A multi-period maximal coverage model for locating simultaneous ground and air emergency medical services facilities

Abstract

In many cases where emergency medical services are required, ground and air ambulances cooperate in a shared activity to transfer the injured from the accident scene to treatment centers. The cooperation becomes necessary, particularly where the air ambulance can not land at the accident scene, and it sometimes seems essential to make use of transfer points during the cooperation. Given that the demand pattern might change during time, it does not seem logical to formulate the problem statically. Hence, in this paper, we have presented a multi-period maximal coverage location model, which simultaneously locates the transfer points, ground ambulance stations, air ambulance stations, and allocates ambulances to them. Also, the available ambulances are moved among the established stations if required subject to demand changes during the planning horizon. Coverage is provided based on the times it takes the ambulance to arrive at the accident scene and to transfer from the accident scene to the nearest treatment center. In this research, four methods of providing service to the injured and transferring them to treatment centers have been considered. In the presented model, the inaccessibility of ambulances has been considered in light of the notion of backup coverage. Given the complexity of the problem and the impossibility to solve it optimally at large-scale problems, a heuristic algorithm with the greedy approach has been presented for solving it. The obtained computational results demonstrate the efficiency of the proposed algorithm in solving different problems as compared to CPLEX.

Appendix: Linearization of constraint 16

As noted, constraint 16 is a non-linear equation that has led to the non-linearization of the model. In this appendix, the linearization of this constraint is given using some standard techniques.

$$ {\text{v}}_{{{\text{ah}}}}^{{\text{t}}} = {\text{S}}_{{{\text{ah}}}}^{{\text{t}}} {{x^{\prime}}}_{{\text{a }}}^{{\text{t}}} ,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{t}} \in {\uptau }, $$

$$ {\text{w}}_{{{\text{ahr}}}}^{{\text{t}}} = {\text{I}}_{{{\text{ahr}}}}^{{\text{t}}} {{x^{\prime}}}_{{\text{a }}}^{{\text{t}}} \;,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{r}} \in {\text{R}},{\text{t}} \in {\uptau }. $$

Auxiliary integer variable ${\text{v}}_{{{\text{ah}}}}^{{\text{t}}}$ was replaced with the multiplication of the two binary variable ${\text{S}}_{{{\text{ah}}}}^{{\text{t}}}$ and integer variable${{x^{\prime}}}_{{\text{a}}}^{{\text{t}}}$. Auxiliary integer variable ${\text{w}}_{{{\text{ahr}}}}^{{\text{t}}}$ is also replaced with the multiplication of the two binary variable ${\text{I}}_{{{\text{ahr}}}}^{{\text{t}}}$ and the integer variable${{x^{\prime}}}_{{\text{a }}}^{{\text{t}}}$. We also need to have the following constraints for the required conditions.

$$ {\text{v}}_{{{\text{ah}}}}^{{\text{t}}} \le {{x^{\prime}}}_{{\text{a }}}^{{\text{t}}} \;,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{t}} \in {\uptau }, $$

$$ {\text{v}}_{{{\text{ah}}}}^{{\text{t}}} \ge {{x^{\prime}}}_{{\text{a }}}^{{\text{t}}} - {2}({1} - {\text{S}}_{{{\text{ah}}}}^{{\text{t}}} ),\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{t}} \in {\uptau }, $$

$$ {\text{v}}_{{{\text{ah}}}}^{{\text{t}}} \le 2{\text{S}}_{{{\text{ah}}}}^{{\text{t}}} ,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{t}} \in {\uptau }, $$

$$ {\text{w}}_{{{\text{ahr}}}}^{{\text{t}}} \le {{x^{\prime}}}_{{\text{a }}}^{{\text{t}}} ,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{r}} \in {\text{R}},{\text{t}} \in {\uptau }, $$

$$ {\text{w}}_{{{\text{ahr}}}}^{{\text{t}}} \ge {{x^{\prime}}}_{{\text{a }}}^{{\text{t}}} - 2 \left( {{1} - {\text{I}}_{{{\text{ahr}}}}^{{\text{t}}} } \right),\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{r}} \in {\text{R}},{\text{t}} \in {\tau ,} $$

$$ {\text{w}}_{{{\text{ahr}}}}^{{\text{t}}} \le 2{\text{I}}_{{{\text{ahr}}}}^{{\text{t}}} \;,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{r}} \in {\text{R}},{\text{t}} \in {\uptau }. $$

After replacing the auxiliary variables, Eq. 16 changes as follows:

$$ \begin{gathered} \mathop \sum \limits_{{{\text{a}} \in {\text{O}}}} {\text{A}}_{{{\text{aj}}}}^{{\text{t}}} {{x^{\prime}}}_{{\text{a }}}^{{\text{t}}} + \mathop \sum \limits_{{{\text{a}} \in {\text{O}}}} \mathop \sum \limits_{{{\text{h}} \in {\text{H}}}} {\text{A}}_{{{\text{ahj}}}}^{{\text{t}}} {\text{v}}_{{{\text{ah}}}}^{{\text{t}}} \left( {\left| {{\text{A}}_{{{\text{aj}}}}^{{\text{t}}} {{x^{\prime}}}_{{\text{a }}}^{{\text{t}}} - 1} \right|} \right) \hfill \\ + \mathop \sum \limits_{{{\text{a}} \in {\text{O}}}} \mathop \sum \limits_{{{\text{h}} \in {\text{H}}}} \mathop \sum \limits_{{{\text{r}} \in {\text{R}}}} {\text{A}}_{{{\text{ahrj}}}}^{{\text{t}}} {\text{w}}_{{{\text{ahr}}}}^{{\text{t}}} \left( {\left| {{\text{A}}_{{{\text{aj}}}}^{{\text{t}}} {{x^{\prime}}}_{{\text{a }}}^{{\text{t}}} - 1} \right|} \right)\left( {\left| {{\text{A}}_{{{\text{ahj}}}}^{{\text{t}}} {\text{v}}_{{{\text{ah}}}}^{{\text{t}}} - 1} \right|} \right) \ge 2{\text{b}}_{{\text{j}}}^{{\text{t}}} - 2{\text{g}}_{{\text{j}}}^{{\text{t}}} \;,\forall {\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }. \hfill \\ \end{gathered} $$

In addition, the following auxiliary binary variables and constraints are introduced to linearize the absolute terms.

$$ \left| {{\text{A}}_{{{\text{aj}}}}^{{\text{t}}} {{x}}_{{\text{a }}}^{{\prime \text{t}}} - 1} \right| = {\overline{\text{x}}}_{{{\text{aj}}}}^{{\text{ t}}} + {\overline{\text{x}}}_{{{\text{aj}}}}^{{\prime \text{ t}}} \;,\forall {\text{a}} \in {\text{O}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }, $$

$$ \left| {{\text{A}}_{{{\text{ahj}}}}^{{\text{t}}} {\text{v}}_{{{\text{ah}}}}^{{\text{t}}} - 1} \right| = {\overline{\text{v}}}_{{{\text{ahj}}}}^{{\text{ t}}} + {\overline{\text{v}}} _{{{\text{ahj}}}}^{\prime {\text{t}}} ,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }, $$

$$ {\text{A}}_{{{\text{aj}}}}^{{\text{t}}} {{x}}_{{\text{a }}}^{\prime {\text{t}}} - 1 = {\overline{\text{x}}}_{{{\text{aj}}}}^{{\text{ t}}} - {\overline{\text{x}}}_{{{\text{aj}}}}^{\prime {\text{ t}}} ,\forall {\text{a}} \in {\text{O}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }, $$

$$ {\overline{\text{x}}}_{{{\text{aj}}}}^{{\text{ t}}} + {\overline{\text{x}}}_{{{\text{aj}}}}^{\prime{\text{ t}}} \le 1,\forall {\text{a}} \in {\text{O}},{\text{j}} \in {\text{D}},{\text{t}} \in {\tau ,} $$

$$ {\text{A}}_{{{\text{ahj}}}}^{{\text{t}}} {\text{v}}_{{{\text{ah}}}}^{{\text{t}}} - 1 = {\overline{\text{v}}}_{{{\text{ahj}}}}^{{\text{ t}}} - {\overline{\text{v}}} _{{{\text{ahj}}}}^{\prime{\text{t}}} ,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }, $$

$$ {\overline{\text{v}}}_{{{\text{ahj}}}}^{{\text{ t}}} + {\overline{\text{v}}} _{{{\text{ahj}}}}^{\prime{\text{t}}} \le 1,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }. $$

