An Improved Saturation Degree-Based Constructive Heuristic for Master Surgery Scheduling Problem: Case Study

The Master Surgery Scheduling Problem (MSSP) can be described as a timetabling problem involving assigning surgery groups to operating theatre (OT) time slots. Previous MSSP optimization models considered throughput, waiting measures, resource utilization, costs, and schedule assignment objectives, but have overlooked consecutive days assignment preferences and surgical equipment-sharing limitations. Furthermore, previous works utilize greedy constructive heuristics to produce solutions, which increases quality but decreases feasibility. Our prior study demonstrated that the saturation degree heuristic enhances feasibility by considering assignment difficulty during event selection. However, its impact on solution quality remained unexplored. Therefore, this study proposes an improved saturation degree-based constructive heuristic that integrates objective function value for event selection to increase both quality and feasibility. The algorithm sorts surgery groups by unit scores, prioritizing higher assignment difficulty and objective value. The highest-scoring group is assigned to its feasible slot with the highest slot score. If no feasible slots exist, the repair mechanism vacates the slot with the highest swap score, which prioritizes lower assignment difficulty and objective value. A new mathematical model is also formulated, incorporating novel objectives regarding consecutive days assignment preference and surgical equipment-sharing limitations. Using real-world data from Hospital Canselor Tuanku Muhriz, the proposed algorithm is evaluated considering repair mechanism usage for feasibility and objective function value for quality. The algorithm is benchmarked against greedy, random, regret-based, and saturation degree-based constructive heuristics. Our algorithm achieved a 14.63% improvement in feasibility compared to the original variant. Its objective function value is over two times better than the closest competitor and 2.6 times superior to the original variant. Comparison with the hospital’s actual plan demonstrates competitive objective function value and a more balanced waiting time distribution among surgical groups. Our study showcases that a saturation degree-based constructive heuristic considering objective function value has increased solution quality while maintaining feasibility.


I. INTRODUCTION
The surgery scheduling problem involves selecting and sequencing surgeries and allocating resources for those surgeries [1].The problem can be divided into three levels: strategic, tactical, and operational [2], [3].The capacity allocation problem (strategic) involves the decisions on the number of operating theatres (OT) to be made available and their allocation to surgery groups such as medical specialties or surgeons [4].Meanwhile, the Master Surgery Scheduling Problem (MSSP) is a tactical problem that produces a master surgery schedule, also called a master plan, which defines the assignment of surgery groups to OT time slots [2].Finally, the operational level determines individual surgeries' time slots and sequence.Good OT planning leads to an efficient hospital in terms of operation management and financial management [5], [6].Hence, it is important to have an effective master plan.
One of the main issues in MSSP is the involvement of numerous stakeholders, each with different and perhaps conflicting objectives [7].The surgery scheduling problem can be represented mathematically as a combinatorial optimization model for distributing hospital resources based on specific objectives [3].Among the common objectives are maximizing the number of surgeries [8], [9], [10], minimizing OT idleness [11], [12], [13] and the number of patients in downstream resources [11], [14], [15].However, no previous works have considered the objectives for the case study hospital, Hospital Canselor Tuanku Muhriz (HCTM), which includes assignment preference and resource-sharing limitations.
Assignment preference refers to the specific preferences of surgery groups for certain OT slots or having repeating patterns in the schedule for high predictability.The model proposed by Penn et al. [16] minimizes surgeons in non-preferred slots and maximizes surgeons in the same slot each week or consecutive slots.Other works, such as Oliveira et al. [17] and Britt et al. [11], address similar objectives.However, these models do not consider preferences for scheduling a surgery group on two consecutive days.In addition, while existing models have included objectives related to various resources such as OTs and upstream and downstream resources [18], limitations in resource sharing that constrain the daily assignment of surgery groups based on equipment availability have not been incorporated as objectives in any MSSP model.
Solution robustness has been linked to a decline in solution quality [19], prompting an exploration into whether increased solution feasibility has a similar adverse impact.Most prior constructive heuristics that address the MSSP employ the greedy approach [5], [20], [21], [22], which can maximize solution quality at the cost of other variables [23].Although alternative approaches such as stochastic methods [24] and regret-based strategies [9] have mitigated this trade-off, they still overlook solution feasibility during generation.To overcome this issue, Razali et al. [25] introduced a saturation degree-based constructive heuristic that enhances feasibility but neglects solution quality.This trend emphasizes the tendency of prior heuristics to prioritize either feasibility or quality, calling for methods to improve both aspects.
The main motivation of this paper is to examine the relationship between solution feasibility and solution quality within the framework of a constructive heuristic to generate high-quality, feasible initial solutions for the MSSP.This study focuses on using constructive heuristics to generate initial solutions for the MSSP.Constructive heuristics can produce high-quality initial solutions, which may reduce the computational effort required to find the optimal solution [26].The objectives of the study can be summarized as follows: • To formulate a new mathematical model tailored for the case study hospital by incorporating variables not previously considered.
• To develop a novel constructive heuristic based on saturation degree, considering both solution feasibility and quality, and investigate the interaction between these factors.This study formulated a new variation of the MSSP mathematical model with a novel objective function that includes assignment preference and resource-sharing limitations.Besides, an improved version of the saturation degree-based constructive heuristic from Razali et al. [25] is proposed to consider both solution feasibility and quality.A constructive heuristic for the MSSP involves an iterative process of selecting a surgery group and an OT slot for the group.The algorithm proposed by Razali et al. [25] selects the surgery group with the fewest feasible OT slots and assigns it to the OT slot with the fewest possible assignments to enhance solution feasibility.In contrast, our proposed algorithm considers a composite value, incorporating the number of feasible slots or surgery groups and potential objective function value for selection, aiming for higher quality solutions while maintaining feasibility.
We hypothesize that if consideration of objective function value in event sorting can affect the solution quality, then a constructive heuristic incorporating saturation degree heuristic based on the objective function value can increase both solution feasibility and quality.The hypothesis is tested by comparing the feasibility and quality of master plans generated using the proposed approach to greedy, random, regret-based, and the original saturation degree-based constructive heuristics.Solution feasibility and quality are measured by repair mechanism usage and objective function value, respectively.Furthermore, a simulation is carried out to assess the operational performance of each master plan.
The remaining sections of this work are structured as follows: Section II provides knowledge about the related works in the problem domain.Next, the mathematical model for the combined capacity allocation and MSSP based on the case study hospital, HCTM, is presented in Section III.Section IV describes our proposed improved saturation degree-based constructive heuristic.Sections V, VI, and VII provide an in-depth discussion of the evaluation results and statistical analysis, focusing on comparing constructive heuristics, improvement algorithms, and performance as initial solutions.Finally, the conclusion and future works are described in Section VIII.
The MSSP is a combinatorial optimization problem that can be solved using exact or approximate methods.Exact methods guarantee optimal solutions within a finite time [40], but they have scaling limitations, making them unsuitable for more complex problems [41].Approximate methods, including heuristic and approximation algorithms, can find good solutions to more complex problems within a reasonable timeframe [42].Heuristic algorithms are categorized into low-level heuristics (commonly referred to as heuristics), metaheuristics, and hyper-heuristics [43].Low-level heuristics modify the solution directly, whereas metaheuristics utilize one or more low-level heuristics to guide the search of the solution space [41].A higher-level technique known as hyper-heuristics integrates multiple low-level heuristics into a framework that searches the heuristic space [44].These heuristics can be divided into constructive and local search (or improvement) heuristics [45].Constructive heuristics build a complete solution incrementally, whereas local search heuris-tics start with a complete solution and make small changes to find improvement.
Within the MSSP domain, constructive heuristics have been employed to obtain initial solutions for the MSSP before being improved by more sophisticated methods.Spratt and Kozan [9] introduced a regret-based constructive heuristic to generate an initial solution for the Simulated Annealing (SA) and Tabu Search hyper-heuristic algorithms.The regret-based approach is applied to select an OT for a chosen surgeon-specialty set in their algorithm.For each feasible OT, the regret value is calculated as the difference between the highest and the second-highest number of patients that could be scheduled.The OT with the largest regret is then chosen.Their approach can produce a master plan with the most scheduled patients.However, solution feasibility cannot be guaranteed, particularly during the OT selection for the next surgeon-specialty set, given that the greedy-based set selection is employed.Mashkani et al. [5] developed two dynamic programming-based heuristic algorithms that prioritize patients and specialties with the highest total profit.Their approach is characterized by its greedy nature, and solution feasibility is not considered during solution construction.Dellaert and Jeunet [24] utilize a constructive heuristic to generate an initial solution for their VNS algorithm.In their constructive heuristic, a day is chosen randomly with a higher probability of selecting days with high under-utilization of OT resources.Then, a patient category with a lower OT resource target deviation on the selected day has a higher chance of being chosen.Their approach introduces stochasticity, departing from the reliance on a purely greedy method.However, like the previous methods, it does not guarantee solution feasibility.Razali et al. [25] implemented a saturation degree-based constructive heuristic that prioritizes surgical units and OTs based on constraint violation to enhance solution feasibility.However, the construction of the solution did not consider solution quality, leading to a suboptimal outcome in terms of solution quality.Similar trends in the popularity of adopting a greedy approach and overlooking solution feasibility have been observed in other works that schedule individual patients through constructive heuristics [20], [21], [22].This trend highlights that prior constructive heuristics typically focus on either solution quality or feasibility.
Various metaheuristic algorithms were employed to solve the MSSP.Schneider et al. [33] utilized SA and conducted a performance comparison with an exact method, Mixed Integer Linear Programming (MILP).Their findings indicated that MILP surpassed SA, and incorporating SA after MILP led to further improvements.Lu et al. [36] addressed a multi-objective problem using an improved Non-dominated Sorting Genetic Algorithm.Their algorithm introduced a novel population initialization process, where constraints are imposed to ensure chromosome feasibility.When solving their three-objective problem, the proposed algorithm outperformed a goal programming approach in two objectives.Marchesi & Pacheco [46] applied the Genetic Algorithm (GA), whereas Britt et al. [11] employed a hybrid VNS-GA to solve the MSSP.Almaneea and Hosny [21] proposed a hybrid Bees Algorithm and SA that demonstrated superiority over the algorithm introduced by Aringhieri et al. [20].Dellaert and Jeunet [24] tested the applicability of the VNS algorithm in solving the MSSP, comparing random VNS (R-VNS) and first descent VNS (FD-VNS) to an exact solver, IBM ILOG CPLEX.Results indicated that R-VNS performed comparably to the exact solver across instances with varying difficulty, whereas FD-VNS performed worse.R-VNS outperformed CPLEX, especially when using random initial solutions for hard-to-solve problems.Spratt and Kozan [10] hybridized SA and VNS to obtain better solutions for the combined MSSP and SCAP, surpassing a decomposition approach in a case study for an Australian public hospital.Zhu et al. [35] proposed a hybrid GWO-VNS to address all three decision levels of the surgical scheduling problem.The proposed algorithm outperformed non-hybrid GWO, VNS, and another swarm intelligence approach, Particle Swarm Optimization (PSO).On the other hand, only one study has employed hyper-heuristics for solving the MSSP, as demonstrated by Spratt and Kozan [9], who employed a hybridized SA and Tabu Search hyper-heuristic algorithm.
Additionally, metaheuristics play a crucial role in solving timetabling and scheduling problems.Li et al. [73] hybridized the Artificial Bee Colony (ABC) algorithm with Q-learning to address the permutation flow-shop problem (PFSP).The Q-learning component rewards effective neighbourhood structures and selects the appropriate action for the current state.Compared to state-of-the-art ABC, a Whale Optimization Algorithm, and ABC with random neighbourhood selection, the hybrid approach demonstrated superior performance across all benchmark instances, with a notable advantage in large-scale ones.Zheng and Wang [74] proposed a hybrid Bat Algorithm and 5NS to solve a variant of the PFSP.The algorithm ensures population diversity by classifying it into two types: search-type population and captive population.Each population exhibits different points of the search space and utilizes a different update method.Comparative evaluations against other metaheuristics, including memetic algorithms, GA, and PSO, demonstrated the superior performance of the proposed method.Other metaheuristic algorithms utilized in this domain include GA [75], Harmony Search [76], Anarchic Society Optimization [77], and Intelligent Water Drops algorithm [78].
In summary, our study is motivated by two research gaps.Firstly, no existing optimization model considers all objectives of the case study hospital, particularly in terms of consecutive days assignment preference and equipment-sharing limitations.Hence, this study will introduce a new mathematical model that combines existing and new objectives.Secondly, we focus on constructive low-level heuristics among the solution approaches to the MSSP optimization model.Despite the acknowledged effectiveness of metaheuristics and hyper-heuristics in tackling the problem, we focus on constructive heuristics because they generate high-quality solutions more rapidly than the other approaches [79].We observed a lack of simultaneous consideration of solution quality and feasibility during solution construction.Our study proposes an improved saturation degree-based constructive heuristic that concurrently addresses both factors.

