Operations Management Session 3 –  Forecasting
Learning Objectives When you complete this chapter you should be able to : <ul><li>Understand the three time horizons and ...
Learning Objectives When you complete this chapter you should be able to : <ul><li>Compute three measures of forecast accu...
Forecasting at Disney World <ul><li>Global portfolio includes parks in Hong Kong, Paris, Tokyo, Orlando, and Anaheim </li>...
Forecasting at Disney World <ul><li>Disney generates daily, weekly, monthly, annual, and 5-year forecasts </li></ul><ul><l...
Forecasting at Disney World <ul><li>20% of customers come from outside the USA </li></ul><ul><li>Economic model includes g...
Forecasting at Disney World <ul><li>Inputs to the forecasting model include airline specials, Federal Reserve policies, Wa...
What is Forecasting? <ul><li>Process of predicting a future event </li></ul><ul><li>Underlying basis of  all business deci...
<ul><li>Short-range forecast </li></ul><ul><ul><li>Up to 1 year, generally less than 3 months </li></ul></ul><ul><ul><li>P...
Distinguishing Differences <ul><li>Medium/long range  forecasts deal with more comprehensive issues and support management...
Types of Forecasts <ul><li>Economic forecasts </li></ul><ul><ul><li>Address business cycle – inflation rate, money supply,...
Seven Steps in Forecasting <ul><li>Determine the use of the forecast </li></ul><ul><li>Select the items to be forecasted <...
The Realities! <ul><li>Forecasts are seldom perfect </li></ul><ul><li>Most techniques assume an underlying stability in th...
Forecasting Approaches <ul><li>Used when situation is vague and little data exist </li></ul><ul><ul><li>New products </li>...
Forecasting Approaches <ul><li>Used when situation is ‘stable’ and historical data exist </li></ul><ul><ul><li>Existing pr...
Overview of Qualitative Methods <ul><li>Jury of executive opinion </li></ul><ul><ul><li>Pool opinions of high-level expert...
Overview of Qualitative Methods <ul><li>Sales force composite </li></ul><ul><ul><li>Estimates from individual salespersons...
Jury of Executive Opinion <ul><li>Involves small group of high-level experts and managers </li></ul><ul><li>Group estimate...
Sales Force Composite <ul><li>Each salesperson projects his or her sales </li></ul><ul><li>Combined at district and nation...
Delphi Method <ul><li>Iterative group process, continues until consensus is reached </li></ul><ul><li>3 types of participa...
Consumer Market Survey <ul><li>Ask customers about purchasing plans </li></ul><ul><li>What consumers say, and what they ac...
Overview of Quantitative Approaches <ul><li>Naive approach </li></ul><ul><li>Moving averages </li></ul><ul><li>Exponential...
<ul><li>Set of evenly spaced numerical data </li></ul><ul><ul><li>Obtained by observing response variable at regular time ...
Time Series Components Trend Seasonal Cyclical Random
Components of Demand Figure 4.1 Demand for product or service | | | | 1 2 3 4 Year Average demand over four years Seasonal...
<ul><li>Persistent, overall upward or downward pattern </li></ul><ul><li>Changes due to population, technology, age, cultu...
<ul><li>Regular pattern of up and down fluctuations </li></ul><ul><li>Due to weather, customs, etc. </li></ul><ul><li>Occu...
<ul><li>Repeating up and down movements </li></ul><ul><li>Affected by business cycle, political, and economic factors </li...
<ul><li>Erratic, unsystematic, ‘residual’ fluctuations </li></ul><ul><li>Due to random variation or unforeseen events </li...
Naive Approach <ul><li>Assumes demand in next  period is the same as  demand in most recent period </li></ul><ul><ul><li>e...
<ul><li>MA is a series of arithmetic means  </li></ul><ul><li>Used if little or no trend </li></ul><ul><li>Used often for ...
Moving Average Example (12 + 13 + 16)/3 = 13  2 / 3 (13 + 16 + 19)/3 = 16 (16 + 19 + 23)/3 = 19  1 / 3 January 10 February...
Graph of Moving Average | | | | | | | | | | | | J F M A M J J A S O N D Shed Sales 30  – 28  – 26  – 24  – 22  – 20  – 18 ...
<ul><li>Used when trend is present  </li></ul><ul><ul><li>Older data usually less important </li></ul></ul><ul><li>Weights...
Weighted Moving 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)...
<ul><li>Increasing n smooths the forecast but makes it less sensitive to changes </li></ul><ul><li>Do not forecast trends ...
Moving Average And  Weighted Moving Average Figure 4.2 30  – 25  – 20  – 15  – 10  – 5  – Sales demand | | | | | | | | | |...
<ul><li>Form of weighted moving average </li></ul><ul><ul><li>Weights decline exponentially </li></ul></ul><ul><ul><li>Mos...
Exponential Smoothing New forecast = Last period’s forecast +    (Last period’s actual demand  –  Last period’s forecast)...
Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant    = .20
Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant    = .20 New fo...
Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant    = .20 New fo...
Impact of Different   225  – 200  – 175  – 150  – | | | | | | | | | 1 2 3 4 5 6 7 8 9 Quarter Demand    = .1 Actual dema...
Impact of Different   225  – 200  – 175  – 150  – | | | | | | | | | 1 2 3 4 5 6 7 8 9 Quarter Demand    = .1 Actual dema...
Choosing   The objective is to obtain the most accurate forecast no matter the technique We generally do this by selectin...
Common Measures of Error Mean Absolute Deviation (MAD) MAD = ∑   |Actual - Forecast| n Mean Squared Error (MSE) MSE = ∑   ...
