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	102	1.0	102.00
1	2	1.0	118	0.69	81.42	216.28	0.00462
	3	1.0	106	0.31	32.86
	1	1.0	102	0.69	70.38
2	2	1.0	118	1.0	118	235.02	0.00425
	3	1.0	106	0.44	46.64
	1	1.0	102	0.31	31.62
3	2	1.0	118	0.44	51.92	189.54	0.00527
	3	1.0	106	1.00	106

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. The detailed computation is given in Table 2.

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	0.00462	98	1.0	0.4523
1	2	0.00425	106	0.694	0.3117	0.9618	1.0397
	3	0.00527	122	0.308	0.1978
	1	0.00462	98	0.69	0.3124
2	2	0.00425	106	1.0	0.4505	1.0458	0.9562
	3	0.00527	122	0.44	0.2829
	1	0.00462	98	0.31	0.1404
3	2	0.00425	106	0.44	0.1982	0.9815	1.0188
	3	0.00527	122	1.00	0.6429

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

$$\left[ \begin{array}{ccc} 1.0&0.69&0.31 \\ 0.69&1.0&0.44 \\ 0.31&0.44&1.0 \\ \end{array} \right]$$ (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. The results are shown in table 3.

Table 3: Step3: Final Table

	1	2	3	$A_i$	$O_i$	$O_i^1$
1	48.01	35.24	15.157	0.00462	98	98.407
2	32.96	50.83	21.40	0.00425	106	105.19
3	21.14	31.919	69.43	0.00527	122	122.489
$B_j$	1.0397	0.9562	1.0188
$D_j$	102	118	106
$D_j^1$	102.11	117.989	105.987

$O_i$ 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 ; $Error = \Sigma{\vert O_i - O_i^1\vert} +\Sigma{\vert D_j - D_j^1\vert}$ $Error = \vert 98-98.407\vert+\vert 106-105.19\vert+\vert 122-122.489\vert+\vert\vert 102 - 102.11\vert+\vert 118-117.989\vert+\vert 106-105.987\vert=2.03$