Research ArticleModel and Resilience Analysis for Handling ChainSystems in Container Ports

Bowei Xu 1 Junjun Li 2 Yongsheng Yang 1

Huafeng Wu2 and Octavian Postolache 3

1 Institute of Logistics Science amp Engineering Shanghai Maritime University Shanghai 201306 China2Merchant Marine College Shanghai Maritime University Shanghai 201306 China3Instituto de Telecomunicacoes ISCTE-IUL Av Das Forcas Armadas Lisbon 1049-001 Portugal

Correspondence should be addressed to Junjun Li jsliljj163com

Received 1 April 2019 Accepted 2 July 2019 Published 22 July 2019

Academic Editor Marcin Mrugalski

Copyright copy 2019 Bowei Xu et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Container ports are critical components of global port and shipping supply chain (PSSC) systems Their handling operationresiliencies can affect their performance along with those of the overall port and shipping supply chain According to thecharacteristics of the container handling operations this study establishes a modeling paradigm for quantifying the resilience ofhandling chain system (HCS) in container ports Considering the nonnegative arrive rate and the container handling completionrate with the upper limit in real container port environments nonlinear links have been added into the model of the HCS Theresilience of the HCS was analyzed according to the different pole distributions of unsatisfied freight requirement transfer functionSimulation results show that the upper limit of the container handling completion rate has a significant impact on the resilienceof the HCS The contributions herein demonstrate a starting point in the development of a quantitative resilience decision makingframework and mitigating the negative impacts for port authorities and other players in the PSSC

1 Introduction

In the global supply chain era container ports are becomingincreasingly important for modern societies As the essentialpoint of intersection between shipping and traffic containerports have evolved from the traditional functions of cargohandling and storage to becoming integral parts of globalport and shipping supply chain (PSSC) (Wang and Cheng2010) [1] They play a significant role in Chinese and worldeconomies serving as the backbone to the import and exporttrade and supply chain networksTheir development not onlyis directly related to the promotion of the ldquoOne Belt and OneRoad Initiativerdquo but is also related to the implementation ofthe national strategy ldquoYangtze River Economic Beltrdquo Chinahas become the world largest container gathering area (Wanget al 2017) [2] In 2017 therewere sevenChinese ports amongthe top ten container ports in the world

The container port and shipping supply chain is a servicesupply chain with the container port enterprise at its core and

smart information technology as the means It has a servicenetwork structure composed of upstream and downstreamenterprises to add value through customs and services (Jianget al 2018) [3] Under global economic integration large-size container vessels and liner company alliance containerports are no longer considered as solitary nodes (Song andPanayides 2007) [4] its competitive position is increasinglydependent on handling operation resilience and synergieswith the transport nodes within port and shipping supplychain networks Correspondingly handling operation of acontainer port should match and collaborate with its freightrequirement for not only providing value-added services toport users (Pettit and Beresford 2009Woo et al 2013) [5 6]but also ensuring the port capacity and performance (Houand Geerlings 2016 Roslashdseth et al 2018) [7 8]

11 Literature Review With respect to port and shippingsupply chain most of previous studies have focused onterminal operating company and customers (Tongzon et al

HindawiComplexityVolume 2019 Article ID 9812651 12 pageshttpsdoiorg10115520199812651

2 Complexity

2009 Hou andGeerlings 2016) [7 9] conceptual discussionsof resilience with several qualitative definitions (Ta et al2009) [10] port supply chain integration strategies andits relationship with port performance (Woo et al 2013)[6] and so on Port resiliency is the ability to resumenormal operations at pre-disruptive performance levels aftera disruptive adverse event and maintain normal operationsand performance over a long period of disruptive adversechange (Gharehgozli et al 2017) [11] Although there isno generally accepted definition of PSSC resilience thereis consensus among many researchers that resilience is thecommon characteristic of efficient PSSCs with expecteddelivery capacity of container ports Such collaboration andcoordination among the container port and its players willalso facilitate the establishment of the resilient port which inturn enables a PSSC more flexible and proactive to uncertainenvironments (Loh andThai 2016) [12] Specifically to PSSCresilience Loh and Thai (2016) [12] introduced a port-related supply chain disruptions management model thatincorporated the application of risk management businesscontinuity management and quality management theorieswith the purpose of increasing port resilience such that supplychain continuity was enhanced Gharehgozli et al (2017) [11]proposed a conceptual framework for evaluating how portscurrently strategized against potential risks and how theyplanned to ensure port resiliency Zavitsas et al (2018) [13]considered the impact of Emission Control Areas and estab-lished a link between environmental and network resilienceperformance for maritime supply chains using operationalcost and SOx emissions cost metrics Despite the importantrole of resilience in enhancing performance improvementsfor PSSCs no studies have been identified to quantify theresilience of handling chain system (HCS) in container portsand examine the real impact of nonlinearities on operationresilience such as control method while including thoseimportant players in the PSSC However this is not an easytask as there are many dynamics at play It is critical forthe success of a PSSC to understand the uncertainties andtheir impact and propose the corresponding measurementsto cope with themThus there is a need for quantitative mea-surement instruments to appropriately evaluate the resilienceof HCS in container ports

System dynamics intervention has been promoted toidentify how structure and decision policies generate systembehavior and to implement structural and policy-orientedsolutions (Saleh et al 2010 Yang et al 2019) [14 15] Numer-ous studies have made efforts to understand and definesystem dynamics in manufacturing supply chains Theyhave designed production and inventory control systems toimprove transportation and production operations as wellas to increase service and financial performance of thosesystems (John et al 1994) [16] Gabbar (2008) [17] proposed amodel-based control mechanism and intelligent control layerto control the operation of chained production enterprises Liet al (2016) [18] developed a fuzzy comprehensive evaluationmodel of feature operation chain selection to realize the selec-tion of possible feature operation chains based on processingrequirements Virginia Spiegler and Mohamed Naim (2017)[19] utilized a nonlinear production and inventory control

model to provide insight into the impact of system constraintsand therefore facilitated a more effective system designAlmaktoom (2017) [20] introduced a method of quantifyingthe reliability of an inventory management system and devel-oped a reliability-based robust design optimization modelto optimally allocate and schedule time while consideringuncertainty associated with inventory movement Ma et al(2018) [21] investigated the impact of lead-times of retailersand manufacture forecasting precision callback index andmarketing share on the bullwhip effect of both retailersand manufacture In this sense it is vastly accepted thatproduction planning and control systems play a crucial rolein improving overall performance (Almaktoom 2017) [20]Understanding how differently they impact on performancewould enable managers to select between different controlstrategies in the resilience operation field However they arefar from being widespread in the practice of PSSC Thisunderscores the need for further exploring the dynamics ofthese PSSC systems

12 Motivation for Work Container ports are mainly cate-gorized as inland and sea container ports Both inland andsea container ports have the similar handling operationsGeneral model of main inland port operations proposedby Pant et al (2014) [22] is also suitable for containerports The landside and seaside operations of a containerterminal are its main container handling operationsThey aredivided into five components as illustrated in Figure 1 (i)deliveryreceipt including the arrival of exported containersand departure of imported containers (ii) landsideseasideyard crane loading and unloading operations defined asthe temporary storage of containers at the yard (iii) yardtruckAGV operations defined as the horizontal transporta-tion of containers between berth and yard (iv) gantry craneloading and unloading operations used to transfer containercargoes to and from port docks and (v) container vesselshipment or the departure of containers for exports andthe arrival of containers for imports (Pant et al 2014) [22]These individual operations are interrelated to form a tightlyconnected container handling chain system (HCS)

The container port and shipping supply chain is adynamic system with bullwhip effect that involves the con-stant flow of information containers services and fundsbetween different stages Currently the research of bullwhipeffect mainly focuses on theoretical study of general modelof manufacturing supply chain while some service supplychain models with practical significance have not beenexplored yet Each stage of the container port HCS performsdifferent processes and interacts with other stages of thesystem As propagated through the HCS in container portsoscillations such as uneven container flow and large-sizecontainer vessels have been associated with serious handlingpressure increased operation costs and poor service levelsand therefore need to be properly understood and controlledThe dynamics of a container port product operation chaincan significantly impact the performance of upstream anddownstream operationsThese dynamics are normally drivenby the application of different handling control policies and

Complexity 3

Container truck Yard

Gantry craneContainer vessel

Import

Export

Delivery and Receipt

Landside Yard crane

Loading and unloading

Horizontal transportationShipment Loading and unloading

AGV Seaside Yard crane

Loading and unloading

Storage

Seaside Operations

Move

Landside Operations

Loading Move

Loading

Unloading

LoadingDelivery

Receipt Unloading

Unloading

Unloading

Loading

Figure 1 General model of container handling operations in automated container ports

have the potential to cause discrepancies between freightrequirements and handling operations

In this regard we have the intention of adapting theAutomated Pipeline Inventory and Order Based ProductionControl System (APIOBPCS) (John et al 1994) [16] inmanu-facturing supply chains to PSSC for improving the operationperformance of container ports This study first identifiescharacteristics of the handling operations by establishing amodeling paradigm for quantifying the resilience of HCS incontainer ports In the next step this study considers both thenonnegative arrive rate of container cargo and the containerhandling capacity with the upper limit explicitly Nonlinearlinks have been added into themodel of theHCS In additionfor the first time this study has conducted complete analysisof the HCS resilience under varying pole distributions ofunsatisfied freight requirement (UFR) transfer functionmoreprecisely Using the proposed model we consistently developfinal values of unsatisfied freight requirement (119906119891119903(119905)) andcontainer handling completion rate (119888119900119898119903119886119905119890(119905)) within thesteady state condition Finally simulation results show thatthe upper limit of the container handling completion ratehas a significant impact on the resilience of HCS Thecontributions herein will not only ensure decision making byport manager to protect ports and make them more resilientbut also promote socially efficient handling chains

2 Models of Container Port HandlingChain System

21 Linear Model The container port and shipping supplychain is a service supply chain system as a network ofsuppliers service providers consumers and other supporting

units that performs the function of the delivery of containersto customers (Wang et al 2015) [23] Container port andshipping services are important in service supply chain asthey significantly influence freight requirement (FR) Obvi-ously proper freight requirement management and relatedoperational measures are crucial to the success of containerport and shipping supply chain A novel model for HCS incontainer ports shown in Figure 2 is adapted fromAPIOBPCSmodel in manufacturing supply chain (John et al 1994)[16] which extended the original Inventory and Order-basedProduction Control System (IOBPCS) archetype (Towill1982) [24] by incorporating an automatic work-in-progress(WIP) feedback loop The APIOBPCS model (John et al1994) [16] a base framework for a production planning andcontrol system can be expressed as follows the order placedis based on the forecast of customer demand plus a fraction(1119879i proportional controller for inventory adjustment) of thediscrepancy between actual and desired inventory levels plusa fraction (1119879w proportional controller forWIP adjustment)of the discrepancy between actualWIP and targetWIP levels(Zhou et al 2017) [25] One of the main objectives whendesigning an APIOBPCS is to obtain a desirable inventorylevel avoiding severe order and inventory variability in theface of exterior disturbances (Almaktoom 2017) [20]

The Laplace transform is utilized in Figure 2 for our con-trol engineering analysis coupled with simulation modelings is a complex frequency in Laplace transform Parametersand segments are detailed in Table 1 In the following workuppercase and lowercase words such as FR and fr areused in the frequency and time domains respectively Theinput and output of this model are freight requirement (FR)and unsatisfied freight requirement (UFR) respectively The

4 Complexity

EFRIP

UFR

FRIP

FR ++ +

+

-

+_

AVFR COMRATEARATE

+ _DFRIP

Lead policy

feedbackUFR

FR policy

UFR policy

FRIP policyFRIP feedback

1

1 + TFs

TQ

1

1 + TGs

1

s

1

s

1

TU

1

TI

Figure 2 Linear model of handling chain system in container ports

Table 1 Parameters and segments in linear model

FR Freight requirement AVFR Average FR ARATE Arrive rate

COMRATE Container handlingcompletion rate UFR Unsatisfied FR FRIP FR in process

DFRIP Desired FRIP EFRIP Error in FRIP 119879119880Adjustment time of

UFR

119879119865Time constant of FR

forecast 119879119866Real lead-time of

handling operations 119879119876Expected lead-time ofhandling operations

119879119868Adjustment time of

EFRIP1

1 + 119879119865119904Freight requirement

policy1

1 + 119879119866119904Lead policy of

handling operation

container arrive rate (ARATE) takes the integrals of averagefreight requirement (AVFR) error in freight requirement inprocess (EFRIP) and unsatisfied freight requirements (UFR)into account 1(1 + 119879119865119904) and 1(1 + 119879119866119904) are employedas the freight requirement policy and the lead policy ofhandling operation in container port respectively Containerhandling completion rate (COMRATE) can be taken as theinput of the next 119879119880 denotes adjustment time of UFR 119879119865denotes time constant of FR forecast 119879119866 denotes real lead-time of handling operations 119879119876 denotes expected lead-timeof handling operations and 119879119868 denotes adjustment time ofEFRIP Controllable decision variables 119879119880 119879119865 119879119866 119879119876 and119879119868 are all positive real numbers

