Unit 1: Introduction to Operations Research (5 Hours)

Operations Research (OR) is a scientific approach to decision-making that applies mathematical models, statistical analyses, and optimization techniques to solve complex problems in management, business, and industrial settings. The aim of OR is to provide decision-makers with quantitative tools and methods to make better decisions in a resource-constrained environment. The concepts introduced in this unit lay the foundation for understanding how OR can be applied across various industries and sectors.

1. Introduction to Operations Research

Operations Research (OR) refers to the application of scientific and quantitative methods to decision-making in business and industry. It involves analyzing complex systems, identifying problems, and finding the optimal solutions through mathematical models, simulations, and algorithms.

The primary goal of OR is to improve efficiency, reduce costs, and enhance decision-making in various organizations. It has wide applications across fields like transportation, manufacturing, healthcare, logistics, military operations, and finance.

Key characteristics of Operations Research include:

• Problem Solving: Identifying and defining the problem in terms of its quantitative aspects.
• Modeling: Translating the problem into a mathematical model.
• Optimization: Finding the best solution using computational techniques.
• Evaluation: Assessing the outcomes and suggesting improvements.

2. History of Operations Research

Operations Research originated during World War II when military leaders faced complex problems related to logistics, resource allocation, and strategic planning. As the war’s demand for efficient decision-making increased, military scientists began developing mathematical models to optimize the use of resources.

Some key milestones in the history of OR:

• 1940s: The term “Operations Research” was coined during World War II when the British military used statistical analysis to improve the efficiency of radar systems and military logistics.
• Post-War Era: Following the war, OR was adopted by industries like manufacturing, transportation, and healthcare to address a broad range of operational issues.
• 1950s-1960s: The development of optimization techniques such as Linear Programming and the Simplex Method revolutionized business decision-making.
• 1970s-Present: OR expanded to cover broader areas such as supply chain management, finance, inventory management, and healthcare operations.

3. Stages of Development of Operations Research

The development of OR can be broadly divided into several stages:

1. Initial Exploration (1940s-1950s):

   • Focus on military operations (e.g., optimizing logistics, military strategies).
   • Introduction of mathematical models for supply chain management and resource allocation.
   • Early use of linear programming, queuing theory, and game theory.

2. Expansion to Business and Industry (1960s-1970s):

   • Industrial Applications: OR techniques expanded to manufacturing, transportation, and finance.
   • Use of inventory management models, production scheduling, and optimization models to streamline operations.
   • Introduction of simulation models to study complex systems and scenarios.

3. Advanced and Integrated Applications (1980s-Present):

   • Integration of OR with information technology and software tools.
   • Use of artificial intelligence (AI), machine learning, and big data analytics for decision support.
   • Applications in healthcare, telecommunications, energy, logistics, and environmental sustainability.

4. Relationship between Manager and OR Specialist

The relationship between a manager and an OR specialist is crucial for the successful application of OR techniques in an organization. While managers provide valuable domain knowledge and decision-making expertise, OR specialists bring in-depth knowledge of mathematical modeling, quantitative analysis, and optimization techniques.

Key aspects of their relationship include:

• Collaboration: Managers need to communicate the problem clearly, while OR specialists help translate it into a quantifiable model.
• Model Development: The OR specialist creates models based on the manager’s inputs, and the manager interprets the model’s results.
• Decision Support: OR specialists provide data-driven insights to support the manager’s decision-making process.
• Implementation: Managers implement the decisions, while OR specialists may assist in adjusting models based on real-world data and feedback.

Successful implementation of OR techniques depends on mutual understanding and communication between managers and OR specialists.

5. OR Tools and Techniques

Operations Research involves various tools and techniques used to model and solve problems. Some of the major OR tools and techniques include:

• Linear Programming (LP): A mathematical method used to find the best outcome (e.g., maximizing profit or minimizing cost) subject to constraints.

• Simplex Method: A popular algorithm for solving LP problems, especially when the number of variables exceeds two.

• Queuing Theory: A mathematical approach to the analysis of waiting lines or queues, used in service industries like telecommunications, healthcare, and banking.

• Game Theory: A mathematical framework used to analyze strategic interactions where the outcome depends on the decisions of multiple participants.

• Inventory Control Models: Techniques like Economic Order Quantity (EOQ) and Just-in-Time (JIT) to manage stock levels and minimize costs.

• Simulation: The use of computer models to replicate the operation of real-world systems, particularly when analytical solutions are not possible.

• Network Analysis: Techniques like PERT (Program Evaluation and Review Technique) and CPM (Critical Path Method) used in project management.

6. Applications of Operations Research

Operations Research has diverse applications across various industries. Some of the most common applications include:

• Manufacturing: Optimizing production schedules, reducing waste, and improving efficiency in manufacturing processes.

• Transportation and Logistics: Optimizing routes, minimizing transportation costs, and improving inventory management.

• Healthcare: Optimizing hospital resource allocation, patient scheduling, and supply chain management in the healthcare industry.

• Finance: Portfolio optimization, risk management, and asset allocation in financial markets.

• Telecommunications: Network design, traffic management, and capacity planning.

• Energy: Optimizing power generation and distribution, reducing costs, and minimizing environmental impact.

• Retail: Inventory management, customer demand forecasting, and store location analysis.

• Military and Defense: Strategy optimization, resource allocation, and logistics.

7. Limitations of Operations Research

Despite its wide-ranging applications, Operations Research also has certain limitations:

• Model Simplification: OR models rely on assumptions and simplifications, which may not always capture the complexity of real-world situations.

• Data Dependence: OR models are heavily dependent on accurate and reliable data. Inaccurate data can lead to suboptimal decisions.

• Time and Cost: Developing and solving OR models can be time-consuming and costly, especially for complex systems.

• Over-Reliance on Quantitative Analysis: OR focuses on quantitative methods and may overlook qualitative factors like human behavior and organizational culture.

• Implementation Challenges: Even after a model is developed, implementing its solution in the real world may be difficult due to organizational constraints or resistance to change.

• Limited Scope: OR techniques are not universally applicable to all types of problems, especially those that are poorly defined or do not have clear objectives.

