Operations Management Chapter 4 - Forecasting PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 6e Operations Management, 8e © 2006 Prentice Hall, Inc.

Outline Global Company Profile: Tupperware Corporation What Is Forecasting? Forecasting Time Horizons The Influence of Product Life Cycle Types Of Forecasts

Outline – Continued The Strategic Importance Of Forecasting Human Resources Capacity Supply-Chain Management Seven Steps In The Forecasting System

Outline – Continued Forecasting Approaches Overview of Qualitative Methods Overview of Quantitative Methods

Outline – Continued Time-series Forecasting Decomposition of a Time Series Naïve Approach Moving Averages Exponential Smoothing Exponential Smoothing with Trend Adjustment Trend Projections Seasonal Variations in Data Cyclical Variations in Data

Outline – Continued Associative Forecasting Methods: Regression And Correlation Analysis Using Regression Analysis to Forecast Standard Error of the Estimate Correlation Coefficients for Regression Lines Multiple-Regression Analysis

Outline – Continued Monitoring And Controlling Forecasts Adaptive Smoothing Focus Forecasting Forecasting In The Service Sector

Learning Objectives When you complete this chapter, you should be able to : Identify or Define: Forecasting Types of forecasts Time horizons Approaches to forecasts

Learning Objectives When you complete this chapter, you should be able to : Describe or Explain: Moving averages Exponential smoothing Trend projections Regression and correlation analysis Measures of forecast accuracy

Forecasting at Tupperware Each of 50 profit centers around the world is responsible for computerized monthly, quarterly, and 12-month sales projections These projections are aggregated by region, then globally, at Tupperware’s World Headquarters Tupperware uses all techniques discussed in text

Tupperware’s Process

Three Key Factors for Tupperware The number of registered “consultants” or sales representatives The percentage of currently “active” dealers (this number changes each week and month) Sales per active dealer, on a weekly basis

Forecast by Consensus Although inputs come from sales, marketing, finance, and production, final forecasts are the consensus of all participating managers The final step is Tupperware’s version of the “jury of executive opinion”

?? What is Forecasting? Process of predicting a future event Underlying basis of all business decisions Production Inventory Personnel Facilities ??

Forecasting Time Horizons 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

Distinguishing Differences Medium/long range forecasts deal with more comprehensive issues and support management decisions regarding planning and products, plants and processes Short-term forecasting usually employs different methodologies than longer-term forecasting Short-term forecasts tend to be more accurate than longer-term forecasts

Influence of Product Life Cycle Introduction – Growth – Maturity – Decline 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

Company Strategy/Issues Drive-through restaurants Product Life Cycle Introduction Growth Maturity Decline Company Strategy/Issues 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 Internet Flat-screen monitors Sales DVD CD-ROM Drive-through restaurants Fax machines 3 1/2” Floppy disks Color printers Figure 2.5

Product Life Cycle Introduction Growth Maturity Decline OM Strategy/Issues Product design and development critical Frequent product and process design changes Short production runs High production costs Limited models Attention to quality Forecasting critical Product and process reliability Competitive product improvements and options Increase capacity Shift toward product focus Enhance distribution Standardization Less rapid 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

Types of Forecasts Economic forecasts Technological 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 product

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 advance

Seven Steps in Forecasting Determine the use of the forecast Select the items to be forecasted Determine the time horizon of the forecast Select the forecasting model(s) Gather the data Make the forecast Validate and implement results

The Realities! Forecasts are seldom perfect Most techniques assume an underlying stability in the system Product family and aggregated forecasts are more accurate than individual product forecasts

Forecasting Approaches Qualitative Methods Used when situation is vague and little data exist New products New technology Involves intuition, experience e.g., forecasting sales on Internet

Forecasting Approaches Quantitative Methods Used when situation is ‘stable’ and historical data exist Existing products Current technology Involves mathematical techniques e.g., forecasting sales of color televisions

Overview of Qualitative Methods Jury of executive opinion Pool opinions of high-level executives, sometimes augment by statistical models Delphi method Panel of experts, queried iteratively

Overview of Qualitative Methods Sales force composite Estimates from individual salespersons are reviewed for reasonableness, then aggregated Consumer Market Survey Ask the customer

