Pre-Positioning Disaster Response Facilities and Relief Items

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Pre-Positioning Disaster Response Facilities and ReliefItemsÇiğdem Renklia & Serhan Durana

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Pre-Positioning Disaster Response Facilities and Relief Items

Çiğdem Renkli and Serhan Duran

Middle East Technical University, Industrial Engineering Department, Ankara, Turkey

Address correspondence to Çiğdem Renkli, Orta Doğu Teknik Üniversitesi Endüstri

Mühendisliği Bölümü Üniversiteler Mahallesi, Dumlupınar Bulvarı No:1 06800 Çankaya

Ankara/TURKEY; E-mail: [email protected]

Running Head: Pre-Positioning Disaster Response Facilities

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ABSTRACT

Large-scale disasters cause enormous damage to people living in the affected areas.

Providing relief quickly to the affected is a critical issue in recovering the effects of a disaster.

Pre-disaster planning has an important role on reducing the arrival time of relief items to the

affected areas and efficiently allocating them. In this study, a mixed integer programming model

is proposed in order to pre-position warehouses throughout a potential affected area and

determine the amount of relief items to be held in those warehouses. Time between the strike of

the disaster and arrival of relief items at the affected areas is aimed to be minimized. In addition,

using probabilistic constraints, the model ensures that relief items arrive at affected areas within

a certain time window with certain reliability. Considering instable fault lines on which Istanbul

is located, the proposed model is applied to the Istanbul case for pre-positioning warehouses a

priori to the possible expected large-scale earthquake.

Key Words: disaster management, humanitarian logistics, mixed integer programming,

probabilistic constraints. Dow

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INTRODUCTION

Disasters are stochastic in their nature; exact time, place, and magnitude of disasters

cannot be known before they strike. They are unpreventable events that cause many deaths and

affect even more people. People who lack survival needs like immediate medical assistance,

food, water, and need immediate assistance are defined as affected people (International

Federation of Red Cross and Red Crescent Societies 2012). Assisting affected people by

responding quickly and effectively is a crucial but difficult task due to these uncertainties. For

quick and effective response, humanitarian logistics plans should be prepared.

It is stated that `humanitarian supply chains are among the most dynamic and complex

supply chains in the world' (Thomas 2005). Challenges due to unpredictability of disasters must

be considered in humanitarian logistics. Balcik and Beamon (2008) describe the challenging

characteristics of humanitarian logistics as `Unpredictability of demand, in terms of timing,

location, type and size; suddenly occurring demand in very large amounts and short lead times

for a variety of supplies; high stakes associated with adequate and timely delivery; and, lack of

resources (supply, people, technology, transportation capacity, and money).' Just the numbers;

9,655 people killed and 124.5 million needing assistance in 2012 due to natural disasters indicate

the colossal needs to be satisfied timely (Guha-Sapir et al. 2013).

In addition to these challenging characteristics, survivability of infrastructure of the affected

areas is another concern (Barbarosoglu and Arda 2004). A large-scale disaster may destroy paths

linking locations on the affected area and block the roads for relief transportation. In this study,

we focus on the infrastructure vulnerability concept and propose a mathematical formulation that

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locates disaster response facilities (DRFs) and relief items while providing certain reliability on

relief item transportation.

The rest of the paper is organized as follows. First, a brief review of the related literature is

given. Then, proposed mathematical models are explained. Later, computational studies and

results are presented using a numerical example and a numerical study on the city of Istanbul. In

the Conclusion, the results are discussed and future studies are mentioned.

LITETATURE REVIEW

Humanitarian logistics is a broad area that includes different stages of disaster

management. Literature on humanitarian logistic can be classified as literature on preparedness,

literature on disaster response, and literature on relief operations (Apte 2010).

