The Versatech 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 531, $29, $32, $28, and $29 in Plants 1, 2, 3,4, and 5, respectively. The unit manufacturing cost of the second product would be S45, S41, 546, $42, and $43 in Plants 1, 2, 3, 4, and 5, respectively. The unit manufacturing cost of the third product would be S38, S35, and $40 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 600, 1,000, and 800 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.

A parameter table was constructed to show the transportation problem and to also obtain an optimal solution.

Explanation:

Solution

(a) the first step is to prepare  the problem as a transportation problem by establishing  the appropriate parameter table

                            1        2          3            Supply

Source    1           31      45      38             400

(Plant)      2          29     41       45             600

               3           32     46      40            400

               4           28     42      M              600

               5           29     43      M              1000

     Demand        600    1000  800

The total supply  is = 400 + 600 + 400+ 600+ 1000= 3000

Total demand is = 600 + 1000 + 800 = 2400

(b) since the problem given is an imbalanced transportation problem,to make it a balance transportation problem we will make use of what is called the dummy destination for this numbers 3000 - 2400 = 600

                  1        2          3                  4 (Dummy)      Supply

Source    1           31      45      38               0                   400

(Plant)      2          29     41       45               0                   600

               3           32     46      40               0                   400

               4           28     42      M                 0                   600

               5           29     43      M                 0                    100

Demand             600    1000  800          600

The positive independent number of  allocations is equal to m+n -1 = 5 + 4-1 =8

This solution is a basic feasible solution called a non -degenerate

The cost of initial transportation is he initial transportation cost=31*400+29*200+41*400+46*400+42*200+M*400+M*400+0*600

=$61400+800M

Note: kindly find an attached document of the part of the solution of the work.