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Review the attached Word document with your homework assignment for this module. Complete the homework by typing your response in the Word document and uploading it to this assignment. Questions may be posted on the Q&A discussion forum.

I attached below :

1- homework qestion

2- Powerpoint module

INFO 564 Homework Assignment 5

This work must be done completely in EXCEL. Answer each question on a separate tab. Label

each tab appropriately. You can copy and paste the data given into an Excel worksheet.

South Shore Construction builds permanent docks and seawalls along the southern shore of

Long Island, New York. The following data show quarterly sales revenues (in $’000s) for the

past 5 years.

Quarter

1

2

3

4

Year 1

20

100

175

13

Year 2

37

136

245

26

Year 3

75

155

326

48

Year 4

92

202

384

82

Year 5

176

282

445

181

Question 1

Plot this data with quarters from years 1-5 on the horizontal axis. What components do you

see in this time series?

Question 2

Ignore any trend or seasonality in the data.

a. Suppose the company uses moving averages to make forecasts. Make forecasts all the

way through Q4 Year 5. Assume the company uses (i) 3-quarterly moving averages and

(ii) 4-quarterly moving averages.

b. Compare the two sets of forecasts from (a) on the basis of Mean Absolute Percent

Deviation. Which is more accurate – 3 quarterly moving average or 4 quarterly moving

average?

c. On a line chart plot the time series along with the forecasts from the method you select

in (b).

Question 3

Ignore any trend or seasonality in the data.

a. Suppose the company uses weighted moving averages to make forecasts. What are the

forecasts starting with Q4 Year 1 all the way through Q4 Year 5? Assume the company

uses (i) 3-quarterly moving averages with weights 0.6, 0.3, and 0.1 and (ii) 4-quarterly

moving averages with weights 0.4, 0.3, 0.2, and 0.1. In both cases the most weight is

given to the most recent quarter and the least to the oldest quarter in the moving

average.

b. Compare the two sets of forecasts from (a) on the basis of Mean Absolute Percent

Deviation. Which is more accurate – 3 quarterly weighted moving average or 4

quarterly weighted moving average?

c. On a line chart plot the time series along with the forecasts from the method you select

in (b).

Question 4

Again ignore any trend or seasonality in the data.

a. Suppose the company uses exponential smoothing to make forecasts. What are the

forecasts for periods Q2 Year 1 through Q4 Year 5 assuming (i) alpha = 0.3 and (ii) alpha

= 0.7? In both cases assume that the forecast for Q1 Year 1 was 25 units.

b. Compare the two sets of forecasts from (a) on the basis of Mean Absolute Percent

Deviation. Which is more accurate – alpha of 0.3 or alpha of 0.7?

c. On a line chart plot the time series along with the forecasts from the method you select

in (b)

Question 5

Now make adjustments for trend and seasonality.

a. Quantify the trend in the time series. What does the trend equation tell you?

b. Quantify the seasonality in the time series by calculating seasonality indexes. What do

these indexes tell you?

c. Using the trend and the seasonality information from (a) and (b) make forecasts from

Q1 Year 1 through Q4 Year 5.

d. Calculate the Mean Absolute Percent Deviation for the forecasts in (c).

e. On a line chart plot the time series along with the forecasts from (c).

Question 6

Using the most accurate method of all of the above,

a. Make forecasts for the four quarters of Year 6.

b. Plot these forecasts on the same line chart as the time series.

c. Summarize in a few lines your findings from your answers to Q1 through Q6b.

INFO 564

Operations & Supply Chain Management

Module 5a: Measuring Forecast Accuracy

Copyright 2017 Montclair State University

Forecast Accuracy

• Measured retrospectively based on past forecasts and

their errors

• Error = Actual – Forecast

• Also referred to as deviation

• Common measures are functions of past errors

• Mean Error (also called bias)

• Mean Absolute Error (MAE)

• Mean Absolute Percent Error (MAPE)

Mean Error

• Suppose we made forecasts for 5 past

periods and wish to measure their

accuracy.

• Error = Actual – Forecast

• Mean error is the average of the errors

in the 5 periods.

• Tells us that on average we are underforecasting by 1.2 units.

• Caveat: Small mean error does not

necessarily mean accurate forecasts

• Large negative errors in some periods could

cancel out large positive errors in others

Period

1

2

3

4

5

Actual

22

29

29

26

26

Forecast

25

26

26

28

21

Mean Error =

Error

-3

3

3

-2

5

1.2

Doesn’t seem to be of same

magnitude as the errors.

