## 4 Hybrid Covering Location Problem Formulation  
In this section, a mixed integer linear programming model for the hybrid covering location problem (HCLP) is presented. The mathematical model integrates the coverage concept of capacitated SCLP and modular MCLP as strategic and tactical planning decisions, respectively. The objective of our model is to maximize the profit obtained from covering demand points and subtracting the cost of facilities location. The constraints of the model are the full coverage of all points by the facilities in a way that the total allocated demands to the facilities do not exceed the capacity of the facilities. We suppose that, as a real fact, the number of demand points and their demands may increase during the time horizon; then, the number of located facilities can expand to guaranty the full coverage of points. Other constraints of the model are the total number of each kind of assigned module to the facilities, the capacity of each module, and not to provide coverage for each point more than its need. Some of the most important decisions in the proposed model are as follows:  
* Location, number, and establishment time of facilities is strategic periods during the planning horizon. * Type and number of each module assigned to the located facilities in tactical periods of each strategic period during the planning horizon. * Percentage of the allocate demand of points to the assigned modules in tactical periods of each strategic period during the planning horizon. The main assumptions to develop the model are as follows:  
* The problem is studied in a multi-period framework. The total planning horizon is classified into two types of periods as the strategic periods and tactical periods. Each strategic period is composed of several tactical periods with different kinds of decisions to be made. * The facilities are supposed to be a piece of land or site, equipped with some initial infrastructures. The locations of these facilities are going to be decided only in strategic periods using the coverage concept of SCLP to determine the minimum number of facilities to be located with the aim of covering all demand points. Once a facility is opened in a strategic period, it cannot be closed and should continue its operation in future periods. In addition, the facilities are supposed to be capacitated, and the number of facilities can be expanded in response to the demand variation in the upcoming strategic periods. * We suppose that there are different kinds of service providing units, namely the modules of facilities that are limited in terms of numbers and capacities. These modules can move to the facilities and should be assigned to the facilities in tactical periods. The optimal decisions of the module assignment to the facilities are supposed to obey the coverage concept provided with multi-period modular MCLP. The arrangement of modules in facilities can be varied in different tactical periods according to the points' demand fluctuation in order to maximize the amount of total covered demands. * Each module comes in different sizes. It can be chosen from different sizes to increase the service quality offered to demand points to overcome the service shortages or having idle units. * The modules are portable, and they can be transferred among the facilities when there is more request in another facility. The transferability is an important specification of modularity design that yields to flexibility in the system and reduces costs. The portability of most modules helps to provide a good level of service to demand points without having to provide more modules. * It is supposed that covering the demand points by the modules obeys the gradual coverage concept using a partial coverage function. In a gradual coverage function, the demand points inside the full coverage radius can be covered completely, but by increasing the coverage radius, the amount of coverage decreases and the points outside the partial coverage radius are supposed not to be covered.