An examination of Benders decomposition approaches in large-scale healthcare optimization problems. Curtiss Luong

Transcription

1 An examination of Benders decomposition approaches in large-scale healthcare optimization problems by Curtiss Luong A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Mechanical and Industrial Engineering University of Toronto c Copyright 2015 by Curtiss Luong

2 Abstract An examination of Benders decomposition approaches in large-scale healthcare optimization problems Curtiss Luong Master of Applied Science Graduate Department of Mechanical and Industrial Engineering University of Toronto 2015 Benders decomposition is an important tool used to solve large-scale optimization problems found in healthcare. Radiation therapy and operating room planning and scheduling are two areas in which Benders decomposition have been applied to solve difficult problems. In radiation therapy, we develop two novel Benders algorithms, including a classical Benders algorithm and a combinatorial Benders algorithm, to solve the sector duration and optimization problem efficiently. In operating room planning and scheduling, we implement an existing logic-based Benders algorithm for tactical operating room planning and scheduling and analyze the effect of changes to the input data on various output statistics. ii

3 Contents 1 Introduction and literature review Stereotactic radiosurgery Tactical operating room planning and scheduling Benders decomposition Contributions List of publications and presentations Publications Presentations Benders algorithm applied to the sector duration and isocenter optimization problem Optimization model Classical decomposition Combinatorial decomposition Evaluation methodology Results Discussion Conclusions Data generation in OR planning and scheduling Optimization model iii

4 3.2 Logic-based Benders decomposition Data generation Evaluation methodology Results Discussion Conclusions Conclusions 57 Bibliography 59 iv

5 List of Tables 2.1 Case information, 7 cases comprising 11 different targets Classical Benders, combinatorial Benders, and branch-and-cut (B&C) convergence results ( for no feasible solution at 10 hours, for not enough memory), bold for best performance Radiosurgical plan quality summary comparing forward and Benders plans, bold numbers are better Forward: forward (manual) plans determined clinically. Benders : inverse plans found by two-phase benders decomposition Variables for MP Sets and parameters for MP SP notation Original data costs Data generation scenarios Solution metrics comparing different data generation strategies against the original data, - for unsolved trials Solution metrics comparing different normalized data generation strategies against one another v

6 List of Figures 2.1 Case 7 convergence Clinical two-phase Benders decomposition results, Case Empirical CDF of surgical lengths Alternative patient-surgeon flexibility matrices Cumulative plot of solved instances of identical and randomized cost instances Cumulative plot of solved instances for different surgical distributions Cumulative plot of solved instances for flexible and block patient-surgeon flexibility vi

7 Chapter 1 Introduction and literature review An important practical challenge in solving healthcare optimization problems is that real problems can be complex, and that complex problems can lead to large-scale mathematical model formulations that are intractable, even with modern optimization solvers and solution techniques. In particular, even though commercial mixed-integer programming software such as Gurobi Optimizer (Gurobi Optimization, Inc.) and IBM CPLEX Optimizer (IBM, Inc.) have seen continued improvement over time, there are many real problems that cannot easily be solved with these software packages. In this thesis, we will study two such problems. First, in the field of radiation therapy optimization, we will examine the sector duration and isocenter optimization problem (SDIO), found in Leskell Gamma Knife R Perfexion (LGK PFX, Elekta, Sweden) treatment plan optimization. Second, in the field of operating room scheduling, we will study the tactical operating room planning and scheduling (TORPS) problem. These two problems seem intractable at first glance, and naive mixed-integer formulations fail to consistently find good solutions to real-sized problem instances. However, we will explore different techniques, including problem decomposition, in order to better understand why these problems are difficult and to discover how to find better solutions. Although SDIO and TORPS are two very different models dealing with completely 1

8 Chapter 1. Introduction and literature review 2 different problems, they do have some similarities. Both can be formulated as large mixed-integer problems, and both have intuitive decompositions. To solve SDIO, we use a Benders decomposition with an upper model that decides where and at what angles to send beams of radiation, and a lower model containing variables deciding how long to apply each beam; this decomposition is analogous to one used in [41] to solve the integral fluence map decomposition problem with rectangular apertures found in intensity-modulated radiation therapy. Similarly, to solve TORPS, we decompose the original model into an upper model that determines a master surgical schedule, and a lower model that finds the optimal sequence of patients within the confines of that schedule, a basic decomposition that has been used previously [17, 37]. Both SDIO and TORPS also help solve significant problems with practical implications in healthcare. In Canada, an estimated 191,300 people will be diagnosed with cancer in 2014 [39]; radiation therapy is a common and effective manner of treating many types of cancers. Specifically, LGK PFX radiation therapy is used to treat many cancers of the head and neck. Since brain cancer alone will kill an estimated 1,950 Canadians in 2014 [39], research in LGK PFX treatment planning is essential. In regards to the importance of good operating rooms planning and scheduling, hospitals are expensive: Hospital-related health expenditure in Canada was above $62 billion in In addition, a 2013 study by the Canadian Institute of Health Information found that wait times for important procedures such as hip replacement, knee replacement, hip fracture repair and cataract surgery exceed the target wait time nationally for over 15% of patients. Clearly, there is work to be done in serving more patients more effectively. With more efficient schedules, hospitals would be able to serve more patients with similar, or even lower, cost as compared to the status quo. With both SDIO and TORPS we extend work existing in the literature; our work improves on what exists already by critically examining and proposing specific improvements to existing research methodology, and by implementing new optimization tech-

