Research ArticleModeling Relief Demands in an Emergency Supply Chain Systemunder Large-Scale Disasters Based on a Queuing Network

Xinhua He1 and Wenfa Hu2

1 School of Economics Management Shanghai Maritime University Shanghai 201306 China2 School of Economics and Management Tongji University Shanghai 200092 China

Correspondence should be addressed to Wenfa Hu wenfahugmailcom

Received 28 August 2013 Accepted 7 November 2013 Published 6 February 2014

Academic Editors R-M Chen F R B Cruz B Naderi and H Wu

Copyright copy 2014 X He and W Hu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a multiple-rescue model for an emergency supply chain system under uncertainties in large-scale affectedarea of disasters The proposed methodology takes into consideration that the rescue demands caused by a large-scale disasterare scattered in several locations the servers are arranged in multiple echelons (resource depots distribution centers and rescuecenter sites) located in different places but are coordinated within one emergency supply chain system depending on the typesof rescue demands one or more distinct servers dispatch emergency resources in different vehicle routes and emergency rescueservices queue inmultiple rescue-demand locationsThis emergency system is modeled as aminimal queuing response timemodelof location and allocation A solution to this complex mathematical problem is developed based on genetic algorithm Finally acase study of an emergency supply chain system operating in Shanghai is discussed The results demonstrate the robustness andapplicability of the proposed model

1 Introduction

In last decades a number of natural or manmade disasterssuch as earthquakes volcanoes floods hurricanes epidem-ics explosions fires and violent attacks have risen threefoldand death counts and property losses are reported in everydisaster [1 2] Although most of those disasters were notavoided efficient delivery of emergency supplies could savelives and reduce lossThose ubiquitous disasters have arousedintensive concerns of emergent relief demanding Emergencyrelief requires coordinated and rapid responses and supplies

The emergency supply chain (ESC) is to locate points ofemergency equipment and supplies and to relieve those inneed with food water shelters and medical care promptly[3] In fact ESC is a network of combined organizationsmutually and cooperatively to plan manage and control theflow of emergency commodities for the purpose of maximiz-ing the affected human survival rate and minimizing the costof the rescue actions after disasters

ESC is a typical three-echelon network supply pointsemergency logistics centers and demand points The key

challenges to ESC as compared to the business supply chainare highlighted as follows demand and route uncertaintiescomplex communication and coordination timely deliveryand limited resources [4 5] Those uncertainty challenges indisaster characteristics were addressed in previous literaturesthrough the use of probabilistic models [6] queuing theory[7] and fuzzy methods [8]

The objectives of the ESC system are to improve the per-formance while minimizing the response time The responsetime includes transportation time waiting time and servicetime of relief commodities in case of congestions Queuingmodels are effective measures to calculate the queue lengthsojourn time and waiting time and probabilities of any delayidleness and turnaways due to insufficient waiting accom-modation [9] Therefore this paper develops a quick-respon-sive ESC system based on the queuing theory where eachdepository in the ESC is considered as a server and waitingcommodities are in a queue to accept the serverrsquos services ina three-echelon network ESC system will be optimized toprovide the relief operations under the consideration of botheconomic loss and response time limitation

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 195053 12 pageshttpdxdoiorg1011552014195053

2 The Scientific World Journal

In this paper we develop a new quick-responsive ESCsystem with a queuing model in a specified three-layer net-work Each depository in the ESC is modeled as a server andwaiting commodities are in a queue By using the queuingmodel we can obtain the classical performances of ESC suchas average queue length and average waitingresidence timeand optimize the relief operations under the consideration ofboth economic loss and response time limitation

Themain contributions of this study are as follows Firstlya new emergency service supply chain model fulfilling therequirement of quick responses and deliveries for disasterrelief is developed Secondly a queuing model is embeddedin this model for the first time to calculate the performanceof ESC Thirdly the physical network of ESC with a queuingmodel is proposed as an effective refined method to solve theintegrated problem including vehicle routing problem andlocation problem of emergency relief centers with multiplelayers

The rest of this paper is organized as follows Section 2presents the literature review on emergency managementA system specification is provided with a physical networkof ESC including a queuing model in Section 3 A queuingminimal response time location-allocation model is formu-lated with a heuristic method in Section 4 A case study isintroduced in Section 5 to demonstrate the performance ofthe proposed approach Finally Section 6 draws conclusionsand provides directions for future research

2 Literature Review

Emergency supply chain management is the key to the suc-cess of relief demand management under the condition oflarge-scale disasters The difficulty of emergency supplychain management is rooted in the uncertainties of abruptrelief demand and collaboration of chaotic condition Unlikedemanders in business logistics the relief demanders are thedisaster-affected people but their locations and their demandsmay not be predicted precisely before disasters happenApparently efficient relief supplying underlines the challengeof collaborative relief demandmanagement in the emergencysupply chain management Despite the urgent necessity ofcollaborating relief demandmanagement in the whole supplychain there is no straightforward collaboration model avail-able for the above issue In contrast with the optimization-based demand and supply models relief demand manage-ment must overcome more issues in disaster uncertaintiesand delivery collaboration

In brief the existing uncertainty based demand modelsappear unsuitable for chaotic relief demand managementaddressed in this study Instead most of the existing reliefdemand management models in the emergency supply chainappear to be limited to general cases for business opera-tions From the literature review we illustrate several relatedsubjects associated with typical models in the following forfurther discussion

By comparing the business supply chain and the human-itarian relief chain Beamon [10] revealed several specificcharacteristics of relief material supply chains including

zero lead times high stakes unreliable incomplete ornonexistent prior information and different demand patternOloruntoba and Gray [11] developed an agile supply chainmodel for humanitarian aid by applying practical elementsof conventional supply chains to the ESC Lodree Jr andTaskin [12] introduced a stochastic inventory control modelto prepare for potential hurricane activity and describeda dynamic programming algorithm to solve the inventoryproblem Bhakoo et al [13] developed an understanding ofthe nature of collaborative arrangements for themanagementof inventories in Australian hospital supply chains

Despite the recent emergence of emergency supply chainmanagement that has increasingly drawn researchersrsquo atten-tionmost previousworks appear to address the issues of reliefsupply and distribution contingent on relief demand assump-tions Yi andKumar [14] decomposed the emergency logisticsproblem into two decision-making phases and proposed anant colony optimization (ACO) model to solve a multicom-modity network flow problem Tzeng et al [15] simplifiedthe disaster context and proposed a fuzzy multiobjectiveprogramming method to optimize multiobjective functionsto avoid the possibility of a severely unfair relief distributionChiu andZheng [16] addressed themultiple emergency trafficflows outbound from the affected areas using a linear celltransmission model (CTM)

In order to deal with the optimization of emergencyman-agement there are a large number of studies on developingoperational research (OR)methods for it [17]Thesemethod-ologies include the use of linear programming techniquesfuzzy methods stochastic programming models probabilis-tic models simulation and decision theory and queuingtheory [5] Stochastic programming has been successfullyused in the area related to disaster studies [18 19] As differentapproaches to handle emergency problems simulation anddecision theory have also been adopted as methodologiesin this field Queuing theory has been applied to exploreemergency facility location problems recently Shavandi andMahlooji [20] combined fuzzy theory and queuing methodto develop a maximal covering location-allocation modelGalvao and Morabito [21] applied the hypercube queuingmodel to solve probabilistic location problems in the emer-gency service system Geroliminis et al [22] presented aqueuing model for locating emergency vehicles on urbannetworks considering both spatial and temporal demandcharacteristics such as the probability that a server is notavailable when required But numerous emergency practicesreveal that chaotic status after disasters worsens rescuecoordination and the relief supply chain is often blocked orcongested in practices Multiple emergency sources improverobustness of supply chains but coordination amongmultipleservers is more critical for emergency management Mean-while organizational skills require that the emergency supplychain operated by local government consists of multipleechelons which forms vertical coordination in emergencycontext All those problems need to be resolved in newmodels and algorithms However no paper indicated aboveembeds the queuingmodels into themultiechelon emergencyservice supply chain system

The Scientific World Journal 3

Therefore to resolve the issues mentioned above wepresent a relief demand management model of emergencysupply chain to address the above issue under the disorderand uncertain conditions in affected areas during the crucialrescue period of a large-scale disaster Rooted in the tech-niques of collaboration in emergency supply chain coupledwith queuing theory and system optimization the proposedmethodology embeds three mechanisms (1) multiechelonsupply chainmodel for disasters (2) dynamic facility locationand vehicle routing selection and (3) rescue systemmanage-ment collaboration

Relative to the previous literature the proposed reliefdemand management methodology has the following twodistinctive features (1)Themodel is capable of collaboratingurgent relief demand management in the large-scale disastercontexts and accelerating rescue efforts to save casualty loss(2) To facilitate dynamic relief allocation and distributionthe proposed model practically groups humanitarian reliefsintomultiechelon resource suppliers and distribution centerswhich form an emergency response system for uncertainlarge disasters

3 System Specification

An ESC system involves selection of sites and vehicle rout-ing decisions which are two major problems in a disasterresponse environment The optimal facilities locations andpath selections can guarantee that the commodities will besent from the supply depots to the demand points in affectedareas as quickly as possible to maximize the survival rate ofwounded persons The above problems arouse our intereststo propose queuing modeling for the ESC system Hence aqueuing network of emergency supply chain is formulated inthis paper as shown in Figure 1

The queuing network of ESC involves a queuing flowformulation where the three supply chain members namelysupply points (SP) emergency logistics centers (ELCs) anddemand points (DP) are treated as servers Emergency com-modities such as food shelter personnel machinery andmedicine are modeled as customers The upstream anddownstream nodes of ESC system constitute some basicactivities that are producing sorting processing packingdelivering and so forth These activities are regarded as theservice for customers Consider the following

(1) Locations of emergency supply points are in 1198781 1198782

1198783 119878

119898

(2) Locations of emergency logistics centers are in 1198711 1198712

1198713 119871

119903

(3) Locations of emergency demand points are in1198631 1198632

1198633 119863

119899

(a) Vehicle routing choices for the commodity flowsin the system are considered in the following

(4) TR1119895 from 119878

1to one of the nodes (119871

1 1198712 1198713

119871119903) TR

2119895 from 119878

2to one of the nodes (119871

1 1198712

1198713 119871

119903) and TR

119898119895 from 119878

119898to one of the

nodes (1198711 1198712 1198713 119871

119903) 119895 = 1 2 119903

(5) TR1119896 from 119871

1to one of the nodes (119863

1 1198632 1198633

119863119899) TR2119896 from 119871

2to one of the nodes (119863

1 1198632 1198633

119863119899) and TR

119903119896 from 119871

119903to one of the nodes

(1198631 1198632 1198633 119863

119899) 119896 = 1 2 119899

Each commodity is considered as a queue where batchesare waiting to be serviced The selection of sites and vehiclerouting decision may be operated under the considerationof the estimated throughput response time from supplydepots to demand depots in affected areas The responsetime comprises not only the transportation times betweenupstream and downstream nodes but also the total waitingtimes and service times in the queuing network

Therefore the above three nodes in the system areassumed to behave as an 1198721198721 queuing where reliefsupplies are treated as customers On the basis of the specifiedESC queuing network we adopt the following hypothesis forsystem operations

(1) The corresponding geographic relationships betweenupstream and downstream nodes are available fromthe existing governmental databases and the reliefdemand needed in a given affected area can be readilyaccessible via advanced disaster detection technolo-gies

(2) The locations of emergency logistics centers are onlyfixed in the given alternative sites

(3) The first customer in the queue receives servicesfirstly namely ldquofirst come first servedrdquo

4 Mathematical Formulation

Based on the aforementioned system specification we pro-pose a queuing theory in this section for facility locationand path selection problems in a multistage emergencysupply chain network Firstly the notations parametersand decision variables for the mathematical formulation areintroduced After that the objective function for the modelis established And then the formulation of the constraints ofthe problems is presented

41 The Parameters and Decision Variables The sets param-eters and decision variables are defined as follows

(1) Notations

119868 set of supply depots 119894 isin 119868 119894 = 1 2 3 119898119869 set of alternative sites of emergency logistics cen-ters 119895 isin 119869 119895 = 1 2 3 ℎ119870 set of depots for handing out relief goods 119896 isin 119870119896 = 1 2 3 119899119877 set of commodities 119903 isin 119877 119903 = 1 2 3 119877

(2) Parameters

120582 the interarrival time of the emergency demandfollowing a negative exponential distribution

4 The Scientific World Journal

Emergency supplypoints

S1

S2

Sm

Transportation

TR1j

TR2j

TRmj

Emergencylogistics centres

Emergencydemand points

L1

L2

Lh

Transportation

TR9984001k

TR9984002k

TR998400hk

D1

D2

Dn

middot middot middotmiddot middot middot middot middot middot middot middot middotmiddot middot middot

Figure 1 The queuing network of ESC

120583 the service rate at eachnode in the queuing networksystem

120575 the parameter of the negative exponential distribu-tion

119888 the lower bound of a uniformly distributed randomvariable that indicates the quantity of resources in arelief request

119889 the upper boundof a uniformly distributed randomvariable that indicates the quantity of resources in arelief request

119905119903

119894119895119896 the response time of the ESC system for com-

modity type 119903 from node 119894 to node 119896 going throughlogistics centers located at nodes 119895

WT119903 the sojourn time of commodity type 119903 in thesystem

WT119903119902 the waiting time of commodity type 119903 in the

queue

TR119903119894119895 the transportation time for commodity type 119903

from node 119894 to node 119895

TR119903119895119896 the transportation time for commodity type 119903

from node 119895 to node 119896

DTR119894119895 the distance between node 119894 and node 119895

DT119896119895 the distance between node 119895 and node 119896

DC the penalty cost for unavailability of commoditiesdemand within the maximum promised responsetime

119886119895 the fixed costs of locating the emergency logistics

center 119895

1199021 the total number of logistics center to be fixed

1199022 the number of supply depots delivering commodi-

ties to the same emergency logistic centers

1199023 the number of demand nodes accepting items

from the same emergency logistics center

V119903119894119895 transport speed for commodity type 119903 from node

119894 to node 119895

V119903119895119896 transport speed for commodity type 119903 from node

119895 to node 119896

(3) Decision Variables

119909119895=

1 emergency logistics center builton the site 119895

0 otherwise

119910119894119895=

1 relief resources from emergency supplypoint 119894 transported to emergencylogistics center 119895

0 otherwise

119911119895119896

=

1 relief resources from emergency logisticscenter 119895 transported to reliefdemand point 119896

0 otherwise

(1)

42 Queuing Minimal Unsatisfied Demand Location-Allocation Model Three types of members (SP ELC andDP) involved in this system are in serial connection Thetransportation routes are necessary to deliver items fromupstream nodes to downstream nodes Based on thisassumption a queuing minimal response location-allocationmodel for the three-stage queuing network is formulated asfollows

Objective function

min119885 = sum119903isin119877

sum119894isin119868

sum119895isin119869

sum119896isin119870

119905119903

119894119895119896 (2)

The Scientific World Journal 5

subject to

119910119903

119894119895le 119909119895

forall119894 isin 119868 119895 isin 119869 119903 isin 119877 (3)

119911119903

119895119896le 119909119895

forall119896 isin 119870 119895 isin 119869 119903 isin 119877 (4)

sum119895isin119869

119910119903

119894119895= 1 forall119894 isin 119868 119903 isin 119877 (5)

sum119895isin119869

119911119903

119895119896= 1 forall119896 isin 119870 119903 isin 119877 (6)

sum119897isin119871|119889119894119897le119889119894119895

119910119903

119894119897ge 119909119895

forall119894 isin 119868 119895 isin 119869 119903 isin 119877 (7)

sum119897isin119871|119889119897119896le119889119895119896

119911119903

119897119896ge 119909119895

forall119896 isin 119870 119895 isin 119869 119903 isin 119877 (8)

sum119886119895119909119895le 119861 forall119895 isin 119869 (9)

sum119909119895= 1199021

forall119895 isin 119869 (10)

sum119894isin119868

119910119903

119894119895= 1199022

forall119895 isin 119869 119903 isin 119877 (11)

sum119896isin119870

119911119903

119895119896= 1199023

forall119895 isin 119869 119903 isin 119877 (12)

119905119903

119894119895119896= WT119903

119894+WT119903

119895+WT119903

119896+ TR119903119894119895+ TR119903119895119896 (13)

119909119903

119895 119910119903

119895119896 119911119903

119894119895= 0 1 forall119896 isin 119870 119894 isin 119868 119895 isin 119869 119903 isin 119877 (14)

The objective aims at minimizing the mean systemresponse time of relief resources including the sojourn timein the queuing network system and transportation timeConstraints (3) and (4) ensure that deliveries can only bemade if emergency logistics centers are fixed Constraint(5) enforces that the items from the supply nodes canonly be delivered to one logistics center and constraint (6)enforces that every demand nodes can obtain items fromjust one logistics center Constraints (7) and (8) representthat the items from upstream nodes are transported to thenearest downstream nodes in the queuing network systemConstraint (9) limits the sum of fixed costs of locating theemergency logistics centers Constraint (10) shows that thetotal number of logistics center to be sited is equal to 119902

1

Constraint (11) forces that the number of supply depots deliv-ering commodities to the same emergency logistic centers isequal to 119902

2to ensure that each emergency logistics is with

enough capacity to deal with these commodities Constraint(12) forces that the number of demand nodes accepting itemsfrom the same emergency logistics center is equal to 119902

3to

ensure that the items from each emergency logistics centerare enough to satisfy the emergency request Constraint (13)represents that the response time is equal to the sojourn timeplus the transporting time Constraint (14) defines all thedecision variables to be binary integer variables

Next we use the queuing theory to compute the objectivefunction The emergency supply chain system is a series-parallel hybrid queuing system consisting of three service

SPi

=1

q2sum yij

TRij ELCj

120582k

DPk

= sum 120582kzjk

120583k

120582998400j120582998400j120582998400998400i

TR998400jk

120583998400998400i 120583998400j

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Figure 2 Equivalent queue of studied ESC network

nodes We can assume that the relief requests for the emer-gency demand depots follow a Poisson distribution withintensity 120582

119896 Each emergency logistics center serves a set

of demand points and therefore the relief requests for anemergency service at the logistics center are the union of therelief requests of the nodes in the set Therefore they can bedepicted as a stochastic process equal to the sum of severalPoisson processes with an intensity 1205821015840

119895equal to the sumof the

intensities of the processes at the nodes served by the logisticscenter We can rewrite parameter 1205821015840

119895by using variables 119911

119895119896

1205821015840

119895=

119899

sum119896=1

120582119896119911119895119896 (15)

The relief request for the supply depots is also assumedto follow a Poisson distribution with intensity 12058210158401015840

119894and also

similar equilibrium equations exist between the arrival rate ofthe relief request for the supply depots and for the emergencylogistics centers For the sake of simplicity we assume thatthe arrival rate of the relief request for the supply depotsthat transport emergency resources to the same emergencylogistics center has the same value Thus the parameter 12058210158401015840

119894

can be rewritten by using variables 119910119894119895and the constant 119902

2

12058210158401015840

119894=

1

1199022

ℎ

sum119895=1

1205821015840

119895119910119894119895=

1

1199022

ℎ

sum119895=1

119899

sum119896=1

120582119896119910119894119895119911119895119896 (16)

Based on the above analysis the equivalent queue of thestudied ESC network is shown in Figure 2

From the affected peoplersquos point of view the ESC systemis equivalent to a queue network that is receiving emergencyrelief orders These relief request orders are waiting to beserved The service is the process of production collec-tion and processing and the results are emergency reliefresources items and so forth

Emergency relief orders are characterized by (i) occur-rence (ii) quantity and (iii) delay Consider the following

119880 random variable indicating the occurrence time ofa relief request

119881 random variable indicating the quantity ofresources in every relief request

119882 UV indicating the occurrence time along with thequantity of resources in every relief request

Assume that119880 follows a negative exponential distributionwith intensify119891

119880(119906) and119881 is a uniformly distributed random

6 The Scientific World Journal

variable with intensify 119891119881(V) between 119888 and 119889 (119889 gt 119888) And

119891119880(119906) and 119891

119881(V) are independent Thus

119891119880(119906) =

120575119890minus120575119906 119906 ge 0

0 119906 lt 0

119891119881(V) =

1

119889 minus 119888 119888 lt V lt 119889

0 otherwise

119864 (119882) = 119864 (119880119881) = 119864 (119880) 119864 (119881) =119888 + 119889

2120575

(17)

Therefore the interarrival times of the emergencydemand (occurrence and quantity) follow a negative expo-nential distribution with intensity 120582 equal to 1119864(119882) Theservice rate at each node in the queuing network system isan independent identically distributed random variable withintensity 120583 and the service time is 1120583

So the interarrival time 120582 and the traffic intensity of thesystem 120588 are represented as

120582 =1

119864 (119882)=

2120575

119888 + 119889 (18)

120588 =120582

120583=

1

120583119864 (119882)=

2120575

120583 (119888 + 119889) (19)

Let us assume that there exits just one server at eachservice node and the servers are independent which meansthat the queuing model at each server is an 1198721198721 Thenthe probability distribution function of the sojourn timeWT (defined as the waiting time plus the service time for acustomer) in an1198721198721 queue can be presented as

119891WT (119905) = (120583 minus 120582) 119890minus(120583minus120582)119905

(20)

From (7) the cumulative distribution function of WT is

119891WT (119905) = 119875 (WT le 119905)

= int119905

0

(120583 minus 120582) 119890minus(120583minus120582)119905

119889119905 = 1 minus 119890minus(120583minus120582)119905

(21)

The average sojourn time WT is given by

WT = 119864 (WT) = intinfin

0

119891WT (119905) 119905 119889119905

= intinfin

0

(120583 minus 120582) 119890minus(120583minus120582)119905

119905 119889119905 =1

120583 minus 120582

(22)

Let WT119902denote the waiting time in the queue The

average waiting time is computed as

WT119902= WT minus

1

120583=

120582

120583 (120583 minus 120582) (23)

From the well-known Littlersquos theorem the average cus-tomers LR in the system including the number of customersboth waiting in the queue and served in the server is given by

LR = 120582WT =120582

120583 minus 120582 (24)

And LR119902denote the queuing length in the system which

is presented as

LR119902= 120582WT

119902=

1205822

120583 (120583 minus 120582) (25)

From (17) (16) (18) (22) (23) (24) and (25) theaverage sojourn time waiting time in the queue the averagecustomers including the number of customers both waitingin the queue and served in the server and the queuing lengthfor the ESC network system are given as

WTsys = sum119903isin119877

sum119894isin119868

WT119877119868+ sum119903isin119877

sum119895isin119869

WT119877119869+ sum119903isin119877

sum119896isin119870

WT119877119896

= sum119903isin119877

sum119894isin119868

1

12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

+ sum119903isin119877

sum119895isin119869

1

1205831015840 minus119899

sum119896=1

(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119896

1

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

WT119902sys = sum

119903isin119877

sum119894isin119868

WT119877119902119868

+ sum119903isin119877

sum119895isin119869

WT119877119902119869+ sum119903isin119877

sum119896isin119870

WT119877119902119896

= sum119903isin119877

sum119894isin119868

(11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

12058310158401015840 (12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)

+ sum119903isin119877

sum119895isin119869

sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

1205831015840 (1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)

+ sum119903isin119877

sum119896isin119896

2120575119903119896 (119888119903119896+ 119888119903119896)

120583 (120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896)))

LRsys = sum119903isin119877

sum119894isin119868

LR119877119868

+ sum119903isin119877

sum119895isin119869

LR119877119869+ sum119903isin119877

sum119896isin119870

LR119877119896

= sum119903isin119877

sum119894isin119868

(11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

+ sum119903isin119877

sum119895isin119869

sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119870

2120575119903119896 (119888119903119896+ 119888119903119896)

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

The Scientific World Journal 7

LR119902sys

= sum119903isin119877

sum119894isin119868

LR119877119902119868+ sum119903isin119877

sum119895isin119869

LR119877119902119869+ sum119903isin119877

sum119896isin119870

LR119877119902119896

= sum119903isin119877

sum119894isin119868

((11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)2

12058310158401015840 (12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)

+ sum119903isin119877

sum119895isin119869

(sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)2

1205831015840 (1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)

+ sum119903isin119877

sum119896isin119896

(2120575119903

119896 (119888119903

119896+ 119888119903

119896))2

120583 (120583 minus 2120575119903119896 (119888119903119896+ 119888119903119896))

(26)

As the response time includes not only the sojourn time ofemergency demand in the queuing network system but alsothe transportation time therefore when the queuing theoryis applied to the present model the above objective functioncan be presented as follows

min119885

= minsum119903isin119877

sum119894isin119868

sum119895isin119869

sum119896isin119870

119905119903

119894119895119896

= min(sum119903isin119877

sum119894isin119868

(1)

times (12058310158401015840minus (

1

1199022

)

ℎ

sum119895=1

119899

sum119896=1

(2120575119903

119896 (119888119903

119896+ 119888119903

119896)) 119910119894119895119911119895119896)

minus1

+ sum119903isin119877

sum119895isin119869

1

1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119896

1

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

+sum119903isin119877

sum119894isin119868

sum119895isin119869

TR119903119894119895119910119894119895+ sum119903isin119877

sum119895isin119869

sum119896isin119870

TR119903119895119896119911119895119896)

(27)

43 GA-Based Approach As the facility location-allocationproblem is NP- (nondeterministic polynomial-) hard wherethe proposed 0-1 nonlinear integer programming model inthis study is used some heuristic algorithms are requiredto solve the problem quickly The genetic algorithms (GAs)have been applied to location problems since 1986 by Hosageand Goodchild [23] GA has exhibited inherent advantagessuch as their robust performancewhen solving combinatorialproblems as well as their ability to incorporate proceduresand logical conditions and to handle discrete variables andnonlinear constraints in a straightforward manner [24]These qualities make GAs particularly attractive for potentialcombination with other methods and external procedures

Wedevelop a refined genetic algorithm to solve the aforemen-tioned problem as indicated belowThe associated frameworkfor embedding the GA with emergency queue network ispresented in Figure 3

431 Methodology of Genetic Algorithm In a genetic algo-rithm a population of candidate solutions to an optimizationproblem is evolved toward better solutions Each candi-date solution has a set of properties (its chromosomes orgenotype) which can be mutated and altered solutions arerepresented in binary as strings of 0 s and 1 s but otherencodings are also possible The evolution usually startsfrom a population of randomly generated individuals andhappens in generations In each generation the fitness ofevery individual in the population is evaluated the more fitindividuals are stochastically selected from the current popu-lation and each individualrsquos genome ismodified (recombinedand possibly randomly mutated) to form a new populationThe new population is then used in the next iteration of thealgorithm Commonly the algorithm terminates when eithera maximum number of generations have been produced or asatisfactory fitness level has been reached for the populationOnce the genetic representation and the fitness function aredefined a GA proceeds to initialize a population of solutionsand then to improve it through repetitive application of themutation crossover inversion and selection operators

432 Procedure of the Refined Genetic Algorithm

Step 1 (encoding) We suppose that the genotype of a chro-mosome with some genes indicates an individual corre-sponding to the location pattern of emergency logisticscenters and paths between nodes in the system network Thegene presented by 1 indicates that an emergency logisticscenter is built at this location or the path between nodes in thesystem network is selectedThe gene presented by 0 indicatesthat no emergency logistics centers is built there or the pathbetween nodes in the system network which is selected is notselected 119873 chromosomes are randomly generated and theinitial population (Pop) is set

Step 2 (fitness evaluation) Once the initial population isavailable evaluation of each chromosome takes place to ex-plore the quality of the corresponding solution For individ-uals with larger fitness value to be transmitted to the nextgeneration with higher probability the chromosome fitnessfunction in this paper can be described as follows and 119888max isthe maximum estimation value of min119885

Fit (min119885) = 119862max minusmin119885 if min119885 lt 119862max

0 otherwise(28)

Step 3 (crossover selection and mutation) 119873 chromosomesare crossed according to a single pointmethodwith crossoverprobability (which is 119875

119888= 09) to generate the offspring

chromosomes There are a total of 2119873 chromosomes forboth parent and offspring chromosomes Then the duplicatechromosomes are deleted to produce the new population(Pop) Select the first 119873 chromosomes of the population on

8 The Scientific World Journal

Step 1 encoding forparameter sets

Initialization ofgenerate population

Step 2 chromosomefitness evaluation

Step 3 geneticmanipulation crossoverselection and mutation

Step 4 offspringgeneration Select the best individual

and output the solution

Yes

No Terminate

(1) Decoding bit string to the parameter(2) Objective function value calculation(3) Mapping function value to the fitness value(4) Fitness value adjustments

Preparation of dataand parameters

Figure 3 Steps of the proposed GA heuristic

(B)(C)

(A)

Figure 4 The location of facilities in an ESC system

the basis of the fitness values in ascending order When thenumber of chromosomes in the new population (Pop) is lessthan 119873 (119873 = 200) 119873-|Pop| chromosomes are randomlygenerated (|Pop| is the number of chromosomes in thepopulation (Pop)) Select a chromosome randomly and applythe mutation operator applied on it with probability 119875

119898=

001

Step 4 Choose new chromosomes by the roulette wheelmethod Repeat Steps 3 and 4 until there are no solutionimprovements within a specified number of iterations

5 A Case Study

A case study example is presented in this section to illustratethe efficiency of our approach to realize the quick-responsiveemergency relief during large-scale natural disastersThepro-posed method is implemented on a scenario where a severetyphoon strikes the southeast coast of Shanghai with a greatprobability within the next 100 years It is assumed that thereare three regions (A B and C) slashed by the typhoon shownin Figure 4 One region is Lingangxincheng District (A)

another one is Fengxian District (B) and the third one isJinshan District (C)

There are ten supply points (SP) (1) disaster relief sup-plies reserve center in Baoshan District (2) Automobile Cityin Jiading (3) Red Cross disaster preparedness warehouse inJiading District (4) disaster relief supplies reserve center inQingqu District (5) disaster relief supplies reserve center inPutuo District (6) disaster relief supplies reserve center inYangpu District (7) disaster relief supplies reserve center inMinhang District (8) disaster relief supplies reserve center inHongkou District (9) disaster relief supplies reserve centerin Pudong District and (10) disaster relief supplies reservecenter in Xuhui District Based on the study of large-scalelogistics center in Shanghai we select ten potential pointsas the emergency logistics centers (ELCs) Four potentialELCs are prepared for the affected areas in LingangxinchengDistrict two potential ELCs are planned to serve FengxianDistrict and two potential ELCs are expected to serve theJinshanDistrictWe assume that a total of thirteen areas (DP)are seriously affected and need emergency rescue urgently inwhich six affected areas are in Lingangxincheng four affectedareas are in Fengxian District and three affected areas are inJinshan District

For simplicity only one type of emergency supply resour-ces will be considered in the queuing network We assumethat the travel speed is fixed to be equal to 40 miles per hourfor all the vehicles travelling between SP and ESC Howeverthe travel speed between ESC and DP is assumed to be 30miles per hour Let us adopt the hypothesis as the abovethat the occurrence time of a relief request follows a negativeexponential distribution with intensify 120575 and the quantity ofresources in every relief request is uniformly distributed withintensify 119891

119881(V) that the lower bound (LB) is 119888 and the upper

bound (UB) is 119889 (119889 gt 119888) The disaster-related informationincluding the population of the affected areas and the distancebetween supply point and emergency logistics center anddistance between emergency logistics center and demandpoint are shown in Tables 1 2 3 and 4 respectively

The Scientific World Journal 9

Table 1 Population and demand data of the affected areas

Affected area Population (119888 119889) 120575

Area 1 (Zhuqiaozhen) 105807 (4 12) 16Area 2 (Huinanzhen) 213845 (4 12) 16Area 3 (Laogangzhen) 37408 (5 14) 18Area 4 (Datuanzhen) 71162 (4 12) 16Area 5 (Xingangzhen) 21475 (6 16) 20Area 6 (Shuyuanzhen) 59831 (6 16) 20Area 7 (Luchaogangzhen) 27850 (6 16) 20Area 8 (Situanzhen) 65389 (5 14) 18Area 9 (Qingcunzhen) 89163 (5 14) 18Area 10 (Nanqiaozhen) 361185 (3 10) 14Area 11 (Zhuanghangzhen) 62388 (3 10) 14Area 12 (Zhelinzhen) 62589 (3 10) 14Area 13 (Caojingzhen) 40722 (2 8) 12Area 14 (Shanyangzhen) 84640 (2 8) 12Area 15 (Jinshanzhen) 70815 (2 8) 12

0 20 40 60 80 100 120 140 160 180 20020

25

30

35

40

45

50

55

60

Interaction number

Tota

l res

pons

e tim

e (ho

urs)

Total response time

Figure 5 Optimization of the fitness function

In this case study the fixed cost to locate the emergencylogistics center is assumed as 119886

1= 85 119886

2= 100 119886

3= 90

1198864= 85 119886

5= 95 119886

6= 110 119886

7= 70 119886

8= 80 119886

9= 115

and 11988610

= 120 The total cost to build the emergency logisticscenters is assumednot exceeding to be119861 = 500 For simplicityof the computation we presume that 119902

1= 5 119902

2= 2 and

1199023= 3According to the background of the above problem a

simplified 10 times 10 times 15 emergency supply chain network isdeveloped in the assumed scenario where the geographicalrelationships among these supply depots emergency logisticscenters and demand points are specified Then we applythe aforementioned model into the emergency supply chainnetwork system And the GA-based approach demonstratedbefore is coded to verify the validity of the model inMatlab2009a

The optimization process of the fitness function isdepicted in Figure 5 where the optimal response time of thequeuing network is 2071 hours

Therefore the average response time of each demandpoint is 138 h which is within a reasonable range The cal-culation result shows that this model can play an importantrole in the emergency supply chain management Differentfrom the general supply chain management the timesavingis the most significant feature of the emergency supply chainsystem The efficient allocation of emergency resources todemand points improves not only the operational perfor-mance of the emergency supply chain system in the emer-gency supply side but also the affected peoplersquos survival ratein the disaster areas From the psychology of point of viewquick responses and deliveries for disaster relief not onlymay improve the governmental efforts to rescue but alsostrengthen the affected peoplersquos willpower to be alive hencestabilizing the affected areas

The emergency supply chain system is composed of threemembers which have different functions in the relief supplydelivery process And the upstream nodes and downstreamnodes in the ESC system are connected by roads If we treateach node in the system as the node in a network considerthe roads as the arcs of the network and treat the length ofeach road as the weight of each arc then the ESC system canbe seen as a directed network diagram with weights And theresponse time of the system includes two parts the first partis the transportation time of the relief supply request on theroads and the second part is the sojourn time of the reliefsupply request in each node

Figure 6 indicates the values of the above two componentsof response time with the total transportation time being1664 h and the total sojourn time being 407 h This iterativeprocess also demonstrates the convergence of the heuristicalgorithm

Queuing models are effective tools for obtaining infor-mation about important performance of each facility in theESC system like sojourn times waiting times queue lengthstotal customers in the system and so on In order to analyzethe efficiency of facilities in the system we calculate thetotal customers and the queue lengths which are shown inFigure 7 We can see that the solution of the above model willnot create queuing delays of relief supply so that the lossesgenerated by the queuing delay can be avoided This iterativeprocess also illustrates the robustness and the potentialapplicability of our model

Based on the built network the coordination between theupstream node and the downstream node in this system canbe enforced and the performance of basic activities includingproducing sorting processing packing and delivering canbe improved tominimize the response time of the systemTheemergency logistics centers for the system should be locatedin DP(6) DP(7) DP(8) DP(9) and DP(10) And the optimalpaths are SP(1) SP(6) rarr ELC(8) rarr DP(1) DP(2)DP(9) SP(2) SP(4) rarr ELC(6) rarr DP(13) DP(14)DP(15) SP(3) SP(8) rarr ELC(7) rarr DP(10) DP(11)DP(12) SP(5) SP(9) rarr ELC(10) rarr DP(3) DP(4)DP(5) and SP(7) SP(9) rarr ELC(9) rarr DP(6) DP(7)DP(8) Overall the examination reveals that the proposedmethod can be a useful tool for a quick response to the urgentneed of relief in the large-scale disaster-affected areas

