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QUANTITATIVE ANALYSISFOR
MANAGEMENT
Forecasting
Barry Render / Ralph M. Stir, JR.
Summarized by - NOPPOL BEOKHAIMOOK
Learning Objectives (1)
Students will be able to:– Understand and know when to use various
families of forecasting models– Compare moving averages, exponential
smoothing, and trend time-series models– Seasonally adjust data.– Understand Delphi and other qualitative
decision-making approaches1
Students will be able to:
– Identify independent and dependent variables and use them in a linear regression model.
– Compute a variety of error measures.
Learning Objectives (2)
2
Topics (1)• Importance of Forecasting
• Steps of Forecasting
• Types of Forecasting
• Scatter Diagrams
• Time-Series Models
• Moving Averages
• Exponential Smoothing
• Trend Projections3
Topics (2)• Seasonal Variations
• Cyclical Variations
• Causal Forecasting Methods
• Regression Analysis
• Monitoring and Controlling Forecasts
• Using the Computer to Forecast
• Key Equations
• Summary4
• #1 - Friday, 22/04/2011 : 18.00 -21.00
• #2 – Saturday, 23/04/2011 : 9.00 -12.00
• #3 - Saturday, 23/04/2011 : 13.00 -16.00
• #4 - Friday, 29/04/2011 : 18.00 -21.00
• #5 - Friday, 06/05/2011 : 18.00 -21.00 (Work-shop)
• #6 - Friday, 20/05/2011 : 18.00 -21.00
Schedule
4.1
• Determine organization : 5W1H- WHY it should do ?- WHERE it should be ? - WHAT it should do ? - WHEN it should do to be done ?- WHO should do ? - HOW it should do ?
Corporate Planning (1)
5
Corporate Planning (2)
• A deliberate and conscious attempt to anticipate and “Forward Look”the environment.
• Looking ahead systematically.
6
• ASSESS - Internal and Environment Factors (SWOT)
• FORMULATE - Corporate Objectives,Strategies and Action Plans
• CASCADE - Corporate Objectives,Strategies, Action Plans down to Lower Levels (One Layer at a time)
Planning Steps (1)
7
Planning Steps (2)
• QUANTIFY - Action Plans &Strategies into Budgets starting with lowest Operating Unit.
• CONSOLIDATE - Budgets upwards until the corporate level is reached.
• IMPLEMENT - Plans
• MONITOR - Feedback and Control8
Desirable Attributes of Good Objective
• ADAPTABLE - Changeable and Flexible.
• MEASURABLE - The results can be readily quantified and evaluated.
• COMMITMENT - Capable of inspiring Commitment
9
Objectives Perspective
• FINANCIAL PERSPECTIVE
• CUSTOMER PERSPECTIVE
• INTERNAL PROCESSPERSPECTIVE
• LEARNING AND GROWTHPERSPECTIVE
10
Importance of Forecasting
Business Planning(Demands for products, Return on Investment,
Sales Volume, Expense, etc.)
Analytical Tools
Mathematical Techniques
Forecasting
11
Steps of Forecasting (1)
1. Determine the forecast objective
2. Select the items or quantities to be forecasted
3. Determined time horizon of forecast- Short term / Medium term / Long term
4. Select Forecasting MODEL (S)
5. Gather data needed to make the forecast
6. Validate forecasting model
7. Make the forecast
8. Implement the results
12
Steps of Forecasting (2)
• Initiating, designing, implementing forecasting system ==> Systematic way and Automatically
• Computer System==> Input (Files / Records)
Processing (Software / Program)Output (Files / Reports)
• No Single method is superior, whatever
(quantitative, subjective, mixed) works best
should be used
13
Types of Forecasting (1)Figure 5.1 - Forecasting Models Discussed
14
Moving Average
Exponential Smoothing
Trend Projections
Time Series Methods
Forecasting Techniques
Delphi Methods
Jury of ExecutiveOpinion
Sales Force Composite
Consumer Market Survey
Qualitative Models Causal Methods
Simple Regression
Multiple Regression
Decomposition (T*S*C*I)
Types of Forecasting (2)
• Time-Series Models– Use past data to make a forecast over a period of time
– Three techniques of time-series models1. Moving Average2. Exponential Smoothing3. Trend Projections
15
Types of Forecasting (3)• Causal Models
– Incorporate the variables that might influence the quantity being forecasted
– Include past data as time-series models do.
