Management of mobile resources in Physical Internet logistic models

This paper deals with the concept of a ‘Physical Internet’, the idea of building large logistics systems like the very successful Digital Internet network. The idea is to handle mobile resources, such as containers, just like Internet data packets. Thus, it is possible to use the principles of encapsulation and routing to optimize the freight. The problem is that mobile resources, such as containers, are not quite similar to data packets, because they are real and not dematerialized. Thus the handling and the storing of mobile resources, such as containers, will create imbalances in the logistics network, leading to starvation or overstocking of logistic network nodes. We propose in this paper a study addressing this problem leading to some solutions.

💡 Research Summary

The paper investigates the “Physical Internet” (PI) concept, which aims to treat physical freight containers much like data packets in the digital Internet, using principles such as encapsulation, routing, and standardized interfaces. While the analogy promises great efficiencies, the authors point out a fundamental difficulty: containers are tangible, costly assets that cannot be created or destroyed at will, unlike digital packets. Consequently, imbalances arise in a logistics network—some nodes accumulate excess containers while others suffer shortages—leading to starvation or overstocking and undermining the PI’s potential benefits.

To formalize the problem, the authors model the logistics area as a directed graph (G=(V,E)). Vertices (V) represent ports, hubs, factories, or consumption sites; edges (E) denote physical routes with known distances (d_{ij}). Each vertex (v) holds an initial number of containers (X_v) and a desired target number (Y_v). The total number of containers (N) is fixed, and the target state (Y) is assumed reachable through a global management policy. The operational decision concerns a single ship with capacity (k) that must relocate containers from the initial distribution (X) to the target distribution (Y) while minimizing total travel distance.

The authors recognize that this relocation problem is NP‑hard, sharing characteristics with the bipartite transportation problem, vehicle routing, and swapping problems. Rather than seeking an exact solution, they adopt a heuristic originally proposed in Benchimol et al. (2011) that runs in (O(\log k + n \log N)) time, where (n) is the number of vertices and (N) the total number of containers. The heuristic proceeds in four stages:

1. Minimum‑cost matching (M) – A b‑matching is computed between “excess” vertices (those with surplus containers) and “deficit” vertices (those lacking containers). The matching minimizes the sum of travel distances, using a nearest‑neighbor rule when multiple candidates exist.

2. Construction of two tours – One tour (C_{ex}) visits all excess vertices, the other tour (C_{def}) visits all deficit vertices. Both tours are built greedily with a nearest‑neighbor heuristic to keep intra‑tour distances low.

3. Subdivision into capacity‑compatible subpaths – Each tour is split into subpaths whose total container demand or surplus is an exact multiple of the ship’s capacity (k). If a vertex’s surplus/deficit is not a multiple of (k), the vertex is conceptually split into several copies so that each subpath can be serviced with whole loads of size (k).

4. Resource transfer via b‑matching – The matching from step 1 is combined with the subpaths to generate concrete ship movements. The ship performs a series of round‑trips: it loads (k) containers at an excess subpath, travels to a deficit subpath, unloads, and possibly returns empty. The sequence of trips is derived directly from the matching edges, ensuring that each excess load is paired with a deficit need.

The paper illustrates the method with a small example of seven vertices (A–G) and a distance matrix. With ship capacity (k=5) and all surplus/deficit values already multiples of five, the heuristic yields a concrete route:
(A, E, A, G, A, G, A, G, C, G, C, G, C, F, D, B, F, B, F, B, F, B, F).
This route moves containers from A to E, then repeatedly shuttles between A and G, transfers from C to G, and finally balances the remaining nodes, achieving the target distribution with a relatively short total distance.

Key insights from the work include:

• Global matching is essential to capture the overall imbalance rather than treating each node in isolation.
• Capacity‑aware subpath decomposition allows a single ship with limited load to service the network efficiently, respecting the indivisible nature of containers.
• Nearest‑neighbor heuristics at both matching and tour‑building stages keep the solution’s travel cost low while preserving computational tractability.

The authors conclude that their heuristic provides a practical, scalable approach for managing mobile resources in a PI‑based logistics system. They acknowledge that the current study assumes a single ship and a single graph, and they outline future research directions: extending the model to multiple ships, handling several interacting logistics regions (each with its own graph), incorporating dynamic demand fluctuations, and testing the algorithm on real‑world data sets. The paper thus contributes both a formal problem definition for container imbalance in the Physical Internet and a concrete algorithmic framework that can be built upon for more complex, realistic logistics scenarios.