Transcript Document

Spring 08-09
MS 401 - Production and Service Systems
Operations
Forecasting
Murat Kaya
FENS, Sabanci University
Murat Kaya, Sabancı Üniversitesi
1
Spring 08-09
Predicting the Future
• “My concern is with the future since I plan to spend the rest
of my life there”
C. F. Kettering
• Hertz: How many cars will be rented during March 2008?
• Apple: How many iPod Nano 8GB will be sold in 2008?
• Why is it important to know the answers to these questions?
Murat Kaya, Sabancı Üniversitesi
2
Spring 08-09
If Forecasting Fails
• Cisco could not forecast the demand for networking
equipment correctly
– result: lost $2.5 billion due to unsold products
• Volvo – Green car example (mid 1990s)
– excessive amount of green color cars in the middle of the year
– to sell these cars, marketing offered special promotions and
discounts
Murat Kaya, Sabancı Üniversitesi
3
Spring 08-09
Forecasts
Forecast: An estimate of the future level of some variable
Characteristics of Forecasts
• They are usually wrong
– the planning systems that use forecasts should be robust
• A good forecast is more than a single number
– include some measure of anticipated error
• Aggregate forecasts are more accurate
• The longer the forecast horizon, the less accurate the
forecast will be
• Forecasts should not be used to the exclusion of known
information
– some information may not be present in the past history
Murat Kaya, Sabancı Üniversitesi
4
Spring 08-09
Time Series Methods
• Time series: A collection of observations of some
economic or physical phenomenon drawn at discrete points
in time
• The idea: Information can be inferred from the pattern of
past observations and can be used to forecast the future
value of the series
• Patterns in time series
– trend: tendency of a time series to exhibit a stable pattern of
growth or decline
– seasonality: having a pattern that repeats in fixed intervals
– cycles: similar to seasonality, but the length and the magnitude of
the cycle may vary
– randomness: when there is no recognizable pattern to the data
Murat Kaya, Sabancı Üniversitesi
5
Spring 08-09
Time Series Patterns
Copyright © 2001 by The McGraw-Hill Companies, Inc
Murat Kaya, Sabancı Üniversitesi
6
Spring 08-09
Evaluating Forecasts
D1 , D2 , D3 , ..., Dt , ... observedvaluesof demand
Ft : forecastmade for periodt in periodt-1
Forecasterror: et  Ft  Dt
Measuresof ForecastAccuracy
1 n
MAD    ei
 n  i 1
1 n 2
MSE    ei
 n  i 1
 1  n ei 
MAPE   
 100
 n  i 1 Di 
Murat Kaya, Sabancı Üniversitesi
7
Spring 08-09
Random versus Biased Forecast Errors
Copyright © 2001 by The McGraw-Hill Companies, Inc
Murat Kaya, Sabancı Üniversitesi
8
Spring 08-09
Forecasting Stationary Time Series
• Stationary time series: Each observation can be represented
by a constant plus a random fluctuation
Dt     t
•
• Two methods
– moving averages (MA)
– exponential smoothing (ES)
Murat Kaya, Sabancı Üniversitesi
9
Spring 08-09
Moving Averages (MA)
• A moving average of order N is the arithmetic average of
the most recent N observations
 1  t 1
Ft     Di
 N i  t  N
1
 Dt 1  Dt 2  ...  Dt  N 
N
• When calculating the forecast for the following period
(period t+1), we do not need to recalculate the N-period
average because
1
Ft 1  Ft   Dt  Dt  N 
N
• Example 2.2
Murat Kaya, Sabancı Üniversitesi
10
Spring 08-09
Moving Average Lags Behind the Trend
Murat Kaya, Sabancı Üniversitesi
11
Spring 08-09
Exponential Smoothing (ES)
• The current forecast is the weighted average of the current
observation of demand and the last forecast
Ft  Dt 1  1   Ft 1
Substituting Ft 1   Dt 2  1    Ft 2 we obtain
Ft   Dt 1   1    Dt 2  1    Ft 2
2
Continuingin thisway, we get

