QA Forecasting

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


• 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


• 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)

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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


• 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


• 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


33


34


• 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


38

60708090

100110120130140150160

1993 1994 1995 1996 1997 1998 1999 2000 2001

TABLE 5.6 Midwestern Manufacturing’s Demand


• 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

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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


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


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


• 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


• 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


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

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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)


- 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


55


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


• 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


• 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


Figure 5.6 - Distribution about the Point Estimate of

$ 600 Million Payroll

59


• 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


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


• 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


• 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 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


• Figure 5.8 - Plot of Tracking Signals

74


• 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

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- 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



77



- 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


• 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


- 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


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


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


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


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

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