A Hybrid Model for the Analysis of a Surgical Pathway

Roberto Aringhieri and Davide Duma

Department of Computer Science, Universit

a degli Studi di Torino, Corso Svizzera 185, 10149, Torino, Italy

Keywords:

Surgical Pathway, Operating Room Management, Bed Management, Simulation, Optimization.

Abstract:

In this paper we focus our attention on the analysis of a surgical pathway from a patient-centred point of view.

The main concern of this work is the introduction of some optimization modules in the management of the

most critical resources in a surgical pathway, that is the stay beds and the operating rooms, and to evaluate

their impact with respect to a set of patient– and facility– centred indices. We propose a hybrid simulation

and optimization model: simulation is used in order to generate a real situation with respect to the inherent

stochasticity of the problem while optimization is used to take the best decisions in different points of the

surgical pathway.

1 INTRODUCTION

The current development of the health care systems

is aimed to recognize the central role of the patient as

opposed to the one of the health care providers. In this

context, Clinical Pathways (CPs) shift the attention

from a single health beneﬁt to the health care chain

that starts to resolve the illness episode. They can be

deﬁned as “health-care structured multidisciplinary

plans that describe spatial and temporal sequences of

activities to be performed, based on the scientiﬁc and

technical knowledge and the organizational, profes-

sional and technological available resources” (Camp-

bell et al., 1998).

The aim of a care pathway is to enhance the qual-

ity of care by improving patient outcomes, promot-

ing patient safety, increasing patient satisfaction, and

optimizing the use of resources as stated by the Eu-

ropean Pathway Association. Moreover, while many

papers show that, appropriately implemented, CPs

have the potential to increase patient outcome, reduce

patient length of stay and limit variability in care,

thereby yielding cost savings (Rotter et al., 2010), lit-

tle attention has been dedicated to study how CP can

optimize the use of resources.

In this paper we focus our attention on the analy-

sis of a surgical pathway from a patient-centred point

of view. The main concern of this work is the intro-

duction of some optimization modules in the manage-

ment of the most critical resources in a surgical path-

way, that is the stay beds and the operating rooms,

and to evaluate their impact with respect to a set of

patient- and facility- centred indices.

Our approach is a hybrid simulation and optimiza-

tion model. Simulation is used in order to generate a

real situation with respect to the inherent stochasticity

of the problem while optimization is used to take the

best decisions in different points of the surgical path-

way. Accordingly to (Magerlein and Martin, 1978),

we consider the operative decisions concerning the

advanced scheduling and allocation scheduling of pa-

tients. Furthermore we consider the real time man-

agement of the operating room planning.

The aims are to reduce the waiting list according

to a prioritized admission system, to operate patients

within a given time limit depending on their level of

urgency, to improve the utilization of the above crit-

ical resources and to minimize the number of can-

cellations. The results show an evident improvement

of the patient-centred indicators without deteriorating

the facility-centred ones.

The paper is organized as follow. Section 2 brieﬂy

review the literature regarding the problem under con-

sideration. The problem is depicted in Section 3 while

the integrated simulation and optimization model is

discussed in Section 4. Model validation and its main

results are discussed in Section 5 and Section 6, re-

spectively. Section 7 closes the paper.

2 LITERATURE REVIEW

A CP can be conceived as an algorithm based on a

ﬂow chart that details all decisions, treatments, and

889

Aringhieri R. and Duma D..

A Hybrid Model for the Analysis of a Surgical Pathway.

DOI: 10.5220/0005148408890900

In Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (HA-2014), pages

889-900

ISBN: 978-989-758-038-3

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

reports related to a patient with a given pathology,

with a logic based on sequential stages (De Bleser

et al., 2006). A CP is therefore ”the path” that a pa-

tient suffering from a disease walks in the National

Health System. This pathway can be analysed at

a single, local level of care (a single hospital, or a

single region) or globally, taking into account every

level of health-care from education and prevention, to

diagnosis of diseases, treatment and recovery. CPs

are speciﬁcally tailored to stimulate continuity and

coordination among the treatments given to the pa-

tient through different disciplines and clinical envi-

ronments. For this reason, they can be considered an

operational tool in the clinical treatment of diseases,

from a patient-focused point of view (Panella et al.,

2003).

As already discussed in the introduction, little at-

tention has been dedicated to study how CP can op-

timize the use of resources. The few papers deal-

ing with the CP resource optimization are reviewed

in (Aringhieri et al., 2012) to which the reader can

refer to deepen this topic. In the remaining of the

section, we brieﬂy report the literature concerning the

optimization solutions applied to the operating room

planning.

In the international literature there is a large num-

ber of papers dealing with the OR planning problem.

