In the injection moulding process, a set-up containing positioning moulds, a cleaning raw material hopper and machine settings is required. The set-up is uncertain owing to random factors, such as the crew's skill levels and unexpected breakdowns. The general version of this problem is known as the single-machine scheduling problem with stochastic sequence-dependent set-up times. In this specialized form of the problem, the objective function is defined so as to minimize the total expected cost, which includes set-up, tardiness and earliness costs. Moreover, the effect of the client-dependent tardiness cost on the optimal machine schedule under uncertainty is explored. As a solution approach, a two-stage stochastic programming method is used. For the solution of large-size problems, the harmony search algorithm is proposed, and its performance is compared with the exact solution approach. Finally, the value of the stochastic solution is assessed by comparing the results with the deterministic approach.