Minimise the total transportation cost while meeting the demand and production constraints.

Solving a Linear Programming Problem using the Graphical Method.

Production Capacity at Plants:

Plant A: 400 units of Product X, 300 units of Product Y

Plant B: 500 units of Product X, 400 units of Product Y

Plant C: 600 units of Product X, 200 units of Product Y

Market Demand at Distribution Centers:

Center 1: 200 units of Product X, 300 units of Product Y

Center 2: 400 units of Product X, 100 units of Product Y

Center 3: 300 units of Product X, 200 units of Product Y

Center 4: 250 units of Product X, 150 units of Product Y

Transportation Costs in usd per unit from Plants to Centers:

	Center 1	Center 2	Center 3	Center 4
Plant A	6	8	5	9
Plant B	9	10	7	6
Plant C	11	7	9	8

Solve this Linear Programming (LP) problem using the graphical method, we will first set up the LP model and then graphically visualize it.

Let:

• X = the number of units of Product X to be transported from each plant to each center.
• Y = the number of units of Product Y to be transported from each plant to each center.

The objective is to minimize the total transportation cost, which is given by the sum of the transportation costs from each plant to each center, multiplied by the number of units transported.

The objective function to minimize the total transportation cost (Z) is as follows:

Z=6XA1+8XA2+5XA3+9XA4+9XB1+10XB2+7XB3+6XB4+11XC1+7XC2+9XC3+8XC4

Subject to the following constraints:

1) Production Capacity Constraints:

• For Product X at Plant A: XA1+XB1+XC1400
• For Product Y at Plant A: XA1+XB1+XC1300
• Similar constraints for Plants B and C.

2) Demand Constraints at Distribution Centers:

• For Product X at Center 1: XA1+XB1+XC1200
• For Product Y at Center 1: YA1+YB1+YC1300
• Similar constraints for Centers 2, 3, and 4.

3) Non-negativity Constraints:

• Xij0 for all i (plants) and j (centers).
• Yij0 for all i (plants) and j (centers).

Now, Help solve graphically represent for this problem:

1) Create a two-dimensional graph with X and Y on the axes?

2) Plot the constraints on the graph to form a feasible region?

3) Calculate the coordinates of the corners of the feasible region?

4) Evaluate the objective function at these corner points to find the optimal solution?