Solving the strategic case mix problem optimally. by using branch-and-price algorithms

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1 Solving the strategic case mix problem optimally by using branch-and-price algorithms Guoxuan Ma 1, Jeroen Beliën 2, Erik Demeulemeester 1, Lu Wang 1 1 K.U.Leuven, Research Center for Operations Management, Leuven (Belgium), guoxuan.ma, erik.demeulemeester, lu.wang@econ.kuleuven.be 2 HUBrussel, Center for Modelling and Simulation, Brussels (Belgium), jeroen.belien@hubrussel.be Abstract: This paper describes a methodology for the case mix problem at the strategic level in the health care sector. Aiming to maximize hospital profits under the given resource capacity, a mathematical model is developed to produce an optimal case mix pattern and a corresponding resource allocation scheme. Two exact methods, integer linear programming and branch-and-price, are deployed to solve the constructed model. For the branch-and-price approach, new formulations based on various decomposition units, namely wards and surgeon groups, are proposed and can be solved efficiently with branch-and-price algorithms. Numerical results are compared and illustrate that the branch-and-price algorithm outperforms the integer program approach and that a large decomposition unit improves the solution both on solution quality and on computational speed. 1 Introduction As people require more and more quality of health services, hospitals faced increasingly unprecedented economic pressures during the last decades, which impels the administrator to organize the system of service delivery according to a more business-like manner. At the same time, the health care sector also has to abide by regulations mandated by governments or insurance agencies. Thus, in order to make a perfect tradeoff between service level and financial contribution, an efficient management of limited resource capacities has been brought forward, where Vissers et al. (2001) developed a five-level planning framework seeking the balance between resource and service at each stage, as well as other planning frameworks appeared (e.g. Butler et al. 1992; Rhyne and Jupp 1988; Roth and Van Dierdonck 1995). In this paper we propose a mathematical model to address this problem at the strategic level and develop a suitable approach to obtain an optimal case mix pattern and a corresponding resource allocation method. The case mix is a critical decision variable incorporated within the health capacity planning problem in order to distinguish the various pathologies or patient groups. It is defined to represent the number of patients of each pathology in which the patients consume similar hospital resources and return equivalent profit, such as based on diagnostic-related groups (DRGs) (Fetter and Freeman 1986). The decision of the case mix will simultaneously affect the capacity planning and resource assignments. The case mix planning problem is to determine how many patients from each pathology are treated by a hospital over the course of one year. Hospitals can choose the case mix based on different rules and targets under the given capacities, so that the case mix decision can be comparable with the product mix problem (Dowling 1976) in a manufacturing system. In addition, hospitals can be categorized into two classes: constrained profit satisfiers and profit maximizers (Simon 1960), according to the opposite goals of health care service providers. The profit satisfiers adhere to the preferred case mix pattern or meet other preset objects on condition that they can manage to break even without violating the capacity constraints, where goal programming is a suitable technique to make the resource planning decisions (Rifai and Pecenka 1990; Blake and Carter 2002). However, the profit maximizer hospital is supposed to be positioned in a competitive environment 1