As shown in Figure 2 the HCS model in container portscan be expressed as follows the container arrive rate (ARATE)placed is based on the forecast of freight requirement (FR)plus a fraction (1119879119880 the proportional controller for unsatis-fied freight requirement adjustment) of the unsatisfied freightrequirement (UFR) plus a fraction (1119879119868 the proportionalcontroller for FR in process adjustment) of the discrepancybetween actual FR in process (FRIP) and desired FRIPlevels The differences between APIOBPCS and HCS incontainer ports are not only with or without a desiredinventory level but also the meaning of each terminologyin HCS has been updated according to the characteristics ofcontainer port and shipping supply chain such as parameter119879119865 related to handling operations planning in container

terminal parameters 119879119866 and 119879119876 related to the concentratedreceipt of export containers ahead of container vessel arrivalparameter 119879119868 related to the fast loading and unloading ofcontainer vessels and parameter 119879119880 related to the detentiontime of containers in port yard Each parameter in the HCSmodel of container ports originates from handling operationpracticeThey undoubtedly have important practical guidingsignificance for container port operation management Themain purpose of HCS in container ports is to obtain anunsatisfied freight requirement (UFR) as small as possible oreven equal to zero and maintain optimal handling operationresilience

As described in Figure 2 it is possible to determine theUFR transfer function the COMRATE transfer function andthe ARATE transfer function in relation to the input FR

119880119865119877119865119877

=(119879119866 + 119879119868 + 119879119868119879119866119904) (1 + 119879119865119904) minus 119879119876 minus 119879119868

119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

(1)

119862119874119872119877119860119879119864119865119877

=119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904

119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

(2)

Complexity 5

119860119877119860119879119864119865119877

=(1 + 119879119866119904) [119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904]

119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

(3)

(1) Steady State Condition One of the poles is easily identified(1199010=-1119879119865) The other two poles are set as 1199011 and 1199012 Then

1199011 + 1199012 =1119879119868

+ 1119879119866

11990111199012 =1

(119879119880119879119866)

(4)

Then it can be seen that the steady state condition is 1119879119868+1119879119866 gt 0 1(119879119880119879119866) gt 0 and 1119879119865 gt 0 Because 119879119880 119879119865 119879119866119879119876 and 119879119868 are all positive real numbers this system is stable

(2) Final Value of 119906119891119903(119905) and 119888119900119898119903119886119905119890(119905) In deriving transferfunctions in complex frequency domain we exploit theclassic step input as it helps to develop insights into dynamicbehavior We use the step response to assess the systemrsquosability to cope with oscillations in the linear HCS The stepresponse is powerful as from this simple FR input thesize and quantity of the subsequent 119906119891119903(infin) 119888119900119898119903119886119905119890(infin)and 119886119903119886119905119890(infin) can be readily determined providing a richunderstanding of the dynamics of the linear HCS If FR is aunit step signal 119906119891119903(infin) 119888119900119898119903119886119905119890(infin) and 119886119903119886119905119890(infin) can beobtained by final value theorem

119906119891119903 (infin) = lim119904997888rarr0

119904 sdot 119880119865119877 (119904)

= lim119904997888rarr0

(119879119866 + 119879119868 + 119879119868119879119866119904) (1 + 119879119865119904) minus 119879119876 minus 119879119868119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

=119879119880 (119879119866 minus 119879119876)

119879119868

(5)

119888119900119898119903119886119905119890 (infin) = lim119904997888rarr0

119904 sdot 119862119874119872119877119860119879119864 (119904)

= lim119904997888rarr0

119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

= 119879119868119879119880119879119868119879119880

= 1

(6)

119886119903119886119905119890 (infin) = lim119904997888rarr0

119904 sdot 119860119877119860119879119864 (119904)

= lim119904997888rarr0

(1 + 119879119866119904) [119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904]119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

= 119879119868119879119865119879119868119879119865

= 1

(7)

It can be seen that 119906119891119903(infin) depends on 119879119880 119879119868 and thedifference between 119879119866 and 119879119876 119906119891119903(infin) would be 0 onlywhen 119879119866 = 119879119876 119888119900119898119903119886119905119890(119905) and 119886119903119886119905119890(119905) will approach 1when 119905 997888rarr infin Therefore when the HCS reaches its steadystate 119888119900119898119903119886119905119890(119905) and119891119903(119905) are balanced and 119906119891119903(119905) no longerchanges

22 Nonlinear Model In real environments of containerports the value of ARATE cannot be negative althoughmathematically it holds The upper value of COMRATEis limited however the linear model in Figure 2 cannotguarantee these two nonlinear natures To reflect them anonlinearHCSmodel consisting of twononlinear segments isgiven in Figure 3 ARATErsquo is the theoretical arrive rate whichequals to 119886V119891119903(119905) + 119906119891119903(119905)119879119880 + 119890119891119903119894119901(119905)119879119868 COMRATErsquo isthe theoretical container handling completion rate which isonly calculated by 119886119903119886119905119890(119905) and the handling operation policy1(1 + 119879119866119904) COMRATEm is the upper limit of COMRATECOMRATEm should not be less than 119891119903(119905) over time other-wise 119906119891119903(119905) will continuously increase

If COMRATEm is equal to 119891119903(119905) or slightly larger than119891119903(119905) the actual container handling completion rate wouldnot process the accumulated freight requirement timelyThen119906119891119903(infin) may be larger than 119879119880(119879119866 minus 119879119876)119879119868 If 119906119891119903(119905) is toolarge it would congest the HCS

3 Resilience for Handling Chain System

To understand the resilience for HCS in container portswe use the integral of time multiplied by the absolute error(ITAE) as a benchmark ITAE emphasizes long-durationerrors It is recommended for the analyses of systems whichrequire a fast settling time The minimum value of ITAEcorresponds to the best response and recoverywith the lowestdeviation from the target or readiness (Virginia et al 2012)[26] The ITAE is given by 119868119879119860119864 = int+infin

0119905 sdot |119890(119905)|119889119905 =

lim120575119905997888rarr0suminfin119905=0 119905|119890(119905)|120575119905 where 119890(119905) = 119906119891119903(119905) minus 119906119891119903(infin) To

facilitate uniform analysis 119906119891119903(infin) = 119879119880(119879119866 minus119879119876)119879119868 whichis consistent with (5)

In linear HCS 119906119891119903(119905) can be calculated in the followingfour cases below where A B and 119862 are coefficients related tothe system poles and119863 is the coefficient of the step input pole(s=0)

Case 1 (distinct poles real andor complex 1199011 = 1199012 = 1199010)(a) Δ 1 gt 0 (Δ 1 = 119879119880(119879119866 + 119879119868)

2 minus 41198792119868119879119866)

119906119891119903 (119905) = 1198601198901199010119905 + 1198611198901199011119905 + 1198621198901199012119905 + 119863 (8)

(b) Δ 1 lt 0

119906119891119903 (119905) = 1198601198901199010119905 + 1198611198901205901119905 (cos1205961119905 minus 119895 sin1205961119905)

+ 1198621198901205902119905 (cos1205962119905 minus 119895 sin1205962119905) + 119863(9)

In Case 1 (a) and (b) the coefficient values of A B and 119862are

119860 =119879119868 + 119879119866 minus 11987911986611987911986811986311990121199011119879119866119879119868 (1199011 minus 1199010) (1199012 minus 1199010)

119861 =119879119868119879119865 + 119879119866119879119865 + 119879119866119879119868 + 119879119866119879119868119879119865 (1199011 minus 11986311990121199010)

119879119866119879119868119879119865 (1199011 minus 1199012) (1199011 minus 1199010)

119862 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199012 minus 11986311990101199011)

119879119866119879119868119879119865 (1199012 minus 1199010) (1199012 minus 1199011)

(10)

6 Complexity

EFRIP

UFR

FRIP

FR ++ +

+

+_

AVFR COMRATEARATE

+ _

DFRIP

0

_

ARATE COMRATE

feedbackUFR

Lead policy

UFR policy

FR policy

FRIP policy

1

1 + TFs

TQ

1

1 + TGs

1

s

1

s

FRIP feedback

COMRATEＧ

1

TU

1

TI

Figure 3 Nonlinear model of handling chain system in container ports

Case 2 (two repeated poles 1199011=1199010 or 1199012=1199010) (a) 1199011 = 11990121199011 = 1199010

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 1198621198901199012119905 + 119863 (11)

where

119860

= minus119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 [1199012 + 1198631199012 (1199012 minus 21199010)]

119879119866119879119868119879119865 (1199010 minus 1199012)2

119861 =119879119866 + 119879119868 minus 11987911986611987911986811986311990101199012

119879119866119879119868 (1199010 minus 1199012)

119862 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199012 minus 11986311990120)

119879119866119879119868119879119865 (1199010 minus 1199012)2

(12)

(b) 1199011 = 1199012 1199012 = 1199010

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 1198621198901199011119905 + 119863 (13)

where

119860

= minus119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 [1199011 + 1198631199011 (1199011 minus 21199010)]

119879119866119879119868119879119865 (1199010 minus 1199011)2

119861 =119879119866 + 119879119868 minus 11987911986611987911986811986311990101199011

119879119866119879119868 (1199010 minus 1199011)

119862 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199011 minus 11986311990120)

119879119866119879119868119879119865 (1199010 minus 1199011)2

(14)

Case 3 (two repeated poles 1199011 = 1199012 = 1199010)

119906119891119903 (119905) = 1198601198901199011119905 + 1198611199051198901199011119905 + 1198621198901199010119905 + 119863 (15)

where

119860 = minus119879119866 + 119879119868 + 1198791198661198791198681198631199010 (1199010 minus 21199011)

119879119866119879119868 (1199011 minus 1199010)2

119861 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199011 minus 11986311990101199011)

119879119866119879119868119879119865 (1199011 minus 1199010)

119862 =119879119866 + 119879119868 minus 11987911986611987911986811986311990121119879119866119879119868 (1199011 minus 1199010)

2

(16)

Case 4 (three repeated poles 1199011 = 1199012 = 1199010)

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 0511986211990521198901199010119905 + 119863 (17)

where119860 = minus119863

119861 = 1 + 1198631199010

119862 = 1119879119866

+ 1119879119868

minus 11986311990120

(18)

In all cases119863 = 119879119880(119879119866 minus 119879119876)119879119868 In all cases except Case1 (b)

1199011 =minus (119879119866 + 119879119868)radic119879119880 + radicΔ 1

2119879119868119879119866radic119879119880

1199012 =minus (119879119866 + 119879119868)radic119879119880 minus radicΔ 1

2119879119868119879119866radic119879119880

(19)

In Case 1 (b)

1199011 = 1205901 + 1198951205961

1199012 = 1205902 + 1198951205962

1205901 = 1205902 =minus (119879119866 + 119879119868)2119879119868119879119866

12059612 =plusmnradicminusΔ 1

2119879119868119879119866radic119879119880

(20)

Complexity 7

+1

+j

+1

+j

+1

+j

Case 1(a) Case 1(b) Case 2(a)

+1

+j

+1

+j

+1

+j

Case 2(b) Case 3 Case 4

p2

p2p2

p1

p1

p1

p0

p0p0

p2 = p0p1 = p2

p1 = p0

p1 = p2 = p0

Figure 4 Pole distributions of UFR transfer function

Thepole distributions ofUFR transfer function are shownin Figure 4

It can be seen that 119906119891119903(119905) approaches119863 in Cases 1 through4 This is consistent with (5) in which 119906119891119903(infin) is equal to119863

In nonlinear HCS if aratersquo(t)lt0 or comratersquo(t)gt COM-RATEm 119906119891119903(119905) would be different from that in a linearHCS Specifically if comratersquo(t)gt COMRATEm ufr(t) wouldincrease because the actual container handling completionrate cannot process the freight requirement in time and119906119891119903(infin)may be larger than D ITAE would be different fromthat of linear HCS If 119906119891119903(infin) gt 119863 ITAE would increaseindefinitely

4 Resilience Analysis of HCS inContainer Ports

The quantitative resilience framework for HCS developedin this work is deployed with an illustrative example of acontainer port in China Its HCS serves an important rolein container flows around the world and its resilience isvital to the larger multimodal transportation system and thePSSC It has four container docks Under normal operationsits annual container throughput would be around 4 millionTEU 119879119876 would be between 0 and 3 days while 119879119868 would bebetween 0 and 1 day For the simulationmodels and resilienceanalysis it is assumed that 119879119876=15 days and 119879119868=05 days 119879119866is 1 day 119879119865 is between 2 and 30 days while 119879119880 is between0 and 10 days Based on the modeling concepts developedin the previous sections resiliencies of two different HCSsare studied separately (i) linear HCS and (ii) nonlinear HCSSimulation code was written in MATLAB If any of the HCSwere to oscillate and become inoperable it might stop theflow of containers handled by that dock and result in portcongestions

