Maximizing the Efficiency of the U.S. Liver Allocation System through Region Design

Transcription

1 Maximizing the Efficiency of the U.S. Liver Allocation System through Region Design Nan Kong Weldon School of Biomedical Engineering, Purdue University West Lafayette, IN Andrew J. Schaefer Department of Industrial Engineering, University of Pittsburgh Pittsburgh, PA Brady Hunsaker Department of Industrial Engineering, University of Pittsburgh Pittsburgh, PA Mark S. Roberts Department of Medicine, University of Pittsburgh School of Medicine Pittsburgh, PA Cadaveric liver transplantation is the only viable therapy for end-stage liver disease patients without a living donor. However, this type of transplantation is hindered in the United States by donor scarcity and rapid viability decay. Given these difficulties, the current U.S. liver transplantation and allocation policy attempts to balance allocation likelihood and geographic proximity by allocating cadaveric livers hierarchically. In this paper, we consider the problem of maximizing the efficiency of intra-regional transplants through the design of liver harvesting regions. We formulate the problem as a set partitioning problem that clusters organ procurement organizations (OPOs) into regions. We define the estimate of viability-adjusted transplant quantity to capture the tradeoff between large and small regions. We utilize branch and price because the set partitioning formulation includes too many potential regions to handle explicitly. We formulate the pricing problem as a mixed-integer program and design a geographic-decomposition heuristic to generate promising columns quickly. As the optimal solution depends on the initial design of geographic decomposition, we develop an iterative procedure that integrates branch and price with local search to alleviate this dependency. Finally, we present computational studies that show the benefit of region design and the efficacy of our solution approach. Our test instances are generated based on real clinical data. These instances can be solved within a reasonable amount of time and the resulting region designs yield an improvement equivalent of over 130 transplants a year. Key words: Set Partitioning; Branch and Price; Column Generation; Clustering; Organ Allocation 1 Introduction In the United States, end-stage liver disease (ESLD) is the twelfth-leading cause of death, which accounted for nearly 30,000 deaths in 2006 alone [14]. At present, the only viable therapy is liver transplantation and most ESLD patients receive liver donations from cadaveric donors. Unfortunately, the supply of cadaveric livers is far smaller than the demand. Furthermore, procured livers are often underutilized due to various clinical factors. Therefore, a major concern is the efficient allocation of cadaveric livers. 1

2 The United Network for Organ Sharing (UNOS) operates the organ transplantation and allocation system in the U.S. Currently, there are nearly 60 local organ procurement organizations (OPOs). Each OPO is responsible for procuring transplantable organs in a designated service area (see Figure 1), matching organs with patients, and allocating organs to matched patients within a reasonable time frame. Figure 1: OPO Service Areas [9, 26] In the current liver allocation process, higher priority is generally given to more sick patients who are biologically matched to the donor liver when it is procured. However, current practice also gives preference to patients living near to donors. Part of the rationale behind this practice is that reduced transport time of donor livers improves liver viability [15, 25] and consequently posttransplant outcomes [25]. Hence, UNOS also takes geographic proximity into account, through the development of local and regional prioritization. With a few exceptions, the allocation process uses the following hierarchy. Once a liver is procured at an OPO, it is first offered to a suitable candidate that is registered on the waiting list of the same OPO. If the liver is not accepted at the procurement OPO, it is offered to a larger area, or region, that consists of one or more OPOs. Currently, the entire country is divided into 11 regions (see Figure 2). If a liver is not accepted in the procurement region, it is offered nationwide. We call the three levels of geographic proximity the local, regional, and national levels. In fact, a small number of most seriously and acutely ill patients (who are termed Status 1 ) sometimes receive livers from other OPOs in their region before other patients in the donor s OPO. However, there are typically no more than 15 such patients awaiting at any one time, so we ignore this possibility in our model. Among the three tiers, transplants at the regional level represent more than 50% of total transplants in many parts of the country, and this percentage has been increasing recently [28]. These regions make a procured liver more likely to find a matched patient within a relatively small geographic area. A critical factor of efficient allocation is the geographic composition of regions. Ideally, regions that have sufficient population create more donor-recipient matches but geographically small enough so that donor liver transport times are not too great. The current regional configuration appears to have been developed through an ad hoc procedure, we are unaware of any quantitative analysis that informed the development. In this paper we develop a mathematical framework that aids in designing an optimal set of regions to improve intra-regional transplant efficiency. We call this problem the optimal region design problem. There are several mathematical and computer simulation models in the literature on liver alloca- 2

