Approach food.

The slime mould spontaneously approaches the food according to the smell of the food, and the following mathematical expression can express its expansion pattern. Fig 1 shows how the slime mold search searches for the optimal solution and the solution to the CVRP problem.

(1) (2) Where is a parameter in the range of [-a, a], and the formula is shown in Eq (2); where the value of t represents the current number of iterations, and max_t represents the maximum iterations as shown in Eq (3); is a linear decreasing process, and the absolute value takes values from 1 to 0; represents the current individual location with the highest food odour; represents the position of slime mould; and represent two individuals randomly selected from all slime mould; represents the specific weight of slime mould.

(3)(4)(5)(6)

The Eq (4) denote the P-value, where S(i) represents the fitness of the mucilaginous individual, i ∈ (1,2,3,……, n); DF represents the best fitness obtained by the slime mould individual in all iterations. The Eq (5) simulates the relationship between food odour concentration and mucor vein width; where the condition represents the S(i) value in the first half of the optimal solution ordering, this condition serves to simulate the search pattern chosen by the mucor based on food concentration; the r is used to simulate the uncertainty in the form of mucor vein contraction, and the value is a random value in the interval [0,1]; the log contributes to reduce the rate of change of the numerical contraction. The bF value represents the optimal iteration value in the current iteration process; wF value is the worst value in the current iteration process; Smelllndex is the ranking of all iterations.

Wrap food.

The Eq (7) simulates the process of slime mould searching for food and updating its position after searching for food based on Eq (1) and combines Eq (5) to simulate the shrinkage of the tissue structure of slime mould veins. In the process of searching for food, the slime mould determines the search weight of the area by judging the concentration of the food odour between the areas. The higher the concentration of food, the stronger the bio-vibration of the bio-vibrator, the faster the flow of cytoplasm, and the thicker the intercellular venous route, the greater the search weight of the area; otherwise, it will turn to search other areas. (7) Where LB and UB represent the size of the upper and lower boundaries of the slime search process; rand denotes a random value in the interval [0,1], and how to determine the parameter z value need to be discussed according to specific experiments.

Oscillation.

The slime mold controls the thickness of their own venous network by their own oscillation fluctuations, the simulation process is shown in Eq (5). And the exploration process is realized by changing parameters such as , , according to the size of food concentration.

The values of and oscillate randomly between the intervals [-a, a] and [–1,1], and eventually converge to zero. In addition, when the slime mold explores the food based on the food concentration, some of the slime will still be separated to explore other unknown areas. It is worth mentioning that the harsh environments encountered during exploration are informative for simulating real situations, such as strong sunlight and dryness.

Cauchy mutation strategy.

In the original slime mould algorithm, the generation of the optimal position depends on the concentration of food odor when the slime mould searches the unknown area. When the population of individuals tends to aggregate due to the increasing number of iterations, the optimal individual will lack the ability to quickly jump out of the optimal local solution, which makes the algorithm prone to the premature phenomenon. Therefore, to strengthen the ability of SMA to jump out of the optimal local solution, this paper introduces the fusion Cauchy mutation strategy to perturb the current optimal solution to ensure that the algorithm has greater population diversity during local optimization.

The Cauchy variation originates from the Cauchy distribution. According to the Cauchy distribution mechanism, the two ends of the Cauchy distribution function curve are longer, indicating that it can make it easier for individuals to escape from local extremes, and a smaller peak will guide individuals to spend less time searching for the optimal location. Therefore, the Cauchy mutation operator is introduced into the slime mould algorithm to fully use its strong perturbation ability to control the current optimal solution. The formula is as follows: (8) (9)

Among them, rand1 is a random probability number that obeys the normal distribution. When rand1>0.5, the algorithm uses the Cauchy operator to mutate the optimal solution, and its powerful perturbation ability can greatly improve the diversity of the population around the optimal solution. When the optimal individual of the local extreme value is found, it can help it escape quickly to ensure the robust optimization of the algorithm; when rand1<0.5, the algorithm retains the current optimal solution.

Annealing mechanism.

The SMA-CSA algorithm proposed in this section is derived from the original SMA, and S2 Algorithm shows the pseudo-code of SMA-CSA. The annealing mechanism is introduced into the original SMA to improve the algorithm’s search and global optimization capabilities, the Metropolis sampling criteria as acceptance criteria, and control the annealing rate and minimum temperature, which form the algorithm’s structure in this paper. Compared with SMA, adding the annealing operator strengthens the robustness of the algorithm and reduces the probability of the algorithm falling into a local optimal solution during the search process. This method has also been extensively validated in solving problems such as TSP problems and 3D/2D fixed-outline floor planning [54, 55].

