Given an undirected graph \(G=(V,E)\) with weighted vertices and edges, a set J of \(n=|V|\) independent jobs and m unrelated machines, each vertex \(v\in V\) corresponds to a job \(J_j\in J\) . The combination of prize-collecting vertex cover and scheduling (CPVCS) is to select a subset \(C\subseteq V\) of vertices and schedule the jobs corresponding to the vertices in C on m unrelated machines, edges that are not covered incur a penalty cost, such that the sum of makespan and the total weight of the uncovered edges by C is minimized, where the makespan is defined as the maximum load of the machines. When the number of machines is an input number, we design a 4-approximation algorithm based on LP-rounding method. When the number of machines is a constant, we design a 3-approximation algorithm by guessing the job with maximum processing time on each machine.