Conclusion

Operations Research is a powerful tool for optimizing decision-making and solving complex problems across various sectors. Understanding its history, tools, techniques, and limitations is essential for successfully applying OR in real-world scenarios. While OR has proven to be a highly effective decision-making approach, it is essential to recognize that its effectiveness depends on the quality of data, model assumptions, and collaborative efforts between managers and OR specialists.

Unit 2: Linear Programming Problem (LPP) – Comprehensive Notes

1. Introduction to Linear Programming

Definition

Linear Programming (LP) is a mathematical technique used to determine the best possible outcome (such as maximum profit or minimum cost) in a given mathematical model with constraints. It involves decision-making in resource allocation where objectives and constraints are linear in nature.

Key Features of LP

• Objective Function: A mathematical expression that needs to be maximized or minimized (e.g., profit maximization or cost minimization).
• Decision Variables: Variables that represent choices available in a decision-making problem.
• Constraints: Restrictions imposed on the decision variables (e.g., resource limitations).
• Linearity: All relationships in the model (objective function and constraints) must be linear.
• Non-Negativity: Decision variables must be zero or positive, meaning negative values are not allowed.

Applications of Linear Programming

• Business and Economics: Profit maximization, cost minimization, production planning.
• Manufacturing: Resource allocation, production scheduling.
• Transportation and Logistics: Optimal route selection, supply chain optimization.
• Finance: Investment portfolio optimization.
• Healthcare: Hospital resource allocation, patient scheduling.

2. Linear Programming Problem Formulation

Formulating an LP problem involves:

1. Defining the Decision Variables – Identify variables that impact the objective function.
2. Formulating the Objective Function – Define a function to maximize or minimize.
3. Setting Constraints – Define restrictions on available resources or conditions.
4. Adding Non-Negativity Constraints – Ensure variables are non-negative.

General Form of an LP Model

(Such that)

Where:

• $Z$ is the objective function,
• $x_1, x_2, …, x_n$ are decision variables,
• $c_1, c_2, …, c_n$ are coefficients of the objective function,
• $a_{ij}$ are coefficients of constraints,
• $b_1, b_2, …, b_m$ are constraint limits.

3. Formulation with Different Types of Constraints

Types of Constraints in LP

1. Less than or equal to (≤) constraints: Represent resource limitations (e.g., labor, material, time).
2. Greater than or equal to (≥) constraints: Represent minimum requirements (e.g., production quotas).
3. Equality (=) constraints: Represent fixed requirements (e.g., specific production levels).

Example:
A company produces two products: P1 and P2. The objective is to maximize profit given the following conditions:

• Each unit of P1 takes 2 hours of labor and P2 takes 3 hours. Total available labor: 60 hours.
• Each unit of P1 requires 1 unit of material and P2 requires 2 units. Total available material: 40 units.
• Profit per unit: P1 = $30, P2 = $50.

LP Formulation:

(Such that)

4. Graphical Analysis of Linear Programming

Graphical methods are useful when solving LP problems with two decision variables.

Steps for Graphical Method

1. Plot the constraints on a 2D graph.
2. Identify the feasible region, the area where all constraints are satisfied.
3. Plot the objective function and move it parallel to find the optimal solution.
4. Find the optimal point at the extreme points (corner points) of the feasible region.

5. Graphical Linear Programming Solution

Example:
Solve the following LP problem using the graphical method:

(Such that)

Solution

1. Plot constraints: Convert inequalities to equalities and plot their lines.
2. Find intersection points: Solve equations for boundaries.
3. Identify the feasible region: The common area satisfying all constraints.
4. Evaluate the objective function: Calculate $Z$ at each corner point.
5. Choose the optimal solution: The point giving the highest $Z$.

6. Special Cases in LP

a) Multiple Optimal Solutions

Occurs when more than one solution gives the same optimal value.

b) Unbounded Solution

Occurs when the objective function can increase indefinitely without violating constraints.

c) Infeasible Solution

Occurs when there is no common feasible region satisfying all constraints.

7. Basics of Simplex Method

Why Simplex Method?

Graphical methods work only for two-variable problems. The Simplex Method is used for larger problems with multiple variables.

Steps of Simplex Method

1. Convert constraints to equalities using slack variables.
2. Set up the Simplex tableau.
3. Perform pivot operations to improve the objective function.
4. Continue iterating until an optimal solution is reached.

8. Simplex Method Computation

1. Identify the pivot column (most negative value in the objective row).
2. Identify the pivot row (smallest positive ratio of right-hand side to pivot column value).
3. Perform row operations to get 1 in the pivot column.
4. Iterate until no negative values remain in the objective row.

9. Simplex Method with More Than Two Variables

For problems with three or more variables, the Simplex Algorithm systematically finds an optimal solution without graphical representation.

10. Primal and Dual Problems

• Primal Problem: The original LP problem.
• Dual Problem: Derived from the primal problem by interchanging constraints and objective coefficients.
• Economic Interpretation: The dual values represent shadow prices, indicating how much the objective function will change with one additional unit of a constraint resource.

Conclusion

Linear Programming is a powerful optimization tool used in decision-making. It provides mathematical approaches for resource allocation, scheduling, and production planning. Understanding graphical methods, Simplex method, and special cases helps in solving real-world business and industrial problems efficiently.

Unit 3: Transportation and Assignment Problem

1. Introduction to Transportation and Assignment Problems

Transportation and assignment problems are special cases of Linear Programming that deal with resource allocation and cost minimization in logistics, supply chain, and workforce management.

Objectives:

• Transportation Problem: Minimize the cost of transporting goods from multiple sources (e.g., factories) to multiple destinations (e.g., warehouses or markets).
• Assignment Problem: Assign tasks, jobs, or workers to minimize costs or maximize efficiency.

2. Transportation Problem

Definition

A Transportation Problem (TP) is a type of LP that involves determining the optimal way to transport goods from multiple sources to multiple destinations while minimizing the total transportation cost.

Mathematical Formulation

Let:

• $x_{ij}$ = number of units transported from source $i$ to destination $j$
• $c_{ij}$ = transportation cost per unit from source $i$ to destination $j$
• $s_i$ = supply at source $i$
• $d_j$ = demand at destination $j$

Objective Function:

Subject to Constraints:

3. Solution Methods for Transportation Problem

(a) North-West Corner Method

• A basic and quick method for obtaining an initial feasible solution.
• Steps:
  1. Allocate as much as possible to the top-left (north-west) corner.
  2. Adjust supply and demand.
  3. Move to the next cell right (or down if exhausted) and repeat.