3 Figure 2: Current Region Map [27] tion. Most of the previous mathematical models are stochastic models that focus on transplantation timing and eligibility. They aim at improving allocation decisions at the operational phase. These works can be classified into two categories: those that consider the centralized decision maker s perspective, seeking optimal strategies of matching multiple livers with a set of candidates [32, 22] and those that consider the potential recipient s perspective in accepting or rejecting a liver offer [1, 3, 4]. [5] provide a recent survey. Most of previous work that considers improving allocation policy uses simulation-based analysis to examine how alternative allocation policies can affect system outcomes. Under a contract from the Health Resources and Services Administration (HRSA), the Scientific Registry of Transplant Recipient (SRTR) has built a family of simulations that can inform the policy process. [24] describe this family of models and initial experience in the application of one member of this family, the Liver Simulated Allocation Model (LSAM). [18] develop a more clinically based simulation model, the Liver Allocation Simulation Model (LASM), that evaluates and compares alternative policies in terms of waiting time and post-transplant survival rates. Similar simulation models have been developed for renal [23] and heart [29] transplants. Although the aforementioned simulations may evaluate various policies, they cannot find optimal policies. This paper addresses liver allocation region design, which is an important issue in policy development. To the best of our knowledge, [20] is the only work that studies optimal liver allocation region design. However, the paper assumes all harvested donor livers are allocated at the regional level and eliminates large potential regions from consideration due to computational intractability. Our paper extends their work in two significant aspects. First, we refine the regional transplant efficiency estimate to incorporate more realism. Second, we apply a branch-and-price method to consider vastly more potential regions. Hence, our main contribution in this paper is developing an appropriate mathematical model that captures the essential features of an ideal region design and enables the comparison of an enormous number of feasible region designs to help determine a more beneficial design. In the next section, we develop a mathematical model of the optimal region design problem that refines the estimate of intra-regional transplant efficiency and enables us to consider many more potential regions via column generation. In Section 3, we give details of how our branchand-price methodology can be practically implemented and improved via an iterative procedure. In Section 4, we describe computational experiments based on the recent UNOS liver procurement 3

4 and transplantation data and compare the solutions provided by our methods with an enumeration method. We conclude the paper and discuss our future research in Section 5. 2 A Mathematical Framework A regional configuration is defined as a partition of the entire country into a number of regions such that each region encompasses one or more OPO service areas. The region design problem can be modeled as a standard set-partitioning problem. An intra-regional transplant is a transplant for which the donor and recipient are from different OPOs in the same region. Let I be the set of OPOs and R be the set of all potential regions. Let x r be a binary decision variable for each r R, where x r = 1 means that region r is chosen in the solution and x r = 0, otherwise. Let the coefficient a ir = 1 if OPO i is contained in region r, and a ir = 0 otherwise. Each OPO i has an associated set-partitioning constraint r R a irx r = 1. The set-partitioning model is thus presented as follows: max r R c rx r s.t. r R a irx r = 1, for all i I; x r {0,1}, for all r R. (1) Each objective coefficient c r, r R, measures some overall benefit of potential region r that is accumulated by all intra-regional transplants taking place in the region and adjusted by the viability of the transplanted livers. 2.1 Regional Benefit Estimation To evaluate the macro-level benefit of clustering OPOs in r, we introduce the concept of viabilityadjusted transplants. It is known that viability decay is one of the main reasons for a patient to reject a liver offer and a transplanted liver fails to function after the transplant [15, 25]. Organ viability is usually measured by cold-ischemia time (CIT), the time interval between when the blood is stopped flowing to the organ in the donor and when the blood flow is restored in the recipient. Prolonged CIT worsens donor liver viability and post-transplant outcomes [15]. In recent years, due to increasing long-distance distribution of livers, it has been observed that travel distance is the main factor that prolongs CIT [25] for donor livers. Hence, higher transplant benefit, in general, is associated with shorter distance traveled by the donor liver. We assume that once a liver is allocated intra-regionally, its transplant benefit depends only on the travel distance between the donor and recipient OPOs. This assumption implies that the transplant benefit is identical for donors with different clinical and demographic characteristics. It also implies that given a donor liver, the transplant benefit is identical for patients with different clinical and demographic characteristics. We define α ij, 0 α ij 1, to be a transport factor, given that the liver harvested at OPO i and a recipient at OPO j, for i,j I r, i j. We interpret α ij as follows. The benefit of transplanting a donor liver that is procured at OPO i to a patient at OPO j is α ij times the benefit of transplanting that liver to an identical patient from OPO i, where we assume there is no viability loss due to transportation. As stated earlier, liver viability loss increases as the transport distance increases. Hence, the inclusion of α ij penalizes geographically large regions in optimal region design. As discussed in Section 1, a three-tier hierarchical model can be used for analyzing the allocation process. In this model, livers are offered to matched patients within each OPO before outside the OPO. Thus, the choice of regions does not affect those transplants that occur at the local level and 4

