Introduction to Operation Research

Introduction of Operation Research

• Operation Research (OR) utilizes mathematical models and analysis to optimize complex systems.
• OR is widely adopted due to its ability to enhance business processes, increase efficiency, minimize costs, and facilitate data-driven decisions.
• Ultimately, OR contributes to increased productivity and a competitive edge.

Origin and Development

• Operation Research originated during World War II, as military leaders sought to optimize resource allocation.
• Post-war, OR gained significant traction and expanded into various industries.

Key Pioneers

• George B. Dantzig: Developed the simplex method for solving linear programming problems.
• Frederick W. Lanchester: Introduced mathematical models for warfare analysis.
• George E. Kimball: Made significant contributions to operations research during World War II.
• Patrick M. Morse: Studied operations research in various fields.

Introduction to Linear Programming

• Linear Programming (LP) is a mathematical technique used to determine the most optimal outcome within a given mathematical model.
• Applications of LP span diverse industries, including:
  • Production planning: Optimizing resource usage and manufacturing schedules.
  • Transportation: Finding the most efficient routes for delivery and logistics.
  • Investment: Maximizing returns while managing risk.
  • Financial planning: Allocating resources for optimal financial performance.
  • Resource allocation: Distributing resources among various projects or departments.

Problem Formulation of Linear Programming

• Components of Linear Programming:
  • Decision Variables: Variables that decision-makers directly control.
  • Objective Function: Represents the primary goal of an LP problem, typically expressed as a linear function of the decision variables. Example: P = c_1x + c_2y
  • Constraints: Represent limitations or conditions placed on the decision variables, often expressed as inequalities.
• Identifying Decision Variables: Examples include:
  • Number of units to produce.
  • Amount of resources to allocate.
• Establishing Constraints: Examples include:
  • Production capacity constraints.
  • Material availability limitations.

Characteristics of Linear Programming Problems

• Linearity: The objective function and constraints are linear equations or inequalities.
• Additivity: The total effect of multiple activities or variables is the sum of their individual effects.
• Certainty: The values of coefficients and parameters are known with certainty.
• Non-negativity: Decision variables are typically restricted to non-negative values (x≥0, y≥0).
• Divisibility: Decision variables are typically continuous and can be divided into fractions.
• Finite number of solutions: The number of feasible solutions is finite.

Application of Linear Programming

• Production planning: Determining the optimal production schedule to meet demand while minimizing costs.
• Transportation: Finding the most cost-effective routes for transporting goods from suppliers to customers.
• Investment: Maximizing returns on investments while managing risks.

Summary

• Linear programming is a powerful optimization method that utilizes linear relationships to achieve optimal outcomes.
• The key aspects of LP are linearity, additivity, certainty, and a focus on finding solutions.
• Problem Solving Examples:
  • Problem #1: An example where the LP framework could address a basic resource allocation scenario.
  • Problem #2: A visual illustration of the relationship between constraints and the feasible region in an LP problem.
  • Problem # 3 & 4: Additional problems where LP principles could be applied.

This quiz covers the fundamentals of Operation Research, including its origins, key pioneers, and the introduction of linear programming. Learn how OR improves efficiency and decision-making in various industries, tracing its development from military applications in World War II to modern business optimization techniques.