# An Integrated Capacity Allocation and Dynamic Pricing Model Designed for Air Cargo Transportation
## 2 Materials, Methods, and Models
### Capacity Allocation  
Air cargo companies can market their capacity through various channels, including free sales and contract sales. Companies can sell their capacities directly to shippers, but they generally sell to forwarders. These forwarders can be allocated capacity on specific routes based on their performance in the previous year. Once a flight is completed, the allocated capacity cannot be used. Therefore, it is imperative to allocate capacity effectively to maximize a company's profits. Following the formulation of a plan, there is a possibility that the capacity will be inadequate to fulfill the order on certain routes, while remaining surplus on others [2]. A multitude of factors, including the presence of multiple capacity constraints, variations in capacity, and uncertainty in demand, influence the management of capacity. The company has the option to cancel demand or enter into capacity agreements to mitigate the impact of forwarders canceling their demands. Nevertheless, it remains impractical to entirely eliminate the risk. Figure 2: Summary of the stages of our study. Capacity allocation assumes particular importance in circumstances where demand exceeds capacity or when a carrier sells its capacity to multiple carriers [2]. The objective of this study was to propose a solution to the capacity allocation problem. To this end, the study first sought to ascertain the distribution of contracted sales and free sales capacity within the total capacity. For the capacity allocation problem, the study utilized two years of data for four different destinations of one of the successful air cargo companies in the sector. The data were disaggregated monthly for each destination. The creation of the data set and the application of the methods are detailed in the study [7], which includes a capacity allocation model for air cargo in 2022. Two methods were used for capacity allocation: artificial neural network (ANN) and conditional value-at-risk (CVaR). Before proceeding to the model results, the details of the NARX and CVaR methods are given below. NARX neural network: In 1996, [PERSON] et al. proposed the use of NARX (nonlinear autoregressive exogenous model) for problems where the desired output depends on past data. NARX consists of multiple feedback and forward computation layers and is a dynamic artificial neural network. Compared to traditional feedback network structures, NARX networks show improved learning efficiency [16]. Unlike other feedback networks, NARX networks are characterized by the fact that the feedback is received from the output layer of the neurons and not from all hidden layers. The NARX model can be expressed by the following Equation (1) [17]. \[y(t)=f(y(t-1),y(t-2),\ldots,y(t-ny),x(t-1),x(t-2),\ldots,x(t-nx)) \tag{1}\]  
In the equation, \(y(t-1),y(t-2),\ldots,y(t-ny)\) represents the network outputs, while \(x(t-1),\)\(x(t-2),\)\(\ldots,\)\(x(t-nx)\) denotes the network inputs. The parameters \(ny\) and \(nx\) specify the number of previous outputs and inputs, respectively, to be utilized for feedback purposes. The value \(Y(t)\), which depends on the output signal, is calculated by combining the signal of the previous output value with the return of the previous independent input signal. CVaR (conditional value at risk): VaR (value at risk) can be defined as the maximum loss that will occur in a financial asset at a specified time [18]. Accurate VaR measurements can enable businesses to protect themselves against financial risks in advance. In 1999, [PERSON] et al. developed the CVaR model in response to criticisms of the VaR method. The CVaR model is fundamentally defined as the expected loss if the potential loss exceeds the value at risk in Equation (2). \[CVaR\ [x]=E\ \{x\ l\ x>VaR[x]\}, \tag{2}\]  
The calculation of VaR is frequently the subject of critique due to its tendency to produce inconsistent risk measurement results in fat-tailed distributions.