Forecasting. Operations Analysis and Improvement Spring

Transcription

1 Forecasting Operations Analysis and Improvement 2015 Spring Dr. Tai-Yue Wang Industrial and Information Management Department National Cheng Kung University

2 1-2 Outline Introduction to Forecasting Subjective Forecasting Methods Objective Forecasting Methods Evaluating Forecasts Methods for Forecasting Stationary Series Trend-Based Methods Methods for Seasonal Series Box-Jenkins Models

3 Overview of Operations Planning Activities 1-3

4 1-4 Introduction to Forecasting What is forecasting? Primary Function is to Predict the Future Why are we interested? Affects the decisions we make today Examples: who uses forecasting in their jobs? forecast demand for products and services forecast availability of manpower forecast inventory and materiel needs daily

5 1-5 Introduction to Forecasting -- What Makes a Good Forecast It should be timely It should be as accurate as possible It should be reliable It should be in meaningful units It should be presented in writing The method should be easy to use and understand in most cases.

6 Introduction to Forecasting The Time Horizon 1-6

7 1-7 Introduction to Forecasting Characteristics of Forecasts They are usually wrong! A good forecast is more than a single number mean and standard deviation range (high and low) Aggregate forecasts are usually more accurate Accuracy erodes as we go further into the future. Forecasts should not be used to the exclusion of known information

8 1-8 Subjective Forecasting Methods Sales Force Composites Aggregation of sales personnel estimates Customer Surveys Jury of Executive Opinion The Delphi Method Individual opinions are compiled and reconsidered. Repeat until and overall group consensus is (hopefully) reached.

9 1-9 Objective Forecasting Methods Two primary methods: causal models and time series methods Causal Models Let Y be the quantity to be forecasted and (X 1, X 2,..., X n ) be n variables that have predictive power for Y. A causal model is Y = f (X 1, X 2,..., X n ). A typical relationship is a linear one. That is, Y = a 0 + a 1 X a n X n.

10 1-10 Time Series Methods A time series is just collection of past values of the variable being predicted. Also known as naïve methods. Goal is to isolate patterns in past data. (See Figures on following pages) Trend Seasonality Cycles Randomness

11 1-11

12 1-12 Time Series Methods -- Notation Conventions Let D 1, D 2,... D n,... be the past values of the series to be predicted (demand). If we are making a forecast in period t, assume we have observed D t, D t-1 etc. Let F t, t + t = forecast made in period t for the demand in period t + t where t = 1, 2, 3, F t -1, t is the forecast made in t-1 for t and F t, t+1 is the forecast made in t for t+1. (one step ahead) Use shorthand notation F t = F t - 1, t.

13 1-13 Evaluation of Forecasts The forecast error in period t, e t, is the difference between the forecast for demand in period t and the actual value of demand in t. For a multiple step ahead forecast: e t = F t - t, t - D t. For one step ahead forecast: e t = F t - D t.

14 1-14 Evaluation of Forecasts MAD = (1/n) S e i MSE = (1/n) S e i 2

15 1-15 Evaluation of Forecasts -- Biases in Forecasts A bias occurs when the average value of a forecast error tends to be positive or negative. Mathematically an unbiased forecast is one in which E (e i ) = 0. See Figure 2.3 on page 64 in text (next slide).

16 Forecast Errors Over Time Figure

17 1-17 Forecasting for Stationary Series A stationary time series has the form: D t = m + e t where m is a constant and e t is a random variable with mean 0 and var s 2. Two common methods for forecasting stationary series are moving averages and exponential smoothing.

18 1-18 Forecasting for Stationary Series -- Moving Averages In words: the arithmetic average of the n most recent observations. For a one-step-ahead forecast: F t = (1/N) (D t D t D t - n )

19 1-19 Forecasting for Stationary Series -- Moving Averages Example: Quarterly data:

20 1-20 Forecasting for Stationary Series -- Moving Averages --Summary Advantages of Moving Average Method Easily understood Easily computed Provides stable forecasts Disadvantages of Moving Average Method Requires saving all past N data points Lags behind a trend Ignores complex relationships in data

21 Forecasting for Stationary Series -- Moving Averages --Summary 1-21

22 1-22 Forecasting for Stationary Series -- Exponential Smoothing Method A type of weighted moving average that applies declining weights to past data. 1. New Forecast = a (most recent observation) + (1 - a) (last forecast) or 2. New Forecast = last forecast - a (last forecast error) where 0 < a < 1 and generally is small for stability of forecasts ( around.1 to.2)

