## 1 Introduction
Those real-world problems and demand variation in different time periods necessitate to develop a multi-period model that optimizes the objective value in different time periods looking for optimal solutions for assigning the modules to the facilities and allocating the demand points to the modules. The proposed mathematical model in this paper locates the capacitated facilities at the beginning of the planning horizon and then in each time period, assigns the optimum number of capacitated modules to the facilities with the objective of maximizing the covered demands and minimizing the module assignment cost. Modularization of the facilities provides the opportunity to develop more cost and energy-efficient smart facilities. One of the main applications of this model is for a disaster relief situation in which government is the decision maker. The service request of demand points is not the same in all disasters and depends on many factors such as the severity of the disaster. Demand fluctuation can be reflected by having the multi-period planning model. In this application, the facilities are the disaster relief centers, the modules can be ambulances, trucks, helicopters, first aid units, food providing units, sleeping tents, shower rooms, etc. and the demand points are the residents or residential areas affected by the disaster and having service requests. The locations of a limited number of disaster relief facilities are first determined and having the assumption that each disaster is happening in one of the time periods of the planning horizon, the modules are assigned to the located facilities. The modules are capacitated and they have different sizes. The optimal number and size of each module's kind that should be assigned to the relief centers are determined by the model in order to have maximized allocated and covered demand points. After terminating the mission of modules in one of the affected areas in one period, in the next period the modules can be transferred to the other affected area to provide service there. Other applications of this model can be in locating and service providing of chain convenient stores as the facilities and decision maker is the operation planning department. There are many product categories available in these stores that can be assumed as the modules and customers are the demand points. Most of the time, there is no need to provide all kinds of products in all stores when there might be not enough demand for them in different seasons. According to our model, the predefined stores can be located using the developed model at the beginning of the planning horizon. Afterwards, according to the demand variation in different areas and time periods, the optimal number and size of modules are assigned to the located stores. The objective of maximizing the covered demands from one hand, and minimizing the cost of module assignments on the other hand creates a good balance between service quality and cost management. The mathematical model of the multi-period modular capacitated maximal covering location problem (MMCMCLP) is a mixed integer linear programming model. The objective is to maximize the profit that is obtained from the income of covering demand points and the cost of module assignment. Accordingly, three kinds of decisions are determined by MMCMCLP as:  
* Location of the predefined number of facilities;
* Type and number of each module assigned to the located facilities in each time period during the planning horizon;
* Percentage of allocated demands of points to the assigned modules in each time period. The small-size test instances can be solved to optimality using general solvers such as CPLEX. As the size of the problems increases, this solver is unable to solve the problems. To overcome this difficulty, a Lagrangian decomposition-based algorithm is developed that obtains an upper value and lower value for the optimal objective value, for which the decision makers can be assured that their profit is a value between the generated bounds and will not exceed these values.