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Journal of the Operations Research Society of China

https://doi.org/10.1007/s40305-018-0215-5

Optimizing Locations and Scales of Emergency Warehouses

Based on Damage Scenarios

Bo-Chen Wang1·Miao Li1·Yi Hu1·Lin Huang1·Shu-Min Lin1

Received: 23 March 2018 / Revised: 7 June 2018 / Accepted: 27 June 2018

© Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-

Verlag GmbH Germany, part of Springer Nature 2018

Abstract

Choosing the locations and the capacities of emergency warehouses for the storage of

relief materials is critical to the quality of services provided in the wake of a large-

scale emergency such as an earthquake. This paper proposes a stochastic programming

model to determine disaster sites’ locations as well as their scales by considering

damaged scenarios of the facility and by introducing seismic resilience to describe the

ability of disaster sites to resist earthquakes. The objective of the model is to minimize

ﬁxed costs of building emergency warehouses, expected total transportation costs

under uncertain demands of disaster sites and penalty costs for lack of relief materials.

A local branching (LB) based solution method and a particle swarm optimization

(PSO) based solution method are proposed for the problem. Extensive numerical

experiments are conducted to assess the efﬁciency of the heuristic according to the

real data of Yunnan province in China.

Keywords Emergency warehouse location ·Damage scenario ·Disaster resilience ·

Stochastic programming

Mathematics Subject Classiﬁcation 90C15 ·91B32 ·90B15 ·90B80

BShu-Min Lin

minmin@shu.edu.cn

Bo-Chen Wang

websterchen@163.com

Miao Li

lioaim@163.com

Yi Hu

huyi616@shu.edu.cn

Lin Huang

linhuang0503@163.com

1School of Management, Shanghai University, Shanghai 200444, China

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B.-C. Wang et al.

1 Introduction

Earthquakes are uncertain and unpredictable and often result in many casualties.

Table 1presents the number of victims and the economic losses caused by earthquakes

of magnitude 6.5 and above in China from 2008 to 2017, according to data provided

by the China Seismological Bureau and Civil Affairs Bureau. To signiﬁcantly reduce

or even eliminate the losses caused by earthquakes, it is necessary to study the ﬁeld of

disaster emergency management. How to reasonably use limited emergency resources

and transport relief materials from emergency warehouses to disaster sites is the key

to improving the efﬁciency of emergency rescue. Moreover, whether the emergency

warehouse location is appropriate affects the operational efﬁciency of the entire relief

materials logistics network directly.

In the traditional facility location problem, it is usually assumed that the facility

will never be damaged or failed once it is built. However, in the real world, the facility

is possibly to be damaged. Thus, this study presented a stochastic mixed-integer pro-

gramming model considering the potential damage situation of emergency warehouses

to improve the reliability of disaster relief materials logistics networks. The commer-

cial linear programming solvers such as CPLEX can only obtain optimal solutions in

reasonable time for small-scale instances. Thus, we designed two solution methods,

the local branching (LB) based solution method and the particle swarm optimization

(PSO) based solution method, to solve the problem within a relatively short time for

large-scale instances.

The rest of this paper is organized as follows. Section 2reviews the existing lit-

erature. The background for the facility location problem is described in Sect. 3.A

stochastic mixed-integer programming model is formulated in Sect. 4.Tosolvethe

model, Sect. 5elaborates on our two solution approaches. Section 6presents the results

Table 1 The statistics of Chinese major earthquakes in 2008–2017

Date Earthquake

disaster

Magnitude Number of victims Immediate

pecuniary loss

(million yuan)

May 12th, 2008 Wenchuan 8.0 69 227 845 215

October 16th, 2008 Xiongxian 6.6 56 411.37

April 14th, 2010 Yushu 7.3 2 698 22 847.41

June 30th, 2012 Xinyuan 6.6 51 1 990.32

April 20th, 2013 Ya’an 7.0 11 470 66 513.7

July 22nd, 2013 Minxian 6.7 2 414 24 416

August 3rd, 2014 Ludian 6.5 3 143 19 849

October 7th, 2014 Pu’er 6.6 331 5 110.2

July 3rd, 2015 Pishan 6.5 260 5 340

November 25th,

2016

Aketao 6.7 1 17

August 8th, 2017 Jiuzhaigou 7.0 525 114.46

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Optimizing Locations and Scales of Emergency Warehouses…

of the numerical experiments. Section 7concludes our research and proposes some

future work.

2 Literature Review

The facility location problem in disaster areas has received considerable attention in

the past few decades. Disasters can be divided into pre-disaster stage and post-disaster

stage [1]. The life cycle of disaster relief includes four stages, namely mitigation,

preparation, response and recovery [2]. A series of measures are taken in the pre-

disaster stage during the mitigation stage and the preparation stage to reduce the

potential losses from a disaster. The papers related to these two stages concern facility

location and inventory management [3–5]. Other works focused on the response stage

and the recovery stage in the post-disaster stage and mainly discussed the distribution

of relief materials in the logistics network [6–8].

