A hybrid scheduling approach for mega event transportation: integrating harmony search and black widow optimization

Designed transportation programs

The study by Khan & Shambour (2023) examined the transportation process for a group of pilgrims traveling to Muzdalifah. They found that the standard unit of time for this process was 90 min. The total duration for assigning all the pilgrims to the Muzdalifah area was divided into seven time slots:

• Three time slots before midnight.

• Three time slots after midnight.

• One time slot after dawn.

This segmentation of the transportation period into different time slots helps facilitate the efficient management and scheduling of pilgrim movement to Muzdalifah. It allows for better coordination and a more organized, systematic assignment process.

Based on the arrival and departure timings at Muzdalifah, the researchers identified five main transportation programs:

• Main Program 1: Pilgrims arrive before midnight and depart before midnight.

• Main Program 2: Pilgrims arrive before midnight and depart after midnight.

• Main Program 3: Pilgrims arrive before midnight and depart after dawn.

• Main Program 4: Pilgrims arrive after midnight and depart after midnight.

• Main Program 5: Pilgrims arrive after midnight and depart after dawn.

From these five main programs, a total of 27 sub-programs were derived based on the distribution of pilgrims across different time slots. Table 1 delineates all sub-programs corresponding to each main program. Specifically, six sub-programs were derived from Main Program 1, nine sub-programs from Main Program 2, three sub-programs from Main Program 3, six sub-programs from Main Program 4, and three sub-programs from Main Program 5. These sub-programs are designed to streamline the scheduling process for pilgrims according to their specific arrival and departure time slots.

Problem formulation

To ensure the effectiveness of the proposed algorithm in achieving its objectives of providing diverse transportation programs to a large number of pilgrims, several constraints were implemented to guide its functionality. These constraints were designed to assist the algorithm in generating programs that offer a wide range of options, enabling pilgrims to select programs that best suit their individual preferences. This approach promotes inclusivity and enhances the overall satisfaction of pilgrims participating in the Hajj event.

There are two forms of constraints that control the program scheduling process: hard constraints, which must be satisfied in the final solution, and soft constraints, which allow for some relaxation or violation.

The problem is encoded using an assignment function (A) that maps various resources—such as pilgrim groups (PG), timeslots (T), Muzdalifah sites (S), and pilgrimage programs (P)—to their respective constraints. The representation scheme captures the relationships between these resources, ensuring that the constraints (both hard and soft) are respected during the solution generation process. The constraints are outlined as follows (Khan & Shambour, 2023):

Hard constraints:

H1. Each group of pilgrims is transported once. This is critical for ensuring fairness and avoiding any group being omitted or duplicated. The formula Eq. (1) expresses that for every pilgrim group PG_j , there is a unique assignment A to a Muzdalifah site S_i, meaning no pilgrim group is left unassigned or assigned more than once. (1)${\displaystyle A_{PG}^S=A_{P{G}_j}^{S_i}\forall j\in PG;i\in S}$ H2. Each timeslot contains only one pilgrim group, avoiding any conflicts or overlaps. This constraint is designed to prevent conflicts and overcrowding by ensuring that no two pilgrim groups are assigned to the same timeslot and site simultaneously. Equation (2) mathematically represents this by ensuring that for any timeslot T_t and site S_i, the assignment of pilgrim group PG_j does not overlap with another group PG_k. (2)${\displaystyle A_{P{G}_j}^{T_t,S_i}\ne A_{P{G}_k}^{T_t,S_i}t\in T;j\ne k;\forall j,k\in PG;i\in S}$ H3. The generated solutions must contain all main transportation programs. To ensure diversity in the transportation programs, this constraint enforces that the final scheduling solution contains a minimum percentage of each main program. Equation (3) demonstrates that each pilgrim group PG_j is assigned to a program P_i, with the assignment percentages meeting the minimum threshold x_i for each program i. The specified values x_i ensure that no program is underrepresented. (3)${\displaystyle A_{PG}^P=A_{P{G}_j}^{P_i}\forall i\in P;j\in PG}$ such that $X\left(A_{PG}^{P_i}\right)\ge x_i\forall i\in P$, where X indicates the assignment percentages of the main pilgrim groups. The x_i represents the percentage of the main program i, such that x_i=1 =0.02, x_i=2 =0.03, x_i=3 =0.05, x_i=4 =0.01, and x_i=5 =0.01.

