The multi-commodity network flow problem (MCNF) consists in routing a set of commodities through a capacitated network at minimum cost and is relevant for routing containers in liner shipping networks. As commodity transit times are often a critical factor, the literature has introduced hard limits on commodity transit times. In practical contexts, however, these hard limits may fail to provide sufficient flexibility since routes with even tiny delays would be discarded. Motivated by a major liner shipping operator, we study an MCNF generalization where transit time restrictions are modeled as soft constraints, in which delays are discouraged using penalty functions of transit time. Similarly, early commodity arrivals can receive a discount in cost. We derive properties that distinguish this model from other MCNF variants and adapt a column generation procedure to efficiently solve it. Extensive numerical experiments conducted on realistic liner shipping instances reveal that the explicit consideration of penalty functions can lead to significant cost reductions compared to hard transit time deadlines. Moreover, the penalties can be used to steer the flow towards slower or faster configurations, resulting in a potential increase in operational costs, which generates a trade-off that we quantify under varying penalty functions.

多商品网络流动问题（MCNF）在于以最小的成本通过一个功能强大的网络运送一组商品，这与在班轮运输网络中运送集装箱有关。由于商品运输时间通常是一个关键因素，因此文献对商品运输时间提出了严格的限制。但是，在实际情况下，这些硬限制可能无法提供足够的灵活性，因为即使是很小的延迟也会被丢弃。受主要班轮运输运营商的激励，我们研究了MCNF概括，其中将运输时间限制建模为软约束，其中使用运输时间的惩罚函数来阻止延误。同样，早期到达的商品也可以享受成本折扣。我们推导了将该模型与其他MCNF变型区分开来的属性，并采用了一种色谱柱生成程序来有效地解决它。在现实的班轮运输实例上进行的大量数值实验表明，与硬运输时间期限相比，对罚款函数的明确考虑可以导致成本的显着降低。此外，罚款可用于将流量引向较慢或较快的配置，从而导致运营成本的潜在增加，这产生了一种折衷，我们可以在不同的惩罚函数下进行量化。