Operations Management

Operations Management
Chapter 4 – Forecasting PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 7e Operations Management, 9e

Outline Global Company Profile: Disney World 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 Naive 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 for Forecasting 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 : Understand the three time horizons and which models apply for each use Explain when to use each of the four qualitative models Apply the naive, moving average, exponential smoothing, and trend methods

Learning Objectives When you complete this chapter you should be able to : Compute three measures of forecast accuracy Develop seasonal indexes Conduct a regression and correlation analysis Use a tracking signal

Forecasting at Disney World
Global portfolio includes parks in Hong Kong, Paris, Tokyo, Orlando, and Anaheim Revenues are derived from people – how many visitors and how they spend their money Daily management report contains only the forecast and actual attendance at each park

Disney generates daily, weekly, monthly, annual, and 5-year forecasts Forecast used by labor management, maintenance, operations, finance, and park scheduling Forecast used to adjust opening times, rides, shows, staffing levels, and guests admitted

20% of customers come from outside the USA Economic model includes gross domestic product, cross-exchange rates, arrivals into the USA A staff of 35 analysts and 70 field people survey 1 million park guests, employees, and travel professionals each year

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 Average forecast error for the 5-year forecast is 5% Average forecast error for annual forecasts is between 0% and 3%

?? 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 search engines Sales Xbox 360 Drive-through restaurants CD-ROMs 3 1/2” Floppy disks LCD & plasma TVs Analog TVs iPods 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 products and services

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

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 experts, 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 experts and 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, no other variables important 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 | | | | 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

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 January sales were 68, then February sales will be 68 Sometimes cost effective and efficient Can be good starting point

∑ 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 ( )/3 = 11 2/3 ( )/3 = 13 2/3 ( )/3 = 16 ( )/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

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

Predicted demand = 142 Ford Mustangs Actual demand = 153 Smoothing constant a = .20 New forecast = (153 – 142) = = ≈ 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 = a =

Impact of Different  Actual demand a = .5 a = .1 225 – 200 – 175 –
225 – 200 – 175 – 150 – | | | | | | | | | Quarter Demand Actual demand a = .5 a = .1

Impact of Different  225 – 200 – 175 – 150 – | | | | | | | | | Quarter Demand Chose high values of  when underlying average is likely to change Choose low values of  when underlying average is stable 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

MAD = ∑ |deviations| n Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded a = .10 a = .10 a = .50 a = .50 = 82.45/8 = 10.31 For a = .10 = 98.62/8 = 12.33 For a = .50

MSE = ∑ (forecast errors)2 n Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded a = .10 a = .10 a = .50 a = .50 = 1,526.54/8 = For a = .10 MAD = 1,561.91/8 = For a = .50

MAPE = ∑100|deviationi|/actuali n i = 1 Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded a = .10 a = .10 a = .50 a = .50 = 44.75/8 = 5.59% For a = .10 MAD MSE = 54.05/8 = 6.76% For a = .50

Rounded Absolute Rounded Absolute Actual Forecast Deviation Forecast Deviation Tonnage with for with for Quarter Unloaded a = .10 a = .10 a = .50 a = .50 MAD MSE MAPE 5.59% 6.76%

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 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 2 17 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10 Table 4.1

Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 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) = = 12.8 units Table 4.1

Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 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)( ) + (1 - .4)(2) = = 1.92 units Table 4.1

Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 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 = = units Table 4.1

Forecast Actual Smoothed Smoothed Including Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt 3 20 4 19 5 24 6 21 7 31 8 28 9 36 10 Table 4.1

| | | | | | | | | Time (month) Product demand 35 – 30 – 25 – 20 – 15 – 10 – 5 – 0 – Actual demand (At) Forecast including trend (FITt) with  = .2 and  = .4 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 (error) 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 ∑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 = (4) = 56.70

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

Least Squares Example Trend line, y = 56.70 + 10.54x ^ 160 – 150 –
| | | | | | | | | 160 – 150 – 140 – 130 – 120 – 110 – 100 – 90 – 80 – 70 – 60 – 50 – Year Power demand

Seasonal Variations In Data
The multiplicative seasonal model can adjust trend data for seasonal variations in demand

Seasonal Variations In Data
Steps in the process: 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 Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec Demand Average Average Seasonal Month Monthly Index

Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec Demand Average Average Seasonal Month Monthly Index 0.957 Seasonal index = average monthly demand average monthly demand = 90/94 = .957

Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec Demand Average Average Seasonal Month Monthly Index

Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec Demand Average Average Seasonal Month Monthly Index Forecast for 2008 Expected annual demand = 1,200 Jan x .957 = 96 1,200 12 Feb x .851 = 85 1,200 12

2008 Forecast 2007 Demand 2006 Demand 2005 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 Month Inpatient Days 9530 9551 9573 9594 9616 9637 9659 9680 9702 9724 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 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 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 ($ millions), y ($ billions), x 2.0 1 3.0 3 2.5 4 2.0 2 3.5 7 4.0 – 3.0 – 2.0 – 1.0 – | | | | | | | Sales Area payroll

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

y = x ^ Sales = (payroll) 4.0 – 3.0 – 2.0 – 1.0 – | | | | | | | Sales Area payroll If payroll next year is estimated to be $6 billion, then: 3.25 Sales = (6) Sales = $3,250,000

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 – | | | | | | | Sales Area payroll 3.25 Figure 4.9

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

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

Sy,x = = ∑y2 - a∑y - b∑xy n - 2 (15) - .25(51.5) 6 - 2 4.0 – 3.0 – 2.0 – 1.0 – | | | | | | | Sales Area payroll 3.25 Sy,x = .306 The standard error of the estimate is $306,000 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 = nSxy - SxSy [nSx2 - (Sx)2][nSy2 - (Sy)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 = x x2 ^ An improved correlation coefficient of r = .96 means this model does a better job of predicting the change in construction sales Sales = (6) - 5.0(.12) = 3.00 Sales = $3,000,000

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

Tracking Signal Example
Cumulative Absolute Absolute Actual Forecast Forecast Forecast Qtr Demand Demand Error RSFE Error Error MAD 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 ones There is no single technique 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% – (Lunchtime) (Dinnertime) Hour of day Percentage of sales Figure 4.12

FedEx Call Center Forecast
12% – 10% – 8% – 6% – 4% – 2% – 0% – Hour of day A.M. P.M. 2 4 6 8 10 12 Figure 4.12

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