Chapter 12, Problem 14P

To determine

The trip distribution using the average growth factor method.

Answer to Problem 14P

The trip distribution has been developed using the average growth factor method.

Explanation of Solution

Given:

For Zone $\text{A}$,

Present total observed trip ends is $100$.

Zonal growth rate is $3.$

For Zone $\text{B}$,

Present total observed trip ends is $250$.

Zonal growth rate is $4$.

For Zone $\text{C}$,

Present total observed trip ends is $400$.

Zonal growth rate is $2$.

For Zone $\text{D}$,

Present total observed trip ends is $300$.

Zonal growth rate is $1$.

Formula used:

Estimated growth total is given by,

  $\text{Estimated growth total}=\left(\text{Present total}\right)\times \left(\text{Growth factor}\right)$ ...... (I)

The number of trips estimated from zone $i$ to zone $j$ is given by,

  ${T^{\prime }}_{\text{ij}}=T_{\text{ij}}\left(\frac{G_{\text{i}}+G_{\text{j}}}{2}\right)$ ...... (II)

Here, $T_{\text{ij}}{}^{\prime }$ is the number of trips estimated from zone $i$ to zone $j$, $T_{\text{ij}}$ is the observed zonal interchange between zone $i$ and zone $j$, $G_{\text{i}}$ is the growth factor of zone $i$ and $G_{\text{j}}$ is the growth factor of zone $j$.

The new growth factor is given as,

  $\text{New Growth Factor}=\frac{\text{Present total}}{\text{New Present total}}$ ...... (III)

Calculation:

The estimated growth total for zone $\text{A}$ is calculated as,

Substitute $100$ for $\text{Present total}$ and $3$ for $\text{Growth factor}$ in equation (I).

  $\begin{array}{c}\text{Estimated growth total}=100\times \text{3}\\ =\text{300}\end{array}$

The estimated growth total for zone $\text{B}$ is calculated as,

Substitute $250$ for $\text{Present total}$ and $4$ for $\text{Growth factor}$ in equation (I).

  $\begin{array}{c}\text{Estimated growth total}=250\times 4\\ =\text{1000}\end{array}$

The estimated growth total for zone $\text{C}$ is calculated as,

Substitute $400$ for $\text{Present total}$ and $2$ for $\text{Growth factor}$ in equation (I).

  $\begin{array}{c}\text{Estimated growth total}=400\times 2\\ =\text{800}\end{array}$

The estimated growth total for zone $\text{D}$ is calculated as,

Substitute $300$ for $\text{Present total}$ and $1$ for $\text{Growth factor}$ in equation (I).

  $\begin{array}{c}\text{Estimated growth total}=300\times 1\\ =\text{300}\end{array}$

The data of $\text{Estimated Growth Total}$, $\text{Present total}$ and $\text{Growth factor}$ for each zone is tabulated below.

Zones	$\text{Present total}$	$\text{Growth factor}$	$\text{Estimated Growth Total}$
$\text{A}$	$100$	$3$	$300$
$\text{B}$	$250$	$4$	$1000$
$\text{C}$	$400$	$2$	$800$
$\text{D}$	$300$	$1$	$300$

Table (1)

The number of trips estimated from zone $\text{A}$ to zone $\text{B}$ is calculated as,

Substitute $25$ for $T_{\text{ij}}$, $3$ for $G_{\text{A}}$ and $4$ for $G_{\text{B}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{AB}}=25\left(\frac{3+4}{2}\right)\\ =25\times 3.5\\ =87.5\end{array}$

The number of trips estimated from zone $\text{A}$ to zone $\text{C}$ is calculated as,

Substitute $50$ for $T_{\text{ij}}$, $3$ for $G_{\text{A}}$ and $2$ for $G_{\text{C}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{AC}}=50\left(\frac{3+2}{2}\right)\\ =50\times 2.5\\ =125\end{array}$

The number of trips estimated from zone $\text{A}$ to zone $\text{D}$ is calculated as,

Substitute $25$ for $T_{\text{ij}}$, $3$ for $G_{\text{A}}$ and $1$ for $G_{\text{D}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{AD}}=25\left(\frac{3+1}{2}\right)\\ =25\times 2\\ =50\end{array}$

