Tsunami Evacuation Mathematical Modelfor the City of Padang

R. Kusdiantara∗, R. Hadianti∗, M. S. Badri Kusuma† and E. Soewono∗

∗Department of Mathematics Institut Teknologi Bandung, Bandung 40132 Indonesia†Department of Civil Engineering Institut Teknologi Bandung, Bandung 40132 Indonesia

rkusdiantara@students.itb.ac.id, {hadianti, esoewono}@math.itb.ac.id, msbadrik@si.itb.ac.id

Abstract. Tsunami is a series of wave trains which travels with high speed on the sea surface. This traveling wave is caused bythe displacement of a large volume of water after the occurrence of an underwater earthquake or volcano eruptions. The speedof tsunami decreases when it reaches the sea shore along with the increase of its amplitudes. Two large tsunamis had occurredin the last decades in Indonesia with huge casualties and large damages. Indonesian Tsunami Early Warning System has beeninstalled along the west coast of Sumatra. This early warning system will give about 10-15 minutes to evacuate people fromhigh risk regions to the safe areas. Here in this paper, a mathematical model for Tsunami evacuation is presented with thecity of Padang as a study case. In the model, the safe areas are chosen from the existing and selected high rise buildings,low risk region with relatively high altitude and (proposed to be built) a flyover ring road. Each gathering points are locatedin the radius of approximately 1 km from the ring road. The model is formulated as an optimization problem with the totalnormalized evacuation time as the objective function. The constraints consist of maximum allowable evacuation time in eachroute, maximum capacity of each safe area, and the number of people to be evacuated. The optimization problem is solvednumerically using linear programming method with Matlab. Numerical results are shown for various evacuation scenarios forthe city of Padang.

Keywords: Linear Programming, Tsunami evacuation model.PACS: 89.60.Gg, 89.40.-a, 02.70.-c

INTRODUCTION

Pacific Earthquake Belt (Ring of Fire) is a horseshoe-shaped area that surrounds the Pacific Ocean coveringthe length of about 40,000 km. Around 90% of earth-quakes occurred in this area, in which 81% of themoccurred along the Ring of Fire. Large portion of In-donesian archipelago which lies in Pacific seismic belthas been causing earthquakes and volcanic eruptions. In-donesia is also a meeting point between the Pacific plate,the Indo-Australian plate and Eurasian plate. As a re-sult, Indonesia is high-risk to earthquakes and tsunamis[1]. Tsunamis occur due to a disorder that causes thedisplacement amount of water, caused by: earthquakes,volcanic eruptions, or meteorite that fell to earth. About90% of tsunamis occurred because of earthquakes un-der the sea as happened in Aceh and Japan recently[1].Here are some of the historical occurrence of large earth-quakes accompanied by tsunamis in the world:

1. March 11, 2011, earthquake in Japan, 373 km fromthe city of Tokyo with magnitude 9.0 on the RichterScale. The earthquake has also caused a tsunamialong the eastern coast of Japan[2].

2. October 26, 2010, large-scale earthquakes in theMentawai with magnitude 7.2 on the Richter Scale.On November 9, 2010, it was reported 156 people

died due to the tsunami[3].3. February 27, 2010, earthquake in Chile with 8.8

on the Richter Scale. On March 30, 2010, it wasreported 432 people died due to the tsunami. Re-sulting tsunamis that reach across the Pacific Oceanto New Zealand, Australia, the Hawaii, archipelagostates in the Pacific and Japan with the effects oflight and medium impact[4].

4. December 26, 2004, devastating earthquake mea-suring 9.0 on the Richter scale rocked Aceh andNorth Sumatra as well as cause a Tsunami in theIndian Ocean. These natural disasters, caused morethan 220,000 people died inhabitants[5].

In order to reduce casualties, Indonesian TsunamiEarly Warning System (InaTEWS) has been developedby the Government of Indonesia. The system is in di-rect control by the Indonesian Meteorogical, Climatol-ogy and Geophysic Agency (BMKG) in Jakarta. Withthis InaTEWS, BMKG can transmit a Tsunami alert incase of a potential Tsunami earthquake[6]. The system,which is still further developed by BMKG, provides onlya short period of time for evacuees to find safe placesfor evacuation. Time available after this warning is about15 minutes. With a very limited time, the main objec-tive here is to find the best scenario for the evacuation ofevacuees [7].

GEOGRAPHY AND DEMOGRAPHYCITY OF PADANG

The City of Padang is the capital city of West Sumatrawhich is located on the west coast of Sumatra whichhas coastline of the length about 84 km and area about694.96 km2. The altitude of the City of Padang variesfrom 0 to 1,853 m above sea level, where 40% ofthe total area has elevation only from 0 to 5 m. ThisTsunami high-risk region is divided into eight kelurahan(the smallest administrative region). The City of Padanghas five major rivers and 16 small rivers, with the longestriver of the length 20 km along Batang Kandis. Theseexisting rivers should be taken into account in selectingthe evacuation routes.