Advantages:

• Simple and easy to apply.
  Disadvantages:
• Does not guarantee an optimal solution.

(b) Least Cost Method

• Considers the least-cost cell first, ensuring lower transportation costs.
• Steps:
  1. Find the lowest cost cell and allocate as much as possible.
  2. Adjust supply and demand.
  3. Move to the next lowest-cost cell and repeat.

Advantages:

• Provides a better initial solution than the North-West Corner Method.
  Disadvantages:
• Still not necessarily optimal.

(c) Vogel’s Approximation Method (VAM)

• Provides a better initial solution than the Least Cost Method.
• Steps:
  1. Compute the penalty for each row and column (difference between the lowest and second-lowest cost).
  2. Select the row/column with the highest penalty and allocate to the lowest-cost cell.
  3. Adjust supply and demand, then repeat.

Advantages:

• Produces an initial solution closer to optimal.
  Disadvantages:
• Slightly more complex than other methods.

4. Optimality Check and Improvement: Modified Distribution Method (MODI Method)

• After obtaining an initial solution, we check for optimality using the MODI (Modified Distribution) Method.
• If the solution is not optimal, MODI adjusts allocations to reduce total transportation costs.

5. Special Cases in Transportation Problem

(a) Degeneracy in Transportation Problem

• Occurs when the number of occupied cells is less than ($m + n – 1$).
• Handled by adding small values (ε) to avoid computational errors.

(b) Unbalanced Transportation Problem

• When total supply ≠ total demand.
• A dummy row or column is added to balance the problem.

(c) Profit Maximization in Transportation Problem

• Instead of minimizing cost, the goal is to maximize profit by choosing routes that yield the highest profit margin.

(d) Transshipment Problems

• Involves an intermediate transfer point between source and destination (e.g., warehouses).
• Handled by adding intermediate nodes to the LP formulation.

6. Assignment Problem

Definition

The Assignment Problem is a special case of LP where tasks (jobs) are assigned to agents (workers/machines) in a way that minimizes total cost or maximizes efficiency.

Mathematical Formulation

Let:

• $x_{ij}$ = 1 if task $i$ is assigned to worker $j$, else 0.
• $c_{ij}$ = cost of assigning task $i$ to worker $j$.

Objective Function:

Subject to:

7. Solution Methods for Assignment Problem

Hungarian Method (Short-Cut Method)

• Optimal method for solving assignment problems.
• Steps:
  1. Subtract row minimum from each row.
  2. Subtract column minimum from each column.
  3. Cover all zeros using minimum number of lines.
  4. If number of lines < $n$, adjust matrix and repeat.

Advantages:

• Guarantees optimal solution.

8. Special Cases in Assignment Problems

(a) Unbalanced Assignment Problem

• When number of tasks ≠ number of workers.
• A dummy row or column is added.

(b) Infeasible Assignment Problem

• Some tasks cannot be assigned to specific workers.
• Represented by high-cost values (M) in the cost matrix.

(c) Maximization in Assignment Problem

• Instead of minimizing cost, we maximize efficiency or profit by converting maximization to minimization using: $$C’_{ij} = \text{Max cost} – C_{ij}$$

(d) Crew Assignment Problem

• A specific assignment problem where workers (crews) are assigned to shifts/jobs based on efficiency.

9. Comparison of Transportation and Assignment Problems

10. Conclusion

• Transportation and Assignment Problems are crucial in logistics, manufacturing, and operations management.
• Transportation Problem helps in efficient distribution of goods.
• Assignment Problem ensures optimal task allocation.
• Methods like VAM, MODI, and Hungarian Method provide optimal solutions.
• Understanding these techniques enhances decision-making and cost efficiency in real-world applications.

Unit 4: Queuing Theory

1. Introduction to Queuing Theory

Queuing theory is a mathematical study of waiting lines or queues. It helps in analyzing and optimizing systems where resources (like servers, counters, or machines) provide services to customers (or jobs, requests, or packets in a network).

Objectives of Queuing Theory:

1. Minimize customer waiting time.
2. Optimize resource utilization.
3. Improve service efficiency.
4. Reduce costs by balancing service capacity with demand.

Applications of Queuing Theory:

• Business and Service Industry: Customer service desks, call centers.
• Healthcare: Patient waiting times in hospitals.
• Manufacturing: Production lines, assembly operations.
• Telecommunications: Internet traffic, data packet transmission.
• Transport & Logistics: Airport check-in, toll booths, ride-sharing.

2. Elements of Queuing Theory

A queuing system consists of six key elements:

1. Arrival Process (Input Process):

   • Describes how customers (or jobs) arrive at the system.
   • Can follow Poisson Distribution (random arrivals).
   • Arrival rate $\lambda$ (lambda) = Average number of arrivals per time unit.

2. Service Mechanism (Servers):

   • The process by which customers are served.
   • Can be single-server or multi-server.
   • Service rate $\mu$ (mu) = Average number of customers served per time unit.

3. Queue Discipline (Service Policy):

   • The rule for selecting customers from the queue.
   • Common types:
     • First Come First Serve (FCFS)
     • Last In First Out (LIFO)
     • Shortest Job First (SJF)
     • Priority Queuing

4. Queue Length (Capacity of System):

   • Finite queue: Only a limited number of customers can wait.
   • Infinite queue: No restriction on waiting line size.

5. System Capacity:

   • The maximum number of customers allowed in the system (queue + service).
   • Can be limited or unlimited.

6. Customer Behavior:

   • Balking: Customer leaves without entering the queue.
   • Reneging: Customer leaves after waiting for some time.
   • Jockeying: Customer switches between queues.

3. Kendall’s Notation

Queuing models are represented using Kendall’s notation, written as:

Where:

• $A$ = Arrival pattern (Poisson, Exponential, Deterministic).
• $B$ = Service pattern (Exponential, General, Deterministic).
• $C$ = Number of servers.
• $D$ = System capacity (finite or infinite).
• $E$ = Queue discipline (FCFS, LIFO, Priority).
• $F$ = Customer population (finite or infinite).