• Quantitative Models– Time-series and Causal Models ==> rely on quantitative data
• Qualitative Models==> rely on judgmental / subjective factors==> opinions by experts, individuals experiences
and judgment, other subjective factors may be considered
16
Types of Forecasting (4)
• Qualitative Models (cont.)
– Four techniques of qualitative models
1. Delphi method- experts in different places- three different types of participants
(a) Decision Making Group of Experts(b) Staff Personnel(c) Respondents
17
Types of Forecasting (5)
• Qualtatative Models (cont.)
2. Jury of Executive Opinion- small group of high-level managers- often in combination with statistical models
3. Sales Force Composite- estimate by salespersons in each region
- ensured / combined at the higher levels to reach overall forecast
4. Consumer Market Survey- input from potential customers about future
purchasing and products demand18
Scatter Diagrams (1)
• Clearly Illustrate relationship between variables
• Two-dimensional graph
- horizontal (X) axis and vertical (Y) axis
- independent variables (X-axis) eg.time- dependent variables (Y-axis) eg. Sale
volumes
19
Scatter Diagrams (2)TABLE 5.1 Annual Sales of Three Products
20
123456789
10
COMPACT DISCS110100120140170150160190200190
RADIOS300310320330340350360370380390
TELEVISION SETS250250250250250250250250250250
YEAR
Scatter Diagrams (3)Figure 5.2 - Scatter Diagram for Sales
21
050
100150200250300350400450
0 2 4 6 8 10 12
Time (Years)
Annu
al S
ales
Time - Series Models (1)• Decomposition of a Time-Series
- Analyzing time-series
==> breaking down components, projecting forward
- Four components of time-series
1. Trend (T) : Movement data over time
2. Seasonal (S) : Fluctuation in a year
3. Cycles (C) : Fluctuation every several years
4. Random Variations (R) : Irregular situations
22
Time - Series Models (2)• Figure 5.3 - Product Demand Charted over Four Years
with Trend and Seasonal Indicated
23
-150-5050
150250350450550650
0 1 2 3 4 5
Time (Years
Dem
and
Time - Series Models (3)• Two general forms of Time-Series
1. Multiplication - most widely used
Demand = T x S x C x R
Unit : T is amount while S, C and R is
percentage / proportion
2. Addition - provide as estimate
Demand = T + S + C + R
Unit : T, S, C and R is amount
- Real world : R is averaged out over time
24
Moving Averages (1)• A technique of time-series
• Assume that market demands will stay fairly steady
over time
• MA smooth out variations when forecasting demandsare fairly steady
• Moving average = ∑ demand in previous n periodsn
whereas n is the number of period in the MA
• Wallace Garden Supply - Example
25
Moving Averages (2)
26
Moving Averages (3)• Weighted moving average- can be used to put more emphasis on recent periods- weighted moving average= Σ of (weight for period n) (demand in period n)
Σ of weights
• Wallace Garden Supply decides to forecast storage shedsales by weighting the past three months as follows :
27
PERIODLast mountTwo months agoThree months ago
1 X Sales three months ago
WEIGHTS APPLIED321
3 X Sales last month + 2 X Sales two months ago +
Sum of the weights6
Moving Averages (4)
28
Moving Averages (5)
• Three problems of moving overages
1. Large number of periods may smooth out real
change
2. They don’t pick up trends
3. Lots of past data must be kept
29
Exponential Smoothing (1)
• A type of moving averages technique but it involvelittle record keeping of past data
• New forecast = last period’s forecast + a (last period actual demand - last period’s forecast); a is a weight (smoothing constant) which has value
between 0 and 1
• Math Formulas : Ft = Ft-1 + a (At-1 - F t-1)
- Ft = new forecast, Ft-1 = previous forecast
- a = smoothing constant (0 ≤ a ≤ 1)
- At-1 = previous period’s actual demand
30
Exponential Smoothing (2)
• The smoothing constant, α , allows managers to assignweight to recent dataeg. α = 0.5 ==> the past three periods
α = 0.1 ==> the past nineteen periods• Examples : The forecast of cases
Feb. forecast = 142Feb. actual = 153, α = 0.20Mar. forecast = 142 + 0.20 (153 - 142)
