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The paper aims to study a multi-period maximal covering location problem with the configuration of different types of facilities, as an extension of the classical maximal covering location problem (MCLP). The proposed model can have applications such as locating disaster relief facilities, hospitals, and chain supermarkets. The facilities are supposed to be comprised of various units, called the modules. The modules have different sizes and can transfer between facilities during the planning horizon according to demand variation. Both the facilities and modules are capacitated as a real-life fact. To solve the problem, two upper bounds&mdash, (LR1) and (LR2)&mdash, and Lagrangian decomposition (LD) are developed. Two lower bounds are computed from feasible solutions obtained from (LR1), (LR2), and (LD) and a novel heuristic algorithm. The results demonstrate that the LD method combined with the lower bound obtained from the developed heuristic method (LD-HLB) shows better performance and is preferred to solve both small- and large-scale problems in terms of bound tightness and efficiency especially for solving large-scale problems. The upper bounds and lower bounds generated by the solution procedures can be used as the profit approximation by the managerial executives in their decision-making process.

This paper presents a proposal of a time domain room acoustic solver using novel fourth-order accurate explicit time domain finite element method (TD-FEM), with demonstration of its applicability for practical room acoustic problems. Although time domain wave acoustic methods have been extremely attractive in recent years as room acoustic design tools, a computationally efficient solver is demanded to reduce their overly large computational costs for practical applications. Earlier, the authors proposed an efficient room acoustic solver using explicit TD-FEM having fourth-order accuracy in both space and time using low-order discretization techniques. Nevertheless, this conventional method only achieves fourth-order accuracy in time when using only square or cubic elements. That achievement markedly impairs the benefits of FEM with geometrical flexibility. As described herein, that difficulty is solved by construction of a specially designed time-integration method for time discretization. The proposed method can use irregularly shaped elements while maintaining fourth-order accuracy in time without additional computational complexity compared to the conventional method. The dispersion and dissipation characteristics of the proposed method are examined respectively both theoretically and numerically. Moreover, the practicality of the method for solving room acoustic problems at kilohertz frequencies is presented via two numerical examples of acoustic simulations in a rectangular sound field including complex sound diffusers and in a complexly shaped concert hall.