Foraging Bee Optimization Algorithm

IJIEM (Indonesian Journal of Industrial Engineering & Management) Vol 4 No 2 June 2023, 99-112 Available online at: http://publikasi.mercubuana.ac.id/index.php/ijiem IJIEM (Indonesian Journal of Industrial Engineering & Management) ISSN (Print) : 2614-7327 ISSN (Online) : 2745-9063 Foraging Bee Optimization Algorithm Ebun Phillip Fasina*, Babatunde Alade Sawyerr, Shuaibu Babangida Alkassim Department of Computer Science, University of Lagos, Lagos, Nigeria ARTICLE INFORMATION A B S T R A C T Article history: Received: 3 May 2023 Revised: 2 June 2023 Accepted: 4 June 2023 Category: Research paper Keywords: Swarm intelligence Nature-inspired metaheuristics Bee-inspired optimization algorithm Numerical optimization Particle swarm optimization DOI: 10.22441/ijiem.v4i2.20275 Honeybees feed on pollen and nectar from flowers. Nectar to meet their energy requirements and pollen for protein and other vital nutrients. The act of searching for these flowers by honeybees is called foraging. The foraging behaviour of bees depends on the profitability of nectar and pollen sources as well as the needs of the colony. This behaviour is modelled by the Foraging Bee Optimization Algorithm (FBA) as metaphor for optimization. After initialization, the algorithm loops through three phases based on bees’ foraging behaviour –work, withdraw, and waggle (3W). Flowers are initialized randomly in the problem space. During the waggle phase, bees are recruited to flowers with profitable nectar sources. In the work phase, new flowers are discovered and memorized by bees. In the withdraw phase bees remove unprofitable flowers from collective memory and recalibrate for recruitment. The proposed FBA is tested on three unimodal and twelve multimodal benchmark functions. The result is compared with two other state-ofthe-art swarm intelligence algorithms, Artificial Bee Colony (ABC) and Particle Swarm Optimization (PSO). Analysis of comparison results shows FBA to be highly competitive, outperforming PSO on all benchmarks and matching ABC in overall performance. *Corresponding Author This is an open access article under the CC–BY-NC license. Ebun Fasina E-mail: [email protected] 1. INTRODUCTION The study of the behavior of social organisms as a swarm in and outside their colonies led to Swarm Intelligence (SI) (Eberhart, Shi, & Kennedy, 2001; Janaki & Geethalakshmi, 2022; Selvaraj & Choi, 2020). SI is a discipline in computer science that mimics the intelligence displayed by social organisms (Kaswan, Dhatterwal, & Kumar, 2023; Schumann, 2020). This intelligence can be self-learning, healing, or optimizing. Researchers model and create algorithms based on this intelligence. These algorithms are classified as Nature-Inspired or Swarm Intelligence Optimization algorithms or metaheuristics and have been applied to solve a diverse range of problems (Fakhermand & Derakhshani, 2023; Tzanetos & Dounias, 2020; Engelbrecht, 2007; Alizadehsani, et al., 2023; Altshuler, 2023; Shahzad, et al., 2023; Kumar, Chatterjee, Payal, & Rathore, 2022; Cruz, Maia, & de Castro, 2021). Nature-Inspired algorithms find approximate solutions to optimization problems, the solution can be local or global optimum depending on the set of constraints the optimization problem How to Cite: Fasina, E. P., Sawyerr, B. A., & Alkassim, S. B. (2023). Foraging Bee Optimization Algorithm. IJIEM (Indonesian Journal of Industrial Engineering & Management), 4(2), 99-112. https://doi.org/10.22441/ijiem.v4i2.20275 99 IJIEM (Indonesian Journal of Industrial Engineering & Management) Vol 4 No 2 June 2023, 99-112 is subjected to. An optimization problem requires an objective function that may be constrained or unconstrained to be maximized or minimized. Optimization algorithms invoke the objective function to determine the fitness of a large and varied selection of solutions to determine the best or near optimum. Optimization techniques are mostly applied to minimize cost or error, maximize profit, and find optimal designs for engineering problems or provide optimal decisions for operational and management problems. Various natured-inspired algorithm has been proposed among which are the Particle Swarm Optimization (PSO) by Kennedy and Eberhart (Kennedy J. and Eberhart, 1995) which is inspired by simulation studies of the social behavior found in schools of fish and flocks of birds. Bee Colony Optimization (BCO) by (Teodorovic & Dell’orco, 2005), Bee Algorithm (BA) by (Pham, et al., 2005), Artificial Bee Colony by (Karaboga, 2005) are all inspired the foraging behavior of bee colonies. Genetic recombination and natural selection inspired the Generic Algorithm (GA) proposing by (Holland, 1975). Studies of ant colonies resulted in the Ant Colony Optimization (ACO) algorithm by (Dorigo, Colorni, & Maniezzo, 1991). Differential Evolution (DE) was proposed by (Storn & Price, 1997), and Glowworm Swarm Optimization was proposed by (Krishnanand & Ghose, 2005). GSO mimics the behavior of luminescent glowworms in nature. In this work a new algorithm called FBA that is inspired by the foraging behavior of bees is proposed and implemented to improve the speed of convergence of bee-inspired algorithms, avoid premature convergence as well as balance exploitation with exploration. 2. LITERATURE REVIEW I. Foraging Bee in Nature Honeybees are social insects or organisms that live together in well-organized colonies and can perform complex tasks in reasonable time with ease. These tasks include controlling the environment, division of labor, defense of nest and queen, nest construction, communications, and foraging for food. The process of foraging for food involves scouting, collection of pollen 100 and nectar from flowers, and conveyance of pollen and nectar to the colony. Bees in charge of foraging are called foragers. Each forager modulates its behaviour in relation to the profitability of the nectar source – the more profitable the source, the higher the intensity of foraging activity around the source, the more repetitive and dancing (or waggle) at the nest pointing to the source, and the lower the probability of abandoning the source. Without comparing sources, bee individually calculate the absolute profitability of a source. The collective nectar and pollen source selection by a colony of bees is decentralized; it is a process of natural selection where foragers from more profitable nectar sources continue to visit these sources over a long period of time and eventually recruit bees from less profitable sources. In a typical foraging season, bees collect roughly 20 – 30 kg of pollen and 125kg nectar which translate to between 1,125,000 and 4,000,000 visits to flowers. II. Bee Colonies as Metaphors for Swarm Intelligence Algorithms Agents in the Bee Algorithm (BA) first proposed by (Pham, et al., 2005) combined randomized search of the problem space with neighborhood search in promising regions of this space. BA is complex and can easily be trapped in a local optimum. The Artificial Bee Colony (ABC) algorithm proposed by (Karaboga, 2005) is less complex when compared with previous bee optimization algorithms (Bolaji, Khader, Al-Betar, & Awadallah, 2013) but converges poorly. (Sato & Hagiwara, 1997) reformulated the Genetic Algorithm (GA) to develop a new algorithm called the Bee System (BS). BS performs global search using GA operators and then improves on local search by introducing new operators such as concentrated crossover and the pseudosimplex method. The mating behavior of bees is the inspiration for the Mating Bee Optimization MBO algorithm (Abbass, 2001). MBO algorithm begins with one queen with no relatives, to a colony of relatives with a single queen or multiple queens. MBO has been modified several to form a new algorithms such as the Honey Bee Optimization (HBO) algorithm by (Curkovic & Jerbic, 2007), Honey Bees Mating IJIEM (Indonesian Journal of Industrial Engineering & Management) Vol 4 No 2 June 2023, 99-112 Optimization (HBMO) algorithm by (Haddad, Afshar, & Mariño, 2006) and the Fast Marriage in Honey Bees Optimization (FMHBO) algorithm by (Yang, Chen, & Tu, 2007). that an intelligent forager forwarding strategy significantly improves the quality of final solutions and the convergence speed of ABC algorithms. (Gao, Liu, & Huang, 2012) modified the ABC algorithm in order to improve its exploitation. The new algorithm called ABC/Best searches only around the fittest bee based on the last best solution. They employed a chaotic system and opposition-based learning for improving the speed of global convergence. (Chen, Tianfield, & Du, 2021) proposed a novel bee-foraging learning PSO (BFL-PSO) algorithm that is inspired by the search mechanism of the artificial bee colony algorithm. The proposed BFL-PSO has three different search phases, namely: employed learning, onlooker learning and scout