ExErcISES
3.1 IntroductIon
3.2.2 Quantitative Forecasting
Quantitative forecasting is based on the assumption that the ‘forces’ that generated the past demand will also generate the future demand, i.e., history will tend to repeat itself. Analysis of the past demand pattern provides a good basis for forecasting the future demand. Majority of quantitative approaches fall in the category of time-series analysis. Quantitative methods of forecasting are divided into two categories: (a) time-series methods and (b) causal methods.
3.3 tIMe-SerIeS ForecaStIng
The time-series forecasting methods are based on the analysis of historical data. Time series may be defined as a set of observations measured at successive times or over successive periods. In the time-series methods, the assumption is that the past patterns in the data can be used to forecast future data points. Here, future data point means the points on the projection of the line using extrapolation. The patterns of demand variations of a time series are shown in Figure 3.1.
Trends are noted by an upward or downward sloping line. There is a linear relationship between time and demand. The demand in the future can be estimated by extending the straight line or extrapolation. The linear trend is shown in Figure 3.1(a).
Seasonality is a data pattern that repeats itself over the period of one year or less. It leads rise or fall in demand among the seasons. In a particular season, demand may increase while in other season demand may decrease. Thus, the demand is dependent on season or quarter of a year. The seasonal variation in demand is shown in Figure 3.1(b).
Cycle is a data pattern that may cover several years before it repeats itself. Cyclic nature in demand variation shows the same style in increasing or decreasing in demand in future. If we draw a linear line through the mean values of the actual demands, we can observe that the upper and lower variation in demands follow the same pattern throughout the straight line as shown in Figure 3.1(c).
Random fluctuation (noise) results from random variation or unexplained causes. There are a number of causes of random fluctuation in demand, for example, anticipation of increasing the price or shortages of product in near future increases the demand. Random fluctuations in demand are shown in Figure 3.1(d).
The following methods are for time-series forecasting:
(a) Naïve method
(b) Simple moving average (SMA) method (c) Weighted moving average (WMA) method (d) Exponential smoothing method
(A) Naive Method
The naive forecasting model is a special case of the moving average forecasting model where the number of periods used for smoothing is 1. Therefore, in the naive method, the forecast for a period, t, is simply the observed value of the previous period, t – 1. Due to the simplistic nature
Time
Increasing linear trend
Demand
(a)
Time (b)
Demand
Seasonal pattern with linear growth
Cyclic pattern
Time
Demand
(c)
Purely random- No recognizable pattern
Time
Demand
(d)
Figure 3-1: (a) Linear variation, (b) seasonal variation, (c) cyclic variation and (d) random variation
of the naive forecasting model, it can only be used to forecast up to one period in the future. It is not at all useful as a medium-/long-range forecasting tool.
The naive method is used partly for completeness and partly for its simplicity. It is unlikely that any one will want to use this model directly because it can be only used for the just next period and also, the forecast for the next period is same as the previous actual demand, i.e. there is no change is considered in forecasting the demand. Instead, consider using either the moving average model or the more general WMA model with a higher (i.e. greater than 1) number of periods, and possibly a different set of weights.
(B) Simple Moving Average Method
The SMA method uses the average value of actual demand for some recent periods. The value of averaging period depends on the nature of variation in the actual demands for last some periods and accuracy of forecasting. Using this method, we can forecast the demand only for the next period in the future, but to know the forecasting errors we have to find the forecast value based on the actual demands for the past periods, also. Suppose t represents the current period and we want to forecast for the period t + 1. Specifically, the forecast for period t + 1 can be calculated at the end of period t (after the actual demand for period t is known) as
F n D D D
F n D
t t t t n
t i
i t n
t
+ − + −
+ = + −
= + + +
⇒ =
∑
1 1 1
1 1
1 1
( )
where D indicates the demand and F indicates the forecast; t is the time period; n is the number of the averaging period.
The assumptions in making the forecast using the SMA method are as follows:
1. All n past observations are treated equally.
2. Observations older than n are not included at all.
3. It requires that n past observations be retained.
example 3.1: The monthly demands for an office furniture (in units) are given in Table 3.2.
Forecast the demand using 3-period and 5-period SMA for the 12th month. Also, show the variation in forecasting graphically.
Table 3-2: Monthly demands for furniture for eleven months
Month (t) 1 2 3 4 5 6 7 8 9 10 11 12
Demand (Dt) 600 628 670 735 809 870 800 708 842 870 739 — Solution:
We use the formula
F n D D D
n D
t t t t n i
i t n
t
+ − + −
= + −
= + + + =
∑
1 1 1
1
1 1
( )
For 3-period moving average method
Here, the average value of last three periods is used as forecast for the fourth period as shown in the formula below:
F4 1 D1 D2 D3 3
1
3 600 628 670 632 66 633
= ( + + )= ( + + )= . ≈ Similarly, F5, F6, …, F12 can be calculated as:
F5 1 D2 D3 D4
3 678
= ( + + )= F6 1 D3 D4 D5
3 738
= ( + + )=
………..