Soft constraints:

S1. Pilgrims should be distributed among the main programs based on the preferred percentage for each main program. This soft constraint aims to align the distribution of pilgrims with preferred percentages for each program. Equation (4) expresses this by ensuring that the assignment percentage function Y for each program P_i closely matches the desired proportion y_i. While this constraint allows for some deviation, the goal is to approach the target distribution as closely as possible. (4)${\displaystyle Y\left(A_{PG}^{P_i}\right)\cong y_i\forall i\in P}$ such that y_i=1 = 0.2, y_i=2 = 0.5, y_i=3 = 0.2, y_i=4 = 0.01,  and y_i=5 = 0.09.

S2. All timeslots are occupied by pilgrim groups across all Muzdalifah sites. This constraint ensures that all available timeslots and sites are utilized, avoiding any unused resources. Equation (5) states that for each timeslot T_t and site S_i , there must be at least one assigned pilgrim group PG_j . This helps maximize the use of all available timeslots and locations, promoting efficiency. (5)${\displaystyle A_{P{G}_j}^{T_{t,}{S}_i}\ne \varnothing \forall t\in T;\forall i\in S;j\in PG}$ S3. Utilize each Muzdalifah site as much as possible by allocating the largest possible number of pilgrim groups. This soft constraint encourages the allocation of as many pilgrim groups as possible to each site. Equation (6) represents this by aiming to maximize the number of pilgrim groups nPG assigned to a site S_i. While some flexibility is allowed, the objective is to use each site to its full capacity. (6)${\displaystyle A_{PG}^{S_i}\cong Max\left(nPG\right)\forall i\in S}$

The minimization objective function serves as a metric to assess the quality of the generated solutions and guides in determining the most suitable distribution for transporting pilgrim groups to Muzdalifah sites. This objective function assigns a numerical value to each solution, reflecting the efficiency of the final solution. Since the objective of this study is to achieve the optimal distribution of transportation programs while adhering to as many constraints as possible, the costs associated with violating these constraints are determined based on their significance in attaining the final solution. The objective function, which accounts for the costs associated with violating both hard and soft constraints, is represented as illustrated in Eqs. (7)–(13): (7)${\displaystyle ObjectiveFunctionCost=Cost\left(HardConstraintsViolations\right)+Cost\left(SoftConstraintsViolations\right)}$ (8)${\displaystyle Cost\left(HardConstraintsViolations\right)=1,000\times \left(Numberofviolatedhardconstraints\right)}$ (9)${\displaystyle Cost\left(SoftConstraintsViolations\right)=Cost\left(S1Violations\right)+Cost\left(S2Violations\right)+Cost\left(S3Violations\right).}$ Where: (10)${\displaystyle Cost\left(S1Violations\right)=10\times \left(abs\left(Actualdistributionpercentageforeachmainprogram-Preferreddistributionpercentageforeachmainprogram\right)\right)}$ (11)${\displaystyle Cost\left(S2Violations\right)=5\times \left(Numberofunoccupiedtimeslots\right)}$ (12)${\displaystyle Cost\left(S3Violations\right)=Numberoftimeslotswithoutanewsub\text{-}programassignment.}$

The fitness function in this problem is defined as a minimization objective function that measures the quality of the solution. The function assigns costs to violations of both hard and soft constraints, with hard constraints carrying a higher penalty (e.g., a multiple of 1,000 per violation). Soft constraints are penalized based on the degree of deviation from the desired distribution or usage, with different weights applied (e.g., 10 for distribution discrepancies, 5 for unoccupied time slots). The fitness function guides the algorithm toward minimizing these costs and thus improving the quality of the solution. Accordingly, the objective function is represented as follows: (13)${\displaystyle Min\sum _{h=1}^3{n}_{v_{H_h}}\times W_{H_h}+\sum _{s=1}^3{n}_{v_{S_s}}\times W_{S_s}}$ where n_vHh represents the violation times for each of the hard constraints (H₁, H₂, H₃), n_vSs denotes the violation times for each of the soft constraints $\left(S_1,S_2,S_3\right),$ W_Hh and W_Ss indicate the violation cost values for the hard and soft constraints, respectively.