Mean Absolute Error

• Very popular measure

• Absolute Error ignores the sign

associated with error.

• Mean Absolute Error averages

the absolute errors.

• More reliable measure of

forecast errors.

• Forecasts are typically off by 3.2

units

• But is 3.2 big or small? MAE

does not tell us

Period

1

2

3

4

5

Actual

22

29

29

26

26

Forecast

25

26

26

28

21

Error

-3

3

3

-2

5

Absolute

Error

3

3

3

2

5

Mean Absolute Error =

Is of same magnitude

as the errors.

3.2

Mean Absolute Percent Error

• Absolute Percent Error = Absolute

Error ÷ Actual

• Mean Absolute Percent Error =

average of all the Absolute

Percent Errors.

• On average, forecasts are off by

about 12.2% of actual.

• Provides estimate of the relative

size of forecast error

• Another popular measure of

forecast accuracy

Period

1

2

3

4

5

Actual

22

29

29

26

26

Forecast

25

26

26

28

21

Error

-3

3

3

-2

5

Absolute

Absolute

Error

Percent Error

3

13.6%

3

10.3%

3

10.3%

2

7.7%

5

19.2%

Mean Absolute Percent Error =

12.2%

Forecasts off typically by

about 12.2% of actual values.

In Conclusion…

• Three very common measures of forecast accuracy

• Mean Error (also called bias)

• Mean Absolute Error (MAE)

• Mean Absolute Percent Error (MAPE)

• Found in all forecasting software

• Other measures available for specialized situations

• Can be used to compare different forecasting methods

• All based on past performance

• No guarantee of future performance of forecasts

INFO 564

Operations & Supply Chain Management

Module 5b: Patterns in Time-Series Data

Closing Price of Stock

Closing Price $

What are Time Series?

80

60

40

20

0

• Data collected over time

2

3

4

5

6

Week

7

8

9

10

Daily High Temperature

Temperature (F)

Monthly energy bills

Yearly college enrollment

Daily closing value of the DJIA

Hourly temperatures in a given

zip-code

• Quarterly earnings of a

company

90

88

86

84

82

80

78

76

74

1

2

3

4

5

6

7

8

Laptop Sales

3500

Sales (Units)

•

•

•

•

1

3000

2500

2000

1500

9 10 11 12

Patterns in Time Series: Randomness

• No pattern

• Seemingly random small ups

and downs in the time series

• Too many small causes that

contribute

• Difficult to forecast

• Impact reduced by averaging

1200

1100

# of Calls

• Movement not too big

compared with general level of

the series

Number of Calls to Help Center

1000

900

800

700

600

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Day

Patterns in Time Series: Trend

• The spikes are random ups

and downs

• Useful for forecasting

• If we can assume trend will

continue

Temperature (F)

• Upward: cloud services, Alexa,

electric cars, battery life

• Downward: compact discs, stickshifts, cash

Daily High Temperature

90

88

86

84

82

80

78

76

74

1

2

3

4

5

6

7

8

9

10

11

12

10

11

12

Day

Bank Balance ($)

6000

5000

Dollars

• Trend: sustained upward or

downward movement

4000

3000

2000

1000

0

1

2

3

4

5

6

7

End of Week

8

9

Patterns in Time Series: Seasonality

Laptop Sales

• Often products and services

exhibit seasonal demand

• Randomness is present.

2500

2000

1500

1000

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

• Patterns can repeat annually,

monthly, weekly, even daily.

• If pattern can be expected to

continue, can use in forecasting.

• Pattern is not perfectly repeated.

3000

# of Laptops

• Some periods are consistently high,

some consistently low

• Christmas trees, school supplies,

vacation travel, business travel,

construction, etc.

3500

Patterns in Time Series: Cycles

•Cycles are like seasonality, but they repeat over much longer periods

•Correspond with business cycles, economic cycles

•Relevant in medium-term (3-5 years) and long-term (5 or more

years) forecasting

•Require lots of past data to recognize these patterns

Year-1

Year-2

Year-3

Year-4

Year-5

Year-6

Year-7

Year-8

Year-9

Using the Patterns for Forecasting

• Time-series can exhibit one or more of these patterns.

• Recognizing patterns – trends, seasonality, cyclicality

– allows us to use them for forecasting

• We have to be able to quantify them.

• Assumption: these patterns will hold in the future.