5 we only need to consider those livers that are not matched locally. Furthermore, we do not model national-level allocation exactly in the estimate. This is because the portion of transplants that occur at the national level is relatively small, e.g., from 1999 to 2002, approximately 5% of total transplants took place at the national level [28]. With viability adjustment, this portion would become even less significant compared to intra-regional transplantation. However, in Section 4 we evaluate our solutions in a simulation model that considers the national throughput. The other factor that affects transplant benefit is allocation likelihood. Intuitively, a liver is more likely to be allocated within a larger pool of patients. We assume that the allocation process is in steady state, a common assumption for a closed-form analysis of dynamic resource allocation models. We also assume that the allocation likelihood depends only on the donor OPO, the recipient OPO, and the potential region containing the two OPOs. This assumption further simplifies the allocation process by aggregating donors and recipients with different clinical and demographic characteristics. These differences are, instead, implicitly incorporated when calibrating several parameters in c r via simulation. With the above two assumptions, we can allocate livers to matched patients at other OPOs within the same region based on a macro-level allocation scheme, proportional allocation. More formally, for any potential region r R, let I r I be the set of OPOs that comprise it. We define z ij to be the likelihood that a liver procured at OPO i is allocated to OPO j, for i,j I r, i j. While z ij depends on the region, we drop the region index for notational simplicity. Considering any potential region r R, for each liver procured at OPO i I r that is available at the regional level, we assume that it is allocated proportionally to any other OPO j I r according to an affinity value between OPOs i and j, denoted by l ij. The affinity value l ij may be interpreted as the percentage of livers procured at OPO i that would be offered to matched patients at OPO j if the allocation hierarchy did not exist. Therefore, the proportional allocation can be formulated as z ij l ij = z ik l ik, for i,j,k I r and distinct. We thus derive z ij as z ij = l ij k I r,k i l ik + l 0 i, (2) where a positive parameter li 0 is included to model the fact that a larger region will find matches at the regional level for a larger portion of livers but not all livers can be matched within the region irrespective of its size. As li 0 increases, more livers procured at OPO i cannot be matched at the regional level. There are two reasons for using li 0 rather than allowing it to vary by region. One is that the model in (2) sufficiently captures the following fact that a larger portion of donor livers would be allocated intra-regionally within a larger region due to increased number of recipient candidates. The second reason is that li 0 allows for an effective application of column generation. Define o i to be the number of organs procured at OPO i I over a specified time horizon that are offered at the regional level. Hence, the expected viability-adjusted transplants whose donor and recipient OPOs are i and j, respectively, i,j I r,i j, are given by o i z ij α ij. If I r = 1, then c r = 0. If I r > 1, summing up the viability-adjusted transplants over all donor and recipient OPOs in I r yields c r = l ij o i i I r j I r,j i k I l r,k i ik + li 0 α ij. (3) As introduced earlier, the inclusion of α ij and li 0 in the estimate favors small and large regions, respectively. On one hand, each donor liver is likely to result in a higher viability-adjusted transplant in smaller regions. On the other hand, more transplants are likely to take place within larger 5

6 regions. Hence, the estimate of c r in (3) reflects the tradeoff between large and small regions. Intuitively, a promising region tends to be relatively more convex, contiguous, and of medium size. 2.2 Region Generation Since there is no clear definition of a feasible region by UNOS, any combination of OPOs could be considered as a region. This leads to an exponentially large number of potential regions. What complicates the problem significantly is that we must design a regional configuration that partitions the entire country. Therefore, we resort to branch and price to generate potential regions as needed to overcome this computational difficulty. Branch and price is often referred to as IP column generation. It embeds column generation into a branch-and-bound framework. At each node of the search tree, columns are generated adaptively by solving the pricing problem and the restricted master problem is solved iteratively as an LP with added columns. The purpose of the pricing problem is to check if the LP-relaxation solution computed over a subset of regions R R is optimal for R. Once no additional columns price out favorably, the column generation process is completed and an upper bound is then obtained for the considered search tree node. Meanwhile the integrality of the relaxation solution is checked and a branching operation may be carried out accordingly. For a recent survey of branch and price, we refer to [12]. As our computational results demonstrate, our branch-and-price approach offers a substantial improvement over an explicit region enumeration approach, which seeks promising regional configurations by explicitly enumerating all potential contiguous regions. Intuitively, contiguous regions tend to result in more viability-adjusted intra-regional transplants. Hence, such an algorithm is likely to provide a near-optimal solution. Unfortunately, to effectively implement the algorithm, one can only explore all contiguous regions with no more than 8 OPOs. In the context of column generation, a restricted master problem, denoted by RMP(R ), is the LP relaxation of the set-partitioning problem over a subset R of all potential regions, i.e., R R. Given a potential region r, the reduced cost c r is then: c r = c r a ir π i = c r π i, (4) i I i I r where π i is an optimal dual in RMP(R ), associated with OPO i. It is clear that an optimal solution to RMP(R ) is a feasible solution to RMP(R). To determine if this feasible solution is optimal to RMP(R), we need to check if there are columns in R\R that price out favorably. This process of dynamically generating columns is called pricing and thus this problem is called the pricing problem. Define y i = 1 if i I r ; y i = 0, otherwise. In addition, we introduce auxiliary binary decision variables w ij = y i y j to linearize the nonlinear relationship between y and z. Given a dual vector π, we present as follows the MIP pricing problem, labeled as RPP(π). subject to RPP(π) : max i α ij z ij i I j I\{i}o π i y i (5) i I j I\{i} z ij + z 0 i = y i, i I; (6) z ij y j, i,j I,i j; (7) 6

