Membrane computing approach to nurse rostering: enhanced scheduling efficiency with tissue-like P system (TLPS) algorithm

Research problem

Management of nurses in hospitals and other kinds of healthcare facilities continue to challenge the delivery of healthcare, particularly in the context of efficient human resources allocation (Wang et al., 2023). NRP is among the most critical challenges of nurse’s management in hospitals, as it is a process which requires distribution of human resources across multiple wards, and time schedules while also adhering to best practices of healthcare delivery (Booker et al., 2024). NRP as a problem is compounded with the need to balance quality of healthcare delivery and also cost or resource efficiency for the delivery (Wynendaele et al., 2025). Manual methods which are often used to solve the NRP challenge are resource inefficient, and often lead to gaps in services delivery and operational costs. Despite the existence of various computational approaches to address the NRP, there remains a need for more effective, adaptable, and scalable solutions that can handle the dynamic and multifaceted nature of hospital environments (Bebien et al., 2024).

Contribution of study

This study seeks to address the gap of NRP by exploring the potential of a novel bio-inspired algorithm. The tissue-like P systems (TLPS), is proposed in this study to optimize nurse rostering, offering a robust solution to this pervasive problem in healthcare management. The complex and NP hard problem challenges of healthcare management are approached using the computational techniques and parallel processing approach of TLPS in this study. The research proposes a novel approach of optimizing hospital nurse scheduling in a manner that considers both hard and soft constraints optimally. The contributions of this study can be outlined as follows:

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  The proposed approach of TLPS for NRP adequately factors both hard and soft constraints of NRP.

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  The proposed approach shows efficiency in its applications in this study, and also provides scalability and adaptability to resource allocation.

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  The study demonstrates the efficiency of bio inspired computing to achieve feasible solutions.

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  The study also demonstrates the performance of the TLPS approach for NRP to be more efficient compared to other approaches, such as GA and HAS on UKMMC dataset

Membrane computing P system

Membrane computing P system is a paradigm of computing that is inspired by the functional and architectural characteristics of living cells, they are often referred to as P systems (Păun & Păun, 2006). P systems computing is characterized by the use of inherent parallel and distributed functions which are synonymous with cellular structures of biological membranes (Jiang et al., 2012). The conceptualization of P systems was introduced by Păun (2000). This is based on tissue like P systems the use of multiple single membrane cells which contain sub modules that interact with one another directly using established and open communication channels (Song, Zhang & Pan, 2017; Martín-Vide et al., 2003). P systems have over the years been applied in computational algorithms which include their use in software development, this is among other several biological inspired computing algorithms (Liu et al., 2023). Both theoretical and applications of P systems suggest the systems remain functional and efficient given their diversity in controlling rule applications such as scheduling and rostering (Dong et al., 2023).

P systems are notably efficient in their enhanced parallelism and their ability to function using a distributed approach of multiple functions at a time (Leporati et al., 2023). P systems generate exponential workspaces using a linear time frame which have efficient modular processes working as a system, as is the case of cell membranes (Song, Zhang & Pan, 2017). P systems offer a versatile framework of modeling discrete processes which are characterized by efficient scalability, inherent compartmentalization, programmability, and discrete data handling (Jiang et al., 2012). P systems have seen successful applications in domains of biology, fault diagnosis, fuzzy reasoning, and computational systems (Zhou et al., 2024).

Research case

This study is conducted using the context of the National University of Malaysia Medical Center (UKMMC), the UKMMC is a healthcare facility which staffs over 1,400 nurses (Hadwan et al., 2013). At UKMMC Medical Center, head nurse is responsible to generate the roster for the nurses, s/he spends a lot of time in addition to the effort that are used to generate this roster, next steps provide more details about generating roster for nurses by head nurse at UKM Medical Center:

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  Nurses write the request and preference in the request book and submit to the head nurse.

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  Head nurse has decision to accept or reject the nurse preference, and then prepare the master plan and initial roster.

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  Department of nursing check the roster and also has decision to approve the roster or to ask the head nurse to repair the roster again.

The head nurse serving at each ward starts generating a master plan before s/he fills up the nurses’ request book. Later, the head nurse constructs the initial roster, but while doing so, s/he either accepts or rejects the nurses’ preferences, which are based on the criteria for acceptance. After that, the head nurse needs approval, so s/he sends the initial roster to the nursing department. If the nursing department approves it, it is distributed among the nurses as the final roster; otherwise, it is revised and amended. After that, the head nurse resends it back for approval.

