Linear Programming Class 12 Notes

• Draw each constraint as a line, shade the side that satisfies the inequality, and find the intersection of all shaded regions.
• Locate the corner (vertex) points of the feasible region.
• Evaluate Z at every corner point.
• Pick the largest value for a maximisation problem, the smallest for minimisation.

• Decide the decision variables — the quantities the problem asks to choose (e.g. number of units of two products).
• Write the objective as a linear function of the variables from the profit or cost data.
• Translate each resource limit and requirement into a linear constraint.
• Include the implicit
• $x\ge 0,y\ge 0$
• constraints.

• Unbounded region: the optimum may not exist for one direction — compare with a test to decide.
• Infeasible: constraints with no common region, so no solution at all.
• Multiple optima: when Z takes the same optimal value on a whole edge of the feasible region, every point of that edge is optimal.

Example: Maximise $Z=3x+2y$ subject to $x+y\le 4$, $x-y\le 2$, $x,y\ge 0$.

Solution: The feasible region has corners (0,0), (2,0), (3,1), (0,4). Evaluating Z: 0, 6, 11 and 8. The maximum is 11 at (3,1).

Example: A machine makes two items A and B with a profit of ₹50 and ₹80, and takes 4 and 6 hours per unit with 60 hours available. Maximise daily profit.

Solution: $\text{Max }Z=50x+80y$ subject to $4x+6y\le 60$, $x,y\ge 0$. Corners give (0,0)=0, (15,0)=750, (0,10)=800, so 10 units of B alone maximise profit at ₹800.

Exercises (NCERT Q1–Q18)

• 1) Maximize Z = 3x + 4y, subject to the constraints: x + y ≤ 4, x ≥ 0, y ≥ 0.
• 2) Minimize Z = -3x + 4y, subject to the constraints: x + 2y ≤ 8, 3x + 2y ≤ 12, x ≥ 0, y ≥ 0.
• 3) Maximize Z = 5x + 3y, subject to the constraints: 3x + 5y ≤ 15, 5x + 2y ≤ 10, x ≥ 0, y ≥ 0.
• 4) Minimize Z = 3x + 5y, subject to the constraints: x + 3y ≥ 3, x + y ≥ 1, x ≥ 0, y ≥ 0.
• 5) Maximize Z = 3x + 5y, subject to the constraints: x + 3y ≤ 3, x + y ≤ 1, x ≥ 0, y ≥ 0.
• 6) Minimize and maximize Z = 5x + 10y, subject to the constraints: x + 2y ≤ 12, x + y ≥ 6, x ≥ 0, y ≥ 0.
• 7) Minimize and maximize Z = x + 2y, subject to the constraints: 2x + y ≥ 3, x + 2y ≤ 6, x ≥ 0, y ≥ 0.
• 8) Maximize Z = -x₁ + 2x₂, subject to the constraints: x₁ + x₂ ≤ 4, x₁ + 3x₂ ≤ 6, x₁ ≥ 0, x₂ ≥ 0.
• 9) A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F₁ and F₂ are available. Food F₁ costs Rs 4 per unit and contains 3 units of vitamin A and 5 units of minerals. Food F₂ costs Rs 6 per unit and contains 5 units of vitamin A and 2 units of minerals. Formulate this as an LPP and find the minimum cost for the diet which meets the nutritional requirements.
• 10) A furniture manufacturer produces tables and chairs. Each table requires 2 hours of carpentry and 1 hour of finishing. Each chair requires 1 hour of carpentry and 1 hour of finishing. The profit on a table is Rs 500 and on a chair is Rs 300. The carpentry workshop has at most 120 hours and the finishing department has at most 80 hours available. Formulate the LPP to maximize profit and solve it graphically.
• 11) A dietician wishes to mix two types of foods in such a way that the mixture contains at least 10 units of vitamin A, at least 8 units of vitamin B, and at least 7 units of vitamin C. Food I contains 2 units of A, 1 unit of B, 1 unit of C per unit. Food II contains 1 unit of A, 2 units of B, 3 units of C per unit. Food I costs Rs 50 per unit and Food II costs Rs 70 per unit. Formulate the LPP and solve it.
• 12) A company manufactures two products A and B on two machines I and II. Product A requires 2 hours on Machine I and 1 hour on Machine II. Product B requires 1 hour on Machine I and 3 hours on Machine II. Machine I is available for 200 hours, Machine II for 300 hours. Profit on A is Rs 30 and on B is Rs 40 per unit. Find the number of units of each product for maximum profit.
• 13) A manufacturer produces two products P and Q. Each unit of P requires 1 hour on Machine A, 2 hours on Machine B, and 1 hour on Machine C. Each unit of Q requires 2 hours on Machine A, 1 hour on Machine B, and 3 hours on Machine C. Machine A is available for 8 hours, Machine B for 10 hours, and Machine C for 12 hours. Profit on P is Rs 200 and on Q is Rs 300 per unit. Find how many units of each to produce to maximize profit.
• 14) A factory has two machines A and B. Machine A can produce at most 120 units of item X and 80 units of item Y per day. Machine B can produce at most 60 units of item X and 140 units of item Y per day. The factory must produce at least 200 units of item X and 160 units of item Y per day. The operating cost of Machine A is Rs 2000 per day and Machine B is Rs 1600 per day. How should the factory allocate production to minimise cost?
• 15) A golfer requires 200 units of A, 500 units of B, and 150 units of C. Brand X costs Rs 40 per pill and contains 20 units of A, 100 units of B, and 30 units of C. Brand Y costs Rs 60 per pill and contains 30 units of A, 150 units of B, and 20 units of C. Find the minimum cost diet.
• 16) A pharmaceutical company produces two types of tonic: T₁ and T₂. T₁ requires 2 hours of processing in department A and 1 hour in department B. T₂ requires 1 hour in department A and 3 hours in department B. Department A has 120 hours and department B has 150 hours available per week. Profit is Rs 80 on T₁ and Rs 120 on T₂ per unit. Find the number of units for maximum profit.
• 17) A company makes two products A and B using two raw materials M₁ and M₂. Product A requires 3 units of M₁ and 2 units of M₂. Product B requires 1 unit of M₁ and 4 units of M₂. The available stock is 120 units of M₁ and 160 units of M₂. Profit is Rs 60 per unit of A and Rs 80 per unit of B. Solve for maximum profit.
• 18) A warehouse supplies two shops from two stores A and B. Store A has 200 quintals and Store B has 300 quintals. Shop I needs 150 quintals and Shop II needs 350 quintals. The transportation cost (in Rs per quintal) from A to I is 5, from A to II is 3, from B to I is 4, and from B to II is 2. Find the transportation schedule that minimizes cost.

