Question

Transportation Problem

A firm has four manufacturing plants and four warehouses. It is trying to determine the optimal way to ship from its plants to the warehouses.

The plant capacities and warehouse requirements are as follows:

     Plant Capacity                      Warehouse requirement

   1 = 1200                                     A = 3100

         2 = 4200                                      B = 1800

         3 = 5000                                      C = 2000

         4 = 2000                                      D = 4400

A union requirement prohibits the firm from shipping to warehouse C from Plant 1 (Hint: Use a large cost in this cell when solving with LP so that the program will not assign any units to this cell).

The freight charges per unit from each plant to each warehouse are as follows:

Plant	Warehouse A                          B                     C                             D
1 2 3 4	$9 6 8 7	$5 8 9 9	---- $11     9     8	$7 6 5 8

Find an initial solution using the Difference method, and find the cost of this solution.

Answer 1

Let us assume the number of units transported from a plant to a warehouse be as below -

Let the transportation cost from Plant 1 to Warehouce C = $999

Objective Function

Min Z = 9a1 + 6a2 + 8a3 + 7a4 + 5b1 + 8b2 + 9b3 + 9b4 + 999c1 + 11c2 + 9c3 + 8c4 + 7d1 + 6d2 + 5d3 + 8d4

Constraints

a1 + a2 + a3 + a4 = 3100 ...Demand Constraint

b1 + b2 + b3 + b4 = 1800 ...Demand Constraint

c1 + c2 + c3 + c4 = 2000 ...Demand Constraint

d1 + d2 + d3 + d4 = 4400 ...Demand Constraint

a1 + b1 + c1 + d1 <= 1200 ...Capacity Constraint

a2 + b2 + c2 + d2 <= 4200 ...Capacity Constraint

a3 + b3 + c3 + d3 <= 5000 ...Capacity Constraint

a4 + b4 + c4 + d4 <= 2000 ...Capacity Constraint

Let us now use excel solver to define these constraints and maximize the cost -

Run the solver to get the minimum cost as $67,400