In SMA-CSA, whether to accept the current solution is determined by Eq (10): (10) (11) (12)

In Eq (10), Tc represents the current temperature, and the Metropolis sampling criterion PA(a) represents the possible probability that solution a is accepted. Each time a new solution is constructed, the current temperature is updated with Eq (11); where ε is a parameter that controls the rate of decrease of the current temperature, and T_n represents the temperature of the next iteration. It should be noted that in order to prevent the temperature from converging too fast to affect the quality of the solution, we need to introduce Eq (12) to adjust the current temperature; in Eq (12), T_p denotes the temperature at the last time when a feasible solution generated by SMA is accepted, where ω is a control parameter for the convergence rate, which is set to 0.998 in this paper.

The improved algorithm structure and steps are shown in Fig 2.

Comparison of benchmark function results between traditional algorithms.

The results of the fitness test between SMA-CSA and the traditional algorithm are shown in Table 5, where the bolded experimental data are the best results for the selected data. Among them, for SMA-CSA, F1-F4 and F7 can achieve the best results and show excellent global convergence performance. For F5, SMA-CSA is second only to HHO; the performance is close. And F6 is worse than EO and HHO, but the result is still satisfactory, so it can be ranked third.

Overall, SMA-CSA can obtain the global optimal solution in most cases and exhibits excellent convergence speed and local search capability.

For F9–F11 and F14, SMA-CSA can achieve optimal global convergence with the smallest standard deviation; for F12, it is worse than EO and HHO, can be ranked third; for F13, the global convergence performance is slightly inferior to HHO, ranking second and showing a better global search capability than the original SMA.

In addition, F15-F17 also can achieve optimal global solution, and have the smallest standard deviation and show excellent stability. For F21-F23, the average performance of SMA-CSA and SMA is equivalent, but SMA-CSA has the smallest standard deviation. Which also fully proves that SMA-CSA can significantly improve the equilibrium exploration ability and global search ability after adding the strategy of Cauchy mutation and simulated annealing. And Fig 3 shows the convergence curve of the traditional algorithm for some CEC2013 benchmark functions.

Comparison of benchmark function results between traditional algorithms with 60 dimensions.

In order to highlight the high-dimensional optimization performance of SMA-CSA, this paper further tests 23 benchmark functions in 60 dimensions, and the results are shown in Table 6. SMA-CSA achieves the number one global value in most cases, such as F1-F5, F7, F8, F10, F11, F15, F16, and F18, and the variance stabilizes. However, SMA-CSA is inferior to HHO in F6 and F13. SMA-CSA still has excellent solution accuracy and global optimization capabilities, even in high-dimensional tests.

Comparison of benchmark function results between improved algorithms.

In order to compare the comprehensive performance of SMA-CSA more fairly, other improved algorithms are compared in this paper, and the results are shown in Table 7. Among the 23 test functions, SMA-CSA can obtain the optimal global value in most cases, such as F1-F3, F7-F11, F14-F19, and F21-F23. Moreover, the variance tends to be stable. However, SMA-CSA is slightly inferior to L-SHADE in the F6, F12, and F13 test functions. Overall, even with the improved algorithm, SMA-CSA still has excellent solution accuracy and global optimization ability.

Problem description.

A mathematical model can describe the capacitated vehicle routing problem (CVRP): A distribution center O provides logistics services for N customers, where the number of logistics vehicles is M and the maximum load capacity of each vehicle is Q.

The logistics vehicles must provide logistics services for customers from the distribution center and return to the center after completing their tasks.

All logistics vehicles must serve customers at all nodes within their capacity Q limits, with no omissions.

The final capacity of the vehicle is linked to the number of goods to be delivered at each node and the number of nodes (customers) to be served by the vehicle on the route.

Building the CVRP mathematical model: (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) Where m is the number of vehicles; q_i represents the quantity demanded by the customer, i ∈ V; (q₀ represent the warehouse); c_ij represents the distance from customer i to customer j; c_0j indicates the distance from the warehouse to customer j; c_i0 represents the distance from customer i to the warehouse; If vehicle k visits client i, let y_ik = 1, otherwise y_ik = 0; If vehicle k continues to visit customer j after visiting customer i, let x_ijk = 1, otherwise x_ijk = 0.

And f₁ is the desired objective function, that is, the shortest total distribution path length; f₂ is the longest path for a single vehicle; f is the individual fitness; the constraint (4.4) indicates that there is one and only one vehicle from one node to another; (Eq 17) indicates that the transportation capacity of a vehicle must not exceed its own maximum carrying capacity; Eq (18) indicates that the starting point of a vehicle and the ending point of a vehicle are warehouses; Eq (19) ensures that each vehicle is visited (except warehouses); Eq (20) is used to eliminate subloops; Eq (21) is a range of parameter values; and finally Eq (22) is the problem of the minimum path we need to solve.

Hybrid SMA-CSA for CVRP.

The main content of this section is to design the application idea of SMA-CSA for the capacitated vehicle routing problem, and the algorithm structure and the steps are shown in Fig 5. Two standard benchmark datasets of CVRP are selected reference [64] and [65], and the results are compared with other algorithms to analyze the advantages and disadvantages of different algorithms for solving the problem. The first of these data sets has the number of instance nodes between 50–199; the second is distributed between 200–483. The distances between customer and customer nodes and between customer and warehouse nodes are measured using Euclidean distance.

