  In this paper, we propose an exact general algorithm for solving non-convex
optimization problems, where the non-convexity arises due to the presence of an
inverse S-shaped function. The proposed method involves iteratively
approximating the inverse S-shaped function through piece-wise linear inner and
outer approximations. In particular, the concave part of the inverse S-shaped
function is inner-approximated through an auxiliary linear program, resulting
in a bilevel program, which is reduced to a single level using KKT conditions
before solving it using the cutting plane technique. To test the computational
efficiency of the algorithm, we solve a facility location problem involving
economies and dis-economies of scale for each of the facilities. The
computational experiments indicate that our proposed algorithm significantly
outperforms the previously reported methods. We solve non-convex facility
location problems with sizes up to 30 potential facilities and 150 customers.
Our proposed algorithm converges to the global optimum within a maximum
computational time of 3 hours for 95 percent of the datasets. For almost 60
percent of the test cases, the proposed algorithm outperforms the benchmark
methods by an order of magnitude. The paper ends with managerial insights on
facility network design involving economies and dis-economies of scale. One of
the important insights points out that it may be optimal to increase the number
of production facilities operating under dis-economies of scale with an overall
decrease in transportation costs.
