# A Multi-Objective Stochastic Model for an Earthquake Relief Network

Authors: Zeren D. Yenice; Funda Samanlioglu
Journal: Journal of Advanced Transportation
 (2020)

---

## Abstract

Earthquake relief network involves storage and distribution of relief aid to people in need. In this paper, a new stochastic multi-objective mixed integer mathematical model is developed and implemented in Kadikoy municipality of Istanbul, Turkey in order to configure part of the earthquake relief network. The aim of the model is to help decision makers decide on the locations of storage areas for shelters pre-earthquake and distribution of shelters from these areas to temporary shelter areas post-earthquake while minimizing earthquake scenario-specific total expected distribution distance, total expected earthquake damage risk factor of storage areas and expected total penalty cost related to unsatisfied demand at temporary shelter areas, simultaneously. In the model, storage area capacity and coverage distance restrictions are taken into consideration. The data related to potential storage areas and shelter locations were obtained from Kadikoy municipality of Istanbul and Istanbul Metropolitan Municipality (IMM). The earthquake damage risk factors were determined based on possible earthquake scenarios given in Japan International Cooperation Agency’s (JICA) report. Four event scenarios with two different earthquake scenario likelihoods were considered and sample efficient solutions from the Pareto frontier were obtained implementing the normalized (scaled) weighted sum method.


