A boosted chimp optimizer for numerical and engineering design optimization challenges

Abstract

Chimp optimization algorithm (ChoA) has a wholesome attitude roused by chimp’s amazing thinking and hunting ability with a sensual movement for finding the optimal solution in the global search space. Classical Chimps optimizer algorithm has poor convergence and has problem to stuck into local minima for high-dimensional problems. This research focuses on the improved variants of the chimp optimizer algorithm and named as Boosted chimp optimizer algorithms. In one of the proposed variants, the existing chimp optimizer algorithm has been combined with SHO algorithm to improve the exploration phase of the existing chimp optimizer and named as IChoA-SHO and other variant is proposed to improve the exploitation search capability of the existing ChoA. The testing and validation of the proposed optimizer has been done for various standard benchmarks and Non-convex, Non-linear, and typical engineering design problems. The proposed variants have been evaluated for seven standard uni-modal benchmark functions, six standard multi-modal benchmark functions, ten standard fixed-dimension benchmark functions, and 11 types of multidisciplinary engineering design problems. The outcomes of this method have been compared with other existing optimization methods considering convergence speed as well as for searching local and global optimal solutions. The testing results show the better performance of the proposed methods excel than the other existing optimization methods.

1 Introduction

Nowadays, artificial intelligence as well as machine learning are rapidly increasing, because it is easy to implement to solve real-life issues which are continuous or discontinuous, constrained or unconstrained [1, 2]. For handling these characteristics using conventional approaches such as the quasi-Newton method, sequential quadratic programming, fast steepest and conjugate gradient, etc. faced difficulties to solve them [3, 4]. In the existing research, all these methods were tested experimentally and noticed that they are not exactly sufficient to obtain effectual solutions to non-continuous, non-differential problems and real-life multi-model problems [5]. Thus, the meta-heuristics algorithm came into the picture which is very simple to understand and easily be implemented to handle several issues. Generally, in optimization, techniques depend on inhabitants to find out the solution on optimal and sub-optimal which is closer to an exact optimal value, located at the nearest point. In this algorithm, the optimization process starts unless the population set of the individuals are generated and then relying on optimization method every individual act for candidate solution for the problem. Thus, by updating the present location with the best position, the population will be up-to-date by reaching maximum iterations. In modern research, the meta-heuristics algorithm which gives better efficiency, less expensive, and successful in implementation is given prior importance to utilize.

With such traits integrated, a new hybrid meta-heuristics optimization approach, ICHIMP-SHO algorithm is suggested in this research that depends on nature-lead and its mathematical formulation of search functions was designed to offer good competitiveness to current existing meta-heuristics optimizers. The intention to design this optimization technique is motivated by individual intelligence and sensual movement of social carnivores, named Chimps for their mass hunting mannerism in targeting the prey [6]. Hence, a stochastic and meta-heuristic mathematical model intended to handle various optimization problems and is verified by testing experimentally in this research work.

It is true that optimization technique is a large field of study, and researchers are rapidly applying new approaches to provide better answers to various issues that target specific obstacles and can succeed in their discoveries. In research, old techniques give way to new approaches, which use a hybrid unique strategy to eliminate inefficient ways from the present. In this suggested study, a collection of research articles is offered in the literature review to enumerate the flaws of modern algorithms.

Broadly speaking, meta-heuristics are of two types, named single solution-based meta-heuristics and population solution-based meta-heuristics. Improved Chimp (ICHIMP) variant belongs to swarm intelligence-based algorithm of the categories of population meta-heuristics, which is combined along with newly introduced swarm intelligence-based algorithm called Spotted Hyena Optimizer algorithm and named as Improved Chimp-Spotted Hyena Optimizer (ICHIMP-SHO) algorithm which is introduced in this paper. On the whole, this algorithm is simple to apply and involves very few operators than other population-based algorithms with minimum computational efforts.

The remaining parts of the present article contain literature review on related algorithms in Sect. 2, and concepts of improved chimp optimizer (ICHIMP) algorithm are discussed in Sect. 3. Sections 4 and 5 describe spotted hyena optimizer (SHO) algorithm and proposed ICHIMP-SHO algorithm, respectively. Standard benchmark functions are described in Sect. 6. Section 7 showcases the outcomes and comparison of results with other existing algorithms. Testing of 11 engineering-based optimization design problems are shown in Sect. 8, and finally, conclusion and future scope of the paper are presented in Sect 9.

2 Literature review

Meta-heuristics approaches have been frequently used in recent years due to their efficiency when compared to other approaches. These algorithms provide a more effective answer to real-world optimization problems. As a result, new meta-heuristics algorithms must be introduced to overcome these optimization challenges. Meta-heuristics optimization algorithms (MOAs) are important in the ever-increasing use of engineering applications. Because of the complexity of today's situations, the need for the most up-to-date MOAs is quickly growing.

It acquires distinct profits as: (i) Its natural algorithmic structure helps to implement it effortlessly; (ii) this suits real-life problems in engineering as it is a derivation-free mechanism; (iii) when compared to traditional optimization algorithms, this has better ability to minimize local optima; (iv) this is flexible in applying on different problems as its structure does not need any particular changes; (v) because of its simplicity and efficiency, this can be applied simultaneously in hardware applications as well as in computing applications. [like Field Programmable Gate Array (FPGA)] [6].

To limit the drawbacks of classical methods, meta-heuristics search algorithms were introduced. Few such algorithms are Biogeography-based optimization (BBO) [7], Artificial Bee Colony (ABC) [8], Differential Evolution (DE) [9], Genetic algorithm (GA) [10], Cuckoo Search algorithm (CSA) [11], Bacterial Foraging algorithm (BFA) [12], Flower pollination algorithm (FPA) [13], Chemical Reaction optimization (CRO) [14], Firefly algorithm (FA) [15], Immune algorithm (IA) [16], Teaching–Learning-based optimization algorithm [17], Particle Swarm optimization algorithm (PSO) [18], Grey wolf optimization (GWO) [19], Social spider for constrained optimization (SSO-C) [20], Gravitational Search algorithm (GSA) [21], and Bat algorithm (BA) [22]. The reasons how meta-heuristics algorithms are classified are explained in [23, 24], and with reference to [25, 26], meta-heuristics algorithms are considered by natural behavior and divided as single solution-based and population-based algorithms. Examples for single-based algorithms and population-based algorithms are: Variable Neighbourhood search (VNS) [27], Vortex search algorithm (VS) [28], whereas Simulated Annealing (SA) [29], Genetic algorithm (GA) [30], and Tabu search (TS) [31] have an emerging way to find a solution for combinatorial real-world problems in covering and scheduling, Cuckoo search algorithm (CSA) [32], Gravitational search algorithm (GSA) [33], Evolutionary programming (EP) [34] are a fast technique and classical evolutionary programmings were performed on real-world problems. Harmony search (HS) [35] is inspired using the music production cycle analogy. HS may not need the initial values of the variables for decision. Forest Optimization Algorithm (FOA) [36] is for finding maximum value and minimum value with a real application and found that the FOA can typically find solutions correctly. Grey Wolf Optimizer Algorithm (GWO) [19] work was inspired by a Swarm intelligence optimization through the grey wolves and the suggested model imitated the grey wolves' social hierarchical and hunting behavior. Moth Flame Optimizer (MFO) [37], the key influence of this optimizer is the moth navigation system called transverse orientation in nature. Moths migrate in darkness by keeping a preset moon angle, a very effective method for long-distance flying in a straight line. However, such fancy insects are stuck around artificial lights in a useless/deadly spiralling course. Stochastic Fractal Search Algorithm (SFS) [38] centered on random fractals to address global optimization problems with continuous variables, both constrained and unconstrained. In the entire optimization, if only one solution carries then it is known as a single solution-based algorithm and if there are many different solutions in the whole optimization phase, then it is a population-based algorithm, and as such, the solution may coincide with the optimum very nearly.

The two main components of meta-heuristics are exploitation and exploration [25]. Exploration extends searching widely to produce many different solutions, whereas exploitation focuses on searching in a specified area, assuming that area is the best for the present. It is very much important and necessary to balance these two components exploitation and exploration in MOA to keep away the fluctuations in the rate of convergence, as well preventing local and global optimum [39, 40]. Exploitation indicates single solution-based meta-heuristics and exploration indicates populated solution-based meta-heuristics.