10 The Scientific World Journal

Table 2 The service rates specifications of different servers in the network

SP 120583101584010158401

120583101584010158402

120583101584010158403

120583101584010158404

120583101584010158405

120583101584010158406

120583101584010158407

120583101584010158408

120583101584010158409

1205831015840101584010

10 9 11 8 11 12 14 16 14 17

ELC 12058310158401

12058310158402

12058310158403

12058310158404

12058310158405

12058310158406

12058310158407

12058310158408

12058310158409

120583101584010

16 18 11 10 11 12 14 16 14 10

DP 1205831

1205832

1205833

1205834

1205835

1205836

1205837

1205838

1205839

12058310

12058311

12058312

12058313

12058314

12058315

8 8 8 8 10 9 11 8 9 12 9 8 12 14 13

Table 3 The distance between supply point and emergency logistics center

SP(1) SP(2) SP(3) SP(4) SP(5) SP(6) SP(7) SP(8) SP(9) SP(10)ELC(1) 1828 1707 1197 2163 614 826 1091 973 1530 879ELC(2) 3073 2147 1624 571 2039 2056 802 1679 2554 1390ELC(3) 2271 2288 1712 1622 1419 1463 998 1102 1842 756ELC(4) 4422 3095 2462 1264 3971 3492 2429 3276 4153 3003ELC(5) 3962 2801 2716 1507 3256 2769 1449 2551 3325 2006ELC(6) 3585 2356 1923 1647 2784 2313 989 2136 2839 1437ELC(7) 2973 2968 2364 2301 2036 2149 1275 1797 2575 1471ELC(8) 2308 3261 2878 2892 1494 1401 1764 1132 1961 1933ELC(9) 2584 3643 3303 3249 2015 1165 2258 941 1600 2419ELC(10) 3038 4666 4240 4139 2831 2039 3008 1742 1405 3105

05

101520253035404550

Tim

e (ho

urs)

Total transportation timeTotal sojourn time

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 6 Optimization of total transportation time and totalsojourn time

6 Conclusions

An emergency supply chain (ESC) system is developed inthis paper for the quick response to the urgent need ofrelief in the large-scale disaster-affected areas which mainlyconsists of three chain members in series supply points(SP) emergency logistics centers (ELCs) and demand points(DP) As the timeliness is the most crucial characteristic ofthe ESC system and queuing theory is a useful tool in theanalysis of the performance of the time-dependent systemwe propose queuingmodeling for the ESC system to optimizethe emergency rescue process Based on the above queuingnetwork system a queuing minimal response time location-allocation model is established to decide the selection ofemergency logistics centers and vehicle routing decision

0

5

10

15

20

25

Total customersQueuing length

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 7 Optimization of the total customers and the queue length

For the complexity of mathematical model the GA-basedapproach is introduced to solve the model

A case study with an assumed severe typhoon striking thesoutheast coast of Shanghai with a great probability withinthe next 100 years is conducted to assess and validate ourmodel The optimal response time of the queuing network is2071 hours and the average response time of each demandpoint is 138 h which is within a reasonable range Theoptimal emergency logistics centers for the system are DP(6)DP(7) DP(8) DP(9) and DP(10) Besides the performanceof queuing modeling for the ESC system illustrates therobustness and the potential applicability of our model

For future research we suggest the following threedirections Firstly we will investigate the queuing modelingunder consideration of the facility blocking problems and

The Scientific World Journal 11

Table 4 The distance between emergency logistics center and demand point

ELC(1) ELC(2) ELC(3) ELC(4) ELC(5) ELC(6) ELC(7) ELC(8) ELC(9) ELC(10)DP(1) 3761 4191 3253 4599 3801 3752 2439 2397 1553 739DP(2) 4271 4702 3761 4794 3945 4152 2818 2822 1898 1075DP(3) 4508 4886 3986 5028 4501 4384 3046 3025 2117 1356DP(4) 4364 4579 3697 4824 4307 4138 2798 2940 1956 1710DP(5) 5283 5416 4386 5466 5435 4986 3836 3271 2693 1917DP(6) 5214 5721 4866 5896 5294 5207 3797 4221 3252 2553DP(7) 5334 5776 5299 5862 5359 5164 3871 3851 2955 2554DP(8) 4689 4581 3763 4983 4306 4173 2818 2997 2200 1968DP(9) 3309 3624 2657 3834 3334 3154 1886 3154 3452 3235DP(10) 2678 3263 1999 2911 2741 2518 1198 2505 2360 3037DP(11) 3369 2912 2386 2349 2182 1858 1894 2392 2915 3596DP(12) 3376 3794 2541 3287 3275 2612 1708 3062 2896 3256DP(13) 3691 3419 2987 2896 2694 2279 2162 3500 3275 4007DP(14) 4230 3273 3402 3011 2555 2141 2606 3906 3749 4448DP(15) 4609 3765 3858 2533 3054 2680 3166 4390 4249 4897

demand priority for the emergency relief points Secondly wewill study the generation of emergency demand and servicetime obeying the other random distributions Thirdly moreefficient multiobjective optimization methodologies for boththe above problem and emergency resources transportationscheduling problem will be considered

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grant nos 71102043 and 71371145) theMinistry of Education of China (Grant no MCM20125021)the Innovative Program of Shanghai Municipal EducationCommission (Grant no 14ZS123) and the ShanghaiMaritimeUniversity (Grant no 20120054)

References

[1] D Guha-Sapir G S Debarati D Hargitt P Hoyois R Belowand D BrechetThirty Years of Natural Disasters 1974ndash2003 theNumbers Presses universitaires de Louvain 2004

[2] J-B Sheu ldquoAn emergency logistics distribution approach forquick response to urgent relief demand in disastersrdquo Trans-portation Research E vol 43 no 6 pp 687ndash709 2007

[3] B S Manoj and A H Baker ldquoCommunication challenges inemergency responserdquo Communications of the ACM vol 50 no3 pp 51ndash53 2007

[4] B Balcik B M Beamon and K Smilowitz ldquoLast mile distribu-tion in humanitarian reliefrdquo Journal of Intelligent TransportationSystems vol 12 no 2 pp 51ndash63 2008

[5] A M Caunhye X F Nie and S Pokharel ldquoOptimizationmodels in emergency logistics a literature reviewrdquo Socio-Economic Planning Sciences vol 46 no 1 pp 4ndash13 2012

[6] S Coles and L Pericchi ldquoAnticipating catastrophes throughextreme valuemodellingrdquo Journal of the Royal Statistical SocietyC vol 52 no 4 pp 405ndash416 2003

[7] J R Artalejo ldquoG-networks a versatile approach for workremoval in queueing networksrdquo European Journal of Opera-tional Research vol 126 no 2 pp 233ndash249 2000

[8] W Jiang L Deng L Chen J Wu and J Li ldquoRisk assessmentand validation of flood disaster based on fuzzy mathematicsrdquoProgress in Natural Science vol 19 no 10 pp 1419ndash1425 2009

[9] V Bhaskar and P Lallement ldquoModeling a supply chain usinga network of queuesrdquo Applied Mathematical Modelling vol 34no 8 pp 2074ndash2088 2010

[10] B M Beamon ldquoHumanitarian relief chains issues and chal-lengesrdquo in Proceedings of the 34th International Conference onComputers and Industrial Engineering San Francisco CalifUSA November 2004

[11] R Oloruntoba and R Gray ldquoHumanitarian aid an agile supplychainrdquo Supply Chain Management vol 11 no 2 pp 115ndash1202006

[12] E J Lodree Jr and S Taskin ldquoSupply chain planning for hur-ricane response with wind speed information updatesrdquo Com-puters and Operations Research vol 36 no 1 pp 2ndash15 2009

[13] V Bhakoo P Singh and A Sohal ldquoCollaborative managementof inventory in Australian hospital supply chains practices andissuesrdquo Supply Chain Management vol 17 no 2 pp 217ndash2302012

[14] W Yi and A Kumar ldquoAnt colony optimization for disaster reliefoperationsrdquo Transportation Research E vol 43 no 6 pp 660ndash672 2007

[15] G-H Tzeng H-J Cheng and T D Huang ldquoMulti-objectiveoptimal planning for designing relief delivery systemsrdquo Trans-portation Research E vol 43 no 6 pp 673ndash686 2007

[16] Y-C Chiu and H Zheng ldquoReal-time mobilization decisionsfor multi-priority emergency response resources and evacua-tion groups model formulation and solutionrdquo TransportationResearch E vol 43 no 6 pp 710ndash736 2007

[17] N Altay andWG Green III ldquoORMS research in disaster oper-ations managementrdquo European Journal of Operational Researchvol 175 no 1 pp 475ndash493 2006

12 The Scientific World Journal

[18] K J Cormican D P Morton and R K Wood ldquoStochasticnetwork interdictionrdquo Operations Research vol 46 no 2 pp184ndash197 1998

[19] HOMete andZ B Zabinsky ldquoStochastic optimization ofmed-ical supply location and distribution in disaster managementrdquoInternational Journal of Production Economics vol 126 no 1 pp76ndash84 2010

[20] H Shavandi and H Mahlooji ldquoA fuzzy queuing locationmodel with a genetic algorithm for congested systemsrdquo AppliedMathematics and Computation vol 181 no 1 pp 440ndash4562006

[21] R D Galvao and R Morabito ldquoEmergency service systemsthe use of the hypercube queueing model in the solution ofprobabilistic location problemsrdquo International Transactions inOperational Research vol 15 no 5 pp 525ndash549 2008

[22] N Geroliminis M G Karlaftis and A Skabardonis ldquoA spatialqueuing model for the emergency vehicle districting andlocation problemrdquo Transportation Research B vol 43 no 7 pp798ndash811 2009

[23] C M Hosage and M F Goodchild ldquoDiscrete space location-allocation solutions from genetic algorithmsrdquo Annals of Opera-tions Research vol 6 no 2 pp 35ndash46 1986

[24] P Chakroborty ldquoGenetic algorithms for optimal urban transitnetwork designrdquo Computer-Aided Civil and Infrastructure Engi-neering vol 18 no 3 pp 184ndash200 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 The Scientific World Journal

In this paper we develop a new quick-responsive ESCsystem with a queuing model in a specified three-layer net-work Each depository in the ESC is modeled as a server andwaiting commodities are in a queue By using the queuingmodel we can obtain the classical performances of ESC suchas average queue length and average waitingresidence timeand optimize the relief operations under the consideration ofboth economic loss and response time limitation

Themain contributions of this study are as follows Firstlya new emergency service supply chain model fulfilling therequirement of quick responses and deliveries for disasterrelief is developed Secondly a queuing model is embeddedin this model for the first time to calculate the performanceof ESC Thirdly the physical network of ESC with a queuingmodel is proposed as an effective refined method to solve theintegrated problem including vehicle routing problem andlocation problem of emergency relief centers with multiplelayers

The rest of this paper is organized as follows Section 2presents the literature review on emergency managementA system specification is provided with a physical networkof ESC including a queuing model in Section 3 A queuingminimal response time location-allocation model is formu-lated with a heuristic method in Section 4 A case study isintroduced in Section 5 to demonstrate the performance ofthe proposed approach Finally Section 6 draws conclusionsand provides directions for future research

2 Literature Review

Emergency supply chain management is the key to the suc-cess of relief demand management under the condition oflarge-scale disasters The difficulty of emergency supplychain management is rooted in the uncertainties of abruptrelief demand and collaboration of chaotic condition Unlikedemanders in business logistics the relief demanders are thedisaster-affected people but their locations and their demandsmay not be predicted precisely before disasters happenApparently efficient relief supplying underlines the challengeof collaborative relief demandmanagement in the emergencysupply chain management Despite the urgent necessity ofcollaborating relief demandmanagement in the whole supplychain there is no straightforward collaboration model avail-able for the above issue In contrast with the optimization-based demand and supply models relief demand manage-ment must overcome more issues in disaster uncertaintiesand delivery collaboration

In brief the existing uncertainty based demand modelsappear unsuitable for chaotic relief demand managementaddressed in this study Instead most of the existing reliefdemand management models in the emergency supply chainappear to be limited to general cases for business opera-tions From the literature review we illustrate several relatedsubjects associated with typical models in the following forfurther discussion

By comparing the business supply chain and the human-itarian relief chain Beamon [10] revealed several specificcharacteristics of relief material supply chains including

zero lead times high stakes unreliable incomplete ornonexistent prior information and different demand patternOloruntoba and Gray [11] developed an agile supply chainmodel for humanitarian aid by applying practical elementsof conventional supply chains to the ESC Lodree Jr andTaskin [12] introduced a stochastic inventory control modelto prepare for potential hurricane activity and describeda dynamic programming algorithm to solve the inventoryproblem Bhakoo et al [13] developed an understanding ofthe nature of collaborative arrangements for themanagementof inventories in Australian hospital supply chains

Despite the recent emergence of emergency supply chainmanagement that has increasingly drawn researchersrsquo atten-tionmost previousworks appear to address the issues of reliefsupply and distribution contingent on relief demand assump-tions Yi andKumar [14] decomposed the emergency logisticsproblem into two decision-making phases and proposed anant colony optimization (ACO) model to solve a multicom-modity network flow problem Tzeng et al [15] simplifiedthe disaster context and proposed a fuzzy multiobjectiveprogramming method to optimize multiobjective functionsto avoid the possibility of a severely unfair relief distributionChiu andZheng [16] addressed themultiple emergency trafficflows outbound from the affected areas using a linear celltransmission model (CTM)

In order to deal with the optimization of emergencyman-agement there are a large number of studies on developingoperational research (OR)methods for it [17]Thesemethod-ologies include the use of linear programming techniquesfuzzy methods stochastic programming models probabilis-tic models simulation and decision theory and queuingtheory [5] Stochastic programming has been successfullyused in the area related to disaster studies [18 19] As differentapproaches to handle emergency problems simulation anddecision theory have also been adopted as methodologiesin this field Queuing theory has been applied to exploreemergency facility location problems recently Shavandi andMahlooji [20] combined fuzzy theory and queuing methodto develop a maximal covering location-allocation modelGalvao and Morabito [21] applied the hypercube queuingmodel to solve probabilistic location problems in the emer-gency service system Geroliminis et al [22] presented aqueuing model for locating emergency vehicles on urbannetworks considering both spatial and temporal demandcharacteristics such as the probability that a server is notavailable when required But numerous emergency practicesreveal that chaotic status after disasters worsens rescuecoordination and the relief supply chain is often blocked orcongested in practices Multiple emergency sources improverobustness of supply chains but coordination amongmultipleservers is more critical for emergency management Mean-while organizational skills require that the emergency supplychain operated by local government consists of multipleechelons which forms vertical coordination in emergencycontext All those problems need to be resolved in newmodels and algorithms However no paper indicated aboveembeds the queuingmodels into themultiechelon emergencyservice supply chain system

The Scientific World Journal 3

Therefore to resolve the issues mentioned above wepresent a relief demand management model of emergencysupply chain to address the above issue under the disorderand uncertain conditions in affected areas during the crucialrescue period of a large-scale disaster Rooted in the tech-niques of collaboration in emergency supply chain coupledwith queuing theory and system optimization the proposedmethodology embeds three mechanisms (1) multiechelonsupply chainmodel for disasters (2) dynamic facility locationand vehicle routing selection and (3) rescue systemmanage-ment collaboration

Relative to the previous literature the proposed reliefdemand management methodology has the following twodistinctive features (1)Themodel is capable of collaboratingurgent relief demand management in the large-scale disastercontexts and accelerating rescue efforts to save casualty loss(2) To facilitate dynamic relief allocation and distributionthe proposed model practically groups humanitarian reliefsintomultiechelon resource suppliers and distribution centerswhich form an emergency response system for uncertainlarge disasters

3 System Specification

An ESC system involves selection of sites and vehicle rout-ing decisions which are two major problems in a disasterresponse environment The optimal facilities locations andpath selections can guarantee that the commodities will besent from the supply depots to the demand points in affectedareas as quickly as possible to maximize the survival rate ofwounded persons The above problems arouse our intereststo propose queuing modeling for the ESC system Hence aqueuing network of emergency supply chain is formulated inthis paper as shown in Figure 1

The queuing network of ESC involves a queuing flowformulation where the three supply chain members namelysupply points (SP) emergency logistics centers (ELCs) anddemand points (DP) are treated as servers Emergency com-modities such as food shelter personnel machinery andmedicine are modeled as customers The upstream anddownstream nodes of ESC system constitute some basicactivities that are producing sorting processing packingdelivering and so forth These activities are regarded as theservice for customers Consider the following

(1) Locations of emergency supply points are in 1198781 1198782

1198783 119878

119898

(2) Locations of emergency logistics centers are in 1198711 1198712

1198713 119871

119903

(3) Locations of emergency demand points are in1198631 1198632

1198633 119863

119899

(a) Vehicle routing choices for the commodity flowsin the system are considered in the following

(4) TR1119895 from 119878

1to one of the nodes (119871

1 1198712 1198713

119871119903) TR

2119895 from 119878

2to one of the nodes (119871

1 1198712

1198713 119871

119903) and TR

119898119895 from 119878

119898to one of the

nodes (1198711 1198712 1198713 119871

119903) 119895 = 1 2 119903

(5) TR1119896 from 119871

1to one of the nodes (119863

1 1198632 1198633

119863119899) TR2119896 from 119871

2to one of the nodes (119863

1 1198632 1198633

119863119899) and TR

119903119896 from 119871

119903to one of the nodes

(1198631 1198632 1198633 119863

119899) 119896 = 1 2 119899

Each commodity is considered as a queue where batchesare waiting to be serviced The selection of sites and vehiclerouting decision may be operated under the considerationof the estimated throughput response time from supplydepots to demand depots in affected areas The responsetime comprises not only the transportation times betweenupstream and downstream nodes but also the total waitingtimes and service times in the queuing network

Therefore the above three nodes in the system areassumed to behave as an 1198721198721 queuing where reliefsupplies are treated as customers On the basis of the specifiedESC queuing network we adopt the following hypothesis forsystem operations

(1) The corresponding geographic relationships betweenupstream and downstream nodes are available fromthe existing governmental databases and the reliefdemand needed in a given affected area can be readilyaccessible via advanced disaster detection technolo-gies

(2) The locations of emergency logistics centers are onlyfixed in the given alternative sites

(3) The first customer in the queue receives servicesfirstly namely ldquofirst come first servedrdquo

4 Mathematical Formulation

Based on the aforementioned system specification we pro-pose a queuing theory in this section for facility locationand path selection problems in a multistage emergencysupply chain network Firstly the notations parametersand decision variables for the mathematical formulation areintroduced After that the objective function for the modelis established And then the formulation of the constraints ofthe problems is presented

41 The Parameters and Decision Variables The sets param-eters and decision variables are defined as follows

(1) Notations

119868 set of supply depots 119894 isin 119868 119894 = 1 2 3 119898119869 set of alternative sites of emergency logistics cen-ters 119895 isin 119869 119895 = 1 2 3 ℎ119870 set of depots for handing out relief goods 119896 isin 119870119896 = 1 2 3 119899119877 set of commodities 119903 isin 119877 119903 = 1 2 3 119877

(2) Parameters

120582 the interarrival time of the emergency demandfollowing a negative exponential distribution

4 The Scientific World Journal

Emergency supplypoints

S1

S2

Sm

Transportation

TR1j

TR2j

TRmj

Emergencylogistics centres

Emergencydemand points

L1

L2

Lh

Transportation

TR9984001k

TR9984002k

TR998400hk

D1

D2

Dn

middot middot middotmiddot middot middot middot middot middot middot middot middotmiddot middot middot

Figure 1 The queuing network of ESC

120583 the service rate at eachnode in the queuing networksystem

120575 the parameter of the negative exponential distribu-tion

119888 the lower bound of a uniformly distributed randomvariable that indicates the quantity of resources in arelief request

119889 the upper boundof a uniformly distributed randomvariable that indicates the quantity of resources in arelief request

119905119903

119894119895119896 the response time of the ESC system for com-

modity type 119903 from node 119894 to node 119896 going throughlogistics centers located at nodes 119895

WT119903 the sojourn time of commodity type 119903 in thesystem

WT119903119902 the waiting time of commodity type 119903 in the

queue

TR119903119894119895 the transportation time for commodity type 119903

from node 119894 to node 119895

TR119903119895119896 the transportation time for commodity type 119903

from node 119895 to node 119896

DTR119894119895 the distance between node 119894 and node 119895

DT119896119895 the distance between node 119895 and node 119896

DC the penalty cost for unavailability of commoditiesdemand within the maximum promised responsetime

119886119895 the fixed costs of locating the emergency logistics

center 119895

1199021 the total number of logistics center to be fixed

1199022 the number of supply depots delivering commodi-

ties to the same emergency logistic centers

1199023 the number of demand nodes accepting items

from the same emergency logistics center

V119903119894119895 transport speed for commodity type 119903 from node

119894 to node 119895

V119903119895119896 transport speed for commodity type 119903 from node

119895 to node 119896

(3) Decision Variables

119909119895=

1 emergency logistics center builton the site 119895

0 otherwise

119910119894119895=

1 relief resources from emergency supplypoint 119894 transported to emergencylogistics center 119895

0 otherwise

119911119895119896

=

1 relief resources from emergency logisticscenter 119895 transported to reliefdemand point 119896

0 otherwise

(1)

42 Queuing Minimal Unsatisfied Demand Location-Allocation Model Three types of members (SP ELC andDP) involved in this system are in serial connection Thetransportation routes are necessary to deliver items fromupstream nodes to downstream nodes Based on thisassumption a queuing minimal response location-allocationmodel for the three-stage queuing network is formulated asfollows

Objective function

min119885 = sum119903isin119877

sum119894isin119868

sum119895isin119869

sum119896isin119870

119905119903

119894119895119896 (2)

The Scientific World Journal 5

subject to

119910119903

119894119895le 119909119895

forall119894 isin 119868 119895 isin 119869 119903 isin 119877 (3)

119911119903

119895119896le 119909119895

forall119896 isin 119870 119895 isin 119869 119903 isin 119877 (4)

sum119895isin119869

119910119903

119894119895= 1 forall119894 isin 119868 119903 isin 119877 (5)

sum119895isin119869

119911119903

119895119896= 1 forall119896 isin 119870 119903 isin 119877 (6)

sum119897isin119871|119889119894119897le119889119894119895

119910119903

119894119897ge 119909119895

forall119894 isin 119868 119895 isin 119869 119903 isin 119877 (7)

sum119897isin119871|119889119897119896le119889119895119896

119911119903

119897119896ge 119909119895

forall119896 isin 119870 119895 isin 119869 119903 isin 119877 (8)

sum119886119895119909119895le 119861 forall119895 isin 119869 (9)

sum119909119895= 1199021

forall119895 isin 119869 (10)

sum119894isin119868

119910119903

119894119895= 1199022

forall119895 isin 119869 119903 isin 119877 (11)

sum119896isin119870

119911119903

119895119896= 1199023

forall119895 isin 119869 119903 isin 119877 (12)

119905119903

119894119895119896= WT119903

119894+WT119903

119895+WT119903

119896+ TR119903119894119895+ TR119903119895119896 (13)

119909119903

119895 119910119903

119895119896 119911119903

119894119895= 0 1 forall119896 isin 119870 119894 isin 119868 119895 isin 119869 119903 isin 119877 (14)

The objective aims at minimizing the mean systemresponse time of relief resources including the sojourn timein the queuing network system and transportation timeConstraints (3) and (4) ensure that deliveries can only bemade if emergency logistics centers are fixed Constraint(5) enforces that the items from the supply nodes canonly be delivered to one logistics center and constraint (6)enforces that every demand nodes can obtain items fromjust one logistics center Constraints (7) and (8) representthat the items from upstream nodes are transported to thenearest downstream nodes in the queuing network systemConstraint (9) limits the sum of fixed costs of locating theemergency logistics centers Constraint (10) shows that thetotal number of logistics center to be sited is equal to 119902

1

Constraint (11) forces that the number of supply depots deliv-ering commodities to the same emergency logistic centers isequal to 119902

2to ensure that each emergency logistics is with

enough capacity to deal with these commodities Constraint(12) forces that the number of demand nodes accepting itemsfrom the same emergency logistics center is equal to 119902

3to

ensure that the items from each emergency logistics centerare enough to satisfy the emergency request Constraint (13)represents that the response time is equal to the sojourn timeplus the transporting time Constraint (14) defines all thedecision variables to be binary integer variables

Next we use the queuing theory to compute the objectivefunction The emergency supply chain system is a series-parallel hybrid queuing system consisting of three service

SPi

=1

q2sum yij

TRij ELCj

120582k

DPk

= sum 120582kzjk

120583k

120582998400j120582998400j120582998400998400i

TR998400jk

120583998400998400i 120583998400j

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Figure 2 Equivalent queue of studied ESC network

nodes We can assume that the relief requests for the emer-gency demand depots follow a Poisson distribution withintensity 120582

119896 Each emergency logistics center serves a set

of demand points and therefore the relief requests for anemergency service at the logistics center are the union of therelief requests of the nodes in the set Therefore they can bedepicted as a stochastic process equal to the sum of severalPoisson processes with an intensity 1205821015840

119895equal to the sumof the

intensities of the processes at the nodes served by the logisticscenter We can rewrite parameter 1205821015840

119895by using variables 119911

119895119896

1205821015840

119895=

119899

sum119896=1

120582119896119911119895119896 (15)

The relief request for the supply depots is also assumedto follow a Poisson distribution with intensity 12058210158401015840

119894and also

similar equilibrium equations exist between the arrival rate ofthe relief request for the supply depots and for the emergencylogistics centers For the sake of simplicity we assume thatthe arrival rate of the relief request for the supply depotsthat transport emergency resources to the same emergencylogistics center has the same value Thus the parameter 12058210158401015840

119894

can be rewritten by using variables 119910119894119895and the constant 119902

2

12058210158401015840

119894=

1

1199022

ℎ

sum119895=1

1205821015840

119895119910119894119895=

1

1199022

ℎ

sum119895=1

119899

sum119896=1

120582119896119910119894119895119911119895119896 (16)

Based on the above analysis the equivalent queue of thestudied ESC network is shown in Figure 2

From the affected peoplersquos point of view the ESC systemis equivalent to a queue network that is receiving emergencyrelief orders These relief request orders are waiting to beserved The service is the process of production collec-tion and processing and the results are emergency reliefresources items and so forth

Emergency relief orders are characterized by (i) occur-rence (ii) quantity and (iii) delay Consider the following

119880 random variable indicating the occurrence time ofa relief request

119881 random variable indicating the quantity ofresources in every relief request

119882 UV indicating the occurrence time along with thequantity of resources in every relief request

Assume that119880 follows a negative exponential distributionwith intensify119891

119880(119906) and119881 is a uniformly distributed random

6 The Scientific World Journal

variable with intensify 119891119881(V) between 119888 and 119889 (119889 gt 119888) And

119891119880(119906) and 119891

119881(V) are independent Thus

119891119880(119906) =

120575119890minus120575119906 119906 ge 0

0 119906 lt 0

119891119881(V) =

1

119889 minus 119888 119888 lt V lt 119889

0 otherwise

119864 (119882) = 119864 (119880119881) = 119864 (119880) 119864 (119881) =119888 + 119889

2120575

(17)

Therefore the interarrival times of the emergencydemand (occurrence and quantity) follow a negative expo-nential distribution with intensity 120582 equal to 1119864(119882) Theservice rate at each node in the queuing network system isan independent identically distributed random variable withintensity 120583 and the service time is 1120583

So the interarrival time 120582 and the traffic intensity of thesystem 120588 are represented as

120582 =1

119864 (119882)=

2120575

119888 + 119889 (18)

120588 =120582

120583=

1

120583119864 (119882)=

2120575

120583 (119888 + 119889) (19)

Let us assume that there exits just one server at eachservice node and the servers are independent which meansthat the queuing model at each server is an 1198721198721 Thenthe probability distribution function of the sojourn timeWT (defined as the waiting time plus the service time for acustomer) in an1198721198721 queue can be presented as

119891WT (119905) = (120583 minus 120582) 119890minus(120583minus120582)119905

(20)

From (7) the cumulative distribution function of WT is

119891WT (119905) = 119875 (WT le 119905)

= int119905

0

(120583 minus 120582) 119890minus(120583minus120582)119905

119889119905 = 1 minus 119890minus(120583minus120582)119905

(21)

The average sojourn time WT is given by

WT = 119864 (WT) = intinfin

0

119891WT (119905) 119905 119889119905

= intinfin

0

(120583 minus 120582) 119890minus(120583minus120582)119905

119905 119889119905 =1

120583 minus 120582

(22)

Let WT119902denote the waiting time in the queue The

average waiting time is computed as

WT119902= WT minus

1

120583=

120582

120583 (120583 minus 120582) (23)

From the well-known Littlersquos theorem the average cus-tomers LR in the system including the number of customersboth waiting in the queue and served in the server is given by

LR = 120582WT =120582

120583 minus 120582 (24)

And LR119902denote the queuing length in the system which

is presented as

LR119902= 120582WT

119902=

1205822

120583 (120583 minus 120582) (25)

From (17) (16) (18) (22) (23) (24) and (25) theaverage sojourn time waiting time in the queue the averagecustomers including the number of customers both waitingin the queue and served in the server and the queuing lengthfor the ESC network system are given as

WTsys = sum119903isin119877

sum119894isin119868

WT119877119868+ sum119903isin119877

sum119895isin119869

WT119877119869+ sum119903isin119877

sum119896isin119870

WT119877119896

= sum119903isin119877

sum119894isin119868

1

12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

+ sum119903isin119877

sum119895isin119869

1

1205831015840 minus119899

sum119896=1

(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119896

1

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

WT119902sys = sum

119903isin119877

sum119894isin119868

WT119877119902119868

+ sum119903isin119877

sum119895isin119869

WT119877119902119869+ sum119903isin119877

sum119896isin119870

WT119877119902119896

= sum119903isin119877

sum119894isin119868

(11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

12058310158401015840 (12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)

+ sum119903isin119877

sum119895isin119869

sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

1205831015840 (1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)

+ sum119903isin119877

sum119896isin119896

2120575119903119896 (119888119903119896+ 119888119903119896)

120583 (120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896)))

LRsys = sum119903isin119877

sum119894isin119868

LR119877119868

+ sum119903isin119877

sum119895isin119869

LR119877119869+ sum119903isin119877

sum119896isin119870

LR119877119896

= sum119903isin119877

sum119894isin119868

(11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

+ sum119903isin119877

sum119895isin119869

sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119870

2120575119903119896 (119888119903119896+ 119888119903119896)

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

The Scientific World Journal 7

LR119902sys

= sum119903isin119877

sum119894isin119868

LR119877119902119868+ sum119903isin119877

sum119895isin119869

LR119877119902119869+ sum119903isin119877

sum119896isin119870

LR119877119902119896

= sum119903isin119877

sum119894isin119868

((11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)2

12058310158401015840 (12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)

+ sum119903isin119877

sum119895isin119869

(sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)2

1205831015840 (1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)

+ sum119903isin119877

sum119896isin119896

(2120575119903

119896 (119888119903

119896+ 119888119903

119896))2

120583 (120583 minus 2120575119903119896 (119888119903119896+ 119888119903119896))

(26)

As the response time includes not only the sojourn time ofemergency demand in the queuing network system but alsothe transportation time therefore when the queuing theoryis applied to the present model the above objective functioncan be presented as follows

min119885

= minsum119903isin119877

sum119894isin119868

sum119895isin119869

sum119896isin119870

119905119903

119894119895119896

= min(sum119903isin119877

sum119894isin119868

(1)

times (12058310158401015840minus (

1

1199022

)

ℎ

sum119895=1

119899

sum119896=1

(2120575119903

119896 (119888119903

119896+ 119888119903

119896)) 119910119894119895119911119895119896)

minus1

+ sum119903isin119877

sum119895isin119869

1

1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119896

1

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

+sum119903isin119877

sum119894isin119868

sum119895isin119869

TR119903119894119895119910119894119895+ sum119903isin119877

sum119895isin119869

sum119896isin119870

TR119903119895119896119911119895119896)

(27)

43 GA-Based Approach As the facility location-allocationproblem is NP- (nondeterministic polynomial-) hard wherethe proposed 0-1 nonlinear integer programming model inthis study is used some heuristic algorithms are requiredto solve the problem quickly The genetic algorithms (GAs)have been applied to location problems since 1986 by Hosageand Goodchild [23] GA has exhibited inherent advantagessuch as their robust performancewhen solving combinatorialproblems as well as their ability to incorporate proceduresand logical conditions and to handle discrete variables andnonlinear constraints in a straightforward manner [24]These qualities make GAs particularly attractive for potentialcombination with other methods and external procedures

Wedevelop a refined genetic algorithm to solve the aforemen-tioned problem as indicated belowThe associated frameworkfor embedding the GA with emergency queue network ispresented in Figure 3

431 Methodology of Genetic Algorithm In a genetic algo-rithm a population of candidate solutions to an optimizationproblem is evolved toward better solutions Each candi-date solution has a set of properties (its chromosomes orgenotype) which can be mutated and altered solutions arerepresented in binary as strings of 0 s and 1 s but otherencodings are also possible The evolution usually startsfrom a population of randomly generated individuals andhappens in generations In each generation the fitness ofevery individual in the population is evaluated the more fitindividuals are stochastically selected from the current popu-lation and each individualrsquos genome ismodified (recombinedand possibly randomly mutated) to form a new populationThe new population is then used in the next iteration of thealgorithm Commonly the algorithm terminates when eithera maximum number of generations have been produced or asatisfactory fitness level has been reached for the populationOnce the genetic representation and the fitness function aredefined a GA proceeds to initialize a population of solutionsand then to improve it through repetitive application of themutation crossover inversion and selection operators

432 Procedure of the Refined Genetic Algorithm

Step 1 (encoding) We suppose that the genotype of a chro-mosome with some genes indicates an individual corre-sponding to the location pattern of emergency logisticscenters and paths between nodes in the system network Thegene presented by 1 indicates that an emergency logisticscenter is built at this location or the path between nodes in thesystem network is selectedThe gene presented by 0 indicatesthat no emergency logistics centers is built there or the pathbetween nodes in the system network which is selected is notselected 119873 chromosomes are randomly generated and theinitial population (Pop) is set

Step 2 (fitness evaluation) Once the initial population isavailable evaluation of each chromosome takes place to ex-plore the quality of the corresponding solution For individ-uals with larger fitness value to be transmitted to the nextgeneration with higher probability the chromosome fitnessfunction in this paper can be described as follows and 119888max isthe maximum estimation value of min119885

Fit (min119885) = 119862max minusmin119885 if min119885 lt 119862max

0 otherwise(28)

Step 3 (crossover selection and mutation) 119873 chromosomesare crossed according to a single pointmethodwith crossoverprobability (which is 119875

119888= 09) to generate the offspring

chromosomes There are a total of 2119873 chromosomes forboth parent and offspring chromosomes Then the duplicatechromosomes are deleted to produce the new population(Pop) Select the first 119873 chromosomes of the population on

8 The Scientific World Journal

Step 1 encoding forparameter sets

Initialization ofgenerate population

Step 2 chromosomefitness evaluation

Step 3 geneticmanipulation crossoverselection and mutation

Step 4 offspringgeneration Select the best individual

and output the solution

Yes

No Terminate

(1) Decoding bit string to the parameter(2) Objective function value calculation(3) Mapping function value to the fitness value(4) Fitness value adjustments

Preparation of dataand parameters

Figure 3 Steps of the proposed GA heuristic

(B)(C)

(A)

Figure 4 The location of facilities in an ESC system

the basis of the fitness values in ascending order When thenumber of chromosomes in the new population (Pop) is lessthan 119873 (119873 = 200) 119873-|Pop| chromosomes are randomlygenerated (|Pop| is the number of chromosomes in thepopulation (Pop)) Select a chromosome randomly and applythe mutation operator applied on it with probability 119875

119898=

001

Step 4 Choose new chromosomes by the roulette wheelmethod Repeat Steps 3 and 4 until there are no solutionimprovements within a specified number of iterations

5 A Case Study

A case study example is presented in this section to illustratethe efficiency of our approach to realize the quick-responsiveemergency relief during large-scale natural disastersThepro-posed method is implemented on a scenario where a severetyphoon strikes the southeast coast of Shanghai with a greatprobability within the next 100 years It is assumed that thereare three regions (A B and C) slashed by the typhoon shownin Figure 4 One region is Lingangxincheng District (A)

another one is Fengxian District (B) and the third one isJinshan District (C)

