Introduction

Optimizing supply chain operations has become essential for organizations looking to remain competitive and better meet customer needs in today’s dynamic business environment. However, traditional optimization methods often fail to meet the challenges of modern supply chains, which are characterized by diverse stakeholders, overlapping processes, and ongoing market dynamics the characteristics of the solution1.

This paper sets the stage for exploring new bio-inspired metaheuristic algorithms specifically designed to solve supply chain optimization problems. The algorithm of supply-chain management provides a promising approach for optimization in various aspects, from inventory control to production scheduling and logistics systematically2.

Optimization problems are a type of problems that have more than one feasible solution. Optimization is the process of finding the best feasible solution among all the available solutions for an optimization problem3. Decision variables, constraints, and objective function are the three main parts in the mathematical model of any optimization problem. The goal in the optimization process is to determine the appropriate values for the decision variables, provided that the objective function becomes optimal (maximum or minimum) by respecting the constraints of the problem4. Numerous optimization problems in science, mathematics, engineering, and real-world applications need to be solved using appropriate techniques. Solving techniques for optimization problems are available in two groups: deterministic and stochastic approaches5. Deterministic approaches in two classes, gradient-based and non-gradient-based, are effective tools for tackling linear, convex, continuous, differentiable, and low-dimensional optimization problems6. Meanwhile, as the optimization problems become more complex and especially the dimensions of the problem increase, the deterministic approaches are stopped by getting stuck in the local optima7. On the other hand, many practical optimization problems are non-linear, non-convex, discontinuous, non-differentiable, and high-dimensional8. The inability and difficulties of deterministic approaches in tackling such optimization problems have led researchers to develop stochastic approaches3.

Metaheuristic algorithms are one of the most effective stochastic approaches to solve optimization problems. Metaheuristic algorithms are able to provide suitable solutions for optimization problems without the need for a derivation process, based on random search in the problem-solving space and using trial and error processes. The optimization process in metaheuristic algorithms starts with randomly generating a certain number of solvable solutions as the population of the algorithm. Then, these initial candidate solutions are improved based on the updating steps of the algorithm and changing their position in the problem solving space. The process of updating and improving candidate solutions continues until the last iteration of the algorithm. At the end, the best candidate solution obtained is presented as the solution for the given problem9. This nature of random search in the performance of metaheuristic algorithms leads to there being no guarantee to achieve the global optimum. However, because the solutions obtained from metaheuristic algorithms are close to the global optimum, they are acceptable as quasi-optimal solutions for optimization problems. Achieving better quasi-optimal solutions for optimization problems has become the main motivation for researchers to develop numerous metaheuristic algorithms10. These metaheuristic algorithms have been used to handle optimization tasks in various fields such as: feature selection11, engineering applications12,13, power engineering14, parallel implementation on the graphics processing unit (GPU)15, data classification16, internet of things (IOT)17, and supply chain management18.

The condition for a metaheuristic algorithm to have a successful performance in optimization is that the random search process mechanism is well managed at both global and local levels. Global search management with the concept of exploration expresses the ability of the metaheuristic algorithm to comprehensively scan the problem solving space in order to prevent the algorithm from getting stuck in local optima and discover the main optimal area. Local search management with the concept of exploitation expresses the ability of the metaheuristic algorithm to accurately scan the problem solving space in promising areas and around the obtained solutions. In addition to exploration and exploitation abilities, balancing them during the search process is the main key in providing an effective search process in the problem solving space in order to achieve a suitable solution for optimization problems19.

The main research question is that despite the numerous metaheuristic algorithms developed so far, is there still a need to design newer metaheuristic algorithms? The No Free Lunch (NFL)20 theorem answers the question that there is no unique metaheuristic algorithm that is the best optimizer for all optimization problems. In fact, the effective performance of a metaheuristic algorithm in handling a set of optimization problems is no guarantee for the similar performance of that algorithm in handling other optimization problems. According to the NFL theorem, there is no assumption about the success or failure of the result of implementing a metaheuristic algorithm on an optimization problem. By keeping the study field of metaheuristic algorithms active, the NFL theorem motivates researchers to be able to achieve better solutions for optimization problems by designing newer algorithms.

Based on the best knowledge obtained from the literature review, no metaheuristic algorithm has been designed so far inspired by the natural behavior of bobcats. Meanwhile, the bobcat hunting strategy during the stages of moving towards the prey and chasing is an intelligent process that has a special potential for designing a new optimizer. In order to address this research gap, a new biomimetics metaheuristic algorithm is introduced in this paper based on the mathematical modeling of the hunting strategy of bobcats in the wild. Therefore, the originality of the proposed approach is guaranteed and it is confirmed that this is the first time that a metaheuristic algorithm inspired by the natural behavior of the bobcat has been introduced and designed.

The innovation and novelty of this paper is in the introduction of a new metaheuristic algorithm called Bobcat Optimization Algorithm (BOA) which is used to solve optimization problems. In order to express the novelty aspects of the Bobcat Optimization Algorithm (BOA) and justify why this proposal is necessary, despite the existence of other wildcat-inspired optimizers, the unique aspects and specific advantages of BOA are highlighted:

Unique hunting strategy emulation

The BOA uniquely models the bobcat’s hunting strategy, which involves distinct phases of tracking and ambushing prey, followed by a chase to catch the prey. This dual-phase approach mirrors both global exploration and local exploitation in optimization, providing a balanced search mechanism.

Adaptive predator behavior

The algorithm mimics the bobcat’s adaptability to different prey and environments. This results in a dynamic balance between exploration and exploitation, where the algorithm adjusts its search behavior based on the problem landscape and current performance, leading to potentially better optimization results.

Phase-specific position updates

The BOA uses different mathematical models for position updates during the exploration and exploitation phases. The exploration phase involves extensive position changes to search broadly, while the exploitation phase makes smaller, refined adjustments to converge on the optimal solution. This phase-specific update mechanism enhances the algorithm’s efficiency and accuracy.

The Bobcat Optimization Algorithm (BOA) presents a novel and justified approach to solving optimization problems by leveraging the unique hunting strategies of bobcats. Its distinct exploration and exploitation phases, adaptive behavior, and empirical validation against benchmark problems establish its novelty and necessity in the field of optimization. By addressing the limitations of existing algorithms and providing a balanced search mechanism, the BOA contributes valuable diversity to optimization techniques, aligning with the principles of the NFL theorem. Focusing on these points, the novelty and justification of BOA is clearly confirmed, and a powerful optimization tool is established to develop and use it in optimization programs. The main contributions of this paper are as follows:

  • BOA is designed based on mimicking bobcat behaviors in the wild.

  • The fundamental inspiration of BOA comes from the bobcat’s hunting strategy.

  • The BOA theory is stated and its mathematical model is presented in two phases (i) exploration based on the simulation of the bobcat’s position change while moving towards the prey and (ii) exploitation based on simulating the bobcat’s position change during the chase process to catch the prey.

  • The performance of BOA in optimization is tested to solve CEC 2017 test suite for problem dimensions of 10, 30, 50, and 100, as well as to address CEC 2020.

  • The performance of BOA is challenged in real-world applications to solve twenty-two constrained optimization problems from the CEC 2011 test suite and four engineering design problems.

  • The performance of BOA is compared with the performance of twelve well-known competitor algorithms.

  • The performance of BOA in handling SCM applications is challenged on sustainable lot size optimization.

The structure of the paper is as follows; literature review is presented in Sect. “Literature review”. Then the proposed BOA is introduced and modeled in Section “Bobcat Optimization Algorithm”. Simulation studies and results are presented in Section “Simulation studies and results”. The effectiveness of BOA in solving real-world applications is investigated in Section “BOA for real-world applications”. The efficiency of BOA to deal with sustainable lot size optimization is evaluated in Section “BOA for Supply Chain Management (SCM)”. Analysis and discussion of the results, advantages and disadvantages of the proposed approach are presented in “Discussion”. Conclusions and suggestions for future research are provided in Sect. “Conclusion and future works”.

Literature review

Companies operating in local and international supply chains (SCs) are increasingly considering sustainability when making decisions21. Plus, the durability they are usually divided into three categories according to area of interest: social, economic, and environmental sustainability22. Complex supply chain management a Multi-channel supply chain. Offline and e-commerce management Supply chain companies have now become increasingly diversified chain processes23. Rapid supply chain management in high-demand markets it is important because supply chain management can be influenced in response to increased demand Quality of goods and services24.

Lot size optimization plays an important role in supply chain management, as it directly affects inventory costs, product quality, and customer satisfaction Over the years, researchers have explored various optimization techniques and strategies to overcome the challenges of in lot size determination is addressed25.