7 l ik z ij l ij z ik + l ik (1 w jk ), i,j,k I,j < k,i j,k; (8) l ij z ik l ik z ij + l ij (1 w jk ), i,j,k I,j < k,i j,k; (9) l 0 i z ij l ij z 0 i + l 0 i (1 w ij ), i,j I,i < j; (10) l ij z 0 i l 0 i z ij + l ij (1 w ij ), i,j I,i < j; (11) l 0 j z ji l ji z 0 j + l0 j (1 w ij), i,j I,i < j; (12) l ji z 0 j l 0 j z ji + l ji (1 w ij ), i,j I,i < j; (13) w ij y i + y j 1, i,j I,i < j; (14) y i IB,0 z 0 i 1, i I,0 z ij 1, i,j I,i j,w ij IB, i,j I,i < j. (15) Consider an OPO i I r. If y j = y k = 1, w jk = 1 by Constraint (14) and the binary restriction on w jk. Hence both Constraints (8) and (9) are satisfied with equality and thus proportional allocation between j and k is enforced. Variable z 0 i is introduced to measure the likelihood that a liver procured in OPO i cannot be matched at the regional level (see Constraint (6)). We construct the proportionality between z 0 i and z ij for any j I r as z0 i l 0 i = z ij l ij. Hence, in a similar manner, Constraints (10) (11) and Constraints (12) (13) are included to ensure the proportionality between z 0 i and z ij. The classic approach is to solve RPP(π) to optimality for any given π so as to find a column with the largest reduced cost. If the optimal objective value of RPP(π) is nonpositive, the current solution to RMP(R) is optimal. Otherwise, a new column is generated based on the optimal values of decision variables y. Some pricing problems arising in column generation, such as the shortest path problem for the multicommodity flow problem [2] and the integer knapsack problem for the cutting stock problem [30], are easy to solve practically and/or theoretically. On the other hand, in column generation applications to many vehicle routing and crew scheduling problems, the pricing problem is NP-hard (e.g., [8, 6]). For the optimal region design problem, the pricing problem has been shown to be NP-hard by reducing from the maximum facility location problem [11]. In addition, preliminary computational results indicated that the full pricing problem is hard to solve directly in practice using off-the-shelf MIP solvers (see Section 4.2.1). Therefore it was impossible to obtain the global optimum, even for the LP relaxation of the optimal region design problem within a reasonable amount of time. We develop a decomposition-based heuristic in the next section to overcome this major bottleneck in the branch-and-price implementation. 3 Computational Considerations In this section, we explore several computational enhancement strategies to address the following two difficulties in our branch-and-price implementation: 1) solving restricted master problems to optimality may not be efficient, e.g., the pricing problem is computationally prohibitive; 2) conventional branching on variables may not be effective. These two difficulties appear to be the fundamental difficulties in LP-based branch and price [10, 7]. In this section, we describe a decomposition-based heuristic, namely geographic decomposition. This heuristic alleviates the difficulty in solving the pricing problem. In addition, we briefly discuss multiple regions generation and initial restricted master problem construction. We next adapt a specialized branching rule that is effective for set-partitioning problems. Although applying the geographic-decompositionbased heuristic in the branch-and-price algorithm saves computational time noticeably, its solution 7

8 quality clearly depends on the design of the decomposition. To improve the solution, we integrate the branch-and-price algorithm with small-scale local search in an iterative procedure. A full description of our algorithms is provided in Appendix A of the electronic companion. 3.1 Pricing via Geographic Decomposition To alleviate the computational difficulty of the pricing problem, we consider solving smallerscale pricing problems constructed over subsets of OPOs (called geographic subsets) instead of solving the full pricing problem over the entire country. Due to the poor scalability of integer programming, solving multiple smaller-scale pricing problems may be computationally beneficial. There are two important features in the design of the geographic subsets. First, the geographic subsets form contiguous regions. Intuitively, it is unlikely that one region in the optimal regional configuration contains two OPOs that are far apart, e.g., the OPOs located in Miami and Seattle. Second, some of the designed geographic subsets should overlap to some degree. It ensures that we are still capable of identifying promising regions even with the decomposition. Figure 3: Illustration of Geographic Decomposition Figure 3 shows an example of the geographic subsets. There are three geographic subsets in the figure, specified by three different line patterns. We call them the dashed, solid, and dotted subsets (also numbered 1, 2, and 3). One important feature of our geographic decomposition is that some of the designed subsets overlap. In Figure 3, the dashed and solid subsets overlap in Minnesota, Iowa, Nebraska, etc. The solid subset overlaps with the dotted subset in the two OPO service areas mainly in Tennessee. Note that to cover the entire country, additional subsets are needed. Let us define RPP(π,I ) to be the pricing problem over I I with respect to duals π. The objective function and all constraints are constructed accordingly. Let I be a collection of geographic subsets I whose union is I. We call such a collection a geographic cover. Given a geographic cover I, we define R i to be the set of potential regions ( given I i I for i = 1,..., I. With geographic I ) decomposition, we in fact solve the problem RMP i=1 R i. Note that the optimality of RMP(R) can be verified only if the full pricing problem is solved. Our computational results show that it is prohibitive to solve a full pricing problem ) (see Section 4.2.1) via standard MIP solvers. Hence, we only solve the problem RMP and leave the exact solution of the full pricing problem to ( I i=1 R i 8

