## 3 System Model and Problem Definition  
This section describes the system model, and then the problems' definitions are presented. Modern consumer product recycling can only be optimized by thoroughly comprehending the whole recycling system as a dynamic feedback mechanism [24]. Governments and businesses invest tens of thousands of dollars in waste management each year, but they never reach 100% efficiency. Sometimes the collection schedules and frequency are not well thought out, or the bins are placed in the incorrect locations. Smart bins, which include the IoT, sensors, and eco-friendly technology to enhance city garbage management systems, have been developed to address this issue. Due to significant ecological concerns, academics have been paying close attention to emission reduction, carbon labeling, energy conservation, trade, and footprint. Leveraging, architecture, and enabling technology are the three ideas that make up Green IoT. Architectural technologies, to begin with, pertain to energy-efficient device designs, network structures, communication protocols, and linkages. Additionally, leverage technologies pertain to reducing carbon emissions and increasing energy efficiency, enabling technologies to assist the IoT to become more efficient by lowering energy usage, resource usage, hazardous emissions, and pollution. Therefore, Green IoT contributes to preserving natural resources, reducing costs, and reducing technology's influence on the ecosystem and human wellness. So, Green IoT focuses on green production, green design, green usage, and green disposal [25]. The most common IoT architectural model comprises three levels, as shown in Figure 1, including network, application, and perception layers. The first layer of the three-layer architectural model is the objects or perception layer, which refers to IoT devices and physical sensors responsible for gathering and processing data and having sensors to detect and obtain info about their surroundings. It recognizes other intelligent objects nearby or picks up on specific physical characteristics. Connecting to other network devices, smart objects, and servers is the responsibility of the network layer. Its capabilities are also utilized in the transmission and processing of sensor data. This layer has two sublayers: a routing layer that manages packet transmission from source to destination, and an encapsulation layer that creates packets. The application layer is responsible for delivering the services that subscribers demand. It describes numerous IoT applications, such as smart cities, homes, recycling, and health [26; 27; 28]. Figure 1: The IoT architecture: three-layer. In the considered scenario, the user equipment (UE) sent service requests to the controller, and controller allocated the recycling service (RS) to provide the service path. Therefore, the set of services, \(\text{Set\_RS}=\{\text{rs}_{1}\cdot\text{rs}_{2}\cdot\ldots\text{rs}_{k}\}\), the set of function, \(\text{Set\_SF}=\{\text{sf}_{1}\cdot\text{sf}_{2}\cdot\ldots\text{sf}_{n}\}\), and the set of equipment, \(\text{Set\_UE}=\{\text{ue}_{1}\cdot\text{ue}_{2}\cdot\ldots\text{ue}_{m}\}\), are defined. Where \(k\) is the total number of \(RS\)s, \(n\) is the total number of \(SF\)s and \(m\) is the total number of UEs. The \(\text{rs}_{i}\) represented \(i\)th RS, \(\text{sf}_{j}\) meat/\(i\)th \(SF\), and \(\text{ue}_{i}\) is \(q\)th UE. Each \(\text{sf}_{j}\) has its own capacity (\(\text{SF\_Cap}_{i}\)). For the definition of \(\text{rs}_{i}\), see Equation (1) [29]. \[RS_{i}= \begin{bmatrix}\left(\text{sf}\_{s(1.1)}\text{sf}\_{o(1.1)}\right)& \cdots&\left(\text{sf}\_{s(1.j)}\text{sf}\_{s(1.j)}\right)\\ \vdots&\ddots&\vdots\\ \left(\text{sf}\_{s(i.1)}\text{sf}\_{o(i.1)}\right)&\cdots&\left(\text{sf}\_ {s(i.j)}\text{sf}\_{o(i.j)}\right)\end{bmatrix}  \tag{1}\]  
Each RS will be stored in a matrix.