Efficient overtime planning in packing shops with lines of identical manning

We consider a capacity planning problem arising at the packaging stage in many process industries. Termed economic manpower shift planning with overtime (EMSP-O), it seeks the manpower and overtime to be planned at each workday shift so that production targets for all packing lines are met at minimum cost. The problem was previously modelled as an ILP and its solution explored. Here we present a new EMSP-O model that relaxes a restrictive assumption of the original model. We show that the relaxed EMSP-O is NP-hard and focus on the special case where all lines have identical manning. For this case, we establish properties of the optimal solution and show that, with no overtime constraints, an O(N log N) algorithm can find an optimal solution. With overtime constraints, the algorithm effectively constitutes a heuristic. Computations and comparisons with a commercial optimizer for two- and three-shift problems (as found in practice) were performed. Results demonstrate the algorithm efficiency which, except for very tight overtime constraints, reaches an optimum at high speed.