The Verstech Corporation has decided to produce three new products. Five branch plants now

have excess product capacity. The unit manufacturing cost of the first product would be $$41,$ $$39,$

$$42,$ $$38,$ and $$39$ in Plants $1,2,3,4,$ and $5,$ respectively. The unit manufacturing cost of the second

product would be $$55,$ $$51,$ $$56,$ $$52,$ and $$53$ in Plants $1,2,3,4,$ and $5,$ respectively. The unit

manufacturing cost of the third product would be $$48,$ $$45,$ and $$50$ in Plants $1,2,$ and $3,$

respectively, whereas Plants $4$ and $5$ do not have the capability for producing this product. Sales

forecasts indicate that $700,1,000,$ and $900$ units of products $1,2,$ and $3,$ respectively, should be

produced per day. Plants $1,2,3,4,$ and $5$ have the capacity to produce $400,600,400,600,$ and $1,000$

units daily respectively, regardless of the product or combination of products involved. Assume

that any plant having the capability and capacity to produce them can produce any combination

of the products in any quantity. Management wishes to know how to allocate the new products

to the plants to minimize total manufacturing cost.

$($a$)$ Formulate this problem as a transportation problem by constructing the appropriate

parameter table.

$($b$)$ Obtain an optimal solution.

Suppose that the sales forecasts have been revised downward to $280,400,$ and $350$ units per day

of products $1,2,$ and $3,$ respectively, and that each plant now has the capacity to produce all that

is required of any one product. Therefore, management has decided that each new product

should be assigned to only one plant and that no plant should be assigned more than one product

$($so that three plants are each to be assigned one product, and two plants are to be assigned none$).$

The objective is to make these assignments so as to minimize the total cost of producing these

amounts of the three products.

$($c$)$ Formulate this problem as an assignment problem by constructing the appropriate cost

table.

$($d$)$ Obtain an optimal solution.

$($e$)$ Reformulate this assignment problem as an equivalent transportation problem by

constructing the appropriate parameter table.

$($f$)$ Starting with Vogel$$s approximation method, interactively apply the transportation

simplex method to solve the problem as formulated in part $($c$).$

$($g$)$ From the context above, run both the transportation and assignment model R specifying

the packages employed and compare your optimal Suppose that the sales forecasts have been revised downward to $280,400,$ and $350$ units per day

of products $1,2,$ and $3,$ respectively, and that each plant now has the capacity to produce all that

is required of any one product. Therefore, management has decided that each new product

should be assigned to only one plant and that no plant should be assigned more than one product

$($so that three plants are each to be assigned one product, and two plants are to be assigned none$).$

The objective is to make these assignments so as to minimize the total cost of producing these

amounts of the three products.

$($c$)$ Formulate this problem as an assignment problem by constructing the appropriate cost

table.

$($d$)$ Obtain an optimal solution.

$($e$)$ Reformulate this assignment problem as an equivalent transportation problem by

constructing the appropriate parameter table.

$($f$)$ Starting with Vogel$$s approximation method, interactively apply the transportation

simplex method to solve the problem as formulated in part $($c$).$