Chapter 12, Problem 9P

To determine

The trip distribution calculation using the gravity model for two iterations.

Answer to Problem 9P

The trip distribution calculation using the gravity model for two iterations is shown below.

Explanation of Solution

Given:

The given table for number of productions and attractions in each district and travel time in minutes is

District	Productions	Attractions	Travel time (min)
			$1$	$2$	$3$	$4$
$1$	$3400$	$2800$	$4$	$11$	$15$	$10$
$2$	$6150$	$6500$	$11$	$6$	$6$	$9$
$3$	$3900$	$2550$	$15$	$6$	$6$	$11$
$4$	$2800$	$4400$	$10$	$9$	$11$	$4$

    Table (1)

The given table for travel time versus friction factor is

Time(min)	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
$F_{ij}$	$1.51$	$1.39$	$1.30$	$1.22$	$1.13$	$1.04$	$0.97$	$0.92$	$0.85$	$0.80$	$0.75$	$0.71$

    Table (2)

Formula used:

The equation to calculate the number of attractions is given by

$A_{jk}=\frac{A_j}{C_{j\left(k-1\right)}}{A}_{j\left(k-1\right)}$   ...... (I)

Here, $A_{jk}$ is the adjusted factor for attraction district, $C_{jk}$ is the actual attraction total for district, $A_j$ is the desired attraction total for district, $j$ is attraction district number, $n$ is number of district s, $k$ is iteration number, and $m$ is the number of iteration..

The equation to calculate the number of trip between each district is given by

$T_{ij}=P_i\left[\frac{A_j{F}_{ij}{K}_{ij}}{{\displaystyle \sum A_j{F}_{ij}{K}_{ij}}}\right]$   ...... (II)

Here, $P_i$ is the number of trips produced in district $i$, $A_j$ is the number of trips attracted to district $j$, $F_{ij}$ is the inverse function of travel time, $K_{ij}$ is the socio-economic adjustment factor for interchange $j$ and $T_{ij}$ is the number of trips that are produced in district $i$ and attracted to district $j$

Calculation:

Trip distribution for district $1$ is calculated as

Substitute $3400$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{1-1}$.

$\begin{array}{c}{T}_{1-1}=3400\left[\frac{2800\times 1.51\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =3400\left[\frac{4228}{16286.5}\right]\\ =883\end{array}$

Substitute $3400$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{1-2}$.

$\begin{array}{c}{T}_{1-2}=3400\left[\frac{6500\times 0.92\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =3400\left[\frac{5980}{16286.5}\right]\\ =1248\end{array}$

Substitute $3400$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{1-3}$.

$\begin{array}{c}{T}_{1-3}=3400\left[\frac{2550\times 0.71\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =3400\left[\frac{1810.5}{16286.5}\right]\\ =378\end{array}$

Substitute $3400$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{1-4}$.

$\begin{array}{c}{T}_{1-4}=3400\left[\frac{4400\times 0.97\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =3400\left[\frac{4268}{16286.5}\right]\\ =891\end{array}$

Trip distribution for district $2$ is calculated as

Substitute $6150$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{2-1}$.

$\begin{array}{c}{T}_{2-1}=6150\left[\frac{2800\times 1.51\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =6150\left[\frac{4228}{16286.5}\right]\\ =1597\end{array}$

Substitute $6150$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{2-2}$.

$\begin{array}{c}{T}_{2-2}=6150\left[\frac{6500\times 0.92\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =6150\left[\frac{5980}{16286.5}\right]\\ =2258\end{array}$

Substitute $6150$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{2-3}$.

$\begin{array}{c}{T}_{2-3}=6150\left[\frac{2550\times 0.71\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =6150\left[\frac{1810.5}{16286.5}\right]\\ =684\end{array}$

Substitute $6150$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{2-4}$.

$\begin{array}{c}{T}_{2-4}=6150\left[\frac{4400\times 0.97\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =6150\left[\frac{4268}{16286.5}\right]\\ =1612\end{array}$

Trip distribution for district $3$ is calculated as

Substitute $3900$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{3-1}$.

$\begin{array}{c}{T}_{3-1}=3900\left[\frac{2800\times 1.51\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =3900\left[\frac{4228}{16286.5}\right]\\ =1013\end{array}$

Substitute $3900$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{3-2}$.