41 Resilience of Linear HCS Linear HCSs are without limitsforARATE andCOMRATE Based on the above data ITAE oflinear HCS is calculated according to Section 3The results ofITAE are calculated and plotted when 119879119865 isin [2 days 30 days]and 119879119880 isin (0 day 10 days] 3D mesh and contour maps ofITAE are shown in Figures 5(a) and 5(b) respectively InFigure 5 it can be seen that ITAE increases gradually withincreases in 119879119865 and 119879119880 in most cases But this change is notcompletely linear When 119879119865 and 119879119880 approach 2 days and10 days respectively ITAE will gradually increase within acertain range We are concerned about the values of 119879119865 and119879119880 when ITAE reaches its minimum value However they arenot easily found in Figure 5 For this reason 3D mesh andcontour maps of 1ITAE are plotted in Figures 6(a) and 6(b)respectively From Figure 6 1ITAE reaches its maximumvalue when 119879119865 = 2 days and 119879119880 997888rarr 0 day (119879119880 = 0)The minimum of ITAE is approximately equal to 52812 Thisimplies that carefully setting the parameters 119879119865 and 119879119880 mayimprove dynamic performance of the linearHCS In additionit suggests that the closer the collaboration of port authoritiesand other players in PSSC is the better the resilience of linearHCS is

42 Resilience of Nonlinear HCS Nonlinear HCS focuses onthe limits for ARATE and COMRATE How the ARATE andCOMRATE influence the nonlinear HCS dynamics is to beexamined In nonlinear HCS the smaller COMRATEm isthe more likely it is that COMRATErsquo(t)gt COMRATEm andthe difference between 119906119891119903(119905) in nonlinear and linear HCSsis larger Correspondingly the difference between ITAE ofnonlinear and linear HCSs is greater In order to facilitatethe obvious distinction COMRATEm =105 here When 119879119865 isin[2 days 30 days] and 119879119880 isin (0 day 10 days] 3D mesh andcontour maps of ITAE are given in Figures 7(a) and 7(b)

8 Complexity

0

5

10

10

20

300

2

4

6

8

TUTF

ITA

E

times104

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 5 ITAE of linear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 6 1 ITAE of linear handling chain system

respectively 3D mesh and contour maps of 1ITAE are givenin Figures 8(a) and 8(b) respectively In Figure 8(a) someareas in which 119906119891119903(infin) gt 119863 and ITAE is infinite are notplotted These areas are the light blue shaded regions inFigure 8(b) From Figure 8 ITAE reaches its minimum value(1ITAEasymp0092 and ITAEasymp1082) when 119879119865 = 2 days 119879119880 =24days From a comparison between Figures 5 and 7 ITAE(approximately 6660) is infinitymore thanhalf of the entirearea It reveals that the impact of COMRATEm on containerport performance is obvious Considering these results canbe determined that if COMRATEm are just applied to thenonlinear HCS modifying the setting of parameter 119879119880 willimprove the resilience of the nonlinear HCS

In Figure 6 there are two regions in which 1ITAEgradually increases and reaches the peaks of these regions

One region is close to the point of 119879119880=0 day and 119879119865=2 daysand the other region is close to the point of 119879119880=24 daysand 119879119865=2 days 1ITAE gradually reaches its maximum whenapproaching the point of 119879119880=0 day and 119879119865=2 days This ismost likely due to the fact that they are dependent processesThe faster the unsatisfied freight requirement is deliveredthe sooner the linear HCS will recover to normal operationHence reasonably setting the parameters119879119865 and119879119880 can solveport congestion problems but it is important to take intoaccount that each modification will affect the resilience oflinear HCS

In Figure 8 1ITAE becomes very small in the region closeto the point of 119879U=0 day and 119879F=2 days From Figure 8(b)ITAE becomes infinite in this region The new maximum of1ITAE is in the point of 119879U=24 days and 119879F=2 days The

Complexity 9

24

68

10

10

20

300

1

2

3

4

TUTF

ITA

E

times104

(a) 3D mesh map

2 4 6 8 10TU

TF

30

25

20

15

10

5

(b) Contour map

Figure 7 ITAE of nonlinear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 8 1ITAE of nonlinear handling chain system

results of the two HCSs discussed above show the usefulnessof the resilience measurement is highlighted in the analysisThese can be used for the PSSCrsquos decisions

43 COMRATEmrsquos Influence on Nonlinear HCS In orderto examine how the COMRATEm influences the nonlinearHCS dynamics the point of 119879119865=2 days and 119879119880=02 day istaken as an example and ufr(t) arate(t) and comrate(t) areplotted in Figure 9 with different COMRATEm (COMRATEm=100 105 110 115 and 120) date isin [0 day 40 days] Inorder to clearly distinguish these curves the horizontal axiscoordinates only take 0sim15 days in Figures 9(a) and 9(b)They eventually reach stable values The stable value of ufr(t)is different with different COMRATEm the stable values ofarate(t) and comrate(t) are always 1

From Figure 9(a) the values of ufr(t) with differentCOMRATEm are the same before it reaches its peak After itspeak the values of ufr(t) with differentCOMRATEm are quitedifferent from one another When COMRATEm =1 ufr(t)maintains its peak value after reaching it When COMRATEm=12 ufr(t) eventually is equal to D(D=02) which is thesame as that in the linear HCS From Figure 9(b) in dates2 through 8 the enhanced ufr(t) makes arate(t) higherwhen COMRATEm is lower From Figure 9(c) comrate(t) islimited by COMRATEm in dates 2sim8 This limitation resultsin the cumulative increase of ufr(t) Due to the limited valueof COMRATE there are cases where dynamic behaviorsof nonlinear HCS cause fr(t) not to be handled timelyeventually resulting in the increased ufr(t) Meanwhile therewould be change noted in arate(t)

10 Complexity

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 15 20 25 30 35 400Date

0

02

04

06

08

1U

ncom

plet

ed F

reig

ht R

equi

rem

ent

(a) ufr(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

05

1

15

2

25

Arr

ive r

ate

(b) arate(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

02

04

06

08

1

12

Com

plet

e rat

e

(c) comrate(t)

Figure 9 States and outputs in nonlinear HCS with different COMRATEm

To investigate the influence of COMRATEm on ITAEthe proportion of ITAE=infin in nonlinear HCS with differentCOMRATEm is plotted in Figure 10 The observation ofdifferent COMRATEm suggests the following all values ofITAE are infinite when COMRATEm le 1 When COM-RATEm = 1sim11 the proportion of ITAE=infin clearly decreasesWhen COMRATEm = 11sim12 the proportion of ITAE=infinis close to 0 and decreases slowly When COMRATEm =12 the proportion of ITAE=infin equals 0 which is the sameas that in the linear HCS These observations show thatthe fluctuation of ufr(t) could be alleviated if the non-linear HCS is carefully designed although it seems to beinevitable

5 Conclusion

The emphasis of container port authorities and other playersin the PSSC has shifted from traditional passive managementto active response highlighting the need to quantify theresilience of HCS for inevitable oscillationsThis work buildsand analyzes linear and nonlinearHCSmodels Resilience forHCS in container ports is measured by ITAE of unsatisfiedfreight requirement It provides a new perspective on PSSCpartners to plan for HCS where resilience could guide theselection of control parameters HCSs in container portsare taken as example to analyze resilience and it is foundthat Comratem has significant influence on resilience for

Complexity 11

095 1 105 11 115 120

02

04

06

08

1

COMRATEm

Prop

ortio

n of

ITA

E =infin

Figure 10 Proportion of ITAE =infin in nonlinear HCS with differentCOMRATEm

nonlinear model These contributions serve as a startingpoint in the development of a quantitative resilience decisionmaking framework for PSSC Future work will consider theresilience and recovery strategies under stochastic freightrequirements given the two HCSs

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Social Science Foun-dation project of China (No 17CGL018) Here the authorswould like to express their gratitude to it

References

[1] J J Wang and M C Cheng ldquoFrom a hub port city to a globalsupply chain management center a case study of Hong KongrdquoJournal of Transport Geography vol 18 no 1 pp 104ndash115 2010

[2] F Wang J Huang and Z Liu ldquoPort management and oper-ations Emerging research topics and progressrdquo Journal ofManagement Sciences in China vol 20 no 05 pp 111ndash126 2017

[3] B Jiang J Li and S Shen ldquoSupply Chain Risk Assessment andControl of Port Enterprises Qingdao port as case studyrdquo AsianJournal of Shipping andLogistics vol 34 no 3 pp 198ndash208 2018

[4] D W Song and P M Panayides ldquoGlobal supply chain andportterminal integration and competitivenessrdquo in Proceedingsof the International Conference on Logistics Shipping and PortManagement Taiwan March 2007

[5] S J Pettit and A K C Beresford ldquoPort development Fromgateways to logistics hubsrdquoMaritime Policy ampManagement vol36 no 3 pp 253ndash267 2009

[6] S-HWoo S J Pettit and A K C Beresford ldquoAn assessment ofthe integration of seaports into supply chains using a structuralequation modelrdquo Supply Chain Management An InternationalJournal vol 18 no 3 pp 235ndash252 2013

[7] L Hou and H Geerlings ldquoDynamics in sustainable port andhinterland operations A conceptual framework and simulationof sustainability measures and their effectiveness based onan application to the Port of Shanghairdquo Journal of CleanerProduction vol 135 pp 449ndash456 2016

[8] K L Roslashdseth P B Wangsness and H Schoslashyen ldquoHow doeconomies of density in container handling operations affectshipsrsquo time and emissions in port Evidence from Norwegiancontainer terminalsrdquo Transportation Research Part D Transportand Environment vol 59 pp 385ndash399 2018

[9] J Tongzon Y Chang and S Lee ldquoHow supply chain oriented isthe port sectorrdquo International Journal of Production Economicsvol 122 no 1 pp 21ndash34 2009

[10] C Ta A V Goodchild and K Pitera ldquoStructuring a definitionof resilience for the freight transportation systemrdquo Transporta-tion Research Record no 2097 pp 19ndash25 2009

[11] A H Gharehgozli J Mileski A Adams and W von ZharenldquoEvaluating a ldquowicked problemrdquo A conceptual frameworkon seaport resiliency in the event of weather disruptionsrdquoTechnological Forecasting amp Social Change vol 121 pp 65ndash752017

[12] H S Loh and V V Thai ldquoManaging port-related supplychain disruptions (PSCDs) amanagementmodel and empiricalevidencerdquo Maritime Policy amp Management vol 43 no 4 pp436ndash455 2016

[13] K Zavitsas T Zis and M G Bell ldquoThe impact of flexibleenvironmental policy on maritime supply chain resiliencerdquoTransport Policy vol 72 pp 116ndash128 2018

[14] M Saleh R Oliva C E Kampmann and P I Davidsen ldquoAcomprehensive analytical approach for policy analysis of systemdynamics modelsrdquo European Journal of Operational Researchvol 203 no 3 pp 673ndash683 2010

[15] B Yang X Wang Y Zhang Y Xu and W Zhou ldquoFinite-time synchronization and synchronization dynamics analysisfor two classes of markovian switching multiweighted complexnetworks from synchronization control rule viewpointrdquo Com-plexity vol 2019 Article ID 1921632 17 pages 2019

[16] S John M M Naim and D R Towill ldquoDynamic analysisof a WIP compensated decision support systemrdquo InternationalJournal of Manufacturing System Design vol 1 pp 283ndash2971994

[17] H A Gabbar ldquoModeling approach for flexible productionchain operationrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 38 no 1 pp 48ndash55 2008

[18] C Li R Mo Z Chang H Yang N Wan and D ZhangldquoDecision-making of feature operation chain considering pro-cessing requirements andmanufacturing stabilityrdquoThe Interna-tional Journal of Advanced Manufacturing Technology vol 87no 5-8 pp 1ndash13 2016

[19] V L Spiegler and M M Naim ldquoInvestigating sustained oscilla-tions in nonlinear production and inventory control modelsrdquoEuropean Journal of Operational Research vol 261 no 2 pp572ndash583 2017

[20] A T Almaktoom ldquoStochastic reliability measurement anddesign optimization of an inventory management systemrdquoComplexity vol 2017 Article ID 1460163 9 pages 2017

[21] J Ma L Zhu Y Yuan and S Hou ldquoStudy of the bullwhip effectin amultistage supply chain with callback structure considering

12 Complexity

two retailersrdquo Complexity vol 2018 Article ID 3650148 12pages 2018

[22] R Pant K Barker J E Ramirez-Marquez and C M RoccoldquoStochastic measures of resilience and their application tocontainer terminalsrdquo Computers amp Industrial Engineering vol70 no 1 pp 183ndash194 2014

[23] Y Wang S W Wallace B Shen and T Choi ldquoService supplychain management a review of operational modelsrdquo EuropeanJournal of Operational Research vol 247 no 3 pp 685ndash6982015

[24] D R Towill ldquoDynamic analysis of an inventory and order basedproduction control systemrdquo International Journal of ProductionResearch vol 20 no 6 pp 671ndash687 1982

