Maximize Z=2x₁+4x₂+4x₃−3x₄ with αx₁+x₂+x₃=4, x₁+βx₂+x₄=8, x≥0. S1: for α=2,β=1, (x₁,x₂) is an optimal basis. S2: for α=1,β=4, (x₃,x₂) is an optimal basis. Choose the truth values.
For S1, setting nonbasic x₃=x₄=0 gives 2x₁+x₂=4 and x₁+x₂=8, hence x₁=−4: infeasible, so false. For S2, x₁=x₄=0 gives x₂=2,x₃=2 and objective 16. First constraint gives Z=16−2x₁−3x₄≤16, so this feasible basis is optimal. **S2 only, B**.
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