Common Measures of Error Mean Absolute Percent Error (MAPE) MAPE = ∑ 100|Actual i  - Forecast i |/Actual i n n i = 1
Comparison of Forecast Error  Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with ...
Comparison of Forecast Error  Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with ...
Comparison of Forecast Error  Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with ...
Comparison of Forecast Error  Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with ...
Comparison of Forecast Error  Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with ...
Exponential Smoothing with Trend Adjustment When a trend is present, exponential smoothing must be modified Forecast  incl...
Exponential Smoothing with Trend Adjustment F t  =   (A t - 1 ) + (1 -   )(F t - 1  + T t - 1 ) T t  =   (F t  - F t - ...
Exponential Smoothing with Trend Adjustment Example Table 4.1 Forecast Actual Smoothed Smoothed Including Month(t) Demand ...
Exponential Smoothing with Trend Adjustment Example Table 4.1 F 2   =   A 1  + (1 -   )(F 1  + T 1 ) F 2   = (.2)(12) + ...
Exponential Smoothing with Trend Adjustment Example Table 4.1 T 2   =   (F 2  - F 1 ) + (1 -   )T 1 T 2   = (.4)(12.8 - ...
Exponential Smoothing with Trend Adjustment Example Table 4.1 FIT 2   = F 2  + T 1 FIT 2   = 12.8 + 1.92 = 14.72 units Ste...
Exponential Smoothing with Trend Adjustment Example Table 4.1 15.18 2.10 17.28 17.82 2.32 20.14 19.91 2.23 22.14 22.51 2.3...
Exponential Smoothing with Trend Adjustment Example Figure 4.3 | | | | | | | | | 1 2 3 4 5 6 7 8 9 Time (month) Product de...
Trend Projections Fitting a trend line to historical data points to project into the medium to long-range Linear trends ca...
Least Squares Method Figure 4.4 Time period Values of Dependent Variable Deviation 1 (error) Deviation 5 Deviation 7 Devia...
Least Squares Method Figure 4.4 Least squares method minimizes the sum of the squared errors (deviations) Time period Valu...
Least Squares Method Equations to calculate the regression variables b =  xy - nxy  x 2  - nx 2 y = a + bx ^ a = y - bx
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 - ...
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 - ...
Least Squares Example | | | | | | | | | 2001 2002 2003 2004 2005 2006 2007 2008 2009 160  – 150  – 140  – 130  – 120  – 11...
Least Squares Requirements <ul><li>We always plot the data to insure a linear relationship </li></ul><ul><li>We do not pre...
Seasonal Variations In Data The multiplicative seasonal model can adjust trend data for seasonal variations in demand
Seasonal Variations In Data <ul><li>Find average historical demand for each season  </li></ul><ul><li>Compute the average ...
Seasonal Index Example Jan 80 85 105 90 94 Feb 70 85 85 80 94 Mar 80 93 82 85 94 Apr 90 95 115 100 94 May 113 125 131 123 ...
Seasonal Index Example 0.957 Jan 80 85 105 90 94 Feb 70 85 85 80 94 Mar 80 93 82 85 94 Apr 90 95 115 100 94 May 113 125 13...
Seasonal Index Example Jan 80 85 105 90 94 0.957 Feb 70 85 85 80 94 0.851 Mar 80 93 82 85 94 0.904 Apr 90 95 115 100 94 1....
Seasonal Index Example Expected annual demand = 1,200 Jan 80 85 105 90 94 0.957 Feb 70 85 85 80 94 0.851 Mar 80 93 82 85 9...
Seasonal Index Example 140  – 130  – 120  – 110  – 100  – 90  – 80  – 70  – | | | | | | | | | | | | J F M A M J J A S O N ...
San Diego Hospital Figure 4.6 Trend Data 10,200  – 10,000  – 9,800  – 9,600  – 9,400  – 9,200  – 9,000  – | | | | | | | | ...
San Diego Hospital Figure 4.7 Seasonal Indices 1.06  – 1.04  – 1.02  – 1.00  – 0.98  – 0.96  – 0.94  – 0.92 – | | | | | | ...
San Diego Hospital Figure 4.8 Combined Trend and Seasonal Forecast 10,200  – 10,000  – 9,800  – 9,600  – 9,400  – 9,200  –...
Associative Forecasting Used when changes in one or more independent variables can be used to predict the changes in the d...
Associative Forecasting Forecasting an outcome based on predictor variables using the least squares technique y = a + bx ^...
Associative Forecasting Example Sales Local Payroll ($ millions), y ($ billions), x 2.0 1 3.0 3 2.5 4 2.0 2 2.0 1 3.5 7 4....
Associative Forecasting Example Sales, y  Payroll, x x 2 xy 2.0 1 1 2.0 3.0 3 9 9.0 2.5 4 16 10.0 2.0 2 4 4.0 2.0 1 1 2.0 ...
Associative Forecasting Example Sales = 1.75 + .25(payroll) If payroll next year is estimated to be $6 billion, then: Sale...
Standard Error of the Estimate <ul><li>A forecast is just a point estimate of a future value </li></ul><ul><li>This point ...
Standard Error of the Estimate where y = y-value of each data point y c = computed value of the dependent variable, from t...
Standard Error of the Estimate Computationally, this equation is considerably easier to use We use the standard error to s...
Standard Error of the Estimate S y,x  =  .306 The standard error of the estimate is $306,000 in sales 4.0  – 3.0  – 2.0  –...