Hongzhong et al. propose a model that optimizes the location of facilities for medical

supplies to address large-scale emergencies in the Los Angeles area (Hongzhong et al. 2005). In

this study, an emergency service location problem is examined considering large-scale

emergency situations and characteristics of a large-scale facility location problem. Proposed

large-scale emergency facility location problem (LEMS) mainly decides on location of the

facilities. Uncertainty of demand is handled by defining a set of possible emergency situations,

and using parameters representing both the `likelihood that a certain emergency situation affects

a demand point and impact that the emergency situation will have on the population of a demand

point' (ibid.). Additional coverage for the demand points is provided by forcing the expected

coverage of a demand point to be greater than a required number.

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Balcik and Beamon (2008) study a variant of the maximal covering location problem that

determines number and location of distribution centers and amount of relief supplies to be stored

considering quick-onset disasters. The problem includes multiple relief items with different

coverage requirements, a pre-disaster budget constraint, an expected post-disaster budget

constraint, capacity constraints, and stepwise partial coverage. Uncertainty is again handled by

using possible scenarios.

Mete and Zabinsky also propose a two-stage stochastic programming problem that

determines locations of the facilities and inventory levels to be held. Their formulation provides

a solution for storage and distribution of medical supplies. Using possible disaster scenarios, the

model handles the uncertainty related with a disaster. They use the model in a case study for

earthquake scenarios in the Seattle area where hospitals use their own warehouses or shared

warehouses to store relief items. In the case study, the authors aim to use pre-positioning

warehouses to store additional items by considering timely delivery.

Another two-stage stochastic programming model is proposed by Rawls and Turnquist

(2010) to decide the location and amounts of various types of emergency supplies to be

positioned a priori to a disaster. Uncertainties in demand and in transportation network

availability are handled using scenarios. In addition, probability that supplies are damaged is

considered in the model. They stated that solving deterministic equivalent of the two-stage

stochastic programming model is problematic with large data and they used a L-shaped method

to solve it. The model is applied to a case study on hurricane threats in southeastern USA. Later,

Rawls and Turnquist (2011) extend their previous study by adding reliability and maximal

service distance constraints. They define a reliable set of scenarios and allow the model to

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endogenously select the reliability set of scenarios. Demand in a reliable set is certainly covered

within the maximal service distance. Sum of occurring probabilities of the reliable scenarios is

forced to be greater than a given reliability so that probability of covering all demand within the

maximal service distance is greater than or equal to that given reliability. After their work in

2011, Rawls and Turnquist extend their study with reliability constraints by adding time periods

(Rawls and Turnquist 2012).

Duran et al. (2011) conduct a study to improve CARE International's emergency response

times. The model finds optimal number and location of pre-positioning warehouses given that

demand for relief supplies can be met from both pre-positioned warehouses and suppliers. They

allow multiple events to occur within a replenishment lead time. Their Mixed Integer

Programming (MIP) model minimizes the average response time over all demand instances and

gives decisions on location of warehouses, quantities of supply from warehouses and from

suppliers, and the quantity of supply held in a warehouse.

Verma and Gaukler (2011) present a two-stage stochastic programming model in order to

locate disaster response facilities. Their model locates the facilities in the first stage, and in the

second stage it determines the amounts transported from an opened facility to demand points.

They use possible scenarios for epicenter of the disaster and their model takes the possible effect

of the disaster on the functioning of a facility into account using a random variable depending on

the distance between the facility and the epicenter of the disaster. They conduct a computational

study for the state of California and present their results.

Klibi et al. (2013) propose a stochastic programming approach for pre-positioning

emergency supplies. They use a scenario-based approach and make decisions on location,

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number, capacity of the facilities, and inventories to be held. They generate scenarios using the

Monte Carlo procedure, solve a two-stage stochastic programming model, and they evaluate

alternative designs using performance measures.

