The foundations of efficient management are laid on transport networks in various scientific and industrial fields. Nonetheless, establishing an optimum transport network design (TND) is complicated due to uncertainty in the operating environment. As a result, an uncertain network may be a more realistic representation of an actual transport network. The present study deals with an uncertain TND problem in which uncertain programming and the greedy randomized adaptive search procedure (GRASP) are used to develop an original optimization framework and propose a solution technique for obtaining cost-efficient designs. To this end, we originally develop the concept of α-shortest cycle (α-SC) employing the pessimistic value criterion, given a user-defined predesignated confidence level α. Employing this concept and the operational law of uncertain programming, a new auxiliary chance-constrained programming model is established for the uncertain TND problem, and we prove the existence of an equivalence relation between TNDs in an uncertain network and those in an auxiliary deterministic network. Specifically, we articulate how to obtain the uncertainty distribution of the overall optimal uncertain network's design cost. After all, the effectiveness and practical performance of the heuristic and optimization model is illustrated by adopting samples with different topology from a case study to show how our approach work in realistic networks and to highlight some of the heuristic's features.