At work I stumbled across this problem of allocating resources:
We are given a set of objects belonging to one of seven possible classes (multiple objects per class are allowed). We distinguish between classes A, B, ... G, because they have 3 possible attributes (weight, interest, size). We also have four boxes X, Y, W, Z. The boxes have three characteristics on their own: importance, strength, size. We want to put the objects into the boxes in order to maximize a metric based on the properties of the classes and the boxes. We are also wondering if, given enough pairs "set of objects"-"allocation in boxes" tweak the characteristics of the classes in order to minimize a cost function representing the difference between the desired assignment and the one computed.
I tried checking standard methods of resource allocation or cost minimization (definitely not my expertise), but I couldn't manage to apply them to this specific problem. Any ideas on how to proceed, or on where could I find info related to this problem?
EDIT: This is the definition of the metric used that must be maximized. After that a simple example.
Let's just define $W_{obj}$, $I_{obj}$, $S_{obj}$ as the characteristics of an object and $I_box$, $St_box$, $Si_box$ as the characteristics of the box. The metric is the sum, over all the boxes, of this measure:
$I_{box} * \sum{I_{obj}} + 1/(St_{box} - \sum{W_{obj}}) + 1/(Si_{box} - \sum{S_{obj}})$
If you have multiple objects you just some over all of them.
As an example, suppose we have Box 1 and Box 2, with properties:
- $St_{b1} = 10$, $I_{b1} = 3$, $Si_{b1} = 9$;
- $St_{b2} = 20$, $I_{b2} = 1$, $Si_{b2} = 10$;
and three objects with properties:
- $I_{o1} = 1$, $W_{o1} = 3$, $S_{o1} = 4$;
- $I_{o2} = 1$, $W_{o2} = 3$, $S_{o2} = 4$;
- $I_{o3} = 3$, $W_{o3} = 8$, $S_{o3} = 1$;
If we put all the objects in Box 2, our score will be:
$I_{b2} * (I_{o1} + I_{o2} + I_{o3}) + 1/(St_{b2} - (W_{o1} + W_{o2} + W_{o3})) + 1/(Si_{b2} - (S_{o1} + S_{o2} + S_{o3}))$ $= (1 * 5) + 1/(20-14) + 1/(10-9)$ $= 6.16$
Since no objects goes in Box 1, its score will be:
$1/St_{b1} + 1/Si_{b1} = 1/10 + 1/9 = 0.21$
Hence the score for this particular configuration is $6.16 + 0.21 = 6.37$
The aim is to find the configuration (i.e. the object distribution) maximising such score.
Hope it's a bit clearer now.