(Cardoen et al., 2010; Guerriero and Guido, 2011)

have recently published exhaustive literature reviews

on the operating room planning and scheduling prob-

lem classes, analysing in detail multiple ﬁelds related

to the problem settings and summarizing signiﬁcant

trends in research areas of future interest.

Problems arising in the OR planning and schedul-

ing are usually classiﬁed into three phases corre-

sponding to three decision levels, that is strategic

(long term), tactical (medium term) and operational

(short term) (Testi et al., 2007). Here, we take into

account the last phase – “surgery process scheduling”

– which is generally separated into two sub-problems

referred to as “advanced scheduling” and “allocation

scheduling” (Magerlein and Martin, 1978). The ﬁrst

sub-problem consists in assigning a speciﬁc surgery

and OR time block to each patient over the plan-

ning horizon, which can range from one week to

one month (Ozkarahan, 2000; Guinet and Chaabane,

2003; Lamiri et al., 2008; Fei et al., 2008; Hans

et al., 2008; Marques et al., 2012; Rizk and Arnaout,

2012). Given this advanced schedule, the second sub-

problem then determines the precise sequence of sur-

gical procedures and the allocation of resources for

each OR time block and day combination (Pham and

Klinkert, 2008; Cardoen et al., 2009; Roland et al.,

2010; Riise and Burke, 2011; Herring and Herrmann,

2012; Meskens et al., 2013) in order to implement

it as efﬁciently as possible. Approaches dealing with

more than one planning level simultaneously are quite

rare (Jebali et al., 2006; Testi et al., 2007; T

anfani and

Testi, 2010).

When dealing with uncertainty, literature usu-

ally considers three main issues, that is the arrival

of patients (see, e.g.,(Lamiri et al., 2008; M et al.,

2009)), the variability of patient length of stays (see,

e.g., (Beli

en and Demeulemeester, 2007; Adan et al.,

2011; Herring and Herrmann, 2011)) and the variabil-

ity of patient operating times (see, e.g., (Hans et al.,

2008; Min and Yih, 2010; T

anfani et al., 2010; Batun

et al., 2011; Shylo et al., 2013)). Methodologies

adopted ranges from montecarlo simulation to sample

average approximation, from probabilistic and chance

constraints programming to stochastic programming

with recourse. Some authors use simulation to com-

pare different scheduling and sequencing strategies

and test the solution robustness against the random-

ness of surgery duration (Testi et al., 2007; Persson

and Persson, 2010; Sobolev et al., 2011).

3 PROBLEM STATEMENT

We consider the problem of managing a single sur-

gical pathway taking into account the optimization

problems arising when dealing with the decision

levels concerning the advanced and the allocation

scheduling. In order to guarantee the execution of

such decisions, we deal with the real time manage-

ment of an operating room planning. It consists in

a sort of centralized surveillance system whose main

task is to supervise the execution of the planning and,

in the case of delays, to take a decision regarding the

patient cancellation or the overtime assignment.

The deﬁnition of the surgical pathway is inspired

to that presented and analysed in (Ozcan et al., 2011)

for the thyroid surgical treatment. The reader can re-

fer to this paper for further details. From a manage-

ment point of view, a surgical pathway can be seen as

made up of three phases.

The ﬁrst phase concerns the pre-admission phase

and it is related to all the activities regarding the pa-

tients before the admission as depicted in Figure 1.

A Diagnosis Related Groups (DRG) deﬁnes a gen-

eral time limit before which the patient should be op-

erated on. In our context, a Urgency Related Group

(URG) is assigned to each patient belonging to the

same DRG: the URG states a more accurate time

limit. In other word, URG allows to deﬁne a partition

of the patients in such a way to prioritize their surgical

intervention. The optimization problem arising in this

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Process

Decision

Data

Predefined Process

Internal Storage

Document

Multidocument

Preparation

Manual Input

Manual Operation

Connector

Off-page Connector

Terminator

Card

Punched Tape

Summing Junction

Collate

Sort

Extract

Merge

Stored Data

Delay

Sequential Access

Storage

Magnetic Disk

Direct Access

Storage

Display

Booking

Center

queue

Arrival

Process

yes

Exit

Booking

Center

Ambulatory

visit queue

Ambulatory

visit queue

Need a

surgery?

Pre-operative

exams queue

Pre-operative

exams

Check list

Need

additional

exams?

Pre-admission

visit queue

Pre-admission

visit

Wait for

scheduling

Has been

successfull

scheduled?

Wait for

admission date

Hospital phase

yes

Figure 1: Pre-admission phase ﬂowchart.

phase – the advanced scheduling problem – is that of

selecting patients from the (usually long) waiting list

and to assign them to an OR session (i.e., an operating

room on a given day) in such a way to satisfy several

operative constraints (number of beds available dur-

ing the patient stay, total time available for the OR

session, and so on). Our objective is to maximize the

utilization of the operating rooms in each day in such

a way to guarantee that each patient is operated within

the time limit deﬁned by the URG. This problem is

well known in literature as Surgical Case Assignment

Problem (SCAP) (T

anfani and Testi, 2010).

