 =previous forecast  =smoothing (or weighting) constant (0 ≤  ≤ 1)

30. 4 - 30© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20

31. 4 - 31© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20 New forecast= 142 +.2(153 – 142)

32. 4 - 32© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  =.20 New forecast= 142 +.2(153 – 142) = 142 + 2.2 = 144.2 ≈ 144 cars

33. 4 - 33© 2011 Pearson Education, Inc. publishing as Prentice Hall Effect of Smoothing Constants Weight Assigned to Most2nd Most3rd Most4th Most5th Most RecentRecentRecentRecentRecent SmoothingPeriodPeriodPeriodPeriodPeriod Constant(  )  (1 -  )  (1 -  ) 2  (1 -  ) 3  (1 -  ) 4  =.1.1.09.081.073.066  =.5.5.25.125.063.031

34. 4 - 34© 2011 Pearson Education, Inc. publishing as Prentice Hall Impact of Different  225 – 200 – 175 – 150 – |||||||||123456789|||||||||123456789 Quarter Demand  =.1 Actual demand  =.5

35. 4 - 35© 2011 Pearson Education, Inc. publishing as Prentice Hall Impact of Different  225 – 200 – 175 – 150 – |||||||||123456789|||||||||123456789 Quarter Demand  =.1 Actual demand  =.5  Chose high values of  when underlying average is likely to change  Choose low values of  when underlying average is stable

36. 4 - 36© 2011 Pearson Education, Inc. publishing as Prentice Hall Choosing  The objective is to obtain the most accurate forecast no matter the technique We generally do this by selecting the model that gives us the lowest forecast error Forecast error= Actual demand - Forecast value = A t - F t

37. 4 - 37© 2011 Pearson Education, Inc. publishing as Prentice Hall Common Measures of Error Mean Absolute Deviation (MAD) MAD = ∑ |Actual - Forecast| n Mean Squared Error (MSE) MSE = ∑ (Forecast Errors) 2 n

38. 4 - 38© 2011 Pearson Education, Inc. publishing as Prentice Hall Common Measures of Error Mean Absolute Percent Error (MAPE) MAPE = ∑ 100|Actual i - Forecast i |/Actual i n i = 1

39. 4 - 39© 2011 Pearson Education, Inc. publishing as Prentice Hall Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62

40. 4 - 40© 2011 Pearson Education, Inc. publishing as Prentice Hall Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62 MAD = ∑ |deviations| n = 82.45/8 = 10.31 For  =.10 = 98.62/8 = 12.33 For  =.50

41. 4 - 41© 2011 Pearson Education, Inc. publishing as Prentice Hall Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62 MAD10.3112.33 = 1,526.54/8 = 190.82 For  =.10 = 1,561.91/8 = 195.24 For  =.50 MSE = ∑ (forecast errors) 2 n

42. 4 - 42© 2011 Pearson Education, Inc. publishing as Prentice Hall Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62 MAD10.3112.33 MSE190.82195.24 = 44.75/8 = 5.59% For  =.10 = 54.05/8 = 6.76% For  =.50 MAPE = ∑ 100|deviation i |/actual i n i = 1

43. 4 - 43© 2011 Pearson Education, Inc. publishing as Prentice Hall Comparison of Forecast Error RoundedAbsoluteRoundedAbsolute ActualForecastDeviationForecastDeviation Tonnagewithforwithfor QuarterUnloaded  =.10  =.10  =.50  =.50 11801755.001755.00 2168175.57.50177.509.50 3159174.7515.75172.7513.75 4175173.181.82165.889.12 5190173.3616.64170.4419.56 6205175.0229.98180.2224.78 7180178.021.98192.6112.61 8182178.223.78186.304.30 82.4598.62 MAD10.3112.33 MSE190.82195.24 MAPE5.59%6.76%

44. 4 - 44© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment When a trend is present, exponential smoothing must be modified Forecast including (FIT t ) = trendExponentially smoothed (F t ) +smoothed(T t ) forecasttrend

45. 4 - 45© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment F t =  (A t - 1 ) + (1 -  )(F t - 1 + T t - 1 ) T t =  (F t - F t - 1 ) + (1 -  )T t - 1 Step 1: Compute F t Step 2: Compute T t Step 3: Calculate the forecast FIT t = F t + T t

