an examination of benders’ decomposition approaches … · abstract an examination of benders’...

An examination of Benders’ decomposition approaches inlarge-scale healthcare optimization problems

by

Curtiss Luong

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Mechanical and Industrial EngineeringUniversity of Toronto

c© Copyright 2015 by Curtiss Luong

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 prob-

lems found in healthcare. Radiation therapy and operating room planning and schedul-

ing 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 schedul-

ing, 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

Contents

1 Introduction and literature review 1

1.1 Stereotactic radiosurgery . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Tactical operating room planning and scheduling . . . . . . . . . . . . . 5

1.3 Benders’ decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 List of publications and presentations . . . . . . . . . . . . . . . . . . . . 10

1.5.1 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5.2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Benders’ algorithm applied to the sector duration and isocenter opti-

mization problem 12

2.1 Optimization model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Classical decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Combinatorial decomposition . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Evaluation methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Data generation in OR planning and scheduling 34

3.1 Optimization model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

iii

3.2 Logic-based Benders’ decomposition . . . . . . . . . . . . . . . . . . . . . 36

3.3 Data generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 Evaluation methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4 Conclusions 57

Bibliography 59

iv

List of Tables

2.1 Case information, 7 cases comprising 11 different targets . . . . . . . . . 25

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 . . . . . . . . . . . . . . . . . . . . . 27

2.3 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. . . . . . . . . . . . . . . . . . . . . . . 30

3.1 Variables for MP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Sets and parameters for MP . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 SP notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Original data costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5 Data generation scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.6 Solution metrics comparing different data generation strategies against the

original data, - for unsolved trials . . . . . . . . . . . . . . . . . . . . . . 53

3.7 Solution metrics comparing different normalized data generation strategies

against one another . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

v

List of Figures

2.1 Case 7 convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Clinical two-phase Benders’ decomposition results, Case 1 . . . . . . . . 28

3.1 Empirical CDF of surgical lengths . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Alternative patient-surgeon flexibility matrices . . . . . . . . . . . . . . . 49

3.3 Cumulative plot of solved instances of identical and randomized cost in-

stances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4 Cumulative plot of solved instances for different surgical distributions . . 52

3.5 Cumulative plot of solved instances for flexible and block patient-surgeon

flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

vi

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 mathe-

matical model formulations that are intractable, even with modern optimization solvers

and solution techniques. In particular, even though commercial mixed-integer program-

ming 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 ex-

amine 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 so-

lutions 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

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 impli-

cations 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 2013. 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 improve-

ments to existing research methodology, and by implementing new optimization tech-


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 algo-

rithm 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 methodol-

ogy 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) PerfexionTM (PFX, Elekta, Stockholm, Sweden) device. LGK PFX is an SRS

device that can accurately deliver high-dose radiation to target structures within a pa-

tient. 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


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 com-

plex treatment plans were impractical. As a result, studies on LGK SRS optimiza-

tion [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 loca-

tions 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 geometry-

based 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 heuristi-

cally 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 opti-

mal 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 demon-

strated 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-


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,


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], ne-

cessitating 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 determin-

ing 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 schedul-

ing 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


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


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 combina-

torial [41] Benders’ decomposition algorithms in radiation therapy were in the context of

the integral fluence map decomposition problem with rectangular apertures (IFR) in in-

tensity 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 two-

phase 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 dual-

ity 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’ de-

composition. 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,


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, Logic-

based 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 Ben-

ders’ 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 devel-

oping an effective two-stage Benders’ decomposition-based algorithm to solve the SDIO

problem. This two-stage Benders’ algorithm was demonstrated to outperform a stan-

dard 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 im-

plementing 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 real-

istic 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 demon-

strate 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


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

1.5.1 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.

1.5.2 Presentations

1. C. Luong, D.M. Aleman. Benders’ decomposition in radiation therapy inverse

planning. IEEE Annual Conference 2014. Montreal, Canada. June 2014.


2. C. Luong, D. M. Aleman. Integer programming approaches to Gamma-Knife ra-

diosurgery planning, Mechanical and Industrial Engineering Research Symposium,

University of Toronto, Canada, June 2013.

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

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 two-

phase 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 be-

tween 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 de-

rived 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


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 λθ as a binary decision variable

representing whether or not to use an isocenter θ, and tθbc 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 Dθbcj. As a result, we can

write the total dose delivered to any voxel j as

∑θ∈Θ

∑b∈B

∑c∈C

Dθbcjtθbc

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 λθ). 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


voxels. Notably, the Dθbcj 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). Vs is the

subset of healthy and target voxels considered after sampling.

The full SDIO formulation is

mintθbc,λθ

∑θ∈Θ

∑b∈B

∑c∈C

(∑s∈S

1

|Vs|∑j∈Vs

Dθbcj

)tθbc +

w

|Θ|∑θ∈Θ

λθ (SDIO)

s.t. T s ≤∑θ∈Θ

∑b∈B

∑c∈C

Dθbcjtθbc ≤ T s ∀j ∈ Vs, s ∈ T

∑b∈B

∑c∈C

tθbc ≤Mλθ ∀θ ∈ Θ

tθbc ≥ 0 ∀θ ∈ Θ, b ∈ B, c ∈ C

λθ ∈ 0, 1 ∀θ ∈ Θ

Even with voxel sampling and isocenter selection resulting in SDIO, traditional branch-

and-cut based algorithms struggle to find optimal solutions. One explanation for the dif-

ficulties is that the big M constraints relating λθ and tθbc produce weak LP relaxations:

If λθ is even slightly greater than zero, it allows the corresponding tθbc variables to vary

significantly. As a result, LP relaxations of the mixed-integer program have optimal so-

lutions with many fractional λθ 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 branch-

and-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.


