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A new unrelated parallel machine scheduling problem with deteriorating effect and the objective of makespan minimization is presented in this paper. The deterioration of each machine (and therefore of the job processing times) is a function of the sequence of jobs that have been processed by the machine and not (as considered in the literature) by the time at which each job is assigned to the machine or by the number of jobs already processed by the machine. It is showed that for a single machine the problem can be solved in polynomial time, whereas the problem is NP-hard when the number of machines is greater or equal than two. For the last case, a set of list scheduling algorithms and simulated annealing meta-heuristics are designed and the effectiveness of these approaches is evaluated by solving a large number of benchmark instances.