Analysis of fire protection efficiency in the United States: a two-stage DEA-based approach

Abstract

Fire-related hazards and incidents are a very common phenomenon that affects human society heavily, so many organisations, over a long period of time, have made efforts to mitigate fires and the caused damages. It is widely acknowledged that an evaluation of fire protection performance is critical for such efforts. In this paper, we propose a new data envelopment analysis-based approach for fire protection efficiency analysis in the United States at the state level. For this purpose, the fire protection system is generalised as an innovative two-stage network process, in which the fire hazard defence subsystem in the first stage is followed by a fire incident fighting subsystem in the second stage. Further, both intermediate outputs and final outputs are all undesirable measures. The fire protection expenditure, a kind of shared resources, is modelled under managerial disposability since it can be intentionally used to reduce fires and fire damages. Based on an empirical study of data from 2010 to 2014, we find that: (1) Texas, California, New York, Florida, and Illinois are the top five states prone to fire incidents. (2) The United States as a whole has an average fire protection efficiency of 0.590, implying relatively low fire protection performances at the state level. (3) Wyoming, Vermont, and Rhode Island are the top three most efficient states, whereas Iowa, Arkansas, and Pennsylvania are ranked the least efficient. Further, (4) the fire incident fighting efficiency is more likely to be higher than the fire hazard defence efficiency for most states. (5) Looking at larger scale by grouping all states into eight areas, the Far West has the highest fire protection efficiency, while Southeast and Plains areas have the lowest efficiency scores. Interestingly, (6) the results show also that a 1% increase in the fire protection expenditure by states per capita will result in a 2.6893% improvement in fire protection efficiency.

Appendix

1.1 Appendix A: Calculation process of model (5)

Substituting Eq. (4) into model (3), we have

$$\begin{aligned}&\mathrm{Max} \frac{\sum _{p=1}^q {\varphi _p z_{pd} } +u^{1}+\sum _{r=1}^s {u_r y_{rd} +u^{2}} }{\sum _{i=1}^m {v_i x_{id} } +\vartheta t_d +\sum _{p=1}^q {\varphi _p z_{pd} } } \nonumber \\&\mathrm{s.t.}\quad E_j^1 = {\frac{\sum _{p=1}^q {\varphi _p z_{pj} } +u^{1}}{\sum _{i=1}^m {v_i x_{ij} } +\vartheta \alpha t_j }} \le 1,j=1,\ldots ,n \nonumber \\&E_j^2 = {\frac{\sum _{r=1}^s {u_r y_{rj} } +u^{2}}{\sum _{p=1}^q {\varphi _p z_{pj} } +\vartheta \left( {1-\alpha } \right) t_j }} \le 1,j=1,\ldots ,n \nonumber \\&u_r ,v_i ,\vartheta ,\varphi _p \ge \varepsilon ,r=1,\ldots ,s;i=1,\ldots ,m;p=1,\ldots ,q;u^{1},u^{2}\, \text {for free.}\qquad \qquad \end{aligned}$$

(A1)

By using C–C transformation (Charnes and Cooper 1962), the above model (A1) is changed into the following model (A2).

$$\begin{aligned} \mathrm{Max}&\sum _{p=1}^q {\varphi _p^{\prime } z_{pd} } +u^{A}+\sum _{r=1}^s {u_r^{\prime } y_{rd} +u^{B}} \nonumber \\ \mathrm{s.t.}&\sum _{i=1}^m {v_i^{\prime } x_{id} } +\vartheta ^{{\prime }}t_d +\sum _{p=1}^q {\varphi _p^{\prime } z_{pd} } =1 \nonumber \\&\sum _{p=1}^q {\varphi _p^{\prime } z_{pj} } -\sum _{i=1}^m {v_i^{\prime } x_{ij} } -\vartheta ^{{\prime }}\alpha t_j +u^{A}\le 0,j=1,\ldots ,n \nonumber \\&\sum _{r=1}^s {u_r^{\prime } y_{rj} } -\sum _{p=1}^q {\varphi _p^{\prime } z_{pj} } -\vartheta ^{{\prime }}\left( {1-\alpha } \right) t_j +u^{B}\le 0,j=1,\ldots ,n \nonumber \\&u_r^{\prime } ,v_i^{\prime } ,\vartheta ^{{\prime }},\varphi _p^{\prime } \ge \varepsilon ,r=1,\ldots ,s;i=1,\ldots ,m;\nonumber \\&\quad p=1,\ldots ,q;u^{A},u^{B}\, \text {for free.} \end{aligned}$$

(A2)