Optimization problems can find solutions by nature-inspired MOAs' physical or biological behavior implementation. They are classified into four main classes (Fig. 1) [24, 41]: Swarm Intelligence based algorithm, Evolutionary algorithms (EAs), Human-based, and Physics-based algorithms. The below is the survey made on the algorithms which fall under these four categories. Among them, first, the Evolutionary algorithms replicate features of biological generation like recombining, mutation, and selecting processes [23]. The famous Evolutionary algorithms are Differential Evolution (DE) which presented the minimization of potentially nonlinear and non-differentiable continuous space functions. It only requires some strong control variables, taken from a perfectly defined number interval, Evolutionary Strategy (ES) [42], Biogeography-based optimization (BBO) made analysis of biological species, that can be used to deduce algorithms suitable for optimization. Evolutionary Programming (EP) and Genetic algorithm (GA) are drawn from Darwinian Theory. Second, as per [41, 43], Physics-based algorithms are analogous to natural physical laws. The famous algorithms are Quantum Mechanics-Based (QMBA) and Gravitational Search (GSA) which were influenced by the Gravitational Law and the theory of mass interaction. GSA utilizes Newtonian mechanics theory, and its search agent is the set of masses. Few more physics-based algorithms are Central Force Optimization (CFO) [44], Charged System Search (CSS) [45], Electromagnetism Like Algorithms (ELA) [46], Lightning Attachment Procedure Optimization (LAPO) [41], Big-Bang Big-Crunch (BBBC) [47], and Adaptive gbest-guided gravitational search algorithm (AGBGSA) [48]. Third, MOAs are inspired by natural human behavior. The best examples of them are Teaching–Learning-based optimization (TLBO) which comprises of two phases, teaching phase and learner phase, Imperialist Competitive Algorithm (ICA) [49], and Socio Evolution and Learning Optimization (SELO) [50]. Fourth, MOAs imitate the social behavior of organisms like swarms, shoals, flocks, or herds [51]. Few algorithms under this class are Particle Swarm optimization (PSO), Bat algorithm (BA), Ant colony optimization (ACO), Improved monarch butterfly optimization algorithm (MBO) [52], Cuckoo Search algorithm (CSA), Krill herd (KH) [53], Grey wolf optimizer (GWO), Multi-Objective Grasshopper optimization algorithm (MOGOA) [54], binary salp swarm algorithm (BSSA) [55], hybrid dragonfly optimization algorithm and MLP (DOA-MLP) [56], and Improved Whale Trainer [57].

Fig. 1

Classifications of population-based meta-heuristics search algorithms

A brief of recently developed algorithms to find solution for optimization problems: Harris Hawks optimizer (HHO) [25] is being introduced to tackle different tasks of optimization. The strategy is influenced by nature's cooperative activities and by the patterns of predatory birds, Harris' hawks. Henry Gas Solubility Optimization Algorithm (HGSO) [58] imitates the procedures of Henry’s rule. HGSO aimed at matching the production and conservation capabilities of check room and overcome local optimum. Photon Search Algorithm (PSA) [59] got inspired by the properties of photons in the field of physics. Chaotic Krill Herd Algorithm (CKH) [60] combined chaos theory with Krill Herd Optimization procedure to speed up global convergence. Bird Swarm Algorithm (BSA) [61] depends on social interactions of swarm intelligence with bird swarm. Lightning Search algorithm (LSA) [62] is a meta-heuristic technique used to resolve problems on constraint optimization by following lightning phenomenon applying the concept of fast-moving particles called projectiles. Multi-Verse Optimizer (MVO) [63], an environment lead heuristic algorithm, relies on three stages named: wormhole, black hole, and white hole. Virus Colony search (VCS) [64] is an environment-inspired method that affects the spreading and infection stages of the host cells followed by the virus for its survival in the cell environment. To find solutions for real-time problems, the Grasshopper Optimization algorithm (GOA) [65] follows grasshopper swarms behavior. Based on the thinking ability of the chicken swarm, the Chicken Swarm Optimization algorithm (CSO) [66] came into existence. Grey Wolf Optimizer-Sine Cosine Algorithm (GWO-SCA) [67] is a meta-heuristics optimizer correlating the nature of wolf with mathematical sine–cosine concepts. Crow Particle Swarm Optimization algorithm (CPO) [68] is a hybrid combination of crow search algorithm and particle swarm optimization. Whale Optimization technique (WOA) [69] is a hybridized combinatorial meta-heuristics technique of Whale and swarm human-based optimizers for finding perfect exploratory and convergence capabilities. Spotted Hyena Optimizer (SHO) [70] is a new meta-heuristic algorithm encouraged by the natural collaborative behavior of spotted hyenas in searching, encircling, and attacking the prey. Multi-Objective Spotted Hyena Optimizer (MOSHO) [71] is developed to reduce multiple objective functions. A modified adaptive butterfly optimization algorithm (BOA) [72] is developed based on butterfly observation that produces its fragrance when traveling in search of food from one place to another place. Binary Spotted Hyena Optimizer (SHO) [73] is a meta-heuristic algorithm introduced based on hunting behavior of spotted hyena which deals with discrete optimization problems. Hybrid Harris Hawks pattern search algorithm (HHO-PS) [74] is a meta-heuristic optimizer developed to figure out a newer version of Harris Hawks for finding a solution in local and global search. The Hybrid Harris Hawks-Sine–Cosine method (HHO-SCA) [75] is influenced by the virtuous behavior of Harris Hawks which added up with mathematical concepts of sine and cosine to increase its ability in exploration and exploitation phases. Bernstrain-Search Differential Evolution algorithm (EBSD) [76] belongs to a family of universal differential evolution algorithms, which is proposed based on mutation and crossover operators. Reliability-based design optimization algorithm (RBDO) [77] deals with the uncertainty factors like global convergence, complicated design variables. Table 1 presents a brief review on population based meta-heuristics.

Table 1 A brief review on few of population meta-heuristics

2.1 Literature survey on CHIMP variants

A specific related study has been provided in this area to investigate information regarding current developments linked to CHIMP variations, and recently developed methods by various researchers are mentioned. As demonstrated by the stated literature studies, the researcher has built a wide range of meta-heuristic and hybrid versions of CHIMP to solve various sorts of stochastic challenges. Various academics evaluated real-time troubles such as data mining, climatic and environment concerns, medication and pharmaceuticals, engineering design issues, picture segmentation, power flow, solar PV modules, and so on using a heuristic technique. The capacity of any algorithm to find a suitable balance between intensification and diversity determines the accuracy of its answer. According to research, slow convergence is a common problem with most heuristic algorithms. As a result, the computational efficiency suffers. As a result, the use of hybrid algorithms to improve solution efficiency is becoming increasingly popular. Various CHIMP approaches have also been successfully employed by many researchers to maximize specific objective functions. The ultimate objective of these methods is to discover the optimal solution to a problem.

Researchers have recently created novel CHIMP versions for a variety of applications one of which is the DCELM-ChOA algorithm; first, ELMs’ parameters are tuned dimensionally, and then, ChOA is applied to acclimatize input layer weights and moreover bias ELM to eventually shoot up the system’s stableness and reliability which was invented to obtain accurate X-ray for detection of COVID-19 positive [110]. RVFL-CHOA [111], the standard CHIMP, was enhanced with Random Vector Functional Link (RVFL); RVFL is used to foretell the instant power outcome of the network and the production of power of a solar dish/stirling power plant in a month. SSC [112] Sine–cosine and Spotted Hyena-based Chimp Optimization algorithm was introduced to fight against the limitations of slow convergence and stuck at local optima of ChoA technique and its efficacy was tested on six real-time engineering problems proving its effectiveness with other techniques. SChoA [113] deputes sine–cosine functions with chimp optimization algorithm to modify the equations of standard CHIMP in its hunting procedure in minimizing various limitations of ChoA technique.

The burning topic is the challenge of discovering solutions to difficulties for optimization. If the number of optimization parameters continues to grow, the complexity of optimization issues will increase. Furthermore, some of the proposed deterministic techniques are vulnerable to local optima entrapment. To solve such issues, meta-heuristic (MA) nature-inspired optimization approaches are used. The lack of starting assumptions and population dependency are two key features of these approaches. Even still, no optimization strategy has yet been discovered that can solve all optimization problems [114]. This inspired to create the Improved Chimp-Spotted Hyena Optimizer, a meta-heuristic hybrid variation optimizer (ICHIMP-SHO).

Chimp Optimization Algorithm (ChoA) [6] is designed based on the intelligence ability of Chimps in group hunts. This algorithm is developed to solve slow convergence speed, trapping in high-dimensional problems. Spotted Hyena optimizer (SHO) is a new upcoming optimizer influenced by the trapping behavior of spotted hyena. This technique benefits upon other meta-heuristics as follows:

(i) implementation of the algorithm is easy because of its simple structure;

(ii) it makes smooth continuous solutions in local optimum;

(iii) it has finer local and global search capability;

(iv) due to the continued diminution of search space, SHO convergence rate is faster. And this solves many types of engineering design problems [70].

Data mining feature selection and unit commitments are the major discrete optimization issues. To solve these problems, SHO is used. Feature selection targets unnecessary features and removes them from the data set and minimizes computation requirement, dimensionality, and results in better accuracy. In practice, real-time problems may have a huge number of features with relevant and irrelevant features. At that time, it is difficult for finding a solution. Then, the characteristic selection is treated as a combinatorial optimization problem. To solve this, selection feature problem binary meta-heuristics algorithms are used. Few examples are Binary Gravitational Search algorithm (BGSA) [115], Binary Grey Wolf optimizer (BGWO) [116], Binary Bat algorithm (BBA) [117, 118], and Binary Particle Swarm optimization (BPSO) [119].

Some of Spotted Hyena optimizer algorithm variants are: HPSSHO algorithm targets in improving hunting tactic of spotted hyena by merging standard SHO with Particle Swarm Optimization and tested on standard benchmark functions to prove its effectiveness in regulating to validate the significance of the proposed HPSSHO performance in assessment with state-of-the art optimization techniques; the parametric tests have been conducted on the benchmark functions [120]. HMOSHSSA [121], hybrid technique, uses MOSHO exploration skill, and SSA updates global search for finding best solution than the standard SHO. MOSHEPO [122] combined Multi-objective Spotted Hyena optimizer and Emperor Penguin Optimizer to contemplate many physical and operational constraints. To reduce heating effect, providing ventilation and air conditioning in the systems, a modification is carried out by merging four different meta-heuristic techniques: salp swarm, spotted hyena, wind-driven, and whale optimization algorithm with multilayer perceptron neural network to conquer computation time [123].

