Average Value of a Function with Linear Constraints

Assume:

x1, …, xn where 0<= xi <= 1

a1, …., an are positive real constants.

A = Another Positive real contant

Where,

Sum {i = 1 to n} (xi) = 1 (i)

Sum {i = 1 to n} (ai * xi) = A (ii)

In other words, A is the weighted average of a1 , … ,an where x1, … ,xn is the weighting.

Now, here’s the problem:

Can I find the Average value of a function F(X) where F has the format:

F(X) = b1*x1 + b2*x2 + … + bn*xn

Where b1, …, bn are positive real constants ?

It seems like there should be a solution.

It’s the average value of a relatively simple function over a Solution space.

The Solution space is defined as all the valid combinations of x1, …, xn that satisfy the conditions (I) and (ii) from above, which themselves are not that complex.

I’m thinking the average is the Integral of F(X) / Integral (Identity Function) over the solution space.

You would have to use a Multiple Integral n-times for each xi , But I have no idea how to construct the Bounds of each integral given that there are Two constraints.

It seems like the problem should be quite common.

It resembles blending problems in Linear Programming - The Total Make-up of your mixture has to equal 100% and each of the contributing elements adds a certain quantity for a desired blend (the Ai values.)

Then the F(X) function would be like an Objective Function where the Bi values are cost weights associated with the contibuting elements of the Blend.

(Blending Example: I'm mixing drinks and I require a certain alcohol % for the blend. N ingredients each with a given alcohol percentage "a" and a price "b")

In LP the objective is generally to find the Optimal solution, either maximizing or minimizing the value of F(X) that satisfies the constraints.

In this case I'm looking for the average of F(X) for all valid solutions of x1, ..., xn

I'm surprised that a google-search hasn't provided a solution so far.

("Average value of a linear multivariable function with linear constraints.")

Is this a lot more complicated than I think it is?