FFA is a metaheuristic method based on agricultural fertility described in [30]. Farmers split their farms into sections based on soil quality, enhancing the soil quality in each region by adding particular materials. Due to their quality, these materials may have different types, and farmers add them to the soil at each visit; this quality is stored in memory next to each section. As a result, the best material is sent to the worst area of the farmland. In this approach, the best quality in each segment is termed local memory, and the best soil quality in all portions is called global memory. In other words, local memory saves the best solution in each segment, but global memory keeps the best solution across all search areas. These memories are employed to enhance the solutions in the ensuing optimization process. As a result, the component with the lowest quality is coupled with one of the global memory solutions. All the solutions in the search space are integrated with the solutions from the other parts. Following this step, each section’s answers are linked to one of the best global or local memory solutions. There are six general steps in the algorithm. The following is the first step.

Step 1: Initializing parameters used in the algorithm

The total population is obtained using Eq. (2) in the FFA algorithm. (2)N=k∗n

In Eq. (2), N is the total population in the problem space, and k represents the sections. Moreover, the number of solutions in each section is described by n., and k must be an integer greater than zero. Of course, k can be a number between 1 and N. However, the authors have stated [30] that if this value is initialized equal to 1 or greater than 8, it decreases the algorithm’s performance. Therefore, this number must be 2 ≤ k ≤ 8, and these results are obtained by trial and error. Eq. (3) is used to generate the search space randomly. (3)xij=Li+rand(0,1)×(Uj−Lj)

In Eq. (3), Ljis the lower bound and Uj is the upper bound of the variable x, and rand is by a function generates random integers between 0 and 1, and j = [1, …, D] represents the dimensions of x, and i equals the total population [1, …, N].

Step 2: Determination of soil quality in each part of the farm

After generating the initial population and evaluating each solution’s fitness, each section’s quality is assessed using Eqs. (4) and (5). (4)Sectionss=x(aj),j={1,2,…,4},a=n∗(s−1):n∗s,s={1,2,…,k}

In Eq. (4), x represents all solutions in the search space, s the number of parts, and j = [1, …, D] the dimension of the variable x. (5)FitSectionss=Mean(allFit(xji)inSections),i={1,2,…,n},s={1,2,…,k}

In Eq. (5), for each section of farmland, Fit_Section display the worth and quality of the solutions.

Step 3: Updating the memory

Some of the best solutions in each section are stored in local memory. In all areas of global memory, the optimal solution is stored. Eq. (6) provides the number of best local memories, whereas Eq. (7) provides the best global memories.

Step 4: Alteration of the soil quality

The worst section is paired with one of the global memory solutions to produce the most significant improvement at this step. It is expressed by Eqs. (8) and (9). (8)h=α∗rand(−1.1) (9)Xnew=h∗(Xij−XMGlobal)+Xijwhere, α is a parameter between 0 and 1. h is a decimal number obtained by Eq. (8) and Xnew is the new solution. XMGlobal is a random solution available in global memory, and Xij is the selected solution in the worst section. After making changes to the explanations of the worst section, the other sections’ solutions are combined with the complete search space solutions using Eqs. (10) and (11). (10)h=β∗rand(0.1) (11)Xnew=h∗(Xij−Xuj)+Xijwhere, β is a parameter between 0 and 1. Xuj is a random solution in the search space. Xij is a selective solution among all sections, except for the worst section. h is a decimal and Xnew is a new solution.

Step 5: Integrating soil

All solutions are integrated with one of the best solutions in the local or global memory, as illustrated by Eq. (12). (12)H={Xnew=Xij+ω1∗(Xij−BestGlobal(b)).Q>randXnew=Xij+rand(0.1)∗(Xij−BestLocal(b)).else

In Eq. (12), the parameter Q determines which of the solutions should be combined with BestGlobal. This parameter must be initialized between 0 and 1 before the start of the algorithm. The value of ω1 parameter must be an integer and must be set before the algorithm begins. This parameter’s value rapidly lowers with each iteration of the method, as seen in Eq. (13). (13)ω1=ω1∗Rv.0<Rv<1

Step 6: Termination conditions

The algorithm’s termination criteria are reviewed at the end of each step of the algorithm iteration. The algorithm ends if the constraints are satisfied. Otherwise, the algorithm will keep working until the termination constraints are satisfied.