2.2 Novelty of proposed research work

(i) The spotted hyena optimizer is used to improve the local search capacity of ICHIMP in the suggested study.

(ii) The specifications of ICHIMP are not changed to preserve the original features of ICHIMP.

(iii) The ICHIMP-SHO method has been successfully applied for seven standard uni-modal benchmark functions, six standard multi-modal benchmark functions, ten standard fixed-dimension benchmark functions, and 11 types of interdisciplinary engineering design challenges.

(iv) The efficacy of the suggested algorithm has been validated by Wilcoxon Rank test.

(v) According to the comparative analysis shown in the results section, the proposed technique performs very well in terms of fitness evaluation and solution precision.

2.3 Background of suggested work

Chimps (Chimpanzees) correspond to a family of African genus of huge chimpanzee. The living style of them is close to humans. Brain-to-body ratio (BBR) of Chimps and Dolphins are alike to humans. It is noticed that mammals along BBR are generally understood to be brilliant [124]. The DNA of human and Chimp are alike as they are from same solitary ancestors that existed a few million years back. Chimps hunt in group. All the chimps in a group are not same according to their ability and brilliance, but they perform their duties as a part of a chimp colony. The hunting procedure entails their natural capacity to communicate among group to drive, chase, and assault in lower canopy. If the prey manages to flee throughout this procedure, the chimps will regroup and launch another attack. In this process, each chimp may switch places. The exhausted victim eventually runs out of energy and is attacked by the chimps. In this procedure, each matching approach has a probability based on the locations of chimps in a group and the prey. Despite a good convergence rate, CHIMP struggles to identify the most optimal solution. As a result, an improved approach is introduced to reduce this effect while increasing its effectiveness.

The literature survey on some newly developed CHIMP variants is: The paper [125] presented ChOA for training artificial neural network and proved best than other existing algorithms. Abbas et al. [126] used a new chimp optimization algorithm to train radial basis function neural network which is the utilized as a detector and further improvised to eradicate exploration and exploitation phases by upgrading ChOA and stood better with outstanding performance when compared with five well-noted algorithms. Heming Jia et al. applied enhanced chimp optimization algorithm (EChOA) in [127] and verified its effectiveness on standard benchmark functions in giving tough competition with other algorithms. Jianhao Wang et al. proposed Binary Chimp Optimization algorithm (BChOA) in [128] as the basic ChOA is not suitable in finding solutions for binary problems because of its continuous hunting nature. To evaluate its efficiency, it has been tested on 43 standard benchmark functions obtaining good results. ICHIMP in [129] is implemented to find solutions for dynamic economic load dispatch problems in single area. To overcome the drawbacks of ChOA to stuck in local optima, Di Wu et al. introduced Enhanced Chimp Optimization Algorithm (EChOA); here, highly disruptive polynomial mutation is involved to multiply the population in space to shoot up the diversity in the population, Spearman’s rank correlation coefficient calculates the highest and lowest fitness among chimps, and later, Beetle Antenna Search Algorithm (BAS) is used to evade local optimum by chimps with lowest fitness. The combination of these three strategies enhances the exploration and exploitation phases and is tested on 17 benchmark datasets to prove its efficacy. Abdul Jabbar et al. [130] proposed a fresh hybrid algorithm by merging chimp optimization with conjugate gradient algorithm and tested on ten optimization functions, proving that the combination noted good results in gaining optimal solutions. Essam et al. [131] introduced opposition-based Levy Flight chimp optimizer (IChOA) in which opposition-based learning is involved in increasing pop in initializing stage of ChOA and Levy Flight is responsible for improving exploitation ability. This combination brought good results when compared with other algorithms in obtaining better thermography images to detect breast cancer. Bismin et al. [132] introduced Chimp-CoCoWa-AODV to enhance the MANET performance.

The recommended calculation aims to increase the local search capacity of CHIMP utilizing Improved Chimp Optimizer; in an effort to speed up ICHIMP, a combination of ICHIMP-SHO is introduced. Seven standard uni-modal benchmark functions, six standard multi-model benchmark functions, ten standard fixed-dimension benchmark functions, and 11 types of interdisciplinary engineering design challenges are all used to evaluate it. The findings are superior to those of other algorithms now in use.

3 Improved chimp optimizer

Chimps hunt very cleverly remembering the previous track of their attacks and are very closely related to swarm intelligence strategy, and based on this behavior, an innovative algorithm known as Chimp Optimization Algorithm (ChoA) is introduced. Chimps hunt in a group very intelligently based on two phases, namely, exploration and exploitation. Chimps are divided into four parties specifically named driver, barrier, chaser, and attacker. They streamline themselves by chasing, driving, blocking, and attacking in trapping the prey.

The mathematical equations [Eqs. (1) and (2)] represent driving and chasing of the prey

$$\vec{D} = \left| {\vec{C}\vec{Y}_{{{\text{Prey}}}} ({\text{iteration}}) - \xi .\vec{Y}_{{{\text{Chimp}}}} ({\text{iteration}})} \right|$$

(1)

$$\vec{Y}_{{{\text{Chimp}}}} ({\text{iteration}} + 1) = \vec{Q}_{k} + \vec{Y}_{{{\text{Prey}}}} ({\text{iteration}}) - \vec{A}.\vec{D}.$$

(2)

Here, $\vec{A}$, $\xi$, and $\vec{C}$ = coefficient vectors, t = number of current iteration, ${\text{Chimp location vector}}\, = \vec{Y}_{\rm Chimp}$, and $\vec{Y}_{\rm Prey} \, = \,{\text{the vector of prey position}}$.

Coefficient vectors $\vec{A}$, $\xi$, and $\vec{C}$ are found out using Eqs. (3), (4), and (5).

In the improved chimp optimizer, Eqs. (1) and (2) have been modified as follows:

$$\vec{Y}_{\rm Chimp} ({\rm iteration} + 1) = \left\{ \begin{array}{ll} \vec{Y}_{\rm Prey} { \text{(iteration)}} - \vec{A}.\vec{D} &\quad {\rm if}\,\, {\xi > 0.5} \\ Chaotic\_value &\quad {\rm if}\,\, {\xi < 0.5} \end{array} \right.,$$

(2-i)

where ran(1) and ran(3) represent the random integer values and can be given by the following mathematical equation:

$${\text{ran(index)}} = \text{randi([1, SAN],1,3),}$$

(2-ii)

where SAN represents the search agent number;

$$\vec{A} = 2\vec{\eta }\nu_{1} - \vec{\eta }$$

(3)

$$\vec{C} = 2\nu_{2}$$

(4)

$$\xi = {\text{chaotic vector}}$$

(5)

$$x_{i + 1} = 1.07x_{i} (7.86x_{i} - 23.31x_{i}^{2} + 28.75x_{i}^{3} - 13.302875x_{i}^{4} ).$$

(6)

$\left| {\mathop A\limits^{ \to } } \right|$ Non-linearly decreases from 2.5 to 0 in both the phases iteratively. The vectors $\nu_{1}$ and $\nu_{2}$ are ranged [0, 1]. $\xi$ the chaotic vector serves chimps in the process of trapping (Fig. 3a).

In this hunting process usually, an attacker chimp leads this operation followed by driver, barrier, and chaser. Mathematically, the actions of Chimps are imitated in the sequence initially starting from an attacker, driver, and then barrier; chaser will give better lead to notice the position of prey. Up till now, the location of Chimps is to be updated immediately and store the best positions of Chimps. This process is reflected mathematically in the Eqs. (7), (8), and (9)

$$\vec{D}_{{\rm Attacker} } = abs\left| {\vec{C}_{1} \vec{Y}_{Attacker} - \vec{Y}} \right|.$$

(7a)

In the modify chimp algorithm, the $\vec{D}_{{\rm Attacker} }$ has been selected with the help of the following equation:

$$\vec{D}_{{\rm Attacker} } = \left\{ \begin{gathered} \left| {\vec{C}\vec{Y}_{{\rm Attacker} } \text{(iteration)} - \xi .\vec{Y} \text{(iteration)}} \right|\begin{array}{*{20}c} ; & {\left| A \right| < 1} \\ \end{array} \hfill \\ \left| {\vec{C}\vec{Y}_{{\rm Attacker} } (ran(1),iteration) - \xi .\vec{Y}(ran(3),iteration)} \right|\begin{array}{*{20}c} ; & {\left| A \right| > 1} \\ \end{array} \hfill \\ \end{gathered} \right.$$

(7.a-i)

$$\vec{D}_{\rm Barrier} = abs\left| {\vec{C}_{2} \vec{Y}_{\rm Barrier} - \vec{Y}} \right|.$$

(7.b)

In the modify chimp algorithm, the $\vec{D}_{\rm Barrier}$ has been selected with the help of the following equation:

$$\vec{D}_{\rm Barrier} = \left\{ \begin{gathered} \left| {\vec{C}\vec{Y}_{\rm Barrier} \text{(iteration)} - \xi .\vec{Y} \text{(iteration)}} \right|\begin{array}{*{20}c} ; & {\left| A \right| < 1} \\ \end{array} \hfill \\ \left| {\vec{C}\vec{Y}_{\rm Barrier} (ran(1),iteration) - \xi .\vec{Y}(ran(3),iteration)} \right|\begin{array}{*{20}c} ; & {\left| A \right| > 1} \\ \end{array} \hfill \\ \end{gathered} \right.$$