We provide a survey of some closely related studies. Cui et al. [9] investigated

the reliable uncapacitated ﬁxed charge location problem. They proposed a contin-

uum approximation model to solve this problem, which is more efﬁcient than the

Lagrangian relaxation algorithm. Rath and Gutjahr [10] studied how to build ware-

houses to meet the material needs of affected people after natural disasters. They

developed a three-objective optimization model and designed an exact solution method

and ‘mathematical heuristic’ technique to solve their model. Abounacer et al. [11] stud-

ied a multiple objective location-routing problem in the response stage. The objective

of the location problem is to determine the number, position and mission of distribu-

tion centers within the disaster region; the objective of the routing problem is related

to the distribution of aid from distribution centers to disaster sites. Zhen et al. [12]

studied disaster relief facility network planning in metropolises, which concerned

the long-term decisions on locating the emergency shelters, the supply and medical

centers, and maintaining ﬂeets of ambulances and transportation vehicles. Moreover,

they developed a Lagrangian relaxation method for solving the integer programming

model. Caunhye et al. [13] proposed a two-stage location-routing model by integrat-

ing the preparation and the response stages. Their model is used for reducing the risk

in disaster situations where there are uncertain demands and facilities. The solution

method is converting this two-stage model into a single-stage counterpart.

Regarding the uncertainty involved in the model, stochastic programming methods

are commonly used in some studies [14]. Salman and Yücel [15] modeled the spatial

impact of a disaster on network paths by stochastic failures with dependency. In order

to obtain better solutions efﬁciently, they designed a tabu search heuristic algorithm.

Verma and Gaukler [16] developed two location models by considering the impact

of disasters. For the stochastic programming model, they designed a solution method

based on Benders decomposition, which can be applied to other two-stage stochastic

programming problems. Yu et al. [17] formulated a scenario-based model that not

only considered the uncapacitated facility location problem but also considered more

realistic factors in rescue activities. They put forward important guidelines for con-

trolling the risk of interruption. Qu et al. [18] studied the relationship between the

speed-density and ﬂow-density in trafﬁc ﬂows, which can be considered in studies on

123

B.-C. Wang et al.

the decision of relief materials transportation. Moreno et al. [19] provided two MIP

models to coordinate facility location, transportation and ﬂeet sizing decisions under

uncertainty. A heuristic algorithm is proposed to solve the problem. Their conclu-

sion showed that the total costs can be reduced by integrating multi-period decision

making and vehicle reuse. Wang and Meng [20] proposed a mixed-integer, nonlin-

ear, non-convex programming model for container shipping networks. Furthermore, a

mixed-integer linear programming model is obtained by applying linearization tech-

niques [21]. Zhang et al. [22] investigated the facility location problem by considering

the disruptions of facilities and the cost savings. They designed an exact algorithm and

a heuristic algorithm based on Lagrangian relaxation to solve this problem effectively.

To sum up, the studies on facility location problem commonly include the multi-

objective, multi-facility and multi-commodity optimization decision problems in

different pre-disaster and post-disaster stages. In addition, decision models have con-

sidered the issues of uncertainty and other realistic factors. The research contribution

is threefold. First, this paper orientates to the emergency warehouse location and the

relief materials distribution. We address a pre-disaster planning problem that seeks

to strengthen the efﬁciency of emergency rescue in the rescue operation network

whose facilities are subject to random damages due to a disaster. Second, this paper

focuses on the emergency warehouse location including multiple relief materials, a

set of suppliers, disaster sites and emergency warehouses in the preparation stage

by considering damaged scenarios under uncertainty. Incorporating damaged scenar-

ios into the model is more realistic and reliable. Furthermore, we introduce a seismic

resilience function to quantize the earthquake resistance of affected areas. To calculate

a reasonable value of seismic resilience, we adopt the fault tree analysis, the analytic

hierarchy process, fuzzy evaluation and neural network. Last, we formulate our emer-

gency warehouse location problem to a stochastic mixed-integer programming model

and developed an LB based solution method and a PSO based solution method to solve

this model.

3 Problem Statement

The problem studied in this paper is the emergency warehouse location problem

with damaged scenarios. Determining the location of emergency facilities is a strategic

decision that directly affects the success of disaster response operations. Setting up

these facilities near vulnerable areas is critical to reducing response time. However, it is

under risk as well because the facilities will be damaged or destroyed in an earthquake.

In addition, the facilities of each area and the abilities of local residents to resist the

disaster are different. Accordingly, the emergency warehouse location problem, given

the different seismic resilience of disaster sites under facility damage scenarios, is

closer to the real world.

Yunnan Province is located on the Yunnan–Guizhou Plateau and situated at the

northeast border of a crush collision zone between the Indian plate and the Eurasian

plate; mountainous areas in Yunnan constitute 94% of its total area. Therefore, Yunnan

Province is in a frequent earthquake area. National Seismological Bureau data show

that in 2008–2017, China had 25 earthquakes (magnitude 5.0 and above), 16 of which

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Optimizing Locations and Scales of Emergency Warehouses…

occurred in Yunnan Province. The abilities of different regions in this province to resist

disasters vary widely. Thus, it is particularly important to set the reasonable location

of emergency warehouses.