= 144.20 (≈144)Mar. actual = 136, α = 0.20Apr. forecast = 144.2 + 0.20 (136 - 144.20)
= 142.56 (≈143) 31
Exponential Smoothing (3)
• Selecting the smoothing constant (α) can make the
difference between accurate and inaccurate forecast
• Forecast error tells us how well the model performed
against itself using past data
• Forecast error = Demand - Forecast
• Mean Absolute Deviation (MAD) = Σ Forecast errorsn
• Port of Baltimore - Example
32
Exponential Smoothing (4)
33
Exponential Smoothing (5)
34
Exponential Smoothing (6)
• Besides MAD, there are three other measures of error
1. Mean Squared Error (MSE)- MSE = Σ (actual - forecast)2
n
2. Mean Absolute Percent Error (MAPE)
- MAPE = ( Σ actual - forecast x 100% ) / nactual
3. Bias- Bias tells whether the forecast is too high or too low and how much
35
Exponential Smoothing (7)• Exponential Smoothing with Trend Adjustment- Forecast including trend (FITt) (second-order smoothing)= new forecast (Ft) + trend correction (Tt)
- Trend correction (Tt)= (1 - β )Tt-1 + β (Ft - Ft-1)where Tt = smoothed trend for period t
β = trend smoothing constant that we selectFt = simple exponential smoothed forecast
for period t (first - order smoothing)- β ‘s responsiveness is like that of α
- a low β gives less weight to more recent trendsthan high β
36
Trend Projections (1)
• Last time-series forecasting method
• Fits a trend line to a series of historical data points
and then projects the line into the future
• Trend Projections
- exponential, quadratic and linear (straight line)
• Midwestern Manufacturing Company-Example
37
Trend Projections (2)
38
60708090
100110120130140150160
1993 1994 1995 1996 1997 1998 1999 2000 2001
TABLE 5.6 Midwestern Manufacturing’s Demand
Trend Projections (3)
• Least squares method- finds a straight line that minimizes the sum of the
vertical differences from the line to each of the data point
• A Least squares lines : Ŷ = a + bx- Ŷ : value of variable to be predicted (dependent variable)
- Y-intercept (a) : The height at which it intercepts the Y-axis
- slope (b) : The angle of line
- x : Independent variable
39
Trend Projections (4)FIGURE 5.4 - Least Square Method for finding the
Best-Fitting Straight Line
40
Trend Projections (5)• To Compute a and b
- slope (b) = Σ XY - nXYΣ X2 - nX2
where X = values of independent variable (time)
Y = values of dependent variable (generator)
X,Y = average value of X and Y respectively
n = number of data points (observations)
- Y-axis intercept (a) = Y - bX
• Transforming time variable
- direct method / short-cut method41
Trend Projections (6)TABLE 5.7 Midwestern Manufacturing’ s Trend Calculations
42
74158240360525852854
Σ XY = 3,063
149
16253649
Σ X 2 = 140
74798090
105142122
Σ Y = 692
1234567
Σ X = 28
1993199419951996199719981999
XYX 2GENERATOR
DEMANDTIME
PERIODYEAR
Trend Projections (7)• Form TABLE 5.7
- We can calculate b = 10.54 and a = 56.70
- The least squares trend equation is Ŷ = 56.70+10.54 X
- To project demand in 2000 = 56.70+10.54 (8)
= 141.02
- To project demand in 2001 = 56.70+10.54 (9)
= 151.56
43
Trend Projections (8)
44
Seasonal Variations (1)
• Recurring variations at certain seasons of the
year make seasonal adjustment in the trend line
forecast necessary
• Seasonal Index
- Calculation of seasonal indices
45
Seasonal Variations (2)
46
TABLE 5.8 Answering Machine Sales and Seasonal Indices
AVERAGESALES DEMAND SEASONALMONTHLYAVERAGE TWO- INDEX bDEMAND aYEAR DEMANDYEAR 2YEAR1MONTH0.957949010080January0.85194808575February0.90494859080March
1.30994123113115May1.0649410011090April
0.95794909585September
0.85194808080December
1.0649410011090August1.11794105110100July1.22394115120110June
0.85194808575November0.85194808575October
Total average demand = 1,128Average monthly demand = 1,128 = 94 Seasonal Index = average two years demand
12 months average monthly demand
Seasonal Variations (3)• From TABLE 5.8, if we expected the third year’s
annual demand to be 1,200, we would forecast the
monthly demand as follows :
47
Jan.
Feb.
Mar.
Apr.