2 and then be prone to choose patient cases that will bring maximum rewards within the given capacity and satisfying certain fundamental requirements from the health care institutions. The literature on the application of optimization techniques to the case mix problem is relatively scarce. Feldstein (1967) and Baligh and Laughhunn (1969) describe linear programming models to maximize the volume of weighted cases. Dowling (1976) proposes a model in which the unweighted number of medical and surgical patients treated is maximized, subject to capacity constraints on functional departments. Hughes and Soliman (1985) and Robbins and Tuntiwongpiboon (1989) relate case mix planning problems in hospital to manufacturing product mix decisions. Both papers describe the application of linear programming to maximize the total contribution of patient groups within the existing capacity constraint. However, not simultaneously considering the contributions and the need for resources of the different patient groups yields naïve and unpractical results. Rifai and Pecenka (1989) illustrate the use of goal programs to allocate resources in a health care system. Blake and Carter (2002) describe two linear goal programming models in order to determine the case mix and case costs in such a way that the institution is able to break even. Adan and Vissers (2002) describes an integer linear programming model to generate an admission profile for a specialty under a given target patient throughput and utilization of resources. Among the various techniques, branch-and-price, a methodology developed to solve huge integer programs (Barnhart et al. 1998), is a candidate to solve the proposed case mix problem efficiently, according to the previous research works (e.g. Bard and Purnomo 2005; Beliën and Demeulemeester 2006, 2007, 2008). The columns in such an approach can represent the efficient tradeoffs between capacity and demand for each surgeon group or department in a hospital, whereas the master problem determines the optimal combination of existing patterns and the subproblem prices out the favorable columns for each unit. In the remainder of this paper, we will firstly describe the case mix problem with an integer linear programming(ilp) model in Section 2. Afterwards the problem is interpreted from a branchand-price perspective and reformulated with two decomposition methods in Section 3, which can be solved efficiently by the branch-and-price algorithm. In Section 4, numerical experiments are implemented to evaluate both proposed approaches of the case mix problem, in which the original model is solved with a commercial ILP solver and the reformulations are solved with the branchand-price algorithms. Computational results are compared and analyzed consequently. Finally, Section 5 draws conclusions and suggests potential areas for the further research. 2 Problem statement The assumption in this paper is that hospitals are profit maximizers, which will select an optimal case mix under given capacities and other constraints. Three main consumed resources, namely surgeons, operation rooms and beds, are considered in the treatment process, and three parameters, i.e., the reward of treatment, the surgery duration time and the length of stay(los), are determined to characterize a patient treated. A patient group is defined such that patients within the same group occupy similar resources and have the same parameters while those from various groups consume different resources or differ from each other for at least one of the parameters above. The hospital is supposedly divided into various departments, each of which consists of a number of surgeon groups and a ward with a fixed number of beds to accommodate patients operated by the associated surgeons. Figure 1 illustrates the relationship among the wards, surgeon groups and patient groups. Each surgeon group can perform surgeries for several patient groups, for each of which the number of surgeries operated per cycle is restricted by the capacity, admission standards and other mandated requirements. In contrast to wards with fixed bed allocation scheme, operation rooms are shared and assigned to various surgeon groups according to the surgery schedules. A block is defined as the smallest time unit for which an operating room can be assigned to a surgeon group. The block length is assumed to 8 hours in our model, but it can be modified if necessary. The case of a surgery lasting longer than the block length is excluded from discussions in this 2

3 paper. Additionally, a block can be only assigned on weekdays, i.e., from Monday to Friday, while beds in the ward are available during the whole week, and the cycle length is also set as one week. Figure 1: Relationship among wards, surgical groups and patient groups Based on the above assumptions, an integer linear programming model is constructed to determine how many patients from each group can be taken care of and how resources are allocated to various specialties in view of maximizing the overall contributions. Given the set of wards W with index w, surgical groups set S indexed by s, patient groups set P indexed by p, the active day set A indexed by a and another day set D with index d, the ILP model is presented as follows: subject to max y w w W r p x pa p P BEDS, (2) x pa y w, w W, d D, (3) p P w pd 1 s S z sa BLOCKS, a A, 4 dur p x pa z sa LENGTH, s S, a A, (5) p P s LB p x pa UB p, p P, 6 x pa, y w, z sa Ν, w W, s S, p P, a A, (7) where the decision variables are x pa : the number of patients of patient group p that receive surgery on day a; y w : the number of beds assigned to ward w; z sa : the number of blocks assigned to surgical group s on day a; and the notations, coefficients and RHS values are S w : subset of surgical groups S whose patients are transferred to ward w after surgery; P w : subset of patient groups P of which the patients occupy the beds of ward w; P s : subset of patient groups P of which the patients are treated by surgical group s; A pd : subset of active days on which patients of group p operated still stay in the hospital on d; r p : the profit generated by treating a patient of patient group p; dur p : the surgery duration time for a patient of patient group p; LB p : the lower bound on the number of patients of group p treated per cycle; UB p : the upper bound on the number of patients of group p treated per cycle; BEDS: the total number of available beds (assumed to be the existing bed capacity); BLOCKS: the total number of available operating room blocks (the existing OR block capacity); LENGTH: operating room block length (assumed to be 480 minutes). 3