<ul><li>How strong is the linear relationship between the variables? </li></ul><ul><li>Correlation does not necessarily im...
Correlation Coefficient r =  n  xy -   x  y  [n  x 2  - (  x) 2 ][n  y 2  - (  y) 2 ]
Correlation Coefficient r =  n  xy -   x  y  [n  x 2  - (  x) 2 ][n  y 2  - (  y) 2 ] y x (a) Perfect positive corr...
<ul><li>Coefficient of Determination, r 2 , measures the percent of change in y predicted by the change in x </li></ul><ul...
Multiple Regression Analysis If more than one independent variable is to be used in the model, linear regression can be ex...
Multiple Regression Analysis In the Nodel example, including interest rates in the model gives the new equation: An improv...
<ul><li>Measures how well the forecast is predicting actual values </li></ul><ul><li>Ratio of running sum of forecast erro...
Monitoring and Controlling Forecasts Tracking signal RSFE MAD = Tracking signal = ∑ (Actual demand in  period i -  Forecas...
Tracking Signal Tracking signal + 0 MADs – Upper control limit Lower control limit Time Signal exceeding limit Acceptable ...
Tracking Signal Example Cumulative Absolute Absolute Actual Forecast Forecast Forecast Qtr Demand Demand Error RSFE Error ...
Tracking Signal Example The variation of the tracking signal between -2.0 and +2.5 is within acceptable limits Cumulative ...
Adaptive Forecasting It’s possible to use the computer to continually monitor forecast error and adjust the values of the ...
Focus Forecasting Developed at American Hardware Supply, focus forecasting is based on two principles: <ul><li>Sophisticat...
Forecasting in the Service Sector <ul><li>Presents unusual challenges </li></ul><ul><ul><li>Special need for short term re...
Fast Food Restaurant Forecast Figure 4.12 20%  – 15%  – 10%  – 5%  – 11-12 1-2 3-4 5-6 7-8 9-10 12-1 2-3 4-5 6-7 8-9 10-11...
FedEx Call Center Forecast Figure 4.12 12%  – 10%  – 8%  – 6%  – 4%   – 2%  – 0%  – Hour of day A.M. P.M. 2 4 6 8 10 12 2 ...
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Session 3

  1. 1. Operations Management Session 3 – Forecasting
  2. 2. Learning Objectives When you complete this chapter you should be able to : <ul><li>Understand the three time horizons and which models apply for each use </li></ul><ul><li>Explain when to use each of the four qualitative models </li></ul><ul><li>Apply the naive, moving average, exponential smoothing, and trend methods </li></ul>
  3. 3. Learning Objectives When you complete this chapter you should be able to : <ul><li>Compute three measures of forecast accuracy </li></ul><ul><li>Develop seasonal indexes </li></ul><ul><li>Conduct a regression and correlation analysis </li></ul><ul><li>Use a tracking signal </li></ul>
  4. 4. Forecasting at Disney World <ul><li>Global portfolio includes parks in Hong Kong, Paris, Tokyo, Orlando, and Anaheim </li></ul><ul><li>Revenues are derived from people – how many visitors and how they spend their money </li></ul><ul><li>Daily management report contains only the forecast and actual attendance at each park </li></ul>
  5. 5. Forecasting at Disney World <ul><li>Disney generates daily, weekly, monthly, annual, and 5-year forecasts </li></ul><ul><li>Forecast used by labor management, maintenance, operations, finance, and park scheduling </li></ul><ul><li>Forecast used to adjust opening times, rides, shows, staffing levels, and guests admitted </li></ul>
  6. 6. Forecasting at Disney World <ul><li>20% of customers come from outside the USA </li></ul><ul><li>Economic model includes gross domestic product, cross-exchange rates, arrivals into the USA </li></ul><ul><li>A staff of 35 analysts and 70 field people survey 1 million park guests, employees, and travel professionals each year </li></ul>
  7. 7. Forecasting at Disney World <ul><li>Inputs to the forecasting model include airline specials, Federal Reserve policies, Wall Street trends, vacation/holiday schedules for 3,000 school districts around the world </li></ul><ul><li>Average forecast error for the 5-year forecast is 5% </li></ul><ul><li>Average forecast error for annual forecasts is between 0% and 3% </li></ul>
  8. 8. What is Forecasting? <ul><li>Process of predicting a future event </li></ul><ul><li>Underlying basis of all business decisions </li></ul><ul><ul><li>Production </li></ul></ul><ul><ul><li>Inventory </li></ul></ul><ul><ul><li>Personnel </li></ul></ul><ul><ul><li>Facilities </li></ul></ul>??