The hospital phase is concerned with all the ac-

tivities involving the admitted patient stay except for

those related to the operating theatre as depicted in

Figure 2. The relevant information in this phase is the

Length Of Stay (LOS) of each patient, that is the num-

ber of days required before the discharge. The opti-

mization problem arising in this phase – the allocation

scheduling problem – is that of ﬁnding a sequence of

patients to decide the order in which they are operated

on. The objective is to minimize the risk of cancel-

lation according to their assigned position in the se-

quence taking into account a patient-centred point of

view (considering waiting time, class of urgency, pos-

sible previous referrals) with respect to the available

operating time.

Figure 3 depicts the operating theatre phase

which is a component of the hospital phase – as high-

lighted in Figure 2. Due to its importance in a surgical

pathway, it requires to be treated separately. Patients

assigned to a given OR session will be operated on

following the sequence previously deﬁned unless de-

lays imposes to deﬁne a new sequence. Patient not op-

erated on will be rescheduled. The optimization prob-

lem arising in this phase is the real time management

of the operating room planning. When the Estimated

Operating Time (EOT) differs from the Real Oper-

ating Time (ROT), we could have a delays. If such

delays become signiﬁcant, that is exceeding the to-

tal operating time allowed, the real time management

should deal with the following possible decision to be

taken every time a patient ﬁnish its intervention:

• to use some overtime reducing the total amount

weekly available;

• to cancel 1 or more patients and to re-schedule

them, if possible;

• to change the sequence of the remaining patients

in order to minimize the number of cancellations.

The ﬁrst two choices are generally non-trivial and

alternatives requiring to consider several aspects.

For instance, the decision of postponing a patient

could violate the so called Maximum Time Before

Treatment (MTBT) determined by its URG. Further,

it determines an increased patient stay lowering the

patient satisfaction and, by consequence, the qual-

ity of the service. These decisions have to take into

account the inherent uncertainty. On the other side,

overtime is a scarce resource. So, it seems crucial to

establish some criteria driving the decisions of using

it to avoid cancellations.

Process

Alternative Process

Decision Data Predefined Process

Internal Storage

Document

Multidocument

Preparation Manual Input Manual Operation

Connector

Off-page Connector

Terminator

Card

Punched Tape Summing Junction

Collate Sort Extract Merge Stored Data

Delay

Sequential Access

Storage

Magnetic Disk

Direct Access

Storage

Display

Area

sì

Alternative Process

Degenza

pre-

operatoria

Fase

pre-ammissione

Coda per

l’ammissione

Sono

disponibili

dei posti

letto?

Attesa per la

liberazione di un

posto letto

Ammissione nel

reparto

Fase del

teatro operativo

È stato

operato?

Degenza post-

operatoria

Dimissione

Uscita

sì

Pre-admission

phase

Admission

queue

There is an

empty bed?

Wait for the

release of a

bed

Admission to

the ward

Pre-operation

hospital stay

Operating

theatre phase

yes

Was

operated?

Post-operation

hospital stay

yes

Discharge

Exit

Figure 2: Hospital phase ﬂowchart.

AHybridModelfortheAnalysisofaSurgicalPathway

891

Process

Alternative Process

Decision Data Predefined Process

Internal Storage

Document

Multidocument

Preparation Manual Input Manual Operation

Connector

Off-page Connector

Terminator

Card

Punched Tape Summing Junction Or

Collate Sort Extract Merge Stored Data

Delay

Sequential Access

Storage

Magnetic Disk

Direct Access

Storage

Display

Area

Alternative Process

Hospital phase

Wait for

operating

room release

Is the

session

open?

yes

Is the next

surgery?

Surgery

Hospital phase

yes

Figure 3: Operating theatre phase ﬂowchart.

3.1 Notation

Let us introduce the notation of the problem used

hereafter in the paper.

N: number of OR sessions,

: duration of j-th OR session,

: day of the week (from Monday to Friday) of the

j-th OR session,

: number of beds available the k-th day of the week

(k = 1, ..., 7 such that k = 1 is Monday, k = 5 is

Friday, k = 7 is Sunday),

I: set of patients in the pre-admission waiting list,

L: set of scheduled patients,

( j)

: set of patients scheduled into the j-th OR ses-

sion,

: waiting time of the i-th patient,

: MTBT of patient i,

: EOT of patient i,

: ROT of patient i,

: LOS of patient i,

Ω: weekly overtime available.

