*4.2. Results*

This section is devoted to present and analyze the results obtained by solving the proposed model for the base case study introduced in the previous section and several variations of it, carried out to allow for a deeper analysis of the performance of the model. The mathematical models are solved by GAMS 23.7 with a relative gap of 5% on an Intel(R) Core(TM) i5-6300HQ CPU 2.30 GH, 8G RAM running Windows 10.

The 4-level lexicographical goal programming model defined in (1)–(25) will be solved as a way to deal with all criteria in a joint way. Each of the four goals considered requires an aspiration level, as detailed hereunder. The aspiration levels will be: evacuating all people that arrive to the pick-up points along the time horizon, both high (8790 people) and normal (7340 people) priority population (related to the first and second lexicographical levels, respectively), within 36 h, that is 50% of the time span (third lexicographical level), and using up to 50% of the total available budget (fourth lexicographical level), that is, 100,000\$. Each level leads to a mixed integer linear programming model whose size (number of constraints, variables, non-zero elements and discrete variables) is given in Table 5, emphasizing the large dimensions of realistic case studies as the one considered here.

**Table 5.** Size of the resulting models level by level.


Solving the four levels lexicographically, ensuring that the deviation from the aspiration level of the previous goal is maintained in the following levels, a solution is obtained in which 98.36% of high priority and 92.89% of normal priority population are evacuated with an operation time of 72 h (twice as desired) and a cost significantly below the aspiration level (54,956.13\$). Nearly all the high priority population is evacuated, as well as most of the normal priority population, and the cost is quite restrained; however, the whole time span considered in the model is required to complete the evacuation. Figure 4 illustrates the location of evacuees at the end of the operation in this solution and the detailed information regarding the amount of people of each category that stay at each node can be found in Table 6. Both representations state that most of evacuees are located at safe nodes at the end of the operation. In particular, Mako Sabhara (S03), in the east of Palu river, and Makorem area (S02), in the west, sheltered a very high number of evacuees due to their location and capacity. Additionally, it must be pointed out that S02 is a normal shelter and S03 a safe one, which means that these shelters accommodate people with both high and normal priorities. The next location hosting the highest number of remaining evacuees is the airport, from where they could be transported to other nearby cities. On the other hand, it is important to highlight the fact that, in practice, the information regarding the people that could not be evacuated and remain at unsafe areas (number of people, location, health condition, etc.) is shared with other agencies, in order to coordinate the complete evacuation of all affected population.

**Figure 4.** Location of evacuees at the end of the operation.

**Table 6.** Distribution of evacuees at the end of the operation by category. SIA: Severely injured adults. PW: pregnan<sup>t</sup> women. UM: unaccompanied minors. GBV: women susceptible to gender violence. RP: rest of the population.


To provide a more detailed example regarding a particular group of affected people, Figure 5 illustrates how the evacuation of women susceptible to GBV was performed. In particular, it shows the time period (or periods) in which women susceptible to GBV start being transported from one node to another and the type of vehicle used for transportation. Square brackets represent different periods where the same path and type of vehicle are used. It can be seen how trucks is the most widely used type of vehicle, even though helicopters, ambulances and rafts are also used at certain points.

**Figure 5.** Evacuation of women susceptible to GBV.

In order to further evaluate the performance of the model in different situations, several specific modifications were performed on the case study and the resulting lexicographical goal programming problems were resolved again. For example, as stated by the International Civil Aviation Organization of OCHA [58], the Palu airport suffered a grea<sup>t</sup> amount of damage and it could not be used during the first hours of the evacuation operation. As a result, it could be interesting to analyze what may happen if the airport is assumed to be unavailable in the network of the case study. In this situation, 8601 people with high priority (97.85%) and 6824 with normal priority (92.94%) could be evacuated, which is quite similar to what could be achieved originally, and within the same operation time; however, without the airport being available, this can only be achieved at a significantly higher cost of 59,019.69\$. A similar situation could arise if the hospitals were unavailable as safe locations for the evacuees. In that case, less high priority population could be evacuated (7764 people, 88.33%), but this would allow the evacuation of more normal priority population (7035 people, 95.84%). The operation time would be unchanged and the cost would be slightly higher (56,204.31\$).

Another usual limitation that is often found during or after the occurrence of a disaster is the lack of certain types of vehicles. Since the number and types of vehicles available for the evacuation operation may influence significantly on the results, we have resolved the case study removing some of them. For example, without helicopters, less high priority population (8527, 97.01%), but slightly more normal priority population (6829, 93.01%), can be evacuated, with the same operation time (72 h) and a higher cost (57,792.54\$). Meanwhile, the solution without rafts shows that only 8422 (95.81%) of the high priority population and 6827 (92.99%) of the normal priority population could be evacuated. This could be done three hours faster, but with a much higher cost of 70,142.55\$. The case in which trucks are not available is especially extreme, because without them very few people could be evacuated: namely, only 40.52% and 2.06% of the high and normal priority population, respectively. Furthermore, 72 h are still necessary to complete the operation, which only costs 5292\$. This happens because most of the available vehicles are actually trucks and, in addition, this type of vehicle has the highest capacity.