learning. The employed learning phase is the one-phasebased PSO search, while the onlooker learning phase exploits the region around promising solutions, and the scout learning phase introduces new diversity by re-initializing stagnant particles. The proposed BFL-PSO is evaluated on the CEC2014 benchmark suite, and compared with state-of-the-art PSO and artificial bee colony algorithms The experimental results show BFL-PSO to be competitive in performance and the accuracy of its solutions. (Mathlouthi & Bouamama, 2016) integrated a centralized and distributed technique called a local optimum detector to an algorithm inspired by marriage in honeybees. The local detector enhanced finding the local optimum. (Li & Yang, 2016) proposed a variant of ABC. They introduced a memory mechanism that aids artificial bees by memorizing their best foraging experience so far. (Pan, 2016) hybridized ABC and GA to develop a self-adaptive algorithm with a dual population of independently evolving bees that exchange information through information entropy that ensures diversity and accelerates convergence. (Pan, 2016) proposed a hybrid, self-adaptive genetic-bee colony algorithm based on information entropy. The algorithm evolved two populations of bees independently but allowed the exchange of information between bees in the two populations using entropy to maintain population diversity and accelerate the evolution process. Under analysis it was found that this strategy accelerated the emergence of fitter individuals by competition between the populations performs better in complex function optimization problems. (Aslan, Karaboga, & Badem, 2020) modeled the complex behavior of foraging bees in detail – how they pass through the dance area and how long they performed their dance to attract onlooker bees – then adapted it to ABC to develop a new variant of ABC, termed the intelligent forager forwarding ABC (iff-ABC). They analyzed the contribution of the intelligent forager forwarding strategy on the performance of ABC algorithms by evaluating its performance on the CEC benchmark suite and comparing it with the performance of different variants of ABC. The results obtained showed It is helpful to study and compare various versions of bee inspired metaheuristics to enable the selection of these algorithms in the optimization tasks and the refinement and development of new variants. (Solgi & Loáiciga, 2021) identifies seven basic or root algorithms applied to solve continuous optimization problems, namely: Bee System (BS), Mating Bee Optimization (MBO), Bee Colony Optimization (BCO), Bee Evolution for Genetic Algorithms (BEGA), Bee Algorithm (BA), Artificial Bee Colony (ABC), and Bee Swarm Optimization. They ranked these algorithms by performance and convergence efficiency and found ABC, BEGA, and MBO to be the most efficient. They discussed the strengths and shortcomings of each algorithm and explained the variations observed in the convergence rate of these algorithms. 3. THE FORAGING BEE OPTIMIZATION ALGORITHM (FBA) The Foraging Bee (Optimization) Algorithm (FBA) is inspired by the foraging behaviour of bees for pollen and nectar, and the collective natural selection of more profitable nectar sources over poor ones. The FBA algorithm is 101 IJIEM (Indonesian Journal of Industrial Engineering & Management) Vol 4 No 2 June 2023, 99-112 developed. In FBA, the colony consists of bees, termed foragers, who scout for flowers that are rich sources of pollen and nectar in a patch of the problem space in the work phase, then return to the colony during the withdraw phase to communicate their findings using dance in the waggle phase. The FBA pseudocode is listed below as follows: Bee A bee 𝑏𝑖 is modeled by the tuple 𝐵 = 𝐵(𝑥𝐵 , 𝑓𝐵 , 𝐷, 𝑃) where 𝑥𝐵 is the vector representing the current position of the bee, 𝑓𝐵 ← 𝑓(𝑥𝐵 ) is the fitness of the current position of the bee, 𝐷 is the direction of the bee and 𝑃 is the patch in which the bee is initialized. Each bee makes a foraging move in time 𝑡 + 1 in dimension 𝑗 as follows: 𝑥𝑗 (𝑡 + 1) = 𝑥𝑗 (𝑡) + 𝑝𝑟1 (𝑑𝑗+ {𝑈𝑗 − 𝑥𝑗 (𝑡)} + 𝑑𝑗− {𝑥𝑗 (𝑡) − 𝐿𝑗 }) (1) where 𝑟1 is a random number between 0 and 1, 𝑈𝑗 𝑎𝑛𝑑 𝐿𝑗 are the upper and lower bounds in dimension 𝑗 of patch 𝑃, 𝑝 is the propensity of the bee. The direction vector 𝐷 is a unit vector indicating the current direction of the foraging bee. Bees make decisions before moving in direction 𝐷 by determining the direction 𝑑𝑗± to move in each dimension 𝑗 using the random variable 𝑟2 ~𝑈(0,1). Assume that the bee is moving in direction 𝑑𝑗+ in time 𝑡. The decision to continue in direction 𝑑𝑗+ is determined by |𝑎𝑗 − 𝑥𝑗 | (2) 𝑈𝑗 − 𝐿𝑗 where 𝐴 = (𝑎1 , 𝑎2 , … , 𝑎𝑛 ) is the bee attractor in each patch. If (2) is true and 𝑐𝑗 is in direction of 𝑈𝑗 , then 𝑑𝑗+ = 1 and 𝑑𝑗− = 0 otherwise 𝑑𝑗− = 1 and 𝑑𝑗+ = 0. 