Jury of Executive Opinion Involves small group of high-level managers Group estimates demand by working together Combines managerial experience with statistical models Relatively quick ‘Group-think’ disadvantage

Sales Force Composite Each salesperson projects his or her sales Combined at district and national levels Sales reps know customers’ wants Tends to be overly optimistic

Delphi Method Iterative group process, continues until consensus is reached 3 types of participants Decision makers Staff Respondents Decision Makers (Evaluate responses and make decisions) Staff (Administering survey) Respondents (People who can make valuable judgments)

Consumer Market Survey Ask customers about purchasing plans What consumers say, and what they actually do are often different Sometimes difficult to answer

Overview of Quantitative Approaches Naive approach Moving averages Exponential smoothing Trend projection Linear regression Time-Series Models Associative Model

Time Series Forecasting Set of evenly spaced numerical data Obtained by observing response variable at regular time periods Forecast based only on past values Assumes that factors influencing past and present will continue influence in future

Time Series Components Trend Cyclical Seasonal Random

Components of Demand Trend component Seasonal peaks Actual demand Demand for product or service | | | | 1 2 3 4 Year Seasonal peaks Actual demand Average demand over four years Random variation Figure 4.1

Trend Component Persistent, overall upward or downward pattern Changes due to population, technology, age, culture, etc. Typically several years duration

Seasonal Component Regular pattern of up and down fluctuations Due to weather, customs, etc. Occurs within a single year 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

Cyclical Component Repeating up and down movements Affected by business cycle, political, and economic factors Multiple years duration Often causal or associative relationships 0 5 10 15 20

Random Component Erratic, unsystematic, ‘residual’ fluctuations Due to random variation or unforeseen events Short duration and nonrepeating M T W T F

Naive Approach Assumes demand in next period is the same as demand in most recent period e.g., If May sales were 48, then June sales will be 48 Sometimes cost effective and efficient

∑ demand in previous n periods Moving Average Method 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 = ∑ demand in previous n periods n

Moving Average Example 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 (12 + 13 + 16)/3 = 13 2/3 (13 + 16 + 19)/3 = 16 (16 + 19 + 23)/3 = 19 1/3

Graph of Moving Average Moving Average Forecast | | | | | | | | | | | | 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

Weighted Moving Average Used when trend is present Older data usually less important Weights based on experience and intuition Weighted moving average = ∑ (weight for period n) x (demand in period n) ∑ weights

Weighted Moving Average Weights Applied Period 3 Last month 2 Two months ago 1 Three months ago 6 Sum of weights Weighted Moving Average January 10 February 12 March 13 April 16 May 19 June 23 July 26 Actual 3-Month Weighted Month Shed Sales Moving Average [(3 x 16) + (2 x 13) + (12)]/6 = 141/3 [(3 x 19) + (2 x 16) + (13)]/6 = 17 [(3 x 23) + (2 x 19) + (16)]/6 = 201/2 10 12 13 [(3 x 13) + (2 x 12) + (10)]/6 = 121/6

Potential Problems With Moving Average Increasing n smooths the forecast but makes it less sensitive to changes Do not forecast trends well Require extensive historical data

Moving Average And Weighted Moving Average 30 – 25 – 20 – 15 – 10 – 5 – Sales demand | | | | | | | | | | | | J F M A M J J A S O N D Actual sales Moving average Figure 4.2

Exponential Smoothing 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 New forecast = last period’s forecast + a (last period’s actual demand – last period’s forecast) Ft = Ft – 1 + a(At – 1 - Ft – 1) where Ft = new forecast Ft – 1 = previous forecast a = smoothing (or weighting) constant (0  a  1)

Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant a = .20

Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant a = .20 New forecast = 142 + .2(153 – 142)

Exponential Smoothing Example Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant a = .20 New forecast = 142 + .2(153 – 142) = 142 + 2.2 = 144.2 ≈ 144 cars

Effect of Smoothing Constants Weight Assigned to Most 2nd Most 3rd Most 4th Most 5th Most Recent Recent Recent Recent Recent Smoothing Period Period Period Period Period Constant (a) a(1 - a) a(1 - a)2 a(1 - a)3 a(1 - a)4 a = .1 .1 .09 .081 .073 .066 a = .5 .5 .25 .125 .063 .031

