The Physical Internet (PI) paradigm, which has gained attention in research and academia in recent years, leverages advanced logistics and interconnected networks to revolutionise the way goods are transported and delivered, thereby enhancing efficiency, reducing costs and delays, and minimising environmental impact. Within this system, PI-hubs function similarly to cross-docks, enabling the splitting of PI-containers into smaller modules for delivery through a network of interconnected hubs. This allows dynamic routing optimisation and efficient consolidation of PI-containers. However, the impact of system parameters and relevant uncertainties on the performance of this innovative logistics framework is still unclear. For this reason, this work proposes a robustness analysis to understand how the PI logistics framework is affected by the handling, consolidation, and processing of PI-containers at PI-hubs. To this end, the considered PI logistics system is represented via a mathematical programming model that determines the best allocation of PI-containers in an intermodal setting with different transportation modes. In doing so, four Key Performance Indicators (KPIs) are separately considered to investigate different aspects of the PI system's performance, and the relevant robustness is assessed with respect to the PI-hub processing times and the number of modules per PI-container. In particular, a Global Sensitivity Analysis (GSA) is performed to evaluate, through a case study, the individual relevance of each input parameter on the resulting performance.

This paper addresses the problem of optimizing the transport of goods in the Physical Internet (PI) framework in a multi-modal setting using a multi-objective mixed-integer linear programming (MILP) approach. The model is specifically designed to meet the requirements related to modular shipments and PI-hubs, and in particular, determines the allocation of modular shipments to each transport mode in an intermodal setting. In doing so, parallel direct connection via road, the delivery times and the transportation costs are minimized. The model is applied to a numerical case study, to test its effectiveness to enhance freight transport efficiency within the PI framework, by exploiting, in particular, all the capacities of the available vehicles. In addition, a sensitivity analysis is conducted on some model parameters, to test its reaction to changes in the supply system and in the objective priorities. Results show that all the shipments are effectively transported between the origin and the destination terminals, they are divided into modules when necessary, and the selected transport modes, allocation strategy, and delivery times vary accordingly to the objective priorities.