FIGURE 1. Tsunami High-risk Areas

The table below shows, the percentage of Tsunami riskin each kelurahan.

TABLE 1. Percentage of Tsunami High-risk Areas

Cluster Kelurahan Persentageof area with

Tsunami high risk

K1 Air Tawar Barat 100%

K2 Air Tawar Timur 100%

K3 Ulak Karang Utara 100%

K4 Gunung Pangilun 20%

K5 Ulak Karang Selatan 100%

K6 Lolong Belanti 70%

K7 Plamboyan Baru 100%

K8 Rimbo Kaluang 50%

Table 2 shows the population of each high-risk area.

EVACUATION SCENARIO

In the evacuation model here, the transportation unit be-ing used is PCU (per car unit), and 1 PCU is equivalent to1 car with a capacity of 8 persons. From the geographicaland demographic we described above, the simplification

TABLE 2. Number of Evacuees in High-risk Area

Clus- Village Population Number ofter People

in High-RiskArea

K1 Air Tawar Barat 17625 17625

K2 Air Tawar Timur 3744 3744

K3 Ulak Karang Utara 10032 10032

K4 Gunung Pangilun 11282 2256

K5 Ulak Karang 11260 11260Selatan

K6 Lolong Belanti 9101 6371

K7 Plamboyan Baru 5924 5924

K8 Rimbo Kaluang 4353 2177

Jumlah 73321 59389

of the problem needs to be done to build a mathematicalmodel. Listed below are the assumptions.

1. In each route vehicles run with the same speed.2. The speed v and the density ρ satisfy the relation

v = v f

(1− 2ρ

ρ f

), where v f is the maximum allow-

able speed, and ρ f is the largest density before con-gestion occurs [8].

3. Evacuees with no vehicle take separate routes.4. Every person and transportation vehicles are ready

at the Muster(gathering) Points when the evacuationstarts.

The evacuation process starts immediately after aTsunami alarm sounds, and evacuees gather at theMuster Point, and then from the Muster Point evacueesare distributed with designated transportation modes intothe Safe Area.

FIGURE 2. Evacuation Process

In FIGURE 2 the small circle represent the peo-ple/vehicles who will be evacuated, gathering at theMuster Point and being directed with designated trans-portation modes toward the Safe Area. In City of Padang,with limited high rise buildings satisfying the building

codes for Tsunami, and relatively no empty lands withhigh altitudes, we find that the total capacity of safeareas is far below the total number of evacuees. In themodel here, the scenario of evacuation will utilize highrise buildings and (proposed to be built) a flyover ringroad.

FIGURE 3. The Flyover

From the illustration above, it is assumed that flyovercan be considered as shared safe area of each cluster.There are five entry points E1-E5 to the flyover with totalcapacity 10000 people. The routes to the entry points aregiven below.

1. Entry point 1 can be accessed by cluster 1 andcluster 3.

2. Entry point 2 can be accessed by cluster 3 andcluster 5.

3. Entry point 3 can be accessed by cluster 5 andcluster 6.

4. Entry point 4 can be accessed by cluster 6 andcluster 7.

5. Entry point 5 can be accessed by cluster 7 andcluster 8.

Here are the distances from each Muster Point to theflyover.

The letter X above states that there is no access routefrom the Muster Point to corresponding safe area.

MODEL FORMULATION

As mention before, the evacuees in a muster point willbe evacuated with selected mode of transportation to anumber safe areas or to the flyover. In the following,we will denote by Mi as the muster point, the S j as thesafe area, and the Ek as the share area (see the followingillustration).

FIGURE 4. Illustration of The Model

Suppose there are n muster points and m safe area,the evacuation time to evacuate Ni people/PCUs inmuster point i to safe area j is

Ti j =

(Nili j

xi j −1)

di j +Li j

vi j, (1)

where xi j is the portion of people/PCUs in muster point-i to be evacuated to the safe area-j along the route (i, j)of the length Li j and with the corresponding speed vi j,li j is the number of lanes in route (i, j) and di j = 1/ρi j.In the following, we derive an optimization model fordetermining xi j, the optimal proportion of the evacueesin muster point i to safe area/entry point j, i= 1, ...,n, j =1, ...,m.The optimization model has the following constraints

1. for every i, ∑mj=1 (xi j) = 1, meaning that all evac-

uees in each muster point must be evacuated2. for every j, ∑n

i=1 (Nixi j) ≤ C j, meaning that thetotal number of evacuees to each safe area/entrypoint must be less than the capacity of the safearea/entry point

3. for every i, j, Ti j ≤ τ , meaning that all evacueesin each route must be evacuated within τ minutes

4. for every i, j, 0 ≤ xi j ≤ 1,

where C j is the capacity of the safe area j.Further we construct a multi-cluster model which allowsevacuees from different clusters share the same safe areaor the entry point of the flyover. Now we extend thenotation of the decision variables

xi jk is the proportion of evacuees from muster point ithat will be evacuated to safe area/entry point jwithin cluster k.

