In this paper we develop a capacities theory connected with the fractional Sobolev spaces with variable exponents. Two kinds of capacities are studied: Sobolev capacity and relative capacity. Basic properties of capacities, including monotonicity, outer capacity and several results, are studies. We prove that both capacities is a Choquet capacity and all borel sets are capacitable.Mathematics Subject Classification: Primary 31B15, 46E35.
Assume that Ω ⊂ R n is a bounded domain. We show that under certain regularity assumptions on Ω there exists a linear extension operator from the space W s,p(.,.) (Ω) to W s,p(.,.) (R n ). As an application we study complemented subspaces in W s,p(.,.) via the trace operator.Keywords extension operator · trace operator · fractional variable exponent Sobolev spaces · complemented subspace problem.
Purpose
The purpose of this paper is to solve the capacitated location routing problem (CLRP), which is an NP-hard problem that involves making strategic decisions as well as tactical and operational decisions, using a hybrid particle swarm optimization (PSO) algorithm.
Design/methodology/approach
PSO, which is a population-based metaheuristic, is combined with a variable neighborhood strategy variable neighborhood search to solve the CLRP.
Findings
The algorithm is tested on a set of instances available in the literature and gave good quality solutions, results are compared to those obtained by other metaheuristic, evolutionary and PSO algorithms.
Originality/value
Local search is a time consuming phase in hybrid PSO algorithms, a set of neighborhood structures suitable for the solution representation used in the PSO algorithm is proposed in the VNS phase, moves are applied directly to particles, a clear decoding method is adopted to evaluate a particle (solution) and there is no need to re-encode solutions in the form of particles after applying local search.
This paper aims to solve a multiperiod location lot-sizing routing problem with deterministic demand in a two-echelon network composed of a single factory, a set of potential depots, and a set of customers. Solving this problem involves making strategic decisions such as location of depots as well as operational and tactical decisions which include customers' assignment to the open depots, vehicle routing organization, and inventory management. A mathematical model is presented to describe the problem and a genetic algorithm combined with a local search procedure is proposed to solve it and is tested over three sets of instances.
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