Forecasting: The First Step in Demand Planning
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1 Forecasting: The First Step in Demand Planning Jayant Rajgopal, Ph.D., P.E. University of Pittsburgh Pittsburgh, PA 15261
2 In a supply chain context, forecasting is the estimation of future demand General approaches to forecasting could be Simple and subjective (predictions based on expert opinions or educated guesses) Statistical and sophisticated (Box- Jenkins and regression models) A compromise: Simple but scientific methods based on TIME SERIES that work well in practice Jayant Rajgopal, 2016
3 Sophisticated methods yield lower errors but tend to be more expensive COSTS Naïve Forecasts E[Cost of error] Desired operating range Sophisticated Causal Models E[Cost of procedure] COST E[Total Cost] FORECAST ACCURACY Jayant Rajgopal, 2016
4 A general forecasting framework Past data Forecasting Procedure Forecast Control Actual Demand Qualitative Analysis Final Forecast Jayant Rajgopal, 2016
5 A good forecasting procedure should yield ACCURATE and UNBIASED forecasts Some things to keep in mind There is no single best forecasting model for any dataset Forecasts are typically wrong! Most planners/managers like a range of values for the forecast rather than one single number Aggregate forecasts are always more accurate than individual ones Forecast accuracy decreases as the forecast horizon increases Jayant Rajgopal, 2016
6 A time series is a set of values measured at equally spaced time intervals (D 1, D 2, D 3, ) For example monthly demand quarterly earnings annual production daily sales hourly power consumption et cetera Jayant Rajgopal, 2016
7 Possible Components of a time series Level (L) - always Level + Trend (T) Level + Trend + Seasonality Level + Seasonality (S) Important: In all cases there is randomness! Jayant Rajgopal, 2016
8 Suppose we are now at the end of period t. Then the forecast for a future period j>t is: E [D j D t, D t-1, D t-2, ]; j>t The most common forecast is the one-step-ahead (o-s-a) forecast, i.e., E [D t+1 D t, D t-1, D t-2, ] However, we might also be interested in looking further into the future with a k-steps-ahead (k-s-a) forecast: E [D t+k D t, D t-1, D t-2, ] Jayant Rajgopal, 2016
9 Notational conventions we will use: Forecast origin t: when (i.e., end of which period) forecast is being made Forecast horizon k: for how far ahead it is being made Then F t,t+k is the forecast made at the end of period t for period t+k: F t,t+k = E [D t+k D t, D t-1, D t-2,.] E.g. If t=10, k=4, then F 10,14 is the 4-step-ahead forecast for t=14 made at the end of t=10 (based on all data up to and including D 10 ): 4 F 10,14 = E [D 14 D 10, D 9, ] Jayant Rajgopal, 2016
10 There are two broad approaches to handle time series data Statistical Approach Assumes that the data can be statistically modeled, i.e., there is some underlying statistical process driving the data e.g., D t = β 1 D t-1 + β 2 D t-2 + ε t where ε t ~N(0,σ 2 ) Heuristic Approach Does not worry about any statistical model just uses sensible rules to forecast Jayant Rajgopal, 2016
11 Level Time Series (no trends or seasonality) Locally Constant Mean Model: D t = L t + ε t Our model will be based on estimating the level L t at time t (let s denote this estimate by L t ). This estimate of the level becomes our forecast at time t for all points in the future: F t, t+1 = F t, t+2 = F t, t+k = = L t one-step-ahead (o-s-a) forecast at time t 2-step-ahead forecast at time t k-step-ahead forecast at time t
12 Since the level varies over time, our estimates of the level must be updated as we get new data Suppose we just observed D t+1 Current estimate L t (=F t, t+1 =F t, t+1 =F t, t+1 = ) t+1 Updated estimate of level L t+1 actual demand D t+1 Then for the future we will now forecast via F t+1, t+2 = F t+1, t+3 = F t+1, t+1+k = = L t+1
13 Here is a visual representation: t-2 t-1 t t+1 t+2 t+3 t+4 D t-2 D t-1 D t F t, t+1 F t, t+2 F t, t+3 F t, t+4 At the end of t F t,t+k = L t t-2 t-1 t t+1 t+2 t+3 t+4 D t-2 D t-1 D t D t+1 F t+1, t+2 F t+1, t+3 F t+1, t+4 F t+1,t+1+k = L t+1 At the end of t+1 t-2 t-1 t t+1 t+2 t+3 t+4 D t-2 D t-1 D t D t+1 D t+2 F t+2, t+3 F t+2, t+4 F t+2,t+2+k = L t+2 At the end of t+2 t-2 t-1 t t+1 t+2 t+3 t+4 D t-2 D t-1 D t D t+1 D t+2 D t+3 F t+3, t+4 F t+3,t+3+k = L t+3 At the end of t+3 Jayant Rajgopal, 2016
14 METHOD 1: Updating via Moving Averages Use the average of the N most recent observations to estimate the current value of the mean and use this to forecast; the procedure is denoted as MA(N): Notes L t = D t+d t 1 + +D t (N 1) N = F t,t+k If there are any changes in the level the forecast will lag (with the lag depending on the value of N) This model 1. considers only the N most recent observations, and 2. gives each of them equal weight Jayant Rajgopal, 2016