Impact of Different  Actual demand a = .5 a = .1 225 – 200 – 175 – 225 – 200 – 175 – 150 – | | | | | | | | | 1 2 3 4 5 6 7 8 9 Quarter Demand Actual demand a = .5 a = .1

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 = At - Ft

Common Measures of Error Mean Absolute Deviation (MAD) MAD = ∑ |actual - forecast| n Mean Squared Error (MSE) MSE = ∑ (forecast errors)2 n

Common Measures of Error Mean Absolute Percent Error (MAPE) MAPE = 100 ∑ |actuali - forecasti|/actuali n i = 1

Comparison of Forecast Error Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded a = .10 a = .10 a = .50 a = .50 1 180 175 5 175 5 2 168 176 8 178 10 3 159 175 16 173 14 4 175 173 2 166 9 5 190 173 17 170 20 6 205 175 30 180 25 7 180 178 2 193 13 8 182 178 4 186 4 84 100

Comparison of Forecast Error MAD = ∑ |deviations| n Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonage with for with for Quarter Unloaded a = .10 a = .10 a = .50 a = .50 1 180 175 5 175 5 2 168 176 8 178 10 3 159 175 16 173 14 4 175 173 2 166 9 5 190 173 17 170 20 6 205 175 30 180 25 7 180 178 2 193 13 8 182 178 4 186 4 84 100 = 84/8 = 10.50 For a = .10 = 100/8 = 12.50 For a = .50

Comparison of Forecast Error MSE = ∑ (forecast errors)2 n Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonage with for with for Quarter Unloaded a = .10 a = .10 a = .50 a = .50 1 180 175 5 175 5 2 168 176 8 178 10 3 159 175 16 173 14 4 175 173 2 166 9 5 190 173 17 170 20 6 205 175 30 180 25 7 180 178 2 193 13 8 182 178 4 186 4 84 100 MAD 10.50 12.50 = 1,558/8 = 194.75 For a = .10 = 1,612/8 = 201.50 For a = .50

Comparison of Forecast Error MAPE = 100 ∑ |deviationi|/actuali n i = 1 Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonage with for with for Quarter Unloaded a = .10 a = .10 a = .50 a = .50 1 180 175 5 175 5 2 168 176 8 178 10 3 159 175 16 173 14 4 175 173 2 166 9 5 190 173 17 170 20 6 205 175 30 180 25 7 180 178 2 193 13 8 182 178 4 186 4 84 100 MAD 10.50 12.50 MSE 194.75 201.50 = 45.62/8 = 5.70% For a = .10 = 54.8/8 = 6.85% For a = .50

Comparison of Forecast Error Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded a = .10 a = .10 a = .50 a = .50 1 180 175 5 175 5 2 168 176 8 178 10 3 159 175 16 173 14 4 175 173 2 166 9 5 190 173 17 170 20 6 205 175 30 180 25 7 180 178 2 193 13 8 182 178 4 186 4 84 100 MAD 10.50 12.50 MSE 194.75 201.50 MAPE 5.70% 6.85%

Exponential Smoothing with Trend Adjustment When a trend is present, exponential smoothing must be modified Forecast including (FITt) = trend exponentially exponentially smoothed (Ft) + (Tt) smoothed forecast trend

Exponential Smoothing with Trend Adjustment Ft = a(At - 1) + (1 - a)(Ft - 1 + Tt - 1) Tt = b(Ft - Ft - 1) + (1 - b)Tt - 1 Step 1: Compute Ft Step 2: Compute Tt Step 3: Calculate the forecast FITt = Ft + Tt

Exponential Smoothing with Trend Adjustment Example Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 1 12 11 2 13.00 2 17 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10 Table 4.1

Exponential Smoothing with Trend Adjustment Example Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 1 12 11 2 13.00 2 17 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10 Step 1: Forecast for Month 2 F2 = aA1 + (1 - a)(F1 + T1) F2 = (.2)(12) + (1 - .2)(11 + 2) = 2.4 + 10.4 = 12.8 units Table 4.1

