After the opening of the said bids, the Council shall revise the estimated cost of the said improvement upon the basis of the bids received or, where the work is to be done by the City, shall revise the estimated cost on the basis of the estimate of the City Manager, and shall direct the Board of Special Assessors to make a special assessment roll of the part of proportion of the costs to be borne by the lands especially benefited as determined by the Council.