Traditional lot-sizing techniques, such as economic order quantity (EOQ) and production order quantity (POQ) have long been used for inventory management. These deterministic models aim to minimize all relative costs by a storage costs and balancing ordering/setup costs. Although EOQ and POQ provide convenience and ease of use, they often fail to account for dynamic demand patterns, timing variations, and other real-world complexities26.

Researchers have developed stochastic lot-size models to deal with the uncertainties in supply chain operations. This model incorporates probabilistic demand forecasting, lead time variability, and other sources of uncertainty into the optimization framework. Stochastic programming, dynamic programming, and simulation-based approaches are often used to solve stochastic lot size problems. By explicitly considering uncertainty, stochastic models provide more robust and realistic lot size decisions. In recent years, bio-inspired metaheuristic algorithms have gained popularity in lot size optimization in supply chain management27.

These metaheuristic algorithms have been developed with inspiration from natural phenomena, swarming behavior of living organisms in nature, genetic and biological sciences, concepts of physical science, human activities, and other evolutionary phenomena. Based on the main source of inspiration used in the design, metaheuristic algorithms are placed in five groups: swarm-based, evolutionary-based, physics-based, and human-based approaches.

Swarm-based metaheuristic algorithms are designed inspired by the lifestyles of insects, birds, aquatic animals, animals, plants, and other living organisms in the wild. Particle Swarm Optimization (PSO)28, Ant Colony Optimization (ACO)29, Artificial Bee Colony (ABC)30, and Firefly Algorithm (FA)31 are among the most prominent swarm-based metaheuristic algorithms. The design idea in PSO comes from the swarming behavior of birds and fish searching for food resources. The behavioral characteristic of ants in identifying the shortest communication path between the colony and food location has been employed as a source of inspiration in ACO design. The source of inspiration for the design of ABC is derived from the hierarchical activities and cooperation of colony bees with the aim of obtaining food resources. The origin of the FA design comes from the ability of fireflies to establish and exchange information through optical communication. The source of inspiration in the design of Golden Jackal Optimization (GJO) comes from the collaborative hunting behavior of the golden jackals in the wild, consisting of three steps: prey searching, enclosing, and pouncing32. African Vultures Optimization Algorithm (AVOA) is proposed inspired by foraging and navigation strategies among African vultures33. Foraging, hunting, chasing, migration, digging, and reproduction are among the most prominent activities of living organisms in the wild, which have been employed as a source of inspiration in the design of algorithms such as: Woodpecker Mating Algorithm (WMA)34, WMA based on opposition-based learning (OWMA)35, Hybrid Sine Cosine Algorithm-Woodpecker Mating (HSCWMA)36, Hybrid Woodpecker Mating Algorithm-Whale Optimization Algorithm (HWMWOA)37, Orca Predation Algorithm (OPA)38, Electric Eel Foraging Optimization (EEFO)39, Elk Herd Optimizer (EHO)40, Reptile Search Algorithm (RSA)41, White Shark Optimizer (WSO)42, Termite Alate Optimization Algorithm (TAOA)43, Fire Hawk Optimizer44, Spider Wasp Optimization (SWO)45, Marine Predator Algorithm (MPA)46, Honey Badger Algorithm (HBA)47, Piranha Foraging Optimization Algorithm (PFOA)48, and Snow Ablation Optimizer (SAO)49.

Evolutionary-based metaheuristic algorithms are designed with inspiration from the sciences of genetics, biology, concepts of natural selection, and survival of the fittest. Genetic Algorithm (GA)50 and Differential Evolution (DE)51 are among the most well-known evolutionary-based metaheuristic algorithms, the source of inspiration in their design comes from the reproduction process, genetic concepts, Darwin’s evolutionary theory, and the evolutionary-random operators of mutation, crossover, and selection. The idea of designing Artificial Immune Systems (AISs) is derived from the confrontation of the human immune system with diseases and microbes52. Some other evolutionary-based metaheuristic algorithms are: Cultural Algorithm (CA)53, Genetic programming (GP)54, and Evolution Strategy (ES)55.

Physics-based metaheuristic algorithms are designed with inspiration from forces, laws, transformations, transitions, processes, phenomena, laws, and other concepts in physics. Simulated Annealing (SA) is one of the most popular Physics-based metaheuristic algorithms, whose design imitates the annealing process of metals, during which metals are melted with high heat, and then, with the aim of achieving an ideal crystal, the metals are cooled slowly56. Physical forces have been the source of design for the development of algorithms such as: Spring Search Algorithm (SSA)57 inspired by the implementation of Hooke’s law and the tensile force of the spring between weights connected to each other with springs, Momentum Search Algorithm (MSA)58 inspired by the force derived from the collision momentum between two balls, and Gravitational Search Algorithm (GSA)59 inspired by from the gravitational force of attraction between masses that are located at different distances from each other. The Water Cycle Algorithm (WCA) design idea comes from the state transformations of water in its natural cycle60. Henry gas solubility optimization (HGSO) is developed inspired by the behavior governed by Henry’s law61. Some other physics-based metaheuristic algorithms are: Young’s Double-Slit Experiment (YDSE) optimizer62, Prism Refraction Search (PRS)63, Chernobyl Disaster Optimizer (CDO)64, Kepler Optimization Algorithm (KOA)65, Thermal Exchange Optimization (TEO)66, Black Hole Algorithm (BHA)67, Nuclear Reaction Optimization (NRO)68, Electro-Magnetism Optimization (EMO)69, Equilibrium Optimizer (EO)70, Multi-Verse Optimizer (MVO)71, Lichtenberg Algorithm (LA)72, and Archimedes Optimization Algorithm (AOA)73.

Human-based metaheuristic algorithms are designed with inspiration from decisions, thoughts, trainings, choices, interactions, collaborations, and other activities of humans in individual and social life. Teaching–Learning Based Optimization (TLBO) is one of the most widely used human-based metaheuristic algorithms, whose design idea comes from the sharing of science and knowledge between teachers and students, as well as students with each other in the classroom74. Mother Optimization Algorithm (MOA) is proposed based on the modeling of Eshrat’s care of her children7. The source of inspiration in the design of Doctor and Patient Optimization (DPO) comes from the therapeutic communication between the medical staff and the patients in the hospital75. War Strategy Optimization (WSO) is proposed by imitating ancient war strategy and strategic movement of army troops during the war76. The idea of designing Election-Based Optimization Algorithm (EBOA) comes from the process of holding elections and voting from people to choose the community leader77. Some other human-based metaheuristic algorithms are: Great Wall Construction Algorithm (GWCA)78, Gaining Sharing Knowledge based Algorithm (GSK)79, Special Forces Algorithm (SFA)80, Coronavirus Herd Immunity Optimizer (CHIO)81, Growth Optimizer (GO)82, and Ali Baba and the Forty Thieves (AFT)83.

Bobcat Optimization Algorithm

In this section, the inspiration source in the design of the proposed Bobcat Optimization Algorithm (BOA) approach is stated, then it is mathematically modeled in order to use it in optimization applications.

Inspiration and main idea of BOA

The bobcat (Lynx rufus), also known as the red lynx, is one of the four extant species within the medium-sized wild cat genus Lynx. The bobcat is native to North America and ranges from southern Canada through most of the contiguous United States to Oaxaca in Mexico84. The coat color in bobcats varies, although it is mostly brown or grayish-brown with dark bars on the tail and forelegs and black streaks on the body. The ears are black-tipped and pointed, with short, black tufts. The ears are pointed and black-tipped, with black, short tufts. The length of an adult bobcat is about 125–47.5 cm from the head to the base of its distinctive stubby tail. The length of the tail of this animal is about 9 to 20 cm. Shoulder size for adult bobcat is around 30 to 60 cm. The weight of the male bobcat is about 6.4–18.3 kg and the female is about 4–15.3 kg85. An image of the bobcat is shown in Fig. 1.

Figure 1
figure 1

Photo of a Bobcat; downloaded from free media Wikimedia Commons.

The bobcat is an adaptable predator that lives in forest areas, forest edge, urban edge, semidesert areas, and swampland environments. This animal can go without food for long periods of time, but it eats more when prey is plentiful. Although the bobcat prefers hares and rabbits, it hunts geese, chickens and other birds, insects, deer, and small rodents. Prey selection depends on habitat and location, abundance, and season. The bobcat’s hunting strategy is to first track the prey. Then by ambushing, it attacks the prey at the right time and hunts it after a chase process86.

