This paper addresses the winner determination problem (WDP) for TL transportation procurement auctions under uncertain shipment volumes and uncertain carriers' capacity. It extends an existing two-stage robust formulation proposed for the WDP with uncertain shipment volumes. The paper identifies and theoretically validates a number of accelerating strategies to speed up the convergence of a basic constraint generation algorithm proposed in the literature. Experimental results prove the high computational performance of the proposed new algorithm and the relevance of considering uncertainty on the carriers' capacity when solving the WDP.