Abstract: As people’s medical concerns gradually shift from treating diseases to preventing ones, vaccination, as one of the most effective ways to prevent serious infectious diseases, has attracted much attention from governments and the public. In recent years, the demands for vaccination have been increasing year by year. However, the vaccine supply chain is characterized by rather long research, development and production cycles. Its high supply-demand uncertainties and strict storage conditions make it difficult to significantly reduce vaccine supply lead-time, which will result in frequent shortages of vaccines. In addition, as a medical health resource, the fairness of vaccine allocation among different regions and groups is particularly important. Ignoring this factor may lead to dissatisfaction and disorder among the public, which can bring about serious social problems. Furthermore, the queuing situation at vaccination stations may lead to a decrease in the number of vaccinators and affect people’s willingness to vaccinate. As a result, low vaccine coverage rate and related problems may occur. Therefore, in the case of limited vaccine supply, it is very essential to construct an optimal network of vaccination station to achieve equitable allocation of vaccines and improve people’s enthusiasm for vaccination.
Regarding the fairness in allocation, decision-makers may wish to adjust the level of fairness in allocation based on the reality situation, such as regional economic status, population distribution, availability of medical resources, and national vaccination policies. To handle the problem, this article proposes a method of using the Gini coefficient as a constraint to ensure the fairness of vaccine allocation. By setting a Gini coefficient threshold, the level of fairness in allocation can be adjusted. Moreover, because the vaccination process is a service system, we apply the M/M/1 queuing system to model the queuing phenomenon in the vaccination service process, and ensure the service quality of vaccine vaccination by adding the queue length as a constraint to the model. As a consequence, the number of vaccination service points can be determined by solving the model. Additionally, when the fairness constraint of vaccine allocation cannot be met, a mechanism of transferring vaccines between vaccination stations to achieve fairness in vaccine allocation is put forward in this paper. Furthermore, since the demands for multiple types of vaccines over periods are uncertain, a stochastic programming method is applied to handle the uncertainty. A series of discrete scenarios is generated to simulate the demand by using SAA(Sample Average Approximation).
To sum up, for the problem of vaccination station location and fairness of vaccine allocation, we employ the Gini coefficient as a constraint to ensure the fairness of vaccine distribution, utilize the M/M/1 queuing system to model the queuing phenomenon in the vaccination service process, and construct a two-stage stochastic programming model with the objective of minimizing total cost. Because solving the expected value of the two-stage stochastic programming model involves the high-dimensional integral of random variables, directly solving the model is very time-consuming. According to the characteristics of the model, we design a Benders decomposition algorithm based on sample average approximation and a series of Benders decomposition acceleration methods to solve the model.
Finally, a series of numerical experiments and sensitivity analysis are conducted. Numerical experimental results show that compared to commercial solution software and the basic Benders decomposition algorithm, the Benders decomposition algorithm combined with a series of acceleration methods has significantly improved the efficiency of solving problems. Moreover, decision-makers can adjust the fairness of distribution by setting Gini coefficient thresholds. In a case where the fairness of vaccine allocation cannot be met, the transfer mechanism can achieve fairness in vaccine distribution and reduce the total cost.

Key words: vaccination station location, Gini coefficient, two-stage stochastic programming, Benders decomposition

摘要： 随着人们预防疾病的意识逐渐提高,疫苗接种需求量逐年提升。但疫苗的供应量很难在短时间内有大量增加,导致疫苗经常出现缺货现象。本文考虑疫苗接种站选址和疫苗分配公平性问题,以基尼系数作为约束来保证疫苗分配的公平性,以M/M/1排队系统来建模接种服务过程中的排队现象,构建了以总成本最小化为目标的两阶段随机规划模型。根据模型特点,设计了基于样本平均近似的Benders分解算法和一系列Benders分解加速方法来对模型求解。数值实验结果表明:结合一系列加速方法的Benders分解算法在求解效率上有明显的提升。此外,在疫苗公平性不能满足的情况下,可以通过在疫苗接种站之间进行转运的机制能来保证疫苗分配的公平性,并能达到降低总成本的目的。

关键词: 接种站选址, 基尼系数, 两阶段随机规划, Benders分解