The research is designed to propose an approach which considers the constraints requirements of nurse rostering which has a total of eight hard constraints labeled H1–H8, and five soft constraints labeled S1–S5 as outlined in Table 1. The constraints outlined in Table 1 are requirements set out for best practices of ensuring rostering and scheduling of nurses are both feasible and qualitative. The study also uses the demands of the minimum required number of nurses for each shift schedule in a day to propose the approach in this study. Table 2 outlines the rostering requirements of the UKMMC for each shift of the day and each department as represented in a sub dataset for a given week. The study also considers the required shift patterns of nurses rostering and schedule, where shift patterns are categorized to ensure there is ethical and policy regulation of the number of shifts of nurses to ensure adequate healthcare delivery. There are three categories of shift patterns which are defined as prohibited patterns, undesirable patterns, and desirable patterns. The prohibited patterns are the shift patterns that violate the hard constraints requirements, undesired adhere to hard constraints but do not have efficient use of soft constraints, and desirable adhere to both hard and soft constraints. The three categories of shift patterns are outlined in Table 3.

The proposed approach of this study primarily focuses on the minimization of the penalty of the rostering for a specific scheduling period, given a coverage requirement with a fixed number of nurses and defined set of constraints. Penalties in scheduling and rostering are characterized as violations of defined required constraints.

Mathematical model

This study uses the notations for mathematical formulation of UKMMC Medical Center nurse rostering problem presented as follows:

$I$: represents the set of available nurses at the scheduling period i $=\mathrm{1...}n$;

$I_s$: represents senior nurses, $I_s$ $\in$ $I$;

$DY$: represents the set of days;

$WT$: represents the set of weights;

$SH$: represents the set of shifts;

$PS$: represents the set of night shift;

$d$: represents the $d^{th}$ day where $d\in DY$;

$s$: represents the $S^{th}$ shift where $s\in SH$;

$w$: represents $w^{th}$ weight where $w\in WT$;

$p$: represents $p^{th}$ possible shifts where $p\in PS$;

$R_{ds}$: represents the demand for day $d$ on shift $s$;

$d_p$: represents the starting date for shifts $p$.

$x_{idx}$: represents a decision variable, $x_{idx}=1$ if nurse $i$has working day ( $d$) on shift ( $s$); or $0$ otherwise;

$y_{pi}$represents a decision variable, $y_{pi}=1$ if shifts $p$ are worked by nurse $i$; or $0$ otherwise.

The study focused about minimizing total violation of soft constraints for NRP at UKM Medical Center as following:

1. Minimize the total violation of (S01) which aims to give a fair number of days off and working days to all the nurses.

2. Minimize the total violation of (S02) which aims to give each nurse a minimum of one off day on the weekend in the scheduled period.

3. Minimize the total violation of (S03) which represents every four consecutive morning shifts of a nurse should be followed by one day off.

4. Minimize the total violation of (S04) which represents every four consecutive evening shifts of a nurse should be followed by one day off.

5. Minimize the total violation of (S05) which is give an evening shift after the day off comes after a night shift.

The model

The study uses the weighted method approach as in Hadwan & Ayob (2010), where the objective function minimizes the total penalty to violations of soft constraints. Nurse rostering problem at UKMMC Medical Center has possible shifts which are: morning shifts, evening shifts, night shifts and day off shifts. These shifts represented by $S=\left\{1,2,3,4\right\}$ respectively. The objective function for this study is to minimize total penalties that occur due to violating the soft constraints during generating the roster.

Objective function is:

$$\begin{array}{rl}\mathrm{M}\mathrm{i}\mathrm{n}f=& WT1\sum _{i\in I}(d{1}_i^{+}+d{1}_i^{-})+WT2\sum _{i\in I}(d{2}_i^{+}+d{1}_i^{-})\\ & +WT3\sum _{i\in I}\sum _{d\in DY}d{3}_{id}^{-}+WT4\sum _{i\in I}\sum _{d\in DY}d{3}_{id}^{-}+WT5\sum _{p\in ps}d{5}_p^{+}\end{array}$$

The WT1, WT2, WT3, WT4 and WT5 are the weights connected with goals (1, 2, 3, 4, 5). Subject to

