Location of a discrete resource and its allocation according to a fractional objective

The problem considered is that of the location of a discrete resource and its allocation to activities with concave return functions in such a way as to maximize the ratio of ‘return’ to ‘cost’, the total cost being the sum of a fixed cost and linearly variable costs. It is assumed that each resource has an effectiveness of 0 or 1 against each activity. It is demonstrated that an optimal solution can be determined by the rounding to integers of the solution of an associated problem in continuous variables. Solutions with objective values arbitrarily close to the optimal value can be generated by resource-wise optimizations. An upper bound of the number of non-zero integer allocations in an optimal solution is derived.