May
June
1,20012
1,20012
1,20012
1,20012
1,20012
1,20012
x 0.957 = 96
x 0.851 = 85
x 0.904 = 90
x 1.064 = 106
x 1.309 = 131
x 1.223 = 122
July
Aug.
Sept.
Oct.
Nov.
Dec.
1,20012
1,20012
1,20012
1,20012
1,20012
1,20012
x 1.117 = 112
x 1.064 = 106
x 0.957 = 96
x 0.851 = 85
x 0.851 = 85
x 0.851 = 85
Seasonal Variations (4)
• Trend line forecasts with seasonal adjustments
• San Diego Hospital-Example
- used 66 months of adult inpatient hospital days
==> Ŷ = 8,091 + 21.5X ; Ŷ = patient days, X= months
- forecasts patient days for next month (period 67)
==> patient days = 8,091 + (21.5)(67) = 9,532
48
Seasonal Variations (5)
• San Diego Hospital-Example (cont.)
- note that 9,532 days which come from trend only
- to correct the time-series extrapolation for
seasonal, the hospital multiplies the monthly
forecast by appropriate seasonal index, i.e. patient
days (period 67) should be (9,532)(1.0436) = 9,948
which come from trend and seasonal and vice versa
for other periods forecast.
49
Seasonal Variations (6)
50
Cyclical Variations (1)
• From multiplication formulas of time-series
Demand = T x C x S x R
- S is average-out since the seasonal fluctuation in a year
- R is average-out since the less probability of irregular events
- Then Demand (Yt) = T x C ; Ct = Yt = Yt
T Ŷ
• ABC Production Company-Example
51
Cyclical Variations (2)
52
ABC Production Company (Millions $)
Year(t)
Year (Xt)Transform
ProductionValue (Yt)
TrendProj-(Yt)
CyclicalIndices (Ct)
1994 -3 4 3.73 1.07
1995 -2 3 4.44 0.68
1996 -1 5 5.15 0.97
1997 0 7 5.86 1.19
1998 1 9 5.57 1.37
1999 2 5 7.28 .69
2000 3 8 7.99 1.00
Causal Forecasting Methods• Consider several variables that are related to thevariable being predicted which can be more powerful than time-series methods that use only the historicvalues of the forecasted variable
• The dependent variable is the item we are trying to
forecast e.g. sale volume
• The independent variable(s) is item(s) we think might
have a causal effect on the dependent variable.e.g. advertising budget, price, competitor’s price,promotional strategies, unemployment rate and etc.
53
Regression Analysis (1)
• Regression Analysis
- the most common quantitative causal forecasting model
which can calculate the best statistical relationship between dependent variable and the set of independentvariables
- use least squares method as trend projections but X is independent variable(s) instead of time
- simple regression analysis formula : Ŷ = a + bXwhere Ŷ = value of dependent var. , a = Y-axis intercept
b = slope of the regression line, X = independent var.
• Triple A Construction Company-Example
54
Regression Analysis (2)
55
Regression Analysis (3)
56
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
ΣY = 15.0 ΣX = 18 ΣX 2 = 80 ΣXY = 51.5
Regression Analysis (4)
• From TABLE 5.10
- we calculate the estimated regression equation in the
same procedure as the trend projections calculation.
- Ŷ= 1.75 + 0.25 X or sales = 1.75 + .25 (payroll)
- if X (payroll) is $600 Million next year
so sales = 1.75 +.25 (6) = 3.25 or $325,000
• One weakness of regression is that we need to know
the values of the independent variable
57
Regression Analysis (5)
• Standard error of the estimate
- from example above, 3.25 is called ‘ point estimated of
Y’ or expected value of sales which may different
from the actual value
- to measure the difference, we can compute the
‘ standard error of the estimate (Sy,x)’ or
‘standard deviation of the regression’
58
Regression Analysis (6)
Figure 5.6 - Distribution about the Point Estimate of
$ 600 Million Payroll
59
Regression Analysis (7)
• Standard error of the estimate (cont.)
- to compute Sy,x = Σ ( Y-Yc)2
n-2Y = Y-value of each data point
Yc = value of the dependent variable from regression
equationn = number of data points
- more simple formula of Sy,x = which has the same result
Sy,x = Σ Y2 - a Σ Y - b Σ XY
n-2
60
Regression Analysis (8)
61
Regression Analysis (9)• Standard error of the estimate (cont.)