Suppliers, frequent earthquake sites and possible candidate locations to establish

emergency warehouses in Yunnan Province are depicted in Fig. 1. As we can see, sup-

pliers are mainly concentrated in the capital and other developed cities, while disaster

sites are comparatively decentralized. Supposing that all relief materials are delivered

directly from suppliers to these decentralized disaster sites in the province, it would be

so prohibitively expensive and time-consuming that the purpose of emergency rescue

would not be realized. Therefore, the best approach is to deliver relief materials from

the supplier to these emergency warehouses and then to the corresponding disaster

sites via emergency warehouses. Remarkably, relief materials can also be delivered

directly from suppliers to the disaster sites.

Fig. 1 Location of disaster sites, emergency warehouses and suppliers in Yunnan Province, China

123

B.-C. Wang et al.

Figure 2illustrates a case of the delivery process for relief materials in a supply

chain network where there are three suppliers, three emergency warehouses and three

disaster sites. A variety of relief materials can be delivered from suppliers to disaster

sites via emergency warehouses. Relief materials should be delivered to the emer-

gency warehouse before the earthquake. Then, these materials will be transported from

warehouses to disaster sites in order to support rescue operations when the earthquake

occurs. In our research, the quantity, location and scale of emergency warehouses that

should be built under the facility damage scenarios are the core decision problem.

Note that a set of candidate sites for building an emergency warehouse are known

beforehand.

The challenges embedded in the above decision problem contain the following

aspects. First, each disaster site can be served by several suppliers or emergency houses;

relief materials can be delivered to each emergency warehouse by several suppliers.

Additionally, each supplier can serve various emergency warehouses or disaster sites;

the relief materials can be delivered by each emergency warehouses to various disaster

sites. The many-to-many mapping relationships as well as the transportation capacity

limitation of trucks and planes and the warehouse scale limitation pose signiﬁcant chal-

lenges to the model formulation of this problem. Second, in terms of computing seismic

resilience value, we employ fault tree analysis, analytic hierarchy process, fuzzy set

theory and neural network to quantize each index of seismic resilience to ﬁnd a rea-

sonable value. Lastly, the demands of disaster sites are stochastic, and the decision of

assigning relief materials to emergency warehouses is a strategic-level decision; there-

fore, a stochastic programming methodology should be adopted to handle this problem.

Fig. 2 Logistics network between disaster sites, emergency warehouses and suppliers

123

Optimizing Locations and Scales of Emergency Warehouses…

4 Model Formulation

A stochastic mixed-integer programming model is formulated for the emergency

warehouse location problem in pre-disaster operations and the materials scheduling

problem in post-disaster operations. The objective function of this model includes four

parts: the ﬁxed cost of building capacitated emergency warehouses, the air and land

transportation cost of relief materials, the penalty cost for the lack of relief materials

and the resilience of each disaster site. By considering the uncertain demand at the

disaster sites, this paper uses the scenario-based method to model the random demands

by redeﬁning the uncertain parameters through adding a new subscript, i.e., the index

of scenarios k. Each scenario is composed of collective random demands of disaster

sites.

4.1 Notions

Indices and sets

sIndex of supplier.

SSet of all suppliers.

iIndex of location where an emergency warehouse is built.

ISet of all candidate locations where an emergency warehouse is built.

jIndex of disaster site.

JSet of all disaster sites.

kIndex of scenario.

KSet of all scenarios.

pIndex of relief material.

PSet of all relief materials.

qIndex of capacity option.

QiSet of all capacity options for an emergency warehouse at location i.

Para meters

tjNumber of victims at disaster site j.

wkProbability of scenario k.

ck

iDamage coefﬁcient of relief materials at location iin each scenario k,ck

i∈[0,1].

ak

jImpact coefﬁcient on victims at disaster site jin each scenario k,ak

j∈[0,1].

dsi Transportation distance from supplier sto location i.

dij Transportation distance from location ito disaster site j.

u1Cost of transporting one truckload of relief materials per kilometer.

u2Cost of delivering one planeload of relief material per kilometer.

gij Percentage of airlift to total relief material from location ito disaster site j.

ep1Fraction of truck capacity (in terms of one truck load) needed by a unit of relief

material p.

ep2Fraction of plane capacity (in terms of one plane load) needed by a unit of relief

material p.

vpVolume of a unit of relief material p.

fiq Fixed cost of an emergency warehouse at location iwith capacity option q,

q∈Qi.

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B.-C. Wang et al.

viq Capacity of an emergency warehouse at location iwith capacity option q,q∈Qi.

osp Capacity of relief material pin supplier s.

nk

jp Demand of relief material pat disaster site jin scenario k.

bpDemand of each victim for relief material p.

rjDisaster resilience coefﬁcient at disaster site j.

hjQuantity of humanitarian relief materials at disaster site j.

mPenalty coefﬁcient, in the case of insufﬁcient supplies.

nResistance coefﬁcient of victims, in the case of insufﬁcient supplies.

tCoefﬁcient of the minimum demand of disaster sites.