(7.b-i)

$$\vec{D}_{\rm Chaser} = abs\left| {\vec{C}_{3} \vec{Y}_{\rm Chaser} - \vec{Y}} \right|.$$

(7c)

In the modify chimp algorithm, the $\vec{D}_{\rm Chaser}$ has been selected with the help of the following equation:

$$\vec{D}_{\rm Chaser} = \left\{ \begin{gathered} \left| {\vec{C}\vec{Y}_{\rm Chaser} \text{(iteration)} - \xi .\vec{Y} \text{(iteration)}} \right|\begin{array}{*{20}c} ; & {\left| A \right| < 1} \\ \end{array} \hfill \\ \left| {\vec{C}\vec{Y}_{\rm Chaser} ({\rm ran}(1), {\rm iteration}) - \xi .\vec{Y}(ran(3),iteration)} \right|\begin{array}{*{20}c} ; & {\left| A \right| > 1} \\ \end{array} \hfill \\ \end{gathered} \right.$$

(7.c-i)

$$\vec{D}_{\rm Driver} = abs\left| {\vec{C}_{4} \vec{Y}_{\rm Driver} - \vec{Y}} \right|.$$

(7d)

In the modify chimp algorithm, the $\vec{D}_{\rm Driver}$ has been selected with the help of the following equation:

$$\vec{D}_{\rm Driver} = \left\{ \begin{gathered} \left| {\vec{C}\vec{Y}_{\rm Driver} \text{(iteration)} - \xi .\vec{Y} \text{(iteration)}} \right|\begin{array}{*{20}c} ; & {\left| A \right| < 1} \\ \end{array} \hfill \\ \left| {\vec{C}\vec{Y}_{\rm Driver} (ran(1),iteration) - \xi .\vec{Y}({\rm ran}(3), {\rm iteration})} \right|\begin{array}{*{20}c} ; & {\left| A \right| > 1.} \\ \end{array} \hfill \\ \end{gathered} \right.$$

(7.d-i)

Equation (2) mentioned above can be used to determine the spot of attacker, barrier, chaser, and driver as per Eqs. (8a)–(8d), respectively

$$\vec{Y}_{1} = \vec{Y}_{{\rm Attacker} } - \vec{A}_{1} .\vec{D}_{{\rm Attacker} }$$

(8a)

$$\vec{Y}_{2} = \vec{Y}_{\rm Barrier} - \vec{A}_{2} .\vec{D}_{\rm Barrier}$$

(8b)

$$\vec{Y}_{3} = \vec{Y}_{\rm Chaser} - \vec{A}_{3} .\vec{D}_{\rm Chaser}$$

(8c)

$$\vec{Y}_{4} = \vec{Y}_{\rm Driver} - \vec{A}_{4} .\vec{D}_{\rm Driver} .$$

(8d)

The overall final positions of all the chimps can be obtained by taking the mean of the attacker, barrier, chaser, and driver positions as per Eq. (9)

$$\vec{Y}({\rm iteration} + 1) = \frac{{(\vec{Y}_{1} + \vec{Y}_{2} + \vec{Y}_{3} + \vec{Y}_{4} )}}{4}.$$

(9)

To generate the initial arbitrary position of search agents, the below mathematical equation can be adopted

$$\vec{Y}_{\rm rand} = LB_{i} + \xi \times (UB_{i} - LB_{i} )\begin{array}{*{20}c} ; & {i \in 1,2,3,...,{\rm Dim}} \\ \end{array} .$$

(10)

The PSEUDO code for calculations of Y1, Y2, Y3, and Y4 are given in Fig. 2a, b.

Fig. 2

a PSEUDO code for calculation of Y1 and Y2. b PSEUDO code for calculation of Y3 and Y4

4 Spotted hyena optimizer

The spotted hyena lives in a group of no less than 100 individuals. They embark on hunting expeditions in groups. Spotted, striped, brown, and aardwolf are the four classifications. These are colossal hunters who know what they are doing. They create a sound that sounds like a human chuckle to communicate with one another. They have spots on their bodies. They devise coordinated arrays to encourage organizational understanding among hyenas.

SHO is mathematically illustrated by three stages, i.e., hunting, encircling, and finally attacking the prey. The present finest solution is prey which is nearer to optimum solution. Remaining hyenas renew their location once that finest solution is determined.

Mathematically, spotted hyenas encircling behavior is formulated using below equations

$$\vec{d}_{h} = \left| {\vec{y}\vec{Q}_{q} } \right.(s) - \vec{Q}(s)\left. {} \right|$$

(11)

$$\vec{Q}(s + 1) = \vec{Q}_{q} (s) - \vec{z}\vec{d}_{h} ,$$

(12)

where $\vec{d}_{h}$ = the gap among prey and hyena. $\vec{y}$ and $\vec{z}$ = coefficient vectors. s = the present iteration. $\vec{Q}_{q}$ = the vector spot of prey. $\vec{Q}$ = the vector spot of hyena. $\vec{y}$ and $\vec{z}$ are compared as follows:

$$\vec{y} = 2.\vec{r}_{1}$$

(13)

$$\vec{z} = 2.\vec{H}\vec{r}_{2} - \vec{H}$$

(14)

$$\vec{H} = 5 - ({\rm Itr} \times (5/{\rm Max}_{\rm itr} )),$$

(15)

where Itr = 1, 2, 3, …, Max_itr.

Here, $\vec{H}$ from 5 to 0 linearly decreases during iteration process, and maintains steadiness between exploration and exploitation. The random vectors $\vec{r}_{1}$, $\vec{r}_{2}$ ranged [0, 1]. The $\vec{y}$ and $\vec{z}$ values are fine tuned, such that hyenas move to other area about the present position. Using Eqs. (11) and (12), hyenas renew their points randomly all over the prey.

To structure the hunting activities of spotted hyenas, we expect finest searching agent has awareness regarding prey position. Remaining search agents designs an array which is of devoted friends and renews the location for the finest search agent.

Mathematically hunting is formulated as

$$\vec{d}_{h} = \left| {\vec{y}.\vec{Q}} \right._{h} - \vec{Q}_{k} \left. {} \right|$$

(16)

$$\vec{Q}_{k} = \vec{Q}_{h} - \vec{z}.\vec{d}_{h}$$

(17)

$$\vec{C}_{h} = \vec{Q}_{k} + \vec{Q}_{k + 1} + \cdots + \vec{Q}_{k + N} ,$$

(18)

where $\vec{Q}_{h}$ = first best position of spotted hyena. $\vec{Q}_{k}$ = the location of remaining spotted hyenas.

N = the count of spotted hyenas can be worked out as

$$N = {\rm count}_{Ns} (\vec{Q}_{h} ,\vec{Q}_{h + 1} ,\vec{Q}_{h + 2} ,...(\vec{Q}_{h} + \vec{M})),$$

(19)

where vector $\vec{M}$ ranges [0.5, 1]. N_s = the number of candidate solutions, related to the superlative optimum solution in search space. $\vec{C}_{h}$ is group of N optimum solutions.

To explain the attacking stage, it is necessary to reduce the value of H. Thus, difference in $\vec{z}$ is also reduced due to change in H value which diminished from 5 to 0 during iteration runs.

The mathematically attacking the prey (exploitation) is prearranged by

$$\vec{Q}(s + 1) = \frac{{\vec{C}_{h} }}{N},$$

(20)

where $\vec{Q}(s + 1)$ accumulates finest solution and further search agents renew their locations by the positions of finest searching agent. SHO permits their hyenas to renew their locations and attack the prey.

The searching behavior explains the exploration ability of an algorithm. SHO algorithm guarantees the ability of using $\vec{z}$ with random values > 1 or < − 1.

$\vec{y}$ takes the responsibility for more randomized behavior of SHO algorithm and avoids local optimal values.

Below Algorithm 2 depicts spotted hyena optimizer.

5 Proposed improved chimp optimizer (ICHIMP-SHO)

This work extends an enhanced version of hunting behavior of Improved Chimp optimizer by means of spotted hyena, as depicted in Fig. 3c. To experience this consequence, the driving and chasing Eqs. (1) and (2) of IChimp along with hunting behavior of spotted hyena in Eq. (17) are considered to modify into Eq. (21). The pseudo code for the suggested ICHIMP-SHO algorithm is discussed in Algorithm 3

$$\vec{Y}_{\rm Chimp} ({\rm iteration} + 1) = \vec{Q}_{k} + \vec{Y}_{\rm Prey} \text{(iteration)} - \vec{A}.\vec{D}.$$

(21)

Fig. 3

a 2D view for the position of prey and chimp, b 3D view for the position of prey and chimp, and c flowchart of proposed ICHIMP-SHO algorithm

The two-dimensional and three-dimensional views for the position of chimp from the respective prey are depicted in Fig. 3a, b, respectively.

The suggested ICHIMP-SHO variant is beneficial above few population-based meta-heuristic techniques mainly in three aspects as follows.

The first aspect refers in combining two conventional techniques to frame a simple new efficient simulation method which executes faster with complex mathematical operations when compared with other existing methods. The features of standard ICHIMP are injected to SHO technique as initial parameters to strengthen its power which excels in processing and endeavors to optimize these values to boost up the ability of ICHIMP to consider the optimal value of optimization issue. This process is done without involving complex operations.