There are ten supply points (SP) (1) disaster relief sup-plies reserve center in Baoshan District (2) Automobile Cityin Jiading (3) Red Cross disaster preparedness warehouse inJiading District (4) disaster relief supplies reserve center inQingqu District (5) disaster relief supplies reserve center inPutuo District (6) disaster relief supplies reserve center inYangpu District (7) disaster relief supplies reserve center inMinhang District (8) disaster relief supplies reserve center inHongkou District (9) disaster relief supplies reserve centerin Pudong District and (10) disaster relief supplies reservecenter in Xuhui District Based on the study of large-scalelogistics center in Shanghai we select ten potential pointsas the emergency logistics centers (ELCs) Four potentialELCs are prepared for the affected areas in LingangxinchengDistrict two potential ELCs are planned to serve FengxianDistrict and two potential ELCs are expected to serve theJinshanDistrictWe assume that a total of thirteen areas (DP)are seriously affected and need emergency rescue urgently inwhich six affected areas are in Lingangxincheng four affectedareas are in Fengxian District and three affected areas are inJinshan District

For simplicity only one type of emergency supply resour-ces will be considered in the queuing network We assumethat the travel speed is fixed to be equal to 40 miles per hourfor all the vehicles travelling between SP and ESC Howeverthe travel speed between ESC and DP is assumed to be 30miles per hour Let us adopt the hypothesis as the abovethat the occurrence time of a relief request follows a negativeexponential distribution with intensify 120575 and the quantity ofresources in every relief request is uniformly distributed withintensify 119891

119881(V) that the lower bound (LB) is 119888 and the upper

bound (UB) is 119889 (119889 gt 119888) The disaster-related informationincluding the population of the affected areas and the distancebetween supply point and emergency logistics center anddistance between emergency logistics center and demandpoint are shown in Tables 1 2 3 and 4 respectively

The Scientific World Journal 9

Table 1 Population and demand data of the affected areas

Affected area Population (119888 119889) 120575

Area 1 (Zhuqiaozhen) 105807 (4 12) 16Area 2 (Huinanzhen) 213845 (4 12) 16Area 3 (Laogangzhen) 37408 (5 14) 18Area 4 (Datuanzhen) 71162 (4 12) 16Area 5 (Xingangzhen) 21475 (6 16) 20Area 6 (Shuyuanzhen) 59831 (6 16) 20Area 7 (Luchaogangzhen) 27850 (6 16) 20Area 8 (Situanzhen) 65389 (5 14) 18Area 9 (Qingcunzhen) 89163 (5 14) 18Area 10 (Nanqiaozhen) 361185 (3 10) 14Area 11 (Zhuanghangzhen) 62388 (3 10) 14Area 12 (Zhelinzhen) 62589 (3 10) 14Area 13 (Caojingzhen) 40722 (2 8) 12Area 14 (Shanyangzhen) 84640 (2 8) 12Area 15 (Jinshanzhen) 70815 (2 8) 12

0 20 40 60 80 100 120 140 160 180 20020

25

30

35

40

45

50

55

60

Interaction number

Tota

l res

pons

e tim

e (ho

urs)

Total response time

Figure 5 Optimization of the fitness function

In this case study the fixed cost to locate the emergencylogistics center is assumed as 119886

1= 85 119886

2= 100 119886

3= 90

1198864= 85 119886

5= 95 119886

6= 110 119886

7= 70 119886

8= 80 119886

9= 115

and 11988610

= 120 The total cost to build the emergency logisticscenters is assumednot exceeding to be119861 = 500 For simplicityof the computation we presume that 119902

1= 5 119902

2= 2 and

1199023= 3According to the background of the above problem a

simplified 10 times 10 times 15 emergency supply chain network isdeveloped in the assumed scenario where the geographicalrelationships among these supply depots emergency logisticscenters and demand points are specified Then we applythe aforementioned model into the emergency supply chainnetwork system And the GA-based approach demonstratedbefore is coded to verify the validity of the model inMatlab2009a

The optimization process of the fitness function isdepicted in Figure 5 where the optimal response time of thequeuing network is 2071 hours

Therefore the average response time of each demandpoint is 138 h which is within a reasonable range The cal-culation result shows that this model can play an importantrole in the emergency supply chain management Differentfrom the general supply chain management the timesavingis the most significant feature of the emergency supply chainsystem The efficient allocation of emergency resources todemand points improves not only the operational perfor-mance of the emergency supply chain system in the emer-gency supply side but also the affected peoplersquos survival ratein the disaster areas From the psychology of point of viewquick responses and deliveries for disaster relief not onlymay improve the governmental efforts to rescue but alsostrengthen the affected peoplersquos willpower to be alive hencestabilizing the affected areas

The emergency supply chain system is composed of threemembers which have different functions in the relief supplydelivery process And the upstream nodes and downstreamnodes in the ESC system are connected by roads If we treateach node in the system as the node in a network considerthe roads as the arcs of the network and treat the length ofeach road as the weight of each arc then the ESC system canbe seen as a directed network diagram with weights And theresponse time of the system includes two parts the first partis the transportation time of the relief supply request on theroads and the second part is the sojourn time of the reliefsupply request in each node

Figure 6 indicates the values of the above two componentsof response time with the total transportation time being1664 h and the total sojourn time being 407 h This iterativeprocess also demonstrates the convergence of the heuristicalgorithm

Queuing models are effective tools for obtaining infor-mation about important performance of each facility in theESC system like sojourn times waiting times queue lengthstotal customers in the system and so on In order to analyzethe efficiency of facilities in the system we calculate thetotal customers and the queue lengths which are shown inFigure 7 We can see that the solution of the above model willnot create queuing delays of relief supply so that the lossesgenerated by the queuing delay can be avoided This iterativeprocess also illustrates the robustness and the potentialapplicability of our model

Based on the built network the coordination between theupstream node and the downstream node in this system canbe enforced and the performance of basic activities includingproducing sorting processing packing and delivering canbe improved tominimize the response time of the systemTheemergency logistics centers for the system should be locatedin DP(6) DP(7) DP(8) DP(9) and DP(10) And the optimalpaths are SP(1) SP(6) rarr ELC(8) rarr DP(1) DP(2)DP(9) SP(2) SP(4) rarr ELC(6) rarr DP(13) DP(14)DP(15) SP(3) SP(8) rarr ELC(7) rarr DP(10) DP(11)DP(12) SP(5) SP(9) rarr ELC(10) rarr DP(3) DP(4)DP(5) and SP(7) SP(9) rarr ELC(9) rarr DP(6) DP(7)DP(8) Overall the examination reveals that the proposedmethod can be a useful tool for a quick response to the urgentneed of relief in the large-scale disaster-affected areas

10 The Scientific World Journal

Table 2 The service rates specifications of different servers in the network

SP 120583101584010158401

120583101584010158402

120583101584010158403

120583101584010158404

120583101584010158405

120583101584010158406

120583101584010158407

120583101584010158408

120583101584010158409

1205831015840101584010

10 9 11 8 11 12 14 16 14 17

ELC 12058310158401

12058310158402

12058310158403

12058310158404

12058310158405

12058310158406

12058310158407

12058310158408

12058310158409

120583101584010

16 18 11 10 11 12 14 16 14 10

DP 1205831

1205832

1205833

1205834

1205835

1205836

1205837

1205838

1205839

12058310

12058311

12058312

12058313

12058314

12058315

8 8 8 8 10 9 11 8 9 12 9 8 12 14 13

Table 3 The distance between supply point and emergency logistics center

SP(1) SP(2) SP(3) SP(4) SP(5) SP(6) SP(7) SP(8) SP(9) SP(10)ELC(1) 1828 1707 1197 2163 614 826 1091 973 1530 879ELC(2) 3073 2147 1624 571 2039 2056 802 1679 2554 1390ELC(3) 2271 2288 1712 1622 1419 1463 998 1102 1842 756ELC(4) 4422 3095 2462 1264 3971 3492 2429 3276 4153 3003ELC(5) 3962 2801 2716 1507 3256 2769 1449 2551 3325 2006ELC(6) 3585 2356 1923 1647 2784 2313 989 2136 2839 1437ELC(7) 2973 2968 2364 2301 2036 2149 1275 1797 2575 1471ELC(8) 2308 3261 2878 2892 1494 1401 1764 1132 1961 1933ELC(9) 2584 3643 3303 3249 2015 1165 2258 941 1600 2419ELC(10) 3038 4666 4240 4139 2831 2039 3008 1742 1405 3105

05

101520253035404550

Tim

e (ho

urs)

Total transportation timeTotal sojourn time

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 6 Optimization of total transportation time and totalsojourn time

6 Conclusions

An emergency supply chain (ESC) system is developed inthis paper for the quick response to the urgent need ofrelief in the large-scale disaster-affected areas which mainlyconsists of three chain members in series supply points(SP) emergency logistics centers (ELCs) and demand points(DP) As the timeliness is the most crucial characteristic ofthe ESC system and queuing theory is a useful tool in theanalysis of the performance of the time-dependent systemwe propose queuingmodeling for the ESC system to optimizethe emergency rescue process Based on the above queuingnetwork system a queuing minimal response time location-allocation model is established to decide the selection ofemergency logistics centers and vehicle routing decision

0

5

10

15

20

25

Total customersQueuing length

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 7 Optimization of the total customers and the queue length

For the complexity of mathematical model the GA-basedapproach is introduced to solve the model

A case study with an assumed severe typhoon striking thesoutheast coast of Shanghai with a great probability withinthe next 100 years is conducted to assess and validate ourmodel The optimal response time of the queuing network is2071 hours and the average response time of each demandpoint is 138 h which is within a reasonable range Theoptimal emergency logistics centers for the system are DP(6)DP(7) DP(8) DP(9) and DP(10) Besides the performanceof queuing modeling for the ESC system illustrates therobustness and the potential applicability of our model

For future research we suggest the following threedirections Firstly we will investigate the queuing modelingunder consideration of the facility blocking problems and

The Scientific World Journal 11

Table 4 The distance between emergency logistics center and demand point

ELC(1) ELC(2) ELC(3) ELC(4) ELC(5) ELC(6) ELC(7) ELC(8) ELC(9) ELC(10)DP(1) 3761 4191 3253 4599 3801 3752 2439 2397 1553 739DP(2) 4271 4702 3761 4794 3945 4152 2818 2822 1898 1075DP(3) 4508 4886 3986 5028 4501 4384 3046 3025 2117 1356DP(4) 4364 4579 3697 4824 4307 4138 2798 2940 1956 1710DP(5) 5283 5416 4386 5466 5435 4986 3836 3271 2693 1917DP(6) 5214 5721 4866 5896 5294 5207 3797 4221 3252 2553DP(7) 5334 5776 5299 5862 5359 5164 3871 3851 2955 2554DP(8) 4689 4581 3763 4983 4306 4173 2818 2997 2200 1968DP(9) 3309 3624 2657 3834 3334 3154 1886 3154 3452 3235DP(10) 2678 3263 1999 2911 2741 2518 1198 2505 2360 3037DP(11) 3369 2912 2386 2349 2182 1858 1894 2392 2915 3596DP(12) 3376 3794 2541 3287 3275 2612 1708 3062 2896 3256DP(13) 3691 3419 2987 2896 2694 2279 2162 3500 3275 4007DP(14) 4230 3273 3402 3011 2555 2141 2606 3906 3749 4448DP(15) 4609 3765 3858 2533 3054 2680 3166 4390 4249 4897

demand priority for the emergency relief points Secondly wewill study the generation of emergency demand and servicetime obeying the other random distributions Thirdly moreefficient multiobjective optimization methodologies for boththe above problem and emergency resources transportationscheduling problem will be considered

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grant nos 71102043 and 71371145) theMinistry of Education of China (Grant no MCM20125021)the Innovative Program of Shanghai Municipal EducationCommission (Grant no 14ZS123) and the ShanghaiMaritimeUniversity (Grant no 20120054)

References

[1] D Guha-Sapir G S Debarati D Hargitt P Hoyois R Belowand D BrechetThirty Years of Natural Disasters 1974ndash2003 theNumbers Presses universitaires de Louvain 2004

[2] J-B Sheu ldquoAn emergency logistics distribution approach forquick response to urgent relief demand in disastersrdquo Trans-portation Research E vol 43 no 6 pp 687ndash709 2007

[3] B S Manoj and A H Baker ldquoCommunication challenges inemergency responserdquo Communications of the ACM vol 50 no3 pp 51ndash53 2007

[4] B Balcik B M Beamon and K Smilowitz ldquoLast mile distribu-tion in humanitarian reliefrdquo Journal of Intelligent TransportationSystems vol 12 no 2 pp 51ndash63 2008

[5] A M Caunhye X F Nie and S Pokharel ldquoOptimizationmodels in emergency logistics a literature reviewrdquo Socio-Economic Planning Sciences vol 46 no 1 pp 4ndash13 2012

[6] S Coles and L Pericchi ldquoAnticipating catastrophes throughextreme valuemodellingrdquo Journal of the Royal Statistical SocietyC vol 52 no 4 pp 405ndash416 2003

[7] J R Artalejo ldquoG-networks a versatile approach for workremoval in queueing networksrdquo European Journal of Opera-tional Research vol 126 no 2 pp 233ndash249 2000

[8] W Jiang L Deng L Chen J Wu and J Li ldquoRisk assessmentand validation of flood disaster based on fuzzy mathematicsrdquoProgress in Natural Science vol 19 no 10 pp 1419ndash1425 2009

[9] V Bhaskar and P Lallement ldquoModeling a supply chain usinga network of queuesrdquo Applied Mathematical Modelling vol 34no 8 pp 2074ndash2088 2010

[10] B M Beamon ldquoHumanitarian relief chains issues and chal-lengesrdquo in Proceedings of the 34th International Conference onComputers and Industrial Engineering San Francisco CalifUSA November 2004

[11] R Oloruntoba and R Gray ldquoHumanitarian aid an agile supplychainrdquo Supply Chain Management vol 11 no 2 pp 115ndash1202006

[12] E J Lodree Jr and S Taskin ldquoSupply chain planning for hur-ricane response with wind speed information updatesrdquo Com-puters and Operations Research vol 36 no 1 pp 2ndash15 2009

[13] V Bhakoo P Singh and A Sohal ldquoCollaborative managementof inventory in Australian hospital supply chains practices andissuesrdquo Supply Chain Management vol 17 no 2 pp 217ndash2302012

[14] W Yi and A Kumar ldquoAnt colony optimization for disaster reliefoperationsrdquo Transportation Research E vol 43 no 6 pp 660ndash672 2007

[15] G-H Tzeng H-J Cheng and T D Huang ldquoMulti-objectiveoptimal planning for designing relief delivery systemsrdquo Trans-portation Research E vol 43 no 6 pp 673ndash686 2007

[16] Y-C Chiu and H Zheng ldquoReal-time mobilization decisionsfor multi-priority emergency response resources and evacua-tion groups model formulation and solutionrdquo TransportationResearch E vol 43 no 6 pp 710ndash736 2007

[17] N Altay andWG Green III ldquoORMS research in disaster oper-ations managementrdquo European Journal of Operational Researchvol 175 no 1 pp 475ndash493 2006

12 The Scientific World Journal

[18] K J Cormican D P Morton and R K Wood ldquoStochasticnetwork interdictionrdquo Operations Research vol 46 no 2 pp184ndash197 1998

[19] HOMete andZ B Zabinsky ldquoStochastic optimization ofmed-ical supply location and distribution in disaster managementrdquoInternational Journal of Production Economics vol 126 no 1 pp76ndash84 2010

[20] H Shavandi and H Mahlooji ldquoA fuzzy queuing locationmodel with a genetic algorithm for congested systemsrdquo AppliedMathematics and Computation vol 181 no 1 pp 440ndash4562006

[21] R D Galvao and R Morabito ldquoEmergency service systemsthe use of the hypercube queueing model in the solution ofprobabilistic location problemsrdquo International Transactions inOperational Research vol 15 no 5 pp 525ndash549 2008

[22] N Geroliminis M G Karlaftis and A Skabardonis ldquoA spatialqueuing model for the emergency vehicle districting andlocation problemrdquo Transportation Research B vol 43 no 7 pp798ndash811 2009

[23] C M Hosage and M F Goodchild ldquoDiscrete space location-allocation solutions from genetic algorithmsrdquo Annals of Opera-tions Research vol 6 no 2 pp 35ndash46 1986

[24] P Chakroborty ldquoGenetic algorithms for optimal urban transitnetwork designrdquo Computer-Aided Civil and Infrastructure Engi-neering vol 18 no 3 pp 184ndash200 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 3

Therefore to resolve the issues mentioned above wepresent a relief demand management model of emergencysupply chain to address the above issue under the disorderand uncertain conditions in affected areas during the crucialrescue period of a large-scale disaster Rooted in the tech-niques of collaboration in emergency supply chain coupledwith queuing theory and system optimization the proposedmethodology embeds three mechanisms (1) multiechelonsupply chainmodel for disasters (2) dynamic facility locationand vehicle routing selection and (3) rescue systemmanage-ment collaboration

Relative to the previous literature the proposed reliefdemand management methodology has the following twodistinctive features (1)Themodel is capable of collaboratingurgent relief demand management in the large-scale disastercontexts and accelerating rescue efforts to save casualty loss(2) To facilitate dynamic relief allocation and distributionthe proposed model practically groups humanitarian reliefsintomultiechelon resource suppliers and distribution centerswhich form an emergency response system for uncertainlarge disasters

3 System Specification

An ESC system involves selection of sites and vehicle rout-ing decisions which are two major problems in a disasterresponse environment The optimal facilities locations andpath selections can guarantee that the commodities will besent from the supply depots to the demand points in affectedareas as quickly as possible to maximize the survival rate ofwounded persons The above problems arouse our intereststo propose queuing modeling for the ESC system Hence aqueuing network of emergency supply chain is formulated inthis paper as shown in Figure 1

The queuing network of ESC involves a queuing flowformulation where the three supply chain members namelysupply points (SP) emergency logistics centers (ELCs) anddemand points (DP) are treated as servers Emergency com-modities such as food shelter personnel machinery andmedicine are modeled as customers The upstream anddownstream nodes of ESC system constitute some basicactivities that are producing sorting processing packingdelivering and so forth These activities are regarded as theservice for customers Consider the following

(1) Locations of emergency supply points are in 1198781 1198782

1198783 119878

119898

(2) Locations of emergency logistics centers are in 1198711 1198712

1198713 119871

119903

(3) Locations of emergency demand points are in1198631 1198632

1198633 119863

119899

(a) Vehicle routing choices for the commodity flowsin the system are considered in the following

(4) TR1119895 from 119878

1to one of the nodes (119871

1 1198712 1198713

119871119903) TR

2119895 from 119878

2to one of the nodes (119871

1 1198712

1198713 119871

119903) and TR

119898119895 from 119878

119898to one of the

nodes (1198711 1198712 1198713 119871

119903) 119895 = 1 2 119903

(5) TR1119896 from 119871

1to one of the nodes (119863

1 1198632 1198633

119863119899) TR2119896 from 119871

2to one of the nodes (119863

1 1198632 1198633

119863119899) and TR

119903119896 from 119871

119903to one of the nodes

(1198631 1198632 1198633 119863

119899) 119896 = 1 2 119899

Each commodity is considered as a queue where batchesare waiting to be serviced The selection of sites and vehiclerouting decision may be operated under the considerationof the estimated throughput response time from supplydepots to demand depots in affected areas The responsetime comprises not only the transportation times betweenupstream and downstream nodes but also the total waitingtimes and service times in the queuing network

Therefore the above three nodes in the system areassumed to behave as an 1198721198721 queuing where reliefsupplies are treated as customers On the basis of the specifiedESC queuing network we adopt the following hypothesis forsystem operations

(1) The corresponding geographic relationships betweenupstream and downstream nodes are available fromthe existing governmental databases and the reliefdemand needed in a given affected area can be readilyaccessible via advanced disaster detection technolo-gies

(2) The locations of emergency logistics centers are onlyfixed in the given alternative sites

(3) The first customer in the queue receives servicesfirstly namely ldquofirst come first servedrdquo

4 Mathematical Formulation

Based on the aforementioned system specification we pro-pose a queuing theory in this section for facility locationand path selection problems in a multistage emergencysupply chain network Firstly the notations parametersand decision variables for the mathematical formulation areintroduced After that the objective function for the modelis established And then the formulation of the constraints ofthe problems is presented

41 The Parameters and Decision Variables The sets param-eters and decision variables are defined as follows

(1) Notations

119868 set of supply depots 119894 isin 119868 119894 = 1 2 3 119898119869 set of alternative sites of emergency logistics cen-ters 119895 isin 119869 119895 = 1 2 3 ℎ119870 set of depots for handing out relief goods 119896 isin 119870119896 = 1 2 3 119899119877 set of commodities 119903 isin 119877 119903 = 1 2 3 119877

(2) Parameters

120582 the interarrival time of the emergency demandfollowing a negative exponential distribution

4 The Scientific World Journal

Emergency supplypoints

S1

S2

Sm

Transportation

TR1j

TR2j

TRmj

Emergencylogistics centres

Emergencydemand points

L1

L2

Lh

Transportation

TR9984001k

TR9984002k

TR998400hk

D1

D2

Dn

middot middot middotmiddot middot middot middot middot middot middot middot middotmiddot middot middot

Figure 1 The queuing network of ESC

120583 the service rate at eachnode in the queuing networksystem

120575 the parameter of the negative exponential distribu-tion

119888 the lower bound of a uniformly distributed randomvariable that indicates the quantity of resources in arelief request

119889 the upper boundof a uniformly distributed randomvariable that indicates the quantity of resources in arelief request

119905119903

119894119895119896 the response time of the ESC system for com-

modity type 119903 from node 119894 to node 119896 going throughlogistics centers located at nodes 119895

WT119903 the sojourn time of commodity type 119903 in thesystem

WT119903119902 the waiting time of commodity type 119903 in the

queue

TR119903119894119895 the transportation time for commodity type 119903

from node 119894 to node 119895

TR119903119895119896 the transportation time for commodity type 119903

from node 119895 to node 119896

DTR119894119895 the distance between node 119894 and node 119895

DT119896119895 the distance between node 119895 and node 119896

DC the penalty cost for unavailability of commoditiesdemand within the maximum promised responsetime

119886119895 the fixed costs of locating the emergency logistics

center 119895

1199021 the total number of logistics center to be fixed

1199022 the number of supply depots delivering commodi-

ties to the same emergency logistic centers

1199023 the number of demand nodes accepting items

from the same emergency logistics center

V119903119894119895 transport speed for commodity type 119903 from node

119894 to node 119895

V119903119895119896 transport speed for commodity type 119903 from node

119895 to node 119896

(3) Decision Variables

119909119895=

1 emergency logistics center builton the site 119895

0 otherwise

119910119894119895=

1 relief resources from emergency supplypoint 119894 transported to emergencylogistics center 119895

0 otherwise

119911119895119896

=

1 relief resources from emergency logisticscenter 119895 transported to reliefdemand point 119896

0 otherwise

(1)

42 Queuing Minimal Unsatisfied Demand Location-Allocation Model Three types of members (SP ELC andDP) involved in this system are in serial connection Thetransportation routes are necessary to deliver items fromupstream nodes to downstream nodes Based on thisassumption a queuing minimal response location-allocationmodel for the three-stage queuing network is formulated asfollows

Objective function

min119885 = sum119903isin119877

sum119894isin119868

sum119895isin119869

sum119896isin119870

119905119903

119894119895119896 (2)

The Scientific World Journal 5

subject to

119910119903

119894119895le 119909119895

forall119894 isin 119868 119895 isin 119869 119903 isin 119877 (3)

119911119903

119895119896le 119909119895

forall119896 isin 119870 119895 isin 119869 119903 isin 119877 (4)

sum119895isin119869

119910119903

119894119895= 1 forall119894 isin 119868 119903 isin 119877 (5)

sum119895isin119869

119911119903

119895119896= 1 forall119896 isin 119870 119903 isin 119877 (6)

sum119897isin119871|119889119894119897le119889119894119895

119910119903

119894119897ge 119909119895

forall119894 isin 119868 119895 isin 119869 119903 isin 119877 (7)

sum119897isin119871|119889119897119896le119889119895119896

119911119903

119897119896ge 119909119895

forall119896 isin 119870 119895 isin 119869 119903 isin 119877 (8)

sum119886119895119909119895le 119861 forall119895 isin 119869 (9)

sum119909119895= 1199021

forall119895 isin 119869 (10)

sum119894isin119868

119910119903

119894119895= 1199022

forall119895 isin 119869 119903 isin 119877 (11)

sum119896isin119870

119911119903

119895119896= 1199023

forall119895 isin 119869 119903 isin 119877 (12)

119905119903

119894119895119896= WT119903

119894+WT119903

119895+WT119903

119896+ TR119903119894119895+ TR119903119895119896 (13)

119909119903

119895 119910119903

119895119896 119911119903

119894119895= 0 1 forall119896 isin 119870 119894 isin 119868 119895 isin 119869 119903 isin 119877 (14)

The objective aims at minimizing the mean systemresponse time of relief resources including the sojourn timein the queuing network system and transportation timeConstraints (3) and (4) ensure that deliveries can only bemade if emergency logistics centers are fixed Constraint(5) enforces that the items from the supply nodes canonly be delivered to one logistics center and constraint (6)enforces that every demand nodes can obtain items fromjust one logistics center Constraints (7) and (8) representthat the items from upstream nodes are transported to thenearest downstream nodes in the queuing network systemConstraint (9) limits the sum of fixed costs of locating theemergency logistics centers Constraint (10) shows that thetotal number of logistics center to be sited is equal to 119902

1

Constraint (11) forces that the number of supply depots deliv-ering commodities to the same emergency logistic centers isequal to 119902

2to ensure that each emergency logistics is with

enough capacity to deal with these commodities Constraint(12) forces that the number of demand nodes accepting itemsfrom the same emergency logistics center is equal to 119902

3to

ensure that the items from each emergency logistics centerare enough to satisfy the emergency request Constraint (13)represents that the response time is equal to the sojourn timeplus the transporting time Constraint (14) defines all thedecision variables to be binary integer variables

Next we use the queuing theory to compute the objectivefunction The emergency supply chain system is a series-parallel hybrid queuing system consisting of three service

SPi

=1

q2sum yij

TRij ELCj

120582k

DPk

= sum 120582kzjk

120583k

120582998400j120582998400j120582998400998400i

TR998400jk

120583998400998400i 120583998400j

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Figure 2 Equivalent queue of studied ESC network

nodes We can assume that the relief requests for the emer-gency demand depots follow a Poisson distribution withintensity 120582

119896 Each emergency logistics center serves a set

of demand points and therefore the relief requests for anemergency service at the logistics center are the union of therelief requests of the nodes in the set Therefore they can bedepicted as a stochastic process equal to the sum of severalPoisson processes with an intensity 1205821015840

119895equal to the sumof the

intensities of the processes at the nodes served by the logisticscenter We can rewrite parameter 1205821015840

119895by using variables 119911

119895119896

1205821015840

119895=

119899

sum119896=1

120582119896119911119895119896 (15)

The relief request for the supply depots is also assumedto follow a Poisson distribution with intensity 12058210158401015840

119894and also

similar equilibrium equations exist between the arrival rate ofthe relief request for the supply depots and for the emergencylogistics centers For the sake of simplicity we assume thatthe arrival rate of the relief request for the supply depotsthat transport emergency resources to the same emergencylogistics center has the same value Thus the parameter 12058210158401015840

119894

can be rewritten by using variables 119910119894119895and the constant 119902

2

12058210158401015840

119894=

1

1199022

ℎ

sum119895=1

1205821015840

119895119910119894119895=

1

1199022

ℎ

sum119895=1

119899

sum119896=1

120582119896119910119894119895119911119895119896 (16)

Based on the above analysis the equivalent queue of thestudied ESC network is shown in Figure 2

From the affected peoplersquos point of view the ESC systemis equivalent to a queue network that is receiving emergencyrelief orders These relief request orders are waiting to beserved The service is the process of production collec-tion and processing and the results are emergency reliefresources items and so forth

Emergency relief orders are characterized by (i) occur-rence (ii) quantity and (iii) delay Consider the following

119880 random variable indicating the occurrence time ofa relief request

119881 random variable indicating the quantity ofresources in every relief request

119882 UV indicating the occurrence time along with thequantity of resources in every relief request

Assume that119880 follows a negative exponential distributionwith intensify119891

119880(119906) and119881 is a uniformly distributed random

6 The Scientific World Journal

variable with intensify 119891119881(V) between 119888 and 119889 (119889 gt 119888) And

119891119880(119906) and 119891

119881(V) are independent Thus

119891119880(119906) =

120575119890minus120575119906 119906 ge 0

0 119906 lt 0

119891119881(V) =

1

119889 minus 119888 119888 lt V lt 119889

0 otherwise

119864 (119882) = 119864 (119880119881) = 119864 (119880) 119864 (119881) =119888 + 119889

2120575

(17)

Therefore the interarrival times of the emergencydemand (occurrence and quantity) follow a negative expo-nential distribution with intensity 120582 equal to 1119864(119882) Theservice rate at each node in the queuing network system isan independent identically distributed random variable withintensity 120583 and the service time is 1120583

So the interarrival time 120582 and the traffic intensity of thesystem 120588 are represented as

120582 =1

119864 (119882)=

2120575

119888 + 119889 (18)

120588 =120582

120583=

1

120583119864 (119882)=

2120575

120583 (119888 + 119889) (19)

Let us assume that there exits just one server at eachservice node and the servers are independent which meansthat the queuing model at each server is an 1198721198721 Thenthe probability distribution function of the sojourn timeWT (defined as the waiting time plus the service time for acustomer) in an1198721198721 queue can be presented as

119891WT (119905) = (120583 minus 120582) 119890minus(120583minus120582)119905

(20)

From (7) the cumulative distribution function of WT is

119891WT (119905) = 119875 (WT le 119905)

= int119905

0

(120583 minus 120582) 119890minus(120583minus120582)119905

119889119905 = 1 minus 119890minus(120583minus120582)119905

(21)

The average sojourn time WT is given by

WT = 119864 (WT) = intinfin

0

119891WT (119905) 119905 119889119905

= intinfin

0

(120583 minus 120582) 119890minus(120583minus120582)119905

119905 119889119905 =1

120583 minus 120582

(22)

Let WT119902denote the waiting time in the queue The

average waiting time is computed as

WT119902= WT minus

1

120583=

120582

120583 (120583 minus 120582) (23)

From the well-known Littlersquos theorem the average cus-tomers LR in the system including the number of customersboth waiting in the queue and served in the server is given by

LR = 120582WT =120582

120583 minus 120582 (24)

And LR119902denote the queuing length in the system which

is presented as

LR119902= 120582WT

119902=

1205822

120583 (120583 minus 120582) (25)

From (17) (16) (18) (22) (23) (24) and (25) theaverage sojourn time waiting time in the queue the averagecustomers including the number of customers both waitingin the queue and served in the server and the queuing lengthfor the ESC network system are given as

WTsys = sum119903isin119877

sum119894isin119868

WT119877119868+ sum119903isin119877

sum119895isin119869

WT119877119869+ sum119903isin119877

sum119896isin119870

WT119877119896

= sum119903isin119877

sum119894isin119868

1

12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

+ sum119903isin119877

sum119895isin119869

1

1205831015840 minus119899

sum119896=1

(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119896

1

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

WT119902sys = sum

119903isin119877

sum119894isin119868

WT119877119902119868

+ sum119903isin119877

sum119895isin119869

WT119877119902119869+ sum119903isin119877

sum119896isin119870

WT119877119902119896

= sum119903isin119877

sum119894isin119868

(11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

12058310158401015840 (12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)

+ sum119903isin119877

sum119895isin119869

sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

1205831015840 (1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)

+ sum119903isin119877

sum119896isin119896

2120575119903119896 (119888119903119896+ 119888119903119896)

120583 (120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896)))

LRsys = sum119903isin119877

sum119894isin119868

LR119877119868

+ sum119903isin119877

sum119895isin119869

LR119877119869+ sum119903isin119877

sum119896isin119870

LR119877119896

= sum119903isin119877

sum119894isin119868

(11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

+ sum119903isin119877

sum119895isin119869

sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119870

2120575119903119896 (119888119903119896+ 119888119903119896)

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

The Scientific World Journal 7

LR119902sys

= sum119903isin119877

sum119894isin119868

LR119877119902119868+ sum119903isin119877

sum119895isin119869

LR119877119902119869+ sum119903isin119877

sum119896isin119870

LR119877119902119896

= sum119903isin119877

sum119894isin119868

((11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)2

12058310158401015840 (12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)

+ sum119903isin119877

sum119895isin119869

(sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)2

1205831015840 (1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)

+ sum119903isin119877

sum119896isin119896

(2120575119903

119896 (119888119903

119896+ 119888119903

119896))2

120583 (120583 minus 2120575119903119896 (119888119903119896+ 119888119903119896))

(26)

As the response time includes not only the sojourn time ofemergency demand in the queuing network system but alsothe transportation time therefore when the queuing theoryis applied to the present model the above objective functioncan be presented as follows

min119885

= minsum119903isin119877

sum119894isin119868

sum119895isin119869

sum119896isin119870

119905119903

119894119895119896

= min(sum119903isin119877

sum119894isin119868

(1)

times (12058310158401015840minus (

1

1199022

)

ℎ

sum119895=1

119899

sum119896=1

(2120575119903

119896 (119888119903

119896+ 119888119903

119896)) 119910119894119895119911119895119896)

minus1

+ sum119903isin119877

sum119895isin119869

1

1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119896

1

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

+sum119903isin119877

sum119894isin119868

sum119895isin119869

TR119903119894119895119910119894119895+ sum119903isin119877

sum119895isin119869

sum119896isin119870

TR119903119895119896119911119895119896)

(27)

43 GA-Based Approach As the facility location-allocationproblem is NP- (nondeterministic polynomial-) hard wherethe proposed 0-1 nonlinear integer programming model inthis study is used some heuristic algorithms are requiredto solve the problem quickly The genetic algorithms (GAs)have been applied to location problems since 1986 by Hosageand Goodchild [23] GA has exhibited inherent advantagessuch as their robust performancewhen solving combinatorialproblems as well as their ability to incorporate proceduresand logical conditions and to handle discrete variables andnonlinear constraints in a straightforward manner [24]These qualities make GAs particularly attractive for potentialcombination with other methods and external procedures

Wedevelop a refined genetic algorithm to solve the aforemen-tioned problem as indicated belowThe associated frameworkfor embedding the GA with emergency queue network ispresented in Figure 3

431 Methodology of Genetic Algorithm In a genetic algo-rithm a population of candidate solutions to an optimizationproblem is evolved toward better solutions Each candi-date solution has a set of properties (its chromosomes orgenotype) which can be mutated and altered solutions arerepresented in binary as strings of 0 s and 1 s but otherencodings are also possible The evolution usually startsfrom a population of randomly generated individuals andhappens in generations In each generation the fitness ofevery individual in the population is evaluated the more fitindividuals are stochastically selected from the current popu-lation and each individualrsquos genome ismodified (recombinedand possibly randomly mutated) to form a new populationThe new population is then used in the next iteration of thealgorithm Commonly the algorithm terminates when eithera maximum number of generations have been produced or asatisfactory fitness level has been reached for the populationOnce the genetic representation and the fitness function aredefined a GA proceeds to initialize a population of solutionsand then to improve it through repetitive application of themutation crossover inversion and selection operators

432 Procedure of the Refined Genetic Algorithm

Step 1 (encoding) We suppose that the genotype of a chro-mosome with some genes indicates an individual corre-sponding to the location pattern of emergency logisticscenters and paths between nodes in the system network Thegene presented by 1 indicates that an emergency logisticscenter is built at this location or the path between nodes in thesystem network is selectedThe gene presented by 0 indicatesthat no emergency logistics centers is built there or the pathbetween nodes in the system network which is selected is notselected 119873 chromosomes are randomly generated and theinitial population (Pop) is set

Step 2 (fitness evaluation) Once the initial population isavailable evaluation of each chromosome takes place to ex-plore the quality of the corresponding solution For individ-uals with larger fitness value to be transmitted to the nextgeneration with higher probability the chromosome fitnessfunction in this paper can be described as follows and 119888max isthe maximum estimation value of min119885

Fit (min119885) = 119862max minusmin119885 if min119885 lt 119862max

0 otherwise(28)

Step 3 (crossover selection and mutation) 119873 chromosomesare crossed according to a single pointmethodwith crossoverprobability (which is 119875

119888= 09) to generate the offspring

chromosomes There are a total of 2119873 chromosomes forboth parent and offspring chromosomes Then the duplicatechromosomes are deleted to produce the new population(Pop) Select the first 119873 chromosomes of the population on