4 3 The branch-and-price approach When studied from the perspective of column generation, the ILP model of the case mix problem can be formulated differently. Clearly, the decision variable x pa is the basic driving force, which will definitely decide the case mix and capacity allocation schemes. Thus, through aggregating the case mix variables x pa on various units, such as wards and surgical groups, rather than determining the value of x pa on an individual basis, we can produce different column generation formulations for the case mix problem. A new variable is introduced to represent the pattern of the group treatment of patients, which is termed as a column for the remainder of this paper. A feasible column should satisfy the capacity constraints and numerical range of admitted patients. By finding a column combination with the highest profits generated, an optimal case mix pattern is obtained, together with an optimal capacity assignment. Enumerating all feasible columns is intimidating due to the huge number of patterns. However, the branch-and-price approach can solve the column generation formulations efficiently with a pool of initial columns. 3.1 Decomposition based reformulation In this section, two reformulations of the case mix problem are produced based on the different decomposition units, wards and surgical groups, which will only affect the definition of decision variables while keeping the target of profit maximization and the corresponding constraints Decomposition on wards Decomposition on wards means that a set of x pa variables for each active day a and each group p of which patients are moved to a ward w for recovery will be aggregated to form a feasible column. In order to reformulate the ILP model, new binary column variables are introduced to represent whether the column is selected for a ward, as well as the column set K indexed by k. Then, the master problem of the column generation formulation decomposed on wards is described as: subject to max w W k K w W k K k K w W k K c wk r wk c wk y wk c wk BEDS, (9) z wak c wk BLOCKS, a A, 10 = 1, w W, 11 c wk 0,1, w W, k K, (12) where the new variables and coefficients introduced are: c wk : indicating whether the column k is selected for ward w; y wk : the number of beds allocated to ward w when column k is selected for w; z wak : the number of blocks assigned to surgical groups of ward w on active day a; r wk : the total profit generated by the patients in ward w if k is selected for w. To avoid enumerating all the feasible columns, the technique of column generation is adopted to produce favorable patterns only. After introducing the dual variable π to constraint (9), dual variables λ a to constraints (10) and dual variables γ w to the one-column selection constraints (11), the reduced cost of column k for ward w can be expressed by (8) RC wk = r wk y wk π z wak λ a γ w = r p x pak p P w y wk π z wak λ a γ w, 4

5 and the subproblem pricing out favorable columns for each ward w is formulated as follows: subject to max p P w r p x pak y wk π z wak λ a x pak y wk, d D, 14 p P w pd z wak = z sak, a A, 15 s S w x pak dur p z sak LENGTH, s S w, a A, 16 p P s LB p x pak UB p, p P w, 17 x pak, y wk, z wak, z sak Ν, s S w, p P w, a A, (18) where the new variables x pak represent the number of patients of group p receiving surgery on day a and variables z sak represent the number of blocks assigned to surgical group s on day a, when column k is selected for ward w Decomposition on surgical groups Similarly, when the decomposition unit is set as surgical groups, a set of x pa variables for each active day a and each patient group p of which the patients are treated by surgical group s are aggregated. After defining the new binary column variables c sk, the master problem of the decomposed formulation on surgical groups is presented in the following: subject to max w W s S k K y w r sk c sk BEDS, 20 y sdk c sk y w, w W, d D, 21 s S w k K s S k K k K c sk z sak c sk BLOCKS, a A, 22 = 1, s S, 23 y w Ν, w W, c sk 0,1, s S, k K, 24 where the new introduced variables are c sk : the binary variable implying whether the column k is selected for surgical group s; y sdk : the number of beds occupied by patients of surgical group s on day d if k is selected for s; r sk : the total profits generated from patients operated by surgical group s if k is selected for s. Through introducing the dual variables π wd to constraints (21), dual variables λ a to constraints (22) and dual variables γ s to the one-column selection constraints (23), the reduced cost of column k for surgical group s can be expressed by RC sk = r sk d D y sdk π wd z sak λ a γ s = p P s r p x pak and the pricing subproblem for each surgical group s is formulated as: d D y sdk π wd z sak λ a γ s, 5

6 subject to max r p x pak y sdk π wd z sak λ a 25 = y sdk D, 26 p P s d D x pak, d p P s pd x pak dur p z sak LENGTH, a A, 27 p P s LB p x pak UB p, p P s, 28 x pak, y sdk, z sak Ν, p P s, a A, d D. (29) 3.2 Branch-and-price implementation Branch-and-price overview The general scheme of the branch-and-price approach is illustrated in Figure 2. The algorithm starts with a pool of initial columns, which are usually generated through heuristic methods and used to initialize the computation. In addition, super columns are also included in the restricted master problem to ensure the solution feasibility at each level of the branch-and-bound tree. At each round, the relaxed master problem is solved with the column generation method, and then the dual prices are obtained and transferred to the pricing subproblems. The restricted master problem is consecutively updated with new columns and solved until no more favorable columns can be produced. If the solution of the relaxed master problem satisfies the integrality, an integer feasible solution is found. Otherwise, the master problem will be constrained with new branching conditions and solved further. Any integer solution to the master problem is recognized as a lower bound and the optimal solution to the relaxed problem is identified as an upper bound. The best bound will be checked against the newly generated ones and be updated if necessary. The iteration loop continues until the branch-and-bound tree is thoroughly explored or when the best upper bound and the best lower bound meet together. Figure 2: Branch-and-price scheme 6