III. PROBLEM DESCRIPTION
The MSSP at the case study hospital, HCTM, is presented in an integrated capacity allocation and MSSP optimization model.The objective function and constraints of the problem are expressed.Master plans are generated using a constructive heuristic inspired by the saturation degree heuristic, later introduced in Section IV.Fig. 1 provides an overview of the study's methodology.

A. PROBLEM AND DEFINITIONS
This study covers the strategic and tactical levels of surgical scheduling, and the optimization is divided into two parts.The first part, called capacity allocation, distributes OT time to medical subspecialties (j ∈ J ) and their surgical units (g ∈ G).Afterwards, the surgical units are assigned to suitable OTs and days, forming a master plan.The relationship between medical subspecialties and surgical units can be described as one-to-many.Correspondingly, the surgery personnel (surgeons, anaesthetists and nurses) responsible for subspecialty j is responsible for all its units (g ∈ G j ).
OT time is divided into full-day slots identified by the indices of the week (w ∈ W ), weekly working days (p ∈ P), and OT (o ∈ O).A planning horizon has a slot total number of slots calculated by multiplying the number of weeks, weekly working days, and OTs.Each slot is hereafter denoted as (w, p, o) ∈ SL, where SL is the set of slots in the planning horizon.Each slot is assigned to only one surgical unit, and only a limited amount of OT time is available for scheduling surgeries from the chosen unit (s ∈ S g ).All slots have equal length µ.
The number of slots assigned to each unit in the master plan is limited by the number allocated in the capacity allocation (slot g ).OT time allocation is determined based on the surgery demand and waiting time.Surgery demand is represented by the cumulative expected duration of surgeries on the waiting list, where the expected duration dur s were given by surgeons.Additionally, the surgery waiting time WT s can be calculated as the difference between the first planning day and the surgery booking date.The assignment of surgical units to OT time slots is subject to OT suitability and surgeon availability, referred to as feasible slots.Contrarily, slots that are not suitable are called blocked slots.The OTs can only be assigned to surgical units that are suitable for them (g ∈ G O ) according to the OT and surgical unit types.The OTs are classified into three types: fixed, ultra-clean and general.Fixed OTs are reserved for specific units, whereas ultra-clean OTs can be used by any unit without dirty surgeries.Surgical units with ultra-clean surgeries must be assigned to ultraclean OTs.General OTs can be used by any unit.
Moreover, some units prefer to be assigned to specific OTs (o ∈ O preferred g ).This preference could be due to the equipment required for their surgeries being stored near or at the OTs.These slots are hereafter called preferred slots.Meanwhile, a surgical unit can only be assigned on days when its surgery team is available (p ∈ P feasible j ).Surgical units from the same subspecialty can be assigned on day p up to the maximum parallel slots (slot parallel j ) due to shortages in surgery personnel.
Besides, the master plan assignments consider the availability of movable equipment (e ∈ E).Each equipment has a limited quantity available for usage on each planning day (qty ewp ), as it is not practical to equip all OTs due to the high cost [17].Despite this, the number of surgical units using equipment e assigned on day p is not limited by the available quantity.The equipment can be shared between several OTs since not all surgeries scheduled in a slot use the same equipment.The movement of the equipment should be limited to avoid time loss and surgery blockings.Therefore, the model aims to assign as few surgical units that use the same equipment on the same day.
Furthermore, there is high variability in the availability of the equipment, such as equipment breakdown.Therefore, it is preferred that an extra quantity of each piece of equipment (qty extra e ) is made available on all days to ensure uninterrupted services.Less expensive equipment may have a lot of extra quantities, whereas costly equipment may not be possible to have any extras.
Schedule stability is a desirable feature of a master plan and can reduce operational planning complexity [28].It could be achieved by minimizing the differences in the assignments.Surgeons from a surgical unit prefer to be assigned to the same OT throughout the planning horizon.Besides, the surgery team will also benefit by reducing the complexity of their schedule.Surgical units should also be assigned on consecutive days so that cancelled surgeries can be moved to the following day if there is available time.Meanwhile, in a multiple-week master plan, assigning the same unit to the same day and OT each week is desirable.
Multiple subspecialties may have clashing resource requirements and should not be assigned on the same day.The clashing units for unit g are defined as C g ⊆ G.
Furthermore, the surgical units can be divided into two types based on their resource consumption: heavy and light.Heavy units (g ∈ G heavy , G heavy ⊂ G) are characterized as having many resource requirements, including requiring anaesthesia.Light units (g ∈ G light , G light ⊂ G) do not require anaesthesia, and hospital stays for their patients are discretional.
On each day, it is preferred that only one OT is assigned to any unit of a subspecialty.However, subspecialties with many surgeries may require more OT time to operate them timely.In this case, having one heavy and one light unit is preferable to two heavy unit slots for the same subspecialty on the same day.Coordinating operations between heavy unit slots will be more difficult.
The case study hospital handles both elective and emergency surgeries.Ultra-clean emergency surgeries had to use elective ultra-clean OTs since the emergency OTs are not suitable for this surgery type.Therefore, one ultra-clean OT should be assigned to a unit with light resource requirements or left unassigned daily to minimize the interruption to the planned elective surgeries.Interruptions to light surgeries are more tolerable as the patients do not require anaesthesia.Table 1 presents the parameters, whereas Table 2 shows the decision variables available in the mathematical model.

B. PROPOSED OPTIMIZATION MODEL FOR CAPACITY ALLOCATION AND MASTER SURGERY SCHEDULING PROBLEM
Based on the problem at the case study hospital, a mathematical model including a novel objective function and general constraints is formulated.The objective function minimizes violations of soft constraints (SC).It comprises eight assignment preference objectives (SC1, 4 − 10) and two resource-sharing limitation objectives (SC2 − 3).The constraints are extracted from various mathematical models, with minor modifications to suit the problem.

1) OBJECTIVE FUNCTION
The objective function (1) minimizes violations of ten soft constraints.The soft constraints can be divided into two types: penalties and rewards.Penalties are given to undesirable violations that do not affect solution feasibility, whereas rewards are awarded when preferred constraints are satisfied.Since the objective function minimizes, penalties increase the objective value, whereas rewards reduce it.SC1, described in (2), rewards a unit each time it is assigned to its preferred slots (preferred OT or preferred days).Meanwhile, SC2, expressed in (3), penalizes the objective value for each unit assigned to a day exceeding the available quantity of movable equipment [14], [80].SC3 presented in (4) rewards the objective value for not using all quantities of equipment on each day [14], [80].
Equations ( 5)-( 7) promote schedule stability by rewarding each time a unit is assigned to the same OT in two consecutive days (SC4), or to any OT in two consecutive days (SC5), or to the same slot in two consecutive weeks (SC6) [16].However, SC4 − 6 do not incentivize assignments to consecutive days or weeks of more than two.Furthermore, SC4 and SC5 do not consider the blocked days of a surgical unit.SC7, described in ( 8), gives out a penalty if a unit is assigned to a day when its clashing unit has been assigned.SC8 and SC9 are concerned with the parallel slots of a unit.Equation ( 9) sums the penalties handed out each time a unit is assigned to more than one slot a day.Meanwhile, (10) penalizes each time heavy units from the same subspecialty are assigned to multiple slots on a day.
Finally, SC10, expressed in (11), gives a reward for each day an ultra-clean OT is assigned to a unit with a light resource requirement or not assigned [80].
2) NORMALIZED VALUES The objectives have different granularity and levels of importance.To address the former, we normalized the values from each objective.The normalized value is obtained using where N is the normalized value, y is the non-normalized value, O maxvalue is the extreme best value if the objective is satisfied, O minvalue is the extreme worst value for the objective (ignoring other objectives) [56], [81], [82].
All soft constraints have minimum values of 0. The maximum values for each objective are given in Appendix A. Afterwards, the normalized objective is multiplied by a weight assigned based on the importance of the objective.

3) CONSTRAINTS
The constraints can be divided into capacity allocation ( 12)-( 15) and MSSP ( 16)- (20).Constraint (12) states that each surgical unit must be allocated at least one slot in the planning horizon.Constraint (13) maintains that the total slots allocated to all units of a subspecialty must not exceed the number of slots allocated to the subspecialty.Constraint (14) guarantees that the total slots allocated to all subspecialties must not exceed the number of slots in the planning horizon [36].The number of slots for subspecialty j must not exceed its total feasible slots in the planning horizon, as stated in constraint (15).For the MSSP, constraint ( 16) requires each slot to be assigned to only one unit [9], [14], [17].Constraint (17) forces unit g to be assigned to a slot with an available surgery team on day p of the week w [9], [31].Constraint ( 18) ensures unit g is only assigned to suitable or predefined OTs [17].For each day, the number of slots assigned to units of subspecialty j must not exceed its maximum parallel slots, as expressed in constraint (19) [31].Finally, constraint (20) ensures that the number of slots assigned to unit g equals the number allocated to the unit [21], [31].
w∈W p∈P o∈O

C. NUMERICAL EXAMPLE
The solution for the MSSP is represented in a two-dimensional array (n × m), where n denotes the number of planning days, and m is the number of OTs.This array is populated by surgical units, signifying that the unit assigned to a particular slot can utilize the designated OT on the specified day.Empty positions within the array indicate unassigned slots.We present a numerical example demonstrating the assignment of 15 surgical units from 11 subspecialties to a master plan, as in Fig. 2. The list of subspecialties and their surgical  units can be found in Table 3.Each surgical unit is allocated a suitable number of slots based on surgery demand and waiting time.During master plan construction, the list of all booked surgeries for each surgical unit is retrieved.Each surgery contains two crucial pieces of information for OT capacity allocation, including the booking date and the estimated surgery duration provided by the medical team.Each surgical unit's cumulative estimated surgery duration is computed, and OT slots are allocated proportionally to the total duration.Surgical units with higher cumulative durations receive more slots in the master plan.The surgery booking date is utilized to calculate the waiting time for each surgery, and surgical units with longer average waiting times receive additional slots in the master plan.The resulting slot allocations are summarized in Table 3.
Subsequently, each surgical unit is assigned to specific OTs and days within the master plan based on the designated slot quantities.The slot assignment process adheres to constraints related to slot suitability and preferability for each surgical unit.For example, consider the unit VAS, which can only be scheduled for surgeries on Mondays, Tuesdays, or Wednesdays due to the availability of surgical personnel.However, there are no restrictions on which OT can be assigned.Consequently, the feasible slots for unit VAS encompass all OTs from Monday to Wednesday, whereas the blocked slots include all OTs on Thursday and Friday.In the overall slot assignment process, each surgical unit is accommodated in the master plan within their feasible slots.Fig. 2 provides a visual representation of a complete master plan, showcasing the assignment of each surgical unit to its allocated slots across specific OTs and days.
We provide examples illustrating the application of soft constraints within the given master plan.The assignment of unit VAS to OT1 on Monday in Week 1 (W1, Mon, OT1) satisfies SC1, as OT1 is the preferred choice for unit VAS.This adherence to the preference results in a reward being added to the objective value.SC4 is fulfilled when unit VAS is consecutively assigned on (W1, Mon, OT1) and (W1, Tue, OT1), earning the associated reward.In the case of SC5, unit PAE (LT) is assigned to different OTs on consecutive days (W1, Wed, OT4) and (W1, Thu, OT5), meeting the conditions and receiving a reward for the objective value.Similarly, for SC6, unit VAS is assigned to the same days and OTs for both (W1, Mon, OT1) and (W2, Mon, OT1), meeting the specified criteria.
For SC7, note that both units, CTC and HEP, utilize the same nursing team.Thus, they should not be assigned on the same day.In the example presented, however, both units are assigned on Monday in Week 1, resulting in a penalty to the objective value.The assignment of unit HEP also violates SC8 on Monday in Week 1, as it prefers to have only one slot per day but is given two slots.On the other hand, OPH assignments on Monday in Week 1 violate SC9 since two heavy units are assigned on the same day, which is avoided on other days.SC10 is fulfilled for assignments on all planning days except Monday in Week 1, when no slots are assigned to a unit with light resource requirements or are left empty.
For equipment sharing constraints, we present an example involving the equipment II machine used by VAS and ART units, where the hospital has only two units available.To avoid equipment sharing, SC2 requires that these units be assigned in just two slots on the same day.The example, however, reveals that the units have been assigned to three slots on Tuesday in Week 1 at OT1, OT4, and OT5, thereby violating the soft constraint.Conversely, SC3 requires that just one slot be assigned to any unit employing the equipment daily.This soft constraint is fulfilled on Monday in Week 1 since only OT1 is assigned to VAS and none to ART, satisfying the soft constraint and rewarding the objective value.