2.2 Classical decomposition

In our Benders’ decomposition, we can imagine a master problem that selects which

isocenters will deliver dose, and a subproblem with isocenters λθ 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

mintθbc

∑θ∈Θ

∑b∈B

∑c∈C

(∑s∈S

1

|Vs|∑j∈Vs

Dθbcj

)tθbc (BSP)

s.t.∑θ∈Θ

∑b∈B

∑c∈C

Dθbcjtθbc ≤ T s ∀j ∈ Vs, s ∈ T

∑θ∈Θ

∑b∈B

∑c∈C

Dθbcjtθbc ≥ T s ∀j ∈ Vs, s ∈ T

∑b∈B

∑c∈C

tθbc ≤Mλθ ∀θ ∈ Θ

tθbc ≥ 0 ∀θ ∈ Θ, b ∈ B, c ∈ C

Benders’ cuts are found from the dual of BSP. Let ej and uj be the dual variables

corresponding to the first two constraints of the primal problem, respectively, and let mθ

be the dual variable corresponding to the third constraint. Then, we can write the dual

subproblem (DSP) as

maxej ,uj ,mθ

∑s∈T

∑j∈Vs

(ejT s + ujT s

)+∑θ∈Θ

Mλθmθ (DSP)

s.t.∑s∈T

∑j∈Vs

(ejDθbcj + ujDθbcj) +mθ ≤∑s∈S

1

Vs

∑j∈Vs

Dθbcj ∀θ ∈ Θ, b ∈ B, c ∈ C

ej ≤ 0 ∀j ∈ Vs, s ∈ T

uj ≥ 0 ∀j ∈ Vs, s ∈ T

mθ ≤ 0 ∀θ ∈ Θ


The feasible region of DSP does not depend on λθ, thus, we can enumerate all finite

optimal solutions to DSP, which are extreme points of the feasible region of DSP. Sim-

ilarly, we can represent unbounded optimal solutions of DSP as the extreme rays of

DSP. Let the extreme points and extreme rays of DSP beepj , u

pj ,m

pθ

∀p ∈ Ip and

erj , urj ,m

rθ

∀r ∈ Ir, respectively, where Ip and Ir 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

minλθ

z (MP)

s.t.w

|θ|∑θ∈Θ

λθ +∑s∈T

∑j∈Vs

(epjT s + upjT s

)+∑θ∈Θ

Mλθmpθ ≤ z ∀p ∈ Ip

∑s∈T

∑j∈Vs

(erjT s + urjT s

)+∑θ∈Θ

Mλθmrθ ≤ 0 ∀r ∈ Ir

λθ ∈ 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


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 λθ ∈ 0, 1

constraint to 0 ≤ λθ ≤ 1; otherwise, LRMP is identical to MP. We call this modified MP

the linearly relaxed master problem (LRMP):

minλθ

z (LRMP)

s.t.w

|θ|∑θ∈Θ

λθ +∑s∈T

∑j∈Vs

(epjT s + upjT s

)+∑θ∈Θ

Mλθmpθ ≤ z ∀p ∈ Ip

∑s∈T

∑j∈Vs

(erjT s + urjT s

)+∑θ∈Θ

Mλθmrθ ≤ 0 ∀r ∈ Ir

0 ≤ λθ ≤ 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 λθ that are greater than a certain threshold α and set all other

λθ 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


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. PLP and RLP 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’ de-

composition, 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 tθbc as artificial variables

meant to imply certain feasibility conditions on λθ 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 λθ constraints much change; therefore, if λθ is the solution to the master


Algorithm 1 Benders’ algorithm, Phase I

Require: ε

1: P = PLP = R = RLP = ∅

2: lb = −∞

3: ub =∞

4: while ub− lb ≥ ε do

5:(λLRMPθ , zLRMP

)← solve LRMP

6: lb = zLRMP

7: solve DSP with λθ = λLRMPθ ∀θ ∈ Θ

8: if DSP has finite optimal solution(epj , u

pj ,m

pθ

)then

9: PLP = PLP ∪(epj , u

pj ,m

pθ

)10: zDSP =

∑s∈T∑

j∈Vs

(epjT s + upjT s

)+∑

θ∈ΘMλθmpθ

11: ub = min

ub, zDSP + w|θ|∑

θ∈Θ λLRMPθ

12: else if DSP is unbounded with extreme ray

(erj , u

rj ,m

rθ

)then

13: RLP = RLP ∪(erj , u

rj ,m

rθ

)14: end if

15: λθ = 1 if λLRMPθ > α, λθ = 0 otherwise∀θ ∈ Θ

16: solve DSP with λθ = λθ ∀θ ∈ Θ


pj ,m

pθ

)then

18: P = P ∪(epj , u

pj ,m

pθ

)19: else if DSP is unbounded with extreme ray

(erj , u

rj ,m

rθ

)then

20: R = R∪(erj , u

rj ,m

rθ

)21: end if

22: CPLP← constraints generated from PLP

23: CRLP← constraints generated from RLP

24: LRMP← LRMP + CPLPand CRLP

25: end while

26: return lb, P , R


Algorithm 2 Benders’ algorithm, Phase II

Require: lb,P ,R from Phase I

Require: ε

1: ub =∞

2: lb = −∞

3: while ub− lb ≥ ε do

4:(λMPθ , zMP

)← solve MP

5: lb = z(MP)

6: solve DSP with λ = λθ ∀θ ∈ Θ


pj ,m

pθ

)then

8: P = P ∪(epj , u

pj ,m

pθ

)9: else if DSP unbounded with unbounded ray

(erj , u

rj ,m

rθ

)then

10: R = R∪(erj , u

rj ,m

rθ

)11: end if

12: ub = min(

ub, zSP + w|Θ|∑

θ∈Θ λθ

)13: CP ← constraints generated from P

14: CR ← constraints generated from R

15: MP← MP + CP and CR

16: end while

17: Z∗ = ub

18: return Z∗


problem,

∑λθ=0

λθ +∑λθ=1

(1− λθ) ≥ 1

must be valid. Observe that only changing λθ from 1 to 0 will never make an infeasible

subproblem feasible. As a result, we can lift the cut and assert that∑

λθ=0 λθ ≥ 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 λθ variables participating in a constraint in the

MIS of type∑

b∈B∑

c∈C tθbc ≤ Mλθ 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∑

λθ=0,λθ∈Λθλθ > 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

minej ,uj ,mθ

∑θ∈Θ

mθ (MISLP)

s.t.∑s∈T

∑j∈Vs

(ejT s + ujT s

)+∑θ∈Θ

Mλθmθ = −1

∑s∈T

∑j∈Vs

(ejDθbcj + ujDθbcj) +mθ ≤ 0 ∀θ ∈ Θ, b ∈ B, c ∈ C

ej ≤ 0, uj ≥ 0 ∀j ∈ Vs, s ∈ T

mθ ≤ 0 ∀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∈Θmθ


because we want the fewest possible λθ variables included in our combinatorial cut. To

find other MISs, we simply add constraints to LRMP, setting mθ 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 OpteronTM

processor with

40GB of RAM. Due to practical time constraints, we run the algorithm for 10 hours, or


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,


Table 2.1: Case information, 7 cases comprising 11 different targets

Case Rx (Gy) Target volume (cm3) Target voxels Total voxels Sampling (%)

