# Dual Market Facility Network Design under Bounded Rationality
## 3 Mathematical Model Formulation
\(D_{j}\): Demand at the market \(j\), following a uniform distribution, \(Dj\)-\(U[aj,bj]\)  
\(a_{j}\): Lower bound on the uniform demand distribution in market \(j\)  
\(b_{j}\): Upper bound on the uniform demand distribution in market \(j\)  
\(Ca_{j}\): Average cost per item incurred at market \(j\) due to various sources from which the items are transported  
\(v_{ij}\): Value addition in terms of per unit profit obtained due to a common component transportation from plant \(i\) and sold at market \(j\)  
\(A_{j}\): Coefficient of x\({}^{2}\) term in the Equation (2) for profit function in market \(j\)  
\(B_{j}\): Coefficient of x term in the Equation (2) for profit function in market \(j\)  
\(C_{j}\): Constant term in the Equation (2) for profit function in market \(j\)  
\(\beta_{j}\): Bounded rationality parameter for market \(j\)  
\(\pi_{j}\): Profit in market \(j\)  
\(n\): Total number of countries in the multi-market network. Plant, \(i=1,2\ldots n\)  
Market \(j=1,2\ldots n\)  
To apply the newsvendor's model to each market, we need to determine the average cost per item at each market. This is taken as the weighted average of the common component manufacturing cost per item from the various source plants from which the items are transported. \[\text{Ca}_{j}=\frac{\sum\limits_{j=1}^{a}x_{ij}(Cm_{i}+Ct_{ij})}{\sum\limits_{ i=1}^{a}x_{ij}} \tag{3}\]  
Note here that \(Ct_{ii}=0\), as there is no transportation cost from plant \(i\) to market \(i\). The value addition in each path in terms of per unit profit obtained due to a common component transportation from plant \(i\) and sold at market \(j\) is given by,  
\[V_{ij}=p_{j}-Cm_{i}-Ct_{ij}\]  
Thus, from Equation (2), the profit in each market \(j\), under uniform demand distribution \(Dj\)-\(U[aj,bj]\) would be given as,  
\[\pi_{j}=A_{j}(\sum\limits_{i=1}^{a}x_{ij})^{2}+B_{j}(\sum\limits_{i=1}^{a}x_{ ij})+C_{j} \tag{4}\]  
where,  
\[A_{j}=\frac{-p_{j}}{2(b_{j}-a_{j})},\;\;B_{j}=\frac{p_{j}b_{j}}{(b_{j}-a_{j})}- Ca_{j},\;C_{j}=\frac{-p_{j}a_{j}^{2}}{2(b_{j}-a_{j})}\]Note here that \(Caj\) is not a constant term. Thus the equations is expanded and taking \(BB_{j}=\frac{p_{j}b_{j}}{(b_{j}-a_{j})}\), we have the expression for the total profit \(\pi\) as,  
\[\pi(x)=p\,E(\min(D,x))-cx \tag{1}\]0  
Thus, we have the objective function to maximize the total profit: max \(\pi\)  
This is subject to capacity constraints in each plant as well as each market for the final assembly,  
Plant capacity constraint: \(\sum\limits_{j=1}^{n}x_{ij}\leq Km_{i}\) \(\forall i=1\ldots n\)  
Final Assembly capacity constraint: \(\sum\limits_{j=1}^{n}x_{ij}\leq Ka_{j}\) \(\forall j=1\ldots n\)  
The formulation in Equation (5) is having a quadratic objective function and linear constraints. We apply [PERSON]'s sequential linearization method, and [PERSON]'s Decomposition method is used for decomposing the linear problem within the [PERSON]'s method. Stepwise procedure for the integrated methodology is given as follows:  
Consider the problem to,  
\[ \begin{array}{l}\text{Minimize }\pi(x)\\ \text{subject to }x\in S;\\ \text{where }S\subset R^{n}\end{array} \]  
Let,\(f\) is a continuously differentiable function. _Step 0_. Choose an initial solution, \(x^{0}\in S\). Let \(k=0\). Here an arbitrary basic feasible solution is chosen, that is, an extreme point. _Step 1_. Determine a search direction, \(p^{k}\). In the [PERSON] algorithm \(p^{k}\) is determined through the solution of the approximation of the problem (1) that is obtained by replacing the function \(f\) with its first-order Taylor expansion around \(x_{k}\). Therefore, solve the problem to minimize:  
\[z_{k}(x)=\pi(x_{k})+\
abla\pi(x_{k})^{T}(x-x_{k})\]  
Subject to \(x\in S\)  
This is a Linear Programming (LP) problem. In large scale problems this would be computationally complex, and would require to be decomposed into smaller problems which can be solved easily. Hence we apply the [PERSON]'s decomposition method at this point to solve the LP. _Step 2_.