📡 AAI ATC Exam Prep · Class 12 Maths · Chapter 12

Linear Programming

Master optimisation problems the smart way — with real ATC applications!

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Class 12 Maths
Chapter 12
NCERT + AAI ATC
🎯 Graphical Method
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Why Linear Programming for AAI ATC?

In Air Traffic Control operations, resources like runways, communication channels, and radar slots are always limited. Linear Programming gives us a mathematical framework to allocate these resources optimally — maximising efficiency while satisfying all operational constraints. This chapter is directly relevant to quantitative aptitude questions in the AAI Junior Executive (ATC) exam.

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Introduction & Key Terminology

What is Linear Programming?

A Linear Programming Problem (LPP) seeks the optimal value (maximum or minimum) of a linear function — called the Objective Function — subject to a set of linear inequalities called constraints. Variables involved are always non-negative (decision variables).

📌 Objective Function

The linear function Z = ax + by to be maximised or minimised (e.g., profit, cost, time).

📌 Constraints

Linear inequalities/equations limiting the decision variables (e.g., budget, capacity).

📌 Decision Variables

Variables x, y (always ≥ 0) whose values we need to find for the optimal solution.

📌 Feasible Region

The set of all points satisfying all constraints simultaneously — always a convex region.

Maximise / Minimise   Z = ax + by
subject to: linear constraints & x ≥ 0, y ≥ 0
General form of a Linear Programming Problem
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ATC Application

An ATC centre must allocate two types of radar scanning modes — Mode A (covers 5 sectors) and Mode B (covers 2 sectors) — within a power budget. The objective is to maximise total sector coverage. This is a classic LPP!

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Image: Add diagram showing Objective Function, Constraints and Feasible Region labels

Recommended size: 800×400 px | Replace this block with your <img> tag

Practice MCQs

Test Your Understanding — Subtopic 1

1 In an LPP, the function Z = 3x + 5y that needs to be maximised or minimised is called the:
✅ Answer: (C) Objective function
The linear function Z = ax + by that we seek to maximise or minimise is defined as the objective function. The variables x and y are the decision variables, while the inequalities form the constraints.
2 An ATC radar system can cover either 4 long-range sectors (type X) or 6 short-range sectors (type Y) per hour. With a total capacity of 24 sectors/hour, which constraint correctly represents this?
✅ Answer: (C) 4x + 6y ≤ 24
Since total sectors covered = 4x + 6y and this cannot exceed 24 sectors/hour, the correct constraint is 4x + 6y ≤ 24. Note: In AAI ATC problems, capacity limits always use ≤ (not strict equality) to allow partial usage.
3 If x = 3 and y = 4 satisfy all constraints of an LPP including x ≥ 0, y ≥ 0, then the point (3, 4) is called a:
✅ Answer: (C) Feasible solution
Any point that satisfies all constraints (including non-negativity) is a feasible solution. It becomes an optimal solution only if it also gives the best (max/min) value of Z. A corner point is specifically a vertex of the feasible region.
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Graphical Method & Feasible Region

Steps to Solve LPP Graphically (Corner Point Method)

Theorem: Optimal value of Z always occurs at a corner point of the feasible region
Fundamental Theorem of Linear Programming (NCERT Theorem 1 & 2)
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ATC Real-World Link

Graphical LPP is used to plan runway utilisation. If runway A handles 3 flights/hour and runway B handles 2 flights/hour, with operational constraints on fuel supply and staff, the feasible region shows all valid flight-schedule combinations. The optimal corner point gives the maximum throughput!