23 1-23 Forecasting for Stationary Series -- Exponential Smoothing Method In symbols: F t+1 = a D t + (1 - a ) F t = a D t + (1 - a ) (a D t-1 + (1 - a ) F t-1 ) = a D t + (1 - a )(a )D t-1 + (1 - a) 2 (a )D t Hence the method applies a set of exponentially declining weights to past data. It is easy to show that the sum of the weights is exactly one. (Or F t + 1 = F t - a (F t - D t ) )

24 Forecasting for Stationary Series -- Exponential Smoothing Method --Weights in Exponential Smoothing 1-24

25 1-25 Forecasting for Stationary Series -- Exponential Smoothing Method Example Quarterly data What is F 3 if a=0.1?

26 Forecasting for Stationary Series -- Exponential Smoothing Method 1-26

27 1-27 Forecasting for Stationary Series -- Comparison of ES and MA Similarities. Both methods are appropriate for stationary series Both methods depend on a single parameter Both methods lag behind a trend One can achieve the same distribution of forecast error by setting a = 2/ ( N + 1).

28 1-28 Forecasting for Stationary Series -- Comparison of ES and MA Differences ES carries all past history. MA eliminates bad data after N periods MA requires all N past data points while ES only requires last forecast and last observation.

29 1-29 Trend-Based Methods Regression Analysis Regression Methods Can be Used When Trend is Present. Model: D t = a + bt.

30 1-30 Trend-Based Methods If t is scaled to 1, 2, 3,..., then the least squares estimates for a and b can be computed as follows: Set S xx = n 2 (n+1)(2n+1)/6 - [n(n+1)/2] 2 Set S xy = n S id i - [n(n + 1)/2] S D i Let b = S xy / S xx and a = D- b (n+1)/2 These values of a and b provide the best fit of the data in a least squares sense.

31 1-31 Trend-Based Methods -- Double exponential smoothing with Holt s method Double exponential smoothing, of which Holt s method is only one example, can also be used to forecast when there is a linear trend present in the data. The method requires separate smoothing constants for slope and intercept. S t = αd t + 1 α G t = β S t S t 1 S t 1 + G t 1 + (1 β) G t 1 where S t =the intercept, G t =slope, D t =demand, a,b = constants

32 1-32 Trend-Based Methods -- Double exponential smoothing with Holt s method So the t step ahead forecast made in period t, denoted by F t, t+t, F t, t+τ = S t + τg t

33 1-33 Trend-Based Methods -- Double exponential smoothing with Holt s method Example Quarterly data What is F 3, 3+1 if a=b=0.1, S 0 =200, and G 0 =10?

34 Methods For Seasonal Series --A Seasonal Demand Series

35 1-35 Methods For Seasonal Series Seasonal Factors for Stationary Series Seasonality corresponds to a pattern in the data that repeats at regular intervals. Multiplicative seasonal factors: c 1, c 2,..., c N where i = 1 is first period of season, i = 2 is second period of the season, etc.. S c i = N. c i = 1.25 implies 25% higher than the baseline on avg. c i = 0.75 implies 25% lower than the baseline on avg.

36 1-36 Methods For Seasonal Series Seasonal Factors for Stationary Series Procedure: 1. Compute the sample mean of all the data 2. Divide each observation by the sample mean factors for each period of observed data 3. Average the factors for like periods within each season average all\factors corresponding to the same period of a season 4. Forecast your time series

37 1-37 Methods For Seasonal Series Seasonal Factors for Stationary Series Example: Week 1 Week 2 Week 3 Week 4 Monday Tuesday Wednesday Thursday Friday

38 1-38 Methods For Seasonal Series Seasonal Factors for Stationary Series Solution: 1. Sample mean= Divide each observation by sample mean Week 1 Week 2 Week 3 Week 4 Monday Tuesday Wednesday Thursday Friday

39 1-39 Methods For Seasonal Series Seasonal Factors for Stationary Series Solution: 3. Average all factors for Mondays thru Friday Average Factors Monday 0.98 Tuesday 0.74 Wednesday 0.84 Thursday 1.04 Friday 1.40

40 1-40 Methods For Seasonal Series Seasonal Factors for Stationary Series Solution: 4. Forecast the numbers by multiplying mean and each of average factor. Average Factors Monday 16.1 Tuesday 12.1 Wednesday 13.8 Thursday 17.1 Friday 23.0

41 Methods For Seasonal Series Seasonal decomposition Using Moving Average 1-41 Steps of Seasonal decomposition Using Moving Average 1. Calculate the moving averages. 2. Center the moving average 3. Center the centered moving average on integervalued periods. 4. Determine the seasonal and irregular factors. 5. Determine the average seasonal factors.