The above model is still nonlinear, by replacing $\vartheta ^{{\prime }}\alpha =\sigma ^{{\prime }}$ we have the following linear mathematical programming,

$$\begin{aligned} E_d^{\mathrm{Add}*} = \mathrm{Max}&\sum _{p=1}^q {\varphi _p^{\prime } z_{pd} } +u^{A}+\sum _{r=1}^s {u_r^{\prime } y_{rd} +u^{B}} \nonumber \\ \mathrm{s.t.}&\sum _{i=1}^m {v_i^{\prime } x_{id} } +\vartheta ^{{\prime }}t_d +\sum _{p=1}^q {\varphi _p^{\prime } z_{pd} } =1 \nonumber \\&\sum _{p=1}^q {\varphi _p^{\prime } z_{pj} } +u^{A}-\sum _{i=1}^m {v_i^{\prime } x_{ij} } -\sigma ^{{\prime }}t_j \le 0,j=1,\ldots ,n \nonumber \\&\sum _{r=1}^s {u_r^{\prime } y_{rj} } +u^{B}-\sum _{p=1}^q {\varphi _p^{\prime } z_{pj} } -\left( {\vartheta ^{{\prime }}-\sigma ^{{\prime }}} \right) t_j \le 0,j=1,\ldots ,n \nonumber \\&u_r^{\prime } ,v_i^{\prime } ,\vartheta ^{{\prime }},\varphi _p^{\prime } \ge \varepsilon ,r=1,\ldots ,s;i=1,\ldots ,m;\nonumber \\&\quad p=1,\ldots ,q;u^{A},u^{B}\,\text {for free.} \end{aligned}$$

(5)

This completes the calculation process of model (5) derived from model (3) and (4). $\square$

1.2 Appendix B: Proof of Theorem 1.

Theorem 1 Model (11) will always be feasible if $t_d \le 1/{\left( {2R^{t}} \right) },z_{pd} \le 1/{\left( {2qR_p^z } \right) }$.

Proof

Consider $\hat{{w}}_i =\hat{{\delta }}=0\left( {i=1,\ldots ,m} \right) ,\hat{{\phi }}=1/{\left( {2t_d } \right) },\hat{{\nu }}_p =1/{\left( {2q\cdot z_{pd} } \right) }$ $\left( {p=1,\ldots ,q} \right)$ and $\hat{{\mu }}_r =R_r^y \left( {r=1,\ldots ,s} \right)$, then there would exist large enough $\hat{{u}}^{A}$ and $\hat{{u}}^{B}$ and according $\left( {\hat{{\zeta }}_1 ,\hat{{\zeta }}_2 } \right)$ that can make the following constraints satisfied.

$$\begin{aligned}&\sum _{i=1}^m {\hat{{w}}_i x_{ij} } -\hat{{\delta }}t_j +\sum _{p=1}^q {\hat{{\nu }}_p z_{pj} } +\hat{{u}}^{A}\ge 0,j=1,\ldots ,n \nonumber \\&\sum _{p=1}^q {\hat{{\nu }}_p z_{pj} } -\left( {\hat{{\phi }}-\hat{{\delta }}} \right) t_j +\sum _{r=1}^s {\hat{{\mu }}_r y_{rj} } +\hat{{u}}^{B}\ge 0,j=1,\ldots ,n \nonumber \\&\sum _{i=1}^m {\hat{{w}}_i x_{id} } +\sum _{p=1}^q {\hat{{\nu }}_p z_{pd} } +\hat{{u}}^{A}+\hat{{u}}^{B}-\hat{{\zeta }}_1 +\hat{{\zeta }}_2 =0. \end{aligned}$$

(B1)

Further, we have $\sum _{i=1}^m {\hat{{w}}_i x_{id} } +\hat{{\phi }}t_d +\sum _{p=1}^q {\hat{{\nu }}_p z_{pd} } =0+1/2+1/2=1$, and

