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E-LIA (Enhanced local improvement algorithm)

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Introduction

Enhanced local improvement algorithm (E-LIA) is a meta-heuristic algorithm that can be used for solving combinatorial optimization problems. The algorithm is based on the concept of local search and employs different strategies for enhancing its performance. E-LIA has been successfully applied to a variety of optimization problems, such as the traveling salesman problem, vehicle routing problem, and facility location problem.

The local search is a well-known approach for solving combinatorial optimization problems, which starts with an initial solution and improves it by iteratively modifying the current solution through local moves. The basic idea of local search is to find a local optimum by exploring the neighborhood of the current solution. However, local search algorithms may suffer from getting stuck in local optima, which is a common problem in optimization.

E-LIA, as a meta-heuristic algorithm, is designed to overcome this limitation of local search by introducing some enhancements to the basic local search procedure. These enhancements include perturbation, diversification, and intensification strategies that aim to improve the exploration and exploitation of the search space.

In this article, we will explain the key concepts of E-LIA and how it works. We will also discuss the strengths and weaknesses of the algorithm and provide some examples of its applications.

Key Concepts of E-LIA

E-LIA is a population-based algorithm, which means that it maintains a set of candidate solutions, called the population, throughout the search process. Each solution in the population is represented as a vector of decision variables that define the problem's solution space. The algorithm iteratively modifies the population by applying local moves to each solution and selecting the best ones to form the new population.

The algorithm's performance depends on the quality of the initial population and the efficiency of the local search procedure. Therefore, E-LIA employs a random initialization procedure to generate an initial population of diverse solutions. The diversity of the population is crucial to ensure that the algorithm explores different regions of the search space.

The local search procedure used in E-LIA is based on the iterative improvement approach, which involves making small modifications to the current solution to obtain a better one. The modifications are performed through local moves, which are typically defined as swapping, inserting, or deleting a single element of the solution vector.

Perturbation Strategy:

The perturbation strategy is used to escape from local optima by introducing randomness into the search process. It involves applying a set of random perturbations to the current solution, such as swapping two randomly selected elements or randomly selecting a subset of elements to be modified. This helps to explore new regions of the search space and avoid getting stuck in local optima.

Diversification Strategy:

The diversification strategy is used to increase the diversity of the population by encouraging exploration of different regions of the search space. It involves maintaining a set of elite solutions that are selected based on their fitness value, and a set of non-elite solutions that are randomly selected from the population. The non-elite solutions are then modified through local moves to create new solutions that may have better fitness values than the elite solutions. This helps to maintain diversity in the population and prevent premature convergence to a sub-optimal solution.

Intensification Strategy:

The intensification strategy is used to exploit the local structure of the search space by focusing the search on promising regions. It involves identifying the best solutions in the current population and applying more intensive local search procedures to improve them further. This helps to refine the solutions and move towards the global optimum.

How E-LIA Works

The basic steps of E-LIA can be summarized as follows:

  1. Initialization: Generate an initial population of candidate solutions randomly.
  2. Local search: Apply local moves to each solution in the population to obtain a set of neighboring solutions.
  3. Perturbation: Apply a set of random perturbations to the solutions in the population to escape from local optima.
  4. Diversification: Select a set of non-elite solutions from the population and apply local moves to them to create new solutions.
  5. Intensification: Identify the best solutions in the population and apply more intensive local search procedures to them.
  6. Selection: Select the best solutions from the current population to form the next population.
  7. Termination: Stop the search when a stopping criterion is met, such as a maximum number of iterations or a minimum improvement threshold.

Strengths and Weaknesses of E-LIA

E-LIA has several strengths that make it a competitive algorithm for solving combinatorial optimization problems. First, it is a flexible algorithm that can be easily adapted to different types of problems and problem constraints. Second, it has been shown to achieve high-quality solutions in a reasonable amount of time, making it suitable for real-world applications. Third, it is a parallelizable algorithm that can be easily implemented on parallel computing architectures, allowing for faster convergence and better scalability.

However, E-LIA also has some weaknesses that need to be considered. First, it may suffer from premature convergence if the diversity of the population is not maintained. Second, it may require a large number of iterations to achieve convergence, making it computationally expensive for large-scale problems. Third, it may not be able to guarantee finding the global optimum due to its reliance on local search procedures.

Applications of E-LIA

E-LIA has been successfully applied to a variety of combinatorial optimization problems, including the traveling salesman problem, vehicle routing problem, facility location problem, graph coloring problem, and others. In the traveling salesman problem, E-LIA has been shown to achieve high-quality solutions that are competitive with other state-of-the-art algorithms. In the vehicle routing problem, E-LIA has been used to optimize the routing of delivery vehicles, resulting in significant cost savings for logistics companies. In the facility location problem, E-LIA has been used to identify the optimal locations for new facilities, reducing transportation costs and improving customer satisfaction.

Conclusion

E-LIA is a meta-heuristic algorithm that combines local search with perturbation, diversification, and intensification strategies to solve combinatorial optimization problems. It has been shown to achieve high-quality solutions in a reasonable amount of time, making it suitable for real-world applications. Its flexibility and parallelizability make it a competitive algorithm for solving a variety of problems.

However, like any algorithm, E-LIA has its limitations and may not be suitable for all types of problems. It may suffer from premature convergence, require a large number of iterations, and not be able to guarantee finding the global optimum.