# Dual Market Facility Network Design under Bounded Rationality
## 3 Mathematical Model Formulation  
The present research investigates the operations policy of a multi-national firm which makes goods for serving geographically disjointed markets with localized final assemblies and joint parts. The joint part can be shipped from the markets with diverse financial and demand features. The cost required for transferring the joint components (intermediate goods between markets), and for foreign trade barriers like tariffs and duties provides the total transportation cost. The generalized operations strategy structure is illustrated in Figure 1 for multiple-product and multiple-market model with commonality. This study dealt with the strategic facility location decision of engine production by taking into account the financial and demand features. The newsvendor network approach is used to evaluate the model. Firstly, we analyze the scenario under deterministic demand. The sensitivity of the trade-off between the manufacturing costs and the transportation costs are examined. Secondly, the situation under demand uncertainty is evaluated. Here, we assume uniformly distributed demand. The profit function is accordingly formulated and similar analysis is carried out. The function is solved using [PERSON]'s sequential linearization technique with [PERSON]'s Decomposition method. Thirdly, the results of optimal decision scenario is compared with the scenario of bounded rationality considering uniformly distributed demand. When we consider the bounded rationality, the best option is not always picked by individuals, however, they opt for better options o  
Figure 1: Generalized multi-product multi-market network design. give a statistical proof of the superiority of the algorithm used over the traditional gradient search method by [PERSON]. Thus, a three-fold comprehensive comparison is made for the multi-market network design. The canonical setting at each market includes a newsvendor setup who has to obtain how much of the commonality to order. Every item costs \(c\) but can be sold at price \(p\), where \(p>c\). For a given demand, the profit is given as the product of price and minimum of demand and ordering quantity minus the procurement cost, which is cost per item multiplied by ordering quantity. Suppose we have a random demand, \(D\) having density \(f\) and cumulative distribution \(F\). The expected profit in any of the selling points in the network under newsvendor model with demand uncertainty is given by:  
\[\pi(x)=p\,E(\min(D,x))-cx \tag{1}\]  
where,  
\(x\): Number of items ordered  
\(p\): Selling price of the item  
\(c\): Cost of item, where \(c<p\)  
\(D\): Demand  
\(E(y)\): Expected value of \(y\)  
Considering that the demand \(D\) is uniformly distributed between \([a,b]\). Then the profit function in Equation (1) can be expressed as [13]:  
\[\pi(x)=Ax^{2}+Bx+C \tag{2}\]  
where,  
\[A=\frac{-p}{2(b-a)},\,B=\frac{pb}{b-a}-c,\,C=\frac{-pa^{2}}{2(b-a)}\]  
Now, we apply the above model to the multi-market scenario. Each of the markets can be considered as a newsvendor model. Consider the multi-market model as depicted in Figure 2.  
Figure 2: Generalized multi-market design. With reference to Figure 2, we use the following notations:  
Notations  
\(x_{ij}\): Quantity of common component manufactured in plant \(i\), and assembled at market \(j\). \(\eta\): Selling price of final assembly at market \(j\). \(Ct_{ij}\): Cost of transportation of common component from plant \(i\) to market \(j\). (\(Ct_{ii}=0\))  
\(Cm_{i}\): Cost of manufacturing the common component in plant \(i\)  
\(Km_{i}\): Capacity limit on the common component manufacturing at plant \(i\)  
\(Ka_{j}\): Capacity limit on the final assembly at market \(j\).