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A Recourse Policy to Improve Number of Successful Transplants in Uncertain Kidney Exchange Programs

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Abstract

Uncertainty of kidney exchange programs (KEP) is one of the main challenges that cause failure in most planned transplants to proceed to surgery. Unfortunately, this challenge usually decreases KEP performance in the real world. To face this challenge, we develop an efficient approach, consisting of scheduling methodology and recourse actions, to recover failing pairs and improve the number of successful transplants in KEPs. The developed policy can be implemented in two different schemes. The first scheme receives an optimal matching as an input and determines recourse actions and the sequence of disjoint cycles and chains for the given optimal matching. However, the second scheme does not need optimal matching. This scheme simultaneously receives the compatibility graph as input and specifies the optimal matching, cycles and chains’ schedule, and corresponding recourse actions. Moreover, we investigate the performance of the developed approach under various circumstances to validate its positive impact on KEP performance. The results demonstrate that this approach increases the number of successful transplants in KEPs and decreases the number of remaining highly sensitive and O blood type patients in the pool.

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Authors and Affiliations

  1. Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran

    Mehdi Zeynivand, Mehdi Najafi & Mohammad Modarres Yazdi

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  1. Mehdi Zeynivand
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Appendices

Appendix A: Upper Bound Estimation for Tiem Slot

Since the smCSRA determines the optimal matching, the number of cycles and chains in the optimal matching is unknown before running the model. However, we know that optimal matching includes a set of disjoint cycles and chains. Therefore, the maximum number of disjoint cycles and chains can be calculated as follows:

  • Case A. |N1| <|N2|., the number of incompatible pairs is smaller than the number of altruistic donors.

In this case, the maximum number of cycles and chains could not be more than a situation in which the smCSRA scheme assigns one altruistic donor to each patient. Therefore, the maximum cycles and chains in the optimal matching could not be larger than |N1|.

  • Case B. |N1| ≥|N2|, the number of incompatible pairs is greater than the number of altruistic donors.

Suppose the number of incompatible pairs is greater than or equal to the number of altruistic donors. In that case, the maximum number of components (i.e., cycles or chains) in the optimal matching is achieved when each altruistic donor triggers a chain with only one incompatible pair and the other incompatible pairs form cycles of length two. Under such a situation, the number of chains in the optimal matching is |N2|. Furthermore, the rest (|N1| -|N2|) pairs can form at most \(\left\lfloor {\frac{{\left| {N_{1} } \right| - \left| {N_{2} } \right|}}{2}} \right\rfloor\) cycles with length 2. Therefore, the maximum components in the optimal matching could not be greater than \(\left\lfloor {\frac{{\left| {N_{1} } \right| - \left| {N_{2} } \right|}}{2}} \right\rfloor + \left| {N_{2} } \right|\). Hence, the upper bound of parameter M is calculated as follows:

\(M^{LB} = \left\{ \begin{gathered} \left| {N_{1} } \right|,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| {N_{1} } \right| < \left| {N_{2} } \right|\, \hfill \\ \left\lfloor {\frac{{\left| {N_{1} } \right| - \left| {N_{2} } \right|}}{2}} \right\rfloor + \left| {N_{2} } \right|,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left| {N_{1} } \right| \ge \left| {N_{2} } \right|\,. \hfill \\ \end{gathered} \right.\).

Appendix B: An Illustrative Example

Suppose that Fig. 11 shows the compatibility graph of a KEP with seven incompatible pairs. There are two different matchings including M1 = {c2 = (6–7), c3 = (3–4–5)} and M2 = {c1 = (4–5), c2 = (6–7), c4 = (1–2–3)}.

Fig. 11
figure 11

Compatibility graph of the KEP with seven incompatible pairs

Full size image

Assume that the BA selects matching M1. If CSRA chooses the M1, it achieves the same as the BA. However, selecting matching M2 with sequence (c4, c1, c2) by the CSRA results in a better outcome because each failure in cycle c4 leads to matching M1 by adding failing pair 3 between pairs 5 and 4 in cycle c1. Therefore, both policies result in the same number of transplants in case of failing cycle c4. On the other hand, if the BA selects matching M2 according to the calculated expected number of transplants, the CSRA certainly would select M2 as the optimal matching due to its objective function. Note that the CSRA considers transplants in both primary cycles and recourse actions. Hence, it prefers matching M2 to matching M1. In this case, pairs 1, 2, and 3 do not have any chance of receiving a kidney in case of failure in cycle c4 if the BA is adopted for the planning. However, the CSRA provides an opportunity for pair 3 by adding it to cycle c1 = (4–5).

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Zeynivand, M., Najafi, M. & Yazdi, M.M. A Recourse Policy to Improve Number of Successful Transplants in Uncertain Kidney Exchange Programs. J Optim Theory Appl 197, 476–507 (2023). https://doi.org/10.1007/s10957-023-02211-6

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