1) Initialization. First, the algorithm initialization parameters are defined, including population size, maximum iterations, and crossover (C-R), and then the algorithm randomly generates the initial population of slime mould.

The formation of the optimal solution (X_i) for each individual slime mould is based on the following steps:

1. (Step1). Create a path that uses the warehouse as the starting point (zero points);
2. (Step2). select a customer from the customer list in a random non-replacement form;
3. (Step3). Add the selected customer to the route;
4. (Step4). Determine whether the total demand of all customers on the path is less than the vehicle capacity;
5. (Step5). Repeat step2 and step3, otherwise add the warehouse to the route;
6. (Step6). Add a route;
7. (Step7). Repeat steps 1–6 until all customers have been traversed;
8. So far, each individual’s solution (X_i) has been calculated, sorting the solution X_i of all slime mould, and calculating the weight of the corresponding solution, finally updating the optimal solution.

2) Crossover. When using the SMA-CSA algorithm to solve CVRP, introduce crossover (C-R) in the process of position updating to improve the quality of the solution. In the crossover, an individual is selected first, and then new positions P1 and P2 are randomly generated between the solution set, and the positions of the newly generated solutions are positioned between P1 and P2.

3) Local search. In order to expand the range of the solutions, three local search strategies are used, including point-swap, 2-opt, and 3-opt.

(1) Point swap strategy. As shown in Fig 6, in the current solution, two customers at different locations are randomly selected and swapped to generate a new domain solution.

(2) 2-opt search. As shown in Fig 7, similar to the point swap strategy, randomly generate two customer points at different locations, flip the path between the two customer points, and add their numbers to the new path; the path number before and after the two customer points remains unchanged.

(3) 3-opt search. As shown in Fig 8, Similarly, the 3-opt search strategy is similar to the 2-opt search strategy, in which three location client points are randomly selected for the path and swapped and flipped in turn.

4) Mutation. In order to ensure the global search ability of the algorithm, this paper uses three mutation operators with different selection probabilities to improve the diversity of the solutions. The first mutation randomly selects two customers in the CVRP solution with two paths and swaps the two customers in each of the two paths. The second mutation randomly selects two routes in the CVRP solution and swaps the customers within the two paths to form a new path. The third mutation is the 2-h-opt mutation and is one of the most effective mutations for complex problems [66].

First, the multi-vehicle path is converted into a single-vehicle path problem using the predefined probability M-R. Then, the 2-h-opt operator is used to find the twisted connections in the original path and reopen them to generate a new multi-vehicle problem.

Result of CVRP.

This section discusses the results of solving CVRP using SMA-CSA and analyzes its performance. Two standard benchmark datasets of CVRP are selected, and the results are compared with other algorithms to analyze the advantages and disadvantages of different algorithms for solving the problem. The algorithms used for comparison include SMA-CSA, ISOS, EACO [67], LNS-ACO [68], ILS–RVND [69, 70], GRELS [71], AGES [72], and HGPSO [73].

The test results of the selected cases are shown in Tables 13 and 14. In order to make the final results converge to the optimal value, two stopping criteria are designed in this paper. The first is that the result of the cases is equal to the optimal result BKS (best known), and the second is that the maximum iteration value of the algorithm is reached.

Secondly, the parameters related to SMA-CSA and the comparison algorithm are set, as shown in Table 12. Each case is executed ten times, respectively, and the optimal value is taken for comparison. The average gap between the actual value and the optimal value of the cases is then calculated.

In Table 13, only in the C1, C12, and C14 cases do the solutions of all variants obtain results consistent with the optimal values. In the C3, C6, and C8 cases, all the algorithms obtain results consistent with BKS, except for ILS-RVND. In C2 and C7, SMA-CSA, EACO, and LNS-ACO outperform ISOS and ILS-RVND. In terms of average solutions, among all algorithms, the average gap of SMA-CSA is significantly smaller than that of other algorithms, showing a better and more stable global search capability than other algorithms.

It can be seen from Table 14 that only in the GWKC5 example does SMA-CSA achieve the best results. In the cases of GWKC3, GWKC6, GWKC9, and GWKC17, actual results that are close to optimal results were obtained. In addition, it can be seen from the table that it seems that when the number of nodes n is large, the actual result of SMA-CSA is not particularly ideal; when n is small, it is close to the optimal result. In terms of average solution, among all algorithms, the average gap size of SMA-CSA is second only to AGES, and its performance is good.

In order to observe the results of SMA and SMA-CSA in multiple operations more intuitively, Fig 9 draws the boxplots of the C5 and GWKC4 examples. In 10 operations, SMA-CSA has better optimization results and a more robust mean than other algorithms.

In order to verify the effect of the article search strategy, the article calculates the results when SMA-CSA does not apply a local search strategy or crossover mutation strategy. It can be seen from Fig 10 that when the crossover and mutation operations are not applied, the difference in the algorithm optimization results is the most obvious; in the local search strategy, when 2-opt is not used, the algorithm optimization results are greatly affected.