This section presents a description of the work related to APLs which helps the reader to understand our model. The use of APLs is an important step forward in the development of the last mile distribution in urban delivery of goods and merchandises. Nowadays, the use of APLs has been revealed as a very convenient experience for many customers [
13,
14], apart from its suitability for parcel urban logistics [
15]. Its success is mainly due to the failed delivery problems observed in the urban distribution of goods [
16]. Furthermore, some authors [
17] has recently pointed out the increasing popularity of APLs among customers, when they have to choose the delivery mode and pick-up system for their purchases. Therefore, an optimum design of the urban network of APLs is needed considering their potential population of users, their associated costs and their efficiency. Thus, developed an initial paper [
18] to optimize the APLs network, which has been complemented by the study [
19], in which authors designed some accurate models to build an APLs distribution structure. Another important aspect of the APLs appropriateness for last mile urban distribution is their contribution to the improvement of pollution in urban centers [
20,
21], due to the fact of the substitution effect in relation to delivery vans. Likewise, the COVID-19 outbreak and its evolution during the years 2020–2021 has accelerated the use of e-commerce and, as a consequence, the number of APLs in the urban areas have mushroomed [
22]. Thus, this boom of e-commerce has involved the need for optimizing the APLs network using different methodologies of simulation-optimization techniques. For instance, research [
23] designed some interesting optimization models to implement the APLs network structure in Singapore. Similarly [
6], built an agent-based simulation model to evaluate the current APLs network in the Brazilian city of Belo Horizonte. The utilization of parcel lockers in a crowdshipping network in case of Melbourne [
24] allows for shorter trip detour and better geographical coverage. Authors [
24] develop a model for locating parcel lockers and allocating delivery task. In research paper [
25], authors consider the use of lockers in parcel delivery. Study of routing problems wherein one or more vehicles are deployed to deliver packages directly to clients or lockers [
25]. Authors [
25] examine the implications of incorporating lockers when these problems include time constraints. Authors propose new formulations for these problems, as well as some valid inequalities and a branch-and-cut algorithm. Furthermore, researchers investigate the distinction between routing problems with lockers and classical routing challenges. Another research paper [
26] scrutinize the impact of parcel lockers on travel distances as well as Carbon dioxide emission. Authors demonstrate that, in certain circumstances, parcel lockers positively contribute to both of the previous mentioned performance indices. In paper [
27] authors present an approach for designing joint delivery networks in urban areas enhanced by employing parcel lockers. This model is divided into two levels, with the lower level dealing with multi-depot capacitated vehicle routing problems (MDCVRP) for a bundle of depots and lockers, while the upper level is a (minimum-cost) parcel network flow problem (PNFP) considering goods supplied among distribution centers and storage areas, as well as the placement and size of storage. A hybrid algorithm integrating a Genetic Algorithm with the Lin-Kernighan Heuristic has been developed. The GA focuses on finding solutions for the PNFP. Once the package flow paths have been resolved, the LKH optimizes vehicle flow. Different study [
28] proposes six mathematical models to different crowdsourced delivery operation modes are quantified. Numerous realistic aspects, such as the most recent service time for each task, task cancellation rate, and task range distribution, are also considered. Authors investigate [
29] a last-mile delivery approach in the augmented system with three service modes: home delivery plus pickup, parcel locker delivery plus pickup, and home or parcel locker delivery plus pickup. Problem considered [
29] outlines further locker pickup and delivery options. The topic is known as the vehicle routing problem with simultaneous pickup and delivery and parcel lockers (VRPSPDPL). The goal is to minimize the total traveling cost [
29]. Paper [
30] examined consumers’ preferences for receiving parcels that are ordered online, and the findings indicate that changes in prices have a considerable significantly affect on utility. Author [
31] investigate the relative performance of the two frequently used material convergence in humanitarian logistics, which corresponds to the movement of relief supplies from donor sites to distribution centers in disaster areas, from which the final deliveries are done to survivors. Considered approaches [
31] in terms of their ability to facilitate effective last-mile delivery. An extended version of the Vehicle Routing Problem with Shipment Consolidation (VRPC) is examined by the authors [
32]. Model considers not only split deliveries, but also shipment transfers between different vehicles at certain customer locations (a process referred to as mid-route shipment consolidation) [
32]. Researchers [
33] create a toolkit for investigating the long-term viability of last-mile logistics and distribution methodologies, employing (1) a centralized distribution network with a click and collect option, (2) a decentralized distribution network with a home-delivery option, and (3) a distributed network due to a crowd logistics concept. A system dynamics [
33] for a case study of a local food cooperative and a logistics service provider in Austria, simulation and a multi-objective decision support were employed to evaluate the sustainability performance of distribution channel options. Results [
33] demonstrate that the most practical and long-lasting solution is to combine the two players into a distributed network strategy built on the idea of crowd logistics.
