Problem $2.(30\%)$ Consider a transportation problem $[$ref$.$ Lecture $2]$ specified by the following figure:

Under the above setting, Factory $1,2,3$ are producing a divisible product to Shop $1,2,$ with the capacities and demands specified as in the figure. The cost of transporting $1$ unit of product from factory $i$ to shop $j$ is as specified by the number next to the line connecting them, e$.$g$.,$ it costs $$5$ per unit to transport the product from factory $3$ to shop $2.$ Answer the following question:

$($a$)(10\%)$ Formulate an optimization problem to minimize the total transportation cost while satisfying the demands raised by the shops and the capacities of the factories.

In the optimization problem formulated, what are the decision variable$($s$)?$ what is the objective function? what is the constraint? what is the type of this optimization problem?

$1$

Homework $1$

$2$

Note: This problem is different from the example shown in the lecture, e$.$g$.,$ Factory $3$ cannot deliver to Shop $1,$ Factory $1$ cannot deliver to Shop $3.$

$($b$)(10\%)$ Suppose that now Factory $2$ can not deliver to Shop $1.$ Formulate an optimization problem to minimize the total transportation cost similar to $($a$).$ Does this problem has a feasible solution?

$($c$)(10\%)$ Suppose that now Shop $2$ can also deliver to Shop $1$ at a cost of $$2$ per unit $($but not vice versa$)$ while Factory $2$ can not deliver to Shop $1.$ Formulate an optimization problem to minimize the total transportation cost similar to $\frac{?}{\text{b}\text{a}\text{r}}((a).$ Does this problem has a feasible solution? $)$