If a graph G(V, E) satisfies the following conditions: if x is an isolated vertex in the subgraph induced by the set of vertices labeled with 1 and 2, then \(f(x) = 1\) ; otherwise, every vertex u for which \(f(u) = 0\) is adjacent to at least one vertex v for which \(f(v) = 2\) . The value of \(\sum _{u\in V} f(u)\) represents the weight of quasi-total Roman domination. The minimum weight of quasi-total Roman dominating function on graph G is called the quasi-total Roman domination number of G indicated by \(\gamma _{qtR}(G)\) . There isn’t a polynomial time algorithm for the same problem because QTRD is NP-hard. Proposing an effective polynomial time solution can aid in overcoming the constraints encountered in applications, since the QTRD problem has applications in supply chain management, network security and defence, social network analysis, and telecommunications. To the best of our knowledge, QTRD problem lacks established meta-heuristic techniques, in contrast to domination problem. To combat this, we present in this work solutions to the QTRD problem based on the genetic algorithm. The suggested algorithm employs heuristics to create population and run them through several stages of algorithm to produce more workable answers. The QTRD problem’s performance of algorithm is evaluated and contrasted on a variety of random graphs created with the Erdős–Rényi model and the well-known graph dataset Harwell–Boeing (HB).