---

## Body

## 1. Introduction

Today’s one of the prominent problems is natural disasters such as earthquakes. An accurate prediction of earthquakes is not yet possible, however it is possible to plan the stages of disaster operations management (mitigation, preparedness, response and recovery) based on several earthquake scenarios. Inefficient disaster management; emergency response and disaster relief may lead to starvation, diseases and eventually loss of lives even after an earthquake. Immediate typical needs post-earthquake are search and rescue operations, medical assistance, food and water procurement, and provision of temporary shelters. In this paper, a multi-objective stochastic model is developed to help decision makers in decisions related to storage and distribution of shelters.Earthquake occurrence is the leading hazard in Turkey since the northern Anatolian fault crosses the country east to west, and the Anatolian fault north to south. 96% of the country’s area is prone to earthquake risk and 66% contains active faults. Turkey went through several earthquake disasters in the century and based on the Japan International Cooperation Agency’s (JICA) report [1], an earthquake is expected to occur in the following years at Marmara Sea which is very close to Istanbul and therefore people need to be prepared for it. An earthquake happened in August 17, 1999 and damaged several cities in the Marmara Region and the center of the earthquake was Golcuk. Estimated number of casualties were 15918, the number of heavily damaged houses were 66403, and the number of heavily damaged business units were 15000 [2]. Around 60% of these houses and business units collapsed immediately during the earthquake. Number of weakly or medium damaged houses were 146493 and number of weakly or medium damaged business units were 21000 [3]. As a result, around 120000 families needed immediate shelter after the earthquake in Golcuk which shows the significance of shelter storage and distribution problem.In the literature, stochastic models are developed to handle mostly location, allocation, and distribution decisions related to logistics. Logendran and Terrell [4] developed a facility location model with stochastic demands and determined allocation of clients to uncapacitated plants. Louveaux and Peeters [5] presented a stochastic facility location model taking into consideration uncertain demands, uncertain transportation and production costs and uncertain selling prices with different event scenarios. Laporte et al. [6] studied location of facilities and allocation of customers to facilities as first-stage, and determination of quantities to send as second-stage decisions. Mousavi and Niaki [7] presented a capacitated location allocation problem where demands and locations of customers were uncertain. They assumed that demands are fuzzy and locations have normal probability distribution, and minimized fuzzy expected cost while finding optimal stochastic location.Stochastic models are also used to handle uncertainties in health services, security systems, and service systems. Carson and Batta [8] constructed a location allocation model for preparing dynamic ambulance position plan for campus emergency service. Demands changed based on the scenarios and ambulances were relocated in each scenario based on minimization of average response time. Mestre et al. [9] developed a stochastic location allocation model for hospital network planning of Portuguese National Health Service, minimizing total expected travel time and total expected cost under demand uncertainty. Michalopoulos et al. [10] developed a stochastic network model to determine the (priority) list of locations of radiation detectors to build in order to prevent nuclear smuggling. They created possible threat scenarios by determining smuggler population. Khoo and Teoh [11] presented a methodology to determine optimal fleet management decisions taking into consideration stochastic travelers’ demand.A number of stochastic models in the disaster relief literature addressed the issues related to storage locations, inventory levels and distributions of relief supplies. Inventory and location decisions are long term strategic resource allocation decisions, typically done for preparedness pre-disaster and distribution decisions are short term allocation decisions, typically done for response post-disaster. However, several models integrate both preparedness and response decisions implementing scenario-based stochastic programming approaches. Pre-disaster operations are listed as evacuation from the potential disaster sites, stock pre-positioning and facility location and post disaster operations are given as evacuation to shelters, relief distribution and casualty transportation. In their literature survey, Altay and Green [12] identified research contributions in different stages of disaster operations management and listed these stages as mitigation, preparedness, response and recovery. Caunhye et al. [13] stated that emergency transportation operations were divided into three parts in the literature: facility locations, relief distribution and casualty transportation. For these three parts, they compared models based on their objective functions and constraints, and showed that location models that plan pre-disaster operations are single-period, whereas models that plan relief transportation post-disaster are mostly multi-period. Extensive literature reviews related to relief distribution networks, two-stage stochastic programs in disaster management with challenges and future research directions and facility locations in humanitarian relief can be found in Anaya-Arenas et al. [14], Grass and Fischer’s [15] and Trivedi and Singh’s [16] research, respectively.Some of the stochastic models in the disaster relief literature include single objectives such as minimization of total cost, total delay, expected response time and maximization of total expected demand. Barbarosoglu and Arda [17] developed a two-stage stochastic multi-commodity, multi-modal network flow model to plan the transportation of vital first-aid commodities to disaster-affected areas, and validated the model with the data of 1997 Istanbul earthquake. In the model, resource requirements were taken as random variables and due to uncertainty of transportation systems in emergency situations, random arc capacities and supply amounts were used. The objective function was the minimization of the total first-stage transportation cost and the expected recourse cost which includes the total flow costs, mode shift costs and the penalty costs of inventory holding and shortage in the second stage. Chia [18] stated that the primary role of the logistic management in disasters is to deliver supplies in good conditions with the required quantities at the right time to the right place, while taking into consideration bottlenecks related to destruction of roads and security. Yi and Özdamar [19] introduced a location and distribution model that coordinates logistics and evacuation operations in emergency response management. The model was a mixed integer multi-commodity network flow model and its objective function was minimizing total delay in delivery of the commodities at aid centers and in the provision of healthcare for the injured survivors. Balcik and Beamon [20] introduced a maximal covering location model to maximize the total expected demand covered by distribution centers, while taking into consideration multiple item types, and budget and capacity restrictions. Rawls and Turnquist [21] presented a two-stage stochastic optimization model to pre-position various kinds of emergency supplies in storage areas pre-hurricane and to ship those to demand points post-hurricane and presented a case study focusing on the hurricane threat in the Gulf coast of US. In their model, they considered scenario-based demand locations, quantities, and transportation capacities, and minimized the total cost which includes fixed cost of opening storage facilities, acquisition cost for items, expected cost of shipment to demand points, expected holding cost of unused items and expected penalty cost for shortage at demand points. Duran et al. [22] developed a stochastic model to increase effectiveness of CARE’s (Cooperative for Assistance and Relief Everywhere) average relief-aid emergency response time. The model included stochastic demands that change according to the disaster type and the objective was to minimize the expected response time. Davis et al. [23] developed a stochastic two stage expected cost minimization model in order to manage relief distribution before and after a hurricane. The first stage included preposition of relief goods between storage areas according to forecasts, and the second stage was the response phase that includes distribution of relief goods between storage points and demand points. Paul and MacDonald [24] developed a stochastic model to determine the location and capacity of distribution centers in order to mitigate the impacts of disasters with (almost) no forewarning such as earthquakes. The objective function in the model was to minimize the sum of expected fatality costs, supply costs and the costs of building distribution centers. Cavdur et al. [25] studied the allocation problem of temporary disaster response facilities and presented an earthquake case-study in Turkey. They developed a two-stage stochastic program, minimizing the total distance traveled, the unmet demand and the total number of facilities. Celik et al. [26] presented a two-stage mixed integer programming model that includes preposition of facility locations and pre-stocking levels of relief supplies along with minimization of total fixed facility locating cost at the first stage. At the second stage, proposed model allocates located distribution centers to affected locations and distributes relief supplies to demand points. They considered uncertainty in demand and minimized the expected cost of satisfying demand based on different scenarios. Maharjan and Hanaoka [27] determined the optimal number and location of warehouses for relief distribution problem in Nepal by modeling it as a modified maximal covering location problem. Mohamadi and Yaghoubi [28] presented a bi-objective stochastic optimization model for the location of transfer points and medical supplies distribution centers, taking into consideration triage system and failure of distribution centers and routes, and utilizing backup distribution centers. They converted the bi-objective model to a single-objective model with e-constraint method and presented a case study about the erathquake disaster in Iran.Some stochastic models in the disaster relief literature include multiple objectives. Sheu [29] presented a dynamic multi-criteria relief demand management model to meet urgent relief demands based on time-varying relief demand and demand urgency of the affected area. Mete and Zabinsky [30] developed a two-stage stochastic programming model for storage and distribution of medical supplies and presented a case study in Seattle area. At the first stage, warehouse locations and inventory levels are determined minimizing the total warehouse operation cost and at the second stage, the amount of medical supplies to be distributed to the hospitals are determined minimizing the total transportation duration and the penalty of unfulfilled demand. Zhang et al. [31] introduced a node-weighted bottleneck Steiner tree based multi-objective location optimization model for emergency response logistics. The weights represented the importance of demand points and the model had two objective functions, minimizing the total distance and minimizing the maximum distance between facilities and demand points. Bozorgi-Amiri et al. [32] developed a bi-objective stochastic model for disaster relief logistics where the demands, supplies and the procurement and transportation costs were considered uncertain. The objectives in the model were minimization of the sum of the expected value and the variance of the total cost and penalty due to infeasibilities and minimization of the sum of the maximum shortages in the affected demand points. Najafi et al. [33] presented a multi-commodity, multi-mode and multi-period stochastic model to manage the logistics of commodities and injured people in the earthquake response. In their model, they minimized the total (weighted) unserved injured people, the total (weighted) unsatisfied demand, and the total number of used vehicles. Liberatore et al. [34] worked on distribution of emergency goods and presented a multi-criteria model called RecHADS with a case study about 2010 Haiti earthquake. Their model included reliability and security attributes as well as cost attribute. Hong et al. [35] developed a goal programming multi-objective facility location model and presented a case study with the data of South Carolina, USA. In the model, the goals were about the minimization of total logistics costs that consist of fixed facility costs and transportation costs and the minimization of the expected number of disrupted relief items in an emergency logistics network. Tofighi et al. [36] presented a two-stage stochastic model for storage and distribution of critical and noncritical relief supplies and presented a case study in Tehran. At the first stage, they determined locations of warehouses and distribution centers along with inventory levels and capacities of warehouses, minimizing the total cost of operation warehouses and distribution centers and inventory costs. At the second stage, they planned the distribution of items, minimizing the total distribution time, the maximum weighted distribution time for the critical items and the total cost of unused inventories and the weighted shortage cost of unmet demands based on different scenarios. Moreno et al. [37] developed a two-stage stochastic location and transportation model which involves social concerns by considering deprivation costs in an emergency distribution network. At the first stage, they tried to obtain locations of relief centers and transportation capacities, minimizing costs of opening and operating relief centers and acquiring transportation capacities. At the second stage, they determined transportation of relief goods, minimizing total costs of acquiring extra transportation capacity, transportation costs, inventory holding costs and monetary cost of depriving victims.In this paper, a multi objective stochastic mixed integer location and distribution model was developed in order to prepare an inventory and distribution plan and solve predicted shelter provision problem. In the model, the criteria considered were simultaneous minimization of total expected distribution distance, total expected earthquake damage risk factor of storage areas and total expected unsatisfied demand penalty cost of demand points. Here, total expected distribution distance was used as one of the objectives instead of total expected distribution cost or response time since in this research these are assumed to be positively correlated with the distribution distance. Here, locations of shelter storage areas (SA) are determined from potential SA and shelters are distributed from the SA to temporary shelter areas (TSA), taking into consideration earthquake scenario dependent demand, earthquake damage risk factor of storage areas, coverage distance and storage capacity restrictions. To the best of authors’ knowledge, none of the stochastic models presented in the disaster relief literature include simultaneously these objectives and coverage distance and capacity restrictions. Kadikoy municipality in Istanbul Turkey was chosen as the pilot application area, and efficient solutions from the Pareto frontier were obtained in order to make decisions related to pre-earthquake shelter storage and post-earthquake shelter distribution to survivors. In Section2, details of the mathematical model are presented, along with the normalized (scaled) weighted sum method implementation. Details of the application of the mathematical model in Turkey is given in Section 3, along with conclusions and suggestions for future directions in Section 4.