15 Demand Example Time Value t D t Time
16 MA(3) MA(7) Time Value o-s-a Forecast Abs. Err. o-s-a Forecast Abs. Err. t D t F t-1,t =L t 1 e t = D t -F t-1,t F t-1,t =L t 1 e t = D t -F t-1,t MAE e.g., for N=3 L t = (D t +D t-1 +D t-2 ) 3 L 24 =( )/3=16.33 is our current forecast for each month in the future. We will of course, update this after we see D 25 Note that IF we had been using MA(3) in the past, then L 23 =( )/3=16.67=F 23,23+k L 22 =( )/3=16.67=F 22,22+k etc. L 11 =( )/3=16.00=F 11,11+k With MA(7), we have L 24 = ( ) 7 =15.57 Jayant Rajgopal, 2016
17 Forecasting with Moving Averages 21 Actual Values (Dt) O-S-A Forecast MA(3) O-S-A Forecast MA(7) Demand/Forecast Time Jayant Rajgopal, 2016
18 A good value for N might be something between 3 and 10 Note that L t = 1 N D t + D t D t (N 1) So, as N increases we consider more data but give less weight to each individual data point this smooths out fluctuations better Smaller values of N emphasize only recent data and give these more weight; it will therefore pick up any changes in the level more quickly Depending on the data, pick some candidate values for N. Then pretend that the model had been in use in the past and simulate on old data to see how each one performs before picking one
19 METHOD 2: Updating via Exponential Smoothing We use the following formula to update the estimate of the level: L t = L t 1 + α(d t L t 1 ) = L t 1 + α(d t F t-1,t ) Our previous estimate correction based upon what we just observed Note that If D t F t-1,t >0, i.e., F t-1,t < D t then we are under-forecasting; so we apply a positive correction to our previous estimate If D t F t-1,t < 0, i.e., F t-1,t > D t then we are over-forecasting; so we apply a negative correction to our previous estimate
20 Here α is called a smoothing constant; larger values provide a bigger correction based on what just happened We may also rearrange the formula to its more common form: L t = L t 1 + α(d t L t 1 ) = F t-1,t + α(d t F t-1,t ) = αd t + (1-α) L t 1 = αd t + (1-α) F t-1,t i.e., a weighted average of (1) what we just saw and (2) our forecast of what we thought we would see
21 Unlike MA, Exponential Smoothing considers ALL data and weights them unequally! We can recursively write: L t = αd t + (1-α)L t 1 etc. etc., i.e., = αd t + (1-α){αD t-1 + (1-α)L t 2 } = αd t + α(1-α)d t-1 + (1-α) 2 L t 2 = αd t + α(1-α)d t-1 + (1-α) 2 {αd t-2 + (1-α) L t 3 } = αd t + α(1-α)d t-1 + α(1-α) 2 D t-2 + (1-α) 3 L t 3 L t = αd t + α(1-α)d t-1 + α(1-α) 2 D t-2 + α(1-α) 3 D t-3 + α(1-α) 4 D t-4 + Since {α + α(1-α) + α(1-α) 2 + α(1-α) 3 + } = α/{1-(1-α)} = 1, the estimate is a weighted average of all prior data with the weights decreasing as we go further back in time
22 Demand Consider the same example Time Value t D t Time
23 Time Value O-S-A Forecast Abs. Err. t D t F t-1,t = L t 1 e t = D t -F t-1,t MAE Assume α=0.1. L t =αd t +(1-α)L t 1 =αd t +(1-α)F t-1,t L 24 =0.1* *16.18 = is our current forecast for each month in the future = F 24,24+k We will of course, update this after we see D 25 Note that L 23 =0.1* *16.09 = L 22 =0.1* *15.88 = L 21 =0.1* *15.97 = etc. L 12 =0.1* *15.87 = etc. Jayant Rajgopal, 2016
24 Forecasting with Single Exponential Smoothing Demand/Forecast Time Actual Values O-S-A Forecasts
25 Smaller α values provide better smoothing; larger values are more responsive L t = αd t + (1-α) L t 1 What just happened What we thought would happen based upon information from all previous data) Larger α implies more weight to D t (i.e., to what happened recently catches changes in the level quickly) Smaller α implies more weight to L t 1 (i.e., to what has been happening in the past smoother and less jumpy ) larger α may be better smaller α may be better Jayant Rajgopal, 2016
26 A suggested range for α is [0.05, 0.30] Use a combination of subjective and objective evaluations to pick a final value Objective: Pretend the model had been in use in the past and simulate on old data with different α values to see how it performs. Subjective: Does the data seem largely random If so, prefer a smaller value Does the data seem to have sudden level changes If so prefer a larger value
27 Time Series with Linear Trends Locally Constant Linear Trend Model: D t+k =L t+k +ε t = L t +kt t + ε t Our model will be based on estimating the level L t and trend T t at time t (via L t and T t ). These estimates are then used to forecast future demand via: F t, t+k = L t+k = L t + kt t k-step-ahead forecast at time t Note that the o-s-a, 2-s-a, 3-s-a, k-s-a forecasts are all different here!
