Linear programming (LP) applicability Which conditions are necessary to apply linear programming to a decision problem?

Introduction / Context:
Linear programming is a quantitative method to optimize an objective (maximize profit or minimize cost) subject to linear constraints. Recognizing when LP is appropriate prevents model misuse and erroneous recommendations.

Given Data / Assumptions:

• Objective function is linear in decision variables.
• Constraints are linear equalities/inequalities reflecting limited resources.
• Non-negativity of decision variables is required.

Concept / Approach:
LP presumes proportionality, additivity, divisibility (unless using integer programming), and certainty within the modeling horizon. When these hold, the feasible region is a convex polytope and the optimum occurs at an extreme point, solvable by simplex or interior-point algorithms.

Step-by-Step Solution:
Define variables and linear objective (e.g., maximize Z = c1 x1 + c2 x2).Express resource limits as linear constraints (e.g., a11 x1 + a12 x2 ≤ b1).Include x ≥ 0 (non-negativity).Solve using LP methods to find optimal values.

Verification / Alternative check:
Sensitivity analysis on shadow prices and reduced costs verifies robustness and identifies bottleneck resources.

Why Other Options Are Wrong:
Requiring integer decisions is not necessary for LP (that is a separate class, integer programming). Each of the first three statements is essential, so “All of the above.”

Common Pitfalls:
Modeling non-linear realities as linear; ignoring uncertainty; omitting critical constraints leading to infeasible or misleading solutions.

Final Answer:
All of the above