4 THE HYBRID MODEL

This section discusses the hybrid simulation opti-

mization model proposed in this paper. Simulation

allows to deal with the inherent stochasticity of the

problem while optimization allows to deal with the

three problems arising in the three phases depicted

in Section 3. In the following, we will brieﬂy de-

scribe the hybrid model through the description of its

main components, that is the Discrete Event Simula-

tion (DES) simulation framework and the three opti-

mization modules.

Note that the hybrid model is implemented using

AnyLogic 6.9 (Borshchev, 2013). The Enterprise Li-

brary is exploited for the implementation of the DES

simulation framework whilst the optimization mod-

ules are implemented from scratch using Java exploit-

ing the fact that AnyLogic is build on the top of an

Eclipse platform.

4.1 The Simulation Framework

The simulation framework is based on a DES method-

ology. It is a straightforward implementation of the

surgical pathway depicted in Figures 1, 2 and 3. The

main parameters of the simulation model and their

distribution are depicted in the Appendix.

4.2 The Advanced Scheduling Problem

As reported in Section 2, several approaches have

been proposed. Here, we proposed a simple meta-

heuristic based on a greedy construction of an initial

solution and then a local search to improve that solu-

tion. Note that we should take into account the fact

that the resources available can be reduced since pa-

tients admitted the previous week are already in the

hospital phase, usually waiting for the discharge but

also for their surgical intervention.

4.2.1 Constructive Greedy Algorithm

The algorithm associates to each patient i ∈I the fol-

lowing values

+ min

1≤j≤N

, (1)

˜w

+ min

1≤j≤N

+ 7

= w

. (2)

measures the ratio of the time elapsed before the

surgical intervention and the MTBT associated to the

URG of the patient i ∈I whilst ˜w

is the same meaning

but referred to the next week.

Patients are ordered by decreasing value of w

such a way to promote the scheduling of those pa-

tients which are close to their MTBT. Then, each pa-

tient is considered for the scheduling. A patient will

be inserted in the current schedule if there exist an

OR session available with enough free operating time

in such a way to satisfy the operative constraints re-

garding the bed occupation and the operating time S

Among different possible OR sessions, the algo-

rithm tries to schedule the patient ﬁrst in a day k such

that k + `

≤5. If it is not possible, the algorithm tries

the insertion in a day k such that k + `

> 5. The ratio-

nale here is to avoid the use of the weekend stay beds

which are usually a limited resource. This rule can be

overridden when ˜w

≥ 1 assigning the patient to the

ﬁrst day k = 1, if possible, or to the second day k = 2,

and so on. In this case, we would like to reduce the

probability to do not satisfy the URG requirements in

case of cancellation.

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Finally, if a patient cannot be scheduled, the al-

gorithm will consider the next patient. The algorithm

terminates when all patient are considered for the in-

sertion in the schedule.

4.2.2 Improvement Local Search Algorithm

The Local Search tries to improve the solution com-

puted by the greedy exchanging pairs of patients al-

ready scheduled in such a way to cluster them in a

reduced number of OR sessions and, by consequence,

to allow the insertion of new patients previously not

scheduled. Le us consider the OR session j

∗

having

the maximum operating time yet available, that is the

one having the minimal utilization. The Local Search

algorithm follows these criteria to select a new solu-

tion when exploring the neighbourhood:

• the new solution will be that providing the maxi-

mal increase of the time yet available of j

∗

;

• otherwise, if the two schedules are equivalent in

∗

, the algorithm will considers the second one

less utilized OR session, and so on;

• otherwise, if the two schedules are equivalent in

all OR sessions, the algorithm selects those solu-

tions having OR sessions less utilized at the end

of the week.

4.3 The Allocation Scheduling Problem

In our settings, the allocation scheduling problem

consists in establishing the order in which patients

i ∈L

( j)

will be operated on. For this reason, it is also

called sequencing problem. The main objective is to

minimize the inefﬁciency due to possible cancella-

tions. Note that the real time management – depicted

in Section 4.4 – will be devoted to the minimization

of the number of cancellations, that is to minimize the

number of patients re-scheduled due to a delay.

Considering a given schedule, there is a set of pa-

tients for which is better to avoid the cancellation of

their surgical intervention, that is those patients whose

˜w

is greater than or equal to 1 and those patients

whose their intervention was already postponed. To

deal with these special cases, let us introduce the fol-

lowing values:

(

˜w

if ˜w

> 1

0 otherwise

, (3)

and D

> 0 is the number of days elapsed after a can-

cellation, 0 otherwise. Finally, we deﬁne the value

= α

+ α

(4)

for each i ∈ L

( j)

where α are parameters. Setting

 α

(

1 case (A)

0 case (B)

the sequencing of patients i ∈ L

( j)

is simply obtained

by ordering them by the decreasing order of the asso-

ciated s

The use of α imposes a three levels of priorities.