8 The Scientific World Journal

Step 1 encoding forparameter sets

Initialization ofgenerate population

Step 2 chromosomefitness evaluation

Step 3 geneticmanipulation crossoverselection and mutation

Step 4 offspringgeneration Select the best individual

and output the solution

Yes

No Terminate

(1) Decoding bit string to the parameter(2) Objective function value calculation(3) Mapping function value to the fitness value(4) Fitness value adjustments

Preparation of dataand parameters

Figure 3 Steps of the proposed GA heuristic

(B)(C)

(A)

Figure 4 The location of facilities in an ESC system

the basis of the fitness values in ascending order When thenumber of chromosomes in the new population (Pop) is lessthan 119873 (119873 = 200) 119873-|Pop| chromosomes are randomlygenerated (|Pop| is the number of chromosomes in thepopulation (Pop)) Select a chromosome randomly and applythe mutation operator applied on it with probability 119875

119898=

001

Step 4 Choose new chromosomes by the roulette wheelmethod Repeat Steps 3 and 4 until there are no solutionimprovements within a specified number of iterations

5 A Case Study

A case study example is presented in this section to illustratethe efficiency of our approach to realize the quick-responsiveemergency relief during large-scale natural disastersThepro-posed method is implemented on a scenario where a severetyphoon strikes the southeast coast of Shanghai with a greatprobability within the next 100 years It is assumed that thereare three regions (A B and C) slashed by the typhoon shownin Figure 4 One region is Lingangxincheng District (A)

another one is Fengxian District (B) and the third one isJinshan District (C)

There are ten supply points (SP) (1) disaster relief sup-plies reserve center in Baoshan District (2) Automobile Cityin Jiading (3) Red Cross disaster preparedness warehouse inJiading District (4) disaster relief supplies reserve center inQingqu District (5) disaster relief supplies reserve center inPutuo District (6) disaster relief supplies reserve center inYangpu District (7) disaster relief supplies reserve center inMinhang District (8) disaster relief supplies reserve center inHongkou District (9) disaster relief supplies reserve centerin Pudong District and (10) disaster relief supplies reservecenter in Xuhui District Based on the study of large-scalelogistics center in Shanghai we select ten potential pointsas the emergency logistics centers (ELCs) Four potentialELCs are prepared for the affected areas in LingangxinchengDistrict two potential ELCs are planned to serve FengxianDistrict and two potential ELCs are expected to serve theJinshanDistrictWe assume that a total of thirteen areas (DP)are seriously affected and need emergency rescue urgently inwhich six affected areas are in Lingangxincheng four affectedareas are in Fengxian District and three affected areas are inJinshan District

For simplicity only one type of emergency supply resour-ces will be considered in the queuing network We assumethat the travel speed is fixed to be equal to 40 miles per hourfor all the vehicles travelling between SP and ESC Howeverthe travel speed between ESC and DP is assumed to be 30miles per hour Let us adopt the hypothesis as the abovethat the occurrence time of a relief request follows a negativeexponential distribution with intensify 120575 and the quantity ofresources in every relief request is uniformly distributed withintensify 119891

119881(V) that the lower bound (LB) is 119888 and the upper

bound (UB) is 119889 (119889 gt 119888) The disaster-related informationincluding the population of the affected areas and the distancebetween supply point and emergency logistics center anddistance between emergency logistics center and demandpoint are shown in Tables 1 2 3 and 4 respectively

The Scientific World Journal 9

Table 1 Population and demand data of the affected areas

Affected area Population (119888 119889) 120575

Area 1 (Zhuqiaozhen) 105807 (4 12) 16Area 2 (Huinanzhen) 213845 (4 12) 16Area 3 (Laogangzhen) 37408 (5 14) 18Area 4 (Datuanzhen) 71162 (4 12) 16Area 5 (Xingangzhen) 21475 (6 16) 20Area 6 (Shuyuanzhen) 59831 (6 16) 20Area 7 (Luchaogangzhen) 27850 (6 16) 20Area 8 (Situanzhen) 65389 (5 14) 18Area 9 (Qingcunzhen) 89163 (5 14) 18Area 10 (Nanqiaozhen) 361185 (3 10) 14Area 11 (Zhuanghangzhen) 62388 (3 10) 14Area 12 (Zhelinzhen) 62589 (3 10) 14Area 13 (Caojingzhen) 40722 (2 8) 12Area 14 (Shanyangzhen) 84640 (2 8) 12Area 15 (Jinshanzhen) 70815 (2 8) 12

0 20 40 60 80 100 120 140 160 180 20020

25

30

35

40

45

50

55

60

Interaction number

Tota

l res

pons

e tim

e (ho

urs)

Total response time

Figure 5 Optimization of the fitness function

In this case study the fixed cost to locate the emergencylogistics center is assumed as 119886

1= 85 119886

2= 100 119886

3= 90

1198864= 85 119886

5= 95 119886

6= 110 119886

7= 70 119886

8= 80 119886

9= 115

and 11988610

= 120 The total cost to build the emergency logisticscenters is assumednot exceeding to be119861 = 500 For simplicityof the computation we presume that 119902

1= 5 119902

2= 2 and

1199023= 3According to the background of the above problem a

simplified 10 times 10 times 15 emergency supply chain network isdeveloped in the assumed scenario where the geographicalrelationships among these supply depots emergency logisticscenters and demand points are specified Then we applythe aforementioned model into the emergency supply chainnetwork system And the GA-based approach demonstratedbefore is coded to verify the validity of the model inMatlab2009a

The optimization process of the fitness function isdepicted in Figure 5 where the optimal response time of thequeuing network is 2071 hours

Therefore the average response time of each demandpoint is 138 h which is within a reasonable range The cal-culation result shows that this model can play an importantrole in the emergency supply chain management Differentfrom the general supply chain management the timesavingis the most significant feature of the emergency supply chainsystem The efficient allocation of emergency resources todemand points improves not only the operational perfor-mance of the emergency supply chain system in the emer-gency supply side but also the affected peoplersquos survival ratein the disaster areas From the psychology of point of viewquick responses and deliveries for disaster relief not onlymay improve the governmental efforts to rescue but alsostrengthen the affected peoplersquos willpower to be alive hencestabilizing the affected areas

The emergency supply chain system is composed of threemembers which have different functions in the relief supplydelivery process And the upstream nodes and downstreamnodes in the ESC system are connected by roads If we treateach node in the system as the node in a network considerthe roads as the arcs of the network and treat the length ofeach road as the weight of each arc then the ESC system canbe seen as a directed network diagram with weights And theresponse time of the system includes two parts the first partis the transportation time of the relief supply request on theroads and the second part is the sojourn time of the reliefsupply request in each node

Figure 6 indicates the values of the above two componentsof response time with the total transportation time being1664 h and the total sojourn time being 407 h This iterativeprocess also demonstrates the convergence of the heuristicalgorithm

Queuing models are effective tools for obtaining infor-mation about important performance of each facility in theESC system like sojourn times waiting times queue lengthstotal customers in the system and so on In order to analyzethe efficiency of facilities in the system we calculate thetotal customers and the queue lengths which are shown inFigure 7 We can see that the solution of the above model willnot create queuing delays of relief supply so that the lossesgenerated by the queuing delay can be avoided This iterativeprocess also illustrates the robustness and the potentialapplicability of our model

Based on the built network the coordination between theupstream node and the downstream node in this system canbe enforced and the performance of basic activities includingproducing sorting processing packing and delivering canbe improved tominimize the response time of the systemTheemergency logistics centers for the system should be locatedin DP(6) DP(7) DP(8) DP(9) and DP(10) And the optimalpaths are SP(1) SP(6) rarr ELC(8) rarr DP(1) DP(2)DP(9) SP(2) SP(4) rarr ELC(6) rarr DP(13) DP(14)DP(15) SP(3) SP(8) rarr ELC(7) rarr DP(10) DP(11)DP(12) SP(5) SP(9) rarr ELC(10) rarr DP(3) DP(4)DP(5) and SP(7) SP(9) rarr ELC(9) rarr DP(6) DP(7)DP(8) Overall the examination reveals that the proposedmethod can be a useful tool for a quick response to the urgentneed of relief in the large-scale disaster-affected areas

10 The Scientific World Journal

Table 2 The service rates specifications of different servers in the network

SP 120583101584010158401

120583101584010158402

120583101584010158403

120583101584010158404

120583101584010158405

120583101584010158406

120583101584010158407

120583101584010158408

120583101584010158409

1205831015840101584010

10 9 11 8 11 12 14 16 14 17

ELC 12058310158401

12058310158402

12058310158403

12058310158404

12058310158405

12058310158406

12058310158407

12058310158408

12058310158409

120583101584010

16 18 11 10 11 12 14 16 14 10

DP 1205831

1205832

1205833

1205834

1205835

1205836

1205837

1205838

1205839

12058310

12058311

12058312

12058313

12058314

12058315

8 8 8 8 10 9 11 8 9 12 9 8 12 14 13

Table 3 The distance between supply point and emergency logistics center

SP(1) SP(2) SP(3) SP(4) SP(5) SP(6) SP(7) SP(8) SP(9) SP(10)ELC(1) 1828 1707 1197 2163 614 826 1091 973 1530 879ELC(2) 3073 2147 1624 571 2039 2056 802 1679 2554 1390ELC(3) 2271 2288 1712 1622 1419 1463 998 1102 1842 756ELC(4) 4422 3095 2462 1264 3971 3492 2429 3276 4153 3003ELC(5) 3962 2801 2716 1507 3256 2769 1449 2551 3325 2006ELC(6) 3585 2356 1923 1647 2784 2313 989 2136 2839 1437ELC(7) 2973 2968 2364 2301 2036 2149 1275 1797 2575 1471ELC(8) 2308 3261 2878 2892 1494 1401 1764 1132 1961 1933ELC(9) 2584 3643 3303 3249 2015 1165 2258 941 1600 2419ELC(10) 3038 4666 4240 4139 2831 2039 3008 1742 1405 3105

05

101520253035404550

Tim

e (ho

urs)

Total transportation timeTotal sojourn time

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 6 Optimization of total transportation time and totalsojourn time

6 Conclusions

An emergency supply chain (ESC) system is developed inthis paper for the quick response to the urgent need ofrelief in the large-scale disaster-affected areas which mainlyconsists of three chain members in series supply points(SP) emergency logistics centers (ELCs) and demand points(DP) As the timeliness is the most crucial characteristic ofthe ESC system and queuing theory is a useful tool in theanalysis of the performance of the time-dependent systemwe propose queuingmodeling for the ESC system to optimizethe emergency rescue process Based on the above queuingnetwork system a queuing minimal response time location-allocation model is established to decide the selection ofemergency logistics centers and vehicle routing decision

0

5

10

15

20

25

Total customersQueuing length

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 7 Optimization of the total customers and the queue length

For the complexity of mathematical model the GA-basedapproach is introduced to solve the model

A case study with an assumed severe typhoon striking thesoutheast coast of Shanghai with a great probability withinthe next 100 years is conducted to assess and validate ourmodel The optimal response time of the queuing network is2071 hours and the average response time of each demandpoint is 138 h which is within a reasonable range Theoptimal emergency logistics centers for the system are DP(6)DP(7) DP(8) DP(9) and DP(10) Besides the performanceof queuing modeling for the ESC system illustrates therobustness and the potential applicability of our model

For future research we suggest the following threedirections Firstly we will investigate the queuing modelingunder consideration of the facility blocking problems and

The Scientific World Journal 11

Table 4 The distance between emergency logistics center and demand point

ELC(1) ELC(2) ELC(3) ELC(4) ELC(5) ELC(6) ELC(7) ELC(8) ELC(9) ELC(10)DP(1) 3761 4191 3253 4599 3801 3752 2439 2397 1553 739DP(2) 4271 4702 3761 4794 3945 4152 2818 2822 1898 1075DP(3) 4508 4886 3986 5028 4501 4384 3046 3025 2117 1356DP(4) 4364 4579 3697 4824 4307 4138 2798 2940 1956 1710DP(5) 5283 5416 4386 5466 5435 4986 3836 3271 2693 1917DP(6) 5214 5721 4866 5896 5294 5207 3797 4221 3252 2553DP(7) 5334 5776 5299 5862 5359 5164 3871 3851 2955 2554DP(8) 4689 4581 3763 4983 4306 4173 2818 2997 2200 1968DP(9) 3309 3624 2657 3834 3334 3154 1886 3154 3452 3235DP(10) 2678 3263 1999 2911 2741 2518 1198 2505 2360 3037DP(11) 3369 2912 2386 2349 2182 1858 1894 2392 2915 3596DP(12) 3376 3794 2541 3287 3275 2612 1708 3062 2896 3256DP(13) 3691 3419 2987 2896 2694 2279 2162 3500 3275 4007DP(14) 4230 3273 3402 3011 2555 2141 2606 3906 3749 4448DP(15) 4609 3765 3858 2533 3054 2680 3166 4390 4249 4897

demand priority for the emergency relief points Secondly wewill study the generation of emergency demand and servicetime obeying the other random distributions Thirdly moreefficient multiobjective optimization methodologies for boththe above problem and emergency resources transportationscheduling problem will be considered

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grant nos 71102043 and 71371145) theMinistry of Education of China (Grant no MCM20125021)the Innovative Program of Shanghai Municipal EducationCommission (Grant no 14ZS123) and the ShanghaiMaritimeUniversity (Grant no 20120054)

References

[1] D Guha-Sapir G S Debarati D Hargitt P Hoyois R Belowand D BrechetThirty Years of Natural Disasters 1974ndash2003 theNumbers Presses universitaires de Louvain 2004

[2] J-B Sheu ldquoAn emergency logistics distribution approach forquick response to urgent relief demand in disastersrdquo Trans-portation Research E vol 43 no 6 pp 687ndash709 2007

[3] B S Manoj and A H Baker ldquoCommunication challenges inemergency responserdquo Communications of the ACM vol 50 no3 pp 51ndash53 2007

[4] B Balcik B M Beamon and K Smilowitz ldquoLast mile distribu-tion in humanitarian reliefrdquo Journal of Intelligent TransportationSystems vol 12 no 2 pp 51ndash63 2008

[5] A M Caunhye X F Nie and S Pokharel ldquoOptimizationmodels in emergency logistics a literature reviewrdquo Socio-Economic Planning Sciences vol 46 no 1 pp 4ndash13 2012

[6] S Coles and L Pericchi ldquoAnticipating catastrophes throughextreme valuemodellingrdquo Journal of the Royal Statistical SocietyC vol 52 no 4 pp 405ndash416 2003

[7] J R Artalejo ldquoG-networks a versatile approach for workremoval in queueing networksrdquo European Journal of Opera-tional Research vol 126 no 2 pp 233ndash249 2000

[8] W Jiang L Deng L Chen J Wu and J Li ldquoRisk assessmentand validation of flood disaster based on fuzzy mathematicsrdquoProgress in Natural Science vol 19 no 10 pp 1419ndash1425 2009

[9] V Bhaskar and P Lallement ldquoModeling a supply chain usinga network of queuesrdquo Applied Mathematical Modelling vol 34no 8 pp 2074ndash2088 2010

[10] B M Beamon ldquoHumanitarian relief chains issues and chal-lengesrdquo in Proceedings of the 34th International Conference onComputers and Industrial Engineering San Francisco CalifUSA November 2004

[11] R Oloruntoba and R Gray ldquoHumanitarian aid an agile supplychainrdquo Supply Chain Management vol 11 no 2 pp 115ndash1202006

[12] E J Lodree Jr and S Taskin ldquoSupply chain planning for hur-ricane response with wind speed information updatesrdquo Com-puters and Operations Research vol 36 no 1 pp 2ndash15 2009

[13] V Bhakoo P Singh and A Sohal ldquoCollaborative managementof inventory in Australian hospital supply chains practices andissuesrdquo Supply Chain Management vol 17 no 2 pp 217ndash2302012

[14] W Yi and A Kumar ldquoAnt colony optimization for disaster reliefoperationsrdquo Transportation Research E vol 43 no 6 pp 660ndash672 2007

[15] G-H Tzeng H-J Cheng and T D Huang ldquoMulti-objectiveoptimal planning for designing relief delivery systemsrdquo Trans-portation Research E vol 43 no 6 pp 673ndash686 2007

[16] Y-C Chiu and H Zheng ldquoReal-time mobilization decisionsfor multi-priority emergency response resources and evacua-tion groups model formulation and solutionrdquo TransportationResearch E vol 43 no 6 pp 710ndash736 2007

[17] N Altay andWG Green III ldquoORMS research in disaster oper-ations managementrdquo European Journal of Operational Researchvol 175 no 1 pp 475ndash493 2006

12 The Scientific World Journal

[18] K J Cormican D P Morton and R K Wood ldquoStochasticnetwork interdictionrdquo Operations Research vol 46 no 2 pp184ndash197 1998

[19] HOMete andZ B Zabinsky ldquoStochastic optimization ofmed-ical supply location and distribution in disaster managementrdquoInternational Journal of Production Economics vol 126 no 1 pp76ndash84 2010

[20] H Shavandi and H Mahlooji ldquoA fuzzy queuing locationmodel with a genetic algorithm for congested systemsrdquo AppliedMathematics and Computation vol 181 no 1 pp 440ndash4562006

[21] R D Galvao and R Morabito ldquoEmergency service systemsthe use of the hypercube queueing model in the solution ofprobabilistic location problemsrdquo International Transactions inOperational Research vol 15 no 5 pp 525ndash549 2008

[22] N Geroliminis M G Karlaftis and A Skabardonis ldquoA spatialqueuing model for the emergency vehicle districting andlocation problemrdquo Transportation Research B vol 43 no 7 pp798ndash811 2009

[23] C M Hosage and M F Goodchild ldquoDiscrete space location-allocation solutions from genetic algorithmsrdquo Annals of Opera-tions Research vol 6 no 2 pp 35ndash46 1986

[24] P Chakroborty ldquoGenetic algorithms for optimal urban transitnetwork designrdquo Computer-Aided Civil and Infrastructure Engi-neering vol 18 no 3 pp 184ndash200 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

Emergency supplypoints

S1

S2

Sm

Transportation

TR1j

TR2j

TRmj

Emergencylogistics centres

Emergencydemand points

L1

L2

Lh

Transportation

TR9984001k

TR9984002k

TR998400hk

D1

D2

Dn

middot middot middotmiddot middot middot middot middot middot middot middot middotmiddot middot middot

Figure 1 The queuing network of ESC

120583 the service rate at eachnode in the queuing networksystem

120575 the parameter of the negative exponential distribu-tion

119888 the lower bound of a uniformly distributed randomvariable that indicates the quantity of resources in arelief request

119889 the upper boundof a uniformly distributed randomvariable that indicates the quantity of resources in arelief request

119905119903

119894119895119896 the response time of the ESC system for com-

modity type 119903 from node 119894 to node 119896 going throughlogistics centers located at nodes 119895

WT119903 the sojourn time of commodity type 119903 in thesystem

WT119903119902 the waiting time of commodity type 119903 in the

queue

TR119903119894119895 the transportation time for commodity type 119903

from node 119894 to node 119895

TR119903119895119896 the transportation time for commodity type 119903

from node 119895 to node 119896

DTR119894119895 the distance between node 119894 and node 119895

DT119896119895 the distance between node 119895 and node 119896

DC the penalty cost for unavailability of commoditiesdemand within the maximum promised responsetime

119886119895 the fixed costs of locating the emergency logistics

center 119895

1199021 the total number of logistics center to be fixed

1199022 the number of supply depots delivering commodi-

ties to the same emergency logistic centers

1199023 the number of demand nodes accepting items

from the same emergency logistics center

V119903119894119895 transport speed for commodity type 119903 from node

119894 to node 119895

V119903119895119896 transport speed for commodity type 119903 from node

119895 to node 119896

(3) Decision Variables

119909119895=

1 emergency logistics center builton the site 119895

0 otherwise

119910119894119895=

1 relief resources from emergency supplypoint 119894 transported to emergencylogistics center 119895

0 otherwise

119911119895119896

=

1 relief resources from emergency logisticscenter 119895 transported to reliefdemand point 119896

0 otherwise

(1)

42 Queuing Minimal Unsatisfied Demand Location-Allocation Model Three types of members (SP ELC andDP) involved in this system are in serial connection Thetransportation routes are necessary to deliver items fromupstream nodes to downstream nodes Based on thisassumption a queuing minimal response location-allocationmodel for the three-stage queuing network is formulated asfollows

Objective function

min119885 = sum119903isin119877

sum119894isin119868

sum119895isin119869

sum119896isin119870

119905119903

119894119895119896 (2)

The Scientific World Journal 5

subject to

119910119903

119894119895le 119909119895

forall119894 isin 119868 119895 isin 119869 119903 isin 119877 (3)

119911119903

119895119896le 119909119895

forall119896 isin 119870 119895 isin 119869 119903 isin 119877 (4)

sum119895isin119869

119910119903

119894119895= 1 forall119894 isin 119868 119903 isin 119877 (5)

sum119895isin119869

119911119903

119895119896= 1 forall119896 isin 119870 119903 isin 119877 (6)

sum119897isin119871|119889119894119897le119889119894119895

119910119903

119894119897ge 119909119895

forall119894 isin 119868 119895 isin 119869 119903 isin 119877 (7)

sum119897isin119871|119889119897119896le119889119895119896

119911119903

119897119896ge 119909119895

forall119896 isin 119870 119895 isin 119869 119903 isin 119877 (8)

sum119886119895119909119895le 119861 forall119895 isin 119869 (9)

sum119909119895= 1199021

forall119895 isin 119869 (10)

sum119894isin119868

119910119903

119894119895= 1199022

forall119895 isin 119869 119903 isin 119877 (11)

sum119896isin119870

119911119903

119895119896= 1199023

forall119895 isin 119869 119903 isin 119877 (12)

119905119903

119894119895119896= WT119903

119894+WT119903

119895+WT119903

119896+ TR119903119894119895+ TR119903119895119896 (13)

119909119903

119895 119910119903

119895119896 119911119903

119894119895= 0 1 forall119896 isin 119870 119894 isin 119868 119895 isin 119869 119903 isin 119877 (14)

The objective aims at minimizing the mean systemresponse time of relief resources including the sojourn timein the queuing network system and transportation timeConstraints (3) and (4) ensure that deliveries can only bemade if emergency logistics centers are fixed Constraint(5) enforces that the items from the supply nodes canonly be delivered to one logistics center and constraint (6)enforces that every demand nodes can obtain items fromjust one logistics center Constraints (7) and (8) representthat the items from upstream nodes are transported to thenearest downstream nodes in the queuing network systemConstraint (9) limits the sum of fixed costs of locating theemergency logistics centers Constraint (10) shows that thetotal number of logistics center to be sited is equal to 119902

1

Constraint (11) forces that the number of supply depots deliv-ering commodities to the same emergency logistic centers isequal to 119902

2to ensure that each emergency logistics is with

enough capacity to deal with these commodities Constraint(12) forces that the number of demand nodes accepting itemsfrom the same emergency logistics center is equal to 119902

3to

ensure that the items from each emergency logistics centerare enough to satisfy the emergency request Constraint (13)represents that the response time is equal to the sojourn timeplus the transporting time Constraint (14) defines all thedecision variables to be binary integer variables

Next we use the queuing theory to compute the objectivefunction The emergency supply chain system is a series-parallel hybrid queuing system consisting of three service

SPi

=1

q2sum yij

TRij ELCj

120582k

DPk

= sum 120582kzjk

120583k

120582998400j120582998400j120582998400998400i

TR998400jk

120583998400998400i 120583998400j

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Figure 2 Equivalent queue of studied ESC network

nodes We can assume that the relief requests for the emer-gency demand depots follow a Poisson distribution withintensity 120582

119896 Each emergency logistics center serves a set

of demand points and therefore the relief requests for anemergency service at the logistics center are the union of therelief requests of the nodes in the set Therefore they can bedepicted as a stochastic process equal to the sum of severalPoisson processes with an intensity 1205821015840

119895equal to the sumof the

intensities of the processes at the nodes served by the logisticscenter We can rewrite parameter 1205821015840

119895by using variables 119911

119895119896

1205821015840

119895=

119899

sum119896=1

120582119896119911119895119896 (15)

The relief request for the supply depots is also assumedto follow a Poisson distribution with intensity 12058210158401015840

119894and also

similar equilibrium equations exist between the arrival rate ofthe relief request for the supply depots and for the emergencylogistics centers For the sake of simplicity we assume thatthe arrival rate of the relief request for the supply depotsthat transport emergency resources to the same emergencylogistics center has the same value Thus the parameter 12058210158401015840

119894

can be rewritten by using variables 119910119894119895and the constant 119902

2

12058210158401015840

119894=

1

1199022

ℎ

sum119895=1

1205821015840

119895119910119894119895=

1

1199022

ℎ

sum119895=1

119899

sum119896=1

120582119896119910119894119895119911119895119896 (16)

Based on the above analysis the equivalent queue of thestudied ESC network is shown in Figure 2

From the affected peoplersquos point of view the ESC systemis equivalent to a queue network that is receiving emergencyrelief orders These relief request orders are waiting to beserved The service is the process of production collec-tion and processing and the results are emergency reliefresources items and so forth

Emergency relief orders are characterized by (i) occur-rence (ii) quantity and (iii) delay Consider the following

119880 random variable indicating the occurrence time ofa relief request

119881 random variable indicating the quantity ofresources in every relief request

119882 UV indicating the occurrence time along with thequantity of resources in every relief request

Assume that119880 follows a negative exponential distributionwith intensify119891

119880(119906) and119881 is a uniformly distributed random

6 The Scientific World Journal

variable with intensify 119891119881(V) between 119888 and 119889 (119889 gt 119888) And

119891119880(119906) and 119891

119881(V) are independent Thus

119891119880(119906) =

120575119890minus120575119906 119906 ge 0

0 119906 lt 0

119891119881(V) =

1

119889 minus 119888 119888 lt V lt 119889

0 otherwise

119864 (119882) = 119864 (119880119881) = 119864 (119880) 119864 (119881) =119888 + 119889

2120575

(17)

Therefore the interarrival times of the emergencydemand (occurrence and quantity) follow a negative expo-nential distribution with intensity 120582 equal to 1119864(119882) Theservice rate at each node in the queuing network system isan independent identically distributed random variable withintensity 120583 and the service time is 1120583

So the interarrival time 120582 and the traffic intensity of thesystem 120588 are represented as

120582 =1

119864 (119882)=

2120575

119888 + 119889 (18)

120588 =120582

120583=

1

120583119864 (119882)=

2120575

120583 (119888 + 119889) (19)

Let us assume that there exits just one server at eachservice node and the servers are independent which meansthat the queuing model at each server is an 1198721198721 Thenthe probability distribution function of the sojourn timeWT (defined as the waiting time plus the service time for acustomer) in an1198721198721 queue can be presented as

119891WT (119905) = (120583 minus 120582) 119890minus(120583minus120582)119905

(20)

From (7) the cumulative distribution function of WT is

119891WT (119905) = 119875 (WT le 119905)

= int119905

0

(120583 minus 120582) 119890minus(120583minus120582)119905

119889119905 = 1 minus 119890minus(120583minus120582)119905

(21)

The average sojourn time WT is given by

WT = 119864 (WT) = intinfin

0

119891WT (119905) 119905 119889119905

= intinfin

0

(120583 minus 120582) 119890minus(120583minus120582)119905

119905 119889119905 =1

120583 minus 120582

(22)

Let WT119902denote the waiting time in the queue The

average waiting time is computed as

WT119902= WT minus

1

120583=

120582

120583 (120583 minus 120582) (23)

From the well-known Littlersquos theorem the average cus-tomers LR in the system including the number of customersboth waiting in the queue and served in the server is given by

LR = 120582WT =120582

120583 minus 120582 (24)

And LR119902denote the queuing length in the system which

is presented as

LR119902= 120582WT

119902=

1205822

120583 (120583 minus 120582) (25)

From (17) (16) (18) (22) (23) (24) and (25) theaverage sojourn time waiting time in the queue the averagecustomers including the number of customers both waitingin the queue and served in the server and the queuing lengthfor the ESC network system are given as

WTsys = sum119903isin119877

sum119894isin119868

WT119877119868+ sum119903isin119877

sum119895isin119869

WT119877119869+ sum119903isin119877

sum119896isin119870

WT119877119896

= sum119903isin119877

sum119894isin119868

1

12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

+ sum119903isin119877

sum119895isin119869

1

1205831015840 minus119899

sum119896=1

(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119896

1

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

WT119902sys = sum

119903isin119877

sum119894isin119868

WT119877119902119868

+ sum119903isin119877

sum119895isin119869

WT119877119902119869+ sum119903isin119877

sum119896isin119870

WT119877119902119896

= sum119903isin119877

sum119894isin119868

(11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

12058310158401015840 (12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)

+ sum119903isin119877

sum119895isin119869

sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

1205831015840 (1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)

+ sum119903isin119877

sum119896isin119896

2120575119903119896 (119888119903119896+ 119888119903119896)

120583 (120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896)))

LRsys = sum119903isin119877

sum119894isin119868

LR119877119868

+ sum119903isin119877

sum119895isin119869

LR119877119869+ sum119903isin119877

sum119896isin119870

LR119877119896

= sum119903isin119877

sum119894isin119868

(11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

+ sum119903isin119877

sum119895isin119869

sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119870

2120575119903119896 (119888119903119896+ 119888119903119896)

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

The Scientific World Journal 7

LR119902sys

= sum119903isin119877

sum119894isin119868

LR119877119902119868+ sum119903isin119877

sum119895isin119869

LR119877119902119869+ sum119903isin119877

sum119896isin119870

LR119877119902119896

= sum119903isin119877

sum119894isin119868

((11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)2

12058310158401015840 (12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)

+ sum119903isin119877

sum119895isin119869

(sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)2

1205831015840 (1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)

+ sum119903isin119877

sum119896isin119896

(2120575119903

119896 (119888119903

119896+ 119888119903

119896))2

120583 (120583 minus 2120575119903119896 (119888119903119896+ 119888119903119896))

(26)

As the response time includes not only the sojourn time ofemergency demand in the queuing network system but alsothe transportation time therefore when the queuing theoryis applied to the present model the above objective functioncan be presented as follows

min119885

= minsum119903isin119877

sum119894isin119868

sum119895isin119869

sum119896isin119870

119905119903

119894119895119896

= min(sum119903isin119877

sum119894isin119868

(1)

times (12058310158401015840minus (

1

1199022

)

ℎ

sum119895=1

119899

sum119896=1

(2120575119903

119896 (119888119903

119896+ 119888119903

119896)) 119910119894119895119911119895119896)

minus1

+ sum119903isin119877

sum119895isin119869

1

1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119896

1

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

+sum119903isin119877

sum119894isin119868

sum119895isin119869

TR119903119894119895119910119894119895+ sum119903isin119877

sum119895isin119869

sum119896isin119870

TR119903119895119896119911119895119896)

(27)

43 GA-Based Approach As the facility location-allocationproblem is NP- (nondeterministic polynomial-) hard wherethe proposed 0-1 nonlinear integer programming model inthis study is used some heuristic algorithms are requiredto solve the problem quickly The genetic algorithms (GAs)have been applied to location problems since 1986 by Hosageand Goodchild [23] GA has exhibited inherent advantagessuch as their robust performancewhen solving combinatorialproblems as well as their ability to incorporate proceduresand logical conditions and to handle discrete variables andnonlinear constraints in a straightforward manner [24]These qualities make GAs particularly attractive for potentialcombination with other methods and external procedures

Wedevelop a refined genetic algorithm to solve the aforemen-tioned problem as indicated belowThe associated frameworkfor embedding the GA with emergency queue network ispresented in Figure 3

431 Methodology of Genetic Algorithm In a genetic algo-rithm a population of candidate solutions to an optimizationproblem is evolved toward better solutions Each candi-date solution has a set of properties (its chromosomes orgenotype) which can be mutated and altered solutions arerepresented in binary as strings of 0 s and 1 s but otherencodings are also possible The evolution usually startsfrom a population of randomly generated individuals andhappens in generations In each generation the fitness ofevery individual in the population is evaluated the more fitindividuals are stochastically selected from the current popu-lation and each individualrsquos genome ismodified (recombinedand possibly randomly mutated) to form a new populationThe new population is then used in the next iteration of thealgorithm Commonly the algorithm terminates when eithera maximum number of generations have been produced or asatisfactory fitness level has been reached for the populationOnce the genetic representation and the fitness function aredefined a GA proceeds to initialize a population of solutionsand then to improve it through repetitive application of themutation crossover inversion and selection operators

432 Procedure of the Refined Genetic Algorithm

Step 1 (encoding) We suppose that the genotype of a chro-mosome with some genes indicates an individual corre-sponding to the location pattern of emergency logisticscenters and paths between nodes in the system network Thegene presented by 1 indicates that an emergency logisticscenter is built at this location or the path between nodes in thesystem network is selectedThe gene presented by 0 indicatesthat no emergency logistics centers is built there or the pathbetween nodes in the system network which is selected is notselected 119873 chromosomes are randomly generated and theinitial population (Pop) is set

Step 2 (fitness evaluation) Once the initial population isavailable evaluation of each chromosome takes place to ex-plore the quality of the corresponding solution For individ-uals with larger fitness value to be transmitted to the nextgeneration with higher probability the chromosome fitnessfunction in this paper can be described as follows and 119888max isthe maximum estimation value of min119885

Fit (min119885) = 119862max minusmin119885 if min119885 lt 119862max

0 otherwise(28)

Step 3 (crossover selection and mutation) 119873 chromosomesare crossed according to a single pointmethodwith crossoverprobability (which is 119875

119888= 09) to generate the offspring

chromosomes There are a total of 2119873 chromosomes forboth parent and offspring chromosomes Then the duplicatechromosomes are deleted to produce the new population(Pop) Select the first 119873 chromosomes of the population on

8 The Scientific World Journal

Step 1 encoding forparameter sets

Initialization ofgenerate population

Step 2 chromosomefitness evaluation

Step 3 geneticmanipulation crossoverselection and mutation

Step 4 offspringgeneration Select the best individual

and output the solution

Yes

No Terminate

(1) Decoding bit string to the parameter(2) Objective function value calculation(3) Mapping function value to the fitness value(4) Fitness value adjustments

Preparation of dataand parameters

Figure 3 Steps of the proposed GA heuristic

(B)(C)

(A)

Figure 4 The location of facilities in an ESC system

the basis of the fitness values in ascending order When thenumber of chromosomes in the new population (Pop) is lessthan 119873 (119873 = 200) 119873-|Pop| chromosomes are randomlygenerated (|Pop| is the number of chromosomes in thepopulation (Pop)) Select a chromosome randomly and applythe mutation operator applied on it with probability 119875

119898=

001

Step 4 Choose new chromosomes by the roulette wheelmethod Repeat Steps 3 and 4 until there are no solutionimprovements within a specified number of iterations

5 A Case Study

A case study example is presented in this section to illustratethe efficiency of our approach to realize the quick-responsiveemergency relief during large-scale natural disastersThepro-posed method is implemented on a scenario where a severetyphoon strikes the southeast coast of Shanghai with a greatprobability within the next 100 years It is assumed that thereare three regions (A B and C) slashed by the typhoon shownin Figure 4 One region is Lingangxincheng District (A)

another one is Fengxian District (B) and the third one isJinshan District (C)

There are ten supply points (SP) (1) disaster relief sup-plies reserve center in Baoshan District (2) Automobile Cityin Jiading (3) Red Cross disaster preparedness warehouse inJiading District (4) disaster relief supplies reserve center inQingqu District (5) disaster relief supplies reserve center inPutuo District (6) disaster relief supplies reserve center inYangpu District (7) disaster relief supplies reserve center inMinhang District (8) disaster relief supplies reserve center inHongkou District (9) disaster relief supplies reserve centerin Pudong District and (10) disaster relief supplies reservecenter in Xuhui District Based on the study of large-scalelogistics center in Shanghai we select ten potential pointsas the emergency logistics centers (ELCs) Four potentialELCs are prepared for the affected areas in LingangxinchengDistrict two potential ELCs are planned to serve FengxianDistrict and two potential ELCs are expected to serve theJinshanDistrictWe assume that a total of thirteen areas (DP)are seriously affected and need emergency rescue urgently inwhich six affected areas are in Lingangxincheng four affectedareas are in Fengxian District and three affected areas are inJinshan District

For simplicity only one type of emergency supply resour-ces will be considered in the queuing network We assumethat the travel speed is fixed to be equal to 40 miles per hourfor all the vehicles travelling between SP and ESC Howeverthe travel speed between ESC and DP is assumed to be 30miles per hour Let us adopt the hypothesis as the abovethat the occurrence time of a relief request follows a negativeexponential distribution with intensify 120575 and the quantity ofresources in every relief request is uniformly distributed withintensify 119891

