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Linear Programming & Simplex Tableau SuiteOptimal Solution Summary
Linear Programming Mechanics: The Primal Simplex Algorithm
Linear Programming (LP) optimizes a linear objective function subject to a set of linear inequality or equality constraints. The fundamental theorem of linear programming asserts that the optimal solution to a convex feasible region always occurs at one of its extreme corner points (vertices).
1. Standard Form Representation
To solve a canonical maximization linear program using the Simplex method, constraints are converted into equality standard form by introducing non-negative slack variables ($s_i \ge 0$):
$$\text{Maximize } Z = \mathbf{c}^T \mathbf{x}$$ $$\text{Subject to } A\mathbf{x} + \mathbf{s} = \mathbf{b}, \quad \mathbf{x} \ge \mathbf{0}, \, \mathbf{s} \ge \mathbf{0}$$| Simplex Operation Step | Mathematical Selection Criteria | Geometrical Significance |
|---|---|---|
| Pivot Column Selection | Most negative coefficient in objective row ($Z$-row). | Identifies the edge direction along the feasible region giving the steepest rate of increase in $Z$. |
| Minimum Ratio Test | $\min_i \left( \frac{b_i}{a_{i, \text{pivot}}} \right)$ for $a_{i, \text{pivot}} > 0$. | Determines the bounding vertex constraint to prevent leaving the non-negative feasible region. |
| Gauss-Jordan Elimination | Normalize pivot row and eliminate column entries to zero. | Algebraically shifts the basic feasible solution (BFS) to an adjacent corner vertex. |
2. Duality and Sensitivity Analysis
Every primal linear program possesses an associated Dual problem. Shadow prices derived from the final Simplex tableau coefficients corresponding to slack variables indicate the marginal gain in the objective function per unit increase in available constraint capacity $b_i$.
$$\text{Primal Max } Z = \mathbf{c}^T \mathbf{x} \quad \Longleftrightarrow \quad \text{Dual Min } W = \mathbf{b}^T \mathbf{y}$$Frequently Asked Questions
When does the Simplex algorithm encounter Unboundedness?
Unboundedness occurs when an entering variable chosen during pivot column selection yields no positive entries in that column during the Minimum Ratio Test. This indicates that the feasible region extends infinitely in the optimization direction.
How are Artificial Variables used in $\ge$ constraints?
When constraints contain greater-than-or-equal ($\ge$) or exact equality ($=$) relations, standard slack variables do not provide an initial identity matrix basis. Methods such as the Two-Phase Simplex or Big-M Method introduce non-negative artificial variables to establish an initial Basic Feasible Solution.
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