## 2. The Mathematical Model

The proposed mathematical model is a stochastic multi-criteria mixed integer model. Expected values are optimized in stochastic models since uncertainty creates possible different probabilistic scenarios. The notation, parameters, and decision variables of the model are presented below.Sets;S
=
n
|
n
=
1
…
N
set of SA.F
⊂
S set of fixed SA.D
=
j
|
j
=
1
…
Mset of TSA.S
c
=
t
|
t
=
1
…
Wset of earthquake scenario in the model.Decision Variables;A
n
:
1
,
if storage area
n
is used for storing shelters
n
S
0
,
otherwiseX
n
j
t
: Amount of shelters sent from SA n to TSA j according to scenario t,j
∈
D
,
t
∈
S
c.U
j
t
: Amount of unsatisfied demand at TSA j according to scenario t, j
∈
D
,
t
∈
S
c.B
n
j
t
1
,
if storage area
n
serves to shelter area
j
according to scenario
t
,
n
S
,
j
D
,
t
S
c
0
,
otherwiseParameters;P
t Probability of earthquake scenario t, t
∈
S
c.C
n
: Potential storage capacity of SA n,
n
∈
S.R
n
t
: Potential earthquake damage risk factor of SA n according to scenario t,
n
∈
S,t
∈
S
c
.D
j
t
: Demand of TSA j according to scenario t, j
∈
D,
t
∈
S
c.T
n
j
: Distance between SA n and TSA j, n
∈
S
,
j
∈
D
.R
j
t
: Potential earthquake damage risk factor of TSA j according to scenario t, j
∈
D
,
t
∈
S
c
.M
j
t
: Penalty cost for unsatisfied demand of TSA j according to scenario t, j
∈
D
,
t
∈
S
c.M
j
t
=
L
∗
R
j
t
∀
j
∈
D
,
t
∈
S
c
, where
L
=
max
n
S
C
n.Useful Definitions;Definition 1: K is the distance limit for shelter allocation and defined as
K
≈
max
n
∈
S
,
j
∈
D
T
n
j
/
2, K value can be changed according to decision maker’s preference.Definition 2: H is a large enough number and defined asH
≈
max
n
∈
S
C
n.The mathematical model is as follows:(1)
Minimize
f
1
x
=
∑
t
∈
S
c
P
t
∑
n
∈
S
∑
j
∈
D
T
n
j
B
n
j
t
.
(2)
Minimize
f
2
x
=
∑
t
∈
S
c
P
t
∑
n
∈
S
R
n
t
A
n
.
(3)
Minimize
f
3
x
=
∑
t
∈
S
c
P
t
∑
j
∈
D
M
j
t
U
j
t
.s.t.(4)
∑
j
∈
D
X
n
j
t
≤
C
n
A
n
∀
n
∈
S
,
t
∈
S
c
.
(5)
∑
n
∈
S
X
n
j
t
+
U
j
t
=
D
j
t
∀
j
∈
D
,
t
∈
S
c
.
(6)
X
n
j
t
≤
H
B
n
j
t
∀
n
∈
S
,
j
∈
D
,
t
∈
S
c
.
(7)
A
n
=
1
∀
n
∈
F
.
(8)
T
n
j
B
n
j
t
≤
K
∀
n
∈
S
,
j
∈
D
,
t
∈
S
c
.
(9)
B
n
j
t
≤
A
n
∀
n
∈
S
,
j
∈
D
,
t
∈
S
c
.
(10)
A
n
∈
0
,
1
∀
n
∈
S
.
(11)
B
n
j
t
∈
0
,
1
∀
n
∈
S
,
j
∈
D
,
t
∈
S
c
.
(12)
X
n
j
t
≥
0
∀
n
∈
S
,
j
∈
D
,
t
∈
S
c
.
(13)
U
j
t
≥
0
∀
j
∈
D
,
t
∈
S
c
.The first objective function (1) is the minimization of total expected distribution distance between SA and TSA. Second objective function (2) is the minimization of total expected earthquake damage risk factor of SA. Third objective function (3) is the minimization of total expected penalty cost related to unsatisfied demand at TSA. Note that, in (3) penalty cost of unsatisfied demand at TSA depends on the potential earthquake damage risk factor of TSA. Constraint (4) limits the distribution of shelters from SA to SA’s storage capacity. Constraint (5) is about the satisfaction of demand at TSA. Constraint (6) is to relateX
n
j
t and B
n
j
t variables and assign “1” to variable B
njt if related SA sends shelter(s) to TSA. Constraint (7) is to mark fixed SA (already decided storage areas). Constraint (8) is the coverage distance restriction for distribution of shelters from SA to TSA. Constraint (9) is to relate B
n
j
t and A
n variables and to ensure that only used SA serves to TSA. Constraints (10) and (11) are for binary decision variables and Constraints (12) and (13) are nonnegativity restrictions.In this paper, a three-objective mathematical model is formulated in order to simultaneously consider three objectives and due to its simplicity, normalized (scaled) weighted sum method is used to obtain representative efficient solutions from the Pareto frontier. The solution of (normalized) weighted sum problem is efficient (Pareto optimal) if all weights are positive [38].A MOP,min
f
x
=
f
1
x
,
f
2
x
,
…
f
k
x
,
s
.
t
.
x
∈
X is assumed to have kcompeting objective functions (f
i
:
R
n
→
R) that are to be minimized simultaneously.Definition 1.
A decision vectorx
′
∈
X is efficient (Pareto optimal) for MOP if there does not exist a x
∈
X
,
x
≠
x
′ such that f
i
x
≤
f
i
x
′
f
o
r
i
=
1
,
…
,
k with strict inequality holding for at least one index i. (x
′
∈
X is efficient, f
i
x
′ is non-dominated.)Definition 2.
A decision vectorx
′
∈
X is weakly efficient (weakly Pareto optimal) for MOP if there does not exist a x
∈
X
,
x
≠
x
′ such that f
i
x
<
f
i
x
′
f
o
r
i
=
1
,
.
.
.
,
k. (x
′
∈
X is weakly efficient, f
i
x
′ is weakly non-dominated.)The normalized weighted sum formulation of this problem is given in (14) as:(14)
min
z
=
∑
i
=
1
i
=
3
w
i
f
i
x
−
f
i
∗
x
f
i
∗
x
−
f
i
n
x
,s.t. (4)–(13).wherew
i
>
0 are the importance weights ∑
i
=
1
i
=
3
w
i
=
1, f
i
∗
x (i = 1, 2, 3) is the utopia point defined as f
i
∗
x
=
min
x
∈
X
f
i
x
−
δ
i= for i = 1, 2, 3 δ
i
>
0 and f
i
n
x (i= 1, 2, 3) is the nadir point defined as the upper bound of the Pareto optimal set. Problem (14) with different weights (w
i
>
0
and
∑
i
w
i
=
1) is solved each time to obtain several representative efficient solutions of the problem.


