03 Forecasting

8/2/2019 03 Forecasting

1/29

1

MD 021 - Operations Management

Forecasting

Outline

Components of demand

Judgment methods

Linear regression

Time series methods

Forecast errors


2/29

2

Time

Qu

antity

(a) Average: Data cluster about a

horizontal line

Time

(b) Linear trend: Data consistently increase

or decrease

Qu

antity

J F M A M J J A S O N D

Months

Year 1

Year 2Quan

tity

(d) Cyclical movements: Data revealgradual increases and decreases overextended periods of time

Components of Demand

Years

1 2 3 4 5 6

Quan

tit

(c) Seasonal influence: Data consistentlyshow peaks and valleys


3/29

3

Judgment Methods

Sales force estimates

Executive opinion

Market research

Delphi method


4/29

4

Linear Regression

Y a bX i i= +

where:

Y = dependent variable

X = independent variable

a = Y-intercept of the line

b = slope of the line


5/29

5

Measures of Forecast Accuracy in Linear Regression

Coefficient of correlation

Coefficient of determination

Standard error of the estimate


6/29

6

Regression Analysis Example

The manager of Als Diner is interested in forecasting the number of potato skin appetizers soldeach week. He believes that the number sold has a linear relationship to the price and uses linearregression to determine if this is the case.

WeekX

(Price)Y

(Appetizers)

1. $2.70 7602. 3.50 510

3. 2.00 9804. 4.20 2505. 3.10 3206. 4.05 480


7/29

7

The Excel output is below:

Regression Statistics

Multiple R 0.843R Square 0.711

Adjusted R Square 0.639Standard Error 165.257Observations 6

ANOVA

df SS MS F SignificanceF

Regression 1 269160 269160 9.856 0.035Residual 4 109239 27309Total 5 378400

Coefficients StandardError

t Stat P-value

Intercept 1454.604 295.939 4.915 0.008Price ($) -277.628 88.434 -3.139 0.035


8/29

8

Linear Regression Example

A professor is interested in determining whether average study hours per week are a good predictorof test scores. The results of her study are:

Hours (X) Score (Y)

3.0 902.1 955.8 653.8 804.2 953.2 60

5.3 854.6 70

A student says: "Professor, what can I do to get a B on the next test?The professor asks, "On average, how many hours do you spend studying for this course per week?"

The student responds, "About 2 hours."

Use linear regression to forecast the student's test score.


9/29

9


Regression StatisticsMultiple R 0.391R Square 0.153Adjusted R Square 0.0121Standard Error 13.544

Observations 8

ANOVA

df SS MS F SignificanceF

Regression 1 199.246 199.246 1.0861 0.3375Residual 6 1100.753 183.458

Total 7 1300

Coefficients Standard Error t Stat P-valueIntercept 97.325 17.301 5.625 0.0013Study hours -4.331 4.156 -1.042 0.3375


10/29

10

Time Series Methods

Naive forecasts

Moving averages

Weighted moving averages

Exponential smoothing

Trend-adjusted exponential smoothing

Regression method

Multiplicative seasonal method


11/29

11

Moving Average Method

n

AMAF

n

iit

nt

== =

1

CustomerMonth arrivals

1 8002 7403 8104 790

Use a 3-month moving average to forecast customer arrivals for month 5.

F5 =

If the actual demand for month 5 is 805 customers, what is the forecast for month 6?

F6 =


12/29

12

Month

Cus

tomerarrivals

0 5 10 15 20 25

6-month MA

forecast

3-month MA

forecast

Comparison of 3-month and 6-month Moving Average Forecasts


13/29

13

Weighted Moving Average Method

11)1(1 ... +++= tntnntnt AwAwAwF


1 8002 7403 8104 790

Let W W W1 2 3050 0 30 0 20= = =. , . , . .andCalculate the forecast for month 5.

F5 =

If the actual demand for month 5 is 805 customers, what is the forecast for month 6?

F6 =


14/29

14

Exponential Smoothing

11)1(F += ttt AF


1 800

2 7403 8104 790

Suppose F3 783 0 20= =customers and . .

