Abstract: The traditional facility location problem, such as the covering problem, mostly assumes that once facilities are established, they will operate smoothly. However, in the real environment, the lifecycle of facility is generally long, and they are inevitably impacted by numerous uncertain factors during operation. These factors include adverse weather conditions, natural disasters, and deliberate human sabotage. As a result, facility interruptions occur, and the opening facilities may not always be reliable. Facility interruptions lead to higher transportation costs, lower service levels, and even the inability of the system to operate normally. For example, in the context of emergency facility location problems, natural disasters like earthquakes and floods can cause emergency facilities to interrupt. As a result, residents assigned to these interrupted facilities are unable to enjoy the corresponding services. More severely, their livelihoods and personal safety can be compromised. Therefore, considering the reliability of facilities is crucial in addressing facility location problems. Facility reliability has attracted widespread attention from both industry and academia.
In this paper, the reliability facility location problem is studied based on the uncertain interruption probability of facility, and the classical set covering model is extended to ensure that the probability of user points being covered is not lower than the given system reliability level. Reliability facility location model under random interruption probability is proposed. Then, we assume that the facility is interrupted with a probability, and the probability of interruption is uncertain, and the value is taken in a symmetrically bounded interval. The reliability facility location problem under uncertain interruption is suggested based on the budget-of-uncertainty robust approach. The model ensures that the probability of user points being covered is not lower than the given system reliability level even if some of the interruption probability values the worst case. The formulation of the reliability facility location problem is clearly nonlinear. By introducing auxiliary variables and using means of strong duality theory, we derive two propositions to reformulate the model as a linear optimization model. The robust counterpart programming is computationally tractable and easy to be applied to solve real-world problems.
Finally, numerical case studies are conducted using data from 88 city nodes in the United States. The results of the numerical experiments demonstrate that the topology of the facility network varies significantly under different combinations of uncertain budgets and system reliability levels. This indicates a significant impact of uncertain budgets on the facility location solution. Furthermore, an analysis is conducted to examine the influence of uncertain budgets on locating costs. The results reveal that locating costs increase as the uncertain budget increases. Similarly, locating costs also increase as the system reliability level improves. When the reliability level is relatively low, the increase in locating costs will be relatively slow. However, when the reliability level is high, the increase in locating costs will be rapid. The proposed model in this study offers a scientific basis and technical support for addressing reliable facility location problems in environments with uncertain facility interruption probabilities, such as emergency facilities and post-disaster medical facilities.
Compared with the existing similar research, the main contributions and novelty of this paper are highlighted as follows: (1)We assume that the facility is interrupted with a probability, where the probability of interruption is uncertain, and the value is taken in a symmetrically bounded interval. The budget-of-uncertainty robust approach is employed to deal with the problem, which can completely control the conservatism of the model and adjust the effect on the objective function. (2)The proposed non-linear model can be reformulated as a linear programming equivalently by introducing auxiliary variables and using means of strong duality theory. The resulting linear programming is computationally tractable and easy to be applied to solve real-world problems. (3)We provide insights into the different optimal locations corresponding to different protection levels and the system reliability level. The impact of uncertain budget on facility location cost is further analyzed, as well as how the facility location cost changes with the system reliability level. The model's contributions are expected to enhance the reliability and effectiveness of facility location strategies in critical sectors, benefiting emergency management, disaster response, and public services.

Key words: facility location, uncertain interruption, covering model, robust optimization

摘要： 设施选址是长期的战略性问题。由于设施的生命周期一般较长,运营过程中难免会受到众多不确定因素的影响而导致设施中断,因此在选址问题中考虑设施的可靠性至关重要。假设设施以一定的概率中断,且中断概率是不确定的,取值于对称有界区间。在经典覆盖选址模型的基础上,考虑即使设施中断,需求点仍以一定的概率被覆盖。基于基约束鲁棒优化方法,提出不确定中断概率情形下的鲁棒选址模型。然后,将鲁棒模型转化为等价的线性规划模型,并利用美国88个城市节点数据进行数值算例分析。结果表明,在不同的不确定预算下,设施网络的拓扑结构截然不同,表明不确定预算对设施选址方案有重大影响。此外,选址成本随着不确定预算的增加而增加。同时,选址成本也会随系统可靠性水平的提高而增加。当可靠性水平相对较低时,选址成本的增加相对缓慢。而当可靠性水平较高时,选址成本会迅速增加。

关键词: 设施选址, 不确定中断, 覆盖模型, 鲁棒优化