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Semester 2: Applied Operations Research
Introduction: Overview of operations research – Origin – Nature, scope & characteristics of OR – Models in OR – Application of operations research in functional areas of management
Introduction to Operations Research
Overview of Operations Research
Operations Research is a discipline that uses advanced analytical methods to help make better decisions. It originated during World War II, where it was used to optimize military logistics and strategy.
Origin of Operations Research
The roots of Operations Research can be traced back to military operations during the 1940s. Researchers utilized mathematical modeling and statistical analysis to solve complex operational problems.
Nature, Scope, & Characteristics of Operations Research
Operations Research is characterized by its interdisciplinary nature, incorporating elements from mathematics, statistics, engineering, and economics. Its scope includes optimization, simulation, and decision analysis.
Models in Operations Research
Operations Research employs various models including deterministic and stochastic models, linear programming, and non-linear programming, to represent and solve real-world problems effectively.
Applications of Operations Research in Functional Areas of Management
Operations Research can be applied across various management functions such as supply chain management, production planning, financial management, and marketing strategies. Its techniques enhance efficiency and effectiveness in decision-making.
Linear Programming Problem: Linear programming problem model – Formulation – Maximization & Minimization problem – Graphical method – Simplex method – Artificial variable – Primal & Dual.
Linear Programming Problem
Formulation
Linear programming involves defining a problem in terms of mathematical equations involving variables that are to be maximized or minimized.
Maximization & Minimization Problems
Maximization problems seek to achieve the highest value of an objective function while minimization problems aim to reduce the value.
Graphical Method
The graphical method is a visual approach to solving linear programming problems with two variables by plotting constraints and identifying feasible regions.
Simplex Method
The simplex method is an algorithm used for larger linear programming problems that can handle multiple variables and constraints efficiently.
Artificial Variable
Artificial variables are added to facilitate the use of the simplex method when the initial solution is not obvious or feasible.
Primal & Dual
The primal refers to the original linear programming problem while the dual represents a related problem that provides insights and solutions to the primal.
Transportation problem: Basic Solution – North / West corner Solution, LCM, VAM, Matrices method – Optimal Solution – Stepping stone method – Vogel‘s approximation method – Modi method – Degeneracy – Imbalance matrix. Assignment model: Hungarian method – Traveling salesmen problem.
Transportation Problem
Basic Solution
The basic solution in a transportation problem is an initial feasible solution obtained through various methods. It establishes a starting point for optimization.
North West Corner Solution
A method for finding a basic feasible solution for the transportation problem. It starts from the top-left corner and allocates as much as possible to the cell until either supply or demand is met.
Least Cost Method (LCM)
A technique that focuses on allocating goods to the least cost cells in the transportation table first, ensuring a cost-effective initial solution.
Vogel's Approximation Method (VAM)
An improved method for finding the initial feasible solution by considering the penalty costs of not using the least cost routes, leading to a more optimal starting solution.
Matrices Method
Utilizes matrix operations to represent the transportation problem, allowing for systematic allocation and optimization of resources.
Optimal Solution
An optimal solution minimizes transportation costs while satisfying supply and demand constraints. It is often found after the initial basic solution is established.
Stepping Stone Method
A method used to explore potential improvement in the current solution by evaluating the costs of moving goods along alternative routes.
Modified Distribution Method (MODI)
An optimization technique that refines the basic feasible solution, ensuring that the total transportation cost is minimized by adjusting allocations.
Degeneracy
A situation in the transportation problem where the number of basic variables is less than m+n-1, leading to multiple optimal solutions or need for special handling.
Imbalance Matrix
Occurs when total supply does not equal total demand, requiring a dummy source or destination to balance the matrix.
Assignment Model
Special case of the transportation problem where the goal is to assign tasks or resources optimally.
Hungarian Method
An algorithm used for solving the assignment problem that guarantees an optimal solution by minimizing cost.
Traveling Salesman Problem (TSP)
A classic optimization problem aiming to find the shortest route that visits each city exactly once and returns to the origin city.
