A novel hybrid hunger games search algorithm with differential evolution for improving the behaviors of non-cooperative animals

Inspired by the behaviors of animals in the state of starvation, hunger games search algorithm (HGS) is proposed. HGS has shown competitive performance among other meta-heuristic (MH) algorithms. However, HGS tends to stagnate in local optimal for some complex optimization problems and remains premature convergence. Therefore, to solve these problems and enhance the diversity of the population, a modified HGS based on the operators of the differential evolution algorithm (DE), chaotic local search (CLS) strategies, and evolutionary population dynamics technique (EPD) is proposed (named DECEHGS). The proposed DECEHGS algorithm consists of two stages: in the first stage, based on the animals’ behaviors, we use different evolutionary methods to update animals’ positions; in the second stage, the CLS strategy and EPD technique are combined to prevent premature convergence and stagnation in a local optimum. The proposed algorithm was evaluated using IEEE CEC2014 and IEEE CEC2017 mathematical functions and four engineering problems. The experimental results demonstrate that DECEHGS has competitive performance in global optimization tasks and engineering problems compared with state-of-the-art algorithms.


I. INTRODUCTION
Many meta-heuristic (MH) algorithms have been proposed to solve real-world problems and obtain better performances, such as neural networks [1], [2], image thresholding [3], task scheduling on cloud computing [4], [5], ice manufacturing industry [6], feature selection [7], [8], data clustering problems [9] and distributed cloud framework [10]. Many phenomena in nature are a source of inspiration for algorithms. The diversity of phenomena leads researchers to propose several optimization approaches derived from swarm intelligence, physical principles, and coevolution between organisms.
MH algorithms can be divided into three categories based on different inspirations as evolutionary computation (EC) [11], swarm intelligence algorithms (SIA), and physical phenomena algorithms (PPA).
The first category EC is inspired by Darwin's principles of nature's capability to evolve the individuals to be well adapted to their environment. The most popular algorithms include genetic algorithm (GA) [12], differential evolution algorithm (DE) [13], biogeography-based optimization (BBO) [14] and evolution strategies (ES) [15].
Inspired by these ideologies, many improved algorithms have been proposed, such as hybrid harmony search algorithm with grey wolf optimizer (GWO-HS) [29], coevolutionary particle swarm optimization with bottleneck objective learning strategy (CPSO) [30], fuzzy gravitational search algorithm (FGSA) [31] and differential evolution algorithm with strategy adaptation and knowledge-based control parameters (SAKPDE) [32].
Hungry games search algorithm (HGS) [33] is a new and efficient meta-heuristic algorithm inspired by the behavior of animals in nature when they are hungry. Like other MH methods, HGS consists of two phases (i.e., cooperative and non-cooperative behaviors), and each phase is determined according to a given value. HGS was reported to be superior to other recognized optimizers, such as PSO, GA, and gravitational search algorithm (GSA) [34], etc. However, HGS will quickly converge on the local optimum when solving complex problems, which is a widely existing problem of the MH algorithms [35], [36]. Another problem is the stagnation in a local optimum. MH methods' success in solving this problem depends on balancing the exploration and exploitation or hybridizing other algorithms. The original operators of non-cooperative animals in HGS fail to jump out of the local optimal, which will significantly degrade the quality of the solutions.
The DE algorithm was first proposed in 1994 by Storn and Price. DE is a simple and powerful evolutionary algorithm. The inherent simplicity of DE has drawn the attention of many researchers. For example, in [37], Pant et al. proposed a hybrid version of DE with PSO, and the results show that the proposed DE-PSO is reasonably competent for solving the benchmark functions and realworld problems. Wu et al. [38] presented a multipopulation-based ensemble DE (MPEDE), in which the whole population is divided into three equally sized subpopulations and one larger subpopulation. Three different mutation strategies named "current-to-pbest/1", "current-to-rand/1", and "rand/1" are applied to the three populations, the controlling parameters of each strategy are made adaptively. Zou et al. [39] proposed a modified DE named MDE, employing Gaussian and uniform distribution to adjust scale factor and crossover rate.
We proposed an improved version of the HGS algorithm (DECEHGS) using the DE algorithm to improve the noncooperative animals of the classic HGS algorithm to overcome these limitations. Moreover, evolutionary population dynamics technique (EPD) [40] and chaotic local search (CLS) [41] strategy are used to update the worst solutions and best solutions during the optimization process, which will reduce the effect of worst solutions on the quality of the population and the possibility of local optimum stagnation.
The main contribution of the current study can be outlined as follows: 1. The operators of DE are used to enhance the noncooperative animals. 2. Apply the EPD technique to update the worst solutions. 3. CLS strategy is employed to update the best solution to avoid falling into local optimum. 4. The performance of the DECEHGS is evaluated using CEC2014 and CEC2017 test suites and four engineering problems.
The remainder of this paper is organized as follows: Section 2 briefly introduces the basic version of the HGS algorithm. We introduce the DE algorithm, EPD technique, and CLS strategy in Section 3. In Section 4, the DECEHGS algorithm is proposed. The comparison results are presented and discussed in Section 5. Finally, conclusions and future work are summarized in Section 6.

