Chapter 3 Operational Demand Analysis
3.5 Daily demand and production analysis
Figure 3.12 gives an explanation as to the reason for the changes in the averaged daily demand pattern for ULG95. The four different colours show the daily pipeline and delivery tank outflows. It is assumed that on average a day has a demand distribution like this. The green colour represents the first delivery tank, the orange the second delivery tank, the cyan the first pipeline tank and finally the magenta the second pipeline tank.
From closer inspection of Figure 3.12, the lowest demand always appears, regardless of the type of tank at the same time of the day, namely in the early morning.
This provides important information for transportation planners, since by 6AM the road delivery trucks have to be loaded and ready to take their daily route. The peak period of daily demand differs slightly from tank to tank but the busiest period of the day is between 7AM and 5PM. This gives the opportunity to reduce pipeline capacity after 5PM and carry out maintenance work.
Figure 3.12 Daily periodic variation in outflow from delivery & pipeline tanks
Figure 3.13 shows the total daily inflow to the system. As it was mentioned earlier, at any time, any given tank is either being filled, emptied or held at a constant level. This aspect of the network is seen in this figure, especially when compared with Figure 3.11.
Figure 3.11 shows the lowest demand interval where the outflow from the tanks is minimal, from approximately 4AM to 6AM, whilst for the same period the peak inflow period, where the inflow to blender tanks is at the maximum, Figure 4.13. The network carries a safety stock buffer to provide sufficient gasoline to fill the delivery or pipeline tanks when the blender tanks are filled but cannot therefore be simultaneously emptied.
0 5 10 15 20 25
-220 -200 -180 -160 -140 -120 0
Daily periodic variation in outflow from delivery and pipeline tanks
Hours of a day Green: delivery tank 1
Orange: delivery tank 2 Cyan: Pipeline tank1 Magenta: Pipeline tank 2
The need for the safety stock is justified again when the outflow is at its peak and the inflow is at its lowest level. The peak intervals of total inflow and total outflow in the two figures are the inverse of each other. So are the off-peak periods.
Figure 3.13 Daily averaged periodic variation in total inflow to the ULG95 network
The next step in the analysis was to identify an appropriate model for the averaged demand over all days.
0 5 10 15 20 25
500 550 600 650 700
750 Daily averaged periodic variation in total inflow to ULG95 network
Hours of a day
Figure 3.14 Autocorrelation function and partial autocorrelation function for total inflow to the network
Since it is difficult to assess the order of an auto-regressive process from the sample auto-correlation function alone (Letchford, 1994), the partial auto-correlation function was applied. The partial autocorrelation function is shown in Figure 3.14. When fitting an auto-regressive model at lag p, the last coefficient α(p) will be denoted by π(p).
It measures the correlation at lag p, which is not accounted for by an auto-regressive (p-1) model. It is called the pth partial auto-correlation coefficient, and when plotted against p, it gives the partial correlation function. It can be shown that the partial auto-correlation function of an auto-regressive p process cuts off at lag p so that the correct order is assessed as the value of p beyond which the sample values of {π(i)} are not significantly different from zero.
Figure 3.14 shows the correlation values and also displays the partial auto-correlation function of the total inflow time series. The plots show the sinusoidal convergence to zero, which is the typical attribute of the second order auto-correlation and partial auto-correlation plots. The most appropriate model for the averaged daily inflow time series was an auto-regressive 2 model. The total inflow at time t is expressed by:
20 15
10 5
1.00.8 0.60.4 0.2 -0.20.0 -0.4-0.6 -0.8
Autocorrelation -1.0
LBQ T Corr Lag LBQ T Corr Lag LBQ T Corr Lag
109.53 108.92 108.09 106.86 105.72 105.03
104.90 104.81 103.88 100.76 94.17 83.79 70.06
56.76 47.57 43.73 43.45 42.45 36.12 21.97
0.11 0.15 0.20 0.21 0.17 0.08
-0.07 -0.24 -0.46 -0.70 -0.95 -1.20 -1.30
-1.18 -0.80 -0.22 0.44 1.19 2.14 4.41
0.06 0.08 0.11 0.11 0.09 0.04
-0.04 -0.13 -0.25 -0.37 -0.48 -0.57 -0.58
-0.50 -0.33 -0.09 0.18 0.46 0.71 0.90
20 19 18 17 16 15
14 13 12 11 10 9 8
7 6 5 4 3 2 1
Autocorrelation Function for Total Inflow
20 15
10 5
1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
Partial Autocorrelation
T PAC Lag T PAC Lag T PAC Lag
-0.12 -1.08 -0.45 -0.16 0.07 0.79
-0.17 -0.85 -0.43 -1.55 0.06 0.62 -0.05
0.43 -0.72 -0.31 -1.50 -1.12 -2.67 4.41
-0.02 -0.22 -0.09 -0.03 0.01 0.16
-0.03 -0.17 -0.09 -0.32 0.01 0.13 -0.01
0.09 -0.15 -0.06 -0.31 -0.23 -0.54 0.90
20 19 18 17 16 15
14 13 12 11 10 9 8
7 6 5 4 3 2 1
Partial Autocorrelation Function for Total Inflow
Total Inflow(t) = α1*Total Inflow(t-1) + α2*Total Inflow(t-2)+Z(t), (3.2)
where {Z(t)} is a purely random process with zero mean and constant variance, and α1 is equal to 1.4762 and α2 equals -0.6146. The calculations were carried out by using statistical analysis software of the Department of Engineering Mathematics at the University of Newcastle upon Tyne.