MA sufﬁciently large positive number.

Decision variables

θiq Binary variable, equals one if an emergency warehouse with capacity option q

is built at the location i,qQi; otherwise, equals zero.

πk

sip Quantity of relief material ptransported from supplier sto location iin scenario

k.

πk

ijp Quantity of relief material ptransported from location ito demand point jin

scenario k.

4.2 Mathematical Model

Min Z

i∈I,q∈Qi

fiqθiq +

k∈K

wkN(θ,k)+m

k∈K

j∈J

p∈P

maxnk

jp −

i∈I

πk

ijp,0

−n

k∈K

j∈J

rj

p∈Pi∈Iπk

ijp +hj

nk

jp

.(4.1)

Here

N(θ,k)Min

s∈S

i∈I

dsi u1⎛

⎝

p∈P

ep1πk

sip ⎞

⎠+

i∈I

j∈J

diju1⎛

⎝

p∈P

ep11−gijπk

ijp⎞

⎠

+

i∈I

j∈J

diju2⎛

⎝

p∈P

ep2gijπk

ijp⎞

⎠(4.2)

s.t.

q∈Q

θiq 1,∀i∈I,(4.3)

i∈I

πk

sip osp,∀s∈S,∀p∈P,∀k∈K,(4.4)

i∈I

πk

ijp t·nk

jp,∀j∈J,∀p∈P,∀k∈K,(4.5)

s∈S

πk

sip 1−ck

i

j∈J

πk

ijp,∀i∈I,∀p∈P,∀k∈K,(4.6)

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Optimizing Locations and Scales of Emergency Warehouses…

s∈S,p∈P

vpπk

sip

q∈Qi

θiqviq,∀i∈I,∀k∈K,(4.7)

s∈S

πk

sip −M·θiq 0,∀i∈I,∀p∈P,∀k∈K,∀q∈Qi,(4.8)

nk

jp tjak

jbp,∀j∈J,∀p∈P,∀k∈K,(4.9)

θiq ∈{0,1},∀i∈I,∀q∈Qi,(4.10)

πk

sip 0∀s∈S,∀i∈I,∀p∈P,∀k∈K,(4.11)

πk

ijp 0∀i∈I,∀j∈J,∀p∈P,∀k∈K.(4.12)

Objectives (4.1) and (4.2) minimize the ﬁxed cost, the transportation cost, the

penalty cost and the unit cost of the seismic resilience function. Constraints (4.3)

represent that at most one emergency warehouse can be established at the emergency

warehouse candidate location. Constraints (4.4) indicate that the overall amount of a

type of relief material transported from a supplier to an emergency warehouse location

cannot exceed the supplier’s supply capacity. Constraints (4.5) require that the overall

amount of a type of relief material transported to the disaster site can satisfy the

minimum demand of the disaster site in each scenario. Constraints (4.6) state that

the overall amount of inputs at an emergency warehouse is no less than the overall

amount of outputs at the emergency warehouse for each type of relief material in each

scenario. Constraints (4.7) are the capacity constraints for emergency warehouses.

Constraints (4.8) guarantee that an emergency warehouse is built if there are materials

delivered through the emergency warehouse. Constraints (4.9) state the demand for

relief material pat disaster site j. Constraints (4.10–4.12) deﬁne the domains of the

decision variables.

5 Solution Approaches

When solving small-scale instances for the problem, existing commonly used opti-

mization solver (e.g., CPLEX) can be directly employed to solve the model. However,

the number of variables in the model grows exponentially with the number of suppli-

ers, candidate locations and disaster sites considered in the problem. It will be hence

rather time-consuming or impractical for a solver to handle even a moderate sized

instance. In this study, in order to overcome the computational difﬁculty, we propose

another two methods, an LB based method and a PSO based method, both of which

are able to tackle large-scale instances for our problem.

5.1 An LB Based Method

In our model, the binary variables πk

sip and πk

ijp depend on the binary variables θiq.

This means only one set of variables θiq exists in this model. Thus, we solve the model

by branching binary variables θiq and using local branch strategy [23] to accelerate

the speed of computation time. Figure 3presents the main process.

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B.-C. Wang et al.

Let node 1 represent the initial point of the LB process (see Fig. 3). At node 1,

we use CPLEX to ﬁnd a feasible solution for the proposed model before initializing

the variables θiq. According to node 1, node 2 and node 3 can be derived. Node 2

indicates that constraint |θ−θ1|ais added to the original model. Similarly, node

3 indicates that constraint |θ−θ1|a+ 1 is added to the original model. Note that

|θ−θ1| represents the radius of θ1’s neighborhood in the solution space. When search-

ing for solutions around the incumbent solution, the parameter adetermines the size

of the search neighborhood, which will cause the process of solution in node 2 is

time-consuming if ais large.

Figure 3illustrates the branching process, where all black nodes are solved by

CPLEX. It is time-consuming to solve some nodes in this process. Therefore, in order

to improve the solution efﬁciency, we set a solution time upper limit in advance when

solving each node. When the solution time reaches this upper limit, CPLEX will

stop solving at the current node and will obtain a feasible but non-optimal solution;

otherwise, an optimal solution would be obtained by CPLEX.