𝑟2 < Algorithm 1: Foraging Bee Optimization Algorithm 1 set the following parameters 𝐵𝑝𝑜𝑝 is the population of bees M is the minimum population of flowers N is the minimum population of newly discovered flowers 102 K number of scouts added as recruits during each waggle phase 𝑃𝑝𝑟𝑜𝑏 is the search space 𝛽 is fraction of resource rich flowers for estimating the attractor of a patch p is the propensity of bees when exploring patches 2 initialize M flowers in patches set 𝑓𝑇 as the fitness of the fittest flower initialize bees randomly in patch 𝑃 termcond ← false n ← 1, k ← 0 3 while true // WORK PHASE for 𝑖 = 1 to T move bee 𝑏𝑖 if 𝑓(𝑏𝑖 ) < 𝑓𝑇 mark 𝑏𝑖 with flower 𝐹𝑀+𝑛 and increment n if 𝑛 < 𝑁 then continue // WITHDRAW PHASE termcond ← GET-TERMCOND( ) if termcond then return fittest flower as optimum n←1 set 𝑓𝑇 as the fitness of the fittest flower // WAGGLE PHASE select best M flowers in 𝑃𝑝𝑟𝑜𝑏 Estimate promising patch using selected flowers 𝑃𝑏𝑒𝑠𝑡 Determine the location of attractors in each patch. increment k initialize 𝑘 bees (recruits) randomly in 𝑃𝑏𝑒𝑠𝑡 initialize other bees 𝐵𝑝𝑜𝑝 − 𝑘 (scouts) in 𝑃𝑝𝑟𝑜𝑏 The propensity 𝑝 determines how bees explore or exploit a patch. Lower values of 𝑝 favors exploitation over exploration. The continuous reduction in the spatial dimensions of 𝑃𝑏𝑒𝑠𝑡 allows the exploitation of promising patches by recruits while scouts continue to explore the entire problem space. The bee search equation guides the bees to forage only within the patch in which they are initialized making exploration and exploitation explicit processes guided by patches 𝑃𝑝𝑟𝑜𝑏 and 𝑃𝑏𝑒𝑠𝑡 . IJIEM (Indonesian Journal of Industrial Engineering & Management) Vol 4 No 2 June 2023, 99-112 Flower A flower 𝐹 is modeled by the tuple 𝐹 = 𝐹(𝑥𝐹 , 𝑓𝐹 ), where 𝑥𝐹 is the vector representing the current position of the flower 𝐹 and 𝑓𝐹 is its fitness. The lower value of 𝑓𝐹 the richer the flower as a source nectar and pollen to bees. Patch Patches are modeled by the tuple 𝑃 = 𝑃(𝐿, 𝑈, 𝐴) where 𝐿 is the vector that represents the lower limit of the patch in all dimensions, 𝑈 is the vector that represents the upper limit of the patch in all dimensions and 𝐴 is the bee attractor. 𝑃𝑝𝑟𝑜𝑏 is initialized with 𝑀 + 𝑁 or more flowers while 𝑃𝑏𝑒𝑠𝑡 is estimated with 𝑀 best flowers. The point attractor of bees in a patch is the centroid the best flower 𝑓𝑇 and a fraction 𝛽 ≅ 0.5 of the other flowers in the patch. Candidate flowers for bee attractor computation selected using the roulette operator []. Unlike GA flowers are selected without replacement, i.e., a candidate flower cannot be selected more than once. It is important that the point attractor of bees in patches 𝑃𝑝𝑟𝑜𝑏 and 𝑃𝑏𝑒𝑠𝑡 are not coincident at the early stages of search. Observe that all points in 𝑃𝑏𝑒𝑠𝑡 are interior points of 𝑃𝑝𝑟𝑜𝑏 . Foraging Bee Algorithm The flowchart in Fig. 1 highlight phases FBA. It begins with the initialization of search parameters and objects such as bees, patches, and flowers. This is followed by the work phase where scout bees and recruits search for new resource rich flowers. During the withdraw phase critical parameters are reset and the GETTERMCOND method determines if an approximate solution has been found or the maximum number of objective function evaluation has been exceeded. If the algorithm does not stop it enters the waggle phase where information is shared; 𝑃𝑏𝑒𝑠𝑡 is initialized or recalibrated and scouts are recruited to exploit the patch. The algorithm repeats the work, withdraw and waggle phases until it terminates in the withdrawal phase. Start Work Phase Withdraw Phase Stop Yes Is Termination Condition? No Waggle Phase Fig. 1. Flowchart of FBA 4. RESULT AND DISCUSSION The FBA algorithm was run on standard benchmark test function; these functions were presented in Table 1 as equations (3) to (17) and their properties a tabulated in Table 2. They were carefully chosen to test FBA’s capacity to solve problems with diverse properties and varying levels of difficulty. 