Chapter 7 — Operations Research
Key Concepts Quick Recap
| Term | Definition |
|---|---|
| Objective function | Linear function to be optimised (Z = ax + by) |
| Constraints | Linear inequalities restricting decision variables |
| Feasible region | Set of all points satisfying all constraints and non-negativity |
| Corner point | Vertex of feasible region; optimal solution lies here |
| Optimal solution | Values of decision variables giving best objective value |
| Unbounded solution | Feasible region extends to infinity; may have no finite optimum |
| Infeasible | No point satisfies all constraints simultaneously |
I. Choose the Best Answer (MCQs)
1. In a Linear Programming Problem, the objective function is always: (a) Non-linear (b) Linear (c) Quadratic (d) Exponential
2. The optimal solution of an LPP always lies at: (a) The origin (b) A corner (extreme) point of the feasible region (c) The centre of the feasible region (d) Any point in the feasible region
3. Maximise Z = 3x + 5y subject to x + y ≤ 6, x ≥ 0, y ≥ 0. The maximum value of Z at (0, 6) is: (a) 18 (b) 30 (c) 24 (d) 6
4. If the feasible region is empty, the LPP is: (a) Unbounded (b) Infeasible (c) Has multiple solutions (d) Has optimal solution at origin
5. The transportation problem involves: (a) Maximising profit only (b) Finding optimal shipping plan from sources to destinations at minimum cost (c) Maximising number of units transported (d) Minimising number of warehouses
6. The assignment problem is used for: (a) Transporting goods (b) Assigning n tasks to n persons at minimum cost (c) Maximising production (d) Inventory control
7. Which of the following is NOT a requirement of an LPP? (a) Objective function must be linear (b) Constraints must be linear (c) Decision variables must be integers (d) Non-negativity conditions
II. Short Answer Questions
Q1. Formulate and solve graphically: A factory produces two products A and B. Profit: ₹3 per unit of A, ₹5 per unit of B. Machine time: A needs 2 hours, B needs 3 hours; total available = 18 hours. Raw material: A needs 1 kg, B needs 2 kg; total available = 10 kg. Maximise profit.
Formulation: Let x = units of A, y = units of B.
Objective: Maximise Z = 3x + 5y
Constraints:
- 2x + 3y ≤ 18 (machine time)
- x + 2y ≤ 10 (raw material)
- x ≥ 0, y ≥ 0
Corner Points: 1. (0, 0): Z = 0 2. (9, 0): Check x + 2y = 9 ≤ 10 ✓; Z = 3(9) + 5(0) = 27 3. (0, 5): Check 2(0) + 3(5) = 15 ≤ 18 ✓; Z = 3(0) + 5(5) = 25 4. Intersection of 2x+3y=18 and x+2y=10: Solve → x=6, y=2; Z = 3(6)+5(2) = 18+10 = 28
Maximum Z = 28 at x = 6, y = 2 (6 units of A, 2 units of B).
Exam Checklist
- LPP: linear objective function + linear constraints + non-negativity.
- Graphical method: plot constraints → find feasible region → evaluate Z at corner points.
- Optimal solution at corner point (Corner Point Theorem).
- Infeasible: no feasible region; unbounded: region extends to infinity.
- Transportation: minimise shipping cost; Assignment: minimise assignment cost (Hungarian method).