Common Kendall’s Notation Models:

1. M/M/1 → Poisson arrivals, Exponential service, 1 server.
2. M/M/c → Poisson arrivals, Exponential service, c servers.
3. M/G/1 → Poisson arrivals, General service time, 1 server.
4. G/G/c → General arrival, General service time, c servers.

4. Operating Characteristics of a Queuing System

Several key performance measures describe the efficiency of a queuing system:

1. $L_q$ = Average number of customers in the queue

   • Expected number of customers waiting in line.

2. $L_s$ = Average number of customers in the system

   • Includes both waiting and being served.

3. $W_q$ = Average waiting time in queue

   • Time a customer spends waiting before service starts.

4. $W_s$ = Average waiting time in system

   • Time a customer spends in the entire system (waiting + service).

5. $\rho$ = Server utilization factor

   • Formula: $\rho = \frac{\lambda}{c \mu}$
   • Determines how busy the servers are.

6. $P_0$ = Probability of zero customers in the system

   • Probability that the system is empty.

7. $P_n$ = Probability of exactly $n$ customers in the system

   • Used to calculate congestion levels.

Little’s Law

A fundamental relationship in queuing theory:

Where:

• $L_s$ = Average number of customers in system.
• $W_s$ = Average time spent in system.
• $L_q$ = Average number of customers in queue.
• $W_q$ = Average time spent in queue.
• $\lambda$ = Arrival rate.

5. Classification of Queuing Models

Queuing models can be classified based on:

(A) Number of Servers

1. Single-Server Model (M/M/1)

   • One server handles all customers.
   • Example: Single ATM machine.

2. Multi-Server Model (M/M/c)

   • Multiple servers provide service simultaneously.
   • Example: Bank counters, call centers.

(B) Arrival and Service Patterns

1. Markovian Queues (M/M/1, M/M/c)

   • Poisson arrivals and exponential service times.

2. General Queues (M/G/1, G/M/1)

   • Service times follow a general distribution.

(C) Queue Capacity

1. Finite Queue Model (M/M/1/N)

   • Queue has a maximum limit.
   • Example: Limited parking space.

2. Infinite Queue Model (M/M/1)

   • No restriction on the number of customers waiting.

(D) Customer Behavior Models

1. Balking Model

   • Customers decide not to join if the queue is long.

2. Reneging Model

   • Customers leave after waiting too long.

3. Jockeying Model

   • Customers switch between queues.

6. Practical Applications of Queuing Theory

1. Banking & Finance

   • Reduce waiting time in teller queues.
   • Optimize ATM placements.

2. Retail & Supermarkets

   • Improve checkout efficiency.
   • Self-service vs. cashier queues.

3. Call Centers

   • Minimize customer wait time.
   • Optimize number of call agents.

4. Manufacturing & Production

   • Reduce machine idle time.
   • Optimize assembly lines.

5. Healthcare & Hospitals

   • Reduce patient waiting time.
   • Improve doctor-to-patient ratios.

7. Conclusion

• Queuing Theory helps analyze service systems and optimize resource allocation.
• Kendall’s notation classifies different queuing models.
• Performance measures like $L_q, W_s, \rho$ help evaluate system efficiency.
• Little’s Law provides a fundamental relation between customers and wait times.
• Practical applications include banking, call centers, manufacturing, and healthcare.

By applying queuing models effectively, organizations can improve efficiency, reduce costs, and enhance customer satisfaction.

Unit 5: Inventory Control

1. Introduction to Inventory Control

Inventory control refers to the management and regulation of stock items to ensure that the right amount of inventory is available at the right time to meet demand while minimizing costs. It is a crucial part of supply chain management that helps businesses maintain smooth operations and customer satisfaction.

Objectives of Inventory Control:

• To ensure a continuous supply of materials.
• To prevent overstocking and understocking.
• To reduce storage costs and wastage.
• To optimize ordering policies and reduce procurement costs.

Importance of Inventory Control:

1. Reduces holding costs and wastage.
2. Helps maintain a steady production process.
3. Prevents stockouts and overstocking.
4. Improves cash flow and working capital.
5. Enhances customer satisfaction by ensuring product availability.

2. Inventory Classification

Inventory is classified based on various factors such as usage, value, and demand patterns.

(A) Based on Usage:

1. Raw Materials:
   • Basic materials used in production (e.g., steel, wood, plastic).
2. Work-in-Progress (WIP):
   • Semi-finished goods undergoing production.
3. Finished Goods:
   • Completed products ready for sale.
4. Maintenance, Repair, and Operations (MRO):
   • Items used for maintenance and support (e.g., lubricants, spare parts).

(B) Based on Demand Patterns:

1. Independent Demand Inventory:
   • Demand is not dependent on other items (e.g., finished goods).
2. Dependent Demand Inventory:
   • Demand depends on the demand for other items (e.g., raw materials for production).

(C) Based on Cost and Value:

1. High-Value Items (A-Class)
2. Medium-Value Items (B-Class)
3. Low-Value Items (C-Class)
   (Discussed in ABC Analysis below)

3. Different Costs Associated with Inventory

Inventory management involves various costs, which can be categorized into:

(A) Ordering Costs

• Costs associated with placing and receiving orders.
• Includes administrative costs, transportation, and procurement costs.

(B) Holding (Carrying) Costs

• Costs incurred for storing unsold inventory.
• Includes warehousing, insurance, depreciation, and obsolescence costs.

(C) Shortage Costs (Stockout Costs)

• Costs arising from running out of stock.
• Includes lost sales, backorder processing, and customer dissatisfaction.

(D) Setup Costs

• Costs incurred when preparing machinery for production.
• Includes labor, tool replacement, and downtime costs.

(E) Purchase Costs

• The actual cost of acquiring inventory.

Total Inventory Cost

4. Economic Order Quantity (EOQ)

EOQ is the optimal order quantity that minimizes total inventory costs (ordering and holding costs).

EOQ Formula

Where:

• D = Annual demand
• S = Ordering cost per order
• H = Holding cost per unit per year

Assumptions of EOQ Model:

1. Demand is constant and known.
2. Ordering cost and holding cost are constant.
3. No stockouts or shortages occur.
4. Lead time is fixed.