𝑓1 to 𝑓3 are simple unimodal functions while 𝑓4 to 𝑓15 are multimodal functions with local minima ranging from a few hundred to millions. The performance of FBA on each test function is benchmarked with 30 trials of 50000 function evaluations. 10 9 8 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 BEST VALUE AVERAGE VALUE WORST VALUE Fig. 2. Graph of FBA results The overall performance based on the best, average, and worst-case error rates, standard deviation, and success rate of each test function 103 IJIEM (Indonesian Journal of Industrial Engineering & Management) Vol 4 No 2 June 2023, 99-112 is tabulated in Table 3 and shown graphically in Fig. 2. The success rate of each function is also shown in Fig. 5. The success of any run is determined by an error of at least four leading zeros (E-04). Table 1. Benchmark test functions 1) Sphere function 2) 𝑓1 (𝑥) = ∑ 𝑥𝑖2 3) 𝑓2 (𝑥) = ∑|𝑥𝑖 | + ∏|𝑥𝑖 | 4) 𝑓3 (𝑥) = ∑{100(𝑥𝑖2 − 𝑥𝑖+1 )2 + (𝑥𝑖 − 1)2 } 5) 1 1 𝑓4 (𝑥) = −20 exp (−0.2√ ∑ 𝑥𝑖2 ) − exp ( ∑ cos(2𝜋𝑥𝑖 )) + 20 + 𝑒 𝑛 𝑛 𝑛 Schwefel P2.22 function 𝑛 𝑛 𝑖=1 𝑖=1 7) (4) Rosenbrock’s function 𝑛−1 6) (3) 𝑖=1 (5) 𝑖=1 Ackley F1 𝑛 𝑛 𝑖=1 𝑖=1 Goldstein-Price 𝑓5 (𝑥) = {1 + (𝑥1 + 𝑥2 + 1)2 (19 − 14𝑥1 + 3𝑥12 − 14𝑥2 + 6𝑥1 𝑥2 + 3𝑥22 )} × {30 + (2𝑥1 − 3𝑥2 )2 (18 − 32𝑥1 + 12𝑥12 + 48𝑥2 − 36𝑥1 𝑥2 + 27𝑥22 )} Penalized Function P8 𝐷−1 𝐷 𝑖=1 𝑖=1 𝜋 𝑓6 (𝑥) = {10 sin2 (𝜋𝑦𝑖 ) + ∑{1 + 10 sin2 (𝜋𝑦𝑖+1 )} + (𝑦𝑑 − 1)2 } + ∑ 𝜇(𝑥𝑖 , 10,100,4) 𝑥 Penalized Function P16 (6) (7) (8) 𝑓7 (𝑥) = 0.1 {sin2 (3𝜋𝑥𝑖 ) 𝑛−1 + ∑(𝑥𝑖 − 1)2 {1 + 10 sin2 (3𝜋𝑥𝑖+1 )} + (𝑥𝑑 − 1)2 {1 + 10 sin2 (2𝜋𝑥𝐷 )}} (9) 𝑖=1 𝐷 8) 9) + ∑ 𝜇(𝑥𝑖 , 5,100,4) 𝑖=1 Schaffer’s F6 function sin2 (√∑𝑛𝑖=1 𝑥𝑖2 ) − 0.5 𝑓8 (𝑥) = 0.5 + {1 + 0.001(∑𝑛𝑖 𝑥𝑖2 )}2 Shekel 5 function 5 10) 𝑓9 (𝑥) = − ∑ 4 𝑖=1 ∑𝑗=1(𝑥𝑖 Shekel 7 function 7 11) 104 (10) 𝑓10 (𝑥) = − ∑ 𝑖=1 1 2 − 𝑎𝑖𝑗 ) + 𝑐𝑖 1 2 ∑4𝑗=1(𝑥𝑖 − 𝑎𝑖𝑗 ) + 𝑐𝑖 Shekel 10 function (11) (12) IJIEM (Indonesian Journal of Industrial Engineering & Management) Vol 4 No 2 June 2023, 99-112 10 𝑓11 (𝑥) = − ∑ 𝑖=1 12) 1 4 1 8 6 𝐴 = [𝑎𝑖𝑗 ] = 3 2 5 8 6 [7 Six-Hump Camelback 4 1 8 6 7 9 5 1 2 3.6 𝑥14 2 ) 𝑥 + 𝑥1 𝑥2 + (−4 + 4𝑥22 )𝑥22 3 1 13) 14) 418.9829𝑛 − ∑ 𝑥𝑖 sin (√|𝑥𝑖 |) Schwefel P2.6 function 𝑛 0.1 0.2 0.2 0.4 0.4 𝐶 = [𝑐𝑖 ] = 0.6 0.3 0.7 0.5 [0.5] 4 4 1 1 8 8 6 6 3 7 2 9 3 3 8 1 6 2 7 3.6] 𝑓12 (𝑥) = (4 − 2.1𝑥12 + 15) (13) 2 ∑4𝑗=1(𝑥𝑖 − 𝑎𝑖𝑗 ) + 𝑐𝑖 (14) (15) 𝑖=1 Griewank’s function 𝑛 𝑛 1 𝑥𝑖 𝑓14 (𝑥) = 1 + ∑ 𝑥𝑖2 − ∏ cos ( ) 4000 √𝑖 Rastrigin’s function 𝑛 𝑖=1 (16) 𝑖=1 𝑓15 (𝑥) = ∑(𝑥𝑖2 − 10 cos(2𝜋𝑥𝑖 ) + 10) 𝑖=1 𝑓1 𝑓2 𝑓3 𝑓4 𝑓5 𝑓6 𝑓7 𝑓8 𝑓9 𝑓10 𝑓11 𝑓12 𝑓13 𝑓14 𝑓15 Table 2. Properties of benchmark test functions Name Sphere Schwefel P2.22 Rosenbrock’s Ackley’s F1 Goldstein-Price Penalized F8 Penalized P16 Schaffer F6 Shekel 5 Shekel 7 Shekel 10 Six-Hump Camel Schwefel P2.6 Griewank Rastrigin (17) Feasible Bounds [−100, 100]𝑛 [−500, 500]𝑛 [−100, 100]𝑛 [−32.768, 32.768] [−2, 2] [−50, 50] [−50, 50] [−100, 100]𝑛 [0, 10]𝑛 [0, 10]𝑛 [0, 10]𝑛 [−5, 5]𝑛 [−500, 500]𝑛 [−600, 600]𝑛 [−5.12, 5.12] 𝑛 5 5 5 5 2 5 5 2 4 4 4 2 5 5 5 Optimum, 𝒙∗ 0𝑛 420.9687𝑛 1𝑛 0𝑛 (0, −1) −1𝑛 1𝑛 0𝑛 4.0𝑛 4.0𝑛 4.0𝑛 (-0.0898, 0.7126), (0.0898, -0.7126) 420.9687𝑛 0𝑛 0𝑛 𝒇(𝒙∗ ) 0 0 0 0 0 0 0 0 -10.1499 -10.3999 -10.5319 -1.0316 0 0 0 105 IJIEM (Indonesian Journal of Industrial Engineering & Management) Vol 4 No 2 June 2023, 99-112 Func. 𝑓1 𝑓2 𝑓3 𝑓4 𝑓5 𝑓6 𝑓7 𝑓8 𝑓9 𝑓10 𝑓11 𝑓12 𝑓13 𝑓14 𝑓15 Table 3. The summary results obtained by the FBA algorithms for 30 runs Best Value Average Value Worst Value Std. Dev. Success Rate 1.0686E-119 1.7687E-16 5.2875E-15 9.6525E-16 100 4.6843E-33 3.7497E-04 7.0707E-03 1.4766E-03 93.33 2.8994E+00 1.8798E+04 2.2862E+04 4.4271E+03 0 0.0000E+00 7.1304E-08 2.1232E-06 3.8756E-07 100 -9.5923E-14 5.4712E-05 1.6414E-03 2.9967E-04 96.67 2.1903E-11 3.8512E-02 9.2391E-01 1.6801E-01 43.33 1.4096E-14 6.2899E-05 1.5010E-03 2.7692E-04 96.67 3.3695E-13 3.7001E-04 2.4989E-03 5.6688E-04 90 -3.8281E-06 2.7995E+00 7.6638E+00 2.8028E+00 33.33 -1.2173E-04 1.0171E+00 6.4562E+00 1.8296E+00 60 -1.2609E-04 1.2683E+00 7.7326E+00 2.2092E+00 66.67 -3.0562E-08 9.4618E-03 1.0871E-01 2.0714E-02 33.33 3.5809E+01 1.4418E+02 2.0573E+02 4.3292E+01 0 2.1281E-02 7.9438E-02 1.2790E-01 2.6691E-02 0 1.7127E+00 3.2470E+00 6.2162E+00 1.1694E+00 0 and 𝑓15) benchmark functions out of the fifteen tested on. Three (𝑓6, 𝑓9 , and 𝑓12) were below fifty percent while the remaining eight ranges from sixty to hundred percent. Fig. 3(a) to (k) show successful runs of FBA on 11 benchmark test functions for which it converges, and successfully returns at least once an approximate solution to the optimum with error rates less the 1E-08. Fig. 4 (a) to (d) on the other hand are unsuccessful runs of FBA on 4 benchmark test functions. Results in Table 5 show that the comparison between FBA and PSO on all 10 benchmark test functions is statistically significant. Results also show that 7 out of the 10 benchmark tests between FBA and ABC are statistically significant. In nine of the ten statistically significant benchmark test FBA performed better than PSO while in two of the seven statistically significant test FBA performed better than ABC. 