msci 709 fall 2005 - university of waterloo › ~sjayaswa › projects › msci709... · 1...

LOGISTICS AND SUPPLY CHAIN MANAGEMENT

MSCI 709 FALL 2005

COURSE INSTRUCTOR

DR. SCOTT HADLEY

PROJECT REPORT

DEMAND FORECASTING FOR TECHWORX INC.

Submitted by

SACHIN JAYASWAL (20186226)

YOGENDRA VASHIST (20157269)

MAHSA TAVASSOLI (20161170)

VADIVANANTHAN VISUVALINGAM (98180712)

Executive Summary

The organization chosen to study for the course project for MSci 709 is TechWorx Inc.

TechWorx is a University of Waterloo Retail Services shop that sells stationery supplies,

specialty papers, and drafting supplies. TechWorx serves the market of students,

professors and other individuals who inhabit the main campus of University of Waterloo.

The major problem TechWorx had was they needed to be able to meet customer demands,

so as not to lose sales and at the same time maintain low levels of inventory. This is

important to the organization because they should not lose revenue in form of lost sales

and at the same time they should not increase the inventory holding cost by keeping

excess inventory. This problem falls in the general category of inventory management

and demand forecasting.

Based on an inventory listing provided by the store management 12 different items were

chosen that were thought to possibly reflect 3 different types of sales patterns. These

would include items which have a seasonal pattern to them, items which have a regular

sales pattern and items which are sold rarely i.e. once in a while sales type items. The

past sales figures were analyzed and demand forecasting models for each item were

determined. The various adaptive forecasting models were explored to see which of them

would be suitable for each of the items. Different time periods were also considered,

these include a 17 week time period (semester), 52 week time period (yearly) and

seasonal time period (distinctive Winter, Spring and Fall).

Data collection was fairly easy since TechWorx is meticulous and collects detailed data.

The group did run into some problems with missing data but these were overcome with

assumptions and approximations. The methodology for the analysis comprised of

running various forecasting models and various time periods to see which is most

appropriate. Once an appropriate model was chosen further parameter refinement was

performed and future forecasting capability of the model was verified.

ii

Most of the items chosen did not reveal any sales patterns that were initially hypothesized

for them. This draws an important conclusion that it may be erroneous to predict the

sales pattern of an item without thorough analysis of its historical sales data. Among all

the models considered under study, Winter’s model with seasonal time period has

provided the best results albeit with roughly 25% deviation for many of the items.

From our study, we recommend further analysis of the sales data using some other

forecasting techniques to come up with improved methods for implementation. The

current study should serve as a guideline for further study of other forecasting techniques

that can better be applied for the items at TechWorx Inc. As we currently do not propose

any new implementation for TechWorx Inc, no organisational change is suggested.

iii

Table of Contents Executive Summary ............................................................................................................ ii Table of Contents............................................................................................................... iv 1 Organizational Situation ............................................................................................. 1 2 Managerial Problem.................................................................................................... 2 3 Problem Formulation .................................................................................................. 3 4 Data Gathering ............................................................................................................ 5 5 Development of Methodology .................................................................................... 6

5.1 Forecasting models ............................................................................................. 6 5.2 Time periods, Parameter Refinement and Future Forecasting............................ 8

5.2.1 Time periods ............................................................................................... 8 5.2.2 Parameter Refinement................................................................................. 9 5.2.3 Future Forecasting ...................................................................................... 9

6 Analysis..................................................................................................................... 11 6.1 Binders/Clippers (Item no. 1060) ..................................................................... 11

6.1.1 17 week time period.................................................................................. 13 6.1.2 52 week time period.................................................................................. 14 6.1.3 Winter seasonal time period...................................................................... 15 6.1.4 Comparison of the Various Models .......................................................... 16 6.1.5 Parameter Refinement and Future Forecasting......................................... 16

6.2 Pens (Item no. 1205) ......................................................................................... 19 6.2.1 17 week time period.................................................................................. 20 6.2.2 52 week time period.................................................................................. 21 6.2.3 Spring Seasonal time period. .................................................................... 22 6.2.4 Winter Seasonal time period..................................................................... 23 6.2.5 Comparison of the Various Models .......................................................... 24 6.2.6 Parameter Refinement and Future Forecasting......................................... 24

6.3 Speakers (Item no. 8207) .................................................................................. 27 6.3.1 17 week time period.................................................................................. 29 6.3.2 52 week time period.................................................................................. 30 6.3.3 Spring Seasonal time period. .................................................................... 31 6.3.4 Winter Seasonal time period..................................................................... 32 6.3.5 Comparison of the Various Models .......................................................... 33 6.3.6 Parameter Refinement and Future Forecasting......................................... 33

6.4 Lead (Item no. 1200)......................................................................................... 36 6.4.1 17 week time period.................................................................................. 37 6.4.2 52 week time period.................................................................................. 38 6.4.3 Spring Seasonal time period. .................................................................... 39 6.4.4 Winter Seasonal time period..................................................................... 40 6.4.5 Comparison of the Various Models .......................................................... 41 6.4.6 Parameter Refinement and Future Forecasting......................................... 41

6.5 Pen Refills (Item no. 1209)............................................................................... 44 6.5.1 4 month time period. ................................................................................. 46 6.5.2 Comparison of the Various Models .......................................................... 47

iv

6.5.3 Parameter Refinement and Future Forecasting......................................... 47 7 Conclusion and Scope for Further Study .................................................................. 50

7.1 Conclusions, Strengths and Weaknesses .......................................................... 50 7.2 Scope for further study...................................................................................... 52

8 References................................................................................................................. 53 Appendix A: Raw Data................................................................................................. 1

v

1 Organizational Situation

The organization chosen to study for the course project for MSci 709 is TechWorx Inc.

TechWorx is a University of Waterloo Retail Services shop that sells stationery supplies,

specialty papers, and drafting supplies. It falls under the jurisdiction of University of

Waterloo Retail Services. It is located in South Campus Hall, very close to the Douglas

Wright Engineering Building and the Grad House. The store manager is Will Russell and

Retail Services Merchandise manager is Darrell Kane. It operates Mondays through

Fridays between 8 am and 5 pm and Saturdays between 12 pm and 4 pm.

TechWorx serves the market of students, professors and other individuals who inhabit the

main campus of University of Waterloo. The campus can be viewed as the only market

served by TechWorx since it only advertises and cater services to those members of the

University of Waterloo community. TechWorx does not have any known competitors

within the University of Waterloo campus as all retail services within the campus are

owned and operated by the University or its affiliated members. As such, they do not

engage in the duplication of services. TechWorx does face indirect competition in that

customers have the option of buying from off campus stores such as Staples.

The general business strategy of the organization is to meet and provide the stationery

requirements of students, professors and other individuals. The store’s cash registers and

customer services are run by individuals, usually students, hired on term by term basis.

Will Russell is responsible for the day to day operations of the store and Darrell Kane is

responsible for long term strategic decisions.

1

2 Managerial Problem

Upon first propositioning our interest in doing a course project based on TechWorx, Will

Russell exhibited interest saying that there was potential but he would have to get further

clarification and details. Thus we drew up and submitted a proposal citing potential areas

of study such as inventory management, demand forecasting and supplier relationship

management. This led to a meeting between the group and the store’s management

which consist of Will Russell and Darrel Kane. One of major outcomes of the meeting

was identification of the problems faced by TechWorx. The major problem TechWorx

had was they needed to be able to meet customer demands in a timely and efficient

manner, so as not to lose sales and at the same time maintain low levels of inventory.

This is important to the organization because they should not lose revenue in form of lost

sales and at the same time they should not increase the inventory holding cost by keeping

excess inventory. This problem falls in the general category of inventory management

and demand forecasting. The store’s management felt that we would probably find some

seasonal patterns in the customer demand but this would have to be verified by the

analysis.

Initially, store management inquired whether the group would be able to do a detailed

level of analysis on how to optimize the store’s inventory, especially on such items as

pens which included literally thousands of different types and kinds of pens. However,

given the time constraints of the project the group opted to do a more aggregate level of

analysis that could provide some insight for management with regards to demand

forecasting. After some discussion with Dr. Scott Hadley, the group decided to choose a

sampling of the store’s items to perform the analysis.

2

3 Problem Formulation Based on an inventory listing provided by the store management different items were

chosen that were thought to possibly reflect different types of sales patterns. The group

hypothesized that there may be potentially 3 different types of sales pattern that we may

see. These would include items which have a seasonal pattern to them, items which have

regular sales pattern and items which are sold rarely i.e. once in a while sales type items.

Considering these three types of patterns, the group decided to analyze the past sales

figures for 12 items (3-5 items from each category) and determine a demand forecasting

model for each item. The 12 items selected are listed below with the sub class code (used

in-house in Techworx) and the initial sales pattern categorization.

Table 1: Items based on initial sales pattern categorization

Category Item Description Sub Class Code

Rarely sold Speakers 8207

Rarely sold Ethernet cards 8208

Rarely sold Printers 8400

Seasonal Binders/clipboards 1060

Seasonal Pens 1205

Seasonal Mechanical pencils 1160

Stable Cassettes and CDs 6066

Stable CDR/CDRW 1081

Stable Pen Refills 1209

Stable LEAD 1200

Stable Batteries 1055

Stable Lamp & Light Bulbs 1155

In the MSci 709 course we were exposed to various time series forecasting models which

included static and adaptive models. For this project we decided that the best type of

models to explore would be adaptive models. The adaptive models were considered

suitable since they take into consideration new data whereas the static models do not.

3

This is especially an important characteristic since new data can exhibit new trends and

patterns that may need to be taken into account. We used different types of adaptive

models (Moving Average, Simple Exponential Smoothing, Holt’s, and Winter’s) to see

which one was the most suitable. Along with determining the suitability of the model we

also determined what parameter values were most suitable.

One of the assumptions we made before proceeding with the analysis is that there would

be seasonality and trends that would be observable in the sales data for each of the

selected items. This assumption would be verified by visual inspection and running of

various forecasting models. Visual inspection can reveal observable cycles and patterns

in the sales data. The various forecasting models will reveal if this is true and also they

will reveal potential trends in the data if there are any. Another assumption we made is

that the sales data reflect the demand pattern for a category of similar items. While this

can not be totally verified for an individual brand i.e. Bic pens, the items we chose and

the sales data we are looking at are the aggregate sales figures, i.e. we look at the sales

figures for pens not just Bic pens, which means that most of the customer demand will be

reflected in the sales figure. This is because even if a customer is unable to find a Bic

pen, they will still find another pen and probably will purchase that. So a vast majority of

customers will have their demand satisfied and only the very selective few will have their

demand unsatisfied.

4

4 Data Gathering

The University of Waterloo Retail Services is very meticulous and performs very detailed

data collection of their operations. Thus the data collection merely consisted of

forwarding a request to Darrell Kane to provide data on a weekly basis for the particular

items that were identified for analysis. The group did encounter some issues with the

data. Some of the items had a few weeks with missing data and some sales quantity had

zero as their value.

Missing data presents a problem so that it becomes very difficult to apply time series

models which expect continuous data for every time period. This issue was resolved one

way by substituting the missing data with the average of the sales quantity of the previous

period (t-1) and the following period (t+1). Another way used to resolve missing data

was to substitute the missing data with the data from the sales quantity for the same

period in previous year.

Having a zero value for the sales figure is also an issue because the time series models

involve stages where there is a division by the sales quantity. If the sales quantity has a

zero value, this causes a “division by zero” error. The group overcame this by

substituting the zero with 0.1 which is sufficiently low to represent low sales but

sufficiently high to not cause a “division by zero” error.

Another problem we encountered in the data was that some of the items had many

periods with zeroes. For this type of data the group decided to aggregate the weekly

figures into monthly figures to perform the analysis. One of the items also had missing

data for several periods. After some deliberations, the group decided to drop this item as

it was not feasible to use the time series models for this item.

5

5 Development of Methodology The methodology for the analysis will comprise of exploring various forecasting models

and various time periods to see which is most appropriate. Once an appropriate model

has been chosen further parameter refinement will be performed as well as the future

forecasting capability of the model will be verified.

5.1 Forecasting models The forecasting techniques that we will be using are adaptive forecasting models (Meindl

and Chopra, 2004). In adaptive forecasting, the estimates of level, trend, and seasonality

are updated after each demand observation. The updated estimates are then used to

forecast demand for the next period.

Let:

tL = Estimate of level at the end of period t

tT = Estimate of trend at the end of period t

tS = Estimate of seasonal factor for period t

tF = Forecast of demand for period t made in period t -1 or earlier

tD = Actual demand observed in period t

tE = Forecast error in period t

In adaptive methods, the forecast for period t + l in period t is given as:

( ) ltttlt SlTLF ++ +=

Adaptive forecasting methods frequently discussed in literature are (1) Moving Average,

(2) Simple Exponential Smoothing, (3) Trend Corrected Exponential Smoothing (Holt’s

Model), (4) Trend and Seasonality Corrected Exponential Smoothing (Winter’s Model)

(Meindl and Chopra, 2004). The method that is most appropriate depends on the

characteristics of demand and the composition of the systematic component of demand.

