In a highly competitive market environment, outsourcing of logistics service is one of the most significant strategies for enterprises to enhance competitiveness. However, how to construct an efficient and reliable logistics network by selecting suitable logistics carriers is a complex but important practical issue. In general, there are multiple carriers to provide freight service along each route of a logistics network. Moreover, the transportation capacity of each carrier has multiple states due to various factors, which can be portrayed by a capacity probability distribution. In addition, delivery spoilage and cost constraint are the key indicators affecting carrier selection. Therefore, this paper studies the logistics carrier selection problem from the perspective of multi-state network reliability under delivery spoilage and cost constraint, and apply the minimal path (MP) method to solve the problem. The practicality of the algorithm is verified by example analysis.