Vol.:(0123456789) 1 3 Evolutionary Intelligence https://doi.org/10.1007/s12065-018-0188-7 SPECIAL ISSUE Particle swarm optimization with adaptive inertia weight based on cumulative binomial probability Ankit Agrawal 1  · Sarsij Tripathi 1 Received: 22 May 2018 / Revised: 20 October 2018 / Accepted: 31 October 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract Particle swarm optimization (PSO) is a population oriented heuristic numerical optimization algorithm, infuenced by the combined behavior of some birds. Since its introduction in 1995, a large number of variants of PSO algorithm have been introduced that improves its performance. The performance of the algorithm mostly rely upon inertia weight and optimal parameter setting. Inertia weight brings equivalence among exploitation and exploration while searching optimal solution within the search region. This paper presents a new improved version of PSO that uses adaptive inertia weight technique which is based on cumulative binomial probability (CBPPSO). The proposed approach along with four other PSO variants are tested over a set of ten well-known optimization test problems. The result confrms that the performance of proposed algorithm (CBPPSO) is better than other PSO variants in most of the cases. Also, the proposed algorithm has been evaluated on three real-world engineering problems and the results obtained are promising. Keywords Inertia weight · Particle swarm optimization (PSO) · Exploration and exploitation · Convergence 1 Introduction The particle swarm optimization is a population oriented meta-heuristic optimization technique which was frst pro- posed by Russell Eberhart and James Kennedy [1]. This algorithm is motivated from the social behavior of some ani- mal groups like fsh schools or bird focks. Similar to other meta-heuristic optimization algorithms, in PSO, a swarm of possible solutions is derived in succeeding iterations. PSO is relatively easy to understand and implement since there are very few parameter settings that are required to tune in comparison to other optimization strategies. In PSO, each particle represents a prospective solution to an optimization problem. The group of particles (or swarm) fy within search space by trailing the current optimum solu- tions. Every particle remembers its best coordinate position in the search region which is related to the best position it has attained so far. This position is called pbest. Similarly, the best current position of the particle within whole swarm, called as gbest is also remembered. In PSO, each particle changes its velocity towards its gbest and pbest location after each time step. The basic PSO [1] is slow in most cases and prematurely converges to local optima. The solution to their problem is use of inertia weight. The inertia weight [2] is the primary parameter of the PSO, and has an important part in balancing the exploration and exploitation. After the introduction of inertia weight, many versions of the PSO algorithm have been presented aiming balanced explora- tion–exploitation trade-of. Shi and Eberhart [2] examined diferent constant values of inertia weight and reached to the conclusion that within specifc range, PSO search the optimum global solution within a acceptable number of iterations. Mostly with lower value of inertia weight, PSO converges in local optima region and with higher value of inertia weight; it diverges and hence performs little exploration in the search region. For fnding an optimum solution in a dynamic environment, a random inertia weight strategy is used by Russell Eberhart and Shi [3] and varies in the range [0.5, 1]. This approach is used to alter exploitation and exploration randomly. A large number of inertia weight variants use time-vary- ing inertia weight method which utilizes iteration number to determine the value of inertia weight. Most famous among them is linear decreasing inertia weight technique [4], in * Ankit Agrawal aagrawal.phd2017.cse@nitrr.ac.in 1 Department of Computer Science and Engineering, National Institute of Technology Raipur, G.E Road, Raipur, Chhattisgarh 492001, India