Harmony Search algorithm

The HS algorithm is an evolutionary algorithm inspired by the principles of biological evolution found in nature, as proposed by Geem, Kim & Loganathan (2001). Its mechanism involves maintaining a set of solutions throughout the search process, where these solutions collaborate to generate a new solution in each iteration.

• HS algorithm operators are (Geem, Kim & Loganathan, 2001; Shambour et al., 2014):

  • Harmony initialization: Producing a starting population of harmonies at random.

  • Harmony memory consideration: Determining the procedure for constructing a new harmony from either the HM or through random search.

  • Pitch adjustment: Adjusting the content or values of specific elements in a new harmony.

  • Harmony memory update: Updating the contents of the HM by including the best harmonies in the HM according to their fitness values.

• Characteristics of the HS algorithm (Alia & Mandava, 2011; Shambour, 2018):

  • Ease of implementation.

  • HS uses a stochastic search technique influenced by music improvisation.

  • Pitch modification and harmony memory consideration are two ways the algorithm strikes a balance between exploration and exploitation.

  • HS is capable of handling both discrete and continuous optimization problems and does not require gradient information.

  • It has been effective in solving a variety of optimization issues.

Black Widow Optimization algorithm

The BWO algorithm is an evolutionary algorithm inspired by the real-life mating process of black widow spiders in nature, as proposed by Hayyolalam & Pourhaji Kazem (2020). Spiders share information, including their positions or attributes, to facilitate collaboration and information exchange. This collaborative effort enables spiders to collectively generate a new solution or update their positions for subsequent iterations.

• The BWO algorithm’s operators are Hayyolalam & Pourhaji Kazem (2020):

  • Population initialization: Generating an initial population of potential solutions randomly, often referred to as “spiders.”

  • Procreation: determining which spiders will be selected for the next round of reproduction or survival in the subsequent generation process.

  • Cannibalism: eliminating fewer fit spiders and promoting the survival of stronger ones.

  • Mutation: introducing small stochastic changes to the genetic makeup of spiders within the population.

• Characteristics of the BWO algorithm (Abu-Hashem & Shambour, 2024; Madhiarasan, Cotfas & Cotfas, 2023):

  • Ease of implementation.

  • Efficiently navigate the search space, allowing for quicker identification of optimal or near-optimal solutions.

  • Ability to avoid local optima by circumventing the problem of getting trapped in local optima.

  • Exhibiting satisfactory accuracy in finding solutions.

  • Reduced complexity.

Proposed HSBWO approach

The proposed approach in this study is a modification of the algorithm presented by Khan & Shambour (2023). This new approach combines the HS algorithm and the BWO algorithm to create a hybrid approach called HSBWO. The primary goal of this hybrid approach is to optimize the scheduling procedures for transportation programs in the Muzdalifah area. By leveraging the strengths of both algorithms, HSBWO aims to deliver a complete solution that accurately represents the scheduled assignment programs for all locations in the Muzdalifah area.

Step 1. Initializing the algorithm and problem parameters

The initialization step involves determining the parameters for the scheduling programs. These parameters include the number of locations, main programs, sub-programs, distribution ratios, timeslots, and other relevant factors. Additionally, specific parameters related to the algorithm itself are initialized. The HSBWO algorithm parameters comprise the following:

• Harmony memory size: refers to the number of solutions stored in the memory of the HSBWO algorithm.

• Harmony memory consideration rate (HMCR): HMCR is a parameter that determines the balance between adopting solutions from HM or through random search. Where 0 ≤ HMCR ≤ 1.

