# PowerPoint Presentation by xzS9z5

VIEWS: 6 PAGES: 40

• pg 1
```									Quantitative Methods

MAT 540
Forecasting

1
Objectives

• Upon completion of this lesson, you will
be able to:
– Apply the most appropriate forecasting
method for the properties of the available
data
– Use technology and information resources
to research and communicate issues in
Management Science.

2
What Is Forecasting?

• Forecasting is the process of predicting the
future. It is an integral part of almost all business
enterprises.
• Examples:
– Manufacturing firms forecast demand for their
product, to schedule manpower and raw material
allocation.
– Service organizations forecast customer arrival
patterns to maintain adequate customer service.

3
Forecasting Components
• Time Frames:
– Short-range (one to two months)
– Medium-range (two months to one years)
– Long-range (more than one or two years)
• Patterns:
–   Trend
–   Random variations
–   Cycles
–   Seasonality

4
Forecasting Methods
• Times Series
– uses historical data assuming the future will be
like the past.
• Regression Methods
– Regression (or causal ) methods that attempt to
develop a mathematical relationship between the
item being forecast and factors that cause it to
behave the way it does.
• Qualitative Methods
– Methods using judgment, expertise and opinion
to make forecasts.
5
PROPERTIES
On passing, 'Finish' button:       Goes to Next Slide
On failing, 'Finish' button:       Goes to Next Slide
Allow user to leave quiz:          At any time
User may view slides after quiz:   At any time
User may attempt quiz:             Unlimited times
Time Series Methods

• Time Series Methods
– Moving average
– Weighted moving average
– Exponential smoothing

7
Moving Average
• Moving average – A technique that averages a
number of recent actual values, updated as new
values become available.
• Formula:
n
 Di
MAn  i1n
where:
n  number of periods in the moving average
D  data in period i
i

8
Moving Average Continued
Example:
Sales of Comfort brand headache medicine for the past
ten weeks at Robert’s Drugs are shown below. If
Robert’s Drugs uses a 3-period moving average to
forecast sales, what is the forecast for Week 11?

Week   Sales       Week     Sales
1      110          6       120
2      115          7       130
3      125          8       115
4      120          9       110
5      125         10       130
9
Moving Average Continued
Solution:
Forecast 11 = (115 + 110 + 130 ) / 3 ≈ 118.3
Robert’s Drugs
Week (t)       Salest        Forecast t+1
1             110              —
2             115              —
3             125              —
4             120            116.7
5             125            120.0
6             120            123.3
7             130            121.7
8             115            125.0
9             110            121.7
10             130            118.8
11            —             118.3          10
Weighted Moving Average
• Weighted Moving Average – More recent
values in a series are given more weight in
computing the forecast.
• Formula
n
WMAn   W D
i1 i i
where W  the weight for period i, between 0% and 100%
i
Wi  1.00

11
Weighted Moving Average
Continued
Example:
The following data summarizes the historical demand for
door sales at Westmore Hardware. Use a weighted
moving average method with weights w1 = 0.2, w2 = 0.3,
w3 = 0.5 and determine the forecasted demand for
August and September.       Month      Actual Demand
March          20
April          25
May            40
June           35
July           30
August          45
12
Weighted Moving Average
Continued
Solution:
Forecast August = 40 • 0.2 + 35 • 0.3 + 30 • 0.5 = 33.5
Forecast September = 35 • 0.2 + 30 • 0.3 + 45 • 0.5 = 38.5
Westmore Hardware
Period   Week (t)    Actual Demand   Forecast t+1
1       March           20             —
2        April          25             —
3        May            40             —
4        June           35            31.5
5        July           30            34.5
6       August          45            33.5
7      September        —             38.5        13
Exponential Smoothing
• Exponential Smoothing
– the forecast for the next period is equal to the
forecast for the current period plus a proportion () of
the forecast error in the current period.
• Formula
Ft + 1 = Dt + (1 - )Ft
where: Ft + 1 = the forecast for the next period
Dt = actual demand in the present period
Ft = the previously determined forecast for the present period
 = a weighting factor (smoothing constant).

14
Exponential Smoothing
Continued
• Example
Make a prediction, given the following data on Gillmore
Hotel check-ins for a 5-month period below. Using the
exponential smoothing factor 0.3, how many check-ins
can be forecasted for January?