First we schedule patients whose intervention was

previously postponed, and then the others. Among

those postponed, ﬁrst we schedule those close to their

MTBT and, in case of same value, those waiting for

more days after the previous cancellation. Finally.

when the ﬁrst two components of s

, that is α

and

, yield to the same value for two different pa-

tients, we break ties by ordering them following a

LPT or a SPT policy (with respect to EOT) in the case

(A) and in the case (B), respectively.

4.4 Operating Room Real Time

Management

During the execution of the operating room plan,

it could be happen that the EOT differs from ROT.

When r

> e

, for a patient i ∈ L

( j)

, the whole plan

will be delayed. When the overall delay could de-

termine the exceeding of the jth OR session duration

, the operating room real time management should

deal with the problem of postponing an intervention

or to use a part of the available overtime. Such a de-

cision poses the problem of evaluating the impact of

consuming overtime or to have a cancellation.

Let us consider the jth OR session on day k = d

having duration S

and a list L

( j)

of scheduled and

sequenced patients. We suppose that m <



( j)



pa-

tients are already operated on. Let ρ

the effective

time elapsed to operate on the m patients, that is

∑

i=i

,...,i

. (5)

Setting ρ

= t, let us introduce the following param-

eter:

1 +

−

Ω

(6)

where N

is the number of OR sessions from the day

after k and Ω

is the remaining overtime after the in-

tervention of patient i

The value β

would like to measure the overtime

still available with respect to the number of OR ses-

sions to be still performed. Actually, β

is closed to

1 when the overtime has been used proportionally; it

is between 0 and 1 or it is greater than 1 when it is

underused or overused, respectively. Because of N

equal to 0, we remark that the last day of the week it

is always less than or equal to 1 hence promoting the

use of overtime.

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The real time management is activated whenever

∑

i=i

,...,i

after operating the last patient. After checking the se-

quencing of the remaining patients, the decision of

postponing or assigning overtime could be taken.

The sequencing is checked in such a way to en-

sure that (i) all the remaining patients having ˜w

> 1

are scheduled prior to the other patients and (ii) those

having ˜w

> 1 are ordered by decreasing value of ˜w

Let i

m+1

be the next patient in the schedule. Then,

m+1

> S

−ρ

the patient i

m+1

could incur in a cancellation. There-

fore, the real time management algorithm checks if

,ρ



m+1

+ ρ



≤ 1 (7)

and if (7) is satisﬁed, the required overtime is assigned

to patient i

m+1

Finally, we implemented an algorithm that runs at

the end of the day and it is responsible to reschedule

the next day all the postponed surgical interventions.

The algorithm tries to insert an intervention with EOT

in the jth OR session planned in the next days in

such a way to minimize the difference between S

and

the sum of the EOTs of the already assigned patients

plus e

. If an insertion is not possible, the patient will

be scheduled next week.

5 MODEL VALIDATION

The validation of a simulation model requires a quite

complex analysis. In our case, we are interested in the

logical correctness of the simulation model represent-

ing the surgical pathway while we are not interested

in the replication of a real system.

To this purpose, we adapted our simulation model

to represent the inspiring case, that is that reported

in (Ozcan et al., 2011). In that paper, the proposed

model dealt with two patient ﬂows having similar

EOT but different LOS. Note that the LOS of the sec-

ond ﬂow is roughly the double of the ﬁrst one while

the number of patients in the ﬁrst ﬂow is roughly the

double of the second ﬂow. Since our model deals with

only one patient ﬂow, we adapted our patient ﬂow

generator in such a way to have, on average, the same

number of patients having the LOS of the ﬁrst ﬂow

which is the most numerous. Furthermore, we turn

off all the optimization during the three phases. The

other parameters are reported in the Appendix.

Let us introduce the following performance in-

dices: u

bed

is the bed utilization whilst u

is the OR

session utilization. Table 1 reports the results of the

comparisons with respect to the measures

of the sys-

tem modelled in (Ozcan et al., 2011). Our results are

the average value over those obtained by running the

hybrid model 30 times with different starting condi-

tions. Each of these computational experiments runs

for a time horizon of 2 years but collecting data only

in the second year.

Table 1: Model validation: comparison with real measures.

bed

Real measures 51.10% 77.33%

Simulation model 49.10% 80.82%

Difference 2.00% 3.49%

The differences in the two performance indices

can be accounted to the different composition of the

patient ﬂow as depicted above. For instances, the gap

of 3.49% for u

expressed in minutes corresponds

to the execution of an intervention having average du-

ration. On the basis of these considerations, the com-

parison is satisfactory with respect to our objective,

that is the validation of the logical correctness of our

simulation model.