1 12 8.56 7 178 49 538 50

2a24 in 3

17.7257 736 129 857

25

2b 11.71 25

3 12 1.28 3 788 27 910 50

4a

24 in 3

0.85

77 936 125 605

50

4b 25.81 5

4c 5.66 10

5 12 5.08 5 037 47 542 50

6 12 13.06 13 159 47 296 25

7a15

0.194 945 40 369

50

7b 2.71 50

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-and-

cut 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.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5155

160

165

170

175

180

185

190

195

200

205

CPU time (hours)

Obje

ctive function

upper bound

lower bound

(a) Two-phase Benders’

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5155

160

165

170

175

180

185

190

195

200

205

CPU time (hours)

Obje

ctive function

upper bound

lower bound

(b) Branch-and-cut

Figure 2.1: Case 7 convergence


Table 2.2: Classical Benders’, combinatorial Benders’, and branch-and-cut (B&C) conver-gence results (– for no feasible solution at 10 hours, × for not enough memory), bold for bestperformance

Gap (%) Computational Time (h)

Case 2-phase 1-phase Combin. B&C 2-phase 1-phase Combin. B&C

1 10 39.6 10 10 1.33 10 1.1 1.69

2 10 – – × 9.9 – – ×

3 9.9 9.6 9.9 10 2.1 1.9 0.8 1.4

4 0.1 – – × 6.1 – – ×

5 9.8 9.5 9.8 10 3.4 3.7 2.5 3.6

6 56 – – 8.3 10 – – 0.5

7 8.4 9.2 8.4 2.6 5.0 5.1 4.1 0.4

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 com-

pared 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 com-

pared 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 nor-

malized such that V100, 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 dose-

volume histogram demonstrates that the GTV receives the required dose of 12Gy, and


0 25 50 75 100 125 150 175 200 225 2500

10

20

30

40

50

60

70

80

90

100

Percent dose (%)

Pe

rce

nt

vo

lum

e (

%)

Case 1

GTVChiasmLLensL_Optic_NL_eyeRLensR_Optic_NR_eye(Brainstem)_minus_(GTV)

(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


that the brainstem is sufficiently spared as the maximum brainstem dose is less than

15Gy (represented by 125% of the prescribed dose in the DVH).

We also evaluate treatment plans using summary statistics shown in Table 2.3. To

evaluate the treatment plans found by our two-phase Benders’ algorithm, we compare the

treatment plans to manual (forward) plans, established clinically. In general, the Paddick

and classic conformity indices are equal or better for the automatically generated plans

compared to the forward plans. However, forward plans are less complex, resulting in

shorter beam-on times. Aside from Case 7, the maximum brainstem dose for all treatment

plans are within clinical guidelines. In Case 7, although the Benders’ plan has a lower

maximum brainstem dose compared to the forward plan, we could not find a solution that

delivers less than 15Gy to the brainstem. Notably, when the one-phase and combinatorial

Benders’ algorithms did find solutions, the solutions found were similar clinically to the

two-phase Benders’ solution; these results are expected as all of the algorithms are solving

the same optimization problem.

2.6 Discussion

The classical Benders’ trials indicate that our two-phase Benders’ decomposition is a vi-

able tool for solving the SDIO problem, and that the two-phase decomposition dominates

a typical one-phase decomposition approach for these types of problems. The two-phase

Benders’ decomposition is able to find solutions within a 10% optimality gap over all seven

clinical test cases, whereas the typical two-phase implementation finds only solutions to

the required gap in three test cases, and fails to find any feasible solutions at all in three

of the seven test cases. The results also show that the two-phase Benders’ decomposition

has computationally competitive results compared to commercial branch-and-cut solvers.

The SDIO instances generated by Cases 2 and 4 are too large to be solved by Gurobi,

whereas we can solve the decomposed problems in a reasonable amount of time. In con-


Table 2.3: Radiosurgical plan quality summary comparing forward and Benders’ plans, boldnumbers are betterForward: forward (manual) plans determined clinically. Benders’: inverse plans found by two-phase

benders’ decomposition.

Case Paddick CI Classic CIBrainstem

dose (Gy)

Beam-on

time (min)

Forward Benders’ Forward Benders’ Forward Benders’ Forward Benders’

1 0.85 0.89 1.14 1.08 14.4 13.1 32.4 98

2a 0.84 0.86 1.17 1.173 2.1 28.3 214

2b 0.80 0.94 1.23 1.02

3 0.81 0.90 1.15 1.05 14.6 12.7 34.3 55

4a 0.77 0.69 1.30 1.45

1.8 4.2 25.2 3404b 0.83 0.88 1.18 1.46

4c 0.82 0.79 1.21 1.22

5 0.82 0.89 1.20 1.08 14.2 13.6 24.1 86

6 0.69 0.70 1.40 1.37 14.9 15.0 60.8 191

7a 0.67 0.71 1.38 1.3716.9 20.2 60.2 98

7b 0.91 0.89 1.07 1.08


trast, Cases 6 and 7 were solved very quickly using branch-and-cut, but were difficult for

the two-phase Benders’ algorithm. All other cases are solved by the branch-and-cut and

the two-phase Benders’ algorithm in similar time frames. As a result, we can see that the

Benders’ algorithm performs better on certain problems, but performs worse on others.

Unfortunately, of the medium-sized cases that can be solved by both branch-and-cut and

two-phase Benders’ algorithms, it is very difficult to distinguish a priori which of the

two algorithms will perform better. One reason for this challenge is that we have seen

that a big part of why the branch-and-cut algorithm performs better than the Benders’

algorithm is that the branch-and-cut algorithm is able to quickly find very good heuristic

solutions to some of the problems, and it is not clear how to determine on which problems

these heuristics work best. Further work is possible to determine how good incumbent

solutions are found quickly by the branch-and-cut algorithm in order to determine which

cases will be solved better using branch-and-cut, or even to incorporate these heuristics

into the Benders’ decomposition.

Clinically, as expected, the two-phase Benders’ algorithm was able to find plans that

had, in general, much better dose-related statistics compared to forward plans. The

graphs and statistics all indicate excellent plan quality, aside from the beam-on-times.

Unfortunately, the beam-on-times of plans generated by the Benders’ algorithm were

worse than forward plans across the board. This discrepancy indicates a weakness in

SDIO: Using an isocenter penalty to reduce beam-on time reduction appears not to be

effective. As beam-on time is an important factor in clinical decision-making, alternative

MILP formulations that explicitly consider beam-on time should be considered in future

work.