9 Chapter 1. Introduction and literature review 3 niques when they are required. In radiation therapy, we have implemented a novel two-phase Benders decomposition and a novel combinatorial Benders decomposition. In operating room scheduling, we not only implement a state-of-the-art solution algorithm for TORPS, but we generate data in full transparency and examine, for the first time, how computational results are sensitive to changes in data generation methodology for operating room scheduling. We offer insight into how existing research methodology could account for data sensitivity when evaluating solution techniques like Benders algorithm. 1.1 Stereotactic radiosurgery Stereotactic radiosurgery (SRS) is a non-invasive alternative to surgery for various types of head and neck disease, including cancer. As opposed to stereotactic radiotherapy, in which smaller doses of radiation are given over a large number of treatment sessions, SRS is a radiation therapy treatment system in which radiation is delivered to the patient in a single treatment session, called a fraction. In SRS, beams of radiation are applied to a target, denoted as the gross tumor volume (GTV), to achieve a specific prescribed dose while minimizing harm to nearby organs at risk (OARs). In this thesis, we will study radiosurgery done using the Leskell Gamma Knife R (LGK) Perfexion TM (PFX, Elekta, Stockholm, Sweden) device. LGK PFX is an SRS device that can accurately deliver high-dose radiation to target structures within a patient. To accomplish this task, the PFX simultaneously produces beams of radiation from eight sectors of radiation sources surrounding the patient. A single collimator array determines the size of each of the beams of radiation, and each sector s beam can be driven independently to a diameter of 4mm, 8mm or 16mm, or deliver no radiation at all. During a treatment plan, the patient is secured to a mechanical couch, and this couch is positioned so that the sectors are aimed at a precise location, called an isocenter, for a

10 Chapter 1. Introduction and literature review 4 planned duration of time. A combination of isocenter location, duration, and collimator size is called a shot. Older LGK devices required manual intervention between shots, meaning that complex treatment plans were impractical. As a result, studies on LGK SRS optimization [18, 19, 38, 47] done using previous models of the LGK assume a small, fixed number of isocenters and focus mostly on the optimization problem of deciding on isocenter locations and durations within a target. More recent research in LGK SRS optimization uses a two-stage approach [21, 46]: First, determine good isocenter locations using geometrybased heuristics; then, run a sector duration optimization to find the treatment time of all of the radiation beams to apply appropriate dose to the GTV. In the first stage, when trying to place isocenters, the quantity and location of shots are chosen heuristically based on the geometry of the target rather than on dosimetric calculations; popular heuristic algorithms incorporate different heuristics such as grassfire [43], skeletonization [14, 45], sphere-covering [29], sphere-packing [44], and genetic [28] algorithms. In the second stage, isocenters are assumed to be fixed, and the resulting problem to find the durations of each radiation beam can be solved. This problem is often cast as a convex optimization problem [21, 32] or, more simply, as a linear program [46]. We have not found any work showing how close the geometry-based approaches come to finding optimal isocenter locations; however, these geometric approaches often yield good practical results. Isocenter selection and sector duration can be combined into a single exact problem formulation called the sector duration and isocenter optimization problem (SDIO). SDIO combines the isocenter location stage and the sector duration optimization stage into a single mixed integer optimization problem [20]. This approach was previously demonstrated to find acceptable treatment plans [20]; however, the technique used to solve the one-stage MILP formulation required some restrictions on the solution space in the form of tight upper and lower bounds on the number of isocenters chosen, and heavy approx-

11 Chapter 1. Introduction and literature review 5 imations in terms of reductions to the number of constraints and decision variables in a process called sampling, to achieve tractability. 1.2 Tactical operating room planning and scheduling One strategy for cost containment in activities surrounding the operating room (OR) is efficient utilization of OR resources through scheduling optimization. As OR-related costs contribute, on average, 8-10 percent of a hospital s total expenses [30], OR scheduling optimization is an extremely important task. Furthermore, inefficient use of ORs through mismanagement of available OR time and surrounding OR resources can lead not only to increased costs, but also to prolonged patient wait times surgical case cancellations, and overall patient dissatisfaction. OR planning and scheduling across multiple ORs has been widely studied in the literature. Two main tactical planning strategies are commonly used: block scheduling [4, 6, 13] and open scheduling [7]. In block scheduling, all available OR time is divided into discrete time intervals, called blocks. Surgeons, or groups of surgeons, are allocated to each block, and each group schedules patients freely within the assigned block. In contrast, in open scheduling, surgeons are not scheduled to work within blocks of time, but instead perform surgeries whenever they are available and the appropriate hospital resources are free. In this paper, we focus on assumptions commonly made in data generated for open scheduling type problems. However, many of these assumptions are also made in block scheduling and some of our conclusions are applicable across planning strategies. In general, each paper in open scheduling optimization chooses a different set of data on which to run experiments, and there is no standard data set that is frequently used. To generate data, researchers will attempt to sample from real data if it is available, but will randomly generate any data that is missing as realistically as possible. Unfortunately,

