Stochastic Programming Model for Elective Surgery Planning: An Effect

of Emergency Surgery

Ryota Akiyama, Mari Ito, Ryuta Takashimaand Kinju Hoshino

Department of Industrial Administration Faculty of Science and Technology, Tokyo University of Science,

2641 Yamazaki, Noda, Chiba 278-8510, Japan

Keywords:

Stochastic Programming, Emergency Surgery, Surgery Planning, Operating Room.

Abstract:

This paper introduces a stochastic programming model for a hospital with two surgery types: elective and

emergency surgeries. We propose a model that decides the number of the elective surgeries per day according

to a scheme that makes best use of the operating rooms. Speciﬁcally, we model when the demand capacity

for emergency surgery in the operating room of one day is uncertain. We created multiple surgery times, per-

formed random sampling, and conducted numerical experiments. In the results, emergency surgery changed

the allocation of elective surgery. In this paper, we report on the proposed model and numerical results, and

discuss these and the future research prospects.

1 INTRODUCTION

Operating room management is one of the critical fac-

tors in the management of hospitals. It is important

both for achieving high patient treatment quality and

suppressing hospital costs. From the viewpoint of

the treatment quality, a long waiting time for patients

caused by a delay relative to the scheduled end time of

a previous surgery is a serious problem. From an eco-

nomic point of view, about 60% of hospital income

comes from surgery (Jacson et al., 2002). In contrast,

about 40% of expenditure comes from surgery (Mar-

cro et al., 1995). Therefore, hospitals are creating op-

erating room schedules to improve the usage rate of

operating rooms and reduce the cost of surgery.

Surgery can be broadly divided into two types:

elective surgery and emergency surgery. Elective

surgery refers to surgery that is scheduled in advance.

Emergency surgery is not scheduled through discus-

sions between the patient and the doctor. Rather,

emergency surgery becomes necessary suddenly due

to a life-threatening condition in the patient. Emer-

gency surgery treatment must be done on the same

day. Many hospitals use the same operating room

for both elective surgery and emergency surgery. At

Erasmus University Medical Center Rotterdam, the

operating room dedicated to emergency surgery was

closed, so both types of surgery are being performed

in the same operating room (van Veen-Berkx et al.,

2016). The closure was due to a published study that

reported that performing the two different types of

surgery in the same operating room is more efﬁcient.

However, in the case of performing elective surgery

and emergency surgery in the same operating room,

operating room management needs to take into ac-

count the uncertainty of emergency surgery. This is

because not considering the uncertainty of emergency

surgery in operating room management would result

in poor service, such as delays in elective surgery.

Therefore, additional costs may be incurred under this

arrangement, including overtime operating room us-

age, labor costs, and hospitalization costs if carried

forward.

The ﬂow until the schedule is set starts with the

surgeon and patient deciding the date of the elec-

tive surgery by mutual agreement. This surgery date

will also depend on the available reserved capacity

for elective surgery on each day. Since when emer-

gency surgery will occur is uncertain, in order to man-

age elective surgery efﬁciently, the manager must aim

at the creation of a robust surgery planning schedule

that considers the timing uncertainty associated with

emergency surgery.

In elective surgery planning, costs associated with

overuse of the operating room and costs associated

with performing elective surgery should be mini-

mized. However, few studies have considered the un-

certainty of emergency surgery (Cardoen et al., 2010,

Zhu et al., 2019). If uncertainty is not taken into ac-

count, service degradation such as delayed procedures

can occur, which incurs associated additional costs.

Akiyama, R., Ito, M. and Hoshino, R.

Stochastic Programming Model for Elective Surgery Planning: An Effect of Emergency Surgery.

DOI: 10.5220/0010901800003117

In Proceedings of the 11th International Conference on Operations Research and Enterprise Systems (ICORES 2022), pages 231-235

ISBN: 978-989-758-548-7; ISSN: 2184-4372

231

Therefore, surgery planning carries forward overtime

use of the operating room and elective surgery. It

is considered necessary to minimize the costs in-

curred by this. Gerchak et al. (1996) proposed a

dynamic programming model for pre-scheduling pa-

tients with single-period elective surgery. Lamiri et

al. (2006, 2009) suggested considering the uncer-

tainty of emergency surgery demand and proposed

optimization models and algorithms for surgery plan-

ning. Their optimization models use the time of emer-

gency surgery as a random variable generated for each

surgery according to an exponential distribution. We

propose elective surgery plans with multiple planning

periods. However, it is unclear whether the best ap-

proach is to follow Lamiri et al. (2006, 2009) and gen-

erate a probabilistic emergency surgery time and cre-

ate a deterministic schedule. Akiyama et al. (2021)

proposed a probabilistic planning model for schedul-

ing surgery. Comparing the propose model and the

Lamiri model, it can be seen that the proposed model

is superior to the Lamiri model because the amounts

of overtime are shorter.