The combinatorial Benders’ results are by no means conclusive; they are only over

four test cases, and the larger cases are not solvable due to overhead issues as the prob-

lem gets larger. However, even in these four cases, it is evident that the combinatorial

Benders’ algorithm can be faster than the standard two-phase method. All four cases


that were solved by combinatorial Benders’ were solved more quickly than the two-phase

classical Benders’ algorithm; Cases 1, 3, and 5 were solved more quickly by the combina-

torial Benders’ algorithm, whereas case 7 was solved more quickly by the branch-and-cut

algorithm. It is unclear why the Benders’ subproblem is not able to be loaded in a rea-

sonable timeframe using the Python-Gurobi interface; a more sophisticated analysis into

this interface is required to further investigate this algorithm.

Due to the nature of the Benders’ decomposition solution process, the upper bounds

produced in our algorithm improve rather slowly and non-monotonously. Although we

find good solutions in the Phase I of our algorithm, it is possible that there are other

solutions feasible solutions that we still not finding. As a result future work could involve

the incorporation of local branching techniques to locate even more feasible solutions; al-

though local branching was originally applied to Benders’ decomposition in the context of

fixed-charge network optimization problems [35], it is certainly applicable to radiosurgery

planning problems.

2.7 Conclusions

In conclusion, with some innovations, Benders’ decomposition appears to be a viable ap-

proach to deal with larger clinical cases in LGK PFX radiosurgery. Our two-phase clas-

sical Benders’ algorithm can generate clinically viable solutions to real-sized problems

larger than can be handled by commercial software. Furthermore, we can potentially

accelerate the classical Benders’ algorithm with combinatorial cuts. However, there is

possible future research in finding better incumbent solutions within a Benders’ algo-

rithm, and to efficiently load a combinatorial Benders’ subproblem into memory.

Although the current model has practical use, there are some approximations that

were made in terms of voxel sampling. The voxel sampling techniques used were rudimen-

tary, and further study on which voxels to consider in the optimization and which voxels


to exclude is necessary. The same criticism could apply to the selection of candidate

isocenter locations: Although the grassfire/sphere-packing algorithm has been shown to

work effectively, there may be more effective ways to select candidate isocenters that

would result in better clinical outcomes.

Chapter 3

Data generation in OR planning and

scheduling

We solve TORPS using a logic-based Benders’ algorithm. TORPS is a simplified ver-

sion of a distributed dual-resource constrained tactical operating room planning and

scheduling model (DTORPS) formulated in [37]. We have implemented the algorithm

found in [37] in order to solve TORPS; it is effective in solving tactical OR planning and

scheduling problems. Our main contribution is to answer one key question: To what

extent does the underlying data over which an operating room planning and scheduling

algorithm is tested effect the conclusions that we make about that algorithm. As the

current data generation practices across different papers in the tactical operating room

planning and scheduling literature are fragmented and inconsistent, the importance of

data generation is an important question that has, so far, been largely ignored. By hold-

ing the Benders’ algorithm constant and varying the data generation procedure, we will

evaluate the impact of data generation on results.

34

Chapter 3. Data generation in OR planning and scheduling 35

3.1 Optimization model

TORPS differs from DTORPS in that TORPS does not explicitly deal with multiple

hospitals, instead reducing the formulation into a single-hospital model. We use a single-

hospital problem for simplicity: We do not need to use the most complex formulation

because our goal is not to evaluate the efficiency of the Benders’ algorithm, but rather

to see how the algorithm performs under different conditions. By solving appropriate,

but still real-sized, problems we can fairly evaluate the difference between alternative

data generation schemes. We make no assumptions on the homogeneity of the ORs or

surgeons in TORPS; however, we do assume that the ORs and surgeons are the only

significant hospital resources that need to be scheduled. Nurses, anesthetists and other

resources are assumed to be scheduled a priori. We also assume that surgical cases have

a deterministic length, are continuous, and are uninterruptible.

We schedule patients at specific times to be operated. Scheduling a patient will come

with a reward, but that reward is balanced by hospital resource costs such as surgeons

and operating rooms. Patients are scheduled over a time horizon of one week. Some

patients are due during this time period; they must be scheduled during the current

time horizon before their due date, otherwise the schedule is not feasible. The remaining

patients may be scheduled during the week, or they may be deferred to a future date.

To solve TORPS, we use a decomposition comprised of an allocation master problem

(MP) and multiple dual-resource constrained scheduling subproblems. The MP decides

which ORs to open, which surgeons to assign (and for how long), and which surgical

cases to execute, for each day within the planning horizon. Each SP determines the

exact surgical schedule for a single day. The SP communicates with the MP through

feasibility cuts and optimality cuts. A feasibility cut is generated when a SP finds that

there is no feasible schedule; an optimality cut is produced when the subproblem finds

that a feasible schedule needs more resources than what was allocated in the MP.


Table 3.1: Variables for MP

Variables

xpsdr 1 if patient p is operated by surgeon s on day d in OR r and 0 otherwise

zsd 1 if surgeon s is working on day d and 0 otherwise

ydr 1 if room r is open on day d and 0 otherwise

vdr Amount of overtime for room r on day d, continuous

3.2 Logic-based Benders’ decomposition

The logic-based Benders’ decomposition that follows is an existing method which can

be found in [37]; we include a description of the decomposition and solution method for

completeness.

There are four decision variables in the MP of our Benders’ decomposition. The

binary variable xpsdr indicates whether or not patient p is operated on by surgeon s

in day d in room r. The binary variables zsd and ydr show whether or not surgeon s

and room r, respectively, are used on day d. The continuous variable vdr represents the

amount of overtime incurred by operating room r on day d (Table 3.1). All sets and

parameters for MP are shown in Table 3.2.

The MP balances the costs associated with OR utilization (including regular and

overtime costs), surgeon availability, and a reward for allocating patients of the next

planning horizon. It is written as follows:

min∑d∈D

∑r∈Rh

Kdrydr +∑s∈S

∑d∈D

Lsdzsd +∑d∈D

∑r∈R

Cdrvdr (MP)

−∑

p∈P|θp>|D|

Up∑s∈Ωp

∑d∈D

∑r∈Qp

xpsdr

s.t.∑s∈Ωp

∑d ≤θp

∑r∈Qp

xpsdr = 1 ∀p ∈ P | (θp ≤ |D|) (3.1)


Table 3.2: Sets and parameters for MP

Sets

P Patients

S Surgeons

R ORs

D Days belonging to the first planning horizon, d ∈ D

Ωp Allowed surgeons for patient p

Qp Allowed ORs for patient p

∆s Days on which surgeon s is available

Pdr Patients that can be operated on in room r with θp ≥ d

Λsd Patients that can be operating on by surgeon s and that θp ≥ d

Parameters

Kdr Fixed opening cost of room r on day d

Cdr Cost of overtime of room r on day d

Lsd Fixed cost of surgeon s’s availability on each day d

Bdr Regular time of each OR r on day d

Tps Total time of preparation, surgery, and cleaning time of patient p by surgeon s

