=Paper= {{Paper |id=Vol-3065/paper3_196 |storemode=property |title=Scheduling Pre-Operative Assessment Clinic via Answer Set Programming |pdfUrl=https://ceur-ws.org/Vol-3065/paper3_196.pdf |volume=Vol-3065 |authors=Simone Caruso, Giuseppe Galatà,Marco Maratea,Marco Mochi,Ivan Porro |dblpUrl=https://dblp.org/rec/conf/aiia/CarusoGMMP21 }} ==Scheduling Pre-Operative Assessment Clinic via Answer Set Programming == https://ceur-ws.org/Vol-3065/paper3_196.pdf

Scheduling Pre-Operative Assessment Clinic via
Answer Set Programming
Simone Caruso1 , Giuseppe Galatà2 , Marco Maratea1 , Marco Mochi1,2 and Ivan Porro2
1
DIBRIS, University of Genova, Genova, Italy
2
SurgiQ srl, Italy

Abstract
The problem of scheduling Pre-Operative Assessment Clinic (PAC) consists of assigning patients to a day
for the exams needed before a surgical procedure, taking into account patients with different priority
levels, due dates, and operators availability. Realizing a satisfying schedule is of upmost importance for
a clinic, since delay in PAC can cause delay in the subsequent phases, causing a decrease in patients’
satisfaction. In this paper, we divide the problem in two sub-problems, and present the results of a
first preliminary analysis of the two problems based on Answer Set Programming (ASP). In the first
sub-problem patients are assigned to a day taking into account a default list of exams; then, the second
sub-problem, having the actual list of exams needed by each patient, use the result of the first sub-problem
to assign a starting time to each exam.

Keywords
Healthcare, Pre-Operative Assessment Clinic Scheduling, Answer Set Programming

1. Introduction
The Pre-Operative Assessment Clinic (PAC) scheduling problem is the task of assigning patients
to a day, in which the patient will be examined and prepared to a surgical operation, taking in
account patients with different priority levels, due dates, and operators availability. The PAC
consists of several exams needed by patients to ensure they are well prepared for their operation.
This allows patients to stay at home until the morning of the surgery, instead of being admitted
to the hospital one or two days before the scheduled operation, moreover, reducing waiting time
between the exams increase patient satisfaction [1] and avoid the cancellation of the surgery
[2].
The problem is divided into two sub-problems [3]: in the first sub-problem, patients are
assigned to a day taking into account a default list of exams, and the solution has to schedule
patients before their due date and prioritizing the assignments to patients with higher priority.
In the second sub-problem, the scheduler assigns a starting time to each exam needed by patients,
considering the available operators and the duration of the exams. A proper solution to the
PAC scheduling problem is vital to improve the degree of patients’ satisfaction and to reduce
surgical complications. Complex combinatorial problems, possibly involving optimizations,

IPS-2021: 9th Italian Workshop on Planning and Scheduling
" 4493864@studenti.unige.it (S. Caruso); giuseppe.galatà@surgiq.com (G. Galatà); marco.maratea@unige.it
(M. Maratea); marco.mochi@unige.it (M. Mochi); ivan.porro@surgiq.com (I. Porro)
© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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such as the PAC problem, are usually the target applications of AI languages such as Answer
Set Programming (ASP). Indeed ASP, thanks to its readability and availability of efficient solvers
([4, 5, 6, 7]) has been successfully employed for solving hard combinatorial problems in several
research areas, and it has been also employed to solve many scheduling problems [8, 9, 10, 11,
12, 13, 14], also in industrial contexts (see, e.g., [15, 16, 17, 18] for detailed descriptions of ASP
applications).
In this paper, we apply ASP to solve the PAC scheduling problem. In the solution of first
sub-problem, the scheduler minimizes the number of unassigned patients. Then, we propose a
solution to the second sub-problem, using as input of the problem the results of the first one and
minimizing the time each patient stays at the hospital. We have finally run a first experimental
analysis on PAC benchmarks with realistic sizes and parameters inspired from data seen in
literature, scenarios for the first sub-problem, varying the number of patients to schedule and
the available operators. Overall, results using the state-of-the-art ASP solver clingo [19] of this
preliminary analysis show that ASP is a suitable solving methodology also for PAC scheduling
problem.