In this study, we propose a stochastic program-

ming model for elective surgery planning. The ef-

fectiveness of the proposed model was veriﬁed by

considering the uncertainty of duration of emergency

surgery, as well as the allocation of elective surgery.

An emergency surgery scenario created using Monte

Carlo sampling was used to analyze how elective

surgery was assigned. Our results were that the

surgery was carried over if the surgical capacity for

a later period was sufﬁcient. The question of whether

the commonly used one-week planning period is cor-

rect was also considered.

The structure of the remainder of the manuscript

is as follows. Section 2 shows the proposed formula-

tion of elective surgery planning. Section 3 describes

the problem setting used for the numerical analysis.

Section 4 presents and discusses the results. Section

5 summarizes the paper and describes future research.

2 THE MODEL

In this section, we introduce proposed formulation of

elective surgery planning under uncertainty in emer-

gency surgery.

2.1 Elective Surgery Planning

The elective surgery planning determines the number

of elective surgeries per day using an uncertain

demand model for emergency surgery. In operating

room scheduling, it is necessary to schedule all

surgeries performed during a given planning period.

In this study, the capacity of all operating rooms

subject to operating room scheduling is available for

surgery. Thus, the total daily capacity of operating

rooms is the same as the time per day that they are

open. The cost carried forward and total capacity are

considered here. We consider overtime, deﬁned as

the amount of time beyond the intended closing time,

of the operating room, but not the penalties for low

operating room usage.

Notation

Index Sets

T : The set of days for scheduling.

I: The set of elective surgery index values.

S: The set of emergency surgery scenarios.

Parameters

: time needed for performing elective surgery i,

which is assumed to be a given constant.

: earliest period for performing elective surgery i.

: cost ratio of performing elective case i within

day t.

: total available regular capacity of all ORs on

day t.

: cost ratio per unit of overtime on day t.

: capacity needed for emergency surgery on day t.

Variables

: expected operating room overtime on day t.

: probability of allocated elective surgery i on

day t.

: 1 if elective surgery i is to be scheduled in

scenario s on day t; 0 otherwise.

Formulation

Minimize

∑

i∈I

|T |+1

∑

t=B

∑

t∈T

(1)

subject to

= E

[(W

∑

i∈I

− T

)], ∀t ∈ T, (2)

|T |+1

∑

t=1

= 1, ∀i ∈ I, (3)

= {0, 1}, ∀i ∈ I, ∀t ∈ |T | + 1, ∀s ∈ S, (4)

= E

], ∀i ∈ I, ∀t ∈ T, ∀s ∈ S, (5)

≥ 0, ∀i ∈ I, ∀t ∈ T. (6)

ICORES 2022 - 11th International Conference on Operations Research and Enterprise Systems

232

In the above formulation, the objective function

(1) minimizes carry-forward costs and overtime ac-

ceptance costs. Constraint (2) deﬁnes overtime from

the closing time of operating room on day t. Con-

straint (3) guarantees that each elective surgery is al-

located only once. Constraint (4) is a binary con-

straint . Constraint (5) is an equation to ﬁnd the al-

located distribution of elective surgery. Constraint (6)

represents a non-negativity constraint on Y

3 ANALYTICAL METHOD

In this study, we conducted numerical experiments

on how elective surgery was assigned for 100 emer-

gency surgery scenarios. The procedure was as fol-

lows. Based on the surgery data of an actual hospital

(Akiyama et al., 2021), we assume ﬁve types of elec-

tive surgery and one type of emergency surgery. We

selected the top ﬁve types with the largest sample size

from about 9,000 actual surgery data. The time of

each of the ﬁve elective surgeries follows a lognormal

distribution, and ten random numbers are generated

for each type of elective surgery to create a total of 50

data. The operation time of emergency surgery is also

lognormal. It is assumed that a normal distribution

is followed, and 10 units are created. In the veriﬁca-

tion of this study, two operating rooms are assumed.

The open time per room is 480 minutes. In addition,

elective surgery is performed every day. It is assumed

that 4 elective surgeries and one emergency surgery

occur every day. The scheduling period is 6 to 8 days.

In the case of 6 days, a total of 24 elective surgeries

will occur and 6 emergency surgeries will occur. For

these 24 elective surgeries, further random sampling

is performed from the created 50 data.

The above data are applied to the proposed model

to determine the optimal number of elective surgery

assignments for each day, considering the uncer-

tainty of emergency surgery. We generate emergency

surgery scenarios by Monte Carlo sampling. The

probability of the occurrence of a scenario follows a

uniform distribution. We assume that a schedule for

emergency surgery will always be accepted. The to-

tal available capacity of the operating rooms here is

about 960 minutes, which is the capacity of the two

operating rooms.