6

Moving Average: This method of forecasting is used when demand has no observable

trend or seasonality. Level in period t is given by the average demand over the most

recent N periods.

tL

( ) NDDDL Ntttt /............. 11 +−− +++=

In absence of trend and seasonality, systematic component of demand = level. Future

demand is forecasted based on updated estimate of level as:

tt LF =+1 and tnt LF =+

Simple Exponential Smoothing: This method of forecasting is also applicable when

demand has no observable trend or seasonality. So, systematic component of demand =

level. The estimates of levels for period 0, ( ) and period t+1, ( ) and forecast for

period t+1, ( ) are defined as:

0L 1+tL

1+tF

∑=

=n

iiDnL

10

1

ttt LDL )1(11 αα −+= ++

tt LF =+1 and tnt LF =+

Trend Corrected Exponential Smoothing (Holt’s Model): This method is appropriate

when demand is assumed to have a level and a trend in the systematic component but no

seasonality. In this case, systematic component of demand = level + trend. The initial

estimates of level and trend are obtained by running a linear regression between demand

and period t. Given Estimates of level and trend in period t, forecast for future

periods are given as:

tD tL tT

ttt TLF +=+1 and ttnt nTLF +=+

Level and trend for period t + 1are then revised as:

( )( )tttt TLDL +−+= ++ αα 111

( ) ( ) tttt TLLT ββ −+−= ++ 111

where, α and β are the smoothing constants for the level and trend, respectively.

7

Trend and Seasonality Corrected Exponential Smoothing (Winter’s Model): This

method is appropriate when the systematic component of demand is assumed to have a

level, trend and seasonal factor. In this case, systematic component of demand = (level +

trend) * seasonal factor. Given the estimates for level ( ), trend ( ) and seasonal factor

( ) in period t, the forecast for future periods are given as:

tL tT

tS

11 )( ++ += tttt STLF and ltttnt SnTLF ++ += )(

Level, trend and seasonal factor are then revised as:

( )( ttt

tt TLS

DL +−+⎟⎠⎞⎜

⎝⎛=

+

++ αα 1

1

11 )

( ) ( ) tttt TLLT ββ −+−= ++ 111

( ) 11

11 1 +

+

+++ −+⎟

⎠⎞⎜

⎝⎛= t

t

tpt SL

DS γγ

where, α , β and γ are smoothing constants for the level, trend and seasonal factor,

respectively.

5.2 Time periods, Parameter Refinement and Future Forecasting

5.2.1 Time periods

The various time periods that will be considered are as follows:

17 week time period: The 17 week time period corresponds to each semester. The

assumption here is that there are patterns every semester and that this pattern repeats

every semester.

52 week time period: The 52 week time period corresponds to each year. The

assumption here is that there are observable patterns to be seen every year and that this

pattern will repeat itself every year.

8

Seasonal (Winter, Spring, Fall) time period: The seasonal time period corresponds to

each semester. However the difference between this and the 17 week time period is that

this time period assumes that the each of the different time periods will be different. So

Winter terms will have one pattern, the Spring terms will have another pattern and the

Fall terms will have another pattern. One thing to note is that due to the lack of data in

the fall term, this term will not be examined.

5.2.2 Parameter Refinement

Once a model is chosen, its important parameters will be adjusted, for example in the

Winter’s model the gamma value will be varied since it has a major impact on the

forecast by the model. However, initially the following parameter values will be used for

the different models (this is based on the parameter values chosen in the examples

presented in Chapter 7 Demand Forecasting in a Supply Chain, in Supply Chain

Management by Meindl and Chopra)

Table 2: Initial values of parameters

Model Alpha (α) Beta (β) Gamma (γ)

Moving Average Not included Not included Not included

Simple Exponential

Smoothing

0.1 Not included Not included

Holt’s 0.1 0.2 Not included

Winter’s 0.05 0.1 0.1

5.2.3 Future Forecasting

Future forecasting capability of a model is important since it indicates how useful the

model will be. How this exactly will be achieved will be as follows:

9

1. A reasonable lead time will be assumed for each of the items.

2. The data for the period being looked at will be rolled back by the lead time. So,

for example, if we assume a 3 week lead time and we have 105 data points

representing each of the weeks, the data will be rolled back to 102 data points.

3. The best model chosen for the item will rerun for the 102 data points.

4. Forecasting will be performed for the time periods that the data was rolled back.

So, for example, if we had original data till Oct 30, 2004 the data would be rolled

back to Oct 9, 2004. So now forecasting will be performed for Oct 16, Oct 23 and

Oct 30.

5. The forecasts will then be compared with the original data and the MAD and

MAPE values will be looked at to see how well the forecast was performed.

10

6 Analysis

The analysis consists of performing the previously described methodology for each of the

various items chosen. For the analysis the items that we look at do not include all the 12

items that we initially mentioned. This was done because it was noticed that some of the

items display similar sales characteristics to each other and mentioning them will consist

of exhibiting redundant material. The following items were chosen for analysis

Binders/Clippers: Binders/Clippers were chosen since they exhibit seasonality and are

very prevalent stationary items. Items that showed similar sales patterns were Batteries,

CDR/CDRW, which we did not show.

Pens: Pens were chosen since they seem to exhibit seasonal patterns but it is not very

clear if they do so. Also the store management was very interested in seeing some

analysis done on this item. Similar items included mechanical pencils.

Speakers: Speakers were chosen since they seem to exhibit stable sales patterns and also

they are a non stationary item that is sold by TechWorx. Similar items included Ethernet

cards, Printers, which we did not show.

Lead: Lead was chosen because they seem to exhibit stable sales patterns but also seem

to exhibit seasonal sales patterns. We were interested to see which of these is really the

case.

Pen Refills: Pen refills were chosen since they exhibit sporadic sales patterns. Items that

showed similar sales patterns were Lamps & Light Bulbs, which we did not show.

Cassettes and CDs are not shown since they had very little data that could be used to do

forecasting.

6.1 Binders/Clippers (Item no. 1060)

The initial intuition was that the demand for binders/clippers would exhibit some

seasonal pattern, with demand surging during the start of each term and then levelling off.

11

The store being in the campus would attract most of the terms demand for

binders/clippers. The sales pattern for the binders/clippers is shown in the following

figure.

Clippers/Binders(Item No. 1060)

0

100

200

300

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700

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04Date (Week of)

Sale

s

SalesData

Figure 1: Sales figures for clippers/binders

The above figure for sales of clippers/binders supports our earlier intuition that the

demand for binders/clippers will exhibit a seasonal pattern. The demand pattern, however,

does not repeat an identical cycle each term. The demand for the item surges during the

start of each term but the surge in demand in spring in general is less that in the other two

terms. In the next section, we look at the four adaptive forecasting techniques to

determine which one best suits the demand pattern. The time periods that will be looked

at would be 17 week (semester), 52 week (yearly) and seasonal (two consecutive winter

terms).

12

6.1.1 17 week time period.

Binders/Clipboards

0

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and

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Binders/Clipboards

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Week

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and

SalesSimple Exponential

Figure 2: Moving Average, 17 week. Figure 3: Simple Exponential Smoothing, 17 week.

Binders/Clipboards

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and

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Binders/Clipboards

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and

SalesWinter's Model

Figure 4: Holt's, 17 week Figure 5: Winter's, 17 week.

13


Clippers/Binders

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70002

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Clippers/Binders

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Clippers/Binders

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Clippers/Binders

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14

6.1.3 Winter seasonal time period.

Binders/Clippers

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3

26-A

pr-0

3

10-J

an-0

4

24-J

an-0

4

07-F

eb-0

4

21-F

eb-0

4

06-M

ar-0

4

20-M

ar-0

4

03-A

pr-0

4

17-A

pr-0

4

Date

Dem

and

SimpleExponentialSales

Figure 10: Moving Average, Winter Seasonal. Figure 11: Simple Exponential Smoothing, Winter Seasonal.

Binders/Clippers

0

100

200

300

400

500

600

04-J

an-0

3

18-J

an-0

3

01-F

eb-0

3

15-F

eb-0

3

01-M

ar-0

3

15-M

ar-0

3

29-M

ar-0

3

12-A

pr-0

3

26-A

pr-0

3

10-J

an-0

4

24-J

an-0

4

07-F

eb-0

4

21-F

eb-0

4

06-M

ar-0

4

20-M

ar-0

4

03-A

pr-0

4

17-A

pr-0

4

Date

Dem

and

Holt's ModelSales

Binders/Clippers

0

100

200

300

400

500

600

04-J

an-0

3

18-J

an-0

3

01-F

eb-0

3

15-F

eb-0

3

01-M

ar-0

3

15-M

ar-0

3

29-M

ar-0

3

12-A

pr-0

3

26-A

pr-0

3

10-J

an-0

4

24-J

an-0

4

07-F

eb-0

4

21-F

eb-0

4

06-M

ar-0

4

20-M

ar-0

4

03-A

pr-0

4

17-A

pr-0

4

Date

Dem

and

Winter's ModelSales

Figure 12: Holt's, Winter Seasonal Figure 13: Winter's, Winter Seasonal.

15

6.1.4 Comparison of the Various Models The demand results presented in the figures above suggests that winter’s model predicts a

demand pattern that matches best with the sales pattern for each of the time periods

considered. However, it needs to be ascertained which time cycle period in winter’s

model best reflects the sales pattern. For this, we compare the results of winter’s model

for various cycle times. The following table indicates the MAD, MAPE and TS range for

Winter’s model for each time period.

Table 3: Error estimate in forecasting

TS Time Period Model MAD MAPE Min Max

17 week Winter’s 44.5379

88.03099922

-15.9526

18.8977


72.49508045

-9.20027

19.8516

Seasonal, Winter

Winter’s 6.54822

25.02731233

-10.59

-0.45554

From the above it can be seen that the Seasonal is the best time period. This is not much

surprising since the seasonal takes into account similar periods during which the market

situation is fairly similar, e.g. the number of people on campus during Winter 2003

should be fairly similar to the number of people on campus during Winter 2004.

6.1.5 Parameter Refinement and Future Forecasting

For the Winter’s model in Seasonal time period, the gamma value has a major impact on

the quality of the model. Let us vary gamma and determine an appropriate value for

gamma. The following figure depicts the variation of the gamma values.

16

Adaptive Forecasting Model: Winter's: (Binders)

0

100

200

300

400

500

600

04-J

an-0

3

18-J

an-0

3

01-F

eb-0

3

15-F

eb-0

3

01-M

ar-0

3

15-M

ar-0

3

29-M

ar-0

3

12-A

pr-0

3

26-A

pr-0

3

10-J

an-0

4

24-J

an-0

4

07-F

eb-0

4

21-F

eb-0

4

06-M

ar-0

4

20-M

ar-0

4

03-A

pr-0

4

17-A

pr-0

4

Date (week of)

Dem

and Actual

Ft - 0.1Ft - 0.5Ft - 0.7

Figure 14: Winter’s Model for varying Gamma values.

From the above figure we note that changing the value of gamma does not have

significant effect in the result. There is, however, a very little deterioration in the result

with increase in gamma. Increasing gamma slightly increases the deviation of the forecast,

thus the most suitable gamma value is 0.1.

Let us now perform future forecasting to see how well the model performs, we will

assume a 3 week lead time.

Adaptive Forecasting Mode: Winter's (Clippers/Binders)

0

5

10

15

20

25

30

35

40

13/0

3/04

20/0

3/04

27/0

3/04

03/0

4/04

10/0

4/04

17/0

4/04

24/0

4/04

Demand

Wee

ks (D

ate

of)

ActualHistorical ForecastFuture Forecast

Figure 14: Future forecasting in Winter Season

17

For the above future forecasting we get a MAD = 6.545059 and a MAPE = 24.98808.

This is very similar to the values we got when performing historical forecasting, thus it

indicates that the model is somewhat reliable for future forecasting with a median

absolute percentage error of 24.9%.

18

6.2 Pens (Item no. 1205) For the pens we thought that the demand would be seasonal since at the beginning of

each term, students tend to buy new pens for their utilisation through the term. The sales

pattern for pens is shown in the following figure.

Pens

0

200

400

600

800

1000

1200

1400

1600

01/1

1/20

02

01/0

1/20

03

01/0

3/20

03

01/0

5/20

03

01/0

7/20

03

01/0

9/20

03

01/1

1/20

03

01/0

1/20

04

01/0

3/20

04

01/0

5/20

04

01/0

7/20

04

01/0

9/20

04

Date

Dem

and

(Uni

ts)

Sales Data

Figure 15: Sales figures for Pens

From the above figure it can be seen that pens are sold seasonal as expected. We will

explore this in the next few pages by running different adaptive forecasting models for

different time periods to determine which would suit the best. The time periods that will

be looked at would be 17 week (semester), 52 week (yearly) and seasonal (fall, winter,

spring). For the seasonal we will not look at the fall component since the data is

incomplete.