  9. 9. <ul><li>Short-range forecast </li></ul><ul><ul><li>Up to 1 year, generally less than 3 months </li></ul></ul><ul><ul><li>Purchasing, job scheduling, workforce levels, job assignments, production levels </li></ul></ul><ul><li>Medium-range forecast </li></ul><ul><ul><li>3 months to 3 years </li></ul></ul><ul><ul><li>Sales and production planning, budgeting </li></ul></ul><ul><li>Long-range forecast </li></ul><ul><ul><li>3 + years </li></ul></ul><ul><ul><li>New product planning, facility location, research and development </li></ul></ul>Forecasting Time Horizons
  10. 10. Distinguishing Differences <ul><li>Medium/long range forecasts deal with more comprehensive issues and support management decisions regarding planning and products, plants and processes </li></ul><ul><li>Short-term forecasting usually employs different methodologies than longer-term forecasting </li></ul><ul><li>Short-term forecasts tend to be more accurate than longer-term forecasts </li></ul>
  11. 11. Types of Forecasts <ul><li>Economic forecasts </li></ul><ul><ul><li>Address business cycle – inflation rate, money supply, housing starts, etc. </li></ul></ul><ul><li>Technological forecasts </li></ul><ul><ul><li>Predict rate of technological progress </li></ul></ul><ul><ul><li>Impacts development of new products </li></ul></ul><ul><li>Demand forecasts </li></ul><ul><ul><li>Predict sales of existing products and services </li></ul></ul>
  12. 12. Seven Steps in Forecasting <ul><li>Determine the use of the forecast </li></ul><ul><li>Select the items to be forecasted </li></ul><ul><li>Determine the time horizon of the forecast </li></ul><ul><li>Select the forecasting model(s) </li></ul><ul><li>Gather the data </li></ul><ul><li>Make the forecast </li></ul><ul><li>Validate and implement results </li></ul>
  13. 13. The Realities! <ul><li>Forecasts are seldom perfect </li></ul><ul><li>Most techniques assume an underlying stability in the system </li></ul><ul><li>Product family and aggregated forecasts are more accurate than individual product forecasts </li></ul>
  14. 14. Forecasting Approaches <ul><li>Used when situation is vague and little data exist </li></ul><ul><ul><li>New products </li></ul></ul><ul><ul><li>New technology </li></ul></ul><ul><li>Involves intuition, experience </li></ul><ul><ul><li>e.g., forecasting sales on Internet </li></ul></ul>Qualitative Methods
  15. 15. Forecasting Approaches <ul><li>Used when situation is ‘stable’ and historical data exist </li></ul><ul><ul><li>Existing products </li></ul></ul><ul><ul><li>Current technology </li></ul></ul><ul><li>Involves mathematical techniques </li></ul><ul><ul><li>e.g., forecasting sales of color televisions </li></ul></ul>Quantitative Methods
  16. 16. Overview of Qualitative Methods <ul><li>Jury of executive opinion </li></ul><ul><ul><li>Pool opinions of high-level experts, sometimes augment by statistical models </li></ul></ul><ul><li>Delphi method </li></ul><ul><ul><li>Panel of experts, queried iteratively </li></ul></ul>
  17. 17. Overview of Qualitative Methods <ul><li>Sales force composite </li></ul><ul><ul><li>Estimates from individual salespersons are reviewed for reasonableness, then aggregated </li></ul></ul><ul><li>Consumer Market Survey </li></ul><ul><ul><li>Ask the customer </li></ul></ul>
  18. 18. Jury of Executive Opinion <ul><li>Involves small group of high-level experts and managers </li></ul><ul><li>Group estimates demand by working together </li></ul><ul><li>Combines managerial experience with statistical models </li></ul><ul><li>Relatively quick </li></ul><ul><li>‘ Group-think’ disadvantage </li></ul>
  19. 19. Sales Force Composite <ul><li>Each salesperson projects his or her sales </li></ul><ul><li>Combined at district and national levels </li></ul><ul><li>Sales reps know customers’ wants </li></ul><ul><li>Tends to be overly optimistic </li></ul>
  20. 20. Delphi Method <ul><li>Iterative group process, continues until consensus is reached </li></ul><ul><li>3 types of participants </li></ul><ul><ul><li>Decision makers </li></ul></ul><ul><ul><li>Staff </li></ul></ul><ul><ul><li>Respondents </li></ul></ul>Staff (Administering survey) Decision Makers (Evaluate responses and make decisions) Respondents (People who can make valuable judgments)
  21. 21. Consumer Market Survey <ul><li>Ask customers about purchasing plans </li></ul><ul><li>What consumers say, and what they actually do are often different </li></ul><ul><li>Sometimes difficult to answer </li></ul>
  22. 22. Overview of Quantitative Approaches <ul><li>Naive approach </li></ul><ul><li>Moving averages </li></ul><ul><li>Exponential smoothing </li></ul><ul><li>Trend projection </li></ul><ul><li>Linear regression </li></ul>Time-Series Models Associative Model
  23. 23. <ul><li>Set of evenly spaced numerical data </li></ul><ul><ul><li>Obtained by observing response variable at regular time periods </li></ul></ul><ul><li>Forecast based only on past values, no other variables important </li></ul><ul><ul><li>Assumes that factors influencing past and present will continue influence in future </li></ul></ul>Time Series Forecasting