[25] L ZhouMM Naim and SM Disney ldquoThe impact of productreturns and remanufacturing uncertainties on the dynamicperformance of a multi-echelon closed-loop supply chainrdquoInternational Journal of Production Economics vol 183 pp 487ndash502 2017

[26] V L M Spiegler M M Naim and J Wikner ldquoA control engi-neering approach to the assessment of supply chain resiliencerdquoInternational Journal of Production Research vol 50 no 21 pp6162ndash6187 2012

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Complexity 3

Container truck Yard

Gantry craneContainer vessel

Import

Export

Delivery and Receipt

Landside Yard crane

Loading and unloading

Horizontal transportationShipment Loading and unloading

AGV Seaside Yard crane

Loading and unloading

Storage

Seaside Operations

Move

Landside Operations

Loading Move

Loading

Unloading

LoadingDelivery

Receipt Unloading

Unloading

Unloading

Loading

Figure 1 General model of container handling operations in automated container ports

have the potential to cause discrepancies between freightrequirements and handling operations

In this regard we have the intention of adapting theAutomated Pipeline Inventory and Order Based ProductionControl System (APIOBPCS) (John et al 1994) [16] inmanu-facturing supply chains to PSSC for improving the operationperformance of container ports This study first identifiescharacteristics of the handling operations by establishing amodeling paradigm for quantifying the resilience of HCS incontainer ports In the next step this study considers both thenonnegative arrive rate of container cargo and the containerhandling capacity with the upper limit explicitly Nonlinearlinks have been added into themodel of theHCS In additionfor the first time this study has conducted complete analysisof the HCS resilience under varying pole distributions ofunsatisfied freight requirement (UFR) transfer functionmoreprecisely Using the proposed model we consistently developfinal values of unsatisfied freight requirement (119906119891119903(119905)) andcontainer handling completion rate (119888119900119898119903119886119905119890(119905)) within thesteady state condition Finally simulation results show thatthe upper limit of the container handling completion ratehas a significant impact on the resilience of HCS Thecontributions herein will not only ensure decision making byport manager to protect ports and make them more resilientbut also promote socially efficient handling chains

2 Models of Container Port HandlingChain System

21 Linear Model The container port and shipping supplychain is a service supply chain system as a network ofsuppliers service providers consumers and other supporting

units that performs the function of the delivery of containersto customers (Wang et al 2015) [23] Container port andshipping services are important in service supply chain asthey significantly influence freight requirement (FR) Obvi-ously proper freight requirement management and relatedoperational measures are crucial to the success of containerport and shipping supply chain A novel model for HCS incontainer ports shown in Figure 2 is adapted fromAPIOBPCSmodel in manufacturing supply chain (John et al 1994)[16] which extended the original Inventory and Order-basedProduction Control System (IOBPCS) archetype (Towill1982) [24] by incorporating an automatic work-in-progress(WIP) feedback loop The APIOBPCS model (John et al1994) [16] a base framework for a production planning andcontrol system can be expressed as follows the order placedis based on the forecast of customer demand plus a fraction(1119879i proportional controller for inventory adjustment) of thediscrepancy between actual and desired inventory levels plusa fraction (1119879w proportional controller forWIP adjustment)of the discrepancy between actualWIP and targetWIP levels(Zhou et al 2017) [25] One of the main objectives whendesigning an APIOBPCS is to obtain a desirable inventorylevel avoiding severe order and inventory variability in theface of exterior disturbances (Almaktoom 2017) [20]

The Laplace transform is utilized in Figure 2 for our con-trol engineering analysis coupled with simulation modelings is a complex frequency in Laplace transform Parametersand segments are detailed in Table 1 In the following workuppercase and lowercase words such as FR and fr areused in the frequency and time domains respectively Theinput and output of this model are freight requirement (FR)and unsatisfied freight requirement (UFR) respectively The

4 Complexity

EFRIP

UFR

FRIP

FR ++ +

+

-

+_

AVFR COMRATEARATE

+ _DFRIP

Lead policy

feedbackUFR

FR policy

UFR policy

FRIP policyFRIP feedback

1

1 + TFs

TQ

1

1 + TGs

1

s

1

s

1

TU

1

TI

Figure 2 Linear model of handling chain system in container ports

Table 1 Parameters and segments in linear model

FR Freight requirement AVFR Average FR ARATE Arrive rate

COMRATE Container handlingcompletion rate UFR Unsatisfied FR FRIP FR in process

DFRIP Desired FRIP EFRIP Error in FRIP 119879119880Adjustment time of

UFR

119879119865Time constant of FR

forecast 119879119866Real lead-time of

handling operations 119879119876Expected lead-time ofhandling operations

119879119868Adjustment time of

EFRIP1

1 + 119879119865119904Freight requirement

policy1

1 + 119879119866119904Lead policy of

handling operation

container arrive rate (ARATE) takes the integrals of averagefreight requirement (AVFR) error in freight requirement inprocess (EFRIP) and unsatisfied freight requirements (UFR)into account 1(1 + 119879119865119904) and 1(1 + 119879119866119904) are employedas the freight requirement policy and the lead policy ofhandling operation in container port respectively Containerhandling completion rate (COMRATE) can be taken as theinput of the next 119879119880 denotes adjustment time of UFR 119879119865denotes time constant of FR forecast 119879119866 denotes real lead-time of handling operations 119879119876 denotes expected lead-timeof handling operations and 119879119868 denotes adjustment time ofEFRIP Controllable decision variables 119879119880 119879119865 119879119866 119879119876 and119879119868 are all positive real numbers

As shown in Figure 2 the HCS model in container portscan be expressed as follows the container arrive rate (ARATE)placed is based on the forecast of freight requirement (FR)plus a fraction (1119879119880 the proportional controller for unsatis-fied freight requirement adjustment) of the unsatisfied freightrequirement (UFR) plus a fraction (1119879119868 the proportionalcontroller for FR in process adjustment) of the discrepancybetween actual FR in process (FRIP) and desired FRIPlevels The differences between APIOBPCS and HCS incontainer ports are not only with or without a desiredinventory level but also the meaning of each terminologyin HCS has been updated according to the characteristics ofcontainer port and shipping supply chain such as parameter119879119865 related to handling operations planning in container

terminal parameters 119879119866 and 119879119876 related to the concentratedreceipt of export containers ahead of container vessel arrivalparameter 119879119868 related to the fast loading and unloading ofcontainer vessels and parameter 119879119880 related to the detentiontime of containers in port yard Each parameter in the HCSmodel of container ports originates from handling operationpracticeThey undoubtedly have important practical guidingsignificance for container port operation management Themain purpose of HCS in container ports is to obtain anunsatisfied freight requirement (UFR) as small as possible oreven equal to zero and maintain optimal handling operationresilience

As described in Figure 2 it is possible to determine theUFR transfer function the COMRATE transfer function andthe ARATE transfer function in relation to the input FR

119880119865119877119865119877

=(119879119866 + 119879119868 + 119879119868119879119866119904) (1 + 119879119865119904) minus 119879119876 minus 119879119868

119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

(1)

119862119874119872119877119860119879119864119865119877

=119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904

119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

(2)

Complexity 5

119860119877119860119879119864119865119877

=(1 + 119879119866119904) [119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904]

119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

(3)

(1) Steady State Condition One of the poles is easily identified(1199010=-1119879119865) The other two poles are set as 1199011 and 1199012 Then

1199011 + 1199012 =1119879119868

+ 1119879119866

11990111199012 =1

(119879119880119879119866)

(4)

Then it can be seen that the steady state condition is 1119879119868+1119879119866 gt 0 1(119879119880119879119866) gt 0 and 1119879119865 gt 0 Because 119879119880 119879119865 119879119866119879119876 and 119879119868 are all positive real numbers this system is stable

(2) Final Value of 119906119891119903(119905) and 119888119900119898119903119886119905119890(119905) In deriving transferfunctions in complex frequency domain we exploit theclassic step input as it helps to develop insights into dynamicbehavior We use the step response to assess the systemrsquosability to cope with oscillations in the linear HCS The stepresponse is powerful as from this simple FR input thesize and quantity of the subsequent 119906119891119903(infin) 119888119900119898119903119886119905119890(infin)and 119886119903119886119905119890(infin) can be readily determined providing a richunderstanding of the dynamics of the linear HCS If FR is aunit step signal 119906119891119903(infin) 119888119900119898119903119886119905119890(infin) and 119886119903119886119905119890(infin) can beobtained by final value theorem

119906119891119903 (infin) = lim119904997888rarr0

119904 sdot 119880119865119877 (119904)

= lim119904997888rarr0

(119879119866 + 119879119868 + 119879119868119879119866119904) (1 + 119879119865119904) minus 119879119876 minus 119879119868119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

=119879119880 (119879119866 minus 119879119876)

119879119868

(5)

119888119900119898119903119886119905119890 (infin) = lim119904997888rarr0

119904 sdot 119862119874119872119877119860119879119864 (119904)

= lim119904997888rarr0

119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

= 119879119868119879119880119879119868119879119880

= 1

(6)

119886119903119886119905119890 (infin) = lim119904997888rarr0

119904 sdot 119860119877119860119879119864 (119904)

= lim119904997888rarr0

(1 + 119879119866119904) [119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904]119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

= 119879119868119879119865119879119868119879119865

= 1

(7)

It can be seen that 119906119891119903(infin) depends on 119879119880 119879119868 and thedifference between 119879119866 and 119879119876 119906119891119903(infin) would be 0 onlywhen 119879119866 = 119879119876 119888119900119898119903119886119905119890(119905) and 119886119903119886119905119890(119905) will approach 1when 119905 997888rarr infin Therefore when the HCS reaches its steadystate 119888119900119898119903119886119905119890(119905) and119891119903(119905) are balanced and 119906119891119903(119905) no longerchanges

22 Nonlinear Model In real environments of containerports the value of ARATE cannot be negative althoughmathematically it holds The upper value of COMRATEis limited however the linear model in Figure 2 cannotguarantee these two nonlinear natures To reflect them anonlinearHCSmodel consisting of twononlinear segments isgiven in Figure 3 ARATErsquo is the theoretical arrive rate whichequals to 119886V119891119903(119905) + 119906119891119903(119905)119879119880 + 119890119891119903119894119901(119905)119879119868 COMRATErsquo isthe theoretical container handling completion rate which isonly calculated by 119886119903119886119905119890(119905) and the handling operation policy1(1 + 119879119866119904) COMRATEm is the upper limit of COMRATECOMRATEm should not be less than 119891119903(119905) over time other-wise 119906119891119903(119905) will continuously increase

If COMRATEm is equal to 119891119903(119905) or slightly larger than119891119903(119905) the actual container handling completion rate wouldnot process the accumulated freight requirement timelyThen119906119891119903(infin) may be larger than 119879119880(119879119866 minus 119879119876)119879119868 If 119906119891119903(119905) is toolarge it would congest the HCS

3 Resilience for Handling Chain System

To understand the resilience for HCS in container portswe use the integral of time multiplied by the absolute error(ITAE) as a benchmark ITAE emphasizes long-durationerrors It is recommended for the analyses of systems whichrequire a fast settling time The minimum value of ITAEcorresponds to the best response and recoverywith the lowestdeviation from the target or readiness (Virginia et al 2012)[26] The ITAE is given by 119868119879119860119864 = int+infin

0119905 sdot |119890(119905)|119889119905 =

lim120575119905997888rarr0suminfin119905=0 119905|119890(119905)|120575119905 where 119890(119905) = 119906119891119903(119905) minus 119906119891119903(infin) To

facilitate uniform analysis 119906119891119903(infin) = 119879119880(119879119866 minus119879119876)119879119868 whichis consistent with (5)

In linear HCS 119906119891119903(119905) can be calculated in the followingfour cases below where A B and 119862 are coefficients related tothe system poles and119863 is the coefficient of the step input pole(s=0)

Case 1 (distinct poles real andor complex 1199011 = 1199012 = 1199010)(a) Δ 1 gt 0 (Δ 1 = 119879119880(119879119866 + 119879119868)

2 minus 41198792119868119879119866)

119906119891119903 (119905) = 1198601198901199010119905 + 1198611198901199011119905 + 1198621198901199012119905 + 119863 (8)

(b) Δ 1 lt 0

119906119891119903 (119905) = 1198601198901199010119905 + 1198611198901205901119905 (cos1205961119905 minus 119895 sin1205961119905)

+ 1198621198901205902119905 (cos1205962119905 minus 119895 sin1205962119905) + 119863(9)

In Case 1 (a) and (b) the coefficient values of A B and 119862are

119860 =119879119868 + 119879119866 minus 11987911986611987911986811986311990121199011119879119866119879119868 (1199011 minus 1199010) (1199012 minus 1199010)

119861 =119879119868119879119865 + 119879119866119879119865 + 119879119866119879119868 + 119879119866119879119868119879119865 (1199011 minus 11986311990121199010)

119879119866119879119868119879119865 (1199011 minus 1199012) (1199011 minus 1199010)

