We develop a variant of the flow interception problem (FIP) in which it is more desirable for travelers to be intercepted as early as possible in their trips. In addition, we consider flows being intercepted probabilistically instead of the deterministic view of coverage assumed in the FIP literature. We call the proposed model the probabilistic minisum FIP (PMFIP); it involves minimizing the sum of the expected distance that each flow travels until intercepted at a facility among placed facilities. This extension allows us to evaluate the effect of facility location under any given value of the interception probability and to apply the model to a variety of situations. We apply the proposed model to an example network by assuming a hypothetical situation in which people gather at a stadium from various nodes on the network, and receive some goods or services on the way to the stadium. We analyze optimal solutions obtained by varying the number of facilities and interception probability. It is shown that the expected travel distance until intercept is greatly reduced by means of a few optimally located facilities under a moderate interception probability.

中文翻译：

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概率最小和流拦截问题：在概率拦截下最小化期望的行进距离，直到拦截为止

我们开发了流量拦截问题（FIP）的变体，其中更希望旅行者在旅途中尽早被拦截。此外，我们认为流量被概率拦截，而不是FIP文献中假设的确定性视图。我们将提出的模型称为概率最小和FIP（PMFIP）；它涉及最小化每个流传播的预期距离之和，直到在放置的设施中的某个设施被拦截为止。此扩展使我们可以在任何给定的拦截概率值下评估设施位置的影响，并将模型应用于各种情况。我们通过假设以下假设情况将提议的模型应用于示例网络，在这种情况下，人们从网络上的各个节点聚集在体育场，并在前往体育场的途中获得一些商品或服务。我们分析了通过改变设施数量和拦截概率获得的最佳解决方案。结果表明，在中等拦截概率下，通过一些最佳布置的设施大大降低了直至拦截的预期行驶距离。