?-Reliable Mean-Excess Regret Model For Emergency Location Routing Problem Under Demand Uncertainty

Every year,occurrences of large-scale emergencies cause severe casualties and economic losses.The losses can be reduced by efficient and timely rescue which relies on emergency logistics that contains two critical issues,i.e.,facility location and vehicle routing.Traditionally,these two problems are solved separately,while from the perspective of system optimization,it is necessary to optimize the integrated location routing problem(LRP)to make for shorter waiting time and lower system cost.This paper proposes an ?-reliable mean-excess regret(?-MER)model for solving the emergency location routing problem(ELRP)under demand uncertainty.The novelty of this study is to consider the regrets caused by high-consequence scenarios via minimizing the ?-MER of waiting time and system cost simultaneously.Given a set of scenarios and their occurrence probability,the regret associated with one scenario under a ELRP solution is defined as the difference between the objective value given by the solution and that given by the optimal solution that can be achieved under the same scenario;the ?-MER values(i.e.,waiting time and system cost)are defined by the expectation of the regrets with respect to the scenarios in the tail whose collective probability of occurrence is less than 1-?.The problem is formulated as a bi-objective mixed integer nonlinear programming problem and solved by a heuristic algorithm.The algorithm relies on CPLEX to obtain the optimal solution for each scenario in a small network,a single objective genetic algorithm to obtain the optimal value associated with each scenario in a medium-sized network,and the non-dominated sorting genetic algorithm(NSGA-II)to obtain the set of Pareto optimal solutions considering all scenarios.To assist the decision maker to select solutions from the Pareto frontier,a trade-off function is introduced to find the Nash bargaining solution(NBS),which captures the decision maker's risk preference.Numerical examples are conducted to demonstrate the accuracy of the model,the performance of the solution method,and the effects of the decision maker's risk preference on the decision making.