^{a}, Yiyong Xiao

^{b}, Siyue Zhang

^{c}, Wenbing Chang

^{d}and Shenghan Zhou

^{e}

^{a}yangpei@buaa.edu.cn

^{b}xiaoyiyong@buaa.edu.cn

^{c}zhang_sy@buaa.edu.cn

^{d}changwenbing@buaa.edu.cn

^{e}zhoush@buaa.edu.cn

We present the r-interdiction p-median problem with uncertain number of attacks which is an extension of the continuous location problem. Various uncertain attacks are supposed to take place at the facilities with estimated possibilities and they may cause the service suspending in local area. Therefore, coping measures are needed at the design stage to improve the network system with stronger reliability and restorability, such that facility services can be fast restored at a minimum loss as possible after the attacks happen. This paper develops a bi-objective optimization model such that minimizes the expected loss of the system under various interdiction events. In our model, there are two opposite objectives, one of them is the attacker side that pursue a maximum loss of the system, and the other one is from the designer side who wishes the system can be restored at a minimum expected loss. The designer is supposed to make a decision for continuous locations of the facilities in a plane region without prespecified candidate sites. To solve the bi-objective model, we convert dual targets into single targets by linear weighted combination method such that the bi-level formulation is converted to a single-level formulation. We also use a linear approximation of the Euclidean distance for continuous location therefore simplifying the nonlinear formulation into a linear one. We use CPLEX by AMPL to solve our model directly for small-sized problems. Finally, we verify our model and solution approaches by computational experiments. The computational results show that our model and methodology yield good solutions and can be used to improve the invulnerability of the system.