Question

4. A company manufactures two products, X and Y by using three machines A, B, and c Each of the products has to be processed on each machines. The Table below shows (a) the hours required on each machine to produce one unit of each of the products (b) the total number of hours available on each machine (c) the profit made on one unit of each product. It is assumed that the profit is directly proportional to the number of units sold. It is required find the weekly output of each product that will maximize profits. (a) Develop the LP model and convert the model to standard LP form. (b) Use the Simplex method to solve the LP model Product X ProductY Total Hours Machine Type Available 70 (hr) (hr) 2 40 90 3 60 40 Profit per Unit ($)

Answer 1

(a)

x₁ be the number of product X

x₂ be the number of product Y

Maxmize z= 40x₁ + 60x₂

2x₁ + x₂ <= 70

x₁ + x₂ <= 40

x₁ +3x₂ <= 90

x₁ >= 0, x₂ > = 0

(b)

Max Z = 40 x_1 + 60 x_2 subject to 2x_1 + x_2 lessthanorequalto 70 x_1 + x_2 lessthanorequalto 40 x_1 + 3 x_2 lessthanorequalto 90 and x_1, x_2 greaterthanorequalto 0: The problem is converted to canonical form by adding slack, surplus and artificial variables as appropriate 1. As the constraint-1 is of type' lessthanorequalto ' we should add slack variable s_1 2. As the constraint-2 is of type �lessthanorequalto' we should add slack variable s_2 3. As the constraint-3 is of type �lessthanorequalto' we should add slack variable s_3 After introducing slack variables Max Z = 40x_1 + 60 x_2 + 0 s_1 + 0s_2 + 0 s_3 subject to 2x_1 + x_2 + s_1 = 70 x_1 + x_2 + s_2 = 40 x_1 + .3x_2 + s_3 = 90 and x_1, x_2, s_1, s_2, s_3 greaterthanorequalto 0 \r\n Negative minimum z,-c, is -60 and its column index is 2. So, the entering variable is x_2.