119881(V) that the lower bound (LB) is 119888 and the upper

bound (UB) is 119889 (119889 gt 119888) The disaster-related informationincluding the population of the affected areas and the distancebetween supply point and emergency logistics center anddistance between emergency logistics center and demandpoint are shown in Tables 1 2 3 and 4 respectively

The Scientific World Journal 9

Table 1 Population and demand data of the affected areas

Affected area Population (119888 119889) 120575

Area 1 (Zhuqiaozhen) 105807 (4 12) 16Area 2 (Huinanzhen) 213845 (4 12) 16Area 3 (Laogangzhen) 37408 (5 14) 18Area 4 (Datuanzhen) 71162 (4 12) 16Area 5 (Xingangzhen) 21475 (6 16) 20Area 6 (Shuyuanzhen) 59831 (6 16) 20Area 7 (Luchaogangzhen) 27850 (6 16) 20Area 8 (Situanzhen) 65389 (5 14) 18Area 9 (Qingcunzhen) 89163 (5 14) 18Area 10 (Nanqiaozhen) 361185 (3 10) 14Area 11 (Zhuanghangzhen) 62388 (3 10) 14Area 12 (Zhelinzhen) 62589 (3 10) 14Area 13 (Caojingzhen) 40722 (2 8) 12Area 14 (Shanyangzhen) 84640 (2 8) 12Area 15 (Jinshanzhen) 70815 (2 8) 12

0 20 40 60 80 100 120 140 160 180 20020

25

30

35

40

45

50

55

60

Interaction number

Tota

l res

pons

e tim

e (ho

urs)

Total response time

Figure 5 Optimization of the fitness function

In this case study the fixed cost to locate the emergencylogistics center is assumed as 119886

1= 85 119886

2= 100 119886

3= 90

1198864= 85 119886

5= 95 119886

6= 110 119886

7= 70 119886

8= 80 119886

9= 115

and 11988610

= 120 The total cost to build the emergency logisticscenters is assumednot exceeding to be119861 = 500 For simplicityof the computation we presume that 119902

1= 5 119902

2= 2 and

1199023= 3According to the background of the above problem a

simplified 10 times 10 times 15 emergency supply chain network isdeveloped in the assumed scenario where the geographicalrelationships among these supply depots emergency logisticscenters and demand points are specified Then we applythe aforementioned model into the emergency supply chainnetwork system And the GA-based approach demonstratedbefore is coded to verify the validity of the model inMatlab2009a

The optimization process of the fitness function isdepicted in Figure 5 where the optimal response time of thequeuing network is 2071 hours

Therefore the average response time of each demandpoint is 138 h which is within a reasonable range The cal-culation result shows that this model can play an importantrole in the emergency supply chain management Differentfrom the general supply chain management the timesavingis the most significant feature of the emergency supply chainsystem The efficient allocation of emergency resources todemand points improves not only the operational perfor-mance of the emergency supply chain system in the emer-gency supply side but also the affected peoplersquos survival ratein the disaster areas From the psychology of point of viewquick responses and deliveries for disaster relief not onlymay improve the governmental efforts to rescue but alsostrengthen the affected peoplersquos willpower to be alive hencestabilizing the affected areas

The emergency supply chain system is composed of threemembers which have different functions in the relief supplydelivery process And the upstream nodes and downstreamnodes in the ESC system are connected by roads If we treateach node in the system as the node in a network considerthe roads as the arcs of the network and treat the length ofeach road as the weight of each arc then the ESC system canbe seen as a directed network diagram with weights And theresponse time of the system includes two parts the first partis the transportation time of the relief supply request on theroads and the second part is the sojourn time of the reliefsupply request in each node

Figure 6 indicates the values of the above two componentsof response time with the total transportation time being1664 h and the total sojourn time being 407 h This iterativeprocess also demonstrates the convergence of the heuristicalgorithm

Queuing models are effective tools for obtaining infor-mation about important performance of each facility in theESC system like sojourn times waiting times queue lengthstotal customers in the system and so on In order to analyzethe efficiency of facilities in the system we calculate thetotal customers and the queue lengths which are shown inFigure 7 We can see that the solution of the above model willnot create queuing delays of relief supply so that the lossesgenerated by the queuing delay can be avoided This iterativeprocess also illustrates the robustness and the potentialapplicability of our model

Based on the built network the coordination between theupstream node and the downstream node in this system canbe enforced and the performance of basic activities includingproducing sorting processing packing and delivering canbe improved tominimize the response time of the systemTheemergency logistics centers for the system should be locatedin DP(6) DP(7) DP(8) DP(9) and DP(10) And the optimalpaths are SP(1) SP(6) rarr ELC(8) rarr DP(1) DP(2)DP(9) SP(2) SP(4) rarr ELC(6) rarr DP(13) DP(14)DP(15) SP(3) SP(8) rarr ELC(7) rarr DP(10) DP(11)DP(12) SP(5) SP(9) rarr ELC(10) rarr DP(3) DP(4)DP(5) and SP(7) SP(9) rarr ELC(9) rarr DP(6) DP(7)DP(8) Overall the examination reveals that the proposedmethod can be a useful tool for a quick response to the urgentneed of relief in the large-scale disaster-affected areas

10 The Scientific World Journal

Table 2 The service rates specifications of different servers in the network

SP 120583101584010158401

120583101584010158402

120583101584010158403

120583101584010158404

120583101584010158405

120583101584010158406

120583101584010158407

120583101584010158408

120583101584010158409

1205831015840101584010

10 9 11 8 11 12 14 16 14 17

ELC 12058310158401

12058310158402

12058310158403

12058310158404

12058310158405

12058310158406

12058310158407

12058310158408

12058310158409

120583101584010

16 18 11 10 11 12 14 16 14 10

DP 1205831

1205832

1205833

1205834

1205835

1205836

1205837

1205838

1205839

12058310

12058311

12058312

12058313

12058314

12058315

8 8 8 8 10 9 11 8 9 12 9 8 12 14 13

Table 3 The distance between supply point and emergency logistics center

SP(1) SP(2) SP(3) SP(4) SP(5) SP(6) SP(7) SP(8) SP(9) SP(10)ELC(1) 1828 1707 1197 2163 614 826 1091 973 1530 879ELC(2) 3073 2147 1624 571 2039 2056 802 1679 2554 1390ELC(3) 2271 2288 1712 1622 1419 1463 998 1102 1842 756ELC(4) 4422 3095 2462 1264 3971 3492 2429 3276 4153 3003ELC(5) 3962 2801 2716 1507 3256 2769 1449 2551 3325 2006ELC(6) 3585 2356 1923 1647 2784 2313 989 2136 2839 1437ELC(7) 2973 2968 2364 2301 2036 2149 1275 1797 2575 1471ELC(8) 2308 3261 2878 2892 1494 1401 1764 1132 1961 1933ELC(9) 2584 3643 3303 3249 2015 1165 2258 941 1600 2419ELC(10) 3038 4666 4240 4139 2831 2039 3008 1742 1405 3105

05

101520253035404550

Tim

e (ho

urs)

Total transportation timeTotal sojourn time

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 6 Optimization of total transportation time and totalsojourn time

6 Conclusions

An emergency supply chain (ESC) system is developed inthis paper for the quick response to the urgent need ofrelief in the large-scale disaster-affected areas which mainlyconsists of three chain members in series supply points(SP) emergency logistics centers (ELCs) and demand points(DP) As the timeliness is the most crucial characteristic ofthe ESC system and queuing theory is a useful tool in theanalysis of the performance of the time-dependent systemwe propose queuingmodeling for the ESC system to optimizethe emergency rescue process Based on the above queuingnetwork system a queuing minimal response time location-allocation model is established to decide the selection ofemergency logistics centers and vehicle routing decision

0

5

10

15

20

25

Total customersQueuing length

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 7 Optimization of the total customers and the queue length

For the complexity of mathematical model the GA-basedapproach is introduced to solve the model

A case study with an assumed severe typhoon striking thesoutheast coast of Shanghai with a great probability withinthe next 100 years is conducted to assess and validate ourmodel The optimal response time of the queuing network is2071 hours and the average response time of each demandpoint is 138 h which is within a reasonable range Theoptimal emergency logistics centers for the system are DP(6)DP(7) DP(8) DP(9) and DP(10) Besides the performanceof queuing modeling for the ESC system illustrates therobustness and the potential applicability of our model

For future research we suggest the following threedirections Firstly we will investigate the queuing modelingunder consideration of the facility blocking problems and

The Scientific World Journal 11

Table 4 The distance between emergency logistics center and demand point

ELC(1) ELC(2) ELC(3) ELC(4) ELC(5) ELC(6) ELC(7) ELC(8) ELC(9) ELC(10)DP(1) 3761 4191 3253 4599 3801 3752 2439 2397 1553 739DP(2) 4271 4702 3761 4794 3945 4152 2818 2822 1898 1075DP(3) 4508 4886 3986 5028 4501 4384 3046 3025 2117 1356DP(4) 4364 4579 3697 4824 4307 4138 2798 2940 1956 1710DP(5) 5283 5416 4386 5466 5435 4986 3836 3271 2693 1917DP(6) 5214 5721 4866 5896 5294 5207 3797 4221 3252 2553DP(7) 5334 5776 5299 5862 5359 5164 3871 3851 2955 2554DP(8) 4689 4581 3763 4983 4306 4173 2818 2997 2200 1968DP(9) 3309 3624 2657 3834 3334 3154 1886 3154 3452 3235DP(10) 2678 3263 1999 2911 2741 2518 1198 2505 2360 3037DP(11) 3369 2912 2386 2349 2182 1858 1894 2392 2915 3596DP(12) 3376 3794 2541 3287 3275 2612 1708 3062 2896 3256DP(13) 3691 3419 2987 2896 2694 2279 2162 3500 3275 4007DP(14) 4230 3273 3402 3011 2555 2141 2606 3906 3749 4448DP(15) 4609 3765 3858 2533 3054 2680 3166 4390 4249 4897

demand priority for the emergency relief points Secondly wewill study the generation of emergency demand and servicetime obeying the other random distributions Thirdly moreefficient multiobjective optimization methodologies for boththe above problem and emergency resources transportationscheduling problem will be considered

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grant nos 71102043 and 71371145) theMinistry of Education of China (Grant no MCM20125021)the Innovative Program of Shanghai Municipal EducationCommission (Grant no 14ZS123) and the ShanghaiMaritimeUniversity (Grant no 20120054)

References

[1] D Guha-Sapir G S Debarati D Hargitt P Hoyois R Belowand D BrechetThirty Years of Natural Disasters 1974ndash2003 theNumbers Presses universitaires de Louvain 2004

[2] J-B Sheu ldquoAn emergency logistics distribution approach forquick response to urgent relief demand in disastersrdquo Trans-portation Research E vol 43 no 6 pp 687ndash709 2007

[3] B S Manoj and A H Baker ldquoCommunication challenges inemergency responserdquo Communications of the ACM vol 50 no3 pp 51ndash53 2007

[4] B Balcik B M Beamon and K Smilowitz ldquoLast mile distribu-tion in humanitarian reliefrdquo Journal of Intelligent TransportationSystems vol 12 no 2 pp 51ndash63 2008

[5] A M Caunhye X F Nie and S Pokharel ldquoOptimizationmodels in emergency logistics a literature reviewrdquo Socio-Economic Planning Sciences vol 46 no 1 pp 4ndash13 2012

[6] S Coles and L Pericchi ldquoAnticipating catastrophes throughextreme valuemodellingrdquo Journal of the Royal Statistical SocietyC vol 52 no 4 pp 405ndash416 2003

[7] J R Artalejo ldquoG-networks a versatile approach for workremoval in queueing networksrdquo European Journal of Opera-tional Research vol 126 no 2 pp 233ndash249 2000

[8] W Jiang L Deng L Chen J Wu and J Li ldquoRisk assessmentand validation of flood disaster based on fuzzy mathematicsrdquoProgress in Natural Science vol 19 no 10 pp 1419ndash1425 2009

[9] V Bhaskar and P Lallement ldquoModeling a supply chain usinga network of queuesrdquo Applied Mathematical Modelling vol 34no 8 pp 2074ndash2088 2010

[10] B M Beamon ldquoHumanitarian relief chains issues and chal-lengesrdquo in Proceedings of the 34th International Conference onComputers and Industrial Engineering San Francisco CalifUSA November 2004

[11] R Oloruntoba and R Gray ldquoHumanitarian aid an agile supplychainrdquo Supply Chain Management vol 11 no 2 pp 115ndash1202006

[12] E J Lodree Jr and S Taskin ldquoSupply chain planning for hur-ricane response with wind speed information updatesrdquo Com-puters and Operations Research vol 36 no 1 pp 2ndash15 2009

[13] V Bhakoo P Singh and A Sohal ldquoCollaborative managementof inventory in Australian hospital supply chains practices andissuesrdquo Supply Chain Management vol 17 no 2 pp 217ndash2302012

[14] W Yi and A Kumar ldquoAnt colony optimization for disaster reliefoperationsrdquo Transportation Research E vol 43 no 6 pp 660ndash672 2007

[15] G-H Tzeng H-J Cheng and T D Huang ldquoMulti-objectiveoptimal planning for designing relief delivery systemsrdquo Trans-portation Research E vol 43 no 6 pp 673ndash686 2007

[16] Y-C Chiu and H Zheng ldquoReal-time mobilization decisionsfor multi-priority emergency response resources and evacua-tion groups model formulation and solutionrdquo TransportationResearch E vol 43 no 6 pp 710ndash736 2007

[17] N Altay andWG Green III ldquoORMS research in disaster oper-ations managementrdquo European Journal of Operational Researchvol 175 no 1 pp 475ndash493 2006

12 The Scientific World Journal

[18] K J Cormican D P Morton and R K Wood ldquoStochasticnetwork interdictionrdquo Operations Research vol 46 no 2 pp184ndash197 1998

[19] HOMete andZ B Zabinsky ldquoStochastic optimization ofmed-ical supply location and distribution in disaster managementrdquoInternational Journal of Production Economics vol 126 no 1 pp76ndash84 2010

[20] H Shavandi and H Mahlooji ldquoA fuzzy queuing locationmodel with a genetic algorithm for congested systemsrdquo AppliedMathematics and Computation vol 181 no 1 pp 440ndash4562006

[21] R D Galvao and R Morabito ldquoEmergency service systemsthe use of the hypercube queueing model in the solution ofprobabilistic location problemsrdquo International Transactions inOperational Research vol 15 no 5 pp 525ndash549 2008

[22] N Geroliminis M G Karlaftis and A Skabardonis ldquoA spatialqueuing model for the emergency vehicle districting andlocation problemrdquo Transportation Research B vol 43 no 7 pp798ndash811 2009

[23] C M Hosage and M F Goodchild ldquoDiscrete space location-allocation solutions from genetic algorithmsrdquo Annals of Opera-tions Research vol 6 no 2 pp 35ndash46 1986

[24] P Chakroborty ldquoGenetic algorithms for optimal urban transitnetwork designrdquo Computer-Aided Civil and Infrastructure Engi-neering vol 18 no 3 pp 184ndash200 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 5

subject to

119910119903

119894119895le 119909119895

forall119894 isin 119868 119895 isin 119869 119903 isin 119877 (3)

119911119903

119895119896le 119909119895

forall119896 isin 119870 119895 isin 119869 119903 isin 119877 (4)

sum119895isin119869

119910119903

119894119895= 1 forall119894 isin 119868 119903 isin 119877 (5)

sum119895isin119869

119911119903

119895119896= 1 forall119896 isin 119870 119903 isin 119877 (6)

sum119897isin119871|119889119894119897le119889119894119895

119910119903

119894119897ge 119909119895

forall119894 isin 119868 119895 isin 119869 119903 isin 119877 (7)

sum119897isin119871|119889119897119896le119889119895119896

119911119903

119897119896ge 119909119895

forall119896 isin 119870 119895 isin 119869 119903 isin 119877 (8)

sum119886119895119909119895le 119861 forall119895 isin 119869 (9)

sum119909119895= 1199021

forall119895 isin 119869 (10)

sum119894isin119868

119910119903

119894119895= 1199022

forall119895 isin 119869 119903 isin 119877 (11)

sum119896isin119870

119911119903

119895119896= 1199023

forall119895 isin 119869 119903 isin 119877 (12)

119905119903

119894119895119896= WT119903

119894+WT119903

119895+WT119903

119896+ TR119903119894119895+ TR119903119895119896 (13)

119909119903

119895 119910119903

119895119896 119911119903

119894119895= 0 1 forall119896 isin 119870 119894 isin 119868 119895 isin 119869 119903 isin 119877 (14)

The objective aims at minimizing the mean systemresponse time of relief resources including the sojourn timein the queuing network system and transportation timeConstraints (3) and (4) ensure that deliveries can only bemade if emergency logistics centers are fixed Constraint(5) enforces that the items from the supply nodes canonly be delivered to one logistics center and constraint (6)enforces that every demand nodes can obtain items fromjust one logistics center Constraints (7) and (8) representthat the items from upstream nodes are transported to thenearest downstream nodes in the queuing network systemConstraint (9) limits the sum of fixed costs of locating theemergency logistics centers Constraint (10) shows that thetotal number of logistics center to be sited is equal to 119902

1

Constraint (11) forces that the number of supply depots deliv-ering commodities to the same emergency logistic centers isequal to 119902

2to ensure that each emergency logistics is with

enough capacity to deal with these commodities Constraint(12) forces that the number of demand nodes accepting itemsfrom the same emergency logistics center is equal to 119902

3to

ensure that the items from each emergency logistics centerare enough to satisfy the emergency request Constraint (13)represents that the response time is equal to the sojourn timeplus the transporting time Constraint (14) defines all thedecision variables to be binary integer variables

Next we use the queuing theory to compute the objectivefunction The emergency supply chain system is a series-parallel hybrid queuing system consisting of three service

SPi

=1

q2sum yij

TRij ELCj

120582k

DPk

= sum 120582kzjk

120583k

120582998400j120582998400j120582998400998400i

TR998400jk

120583998400998400i 120583998400j

middot middot middot

middot middot middot

middot middot middot

middot middot middot

Figure 2 Equivalent queue of studied ESC network

nodes We can assume that the relief requests for the emer-gency demand depots follow a Poisson distribution withintensity 120582

119896 Each emergency logistics center serves a set

of demand points and therefore the relief requests for anemergency service at the logistics center are the union of therelief requests of the nodes in the set Therefore they can bedepicted as a stochastic process equal to the sum of severalPoisson processes with an intensity 1205821015840

119895equal to the sumof the

intensities of the processes at the nodes served by the logisticscenter We can rewrite parameter 1205821015840

119895by using variables 119911

119895119896

1205821015840

119895=

119899

sum119896=1

120582119896119911119895119896 (15)

The relief request for the supply depots is also assumedto follow a Poisson distribution with intensity 12058210158401015840

119894and also

similar equilibrium equations exist between the arrival rate ofthe relief request for the supply depots and for the emergencylogistics centers For the sake of simplicity we assume thatthe arrival rate of the relief request for the supply depotsthat transport emergency resources to the same emergencylogistics center has the same value Thus the parameter 12058210158401015840

119894

can be rewritten by using variables 119910119894119895and the constant 119902

2

12058210158401015840

119894=

1

1199022

ℎ

sum119895=1

1205821015840

119895119910119894119895=

1

1199022

ℎ

sum119895=1

119899

sum119896=1

120582119896119910119894119895119911119895119896 (16)

Based on the above analysis the equivalent queue of thestudied ESC network is shown in Figure 2

From the affected peoplersquos point of view the ESC systemis equivalent to a queue network that is receiving emergencyrelief orders These relief request orders are waiting to beserved The service is the process of production collec-tion and processing and the results are emergency reliefresources items and so forth

Emergency relief orders are characterized by (i) occur-rence (ii) quantity and (iii) delay Consider the following

119880 random variable indicating the occurrence time ofa relief request

119881 random variable indicating the quantity ofresources in every relief request

119882 UV indicating the occurrence time along with thequantity of resources in every relief request

Assume that119880 follows a negative exponential distributionwith intensify119891

119880(119906) and119881 is a uniformly distributed random

6 The Scientific World Journal

variable with intensify 119891119881(V) between 119888 and 119889 (119889 gt 119888) And

119891119880(119906) and 119891

119881(V) are independent Thus

119891119880(119906) =

120575119890minus120575119906 119906 ge 0

0 119906 lt 0

119891119881(V) =

1

119889 minus 119888 119888 lt V lt 119889

0 otherwise

119864 (119882) = 119864 (119880119881) = 119864 (119880) 119864 (119881) =119888 + 119889

2120575

(17)

Therefore the interarrival times of the emergencydemand (occurrence and quantity) follow a negative expo-nential distribution with intensity 120582 equal to 1119864(119882) Theservice rate at each node in the queuing network system isan independent identically distributed random variable withintensity 120583 and the service time is 1120583

So the interarrival time 120582 and the traffic intensity of thesystem 120588 are represented as

120582 =1

119864 (119882)=

2120575

119888 + 119889 (18)

120588 =120582

120583=

1

120583119864 (119882)=

2120575

120583 (119888 + 119889) (19)

Let us assume that there exits just one server at eachservice node and the servers are independent which meansthat the queuing model at each server is an 1198721198721 Thenthe probability distribution function of the sojourn timeWT (defined as the waiting time plus the service time for acustomer) in an1198721198721 queue can be presented as

119891WT (119905) = (120583 minus 120582) 119890minus(120583minus120582)119905

(20)

From (7) the cumulative distribution function of WT is

119891WT (119905) = 119875 (WT le 119905)

= int119905

0

(120583 minus 120582) 119890minus(120583minus120582)119905

119889119905 = 1 minus 119890minus(120583minus120582)119905

(21)

The average sojourn time WT is given by

WT = 119864 (WT) = intinfin

0

119891WT (119905) 119905 119889119905

= intinfin

0

(120583 minus 120582) 119890minus(120583minus120582)119905

119905 119889119905 =1

120583 minus 120582

(22)

Let WT119902denote the waiting time in the queue The

average waiting time is computed as

WT119902= WT minus

1

120583=

120582

120583 (120583 minus 120582) (23)

From the well-known Littlersquos theorem the average cus-tomers LR in the system including the number of customersboth waiting in the queue and served in the server is given by

LR = 120582WT =120582

120583 minus 120582 (24)

And LR119902denote the queuing length in the system which

is presented as

LR119902= 120582WT

119902=

1205822

120583 (120583 minus 120582) (25)

From (17) (16) (18) (22) (23) (24) and (25) theaverage sojourn time waiting time in the queue the averagecustomers including the number of customers both waitingin the queue and served in the server and the queuing lengthfor the ESC network system are given as

WTsys = sum119903isin119877

sum119894isin119868

WT119877119868+ sum119903isin119877

sum119895isin119869

WT119877119869+ sum119903isin119877

sum119896isin119870

WT119877119896

= sum119903isin119877

sum119894isin119868

1

12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

+ sum119903isin119877

sum119895isin119869

1

1205831015840 minus119899

sum119896=1

(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119896

1

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

WT119902sys = sum

119903isin119877

sum119894isin119868

WT119877119902119868

+ sum119903isin119877

sum119895isin119869

WT119877119902119869+ sum119903isin119877

sum119896isin119870

WT119877119902119896

= sum119903isin119877

sum119894isin119868

(11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

12058310158401015840 (12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)

+ sum119903isin119877

sum119895isin119869

sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

1205831015840 (1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)

+ sum119903isin119877

sum119896isin119896

2120575119903119896 (119888119903119896+ 119888119903119896)

120583 (120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896)))

LRsys = sum119903isin119877

sum119894isin119868

LR119877119868

+ sum119903isin119877

sum119895isin119869

LR119877119869+ sum119903isin119877

sum119896isin119870

LR119877119896

= sum119903isin119877

sum119894isin119868

(11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

+ sum119903isin119877

sum119895isin119869

sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119870

2120575119903119896 (119888119903119896+ 119888119903119896)

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

The Scientific World Journal 7

LR119902sys

= sum119903isin119877

sum119894isin119868

LR119877119902119868+ sum119903isin119877

sum119895isin119869

LR119877119902119869+ sum119903isin119877

sum119896isin119870

LR119877119902119896

= sum119903isin119877

sum119894isin119868

((11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)2

12058310158401015840 (12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)

+ sum119903isin119877

sum119895isin119869

(sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)2

1205831015840 (1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)

+ sum119903isin119877

sum119896isin119896

(2120575119903

119896 (119888119903

119896+ 119888119903

119896))2

120583 (120583 minus 2120575119903119896 (119888119903119896+ 119888119903119896))

(26)

As the response time includes not only the sojourn time ofemergency demand in the queuing network system but alsothe transportation time therefore when the queuing theoryis applied to the present model the above objective functioncan be presented as follows

min119885

= minsum119903isin119877

sum119894isin119868

sum119895isin119869

sum119896isin119870

119905119903

119894119895119896

= min(sum119903isin119877

sum119894isin119868

(1)

times (12058310158401015840minus (

1

1199022

)

ℎ

sum119895=1

119899

sum119896=1

(2120575119903

119896 (119888119903

119896+ 119888119903

119896)) 119910119894119895119911119895119896)

minus1

+ sum119903isin119877

sum119895isin119869

1

1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119896

1

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

+sum119903isin119877

sum119894isin119868

sum119895isin119869

TR119903119894119895119910119894119895+ sum119903isin119877

sum119895isin119869

sum119896isin119870

TR119903119895119896119911119895119896)

(27)

43 GA-Based Approach As the facility location-allocationproblem is NP- (nondeterministic polynomial-) hard wherethe proposed 0-1 nonlinear integer programming model inthis study is used some heuristic algorithms are requiredto solve the problem quickly The genetic algorithms (GAs)have been applied to location problems since 1986 by Hosageand Goodchild [23] GA has exhibited inherent advantagessuch as their robust performancewhen solving combinatorialproblems as well as their ability to incorporate proceduresand logical conditions and to handle discrete variables andnonlinear constraints in a straightforward manner [24]These qualities make GAs particularly attractive for potentialcombination with other methods and external procedures

Wedevelop a refined genetic algorithm to solve the aforemen-tioned problem as indicated belowThe associated frameworkfor embedding the GA with emergency queue network ispresented in Figure 3

431 Methodology of Genetic Algorithm In a genetic algo-rithm a population of candidate solutions to an optimizationproblem is evolved toward better solutions Each candi-date solution has a set of properties (its chromosomes orgenotype) which can be mutated and altered solutions arerepresented in binary as strings of 0 s and 1 s but otherencodings are also possible The evolution usually startsfrom a population of randomly generated individuals andhappens in generations In each generation the fitness ofevery individual in the population is evaluated the more fitindividuals are stochastically selected from the current popu-lation and each individualrsquos genome ismodified (recombinedand possibly randomly mutated) to form a new populationThe new population is then used in the next iteration of thealgorithm Commonly the algorithm terminates when eithera maximum number of generations have been produced or asatisfactory fitness level has been reached for the populationOnce the genetic representation and the fitness function aredefined a GA proceeds to initialize a population of solutionsand then to improve it through repetitive application of themutation crossover inversion and selection operators

432 Procedure of the Refined Genetic Algorithm

Step 1 (encoding) We suppose that the genotype of a chro-mosome with some genes indicates an individual corre-sponding to the location pattern of emergency logisticscenters and paths between nodes in the system network Thegene presented by 1 indicates that an emergency logisticscenter is built at this location or the path between nodes in thesystem network is selectedThe gene presented by 0 indicatesthat no emergency logistics centers is built there or the pathbetween nodes in the system network which is selected is notselected 119873 chromosomes are randomly generated and theinitial population (Pop) is set

Step 2 (fitness evaluation) Once the initial population isavailable evaluation of each chromosome takes place to ex-plore the quality of the corresponding solution For individ-uals with larger fitness value to be transmitted to the nextgeneration with higher probability the chromosome fitnessfunction in this paper can be described as follows and 119888max isthe maximum estimation value of min119885

Fit (min119885) = 119862max minusmin119885 if min119885 lt 119862max

0 otherwise(28)

Step 3 (crossover selection and mutation) 119873 chromosomesare crossed according to a single pointmethodwith crossoverprobability (which is 119875

119888= 09) to generate the offspring

chromosomes There are a total of 2119873 chromosomes forboth parent and offspring chromosomes Then the duplicatechromosomes are deleted to produce the new population(Pop) Select the first 119873 chromosomes of the population on

8 The Scientific World Journal

Step 1 encoding forparameter sets

Initialization ofgenerate population

Step 2 chromosomefitness evaluation

Step 3 geneticmanipulation crossoverselection and mutation

Step 4 offspringgeneration Select the best individual

and output the solution

Yes

No Terminate

(1) Decoding bit string to the parameter(2) Objective function value calculation(3) Mapping function value to the fitness value(4) Fitness value adjustments

Preparation of dataand parameters

Figure 3 Steps of the proposed GA heuristic

(B)(C)

(A)

Figure 4 The location of facilities in an ESC system

the basis of the fitness values in ascending order When thenumber of chromosomes in the new population (Pop) is lessthan 119873 (119873 = 200) 119873-|Pop| chromosomes are randomlygenerated (|Pop| is the number of chromosomes in thepopulation (Pop)) Select a chromosome randomly and applythe mutation operator applied on it with probability 119875

119898=

001

Step 4 Choose new chromosomes by the roulette wheelmethod Repeat Steps 3 and 4 until there are no solutionimprovements within a specified number of iterations

5 A Case Study

A case study example is presented in this section to illustratethe efficiency of our approach to realize the quick-responsiveemergency relief during large-scale natural disastersThepro-posed method is implemented on a scenario where a severetyphoon strikes the southeast coast of Shanghai with a greatprobability within the next 100 years It is assumed that thereare three regions (A B and C) slashed by the typhoon shownin Figure 4 One region is Lingangxincheng District (A)

another one is Fengxian District (B) and the third one isJinshan District (C)

There are ten supply points (SP) (1) disaster relief sup-plies reserve center in Baoshan District (2) Automobile Cityin Jiading (3) Red Cross disaster preparedness warehouse inJiading District (4) disaster relief supplies reserve center inQingqu District (5) disaster relief supplies reserve center inPutuo District (6) disaster relief supplies reserve center inYangpu District (7) disaster relief supplies reserve center inMinhang District (8) disaster relief supplies reserve center inHongkou District (9) disaster relief supplies reserve centerin Pudong District and (10) disaster relief supplies reservecenter in Xuhui District Based on the study of large-scalelogistics center in Shanghai we select ten potential pointsas the emergency logistics centers (ELCs) Four potentialELCs are prepared for the affected areas in LingangxinchengDistrict two potential ELCs are planned to serve FengxianDistrict and two potential ELCs are expected to serve theJinshanDistrictWe assume that a total of thirteen areas (DP)are seriously affected and need emergency rescue urgently inwhich six affected areas are in Lingangxincheng four affectedareas are in Fengxian District and three affected areas are inJinshan District

For simplicity only one type of emergency supply resour-ces will be considered in the queuing network We assumethat the travel speed is fixed to be equal to 40 miles per hourfor all the vehicles travelling between SP and ESC Howeverthe travel speed between ESC and DP is assumed to be 30miles per hour Let us adopt the hypothesis as the abovethat the occurrence time of a relief request follows a negativeexponential distribution with intensify 120575 and the quantity ofresources in every relief request is uniformly distributed withintensify 119891

119881(V) that the lower bound (LB) is 119888 and the upper

bound (UB) is 119889 (119889 gt 119888) The disaster-related informationincluding the population of the affected areas and the distancebetween supply point and emergency logistics center anddistance between emergency logistics center and demandpoint are shown in Tables 1 2 3 and 4 respectively

The Scientific World Journal 9

Table 1 Population and demand data of the affected areas

Affected area Population (119888 119889) 120575

Area 1 (Zhuqiaozhen) 105807 (4 12) 16Area 2 (Huinanzhen) 213845 (4 12) 16Area 3 (Laogangzhen) 37408 (5 14) 18Area 4 (Datuanzhen) 71162 (4 12) 16Area 5 (Xingangzhen) 21475 (6 16) 20Area 6 (Shuyuanzhen) 59831 (6 16) 20Area 7 (Luchaogangzhen) 27850 (6 16) 20Area 8 (Situanzhen) 65389 (5 14) 18Area 9 (Qingcunzhen) 89163 (5 14) 18Area 10 (Nanqiaozhen) 361185 (3 10) 14Area 11 (Zhuanghangzhen) 62388 (3 10) 14Area 12 (Zhelinzhen) 62589 (3 10) 14Area 13 (Caojingzhen) 40722 (2 8) 12Area 14 (Shanyangzhen) 84640 (2 8) 12Area 15 (Jinshanzhen) 70815 (2 8) 12

0 20 40 60 80 100 120 140 160 180 20020

25

30

35

40

45

50

55

60

Interaction number

Tota

l res

pons

e tim

e (ho

urs)

Total response time

Figure 5 Optimization of the fitness function

In this case study the fixed cost to locate the emergencylogistics center is assumed as 119886

1= 85 119886

2= 100 119886

3= 90

1198864= 85 119886

5= 95 119886

6= 110 119886

7= 70 119886

8= 80 119886

9= 115

and 11988610

= 120 The total cost to build the emergency logisticscenters is assumednot exceeding to be119861 = 500 For simplicityof the computation we presume that 119902

1= 5 119902

2= 2 and

1199023= 3According to the background of the above problem a

simplified 10 times 10 times 15 emergency supply chain network isdeveloped in the assumed scenario where the geographicalrelationships among these supply depots emergency logisticscenters and demand points are specified Then we applythe aforementioned model into the emergency supply chainnetwork system And the GA-based approach demonstratedbefore is coded to verify the validity of the model inMatlab2009a

The optimization process of the fitness function isdepicted in Figure 5 where the optimal response time of thequeuing network is 2071 hours

Therefore the average response time of each demandpoint is 138 h which is within a reasonable range The cal-culation result shows that this model can play an importantrole in the emergency supply chain management Differentfrom the general supply chain management the timesavingis the most significant feature of the emergency supply chainsystem The efficient allocation of emergency resources todemand points improves not only the operational perfor-mance of the emergency supply chain system in the emer-gency supply side but also the affected peoplersquos survival ratein the disaster areas From the psychology of point of viewquick responses and deliveries for disaster relief not onlymay improve the governmental efforts to rescue but alsostrengthen the affected peoplersquos willpower to be alive hencestabilizing the affected areas

The emergency supply chain system is composed of threemembers which have different functions in the relief supplydelivery process And the upstream nodes and downstreamnodes in the ESC system are connected by roads If we treateach node in the system as the node in a network considerthe roads as the arcs of the network and treat the length ofeach road as the weight of each arc then the ESC system canbe seen as a directed network diagram with weights And theresponse time of the system includes two parts the first partis the transportation time of the relief supply request on theroads and the second part is the sojourn time of the reliefsupply request in each node

Figure 6 indicates the values of the above two componentsof response time with the total transportation time being1664 h and the total sojourn time being 407 h This iterativeprocess also demonstrates the convergence of the heuristicalgorithm

Queuing models are effective tools for obtaining infor-mation about important performance of each facility in theESC system like sojourn times waiting times queue lengthstotal customers in the system and so on In order to analyzethe efficiency of facilities in the system we calculate thetotal customers and the queue lengths which are shown inFigure 7 We can see that the solution of the above model willnot create queuing delays of relief supply so that the lossesgenerated by the queuing delay can be avoided This iterativeprocess also illustrates the robustness and the potentialapplicability of our model

Based on the built network the coordination between theupstream node and the downstream node in this system canbe enforced and the performance of basic activities includingproducing sorting processing packing and delivering canbe improved tominimize the response time of the systemTheemergency logistics centers for the system should be locatedin DP(6) DP(7) DP(8) DP(9) and DP(10) And the optimalpaths are SP(1) SP(6) rarr ELC(8) rarr DP(1) DP(2)DP(9) SP(2) SP(4) rarr ELC(6) rarr DP(13) DP(14)DP(15) SP(3) SP(8) rarr ELC(7) rarr DP(10) DP(11)DP(12) SP(5) SP(9) rarr ELC(10) rarr DP(3) DP(4)DP(5) and SP(7) SP(9) rarr ELC(9) rarr DP(6) DP(7)DP(8) Overall the examination reveals that the proposedmethod can be a useful tool for a quick response to the urgentneed of relief in the large-scale disaster-affected areas