Table 1 shows the expected value and standard

deviation of the surgery duration, and the end time,

Table 1: Duration for elective and emergency surgery [min].

Surgery ID A B C D E Emergency

Expected value 154 177 293 215 235 91

Standard deviation 95 77 78 87 64 79

which is the time when the surgery needs to be com-

pleted.

4 RESULTS AND DISCUSSION

In this study, we performed and analyzed multiple nu-

merical experiments. Here, we mention the analysis

of one of these in detail as an example.

Table 2 shows the elective surgery requests. As

described in Section 3, we assumed in the created

surgical data that randomly occurring elective surgery

was performed. The values in the table are the dura-

tion of each elective surgery in minutes and the letters

in parentheses refer to the surgery ID.

Figures 1–3 show which surgery was carried

over and how frequently among the 100 emergency

surgery scenarios for each day. In the ﬁgures 1–3,

the vertical axis gives the execution rate, which is

the ratio of which day the elective surgery was as-

signed among 100 emergency surgery scenarios.The

horizontal axis shows the “elective surgery number”.

The depth axis shows the day. Regarding the surgery

numbers on the horizontal axis, theses represent the

dates, and A to D indicate the four elective surgeries

scheduled for that day. The longer the period, the

more often elective surgery will be carried over. The

results show that this carrying over occurred when

there was unused capacity of the operating room in a

later period. This implies that as a result of extending

the planning period, the number of carry-overs has in-

creased because the number of days with unused sur-

gical capacity has increased. On the other hand, how-

ever, it would be represented that even if the planning

period is extended, overtime will increase if the num-

Table 2: Elective surgery requests: surgery duration

(surgery ID) [min].

Surgery Day 1 Day 2 Day 3 Day 4 Day 5 Day 6

1(Elective) 332(C) 126(B) 383(C) 353(D) 212(B) 33(A)

2(Elective) 309(E) 209(B) 263(B) 212(A) 256(B) 286(B)

3(Elective) 214(C) 269(C) 269(E) 231(B) 277(D) 59(B)

4(Elective) 71(A) 253(E) 253(A) 335(C) 189(A) 404(C)

Figure 1: Execution rate for 6 days.

Stochastic Programming Model for Elective Surgery Planning: An Effect of Emergency Surgery

233

Figure 2: Execution rate for 7 days.

Figure 3: Execution rate for 8 days.

Figure 4: Overtime for 6 days.

ber of days exceeding the surgical capacity increases.

Overtime will also increase if surgical capacity is ex-

ceeded on the last day of the planning period.

Figures 4–6 show overtime as a function of day. In

these ﬁgures, the vertical axis shows the time and the

horizontal axis shows the day. In this result, we found

that the length of overtime decreases in proportion to

the length of the period. Operating room capacity for

later days must be available in order to accept elective

surgery.

Most hospitals create operating room schedules

for a period of one week, but the results here indicate

that a longer planning period reduces overtime. This

Figure 5: Overtime for 7 days.

Figure 6: Overtime for 8 days.

raises the question of whether the planning period cur-

rently in use is appropriate. However, although there

is a possibility that overtime can be shortened by ex-

tending the planning period, there is also a overtime

that it will increase. Three cases were carried forward.

We found that the surgery with the longest operation

time was selected among these three cases during the

scheduling period, and that these surgeries could be

signiﬁcantly delayed if performed on time.

5 CONCLUDING REMARKS

In this study, we developed an elective surgery plan-

ning model for establishing a surgical plan for multi-

period elective surgery considering the uncertainty of

the occurrence of emergency surgery. The proposed

model considers the uncertainty of emergency surgery

by scenario. Considering the uncertainty duration of

emergency surgery for the elective surgery schedul-

ing, the number of elective surgery allocations sched-

uled during the period was determined. In the re-

sults, the elective surgery was assigned to another

day to avoid overtime in the operating room due to

elective surgery during the scheduling period. Cur-

rently, most hospitals create operating room schedules

ICORES 2022 - 11th International Conference on Operations Research and Enterprise Systems

234

for one week. However the results here indicate that

a longer planning period reduces overtime. There-

fore, there is room to consider whether one week is

appropriate for the length of the planning period for

scheduling.

As potential future research, the uncertainty in the

number of emergency surgeries should be taken into

consideration. In addition, the operating room capac-

ity of each operating room should be separately con-

sidered, rather than just the total operating room ca-

pacity. Therefore, we would like to extend the pro-

posed model to a model of multiple operating rooms.

We would also like to further examine whether the

appropriate planning period is one week.

ACKNOWLEDGEMENTS

This work was supported by JSPS KAKENHI Grant

Number JP21K14371.

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