Supply chain (SC) network design problems are complex problems with multi-layer levels and dynamic relationships which involve a considerable amount of uncertainty concerning customer demand, facility capacity, or lead times, among others. A large number of optimization methods (i.e., fuzzy mathematical programming, stochastic programming, and interval mathematical programming) have been proposed to cope with the uncertainties in SC network design problems. We propose a fuzzy bi-objective mixed-integer linear programming (MILP) model to enhance the material flow in dual-channel, multi-item, and multi-objective SCs with multiple echelons under both ambiguous and vague conditions, concurrently. We use a computationally efficient ranking method to resolve the ambiguity of the parameters and propose two methods for resolving the vagueness of the objective functions in the proposed fuzzy MILP model. The preferences of the decision makers (DMs) on the priority of the fuzzy goals are represented with crisp importance weights in the first method and fuzzy preference relations in the second method. The fuzzy preference relations in the second method present a unique practical application of type-II fuzzy sets. The performance of the two methods is compared using comprehensive statistical analysis. The results show the perspicuous dominance of the method which uses fuzzy preference relations (i.e., type-II fuzzy sets). We present a case study in the food industry to demonstrate the applicability of the proposed model and exhibit the efficacy of the procedures and algorithms. To the best of our knowledge, a concurrent interpretation of both ambiguous and vague uncertainties, which is applicable to many real-life problems, is novel and has not been reported in the literature.