Cutting Plane Approaches for the Robust Kidney Exchange Problem

ObjectivesLiving donor kidney transplantation (LTx) is the preferred treatment for patients with end-stage renal disease. Kidney exchange programs (KEPs) promote LTx by facilitating exchange of donors among patients who are not compatible with their donors. We analyze and maximize the efficacy and effectiveness of KEPS from a health value perspective and the health value of altruistic donation in KEPs. MethodsWe developed a Markov model for the health outcomes of patients, which was embedded in a discrete event simulation model to assess the effectiveness of allocation policies in KEPs. A new allocation policy to maximize health value was developed on the basis of integer programing techniques. The evidence-based transition probabilities in the Markov model were based on data from the Dutch KEP using a variety of econometric models. Scenarios analysis was presented to improve robustness. ResultsThe efficacy of the Dutch KEP without altruistic donation is reflected by the increase in expected discounted quality-adjusted life-years (QALYs) by 3.23 from 6.42 to 9.65. The present Dutch policy and the policy to maximize the number of transplants achieve 63% of the potential efficacy gain (2.11 discounted QALYs). The new policy achieves 69% of this gain (2.33 discounted QALYs). When systematically enrolling altruistic donors in the KEP, the new policy increased expected discounted QALYs by 4.05 to 10.27 and reduced inequities for patients with blood type O. ConclusionsThe Dutch KEP can increase health value for patients by more than half. An allocation policy that maximizes health outcomes and maximally allows altruistic donation can yield significant further improvements.

In kidney exchange programs, multiple patient-donor pairs each of whom are otherwise incompatible, exchange their donors to receive compatible kidneys. The Kidney Exchange problem is typically modelled as a directed graph where every vertex is either an altruistic donor or a pair of patient and donor; directed edges are added from a donor to its compatible patients. The computational task is to find if there exists a collection of disjoint cycles and paths starting from altruistic donor vertices of length at most $\ell_c$ and $\ell_p$ respectively that covers at least some specific number t of non-altruistic vertices (patients). We study parameterized algorithms for the kidney exchange problem in this paper. Specifically, we design FPT algorithms parameterized by each of the following parameters: (1) the number of patients who receive kidney, (2) treewidth of the input graph + $\max\\ell_p,\ell_c\ $, and (3) the number of vertex types in the input graph when $\ell_płeq \ell_c$. We also present interesting algorithmic and hardness results on the kernelization complexity of the problem. Finally, we present an approximation algorithm for an important special case of Kidney Exchange.

In kidney exchange programs, multiple patient-donor pairs each of whom are otherwise incompatible, exchange their donors to receive compatible kidneys. The Kidney Exchange problem is typically modelled as a directed graph where every vertex is either an altruistic donor or a pair of patient and donor; directed edges are added from a donor to its compatible patients. The computational task is to find if there exists a collection of disjoint cycles and paths starting from altruistic donor vertices of length at most l_c and l_p respectively that covers at least some specific number t of non-altruistic vertices (patients). We study parameterized algorithms for the kidney exchange problem in this paper. Specifically, we design FPT algorithms parameterized by each of the following parameters: (1) the number of patients who receive kidney, (2) treewidth of the input graph + max{l_p, l_c}, and (3) the number of vertex types in the input graph when l_p <= l_c. We also present interesting algorithmic and hardness results on the kernelization complexity of the problem. Finally, we present an approximation algorithm for an important special case of Kidney Exchange.

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).

In kidney transplantation, immunosuppressants are being used to relax all immunological constraints, allowing patients to receive transplants from biologically incompatible donors. We propose an efficient way of using immunosuppressants in combination with a kidney exchange program. We introduce the top‐trading cycles and chains solution, a modification of the top‐trading cycles rule for the standard kidney exchange problem, and investigate its properties.

