The out-of-kilter algorithm applied to traffic congestion

The out-of-kilter algorithm finds a minimum cost assignment of flows to a network defined in terms of one-way arcs, each with upper and lower bounds on flow, and a cost. It is a mathematical programming algorithm which exploits the network structure of the data. The objective function, being the sum taken over all the arcs of the products, cost×flow, is linear. The algorithm is applied in a new way to minimise a series of linear functions in a heuristic method to reduce the value of a non-convex quadratic function which is a measure of traffic congestion. The coefficients in these linear functions are chosen in a way which avoids one of the pitfalls occurring when Beale's method is applied to such a quadratic function. The method does not guarantee optimality but has produced optimal results with networks small enough for an integer linear programming method to be feasible.