9 future research. As a result, the design of the geographic cover is critical ( to obtaining a good solution. It is ideal I ) to design a geographic cover such that the problem RMP i=1 R i can be solved within a reasonable amount of time, and the obtained suboptimal solution is close to the true optimal solution of RMP(R). One consideration is to start with regions in a good feasible regional configuration obtained through heuristics or exact solutions over a reduced solution space. The other consideration is to find appropriate cardinalities of the geographic subsets and an appropriate number of the subsets that form a geographic cover. On one hand, we intend to design many large geographic subsets so that it is more likely to identify promising regions that are potentially in the optimal basis of RMP(R). Intuitively the larger a subset I I, the more likely it contains a region with the largest reduced cost over R. Moreover, the more geographic subsets there are to be considered at each iteration of column generation, the more likely there is a region with the largest reduced cost over R. On the other hand, the computational burden of pricing problems may be caused by a large number of geographic subsets or geographic subsets with large cardinality. Another undesirable feature of having large geographic subsets and many geographic subsets is that it may result in similar pricing problems. To further alleviate the computational difficulty in column generation, we consider adding multiple feasible columns to the restricted master problem when solving each pricing problem. Generating multiple good feasible columns may be beneficial when solving one pricing problem. Computational experience in the literature has made such suggestion for cases where the pricing problem is computationally expensive to solve [31, 7]. We specify the maximum number of columns to be generated from each pricing problem. If the specified number of columns have been generated but the optimality has not been reached, the pricing problem solution terminates prematurely and only the generated feasible columns are added to the restricted master problem. [19] shows that multiple pricing may be advantageous for set-partitioning problems. To initialize the restricted master problem for column generation, we need to construct a feasible basis, which is easy in our problem since the solution formed by I regions with a single OPO in each region is feasible. However, the performance of column generation appears to depend on the initial solution, so we construct and test two alternative sets of initial columns. First, we consider a set of presumably beneficial regions generated a priori. For example, we may generate all contiguous regions with a specified number of OPOs. Second, we use a known regional configuration. For example, we may use the current regional configuration or a good regional configuration constructed by heuristics or exact solutions over a reduced solution space. In Section we describe our computational experiments that explore the effects of geographic cover design, multiple columns generation, and initial restricted master problem construction. With these experiments, we specify the best algorithmic parameter setting for column generation. 3.2 OPO-Pair Branching The usual branching strategy, branching on variables, is not suitable for solving set-partitioning problems. This is because the branching strategy may regenerate the same region on one branch and result in an unbalanced search tree. To prevent this normally requires unaffordable computation effort. For more details, we refer to [13]. To overcome this difficulty, we use Ryan-Foster branching [17], which has proved effective for set-partitioning applications in the literature, including crew scheduling (both airline and urban transit), vehicle routing, and constrained graph partitioning. For a recent discussion of these applications, we refer to [12]. [17] prove that in a set-partitioning constraint matrix {a ij }, one can always identify a pair of rows s and t such that 0 < j:a sj =1,a tj =1 x j < 1 as long as the basic solution to the restricted 9

10 master problem is fractional. Hence, for s and t, the pair of branching constraints are j:a sj =1,a tj =1 x j = 1 and j:a sj =1,a tj =1 x j = 0. In our case, we can interpret the branching constraints as follows. On the first branch, we group the two OPOs (OPOs s and t) together; whereas on the second branch, we separate the two OPOs. Therefore, we call this specialized branching strategy branching on OPO pairs. We call the first branch the together branch, and the second branch the separate branch. In our implementation, we enforce the branching constraints in the pricing problem, i.e., we add y s = y t and y s + y t 1 to the pricing problem along the together and separate branches, respectively. This has the advantage of not introducing new dual variables to the pricing problem. With this branching scheme, branch and price must terminate finitely since there are only a finite number of pairs of OPOs. 3.3 Integrating Local Search Due to the application of geographic decomposition, the solution of the branch-and-price algorithm is sensitive to the design of the geographic cover. As discussed earlier in this section, a straightforward implementation to guarantee good solution quality is to consider more geographic subsets and larger geographic subsets. However, this approach is likely computationally prohibitive. In addition, we cannot easily predict a priori the contribution from each subset to the construction of an optimal regional configuration. To compensate for the dependence of the branch-and-price algorithm on the geographic cover, we perform local search on the suboptimal regional configuration obtained from the branch-andprice algorithm, and alternate branch and price and local search iteratively. Integrating local search helps to find good regions that cannot be enumerated with the given geographic cover. The main idea of the iterative procedure is to adaptively update the geographic cover design and improve the design only in those critical areas. We consider two local search operations, swapping and moving. We select two regions from a given regional configuration. The swapping operation selects an OPO from each region and exchanges their memberships. The moving operation selects an OPO from one region and adds it to the other region. The neighborhood of the local search is defined as the set of regional configurations resulted from all possible swapping and moving operations. At each iteration, a regional configuration is obtained with the application of the branch-andprice algorithm. The two local search operations are performed in the defined neighborhood as above. If the solution is improved, we select the operation that yields the largest improvement and expand the geographic cover as follows. If the selected operation is swapping, two regions and two OPOs are identified as the modification from the incumbent regional configuration. We thus expand each geographic subset in the incumbent geographic cover that contains either identified region, by adding the corresponding OPO that is inserted to the region in the improved regional configuration. If the selected operation is moving, the augmentation is similar. In this case, one region and one OPO are identified and we only expand each geographic subset that contains the identified region accordingly. With the modified geographic cover, we run the branch-andprice algorithm again. This iterative procedure continues until some stopping criterion is met. Our computational results in Section 4.3 show that the iterative procedure improves the solution significantly within a reasonable amount of time. See the Appendix B of the electronic companion for a complete statement of the procedure. 10