• Cyclicality is hard to recognize and quantity

• Only occasionally used in time-series forecasting

INFO 564

Operations & Supply Chain Management

Module 5c: Forecasting with Moving Averages

Example

Month

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

Apr

Demand

988

978

1059

1013

1092

948

1002

952

958

1029

978

917

944

955

998

1017

• Past demand for a product is given in the time-series on the left.

• A graph of the series is shown below:

Demand

1200

1100

1000

900

800

700

600

500

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr

• The series shows no trend or seasonality

• Only random ups and downs.

• What is our forecast for the next period, May?

Approach – Moving Averages

• Forecast for May based on a moving average of recent data

• Example:

• 3-month moving average: Average of demand in February, March,

and April

• = (955+998+1017)/3 = 990 units

• 6-month moving average: Average of demand from November

through April

• = (978+917+…+1017)/6 = 968 units

• Moving averages assume only recent periods are relevant.

• Older periods may be ignored safely

• Reasonable assumption in real life

Moving Averages: Impact of Period

• Graph shows demand, 3-month, and

6-month moving average forecasts

• 3-month moving average forecasts more

responsive to actual demand

• 6-month moving average forecasts less

responsive to demand

• Which is better?

• Longer periods dampen random fluctuations

(good) but also dampen trends (bad)

• Shorter periods respond to random

fluctuations (bad) and to trends (good)

Demand & Forecasts

1050

1000

950

900

850

800

750

700

Jan

Feb

Mar

Apr

Demand

• We want forecasts to ignore random

fluctuations but highlight trends

May

Jun

3-Mth MA

Jul

Aug

6-Mth MA

Sep

Oct

How to pick a period for moving averages?

• Experience, knowledge, instinct

OR…

• Use past data to experiment

• 3-month moving average forecasts

have a MAD of 37

• 6-month moving average forecasts

have a MAD of 36

• 6-month moving averages seem

slightly superior

• May be the choice going forward

Month

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

Apr

Demand

1002

952

958

1029

978

917

944

955

998

1017

3-Mth MA Abs Dev

1017

15

1014

62

967

9

971

58

980

1

988

72

975

31

947

9

939

59

966

52

37

MAD

6-Mth MA Abs Dev

1013

11

1015

64

1011

53

994

35

997

18

978

61

973

28

963

8

964

34

970

47

36

MAD

Weighted Moving Averages

• Simple moving averages assume equal importance (weight) of

each period used to compute the moving average.

• We could give different weights to different periods

• In a 3-month weighted moving average, 60% to the most recent

period, 30% to the one before, and 10% to the oldest period.

May forecast = 0.6*1017+0.3*998+0.1*955 = 1005 units

• Weights are subjective

• Most recent period is considered the most important and gets most

weight

• Oldest period is least important and gets the least weight

• Add up to 1

Summary

• Moving averages are appropriate when the time-series shows

no trend or seasonality

• Subjective considerations

• Averaging period

• Weighting if any

• Moving averages are reactive

• When there is trend moving averages will always lag behind

• Easy to understand

• Easy to implement on a spreadsheet

INFO 564

Operations & Supply Chain Management

Module 5d: Forecasting with Exponential Smoothing

Exponential Smoothing

• A weighted-average forecasting method

• Forecasts are a series of adjustments to previous forecasts

• New Forecast = Old Forecast + Adjustment

• Adjustment depends on forecast error

• All past periods are used in calculating the new forecast

• Unlike moving averages

• Given declining weights; most recent period the most.

• A subjective parameter, denoted α, is used to perform the

weighting

• α is between 0 and 1

Basic Idea

• Ft+1 = Forecast for period t+1

(upcoming period)

• Ft= Forecast for period t

(period that just ended)

• At= Actual demand for period t

Now

Ft

Period t+1

Period t

Ft+1

At

Adjustment

Ft+1= Ft + α(At – Ft)

Thus Ft+1 is Ft plus a portion of

the forecast error.

