Module 2: Transportation planning

Lecture 8 Trip distribution

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Solution

The first step is given in Table 1

Table 1: Step1: Computation of parameter $A_i$

$i$	$j$	$B_j$	$D_J$	$f(c_{ij})$	${B_j D_j f(c_{ij})}$	${\Sigma{B_j D_j f(c_{ij})}}$	$A_i=\frac{1}{{\Sigma{B_j D_j f(c_{ij})}}}$
	1	1.0	120	1.0	120.00
1	2	1.0	108	0.833	89.964	275.454	0.00363
	3	1.0	118	0.555	65.49
	1	1.0	120	0.833	99.96
2	2	1.0	108	1.0	108	286.66	0.00348
	3	1.0	118	0.667	78.706
	1	1.0	120	0.555	66.60
3	2	1.0	108	0.667	72.036	256.636	0.00389
	3	1.0	118	1.00	118

The second step is to find $B_j$. This can be found out as $B_j = 1/{\Sigma{A_i O_i f(c_{ij})}}$, where $A_i$ is obtained from the previous step.

Table 2: Step2: Computation of parameter $B_j$

$j$	$i$	$A_i$	$O_i$	$f(c_{ij})$	${A_i O_i f(c_{ij})}$	$\Sigma{A_i O_i f(c_{ij})}$	$B_j = 1/{\Sigma{A_i O_i f(c_{ij})}}$
	1	0.00363	110	1.0	0.3993
1	2	0.00348	122	0.833	0.3536	0.9994	1.048
	3	0.00389	114	0.555	0.2465
	1	0.00363	110	0.833	0.3326
2	2	0.00348	122	1.0	0.4245	1.05	0.9494
	3	0.00389	114	0.667	0.2962
	1	0.00363	110	0555	0.2216
3	2	0.00348	122	0.667	0.2832	0.9483	1.054
	3	0.00389	114	1.00	0.44346

The function $f(c_{ij})$ can be written in the matrix form as:

$$\left[ \begin{array}{ccc} 1.0 & 0.833 &0.555 \\ 0.833 &1.0 & 0.667 \\ 0.555 & 0.667 & 1.0 \\ \end{array} \right]$$ (1)

Then $T_{ij}$ can be computed using the formula

$$T_{ij} = A_i O_i B_j D_jf(c_{ij})$$ (2)

For eg, $T_{11} = 102\times 1.0397\times 0.00462 \times 98 \times 1 = 48.01$. $O_i$ is the actual productions from the zone and $O_i^1$ is the computed ones. Similar is the case with attractions also. This step is given in Table 3

Table 3: Step 3: Final Table

	1	2	3	$A_i$	$O_i$	$O_i^1$
1	48.01	34.10	27.56	0.00363	110	109.57
2	42.43	43.53	35.21	0.00348	122	121.17
3	29.53	30.32	55.15	0.00389	114	115
$B_j$	1.048	0.9494	1.054
$D_j$	120	108	118
$D_j^1$	119.876	107.95	117.92

$O_i$ is the actual productions from the zone and $O_i^1$ is the computed ones. Similar is the case with attractions also.

Therefore error can be computed as ; $Error = \Sigma{\vert O_i - O_i^1\vert} +\Sigma{\vert D_j - D_j^1\vert}$ $Error = \vert 110-109.57\vert+\vert 122-121.17\vert+\vert 114-115\vert+\vert 120-119.876+\vert 108-107.95\vert+\vert 118-117.92\vert=2.515$