There are four cases of each black node in solving process (see Fig. 3). In case

1, CPLEX obtains the improved objective value within time limit. In this situation,

CPLEX can solve this node optimally and update the current best solution. In case 2,

the objective value is improved, but the previously set solution time limit is reached. In

this situation, although CPLEX cannot obtain the optimal solution, it can still obtain

a better solution than the current best solution. In case 3, the preset time limit is not

reached in the solution process of CPLEX, and the objective value is not improved. In

this situation, CPLEX can ﬁnd the optimal solution, but it will be not better than the

current best solution. In case 4, the preset time limit is reached, but the objective value

has not been improved. In this situation, CPLEX can obtain a feasible but non-optimal

solution, which is not better than the current best solution.

We provide the handling strategies for the above four cases as follows (see Fig. 4).

For case 1, it is the standard process in the LB solution procedure. For case 2, it results

Fig. 3 The main steps of the LB solution method

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Optimizing Locations and Scales of Emergency Warehouses…

Fig. 4 The handling strategies for different cases

in updating the neighborhood-related constraint θ−θ4a, which is changed to

θ−θ6abecause the solution obtained by node 4 is replaced by the solution

obtained by node 6. For case 3, since the current best solution is better than the

optimal solution obtained by CPLEX, node 6 will be completely pruned. At this node,

the solution obtained by node 7 is the solution of the original model. For case 4, the size

of the search neighborhood should be reduced in order to obtain an optimal solution

for this node or a better solution than the current best solution.

The neighborhood involved in this method was obtained by deﬁning some linear

inequalities (or branch cuts) for variables θiq. It should be noted that if the current

best value cannot be improved within a preset number of iterations, the entire solution

process is terminated.

5.2 A PSO Based Method

The PSO is an intelligent optimization algorithm by simulating the foraging process

of birds, which is ﬁrst proposed by Eberhart and Kennedy [24]. It has been successfully

applied in facility location problems [25].

5.2.1 Solution Representation and Velocity Updating

In the PSO-based solution method, each particle represents a feasible solution

within the search neighborhood. The status of a particle includes its position and

velocity. The variables πk

sip and πk

ijp in the formulated model depend on the vari-

ables θiq. Thus, if variables θiq are known, others (πk

sip,πk

ijp) will be ﬁxed. The

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B.-C. Wang et al.

core variables θiq determine whether an emergency warehouse with capacity option

qis built at the location i. Each particle with |I| dimension is represented to

Aa1,a2,···,ai,···,a|I|. We code each dimension aias a positive number

of which the integral part airecords that an emergency warehouse is built at candi-

date location iwith capacity ai+ 1. For example, ai+12 means that an emergency

warehouse is built at candidate location 1 with capacity 2.

Suppose that a swarm has mparticles in the PSO solution method. Let Pn

m

pn

miq be the position of particle mat iteration n, and Vn

mvn

miq be the velocity

of particle mat iteration n. The position and velocity of the particles will be updated

as follows:

vn+1

miq wn·vn

miq +c1r1PLBn

miq −pn

miq +c2r2PGBn

miq −pn

miq ,(5.1)

pn+1

miq pn

miq +vn+1

miq .(5.2)

Here wnis an inertia weight parameter, which affects the convergence procedure of

the optimal solution. c1and c2are learning factors. PLBn

miq denotes the best position

for particle mto iterate ntimes on dimensions i,q.PGB

n

miq denotes the global best

position of the swarm to iterate ntimes on dimensions i,q.r1and r2are random

numbers generated between [0, 1].

In addition, as decision variables are binary variables, the particle’s position pn+1

miq

should be changed to the decision variable θiq. It is modiﬁed as follows:

θiq 1,when pn+1

miq <0.5,

0,when pn+1

miq 0.5.(5.3)

5.2.2 Main Framework of the PSO Solution Method

According to the above settings, the main process of the PSO solution method is

provided in Algorithm 1.

Algorithm 1 Main framework of the PSO solution method

Step 1. Set the iteration number n1. Initialize the particle group m. Particles’ posi-

tions and velocities are randomly generated, which determine their qualities.

Step 2. Evaluate the ﬁtness value of each particle. CPLEX is used to solve the original

model with the variables θiq that are ﬁxed and represented by the particles.

Step 3. For each particle, compare its ﬁtness value and the global best optimal solution

PGBestn

miq, if PLBestn

miq >PGBestn

miq , replace it.

Step 4. According to formulas (5.1) and (5.2), update the particle velocity and posi-

tion.

Step 5. If the iteration number reaches the preset maximum value or the global opti-

mal solution has remained unchanged during a preset number of iterations,

stop the procedure; otherwise, set n:=n+ 1 and then go to Step 2.