II. HUNGER GAMES SEARCH ALGORITHM
The hunger games search algorithm is a new MH algorithm proposed by Yang (2021) [33], which is inspired by the activities and behaviors of animals in a state of starvation. In HGS, animals mainly have two behaviors, cooperation and non-cooperation. The formula of cooperative behavior is shown as follows: Where X(t) represents the position of the current individual, randn is a random number that satisfies a normal distribution.
The formula of non-cooperative behavior is shown as follows: Where Xb represents the best individual at the current iteration, W1 and W2 represent the weight of hungry, r2 is a random number in the range of [0,1], R is a ranging controller, and E represents the variable number that controls the global location, and their formulas are shown as follows: Where F(i) represents the fitness value of each individual, BF is the best fitness at the current iteration, and the formula of sech is shown as follows: Where rand is a random number in the range of [0, 1], t is the current iteration, T is the maximum number of iterations, hungry represents the hunger for each individual, N means the swarm size of the individuals, SHungry is the sum of hungry feelings of all individuals, and r3, r4, r5 are random numbers in the range of [0, 1]. The formula of hungry(i) is calculated as below: Where AllFitness(i) preserves each individual's fitness in the current iteration, and the best individual's hunger value is set to 0 in the current iteration. For the other individuals, a new hunger value H is added based on the former hunger value. The corresponding parameter H of each individual is different.
The formula of H is shown as follows: Where r6 and r7 represent a random number in the range of [0, 1], LH is a limited parameter, F(i) represents the fitness value of each individual, BF and WF are the best fitness and worst fitness value obtained from the current iteration, respectively, ubi and lbi represent the upper and lower bounds of the search space.

III. PRELIMINARIES
The DE algorithm is applied to enhance the behaviors of non-cooperative animals. Moreover, it can be found that the worst solution in the population may influence the performance of the algorithm and prevent it from converging on the global optimum. Therefore, we integrate the updated solutions with the concept of CLS and EPD to avoid premature convergence and stagnation in the local optimum.
The EPD technique replaces the worst agents by generating new agents in the neighborhood of the better ones. The CLS strategy creates a new agent randomly and chaotically in the neighborhood of the old agent.

A. DIFFERENTIAL EVOLUTION ALGORITHM
DE algorithm has been proved to be a simple and efficient algorithm. DE works in two phases: initialization and evolution. In the first phase, the population is generated randomly. The generated population mainly goes through three processes in the second phase: mutation, crossover, and selection. DE repeats the second process until the termination criteria are satisfied. During initialization, each individual in the current iteration is generated as follows: Where rand  [0, 1], lbi and ubi are lower and upper bounds of search space, respectively.
In the mutation phase, a mutant vector Vi is generated for each individual as below: Where F is the scaling factor, and the value of F is varied from 1 to 0, r8, r9, r10 are random integer numbers in the range of [1, N].
After the mutation phase, the crossover phase generates a new vector called trial vector Ui, which is denoted as follows: In the selection phase, a comparison between the target and trial vector is made according to their fitness value. This operation performs as below: Where f denotes the fitness function.