Exponential Smoothing with Trend Adjustment Example Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 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 Step 2: Trend for Month 2 T2 = b(F2 - F1) + (1 - b)T1 T2 = (.4)(12.8 - 11) + (1 - .4)(2) = .72 + 1.2 = 1.92 units Table 4.1

Exponential Smoothing with Trend Adjustment Example Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 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 Step 3: Calculate FIT for Month 2 FIT2 = F2 + T1 FIT2 = 12.8 + 1.92 = 14.72 units Table 4.1

Exponential Smoothing with Trend Adjustment Example Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 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 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 Table 4.1

Exponential Smoothing with Trend Adjustment Example | | | | | | | | | 1 2 3 4 5 6 7 8 9 Time (month) Product demand 35 – 30 – 25 – 20 – 15 – 10 – 5 – 0 – Actual demand (At) Forecast including trend (FITt) Figure 4.3

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 ^

Actual observation (y value) Least Squares Method Time period Values of Dependent Variable Deviation1 Deviation5 Deviation7 Deviation2 Deviation6 Deviation4 Deviation3 Actual observation (y value) Trend line, y = a + bx ^ Figure 4.4

Actual observation (y value) Least Squares Method Time period Values of Dependent Variable Actual observation (y value) Deviation7 Deviation5 Deviation6 Deviation3 Least squares method minimizes the sum of the squared errors (deviations) Deviation4 Trend line, y = a + bx ^ Deviation1 Deviation2 Figure 4.4

Least Squares Method Equations to calculate the regression variables y = a + bx ^ b = Sxy - nxy Sx2 - nx2 a = y - bx

Least Squares Example Time Electrical Power Year Period (x) Demand x2 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 ∑x2 = 140 ∑xy = 3,063 x = 4 y = 98.86 b = = = 10.54 ∑xy - nxy ∑x2 - nx2 3,063 - (7)(4)(98.86) 140 - (7)(42) a = y - bx = 98.86 - 10.54(4) = 56.70

Least Squares Example The trend line is y = 56.70 + 10.54x ^ Time Electrical Power Year Period (x) Demand x2 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 Sx = 28 Sy = 692 Sx2 = 140 Sxy = 3,063 x = 4 y = 98.86 The trend line is y = 56.70 + 10.54x ^ b = = = 10.54 Sxy - nxy Sx2 - nx2 3,063 - (7)(4)(98.86) 140 - (7)(42) a = y - bx = 98.86 - 10.54(4) = 56.70

Least Squares Example Trend line, y = 56.70 + 10.54x ^ 160 – 150 – | | | | | | | | | 1999 2000 2001 2002 2003 2004 2005 2006 2007 160 – 150 – 140 – 130 – 120 – 110 – 100 – 90 – 80 – 70 – 60 – 50 – Year Power demand

Seasonal Variations In Data The multiplicative seasonal model can modify trend data to accommodate seasonal variations in demand Find average historical demand for each season Compute the average demand over all seasons Compute a seasonal index for each season Estimate next year’s total demand Divide this estimate of total demand by the number of seasons, then multiply it by the seasonal index for that season

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 2003 2004 2005 2003-2005 Monthly Index

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 2003 2004 2005 2003-2005 Monthly Index 0.957 Seasonal index = average 2003-2005 monthly demand average monthly demand = 90/94 = .957

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 2003 2004 2005 2003-2005 Monthly Index

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 2003 2004 2005 2003-2005 Monthly Index Forecast for 2006 Expected annual demand = 1,200 Jan x .957 = 96 1,200 12 Feb x .851 = 85 1,200 12

Seasonal Index Example 2006 Forecast 2005 Demand 2004 Demand 2003 Demand 140 – 130 – 120 – 110 – 100 – 90 – 80 – 70 – | | | | | | | | | | | | J F M A M J J A S O N D Time Demand

San Diego Hospital 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 9723 9745 9766 Figure 4.6

San Diego Hospital Seasonal Indices 1.06 – 1.04 – 1.04 1.02 – 1.03 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 1.00 0.98 0.97 0.96 Figure 4.7

San Diego Hospital 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 Figure 4.8