$\begin{array}{c}{T}_{3-2}=3900\left[\frac{6500\times 0.92\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =3900\left[\frac{5980}{16286.5}\right]\\ =1432\end{array}$

Substitute $3900$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{3-3}$.

$\begin{array}{c}{T}_{3-3}=3900\left[\frac{2550\times 0.71\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =3900\left[\frac{1810.5}{16286.5}\right]\\ =434\end{array}$

Substitute $3900$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{3-4}$.

$\begin{array}{c}{T}_{3-4}=3900\left[\frac{4400\times 0.97\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =3900\left[\frac{4268}{16286.5}\right]\\ =1022\end{array}$

Trip distribution for district $4$ is calculated as

Substitute $2800$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{4-1}$.

$\begin{array}{c}{T}_{4-1}=2800\left[\frac{2800\times 1.51\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =2800\left[\frac{4228}{16286.5}\right]\\ =727\end{array}$

Substitute $2800$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{4-2}$.

$\begin{array}{c}{T}_{4-2}=2800\left[\frac{6500\times 0.92\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =2800\left[\frac{5980}{16286.5}\right]\\ =1028\end{array}$

Substitute $2800$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{4-3}$.

$\begin{array}{c}{T}_{4-3}=2800\left[\frac{2550\times 0.71\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =2800\left[\frac{1810.5}{16286.5}\right]\\ =311\end{array}$

Substitute $2800$ for $P_i$, $2800$, $6500$, $2550$, $4400$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{4-4}$.

$\begin{array}{c}{T}_{4-4}=2800\left[\frac{4400\times 0.97\times 1}{2800\times 1.51\times 1+6500\times 0.92\times 1+2550\times 0.71\times 1+4400\times 0.97\times 1}\right]\\ =2800\left[\frac{4268}{16286.5}\right]\\ =734\end{array}$

The table for the above calculated values is shown below.

District	$1$	$2$	$3$	$4$	Computed P	Given P
$1$	$883$	$1248$	$378$	$891$	$3400$	$3400$
$2$	$1597$	$2258$	$684$	$1612$	$6151$	$3400$
$3$	$1013$	$1432$	$434$	$1022$	$3901$	$3900$
$4$	$727$	$1028$	$311$	$734$	$2800$	$2800$
Computed A	$189$	$5966$	$1807$	$4263$	$16252$	$16250$
Given A	$2800$	$6500$	$2550$	$4400$	$16250$

    Table (3)

The adjusted value for district $1$ is calculated as

Substitute $2800$ for $A_j$, $2800$ for $A_{j\left(k-1\right)}$, and $4220$ for $C_{j\left(k-1\right)}$ in equation (I).

$\begin{array}{c}{A}_{jk}=\frac{2800}{4220}\times 2800\\ =1858\end{array}$

The adjusted value for district $2$ is calculated as

Substitute $6500$ for $A_j$, $6500$ for $A_{j\left(k-1\right)}$, and $5966$ for $C_{j\left(k-1\right)}$ in equation (I).

$\begin{array}{c}{A}_{jk}=\frac{6500}{5966}\times 6500\\ =7082\end{array}$

The adjusted value for district $3$ is calculated as

Substitute $2550$ for $A_j$, $2550$ for $A_{j\left(k-1\right)}$, and $1807$ for $C_{j\left(k-1\right)}$ in equation (I).

$\begin{array}{c}{A}_{jk}=\frac{2550}{1807}\times 2550\\ =3599\end{array}$

The adjusted value for district $4$ is calculated as

Substitute $4400$ for $A_j$, $4400$ for $A_{j\left(k-1\right)}$, and $4263$ for $C_{j\left(k-1\right)}$ in equation (I).

$\begin{array}{c}{A}_{jk}=\frac{4400}{4263}\times 4400\\ =4541\end{array}$

Carry out second iteration for each district with adjusted values.

Trip distribution for district $1$ is calculated as

Substitute $3400$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{1-1}$.

$\begin{array}{c}{T}_{1-1}=3400\left[\frac{1858\times 1.51\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =3400\left[\frac{2805.58}{16290.28}\right]\\ =585\end{array}$

Substitute $3400$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{1-2}$.

$\begin{array}{c}{T}_{1-2}=3400\left[\frac{7092\times 0.92\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =3400\left[\frac{6524.64}{16290.28}\right]\\ =1362\end{array}$

Substitute $3400$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{1-3}$.