Gillmore Hotel
Month               Room Occupied
August                      40
September                    30
October                     45
November                     60
December                     55
15
Exponential Smoothing
Continued
• Solution
F2 = 0.3 • 40 + 0.7 • 40 = 40
F3 = 0.3 • 30 + 0.7 • 40 = 37
F6 = 0.3 • 55 + 0.7 • 45.6 = 48.4
Gillmore Hotel
Period    Month      Room Occupied    Forecast , Ft+1
1       August           40               —
2      September         30            F2 = 40
3       October          45            F3 = 37
4      November          60           F4 = 39.4
5      December          55           F5 = 45.6
6       January          —            F6 = 48.4
16
Smoothing
– exponential smoothing with a trend
Formula:
AFt + 1 = Ft + 1 + Tt+1
where: T = an exponentially smoothed trend factor
Tt + 1 + (Ft + 1 - Ft) + (1 - )Tt
Tt = the last period trend factor
 = smoothing constant for trend ( a value
between zero and one).        17
Smoothing Continued
• Example
Given the following data on Gillmore Hotel check-ins for
a 6-month period below, with alpha = 0.2 and beta = 0.1,
what is the adjusted exponentially smoothed forecast for
September?
Gillmore Hotel
Month             Room Occupied
July                      45
August                     36
September                    52
October                     48
November                    56
December                    60
18
Smoothing Continued
• Solution
T3 = 0.1 (43.20 – 45.00) + (1 – 0.1) (0.00) = – 0.18 + 0 = – 0.18
AF3 = 43.20 + (– 0.18) = 43.02
Gillmore Hotel
Period    Month       Room      Forecast       Trend     Adjusted Forecast
Occupied    (Ft + 1)      (Tt+1 )       ( AFt + 1 )
1        July       45.00       45.00          —              —
2       August      36.00       45.00        0.00            45.00
3      September    52.00       43.20        -0.18          43.02
4       October     48.00       44.96        0.01            44.97
5      November     56.00       45.57        0.07            45.64
6      December     60.00       47.65        0.27            47.93

19
Linear Trend Line
• Linear Trend Line
– A linear regression model that relates demand to
time.

20
by multiplying the normal forecast by a
seasonal factor.
• Formula:
– A seasonal factor can be determined by
dividing the actual demand for each seasonal
period by total annual demand:
Si =Di/D
21
Continued
• Example:
Tony’s Shoe Store sells various types of shoes.
Quarterly sales is given by the table below for the past
three years, determine the seasonal factors for each
quarter. If the forecast for the forth year is 25,000
shoes demanded, determine the seasonally adjusted
forecasts.
Tony’s Shoe Store
Year     Winter       Spring    Summer      Fall
1       4800         4500          4100   5500
2       5700         3800          4500   6000
3       6000         4600          4900   6500
22
Continued
• Solution
–   Swinter = D1/ D = 16500/60900 ≈ 0.2709
–   Sspring = D2/ D = 12900/609000 ≈ 0.2118
–   Ssummer = D3/ D = 13500/609000 ≈ 0.2217
–   Sfall = D4/ D = 18000/609000 ≈ 0.2956
Tony’s Shoe Store
Year      Winter   Spring    Summer   Fall    Total
1        4800      4500      4100    5500    18900
2        5700      3800      4500    6000    20000
3        6000      4600      4900    6500    22000
Total     16500    12900      13500   18000   60900
23
Continued
• Solution Continued:
The forecast for 4th year is 25,000 shoes demanded.
F4 = 25,000
SFwinter = (Swinter)(F4) = (0.2709)(25000) = 6773
SFspring = (Sspring)(F4) = (0.2118)(25000) = 5295
SFsummer = (Ssummer)(F4) = (0.2217)(25000) = 5543
SFfall = (Sfall)(F4) = (0.2956)(25000) = 7390

24
PROPERTIES
On passing, 'Finish' button:       Goes to Next Slide
On failing, 'Finish' button:       Goes to Next Slide
Allow user to leave quiz:          At any time
User may view slides after quiz:   At any time
User may attempt quiz:             Unlimited times
PROPERTIES
On passing, 'Finish' button:       Goes to Next Slide
On failing, 'Finish' button:       Goes to Next Slide
Allow user to leave quiz:          At any time
User may view slides after quiz:   At any time
User may attempt quiz:             Unlimited times
Forecasting Accuracy
Forecast error
– the average, absolute difference between the forecast
and the demand
– Formula:             D F
n
where:
t  the period number
D  demand in period t
t
F  the forecast for period t
t
n  the total number of periods    27
Forecasting Accuracy
Continued
• Mean Absolute Percent Deviation (MAPD)
– Absolute error as a percentage of demand
– Formula:             Dt  Ft
MAPD 
 Dt
• Cumulative Error (E)
– the sum of the forecast errors
– Formula: E =et
• Average Error ( E )
– The per-period average of cumulative error
– Formula:        et
E n                              28
Forecasting Accuracy
Continued
• Example:
Freeman’s Electronics recorded recent past demand for
large screen LCD TVs in the following table.
