At present, most fuzzy relation entropies are constructed by means of general fuzzy binary relations, which cannot effectively evaluate the fuzzy relation families with order attributes. This limits their application in multi-attribute decision-making. For this reason, a new superior-inferior relationship entropy is presented in this paper. First, a parameterized fuzzy superior-inferior relation is studied to characterize the differences between samples, and some fuzzy classes are then proposed with sigma superior-inferior relation. On this basis, a new sigma superior-inferior relation entropy are proposed, and some of its derived entropy, such as sigma superior-inferior relation conditional entropy, sigma superior-inferior relation joint entropy and sigma superior-inferior relation mutual information, are then introduced. We discuss the relationship between them and some important properties are explored. Finally, two multi-attribute decision-making methods with sigma superior-inferior relation entropy are developed, and the effectiveness and feasibility of the presented method are verified by data examples. Comparison and sensitivity analysis show that the ranking results of the proposed model and some classical methods are highly consistent. In particular, the proposed method has wider applicability in the multiple expert evaluation environment.