  24. 24. Time Series Components Trend Seasonal Cyclical Random
  25. 25. Components of Demand Figure 4.1 Demand for product or service | | | | 1 2 3 4 Year Average demand over four years Seasonal peaks Trend component Actual demand Random variation
  26. 26. <ul><li>Persistent, overall upward or downward pattern </li></ul><ul><li>Changes due to population, technology, age, culture, etc. </li></ul><ul><li>Typically several years duration </li></ul>Trend Component
  27. 27. <ul><li>Regular pattern of up and down fluctuations </li></ul><ul><li>Due to weather, customs, etc. </li></ul><ul><li>Occurs within a single year </li></ul>Seasonal Component Number of Period Length Seasons Week Day 7 Month Week 4-4.5 Month Day 28-31 Year Quarter 4 Year Month 12 Year Week 52
  28. 28. <ul><li>Repeating up and down movements </li></ul><ul><li>Affected by business cycle, political, and economic factors </li></ul><ul><li>Multiple years duration </li></ul><ul><li>Often causal or associative relationships </li></ul>Cyclical Component 0 5 10 15 20
  29. 29. <ul><li>Erratic, unsystematic, ‘residual’ fluctuations </li></ul><ul><li>Due to random variation or unforeseen events </li></ul><ul><li>Short duration and nonrepeating </li></ul>Random Component M T W T F
  30. 30. Naive Approach <ul><li>Assumes demand in next period is the same as demand in most recent period </li></ul><ul><ul><li>e.g., If January sales were 68, then February sales will be 68 </li></ul></ul><ul><li>Sometimes cost effective and efficient </li></ul><ul><li>Can be good starting point </li></ul>
  31. 31. <ul><li>MA is a series of arithmetic means </li></ul><ul><li>Used if little or no trend </li></ul><ul><li>Used often for smoothing </li></ul><ul><ul><li>Provides overall impression of data over time </li></ul></ul>Moving Average Method Moving average = ∑ demand in previous n periods n
  32. 32. Moving Average Example (12 + 13 + 16)/3 = 13 2 / 3 (13 + 16 + 19)/3 = 16 (16 + 19 + 23)/3 = 19 1 / 3 January 10 February 12 March 13 April 16 May 19 June 23 July 26 Actual 3-Month Month Shed Sales Moving Average 10 12 13 ( 10 + 12 + 13 )/3 = 11 2 / 3
  33. 33. Graph of Moving Average | | | | | | | | | | | | J F M A M J J A S O N D Shed Sales 30 – 28 – 26 – 24 – 22 – 20 – 18 – 16 – 14 – 12 – 10 – Actual Sales Moving Average Forecast
  34. 34. <ul><li>Used when trend is present </li></ul><ul><ul><li>Older data usually less important </li></ul></ul><ul><li>Weights based on experience and intuition </li></ul>Weighted Moving Average Weighted moving average = ∑ (weight for period n) x (demand in period n) ∑ weights
  35. 35. Weighted Moving 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 January 10 February 12 March 13 April 16 May 19 June 23 July 26 Actual 3-Month Weighted Month Shed Sales Moving Average 10 12 13 [(3 x 13 ) + (2 x 12 ) + ( 10 )]/6 = 12 1 / 6 Weights Applied Period 3 Last month 2 Two months ago 1 Three months ago 6 Sum of weights
  36. 36. <ul><li>Increasing n smooths the forecast but makes it less sensitive to changes </li></ul><ul><li>Do not forecast trends well </li></ul><ul><li>Require extensive historical data </li></ul>Potential Problems With Moving Average
  37. 37. Moving Average And Weighted Moving Average Figure 4.2 30 – 25 – 20 – 15 – 10 – 5 – Sales demand | | | | | | | | | | | | J F M A M J J A S O N D Actual sales Moving average Weighted moving average
  38. 38. <ul><li>Form of weighted moving average </li></ul><ul><ul><li>Weights decline exponentially </li></ul></ul><ul><ul><li>Most recent data weighted most </li></ul></ul><ul><li>Requires smoothing constant (  ) </li></ul><ul><ul><li>Ranges from 0 to 1 </li></ul></ul><ul><ul><li>Subjectively chosen </li></ul></ul><ul><li>Involves little record keeping of past data </li></ul>Exponential Smoothing
  39. 39. 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 ) where F t = new forecast F t – 1 = previous forecast  = smoothing (or weighting) constant (0 ≤  ≤ 1)
  40. 40. Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  = .20
  41. 41. Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant  = .20 New forecast = 142 + .2(153 – 142)
  42. 42. 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
  43. 43. Impact of Different  225 – 200 – 175 – 150 – | | | | | | | | | 1 2 3 4 5 6 7 8 9 Quarter Demand  = .1 Actual demand  = .5
  44. 44. Impact of Different  225 – 200 – 175 – 150 – | | | | | | | | | 1 2 3 4 5 6 7 8 9 Quarter Demand  = .1 Actual demand  = .5 <ul><li>Chose high values of  when underlying average is likely to change </li></ul><ul><li>Choose low values of  when underlying average is stable </li></ul>
  45. 45. 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
  46. 46. Common Measures of Error Mean Absolute Deviation (MAD) MAD = ∑ |Actual - Forecast| n Mean Squared Error (MSE) MSE = ∑ (Forecast Errors) 2 n
