# Operations Research Question Paper for Practice

Practice makes perfect, especially in operations research. Delve into a comprehensive set of practice questions covering various operations research concepts and techniques. Strengthen your grasp of this discipline and prepare yourself for real-world problem-solving scenarios.

In the previous articles, we briefly discussed the history, syllabus, importance, limitations, applications, and models of operations research. We also discussed some books to gain a deeper understanding of operations research. Now, in this article, we will discuss some questions based on operations research to help you master the concepts of operations research.

## Basic Operations Research Questions

- What is Operations Research (OR)? Explain its importance and applications.
- Define linear programming and its components (decision variables, objective function, and constraints).
- Explain the graphical method for solving linear programming problems with two decision variables.
- What is the simplex method, and how is it used to solve linear programming problems?
- Describe the transportation problem and its applications.
- What is the assignment problem, and how is it different from the transportation problem?
- Explain the concept of queuing theory and its significance in Operations Research.
- Define the terms service rate, arrival rate, and utilization factor in queuing models.
- What is the importance of game theory in Operations Research?
- Explain the concept of decision-making under uncertainty and its applications.

## Real-Life Industry-Based Operations Research Questions

Sure, here are some real-life industry-based questions related to Operations Research:

**Supply Chain Optimization**: A multinational company has several manufacturing plants and distribution centers located in different countries. The company needs to optimize its supply chain network to minimize the total cost of production, transportation, and inventory while meeting customer demand. Formulate a mathematical model to solve this problem and determine the optimal production quantities, transportation routes, and inventory levels.**Workforce Scheduling**: A call center operates 24/7 and receives varying call volumes throughout the day. The management wants to develop an efficient workforce scheduling plan to ensure adequate staffing levels while minimizing labor costs. Use queuing theory and optimization techniques to determine the optimal number of agents required during different time periods and create a cost-effective shift schedule.**Airline Route Planning**: An airline company wants to optimize its route network to maximize revenue while considering factors such as fuel costs, passenger demand, and aircraft availability. Develop a model using network optimization techniques to determine the optimal flight routes, frequencies, and aircraft assignments for the airline's domestic and international operations.**Portfolio Optimization**: An investment firm manages a diverse portfolio of assets, including stocks, bonds, and real estate. The firm aims to maximize the portfolio's expected return while minimizing risk. Formulate a portfolio optimization model using techniques such as the Markowitz model or the Capital Asset Pricing Model (CAPM) to determine the optimal asset allocation strategy.**Facility Location**: A retail chain plans to open new stores in a specific region. The company needs to identify the optimal locations for these stores while considering factors such as customer demand, competition, and transportation costs. Develop a location-allocation model using techniques like the p-median problem or the p-center problem to determine the best locations for the new stores.**Production Planning**: A manufacturing company produces multiple products using various machines and resources. The company needs to develop a production plan that maximizes profit while considering factors such as machine capacities, setup times, and inventory costs. Formulate a mixed-integer programming model or use techniques like material requirements planning (MRP) to optimize the production schedule.**Traffic Signal Optimization**: A city's transportation department wants to optimize the timing of traffic signals at intersections to minimize delays and reduce congestion. Develop a model using techniques such as dynamic programming or queuing theory to determine the optimal signal timing patterns for different intersections and traffic conditions.**Project Scheduling**: A construction company has been awarded a large-scale infrastructure project with multiple interdependent tasks and resource constraints. Use the Critical Path Method (CPM) or the Program Evaluation and Review Technique (PERT) to create a project schedule that minimizes the project duration while optimizing resource allocation.**Vehicle Routing**: A delivery company needs to plan routes for its fleet of vehicles to serve customers in a specific area. The objective is to minimize the total distance traveled while ensuring that all customer orders are delivered on time. Develop a vehicle routing model using techniques like the Traveling Salesman Problem (TSP) or the Vehicle Routing Problem (VRP) to determine the optimal routes for the delivery vehicles.**Inventory Management**: A retailer sells various products with different demand patterns and lead times. The company wants to optimize its inventory levels to minimize holding costs while ensuring sufficient stock to meet customer demand. Develop an inventory management model using techniques such as the Economic Order Quantity (EOQ) or the Newsvendor model to determine the optimal reorder points and order quantities for each product.

**Numerical-Based Operations Research Questions**

**Linear Programming**: A company manufactures two products, A and B, using two resources, X and Y. The profit per unit of product A is $5, and the profit per unit of product B is $8. The resource requirements and availability are given in the table below:

Resource X | Resource Y | |
---|---|---|

Product A | 2 units | 3 units |

Product B | 4 units | 1 unit |

Availability | 24 units | 18 units |

Formulate a linear programming model to maximize the company's total profit and solve it using the simplex or graphical methods.