May also be written as:

Ft+1 = αAt + (1-α)Ft

Easier for calculation

Example

Suppose α = 0.4

Forecast for

April=30

• FMay = FApr + α(AApr-FApr)

= 30+0.4*(25-30) = 28

Actual for

April=25

• FJun = FMay + α(AMay-FMay)

Forecast for

May=28

= 28+0.4*(29-28) = 28.4

Actual for

May=29

= 28.4+0.4*(32-28.4) = 29.84

Forecast for

June=28.4

Actual for

June=32

Forecast for

July=29.84

• FJul = FJun + α(AJun-FJun)

• And so on…

Effect of α

• Alternate form Ft+1 = αAt + (1-α)Ft

• Can interpret Ft+1 as a weighted average of At and Ft

• α is the weight given to At, 1-α the weight given to Ft

• Large values of α give more weight to actual demand At

• Forecasts become more responsive to actual demand

• Small values of α give less weight to At

• Forecasts less responsive to actual demand

Effect of α

Forecasts

Month

Exponential Smoothing Forecasts

Demand Alpha=0.2 Alpha=0.8

988

1000

1000

1150

Feb

978

998

990

1100

Mar

1059

994

980

1050

Apr

1013

1007

1043

May

1092

1008

1019

Jun

948

1025

1077

Jul

1002

1009

974

Aug

952

1008

996

Sep

958

997

960

Oct

1029

989

959

Nov

978

997

1015

Dec

917

993

986

Jan

944

978

931

Feb

955

971

942

Mar

998

968

953

Apr

1017

974

989

* Forecasts for this January are

assumed numbers

Units Demand

Jan*

1000

950

900

850

800

750

700

Jan

Feb

Mar

Apr

May

Jun

Demand

Jul

Aug

Sep

Alpha = 0.2

Oct

Nov

Dec

Jan

Feb

Mar

Alpha=0.8

Forecasts with the smaller value of α are much steadier than with the larger

value of α

Apr

Picking a value of α

• Judgment, experience, intuition

• Using value of α that works well on past

data

• Table on right, forecasts for past periods

using α=0.2 and α=0.8.

• α=0.2 provides more accurate forecasts

(smaller MAD)

• Going forward, α=0.2 may be a better value

than α=0.8.

• Can also experiment with other values to

obtain best value.

• No guarantee that this value will work

well in the future

Month

Demand

Alpha=0.2

Abs.Dev Alpha=0.8

Abs.Dev

Jan

988

1000

12.4

1000

12.4

Feb

978

998

20.0

990

12.5

Mar

1059

994

65.6

980

79.1

Apr

1013

1007

6.1

1043

30.6

May

1092

1008

83.8

1019

72.8

Jun

Jul

948

1002

1025

1009

77.0

7.3

1077

974

129.4

28.4

Aug

952

1008

56.3

996

44.8

Sep

958

997

38.4

960

2.3

Oct

1029

989

40.0

959

70.2

Nov

978

997

18.4

1015

36.3

Dec

917

993

76.3

986

68.8

Jan

944

978

33.7

931

13.5

Feb

955

971

15.8

942

13.8

Mar

998

968

30.1

953

45.6

Apr

1017

974

43.5

989

28.5

39.0

43.1

MAD

MAD

Advantages of Exponential Smoothing

• More accurate than more sophisticated methods

• Easy to use and understand

• Easy to adjust importance given to actual demand through α

• Nested mechanism means that all past periods are used in making a

forecast

• FNov depends on AOct and FOct. FOct depends on ASep and FSep. FSep depends on

AAug and FAug, and so on.

• Thus FNov depends on AOct, ASep, AAug, and so on.

• No period is ignored

• More importance is given to more recent data

• Included in all popular forecasting packages

INFO 564

Operations & Supply Chain Management

Module 5e: Trend in Time Series

Example – # of Passengers

• Weekly number of passengers

carried by a bus service

reveals an upward trend.

• How to quantify this trend?

• Trend in this instance seems

linear

• A straight line with random

departures from it

#

Week Pass

1

305

2

302

3

380

4

372

5

452

6

404

7

424

8

408

9

533

10

522

11

510

12

588

13

604

14

581

15

585

16

617

# of Passengers

700

600

500

400

300

200

100

0

1

2

3

4

5

6

7

8

9

Week

10

11

12

13

14

15

16

Example – # of Passengers

• Weekly number of passengers

carried by a bus service

reveals an upward trend.

• How to quantify this trend?

• Trend in this instance seems

linear

• A straight line with random

departures from it

#

Week Pass

1

305

2

302

3

380

4

372

5

452

6

404

7

424

8

408

9

533

10

522

11

510

12

588

13

604

14

581

15

585

16

617

# of Passengers

700

600

500

400

300

200

100

0

1

2

3

4

5

6

7

8

9

Week

10

11

12

13

14

15

16

Quantifying Trend

• A line is described by a slope and

an intercept

• Y = bX + a

• b: slope, a: intercept

• The slope b measures the rate at

which the line climbs or falls – trend

• How to find slope and intercept?