Gormez and Koksalan (2010) address the disaster response and relief facility location

problem for Istanbul. In terms of purposes, it is very similar to our study. However, the

formulations that are proposed are different. The data used are also obtained from the same

report (JICA and IMM 2002), however, the proposed models are single-item models and they are

deterministic. Stochastic elements are not included in the formulations that distinguish our study.

The preparation stage of humanitarian logistics, which is the focus of this work, includes

pre-positioning disaster response facilities and relief items a priori to a disaster. Facilities are

pre-positioned in order to reduce the response time. Although the exact time, magnitude, and the

place of a disaster cannot be known, its effect can be predicted. Therefore, in the literature,

stochasticity of the situation is mostly covered by considering possible scenarios and the

expected value of an objective function with respect to the scenarios. For example, when the

number of affected people in a population center is not known due to unpredictable magnitude of

the disaster, different scenarios with different disaster magnitude and thus with different number

of affected people can be utilized. Different probabilistic elements are present in disaster

scenarios such as survivability of the infrastructure. A disaster response facility can be pre-

positioned and some affected areas can be assigned to that facility to be served; however,

assuming that relief item transportation roads are not blocked, the aftermath of a disaster is not

realistic. This is why, in this study, probabilistic constraints are used to ensure a certain

reliability in transportation.

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Probabilistic constraints, which are similar to the ones used in emergency facility location

problems in order to provide reliability in serving a demand point, are modified and adapted to

humanitarian logistics area for serving the purpose of ensuring reliability for an affected area to

get its relief items. In an emergency vehicle problem ReVelle and Hogan (1989) introduce

probabilistic constraints in order to provide a certain service reliability for demand points. In

other words, they ensure that every demand point is served by at least one emergency vehicle

with a certain probability. In this study, we modify their probabilistic constraints and adapt them

for our humanitarian problem in order to ensure that whole demand of every affected area is

satisfied with a certain probability. So, the problem of pre-positioning disaster response facilities

and relief items is formulated with the probabilistic constraints, which are also called as chance

constraints. Effects of the chance constraints on the solutions are discussed and the results of

chance-constrained formulation and scenario-based formulation are compared on a simulated

data consisting of 16 affected areas and 65 candidate disaster response facility locations.

Afterwards, the related data of Istanbul European Side is introduced and the chance-constrained

formulation is also used on this real life case.

In humanitarian logistics literature as explained so far, formulations that capture the

stochastic properties of a disaster by considering a set of possible scenarios are frequently used.

On the other hand in this study, a chance constrained formulation is used to handle the

uncertainties imposed by a disaster.

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MATHEMATICAL MODELS

Firstly, we introduce a deterministic location-allocation formulation and name it as the

uncapacitated location problem (UNLP). UNLP chooses the optimal locations of disaster

response facilities (DRFs) among candidate sites, assigns the affected areas to DRFs, and

determines the inventories of relief items in the DRFs. UNLP’s objective is minimizing the total

weighted distance between the DRFs and their assigned affected areas. This formulation does not

capture the vulnerability of the infrastructure due to the disaster; therefore, we introduce chance

constraints to UNLP to have more reliable transportation. The new model in which the chance

constraints are included is called the uncapacitated location problem with chance constraints

(UNLP-C). First, UNLP formulation is introduced below. The aim is to minimize response time

by locating disaster response facilities and positioning relief items in those facilities a priori to a

possible disaster.

In the following formulation indices i, j, and k are used for indicating a set of candidate

locations, affected areas, and relief items, respectively.