  47. 47. Common Measures of Error Mean Absolute Percent Error (MAPE) MAPE = ∑ 100|Actual i - Forecast i |/Actual i n n i = 1
  48. 48. Comparison of Forecast Error Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded  = .10  = .10  = .50  = .50 1 180 175 5.00 175 5.00 2 168 175.5 7.50 177.50 9.50 3 159 174.75 15.75 172.75 13.75 4 175 173.18 1.82 165.88 9.12 5 190 173.36 16.64 170.44 19.56 6 205 175.02 29.98 180.22 24.78 7 180 178.02 1.98 192.61 12.61 8 182 178.22 3.78 186.30 4.30 82.45 98.62
  49. 49. Comparison of Forecast Error Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded  = .10  = .10  = .50  = .50 1 180 175 5.00 175 5.00 2 168 175.5 7.50 177.50 9.50 3 159 174.75 15.75 172.75 13.75 4 175 173.18 1.82 165.88 9.12 5 190 173.36 16.64 170.44 19.56 6 205 175.02 29.98 180.22 24.78 7 180 178.02 1.98 192.61 12.61 8 182 178.22 3.78 186.30 4.30 82.45 98.62 MAD = ∑ |deviations| n = 82.45/8 = 10.31 For  = .10 = 98.62/8 = 12.33 For  = .50
  50. 50. Comparison of Forecast Error Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded  = .10  = .10  = .50  = .50 1 180 175 5.00 175 5.00 2 168 175.5 7.50 177.50 9.50 3 159 174.75 15.75 172.75 13.75 4 175 173.18 1.82 165.88 9.12 5 190 173.36 16.64 170.44 19.56 6 205 175.02 29.98 180.22 24.78 7 180 178.02 1.98 192.61 12.61 8 182 178.22 3.78 186.30 4.30 82.45 98.62 MAD 10.31 12.33 = 1,526.54/8 = 190.82 For  = .10 = 1,561.91/8 = 195.24 For  = .50 MSE = ∑ (forecast errors) 2 n
  51. 51. Comparison of Forecast Error Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded  = .10  = .10  = .50  = .50 1 180 175 5.00 175 5.00 2 168 175.5 7.50 177.50 9.50 3 159 174.75 15.75 172.75 13.75 4 175 173.18 1.82 165.88 9.12 5 190 173.36 16.64 170.44 19.56 6 205 175.02 29.98 180.22 24.78 7 180 178.02 1.98 192.61 12.61 8 182 178.22 3.78 186.30 4.30 82.45 98.62 MAD 10.31 12.33 MSE 190.82 195.24 = 44.75/8 = 5.59% For  = .10 = 54.05/8 = 6.76% For  = .50 MAPE = ∑ 100|deviation i |/actual i n n i = 1
  52. 52. Comparison of Forecast Error Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded  = .10  = .10  = .50  = .50 1 180 175 5.00 175 5.00 2 168 175.5 7.50 177.50 9.50 3 159 174.75 15.75 172.75 13.75 4 175 173.18 1.82 165.88 9.12 5 190 173.36 16.64 170.44 19.56 6 205 175.02 29.98 180.22 24.78 7 180 178.02 1.98 192.61 12.61 8 182 178.22 3.78 186.30 4.30 82.45 98.62 MAD 10.31 12.33 MSE 190.82 195.24 MAPE 5.59% 6.76%
  53. 53. Exponential Smoothing with Trend Adjustment When a trend is present, exponential smoothing must be modified Forecast including (FIT t ) = trend Exponentially Exponentially smoothed (F t ) + (T t ) smoothed forecast trend
  54. 54. 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
  55. 55. Exponential Smoothing with Trend Adjustment Example Table 4.1 Forecast Actual Smoothed Smoothed Including Month(t) Demand (A t ) Forecast, F t Trend, T t Trend, FIT t 1 12 11 2 13.00 2 17 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10
  56. 56. Exponential Smoothing with Trend Adjustment Example 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 Forecast Actual Smoothed Smoothed Including Month(t) Demand (A t ) Forecast, F t Trend, T t Trend, FIT t 1 12 11 2 13.00 2 17 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10
  57. 57. Exponential Smoothing with Trend Adjustment Example 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 Forecast Actual Smoothed Smoothed Including Month(t) Demand (A t ) Forecast, F t Trend, T t Trend, FIT t 1 12 11 2 13.00 2 17 12.80 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10
  58. 58. Exponential Smoothing with Trend Adjustment Example Table 4.1 FIT 2 = F 2 + T 1 FIT 2 = 12.8 + 1.92 = 14.72 units Step 3: Calculate FIT for Month 2 Forecast Actual Smoothed Smoothed Including Month(t) Demand (A t ) Forecast, F t Trend, T t Trend, FIT t 1 12 11 2 13.00 2 17 12.80 1.92 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10
  59. 59. Exponential Smoothing with Trend Adjustment Example Table 4.1 15.18 2.10 17.28 17.82 2.32 20.14 19.91 2.23 22.14 22.51 2.38 24.89 24.11 2.07 26.18 27.14 2.45 29.59 29.28 2.32 31.60 32.48 2.68 35.16 Forecast Actual Smoothed Smoothed Including Month(t) Demand (A t ) Forecast, F t Trend, T t Trend, FIT t 1 12 11 2 13.00 2 17 12.80 1.92 14.72 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10
  60. 60. Exponential Smoothing with Trend Adjustment Example Figure 4.3 | | | | | | | | | 1 2 3 4 5 6 7 8 9 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
  61. 61. 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 ^
  62. 62. Least Squares Method Figure 4.4 Time period Values of Dependent Variable Deviation 1 (error) Deviation 5 Deviation 7 Deviation 2 Deviation 6 Deviation 4 Deviation 3 Actual observation (y value) Trend line, y = a + bx ^
  63. 63. Least Squares Method Figure 4.4 Least squares method minimizes the sum of the squared errors (deviations) Time period Values of Dependent Variable Deviation 1 Deviation 5 Deviation 7 Deviation 2 Deviation 6 Deviation 4 Deviation 3 Actual observation (y value) Trend line, y = a + bx ^
