Least Expensive Circuit

There are two aspects to this problem:
1) Search the space of possible trunk/branch junction coordinates
2) Evaluate candidate junction coordinates to determine the cost function

Item (1) can be accomplished with various methods, such as grid scanning or hill-climbing algorithms. It would be prohibitively difficult to do this problem in a completely formal manner.

Item (2) analysis:

Suppose we have a candidate junction location $(x,y)$ . Find the trunk and branch lengths. $(x_1,y_1,x_2,y_2,x_3,y_3)$ are coordinates of load resistor connection points:

$L_T = \sqrt{(x-0)^2 + (y-0)^2} \\ L_{B1} = \sqrt{(x-x_1)^2 + (y-y_1)^2} \\ L_{B2} = \sqrt{(x-x_2)^2 + (y-y_2)^2} \\ L_{B3} = \sqrt{(x-x_3)^2 + (y-y_3)^2}$

Call the resistance per unit length of the red wire $R'$ . Find the trunk and branch resistances:

$R_T = R' L_T \\ R_{B1} = R' L_{B1} \\ R_{B2} = R' L_{B2} \\ R_{B3} = R' L_{B3}$

Find the combined branch / load resistances:

$R_{C1} = R_{B1} + R_1 \\ R_{C2} = R_{B2} + R_2 \\ R_{C3} = R_{B3} + R_3$

Define a "parallel resistance":

$R_P = \frac{1}{\frac{1}{R_{C1}} + \frac{1}{R_{C2}} + \frac{1}{R_{C3}}}$

Junction voltage:

$V_J = 5 \frac{R_P}{R_T + R_P}$

Load resistor voltages:

$V_{1} = V_J \frac{R_1}{R_{C1}} \\ V_{2} = V_J \frac{R_2}{R_{C2}} \\ V_{3} = V_J \frac{R_3}{R_{C3}}$

Total red wire length:

$L = L_T + L_{B1} + L_{B2} + L_{B3}$

We now have everything we need to evaluate the cost function. As it turns out, the optimal junction location is at approximately $(x,y) = (3.7796,-0.1236)$ , with an associated minimum cost of approximately $77.5$ .