The second aspect is the proposed new method succeeded in obtaining best results than the solution drawn by ICHIMP. The experimental result stands as proof in the result section displaying its performance in terms of numerically and experimentally. This makes difference between the suggested techniques with other techniques. Majorly, most of the techniques suffer to attain optimum solution with increasing number of iterations due to downside inability. The suggested method develops a vital and standard method to solve this issue which can be practiced by the other methods in optimization by considering the operating phases of this method.

The third aspect is the idea behind the ICHIMP-SHO method is to enhance the optimization strength of ICHIMP to attain the optimized values, but not the complexity of the algorithm. The suggested optimization technique is developed with incorporating SHO algorithm functionality to the ICHIMP. The above two mathematical models have independent structures for managing optimization. To combine them, the computational methods are utilized to transform the principles of one algorithm into the other algorithm. As such, in this research work, ICHIMP pattern is mapped into the SHO parameters and translating SHO attributes back to ICHIMP. Along with this procedure, new operators have been introduced to improve the sophistication of hybrid variants. To examine the proposed hybrid variant ICHIMP-SHO, 16 benchmark functions and 11 constrained engineering optimal issues are considered to verify with different types of parameter settings.

6 Standard benchmark functions

A cluster of unique benchmark functions [30, 105] is used to put the proposed ICHIMP-SHO optimization approach to the test. The standard benchmarks are categorized into uni-modal (UM), multi-modal (MM), and fixed dimensions (FD). Based on objective fitness, these standard benchmark functions are defined such as dimension, range limit, and optimum value (f_min). The mathematical formulations for UM, MM, and FD are displayed in Tables 2, 3, and 4, and their results are described in outcomes and discussion section. Thirty trial runs are used to test the performance of standard benchmark functions. Table 5 illustrates the proposed algorithm’s details of parameter setting.

The complete study is considered by 30 search agents, and maximum iterations of 500. The suggested ICHIMP-SHO was tested using the MATLAB R2016a software on an Intel corei3 processor laptop with a 7th generation CPU and 8GB RAM.

Table 2 Uni-modal (UM) standard benchmark functions

Table 3 Multi-modal (MM) standard functions

Table 4 Fixed-Dimension (FD) standard functions

Table 5 Parameter constraints for the proposed search method

The aforementioned parametric settings are the ideal choice for testing the proposed optimizer for standard benchmarks and engineering design challenges.

7 Outcomes and discussion

In this research work, the introduced Improved Chimp-Spotted Hyena Optimizer algorithm is tested on three major classes of standard benchmark functions to verify the presentation of the developed ICHIMP-SHO technique. The exploitation and convergence rate of ICHIMP-SHO is tested by uni-modal benchmark functions which have a single minimum. As the name multi-modal replicates which have more than one minimum, hence, these functions are utilized to test for exploration and avoid local optimum. The design variables are obtained by the difference between multi-modal and fixed-dimension benchmark functions. The fixed-dimension benchmark functions will store these design variables, and maintain a chart of previous data of search space and compare with multi-modal benchmark functions.

For comprehensive comparison analysis, a record of results of the developed ICHIMP-SHO algorithm was framed which were tabulated in the criteria of mean value, standard deviation, median value, the best value, worst value, and parametric tests by performing with 500 iterations and maximum runs of 30.

7.1 Evaluation of (F1–F7) functions (exploitation)

The test results for uni-modal (F1–F7) benchmark functions of suggested technique are illustrated in Tables 6, 7. The mean value and standard deviation were considered for evaluation of the test results with few newly developed meta-heuristic algorithms named LSA [62], BRO [133], OEGWO [134], PSA [59], HHO-PS [74], SHO [70], HHO [100], ECSA [135], and TSO [136], and are presented in Table 8. Its characteristic curves, trial runs, and convergence comparative curves with other algorithms are depicted in Figs. 4, 5, 6.

Table 6 Test observations of (F1–F7) functions using ICHIMP-SHO algorithm

Table 7 Execution Time for Uni-modal Benchmark Problems using ICHIMP-SHO algorithm

Table 8 Evaluation for (F1–F7) problems

Fig. 4

3D view of uni-modal (UM) standard benchmark problems

Fig. 5

Comparative curve of ICHIMP-SHO with GWO, DA, ALO, MVO, SSA, and PSO for UM standard bench mark functions

Fig. 6

Trial runs of ICHIMP and ICHIMP-SHO for UM standard bench mark functions

7.2 Evaluation of (F8–F13) functions (exploration)

The multi-modal benchmark functions (F8–F13) show the design variables in the desired number in the exploration phase. The test results are tabulated in Tables 9, 10. As well, the comparison of results was done considering mean value and standard deviation with other algorithms, such as LSA [55], BRO [106], OEGWO [107], PSA [40], hHHO-PS [67], SHO [63], HHO [51], ECSA [108], and TSO [109], and is recorded in Table 11. Also, its characteristics curves, trial runs, and convergence comparative curves with other algorithms are depicted in Figs. 7, 8, 9.

Table 9 Test results of multi-modal benchmark functions using ICHIMP-SHO algorithm

Table 10 Execution time for multi-modal benchmark problems using ICHIMP-SHO algorithm

Table 11 Comparison for multi-modal benchmark functions

Fig. 7

3D view of multi-modal (MM) standard benchmark problem

Fig. 8

Comparative curve of ICHIMP-SHO with GWO, DA, ALO, MVO, SSA, and PSO for MM standard bench mark functions

Fig. 9

Trial Runs of ICHIMP and ICHIMP-SHO for MM standard bench mark functions

7.3 Evaluation of (F14–F23) functions

The fixed-dimensional benchmark (F14–F23) functions do not manipulate the design variables, but prepare the previous search space record of multi-modal benchmark functions. Tables 12, 13 show the test results of proposed algorithm and Table 14 showcases the comparative analysis of mean value and standard deviation with LSA [55], ECSA [108], TSO [109], PSA [40], hHHO-PS [67], SHO [63], and HHO [51]. Figures 10, 11, 12 show characteristics curves, trial runs, and convergence comparative curves with other algorithms

Fig. 10

3D view of fixed-dimension (FD) modal standard benchmark functions

Fig. 11

Comparative curve of ICHIMP-SHO with GWO, DA, ALO, MVO, SSA, and PSO for fixed standard

Fig. 12

Trial Runs of ICHIMP and ICHIMP-SHO for fixed-dimension standard bench mark functions

.

Table 12 Test observations for Fixed Dimensions Functions using ICHIMP-SHO algorithm

Table 13 Execution time for fixed dimensions benchmark problems using ICHIMP-SHO algorithm

Table 14 Comparison for fixed-dimension benchmark functions

Hence, the test results for UM, MM, and FD benchmarks problems are tabulated in Tables 6, 7, 8, 9, 10, 11, 12, 13, 14, and the assessment of the proposed optimizer with other meta-heuristics search algorithms for UM, MM, and FD benchmark problems is given in Figs. 5, 8 and 11 and trial run solutions for UM, MM, and FD benchmarks problems are shown in Figs. 6, 9, and 12. The above result clearly shows that the proposed optimizer presents much better than other algorithms. In subsequent sections, the proposed optimizers have been applied to 11 engineering optimization problems.

8 Engineering-based optimization design problems

To validate the efficacy of the suggested ICHIMP-SHO algorithm, 11 types of engineering-based optimization designs are considered: pressure vessel, Speed reducer problem, Three-bar truss problem, welded beam, gear train design problem, belleville spring problem, cantilever beam design, rolling element bearing, (discrete variables), I-beam design, Multi-disk clutch break, and Tension/compression spring design problem. The results for engineering-based optimization design issues were examined using several meta-heuristic optimizers, and convergence curves were compared to the standard CHIMP method, as shown in Fig. 24. Table 15 describes the engineering-based optimization design problems; Table 16 presents best values (Best fit), the average value (Ave), median value (Median), standard deviation (SD), and worst value (Worst fit); Table 17 shows Wilcoxon P value and t test values and the computation time of engineering-based optimization design problems is shown in Table 18.

Table 15 Basic information of (SPECIAL1—SPECIAL11) engineering-based designs

Table 16 ICHIMP-SHO results for engineering design issues

Table 17 Parametric results using proposed ICHIMP-SHO Algorithm

Table 18 Results of computational time using proposed ICHIMP-SHO algorithm

8.1 Pressure vessel design

One of the multidisciplinary engineering optimization problems is depicted in Fig. 13, which is named Pressure Vessel design problem [137, 138]. The important aspect of this issue in engineering optimization design is to minimize or decrease the overall price, which includes material quality, welding, and the vessel's cylindrical form, as illustrated in Fig. 13. While, there are four types of factors utilized to create the pressure vessel issue (q1, q2, q3, and q4), such as shell thickness (T_s), head thickness (T_h), internal radius ®, and cylindrical unit length (L) which are taken into account. This vessel has end caps on either sides, and the structure's head is hemispherical in form. The four types of constraints described above are the topic of a design problem, and the mathematical specification issue for the pressure vessel is represented in Eqs. (22)–(23d). Table 19 summarizes the conclusions of the analysis. The following are the results of ICHIMP-SHO compared with various algorithms.