IV. PROPOSED IMPROVED SATURATION DEGREE-BASED CONSTRUCTIVE HEURISTIC
The MSSP constructive heuristic consists of three main tasks.The first task is selecting a surgical unit to assign.Then, a slot is selected among the feasible slots for the unit.Finally, if a suitable slot is not found, a unit already assigned in the incomplete master plan is selected to be swapped out.
This section first provides an overview of the main procedure involved in the proposed improved saturation degree-based constructive heuristic.This procedure comprises two key tasks: slot allocation and slot assignment, which are subsequently detailed in separate subsections.The proposed constructive heuristic, encompassing both tasks, is presented as Algorithm 1.Additionally, the algorithm governing the allocation of OT slots to surgery groups is extracted as Algorithm 2 and discussed in its corresponding subsection.The intentional separation of Algorithm 2 from Algorithm 1 aims to enhance clarity and facilitate a better understanding of the distinct tasks involved.

A. MAIN ALGORITHM
The algorithm requires the number of planning days and OTs to define the dimension of the master plan.Multiplying the number of weeks by the number of weekly working days yields the number of planning days.Besides, the list of medical subspecialties and their surgical units with information on constraints is also required.Information on the types of movable equipment and their quantities is also required.Finally, the algorithm requires a list of surgeries with their booking date, estimated duration and associated surgical unit for slot allocation.The pseudocode for the proposed improved saturation degree-based constructive heuristic is provided in Algorithm 1.The detailed pseudocode using programming syntax is given in Appendix B.
The algorithm is divided into two tasks: determining the number of slots allocated to each surgical unit (slot allocation) and assigning them to the master plan (slot assignment).The slot allocation is determined for each medical subspecialty j ∈ J and then their surgical units g ∈ G j , ∀j ∈ J (line 1).The pseudocode for the slot allocation process is presented in Algorithm 2 and discussed in Section IV-B.
Afterwards, the surgical units are assigned to slots in the master plan.During slot assignment, conflicts may arise when a unit cannot be assigned due to its feasible slots having been taken by other units.The saturation degree heuristic is employed in sorting the surgical units to minimize conflicts (line 2).In addition to the number of feasible slots, two other factors were considered in the basic saturation degree heuristic.They are the number of preferred slots and the ratio between the slots allocated to the maximum possible slot allocation of each surgical unit.The former is added to increase the reward for assigning preferred slots.The saturation degree surgical unit sorting is explained in Section IV-C.
The algorithm then assigns surgical units with predefined OTs to their respective slots (line 3).In the slot assignment, units assigned to their allocated slots are removed from the unassigned list.Besides, the saturation degree's heuristic value, such as the number of feasible slots, changes with each assignment in the partial solution [66].The list is re-sorted after each assignment.Re-sort the unassigned list.5 Iterate the unassigned list and assign the first unit (g).
Stop when all units are assigned.

6
If unit g has feasible slots in the partial master plan.If the repair mechanism has not reached its usage limit.Mark the solution as infeasible.13 End (return a complete master plan) Subsequently, the algorithm assigns surgical units to their preferred slots if available (line 4).Then, the remaining units are iterated and assigned to a feasible slot at each iteration (lines 5-12).If feasible slots are available for the unit, the slot with the fewest feasible unassigned units and the highest potential objective value is chosen (lines 6-7).The slot selection process based on saturation degree is described in Section IV-C.
If no feasible slot can be found for the unit, the algorithm will execute a repair mechanism in which a unit is unassigned from the partial solution (lines 8-12).The saturation degree repair mechanism is discussed in Section IV-C.A maximum number of repair mechanism invocations is set to prevent cycling between two surgical units.The solution will be deemed infeasible if this limit is reached during the slot assignment (lines [11][12]. The algorithm stops when all slots have been assigned to the master plan.The algorithm's output includes a complete master plan satisfying all hard constraints, provided the repair mechanism does not exceed its allowed usage.Fig. 3 summarizes the proposed constructive heuristic in a flowchart.The green-shaded boxes represent the slot allocation (Algorithm 2), whereas the blue-shaded boxes represent the repair mechanism.The computational complexity of the proposed improved saturation degree-based constructive heuristic can be derived as O(n log n), where n depends on the number of surgical units.The algorithm sorts all surgical units in line 2, which takes O(n log n) computation.Assigning surgical units with predefined OTs and preferred slots involves nested loops iterating through surgical units, weeks, fixed OTs or preferred slots, and weekly days.The time complexity is influenced by the number of surgical units with fixed OTs or preferred slots.The main loop (lines 5-12) has a complexity that depends on the number of surgical units and the execution of assigning and swapping units.In both cases, score calculation involves operations dependent on the number of feasible slots and sorting that involves O(m log m), where m is the number of feasible slots.The overall time complexity of the entire algorithm is dominated by the main loop and the operations within each loop, which depend on the number of surgical units and slots.The exact complexity may vary based on specific conditions and the input data size.

B. SLOTS ALLOCATION
The slot allocation is determined based on the surgery demand represented by the estimated surgery duration.The number of surgeries was not considered to eliminate bias, such as surgical units with many short surgeries would be prioritized over units with fewer long surgeries.Slot allocation occurs in two phases, as described in Algorithm 2.
In the first phase, all slots in the planning horizon (slot total ) are distributed to the medical subspecialties (lines 1-5).The second phase involves distributing each subspecialty's allotted slots (slot j ) to their respective surgical units, following the same procedure as the first, where slot total is substituted to slot j , dur j to dur g and dur total to dur j (line 6).Firstly, the suggested slots are calculated from the ratio of the total estimated surgical times of the subspecialty/unit over the total

Algorithm 2 Algorithm to Allocate the Number of Slots for the Master Plan
Input: Subspecialties (J ), Surgical units (G), Surgeries list with estimated duration and waiting time for each unit (S j ), number of slots in the planning horizon (slot total ) Output: Slots allocation to each surgical unit (slot g ) 1 For each subspecialty j, calculate the suggested number of slots (slot j ) based on the total estimated surgery duration using slot j = ⌊slot total * dur j dur total ⌋, where slot j is rounded to the lower integer value, dur j = g∈G j s∈S g dur s and dur total = j∈J dur j .
2 Repair slot j for each subspecialty to avoid invalid values.
3 Calculate total residual slots (slot residual ) where slot residual = slot total − j∈J slot j .
4 Sort the subspecialties by the average waiting time in decreasing order.5 Iterate the subspecialties list and redistribute slot residual where slot j = slot j + 1 at each valid iteration until slot residual is exhausted 6 Allocate slot j to the surgical units of subspecialty j using the same method as in lines 1-5, ensuring slot j = g∈G j slot g 7 End (return number of slots allocated to all surgical units) estimated surgical times of all subspecialties/units (line 1).The calculation will produce a decimal value.As the OT slots are non-shareable, the decimal number is converted to discrete values by rounding the values to the lowest integer.Then, the algorithm fixes invalid slot allocation (line 2).Next, the number of residual slots is then calculated (line 3).A redistribution mechanism is introduced to allocate the residual slots based on waiting time (lines 4-5).Subspecialties/surgical units are sorted by average waiting time (line 4).The algorithm iterates through the sorted list, adding one slot to each subspecialty/unit subject to the hard constraints until there are no more residual slots (line 5).At the end of the algorithm, each surgical unit has been allocated a fair number of slots in the master plan based on surgery demand and waiting time.

C. SLOTS ASSIGNMENT
This section describes the saturation degree-based sorting for the surgical unit selection, slot selection and repair mechanism.The MSSP can be described as a timetabling problem involving allocating events and resources to space and time [47].In the MSSP, surgery groups (events) and their resources, such as surgical equipment, are assigned to a set of OTs (space) and days (time).The saturation degree heuristic prioritizes events with fewer remaining feasible space and time, and the order dynamically changes after every assignment [66].