119862 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199012 minus 11986311990101199011)

119879119866119879119868119879119865 (1199012 minus 1199010) (1199012 minus 1199011)

(10)

6 Complexity

EFRIP

UFR

FRIP

FR ++ +

+

+_

AVFR COMRATEARATE

+ _

DFRIP

0

_

ARATE COMRATE

feedbackUFR

Lead policy

UFR policy

FR policy

FRIP policy

1

1 + TFs

TQ

1

1 + TGs

1

s

1

s

FRIP feedback

COMRATEＧ

1

TU

1

TI

Figure 3 Nonlinear model of handling chain system in container ports

Case 2 (two repeated poles 1199011=1199010 or 1199012=1199010) (a) 1199011 = 11990121199011 = 1199010

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 1198621198901199012119905 + 119863 (11)

where

119860

= minus119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 [1199012 + 1198631199012 (1199012 minus 21199010)]

119879119866119879119868119879119865 (1199010 minus 1199012)2

119861 =119879119866 + 119879119868 minus 11987911986611987911986811986311990101199012

119879119866119879119868 (1199010 minus 1199012)

119862 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199012 minus 11986311990120)

119879119866119879119868119879119865 (1199010 minus 1199012)2

(12)

(b) 1199011 = 1199012 1199012 = 1199010

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 1198621198901199011119905 + 119863 (13)

where

119860

= minus119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 [1199011 + 1198631199011 (1199011 minus 21199010)]

119879119866119879119868119879119865 (1199010 minus 1199011)2

119861 =119879119866 + 119879119868 minus 11987911986611987911986811986311990101199011

119879119866119879119868 (1199010 minus 1199011)

119862 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199011 minus 11986311990120)

119879119866119879119868119879119865 (1199010 minus 1199011)2

(14)

Case 3 (two repeated poles 1199011 = 1199012 = 1199010)

119906119891119903 (119905) = 1198601198901199011119905 + 1198611199051198901199011119905 + 1198621198901199010119905 + 119863 (15)

where

119860 = minus119879119866 + 119879119868 + 1198791198661198791198681198631199010 (1199010 minus 21199011)

119879119866119879119868 (1199011 minus 1199010)2

119861 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199011 minus 11986311990101199011)

119879119866119879119868119879119865 (1199011 minus 1199010)

119862 =119879119866 + 119879119868 minus 11987911986611987911986811986311990121119879119866119879119868 (1199011 minus 1199010)

2

(16)

Case 4 (three repeated poles 1199011 = 1199012 = 1199010)

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 0511986211990521198901199010119905 + 119863 (17)

where119860 = minus119863

119861 = 1 + 1198631199010

119862 = 1119879119866

+ 1119879119868

minus 11986311990120

(18)

In all cases119863 = 119879119880(119879119866 minus 119879119876)119879119868 In all cases except Case1 (b)

1199011 =minus (119879119866 + 119879119868)radic119879119880 + radicΔ 1

2119879119868119879119866radic119879119880

1199012 =minus (119879119866 + 119879119868)radic119879119880 minus radicΔ 1

2119879119868119879119866radic119879119880

(19)

In Case 1 (b)

1199011 = 1205901 + 1198951205961

1199012 = 1205902 + 1198951205962

1205901 = 1205902 =minus (119879119866 + 119879119868)2119879119868119879119866

12059612 =plusmnradicminusΔ 1

2119879119868119879119866radic119879119880

(20)

Complexity 7

+1

+j

+1

+j

+1

+j

Case 1(a) Case 1(b) Case 2(a)

+1

+j

+1

+j

+1

+j

Case 2(b) Case 3 Case 4

p2

p2p2

p1

p1

p1

p0

p0p0

p2 = p0p1 = p2

p1 = p0

p1 = p2 = p0

Figure 4 Pole distributions of UFR transfer function

Thepole distributions ofUFR transfer function are shownin Figure 4

It can be seen that 119906119891119903(119905) approaches119863 in Cases 1 through4 This is consistent with (5) in which 119906119891119903(infin) is equal to119863

In nonlinear HCS if aratersquo(t)lt0 or comratersquo(t)gt COM-RATEm 119906119891119903(119905) would be different from that in a linearHCS Specifically if comratersquo(t)gt COMRATEm ufr(t) wouldincrease because the actual container handling completionrate cannot process the freight requirement in time and119906119891119903(infin)may be larger than D ITAE would be different fromthat of linear HCS If 119906119891119903(infin) gt 119863 ITAE would increaseindefinitely

4 Resilience Analysis of HCS inContainer Ports

The quantitative resilience framework for HCS developedin this work is deployed with an illustrative example of acontainer port in China Its HCS serves an important rolein container flows around the world and its resilience isvital to the larger multimodal transportation system and thePSSC It has four container docks Under normal operationsits annual container throughput would be around 4 millionTEU 119879119876 would be between 0 and 3 days while 119879119868 would bebetween 0 and 1 day For the simulationmodels and resilienceanalysis it is assumed that 119879119876=15 days and 119879119868=05 days 119879119866is 1 day 119879119865 is between 2 and 30 days while 119879119880 is between0 and 10 days Based on the modeling concepts developedin the previous sections resiliencies of two different HCSsare studied separately (i) linear HCS and (ii) nonlinear HCSSimulation code was written in MATLAB If any of the HCSwere to oscillate and become inoperable it might stop theflow of containers handled by that dock and result in portcongestions

41 Resilience of Linear HCS Linear HCSs are without limitsforARATE andCOMRATE Based on the above data ITAE oflinear HCS is calculated according to Section 3The results ofITAE are calculated and plotted when 119879119865 isin [2 days 30 days]and 119879119880 isin (0 day 10 days] 3D mesh and contour maps ofITAE are shown in Figures 5(a) and 5(b) respectively InFigure 5 it can be seen that ITAE increases gradually withincreases in 119879119865 and 119879119880 in most cases But this change is notcompletely linear When 119879119865 and 119879119880 approach 2 days and10 days respectively ITAE will gradually increase within acertain range We are concerned about the values of 119879119865 and119879119880 when ITAE reaches its minimum value However they arenot easily found in Figure 5 For this reason 3D mesh andcontour maps of 1ITAE are plotted in Figures 6(a) and 6(b)respectively From Figure 6 1ITAE reaches its maximumvalue when 119879119865 = 2 days and 119879119880 997888rarr 0 day (119879119880 = 0)The minimum of ITAE is approximately equal to 52812 Thisimplies that carefully setting the parameters 119879119865 and 119879119880 mayimprove dynamic performance of the linearHCS In additionit suggests that the closer the collaboration of port authoritiesand other players in PSSC is the better the resilience of linearHCS is

42 Resilience of Nonlinear HCS Nonlinear HCS focuses onthe limits for ARATE and COMRATE How the ARATE andCOMRATE influence the nonlinear HCS dynamics is to beexamined In nonlinear HCS the smaller COMRATEm isthe more likely it is that COMRATErsquo(t)gt COMRATEm andthe difference between 119906119891119903(119905) in nonlinear and linear HCSsis larger Correspondingly the difference between ITAE ofnonlinear and linear HCSs is greater In order to facilitatethe obvious distinction COMRATEm =105 here When 119879119865 isin[2 days 30 days] and 119879119880 isin (0 day 10 days] 3D mesh andcontour maps of ITAE are given in Figures 7(a) and 7(b)

8 Complexity

0

5

10

10

20

300

2

4

6

8

TUTF

ITA

E

times104

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 5 ITAE of linear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 6 1 ITAE of linear handling chain system

respectively 3D mesh and contour maps of 1ITAE are givenin Figures 8(a) and 8(b) respectively In Figure 8(a) someareas in which 119906119891119903(infin) gt 119863 and ITAE is infinite are notplotted These areas are the light blue shaded regions inFigure 8(b) From Figure 8 ITAE reaches its minimum value(1ITAEasymp0092 and ITAEasymp1082) when 119879119865 = 2 days 119879119880 =24days From a comparison between Figures 5 and 7 ITAE(approximately 6660) is infinitymore thanhalf of the entirearea It reveals that the impact of COMRATEm on containerport performance is obvious Considering these results canbe determined that if COMRATEm are just applied to thenonlinear HCS modifying the setting of parameter 119879119880 willimprove the resilience of the nonlinear HCS

In Figure 6 there are two regions in which 1ITAEgradually increases and reaches the peaks of these regions

One region is close to the point of 119879119880=0 day and 119879119865=2 daysand the other region is close to the point of 119879119880=24 daysand 119879119865=2 days 1ITAE gradually reaches its maximum whenapproaching the point of 119879119880=0 day and 119879119865=2 days This ismost likely due to the fact that they are dependent processesThe faster the unsatisfied freight requirement is deliveredthe sooner the linear HCS will recover to normal operationHence reasonably setting the parameters119879119865 and119879119880 can solveport congestion problems but it is important to take intoaccount that each modification will affect the resilience oflinear HCS

In Figure 8 1ITAE becomes very small in the region closeto the point of 119879U=0 day and 119879F=2 days From Figure 8(b)ITAE becomes infinite in this region The new maximum of1ITAE is in the point of 119879U=24 days and 119879F=2 days The

Complexity 9

24

68

10

10

20

300

1

2

3

4

TUTF

ITA

E

times104

(a) 3D mesh map

2 4 6 8 10TU

TF

30

25

20

15

10

5

(b) Contour map

Figure 7 ITAE of nonlinear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 8 1ITAE of nonlinear handling chain system

results of the two HCSs discussed above show the usefulnessof the resilience measurement is highlighted in the analysisThese can be used for the PSSCrsquos decisions

43 COMRATEmrsquos Influence on Nonlinear HCS In orderto examine how the COMRATEm influences the nonlinearHCS dynamics the point of 119879119865=2 days and 119879119880=02 day istaken as an example and ufr(t) arate(t) and comrate(t) areplotted in Figure 9 with different COMRATEm (COMRATEm=100 105 110 115 and 120) date isin [0 day 40 days] Inorder to clearly distinguish these curves the horizontal axiscoordinates only take 0sim15 days in Figures 9(a) and 9(b)They eventually reach stable values The stable value of ufr(t)is different with different COMRATEm the stable values ofarate(t) and comrate(t) are always 1

From Figure 9(a) the values of ufr(t) with differentCOMRATEm are the same before it reaches its peak After itspeak the values of ufr(t) with differentCOMRATEm are quitedifferent from one another When COMRATEm =1 ufr(t)maintains its peak value after reaching it When COMRATEm=12 ufr(t) eventually is equal to D(D=02) which is thesame as that in the linear HCS From Figure 9(b) in dates2 through 8 the enhanced ufr(t) makes arate(t) higherwhen COMRATEm is lower From Figure 9(c) comrate(t) islimited by COMRATEm in dates 2sim8 This limitation resultsin the cumulative increase of ufr(t) Due to the limited valueof COMRATE there are cases where dynamic behaviorsof nonlinear HCS cause fr(t) not to be handled timelyeventually resulting in the increased ufr(t) Meanwhile therewould be change noted in arate(t)

10 Complexity

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 15 20 25 30 35 400Date

0

02

04

06

08

1U

ncom

plet

ed F

reig

ht R

equi

rem

ent

(a) ufr(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

05

1

15

2

25

Arr

ive r

ate

(b) arate(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

02

04

06

08

1

12

Com

plet

e rat

e

(c) comrate(t)

Figure 9 States and outputs in nonlinear HCS with different COMRATEm

To investigate the influence of COMRATEm on ITAEthe proportion of ITAE=infin in nonlinear HCS with differentCOMRATEm is plotted in Figure 10 The observation ofdifferent COMRATEm suggests the following all values ofITAE are infinite when COMRATEm le 1 When COM-RATEm = 1sim11 the proportion of ITAE=infin clearly decreasesWhen COMRATEm = 11sim12 the proportion of ITAE=infinis close to 0 and decreases slowly When COMRATEm =12 the proportion of ITAE=infin equals 0 which is the sameas that in the linear HCS These observations show thatthe fluctuation of ufr(t) could be alleviated if the non-linear HCS is carefully designed although it seems to beinevitable

5 Conclusion

The emphasis of container port authorities and other playersin the PSSC has shifted from traditional passive managementto active response highlighting the need to quantify theresilience of HCS for inevitable oscillationsThis work buildsand analyzes linear and nonlinearHCSmodels Resilience forHCS in container ports is measured by ITAE of unsatisfiedfreight requirement It provides a new perspective on PSSCpartners to plan for HCS where resilience could guide theselection of control parameters HCSs in container portsare taken as example to analyze resilience and it is foundthat Comratem has significant influence on resilience for

Complexity 11

095 1 105 11 115 120

02

04

06

08

1

COMRATEm

Prop

ortio

n of

ITA

E =infin

Figure 10 Proportion of ITAE =infin in nonlinear HCS with differentCOMRATEm

nonlinear model These contributions serve as a startingpoint in the development of a quantitative resilience decisionmaking framework for PSSC Future work will consider theresilience and recovery strategies under stochastic freightrequirements given the two HCSs