Fig. 13

Design of pressure vessel

Table 19 Comparative observations of ICHIMP-SHO for pressure vessel optimisation design issue with other algorithms

We consider

$$\vec{q} = \left[ {q_{1} q_{2} q_{3} q_{4} } \right] = \left[ {T_{s} T_{h} RL_{h} } \right].$$

(22)

To minimize

$$f(\vec{q}) = 0.6224q_{1} q_{3} q_{4} + 1.7781q_{2} q_{3}^{2} + 3.1661q_{1}^{2} q_{4} + 19.84q_{1}^{2} q_{3} .$$

(23)

Here

$$g_{1} \left( {\vec{q}} \right) = - q_{1} + 0.0193q_{3} \le 0$$

(23a)

$$g_{2} \left( {\vec{q}} \right) = q_{3} + 0.00954q_{3} \le 0$$

(23b)

$$q_{3} \left( {\vec{p}} \right) = - \pi q_{3}^{2} q_{4} - \frac{4}{3}\pi q_{3}^{3} + 1296000 \le 0$$

(23c)

$$g_{4} \left( {\vec{q}} \right) = q_{4} - 240 \le 0;$$

(23d)

Variable range, $0 \le q_{1} \le 99$

$$0 \le q_{2} \le 99$$

$$10 \le q_{3} \le 200$$

$$10 \le q_{4} \le 20.$$

8.2 Speed reducer

As illustrated in Fig. 14 [110], this type of design issue has seven variables. It is made up of the face width × 1, the teeth module × 2, the pinion teeth number × 3, the first shaft length bearings × 4, the second shaft length bearings × 5, the 1st shaft diameter × 6, and the 2nd shaft diameter × 7. The weight of the velocity reducer must be reduced first which is the main aim of this issue. Figure 14 depicts the engineering design of a speed reducer. Table 20 summarizes the results of the analysis. GSA [61], hHHO-SCA [68], PSO [18], OBSCA, MFO [122], SCA, HS [31], and GA [10] are compared to the analytical findings of ICHIMP-SHO. Equations (24)–(24k) show the mathematical framework of the pressure vessel optimization design issue. The following is how the equations are written

Fig. 14

Speed reducer design of engineering problem

.

Table 20 Comparative results of ICHIMP-SHO for speed reducer optimisation design issue with other algorithms

Minimizing

$$\begin{aligned} f(x) & = 0.7854x_{1} x_{2} (3.3333x_{3}^{2} + 14.9334x_{3} - 43.0934) - 1.508x_{1} (x_{6}^{2} + x_{7}^{2} ) \\ & \quad + 7.4777(x_{6}^{3} + x_{7}^{3} ) + 0.7854(x_{4} x_{6}^{2} + x_{5} x_{7}^{2} ). \\ \end{aligned}$$

(24)

Subjected to

$$g_{1} (\vec{x}) = \frac{27}{{x_{1} x_{2}^{2} x_{3} }} - 1 \le 0$$

(24a)

$$g_{2} (\vec{x}) = \frac{397.5}{{x_{1} x_{2}^{2} x_{3}^{2} }} - 1 \le 0$$

(24b)

$$g_{3} (\vec{x}) = \frac{{1.93x_{4}^{3} }}{{x_{2} x_{3} x_{6}^{4} }} - 1 \le 0$$

(24c)

$$g_{4} (\vec{x}) = \frac{{1.93x_{5}^{3} }}{{x_{2} x_{3} x_{7}^{4} }} - 1 \le 0$$

(24d)

$$g_{5} (\overrightarrow {x} ) = \frac{1}{{110x_{6}^{3} }}\sqrt {(\frac{{745.0x_{4} }}{{x_{2} x_{3} }}} )^{2} + 16.9 \times 10^{6} - 1 \le 0$$

(24e)

$$g_{6} (\overrightarrow {x} ) = \frac{1}{{85x_{7}^{3} }}\sqrt {(\frac{{745.0x_{5} }}{{x_{2} x_{3} }}} )^{2} + 157.5 \times 10^{6} - 1 \le 0$$

(24f)

$$g_{7} (\vec{x}) = \frac{{x_{2} x_{3} }}{40} - 1 \le 0$$

(24g)

$$g_{8} (\vec{x}) = \frac{{5x_{2} }}{{x_{1} }} - 1 \le 0$$

(24h)

$$g_{9} (\vec{x}) = \frac{{x_{1} }}{{12x_{2} }} - 1 \le 0$$

(24i)

$$g_{10} (\vec{x}) = \frac{{1.5x_{6} + 1.9}}{{12x_{2} }} - 1 \le 0$$

(24j)

$$g_{11} (\vec{x}) = \frac{{1.1x_{7} + 1.9}}{{x_{5} }} - 1 \le 0.$$

(24k)

Here

$$2.6 \le x_{1} \le 3.6,0.7 \le x_{2} \le 0.8,17 \le x_{3} \le 28,7.3 \le x_{4} \le 8.3,7.8 \le x_{5} \le 8.3,2.9 \le x_{6} \le 3.9and5 \le x_{7} \le 5.5.$$

8.3 Three-bar truss engineering design problem

To test the suggested ICHIMP-SHO algorithm output, this engineering design is considered which is figured in Fig. 15. The idea is to reduce the fitness value of the weight. It is imbibed with three constraints, namely, deflection constraint, buckling constraint, and stress constraint. Equations (25–26c) expose the three-bar truss problem numerically and its comparison results are tabulated in Table 21

Fig. 15

Three-bar truss engineering design issue

.

$${\text{Consider}}\;\mathop x\limits^{ \to } = [x_{1} ,x_{2} ] = [A_{1} ,A_{2} ]$$

(25)

$${\text{Minimize}}\;f(\mathop x\limits^{ \to } ) = (2\sqrt 2 x_{1} + x_{2} )*l$$

(26)

$${\text{Subject to}}\;g_{1} (\mathop x\limits^{ \to } ) = \frac{{\sqrt 2 x_{1} + x_{2} }}{{\sqrt 2 x^{2}_{1} + 2x_{1} x_{2} }}P - \sigma \le 0$$

(26a)

$$g_{2} (\mathop x\limits^{ \to } ) = \frac{{x_{2} }}{{\sqrt 2 x^{2}_{1} + 2x_{1} x_{2} }}P - \sigma \le 0$$

(26b)

$$g_{3} (\mathop x\limits^{ \to } ) = \frac{1}{{\sqrt 2 x_{2} + x_{1} }}P - \sigma \le 0.$$

(26c)

Table 21 Comparative observations of ICHIMP-SHO for three-bar truss optimisation design issue with other algorithms

Variable range $0 \le x_{1} ,x_{2} \le 1$.

Here, l = 100 cm, P = 2KN/cm², and σ = 2KN/cm².

8.4 Welded beam

In Fig. 16 [110, 111], this problem is depicted. The main focus is on lowering the welded beam's manufacturing costs: (i) bar height (h), (ii) weld thickness (h), (iii) bar length (l), and (iv) bar thickness (b) are the four variables which are all constrained by things like Buckling bar (Pc), End beam deflection (d), Side restrictions and shear stress (s), and Bending beam stress (h). The welded beam optimization design equations are presented in Eqs. (27)–(29f). In Table 22, the results of ICHIMP-SHO are compared to those of hHHO-SCA [68] and other algorithms.

Fig. 16

Welded mechanical beam model

Table 22 Comparative observations of ICHIMP-SHO for welded beam optimisation design issue with other algorithms

Let us consider

$$\vec{z} = \left[ {z_{1} z_{2} z_{3} z_{4} } \right] = \left[ {hltb} \right]$$

(27)

$$f(\vec{z}) = 1.10471z_{1}^{2} z_{2} + 0.04811z_{3} z_{4} \left( {14.0 + z_{2} } \right).$$

(28)

By addressing

$$g_{1} (\vec{z}) = \tau (\vec{z}) - \tau_{{\text{maxi}}} \le 0,$$

(28a)

$$g_{2} (\vec{z}) = \sigma (\vec{z}) - \sigma_{{\text{maxi}}} \le 0$$

(28b)

$$g_{3} (\vec{z}) = \delta (\vec{z}) - \delta_{{\text{maxi}}} \le 0$$

(28c)

$$g_{4} (\vec{z}) = z_{1} - z_{4} \le 0$$

(28d)

$$g_{5} (\vec{z}) = P_{i} - P_{c} (\vec{z}) \le 0$$

(28e)

$$g_{6} (\vec{z}) = 0.125 - z_{1} \le 0$$

(28f)

$$g_{7} (\vec{z}) = 1.10471z_{1}^{2} + 0.04811z_{3} z_{4} (14.0 + z_{2} ) - 5.0 \le 0.$$

(28g)

Range of variables: $0.1 \le z_{1} \le 2,0.1 \le z_{2} \le 10,0.1 \le z_{3} \le 10,0.1 \le z_{4} \le 2.$

Here

$$\tau (\vec{z}) = \sqrt {(\tau^{/} )^{2} + 2\tau^{/} \tau^{//} \frac{{z_{2} }}{2R} + (\tau^{//} )^{2} ,}$$

(29a)

$$\tau^{/} = \frac{{P_{i} }}{{\sqrt 2 z_{1} z_{2} }},\tau^{//} = \frac{MR}{J},M = P_{i} \left( {L + \frac{{z_{2} }}{2}} \right),$$

(29b)

$$R = \sqrt {\frac{{z_{2}^{2} }}{4} + \left( {\frac{{z_{1} + z_{3} }}{2}} \right)^{2} }$$

(29c)

$$J = 2\left\{ {\sqrt 2 z_{1} z_{2} \left[ {\frac{{z_{2}^{2} }}{4} + \left( {\frac{{z_{1} + z_{3} }}{2}} \right)^{2} } \right]} \right\}$$