6 IMPACT OF THE

OPTIMIZATION

In this Section we would like to evaluate the impact of

the optimization modules integrated in the simulation

model. To this purpose, we need to deﬁne a baseline

conﬁguration with respect to the three phases as fol-

lows:

Phase 1: advanced scheduling performed by a ﬁrst-

ﬁt algorithm, that is (i) consider patients by de-

creasing order of w

, (ii) place a patient in the ﬁrst

OR session available from Monday to Friday;

Phase 2: patients are sequenced in a random order;

Phase 3: overtime is assigned a priori uniformly to

all OR sessions;

Phase 3: interventions rescheduled only at the end of

the day using a ﬁrst-ﬁt algorithm.

The simulation parameters are depicted in the Ap-

pendix.

Taken from the presentation made by the authors at the

ORAHS 2012 conference.

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Table 2: Performance indices for each test conﬁgurations.

Option(s) Performance indices

id 1 2 3 seq. A B C f

MTBT

avg

bed

avg

max

(0) baseline conﬁguration 234 (2348) 32.63% 338 55 63.61% 89.88% 1.17 4.05

(1)

√

235 (2347) 31.85% 346 56 60.20% 89.78% 1.11 3.29

(2)

√ √

226 (2340) 25.97% 360 58 60.57% 89.32% 1.16 3.27

(3)

√

252 (2346) 35.95% 324 52 60.35% 89.60% 1.12 3.61

(4)

√ √

246 (2349) 35.34% 330 53 60.26% 89.78% 1.06 3.41

(5)

√ √ √

230 (2338) 27.17% 355 58 60.79% 89.55% 1.17 3.10

(6) LPT 236 (2367) 47.85% 292 48 60.52% 90.76% 1.03 3.79

(7) SPT 240 (2261) 12.14% 452 72 58.57% 86.35% 1.51 4.91

(8)

√

197 (2384) 74.62% 213 35 59.26% 91.34% 0.80 2.64

(9)

√

236 (2315) 30.70% 339 55 72.56% 88.79% 1.18 3.79

(10)

√ √

222 (2372) 72.97% 223 37 64.16% 90.65% 0.83 2.68

(11)

√

LPT

√

239 (2389) 79.91% 192 32 60.27% 91.75% 0.73 2.62

(12)

√ √

LPT

√

248 (2390) 85.53% 207 34 60.58% 91.78% 0.71 1.87

6.1 Test Conﬁgurations and

Performance Indices

Here we deﬁne the various conﬁgurations we used to

evaluate the impact of the optimization tools. Each

conﬁguration is deﬁned with respect to the baseline

conﬁguration previously deﬁned.

• Phase 1:

option 1: computing w

w.r.t Monday instead of

the previous Friday;

option 2: adopting the greedy depicted in Sec-

tion 4.2.1 (instead of the ﬁrst-ﬁt algorithm);

option 3: adopting the Local Search depicted in

Section 4.2.2;

• Phase 2:

LPT/SPT: use LPT or SPT rules in sequencing

(case (A) or (B) in Section 4.3);

• Phase 3:

option A: adopting real time management after

each interventions;

option B: adopting the algorithm depicted in

Section 4.4 for rescheduling patients at the end

of the day (instead of the ﬁrst-ﬁt algorithm).

Finally, we introduce all the performance indices used

in the following analysis, that is the patient-centred

indices:

C: number of cancellations,

MTBT

: percentage of patients operated within the

MTBT,

avg

: average length (number of patients) of the wait-

ing list,

avg

: average waiting time spent in the waiting list,

avg

: average value of patient’s w

at the time of their

surgery,

max

: maximum value of patient’s w

at the time of

their surgery,

and the facility-centred indices:

bed

: bed utilization,

: OR session utilization.

It is quite evident that different indices can affect each

other. For instance, the increase of the number of

cancellations can affect the bed utilization and, in its

turn, can reduce the percentage of patients operated

within the MTBT. Our aim is to identify a test con-

ﬁguration which increases the patient-centred indices

without deteriorating the facility-centred ones.

6.2 Quantitative Analysis

As reported for the model validation, the reported re-

sults are the average value among those obtained by

running the hybrid model 30 times on a given conﬁg-

uration and, each time, starting from a different initial

condition.

First, the impact of each optimization modules is

evaluated through the quantitative analysis. Based on

these results, two further conﬁgurations are studied.

The results of such analysis are summarized in Ta-

ble 2 which reports the value of the performance in-

dices for each test conﬁgurations denoted by the value

in the ﬁrst column “id”. Note that the column report-

ing the number of cancellations also reports in brack-

ets the total number of patients operated on. All the

analysis are compared with the baseline conﬁgura-

tion.