2. Problem Description
Since the schedule of the PAC day must be scheduled as soon as possible, this problem is
typically divided in two phases, in the first phase, dealing with this sub-problem before the
actual PAC day, the clinics don’t know which exam each patient will require, thus, in the first
sub-problem, as typically done in hospitals, we consider that each patient requires a default
list of exams according to his specialty, the lists of exams are equal for patients requiring the
same specialty but differ among specialties, and the scheduler assigns the day of PAC without
considering the starting time of the exams. In the first sub-problem the solution assigns patients
overestimating the duration and the number of exams needed. In particular, all the optional
exams, such as exams required by smokers or patients with diabetes, are assigned to all the
patients in the first phase. The overestimation is important to have a second sub-problem that is
not unsatisfiable. Then, when the operation day is closer, the hospital knows exactly the exams
needed by each patient and can assign the starting time of each exam.
Going in more details, the first sub-problem consists of scheduling appointments in a range
of days for patients requiring surgical operation. Each patient is linked to a due date, a target
day, and to a priority level: the due date is the maximum day in which (s)he can be assigned, the
target day is the optimal day in which schedule the appointment, while the solution prioritizes
patients with higher priority level. There are several exam areas, corresponding to the locations
in which patients will be examined. Each exam area needs operators to be activated and has
a limit time of usage. Each operator can activate three different exam areas, but they can be
assigned to just one exam area for each day. The solution must assign the operators to the
exam areas, to activate them, and assign the day of PAC to patients, ensuring that the total
time of usage of each exam location is lower than its limit. Since in this first sub-problem the
list of exams needed by patients is not the final list, i.e., just the first and the last exam are the
same for every patient, and in the second sub-problem some exams could be added, the solution
schedules patients leaving some unused time to each exam area. An optimal solution minimizes
1 {x(RID,PR,TOTDUR,DAY) : day(DAY), DAY < DUEDATE} 1 :- reg(RID,PR,TARGET,TOTDUR,DUEDATE).
2 :- x(RID,_,_,DAY), exam(RID,FORNID,_), not examLoc(FORNID,_,_,DAY,_).
3 res(RID,FORNID,DAY,DUR) :- x(RID,PR,_,DAY), exam(RID,FORNID,DUR).
4 :- N1 = #count{FORNID: res(RID,FORNID,_,_)},N2 = #count{FORNID: exam(RID,FORNID,_,_)},
x(RID,_,_,_), N1 != N2.
5 :- #sum{DUR, RID: res(RID,FORNID,DAY,DUR)} > NHOURS/2, examLoc(FORNID,_,NHOURS,DAY,N).
6 :- #sum{TOTDUR, RID: x(RID,_,TOTDUR,DAY)} = M, #count{RID: x(RID,_,_,DAY)} = N, day(DAY), M >
((60*N)-(5*N*(N+1))), N>1.
7 {operator(ID, FORNID, DAY) : operators(ID, FORNID, DAY)} == NOP :- examLoc(FORNID, NOP, _,
DAY,_), res(REGID, FORNID, DAY, _).
8 :- operator(ID,FORNID1,DAY), operator(ID,FORNID2,DAY), FORNID1 < FORNID2.
9 unassignedP1(N) :- M = #count {RID: x(RID,1,_,_)}, N = totRegsP1 - M.
10 unassignedP2(N) :- M = #count {RID: x(RID,2,_,_)}, N = totRegsP2 - M.
11 unassignedP3(N) :- M = #count {RID: x(RID,3,_,_)}, N = totRegsP3 - M.
12 unassignedP4(N) :- M = #count {RID: x(RID,4,_,_)}, N = totRegsP4 - M.
13 :∼ unassignedP1(N). [N@8]
14 :∼ unassignedP2(N). [N@7]
15 :∼ unassignedP3(N). [N@6]
16 :∼ unassignedP4(N). [N@5]
17 :∼ x(RID,1,_,DAY), reg(RID,_,TARGET,_,_). [|DAY-TARGET|@4,RID]
18 :∼ x(RID,2,_,DAY), reg(RID,_,TARGET,_,_). [|DAY-TARGET|@3,RID]
19 :∼ x(RID,3,_,DAY), reg(RID,_,TARGET,_,_). [|DAY-TARGET|@2,RID]
20 :∼ x(RID,4,_,DAY), reg(RID,_,TARGET,_,_). [|DAY-TARGET|@1,RID]

Figure 1: ASP encoding of the first sub-problem

the number of unassigned patients, giving priority to patients with higher priority levels, and
ties are broken by minimizing the difference between the day assigned and the target day of
each patient, giving again priority to patients with higher priority.
In the second sub-problem, patients are linked to their real exams, so the solution has to
assign the starting time of each exam, having the first sub-problem already assigned the day.
The input consists of registrations, exams needed by patients and the exam areas activated.
Exams are ordered, so the solution must assign the starting time of each exam respecting the
order and their duration, considering that each exam area can be used by one patient at a time.
Moreover, the solution minimizes the difference between the starting time of the first exam and
the last exam of each patient.

3. ASP Encoding
In this section we first present the ASP encoding for the first sub-problem, and then the main
elements of the second sub-problem, in two separate sub-sections.

3.1. ASP Encoding for the first PAC sub-problem
We assume the reader is familiar with syntax and semantics of ASP. Starting from the specifica-
tions in the previous section, here we present the ASP encoding for the first sub-problem, based
on the input language of clingo [20]. For details about syntax and semantics of ASP programs
we refer the reader to [21].
Data Model. The input data is specified by means of the following atoms:
• Instances of reg(RID,PR,TARGET,TOTDUR,DUEDATE) represent the registrations, char-
acterized by an id (RID), the priority level (PR), the ideal day in which assign the patient
(TARGET), the sum of the durations of the exams needed by the patient, and the due date
(DUEDATE).
• Instances of exam(RID,FORNID,DUR) represents the exams needed by the patients
identified by an id (RID), the exam area (FORNID), and the duration (DUR).
• Instances of examLoc(FORNID,NOP,NHOURS,DAY,N) represent the exam areas, char-
acterized by an id (FORNID), which requires (NOP) operators to be activated, which is
active for a certain value of time (NHOURS) in the day (DAY), and can be concurrently
assigned up to n (N) patients.
• Instances of operators(ID,FORNID,DAY) represent the operators, characterized by
an id (ID), who can be assigned to the exam ares (FORNID) in the day day(DAY).
• Instances of day(DAY) represent the available days.
The output is an assignment represented by atom of the form:
x(RID,PR,TOTDUR,DAY)

where the intuitive meaning is that the exams of registration with id RID and priority level PR
is assigned to the day DAY and has a total duration of his/her exams equal to TOTDUR.

Encoding. The related encoding is shown in Figure 1, and is described in the following. To
simplify the description, we denote as 𝑟𝑖 the rule appearing at line 𝑖 of Figure 1.
Rule 𝑟1 assigns registrations to a day. The assignment is made assigning a day that is before
the due date. Rule 𝑟2 checks that every registration is assigned to a day with all the exams
area needed activated. Rule 𝑟3 derives an auxiliary atom that is used later in other rules. In
particular, the new atom is used to get the duration of the visit for each patient and for each
exam area. Then, rule 𝑟4 checks that the number of needed exams and the number of res
atoms created are the same. Rule 𝑟5 is used to ensure that each exam area is used for a total
amount of time that is lower than its limit minus 30 time slots, in order to overestimate the
required time for the visits. Rule 𝑟6 is used to be sure to not assign too many patients in the
first sub-problem to a particular day. So, it overestimates the time needed by each patient. Rule
𝑟7 assigns operators to the required exam areas. Rule 𝑟8 checks that each operator is assigned
to just one exam area in every day. Rules from 𝑟9 to 𝑟12 are needed to derive auxiliary atoms
that are used later on in optimization. In particular, they are used to count how many patients
with different priorities are not be assigned to a day. Weak constraints from 𝑟13 and 𝑟16 are
used to minimize the number of unassigned registrations according to their priority. Finally,
weak constraints from 𝑟17 and 𝑟20 minimizes the difference between the assigned day and the
target day of each patient, prioritizing the difference of patients with higher priority.

3.2. ASP Encoding for the second PAC sub-problem
Data Model. The input data is the same of the first sub-problem for the atoms exam and
time, while other atoms are changed:
1 {x(RID,FORNID,ST,ST+DURATA,DAY) : examLoc(FORNID,DAY,FORNST,FORNET,_), time(ST), ST >= FORNST,
ST <= FORNET-DURATA} = 1 :- reg(RID,DAY), esame(RID,FORNID,DURATA).
2 :- x(RID,FORNID1,ST1,_,_), x(RID,FORNID2,ST2,_,_), phase(FORNID1,ORD1), phase(FORNID2,ORD2),
ORD2 < ORD1, ST1 < ST2.
3 :- #count{FORNID: x(RID,FORNID,ST,ET,DAY), T >= ST, T < ET} > 1, reg(RID,DAY), time(T).
4 :- #count{FORNID: x(RID,FORNID,ST,ET,DAY), T >= ST, T < ET} > N, examLoc(FORNID,DAY,_,_,N),
time(T).
5 :∼ reg(RID,_), x(RID,0,ST,_,_), x(RID,23,_,ET,_). [ET-ST@1, RID]

Figure 2: ASP encoding of the second sub-problem

• Instances of reg(RID,DAY) represent the registrations, characterized by an id (RID)
assigned to the day (DAY).
• Instances of examLoc(FORNID,DAY,FORNST,FORNET,N) represent the exam areas,
characterized by an id (FORNID), which in the day (DAY) has a starting time and closing
time respectively equals to (FORNST) and (FORNET), which is active for a certain value of
time (NHOURS) in the day (DAY), and can provide the exam to a value (N) of patients.
• Instances of phase(FORNID, ORD) represent the order (ORD) of the exams provided by
the exam area characterized by an id (FORNID).
The output is represented by atom of the form:

x(RID,FORNID,ST,ET,DAY)

where the intuitive meaning is that the exam of the registration with id RID is in the exam area
FORNID, starts at the starting time ST and ends at the ending time ET, on the day DAY.

Encoding. The encoding consists of the rules reported in Figure 2. Rule 𝑟1 assigns a starting
and an ending time to each exam needed by every patient, checking that the time in which is
assigned is inside the opening time of the required exam area. Rule 𝑟2 ensures that the order
between the exams is respected. Rules 𝑟3 checks that each patient is assigned to at most one
exam for every time slot. Then, rule 𝑟4 checks that each exam area provides the exam to at
most N patients for every time slot. Finally, rule 𝑟5 minimizes the difference between the ending
time of the last exam and the starting time of the first exam of each patient.

4. Experimental Results
In this section we report the results of an empirical analysis of the PAC scheduling problem via
ASP. For the first sub-problem, data have been randomly generated using parameters inspired
by literature and real world data, then the results of the first sub-problem have been used as
input for the second sub-problem. The experiments were run on a AMD Ryzen 5 2600 CPU
@ 3.40GHz with 16 GB of physical RAM. The ASP system used was clingo [20] 5.1.0, using
parameters --restart-on-model for faster optimization and --parallel-mode 8 for parallel execution.
This setting is the result of a preliminary analysis done on the same instances where we tested
also other parameters, e.g., –opt-strategy=usc for optimization. The time limit was set to
300 seconds for the first and the second sub-problems.
Table 1
Percentage of assigned patients according to their priority level
Total #Patients P1 P2 P3 P4
40 94% 85% 77% 66%
60 90% 69% 20% 13%
80 67% 22% 14% 10%

4.1. PAC benchmarks
Data are based on the sizes and parameters of a typical middle sized hospital, with 24 different
exam areas. The solution schedules patients in a range of 14 days, for each day there are 60
time slots, corresponding to 5 minute per time slot. To test scalability we generated 3 different
benchmarks of different dimensions. Each benchmark was tested 5 times with different randomly
generated input.
In particular, each patient is linked to a surgical specialty, and needs a number of exams
between 5 and 13, according to the specialty, while the duration of each exam varies between 3
and 6 time slots. The priorities of the registrations have been generated from an even distribution
of four possible values (with weights of 0.25 for registrations having priority 1, 2, 3, and 4,
respectively). For all the benchmarks, there are 24 exam areas and the operators, that are 35
in all the benchmarks, can be assigned to 3 different exam areas. So, increasing the number of
patients while maintaining fixed the number of operators, we tested different scenarios with
low, medium and high requests.
For the second sub-problem, we used the results of the first sub-problem as input. Thus, the
number of patients and the exam locations activated depend on the assignment of the solution
of the first sub-problem. Patients require all the same first and last exam, while the other exams
required by each patient are linked to an order that is randomly assigned and that must be
respected by the scheduler. In the second sub-problem clinics know the actual list of exams
needed by patients, to simulate this scenario, we randomly added and discarded the optional
exams assigned to patients in the first sub-problem. For example, optional exams are needed by
patients that are over 65 years old or smokers. There are 5 instances for each benchmark, each
corresponding to the assignments of 14 days.