Benefits of EOQ:

• Minimizes total inventory cost.
• Prevents excessive stock accumulation.
• Ensures efficient use of storage space.

5. Inventory Models with Deterministic Demand

Deterministic inventory models assume that demand is known and constant.

(A) Basic EOQ Model

• Used when demand is steady and predictable.

(B) EOQ with Discounts

• Used when suppliers offer price discounts for bulk purchases.

(C) Production Order Quantity Model

• Used when inventory is produced rather than ordered.

(D) Inventory Replenishment Model

• Used when inventory is replenished gradually over time rather than instantly.

6. ABC Analysis

ABC analysis is a categorization technique used to classify inventory based on value and importance.

Categories in ABC Analysis:

1. A-Class Items (High-Value, Low-Quantity)

   • Account for 70-80% of total inventory value.
   • Require strict control and monitoring.
   • Example: Expensive raw materials.

2. B-Class Items (Moderate-Value, Moderate-Quantity)

   • Account for 15-25% of total inventory value.
   • Require moderate control.
   • Example: Mid-range electronic components.

3. C-Class Items (Low-Value, High-Quantity)

   • Account for 5-10% of total inventory value.
   • Require minimal control and monitoring.
   • Example: Office supplies, screws, and bolts.

Steps in ABC Analysis:

1. List all inventory items with annual consumption value.
2. Rank items from highest to lowest value.
3. Divide items into A, B, and C categories based on cumulative percentage.

Benefits of ABC Analysis:

• Helps focus on high-value items.
• Reduces excessive spending on low-value items.
• Improves overall inventory efficiency.

7. Conclusion

• Inventory control is essential for maintaining a balance between demand and supply while minimizing costs.
• Different cost components influence inventory decisions, including ordering, holding, and shortage costs.
• The EOQ model provides an optimal ordering strategy for minimizing costs.
• Deterministic inventory models help manage predictable demand efficiently.
• ABC Analysis helps prioritize inventory management based on value and usage.

By applying inventory control techniques, businesses can reduce waste, optimize storage, and improve overall supply chain efficiency. 🚀

Unit 6: Replacement Theory

1. Introduction to Replacement Theory

Replacement Theory is a branch of Operations Research that deals with the problem of replacing equipment, machinery, or assets that deteriorate over time. The objective is to determine the optimal time to replace an asset to minimize total costs (maintenance, operation, and replacement costs) while ensuring efficiency and productivity.

Importance of Replacement Theory:

• Helps in minimizing costs and maximizing efficiency.
• Ensures that equipment operates at optimal performance.
• Reduces unexpected failures and downtime.
• Balances the trade-off between maintenance and replacement.

Types of Replacement Situations:

1. Replacement due to Deterioration (e.g., machines, tools, vehicles).
2. Replacement due to Obsolescence (e.g., outdated computers, software).
3. Replacement due to Breakdown (e.g., sudden failure of a machine).

2. Replacement of Capital Equipment Which Depreciates Over Time

(A) Reasons for Depreciation of Capital Equipment:

1. Physical Deterioration – Wear and tear over time.
2. Technological Obsolescence – New, more efficient models available.
3. Economic Factors – Rising operating and maintenance costs.

(B) Cost Components Involved in Replacement Decisions:

• Initial cost of the asset.
• Operating and maintenance costs over time.
• Resale or salvage value at the end of its life.
• Interest rates affecting investment decisions.

(C) Methods to Determine Optimal Replacement Time:

1. Average Cost Method:

   • Calculate the total cost per year (including purchase, maintenance, and salvage value).
   • Replace when the average annual cost starts increasing.

2. Present Worth Method:

   • Compares the present worth of keeping the existing equipment vs. purchasing a new one.
   • Replacement occurs when a new option has a lower present worth of total costs.

3. Equivalent Annual Cost (EAC) Method:

   • Converts total costs into equal annual payments.
   • Replace when a new alternative has a lower EAC.

3. Replacement by Alternative Equipment

Sometimes, instead of replacing an asset with a similar model, a completely different type of equipment is chosen.

Scenarios for Alternative Equipment Replacement:

1. Technological Advancements:
   • Newer models are more efficient, faster, and cost-effective.
2. Change in Business Needs:
   • Demand increases, requiring higher-capacity machines.
3. Environmental Regulations:
   • Switching to energy-efficient or eco-friendly equipment.
4. Market Trends:
   • Adopting new technologies to remain competitive.

Key Considerations for Choosing Alternative Equipment:

• Initial and operational cost differences.
• Return on Investment (ROI) for the new equipment.
• Compatibility with existing infrastructure.
• Ease of transition and training for employees.

4. Group and Individual Replacement Policy

(A) Individual Replacement Policy

• Each item is replaced independently when it fails or becomes uneconomical.
• Common for high-value assets like machines, vehicles, or computers.

Advantages:

✔ Suitable when failures are random.
✔ Avoids unnecessary replacements.
✔ Cost-effective for expensive, durable items.

Disadvantages:

✘ Higher administrative effort in tracking individual replacements.
✘ May cause unplanned downtime if failures occur suddenly.

(B) Group Replacement Policy

• Items are replaced together at fixed intervals, regardless of individual failure.
• Common for low-cost, high-usage items like batteries, bulbs, and electronic components.

Advantages:

✔ Reduces maintenance effort and tracking costs.
✔ Ensures consistent performance by avoiding random failures.
✔ More effective when failure rates increase over time.

Disadvantages:

✘ Some items are replaced before they fail, leading to higher initial costs.
✘ Not suitable for high-value equipment.

Comparison of Individual vs. Group Replacement

5. Practical Applications of Replacement Theory

• Manufacturing Industries: Machine replacement to reduce breakdowns.
• Transport Companies: Fleet replacement to reduce fuel and maintenance costs.
• IT Industry: Upgrading servers, laptops, and software for efficiency.
• Energy Sector: Replacing old power plants with modern, eco-friendly systems.
• Healthcare: Upgrading medical equipment for better diagnosis and treatment.