Among the natural behaviors of the bobcat in the wild, the strategy of this animal during hunting is much more prominent. This hunting strategy can be expressed in two processes: (i) tracking and moving towards the prey and (ii) chasing and catching the prey. Mathematical modeling of these intelligent processes is employed to design the proposed Bobcat Optimization Algorithm (BOA) approach, which is discussed below.

Mathematical model of BOA

In the design of the proposed BOA approach, in order to update the population of the algorithm in the problem solving space, it is inspired by the hunting strategy of bobcats in the wild. In this strategy, the bobcat first tracks the position of the prey and moves towards it. Then it ambushes and attacks the prey at the right time and finally catches it after a chasing process. According to this, changes in the position of the bobcat in its habitat during the hunting process can be considered in two parts: (i) tracking and moving towards the prey and (ii) chasing and catching the prey. Inspired by this natural strategy in the lifestyle of bobcats, in the BOA design, the position of the population members is updated in each iteration in two phases (i) exploration based on the simulation of the bobcat’s position change while moving towards the prey and (ii) exploitation based on the simulation of the bobcat’s position change during the chase process to catch the prey. In the following, each of these BOA update phases is described in detail.

Initialization

The proposed BOA approach is a population-based optimizer that can achieve suitable solutions for optimization problems in an iteration-based process by benefiting from the searching power of its members in the problem solving space. According to BOA’s design inspiration, the wildlife habitat of the bobcats corresponds to the problem-solving space, and the location of the bobcats in this habitat corresponds to the position of the BOA members in the problem-solving space. Therefore, in BOA, each bobcat as a member of the population, according to the position it creates in the problem solving space, determines the values for the decision variables. Hence, the position of each bobcat represents a candidate solution to the problem, which can be modeled from a mathematical point of view using a vector, where each element of this vector represents a decision variable. Together, bobcats form the population of the algorithm, which can be modeled from a mathematical point of view using a matrix according to Eq. (1). The primary position of bobcats in the problem-solving space is initialized randomly using Eq. (2).

$$X = \left[ {\begin{array}{*{20}c} {X_{1} } \\ \vdots \\ {X_{i} } \\ \vdots \\ {X_{N} } \\ \end{array} } \right]_{N \times m} = \left[ {\begin{array}{*{20}c} {x_{1,1} } & \cdots & {x_{1,d} } & \cdots & {x_{1,m} } \\ \vdots & \ddots & \vdots & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & \vdots \\ {x_{i,1} } & \cdots & {x_{i,d} } & \cdots & {x_{i,m} } \\ \vdots & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & \vdots & \ddots & \vdots \\ {x_{N,1} } & \cdots & {x_{N,d} } & \cdots & {x_{N,m} } \\ \end{array} } \right]_{N \times m} ,$$
(1)
$${x}_{i,d}=l{b}_{d}+r\cdot \left(u{b}_{d}-l{b}_{d}\right).$$
(2)

Here, \(X\) is the BOA population matrix, \({X}_{i}\) is the \(i\) th bobcat (candidate solution), \({x}_{i,d}\) is its \(d\) th dimension in search space (decision variable), \(N\) is the number of bobcats, \(m\) is the number of decision variables, \(r\) is a random number in interval \(\left[\text{0,1}\right]\), \(l{b}_{d}\), and \(u{b}_{d}\) are the lower bound and upper bound of the \(d\) th. decision variable, respectively.

As mentioned, the position of each bobcat represents a candidate solution for the problem, corresponding to which the objective function of the problem can be evaluated. The set of evaluated values for the objective function can be represented using a vector according to Eq. (3).

$$F={\left[\begin{array}{c}{F}_{1}\\ \vdots \\ {F}_{i}\\ \vdots \\ {F}_{N}\end{array}\right]}_{N\times 1}={\left[\begin{array}{c}F({X}_{1})\\ \vdots \\ F({X}_{i})\\ \vdots \\ F({X}_{N})\end{array}\right]}_{N\times 1}.$$
(3)

Here, \(F\) is the vector of evaluated objective function and \({F}_{i}\) is the evaluated objective function based on the \(i\) th bobcat.

These evaluated values for the objective function of the problem are suitable criteria to measure the quality of each BOA member in providing candidate solutions. According to this, the best evaluated value for the objective function corresponds to the best BOA member and similarly the worst evaluated value for the objective function corresponds to the worst BOA member. In BOA design, in each iteration, the position of the bobcats in the problem-solving space is updated, and accordingly, the candidate solutions and the objective function values are also updated. Therefore, in each iteration, based on the comparison of objective function values, the best BOA member should also be updated.

PHASE 1: Tracking and moving towards prey (exploration phase)

In the first phase of BOA, the position of the population members in the problem solving space is updated based on the simulation of tracking and movement of bobcats towards prey during hunting. Modeling the movement of bobcat towards the prey leads to extensive changes in the position of the population members in the problem solving space and thus increases the exploration ability of BOA in order to manage the global search.

In BOA design, for each bobcat, the position of other population members who have a better value for the objective function is considered as the prey position. The candidate prey set for each bobcat is determined using Eq. (4).

$$C{P}_{i}=\left\{{X}_{k}:{F}_{k}<{F}_{i} \,and \,k\ne i\right\}, \text{where} \,i=\text{1,2}, \dots , N \,\text{and} \,k\in \left\{\text{1,2}, \dots , N\right\}.$$
(4)

Here, \(C{P}_{i}\) is the set of candidate preys’ locations for the \(i\) th bobcat, \({X}_{k}\) is the is the population member with a better objective function value than \(i\) th bobcat, and \({F}_{k}\) is the its objective function value.

In the BOA design, it is assumed that each bobcat randomly selects one of these preys and attacks it. Based on the modeling of the bobcat’s position change while moving towards the prey in this strategy, a new position is calculated for each BOA member using Eq. (5). This new position replaces the previous position of the corresponding member if it improves the value of the objective function according to Eq. (6).

$${x}_{i,j}^{P1}={x}_{i,j}+\left(1-2 {r}_{i,j}\right) \cdot (S{P}_{i,j}-{I}_{i,j}\cdot {x}_{i,j}),$$
(5)
$${X}_{i}=\left\{\begin{array}{c}{X}_{i}^{P1}, {F}_{i}^{P1}\le {F}_{i},\\ {X}_{i}, else ,\end{array}\right.$$
(6)

Here, \(S{P}_{i}\) is the selected prey by \(i\) th bobcat, \(S{P}_{i,j}\) is its \(j\) th dimension, \({X}_{i}^{P1}\) is the new position calculated for the \(i\) th bobcat based on exploration phase of the proposed BOA, \({x}_{i,j}^{P1}\) is its \(j\) th dimension, \({F}_{i}^{P1}\) is its objective function value, \({r}_{i,j}\) are random numbers from the interval \(\left[0, 1\right]\), and \({I}_{i,j}\) are numbers which are randomly selected as 1 or 2.

PHASE 2: Chasing to catch prey (exploitation phase)

In the second phase of BOA, the position of the population members in the problem solving space is updated based on the chase simulation between the bobcat and the prey during hunting. This process of chasing happens near the hunting place so that finally the bobcat catches the prey. Modeling the movement of bobcat during the process of chasing and catching prey leads to small changes in the position of population members in the problem solving space and thus increases the exploitation ability of BOA in order to manage local search.

In BOA design, based on the modeling of bobcat position change during the chase process, a new position for each BOA member near the hunting place is calculated using Eq. (7). Then, this new position, if it improves the value of the objective function, replaces the previous position of the corresponding member according to Eq. (8).

$${x}_{i,j}^{P2}={x}_{i,j}+\frac{1-2 {r}_{i,j}}{1+t} \cdot {x}_{i,j}$$
(7)
$${X}_{i}=\left\{\begin{array}{c}{X}_{i}^{P2}, {F}_{i}^{P2}\le {F}_{i}\\ {X}_{i}, else \end{array}\right..$$
(8)

Here, \({X}_{i}^{P2}\) is the new position calculated for the \(i\) th bobcat based on exploitation phase of the proposed BOA, \({x}_{i,j}^{P2}\) is its \(j\) th dimension, \({F}_{i}^{P2}\) is its objective function value, \({r}_{i,j}\) are random numbers from the interval \(\left[0, 1\right]\), and \(t\) is the iteration counter.

Repetition process, pseudo-code, and flowchart of BOA

The first iteration in the proposed BOA approach is completed after updating the position of all bobcats in the problem solving space based on exploration and exploitation phases. After that, the algorithm enters the next iteration with updated values for the position of bobcats and the objective function, and the process of updating bobcats continues until the last iteration of the algorithm based on Eqs. (4)–(8). In each iteration, the best solution is obtained until that iteration is updated and saved. After the full implementation of the algorithm, the best candidate solution obtained during the iterations of the algorithm is presented as the BOA solution for the given problem. The implementation steps of BOA are shown as a flowchart in Fig. 2 and its pseudocode is presented in Algorithm 1.

