Multi-objective allocation of multi-function workers with lower bounded capacity

This paper deals with the assignment of a type of task to each member of a multi-function staff (each worker is able to perform a given subset of types of tasks, possibly with a priority index associated to each element of the subset). The resulting number of workers for each type of task must be not less than a given lower bound and as close as possible to another given value. The objectives are to minimise a function of the slacks and the surpluses of capacity, to distribute the slacks and the surpluses homogeneously among the types of task and to maximise the sum of priority indexes of the assignments. The problem is modelled as a nonlinear mixed integer program and is transformed and solved as a minimum cost flow problem.