Market clearing is the process of matching buy and sell bids in securities markets. The allocative efficiency of such algorithms is important, as the Auctioneer is typically paid a commission on the number of bids matched and the volume of quantity traded. Previous algorithms have concentrated on price issues. This paper presents several market clearing algorithms that focus solely on allocating quantity among matching buy and sell bids. The goal is to maximise the number of bids matched, while at the same time minimise the amount of unmatched quantity. The algorithms attempt to avoid situations resulting in unmarketable quantities (i.e., quantities too small to sell). Algorithmic performance is tested using simulated data designed to emulate the Australian Stock Exchange (ASX) and other world stock markets. Our results show that it is difficult to avoid partial matchings as the complexity of doing so is NP-complete. The optimal offline algorithm for partial quantity matching is used as a benchmark to compare online matching strategies. We present three algorithms that outperform the ASX's strategy by increasing the number of bids matched, the amount of quantity matched, and the number of bids fully matched.