Partial ranking in sets of partial solutions of problems of linear and quadratic assignment

We present an approach to Quadratic Assignment Problem, QAP, via a linear relaxation, through the use of the Linear Assignment Problem, LAP. We define one ordering an LAP-solution-set that makes us compare QAP costs. From here, this poset, partial ordering set, considers coordinates of instances as free variables. We have managed to build a capable algorithm to determine freely comparable permutation pairs. A theorem shows that the order mentioned above is compatible to the one that involves inversion numbers of permutations. We can apply linear and quadratic solutions to permutations and empiric tests are presented to show that the inversion numbers can be used to measure the quality of QAP-solutions.