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📘 Lesson 5 of 7 · Applied Mathematics

🏭 Operations Research

Operations research applies mathematical modeling and optimization to real-world decision-making — helping organizations run more efficiently by finding the best way to allocate limited resources.

Course progress: 71%

01 Key Concepts

What is Operations Research?

The application of mathematical methods (especially optimization and modeling) to help organizations make better decisions about resources, logistics, and scheduling.

Linear Programming in OR

A core tool: maximizing or minimizing an objective (like cost or profit) subject to a set of linear constraints (like limited labor hours or materials).

The Feasible Region

The set of all possible solutions that satisfy every constraint simultaneously, often visualized as a shaded region on a graph.

Queuing Theory

The mathematical study of waiting lines, used to analyze and improve systems like customer service lines, call centers, or network traffic.

The Transportation Problem

A classic OR problem: finding the cheapest way to ship goods from multiple suppliers to multiple destinations while meeting demand and supply limits.

02 Solved Examples

Example 1 A factory can make at most 40 chairs and 30 tables per week due to labor limits: chairs take 2 hours, tables take 3 hours, with 100 hours available. Write the labor constraint as an inequality.
  1. Let c = chairs, t = tables.
  2. Labor used: 2c + 3t, which must not exceed 100 hours.
Answer: 2c + 3t <= 100
Example 2 A company earns $30 profit per chair and $50 profit per table. Write the objective function to maximize profit.
  1. Profit = (profit per chair)*chairs + (profit per table)*tables.
Answer: Maximize P = 30c + 50t
Example 3 Why does the optimal solution to a linear programming problem occur at a corner of the feasible region rather than somewhere in the middle?
  1. The objective function is linear, so its value changes at a constant rate across the feasible region.
  2. This means the maximum or minimum value is always pushed to an extreme point (a corner), never trapped in the interior.
Answer: Because a linear objective function's extreme values always occur at the boundary/corners of a convex feasible region

03 Practice Questions

1A constraint says 3x+2y <= 60. Is the point (10,10) feasible?
Yes, 3(10)+2(10)=50, which is <= 60
2What does 'operations research' primarily apply mathematics to?
Improving real-world decision-making about resources and logistics
3What kind of region contains all valid solutions to a set of constraints?
The feasible region
4What OR topic studies waiting lines?
Queuing theory
5What classic OR problem finds the cheapest way to ship goods between suppliers and destinations?
The transportation problem

📄 Operations Research — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet