Emergency relief routing models for injured victims considering equity and priority

Abstract

In humanitarian aid, emergency relief routing optimization needs to consider equity and priority issues. Different from the general path selection optimization, this paper builds two models differentiated by considerations on the identical and diverse injured degrees, where the relative deprivation cost is proposed as one of the decision-making objectives to emphasize equity, and the in-transit tolerable suffering duration is employed as a type of time window constraint to highlight rescue priority. After proving the NP-hardness of our models, we design a meta-heuristic algorithm based on the ant colony optimization to accelerate the convergence speed, which is more efficient than the commonly-used genetic algorithm. Taking 2017 Houston Flood as a case, we find results by performing the experimental comparison and sensitivity analysis. First, our models have advantages in the fairness of human sufferings mitigation. Second, the role of the in-transit tolerable suffering time window cannot be ignored in humanitarian relief solutions. Various measures are encouraged to extend this type of time window for achieving better emergency relief. Finally, our proposed hybrid transportation strategy aiming at diverse injured degrees stably outperforms the separated strategy, both in operational cost control and psychological sufferings alleviation, especially when relief supplies are limited.

Appendices

Appendix A: NP-hardness proof of Models I and II

By the following proposition, we prove the relationship between our Models (VRPTW) and the travelling salesman problem (TSP).

Proposition A.1

The VRPTW is at least as hard as TSP.

Proof

VRPTW is considered as a problem with additional time window constraints based on the general vehicle routing problem (VRP). VRPTW is equivalent to VRP in the case of unlimited time windows. In this sense, VRP is a special case of VRPTW.

Then, we review the definitions of VRP and TSP. VRP tries to answer “what are the optimal routes for a group of vehicles when serving a given set of customers, where vehicles initially-located at a depot are dispatched to customers and return to the origin depot?”, while TSP refers to “given a list of customers and a starting depot, what is the optimal route for one vehicle when services each customer exactly once and returns to the origin depot?”. So, VRP can be treated as a generalization of TSP. When the number of vehicles is 1, VRP becomes TSP. In other words, TSP is a special case of VRP.

Therefore, if TSP is NP-hard, both VRPTW and VRP are naturally NP-hard. That is to say, VRPTW and VRP are at least as hard as TSP.□

It is well-known that TSP is NP-hard, since the Hamiltonian Cycle (HC) that is NP-complete can be reducible to TSP in polynomial time (Rahman and Kaykobad 2005). Thereby, we prove that Models I and II are NP-hard according to Proposition A.1.

Appendix B: Comparison analysis based on Model I and Model II

See Figs. 6 and 7.

Fig. 6

The optimal paths of traditional path selection models. a No deprivation cost objective. b No tolerable suffering time constraint

Fig. 7

The optimal paths under different transportation strategies. a1 The 1st stage paths in separated strategy. a2 The 2nd stage paths in separated strategy. b1 The 1st stage paths in hybrid strategy. b2 The 2nd stage paths in hybrid strategy

Appendix C: Sensitivity analysis based on Model I and Model II

See Figs. 8 and 9.

Fig. 8

The optimal paths under different changes of $u_{i}$ and $b_{i}$ in Model I. a1 Both $u_{i}$ and $b_{i}$ decreased by 20%. a2 Both $u_{i}$ and $b_{i}$ increased by 20%. b1 Only $u_{i}$ decreased by 20%. b2 Only $u_{i}$ increased by 30%. c1 Only $b_{i}$ decreased by 20%. c2 Only $b_{i}$ increased by 30%

Fig. 9

The optimal paths under different changes of $u_{i}$ and $b_{i}$ using two transportation strategies in Model II. a1 The 1st stage paths in separated strategy with 20% decrease of $u_{i}$ and $b_{i}$. a2 The 2nd stage paths in separated strategy with 20% decrease of $u_{i}$ and $b_{i}$. b1 The 1st stage paths in separated strategy with 20% increase of $u_{i}$ and $b_{i}$. b2 The 2nd stage paths in separated strategy with 20% increase of $u_{i}$ and $b_{i}$. c1 The 1st stage paths in hybrid strategy with 20% decrease of $u_{i}$ and $b_{i}$. c2 The 2nd stage paths in hybrid strategy with 20% decrease of $u_{i}$ and $b_{i}$. d1 The 1st stage paths in hybrid strategy with 20% increase of $u_{i}$ and $b_{i}$. d2 The 2nd stage paths in hybrid strategy with 20% increase of $u_{i}$ and $b_{i}$