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Social Science Foun-dation project of China (No 17CGL018) Here the authorswould like to express their gratitude to it

References

[1] J J Wang and M C Cheng ldquoFrom a hub port city to a globalsupply chain management center a case study of Hong KongrdquoJournal of Transport Geography vol 18 no 1 pp 104ndash115 2010

[2] F Wang J Huang and Z Liu ldquoPort management and oper-ations Emerging research topics and progressrdquo Journal ofManagement Sciences in China vol 20 no 05 pp 111ndash126 2017

[3] B Jiang J Li and S Shen ldquoSupply Chain Risk Assessment andControl of Port Enterprises Qingdao port as case studyrdquo AsianJournal of Shipping andLogistics vol 34 no 3 pp 198ndash208 2018

[4] D W Song and P M Panayides ldquoGlobal supply chain andportterminal integration and competitivenessrdquo in Proceedingsof the International Conference on Logistics Shipping and PortManagement Taiwan March 2007

[5] S J Pettit and A K C Beresford ldquoPort development Fromgateways to logistics hubsrdquoMaritime Policy ampManagement vol36 no 3 pp 253ndash267 2009

[6] S-HWoo S J Pettit and A K C Beresford ldquoAn assessment ofthe integration of seaports into supply chains using a structuralequation modelrdquo Supply Chain Management An InternationalJournal vol 18 no 3 pp 235ndash252 2013

[7] L Hou and H Geerlings ldquoDynamics in sustainable port andhinterland operations A conceptual framework and simulationof sustainability measures and their effectiveness based onan application to the Port of Shanghairdquo Journal of CleanerProduction vol 135 pp 449ndash456 2016

[8] K L Roslashdseth P B Wangsness and H Schoslashyen ldquoHow doeconomies of density in container handling operations affectshipsrsquo time and emissions in port Evidence from Norwegiancontainer terminalsrdquo Transportation Research Part D Transportand Environment vol 59 pp 385ndash399 2018

[9] J Tongzon Y Chang and S Lee ldquoHow supply chain oriented isthe port sectorrdquo International Journal of Production Economicsvol 122 no 1 pp 21ndash34 2009

[10] C Ta A V Goodchild and K Pitera ldquoStructuring a definitionof resilience for the freight transportation systemrdquo Transporta-tion Research Record no 2097 pp 19ndash25 2009

[11] A H Gharehgozli J Mileski A Adams and W von ZharenldquoEvaluating a ldquowicked problemrdquo A conceptual frameworkon seaport resiliency in the event of weather disruptionsrdquoTechnological Forecasting amp Social Change vol 121 pp 65ndash752017

[12] H S Loh and V V Thai ldquoManaging port-related supplychain disruptions (PSCDs) amanagementmodel and empiricalevidencerdquo Maritime Policy amp Management vol 43 no 4 pp436ndash455 2016

[13] K Zavitsas T Zis and M G Bell ldquoThe impact of flexibleenvironmental policy on maritime supply chain resiliencerdquoTransport Policy vol 72 pp 116ndash128 2018

[14] M Saleh R Oliva C E Kampmann and P I Davidsen ldquoAcomprehensive analytical approach for policy analysis of systemdynamics modelsrdquo European Journal of Operational Researchvol 203 no 3 pp 673ndash683 2010

[15] B Yang X Wang Y Zhang Y Xu and W Zhou ldquoFinite-time synchronization and synchronization dynamics analysisfor two classes of markovian switching multiweighted complexnetworks from synchronization control rule viewpointrdquo Com-plexity vol 2019 Article ID 1921632 17 pages 2019

[16] S John M M Naim and D R Towill ldquoDynamic analysisof a WIP compensated decision support systemrdquo InternationalJournal of Manufacturing System Design vol 1 pp 283ndash2971994

[17] H A Gabbar ldquoModeling approach for flexible productionchain operationrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 38 no 1 pp 48ndash55 2008

[18] C Li R Mo Z Chang H Yang N Wan and D ZhangldquoDecision-making of feature operation chain considering pro-cessing requirements andmanufacturing stabilityrdquoThe Interna-tional Journal of Advanced Manufacturing Technology vol 87no 5-8 pp 1ndash13 2016

[19] V L Spiegler and M M Naim ldquoInvestigating sustained oscilla-tions in nonlinear production and inventory control modelsrdquoEuropean Journal of Operational Research vol 261 no 2 pp572ndash583 2017

[20] A T Almaktoom ldquoStochastic reliability measurement anddesign optimization of an inventory management systemrdquoComplexity vol 2017 Article ID 1460163 9 pages 2017

[21] J Ma L Zhu Y Yuan and S Hou ldquoStudy of the bullwhip effectin amultistage supply chain with callback structure considering

12 Complexity

two retailersrdquo Complexity vol 2018 Article ID 3650148 12pages 2018

[22] R Pant K Barker J E Ramirez-Marquez and C M RoccoldquoStochastic measures of resilience and their application tocontainer terminalsrdquo Computers amp Industrial Engineering vol70 no 1 pp 183ndash194 2014

[23] Y Wang S W Wallace B Shen and T Choi ldquoService supplychain management a review of operational modelsrdquo EuropeanJournal of Operational Research vol 247 no 3 pp 685ndash6982015

[24] D R Towill ldquoDynamic analysis of an inventory and order basedproduction control systemrdquo International Journal of ProductionResearch vol 20 no 6 pp 671ndash687 1982

[25] L ZhouMM Naim and SM Disney ldquoThe impact of productreturns and remanufacturing uncertainties on the dynamicperformance of a multi-echelon closed-loop supply chainrdquoInternational Journal of Production Economics vol 183 pp 487ndash502 2017

[26] V L M Spiegler M M Naim and J Wikner ldquoA control engi-neering approach to the assessment of supply chain resiliencerdquoInternational Journal of Production Research vol 50 no 21 pp6162ndash6187 2012

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4 Complexity

EFRIP

UFR

FRIP

FR ++ +

+

-

+_

AVFR COMRATEARATE

+ _DFRIP

Lead policy

feedbackUFR

FR policy

UFR policy

FRIP policyFRIP feedback

1

1 + TFs

TQ

1

1 + TGs

1

s

1

s

1

TU

1

TI

Figure 2 Linear model of handling chain system in container ports

Table 1 Parameters and segments in linear model

FR Freight requirement AVFR Average FR ARATE Arrive rate

COMRATE Container handlingcompletion rate UFR Unsatisfied FR FRIP FR in process

DFRIP Desired FRIP EFRIP Error in FRIP 119879119880Adjustment time of

UFR

119879119865Time constant of FR

forecast 119879119866Real lead-time of

handling operations 119879119876Expected lead-time ofhandling operations

119879119868Adjustment time of

EFRIP1

1 + 119879119865119904Freight requirement

policy1

1 + 119879119866119904Lead policy of

handling operation

container arrive rate (ARATE) takes the integrals of averagefreight requirement (AVFR) error in freight requirement inprocess (EFRIP) and unsatisfied freight requirements (UFR)into account 1(1 + 119879119865119904) and 1(1 + 119879119866119904) are employedas the freight requirement policy and the lead policy ofhandling operation in container port respectively Containerhandling completion rate (COMRATE) can be taken as theinput of the next 119879119880 denotes adjustment time of UFR 119879119865denotes time constant of FR forecast 119879119866 denotes real lead-time of handling operations 119879119876 denotes expected lead-timeof handling operations and 119879119868 denotes adjustment time ofEFRIP Controllable decision variables 119879119880 119879119865 119879119866 119879119876 and119879119868 are all positive real numbers

As shown in Figure 2 the HCS model in container portscan be expressed as follows the container arrive rate (ARATE)placed is based on the forecast of freight requirement (FR)plus a fraction (1119879119880 the proportional controller for unsatis-fied freight requirement adjustment) of the unsatisfied freightrequirement (UFR) plus a fraction (1119879119868 the proportionalcontroller for FR in process adjustment) of the discrepancybetween actual FR in process (FRIP) and desired FRIPlevels The differences between APIOBPCS and HCS incontainer ports are not only with or without a desiredinventory level but also the meaning of each terminologyin HCS has been updated according to the characteristics ofcontainer port and shipping supply chain such as parameter119879119865 related to handling operations planning in container

terminal parameters 119879119866 and 119879119876 related to the concentratedreceipt of export containers ahead of container vessel arrivalparameter 119879119868 related to the fast loading and unloading ofcontainer vessels and parameter 119879119880 related to the detentiontime of containers in port yard Each parameter in the HCSmodel of container ports originates from handling operationpracticeThey undoubtedly have important practical guidingsignificance for container port operation management Themain purpose of HCS in container ports is to obtain anunsatisfied freight requirement (UFR) as small as possible oreven equal to zero and maintain optimal handling operationresilience

As described in Figure 2 it is possible to determine theUFR transfer function the COMRATE transfer function andthe ARATE transfer function in relation to the input FR

119880119865119877119865119877

=(119879119866 + 119879119868 + 119879119868119879119866119904) (1 + 119879119865119904) minus 119879119876 minus 119879119868

119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

(1)

119862119874119872119877119860119879119864119865119877

=119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904

119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

(2)

Complexity 5

119860119877119860119879119864119865119877

=(1 + 119879119866119904) [119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904]

119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

(3)

(1) Steady State Condition One of the poles is easily identified(1199010=-1119879119865) The other two poles are set as 1199011 and 1199012 Then

1199011 + 1199012 =1119879119868

+ 1119879119866

11990111199012 =1

(119879119880119879119866)

(4)

Then it can be seen that the steady state condition is 1119879119868+1119879119866 gt 0 1(119879119880119879119866) gt 0 and 1119879119865 gt 0 Because 119879119880 119879119865 119879119866119879119876 and 119879119868 are all positive real numbers this system is stable

(2) Final Value of 119906119891119903(119905) and 119888119900119898119903119886119905119890(119905) In deriving transferfunctions in complex frequency domain we exploit theclassic step input as it helps to develop insights into dynamicbehavior We use the step response to assess the systemrsquosability to cope with oscillations in the linear HCS The stepresponse is powerful as from this simple FR input thesize and quantity of the subsequent 119906119891119903(infin) 119888119900119898119903119886119905119890(infin)and 119886119903119886119905119890(infin) can be readily determined providing a richunderstanding of the dynamics of the linear HCS If FR is aunit step signal 119906119891119903(infin) 119888119900119898119903119886119905119890(infin) and 119886119903119886119905119890(infin) can beobtained by final value theorem

119906119891119903 (infin) = lim119904997888rarr0

119904 sdot 119880119865119877 (119904)

= lim119904997888rarr0

(119879119866 + 119879119868 + 119879119868119879119866119904) (1 + 119879119865119904) minus 119879119876 minus 119879119868119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

=119879119880 (119879119866 minus 119879119876)

119879119868

(5)

119888119900119898119903119886119905119890 (infin) = lim119904997888rarr0

119904 sdot 119862119874119872119877119860119879119864 (119904)

= lim119904997888rarr0

119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

= 119879119868119879119880119879119868119879119880

= 1

(6)

119886119903119886119905119890 (infin) = lim119904997888rarr0

119904 sdot 119860119877119860119879119864 (119904)

= lim119904997888rarr0

(1 + 119879119866119904) [119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904]119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

= 119879119868119879119865119879119868119879119865

= 1

(7)

It can be seen that 119906119891119903(infin) depends on 119879119880 119879119868 and thedifference between 119879119866 and 119879119876 119906119891119903(infin) would be 0 onlywhen 119879119866 = 119879119876 119888119900119898119903119886119905119890(119905) and 119886119903119886119905119890(119905) will approach 1when 119905 997888rarr infin Therefore when the HCS reaches its steadystate 119888119900119898119903119886119905119890(119905) and119891119903(119905) are balanced and 119906119891119903(119905) no longerchanges

22 Nonlinear Model In real environments of containerports the value of ARATE cannot be negative althoughmathematically it holds The upper value of COMRATEis limited however the linear model in Figure 2 cannotguarantee these two nonlinear natures To reflect them anonlinearHCSmodel consisting of twononlinear segments isgiven in Figure 3 ARATErsquo is the theoretical arrive rate whichequals to 119886V119891119903(119905) + 119906119891119903(119905)119879119880 + 119890119891119903119894119901(119905)119879119868 COMRATErsquo isthe theoretical container handling completion rate which isonly calculated by 119886119903119886119905119890(119905) and the handling operation policy1(1 + 119879119866119904) COMRATEm is the upper limit of COMRATECOMRATEm should not be less than 119891119903(119905) over time other-wise 119906119891119903(119905) will continuously increase

If COMRATEm is equal to 119891119903(119905) or slightly larger than119891119903(119905) the actual container handling completion rate wouldnot process the accumulated freight requirement timelyThen119906119891119903(infin) may be larger than 119879119880(119879119866 minus 119879119876)119879119868 If 119906119891119903(119905) is toolarge it would congest the HCS