Fp Preparation time of patient p

Gp Cleaning time for patient p

Eps Time required for executing surgical procedure of patient p by surgeon s

Asd Available time of surgeon s on day d

θp Deadline of patient p

Up Reward assigned to patient p if operated in the first week

Vdr Maximum allowable amount of overtime on day d for room r


∑s∈Ωp

∑d∈D

∑r∈Qp

xpsdr ≤ 1 ∀p ∈ P | (θp > |D|) (3.2)

xpsdr ≤ zsd ∀p ∈ P ; s ∈ Ωp; d ∈ ∆s; r ∈ Qp (3.3)

xpsdr ≤ ydr ∀p ∈ P ; s ∈ Ωp; d ∈ ∆s; r ∈ Qp (3.4)

0 ≤ vdr ≤ Vdr ∀d ∈ D; r ∈ R (3.5)∑p∈Λsd

∑r∈Qp

Epsxpsdr ≤ Asdzsd ∀s ∈ S; d ∈ ∆s (3.6)

∑p∈Pdr

∑s∈Ωp

Tpsxpsdr ≤ Bdrydr + vdr ∀d ∈ ∆s; r ∈ R (3.7)

xpsdr, ydr, zsd ∈ 0, 1 ∀p ∈ P ; s ∈ Ωp; d ∈ ∆s; r ∈ Qp

It is notable that Equation (3.6) enforces a lower bound on zsd: For any given day

d and surgeon s, the sum of the surgical times of all of the allocated patients must

be less than the availability time of that particular surgeon. However, the inverse is

not true; even though a surgeon may seem to be available for enough time, there may

not exist a viable sequencing of patients that would allow all the allocated patients

to be scheduled. In the same way, Equation (3.7) enforces a lower bound on ydr and

vdr. As a result, we must solve a scheduling subproblem for each day in the planning

horizon to ensure that the MP solution found is exact. As a result, we extract the

MP-optimal solution (xpsdr, ydr, zsd, vsd), and pass it to SPs to determine the minimum-

overtime feasible schedule for each day, if one exists.

From the master problem, we write a subproblem for each day. Each subproblem

determines the order in which the patients assigned to that day are operated on. Notation

for the scheduling subproblem is shown in Table 3.3. The intermediate variable ypk

ensures that patient sequencing is enforced, allowing SP to calculate needed surgeon

availability (es − is) and OR completion time (cr) and overtime (vr) exactly. The SP is


written as follows:

min v(i)dr = cv

∑r∈R

vr (SP)

s.t.∑s∈Ωp

∑r∈Qp

xpsr = 1 ∀p ∈ P (3.8)

fp ≥ Fp +∑s∈Ωp

∑r∈Qp

Epsxpsr ∀p ∈ P (3.9)

fp ≥ fk +Gk + Fp +∑s∈Ωp

Epsxpsr −M(3− ypk −∑s∈Ωp

xpsr −∑s∈Ωk

xksr)

∀p = 1, . . . , |P| − 1; p < k ≤ |P|; r ∈ Qpk

(3.10)

fk ≥ fp +Gp + Fk +∑s∈Ωk

Eksxksr −M(2 + ypk −∑s∈Ωp

xpsr −∑s∈Ωk

xksr)

∀p = 1, . . . , |P| − 1; p < k ≤ |P|; r ∈ Qpk

(3.11)

fp ≥ fk + Eps −M(3− ypk −∑r∈Qp

xpsr −∑r∈Qk

xksr)

∀p = 1, . . . , |P| − 1; p < k ≤ |P|; s ∈ Ωpk

(3.12)

fk ≥ fp + Eks −M(2 + ypk −∑r∈Qp

xpsr −∑r∈Qk

xksr)

∀p = 1, . . . , |P| − 1; p < k ≤ |P|; s ∈ Ωpk

(3.13)

fp +Gp −M(1−∑s∈Ωp

xpsr) ≤ Br + vr ∀p ∈ P ; r ∈ Qp (3.14)

es ≥ fp −M(1−∑r∈Qp

xpsr) ∀s ∈ S; p ∈ Λs (3.15)

is ≤ fp − Eps +M(1−∑r∈Qp

xpsr) ∀s ∈ S; p ∈ Λs (3.16)

es − is ≤ As ∀s ∈ S (3.17)


∑p∈P

∑r∈Qp

Epsxpshdr ≥ (es − is)Ψ ∀s ∈ Ωp; h ∈ H; d ∈ D (3.18)

cr ≥ fp +Gp −M(1−∑s∈Ωp

xpsr) ∀p ∈ P ; r ∈ Qp (3.19)

0 ≤ vr ≤ Vr ∀r ∈ R (3.20)

vr ≥ cr −Br ∀r ∈ R (3.21)

xpsr, ypk ∈ 0, 1 ∀p ∈ P ; p = 1, . . . |P| − 1; p < k ≤ |P|;

s ∈ S; r ∈ R

fp, es, is, cr, vr ≥ 0 ∀p ∈ P ; s ∈ S; r ∈ R

Equation (3.8) ensures that all assigned patients are scheduled. Equations (3.9)-

(3.14) enforce patient sequencing. Equations (3.15)-(3.18) imply that surgeons should not

work more than their availability time. Equations (3.19)-(3.21) manage the relationship

between surgeries, OR availability, and overtime.

For each SP that is solved, we look for a Benders cut. Two types of Benders’ cuts

can be generated: feasibility and optimality. If the SP is found infeasible for any given

day d, we know that we must reduce the number of patients on that day by at least one,

or use at least one of the surgeons or ORs that was not previously allocated to day d.

