A dynamic dispatching and routing model to plan/ re-plan logistics activities in response to an earthquake

Abstract

The unpredictable nature and devastating impact of earthquakes enforce governments of disaster-prone regions to provide practical response plans to minimize damage and losses resulting from earthquakes. Logistics management is one of the key issues that should be considered for an appropriate response, in particular, the planning of the transport of commodities required during response and the evacuation of injured people. This paper develops a dynamic model for dispatching and routing vehicles in response to an earthquake. We focus on the transport of both commodity towards affected areas and injured people to hospitals. The proposed model is capable of receiving updated information at any time and adjusting plans accordingly. Since speed is a key to a successful earthquake response, the model hierarchically minimizes the total time until arrival at a hospital for injured people, as well as the total lead-time to fulfill commodity needs. We designed experiments to investigate the effect of the network topology on earthquake response rapidity to improve the quality of earthquake response.

Appendix

Proposition

the total (weighted) waiting time of injured people during the planning horizon equals the total (weighted) number of injured person-periods not picked up from the affected area. Related to this, the total (weighted) lead-time for satisfaction of commodity needs during the planning horizon equals the summation of commodity need-period unsatisfied during the planning.

Proof:

to prove these equalities, consider a specific demand node $(m)$ and a specific wounded type $(h).$ In addition, define two variables $\mathrm{TWT}_{hm}^i$ and $\mathrm{TUW}_{hm}^i$ as follows:

$\mathrm{TWT}_{hm}^i$ :

Total waiting time of injury type $h$ achieved until serving the $i$th injured person of type $h$ in node $m$ during the planning horizon.

$\mathrm{TUW}_{hm}^i$ :

Total number of person-period of injury type $h$ until serving the $i$th injured person of type $h$ in node $m$ during the planning horizon.

Obviously, $\mathrm{TWT}_{hm}^0 =\mathrm{TUW}_{hm}^0 =0.$ Now, suppose that the $i$th injured person with injury of type $h$ was evacuated at time $t_{i }$ from node $m$ and will be delivered to a hospital at time $s_{i}.$ Hence,

$$\begin{aligned} \mathrm{TWT}_{hm}^i =\mathrm{TWT}_{hm}^{i-1} +( {s_i -t_i })=\mathrm{TWT}_{hm}^{i-1} +\Delta \mathrm{TWT}_{hm}^i \end{aligned}$$

(40)

In addition, since the $i$th injured person will not be served until $s_{i}$ it will add one unit to the total number of unserved wounded people of type $h$ in node $m$ at times $t_{i},\,t_{i+1}, {\ldots },s_{i-1}.$ Therefore, the increase of TUW is$( {( {s_i -1})-t_i +1})=( {s_i -t_i }).$ As a result:

$$\begin{aligned} \mathrm{TUW}_{hm}^i =\mathrm{TUW}_{hm}^{i-1} +( {s_i -t_i })=\mathrm{TUW}_{hm}^{i-1} +\Delta \mathrm{TUW}_{hm}^i \end{aligned}$$

(41)

Since $\mathrm{TWT}_{hm}^0 =\mathrm{TUW}_{hm}^0$ and $\Delta \mathrm{TUW}_{hm}^i =\Delta \mathrm{TWT}_{hm}^i,$ it can be claimed that $\mathrm{TUW}_{hm}^i =\mathrm{TTW}_{hm}^i .$ According to the priorities of various injured people and all demand nodes, equation (21) can be obtained.

$$\begin{aligned} \sum \limits _{m=1}^M {\sum \limits _{h=1}^H {p_h .\mathrm{TWT}_{hm}^i } } =\sum \limits _{m=1}^M {\sum \limits _{h=1}^H {p_h .\mathrm{TUW}_{hm}^i } } \end{aligned}$$

(42)

Similarly, the proof presented above can be used to prove the equality of the total (weighted) unsatisfied commodities needs and the total (weighted) lead-time before satisfying commodity needs. For this purpose, $\mathrm{TDT}_{am}^k$ and $\mathrm{TUD}_{am}^k$ are, respectively, defined as total lead-time of commodity need type $a$ for satisfaction from the moment demand materializes until the $k$th request of this commodity will be met at node $m,$ and total unsatisfied need-periods of commodity type $a$ until satisfaction of $k$th request of commodity type $a$ from node $m.$ Clearly, $\mathrm{TDT}_{am}^0 =\mathrm{TUD}_{am}^0 =0.$ Now, assume that the $k$th commodity type $a$ was requested in node $m$ at time $t_{i}$ and will be delivered to this node at time $s_{i}.$ Hence, $\mathrm{TDT}_{am}^k =\mathrm{TDT}_{am}^{k-1} +( {s_i -t_i })=\mathrm{TDT}_{am}^{k-1} +\Delta \mathrm{TDT}_{am}^k .$ Due to the inability to satisfy this need before $s_{i},$ it will add one unit to the total unsatisfied needs of node $m$ at times $t_{i},\,t_{i+1}, \ldots ,s_{i-1}.$ Therefore, the increase of TUD is $( {( {s_i -1})-t_i +1})=( {s_i -t_i }).$ That is, $\mathrm{TUD}_{am}^k =\mathrm{TUD}_{am}^{k-1} +( {s_i -t_i })=\mathrm{TUD}_{am}^{k-1} +\Delta \mathrm{TUD}_{am}^k .$ Since $\mathrm{TDT}_{am}^0 =\mathrm{TUD}_{am}^0$ and $\Delta \mathrm{TUD}_{hm}^k =\Delta \mathrm{TDT}_{am}^k$; it is concluded that $\mathrm{TUD}_{am}^k =\mathrm{TTD}_{am}^k .$ Finally, according to the different commodity needs and all demand nodes, the following holds: $\sum _{m=1}^M {\sum _{a=1}^A {p_a .\mathrm{TDT}_{am}^k } } =\sum _{m=1}^M {\sum _{a=1}^A {p_a .\mathrm{TUD}_{am}^k } } .$