Answers at a glance

1. 1
   Q1 → answer not in source — verify against the textbook
2. 2
   Q2 → answer not in source — verify against the textbook
3. 3
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4. 4
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5. 5
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9. 9
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10. 10
    Q10 → answer not in source — verify against the textbook
11. 11
    Q11 → answer not in source — verify against the textbook
12. 12
    Q12 → answer not in source — verify against the textbook
13. 13
    Q13 → answer not in source — verify against the textbook
14. 14
    Q14 → answer not in source — verify against the textbook
15. 15
    Q15 → answer not in source — verify against the textbook
16. 16
    Q16 → answer not in source — verify against the textbook
17. 17
    Q17 → answer not in source — verify against the textbook
18. 18
    Q18 → answer not in source — verify against the textbook

MCQ Questions

1 Mark Questions

1. 1
   Define the feasible region of an LP problem.
2. 2
   State the fundamental theorem about the optimum value of Z for a bounded LPP.
3. 3
   What name is given to a feasible solution that maximises or minimises Z?
4. 4
   Write the non-negativity constraints of a factory LPP.

2 Mark Questions

1. 1
   Find the corner points of the feasible region x + y ≤ 4, x ≥ 0, y ≥ 0. (Ans: (0,0), (4,0), (0,4))
2. 2
   Maximise Z = x + y subject to x ≤ 2, y ≤ 3, x, y ≥ 0. (Ans: Z = 5 at (2,3))
3. 3
   Show that the region x + y ≤ 1, x ≥ 0, y ≥ 0 is bounded while x + y ≤ 1 with no non-negativity constraints is unbounded.

3 Mark Questions

1. 1
   Solve graphically: minimise Z = 3x + 2y subject to x + 2y ≥ 4, x, y ≥ 0. (Ans: minimum Z = 4 at (4,0) and (0,2))
2. 2
   A company makes two products; the LPP is Z = 5x + 4y, x ≥ 0, y ≥ 0, 2x + 3y ≤ 12, x + y ≤ 5. Find the corner points and the maximum Z. (Ans: corners (0,0),(5,0),(3,2),(0,4); Z max = 23 at (3,2))
3. 3
   Explain the graphical method for a bounded LPP in steps: graph, identify feasible region, list corner points, evaluate Z.

4 Mark Questions

1. 1
   A manufacturer produces two types of products A and B, needing 2 and 3 machine-hours respectively, with at most 12 machine-hours daily; demand of A ≤ 5 units and non-negativity. Formulate the LPP to maximise profit at ₹5 and ₹4 per unit and solve it graphically, stating the feasible region and the optimal product mix. (Ans: Z max = 23 at x = 3, y = 2)
2. 2
   Formulate and solve: minimise Z = 3x + 2y subject to x + 2y ≥ 6, 3x + y ≥ 7, x, y ≥ 0 (diet problem style). Find the minimum cost and the optimum point. (Ans: optimum at (2, 2): Z = 10)