## 3 System Methodology
### Multicast Routing and QoS Constraints  
Multicast routing involves the concurrent transmission of information from a single source to multiple destinations in a network. In the context of multimodal transportation, this concept extends beyond data to encompass the efficient routing of goods via various transportation modes. The significance of multicast routing lies in its ability to streamline the delivery process, reducing costs and improving overall system efficiency. In this section, we delve into the key concepts of multicast routing, explore the challenges posed by the Steiner tree problem, and emphasize the importance of addressing quality of service (QoS) constraints. Mathematically, let \(G=(V,\ E)\) represent the transportation network graph, where V is the set of cities and E is the set of arcs connecting these cities, as shown in Figure 1. The multicast routing problem can be formalized as finding a tree that spans a subset of V and minimizes the overall cost or distance. The business has access to four modes of transportation in Iraq: truck, rail, ship, and air, which can be utilized for product transportation. Each arc \((u,\ v)\ \in\ E\) in the transportation network may have multiple transportation modes available for use, and for each mode \(m\ \in\ M\) on arc \((u,\ v)\), a number of possible departure times are listed by \((u,\ v,\ m)\) as shown in Figure 1. The transportation capacity of each arc is mode-dependent, and upload and download times at switch points are taken into account to estimate transportation time. It is assumed that no return transportation is considered in this study [18; 23].  
The received customer orders are represented by \(|\)K\(|\), with varying demand due dates, quantities, and priority levels. While maintaining the service quality for urgent orders and VIP customers, the corporation may be willing to delay the delivery of some non-urgent orders during periods of high demand. The goal is to reduce the overall transportation, carbon emissions, and delay costs subject to the supply and demand constraints. Historical data collected from the Ministry of Transportation, Iraq are used to calculate the transportation costs and CO\({}_{2}\) emissions for each arc and mode combination.  
The Steiner tree problem is a fundamental challenge in multicast routing. It involves finding the minimal tree that connects a subset of vertices, known as terminal nodes, in a network. In the context of multimodal transportation, the Steiner tree problem corresponds to determining the optimal routes that connect various cities while considering QoS parameters. Introducing QoS constraints adds a layer of complexity to the Steiner tree problem. QoS parameters may include constraints on delay, bandwidth, packet loss, and other factors crucial for efficient transportation. Mathematically, the constrained Steiner tree problem can be expressed as follows:  
\[Min\sum_{(i,j)\in A}c_{ij}\cdot x_{ij} \tag{1}\]  
Subject to constraints such as \(\sum_{j}x_{ij}=1\) and \(\sum_{(i,j)\in A}q_{ij}\cdot x_{ij}\leq Q\), where \(c_{ij}\) represents the cost of traveling from city \(i\) to city \(j\), \(q_{ij}\) denotes the QoS parameter associated with the arc \((i,j)\), and \(Q\) is the specified limit for the QoS parameter. Traditional multicast algorithms often fall short in meeting the demands of modern multimodal transportation networks. These algorithms may neglect QoS constraints or fail to consider the dynamic nature of transportation systems. As a result, there is a pressing need for a more sophisticated approach that integrates advanced techniques such as DEA and HMM to enhance the accuracy and efficiency of multicast routing. In the following sections, we will explore how DEA and HMM contribute to addressing these limitations, providing a robust framework for optimizing the multimodal transportation network under various constraints and uncertainties.