The "lowest tender (LT)" has been widely adopted in construction industry due to its simplicity, objectivity and less human involvement in the contractor selection process. However, the LT also suffers in its inevitable drawback of lowest price competitiveness that usually results in inferior quality compared with other methods. To tackle the disadvantages of traditional LT, this paper proposes an A + Q Contractor Selection Model (A + Q) based on previous model of Market Price Elasticity of Performance (E_m) and Historical Contract Performance Evaluation Data (HCPED) of the contractors to form a new contractor selection method for project owner. Based on the results of a case study with the HCPED of building projects and a set of simulated bidding data, the A + Q is shown to select the most advantageous contractor for the owner. The proposed A + Q does not only include the strengths of fast and streamline selection process of traditional LT, but also retain the benefits of the BV method. Moreover, the proposed A + Q tends to encourage the contractor in improving their quality and schedule performance by incorporating the past performance of the contractors into the selection process. As a result, the proposed A + Q has the potential to enhance traditional LT not only in terms of the benefit of the owner, but also for a healthier construction procurement market in the long run.