• Pitch adjustment rate (PAR): The PAR is a parameter used to facilitate the modification of previously chosen pilgrim groups from the assigned programs stored in the HM. It provides the possibility of generating new harmony that is adjacent to the previous solutions, thereby intensifying the search in the vicinity of the current solutions. The pitch adjustment operator is defined as follows:

where $U\left(0,1\right)$ represents a uniform random distribution ranging between 0 and 1. The procedures P1, P2, and P3 are detailed as follows:

• P1: Choosing two random timeslots from the currently scheduled programs and identifying the subprograms associated with the chosen timeslots.

• P2: Removing the selected subprograms from the current schedule and making their timeslots empty.

• P3: Developing one or more subprograms that cover empty timeslots.

Step 2. Initialization of HM

During the initialization of the HM, the algorithm generates initial solutions (denoted as H) for assigned programs using the greedy search technique. These solutions are then stored in HM as the starting set of solutions as shown below: ${\displaystyle HM=\left[\begin{array}{cccc}\hfill h_1^1\hfill & \hfill h_2^1\hfill & \hfill .....\hfill & \hfill h_N^1\hfill \\ \hfill h_1^2\hfill & \hfill h_2^2\hfill & \hfill .....\hfill & \hfill h_N^2\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill h_1^{HMS}\hfill & \hfill h_2^{HMS}\hfill & \hfill .....\hfill & \hfill h_N^{HMS}\hfill \end{array}\right]\left[\begin{array}{c}\hfill f\left(H^1\right)\hfill \\ \hfill f\left(H^2\right)\hfill \\ \hfill ⋮\hfill \\ \hfill f\left(H^{HMS}\right)\hfill \end{array}\right]}$

Step 3. Improvisation process

The improvisation process comprises the following stages:

Stage 1. Applying the conventional procedure of the HS algorithm to generate a New Harmony (NH) by utilizing the HS algorithm operators (i.e., HMC and PA).

Stage 2. Applying the cannibalism operation of the BWO algorithm that relies on eating members of the same species. The following are the steps of the cannibalism process in detail:

• Selection of harmonies: In this step, two solutions (harmonies) are randomly selected from the population. These harmonies are labeled as H_Y and H_Z.

• Swap mutation process: A swap mutation is performed between the selected harmonies where some components of these two harmonies are exchanged. The number of times this swap occurs depends on a predefined parameter called the number of cannibalism (nCann).

• Generation of new harmonies: After performing the swap process, two new harmonies H_Y′ and H_Z′ are created. This process introduces variability into the population and helps prevent the algorithm from getting stuck in local optima.

Mathematically, the swap mutation between H_Y and H_Z for a specified nCann, is represented as follows: ${\displaystyle \left[H_Y^{\prime },H_Z^{\prime }\right]=Mutate\left(H_Y,H_Z,nCann\right)}$

This swapping procedure of harmony memory (HM) components aims to maintain diversity among harmonies and reduce the likelihood of premature convergence by introducing differences into the solution space. The proposed combination of the HS and BWO algorithms achieves a balance between intensification and diversification in the search process, resulting in more robust optimization performance.

Step 4. Update HM

Newly created harmonies (NHs) undergo an evaluation process using an objective function (F). Only harmonies with good qualities will be kept in the HM. The updated rule is given as follows: ${\displaystyle HM=best\left(\left[HM,F\left(HM\right)\right]\cup \left[NHs,F\left(NHs\right)\right]\right)}$ where best selects the harmonies with the highest fitness values.

Step 5. Stopping Criterion

Indicating whether to continue or end the improvisation cycle based on a predetermined condition, such as the number of evaluations (NE). The basic flowchart of the proposed HSBWO algorithm is given in Fig. 1, and the pseudo-code algorithm is provided in Algorithm 1 .