Algorithm 1
figure a

Pseudo-code of BOA.

Figure 2
figure 2

Flowchart of BOA.

Computational complexity of BOA

In this subsection, the computational complexity of the proposed BOA approach is analyzed.

Initialization phase

Population initialization
  1. o

    Each bobcat’s position is initialized randomly within the bounds.

  2. o

    Complexity: \(O(N\times m)\), where \(N\) is the number of bobcats (population size) and \(m\) is the number of decision variables.

Objective function evaluation
  1. o

    Each bobcat’s initial position is evaluated using the objective function.

  2. o

    Complexity: \(O(N)\), assuming the objective function evaluation is \(O(1)\).

Iterative phases

Each iteration consists of two main phases: Exploration and Exploitation.

Phase 1: Exploration (tracking and moving towards prey)
Determining candidate prey set
  1. o

    For each bobcat, identify all other bobcats with better objective function values.

  2. o

    Complexity: \(O({N}^{2})\), as each bobcat must compare its value with every other bobcat.

Updating positions
  1. o

    For each bobcat, update its position based on the selected prey and the defined rules.

  2. o

    Complexity: \(O(N\times m)\), as each bobcat’s position is updated in the \(m\)-dimensional space.

Objective function evaluation
  1. o

    Evaluate the new positions of the bobcats.

  2. o

    Complexity: \(O(N)\).

Phase 2: Exploitation (chasing and catching prey)
Updating positions
  1. o

    For each bobcat, update its position based on the chasing process near the hunting place.

  2. o

    Complexity: \(O(N\times m)\).

Objective function evaluation
  1. o

    Evaluate the new positions of the bobcats.

  2. o

    Complexity: \(O(N)\).

Total complexity per iteration

  • Exploration phase: \(O\left({N}^{2}\right)+O\left(N\times m\right)+O\left(N\right)=O({N}^{2}+N\times m)\)

  • Exploitation phase: \(O\left(N\times m\right)+O\left(N\right)=O(N\times m)\)

Overall complexity

Assuming the algorithm runs for \(T\) iterations, the total computational complexity can be summarized as: \(O(T\times ({N}^{2}+N\times m))\)

In this complexity analysis:

  • \(N\) is the number of bobcats (population size).

  • \(m\) is the number of decision variables.

  • \(T\) is the number of iterations.

Explanation

  1. 1

    Initialization: The initialization phase involves setting up the initial positions of the bobcats, which scales linearly with both the population size \(N\) and the number of decision variables \(m\).

  2. 2

    Exploration phase: During each iteration, each bobcat must compare itself with every other bobcat to determine the candidate prey set, leading to a quadratic complexity \(O({N}^{2})\). Updating the positions of the bobcats involves linear operations with respect to the number of decision variables, resulting in \(O(N\times m)\).

  3. 3

    Exploitation phase: The exploitation phase involves updating each bobcat’s position based on a local search strategy, which also scales linearly with the population size and the number of decision variables.

The quadratic term \(O({N}^{2})\) arises from the need to compare each bobcat with every other bobcat to determine the candidate prey set. This could potentially be optimized using advanced data structures or parallel processing techniques, but in its basic form, it dominates the complexity, especially for large populations.

Population diversity, exploration, and exploitation analysis

Population diversity of BOA refers to how population members are spread out within the problem space, which is crucial for tracking the algorithm’s search processes. This measure shows if the population is more geared towards exploring new possibilities or focusing on optimizing known solutions. Evaluating the diversity within the BOA population allows for assessing and adjusting the algorithm’s ability to effectively explore and exploit as a group. Researchers have proposed various definitions of diversity. Pant87 described diversity using Eqs. (9), (10).

$$Diversity=\frac{1}{N}\sum_{i=1}^{N}\sqrt{{{\sum }_{d=1}^{m}\left({x}_{i,d}-{\overline{x} }_{d}\right)}^{2}},$$
(9)
$${\overline{x} }_{d}=\frac{1}{N}{\sum }_{i=1}^{N}{x}_{i,d}.$$
(10)

Here, \(N\) represents the number of population members, \(m\) is the number of problem dimensions, and \({\overline{x} }_{d}\)​ is the mean of the population in the dth dimension. Therefore, the extent of exploration and exploitation within the population for each iteration can be defined by Eqs. (11), (12),

$$Exploration=\frac{Diversity}{Diversit{y}_{max}},$$
(11)
$$Exploitation=\frac{\left|Diversity-Diversit{y}_{max}\right|}{Diversit{y}_{max}}.$$
(12)

In this segment, the diversity, exploration, and exploitation of the population have been assessed using twenty-three standard benchmark functions, which include seven unimodal functions (F1 to F7) and sixteen multimodal functions (F8 to F23). A detailed description of these benchmark functions can be found in88.

Figure 3 demonstrates the exploration–exploitation ratio of the BOA method over the iteration process, providing a visual tool for understanding how the algorithm balances global and local search strategies. Additionally, Table 1 presents the analysis results of population diversity, exploration, and exploitation. The simulation results indicate that the BOA maintains a favorable population diversity, with high values at the initial iteration and lower values towards the final iteration. Furthermore, the results generally show that the exploration–exploitation ratio of BOA tends to be close to 0.00%:100%. These findings validate that the proposed BOA approach, by fostering suitable population diversity throughout the algorithm’s iterations, achieves effective performance in managing and balancing exploration and exploitation during the search process.

Figure 3
figure 3

Exploration and exploitation of the BOA.

Table 1 Population diversity, exploration, and exploitation percentage results.

Qualitative analysis of BOA

This section delves into the qualitative analysis of the proposed Bobcat Optimization Algorithm (BOA) when tackling optimization problems. To facilitate a visual examination of BOA, four metrics are utilized: (i) search history, (ii) the trajectory of the leading bobcat in the first dimension, (iii) the average fitness of population members, and (iv) the convergence curve of BOA as it progresses towards a solution over successive iterations. Thirteen standard benchmark functions (F1 to F13) serve as the basis for this evaluation, with detailed descriptions of these functions provided in88. The results from the qualitative analysis of BOA on these functions are illustrated in Fig. 4.

Figure 4
figure 4figure 4

Qualitative analysis of BOA.

The search history metric tracks the positional changes of bobcats within the search space throughout the algorithm’s iterations. The trajectory of the leading bobcat metric maps the changes in the first dimension of the primary bobcat across iterations. The average fitness metric monitors the variations in the average fitness of the bobcat population during the algorithm’s execution. Lastly, the convergence curve metric illustrates the BOA's progression towards the optimal solution during the algorithm’s iterations.

Simulation studies and results

In this section, the performance of the proposed approach BOA is evaluated to handle the optimization tasks. With this aim, BOA is implemented on the CEC 2017 test suite for problem dimensions equal to 10, 30, 50, and 100, as well as CEC 2020 test suite.

Performance comparison and experimental setting

The quality of the results obtained from BOA is evaluated in comparison with the performance of twelve well-known metaheuristic algorithms: CMA-ES89, EBOwithCMAR90, SPS_L_SHADE_EIG91, LSHADE_cnEpSi92, SHADE93, GWO94, WOA95, MPA46, TSA96, RSA41, AVOA33, and WSO42. As evident from the literature review, countless metaheuristic algorithms have been introduced and designed so far. Therefore, it is not reasonable to compare the performance of the proposed BOA approach with all these algorithms. For this reason, the authors have used these twelve metaheuristic algorithms in order to compare with the performance of BOA. The reasons and criteria for choosing these twelve algorithms are stated below:

Wide range of applications

The selected algorithms have been successfully applied to various real-world and benchmark optimization problems, showcasing their versatility and robustness.

State-of-the-art performance

The chosen algorithms are well-regarded in the optimization community for their competitive performance in terms of convergence speed, solution quality, and robustness.

Representation of different metaheuristic families

The algorithms cover a broad spectrum of metaheuristic families, including evolutionary algorithms, swarm intelligence, physics-based algorithms, and bio-inspired algorithms. This ensures a comprehensive comparison across different optimization strategies.

Popularity and citations

The selected algorithms are highly cited in academic literature, indicating their acceptance and relevance in the field of optimization.

Six statistical indicators mean, best, worst, standard deviation (std), median, and rank are used to report the simulation results. The mean index is used as a criterion for ranking metaheuristic algorithms in dealing with each of the optimization problems.