## 3. Application in Turkey

The mathematical model is applied in Kadikoy municipality of Istanbul, Turkey. The data related to the potential SA and TSA were obtained from Kadikoy municipality of Istanbul and Istanbul Metropolitan Municipality (IMM). Kadikoy municipality has 21 neighborhoods and in the paper for simplicity these neighborhoods are given numbers as: Caferaga (1), Osmanaga (2), Rasimpasa (3), Kosuyolu (4), Acibadem (5), Hasanpasa (6), Zuhtupasa (7), Egitim (8), Fikirtepe (9), Dumlupinar (10), Feneryolu (11), Merdivenkoy (12), Goztepe (13), Fenerbahce (14), Caddebostan (15), Erenkoy (16), Suadiye (17), Bostanci (18), Kozyatagi (19), 19 Mayis (20) and Sahrayicedit (21). The earthquake damage risks were determined based on possible earthquake scenarios presented in JICA report [1]. JICA team created four possible earthquake scenarios for fault line breakdowns of the expected earthquake in Turkey. These four earthquake scenarios were created based on historical earthquakes and North Anatolian fault line and listed below.Model A: Approximately 120 km long fault line of 1999 Izmit earthquake causes disaster and seismic activity goes from east to west. Moment magnitude was assumed to be 7.5 Mw.Model B: Approximately 110 km long fault line of 1912 Murefte-Sarkoy earthquake causes disaster and its moment magnitude was expected to be 7.4 Mw.Model C: This scenario assumes that 170 km long North Anatolian fault line in Marmara Sea will break at the same time. The moment magnitude is expected to be 7.7 Mw. It is the biggest magnitude that will occur in this area. The largest earthquake that has occurred around the Marmara Sea in the history has 7.6 Mw moment magnitude.Model D: This scenario assumes that the fault line in the north of Marmara Sea intersects with Cinarcik Graben and creates earthquake which has 6.7 Mw moment magnitude.Among these 4 earthquake scenarios, Model A has the greatest probability to take place and Model C is the worst-case scenario to happen. In the JICA report [1], detailed data were available only for these two earthquake scenarios, so in this application only Model A (earthquake scenario 1) and Model C (earthquake scenario 2) were considered. In Tables 1, 4 event scenarios were created taking into consideration that Model A (earthquake scenario 1) is more likely to occur than Model C (earthquake scenario 2) based on the JICA report [1].Table 1
Event scenarios.
Event scenario no
Probability of model A (earthquake scenario 1)
Probability of model C (earthquake scenario 2)
1
0.9
0.1
2
0.8
0.2
3
0.7
0.3
4
0.6
0.4JICA report [1] contains maps of Istanbul with color-coded areas that are based on the expected number of heavily damaged buildings and seismic intensity for both Model A and C. These maps are given in Figures 1–4. Red and orange colored areas will be affected heavily, yellow colored areas will suffer moderate level damage, green and blue colored areas will be affected lightly from the earthquake. From these maps, Kadiköy municipality is enlarged and related maps of Kadiköy municipality are given in Figures 5–8. In Figure 5, previously listed neighborhood numbers of Kadiköy municipality are also presented to show the readers the locations of these neighborhoods and potential SA and TSA. These maps are used to determine the potential earthquake damage risk factors of SA and TSA. For model A (earthquake scenario 1) and C (earthquake scenario 2), risk factor of each SA and TSA was calculated, depending on the neighborhood it is located at, as the average of damaged building factor, seismic damage factor and closeness to sea side factor due to possibility of tsunami. The neighborhoods of Kadiköy and related risk factors are given in Table 2. Damaged building factor and seismic damage factor were determined from color-coded hazard maps of JICA report [1], which are given in Figures 1–8. For these factors, numerical values 1–5 were assigned, 5 for very high risk and 1 for very low risk. Areas close to sea could be damaged more than far ones due to tsunami and south shores of Istanbul might be damaged as a result of an earthquake affecting Marmara Sea. To determine closeness to sea side factor of each SA and TSA, numerical values 1–5 were assigned, 5 for very close and 1 for very far.Figure 1
Number of heavily damaged buildings in model A [1].Figure 2
Number of heavily damaged buildings in model C [1].Figure 3
Seismic intensity in model A [1].Figure 4
Seismic intensity in model C [1].Figure 5
Map of number of damaged buildings of Kadiköy for model A [1].Figure 6
Seismic intensity map of Kadiköy for model A [1].Figure 7
Map of number of damaged building of Kadiköy for model C [1].Figure 8
Seismic intensity map of Kadiköy for model C [1].Table 2
Risk factors of neighborhoods of Kadiköy municipality for model A (earthquake scenario 1) and C (earthquake scenario 2).
Model A (earthquake scenario 1)
Model C (earthquake scenario 2)
Neighborhood
Damaged building factor
Seismic damage factor
Closeness to sea side factor
Risk factor
Damaged building factor
Seismic damage factor
Closeness to sea side factor
Risk factor
Caferaga (1)
4
2
4
3.33
4
3
4
3.67
Osmanaga (2)
3
2
3
2.67
4
3
3
3.33
Rasimpasa (3)
4
2
4
3.33
4
3
4
3.67
Kosuyolu (4)
3
2
2
2.33
4
2
2
2.67
Acibadem (5)
2
2
2
2.00
3
2
2
2.33
