Efficiency measurement
In cases where there is an input and output, the efficiency score is simply obtained from the ratio of output to input. In occasions with more than input and output, it is required to assign weights to variables and to insert sum of weighted outputs to inputs. The weakness of this method refers to the disagreements between experts about values of assigned weights.
Due to some constraints and through solving mathematical programming, in DEA, as a nonparametric method, the optimum weights were assigned to inputs and outputs for every DMU. The assigned weights are obtained the maximum possible efficiency score of DMU. Having combined the weighted input and output variables, a virtual DMU is made that with respect to the input-output oriented approach, either uses minimum level of inputs whilst keeping outputs constant or produces maximum level of outputs whilst keeping inputs constant. The ratio of virtual DMU to considered DMU determines the efficiency score [
10]. Selecting input/output-oriented approach refers to the possibility of reducing input usage or increasing output production. For instance, to measure the efficiency of hospitals, the outputs are the treated patients and cannot be changed, arbitrary. While it is possible to reduce input usage, man powers (doctors, nurses). In this case, it is preferred to apply input-oriented DEA model. For traffic police, the police stations try to reduce crashes as much as possible with the available resources. Therefore, it is suitable to conduct output-oriented DEA model.
In addition to input/output-oriented approaches, there are two CCR (Charnes, Cooper and Rhodes) and BCC (Banker, Charnes and Cooper) models [
6,
9]. CCR model is based on constant return to scale (CRS) assumption. It means input augmentation produces output increment at the same ratio. The CCR-Output oriented (CCR-O) efficiency score is obtained by solving the below equations and constraints.
$$ {\displaystyle \begin{array}{l}\left(\mathrm{CCR}-{\mathrm{O}}_{\mathrm{o}}\right){\mathrm{max}}_{\upeta \kern0.1em \upmu}\kern2em \upeta \\ {}\kern0.5em \mathrm{subject}\ \mathrm{to}\ {\mathrm{x}}_{\mathrm{o}}-\mathrm{X}\upmu \ge 0\kern5em \\ {}\kern4.25em {\upeta \mathrm{y}}_{\mathrm{o}}-\mathrm{Y}\upmu \le 0\kern7.5em \\ {}\kern7em \upmu \ge 0\kern1.75em \end{array}} $$
(1)
Where (ɳ) is the efficiency score of CCR-O model, (μ) is the output and input weights, (o) is the number of DMU, (X) is the input variable, (Y) is the output variable, (xo) and (yo) are the input and output variables of under measurement DMU.
In BCC model, with respect to the variable return to scale assumption (VRS), input augmentation cannot increase the output as the same ratio. According to the scale size of DMU, the output increment could be more, less or equal to the input increment. This could be due to non-coordination between various sectors of DMU
s with big scale sizes or requiring minimum inputs to starting performance in DMU
s with small scale sizes. In this study the BCC-O model was applied to extract pure technical efficiency from scale efficiency [
6].
The BCC-O model is solving by eq. (
2).
$$ {\displaystyle \begin{array}{l}\left(\mathrm{BCC}-{\mathrm{O}}_{\mathrm{o}}\right){\mathrm{max}}_{\upeta \mathrm{B},\uplambda}{\upeta}_{\mathrm{B}}\\ {}\kern0.5em \mathrm{subject}\ \mathrm{to}\kern0.75em \mathrm{X}\uplambda \le {\mathrm{x}}_{\mathrm{o}}\\ {}\kern2.25em {\upeta}_{\mathrm{B}}{\mathrm{y}}_{\mathrm{o}}-\kern0.5em \mathrm{Y}\uplambda \le 0\kern4.5em \\ {}\kern4.75em \sum \uplambda =1\\ {}\kern5.75em \uplambda \ge 0\end{array}} $$
(2)
Where (ɳ
B) is the efficiency score of BCC-O model, (λ) is the output and input weights, (o) is the number of DMU, (X) and (Y) are the input and output variable of DMU
s, (x
o) and (y
o) are the input and output of the under measurement DMU [
10]. The goal of solving eq. (
2) is obtaining efficiency score aimed to increase output production as much as possible while keeping input values at the same level.
In eq. (
2), the outcomes are desirable. Since in this study the outcomes were non- desirable (the number of crashes caused damage, injury and death), therefore, the values of outcomes were inversed with subtracting from maximum value. To avoid zero value of DMU(s) with maximum value, one is added to all values [
1].
Subtracting (Yλ) from (ɳ
By
o) in eq. (
2) provides output slacks. It means, for inefficient DMUs with efficiency score less than one, it is possible to increase outputs to (ɳ
B) times while keeping inputs at the same level. This capability of DEA model could provide opportunity to set quantitative targets for every inefficient DMU. These targets are determined with benchmarking from peer efficient unit(s). Since transformed value of non-desirable outcomes have been used in this study; hence, instead of adding slacks to the outcomes of inefficient DMU
s as the target, they are subtracted.