1) SURGICAL UNIT SORTING USING SATURATION DEGREE
Drawing inspiration from the heuristic hybridization from Sabar et al. [65], the surgical units are sorted based on multi-ple variables, and their ranks from each variable are combined to produce the unit score.Three variables are considered: the number of feasible slots, preferred slots, and the ratio slot allocation, hereafter called ratio max.The ratio max (rm) is calculated using rm = Furthermore, the surgical units are sorted separately before being recombined from three smaller exclusive lists: constrained units (units with blocked OTs), unconstrained units (units without blocked OTs), and fixed units (units with predefined OTs).The surgical units are segregated as described because each list has different priorities.
Constrained units are sorted by all three variables.Firstly, the list is sorted by the number of feasible slots in increasing order, with ties broken by the number of preferred slots and random selection.Surgical units with fewer feasible slots are prioritized to ensure solution feasibility.Secondly, the units are sorted according to the number of preferred slots in increasing order, where units without any preferred slots are placed at the end of the list.Ties are broken by the number of feasible slots and random selection.Surgical units with fewer preferred slots have precedence in order to maximize the preferred slots' reward.Finally, the list is sorted by the decreasing ratio max, where ties are broken by random selection to increase solution feasibility.
In each sorted list, sorting scores are given inverse to the sorted placement, with the unit at the first position receiving the maximum score equivalent to the list size.In preferred slots sorting, units without preferred slots do not receive scores while not affecting the scores for other units.Afterwards, the scores from each sorted list are aggregated to obtain the unit score.Each score is multiplied by its respective weight coefficient: θ preferred = 1, θ ratio = 2, θ feasible = 3.The units are sorted by the unit score in decreasing order, where the number of feasible slots, preferred slots, and random selection breaks ties.
The unconstrained and fixed units are sorted considering only the number of preferred slots and the ratio max, following steps similar to those of the constrained units.The three smaller exclusive lists are then recombined, where the constrained units are placed before the unconstrained units to avoid infeasible solutions.Fixed units are added to the end of the final sorted list as they will not affect other assignments.Fig. 4 represents an example of the surgical unit sorting method, where yellow-shaded boxes denote constrained units, green-shaded boxes denote unconstrained units, and blue-shaded boxes denote fixed units.The sorting scores are indicated in brackets.

2) SLOT SELECTION USING SATURATION DEGREE
The algorithm iterates through unassigned units, assigning the unit with the highest priority to a slot selected based on the saturation degree strategy.All feasible slots for the unit are given slot scores calculated based on two factors, including: 1.The number of feasible unassigned units for the slot.

The potential objective function value if the chosen unit
is assigned to the slot.
The potential objective function value is derived from the equations presented in Section III-B, limited to the chosen unit and slot.The slot with fewer feasible unassigned units and lower potential objective function value, as the problem is minimization, is given a higher score.The score values are normalized, where the number of feasible unassigned units is divided by the total number of unassigned units, and the potential objective function values are divided by their respective upper boundaries, given in Appendix C. The chosen unit is assigned to the slot with the highest slot score calculated by summing the weighted normalized score values.

3) REPAIR MECHANISM USING SATURATION DEGREE
A repair mechanism is called when all feasible slots of the chosen unit have already been taken.The repair mechanism vacates one of the feasible slots chosen based on its swap score.The swap score for a slot is calculated by adding the normalized values of the repair unit score of its original unit and its repair slot score.
The repair unit score is calculated following the unit sorting method based on the saturation degree discussed in Section IV-C, where surgical units are sorted considering three variables.The difference is that the number of feasible slots is measured when the master plan is empty (initial state) and at the partial master plan (current state).The following weights are used to obtain the unit score: θ preferred = 1, θ ratio = 2, θ feas.init= 3, θ feas.cur= 4.
Contrary to unit sorting in Section IV-C, a higher repair unit score is given to units with a higher feasible and preferred slots count and a lower ratio max.This method is employed because the repair mechanism aims to unassign a unit with the lowest assignment difficulty.The repair unit score is also normalized by dividing it by the number of unassigned units.
Meanwhile, the repair slot score is calculated similarly to the slot score in Section IV-C.The potential objective function value is extended to consider the potential gains if the chosen unit is assigned to the slot and the potential loss if the original unit is unassigned.A slot with fewer feasible unassigned units and higher objective value gain and lower objective value loss is preferred.
The slot with the highest swap score is selected.The original unit from the slot is re-inserted into the list of unassigned surgical units.Since the unit with the lowest assignment difficulty is selected, finding a replacement slot in the following iterations should be easier.The swap is performed by considering the objective function value so that the solution quality is unaffected.

V. COMPUTATIONAL EXPERIMENT A. DATASET
The experiment utilizes the post-operative dataset from the case study hospital, HCTM, which includes surgeries booked between 1 January 2023 and 31 December 2023.Each surgery includes details on its booking date, estimated surgery duration, associated subspecialty, surgical unit, and type of resource requirement.The booking date is crucial for calculating waiting time, whereas the estimated surgery duration is used for slot allocation.These variables will be utilized in a simulation to assess the operational performance of the generated master plan.Information on subspecialty, surgical unit, and type of resource requirement determine the surgery's group affiliation to guide the assignment to the appropriate slot.Table 4 briefly summarizes the dataset employed in the experiment.
HCTM has 13 OTs and utilizes a two-week cyclic master plan with five weekly working days.25 medical subspecialties comprised of 29 surgical units, where several subspecialties have two units, were considered.Surgeons from a subspecialty can only operate surgeries from its subspecialty.Meanwhile, anaesthetists and nurses are shared among the subspecialties and are assumed to be always available.

B. EXPERIMENTAL SETUP
Master plans for the case study hospital are generated by executing the proposed constructive heuristic written in Java.An additional Java program is developed to simulate surgery assignments to the master plans produced.The simulation gives an operation date for each surgery in the dataset from the first simulated planning day (1 January 2024) until the dataset is exhausted, following the cyclic master plan.No holidays on workdays throughout the period are assumed.
The algorithm's performance is assessed using three metrics: solution feasibility, quality, and operational-level performance.Solution feasibility is measured by the number of times a feasible solution is found after 100 executions and the average repair mechanism calls.Solution quality is determined by the objective function value.The operational-level performance is measured by the average waiting time, waiting time standard deviation or variability among surgical units, and the number of days required to schedule all surgeries from the dataset.Each algorithmic run starts with an empty solution and stops when a complete solution is produced or the limit for repair mechanism usage is reached, resulting in an infeasible run.The experiment was executed on a computer with an Apple M1 chip with an 8-core CPU and 8GB of RAM.

C. SENSITIVITY ANALYSIS
The objective function for the optimization model consists of ten objectives aggregated using the weighted sum method.Each objective is assigned a weight value based on its importance.Ideally, the master plan generated should differ as the weightings are modified.The modifiable weightings will enable hospital planners to evaluate the effect of each objective on the solution quality and simulated operational performance of the master plan.Adapting the method by Penn et al. [16], the analysis is designed to explore the effect of small changes on each objective weighting.
Table 5 shows that the objective values of the master plans produced vary based on the different weightings used.On the other hand, the operational performance, assessed through the average waiting time of surgeries scheduled in a simulation, is only minimally influenced by the different master plans.
Additionally, the proposed constructive heuristic incorporates weight coefficients in calculating the unit score and repair unit score.These weight coefficients are lexicographical, leading to the examination of different permutations.Table 6 illustrates that the best objective function value is obtained using θ preferred = 1, θ ratio = 2, and θ feasible = 3 as the weight coefficients for unit scores.Table 7 indicates that using θ preferred = 1, θ ratio = 2, θ feas.init= 3, and θ feas.cur= 4 weight coefficients for repair unit scores lead to the best objective function value.Consequently, these values are utilized in our experiment.

D. BENCHMARK CONSTRUCTIVE ALGORITHMS
The proposed improved saturation degree-based constructive heuristic is compared to two basic approaches (greedy and random constructive heuristics) and two algorithms from the literature (regret-based and saturation degree-based constructive heuristics).Benchmarking is conducted against only constructive heuristics and not to improvement heuristics.The slot allocation is obtained from Algorithm 2 and remains constant for all benchmark algorithms.Algorithmic runs for each benchmark algorithm utilize the same execution environment as those employed in the proposed algorithm to ensure fairness.In addition, the generated master plans are compared to a manual master plan from the case study hospital.
A greedy approach is very popular due to its simplicity and efficacy as an initial solution for perturbative algorithms [83].The greedy constructive heuristic in the experiment minimizes waiting time by assigning the highest average waiting time unit to the earliest feasible slot available.Ties are broken by selecting the unit with the higher total estimated surgery duration.Further ties are broken with random selection.The repair mechanism vacates the slot with the lowest average waiting time unit.However, this strategy may not return a feasible solution.Therefore, it is later replaced with random swaps.
The random constructive heuristic randomly selects a surgical unit at each iteration.Then, the unit is assigned to a random slot that does not violate hard constraints.If a feasible slot is unavailable, the heuristic vacates a random feasible slot in the partially constructed master plan.
The regret-based constructive heuristic [9] is adapted to prioritize surgical units with a higher waiting time.Ties are broken by selecting the unit with the higher total surgical times and random selection.Then, the regret value for each feasible slot is calculated by finding the difference between the highest and second-highest average waiting time of its feasible units.The slot with the smallest regret value is chosen, while ties are broken by considering the third highest.Infeasibility is repaired by swapping out the unit with the lowest average waiting time.However, the repair mechanism may not be able to guarantee feasibility and later switch to random swap.
The saturation degree-based constructive heuristic from Razali et al. [25] only considers the solution feasibility in surgical units and OT sorting.At each iteration, the surgical unit with the fewest feasible slots is assigned to a slot with the least feasible units.The repair removes the unit with the most feasible slots in the partial master plan.
The master plan utilized in the case study hospital, HCTM, for 2023 and 2024 is taken to compare the solution quality and operational-level performance.The comparison will demonstrate the potential improvements of using intelligent systems to generate the hospital's timetable.

