Linear Programming

Terminology and Formulation

Understand

A linear programming problem (LPP) asks for the maximum or minimum of a linear function \(Z=px+qy\) — the objective function — where the decision variables \(x,y\) must satisfy linear inequalities called constraints. Quantities that cannot be negative give the non-negativity constraints \(x\ge0,\ y\ge0\).

Term	Meaning
Decision variables	The quantities you choose (e.g. number of chairs and tables)
Objective function	The linear expression to be maximised (profit) or minimised (cost)
Constraints	Linear inequalities from limited resources or requirements
Feasible solution	Any point satisfying all constraints
Optimal solution	A feasible point giving the best value of \(Z\)

Formulating a word problem

Name the decision variables (with units).
Write the objective function from the profit/cost per unit.
Write one inequality per limited resource (≤) or requirement (≥).
Add x ≥ 0, y ≥ 0.

Example. A potter makes mugs (\(x\)) and bowls (\(y\)). A mug needs 1 h on the wheel and a bowl 2 h; the wheel is free for at most 16 h. Kiln space allows at most 10 pieces. Profit is ₹60 per mug and ₹90 per bowl. Maximise \(Z=60x+90y\) subject to \(x+2y\le16,\ x+y\le10,\ x,y\ge0\).

Key Concepts

Identify variables, objective, constraints.
“At most / not more than / available” ⇒ ≤; “at least / minimum requirement” ⇒ ≥.
Never forget x ≥ 0, y ≥ 0.

Formula Bank

Standard form of an LPP

\[\text{Optimise } Z=px+qy\ \text{ subject to }\ a_ix+b_iy\ (\le,\ \ge)\ c_i,\quad x\ge0,\ y\ge0\]

Common Mistakes

Reversing inequalities

“At least” means ≥ and “at most” means ≤. Reversing one inequality changes the whole feasible region.

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Topic Summary

Formulation is translation: variables, one objective, one inequality per restriction, plus non-negativity.

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