11 4 Numerical Study We considered the aggregation of viability-adjusted intra-regional transplants nationwide from 1999 to 2002 and call this efficiency outcome intra-regional transplant cardinality. The branch-andprice algorithm was implemented within the COIN/BCP framework, an open-source branch, cut, and price project within the COmputational INfrastructure for Operations Research (COIN-OR). For a comprehensive description of COIN/BCP, we refer to [16]. All tests were conducted on a Linux machine (AMD Opteron 240 processor and 4GB RAM). CPLEX 9.0 LP and MIP solvers were used to solve linear programs and mixed-integer programs encountered in the branch-and-price solution process. 4.1 Data Acquisition and Parameter Estimation The liver donation and patient registration data at each OPO were acquired from the UNOS website [28]. Several parameters in the objective function are dependent upon donors and patients clinical and demographic characteristics and/or the potential regional configuration. Rather than modeling these factors explicitly, we estimated them via the simulation model LASM [18] and calibrated the analytic model via the simulation for the current regional configuration. We first simulated the allocation system without any geographic preference, i.e., all patients nationwide were ranked based on their illness severity. The affinity value l ij was thus estimated as the average percentage of organs that are procured at OPO i and allocated to OPO j, over 100 replications. We then randomly selected 1000 regional configurations and simulated the allocation system with each of them for 100 replications. For each OPO pair i and j, we used z ij (r, R) to measure the average percentage of organs that are procured at OPO i and allocated to OPO j, given a simulated regional configuration R such that r R and i,j I r. Given an OPO i, for each simulated regional configuration R, we let li 0(r, R) = 1 z ii(r, R) j I r z ij (r, R) where i,j I r and r R. Then li 0 was estimated as the average of li 0 (r, R) over all simulated regional configurations R. We are aware that li 0 (r, R) varies with r and R. However, for computational tractability, we took the average. In Appendix C of the electronic companion, we discuss the variance of li 0(r, R). For each o i, i I, we let o i = o i β i, where o i is the number of procured organs at OPO i over the studied 4-year period, and β i is the average proportion of organs procured at OPO i that are available at the regional level over the same duration. Using the same set of simulation data as above, β i was estimated to be 1 z ii (r i, R c ), where R c represents the current regional configuration and r i is the region that contains OPO i in the current configuration. We modeled α ij as the percentage of graft primary nonfunction (PNF) given that the liver is procured at OPO i and allocated to OPO j. PNF is a condition in which a transplanted liver graft fails to function soon after being transplanted into a recipient. The incorporation of other reasons for graft failure, such as due to multiple rejections of a graft, are left for future research. To estimate α ij, we considered two components to construct a functional relationship between PNF and organ transport distance (OTD). One component is a linear ( linear decay ) or cubic ( cubic decay ) function modeling PNF vs. CIT [21], denoted by f; the other is a linear function modeling CIT vs. OTD, denoted by g. Function g was obtained from a regression analysis of UNOS data [28]. We used the straight line distance between the administrative locations of two OPOs to estimate the OTD between the two OPO service areas. Let D(i,j) be the OTD between OPOs i and j. Hence α ij = 1 f(g(d(i,j))). We modified the simulation model LASM [18] to incorporate liver quality decay as described by these functions. Many studies have concluded that prolonged organ travel/cold ischemia time is a significant predictor of long-term organ loss and patient survival rates [25]. Unfortunately, the only systematic clinical study (to our knowledge) that provides close-form 11

12 functional relationships of cold-ischemia time on liver transplantation considered only PNF [21]. 4.2 Performance of Branch and Price Alone With the acquired data and estimated parameters mentioned above, we constructed two instances in our numerical study that correspond to the two viability cases. In this section we first show the necessity of solving smaller-scale pricing problems. We then discuss how we tuned the branch-and-price algorithm with regard to the proposed algorithmic performance improvement strategies. At the end of the section, we report the solution improvement with the fine-tuned branch-and-price algorithm Solution of the Pricing Problems Our preliminary experiments indicated that it is intractable to consider all 59 OPOs of the entire country in a pricing problem. For example, we imposed a 15-hour time limit on the very first pricing problem with all initial duals being 0. For the linear and cubic viability cases, the first pricing problem had a 13.37% and a 10.48% gap, respectively, after 15 hours. Thus solving the LP relaxation of the initial formulation is intractible due to these difficult full pricing problems. Therefore, we applied geographic decomposition. We designed geographic covers in which each geographic subset contains 13 to 15 OPOs. Note that the design of these covers is partly based on promising regional configurations obtained through explicit enumeration of contiguous regions. The computational results, however, showed that it was still impractical to solve the resultant pricing problems to optimality. Hence, we decided not to create geographic subsets with more than 12 OPOs. We in Section further compared geographic cover designs with fewer OPOs in each geographic subset Tuning the Branch-and-Price Algorithm The purpose of our experiments was to find a promising set of algorithmic parameters for branch and price. We terminated branch and price once an integer optimal solution was found with respect to the given geographic cover. We therefore call the regional configuration corresponding to such a solution a terminating regional configuration. In all the computational exeriments, we applied OPO-pair branching. Here we only describe our experiments and draw necessary conclusions for tuning the algorithm. We present the relevant computational results in Appendix D of the electronic companion. Among the three strategies, geographic decomposition is the critical one. We thus first discuss the effects of feasible column generation and initial master problem construction. Feasible Columns Generation. We tested several column generation strategies in our experiments. For example, we considered adding all integer feasible solutions to the restricted master problem. We also considered the cases where we terminated the solution of the pricing problem before reaching optimality and only added the feasible solutions found so far. As for other experimental settings, we included all I single-opo regions in the set of initial columns. We chose to use a geographic cover design with 20 geographic subsets and 10 OPOs in each subset. Our computational results suggest that allowing 3 or more columns per pricing problem per iteration is likely to generate columns that are potentially important for constructing the optimal basis. We concluded that it tends to be preferable to add all columns that can be generated in the solution of each pricing problem. Restricted Master Problem Initialization. We tested several initial restricted master problems in our experiments. For example, we used all I single-opo regions. We also included 12