  64. 64. Least Squares Method Equations to calculate the regression variables b =  xy - nxy  x 2 - nx 2 y = a + bx ^ a = y - bx
  65. 65. 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 Time Electrical Power Year Period (x) Demand x 2 xy 2001 1 74 1 74 2002 2 79 4 158 2003 3 80 9 240 2004 4 90 16 360 2005 5 105 25 525 2005 6 142 36 852 2007 7 122 49 854 ∑ x = 28 ∑ y = 692 ∑ x 2 = 140 ∑ xy = 3,063 x = 4 y = 98.86
  66. 66. 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 Time Electrical Power Year Period (x) Demand x 2 xy 1999 1 74 1 74 2000 2 79 4 158 2001 3 80 9 240 2002 4 90 16 360 2003 5 105 25 525 2004 6 142 36 852 2005 7 122 49 854  x = 28  y = 692  x 2 = 140  xy = 3,063 x = 4 y = 98.86 The trend line is y = 56.70 + 10.54x ^
  67. 67. Least Squares Example | | | | | | | | | 2001 2002 2003 2004 2005 2006 2007 2008 2009 160 – 150 – 140 – 130 – 120 – 110 – 100 – 90 – 80 – 70 – 60 – 50 – Year Power demand Trend line, y = 56.70 + 10.54x ^
  68. 68. Least Squares Requirements <ul><li>We always plot the data to insure a linear relationship </li></ul><ul><li>We do not predict time periods far beyond the database </li></ul><ul><li>Deviations around the least squares line are assumed to be random </li></ul>
  69. 69. Seasonal Variations In Data The multiplicative seasonal model can adjust trend data for seasonal variations in demand
  70. 70. Seasonal Variations In Data <ul><li>Find average historical demand for each season </li></ul><ul><li>Compute the average demand over all seasons </li></ul><ul><li>Compute a seasonal index for each season </li></ul><ul><li>Estimate next year’s total demand </li></ul><ul><li>Divide this estimate of total demand by the number of seasons, then multiply it by the seasonal index for that season </li></ul>Steps in the process:
  71. 71. Seasonal Index Example Jan 80 85 105 90 94 Feb 70 85 85 80 94 Mar 80 93 82 85 94 Apr 90 95 115 100 94 May 113 125 131 123 94 Jun 110 115 120 115 94 Jul 100 102 113 105 94 Aug 88 102 110 100 94 Sept 85 90 95 90 94 Oct 77 78 85 80 94 Nov 75 72 83 80 94 Dec 82 78 80 80 94 Demand Average Average Seasonal Month 2005 2006 2007 2005-2007 Monthly Index
  72. 72. Seasonal Index Example 0.957 Jan 80 85 105 90 94 Feb 70 85 85 80 94 Mar 80 93 82 85 94 Apr 90 95 115 100 94 May 113 125 131 123 94 Jun 110 115 120 115 94 Jul 100 102 113 105 94 Aug 88 102 110 100 94 Sept 85 90 95 90 94 Oct 77 78 85 80 94 Nov 75 72 83 80 94 Dec 82 78 80 80 94 Demand Average Average Seasonal Month 2005 2006 2007 2005-2007 Monthly Index Seasonal index = average 2005-2007 monthly demand average monthly demand = 90/94 = .957
  73. 73. Seasonal Index Example Jan 80 85 105 90 94 0.957 Feb 70 85 85 80 94 0.851 Mar 80 93 82 85 94 0.904 Apr 90 95 115 100 94 1.064 May 113 125 131 123 94 1.309 Jun 110 115 120 115 94 1.223 Jul 100 102 113 105 94 1.117 Aug 88 102 110 100 94 1.064 Sept 85 90 95 90 94 0.957 Oct 77 78 85 80 94 0.851 Nov 75 72 83 80 94 0.851 Dec 82 78 80 80 94 0.851 Demand Average Average Seasonal Month 2005 2006 2007 2005-2007 Monthly Index
  74. 74. Seasonal Index Example Expected annual demand = 1,200 Jan 80 85 105 90 94 0.957 Feb 70 85 85 80 94 0.851 Mar 80 93 82 85 94 0.904 Apr 90 95 115 100 94 1.064 May 113 125 131 123 94 1.309 Jun 110 115 120 115 94 1.223 Jul 100 102 113 105 94 1.117 Aug 88 102 110 100 94 1.064 Sept 85 90 95 90 94 0.957 Oct 77 78 85 80 94 0.851 Nov 75 72 83 80 94 0.851 Dec 82 78 80 80 94 0.851 Demand Average Average Seasonal Month 2005 2006 2007 2005-2007 Monthly Index Jan x .957 = 96 1,200 12 Feb x .851 = 85 1,200 12 Forecast for 2008
  75. 75. Seasonal Index Example 140 – 130 – 120 – 110 – 100 – 90 – 80 – 70 – | | | | | | | | | | | | J F M A M J J A S O N D Time Demand 2008 Forecast 2007 Demand 2006 Demand 2005 Demand
  76. 76. San Diego Hospital Figure 4.6 Trend Data 10,200 – 10,000 – 9,800 – 9,600 – 9,400 – 9,200 – 9,000 – | | | | | | | | | | | | Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec 67 68 69 70 71 72 73 74 75 76 77 78 Month Inpatient Days 9530 9551 9573 9594 9616 9637 9659 9680 9702 9724 9745 9766
  77. 77. San Diego Hospital Figure 4.7 Seasonal Indices 1.06 – 1.04 – 1.02 – 1.00 – 0.98 – 0.96 – 0.94 – 0.92 – | | | | | | | | | | | | Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec 67 68 69 70 71 72 73 74 75 76 77 78 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
  78. 78. San Diego Hospital Figure 4.8 Combined Trend and Seasonal Forecast 10,200 – 10,000 – 9,800 – 9,600 – 9,400 – 9,200 – 9,000 – | | | | | | | | | | | | Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec 67 68 69 70 71 72 73 74 75 76 77 78 Month Inpatient Days 9911 9265 9764 9520 9691 9411 9949 9724 9542 9355 10068 9572
  79. 79. 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
  80. 80. 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 ^
