Excel Solver: Is there a way to iterate over 2 changing variables? - excel

Question

I have an issue with solver as follows (simplified version):
So I have a nested If statement that describes condition for 2 changing variables(x,y). For example:
In one cell: IF(AND((x<=2),(x>=0.5),(y<=10),(y>=5)),1,0
The cell below it: IF(AND((x<=2.5),(x>=1.9),(y<=11),(y>=9)),1,0
The objective function is the sum of these 2 variables
Solver or goal seek (unless i give it the awnser) can't seem to get an awnser other than 0,0.
My actual problem is that i have 6 of these IF cells and I'm trying to find an (x,y) that maximizes my objective function. I want excel to go through as many combinations it can.
Any thoughts or other ways to do this? Thanks.

Answer 1

The reason that the Solver does not find the optimal solution in this toy problem is because the use of IF and AND statements make the problem non convex. For non-convex problems, the GRG Nonlinear solution method (the default used by solver) does not guarantee an optimal solution, as it can be trapped in locally best solutions which are not optimal.
Having said that, there is a way to formulate your problem as a mixed integer program, which, although still non-convex, can be solved with the "Simplex LP" method of Solver, and give a guaranteed maximum.
Model Setup
Here is a screenshot of the spreadsheet setup:
For convenience, I have used named ranges for the several quantities.
In particular:
- B2 --> x_var
- C2 --> x_UB1
- D2 --> x_LB1
- E2 --> x_UB2
- F2 --> x_LB2
and for row 3 I use the same convention, but instead of x_ we have y_.
The red cells (B4 and E4) have the conditions you described, and the blue cell (B5) has their sum.
For example, the condition for B4 reads
=IF(AND(x_var<=x_UB1,x_var>=x_LB1,y_var<=y_UB1,y_var>=y_LB1),1,0)
We are going to replace these expressions with two binary variables, which equal one if each expression is satisfied and zero otherwise.
The logic is that instead of an IF expression we can impose the constraints:
LB_x * z <= x <= UB_x * z
LB_y * z <= y <= UB_y * z
z is binary
then z = 1 ==> LB_x <= x <= UB_x
LB_y <= y <= UB_y
and because we maximize the sum of the two z variables, the x and y will try to fit i the corresponding ranges so that as many z as possible equal 1.
The green cells H2, J2 have the two new binary varibles, called cond1_true, cond2_true respectively. The other cells have the constraints described above:
For example, for the first expression:
J2: =x_var-cond1_true*x_UB1
J3: =y_var-cond1_true*y_UB1
K2: =x_LB1*cond1_true-x_var
K3: =y_LB1*cond1_true-y_var
All these cells need to be <= 0 in the solver model.
Solver Model:
In the mode, the objective function cell is the sum of the binary variables. The decision variables are x_var, y_yar, cond1_true, cond2_true. The constraints are all in expression <= 0 format. Here is the worksheet: https://www.dropbox.com/s/uek2k9gownhh3ni/excel-solver-is-there-a-way-to-iterate-over-2-changing-variables.xlsx?dl=0
Using this formulation, the solver goes through many combinations of variables and tries to pick up the best one. It can often guarantee an optimal solution (which is almost always the case for small problems)
UPDATE
If the intervals are non overlapping we need to modily
LB_x * z <= x <= UB_x * z
to
min(LB_x) * (1-z) + LB_x * z <= x <= UB_x * z + max(UB_x) * (1-z)
Where min(LB_x) is the minimum lower bound across all intervals (likewise for UB and for y). This way, if an x does not fall into the interval (z=0) it is only forced to fall in some other interval.
I hope this helps!