Matching problems typically involve assigning agents to commodities, possibly on the basis of ordinal preferences or other metrics. These problems have large-scale applications to centralised matching schemes in many countries and contexts. In this talk I will describe the matching problems featuring in two such schemes in the UK that have involved collaborations between the National Health Service and the University of Glasgow. One of these dealt with the allocation of junior doctors to Scottish hospitals (1999-2012), and the other is concerned with finding kidney exchanges among incompatible donor-patient pairs across the UK (2007-date). In each case I will describe the applications, present the underlying algorithmic problems, outline the integer programming models for their solution and give an overview of results arising from real data connected with the matching schemes in recent years.
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