19


Simple Exponential Smoothing (Pens)

0

200

400

600

800

1000

1200

1400

1600

01/1

1/20

02

01/0

1/20

03

01/0

3/20

03

01/0

5/20

03

01/0

7/20

03

01/0

9/20

03

01/1

1/20

03

01/0

1/20

04

01/0

3/20

04

01/0

5/20

04

01/0

7/20

04

01/0

9/20

04

Date

Dem

and

(Uni

ts)

ActualForecast

Moving Average (Pens)

0

200

400

600

800

1000

1200

1400

160001

/11/

2002

01/0

1/20

03

01/0

3/20

03

01/0

5/20

03

01/0

7/20

03

01/0

9/20

03

01/1

1/20

03

01/0

1/20

04

01/0

3/20

04

01/0

5/20

04

01/0

7/20

04

01/0

9/20

04

Date

Dem

and

(Uni

ts)

ActualForecast

Figure 16: Moving Average, 17 week. ing 7 week. Figure 17: Simple Exponential Smooth , 1

Winter's (Pens)

0

200

400

600

800

1000

1200

1400

1600

01/1

1/20

02

01/0

1/20

03

01/0

3/20

03

01/0

5/20

03

01/0

7/20

03

01/0

9/20

03

01/1

1/20

03

01/0

1/20

04

01/0

3/20

04

01/0

5/20

04

01/0

7/20

04

01/0

9/20

04

Date

Dem

and

(Uni

ts)

ActualForecast

Holt's (Pens)

0

200

400

600

800

1000

1200

1400

1600

01/1

1/20

02

01/0

1/20

03

01/0

3/20

03

01/0

5/20

03

01/0

7/20

03

01/0

9/20

03

01/1

1/20

03

01/0

1/20

04

01/0

3/20

04

01/0

5/20

04

01/0

7/20

04

01/0

9/20

04

Date

Dem

and

(Uni

ts)

ActualForecast

Figure 18: Holt’s , 17 week. Figure 19: Winter’s, 17 week.

20

21


Simple Exponential SMoving Average

0

200

400

600

800

1000

1200

1400

1600

01/1

1/20

02

01/0

1/20

03

01/0

3/20

03

01/0

5/20

03

01/0

7/20

03

01/0

9/20

03

01/1

1/20

03

01/0

1/20

04

01/0

3/20

04

01/0

5/20

04

01/0

7/20

04

01/0

9/20

04

Date

Dem

and

(Uni

ts)

moothing

0

200

400

600

800

1000

1200

1400

1600

01/1

1/20

02

01/0

1/20

03

01/0

3/20

03

01/0

5/20

03

01/0

7/20

03

01/0

9/20

03

01/1

1/20

03

01/0

1/20

04

01/0

3/20

04

01/0

5/20

04

01/0

7/20

04

01/0

9/20

04

Date

Dem

and

(Uni

ts)

Actual ActualForecast

Forcast


Holt's

0

200

400

600

800

1000

1200

1400

1600

01/1

1/20

02

01/0

1/20

03

01/0

3/20

03

01/0

5/20

03

01/0

7/20

03

01/0

9/20

03

01/1

1/20

03

01/0

1/20

04

01/0

3/20

04

01/0

5/20

04

01/0

7/20

04

01/0

9/20

04

Date

Dem

and

(Uni

ts)

ActualForecast

Winter's

0

200

400

600

800

1000

1200

1400

1600

01/1

1/20

02

01/0

1/20

03

01/0

3/20

03

01/0

5/20

03

01/0

7/20

03

01/0

9/20

03

01/1

1/20

03

01/0

1/20

04

01/0

3/20

04

01/0

5/20

04

01/0

7/20

04

01/0

9/20

04

Date

Dem

and

(Uni

ts)

Actual

Forecast


22

6.2.3 Spring Seasonal time period.

Moving Average

0

100

200

300

400

500

600

70010

/05/

2003

24/0

5/20

03

07/0

6/20

03

21/0

6/20

0305

/07/

2003

19/0

7/20

0302

/08/

2003

16/0

8/20

03

30/0

8/20

0315

/05/

2004

29/0

5/20

0412

/06/

2004

26/0

6/20

04

10/0

7/20

0424

/07/

2004

07/0

8/20

0421

/08/

2004

Date

Dem

and

(Uni

ts) Simple Exponential Smoothing

0

100

200

300

400

500

600

700

10/0

5/20

03

24/0

5/20

03

07/0

6/20

03

21/0

6/20

03

05/0

7/20

03

19/0

7/20

03

02/0

8/20

03

16/0

8/20

03

30/0

8/20

03

15/0

5/20

04

29/0

5/20

04

12/0

6/20

04

26/0

6/20

04

10/0

7/20

04

24/0

7/20

04

07/0

8/20

04

21/0

8/20

04

Date

Dem

and

(Uni

ts)

ActualForecast

Figure 25: Simple Exponential Smoothing, Spring Seasonal.

ActualForecast

Figure 24: Moving Average, Spring Seasonal.

Holt's Model

0

100

200

300

400

500

600

700

10/0

5/20

03

24/0

5/20

03

07/0

6/20

03

21/0

6/20

03

05/0

7/20

0319

/07/

2003

02/0

8/20

03

16/0

8/20

03

30/0

8/20

0315

/05/

2004

29/0

5/20

0412

/06/

2004

26/0

6/20

04

10/0

7/20

04

24/0

7/20

04

07/0

8/20

04

21/0

8/20

04

Date

Dem

and

(Uni

ts)

ActualForecast

Winter's

0

100

200

300

400

500

600

700

10/0

5/20

0324

/05/

2003

07/0

6/20

03

21/0

6/20

0305

/07/

2003

19/0

7/20

0302

/08/

2003

16/0

8/20

03

30/0

8/20

0315

/05/

2004

29/0

5/20

0412

/06/

2004

26/0

6/20

04

10/0

7/20

0424

/07/

2004

07/0

8/20

0421

/08/

2004

Date

Dem

and

(Uni

ts)

ActualForecast

Figure 26: Holt's, Spring Seasonal Figure 27: Winter's, Spring Seasonal.

23

6.2.4 Winter Seasonal time period. Simple Exponential Smoothing (Pens)

0

200

400

600

800

1000

1200

1400

04/0

1/20

03

18/0

1/20

0301

/02/

2003

15/0

2/20

0301

/03/

2003

15/0

3/20

03

29/0

3/20

0312

/04/

2003

26/0

4/20

0310

/01/

2004

24/0

1/20

0407

/02/

2004

21/0

2/20

0406

/03/

2004

20/0

3/20

0403

/04/

2004

17/0

4/20

04

Date

Dem

and

(Uni

ts)

ActualForecast

Moving Average (Pens)

0

200

400

600

800

1000

1200

14000 24/

01/

5/01

/

5/02

/

8/03

/

9/03

/

9/04

/

0/01

2003

2003

120

03

020

03

220

03

120

03

1/2

004

31/0

1/20

04

21/0

2/20

04

13/0

3/20

04

03/0

4/20

04

24/0

4/20

04

Date

Dem

and

(Uni

ts)

ActualForecast


Holt's Model (Pens)

0

200

400

600

800

1000

1200

1400

04/0

1/20

03

25/0

1/20

03

15/0

2/20

03

08/0

3/20

03

29/0

3/20

03

19/0

4/20

03

10/0

1/20

04

31/0

1/20

04

21/0

2/20

04

13/0

3/20

04

03/0

4/20

04

24/0

4/20

04

Date

Dem

and

(Uni

ts)

ActualForecast

Winter's (Pens)

0

200

400

600

800

1000

1200

1400

04/0

1/20

03

25/0

1/20

03

15/0

2/20

03

08/0

3/20

03

29/0

3/20

03

19/0

4/20

03

10/0

1/20

04

31/0

1/20

04

21/0

2/20

04

13/0

3/20

04

03/0

4/20

04

24/0

4/20

04

Date

Dem

and

(Uni

ts)

ActualForecast


6.2.5 Comparison of the Various Models When we look at the various time periods it can be seen that the Winter’s Model

performs best for each of the time periods. However, we still need to know which time

period shows the best result. For this, we examine the individual time periods’ Winter’s

model against each other. The following table indicates the MAD, MAPE and TS range

for each Winter’s model for each time period:


TS Time Period Mode MAD MAPE Min Max

l

17 week Winter 119 52 -14 19 ’s

52 week Winter 53 23 -11 16 ’s

Seasonal, Spring Winter’s 20 11 -3 4

Seasonal, Winter Winter’s 45 21 -6 2



situation is fairly similar, e.g. the number of people on campus during winter 2003 should

be fairly similar to the number of people on campus during Winter 2004.


For the Winter’s model, the gamma value has a major impact on the quality of the model.

Let us vary gamma and deter e an appropriate value for gamma, we will use the winter

season for this variation. The following figure depicts the variation of the gamma values.

min

24

Winter's (Pens)

400

600

800

1000

1200

1400

Dem

and

0

200

1600

04/0

1/20

0318

/01/

2003

01/0

2/20

0315

/02/

2003

01/0

3/20

0315

/03/

2003

29/0

3/20

0312

/04/

2003

26/0

4/20

0310

/01/

2004

24/0

1/20

0407

/02/

2004

21/0

2/20

0406

/03/

2004

20/0

3/20

0403

/04/

2004

17/0

4/20

04

Date

ActualFt - 0.1Ft - 0.5Ft - 0.7

Figure 32: Varying Gam

rom ve figure we note that increasing gamma increases the deviation of the

ma values.

F the abo

forecast, thus the most suitable gamma value is 0.1.



25

Winter's (Pens)

0

50

100

150

200

250

300

350

20/03

/2004

27/03

/2004

03/04

/2004

10/04

/2004

17/04

/2004

24/04

/2004

Date

Dem

and Actual

Historical ForecastFuture Forecast


or the above future forecasting we get a MAD = 45 and a MAPE = 21 and TS between 2

odel

dian absolute percentage error of

1%.

F

and -6, which are exactly the same as historical forecast. Thus, it indicates that the m

is somewhat reliable for future forecasting with a me

2

26

6.3 Speakers (Item no. 8207) For the speakers, the group felt that the item would exhibit a “rarely sold” sales pattern.

“Rarely sold” sales items would be items that are unique and most likely not seasonal.

This profile fits the speakers very well since speakers, although common, would be a

unique item in the store and can generally be expected to be sold once in a while since

customers are more likely to try purchase such items in well known stores such as Future

Shop rather than from an on campus retailers. The sales pattern for the speakers is shown

in the following figure.

Speakers (Item no. 8207)

0

2

4

6

8

10

Dem

and

12

14

11/2

/200

2

12/2

/200

2

1/2/

2003

2/2/

2003

3/2/

2003

4/2/

2003

5/2/

2003

6/2/

2003

7/2/

2003

8/2/

2003

9/2/

2003

10/2

/200

3

11/2

/200

3

12/2

/200

3

1/2/

2004

2/2/

2004

3/2/

2004

4/2/

2004

5/2/

2004

6/2/

2004

7/2/

2004

8/2/

2004

9/2/

2004

10/2

/200

4

Date (week of)

Sales Data

Figure 34: Sales figures for Speakers.

From the above figure it can be seen that speakers are not “rarely sold” items. In fact

they seem to have regular sales patterns although it is a bit difficult to discuss if there is a

seasonal aspect to them. We will explore this in the next few pages by running different

adaptive forecasting models for different time periods to determine which would be best

suitable. The time periods that will be looked at would be 17 week (semester), 52 week

27

(yearly) and seasonal (fall, winter, spring). For the seasonal we will not look at the fall

component since the data is incomplete.

28

29


Adaptive Forecasting Model: Moving Average (Speakers)

0

2

4

6

8

10

12

14

3/1/

2003

4/1/

2003

5/1/

2003

6/1/

2003

7/1/

2003

8/1/

2003

9/1/

2003

10/1

/200

3

11/1

/200

3

12/1

/200

3

1/1/

2004

2/1/

2004

3/1/

2004

4/1/

2004

5/1/

2004

6/1/

2004

7/1/

2004

8/1/

2004

9/1/

2004

10/1

/200

4

Date (week of)

Dem

and

ActualForecast

Adaptive Forecasting Model: Simple Exponential Smoothing (Speakers)

0

2

4

6

8

10

12

14

11/2

/200

2

1/2/

2003

3/2/

2003

5/2/

2003

7/2/

2003

9/2/

2003

11/2

/200

3

1/2/

2004

3/2/

2004

5/2/

2004

7/2/

2004

9/2/

2004

Date (week of)

Dem

and

ActualFore


cast

Adaptive Forecasting Model: Holt's (Speakers)

0

2

4

6

8

10

12

14

11/2

/200

2

1/2/

2003

3/2/

2003

5/2/

2003

7/2/

2003

9/2/

2003

11/2

/200

3

1/2/

2004

3/2/

2004

5/2/

2004

7/2/

2004

9/2/

2004

Date (week of)

Dem

and

ActualForecast

Adaptive Forecasting Model: Winter's (Speakers)

0

2

4

6

8

10

12

14

1/4/

2003

2/4/

2003

3/4/

2003

4/4/

2003

5/4/

2003

6/4/

2003

7/4/

2003

8/4/

2003

9/4/

2003

10/4

/200

3

11/4

/200

3

12/4

/200

3

Date (week of)

Dem

and

ActualForecast


30



0

2

4

6

8

10

1211

/1/2

003

12/1

/200

3

1/1/

2004

2/1/

2004

3/1/

2004

4/1/

2004

5/1/

2004

6/1/

2004

7/1/

2004

8/1/

2004

9/1/

2004

10/1

/200

4

Date (week of)

Dem

and

ActualForecast


0

2

4

6

8

10

12

14

11/2

/200

2

1/2/

2003

3/2/

2003

5/2/

2003

7/2/

2003

9/2/

2003

11/2

/200

3

1/2/

2004

3/2/

2004

5/2/

2004

7/2/

2004

9/2/

2004

Date (week of)

Dem

and

ActualForecast



0

2

4

6

8

10

12

14

11/2

/200

2

1/2/

2003

3/2/

2003

5/2/

2003

7/2/

2003

9/2/

2003

11/2

/200

3

1/2/

2004

3/2/

2004

5/2/

2004

7/2/

2004

9/2/

2004

Date (week of)

Dem

and

ActualForecast


0

2

4

6

8

10

12

14

11/2

/200

2

1/2/

2003

3/2/

2003

5/2/

2003

7/2/

2003

9/2/

2003

11/2

/200

3

1/2/

2004

3/2/

2004

5/2/

2004

7/2/

2004

9/2/

2004

Date (week of)

Dem

and

ActualForecast


31

6.3.3 Spring Seasonal time period.


0

1

2

3

4

5

6

30-A

ug-0

3

1-M

ay-0

4

8-M

ay-0

4

15-M

ay-0

4

22-M

ay-0

4

29-M

ay-0

4

5-Ju

n-04

12-J

un-0

4

19-J

un-0

4

26-J

un-0

4

3-Ju

l-04

10-J

ul-0

4

17-J

ul-0

4

24-J

ul-0

4

31-J

ul-0

4

7-A

ug-0

4

14-A

ug-0

4

21-A

ug-0

4

28-A

ug-0

4

Date (week of)

Dem

and

ActualForecast


0

1

2

3

4

5

6

7

3-M

ay-0

3

17-M

ay-0

3

31-M

ay-0

3

14-J

un-0

3

28-J

un-0

3

12-J

ul-0

3

26-J

ul-0

3

9-A

ug-0

3

23-A

ug-0

3

1-M

ay-0

4

15-M

ay-0

4

29-M

ay-0

4

12-J

un-0

4

26-J

un-0

4

10-J

ul-0

4

24-J

ul-0

4

7-A

ug-0

4

21-A

ug-0

4

Date (week of)

Dem

and

ActualForecast

Figure 43: Moving Average, Spring Seasonal. Figure 44: Simple Exponential Smoothing, Spring Seasonal.


