Question

Q1. By using the Simplex method solves the problem below: The company wants to Maximize the profit, the analysis and the summary of products are based on the following equations: XI x2 Operation I 40 Operation 2 120 Profit 40s 5os

Answer 1

Solving using the simplex method as follows:

Max Z = 405X1 + 505x2 subject to *1 + 2X2540 4X1 + 3X25 120 and X1, X2, X3 2 0; The problem is converted to canonical form by adding slack, surplus and artificial variables as appropiate 1. As the constraint-1 is of type's' we should add slack variable Si 2. As the constraint-2 is of type's' we should add slack variables, After introducing slack variables Max Z = 405X1 + 505X2 + OS1 + OS2 subject to X1 + 2X2 + Si = 40 481 + 3x2 + S2 = 120 and X1, X2, S1, S220

Iteration-1 MinRatio 5 = 20 130 – 40 3-9 -405 -5051 0 Negative minimum 2;-C; is - 505 and its column index is 2. So, the entering variable is xz. Minimum ratio is 20 and its row index is 1. So, the leaving basis variable is S. · The pivot element is 2. Entering = X2, Departing = Si, Key Element = 2 + Ri(new) = R (old) = 2 + Rz(new) = Rz(old) - 3R, (new)

Iteration-2 405 505 MinRatio 505 0.5 (2.5) -1.5 Z = 10100 252.5 505 252.5 Z; - O; - 152.51 252.5 Negative minimum Z;- C, is - 152.5 and its column index is 1. So, the entering variable is x1. Minimum ratio is 24 and its row index is 2. So, the leaving basis variable is Sz. : The pivot element is 2.5. Entering =Xz, Departing = Sz, Key Element = 2.5 + Rz(new) = Rz(old) - 2.5 + Ri(new) = R1 (old) - 0.5R (new)

Iteration-3 MinRatio 505 0.8 doc 405 -0.6 Z = 13760 161 Z;- 0 161 Since all Z;-C;20 Hence, optimal solution is arrived with value of variables as: x1 = 24, x2 = 8 Max Z = 13760