This paper introduces the transit time constrained fixed charge multi-commodity network design problem. Transit times are origin-to-destination time limits for the commodities, which appear for example in transport systems with perishable goods. We discuss how to model the problem and present three different formulations of it. The first formulation is an exponential size path formulation, which we solve with a branch-and-price algorithm. Several speed up techniques from the literature on fixed charge multi-commodity network design problems are implemented, such as lifted cover inequalities and the recently proposed deep dual-optimal inequalities. In an extensive set of computational experiments, we show that these inequalities significantly improve the performance of the algorithm. The other two formulations are of polynomial size: one uses path indices and the other uses time indices. While the branch-and-price algorithm outperforms solving the compact formulations with a general-purpose mixed-integer programming solver, the study of compact models helps better understand the problem, and we can use them as benchmarks. A detailed sensitivity analysis of the branch-and-price algorithm shows that longer transit times and an increased ratio of fixed charge to flow cost increase the difficulty of solving the problem whereas the arc capacity has less impact. We further discuss in-depth implementational details.

本文介绍了中转时间约束固定收费多商品网络设计问题。运输时间是商品的始发地到目的地的时间限制，例如出现在带有易腐烂货物的运输系统中。我们讨论如何对问题建模并提出它的三种不同表述。第一个公式是指数大小的路径公式，我们用分支和价格算法解决它。实施了关于固定收费多商品网络设计问题的文献中的几种加速技术，例如提升覆盖不等式和最近提出的深度对偶最优不等式。在大量的计算实验中，我们表明这些不等式显着提高了算法的性能。另外两个公式是多项式大小：一种使用路径索引，另一种使用时间索引。虽然分支和价格算法优于使用通用混合整数编程求解器求解紧凑公式，但对紧凑模型的研究有助于更好地理解问题，我们可以将它们用作基准。分支价格算法的详细敏感性分析表明，较长的运输时间和增加的固定费用与流量成本的比率增加了解决问题的难度，而电弧容量的影响较小。我们进一步讨论深入的实现细节。我们可以将它们用作基准。分支价格算法的详细敏感性分析表明，较长的运输时间和增加的固定费用与流量成本的比率增加了解决问题的难度，而电弧容量的影响较小。我们进一步讨论深入的实现细节。我们可以将它们用作基准。分支价格算法的详细敏感性分析表明，较长的运输时间和增加的固定费用与流量成本的比率增加了解决问题的难度，而电弧容量的影响较小。我们进一步讨论深入的实现细节。