3 Resilience for Handling Chain System

To understand the resilience for HCS in container portswe use the integral of time multiplied by the absolute error(ITAE) as a benchmark ITAE emphasizes long-durationerrors It is recommended for the analyses of systems whichrequire a fast settling time The minimum value of ITAEcorresponds to the best response and recoverywith the lowestdeviation from the target or readiness (Virginia et al 2012)[26] The ITAE is given by 119868119879119860119864 = int+infin

0119905 sdot |119890(119905)|119889119905 =

lim120575119905997888rarr0suminfin119905=0 119905|119890(119905)|120575119905 where 119890(119905) = 119906119891119903(119905) minus 119906119891119903(infin) To

facilitate uniform analysis 119906119891119903(infin) = 119879119880(119879119866 minus119879119876)119879119868 whichis consistent with (5)

In linear HCS 119906119891119903(119905) can be calculated in the followingfour cases below where A B and 119862 are coefficients related tothe system poles and119863 is the coefficient of the step input pole(s=0)

Case 1 (distinct poles real andor complex 1199011 = 1199012 = 1199010)(a) Δ 1 gt 0 (Δ 1 = 119879119880(119879119866 + 119879119868)

2 minus 41198792119868119879119866)

119906119891119903 (119905) = 1198601198901199010119905 + 1198611198901199011119905 + 1198621198901199012119905 + 119863 (8)

(b) Δ 1 lt 0

119906119891119903 (119905) = 1198601198901199010119905 + 1198611198901205901119905 (cos1205961119905 minus 119895 sin1205961119905)

+ 1198621198901205902119905 (cos1205962119905 minus 119895 sin1205962119905) + 119863(9)

In Case 1 (a) and (b) the coefficient values of A B and 119862are

119860 =119879119868 + 119879119866 minus 11987911986611987911986811986311990121199011119879119866119879119868 (1199011 minus 1199010) (1199012 minus 1199010)

119861 =119879119868119879119865 + 119879119866119879119865 + 119879119866119879119868 + 119879119866119879119868119879119865 (1199011 minus 11986311990121199010)

119879119866119879119868119879119865 (1199011 minus 1199012) (1199011 minus 1199010)

119862 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199012 minus 11986311990101199011)

119879119866119879119868119879119865 (1199012 minus 1199010) (1199012 minus 1199011)

(10)

6 Complexity

EFRIP

UFR

FRIP

FR ++ +

+

+_

AVFR COMRATEARATE

+ _

DFRIP

0

_

ARATE COMRATE

feedbackUFR

Lead policy

UFR policy

FR policy

FRIP policy

1

1 + TFs

TQ

1

1 + TGs

1

s

1

s

FRIP feedback

COMRATEＧ

1

TU

1

TI

Figure 3 Nonlinear model of handling chain system in container ports

Case 2 (two repeated poles 1199011=1199010 or 1199012=1199010) (a) 1199011 = 11990121199011 = 1199010

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 1198621198901199012119905 + 119863 (11)

where

119860

= minus119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 [1199012 + 1198631199012 (1199012 minus 21199010)]

119879119866119879119868119879119865 (1199010 minus 1199012)2

119861 =119879119866 + 119879119868 minus 11987911986611987911986811986311990101199012

119879119866119879119868 (1199010 minus 1199012)

119862 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199012 minus 11986311990120)

119879119866119879119868119879119865 (1199010 minus 1199012)2

(12)

(b) 1199011 = 1199012 1199012 = 1199010

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 1198621198901199011119905 + 119863 (13)

where

119860

= minus119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 [1199011 + 1198631199011 (1199011 minus 21199010)]

119879119866119879119868119879119865 (1199010 minus 1199011)2

119861 =119879119866 + 119879119868 minus 11987911986611987911986811986311990101199011

119879119866119879119868 (1199010 minus 1199011)

119862 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199011 minus 11986311990120)

119879119866119879119868119879119865 (1199010 minus 1199011)2

(14)

Case 3 (two repeated poles 1199011 = 1199012 = 1199010)

119906119891119903 (119905) = 1198601198901199011119905 + 1198611199051198901199011119905 + 1198621198901199010119905 + 119863 (15)

where

119860 = minus119879119866 + 119879119868 + 1198791198661198791198681198631199010 (1199010 minus 21199011)

119879119866119879119868 (1199011 minus 1199010)2

119861 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199011 minus 11986311990101199011)

119879119866119879119868119879119865 (1199011 minus 1199010)

119862 =119879119866 + 119879119868 minus 11987911986611987911986811986311990121119879119866119879119868 (1199011 minus 1199010)

2

(16)

Case 4 (three repeated poles 1199011 = 1199012 = 1199010)

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 0511986211990521198901199010119905 + 119863 (17)

where119860 = minus119863

119861 = 1 + 1198631199010

119862 = 1119879119866

+ 1119879119868

minus 11986311990120

(18)

In all cases119863 = 119879119880(119879119866 minus 119879119876)119879119868 In all cases except Case1 (b)

1199011 =minus (119879119866 + 119879119868)radic119879119880 + radicΔ 1

2119879119868119879119866radic119879119880

1199012 =minus (119879119866 + 119879119868)radic119879119880 minus radicΔ 1

2119879119868119879119866radic119879119880

(19)

In Case 1 (b)

1199011 = 1205901 + 1198951205961

1199012 = 1205902 + 1198951205962

1205901 = 1205902 =minus (119879119866 + 119879119868)2119879119868119879119866

12059612 =plusmnradicminusΔ 1

2119879119868119879119866radic119879119880

(20)

Complexity 7

+1

+j

+1

+j

+1

+j

Case 1(a) Case 1(b) Case 2(a)

+1

+j

+1

+j

+1

+j

Case 2(b) Case 3 Case 4

p2

p2p2

p1

p1

p1

p0

p0p0

p2 = p0p1 = p2

p1 = p0

p1 = p2 = p0

Figure 4 Pole distributions of UFR transfer function

Thepole distributions ofUFR transfer function are shownin Figure 4

It can be seen that 119906119891119903(119905) approaches119863 in Cases 1 through4 This is consistent with (5) in which 119906119891119903(infin) is equal to119863

In nonlinear HCS if aratersquo(t)lt0 or comratersquo(t)gt COM-RATEm 119906119891119903(119905) would be different from that in a linearHCS Specifically if comratersquo(t)gt COMRATEm ufr(t) wouldincrease because the actual container handling completionrate cannot process the freight requirement in time and119906119891119903(infin)may be larger than D ITAE would be different fromthat of linear HCS If 119906119891119903(infin) gt 119863 ITAE would increaseindefinitely

4 Resilience Analysis of HCS inContainer Ports

The quantitative resilience framework for HCS developedin this work is deployed with an illustrative example of acontainer port in China Its HCS serves an important rolein container flows around the world and its resilience isvital to the larger multimodal transportation system and thePSSC It has four container docks Under normal operationsits annual container throughput would be around 4 millionTEU 119879119876 would be between 0 and 3 days while 119879119868 would bebetween 0 and 1 day For the simulationmodels and resilienceanalysis it is assumed that 119879119876=15 days and 119879119868=05 days 119879119866is 1 day 119879119865 is between 2 and 30 days while 119879119880 is between0 and 10 days Based on the modeling concepts developedin the previous sections resiliencies of two different HCSsare studied separately (i) linear HCS and (ii) nonlinear HCSSimulation code was written in MATLAB If any of the HCSwere to oscillate and become inoperable it might stop theflow of containers handled by that dock and result in portcongestions

41 Resilience of Linear HCS Linear HCSs are without limitsforARATE andCOMRATE Based on the above data ITAE oflinear HCS is calculated according to Section 3The results ofITAE are calculated and plotted when 119879119865 isin [2 days 30 days]and 119879119880 isin (0 day 10 days] 3D mesh and contour maps ofITAE are shown in Figures 5(a) and 5(b) respectively InFigure 5 it can be seen that ITAE increases gradually withincreases in 119879119865 and 119879119880 in most cases But this change is notcompletely linear When 119879119865 and 119879119880 approach 2 days and10 days respectively ITAE will gradually increase within acertain range We are concerned about the values of 119879119865 and119879119880 when ITAE reaches its minimum value However they arenot easily found in Figure 5 For this reason 3D mesh andcontour maps of 1ITAE are plotted in Figures 6(a) and 6(b)respectively From Figure 6 1ITAE reaches its maximumvalue when 119879119865 = 2 days and 119879119880 997888rarr 0 day (119879119880 = 0)The minimum of ITAE is approximately equal to 52812 Thisimplies that carefully setting the parameters 119879119865 and 119879119880 mayimprove dynamic performance of the linearHCS In additionit suggests that the closer the collaboration of port authoritiesand other players in PSSC is the better the resilience of linearHCS is

42 Resilience of Nonlinear HCS Nonlinear HCS focuses onthe limits for ARATE and COMRATE How the ARATE andCOMRATE influence the nonlinear HCS dynamics is to beexamined In nonlinear HCS the smaller COMRATEm isthe more likely it is that COMRATErsquo(t)gt COMRATEm andthe difference between 119906119891119903(119905) in nonlinear and linear HCSsis larger Correspondingly the difference between ITAE ofnonlinear and linear HCSs is greater In order to facilitatethe obvious distinction COMRATEm =105 here When 119879119865 isin[2 days 30 days] and 119879119880 isin (0 day 10 days] 3D mesh andcontour maps of ITAE are given in Figures 7(a) and 7(b)

8 Complexity

0

5

10

10

20

300

2

4

6

8

TUTF

ITA

E

times104

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 5 ITAE of linear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 6 1 ITAE of linear handling chain system

respectively 3D mesh and contour maps of 1ITAE are givenin Figures 8(a) and 8(b) respectively In Figure 8(a) someareas in which 119906119891119903(infin) gt 119863 and ITAE is infinite are notplotted These areas are the light blue shaded regions inFigure 8(b) From Figure 8 ITAE reaches its minimum value(1ITAEasymp0092 and ITAEasymp1082) when 119879119865 = 2 days 119879119880 =24days From a comparison between Figures 5 and 7 ITAE(approximately 6660) is infinitymore thanhalf of the entirearea It reveals that the impact of COMRATEm on containerport performance is obvious Considering these results canbe determined that if COMRATEm are just applied to thenonlinear HCS modifying the setting of parameter 119879119880 willimprove the resilience of the nonlinear HCS

In Figure 6 there are two regions in which 1ITAEgradually increases and reaches the peaks of these regions

One region is close to the point of 119879119880=0 day and 119879119865=2 daysand the other region is close to the point of 119879119880=24 daysand 119879119865=2 days 1ITAE gradually reaches its maximum whenapproaching the point of 119879119880=0 day and 119879119865=2 days This ismost likely due to the fact that they are dependent processesThe faster the unsatisfied freight requirement is deliveredthe sooner the linear HCS will recover to normal operationHence reasonably setting the parameters119879119865 and119879119880 can solveport congestion problems but it is important to take intoaccount that each modification will affect the resilience oflinear HCS

In Figure 8 1ITAE becomes very small in the region closeto the point of 119879U=0 day and 119879F=2 days From Figure 8(b)ITAE becomes infinite in this region The new maximum of1ITAE is in the point of 119879U=24 days and 119879F=2 days The

Complexity 9

24

68

10

10

20

300

1

2

3

4

TUTF

ITA

E

times104

(a) 3D mesh map

2 4 6 8 10TU

TF

30

25

20

15

10

5

(b) Contour map

Figure 7 ITAE of nonlinear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 8 1ITAE of nonlinear handling chain system

results of the two HCSs discussed above show the usefulnessof the resilience measurement is highlighted in the analysisThese can be used for the PSSCrsquos decisions

43 COMRATEmrsquos Influence on Nonlinear HCS In orderto examine how the COMRATEm influences the nonlinearHCS dynamics the point of 119879119865=2 days and 119879119880=02 day istaken as an example and ufr(t) arate(t) and comrate(t) areplotted in Figure 9 with different COMRATEm (COMRATEm=100 105 110 115 and 120) date isin [0 day 40 days] Inorder to clearly distinguish these curves the horizontal axiscoordinates only take 0sim15 days in Figures 9(a) and 9(b)They eventually reach stable values The stable value of ufr(t)is different with different COMRATEm the stable values ofarate(t) and comrate(t) are always 1

From Figure 9(a) the values of ufr(t) with differentCOMRATEm are the same before it reaches its peak After itspeak the values of ufr(t) with differentCOMRATEm are quitedifferent from one another When COMRATEm =1 ufr(t)maintains its peak value after reaching it When COMRATEm=12 ufr(t) eventually is equal to D(D=02) which is thesame as that in the linear HCS From Figure 9(b) in dates2 through 8 the enhanced ufr(t) makes arate(t) higherwhen COMRATEm is lower From Figure 9(c) comrate(t) islimited by COMRATEm in dates 2sim8 This limitation resultsin the cumulative increase of ufr(t) Due to the limited valueof COMRATE there are cases where dynamic behaviorsof nonlinear HCS cause fr(t) not to be handled timelyeventually resulting in the increased ufr(t) Meanwhile therewould be change noted in arate(t)