(H01) $$\sum _{i\in I}{\mathrm{X}}_{id\left(1\right)}\ge {\mathrm{R}}_{d\left(1\right),}\mathrm{\forall }d\in DY$$

(H02) $$\sum _{s\in S}{\mathrm{X}}_{ids}=1fori\in I,d\in DY$$

(H03) $$\sum _{i\in I}{\mathrm{X}}_{ids}\ge ford\in DY,SH=\mathrm{1..3}$$

(H04) $$\begin{array}{rl}& {\mathrm{X}}_{id\left(4\right)}+{\mathrm{X}}_{i\left(d+1\right)\left(1\right)}+{\mathrm{X}}_{i\left(d+1\right)\left(2\right)}+{\mathrm{X}}_{i\left(d+1\right)\left(3\right)}+{\mathrm{X}}_{i\left(d+1\right)\left(4\right)}\le 2fori\in I,\\ & d=\mathrm{1..}\left|DY\right|-2\end{array}$$

(H05(a)) $$\sum _{d\in DY}\sum _{s\in SH-\left\{4\right\}}{\mathrm{X}}_{ids}\le 12fori\in I$$

(H05(b)) $$\sum _{d\in DY}\sum _{s\in SH-\left\{4\right\}}{\mathrm{X}}_{ids}\ge 10fori\in I$$

(H06) $$\begin{array}{rl}& {\mathrm{X}}_{id\left(4\right)}+{\mathrm{X}}_{i\left(d+1\right)\left(4\right)}+{\mathrm{X}}_{i\left(d+2\right)\left(4\right)}+{\mathrm{X}}_{i\left(d+3\right)\left(4\right)}+{\mathrm{X}}_{i\left(d+4\right)\left(4\right)}\ge 1fori\in I,\\ & d=\mathrm{1..}\left|DY\right|-4\end{array}$$

(H07a) $$\sum _{i\in I}{\mathrm{y}}_{pi}=1,p\in PN$$

(H07b) $$\sum _{p\in PS}{\mathrm{y}}_{pi}\le 1,i\in I$$

(H08) $$\sum _{d\in DY}{\mathrm{X}}_{id4}\ge 2,fori\in I$$

(S01) $$\sum _{d\in DY}\sum _{s\in SH-\left\{4\right\}}{\mathrm{X}}_{ids}+\left(d{1}_i^{+}-d{1}_i^{-}\right)=11\mathrm{\forall }i\in I$$

(S02) $${\mathrm{X}}_{i\left(6\right)\left(4\right)}+{\mathrm{X}}_{i\left(7\right)\left(4\right)}+{\mathrm{X}}_{i\left(13\right)\left(4\right)}+{\mathrm{X}}_{i\left(14\right)\left(4\right)}+{\mathrm{X}}_{i\left(d+4\right)\left(4\right)}+\left(d{2}_i^{+}-d{2}_i^{-}\right)=1\mathrm{\forall }i\in I$$

(S03) $${\mathrm{X}}_{id\left(1\right)}+{\mathrm{X}}_{i\left(d+1\right)\left(2\right)}+\left(d{3}_{id}^{+}-d{3}_{id}^{-}\right)=1\mathrm{\forall }i\in I,d=\mathrm{1..}\left|DY\right|-1$$

(S04) $${\mathrm{X}}_{id\left(2\right)}+{\mathrm{X}}_{i\left(d+1\right)\left(1\right)}+\left(d{4}_{id}^{+}-d{4}_{id}^{-}\right)=1\mathrm{\forall }i\in I,d=\mathrm{1..}\left|DY\right|-1$$

(S05a) $$(\sum _{i\in I}[{\mathrm{Y}}_{pi}({\mathrm{X}}_{i(7)(2)}+{\mathrm{X}}_{i(7)(4)})])+(d{5}_p^{+}-d{5}_p^{-})=1,p\in \{1,2,3\}$$

(S05b) $$(\sum _{i\in I}[{\mathrm{Y}}_{pi}({\mathrm{X}}_{i(11)(2)}+{\mathrm{X}}_{i(11)(4)})])+(d{5}_p^{+}-d{5}_p^{-})=1,p\in \{4,5,6\}).$$