46. 4 - 46© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 217 320 419 524 621 731 828 936 10 Table 4.1

47. 4 - 47© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 217 320 419 524 621 731 828 936 10 Table 4.1 F 2 =  A 1 + (1 -  )(F 1 + T 1 ) F 2 = (.2)(12) + (1 -.2)(11 + 2) = 2.4 + 10.4 = 12.8 units Step 1: Forecast for Month 2

48. 4 - 48© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 21712.80 320 419 524 621 731 828 936 10 Table 4.1 T 2 =  (F 2 - F 1 ) + (1 -  )T 1 T 2 = (.4)(12.8 - 11) + (1 -.4)(2) =.72 + 1.2 = 1.92 units Step 2: Trend for Month 2

49. 4 - 49© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 21712.801.92 320 419 524 621 731 828 936 10 Table 4.1 FIT 2 = F 2 + T 2 FIT 2 = 12.8 + 1.92 = 14.72 units Step 3: Calculate FIT for Month 2

50. 4 - 50© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Forecast ActualSmoothedSmoothedIncluding Month(t)Demand (A t )Forecast, F t Trend, T t Trend, FIT t 11211213.00 21712.801.9214.72 320 419 524 621 731 828 936 10 Table 4.1 15.182.1017.28 17.822.3220.14 19.912.2322.14 22.512.3824.89 24.112.0726.18 27.142.4529.59 29.282.3231.60 32.482.6835.16

51. 4 - 51© 2011 Pearson Education, Inc. publishing as Prentice Hall Exponential Smoothing with Trend Adjustment Example Figure 4.3 |||||||||123456789|||||||||123456789 Time (month) Product demand 35 – 30 – 25 – 20 – 15 – 10 – 5 – 0 – Actual demand (A t ) Forecast including trend (FIT t ) with  =.2 and  =.4

52. 4 - 52© 2011 Pearson Education, Inc. publishing as Prentice Hall Trend Projections Fitting a trend line to historical data points to project into the medium to long-range Linear trends can be found using the least squares technique y = a + bx ^ where y= computed value of the variable to be predicted (dependent variable) a= y-axis intercept b= slope of the regression line x= the independent variable ^

53. 4 - 53© 2011 Pearson Education, Inc. publishing as Prentice Hall Least Squares Method Time period Values of Dependent Variable Figure 4.4 Deviation 1 (error) Deviation 5 Deviation 7 Deviation 2 Deviation 6 Deviation 4 Deviation 3 Actual observation (y-value) Trend line, y = a + bx ^

54. 4 - 54© 2011 Pearson Education, Inc. publishing as Prentice Hall Least Squares Method Time period Values of Dependent Variable Figure 4.4 Deviation 1 (error) Deviation 5 Deviation 7 Deviation 2 Deviation 6 Deviation 4 Deviation 3 Actual observation (y-value) Trend line, y = a + bx ^ Least squares method minimizes the sum of the squared errors (deviations)

55. 4 - 55© 2011 Pearson Education, Inc. publishing as Prentice Hall Least Squares Method Equations to calculate the regression variables b =  xy - nxy  x 2 - nx 2 y = a + bx ^ a = y - bx

56. 4 - 56© 2011 Pearson Education, Inc. publishing as Prentice Hall Least Squares Example b = = = 10.54 ∑xy - nxy ∑x 2 - nx 2 3,063 - (7)(4)(98.86) 140 - (7)(4 2 ) a = y - bx = 98.86 - 10.54(4) = 56.70 TimeElectrical Power YearPeriod (x)Demandx 2 xy 2003174174 20042794158 20053809240 200649016360 2007510525525 2008614236852 2009712249854 ∑x = 28∑y = 692∑x 2 = 140∑xy = 3,063 x = 4y = 98.86