Due to the geographical aspects, we see that evacuees ina muster point can only be evacuated to a safe area/entrypoint within its cluster and its neighborhood.The optimization model has the following constraints

m(k)

∑j=1

(xi jk

)= 1, 1 ≤ i ≤ n,1 ≤ k ≤ K, (2)

n(k)

∑i=1,(i, j,k)/∈SS

(Nikxi jk

)≤C jk,1 ≤ j ≤ m,1 ≤ k ≤ K, (3)

SS = ∪Aa=1SSa, (4)

∑(i, j,k)∈SS

(Nikxi jk

)≤Ca, a = 1,2, ...,A (5)

(6)Ti jk ≤ τ, for every i, j,k, (7)

0 ≤ xi jk ≤ 1, for every i, j,k, (8)(9)

For the multi-cluster model, we induce the evacuationtime

Ti jk =

(Nikli jk

xi jk −1)

di jk +Li jk

vi jk(10)

We can choose the total evacuation time

K

∑k=1

n(k)

∑i=1

m(k)

∑j=1

(Ti jk

)(11)

as the objective function of our optimization model. Wethen have a linear programming, which is easily to solve.But the solution unrealistic since the parameter Li jk willbe regarded as a constant. Where as Li jk must have animportant role in obtaining the optimal solution. Since,we prefer to evacuate the evacuees from muster point ito a safe area j in cluster k with small distance Li jk. Forthat, we choose

K

∑k=1

n(k)

∑i=1

m(k)

∑j=1

(Ti jk

)Li jk (12)

as the objective function.In the optimization model (12) there is a development inparts related to capacity constraints in which some safeareas are shared by more than one cluster. Here A is thenumber of joint-safe areas, and for each 1 ≤ a ≤ A, Ca isthe capacity of the safe area which can be jointly sharedby the corresponding clusters.

The list of parameters are given in the followingtable.

Nik is the number of people from muster point i tothe safe area j in cluster k,

di jk is the distance between the two unit fromthe muster point i to the safe area j in cluster k,

vi jk is the speed of the unit from the muster point ito the safe area j in cluster k,

li jk is the number of evacuation lines fromthe muster point i to the safe area j in cluster k,

Li jk is the distance from the muster point i tothe safe area j in cluster k,

SS is the share area who can be access bycluster k,

A is the number of share area,C jk is the capacity of the safe area j in

cluster k,Ca is the capacity of the safe area at cluster k that

had been drop and become share area andbecome member of the SS or the combinationof cluster that can access Ca

K is the number of the clustern(k) is the number of the muster point in cluster k,m(k) is the number of the safe area in cluster k

NUMERICAL SIMULATION

In the following simulation, we use only a single modeof transportation i.e. with cars. Here are the data to be

used in the simulation for 8 clusters. We mean by theshortage as the total capacity at safe area minus the totalpopulation.

Cluster 1

Cluster 2

Cluster 3

Cluster 4

Cluster 5

Cluster 6

Cluster 7

Cluster 8

The optimal solution of the optimization problem aregiven.

Cluster 1

Cluster 2

Cluster 3

Cluster 4

Cluster 5

Cluster 6

Cluster 7

Cluster 8

The numerical solution shows that all evacuees in cluster2 and cluster 8 will be evacuated to safe areas withinthe cluster. This makes sense since the shortages in theseclusters are negative. Meanwhile, although the shortageaf cluster 7 is also negative, a number of evacuees in thiscluster will be evacuated to share area 5. Since certainmuster points in this cluster are closer to the share arearather than to the safe areas.

CONCLUSION

We derive an optimization model for determining op-timal tsunami evacuation plan for the City of Padang,where the evacuation is performed by using cars as the

transportation mode. We propose a flyover as a verticalstructure that can be used for evacuation. The solution ofthe optimization model shows that the flyover will givesignificant contribution evacuation in case of tsunami.The optimization model can be generalized to a more re-alistic model with more than one transportation modes.

ACKNOWLEDGEMENT

Part of this research is funded by Hibah Penelitian Ker-jasama Luar Negeri dan Publikasi Internasional DIKTI2010

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