SBCO Forecasting HB Student

D. Anthony CheversSBCO 6240 - Production and Operations Management

Lecture 6 – Forecasting |

Lecture #4 – Forecasting

Definition & purposeForecasting models

Simple moving averageWeighted moving averageExponential smoothingTrend projectionsRegression analysis

Forecast errorsExercises

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Lecture 6 – Forecasting | 3

My interest is in the future because I am going to spend the rest of my life there.

—Charles F. Kettering


Some Reasons WhyForecasting is Essential

in OMNew Facility Planning – It can take 5 years to design

and build a new factory or design and implement a new production process.

Production Planning – Demand for products vary from month to month and it can take several months to change the capacities of production processes.

Workforce Scheduling – Demand for services (and the necessary staffing) can vary from hour to hour and employees weekly work schedules must be developed in advance.

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Forecasting - Defined

Forecasting is the prediction/estimation of future activities. Types of forecast – Economic, Technological and Demand• The first step in planning is forecasting, or estimating the

future demand for products and services and the resources necessary to produce these outputs. E.g. Rising Star finals (2005); Craft Village –New Kgn for World Cup Cricket 2007 -closure & Diana Ross (Jazz Festival, Jan 2008); Girl you have Intimate potential – Gregory Isaacs (2008)

• Operations managers need long range forecasts to make strategic decisions about products, processes and facilities.

• Operations managers need short range forecasts to assist them in making decisions about production issues that span only the next few weeks.

Lecture 6 – Forecasting | 6

Factors to Consider

Costs associated with a modelRequired accuracyRelevance of past data & availabilityForecast horizonPattern of dataTime needed for analysis


Introduction

Eight steps to forecasting:

• Determine the use of the forecast

• Select the items or quantities to be forecasted

• Determine the time horizon of the forecast

• Select the forecasting model or models

• Gather the data needed to make the forecast

• Validate the forecasting model

• Make the forecast

• Implement the results


Forecasting Models

MovingAverage

Exponential Smoothing

Trend Projections

Time Series Methods

Forecasting Techniques

Delphi Methods

Jury of Executive Opinion

Sales ForceComposite

Consumer Market Survey

Qualitative Models

Causal Methods

Regression Analysis

Multiple Regression


Method Data Required Relative Cost

Forecast Horizon

Application

Subjective Models Delphi Method Cross-impact analysis Historical analogy

Survey Result Correlation between events Years of data for similar situation

High High High

Long Term Long Term Medium-Long

Technological Forecasting Technological Forecasting Life-Cycle Demand

Causal Models Regression Econometric

Past data for variables Past data for variables

Moderate Moderate-High

Medium-Term Medium-Long

Demand Forecasting Economic Condition

Time Series Models Moving Average Exponential Smoothing

N most recent observations Previous forecast and recent observation

Very Low Very Low

Short-Term Short-Term

Demand Forecasting Demand Forecasting


Some Time Series Models

Simple Moving AverageK-Period MA =

∑(Actual Value in previous k periods) / k

Weighted moving averageK-Period WMA =

∑(weight for period i) (actual value in period i)

∑(weights)

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k

i=1 k

i=1



Exponential Smoothing method: A sophisticated weighted moving average method that calculates the average of a time series by giving recent demands more weight than earlier demands. Exponential smoothing is the most frequent used formal forecasting method because of its simplicity and the small amount of date needed to support it.

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SES: Ft+1 = α + (1-α)

(Demand this period) (Demand this period) (Forecast calculated last period)(Forecast calculated last period)

= α Dt + (1-α)Ft

Where Where αα is a smoothing parameter (0 ≤ is a smoothing parameter (0 ≤ αα ≤ 1). ≤ 1).

Or an equivalent equation: Ft+1 = Ft +α(Dt-Ft)


