I am working on an optimization problem.
Let's say we have a grid of bins where solid metal cubes could be placed. We have a number of colored metal cubes to be arranged in the bins.
Each of these cubes weighs
1kg. But the weight could change slightly over time due to natural phenomena and becomes
1kg + delta(x,y). We are interested in making sure the ratio between the total weight of one color to another remains same over time.
Observations say that
delta(x,y) is not completely random. It can be thought of as
k0 + k1*x + k2*y + k3*x^2 + k4*y^2 + k5*x*y where the constants
k0 k1 k2 ... are unknown at the time of arranging the cubes. But in all cases
k0 > k1 > k3 > k5 and
k0 > k2 > k4 > k5 We can assume
cn > 1 for all n.
k2 are unrelated.
This observation lets us cancel out certain terms in
delta(x, y) by arranging carefully. For example, if we arrange two red balls and two blue balls as
R B B R, we can expect the linear terms to cancel.
Take for example a grid of
3x3. Initially, we have the empty grid:
[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]
and a number of metal cubes to be placed. As an example, let's say we are to arrange one red cube, two blue cubes, four green cubes and two dummy cubes.
Metal cubes to be placed:
1 x R 2 x B 4 x G 2 x *
One possible arrangement could be:
B * G G R G G * B
Given a set of values for c1, c2 etc, there is one arrangement that must be the best. I am trying to model a cost function which indicates the expected variation between the two colors. Then choose the arrangement that has minimum cost by iterating through all possible arrangement.
I am looking for some advice on how to model the expected variation between two colors.
Also, any advice on reducing the search space.