For some hard-to-fill jobs the Navy awards Assignment Incentive Pay using an auction-like format. With respect to the optimal job assignment auction format, however, there is only a very limited academic literature. Furthermore, the extant literature assumes all bidders are equally qualified. In the Navy assignment context, this is not a tenable assumption, as other considerations, such as relocation and en-route training costs, must be considered when making an assignment. The lower the weight on the bid, the greater the weight that can be attached to the qualification component in the objective function. The lower the weight, however, the weaker the incentive to bid near one's reservation wage. The consideration of such other criteria precludes the implementation of the incentive-compatible, Vickery-Leonard assignment auction. We relax the assumption that bid amounts alone determine the assignment set and experimentally estimate the efficiency reductions associated with decreased bid-weights. The estimated elasticity of the value of the bids to changes in the bid-weight in low contention, first price auctions vary by bid-weight. Nonetheless, an increase from a 10 to a 50% weight on the bids decreases the level of the submitted bids by approximately 28%.