Equations - Simplified

1 SMA = Ʃ Demand# of periods

2 WMA = Ʃ (Demand x Weight) # of period

3 Expo.Smooth: Ft = Ft-1 + α (At-1 - Ft-1)


Calculation of 3-Month Simple Moving Average

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MonthActual Shed

Sales3-Month Moving Average

January 10

February 12

March 13

April 16 (10+12+13)/3 = 11 2/3

May 19 (12+13+16)/3 = 13 2/3

June 23 (13+16+19)/3 = 16

July 26 (16+19+23)/3 = 19 1/3


Limitations of Simple MA

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MonthActual Sales

3 Month MA

January 13

February 12

March 10

April 16 (13 + 12 + 10)/3 = 11.67

Simple MA does not capture trends nor seasonalities


Calculating Weighted Moving Averages

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Weights Applied

Period

3 Last month

2 Two months ago

1 Three months ago3 * Sales last moth + 2 * Sales two months ago + 1 * Sales three months ago

6 Sum of weights


Calculation of 3-Month Weighted-Moving Average

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MonthActual Shed Sales

3 Month MA

January 10

February 12

March 13

April 16 [(3*13) + (2*12) + (1*10)] 6 = 12

May 19 [(3*16) + (2*13) + (1*12)] = 14

June 23 [(3*19) + (2*16) + (1*14)] = 17

July 26 [(3*23) + (2*19) + (1*16)] = 20 ½

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61



New forecast =

previous forecast + (previous actual - previous)

or:

where

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tttt FAFF

actual period previous

constant smoothing

forecast previous

forecast new

t

t-

t

A

F

F


Equation [Ft = Ft-1 + α (At-1 – Ft-1)]

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Port of Baltimore Exponential Smoothing Forecasts for a = 0.10 and = 0.50

QUARTERACTUAL

TONNAGE UNLOADED

ROUNDED FORECAST USING =0.10*=0.10*

ROUNDED FORECAST

USING =0.50=0.50**

1 180 175 175

2 168 176 = 175.00 + 1.10(180 – 175) 178

3 159 175 =175.50 + 0.10(168-175.50) 173

4 175 173 =174.75 + 0.10(159-174.75) 166

5 190 173 =173.18 + 0.10(175-173.18) 170

6 205 175 =173.36 + 0.10(190-173.36) 180

7 180 178 =175.02 + 0.10(205-175.02) 193

8 182 178 =178.02 + 0.10(180-178.02) 186

9 ? 179= 178.22 + 0.10(182-178.22) 184

* Forecasts rounded to the nearest ton

TABLE 5.4


Selecting the Smoothing Constant ()

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n||

MAD errorsforecast deviation absolute Mean

n)(

MSE errorsforecast

Error Square Mean

actual|errorforecast |

n1

MAPEError

Percent AbsoluteMean

errorsforecast Bias

Select to minimize:


Table – Detailed Calculation

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[84/8 = 10.50 & 100/8 = 12.50; Select: α = 0.10]

Absolute Deviations and MADs for Port of Baltimore Example

QUARTERACTUAL

TONNAGE UNLOADED

ROUNDED FORECAST

USING =0.10*=0.10*

ABSOLUTE DEVIATIONS

FOR=0.10*=0.10*

ROUNDED FORECAST

USING=0.50=0.50**

ABSOLUTE FORECAST

USING =0.50=0.50**

1 180 175 5 175 5

2 168 176 8 178 10

3 159 175 16 173 14

4 175 173 2 166 9

5 190 173 17 170 20

6 205 175 30 180 25

7 180 178 2 193 13

8 182 178 4 186 4

Sum of absolute deviations 84 100

TABLE 5.5

MAD = 12.50 MAD = ●| deviations |

n= 10.50


Forecast Error – MSE[n = 4 (# of periods)]

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MSE = forecast errors2 87.25 ? n

As a practice, find the MSE for = 0.50 and conclude findings.

Quarter

Actual Tonnage Unloaded

Rounded Forecast with 0.10 constant Error @ 0.10 (Error)2

Rounded Forecast with 0.50 constant

Error @ 0.50 (Error)2

1 180 175 5 25 ? ? ?

2 168 176 8 64 ? ? ?

3 159 175 16 256 ? ? ?

4 175 173 2 4 ? ? ?

Sum of errors squared = 349 ? ? ?


Trend Projection

A time series forecasting method that fits a trend line to a series of historical data points and then projects the line into the future for medium-to-long range forecasts.

Look at linear (straight-line) trends only

Develop a linear trend line by a precise statistical method, we can apply the least squares method

A straight line that minimizes the sum of the squares of the vertical differences or deviations from the line to each of the actual observations.