1) SOLUTION FEASIBILITY
Due to the poor results on solution feasibility when using the greedy repair mechanism, additional runs using random swaps are executed.The solution feasibility for the hospital's master plan cannot be measured as it was produced manually by the hospital planners.Table 8 the results of each algorithm to produce feasible solutions over 100 runs.It indicated that algorithms employing a greedy repair mechanism failed in producing a solution.Infeasibility could be due to cycling, which occurs when the same pair of surgical units are exchanged with each other.Greedy approaches are more prone to suffer from this issue since they cannot select a different surgical unit when cycling occurs.On the other hand, random repair and saturation degree-based repair mechanisms produced feasible solutions in every run.Afterwards, we observe the number of repair mechanism usage required to obtain the feasible solutions among these algorithms.Fig. 5 shows the box-plot graph of the repair calls, where a lower value is preferred.It shows that both saturation degree-based constructive heuristics outperform other algorithms in minimizing the repair mechanism usage.The improved variant performs best at an average of 2.80 calls over 100 runs.Furthermore, both saturation degree-based constructive heuristics have a smaller deviation and lower maximum value compared to the other algorithms using random swaps.Fewer requirements for the repair mechanism demonstrate the efficacy of the saturation degree heuristic for choosing a surgical unit and slot.Next, the difference in execution time between the basic approaches (greedy and random) and the designed approaches (saturation degree-based and Spratt and Kozan [9]'s regret-based) is investigated.
Fig. 6 represents the box-plot graph of the execution times, in seconds, for the benchmark configurations.It demonstrates that the improved saturation degree-based constructive heuristic suffers a longer average execution time that is arguably unnoticeable.Notably, extreme outliers exist for the proposed algorithm, with runs between 0.06 and 0.10 seconds.Nevertheless, the difference is negligible as all values are small (less than one second).

2) SOLUTION QUALITY
Runs using the greedy repair mechanism are excluded from hereafter.Table 9 shows the normalized rewarding objectives values.The improved saturation degree-based constructive heuristic recorded the most rewards for the preferred 44760 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.slot (SC1), same OT (SC4), and consecutive days (SC5) objectives.Meanwhile, the original saturation degree-based constructive heuristic outputs the best results for the extra equipment (SC3) objective.The hospital's master plan achieved the highest same weekly slot (SC6), indicating the master plans from the first and second weeks have high schedule stability.Besides, the plan was able to reserve an ultra-clean OT (SC10) on each planning day, which is not achieved by any of the intelligent approaches.
Table 10 shows the values for penalizing objectives.Based on the results shown, the movable shar- ing (SC2), clashing subspecialty (SC7), and parallel heavy resource requirements slots (SC9) were minimized best by the proposed improved saturation degree-based constructive heuristic.This observation highlights the effectiveness of the proposed algorithm in reducing resource-sharing among scheduled surgeries, contributing to a lower chance of scheduling disruption.The best value for exceeding the soft limit on the parallel slots (SC8) is obtained by the hospital's master plan.This could happen because some slots are not utilized in the hospital's master plan, potentially affecting the surgery scheduling performance since less OT time is available.
The weighted sum of the objectives is demonstrated in Fig. 7.It indicates the hospital's master plan as the best performer (at -2.7411), followed closely by the proposed improved saturation degree-based constructive heuristic (with -2.2819).The hospital's plan excelled in achieving the ultra-clean OT objective (SC10), significantly increasing its weighted sum of objectives.In contrast, the improved saturation degree-based constructive heuristic did not fulfil this soft constraint on any planning days.Despite this, we contend that the proposed constructive heuristic produces high-quality solutions, as evidenced by its superior performance in six of the ten objectives.Other benchmarked algorithms performed significantly worse than the two.We theorize that there could be a trade-off between several objectives and operational scheduling performance.The objective values for SC2, SC3, SC7, SC8 and SC9 could be increased if fewer surgical units are assigned on the same day or fewer OTs are in use.

3) SIMULATED OPERATIONAL LEVEL PERFORMANCE
The operational-level performance is evaluated in terms of surgeries' waiting time, the standard deviation or variability for waiting times among surgical units, and the number of days required to operate all surgeries.The results of the surgery scheduling simulation are presented in Table 11.Based on the results, the hospital's plan obtained the best average waiting time (with an average of 300.53 days).Results from the constructive heuristics are nearly identical, nearly three days longer than the hospital's master plan (ranging from 302.97 to 303.26 days).This increase may be attributed to the slot allocation from the constructive heuristic, which failed to allocate sufficient slots to surgical units with many surgeries.Consequently, the days to complete the surgery list are lower for the manually generated master plan (309 days) than for the intelligent approaches (370 days).Despite this, the standard deviation for waiting times among surgical units is smaller for the master plans produced by constructive heuristics compared to the hospital's plan, at 67.772 compared to 70.720.This suggests a more distribution of time values among the surgical units.

F. STATISTICAL ANALYSIS
To support the conclusions drawn from the results, we conducted statistical analysis on all results acquired, including normality tests and non-parametric tests with pairwise comparison.The hospital's master plan is excluded from the analysis as only one master plan exists, and each run of the simulation will produce the same result.
Normality Tests (Shapiro-Wilk Royston Test): We tested the normality of the three performance measures (repair calls, average waiting time and objective value) from the five benchmarked algorithms using the Shapiro-Wilk Royston test [84].According to the results, at least one sample is not normally distributed for every performance measure.Therefore, non-parametric tests are carried out next.The claim that the proposed algorithm outperformed the benchmark algorithms in repair calls has been statistically proven, except for the original saturation degree-based constructive heuristic, where no statistical difference was observed between the improved and original variants.The regret-based constructive heuristic exhibits a statistically significant advantage over other algorithms regarding average waiting time but shows no significant difference in objective value compared to the random approach.Importantly, the improved saturation degree-based algorithm demonstrates a statistically significant superiority in objective value compared to other algorithms.

VI. PERFORMANCE AGAINST IMPROVEMENT HEURISTICS A. BENCHMARK IMPROVEMENT ALGORITHMS
Comparison of our proposed constructive heuristic to improvement heuristics is unfair as the former is only run for one iteration.In contrast, the latter executes multiple iterations to find a higher-quality solution.Despite this, we are interested in observing the performance of our proposed algorithm against state-of-the-art improvement heuristics for the MSSP, including SA metaheuristics and two SA-based hyper-heuristics.Our proposed algorithm does not necessarily have to beat the improvement heuristics to be presented as a good solution method.
The improved saturation degree-based constructive heuristic is compared against SA, Hyper SA, and Hyper SA-TS, which were introduced by Spratt and Kozan [9].The algorithms utilize five types of moves involving assignment swaps and modifications to the schedule structure, particularly regarding half-day and full-day slots.When adapting their approach to our problem, we incorporated the assignment swap moves while omitting the schedule structure modifications.This is because our case study does not utilize half-day slots, such modifications not possible.We adapted the assignment swap moves to: 1. Swap two subspecialties.2. Swap two surgical units.3. Shuffle three subspecialties.4. Shuffle three surgical units.5. Swap surgical units until a feasible solution is obtained, up to a maximum number of swaps.
SA employs a randomly selected move to obtain a candidate solution.The acceptance of this candidate solution as the current working solution is determined by a probability based on the change in the objective function value and the solution temperature.The solution temperature decreases upon accepting a solution, calculated using the formula t * = t 1+εt , where t * is the new value, t is the previous value, and ε is the temperature decrease factor.
In Hyper SA, a move is selected based on heuristic rankings, guided by a Tabu search strategy.The algorithm chooses the move with the highest heuristic rank, which is not in the Tabu list.It applies the same move for multiple iterations, referred to as a block, before updating the heuristic ranks.A heuristic's rank increases when the solution from a block improves the previous solution; otherwise, the rank is reduced, and the heuristic is added to the Tabu list.The SA approach is employed for move acceptance, where the temperature decreases when the candidate solution is accepted and increases otherwise.The solution temperature increases at a rate of t * = t 1−γ t , where t * is the new value, t is the previous value, and γ is the temperature increase factor.Hyper SA-TS extends Hyper SA by incorporating a Tabu mechanism to restrict neighbourhood moves on forbidden assignments (combinations of OT time block and subspecialties or surgical units).Specifically, the algorithm is prohibited from performing swaps on assignments that have previously led to worsening solutions.The algorithm maintains two Tabu lists: one for the neighbourhood moves and the other for the assignments.The Tabu length for the latter is dynamic and decreases at iteration intervals.Other mechanisms, including the move acceptance strategy, remain similar to the implementation of Hyper SA.
The improvement algorithms undergo 31 runs to ensure robust and representative results across multiple iterations.The first 31 runs of the improved saturation degree-based constructive heuristic, as used in the constructive heuristics comparison, are employed for the comparison of improvement algorithms.Based on the configuration provided by Spratt and Kozan [9], each algorithm undergoes 16,000 iterations from an initial solution produced by the improved saturation degree-based constructive heuristic.The parameter settings align with the specifications outlined in the original publication, as illustrated in Table 12.Readers are encouraged to consult the source publication for a comprehensive understanding of each algorithm.