6. Conclusion

Replacement Theory helps businesses decide when and how to replace assets to minimize costs and maximize efficiency. The decision depends on factors like deterioration, obsolescence, operational costs, and business needs. By implementing the right replacement policy (individual or group), organizations can optimize resources and improve performance. 🚀

Unit 7: Game Theory

1. Introduction to Game Theory

Game Theory is a mathematical framework for analyzing strategic interactions where the outcome for one participant depends not only on their own decisions but also on the decisions of others. It is widely used in economics, business, politics, and military strategy to model competitive and cooperative scenarios.

Key Elements of Game Theory:

1. Players – Individuals or entities making decisions.
2. Strategies – A set of possible actions available to each player.
3. Payoffs – The outcomes or rewards received based on strategy choices.
4. Rules of the Game – Constraints and conditions governing the interaction.
5. Rationality – Assumption that each player aims to maximize their own benefit.

2. Characteristics of Game Theory

Game Theory models different types of strategic interactions, each with distinct features:

1. Interdependence – The outcome for one player depends on the actions of others.
2. Multiple Players – Games can involve two or more players.
3. Strategic Decision-Making – Players choose strategies based on expected reactions of opponents.
4. Payoff Matrix Representation – Used to visualize possible outcomes.
5. Assumption of Rationality – Players make logical and optimal choices.
6. Perfect vs. Imperfect Information – Some games provide full knowledge of opponent moves, while others do not.
7. Zero-Sum vs. Non-Zero-Sum Games – In zero-sum games, one player’s gain is another’s loss, while in non-zero-sum games, both players can benefit.

3. Two-Person, Zero-Sum Games

A two-person, zero-sum game is a competitive scenario where one player’s gain is exactly equal to the other player’s loss.

Mathematical Representation:

• A payoff matrix represents the outcomes.
• If Player A gains +5, then Player B loses -5.
• The total sum of payoffs is zero.

Example of a Zero-Sum Game:

Applications:

• Military – War strategies where one country’s gain is another’s loss.
• Sports – One team’s win is the opponent’s loss.
• Stock Market Trading – Profit for one trader means loss for another.

4. Pure Strategy in Game Theory

A pure strategy is when a player selects a single best response and follows it consistently, regardless of the opponent’s actions.

Characteristics:

1. Each player chooses one strategy with certainty.
2. There is no randomness in decision-making.
3. The game has a deterministic outcome.
4. Works well if there is a dominant strategy (one that is always the best).

Example:

• In Rock-Paper-Scissors, choosing Rock every time is a pure strategy.
• In Chess, always opening with the same move (e.g., King’s Pawn) is a pure strategy.

5. Dominance Theory

Dominance Theory helps eliminate inferior strategies in decision-making. A strategy is dominant if it always provides a better or equal outcome compared to other strategies.

Types of Dominance:

1. Strictly Dominant Strategy: Always better than any other option.
2. Weakly Dominant Strategy: Sometimes better, but never worse.
3. Dominated Strategy: Always worse and should be eliminated.

Example:

• Strategy X dominates Strategy Y because 5 > 2 and 3 > 1.
• Player A should always choose X.

Application:

Used in business pricing models, advertising strategies, and political campaigns to eliminate weak options.

6. Mixed Strategies (2×2, m×2)

A mixed strategy occurs when a player randomizes their choices instead of always choosing one strategy. This is useful when there is no pure dominant strategy.

Characteristics of Mixed Strategies:

1. Players assign probabilities to each strategy.
2. Ensures unpredictability in games.
3. Used when pure strategies lead to weak outcomes.

Example (2×2 Mixed Strategy):

• Rock-Paper-Scissors: A player might choose Rock 50%, Paper 25%, Scissors 25% to remain unpredictable.

Example (m×2 Mixed Strategy):

• In business competition, a company may choose advertising on TV 70% and online 30% to balance effectiveness and cost.

Applications of Mixed Strategy:

• Auction Theory – Bidders randomize their bids to gain an advantage.
• Poker and Gambling – Unpredictable play prevents opponents from predicting moves.
• Stock Market Trading – Investors diversify portfolios to reduce risk.

7. Algebraic and Graphical Methods in Game Theory

Algebraic Method:

This method is used to find equilibrium points (where neither player benefits from changing their strategy).

Steps:

1. Formulate the payoff matrix.
2. Identify dominant strategies (if any).
3. Solve using equations to determine probabilities for mixed strategies.
4. Apply minimax theorem (minimizing maximum losses).

Example:
If two strategies are used with probabilities p and (1-p), we solve for p where expected payoffs are equal.

Graphical Method (for 2×2 Games):

The graphical method is useful when one player has two strategies and the other has multiple strategies.

Steps:

1. Plot the payoffs of each strategy on a graph.
2. Identify the intersection point, which represents an equilibrium solution.
3. The optimal mixed strategy is found at the point of intersection.

Example:

• Used in pricing strategies where companies adjust prices based on competitor actions.
• Applied in military resource allocation for finding the best attack-defense strategy.

8. Conclusion

Game Theory is a powerful tool for decision-making in competitive environments. It helps in understanding conflicts, negotiations, and optimal choices in various fields like business, economics, military, and politics.

Key Takeaways:

✔ Pure strategies are straightforward but may not always work.
✔ Mixed strategies provide unpredictability.
✔ Dominance theory eliminates weak strategies.
✔ Algebraic and graphical methods help determine equilibrium points.

By mastering these concepts, individuals and organizations can make strategic, informed decisions in competitive scenarios. 🚀

Below is the syllabus in a systematic format for the Operational Research (CAOR451) course:

Course Title: Operational Research

Course Code: CAOR451
Year/Semester: IV/VIII
Class Load: 4 Hrs./Week
(Theory: 3 Hrs, Tutorial: 1 Hrs)
Credits: 3

Course Description:

Operational Research (OR) is the scientific study of decision-making using mathematical models and optimization techniques. It helps in improving complex systems through quantitative methods, either deterministic or stochastic, depending on the system’s nature. The course covers powerful modeling and solution techniques like linear programming, transportation, assignment problems, inventory control, replacement theory, and game theory. Real-world decision-making challenges in business management are addressed with analytical techniques and software tools.

