NATIONAL ECONOMICS UNIVERSITY

OPERATIONS MANAGEMENT
FORECASTING

PhD. Le Phan Hoa

 CONTENT

1. DEFINITION
2. TYPES OF FORECASTING
3. IMPORTANCE OF FORECASTING
4. FORECASTING APPROACH
5. MONITORING AND CONTROLLING FORECAST
DEFINITION
WHAT IS FORECASTING?
 Process of predicting a
future event
??
 Underlying basis of
all business decisions
 Production
 Inventory
 Personnel
 Facilities
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
INFLUENCE OF PRODUCT LIFE CYCLE

Introduction Growth Maturity Decline

Best period to Practical to change Poor time to Cost control
Company Strategy/Issues increase market price or quality change image, critical
share image price, or quality

R&D engineering is Strengthen niche Competitive costs

critical become critical
Defend market
position
CD-ROMs
Internet search engines
Analog TVs
Drive-through
LCD & plasma TVs restaurants

Sales iPods

3 1/2”
Xbox 360 Floppy
disks

Figure 2.5
THE REALITY

 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
TYPES OF FORECASTING
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
TYPES OF FORECASTING
 Economic forecasts
Address business cycle – inflation rate, money supply, housing
starts, etc.
 Technological forecasts
Predict rate of technological progress
Impacts development of new products
 Demand forecasts
Predict sales of existing products and services
IMPORTANCE OF FORECASTING
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
FORECASTING APPROACHES
 FORECASTING APPROACHES
Qualitative Methods Quantitative Methods
 Used when situation is  Used when situation is
vague and little data exist ‘stable’ and historical data
 New products
exist
 New technology  Existing products
 Current technology
 Involves intuition,
experience  Involves mathematical
 e.g., forecasting sales on
techniques
Internet  e.g., forecasting sales of
color televisions
QUALITATIVE METHOD
OVERVIEW OF QUALITATIVE METHOD
 Jury of executive opinion
 Pool opinions of high-level experts, sometimes
augment by statistical models
 Delphi method
 Panel of experts, queried iteratively
 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
SALE 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, Staff
continues until consensus is (Administering
reached survey)
Decision Makers
 3 types of participants (Evaluate
 Decision makers Respondents responses and
 Staff (People who can make decisions)
 Respondents 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
QUANTITATIVE METHOD
QUANTITATIVE METHODS
1. Naive approach
2. Moving averages
Time-Series
3. Exponential Models
smoothing
4. Trend projection
Associative
5. Linear regression Model

HOW TO FORECAST USING DIFFERENT METHODS?

WHICH METHOD IS BETTER?
1. 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
2. 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

∑ demand in previous n periods

Moving average = n
Moving Average Example
Actual 3-Month
Month Shed Sales Moving Average
January 10
10
February 12
March 12
13
April 13
16
May 19 (10 + 12 + 13)/3 = 11 2/3
June 23
July 26 (12 + 13 + 16)/3 = 13 2/3
(13 + 16 + 19)/3 = 16
(16 + 19 + 23)/3 = 19 1/3
Graph of Moving Average
Moving
30 –
Average
28 –
Forecast
26 – Actual
24 – Sales
Shed Sales

22 –
20 –
18 –
16 –
14 –
12 –
10 –
| | | | | | | | | | | |
J F M A M J J A S O N D
WEIGHTED MOVING AVERAGE
 Used when trend is present
Older data usually less important
 Weights based on experience and intuition

∑ (weight for period n)

Weighted x (demand in period n)
moving average = ∑ weights
 Weights Applied Period
Weighted Moving Average
3 Last month
2 Two months ago
1 Three months ago
6 Sum of weights

Actual 3-Month Weighted

Month Shed Sales Moving Average
January 10
February 12
March 13
April 16 [(3 x 13) + (2 x 12) + (10)]/6 = 121/6
May 19 [(3 x 16) + (2 x 13) + (12)]/6 = 141/3
June 23 [(3 x 19) + (2 x 16) + (13)]/6 = 17
July 26 [(3 x 23) + (2 x 19) + (16)]/6 = 201/2
 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
Weighted
30 – moving
average
25 –
Sales demand

20 – Actual
sales
15 –
Moving
10 – average

5 –
| | | | | | | | | | | |
J F M A M J J A S O N D
Figure 4.2
 COMMON MEASURES OF ERROR
Mean Absolute Deviation (MAD)
∑ |Actual - Forecast|
MAD =
n
Mean Squared Error (MSE)
∑ (Forecast Errors)2
MSE =
n
Mean Absolute Percent Error (MAPE)

n
∑100|Actuali - Forecasti|/Actuali
MAPE = i=1
n
Use quantitative Month Sales
methods (MA, 1 40
WMA) to forecast
for September 2 42
sales? 3 38
Weighted: 0.1; 0.3; 4 44
0.2 5 45
Which method is 6 49
better?
7 48
8 50
3. 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

