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Facility Location problems are one of the most prominent optimization problems in computer science, operations research, and combinatorial optimization. Their simple yet intrinsic structure has led to their widespread application in diverse fields, such as leasing, clustering, and urban planning. In facility location problems, potential sites for opening facilities, like warehouses, each have an associated opening cost. Client locations, such as stores, require service from a facility, incurring a cost proportional to their distance from them when assigned. Identifying optimal facility locations to minimize total costs associated with opening facilities and client assignment is the aim. This paper focuses on the online setting, in which client locations are revealed to the algorithm incrementally, unlike the classical offline setting, in which all client locations are given in advance. In this dynamic context, often referred to as the online algorithm scenario, immediate decisions are made to determine whether the newly disclosed client should be assigned to an existing facility or necessitate the establishment of a new facility. To gauge the effectiveness of online algorithms, the concept of competitive analysis is employed. This approach assesses the worst-case performance of online algorithms by comparing them to an optimal algorithm equipped with complete knowledge of forthcoming input sequences. Despite the considerable significance of the Facility Location problem and the substantial body of literature on it within the online algorithm framework, this article represents the first comprehensive summary of these research findings. It explores the algorithmic techniques employed, delves into practical applications, and identifies pertinent research gaps within this domain. |
