Nowadays, RFID technology is increasingly become popular and begin to enter many spheres of everyday life and industrial sectors, supply chain management is one of the most significant applications using RFID solutions. Recently, the path-checking scheme in which readers can verify the validity of the product's path in supply chain is receiving more and more attentions. In this paper, we first review some existing RFID path-checking protocols and point out the defections of their constructions. Then we propose a path-checking solution based on Boldyreva's ordered multisignatures (OMS), a reader will provide its own signature when a tag passes by in order to record the path information, the scheme achieves that each reader in the supply chain can verify the validity of the path taken by the tag. However, OMS can also be verified by adversaries who have the public keys of the readers, so we improve Boldyreva's scheme to implement verification by designated readers, which also protect the path privacy of the tags. In order to resist impersonate attacks, we add mutual authentication into our path-checking scheme by extending Ma's protocol using only pseudorandom number generator. Finally, we do some security and privacy analysis and compare our solution with recently proposed CHECKER.