12 Chapter 1. Introduction and literature review 6 there are at least two potential problems with the current data generation practice in the literature. Firstly, since researchers are working with different data sets, it is difficult to assess computational results across different algorithms in the literature. Secondly, it is challenging to assess the integrity of the models as compared to real scheduling situations found in hospitals. In general, assumptions made in the data generation process can impact the conclusions made in any single test. In this paper, we identify some important decisions commonly made in the data generation process, and seek to quantify the impact of these decisions. We will shed some light on which assumptions in the literature are justified and which elements should be highlighted and reworked. Many different types of data are needed to simulate a typical hospital. Cost-based objective functions are commonplace in the open scheduling literature [17, 26, 27], necessitating data regarding OR and surgeon-related expenses. This information is often unavailable to researchers, subjective, and subject to change. As a result, although these numbers sometimes originate from consultations with OR scheduling decision-makers [12], many papers choose not to delve too deeply into accurate costing, instead determining costs based on averages determined by past studies [26, 27]. Various surgery-related durations such as surgical time, OR cleaning time, and OR preparation time for each operation are also needed for OR scheduling models. In the OR planning and scheduling literature, researchers have sampled directly from real data [7, 10, 12, 17], or have simulated data based on uniform [16, 23, 27, 36], lognormal [26], normal [37] or Pearson III [16] distributions. In the OR simulation literature, the lognormal distribution [11] has been used. Although there is no consensus on the correct distribution to use, the lognormal distribution has been shown to be a reasonable choice to simulate surgical times in at least one study [40]. In addition to surgical durations, any data generation scheme must decide on allowable patient-resource assignments; in particular, allowable patient-or and patient-surgeon assignments must be determined. Surgical cases are often considered to be assigned to a

13 Chapter 1. Introduction and literature review 7 single surgeon and able to be operated on in any room [3, 17]; however, general models that include opportunity for partial patient-surgeon and partial patient-room flexibility have been proposed [37], and accurately represent realities within some hospitals. Data sets must also consider realistic resource availability schedules. Almost always, ORs are allowed to be open for 8 hours each day, whereas surgeons are available on certain days of the week on a rotating schedule. Since we are working in tactical OR planning, we deal with a rotating schedule based on a week-long planning horizon. Some patients must be scheduled within the current planning horizon, others can be pushed to the next planning horizon. If data is not available on the percentage of patients that must be scheduled in the current planning horizon, this data must also be estimated. In our case, it is set at 50%, assuming half of the patients should be scheduled during this planning horizon. Furthermore, each patient of this planning horizon should have a deadline before which patients must be scheduled. One important implication of this hard deadline is that a naively generated data set is not necessarily feasible, and any data generation scheme must ensure that there is at least a single feasible schedule. It is not obvious how many papers in the literature ensure schedule feasibility. In general, data generation is not simple, and there are many assumptions that must be made to create reasonable input data for our models. Thus, it is unsurprising that the data sets used in the literature vary widely. Our research attempts to assess the impact of using different data generation methodologies. 1.3 Benders decomposition Classical Benders algorithm has been applied to many areas including network design [9], integrated aircraft routing and crew scheduling [31], and production management [2]. Originally conceived by J. F. Benders in 1962 [5], Benders decomposition is a technique

14 Chapter 1. Introduction and literature review 8 designed to exploit the structure of large linear or mixed-integer optimization problems. Classical Benders decomposition is often used on problems with many continuous variables, lending itself well to SDIO, as one of the challenges of SDIO is the enormous number of continuous variables, each associated with the duration of a potential shot. However, the literature on Benders algorithm in radiation therapy treatment planning is sparse. The only such previous studies apply classical [42] and subsequently combinatorial [41] Benders decomposition algorithms in radiation therapy were in the context of the integral fluence map decomposition problem with rectangular apertures (IFR) in intensity modulated radiation therapy; however, SDIO is quite different from IFR because of the smaller problem size, rectangular apertures and lack of continuous variables in the objective function found in IFR. One notable development to classical Benders decomposition is the use of local branching on potential solutions by exploring the neighborhood of any solution using local branching [35] at each Benders iteration, generating many incumbent solutions (and optimality cuts). This strategy differs from the classical Benders algorithms found in this thesis as it performs additional work at each Benders iteration to find promising feasible solutions in the neighboorhood of the current solution instead of locating feasible solutions in a Phase I step. It is possible to incorporate local branching into our twophase classical algorithm, as well as our combinatorial algorithm, although this strategy has not been attempted in this thesis. Unforunately, one weakness of classical Benders decomposition is that it requires all subproblem variables to be continuous, and the subproblem objective and constraints to be linear so that the subproblem is a linear program, and linear programming duality theory can be used to develop valid cuts. To deal with this limitation, logic-based Benders decomposition was developed [25] as a generalization of classical Benders decomposition. Logic-based Benders is similar to classical Benders in that it decomposes a large-scale optimization problem into a master problem and one or many subproblems,