These operators are applied to get the best results. These operators are also classified into two categories: point-to-point and subsequence. One city is chosen to change in point-to-point operators, but many cities are selected to vary in sequence operators. Point-to-point operators include RS and RI; subsequence operators include RSS, RIS, RRS, RRIS, SS, RSIS, RSSS, and RRSS. The proposed algorithm incorporates these ten operators as 10 LLHs composed of four primary operations-insert, shuffle, swap, and reverse. RRS performs specifically a function to change the subsequence. RI and RIS perform an insert operation. The swap and shuffle operations are performed by RS and RSS, whereas SS. RRIS performs the shuffle operation; RRSS, RSIS, and RSSS are four of the ten LLHs that are a hybridization of two operators. Note that RRS, RIS, RSS, SS, and the four LLHs with combined operations consider a subsequence of a TSP solution in size range [2: dim], where dim represents the TSP dimension. These operators have been listed in Table 1.

A version of the choice function is MCF heuristic selection. Within the heuristic search process, it promotes intensity above diversification. A choice-function-based hyper-heuristic with a score-based method is provided in [56]. Each LLH’s rating model is evaluated based on the LLH’s prior performance. Each LLH’s score is made up of three different criteria: f1,f2 & f3. In this model, the criterion of first measurement is f1 and the recent performance of each LLH is calculated by Eq. (14). (14)f1(hj)=∑nαn−1In(hj)Tn(hj)

In Eq. (14), α is a parameter between zero and one that prioritizes recent performance, hj is the same LLHj, In(hj) shows the nth use of hj displays the value of the difference between the current and new solutions. Tn(hj) expresses the amount of time spent by nth use of hj to propose a new solution. (15)f2(hk,hj)=∑nβn−1In(hk,hj)Tn(hk,hj)

In Eq. (15), f2 shows the dependence between a successive pair of LLHs. In(hk,hj) shows the value of the difference between the new solution using nth successive use of hk and hj and the current solution. Tn(hk,hj) shows the amount of time spent by nth sequential use of hk and hj to propose a new solution, β is a parameter between zero and one that prioritizes recent performance. Two criteria, f1 and f2, increase the selection of LLHs with high performance are components of intensification in the selection function. The third criterion, f3, records the time spent for the last implementation of a particular LLH (Eq. (16)). (16)f3(hj)=τ(hj)

In Eq. (16), τ(hj) indicates the time spent for the last execution of hj (in seconds). Note that f3 performs its work as a component of diversification in the selection function. It prioritizes those LLHs that have not been used for a long time. The score of each LLH is calculated using the sum of the weights of three criteria (f1,f2&f3), as in Eq. (17) is shown. (17)F(hj)=αf1(hj)+βf2(hk,hj)+δf3(hj)

In Eq. (17), the weight of the criteria is as follows (f1,f2&f3) represented by the α, β, and δ parameters. These parameters are constant values in the initial model. In [57], they have proposed an improved version of the hyper-heuristic to increase performance. In this version, if an LLH makes the solution better, the values of parameters β and α increase the new solution’s improvement rate compared to the previous solution. However, if the selected LLH does not improve the solution, the parameters α and β are reduced due to the difference between the new solution’s costs and the previous solution. Therefore, it can earn many rewards. Moreover, improving solutions in subsequent optimization phases may be reduced due to the convergence of one of the optimal solutions. As a result, much fewer rewards will be gained. Improvements in the subsequent optimization steps are far more essential than improvements in earlier ones. Therefore, the reward-penalty mechanism is not the correct mechanism. In addition to these limitations, the LLH selection mechanism must be randomly performed if solutions are not improved after several repetitions. Because the values of parameters α and β are reduced due to the reward-penalty mechanism. Also, the criterion f3 that is, a diversification component prevails over other criteria. Due to the limitations stated in [57] an improved version of the selection function. In this selection function version, parameters α and β are combined into a single parameter called μ. Finally, Eq. (18) computes each LLH’s score. (18)Ft(hj)=μt[f1(hj)+f2(hk,hj)]+δf3(hj)