Kidney exchanges were developed to match kidney failure patients with willing but incompatible donors to other donor-patient pairs. Finding a match in a large candidate pool can be modeled as an integer program. However, these exchanges accumulate participants with characteristics that increase the difficulty of finding a match and, therefore, increase patients' waiting time. Therefore, we sought to fine-tune the formulation of the integer program by more accurately assigning priorities to patients based on their difficulty of matching. We provide a detailed formulation of prioritized kidney exchange and propose a novel prioritization algorithm. Our approach takes advantage of the global knowledge of the donor-patient compatibility within a pool of pairs and calculates an iterative, paired match power (iPMP) to represent the donor-patient pairs' abilities to match. Monte Carlo simulation shows that an algorithm using the iPMP reduces the waiting time more than using paired match power (PMP) for the difficult-to-match pairs with hazard ratios of 1.3480 and 1.1100, respectively. Thus, the iPMP may be a more accurate assessment of the difficulty of matching a pair in a pool than PMP is, and its use may improve matching algorithms being used to match donors and recipients.

In Kidney exchange programs patients with chronic kidney disease and a willing but incompatible donor are joined in a pool and preliminary compatibility between patient in one pair and donors in other pairs is evaluated. Based on this information a set of transplants is planned. After a transplants plan is set, additional compatibility tests are performed for each pair selected for transplant. These tests can elicit extra incompatibilities and lead to cancellation of planned transplants. In this paper we propose to conduct some crossmatch tests before a transplantation plan is set and address the issue of how many additional transplants can be performed for a given number of tests. Three different policies are proposed to decide which pairs should be selected for crossmatch. They differ in the rules used to update the current plan, whenever a new incompatibility is found. Comparison of results, in terms of transplants performed with and without a priori crossmatch shows that more flexible policies allow for substantial improvement on the actual number of transplants for a small number of tests.

Living donor kidney transplantation is the preferred treatment for patients suffering from end-stage renal disease. To alleviate the shortage of kidney donors, many advances have been made to improve the utilization of living donors deemed incompatible with their intended recipient. The most prominent of these advances is kidney paired donation (KPD), which matches incompatible patient-donor pairs to facilitate a kidney exchange. This review discusses the various approaches to matching and allocation in KPD. In particular, it focuses on the underlying principles of matching and allocation approaches, the combination of KPD with other strategies such as ABO incompatible transplantation, the organization of KPD, and important future challenges. As the transplant community strives to balance quantity and equity of transplants to achieve the best possible outcomes, determining the right long-term allocation strategy becomes increasingly important. In this light, challenges include making full use of the various modalities that are now available through integrated and optimized matching software, encouragement of transplant centers to fully participate, improving transplant rates by focusing on the expected long-run number of transplants, and selecting uniform allocation criteria to facilitate international pools.

Surrogate duality for robust optimization

This paper presents a new approach aims to solve robust multidisciplinary design optimization MDO problem called Improved Multi-objective Robust Collaborative Optimization . This method combines the Multi-objective Robust Collaborative Optimization method, the Worst Possible Point constraint cuts and the Genetic algorithm NSGA-II type as an optimizer to solve the robust optimization problem of complex structure named Y-stiffened panel under interval uncertainty . The proposed approach hierarchically decomposes the optimization problem into a structure level considered as an upper level in the Y-stiffened panel and a second level considered as a lower level of the studied panel. A robust multi-objective optimization problem intended to optimize the eigenfrequency, the global mass and the displacement at a fixed point of the Y-stiffened panel at the first level and each structure’s robust optimization problem allows optimizing its eigenfrequency and mass limited by their local constraint functions at the second one. Tor demonstrate our method, an engineering example of Y-stiffened panel is treated. A good performance of proposed method is proved by a comparison between obtained results and Non-Distributed Multi-objective Robust Optimization .

