A method for computing the reliability of a multi-state distribution network under considerations of transportation damage and delivery cost was presented. The arc spoilage rate was used to characterize transportation damage. Subsequently, the network flow allocation strategy was constructed in terms of the demand level and transportation damage. Based on the flow allocation strategy, arc capacity constraint and delivery cost constraint, a minimal capacity vector method was proposed to compute the reliability of a distribution network. The effectiveness of the proposed method was demonstrated through numerical examples, and a method for the identification of critical arcs was explored from the reliability viewpoint. The research results can provide guidelines for managers in the distribution operation and maintenance.