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 FITNESS FITNESS FBA is compared ABC and PSO using T-test. Table 4 shows the mean and standard error of FBA, ABC and PSO on the test function while Table 5 tabulates the results of the T-test and indicates test functions in which the performance of FBA is statistically significant when compared with both ABC and PSO. FBA did not return any success for four (𝑓3, 𝑓13, 𝑓14, 0 250 500 750 FUNCTION EVALUATION Fig. 3(a). Sphere 106 1000 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0 1000 2000 3000 4000 FUNCTION EVALUATION Fig. 3(b). Schwefel P2.22 5000 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 FITNESS FITNESS IJIEM (Indonesian Journal of Industrial Engineering & Management) Vol 4 No 2 June 2023, 99-112 0 250 500 750 1000 1250 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1500 0 100 200 300 400 500 600 700 800 FUNCTION EVALUATION FUNCTION EVALUATION FITNESS Fig. 3(c). Ackley Fig. 3(d). Goldstein-price 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 FUNCTION EVALUATION 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0,5 0,4 FITNESS 0,3 0,2 0,1 0 0 250 500 750 1000 1250 1500 1750 2000 0 FUNCTION EVALUATION Fig. 3(f). Penalized function P16 -9 250 500 750 1000 1250 1500 1750 2000-9,1 -9,2 -9,3 -9,4 -9,5 -9,6 -9,7 -9,8 -9,9 -10 -10,1 -10,2 -10,3 FUNCTION EVALUATION Fig. 3(h). Shekel 5 Fig. 3(g). Schaffer F6 0 250 -10 500 750 1000 1250 1500 1750 2000 -10,1 -10,2 FITNESS 0 250 500 750 1000 1250 1500 1750 2000 FUNCTION EVALUATION -10,3 -10,4 -10,5 FUNCTION EVALUATION Fig. 3(i). Shekel 7 107 FITNESS FITNESS Fig. 3(e). Penalized function P8 IJIEM (Indonesian Journal of Industrial Engineering & Management) Vol 4 No 2 June 2023, 99-112 0 -10 250 500 750 1000 1250 1500 1750 2000 -10,1 0 FITNESS -10,3 FITNESS -10,2 -10,4 -10,5 -10,6 FUNCTION EVALUATION FUNCTION EVALUATION 50 100 150 200 250 300 350 400 450 500 0 -0,1 -0,2 -0,3 -0,4 -0,5 -0,6 -0,7 -0,8 -0,9 -1 -1,1 Fig. 3(j). Shekel 10 Fig. 3(k). Six-Hump Camel 30 FITNESS 25 20 15 10 5 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 FUNCTION EVALUATION FITNESS Fig. 4(a). Rosenbrock 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 FUNCTION EVALUATION Fig. 4(b). Griewank 0 10 20 30 40 50 FITNESS FITNESS -50 0 -100 -150 -200 -250 FUNCTION EVALUATION 100 90 80 70 60 50 40 30 20 10 0 0 500 1000 1500 2000 FUNCTION EVALUATION Fig. 4(c). Schwefel P2.6 108 Fig. 4(d). Rastrigin 2500 3000 IJIEM (Indonesian Journal of Industrial Engineering & Management) Vol 4 No 2 June 2023, 99-112 Fig. 5. Histogram of success rates Function Sphere Schwefel P2.22 Rosenbrock Ackley F1 Penalized F. P8 Penalized P16 Schaffer F6 Shekel 7 Shekel 10 Griewank Six-Hump Camel Table 4: The Comparison between FBA, PSO, and ABC FBA ABC PSO Mean Std. Error Mean Std. Error Mean Std. Error 1.77E-16 3.75E-04 1.87E-03 7.13E-08 3.85E-02 6.29E-05 3.70E-04 -9.38E+00 -9.26E+00 7.94E-02 9.46E-03 ±1.76E-16 ±2.70E-04 ±8.08E+02 ±7.08E-08 ±3.07E-02 ±5.06E-05 ±1.03E-04 ±3.34E-01 ±4.03E-01 ±4.87E-03 ±2.07E-02 6.99E-10 2.36E-06 3.93E-02 1.02E-05 1.60E-11 3.72E-09 2.07E-03 -1.04E-01 -1.05E-01 8.73E-09 - ±1.08E-10 ±1.52E-07 ±5.68E-03 ±4.45E-03 ±3.56E-12 ±3.27E-10 ±7.14E-04 ±2.88E-16 ±6.19E-15 ±2.68E-09 - 2.75E+00 5.45E+00 5.46E+01 2.02E+01 8.86E+00 1.22E-01 5.31E+00 5.40E+00 1.01E+00 1.77E+00 ±4.48E-02 ±1.43E-01 ±2.85E+00 ±8.20E-03 ±3.83E-01 ±2.10E-03 ±6.05E-06 ±6.19E-06 ±3.10E-03 ±2.89E-01 Table 5: The T-test between FBA/PSO and FBA/ABC Function Sphere Schwefel P2.22 Rosenbrock Ackley F1 Penalized F. P8 Penalized P16 Schaffer F6 Shekel 7 Shekel 10 Griewank Six-Hump Camel T-test FBA/PSO Critical Value Significant Value 61.5223 <0.00001 YES 38.2318 <0.00001 YES 2.2662 0.0154 YES 2472.81 <0.00001 YES 25.0648 <0.00001 YES 59.4977 <0.00001 YES -44.8228 <0.00001 YES -36.3861 <0.00001 YES -525.4461 <0.00001 YES 6.5879 <0.00001 YES Value 6.4781 1.3829 2.3258 0.0023 1.2555 1.2440 2.7852 