Experimental design

As shown in Table 2, eight scenarios with various parameter settings were constructed for the comparison algorithm. Thirty trials were carried out for each scenario with different population sizes of 5, 20, 50, and 100. It is worth noting that the parameter setting for the BWO algorithm “cannibalism rate” was set to 0.44, as recommended in Hayyolalam & Pourhaji Kazem (2020). These experimental settings were chosen to examine the exploration and exploitation search efficacy compared to HS (Khan & Shambour, 2023), BWO (Hayyolalam & Pourhaji Kazem, 2020), and HSBWO algorithms. Several statistical measurements were employed in the comparison of the studied algorithms, including the mean, standard deviation, and best and worst fitness values. All experiments were coded using MATLAB 2020b, which was run on Windows 11 64-bit 12th Gen Intel(R) Core (TM) i7-12650H 2.30 GHz and 24 GB of RAM.

Experimental results

Simulation results with a population size of five

The experimental outcomes for a population size of five, as detailed in Table 3, reveal that the proposed HSBWO algorithm ( Algorithm 1 ) consistently outperforms other algorithms. It achieves the highest average fitness scores across all test scenarios (highlighted in bold), demonstrating its superior optimization capabilities. Notably, the HS algorithm also surpasses the performance of the BWO algorithm in all scenarios, reinforcing its effectiveness within the hybrid approach.

Table 3:

Statistical results of fitness values for 30 experimental tests with a population size of five.

Scenarios		Sc1	Sc2	Sc3	Sc4	Sc5	Sc6	Sc7	Sc8
HSBWO	Mean	1,108	1,092.3	1,133.8	1,248.6	1,255.2	1,326.3	1,689.6	2,037.6
	Std	335.1	309.7	328.9	447.2	461.1	445.7	504.5	375.8
	Best	866	893	925	931	897	973	1,068	1,173
	Worst	1,995	2,065	1,971	2,063	2,095	2,136	2,198	2,268
HS	Mean	2,420.5	2,437.3	2,410.1	2,430.7	2,382.1	2,422.5	2,349.3	2,376.1
	Std	14.701	16.735	16.8	11.4	15.0	14.7	43.3	111.4
	Best	2,389	2,397	2,384	2,410	2,349	2,399	2,289	1,792
	Worst	2,445	2,465	2,441	2,449	2,407	2,454	2,553	2,430
BWO	Mean	2,517.6	–	2,646.9	–	2,581.8	–	2,706.2	–
	Std	25.8	–	37.3	–	41.9	–	31.7	–
	Best	2,467	–	2,589	–	2,488	–	2,646	–
	Worst	2,563	–	2,720	–	2,655	–	2,759	–

DOI: 10.7717/peerjcs.2526/table-3

Notes:

The best mean results are highlighted in bold font.

Figure 2 provides a visual comparison of the best convergence trends across various parameter settings for the algorithms. These results highlight the efficiency and reliability of the HSBWO in providing higher solution qualities.

To further validate these findings, an ANOVA statistical analysis was conducted. The goal was to determine if there were statistically significant differences between the fitness results of the compared algorithms. The hypotheses for the ANOVA test were as follows:

h0: The average fitness values of the compared algorithms are equal (i.e., μ1 = μ2 = μ3, where μ is the mean).

h1: At least one algorithm’s average fitness value is significantly different from the others.

The ANOVA results, shown in Table 4, indicate a significant difference in the mean fitness values, as the P-values are consistently lower than the 0.05 significance threshold. This statistical evidence leads to the rejection of the null hypothesis (h0) and supports the alternative hypothesis (h1), confirming that the fitness results differ significantly between the algorithms.

• Simulation results with a population size of 20

When the population size is increased to 20, the experimental results, as presented in Table 5, continue to demonstrate the superiority of the HSBWO algorithm. It achieves the highest average fitness scores in all scenarios (highlighted in bold), further solidifying its robustness and effectiveness. The HS algorithm also maintains its advantage over the BWO algorithm, consistently ranking second.

Table 4:

ANOVA test results for the compared algorithms with a 5-population size.