One of the limitation of DEA method refers to deterministic property. Therefore, the effect of statistical noise on efficiency score is not accounted. In other words, obtained efficiency score may be due to measurement errors, or sampling variations. Bootstrap has been proposed as a method to consider sampling variations and stability of measured efficiency score [
22]. Since this study aimed to measure efficiency score of total Iran’s police stations, the sampling variations were not a problem. While measurement error remained as a limitation of DEA method in this study. To solve this problem and also to extract the effect of environmental factors on police efficiency (non-discretionary), a method should be adopted that treat both of them.
Using three-stage DEA model, the pure technical efficiency is obtained without environmental and random error effects [
10]. At the first stage crude efficiency score is measured using the original inputs and outputs. Next, after standardization, the slacks of stage one are regressed to environmental variables with applying Stochastic Frontier Analysis (SFA) model [
7]. See eq. (
3).
$$ \frac{{\mathrm{s}}_{\mathrm{rj}}^{+}}{{\mathrm{y}}_{\mathrm{rj}}}={\upbeta}_{\mathrm{ro}}+\sum_{\mathrm{k}=1}^{\mathrm{K}}{\upbeta}_{\mathrm{rk}}{\mathrm{lnZ}}_{\mathrm{k}\mathrm{j}}+{\mathrm{v}}_{\mathrm{rj}}+{\upupsilon}_{\mathrm{rj}}\kern2.75em \left(\mathrm{r}=1,\dots, \mathrm{s}\right) $$
(3)
Where
\( \frac{{\mathrm{s}}_{\mathrm{rj}}^{+}}{{\mathrm{y}}_{\mathrm{rj}}} \) is standardized slack (ratio of slack to output), β
ro is intercept of r
th equation, β
rk is the coefficient of environment variable, k is the number of environmental variable, Z is the value of environmental variable, v
rj is random error, υ
rj is technical inefficiency, r (
r = 1, 2,3, …, N) is the number of outputs, j (j = 1, 2,3, …, N) is the number of DMUs, k (k = 1, 2,3, …, N) is the number of environmental variables and o (o = 1, 2, 3, …, N) is the number of DMU under evaluation. In eq.
(3) standardized slack regresses to environmental variables, random error and technical inefficiency.
To adjust the effect of random error and environmental conditions on technical efficiency, the following formula is used to modify all outputs according to similar environmental and random error condition (Eq.
4) [
10].
$$ {y}_{rj}^a={y}_{rj}\left(1+{\upbeta}_{\mathrm{ro}}+\sum_{\mathrm{k}=1}^{\mathrm{K}}{\upbeta}_{\mathrm{rk}}{\mathrm{lnZ}}_{\mathrm{k}\mathrm{j}}+{\upnu}_{\mathrm{rj}}\right)={\mathrm{y}}_{\mathrm{rj}}\left(1+\frac{{\mathrm{s}}_{\mathrm{rj}}^{+}}{{\mathrm{y}}_{\mathrm{rj}}}-{\upupsilon}_{\mathrm{rj}}\right) $$
(4)
Where \( {\mathrm{y}}_{\mathrm{rj}}^{\mathrm{a}} \) is the adjusted output,, yrjis the original output, βro is intercept of rth equation, Z is the value of the environmental variable, ν
rj
is random error, \( \frac{{\mathrm{s}}_{\mathrm{rj}}^{+}}{{\mathrm{y}}_{\mathrm{rj}}} \) is standardized slack (ratio of slack to output), υrj is technical inefficiency of output, r (r = 1, 2,3, …, N) is the number of outputs, j (j = 1, 2,3, …, N) is the number of DMUs, k (k = 1, 2,3, …, N) is the number of environmental variables and o (o = 1, 2, 3, …, N) is the number of DMU under evaluation.
In stage three, the adjusted outputs were used in BBC-O model again. The Results of this stage showed pure technical efficiency.
In addition to exploring the most effective input variable on police efficiency, sensitivity analysis was also conducted through removing each variable in model and measuring efficiency score changes.
Study setting and applied variables
This study was conducted using RAHVAR data from March 20, 2013 to March 20, 2014, according to Iranian calendar. Unit of analysis (DMU) was the rural police stations of all country. Put briefly, RAHVAR is a vice chancellor of Iran’s police (NAJA) at national level. It has sub directories in every province which are divided into two directories. One directory is responsible for traffic supervision in urban area and the other is about rural area that perform its tasks in police stations placed alongside the main roads.
Selection of Input and variables were based on the review of related studies and availability of valid data. Those variables were collected in a review article [
21].
Input variables used in model were the number of police, patrol cars, police motorcycles, patrol teams, drivers penalized and punished, speed cameras and score of educational and cultural activity of police. The score of educational and cultural activity was derived according to annual evaluation of the send reports of provinces to RAHVAR. The assigned score was ranged 0–100.
The output variables were the number of crashes lead to damage, injury or death, separately. It is notable that if more than one outcome was occurred in a crash, the worst outcome would be considered. To exclude the effect of pre-hospital facilities on death rates, only the number of occurred deaths on crash scene was applied.
The environmental variables were in three main categories. They were the road infrastructure category; length of freeway/highway, length of one-way main roads, length of by-way roads, ratio of main roads to the total roads, the socioeconomic and demographic category; literacy rate, population of young people (aged 18–24), the number of household owning motorcycle, ratio of household owning car and traffic volume category; traffic count on main road (per minutes) in every traffic police jurisdiction.