Optimization fundamentals: in linear programming (LP), what does solving the model identify for a well-posed objective and constraints?

Introduction / Context:
Linear programming (LP) is a cornerstone of operations research used to allocate scarce resources optimally under linear constraints. A correct grasp of LP outputs helps managers move from qualitative judgment to quantitative, defensible decisions for production planning, logistics, staffing, and more.

Given Data / Assumptions:

• An objective function (maximize profit or minimize cost) is linear.
• Decision variables and constraints are linear with feasible solutions.
• Standard solvers (simplex or interior point) can find an optimal basic feasible solution if one exists.

Concept / Approach:
Solving an LP yields both the optimal decision vector (variable quantities) and the optimal objective value (best achievable profit or cost). Dual information (shadow prices) and reduced costs also emerge, supporting sensitivity analysis. But the two headline results remain the variable levels and the associated optimal objective value.

Step-by-Step Solution:

Verification / Alternative check:
Solver reports list variable levels (x*) and the optimal objective (z*), often with dual values and sensitivity ranges, confirming that both outcomes are identified by LP.

Why Other Options Are Wrong:

• Only (a) or only (b) is incomplete because LP simultaneously returns both.
• “Neither” is incorrect because LP solutions explicitly provide these results when feasible and bounded.

Common Pitfalls:
Ignoring feasibility/boundedness checks; mis-specified constraints can yield infeasible or unbounded models, in which case optimal values do not exist.

Final Answer:
both (a) and (b)