F12 1 D9 D10 D11
3 817
= ( + + )= For 5-period moving average method
Here, the average value of last five periods is used as forecast for the sixth period as shown in the formula below:
F6 1 D1 D2 D3 D4 D5 5
1
5 600 628 670 735 809 688 4 689
= + + + +
= + + + + = ≈
( )
( ) .
Similarly, F7, F8, …, F12 can be calculated as:
F7 1 D2 D3 D4 D5 D6
5 743
= ( + + + + )= F8 1 D3 D4 D5 D6 D7
5 777
= ( + + + + )=
………
F12 1 D7 D8 D9 D10 D11
5 792
= ( + + + + )=
The forecasting of the demand for the 12 months is shown in Table 3.3 using 3-period simple average and 5-period SMA.
Table 3-3: Forecasting of furniture demand using simple moving average (SMA) Month Demand 3-Period moving average 5-Period moving average
1 600 — —
2 628 — —
(Continued )
Month Demand 3-Period moving average 5-Period moving average
3 670 — —
4 735 633 —
5 809 678 —
6 870 738 689
7 800 805 743
8 708 827 777
9 842 793 785
10 870 784 806
11 739 807 818
12 – 817 792
It can be observed from the graph in Figure 3.2 that the variation in actual demand is very high and it is difficult to forecast using any trend line. Thus, the moving average method is suitable for this kind of data. The 5-period moving average graph is smoother than the 3-period moving average graph, but the limitation is forecasting error. A larger periods of averaging show large variation from the actual demand curve as shown in Figure 3.2. Three-period averaging is closer the actual demand capered to five-period averaging.
900 850 800
Demands (in units)
750 700 650 600
1 2 3 4 5 6 7
Months
Demand
3-Period moving average 5-Period moving average
8 9 10 11 12
Figure 3-2: Variation in furniture demand with time
(C) Weighted Moving Average Method
The WMA method is very similar to SMA method, but in the former method different weights are provided for the periods. The largest weight is provided with most recent period and the weights are decreasing as we move to the previous period in the past. A WMA is a moving average where each historical demand may be weighted as:
Ft+1=W D1 t +W D2 t−1+ + W Dn t+ −1 n
Table 3-3: Forecasting of furniture demand using simple moving average (Continued)
where n is the total number of periods in the average, Wt is the weight applied to period t’s demand, W1 > W2 >…> Wn, sum of all the weights = 1, Forecast Ft + 1= forecast for period t + 1.
The assumptions in making the forecast using the SMA method are as follows:
1. Adjustments in the moving average to more closely reflect fluctuations in the data.
2. Weights are assigned to the most recent data.
3. Requires some trial and error to determine precise weights.
example 3.2: Using the data shown in Table 3.3, forecast the demand for the periods with the three-period WMA method. The most recent data should be given 50 per cent weightage, second year past data, 30 per cent, and third year past data, 20 per cent. Also, compare the result with the result of three-period SMA graphically.
Solution:
F W D W D W D
F W D W D W D
t+ = t+ t− + + n t+ −n
= + + = × + ×
1 1 2 1 1
4 1 3 2 2 3 1 0 5 670 0 3 6
. . 228 0 2 600 643 4 644+ . × = . ≈ Similarly, F5, F6, …, F12 can be calculated as:
F5 =W D1 4+W D2 3+W D3 2 =695 F6 =W D1 5+W D2 4+W D3 3=759
……….
F12=W D1 11+W D2 10+W D3 9=799 The forecasting of the demand for the 12th month is shown in Table 3.4.
Table 3-4: Forecasting of furniture demand using weighted moving average (WMA) method
Month Demand 3-Period WMA
1 600 —
2 628 —
3 670 —
4 735 644
5 809 695
6 870 759
7 800 825
8 708 823
9 842 768
10 870 794
11 739 830
12 — 799
900 850 800
Demands (in units)
750 700 650 600
1 2 3 4 5 6 7
Months
Demand 3-Period moving average
3-Period weighted moving average 8 9 10 11 12
Figure 3-3: Variation in furniture demand using 3-period SMA and WMA methods
In Figure 3.3, we observe that the WMA forecast closer to the actual demand compared to the SMA method, i.e. the forecasts from WMA pursuing the actual demand closely since the recent period is given more weightage.
(D) Exponential Smoothing Method
Exponential smoothing is the most popular and cost effective of the statistical methods. It is based on the principle that the latest data should be weighted more heavily and ‘smoothers’ out cyclical variations to forecast the trend (Armstrong et al. 2005). It assumes that the data gets older, becomes less relevant and should be given less weight. In order to calculate the forecasting, the old average, the actual new demand and a weighting factor are needed.
Exponential smoothing gives greater weight to demand in more recent periods, and less weight to demand in earlier periods as shown in Figure 3.4. It is a sophisticated WMA method that calculates the average of a time series by giving recent demands more weight than earlier demands. In this method, both the actual demands and forecast of the past demands are used in the calculation of forecasting, i.e. all the past data are used to estimate the demand in the future.
Here, α is a smoothening factor, its value is decreasing as we move towards the past. Figure 3.4 shows the decreasing pattern for the value of α as we move towards the past; this decrease follows an exponential pattern. Thus, this method is known as exponential smoothing method.