+  (Last period’s actual demand
– Last period’s forecast)

Ft = Ft – 1 + (At – 1 - Ft – 1)

where Ft = new forecast

Ft – 1 = previous forecast
 = smoothing (or weighting)
constant (0 ≤  ≤ 1)
EXPONENTIAL SMOOTHING

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 forecast = 142 + .2(153 – 142)

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
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
 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
 Comparison of Forecast Error
∑ |deviations|
Rounded Absolute Rounded Absolute
MADActual
= Forecast Deviation Forecast Deviation
Tonnage n
with for with for
Quarter Unloaded  = .10  = .10  = .50  = .50
1
For 180
= .10 175 5.00 175 5.00
2 168 = 82.45/8
175.5 = 10.31
7.50 177.50 9.50
3 159 174.75 15.75 172.75 13.75
4 For 175
= .50 173.18 1.82 165.88 9.12
5 190 173.36 16.64 170.44 19.56
6 205 = 98.62/8
175.02 = 12.33
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
 Comparison of Forecast Error
∑ (forecast errors)2
Rounded Absolute Rounded Absolute
MSE = Actual Forecast Deviation Forecast Deviation
Tonnage
n
with for with for
Quarter Unloaded  = .10  = .10  = .50  = .50
1
For 180
= .10 175 5.00 175 5.00
2 = 1,526.54/8
168 175.5 = 190.82
7.50 177.50 9.50
3 159 174.75 15.75 172.75 13.75
4 For 175
= .50 173.18 1.82 165.88 9.12
5 190 173.36 16.64 170.44 19.56
6 = 1,561.91/8
205 175.02 = 195.24
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
 Comparison
n
of Forecast Error
∑100|deviationi|/actuali
Rounded Absolute Rounded Absolute
MAPE =Actual
i=1
Forecast Deviation Forecast Deviation
Tonnage with n for with for
Quarter Unloaded  = .10  = .10  = .50  = .50
1
 = .10 175
For 180 5.00 175 5.00
2 168 = 44.75/8
175.5 = 7.50
5.59% 177.50 9.50
3 159 174.75 15.75 172.75 13.75
4 =
For 175 .50 173.18 1.82 165.88 9.12
5 190 173.36 16.64 170.44 19.56
6 205 = 54.05/8
175.02 =29.98
6.76% 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
 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%
 Exponential Smoothing with Trend
Adjustment
When a trend is present, exponential
smoothing must be modified

Forecast Exponentially Exponentially

including (FITt) = smoothed (Ft) + (Tt) smoothed
trend forecast trend
Exponential Smoothing with Trend
Adjustment

Ft = (At - 1) + (1 - )(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 Step 1: Forecast for Month 2
6 21
7 31 F2 = A1 + (1 - )(F1 + T1)
8 28 F2 = (.2)(12) + (1 - .2)(11 + 2)
9 36
10 = 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 Step 2: Trend for Month 2
6 21
7 31 T2 = b(F2 - F1) + (1 - b)T1
8 28 T2 = (.4)(12.8 - 11) + (1 - .4)(2)
9 36
10 = .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 Step 3: Calculate FIT for Month 2
6 21
7 31 FIT2 = F2 + T1
8 28 FIT2 = 12.8 + 1.92
9 36
10 = 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 15.18 2.10 17.28
4 19 17.82 2.32 20.14
5 24 19.91 2.23 22.14
6 21 22.51 2.38 24.89
7 31 24.11 2.07 26.18
8 28 27.14 2.45 29.59
9 36 29.28 2.32 31.60
10 32.48 2.68 35.16

Table 4.1
Exponential Smoothing with Trend
Adjustment Example
35 –

Product demand 30 – Actual demand (At)

25 –

20 –

15 –

10 – Forecast including trend (FITt)

with  = .2 and b = .4
5 –

0 – | | | | | | | | |
1 2 3 4 5 6 7 8 9
Figure 4.3
Time (month)
4. 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
Least Squares Method
Equations to calculate the regression variables

y^ = a + bx

Sxy - nxy
b=
Sx2 - nx2

a = y - bx
Least Squares Example
Time Electrical Power
Year Period (x) Demand x2 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 ∑x2 = 140 ∑xy = 3,063
x=4 y = 98.86

∑xy - nxy 3,063 - (7)(4)(98.86)

b= = = 10.54
∑x - nx
2 2 140 - (7)(4 2)

a = y - bx = 98.86 - 10.54(4) = 56.70

Least Squares Example
Time Electrical Power
Year Period (x) Demand x2 xy
1999 1 74 1 74
2000 2 79 4 158
2001The trend
3 line is 80 9 240
2002 4 90 16 360
2003 y^ 5= 56.70 + 10.54x
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

Sxy - nxy 3,063 - (7)(4)(98.86)

b= = = 10.54
Sx - nx
2 2 140 - (7)(4 2)

a = y - bx = 98.86 - 10.54(4) = 56.70

 Least Squares Example
Trend line,
160 – y^= 56.70 + 10.54x
150 –
140 –
Power demand