The intensification component is prioritized if an LLH improves the solution and the parameter approaches its maximum static value around 1. The parameter δ is reduced to a minimum static value close to zero at the exact moment. If LLH does not improve the solution, the μ parameter will be penalized linearly and lower (0.01). This mechanism causes the parameter δ to grow with uniform and low speed. The parameters μ and δ are calculated using Eqs. (19) and (20). Eq. (19) shows the difference between its cost and the previous solution’s cost. (19)μt(hj)={0.99,d>0max[0.01,μt−1(hj)−0.01],d≤0 (20)δt(hj)=1−μt(hj)

The proposed algorithm uses MCF to select the best automatic LLHs during optimization implementation to increase the proposed algorithm’s performance and efficiency.

Local search solves many hybrid optimization problems because it increases the ability to solve them. Researchers have proposed various types of local search such as Opt-2, Opt-3, and local search such as LK. Local search can find optimal solutions; therefore, local search capability is limited to intensification. A method increases the probability of finding a global optimum that restarts the search after exploiting a particular search area. This type of local search that uses a re-search mechanism is called the Multi-start Local Search (MSLS) [58]. In MSLS, it is only possible to search for different primary solutions. Finally, we can obtain a set of locally optimal solutions in a globally optimal solution or find a solution close to the global optimum to collect locally optimal solutions. In [59,60], Chained LK (CLK) heuristic has used the ILS mechanism to solve the TSP problem. It is followed in CLK by a double-bridge motion to exchange the four arcs of the solution with the other four arcs, and this operation is done repeatedly on the solution. According to the importance of the balance of exploration and exploitation and the performance improvement in combining metaheuristic and local search algorithms, the proposed MCF-FFA model has been integrated with LK local search. A local search is carried out after each Neighborhood operator, performed on the solutions in bad and other parts before applying the acceptance condition. Due to local search use in the proposed algorithm, this model is very similar to the ILS and MSLS models. Based on the previous explanations, it has many similarities to the ILS and MSLS models.

A computer with an Intel Core i7-7700k processor 4.50 GHz (CPU) and 16 GB RAM has been used to test the proposed algorithm. It is implemented using the C programming language, and the Concorde [59] is used to implement the LK local search in the proposed algorithm. Seventy-one instances from the TSPLIB [32] with dimensions of 100 to 85,900 cities were used to evaluate the proposed algorithm’s performance. The number of instance names represents the number of cities in an instance. For example, the number of cities equals 100 in the instance kroA100. Two criteria, such as percentage deviation from optimal solution and computational time, were used to obtain the best tour (measured in seconds) to evaluate the performance of the proposed algorithm.

Each of the TSP instances has been investigated for 30 iterations. It was placed the best tour per implementation in the set X = {c1, c2, …, c30} and the computational time to obtain the best tour was placed in another set called T = {t1, t2, …, t30}, where X is the set of best tours, and T is the set of computational time to obtain the best tours, and finally, an average is obtained for these two sets, which µT is used to represent the average of set T. Average represents the mean of the set X. Finally, Eq. (21) is used to describe the percentage deviations of the average solution (PDav) (%). In this equation, BKS is the optimal tour length. (21)PDav(%)=Average−BKSBKS×100

In the proposed algorithm, two main POP and K-Value parameters must be adjusted to obtain optimal performance before starting optimization. POP is the population, and K-Value is the number of segments. All the instances are grouped into four categories in terms of their dimensions: Instances with dimensions [1: 500] in category X and instances with dimensions [501: 1000] in category Y, and instances with dimensions [1001: 10000] in category Z and instances with dimensions larger than 10000 in category E. Then an instance is selected as representative of each class. Instances gr202, gr666, rl5915, and rl11849 are chosen to represent the X, Y, Z, and E categories, respectively. Finally, we apply settings related to representatives for all other instances about the best performance results. A decrease in repeats occurs with the increasing population to make a fair comparison. For example, repetition terminates testing the population (10 × 400). Moreover, the replay ends for testing the population size (20 × 200). Table 2 shows the results of different settings for parameters.

