Excel Solver finds an incorrect answer

Check my logic: K9 is an ABS() function, so the minimum of this function should be 0 -- when all four values are equal, is that correct?

Why is this set of values not The Correct (TM) solution? What makes the solution you posted more correct than this one? It appears to me that both solutions yield a minimum (0) in K9, which is what you have asked it to do?

There are no shortage of numeric problems like this where there is more than one solution. Much of the time, if you want to find a specific solution when multiple solutions are possible, you need to know enough about your desired solution to give Excel/Solver a good starting guess so that it will converge on the correct solution. In this case, it looks like you are starting from the "all values are 0" starting point, and it is converging on the wrong solution. I have not studied your problem in detail, but I would suggest that you need to study this so that you will know what starting values to give Solver so it can converge on the correct solution.

I don't think there is any way to force Solver to use a "brute force" algorithm like that. You could simply program that brute force algorithm into the spreadsheet and sidestep Solver, but that seems overkill to me.

Looking at your problem more carefully, it almost looks like a basic "solve a system of equations" kind of problem, where you have 4 equations and 7 unknowns. This kind of situation often yields a scenario where there are infinite solutions (even limiting the variables to narrow ranges). You say that there is another set of data that somehow determines which solution is correct. It seems to me that the best way to get your solution to be correct is to incorporate this additional data into your Solver model (or other algorithm).

Since it looks like a "solve a system of equations" kind of problem, it might be of value to research other possible solutions. For example, one can use matrix algebra (and Excel's built in matrix functions like MINVERSE()) to solve systems of linear equations: http://www.excel-easy.com/examples/s...equations.html
A couple of algebra lessons might also be useful:
Solving systems of non-linear equations: http://www.purplemath.com/modules/syseqgen.htm
solving systems of linear equations: http://www.purplemath.com/modules/systlin1.htm

Whether you completely change tactics or not, I think the key to a good solution will be to bring the rest of the information from the other data set that decides which solution is correct and incorporate that into your algorithm.