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									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
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       Time Series Methods

• Time Series Methods
  – Moving average
  – Weighted moving average
  – Exponential smoothing
  – Adjusted exponential smoothing
  – Seasonal adjustment



                                     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
        Adjusted Exponential
             Smoothing
• Adjusted Exponential Smoothing
  – exponential smoothing with a trend
  adjustment factor added.
  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
         Adjusted Exponential
         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
             Adjusted Exponential
             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
       Seasonal Adjustment
• Seasonal Adjustment
  – An adjustment can be made to the forecast
   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
         Seasonal Adjustment
              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
              Seasonal Adjustment
                   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
        Seasonal Adjustment
             Continued
• Solution Continued:
  The forecast for 4th year is 25,000 shoes demanded.
  Seasonally adjusted forecasts:
  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
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        Forecasting Accuracy
Forecast error
• Mean Absolute Deviation (MAD)
  – the average, absolute difference between the forecast
    and the demand
  – Formula:             D F
                 MAD       t   t
                            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
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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
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                 Summary

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




                                               40

								
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