# Modeling random variation from partial information

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 k0/k1 as c1, k1/k3 as c2... and cn > 1 for all n. k1 and 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.

• Shat are you defining $x$ and $y$ to be? The position of each cube? Please update your question. May 20 '17 at 10:54
• Also your example has 4 x G in the answer but 3 x G in the definition.. May 20 '17 at 10:55