What is the forecast for month 5?

F4 =

F5 =

IfD5 805= , what is the forecast for month 6?

=6F


15/29

15

Trend-Adjusted Exponential Smoothing

ttt

ttttt

tttt

TSTAF

TTAFTAFTT

TAFATAF

+=

+=

+=

+

1

111 )(

)(S

Month12345

Customer Arrivals4852505455

Using months 1-4, an initial estimate of the trend is 2 [(4-2+4)/3 = 2]. The startingforecast for month 5 is 54+2 = 56. Using 3.0= and 4.0= , forecast the number of

patients in month 6.

=5S

=5T

=6TAF


16/29

16

If the actual number of patients in month 6 is 58, what is the forecast for month 7?

=6S

=6T

=7TAF


17/29

17

Regression Method

Example: Garcia Garage

Month (t) Number oftime periods

from t = 0

Number of OilChanges (Y)

Jan. 1 41Feb. 2 46Mar. 3 57Apr. 4 52May 5 59

Jun. 6 51Jul. 7 60Aug. 8 62

1. Forecast the numbers of oil changes in September, October, and November.

2. What is the average value of the trend?


18/29

18


Regression StatisticsMultiple R 0.817R Square 0.668Adjusted RSquare 0.613

StandardError 4.572Observations 8.000

ANOVA

df SS MS FSignificance

F

Regression 1.000 252.595 252.595 12.085 0.013Residual 6.000 125.405 20.901Total 7.000 378.000

CoefficientsStandard

Error t Stat P-value Lower 95%Upper95%

Intercept 42.464 3.562 11.921 0.000 33.748 51.181X Variable 1 2.452 0.705 3.476 0.013 0.726 4.179


19/29

19

Multiplicative Seasonal Method

Step 1: Calculate the trend line based on the available data using regression.

Step 2: Calculate the centered moving average, with the number of periodsequal to the number of seasons.

Step 3: Calculate the seasonal relative for a period by dividing the actualdemand for the period by the corresponding centered moving average.

Step 4: Calculate the overall estimated seasonal relative by averaging theseasonal relatives from the same periods over the cycle.

Step 5: Calculate the trend values for each of the periods to be forecast basedon the trend line determined in Step 1.

Step 6: To get a forecast for a given period in a future cycle, multiply theseasonal factor by the trend values.


20/29

20

Multiplicative Seasonal Method ApplicationQuarter Demand CMA (4 seasons) MA (2 periods) Seasonal Relatives Normalized S.R.

1 100

2 400250

3 300 261.5 1.147227533 1.171002862

2734 200 274 0.729927007 0.745054133

2755 192 285.5 0.672504378 0.686441468

2966 408 298 1.369127517 1.397501537

3007 384 Total 3.918786436 4

8 216

9 331 (trend value*) 227 (forecast)

10 344 (trend value*) 480 (forecast)

11 356 (trend value*) 417 (forecast)12 369 (trend value*) 275 (forecast)

* Using regression, the trend line is 218.86 + 12.48t.


21/29

21

Forecast ErrorstttFAe =

Systematic errors --- Bias

Random errors --- Variability

Example:Day 1 Day 2 Day 3 Day 4

ActualDemand 100 100 100 100

Forecast 1 105 105 105 105Forecast 2 50 150 50 150


22/29

22

Forecast Error Measures

Bias:

Average errorn

en

tt

= =1

Variability:

Mean squared error MSE1

1

2

= =

n

en

tt

Standard deviation s = MSE

Mean absolute error MADn

en

tt

= =1

Mean percent absolute error MAPEn

A

en

tt

t

==1

)]100([


23/29

23

Control Chart for Forecast Errors

Upper Control Limit: 0 MSEzUCL +=

Lower Control Limit: 0 MSEzLCL =

z = the number of standard deviations from the mean

Where to find z given the percentage of the control chart, 0P ?

Where to find z given the probability for type I error, ?