15 Chapter 1. Introduction and literature review 9 and uses constraint generation to gradually decrease the solution space of the master problem. However, instead of using a linear programming dual to generate cuts, Logicbased Benders decomposition uses the broader concept of an inference dual, which can be defined as an optimization problem that finds the best possible bound implied by a set of master problem variables, also called primary variables. This best possible bound is used to generate cuts that are passed back to the master problem. Logic-based Benders decomposition has been applied frequently in the past to various difficult planning and scheduling problems [15, 24]. However, logic-based Benders decomposition has been used in the operating rooms planning and scheduling field only in [37]. 1.4 Contributions This thesis contributes to the improvement of the radiation therapy literature by developing an effective two-stage Benders decomposition-based algorithm to solve the SDIO problem. This two-stage Benders algorithm was demonstrated to outperform a standard implementation of Benders algorithm over seven clinical test cases, and is able to solve larger test cases than Gurobi, a commercial branch-and-cut solver. This thesis also proposes a combinatorial Benders algorithm that has shown promising computational results for smaller sized test cases. We also contribute to the operating room planning and scheduling literature by implementing and documenting a complete, standard, transparent data generation scheme; currently, no paper has detailed a well-defined procedure to follow to generate data for operating room planning and scheduling. Our data ensures that there is at least one realistic feasible schedule embedded in the data, and that flexibility, as well as other features of the data, can be easily modified without compromising data feasibility. We demonstrate the power of this data generation scheme by implementing TORPS, a solution technique similar to a model found in [37], and perform novel analysis on the significance

16 Chapter 1. Introduction and literature review 10 of different data sets run using the same algorithm. We show that small changes in the data generation technique have a potentially large impact on overall computation time, a finding that has not previously been discussed in the existing operating room planning and scheduling literature. 1.5 List of publications and presentations Publications 1. C. Luong, D. M. Aleman. A two-phase Benders algorithm applied to the sector duration and isocenter optimization problem. Work in progress. 2. C. Luong, D. M. Aleman. A hybrid combinatorial Benders algorithm applied to the sector duration and isocenter optimization problem. Work in progress. 3. C. Luong, D. M. Aleman. Data generation for tactical operating room planning and scheduling. Work in progress. 4. V. Roshanaei, C. Luong, D. M. Aleman, D. Urbach. Distributed operating room scheduling via a logic-based Benders decomposition approach. Work in progress. 5. V. Roshanaei, C. Luong, D. M. Aleman, D. Urbach. Distributed integrated master surgical scheduling and surgical case scheduling using a bi-cut logic-based Benders decomposition. Work in progress. 6. S. Kulkarni-Thaker, C. Luong, D. M. Aleman, A. Fenster. Inverse planning for focal ablation in cancer treatment using approximations. Work in progress Presentations 1. C. Luong, D.M. Aleman. Benders decomposition in radiation therapy inverse planning. IEEE Annual Conference Montreal, Canada. June 2014.

17 Chapter 1. Introduction and literature review C. Luong, D. M. Aleman. Integer programming approaches to Gamma-Knife radiosurgery planning, Mechanical and Industrial Engineering Research Symposium, University of Toronto, Canada, June 2013.

18 Chapter 2 Benders algorithm applied to the sector duration and isocenter optimization problem We describe two different Benders -type algorithms to efficiently solve SDIO. We start by showing that SDIO can be decomposed into an integer master problem and a linear subproblem. Once decomposed, we solve SDIO using a classical two-stage Benders algorithm, and show that the two-stage Benders algorithm is an efficient method to solve this problem. We test this two-stage Benders approach on seven different clinical cases, and show that it finds acceptable clinical plans, that it outperforms a standard one-stage Benders algorithm, and that it is capable of solving larger problems than commercial branch-and-cut solvers. We also propose and implement a combinatorial Benders algorithm, and show that the combinatorial Benders algorithm is faster than the classical Benders algorithm when it is able to run, but is not able to even begin the solution process for the larger test cases because of large overhead times. As mentioned before, SDIO combines the isocenter location stage and the sector duration optimization stage of a typical two-stage radiation therapy inverse planning 12

19 Chapter 2. Benders algorithm applied to SDIO 13 algorithm into a single mixed integer optimization problem [20, 21]. Instead of using a traditional branch-and-cut algorithm, we implement Benders -type approaches designed for large-scale optimization problems. Classical Benders algorithm uses a decomposition that divides the mixed-integer linear program into one or many subproblems with only continuous variables, and a master problem with an exponential number of constraints, each corresponding to an extreme point or an extreme ray of the dual of the subproblem. Although there are an exponential number of potential master problem constraints, a Benders algorithm is used, expecting that an optimal solution can be found with only a subset of the complete set of constraints. Our classical Benders decomposition implementation has two significant differences from a standard implementation. Firstly, the solver that is used to solve the Benders linear programming subproblems use an interior-point method with the previous subproblems solution as a starting point. As each subproblem is only slightly different than the previous one, using the previous subproblem s starting point has been shown to accelerate the subproblem solution process [1, 22]. Secondly, we use a twophase technique to accelerate the solution process. In Phase I, we simply solve the linear relaxation of SDIO using Benders decomposition. With each iteration of our Phase I, we use a rounding heuristic to find an incumbent solution to the original SDIO problem, and generate cuts to use in Phase II. The optimal solution to Phase I is also used as a lower bound to start Phase II. Using these techniques to accelerate our Benders decomposition implementation, we are able to efficiently solve SDIO. A second algorithm that was implemented incorporates combinatorial Benders cuts into the classical Benders algorithm. Our implementation can be seen as a hybrid between the combinatorial benders algorithm developed in [8] and a classical Benders approach. In [8], the authors formulate a special kind of logic based no-good cut derived from a feasibility subproblem, this cut can be called a combinatorial constraint. In practice, we cannot simply adopt a pure combinatorial Benders decomposition as both