The hard constraints for UKMMC are Eqs. (H01) to (H08); where Eq. (H01) is used to ensure that the demand is satisfied all times. Constraint Eq. (H02) is used ensure that each nurse works one shift per day at the most, to cover constraints Eq. (H03) which is one senior nurse should be scheduled at each shift. To ensure that each nurse not to give an isolated working day, constraint Eq. (H04) is used. Constraint Eq. (H05a) used to ensure that the maximum number of working days for each nurse is 12 days during the whole rostering period. Furthermore, constraint Eq. (H05b) to ensures that the minimum number of working days for each nurse is not more than 10 days during the whole rostering period. Meanwhile, to ensure the maximum number of consecutives working days for each nurse is 4 days, a constraint Eq. (H06) is used. Constraints (Eqs. (H07a), (H07b)) ensure that the shifts are covered exactly for each nurse. According to each nurse should have at least two days off during the scheduling period, Constraint Eq. (H08) is stated. To give a fair number of working days and days off for each nurse, constraint Eq. (S01) is used. Constraint Eq. (S02) is used to give one day off in the weekend for each nurse during the scheduling period. Constraints Eqs. (S03) and (S04) try to give consecutive morning or evening shifts. Constraints (Eqs. (S05a) and (S05b)) try to give an evening shift after the day off followed by the four consecutive night shifts. The constraints Eqs. (S01) to (S05b) are soft constraints; $(d{1}_i^{+}+d{1}_i^{-}$) represents the amount of negative deviations for any nurse which is represented by i from goal 1, likewise for each goal positive deviations are penalized.

Tissue like P system (TLPS) algorithm

Several single-membrane or cells evolve in a common environment that provides the basis for TLPS. They have object multi-sets, while some cells directly communicate through channels between them. All cells communicate through their environment. For inter- cellular communication, channels are already given, or it is possible to establish them dynamically.

A structure of a TLPS (degree $d$ ≥ 1):

(1) $$\pi =(\mathrm{\Gamma },O,w_1,\dots \dots .,R_{1\dots \dots ..}{R}_d,i_{out})$$where:

Γ is an object represents non-empty set of symbols;

$O$ $\subseteq$ Γ is the set of objects, that are initially located in the environment, with arbitrary many copies;

$w_1$, …., $w_d$ (1 ≤ $i$ ≤ $d$) are strings representing multisets over Γ, describing the initial contents of the d cells;

$R_{1\dots \dots ..}{R}_d$ (1 ≤ $i$ ≤ $d$) are a set of rules related to the region;

$i_{out}$ is the output membrane structure of $\pi$ to save the output of the result.

The membrane structure of a TLPS is inspired by the constructing of tissues and the communicating substances between two or more cells (membranes), it is represented by a graph of nodes the cells, by applying communication rules the arcs will be existing between two membranes of cells, then they are communicated by (symport/antiport) rules.

In symport rules, a group of objects is transported simultaneously from one membrane (or cell) to another in a single direction. Conversely, antiport rules involve a bidirectional exchange, where objects move in opposite directions between two membranes. This work examines TLPS utilizing communication rules based on symport and antiport mechanisms. These rules are denoted as (i, h/k, j), where i and j range over the set {0, …, d} with i ≠ j, and h, k are multisets over the object alphabet Γ, with |hk| > 0. If both h and k are non-empty, the rule is classified as an antiport rule; otherwise, it is a symport rule. A rule is applicable when multiset h exists in membrane i and k exists in membrane j.

Symport and antiport rules are expressed as follows:

Symport rule:

(2) $$R_i={\left[u\right]}_i{\left[\right]}_j\to {\left[\right]}_i{\left[u\right]}_i$$where $\mathrm{u}$ is an object in membrane or cell $i$ is transferred to membrane or cell $\mathrm{j}$. $R_i$ represent rule number.

Antiport rule:

(3) $$R_i={\left[u\right]}_i{\left[v\right]}_j\to {\left[v\right]}_i{\left[u\right]}_i.$$In this scenario, object u is transferred from membrane i to j, while simultaneously object v moves from membrane j to i. This mechanism enables a bidirectional transfer of objects between membranes.

The proposed TPLS algorithm used in this study is based on the biological technique of modular processing. The proposed TLPS in this study is designed to use five primary membranes, where each membrane contains a key element representing shift schedules. The five membranes interact with each other based on a set of predefined constraints as presented in Table 1. The interactions between the five membranes ensures a final feasible and best case scheduling or roster of nurses. Figure 1 illustrates the flow chart of the TLPS algorithm proposed in this study.