0

1

2

3

4

5

6

7

3-M

ay-0

3

17-M

ay-0

3

31-M

ay-0

3

14-J

un-0

3

28-J

un-0

3

12-J

ul-0

3

26-J

ul-0

3

9-A

ug-0

3

23-A

ug-0

3

1-M

ay-0

4

15-M

ay-0

4

29-M

ay-0

4

12-J

un-0

4

26-J

un-0

4

10-J

ul-0

4

24-J

ul-0

4

7-A

ug-0

4

21-A

ug-0

4

Date (week of)

Dem

and

ActualForecast


0

1

2

3

4

5

6

7

3-M

ay-0

3

17-M

ay-0

3

31-M

ay-0

3

14-J

un-0

3

28-J

un-0

3

12-J

ul-0

3

26-J

ul-0

3

9-A

ug-0

3

23-A

ug-0

3

1-M

ay-0

4

15-M

ay-0

4

29-M

ay-0

4

12-J

un-0

4

26-J

un-0

4

10-J

ul-0

4

24-J

ul-0

4

7-A

ug-0

4

21-A

ug-0

4

Date (week of)

Dem

and

ActualForecast

Figure 45: Holt's, Spring Seasonal Fi 6: Winter's, Spring Seasonal. gure 4

32

.3.4 Winter Seasonal time period. 6

Adaptive Forecasting Model: Moving (Speakers)

Average

0

2

4

6

8

10

12

3-Ja

n-04

10-J

an-0

4

17-J

an-0

4

24-J

an-0

4

31-J

an-0

4

7-Fe

b-04

14-F

eb-0

4

21-F

eb-0

4

28-F

eb-0

4

6-M

ar-0

4

13-M

ar-0

4

20-M

ar-0

4

27-M

ar-0

4

3-A

pr-0

4

10-A

pr-0

4

17-A

pr-0

4

24-A

pr-0

4

Date (week of)

Dem

and


0

2

4

6

8

10

12

4-Ja

n-03

18-J

an-0

3

1-Fe

b-03

15-F

eb-0

3

1-M

ar-0

3

15-M

ar-0

3

29-M

ar-0

3

12-A

pr-0

3

26-A

pr-0

3

10-J

an-0

4

24-J

an-0

4

7-Fe

b-04

21-F

eb-0

4

6-M

ar-0

4

20-M

ar-0

4

3-A

pr-0

4

17-A

pr-0

4

Date (week of)

Dem

and

ActualForecast

ActualForecast



0

2

4

6

8

10

12

4-Ja

n-03

18-J

an-0

3

1-Fe

b-03

15-F

eb-0

3

1-M

ar-0

3

15-M

ar-0

3

29-M

ar-0

3

12-A

pr-0

3

26-A

pr-0

3

10-J

an-0

4

24-J

an-0

4

7-Fe

b-04

21-F

eb-0

4

6-M

ar-0

4

20-M

ar-0

4

3-A

pr-0

4

17-A

pr-0

4

Date (week of)

Dem

and


0

2

4

6

8

10

12

4-Ja

n-03

18-J

an-0

3

1-Fe

b-03

15-F

eb-0

3

1-M

ar-0

3

15-M

ar-0

3

29-M

ar-0

3

12-A

pr-0

3

26-A

pr-0

3

10-J

an-0

4

24-J

an-0

4

7-Fe

b-04

21-F

eb-0

4

6-M

ar-0

4

20-M

ar-0

4

3-A

pr-0

4

17-A

pr-0

4

Date (week of)

Dem

and

Actual ActualForecast Forecast


6.3.5 Comparison of the Various Models When we look at the various time periods it can be seen that the Winter’s Model

performs best for each of the time periods. However, we still need to know which time

period shows the best result. For this s examine the individual time period’s

Winter’s model against each other. The following table indicates the MAD, MAPE and

TS range for each Winter’s model for each time period


TS

let u

Time Period Model MAD MAPE Min Max


1.129782

-9.02 13.96


1.036051

-13.07 8.33

Seasonal, Spring

Winter’s 0.636528

1.218914

-5.36 6.62

Seasonal, Winter

Winter’s 0.945711

1.050965

-2.25 5.69





6.3.6 Parameter Refinement a ecasting


Let us vary gamma and determine an appropriate value for gamma, we will use the wint

season for this variation. The following figure depicts the variation of the gamma value

nd Future For

er

s.

33


2

4

6

8

10

Dem

and

0

3 4

12

15-F

eb-0

3

12-A

pr-0

3

26-A

pr-0

3

10-J

an-0

24-J

an-0

4

21-F 6-M

ar-0

4

20-M

ar-0

4

3-A

pr-0

4

17-A

pr-0

4

e (week of)

ActualFt - 0.1Ft - 0.5Ft - 0.7

4-Ja

n-03

18-J

an-0

1-Fe

b-03

1-M

ar-0

3

15-M

ar-0

3

29-M

ar-0

3

7-Fe

b-04

eb-0

4

Dat

Varying G es.

From the above figure we note that increasing gamma increases the deviation of the

us the m le gamm 0.1.

ow perform future forecastin to see how well the model performs, we will

Figure 51: amma valu

forecast, th ost suitab a value is

Let us n g



7

0

1

13-M

ar-0

4

20-M

ar-0

4

27-M

ar-0

4

3-A

pr-0

4

10-A

pr-0

4

17-A

pr-0

4

24-A

pr-0

4

2

3

5

6

Date (week of)

De

4

man

d ActualHistorical ForecastFuture Forecast


For the above future forecasting we get a MAD = 1.233825 and a MAPE = 1.569127.

This is very similar to the values we got when performing historical forecasting, thus it

34

indicates that the model is fairly reliable for future forecasting with a median absolute

percentage error of 1.56%.

35

6.4 Lead (Item no. 1

200)

r lead, the group felt that the item would exhibit a “Stable” sales pattern. “Stable”

sales items would be the items that have a stable sales pattern i.e. they have consistent

sales. This profile fits the Pencil Lead very well since lead is a consumable item.

Customers are expected to buy lead more often throughout the year. The sales pattern for

the Lead is shown in the following figure.

Fo

LEAD (ITEM NO. 1200)

0

50

100

150

200

250

02/1

1/20

02

02/0

1/20

03

02/0

3/20

03

02/0

5/20

03

02/0

7/20

03

02/0

9/20

03

02/1

1/20

03

02/0

1/20

04

02/0

3/20

04

02/0

5/20

04

02/0

7/20

04

02/0

9/20

04

DATE (WEEK OF )

DEM

AND

Series1

Figure 53: Sales figures for Pencil Leads.

From the above figure it can be seen that leads seem to have regular sales patterns

although it is a bit difficult to discuss if there is a seasonal aspect to them. We will

explore this in the next few pages by running different adaptive forecasting models for

different time periods to determine which would be best suitable. The time periods that

will be looked at would be 17 week (semester), 52 week (yearly) and seasonal (fall,

winter, spring). For the seasonal we will not look at the fall component since the data is

insufficient..

36

37


Adaptive Forecasting Model : Moving Average (Lead)

0

50

100

150

200

25002

/11/

02

02/0

1/03

02/0

3/03

02/0

5/03

02/0

7/03

02/0

9/03

02/1

1/03

02/0

1/04

02/0

3/04

02/0

5/04

Period02

/07/

04

Dem

and

02/0

9/04

Actual Sale

Forecast

Adaptive Forecasting Model : Simple Exponential Smoothing , (Lead)

0

50

100

150

200

250

02/1

1/02

02/0

1/03

02/0

3/03

02/0

5/03

02/0

7/03

02/0

9/03

02/1

1/03

02/0

1/04

02/0

3/04

02/0

5/04

02/0

7/04

02/0

9/04

Period

Dem

and

Actu eal Sal

Forecast

Figure 54: Movi Figure 55: Simple Exponential Smoothng Average, 17 week. ing, 17 week.

Ada Adaptive Forecasting Model : Wptive Forecasting Mode (Lead)

02/0

3/03

02/0

5/03

02/0

7/03

02/0

9/03

02/1

1/03

02/0

1/04

02/0

3/04

05/0

4

period

l : Holt's Model

0

50

100

150

200

250

02/1

1/02

02/0

1/03

02/

02/0

7/04

02/0

9/04

Dem

and

ActualSaleForecast

inter's Model (Lead)

0

50

100

150

200

250

02/1

1/02

02/0

1/03

02/0

3/03

02/0

5/03

02/0

7/03

02/0

9/03

02/1

1/03

02/0

1/04

02/0

3/04

02/0

5/04

02/0

7/04

02/0

9/04

Period

Dem

and

ActuSale

al

Forecast


38



0

50

100

150

200

25002

/11/

0202

/01/

0302

/03/

0302

/05/

0302

/07/

0302

/09/

0302

/11/

0302

/01/

0402

/03/

0402

/05/

0402

/07/

0402

/09/

04

Period

Dem

and

ActualSaleForecas


0

50

100

150

200

250

02/1

1/02

02/0

1/03

02/0

3/03

02/0

5/03

02/0

7/03

02/0

9/03

02/1

1/03

02/0

1/04

02/0

3/04

02/0

5/04

02/0

7/04

02/0

9/04

Period

Dem

and

ActualSaleForecast


Adaptive Forecasting Model : Holt's Model (Lead)

0

50

100

150

200

250

02/1

1/02

02/0

1/03

02/0

3/03

02/0

5/03

02/0

7/03

02/0

9/03

02/1

1/03

02/0

1/04

02/0

3/04

02/0

5/04

02/0

7/04

02/0

9/04

Period

Dem

and

ActualSaleForeca

Adaptive Forecasting Model : Winter's Model (Lead)

0

50

100

150

200

250

02/1

1/02

02/0

1/03

02/0

3/03

02/0

5/03

02/0

7/03

02/0

9/03

02/1

1/03

02/0

1/04

02/0

3/04

02/0

5/04

02/0

7/04

02/0

9/04

Period

Dem

and

ActualSaleForecast


39

period. 6.4.3 Spring Seasonal time


0

20

40

60

80

100

120

03/0

5/03

17/0

5/03

31/0

5/03

14/0

6/03

28/0

6/03

12/0

7/03

26/0

7/03

09/0

8/03

23/0

8/03

01/0

5/04

15/0

5/04

29/0

5/04

12/0

6/04

26/0

6/04

10/0

7/04

24/0

7/04

07/0

8/04

21/0

8/04

Period

Dem

and

ActualSaleForeca


020406080

100120

03/0

5/03

17/0

5/03

31/0

5/03

14/0

6/03

28/0

6/03

12/0

7/03

26/0

7/03

09/0

8/03

23/0

8/03

01/0

5/04

15/0

5/04

29/0

5/04

12/0

6/04

26/0

6/04

10/0

7/04

24/0

7/04

07/0

8/04

21/0

8/04

Period

Dem

and Actual

SaleForeca

Figure 62: Moving Average, Spring Seasonal. Figure 63: Simple Exponential Smoothing, Spring Seasonal.

Adaptive Forecasting Model : Holt's (Lead)

020406080

100120

03/0

5/03

17/0

5/03

31/0

5/03

14/0

6/03

28/0

6/03

12/0

7/03

26/0

7/03

09/0

8/03

23/0

8/03

01/0

5/04

15/0

5/04

29/0

5/04

12/0

6/04

26/0

6/04

10/0

7/04

24/0

7/04

07/0

8/04

21/0

8/04

Period

Dem

and

ActualSaleForecas


020406080

100120

03/0

5/03

17/0

5/03

31/0

5/03

14/0

6/03

28/0

6/03

12/0

7/03

26/0

7/03

09/0

8/03

23/0

8/03

01/0

5/04

15/0

5/04

29/0

5/04

12/0

6/04

26/0

6/04

10/0

7/04

24/0

7/04

07/0

8/04

21/0

8/04

Period

Dem

and

ActualSaleForecas

Figure 64: Holt's, Spring Se Figure 65: Winter's, Spring Seasasonal onal.

