Dynamic Model for Determining Disaster Evacuation Locations with Game Theory

Disaster management can be grouped into three stages with several activities that can be carried out starting from pre-disaster, during emergency response and post-disaster. In the pre-disaster stage one of which is mitigation, which is a series of efforts to reduce disaster risk, both through physical development and awareness raising and capacity to face the threat of disaster. In mitigation one important factor is about evacuation. Evacuation locations in disaster transportation management are urgently needed for an effective evacuation process. Evacuation location determination is a benchmark in handling evacuation that can minimize the impact of losses from the disaster. To determine the evacuation location dynamically using game theory can solve the problem by providing several alternative locations by showing the dominant factor in the variables in the selection of the evacuation site. Game theory will use pure and mixed strategies to provide the most optimum choice of criteria. With this method will produce a decision support system to determine the optimal evacuation location or safe area in disaster mitigation


Introduction
Disaster is an event or series of events that threaten and disrupt people's lives and livelihoods caused, both by natural factors and / or non-natural factors as well as human factors resulting in human casualties, environmental damage, property losses and psychological impacts (Law No. 24 of 2007 concerning Disaster Management). Evacuation is the process by which people are placed from dangerous places to safer places to reduce health problems and people's lives that are vulnerable to being affected (Saadatseresht et al., 2009) (Mei et al., 2016) (Coutinho-Rodrigues et al., 2012). Disaster management as a process of evacuation of victims and an early warning system for disaster arrivals is slow due to difficulties in determining the location of the evacuation. The selection of evacuation sites in mitigation must consider the places that are available and based on geographical data from the disaster area. Each disaster affected area has different characteristics and this needs to be considered during evacuation. Considerations in the evacuation process are calculations of the number of people who can be mobilized, the capacity that can be carried, and the route chosen to go to safety (Campos et al., 2012). Bringing and directing people to safe areas is a major challenge in managing evacuation areas (Kendra et al., 2008). Evacuation is characterized by an immediate and rapid movement of people who  (Hamidazada et al., 2019). The important thing to note is that there is no congestion and chaos during the evacuation process (LIM et al., 2013).
In (Rogers & Luna, 2004) One of the anticipatory actions that can be taken before a disaster comes is to determine the location of the evacuation or a place to stop for disaster relief and disseminated to the wider community to accelerate the process of evacuation of disaster victims so as to minimize losses from the disaster, Geographical Information Systems (GIS) will be utilized with use the Overlay and Buffer functions and proceed with calculations using the Simple Additive Weighting (SAW) method which is expected to determine the best evacuation location so that it can be taken into consideration and help related agencies such as BPBD in order to improve mitigation. Game Theory is a mathematical model used in situations of conflict or competition between various interests that face each other as competitors (Kondratev & Mazalov, 2020). This theory was developed to analyze the decision making process of different competitive situations, and involves two or more interests (Bennett, 1995). Applying game theory will produce an optimal evacuation location in the decision support system in disaster mitigation (Li et al., 2014).

Game Theory with Linear Programs
To solve mixed strategy games with dimensions or larger, one can use a linear program by transforming the maximal and minimax equations into a linear programming form (Singh & Tech, 2010). The method that will be used in this research is a linear program method which can be formulated generally as follows: In solving the problem of game theory using a linear program using the simplex method, namely forming a linear program and finding the optimum solution (Kirtiwant; P. Ghadle, 2014). The steps in linear programming game theory are as follows:

For P1 Players (Line Players)
Player I is a line player (maximizing player), so it can be expressed hope of winning player I in a sign of greater inequality. This means that player I might get more wins than V if player II By dividing the above inequality with V, we get: Because player I is the maximizing player, the objective function is to maximize the value of V or equal to minimize 1 V , then the linear program for player I can be formulated as follows:

For P2 Players (Column Players)
Player II is a column player (minimizing player), so it can be expressed hope of winning Player II in a smaller sign of inequality. This means that Player II may suffer less than V if Player I uses a weak strategy. So the expected value of winning player II is: By dividing the above inequality with V, we get: Because Player II is a minimizing player, the objective function is to minimize the value of V or equal to maximize 1 v , then the linear program can be formulated for player II as follows: The linear player program for player II is: Min g0 = Y1 + Y2 + ... + Yn With restrictions: The linear program formulation in player I is dual from player II and vice versa. Obtained an optimal strategy for player I to be the optimal strategy for player II. Furthermore, to solve this linear program problem can be solved by using the simplex method. Players and strategies are used to determine the results of the calculation of pay-offs that show the optimal solution for the strategies used in evacuation to achieve benefits in determining the location of the evacuation. The count of each player is explained by comparing the evacuation location 1 (L1) with evacuation location 3 (L3), evacuation location 2 (L2) with evacuation location 3 (L3), and evacuation location 1 (L1) with evacuation location 2 (L2). At each evacuation site get a strategy that can be used to achieve profits in evacuation, with the acquisition value of the player comparison game as follows: In the table above, it can be seen that the smallest value (maximin) in the row is -20 while in the column player the greatest value (minimaks) is 35. The maximin value is not the same, so the game does not have a saddle point so the next step will be to find the balance point with rules of domination. For Location 1 (L1) can be formulated into a linear program for line players as follows:

Evacuation Location 1 (L1) with Evacuation Location 3 (L3)
Because the acquisition matrix elements above have been added with K = 20, the game value becomes V = 25-20 = 5, so the optimal solution is 5. Obtained optimal strategy for Location 3, namely strategy (Height of Place), strategy (Number of Population), strategy (Length of Evacuation Path), strategy (Hierarchy of Roads), Strategy (Type of Transportation) and strategy (Area of Evacuation Place) with large game value ( value of games) of 5. For Location 3 (L3) can be formulated into a linear program for line players as follows: Maximize: With restrictions: Because the acquisition matrix elements above have been added with K = 20, the game value becomes V = 25-20 = 5, so the optimal solution is 5. Obtained optimal strategy for Location 3, namely strategy (Height of Place), strategy (Number of Population), strategy (Length of Evacuation Path), strategy (Hierarchy of Roads), Strategy (Type of Transportation) and strategy (Area of Evacuation Place) with large game value ( value of games) of 5. In the table above, it can be seen that the smallest value (maxima) in the row is -31 while in the column player, the biggest value(minimax) is 63. The maximal value with the minimax value is not the same, so the game does not have a saddle point, so the next step will be to find the balance point with rules of domination.  In the table above it can be seen that there is a similar value of 7, then the game is optimal because it has saddle points. In the table above, it can be seen that the smallest value (maximum) in the row is -40 while in the column player the biggest value (minimax) is 22. The maximum value with the minimum value is not the same so the game does not have a saddle point so the next step will be to find the balance point with rules of domination. For Location 2 (L2) Can be formulated into a linear program for line players as follows:  10 The following optimal solution is obtained:  Since the acquisition matrix elements above have been added K = 40, the game value is V = 25-40 = -15. So the optimal strategy is -15. Obtained optimal strategy for Location 2, namely strategy (Height of Place), strategy (Number of Population), strategy (Length of Evacuation Path), strategy (Road Hierarchy), Strategy (Type of Transportation) and strategy (Area of Evacuation Place) with large game value ( value of games) of -15.