The number of trips estimated from zone $\text{B}$ to zone $\text{A}$ is calculated as,

Substitute $25$ for $T_{\text{ij}}$, $3$ for $G_{\text{A}}$ and $4$ for $G_{\text{B}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{BA}}=25\left(\frac{3+4}{2}\right)\\ =25\times 3.5\\ =87.5\end{array}$

The number of trips estimated from zone $\text{B}$ to zone $\text{C}$ is calculated as,

Substitute $150$ for $T_{\text{ij}}$, $4$ for $G_{\text{B}}$ and $2$ for $G_{\text{C}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{BC}}=150\left(\frac{4+2}{2}\right)\\ =150\times 3\\ =450\end{array}$

The number of trips estimated from zone $\text{B}$ to zone $\text{D}$ is calculated as,

Substitute $75$ for $T_{\text{ij}}$, $4$ for $G_{\text{B}}$ and $1$ for $G_{\text{D}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{BD}}=75\left(\frac{4+1}{2}\right)\\ =75\times 2.5\\ =187.5\end{array}$

The number of trips estimated from zone $\text{C}$ to zone $\text{A}$ is calculated as,

Substitute $50$ for $T_{\text{ij}}$, $3$ for $G_{\text{A}}$ and $2$ for $G_{\text{C}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{CA}}=50\left(\frac{3+2}{2}\right)\\ =50\times 2.5\\ =125\end{array}$

The number of trips estimated from zone $\text{C}$ to zone $\text{B}$ is calculated as,

Substitute $150$ for $T_{\text{ij}}$, $2$ for $G_{\text{C}}$ and $4$ for $G_{\text{B}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{CB}}=150\left(\frac{2+4}{2}\right)\\ =150\times 3\\ =450\end{array}$

The number of trips estimated from zone $\text{C}$ to zone $\text{D}$ is calculated as,

Substitute $200$ for $T_{\text{ij}}$, $2$ for $G_{\text{C}}$ and $1$ for $G_{\text{D}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{CD}}=200\left(\frac{2+1}{2}\right)\\ =200\times 1.5\\ =300\end{array}$

The number of trips estimated from zone $\text{D}$ to zone $\text{A}$ is calculated as,

Substitute $25$ for $T_{\text{ij}}$, $3$ for $G_{\text{A}}$ and $1$ for $G_{\text{D}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{DA}}=25\left(\frac{3+1}{2}\right)\\ =25\times 2\\ =50\end{array}$

The number of trips estimated from zone $\text{D}$ to zone $\text{B}$ is calculated as,

Substitute $75$ for $T_{\text{ij}}$, $4$ for $G_{\text{B}}$ and $1$ for $G_{\text{D}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{DB}}=75\left(\frac{4+1}{2}\right)\\ =75\times 2.5\\ =187.5\end{array}$

The number of trips estimated from zone $\text{D}$ to zone $\text{C}$ is calculated as,

Substitute $200$ for $T_{\text{ij}}$, $2$ for $G_{\text{C}}$ and $1$ for $G_{\text{D}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{DC}}=200\left(\frac{2+1}{2}\right)\\ =200\times 1.5\\ =300\end{array}$

The calculated trips from zone $i$ to zone $j$ are tabulated below.

Zones	${T^{\prime }}_{\text{ij}}$
$\text{A-B}$	$87.50$
$\text{A-C}$	$125.00$
$\text{A-D}$	$50.00$
$\text{B-A}$	$87.50$
$\text{B-C}$	$450.00$
$\text{B-D}$	$187.50$
$\text{C-A}$	$125.00$
$\text{C-B}$	$450.00$
$\text{C-D}$	$300.00$
$\text{D-A}$	$50.00$
$\text{D-B}$	$187.50$
$\text{D-C}$	$300.00$

Table (2)

The $\text{New Present total}$ for zone $\text{A}$ will be,

  $\begin{array}{c}\text{New Present Total}=T_{\text{A-B}}+T_{\text{A-C}}+T_{\text{A-D}}\\ =87.5+125+50\\ =262.5\end{array}$

The $\text{New growth factor}$ for zone $\text{A}$ is calculated as,

Substitute $300$ for $\text{Present total}$ and $262.5$ for $\text{New Present total}$ in equation (III).

  $\begin{array}{c}\text{New Growth Factor}=\frac{\text{300}}{262.5}\\ =1.143\end{array}$

The $\text{New Present total}$ for zone $\text{B}$ will be,

  $\begin{array}{c}\text{New Present Total}=T_{\text{B-A}}+T_{\text{B-C}}+T_{\text{B-D}}\\ =87.5+450+187.5\\ =725\end{array}$

The $\text{New growth factor}$ for zone $\text{B}$ is calculated as,

Substitute $1000$ for $\text{Present total}$ and $725$ for $\text{New Present total}$ in equation (III).

  $\begin{array}{c}\text{New Growth Factor}=\frac{\text{1000}}{725}\\ =1.379\end{array}$

The $\text{New Present total}$ for zone $\text{C}$ will be,

  $\begin{array}{c}\text{New Present Total}=T_{\text{C-A}}+T_{\text{C-B}}+T_{\text{C-D}}\\ =125+450+300\\ =875\end{array}$

The $\text{New growth factor}$ for zone $\text{C}$ is calculated as,

Substitute $800$ for $\text{Present total}$ and $875$ for $\text{New Present total}$ in equation (III).

  $\begin{array}{c}\text{New Growth Factor}=\frac{\text{800}}{875}\\ =0.914\end{array}$

The $\text{New Present total}$ for zone $\text{D}$ will be,

  $\begin{array}{c}\text{New Present Total}=T_{\text{D-A}}+T_{\text{D-B}}+T_{\text{D-C}}\\ =50+187.5+300\\ =437.5\end{array}$

The $\text{New growth factor}$ for zone $\text{D}$ is calculated as,

Substitute $300$ for $\text{Present total}$ and $437.5$ for $\text{New Present total}$ in equation (III).

  $\begin{array}{c}\text{New Growth Factor}=\frac{\text{300}}{437.5}\\ =0.686\end{array}$

The remaining movement between zones, new present total and new growth factor for zone s $\text{A,}\text{B,}\text{C}$ and $\text{D}$ are tabulated below.

Zones	New Present total	New Growth factor
$\text{A-B}$	$262.5$	$1.143$
$\text{A-C}$
$\text{A-D}$
$\text{B-A}$	$725$	$1.379$
$\text{B-C}$
$\text{B-D}$
$\text{C-A}$	$875$	$0.914$
$\text{C-B}$
$\text{C-D}$
$\text{D-A}$	$437.5$	$0.686$
$\text{D-B}$
$\text{D-C}$

Table (3)

The estimated growth total for zone $\text{A}$ is calculated as,

Substitute $262.5$ for $\text{Present total}$ and $1.143$ for $\text{Growth factor}$ in equation (I).

  $\begin{array}{c}\text{Estimated growth total}=262.5\times 1.143\\ =\text{300}\end{array}$

The estimated growth total for zone $\text{B}$ is calculated as,

Substitute $725$ for $\text{Present total}$ and $1.379$ for $\text{Growth factor}$ in equation (I).

  $\begin{array}{c}\text{Estimated growth total}=725\times 1.379\\ =\text{1000}\end{array}$

The estimated growth total for zone $\text{C}$ is calculated as,

Substitute $875$ for $\text{Present total}$ and $0.914$ for $\text{Growth factor}$ in equation (I).

  $\begin{array}{c}\text{Estimated growth total}=875\times 0.914\\ =\text{800}\end{array}$

The estimated growth total for zone $\text{D}$ is calculated as,

Substitute $437.5$ for $\text{Present total}$ and $0.686$ for $\text{Growth factor}$ in equation (I).

  $\begin{array}{c}\text{Estimated growth total}=437.5\times 0.686\\ =\text{300}\end{array}$

The data of $\text{Estimated Growth Total}$, $\text{Present total}$ and $\text{Growth factor}$ for each zone is tabulated below.

Zones	$\text{Present total}$	$\text{Growth factor}$	$\text{Estimated Growth Total}$
$\text{A}$	$262.5$	$1.143$	$300$
$\text{B}$	$725$	$1.379$	$1000$
$\text{C}$	$875$	$0.914$	$800$
$\text{D}$	$437.5$	$0.686$	$300$

Table (4)

The number of trips estimated from zone $\text{A}$ to zone $\text{B}$ is calculated as,

Substitute $87.5$ for $T_{\text{ij}}$, $1.143$ for $G_{\text{A}}$ and $1.379$ for $G_{\text{B}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{AB}}=87.5\left(\frac{1.143+1.379}{2}\right)\\ =87.5\times 1.261\\ =110.34\end{array}$

The number of trips estimated from zone $\text{A}$ to zone $\text{C}$ is calculated as,

Substitute $125$ for $T_{\text{ij}}$, $1.143$ for $G_{\text{A}}$ and $0.914$ for $G_{\text{C}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{AC}}=125\left(\frac{1.143+0.914}{2}\right)\\ =125\times 1.0285\\ =128.57\end{array}$

The number of trips estimated from zone $\text{A}$ to zone $\text{D}$ is calculated as,

Substitute $50$ for $T_{\text{ij}}$, $1.143$ for $G_{\text{A}}$ and $0.686$ for $G_{\text{D}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{AD}}=50\left(\frac{1.143+0.686}{2}\right)\\ =50\times 0.9145\\ =45.72\end{array}$

The number of trips estimated from zone $\text{B}$ to zone $\text{A}$ is calculated as,

Substitute $87.5$ for $T_{\text{ij}}$, $1.143$ for $G_{\text{A}}$ and $1.379$ for $G_{\text{B}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{BA}}=87.5\left(\frac{1.143+1.379}{2}\right)\\ =87.5\times 1.261\\ =110.34\end{array}$

The number of trips estimated from zone $\text{B}$ to zone $\text{C}$ is calculated as,

Substitute $450$ for $T_{\text{ij}}$, $1.379$ for $G_{\text{B}}$ and $0.914$ for $G_{\text{C}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{BC}}=450\left(\frac{1.379+0.914}{2}\right)\\ =450\times 1.1465\\ =515.93\end{array}$

The number of trips estimated from zone $\text{B}$ to zone $\text{D}$ is calculated as,

Substitute $187.5$ for $T_{\text{ij}}$, $1.379$ for $G_{\text{B}}$ and $0.686$ for $G_{\text{D}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{BD}}=187.5\left(\frac{1.379+0.686}{2}\right)\\ =187.5\times 1.0325\\ =193.59\end{array}$

The number of trips estimated from zone $\text{C}$ to zone $\text{A}$ is calculated as,

Substitute $125$ for $T_{\text{ij}}$, $1.143$ for $G_{\text{A}}$ and $0.914$ for $G_{\text{C}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{CA}}=125\left(\frac{1.143+0.914}{2}\right)\\ =125\times 1.0285\\ =128.57\end{array}$

The number of trips estimated from zone $\text{C}$ to zone $\text{B}$ is calculated as,

Substitute $450$ for $T_{\text{ij}}$, $1.379$ for $G_{\text{B}}$ and $0.914$ for $G_{\text{C}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{CB}}=450\left(\frac{1.379+0.914}{2}\right)\\ =450\times 1.1465\\ =515.93\end{array}$

The number of trips estimated from zone $\text{C}$ to zone $\text{D}$ is calculated as,

Substitute $300$ for $T_{\text{ij}}$, $0.914$ for $G_{\text{C}}$ and $0.686$ for $G_{\text{D}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{CD}}=300\left(\frac{0.914+0.686}{2}\right)\\ =300\times 0.8\\ =240\end{array}$

The number of trips estimated from zone $\text{D}$ to zone $\text{A}$ is calculated as,

Substitute $50$ for $T_{\text{ij}}$, $1.143$ for $G_{\text{A}}$ and $0.686$ for $G_{\text{D}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{DA}}=50\left(\frac{1.143+0.686}{2}\right)\\ =50\times 0.9145\\ =45.72\end{array}$

The number of trips estimated from zone $\text{D}$ to zone $\text{B}$ is calculated as,

Substitute $187.5$ for $T_{\text{ij}}$, $1.379$ for $G_{\text{B}}$ and $0.686$ for $G_{\text{D}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{DB}}=187.5\left(\frac{1.379+0.686}{2}\right)\\ =187.5\times 1.0325\\ =193.59\end{array}$

The number of trips estimated from zone $\text{D}$ to zone $\text{C}$ is calculated as,

Substitute $300$ for $T_{\text{ij}}$, $0.914$ for $G_{\text{C}}$ and $0.686$ for $G_{\text{D}}$ in equation (II).

  $\begin{array}{c}{T^{\prime }}_{\text{DC}}=300\left(\frac{0.914+0.686}{2}\right)\\ =300\times 0.8\\ =240\end{array}$

The calculated trips from zone $i$ to zone $j$ are tabulated below.

Zones	${T^{\prime }}_{\text{ij}}$
$\text{A-B}$	$110.34$
$\text{A-C}$	$128.57$
$\text{A-D}$	$45.72$
$\text{B-A}$	$110.34$
$\text{B-C}$	$515.93$
$\text{B-D}$	$193.59$
$\text{C-A}$	$128.57$
$\text{C-B}$	$515.93$
$\text{C-D}$	$240$
$\text{D-A}$	$45.72$
$\text{D-B}$	$193.59$
$\text{D-C}$	$240$

Table (5)

The $\text{New Present total}$ for zone $\text{A}$ will be,

  $\begin{array}{c}\text{New Present Total}=T_{\text{A-B}}+T_{\text{A-C}}+T_{\text{A-D}}\\ =110.34+128.57+45.72\\ =284.63\end{array}$

The $\text{New growth factor}$ for zone $\text{A}$ is calculated as,

Substitute $300$ for $\text{Present total}$ and $284.63$ for $\text{New Present total}$ in equation (III).

  $\begin{array}{c}\text{New Growth Factor}=\frac{\text{300}}{284.63}\\ =1.054\end{array}$

The $\text{New Present total}$ for zone $\text{B}$ will be,

  $\begin{array}{c}\text{New Present Total}=T_{\text{B-A}}+T_{\text{B-C}}+T_{\text{B-D}}\\ =110.34+515.93+193.59\\ =819.86\end{array}$

The $\text{New growth factor}$ for zone $\text{B}$ is calculated as,

Substitute $1000$ for $\text{Present total}$ and $819.86$ for $\text{New Present total}$ in equation (III).

  $\begin{array}{c}\text{New Growth Factor}=\frac{\text{1000}}{819.86}\\ =1.220\end{array}$

The $\text{New Present total}$ for zone $\text{C}$ will be,

  $\begin{array}{c}\text{New Present Total}=T_{\text{C-A}}+T_{\text{C-B}}+T_{\text{C-D}}\\ =128.57+515.93+240\\ =884.5\end{array}$

The $\text{New growth factor}$ for zone $\text{C}$ is calculated as,

Substitute $800$ for $\text{Present total}$ and $884.5$ for $\text{New Present total}$ in equation (III).

  $\begin{array}{c}\text{New Growth Factor}=\frac{\text{800}}{884.5}\\ =0.905\end{array}$

The $\text{New Present total}$ for zone $\text{D}$ will be,

  $\begin{array}{c}\text{New Present Total}=T_{\text{D-A}}+T_{\text{D-B}}+T_{\text{D-C}}\\ =45.72+193.59+240\\ =479.31\end{array}$

The $\text{New growth factor}$ for zone $\text{D}$ is calculated as,

Substitute $300$ for $\text{Present total}$ and $479.31$ for $\text{New Present total}$ in equation (III).

  $\begin{array}{c}\text{New Growth Factor}=\frac{\text{300}}{479.31}\\ =0.626\end{array}$

The remaining movement between zones, new present total and new growth factor for zone $\text{A,}\text{B,}\text{C}$ and $\text{D}$ are tabulated below:

Zones	New Present total	New Growth factor
$\text{A-B}$	$284.63$	$1.054$
$\text{A-C}$
$\text{A-D}$
$\text{B-A}$	$819.86$	$1.220$
$\text{B-C}$
$\text{B-D}$
$\text{C-A}$	$884.5$	$0.905$
$\text{C-B}$
$\text{C-D}$
$\text{D-A}$	$479.31$	$0.626$
$\text{D-B}$
$\text{D-C}$

Table (6)

Conclusion:

The trip distribution has been done using the average growth factor method.