Workload Assessment (Forecasting)
3.3 Forecasting Methods for Time-Series Analysis
3.3.3 Exponential Smoothing
The exponential smoothing (ES) method, like the WMA method, calculates an average demand (forecast). ES methodology remembers the last estimate of the aver- age value of demand and combines it with the most recent observed, actual value to form a new estimated average. ES forecasts the demand for a given period t by combining the forecast of the previous period (t − 1) and the actual demand of the previous period (t − 1). The actual demand for the previous period is given a weight of α and the forecast of the prior period is given a weight of (1 − α), where α is a smoothing constant whose value lies between 0 and 1. The equation for the forecast for period t is
Forecast( )t = ∗Actual demand(t − + −) ( )∗Forecast(t − ),
≤ ≤
a a
a
1 1 1
0 1..
Table 3.2 An Example of the WMA Method (n = 3)
Month Sales Forecast Calculation Weights = 0.2, 0.3, 0.5
1 100
2 80
3 90
4 110 89.00 = 0.2 * 100 + 0.3 * 80 + 0.5 * 90 5 100 98.00 = 0.2 * 80 + 0.3 * 90 + 0.5 * 110 6 110 101.00 = 0.2 * 90 + 0.3 * 110 + 0.5 * 100 7 95 107.00 = 0.2 * 110 + 0.3 * 100 + 0.5 * 110 8 115 100.50 = 0.2 * 100 + 0.3 * 110 + 0.5 * 95 9 120 108.00 = 0.2 * 110 + 0.3 * 95 + 0.5 * 115 10 90 113.50 = 0.2 * 95 + 0.3 * 115 + 0.5 * 120 11 105 104.00 = 0.2 * 115 + 0.3 * 120 + 0.5 * 90 12 110 103.50 = 0.2 * 120 + 0.3 * 90 + 0.5 * 105
By rearranging the terms, this equation can also be written as Forecast Forecast
Forecast
Actual Demand
( ) ( ) { ( )
(
t t t
t
= − + ∗ −
− −
1 a 1
11)}, 0≤a ≤1.
Example: Consider the data given in Table 3.3. The ES method makes a forecast starting from period 2. The forecast for the first period is generally set equal to the actual demand in that period to get the forecasting process started. Forecasts for all periods 2–10 are given in Table 3.3. For example, the forecast calculations for periods 2 and 3 are shown below:
F( )2 = F( )1 + ∗a { ( )A1 −F( )}1 =100 0 2 100 100+ . ∗( − ) =100, F( )3 = F( )2 + ∗a { ( )A 2 −F( )}2 =100 0 2+ . ∗(80 100− )=96. ES is a simpler method, requiring fewer calculations than WMA, which needs n weights and n periods of data for each forecast estimate. ES needs only three pieces of data. Also, it can be more effective because only one weight, alpha (α), has
Table 3.3 An Example of the ES Method (α = 0.2)
Month Sales Forecast Comment and Calculation 1 100 100 Forecast for period 1 should be available
before starting the calculations. If it is not given, then set it equal to the sales of period 1 2 100.00 = (100 + 0.2(100 − 100))
3 90 96.00 = (100 + 0.2(80 − 100)) 4 110 94.80 = (96 + 0.2(90 − 96)) 5 100 97.84 = (94.8 + 0.2(110 − 94.8)) 6 110 98.27 = (97.84 + 0.2(100 − 97.84)) 7 95 100.62 = (98.27 + 0.2(110 − 98.27)) 8 115 99.50 = (100.62 + 0.2(95 − 100.62)) 9 120 102.60 = (99.5 + 0.2(115 − 99.5)) 10 90 106.08 = (102.6 + 0.2(120 − 102.6)) 11 105 102.86 = (106.08 + 0.2(90 − 106.08)) 12 110 103.29 = (102.86 + 0.2(105 − 102.86))
to be chosen. This makes it easier to experiment with past data to see which value of α provides the least forecast error. Often, the use of the ES method for forecasting is preferred to the WMA method.
