Abstract
Kidney exchange programs (KEPs) allow the exchange of kidneys between incompatible donor-recipient pairs. Optimization approaches can help KEPs in defining which transplants should be made among all incompatible pairs according to some objective. The most common objective is to maximize the number of transplants. In this paper, we propose an integer programming model which addresses the objective of maximizing the expected number of transplants, given that there are equal probabilities of failure associated with vertices and arcs. The model is compact, i.e. has a polynomial number of decision variables and constraints, and therefore can be solved directly by a general purpose integer programming solver (e.g. Cplex).
Highlights
Kidney transplantation is essential for survival of many patients suffering from chronic kidney disease for which there is no known cure
The known compact formulations for the Kidney exchange programs (KEPs) [3] are not suitable for solving it, as the presented problem has the weights associated to the types of cycles which may not be transformed into the weights of arcs and vertices
Integer programming formulation we present a new compact, i.e. with a polynomial number of decision variables and constraints, integer programming formulation for maximizing the expected number of transplants in a KEP when equal probabilities of failure are associated to arcs and vertices
Summary
Kidney transplantation is essential for survival of many patients suffering from chronic kidney disease for which there is no known cure. The kidney exchange problem (KEP) for maximizing the number of planned transplants can be defined as follows: find a packing of vertex-disjoint cycles with length at most K with maximum number of arcs. The expected number of transplants for this cycle is (see [4]): E(c) = 3(1 − p1)(1 − p2)(1 − p3) + 2(1 − p1)(1 − p2)p3 and in case of equal probabilities of vertices failure resumes to: E(c) = 3(1 − pv)3 + 2pv(1 − pv)2
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