Ft    1    Dt i 1
i
i 0
• High α: forecast reacts better, however it is less stable
–
Murat Kaya, Sabancı Üniversitesi
12
Spring 08-09
Weights in Exponential Smoothing
Copyright © 2001 by The McGraw-Hill Companies, Inc
Murat Kaya, Sabancı Üniversitesi
13
Spring 08-09
Exponential Smoothing with Different α Values
Copyright © 2001 by The McGraw-Hill Companies, Inc
Murat Kaya, Sabancı Üniversitesi
14
Spring 08-09
Example 2.3 from Nahmias
• Observed number of failures:
– 200, 250, 175, 186, 225, 285, 305 190
• Assume F1 was 200
• Using α=0.1
(we need a starting value)
F2  D1  1   F1  200
The forecasts are
quite stable due to
low α
Murat Kaya, Sabancı Üniversitesi
15
Spring 08-09
In-Class Exercise
Handy, Inc. produces a calculator that experienced the
following monthly sales history for the first four months
of the year:
Jan:23.3;
Feb: 72.3;
March: 30.3; April: 15.5
a) If the forecast for January was 25, determine the one-stepahead forecasts for February, March, April and May using
exponential smoothing with α=0.15
b) Repeat the calculations using α=0.40
c) Compute the MSEs for the forecasts in parts (a) and (b)
Murat Kaya, Sabancı Üniversitesi
16
Spring 08-09
Solution - 1
Ft = Dt-1 + (1-)Ft-1
Murat Kaya, Sabancı Üniversitesi
17
Spring 08-09
Solution - 2
Murat Kaya, Sabancı Üniversitesi
18
Spring 08-09
Similarities Between Moving Averages and
Exponential Smoothing
• Stationary demand assumption
– can also handle shifts in demand (will adjust)
• Single parameter: N, α
– small N or large α results in
• greater weight on current data
• more responsive forecasts
• Not effective in catching trends
– both lag behind trends
Murat Kaya, Sabancı Üniversitesi
19
Spring 08-09
Differences Between Moving Averages and
Exponential Smoothing
•
– ES assigns weight to all past data points
– MA uses only the latest N
•
– ES requires only the latest data point
– MA requires to save N past data points
Murat Kaya, Sabancı Üniversitesi
20
Spring 08-09
Forecasting Time Series with Trend
• Two methods
– regression analysis (we will not cover)
• fits a straight line to a set of data
– double exponential smoothing (Holt’s method)
• simultaneous smoothing on the series and the trend
Murat Kaya, Sabancı Üniversitesi
21
Spring 08-09
Double Exponential Smoothing
Using Holt’s Method
T heτ - step- aheadforecastmade in period t :
Intercept
Slope
Ft ,t   St  Gt
St  Dt  1   St 1  Gt 1 
Gt   St  St 1   1   Gt 1
• Initialization issue: The best way is to use some initial period
data to estimate the initial intercept (S0) and slope (G0)
Murat Kaya, Sabancı Üniversitesi
22
Spring 08-09
Example 2.5 from Nahmias
• Observed number of failures: 200, 250, 175, 186, 225, 285, 305, 190
• Assume S0 = 200, G0 = 10. Use α=0.1, β=0.1
t
Ft-1,t (forecasted)
Dt (actual)
St (intercept)
Gt (slope)
0
---
---
200.0
10.0
1
210.0
200
209.0
9.9
2
218.9
250
222.0
10.2
3
232.2
175
226.5
9.6
4
236.1
186
…
…
5
240.3
225
…
…
6
247.7
285
…
…
• Multi-step ahead forecast: F2,5=S2+(3)G2=222+(3)(10.2)=252.6
Murat Kaya, Sabancı Üniversitesi
23
Spring 08-09
Forecasting Seasonal Series
• A seasonal series is a series that has a pattern repeating
every N periods (length of the season)
• Note that this is different than using “season” to refer to a
time of the year
• To model seasonality, use seasonal factors:
c1, c2 ,...,cN
where
c
t
N
• ct represents the average amount that the demand in the tth
period of the season is above or below the overall average
• We will study the Winter’s method
– triple exponential smoothing
Murat Kaya, Sabancı Üniversitesi
24
Spring 08-09
Winter’s Method:
Seasonal Series with Increasing Trend
Copyright © 2001 by The McGraw-Hill Companies, Inc
Murat Kaya, Sabancı Üniversitesi
25
Spring 08-09
Winter’s Method
• Assume a model of the form Dt    Gt ct   t
Ft ,t   St  Gt ct   N
Trend
Seasonal factors
Murat Kaya, Sabancı Üniversitesi
 Dt 
  1   St 1  Gt 1 
St   
 ct  N 
Gt   St  St 1   1   Gt 1
 Dt
ct   
 St

  1   ct  N

26
Spring 08-09
Winter’s Method: Initialization Procedure
•
•
•
•
Check Nahmias, page 85 for details
Use at least two seasons of data (2N data)
Calculate the sample means for the two seasons V1, V2
Calculate the initial slope estimate G0
–
• Calculate the initial intercept estimate S0
–
• Calculate the initial seasonal factors
–
– find the average of each seasonal factor
– normalize the seasonal factors (so that they sum up to 1)
Murat Kaya, Sabancı Üniversitesi
27
Spring 08-09
Example 2.8 from Nahmias
• The data set: 10, 20, 26, 17, 12, 23, 30, 22
Season 1
Season 2
• Initialize
• Suppose that at time t=1, we observe D1=16. Update the
equations using α=0.2, β=0.1, γ=0.1
• Suppose that we observe one full year of demand given by
D1=16, D2=33, D3=34, D4=26. Update the equations again
Murat Kaya, Sabancı Üniversitesi
28
Spring 08-09
Seasonal Demand, No Trend
Copyright © 2001 by The McGraw-Hill Companies, Inc
Murat Kaya, Sabancı Üniversitesi
29
Spring 08-09
Affecting the Demand
• “The best way to forecast the future is to create it”
Peter Drucker
• “Forecasting the demand” versus “demand planning”, or
“demand management”
• Firms can “affect” their demand through their actions
– promotions
– sales effort
• Encourages the retailers / wholesalers to “forward buy”
• What are the effects of past promotions in the health of
forecasting data?
Murat Kaya, Sabancı Üniversitesi
30
Spring 08-09
Some Practical Issues
• Sales data versus demand data
– how can a firm capture “lost sales” ?
• Forecasting demand for a new product is difficult
– will it generate demand, or will it steal demand from existing
products?
• Forecasting assumes that history represents future. What if
there are some external changes?
– a new competitor
• Slow-moving items are hard to forecast
– sparse data
Murat Kaya, Sabancı Üniversitesi
31