Evaluation CEC 2017 test suite

In this subsection, the ability of BOA and competitor algorithms in facing the CEC 2017 test suite has been analyzed for problem dimensions equal to 10, 30, 50, and 100. CEC 2017 test suite has thirty standard benchmark functions consisting of: (i) three unimodal functions of C17-F1 to C17-F3, (ii) seven multimodal functions of C17-F4 to C17-F10, (iii) ten hybrid functions of C17-F11 to C17-F20, and (iv) ten composition functions of C17-F21 to C17-F30. Among these benchmark functions, C17-F2 is excluded from simulation studies due to its unstable behavior. Full description and details of CEC 2017 test suite are available at97.

Tables 25 report the implementation results of BOA and competitor algorithms on CEC 2017 test suite for problem dimensions equal to 10, 30, 50, and 100. Figures 58 show the convergence curves resulting from the performance of metaheuristic algorithms in this implementation. Analysis of simulation results shows that BOA in handling the CEC 2017 test suite, for problem dimension equal to 10 (m = 10), is the first best optimizer for functions C17-F1, C17-F3 to C17-F17, C17-F19 to C17-F23, and C17-F26 to C17-F30 (26 functions from 29 functions). For problem dimension equal to 30 (m = 30), proposed GAO approach is the first best optimizer for functions C17-F1, C17-F3 to C17-F6, C17-F9, C17-F10, C17-F12 to C17-F22, C17-F24, C17-F25, C17-F27, C17-F28, and C17-F30 (23 functions from 29 functions). For problem dimension equal to 50 (m = 50), proposed GAO approach is the first best optimizer for functions C17-F1, C17-F3, C17-F5 to C17-F25, and C17-F27 to C17-F30 (27 functions from 29 functions). For problem dimension equal to 100 (m = 100), proposed GAO approach is the first best optimizer for functions C17-F1, C17-F3 to C17-F22, and C17-F24 to C17-F28 (26 functions from 29 functions).

Table 2 Optimization results of CEC 2017 test suite (dimension = 10).
Table 3 Optimization results of CEC 2017 test suite (dimension = 30).
Table 4 Optimization results of CEC 2017 test suite (dimension = 50).
Table 5 Optimization results of CEC 2017 test suite (dimension = 100).
Figure 5
figure 5figure 5

Convergence curves of algorithms performances on CEC 2017 test suite (dimension = 10).

Figure 6
figure 6figure 6

Convergence curves of algorithms performances on CEC 2017 test suite (dimension = 30).

Figure 7
figure 7figure 7

Convergence curves of algorithms performances on CEC 2017 test suite (dimension = 50).

Figure 8
figure 8figure 8

Convergence curves of algorithms performances on CEC 2017 test suite (dimension = 100).

The findings obtained from the optimization results are that BOA is able to find suitable solutions for optimization problems with its high quality in exploration, exploitation, and balancing them during the search process in the problem solving space. The findings obtained from the analysis of the simulation results are that BOA is presented superior performance in order to solve the CEC 2017 test suite for problem dimensions of 10, 30, 50, and 100, by providing better results for most of the benchmark functions and ranking as the first best optimizer overall.

Evaluation of CEC 2020 test suite

This subsection provides an in-depth assessment of the proposed methodology’s effectiveness in solving the CEC 2020 benchmark problems. This particular test suite encompasses ten numerical optimization functions, each with specific boundary constraints. The ten functions are categorized as follows: C20-F1 is unimodal, which means it has a single global optimum. C20-F2 to C20-F4 are basic functions, which are relatively simpler and often used to test the fundamental capabilities of optimization algorithms. C20-F5 to C20-F7 are hybrid functions, combining multiple basic functions to create a more complex landscape. Finally, C20-F8 to C20-F10 are composition functions, which are even more intricate, blending several different types of functions to challenge the optimization process further. For a comprehensive understanding, detailed descriptions and formulations of these benchmark functions are thoroughly documented in source98.

The results of the experiments, where the proposed BOA approach was tested against other competing algorithms, are meticulously detailed in Table 6. This table includes various performance metrics that showcase how each algorithm performed across the different benchmark functions. Additionally, boxplot diagrams, which visually represent the statistical performance data of these metaheuristic algorithms, are presented in Fig. 9. These diagrams help in comparing the consistency and reliability of the algorithms’ performances.

Table 6 Optimization results of CEC 2020 test suite.
Figure 9
figure 9

Boxplot diagrams of BOA and competitor algorithms performances on CEC 2020 test suite.

From the collected data and analysis, it is evident that the BOA method stands out as the most effective optimizer among the tested algorithms. It achieved the best performance across all ten functions, C20-F1 through C20-F10. The simulation findings clearly demonstrate that the BOA method provided significantly better results in handling the CEC 2020 benchmarks when compared to its competitors. This superior performance is indicative of the BOA’s robustness and efficiency in navigating and solving complex optimization problems, thereby validating the proposed approach’s capability in a comprehensive range of optimization scenarios.

Statistical analysis

In this subsection, by applying statistical analysis on the obtained results, it has been checked whether the superiority of BOA against competitor algorithms is significant from a statistical point of view. In this regard, the Wilcoxon rank sum test99 is employed, which is a non-parametric test and has application to determine the existence of a significant difference between the means of two data samples. In the Wilcoxon rank sum test, using a measure called p-value, the presence or absence of a significant difference between the performance of two metaheuristic algorithms is determined.

The results of applying the Wilcoxon rank sum test on the performance of BOA in competition with each of the corresponding compared algorithms are reported in Table 7. Based on the obtained results, in cases where p-value is less than 0.05, BOA has a statistical superiority in competition with the corresponding algorithm. The findings obtained from the statistical analysis are that BOA has a significant statistical superiority compared to all twelve competitor algorithms in order to handle the CEC 2017 test suite for problem dimensions equal to 10, 30, 50, and 100, as well as CEC 2020 test suite.

Table 7 Wilcoxon signed-rank test results.

In order to further analyze the results and performance of the optimization algorithms, another test, called the Friedman rank test100, is used. The results of this test are presented in Table 8. Based on the results of the Friedman test, the proposed BOA ranks first in optimizing CEC 2017 test suite for all dimensions of 10, 30, 50, and 100, as well as CEC 2020 test suite compared to all twelve competitor algorithms.

Table 8 Friedman test results.

At this stage, using the t-test101, the superiority of BOA compared to competitor algorithms is checked. In a t-test, the two algorithms are taken into account simultaneously in order to compute the t-values for each function. The results of t-test implementation on the performance of metaheuristic algorithms to handle CEC 2017 test suite for different dimensions are reported in Table 9. Based on the obtained results, the proposed BOA approach has a significant superiority in the competition with all twelve corresponding algorithms to handle the CEC 2017 test suite in the dimensions of 10, 30, 50, and 100, as well as CEC 2020 test suite.

Table 9 T-test results.

BOA for real-world applications

In this section, the performance of BOA and competitor algorithms for handsling optimization tasks in real-world applications is analyzed. In this regard, twenty-two constrained optimization problems from CEC 2011 test suite and four engineering design problems are selected from real-world applications.

CEC 2011 test suite

In this subsection, the performance of BOA and competitor algorithms in handling the CEC 2011 test suite is tested. CEC 2011 test suite consists of twenty-two constrained optimization prsoblems from real-world applications, the full description and details of which are available in102.

Table 10 reports the implementation results of BOA and competitor algorithms on CEC 2011 test suite. Figure 10 shows the convergence curves obtained from the performance of metaheuristic algorithms in this implementation. The findings from the optsimization results are that BOA with high quality in balancing exploration and exploitation has provided good results for all twenty-two optimization problems C11-F1 to C11-F22. The findings obtained from the simulation results are that BOA has provided superior performance in comparison with competitor algorithms in order to handle CEC 2011 test suite by providing better results and obtaining the rank of the first best optimizer in C11-F1, C11-F2, C11-F4 to C11-F12, C11-F14 to C11-F22 (20 problems from 22 problems) problems. In addition, the results of the statistical analysis obtained from the Wilcoxon rank sum test reported in Table 8 confirm that BOA has significant statistical superiority in comparison with all twelve competitor algorithms for solving the CEC 2011 test suite.

Table 10 Optimization results of CEC 2011 test suite.
Figure 10
figure 10

Convergence curves of BOA and competitor algorithms performances on CEC 2011 test suite.

Pressure vessel design problem

Pressure vessel design is presented as a challenge in real world applications with the schematic shown in Fig. 11. Minimizing construction cost is the main goal in this design. The mathematical model of this design is as follows103:

Figure 11
figure 11

Schematics of the pressure vessel design103.