- from the calculation in table 5.11
Sy,x = Σ Y2 - a Σ Y - b Σ XY
n-2
= 39.5 - (1.75)(15.0) - (0.25)(51.5)6-2
= 0.09375
= 0.306 or $ 30,600 in sales
62
Regression Analysis (10)• Correlation Coefficients (r)
- another way to evaluate the relationship between two
variables
- r measures the degree or strength of the linear
relationship and the value is between -1 and +1
- to compute : r = n ΣXY - ΣX Σ Y
√ [n Σx2 - (Σx)2] [n ΣY2 - (ΣY)2]
- using the data in table 5.11 for Triple A Construction
Company we can compute r = 0.901 which can confirm
the strength relationship between sales volume and
local payroll
63
Regression Analysis (11)
• Figure 5.7 - Four Values of the Correlation Coefficient
64
Regression Analysis (12)• Coefficient of determination (r2)
- is the percent of variation in the dependent (Y) that is
explained by the regression equation and the value is
between 0 and 1- to compute : r2 = b (ΣXY – n X Y)
ΣY2 – n Y2
-in the above example we can compute r2 = 0.81 indicating
that 81% of the total variation is explained by the
regression equation or 81% of sale volumes depend on
the payroll
65
Regression Analysis (13)• Position Correlation Coefficients (r’)
- another way to evaluate the relationship between two
variables regarding to variables position
- r’ measures the degree or strength of the linear position
relationship and the value is between -1 and +1
- to compute : r’ = 1 - (6Σd2/(n(n2)-1))
whereas ‘d’ is difference between rank
66
Regression Analysis (14)• Sample of r’ Calculation
- Sample - 1 : Favourites of Political PartiesParty : TRT DM CHT MHC LBR Red Fav. 1 5 3 4 2Yellow Fav. 3 2 1 4 5
d -2 3 2 0 -3
d2 4 9 4 0 9
Total d2 = 26
r’ = 1 - (6(26) / 5(25 - 1))= - 0.30
67
Regression Analysis (15)• Sample of r’ Calculation
- Sample - 2 : Favourites of Female Characteristics
Char. : สวย นารกั สูงยาวขาวดี เตี้ยล่ําบึ๊ก SEXY
เจาอารมณ ขี้งอน ดั้ งหัก นสิัยดี แกแลวขี้บนX Fav. 4 1 3 2 5Y Fav. 3 2 5 1 4
d 1 -1 -2 1 1d2 1 1 4 1 1
Total d2 = 8r’ = 1 – (6(8) / 5(25 – 1))
= 1 – 0.4 = 0.668
Regression Analysis (16)• Multiple Regression Analysis
- more than one independent variable (X) turns a simple
regression model into a multiple regression model- formula : Ŷ = a + b 1X1 + b2X2
- from the Triple A Construction Company if we add
another var. (interest rate) to be X2 and use computer
calculation the regression equation is
Ŷ = 1.80 + 0.30X1 - 5.0 X2
and the coefficient correlation (r) increase from 0.90
to be 0.96 which implies that the inclusion of interestrate add more strength to the linear relationship
69
Regression Analysis (17)
• Multiple Regression Analysis (cont.)
- to forecast sale if next year payroll is $600 million
and interest rate is 12% (0.12), the sales will be
sales = 1.80 + (0.30) (6) - (5.0) (0.12)
= 3.00 or $ 300,000
- from the sales forecast for simple regression which
is 3.20, if we can decrease interest rate to be 8%
then the sales will also increase to be 3.20
70
Regression Analysis (18)
• Regression Analysis Recommendations
1. The number of data should be much enough
2. If independent var. is more than one, have to us
multiple regression analysis
3. The data have to be quantitative
4. The relationship between independent var. and
dependent var. have to be appropriate in analysis
eg. if the relationship is linear we have to use linear
regression analysis (not exponential)
71
Regression Analysis (19)• Regression Analysis Recommendations (cont.)
5. Regression equation of one sample group cannot
forecast for another sample group despite the same
type of variable
6. In the forecast from regression equation, we can
forecast only the dependent variable which effect
from the independent variables
7. In multiple regression analysis, be careful the
relationship among the independent variable
8. In the forecast from regression equation, we can not
forecast if the characteristics of data is out of rules
of statistical sampling72
Monitoring and Controlling Forecasts (1)
• Tracking Signal
- is a measurement of how well the forecast is predicting
actual values
- to calculate Tracking Signal
- Tracking signal = running sum of the forecast errors (RSFE)
mean absolute deviation (MAD)- RSFE
= Σ (actual demand in period i - forecast demand in period i)
- MAD = Σ forecast errorsn
73
Monitoring and Controlling Forecasts (2)
• Figure 5.8 - Plot of Tracking Signals
74
Monitoring and Controlling Forecasts (3)
• Tracking Signal (cont.)
