Sensitivity Analysis

In this section, we conduct a sensitivity analysis with respect to increasing demand priori- ties and fortification costs. In addition, we assess the impact of different distance calcula- tion methods (e.g. Euclidean vs. Geographic) on the generated solutions.

3.4.8.1 Demand Priority & Fortification Cost

We discuss next the effect of preferred establishment locations on the heuristic solution with respect to demand priority and transport cost.This relates to practical situations where certain locations may be considered as having more priority to access their supplying facil- ities. Some nodes may represent the locations of various critical infrastructure objectives (e.g. power plants, telecommunications relay hubs, etc.) which can have demands that need to be served faster from a strategic perspective. In this setting, we employ a priority factor over the existing demands specific to different number of nodes and assess the impact on the corresponding solutions. We assume the presence of strategic objectives close to locations with larger population since the locations in the used data set represent important urban centers. While this abstraction is somewhat coarse (e.g. conventional power plants are not necessarily adjacent to major urban centers but also not very distant), it serves nevertheless to illustrate the effect on the heuristically obtained solutions.

We use, as example, the 100-node RPMP instance used in the benchmarks. Figure 8 contains two graphs that depict the increase in solution cost with respect to various priority factors.

(a) Increasing demand priority (b) Increasing demand priority and fortif. cost Figure 8: Cost comparison for 100-node RPMP with fortification budget=240

We increase the priority factor as follows: 1.5, 2, 2.5 and 3 for the top 20 most pop- ulated cities in the data set. The original demands are multiplied with the priority factor. The total fortification budget is 240 and p = 8. Figure 8(a) shows the effect on the solution cost with increasing demand priority. Figure 8(b) shows the effect when increasing both demand priority and fortification cost by the same factor. The application of the factor naturally results in more costly solutions. However, since the transport (i.e. the distances among nodes) does not change, we reassess the solutions (after obtaining the initial lo- cation of the facilities and the corresponding customer assignment) without applying the factor. This way, we employ a form of a penalty method over the cost. The actual solu- tion cost still increases albeit to a lesser extent since the initial solution differs from the optimal/near optimal solution in terms of where the facilities are located (i.e. in locations more favourable for the prioritized nodes). The corresponding transport cost and penalty are separately shown in Figure 8(a) and Figure 8(b). We use the Euclidean metric for cost calculation in order to have the means to assess the gap between the solution obtained for preferred establishment locations and the near-optimal solution (as provided in the bench- mark results which employ reference values obtained using the Euclidean metric). The gap can provide an important insight to decision makers with respect to the relative increase in cost (as percentage) compared to the optimal/near-optimal solution.

Figure 9 illustrates the effects of demand priority on a geographic information system. We apply the priority factor on demands and fortification costs over the same 100-node RPMP instance as used in the benchmark results with budget 240 and P = 8.

(a) 2 × Demand Priority (b) 3 × Demand Priority

Figure 9: Solutions for 100-node RPMP with budget=240 and demand priority for 20% nodes

Figure 9(a) and Figure 9(b) show the results obtained by prioritizing the demands of 20% nodes with a factor of 2 and respectively 3. We note that compared to the case where no demand priority is considered (as in Figure 10(a) later), the results show depots located at more populated United States cities. Thus, Figure 9 shows depots established respec- tively at Houston, Jacksonville and Detroit instead of Garland (node 87), Montgomery (node 82) and Dayton (node 15) depicted in Figure 10(a). Also, with increased priority factor from double to triple, Seatle, with more population, is chosen as a facility instead of Oakland.

3.4.8.2 Euclidean vs. Geographic Distance

Figure 10 contrasts the solutions obtained using Euclidean and respectively geographical distances between nodes for the same 100-node RPMP problem instance.

(a) Using Euclidean distance between nodes (b) Using geographic distance between nodes Figure 10: Solution comparison for 100-node RPMP instance (p = 8 and budget = 240)

We assess the solutions for p = 8 facilities under a fortification budget of 240. Figure 10(a) shows that when using the Euclidean distance, the selected facilities are 0,1,2,15,37,50,82 and 87. Among these selected facilities, 0,1,37 and 87 are fortified. Figure 10(b) shows that when using the geographical distance the selected facilities become 0,1,2,20,50,58,82 and 93. Among these selected facilities, 0,2 and 58 are fortified. Thus, the solutions are notably different according to the way of measuring the distance between nodes and sub- sequently the solution cost. In case of using the Euclidean distance calculated based on the latitude and longitude values taken as Cartesian coordinates, as by [54], the solution cost is 12311.9. In contrast, the solution cost is 1228228.4 in case of using the latitude and longitude values to compute the geographical distance, in kilometers. Thus, the Euclidean abstraction used by [54] can lead to different solution both in terms of facility establishment and fortification as well as transport cost. This aspect is even more important in the case of RUFL where the objective function is used to minimize the combined cost of transport and facility establishment.

Figure 11 depicts a solution comparison between the Euclidean and geographical dis- tances for the 100-node RUFL problem instance under a fortification budget of 180. Since the calculated geographical distances are approximately two orders of magnitude greater than in the Euclidean case, we apply a corresponding magnification factor for the facility

establishment cost. Without such factor, the best solution would be to establish facilities at every node since the savings in transport cost would cover the establishment cost.

(a) Using Euclidean distance between nodes (b) Using geographic distance between nodes Figure 11: Solution comparison for 100-node RUFL problem with fortification budget=180

Figure 11(a) shows that when using the Euclidean distance, the solution has 13 estab- lished facilities (2,8,10,14,18,26,46,49,53, 54,57,68 and 90). Among the selected facilities 2,26,49 and 53 are fortified. Figure 11(b) shows that when using the geographical distance, (along with multiplying the establishment cost of all facilities by 100), the solution has 12 established facilities (2,8,10,14,18,26,46,49,53,57,68 and 90). Among these selected facilities, 2,26,49 and 53 are fortified. The solution is different according to the way of measuring the distance between nodes. In case of using the Euclidean distance as men- tioned by [54], the solution cost is 17140.5. In contrast, the solution cost in case of using the geographical distance, is 1709895.2. Also, in this case, the use of geographical distance is more appropriate since only 12 facilities are required to be established.