After replacing the auxiliary variables, the equation changes as follows:

$$ \begin{gathered} \mathop \sum \limits_{{{\text{a}} \in {\text{O}}}} {\text{A}}_{{{\text{aj}}}}^{{\text{t}}} {{x}}_{{\text{a }}}^{\prime{\text{t}}} + \mathop \sum \limits_{{{\text{a}} \in {\text{O}}}} \mathop \sum \limits_{{{\text{h}} \in {\text{H}}}} {\text{A}}_{{{\text{ahj}}}}^{{\text{t}}} ({\text{v}}_{{{\text{ah}}}}^{{\text{t}}} )\left( {{\overline{\text{x}}}_{{{\text{aj}}}}^{{\text{ t}}} + {\overline{\text{x}}}_{{{\text{aj}}}}^{\prime{\text{ t}}} } \right) \hfill \\ + \mathop \sum \limits_{{{\text{a}} \in {\text{O}}}} \mathop \sum \limits_{{{\text{h}} \in {\text{H}}}} \mathop \sum \limits_{{{\text{r}} \in {\text{R}}}} {\text{A}}_{{{\text{ahrj}}}}^{{\text{t}}} \left( {{\text{w}}_{{{\text{ahr}}}}^{{\text{t}}} } \right)\left( {{\overline{\text{x}}}_{{{\text{aj}}}}^{{\text{ t}}} + {\overline{\text{x}}}_{{{\text{aj}}}}^{\prime{\text{ t}}} } \right)\left( {{\overline{\text{v}}}_{{{\text{ahj}}}}^{{\text{ t}}} + {\overline{\text{v}}} _{{{\text{ahj}}}}^{\prime{\text{t}}} } \right) \ge 2{\text{b}}_{{\text{j}}}^{{\text{t}}} - 2{\text{g}}_{{\text{j}}}^{{\text{t}}} ,\forall {\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }. \hfill \\ \end{gathered} $$

In the following, the auxiliary integer variable ${{\varphi }}_{{{\text{ahj}}}}^{{\text{t}}}$ is defined to linearize the multiplication of integer variable $({\text{v}}_{{{\text{ah}}}}^{{\text{t}}} )$ and binary phrase$\left( {{\overline{\text{x}}}_{{{\text{aj}}}}^{{\text{ t}}} + {\overline{\text{x}}}_{{{\text{aj}}}}^{\prime{\text{ t}}} } \right)$. The auxiliary binary variable ${\overline{{\varphi }}}_{{{\text{ahj}}}}^{{\text{ t}}}$ is also used to linearize the multiplication of two terms $\left( {{\overline{\text{v}}}_{{{\text{ahj}}}}^{{\text{ t}}} + {\overline{\text{v}}} _{{{\text{ahj}}}}^{\prime{\text{t}}} } \right)$ and $\left( {{\overline{\text{x}}}_{{{\text{aj}}}}^{{\text{ t}}} + {\overline{\text{x}}}_{{{\text{aj}}}}^{\prime{\text{ t}}} } \right)$ with binary values.

$$ {{\varphi }}_{{{\text{ahj}}}}^{{\text{t}}} = ({\text{v}}_{{{\text{ah}}}}^{{\text{t}}} )\left( {{\overline{\text{x}}}_{{{\text{aj}}}}^{{\text{ t}}} + {\overline{\text{x}}}_{{{\text{aj}}}}^{\prime{\text{ t}}} } \right),\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{t}} \in {\tau ,} $$