Hasanpasa (6)
4
3
3
3.33
4
3
3
3.33
Zuhtupasa (7)
3
3
4
3.33
3
3
4
3.33
Egitim (8)
3
3
2
2.67
4
3
2
3.00
Fikirtepe (9)
4
3
1
2.67
4
3
1
2.67
Dumlupinar (10)
3
2
1
2.00
3
2
1
2.00
Feneryolu (11)
2
2
3
2.33
2
3
3
2.67
Merdivenkoy (12)
3
2
1
2.00
3
3
1
2.33
Goztepe (13)
2
2
3
2.33
3
2
3
2.67
Fenerbahce (14)
2
3
4
3.00
3
3
4
3.33
Caddebostan (15)
2
2
4
2.67
2
3
4
3.00
Erenkoy (16)
2
2
3
2.33
3
2
3
2.67
Suadiye (17)
2
3
4
3.00
3
3
4
3.33
Bostanci (18)
2
2
4
2.67
3
3
4
3.33
Kozyatagi (19)
2
2
3
2.33
2
2
3
2.33
19 Mayis (20)
2
2
2
2.00
2
2
2
2.00
Sahrayicedit (21)
2
2
1
1.67
2
2
1
1.67In Table3, 12 TSA and related shelter demands and earthquake damage risk factors are given for both earthquake scenario 1 (Model A) and 2 (Model C). Amount of required shelters (demands) were calculated by IMM Disaster Coordination Center (AKOM) according to the sizes of TSA and those quantities were used as demands for the worst-case earthquake scenario (model C, earthquake scenario 2). Based on the JICA report [1], Model C has 2313 units of heavily buildings and model A has 1944 units. There is about 17% difference between these scenarios in terms of heavily damaged buildings and seismic intensity will also be lower in model A. So, model A’s (earthquake scenario 1) demands were assumed to be approximately 20% lower than Model C’s (earthquake scenario 2) demands.Table 3
Data of temporary shelter areas (TSA).
ID
Place and (Neighborhood)
Area (m2)
Shelter demands (model A)
Shelter demands (model C)
Risk factors (Rj1) of model A
Risk factors (Rj2) of model C
100
Kosuyolu park, (4)
12000
192
240
2.33
2.67
101
Kadikoy Municipality parking area, (6)
17000
272
340
3.33
3.33
102
Yogurtcu park, (2)
25000
400
500
2.67
3.33
103
Kadikoy Anadolu Lisesi, (1)
22000
352
440
3.33
3.67
104
Moda bazaar area and parking area, (1)
8000
128
160
3.33
3.67
105
Fenerbahce park, (14)
60000
960
1200
3.00
3.33
106
IETT bus station, (8)
8000
128
160
2.67
3
107
Fenerbahce sport facilities, (10)
19000
304
380
2.00
2
108
Selamicesme Ozgurluk park, (13)
120000
1920
2400
2.33
2.67
109
Goztepe park, (15)
90000
1440
1800
2.67
3
110
Cebe sokak, (21)
70000
1120
1400
1.67
1.67
111
Bostanci bazaar area, (18)
30000
480
600
2.67
3.3388 potential SA were considered in this research, and these were available public schools in Kadikoy. Also, in Chen et al. [39] research, potential shelter locations in a pilot area in China were analyzed and 41% of potential shelter locations were determined to be school playgrounds. For model A (earthquake scenario 1) and C (earthquake scenario 2), risk factor of each SA was calculated, depending on the neighborhood it is located at, using the risk factor of neighborhoods in Table 2. Shelter capacity of each SA was determined based on the size of its’ garden, since shelters are planned to be stored in standard transportation containers at the gardens of schools for storage and transportation convenience. In Table 4, 88 SA are presented along with their shelter capacities and calculated earthquake damage risk factors based on Model A and C. Shortest road distances between each SA and TSA (in km) were obtained by utilizing Arcview 9.3 GIS software and Istanbul geographical database.Table 4
Earthquake damage risk factors (RF) of storage areas (SA) based on model A (RF-A) and model C (RF-C) and shelter capacities in units.
ID
SA and (neigborhood)
RF-A
RF-C
Shelter capacity
ID
SA and (neighborhood)
RF-A
RF-C
Shelter capacity
1
29 Ekim Ilkokulu, (13)
2.33
2.67
250
45
Ilhami Ahmed Ornekal Ortaokulu, (13)
2.33
2.67
500
2
30 Agustos Ilkokulu, (20)
2.00
2.00
250
46
Ilhami Ertem Ortaokulu, (19)
2.33
2.33
500
3
60.Yil Anadolu Ilkokulu, (5)
2.00
2.33
500
47
Kazim Karabekir Ortaokulu, (20)
2.00
2.00
250
4
Bostanci Ilkokulu, (18)
2.67
3.33
250
48
Mehmet Sait Aydoslu Isitme Engelliler Ortaokulu, (13)
2.33
2.67
500
5
Cemal Diker Ilkokulu, (19)
2.33
2.33
250
49
Melahat Sefizade Ortaokulu, (11)
2.33
2.67
250
6
Cenap Sehabettin Ilkokulu, (4)
2.33
2.67
500
50
Melehat Akkutlu Ortaokulu, (12)
2.00
2.33
500
7
Dr. Sait Darga Ilkokulu, (5)
2.00
2.33
250
51
Mustafa Mihriban Boysan Ortaokulu, (17)
3.00
3.33
250
8
Erenkoy Ilkokulu, (16)
2.33
2.67
500
52
Nevzad Ayasbeyoglu Ortaokulu, (21)
1.67
1.67
250
9
Gazi Mustafa Kemal Pasa Ilkokulu, (2)
2.67
3.33
250
53
Nurettin Teksan Ortaokulu, (14)
3.00
3.33
500
10
Goztepe Ilkokulu, (13)
2.33
2.67
250
54
Resat Nuri Guntekin Ortaokulu, (4)
2.33
2.67
500
11
Halil Turkkan Ilkokulu, (4)
2.33
2.67
250
55
Yesilbahar Ortaokulu, (13)
2.33
2.67
500
12
Ibrahim Oktem Ilkokulu, (7)
3.33
3.33
250
56
Zuhtupasa Ortaokulu, (7)
3.33
3.33
250
13
Ihsan Sungu Ilkokulu, (3)
3.33
3.67
500
57
Istanbul Avni Akyol Guzel Sanatlar Lisesi, (13)
2.33
2.67
750
14
Ilhami Ahmed Ornekal Ilkokulu, (14)
3.00
3.33
500
58
Erenkoy Kiz Anadolu Lisesi, (16)
2.33
2.67
500
15
Inonu Ilkokulu, (8)
2.67
3.00
750
59
Fenerbahce Anadolu Lisesi, (13)
2.33
2.67
750
16
Kalamis Sehit Murat Ozyalcin Ilkokulu, (14)
3.00
3.33
250
60
Goztepe Ihsan Kursunoglu Anadolu Lisesi, (13)
2.33
2.67
500
17
Kaptan Hasanpasa Ilkokulu, (6)
3.33
3.33
250
61
Hayrullah Kefoglu Anadolu Lisesi, (18)
2.67
3.33
250
18
Kozyatagi Sukran Karabelli Ilkokulu, (19)
2.33
2.33
500
62
Istanbul Anadolu Lisesi, (7)
3.33
3.33
1000
19
Leman Kaya Ilkokulu (18)
2.67
3.33
250
63
Istanbul Kadikoy Lisesi, (1)
3.33
3.67
250
20
