The Integer Linear Programming (ILP), where variables are constrained to take integer values, differing from the standard LP where variables can be any real number within a range.

Special cases and variations of linear programming include Integer Linear Programming (ILP), Binary Integer Linear Programming (0-1 ILP), Mixed-Integer Linear Programming (MILP), Fractional Linear Programming, Degeneracy, Unbounded Linear Programming, Infeasible Linear Programming, and Multiple Objective Linear Programming (MOLP).

These variations differ from the standard linear programming problem by introducing constraints or conditions that require integer solutions, binary decisions, mixed integer and continuous variables, fractional coefficients, or multiple objectives. Each variation poses unique challenges and often demands specific solution approaches or modifications to standard linear programming algorithms.

Special cases and variations of linear programming include Integer Linear Programming (ILP), Binary Linear Programming, Mixed Integer Linear Programming (MILP), Network Flow Problems, degeneracy, unboundedness, infeasibility, redundant constraints, and post-optimality analysis. These cases introduce complexities beyond the standard linear programming problem, involving discrete variables, network structures, degenerate solutions, and specific problem characteristics. It's important to understand these variations and their challenges when tackling optimization problems.