- positive TS indicate that demand is greater than
forecast and vice versa for negative TS
- good TS : small deviation, positive and negative
deviations should be balance and closely around zero
- signal tripped is a point that tracking signal exceeds an
upper or lower limit which means that there is problems
with the forecasting method and the management should
reevaluate the forecasting method
75
Monitoring and Controlling Forecasts (4)
• Tracking Signal (cont.)
- how do firms decide what the upper and lower tracking
limits should be?
- setting tracking limits is a matter of setting reasonable
values for upper and lower limits
- Kimball’s Bakery - Example
76
Monitoring and Controlling Forecasts (5)
• Tracking Signal (cont.)
77
Monitoring and Controlling Forecasts (6)
• Tracking Signal (cont.)
- from the example, we can calculate tracking signal in
each quarter and accumulate every quarter
- then MAD = Σ forecast errors = 85n 6
= 14.2- then tracking signal = RSFE = 35
MAD 14.2= 2.5 MADs
- the limits of tracking signal are between -2.0 to +2.5
which is in acceptable limits
78
Monitoring and Controlling Forecasts (7)
• Adaptive Smoothing
- computer can control the tracking signal limit and
automatic adjustment if a signal is over or under limit
(signal tripped)
- to minimize eror forecast, some smoothing constant
eg. α and β in exponential smoothing have to be
adjusted if the computer notes an errant trackingsignal. This is called “Adaptive Smoothing”
79
Using the Computer to Forecast
• Forecasted by Computer- forecast calculation are seldom performed by hand
in this day of computers
- there are many computer program or software package
for forecasting
eg. SPSS, SAS, BIOMED, SYSTAB and Mini tab,
EXCEL QM (Spreadsheet) which we should learn
how to use some of these software with efficiency
- the most important is how to interpret and analyzethe output from our computer to be the most meaningful for our business
80
Key Equations (1)
81
5.1 Moving average = Σ demand in previous n periodsn
An equation for computing a moving average forecast
5.2 Weighted moving average = Σ (weight for period n) (demand in period n)
Σ weightsAn equation for computing a weighted moving average forecast
5.3 New forecast = last period’s forecast + α (last period’sactual demand - last period’s forecast)
An equation for computing an exponential smoothing forecast
5.4 Ft = Ft-1 + α (At-1 - Ft-1)Equation 5.3 rewritten mathematically
Key Equations (2)
82
5.5 MAD = Σ forecast errorsn
A measure of overall forecast error called mean absolute deviation
5.6 Tt = (1 - β )Tt-1 + β (Ft - Ft-1)
Trend component of an exponential smoothing model
5.7 Ŷ = a + bX
A least squares straight line used in trend projection andregression analysis forecasting
5.8 b = Σ XY – nX Y
Σ X2 - nX2
An equation used to compute the slope, b, of a regression line
Key Equations (3)
83
5.9 a = Y - bXAn equation used to compute the Y-intercept, a, of a regression line
5.10 SYX = Σ (Y – Yc)2
n - 2Standard error of the estimate
5.11 SYX = ΣY2 - a ΣY - b ΣXYn-2
Another way to express Equation 5.10
5.12 r = n Σ XY - Σ X Σ Y
[nΣX2 - (ΣX)2] [nΣY2 - (ΣY)2]Correlation coefficient, Coefficient of Determination (R2 = r2)
Key Equations (4)
84
5.13 Ŷ = a + b1X1 + b2X2
The least squares line used in multiple regression
5.14 Tracking signal = RSFE
MAD
= Σ (actual demand in period i - forecast demand in period i)MAD
An equation for monitoring forecasts
Summary (1)
• Importance of Forecasting
• Steps of Forecasting
• Types of Forecasting
• Scatter Diagrams
• Time-Series Models
• Moving Averages
• Exponential Smoothing
• Trend Projections
• Seasonal Variations
• Cyclical Variations85
Summary (2)• Causal Forecasting Methods
• Regression Analysis
- Standard Error of the Estimate
- Correlation Coefficient
- Coefficient of Determination
- Multiple Regression Analysis
- Regression Analysis Recommendations
• Monitoring and Controlling Forecasts
- Tracking Signal
• Using the Computer to Forecast
• Key Equations 86
Exercise (1)
87
• The demand for coal seems to be tied to an index of weather
severity over the past five years and the experts proposes thatthe forecast for next year coal demand could be made by developing a trend projections and the relation between coal demand and weather index could be made by developing a regressionequation
YEAR WEATHER INDEX COAL SALES (MILLION TONS)
1996 2 4
1997 1 1
1998 4 4
1999 5 62000 3 5
Exercise (2)
88
A) What is trend projections equation? How many coal sales inYear 2001?