The response rate of the ES model is a function of the α value that is used. The response rate of a forecasting system is the speed with which the forecasting sys- tem makes adjustments to the forecasts if the data in the series show an upward or downward trend. A review of the two equations stated above provides a perspective on the response rate. Consider the first equation, Forecast (t) = α * Actual demand (t − 1) + (1 − α) * Forecast (t − 1). When α is large, the actual demand in the prior period is given great weight in the forecast for the next period. If α is 1, the forecast for the next period is the actual value of the prior period (same as n = 1 in MA). The forecast results are markedly affected by the α that is used.
The second equation, Forecast (t) = Forecast (t − 1) + α * {Actual demand (t − 1)–Forecast (t − 1)}, provides a different perspective on the ES technique. The forecast in period t is equal to the forecast in the previous period plus a fraction (specified by α) of the error in the previous period. The error in the previous period is the difference between that actual demand and the forecast, which can be posi- tive or negative. If α is close to 0, the new forecast does not show much adjustment for the error, and in the extreme case, if α = 0, the new forecast is equal to the old forecast with no adjustments.
Small values of α (which will be analogous to large values of n in MA) are used for stable systems where there is, at most, a minimum amount of random fluctua- tion. Large values of α are used for changing and evolving systems where much reli- ance is placed on the last observation. New products, as they move through their life-cycle stages, start with a large α value, which gradually diminishes as the prod- uct enters its maturation stage. For most production scheduling systems in both the job shop and the flow shop, α is kept small, in the neighborhood of 0.050–0.150, to decrease the system’s response to random fluctuations.
The ES method has a clear advantage over the MA method in terms of the amount of memory required for storing the data. In addition to the smoothing con- stant α, at any time we need to store only the exponentially smoothed average and the actual demand for the previous period. For the MA method, actual demands for the last n periods have to be stored.
The different values of α give different forecast just like different forecasts are obtained by changing the values of n in the MA method. A small value of α makes minor adjustments to the forecast results as discussed above—giving smoother forecasts. A smooth forecast is obtained in the MA method if n is large. Therefore, smaller values of α and large values of n tend to give similar forecasts. Similarly, large values of α and small values of n tend to give similar results. In fact, when α = 1, or n = 1, the forecast in a given period is equal to the demand in the previ- ous period and the forecasting system is extremely responsive to changes in the demand. In this case, the forecast fluctuates as actual demand fluctuates. However, forecasts are lagging behind the demand by one period.
If sudden change occurs, then the response will be sluggish irrespective of whether change is genuine or noise is present when α is small. On the other hand, if α is close to 1, there will be substantial adjustment for any error. With a large value of α, the bona fide changes will be reflected immediately in the new average.
It is difficult to say beforehand which value of α will serve the purpose best.
Different values of α starting with a small value may be chosen to make a forecast.
The actual observed data and the forecast using different values of α may be plot- ted on a graph and by examination that value of α which tracks the actual demand best is chosen. Alternatively, for different values of α, the errors—mean absolute deviation (MAD) and mean squared error—may be calculated and that value of α which gives a lower error may be chosen. See appropriate details in the section ahead on forecast errors.
As in the case of MAs, if there is a fundamental systems change, then prior his- tory of patterns is not a help, and ES should be applied as if to a new startup system for the first time.
ES has been found to be very effective in a variety of situations. Many forecast- ing and control systems employ ES because it works better than the older methods of MAs and WMAs. It has proved effective for diverse applications. Fighter aircraft use ES to aim their guns at moving targets. In effect, they forecast the location of enemy jets during flying missions. This application shows how fast the ES method can track a consistent, but dynamically changing pattern. Manufacturers use ES to forecast demand levels, which experience the same kind of nonrandom but volatile shifts from time to time. In such cases, the recent past has the most information about the near future. ES can catch these shifts and make rapid adjustments to inventory levels. Other manufacturers and service organizations use it because it requires less computational work and is readily understood.