Based in the three month           Freeman’s Electronics
moving average, forecasted          Month   Actual Demand
February       20
demand for May, June, July,
March         22
August and September are             April        33
May          35
25, 30, 33, 38, 40 respectively,
June         31
Find the MAD, MAPD, E, and E .       July         48
August      41
29
Forecasting Accuracy
Continued
• Solution:      MAD = 29/4 = 7.25; MAPD = 29/155 = 0.187 = 18.7%
E = 29; E = 29/4 = 7.25
Freeman’s Electronics
Month       Demand, Dt   Forecast, Ft   Error, (Dt – Ft )   │Dt – Ft │

February        20            —                —               —
March          22            —                —               —
April          33            —                —               —
May           35           25                10              10
June           31           30                1                1
July          48           33                15              15
August          41           38                3                3
September        —            40                —               —
Total         155                            29               29
30
PROPERTIES
On passing, 'Finish' button:       Goes to Next Slide
On failing, 'Finish' button:       Goes to Next Slide
Allow user to leave quiz:          At any time
User may view slides after quiz:   At any time
User may attempt quiz:             Unlimited times
Time Series Forecasting
Using Excel

32
Time Series Forecasting
Using QM

33
Regression Methods
• Linear regression
– relates demand (dependent variable) to an
independent variable.
• Regression method
– technique for fitting a line to a set of points
• Correlation
– A measure of the strength of the relationship between
independent and dependent variables
• Coefficient of determination
– the percentage of the variation in the dependent
variable that results from independent variable      34
Regression Methods
Continued
• Least Square Method:                 • Formula for the Correlation
y  a  bx
Coefficient:
a  y bx                             r          n xy   x y
                                       
x2    x2  n y2   y 2 

 
b   xy  n x y
n

                   
                    

2
 x2  n x
where :
x   x  mean of the x data
n
• Coefficient of Determination is
y   y  mean of the y data
n                           computed by r squared, r2.

35
Regression Methods
Continued
• Example:
Infoworks is a large computer discount store that sells
computers and ancillary equipment. Infoworks has
collected historical data on computer sales and printer
sales for the past 10 years .
a. Develop a linear regression model relating printer
sales to computer sales in order to forecast printer
demand in year 11 if 1,300 computers are sold.
b. Determine the correlation coefficient and the
coefficient of determination.

36
Regression Methods
Continued
• Example:
Infoworks
Year   PCs sold   Printers   Year   PCs sold   Printers
Sold                         Sold
1      1,045       326       6      1,117     506
2      1,610       510       7      1,066     612
3       860        296       8      1,310     560
4      1,211       478       9      1,517     590
5       975        305      10      1,246     676

37
Regression Methods
Continued
• Solution:
(a)   a = 74.77
PC     Printers
b = 0.34
Year   Sales    Sold
y = 74.44 + 0.34x
1     1,045     326
2     1,610     510
x= 1300
3      860      296
y = 74.44 + 0.34x
4     1,211     478
y = 74.44 + 0.34 (1300)
5      975      305
y = 516.77
6     1,117     506
7     1,066     612
(b)
8     1,310     560
r = 0.599
9     1,517     590
r2 = 0.359
10    1,246     676
11     1,300     517
38
PROPERTIES
On passing, 'Finish' button:       Goes to Next Slide
On failing, 'Finish' button:       Goes to Next Slide
Allow user to leave quiz:          At any time
User may view slides after quiz:   At any time
User may attempt quiz:             Unlimited times
Summary

•   Forecasting components
•   Time series methods
•   Forecast accuracy
•   Time series forecasting using Excel & QM
•   Regression methods

40

```
To top