Decreasing weight given to older observations
Present 0 < α < 1
α (1 – α) α (1 – α)2 α (1 – α)3 α
Figure 3-4: Decreasing weight to the older observations
The forecast for the period t + 1 is equal to the actual demand for the period t plus the α times of the difference of the actual and forecasting value for the period t. Here, α tries to smoothen the variation in the previous period actual and forecasting values of the demand.
The forecast for tth period can be given as
Ft+1=αDt + −(1 α)Ft =Ft+α(Dt−Ft) Here, α is smoothening factor.
example 3.3: Using the data shown in Table 3.3, forecast the demand for the periods using the exponential smoothing method (α = 0.3 and α = 0.5). Also, compare the results graphically.
Solution:
Using the formula
Ft+1=αDt + −(1 α)Ft =Ft+α(Dt−Ft) The demand forecast data is prepared (see Table 3.5).
Table 3-5: Furniture demand forecasts using exponential smoothing
Month Demand Forecast, Ft
α = 0.3 α = 0.5
1 600 — —
2 628 600 + 0.3(600 – 600) = 600 600
3 670 600 + 0.3(628 – 600) = 608.4 614
4 735 608.4 + 0.3(670 – 608.4) = 626.8 642
5 809 626.8 + 0.3(735 – 626.8) = 659.3 688.5
6 870 659.3 + 0.3(809 – 659.3) = 704.2 748.7
7 800 704.2 + 0.3(870 – 704.2) = 753.9 809.3
8 708 753.9 + 0.3(800 – 753.9) = 767.7 804.6
9 842 767.7 + 0.3(708 – 767.7) = 749.8 756.3
10 870 749.8 + 0.3(842 – 749.8) = 777.4 799.1
11 739 777.4 + 0.3(870 – 777.4) = 805.2 834.5
12 – 805.2 + 0.3(739 – 805.2) = 785.3 786.2
For smaller values of α, we get a smoother curve. In Figure 3.5, it can be observed that the curve for α = 0.3, is smoother than that of the curve for α = 0.5. For smooth curve, forecasting is easy, but the forecasting error may be large because there is a large deviation of the forecast of the actual value. This type of forecasting model is used for the demand fluctuates continuously and there is requirement of smoothening the curve for ease of forecasting.
Features of Exponential Smoothing Method
Some of the features of exponential smoothing method are enumerated as follows.
1. The emphasis given to the most recent demand levels can be adjusted by changing the smoothing parameter.
2. Exponential smoothing is simple and requires minimal data.
3. Larger α values emphasize recent levels of demand and result in forecasts more responsive to changes in the underlying average.
4. Smaller α value treats past demand more uniformly and result in more stable forecasts.
5. When the underlying average is changing, results will lag actual changes.
6. The new forecast is the weighted sum of old forecast and actual demand.
7. Only two values, i.e. actual demand and forecast for just previous period (Dt and Ft), are required, compared with actual demands for n period in moving average method.
8. Parameter α is determined empirically (whatever works best). But, the rule of thumb can be used as: α < 0.5.
9. Typically, α = 0.2 or α = 0.3 works well.
Adjusted Exponential Smoothing
Double exponential smoothing is also called exponential smoothing with trend. If trend exists, single exponential smoothing may need adjustment. There is a need to add a second smoothing constant to account for the trend. It is similar to a single exponential smoothing. The basic idea is to introduce a trend estimator that changes over time. If the underlying trend changes, over- shoots may happen. Issues to choose two smoothing rates, α and b are very important.
The following points may help in choosing the value of α and b:
• b close to 1 means quicker responses to trend changes, but may over-respond to random fluctuations.
• α close to 1 means quicker responses to level changes, but again may over-respond to random fluctuations.
900 850 800
Demands (in units)
750 700 650 600
1 2 3 4 5 6 7
Months
Demand α = 0.3 α = 0.5
8 9 10 11 12
Figure 3-5: Variation in forecasting using exponential smoothing factor 0.3 and 0.5
Adjusted forecasting is given by
AFt+1=Ft+1+Tt+1
Here, T is an exponentially smoothed trend factor given by Tt+1 =β(Ft+1−Ft) (+ −1 β)Tt
where Tt is the last period trend factor and b is a smoothing constant for trend. The formula for the trend factor reflects a weighted measure of the increase or decrease between the next forecast, Ft + 1, and the current forecast, Ft. b is a value between 0.0 and 1.0 and reflects the weight given to the most recent trend data. It is usually determined and subjectively based on the judgement of the forecasters. High b reflects that trend changes more than in the case of low b. It is common for b to equal α in this method.
example 3.4: Using the data shown in Table 3.6, forecast the demand for the periods with adjusted exponential smoothing method (α = 0.5 and b = 0.3). Also, compare the result graphically with simple exponential smoothing (α = 0.5).
Solution:
Adjusted forecast for period 3:
We have
T3 F3 F2 1 T2
0 3 614 600 0 7 0 4 2
= − + −
=
( ) (
−)
+( )( )
=β( ( β)
. . .
)
Therefore,
. .
AF3 = F3+ =T3 614 4 2 618 2+ =