Database Reference
In-Depth Information
[We used C++ programs (for steps 1 - 5 above) to process the input retail market
basket dataset and produced the output in appropriate format. As our ILP formulation
assumes data mining activities as a pre-step, discussions regarding the preprocessing
done by these programs are unnecessary.]
4.2 Sample Optimal Package Selection Problem
To help explain our methodology, we use an example problem and work it through
the different stages of finding the optimal profit from the given dataset. Consider the
following dataset consisting of 5 sales transactions involving 7 items.
X
1
= {7}, X
2
= { 1, 2}, X
3
= {5, 6}, X
4
= {12, 13} and X
5
= {2, 6, 12, 13}
Table 1 below, shows the characteristics of various items (sp
i
- selling price, prof
i
-
profit per unit); while Table 2 presents the details of each item package.
Table 1.
Characteristics of items in the sample dataset
Item
1
2
5
6
7
12
13
s
i
0.2
0.3
0.25
0.3
0.15
0.3
0.2
c
i
2.5
3.1
4.5
3.7
2.1
3.5
2.5
s.p.
i
3.2
3.9
6.7
4.9
2.6
4.0
3.1
prof
i
0.7
0.8
2.2
1.2
0.5
0.5
0.6
Table 2.
Processed sample dataset for creating the ILP model
Count of
items (n
j
)
Number of
transactions (f
j
)
Item
package (X
j
)
Package
revenue
(p
j*
f
j
)
Package
storage
(∑s
i*
f
j
)
Package
cost
(∑c
i*
f
j
)
1
200
7
520
30
420
2
231
1 2
1640.1
115.5
1293.6
2
34
5 6
394.4
18.7
278.8
2
341
12 13
2421
170.5
2046
4
11
2 6 12 13
174.9
12.1
140.8
The first column shows the number of items in the packages viz. n
j
; the second
shows the frequency (f
j
); while the third shows the individual items that make up
each item package. The last three columns show the computed aggregates for each
package.
The createLP program, outlined in step 5 above, processes the formatted dataset
(steps 1-4) and produces the corresponding ILP model (Fig. 1) to the sample dataset.
This model is then solved using CPLEX, a commercial package for solving all kind of
linear programs. Fig. 2 presents the output from the package.
We notice (Fig. 2) that the optimal value i.e. the maximal profit, obtained under the
given constraints of 100 units of storage space and satisfying the minimum revenue of
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