10 The Scientific World Journal

Table 2 The service rates specifications of different servers in the network

SP 120583101584010158401

120583101584010158402

120583101584010158403

120583101584010158404

120583101584010158405

120583101584010158406

120583101584010158407

120583101584010158408

120583101584010158409

1205831015840101584010

10 9 11 8 11 12 14 16 14 17

ELC 12058310158401

12058310158402

12058310158403

12058310158404

12058310158405

12058310158406

12058310158407

12058310158408

12058310158409

120583101584010

16 18 11 10 11 12 14 16 14 10

DP 1205831

1205832

1205833

1205834

1205835

1205836

1205837

1205838

1205839

12058310

12058311

12058312

12058313

12058314

12058315

8 8 8 8 10 9 11 8 9 12 9 8 12 14 13

Table 3 The distance between supply point and emergency logistics center

SP(1) SP(2) SP(3) SP(4) SP(5) SP(6) SP(7) SP(8) SP(9) SP(10)ELC(1) 1828 1707 1197 2163 614 826 1091 973 1530 879ELC(2) 3073 2147 1624 571 2039 2056 802 1679 2554 1390ELC(3) 2271 2288 1712 1622 1419 1463 998 1102 1842 756ELC(4) 4422 3095 2462 1264 3971 3492 2429 3276 4153 3003ELC(5) 3962 2801 2716 1507 3256 2769 1449 2551 3325 2006ELC(6) 3585 2356 1923 1647 2784 2313 989 2136 2839 1437ELC(7) 2973 2968 2364 2301 2036 2149 1275 1797 2575 1471ELC(8) 2308 3261 2878 2892 1494 1401 1764 1132 1961 1933ELC(9) 2584 3643 3303 3249 2015 1165 2258 941 1600 2419ELC(10) 3038 4666 4240 4139 2831 2039 3008 1742 1405 3105

05

101520253035404550

Tim

e (ho

urs)

Total transportation timeTotal sojourn time

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 6 Optimization of total transportation time and totalsojourn time

6 Conclusions

An emergency supply chain (ESC) system is developed inthis paper for the quick response to the urgent need ofrelief in the large-scale disaster-affected areas which mainlyconsists of three chain members in series supply points(SP) emergency logistics centers (ELCs) and demand points(DP) As the timeliness is the most crucial characteristic ofthe ESC system and queuing theory is a useful tool in theanalysis of the performance of the time-dependent systemwe propose queuingmodeling for the ESC system to optimizethe emergency rescue process Based on the above queuingnetwork system a queuing minimal response time location-allocation model is established to decide the selection ofemergency logistics centers and vehicle routing decision

0

5

10

15

20

25

Total customersQueuing length

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 7 Optimization of the total customers and the queue length

For the complexity of mathematical model the GA-basedapproach is introduced to solve the model

A case study with an assumed severe typhoon striking thesoutheast coast of Shanghai with a great probability withinthe next 100 years is conducted to assess and validate ourmodel The optimal response time of the queuing network is2071 hours and the average response time of each demandpoint is 138 h which is within a reasonable range Theoptimal emergency logistics centers for the system are DP(6)DP(7) DP(8) DP(9) and DP(10) Besides the performanceof queuing modeling for the ESC system illustrates therobustness and the potential applicability of our model

For future research we suggest the following threedirections Firstly we will investigate the queuing modelingunder consideration of the facility blocking problems and

The Scientific World Journal 11

Table 4 The distance between emergency logistics center and demand point

ELC(1) ELC(2) ELC(3) ELC(4) ELC(5) ELC(6) ELC(7) ELC(8) ELC(9) ELC(10)DP(1) 3761 4191 3253 4599 3801 3752 2439 2397 1553 739DP(2) 4271 4702 3761 4794 3945 4152 2818 2822 1898 1075DP(3) 4508 4886 3986 5028 4501 4384 3046 3025 2117 1356DP(4) 4364 4579 3697 4824 4307 4138 2798 2940 1956 1710DP(5) 5283 5416 4386 5466 5435 4986 3836 3271 2693 1917DP(6) 5214 5721 4866 5896 5294 5207 3797 4221 3252 2553DP(7) 5334 5776 5299 5862 5359 5164 3871 3851 2955 2554DP(8) 4689 4581 3763 4983 4306 4173 2818 2997 2200 1968DP(9) 3309 3624 2657 3834 3334 3154 1886 3154 3452 3235DP(10) 2678 3263 1999 2911 2741 2518 1198 2505 2360 3037DP(11) 3369 2912 2386 2349 2182 1858 1894 2392 2915 3596DP(12) 3376 3794 2541 3287 3275 2612 1708 3062 2896 3256DP(13) 3691 3419 2987 2896 2694 2279 2162 3500 3275 4007DP(14) 4230 3273 3402 3011 2555 2141 2606 3906 3749 4448DP(15) 4609 3765 3858 2533 3054 2680 3166 4390 4249 4897

demand priority for the emergency relief points Secondly wewill study the generation of emergency demand and servicetime obeying the other random distributions Thirdly moreefficient multiobjective optimization methodologies for boththe above problem and emergency resources transportationscheduling problem will be considered

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grant nos 71102043 and 71371145) theMinistry of Education of China (Grant no MCM20125021)the Innovative Program of Shanghai Municipal EducationCommission (Grant no 14ZS123) and the ShanghaiMaritimeUniversity (Grant no 20120054)

References

[1] D Guha-Sapir G S Debarati D Hargitt P Hoyois R Belowand D BrechetThirty Years of Natural Disasters 1974ndash2003 theNumbers Presses universitaires de Louvain 2004

[2] J-B Sheu ldquoAn emergency logistics distribution approach forquick response to urgent relief demand in disastersrdquo Trans-portation Research E vol 43 no 6 pp 687ndash709 2007

[3] B S Manoj and A H Baker ldquoCommunication challenges inemergency responserdquo Communications of the ACM vol 50 no3 pp 51ndash53 2007

[4] B Balcik B M Beamon and K Smilowitz ldquoLast mile distribu-tion in humanitarian reliefrdquo Journal of Intelligent TransportationSystems vol 12 no 2 pp 51ndash63 2008

[5] A M Caunhye X F Nie and S Pokharel ldquoOptimizationmodels in emergency logistics a literature reviewrdquo Socio-Economic Planning Sciences vol 46 no 1 pp 4ndash13 2012

[6] S Coles and L Pericchi ldquoAnticipating catastrophes throughextreme valuemodellingrdquo Journal of the Royal Statistical SocietyC vol 52 no 4 pp 405ndash416 2003

[7] J R Artalejo ldquoG-networks a versatile approach for workremoval in queueing networksrdquo European Journal of Opera-tional Research vol 126 no 2 pp 233ndash249 2000

[8] W Jiang L Deng L Chen J Wu and J Li ldquoRisk assessmentand validation of flood disaster based on fuzzy mathematicsrdquoProgress in Natural Science vol 19 no 10 pp 1419ndash1425 2009

[9] V Bhaskar and P Lallement ldquoModeling a supply chain usinga network of queuesrdquo Applied Mathematical Modelling vol 34no 8 pp 2074ndash2088 2010

[10] B M Beamon ldquoHumanitarian relief chains issues and chal-lengesrdquo in Proceedings of the 34th International Conference onComputers and Industrial Engineering San Francisco CalifUSA November 2004

[11] R Oloruntoba and R Gray ldquoHumanitarian aid an agile supplychainrdquo Supply Chain Management vol 11 no 2 pp 115ndash1202006

[12] E J Lodree Jr and S Taskin ldquoSupply chain planning for hur-ricane response with wind speed information updatesrdquo Com-puters and Operations Research vol 36 no 1 pp 2ndash15 2009

[13] V Bhakoo P Singh and A Sohal ldquoCollaborative managementof inventory in Australian hospital supply chains practices andissuesrdquo Supply Chain Management vol 17 no 2 pp 217ndash2302012

[14] W Yi and A Kumar ldquoAnt colony optimization for disaster reliefoperationsrdquo Transportation Research E vol 43 no 6 pp 660ndash672 2007

[15] G-H Tzeng H-J Cheng and T D Huang ldquoMulti-objectiveoptimal planning for designing relief delivery systemsrdquo Trans-portation Research E vol 43 no 6 pp 673ndash686 2007

[16] Y-C Chiu and H Zheng ldquoReal-time mobilization decisionsfor multi-priority emergency response resources and evacua-tion groups model formulation and solutionrdquo TransportationResearch E vol 43 no 6 pp 710ndash736 2007

[17] N Altay andWG Green III ldquoORMS research in disaster oper-ations managementrdquo European Journal of Operational Researchvol 175 no 1 pp 475ndash493 2006

12 The Scientific World Journal

[18] K J Cormican D P Morton and R K Wood ldquoStochasticnetwork interdictionrdquo Operations Research vol 46 no 2 pp184ndash197 1998

[19] HOMete andZ B Zabinsky ldquoStochastic optimization ofmed-ical supply location and distribution in disaster managementrdquoInternational Journal of Production Economics vol 126 no 1 pp76ndash84 2010

[20] H Shavandi and H Mahlooji ldquoA fuzzy queuing locationmodel with a genetic algorithm for congested systemsrdquo AppliedMathematics and Computation vol 181 no 1 pp 440ndash4562006

[21] R D Galvao and R Morabito ldquoEmergency service systemsthe use of the hypercube queueing model in the solution ofprobabilistic location problemsrdquo International Transactions inOperational Research vol 15 no 5 pp 525ndash549 2008

[22] N Geroliminis M G Karlaftis and A Skabardonis ldquoA spatialqueuing model for the emergency vehicle districting andlocation problemrdquo Transportation Research B vol 43 no 7 pp798ndash811 2009

[23] C M Hosage and M F Goodchild ldquoDiscrete space location-allocation solutions from genetic algorithmsrdquo Annals of Opera-tions Research vol 6 no 2 pp 35ndash46 1986

[24] P Chakroborty ldquoGenetic algorithms for optimal urban transitnetwork designrdquo Computer-Aided Civil and Infrastructure Engi-neering vol 18 no 3 pp 184ndash200 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 The Scientific World Journal

variable with intensify 119891119881(V) between 119888 and 119889 (119889 gt 119888) And

119891119880(119906) and 119891

119881(V) are independent Thus

119891119880(119906) =

120575119890minus120575119906 119906 ge 0

0 119906 lt 0

119891119881(V) =

1

119889 minus 119888 119888 lt V lt 119889

0 otherwise

119864 (119882) = 119864 (119880119881) = 119864 (119880) 119864 (119881) =119888 + 119889

2120575

(17)

Therefore the interarrival times of the emergencydemand (occurrence and quantity) follow a negative expo-nential distribution with intensity 120582 equal to 1119864(119882) Theservice rate at each node in the queuing network system isan independent identically distributed random variable withintensity 120583 and the service time is 1120583

So the interarrival time 120582 and the traffic intensity of thesystem 120588 are represented as

120582 =1

119864 (119882)=

2120575

119888 + 119889 (18)

120588 =120582

120583=

1

120583119864 (119882)=

2120575

120583 (119888 + 119889) (19)

Let us assume that there exits just one server at eachservice node and the servers are independent which meansthat the queuing model at each server is an 1198721198721 Thenthe probability distribution function of the sojourn timeWT (defined as the waiting time plus the service time for acustomer) in an1198721198721 queue can be presented as

119891WT (119905) = (120583 minus 120582) 119890minus(120583minus120582)119905

(20)

From (7) the cumulative distribution function of WT is

119891WT (119905) = 119875 (WT le 119905)

= int119905

0

(120583 minus 120582) 119890minus(120583minus120582)119905

119889119905 = 1 minus 119890minus(120583minus120582)119905

(21)

The average sojourn time WT is given by

WT = 119864 (WT) = intinfin

0

119891WT (119905) 119905 119889119905

= intinfin

0

(120583 minus 120582) 119890minus(120583minus120582)119905

119905 119889119905 =1

120583 minus 120582

(22)

Let WT119902denote the waiting time in the queue The

average waiting time is computed as

WT119902= WT minus

1

120583=

120582

120583 (120583 minus 120582) (23)

From the well-known Littlersquos theorem the average cus-tomers LR in the system including the number of customersboth waiting in the queue and served in the server is given by

LR = 120582WT =120582

120583 minus 120582 (24)

And LR119902denote the queuing length in the system which

is presented as

LR119902= 120582WT

119902=

1205822

120583 (120583 minus 120582) (25)

From (17) (16) (18) (22) (23) (24) and (25) theaverage sojourn time waiting time in the queue the averagecustomers including the number of customers both waitingin the queue and served in the server and the queuing lengthfor the ESC network system are given as

WTsys = sum119903isin119877

sum119894isin119868

WT119877119868+ sum119903isin119877

sum119895isin119869

WT119877119869+ sum119903isin119877

sum119896isin119870

WT119877119896

= sum119903isin119877

sum119894isin119868

1

12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

+ sum119903isin119877

sum119895isin119869

1

1205831015840 minus119899

sum119896=1

(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119896

1

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

WT119902sys = sum

119903isin119877

sum119894isin119868

WT119877119902119868

+ sum119903isin119877

sum119895isin119869

WT119877119902119869+ sum119903isin119877

sum119896isin119870

WT119877119902119896

= sum119903isin119877

sum119894isin119868

(11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

12058310158401015840 (12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)

+ sum119903isin119877

sum119895isin119869

sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

1205831015840 (1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)

+ sum119903isin119877

sum119896isin119896

2120575119903119896 (119888119903119896+ 119888119903119896)

120583 (120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896)))

LRsys = sum119903isin119877

sum119894isin119868

LR119877119868

+ sum119903isin119877

sum119895isin119869

LR119877119869+ sum119903isin119877

sum119896isin119870

LR119877119896

= sum119903isin119877

sum119894isin119868

(11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896

+ sum119903isin119877

sum119895isin119869

sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119870

2120575119903119896 (119888119903119896+ 119888119903119896)

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

The Scientific World Journal 7

LR119902sys

= sum119903isin119877

sum119894isin119868

LR119877119902119868+ sum119903isin119877

sum119895isin119869

LR119877119902119869+ sum119903isin119877

sum119896isin119870

LR119877119902119896

= sum119903isin119877

sum119894isin119868

((11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)2

12058310158401015840 (12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)

+ sum119903isin119877

sum119895isin119869

(sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)2

1205831015840 (1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)

+ sum119903isin119877

sum119896isin119896

(2120575119903

119896 (119888119903

119896+ 119888119903

119896))2

120583 (120583 minus 2120575119903119896 (119888119903119896+ 119888119903119896))

(26)

As the response time includes not only the sojourn time ofemergency demand in the queuing network system but alsothe transportation time therefore when the queuing theoryis applied to the present model the above objective functioncan be presented as follows

min119885

= minsum119903isin119877

sum119894isin119868

sum119895isin119869

sum119896isin119870

119905119903

119894119895119896

= min(sum119903isin119877

sum119894isin119868

(1)

times (12058310158401015840minus (

1

1199022

)

ℎ

sum119895=1

119899

sum119896=1

(2120575119903

119896 (119888119903

119896+ 119888119903

119896)) 119910119894119895119911119895119896)

minus1

+ sum119903isin119877

sum119895isin119869

1

1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119896

1

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

+sum119903isin119877

sum119894isin119868

sum119895isin119869

TR119903119894119895119910119894119895+ sum119903isin119877

sum119895isin119869

sum119896isin119870

TR119903119895119896119911119895119896)

(27)

43 GA-Based Approach As the facility location-allocationproblem is NP- (nondeterministic polynomial-) hard wherethe proposed 0-1 nonlinear integer programming model inthis study is used some heuristic algorithms are requiredto solve the problem quickly The genetic algorithms (GAs)have been applied to location problems since 1986 by Hosageand Goodchild [23] GA has exhibited inherent advantagessuch as their robust performancewhen solving combinatorialproblems as well as their ability to incorporate proceduresand logical conditions and to handle discrete variables andnonlinear constraints in a straightforward manner [24]These qualities make GAs particularly attractive for potentialcombination with other methods and external procedures

Wedevelop a refined genetic algorithm to solve the aforemen-tioned problem as indicated belowThe associated frameworkfor embedding the GA with emergency queue network ispresented in Figure 3

431 Methodology of Genetic Algorithm In a genetic algo-rithm a population of candidate solutions to an optimizationproblem is evolved toward better solutions Each candi-date solution has a set of properties (its chromosomes orgenotype) which can be mutated and altered solutions arerepresented in binary as strings of 0 s and 1 s but otherencodings are also possible The evolution usually startsfrom a population of randomly generated individuals andhappens in generations In each generation the fitness ofevery individual in the population is evaluated the more fitindividuals are stochastically selected from the current popu-lation and each individualrsquos genome ismodified (recombinedand possibly randomly mutated) to form a new populationThe new population is then used in the next iteration of thealgorithm Commonly the algorithm terminates when eithera maximum number of generations have been produced or asatisfactory fitness level has been reached for the populationOnce the genetic representation and the fitness function aredefined a GA proceeds to initialize a population of solutionsand then to improve it through repetitive application of themutation crossover inversion and selection operators

432 Procedure of the Refined Genetic Algorithm

Step 1 (encoding) We suppose that the genotype of a chro-mosome with some genes indicates an individual corre-sponding to the location pattern of emergency logisticscenters and paths between nodes in the system network Thegene presented by 1 indicates that an emergency logisticscenter is built at this location or the path between nodes in thesystem network is selectedThe gene presented by 0 indicatesthat no emergency logistics centers is built there or the pathbetween nodes in the system network which is selected is notselected 119873 chromosomes are randomly generated and theinitial population (Pop) is set

Step 2 (fitness evaluation) Once the initial population isavailable evaluation of each chromosome takes place to ex-plore the quality of the corresponding solution For individ-uals with larger fitness value to be transmitted to the nextgeneration with higher probability the chromosome fitnessfunction in this paper can be described as follows and 119888max isthe maximum estimation value of min119885

Fit (min119885) = 119862max minusmin119885 if min119885 lt 119862max

0 otherwise(28)

Step 3 (crossover selection and mutation) 119873 chromosomesare crossed according to a single pointmethodwith crossoverprobability (which is 119875

119888= 09) to generate the offspring

chromosomes There are a total of 2119873 chromosomes forboth parent and offspring chromosomes Then the duplicatechromosomes are deleted to produce the new population(Pop) Select the first 119873 chromosomes of the population on

8 The Scientific World Journal

Step 1 encoding forparameter sets

Initialization ofgenerate population

Step 2 chromosomefitness evaluation

Step 3 geneticmanipulation crossoverselection and mutation

Step 4 offspringgeneration Select the best individual

and output the solution

Yes

No Terminate

(1) Decoding bit string to the parameter(2) Objective function value calculation(3) Mapping function value to the fitness value(4) Fitness value adjustments

Preparation of dataand parameters

Figure 3 Steps of the proposed GA heuristic

(B)(C)

(A)

Figure 4 The location of facilities in an ESC system

the basis of the fitness values in ascending order When thenumber of chromosomes in the new population (Pop) is lessthan 119873 (119873 = 200) 119873-|Pop| chromosomes are randomlygenerated (|Pop| is the number of chromosomes in thepopulation (Pop)) Select a chromosome randomly and applythe mutation operator applied on it with probability 119875

119898=

001

Step 4 Choose new chromosomes by the roulette wheelmethod Repeat Steps 3 and 4 until there are no solutionimprovements within a specified number of iterations

5 A Case Study

A case study example is presented in this section to illustratethe efficiency of our approach to realize the quick-responsiveemergency relief during large-scale natural disastersThepro-posed method is implemented on a scenario where a severetyphoon strikes the southeast coast of Shanghai with a greatprobability within the next 100 years It is assumed that thereare three regions (A B and C) slashed by the typhoon shownin Figure 4 One region is Lingangxincheng District (A)

another one is Fengxian District (B) and the third one isJinshan District (C)

There are ten supply points (SP) (1) disaster relief sup-plies reserve center in Baoshan District (2) Automobile Cityin Jiading (3) Red Cross disaster preparedness warehouse inJiading District (4) disaster relief supplies reserve center inQingqu District (5) disaster relief supplies reserve center inPutuo District (6) disaster relief supplies reserve center inYangpu District (7) disaster relief supplies reserve center inMinhang District (8) disaster relief supplies reserve center inHongkou District (9) disaster relief supplies reserve centerin Pudong District and (10) disaster relief supplies reservecenter in Xuhui District Based on the study of large-scalelogistics center in Shanghai we select ten potential pointsas the emergency logistics centers (ELCs) Four potentialELCs are prepared for the affected areas in LingangxinchengDistrict two potential ELCs are planned to serve FengxianDistrict and two potential ELCs are expected to serve theJinshanDistrictWe assume that a total of thirteen areas (DP)are seriously affected and need emergency rescue urgently inwhich six affected areas are in Lingangxincheng four affectedareas are in Fengxian District and three affected areas are inJinshan District

For simplicity only one type of emergency supply resour-ces will be considered in the queuing network We assumethat the travel speed is fixed to be equal to 40 miles per hourfor all the vehicles travelling between SP and ESC Howeverthe travel speed between ESC and DP is assumed to be 30miles per hour Let us adopt the hypothesis as the abovethat the occurrence time of a relief request follows a negativeexponential distribution with intensify 120575 and the quantity ofresources in every relief request is uniformly distributed withintensify 119891

119881(V) that the lower bound (LB) is 119888 and the upper

bound (UB) is 119889 (119889 gt 119888) The disaster-related informationincluding the population of the affected areas and the distancebetween supply point and emergency logistics center anddistance between emergency logistics center and demandpoint are shown in Tables 1 2 3 and 4 respectively

The Scientific World Journal 9

Table 1 Population and demand data of the affected areas

Affected area Population (119888 119889) 120575

Area 1 (Zhuqiaozhen) 105807 (4 12) 16Area 2 (Huinanzhen) 213845 (4 12) 16Area 3 (Laogangzhen) 37408 (5 14) 18Area 4 (Datuanzhen) 71162 (4 12) 16Area 5 (Xingangzhen) 21475 (6 16) 20Area 6 (Shuyuanzhen) 59831 (6 16) 20Area 7 (Luchaogangzhen) 27850 (6 16) 20Area 8 (Situanzhen) 65389 (5 14) 18Area 9 (Qingcunzhen) 89163 (5 14) 18Area 10 (Nanqiaozhen) 361185 (3 10) 14Area 11 (Zhuanghangzhen) 62388 (3 10) 14Area 12 (Zhelinzhen) 62589 (3 10) 14Area 13 (Caojingzhen) 40722 (2 8) 12Area 14 (Shanyangzhen) 84640 (2 8) 12Area 15 (Jinshanzhen) 70815 (2 8) 12

0 20 40 60 80 100 120 140 160 180 20020

25

30

35

40

45

50

55

60

Interaction number

Tota

l res

pons

e tim

e (ho

urs)

Total response time

Figure 5 Optimization of the fitness function

In this case study the fixed cost to locate the emergencylogistics center is assumed as 119886

1= 85 119886

2= 100 119886

3= 90

1198864= 85 119886

5= 95 119886

6= 110 119886

7= 70 119886

8= 80 119886

9= 115

and 11988610

= 120 The total cost to build the emergency logisticscenters is assumednot exceeding to be119861 = 500 For simplicityof the computation we presume that 119902

1= 5 119902

2= 2 and

1199023= 3According to the background of the above problem a

simplified 10 times 10 times 15 emergency supply chain network isdeveloped in the assumed scenario where the geographicalrelationships among these supply depots emergency logisticscenters and demand points are specified Then we applythe aforementioned model into the emergency supply chainnetwork system And the GA-based approach demonstratedbefore is coded to verify the validity of the model inMatlab2009a

The optimization process of the fitness function isdepicted in Figure 5 where the optimal response time of thequeuing network is 2071 hours

Therefore the average response time of each demandpoint is 138 h which is within a reasonable range The cal-culation result shows that this model can play an importantrole in the emergency supply chain management Differentfrom the general supply chain management the timesavingis the most significant feature of the emergency supply chainsystem The efficient allocation of emergency resources todemand points improves not only the operational perfor-mance of the emergency supply chain system in the emer-gency supply side but also the affected peoplersquos survival ratein the disaster areas From the psychology of point of viewquick responses and deliveries for disaster relief not onlymay improve the governmental efforts to rescue but alsostrengthen the affected peoplersquos willpower to be alive hencestabilizing the affected areas

The emergency supply chain system is composed of threemembers which have different functions in the relief supplydelivery process And the upstream nodes and downstreamnodes in the ESC system are connected by roads If we treateach node in the system as the node in a network considerthe roads as the arcs of the network and treat the length ofeach road as the weight of each arc then the ESC system canbe seen as a directed network diagram with weights And theresponse time of the system includes two parts the first partis the transportation time of the relief supply request on theroads and the second part is the sojourn time of the reliefsupply request in each node

Figure 6 indicates the values of the above two componentsof response time with the total transportation time being1664 h and the total sojourn time being 407 h This iterativeprocess also demonstrates the convergence of the heuristicalgorithm

Queuing models are effective tools for obtaining infor-mation about important performance of each facility in theESC system like sojourn times waiting times queue lengthstotal customers in the system and so on In order to analyzethe efficiency of facilities in the system we calculate thetotal customers and the queue lengths which are shown inFigure 7 We can see that the solution of the above model willnot create queuing delays of relief supply so that the lossesgenerated by the queuing delay can be avoided This iterativeprocess also illustrates the robustness and the potentialapplicability of our model

Based on the built network the coordination between theupstream node and the downstream node in this system canbe enforced and the performance of basic activities includingproducing sorting processing packing and delivering canbe improved tominimize the response time of the systemTheemergency logistics centers for the system should be locatedin DP(6) DP(7) DP(8) DP(9) and DP(10) And the optimalpaths are SP(1) SP(6) rarr ELC(8) rarr DP(1) DP(2)DP(9) SP(2) SP(4) rarr ELC(6) rarr DP(13) DP(14)DP(15) SP(3) SP(8) rarr ELC(7) rarr DP(10) DP(11)DP(12) SP(5) SP(9) rarr ELC(10) rarr DP(3) DP(4)DP(5) and SP(7) SP(9) rarr ELC(9) rarr DP(6) DP(7)DP(8) Overall the examination reveals that the proposedmethod can be a useful tool for a quick response to the urgentneed of relief in the large-scale disaster-affected areas

10 The Scientific World Journal

Table 2 The service rates specifications of different servers in the network

SP 120583101584010158401

120583101584010158402

120583101584010158403

120583101584010158404

120583101584010158405

120583101584010158406

120583101584010158407

120583101584010158408

120583101584010158409

1205831015840101584010

10 9 11 8 11 12 14 16 14 17

ELC 12058310158401

12058310158402

12058310158403

12058310158404

12058310158405

12058310158406

12058310158407

12058310158408

12058310158409

120583101584010

16 18 11 10 11 12 14 16 14 10

DP 1205831

1205832

1205833

1205834

1205835

1205836

1205837

1205838

1205839

12058310

12058311

12058312

12058313

12058314

12058315

8 8 8 8 10 9 11 8 9 12 9 8 12 14 13

Table 3 The distance between supply point and emergency logistics center

SP(1) SP(2) SP(3) SP(4) SP(5) SP(6) SP(7) SP(8) SP(9) SP(10)ELC(1) 1828 1707 1197 2163 614 826 1091 973 1530 879ELC(2) 3073 2147 1624 571 2039 2056 802 1679 2554 1390ELC(3) 2271 2288 1712 1622 1419 1463 998 1102 1842 756ELC(4) 4422 3095 2462 1264 3971 3492 2429 3276 4153 3003ELC(5) 3962 2801 2716 1507 3256 2769 1449 2551 3325 2006ELC(6) 3585 2356 1923 1647 2784 2313 989 2136 2839 1437ELC(7) 2973 2968 2364 2301 2036 2149 1275 1797 2575 1471ELC(8) 2308 3261 2878 2892 1494 1401 1764 1132 1961 1933ELC(9) 2584 3643 3303 3249 2015 1165 2258 941 1600 2419ELC(10) 3038 4666 4240 4139 2831 2039 3008 1742 1405 3105

05

101520253035404550

Tim

e (ho

urs)

Total transportation timeTotal sojourn time

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 6 Optimization of total transportation time and totalsojourn time

6 Conclusions

An emergency supply chain (ESC) system is developed inthis paper for the quick response to the urgent need ofrelief in the large-scale disaster-affected areas which mainlyconsists of three chain members in series supply points(SP) emergency logistics centers (ELCs) and demand points(DP) As the timeliness is the most crucial characteristic ofthe ESC system and queuing theory is a useful tool in theanalysis of the performance of the time-dependent systemwe propose queuingmodeling for the ESC system to optimizethe emergency rescue process Based on the above queuingnetwork system a queuing minimal response time location-allocation model is established to decide the selection ofemergency logistics centers and vehicle routing decision

0

5

10

15

20

25

Total customersQueuing length

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 7 Optimization of the total customers and the queue length

For the complexity of mathematical model the GA-basedapproach is introduced to solve the model

A case study with an assumed severe typhoon striking thesoutheast coast of Shanghai with a great probability withinthe next 100 years is conducted to assess and validate ourmodel The optimal response time of the queuing network is2071 hours and the average response time of each demandpoint is 138 h which is within a reasonable range Theoptimal emergency logistics centers for the system are DP(6)DP(7) DP(8) DP(9) and DP(10) Besides the performanceof queuing modeling for the ESC system illustrates therobustness and the potential applicability of our model

For future research we suggest the following threedirections Firstly we will investigate the queuing modelingunder consideration of the facility blocking problems and

The Scientific World Journal 11

Table 4 The distance between emergency logistics center and demand point

ELC(1) ELC(2) ELC(3) ELC(4) ELC(5) ELC(6) ELC(7) ELC(8) ELC(9) ELC(10)DP(1) 3761 4191 3253 4599 3801 3752 2439 2397 1553 739DP(2) 4271 4702 3761 4794 3945 4152 2818 2822 1898 1075DP(3) 4508 4886 3986 5028 4501 4384 3046 3025 2117 1356DP(4) 4364 4579 3697 4824 4307 4138 2798 2940 1956 1710DP(5) 5283 5416 4386 5466 5435 4986 3836 3271 2693 1917DP(6) 5214 5721 4866 5896 5294 5207 3797 4221 3252 2553DP(7) 5334 5776 5299 5862 5359 5164 3871 3851 2955 2554DP(8) 4689 4581 3763 4983 4306 4173 2818 2997 2200 1968DP(9) 3309 3624 2657 3834 3334 3154 1886 3154 3452 3235DP(10) 2678 3263 1999 2911 2741 2518 1198 2505 2360 3037DP(11) 3369 2912 2386 2349 2182 1858 1894 2392 2915 3596DP(12) 3376 3794 2541 3287 3275 2612 1708 3062 2896 3256DP(13) 3691 3419 2987 2896 2694 2279 2162 3500 3275 4007DP(14) 4230 3273 3402 3011 2555 2141 2606 3906 3749 4448DP(15) 4609 3765 3858 2533 3054 2680 3166 4390 4249 4897

demand priority for the emergency relief points Secondly wewill study the generation of emergency demand and servicetime obeying the other random distributions Thirdly moreefficient multiobjective optimization methodologies for boththe above problem and emergency resources transportationscheduling problem will be considered

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grant nos 71102043 and 71371145) theMinistry of Education of China (Grant no MCM20125021)the Innovative Program of Shanghai Municipal EducationCommission (Grant no 14ZS123) and the ShanghaiMaritimeUniversity (Grant no 20120054)

References

[1] D Guha-Sapir G S Debarati D Hargitt P Hoyois R Belowand D BrechetThirty Years of Natural Disasters 1974ndash2003 theNumbers Presses universitaires de Louvain 2004

[2] J-B Sheu ldquoAn emergency logistics distribution approach forquick response to urgent relief demand in disastersrdquo Trans-portation Research E vol 43 no 6 pp 687ndash709 2007

[3] B S Manoj and A H Baker ldquoCommunication challenges inemergency responserdquo Communications of the ACM vol 50 no3 pp 51ndash53 2007

[4] B Balcik B M Beamon and K Smilowitz ldquoLast mile distribu-tion in humanitarian reliefrdquo Journal of Intelligent TransportationSystems vol 12 no 2 pp 51ndash63 2008

[5] A M Caunhye X F Nie and S Pokharel ldquoOptimizationmodels in emergency logistics a literature reviewrdquo Socio-Economic Planning Sciences vol 46 no 1 pp 4ndash13 2012

[6] S Coles and L Pericchi ldquoAnticipating catastrophes throughextreme valuemodellingrdquo Journal of the Royal Statistical SocietyC vol 52 no 4 pp 405ndash416 2003

[7] J R Artalejo ldquoG-networks a versatile approach for workremoval in queueing networksrdquo European Journal of Opera-tional Research vol 126 no 2 pp 233ndash249 2000

[8] W Jiang L Deng L Chen J Wu and J Li ldquoRisk assessmentand validation of flood disaster based on fuzzy mathematicsrdquoProgress in Natural Science vol 19 no 10 pp 1419ndash1425 2009

[9] V Bhaskar and P Lallement ldquoModeling a supply chain usinga network of queuesrdquo Applied Mathematical Modelling vol 34no 8 pp 2074ndash2088 2010

[10] B M Beamon ldquoHumanitarian relief chains issues and chal-lengesrdquo in Proceedings of the 34th International Conference onComputers and Industrial Engineering San Francisco CalifUSA November 2004

[11] R Oloruntoba and R Gray ldquoHumanitarian aid an agile supplychainrdquo Supply Chain Management vol 11 no 2 pp 115ndash1202006

[12] E J Lodree Jr and S Taskin ldquoSupply chain planning for hur-ricane response with wind speed information updatesrdquo Com-puters and Operations Research vol 36 no 1 pp 2ndash15 2009

[13] V Bhakoo P Singh and A Sohal ldquoCollaborative managementof inventory in Australian hospital supply chains practices andissuesrdquo Supply Chain Management vol 17 no 2 pp 217ndash2302012

[14] W Yi and A Kumar ldquoAnt colony optimization for disaster reliefoperationsrdquo Transportation Research E vol 43 no 6 pp 660ndash672 2007

[15] G-H Tzeng H-J Cheng and T D Huang ldquoMulti-objectiveoptimal planning for designing relief delivery systemsrdquo Trans-portation Research E vol 43 no 6 pp 673ndash686 2007

[16] Y-C Chiu and H Zheng ldquoReal-time mobilization decisionsfor multi-priority emergency response resources and evacua-tion groups model formulation and solutionrdquo TransportationResearch E vol 43 no 6 pp 710ndash736 2007

[17] N Altay andWG Green III ldquoORMS research in disaster oper-ations managementrdquo European Journal of Operational Researchvol 175 no 1 pp 475ndash493 2006

12 The Scientific World Journal

[18] K J Cormican D P Morton and R K Wood ldquoStochasticnetwork interdictionrdquo Operations Research vol 46 no 2 pp184ndash197 1998

[19] HOMete andZ B Zabinsky ldquoStochastic optimization ofmed-ical supply location and distribution in disaster managementrdquoInternational Journal of Production Economics vol 126 no 1 pp76ndash84 2010

[20] H Shavandi and H Mahlooji ldquoA fuzzy queuing locationmodel with a genetic algorithm for congested systemsrdquo AppliedMathematics and Computation vol 181 no 1 pp 440ndash4562006

[21] R D Galvao and R Morabito ldquoEmergency service systemsthe use of the hypercube queueing model in the solution ofprobabilistic location problemsrdquo International Transactions inOperational Research vol 15 no 5 pp 525ndash549 2008

[22] N Geroliminis M G Karlaftis and A Skabardonis ldquoA spatialqueuing model for the emergency vehicle districting andlocation problemrdquo Transportation Research B vol 43 no 7 pp798ndash811 2009

[23] C M Hosage and M F Goodchild ldquoDiscrete space location-allocation solutions from genetic algorithmsrdquo Annals of Opera-tions Research vol 6 no 2 pp 35ndash46 1986

[24] P Chakroborty ldquoGenetic algorithms for optimal urban transitnetwork designrdquo Computer-Aided Civil and Infrastructure Engi-neering vol 18 no 3 pp 184ndash200 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 7

LR119902sys

= sum119903isin119877

sum119894isin119868

LR119877119902119868+ sum119903isin119877

sum119895isin119869

LR119877119902119869+ sum119903isin119877

sum119896isin119870

LR119877119902119896

= sum119903isin119877

sum119894isin119868

((11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)2

12058310158401015840 (12058310158401015840 minus (11199022)sumℎ

119895=1sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119910119894119895119911119895119896)