Mehmet Karamanci Ilkokulu, (17)
2.33
2.67
500
64
Kadikoy Anadolu Lisesi, (1)
3.33
3.67
1250
21
Mehmet Sait Aydoslu Isitme Engelliler Ilkokulu, (13)
2.33
2.67
500
65
Kazim Ismen Anadolu Lisesi, (4)
2.33
2.67
500
22
Moda Ilkokulu, (1)
3.33
3.67
250
66
Kemal Ataturk Anadolu Lisesi, (2)
2.67
3.33
750
23
Mustafa Aykin Ilkokulu, (11)
2.33
2.67
750
67
Mustafa Saffet Anadolu Lisesi, (12)
2.00
2.33
250
24
Nihat Isik Ilkokulu, (2)
2.67
3.33
500
68
Suadiye Haci Mustafa Tarman Anadolu Lisesi, (17)
3.00
3.33
500
25
Ogretmen Harun Resit Ilkokulu, (13)
2.33
2.67
250
69
Istanbul Ataturk Fen Lisesi, (8)
2.67
3.00
500
26
Osmangazi Ilkokulu, (3)
3.33
3.67
250
70
50.Yil Cumhuriyet Feridun Tumer Cok Programli Lisesi, (14)
3.00
3.33
250
27
Sener Birsoz Ilkokulu, (21)
1.67
1.67
250
71
Ahmet Sani Gezici Lisesi Kiz Teknik ve Meslek Lisesi, (5)
2.00
2.33
250
28
Turhan Mediha Tansel Ilkokulu, (17)
2.33
2.67
250
72
General Ali Riza Ersin Teknik ve Endustri Meslek Lisesi, (5)
2.00
2.33
250
29
Zihnipasa Ilkokulu, (16)
2.33
2.67
250
73
Kadikoy Kiz Teknik ve Meslek Lisesi, (1)
3.33
3.67
250
30
Zuhtupasa Ilkokulu, (7)
3.33
3.33
250
74
Kadikoy Muhsin Adil Binal Ticaret Meslek Lisesi, (1)
3.33
3.67
250
31
23 Nisan Zehra Hanim Imam Hatip Ortaokulu, (18)
2.67
3.33
500
75
Kadikoy Ticaret Meslek Lisesi, (13)
2.33
2.67
250
32
Mehmet Akif Imam Hatip Ortaokulu, (20)
2.00
2.00
250
76
Mehmet Beyazid Anadolu Saglik Meslek Lisesi, (10)
2.00
2.00
250
33
Bahariye Ortaokulu, (1)
3.33
3.67
500
77
Hamit Ibrahimiye Otistik Cocuklar Egt. Merkezi, (12)
2.00
2.33
250
34
Bostanci Ataturk Ortaokulu, (18)
2.67
3.33
500
78
Hayriye-Kemal Kusun Egt Uyg.Ok.ve Is Egt. M, (1)
3.33
3.67
250
35
Erenkoy Mehmet Sait Aydoslu Ortaokulu, (16)
2.33
2.67
250
79
Mediha-Turhan Tansel Egt Uyg.Ok.ve Is Eg. M, (16)
2.33
2.67
250
36
Faik Resit Unat Ortaokulu, (13)
2.33
2.67
250
80
Sohret Kursunoglu Ilkogretim Okulu ve Is Okulu, (20)
2.00
2.00
250
37
Fehmi Eksioglu Ortaokulu, (16)
2.33
2.67
500
81
Gozcubaba Imam Hatip Lisesi, (12)
2.00
2.33
250
38
Gazi Mustafa Kemal Pasa Ortaokulu, (2)
2.67
3.33
750
82
Intas Imam Hatip Lisesi, (16)
2.33
2.67
250
39
Goztepe H. Halil Turkkan Ortaokulu, (12)
2.00
2.33
250
83
Kadikoy Erkek Anadolu Imam Hatip lisesi, (6)
3.33
3.33
250
40
Goztepe Ortaokulu, (13)
2.33
2.67
250
84
Kadikoy Kiz Anadolu Imam Hatip Lisesi, (5)
2.00
2.33
250
41
Hakki Deger Ortaokulu, (19)
2.33
2.33
250
85
Marmara University Campus 1, (5)
2.00
2.33
1000
42
Halil Turkkan Ortaokulu, (13)
2.33
2.67
500
86
Marmara University Campus 2, (13)
2.33
2.67
2000
43
Huseyin Ayaz Ortaokulu, (10)
2.00
2.00
500
87
Marmara University Campus 3, (6)
3.33
3.33
1250
44
Ikbaliye Erdogan Yuksel Ortaokulu, (6)
3.33
3.33
250
88
Istanbul Medeniyet Universitesi, (10)
2.00
2.00
2000The mathematical model was written in “zpl” format with ZIMPL programming language, and solved with SCIP (Solving Constraint Integer Programs) Solver version 3.1, on an AMD 8 core 3.00 Gigahertz computer with 8 gigabyte RAM. SCIP is a noncommercial solver for mixed integer programming and mixed integer nonlinear programming and it is also a framework for constraint integer programming and branch-cut-and-price (SCIP 2018). Branch-cut-and-price method is a combination of branch-and-price and branch-and-cut methods. In branch-and-cut, the linear program (LP) relaxation of the sub-problems of branch-and bound is strengthened by cutting planes. In branch-and-price, at each node, the LP relaxation is solved with a column generation approach and whenever one of the implicitly given variables improve the current LP solution, it is added explicitly to the problem [40, 41]. For each event scenario, each objective function was individually minimized to obtain utopia and nadir points as seen in Table 5.Table 5
Utopia and nadir points for each event scenario.
Scenario 1
min
f
1
x
min
f
2
x
min
f
3
x
Scenario 2
min
f
1
x
min
f
2
x
min
f
3
x
f
1
x
0
15.25
1450.63
f
1
x
0
15.25
1450.63
f
2
x
225.087
3.362
225.087
f
2
x
227.634
3.398
227.634
f
3
x
40179424
40179424
0
f
3
x
41719488
41719488
0
f
i
∗
x
=
0
3.362
0
f
i
∗
x
=
0
3.398
0
f
i
n
x
=
1450.63
225.087
40179424
f
i
n
x
=
1450.63
227.634
41719488
Scenario 3
min
f
1
x
min
f
2
x
min
f
3
x
Scenario 4
min
f
1
x
min
f
2
x
min
f
3
x
f
1
x
0
15.25
1450.63
f
1
x
0
15.25
1450.63
f
2
x
230.181
3.432
230.181
f
2
x
232.728
3.466
232.728
f
3
x
43259552
43259552
0
f
3
x
44799616
44799616
0
f
i
∗
x
=
0
3.432
0
f
i
∗
x
=
0
3.466
0
f
i
n
x
=
1450.63
230.181
43259552
f
i
n
x
=
1450.63
232.728
44799616To determine representative efficient solutions of the problem from the Pareto frontier, a group of 16 dispersed weight vectors were generated in Table6, where w
i
>
0 are the weights (∑
i
w
i
=
1 ). Same set of dispersed weight vectors were also used in Samanlioglu’s research [42] to find representative efficient solutions. Readers can find methods for generating dispersed weight vectors in Steuer’s research [43].Table 6
16 dispersed weight vectors.
Solution
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
w
1
1/3
0.4
0.4
0.2
0.5
0.25
0.25
0.6
0.1
0.3
0.7
0.2
0.1
0.8
0.1
0.1
w
2
1/3
0.4
0.2
0.4
0.25
0.5
0.25
0.3
0.6
0.1
0.2
0.1
0.7
0.1
0.8
0.1
w
3
1/3
0.2
0.4
0.4
0.25
0.25
0.5
0.1
0.3
0.6
0.1
0.7
0.2
0.1
0.1