B) What is regression analysis equation? How many coal sales
in Year 2001 if the weather index is 2?
C) Compute the coefficient correlation. What does it mean?
D) Compute the standard error of the estimate
Exercise (3)
89
• Sales of Cool-Man air conditioners have grown
steadily during the past five years
YEAR SALES
1 4502 4953 5184 5635 584
6 ?
Exercise (4)
90
A) The sales manager had predicted, before the business started,that year 1’s sales would be 410 air conditioners. Using exponential smoothing with smoothing constants of 0.3 and 0.6to develop forecasts for year 2 through 6
B) What effect did the smoothing constants have on this forecasts?Which smoothing constant gives the most accurate forecast?
C) Use a three-year moving average forecasting model to forecastthe sales of Cool-Man air conditioners
D) Use a trend projection method, develop a forecasting model forthe sales of Cool-Man air conditioners. How many sales forecastsin year 6 and 7?
E) Compare to method of exponential smoothing with smoothing constant = 0.3, method of a three-year moving average and method of trend projections above, which method which you should use?
Exercise (5)
91
• Solutions of Coal sales forecasting
YEAR Tt Weather Index (X)
Coal Sales (Y)
Tt2 TtY X2 XY Y2
1996 -2 2 4 4 -8 4 8 16
1997 -1 1 1 1 -1 1 1 1
1998 0 4 4 0 0 16 16 16
1999 1 5 6 1 6 25 30 36
2000 2 3 5 4 10 9 15 25
Tt = 0 X = 15 Y = 20 Tt2 TtY X2 XY Y2
Tt = 0 X = 3 Y = 4 = 10 = 7 = 55 = 70 = 94
Σ Σ Σ Σ Σ Σ Σ Σ
Exercise (6)
92
• Solutions of Coal sales forecasting (cont.)
A) Trend projections Ŷt = a + bX ; X = Tt
b = Σ TtY – nTtY = 7 - (5)(0)(4) = 7 = 0.7
Σ Tt2- nTt
2 10 - (5)(0) 10
a = Y – bTt = 4 - (0.7)(0) = 4
Ŷt = 4 + 0.7Tt , Ŷ2001 = 4 + 0.7 (3) = 6.1
B) Regression Analysis Ŷ = a + bX
b = Σ XY – nXY = 70 - (5)(3)(4) = 10 = 1
Σ X2 - nX2 55 - (5)(3)2 10
a = Y – bX = 4 - (1)(3) = 1
Ŷ = 1 + 1X , Ŷ = 1 + (1)(2) = 3 if X = 2
Exercise (7)
93
• Solutions of Coal sales forecasting (cont.)
C) Coefficient Correlation (r) = n ΣXY - ΣXΣY
√[ nΣX2 - (ΣX)2 ] [ nΣY2 - (ΣY)2 ]
= (5)(70) - (15)(20)
√[ (5)(55) - (15)2] [ (5)(94) - (20)2]
= 350 - 300
√[ 275 - 225 ] [ 470 - 400 ]
= 350 - 300 = 50 = 0.85
√[ 50 ] [ 70 ] √ 3500
Exercise (8)
94
• Solutions of Coal sales forecasting (cont.)
D) Standard error of the estimate (SY,X)
= Σ Y2 - a Σ Y - b Σ XYn - 2
= 94 - (1)(20) - (1)(70)
5 - 2
= 4 = √1.333
= 1.1547
Exercise (9)
95
• Solutions of Cool-Man air conditioners
A) Year Sales Forecasts
α = 0.3 α = 0.6
1 450 410.00 410.00
2 495 410.00 + (0.3)(450.00 -410.00) = 422.00 434.00
3 518 422.00 + (0.3)(495.00-422.00) = 443.90 470.60
4 563 443.90 + (0.3)(518.00-443.90) = 466.13 499.04
5 584 466.13 + (0.3)(563.00-466.13) = 495.19 537.42
6 ? 495.19 + (0.3)(584.00-495.19) = 521.83 565.37
Exercise (10)
96
• Solutions of Cool-Man air conditioners (cont.)