+ sum119903isin119877

sum119895isin119869

(sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)2

1205831015840 (1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896)

+ sum119903isin119877

sum119896isin119896

(2120575119903

119896 (119888119903

119896+ 119888119903

119896))2

120583 (120583 minus 2120575119903119896 (119888119903119896+ 119888119903119896))

(26)

As the response time includes not only the sojourn time ofemergency demand in the queuing network system but alsothe transportation time therefore when the queuing theoryis applied to the present model the above objective functioncan be presented as follows

min119885

= minsum119903isin119877

sum119894isin119868

sum119895isin119869

sum119896isin119870

119905119903

119894119895119896

= min(sum119903isin119877

sum119894isin119868

(1)

times (12058310158401015840minus (

1

1199022

)

ℎ

sum119895=1

119899

sum119896=1

(2120575119903

119896 (119888119903

119896+ 119888119903

119896)) 119910119894119895119911119895119896)

minus1

+ sum119903isin119877

sum119895isin119869

1

1205831015840 minus sum119899

119896=1(2120575119903119896 (119888119903119896+ 119888119903119896)) 119911119895119896

+ sum119903isin119877

sum119896isin119896

1

120583 minus (2120575119903119896 (119888119903119896+ 119888119903119896))

+sum119903isin119877

sum119894isin119868

sum119895isin119869

TR119903119894119895119910119894119895+ sum119903isin119877

sum119895isin119869

sum119896isin119870

TR119903119895119896119911119895119896)

(27)

43 GA-Based Approach As the facility location-allocationproblem is NP- (nondeterministic polynomial-) hard wherethe proposed 0-1 nonlinear integer programming model inthis study is used some heuristic algorithms are requiredto solve the problem quickly The genetic algorithms (GAs)have been applied to location problems since 1986 by Hosageand Goodchild [23] GA has exhibited inherent advantagessuch as their robust performancewhen solving combinatorialproblems as well as their ability to incorporate proceduresand logical conditions and to handle discrete variables andnonlinear constraints in a straightforward manner [24]These qualities make GAs particularly attractive for potentialcombination with other methods and external procedures

Wedevelop a refined genetic algorithm to solve the aforemen-tioned problem as indicated belowThe associated frameworkfor embedding the GA with emergency queue network ispresented in Figure 3

431 Methodology of Genetic Algorithm In a genetic algo-rithm a population of candidate solutions to an optimizationproblem is evolved toward better solutions Each candi-date solution has a set of properties (its chromosomes orgenotype) which can be mutated and altered solutions arerepresented in binary as strings of 0 s and 1 s but otherencodings are also possible The evolution usually startsfrom a population of randomly generated individuals andhappens in generations In each generation the fitness ofevery individual in the population is evaluated the more fitindividuals are stochastically selected from the current popu-lation and each individualrsquos genome ismodified (recombinedand possibly randomly mutated) to form a new populationThe new population is then used in the next iteration of thealgorithm Commonly the algorithm terminates when eithera maximum number of generations have been produced or asatisfactory fitness level has been reached for the populationOnce the genetic representation and the fitness function aredefined a GA proceeds to initialize a population of solutionsand then to improve it through repetitive application of themutation crossover inversion and selection operators

432 Procedure of the Refined Genetic Algorithm

Step 1 (encoding) We suppose that the genotype of a chro-mosome with some genes indicates an individual corre-sponding to the location pattern of emergency logisticscenters and paths between nodes in the system network Thegene presented by 1 indicates that an emergency logisticscenter is built at this location or the path between nodes in thesystem network is selectedThe gene presented by 0 indicatesthat no emergency logistics centers is built there or the pathbetween nodes in the system network which is selected is notselected 119873 chromosomes are randomly generated and theinitial population (Pop) is set

Step 2 (fitness evaluation) Once the initial population isavailable evaluation of each chromosome takes place to ex-plore the quality of the corresponding solution For individ-uals with larger fitness value to be transmitted to the nextgeneration with higher probability the chromosome fitnessfunction in this paper can be described as follows and 119888max isthe maximum estimation value of min119885

Fit (min119885) = 119862max minusmin119885 if min119885 lt 119862max

0 otherwise(28)

Step 3 (crossover selection and mutation) 119873 chromosomesare crossed according to a single pointmethodwith crossoverprobability (which is 119875

119888= 09) to generate the offspring

chromosomes There are a total of 2119873 chromosomes forboth parent and offspring chromosomes Then the duplicatechromosomes are deleted to produce the new population(Pop) Select the first 119873 chromosomes of the population on

8 The Scientific World Journal

Step 1 encoding forparameter sets

Initialization ofgenerate population

Step 2 chromosomefitness evaluation

Step 3 geneticmanipulation crossoverselection and mutation

Step 4 offspringgeneration Select the best individual

and output the solution

Yes

No Terminate

(1) Decoding bit string to the parameter(2) Objective function value calculation(3) Mapping function value to the fitness value(4) Fitness value adjustments

Preparation of dataand parameters

Figure 3 Steps of the proposed GA heuristic

(B)(C)

(A)

Figure 4 The location of facilities in an ESC system

the basis of the fitness values in ascending order When thenumber of chromosomes in the new population (Pop) is lessthan 119873 (119873 = 200) 119873-|Pop| chromosomes are randomlygenerated (|Pop| is the number of chromosomes in thepopulation (Pop)) Select a chromosome randomly and applythe mutation operator applied on it with probability 119875

119898=

001

Step 4 Choose new chromosomes by the roulette wheelmethod Repeat Steps 3 and 4 until there are no solutionimprovements within a specified number of iterations

5 A Case Study

A case study example is presented in this section to illustratethe efficiency of our approach to realize the quick-responsiveemergency relief during large-scale natural disastersThepro-posed method is implemented on a scenario where a severetyphoon strikes the southeast coast of Shanghai with a greatprobability within the next 100 years It is assumed that thereare three regions (A B and C) slashed by the typhoon shownin Figure 4 One region is Lingangxincheng District (A)

another one is Fengxian District (B) and the third one isJinshan District (C)

There are ten supply points (SP) (1) disaster relief sup-plies reserve center in Baoshan District (2) Automobile Cityin Jiading (3) Red Cross disaster preparedness warehouse inJiading District (4) disaster relief supplies reserve center inQingqu District (5) disaster relief supplies reserve center inPutuo District (6) disaster relief supplies reserve center inYangpu District (7) disaster relief supplies reserve center inMinhang District (8) disaster relief supplies reserve center inHongkou District (9) disaster relief supplies reserve centerin Pudong District and (10) disaster relief supplies reservecenter in Xuhui District Based on the study of large-scalelogistics center in Shanghai we select ten potential pointsas the emergency logistics centers (ELCs) Four potentialELCs are prepared for the affected areas in LingangxinchengDistrict two potential ELCs are planned to serve FengxianDistrict and two potential ELCs are expected to serve theJinshanDistrictWe assume that a total of thirteen areas (DP)are seriously affected and need emergency rescue urgently inwhich six affected areas are in Lingangxincheng four affectedareas are in Fengxian District and three affected areas are inJinshan District

For simplicity only one type of emergency supply resour-ces will be considered in the queuing network We assumethat the travel speed is fixed to be equal to 40 miles per hourfor all the vehicles travelling between SP and ESC Howeverthe travel speed between ESC and DP is assumed to be 30miles per hour Let us adopt the hypothesis as the abovethat the occurrence time of a relief request follows a negativeexponential distribution with intensify 120575 and the quantity ofresources in every relief request is uniformly distributed withintensify 119891

119881(V) that the lower bound (LB) is 119888 and the upper

bound (UB) is 119889 (119889 gt 119888) The disaster-related informationincluding the population of the affected areas and the distancebetween supply point and emergency logistics center anddistance between emergency logistics center and demandpoint are shown in Tables 1 2 3 and 4 respectively

The Scientific World Journal 9

Table 1 Population and demand data of the affected areas

Affected area Population (119888 119889) 120575

Area 1 (Zhuqiaozhen) 105807 (4 12) 16Area 2 (Huinanzhen) 213845 (4 12) 16Area 3 (Laogangzhen) 37408 (5 14) 18Area 4 (Datuanzhen) 71162 (4 12) 16Area 5 (Xingangzhen) 21475 (6 16) 20Area 6 (Shuyuanzhen) 59831 (6 16) 20Area 7 (Luchaogangzhen) 27850 (6 16) 20Area 8 (Situanzhen) 65389 (5 14) 18Area 9 (Qingcunzhen) 89163 (5 14) 18Area 10 (Nanqiaozhen) 361185 (3 10) 14Area 11 (Zhuanghangzhen) 62388 (3 10) 14Area 12 (Zhelinzhen) 62589 (3 10) 14Area 13 (Caojingzhen) 40722 (2 8) 12Area 14 (Shanyangzhen) 84640 (2 8) 12Area 15 (Jinshanzhen) 70815 (2 8) 12

0 20 40 60 80 100 120 140 160 180 20020

25

30

35

40

45

50

55

60

Interaction number

Tota

l res

pons

e tim

e (ho

urs)

Total response time

Figure 5 Optimization of the fitness function

In this case study the fixed cost to locate the emergencylogistics center is assumed as 119886

1= 85 119886

2= 100 119886

3= 90

1198864= 85 119886

5= 95 119886

6= 110 119886

7= 70 119886

8= 80 119886

9= 115

and 11988610

= 120 The total cost to build the emergency logisticscenters is assumednot exceeding to be119861 = 500 For simplicityof the computation we presume that 119902

1= 5 119902

2= 2 and

1199023= 3According to the background of the above problem a

simplified 10 times 10 times 15 emergency supply chain network isdeveloped in the assumed scenario where the geographicalrelationships among these supply depots emergency logisticscenters and demand points are specified Then we applythe aforementioned model into the emergency supply chainnetwork system And the GA-based approach demonstratedbefore is coded to verify the validity of the model inMatlab2009a

The optimization process of the fitness function isdepicted in Figure 5 where the optimal response time of thequeuing network is 2071 hours

Therefore the average response time of each demandpoint is 138 h which is within a reasonable range The cal-culation result shows that this model can play an importantrole in the emergency supply chain management Differentfrom the general supply chain management the timesavingis the most significant feature of the emergency supply chainsystem The efficient allocation of emergency resources todemand points improves not only the operational perfor-mance of the emergency supply chain system in the emer-gency supply side but also the affected peoplersquos survival ratein the disaster areas From the psychology of point of viewquick responses and deliveries for disaster relief not onlymay improve the governmental efforts to rescue but alsostrengthen the affected peoplersquos willpower to be alive hencestabilizing the affected areas

The emergency supply chain system is composed of threemembers which have different functions in the relief supplydelivery process And the upstream nodes and downstreamnodes in the ESC system are connected by roads If we treateach node in the system as the node in a network considerthe roads as the arcs of the network and treat the length ofeach road as the weight of each arc then the ESC system canbe seen as a directed network diagram with weights And theresponse time of the system includes two parts the first partis the transportation time of the relief supply request on theroads and the second part is the sojourn time of the reliefsupply request in each node

Figure 6 indicates the values of the above two componentsof response time with the total transportation time being1664 h and the total sojourn time being 407 h This iterativeprocess also demonstrates the convergence of the heuristicalgorithm

Queuing models are effective tools for obtaining infor-mation about important performance of each facility in theESC system like sojourn times waiting times queue lengthstotal customers in the system and so on In order to analyzethe efficiency of facilities in the system we calculate thetotal customers and the queue lengths which are shown inFigure 7 We can see that the solution of the above model willnot create queuing delays of relief supply so that the lossesgenerated by the queuing delay can be avoided This iterativeprocess also illustrates the robustness and the potentialapplicability of our model

Based on the built network the coordination between theupstream node and the downstream node in this system canbe enforced and the performance of basic activities includingproducing sorting processing packing and delivering canbe improved tominimize the response time of the systemTheemergency logistics centers for the system should be locatedin DP(6) DP(7) DP(8) DP(9) and DP(10) And the optimalpaths are SP(1) SP(6) rarr ELC(8) rarr DP(1) DP(2)DP(9) SP(2) SP(4) rarr ELC(6) rarr DP(13) DP(14)DP(15) SP(3) SP(8) rarr ELC(7) rarr DP(10) DP(11)DP(12) SP(5) SP(9) rarr ELC(10) rarr DP(3) DP(4)DP(5) and SP(7) SP(9) rarr ELC(9) rarr DP(6) DP(7)DP(8) Overall the examination reveals that the proposedmethod can be a useful tool for a quick response to the urgentneed of relief in the large-scale disaster-affected areas

10 The Scientific World Journal

Table 2 The service rates specifications of different servers in the network

SP 120583101584010158401

120583101584010158402

120583101584010158403

120583101584010158404

120583101584010158405

120583101584010158406

120583101584010158407

120583101584010158408

120583101584010158409

1205831015840101584010

10 9 11 8 11 12 14 16 14 17

ELC 12058310158401

12058310158402

12058310158403

12058310158404

12058310158405

12058310158406

12058310158407

12058310158408

12058310158409

120583101584010

16 18 11 10 11 12 14 16 14 10

DP 1205831

1205832

1205833

1205834

1205835

1205836

1205837

1205838

1205839

12058310

12058311

12058312

12058313

12058314

12058315

8 8 8 8 10 9 11 8 9 12 9 8 12 14 13

Table 3 The distance between supply point and emergency logistics center

SP(1) SP(2) SP(3) SP(4) SP(5) SP(6) SP(7) SP(8) SP(9) SP(10)ELC(1) 1828 1707 1197 2163 614 826 1091 973 1530 879ELC(2) 3073 2147 1624 571 2039 2056 802 1679 2554 1390ELC(3) 2271 2288 1712 1622 1419 1463 998 1102 1842 756ELC(4) 4422 3095 2462 1264 3971 3492 2429 3276 4153 3003ELC(5) 3962 2801 2716 1507 3256 2769 1449 2551 3325 2006ELC(6) 3585 2356 1923 1647 2784 2313 989 2136 2839 1437ELC(7) 2973 2968 2364 2301 2036 2149 1275 1797 2575 1471ELC(8) 2308 3261 2878 2892 1494 1401 1764 1132 1961 1933ELC(9) 2584 3643 3303 3249 2015 1165 2258 941 1600 2419ELC(10) 3038 4666 4240 4139 2831 2039 3008 1742 1405 3105

05

101520253035404550

Tim

e (ho

urs)

Total transportation timeTotal sojourn time

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 6 Optimization of total transportation time and totalsojourn time

6 Conclusions

An emergency supply chain (ESC) system is developed inthis paper for the quick response to the urgent need ofrelief in the large-scale disaster-affected areas which mainlyconsists of three chain members in series supply points(SP) emergency logistics centers (ELCs) and demand points(DP) As the timeliness is the most crucial characteristic ofthe ESC system and queuing theory is a useful tool in theanalysis of the performance of the time-dependent systemwe propose queuingmodeling for the ESC system to optimizethe emergency rescue process Based on the above queuingnetwork system a queuing minimal response time location-allocation model is established to decide the selection ofemergency logistics centers and vehicle routing decision

0

5

10

15

20

25

Total customersQueuing length

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 7 Optimization of the total customers and the queue length

For the complexity of mathematical model the GA-basedapproach is introduced to solve the model

A case study with an assumed severe typhoon striking thesoutheast coast of Shanghai with a great probability withinthe next 100 years is conducted to assess and validate ourmodel The optimal response time of the queuing network is2071 hours and the average response time of each demandpoint is 138 h which is within a reasonable range Theoptimal emergency logistics centers for the system are DP(6)DP(7) DP(8) DP(9) and DP(10) Besides the performanceof queuing modeling for the ESC system illustrates therobustness and the potential applicability of our model

For future research we suggest the following threedirections Firstly we will investigate the queuing modelingunder consideration of the facility blocking problems and

The Scientific World Journal 11

Table 4 The distance between emergency logistics center and demand point

ELC(1) ELC(2) ELC(3) ELC(4) ELC(5) ELC(6) ELC(7) ELC(8) ELC(9) ELC(10)DP(1) 3761 4191 3253 4599 3801 3752 2439 2397 1553 739DP(2) 4271 4702 3761 4794 3945 4152 2818 2822 1898 1075DP(3) 4508 4886 3986 5028 4501 4384 3046 3025 2117 1356DP(4) 4364 4579 3697 4824 4307 4138 2798 2940 1956 1710DP(5) 5283 5416 4386 5466 5435 4986 3836 3271 2693 1917DP(6) 5214 5721 4866 5896 5294 5207 3797 4221 3252 2553DP(7) 5334 5776 5299 5862 5359 5164 3871 3851 2955 2554DP(8) 4689 4581 3763 4983 4306 4173 2818 2997 2200 1968DP(9) 3309 3624 2657 3834 3334 3154 1886 3154 3452 3235DP(10) 2678 3263 1999 2911 2741 2518 1198 2505 2360 3037DP(11) 3369 2912 2386 2349 2182 1858 1894 2392 2915 3596DP(12) 3376 3794 2541 3287 3275 2612 1708 3062 2896 3256DP(13) 3691 3419 2987 2896 2694 2279 2162 3500 3275 4007DP(14) 4230 3273 3402 3011 2555 2141 2606 3906 3749 4448DP(15) 4609 3765 3858 2533 3054 2680 3166 4390 4249 4897

demand priority for the emergency relief points Secondly wewill study the generation of emergency demand and servicetime obeying the other random distributions Thirdly moreefficient multiobjective optimization methodologies for boththe above problem and emergency resources transportationscheduling problem will be considered

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grant nos 71102043 and 71371145) theMinistry of Education of China (Grant no MCM20125021)the Innovative Program of Shanghai Municipal EducationCommission (Grant no 14ZS123) and the ShanghaiMaritimeUniversity (Grant no 20120054)

References

[1] D Guha-Sapir G S Debarati D Hargitt P Hoyois R Belowand D BrechetThirty Years of Natural Disasters 1974ndash2003 theNumbers Presses universitaires de Louvain 2004

[2] J-B Sheu ldquoAn emergency logistics distribution approach forquick response to urgent relief demand in disastersrdquo Trans-portation Research E vol 43 no 6 pp 687ndash709 2007

[3] B S Manoj and A H Baker ldquoCommunication challenges inemergency responserdquo Communications of the ACM vol 50 no3 pp 51ndash53 2007

[4] B Balcik B M Beamon and K Smilowitz ldquoLast mile distribu-tion in humanitarian reliefrdquo Journal of Intelligent TransportationSystems vol 12 no 2 pp 51ndash63 2008

[5] A M Caunhye X F Nie and S Pokharel ldquoOptimizationmodels in emergency logistics a literature reviewrdquo Socio-Economic Planning Sciences vol 46 no 1 pp 4ndash13 2012

[6] S Coles and L Pericchi ldquoAnticipating catastrophes throughextreme valuemodellingrdquo Journal of the Royal Statistical SocietyC vol 52 no 4 pp 405ndash416 2003

[7] J R Artalejo ldquoG-networks a versatile approach for workremoval in queueing networksrdquo European Journal of Opera-tional Research vol 126 no 2 pp 233ndash249 2000

[8] W Jiang L Deng L Chen J Wu and J Li ldquoRisk assessmentand validation of flood disaster based on fuzzy mathematicsrdquoProgress in Natural Science vol 19 no 10 pp 1419ndash1425 2009

[9] V Bhaskar and P Lallement ldquoModeling a supply chain usinga network of queuesrdquo Applied Mathematical Modelling vol 34no 8 pp 2074ndash2088 2010

[10] B M Beamon ldquoHumanitarian relief chains issues and chal-lengesrdquo in Proceedings of the 34th International Conference onComputers and Industrial Engineering San Francisco CalifUSA November 2004

[11] R Oloruntoba and R Gray ldquoHumanitarian aid an agile supplychainrdquo Supply Chain Management vol 11 no 2 pp 115ndash1202006

[12] E J Lodree Jr and S Taskin ldquoSupply chain planning for hur-ricane response with wind speed information updatesrdquo Com-puters and Operations Research vol 36 no 1 pp 2ndash15 2009

[13] V Bhakoo P Singh and A Sohal ldquoCollaborative managementof inventory in Australian hospital supply chains practices andissuesrdquo Supply Chain Management vol 17 no 2 pp 217ndash2302012

[14] W Yi and A Kumar ldquoAnt colony optimization for disaster reliefoperationsrdquo Transportation Research E vol 43 no 6 pp 660ndash672 2007

[15] G-H Tzeng H-J Cheng and T D Huang ldquoMulti-objectiveoptimal planning for designing relief delivery systemsrdquo Trans-portation Research E vol 43 no 6 pp 673ndash686 2007

[16] Y-C Chiu and H Zheng ldquoReal-time mobilization decisionsfor multi-priority emergency response resources and evacua-tion groups model formulation and solutionrdquo TransportationResearch E vol 43 no 6 pp 710ndash736 2007

[17] N Altay andWG Green III ldquoORMS research in disaster oper-ations managementrdquo European Journal of Operational Researchvol 175 no 1 pp 475ndash493 2006

12 The Scientific World Journal

[18] K J Cormican D P Morton and R K Wood ldquoStochasticnetwork interdictionrdquo Operations Research vol 46 no 2 pp184ndash197 1998

[19] HOMete andZ B Zabinsky ldquoStochastic optimization ofmed-ical supply location and distribution in disaster managementrdquoInternational Journal of Production Economics vol 126 no 1 pp76ndash84 2010

[20] H Shavandi and H Mahlooji ldquoA fuzzy queuing locationmodel with a genetic algorithm for congested systemsrdquo AppliedMathematics and Computation vol 181 no 1 pp 440ndash4562006

[21] R D Galvao and R Morabito ldquoEmergency service systemsthe use of the hypercube queueing model in the solution ofprobabilistic location problemsrdquo International Transactions inOperational Research vol 15 no 5 pp 525ndash549 2008

[22] N Geroliminis M G Karlaftis and A Skabardonis ldquoA spatialqueuing model for the emergency vehicle districting andlocation problemrdquo Transportation Research B vol 43 no 7 pp798ndash811 2009

[23] C M Hosage and M F Goodchild ldquoDiscrete space location-allocation solutions from genetic algorithmsrdquo Annals of Opera-tions Research vol 6 no 2 pp 35ndash46 1986

[24] P Chakroborty ldquoGenetic algorithms for optimal urban transitnetwork designrdquo Computer-Aided Civil and Infrastructure Engi-neering vol 18 no 3 pp 184ndash200 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 The Scientific World Journal

Step 1 encoding forparameter sets

Initialization ofgenerate population

Step 2 chromosomefitness evaluation

Step 3 geneticmanipulation crossoverselection and mutation

Step 4 offspringgeneration Select the best individual

and output the solution

Yes

No Terminate

(1) Decoding bit string to the parameter(2) Objective function value calculation(3) Mapping function value to the fitness value(4) Fitness value adjustments

Preparation of dataand parameters

Figure 3 Steps of the proposed GA heuristic

(B)(C)

(A)

Figure 4 The location of facilities in an ESC system

the basis of the fitness values in ascending order When thenumber of chromosomes in the new population (Pop) is lessthan 119873 (119873 = 200) 119873-|Pop| chromosomes are randomlygenerated (|Pop| is the number of chromosomes in thepopulation (Pop)) Select a chromosome randomly and applythe mutation operator applied on it with probability 119875

119898=

001

Step 4 Choose new chromosomes by the roulette wheelmethod Repeat Steps 3 and 4 until there are no solutionimprovements within a specified number of iterations

5 A Case Study

A case study example is presented in this section to illustratethe efficiency of our approach to realize the quick-responsiveemergency relief during large-scale natural disastersThepro-posed method is implemented on a scenario where a severetyphoon strikes the southeast coast of Shanghai with a greatprobability within the next 100 years It is assumed that thereare three regions (A B and C) slashed by the typhoon shownin Figure 4 One region is Lingangxincheng District (A)

another one is Fengxian District (B) and the third one isJinshan District (C)

There are ten supply points (SP) (1) disaster relief sup-plies reserve center in Baoshan District (2) Automobile Cityin Jiading (3) Red Cross disaster preparedness warehouse inJiading District (4) disaster relief supplies reserve center inQingqu District (5) disaster relief supplies reserve center inPutuo District (6) disaster relief supplies reserve center inYangpu District (7) disaster relief supplies reserve center inMinhang District (8) disaster relief supplies reserve center inHongkou District (9) disaster relief supplies reserve centerin Pudong District and (10) disaster relief supplies reservecenter in Xuhui District Based on the study of large-scalelogistics center in Shanghai we select ten potential pointsas the emergency logistics centers (ELCs) Four potentialELCs are prepared for the affected areas in LingangxinchengDistrict two potential ELCs are planned to serve FengxianDistrict and two potential ELCs are expected to serve theJinshanDistrictWe assume that a total of thirteen areas (DP)are seriously affected and need emergency rescue urgently inwhich six affected areas are in Lingangxincheng four affectedareas are in Fengxian District and three affected areas are inJinshan District

For simplicity only one type of emergency supply resour-ces will be considered in the queuing network We assumethat the travel speed is fixed to be equal to 40 miles per hourfor all the vehicles travelling between SP and ESC Howeverthe travel speed between ESC and DP is assumed to be 30miles per hour Let us adopt the hypothesis as the abovethat the occurrence time of a relief request follows a negativeexponential distribution with intensify 120575 and the quantity ofresources in every relief request is uniformly distributed withintensify 119891

119881(V) that the lower bound (LB) is 119888 and the upper

bound (UB) is 119889 (119889 gt 119888) The disaster-related informationincluding the population of the affected areas and the distancebetween supply point and emergency logistics center anddistance between emergency logistics center and demandpoint are shown in Tables 1 2 3 and 4 respectively

The Scientific World Journal 9

Table 1 Population and demand data of the affected areas

Affected area Population (119888 119889) 120575

Area 1 (Zhuqiaozhen) 105807 (4 12) 16Area 2 (Huinanzhen) 213845 (4 12) 16Area 3 (Laogangzhen) 37408 (5 14) 18Area 4 (Datuanzhen) 71162 (4 12) 16Area 5 (Xingangzhen) 21475 (6 16) 20Area 6 (Shuyuanzhen) 59831 (6 16) 20Area 7 (Luchaogangzhen) 27850 (6 16) 20Area 8 (Situanzhen) 65389 (5 14) 18Area 9 (Qingcunzhen) 89163 (5 14) 18Area 10 (Nanqiaozhen) 361185 (3 10) 14Area 11 (Zhuanghangzhen) 62388 (3 10) 14Area 12 (Zhelinzhen) 62589 (3 10) 14Area 13 (Caojingzhen) 40722 (2 8) 12Area 14 (Shanyangzhen) 84640 (2 8) 12Area 15 (Jinshanzhen) 70815 (2 8) 12

0 20 40 60 80 100 120 140 160 180 20020

25

30

35

40

45

50

55

60

Interaction number

Tota

l res

pons

e tim

e (ho

urs)

Total response time

Figure 5 Optimization of the fitness function

In this case study the fixed cost to locate the emergencylogistics center is assumed as 119886

1= 85 119886

2= 100 119886

3= 90

1198864= 85 119886

5= 95 119886

6= 110 119886

7= 70 119886

8= 80 119886

9= 115

and 11988610

= 120 The total cost to build the emergency logisticscenters is assumednot exceeding to be119861 = 500 For simplicityof the computation we presume that 119902

1= 5 119902

2= 2 and

1199023= 3According to the background of the above problem a

simplified 10 times 10 times 15 emergency supply chain network isdeveloped in the assumed scenario where the geographicalrelationships among these supply depots emergency logisticscenters and demand points are specified Then we applythe aforementioned model into the emergency supply chainnetwork system And the GA-based approach demonstratedbefore is coded to verify the validity of the model inMatlab2009a

The optimization process of the fitness function isdepicted in Figure 5 where the optimal response time of thequeuing network is 2071 hours

Therefore the average response time of each demandpoint is 138 h which is within a reasonable range The cal-culation result shows that this model can play an importantrole in the emergency supply chain management Differentfrom the general supply chain management the timesavingis the most significant feature of the emergency supply chainsystem The efficient allocation of emergency resources todemand points improves not only the operational perfor-mance of the emergency supply chain system in the emer-gency supply side but also the affected peoplersquos survival ratein the disaster areas From the psychology of point of viewquick responses and deliveries for disaster relief not onlymay improve the governmental efforts to rescue but alsostrengthen the affected peoplersquos willpower to be alive hencestabilizing the affected areas

The emergency supply chain system is composed of threemembers which have different functions in the relief supplydelivery process And the upstream nodes and downstreamnodes in the ESC system are connected by roads If we treateach node in the system as the node in a network considerthe roads as the arcs of the network and treat the length ofeach road as the weight of each arc then the ESC system canbe seen as a directed network diagram with weights And theresponse time of the system includes two parts the first partis the transportation time of the relief supply request on theroads and the second part is the sojourn time of the reliefsupply request in each node

Figure 6 indicates the values of the above two componentsof response time with the total transportation time being1664 h and the total sojourn time being 407 h This iterativeprocess also demonstrates the convergence of the heuristicalgorithm

Queuing models are effective tools for obtaining infor-mation about important performance of each facility in theESC system like sojourn times waiting times queue lengthstotal customers in the system and so on In order to analyzethe efficiency of facilities in the system we calculate thetotal customers and the queue lengths which are shown inFigure 7 We can see that the solution of the above model willnot create queuing delays of relief supply so that the lossesgenerated by the queuing delay can be avoided This iterativeprocess also illustrates the robustness and the potentialapplicability of our model

Based on the built network the coordination between theupstream node and the downstream node in this system canbe enforced and the performance of basic activities includingproducing sorting processing packing and delivering canbe improved tominimize the response time of the systemTheemergency logistics centers for the system should be locatedin DP(6) DP(7) DP(8) DP(9) and DP(10) And the optimalpaths are SP(1) SP(6) rarr ELC(8) rarr DP(1) DP(2)DP(9) SP(2) SP(4) rarr ELC(6) rarr DP(13) DP(14)DP(15) SP(3) SP(8) rarr ELC(7) rarr DP(10) DP(11)DP(12) SP(5) SP(9) rarr ELC(10) rarr DP(3) DP(4)DP(5) and SP(7) SP(9) rarr ELC(9) rarr DP(6) DP(7)DP(8) Overall the examination reveals that the proposedmethod can be a useful tool for a quick response to the urgentneed of relief in the large-scale disaster-affected areas

10 The Scientific World Journal

Table 2 The service rates specifications of different servers in the network

SP 120583101584010158401

120583101584010158402

120583101584010158403

120583101584010158404

120583101584010158405

120583101584010158406

120583101584010158407

120583101584010158408

120583101584010158409

1205831015840101584010

10 9 11 8 11 12 14 16 14 17

ELC 12058310158401

12058310158402

12058310158403

12058310158404

12058310158405

12058310158406

12058310158407

12058310158408

12058310158409

120583101584010

16 18 11 10 11 12 14 16 14 10

DP 1205831

1205832

1205833

1205834

1205835

1205836

1205837

1205838

1205839

12058310

12058311

12058312

12058313

12058314

12058315

8 8 8 8 10 9 11 8 9 12 9 8 12 14 13

Table 3 The distance between supply point and emergency logistics center

SP(1) SP(2) SP(3) SP(4) SP(5) SP(6) SP(7) SP(8) SP(9) SP(10)ELC(1) 1828 1707 1197 2163 614 826 1091 973 1530 879ELC(2) 3073 2147 1624 571 2039 2056 802 1679 2554 1390ELC(3) 2271 2288 1712 1622 1419 1463 998 1102 1842 756ELC(4) 4422 3095 2462 1264 3971 3492 2429 3276 4153 3003ELC(5) 3962 2801 2716 1507 3256 2769 1449 2551 3325 2006ELC(6) 3585 2356 1923 1647 2784 2313 989 2136 2839 1437ELC(7) 2973 2968 2364 2301 2036 2149 1275 1797 2575 1471ELC(8) 2308 3261 2878 2892 1494 1401 1764 1132 1961 1933ELC(9) 2584 3643 3303 3249 2015 1165 2258 941 1600 2419ELC(10) 3038 4666 4240 4139 2831 2039 3008 1742 1405 3105

05

101520253035404550

Tim

e (ho

urs)

Total transportation timeTotal sojourn time

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 6 Optimization of total transportation time and totalsojourn time

6 Conclusions

An emergency supply chain (ESC) system is developed inthis paper for the quick response to the urgent need ofrelief in the large-scale disaster-affected areas which mainlyconsists of three chain members in series supply points(SP) emergency logistics centers (ELCs) and demand points(DP) As the timeliness is the most crucial characteristic ofthe ESC system and queuing theory is a useful tool in theanalysis of the performance of the time-dependent systemwe propose queuingmodeling for the ESC system to optimizethe emergency rescue process Based on the above queuingnetwork system a queuing minimal response time location-allocation model is established to decide the selection ofemergency logistics centers and vehicle routing decision

0

5

10

15

20

25

Total customersQueuing length

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 7 Optimization of the total customers and the queue length

For the complexity of mathematical model the GA-basedapproach is introduced to solve the model

A case study with an assumed severe typhoon striking thesoutheast coast of Shanghai with a great probability withinthe next 100 years is conducted to assess and validate ourmodel The optimal response time of the queuing network is2071 hours and the average response time of each demandpoint is 138 h which is within a reasonable range Theoptimal emergency logistics centers for the system are DP(6)DP(7) DP(8) DP(9) and DP(10) Besides the performanceof queuing modeling for the ESC system illustrates therobustness and the potential applicability of our model

For future research we suggest the following threedirections Firstly we will investigate the queuing modelingunder consideration of the facility blocking problems and

The Scientific World Journal 11

Table 4 The distance between emergency logistics center and demand point

ELC(1) ELC(2) ELC(3) ELC(4) ELC(5) ELC(6) ELC(7) ELC(8) ELC(9) ELC(10)DP(1) 3761 4191 3253 4599 3801 3752 2439 2397 1553 739DP(2) 4271 4702 3761 4794 3945 4152 2818 2822 1898 1075DP(3) 4508 4886 3986 5028 4501 4384 3046 3025 2117 1356DP(4) 4364 4579 3697 4824 4307 4138 2798 2940 1956 1710DP(5) 5283 5416 4386 5466 5435 4986 3836 3271 2693 1917DP(6) 5214 5721 4866 5896 5294 5207 3797 4221 3252 2553DP(7) 5334 5776 5299 5862 5359 5164 3871 3851 2955 2554DP(8) 4689 4581 3763 4983 4306 4173 2818 2997 2200 1968DP(9) 3309 3624 2657 3834 3334 3154 1886 3154 3452 3235DP(10) 2678 3263 1999 2911 2741 2518 1198 2505 2360 3037DP(11) 3369 2912 2386 2349 2182 1858 1894 2392 2915 3596DP(12) 3376 3794 2541 3287 3275 2612 1708 3062 2896 3256DP(13) 3691 3419 2987 2896 2694 2279 2162 3500 3275 4007DP(14) 4230 3273 3402 3011 2555 2141 2606 3906 3749 4448DP(15) 4609 3765 3858 2533 3054 2680 3166 4390 4249 4897

demand priority for the emergency relief points Secondly wewill study the generation of emergency demand and servicetime obeying the other random distributions Thirdly moreefficient multiobjective optimization methodologies for boththe above problem and emergency resources transportationscheduling problem will be considered

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grant nos 71102043 and 71371145) theMinistry of Education of China (Grant no MCM20125021)the Innovative Program of Shanghai Municipal EducationCommission (Grant no 14ZS123) and the ShanghaiMaritimeUniversity (Grant no 20120054)

References

[1] D Guha-Sapir G S Debarati D Hargitt P Hoyois R Belowand D BrechetThirty Years of Natural Disasters 1974ndash2003 theNumbers Presses universitaires de Louvain 2004

[2] J-B Sheu ldquoAn emergency logistics distribution approach forquick response to urgent relief demand in disastersrdquo Trans-portation Research E vol 43 no 6 pp 687ndash709 2007

[3] B S Manoj and A H Baker ldquoCommunication challenges inemergency responserdquo Communications of the ACM vol 50 no3 pp 51ndash53 2007

[4] B Balcik B M Beamon and K Smilowitz ldquoLast mile distribu-tion in humanitarian reliefrdquo Journal of Intelligent TransportationSystems vol 12 no 2 pp 51ndash63 2008

[5] A M Caunhye X F Nie and S Pokharel ldquoOptimizationmodels in emergency logistics a literature reviewrdquo Socio-Economic Planning Sciences vol 46 no 1 pp 4ndash13 2012