13 all contiguous 4-OPO regions into the set of single-opo regions. Additionally, we considered the current regional configuration and two configurations obtained by solving the region design problem through explicit contiguous region enumeration. As for other experimental settings, we used the same geographic cover as in measuring the effect of multiple columns generation. We added all columns that are generated when solving each pricing problem. Our computational results indicate that the solution time is relatively insensitive to the initialization scheme. Including all single-opo and contiguous 4-OPO regions seems to be the most beneficial among the studied initialization schemes in terms of both solution quality and time. Geographic Cover Design. We performed extensive tests on various geographic covers for the branch and price algorithm. We applied the best algorithmic setting found so far. The number of geographic subsets used in a cover ranged from 10 to 30 and the number of OPOs in each subset ranged from 8 to 12. From our computational experiments we concluded that a good geographic cover typically provides a fast solution and a large objective value at the same time. It appeared that several geographic cover designs with 20 geographic subsets and 10 OPOs in each subset were promising (covers 1, 2, 3 in Table 7 in Appendix D). However we also concluded that the ultimate performance of the branch-and-price algorithm depended heavily on the construction of the initial geographic covers. When at least 20 covers were used, each with no fewer than 10 OPOs, the improvement ranged from 8% to 11%, and the performance was more sensitive to the size of the covers than to the number of covers. We were unable to predict which types of covers would perform well. This motivated us to develop an iterative method for adaptively generating geographic covers, which we describe in Section The Performance of Branch and Price To test the overall improvement with the branch-and-price algorithm, we used the best algorithmic choices specified from earlier experiments. These choices are 1) applying geographic decomposition with a geographic cover that has 20 geographic subsets and each subset has 10 OPOs; 2) adding to the restricted master problem all feasible columns that price out favorably in each pricing problem solution; and 3) generating all regions with 4 contiguous OPOs as the set of initial columns in addition to all single-opo regions. Table 1 reports the absolute improvements in intra-regional transplant cardinality compared to the current regional configuration, the relative improvements, the number of regions in each terminating regional configuration, and the CPU time. For comparison, the last two rows in the table report those items for the explicit contiguous region enumeration approach when any of the contiguous regions includes no more than 8 OPOs and 7 OPOs, respectively. Our results show an increase of around 350 viability-adjusted intra-regional transplants over the four-year period. These results indicate significant improvements with the branch-and-price algorithm even though we did not verify whether the solution was also optimal for the problem over the entire set of all potential regions. Table 1 shows that the branch-and-price approach provides better solutions in less time. 4.3 Integrating Local Search with Branch and Price There are two concerns with applying branch and price alone. The first one is that it may result in highly nonconvex or even discontiguous regions. The second is that the solutions depend on the geographic cover design. We also noticed that the nationwide intra-regional transplant cardinality increased with simple local search operations. We thus used local search to design the geographic 13

14 Table 1: Solution Approach Comparison Linear Decay Cubic Decay Solution Abs. Rel. # of CPU Abs. Rel. # of CPU Approach Improv. Improv. Reg. Time Improv. Improv. Reg. Time Branch and Price % 7 0:15: % 8 0:17:34 Contig. Enum. ( r 8) % 8 2:37: % 9 3:34:11 Contig. Enum. ( r 7) % 9 0:47: % 10 0:37:26 cover adaptively and applied the updated geographic cover design in the next round of branch and price. Following the description of the iterative procedure in Section 3.3, we sought further improvement on the solution as well as alleviation of the two concerns stated above. Iteration 0 of the iterative procedure is identical to branch and price alone. In each of the subsequent iterations, we first applied the local search operations to find an alternative regional configuration that offers the largest improvement from the incumbent regional configuration (if possible). Meanwhile we identified the modified region(s) through local search. We then augmented the incumbent geographic cover accordingly to contain the modified regions. At the end of each iteration we solved the optimal region design problem with the incumbent geographic cover. We applied two stopping criteria to the iterative procedure: 1. the iterative solution is locally optimal, i.e., optimal both in terms of branch and price given the incumbent geographic cover and local search given the neighborhood associated with the incumbent cover design; 2. the absolute improvement is below 10 for 5 consecutive iterations. With the iterative procedure, we first investigated its solution robustness on the design of the initial geographic cover. We selected three geographic covers (covers 1, 2, 3 in Table 7 in Appendix D). Table 2 reports the relative improvements at iteration 0 of the procedure, the relative improvements at the end of the procedure, and the number of iterations when the procedure terminated. For the linear case with covers 1 and 2, stopping criterion 1 applied. For the other instances, the algorithm terminated under criterion 2. Therefore, for all cases, approximately 20 iterations were required. We observed from the table that the iterative procedure appears to greatly alleviate the dependency of the solution quality on the selection of the initial geographic cover. Figure 4 also shows that the solution improvement over iterations appears to become insensitive to the selection of the initial geographic cover. Table 2: Performance of the Iterative Procedure Linear Decay Cubic Decay Cover Iter. 0 Term. Iter. Iter. Num. Iter. 0 Term. Iter. Iter. Num % 13.3% % 14.1% % 13.3% % 14.2% % 13.5% % 14.1% 28 Figure 5 shows the corresponding best regional configurations, i.e., cover 3 for the linear case and cover 2 for the cubic case. Our computational experiments show that the iterative procedure further improves the solution quality within a reasonable amount of time. The local search operations took little time and appeared more effective for the solution improvement as the number of iterations increased. However, the branch-and-price algorithm still made about 95% of the overall 14

