A Robust Optimization for a Single Operating Room Scheduling

Problem with Uncertain Durations

Yoshito Namba

, Mari Ito

and Ryuta Takashima

Department of Industrial Administration, Tokyo University of Science, Chiba, Japan

Center for Mathematical and Data Sciences, Kobe University, Hyogo, Japan

Keywords: Operating Room, Robust Optimization, Scheduling, Surgical Management.

Abstract: In order to improve the quality of patient care, efficient surgical management is significant for overall hospital

management. This study proposes a robust optimization model that minimizes delay in surgery by taking the

surgical sequence into account. We verified an influence of the risk-averse tendency on the schedule. In the

numerical analysis, the schedule created by the robust optimization model was compared with that of the

stochastic programming model. The results suggest that robust optimization models tend to avoid long delays.

1 INTRODUCTION

Efficient surgical management is important for the

quality of patient care and hospital management. The

quality of patient care is affected because of the long

waiting time of patients owing to the delay from the

scheduled end time of surgery. In terms of hospital

management, surgeries account for most of the

hospital revenue and expenditure (Jackson, 2002;

Macario

et al.; 1995). Therefore, an operating room

schedule is created to improve its operating rate and

reduce the cost of surgery.

In the scheduling flow of the operating room, the

surgeon and patient decide the surgery date through

mutual agreement. The surgeon then reports the

estimated duration of surgery to the operating room

manager. The manager decides when and in which

operating room to perform the surgery, based on

information such as the estimated duration of surgery.

However, there is uncertainty regarding the duration

of the surgery. Factors of uncertainty include the

patient's condition, lack of information on the

preoperative diagnosis, and the surgeon's skill.

Surgery is often not performed according to the

scheduled end time based on the reported duration. In

addition, there may be a risk of delay, with surgery

being delayed significantly from the scheduled end

time. Long delays lead to increased overtime for

surgical staff, not only increasing costs, but also

https://orcid.org/0000-0001-5590-5008

reducing staff satisfaction. Therefore, to manage the

operating room efficiently, robust scheduling that

considers the uncertainty of the surgical duration is

required. In the operating room scheduling, it is

necessary to consider decision-making to avoid the

risk of delay.

Operating room scheduling has been studied

extensively (Cardoen et al., 2010; Gerchak et al.,

1996; Lamiri et al., 2008). For example, Addis et al.

(2016) proposed the operating room rescheduling by

considering the uncertainty of patient arrival and the

duration of surgery. Ito et al. (2019) formulated a

single operating room scheduling problem that

considers the uncertainty of the surgical duration. A

risk measure called conditional value-at-risk (CVaR)

was used to reflect the tendency toward delayed risk

aversion. Another technique that reflects this

scheduling trend is robust optimization. Aslani et al.

(2021) proposed a robust optimization model with a

radix constraint for the first-time and repeat patients

in urology, considering the risk of a significant

increase in the arrival of a number of first-time

patients. Shi et al. (2019) formulated a robust

optimization model for a home health care routing

and scheduling problem with considering uncertain

travel and service times. The authors compared the

solutions obtained by the stochastic programming

model and the robust optimization model. Denton et

al. (2010) proposed an operating room scheduling

model with robust optimization to address the

180

Namba, Y., Ito, M. and Takashima, R.

A Robust Optimization for a Single Operating Room Scheduling Problem with Uncertain Durations.

DOI: 10.5220/0011733600003396

In Proceedings of the 12th International Conference on Operations Research and Enterprise Systems (ICORES 2023), pages 180-184

ISBN: 978-989-758-627-9; ISSN: 2184-4372

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

uncertainty of the surgical duration. However, the

previous study did not consider the sequence of

surgeries in the operating room. When scheduled, in

practice, it is necessary to consider the sequence of

surgeries within the operating room, because it is

more convenient to perform surgeries belonging to

the same department consecutively when arranging

surgical equipment and adjusting schedules.

In this study, we propose a robust optimization

model that considers the sequence of surgeries and

minimizes the delay. In the numerical analysis, the

delay was calculated for uncertain surgical duration

parameter sets. We compared it with a stochastic

programming model to verify whether the risk-averse

tendency is reflected in the schedule.

2 MATHMATICAL MODEL

2.1 Single Operating Room Scheduling

We propose a robust optimization model for the

single-operating-room scheduling problem under

uncertain parameter sets, the worst-case that results in

maximum total surgical duration. Single operating

room scheduling determines the procedures for

surgeries in an operating room. The operating room

scheduling model and the formulation of the

maximum surgical duration problem, which is

considered the main problem, is presented below.

Note that the stochastic programming model is the

model from Ito et al. (2022).

Notation

Index Sets

𝐽: Set of surgeries.

𝐷: Set of departments.

𝐸



: Set of surgeries belonging to the same department

𝑑, 𝑑∈𝐷.

Parameters

𝑤



: Weight of surgery 𝑗, 𝑗∈𝐽.

𝑑



: The time from the operating room opening to the

time when surgery 𝑗 should be completed, 𝑗∈𝐽.

𝑝



,𝑝



: Upper and lower bounds on the duration of

surgery 𝑗, 𝑗∈𝐽.

𝜏 : Constant control conservative. Set how

conservatively you want to control the worst-case

scenario from the decision -maker’s perspective. This

represents the number of surgeries for which the

upper bound of the surgical duration is reached.

Variables

𝑝



: Duration of surgery 𝑗, 𝑗∈𝐽.

𝑐



: Finishing time of surgery 𝑗, 𝑗∈𝐽.

𝑡



: Delay in surgery 𝑗 from the expected end time, 𝑗∈

𝐽.

𝑧



: Surgery precedence binary variable, where 𝑧



1 if surgery 𝑖 is processed before surgery 𝑗, 𝑧



otherwise, 𝑖,𝑗 ∈ 𝐽,𝑖≠ 𝑗.

𝛼,𝛽



: dual variables, 𝑗∈𝐽.

Formulation

Minimize 𝑤



𝑡



∈



(1)

Subject to

𝑝



𝑧



∈









+𝑝



≤𝑐



,∀

𝑗

∈

𝐽

(2)

𝑡



+𝑑



≥𝑐



,∀

𝑗

∈

𝐽

(3)

𝑧



+𝑧



=1,∀𝑖≠

𝑗

∈

𝐽

(4)

𝑧



+𝑧



+𝑧



≤2,∀𝑖≠𝑘≠

𝑗

∈

𝐽

(5)

𝑧



∈

−𝑧







∈

=1,

∀𝑖≠𝑖



∈𝐸



,∀𝑑∈𝐷,

(6)





𝑝



−𝑝





∈



≥α𝜏+



𝑝



−𝑝





𝛽



∈

(7)

𝑝



−𝑝



𝛼+𝛽



≥1,∀

𝑗

∈

𝐽

(8)

𝑝



≤𝑝



≤𝑝



,∀

𝑗

∈

𝐽

(9)

𝛼,𝛽



,𝑐



,𝑡



≥0,∀

𝑗

∈

𝐽

(10)

𝑧



∈



0,1



,𝑖≠

𝑗

∈

𝐽

(11)

In the formulation above, the objective function

(1) minimizes the delay in surgery 𝑗 from the

expected end time. Constraint (2) defines the surgery

completion time according to the surgery sequencing

relationships. Constraint (3) determines the delay in

surgery. Constraints (4) and (5) ensure the feasibility

of the surgery sequence by eliminating cyclic

sequences. Constraint (6) sequentially allocates

surgeries 𝑖 and 𝑖



because surgeries 𝑖 and 𝑖



are in the

same department, and hence, it is more convenient to

perform surgeries in the same department

consecutively when arranging surgical equipment

A Robust Optimization for a Single Operating Room Scheduling Problem with Uncertain Durations

181

and adjusting schedules. Constraints (7) and (8) are

the objective function values for the dual problem.

Constraint (9) bounds the surgical duration using

upper and lower bounds on the duration of surgery 𝑗.

Constraint (10) is a non-negative constraint.

Constraint (11) is a binary constraint:

2.2 Surgical Duration Uncertainty

As discussed in the Introduction, real-world surgical

durations are often subject to uncertainties. A robust

optimization model that considers uncertainty may be

more suitable and reasonable for decision making.

Our study involved uncertainty regarding surgical

duration. We assumed that the uncertain surgical

duration 𝑞



for each surgery 𝑗 is with respect to the

uncertainty set, without assumptions on distribution.

The formulations are as follows:

Formulation

Maximaize𝑞





∈

(12)

Subject to

𝑞





=𝑝



−𝑝



,∀

𝑗

∈

𝐽

(13)



𝑞





𝑝



−𝑝





∈

≤𝜏,

(14)

0≤𝑞





≤𝑝



−𝑝



, ∀

𝑗

∈

𝐽

(15)

In the above formulation, the objective function (12)

defines the maximum surgical duration. Constraint

(13) sets the left side of constraint (9) to zero and

makes constraint (15) a nonnegative constraint to

create a dual problem. The left side expresses an

upper bound on the number of surgeries that will

achieve their worst-case upper bound on surgical

duration. Constraint (14) controls excessively

conservatively, which is a weakness of robust

optimization.

