The following table represent a general cu$($t$)/($f$)$ill problem, where the a$\_($i$)$ are the supply, b$\_($j$)$ are the demand, c$\_($ij$)$

are the costs per unit of flow and x$\_($ij$)$ are the flow variables. Assume the following values: a$\_(1)=20,$a$\_(2)=30,$b$\_(1)$

$=10,$b$\_(2)=25,$b$\_(3)=20$ and c$\_(17)=$i$+2$j$.$

TABLE $2\mathrm{.}3$ REQUIREMENTS AND DECISIONS IN THE CUT$-$AND$-$FILL PROBLEM

Use the formulation done in class and solve it with EXCEL $($Upload the solved Excel file with the model and

answer report$)$

Model Formulation. The problem we have just described is a form of the Transportation Problem, one of the very earliest linear programming problems to be formulated and solved. It is a problem on which both Koopmans and Kantorovitch, as well as others, worked in the earliest days of mathematical programming.

The problem can now be stated, as read from Table $2\mathrm{.}3,$ as follows:

Minimize Z$=$c$\_(11)$x$\_(11)+$c$\_(12)$x$\_(12)+$c$\_(13)$x$\_(13)+$c$\_(21)$x$\_(21)+$c$\_(22)$x$\_(22)+$c$\_(23)$x$\_(23)$

Subject to: x$\_(11)+$x$\_(12)+$x$\_(13)$

$\{($:x$\_(11)),($x$\_(12)),($x$\_(13)),($x$\_(21)),($x$\_(22)),($x$\_(23)>=0.)$:$\}$