IEEE Robotics & Automation Magazine - June 2015 - 59
Numerical Experiments
This section presents numerical experiments to evaluate the
performance of the patient pooling heuristic and to study the
properties of optimal solutions for capacity allocation, patient
pooling, and case mix. All numerical experiments are performed on a workstation with an Intel Core 16 processor with
32 central processing units (CPUs) at 2.6 GHz. All mathematical programming models are solved with the commercial
solver CPLEX12.2.
Data Setting
We first present a base case from which the numerical
experiments will be defined. This is derived from [3] considering 18 basic types of patients with different cancer
locations (lung, prostate, breast, head, and neck), treatment
intents (palliative and radical), and emergency status. Two
types of LINACs are considered: 1) advanced LINACs
(ALINACs), which can treat all patient types and 2) regular
LINACs (RLINACs), which cannot treat patient types 13
and 14. All LINACs work from 8:00 a.m. to 4:00 p.m. with a
daily capacity of 480 min. The default priority rule is FIFO.
Table 2 provides other patient information where the
arrival rate ^m ih and WTT ^~ ih are measured in days, the
durations ^t ijh are given in minutes on the ALINAC/RLINAC,
and "slot nb" is the minimal number n i of slot servers needed
for each patient type without pooling, BP. The total arrival rate
is 8.25 patients per day. The default maximal BP a i is 5%. The
default weighting factor ε of total breaching rate in patient
pooling is 1, as our main objective is to minimize the number
of slots. According to Assumption 5, patient types 6 and 14
with session durations of 24 min cannot be merged with other
patient types with session durations of 12 min. Patient type 13
can be merged with others if only ALINACs are considered,
and it should be kept alone if both ALINACs and RLINACs
are considered.
Is Patient Pooling Always Beneficial?
Let us consider the simulation results of pooling patient
types 1, 3, 4, and 8 in Table 2. Pooling patients 1 and 3
(with similar WTTs and service times) requires three slots
and saves one slot. Pooling patients 1 and 4 (with different
WTTs and service times) requires 28 slots, which is two
slots more than without pooling. Pooling patients 1 and 8
(with similar service times but different WTTs) requires
nine slots, the same as without pooling, but leads to a
higher BP (0.036) for type 1 patients and a lower BP
(0.000) for type 8 patients. To conclude, whether patient
pooling is beneficial is not a trivial issue, and it depends on
the service times (i.e., the number of sessions), WTTs, and
priority rules.
Efficiency of the Patient Pooling Heuristic
This section compares the pairwise merging heuristic and
the exact method by exhaustive enumeration and standard
mixed-integer programming solvers. Since the size of the
problem instances of the base case is too large for the exact
Table 2. The patient information.
Session Arrival
Slot
Type Number Rate
WTT Duration Number BP
1
5
0.19
5
12/15
2
0.025
2
1
0.11
5
12/15
1
0
3
4
0.11
5
12/15
2
0
4
16
1.43
40
12/15
24
0.002
5
20
0.59
40
12/15
13
0.005
6
16
0.45
40
24/30
8
0.012
7
1
1.42
40
12/15
2
0
8
5
1.36
40
12/15
7
0.034
9
10
0.57
40
12/15
7
0
10
4
0.38
40
12/15
2
0
11
15
0.18
40
12/15
4
0
12
1
0.18
40
12/15
1
0
13
20
0.29
10
12/-
8
0.035
14
35
0.21
10
24/-
11
0.026
15
33
0.3
40
12/15
11
0.043
16
37
0.29
40
12/15
12
0.038
17
37
0.15
40
12/15
7
0.018
18
33
0.04
40
12/15
3
0.002
Table 3. The problem instances of ten patient types.
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
1
2
2
2
2
1
1
1
1
1
2
3
4
3
3
2
6
2
3
2
3
5
6
4
4
3
7
3
6
3
7
6
7
5
5
4
9
4
7
5
8
8
8
7
8
5
10
5
9
6
9
9
9
10
9
6
13
10
11
9
13
10
10
13
11
7
15
12
13
10
14
14
11
14
12
8
16
13
16
13
16
16
12
15
15
12
17
14
17
14
18
18
17
16
18
15
18
16
18
16
method, the comparison is carried out on ten problem
instances with only ALINACs and ten patient types sampled
from the initial 18 types. The list of patient types sampled for
each instance is provided in Table 3.
Table 4 provides for each instance and for each method
(Opt=Optimal, Heu=Heuristic) the total number of slots, the
total breaching rate, and the patient pooling. For each group of
patients, the number of slots needed is shown in parentheses.
The total number of slots is exactly the same for both
methods except for problem instance S8 with a deviation of
one slot. The total breaching rate deviates by less than 0.056
JUNE 2015
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IEEE ROBOTICS & AUTOMATION MAGAZINE
*
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