3 NUMERICAL ANALYSES

3.1 Data and Analysis Procedures

We solve the single operating room scheduling

problem using Gurobi 9.5.1. The computational

equipment is an Intel(R) Core (TM) i7-7500U CPU

@ 2.90 GHz 8.00 GB. Specifically, there is one

operating room, five surgeries, and the lower bound

𝑝



of the surgical duration is defined as 𝔼𝑝



−𝜎



and the upper bound 𝑝



is defined as 𝔼𝑝



+𝜎



Table 1: Two types of instances

Instance 1

Surgery

𝑗

1 2 3 4 5

𝔼𝑝



(min) 120 120 120 120 120

𝜎



(min) 20 40 60 80 100

Instance 2

Surgery

𝑗

1 2 3 4 5

𝔼𝑝



(min) 160 140 120 100 80

𝜎



(min) 20 40 60 80 100

Then, 𝔼𝑝



 and 𝜎



represent the expected value and

standard deviation of the duration of surgery 𝑗,

respectively. The conservative 𝜏 is varies from 1 to

0—5. These two types of instances are listed in Table

1. As shown in Table 1, instance 1 has the same

expected surgical duration for all surgeries. In

contrast, the standard deviations were different. In

instance 2, the standard deviation is the same as that

in instance 1, but the expected value is different. All

weights 𝑤



are 1. The time from the operating room

opening to the time when surgery 𝑗 should be

completed, 𝑑



is 8 h or 480 min. Here, 𝑑



means the

regular opening time of the operating room; it is

desirable that all surgeries be completed within the

closing time.

We compared the schedule created using the robust

optimization model with that derived using the

stochastic programming model. The occurrence

probability of the 1000 scenarios used in the

stochastic programming model was assumed to

follow a uniform distribution. The surgical duration

in each scenario followed a log-normal distribution.

3.2 Results

The results of comparing the two models for each

instance are shown in Tables 2 and 3. All instances

are solved within 10 seconds of CPU time. Tables 2

and 3 show the sequence of surgeries, expected delay,

and number of scenarios in which the delay is greater

than or equal to 1000 min for schedules created using

ICORES 2023 - 12th International Conference on Operations Research and Enterprise Systems

182

Table: 2 Results of instance 1.

Model

Constant

controlling

conservative, 𝜏

Surgical sequence

Expected delay

(min)

Number of parameters

with significant delays

Stochastic

rammin

- 2, 1, 3, 4, 5 170.07 6

Robust

optimization

0 5, 4, 3, 2, 1 206.17 23

1 5, 1, 2, 3, 4 191.26 15

2 4, 1, 2, 3, 5 172.46 6

3 4, 1, 2, 3, 5 172.46 6

4 2, 1, 3, 4, 5 170.07 6

5 2, 1, 3, 4, 5 170.07 6

Table 3: Results of instance 2.

Model

Constant

controlling

conservative, 𝜏

Surgical sequence

Expected delay

time(min)

Number of parameters

with significant delays

Stochastic

rammin

- 3, 4, 5, 2, 1 181.17 25

Robust

optimization

0 5, 4, 3, 1, 2 190.69 25

1 5, 4, 3, 2, 1 185.08 24

2 5, 4, 3, 2, 1 185.08 24

3 5, 4, 3, 2, 1 185.08 24

4 5, 4, 3, 2, 1 185.08 24

5 2, 1, 3, 4, 5 205.78 8

the stochastic programming model and robust

optimization model at each control conservative τ.

From Table 2, the expected delay of the schedule

created using the stochastic programming model and

robust optimization model when τ = 5 is the lowest.

The scenario in which the delay was more than

1000 minutes was also the lowest. The number of

parameters with a significant delay of more than 1000

min reached a maximum at τ = 0. In summary, as τ

increases, the results of the robust optimization model

approach those of the stochastic programming model.

This indicates that the robust optimization model

without distribution assumptions performs as well as

the stochastic programming model, depending on the

setting of the control conservative τ.

According to Table 3, the schedule created using

the stochastic programming model exhibits the lowest

expected delay. The number of parameter sets with a

delay of more than 1000 min were the minimum in

the schedule created by the robust optimization model

when τ = 5. In addition, while the expected delay

increases as 𝜏 increases, the number of parameter sets

in which a delay of more than 1000 minutes occurs

decreases.

These results suggest that the robust optimization

model performs as well as the stochastic

A Robust Optimization for a Single Operating Room Scheduling Problem with Uncertain Durations

183

programming model, and tends to avoid significant

delays under certain conditions. Thus, it is suggested

that robust optimization models may be able to reflect

the risk-averse tendencies of operating room

managers in their schedules.

4 CONCLUDING REMARKS

In this study, we proposed a robust optimization

model that minimizes the delay in surgery by

considering the sequence of surgery. We also verified

whether the risk-averse tendency is reflected in the

schedule. The numerical analysis suggests that robust

optimization models tend to avoid long delays. From

the numerical analysis, compared to stochastic

programming models, the robust optimization model

is more effective for operating room managers who

desire to avoid long delays.

In future work, we will consider the relationship

between conservatism, delay and duration of surgery

set in a robust optimization model. We will clarify

this relationship by performing a numerical analysis

by increasing the set of surgical durations, which is

the input. We will expand the settings from a single

operating room to multiple operating rooms and use

real data to refine the schedules.

ACKNOWLEDGEMENTS

This work was supported by the Japan Society for the

Promotion of Science KAKENHI Grant [number JP

21K14371]. The authors would like to thank Manabu

Hashimoto of National Cancer Center Hospital East

and Hirofumi Fujii of National Cancer Center for

their valuable comments.

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