$$ {\overline{{\varphi }}}_{{{\text{ahj}}}}^{{\text{ t}}} = \left( {{\overline{\text{x}}}_{{{\text{aj}}}}^{{\text{ t}}} + {\overline{\text{x}}}_{{{\text{aj}}}}^{\prime{\text{ t}}} } \right)\left( {{\overline{\text{v}}}_{{{\text{ahj}}}}^{{\text{ t}}} + {\overline{\text{v}}}_{{{\text{ahj}}}}^{\prime{\text{t}}} } \right),\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }, $$

$$ {{\varphi }}_{{{\text{ahj}}}}^{{\text{t}}} \le {\text{v}}_{{{\text{ah}}}}^{{\text{t}}} \;,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }, $$

$$ {{\varphi }}_{{{\text{ahj}}}}^{{\text{t}}} \ge {\text{v}}_{{{\text{ah}}}}^{{\text{t}}} - 2\left( {1 - \left( {{\overline{\text{x}}}_{{{\text{aj}}}}^{{\text{ t}}} + {\overline{\text{x}}}_{{{\text{aj}}}}^{\prime{\text{ t}}} } \right)} \right),\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{t}} \in {\tau ,} $$

$$ {{\varphi }}_{{{\text{ahj}}}}^{{\text{t}}} \le 2\left( {{\overline{\text{x}}}_{{{\text{aj}}}}^{{\text{ t}}} + {\overline{\text{x}}}_{{{\text{aj}}}}^{\prime{\text{ t}}} } \right),\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }, $$

$$ {\overline{{\varphi }}}_{{{\text{ahj}}}}^{{\text{ t}}} \le {\overline{\text{x}}}_{{{\text{aj}}}}^{{\text{ t}}} + {\overline{\text{x}}}_{{{\text{aj}}}}^{\prime{\text{ t}}} \;,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }, $$

$$ {\overline{{\varphi }}}_{{{\text{ahj}}}}^{{\text{ t}}} \le {\overline{\text{v}}}_{{{\text{ahj}}}}^{{\text{ t}}} + {\overline{\text{v}}} _{{{\text{ahj}}}}^{\prime{\text{t}}} ,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }, $$

$$ {\overline{{\varphi }}}_{{{\text{ahj}}}}^{{\text{ t}}} \ge (\left( {{\overline{\text{x}}}_{{{\text{aj}}}}^{{\text{ t}}} + {\overline{\text{x}}}_{{{\text{aj}}}}^{\prime{\text{ t}}} } \right) + \left( {{\overline{\text{v}}}_{{{\text{ahj}}}}^{{\text{ t}}} + {\overline{\text{v}}}_{{{\text{ahj}}}}^{\prime{\text{t}}} } \right)) - 1,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }. $$

Now, the constraint changes as follows by replacing the auxiliary variables:

$$ \mathop \sum \limits_{{{\text{a}} \in {\text{O}}}} {\text{A}}_{{{\text{aj}}}}^{{\text{t}}} {{x}}_{{\text{a }}}^{\prime{\text{t}}} + \mathop \sum \limits_{{{\text{a}} \in {\text{O}}}} \mathop \sum \limits_{{{\text{h}} \in {\text{H}}}} {\text{A}}_{{{\text{ahj}}}}^{{\text{t}}} {{\varphi }}_{{{\text{ahj}}}}^{{\text{t}}} + \mathop \sum \limits_{{{\text{a}} \in {\text{O}}}} \mathop \sum \limits_{{{\text{h}} \in {\text{H}}}} \mathop \sum \limits_{{{\text{r}} \in {\text{R}}}} {\text{A}}_{{{\text{ahrj}}}}^{{\text{t}}} \left( {{\text{w}}_{{{\text{ahr}}}}^{{\text{t}}} } \right)\left( {{\overline{{\varphi }}}_{{{\text{ahj}}}}^{{\text{ t}}} } \right) \ge 2{\text{b}}_{{\text{j}}}^{{\text{t}}} - 2{\text{g}}_{{\text{j}}}^{{\text{t}}} ,\forall {\text{j}} \in {\text{D}},\forall {\text{t}} \in {\uptau }{.} $$

