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Understanding Infeasible Linear Programming Solutions and Two-Phase Method

This document explores the challenges of infeasible linear programming solutions, focusing on maximizing the objective function z = 3x1 + 2x2 under given constraints. It details the Two-Phase Method, including the first phase aimed at minimizing a surplus variable R. The analysis identifies critical points such as entering and leaving variables, the significance of pseudo-optimal solutions, and the absence of feasible solutions. It emphasizes the importance of recognizing when constraints prevent optimality, enhancing understanding of linear programming difficulties.

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Understanding Infeasible Linear Programming Solutions and Two-Phase Method

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  1. Infeasible Solution Maximize z=3x1+2x2 Subject to 2x1+x2 ≤ 2 3x1+4x2 ≥12 x1,x2≥0

  2. Max z=x1+2x2-MR 2x1+x2 +x3 =2 3x1+4x2 -x4+R=12 x1,x2,x3,x4,R≥0

  3. X2 enters, x3 leaves

  4. X2 enters, x3 leaves

  5. X2 enters, x3 leaves

  6. X2 enters, x3 leaves

  7. Pseudo-optimum The table is pseudo-optimum because R=4 which is not zero. There is no feasible solution exist

  8. Constraint 2 Constraint 1

  9. Two-Phase Method Phase-1 Min z=R1 S.T. 2x1+x2 +x3 =2 3x1+4x2 -x4+R=12 x1,x2,x3,x4,R≥0

  10. X2 entering and x3 leaving

  11. X2 entering and x3 leaving

  12. X2 entering and x3 leavingno feasible solution exist

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