Consider: \(X=\left[{x}_{1}, {x}_{2}, {x}_{3}, {x}_{4}\right]=\left[{T}_{s}, {T}_{h}, R, L\right]\).

Minimize: \(f\left(x\right)=0.6224{x}_{1}{x}_{3}{x}_{4}+1.778{x}_{2}{x}_{3}^{2}+3.1661{x}_{1}^{2}{x}_{4}+19.84{x}_{1}^{2}{x}_{3}.\)

Subject to:

$${g}_{1}\left(x\right)= -{x}_{1}+0.0193{x}_{3} \le 0, \,\,\,\,\,\,\,{g}_{2}\left(x\right)=-{x}_{2}+0.00954{x}_{3}\le 0,$$
$${g}_{3}\left(x\right)=-\pi {x}_{3}^{2}{x}_{4}-\frac{4}{3}\pi {x}_{3}^{3}+1296000\le 0, \,\,\,\,\,\,\,{g}_{4}\left(x\right)={x}_{4}-240 \le 0.$$

With

$$0\le {x}_{1},{x}_{2}\le 100\, {\text{and} \,10\le x}_{3},{x}_{4}\le 200.$$

Tables 11, 12 report the implementation results of BOA and metaheuristic algorithms on the pressure vessel design problem. Figure 12 shows the convergence curve of BOA while achieving the solution for pressure vessel design. Based on the obtained results, BOA has provided the optimal design for the pressure vessel with the values of the design variables equal to (0.7780271, 0.3845792, 40.312284, 200) and the value of the objective function equal to (5882.8955). The finding of the simulation results is that BOA has provided superior performance in comparison with competitor algorithms, by providing better results compared to competitor algorithms for handling pressure vessel design.

Table 11 Performance of optimization algorithms on pressure vessel design problem.
Table 12 Statistical results of optimization algorithms on pressure vessel design problem.
Figure 12
figure 12

Convergence curves of BOA on the pressure vessel design.

Speed reducer design problem

Speed reducer design is presented as a challenge in real world applications with the schematic shown in Fig. 13, where minimizing the weight of the speed reducer is the main goal in this design. The mathematical model of this design is as follows104,105:

Figure 13
figure 13

Schematics of the speed reducer design104,105.

Consider: \(X=\left[{x}_{1,} {x}_{2}, {x}_{3}, {x}_{4}, {x}_{5}{ ,x}_{6} ,{x}_{7}\right]=\left[b, m, p, {l}_{1}, {l}_{2}, {d}_{1}, {d}_{2}\right]\).

Minimize: \(f\left(x\right)=0.7854{x}_{1}{x}_{2}^{2}\left(3.3333{x}_{3}^{2}+14.9334{x}_{3}-43.0934\right)-1.508{x}_{1}\left({x}_{6}^{2}+{x}_{7}^{2}\right)+7.4777\left({x}_{6}^{3}+{x}_{7}^{3}\right)+0.7854({x}_{4}{x}_{6}^{2}+{x}_{5}{x}_{7}^{2})\).

Subject to:

$${g}_{1}\left(x\right)=\frac{27}{{x}_{1}{x}_{2}^{2}{x}_{3}}-1 \le 0, \,\,\,\,\,\,\,{g}_{2}\left(x\right)=\frac{397.5}{{x}_{1}{x}_{2}^{2}{x}_{3}}-1\le 0,$$
$${g}_{3}\left(x\right)=\frac{1.93{x}_{4}^{3}}{{x}_{2}{x}_{3}{x}_{6}^{4}}-1\le 0, \,\,\,\,\,\,\,{g}_{4}\left(x\right)=\frac{1.93{x}_{5}^{3}}{{x}_{2}{x}_{3}{x}_{7}^{4}}-1 \le 0,$$
$${g}_{5}\left(x\right)=\frac{1}{110{x}_{6}^{3}}\sqrt{{\left(\frac{745{x}_{4}}{{x}_{2}{x}_{3}}\right)}^{2}+16.9\times {10}^{6}}-1\le 0,$$
$${g}_{6}(x) = \frac{1}{85{x}_{7}^{3}}\sqrt{{\left(\frac{745{x}_{5}}{{x}_{2}{x}_{3}}\right)}^{2}+157.5\times {10}^{6}}-1 \le 0,$$
$${g}_{7}\left(x\right)=\frac{{x}_{2}{x}_{3}}{40}-1 \le 0, \,\,\,\,\,\,\,{g}_{8}\left(x\right)=\frac{{5x}_{2}}{{x}_{1}}-1 \le 0,$$
$${g}_{9}\left(x\right)=\frac{{x}_{1}}{12{x}_{2}}-1 \le 0, \,\,\,\,\,\,\,{g}_{10}\left(x\right)=\frac{{1.5x}_{6}+1.9}{{x}_{4}}-1 \le 0,$$
$${g}_{11}\left(x\right)=\frac{{1.1x}_{7}+1.9}{{x}_{5}}-1 \le 0.$$

With

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

Tables 13, 14 present the results of employing BOA and metaheuristic algorithms to deal with the speed reducer design problem. Figure 14 shows the convergence curve of BOA while achieving the solution for speed reducer design. Based on the obtained results, BOA has provided the optimal design for speed reducer with the values of the design variables equal to (3.5, 0.7, 17, 7.3, 7.8, 3.3502147, 5.2866832) and the value of the objective function equal to (2996.3482). The finding of the simulation results is that BOA has provided superior performance by achieving better results compared to competitor algorithms for addressing speed reducer design.

Table 13 Performance of optimization algorithms on speed reducer design problem.
Table 14 Statistical results of optimization algorithms on speed reducer design problem.
Figure 14
figure 14

Convergence curves of BOA on the speed reducer design.

Welded beam design problem

Welded beam design is presented as a challenge in real world applications with the schematic shown in Fig. 15, where minimizing the fabrication cost of the welded beam is the main goal in this design. The mathematical model of this design is as follows95:

Figure 15
figure 15

Schematics of the welded beam design95.

Consider: \(X=\left[{x}_{1}, {x}_{2}, {x}_{3}, {x}_{4}\right]=\left[h, l, t, b\right]\).

Minimize: \(f(x)=1.10471{x}_{1}^{2}{x}_{2}+0.04811{x}_{3}{x}_{4} (14.0+{x}_{2})\).

Subject to:

$${g}_{1}\left(x\right)= \tau \left(x\right)-13600 \le 0, \,\,\,\,\,\,\,{g}_{2}\left(x\right)= \sigma \left(x\right)-30000 \le 0,$$
$${g}_{3}\left(x\right)= {x}_{1}-{x}_{4}\le 0, {g}_{4}(x) = 0.10471{x}_{1}^{2}+0.04811{x}_{3}{x}_{4} (14+{x}_{2})-5.0 \le 0,$$
$${g}_{5}\left(x\right)= 0.125 - {x}_{1}\le 0, \,\,\,\,\,\,\,{g}_{6}\left(x\right)= \delta \left(x\right)- 0.25 \le 0,$$
$${g}_{7}\left(x\right)= 6000 - {p}_{c} \left(x\right)\le 0.$$

where

$$\tau \left(x\right)=\sqrt{{\left({\tau }{\prime}\right)}^{2}+\left(2\tau {\tau }{\prime}\right)\frac{{x}_{2}}{2R}+{\left(\tau "\right)}^{2} }, {\tau }^{\prime}=\frac{6000}{\sqrt{2}{x}_{1}{x}_{2}}, \tau "=\frac{MR}{J},$$
$$M=6000\left(14+\frac{{x}_{2}}{2}\right), R=\sqrt{\frac{{x}_{2}^{2}}{4}+{\left(\frac{{x}_{1}+{x}_{3}}{2}\right)}^{2}},$$
$$J=2\sqrt{2}{x}_{1}{x}_{2}\left(\frac{{x}_{2}^{2}}{12}+{\left(\frac{{x}_{1}+{x}_{3}}{2}\right)}^{2}\right) , \,\,\,\,\,\sigma \left(x\right)=\frac{504000}{{x}_{4}{x}_{3}^{2}},$$
$$\delta \left(x\right)=\frac{65856000}{\left(30\cdot 1{0}^{6}\right){x}_{4}{x}_{3}^{3}} , {p}_{c} \left(x\right)=\frac{4.013\left(30\cdot 1{0}^{6}\right){x}_{3}{x}_{4}^{3}}{1176}\left(1-\frac{{x}_{3}}{28}\sqrt{\frac{30\cdot 1{0}^{6}}{4(12\cdot 1{0}^{6})}}\right) .$$

With

$$0.1\le {x}_{1}, {x}_{4}\le \,2 \,\text{and }0.1\le {x}_{2}, {x}_{3}\le \,10.$$

Tables 15, 16 have published the results of using BOA and metaheuristic algorithms to address the welded beam design problem. Figure 16 shows the convergence curve of BOA while achieving the solution for welded beam design. Based on the obtained results, BOA has provided the optimal design for welded beam with the values of design variables equal to (0.2057296, 3.4704887, 9.0366239, 0.2057296) and the value of the objective function equal to (1.7246798). The finding of the simulation results is that BOA has provided superior performance by achieving better results compared to competitor algorithms in order to deal with welded beam design.

