## 6 Experiments
### Evaluation  
As discussed in Section 5.2, we computed the criteria values for the five criteria mentioned in Section 6.1. The radius values 1000, 300, 1000, 1000, and 1000 m were used for the criteria TStations, Buildings, EVenues, Shops, and PPlaces, respectively. The reason for using a smaller radius for the criterion Buildings compared to the other criteria is that the Buildings dataset is quite large and we obtain a uniform and a large number of buildings for each candidate for radius 1000 m. Thus, limiting the radius to a smaller value, in this case to 300 m, help us identify the nearby population of a convenience store. The same is not true for the other criteria, i.e., TStations, EVenues, Shops, and PPlaces. Thus radius values of 1000 m are used for them. Table 7 shows the pairwise comparison matrix for the five criteria. In the criteria computation, we assume that the criterion TStations, i.e., transportation stations, is the most important criteria as a large number of people visit convenience stores during their commute or travel. The second most important criterion that we identified is buildings. A large number of buildings around a convenience store means a large number of people either living or working there. Criteria TStations and Buildings are followed by criteria EVenues, Shops, and PPlaces, which are comparatively less significant compared to the first two criteria. In addition to the pairwise comparison matrix, Table 7 shows the computed criteria weights. The details of its computation are discussed in Section 5.2. Tables 8 and 9 show the top 20 stores ranked based on FourSquare visitor count (ground truth) and the top 20 stores ranked by our AHP/TOPSIS approach, respectively. Figure 8 shows the NYC map containing the convenience stores in Tables 8 and 9. Bold tuples in Table 9 are the ones ranked by the ground truth in Table 8 as well. Based on Tables 8 and 9, we computed the True Positive Rate (TPR). We counted the records predicted by AHP/TOPSIS, i.e., the record that is present in Table 9 is a True Positive (TP) if it also appears in the ground truth table i.e., Table 8, irrespective of its rank. On the other hand, a record is counted as a False Negative (FN) if it is predicted by AHP/TOPSIS but it does not appear in Table 8. Hence, the TPR for the criteria weights computed in Table 7 is given as follows: \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Criteria** & **TStations** & **Buildings** & **EVenues** & **Shops** & **PPlaces** & **Weight** \\ \hline TStations & 1 & 3 & 7 & 9 & 7 & 0.495486996 \\ Buildings & 0.3333 & 1 & 7 & 9 & 7 & 0.327448046 \\ EVenues & 0.1428 & 0.1428 & 1 & 3 & 1 & 0.071501752 \\ Shops & 0.1111 & 0.1111 & 0.3333 & 1 & 0.3333 & 0.034061454 \\ PPlaces & 0.1428 & 0.1428 & 1 & 3 & 1 & 0.071501752 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Pairwise comparison matrix and computed criteria weights. \[TPR=\frac{TP}{TP+FN}=\frac{11}{20}=0.55\]  
The TPR is heavily dependent on the criteria weights, and the derivation of the right criteria weights is important to obtain an optimal or desired prediction. To prove this, we performed experiments with a number of manual weight assignments to the criteria vector [TStations, Buildings, EVenues, Shops, PPlaces] as follows:  
* uniform = [0.2, 0.2, 0.2, 0.2, 0.2]
* increasing = [0.05, 0.1, 0.2, 0.3, 0.35]
* decreasing = [0.35, 0.3, 0.2, 0.1, 0.05]
* oneCriteriaZero = [0.4, 0.3, 0.2, 0.1, 0]  
For the random weight vectors uniform, increasing, decreasing, and oneCriteriaZero above, we obtained the TPRs 0.25, 0.25, 0.4, and 0.35, respectively. The TPRs are far lower than the one obtained using carefully derived weights, i.e., 0.55. Figure 9 shows the placement of ground truth convenience stores (green triangles) and the AHP/TOPSIS-predicted convenience stores using the manual weights (red circles). As can be observed from the obtained TPR values of the different weight vectors, the AHP/TOPSIS approach is heavily dependent on the criteria weight computation.