# Test Center Location Problem: A Bi-Objective Model and Algorithms
## 2 Related Work  
The FL problem requires the determination of appropriate locations (centers or hubs) of a set of facilities among a set of demand points (customers or clients) [1]. This problem has numerous real-world applications and has been widely studied in the literature of operations research, industrial engineering, applied mathematics, and computer science [1; 2; 3]. There are several parameters, constraints, and objectives in the FL problem, and consequently, many variations of it have been studied [1; 2]. For example, the demand set may be discrete or continuous, weighted or unweighted, static or dynamic, certain or uncertain. The potential facility set can be discrete or continuous, and capacitated or incapacitated. Furthermore, several definitions for the objective function have been considered [1; 2; 4]. The objective function in FL problems, which is usually determined with regard to the type of application, is very important in the complexity class of the problems [1; 2; 3]. For example, _k-median_ and _k-center_ are two well-known types of FL problems for public FL and emergency FL with the objectives _min-sum_ and _min-max_, respectively. The NP-completeness of both of the problems (and some variations of them) has been proved [4], and many approximations and heuristic approaches have been proposed for solving them (e.g., see [5; 6; 7; 8]). In both \(k\)-median and \(k\)-center problems, the goal is optimizing the process for the client side, e.g., minimizing the average and maximum distance of each client from its closest center, which is useful for both public and emergency facilities. Both of these objectives belong to the client side, that is, objectives to emphasize the service quality that the clients receive. However, there are objectives such as the recently proposed \(k\)-_balanced_ objective that enhance the quality or eligibility in the center side [9]. The \(k\)-balanced objective focuses on the fair distribution of accessibility among the clients [10]. For example, consider the problem of placing some congruent antennas in a wireless network [11]. For some technical reasons, and to have a good connection quality, usually each client is assigned to its closest antenna(s). Thus, to manage the traffic in the network, it is necessary for the antennas to have almost the same network load. As another example, assume locating \(k\) voting stations under the assumption that each person goes to their closest voting station. So to balance the crowding in the stations, the stations' workloads need to be balanced. These considerations may also apply in placing banks, stores, educational, cultural, and sports centers, and are very important in Territory Design [12]. Note that in the \(k\)-balanced problem, each client is served by the closest center; consequently, it is not an assignment problem [10]. Similar to the FL problems, there are many parameters and constraints in the \(k\)-balance problem, and different variations of it can be presented. In addition to the discrete or continuous potential facility centers and different metrics, the definition of the term \"maximum balance\" is not unique and can be determined by the type of application. [PERSON] [13] originally proposed the \(k\)-balance problem in 2011. He studied the discrete version of the problem and constructed integer programming formulations of a variation of the problem and proposed a branch-and-cut algorithm for solving them [13]. Finally, he evaluated the algorithm by some simulations that used computational time as the efficiency factor [13]. He noted that the number of valid inequalities in the formulations of the problem is exponential. [PERSON] et al. [14] proposed a combined heuristic method consisting of a genetic algorithm with an interchange heuristic for the balanced allocation problem [14]. This combined method was a variable neighborhood search heuristic that utilizes a technique called shaking neighborhood in order to avoid becoming stuck in local optima, which has subsequently been improved by [PERSON] et al. [15].