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Optimizing Locations and Scales of Emergency Warehouses Based on Damage Scenarios

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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.
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 [35]. 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 [68].
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
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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
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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
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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,
qQi.
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B.-C. Wang et al.
viq Capacity of an emergency warehouse at location iwith capacity option q,qQi.
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
iI,qQi
fiqθiq +
kK
wkN(θ,k)+m
kK
jJ
pP
maxnk
jp
iI
πk
ijp,0
n
kK
jJ
rj
pPiIπk
ijp +hj
nk
jp
.(4.1)
Here
N(θ,k)Min
sS
iI
dsi u1
pP
ep1πk
sip
+
iI
jJ
diju1
pP
ep11gijπk
ijp
+
iI
jJ
diju2
pP
ep2gijπk
ijp
(4.2)
s.t.
qQ
θiq 1,iI,(4.3)
iI
πk
sip osp,sS,pP,kK,(4.4)
iI
πk
ijp t·nk
jp,jJ,pP,kK,(4.5)
sS
πk
sip 1ck
i
jJ
πk
ijp,iI,pP,kK,(4.6)
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Optimizing Locations and Scales of Emergency Warehouses…
sS,pP
vpπk
sip
qQi
θiqviq,iI,kK,(4.7)
sS
πk
sip M·θiq 0,iI,pP,kK,qQi,(4.8)
nk
jp tjak
jbp,jJ,pP,kK,(4.9)
θiq {0,1},iI,qQi,(4.10)
πk
sip 0sS,iI,pP,kK,(4.11)
πk
ijp 0iI,jJ,pP,kK.(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 (4.104.12) 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 |θθ1|ais added to the original model. Similarly, node
3 indicates that constraint |θθ1|a+ 1 is added to the original model. Note that
|θθ1| 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 θθ4a, which is changed to
θθ6abecause 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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Optimizing Locations and Scales of Emergency Warehouses…
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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B.-C. Wang et al.
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 (ZLZC)/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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Optimizing Locations and Scales of Emergency Warehouses…
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(ZPOBJopt)/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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B.-C. Wang et al.
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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Optimizing Locations and Scales of Emergency Warehouses…
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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