Scenarios	F-Value	P-Value	F-Critical
Sc1	4.93E+02	3.50E−48	3.10E+00
Sc2	5.64E+02	1.43E−31	4.01E+00
Sc3	5.43E+02	7.35E−50	3.10E+00
Sc4	2.56E+02	3.37E−37	3.10E+00
Sc5	2.15E+02	2.32E−34	3.10E+00
Sc6	2.15E+02	2.06E−34	3.10E+00
Sc7	9.30E+01	2.47E−22	3.10E+00
Sc8	6.59E+01	3.72E−18	3.10E+00

DOI: 10.7717/peerjcs.2526/table-4

Table 5:

Statistical results of fitness values for 30 experimental tests with a population size of 20.

Scenarios		Sc1	Sc2	Sc3	Sc4	Sc5	Sc6	Sc7	Sc8
HSBWO	Mean	2,074.9	2,136.2	2,146.6	2,064.0	2,122.8	2,133.6	1,910.9	2,183.3
	Std	276.37	240.12	186.0	307.0	280.8	248.6	372.1	177.5
	Best	1,217	1,223	1,783	1,268	1,494	1,274	1,405	1,720
	Worst	2,336	2,337	2,353	2,358	2,337	2,379	2,387	2,428
HS	Mean	2,431.1	2,440	2,416.3	2,437.1	2,399.4	2,433.8	2,369.4	2,417.7
	Std	13.232	18.048	17.296	12.213	12.207	16.65	21.248	23.783
	Best	2,404	2,397	2,373	2,418	2,376	2,382	2,325	2,395
	Worst	2,454	2,468	2,454	2,460	2,424	2,457	2,427	2,488
BWO	Mean	2,679	–	2,660.1	–	2,656.5	–	2,658.3	–
	Std	31.3	–	30.847	–	28.717	–	26.361	–
	Best	2,620	–	2,565	–	2,591	–	2,592	–
	Worst	2,752	–	2,705	–	2,726	–	2,725	–

DOI: 10.7717/peerjcs.2526/table-5

Notes:

The best mean results are highlighted in bold font.

Figure 3 illustrates the best convergence curves for the algorithms under various parameter configurations, showcasing the HSBWO’s capability to navigate the solution space efficiently.

Again, ANOVA statistical analysis was employed to verify the significance of differences between the algorithms:

h0: The average fitness values of the compared algorithms are equal (i.e., μ1 = μ2 = μ3, where μ is the mean).

h1: At least one algorithm’s average fitness value is significantly different from the others.

The results, shown in Table 6, confirm significant differences in the mean fitness values, as indicated by P-values below 0.05. This reinforces the conclusion that the algorithms differ significantly in performance, leading to the rejection of the null hypothesis (h0) and acceptance of the alternative hypothesis (h1).

Simulation results with a population size of 50

As the population size increases to 50, the experimental data in Table 7 continue to validate the dominance of the HSBWO algorithm. It consistently achieves the highest average scores across all tested scenarios (highlighted in bold), demonstrating its scalability and effectiveness. The HS algorithm maintains its second-place performance, further validating the hybrid approach.

Figure 4 shows the convergence performance for the algorithms across different parameter settings, illustrating the HSBWO’s strong performance in navigating solution space.

To assess the statistical significance of these results, ANOVA analysis was again applied:

h0: The average fitness values of the compared algorithms are equal (i.e., μ1 = μ2 = μ3, where μ is the mean).

h1: At least one algorithm’s average fitness value is significantly different from the others.

The ANOVA results in Table 8 indicate significant differences in the mean fitness values, as evidenced by P-values below 0.05. This leads to the rejection of the null hypothesis (h0) and supports the alternative hypothesis (h1), confirming that the algorithms produce significantly different fitness results.

Table 6:

ANOVA test results for the compared algorithms with a 20-population size.

