## 2 Related Work
### Facility Location Problems  
Facility location is an important and fundamental strategic decision that strongly influences tactical and operational aspects in any organization. Such location decisions are common at various levels of human organizations, from individuals and households to corporations, government agencies, and international organizations. The study of location decisions has a long tradition in the literature. The FLP emerged in the 1960s in the field of operations research [24] and was initially called the plant location problem. The FLPs represent classical optimization problems that involve finding optimal locations for facilities. These problems involve determining the most appropriate locations for one or many facilities to efficiently serve a set of demand points. The definition of the \"best\" location varies depending on the nature of the specific problem, taking into account constraints and optimality criteria [25]. FLPs have applications in a wide variety of settings where strategic placement is required, including hospitals, fire stations, bus stops, train stations, truck terminals, gas stations, blood bank centers, retail stores, libraries, parks, post offices, airports, and landfills. There are sufficient detailed descriptions of FLP algorithms in the literature. A concise but detailed explanation, with an additional investigation of uncertainty and a focus on FLP for supply chains, is provided by [26]. The versatility of FLPs makes them a mandatory tool for optimizing the location of critical facilities and services in a variety of real-world scenarios. Various applications of FLPs can be found in the context of LML [27; 28; 29; 30; 31; 32; 33; 34]. [PERSON] et al. [35] have provided a comprehensive overview of the applications of FLPs and explored future developments in this field. In [5], the authors presented a remarkable application of FLP in APLs for static situations. However, the dynamic aspects of the problem were largely ignored, such as variability in customer willingness to use the service and operational setup costs. Ref. [36] explored healthcare logistics and devised a solution that combines a facility location problem with routing to efficiently supply medications to patients, while minimizing delivery costs. Their study focused on serving patients within the service area of APLs, leaving the remaining demand to be fulfilled through home delivery. Another study by [37] explored using existing infrastructure like supermarkets and gas stations as potential APL locations. While they determined optimal locations based on customer density, they overlooked important factors such as costs and environmental aspects. In [38], the study focused on maximizing profit by determining suitable APL locations. They used a threshold Luce model to represent discrete customer choice, where APL users choose the closest facility or select from a set of APLs. However, they overlooked shipping costs in their analysis, which is a crucial aspect of efficient network design. A dynamic approach was taken by [22]. They optimized the locations of mobile parcel lockers that can adapt to recipients' whereabouts during different time periods. Despite their valuable contributions, these studies failed to consider a holistic LML network encompassing both home and APL deliveries, which is essential for accurately representing the perspective of a logistics service provider. Moreover, an important aspect overlooked in previous research was the external impact, specifically CO\({}_{2}\) emissions generated during the delivery process and APL usage. In recent times, e-commerce retailers have been reshaping their strategies to minimize CO\({}_{2}\) emissions, making it crucial to incorporate environmental considerations when designing APL networks. For example, the e-commerce retailer Zalando has already achieved carbon-neutral in-house operations and carbon-neutral shipping and return processes [39]. The authors of [40] investigate APLs in two areas (rural and suburban) in an experimental set-up.