B. CHAOTIC LOCAL SEARCH
Chaos is a typical nonlinear phenomenon in nature, which is characterized by randomness, ergodicity, and sensitivity to initial conditions. Because of the randomness and ergodicity, chaos optimization works well in the small search space, but it requires unacceptable optimization time in the ample search space. Therefore, chaotic search is frequently incorporated into other global optimizers such as GA, PSO, and WOA to enhance their searching capacities.
The mechanism of CLS can not only make the algorithm more capable of avoiding falling into the local optimum but also enhance the searching capacities and make a better harmony between exploration and exploitation.
In [42], [43], Wei and Mascia et al. used local search to construct the solutions to the objective function. However, standard local search may inevitably lead to rapid convergence or stagnation in local optima. Owing to its randomness and ergodicity, CLS can effectively overcome these shortcomings.

C. EVOLUTIONARY POPULATION DYNAMICS
The evolutionary population dynamics technique is based on self-organized criticality theory (SOC) [44], [45]. SOC theory indicates that the local changes in the population may influence the entire population without the intervention of any external [46]. The EPD aims to improve the quality of the solutions by removing the worst solutions at the current iteration of the solution and replacing the worst solution by generating new individuals around the best solution. EPD has been successfully incorporated in popular MH algorithms, such as extremal optimization (EO) [47] and evolutionary programming with self-organizing criticality (EPSCO) [48].
In [49], [50], the EPD technique is applied to a single population. In this paper, we proposed a modified EPD. We separate the population into three subpopulations, and each subpopulation evolves separately. In this way, we can improve the diversity of the population and improve the quality of the solution. After each iteration, the EPD technique updates the worst individual preventing interference from the worst individual on the present subpopulation. The primary process of EPD is shown in Figure 1.

IV. THE PROPOSED ALGORITHM
In this section, the structure of the proposed algorithm is explained. Algorithm 1 shows the main steps of the proposed algorithm. The flowchart of DECEHGS is presented in Figure 2.
In order to avoid the phenomenon of decreasing diversity when the population iterates to a specific region, the DE algorithm is adopted to maintain the diversity of the population. Moreover, the worst solution to the population may influence the algorithm's performance and prevent it from converging on the global optimum. Therefore, we integrate the updated solutions with the concept of CLS and EPD to avoid premature convergence and stagnation in the local optimum. The EPD technique replaces the worst agents by generating new agents in the neighborhood of the better ones. The CLS strategy creates a new agent randomly and chaotically in the neighborhood of the old agent.

1) Initial stage
The objective of this stage aims to produce the initial population X using the following equation.
Where lbi and ubi are lower and upper bounds of the search space, and i represents the current individual.

2) Updating population using DE
The current solution Xi will be updated either using HGS or DE according to the fixed parameter L. For example, if a random number r1 < L (where r1  [0, 1]), then the DE algorithm is applied to update the current solution Xi by Eq. (13). Whereas, if the r1  L then the Xi will be calculated by Eq. (2). The mathematical formulation of this stage is shown as follows: Where r1 and r2  [0, 1], Xr3 and Xr4 are randomly selected from the main population, F is the scaling factory, W1 and W2 represent the hunger weight, R is a ranging controller, Xb is the best individual in the whole population and E is a variation control for all positions.

3) Updating population using CLS
CLS strategy can avoid falling into the local optimum and improve the quality of the solution owing to its randomness and ergodicity. The equation of CLS is defined as: βi is calculated by the following equation: When μ = 4, the logistic function comes into a thorough chaotic state. L is the contraction factor, which is determined as follows [51]: Where t represents the current iteration and m controls the shrinking speed, the higher m value, the slower the shrinking rate.

