I have x number of items to sell. Each item must be sold as part of a bundle and must not be sold alone. Each bundle contains three items which must be different from each other. Therefore if I had four items (listed as: A, B, C and D) to sell (x = 4) I would have four different bundles (A,B,C; A,B,D; A,C,D and B,C,D) that I could potentially sell. It is important to me that all of my stock sells out and I am not left with any surplus. Each item will have its own level of inventory (say in this example: A = 100, B = 120, C = 85 and B =92).
So I would like to construct an optimisation problem whereby I can model exactly how many of each specific bundle I should sell in order to sell all of my stock. I am seeking a general model and approach as an answer but am unsure whether this can realistically be obtained.
Any help greatly appreciated!
Let the specific bundles be $x_1 (ABC), x_2(ABD), x_3(ACD),x_4(BCD)$ respectively.
Then the optimization problem is
Max $x_1+ x_2+ x_3+x_4$
s.t
$x_1+ x_2+ x_3=100$
$x_1+ x_2+ x_4=120$
$x_1+ x_3+x_4 = 85$
$x_2+ x_3+x_4 = 92$
$x_i$ is integer
$x_i\ge 0$
I ran it in EXCEL, you don't have a feasible solution for you not to have any surplus. But if you changed the A to be $=99$, then the optimal solution is
$x_1=40, x_2=47, x_3=12,x_4=33$
Max bundles $= 132$