Mining Value-Based Item Packages – An Integer Programming Approach - Data Mining: Theory, Methodology, Techniques, and Applications - page 86

Given a certain maximum storage space, the retailer might like to find out the
optimum profit (and item packages) against a maximum number of items to be put on
the shelves. He might also be curious as to how the profit varies if he is able to
acquire more storage space. To show how easily this can be achieved using our ILP
formulation, we varied the values for S, the maximum storage space parameter, from
1000 to 4000 and varied the upper limit for the number of items to be shelved i.e. N U
from 20 to 500. The resulting ILP was then submitted to CPLEX 9.0 to calculate the
value of the net profit function (z). Table 3 shows the effect of changing the
maximum number of items (N U ) has on the objective.

Table 3. Profit function and time (in seconds) for varying storage space (S) and number of
items (N U )

N U

20 50 100 150 200 300 400 450 500

S=4000

Profit 22,697 27,996 32,323 34,646 36,320 39,281 39,384 39,381 39,384

Time 0.24 0.24 0.23 0.26 0.23 0.24 0.22 0.12 0.11

S=3000 Profit 22,697 27,996 32,323 34,646 36,320 39,281 39,281 39,328 39,328

Time 0.25 0.22 0.23 0.26 0.22 0.24 0.22 0.35 0.34

S=2000

22,697 27,996 32,323 34,646 36,320 38,315 38,423 38,423 38,423

Profit

Time 0.25 0.22 0.23 0.26 0.21 0.22 0.23 0.23 0.23

S=1000 Profit 22,697 27,996 32,323 34,556 35,564 35,636 35,636 35,636 35,636

Time 0.25 0.22 0.23 0.38 0.41 0.23 0.23 0.23 0.23

We then chart (Figure 3) the observations to visualize the effects of max. storage
and N u on the value of the objective, z. We observe that while increasing the number
of items does increase the net profit quite substantially, after a certain stage the rate or
amount of change in the same is not significant, eventually peaking and remaining so
in spite of increasing resources (storage space or number of items stored). This
observation could be of value to the retailer as he can clearly visualize the expected
changes in profit by changing certain parameters as need be. Similarly, one can study
the effect of varying the limits of other resources and study their effects on the
profitability function.
For our experiments, we used an AMD Athlon XP2100 PC with a CPU clock of
2.1 GHz having 512 MB of RAM running Windows 2000. Our experiments show
very encouraging results as all of them are achieved in a sub-second response time.
This proves that our method of solving such problems is very much viable.

Limitations. The model presented in the previous sections has been tested with a
reasonable size dataset. However, it is not without its limitations. While the number of
transactions (T) could be very large (limited by how large an integer can be on a specific
system), for the given data varying the minimum number of items (N L ) could increase
the number of possible combinations of items and thereby could affect the solving time.
This is dependant on environmental factors like available memory, storage and CPU
speed. CPLEX could not solve this problem, using the above PC configuration, when
all non-frequent transactions were included within a reasonable time viz. 8 hours.