1) SOLUTION QUALITY
The normalized rewarding objective values in Table 13 reveal that the improved saturation degree-based constructive heuristic excelled in SC1, SC4, and SC5, surpassing the improvement algorithms.Assignment swap moves were ineffective in enhancing these objectives.However, they successfully improved SC6 and SC10, where SA increased the rewards.Hyper SA exhibited the best performance for SC3, albeit with only a slight difference compared to the proposed algorithm.
Regarding penalizing objective values, as shown in Table 14, the proposed algorithm exhibits the best value only for SC2.Assignment swap moves were effective in improving SC7, SC8, and SC9.SA leads to improvements in SC8 and SC9, whereas SA-TS enhances SC7 compared to the proposed constructive heuristic.
Fig. 8 presents a box-plot graph depicting the weighted sum of the objectives, comparing the proposed improved saturation degree-based constructive heuristic to the benchmark improvement algorithms.The graph demonstrates that the improvement algorithm, particularly SA, has improved solution quality by 17.50% (-2.686 compared to -2.286).This improvement is achieved through the application of assignment swap moves to a complete initial solution.SA results suggest that the objectives can be increased at the cost of other objectives.However, the other improvement algorithms, Hyper SA and Hyper SA-TS, fall short of outperforming the proposed constructive heuristic.We observed that they failed to improve their initial solutions, possibly due to ineffective pairing of the swap moves and algorithmic strategies.

2) SIMULATED OPERATIONAL LEVEL PERFORMANCE
Table 15 indicates that average waiting times are improved when simulating scheduling surgeries into master plans produced by the improvement algorithms.Specifically, Hyper SA generates master plans with the lowest average waiting time.However, the variability for waiting time among surgical units worsens compared to the master plans from the proposed improved saturation degree-based constructive heuristic.The days required to schedule all surgeries remain the same for all algorithms because they utilize the same slot allocation.

C. STATISTICAL ANALYSIS
Normality Tests (Shapiro-Wilk Test): The normality of results for three performance metrics (objective value, average wait-  ing time, and waiting time standard deviation are assessed for the improvement algorithms and the proposed constructive heuristic.The Shapiro-Wilk test is employed, considering a sample size of 31.The test results indicated that at least one sample in each performance metric is non-normally distributed.Consequently, non-parametric tests are utilized for hypothesis testing. Non-Parametric Tests (Kruskal-Wallis Test): Kruskal-Wallis tests at a 0.05 significance level indicate statistically significant differences among the results from the four algorithms across all three performance metrics: objective value (p-value = 9.14725E-19), average waiting time (pvalue = 2.39033E-05), and waiting time standard deviation (p-value = 1.68388E-18).Subsequent post-hoc tests are conducted to identify specific pairs of algorithms with significant differences.
Post-hoc Tests (Dunn's Test): Post-hoc Dunn's tests revealed significant differences in the objective value between SA and other algorithms, including the proposed constructive heuristic.Only the pair of Hyper SA and Hyper SA-TS shows an insignificant difference in objective value (p-value = 0.591).Regarding average waiting time, Hyper SA exhibited a significant difference from the proposed algorithm (p-value = 8.70127E-06), whereas there was no significant difference with Hyper SA-TS (p-value = 0.631).For the waiting time standard deviation, all pairings demonstrated significant differences except for Hyper SA-Hyper SA-TS (p-value = 0.896), indicating the statistically proven superior performance of the proposed constructive heuristic in this metric.

VII. PERFORMANCE AS INITIAL SOLUTIONS
We aim to investigate the impact of employing solutions generated from various constructive heuristics on the overall effectiveness of improvement heuristics.In this experiment, we utilize solutions generated by greedy, random, regret-based, saturation degree-based, and improved saturation degree-based constructive heuristics.The improvement heuristic SA, identified as the most effective in Section VI, is applied to enhance solutions produced by each constructive heuristic.We generate 100 initial solutions and improve them using SA to obtain the final solution, employing the same experimental setup as in previous sections.The outcomes are presented below.

1) SOLUTION QUALITY
The SA algorithm produced the best solutions for the preferred slot (SC1), same OT (SC4), and same weekly slot (SC6) objectives when utilizing initial solutions from the proposed improved saturation degree-based constructive heuristic, as indicated in Table 16.The large difference in the objective value of SC1 compared to other initial solutions is attributed to the proposed heuristic's strategy of assigning surgery groups to their preferred slots when the master plan is empty.In contrast, other solutions were improved through iterative swap moves on a complete solution.The changes are more restricted, leading to lower improvements in the objective value.The objectives related to extra equipment (SC3) and consecutive days (SC5) are optimized most effectively when utilizing regret-based constructed initial solutions, albeit with small differences compared to other initial solutions.The original saturation degree-based constructive heuristic can reach solutions with a higher average for the ultra-clean OT (SC10) objective, where the proposed algorithm performed less favourably.
Table 17 highlights that random initial solutions exhibit optimal minimization of penalties for parallel slots (SC8) and parallel heavy resource requirements slots (SC9).The lowest penalty for clashing subspecialty (SC7) is observed when utilizing initial solutions generated through the regret-based constructive heuristic.The combination of the proposed constructive heuristic with the SA improvement heuristic achieved the most favourable outcome for the movable equipment sharing objective (SC2).It is crucial to emphasize that all observed differences are minimal, suggesting that the improvement heuristic could potentially be the bottleneck in terms of performance.The box-plot graph in Fig. 9 demonstrates that employing initial solutions generated by the improved saturation degree-based constructive heuristic yields the overall best solution an average of -2.6793).This aligns with the notion that the initial solution quality impacts the final solution, with high-quality initial solutions leading to better final solutions [85].Wang et al. [86] suggested that high-quality initial solutions can enhance computational efficiency.Our results support this idea, as achieving a solution of comparable quality is possible with lower computational time when starting from initial solutions already close to a specific point, compared to less optimal initial solutions.
Apart from the improved saturation degree-based constructive heuristic, other configurations achieved nearly identical final solution quality despite differing initial fitness, as shown in Table 18.This similarity in outcomes suggests the possibility that these configurations have reached a local optimum.This implies that the effectiveness of an improvement algorithm could be constrained by the starting point [87], especially those without effective diversification strategies.

2) SIMULATED OPERATIONAL LEVEL PERFORMANCE
Table 19 shows that the regret-based constructive heuristic combined with the SA improvement algorithm obtained the best operational-level performance.This is evidenced by achieving the lowest waiting time average and variability across surgical units.

B. STATISTICAL ANALYSIS
Normality Tests (Shapiro-Wilk Royston Test): We assessed the normality of distributions for objective value, average waiting time, and waiting time variability samples using the Shapiro-Wilk Royston test.The results revealed non-normal distribution only for the objective value of the improved saturation degree-based constructive heuristic (p-value = 1.65462E-4) and the waiting time variability of the ran-dom (p-value = 2.01878E-2) and regret-based approaches (p-value = 1.81908E-2).Consequently, non-parametric tests are applied to identify significant differences in objective value and waiting time variability.As all samples for average waiting time exhibit normal distribution, parametric tests are employed.
Hypothesis Testing (Kruskal-Wallis Test & One-Way ANOVA): Kruskal-Wallis tests were performed on nonnormally distributed samples for objective value and waiting time variability, revealing a significant difference in objective value (p-value = 4.9351E-50).However, no statistical significance was observed for waiting time variability (p-value = 0.77705).The parametric one-way analysis of variance (ANOVA) [88] is applied to the normally distributed samples of average waiting time, indicating a significant difference with a p-value of 0.02065.Subsequent pairwise comparisons are conducted using suitable post-hoc tests.
Post-hoc Tests (Dunn's Test & Tukey's Test): Post-hoc tests were applied to the significantly different samples of objective value and average waiting time, with ten pairwise comparisons across five algorithms.Dunn's test, employed for non-normally distributed objective value samples, revealed significant differences between the improved saturation degree-based constructive heuristic and all other heuristics (p-value = 3.12961E-11 to greedy, p-value = 6.70597E-12 to random, p-value = 5.91296E-11 to regretbased, and p-value = 2.77667E-12 to saturation degreebased).No significant differences were observed among the remaining four heuristics.Conversely, Tukey's test, conducted on normally distributed average waiting time samples, displayed no significant differences between any heuristic pairs, contrary to the parametric one-way ANOVA's result.This discrepancy may arise from differing sensitivities of one-way ANOVA and Tukey's test, along with the latter's multiple comparison adjustment, rendering it less conservative and more sensitive [89].Additionally, two heuristic pairings approached significance (p = 0.05969 for Greedy-Random and 0.05381 for Random-Regret-based), suggesting potential sensitivity to sample variations [90].

VIII. CONCLUSION
This study has addressed the issues in MSSP by proposing a mathematical model variant with a novel objective function and an improved saturation degree-based constructive heuristic that enhances both solution quality and feasibility.Compared to the original saturation degree-based variant, the improved heuristic reduces repair mechanism requirements by 14.63% and increases objective function value by 2.6 times, supported by statistical tests that demonstrated significant differences with small p-values.The results indicate that solution quality and feasibility can be increased concurrently.Moreover, a simulation study suggested that the proposed heuristic could yield a more balanced waiting time distribution among surgical units compared to the hospital's plan.Initial solutions generated by the proposed algorithm can also result in superior solutions discovered by improvement algorithms.
We attribute the increased solution quality to the proposed heuristic's ability to maximize potential objective value while ensuring feasibility at each step of solution construction.In contrast, the original variant prioritized slot feasibility, compromising quality, whereas the greedy heuristic focused on objective value, potentially reducing both feasibility and quality.The regret-based approach failed to enhance solution feasibility or quality consistently, whereas the random approach lacks intelligent rules during solution construction.Given the superior performance of the improved saturation degree-based constructive heuristic, it holds potential for application in other timetabling problems, such as conference and event scheduling, sports scheduling, and employee shift scheduling.
Despite its performance improvement, the proposed method has shortcomings, including subjectivity in solution quality and the need for a repair mechanism.Implementation may require adjustments to align with specific objectives in different contexts.Besides, the model formulated in this study has several limitations, such as uncertainty in the surgery duration and demand not being considered.Future works can extend the model and constructive heuristics to include stochastic and dynamic aspects.

A. UPPER BOUNDARY FOR THE NORMALIZATION OF MASTER PLAN'S OBJECTIVE VALUES
The maximum values for each soft constraint are detailed in Table 20.

B. PROGRAMMING SYNTAX PSEUDOCODE OF THE IMPROVED SATURATION DEGREE-BASED CONSTRUCTIVE HEURISTIC
The proposed algorithm is written in Java.Procedures within the algorithm are explained in Algorithm 3 -8.

FIGURE 1 .
FIGURE 1. Overview of the methodology.

FIGURE 2 .
FIGURE 2. Example of a complete master plan generated based on the optimization model.

Algorithm 1
Proposed Improved Saturation Degree-Based Constructive Heuristic Input: Number of planning days and OTs, subspecialties (J ) and surgical units (G) with their constraints information, movable equipment (E) type and quantity, surgeries list (S) with booking date and estimated duration.Output: OT slot allocation and assignment of each surgical unit to a master plan. 1 Determine the suitable slot allocation for each surgical unit (slot g ) [see Algorithm 2]. 2 Sort the surgical units using the saturation degree heuristic considering assignment difficulty and potential objective value [see Section IV-C.1]. 3 Assign surgical units with predefined OTs.Re-sort the unassigned list.4 Assign surgical units with preferred slots, where possible.

7 8 9
Assign unit g to the slot with the fewest feasible unassigned units and highest potential objective value [see Section IV-C.2].Re-sort the unassigned list.Otherwise

10
Swap unit g with unit g ′ from the partial master plan selected using the saturation degree heuristic [see Section IV-C.3].Re-sort the unassigned list.11 Otherwise 12

FIGURE 3 .
FIGURE 3. Flowchart of the improved saturation degree-based constructive heuristic.
g is the slots allocated for unit g and |W | * P feasible g * slot parallel g is unit g's maximum slots allocation.

FIGURE 4 .
FIGURE 4. Illustration of the surgical unit sorting method.

TABLE 8 .
Number of feasible runs by the benchmarked constructive algorithms (over 100 runs).

FIGURE 5 .
FIGURE 5. Box-plot graph for the repair mechanism calls of the benchmarked constructive algorithms (lower is better).

FIGURE 6 .
FIGURE 6. Box-plot graph for the execution time of the benchmarked constructive algorithms (lower is better).

FIGURE 7 .
FIGURE 7. Box-plot graph for the objective function value of the benchmarked constructive algorithms (lower is better).
Non-Parametric Tests (Kruskal-Wallis Test): We performed the Kruskal-Wallis test (α = 0.05) to compare five independent samples for each performance measure.The results showed a significant difference between the algorithms in all performance measures (p-value = 7.63057E-78 for repair calls; p-value = 1.09357E-35 for average waiting time and p-value = 1.78344E-73 for objective value).Post-hoc Tests (Dunn's Test): Post-hoc pairwise comparisons are carried out on ten algorithm pairings for each performance measure.The test has proven the significance of the differences in repair calls for all pairs except Greedy-Random (p-value = 0.247), Greedy-Regret-based (p-value = 0.257), and Saturation degree-based-Improved saturation degree-based (p-value = 0.264).For average waiting time, only Random-Saturation degree-based (p-value = 0.895) indicates non-significance.For objective value, the null hypothesis was rejected only for Random-Regret-based (pvalue = 0.898), indicating no difference between the samples.

FIGURE 8 .
FIGURE 8. Box-plot graph for the objective function value comparing the proposed constructive heuristic to improvement algorithms (lower is better).

FIGURE 9 .
FIGURE 9. Box-plot graph for the objective function value comparing the performance using different initial solutions (lower is better).

TABLE 1 .
Parameters for the mathematical model.

TABLE 2 .
Decision variables for the mathematical model.

TABLE 3 .
Example of capacity allocation.

TABLE 4 .
Details of the dataset.

TABLE 5 .
Sensitivity analysis of the objective function weightings.

TABLE 6 .
Sensitivity analysis of the weight coefficient for the unit score.

TABLE 7 .
Sensitivity analysis of the weight coefficient for the repair unit score.

TABLE
Rewarding objective value results for the benchmarked constructive algorithms (bold represents best).

TABLE 10 .
Penalizing objective value results for the benchmarked constructive algorithms (bold represents best).

TABLE 11 .
Simulation results for the benchmarked constructive algorithms (bold represents best).

TABLE 12 .
Parameter settings for the benchmarked improvement algorithms.

TABLE 13 .
Rewarding objective value results for the benchmarked improvement algorithms (bold represents best).

TABLE 14 .
Penalizing objective value results for the benchmarked improvement algorithms (bold represents best).

TABLE 15 .
Simulation results for the benchmarked improvement algorithms (bold represents best).

TABLE 16 .
Rewarding objective value results comparing different initial solutions (bold represents best).

TABLE 17 .
Penalizing objective value results comparing different initial solutions (bold represents best).

TABLE 18 .
Rate of improvements from using different initial solutions.

TABLE 19 .
Simulation results comparing different initial solutions (bold represents best).

TABLE 20 .
Best-or worst-case scenario for each assignment objective.

TABLE 20 .
(Continued.) Best-or worst-case scenario for each assignment objective.

TABLE 21 .
Upper boundary for the normalization of the objective values for each slot.