57. 4 - 57© 2011 Pearson Education, Inc. publishing as Prentice Hall b = = = 10.54 ∑xy - nxy ∑x 2 - nx 2 3,063 - (7)(4)(98.86) 140 - (7)(4 2 ) a = y - bx = 98.86 - 10.54(4) = 56.70 TimeElectrical Power YearPeriod (x)Demandx 2 xy 2003174174 20042794158 20053809240 200649016360 2007510525525 2008614236852 2009712249854 ∑x = 28∑y = 692∑x 2 = 140∑xy = 3,063 x = 4y = 98.86 Least Squares Example The trend line is y = 56.70 + 10.54x ^

58. 4 - 58© 2011 Pearson Education, Inc. publishing as Prentice Hall Least Squares Example ||||||||| 200320042005200620072008200920102011 160 – 150 – 140 – 130 – 120 – 110 – 100 – 90 – 80 – 70 – 60 – 50 – Year Power demand Trend line, y = 56.70 + 10.54x ^

59. 4 - 59© 2011 Pearson Education, Inc. publishing as Prentice Hall Least Squares Requirements 1.We always plot the data to insure a linear relationship 2.We do not predict time periods far beyond the database 3.Deviations around the least squares line are assumed to be random

60. 4 - 60© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Variations In Data The multiplicative seasonal model can adjust trend data for seasonal variations in demand

61. 4 - 61© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Variations In Data 1.Find average historical demand for each season 2.Compute the average demand over all seasons 3.Compute a seasonal index for each season 4.Estimate next year’s total demand 5.Divide this estimate of total demand by the number of seasons, then multiply it by the seasonal index for that season Steps in the process:

62. 4 - 62© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Index Example Jan80851059094 Feb7085858094 Mar8093828594 Apr909511510094 May11312513112394 Jun11011512011594 Jul10010211310594 Aug8810211010094 Sept8590959094 Oct7778858094 Nov7572838094 Dec8278808094 DemandAverageAverage Seasonal Month2007200820092007-2009MonthlyIndex

63. 4 - 63© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Index Example Jan80851059094 Feb7085858094 Mar8093828594 Apr909511510094 May11312513112394 Jun11011512011594 Jul10010211310594 Aug8810211010094 Sept8590959094 Oct7778858094 Nov7572838094 Dec8278808094 DemandAverageAverage Seasonal Month2007200820092007-2009MonthlyIndex 0.957 Seasonal index = Average 2007-2009 monthly demand Average monthly demand = 90/94 =.957

64. 4 - 64© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Index Example Jan808510590940.957 Feb70858580940.851 Mar80938285940.904 Apr9095115100941.064 May113125131123941.309 Jun110115120115941.223 Jul100102113105941.117 Aug88102110100941.064 Sept85909590940.957 Oct77788580940.851 Nov75728380940.851 Dec82788080940.851 DemandAverageAverage Seasonal Month2007200820092007-2009MonthlyIndex

65. 4 - 65© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Index Example Jan808510590940.957 Feb70858580940.851 Mar80938285940.904 Apr9095115100941.064 May113125131123941.309 Jun110115120115941.223 Jul100102113105941.117 Aug88102110100941.064 Sept85909590940.957 Oct77788580940.851 Nov75728380940.851 Dec82788080940.851 DemandAverageAverage Seasonal Month2007200820092007-2009MonthlyIndex Expected annual demand = 1,200 Janx.957 = 96 1,200 12 Febx.851 = 85 1,200 12 Forecast for 2010

66. 4 - 66© 2011 Pearson Education, Inc. publishing as Prentice Hall Seasonal Index Example 140 – 130 – 120 – 110 – 100 – 90 – 80 – 70 – ||||||||||||JFMAMJJASOND||||||||||||JFMAMJJASOND Time Demand 2010 Forecast 2009 Demand 2008 Demand 2007 Demand

67. 4 - 67© 2011 Pearson Education, Inc. publishing as Prentice Hall San Diego Hospital 10,200 – 10,000 – 9,800 – 9,600 – 9,400 – 9,200 – 9,000 – |||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNovDec 676869707172737475767778 Month Inpatient Days 9530 9551 9573 9594 9616 9637 9659 9680 9702 9724 9745 9766 Figure 4.6 Trend Data