28 Visual representation: t-2 t-1 t t+1 t+2 t+3 t+4 D t-2 D t-1 D t F t, t+1 F t, t+2 F t, t+3 F t, t+4 t-2 t-1 t t+1 t+2 t+3 t+4 D t-2 D t-1 D t D t+1 F t+1, t+2 F t+1, t+3 F t+1, t+4 t-2 t-1 t t+1 t+2 t+3 t+4 End of time t: F t,t+1 = L t +T t, F t,t+2 = L t +2T t, F t,t+3 = L t +3T t End of time t+1: F t+1,t+2 = L t+1 +T t+1, F t+1,t+3 = L t+1 +2T t+1, F t+1,t+4 = L t+1 +3T t+1 End of time t+2: F t+2,t+3 = L t+2 +T t+2, F t+2,t+4 = L t+2 +2T t+2, F t+2,t+5 = L t+2 +3T t+2 D t-2 D t-1 D t D t+1 D t+2 F t+2, t+3 F t+2, t+4 t-2 t-1 t t+1 t+2 t+3 t+4 D t-2 D t-1 D t D t+1 D t+2 D t+3 F t+3, t+4 End of time t+3: F t+3,t+4 = L t+3 +T t+3, F t+3,t+5 = L t+3 +2T t+3, F t+3,t+6 = L t+3 +3T t+3
29 Updating via Double Exponential Smoothing We use the following formulae to update estimates of L t and T t : L t = αd t + (1-α)F t-1,t = αd t + (1-α)(L t 1 +T t 1 ) Observed level Forecasted level T t = β(l t -L t 1 ) + (1- β) T t 1 Observed trend Forecasted trend Again α and β are smoothing constants with values between 0 and 1; typically α [0.05, 0.3 ] and β [0.01, 0.2 ] As before larger values stress recent data and smaller values provide better smoothing
30 Example Time t Value D t
31 Time Value O-S-A Forecast Abs. Err. Mean Trend t D t F t-1,t = L t 1 +T t 1 e t = D t -F t L t T t Smoothing constants α=0.3 and β=0.1 Initialization: Regression through first 6 points yields an intercept c=94.8 and a slope m= Thus, estimates of mean level and trend at time t=6 are: L 6 = *1.629 = and T 6 = L t = αd t +(1-α)F t-1,t; T t = β(l t -L t 1 )+(1- β) T t 1 Note that L 7 =0.3* * = T 7 =0.1*( ) + 0.9*1.629 = L 8 =0.3* * = T 8 =0.1*( ) + 0.9*1.623 = etc. L 23 =0.3* * = T 23 =0.1*( ) + 0.9*1.758 = L 24 =0.3* * = T 24 =0.1*( ) + 0.9*1.916 = These will be used to forecast each month in the future via F 24,24+k = k(1.819) Jayant Rajgopal, 2016
32 Forecasting with Double Exponential Smoothing
33 Time Series with Seasonality A repetitive pattern over some seasonal cycle, e.g., monthly (or quarterly) seasons in a yearly cycle, daily seasons in a weekly cycle, hourly seasons in a daily cycle, etc. The level in each time period is systematically changed because that time period corresponds to some specific season within a repetitive seasonal cycle. So if the raw or de-seasonalized level at time t is L t, and time t is the j th season of the seasonal cycle, then the actual level of the process is L t S j where S j is a seasonality index associated with season j. So D t =L t S j + ε t Jayant Rajgopal, 2016
34 The first step is to estimate the seasonality index S j for each season j in the cycle Let N=No. of seasons in a seasonal cycle and S j = seasonality index for season j, j=1,2,,n The values of S j are such that j S j = N Week 1 Week 2 Time t Season j D t CMA t Index S j = D t /CMA t Mon 4 Tue 7 Wed Thu Fri Mon Tue Wed Thu 15 Fri 10 Thursday s demand is 71% higher than normal because it is Thursday Tuesday s demand is 39% Lower than normal because it is Tuesday etc.
