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This paper presents and examines a newly improved linear technique for solving the equilibrium problem of a pseudomonotone operator and the fixed point problem of a nonexpansive mapping within a real Hilbert space framework. The technique relies two modified mildly inertial methods and the subgradient extragradient approach. In addition, it can be viewed as an advancement over the previously known inertial subgradient extragradient approach. Based on common assumptions, the algorithm's weak convergence has been established. Finally, in order to confirm the efficiency and benefit of the proposed algorithm, we present a few numerical experiments.

This article proposes an iteration algorithm with double inertial extrapolation steps for approximating a common solution of split equilibrium problem, fixed point problem and variational inequity problem in the framework of Hilbert spaces. Unlike several existing methods, our algorithm is designed such that its implementation does not require the knowledge of the norm of the bounded linear operator and the value of the Lipschitz constant. The proposed algorithm does not depend on any line search rule. The method uses a self-adaptive step size which is allowed to increase from iteration to iteration. Furthermore, using some mild assumptions, we establish a strong convergence theorem for the proposed algorithm. Lastly, we present a numerical experiment to show the efficiency and the applicability of our proposed iterative method in comparison with some well-known methods in the literature. Our results unify, extend and generalize so many results in the literature from the setting of the solution set of one problem to the more general setting common solution set of three problems.

In this paper, we introduce a new type of modified generalized $\alpha$-nonexpansive mapping and establish some fixed point properties and demiclosedness principle for this class of mappings in the framework of uniformly convex Banach spaces. We further propose a new iterative method for approximating a common fixed point of two modified generalized $\alpha$-nonexpansive mappings and present some weak and strong convergence theorems for these mappings in uniformly convex Banach spaces. In addition, we apply our result to solve a convex-constrained minimization problem, variational inequality and split feasibility problem and present some numerical experiments in infinite dimensional spaces to establish the applicability and efficiency of our proposed algorithm. The obtained results in this paper improve and extend some related results in the literature.

Research in the area of unrelated parallel machine scheduling problem (UPMSP) with sequence-dependent setup times has received little attention from the research community. However, this problem is NP-hard even without considering the setup times, and when sequence-dependent setup times are included, finding optimal solutions becomes very difficult, especially for the problems with large dimension. In this paper, a firefly algorithm (FA) which is refined with a local search solution improvement mechanism is presented to solve this problem, with the objective of reaching a near-optimum solution. Since the classical FA was originally designed for continuous optimization problems, a new solution representation scheme is designed to make the FA suitable for a combinatorial optimization problem such as the UPMSP. Three different popular metaheuristic algorithms are developed in parallel to verify and measure the effectiveness of the proposed algorithm. More so, the success of the novel firefly scheduling method is measured by comparing the quality of its solutions against the best-known methods from the literature. An exhaustive computational and statistical analysis is carried out to show an excellent performance of the new method on a large set of problem instances. The numerical results show that the improved FA is competitive, fast, and efficient and provide good quality solutions for both small and large problem instances.

The intelligent water drop algorithm is a swarm-based metaheuristic algorithm, inspired by the characteristics of water drops in the river and the environmental changes resulting from the action of the flowing river. Since its appearance as an alternative stochastic optimization method, the algorithm has found applications in solving a wide range of combinatorial and functional optimization problems. This paper presents an improved intelligent water drop algorithm for solving multi-depot vehicle routing problems. A simulated annealing algorithm was introduced into the proposed algorithm as a local search metaheuristic to prevent the intelligent water drop algorithm from getting trapped into local minima and also improve its solution quality. In addition, some of the potential problematic issues associated with using simulated annealing that include high computational runtime and exponential calculation of the probability of acceptance criteria, are investigated. The exponential calculation of the probability of acceptance criteria for the simulated annealing based techniques is computationally expensive. Therefore, in order to maximize the performance of the intelligent water drop algorithm using simulated annealing, a better way of calculating the probability of acceptance criteria is considered. The performance of the proposed hybrid algorithm is evaluated by using 33 standard test problems, with the results obtained compared with the solutions offered by four well-known techniques from the subject literature. Experimental results and statistical tests show that the new method possesses outstanding performance in terms of solution quality and runtime consumed. In addition, the proposed algorithm is suitable for solving large-scale problems.