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Image: Add a graph showing feasible region OABC with corner points labelled, constraint lines 5x+y=100 and x+y=60

Recommended size: 700×500 px | Use graph paper or Desmos screenshot

Practice MCQs

Test Your Understanding — Subtopic 2

4 For the LPP: Maximise Z = 4x + y, subject to x + y ≤ 50, 3x + y ≤ 90, x ≥ 0, y ≥ 0 — what is the maximum value of Z?
✅ Answer: (B) 120
Corner points: O(0,0) → Z=0, A(30,0) → Z=120, B(20,30) → Z=110, C(0,50) → Z=50.
Maximum Z = 120 at point A(30, 0). Solving 3x+y=90 and y=0 gives x=30.
5 An ATC scheduling problem has the feasible region as a bounded convex polygon with corner points P(0,5), Q(4,3) and R(0,6). If Z = 200x + 500y, the minimum value of Z is:
✅ Answer: (C) 2300
Evaluating Z = 200x + 500y:
At P(0,5): Z = 0 + 2500 = 2500
At Q(4,3): Z = 800 + 1500 = 2300 ← Minimum
At R(0,6): Z = 0 + 3000 = 3000
Minimum Z = 2300 at Q(4, 3).
6 The feasible region for an LPP is always:
✅ Answer: (C) A convex region
The intersection of half-planes (linear constraints) always produces a convex region. This is a fundamental property used in Theorem 1 & 2 of LPP — the optimal solution must occur at a corner (vertex) of this convex region. A concave region would violate this theorem.
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Special Cases in LPP

Case 1 — Unbounded Feasible Region

When the feasible region extends infinitely, the optimal value may or may not exist. After finding the corner point giving the extreme value M, check:

Graph: ax + by > M (for maximum check)
If this open half-plane has NO point in the feasible region → M is the maximum.
Unbounded Region Test (Step 3-ii of Corner Point Method)

Case 2 — No Feasible Solution

If the constraints are contradictory (e.g., x + y ≥ 8 and 3x + 5y ≤ 15), there is no common region satisfying all constraints simultaneously → the LPP has no feasible solution and hence no optimal solution.

Case 3 — Multiple Optimal Solutions

When two adjacent corner points give the same optimal value of Z, every point on the line segment joining them is also an optimal solution (infinitely many optimal solutions).

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ATC Exam Alert

Questions on unbounded regions are a favourite in AAI ATC and other government exams. Always check: "Is the feasible region bounded or unbounded?" before concluding the optimal value. If unbounded, apply the open half-plane test — don't skip it!

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Image: Add side-by-side diagrams — (Left) Unbounded feasible region, (Right) No feasible region example

Recommended size: 800×380 px | Label corner points clearly

Practice MCQs

Test Your Understanding — Subtopic 3

7 For Z = 3x + 9y with feasible region ABCD having corner points A(0,10), B(5,5), C(15,15) and D(0,20), what is the maximum value of Z?
✅ Answer: (C) 180
Evaluating Z = 3x + 9y:
A(0,10) → Z = 90  |  B(5,5) → Z = 60 (Minimum)  |  C(15,15) → Z = 180  |  D(0,20) → Z = 180
Maximum = 180 at both C and D. This is a case of multiple optimal solutions — every point on segment CD also gives Z = 180.
8 An LPP has constraints x + y ≥ 8 and 3x + 5y ≤ 15, with x, y ≥ 0. What can you conclude?
✅ Answer: (B) There is no feasible solution
The constraint x + y ≥ 8 requires points above the line x+y=8, while 3x + 5y ≤ 15 restricts points below the line 3x+5y=15. These two regions do not overlap in the first quadrant (where x, y ≥ 0). Hence, no common feasible region exists → no feasible or optimal solution.
9 For objective function Z = −50x + 20y with an unbounded feasible region, the smallest Z value at a corner point is −300 at (6, 0). What additional step must you perform?
✅ Answer: (B) Check if −5x + 2y < −30 has common points with feasible region
For unbounded regions, we must apply the open half-plane test. Graph −50x + 20y < −300 (i.e., −5x + 2y < −30). If this half-plane shares points with the feasible region, then Z has no minimum. If they have no common points, then −300 is confirmed as the minimum value of Z.
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