Solve by Steps for Two-Phase Method

Max Z = 5x₁ + 8x₂

Subject to

3x₁ + 2x₂ ≥ 3

x₁ + 4x₂ ≥ 4

x₁ + x₂ ≤ 5

    &     x₁ ≥ 0, x₂  ≥ 0

Answer

Standard LPP

Max Z = 5x₁ + 8x₂  

    Subject to

                        3x₁ + 2x₂ - s₁+ a₁ = 3

                        x₁ + 4x₂ - s₂+ a₂  = 4

                        x₁ + x₂ + s₃ = 5

                        x₁ , x₂ , s₁, s₂, s₃, a₁, a₂  ≥ 0

Auxiliary LPP

Max Z* = 0x₁ + 0x₂ + 0s₁ + 0s₂ + 0s₃ -1a₁ -1a₂

    Subject to

                        3x₁ + 2x₂ - s₁+ a₁ = 3

                        x₁ + 4x₂ - s₂+ a₂  = 4

                        x₁ + x₂ + s₃ = 5

                        x₁ , x₂ , s₁, s₂, s₃, a₁, a₂  ≥ 0

As all Δ_j ≥ 0, Max Z* = 0 and no artificial vector appears in the basis, we move to phase II. 

Phase II

As all Δ_j ≥ 0, optimal basic feasible solution is achieved. Thus the solution is Max Z = 40, x₁ = 0, x₂ = 5