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

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 ^

Associative Forecasting Example Sales Local Payroll ($000,000), y ($000,000,000), x 2.0 1 3.0 3 2.5 4 2.0 2 3.5 7 4.0 – 3.0 – 2.0 – 1.0 – | | | | | | | 0 1 2 3 4 5 6 7 Sales Area payroll

Associative Forecasting Example Sales, y Payroll, x x2 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 3.5 7 49 24.5 ∑y = 15.0 ∑x = 18 ∑x2 = 80 ∑xy = 51.5 b = = = .25 ∑xy - nxy ∑x2 - nx2 51.5 - (6)(3)(2.5) 80 - (6)(32) x = ∑x/6 = 18/6 = 3 y = ∑y/6 = 15/6 = 2.5 a = y - bx = 2.5 - (.25)(3) = 1.75

Associative Forecasting Example y = 1.75 + .25x ^ Sales = 1.75 + .25(payroll) 4.0 – 3.0 – 2.0 – 1.0 – | | | | | | | 0 1 2 3 4 5 6 7 Sales Area payroll If payroll next year is estimated to be $600 million, then: 3.25 Sales = 1.75 + .25(6) Sales = $325,000

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 4.0 – 3.0 – 2.0 – 1.0 – | | | | | | | 0 1 2 3 4 5 6 7 Sales Area payroll 3.25 Figure 4.9

Standard Error of the Estimate Sy,x = ∑(y - yc)2 n - 2 where y = y-value of each data point yc = computed value of the dependent variable, from the regression equation n = number of data points

Standard Error of the Estimate Computationally, this equation is considerably easier to use Sy,x = ∑y2 - a∑y - b∑xy n - 2 We use the standard error to set up prediction intervals around the point estimate

Standard Error of the Estimate Sy,x = = ∑y2 - a∑y - b∑xy n - 2 39.5 - 1.75(15) - .25(51.5) 6 - 2 4.0 – 3.0 – 2.0 – 1.0 – | | | | | | | 0 1 2 3 4 5 6 7 Sales Area payroll 3.25 Sy,x = .306 The standard error of the estimate is $30,600 in sales

Correlation 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 Coefficient nSxy - SxSy [nSx2 - (Sx)2][nSy2 - (Sy)2]

Correlation Coefficient y x (a) Perfect positive correlation: r = +1 y x (b) Positive correlation: 0 < r < 1 Correlation Coefficient r = n∑xy - ∑x∑y [n∑x2 - (∑x)2][n∑y2 - (∑y)2] y x (c) No correlation: r = 0 y x (d) Perfect negative correlation: r = -1

For the Nodel Construction example: Correlation Coefficient of Determination, r2, measures the percent of change in y predicted by the change in x Values range from 0 to 1 Easy to interpret For the Nodel Construction example: r = .901 r2 = .81

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 + b1x1 + b2x2 … ^ Computationally, this is quite complex and generally done on the computer

Multiple Regression Analysis In the Nodel example, including interest rates in the model gives the new equation: y = 1.80 + .30x1 - 5.0x2 ^ 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 = $300,000

Monitoring and Controlling Forecasts Tracking Signal Measures how well the forecast is predicting actual values Ratio of running sum of forecast errors (RSFE) 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 RSFE MAD = Tracking signal = ∑(actual demand in period i - forecast demand in period i) (∑|actual - forecast|/n)

Tracking Signal Signal exceeding limit Tracking signal + 0 MADs – Upper control limit Lower control limit Time Acceptable range

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

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 Tracking Signal (RSFE/MAD) -10/10 = -1 -15/7.5 = -2 0/10 = 0 +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

Adaptive Forecasting It’s possible to use the computer to continually monitor forecast error and adjust the values of the a and b coefficients used in exponential smoothing to continually minimize forecast error This technique is called adaptive smoothing

Focus Forecasting Developed at American Hardware Supply, focus forecasting is based on two principles: Sophisticated forecasting models are not always better than simple models There is no single techniques that should be used for all products or services 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

Forecasting in the Service Sector Presents unusual challenges Special need for short term records Needs differ greatly as function of industry and product Holidays and other calendar events Unusual events

Fast Food Restaurant Forecast 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 Figure 4.12