(29d)

$$\sigma (\vec{y}) = \frac{{6P_{i} L}}{{z_{4} z_{3}^{2} }},\delta (\vec{y}) = \frac{{6P_{i} L^{3} }}{{Ez_{2}^{2} z_{4} }}$$

(29e)

$$P_{c} (\vec{z}) = \frac{{4.013E\frac{{\sqrt {z_{3}^{2} z_{4}^{6} } }}{36}}}{{L^{2} }}\left( {1 - \frac{{z_{3} }}{2L}\sqrt{\frac{E}{4G}} } \right)$$

(29f)

$$\begin{gathered} L = 14in,\delta_{\max i} = 0.25in,E = 30 \times 1^{6} psi,G = 12 \times 10^{6} psi, \hfill \\ \tau_{{\text{maxi}}} = 13600psi,\sigma_{{\text{maxi}}} = 3000psi,P = 6000lb. \hfill \\ \end{gathered}$$

8.5 Gear train design

Another form of engineering-based design optimization issue is the Gear Train Design problem, which includes four parameter categories, as shown in Fig. 17 [110]. The general objective of the architectural design is to minimize the scalar value of the gears and the teeth ratio. As a result, the teeth of each gear are considered in the decision variable. For the comparative study of ICHIMP-SHO, the analytical data are given in Table 23. The model for the relevant formulae is as follows:

Fig. 17

Design of gear train optimization design

Table 23 Comparative observations of ICHIMP-SHO for gear train optimisation design issue with other algorithms

Let us consider

$$\vec{G}e = \left[ {Ge_{1} Ge_{2} Ge_{3} Ge_{4} } \right] = \left[ {M_{A} M_{B} M_{C} M_{D} } \right].$$

(30)

To minimize

$$f(\overrightarrow {Ge} ) = \left( {\frac{1}{6.931} - \frac{{Ge_{3} Ge_{4} }}{{Ge_{1} Ge_{4} }}} \right)^{2} ;$$

(30a)

subjected to

$$12 \le Ge_{1} ,Ge_{2} ,Ge_{3} ,Ge_{4} \le 60.$$

(30b)

8.6 Belleville spring

This issue is depicted in Fig. 18. This is a technique used to reduce the problem by selecting a parameter that exists already in the constraints to the designed variable ratios. Belleville spring is designed with minimum weight in such a way to suit many designed variables, such as spring height (S_H), external part diameter (DIM_E), internal part diameter (DIM_I), and Belleville spring (S_T) thickness. Table 24 presents the comparison results. The constraints when subjected will be affected in deflection, deflection height, the internal and external portion of diameter, compressive types of stresses, and slope. The below equations are the mathematical expressions

$${\text{Minimizing}}; f(w) = 0.07075\pi (DIM_{E}^{2} - DIM_{I}^{2} )t;$$

(31)

$${\text{subjected to}};\;b_{1} (w) = G - \frac{{4P\lambda_{\max } }}{{(1 - \delta^{2} )\alpha DIM_{E} }}\left[ {\delta (S_{H} - \frac{{\lambda_{\max } }}{2}) + \mu t} \right] \ge 0$$

(32)

$$b_{2} (w) = (\frac{{4P\lambda_{\max } }}{{(1 - \delta^{2} )\alpha DIM_{E} }}\left[ {(S_{H} - \frac{{\lambda }}{2})(S_{H} - \lambda )t + t^{3} } \right])\lambda_{\max } - P_{MAX} \ge 0$$

(32a)

$$b_{3} (w) = \lambda_{1} - \lambda_{\max } \ge 0$$

(32b)

$$b_{4} (w) = H - S_{H} - t \ge 0$$

(32c)

$$b_{5} (w) = DIM_{MAX} - DIM_{E} \ge 0$$

(32d)

$$b_{6} (w) = DIM_{E} - DIM_{I} \ge 0$$

(32e)

$$b_{7} (w) = 0.3 - \frac{{S_{H} }}{{DIM_{E} - DIM_{I} }} \ge 0,$$

(32f)

where

$$\alpha = \frac{6}{\pi \ln J}\left( {\frac{J - 1}{{\ln J}} - 1} \right)^{2}$$

$$\delta = \frac{6}{\pi \ln J}\left( {\frac{J - 1}{{\ln J}} - 1} \right)$$

$$\mu = \frac{6}{\pi \ln J}\left( {\frac{J - 1}{2}} \right)$$

Fig. 18

Belleville spring engineering design

Table 24 Comparative results of ICHIMP-SHO for Belleville spring optimisation design problem with other algorithms

P_MAX = 5400 lb.

P = 30e6 psi, $\lambda_{\max }$ = 0.2 in, $\delta$ = 0.3, G = 200 Kpsi,

H = 2 in, DIM_MAX = 12.01 in, $J = \frac{{{\text{DIM}}_{E} }}{{{\text{DIM}}_{I} }}$, $\lambda_{1} = f(a)a,a = \frac{{S_{H} }}{t}$.

8.7 Cantilever beam design

As shown in Fig. 19, the goal of this civil-based engineering problem is to reduce beam weight. This is made up of five different sorts of shapes [111]. The final goal is to minimize the weight of the beam, as illustrated in Fig. 19. It is also granted upon by any single variable, and the entire design configuration comprises structural characteristics of five types, with the beam thickness being kept constant. To avoid infringing on Eqs. (33)–(34) for the design of the final optimum solution, the location of the vertical constraint should be calculated throughout the design procedure confront. Table 25 compares the results to those of other techniques. ICHIMP-SHO observations fared better than other algorithms. The following is the design formula:

Fig. 19

Design of cantilever beam design

Table 25 Comparative results of ICHIMP-SHO for cantilever beam optimisation design issue with other algorithms

Let us consider $\vec{L} = [L_{1} L_{2} L_{3} L_{4} ]$

$$f(\vec{L}) = 0.6224(L_{1} + L_{2} + L_{3} + L_{4} + L_{5} ).$$

(33)

By addressing

$$g(\vec{L}) = \frac{61}{{L_{1}^{3} }} + \frac{37}{{L_{2}^{3} }} + \frac{19}{{L_{3}^{3} }} + \frac{7}{{L_{4}^{3} }} + \frac{1}{{L_{5}^{3} }} \le 1.$$

(34)

Ranges of variables are $0.01 \le L_{1} ,L_{2} ,L_{3} ,L_{4} ,L_{5} \le 100$.

8.8 Rolling element bearing

The main aim of this design issue is to improve the rolling part's dynamic bearing ability, as shown in Fig. 20 [110, 147]. This problem in engineering design has ten choice variable numbers: (i) pitch diameter (DIM_P), (ii) ball diameter (DIM_B), (iii) ball numbers (N_b), (iv) outer raceway curvature coefficient, and (v) inner raceway curvature coefficient. The following five variables (KD_min, KD_max, ℇ, e, and f), which are only evaluated for discrete integers, have an impact on the interior section of the geometry. On kinematic circumstances and specifications, a total of nine nonlinear restrictions are challenged. Table 26 compares the results of ICHIMP-SHO with other known methods for the rolling bearing design problem. From Eqs. (35a) through (35c), the mathematical formulation for the tendered engineering design is shown.

Fig. 20

Problem of rolling bearing design

Table 26 Comparative results of ICHIMP-SHO for rolling element beam optimisation design problem with other algorithms

For maximizing

$$C_{D} = f_{c} N^{2/3} {\text{DIM}}_{B}^{1.8} .$$

(35a)

If ${\text{DIM}} \le 25.4\;{\text{mm}}$

$$C_{D} = 3.647f_{C} N^{2/3} {\text{DIM}}_{B}^{1.4} .$$

(35b)

If ${\text{DIM}} \ge 25.4\;{\text{mm}}$.

Addressing

$$r_{1} (y) = \frac{{\theta_{0} }}{{2\sin^{ - 1} \left( {\frac{{DIM_{B} }}{{DIM_{MAX} }}} \right)}} - N + 1 \ge 0$$

(36)

$$r_{2} (y) = 2DIM_{B} - K_{{DIM_{MIN} }} (DIM - \dim ) \ge 0$$

(36a)

$$r_{3} (y) = K_{{DIM_{MAX} }} (DIM - \dim ) \ge 0$$

(36b)

$$r_{4} (y) = \beta B_{W} - DIM_{B} \le 0$$

(36c)

$$r_{5} (y) = DIM_{MAX} - 0.5(DIM + \dim ) \ge 0$$

(36d)

$$r_{6} (y) = DIM_{MAX} - 0.5(DIM + \dim ) \ge 0$$

(36e)

$$r_{7} (y) = (0.5 + re)(DIM + \dim ) \ge 0$$

(36f)

$$r_{8} (y) = 0.5(DIM - DIM_{MAX} - DIM_{B} ) - \alpha DIM_{B} \ge 0$$

(36g)

$$r_{9} (y) = f_{I} \ge 0.515$$

(36h)

$$r_{10} (y) = f_{0} \ge 0.515.$$

(36i)