The proposed algorithm in the instance gr202 achieved the optimum for all possible situations in setting the parameters. However, the proposed algorithm’s best performance occurs with POP = 10 and K-Value = 3 in gr666, rl5915, and rl11849. Hence, POP values = ten and K = 3 are considered for solving the other instances.

To investigate the performance of LLHs and the effect of integrating LLHs with MCF in this section, three modes of the proposed algorithm are presented below: Random selection of LLHs and integration of the four main LLHs (SS, PRS, RIS, and SS) with MCF, and integration of all LLHs with MCF have been compared to each other. Also, this comparison is performed to evaluate MCF’s performance when integrated with the FFA. Termination condition (400 iterations) is considered for all three modes (MCF-FFA, (LLH4) MCF-FFA and Random-FFA). In Table 3, to evaluate the performance of these three algorithms, we show the average tour length, PDav, and average computational time to obtain the best solution (μ_T).

According to Table 3, MCF-FFA has the best performance in finding the best tour. MCF-FFA achieved an average of 0.070 PDav (%), while (LLH4) MCF-FFA and Random-FFA models obtained scores equal to 0.292 and 0.217, respectively, performed worse than the MCF-FFA. The MCF-FFA model solved 44 of 71 instances in 30 iterations. Moreover, in most cases, the model MCF-FFA had a better performance than other models. Of course, in some cases, the two models (LLH4), MCF-FFA and Random-FFA, performed better than others, some of which are shown in Fig. 2.

The most notable instance performed convergence analysis in a TSPLIB (pla85900). The best-so-far was calculated using Eq. (21), and its value in each iteration of three modes (Random-FFA and MCF-FFA and (LLH4) MCF-FFA) has been shown in Fig. 3. (LLH4) MCF-FFA has a fast convergence in the analysis of this chart. However, after a few iterations, it falls into the trap of local optima, but the MCF-FFA and Random-FFA modes are more capable of escaping local optimality. Finally, MCF-FFA has more potential to improve the solution.

Wilcoxon signed-rank test [61] with 95% confidence interval was used for statistical comparison of the performance of three models (MCF-FFA, (LLH4) MCF-FFA, and Random-FFA). Wilcoxon signed-rank test has used the difference between the value of PDav (%) in two algorithms for comparison and ranking. Instances with similar values in the two algorithms are not used, regardless of similar cases. N denotes the number of sufficient cases, and R+ represents the scores for the cases with the best performance in the proposed algorithm. Whereas R− represents the sum of scores for the instances that the proposed algorithm performs worse than the comparative algorithm, and the value of W is compared with a critical value WCri,N in the Wilcoxon signed-rank test. If  W≤WCri,N, it indicates a significant difference in the performance of the two algorithms. But if W>WCri,N then there is no significant difference in the performance of the two algorithms. Table 4 presents the Wilcoxon signed-rank test for comparison of three models (MCF-FFA, (LLH4), MCF-FFA and Random-FFA) and the results of the Table show the remarkable performance of the MCF-FFA model.

For further evaluation and investigation of the MCF-FFA model, the average distribution of the implementation of each LLH selected by the MCF is recorded in each model’s performance. The 71 TSP instances used to evaluate these are classified into four categories according to their dimensions. Table 5 shows the average distribution of each LLH implementation by MCF when the cases are solved in each class.

According to the results shown in Table 5, LH (RSS) is selected by MCF over other LLHs in Category X, and MCF prefers LLH (RIS) in comparison with other LLHs in Category Y. Moreover, LLH (RRS) is chosen by MCF in contrast with other LLHs in Category Z. Finally, MCF selects LLH (RRS) in comparison with other LLHs in Category E. In the proposed algorithm, LLHs (such as RRS, RI, RIS, RS, and RSS) are frequently selected by MCF for obtaining good scores from the criteria f1, f2. Because these LLHs create more improvements than other LLHs in a shorter computational time. If the control mechanism in the MCF causes the parameter δ to obtain more value and grow faster in the event of lousy movement, in this case, the diversification component f3 will be prioritized, and LLHs will most likely be selected that were not previously selected. This section has tested the proposed algorithm for further investigation without using LK local search. The results are shown in Table 6 indicate that if the LK local search is deleted, the MCF will select LLHs (RI and RRS) more than other LLHs.

