For most problems that individuals face each day, linear programs are fairly simple to model and solve. However, there are lots of complex problems that are solved each day.
A few examples of large-scale linear programming problems: - For Amazon.com: what quantity of each item should be stocked at each warehouse each day to minimize inventory while also optimizing for shipping time and cost to demand nodes (customers). - For an airline: how to plan and schedule flights to all domestic and international airports to maximize profit - In shipping logistics: how to allocate trucks and set routes to minimize fuel cost while satisfying delivery time.
For complex systems, you can easily run up a linear program with millions of independent variables (producing millions of rows in the linear system).