Cost Allocation for Less-Than-Truckload Collaboration via Shipper Consortium

We study the problem of collaborative less-than-truckload (LTL) transportation in the form of a shipper consortium, which is operated by a third-party logistics provider (3PL) through a cross-dock/pooling network. The 3PL has responsibility for planning the combined loads prior to actual shipments, hiring and routing carriers to execute shipping, and allocating the cost to the shippers in the consortium. Shippers receive substantial cost savings from combined truckload shipments. However, achieving consolidation and realizing this benefit requires addressing two essential issues: (i) how to find an approximately optimal consolidation solution in a large network with many freights and (ii) how to determine a fair cost allocation rule among the shippers’ consolidated freights that ensures budget balance while minimally violating coalitional stability. Our work resolves these two issues. We formulate a time-space network flow model of the problem under both incremental and all-unit discount structures of LTL rates and propose a computationally efficient algorithm based on local search heuristics. We model the problem of allocating cost to the shippers as a cooperative game. We decompose and linearize the Lagrangian dual problem by using total unimodularity and concavity. We propose an efficiently computable cost allocation rule from the linearized dual models. The dual rule ensures stable cooperation but may have underallocation equal to a duality gap. To cover the underallocation, we further develop a budget covering procedure and define an ϵ-core allocation with desirable properties. Through extensive computational experiments, we find that the shipper consortium reduces total shipping costs by more than 40% in most cases; meanwhile, the ϵ-core allocation is typically in the core for small-scale networks while violating stability by at most 5% for large-scale networks and provides consolidated freights with more than 50% individual cost savings on average.