In this paper we study two formulation reductions for the quadratic assignment problem (QAP). In particular we apply these reductions to the well known Adams and Johnson  integer linear programming formulation of the QAP, which we call formulation IPQAP-I. We analyze two cases: In the first case, we study the effect of constraint reduction. In the second case, we study the effect of variable reduction in the case of a sparse cost matrix. Computational experiments with a set of 32 QAPLIB instances, which range from 12 to 32 locations, are presented. The proposed reductions turned out to be very effective: By applying the new constraint reduction or the new variable reduction to the IPQAP-I formulation, we solved 13 and 23 instances, respectively, compared to the 7 instances solved by formulation IPQAP-I.