$\begin{array}{c}{T}_{1-3}=3400\left[\frac{3599\times 0.71\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =3400\left[\frac{2555.29}{16290.28}\right]\\ =533\end{array}$

Substitute $3400$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{1-4}$.

$\begin{array}{c}{T}_{1-4}=3400\left[\frac{4541\times 0.9\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =3400\left[\frac{4404.77}{16290.28}\right]\\ =919\end{array}$

Trip distribution for district $2$ is calculated as

Substitute $6150$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{2-1}$.

$\begin{array}{c}{T}_{2-1}=6150\left[\frac{1858\times 1.51\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =6150\left[\frac{2805.58}{16290.28}\right]\\ =1059\end{array}$

Substitute $6150$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{2-2}$.

$\begin{array}{c}{T}_{2-2}=6150\left[\frac{7092\times 0.92\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =6150\left[\frac{6524.64}{16290.28}\right]\\ =2463\end{array}$

Substitute $6150$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{2-3}$.

$\begin{array}{c}{T}_{2-3}=6150\left[\frac{3599\times 0.71\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =6150\left[\frac{2555.29}{16290.28}\right]\\ =965\end{array}$

Substitute $6150$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{2-4}$.

$\begin{array}{c}{T}_{2-4}=6150\left[\frac{4541\times 0.9\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =6150\left[\frac{4404.77}{16290.28}\right]\\ =1663\end{array}$

Trip distribution for district $3$ is calculated as

Substitute $3900$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{3-1}$.

$\begin{array}{c}{T}_{3-1}=3900\left[\frac{1858\times 1.51\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =3900\left[\frac{2805.58}{16290.28}\right]\\ =671\end{array}$

Substitute $3900$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{3-2}$.

$\begin{array}{c}{T}_{3-2}=3900\left[\frac{7092\times 0.92\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =3900\left[\frac{6524.64}{16290.28}\right]\\ =1562\end{array}$

Substitute $3900$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{3-3}$.

$\begin{array}{c}{T}_{3-3}=3900\left[\frac{3599\times 0.71\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =3900\left[\frac{2555.29}{16290.28}\right]\\ =612\end{array}$

Substitute $3900$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{3-4}$.

$\begin{array}{c}{T}_{3-4}=3900\left[\frac{4541\times 0.9\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =3900\left[\frac{4404.77}{16290.28}\right]\\ =1055\end{array}$

Trip distribution for district $4$ is calculated as

Substitute $2800$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{4-1}$.

$\begin{array}{c}{T}_{4-1}=2800\left[\frac{1858\times 1.51\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =2800\left[\frac{2805.58}{16290.28}\right]\\ =482\end{array}$

Substitute $2800$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{4-2}$.

$\begin{array}{c}{T}_{4-2}=2800\left[\frac{7092\times 0.92\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =2800\left[\frac{6524.64}{16290.28}\right]\\ =1122\end{array}$

Substitute $2800$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{4-3}$.

$\begin{array}{c}{T}_{4-3}=2800\left[\frac{3599\times 0.71\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =2800\left[\frac{2555.29}{16290.28}\right]\\ =439\end{array}$

Substitute $2800$ for $P_i$, $1858$, $7092$, $3599$, $4541$ for $A_{ij}$, $1.51$, $0.92$, $0.71$, $0.97$ for $F_{ij}$ and $1$ for $K_{ij}$ in equation (II) to calculate $T_{4-4}$.

$\begin{array}{c}{T}_{4-4}=2800\left[\frac{4541\times 0.9\times 1}{1858\times 1.51\times 1+7092\times 0.92\times 1+3599\times 0.71\times 1+4541\times 0.97\times 1}\right]\\ =2800\left[\frac{4404.77}{16290.28}\right]\\ =757\end{array}$

The table for the above calculated values is shown below.

District	$1$	$2$	$3$	$4$	Computed P	Given P
$1$	$585$	$1362$	$533$	$919$	$3399$	$3400$
$2$	$1059$	$2463$	$965$	$1663$	$6150$	$3400$
$3$	$671$	$1562$	$612$	$1055$	$3900$	$3900$
$4$	$482$	$1122$	$439$	$757$	$2800$	$2800$
Computed A	$2797$	$6509$	$2549$	$4394$	$16249$	$16250$
Given A	$2800$	$6500$	$2550$	$4400$	$16250$

    Table (4)

Conclusion:

Therefore, the trip distribution calculation using the gravity model for two iterations is shown above.