As a result, we generate the feasibility cut

∣∣Pd∣∣−∑p∈Pd

∑s∈Sd

∑r∈Rd

xpsdr

+∑r∈R′

d

ydr +∑s∈S′d

zsd

≥ 1 (3.22)

where Pd, Sd, and Rd are the set of patients, surgeons, and ORs allocated to day d; and

P ′d, S ′d, and R′d are the set of patients, surgeons, and ORs not allocated to day d. On

the other hand, if there is a feasible sequencing on day d, but that sequencing uses more

overtime than was allocated in the MP it is impossible to schedule the current set of


Table 3.3: SP notation

Sets

Qp Set of qualified ORs for patient p

Qpk Set of qualified shared ORs between patient p and patient k

Ωp Set of qualified surgeons for patient p

Ωpk Set of shared surgeons between patient p and patient k

Parameters

As Available time of surgeon s

Br Available time of OR r

Ψ Maximum idle time between starting and finishing time of a surgeon (%)

Binary variables

xpsr 1 if patient p is operated by surgeon s in OR r and 0 otherwise

ypk 1 if patient p is operated after patient k and 0 otherwise

Continuous variables

cr Completion time of OR r

is Initiation time of surgeon s

es Ending time of surgeon s

vr Overtime of OR r


patients on the given day using the amount of overtime that was allocated. As a result,

we generate the optimality cut

∑r∈Rd

vdr ≥∑r∈Rd

vdr − vdr∣∣Pd∣∣−∑

p∈Pd

∑s∈Sd

∑r1∈Rd

xpsdr1

+∑r1∈R′

d

ydr1 +∑s∈S′d

zsd

(3.23)

mandating, if all other relevant variables stay the same, an increase of overtime on day

d. For more detail in the justification of these Benders cuts, see [37].

3.3 Data generation

To measure the impact of data generation, we first develop what we consider a standard

data generation scheme. As much as possible, we draw from our experiences with the

University Health Network in Toronto, Canada to inform the data generation procedure.

In particular, costs and average surgical times are derived from general surgery data

between the years of 2011 and 2013. However, many of the elements of our standard

procedure are subjective, and the methods described in this section are simply our best

attempt at generating realistic data. After deciding on a standard implementation, we

then identify steps in the data generation procedure that could reasonably be done in a

different manner; these steps will be the factors that we will vary in the data generation

process. We will first consider a standard data generation method; we will use this

method as a benchmark against which to compare other possibilities.

Costs are generated in specific ranges using the uniform distribution (Table 3.4). Daily

OR fixed costs are generated from a uniform distribution between $4500 and $6500;

surgeon costs are generated between $3500 and $5500. The reward for scheduling a

patient from the next planning horizon is distributed uniformly between $2000 and $5000,

and the OR overtime hourly cost is set as (OR fixed cost× 1.5)/8.


Table 3.4: Original data costs

Cost type Cost ($)

OR fixed cost Uniform [4500, 6500]

OR hourly cost OR fixed cost ×1.5/8

Surgeon cost Uniform [3500, 5500]

Patient reward Uniform [2000, 5000]

Surgical times are generated for all patient cases (both for cases of this and next

planning horizon) from a lognormal distribution with a mean surgical time of 180 minutes

and a standard deviation of 60. Once surgical times are generated, we separate cases

into three categories: short (less than 180 minutes), medium (180-300 minutes), and long

(greater than 300 minutes). Short surgical cases require 15 minutes of preparation time

and 10 minutes of cleaning time, medium cases require 20 and 15, long cases require 25

and 20.

To ensure data feasibility, we construct an initial feasible schedule (IFS) using only

patients from the current planning horizon, these are the patients that must be scheduled

in any feasible schedule. Patients that are due in the future do not impact feasibility; as

a result, they are not included in the IFS. To construct an IFS, we divide each planning

horizon (of five days) into 10 shifts, and we create a surgeon schedule by determining

what days each surgeon works, and which shifts that they work during that day. In the

IFS, all surgeons are made to be available for the same number of hours. We also assign

surgeons randomly to ORs for each shift. We then assign each patient to a surgeon.

When a patient is assigned to a surgeon, that patient is put into the earliest available

time slot that is available for that surgeon. If that patient cannot fit anywhere, that

surgeon’s week is considered full and we start scheduling patients for the next surgeon.

Using this procedure, we construct an IFS as a complete, shift-based feasible schedule

that uses no overtime.


From the IFS, we extract initial patient-surgeon and patient-OR flexibility matrices,

as well as initial surgeon schedules. We then expand on the initial extraction to make it

more flexible. We add patient-surgeon flexibility by adding allowable surgeons for each

patient at random until each patient can be operated on by at least 10% of surgeons. We

then add additional patient-OR flexibility. For the purposes of this experiment, we set

patient-OR flexibility to 100%, but any other value would be possible. At this point, we

also randomly generate surgeon and room flexibility matrices for each patient of the next

planning horizon so that the same 10% surgeon flexibility and 100% room flexibility is

reflected in those patients.

We also extract the due date of each patient from this planning horizon as the sched-

uled operating date in the IFS. Finally, we extract each surgeon’s availability time from

the IFS as the amount of time that surgeon spends working each day.

This data generation method is general enough so that it can be used in a wide variety

of situations, but specific enough that it can easily be reproduced exactly. Importantly,

it is guaranteed to have at least one feasible solution, the IFS.

3.4 Evaluation methodology

We consider data generation inputs as independent variables that affect our dependent

(output) variables, illustrating the consequences of data generation decisions.

Our algorithm is implemented in MATLAB 2008b (The Mathworks, Inc.) on a 4 Dual-

Core AMD OpteronTM

Processor 2.2.Ghz in a CentOS 2.6 platform with 40GB of RAM.

Gurobi 5.5 (Gurobi Optimization, Inc.) is used to solve all mixed integer programs. Data

was also generated using MATLAB2008b, and all random distributions that were needed

are generated using the MATLAB built-in Mersenne twister random number generator.

For each data generation alternative, we create 50 test cases, each using a unique random

seed.


Output variables

Output variables are the variables that we will measure as evidence that chances in data

generation will have a significant impact. Our output variables can be divided into two

distinct categories: computational difficulty and solution quality.

First, the computational difficulty will be measured in terms of the number of test

cases solved over time. We run trials up to a time limit of 2 hours (for practical consider-

ations), and will track how many problems are solved, as well as how long each case takes

to find an optimal solution. We do not use any sub-optimal solutions in the computation

of our average optimal solution statistics. Computational difficulty is important as most

researchers will use to it compare algorithms. If simple changes to data generation affects

computation time, then careful attention must be paid to data generation in any paper

that is evaluating a new algorithm.

Second, the solution quality of optimal solutions from different data sets will be

evaluated using the average number of patients, number of ORs and total cost. The

optimal solution is essential, as it will impact how results from generated data will hold

if they are implemented in real hospitals. Many papers give suggestions in terms of which

algorithms should be implemented in real systems; we will attempt to evaluate whether

these suggestions hold in a wide variety of different situations.