42 Methods For Seasonal Series Seasonal decomposition Using Moving Average 1-42 Steps of Seasonal decomposition Using Moving Average 6. Scale the seasonal factors. 7. Determine the deseasonalized data. 8. Determine a trend line of the deseasonalized data. 9. Determine the deseasonalized predictions. 10. Take into account the seasonality.

43 Methods For Seasonal Series Seasonal decomposition Using Moving Average 1-43 Example: Demands for each periods Period Demand MA(4)

44 Methods For Seasonal Series Seasonal decomposition Using Moving Average 1-44 Example: Demands for each periods

45 Methods For Seasonal Series Seasonal decomposition Using Moving Average 1-45 Solution: 1. Moving average with k=3 Period Demand MA(4)

46 Methods For Seasonal Series Period Demand MA(4) Centered MA Seasonal decomposition Using Moving 1 10 Average 1-46 Solution: 2. Center the moving average

47 Methods For Seasonal Series Peri Dema MA(4) Centered od Seasonal decomposition Using Moving 1 10 Average nd MA Centered MA on Integer periods Center the centered moving average on integer-valued periods

48 Methods For Seasonal Series Seasonal decomposition Period Demand (B) MA(4) Using CenteredMoving MA Average 4. Determine the seasonal and irregular factors by dividing (B) by (C) Centered MA on Integer periods (C) 1-48 Seasonal and irregular factors (B/C)

49 Methods For Seasonal Series Seasonal decomposition Period Demand (B) MA(4) Using CenteredMoving MA Average 5. Determine the average seasonal factors. (not in this example) Centered MA on Integer periods (C) 1-49 Seasonal and irregular factors (B/C)

50 Methods For Seasonal Series Seasonal decomposition Perio Demand Using Moving d (B) Average 6. Scale the seasonal factors Centered MA on Integer periods (C) Seasonal and irregular factors (B/C) Scaled seasonal factors *(4/3.982) (D) * =

51 Methods For Seasonal Series Seasonal decomposition Using Moving Average Determine the deseasonalized data. Period Demand (B) Scaled seasonal factors *(4/3.982) (D) Deseasonal Data (B/D) /0.6027= /1.094= * = /1.4113= /0.892=

52 Methods For Seasonal Series Seasonal decomposition Using Moving Average Determine a trend line of the deseasonalized data. By regression, D t = t 9. Determine the deseasonalized predictions Period Deseasonalized predictions

53 Methods For Seasonal Series Seasonal decomposition Using Moving Average Take into account the seasonality Period Deseasonali zed predictions Seasonal Factors Forecast

54 1-54 Methods For Seasonal Series Winter s Method Moving average with or without trend can be used to predict seasonal data, however, one needs to recalculate all factors if new data comes in. Winter s method is a triple exponential smoothing. D t = μ + G t c t + ε t D t : demand at time t; μ: intercept; G t :Slope c t : seasonal factor; ε t : error term

55 Methods For Seasonal Series Winter s Method 1-55

56 1-56 Methods For Seasonal Series Winter s Method Assuming: Length of the season, N Seasonal factors are the same each season and c t = N Three exponential smoothing equation are used: The series The trend The seasonal factors

57 1-57 Methods For Seasonal Series Winter s Method 1. The series the current level of the deseasonalized series, S t D t S t = α + (1 α)(s c t 1 + G t 1 ) t N 2. The trend G t = β[s t S t 1 ] + (1 β)g t 1 3. The seasonal factor c t = γ( D t S t ) + (1 γ)c t N

58 1-58 Methods For Seasonal Series Winter s Method The forecast made in period t for any future period t+t is F t,t+τ = S t + τg t c t+τ N if t N Note, appropriate seasonal factor would be: c t+τ 2N c t+τ 3N if N< τ 2N if 2N< τ 3N

59 1-59 Methods For Seasonal Series Winter s Method Initialization procedure In order to get the method started, one needs to obtain initial estimates of the series Winters suggest minimum 2 seasons of data available Assuming exactly two season are considered and the time now, t=0, so the past demands are labeled: D -2N+1, D -2N+2,.., D 0