20 Chapter 2. Benders algorithm applied to SDIO 14 the master problem and the subproblem of our decomposition have a non-zero objective function. As a result, we create a hybrid algorithm that uses no-good style cuts when the master problem is found to be infeasible, but that reverts to classical Benders cuts otherwise. 2.1 Optimization model We base the SDIO formulation on the mixed integer programming model proposed in [20]. In this model, the treated area has been conceptually divided into cubes, or voxels, sized approximately 1mm 1mm 1mm. We define λ θ as a binary decision variable representing whether or not to use an isocenter θ, and t θbc as a continuous variable representing the treatment duration at isocenter θ from sector b and collimator size c. We refer to the set of source banks as B, the set of collimators sizes as C, and the set of isocenters as Θ. For tractability, as in [20], we do not allow isocenters to be located at any voxel, and instead select a subset of approximately 200 voxels as candidate isocenter locations using a grassfire and sphere-packing algorithm [21]. The radiation dose rate from a beam directed at isocenter θ Θ from sector b B with collimator size c C delivered to voxel j is denoted as D θbcj. As a result, we can write the total dose delivered to any voxel j as D θbcj t θbc θ Θ b B c C The objective function of SDIO is a weighted combination of the dose applied to voxels within healthy structures s S and the number of isocenters used ( θ Θ λ θ). The constraints ensure that voxels within target structures s T are within a range of prescribed dose between T s and T s. For many of the cases being considered, the problem as stated is too large to be solved completely, or even to be held in memory, as a single test case can contain up to 129,000

21 Chapter 2. Benders algorithm applied to SDIO 15 voxels. Notably, the D θbcj coefficients used to calculate dose for every voxel are mostly non-zero, necessitating an extremely large sparse constraint matrix to constrain dose for each voxel. To reduce the problem size, we consider only a subset of target voxels in the constraints, sampled uniformly from the treated area, and a subset of healthy voxels in the objective function that form a contour around the organs at risk (OARs). V s is the subset of healthy and target voxels considered after sampling. The full SDIO formulation is min t θbc,λ θ s.t. ( θ Θ b B c C s S ) 1 D θbcj t θbc + w V s Θ j V s T s D θbcj t θbc T s θ Θ b B c C t θbc Mλ θ b B c C θ Θ λ θ j V s, s T θ Θ (SDIO) t θbc 0 λ θ {0, 1} θ Θ, b B, c C θ Θ Even with voxel sampling and isocenter selection resulting in SDIO, traditional branchand-cut based algorithms struggle to find optimal solutions. One explanation for the difficulties is that the big M constraints relating λ θ and t θbc produce weak LP relaxations: If λ θ is even slightly greater than zero, it allows the corresponding t θbc variables to vary significantly. As a result, LP relaxations of the mixed-integer program have optimal solutions with many fractional λ θ values. Another limitation of traditional techniques is that solving mixed-integer programs with extremely large and dense constraint matrices requires enough memory to store the problem and to track the progress of the branchand-cut algorithm, which is problematic as the problem size increases. As a result, we will introduce decomposition methods designed to deal with larger problem sizes.

22 Chapter 2. Benders algorithm applied to SDIO Classical decomposition In our Benders decomposition, we can imagine a master problem that selects which isocenters will deliver dose, and a subproblem with isocenters ˆλ θ fixed from the master problem that decide how much dose to deliver from each of the selected isocenters. The Benders subproblem BSP is formulated as min t θbc s.t. ( θ Θ b B c C s S D θbcj t θbc T s θ Θ b B c C D θbcj t θbc T s θ Θ b B c C t θbc M ˆλ θ b B c C ) 1 D θbcj t θbc V s j V s j V s, s T j V s, s T θ Θ (BSP) t θbc 0 θ Θ, b B, c C Benders cuts are found from the dual of BSP. Let e j and u j be the dual variables corresponding to the first two constraints of the primal problem, respectively, and let m θ be the dual variable corresponding to the third constraint. Then, we can write the dual subproblem (DSP) as max e j,u j,m θ s.t. ( ) ej T s + u j T s + M ˆλ θ m θ s T j V s θ Θ 1 (e j D θbcj + u j D θbcj ) + m θ D θbcj V s s T j V s s S j V s e j 0 u j 0 m θ 0 (DSP) θ Θ, b B, c C j V s, s T j V s, s T θ Θ

