The Integrated Healthcare Timetabling Competition 2024: Context and Analysis

Cristian Belfiore1, Marco Chiarandini2, Rosita Guido1, Domenico Conforti1

1 de-Health Lab – Laboratory of Decision Engineering for Health Care Services, Department of Mechanical, Energy and Management Engineering, University of Calabria, Rende (CS), Italy
2 Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark

This page provides supplementary material for our investigation of the IHTC 2024 benchmark instances. The interactive figures are organized into thematic sections, allowing the reader to explore instance-space patterns, the effect of weighting choices, and the role of critical resource saturation.

Instance Space Projection (Section 4.1, Equation 1 and Fig. 3)

The projection onto the two-dimensional instance space is obtained using PILOT (Projecting Instances with Linearly Observable Trends), which maps high-dimensional feature data into a two-dimensional space by identifying projection directions that highlight trends in both instance features and algorithm performance. The six features identified by MATILDA as most relevant to explaining algorithm performance, together with the corresponding projection matrix, are reported below.

\[ \begin{bmatrix} Z_1 \\ Z_2 \end{bmatrix} = \begin{bmatrix} 0.477 & 0.1659 \\ 0.265 & 0.391 \\ 0.6594 & -0.7546 \\ 0.1044 & 0.7173 \\ -0.0357 & -0.4106 \\ -0.1202 & -0.0958 \end{bmatrix}^{\!\top} \begin{bmatrix} \text{Incompatible rooms per patient (median)} \\ \text{Total operating theatre availability duration} \\ \text{Total surgeon availability duration} \\ \text{Surgery utilization ratio} \\ \text{Weight: continuity of care} \\ \text{Weight: nurse excessive workload} \end{bmatrix} \]

The figure below shows the projection of the instances in the \((Z_1, Z_2)\) space. The separating boundary between easy and hard instances is approximated through logistic regression, allowing a clearer interpretation of instance difficulty across the projected space.

Feature Distributions in the Instance Space (Section 4.1, Fig. 4)

In the figures below, it is possible to observe the trends over the instances of three selected features: (a) median number of incompatible rooms per patient, (b) total operating theatre availability and (c) total surgeon availability. These features are among the most relevant ones identified by MATILDA. For the remaining features highlighted during the dimensionality reduction process, no simple or immediately interpretable relationships emerge within the instance space.

In each plot, all instances are projected onto the two-dimensional space defined by the coordinates (Z1, Z2). The values of the considered features are encoded through a color scale, ranging from lower values (in purple) to higher values (in yellow).

(a) Median number of incompatible rooms per patient across the instance space.
(b) Operating theatre availability across the instance space (minutes).
(c) Surgeon availability across the instance space (minutes).

Best Performing Algorithm per Instance (Section 4.1, Fig. 5)

In the figure below, each instance is represented in the (Z1, Z2) space and colored according to the algorithm achieving the best performance. This representation highlights how different methods dominate in different regions of the instance space, providing insight into the relationship between instance characteristics and algorithm effectiveness.

Instance Space by Dataset (Section 4.1, Fig. 6)

In the figures below, the instance space is shown separately for the different datasets considered in the analysis. In each plot, one dataset is highlighted while the others are displayed in grey, making it easier to observe how the corresponding instances are distributed in the two-dimensional space defined by the coordinates (Z1, Z2).

First phase dataset
Second phase dataset
Small dataset
Long horizon dataset

Critical Resource Saturation Analysis (Section 4.3, related to Fig. 7)

This section illustrates the values of the soft constraints for the best found solution. The three upper panels report the weights characterizing an instance, the number of violations for each soft constraint, and the corresponding costs, computed as the product of weights and violations. The bottom panel shows the distribution of nurse overload values, with bars indicating the number of nurses at each overload level.

Ideal Bounds and Weighting Effects (Section 4.4, related to Fig. 8)

This section reports the comparison between ideal objective values, lexicographic bounds, and the best solutions obtained during the competition under the weighted-sum objective. Select an instance from the menu below to explore the corresponding results.