40

6.4.4 Winter Seasonal time period.


0

50

100

150

20004

/01/

0318

/01/

0301

/02/

0315

/02/

0301

/03/

0315

/03/

0329

/03/

0312

/04/

0326

/04/

0310

/01/

0424

/01/

0407

/02/

0421

/02/

0406

/03/

0420

/03/

0403

/04/

0417

/04/

04

Period

Dem

and Actual

SaleForecast


0

50

100

150

200

04/0

1/03

18/0

1/03

01/0

2/03

15/0

2/03

01/0

3/03

15/0

3/03

29/0

3/03

12/0

4/03

26/0

4/03

10/0

1/04

24/0

1/04

07/0

2/04

21/0

2/04

06/0

3/04

20/0

3/04

03/0

4/04

17/0

4/04

Period

Dem

and

ActualSaleForecast


Adaptive Forecasting Model : Holt's (Lead)

0

50

100

150

200

04/0

1/03

18/0

1/03

01/0

2/03

15/0

2/03

01/0

3/03

15/0

3/03

29/0

3/03

12/0

4/03

26/0

4/03

10/0

1/04

24/0

1/04

07/0

2/04

21/0

2/04

06/0

3/04

20/0

3/04

03/0

4/04

17/0

4/04

Period

Dem

and

ActualSaleForecast


0

50

100

150

200

04/0

1/03

18/0

1/03

01/0

2/03

15/0

2/03

01/0

3/03

15/0

3/03

29/0

3/03

12/0

4/03

26/0

4/03

10/0

1/04

24/0

1/04

07/0

2/04

21/0

2/04

06/0

3/04

20/0

3/04

03/0

4/04

17/0

4/04

Period

Dem

and

ActualSaleForecast


6.4.5 Comparison of the Various Models When we look at the various time periods it can observed that the Winter’s Model

performs best for most of the time periods. wever, we still need to know which time

period shows the best result. For this let us examine the individual time period’s

Winter’s model against each other. The following table indicates the MAD, MAPE and

TS range for each Winter’s model for each period.


LEAD ( 1200)

Ho

time

Time Period Best Fitting

Model

MAD MAPE TS Range

17 Week Exponential 31.46 111 - 7.12 to 11.87

52 Week Winter's Model 12.21 28.32 - 14.28 to 12.72

Spring Season Winter's Model 14.23 49.98 - 2.88 to 7.22

Winter Season Winter's Model 12.39 32.4 - 6.90 to 1.00

From the above it can be seen that the Seasonal and 52 Week are the best time period.

They have the lowest value of MAD which signifies lower standard deviation. Between

the two, the seasonal time period has lower ad of TS Range. Therefore we chose

Winter’s Model with seasonal time period as the most suitable model. This is not much






Let us vary gamma and determine an appropriate value for gamma, we will use the winter

season for this variation. The following figure depicts the variation of the gamma values.

spre

41


04/

18/0

1 /03

/03

/03

15/

/03

29/0

3/03

12/0

4/03

26/

/03

10/

/04

24/0

1/04

07/0

2/04

21/0

2/04

06/

/04

20/

/04

03/

/04

17/0

4/04

d

50

100

150

200

250

Dem

and

Actual Sale

F t- 0.1

Ft - 0.5

Ft - 0.7

0

01/0

3/0

301

/02

15/0

201

/03 03 04 01 03 03 04

Perio

arying G

fi i g gamma s the dev

e m m e is 0.1.

Figure 70: V amma values.

From the above gure we note that ncreasin increase iation of the

forecast, thus th ost suitable gam a valu



42


0

20

40

60

80

100

120

140

160

10/04/04 17/04/04 24/04/04

Date

Dem

and

Actual

HistoricalForecastFuture forecast


s

storical forecasting, thus it indicates

at the model is somewhat reliable for future forecasting with a median absolute

For the above future forecasting we get a MAD = 12.41 and a MAPE = 32.45. This i

very similar to the values we got when performing hi

th

percentage error of 32.45%.

43

6.5 Pen Refills (Item no. 1209) Before observing the sales figure of the pen refills, we assumed that the sales would be

stable for this item. However, the actual sales figures showed no specific pattern for the

sales and the demand was scattering. The sales pattern for pen refills is shown in the

following figure.

Pen Refills

0

2

3

4

5

6

Dem

and

(Uni

ts)

1

01/1

1/20

02

01/0

1/20

03

01/0

3/20

03

01/0

5/20

03

01/0

7/20

03

01/0

9/20

03

01/1

1/20

03

01/0

1/20

04

01/0

3/20

04

01/0

5/20

04

01/0

7/20

04

01/0

9/20

04

Date

OriginalDemand

gure 72: Sales figures for Pen Refills

rom the above figure it can be seen that pen refills are not sold in a stable pattern as

xpected. In addition, since there were many zero values of sales, we decided to

ggregate the demands of each month and run different adaptive forecasting models for

me period of 4 months (one semester). The following figure shows the monthly

ggregated sales pattern.

Fi

F

e

a

ti

a

44

Pen Refills

12

0

2

4

6

8

10

Dem

and

(Uni

ts)

Actual

Nov

-02

Jan-

03

Mar

-03

May

-03

Jul-0

3

Sep

-03

Nov

-03

Jan-

04

Mar

-04

May

-04

Jul-0

4

Sep

-04

Date

Figure 73: Monthly aggregated sales figures for pen refills

We will run the various adaptive forecasting models for four month time periods.

45

46

6.5.1 4 month time period. Moving Average (Pen Refills)

0

2

4

6

8

10

12N

ov-0

2

Jan-

03

Mar

-03

May

-03

Jul-0

3

Sep

-03

Nov

-03

Jan-

04

Mar

-04

May

-04

Jul-0

4

Sep

-04

Date

Dem

and

(Uni

ts)

ActualForecast

Simple Exponential Smoothing (Pen Refills)

0

2

4

6

8

10

12

Nov

-02

Jan-

03

Mar

-03

May

-03

Jul-0

3

Sep

-03

Nov

-03

te

Jan-

04

Mar

-04

May

-04

Jul-0

4

Sep

-04

Da

Dem

and

(Uni

ts)

ActualForecast

Figure 74: Moving Average, 4 month. Figure 75: Simple Expo al S othing, 4 month. nenti mo

Holt's (Pen Refills)

0

2

4

6

8

10

12

Nov

-02

Jan-

03

Mar

-03

May

-03

Jul-0

3

Sep

-03

Nov

-03

Jan-

04

Mar

-04

May

-04

Jul-0

4

Sep

-04

Date

Dem

and

(Uni

ts)

ActualForecast

Winte en Refills)

0

2

4

6

8

10

12

Nov

-02

Jan-

03

Mar

-03

May

-03

Jul-0

3

Sep

-03

Nov

-03

Jan-

04

Mar

-04

May

-04

Jul-0

4

Sep

-04

Date

Dem

and

(Uni

ts)

r's (P

ActualForecast

Figure 76: Holt's, 4 month. Figure 77: Winter's, 4 month.

6.5.2 Comparison of the Various Models When we look at the various models, we see that there is not a close estimate of demand

for this item. That is because the demand is scattering and follows no specific pattern

with a seasonality factor or a trend. However, it seems that Winter’s model estimates the

closest forecast value to the actual dema

MAPE and TS range for each model:


TS

nd. The following table indicates the MAD,

Model MAD MAPE Min Max

Moving Average 2 512 -6 -1

Simple Exponential Smoothing

2 344 -11 4

Holt’s 2 457 -4 3

Winter’s 2 225 -3 3



Therefore, we vary gamma to determine an appropriate value for it. The following figure

depicts the variation of the gamma values.

47

Winter's (Pen Refills)

8

-03

Mar-03

Ma Jul-0

3

Sep-03

Nov Ja-04

Mar-04

Ma Jul-0

4

Se4

10

12

14

16

18D

eman

d Ft - 0.1Ft - 0.5Ft - 0.7

6

0

2

4

Nov-02

Jan y-0

3 -03 n y-04

p-0

Date

Actual

Figure 78: Varying Gamma

the above figure we note that increasing gamma increases the deviation of the

recast, thus the most suitable gamma value is 0.1.

forms. For pen refills

e reasonable lead time is one or two weeks, however since we aggregated the demand,

values.

From

fo

Let us now perform future forecasting to see how well the model per

th

weekly lead times can not be considered for the calculation. We assume a 2 month lead

time and perform the forecast.

48

Winter's (Pen Refills)

0

1

2

3

4

5

6

7

8

9

10

May-04 Jun-04 Jul-04 Aug-04 Sep-04 Oct-04

Date

Dem

and Actual

Historical ForecastFuture Forecast


For the above future forecasting we get a MAD = 2 and a MAPE = 225 and TS between 3

and -3, which are exactly the same as historical forecast. Thus it indicates that the model

asting with a median absolute percentage error of 225%.

is barely reliable for future forec

49

7 Conclusion and Scope for Further Study We now discuss our conclusions from our analysis, the strengths and weaknesses of our

methodology. We also discuss scope for further study to improve and refine the

forecasting techniques so far determined from the study of items from TechWorx Inc.

7.1 Conclusions, Strengths and Weaknesses Most of the items chosen for study did not reveal any sales patterns that we initially

hypothesized for them. The sales items showed the following patterns as described

below.

Table 8: Hypothesized vs observed pattern

Pattern Observed Pattern Item Hypothesized

Binders/Clippers Seasonal Seasonal

Pens Stable Seasonal

Speakers Rarely Sold Stable

Lead Stable Stable/Seasonal

Pen refills Stable Rarely Sold

The table above indicates that the characteristics of an item may not necessarily reveal

e sales pattern. This draws an important conclusion that it may be erroneous to predict

e sales pattern of an item without thorough analysis of its historical sales data.

mong all the models considered under study, Winter’s model with seasonal time period

as provided the best results albeit with some deviation for many of the items. For some

f the items we found that aggregating the sales data at monthly levels helped to perform

nalysis on them. The following is a summary of the future forecasting capability for the

odel chosen for each item.

th

th

A

h

o

a

m

50

Table 9: Error estimate in future forecasting

Item Model (Type and MAD M

Time Period)

APE

Binders/Clippers Winter’s, Seasonal

Time Period

6.545059 24.98808

Pens Winter’s, Seasonal 45 21

Time Period

Speakers Winter’s, Seasonal

Time Period

1.233825 1.569127

Lead Winter’s, Seasonal 12.41 32.45

Time Period

Pen refills Winter

Time P

’s, Seasonal 2

eriod

225

As can be seen from the above n absolute percen or is quite high for

as low as 1.57% for speakers to as high as

further work is needed to find forecasting

e managerial problem posed.

s, i.e. the

fact that the Winter’s models showed the best result indicates the sales pattern

e periods and a variety of adaptive forecasting

techniques.

4. We varied our base time period from weekly to monthly for some items.

data the mea tage err

some of the future forecasts ranging from

225% for pen refills. This indicates that

techniques to better address th

The strengths of our methodology are as follows:

1. The models under study provide insights into the sales pattern of the item

have trends and seasonality in them.

2. The present study helps us find the time periods to consider for each item, i.e. the

fact that the Seasonal Time period showed the best result indicates the sales

patterns were different for each of the three terms of Winter, Spring and Fall.

3. We considered a variety of tim

51

The weaknesses of our methodology are as follows:

The models use line ssion to find the e of fit to

the sales data, w be justified in case when the sales data follow some

r pa r echniques such as lin-log, log-

log and other a techniques may result in better approximations of the

deseasonalized

2. In order to just iner regression to approximate the sales data, study

the statistica (re of squares) and t-stat is required.

3. Also to justify the use of linear regression, it needs to be ascertained there is no

evidence of het to tion in the data, which the current

study has not c

y of sales f w time period may have produced better

results. However, due to lack of data beyond two years this was not explored for a

bigger time horizon to come up with a better forecast.

he figures of error for all the items under study are fairly large, which suggests a need

emand forecasting. Our study has

determ

items.

(apart f ting methods should also be explored to see

how

From o

fore

current study should serve as a guideline for further study of other forecasting techniques

that ropose

ny new implementation for TechWorx Inc, no organisational change is suggested.

1. d in this study use

hich may not

ar regre best lin

non - linea ttern. Exploring othe

pproximation

non linear t

demand.

ify the use of l

of l parameters like R2 sidual sum

eroscadascity and au

onsidered.

correla

4. Stud igures based on a 52 eek

5. The current study uses time series methods to perform forecasting. Others

methods like Causal and simulation methods may provide better results.

7.2 Scope for further study T

for refinement in the current techniques used for d

ined what are the suitable time periods and sales patterns for particular types of

Further study needs to be performed to better estimate deseasonalized demand

rom linear regression). Other forecas

they fare against the currently discussed forecasting techniques.

ur study, we recommend further analysis of the sales data using some other

casting techniques to come up with improved methods for implementation. The

can better be applied for the items at TechWorx Inc. As we currently do not p

a

52

8

, 2004)

References

1. Meindl, P. and Chopra, S. Supply Chain Management 2nd Edition. New

Jersey: Prentice Hall, 2004.