15 Figure 4: Solution Improvement via the Iterative Procedure (Left: Linear Case; Right: Cubic Case) improvement. We also tested the algorithm that applies only local search iteratively and uses the same stopping criteria, and found that its performance was about the same as branch and price alone. Figure 5: Best Regional Configurations with the Iterative Procedure (Left: Linear Case; Right: Cubic Case) In the best regional configurations found so far, there are fewer regions. In each of them, there is a region that has much larger cardinality than that of other regions. For the linear case, the region spanning from the Southeast and the South contains 22 OPOs. For the cubic case, the region covering the central plain includes 21 OPOs. Note that having the ability to enumerate such large regions in the solution is due to the adaptive geographic cover redesign. In these configurations, the two regions in the Northeast are much smaller. In both of them, there is the same region that contains 3 OPOs in Maryland, District of Columbia, and Eastern Pennsylvania. The further improvements via the iterative procedure were shown to be statistically significant in the simulation model LASM. Finally, we summarize the improvements in Table 3. For comparison, we consider the system that does not implement hierarchical allocation. In other words, in such a system donor livers are directly offered to patients nationwide after they cannot find matches within their donor OPOs. We evaluate the overall allocation efficiency of transplants for which the donor and recipient do not reside in the same OPO. We report the relevant results in row Single Waiting List. The results show that it is not beneficial to apply the approach of equally sharing donor livers in the entire 15

16 country from an efficiency viewpoint. To conclude, 1) the branch-and-price algorithm significantly improves the organ allocation system efficiency. With the iterative procedure, the efficiency is further improved; 2) it is beneficial to include organ allocation at the regional level between the local level and the national level. Table 3: Overall Improvement with Hierarchical Allocation Linear Decay Cubic Decay Absolute Relative Absolute Relative Improvement Improvement Improvement Improvement Iterative Procedure % % Branch and Price Alone % % Contig. Enum. ( r 8) % % Contig. Enum. ( r 7) % % Single Waiting List % % 5 Conclusions We develop a set-partitioning model and apply branch and price to assist policy makers in redesigning the geographic composition of the U.S. liver allocation system to improve transplant efficiency. To estimate the transplant efficiency for each potential region, we take a macro-level viewpoint and introduce the notion of proportional allocation. In the estimation, we introduce a simple parameter to capture the tradeoff between large and small regions. To solve the optimal region design problem, we apply branch and price with an NP-hard pricing problem. Because solving the pricing problem is the bottleneck in the branch-and-price solution process, we develop and apply several computational enhancement methods including geographic decomposition. To alleviate the dependency of the solution on the geographic cover design, we integrate local search with branch and price. Our computational experiments show that the regional configurations obtained by branch and price within a reasonable amount of time make significant increases compared to the current configuration, with respect to a transplant efficiency related outcome. Our carefully calibrated numerical results suggest that the country should be divided into fewer and larger regions. The results also indicate an increase of equivalent to nearly 90 transplants every year, compared to the current regional configuration. The iterative algorithm that integrates branch and price and local search leads to further improvement of around 50 more intra-regional transplants annually. It also shows that the initial geographic cover design is not critical to the solution quality of the iterative algorithm. There are several possible modeling and computational extensions. One future modeling extension is to refine the outcome estimate while relaxing the steady-state assumption. Another is to consider equity, simulatenously with efficiency. Computational extensions include more efficient solution techniques for the pricing problem, and the integration of branch and price with other heuristic techniques for improving the geographic cover design. Acknowledgments This work was supported by National Science Foundation Grant DMI and Air Force Office of Scientific Research Grant FA The authors thank the two anonymous 16

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20 Electronic Companion Appendix A A Generic Flow Chart of Branch and Price Figure 6: Flow Chart of B&P An Algorithmic Description of Branch and Price Algorithm 1. A Branch-and-Price Algorithm Input: An optimal region design problem instance; a geographic cover I; and a set of columns R 0 that contains a feasible regional configuration. 20