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Optimizing Locations and Scales of Emergency Warehouses…

6 Numerical Experiments

In this section, we conduct numerical experiments by using a PC (Intel Core i5,

2.60 GHz; Memory, 8.00 GB) to validate the effectiveness of the proposed models

and the efﬁciency of the developed solution methods, and these models and solution

methods, including the LB and the PSO, are implemented by CPLEX 12.6.2 with

concert technology of C# (VS2015). Section 6.1 mainly provides the setting of seismic

resilience value. Section 6.2 provides the setting of appropriate number of scenarios.

Section 6.3 validates the availability of the model and the efﬁciency of the two solution

methods using real data from Yunnan Province in China.

6.1 Experimental Settings

This paper introduces the seismic resilience function in the model to elaborate

resilience of different disaster sites. Fault tree analysis, analytic hierarchy process,

fuzzy set theory and neural network are used to calculate appropriate resilience values

for different disaster sites. Seismic resilience is deﬁned as the ability of social units

(e.g., organizations, communities) to alleviate hazards, which includes the effects of

disasters when they occur, and fulﬁll recovery activities in ways that minimize social

disruption and alleviate the impacts of future earthquakes [26]. A resilient system has

three features: reduced failure probabilities, reduced consequences from failures (i.e.,

lives lost, damage, negative economic and social consequences) and reduced time to

recovery.

6.1.1 Fault Tree Analysis for Disaster Sites

The fault tree of candidate emergency warehouses is presented in Fig. 5.

In Fig. 5, event T indicates that a disaster site cannot resist earthquakes; TA, TB,

TC and TD represent negative states of economic factors, physical factors, institu-

tional factors and demographic factors, respectively. TAA and TAB represent lacking

social resources and inadequate facilities. TBA, TBB, TBC and TBD refer to dam-

aged infrastructure, transportation, emergency warehouses and urban buildings. TCA,

TCB and TCC denote imperfect emergency regime, professional publicity and govern-

Fig. 5 The fault tree analysis of disaster sites

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B.-C. Wang et al.

ment support. TDA, TDB and TDC are population density, educational standards and

age structure of disaster sites. B1–B6 represent damaged medical facilities, shelters,

trafﬁc ﬂow, relief materials transportation, emergency warehouse and failure emer-

gency warehouse. C1–C5 stand for lacking pre-disaster relief knowledge, disaster

professional medical ambulance personnel, post-disaster professional psychological

counseling, resource reserves and external humanitarian relief.

6.1.2 Seismic Resilience Computing

According to the above fault tree analysis, core seismic resilience factors are

selected and handled by the analytic hierarchy process and fuzzy evaluation. Fur-

thermore, the seismic resilience of disaster sites is obtained by a T–S neural network

[27]. Fuzzy set theory is used to calculate the membership degree of the inﬂuencing

factors as the input of the neural network. There are qualitative factors and quantitative

factors. Population density and ﬁnancial revenue are two qualitative factors; infras-

tructure situation, trafﬁc situation, urban buildings situation, educational standards,

age structure, emergency regime, professional publicity and government support are

quantitative factors. First, classiﬁcations of each index of quantitative factors are listed.

Then, the classiﬁcation by natural language is converted into the corresponding mem-

bership, as shown in Table 2.

The memberships of different indexes in Table 2are the input of the fuzzy neural

network, and the output is the expected value of experts. Each parameter of the earth-

quake disaster sites is quantiﬁed and handled by the T–S neural network, as shown

in Table 3. Finally, the seismic resilience value βjof each disaster site is obtained, as

listed in Table 4.

6.2 Numerical Results Under Different Numbers of Scenarios

As previously mentioned, the scenario-based method is used in modeling, which

means that the number of scenarios has remarkable effect on computational results of

the model. Thus, we conduct seven groups of experiments based on the scale involving

eight suppliers, ten candidate emergency warehouses and ﬁve disaster sites in order to

Table 2 The membership functions of qualitative indexes

Indexes Classiﬁcations Memberships

Infrastructure situation Excellent; good; satisfactory; fair; poor A1{1,0.8,0.5,0.2,0}

Trafﬁc situation Strong; normal; weak A2{1,0.5,0}

Buildings situation Stable; normal; unstable A3{1,0.5,0}

Educational standards High; normal; low A4{1,0.5,0}

Age structure High; normal; low (the young/the old) A5{1,0.5,0}

Emergency regime Excellent; good; satisfactory; fair; poor A6{1,0.8,0.5,0.2,0}

Professional publicity High; normal; low A7{1,0.5,0}

Government support Excellent; good; satisfactory; fair; poor A8{1,0.8,0.5,0.2,0}

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Table 3 The resilience indexes of each disaster site