Scenarios	F-Value	P-Value	F-Critical
Sc1	1.04E+02	1.74E−23	3.11E+00
Sc2	4.37E+01	1.51E−08	4.01E+00
Sc3	1.66E+02	2.14E−30	3.10E+00
Sc4	8.78E+01	1.36E−21	3.10E+00
Sc5	1.86E+02	3.47E−32	3.10E+00
Sc6	1.00E+02	2.60E−23	3.10E+00
Sc7	1.35E+02	2.10E−27	3.10E+00
Sc8	9.16E+01	3.91E−22	3.10E+00

DOI: 10.7717/peerjcs.2526/table-6

Table 7:

Statistical results of fitness values for 30 experimental tests with a population size of 50.

Scenarios		Sc1	Sc2	Sc3	Sc4	Sc5	Sc6	Sc7	Sc8
HSBWO	Mean	2,277	2,258.9	2,263.8	2,269.8	2,192.3	2,232.7	1,969.8	2,198.5
	Std	101.8	87.28	126.3	85.6	226.2	201.8	334.9	245.9
	Best	2,066	2,099	1,732	2,131	1,697	1,725	1,490	1,649
	Worst	2,432	2,434	2,439	2,434	2,477	2,465	2,500	2,493
HS	Mean	2,431.5	2,441.5	2,429.6	2,450	2,417.9	2,446.6	2,417.1	2,449.2
	Std	19.58	15.22	19.01	16.25	15.65	21.84	15.72	13.2
	Best	2,389	2,398	2,365	2,406	2,390	2,375	2,390	2,422
	Worst	2,458	2,470	2,460	2,469	2,454	2,474	2,468	2,471
BWO	Mean	2,730.2	–	2,715	–	2,701.4	–	2,700.3	–
	Std	17.86	–	18.84	–	24.13	–	21.48	–
	Best	2,684	–	2,681	–	2,659	–	2,661	–
	Worst	2,761	–	2,762	–	2,762	–	2,749	–

DOI: 10.7717/peerjcs.2526/table-7

Notes:

The best mean results are highlighted in bold font.

Simulation results with a population size of 100

Finally, with a population size of 100, the experimental results displayed in Table 9 confirm the superior performance of the HSBWO algorithm. It consistently achieves the highest average fitness scores across all scenarios (highlighted in bold), highlighting its capability to manage larger populations effectively. The HS algorithm also continues to outperform the BWO algorithm, ranking second in all tests.

Figure 5 shows the best convergence results of the comparison algorithms across different parameter-setting scenarios.

Furthermore, ANOVA statistical analysis is utilized to check whether the results obtained from the compared algorithms differ significantly, as follows:

h0: The average fitness values of the compared algorithms are equal (i.e., μ1 = μ2 = μ3, where μ is the mean).

h1: At least one algorithm’s average fitness value is significantly different from the others.

The ANOVA results in Table 10 demonstrate that the mean fitness values are significantly different from one another since the resulting P-values are less than the significance level of 0.05. This indicates that there is a significant difference in the fitness mean results and that not all mean fitness values are equal, leading to the rejection of the null hypothesis (h0) and acceptance of the alternative hypothesis (h1).

Table 8:

ANOVA test results for the compared algorithms with a 50-population size.

Scenarios	F-Value	P-Value	F-Critical
Sc1	4.57E+02	7.30E−46	3.11E+00
Sc2	1.47E+02	2.71E−17	4.01E+00
Sc3	1.77E+02	1.00E−36	2.71E+00
Sc4	6.61E+02	2.52E−53	3.10E+00
Sc5	4.20E+02	1.99E−45	3.10E+00
Sc6	9.26E+01	2.82E−22	3.10E+00
Sc7	6.69E+02	1.51E−53	3.10E+00
Sc8	9.29E+02	2.00E−59	3.10E+00

DOI: 10.7717/peerjcs.2526/table-8

Table 9:

Statistical results of fitness values for 30 experimental tests with a population size of 100.