Course Objectives:

• Provide a broad orientation in the field of optimization.
• Focus on basic theory and methods for continuous and discrete optimization in finite dimensions.
• Explore the use of OR techniques to solve practical optimization problems for business decision-making.

Unit-wise Breakdown:

Unit 1: Introduction to Operations Research (5 Hrs)

• Introduction to Operations Research
  • Definition and Scope.
• History of Operations Research
  • Origin and development of OR over time.
• Stages of Development of Operations Research
  • Evolution of techniques and applications in business and industry.
• Relationship between Manager and OR Specialist
  • Role of OR professionals in managerial decision-making.
• OR Tools and Techniques
  • Key methods like Linear Programming, Simulation, Game Theory, etc.
• Applications of Operations Research
  • Examples in industry, logistics, finance, etc.
• Limitations of Operations Research
  • Constraints in real-world applications and model simplifications.

Unit 2: Linear Programming Problem (10 Hrs)

• Introduction to Linear Programming (LP)
  • Basic concepts and significance in optimization.
• LP Problem Formulation
  • Steps to model real-world problems using linear equations.
• Formulation with Different Types of Constraints
  • Constraints like equality, inequality, etc.
• Graphical Analysis of Linear Programming
  • Use of graphical methods to solve 2-variable problems.
• Graphical LP Solution
  • Steps for obtaining optimal solutions graphically.
• Multiple Optimal Solutions
  • Conditions where multiple solutions exist.
• Unbounded Solution
  • Conditions leading to an unbounded objective function.
• Infeasible Solution
  • Understanding situations where no feasible solution exists.
• Basics of Simplex Method
  • Introduction to the Simplex algorithm for solving LP problems.
• Simplex Method Computation
  • Step-by-step method to compute the optimal solution using Simplex.
• Simplex Method with More Than Two Variables
  • Handling LP problems with more than two variables.
• Primal and Dual Problems
  • The relationship between primal and dual linear programming problems.
• Economic Interpretation
  • Interpretation of the solution in economic terms.

Unit 3: Transportation and Assignment Problem (8 Hrs)

• Transportation Problem
  • Definition and linear formulation of transportation problems.
• Solution Methods:
  • North West Corner Method
  • Least Cost Method
  • Vogel’s Approximation Method
• Degeneracy in Transportation
  • Handling degenerate solutions in transportation problems.
• Modified Distribution Method
  • A more efficient method for solving transportation problems.
• Unbalanced Problems and Profit Maximization
  • Solutions to unbalanced transportation problems and profit-maximization scenarios.
• Transshipment Problems
  • Solving transportation problems with intermediate transshipment nodes.
• Assignment Problem
  • Structure and formulation of assignment problems.
• Solution Methods:
  • Hungarian Method (Short-Cut Method)
  • Unbalanced Assignment Problem
  • Infeasible Assignment Problem
• Maximization in an Assignment Problem
  • Approaches for solving maximization versions of assignment problems.
• Crew Assignment Problem
  • Application of assignment problems in the scheduling of crew members.

Unit 4: Queuing Theory (6 Hrs)

• Basis of Queuing Theory
  • Introduction to queuing models and their relevance to system design.
• Elements of Queuing Theory
  • Arrival rate, service rate, queue length, and number of servers.
• Kendall’s Notation
  • Understanding the standard notation for queuing systems (A/B/C).
• Operating Characteristics of a Queuing System
  • Key performance indicators like waiting time, queue length, and service utilization.
• Classification of Queuing Models
  • Types of queuing models (e.g., M/M/1, M/G/1, etc.) and their applications.

Unit 5: Inventory Control (6 Hrs)

• Inventory Classification
  • Types of inventory: Raw materials, work-in-progress, finished goods.
• Different Costs Associated with Inventory
  • Holding costs, ordering costs, shortage costs.
• Economic Order Quantity (EOQ)
  • Model for determining the optimal order quantity that minimizes total inventory costs.
• Inventory Models with Deterministic Demands
  • Models assuming constant demand rate over time.
• ABC Analysis
  • Classification of items based on their importance and value in inventory.

Unit 6: Replacement Theory (6 Hrs)

• Introduction to Replacement Theory
  • The theory behind replacing outdated or inefficient assets.
• Replacement of Capital Equipment
  • Decision-making process for replacing capital equipment that depreciates over time.
• Replacement by Alternative Equipment
  • Cost-benefit analysis of replacing equipment with more efficient alternatives.
• Group and Individual Replacement Policy
  • Approaches to replacing equipment for a group of assets versus individually.

Unit 7: Game Theory (7 Hrs)

• Introduction to Game Theory
  • Basic concepts of strategic decision-making in competitive environments.
• Characteristics of Game Theory
  • Interaction between rational decision-makers with conflicting interests.
• Two-Person, Zero-Sum Games
  • Games in which one player’s gain equals the other player’s loss.
• Pure Strategy
  • A strategy where a player consistently chooses one option.
• Dominance Theory
  • Dominated strategies and how to eliminate them.
• Mixed Strategies
  • Randomized strategies and their applications in decision-making.
• (2×2, m×2) Game Models
  • Analysis of simple games using matrices.
• Algebraic and Graphical Methods
  • Methods to solve game theory problems algebraically and graphically.

Teaching Methods:

• Lectures
• Presentations
• Group Work & Case Studies
• Guest Lectures
• Research and Project Work
• Assignments (Theoretical and Practical)

References/Suggested Readings:

1. Hillier, F.S. & Lieberman, G.J. (1995) – Introduction to Operations Research, 7th edition. The McGraw-Hill Companies, Inc.
2. Natarajan, A. M.; Balasubramani, P. & Tamilarasi, A. (2007) – Operations Research. Pearson Education Inc.
3. Sharma, J.K. (2009) – Operational Research: Theory and Application. Macmillan Publishers India Ltd.
4. Taha, H.A. (2017) – Operations Research: An Introduction, 10th edition. Pearson Education, Inc.
5. Wagner, H. N. (2003) – Operations Research. Prentice Hall.
6. Winston, L.W. (2004) – Operations Research: Applications and Algorithms, 4th edition. Indian University.