The residuals versus fitted values plot in Figure 3.15 indicates no structure and therefore indicates the goodness of fit of the second order auto-regressive model to the total inflow time series of the gasoline network.
Figure 3.15 Residuals versus fitted values of total inflow
The final step in the analysis was to identify an appropriate model for the averaged over all days total demand. Figure 3.16 shows the auto-correlation and the partial auto-correlation function of the total outflow time series of the gasoline distribution network at. The plots again display the sinusoidal convergence to zero, which is the typical attribute of the second order auto-correlation and partial auto-correlation plots.
400 350
300 250
20
10
0
-10
-20
Fitted Value
Residual
Residuals Versus the Fitted Values
(response is total inflow)
Figure 3.16 Autocorrelation and partial autocorrelation function of total outflow
The residuals versus fitted values plot in Figure 3.17 indicates no structure in residuals and therefore indicates the goodness of fit of the second order auto-regressive model (Erickson, et al, 1992).
Figure 3.17 Residuals versus fitted values of total outflow
20 15
10 5
1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8
Autocorrelation -1.0
LBQ T Corr Lag LBQ T Corr Lag LBQ T Corr Lag
30.39 30.31 30.16 30.15 30.04 29.65
28.15 23.14 16.13 14.71 14.47 14.17 14.09
13.73 13.54 12.97 12.58 12.49 12.46 11.71
0.07 0.11 0.04 -0.11 -0.22 -0.46
-0.91 -1.19 -0.57 0.24 0.29 -0.15 -0.33
-0.25 -0.45 -0.39 -0.19 -0.13 0.58 3.22
0.02 0.03 0.01 -0.03 -0.07 -0.15
-0.28 -0.35 -0.17 0.07 0.08 -0.04 -0.10
-0.07 -0.13 -0.11 -0.05 -0.04 0.16 0.66
20 19 18 17 16 15
14 13 12 11 10 9 8
7 6 5 4 3 2 1
Autocorrelation Function for Total Outflow
20 15
10 5
1.0 0.8 0.6 0.40.2 0.0 -0.2 -0.4 -0.6 -0.8
Partial Autocorrelation -1.0
T PAC Lag T PAC Lag T PAC Lag
-0.53 0.04 -0.36 0.30 -0.46 -0.17
0.27 -0.48 -1.02 -0.47 -1.17 1.88 -0.92
-0.36 0.50 -0.48 -0.86 1.32 -2.32 3.22
-0.11 0.01 -0.07 0.06 -0.09 -0.03
0.06 -0.10 -0.21 -0.10 -0.24 0.38 -0.19
-0.07 0.10 -0.10 -0.17 0.27 -0.47 0.66
20 19 18 17 16 15
14 13 12 11 10 9 8
7 6 5 4 3 2 1
Partial Autocorrelation Function for Total Outflow
-60 0 -65 0
-7 0 0 40 30 20 10 0 -10 -20 -30 -40 -50 -60
F itted V alue R e
s id u al
R e sid u als V ers us th e F itted V a lu es
(res p o nse is to tal o u tflo w )
The most appropriate model for the averaged daily total demand or outflow time series is again an auto-regressive 2 model. The total outflow can be described as:
Total Outflow(t) = α1*Total Outflow(t-1) + α2*Total Outflow(t-2)+Z(t), (3.3) where {Z(t)} is a purely random process with zero mean and constant variance, and α1 is equal to 0.9924 and α2 equals -0.4946.