7 3.2.2 Column initialization Generating the initial columns is combined with checking the problem feasibility in this work. The number of patient treatments of each group is restricted by the capacity structure and the admission standards. By changing the numerical range (6) of admitted patients to the fixed lowest possible value in the original model: x pa = LB p, p P, (30) it can be examined whether the given capacity satisfies the minimum requirement. As the number of admitted patients of each group is now fixed, the original ILP model is simplified into a resource-constrained assignment problem, which is easily solved as long as there are enough capacities to accommodate admitted patients. Any feasible solution to the reduced problem is also feasible to the original case mix model and can be included in the initial column pool for the branch-and-price algorithm. To find more feasible solutions to initialize the master computation, we relax the number of admitted patients of a certain group p to its upper bound (i.e., UB p ) while fixing the number of other groups to their lower bounds and solve the revised problem. Repeating this procedure for all patient groups, more initial columns are produced Pricing out The pricing subproblem is solved with the commercial ILP solver ILOG CPLEX in our work. The major concern in this step is to determine which and how many columns are to be added to the master at each round of solving the relaxed problem. Theoretically, any column with a favorable reduced cost can price out to the master problem, and then result in better solutions. However, finding and adding all most favorable columns is practically difficult and not necessary in general. Therefore, an alternative method is adopted to add only one favorable column at each round. A time limit is imposed on searching the favorable column. If no column with a favorable reduced cost is found, more time will be allocated to the subproblem. Otherwise, the best column solved within the time limit, not necessarily the most favorable one, will be retrieved for use Branching The solution of the relaxed master problem is not necessarily integral and applying a standard branch-and-bound procedure to the restricted master will not guarantee an optimal or even feasible solution. Therefore, in order to obtain an optimal case mix pattern, new favorable columns must be generated after branching and the branching strategy cannot destroy the solvability of subproblems. Branching on the column variable c k will produce uneven partitioning of the solution space, also increasing the difficulty of solving the pricing problem. Branching on the case mix variable x pa cannot effectively separate the solution space, since two different case mix patterns might correspond to an identical resource allocation scheme and generate equal profits, i.e., the same columns in the master problem. Thus, we adopt the branching strategy according to the capacity assignment variables y w and z wa in the first reformulation. If the solved optimal solution to the relaxed master problem is not integral, there are at least two fractional column variables related to one ward w, for instance c wk and c wk. Then, there is at least one value of the capacity assignment variables different from each other in the two columns, for example z wak z wak. Then, the branching condition can be defined as z wa z wak and z wa z wak + 1, and the solution space is divided into two branches. Not only branching on capacity assignment variables excludes the fractional optimal solution, but also partitions the solution space evenly. Moreover, the pricing problem is not changed structurally and still tractable after the branching. A similar branching mechanism is designed according to the variables of y sd and z sa for the second reformulation. In addition, the beam search technique is employed in the second branch-and-price implementation in order to gain a better integer solution earlier, while the skip tracking is used in solving the ward decomposed formulation in which the nodes can always be explored quickly. 7