4) Updating population using EPD
The swarm of animals is divided into three subpopulations randomly to equip the HGS algorithm with the EPD technique. The worst animals are eliminated and reinitialized based on the good animals of each subpopulation. The formula for EPD is defined as follows: Where ubi and lbi represent the boundaries of the search space of the best solution Xb, rand is a random number in the range of [0, 1], c indicates the current subpopulation, Ni represents the number of the worst agents, and its formula is defined as follows: Where rand is a random number in the interval [0, 1], N is the swarm size of the subpopulation, and round() is a function used to convert natural numbers to integers.

5) Termination criterion
The steps of the 2), 3), and 4) stages repeat until the stopping condition is satisfied. The number of fitness evaluations (FES) is used as a termination condition in this paper. When the DECEHGS reaches the stop condition, the best solution Xb is returned.

6) Computational complexity of DECEHGS
The population size and subpopulation size are set to be N and SN, respectively. The dimension of the problem is set to be D, the maximum number of fitness evaluations is set to be T, and the swarm size of the worst individuals is set to be Nw. The complexity of DECEHGS depends on the complexity of HGS, DE, EPD, and CLS. So, the complexity of DECEHGS is defined as:

V. EXPERIMENTAL STUDIES
In this section, comprehensive experiments were performed to validate the effectiveness of the proposed DECEHGS. All experiments were carried out using python 3.6 under Windows 10 with Intel (R) Core (TM) i5-1135G7 @ 2.40GHz and 16.0GB of RAM.

A. THE EFFECTIVENESS OF DE, CLS, AND EPD
The DECEHGS algorithm consists of three improved methods: the operators of DE, CLS strategy, and EPD technique. This section mainly aims to verify the effectiveness of these methods. We compared DECEHGS and its variants on the 30-dimensional CEC2014 test suites. The variants of which are DECHGS, DEEHGS, EHGS, CHGS, CEHGS, and DEHGS. The classic HGS algorithm uses DE and CLS strategies, but without the EPD technique, it is named DECHGS. The CEHGS algorithm uses only the CLS strategy and EPD technique. EHGS, CHGS, and DEHGS use only three improvement methods: EPD, CLS, and DE algorithm.
For the fairness of tests, all the involved algorithms were tested under the same conditions. The population size N and parameter settings are shown in Table 1. The maximum number of fitness evaluation MAX_FES was set to 1E5. It is noteworthy that each algorithm was executed according to the average results over 30 runs to reduce stochastic error in this paper.
The numerical results of these algorithms in terms of the average value (AVG), standard deviation (STD) of the function error rates, and rank of the algorithms (RANK) were obtained to assess the potentials of related techniques, and the best results of each task are marked in boldface.
It can be seen from Table 2 that among these seven HGS variants and the original HGS algorithm, DECEHGS ranks 1. The results show that DECEHGS has reached the global optimum value on F26, F29, and F30, but HGS has not achieved the global optimum value. CLS strategy can avoid local optimum and speed up the algorithm's convergence, and the EPD technique can enhance the performance of this algorithm. CEHGS successfully combines the advantages of these strategies and can reach the global optimum value of complex functions. On F1 to F9, DECEHGS, DECHGS, and DEEHGS are better than other variants on the test functions. This is because the DE algorithm is added, which can improve the diversity of the population allow the algorithms to find the global optimum. In summary, each component can enhance the performance of the original HGS, and our proposed algorithm can improve the performance of the original HGS much better by taking advantage of the three components (DE, CLS, and EPD). VOLUME XX, 2017
The experiment was conducted under the same conditions as the previous experiment to make the comparison results more impartial. The parameter settings for the involved algorithms are shown in Table 1. The maximum number of fitness evaluation MAX_FES was set to 1E5. Table 3 shows the average value, standard deviation, and rank of DECEHGS and other well-known algorithms on 30dimensional CEC2014 test suites. As shown in Table 3, the DECEHGS algorithm has achieved minimum optimization on 15 test functions which shows great competitiveness over these advanced algorithms on CEC2014 test suites. The overall rank of DECEHGS is first among these combinations of DE, CLS, and EPD. EPD eliminated the worst solution and reinitialized the solution around the current best solution, DE algorithm increases the diversity of the population, and the CLS strategy can accelerate the algorithm's convergence rate. The convergence curves of the DECEHGS and other advanced algorithms are shown in Figure 3. It can be observed that the DECEHGS converges faster than other advanced algorithms in most cases. On F1, F3, F4, F7