35 The procedure follows four steps 1. Compute seasonality index values for each season using the centeredmoving-average approach 2. De-seasonalize all data via D t* =D t /S j (where j is the index of the season corresponding to time t) this removes the effect of the seasonality 3. Work on the de-seasonalized data D t * as usual (i.e., single or double exponential smoothing depending on whether or not a trend is present) 4. Build back the seasonality into the final forecast by multiplying the forecast by S j where j is the index of the season corresponding to the time for which we are making the forecast: F t,t+k = (F * t,t+k) S j
36 Example Consider the following quarterly data over three years Quarter Year 1 Year 2 Year
37 Find centered-moving-averages and then the seasonality indices S j t Quarter Demand t Quarter C-M-A S j =D t /CMA t Note that with an even no. of seasons in the cycle (=4) we have to do our centering more carefully = { (43+50)} 4, = { (57+61)} 4 etc. etc. Averaging multiple estimates and normalizing to ensure that they add up to 4.00, we get the final index estimates: Quarter Quarter Quarter Quarter SUM=
38 Next, remove the effect of seasonality by de-seasonalizing the data D t* = D t S j t Quarter S j Demand D t D t* =D t /S j
39 Note how the seasonal swings are gone, and an increasing trend seems to be present Values/Forecasts Time Actual Values Deseasonalized Values
40 Now use double exponential smoothing on the de-seasonalized data t Quarter D t* =D t /S j F * t-1,t L * t T * t Smoothing constants α=0.2 and β=0.1 Initialization: An approximate trend line was obtained by eyeballing the data. Specifically, the line chosen passed through D 3 * and D 10 * as displayed on the graph. Corresponding to this line we have. L 3 =D 3* =53.598, and T 3 * = (D 10* - D 3* ) (10-3) = ( )/7 = Now update as usual starting with t=4 At the end of t=12, we get the current estimates of level and trend: L 12 =70.366, and T 12 = 1.866
41 Finally, forecast t Quarter D t* =D t /S j F * t-1,t L * t T * t D t F * t-1,t=f * t-1,t S j o-s-a s-a s-a s-a s-a
42 Seems to be working quite well
43 Forecasts should be assessed w.r.t. accuracy and bias Given errors e 1, e 2, e N Mean error = ( j e j )/N ( should be close to zero if unbiased) Mean Absolute Deviation = MAD = ( j e j )/N ( should be small) Mean Squared Error = MSE = ( j (e j ) 2 )/N ( should be small) Mean Absolute Percentage Error=MAPE= 100*( j e j /D j )/N ( should be small) Often instead of just taking raw error averages, we might choose to smooth the errors (or functions of the error): Smoothed mean error E t = ω(e t ) + (1-ω)E t-1 (should eventually be close to zero) Smoothed Mean Absolute Deviation MAD t = ω( e t ) + (1-ω)(MAD t-1 ). Tracking Signal = E t /MAD t (close to zero if the forecasts are unbiased) Average % error = MAD/Forecast Jayant Rajgopal, 2016
44 Assume α=0.3 and β=0.1 and suppose we had initial estimates at the end of t=3 of L 3 = and T 3 =3.166 E t t D t F t L t T t e t E t e t MAD t MAD t (59.075) (3.166) Average = Jayant Rajgopal, 2016
45 1. Average error = Smoothed Average Error (using ω=0.1) = Mean Absolute Deviation = [ ] 21 = Smoothed Mean Absolute Deviation (absolute errors smoothed exponentially with ω=0.1) = Mean squared error (MSE) = [(-3.241) 2 +(-4.338) 2 + +(0.27) 2 ] 21 = Standard Deviation of forecast errors = (1.25)*(2.136) = 2.67 (or perhaps 1.25*2.442 = , or perhaps Sqrt(8.584) = 2.93) NOTE: If e t is Normally distributed then σ = Std. Dev. of error = (π/2) 0.5 *MAD 1.25MAD So we would expect that 95% of the time e t 2σ, i.e., e t 2.5MAD 98% of the time e t 2.4σ, i.e., e t 3MAD So if e t exceed 2.5 or 3 times the MAD it would signal some possible warning Jayant Rajgopal, 2016
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