  81. 81. Associative Forecasting Example Sales Local Payroll ($ millions), y ($ billions), x 2.0 1 3.0 3 2.5 4 2.0 2 2.0 1 3.5 7 4.0 – 3.0 – 2.0 – 1.0 – | | | | | | | 0 1 2 3 4 5 6 7 Sales Area payroll
  82. 82. Associative Forecasting Example Sales, y Payroll, x x 2 xy 2.0 1 1 2.0 3.0 3 9 9.0 2.5 4 16 10.0 2.0 2 4 4.0 2.0 1 1 2.0 3.5 7 49 24.5 ∑ 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
  83. 83. Associative Forecasting Example 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 – | | | | | | | 0 1 2 3 4 5 6 7 Sales Area payroll y = 1.75 + .25x ^ 3.25
  84. 84. Standard Error of the Estimate <ul><li>A forecast is just a point estimate of a future value </li></ul><ul><li>This point is actually the mean of a probability distribution </li></ul>Figure 4.9 4.0 – 3.0 – 2.0 – 1.0 – | | | | | | | 0 1 2 3 4 5 6 7 Sales Area payroll 3.25
  85. 85. Standard Error of the Estimate where y = 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
  86. 86. 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
  87. 87. Standard Error of the Estimate S y,x = .306 The standard error of the estimate is $306,000 in sales 4.0 – 3.0 – 2.0 – 1.0 – | | | | | | | 0 1 2 3 4 5 6 7 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
  88. 88. <ul><li>How strong is the linear relationship between the variables? </li></ul><ul><li>Correlation does not necessarily imply causality! </li></ul><ul><li>Coefficient of correlation, r, measures degree of association </li></ul><ul><ul><li>Values range from -1 to +1 </li></ul></ul>Correlation
  89. 89. Correlation Coefficient r = n  xy -  x  y [n  x 2 - (  x) 2 ][n  y 2 - (  y) 2 ]
  90. 90. 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
  91. 91. <ul><li>Coefficient of Determination, r 2 , measures the percent of change in y predicted by the change in x </li></ul><ul><ul><li>Values range from 0 to 1 </li></ul></ul><ul><ul><li>Easy to interpret </li></ul></ul>Correlation For the Nodel Construction example: r = .901 r 2 = .81
  92. 92. 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 Computationally, this is quite complex and generally done on the computer y = a + b 1 x 1 + b 2 x 2 … ^
  93. 93. Multiple Regression Analysis 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 y = 1.80 + .30x 1 - 5.0x 2 ^
  94. 94. <ul><li>Measures how well the forecast is predicting actual values </li></ul><ul><li>Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD) </li></ul><ul><ul><li>Good tracking signal has low values </li></ul></ul><ul><ul><li>If forecasts are continually high or low, the forecast has a bias error </li></ul></ul>Monitoring and Controlling Forecasts Tracking Signal
  95. 95. Monitoring and Controlling Forecasts Tracking signal RSFE MAD = Tracking signal = ∑ (Actual demand in period i - Forecast demand in period i)  ∑ |Actual - Forecast|/n)
  96. 96. Tracking Signal Tracking signal + 0 MADs – Upper control limit Lower control limit Time Signal exceeding limit Acceptable range
  97. 97. Tracking Signal Example Cumulative Absolute Absolute Actual Forecast Forecast Forecast Qtr Demand Demand Error RSFE Error Error MAD 1 90 100 -10 -10 10 10 10.0 2 95 100 -5 -15 5 15 7.5 3 115 100 +15 0 15 30 10.0 4 100 110 -10 -10 10 40 10.0 5 125 110 +15 +5 15 55 11.0 6 140 110 +30 +35 30 85 14.2
  98. 98. Tracking Signal Example The variation of the tracking signal between -2.0 and +2.5 is within acceptable limits Cumulative Absolute Absolute Actual Forecast Forecast Forecast Qtr Demand Demand Error RSFE Error Error MAD 1 90 100 -10 -10 10 10 10.0 2 95 100 -5 -15 5 15 7.5 3 115 100 +15 0 15 30 10.0 4 100 110 -10 -10 10 40 10.0 5 125 110 +15 +5 15 55 11.0 6 140 110 +30 +35 30 85 14.2 Tracking Signal (RSFE/MAD) -10/10 = -1 -15/7.5 = -2 0/10 = 0 -10/10 = -1 +5/11 = +0.5 +35/14.2 = +2.5
  99. 99. 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
  100. 100. Focus Forecasting Developed at American Hardware Supply, focus forecasting is based on two principles: <ul><li>Sophisticated forecasting models are not always better than simple ones </li></ul><ul><li>There is no single technique that should be used for all products or services </li></ul>This approach uses historical data to test multiple forecasting models for individual items The forecasting model with the lowest error is then used to forecast the next demand
  101. 101. Forecasting in the Service Sector <ul><li>Presents unusual challenges </li></ul><ul><ul><li>Special need for short term records </li></ul></ul><ul><ul><li>Needs differ greatly as function of industry and product </li></ul></ul><ul><ul><li>Holidays and other calendar events </li></ul></ul><ul><ul><li>Unusual events </li></ul></ul>
  102. 102. Fast Food Restaurant Forecast Figure 4.12 20% – 15% – 10% – 5% – 11-12 1-2 3-4 5-6 7-8 9-10 12-1 2-3 4-5 6-7 8-9 10-11 (Lunchtime) (Dinnertime) Hour of day Percentage of sales
  103. 103. FedEx Call Center Forecast Figure 4.12 12% – 10% – 8% – 6% – 4% – 2% – 0% – Hour of day A.M. P.M. 2 4 6 8 10 12 2 4 6 8 10 12
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