Sets

I set of candidate DRF locations

J set of affected areas

K set of relief items

Parameters

ijd j J shortest distance between DRF i and affected area j

jkn j J , k K need (demand) for relief item k in affected area j

km k K maximal service distance of relief item k

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w number of DRFs to be located

kQ k K total number of relief item k to be allocated

k k K reliability for relief item k

Variables

iy i I 1, if a DRF is located on candidate location i; 0, otherwise

ijkt i I j J k K 1, if a DRF is located on candidate location i sends relief

item k to affected area j; 0, otherwise

ijkx i I j J k K amount of relief item k sent from DFR located on candidate

location I to affected area j

ikh i ,I k K amount of relief item k held at DRF located in candidate location i

Objective

,

min /ij ijk jk

i I j J k K j J k K

d x n

(1)

Constraints

ijk jk

i I

x n

,j J k K (2)

ijk ik

j J

x h

,i I k K (3)

1 ijk i

j J k K

t M y

i I (4)

2ijk ijkx M t i I ,j J k K (5)

ijk ijkt x i I ,j J k K (6)

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ij ijk k id t m y i I ,j J k K (7)

i

i I

y w

(8)

ik k

i I k K

h Q

(9)

0, 0 ik ijkh x i I ,j J k K (10)

, 0,1i ijky t i I ,j J k K (11)

Objective function (1) minimizes total weighted distance between affected areas and their

assigned DRFs. By this way, the model minimizes total response time after a disaster strikes.

Weights are the fractions of total items sent from DRFs to affected areas. Constraint set (2)

ensures that demand of all affected areas for all types of relief items are met. Constraint set (3)

forces that total amount of a relief item type held at a DRF equals to total amount of this relief

item type sent from the DRF to its assigned affected areas. Constraint set (4) guarantees that no

relief item can be sent to any affected area from a DRF at a location if a DRF is not located there.

1M stands for a big number in this constraint set. By Constraint set (5), a DRF cannot send the

type of relief item to an affected area if that DRF is not assigned to that affected area for sending

that type of relief item. 2M stands for a big number in this constraint set. If any relief item k is

not sent from a DRF on candidate location i to affected area j, then the corresponding binary

variable takes the value of zero due to the Constraint set (6) for all DRFs, affected areas and

relief item types. With the Constraint set (7), a DRF on candidate location i cannot be assigned to

affected area j for sending relief item k, if the distance between the DFR and the affected area is

greater than maximal service distance of that relief item. Maximal service distance increases as

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criticality of relief item decreases. Constraint sets (8-9) are budget-related constraints. Number of

DRFs to be pre-positioned is limited by Constraint set (8) and total amount of relief items to be

allocated is limited by Constraint set (9). Constraint sets (10) are non-negativity and (11) are

integrality constraints.

The UNLP model simply finds the optimal locations of the disaster response facilities

minimizing the weighted distance. Because the DRFs are uncapacitated, the number of affected

areas that can be assigned to an open DRF is not limited. Utilizing maximum service distance

constraints, Constraint set (7), UNLP distinguishes between the relief items. Other than its

objective function and Constraint set (7), UNLP is a classical location-allocation problem that

does not have any humanitarian perspective.

With humanitarian purposes in mind, we introduce the reliability constraints to UNLP.

Survivability of the infrastructure is an important concern in transportation of the relief items and

we want to provide a certain reliability for meeting the demand for the relief items through

transportation means. New parameters must be introduced in order to formulate the reliability

constraints. Let

k k K minimum service reliability for item k

ijv i I j J probability that path linking DRF i and affected area j is

blocked

We point out that probabilistic constraints are used before at emergency vehicle location

problem (ReVelle and Hogan 1989).Probabilistic constraints are used to provide a reliability

value for at least one vehicle being available to serve to a demand point, and this condition is

given as:

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1 ijkt

ij k

i I

v

j J .k K (12)

These reliability constraints are linearized by taking the logarithms of both sides of the

Inequality (12) and the Constraint Set (13) is obtained.

log log(1 )ij ijk k

i I

v t

j J .k K (13)

However, ensuring an affected area being served by at least one DRF is not totally

satisfying in disaster response since the amount of relief items arrived to the affected area is also

important for reducing human suffering. Therefore, the chance constraints are modified

accordingly. Let

(1 ) ijkt

jk ij k

i I

P C v

j J ,k K (14)

where the event jkC is defined as all of the relief item k sent to affected area j reached to the

affected area j. Left hand side of Inequality (14) is the probability that event jkC occurs

while right hand side is the service reliability of item k. Constraint set (14) is linearized taking

the logarithms of both sides.

log 1 logij ijk k

i I

v t

j J k K (15)

The formulation consisting of (1‒11) and (15) is the proposed UNLP-C model in this

study. Since, stochastic nature of disasters is mostly included in models by scenario-based

formulations in humanitarian logistics literature; we also propose a scenario-based formulation

(SBP). The final model considered in this study is the scenario-based model (SBP). SBP’s

constraints are similar to UNLP except in SBP, the expected average distance traveled is

minimized over all scenarios considered.

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Objective functions in all three models mentioned so far minimize the total weighted

distance between the affected areas and the DRFs, which helps minimizing the total average

response time. However, as well as the arrival time of the relief items, it is important that the

relief items actually arrive at the affected areas and the expected unsatisfied demand is

minimized. As a performance measure, we define / ij ijk jk

i I j J k K i I k K

E v x n

and use it in computations.

COMPUTATIONAL STUDIES AND RESULTS

We use the formulations introduced in the previous section in numerical examples created

by simulated data and in a numerical study that represents the pre-positioning problem for

European side of Istanbul city. In the numerical example with simulated data, we use different

disaster progression schemes with different intensities. In the numerical study, first, we introduce

the data for the Istanbul case and then use the proposed formulations to solve the problem.

Numerical Example

In the numerical example, we create an experimental setting where candidate DRF

locations and demand points are evenly distributed over a square area. In this experiment, we use

different disaster scenarios in which the center, intensity, and dissemination of the effect of a

disaster vary. There are 16 affected areas and 65 candidate DRF locations. We consider the four

disaster progression schemes that can be seen in Figure 1. In addition, in each of the disaster

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progression schemes, center of the earthquake can be thought to be at different points. The colors

represent the intensity of the disaster, which is also explained in Figure 1.

Black dots are the affected areas and white dots are the candidate DRF locations. Every

grid is a square that has a perimeter of 400 meters. We use shortest distance paths and rectilinear

distance between the locations. In this setting, the scenarios are defined in terms of their

progression schemes and the intensity levels. Population of each affected area is taken as the

same, 1,000 people, and it is assumed that the number of affected people is directly proportional

to the vulnerability factor of the area in which the population center is located. For example, if a

population center is located in an area whose vulnerability factor is 0.8, the number of affected

people at that location is 800.

Vulnerability factors of the paths are calculated as the weighted average of vulnerability

factors of intensity areas where weights are the distances traveled in the areas. Let the path

between candidate DRF location i and population center j consists of 10 grids, four of which are

in strong-intensity area, four of which are in very-strong intensity area, and the remaining two

are in the weak area. Given that vulnerability factors are 0.8, 0.6, and 0.4 in very strong-

intensity, strong-intensity and weak-intensity areas, respectively, then the vulnerability factor of

the path is; (4 x 0.8+4 x 0.6+2 x 0.4)/10 = 0.6.

For the sake of simplicity, one type of relief item is located at DRFs and it is assumed

that one affected person requires one relief item. In addition, maximum service distance is set to

7,000 meters and total number of relief items to be allocated equals to the total demand.

We assume four progression schemes and four different intensity levels; therefore, total

of 16 different problems are solved both using the chance constraints and without using them.

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Results are compared in terms of the average distance and total expected unsatisfied percentage

of demand. Introducing the chance constraint to the formulation always causes an increase in the

optimal objective function value, which is the total weighted distance in all of the scenarios. This

is because the chance constraints require opening DRFs in safer places even if the weighted

distance increases. To capture the differences between UNLP-C, UNLP, and SBP formulations a

small example is considered. A single intensity level and four disaster progression schemes are

used in order to create four different scenarios; Scenarios 1, 2, 3, and 4 use progression schemes

1, 2, 3, and 4, respectively. Intensity level is 0.8, 0.8, 0.4, and 0.2 for very strong, strong, weak,

and very weak intensity areas, respectively. UNLP and UNLP-C are solved for each scenario

(progression scheme) and locations chosen by them for each progression schemes are given in

Figure 2. SPB assumes all four scenarios are equally likely and are solved once, thus locations

chosen for SPB is the same for all four progression schemes in Figure 2.

It can be seen that locations chosen by UNLP and SBP are in higher intensity areas than

the DRF locations of UNLP-C. When UNLP-C is utilized, safer locations are chosen and thus

expected unsatisfied percentage of demand is decreased as observed in Table 1. Since chance

constraints are not included in scenario-based formulation, although it considers all possible

scenarios, it does not capture the risk of relief transportation blockage. Results in Table 1 help in

observing the tradeoff between the time and reliability concepts while providing a reliability for

relief transportation.

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Numerical Study

Data related to Istanbul region

The proposed model in the previous section is applied to the European Side of Istanbul.

All data are obtained from The Study on A Disaster Prevention/Mitigation Basic Plan in Istanbul

Including Seismic Microzonation in the Republic of Turkey (JICA and IMM 2002) prepared by

the Japan International Cooperation Agency (JICA) and the Istanbul Metropolitan Municipality

(IMM). In this report of JICA and IMM, a detailed investigation on disaster management in

Istanbul is given and we use only the relevant data. We use population data of all districts of

Istanbul, and using population density in a district we find the population center of each district

and take them as demand points. Again, in that research, four possible earthquake models are

proposed namely Model A, Model B, Model C, and Model D. They state that Model D resembles

Model A and Model B resembles Model C; thus, they estimate damages for only two of the

models: Model A and Model C. Because Model C is the worst case, in this study estimations for

Model C are used. There are studies in the literature on disaster prediction and risk assessment

for disasters like the studies of Zhang (2009), Li et al. (2013), Huang et al. (2011) and Ding et al.

(2012). However, this topic is not the focus of our study. In order to assess the risk, we use the

report of JICA and IMM and we estimate the parameters of the problem using that source. As an

example, the report of JICA and IMM has building damage percentages for each districts of

Istanbul. To illustrate, they estimate that 14.1% of buildings in Avcılar district will be heavily

damaged with Model A and 16.5% of them will be heavily damaged with Model C. Using

building damage percentages of Model C and population data for the Avcılar district, we

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estimate the demand of the Avcılar district for relief items, which is explained in detail later.

Also, the report of JICA and IMM includes a road risk evaluation. They provide blockage

probabilities of the roads with a map that can be seen in Figure 3. Different colors on the map

represent different magnitudes of risk and we assign each color a relative blockage probability.

Basically, we estimate three types of parameters in our study using the research of JICA and

IMM; demand for relief items, population centers of districts, and blockage probabilities of the

roads. The rest of this paper includes detailed information on estimation of these parameters.

There are 18 districts in Istanbul at the European Side. Weighted population centers of

the districts are used as demand points (affected areas) in the model. Since two of the districts,

Şişli and Avcılar, are divided into two due to their geographical shapes, there are 20 affected

areas in need of relief items.

The amounts of relief items needed by the affected areas are estimated based on the JICA

and IMM report. Most notable cause of damage is claimed to be building collapse. They evaluate

buildings located in each district in terms of their construction year, floor number, and structure;

and they group the buildings into three: heavily damaged buildings, moderately damaged

buildings, and partly damaged buildings. Heavily and moderately damaged buildings are not

usable for living until they are repaired or rebuilt. Therefore, people living in those types of

buildings will need certain items like tents, hygiene kits, food and water supplies. Additionally,

the need for medical equipment kits can be estimated using number of people living in these

types of buildings. The JICA and IMM report provides number and percentage of heavily,

moderately, and partly damaged buildings in each district together with population data of them.