130 –
120 –
110 –
100 –
90 –
80 –
70 –
60 –
50 –
| | | | | | | | |
2001 2002 2003 2004 2005 2006 2007 2008 2009
Year
 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:
1. Find average historical demand for each
season
2. Compute the average demand over all
seasons
3. Compute a seasonal index for each season
4. Estimate next year’s total demand
5. Divide this estimate of total demand by the
number of seasons, then multiply it by the
seasonal index for that season
 Seasonal Index Example
Demand Average Average Seasonal
Month 2005 2006 2007 2005-2007 Monthly Index
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
 Seasonal Index Example
Demand Average Average Seasonal
Month 2005 2006 2007 2005-2007 Monthly Index
Jan 80 85 105 90 94 0.957
Feb 70 85 85 80 94
Mar 80 93 average
82 85 monthly demand
2005-2007 94
Seasonal90index95= 115
Apr 100 94
average monthly demand
May 113 125 131 123 94
= 90/94 = .957
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
 Seasonal Index Example
Demand Average Average Seasonal
Month 2005 2006 2007 2005-2007 Monthly Index
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
 Seasonal Index Example
Demand Average Average Seasonal
Month 2005 2006 2007 2005-2007 Monthly Index
Jan 80 85 105 90 94 0.957
Feb 70 85 Forecast
85 for802008 94 0.851
Mar 80 93 82 85 94 0.904
Apr 90Expected
95 115annual demand
100 = 1,200
94 1.064
May 113 125 131 123 94 1.309
Jun 110 115 120 1,200 115 94 1.223
Jul Jan 113
100 102 x .957 = 96 94
105 1.117
12
Aug 88 102 110 100 94 1.064
1,200
Sept 85 90
Feb 95 x90.851 = 85 94 0.957
Oct 77 78 85 12 80 94 0.851
Nov 75 72 83 80 94 0.851
Dec 82 78 80 80 94 0.851
Seasonal Index Example
2008 Forecast
140 – 2007 Demand
130 – 2006 Demand
2005 Demand
120 –
Demand

110 –
100 –
90 –
80 –
70 –
| | | | | | | | | | | |
J F M A M J J A S O N D
Time
5. 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
^ = computed value of the variable to
where y
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
Least Squares Method
Equations to calculate the regression variables

y^ = a + bx

Sxy - nxy
b=
Sx2 - nx2

a = y - bx
 Associative Forecasting Example
Sales Local Payroll
($ millions), y ($ billions), x
2.0 1
3.0 3
2.5 4 4.0 –
2.0 2
2.0 1 3.0 –

Sales
3.5 7
2.0 –

1.0 –

| | | | | | |
0 1 2 3 4 5 6 7
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
2.0 1 1 2.0
3.5 7 49 24.5
∑y = 15.0 ∑x = 18 ∑x2 = 80 ∑xy = 51.5

∑xy - nxy 51.5 - (6)(3)(2.5)

x = ∑x/6 = 18/6 = 3 b=
∑x2 - nx2
= 80 - (6)(32) = .25

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)

If payroll next year

4.0 –
is estimated to be
$6 billion, then: 3.25
3.0 –

Sales
2.0 –
Sales = 1.75 + .25(6)
Sales = $3,250,000 1.0 –

| | | | | | |
0 1 2 3 4 5 6 7
Area payroll
Standard Error of the Estimate
 A forecast is just a point estimate of a
future value
 This point is 4.0 –
actually the 3.25
mean of a 3.0 –

Sales
probability 2.0 –
distribution
1.0 –

| | | | | | |
0 1 2 3 4 5 6 7
Area payroll
Figure 4.9
Standard Error of the Estimate
Computationally, this equation is
considerably easier to use

∑y2 - a∑y - b∑xy

Sy,x =
n-2

We use the standard error to set up

prediction intervals around the
point estimate
 Standard Error of the Estimate
∑y2 - a∑y - b∑xy 39.5 - 1.75(15) - .25(51.5)
Sy,x = = 6-2
n-2

Sy,x = .306 4.0 –

3.25
3.0 –

Sales
The standard error
2.0 –
of the estimate is
$306,000 in sales 1.0 –

| | | | | | |
0 1 2 3 4 5 6 7
Area payroll
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
r=
[nSx2 - (Sx)2][nSy2 - (Sy)2]
Correlation Coefficient
nSxy - SxSy
r=
[nSx2 - (Sx)2][nSy2 - (Sy)2]
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 = $3,000,000
MONITORING AND CONTROLLING
FORECAST
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 RSFE
signal =
MAD

∑(Actual demand in
period i -
Forecast demand
Tracking in period i)
signal = (∑|Actual - Forecast|/n)
 Tracking Signal
Signal exceeding limit
Tracking signal
Upper control limit
+

0 MADs Acceptable
range

–
Lower control limit

Time
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