C. VALIDATION ON IEEE CEC2017 TEST SUITES WITH DIFFERENT DIMENSIONS
In this part, to further verify the scalability of the proposed algorithm, DECEHGS was tested with different dimensional problems (i.e., Dim = 10, 30, 50, and 100). The experimental results obtained from CEC2017 test suites are presented in Table 4. Some state-of-the-art algorithms are employed to test the performance of DECEHGS, such as enhanced salp swarm algorithm (ESSA) [60], modified global flower pollination algorithm (MGFPA) [61], memory-based grey wolf optimizer (mGWO) [3], hybrid improved whale optimization algorithm (HIWOA) [62] and phasor particle swarm optimization (PPSO) [63]. In this experiment, the population size N and parameter settings are shown in Table  1, the maximum number of fitness evaluations was set to 1E5. Each algorithm was executed 30 times randomly. Table 4 shows the comparison between DECEHGS and the advanced algorithms with different dimensions. As shown in Table 4, DECEHGS ranked first across different dimensions. For unimodal functions (F1 to F3), DECEHGS converges very quickly, and the quality of its solution is very high. For multi-modal functions, DECEHGS performs better at higher dimensions. It is worth noting that for composition functions (F23 to F30), the performance of DECEHGS did not deteriorate seriously as the dimension increased. These results indicate that the searchability of DECEHGS is effective. Moreover, DECEHGS can avoid falling into local optimum, and the optimization performance of solving high dimensional functions is strong.
The convergence curves of DECEHGS and other algorithms on CEC2017 test suites are shown in Figure 4. In Figure 4, F1 and F3 are unimodal functions, F4, F5, F8 to F12, and F16 are multi-modal functions, F17 is a hybrid function, and F27 to F30 are composition functions. The first row of the figure shows the results of the involved algorithms on a 10-dimensional test set. The second row displays the results of the 30-dimensional test set. The third row represents the results of 50-dimensional benchmark functions. The graph in the fourth row shows the results of the 100-dimensional test set.
It is not difficult to see from Figure 4 that of these advanced algorithms, DECEHGS has the best optimization effect. As shown in Figure 4, DECEHGS converges very fast and outperforms most of the other algorithms. In dealing with F17 and F29, there is close competition among all algorithms, but DECEHGS can find a better solution during the overall steps. Although PPSO's convergence rate is faster than DECEHGS on F29, PPSO's optimization effect is worse than DECEHGS. It is because the EPD technique in DECEHGS helps the population deviate from the local optimum. The DE algorithm can increase the diversity of the population, and the CLS strategy can accelerate the algorithm's convergence rate. All in all, the experimental results of DECEHGS are superior to other advanced algorithms.

D. ENGINEERING PROBLEMS
The proposed algorithm was validated using four engineering optimization problems: the welded beam design problem [64], tension/compression spring design problem [65], pressure vessel design problem, and three bar truss design problem. The proposed methods were implemented for 30 independent runs. The population size N and parameter settings were set the same as the original paper, and the maximum number of fitness evaluation MAX_FES is set to 1E4. The results are shown in Table 5, Table 6, Table 7, and Table 8.
The penalty function approach was utilized to handle optimization constraints [66], [67]. The formula is stated as: Where λ and Ne are the penalty coefficients and the number of constraints, respectively. f(x) and ge(x) denote the objective function and their constraints, respectively.

