Adjusting an Infeasible Network by Minimizing the Sum of the Violation Costs

Abstract

KEYWORDS Infeasible network;The cost scaling algorithm; The minimum cost ow problem; Parallel arcs; Convex cost.Abstract. In this paper, for a given infeasible network, we change the lower and upper bounds, such that the sum of the violations costs from the lower and upper bounds is minimum. We call this problem as the adjusting problem and show that it is transformed to a minimum cost ow problem on a special parallel network. Thus, the adjusting problem is solved using minimum cost ow algorithms. Solving a minimum cost ow problem with parallel arcs, in practice, is complicated and needs more time in comparison with a minimum cost ow problem without parallel arcs. If the parallel arcs are eliminated, then, we achieve substantial saving in the storage requirements, which translates into enhanced speed of algorithms. One of the best algorithms to solve the minimum cost ow problem is the cost scaling algorithm of Goldberg and Tajan (1990). In this paper, we present two modi ed versions of their algorithm to solve the adjusting problem. In the rst modi cation, in order to achieve an enhanced speed of algorithm, the parallel arcs are eliminated using an especial residual network. In the second modi cation, the adjusting problem is transformed to a convex cost ow problem, and the cost scaling algorithm is modi ed in a way which performs fewer operations than our rst implementation.