68. 4 - 68© 2011 Pearson Education, Inc. publishing as Prentice Hall San Diego Hospital 1.06 – 1.04 – 1.02 – 1.00 – 0.98 – 0.96 – 0.94 – 0.92 – |||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNovDec 676869707172737475767778 Month Index for Inpatient Days 1.04 1.02 1.01 0.99 1.03 1.04 1.00 0.98 0.97 0.99 0.97 0.96 Figure 4.7 Seasonal Indices

69. 4 - 69© 2011 Pearson Education, Inc. publishing as Prentice Hall San Diego Hospital 10,200 – 10,000 – 9,800 – 9,600 – 9,400 – 9,200 – 9,000 – |||||||||||| JanFebMarAprMayJuneJulyAugSeptOctNovDec 676869707172737475767778 Month Inpatient Days Figure 4.8 9911 9265 9764 9520 9691 9411 9949 9724 9542 9355 10068 9572 Combined Trend and Seasonal Forecast

70. 4 - 70© 2011 Pearson Education, Inc. publishing as Prentice Hall Associative Forecasting Used when changes in one or more independent variables can be used to predict the changes in the dependent variable Most common technique is linear regression analysis We apply this technique just as we did in the time series example

71. 4 - 71© 2011 Pearson Education, Inc. publishing as Prentice Hall Associative Forecasting Forecasting an outcome based on predictor variables using the least squares technique y = a + bx ^ where y= computed value of the variable to be predicted (dependent variable) a= y-axis intercept b= slope of the regression line x= the independent variable though to predict the value of the dependent variable ^

72. 4 - 72© 2011 Pearson Education, Inc. publishing as Prentice Hall Associative Forecasting Example SalesArea Payroll ($ millions), y($ billions), x 2.01 3.03 2.54 2.02 2.01 3.57 4.0 – 3.0 – 2.0 – 1.0 – |||||||01234567|||||||01234567 Sales Area payroll

73. 4 - 73© 2011 Pearson Education, Inc. publishing as Prentice Hall Associative Forecasting Example Sales, y Payroll, xx 2 xy 2.0112.0 3.0399.0 2.541610.0 2.0244.0 2.0112.0 3.574924.5 ∑y = 15.0∑x = 18∑x 2 = 80∑xy = 51.5 x = ∑x/6 = 18/6 = 3 y = ∑y/6 = 15/6 = 2.5 b = = =.25 ∑xy - nxy ∑x 2 - nx 2 51.5 - (6)(3)(2.5) 80 - (6)(3 2 ) a = y - bx = 2.5 - (.25)(3) = 1.75

74. 4 - 74© 2011 Pearson Education, Inc. publishing as Prentice Hall Associative Forecasting Example y = 1.75 +.25x ^ Sales = 1.75 +.25(payroll) If payroll next year is estimated to be $6 billion, then: Sales = 1.75 +.25(6) Sales = $3,250,000 4.0 – 3.0 – 2.0 – 1.0 – |||||||01234567|||||||01234567 Nodel’s sales Area payroll 3.25

75. 4 - 75© 2011 Pearson Education, Inc. publishing as Prentice Hall Standard Error of the Estimate  A forecast is just a point estimate of a future value  This point is actually the mean of a probability distribution Figure 4.9 4.0 – 3.0 – 2.0 – 1.0 – |||||||01234567|||||||01234567 Nodel’s sales Area payroll 3.25

76. 4 - 76© 2011 Pearson Education, Inc. publishing as Prentice Hall Standard Error of the Estimate wherey=y-value of each data point y c =computed value of the dependent variable, from the regression equation n=number of data points S y,x = ∑(y - y c ) 2 n - 2

77. 4 - 77© 2011 Pearson Education, Inc. publishing as Prentice Hall Standard Error of the Estimate Computationally, this equation is considerably easier to use We use the standard error to set up prediction intervals around the point estimate S y,x = ∑y 2 - a∑y - b∑xy n - 2

78. 4 - 78© 2011 Pearson Education, Inc. publishing as Prentice Hall Standard Error of the Estimate 4.0 – 3.0 – 2.0 – 1.0 – |||||||01234567|||||||01234567 Nodel’s sales Area payroll 3.25 S y,x = = ∑y 2 - a∑y - b∑xy n - 2 39.5 - 1.75(15) -.25(51.5) 6 - 2 S y,x =.306 The standard error of the estimate is $306,000 in sales