A kidney exchange is an organized barter market where patients in need of a kidney swap willing but incompatible donors. Determining an optimal set of exchanges is theoretically and empirically hard. Traditionally, exchanges took place in cycles, with each participating patient-donor pair both giving and receiving a kidney. The recent introduction of chains, where a donor without a paired patient triggers a sequence of donations without requiring a kidney in return, increased the efficacy of fielded kidney exchanges---while also dramatically raising the empirical computational hardness of clearing the market in practice. While chains can be quite long, unbounded-length chains are not desirable: planned donations can fail before transplant for a variety of reasons, and the failure of a single donation causes the rest of that chain to fail, so parallel shorter chains are better in practice. In this paper, we address the tractable clearing of kidney exchanges with short cycles and chains that are long but bounded. This corresponds to the practice at most modern fielded kidney exchanges. We introduce three new integer programming formulations, two of which are compact. Furthermore, one of these models has a linear programming relaxation that is exactly as tight as the previous tightest formulation (which was not compact) for instances in which each donor has a paired patient. On real data from the UNOS nationwide exchange in the United States and the NLDKSS nationwide exchange in the United Kingdom, as well as on generated realistic large-scale data, we show that our new models are competitive with all existing solvers---in many cases outperforming all other solvers by orders of magnitude. Finally, we note that our position-indexed chain-edge formulation can be modified in a straightforward way to take post-match edge failure into account, under the restriction that edges have equal probabilities of failure. Post-match edge failure is a primary source of inefficiency in presently-fielded kidney exchanges. We show how to implement such failure-aware matching in our model, and also extend the state-of-the-art general branch-and-price-based non-compact formulation for the failure-aware problem to run its pricing problem in polynomial time.

Mars entry trajectory planning using robust optimization and uncertainty quantification

In Chapter 4, several polynomially solvable cases of robust discrete optimization problems are discussed in detail. However, by results presented in Chapter 3 we also know that most robust discrete optimization problems belong to the NP-hard class. In this chapter, we present our approach for solving these difficult robust discrete optimization problems. We are in this chapter restricting our attention to robust discrete optimization problems with equivalent single scenario problems that can be efficiently solved with a polynomial or pseudo-polynomial procedure. The solution procedures are based on branch-and-bound with both upper and lower bounds generated by surrogate relaxation. To be exact, the upper bound (for a maximization problem) is obtained from surrogate relaxation and the lower bound is obtained as a by-product via a heuristic based on the surrogate relaxation result. In the case when input data satisfies bounded percentage deviation condition (to be defined in Section 5.2), the heuristic is shown to provide a constant approximation. Computational results in Section 5.3 demonstrate the effectiveness of the bounds and the solution procedure.

We study the stochastic matching problem with the goal of finding a maximum matching in a graph whose edges are unknown but can be accessed via queries. This is a special case of stochastic k-cycle packing, in which the problem is to find a maximum packing of cycles, each of which exists with some probability. We provide polynomial-time adaptive and nonadaptive algorithms that provably yield a near-optimal solution, using a number of edge queries that is linear in the number of vertices. We are especially interested in kidney exchange, with which pairs of patients with end-stage renal failure and their willing but incompatible donors participate in a mechanism that performs compatibility tests between patients and donors and swaps the donors of some patients so that a large number of patients receive compatible kidneys. Because of the significant cost of performing compatibility tests, currently, kidney exchange programs perform at most one compatibility test per patient. Our theoretical results applied to kidney exchange show that, by increasing the number of compatibility tests performed per patient from one to a larger constant, we effectively get the full benefit of exhaustive testing at a fraction of the cost. We show, on both generated and real data from the UNOS nationwide kidney exchange, that even a small number of nonadaptive edge queries per vertex results in large gains in expected successful matches.

BackgroundLiving donor kidney transplantation (LDKT) is the preferred treatment for patients with end-stage renal disease and unspecified living kidney donation is morally justified. Despite the excellent outcomes of LDKT, unspecified kidney donation (UKD) is limited to a minority of European countries due to legal constraints and moral objections. Consequently, there are significant variations in practice and approach between countries and the contribution of UKD is undervalued. Where UKD is accepted as routine, an increasing number of patients in the kidney exchange programme are successfully transplanted when a ‘chain’ of transplants is triggered by a single unspecified donor. By expanding the shared living donor pool, the benefit of LDKT is extended to patients who do not have their own living donor because a recipient on the national transplant list always completes the chain. Is there a moral imperative to increase the scope of UKD and how could this be achieved?MethodsAn examination of the literature and individual country practices was performed to identify the limitations on UKD in Europe and recommend strategies to increase transplant opportunities.ResultsPrimary limitations to UKD, key players and their roles and responsibilities were identified.ConclusionsRaising awareness to encourage the public to volunteer to donate is appropriate and desirable to increase UKD. Recommendations are made to provide a framework for increasing awareness and engagement in UKD. The public, healthcare professionals, policy makers and society and religious leaders have a role to play in creating an environment for change.