0.8These weight vectors were then used in normalized (scaled) weighted sum formulation of this problem (14) to obtain sample efficient solutions of the Pareto front. For each event scenario, problem (14) was solved 16 times, each with a different weight vector to obtain 16 representative efficient (Pareto optimal) solutions of the problem from the Pareto frontier. In Table7, these solutions are presented along with CPU times in seconds.Table 7
16 representative efficient solutions from the Pareto frontier for each event scenario.
Scenario 1
Scenario 2
Sol. no
z in (14)
f
1
x
f
2
x
f
3
x
CPU
z in (14)
f
1
x
f
2
x
f
3
x
CPU
1
0.028
22.724
15.421
589526
32
0.029
16.528
20.318
146560
4
2
0.031
22.724
15.421
589526
284
0.033
22.748
15.522
1164400
58
3
0.020
13.97
20.55
73280
24
0.021
14.948
20.79
146560
67036
4
0.030
22.724
15.421
589526
6
0.034
16.528
20.318
146560
4
5
0.025
17.114
17.225
490780
205
0.025
14.948
20.79
146560
8
6
0.035
22.724
15.421
589526
79
0.038
22.748
15.522
1164400
7
7
0.023
15.554
20.149
73.280
4
0.023
15.554
20.149
146560
8010
8
0.027
18.604
16.158
697556
214
0.028
16.618
17.46
1084400
237
9
0.039
22.72
15.421
582200
5
0.042
22.748
15.522
1084400
3
10
0.011
14.25
22.225
0
4620
0.012
15.5
22.46
0
5521
11
0.022
16.66
17.225
589526
84
0.023
14.24
20.79
246760
42
12
0.010
14.25
22.225
0
5642
0.011
15.5
22.46
0
23325
13
0.043
22.724
15.421
582200
17
0.045
22.748
15.522
1084400
12
14
0.015
11.747
20.886
296380
4
0.016
11.198
23.858
165240
8
15
0.041
19.275
9.761
2129496
2
0.043
18.89
9.862
4258992
2
16
0.009
15.871
21.819
0
33447
0.009
17.134
21.819
0
1218
Scenario 3
Scenario 4
Sol. no
z in (14)
f
1
x
f
2
x
f
3
x
CPU
z in (14)
f
1
x
f
2
x
f
3
x
CPU
1
0.031
17.502
20.487
219840
34
0.031
18.476
20.656
293120
546
2
0.036
22.772
15.623
1746600
31
0.036
18.476
20.656
293120
5
3
0.022
17.571
22.157
20040
29774
0.021
18.568
22.326
0
26293
4
0.035
17.502
20.487
219840
169
0.035
18.476
20.656
293120
117
5
0.026
17.502
20.487
219840
534
0.027
18.476
20.656
293120
677
6
0.041
22.772
15.623
1746600
12
0.042
18.476
20.656
293120
3
7
0.024
18.33
22.157
0
14146
0.024
19.58
22.326
0
1493
8
0.029
18.152
17.157
1626600
53
0.030
18.602
18.19
1540944
90
9
0.046
22.772
15.623
1746600
6
0.048
18.88
20.656
293120
4
10
0.012
16.75
22.695
0
5382
0.012
18
22.93
0
25839
11
0.024
14.86
21.025
370140
10
0.024
15.48
21.26
493520
7
12
0.011
18.33
22.157
0
21512
0.011
19.688
22.326
0
31976
13
0.047
22.772
15.623
1746600
3
0.050
22.796
15.724
2328800
3
14
0.016
11.526
24.688
199800
42
0.017
12.168
24.924
266400
100
15
0.044
18.505
9.963
6398508
2
0.046
18.12
10.064
8517984
2
16
0.010
18.33
22.157
0
16030
0.010
19.58
22.326
0
5340In Tables8 and 9, shelter distribution plan of a sample efficient solution (solution number 10 of event scenario 3 in (Table 7) is presented for model A (earthquake scenario 1) and C (earthquake scenario 2). In these tables, expected quantities of shelters that will be shipped from selected SA and TSA are given, and based on this sample efficient solution, all the shelter demands of TSA are satisfied for both earthquake scenarios, which can also be confirmed by the 0 value of f
3
x in Table 7. Also, Tables 8 and 9 show that changes in demand between earthquake scenarios (model A and C) affect the distribution process. The number of links between supply and demand points increase from 15 (model A) to 19 (model C).Table 8
Expected distribution plan of 10th solution of scenario 3 if model A occurs (X
n
j
1).
Temporary shelter areas
Storage areas
100
101
102
103
104
105
106
107
108
109
110
111
27
—
—
—
—
—
—
—
—
—
—
250
—
34
—
—
—
—
—
—
—
—
—
—
—
480
57
—
—
—
—
—
—
—
—
750
—
—
—
59
—
—
—
—
—
—
—
—
610
—
—
—
64
—
—
400
352
128
—
—
—
—
—
—
—
85
—
—
—
—
—
960
—
—
—
—
—
—
86
—
—
—
—
—
—
—
—
560
1440
—
—
87
192
272
—
—
—
—
128
—
—
—
—
—
88
—
—
—
—
—
—
—
304
—
—
870
—Table 9
Expected distribution plan of 10th solution of scenario 3 if model C occurs (X
n
j
2).
Temporary shelter areas
Storage areas
100
101
102
103
104
105
106
107
108
109
110
111
27
—
—
—
—
—
—
—
—
—
—
150
100
34
—
—
—
—
—
—
—
—
—
—
—
500
57
—
—
—
—
—
—
—
—
750
—
—
—
59
—
—
—
—
—
—
—
—
750
—
—
—
64
—
—
—
440
160
650
—
—
—
—
—
—
85
240
—
500
—
—
—
160
—
—
—
—
—
86
—
—
—
—
—
—
—
—
200
1800
—
—
87
—
340
—
—
—
550
-
330
—
—
—
—
88
—
—
—
—
—
—
—
50
700
—
1250
—Due to the nature of stochastic multi-criteria decision problem, the decision process has strategic importance. Proposed results show that while some of the solutions remain persistent along changing event scenarios, some of them may change dramatically. For instance, solution 7 in Table7 provides solutions with fully satisfied demand for event scenarios 3 and 4 but there is unsatisfied demand for event scenarios 1 and 2. Figure 9 shows box plots and means of each objective function against event scenario probabilities. Objective function 2 (total expected earthquake damage risk factor) is sensitive to scenario probabilities since there is an increase in mean and median values with respect to event scenarios. On the other hand, for objective function 1, it can be concluded that the mean of the total expected distribution distance is not sensitive to scenario probabilities. Additionally, all of the objective functions are sensitive to event scenario probabilities in terms of variance within the results of objective functions.Figure 9
Box plots and means of objective functions according to event scenarios.


## 4. Conclusions

In this study, a stochastic multi-objective mixed integer mathematical model for the location and distribution decisions in an earthquake relief network was developed. The model includes some aspects that can be seen in the literature; however, none of the existing models in the relief network literature simultaneously include decisions related to: pre-earthquake shelter storage at SA and post-earthquake shelter distribution from SA to TSA while taking into consideration storage capacity restrictions, earthquake scenario dependent demand, earthquake damage risk factor of SA, and coverage distance restrictions.In this paper, three potentially conflicting significant criteria that needs to be minimized simultaneously were presented. These are; minimization of total expected distribution distance between SA and TSA, minimization of total expected earthquake damage risk factor of SA, and minimization of total expected unsatisfied demand penalty cost of TSA. To obtain efficient solutions from the Pareto frontier, as the multi-criteria decision making method, normalized (scaled) weighted summethod was implemented for its simplicity. For each event scenario, representative Pareto optimal solutions for the problem were obtained, by generating dispersed weight vectors and solving the problem each time with different weight vectors. In reality, these weights are assigned to each objective function depending on decision-makers’ preferences, specifically importance they give to each objective.The model was implemented in a pilot area; Kadikoy municipality of Istanbul, Turkey and based on these results, suggestions were made to the municipality. While some assumptions were made due to a lack of some information, in the implementation, many real-life aspects of the earthquake relief network were considered and realistically implemented in the model. In the implementation, 12 TSA and 88 SA were taken into consideration and related data was obtained from Kadikoy municipality of Istanbul, IMM, and JICA report [1]. In the application, four event scenarios with two earthquake scenarios (model A and C presented in JICA report [1]), having different likelihoods, were created. Mathematical model was written with ZIMPL programming language and solved by SCIP Solver. The computational effort is reasonable since this problem is a multi-criteria strategic decision making problem which will be solved infrequently.For future research, to solve larger problems in a shorter time, an efficient heuristic may be developed. In the mathematical model, minimization of total storage and distribution costs were not taken into consideration since for relief networks, generally priority is given to improving the temporary life quality of survivors rather than budget issues, however if there are also strict budget limitations, storage and distribution costs should be included in the model besides penalty cost for unsatisfied demand. Also, as part of future research direction, in the model, minimization of total distribution time, and the effects of possible traffic and road closures may be taken into consideration.



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*Source: 1910632-2020-01-15.xml*