In this study, the need for the five types of relief items are estimated for each district by

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multiplying district population with the total percentage of heavily and moderately damaged

buildings in that district. Using CARE International's specifications, demands for relief items for

all districts are calculated. According to CARE's specifications, one tent is for 5 people, one

medical equipment kit is for 50 people, one food kit is for 8 people, one hygiene kit is for 8

people, and a person needs 3 liters of water daily.

We assume that the DRFs are not damaged by an earthquake because they can be built as

disaster-resisting buildings. However, they can be isolated due to the fact that the paths linking

them to affected areas can be damaged. Therefore, 25 candidate DRF locations are chosen

considering isolation risk caused by road blockage after an earthquake.

In the report of JICA and IMM, a proposed emergency road network is presented. Roads

are evaluated according to route division, connection, factor on traffic characteristics, building

collapse risk, and crossing large bridges and viaducts in order to propose most suitable road map

to be used in emergency situations. In this study, we use shortest paths over the proposed

emergency road network (see JICA and IMM 2002).

In the report of JICA and IMM, blockage probabilities of the roads in the proposed road

network are determined considering width of roads and building collapse risk (JICA and IMM

2002). Roads are categorized into three groups: wide width roads (16 meters and over), medium

width roads (7‒15 meters), and narrow width roads (2‒6 meters) and for these three types of

roads, blockage probabilities are presented. The average value of a range is used as building

collapse risk of a road passing through an area. A path between a DRF location and an affected

area can pass through the areas with different blockage probabilities. As explained in the

previous section, a weighted average probability of blockage for the path is calculated where

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weights are the distances of parts passing through areas with different blockage probabilities. All

probabilities are less than 1 in the Istanbul case, which means there is no path that will be

definitely blocked, and therefore the explained calculation of blockage probabilities, finding an

average probability, is reasonable.

However, it is possible to calculate blockage probabilities of the paths in another way.

Again, we separate the path into the parts passing through the areas with different blockage

probabilities. Then, we multiply the probabilities of the parts not being blocked and extract the

multiplication from one. In this way, we find the probability that at least one of the parts is

blocked, which is named as overall blockage probability. Therefore, two different calculations

are possible for finding blockage probabilities; we can use average probabilities or overall

probabilities. Average probability calculation ignores the definitely blocked parts of the paths by

using distance weights and decreasing the blockage probability. On the other hand, overall

probability calculation ignores the distances of path parts. Finding the appropriate calculation is

not the main concern of this study; and since in Istanbul case, we do not have any parts that will

definitely be blocked, we prefer to use average probabilities initially.

Numerical results

Using data for the European Side of Istanbul, the problem of finding the locations for pre-

positioning DRFs and relief items a priori to a large-scale earthquake is solved. We set minimum

service reliabilities of relief items to 0.9, 0.85, 0.85, 0.7, and 0.7 for medical equipment, water,

food, hygiene kits, and tents, respectively. The formulation without the chance constraints

(UNLP) is used first. Then, chance-constrained formulation (UNLP-C) is used in order to obtain

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results with smaller expected unsatisfied percentage of demand than those of UNLP. Results of

UNLP and UNLP-C are presented in Table 2.

The minimum number of DRFs is four in order to maintain feasibility in Table 2.

Maximum number of DRFs is chosen arbitrarily as eight while preparing Table 2, since the

results are very similar when more DRFs are opened. These results are expected considering the

analysis done on the simulated data. The chance constraints cause the average distance to

increase and the expected loss of demand to decrease. This is why DRFs are opened at safer but

more distant locations from their assigned affected areas than those of the formulation without

the chance constraints. As an illustration, an examination of Figure 3 shows the optimal locations

when the number of total DRFs is five. Moreover, presented in Table 3 are the corresponding

results on the expected unsatisfied demand percentage of all relief items individually.