[6] S Coles and L Pericchi ldquoAnticipating catastrophes throughextreme valuemodellingrdquo Journal of the Royal Statistical SocietyC vol 52 no 4 pp 405ndash416 2003

[7] J R Artalejo ldquoG-networks a versatile approach for workremoval in queueing networksrdquo European Journal of Opera-tional Research vol 126 no 2 pp 233ndash249 2000

[8] W Jiang L Deng L Chen J Wu and J Li ldquoRisk assessmentand validation of flood disaster based on fuzzy mathematicsrdquoProgress in Natural Science vol 19 no 10 pp 1419ndash1425 2009

[9] V Bhaskar and P Lallement ldquoModeling a supply chain usinga network of queuesrdquo Applied Mathematical Modelling vol 34no 8 pp 2074ndash2088 2010

[10] B M Beamon ldquoHumanitarian relief chains issues and chal-lengesrdquo in Proceedings of the 34th International Conference onComputers and Industrial Engineering San Francisco CalifUSA November 2004

[11] R Oloruntoba and R Gray ldquoHumanitarian aid an agile supplychainrdquo Supply Chain Management vol 11 no 2 pp 115ndash1202006

[12] E J Lodree Jr and S Taskin ldquoSupply chain planning for hur-ricane response with wind speed information updatesrdquo Com-puters and Operations Research vol 36 no 1 pp 2ndash15 2009

[13] V Bhakoo P Singh and A Sohal ldquoCollaborative managementof inventory in Australian hospital supply chains practices andissuesrdquo Supply Chain Management vol 17 no 2 pp 217ndash2302012

[14] W Yi and A Kumar ldquoAnt colony optimization for disaster reliefoperationsrdquo Transportation Research E vol 43 no 6 pp 660ndash672 2007

[15] G-H Tzeng H-J Cheng and T D Huang ldquoMulti-objectiveoptimal planning for designing relief delivery systemsrdquo Trans-portation Research E vol 43 no 6 pp 673ndash686 2007

[16] Y-C Chiu and H Zheng ldquoReal-time mobilization decisionsfor multi-priority emergency response resources and evacua-tion groups model formulation and solutionrdquo TransportationResearch E vol 43 no 6 pp 710ndash736 2007

[17] N Altay andWG Green III ldquoORMS research in disaster oper-ations managementrdquo European Journal of Operational Researchvol 175 no 1 pp 475ndash493 2006

12 The Scientific World Journal

[18] K J Cormican D P Morton and R K Wood ldquoStochasticnetwork interdictionrdquo Operations Research vol 46 no 2 pp184ndash197 1998

[19] HOMete andZ B Zabinsky ldquoStochastic optimization ofmed-ical supply location and distribution in disaster managementrdquoInternational Journal of Production Economics vol 126 no 1 pp76ndash84 2010

[20] H Shavandi and H Mahlooji ldquoA fuzzy queuing locationmodel with a genetic algorithm for congested systemsrdquo AppliedMathematics and Computation vol 181 no 1 pp 440ndash4562006

[21] R D Galvao and R Morabito ldquoEmergency service systemsthe use of the hypercube queueing model in the solution ofprobabilistic location problemsrdquo International Transactions inOperational Research vol 15 no 5 pp 525ndash549 2008

[22] N Geroliminis M G Karlaftis and A Skabardonis ldquoA spatialqueuing model for the emergency vehicle districting andlocation problemrdquo Transportation Research B vol 43 no 7 pp798ndash811 2009

[23] C M Hosage and M F Goodchild ldquoDiscrete space location-allocation solutions from genetic algorithmsrdquo Annals of Opera-tions Research vol 6 no 2 pp 35ndash46 1986

[24] P Chakroborty ldquoGenetic algorithms for optimal urban transitnetwork designrdquo Computer-Aided Civil and Infrastructure Engi-neering vol 18 no 3 pp 184ndash200 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 9

Table 1 Population and demand data of the affected areas

Affected area Population (119888 119889) 120575

Area 1 (Zhuqiaozhen) 105807 (4 12) 16Area 2 (Huinanzhen) 213845 (4 12) 16Area 3 (Laogangzhen) 37408 (5 14) 18Area 4 (Datuanzhen) 71162 (4 12) 16Area 5 (Xingangzhen) 21475 (6 16) 20Area 6 (Shuyuanzhen) 59831 (6 16) 20Area 7 (Luchaogangzhen) 27850 (6 16) 20Area 8 (Situanzhen) 65389 (5 14) 18Area 9 (Qingcunzhen) 89163 (5 14) 18Area 10 (Nanqiaozhen) 361185 (3 10) 14Area 11 (Zhuanghangzhen) 62388 (3 10) 14Area 12 (Zhelinzhen) 62589 (3 10) 14Area 13 (Caojingzhen) 40722 (2 8) 12Area 14 (Shanyangzhen) 84640 (2 8) 12Area 15 (Jinshanzhen) 70815 (2 8) 12

0 20 40 60 80 100 120 140 160 180 20020

25

30

35

40

45

50

55

60

Interaction number

Tota

l res

pons

e tim

e (ho

urs)

Total response time

Figure 5 Optimization of the fitness function

In this case study the fixed cost to locate the emergencylogistics center is assumed as 119886

1= 85 119886

2= 100 119886

3= 90

1198864= 85 119886

5= 95 119886

6= 110 119886

7= 70 119886

8= 80 119886

9= 115

and 11988610

= 120 The total cost to build the emergency logisticscenters is assumednot exceeding to be119861 = 500 For simplicityof the computation we presume that 119902

1= 5 119902

2= 2 and

1199023= 3According to the background of the above problem a

simplified 10 times 10 times 15 emergency supply chain network isdeveloped in the assumed scenario where the geographicalrelationships among these supply depots emergency logisticscenters and demand points are specified Then we applythe aforementioned model into the emergency supply chainnetwork system And the GA-based approach demonstratedbefore is coded to verify the validity of the model inMatlab2009a

The optimization process of the fitness function isdepicted in Figure 5 where the optimal response time of thequeuing network is 2071 hours

Therefore the average response time of each demandpoint is 138 h which is within a reasonable range The cal-culation result shows that this model can play an importantrole in the emergency supply chain management Differentfrom the general supply chain management the timesavingis the most significant feature of the emergency supply chainsystem The efficient allocation of emergency resources todemand points improves not only the operational perfor-mance of the emergency supply chain system in the emer-gency supply side but also the affected peoplersquos survival ratein the disaster areas From the psychology of point of viewquick responses and deliveries for disaster relief not onlymay improve the governmental efforts to rescue but alsostrengthen the affected peoplersquos willpower to be alive hencestabilizing the affected areas

The emergency supply chain system is composed of threemembers which have different functions in the relief supplydelivery process And the upstream nodes and downstreamnodes in the ESC system are connected by roads If we treateach node in the system as the node in a network considerthe roads as the arcs of the network and treat the length ofeach road as the weight of each arc then the ESC system canbe seen as a directed network diagram with weights And theresponse time of the system includes two parts the first partis the transportation time of the relief supply request on theroads and the second part is the sojourn time of the reliefsupply request in each node

Figure 6 indicates the values of the above two componentsof response time with the total transportation time being1664 h and the total sojourn time being 407 h This iterativeprocess also demonstrates the convergence of the heuristicalgorithm

Queuing models are effective tools for obtaining infor-mation about important performance of each facility in theESC system like sojourn times waiting times queue lengthstotal customers in the system and so on In order to analyzethe efficiency of facilities in the system we calculate thetotal customers and the queue lengths which are shown inFigure 7 We can see that the solution of the above model willnot create queuing delays of relief supply so that the lossesgenerated by the queuing delay can be avoided This iterativeprocess also illustrates the robustness and the potentialapplicability of our model

Based on the built network the coordination between theupstream node and the downstream node in this system canbe enforced and the performance of basic activities includingproducing sorting processing packing and delivering canbe improved tominimize the response time of the systemTheemergency logistics centers for the system should be locatedin DP(6) DP(7) DP(8) DP(9) and DP(10) And the optimalpaths are SP(1) SP(6) rarr ELC(8) rarr DP(1) DP(2)DP(9) SP(2) SP(4) rarr ELC(6) rarr DP(13) DP(14)DP(15) SP(3) SP(8) rarr ELC(7) rarr DP(10) DP(11)DP(12) SP(5) SP(9) rarr ELC(10) rarr DP(3) DP(4)DP(5) and SP(7) SP(9) rarr ELC(9) rarr DP(6) DP(7)DP(8) Overall the examination reveals that the proposedmethod can be a useful tool for a quick response to the urgentneed of relief in the large-scale disaster-affected areas

10 The Scientific World Journal

Table 2 The service rates specifications of different servers in the network

SP 120583101584010158401

120583101584010158402

120583101584010158403

120583101584010158404

120583101584010158405

120583101584010158406

120583101584010158407

120583101584010158408

120583101584010158409

1205831015840101584010

10 9 11 8 11 12 14 16 14 17

ELC 12058310158401

12058310158402

12058310158403

12058310158404

12058310158405

12058310158406

12058310158407

12058310158408

12058310158409

120583101584010

16 18 11 10 11 12 14 16 14 10

DP 1205831

1205832

1205833

1205834

1205835

1205836

1205837

1205838

1205839

12058310

12058311

12058312

12058313

12058314

12058315

8 8 8 8 10 9 11 8 9 12 9 8 12 14 13

Table 3 The distance between supply point and emergency logistics center

SP(1) SP(2) SP(3) SP(4) SP(5) SP(6) SP(7) SP(8) SP(9) SP(10)ELC(1) 1828 1707 1197 2163 614 826 1091 973 1530 879ELC(2) 3073 2147 1624 571 2039 2056 802 1679 2554 1390ELC(3) 2271 2288 1712 1622 1419 1463 998 1102 1842 756ELC(4) 4422 3095 2462 1264 3971 3492 2429 3276 4153 3003ELC(5) 3962 2801 2716 1507 3256 2769 1449 2551 3325 2006ELC(6) 3585 2356 1923 1647 2784 2313 989 2136 2839 1437ELC(7) 2973 2968 2364 2301 2036 2149 1275 1797 2575 1471ELC(8) 2308 3261 2878 2892 1494 1401 1764 1132 1961 1933ELC(9) 2584 3643 3303 3249 2015 1165 2258 941 1600 2419ELC(10) 3038 4666 4240 4139 2831 2039 3008 1742 1405 3105

05

101520253035404550

Tim

e (ho

urs)

Total transportation timeTotal sojourn time

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 6 Optimization of total transportation time and totalsojourn time

6 Conclusions

An emergency supply chain (ESC) system is developed inthis paper for the quick response to the urgent need ofrelief in the large-scale disaster-affected areas which mainlyconsists of three chain members in series supply points(SP) emergency logistics centers (ELCs) and demand points(DP) As the timeliness is the most crucial characteristic ofthe ESC system and queuing theory is a useful tool in theanalysis of the performance of the time-dependent systemwe propose queuingmodeling for the ESC system to optimizethe emergency rescue process Based on the above queuingnetwork system a queuing minimal response time location-allocation model is established to decide the selection ofemergency logistics centers and vehicle routing decision

0

5

10

15

20

25

Total customersQueuing length

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 7 Optimization of the total customers and the queue length

For the complexity of mathematical model the GA-basedapproach is introduced to solve the model

A case study with an assumed severe typhoon striking thesoutheast coast of Shanghai with a great probability withinthe next 100 years is conducted to assess and validate ourmodel The optimal response time of the queuing network is2071 hours and the average response time of each demandpoint is 138 h which is within a reasonable range Theoptimal emergency logistics centers for the system are DP(6)DP(7) DP(8) DP(9) and DP(10) Besides the performanceof queuing modeling for the ESC system illustrates therobustness and the potential applicability of our model

For future research we suggest the following threedirections Firstly we will investigate the queuing modelingunder consideration of the facility blocking problems and

The Scientific World Journal 11

Table 4 The distance between emergency logistics center and demand point

ELC(1) ELC(2) ELC(3) ELC(4) ELC(5) ELC(6) ELC(7) ELC(8) ELC(9) ELC(10)DP(1) 3761 4191 3253 4599 3801 3752 2439 2397 1553 739DP(2) 4271 4702 3761 4794 3945 4152 2818 2822 1898 1075DP(3) 4508 4886 3986 5028 4501 4384 3046 3025 2117 1356DP(4) 4364 4579 3697 4824 4307 4138 2798 2940 1956 1710DP(5) 5283 5416 4386 5466 5435 4986 3836 3271 2693 1917DP(6) 5214 5721 4866 5896 5294 5207 3797 4221 3252 2553DP(7) 5334 5776 5299 5862 5359 5164 3871 3851 2955 2554DP(8) 4689 4581 3763 4983 4306 4173 2818 2997 2200 1968DP(9) 3309 3624 2657 3834 3334 3154 1886 3154 3452 3235DP(10) 2678 3263 1999 2911 2741 2518 1198 2505 2360 3037DP(11) 3369 2912 2386 2349 2182 1858 1894 2392 2915 3596DP(12) 3376 3794 2541 3287 3275 2612 1708 3062 2896 3256DP(13) 3691 3419 2987 2896 2694 2279 2162 3500 3275 4007DP(14) 4230 3273 3402 3011 2555 2141 2606 3906 3749 4448DP(15) 4609 3765 3858 2533 3054 2680 3166 4390 4249 4897

demand priority for the emergency relief points Secondly wewill study the generation of emergency demand and servicetime obeying the other random distributions Thirdly moreefficient multiobjective optimization methodologies for boththe above problem and emergency resources transportationscheduling problem will be considered

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grant nos 71102043 and 71371145) theMinistry of Education of China (Grant no MCM20125021)the Innovative Program of Shanghai Municipal EducationCommission (Grant no 14ZS123) and the ShanghaiMaritimeUniversity (Grant no 20120054)

References

[1] D Guha-Sapir G S Debarati D Hargitt P Hoyois R Belowand D BrechetThirty Years of Natural Disasters 1974ndash2003 theNumbers Presses universitaires de Louvain 2004

[2] J-B Sheu ldquoAn emergency logistics distribution approach forquick response to urgent relief demand in disastersrdquo Trans-portation Research E vol 43 no 6 pp 687ndash709 2007

[3] B S Manoj and A H Baker ldquoCommunication challenges inemergency responserdquo Communications of the ACM vol 50 no3 pp 51ndash53 2007

[4] B Balcik B M Beamon and K Smilowitz ldquoLast mile distribu-tion in humanitarian reliefrdquo Journal of Intelligent TransportationSystems vol 12 no 2 pp 51ndash63 2008

[5] A M Caunhye X F Nie and S Pokharel ldquoOptimizationmodels in emergency logistics a literature reviewrdquo Socio-Economic Planning Sciences vol 46 no 1 pp 4ndash13 2012

[6] S Coles and L Pericchi ldquoAnticipating catastrophes throughextreme valuemodellingrdquo Journal of the Royal Statistical SocietyC vol 52 no 4 pp 405ndash416 2003

[7] J R Artalejo ldquoG-networks a versatile approach for workremoval in queueing networksrdquo European Journal of Opera-tional Research vol 126 no 2 pp 233ndash249 2000

[8] W Jiang L Deng L Chen J Wu and J Li ldquoRisk assessmentand validation of flood disaster based on fuzzy mathematicsrdquoProgress in Natural Science vol 19 no 10 pp 1419ndash1425 2009

[9] V Bhaskar and P Lallement ldquoModeling a supply chain usinga network of queuesrdquo Applied Mathematical Modelling vol 34no 8 pp 2074ndash2088 2010

[10] B M Beamon ldquoHumanitarian relief chains issues and chal-lengesrdquo in Proceedings of the 34th International Conference onComputers and Industrial Engineering San Francisco CalifUSA November 2004

[11] R Oloruntoba and R Gray ldquoHumanitarian aid an agile supplychainrdquo Supply Chain Management vol 11 no 2 pp 115ndash1202006

[12] E J Lodree Jr and S Taskin ldquoSupply chain planning for hur-ricane response with wind speed information updatesrdquo Com-puters and Operations Research vol 36 no 1 pp 2ndash15 2009

[13] V Bhakoo P Singh and A Sohal ldquoCollaborative managementof inventory in Australian hospital supply chains practices andissuesrdquo Supply Chain Management vol 17 no 2 pp 217ndash2302012

[14] W Yi and A Kumar ldquoAnt colony optimization for disaster reliefoperationsrdquo Transportation Research E vol 43 no 6 pp 660ndash672 2007

[15] G-H Tzeng H-J Cheng and T D Huang ldquoMulti-objectiveoptimal planning for designing relief delivery systemsrdquo Trans-portation Research E vol 43 no 6 pp 673ndash686 2007

[16] Y-C Chiu and H Zheng ldquoReal-time mobilization decisionsfor multi-priority emergency response resources and evacua-tion groups model formulation and solutionrdquo TransportationResearch E vol 43 no 6 pp 710ndash736 2007

[17] N Altay andWG Green III ldquoORMS research in disaster oper-ations managementrdquo European Journal of Operational Researchvol 175 no 1 pp 475ndash493 2006

12 The Scientific World Journal

[18] K J Cormican D P Morton and R K Wood ldquoStochasticnetwork interdictionrdquo Operations Research vol 46 no 2 pp184ndash197 1998

[19] HOMete andZ B Zabinsky ldquoStochastic optimization ofmed-ical supply location and distribution in disaster managementrdquoInternational Journal of Production Economics vol 126 no 1 pp76ndash84 2010

[20] H Shavandi and H Mahlooji ldquoA fuzzy queuing locationmodel with a genetic algorithm for congested systemsrdquo AppliedMathematics and Computation vol 181 no 1 pp 440ndash4562006

[21] R D Galvao and R Morabito ldquoEmergency service systemsthe use of the hypercube queueing model in the solution ofprobabilistic location problemsrdquo International Transactions inOperational Research vol 15 no 5 pp 525ndash549 2008

[22] N Geroliminis M G Karlaftis and A Skabardonis ldquoA spatialqueuing model for the emergency vehicle districting andlocation problemrdquo Transportation Research B vol 43 no 7 pp798ndash811 2009

[23] C M Hosage and M F Goodchild ldquoDiscrete space location-allocation solutions from genetic algorithmsrdquo Annals of Opera-tions Research vol 6 no 2 pp 35ndash46 1986

[24] P Chakroborty ldquoGenetic algorithms for optimal urban transitnetwork designrdquo Computer-Aided Civil and Infrastructure Engi-neering vol 18 no 3 pp 184ndash200 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 The Scientific World Journal

Table 2 The service rates specifications of different servers in the network

SP 120583101584010158401

120583101584010158402

120583101584010158403

120583101584010158404

120583101584010158405

120583101584010158406

120583101584010158407

120583101584010158408

120583101584010158409

1205831015840101584010

10 9 11 8 11 12 14 16 14 17

ELC 12058310158401

12058310158402

12058310158403

12058310158404

12058310158405

12058310158406

12058310158407

12058310158408

12058310158409

120583101584010

16 18 11 10 11 12 14 16 14 10

DP 1205831

1205832

1205833

1205834

1205835

1205836

1205837

1205838

1205839

12058310

12058311

12058312

12058313

12058314

12058315

8 8 8 8 10 9 11 8 9 12 9 8 12 14 13

Table 3 The distance between supply point and emergency logistics center

SP(1) SP(2) SP(3) SP(4) SP(5) SP(6) SP(7) SP(8) SP(9) SP(10)ELC(1) 1828 1707 1197 2163 614 826 1091 973 1530 879ELC(2) 3073 2147 1624 571 2039 2056 802 1679 2554 1390ELC(3) 2271 2288 1712 1622 1419 1463 998 1102 1842 756ELC(4) 4422 3095 2462 1264 3971 3492 2429 3276 4153 3003ELC(5) 3962 2801 2716 1507 3256 2769 1449 2551 3325 2006ELC(6) 3585 2356 1923 1647 2784 2313 989 2136 2839 1437ELC(7) 2973 2968 2364 2301 2036 2149 1275 1797 2575 1471ELC(8) 2308 3261 2878 2892 1494 1401 1764 1132 1961 1933ELC(9) 2584 3643 3303 3249 2015 1165 2258 941 1600 2419ELC(10) 3038 4666 4240 4139 2831 2039 3008 1742 1405 3105

05

101520253035404550

Tim

e (ho

urs)

Total transportation timeTotal sojourn time

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 6 Optimization of total transportation time and totalsojourn time

6 Conclusions

An emergency supply chain (ESC) system is developed inthis paper for the quick response to the urgent need ofrelief in the large-scale disaster-affected areas which mainlyconsists of three chain members in series supply points(SP) emergency logistics centers (ELCs) and demand points(DP) As the timeliness is the most crucial characteristic ofthe ESC system and queuing theory is a useful tool in theanalysis of the performance of the time-dependent systemwe propose queuingmodeling for the ESC system to optimizethe emergency rescue process Based on the above queuingnetwork system a queuing minimal response time location-allocation model is established to decide the selection ofemergency logistics centers and vehicle routing decision

0

5

10

15

20

25

Total customersQueuing length

0 20 40 60 80 100 120 140 160 180 200Interaction number

Figure 7 Optimization of the total customers and the queue length

For the complexity of mathematical model the GA-basedapproach is introduced to solve the model

A case study with an assumed severe typhoon striking thesoutheast coast of Shanghai with a great probability withinthe next 100 years is conducted to assess and validate ourmodel The optimal response time of the queuing network is2071 hours and the average response time of each demandpoint is 138 h which is within a reasonable range Theoptimal emergency logistics centers for the system are DP(6)DP(7) DP(8) DP(9) and DP(10) Besides the performanceof queuing modeling for the ESC system illustrates therobustness and the potential applicability of our model

For future research we suggest the following threedirections Firstly we will investigate the queuing modelingunder consideration of the facility blocking problems and

The Scientific World Journal 11

Table 4 The distance between emergency logistics center and demand point

ELC(1) ELC(2) ELC(3) ELC(4) ELC(5) ELC(6) ELC(7) ELC(8) ELC(9) ELC(10)DP(1) 3761 4191 3253 4599 3801 3752 2439 2397 1553 739DP(2) 4271 4702 3761 4794 3945 4152 2818 2822 1898 1075DP(3) 4508 4886 3986 5028 4501 4384 3046 3025 2117 1356DP(4) 4364 4579 3697 4824 4307 4138 2798 2940 1956 1710DP(5) 5283 5416 4386 5466 5435 4986 3836 3271 2693 1917DP(6) 5214 5721 4866 5896 5294 5207 3797 4221 3252 2553DP(7) 5334 5776 5299 5862 5359 5164 3871 3851 2955 2554DP(8) 4689 4581 3763 4983 4306 4173 2818 2997 2200 1968DP(9) 3309 3624 2657 3834 3334 3154 1886 3154 3452 3235DP(10) 2678 3263 1999 2911 2741 2518 1198 2505 2360 3037DP(11) 3369 2912 2386 2349 2182 1858 1894 2392 2915 3596DP(12) 3376 3794 2541 3287 3275 2612 1708 3062 2896 3256DP(13) 3691 3419 2987 2896 2694 2279 2162 3500 3275 4007DP(14) 4230 3273 3402 3011 2555 2141 2606 3906 3749 4448DP(15) 4609 3765 3858 2533 3054 2680 3166 4390 4249 4897

demand priority for the emergency relief points Secondly wewill study the generation of emergency demand and servicetime obeying the other random distributions Thirdly moreefficient multiobjective optimization methodologies for boththe above problem and emergency resources transportationscheduling problem will be considered

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grant nos 71102043 and 71371145) theMinistry of Education of China (Grant no MCM20125021)the Innovative Program of Shanghai Municipal EducationCommission (Grant no 14ZS123) and the ShanghaiMaritimeUniversity (Grant no 20120054)

References

[1] D Guha-Sapir G S Debarati D Hargitt P Hoyois R Belowand D BrechetThirty Years of Natural Disasters 1974ndash2003 theNumbers Presses universitaires de Louvain 2004

[2] J-B Sheu ldquoAn emergency logistics distribution approach forquick response to urgent relief demand in disastersrdquo Trans-portation Research E vol 43 no 6 pp 687ndash709 2007

[3] B S Manoj and A H Baker ldquoCommunication challenges inemergency responserdquo Communications of the ACM vol 50 no3 pp 51ndash53 2007

[4] B Balcik B M Beamon and K Smilowitz ldquoLast mile distribu-tion in humanitarian reliefrdquo Journal of Intelligent TransportationSystems vol 12 no 2 pp 51ndash63 2008

[5] A M Caunhye X F Nie and S Pokharel ldquoOptimizationmodels in emergency logistics a literature reviewrdquo Socio-Economic Planning Sciences vol 46 no 1 pp 4ndash13 2012

[6] S Coles and L Pericchi ldquoAnticipating catastrophes throughextreme valuemodellingrdquo Journal of the Royal Statistical SocietyC vol 52 no 4 pp 405ndash416 2003

[7] J R Artalejo ldquoG-networks a versatile approach for workremoval in queueing networksrdquo European Journal of Opera-tional Research vol 126 no 2 pp 233ndash249 2000

[8] W Jiang L Deng L Chen J Wu and J Li ldquoRisk assessmentand validation of flood disaster based on fuzzy mathematicsrdquoProgress in Natural Science vol 19 no 10 pp 1419ndash1425 2009

[9] V Bhaskar and P Lallement ldquoModeling a supply chain usinga network of queuesrdquo Applied Mathematical Modelling vol 34no 8 pp 2074ndash2088 2010

[10] B M Beamon ldquoHumanitarian relief chains issues and chal-lengesrdquo in Proceedings of the 34th International Conference onComputers and Industrial Engineering San Francisco CalifUSA November 2004

[11] R Oloruntoba and R Gray ldquoHumanitarian aid an agile supplychainrdquo Supply Chain Management vol 11 no 2 pp 115ndash1202006

[12] E J Lodree Jr and S Taskin ldquoSupply chain planning for hur-ricane response with wind speed information updatesrdquo Com-puters and Operations Research vol 36 no 1 pp 2ndash15 2009

[13] V Bhakoo P Singh and A Sohal ldquoCollaborative managementof inventory in Australian hospital supply chains practices andissuesrdquo Supply Chain Management vol 17 no 2 pp 217ndash2302012

[14] W Yi and A Kumar ldquoAnt colony optimization for disaster reliefoperationsrdquo Transportation Research E vol 43 no 6 pp 660ndash672 2007

[15] G-H Tzeng H-J Cheng and T D Huang ldquoMulti-objectiveoptimal planning for designing relief delivery systemsrdquo Trans-portation Research E vol 43 no 6 pp 673ndash686 2007

[16] Y-C Chiu and H Zheng ldquoReal-time mobilization decisionsfor multi-priority emergency response resources and evacua-tion groups model formulation and solutionrdquo TransportationResearch E vol 43 no 6 pp 710ndash736 2007

[17] N Altay andWG Green III ldquoORMS research in disaster oper-ations managementrdquo European Journal of Operational Researchvol 175 no 1 pp 475ndash493 2006

12 The Scientific World Journal

[18] K J Cormican D P Morton and R K Wood ldquoStochasticnetwork interdictionrdquo Operations Research vol 46 no 2 pp184ndash197 1998

[19] HOMete andZ B Zabinsky ldquoStochastic optimization ofmed-ical supply location and distribution in disaster managementrdquoInternational Journal of Production Economics vol 126 no 1 pp76ndash84 2010

[20] H Shavandi and H Mahlooji ldquoA fuzzy queuing locationmodel with a genetic algorithm for congested systemsrdquo AppliedMathematics and Computation vol 181 no 1 pp 440ndash4562006

[21] R D Galvao and R Morabito ldquoEmergency service systemsthe use of the hypercube queueing model in the solution ofprobabilistic location problemsrdquo International Transactions inOperational Research vol 15 no 5 pp 525ndash549 2008

[22] N Geroliminis M G Karlaftis and A Skabardonis ldquoA spatialqueuing model for the emergency vehicle districting andlocation problemrdquo Transportation Research B vol 43 no 7 pp798ndash811 2009

[23] C M Hosage and M F Goodchild ldquoDiscrete space location-allocation solutions from genetic algorithmsrdquo Annals of Opera-tions Research vol 6 no 2 pp 35ndash46 1986

[24] P Chakroborty ldquoGenetic algorithms for optimal urban transitnetwork designrdquo Computer-Aided Civil and Infrastructure Engi-neering vol 18 no 3 pp 184ndash200 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 11

Table 4 The distance between emergency logistics center and demand point

ELC(1) ELC(2) ELC(3) ELC(4) ELC(5) ELC(6) ELC(7) ELC(8) ELC(9) ELC(10)DP(1) 3761 4191 3253 4599 3801 3752 2439 2397 1553 739DP(2) 4271 4702 3761 4794 3945 4152 2818 2822 1898 1075DP(3) 4508 4886 3986 5028 4501 4384 3046 3025 2117 1356DP(4) 4364 4579 3697 4824 4307 4138 2798 2940 1956 1710DP(5) 5283 5416 4386 5466 5435 4986 3836 3271 2693 1917DP(6) 5214 5721 4866 5896 5294 5207 3797 4221 3252 2553DP(7) 5334 5776 5299 5862 5359 5164 3871 3851 2955 2554DP(8) 4689 4581 3763 4983 4306 4173 2818 2997 2200 1968DP(9) 3309 3624 2657 3834 3334 3154 1886 3154 3452 3235DP(10) 2678 3263 1999 2911 2741 2518 1198 2505 2360 3037DP(11) 3369 2912 2386 2349 2182 1858 1894 2392 2915 3596DP(12) 3376 3794 2541 3287 3275 2612 1708 3062 2896 3256DP(13) 3691 3419 2987 2896 2694 2279 2162 3500 3275 4007DP(14) 4230 3273 3402 3011 2555 2141 2606 3906 3749 4448DP(15) 4609 3765 3858 2533 3054 2680 3166 4390 4249 4897

demand priority for the emergency relief points Secondly wewill study the generation of emergency demand and servicetime obeying the other random distributions Thirdly moreefficient multiobjective optimization methodologies for boththe above problem and emergency resources transportationscheduling problem will be considered

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grant nos 71102043 and 71371145) theMinistry of Education of China (Grant no MCM20125021)the Innovative Program of Shanghai Municipal EducationCommission (Grant no 14ZS123) and the ShanghaiMaritimeUniversity (Grant no 20120054)

References

[1] D Guha-Sapir G S Debarati D Hargitt P Hoyois R Belowand D BrechetThirty Years of Natural Disasters 1974ndash2003 theNumbers Presses universitaires de Louvain 2004

[2] J-B Sheu ldquoAn emergency logistics distribution approach forquick response to urgent relief demand in disastersrdquo Trans-portation Research E vol 43 no 6 pp 687ndash709 2007

[3] B S Manoj and A H Baker ldquoCommunication challenges inemergency responserdquo Communications of the ACM vol 50 no3 pp 51ndash53 2007

[4] B Balcik B M Beamon and K Smilowitz ldquoLast mile distribu-tion in humanitarian reliefrdquo Journal of Intelligent TransportationSystems vol 12 no 2 pp 51ndash63 2008

[5] A M Caunhye X F Nie and S Pokharel ldquoOptimizationmodels in emergency logistics a literature reviewrdquo Socio-Economic Planning Sciences vol 46 no 1 pp 4ndash13 2012

[6] S Coles and L Pericchi ldquoAnticipating catastrophes throughextreme valuemodellingrdquo Journal of the Royal Statistical SocietyC vol 52 no 4 pp 405ndash416 2003

[7] J R Artalejo ldquoG-networks a versatile approach for workremoval in queueing networksrdquo European Journal of Opera-tional Research vol 126 no 2 pp 233ndash249 2000

[8] W Jiang L Deng L Chen J Wu and J Li ldquoRisk assessmentand validation of flood disaster based on fuzzy mathematicsrdquoProgress in Natural Science vol 19 no 10 pp 1419ndash1425 2009

[9] V Bhaskar and P Lallement ldquoModeling a supply chain usinga network of queuesrdquo Applied Mathematical Modelling vol 34no 8 pp 2074ndash2088 2010

[10] B M Beamon ldquoHumanitarian relief chains issues and chal-lengesrdquo in Proceedings of the 34th International Conference onComputers and Industrial Engineering San Francisco CalifUSA November 2004

[11] R Oloruntoba and R Gray ldquoHumanitarian aid an agile supplychainrdquo Supply Chain Management vol 11 no 2 pp 115ndash1202006

[12] E J Lodree Jr and S Taskin ldquoSupply chain planning for hur-ricane response with wind speed information updatesrdquo Com-puters and Operations Research vol 36 no 1 pp 2ndash15 2009

[13] V Bhakoo P Singh and A Sohal ldquoCollaborative managementof inventory in Australian hospital supply chains practices andissuesrdquo Supply Chain Management vol 17 no 2 pp 217ndash2302012

[14] W Yi and A Kumar ldquoAnt colony optimization for disaster reliefoperationsrdquo Transportation Research E vol 43 no 6 pp 660ndash672 2007

[15] G-H Tzeng H-J Cheng and T D Huang ldquoMulti-objectiveoptimal planning for designing relief delivery systemsrdquo Trans-portation Research E vol 43 no 6 pp 673ndash686 2007

[16] Y-C Chiu and H Zheng ldquoReal-time mobilization decisionsfor multi-priority emergency response resources and evacua-tion groups model formulation and solutionrdquo TransportationResearch E vol 43 no 6 pp 710ndash736 2007

[17] N Altay andWG Green III ldquoORMS research in disaster oper-ations managementrdquo European Journal of Operational Researchvol 175 no 1 pp 475ndash493 2006

12 The Scientific World Journal

[18] K J Cormican D P Morton and R K Wood ldquoStochasticnetwork interdictionrdquo Operations Research vol 46 no 2 pp184ndash197 1998

[19] HOMete andZ B Zabinsky ldquoStochastic optimization ofmed-ical supply location and distribution in disaster managementrdquoInternational Journal of Production Economics vol 126 no 1 pp76ndash84 2010

[20] H Shavandi and H Mahlooji ldquoA fuzzy queuing locationmodel with a genetic algorithm for congested systemsrdquo AppliedMathematics and Computation vol 181 no 1 pp 440ndash4562006

[21] R D Galvao and R Morabito ldquoEmergency service systemsthe use of the hypercube queueing model in the solution ofprobabilistic location problemsrdquo International Transactions inOperational Research vol 15 no 5 pp 525ndash549 2008

[22] N Geroliminis M G Karlaftis and A Skabardonis ldquoA spatialqueuing model for the emergency vehicle districting andlocation problemrdquo Transportation Research B vol 43 no 7 pp798ndash811 2009

[23] C M Hosage and M F Goodchild ldquoDiscrete space location-allocation solutions from genetic algorithmsrdquo Annals of Opera-tions Research vol 6 no 2 pp 35ndash46 1986

[24] P Chakroborty ldquoGenetic algorithms for optimal urban transitnetwork designrdquo Computer-Aided Civil and Infrastructure Engi-neering vol 18 no 3 pp 184ndash200 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 The Scientific World Journal

[18] K J Cormican D P Morton and R K Wood ldquoStochasticnetwork interdictionrdquo Operations Research vol 46 no 2 pp184ndash197 1998

[19] HOMete andZ B Zabinsky ldquoStochastic optimization ofmed-ical supply location and distribution in disaster managementrdquoInternational Journal of Production Economics vol 126 no 1 pp76ndash84 2010

[20] H Shavandi and H Mahlooji ldquoA fuzzy queuing locationmodel with a genetic algorithm for congested systemsrdquo AppliedMathematics and Computation vol 181 no 1 pp 440ndash4562006

[21] R D Galvao and R Morabito ldquoEmergency service systemsthe use of the hypercube queueing model in the solution ofprobabilistic location problemsrdquo International Transactions inOperational Research vol 15 no 5 pp 525ndash549 2008

[22] N Geroliminis M G Karlaftis and A Skabardonis ldquoA spatialqueuing model for the emergency vehicle districting andlocation problemrdquo Transportation Research B vol 43 no 7 pp798ndash811 2009

[23] C M Hosage and M F Goodchild ldquoDiscrete space location-allocation solutions from genetic algorithmsrdquo Annals of Opera-tions Research vol 6 no 2 pp 35ndash46 1986

[24] P Chakroborty ldquoGenetic algorithms for optimal urban transitnetwork designrdquo Computer-Aided Civil and Infrastructure Engi-neering vol 18 no 3 pp 184ndash200 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of