Lagrange multipliers have a definite meaning in load balancing for electric network problems. Consider the generators that can output xi megawatts, i = 1, 2, 3 each generator cost

Ci = 3xi + ( i /40) xi^2 i .

If the total power needed is 1000MW. What load balance (x1, x2, x3) minimizes cost?

The load balance Mw minimizes the total cost

Step-by-step explanation:

Optimizing With Lagrange Multipliers

When a multivariable function f is to be maximized or minimized, the Lagrange multipliers method is a pretty common and easy tool to apply when the restrictions are in the form of equalities.

Consider three generators that can output xi megawatts, with i ranging from 1 to 3. The set of unknown variables is x1, x2, x3.

The cost of each generator is given by the formula

It means the cost for each generator is expanded as

The total cost of production is

Simplifying and rearranging, we have the objective function to minimize:

The restriction can be modeled as a function g(x)=0:

Or

We now construct the auxiliary function

We find all the partial derivatives of f and equate them to 0

Solving for \lambda in the three first equations, we have

Equating them, we find:

Replacing into the restriction (or the fourth derivative)

And also

The load balance Mw minimizes the total cost

use the power rule  

( a b ) n = a n b n

to distribute the exponent.

apply the product rule to  

3 x 2 y 3 . ( 3 x 2 ) 2 ( y 3 ) 2

apply the product rule to  

3 x 2 . 3 2 ( x 2 ) 2 ( y 3 ) 2

raise  

3

to the power of  

2 . 9 ( x 2 ) 2 ( y 3 ) 2 multiply the exponents in   ( x 2 ) 2

.

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 2 ⋅ 2 ( y 3 ) 2

multiply  

2   by   2 . 9 x 4 ( y 3 ) 2

multiply the exponents in  

( y 3 ) 2 .

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 4 y 3

⋅

2

multiply  

3   by   2 . 9 x 4 y 6

step-by-step explanation: