Chapter 5, Problem 23P

To determine

The expected reductions inthe number of crashes during the first three years after implementation of the countermeasures.

Answer to Problem 23P

The expected reductions in number of crashes during the first three years after implementation of the countermeasures are $25\text{crashes,}\text{26}\text{crashes}\text{and}27\text{crashes}$, respectively.

Explanation of Solution

Given:

The number of countermeasures is $4$.

The countermeasure factors $\left(\text{CMFs}\right)$ for the proposed four countermeasures are $0.25,0.30,0.17\text{and}0.28$.

The number of crashes per year at the identified intersection during the past three years are $28,30\text{and}31$.

The average annual daily traffic $\left(AADT\right)$ at the intersection for the last three years respectively the number of crashes per yearsare $8450,9150\text{and}9850$.

The average annual daily traffic $\left(AADT\right)$ at the intersection during the first implementation is $10,850$.

The estimated traffic growth rate per annum after implementation is $4\%$.

Formula used:

The crash reduction factor is given by,

  $CR=\left[\begin{array}{c}C{R}_1+C{R}_2\left(1-C{R}_1\right)+C{R}_3\left(1-C{R}_1\right)\left(1-C{R}_2\right)+\\ C{R}_4\left(1-C{R}_1\right)\left(1-C{R}_2\right)\left(1-C{R}_3\right)\end{array}\right]$ ...... (I)

Here, $C{R}_1$ is the reduction factor for the first countermeasure and $C{R}_2$ is the reduction factor for the second countermeasure, $C{R}_3$ is the reduction factor for the third countermeasure and $C{R}_4$ is the reduction factor for the fourth countermeasure,

The average number of crashes per year is given by,

  $N=\frac{\text{Crash}\text{rate}\text{for}\text{given}\text{numer}\text{of}\text{years}}{\text{Total}\text{number}\text{of}\text{years}}$ ...... (II)

The $ADT\text{before}\text{improvement}$ is given by,

  $N=\frac{\text{Annual}\text{crash}\text{rate}\text{for}\text{given}\text{numer}\text{of}\text{years}}{\text{Total}\text{number}\text{of}\text{years}}$ ...... (III)

The crashes prevented first year is given by,

  $\text{Crashes}\text{prevented}=N\times CR\times \left(\frac{ADT\text{after}\text{improvement}}{ADT\text{before}\text{improvement}}\right)$ ...... (IV)

Here, $N$ is the expected number of crashes if countermeasure is implemented and the traffic volume remains the same, $CR$ is the crash reduction factor and $AADT$ is the average annual daily traffic.

The crashes prevented second year is given by,

  $\text{Crashes}\text{prevented}=\left(1+0.04\right)N\times CR\times \left(\frac{ADT\text{after}\text{improvement}}{ADT\text{before}\text{improvement}}\right)$ ...... (V)

The crashes prevented third year is given by,

  $\text{Crashes}\text{prevented}={\left(1+0.04\right)}^{2}N\times CR\times \left(\frac{ADT\text{after}\text{improvement}}{ADT\text{before}\text{improvement}}\right)$ ...... (VI)

Calculation:

Substitute $0.25$ for $C{R}_1$, $0.30$ for $C{R}_2$, $0.17$ for $C{R}_3$ and $0.28$ for $C{R}_4$ in equation (I).

  $\begin{array}{c}CR=\left[\begin{array}{c}0.25+0.30\left(1-0.25\right)+0.17\left(1-0.25\right)\left(1-0.30\right)+\\ 0.28\left(1-0.25\right)\left(1-0.30\right)\left(1-0.17\right)\end{array}\right]\\ =0.686\\ \approx 0.69\end{array}$

Substitute $\left(28+30+31\right)$ for $\text{Crash}\text{rate}\text{for}\text{given}\text{numer}\text{of}\text{years}$ and $3$ for $\text{Total}\text{number}\text{of}\text{years}$ in equation (II).

  $\begin{array}{c}N=\frac{28+30+31}{3}\\ =29.70\end{array}$

Substitute $\left(8450+9150+9850\right)$ for $\text{Annual}\text{crash}\text{rate}\text{for}\text{given}\text{numer}\text{of}\text{years}$ and $3$ for $\text{Total}\text{number}\text{of}\text{years}$ in equation (III).

  $\begin{array}{c}N=\frac{8450+9150+9850}{3}\\ =9150\end{array}$

Substitute $29.70$ for $N$, $0.69$ for $CR$, $10,850$ for $ADT\text{after}\text{improvement}$ and $9,150$ for $ADT\text{before}\text{improvement}$ in equation (IV).

  $\begin{array}{c}\text{Crashes}\text{prevented}=\left(29.70\right)\left(0.69\right)\left(\frac{10,850}{9,150}\right)\\ =24.30\\ \approx 25\end{array}$

Substitute $29.70$ for $N$, $0.69$ for $CR$, $10,850$ for $ADT\text{after}\text{improvement}$ and $9,150$ for $ADT\text{before}\text{improvement}$ in equation (V).

  $\begin{array}{c}\text{Crashes}\text{prevented}=\left(1+0.04\right)\left(29.70\right)\left(0.69\right)\left(\frac{10,850}{9,150}\right)\\ =\left(1+0.04\right)\left(24.30\right)\\ =25.27\\ \approx 26\end{array}$

Substitute $29.70$ for $N$, $0.69$ for $CR$, $10,850$ for $ADT\text{after}\text{improvement}$ and $9,150$ for $ADT\text{before}\text{improvement}$ in equation (VI).

  $\begin{array}{c}\text{Crashes}\text{prevented}={\left(1+0.04\right)}^2\left(29.70\right)\left(0.69\right)\left(\frac{10,850}{9,150}\right)\\ ={\left(1+0.04\right)}^2\left(24.30\right)\\ =26.28\\ \approx 27\end{array}$

Conclusion:

Therefore, theexpected reductions in number of crashes during the first three years after implementation of the countermeasures are $25\text{crashes,}\text{26}\text{crashes}\text{and}27\text{crashes}$, respectively.