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The Least Squares Method

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for Finding the Best-Fitting Straight Line [Time on x-axis]

Time period

Val

ues

of d

epen

dent

var

iabl

e (v

alue

s)

Deviation Deviation

11(error(error))

Deviation Deviation

55

Deviation Deviation

77

Deviation Deviation

66

Deviation Deviation

44

Deviation Deviation

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Deviation Deviation

22

Actual observation (y value)

1 2 3 4 5 6 7

Trend Line, y=a + bx


Least Squares Line

Described in terms of its y-intercept (the height at which it intercepts the y-axis) and its slope (the angle of the line)

If we compute the y-intercept and slope, we can express the line with the following equation: y = a + bx where y (called “y hat”) = computed value of the variable to

be predicted (dependent variable) a = y-axis intercept b = slope of the regression line (or the rate of change in y for

given change in x) [Slope = y/x]

x = the independent variable (which in this case is time)

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Trend Projection – General Regression equation

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X nX

YXnXYb

Ya

Ywhere

bXaY

Intercept [ a = y – bx ] axis-variable) (dependent predicted be

to variable the of value computed


Notations

ƅ = slope of the regression line

∑ = summation sign

x = known values of the independent variable

y = known values of the dependent variable

x = average of the value of the x’s

ӯ = average of the values of the y’s

n = number of data points or observations

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Exercise - BurkeCampbellChevers

The demand for electrical power at BurkeCampbellChevers Inc. Over the period 1999 to 2005 is shown below, in megawatts. Let’s forecast 2006 demand by fitting a straight-line trend to these data:

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Year Electrical Power Demand

1999 74

2000 79

2001 80

2002 90

2003 105

2004 142

2005 122

We can designate 1999 as year 1, 2000 as year 2, etc.


Solution: b = (∑XY – nXY) / (∑X2 – nX2)

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Year Time Period (x) Power Demand (y) x2 xy1999 1 74 1 742000 2 79 4 1582001 3 80 9 2402002 4 90 16 3602003 5 105 25 5252004 6 142 36 8522005 7 122 49 854

Sum 28 692 140 3063Average 4 98.86

x = 28/7 = 4 y = 692/7 = 98.86

b = 3,063 - (7)(4)(98.86) = 295 = 10.54

140 - (7)(42) 28

a = Y - b X

a = 98.86 - (10.54 x 4)

a = 98.86 - 42.16

a = 56.7


Least Square Trend Equation

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Thus, the least square trend equation is: y = 56.70 + 10.54 x

To project demand in 2006, we first denote the year 2006 as x=8

Demand in 2006 = 56.70 + 10.54 (8)141.02 or 141 megawatts

Estimate for 2007 – Year=9 (x=9)Demand in 2006 = 56.70 + 10.54 (9)151.56 or 152 megawatts


Electrical Power and the Computed Trend Line

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Pow

er d

eman

d

1997

160

150

140

130

120

110

100

90

80

70

60

50

1998 1999 2000 2001 2002 2003 2004 2005

Year

Trend Line, y= 56.70 + 10.54x


Regression Analysis

Causal forecasting models usually consider several variables that are related to the quantity being predicted

This approach is more powerful than the time-series methods that use only the historical values for the forecasting variable

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Regression Analysis For example, the sales of PC’s might be related to

advertising budget, the price charged, competitors’ prices, promotional strategies, the economy, disposable income, unemployment rates or time.

In this case, PC sales would be called the dependent variable (Y) and the other variables would be called independent variables (X).

We will use Y with one other variable (X)

The most common quantitative casual forecasting model is linear-regression analysis

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Linear Regression Analysis

A straight-line mathematical model to describe the functional relationships between independent and dependent variables

Can use the same mathematical model employed in least squares method of trend projection to perform a linear regression analysis

The dependent variable that we want to forecast is still y-hat but the independent variable, x, need no longer be time.

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Using Regression Analysis to Forecast

Question: If the Local Payroll is 6 in the 7th month, what is the Sales?

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YTriple A' Sales

($100,000's)

XLocal Payroll($100,000,000)

2.0 13.0 32.5 42.0 22.0 13.5 7

Y = 1.75 + 0.25X


Solution b = (∑XY – nXY) / (∑X2 – nX2)

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Y X XY2 1 2 13 3 9 9

2.5 4 10 162 2 4 42 1 2 1

3.5 7 24.5 49Total 15 18 51.5 80

Average 2.50 3.00If X is 6

b = 51.5 - (6 x 3 x 2.5)80 - (6 x 3 x 3)51.5 - 4580 - 546.50 = 0.2526.0

a = 2.5 - (0.25 x 3)2.5 - 0.75 = 1.75

Y = = 1.75 + (0.25X) [Where X is 6]

= 1.75 + (0.25 x 6)

= 1.75 + 1.5 = 3.25

Y X XY X2

2 1 2 1

3 3 9 9

2.5 4 10 16

2 2 4 4

2 1 2 1

3.5 7 24.5 49

15 18 51.5 80

2.50 3.00

Total Average

If X is 6


Solution - Projection

If the local Chamber of Commerce predicts that the payroll will be $600 million next period, we can estimate sales with the regression equation:

• 1.75 + 0.25(6)• 1.75 + 1.50 = 3.25

Thus sales = $325,000

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Exercise 1Demand for heart transplant surgery at Washington General Hospital has increased steadily in the past few years, as seen in the table.

