In this paper we propose three modified versions of the Greedy SmAlleSt-cost Path first (GRASP) algorithm for minimizing the total transmission cost in a digital microgrid (DMG). The Simple Dynamic GRASP (SDG) is a cached-version of GRASP that dynamically updates the available capacity of links. The total path Transmission cost Dynamic GRASP (TDG) is a cached version of GRASP that uses the path's transmission cost as the criteria to select the smallest-cost paths. The Brute Force TDG (BFT) is a cached version of GRASP that explores all paths between loads and sources and selects the smallest-cost paths by evaluating the path's transmission costs. Our results show that SDG achieves the smallest total transmission cost and the fewest unsatisfied loads. Although TDG and BFT may achieve similar total transmission costs to those of SDG and GRASP, our results show that using the path's transmission costs in the selection of the smallest-cost path is not as effective as using the path's link costs. However, in more complex power networks, we show that the cached TDG and BFT yield fewer unsatisfied loads than that of a GRASP implementation without a cache.