= 0.3 = 0.6
B) Year Sales Rounded Forecast
Absolute Deviation
Rounded Forecast
Absolute Deviation
1 450 410 40 410 40
2 495 422 73 434 61
3 518 444 74 471 47
4 563 466 97 499 64
5 584 495 89 537 47
Sum = 373 Sum = 256
MAD = deviation = 373 = 74.60 n 5
MAD = 256 = 51.20
5
α α
∑
Exercise (11)
97
• Solutions of Cool-Man air conditioners (cont.)
C) Year Sales Three-Year Moving Average
1 450
2 495
3 518
4 563 (450 + 495 + 518)/3 = 487.67
5 584 (495 + 518 + 563)/3 = 525.33
6 ? (518 + 563 + 584)/3 = 555.00
Exercise (12)
98
• Solutions of Cool-Man air conditioners (cont.)
D) Year X Sales (Y) X2 XY 1 -2 450 4 -9002 -1 495 1 -4953 0 518 0 04 1 563 1 5635 2 584 4 1168
Σ X = 0 Σ Y = 2610 Σ X2 Σ XYX = 0 Y = 522 = 10 = 336
Ŷt = a + bX b = Σ XY – nXY = 336 - (5)(0)(522) = 33.6
ΣX2 - nX2 10 - (5)(0)2
a = Y – bX = 522 - (33.6)(0) = 522
Ŷt = 522 + 33.6X ; Y6 = 522 + (33.6)(3) = 622.80Y7 = 522 + (33.6)(4) = 656.40
Exercise (13)
99
• Jerilyn Ross, a New York City psychologist, specializes in treatingpatients who are phobic and afraid to leave their homes. Thefollowing table indicates how many patients Dr. Ross has seen eachyear for the past ten years. It also indicates what the robberyrate was in New York City during the same year.
Crime RateYear Patients (Robberies per 1,000 population)
1 36 58.32 33 61.63 40 73.44 41 75.75 40 81.16 55 89.07 60 101.18 54 94.89 58 103.3
10 61 116.2
Exercise (14)
100
• Using trend analysis, how many patients do you think Dr. Ross will see in years 11, 12 and 13? How well doesthe model fit the data?
• Apply linear regression to study the relationship between
the crime rate and Dr. Ross’s patient load. If the robberyrate increases to 131.2 in year 11, how many phobic patients will Dr. Ross treat? If the crime rate drops to 90.6, what is the patient projection?
Exercise (15)
101
• Case Study• North-South Airline
North-South Airline Data for Boeing 727-300 Jets
Northern Airline Data
Airframe Cost Engine Cost AverageYear Per Aircraft Per Aircraft Age (Hours)1992 $51.80 $43.49 6,5121993 54.92 38.58 8,4041994 69.70 51.48 11,0771995 68.90 58.72 11,7171996 63.72 45.47 13,2751997 84.73 50.26 15,2151998 78.74 79.60 18,390
Exercise (16)
102
• Case Study• North-South Airline (Cont.)
North-South Airline Data for Boeing 727-300 Jets
Southeast Airline Data
Airframe Cost Engine Cost AverageYear Per Aircraft Per Aircraft Age (Hours)1992 $13.29 $18.86 5,1071993 25.15 31.55 8,1451994 32.18 40.43 7,3601995 31.78 22.10 5,7731996 25.34 19.69 7,1501997 32.78 32.58 9,3641998 35.56 38.07 8,259
Exercise (17)
103
• Case Study• Akron Zoological Park
Admission Fee ($)
Year Attendance Adult Child Group
1998 117,874 4.00 2.50 1.501997 125,363 3.00 2.00 1.001996 126,853 3.00 2.00 1.501995 108,363 2.50 1.50 1.001994 133,762 2.50 1.50 1.001993 95,504 2.00 1.00 0.501992 63,034 1.50 0.75 0.501991 63,853 1.50 0.75 0.501990 61,417 1.50 0.75 0.501989 53,353 1.50 0.75 0.50