This paper proposes a novel multiple chaos embedded gravitational search algorithm (MCGSA) that simultaneously utilizes multiple different chaotic maps with a manner of local search. The embedded chaotic local search can exploit a small region to refine solutions obtained by the canonical gravitational search algorithm (GSA) due to its inherent local exploitation ability. Meanwhile it also has a chance to explore a huge search space by taking advantages of the ergodicity of chaos. To fully utilize the dynamic properties of chaos, we propose three kinds of embedding strategies. The multiple chaotic maps are randomly, parallelly, or memory-selectively incorporated into GSA, respectively. To evaluate the effectiveness and efficiency of the proposed MCGSA, we compare it with GSA and twelve variants of chaotic GSA which use only a certain chaotic map on a set of 48 benchmark optimization functions. Experimental results show that MCGSA performs better than its competitors in terms of convergence speed and solution accuracy. In addition, statistical analysis based on Friedman test indicates that the parallelly embedding strategy is the most effective for improving the performance of GSA.

In Recent years, a paradigm of optimization algorithms referred to as “meta-heuristics” have been gaining attention as a means of obtaining approximate solutions to optimization problems quickly without any special prior knowledge of the problems. Meta-heuristics are characterized by flexibility in implementation. In practical applications, we can make use of not only existing algorithms but also revised algorithms that reflect the prior knowledge of the problems. Most meta-heuristic algorithms lack mathematical grounds, however, and therefore generally require a process of trial and error for the algorithm design and its parameter adjustment. For one of the resolution of the problem, we propose an approach to design algorithms with mathematical grounds. The approach consists of first constructing a “framework” of which dynamic characteristics can be derived theoretically and then designing concrete algorithms within the framework. In this paper, we propose such a framework that employs two following basic strategies commonly used in existing meta-heuristic algorithms, namely, (1) multipoint searching, and (2) stochastic searching with pseudo-random numbers. In the framework, the update-formula of search point positions is given by a linear combination of normally distributed random numbers and a fixed input term. We also present a stability theory of the search point distribution for the proposed framework, using the variance of the search point positions as the index of stability. This theory can be applied to any algorithm that is designed within the proposed framework, and the results can be used to obtain a control rule for the search point distribution of each algorithm. We also verify the stability theory and the optimization capability of an algorithm based on the proposed framework by numerical simulation.

This paper treats meta-heuristics for combinatorial optimization problems. The parallelization of meta-heuristics is then discussed in which we show that parallel processing has possibility of not only speeding up but also improving solution quality. Finally we extend the discussion of the combinatorial optimization into autonomous decentralized systems, say autonomous decentralized optimization. This notion becomes very important with the advancement of the network-connected system architecture.

The vehicle routing and facility location fields are well-developed areas in management science and operations research application. There is an increasing recognition that effective decision-making in these fields requires the adoption of optimization software that can be embedded into a decision support system. In this paper, we describe the implementation details of our software components for solving the vehicle routing and facility location problems.