79. 4 - 79© 2011 Pearson Education, Inc. publishing as Prentice Hall  How strong is the linear relationship between the variables?  Correlation does not necessarily imply causality!  Coefficient of correlation, r, measures degree of association  Values range from -1 to +1 Correlation

80. 4 - 80© 2011 Pearson Education, Inc. publishing as Prentice Hall Correlation Coefficient r = n  xy -  x  y [n  x 2 - (  x) 2 ][n  y 2 - (  y) 2 ]

81. 4 - 81© 2011 Pearson Education, Inc. publishing as Prentice Hall Correlation Coefficient r = n  xy -  x  y [n  x 2 - (  x) 2 ][n  y 2 - (  y) 2 ] y x (a)Perfect positive correlation: r = +1 y x (b)Positive correlation: 0 < r < 1 y x (c)No correlation: r = 0 y x (d)Perfect negative correlation: r = -1

82. 4 - 82© 2011 Pearson Education, Inc. publishing as Prentice Hall  Coefficient of Determination, r 2, measures the percent of change in y predicted by the change in x  Values range from 0 to 1  Easy to interpret Correlation For the Nodel Construction example: r =.901 r 2 =.81

83. 4 - 83© 2011 Pearson Education, Inc. publishing as Prentice Hall Multiple Regression Analysis If more than one independent variable is to be used in the model, linear regression can be extended to multiple regression to accommodate several independent variables y = a + b 1 x 1 + b 2 x 2 … ^ Computationally, this is quite complex and generally done on the computer

84. 4 - 84© 2011 Pearson Education, Inc. publishing as Prentice Hall Multiple Regression Analysis y = 1.80 +.30x 1 - 5.0x 2 ^ In the Nodel example, including interest rates in the model gives the new equation: An improved correlation coefficient of r =.96 means this model does a better job of predicting the change in construction sales Sales = 1.80 +.30(6) - 5.0(.12) = 3.00 Sales = $3,000,000

85. 4 - 85© 2011 Pearson Education, Inc. publishing as Prentice Hall  Measures how well the forecast is predicting actual values  Ratio of cumulative forecast errors to mean absolute deviation (MAD)  Good tracking signal has low values  If forecasts are continually high or low, the forecast has a bias error Monitoring and Controlling Forecasts Tracking Signal

86. 4 - 86© 2011 Pearson Education, Inc. publishing as Prentice Hall Monitoring and Controlling Forecasts Tracking signal Cumulative error MAD = Tracking signal = ∑(Actual demand in period i - Forecast demand in period i)  ∑|Actual - Forecast|/n)

87. 4 - 87© 2011 Pearson Education, Inc. publishing as Prentice Hall Tracking Signal Tracking signal + 0 MADs – Upper control limit Lower control limit Time Signal exceeding limit Acceptable range

88. 4 - 88© 2011 Pearson Education, Inc. publishing as Prentice Hall Tracking Signal Example CumulativeAbsolute ActualForecastCummForecastForecast QtrDemandDemandErrorErrorErrorErrorMAD 190100-10-10101010.0 295100-5-155157.5 3115100+150153010.0 4100110-10-10104010.0 5125110+15+5155511.0 6140110+30+35308514.2

89. 4 - 89© 2011 Pearson Education, Inc. publishing as Prentice Hall CumulativeAbsolute ActualForecastCummForecastForecast QtrDemandDemandErrorErrorErrorErrorMAD 190100-10-10101010.0 295100-5-155157.5 3115100+150153010.0 4100110-10-10104010.0 5125110+15+5155511.0 6140110+30+35308514.2 Tracking Signal Example Tracking Signal (Cumm Error/MAD) -10/10 = -1 -15/7.5 = -2 0/10 = 0 -10/10 = -1 +5/11 = +0.5 +35/14.2 = +2.5 The variation of the tracking signal between -2.0 and +2.5 is within acceptable limits

90. 4 - 90© 2011 Pearson Education, Inc. publishing as Prentice Hall Adaptive Forecasting  It’s possible to use the computer to continually monitor forecast error and adjust the values of the  and  coefficients used in exponential smoothing to continually minimize forecast error  This technique is called adaptive smoothing