Lesson 8.8: Operations Research (LPP, Queuing, Simulation, Network Models)

Operations Research (OR) is vital for decision-making in engineering and production. GATE and PSU exams often test optimization, queuing theory, simulation techniques, and network models.

🔹 1. Introduction

• Definition: Operations Research is the scientific approach to decision-making using mathematical models and analytical methods.

• Applications: Production planning, project scheduling, inventory control, transportation, and logistics

• Key Concepts: Linear programming, queuing systems, simulation, network optimization

🔹 2. Linear Programming Problem (LPP)

• Definition: Optimization of a linear objective function subject to linear constraints

• Formulation:

  • Objective function: Maximize/Minimize Z=c1x1+c2x2+…+cnxnZ = c_1x_1 + c_2x_2 + … + c_nx_nZ=c1​x1​+c2​x2​+…+cn​xn​

  • Constraints: a1x1+a2x2+…+anxn≤ba_1x_1 + a_2x_2 + … + a_nx_n \le ba1​x1​+a2​x2​+…+an​xn​≤b

  • Non-negativity: xi≥0x_i \ge 0xi​≥0

• Solution Methods:

  1. Graphical method (2 variables)

  2. Simplex method (n variables)

• Applications: Resource allocation, production planning, cost minimization

🔹 3. Queuing Theory

• Definition: Study of waiting lines to optimize service efficiency

• Parameters: Arrival rate ($\lambda$), Service rate ($\mu$), Number in system, Waiting time

• Types of Queues:

  1. Single server, infinite capacity

  2. Multi-server systems

• Applications: Call centers, production lines, hospitals, banks

🔹 4. Simulation

• Definition: Imitation of real-world processes using mathematical and computational models

• Types:

  1. Monte Carlo simulation

  2. Discrete-event simulation

• Applications: Inventory management, production scheduling, risk analysis

🔹 5. Network Models

• Definition: Graphical representation of projects/processes to analyze time and cost

• Techniques:

  1. CPM (Critical Path Method) – Deterministic activity times

  2. PERT (Program Evaluation & Review Technique) – Probabilistic activity times

• Applications: Project scheduling, resource allocation, time-cost optimization

🔹 6. Solved Examples (PYQ Style)

1. Formulate and solve an LPP using graphical method

2. Compute average waiting time in a single-server queue

3. Simulate production schedule using Monte Carlo technique

4. Determine critical path and project duration using CPM

🔹 7. Practice Exercises

1. Solve LPP with 2 variables

2. Analyze a queuing system with given $\lambda$ and $\mu$

3. Perform simple simulation for inventory management

4. Draw network diagram and calculate slack for project activities

5. Compare CPM and PERT results for project planning

🔹 8. Summary

• Operations Research: Decision-making using models

• LPP: Optimize linear objective function under constraints

• Queuing: Analyze waiting lines and service efficiency

• Simulation: Model real-world processes computationally

• Network Models: CPM & PERT for project scheduling

• Exam Tip: Focus on formulation, calculation, and interpretation of results