Table 15 Performance of optimization algorithms on welded beam design problem.
Table 16 Statistical results of optimization algorithms on welded beam design problem.
Figure 16
figure 16

Convergence curves of BOA on the welded beam design.

Tension/compression spring design problem

Tension/compression spring design is presented as a challenge in real-world applications with the schematic shown in Fig. 17, where minimizing the weight of tension/compression spring is the main goal in this design. The mathematical model of this design is as follows95:

Figure 17
figure 17

Schematics of the tension/compression spring design95.

Consider: \(X=\left[{x}_{1}, {x}_{2}, {x}_{3} \right]=\left[d, D, P\right].\)

Minimize: \(f\left(x\right)=\left({x}_{3}+2\right){x}_{2}{x}_{1}^{2}.\)

Subject to:

$${g}_{1}\left(x\right)= 1-\frac{{x}_{2}^{3}{x}_{3}}{71785{x}_{1}^{4}} \le 0, {g}_{2}\left(x\right)=\frac{4{x}_{2}^{2}-{x}_{1}{x}_{2}}{12566({x}_{2}{x}_{1}^{3})}+\frac{1}{5108{x}_{1}^{2}}-1\le 0,$$

\({g}_{3}\left(x\right)= 1-\frac{140.45{x}_{1}}{{x}_{2}^{2}{x}_{3}}\le 0\), \({g}_{4}\left(x\right)=\frac{{x}_{1}+{x}_{2}}{1.5}-1 \le 0\).

With

$$0.05\le {x}_{1}\le 2, {0.25\le x}_{2}\le 1.3 \text{and} 2\le {x}_{3}\le 15$$

Tables 17, 18 report the results of addressing the tension/compression spring design problem using BOA and metaheuristic algorithms. Figure 18 shows the convergence curve of BOA while achieving the solution for tension/compression spring design. Based on the obtained results, BOA has provided the optimal design for tension/compression spring with the values of the design variables equal to (0.0516891, 0.3567177, 11.288966) and the value of the objective function equal to (0.0126019). The finding of the simulation results is that BOA has provided superior performance in order to deal with tension/compression spring design, by providing better results in competition with the compared algorithms.

Table 17 Performance of optimization algorithms on tension/compression spring design problem.
Table 18 Statistical results of optimization algorithms on tension/compression spring design problem.
Figure 18
figure 18

Convergence curves of BOA on the tension/compression spring design.

BOA for supply chain management (SCM)

To formulate the sustainable lot size optimization problem, we adopted mathematical models that considered economic costs and environmental impacts. The objective was to determine the optimal lot size at each stage of the supply chain that reduced both CO2 emissions and costs. The model incorporates various constraints such as production, inventory constraints, and demand satisfaction requirements. In addition, specific sustainability limits were added to limit the maximum CO2 allowed Disposals associated with manufacturing, transportation, and storage activities. The accounting system looks like a standard solution. Intelligence strikes a balance between economic prosperity and environmental sustainability. The company must minimize scarcity, maximize surplus, and choose the optimal quantity for the lot size. When customer needs arise, they must review the materials needed to start production or decide how much to order. Outstanding inventory represents a backlog, which should be checked to avoid excess inventory and, if attempted, actions should be planned to reduce it. The flowchart in Algorithm 1 represents an example of an optimized listing system.

TC = Cc  D + Cp  P  (Q + SS) + p  A  D + Ce  D

F [TC] = Total Cost [Cc, Q, D, Cp, SS, A, Ce, P, p]

where: Cc: Order cost/unit; Cp: Holding cost/unit; P: Price; p: Shortage cost/unit; A: Expected shortage/cycle; D: Annual demand; Ce: Footprint emission cost; Q: Quantity; SS: Shortage; TC: Total cost; F [TC]: Objective function for lot size.

Lot size optimization is influenced by demand data, which determines the number of items required to satisfy customer demand. The variability and uncertainty in demand patterns influence the lot sizing decision process. The optimal lot size can be identified to balance customer satisfaction and inventory costs by considering changes in demand and forecast accuracy. The optimal lot size is considered an important decision in terms of travel expenses. Lot sizes can increase economies and reduce transportation costs per unit, resulting in cost savings, but lot sizes can increase storage and warehousing costs and drive transportation cost savings equal lot The goal of size optimization is to minimize inventory costs associated with storage and maintenance at the inventory level. Higher lot sizes can increase inventory costs due to longer storage times. Small lot sizes ordered, there are often setup costs to consider. Regularly increasing lot sizes incorporates environmental considerations such as CO2 emissions. Transport demand increased.

Lot size can increase CO2 emissions. By optimizing lot sizes, a balance can be achieved between inventory costs, transportation costs and CO2 emissions.

The implementation results of BOA and competitor algorithms for sustainable lot size optimization are reported in Table 19. The results and analysis showed the indisputable superiority of the Bobcat algorithm in optimizing lot sizes in supply chain management. Comparative analysis with conventional methods such as EOQ and other algorithms revealed a consistent reduction in total relative costs obtained by the Bobcat system in supply chain planning and demand models. Convergence analysis showed its fast convergence behavior, and it outperformed genetic algorithms and simulated annealing. Furthermore, scalability testing in large scale demonstrated its robust performance, demonstrating linear scalability and suitability for complex supply chain networks including disruption of supply. The comparative analysis with benchmark solutions confirmed the effectiveness of the Bobcat algorithm in finding a nearly always optimal lot-size policy. Finally, sensitivity analysis confirmed its versatility and ease of implementation in parameter settings. Overall, these findings firmly establish the Bobcat algorithm as the first choice for lot size optimization, providing better solutions, convergence speed, scalability, and robustness to uncertainties in real-world supply in chain scenarios.

Table 19 Optimization results of the sustainable lot size optimization.

Discussion

This section delves into a comprehensive analysis of the experiments conducted in this paper, evaluating the advantages, disadvantages, and specific shortcomings of the Bobcat Optimization Algorithm (BOA) compared to existing methods. The experiments conducted in this study are designed to evaluate the performance of the Bobcat Optimization Algorithm (BOA) across a range of optimization problems. These has been included the CEC 2017 test suite for various problem dimensions, the CEC 2011 constrained optimization problems, and several engineering design problems. Additionally, BOA is tested on sustainable lot size optimization in supply chain management (SCM) applications.

CEC 2017 test suite

Benchmark functions in CEC 2017 test suite are of four types: unimodal, multimodal, hybrid, and composition. Unimodal functions are suitable criteria for measuring the ability to exploit metaheuristic algorithms to manage local search in the problem space. The findings show that BOA, with its ability in exploitation, has been able to provide suitable solutions for dealing with unimodal functions. Multimodal functions challenge the discovery ability of metaheuristic algorithms due to having multiple local optima. The findings show that BOA with a high ability in discovery in order to manage global search has provided an effective performance for handling multi-model functions. Hybrid functions and composition are complex optimization problems that challenge the ability of metaheuristic algorithms to balance exploration and exploitation. The findings obtained from the optimization results of these functions show that BOA, with its high ability to balance exploration and exploitation during the search process, has been able to provide an effective performance for handling hybrid and composition functions. The findings obtained from the CEC 2017 test suite optimization confirm that the proposed BOA approach has a high capability in exploring, exploiting, and balancing them during the search process.

The results obtained from BOA have been compared with the performance of twelve well-known algorithms. Based on the analysis of simulation results, BOA has shown high efficiency in solving optimization problems in different dimensions, with a success rate of 89.65% for dimensions 10 and 100, 79.31% for dimension 30, and 93.10% for dimension 50. This shows BOA’s strong ability to maintain a balance between exploration and exploitation.