23 Chapter 2. Benders algorithm applied to SDIO 17 The feasible region of DSP does not depend on ˆλ θ, thus, we can enumerate all finite optimal solutions to DSP, which are extreme points of the feasible region of DSP. Similarly, we can represent unbounded optimal solutions of DSP as the extreme rays of DSP. Let the extreme points and extreme rays of DSP be { e p j, up j, θ} mp p Ip and { } e r j, u r j, m r θ r Ir, respectively, where I p and I r are the finite sets of all extreme points and rays of DSP, respectively. Using these extreme points and extreme rays, we can write the master problem (MP) as min λ θ s.t. z w λ θ + ( e p j θ T s + u p j T ) s + Mλ θ m p θ z θ Θ s T j V s θ Θ ( ) e r j T s + u r jt s + Mλ θ m r θ 0 s T j V s θ Θ p I p r I r (MP) λ θ {0, 1} θ Θ In general, a linear program can have an exponential number of extreme points and extreme rays; however, in practice, we do not generate all of them to find an optimal solution. Instead, we generate constraints one at at time, iterating between MP and DSP. Each solution to DSP is an extreme point (if DSP has an optimal solution) or an extreme ray (if DSP is unbounded) to add to the master problem. We now develop a two-phase Benders algorithm to solve SDIO. In Phase I, the linear relaxation of SDIO is solved using a classical Benders algorithm. During this phase, we use a rounding heuristic at every Phase I incumbent solution to generation solutions that are feasible to the full SDIO model, and we pass them to BSP to generate cuts that are valid for the full model. In Phase II, we solve SDIO normally using Benders decomposition, except that we have incumbent solutions and cuts already found in Phase I. One motivation behind the two-phase approach is to generate some good incumbent solutions quickly; each Phase I solution is heuristically repaired to satisfy global feasibility

24 Chapter 2. Benders algorithm applied to SDIO 18 conditions, resulting in an incumbent solution. With this approach, we can generate several potential feasible solutions, resulting in a good incumbent to start the Phase II solution process. The Phase I master problem is a relaxed version of MP. We loosen the λ θ {0, 1} constraint to 0 λ θ 1; otherwise, LRMP is identical to MP. We call this modified MP the linearly relaxed master problem (LRMP): min λ θ s.t. z w λ θ + ( e p j θ T s + u p j T ) s + Mλ θ m p θ z θ Θ s T j V s θ Θ ( ) e r j T s + u r jt s + Mλ θ m r θ 0 s T j V s θ Θ p I p r I r (LRMP) 0 λ θ 1 θ Θ The Phase I dual subproblem is DSP without changes. To solve the linear relaxation of SDIO, we iterate between DSP and LRMP in a Benders algorithm until convergence, which is guaranteed as DSP has a finite number of extreme points and extreme rays. Phase I serves two purposes. Firstly, the optimal solution to LRMP provides a starting point and lower bound for the overall problem. Secondly, we generate cuts for Phase II from the repaired solutions of LRMP that are feasible to MP. To repair the LRMP solution, we use an extremely simple rounding heuristic: We round up all values for λ θ that are greater than a certain threshold α and set all other λ θ values to zero. Testing values for α on Case 1, values between 0.1 and 0.5 seems to work equally well in practice, and we chose a value of 0.2 without extensive tuning as to avoid over-fitting the solution technique to our problem set. Any heuristic that produces solutions feasible to MP can be used to generate a cut that is valid for SDIO by passing the MP-feasible solution to DSP, and solving DSP to optimality. The resulting cut is not a Benders cut found using the optimal solution to

25 Chapter 2. Benders algorithm applied to SDIO 19 MP, but the cut is easier to find because it does not require the solution of an integer program to optimality. Algorithm 1 shows Phase I of the Benders algorithm. P LP and R LP are the set of extreme points and extreme rays that have been found so far from solving LRMP in the Phase I algorithm, and P and R are the set of extreme points and extreme rays that have been found so far that we will carry through to Phase II of the algorithm. In Phase II, Benders algorithm is applied using MP as the master problem and using DSP as the subproblem. This algorithm will converge to the optimal solution as DSP constraints do not change within the algorithm, and DSP has a finite number of extreme points. The Phase II algorithm is shown in Algorithm 2. We solve DSP in each iteration using the solution to the previous DSP as an initial point. 2.3 Combinatorial decomposition While attempting to solve the problem using classical Benders, we observed that some problems require many feasibility cuts to solve. As a result, in some test cases the master problem generates many solutions, resulting in poor algorithm performance. With that in mind, instead of solving each subproblem exactly at every iteration of the Benders decomposition, we use a fast approximate algorithm to detect some of the subproblems that are infeasible, and instead of finding classical Benders cuts, we generate combinatorial Benders s cuts. To understand combinatorial Benders cuts, we can view t θbc as artificial variables meant to imply certain feasibility conditions on λ θ variables. In our case, subproblem infeasibility implies that the isocenters selected in the master problem are not able to deliver the correct amount of radiation to all of the voxels. Naturally, if BSP is infeasible, one of the λ θ constraints much change; therefore, if ˆλ θ is the solution to the master

