<p>The problem of scheduling non-simultaneously released jobs with due-dates on a single machine with the objective to minimize the maximum job lateness is known to be strongly NP-hard. Here we consider an extended model in which the compression of the job processing times is allowed. The compression is accomplished at the cost of involving additional emerging resources. With a given upper limit <i>U</i> on the total allowable cost, one wishes to minimize the maximum job lateness. By using the available resources, some jobs may complete earlier and the objective function value may respectively be decreased. As we show here, by shortening the processing time of some specially determined jobs, the objective function value can be decreased. Although the extended problem is harder than the generic non-compressible setting, given a “sufficient amount” of additional resources, we can solve the problem optimally. We determine the compression rate for some specific jobs and develop an algorithm that obtains an optimal solution. Such an approach can be beneficial in practice since the manufacturer is provided with an information about the required amount of additional resources to solve the problem optimally. In the case the amount of the available additional resources is less than used in the above solution, i.e., it is not feasible, this solution is transformed into a feasible solution with a tight compression rate.</p>