In this subsection, we have compared the MCF-FFA model’s performance with other models. Due to the comparative models’ lack of sources, the published articles’ results were directly extracted and compared. The PDav (%) was used for comparison. In total, ten different models were used for this comparison. Finally, the proposed algorithm has been compared with CLK to evaluate the performance. The selected models for comparison include ESACO [62], written by C++ programming language and executed using an Intel CPU 2.0 GHz processor and 1.0G RAM. These models end with tenants and 300 reps. The SOS-SA model [12] was programmed using Matlab R2014b and run on a computer with a 2.83 GHz CPU Desktop with 2 GB of RAM. MCF-ABC [38] model has been written in C programming language and runs on a system with an Intel i7-3930K 3.20 GHz processor and 15.6 GB of memory. The pop-Size parameter is set to 10, the limit parameter is set to 200, and these parameters are terminated in 1000 iterations. The ECSDSOS model [44] has been implemented using a computer with Intel (R) Core (TM) CPU 2.6 GHz CPU and 4G RAM, and the size of the ecosystem equals 50, and it is terminated in 1,000 reps. The HDABC model [53] is written using Java programming language and is implemented on a computer with Intel Core i7-5600 2.60 GHz CPU and 8 GB RAM and is completed in 500 iterations. The SOM model [52] is implemented on the GPU using the CUDA platform and 1280 MBytes of memory. The PACO-3Opt [17] is written by Java programming language and is implemented on PC with Pentium i5 3.10 GHz processor, and 2 GB RAM and migration interval is set to 5, finished in 1000 iterations. The D-CLPSO [42] is finished in 1000 iterations, and the length of the temperature list and the swarm size are 140 and 20, respectively. The DSCA model [63] has been coded using MATLAB R2014a and implemented using a the computer with a CORE i3 CPU with 2.27 GHz.

Moreover, the population has been considered equal to 15, and the number of generations equals 200. Java programming language is used for coding the DPIO model [64] and is implemented on a computer with Intel Core i5-3470 CPU, with 3.2 GHz and a RAM of 4 GB, and is finished in 1000 iterations. The source of CLK [59] model code available in the Concorde TSP solver software is used to compare the proposed algorithm with CLK. Instances Z and E were used for comparison. CLK is a single solution-based model, and the number population is equal to 1 in this model; a pre-determined number of repetitions are equivalent to 10,000. Default settings in Concorde are maintained for execution, which are: initialization method (i.e., Quick-Boruvka), choice of the kick (i.e., 50-step random-walk kick) and the level of backtracking (i.e., (4, 3, 3, 2)-breadth).

Table 7 has compared the proposed algorithm’s performance with the other models. The proposed algorithm has shown the best performance compared to the different models and has been able to find the optimal tour in most cases. HDABC and MCF-ABC performed better than the proposed algorithm in some instances. MCF-ABC had similar performance in small-scale cases and was able to find optimal tours, and in some cases with different scales, it was able to find better tours than the proposed algorithm. Because the MCF-ABC model employs a strategy similar to the proposed algorithm, the HDABC model only performs well in small-scale instances. To further evaluate the proposed algorithm, a comparison with CLK [59] has been made in Table 8.

Table 8 compares the proposed algorithm with CLK. Both models use an LK local search strategy. By examining this comparison, it can be concluded that the proposed algorithm performs well in small-scale instances, while the CLK model performs better in large-scale cases. Wilcoxon signed-rank statistic test with a 95% confidence interval to evaluate the proposed algorithm’s performance compared to other algorithms.

The results of this test are shown in Table 9 based on the 95% confidence interval. These results show that the proposed algorithm performs well and significantly against the other 11 models: R+>R− and W≤WCri,N. Whereas we compared it with CLK, the result is as follows: W>WCri,N.