26 Chapter 2. Benders algorithm applied to SDIO 20 Algorithm 1 Benders algorithm, Phase I Require: ɛ 1: P = P LP = R = R LP = 2: lb = 3: ub = 4: while ub lb ɛ do 5: ( λ LRMP θ, z LRMP) solve LRMP 6: lb = z LRMP 7: solve DSP with ˆλ θ = λ LRMP θ θ Θ 8: if DSP has finite optimal solution ( e p j, up j, θ) mp then 9: P LP = P LP ( e p j, up j, ) mp θ 10: z DSP = ( s T j V s e p j T s + u p j T ) s + θ Θ M ˆλ θ m p θ { } 11: ub = min ub, z DSP + w θ θ Θ λlrmp θ 12: else if DSP is unbounded with extreme ray ( e r j, u r j, mθ) r then 13: R LP = R LP ( ) e r j, u r j, m r θ 14: end if 15: λ θ = 1 if λ LRMP θ > α, λ θ = 0 otherwise θ Θ 16: solve DSP with ˆλ θ = λ θ θ Θ 17: if DSP has finite optimal solution ( e p j, up j, θ) mp then 18: P = P ( e p j, up j, ) mp θ 19: else if DSP is unbounded with extreme ray ( e r j, u r j, mθ) r then 20: R = R ( ) e r j, u r j, m r θ 21: end if 22: C PLP constraints generated from P LP 23: C RLP constraints generated from R LP 24: LRMP LRMP + C PLP and C RLP 25: end while 26: return lb, P, R

27 Chapter 2. Benders algorithm applied to SDIO 21 Algorithm 2 Benders algorithm, Phase II Require: lb, P, R from Phase I Require: ɛ 1: ub = 2: lb = 3: while ub lb ɛ do 4: ( λ MP θ, z MP) solve MP 5: lb = z (MP) 6: solve DSP with ˆλ = λ θ θ Θ 7: if DSP has finite optimal solution ( e p j, up j, θ) mp then 8: P = P ( e p j, up j, ) mp θ 9: else if DSP unbounded with unbounded ray ( e r j, u r j, mθ) r then 10: R = R ( ) e r j, u r j, m r θ 11: end if 12: ub = min ( ) ub, z SP + w Θ θ Θ λ θ 13: C P constraints generated from P 14: C R constraints generated from R 15: MP MP + C P and C R 16: end while 17: Z = ub 18: return Z

28 Chapter 2. Benders algorithm applied to SDIO 22 problem, λ θ + ˆλ (1 λ θ ) 1 ˆλ θ =0 θ =1 must be valid. Observe that only changing ˆλ θ from 1 to 0 will never make an infeasible subproblem feasible. As a result, we can lift the cut and assert that ˆλθ =0 λ θ 1 is valid. We can further strengthen this valid cut using the minimally infeasible sets (MIS) associated with DSP. An MIS is an infeasible subset of the constraints of DSP such that the removal of any one of the constraints will result in a feasible system of constraints. A key insight is that at least one of the λ θ variables participating in a constraint in the MIS of type b B c C t θbc M ˆλ θ must be changed in order for the subproblem to be feasible. As a result, letting Λ be the set of isocenters that must be changed, we can say that the corresponding combinatorial Benders cut ˆλθ =0,ˆλ θ ˆΛ θ λ θ > 1 is valid. We can find MIS using standard mixed-integer programming solvers. However, they generally return only a single MIS, resulting in a single cut. In addition to using the MIS finder within Gurobi, we also implement the MIS technique found in [41] to quickly generate many MIS cuts, using the auxiliary linear program min e j,u j,m θ s.t. θ Θ m θ ( ) ej T s + u j T s + M ˆλ θ m θ = 1 s T j V s θ Θ (e j D θbcj + u j D θbcj ) + m θ 0 θ Θ, b B, c C s T j V s (MISLP) e j 0, u j 0 m θ 0 j V s, s T θ Θ This subproblem is based on previous work [34, 41] on MISs; every extreme point in the feasible region of MISLP corresponds to an MIS of BSP. We use an objective of θ Θ m θ

29 Chapter 2. Benders algorithm applied to SDIO 23 because we want the fewest possible λ θ variables included in our combinatorial cut. To find other MISs, we simply add constraints to LRMP, setting m θ variables corresponding to our previously found MISs to zero, and solve LRMP again. The combinatorial Benders algorithm follows the one-phase classical algorithm that we have already shown, with two differences. Firstly, the new algorithm differs in how it generates cuts. In our combinatorial algorithm, instead of solving the full BSP and generating either an optimality or feasibility cut, we solve BSP with a zero objective, and constraining one out of every 10 constraints of type (2.1) and (2.2). In other words, we solve a feasibility problem that considers only 10% of the voxels considered in BSP, sampled uniformly. In this way we can quickly detect subproblem infeasibility; if the subproblem is detected infeasible we develop a combinatorial cut, otherwise we write a classical Benders cut from the full BSP, as described in Section 2.2. Secondly, the new algorithm differs in when it generates cuts. In the classical Benders algorithm, we solved the master problem to optimality before generating cuts; then, with the new cuts, the master problem was solved again. Instead, in the combinatorial Benders algorithm, we add lazy cuts at each incumbent solution through a Gurobi application programming interface as it is solving; as an alternative to stopping the master problem each time when an incumbent solution is found, we simply pause the master problem to solve our subproblem, generate our cut, insert the cut into the master problems solution process and continue the master problem within the new cuts. Since the master problem is difficult and the subproblems are easy (as they are linear programs), using lazy constraints is more effective than solving each master problem to optimality. 2.4 Evaluation methodology We implement and run all algorithms on a Dual-Core AMD Opteron TM processor with 40GB of RAM. Due to practical time constraints, we run the algorithm for 10 hours, or

