A novel probabilistic formulation for locating and sizing emergency medical service stations

Abstract

The paper proposes a novel probabilistic model with chance constraints for locating and sizing emergency medical service stations. In this model, the chance constraints are approximated as second-order cone constraints to overcome computational difficulties for practical applications. The proposed approximations associated with different estimation accuracy of the stochastic nature are meaningful on a practical uncertainty environment. Then, the model is transformed into a conic quadratic mixed-integer program by employing a conic transformation. The resulting model can be efficiently addressed by a commercial optimization package. A special case is also considered and a class of valid inequalities is introduced to improve computational efficiency. Lastly, computational experiences on real data and randomly generated data are reported to illustrate the validity of the program.

Appendices

The proof of Theorem 1

Proof

$D_j=\sum _{i\in I}d_iX_{ij}$ and it follows probability distribution with mean $\mu ^T \bar{X}_j$ and variance $\bar{X}_j^T \varSigma \bar{X}_j$.

(1) For arbitrary random variable $d_i$, applying the following inequality (Popescu 2005)

$$\begin{aligned} \fancyscript{P}(D_j> N_j)\le \left\{ \begin{array}{ll} \frac{\bar{X}_j^T \varSigma \bar{X}_j}{\bar{X}_j^T\varSigma _j \bar{X}_j+(N_j-\mu ^T \bar{X}_j)^2}, &{}\quad if N_j-\mu ^T \bar{X}_j> 0,\\ 1,&{}\quad otherwise, \end{array}\right. \end{aligned}$$

we obtain

$$\begin{aligned} 1-\fancyscript{P}(D_j\le N_j)&\le \frac{\bar{X}_j^T \varSigma _j \bar{X}_j}{\bar{X}_j^T\varSigma _j \bar{X}_j+(N_j-\mu ^T \bar{X}_j)^2},\\ \fancyscript{P}(D_j\le N_j)&\ge 1-\frac{\bar{X}_j^T \varSigma _j \bar{X}_j}{\bar{X}_j^T\varSigma _j \bar{X}_j+(N_j-\mu ^T \bar{X}_j)^2}. \end{aligned}$$

Therefore,

$$\begin{aligned} 1-\frac{\bar{X}_j^T \varSigma _j \bar{X}_j}{\bar{X}_j^T\varSigma _j \bar{X}_j+(N_j-\mu ^T \bar{X}_j)^2}\ge \alpha \end{aligned}$$

is sufficient for constraint (9) to hold. The expression above can be rewritten as

$$\begin{aligned}&\alpha \bar{X}_j^T\varSigma _j \bar{X}_j\le (N_j-\mu ^T \bar{X}_j)^2 (1-\alpha ),\\&\mu ^T \bar{X}_j+\sqrt{\frac{\alpha }{1-\alpha }}\sqrt{\bar{X}_j^T \varSigma _j \bar{X}_j}\le N_j,\\&\mu ^T \bar{X}_j+\sqrt{\frac{\alpha }{1-\alpha }}\parallel \varSigma _j^{\frac{1}{2}} \bar{X}_j\parallel _2 \le N_j. \end{aligned}$$

(2) For symmetric random variables $d_i$, the following inequalities are hold (Popescu 2005):

$$\begin{aligned} \fancyscript{P}(D_j> N_j)\le \left\{ \begin{array}{l@{\quad }l} \frac{1}{2}min\left[ 1,\frac{\bar{X}_j^T\varSigma _j \bar{X}_j}{(N_j-\mu ^T \bar{X}_j)^2}\right] , &{}\quad if N_j-\mu ^T \bar{X}_j> 0,\\ 1,&{}\quad otherwise. \end{array}\right. \\ \end{aligned}$$

Because $\alpha \in [0.5, 1)$, the following inequalities are satisfied in order to hold constraint (9).

$$\begin{aligned} \fancyscript{P}(D_j\le N_j)\ge 1-\frac{1}{2}\frac{\bar{X}_j^T\varSigma _j \bar{X}_j}{(N_j-\mu ^T \bar{X}_j)^2}\ge \alpha .\\ \end{aligned}$$

Then,

$$\begin{aligned} \mu ^T\bar{X}_j+\sqrt{\frac{1}{2(1-\alpha )}}\parallel \varSigma ^{\frac{1}{2}} \bar{X}_j\parallel _2\le N_j. \end{aligned}$$

(3) For unimodal symmetric random variables $d_i$, the following inequalities are hold (Popescu 2005):

$$\begin{aligned} \fancyscript{P}(D_j> N_j)\le \left\{ \begin{array}{l@{\quad }l} \frac{1}{2}min\left[ 1,\frac{4}{9}\frac{\bar{X}_j^T\varSigma _j \bar{X}_j}{(N_j-\mu ^T \bar{X}_j)^2}\right] , &{}if N_j-\mu ^T \bar{X}_j> 0,\\ 1,&{}otherwise. \end{array}\right. \\ \end{aligned}$$

Using the same approach as that of the symmetric random variable, we can show the result for the unimodal symmetric random variables:

$$\begin{aligned} \mu ^T\bar{X}_j+\sqrt{\frac{2}{9(1-\alpha )}}\parallel \varSigma ^{\frac{1}{2}}\bar{X}_j \parallel _2\le N_j. \end{aligned}$$

Constraint (10) is a second-order cone constraint since $\parallel \varSigma _j^\frac{1}{2}\bar{X}_j\parallel _2$ is convex. $\square$

The proof of Property 1

Proof

The total number of emergency vehicles at EMS station $j$ satisfies the following inequality

$$\begin{aligned} \sum _{i\in I_j}X_{ij}(\mu _i+\hat{\alpha }\sigma _i)\le \sum _{i\in I_j}X_{ij}\sum _{j^{'}\in J_i}n_{ij^{'}}=\sum _{i\in I_j}n_{ij}=N_j. \end{aligned}$$

By Cauchy inequality we know that $cov(\xi \eta )\le [D(\xi )]^{\frac{1}{2}}[D(\eta )]^{\frac{1}{2}}$, then

$$\begin{aligned} \mu ^T \bar{X}_j+\hat{\alpha }\parallel \varSigma _j^{\frac{1}{2}}\bar{X}_j\parallel _2\le \mu ^T \bar{X}_j+\hat{\alpha }\sigma ^T\bar{X}_j \le N_j. \end{aligned}$$

$\square$

The computational results for three classes of the random MNCD

See Figs. 7, 8 and 9.

Fig. 7

Number of EMS stations and emergency vehicles and optimal objective value: $d_i$ is a arbitrary random variable

Fig. 8

Number of EMS stations and emergency vehicles and optimal objective value: $d_i$ is a symmetric random variable

Fig. 9

Number of EMS stations and emergency vehicles and optimal objective value: $d_i$ is an unimodal symmetric random variable