60 1-60 Methods For Seasonal Series Winter s Method Initialization procedure 1. Calculate the sample means for the two separate seasons of data V 1 = 1 N j= 2N+1 N D j V 2 = 1 N j= N+1 0 D j

61 1-61 Methods For Seasonal Series Winter s Method Initialization procedure 2. Define G 0 =(V 2 -V 1 )/N as the initial slope estimate If m>2 (m:# of seasons), compute V 1,,,,V m as step 1 shows and define G 0 =(V m -V 1 )/[(m-1)n] 3. Set S 0 =V 2 + G 0 [(N-1)/2]

62 1-62 Methods For Seasonal Series Winter s Method Initialization procedure 4. a. compute initial seasonal factor c t = D t V i [ N+1 2 j]g 0 for 2N + 1 t 0 where i=1 for the first season, i=2 for the second season j: the period of the season, j=1 for t=-2n+1and t=-n+1, j=2 for t=-2n+2 and t=-n+2 b. average the seasonal factors by c N+1 = c 2N+1+c N+1 2,, c 0 = c N+c 0 2 for exactly two seasons

63 1-63 Methods For Seasonal Series Winter s Method Initialization procedure 4. c. Normalize the seasonal factor c j = i=0 N+1 c i *Note: The above initialization method is only one of methods. Moving average, linear regression can be also used. c j

64 1-64 Methods For Seasonal Series Winter s Method --example Initial data set: 1. V 1 = 1 N j= 2N+1 N 0 D j =18.25 V 2 = 1 D N j = j= N+1 2. G 0 =(V 2 -V 1 )/N= S 0 =V 2 + G 0 [(N-1)/2]= Initialization c t = D t V i [ N+1 2 j]g 0 for 7 t 0

65 1-65 Methods For Seasonal Series Winter s Method --example Initial data set: 4. Initialization a. For c 7, c 6, c 5, c 4, i=1; For c 3, c 2, c 1, c 0, i=2 For c 7, c 5, c 3, c 1, j=1; c 7 = D 7 V 1 [ 5 = 10 = ](0.875) [1.5](0.875) For c 6, c 4, c 2, c 0, j=2; c 6 = D 6 V 1 [ 5 = 10 = ](0.875) [1.5](0.875) And so on

66 1-66 Methods For Seasonal Series Winter s Method --example Initial data set: 4. Initialization b. Average seasonal factors: Average c 7 and c 3 c 3 =0.588 Average c 6 and c 2 c 2 =1.010 c. Normalize seasonal factors c 3 =0.5900, c 2 =1.1100,

67 1-67 Methods For Seasonal Series Winter s Method --example Forecasting equations: F t,t+τ = S t + τg t c t+τ N if t N F 0,1 = S 0 + G 0 c 3 if t =0 andτ=1 F 0,2 = S 0 + G 0 c 2 if t =0 and τ=2 For t=0, F 0,1 =14.12, F 0,2 =27.54, F 0,3 = 35.44, F 0,3 =24.38

68 1-68 Methods For Seasonal Series Winter s Method --example For t=1, D 1 is available, and assume a=0.2, b=0.1, =0.1 D t S t = α + (1 α)(s c t 1 + G t 1 ) t N D 1 S 1 = S c 0 + G 0 1 N G t = β[s t S t 1 ] + (1 β)g t 1 G 1 = 0.1 S 1 S G 0 c t = γ( D t S t ) + (1 γ)c t N c 1 = 0.1( D 1 S 1 ) + (1 0.1)c 1 N So S 1 = 24.57, G 1 =0.9385, c 1 =0.5961

69 1-69 Methods For Seasonal Series Winter s Method --example For t=1, D 1 is available, and assume a=0.2, b=0.1, =0.1 Renorm: c 2, c 1, c 0, andc 1, Forecasting for t=1 F 1,2 = S 1 + G 1 c 2 if t =1 andτ=1 F 1,3 = S 1 + G 1 c 1 if t =1 and τ=2 And so on repeat for the rest of data..

70 1-70 Practical Considerations Overly sophisticated forecasting methods can be problematic, especially for long term forecasting. (Refer to Figure on the next slide.) Tracking signals may be useful for indicating forecast bias. TS= i=1 n MAD ei Box-Jenkins methods require substantial data history, use the correlation structure of the data, and can provide significantly improved forecasts under some circumstances.

71 The Difficulty with Long-Term Forecasts