CEC 2020 test suite

The evaluation of the Bobcat Optimization Algorithm (BOA) on the CEC 2020 test suite revealed exceptional performance. This test suite consists of ten numerical optimization functions with specific boundary constraints, ranging from unimodal (C20-F1) to more complex hybrid and composition functions (C20-F2 to C20-F10). BOA successfully handled 100% of these functions and outperformed twelve other algorithms, securing the top rank across all benchmark problems. The experimental results demonstrate BOA’s robustness, efficiency, and superiority in solving diverse and intricate optimization scenarios. Consequently, BOA’s outstanding performance validates its effectiveness as a powerful optimizer for complex numerical problems.

CEC 2011 constrained optimization problems

In order to evaluate the efficiency of BOA to deal with real-world applications, CEC 2011 test suite has been used, which has twenty-two constrained optimization problems. The findings show that BOA, by maintaining the balance between exploration and exploitation, has provided an effective framework for dealing with real-world applications from the CEC 2011 test suite. Also, in the competition with the compared algorithms, BOA has successfully managed 90.90% of these problems, demonstrating its robustness in dealing with real-world constraints.

Engineering design problems

In addition to the CEC 2011 test suite, the efficiency of BOA to address four engineering design challenges has also been evaluated. These design challenges are: pressure vessel design, speed reducer design, welded beam design, and tension/compression spring design. BOA has been achieved a 100% success rate in the selected engineering design problems, underscoring its practical applicability and effectiveness.

Sustainable lot size optimization

The capability of the proposed BOA approach has been evaluated in SCM applications to deal with sustainable lot size optimization. Findings show that BOA is very efficient to solve this challenge of SCM applications. What is evident from the analysis of simulation results, BOA with superior performance in all ten study cases has been 100% successful in dealing with sustainable lot size optimization in competition with compared algorithms.

The proposed BOA offers several advantages for addressing global optimization problems. One notable benefit of BOA is its parameter-free design, which means there are no control parameters to adjust or fine-tune, simplifying its implementation and reducing the potential for human error during setup. Another key advantage of BOA is its high efficiency in tackling a wide array of optimization problems across various scientific disciplines. It is particularly effective when dealing with complex, high-dimensional challenges, making it a versatile tool for many types of optimization scenarios. Additionally, BOA excels in balancing exploration and exploitation within the search process. This balance enables the algorithm to converge rapidly, effectively identifying suitable values for decision variables. This capability is especially beneficial in complex optimization tasks, where swift and accurate convergence is crucial. Moreover, BOA demonstrates robust performance in real-world optimization applications. Whether applied to theoretical problems or practical, real-world scenarios, BOA consistently delivers reliable and powerful results, making it a valuable tool across various industries and research fields.

However, despite these advantages, BOA has several disadvantages and specific shortcomings. Firstly, like all metaheuristic algorithms, BOA’s performance is based on random search principles, and thus, there is no guarantee of achieving the global optimum. This inherent uncertainty is a common drawback of metaheuristic approaches. Secondly, according to the No Free Lunch (NFL) theorem, it cannot be claimed that BOA is the best optimizer for all optimization applications. The performance of any optimization algorithm is problem-dependent, and no single algorithm excels in every scenario. Lastly, there is always the possibility that newer metaheuristic algorithms will be developed by researchers that outperform BOA. The field of optimization is continually evolving, and advancements may lead to the creation of more efficient algorithms in the future, potentially surpassing the capabilities of BOA.

Open research questions (ORQ)

In this paper, a new metaheuristic algorithm called Bobcat Optimization Algorithm (BOA) is introduced and designed to deal with optimization tasks in various sciences. The efficiency of BOA has been evaluated to solve CEC 2017 tests suite, CEC 2011 test suite, four engineering design problems, and sustainable lot size optimization. While BOA has successfully achieved good results in these implementations and has provided superior performance in competition with the compared algorithms, several open research questions (ORQ) are raised as follows:

  • ORQ 1: How can the Bobcat Optimization Algorithm (BOA) be further improved to handle even more complex and higher-dimensional optimization problems?

  • ORQ 2: What are the limitations of BOA when applied to different types of real-world problems outside of the current test suites and engineering design problems?

  • ORQ 3: How does BOA perform in dynamic and uncertain environments where problem parameters change over time?

  • ORQ 4: What are the comparative advantages and disadvantages of BOA compared to other state-of-the-art bio-inspired algorithms in terms of computational efficiency and solution quality?

  • ORQ 5: How can BOA be integrated into decision-support systems for real-time optimization in industrial applications?

  • ORQ 6: Can the principles of BOA be extended or modified to create new variants of the algorithm that are tailored for specific problem domains?

  • ORQ 7: How does BOA perform when applied to multi-objective optimization problems, and what modifications are necessary to handle multiple conflicting objectives effectively?

  • ORQ 8: What is the impact of different initialization strategies on the performance and convergence rate of BOA?

  • ORQ 9: How can BOA be adapted to improve its scalability for large-scale optimization problems involving thousands of variables and constraints?

  • ORQ 10: How does BOA handle noisy and imprecise data in optimization problems, and what techniques can be integrated to improve its robustness in such conditions?

  • ORQ 11: What are the impacts of different fitness landscape characteristics on the performance of BOA, and how can it be tuned to perform better on various types of landscapes?

  • ORQ 12: How can BOA be combined with machine learning models to predict optimal solutions or guide the search process more intelligently?

  • ORQ 13: How does BOA perform in comparison to other bio-inspired algorithms in terms of adaptability to different problem domains and flexibility in handling constraints?

  • ORQ 14: How can BOA be extended or modified to incorporate learning mechanisms that adaptively improve its performance over time based on previous optimization experiences?

  • ORQ 15: What are the implications of using BOA in real-time optimization scenarios where quick decision-making is critical, and how can its response time be optimized?

By addressing these open research questions, researchers can further expand the understanding and application of the Bobcat Optimization Algorithm (BOA), pushing the boundaries of its capabilities and effectiveness in solving a wide range of optimization problems.

Conclusion and future works

In this paper, a new biomimetics metaheuristic algorithm named Bobcat Optimization Algorithm (BOA) was introduced, which imitates the behavior of bobcats in nature. The basic inspiration of BOA is derived from the hunting strategy of bobcats, which has two parts: tracking prey and chasing. The theory of BOA was expressed and then mathematically modeled in two phases (i) exploration based on the simulation of the bobcat’s position change while moving towards the prey and (ii) exploitation based on simulating the bobcat’s position change during the chase process to catch the prey. The performance of BOA in handling optimization applications was evaluated on the CEC 2017 test suite for problem dimensions of 10, 30, 50, and 100, as well as CEC 2020 test suite. The finding of the optimization results was that BOA has a high quality in exploration, exploitation and balance between them during the search process in the problem solving space. The performance of BOA was compared with the performance of twelve competitor metaheuristic algorithms. The finding from the simulation results was that BOA has provided superior performance compared to competitor algorithms, achieving better results for most of the benchmark functions and being ranked as the first best optimizer overall. The finding obtained from the statistical analysis was that BOA has a significant statistical superiority over the corresponding compared algorithms. The performance of BOA and competitor algorithms to address real-world applications was evaluated on twenty-two constrained optimization problems from the CEC 2011 test suite and four engineering design problems. The finding of this implementation was that BOA has an effective performance to tackle optimization tasks in real-world applications.

Following the introduction of BOA, several research suggestions for future work are presented which are listed below:

  • Multi-objective version development: Developing a multi-objective version of BOA and using it to address multi-objective optimization applications is one of the key research potentials of this paper.

  • Binary version development: The design of the binary version of BOA makes it suitable for handling optimization applications that require handling using the binary version.

  • Exploring hybrid approaches: Combining BOA with other optimization techniques or machine learning methods could potentially enhance its performance and applicability to more complex problems.

  • Real-world applications: Extending the application of BOA to a wider array of real-world problems in diverse industries will help validate its effectiveness and identify any potential limitations.

  • Dynamic optimization: Adapting BOA for dynamic and uncertain environments where problem parameters evolve over time will be critical for applications in rapidly changing fields.

  • Benchmarking and comparison: Conducting comprehensive benchmarking studies against other state-of-the-art algorithms to provide a clearer picture of BOA’s relative strengths and weaknesses.

  • Integration into decision-support systems: Developing practical frameworks for integrating BOA into decision-support systems to facilitate real-time optimization in industrial and commercial settings.

  • Algorithm variants: Creating and testing new variants of BOA tailored for specific problem domains could unlock further potential and specialized applications.

  • Automated parameter tuning: Investigating and developing automated parameter tuning mechanisms to enhance BOA’s performance across different problem instances without extensive manual intervention.

By addressing mentioned open research questions and pursuing the suggested future research directions, the scientific community can continue to refine and enhance the capabilities of the Bobcat Optimization Algorithm and its applications in various fields.