1) Welded beam design (WBD) problem
The main objective of designing welded beams is to find the lowest consumption of welded beams under the four constraints of bucking load (Pc), shear stress (τ), bending stress in the beam (θ), and deflection rate (δ). The problem includes four variables: welding thickness (h), length of steel joint (l), the thickness of steel (b), and height of steel (t). The mathematical model is defined as follows: On this subject, DECEHGS was compared with selfadaptive differential evolution algorithm (SADE) [68], success-history-based parameter adaptation for differential evolution (SHADE) [69], GWO, GA, MVO, HHO, BBO, HS, sine cosine algorithm (SCA) [70], HGS and davidonfletcher-powell (DAVID) [71].
The comparison results from Table 5 show that DECEHGS is the best algorithm for the WBD problem. Four parameters are set to be 0.20521, 3.470466, 9.036321, and 0.21524, respectively, and the manufacturing cost of WBD can reach 1.724745.

2) tension/compression spring design (TCSD) problem
The tension/compression spring design problem is described in Arora, for which the aim is to determine the optimal values of active coils (N), the wire diameter (d), and mean coil diameter (D). The mathematical model is as follows:  On this subject, the proposed algorithm is compared with other state-of-the-art techniques such as LSHcEpS, SHADE, a mathematical programming method by Belegundu and Arora [72] (named as BAA), BBO, moth flame optimization (MFO) [73], HGS, MVO, GA, and GSA methods.
It can be seen from Table 6 that DECEHGS exposes the best overall performance. The MFO, SHADE, HGS algorithms rank the 2nd, 3rd, and 4th positions, respectively.

3) Pressure vessel design (PVD) problem
The third engineering design problem aims to solve the cylindrical pressure vessel and determine the optimal values of the depth of the shell (Ts), head (Th), the internal radius (R), and the extent of the section, minus the head (L). The proposed algorithm was compared with referenced methods such as LSHcEpS, SADE, ES, GA, PSO, GSA, HGS, and JAYA [74] Table 7, it can be seen that the DECEHGS and LSHcEpS occupied the first and the second ranks for solving this function. The cylindrical PV obtains the lowest total cost when Ts, Th, R, and L are set to be 0.774531, 0.383204, 40.31962, and 198.9731. Among all of these algorithms, DECEHGS can find a feasible optimal design.

4) Three bar truss design (TBTD) problem
Three bar truss design is a structural optimization problem in the field of civil engineering. Two parameters x1 and x2 are manipulated to minimize the weight subject to stress, deflection, and buckling constraints. This problem can be formulated as follows:

VI. CONCLUSIONS
Hunger games search algorithm is a simple and efficient algorithm but tends to stagnate in local optimal and remains premature convergence. In this paper, we enhance the noncooperative individuals by using the operators of differential evolution. For the traditional evolutionary population dynamics strategy, we propose a modified EPD technique to increase the diversity of the population, improve the quality of the solution and avoid local optimal. Furthermore, a chaotic local search strategy is introduced to prevent premature convergence. However, DECEHGS also has its limitations. With the improvement of computing performance, the time complexity is correspondingly higher.
In the end, we use three groups of experiments to verify the performance of DECEHGS. IEEE CEC2014 and IEEE CEC2017 test suites were chosen to compare DECEHGS to the state-of-the-art algorithms, including ILSHADE, LSHADE_cnEpSi, JSO, CoDE, BBPSO, ISCA, OBWOA, TVBSSA, ESSA, MGFPA, mGWO, HIWOA, and PPSO. The experimental results indicate that the DECEHGS algorithm outperforms other methods. In addition, four realworld engineering problems were used to verify the performance of DECEHGS. The comparison results reveal that the performance of DECEHGS is better than many other methods.
In future works, DECEHGS can be designed as distributed or parallel algorithms to solve large-scale problems such as cloud workflow scheduling problems, power electronic circuit problems, and airline crew rostering problems. Moreover, it is interesting to extend DECEHGS to a multiobjective version to solve multi-objective optimization problems, including job shop scheduling, supply chain configuration, and vehicle dispatch problems.