Input variables

Our input variables are different data generation methods. Decision points that affect

computational difficulty or solution quality are used as independent variables that we

vary, while holding all other elements of the data generation strategy constant. With

more and better data these decisions would be easier to make; however, there are limits

to how much data is available, and to how accurate that data will be. Furthermore,

there is always some compromise between the accuracy of the desired information and

the amount of time that it takes to gather that information. Sometimes, we may not


want to spend a lot of time and money collecting additional data, when the data available

is good enough. Input variables that we consider are OR cost, OR avilability time, and

patient-surgeon assignment flexibility. A summary of the choices that we will evaluate is

shown in Table 3.5.

In the original data generation scheme, described in Section 3.3, it is assumed that

each OR has a different fixed cost. This assumption is fair, as when the cost of equip-

ment, room preparation, room cleaning, anesthetists and other hospital support staff are

calculated for each room independently, the resulting fixed daily cost will be significantly

different for each OR. However, an alternative viewpoint is to spread ancillary costs dis-

tributed among ORs evenly, resulting in identical OR costs. This cost structure can be

justified, as it is often difficult to ascertain what the actual cost should be, and it may

depend highly on the opinions of the decision-maker. Thus, in addition to the variable

fixed OR cost data, we also evaluate an alternative cost structure in which fixed OR

costs are identical across the different ORs at 4500. We expect that the computation

time needed to solve these cases will be more difficult, as ORs will be more similar to

one another and thus our algorithm may spend some time dealing with the increased

symmetry. We also believe that total cost will be slightly higher, as the optimal schedule

will not be able to prefer the ORs with lower cost and avoid the ORs with higher cost.

Another key assumption in the original data generation scheme is the distribution of

surgical lengths. The problem with surgical lengths is that there is no standard in how

to model surgical times in general surgery. As a result, for our standard data generation

scheme we chose to assume surgical times are drawn from a lognormal distribution with

mean 180 and a standard deviation of 60, similar to [26]. However, in the literature we

can also find papers generating surgical times using the Pearson III distribution [16] and

the uniform distribution [23]. To mimic these studies, we generate alternative data sets

using Pearson III (mean 90, standard deviation 15, lower bound 40) and uniform (lower

bound 60, upper bound 240) distributions to compare with our standard lognormal data


set. Cumulative plots of all the surgical times generated are shown in Figure 3.1a. We

hypothesize that the data with Pearson III surgical times will be the most difficult to

solve. Since mean surgical times are much shorter, we can fit more patients into one day,

resulting in a difficult sequencing problem. We also anticipate that the data using the

uniform distribution (with mean value of 150) will be slightly easier to solve compared

to the lognormal distribution data for the same reason. In terms of optimal solution, the

shorter mean surgical times should result in lower objective function values.

In order to isolate the effect of using different distributions, we will also generate

data sets using lognormal, Pearson III, and uniform distributions with the same means

and standard deviations. We simply generate data for the lognormal and Pearson III

distributions with a mean of 150 and a standard deviation of 51.96, matching the statistics

of our previously generated uniform distribution. Empirical cumulative density function

plots of the generated surgical times are shown in Figure 3.1b. In general, we expect to see

the differences in computation time and optimal solution disappear when we normalize

the mean and standard deviation.

One last assumption that we consider is that patient-surgeon eligibilities are generated

at random. However, it is likely that the patient-surgeon flexibility matrices exhibit some

sort of structure. For example, two surgeons could have similar experience levels, resulting

in a large overlap in their potential patients. As another example, two surgeons could

be physically working in separate offices with completely different clientele, resulting in

a smaller overlap in their respective patient lists compared to two surgeons at the same

location. As a result, we develop an alternative data set where surgeons are divided into

three “blocks”. Each block of surgeons has a set of patients, and flexibility is generated

randomly within the blocks. The resulting patient-surgeon flexibility matrix has a block-

angular structure, as illustrated in Figure 3.2. We hypothesize the block structure should

not have a large effect on computational results or overall cost because the overall density

of the patient-surgeon eligibility matrix stays the same.


0 50 100 150 200 250 300 350 400 4500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Surgical length

Pro

babili

ty

Lognormal

Pearson

Uniform

(a) Non-normalized data

0 50 100 150 200 250 300 350 400 4500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Surgical length

Pro

babili

ty

Lognormal

Pearson

Uniform

(b) Normalized data

Figure 3.1: Empirical CDF of surgical lengths


(a) Block (b) Flexible

Figure 3.2: Alternative patient-surgeon flexibility matrices


Table 3.5: Data generation scenarios

Choice Parameter Original choice Alternative choices

OR fixed cost Kdr uniform [lb = 4500, ub = 6500] all identical [5500]

Surgical time Eps lognormal [µ = 180, σ = 60]uniform [lb = 60, ub = 240]

Pearson III [µ = 90, σ = 15]

Normalized surgical time Eps -

lognormal [µ = 150, σ = 51.96]

uniform [lb = 60, ub = 240]

Pearson III [µ = 150, σ = 51.96]

Patient-surgeon flexibility Λsd random [10%] block [10%]

The choices in data generation that we are evaluating is not exhaustive, but should

illustrate the impact of data generation on computational difficulty and solution quality.

With this analysis, we look to quantify the impact of these decisions so that researchers

will be able to make more informed decisions about data generation in the future.

3.5 Results

When we fix the OR costs to a constant value of $5500, the problem gets slightly easier,

as expected, although the effect is small (Figure 3.3). Notably, 1 out of the 50 cases with

identical costs was not solved within the 2 hour time limit, whereas all of the cases from

the uniform cost instance were solved within 40 minutes.

The non-adjusted surgical time cases show a much larger effect (Figure 3.4a). As

expected, the lognormal test instance was the easiest to solve by far. In contrast, none

of the cases in the Pearson III set were solved in the 2-hour time limit. Even when

we normalize the distributions to have the same mean and standard deviation, we still

see some differences among the three curves (Figure 3.4b). Although the lognormal and

Pearson III computation times are similar, they are significantly different compared to the

uniformly distributed data, and the discrepancy between the uniform instance and the

other two instances seems to widen over time. The magnitude of this difference indicates


0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

Time (min)

Num

ber

of solv

ed insta

nces

identical OR costs

randomized OR costs

Figure 3.3: Cumulative plot of solved instances of identical and randomized cost instances

that the chosen surgical distribution has a potentially large effect on computation time.

For patient-surgeon flexibility test instances (Figure 3.5), there is again separation

between the cumulative plot of the flexible instance and the block instance, with the

flexible instance being easier to solve.

If we compare the mean of the solution metrics of the original data and the data

with block patient surgeon assignment structure or static OR cost (Table 3.6), we see

that all differences in means are insignificant. However, there was a large difference

between solution metrics of the original data compared to the non-normalized uniform

distribution data. This difference was expected, as the patients generated under the

uniform distribution had much shorter surgical times, meaning that many more of them

could be scheduled for the same cost. When we correct for different surgical times using

normalization, this difference largely disappears (Table 3.7). As a result of these trials,

we can say that the nature of the optimal solutions found is robust to the small differences

in the input data that we measured.