Year Outpatient Surgeries Performed

1 452 503 524 565 586 ?

_____________________________

The director of medical services predicted six years ago that the demand in year 1 would be for 41 surgeries

(a) Use exponential smoothing, first with a smoothing constant of 0.6 and then with one of 0.9, to develop forecasts for years 2 through 6.

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Exercise 2

Tongren Construction Company renovates old homes in Balvenie, Mandeville. Over time, the company has found that its dollar volume of renovation work is dependent on the Balvenie area payroll. The following table lists Tongren’s revenues and the amount of money earned by wage earners in Balvenie during the following six years.

Tongren’s Sales Local Payroll ($000,000) ($000,000,000)

2 13 32.5 42 22 13.5 7

Forecast sales if the payroll in period #7 is 8.

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Solution – Tutorial #1

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Exponential smoothing, α = 0.6:

Year Demand 3-yr Moving Average Abs. Dev.1 45 412 50 41.0 + 0.6(45-41) = 43.4 6.63 52 43.4 + 0.6(50-43.4) = 47.4 4.64 56 47.4 + 0.6(52-47.4) = 50.2 5.805 58 50.2 + 0.6(56-50.2) = 53.7 4.306 ? 53.7 + 0.6(58-53.7) = 56.3

∑ = 21.3

MAD = 5.3

Exponential smoothing, α = 0.9:

Year Demand 3-yr Moving Average Abs. Dev.1 45 412 50 41.0 + 0.9(45-41) = 44.6 5.43 52 44.6 + 0.9(50-44.6) = 49.5 2.54 56 49.5 + 0.9(52-49.5) = 51.8 4.205 58 51.8 + 0.9(56-51.8) = 55.6 2.406 ? 55.6 + 0.9(58-55.6) = 57.8

∑ = 14.5

MAD = 3.6

3-Year Moving AverageYear Demand 3-yr Moving Average Abs. Dev.

1 452 503 524 56 (45 + 50 + 52)3 = 49 7.005 58 (50 + 52 + 56)3 = 52.7 5.306 ? (52 + 56 + 58)3 = 55.3

∑ = 12.3

MAD = 6.15


Solution – Tutorial #2

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Y X XY X2

2 1 2 13 3 9 9

2.5 4 10 162 2 4 42 1 2 1

3.5 7 24.5 49Total 15 18 51.5 80

Average 2.50 3.00If X is 8?

b = 6(51.5) - (18 x 15)6(80) - (18 x 18)309 - 270480 - 32439.00 = 0.25156.0

a = 2.50 - (0.25 x 3.0)2.50 - 0.75 = 1.75

Y = = 1.75 + (0.25 x 8)= 1.75 + 2.0 = 3.75

X Y XY X2

1 17 17 12 16 32 43 17 51 94 21 84 165 20 100 256 20 120 367 23 161 498 25 200 649 24 216 81

Total 45 183 981 285Average 5 20.33

b = 9(981) - (45 x 183) = (8829) - (8235)(9 x 285) - (45 x 45) (2565) - (2025)594 =540 1.1

a = 20.33 - (1.1 x 5) = 20.33 - (5.5)14.83

Y = 14.83 + (1.1 x 10) [Next year is 10]

14.83 + (11) = 25.83


Normal Distribution

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.00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359 .1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753 .2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141 .3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517 .4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879

.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 .6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 .7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7828 .7852 .8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133. 9 .8159 .8186 .212 .8238 .8264 .8289 .8315 .8340 .8365 .8389

1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .86211.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .88301.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .90151.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .91771.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319

1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .94411.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .95451.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .96331.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .97061.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767

2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .98172.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .98572.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .98902.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .99162.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936

2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .99522.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .99642.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .99742.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .99812.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986

3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .99903.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .99933.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .99953.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .99973.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998

∞ 0 z +- ∞

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NEXT LECTURE:Inventory Management

D. Anthony Chevers

[email protected], Room #28

SBCO Forecasting HB Student

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