Evaluation:

The course evaluation will be based on:

• Theory Exams
• Assignments
• Project Work
• Case Studies
• Class Participation

Tribhuvan University

Faculty of Humanities & Social Sciences
OFFICE OF THE DEAN
2021

Bachelor in Computer Applications
Course Title: Operational Research
Code No.: CAQ451
Semester: VII

Full Marks: 60
Pass Marks: 24
Time: 3 hours

Group B

Attempt any SIX questions.
[6×5 = 30]

1. What is Operational Research? Explain the general methods for solving OR models. [1+4]

2. What do you mean by the mathematical formulation of LPP? A firm manufactures three products: A, B, and C. The time required to manufacture product A is twice that for B and thrice that for C. The products must be produced in a ratio of 3:4:5. The relevant data is given in the table below. If all raw materials are used to manufacture product A, 1600 units can be produced. The demand for the products is at least 300, 250, and 200 units for A, B, and C, respectively. The profit per unit is Rs. 50, Rs. 40, and Rs. 70 for A, B, and C. Formulate this as a linear programming problem. [1+4]

   Raw Material Requirements per Unit of Product (kg):

   Raw Material	A	B	C	Total Availability (kg)
   P	6	5	9	5000
   Q	4	7	3	6000

3. Write an algorithm to maximize the solution of LPP using the Simplex method.

4. Find the optimal solution for the following transportation problem using any method.

   1	2	1	4	30
   3	3	2	1	50
   4	2	5	9	20
   20	40	30	9	100

5. Write the Hungarian algorithm to solve the assignment problem.

6. Classify queueing models with examples. [2.5+2.5]

7. Write short notes on any two of the following:
   a) EOQ
   b) Kendall’s Notation for Queueing Model
   c) Duality Theorem

1. What is Operational Research? Explain the general methods for solving OR models. [1+4]

Definition of Operational Research (OR):

Operational Research (OR) is a scientific approach to decision-making that involves mathematical modeling, statistical analysis, and optimization techniques to solve complex problems in business, industry, government, and other fields. It aims to improve efficiency, maximize profits, and minimize costs.

General Methods for Solving OR Models:

1. Formulation of the Problem – Clearly define the objective, constraints, and variables.
2. Mathematical Modeling – Represent the problem using mathematical equations and inequalities.
3. Solution Techniques – Apply appropriate methods such as Linear Programming, Simplex Method, Dynamic Programming, Game Theory, etc.
4. Validation & Testing – Verify the model by comparing it with real-world scenarios.
5. Implementation & Monitoring – Apply the solution and make necessary adjustments based on feedback.

2. Mathematical Formulation of LPP (Linear Programming Problem) [1+4]

Given Data:

• Three products: A, B, and C
• Production Ratio: A:B:C = 3:4:5
• Raw Material Constraints:

• Maximum production of A using total raw material = 1600 units
• Demand constraints:
  • A ≥ 300, B ≥ 250, C ≥ 200
• Profit per unit:
  • A = Rs. 50, B = Rs. 40, C = Rs. 70

Formulating the LPP:

Decision Variables:
Let x₁, x₂, and x₃ be the number of units produced of products A, B, and C respectively.

Objective Function (Maximize Profit):

Constraints:

1. Production Ratio Constraint:

Let $x₁ = 3k$, $x₂ = 4k$, $x₃ = 5k$.

2. Raw Material Constraints:

For P:

For Q:

3. Demand Constraints:

4. Non-Negativity Constraints:

This forms the Linear Programming Problem (LPP).

3. Algorithm to Maximize the Solution of LPP Using the Simplex Method

1. Formulate the Objective Function (Maximize/Minimize Z).
2. Convert Constraints into Standard Form by introducing slack variables.
3. Construct the Initial Simplex Table.
4. Identify the Pivot Column (largest positive coefficient in the objective row).
5. Identify the Pivot Row (smallest positive ratio of RHS to pivot column).
6. Perform Row Operations to make the pivot element 1 and other values 0 in the pivot column.
7. Iterate Until Optimality is Reached (when there are no positive values in the objective row).
8. Extract the Final Solution.

4. Solving the Given Transportation Problem

Cost Matrix:

Steps to Solve Using the Least Cost Method:

1. Identify the lowest cost cell and allocate as much supply as possible.
2. Adjust the supply and demand.
3. Repeat until all supply and demand are allocated.
4. Calculate the total transportation cost.

(Solution will depend on computation steps)

5. Hungarian Algorithm to Solve the Assignment Problem

Steps of Hungarian Method:

1. Subtract row minima from each row.
2. Subtract column minima from each column.
3. Cover all zeros using the minimum number of horizontal/vertical lines.
4. If the number of lines equals the number of rows/columns, optimal assignment is found. If not, adjust the table by:
   • Finding the smallest uncovered element.
   • Subtracting it from uncovered elements and adding it to double-covered elements.
5. Repeat the process until an optimal assignment is reached.

6. Classification of Queueing Models with Examples [2.5+2.5]

Types of Queueing Models:

1. Single-Server Queue (M/M/1) – A single server with Poisson arrival and exponential service (e.g., ATM line).
2. Multi-Server Queue (M/M/c) – Multiple servers handling requests (e.g., bank counters).
3. Finite Queue Capacity (M/M/1/K) – Limited space for waiting customers (e.g., parking lot).
4. Priority Queueing Model – Customers served based on priority (e.g., emergency room).

7. Short Notes on Any Two of the Following:

a) Economic Order Quantity (EOQ):

EOQ is the optimal order quantity that minimizes the total cost of inventory, including ordering and holding costs.

Formula:

where:

• $D$ = Demand
• $S$ = Ordering Cost
• $H$ = Holding Cost

b) Kendall’s Notation for Queueing Model:

Kendall’s notation describes queueing models as A/B/C/D/E, where:

• A = Arrival process (M = Poisson, D = Deterministic).
• B = Service time distribution (M = Exponential, G = General).
• C = Number of servers.
• D = System capacity.
• E = Queue discipline (FIFO, LIFO, etc.).

Example: M/M/1 – Poisson arrivals, exponential service, single server.

c) Duality Theorem in Linear Programming:

The Duality Theorem states that every linear programming problem (Primal) has a corresponding Dual problem, and the optimal solutions of both problems yield the same objective function value.

Example: If we minimize cost in the primal, the dual will maximize profit under the same constraints.