# Dual Market Facility Network Design under Bounded Rationality
## 4 Test Results and Comparative Analysis
### Comparative Analysis  
For a problem size of 2 (dual market scenario), there are various possible cases and scenarios that might arise due to the tradeoffs between the prices and the costs. In order to evaluate the financial desirability of the options processing actions for every product, the investment decisions declined. For the analysis of various possible cases in the dual market scenario, we take the value addition into consideration, which represents the per-unit profit of the company, due to the particular path taken in the production allocation. Let \(v1\) denote the value addition due to path from plant 1 to market 1, similarly, \(v2\) be for plant 1 to market 2, \(v3\) for plant 2 to market 1 and v4 for plant 2 to market 2. The probable orderings of the four net values are 24 (=4!) without considering any assumption, but their definitions suggest an interdependence, \(v4\geq v1\Rightarrow v2\geq v3\), which removes 6 orderings (where, as mentioned earlier \(vi\) represents the per unit profit value addition in any path). The outstanding 18 are distributed into two clusters, one of which is a replica of the other. The following study emphasizes the 9 orderings because of symmetry in results. These result in 4 possible allocation strategies of market focused, centralized with two possible centralization and hybrid type of model as indicated in Figure 6.  
Table 7 depicts 1 of the 9 possible scenarios which is analyzed under cases of deterministic demand, uniform demand and uniform demand under bounded rationality, and the total profits (T.P) in each case is found. For the case of uniform distribution, the [PERSON]'s Bender decomposition method is compared with the [PERSON]'s gradient search method which is depicted in the Figure 7.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Cases for Bounded Rationality Conditions with Varying \(\beta\) Parameter** \\ \hline \(\beta\) & **Total profits** & **\% Deviation from Achievable Profits** \\ \hline
0 & 723,908 & 0.53 \\
10 & 615,322 & 2.19 \\
100 & 553,790 & 2.85 \\
1000 & 498,411 & 2.68 \\
10,000 & 448,570 & 2.61 \\
100,000 & 723,908 & 3.45 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Profits under bounded rationality for test case 3.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & **Market 1** & **Market 2** & **Market 3** & **Market 4** & **Market 1** & **Market 2** & **Market 3** & **Market 4** & **Market 9** & **Market 10** \\ \hline Plant 8 & 0 & 0 & 0 & 0 & 0 & 1836 & 0 & 0 & 0 & 0 \\ Plant 9 & 0 & 0 & 0 & 1566 & 0 & 0 & 0 & 0 & 0 & 0 \\ Plant 10 & 1724 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 5: _Cont._Figure 6: Different possible scenarios of production allocation for dual market case.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scenario 1** & **Case of Deterministic Demand** &
\begin{tabular}{c} **Case of Uniform Demand** \\ **Distribution D \(\sim\) [1000, 200]** \\ \end{tabular} \\ \hline \multirow{6}{*}{\(v_{1}>v_{2}>v_{3}>v_{4}\)} & \(p_{1}=100\) & \(x_{1}=D_{1}\) & \(x_{1}=1565\) \\  & \(p_{2}=80\) & \(x_{2}=0\) & \(x_{2}=0\) \\ \cline{1-1}  & \(c_{m,1}=40\) & \(x_{3}=0\) & \(x_{3}=0\) \\ \cline{1-1}  & \(c_{m,2}=65\) & \(x_{4}=D_{2}\) & \(x_{4}=1142\) \\ \cline{1-1}  & \(c_{t,1}>\Delta c_{m}\) & \(c_{t,2}>\Delta c_{m}\) & \(c_{t,1}=30\) & \(D=D_{1}=D_{2}\) & T.P \\ \cline{1-1}  & & 1000 & 85,000 & T.P. = 105,951.9 \\ \cline{1-1}  & \(c_{t,2}=25\) & 2000 & 170,000 & T.P. = 105,951.9 \\ \cline{1-1}  & & 1500 & 127,500 & \\ \hline \hline \end{tabular}
\end{table}
Table 7: Profit results of Scenario 1.  
Figure 7: Graphical comparison of the two algorithms for scenario 1.