Abstract: Frequent disasters pose a great threat to human survival and social development, and pose challenges to the government's emergency response and disaster rescue. The key to effectively reduce the post-disaster losses and casualties lies in the scientific and equitable allocation of emergency relief materials. Therefore, how to quickly meet emergency material needs of disaster areas and reduce disaster losses through an effective emergency material allocation scheme is a realistic problem in the field of emergency management. Disasters usually cause multiple disaster sites, and their demand for different materials in different emergency stages changes dynamically in real time, resulting in the traditional static single-stage model may not meet the dynamic multi-stage allocation requirements of emergency materials. Based on this, a dynamic multi-stage allocation model for emergency materials based on multiple distribution centers, multiple supply sites and multiple disaster-affected sites is constructed by introducing an exponential utility function, considering the dynamic change characteristics of demands for different emergency materials at different stages, and combining the principle of efficiency and equity. The efficiency target minimizes the total costs that are produced by allocating all materials to all disaster sites during all stages, and the equity target minimizes the total loss that is produced by unsatisfied material demands during all stages, for achieving equitable allocation of materials among multiple affected sites. Then, the objective transformation and linear approximation method is used to solve the proposed model. For each disaster site m∈M, material n∈N and stage k∈K, create a set of "breakpoints" B_mnk, which are values at which the utility function will be (under)-approximated. A new variable μ_mn^k is introduced to represent the value of the utility function. Then the nonlinearity can be replaced/approximated in the objective function with the new objective function, as long as the following constraints are added:μ_mn^k ≥ γ(g)+γ'(g)(P_mn^k-g), ∀ m∈M,n∈N,k∈K,g∈B_mn^k. Finally, the validity and feasibility of the proposed model is verified through designing numerical examples and solving. The results indicate that the proposed model can consider both efficiency and fairness of material allocation, and can minimize the delay loss of material shortage and the total cost of material allocation. Moreover, quantifying the equity by using the proportional shortage of material demand of disaster victims, can avoid the impact on the fair allocation caused by the difference of demand among the disaster sites. This model can make the disaster sites obtain a certain ratio of materials needed at each stage even the supplies are limited or supplies are in short supply in the initial stage of emergency relief. Thus avoiding large material shortfalls and losses, ensuring fairness of multi-stage allocation of emergency materials among multiple disaster-stricken points are more in line with the reality of disaster relief. In addition, the proposed model, results and insights can provide decision support for the allocation of emergency relief materials in large-scale disasters, thereby developing effective allocation strategies according to different contexts.

Key words: emergency material allocation, distribution centers, supply sites, disaster-affected sites, multi-stage, percentage shortfall