Problem $2$

Consider the following transportation problem with the goal of minimizing total cost:

$\backslash$table$[[,$Dest. $1,$Dest. $2,$Dest. $3,$Dest. $4,$Supply$],[$Source $1,$$$5,$$$1,$$$5,$$$3,300],[$Source $2,$$$8,$$$9,$$$10,$$$4,800],[$Source $3,$$$4,$$$3,$$$8,$$$2,500],[$Demand$,500,500,200,400,]]$

The cost $($$$/$ unit$)$ for transporting from source $i$ to destination $j$ is given in the table. The demand for each destination and the supply from each source is also given in the demand row and the supply column, respectively.

a$)$ Formulate an LP model for the problem.

b$)$ Is this a balanced transportation problem? Why? If it is not balanced, modify the transportation table to make it balanced.

c$)$ Use the Least Cost method to find an initial solution to the problem. What is the total cost?

d$)$ Use the Northwest Corner method to find an initial solution to the problem. What is the total cost?

e$)$ Use Vogel's Approximation method to find an initial solution to the problem. What is the total cost in this case?