Disaster sites Kunming Honghe Pu’er Lincang Chuxiong Baoshan Lijiang Wenshan

Physical factors

Infrastructure

situation

0.93 0.21 0.65 0.50 0.72 0.43 0.41 0.28

Trafﬁc situation 0.56 0.45 0.64 0.68 0.65 0.72 0.49 0.42

Buildings situation 0.83 0.33 0.72 0.70 0.75 0.65 0.62 0.42

Demographic factors

Population density 0.88 0.17 0.65 0.62 0.68 0.52 0.42 0.21

Educational standards 0.80 0.51 0.64 0.62 0.75 0.62 0.65 0.59

Age structure 0.60 0.89 0.58 0.68 0.51 0.65 0.57 0.88

Institutional factors

Emergency regime 0.75 0.21 0.70 0.60 0.74 0.56 0.62 0.31

Professional publicity 0.62 0.15 0.52 0.51 0.56 0.38 0.35 0.11

Government support 0.68 0.12 0.65 0.62 0.65 0.58 0.46 0.15

Economic factors

Financial revenue 0.87 0.35 0.68 0.67 0.69 0.72 0.56 0.39

Table 4 The seismic resilience of each disaster site

Disaster sites Kunming Honghe Pu’er Lincang Chuxiong Baoshan Lijiang Wenshan

Seismic resilience 0.78 0.23 0.55 0.42 0.71 0.41 0.38 0.25

select a more reasonable number of scenarios. The number of scenarios in the seven

experiments is set to 10, 20, 50, 100, 200, 500 and 800, respectively, and each group

covers ten different cases. Note that the probability of each scenario occurring is the

same, for example, if the number of scenarios is K, the probability of each scenario is

1/K.

The results of the above experiments are shown in Table 5, along with the rise of the

number of scenarios, the gap between the maximum and the minimum values gradually

narrows and the standard deviation declines, while the computing time witnesses an

opposite tread. There is a noticeable increase in the computing time and a steady

decrease in the standard deviation when the number of scenarios exceeds 100. This

indicates that it is reasonable to set the number of scenarios to 100, which we will use

in the experiments in Sect. 6.3.

6.3 Performance of the Presented Solution Methods

First, we conduct small- and large-scale experiments according to the real data from

Yunnan Province in China to verify the effectiveness of the model and the performance

of the solution methods. Table 6shows the comparison between the LB and CPLEX

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Table 5 Numerical results of the presented model under different number of scenarios

Number of

scenarios

Max. Min. Gap Avg. SD Avg. CPU

time(s)

10 28 565 402 22 633 722 5 931 680 25 526 961 2 272 251 5

20 28 953 547 23 655 061 5 298 486 26 700 975 2 142 263 10

50 27 845 781 22 583 627 5 262 154 25 046 110 1 806 780 23

100 27 653 607 22 568 704 5 084 903 25 096 181 1 758 476 52

200 26 812 575 21 875 352 4 937 223 23 903 124 1 703 457 125

500 26 852 505 21 965 634 4 886 871 23 640 704 1 698 530 356

800 27 135 684 22 455 876 4 679 808 23 452 375 1 696 258 1368

(1) ‘Max’ and ‘Min’ represent the maximum and minimum values of each case, respectively. (2) Gap

Max–Min. (3) Avg. and SD represent the average and standard deviation of each case, respectively. (4)

‘Avg. CPU Time’ represents the average CPU time of each case

Table 6 Comparison between the LB and CPLEX solver for small-scale problems

Instance id CPLEX LB OBJ

Gap/%

S-W-D ZCTC/s ZLTL/s

8-10-5-1 26 349 692 47 26 349 692 165 0.00

8-10-5-2 25 850 114 40 25 850 114 176 0.00

8-10-5-3 26 298 508 65 26 300 134 218 0.01

10-10-5-1 24 853 507 272 24 853 507 307 0.00

10-10-5-2 25 354 415 231 25 354 415 335 0.00

10-10-5-3 24 350 362 382 24 350 362 323 0.00

10-10-8-1 30 678 340 856 30 678 340 389 0.00

10-10-8-2 28 264 686 735 28 311 122 428 0.16

10-10-8-3 28 746 155 752 28 832 652 465 0.30

Avg. 0.05

(1) ZCand TCrepresent optimal objective values and CPU times obtained by CPLEX solver, respectively.

(2) ZLand TLrepresent objective values and CPU times obtained by the LB-based method, respectively.

(3) OBJ Gap (ZL−ZC)/ZC

solver for small-scale instances; and Table 7shows the comparison between the LB,

the PSO and CPLEX solver for large-scale instances.

Observing the results in Tables 6and 7, we can ﬁnd that CPLEX can only obtain

the optimal solutions for small-scale instances. When the scale is up to ‘10-15-15,’

the proposed model becomes too intractable to be directly solved by the CPLEX. In

terms of small-scale instances, the LB can obtain the optimal solutions in most cases;

the average gap between the objective values of the LB and the optimal solutions

is only 0.05%. As the scale increases, the LB has more advantages than CPLEX in

computing time. On large-scale instances where CPLEX cannot obtain solutions, we

use the feasible solutions obtained by CPLEX within 2 h as a comparison. The average

gap between the objective values of the LB and the feasible solutions is 0.87%, and

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Table 7 Comparison between the LB, the PSO and CPLEX solver for large-scale problems