$$\begin{aligned} \hat{{w}}_i= & {} 0\ge 0=\left( {\sum _{i=1}^m {\hat{{w}}_i x_{id} } +\hat{{\delta }}t_d } \right) R_i^x ,i=1,\ldots ,m \nonumber \\ \hat{{\phi }}= & {} 1/{\left( {2t_d } \right) }\ge 0=\left( {\sum _{i=1}^m {w_i x_{id} } +\delta t_d } \right) R^{t} \nonumber \\ \hat{{\phi }}= & {} 1/{\left( {2t_d } \right) }\ge R^{t}=\left( {\sum _{p=1}^q {\hat{{\nu }}_p z_{pd} } +\left( {\hat{{\phi }}-\hat{{\delta }}} \right) t_d } \right) R^{t} \nonumber \\ \hat{{\nu }}_p= & {} 1/{\left( {2q\cdot z_{pd} } \right) }\ge 0=\left( {\sum _{i=1}^m {\hat{{w}}_i x_{id} } +\hat{{\delta }}t_d } \right) R_p^z ,p=1,\ldots ,q \nonumber \\ \hat{{\nu }}_p= & {} 1/{\left( {2q\cdot z_{pd} } \right) }\ge R_p^z \ge \left( {\sum _{p=1}^q {\hat{{\nu }}_p z_{pd} } +\left( {\hat{{\phi }}-\hat{{\delta }}} \right) t_d } \right) R_p^z ,p=1,\ldots ,q \nonumber \\ \hat{{\mu }}_r= & {} R_r^y \ge \left( {\sum _{p=1}^q {\hat{{\nu }}_p z_{pd} } +\left( {\hat{{\phi }}-\hat{{\delta }}} \right) t_d } \right) R_r^y ,r=1,\ldots ,s. \end{aligned}$$

(B2)

Therefore, we can conclude that $\left( {\hat{{w}}_i ,\hat{{\delta }},\hat{{\phi }},\hat{{\nu }}_p ,\hat{{\mu }}_r ,\hat{{u}}^{A},\hat{{u}}^{B},\hat{{\zeta }}_1 ,\hat{{\zeta }}_2 } \right)$ can make all constraints in model (11) satisfied. Therefore, it is a feasible solution to model (11).    $\square$

1.3 Appendix C: Proof of Theorem 2.

Theorem 2 The objective function of model (11) is unit invariant.

Proof

Suppose $\left( {\hat{{w}}_i ,\hat{{\delta }},\hat{{\phi }},\hat{{\nu }}_p ,\hat{{\mu }}_r ,\hat{{u}}^{A},\hat{{u}}^{B},\hat{{\zeta }}_1 ,\hat{{\zeta }}_2 } \right)$ is an optimal solution to model (11), thus

$$\begin{aligned} S_d^*=\sum _{p=1}^q {\nu _p^*z_{pd} } -\phi ^{*}t_d +\sum _{r=1}^s {\mu _r^*y_{rd} } +U\zeta _1^*-L\zeta _2^*\end{aligned}$$

(C1)

Now without loss of generality, suppose that these input and output measures are multiply by a series of constants $\left( {\beta _i ,\chi ,\pi _p ,\psi _r } \right)$, respectively. It is easy to demonstrate that $\left( {{\hat{{w}}_i }/{\beta _i },{\hat{{\delta }}}/\chi ,{\hat{{\phi }}}/\chi ,{\hat{{\nu }}_p }/{\pi _p },{\hat{{\mu }}_r }/{\psi _r },\hat{{u}}^{A},\hat{{u}}^{B},\hat{{\zeta }}_1 ,\hat{{\zeta }}_2 } \right)$ is an optimal solution to model (11) when new data are used. Here we have

$$\begin{aligned} {\tilde{S}}_d^*= & {} \sum _{p=1}^q {\left( {{\hat{{\nu }}_p^*}/{\pi _p }} \right) \left( {\pi _p z_{pd} } \right) } -\left( {{\hat{{\phi }}^{*}}/\chi } \right) \chi t_d \nonumber \\&\quad +\sum _{r=1}^s {\left( {{\hat{{\mu }}_r^*}/{\psi _r }} \right) \left( {\psi _r y_{rd} } \right) } +U\zeta _1^*-L\zeta _2^*. \end{aligned}$$

(C2)

By simple algebraic calculation, we have ${\tilde{S}}_d^*=S_d^*$. This completes the proof. $\square$

1.4 Appendix D: Proof of Theorem 3.

Theorem 3 $\mathrm{DMU}_d \left( {d=1,\ldots ,n} \right)$ is overall efficient if and only if both subsystems are efficient.

Proof

Take the situation in which the first stage dominates the overall two-stage process for example. First, if $\mathrm{DMU}_d \left( {d=1,\ldots ,n} \right)$ is efficient, then the inefficiencies in the two stages can be minimised to be zero simultaneously. As a result, the inefficiency to each stage would also be zero. And $E_{1d}^{1*} =E_{2d}^{1*} =1$ would be held.

Next, if two subsystems are efficient with an efficiency score of unity, namely $E_{1d}^{1*} =E_{2d}^{1*} =1$, then $E_d^*=E_d^{1*} +E_d^{2*} -1=1$, which indicates that the whole system is efficient. $\square$