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In this paper, we analyse the total cost minimization scheduling problem on a single machine, where jobs can have different release dates and the cost of a job is expressed as the increasing power function dependent on its completion time. We prove that this problem is at least NP-hard even if the cost weight of each job is equal to its processing time and release dates of all jobs are the same. Therefore, to solve the problem we implement parallel algorithms that are based on NEH, tabu search and simulated annealing. Numerical analysis shows the algorithms not only find solutions that are close to optimum, but the decreasing ratio of their running times (parallel tabu search and simulated annealing) is close to the number of threads, thereby multi-core CPU are efficiently utilized by these algorithms.