Interpretacja mnożników Lagrange'a
Lagrange multipliers technique, quick recap
- Step 1: Introduce a new variable , and define a new function as follows:This function is called the "Lagrangian", and the new variable is referred to as a "Lagrange multiplier"
- Step 2: Set the gradient of equal to the zero vector.In other words, find the critical points of .
- Step 3: Consider each solution, which will look something like . Plug each one into . Or rather, first remove the component, then plug it into , since does not have as an input. Whichever one gives the greatest (or smallest) value is the maximum (or minimum) point your are seeking.
Budgetary constraints, revisited
- Problem: Suppose you are running a factory, producing some sort of widget that requires steel as a raw material. Your costs are predominantly human labor, which is per hour for your workers, and the steel itself, which runs for per ton. Suppose your revenue is loosely modeled by the equationWhere
If your budget is , what is the maximum possible revenue?
- represents hours of labor
- represents tons of steel
- We start by writing the Lagrangian based on the function and the constraint .
- Then we find the critical points of , meaning the solutions to
- There might be several solutions to this equation,so for each one you plug in the and components to the revenue function to see which one actually corresponds with the maximum.