Step 1: Five membranes are created, each corresponding to a table: the Main membrane, Membrane of off shift, Membrane of morning shift, Membrane of evening shift, and Membrane of night shift following the TLPS framework. These tables collaborate to develop a feasible nurse schedule for the Malaysia Medical Center.

Step 2: The membranes of shifts or tables are populated with objects representing different shifts as inputs, while the Main membrane generates the final output. night shifts, off-day shifts, morning shifts, and evening shifts are randomly allocated to the membranes as in Fig. 2.

Step 3: Communication rule1 is applied between the membranes or tables, considering specific conditions (hard constraints). Connections between membranes are managed through antiport, symport, or uniport mechanisms, enabling the transfer of shifts between tables according to the defined rules. While communication rule 2 is applied between the membranes to satisfy soft constraints as much as possible as in Fig. 3.

Each gap junction is connected through antiport, symport, or uniport mechanisms, enabling the transfer of shifts between membranes or tables according to defined rules and conditions. These mechanisms play a crucial role in facilitating the redistribution of shifts within the system.

Move acceptance strategy

TLPS algorithm uses an implicit and deterministic move acceptance approach, this is guided strictly by the rules of constraint satisfaction and penalty minimization. Each move is represented by the transfer of shifts between membranes, using the antiport, symport, or uniport mechanisms. The transfers of shifts between membranes is not random, and are executed based on the defined conditions of the hard and soft constraints. All shifts that satisfy all the hard constraints are accepted and maintained. All shifts that lead to a reduction in penalty cost are implicitly accepted. The algorithm continuously checks for constraints compliance. The system effectively accepts moves that improve the roster’s quality by reducing its penalty score.

This study applies the proposed TLPS algorithm on a schedule requirement of the UKMMC for a total of five different departments, these are the surgery, CICU (Cardiac Intensive Care Unit), MD1 (Medical Ward 1), GICU (General Intensive Care Unit), and N50 departments. The required nurses and shifts for each department are ascertained, and schedules are generated for the required nurses in each department accordingly. The generated shifts are cross examined to ensure if each shift schedule covers the demand of the departments as required. Where the schedules are lacking in meeting the demand requirements, the shifts are reiterated until the schedule allocation meets the requirement. Once a schedule is determined to meet the required number of nurses in every shift, the schedule is checked to ensure compliance with hard and soft constraints. Figure 4 illustrates the flowchart of the proposed TLPS algorithm process for NRP scheduling. Table 4 shows the penalty weight assigned to each constraint. Also, following is a detailed pseudocode approach to the NRP scheduling algorithm.

Table 4:

Penalty weight assigned to each constraint.

Soft constraint	Weight	Penalty function	Violation measurement factor
Give fair number of working days and days off to all the nurses.	100	Linear	Number of days when there was an insufficient number of nurses to meet the minimum workday requirements
Give each nurse at least one day off on the weekends during the scheduled period.	100	Linear	Number of days when there is no day off on weekends
Give four consecutive morning shifts followed by one day off.	10	Linear	Four consecutive morning shifts
Give four consecutive evening shifts followed by one day off.	10	Linear	Four consecutive evening shifts
Give an evening shift after the day off, followed by a night shift.	1	Linear	Number of morning shifts after four consecutive night shifts followed by two days off

DOI: 10.7717/peerj-cs.3492/table-4

Input: Days, Nurses, Shifts, Constraints (Hard and Soft)

Output: Feasible and optimized nurse schedule

Begin

  Step 1: Initialization

  Initialize membrane structure M

  Initialize the population of candidate schedules within membranes

  Define communication rules for information exchange between membranes

  Set termination condition ← False

  Step 2: Evolution Process

  While (termination condition == False) do

    Scatter initial shifts randomly into elementary membranes

    For each elementary membrane mi in M do

      Determine the number of iterations for mi

      For i = 1 to d do

        Execute communication rules between membranes

        Update local schedules within mi

     End For

   End For

   Step 3: Combine results

   Collect and output the best main schedule candidate

End While

Step 4: Feasibility Phase—Satisfy Hard Constraints

While (no feasible solution exists) do

  For i = 1 to 8 do

    Apply communication rules that address hard constraints

    Calculate penalties due to hard constraint violations

    If (all hard constraints are satisfied) then

      feasible solution ← True

    End If

  End For

End While

Step 5: Optimization Phase—Minimize Soft Constraint Violations

For j = 1 to 5 do

   While (penalties due to soft constraint violations remain high) do

     Apply communication rules that reduce soft constraint penalties

     Check that all hard constraints remain satisfied

     Compute new penalties for current schedule

     If (new penalties < previous penalties) then

       penalties ← new penalties

     Else

       penalties ← previous penalties

     End If

   End While

 End For

 Step 6: Final Output

 Output the optimized main schedule with minimal soft constraint violations

End

Experimental setup

The proposed TLPS algorithm is designed and implemented on nurse scheduling for five departments of the UKMMC. The scheduling experiment was carried out for an 11 to 73 nurses shift for a period of 2 weeks. Performance of the proposed algorithm is evaluated based on the penalties of the performance of the algorithm. The TPLS algorithm is implemented using the C# programming language, and experiments are carried out using Intel platform with a 2.66 GHz Core TM processor and 4 GB of RAM. The penalty weights used range from 194 to 1,280, with execution time between 0.005 and 0.041 s.

Comparison of genetic algorithm, harmony search algorithm, and TLPS using the UKMMC dataset

The genetic algorithm (GA) is a metaheuristic inspired by the principles of natural selection, operating on a population of candidate solutions that evolve through selection, crossover, and mutation. In contrast, the harmony search algorithm (HSA) is a more recent bio-inspired technique that draws its analogy from the musical process of improvisation, where harmony is iteratively refined to achieve an optimal composition. In this study, both GA and HSA, along with the TLPS, are evaluated on the UKMMC dataset as proposed by Hadwan et al. (2013).

In this study the performance of HSA and GA on the UKMMC dataset was compared with TLPS algorithm performance. Table 6 shows the main details about these comparisons.

The results of the experiments indicate a significant performance improvement using the TLPS algorithm, especially compared to previously proposed algorithms. The resulting penalty weights as seen using the TLPS algorithm indicate good and feasible performance of the TLPS algorithm for NRP. Also, not only is there a significant lower total penalty weight, but the time taken for the construction of the schedules is significantly lower and desirable patterns obtained from applying TLPS are improved than both of the desirable patterns obtained by GA and HSA. Figure 7 illustrates the relation between number of nurses and the objective functions of the compared algorithms and the proposed approach. Figure 8 illustrates the relation between number of nurses and the construction time of the proposed approach compared to other algorithms: GA and HSA. Figure 9 illustrates the relation between number of nurses and the desirable patterns for the proposed approach and compared algorithms; GA and HSA

Based on the results presented in Table 6, TLPS demonstrates the best performance across the critical metrics; construction time, total penalty, and desirable patterns. The TLPS approach achieved a total penalty of 3,112 across all the tested departments, this is significantly lower than the penalties incurred by both GA and HSA which have 13,561 and 3,127 respectively. A lower penalty indicates a roster adheres to soft constraints effectively which leads to efficient nurse schedules. The proposed TLPS algorithm also significantly reduced the construction time, where it constructed the roster in a total time of 0.092 s for all the required departments. Compared to the construction time of both GA and HSA having 1,397.8 and 897.6 s, this is a significantly improved construction time.

Desirable patterns (DP) relate to the quality and optimization of generated nurse rosters. DP considers the context of three categories of shift patterns, these are prohibited, undesirable, and desirable patterns. Prohibited patterns are the shift patterns which violate hard constraints, undesirable patterns adhere to hard constraints but do not efficiently adhere to soft constraints, and DP adhere to both hard and soft constraints. DP are generated or calculated with each successful minimization of soft constraints violations after hard constraints have been successfully adhered to. The more soft constraints violations have been minimized the more DPs are created and counted, as illustrated in Table 3. The proposed TLPS algorithm also resulted in a total of 167 desirable patterns (DP) across all the departments. This DP figure is significantly higher than the DP achieved in both GA and HSA which have 52 and 70 respectively. A higher number of DP indicates a better quality of generated roster; adhering to preferred requirements of scheduling.

Recommendation and future work

This study recommends further exploration of the proposed TLPS algorithm, further applications should include experimenting on other datasets and healthcare facility contexts. This will ensure the scalability of the TLPS algorithm is tested for suitability in different NRP. This study also recommends researching and investigating the applicability and feasibility of the TLPS algorithm on other scheduling problem contexts outside of NRP. Ultimately, the potential of the TLPS algorithm could be enhanced through more extensive testing across different datasets, leading to even more robust and scalable solutions for workforce management.