1.ABC a manufacturer of large size pressure cookers, produces and sells three models of cookers. While market demands pose no restriction, the capacity to produce is currently constrained by the limited supplies of special grade aluminium limited to 1500kg per week and machine processing time limited to 1200 hours per week. In order to determine the optimal product mix to maximize weekly contribution, a linear programming model was formulated as below.

MaximizeZ= 60X₁+ 40X₂+ 80X₃

Subject to

6X₁+ 3X₂+ 5X₃ 1500 (Aluminium)

3X₁+ 4X₂+ 5X₃ 1200 (Machine Hours)

X₁, X₂, X₃ 0

Using the simplex method, the following tableau was obtained

Basic VC_j 60408000 RHS

X₁X₂X₃ S₁ S₂

X₁ _ 1-1/301/3-1/3 100

X₃ _ 011-1/52/5 180

Z_i ______

C_j-Zi_ _ _ _ _ _

a)Fill in all the numerical values in the above table

b)Is the current solution optimal? If not, carry out the iterations until an optimal solution is reached

c)Analyse the sensitivity of the optimal solution to the following changes

d)Due to a machine breakdown, the machine hours available gets reduced to 1050 hours

e)An additional quantity of 150 kg of aluminium can be obtained

f)Formulate the dual of the problem and determine the dual solution from the primal optimal tableau