The considered problem is how to optimally allocate a set of jobs to technicians of different skills such that the number of technicians of each skill does not exceed the number of persons with that skill designation. The key motivation is the quick sensitivity analysis in terms of the workforce size which is quite necessary in many industries in the presence of unexpected work orders. A time-indexed mathematical model is proposed to minimize the total weighted completion time of the jobs. The proposed model is decomposed into a number of single-skill sub-problems so that each one is a combination of a series of nested binary Knapsack problems. A heuristic procedure is proposed to solve the problem. Our experimental results, based on a real-world case study, reveal that the proposed method quickly produces a schedule statistically close to the optimal one while the classical optimal procedure is very time-consuming.