4. Conclusion

data from a commercial territory design problem of different sizes. For each instance, we obtained the optimal solutions for both the PC and the PM models using the CPLEX solver. For each instance, the optimal solution (the values of the decision variables xij) of the p-center problem ZC\* was evaluated with the objective function of the p-median problem. The resulting value PM(ZC\*) was compared

RDPM <sup>¼</sup> PM ZC ð Þ� PM ZMð Þ

RDPC <sup>¼</sup> PC ZMð Þ� PC ZC ð Þ

We name these measures as "relative differences" for each dispersion measure (RDPM for the p-median and RDPC for the p-center measures). Relative differences describe how far the solution of one model is from the other under each dispersion

The instances tested had two activity measures and five districts to be formed and ranged in sizes of 60, 80, 100, 120, and 150 BUs. We tested 20 instances of each size. The results are shown in the following figure (Figure 1). The horizontal axis shows the test instance numbers. Test instances are numbered with respect to their sizes. That is, instance numbers 1–20 correspond to the test instances with 60 BUs, instance numbers 21–40 correspond to the test instances with 80 BUs, and so on. In

As can be seen from the figure, relative differences RDPM (green) are generally better (closer to zero) than RDPC (blue). (Only on five instances, the RDPC (blue) values were very close to zero.) What this means is that ZC, the optimal solution from the PC model, is generally closer to the optimal solution to the PM model.

was evaluated with the objective function of the p-center problem resulting in PC(ZM\*). Accordingly, this value was compared with the optimal value of the

In contrast, for each instance, the optimal solution of the p-median problem ZM

PM ZMð Þ (10)

PC ZC ð Þ (11)

with the optimal value of the p-median problem PM(ZM\*) as follows:

p-center problem PC(ZC\*) as follows:

Technology, Science and Culture - A Global Vision

green, we show RDPM, and in blue, we show RDPC.

measure.

Figure 1.

116

Relative difference in dispersion metrics.

In this chapter, we have shown a comparison of two models for a territory design problem with different dispersion measures for dispersion: the p-center and the p-median. Both models were tested with 100 artificially generated instances, and the optimal solutions obtained were evaluated with the corresponding dispersion measure to compare both models. The optimal solution of the PM model ZM\* was evaluated with the objective function of the PC model. Its value was compared with the optimal solution of the PC model with the defined measure RDPC. The same was done for the optimal solution of the PC model ZP\*, and the results were compared with RDPM. The relative differences RDPC were lower than RDPM for most instances. A relative difference closer to zero means that the assignment of BUs to territory centers obtained from the optimal solution of one model gave a dispersion value that is very close to that of the optimal solution of the other model. These results show that the PC model is more robust compared to the PM model. Future work will consider the comparison of models with other dispersion measures which are not center-based since in practice the definition of a territory center has no practical meaning.