**Transportation Problem**: A company has three factories (F1, F2, and F3) and four warehouses (W1, W2, W3, and W4). The table below shows the supply capacities of the factories and the demand requirements of the warehouses, along with the transportation costs per unit between each factory-warehouse pair.

W1 | W2 | W3 | W4 | Supply | |
---|---|---|---|---|---|

F1 | 4 | 6 | 3 | 5 | 200 |

F2 | 2 | 4 | 7 | 3 | 300 |

F3 | 5 | 3 | 6 | 4 | 400 |

Demand | 250 | 150 | 200 | 300 |

Determine the optimal transportation plan that minimizes the total transportation cost.

**Assignment Problem**: A company needs to assign five jobs (J1, J2, J3, J4, and J5) to five workers (W1, W2, W3, W4, and W5). The cost matrix representing the cost of assigning each job to each worker is given below:

W1 | W2 | W3 | W4 | W5 | |
---|---|---|---|---|---|

J1 | 10 | 15 | 12 | 18 | 20 |

J2 | 14 | 16 | 11 | 13 | 17 |

J3 | 18 | 12 | 20 | 16 | 14 |

J4 | 22 | 25 | 18 | 21 | 24 |

J5 | 15 | 20 | 25 | 22 | 18 |

Use the Hungarian or other appropriate technique to find the optimal assignment that minimizes the total cost.

**Queuing Theory**: A bank has three tellers serving customers in a single queue. Customers arrive 20 per hour, and each teller serves 10 customers per hour. Assume that the arrivals follow a Poisson distribution and the service times are exponentially distributed.

Calculate the following performance measures:

a) Average number of customers in the system

b) Average waiting time in the queue

c) Probability that an arriving customer will have to wait in the queue

**Game Theory**: Consider a two-person, zero-sum game with the following payoff matrix:

Player B | ||
---|---|---|

Strategy 1 | Strategy 2 | |

Player A |
Strategy 1 |
2 |

Strategy 2 |
-1 |

Determine the optimal strategies for both players and the value of the game using the principles of game theory.

**Project Management**: A construction project consists of seven activities, labelled A to G, with the following precedence relationships and duration estimates (in weeks):

Activity | Precedence | Duration |
---|---|---|

A | - | 3 |

B | - | 4 |

C | A | 5 |

D | A | 6 |

E | B, C | 3 |

F | D | 4 |

G | E, F | 2 |

a) Draw the project network diagram. b) Identify the critical path and the project completion time. c) If the project must be completed within 16 weeks, what is the latest start time for activity G?

**Inventory Management**: A retailer faces an annual demand of 10,000 units for a particular product. The ordering cost is $50 per order, and the holding cost is $2 per unit per year. The purchase cost of the product is $10 per unit.

Determine the optimal order quantity and the total annual cost using the Economic Order Quantity (EOQ) model.

**Non-linear Programming**: A company produces two products, X and Y, using two resources, R1 and R2. The profit functions for the two products are given by:

Profit from product X = 10x - x^2 Profit from product Y = 12y - y^2

The resource availabilities and requirements are:

Resource | Availability | Requirement for X | Requirement for Y |
---|---|---|---|

R1 | 20 | 1 | 2 |

R2 | 24 | 2 | 1 |

Formulate a non-linear programming model to maximize the total profit, and solve it using an appropriate technique.

**Simulation**: A fast-food restaurant has two service counters. Customers arrive at the restaurant according to a Poisson process with a rate of 30 customers per hour. The service time at each counter follows an exponential distribution with a mean of 2 minutes.

Use simulation techniques to estimate the following performance measures: a) Average waiting time in the queue b) Probability that the number of customers in the system exceeds 10 c) Average utilization of each service counter

**Integer Programming**: A company manufactures two types of products, A and B, using three machines. The profit per unit of product A is $10, and the profit per unit of product B is $8. The machine time requirements and availability are given in the table below:

Machine 1 | Machine 2 | Machine 3 | |
---|---|---|---|

Product A | 2 hours | 1 hour | 3 hours |

Product B | 1 hour | 2 hours | 2 hours |

Availability | 40 hours | 60 hours | 80 hours |

Formulate an integer programming model to maximize the total profit and solve it using an appropriate technique.

**Conclusion**

Incorporating practice questions into your operations research study regimen is an effective way to reinforce your learning and refine your problem-solving capabilities. By engaging with diverse questions and scenarios, you can fortify your grasp of operations research concepts and cultivate the expertise needed to tackle real-world challenges in this dynamic field.

**About the Author**

Vikram has a Postgraduate degree in Applied Mathematics, with a keen interest in Data Science and Machine Learning. He has experience of 2+ years in content creation in Mathematics, Statistics, Data Science, and Mac... Read Full Bio