Question $3$: Consider the following supply chain network, where $\backslash ($ x $\backslash )$ represents the number of items being sent between suppliers and distribution centers, and $\backslash ($ y $\backslash )$ represents the number of items sent from distribution centers to retailers. The capacities of the suppliers and distribution centers, as well as the demands of the retailers, are indicated inside the respective shapes. The transportation costs between each pair of nodes are shown on the connecting lines. First, verify that all constraints are satisfied, including supplier capacity, distribution center capacity, and retailer demand. Then, using the values of the decision variables, that is $\backslash ($ x $\backslash )$ and $\backslash ($ y $\backslash ),$ calculate the objective function value based on the corresponding transportation costs.

Relailers

Table $1.$ How many items are sent from each supplier to each distribution center $($decision variable $\backslash ($ x $\backslash ))$

$\backslash$begin$\{$tabular$\}\{|$l$|$l$|$l$|\}$

$\backslash$cline $\{2-3\}\backslash$multicolumn$\{1\}\{$c$|\}\{\}$ & variable $\backslash ($ x $\backslash ))\backslash \backslash$

$\backslash$cline $\{2-3\}\backslash$multicolumn$\{1\}\{$c$|\}\{\}$ & Distribution Center $1$ & Distribution center $2\backslash \backslash$

$\backslash$hline Supplier $1$ & $200$ & $\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$\backslash$begin$\{$tabular$\}\{|$l$|$l$|$l$|\}$

$\backslash$hline Supplier $2$ & $0$ & $200\backslash \backslash$

$\backslash$hline Supplier $3$ & $0$ & $400\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

Table $2.$ How many items are sent from each distribution center to each retailer $($decision variable $\backslash ($ y $\backslash ))$

$\backslash$begin$\{$tabular$\}\{|$l$|$l$|$l$|$l$|$l$|\}$

$\backslash$cline $\{2-5\}\backslash$multicolumn$\{1\}\{$c$|\}\{\}$ & Retailer $1$ & Retailer $2$ & Retailer $3$ & Retailer $4\backslash \backslash$

$\backslash$hline Distribution Center $1$ & $100$ & $0$ & $0$ & $0\backslash \backslash$

$\backslash$hline Distribution Center $2$ & $0$ & $100$ & $50$ & $100\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$