The obtained constraint still have non-linear terms, multiplication integer variable $\left( {{\text{w}}_{{{\text{ahr}}}}^{{\text{t}}} } \right)$ and binary variable $\left( {{\overline{{\varphi }}}_{{{\text{ahj}}}}^{{\text{ t}}} } \right)$. Given the auxiliary integer variable ${\overline{\text{w}}}_{{{\text{ahrj}}}}^{{\text{t}}}$, this equation linearized using the below equations.

$$ {\overline{\text{w}}}_{{{\text{ahrj}}}}^{{\text{t}}} = \left( {{\text{w}}_{{{\text{ahr}}}}^{{\text{t}}} } \right)\left( {{\overline{{\varphi }}}_{{{\text{ahj}}}}^{{\text{ t}}} } \right),\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }, $$

$$ {\overline{\text{w}}}_{{{\text{ahrj}}}}^{{\text{t}}} \le {\text{w}}_{{{\text{ahr}}}}^{{\text{t}}} ,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{r}} \in {\text{R}},{\text{t}} \in {\uptau }, $$

$$ {\overline{\text{w}}}_{{{\text{ahrj}}}}^{{\text{t}}} \ge {\text{w}}_{{{\text{ahr}}}}^{{\text{t}}} - 2\left( {1 - {\overline{{\varphi }}}_{{{\text{ahj}}}}^{{\text{ t}}} } \right),\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{r}} \in {\text{R}},{\text{t}} \in {\uptau },{ } $$

$$ {\overline{\text{w}}}_{{{\text{ahrj}}}}^{{\text{t}}} \le 2{\overline{{\varphi }}}_{{{\text{ahj}}}}^{{\text{ t}}} ,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{r}} \in {\text{R}},{\text{t}} \in {\uptau }. $$

Finally, the linear constraint obtained is as follows:

$$ \mathop \sum \limits_{{{\text{a}} \in {\text{O}}}} {\text{A}}_{{{\text{aj}}}}^{{\text{t}}} {{x}}_{{\text{a }}}^{\prime{\text{t}}} + \mathop \sum \limits_{{{\text{a}} \in {\text{O}}}} \mathop \sum \limits_{{{\text{h}} \in {\text{H}}}} {\text{A}}_{{{\text{ahj}}}}^{{\text{t}}} {{\varphi }}_{{{\text{ahj}}}}^{{\text{t}}} + \mathop \sum \limits_{{{\text{a}} \in {\text{O}}}} \mathop \sum \limits_{{{\text{h}} \in {\text{H}}}} \mathop \sum \limits_{{{\text{r}} \in {\text{R}}}} {\text{A}}_{{{\text{ahrj}}}}^{{\text{t}}} \left( {{\overline{\text{w}}}_{{{\text{ahrj}}}}^{{\text{t}}} } \right) \ge 2{\text{b}}_{{\text{j}}}^{{\text{t}}} - 2{\text{g}}_{{\text{j}}}^{{\text{t}}} ,\forall {\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }. $$

All auxiliary variables used in the linearization process are as below:

$$ {\overline{\text{x}}}_{{{\text{aj}}}}^{{\text{ t}}} ,{\overline{\text{x}}}_{{{\text{aj}}}}^{\prime{\text{ t}}} \in \left\{ {0,1} \right\},\forall {\text{a}} \in {\text{O}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }, $$

$$ {\overline{\text{v}}}_{{{\text{ahj}}}}^{{\text{ t}}} , {\overline{\text{v}}} _{{{\text{ahj}}}}^{\prime{\text{t}}} \in \left\{ {0,1} \right\},\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }, $$

$$ {\overline{{\varphi }}}_{{{\text{ahj}}}}^{{\text{ t}}} \in \left\{ {0,1} \right\},\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{t}} \in {\uptau }, $$

$$ {\text{v}}_{{{\text{ah}}}}^{{\text{t}}} ,{\text{w}}_{{{\text{ahr}}}}^{{\text{t}}} ,{\overline{\text{w}}}_{{{\text{ahrj}}}}^{{\text{t}}} \;{\text{integer}}\;,\forall {\text{a}} \in {\text{O}},{\text{h}} \in {\text{H}},{\text{j}} \in {\text{D}},{\text{r}} \in {\text{R}},{\text{t}} \in {\uptau }. $$