• Line of best fit

• Formula for b and a

• Software

• Spreadsheet

# of Passengers

700

600

500

400

300

200

100

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Week

• The line of best fit above is

# of Passengers = b(Week) + a

• b captures the trend: the rate at which the

number of passengers is increasing each

week.

16

Trendline with EXCEL

Put mouse cursor on plot

and Left Click.

Trendline with EXCEL

Right Click and select Add

Trendline…

Trendline with EXCEL

Check

these.

Trend Line

• y = 21.41x + 292.2

• # of Passengers = 21.41*Week # + 292.2

• Interpretation

• The number of passengers increases at the rate of 21.4 (slope) each

week starting from a base of about 292 passengers

• The line can be used for forecasting by “projecting the trend”

• Simply substitute the future Week # into the equation.

Forecasts for Weeks 17-20

Week Forecast

17

656

18

678

19

699

20

720

=21.41*18+292.2

= 678 (rounded)

# of Passengers

800

700

y = 21.41x + 292.2

600

500

400

300

200

100

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

Week

• Careful! You shouldn’t use the equation

to forecast too far into the future.

• Large forecasting errors can result.

• The estimates of slope and intercept are

valid only for a few periods beyond the

data

• Recalculate slope and intercept as fresh

data arrives

Summary

• Quantify trend using the slope of the line of best fit

• Spreadsheets can do this for us easily

• Trendline feature in Excel; also Excel functions: SLOPE and

INTERCEPT

• Statistical software

• Use estimate of slope and intercept to make forecasts

• Trends need not always be linear

• Spreadsheets provide other options: logarithmic, exponential,

quadratic, etc.

• Shouldn’t project trend too far

INFO 564

Operations & Supply Chain Management

Module 5f: Quantifying Seasonality

Seasonality

• Patterns that repeat every week, month, quarter, or year

• Christmas trees, vacation travel, construction, clothing, etc.

• Usually associated with seasons, not always

• “Back to school” supplies, # of defects in cars built on Mondays,

absenteeism on Fridays, demand for tax accountants

• Extent of seasonality measured through a seasonality index

• Seasonality index of 1.30 for December means December sales are

30% higher than average monthly sales.

• Ignoring seasonality can lead to large forecasting errors

Example: Monthly Sales of Laptops

Month

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Year1

2452

2272

2725

2637

2626

2790

2951

2986

2398

2522

2317

2518

Year2

2348

2203

2842

2491

2552

2856

2937

2976

2452

2588

2319

2480

Year3

2233

2114

2673

2543

2632

2857

2924

3119

2488

2566

2332

2445

Monthly Sales of Laptops

3500

3000

2500

2000

1500

1000

Jan

Feb

Mar

Apr

May

Year1

Jun

Jul

Year2

Aug

Sep

Oct

Nov

Dec

Year3

Clearly laptop sales are seasonal with high sales in August, and low sales

in February

Calculating Seasonality Indexes

Month

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Year1

Year2

2452

2348

2272

2203

2725

2842

2637

2491

2626

2552

2790

2856

2951

2937

2986

2976

2398

2452

2522

2588

2317

2319

2518

2480

Overall Average =

Year3

2233

2114

2673

2543

2632

2857

2924

3119

2488

2566

2332

2445

2588

Monthly

Average

2344

2196

2747

2557

2603

2834

2937

3027

2446

2559

2323

2481

Sales in August are 17% higher than average. Sales in

February are 15% lower than average.

Seasonality

Index

0.91

0.85

1.06

0.99

1.01

1.10

1.14

1.17

0.95

0.99

0.90

0.96

• Calculate overall average sales per

month – average of 36 numbers

• Calculate monthly average sales for

each month – average of 3 numbers

• Seasonality Index = monthly average ÷

overall average

• Seasonality indexes must add up to the

number of periods – 12 in this

example.

• One of many ways of calculating

seasonality indexes

Seasonality Indexes in Forecasting

• Forecasts are usually forecasts of average sales for a

particular period

• When seasonality is present, averages can be very misleading

• Months of typically high sales will see low forecasts – shortages of

product, labor, capacity, and other resources

• Months of typically low sales will see high forecasts – excess of

product, labor, capacity, and other resources

• Forecasts of average sales must be adjusted for seasonality

• Average Forecast * Seasonality Index

INFO 564

Operations & Supply Chain Management

Module 5g: Forecasting When Trend

and Seasonality Are Present

Example – Sales of Laptops

Year-1

Year-2

Year-3

Jan

2473

2624

2764

Feb

2314

2500

2666

Mar

2789

3160

3246

Apr

2722

2831

3137

May

2732

2913

3248

Jun

2917

3238

3494

Jul

3100

3340

3582

Aug

3156

3401

3798

Sep

2589

2898

3189

Oct

2734

3055

3288

Nov

2551

2807

3075

Dec

2773

2990

3209

Sales of Laptops

4000

3500

3000

Units

Month

2500

Increasing trend

Seasonality

Randomness

2000

1500

1000

Jan

Feb

Mar

Apr

May

Year-1

Jun

Year-2

Jul

Aug

Year-3

Sep

Oct

Nov

Dec

2473

2314

2789

2722

2732

2917

3100

3156

2589

2734

2551

2773

2624

2500

3160

2831

2913

3238

3340

3401

2898

3055

2807

2990

2764

2666

3246

3137

3248

3494

3582

3798

3189

3288

3075

3209

Example – Sales of Laptops Continued

• Another way of identifying the trend and

seasonality is to plot the data as one time

series.

• Trend and patterns that repeat each year are

clearly seen

Laptop Sales

4000

3500

3000

2500

2000

1500

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Units

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Sales (Units)

Month

Multiplicative Decomposition – A 4-Step

Process

• Calculate seasonality indexes

• Quantifies seasonality

• Draw a trend-line through the data

• Estimates slope (trend) and intercept

• Project the trend into the future periods

• Trend forecast

• Adjust trend forecast with seasonality index

• Final forecast

Calculating Seasonality Indexes

Month

Year-1

Year-2

Year-3

Jan

2473

2624

2764

Feb

2314

2500

2666

Mar

2789

3160

3246

Apr

2722

2831

3137

May

2732

2913

3248

Jun

2917

3238

3494

Jul

3100

3340

3582

Aug

3156

3401

3798

Sep

2589

2898

3189

Oct

2734

3055

3288

Nov

2551

2807

3075

Dec

2773

2990

3209

Overall Average =

2981

Monthly

Average

2620

2493

3065

2897

2964

3216

3341

3452

2892

3026

2811

2991

Seasonality

Index

0.88

0.84

1.03

0.97

0.99

1.08

1.12

1.16

0.97

1.02

0.94

1.00

12.00

Estimating Trend

Laptop Sales

4000

y = 21.231x + 2587.9

3500

3000

2500

Slope = 31.23 units per month

Intercept = 2588 units

2000

Dec

Oct

Nov

Sep

Aug

Jul

Jun

Apr

May

Mar

Jan

Feb

Dec

Nov

Oct

Sep

Jul

Aug

Jun

May

Apr

Mar

Feb

Jan

Dec

Oct

Nov

Sep

Aug

Jul

Jun

Apr

May

1500

Mar

2473

2314

2789

2722

2732

2917

3100

3156

2589

2734

2551

2773

2624

2500

3160

2831

2913

3238

3340

3401

2898

3055

2807

2990

2764

2666

3246

3137

3248

3494

3582

3798

3189

3288

3075

3209

Jan

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Feb

Units

Sales (Units)

Month

Project Trend

• Laptop Sales = 21.23*Period + 2588

• Suppose we want to make forecasts for the next four months.

• Project trend to the first four months of year 4 (periods 37 – 40)

Period Month

Trend

Forecast

37

January

3374

38

February

3395

39

March

3416

40

April

3437

=21.23*37 + 2588

Final Forecasts

• Adjust trend forecasts with seasonality indexes

Period Month

Trend Seasonality

Final

Forecast

Index Forecast

37

January

3373

0.88

2968

38

February

3395

0.84

2852

39

March

3416

1.03

3518

40

April

3437

0.97

3334

=3395*0.84

Final Forecasts

Laptop Sales

4000

y = 18.128x + 2627.4

• The dotted line represents

trend projections for those

four periods.

3000

2500

2000

1500

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan

Feb

Mar

April

Sales (Units)

3500

• The last four points (in red)

are forecasts.

Summary

• When trend and seasonality are present we estimate

each component separately

• The final forecast is a product of the trend component

and the seasonality adjustment

• Randomness is handled through the averaging

process

• Many versions of multiplicative decomposition

• This is the simplest

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