In this paper, a districting problem with stochastic demands is investigated. The goal is to divide a geographic area into $p$ contiguous districts such that, with some given probability, the districts are balanced with respect to some given lower and upper thresholds. The problem is cast as a $p$-median problem with contiguity constraints that is further enhanced with chance-constrained balancing requirements. The total assignment cost of the territorial units to the representatives of the corresponding districts is used as a surrogate compactness measure to be optimized. Due to the tantalizing purpose of deriving a deterministic equivalent for the problem, a two-phase heuristic is developed. In the first phase, the chance-constraints are ignored and a feasible solution is constructed for the relaxed problem; in the second phase, the solution is corrected if it does not meet the chance-constraints. In this case, a simulation procedure is proposed for estimating the probability of a given solution to yield a balanced districting. That procedure also provides information for guiding the changes to make in the solution. The results of a series of computational tests performed are discussed based upon a set of testbed instances randomly generated. Different families of probability distributions for the demands are also investigated, namely: Uniform, Log-normal, Exponential, and Poisson.

本文研究了具有随机需求的分区问题。目标是将地理区域划分为$p$连续的区域，这样就可以在一定的给定下限和上限阈值之间使区域具有一定的平衡性。问题被铸成$p$-具有连续性约束的中间问题，随着机会受限的平衡要求而进一步增强。区域单位对相应地区代表的总分配成本用作要优化的替代紧凑性度量。由于得出问题的确定性等价物的诱人目的，因此开发了两阶段启发式方法。在第一阶段，机会约束被忽略，为松弛问题构造了可行的解决方案。在第二阶段，如果解决方案不符合机会约束条件，则可以对其进行更正。在这种情况下，提出了一种仿真程序，用于估计给定解决方案产生平衡分区的概率。该过程还提供了用于指导解决方案中进行更改的信息。基于随机生成的一组测试平台实例，讨论了执行的一系列计算测试的结果。还研究了需求的不同概率分布族，即：均匀分布，对数正态分布，指数分布和泊松分布。