4.2. Results for the first sub-problem
The first optimization criteria in the PAC scheduling sub-problem is to assign as many patients
as possible, starting from patients with higher priority. Our solution is able to assign a day to
201 patients out of 217 patients with highest priority; moreover, the scheduler is able to assign
all or all but one patients with the highest priority in 12 out of 15 instances tested. The solution
has more difficulties on the instances with 80 patients, since in this scenario the number of
operators is not enough to deal with the high number of patients.
In Table 1 are summarized the results obtained in this first sub-problem, in particular, the
table shows the average number of patients assigned with 40, 60, and 80 patients according to
their priority level.
The second optimization criteria is to have an assigned day that is as much near as possible
6

Patients 2

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Day

Figure 3: Number of patients assigned to each day by the scheduler with 40 patients as input

to the target day. This optimization criteria is able to assign patients with higher priority near
their target day while, for patients with lower priorities, the scheduler is not able to reach great
results. This is due to two reasons, the first one is that there are more optimization criteria
with higher priorities, then, the scheduler already tries to assign as many patients as possible,
without taking into account the target days of the patients, and the second one is that some
patients have a target day in a day without their exam locations available, so, even in an optimal
solution the assigned day to some patients wouldn’t be in the target day.
Figure 3 reports the result obtained by the scheduler with one of the instances with 40 patients
as input. What can be seen by the graph is that some patients are assigned in days 11 and 12,
while in days 9 and 10 there are no patients assigned. This can be explained by the fact that the
scheduler tries to assign as many patients as possible and do not try to assign as soon as possible
the patients. Moreover, in some days patients can not be assigned due to the unavailability
of the exam locations required. In particular, in the assignments in Figure 3, patients that are
assigned in day 11 could not be assigned in another day, because that day is the only day with
the exam locations they required.

4.3. Results for the second sub-problem
In the second sub-problem the solution assigns the starting time of each exam of the patients.
The input is taken from the results obtained in the first sub-problem. The solution minimizes
the difference between the ending time of the last exam and the starting time of the first exam.
While minimizing this value the solution tries to minimize the time spent in the hospital by all
patients. In this sub-problem the solution is able to reach an optimal solution in 13 out of 15
instances tested. While the average total duration of the exams for each patient is 37 time slots,
the solution find a schedule that allows patients to have an average time in hospital that is just
Figure 4: Representations of the assignments of the starting time of the different exam location of each
patient in a single day

38.5 time slots.
The results are obtained on average in 152, 186, 220 seconds respectively in the instances
with 40, 60 and, 80 patients. In Figure 4 there are all the starting times assigned to each patient
in a particular day. The patients that must be scheduled are the same that are assigned by the
first sub-problem in this day; the scheduler of the second sub-problem minimizes the waiting
times of each patient.
As can be seen in Figure 4 the scheduler is able to assign all the patients optimally; indeed,
all patients have no waiting time between each exam and thus the time spent in the hospital is
reduced to the minimum, while respecting the constraints of the sub-problem. As an example,
each exam location is assigned to at most one patient for each time slot.

5. Related Work
The work in [3] used two simulation models to analyse the difficulties of planning in the context
of PAC and to determine the resource needed to reduce waiting times and long access times.
The models were tested in a large university hospital and the results were validated measuring
the level of patient satisfaction. [22] used a Lean quality improvement process changing the
process and the standard routine. For example, patients were not asked to move from a room to
another for the visits, but patients were placed in a room, and remained there for the duration
of their assessment. This and other changes to the processes led to the decrease of the average
lead time for patients and to the number of patients required to return the next day to complete
the visits. [1], [2], [23], and, [24] studied the importance of implementing the PAC and the
positive results obtained by having less waiting time between the exams and for the visit to the
hospital. In particular, while different clinics follow different guidelines for PAC, implementing
it has proved to be an important tool to avoid the cancellation of the surgeries for medical
reasons and to significantly reduce the risk associated with the surgery.
ASP has been already used as a tool for solving scheduling problems [15, 25], also in the
healthcare domain [18]. In this paper, we focus on a problem that was not addressed in previous
works by using ASP.

6. Conclusions and Current Work
In this paper, we have presented a preliminary analysis of the PAC scheduling problem modeled
and solved with ASP. We started from the problem as presented in other works and utilized
parameters that can be found in other works. Results on synthetic data shows that the solution
is able, considering the different constraints, to assign a high number of patients with higher
priority. We are currently working on extending our preliminary experiments. Moreover, we
would like also to implement and test other solving procedures, e.g., [26, 27, 28, 29], considering
the relation between ASP and SAT procedures [30, 31], whose goal would be to improve the
current results. Finally, we plan to add this solution into a platform of solutions for scheduling
problems in healthcare, similarly to, e.g., [32] in the context of SMT solving.

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