2.2. Studies on the Optimization of Automated Parcel Lockers (APLs)
With the burst onto the scene of automated parcel lockers in the logistics business a new paradigm has emerged. So far, customers were accommodated as they were used to receive the deliveries at home. However, it is no longer like that, parcel lockers pose the alternative to clients to go to a point to collect their bundles. For this reason, the APL optimal location problem acts as the key factor in consumer willingness to use this service. Although the subject of APLs is relatively new, this study is not the first one that tries to give an answer to this problem. There exists research [
39] that propose solution based on the crowd. This consists of collecting and delivering parcels using individuals (neighbours). The approach utilizes circle packing to compute the number of neighbours needed, the number of parcels they must manage and the corresponding reward. A research study was carried out on the 12th district of Paris (France) dataset. This study makes three important contributions. To begin, they investigated the ability of crowd logistics to reduce delivery malfunction and improve service level in urban deliveries, with an emphasis on environmental and economic concerns. Second, they present a novel method for calculating the number of neighbours and parcels to deliver using a new way to modelling parcel delivery by individuals in metropolitan environments. Finally, they provide a new simulation environment after adjusting the model’s parameters. Their findings indicate the potential of building a network of neighbours to avoid delivery failure using data on parcel delivery in Paris’s 12th district and population density. In paper [
40] authors present development of research addressing this topic. In this study, an agent-based simulation model (ABSM) is used to evaluate the deployment of DL and the exclusion of the third delivery attempt in the city of Belo Horizonte, Brazil. By adjusting delivery locker implementation and excluding the third attempt, four scenarios were created, allowing for comparable results in terms of gains and operational and external expenses for each agent (emission, noise, and congestion). The scenario that was most similar to the existing reality produced the most negative outcomes. The introduction of the lockers improves this scenario by minimising re-delivery and truck distance travelled. The advantages are significantly greater when the potential of three delivery attempts is eliminated. Lockers also facilitate carriers to reduce the number of trucks required for delivery, enabling them to increase earnings. In summarised study [
41] author presents findings of a demand assessment for an automated parcel locker (APL) system that is currently in use. The potential demand for APL was assessed using a two-stage approach. Based on Random Utility Modelling, they disclose the current demand for both attended and unattended deliveries in the first step. Then they calculate the time spent on grocery shopping and see how much time could be saved if the shopping and pickup tasks were combined. They found out what consumers’ current tastes and behaviours are when it comes to using post office channels (for instance: from 71% for perfumes and cosmetics to 88% for clothing). The discrete choice modelling results revealed that when online customers choose a delivery channel, both economic and spatial factors are important. The suggested APL system’s location, i.e., 24/7 minimarkets, would minimize walking time for the pick-up procedure by 20% to 47%, depending on post office locations in relation to online shoppers’ dwellings. Many new challenges emerged as a consequence of the SARS-CoV-2 pandemic, for example, governments have sanctioned restrictions such as lockdowns, causing supply chain shocks at multiple levels [
42]. Additionally, due to the temporary spike in mail order demand during the lockdowns, delivery services reached their capacity limits [
42]. Utilizing (outdoor) parcel lockers where customers can pick up their orders whenever they want while maintaining physical distance is one way to support supply chain viability at the last-mile delivery tier [
42]. It is well known that the location selection of such lockers is crucial to their success. Another crucial issue is the need to accommodate the (uncertain) customer demand for various commodities in the compartment structure of the parcel lockers [
42]. One optimization problem is created by combining the planning problems [
42]. Given a budget in which to invest, the goal is to maximize a linear function (e.g., expected profits) of the covered demand. A formulation for integer linear programming is proposed, and a reformulation based on Benders decomposition is derived. Benders cuts can be separated in linear time. A performance analysis of computational experiments shows that the developed algorithms are capable of solving large-scale problem instances. A sensitivity analysis demonstrates the effect of different problem parameters on the obtained solutions [
42]. The authors also present a case study based on real-world data from Austria. The findings indicate that using parcel lockers can help sustain supply chain viability at the last-mile distribution tier. Furthermore, the low investment cost yields promising returns. The findings also suggest that small and medium-sized compartments should be preferred over large and x-large compartments in dispatch locker compartment design [
42]. In recently published research [
43] authors explained an approach to the last mile delivery problem in which parcel lockers, in addition to serving as order collection points for customers, are also used as transshipment nodes in a 2-echelon delivery system. Taken into account, that a customer (irregular courier) visiting a locker may accept reimbursement to make a delivery to another client on their regular travel path [
43]. Research recommend that customers who prefer to self-pick up their orders, as well as customers who prefer home delivery, share locker facilities to make better use of the existing available storage space [
43].