Schooling · Class 12 · Mathematics · Linear Programming · Chapter on a page
CBSE Class 12 · Mathematics · NCERT Mathematics Part-II
Linear Programming: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- Objective function
- Decision variables
- Constraints
- Non-negativity restrictions
- Feasible region
- Corner-point (vertex) method
- Bounded vs unbounded feasible region
- Formulas and facts
- General form of objective function
- Standard constraints
- Fundamental theorem (corner-point)
- Unbounded region rule
- Graphical solution steps
- Worked examples
- Maximising profit in a small workshop
- Minimising cost of a diet
- Recognising an unbounded optimum
- Watch out for
- Shading the wrong side of a…
- Forgetting non-negativity restrictions
- Evaluating Z at points that lie…
- Assuming an optimum always exists
- Giving fractional answers when…
What this chapter is about
Linear programming is a mathematical method for finding the best possible outcome—maximum profit, minimum cost, or optimal resource use—when you face several restrictions at once. The word "linear" means all relationships involved are straight-line equations or inequalities; "programming" here means planning, not computer code.
Key ideas
- Objective function: A linear expression Z = ax + by (or with more variables) that you want to maximise or minimise.
- Decision variables: The unknowns (usually x and y) whose values you must choose; they typically represent quantities like number of items or amounts of resources.
- Constraints: Linear inequalities (and sometimes equations) that the decision variables must satisfy, arising from limits such as budget, time, raw material, or capacity.
- Non-negativity restrictions: Almost always x ≥ 0 and y ≥ 0, because negative quantities of goods or time make no practical sense.
- Feasible region: The set of all points (x, y) that satisfy every constraint simultaneously; on a graph it appears as a polygon (or unbounded region).
- Corner-point (vertex) method: The optimal value of a linear objective over a bounded convex polygon always occurs at one of the vertices; check Z at each corner to find the best.
- Bounded vs unbounded feasible region: When the region extends to infinity in some direction, a maximum or minimum may not exist; you must examine whether Z can grow (or shrink) without limit.
Formulas and facts to remember
- Item: General form of objective function · Statement: Z = c₁x₁ + c₂x₂ + … + cₙxₙ (linear in decision variables)
- Item: Standard constraints · Statement: a₁x + b₁y ≤ k₁, a₂x + b₂y ≥ k₂, etc., where a, b, k are constants
- Item: Fundamental theorem (corner-point) · Statement: If the feasible region is non-empty and bounded, Z attains its optimal value at a vertex of the region
- Item: Unbounded region rule · Statement: If feasible region is unbounded, check whether Z can increase (or decrease) without limit; if not, the optimum still occurs at a corner
- Item: Graphical solution steps · Statement: (1) Plot each constraint as a line, shade the correct half-plane, (2) identify the common region, (3) list vertices, (4) evaluate Z at each vertex, (5) pick the best
Common mistakes
- Shading the wrong side of a constraint line → substitute a test point (often the origin) to check which half-plane satisfies the inequality.
- Forgetting non-negativity restrictions → always include x ≥ 0, y ≥ 0 when quantities cannot be negative.
- Evaluating Z at points that lie outside the feasible region → verify every candidate vertex satisfies all constraints before using it.
- Assuming an optimum always exists → in an unbounded region, first check whether Z can grow (or shrink) indefinitely.
- Giving fractional answers when whole-number solutions are needed → re-examine integer lattice points near the optimal corner.
Quick revision
- Objective function = what you optimise; constraints = limits you must respect.
- Feasible region = intersection of all half-planes defined by constraints.
- Optimal value of a linear objective on a bounded polygon occurs at a vertex.
- Always test Z at every corner; compare to find max or min.
- If the region is unbounded, verify whether a finite optimum exists before concluding.