0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

45

50

Time (min)

Num

ber

of solv

ed c

ases

Lognormal

Uniform

Pearson

(a) Non-normalized

0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40

45

50

Time (min)

Num

ber

of solv

ed c

ases

Lognormal

Uniform

Pearson

(b) Normalized

Figure 3.4: Cumulative plot of solved instances for different surgical distributions


0 10 20 30 40 50 60 70 800

5

10

15

20

25

30

35

40

45

50

Time (min)

Num

ber

of solv

ed insta

nces

flexible

block

Figure 3.5: Cumulative plot of solved instances for flexible and block patient-surgeon flexibility

Table 3.6: Solution metrics comparing different data generation strategies against the originaldata, - for unsolved trials

Cost ($1000s) Patients Surgeons Rooms

Mean SD Mean SD Mean SD Mean SD

Original data 145.78 11.40 34.18 1.80 18.42 2.07 13.84 1.04

Block assignment 133.22 9.25 35.46 2.06 16.84 1.82 13.54 1.01

Static OR cost 147.93 11.27 35.02 2.17 18.06 2.05 14.24 1.02

Uniform dist. 92.59 9.01 46.49 3.86 16.93 1.59 11.74 1.00

Pearson dist. - - - - - - - -


Table 3.7: Solution metrics comparing different normalized data generation strategies againstone another

Cost ($1000s) Patients Surgeons Rooms

Mean SD Mean SD Mean SD Mean SD

Logn. (normalized) 81.50 9.66 48.43 3.00 14.28 1.84 12.75 0.84

Unif. (normalized) 92.59 9.01 46.49 3.86 16.93 1.59 11.74 1.00

Pear. (normalized) 81.84 7.79 47.94 3.51 14.03 1.22 12.77 1.05

3.6 Discussion

Our results indicate that input data can have a significant impact on computation time.

Even seemingly innocuous data choices, such as how to cost different ORs, may have an

impact on computation time results.

In some cases, there is a clear justification for the difference in computation time

results. For example, in the case of non-normalized surgical time data variations, shorter

average surgical times resulted in longer computation times; this result is easily explained

by the tremendous difficulty in sequencing many small surgeries in one operating room.

However, in many other cases, it is more difficult to explain where the differences come

from. For example, there were big computational differences between flexible and block

patient assignment flexibility, but it is not simple to explain why. As a result, we can

say that it would be very difficult, for an arbitrary data generation decision, to predict a

priori whether or not that decision will have a large effect on computation time. Because

of this unpredictability, care must be taken when trying to compare computational results

across papers; in many papers it is almost impossible to situate the results within the

context of the larger body of literature as not enough detail has been given in the data

generation section, or because is it simply difficult to assess the impact of important

data assumptions on the results. Even within a paper, researchers must be careful when

concluding that one algorithm is better than another: It may be that the results that


were obtained were entirely due to idiosyncrasies in the data generation process, and

would not be obtained for some other instance of the same problem.

Our results are different when we look at the optimal solution metrics. It seems that

most changes evaluated were insignificant from a solution quality standpoint; when we

made small changes to the input data, the solutions looked similar. When differences

did appear, such as in the cases of non-normalized surgical times, we could easily explain

those differences. In general, changes in data generation produced a small predictable

result on solution metrics.

As a result, we cannot say that when we interpret results from studies done using

simulated data, the data generation procedure will have an impact on the quality of the

solution. This result is encouraging for researchers trying to justify policy decisions in

hospitals: Clinical results found from scheduling optimization models seem to be reliable,

independent of the data generation technique. However, note that we cannot say with

certainty that small changes in the data will have no impact, but only that we did not

find any effect in the dimensions that we tested, and future research is possible to further

confirm our results are generalizable.

3.7 Conclusions

It is important for researchers to publish exactly how their data was generated so that

their research is reproducible. As shown in the results, giving simple numbers about how

many patients and surgeons are in the system is not sufficient to communicate the data

that was tested on. Although it is more difficult to describe data generation procedures

in detail, our work has shown that it is necessary.

Future work could explore in more depth the factors that are the most important

within data generation in OR planning and scheduling. Furthermore, a standard test

set, including data generated using various methodologies, needs to be developed so that


researchers can easily compare algorithms against one another.

Chapter 4

Conclusions

Large-scale optimization models are frequently formulated and analyzed by operations

researchers to address problems in healthcare. To solve their models, researchers will

employ efficient solution methods such as Benders’ decomposition. We have seen two

important large-scale optimization models in this thesis.

In LGK PFX optimization, the SDIO model is used to find good treatment plans

for the LGK PFX in a single stage, instead of the previously attempted inexact two

step approaches that first locate isocenters using geometric heuristics, and only decide

on correct sector durations once isocenters have been fixed. We showed that we can

formulate a Benders’ decomposition that takes advantage of the natural decomposition

of SDIO into an isocenter selection upper model, and a sector duration subproblem,

while maintaining the exact nature of the SDIO formulation, and that the resulting

algorithm could solve instances that the standard branch-and-cut solver could not. The

results showed that that our two-phase Benders’ decomposition had some compuational

advantages over a standard branch-and-cut approach, in particular in terms of solving

extremely large problems, although branch-and-cut was still more effective in some cases.

We also developed a combinatorial Benders’ decomposition that showed some promise

compared to the classical Benders’ approach.

57

Chapter 4. Conclusions 58

In operating room planning and scheduling, we implement and solve TORPS using

a logic-based Benders’ algorithm. We develop a novel data generation procedure and

provided enough detail to guarantee reproducibility. We showed that the specific imple-

mentation of the data generation algorithm can have a large impact on solution time. As

a result, researchers need to develop and use standard test sets, or at least take care in

defining exactly how and why their custom data sets are used.

Overall, our research in this thesis contributes to the large-scale optimization in

healthcare community by enhancing our understanding of the characteristics of two im-

portant mixed-integer models. Although we do not develop any completely novel models,

we reproduce models and algorithms found elsewhere in the literature, create new algo-

rithms, and we attempt to deepen the knowledge of existing algorithms by modifying

certain aspects of the models, and examining the effects of those modifications. Our

findings in radiation therapy suggest novel, efficient, algorithms to apply to real prob-

lems, and our results in operating room planning and scheduling highlight the need to

consider data generation methods in evaluating novel algorithms.

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