8 3.3 Acceleration techniques The integrity of a branch-and-price algorithm is ensured by a proper column generation method together with an effective branching strategy. In order to further enhance the performance of algorithms, a few acceleration techniques are introduced. 1. To overcome the tailing-off effect (the time spent on proving optimality may be comparable or even considerably longer than finding an optimum) the Lagrangian bound is computed to check the potential of the current node outperforming the best bound, which can help fathom the node earlier without any risk of missing the optimal solutions. 2. In order to get better integer feasible solutions earlier and improve the lower bound quickly, the restricted master is solved as an integer problem to actively generate integer feasible solutions at each round rather than waiting for the relaxed master problem to produce integer solutions only, and the best lower bound will be checked and updated if necessary. 3. A cutoff value is applied to terminate the computation process earlier when solving integer programs, since a rough judgment of whether the objective value can outperform the cutoff is often less time-consuming than to calculate an optimal or feasible solution. So, the dual price of the one-column selection constraint is defined as the cutoff value for each pricing subproblem, as well as, the current best lower bound is set as the cutoff for the restricted master. 4 Computational results The basic case mix problem is solved with two exact approaches: the original ILP model solved by the solver ILOG CPLEX and the column generation formulations solved with branch-and-price algorithms. The computational performances are compared in terms of efficiency and effectiveness through numerical experiments, which are all carried out on a personal computer equipped with a Duo Core2 CPU 3.16GHz and Windows XP operating system together with coding in Microsoft Visual Studio 2005 linked to the CPLEX 10.2 optimization library. 4.1 Original ILP formulation solved with CPLEX To study the impact of the problem dimensions on the algorithm performance, the ILP model is solved by CPLEX 10.2 with standard settings for small instances, which involve no more than 5 departments and each department has three surgical groups with each handling three patient groups, as well as other dummy data, such as, the lower bound fixed as 1 for each pathology, the upper bounds generated randomly with the distribution between 3 and 6 for various groups, and more parameters also produced by random functions. Table 1 shows the computational results for three problem dimensions, and clearly indicates that the ILP solver CPLEX fails to solve the integer model efficiently for even small or medium-sized instances within a reasonable time. No. Best sln. Optimal? LP relaxation Node explored Node left Comp. time(s) 1 unit 240 Y units 680 Y units 1357 N Table 1:Computational results of ILP 4.2 Branch-and-price solving decomposition models Computational results of two decomposition methods are compared and discussed in this section. Table 2 illustrates an example of five instances with various problem dimensions, in which part of the associated parameters is listed accordingly while others are generated randomly. Performance indicators include the best solution found, the computational time and the number of nodes explored/left. For all tested instances, the ward-decomposed approach manages to find the optimal solution more quickly and proves optimality with fewer nodes explored. While the efficiency of 8

9 the surgeon-decomposed approach drastically decreases as the problem dimension grows, and the best solution found within a reasonable computation time is usually 5 to 10 percent lower than the optimum for the larger cases. Parameters Ward-decomposed Surg-decomposed ward surg. pat. bed OR best sln. time(s) nodes best sln. time(s) nodes(e/l) / / / / / / / / / /4037 Table 2: Computational efficiency and effectiveness In contrast to the ward-decomposed method, the model of decomposition on surgical groups suffers from two factors: a slow searching process of the optimal solution and a time-consuming process of the optimality testing. The former factor is attributed to a large number of pricing steps involved at each iteration and more variables generated. Since our testing assumes that the surgical groups outnumber the wards by a multiplier of 3, the number of subproblems to be solved at each round of the surgeon-decomposed approach is three times more than that of the ward-decomposed approach. In addition, the surgeon-decomposed method generally produces more columns for the small decomposition unit, which also makes the master problem more difficult to be solved. It is observed that the time needed to explore one node increases significantly with the problem size. For the surgeon-decomposed approach, a node is usually explored within 1 second for a five-ward instance, but it requires more than 10 seconds on the average for the cases of ten wards or above. The latter factor, longer optimality testing, can be explained by the weak relaxation bound of the surgeon-decomposed approach. Table 4 compares the quality of the relaxation bound for both decomposition methods. The relative relaxation efficiency of the surgeon-decomposed approach mostly falls into the range of 5-10%, while the decomposition method on wards gives a much tighter bound (usually within 0.01% for the tested data). Even if the optimal value can be found within a short time, a loose relaxation bound still imposes a demand for a long optimality testing, which prolongs the whole computational process. Moreover, further analysis demonstrates that the lower relaxation efficiency of the surgeon-decomposed approach comes from its loose variable bound in the problem formulation. Parameters Optimum Relaxation bound Relaxation efficiency ward surg pat. bed OR Opt. sln. Ward-D Surg-D Ward-D Surg-D % 4.28% % 7.14% % 9.62% % 6.81% % 7.25% % 5.41% Table 3: Comparison of the relaxation efficiency of two decompositions 5 Conclusions The strategic case mix problem is formulated and solved optimally with the column generation method, in which the hospital is assumed as a profit maximizer aiming to maximize the overall profits by selecting the patient mix under the constraints of capacity and admission limitations. The branch-and-price approach is implemented and exhibits much better performance than the ILP solver ILOG CPLEX. This work is focused on the profit indicator, but it can be easily extended to other purpose by modifying the objective coefficient. However, the optimization model adopts the 9

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