It can be seen in Table 3 that expected unsatisfied demand percentage of more critical

items are in less amounts than the less critical ones when the chance constraints are used. UNLP

treats as if all items have the same importance.

Shown in Figure 4 are the locations of DRFs with both blockage probability calculation

methods mentioned before. Minimum service reliabilities of relief items are taken as 0.5, 0.5,

0.4, 0.4, and 0.3 for medical equipment, food supply, water supply, hygiene kits, and tents,

respectively.

Dashed lines in Figure 4 show how the location of a DRF is changed when average

probabilities are used. As we can see from Figure 4, optimal DRF locations are located at safer

but further locations when overall probability calculation is used.

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Solution times of computational experiments are negligible in this study. Most of the

experiments can be solved in less than 2 minutes using Cplex 12.4 solver and none of the

experiments are solved in more than 4 minutes.

CONCLUSION

The preparation stage of humanitarian logistics is a critical stage for reducing human

suffering and minimizing damage caused by disasters. In the literature, mathematical models

proposed handle the stochastic features of disasters by using expected values with respect to

possible disaster scenarios. In the proposed model, instead of scenarios, probabilistic constraints

are used.

First, an MIP model is developed without chance constraints. Then, the chance

constraints are introduced and implemented into the formulation. Moreover, a scenario-based

formulation is also considered to enable comparisons. In all experiments, the results are similar.

The main effect of chance constraints is that optimal disaster response facility locations are

getting out of risky areas; thus, chance and reliability constraints provide a minimum reliability

for relief item transportation and give better results in terms of expected unsatisfied demand than

the formulations without chance constraints. However, inclusion of chance constraints in the

formulation increases average distance. Therefore, there is a tradeoff between minimizing

average distance and minimizing expected unsatisfied demand, which can be interpreted as the

tradeoff between the response time and the risk.

As a future study, a risk-averse objective function, which is suitable for humanitarian

purposes, can be formulated and used. In addition, for the cases in which it is impossible to

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satisfy all demand, a variation of maximal coverage problem together with the probabilistic

constraints can be developed. Also, considering our findings, there is a significant tradeoff

between the response time and risk in the Istanbul case, and risk assessment has an important

effect on the results. Although it is not the focus of this study, before using risk as a parameter in

such a study, a comprehensive risk assessment study is suggested to be carried out.

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Table 1. Average distance and Expected loss w.r.t. to optimal DRF locations found by three

formulations when w is four.

Results of UNLP-C Results of UNLP Results of SBP

Scenario 1

Average distance (m) 1,666 944 944

Expected loss (%) 44 51 51

Scenario 2

Average distance (m) 982 879 914


Scenario 3



Scenario 4



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Table 2. Results of UNLP-C and UNLP for different w values.

Results of UNLP - C Results of UNLP Difference over UNLP

w Avg. D.* E.L.** Avg. D. E.L. Avg. D. E.L.

4 6,155 4.2% 3,985 9.3% 54% -55%

5 4,950 5.2% 3,508 9.3% 41% -44%

6 4,357 5.1% 3,095 8.1% 41% -37%

7 3,777 5% 2,719 14.1% 39% -64%

8 3,403 5.5% 2,469 18.8% 38% -71%

*Avg. D. : average distance (meters); **E.L. : percent expected loss

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Table 3. Expected unsatisfied demand percentage of all relief items whenw is five.

Relief items

Medical

equipment kit

Water supply Food supply Hygiene Kit Tent

UNLP-C 4.1% 5.1% 5.1% 5.4% 5.4%

UNLP 9.3% 9.2% 9.3% 9.3% 9.3%

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Figure 1. Disaster progression schemes.

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Figure 2. Optimal DRF locations found by three formulations when w is four.

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Figure 3. Optimal DRF locations when w is five.

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Figure 4. Optimal DRF locations obtained by different probability calculations when w is five.

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