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In recent years, multi-access edge computing (MEC) has become a promising technology used in 5G networks based on its ability to offload computational tasks from mobile devices (MDs) to edge servers in order to address MD-specific limitations. Despite considerable research on computation offloading in 5G networks, this activity in multi-tier multi-MEC server systems continues to attract attention. Here, we investigated a two-tier computation-offloading strategy for multi-user multi-MEC servers in heterogeneous networks. For this scenario, we formulated a joint resource-allocation and computation-offloading decision strategy to minimize the total computing overhead of MDs, including completion time and energy consumption. The optimization problem was formulated as a mixed-integer nonlinear program problem of NP-hard complexity. Under complex optimization and various application constraints, we divided the original problem into two subproblems: decisions of resource allocation and computation offloading. We developed an efficient, low-complexity algorithm using particle swarm optimization capable of high-quality solutions and guaranteed convergence, with a high-level heuristic (i.e., meta-heuristic) that performed well at solving a challenging optimization problem. Simulation results indicated that the proposed algorithm significantly reduced the total computing overhead of MDs relative to several baseline methods while guaranteeing to converge to stable solutions.