Here

$$f_{c} = 37.91\left[ {1 + \left\{ {1.04\left( {\frac{1 - \varepsilon }{{1 + \varepsilon }}} \right)^{1.72} \left( {\frac{{f_{I} \left( {2f_{0} - 1} \right)}}{{f_{0} \left( {2f_{I} - 1} \right)}}} \right)^{0.41} } \right\}^{10/3} } \right]^{ - 0.3} \times \left[ {\frac{{\varepsilon^{0.3} \left( {1 - \varepsilon } \right)^{1.39} }}{{\left( {1 + \varepsilon } \right)^{1/3} }}} \right]\left[ {\frac{{2f_{I} }}{{2f_{I} - 1}}} \right]^{0.41}$$

$$\theta_{0} = 2\pi - 2\cos^{ - 1} \left( {\frac{{\left[ {\left\{ {(DIM - \dim )/2 - 3(t/4)} \right\}^{2} + \left( {DIM/2 - t/4 - DIM_{B} } \right)^{2} - \left\{ {\dim /2 + t/4} \right\}^{2} } \right]}}{{2\left\{ {(DIM - \dim )/2 - 3(t/4)} \right\}\left\{ {D/2 - t/4 - DIM_{B} } \right\}}}} \right)$$

$$\varepsilon = \frac{{DIM_{B} }}{{DIM_{MAX} }},f_{I} = \frac{{R_{I} }}{{DIM_{B} }},f_{0} = \frac{{R_{0} }}{{DIM_{B} }},t = DIM - \dim - 2DIM_{B}$$

$$DIM = 160,\dim = 90,B_{W} = 30,R_{I} = R_{0} = 11.033$$

$$0.5\left( {DIM + \dim } \right) \le DIM_{MAX} \le 0.6\left( {DIM + \dim } \right),0.15\left( {DIM - \dim } \right) \le DIM_{B} \le 0.45\left( {DIM - \dim } \right),4 \le N \le 50$$

$0.515 \le f_{I}$ and $f_{0} \le 0.6$

$$0.4 \le K_{{DIM_{MIN} }} \le 0.5,0.6 \le K_{{DIM_{MAX} }} \le 0.7,0.3 \le re \le 0.1,0.02 \le re \le 0.1,0.6 \le \beta \le 0.85.$$

8.9 I-beam design

By altering the four parameters of the vertical I-beam, this engineering issue attempts to minimize vertical I-beam deviation. The four parameters b, h, t_w, t_f are shown in Fig. 21. In [150], it is stated that to obtain the dimensions of the beam shown in the figure, it has to satisfy geometric and strength constraints to optimize with the criteria: (1) cross-section of beam reduces its volume for given length; (2) static deflection to be noted when the beam is displaced on applying force. The mathematical formulations are given in Eqs. (37–39). Table 27 compares the analytical findings of ICHIMP-SHO with those of other well-known techniques.

Fig. 21

I beam design and structure

Table 27 Comparative results of ICHIMP-SHO for I-beam optimisation design problem with other algorithms

Consider

$$\vec{x} = [\begin{array}{*{20}c} {x_{1} } & {x_{2} } & {x_{3} } \\ \end{array} \begin{array}{*{20}c} {} & {x_{4} } & {x_{5} ]} \\ \end{array} = [\begin{array}{*{20}c} b & h & {t_{w} } \\ \end{array} \begin{array}{*{20}c} {} & {t_{f} } \\ \end{array} ],$$

(37)

$${\text{minimize}}\;\;f(\vec{x}) = \frac{5000}{{\frac{{t{}_{w}(h - 2t_{f} )^{3} }}{12} + \frac{{bt_{f}^{3} }}{6} + 2bt_{f} (\frac{{h - t_{f} }}{2})^{2} }},$$

(38)

$${\text{subjected to}} \;g(x) = 2bt_{w} + t_{w} (h - 2t_{f} ) \le 0,$$

(39)

${\text{variable range}}\;10 \le x_{1} \le 50,10 \le x_{2} \le 80,0.9 \le x_{3} \le 5,0.9 \le x_{4} \le 5.$

8.10 Tension/compression spring design problem

This is a component of the mechanical engineering problem [110, 111], and is one of the engineering designs constraints shown in Fig. 22. The proposal's main characteristic is that it reduces the spring weight. To solve the Spring Model Tension/Compression problem, three types of variable designs are needed: wire diameter (dwr), mean coil diameter (D_c), and active coil number (N). The amount of the surge, the minimal variance, and the limitations centered on the shear stress all play a role in the design. Equations (40)–(41d) show the numerical equations for the suggested engineering optimization design issue. The results of ICHIMP-SHO are compared to those of other techniques, as shown in Table 28.

Fig. 22

The spring engineering tension/compression problem

Table 28 Comparative results of ICHIMP-SHO for the spring engineering tension/compression problem with other algorithms

Let us consider

$$\vec{S} = \left[ {S_{1} S_{2} S_{3} } \right] = \left[ {dwrD_{c} N} \right].$$

(40)

But to minimize

$$f(\vec{S}) = \left( {S_{3} + 2} \right)S_{2} S_{1}^{2}$$

(41)

$$g_{1} (\vec{S}) = 1 - \frac{{S_{2}^{3} S_{3} }}{{71785S_{1}^{4} }} \le 0$$

(41a)

$$g_{2} (\vec{S}) = \frac{{4S_{2}^{2} - S_{1} S_{2} }}{{12566\left( {S_{2} S_{1}^{3} - S_{1}^{4} } \right)}} + \frac{1}{{5108S_{1}^{2} }} \le 0$$

(41b)

$$g_{3} (\vec{S}) = 1 - \frac{{140.4S_{1} }}{{S_{2}^{2} S_{3} }} \le 0$$

(41c)

$$g_{4} (\vec{S}) = \frac{{S_{1} + S_{2} }}{1.5} - 1 \le 0.$$

(41d)

Ranges of variables are $0.005 \le S_{1} \le 2.00,0.25 \le S_{2} \le 1.3,2.00 \le S_{3} \le 1$.

8.11 Multi-disk clutch break (discrete variables)

The multi-disk clutch brake design challenge [179] is one of the most critical technical difficulties highlighted in Fig. 23. The technique of optimization’s main purpose is to reduce or increase weight; however, it is made up of five discrete variables: friction surface number (S_fn), disk thickness (T_h), outer surface radius (O_sr), actuating force form (F_ac), and inner surface radius (I_sr). From Eqs. (42)–(43g), the mathematical formulas for this design are shown. Table 29 compares the findings of ICHIMP-SHO with those of other techniques.

Fig. 23

Multiple clutch break design

Table 29 Comparative observations of ICHIMP-SHO for multiple clutch optimisation design problem with other algorithms

Mathematical formulas for optimization design are provided below as follows:

$$f\left( {O_{sr} ,I_{sr} ,S_{fn} ,T_{h} } \right) = \pi Th\gamma \left( {O_{sr}^{2} - I_{sr}^{2} } \right)\left( {S_{fn} + 1} \right),$$

(42)

where,

$$I_{sr} \in 60,61,62....80;O_{sr} \in 90,91,.....110;T_{h} \in 1,1.5,2,2.5,3;F_{ac} \in 600,610,620,1000;S_{fn} \in 2,3,4,5,6,7,8,9,$$

subjected to

$$cb_{1} = D_{0} - D_{in} - \Delta D \ge 0$$

(43)

$$cb_{2} = L_{MAX} - (S_{f} + 1)(Th + \alpha ) \ge 0$$

(43a)

$$cb_{3} = PM_{MAX} - PM_{\pi } \ge 0$$

(43b)

$$cb_{4} = PM_{MAX} Z_{MAX} + PM_{\pi } Z_{SR} \ge 0$$

(43c)

$$cb_{5} = Z_{{SR_{MAX} }} - Z_{SR} \ge 0$$

(43d)

$$cb_{6} = t_{MAX} - t \ge 0$$

(43e)

$$cb_{7} = RC_{h} - RC_{f} \ge 0$$

(43f)

$$cb_{8} = t \ge 0$$

(43g)

Here,

$PM_{\pi } = \frac{{F_{ac} }}{{\Pi \left( {D_{0}^{2} - D_{in}^{2} } \right)}}$

$$Z_{SR} = \frac{{2\pi n\left( {D_{0}^{3} - D_{in}^{3} } \right)}}{{90\left( {D_{0}^{2} - D_{in}^{2} } \right)}}$$

$$t = \frac{{i_{x} \pi n}}{{30\left( {RC_{h} + RC_{f} } \right)}}.$$

9 Conclusion

In the proposed research, two hybrid variants of chimp optimizers have been successfully developed and named as Imp-Chimp and Imp-Chimp-SHO, which are based on a wholesome attitude roused by amazing thinking and hunting ability with a sensual movement for finding the optimal solution in the global search region. The newly developed improved variant of Chimp optimizer has been successfully tested for various engineering design and standard benchmark optimization problems, which includes uni-modal, multi-modal, and fixed dimensions benchmark problems. After validating the efficiency of the proposed optimizers for standard benchmarks and engineering design problems, it has been experimentally observed that both the variants are competitive for finding the solution within the global search space. Based on experimental results and comparative analysis with other methodologies, it has been recommended that the proposed hybrid variants can be universally accepted to solve any of the hard engineering design challenges in the global search space. However, while dealing with these two variants as compared to the standard ChoA, both the algorithms are slow with respect to computational complexity due to sequential hybridized nature of the algorithm (Fig. 24).

Fig. 24

Convergence curve and Trial runs for multidisciplinary engineering design problem with ICHIMP and ICHIMP-SHO

Furthermore, these hybrid variants can be applied to solve the single and multi-area economic load dispatch problem with renewable energy sources, charging and discharging of PEVs/BEVs, storage strategies, automatic generation, and monitoring functions of the realistic power system. Furthermore, the developed hybrid algorithm versions will aid various academics and upcoming analysts working on new population-based approaches, unique optimization strategies, and the development of hybrid optimization algorithms.