Regarding the impact of the advanced scheduling

optimization module, we can observe a lower wait-

ing time in the waiting list and an improvement of the

performance indices related to MTBT in test conﬁg-

urations (3) and (4). On the other side, the minimal

number of cancellations is obtained with conﬁgura-

tion (2) but, at the same time, the percentage of pa-

tients operated on before their MTBT decreases con-

sistently. Note that the use of Local Search allows

to insert more patients determining the improvement

measured in (3) and (4).

AHybridModelfortheAnalysisofaSurgicalPathway

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0 20 40

80 100 120 140

160

180 200 220 240

260

280 300 320 340

360

200

400

baseline

LPT

SPT

Figure 4: Trend of I

avg

(data referred to the 2nd year, days on x-axis, patients on y-axis).

0 20 40

80 100 120 140

160

180 200 220 240

260

280 300 320 340

360

200

400

baseline

conﬁguration (8)

Figure 5: Trend of I

avg

(data referred to the 2nd year, days on x-axis, patients on y-axis).

0 20 40

80 100 120 140

160

180 200 220 240

260

280 300 320 340

360

0.5

1.5

baseline

conﬁguration (8)

Figure 6: Trend of w

avg

(data referred to the 2nd year, days on x-axis, patients on y-axis).

0 20 40

80 100 120 140

160

180 200 220 240

260

280 300 320 340

360

0.5

1.5

baseline

conﬁguration (11)

conﬁguration (12)

Figure 7: Trend of w

avg

(data referred to the 2nd year, days on x-axis, patients on y-axis).

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Regarding the impact of the allocation schedule

optimization module, we can observe a signiﬁcant

better performances when LPT policy is adopted. Fig-

ure 4 shows the trend of I

avg

under the baseline, (6)

and (7) conﬁgurations.

Regarding the impact of the operating room real

time management module, we observe a remarkable

improvement of all the performance indices (see con-

ﬁgurations (8) and in particular f

MTBT

). On the other

side, we observe the negligible impact of the algo-

rithm for the rescheduling postponed patients at the

end of the day (see conﬁgurations (9) and (10)). Fig-

ure 5 and 6 show respectively the trend of I

avg

and

avg

under the baseline and (8) conﬁgurations. Note

that it is positive when w

avg

< 1 which means that, on

average, all the patients are operated on before their

MTBT.

Finally, conﬁgurations (11) and (12) report about

the combination of the different options. We note a

further improvement of the performance indices ex-

cept for that related to the number of cancellations if

compared with conﬁguration (8). This is due to the

fact that Local Search allows to insert more patients

in the advanced scheduling thus reducing the wait-

ing time in the waiting list but increasing the proba-

bility of incurring in a cancellation. Figure 7 shows

the trend of w

avg

under the baseline, (11) and (12)

conﬁgurations. While baseline conﬁguration shows

a value of w

avg

always greater than 1, we remark that

both conﬁgurations (11) and (12) tend to be less than

1. Further, conﬁguration (12) seems more stable and

powerful in reducing this indices.

7 CONCLUSIONS

In this paper we proposed a hybrid simulation and op-

timization model for the analysis of a surgical path-

way from a patient-centred point of view: simulation

is used to generate a real situation with respect to

the inherent stochasticity of the problem while opti-

mization is used to take the best decisions in different

points of the surgical pathway.

The quantitative analysis discussed in Section 6

shown the positive impact of the optimization in the

management of the surgical pathway. In particular,

the most effective optimization module is the operat-

ing room real time management determining a general

improvement of all the performance indices with re-

spect to a baseline conﬁguration of the surgical path-

way.

Comparing the baseline conﬁguration with con-

ﬁgurations (11) and (12) we can observe a great im-

provement of the performance indices related to the

waiting list in terms of its length and the waiting

time. This allow to double (at least) the percent-

age of the patients operated on before their MTBT

time limit. These improvements can determine a gen-

eral improvement of the quality of service from a

patient-centred point of view without deteriorating the

facility-centred performance indices.

The quantitative analysis conﬁrms the trade-off

between the number of cancellations and the number

of operated patients (or, equivalently, the OR session

utilization) as discussed in (Beaulieu et al., 2012).

From this point of view, the proposed hybrid model

could help the hospital management in the evaluation

of this trade-off.

ACKNOWLEDGEMENTS

Authors acknowledge support from the Italian Min-

istry of Education, University and Research (MIUR),

under the grant n. RBFR08IKSB, “Firb – Futuro in

Ricerca 2008”. Finally, the authors wish to thank the

anonymous referees for their comments which helped

in improving the paper.

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APPENDIX

In this appendix, we report the parameters of the

simulation framework and their setting for the model

validation (Section 5) and for the quantitative analysis

(Section 6). In brackets, the unit of measure.