Instance id Optimal OBJ CPU time/s TP/TLOBJ Gap

S-W-D OBJopt ZLZPTLTPGAPL/% GAPP/%

10-15-8-1 26 387 052 26 539 048 26 506 091 986 549 0.56 0.58 0.45

10-15-8-2 27 142 587 27 273 562 27 268 741 1 104 563 0.51 0.48 0.46

10-15-8-3 26 280 527 26 459 867 26 481 068 1 254 649 0.52 0.68 0.76

15-15-8-1 25 670 431 25 961 291 25 850 895 1 567 707 0.45 1.13 0.70

15-15-8-2 25 750 234 26 049 714 25 890 416 1 842 735 0.40 1.16 0.54

15-15-8-3 25 793 212 26 163 123 25 951 651 1 648 752 0.46 1.43 0.61

10-15-15-1 – 41 656 856 41 245 872 2 320 958 0.41 – –

10-15-15-2 – 42 151 258 41 835 348 2 852 1 012 0.35 – –

10-15-15-3 – 41 267 561 40 645 547 2 682 986 0.36 – –

Avg. 1 806 768 0.45 0.91 0.59

(1) OBJopt represents the feasible solution values solved by CPLEX solver in 2 h. (2) ZPand TPrepresent

objective values and CPU times obtained by the PSO-based method, respectively. (3) GAPL(ZL−

OBJopt)/OBJopt, GAPP(ZP−OBJopt)/OBJopt

the average gap between the objective values of the PSO and the feasible solutions is

only 0.59%. In addition, the LB and the PSO can provide the feasible solutions within

a short time, it only takes an average of 1 806 (768) s of computation time. It means

the PSO performs better either in the solution results or the computing time than the

LB.

Moreover, we provide the locations of emergency facilities for small-scale instances

which can further present the practicability of the model. In this experiment, ten cities

in Yunnan Province, Jingdong, Yimen, E’shan, Xinping, Zhenyuan, Jinggu, Mojiang,

Yuanjiang, Shiping and Shuangbai, are chosen as the candidate locations. There are

three scale choices, namely 2 000, 2 500 and 3 000 m3. The number zero indicates

that the city was not chosen to establish an emergency warehouse. It can be seen

from Table 8that Shuangbai, Yimen and Xinping are suitable for establishing a large-

scale emergency warehouse, and Jingdong, Mojiang and Yuanjiang are suitable for

establishing a medium-scale emergency warehouse. However, E’shan, Jinggu and

Shiping are not suitable for establishing emergency warehouses.

7 Conclusion

In emergency logistics networks, determining the location for building emergency

warehouses and allocating relief materials are important strategic-level decision prob-

lems. This paper studies the emergency warehouse location problem by considering

the factors of damage scenario, different scales of emergency warehouses and uncer-

tain relief material demands. This problem is formulated to a stochastic mixed-integer

programming model and two solution methods are designed to solve the model. The

applicability of the formulated model and the efﬁciency of the proposed methods are

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Table 8 The construction programs of emergency warehouse in each disaster site

Instance id

S-W-D

Jingdong Yimen Eshan Xinping Zhenyuan Jinggu Mojiang Yuanjiang Shiping Shuangbai

8-10-5-1 I II 0 II 0 0 I I I III

8-10-5-2 I II 0 III 0 0 III I 0 III

8-10-5-3 I III 0 III 0 0 I I I 0

10-10-5-1 I III 0 0 II 0 III I II III

10-10-5-2 III III 0 II 0 0 II I 0 III

10-10-5-3 I III 0 III 0 0 II II 0 III

10-10-8-1 III I 0 III 0 III III III 0 III

10-10-8-2 II III III III 0 0 III II 0 III

10-10-8-3 II III 0 III I 0 III III 0 III

Three warehouse scales 2 000, 2 500, 3 000 m3are denoted by I, II, III

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validated by conducting numerical experiments using cases from Yunnan Province in

China. This work makes the following substantial contributions:

(1) Most relevant studies have not taken into account the many-to-many mapping

relationships between suppliers, emergency warehouses and disaster sites, as well

as different scales of emergency warehouses under uncertainty. This paper has

conducted exploratory research on this area of emergency warehouses location

problem including multiple relief materials, suppliers, disaster sites and emer-

gency warehouses.

(2) Since emergency warehouses should be located near disaster sites, we consider

the damage scenarios and construct an emergency warehouse location model

to minimize the cost of the entire response stage under uncertainty. To ensure

the minimum response time of the rescue, we add to the demand constraints of

disaster sites and the penalty function in the model. Moreover, to reasonably

allocate relief materials, seismic resilience is introduced and calculated by fault

tree analysis, the analytic hierarchy process, fuzzy evaluation and neural network.

(3) We develop an LB-based method and a PSO-based method for solving the formu-

lated model. Extensive numerical experiments on real instances involving Yunnan

Province in China show the two methods have excellent solution efﬁciency. The

PSO-based solution method outperforms the LB-based solution method for solv-

ing extremely large-scale instances.

However, there are still some limitations in this paper. For example, although the

damage scenarios of the facility are taken into account, the damage scenarios in the

transportation ﬂow are not. In addition, the paper is only concerned with the preparation

and response stages. Facility location problems can be further studied by combining

them with other problems, such as evacuation problems and inventory problems, in

the four stages of disaster (mitigation, preparation, response and recovery). These

limitations will be the possible directions in our future studies.

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