An additional interesting experiment regards splitting the network of the original problem into two or more subgraphs, in such a way that a solution to the original problem could be obtained from solutions of the subproblems associated to the subgraphs. The solution obtained by merging the solutions obtained independently from each subproblem may not be optimal, but this could simplify the resolution of the original problem by reducing it to simpler subproblems. One way to do this would be to use the partition illustrated in Figure 6, which splits the area of interest into two parts, east and

west of the Palu river. In each subproblem, potential evacuees are located only at the pick-up nodes of that particular partition. Similarly, only secure nodes and vehicles of the corresponding partition can be used to perform the evacuation operations. After solving these two subproblems and merging the solutions obtained, the resulting plan allows the evacuation of only 75.92% and 51.23% of the critical and non critical population, respectively, in comparison with 98.36% and 92.89%, that can be achieved with the plan provided by the original model. This shows a very big difference in the effectiveness of the two approaches. On the other hand, the lower number of evacuees that are to be moved allows the operation to be completed in only 39 h and with a significantly lower cost of 34,992.86\$ (in comparison with 72 h and 54,956.13\$). However, we use a lexicographical approach because we believe there should not be any trade off between the number of evacuated people and other attributes such as time or cost, and as a result, these two solutions are not comparable. This experiment highlights that the coordination of the operations is crucial to obtain effective global evacuation plans.

The previous experiments are all performed with very high aspiration levels for the amount of evacuated population, because this should be the main concern in this type of operations. However, they impose quite strong constraints on the third (operation time) and fourth (cost) lexicographical levels, which cannot vary much. In order to test if it would be possible to obtain significantly faster or cheaper operations at the expense of evacuating less people, we have resolved the model with lower aspiration levels of 75% and 50% for the evacuated people with high and normal priority, respectively (75–50). These aspiration levels are quite similar to the results that could be achieved by solving independently the two sub-problems defined by Figure 6. The results we obtained with these alternative aspiration levels indicate that the stated amount of people could be evacuated much faster, in only 30 h, but with a much higher cost of 151,340.09\$. This shows how slight operation time reductions can be extremely costly, even with a reduced number of total evacuees. Additionally, we also solved the case with aspiration levels of 50% and 75% for high and normal priority population, respectively (50–75), obtaining an evacuation plan that could be implemented in only 36 h. However, as it happened earlier, the cost skyrockets to 171,516.01\$, stressing the high cost of reducing the operation times in this context.

**Figure 6.** Splitting the network into two parts.

As a summary of the results obtained in this section, the solutions obtained by solving the case study under the different assumptions considered is given in Table 7.


**Table 7.** Results of the case study.

#### **5. Concluding Remarks**

The main objective of the presented work has been palliating the effects of a disaster by focusing on the supported evacuation of the affected population, in order to protect them. The model provides a feasible evacuation planning to help the authorities facing the consequences of a disaster make decisions about how to proceed operationally. The main novelties introduced with respect to existing approaches have been the consideration of dynamism on the arrival of evacuees to the pick up points, the classification of the population according to their health condition and the joint consideration of conflicting objectives such us the number of people to evacuate, the operation time and the total cost. In order to include all these relevant elements, we have followed a lexicographic goal programming approach, in such a way that there is no possible trade off among the considered criteria: the most important ones are the amount of people evacuated (high priority, first level, and normal priority, second level), followed by the operation time (third level) and finally the cost (last level).

The computational results obtained by solving a realistic case study based on the earthquake and tsunami that hit Indonesia in September 2018 has shown how the proposed model is able to provide detailed feasible evacuation plans that could be implementable in real operations. In addition, the computational experience illustrated how the model could also be used to evaluate the consequences of possible changes in the available resources or infrastructures, as for example if the airport, certain shelters or a hospital were not available, or several vehicles could not be used. Another interesting insight that we have obtained from the results of the case study regards the high conflict among the different criteria considered: in order to be able to evacuate most of the affected population, long operation times are required, even using the whole time span, with reasonably low costs; the only way to obtain faster operations is by reducing the number of people evacuated, and even in that case, the cost increases a lot (up to three times) to achieve this time reduction. If the situation were that of a very high urgency, it could be possible to use the model to design several fast evacuation plans for reduced groups of affected people that could be carried out through the collaboration of several agencies.

One interesting line of future work could be the extension of the proposed evacuation model so that the planning could also take into account other important elements in the response to a disaster, as for example the location of temporal shelters or depots, the pre-positioning of stocks, the distribution of commodities required by the evacuated people, etc. In addition, the introduction of some stochastic variables into the model to represent the arrival of potential evacuees or changes in the status of the infrastructures could also be worth considering.

**Author Contributions:** Conceptualization, I.F., M.T.O., G.T. and B.V.; Data curation, I.F.; Formal analysis, I.F., M.T.O., G.T. and B.V.; Funding acquisition, B.V.; Investigation, I.F., M.T.O., G.T. and B.V.; Methodology, I.F., M.T.O., G.T. and B.V.; Software, I.F.; Supervision, M.T.O., G.T.; Validation, I.F., M.T.O. and G.T.; Visualization, I.F., M.T.O., G.T. and B.V.; Writing—original draft, I.F., M.T.O. and G.T.; Writing—review & editing, I.F., M.T.O., G.T. and B.V. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Marie Skłodowska-Curie RISE H2020 project GEO-SAFE (691161) and the Government of Spain (MTM2015-65803-R).

**Acknowledgments:** The authors would like to thank Belén Fernández Fernández, for her valuable collaboration. **Conflicts of Interest:** The authors declare no conflict of interest.