Project Scheduling and Resource Management: Deterministic Inventory models – Purchasing & Manufacturing models – Probabilistic inventory models – Replacement model – Sequencing – Brief Introduction to Queuing models. Networking – Programme Evaluation and Review Technique (PERT) and Critical Path Method (CPM) for Project Scheduling- Crashing – Resource allocation and Resource Scheduling.
Project Scheduling and Resource Management
Deterministic Inventory Models
Deterministic inventory models focus on known variables and parameters that dictate inventory levels and ordering. These models are based on predictable demand and lead times, making them suitable for environments where variability is low. Key types include Economic Order Quantity (EOQ) and Reorder Point (ROP) models.
Purchasing and Manufacturing Models
These models analyze the procurement process and production schedules to optimize inventory levels. They consider factors such as supplier lead times, order quantities, production rates, and storage costs to minimize overall costs while meeting demand.
Probabilistic Inventory Models
These models account for uncertainties in demand and supply. They use statistical methods to predict inventory needs and incorporate safety stock to mitigate stockouts. Key examples include the Newsvendor model and the Continuous Review Inventory System.
Replacement Models
Replacement models evaluate the optimal time to replace or repair items to minimize costs related to failures and downtime. They guide decisions on both preventive maintenance and complete replacements based on cost-benefit analyses.
Sequencing
Sequencing involves determining the optimal order of operations in production and project tasks. Techniques such as Johnson's rule and priority-based sequencing help minimize time and costs by optimizing workflow and resource usage.
Introduction to Queuing Models
Queuing models analyze the behavior of queues in systems to optimize service efficiency. They help in understanding customer arrival patterns, service rates, and waiting times, which is critical in resource allocation.
Networking Techniques
Network planning techniques like PERT and CPM are essential in project scheduling. PERT focuses on estimating project timeframes with uncertain task durations, while CPM provides a deterministic approach for planning tasks with known durations.
Crashing
Crashing involves reducing project duration by allocating additional resources. An analysis of cost versus time savings is crucial to determine the most effective resource allocation without exceeding budget.
Resource Allocation and Scheduling
Effective resource allocation and scheduling ensure that project tasks are completed using the available resources optimally. Tools like Gantt charts and resource leveling are commonly used to visualize and manage these processes.
Game Theory and Strategies: Games theory – two player zero sum game theory – Saddle Point – Mixed Strategies for games without saddle points – Dominance method – Graphical and L.P Solutions- Goal Programming; Simulation; Integer programming and Dynamic programming.
Game Theory and Strategies
Two Player Zero Sum Game Theory
In a two player zero sum game, the gain of one player is exactly equal to the loss of the other player. The strategies can be analyzed using payoff matrices. The objective is to minimize losses while maximizing gains.
Saddle Point
A saddle point occurs in a payoff matrix when the strategy chosen by players leads to an equilibrium point. Here, neither player has an incentive to change their strategy, resulting in a stable outcome. Identifying a saddle point is crucial for determining optimal strategies in zero sum games.
Mixed Strategies for Games Without Saddle Points
When no saddle point exists, players can employ mixed strategies. This involves randomizing their choices to prevent opponents from predicting their moves. Mixed strategies can be determined through methods such as linear programming.
Dominance Method
The dominance method helps simplify the analysis of games. A strategy is considered dominant if it yields a higher payoff regardless of the opponent's choice. By eliminating dominated strategies, players can focus on optimal strategies.
Graphical and L.P. Solutions
Graphical methods are useful for solving two-variable linear programming problems, representing constraints and objectives visually. Linear programming solutions can also be applied to game theory, particularly for mixed strategies and optimal solutions.
Goal Programming
Goal programming is an extension of linear programming. It focuses on achieving multiple goals within decision making, aligning with real-world scenarios where trade-offs are necessary.
Simulation
Simulation is a technique used to model complex systems and evaluate the impact of different strategies. This is particularly useful in game theory to test outcome probabilities under various scenarios.
Integer Programming
Integer programming involves optimization problems where some or all variables are required to take on integer values. This is critical in game theory for situations involving discrete decisions.
Dynamic Programming
Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. In game theory, it can be used to determine optimal strategies over time, considering the implications of previous decisions.
