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A multi-commodity flow problem consists of moving several commodities from their respective sources to their sinks through a network where each edge has different costs and capacity constraints. This paper explores different linear programming algorithms and their performance regarding finding an optimal solution for multi-commodity flow problems. By testing several of different network constraints, we examine which algorithms are most suitable for specific network and problem structures. Furthermore, we implement our own multi-commodity solver and compare its performance against state-of-the-art linear programming solvers. The results show that for the methods we tested it is difficult to discern which class of linear programming methods are optimal solvers for multi-commodity flow problems and that their performance depends on how the network and commodities are structured.

A multi-commodity flow problem consists of moving several commodities from their respective sources to their sinks through a network where each edge has different costs and capacity constraints. This paper explores different linear programming algorithms and their performance regarding finding an optimal solution for multi-commodity flow problems. By testing several of different network constraints, we examine which algorithms are most suitable for specific network and problem structures. Furthermore, we implement our own multi-commodity solver and compare its performance against state-of-the-art linear programming solvers. The results show that for the methods we tested it is difficult to discern which class of linear programming methods are optimal solvers for multi-commodity flow problems and that their performance depends on how the network and commodities are structured.

The maritime industry is one of the greenest modes of transportation, taking care of almost 90% of the global trade. The maritime container business revolves around liner shipping, which consists of container vessels sailing on fixed itineraries. For the last 20 years, there has been an increasing number of publications regarding how to design such fixed routes (services), to ensure a high level of service while minimizing operational costs and environmental impact. The liner shipping network design problem can briefly be described as follows: Given a set of demands (defined by origin, destination, time limit) and a set of vessels with variable capacity, the task is to design a set of weekly services, assign vessels to the services, and flow the demand through the resulting network such that it arrives within the stated time constraints. The objective is to maximize revenue of transported demand subtracting the operational costs. We present an in-depth literature overview of existing models and solution methods for liner shipping network design, and discuss the four main families of solution methods: integrated mixed integer programming models; two-stage algorithms designing services in the first step and flowing containers in the second step; two-stage algorithms first flowing containers and then designing services; and finally algorithms for selecting a subset of proposed candidate services. We end the presentation by comparing the performance of leading algorithms using the public LINER-LIB instances. The paper is concluded by discussing future trends in liner shipping, indicating directions for future research.