Scenarios		Sc1	Sc2	Sc3	Sc4	Sc5	Sc6	Sc7	Sc8
HSBWO	Mean	2,363.3	2,348.2	2,346.1	2,352.0	2,269.2	2,368.1	2,032.6	2,272.7
	Std	57.913	61.672	116.9	67.7	233.7	64.5	177.6	202.9
	Best	2,210	2,193	1,829	2,175	1,722	2,242	1,803	1,772
	Worst	2,474	2,455	2,471	2,462	2,517	2,490	2,502	2,510
HS	Mean	2,438.1	2,449.8	2,434.1	2,453.7	2,437.5	2,462.4	2,447.1	2,480.5
	Std	13.849	16.138	27.56	19.27	21.27	20.385	91.15	24.98
	Best	2,402	2,415	2,345	2,375	2,395	2,405	1,978	2,405
	Worst	2,461	2,474	2,466	2,482	2,464	2,487	2,499	2,513
BWO	Mean	2,767.1	–	2,748	–	2,741.3	–	2,728.5	–
	Std	30.23	–	21.98	–	23.65	–	21.52	–
	Best	2,689	–	2,711	–	2,680	–	2,683	–
	Worst	2,816	–	2,781	–	2,783	–	2,768	–

DOI: 10.7717/peerjcs.2526/table-9

Notes:

The best mean results are highlighted in bold font.

Table 10:

ANOVA test results for the compared algorithms with a 100-population size.

Scenarios	F-Value	P-Value	F-Critical
Sc1	8.68E+02	7.80E−57	3.11E+00
Sc2	7.19E+01	1.26E−11	4.01E+00
Sc3	1.66E+02	2.14E−30	3.10E+00
Sc4	8.78E+01	1.36E−21	3.10E+00
Sc5	1.86E+02	3.47E−32	3.10E+00
Sc6	8.03E+01	1.71E−20	3.10E+00
Sc7	1.00E+02	2.60E−23	3.10E+00
Sc8	1.35E+02	2.10E−27	3.10E+00

DOI: 10.7717/peerjcs.2526/table-10

Results discussion

The proposed HSBWO algorithm demonstrates superior performance in achieving lower fitness values across various scenarios and population sizes, highlighting its potential for optimizing convergence rates and generating high-quality solutions compared to HS and BWO. The success of the HSBWO algorithm can be attributed to its combination of strong exploration abilities inherited from the HS algorithm and robust exploitation capabilities utilized from the cannibalism process of the BWO algorithm, leading to more accurate and robust scheduling solutions for the transportation problem being optimized. Additionally, the results are further validated by an ANOVA test, which confirmed that the improvements achieved by the HSBWO algorithm are statistically significant.

Notably, in experiments with a population size of 5, the large cost reduction typically happens between 650 to 800 iterations. This is likely because the algorithm may enter a critical phase, where the algorithm transitions from exploration to exploitation, leading to rapid convergence to a feasible area of the solution space.

The effectiveness of the proposed HSBWO algorithm is underscored by the significant percentage improvements it demonstrates. Specifically, for a population size of 5, enhancements over the HS algorithm ranged from approximately 14.2% to 55.2%, while improvements over the BWO algorithm ranged from 37.6% to 56.0%. For population size 20, improvements over HS ranged from 9.7% to 19.4%, and over BWO from 6.4% to 28.1%. For population size 50, improvements over HS ranged from 6.3% to 18.5%, and over BWO from 16.6% to 27.0%. Lastly, for population size 100, improvements over HS ranged from 3.1% to 18.5%, while improvements over BWO ranged from 14.6% to 25.5%. These findings demonstrate that the HSBWO algorithm consistently outperforms both the HS and BWO algorithms across different population sizes and scenarios.

However, it is worth noting that all algorithms exhibit degraded performance in optimizing solutions as population sizes increase. This phenomenon occurs because the search space becomes larger, negatively impacting the convergence speed towards promising areas of the solution space. Furthermore, the results reveal that both HSBWO and HS algorithms perform better under scenario 5 (with settings of 0.9 HMCR, 0.3 PAR, and 0.1 mutation rate) compared to other scenarios, indicating the effectiveness of these parameters in achieving better solution costs. Lastly, despite slight variations observed across different scenarios, all algorithms demonstrate stability in generating their solutions, indicating their reliability and consistency in addressing optimization problems.