2. Hadley, S. “Msci 709 – Logistics and Supply Chain Management”

(November 26

http://www.mansci.uwaterloo.ca/msci709/index.htm

53

Appendix A: Raw Data Table A-1: Sales data

SubClass W1 1055 409.351055 16-Nov-02 1055 1055 30-Nov-02 73 440.851055 7-Dec-02 59 357.311055 14-Dec-02 40 230.881055 21-Dec-02 17 110.931055 4-Jan-03 5 40.751055 11-Jan-03 123 862.611055 18-Jan-03 98 630.661055 25-Jan-03 69 427.771055 1-Feb-03 50 314.91055 8-Feb-03 61 327.591055 15-Feb-03 55 406.011055 22-Feb-03 23 135.771055 1-Mar-03 51 320.891055 8-Mar-03 64 365.861055 15-Mar-03 38 225.61055 22-Mar-03 54 356.561055 29-Mar-03 42 292.761055 5-Apr-03 52 339.581055 12-Apr-03 52 346.881055 19-Apr-03 23 201.371055 26-Apr-03 13 66.271055 3-May-03 13 73.471055 10-May-03 63 375.171055 17-May-03 36 187.441055 24-May-03 28 148.521055 31-May-03 29 178.911055 7-Jun-03 35 250.931055 14-Jun-03 35 198.951055 21-Jun-03 33 151.971055 28-Jun-03 17 127.431055 5-Jul-03 8 36.821055 12-Jul-03 18 97.021055 19-Jul-03 18 114.921055 26-Jul-03 36 185.941055 2-Aug-03 22 120.981055 9-Aug-03 21 117.491055 16-Aug-03 13 77.771055 23-Aug-03 14 74.361055 30-Aug-03 17 90.031055 6-Sep-03 70 470.21055 13-Sep-03 125 800.611055 20-Sep-03 87 505.611055 27-Sep-03 77 447.25

eekend date Sale Qty Total Dollar Sales 055 2-Nov-02 45 337.85

9-Nov-02 6560 410.8

23-Nov-02 67 422.93

A-1

1055 4-Oct-03 108 507.451055 11-Oct-03 85 451.85

403.04428.02

1055 1-Nov-03 62 403.981055 8-Nov-03 73 414.05

5-Nov-03 65 409.35

1

122 2

1122

12 101 6

5

1055 18-Oct-03 641055 25-Oct-03 76

1055 11055 22-Nov-03

271 360.89

1055 9-Nov-03 74 321.861055 6-Dec-03 60 270.4

21055 13-Dec-03 57 64.091055 20-Dec-03 41 286.491055 27-Dec-03

17 52.43

1055 0-Jan-04 08 681.321055 17-Jan-04 82 435.381055 24-Jan-04

3825

437.3431055 1-Jan-04 1 04.09

1055 7-Feb-04 77 408.391055 14-Feb-04 65 365.831055 21-Feb-04

243 221.47

41055 8-Feb-04 79 98.291055 6-Mar-04 54 277.021055 13-Mar-04 57 328.811055 20-Mar-04

253 294.87

1055 7-Mar-04 46 325.241055 3-Apr-04 54 301.66

21055 10-Apr-04 38 33.121055 17-Apr-04 41 253.891055 24-Apr-04

114 104.16

1055 -May-04 4 25.461055 8-May-04 44 297.361055 5-May-04 45 317.351055 2-May-04 38 240.71055 9-May-04 43 77.171055 5-Jun-04 41 272.171055 12-Jun-04 39 241.711055 19-Jun-04 27 173.331055 26-Jun-04 30 202.91055 3-Jul-04 22 130.281055 10-Jul-04 26 139.841055 17-Jul-04 32 198.741055 24-Jul-04

3264

210.2441055 1-Jul-04 3 51.67

1055 7-Aug-04 18 30.921055 4-Aug-04 22 143.581055 1-Aug-04 24 209.661055 8-Aug-04 19 147.411055 4-Sep-04 36 157.3

61055 11-Sep-04 92 13.341055 18-Sep-04 64 18.281055 25-Sep-04 12 96.261055 2-Oct-04 88 39.741055 9-Oct-04 63 456.731055 16-Oct-04 61 402.591055 1055

23-Oct-04 30-Oct-04

8175

441.79709.55

A-2

1060 2-Nov-02 29 98.811060 9-Nov-02 30 132.41060 16-Nov-02 36 155.941060 23-Nov-02 17 60.631060 3

1 5 191

7 3

1 2123 2 1

1

2

123

15 111

2

0-Nov-02 19 71.211060 7-Dec-02 60 168.91060 14-Dec-02 32 133.381060 21-Dec-02 17 92.831060 4-Jan-03 32 140.681060 1-Jan-03 30 39.381060 18-Jan-03 80 590.11060 25-Jan-03 98 348.121060 1-Feb-03 6 07.141060 8-Feb-03 49 202.411060 15-Feb-03 38 123.921060 22-Feb-03 22 99.981060 1-Mar-03 34 139.361060 8-Mar-03 32 125.481060 15-Mar-03 21 81.391060 22-Mar-03 35 111.631060 29-Mar-03 22 94.381060 5-Apr-03 30 117.41060 12-Apr-03 15 65.651060 19-Apr-03 17 70.311060 26-Apr-03 3 9.071060 3-May-03 15 68.251060 0-May-03 19 873.791060 7-May-03 77 301.431060 4-May-03 30 133.881060 1-May-03 8 08.821060 7-Jun-03 20 80.71060 14-Jun-03 26 104.341060 21-Jun-03 31 30.271060 28-Jun-03 11 48.791060 5-Jul-03 15 63.951060 12-Jul-03 10 42.11060 19-Jul-03 8 40.621060 26-Jul-03 18 59.221060 -Aug-03 23 101.351060 9-Aug-03 8 56.781060 6-Aug-03 9 42.811060 3-Aug-03 20 58.11060 0-Aug-03 25 126.351060 6-Sep-03 41 535.991060 13-Sep-03 20 950.081060 20-Sep-03 78 657.361060 27-Sep-03 00 387.41060 4-Oct-03 82 82.581060 11-Oct-03 41 236.931060 18-Oct-03 34 130.861060 25-Oct-03 53 197.471060 1-Nov-03 22 84.181060 8-Nov-03 56 209.141060 15-Nov-03 23 101.571060 22-Nov-03 27 130.031060 29-Nov-03 22 86.88

A-3

1060 6-Dec-03 17 77.631060 1

4 181 1 6

11

1

1

2 81 22 12 1 5

3

71 322

1 13 460 212 721

11222

3 2 8123

3-Dec-03 20 70.51060 20-Dec-03 16 92.041060 27-Dec-03 4 15.261060 10-Jan-04 95 85.151060 7-Jan-04 98 78.221060 24-Jan-04 06 438.941060 31-Jan-04 52 98.561060 7-Feb-04 40 115.11060 14-Feb-04 35 151.651060 21-Feb-04 44 59.161060 28-Feb-04 38 170.51060 6-Mar-04 40 139.631060 3-Mar-04 34 92.461060 20-Mar-04 25 96.751060 27-Mar-04 27 95.831060 3-Apr-04 17 72.811060 10-Apr-04 27 88.131060 17-Apr-04 14 60.461060 24-Apr-04 23 88.571060 1-May-04 18 54.521060 8-May-04 47 66.331060 5-May-04 67 15.641060 2-May-04 47 67.641060 9-May-04 6 4.751060 5-Jun-04 18 77.741060 12-Jun-04 30 102.641060 19-Jun-04 20 82.631060 26-Jun-04 24 67.761060 3-Jul-04 20 751060 10-Jul-04 11 3.991060 17-Jul-04 17 60.031060 24-Jul-04 45 65.061060 31-Jul-04 26 74.341060 -Aug-04 10 36.91060 4-Aug-04 13 7.671060 1-Aug-04 8 30.221060 8-Aug-04 10 36.11060 4-Sep-04 25 81.551060 1-Sep-04 2 82.281060 18-Sep-04 0 36.281060 25-Sep-04 18 3.361060 2-Oct-04 31 423.471060 9-Oct-04 33 113.451060 16-Oct-04 29 115.511060 23-Oct-04 35 114.451060 30-Oct-04 35 23.651081 2-Nov-02 39 372.591081 9-Nov-02 47 461.191081 16-Nov-02 15 424.871081 23-Nov-02 00 699.841081 0-Nov-02 07 98.691081 7-Dec-02 34 603.561081 14-Dec-02 12 790.31081 21-Dec-02 18 742.021081 4-Jan-03 37 66.49

A-4

1081 1 22 79129 13

21 5111 722 82 61 41 3

1 1 51 1 62 1 33 1 3

21 2

31

12

2 2221 4

1 12 1 33 1 3

1 663 1124 84 1102 8

1 11 11

1 1

11

18

2 12

1-Jan-03 50 600.71081 18-Jan-03 25 8.391081 25-Jan-03 82 6351081 1-Feb-03 8 10.741081 8-Feb-03 170 668.41081 15-Feb-03 271 851.151081 22-Feb-03 132 350.461081 1-Mar-03 38 629.31081 8-Mar-03 68 51.721081 15-Mar-03 68 329.421081 22-Mar-03 65 545.111081 29-Mar-03 98 40.821081 5-Apr-03 97 939.251081 12-Apr-03 24 44.821081 19-Apr-03 55 93.451081 26-Apr-03 00 85.481081 3-May-03 79 80.911081 0-May-03 94 78.561081 7-May-03 89 79.551081 4-May-03 28 17.921081 1-May-03 97 08.331081 7-Jun-03 106 331.041081 14-Jun-03 86 346.881081 21-Jun-03 38 577.921081 28-Jun-03 36 78.221081 5-Jul-03 61 05.391081 12-Jul-03 00 263.71081 19-Jul-03 41 12.191081 26-Jul-03 93 74.551081 -Aug-03 83 .371081 9-Aug-03 24 73.961081 6-Aug-03 30 468.81081 3-Aug-03 28 62.621081 0-Aug-03 15 43.351081 6-Sep-03 53 6.471081 13-Sep-03 47 48.951081 20-Sep-03 3 35.131081 27-Sep-03 71 9.821081 4-Oct-03 34 97.441081 11-Oct-03 187 667.531081 18-Oct-03 140 410.741081 25-Oct-03 167 759.491081 1-Nov-03 180 907.71081 8-Nov-03 97 561.951081 15-Nov-03 60 042.661081 22-Nov-03 35 135.031081 29-Nov-03 55 679.351081 6-Dec-03 126 446.081081 3-Dec-03 217 137.531081 20-Dec-03 111 664.891081 27-Dec-03 127 660.631081 10-Jan-04 132 876.341081 7-Jan-04 121 857.491081 24-Jan-04 114 90.741081 31-Jan-04 123 759.731081 7-Feb-04 13 92.95

A-5

1081 14-Feb-04 123 9

124

1 8

7

1

122

7 11 322 1

115

1

25.871081 21-Feb-04 123 700.331081 28-Feb-04 201 5.391081 6-Mar-04 243 826.071081 3-Mar-04 127 08.771081 20-Mar-04 153 678.931081 27-Mar-04 158 771.141081 3-Apr-04 175 33.631081 10-Apr-04 122 655.781081 17-Apr-04 217 996.791081 24-Apr-04 201 108.691081 1-May-04 75 821.611081 8-May-04 117 791.651081 5-May-04 91 710.551081 2-May-04 129 689.071081 9-May-04 127 673.391081 5-Jun-04 170 679.981081 12-Jun-04 233 527.951081 19-Jun-04 102 450.341081 26-Jun-04 83 295.171081 3-Jul-04 76 294.11081 10-Jul-04 82 400.461081 17-Jul-04 65 485.851081 24-Jul-04 150 468.11081 31-Jul-04 90 389.91081 -Aug-04 05 458.171081 4-Aug-04 72 90.361081 1-Aug-04 66 412.121081 8-Aug-04 33 562.191081 4-Sep-04 93 505.171081 1-Sep-04 113 978.051081 18-Sep-04 320 70.581081 25-Sep-04 287 973.251081 2-Oct-04 105 648.891081 9-Oct-04 142 642.721081 16-Oct-04 100 520.51081 23-Oct-04 121 725.951081 30-Oct-04 72 470.441155 2-Nov-02 2 17.981155 9-Nov-02 2 22.981155 16-Nov-02 1 8.991155 21-Dec-02 5 44.951155 4-Jan-03 2 17.981155 1-Jan-03 2 46.961155 18-Jan-03 2 17.981155 15-Feb-03 1 13.991155 22-Feb-03 -1 13.991155 8-Mar-03 10 69.91155 15-Mar-03 1 21.991155 26-Apr-03 1 8.991155 10-May-03 2 26.981155 7-Jun-03 2 44.961155 14-Jun-03 1 19.991155 21-Jun-03 2 17.981155 28-Jun-03 1 8.991155 19-Jul-03 1 8.99

