## 1 Introduction
For MMCMCLP that is a maximization problem, the Lagrangian relaxation (LR) can generate an upper bound on the optimal objective value. Furthermore, many possible relaxations have been examined to figure out the relaxation of which constraints yields to the most time-efficient subproblem. In order to be able to provide sufficient evaluations on the possible methods, two procedures are proposed. In both approaches, the upper bounds are calculated by using two different sets of relaxed constraints and a decomposed problem. We also have developed two procedures to obtain a lower bound for the optimal objective value of the proposed model. In the first approach, the lower bound is computed by generating feasible solutions. In the second procedure, the lower bound is obtained from a developed heuristic method that locates the predefined number of facilities according to the criteria of more capacity and coverage of demands. Afterward, a sufficient number of modules with lower cost and more capacity would be assigned to the located facilities to maximize the number of allocated demands. Two relaxation problems and a decomposed problem provide three different upper bounds, which in addition to two lower bounds obtained from constructed feasible solutions and a heuristic method provide various bounds. The best method is the one that can produce the tightest upper and lower bounds, for which the bounds obtained from different approaches are analyzed to investigate the tightest bounds to select the best method. The main contributions and innovations of this study are summarized as follows:  
* We propose a new model called multi-period modular capacitated maximal covering location problem, which in addition to the facility location decisions, it determines module assignment and demand allocation decisions in different periods of the planning horizon. * The configuration of the facilities using the modularity concept results in having three-level facilities. Different size of modules (first level) along with having a different kind of modules (second level) makes it possible to have different facilities (third level). An efficient Lagrangian decomposition-based method is developed to derive the upper bound. Furthermore, two different lower bounds are developed and compared to synthesize the proposed Lagrangian decomposition (LD) method. * Different upper bounds and lower bounds from various approaches are compared to evaluate the efficiency and tightness of the bounds. Our findings approve the superiority of the bounds obtained from Lagrangian decomposition-based (upper bound) and heuristic method (lower bound) over large-scale instances. * Sensitivity analysis for the problem is conducted to investigate the validity of the proposed model under different parameters. The remainder of the paper is organized as follows. Section 2 reviews the related studies. The mathematical model for the MMCMCLP is provided in Section 3. In Section 4, the solution approach to obtain Lagrangian upper bound integrated with a heuristic method to achieve the lower bound of the problem is proposed. Then numerical examples are analyzed in Section 5 and finally, conclusions are discussed in Section 6.