need to develop a hill climbing algorithm by hand.  How?

There is lots of info on how to do it in various computer languages
but not much i can find on actually doing it by hand.

Its not a complex equation
cost min(a+u)= 400 * a + 300 * u   where a >=4000, u >=5000 total(a+u) >=10000

The following can help you get started:

The idea behind hill climbing is as follows.
1. Pick a random point in the search space.
2. Consider all the neighbours of the current state.
3. Choose the neighbour with the best quality and move to that state.
4. Repeat 2 thru 4 until all the neighbouring states are of lower quality.
5. Return the current state as the solution state.
http://www.cs.nott.ac.uk/~gxk/courses/g5baim/Hill%20Climbing/Hill01-introduction.html

For picking the "neighbors" of a point, you can use a delta of 1000
for "a" and "u".
Once you found a "min" using 1000, reduce the delta to 100, then to 10, 1,...
This is somewhat like the cooling in simulated annealing.
If you look at the partial derivatives, you see that reducing "a" will
be a priority.

Some researcher can give you a more detailed answer.

Method of Steepest Descent
http://mathworld.wolfram.com/MethodofSteepestDescent.html

Gradient Search Method
http://www.math.uncc.edu/~droyster/courses/fall01/classnotes/gradsearch.pdf

Genetic Algorithms vs. Traditional Methods
http://www.brunel.ac.uk/depts/AI/alife/ga-versu.htm