A-6

1155 2-Aug-03 1 19.991155 23-Aug-03 1 6.991155 6-Sep-03 1 21.991155 1

1

22

1

29

11

1

2

2

10 311 413 47

4 110 3

7 29 36 2

332

7 24 1821

11 45 26 25

3 6 25

3-Sep-03 3 63.971155 20-Sep-03 1 8.991155 27-Sep-03 1 6.991155 4-Oct-03 1 8.991155 8-Oct-03 1 8.991155 25-Oct-03 1 21.991155 8-Nov-03 1 6.991155 22-Nov-03 1 21.991155 9-Nov-03 1 6.991155 0-Dec-03 0 01155 27-Dec-03 0 01155 10-Jan-04 1 6.991155 7-Feb-04 1 8.991155 4-Feb-04 1 6.991155 6-Mar-04 16 129.91155 20-Mar-04 0 01155 27-Mar-04 0 01155 -May-04 1 6.991155 26-Jun-04 1 6.991155 24-Jul-04 3 61155 1-Sep-04 2 13.981155 8-Sep-04 1 6.991155 2-Oct-04 1 6.991160 2-Nov-02 81 270.191160 9-Nov-02 18 458.421160 16-Nov-02 101 372.771160 3-Nov-02 85 334.451160 30-Nov-02 76 265.821160 7-Dec-02 121 439.891160 14-Dec-02 84 253.941160 1-Dec-02 47 194.531160 4-Jan-03 14 55.461160 11-Jan-03 240 855.91160 18-Jan-03 121 442.791160 25-Jan-03 94 356.761160 1-Feb-03 8 86.521160 8-Feb-03 4 61.061160 15-Feb-03 7 3.931160 22-Feb-03 2 83.061160 1-Mar-03 3 83.971160 8-Mar-03 7 88.831160 15-Mar-03 7 55.011160 22-Mar-03 8 39.121160 29-Mar-03 81 23.391160 5-Apr-03 71 3.391160 12-Apr-03 4 80.761160 19-Apr-03 5 8.131160 26-Apr-03 0 93.91160 3-May-03 5 65.951160 10-May-03 6 18.421160 17-May-03 7 34.131160 24-May-03 7 7.831160 1-May-03 3 0.95

A-7

1160 7-Jun-03 5 26 25

2 4 14 12 73 124 195 206 24 12 11 31 58.1

12 486.026 10

2 17 716 571 410 352.911 426.815 6311 4612 450.1

8 310 45

2 11 451

1

2 11

1

1 44

1

1

1

1122 20

3

5 41.131160 14-Jun-03 3 0.151160 1-Jun-03 3 66.771160 28-Jun-03 4 88.261160 5-Jul-03 1 0.491160 12-Jul-03 3 6.971160 19-Jul-03 8 3.221160 26-Jul-03 0 9.981160 2-Aug-03 7 34.731160 9-Aug-03 0 50.61160 16-Aug-03 6 14.741160 23-Aug-03 1 8.391160 30-Aug-03 8 21160 6-Sep-03 4 61160 13-Sep-03 8 23.51160 0-Sep-03 8 09.21160 27-Sep-03 0 6.421160 4-Oct-03 37 12.991160 11-Oct-03 5 51160 18-Oct-03 7 11160 25-Oct-03 9 0.631160 1-Nov-03 9 1.411160 8-Nov-03 3 71160 15-Nov-03 5 50.231160 22-Nov-03 4 7.761160 9-Nov-03 6 0.241160 6-Dec-03 57 590.291160 3-Dec-03 104 402.621160 20-Dec-03 30 109.21160 27-Dec-03 3 14.571160 10-Jan-04 85 018.551160 7-Jan-04 133 491.551160 24-Jan-04 02 400.061160 31-Jan-04 99 340.311160 7-Feb-04 17 73.811160 14-Feb-04 125 70.751160 21-Feb-04 43 168.671160 28-Feb-04 17 417.711160 6-Mar-04 81 334.071160 3-Mar-04 82 348.441160 20-Mar-04 72 269.461160 27-Mar-04 91 349.791160 3-Apr-04 112 411.861160 10-Apr-04 17 379.931160 17-Apr-04 89 310.311160 24-Apr-04 27 123.611160 1-May-04 15 56.251160 8-May-04 51 585.691160 5-May-04 74 288.561160 2-May-04 70 246.61160 9-May-04 54 7.961160 5-Jun-04 70 02.281160 12-Jun-04 76 282.741160 19-Jun-04 74 244.561160 26-Jun-04 42 157.581160 3-Jul-04 21 91.49

A-8

1160 10-Jul-04 50 181.21160 17-Jul-04 53 205.671160 24-Jul-04 66 264.11160 31-Jul-04 58 241.521160 7 3

122

1 32 11 41 5

6

3

1

1

1 27 1

1

1

1123 1

2 1

-Aug-04 91 56.491160 4-Aug-04 48 186.021160 1-Aug-04 9 45.711160 8-Aug-04 26 109.241160 4-Sep-04 29 127.211160 1-Sep-04 94 70.561160 18-Sep-04 72 030.661160 25-Sep-04 31 45.491160 2-Oct-04 54 50.261160 9-Oct-04 129 483.311160 16-Oct-04 116 431.981160 23-Oct-04 185 41.211160 30-Oct-04 153 545.271200 2-Nov-02 95 211.551200 9-Nov-02 89 186.111200 16-Nov-02 97 214.031200 23-Nov-02 102 222.581200 0-Nov-02 114 244.441200 7-Dec-02 148 285.221200 14-Dec-02 15 245.751200 21-Dec-02 36 95.941200 4-Jan-03 6 12.941200 1-Jan-03 174 380.761200 18-Jan-03 88 163.121200 25-Jan-03 06 15.921200 1-Feb-03 9 58.011200 8-Feb-03 84 171.441200 15-Feb-03 127 211.031200 22-Feb-03 52 81.081200 1-Mar-03 22 236.141200 8-Mar-03 88 171.621200 15-Mar-03 115 221.451200 22-Mar-03 80 147.71200 29-Mar-03 73 130.371200 5-Apr-03 79 149.911200 12-Apr-03 57 298.131200 19-Apr-03 41 72.491200 26-Apr-03 10 16.51200 3-May-03 14 30.461200 0-May-03 80 154.181200 7-May-03 38 70.221200 4-May-03 39 91.491200 1-May-03 53 01.471200 7-Jun-03 59 122.211200 14-Jun-03 71 141.891200 21-Jun-03 50 107.31200 28-Jun-03 37 75.331200 5-Jul-03 16 36.241200 12-Jul-03 34 72.461200 19-Jul-03 33 72.671200 26-Jul-03 32 66.881200 -Aug-03 60 35.88

A-9

1200 9-Aug-03 47 94.731200 1

23

112 211

11 1

1

18 1

1

1

1

1122 6

17 1 2

122

6-Aug-03 16 30.741200 3-Aug-03 9 14.711200 0-Aug-03 14 30.461200 6-Sep-03 64 147.461200 13-Sep-03 99 430.411200 20-Sep-03 4 59.461200 27-Sep-03 25 253.171200 4-Oct-03 19 210.921200 11-Oct-03 95 196.951200 18-Oct-03 101 216.691200 25-Oct-03 143 284.771200 1-Nov-03 102 203.881200 8-Nov-03 110 236.31200 15-Nov-03 76 149.841200 22-Nov-03 101 246.571200 29-Nov-03 119 248.211200 6-Dec-03 59 324.011200 3-Dec-03 15 216.931200 20-Dec-03 24 47.361200 27-Dec-03 4 9.861200 10-Jan-04 153 328.151200 7-Jan-04 97 181.031200 24-Jan-04 90 173.91200 31-Jan-04 81 65.991200 7-Feb-04 5 63.451200 14-Feb-04 153 320.771200 21-Feb-04 53 97.271200 28-Feb-04 99 188.011200 6-Mar-04 98 179.421200 3-Mar-04 98 182.521200 20-Mar-04 74 133.461200 27-Mar-04 84 24.861200 3-Apr-04 99 144.311200 10-Apr-04 02 155.081200 17-Apr-04 113 188.271200 24-Apr-04 30 44.21200 1-May-04 12 21.381200 8-May-04 05 158.251200 5-May-04 48 87.321200 2-May-04 70 126.61200 9-May-04 39 3.511200 5-Jun-04 60 102.81200 12-Jun-04 84 132.761200 19-Jun-04 49 77.411200 26-Jun-04 52 90.981200 3-Jul-04 30 63.41200 10-Jul-04 42 83.481200 17-Jul-04 57 110.911200 24-Jul-04 60 115.91200 31-Jul-04 66 42.921200 -Aug-04 11 30.291200 4-Aug-04 45 93.851200 1-Aug-04 9 26.711200 8-Aug-04 17 28.331200 4-Sep-04 13 31.57

A-10

1200 1 11 3

9 11

3

37

2

1 1 21

439 76

31 233

3 62221

112 1 323 2 3

1 31 311 221 31 3

2 2 41

1 12 1 23 1

13 2661 135 115 9

1-Sep-04 55 29.551200 18-Sep-04 69 80.211200 25-Sep-04 1 91.891200 2-Oct-04 98 96.321200 9-Oct-04 78 159.421200 16-Oct-04 70 140.081200 23-Oct-04 98 200.021200 30-Oct-04 116 235.641205 2-Nov-02 278 598.341205 9-Nov-02 70 739.41205 16-Nov-02 364 809.821205 23-Nov-02 316 607.841205 0-Nov-02 290 604.121205 7-Dec-02 379 39.891205 14-Dec-02 47 712.691205 21-Dec-02 171 356.791205 4-Jan-03 84 108.561205 1-Jan-03 212 283.821205 18-Jan-03 651 152.071205 25-Jan-03 14 803.31205 1-Feb-03 2 3.561205 8-Feb-03 379 814.051205 15-Feb-03 84 718.941205 22-Feb-03 12 18.481205 1-Mar-03 20 691.31205 8-Mar-03 48 674.521205 15-Mar-03 322 637.161205 22-Mar-03 24 21.941205 29-Mar-03 92 592.461205 5-Apr-03 92 601.581205 12-Apr-03 32 509.081205 19-Apr-03 32 263.381205 26-Apr-03 50 82.51205 3-May-03 71 162.691205 0-May-03 516 866.941205 7-May-03 218 398.921205 4-May-03 81 6.671205 1-May-03 07 73.931205 7-Jun-03 179 353.191205 14-Jun-03 87 47.831205 21-Jun-03 58 32.921205 28-Jun-03 40 345.81205 5-Jul-03 01 41.391205 12-Jul-03 45 508.671205 19-Jul-03 86 75.641205 26-Jul-03 67 86.731205 -Aug-03 18 13.61205 9-Aug-03 58 334.81205 6-Aug-03 00 212.911205 3-Aug-03 08 25.621205 0-Aug-03 54 294.741205 6-Sep-03 423 721.571205 13-Sep-03 49 13.851205 20-Sep-03 4 82.761205 27-Sep-03 70 44.721205 4-Oct-03 80 10.01

A-11

1205 11-Oct-03 3

44433

1 7

1 21 1

4

8

1

6

1122 1 29

1

37

122

41 7

1 211

67 783.331205 18-Oct-03 333 757.271205 25-Oct-03 03 851.551205 1-Nov-03 57 968.851205 8-Nov-03 15 877.711205 15-Nov-03 73 806.41205 22-Nov-03 20 667.611205 29-Nov-03 407 945.131205 6-Dec-03 300 682.281205 3-Dec-03 261 75.161205 20-Dec-03 156 413.361205 27-Dec-03 16 45.041205 10-Jan-04 260 377.281205 7-Jan-04 655 272.331205 24-Jan-04 495 936.921205 31-Jan-04 324 666.941205 7-Feb-04 40 907.961205 14-Feb-04 449 990.061205 21-Feb-04 162 297.981205 28-Feb-04 411 57.231205 6-Mar-04 280 587.581205 3-Mar-04 259 493.321205 20-Mar-04 282 585.641205 27-Mar-04 333 697.921205 3-Apr-04 317 80.451205 10-Apr-04 221 490.581205 17-Apr-04 232 511.31205 24-Apr-04 132 277.261205 1-May-04 155 293.151205 8-May-04 588 061.941205 5-May-04 243 426.271205 2-May-04 182 315.881205 9-May-04 71 6.371205 5-Jun-04 93 420.771205 12-Jun-04 153 315.871205 19-Jun-04 224 449.571205 26-Jun-04 157 311.911205 3-Jul-04 138 244.821205 10-Jul-04 139 262.491205 17-Jul-04 334 645.381205 24-Jul-04 211 482.731205 31-Jul-04 192 68.981205 -Aug-04 180 3091205 4-Aug-04 115 260.451205 1-Aug-04 121 269.991205 8-Aug-04 114 233.661205 4-Sep-04 233 36.871205 1-Sep-04 429 41.211205 18-Sep-04 205 329.351205 25-Sep-04 547 027.581205 2-Oct-04 546 099.431205 9-Oct-04 414 890.661205 16-Oct-04 379 751.231205 23-Oct-04 417 919.691205 30-Oct-04 351 705.691209 9-Nov-02 1 5.99

A-12

1209 16-Nov-02 0 01209 30-Nov-02 1 2.991209 7-Dec-02 1 2.991209 1

2

2

1422

12

1

11

8-Jan-03 1 5.991209 8-Feb-03 1 5.991209 22-Feb-03 1 5.991209 8-Mar-03 1 5.991209 22-Mar-03 1 2.991209 5-Apr-03 2 8.981209 19-Apr-03 1 5.991209 3-May-03 1 5.991209 10-May-03 3 11.971209 17-May-03 1 5.991209 24-May-03 0 5.981209 31-May-03 1 2.991209 14-Jun-03 0 01209 1-Jun-03 1 5.991209 28-Jun-03 1 5.991209 5-Jul-03 1 2.991209 12-Jul-03 1 5.991209 19-Jul-03 1 2.991209 9-Aug-03 1 2.991209 30-Aug-03 0 01209 13-Sep-03 1 5.991209 0-Sep-03 1 2.991209 27-Sep-03 2 8.981209 4-Oct-03 2 8.981209 11-Oct-03 0 01209 18-Oct-03 0 01209 25-Oct-03 1 5.991209 1-Nov-03 3 11.971209 15-Nov-03 5 17.951209 22-Nov-03 2 5.181209 6-Dec-03 0 5.981209 10-Jan-04 1 2.991209 17-Jan-04 3 8.171209 24-Jan-04 3 11.971209 -Feb-04 2 5.981209 1-Feb-04 1 2.591209 8-Feb-04 1 5.991209 6-Mar-04 1 2.991209 3-Mar-04 2 8.981209 0-Mar-04 3 8.971209 27-Mar-04 3 11.971209 1-May-04 1 11.171209 8-May-04 1 2.591209 5-May-04 1 5.991209 22-May-04 0 01209 29-May-04 0 01209 5-Jun-04 2 5.981209 2-Jun-04 0 01209 9-Jun-04 3 8.971209 26-Jun-04 3 11.971209 3-Jul-04 2 8.981209 10-Jul-04 0 01209 17-Jul-04 0 0

A-13

1209 24-Jul-04 2 6.8

1

2 21

1 1

11

22 25

121 1

1 15

111

23

281217

3 4133

417247662819

1

11 1

10109.9

44.91 4

14.913

10 231 217.92 14.9

81209 31-Jul-04 0 01209 7-Aug-04 2 3.981209 28-Aug-04 2 9.881209 4-Sep-04 2 5.581209 18-Sep-04 1 3.891209 5-Sep-04 5 0.951209 2-Oct-04 3 1.971209 6-Oct-04 3 4.971209 23-Oct-04 2 8.581209 30-Oct-04 1 2.596066 2-Nov-02 4 1466066 9-Nov-02 4 1446066 16-Nov-02 8 846066 23-Nov-02 0 2066066 30-Nov-02 5 26066 7-Dec-02 9 946066 14-Dec-02 9 1966066 -Dec-02 6 1606066 25-Jan-03 0 06066 8-Feb-03 9 926066 5-Feb-03 0 1006066 22-Feb-03 5 06066 1-Mar-03 6 1686066 8-Mar-03 2 1286066 15-Mar-03 2 1286066 2-Mar-03 9 926066 29-Mar-03 3 26066 26-Apr-03 0 08207 2-Nov-02 3 8.978207 9-Nov-02 3 05.978207 16-Nov-02 3 79.978207 23-Nov-02 2 1.988207 0-Nov-02 4 9.968207 7-Dec-02 2 4.988207 14-Dec-02 1 3.998207 21-Dec-02 3 57.978207 11-Jan-03 7 3.898207 18-Jan-03 8 9.928207 25-Jan-03 5 4.958207 1-Feb-03 1 4.998207 15-Feb-03 1 0.978207 22-Feb-03 1 4.998207 1-Mar-03 1 87.998207 8-Mar-03 3 54.978207 5-Mar-03 1 4.998207 22-Mar-03 2 2.988207 29-Mar-03 1 98207 5-Apr-03 3 78207 2-Apr-03 2 9.988207 19-Apr-03 1 98207 26-Apr-03 1 6.998207 3-May-03 1 55.998207 -May-03 4 6.968207 7-May-03 6 48207 4-May-03 1 9

A-14

8207 343.9

1 25

2 61

3

2 133 10

1 9466420

2 1542624

27913119

2 62 21

20134129

69.91 11

1

1

1-May-03 0 08207 7-Jun-03 1 98207 4-Jun-03 2 49.988207 21-Jun-03 2 8.988207 8-Jun-03 1 9.998207 5-Jul-03 1 4.998207 12-Jul-03 1 14.998207 26-Jul-03 0 08207 2-Aug-03 2 29.988207 9-Aug-03 1 9.998207 16-Aug-03 0 08207 3-Aug-03 2 9.988207 0-Aug-03 1 9.998207 6-Sep-03 3 .878207 13-Sep-03 6 5.948207 20-Sep-03 6 4.948207 7-Sep-03 3 .978207 4-Oct-03 6 4.958207 18-Oct-03 2 6.988207 25-Oct-03 0 08207 1-Nov-03 4 .968207 8-Nov-03 2 9.988207 15-Nov-03 3 .978207 2-Nov-03 1 9.998207 9-Nov-03 3 9.978207 6-Dec-03 3 9.978207 13-Dec-03 1 .998207 20-Dec-03 4 .968207 27-Dec-03 1 98207 0-Jan-04 1 479.898207 7-Jan-04 1 1558207 24-Jan-04 5 299.958207 31-Jan-04 4 239.968207 7-Feb-04 3 314.978207 4-Feb-04 1 19.998207 28-Feb-04 1 19.998207 6-Mar-04 5 186.958207 13-Mar-04 0 08207 20-Mar-04 2 31.988207 27-Mar-04 6 149.948207 3-Apr-04 1 15.998207 10-Apr-04 0 08207 17-Apr-04 3 47.978207 24-Apr-04 2 85.988207 1-May-04 1 69.998207 8-May-04 2 85.988207 22-May-04 1 69.998207 29-May-04 3 197.978207 5-Jun-04 2 85.988207 19-Jun-04 1 15.998207 3-Jul-04 1 64.998207 10-Jul-04 1 15.998207 17-Jul-04 0 31.988207 24-Jul-04 2 184.988207 7-Aug-04 2 75.988207 21-Aug-04 2 19.98

A-15

8207 28-Aug-04 5 629.9

1332

15.9

11072.8

1 11

2

1

2

2 1883.71 2 22 2

1 11 131 1

21

1 11

1 112

52 29

1 1 12 2124.21 11 1

51 617.8

642

11

133

16 32 33 4

4 21 7 38

5 3

58207 4-Sep-04 1 59.998207 11-Sep-04 1 963.898207 18-Sep-04 8 25.928207 25-Sep-04 5 71.958207 2-Oct-04 2 49.988207 9-Oct-04 2 75.988207 16-Oct-04 1 98207 23-Oct-04 2 75.988207 30-Oct-04 2 05.988208 2-Nov-02 8 98208 9-Nov-02 2 085.888208 16-Nov-02 0 757.928208 3-Nov-02 11 831.918208 30-Nov-02 3 519.968208 7-Dec-02 1 1520.98208 14-Dec-02 4 651.948208 1-Dec-02 5 803.938208 4-Jan-03 4 543.968208 11-Jan-03 3 98208 8-Jan-03 4 775.768208 5-Jan-03 1 2102.88208 1-Feb-03 0 811.838208 8-Feb-03 6 61.858208 15-Feb-03 5 498.868208 2-Feb-03 7 807.918208 1-Mar-03 5 891.858208 8-Mar-03 6 694.948208 15-Mar-03 8 929.88208 22-Mar-03 8 569.888208 29-Mar-03 1 0.888208 5-Apr-03 6 801.928208 12-Apr-03 2 309.988208 19-Apr-03 9 713.928208 26-Apr-03 6 54.948208 3-May-03 7 88.758208 0-May-03 7 171.858208 17-May-03 7 28208 24-May-03 2 891.88208 31-May-03 4 005.358208 7-Jun-03 7 72.918208 14-Jun-03 1 98208 21-Jun-03 7 93.918208 28-Jun-03 2 12.948208 5-Jul-03 4 35.968208 12-Jul-03 1 827.48208 19-Jul-03 0 05.988208 26-Jul-03 0 742.418208 2-Aug-03 5 12.468208 9-Aug-03 3 62.468208 -Aug-03 5 22.468208 3-Aug-03 3 99.958208 0-Aug-03 4 25.978208 6-Sep-03 1 089.718208 3-Sep-03 5 89.198208 20-Sep-03 0 191.36

A-16

8208 27-Sep-03 1 12 13

11

1

1 1015

1

3 21 1

111 1

1

1

1

1 11 122

1

7 11 722

1 4 34 33 21 1

9 416.778208 4-Oct-03 5 43.778208 11-Oct-03 10 133.848208 18-Oct-03 9 074.898208 25-Oct-03 4 892.868208 1-Nov-03 7 503.918208 8-Nov-03 1 26.788208 15-Nov-03 4 85.968208 22-Nov-03 4 97.948208 29-Nov-03 7 525.938208 6-Dec-03 9 691.928208 3-Dec-03 7 352.948208 20-Dec-03 14 947.878208 27-Dec-03 4 299.968208 10-Jan-04 5 279.638208 7-Jan-04 15 198.798208 24-Jan-04 8 863.88208 31-Jan-04 5 882.838208 7-Feb-04 2 058.868208 14-Feb-04 20 1319.88208 21-Feb-04 9 976.898208 28-Feb-04 13 001.858208 6-Mar-04 14 1136.868208 3-Mar-04 6 498.948208 20-Mar-04 5 418.958208 27-Mar-04 5 485.958208 3-Apr-04 8 954.868208 10-Apr-04 3 212.978208 17-Apr-04 7 591.948208 24-Apr-04 4 2600.828208 1-May-04 6 527.928208 8-May-04 9 381.818208 5-May-04 6 924.448208 2-May-04 7 426.738208 9-May-04 6 339.948208 5-Jun-04 4 260.968208 12-Jun-04 8 509.728208 19-Jun-04 13 1074.678208 26-Jun-04 7 467.738208 3-Jul-04 9 512.918208 10-Jul-04 4 522.928208 17-Jul-04 8 592.928208 24-Jul-04 6 2458.68208 31-Jul-04 5 363.758208 -Aug-04 0 550.58208 4-Aug-04 9 93.718208 1-Aug-04 9 757.718208 8-Aug-04 6 952.928208 4-Sep-04 8 711.58208 1-Sep-04 9 783.658208 18-Sep-04 2 239.288208 25-Sep-04 1 357.238208 2-Oct-04 9 551.578208 9-Oct-04 12 856.488208 16-Oct-04 9 805.698208 23-Oct-04 19 1439.41

A-17

8208 30-Oct-04 14 1040.668400 2-Nov-02 3 692.978400 9-Nov-02 2 184.988400 16-Nov-02 5 1609.958400 23-Nov-02 4 1

3

1 4

1

1

1 123

3

16 12 13 1 2

212 16

1 11 1

4

12

286.948400 0-Nov-02 4 624.968400 7-Dec-02 4 2043.978400 14-Dec-02 1 189.998400 21-Dec-02 1 189.998400 4-Jan-03 2 311.988400 1-Jan-03 6 126.928400 18-Jan-03 3 549.978400 25-Jan-03 4 589.968400 1-Feb-03 5 933.958400 8-Feb-03 1 99.998400 15-Feb-03 6 1023.948400 22-Feb-03 0 08400 1-Mar-03 5 630.958400 8-Mar-03 3 669.978400 15-Mar-03 0 08400 29-Mar-03 -4 703.928400 5-Apr-03 2 244.978400 12-Apr-03 1 69.998400 19-Apr-03 2 976.988400 3-May-03 1 69.998400 10-May-03 3 209.978400 7-May-03 6 149.948400 4-May-03 1 69.998400 1-May-03 1 446.998400 7-Jun-03 4 656.968400 14-Jun-03 2 408.998400 21-Jun-03 1 944.988400 28-Jun-03 5 811.958400 5-Jul-03 1 285.998400 12-Jul-03 0 08400 19-Jul-03 2 854.998400 26-Jul-03 5 619.958400 2-Aug-03 2 219.988400 9-Aug-03 6 819.948400 -Aug-03 7 099.948400 3-Aug-03 9 349.918400 0-Aug-03 7 369.858400 6-Sep-03 14 091.878400 3-Sep-03 5 459.968400 0-Sep-03 16 39.848400 27-Sep-03 5 929.868400 4-Oct-03 2 239.918400 11-Oct-03 3 289.978400 18-Oct-03 4 69.978400 25-Oct-03 3 1149.988400 1-Nov-03 6 779.948400 8-Nov-03 3 229.978400 5-Nov-03 3 369.978400 9-Nov-03 2 219.988400 6-Dec-03 1 149.998400 13-Dec-03 1 149.99

A-18

8400 20-Dec-03 3 369.978400 2

1

1

11

243.9

199.9

7

1

1 1182 1139.9

2

7-Dec-03 0 08400 0-Jan-04 5 429.958400 17-Jan-04 2 219.988400 24-Jan-04 5 429.958400 31-Jan-04 0 08400 7-Feb-04 2 369.988400 21-Feb-04 1 69.998400 28-Feb-04 3 467.978400 6-Mar-04 2 139.988400 20-Mar-04 2 319.988400 27-Mar-04 4 754.968400 3-Apr-04 1 364.998400 10-Apr-04 0 08400 17-Apr-04 3 28.988400 24-Apr-04 1 129.998400 1-May-04 2 88400 8-May-04 3 449.978400 5-May-04 0 08400 22-May-04 1 98400 29-May-04 2 486.988400 5-Jun-04 2 279.988400 12-Jun-04 1 99.998400 19-Jun-04 0 08400 26-Jun-04 4 35.968400 3-Jul-04 1 79.998400 10-Jul-04 1 179.998400 17-Jul-04 0 08400 24-Jul-04 1 69.998400 4-Aug-04 3 239.978400 28-Aug-04 4 309.968400 4-Sep-04 6 569.948400 11-Sep-04 3 418.868400 -Sep-04 6 449.948400 5-Sep-04 6 48400 2-Oct-04 7 559.938400 9-Oct-04 2 358.998400 16-Oct-04 3 269.978400 23-Oct-04 3 289.978400 30-Oct-04 0 0

A-19

msci 709 fall 2005 - university of waterloo › ~sjayaswa › projects › msci709... · 1...

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