30 Chapter 2. Benders algorithm applied to SDIO 24 until an optimality gap of less than 10% is reached. The computational effectiveness is measured by computation time and optimality gap at termination. The classical Benders algorithm is solved using MATLAB 2008b (The Mathworks, Inc.) and Gurobi Optimizer, version 5.6 (Gurobi Optimization, Inc.). Due to limitations of the Gurobi interface with relation to callbacks, the combinatorial Benders algorithm is implemented using Python 2.7 and Gurobi 5.6. Although our Python implementations seemed to run slightly slower than the MATLAB equivalent code, there was no way to implement the combinatorial algorithm in MATLAB. We also evaluate the clinical viability of solutions that we have found using our algorithms. The algorithms are applied to seven radiosurgery patient cases comprising 11 different targets, as shown in Table 2.1. Clinically, we use four main metrics over which to measure plan quality: Paddick and classic conformity indices [33], maximum dose applied to the brainstem (in Gy), and beam-on time (in minutes). The classic conformity index is the fraction of the volume encompassed by the prescription isodose line over the total target volume, whereas the Paddick conformity index is an alternative measure that accounts explicitly for how much the target volume and the prescription isodose line overlaps. Both measures are used clinically to ensure effective removal of the target volume. For both the Paddick and classic conformity indices, a value of 1 is ideal. The maximum brainstem dose is reported as the main metric for organ sparing, a dose of less than 15Gy is desired clinically. Finally, the beam-on time is a significant factor for many reasons, including minimization of inaccuracies in the treatment due to movement, and prevention of patient discomfort. 2.5 Results We first discuss the computational performance of the two classical Benders algorithms compared to Gurobi s branch-and-cut based solver (Table 2.2). Of the seven test cases,

31 Chapter 2. Benders algorithm applied to SDIO 25 Table 2.1: Case information, 7 cases comprising 11 different targets Case Rx (Gy) Target volume (cm 3 ) Target voxels Total voxels Sampling (%) a in b a b 24 in c a b the two-phase classical Benders algorithm was able to find feasible solutions for all of them, while the one-phase algorithm only found feasible solutions in four of the six cases. Furthermore, the two-phase Benders decomposition was able to find solutions with less than 10% gap in six out of the seven cases, compared to only three cases for the one-phase method. This discrepancy indicates a large performance gap between a typical one-phase method, and the novel two-phase method. In terms of computation time, branch-andcut outperformed the two-phase Benders decomposition by a significant margin in three cases, and performed slightly worse in two cases. However, branch-and-cut was not able to solve the biggest cases, Case 2 and Case 4, as the memory requirements were too high. To understand why branch-and-cut performs better than Benders decomposition in some test cases, we show the convergence of one such case (Case 7) in detail (Figure 2.1). The Benders algorithm finds a similar lower bound to the branch-and-cut algorithm, however, branch-and-cut quickly finds a much better incumbent solution compared to Benders algorithm. This pattern is repeated for the cases in which branch-and-cut is better than the decomposition approach.

32 Chapter 2. Benders algorithm applied to SDIO upper bound lower bound Objective function CPU time (hours) (a) Two-phase Benders upper bound lower bound Objective function CPU time (hours) (b) Branch-and-cut Figure 2.1: Case 7 convergence

33 Chapter 2. Benders algorithm applied to SDIO 27 Table 2.2: Classical Benders, combinatorial Benders, and branch-and-cut (B&C) convergence results ( for no feasible solution at 10 hours, for not enough memory), bold for best performance Gap (%) Computational Time (h) Case 2-phase 1-phase Combin. B&C 2-phase 1-phase Combin. B&C The combinatorial Benders results, also presented in Table 2.2, show that, for the cases it is able to solve, our combinatorial Benders decomposition performs well compared to classical Benders decomposition. For Cases 1, 3, 5 and 7, combinatorial Benders decomposition is able to find solutions with a similar optimality gap in less time compared to the 2-phase classical Benders decomposition. However, there exists significant overhead in terms of the loading time it takes to read formulations using the Python- Gurobi interface that is not a problem with MATLAB-Gurobi interface, or with reading models straight from text files. As a result, and the larger cases (2, 4 and 6) failed to load at all into the Gurobi interface in the allotted time. As a result, combinatorial Benders could only be run for the 4 smaller cases. We also evaluate the plans found by the two-phase Benders decomposition clinically to ensure that the treatment plans found by our algorithm are effective. As is standard in treatment plan evaluation, plans have been scaled such that the dose coverage is normalized such that V 100, the fraction of tumor volume that receives 100% of the prescribed dose, is 98%. The dose-volume histogram after normalization for a representative case (Case 1) is shown in Figure 2.2a, and several slices are shown in Figure 2.2b. The dosevolume histogram demonstrates that the GTV receives the required dose of 12Gy, and

34 Chapter 2. Benders algorithm applied to SDIO 28 Percent volume (%) Case 1 GTV Chiasm LLens L_Optic_N L_eye RLens R_Optic_N R_eye (Brainstem)_minus_(GTV) Percent dose (%) (a) Dose-volume histogram (b) Case 1 slices with 100% and 50% isodose lines Figure 2.2: Clinical two-phase Benders decomposition results, Case 1