Normal Distribution Table (page 882-883, Table B)

Look for z corresponds to the probability:

Pr{Z


24/29

24

Summarizing Forecast Accuracy

Period Actual (A) Forecast (F) Error (e=A-F) Abs Error Error Sq [(Abs E)/A] x 100

1 113 95 18 18 324 15.93

2 85 80 5 5 25 5.88

3 96 103 -7 7 49 7.294 86 119 -33 33 1089 38.37

5 121 117 4 4 16 3.31

6 100 125 -25 25 625 25.00

7 142 67 75 75 5625 52.82

8 92 96 -4 4 16 4.35

9 72 116 -44 44 1936 61.11

Total -11 215 9705 214.06

MAD = 23.9

MSE = 1213.1

s = 34.8

MAPE = 23.8%


25/29

25

Tracking and Analyzing Forecast Errors

Period Actual (A) Forecast (F) Error (e=A-F)

10 102 130 -28

11 107 102 5

12 112 89 23

13 118 97 21 Average error (periods 1-18)= -0.3914 89 115 -26 Standard deviation (periods 1-9) = 34.8

15 142 82 60

16 100 130 -302s control limits: 0 +/- 2(34.8) = 0 +/- 69.6

17 94 137 -43

18 111 89 22

Total

4

-80

-60

-40

-20

0

20

40

60

80

10 11 12 13 14 15 16 17 18

UCL = 6 .6

LCL = -6 .6


26/29

26

UCL

LCL

Examples of Nonrandomness

(a) Point outside control limits

UCL

LCL

(b) Trend

UCL

LCL

(c) Cycling

UCL

LCL

(d) Bias

Source: Figure 3.12, Page 100, from Stevenson


27/29

27

Forecasting Summary NotesChoosing a Forecasting Method

General considerations:Method Pros Cons

Judgment Can be used in the absence of historicaldata (e.g. new product)

Helpful in identifying turning points andpreparing medium- and long-term forecasts

Subjective estimates are subject to the biases andmotives of the estimators

Causal Most sophisticated method Very good for predicting turning points and

preparing medium- and long-term forecasts

Must have historical data on independent anddependent variables

Relationships can be difficult to specify

Timeseries

Easy to implement Work well when demand is relatively stable

Rely exclusively on past demand data Only useful for short-term estimates

Specific considerations for time series methods:Method Pros Cons

Naive forecast Easiest method, low cost Works well when random

errors are small

Results in highly variable forecasts if the randomerrors are large

Simple movingaverage Easiest moving averagemethod To some extent, controls for

random error

Data must be retained for n periods Forecast lags changes in the underlying average of

demand


28/29

28

Weighted moving

average

Weights can be varied to be

responsive to demand pattern To some extent, controls for

random error

Data must be retained for n periods

Forecast lags changes in the underlying average ofdemand

Exponential smoothing Requires little data can be varied to be

responsive to demand pattern To some extent, controls for

random error

Forecast lags changes in the underlying average ofdemand

In general, emphasize recent demand (i.e. small n, large weights for recent observations, large ) for dynamic (i.e.uncertain) demand patterns. Emphasize historical experience for stable demand patterns. If a trend is present,simple moving average, weighted moving average, and exponential smoothing estimates will always lag actualdemand.


29/29

29

Forecasting NotesChoosing a Time Series Forecasting Method

Evaluating forecast performance: Forecast errors can be classified as either bias errors or random errors. Biaserrors are the results of systematic over- or underestimation. Random errors are unpredictable. Ideally, a forecastshould minimize both bias and random errors.

Method PurposeMean forecasterrors

Measures bias

Mean squared error(MSE)

Measures the dispersion of forecast errors; large errors get more weight than whenusing MAD

Mean absolutedeviation

(MAD)

Measures the dispersion of forecast errors; method is intuitive

Mean absolute percenterror (MAPE)

Measures the dispersion of forecast errors relative to the level of demand

Forecast error controlchart

Determines whether the method of forecasting is accurately predicting actual changesin demand

03 Forecasting

Documents

Transcript of 03 Forecasting