# An Asymmetric Bargaining Model for Natural-Gas Distribution
## 4 An Asymmetric Hybrid Bankruptcy and Nash Bargaining Model for Gas Distribution
\begin{split}\text{{Maximize}}\ N^{wi}=\left(x_{1}-\left(E- \sum\limits_{i\in N/\{1\}}c_{i}\right)\right)^{w_{1}}\left(x_{2}-\left(E- \sum\limits_{i\in N/\{2\}}c_{i}\right)\right)^{w_{2}}\left(x_{3}\\ -(E-\sum\limits_{i\in N/\{3\}}c_{i})\right)^{w_{3}}\ldots\ \left(x_{n}-\left(E-\sum\limits_{i\in N/\{n\}}c_{i}\right)\right)^{w_{2n}} \end{split}  \tag{7}\]  
The model presented in Equation (7) is constraint by individual feasibility and rationality. As discussed above, the disagreement points and province claims serve as the lower and upper bounds, respectively. Therefore, Equation (7) can be reformulated as:  
\[ \begin{split}\text{{Maximize}}\ N^{wi}=\left(x_{p}-\left(E- \sum\limits_{i\in N/\{p\}}c_{i}\right)\right)^{w_{p}}\cdot\left(x_{s}-\left(E -\sum\limits_{i\in N/\{s\}}c_{i}\right)\right)^{w_{s}}\cdot(x_{B}\\ -(E-\sum\limits_{i\in N/B}c_{i}))^{w_{B}}\cdot\left(x_{K}-\left(E -\sum\limits_{i\in N/K}c_{i}\right)\right)^{w_{K}}\cdot\left(x_{ot}-\left(E- \sum\limits_{i\in N/\{OT\}}c_{i}\right)\right)^{w_{OT}}\end{split}  \tag{8}\]  
Here, \(\sum\limits_{i=1}^{n}w_{i}=1\)  
In Equation (8):  
\(x_{p}=\) Punjab's gas optimized distribution value. \(I_{P}=\) Punjab's minimum gas requirement. \(x_{s}=\) Sindh's gas optimized distribution value. \(I_{S}=\) Sindh's minimum gas requirement. \(x_{B}=\) Baluchistan's gas optimized distribution value. \(I_{B}=\) Baluchistan's minimum gas requirement. \(x_{K}=\) Khyber Pakhtunkhwa's (KPK) gas optimized distribution value. \(I_{K}=\) KPK's minimum gas requirement. \(x_{ot}=\) Other area's gas optimized distribution value. \(I_{ot}=\) Other area's minimum gas requirement. For the above gas-allocation model, the following constraints are set:  
1. The gas allocation of each province must be equal to or more than its lower threshold value. \[x_{i}\geq\ m_{i}\ i=1,2,\ldots\ldots..,n\] (9)
2. The gas allocation of each province must be equal to or less than its upper threshold value (claim) but more than or equal to its lower threshold value. \[m_{i}\leq x_{i}\leq c_{i}\] (10)
3. The total gas allocation must be equal to or less than the total available gas. \[\sum\limits_{i=1}^{n}x_{i}\ \leq\ E\] (11)After defining the objective function (Equation (8)) and setting the constraints (Equations (9)-(11)), the allocation of gas among the agents (provinces and administrative units) is done using the Nash bargaining solution. Figure 2 illustrates the working of the proposed approach under a gas shortage period. When the gas demand and its availability change with time, the disagreement points are defined again, using Nash bargaining theory, and resource allocation is done again. Nine different scenarios (from 2020 to 2028) are considered for gas allocation, as mentioned in Table 3. First, the gas allocation for each year is computed using the Nash bargaining solution with equal weights. After that, the reallocation of gas is done using higher 'gas production'. The province which generates/produces more gas would be given higher priority in allocation. The core solutions are defined by rationality, efficiency, and marginality in a cooperative game, which are already defined in Equations (1)-(3). The rationality principle defines the lower bound of the user, whereas the marginality principle defines the upper bound. The lower core values are defined using Equation (6). Results are summarized in Table 3, in which \(x\)(i) represents the gas allocation for the \(i\)th user expressed in \(\text{Mm}^{3}\)/y. The upper and lower core bounds, as stated in Table 3, can be considered the limits of \"feasible values\" that each stakeholder could accept.