To convert a "≥" type inequality constraint into an equality constraint for a linear programming problem (LPP) in standard form, we introduce a slack variable. A slack variable is a new variable that represents the amount by which the left-hand side of the inequality exceeds the right-hand side.
For example, suppose we have the inequality constraint:
2x + 3y ≥ 5
We can introduce a slack variable s to obtain the equation:
2x + 3y - s = 5
Here, s represents the amount by which 2x + 3y exceeds 5. If the optimal solution to the LPP has s = 0, then the original inequality is satisfied as an equality. If s > 0, then the original inequality is satisfied as an inequality with a surplus.
In general, to convert a "≥" type inequality constraint of the form:
a₁x₁ + a₂x₂ + ... + aₙxₙ ≥ b
into an equality constraint of the form:
a₁x₁ + a₂x₂ + ... + aₙxₙ - s = b
we introduce a new slack variable s and subtract it from the left-hand side of the inequality. The slack variable is constrained to be non-negative, i.e., s ≥ 0, since it represents the amount by which the inequality is satisfied as an equality.