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Maximize ( z = 3x_1 + 2x_2 ) Subject to: ( 2x_1 + x_2 \le 2 ) ( 3x_1 + 4x_2 \ge 12 ) ( x_1, x_2 \ge 0 ) Step 1: Convert constraints to standard form with slack, surplus, and artificial variables. 1. ( 2x_1 + x_2 + s_1 = 2, \ s_1 \ge 0 ) (slack for ≤) 2. ( 3x_1 + 4x_2 - s_2 + A_1 = 12, \ s_2 \ge 0, A_1 \ge 0 ) (surplus and artificial for ≥) Step 2: Phase 1 – Minimize ( W = A_1 ) Initial basic variables: ( s_1 = 2 ), ( A_1 = 12 ) Phase 1 tableau (variables: ( x_1, x_2, s_1, s_2, A_1 )): | BV | (x_1) | (x_2) | (s_1) | (s_2) | (A_1) | RHS | | --- | ----- | ----- | ----- | ----- | ----- | --- | | s_1 | 2 | 1 ___ ____ ______ ________ ______ ________ ___ _______ _____.
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