Flow and patient characteristics:

: patient interarrival rate [patients/minutes],

: initial length of the pre-admission waiting list

[patients],

: patient probability to require a surgical treat-

ment during the ambulatory visit (see Fig. 1),

: patient probability to do not require a surgi-

cal treatment but requiring further exams dur-

ing the ambulatory visit (see Fig. 1),

,. .. , p

: patient probability to belong into the

urgency class A,. . .,G.

Duration of activities:

min,avg,mod

A,...,F,I

: minimum, average and modal time

for the execution of A,. . .,F,I [minutes] (see

Figures 1–3),

min, max, mod; A,.. .,G

: minimum, maximum and

modal LOS for patients belonging to the ur-

gency class A,...,G [days],

A,...,G

: average EOT for the surgical interven-

tion of a patient belonging to the urgency class

A,.. .,G [minutes],

max

: maximum duration of a surgery [minutes],

A,...,G

: EOT standard deviation [minutes],

σ: ROT standard deviation for each patient [min-

utes],

τ: tolerance time within which the surgical team

operates a patient at the end of OR session with-

out resorting to the overtime [minutes].

Table 3 shows the distributions used to generate

the required time for the execution of the activities

A,.. .,J. Table 4 reports the values assigned to the pa-

rameters for the model validation and for the quanti-

tative analysis.

The parameters k and ϑ for the Gamma distribu-

tions (see Table 3) were obtained from the empirical

data reported (Ozcan et al., 2011), requiring that the

expected and the modal values of these distributions

coincide with the empirical values reported in that pa-

per. Further, we compute the value of the survival

function on the maximum time for the execution of

activities (always reported in the paper), obtaining a

value less than 10%.

The choice to use a lognormal distribution derives

from the literature (see, e.g., (Strum et al., 2000; May

et al., 2000; Spangler et al., 2004)).

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Table 3: Distribution of the activity durations.

Activities Durations Parameters

A,.. ., F, I T

A,.. .,F,I

min

+ T, k = T

A,...,F,I

avg

−T

A,...,F,I

mod

T ∼Gamma(k,ϑ) ϑ =

A,...,F,I

avg

−T

A,...,F,I

min

A,...,F,I

avg

−T

A,...,F,I

mod

H (LOS) bTriangular(l

min; A,.. .,G

max; A,.. .,G

mod; A,.. .,G

) +

J (EOT) min



max



cu,0



max



, µ = log ε

A,. . . ,G

−

log



A,...,G

+ 1



T ∼Lognormal(µ, s

) s =

log



A,...,G

+ 1



J (ROT) min

{

max

{

0,T

}

max

}

T ∼Gaussian(EOT, σ

)

Table 4: Parameters used in the simulation framework.

Parameters unit of measure Validation Quantitative analysis

patients/minutes 5.8 ·10

−3

2.0 ·10

−2

patients 140 420

, p

0.2, 0.1 0.2, 0.1

,. .. , p

0.0245,0.1401, 0.4136,0.1785 0.0245, 0.1401,0.4136, 0.1785

0.1140,0.0749, 0.0544 0.1140,0.0749, 0.0544

A,...,F,I

min

minutes 5,25,25, 25, 40, 25, 35 5,25, 25,25, 40,25, 35

A,...,F,I

avg

minutes 7.5,31.5, 31,28, 62.5,32, 41 7.5,31.5, 31,28, 62.5,32, 41

A,...,F,I

mod

minutes 6,30,26, 25, 50, 30, 40 6,30, 26,25, 50,30, 40

min; A,.. .,G

days 2,1, 1,1, 1,1, 1 2,1, 1,1, 1,1, 1

max; A,.. .,G

days 29,16, 7,9, 5,5, 5 29,16, 7,9, 5,5, 5

mod; A,.. .,G

days 3,2, 2,2, 2,2, 2 3,2, 2,2, 2,2, 2

max

minutes 360 420

A,...,G

minutes 145,171, 149,153, 171,164, 166 145,171, 149,153, 171,164, 166

A,...,G

minutes 85,85, 66,60, 61,51, 60 85,85, 66,60, 61,51, 60

σ minutes 0 30

τ minutes 30 10

Ω minutes 0 300

u minutes 30 30

N 7 21

,. .. ,S

minutes 360,360, 360,360, 360,360, 360 300,360, 420,420, 420,300, 360

420,420, 300,360, 420,300, 360

420,420, 420,300, 360,420, 420

,. .. ,d

2,2, 3,3, 4,4, 5 1,1, 1,1, 1,2, 2,2, 2,3, 3,3

4,4, 4,4, 4,5, 5,5, 5

,. .. ,B

beds 18,18, 18,18, 18,18, 18 50,50, 50,50, 50,35, 35

URG A,...,URG G

days 8,15, 30,60, 90,120, 180 8,15, 30,60, 90,120, 180

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