Developing equation to satisfy specific parameters (potentially partial derivatives?)

[jbaldwin156]

I'd like to start out by stating that I am an engineer, but would not consider myself a math expert. Which is why I am posting here; so I can pick the brains of those that eat and breathe this stuff (and are likely much smarter than me)

I have a unique problem that I am trying to solve by developing an equation (or system of equations) to satisfy some specific parameters.

1. The equation will contain four variables (categories) that will need to be simultaneously solved in order to output a maximum condition based on given input.

2. Input for each category will range between 0 and 1 and will be multiplied by its corresponding variable. You could consider each categorical input as an individual and independent rating. The purpose of the equation is to take this input, combine them with factors and output a maximum combined rating.

3. The variables will have individually specified ranges (for example: 0.2 <= x1 <= 0.4; 0.2 <= x2 <= 0.4; 0.1 <= x3 <= 0.2; 0.1 <= x4 <= 0.2). The sum of all the variables must be 1.

Example problem:

Ratings for Test 1:
Category 1 rating: C1 = 0.75
Category 2 rating: C2 = 0.90
Category 3 rating: C3 = 0.75
Category 4 rating: C4 = 1.00

Fixed variable ranges for all Tests:
0.2 <= X1 <= 0.4
0.2 <= X2 <= 0.4
0.1 <= X3 <= 0.2
0.1 <= X4 <= 0.2

(C1*X1) + (C2*X2) + (C3*X3) + (C4*X4) = MAX

I look forward to everybody's input!

Thanks!