3.0444 3.1452 16.3014 - T-test FBA/ABC Critical Significant Value <0.00001 YES 0.0885 YES 0.0135 YES 0.4991 NO 0.1095 NO 0.1116 NO 0.0046 YES 0.0024 YES 0.0019 YES <0.00001 YES - 109 IJIEM (Indonesian Journal of Industrial Engineering & Management) Vol 4 No 2 June 2023, 99-112 V. CONCLUSION The Foraging Bee Optimization (FBA) algorithm is a swarm intelligence optimization algorithm, which has a new unique approach inspired by the characteristics and intelligent behavior displayed by the swarm of foraging bees for solving optimization problems. The algorithm mimics bee colonies by organizing search into three phases: work – when bees forage in patches and discover and exploit new resource-rich flowers; withdraw – when bees return to the colony and reset for the next work phase; and waggle – when bees share information about locations containing resource rich flowers. The algorithm increased the speed of convergence and balance exploration and exploitation by positioning flowers at the extreme of a rectangular workspace that must be scooped by the bees with a propensity that ensures thorough exploration. Exploitation is achieved by a spatial reduction of the best patch subspace over several 3W cycles. Unlike PSO, FBA avoids stagnation and minimizes the possibility of premature convergence that occurs when algorithms are guided by exemplars, honeybees in FBAs are guided by attractors which shift as new flowers are discovered. The proposed algorithm was tested on fifteen standards benchmarks of which three are unimodal while the remaining are complex multimodal spaces with millions of local optima. The algorithm was compared with two state-ofthe-art algorithms PSO and ABC, and the statistically significant result shows that FBA is more efficient than PSO while being competitive with ABC. FBA has been tested extensively in this work, but more experiments need to be done to tune the parameters of FBA for solving more complex test functions at higher dimensions. In addition, adaptive variants of FBA should be developed to reduce the number of parameters that require tuning for high performance. Finally, an informationsharing mechanism will be developed to reduce the overall complexity of the search algorithm. 110 REFERENCES Abbass, H. A. (2001). MBO: Marriage in Honey Bees Optimization-A Haplometrosis Polygynous Swarming Approach. Proceedings of the 2001 IEEE Congress on Evolutionary Computation (IEEE Cat. No. 01TH8546) Vol. 1., 207-214. Alizadehsani, R., Roshanzamir, M., Izadi, N. H., Gravina, R., Kabir, H. D., Nahavandi, D., . . . Fortino, G. (2023). Swarm Intelligence in Internet of Medical Things: A Review. Sensors, 23(3), 1466. doi:https://doi.org/10.3390/s2 Altshuler, Y. (2023). Recent Developments in the Theory and Applicability of Swarm Search. Entropy, 25(5). Aslan, S., Karaboga, D., & Badem, H. (2020). A New Artificial Bee Colony Algorithm employing Intelligent Forager Forwarding Strategies. Applied Soft Computing, 96. Belgrade, U. (2015, September 16). Bee Colony Optimization. Retrieved September 16, 2015, from http://www.sf.bg.ac.rs/index.php/enGB/research-fields/814-bee-colonyoptimization-bco Bolaji, A. L., Khader, A. T., Al-Betar, M. A., & Awadallah, M. A. (2013). Artificial Bee Colony Algorithm, Its Variants and Applications. A Survey, Journal of Theoretical and Applied Information Technology, 47(2), 434-459. Chen, X., Tianfield, H., & Du, W. (2021). Beeforaging Learning Particle Swarm Optimization. Applied Soft Computing, 102. doi:10.1016/j.asoc.2021.107134 Cruz, D. P., Maia, R. D., & de Castro, L. N. (2021). A Framework for the Analysis and Synthesis of Swarm Intelligence Algorithms. Journal of Experimental & Theoretical Artificial Intelligence, 33, 659-681. IJIEM (Indonesian Journal of Industrial Engineering & Management) Vol 4 No 2 June 2023, 99-112 Curkovic, P., & Jerbic, B. (2007). Honey-bees optimization algorithm applied to path planning problem. International journal of simulation modelling, 6(3), 154-165. Dorigo, M., Colorni, A., & Maniezzo, V. (1991). Positive feedback as a search strategy. Technical Report 91-016, Politecnico di Milano, Dipartimento di Elettronica, Milan, Italy. Eberhart, R. C., Shi, Y., & Kennedy, J. (2001). Swarm intelligence. Elsevier. Engelbrecht, A. P. (2007). Computational intelligence: an introduction (2nd ed.). Pretoria, South Africa: John Wiley & Sons. Fakhermand, S. M., & Derakhshani, A. (2023). Design Optimization of Soil-Metal Composite Arch Bridges: Recent Swarm Intelligence Applications. Iranian Journal of Science and Technology, Transactions of Civil Engineering, 47(1), 373-387. Gao, W., Liu, S., & Huang, L. (2012). A global best artificial bee colony algorithm for global optimization. Journal of Computational and Applied Mathematics, 236(11), 2741-2753. Haddad, O. B., Afshar, A., & Mariño, M. A. (2006). Honey-Bees Mating Optimization (HBMO) Algorithm. A New Heuristic Approach for Water Resources Optimization. Water Resources Management, 20(5), 661680. Holland, J. H. (1975). Adaption in Natural and Artificial System. . MIT Press. Janaki, M., & Geethalakshmi, S. N. (2022). A Review of Swarm Intelligence-Based Feature Selection Methods and Its Application. Soft Computing for Security Applications: Proceedings of ICSCS 2022, 435-447. Karaboga, D. (2005). An Idea Based on Honey Bee Swarm for Numerical Optimization. Technical report-tr06, Erciyes University, engineering faculty, computer engineering department., Kayseri, Turkiye. Kaswan, K. S., Dhatterwal, J. S., & Kumar, A. (2023). Swarm Intelligence: An Approach from Natural to Artificial. John Wiley & Sons. Kennedy J. and Eberhart, R. (1995). Particle swarm optimization. Proceedings of the IEEE International Conference on Neural Networks, 4, pp. 1942–1948. Krishnanand , K. N., & Ghose, D. (2005). Detection of multiple source locations using a glowworm metaphor with applications to collective robotics. in Proceedings of the IEEE Swarm Intelligence Symposium (SIS ’05), (pp. 84–94). Pasadena, California. Kumar, A., Chatterjee, J. M., Payal, M., & Rathore, P. S. (2022). Revolutionizing the Internet of Things with Swarm Intelligence. System Assurances, 403436. doi:10.1016/B978-0-323-902403.00023-0 Li, X., & Yang, G. (2016). Artificial bee colony algorithm with memory. Applied Soft Computing, 41, 362-372. Mathlouthi , I., & Bouamama, S. (2016). A family of honey-bee optimization algorithms for Max-CSPs. International Journal of Knowledgebased and Intelligent Engineering Systems, 19(4), 215-224. Pan, X. (2016). Genetic-bee Colony Dualpopulation Self-adaptive Hybrid Algorithm Based on Information Entropy. Scientific Bulletin of National Mining University, 1(1), 116. Pham, D. T., Ghanbarzadeh, A., Koç, E., Otri, S., Rahim, S., & Zaidi, M. (2005). The Bees Algorithm – A Novel Tool for Complex Optimization Problem. In D. T. Pham, E. E. Eldukhri, & A. J. Soroka (Ed.), Intelligent Production Machines and Systems (p. 454). Elsevier Science Ltd. 111 IJIEM (Indonesian Journal of Industrial Engineering & Management) Vol 4 No 2 June 2023, 99-112 Sato, T., & Hagiwara, M. (1997). Bee System: Finding Solution by a Concentrated Search. IEEE International Conference on Computational Cybernetics and Simulation (pp. 3954395). Orlando, FL, USA: IEEE. Schumann, A. (. (2020). Swarm Intelligence: From Social Bacteria to Humans. CRC Press. Selvaraj, S., & Choi, E. (2020). Survey of swarm intelligence algorithms. 3rd International Conference on Software Engineering and Information Management, (pp. 69-73). Shahzad, M. M., Saeed, Z., Akhtar, A., Munawar, H., Yousaf, M. H., Baloach, N. K., & F, H. (2023). A Review of Swarm Robotics in a NutShell. Drones, 7(4), 269. Solgi, R., & Loáiciga, H. A. (2021). BeeInspired Metaheuristics for Global Optimization: A Performance Comparison. Artificial Intelligence Review, 54(7), 4967-4996. doi:10.1007/s10462-021-10015-1 Storn, R., & Price, K. V. (1997). Differential evolution—a simple and efficient heuristic for global optimization over 112 continuous spaces. Journal of Global Optimization, 11(4), 341–359. Teodorovic, D., & Dell’orco, M. (2005). Bee Colony Optimization—A Cooperative Learning Approach to Complex Transportation Problems. Proceedings of the 16th Mini-EURO Conference on Advanced OR and AI Methods in Transportation, (pp. 51-60). Poznan. Retrieved September 13-16, 2005 Tzanetos, A., & Dounias, G. (2020). A Comprehensive Survey on the Applications of Swarm Intelligence and Bio-Inspired Evolutionary Strategies. In G. Tsihrintzis, & L. Jain, Machine Learning Paradigms. Learning and Analytics in Intelligent Systems (Vol. 18, pp. 337-378). Cham: Springer. doi:https://doi.org/10.1007/978-3-03049724-8_15 Yang, C., Chen, J., & Tu, X. (2007). Algorithm of Fast Marriage in Honey Bees Optimization and Convergence Analysis. In Proceedings of the IEEE International Conference on Automation and Logistics (pp. 1794– 1799). Jinan, China: ICAL.