10 Complexity

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 15 20 25 30 35 400Date

0

02

04

06

08

1U

ncom

plet

ed F

reig

ht R

equi

rem

ent

(a) ufr(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

05

1

15

2

25

Arr

ive r

ate

(b) arate(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

02

04

06

08

1

12

Com

plet

e rat

e

(c) comrate(t)

Figure 9 States and outputs in nonlinear HCS with different COMRATEm

To investigate the influence of COMRATEm on ITAEthe proportion of ITAE=infin in nonlinear HCS with differentCOMRATEm is plotted in Figure 10 The observation ofdifferent COMRATEm suggests the following all values ofITAE are infinite when COMRATEm le 1 When COM-RATEm = 1sim11 the proportion of ITAE=infin clearly decreasesWhen COMRATEm = 11sim12 the proportion of ITAE=infinis close to 0 and decreases slowly When COMRATEm =12 the proportion of ITAE=infin equals 0 which is the sameas that in the linear HCS These observations show thatthe fluctuation of ufr(t) could be alleviated if the non-linear HCS is carefully designed although it seems to beinevitable

5 Conclusion

The emphasis of container port authorities and other playersin the PSSC has shifted from traditional passive managementto active response highlighting the need to quantify theresilience of HCS for inevitable oscillationsThis work buildsand analyzes linear and nonlinearHCSmodels Resilience forHCS in container ports is measured by ITAE of unsatisfiedfreight requirement It provides a new perspective on PSSCpartners to plan for HCS where resilience could guide theselection of control parameters HCSs in container portsare taken as example to analyze resilience and it is foundthat Comratem has significant influence on resilience for

Complexity 11

095 1 105 11 115 120

02

04

06

08

1

COMRATEm

Prop

ortio

n of

ITA

E =infin

Figure 10 Proportion of ITAE =infin in nonlinear HCS with differentCOMRATEm

nonlinear model These contributions serve as a startingpoint in the development of a quantitative resilience decisionmaking framework for PSSC Future work will consider theresilience and recovery strategies under stochastic freightrequirements given the two HCSs

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Social Science Foun-dation project of China (No 17CGL018) Here the authorswould like to express their gratitude to it

References

[1] J J Wang and M C Cheng ldquoFrom a hub port city to a globalsupply chain management center a case study of Hong KongrdquoJournal of Transport Geography vol 18 no 1 pp 104ndash115 2010

[2] F Wang J Huang and Z Liu ldquoPort management and oper-ations Emerging research topics and progressrdquo Journal ofManagement Sciences in China vol 20 no 05 pp 111ndash126 2017

[3] B Jiang J Li and S Shen ldquoSupply Chain Risk Assessment andControl of Port Enterprises Qingdao port as case studyrdquo AsianJournal of Shipping andLogistics vol 34 no 3 pp 198ndash208 2018

[4] D W Song and P M Panayides ldquoGlobal supply chain andportterminal integration and competitivenessrdquo in Proceedingsof the International Conference on Logistics Shipping and PortManagement Taiwan March 2007

[5] S J Pettit and A K C Beresford ldquoPort development Fromgateways to logistics hubsrdquoMaritime Policy ampManagement vol36 no 3 pp 253ndash267 2009

[6] S-HWoo S J Pettit and A K C Beresford ldquoAn assessment ofthe integration of seaports into supply chains using a structuralequation modelrdquo Supply Chain Management An InternationalJournal vol 18 no 3 pp 235ndash252 2013

[7] L Hou and H Geerlings ldquoDynamics in sustainable port andhinterland operations A conceptual framework and simulationof sustainability measures and their effectiveness based onan application to the Port of Shanghairdquo Journal of CleanerProduction vol 135 pp 449ndash456 2016

[8] K L Roslashdseth P B Wangsness and H Schoslashyen ldquoHow doeconomies of density in container handling operations affectshipsrsquo time and emissions in port Evidence from Norwegiancontainer terminalsrdquo Transportation Research Part D Transportand Environment vol 59 pp 385ndash399 2018

[9] J Tongzon Y Chang and S Lee ldquoHow supply chain oriented isthe port sectorrdquo International Journal of Production Economicsvol 122 no 1 pp 21ndash34 2009

[10] C Ta A V Goodchild and K Pitera ldquoStructuring a definitionof resilience for the freight transportation systemrdquo Transporta-tion Research Record no 2097 pp 19ndash25 2009

[11] A H Gharehgozli J Mileski A Adams and W von ZharenldquoEvaluating a ldquowicked problemrdquo A conceptual frameworkon seaport resiliency in the event of weather disruptionsrdquoTechnological Forecasting amp Social Change vol 121 pp 65ndash752017

[12] H S Loh and V V Thai ldquoManaging port-related supplychain disruptions (PSCDs) amanagementmodel and empiricalevidencerdquo Maritime Policy amp Management vol 43 no 4 pp436ndash455 2016

[13] K Zavitsas T Zis and M G Bell ldquoThe impact of flexibleenvironmental policy on maritime supply chain resiliencerdquoTransport Policy vol 72 pp 116ndash128 2018

[14] M Saleh R Oliva C E Kampmann and P I Davidsen ldquoAcomprehensive analytical approach for policy analysis of systemdynamics modelsrdquo European Journal of Operational Researchvol 203 no 3 pp 673ndash683 2010

[15] B Yang X Wang Y Zhang Y Xu and W Zhou ldquoFinite-time synchronization and synchronization dynamics analysisfor two classes of markovian switching multiweighted complexnetworks from synchronization control rule viewpointrdquo Com-plexity vol 2019 Article ID 1921632 17 pages 2019

[16] S John M M Naim and D R Towill ldquoDynamic analysisof a WIP compensated decision support systemrdquo InternationalJournal of Manufacturing System Design vol 1 pp 283ndash2971994

[17] H A Gabbar ldquoModeling approach for flexible productionchain operationrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 38 no 1 pp 48ndash55 2008

[18] C Li R Mo Z Chang H Yang N Wan and D ZhangldquoDecision-making of feature operation chain considering pro-cessing requirements andmanufacturing stabilityrdquoThe Interna-tional Journal of Advanced Manufacturing Technology vol 87no 5-8 pp 1ndash13 2016

[19] V L Spiegler and M M Naim ldquoInvestigating sustained oscilla-tions in nonlinear production and inventory control modelsrdquoEuropean Journal of Operational Research vol 261 no 2 pp572ndash583 2017

[20] A T Almaktoom ldquoStochastic reliability measurement anddesign optimization of an inventory management systemrdquoComplexity vol 2017 Article ID 1460163 9 pages 2017

[21] J Ma L Zhu Y Yuan and S Hou ldquoStudy of the bullwhip effectin amultistage supply chain with callback structure considering

12 Complexity

two retailersrdquo Complexity vol 2018 Article ID 3650148 12pages 2018

[22] R Pant K Barker J E Ramirez-Marquez and C M RoccoldquoStochastic measures of resilience and their application tocontainer terminalsrdquo Computers amp Industrial Engineering vol70 no 1 pp 183ndash194 2014

[23] Y Wang S W Wallace B Shen and T Choi ldquoService supplychain management a review of operational modelsrdquo EuropeanJournal of Operational Research vol 247 no 3 pp 685ndash6982015

[24] D R Towill ldquoDynamic analysis of an inventory and order basedproduction control systemrdquo International Journal of ProductionResearch vol 20 no 6 pp 671ndash687 1982

[25] L ZhouMM Naim and SM Disney ldquoThe impact of productreturns and remanufacturing uncertainties on the dynamicperformance of a multi-echelon closed-loop supply chainrdquoInternational Journal of Production Economics vol 183 pp 487ndash502 2017

[26] V L M Spiegler M M Naim and J Wikner ldquoA control engi-neering approach to the assessment of supply chain resiliencerdquoInternational Journal of Production Research vol 50 no 21 pp6162ndash6187 2012

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Complexity 5

119860119877119860119879119864119865119877

=(1 + 119879119866119904) [119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904]

119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

(3)

(1) Steady State Condition One of the poles is easily identified(1199010=-1119879119865) The other two poles are set as 1199011 and 1199012 Then

1199011 + 1199012 =1119879119868

+ 1119879119866

11990111199012 =1

(119879119880119879119866)

(4)

Then it can be seen that the steady state condition is 1119879119868+1119879119866 gt 0 1(119879119880119879119866) gt 0 and 1119879119865 gt 0 Because 119879119880 119879119865 119879119866119879119876 and 119879119868 are all positive real numbers this system is stable

(2) Final Value of 119906119891119903(119905) and 119888119900119898119903119886119905119890(119905) In deriving transferfunctions in complex frequency domain we exploit theclassic step input as it helps to develop insights into dynamicbehavior We use the step response to assess the systemrsquosability to cope with oscillations in the linear HCS The stepresponse is powerful as from this simple FR input thesize and quantity of the subsequent 119906119891119903(infin) 119888119900119898119903119886119905119890(infin)and 119886119903119886119905119890(infin) can be readily determined providing a richunderstanding of the dynamics of the linear HCS If FR is aunit step signal 119906119891119903(infin) 119888119900119898119903119886119905119890(infin) and 119886119903119886119905119890(infin) can beobtained by final value theorem

119906119891119903 (infin) = lim119904997888rarr0

119904 sdot 119880119865119877 (119904)

= lim119904997888rarr0

(119879119866 + 119879119868 + 119879119868119879119866119904) (1 + 119879119865119904) minus 119879119876 minus 119879119868119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

=119879119880 (119879119866 minus 119879119876)

119879119868

(5)

119888119900119898119903119886119905119890 (infin) = lim119904997888rarr0

119904 sdot 119862119874119872119877119860119879119864 (119904)

= lim119904997888rarr0

119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

= 119879119868119879119880119879119868119879119880

= 1

(6)

119886119903119886119905119890 (infin) = lim119904997888rarr0

119904 sdot 119860119877119860119879119864 (119904)

= lim119904997888rarr0

(1 + 119879119866119904) [119879119868 + (119879119868119879119865 minus 119879119880119879119876 minus 119879119880119879119868) 119904]119879119880119879119868119879119866119879119865 [1119879119880119879119866 + (1119879119868 + 1119879119866) 119904 + 1199042] (119904 + 1119879119865)

= 119879119868119879119865119879119868119879119865

= 1

(7)

It can be seen that 119906119891119903(infin) depends on 119879119880 119879119868 and thedifference between 119879119866 and 119879119876 119906119891119903(infin) would be 0 onlywhen 119879119866 = 119879119876 119888119900119898119903119886119905119890(119905) and 119886119903119886119905119890(119905) will approach 1when 119905 997888rarr infin Therefore when the HCS reaches its steadystate 119888119900119898119903119886119905119890(119905) and119891119903(119905) are balanced and 119906119891119903(119905) no longerchanges

22 Nonlinear Model In real environments of containerports the value of ARATE cannot be negative althoughmathematically it holds The upper value of COMRATEis limited however the linear model in Figure 2 cannotguarantee these two nonlinear natures To reflect them anonlinearHCSmodel consisting of twononlinear segments isgiven in Figure 3 ARATErsquo is the theoretical arrive rate whichequals to 119886V119891119903(119905) + 119906119891119903(119905)119879119880 + 119890119891119903119894119901(119905)119879119868 COMRATErsquo isthe theoretical container handling completion rate which isonly calculated by 119886119903119886119905119890(119905) and the handling operation policy1(1 + 119879119866119904) COMRATEm is the upper limit of COMRATECOMRATEm should not be less than 119891119903(119905) over time other-wise 119906119891119903(119905) will continuously increase

If COMRATEm is equal to 119891119903(119905) or slightly larger than119891119903(119905) the actual container handling completion rate wouldnot process the accumulated freight requirement timelyThen119906119891119903(infin) may be larger than 119879119880(119879119866 minus 119879119876)119879119868 If 119906119891119903(119905) is toolarge it would congest the HCS

3 Resilience for Handling Chain System

To understand the resilience for HCS in container portswe use the integral of time multiplied by the absolute error(ITAE) as a benchmark ITAE emphasizes long-durationerrors It is recommended for the analyses of systems whichrequire a fast settling time The minimum value of ITAEcorresponds to the best response and recoverywith the lowestdeviation from the target or readiness (Virginia et al 2012)[26] The ITAE is given by 119868119879119860119864 = int+infin

0119905 sdot |119890(119905)|119889119905 =

lim120575119905997888rarr0suminfin119905=0 119905|119890(119905)|120575119905 where 119890(119905) = 119906119891119903(119905) minus 119906119891119903(infin) To

facilitate uniform analysis 119906119891119903(infin) = 119879119880(119879119866 minus119879119876)119879119868 whichis consistent with (5)

In linear HCS 119906119891119903(119905) can be calculated in the followingfour cases below where A B and 119862 are coefficients related tothe system poles and119863 is the coefficient of the step input pole(s=0)

Case 1 (distinct poles real andor complex 1199011 = 1199012 = 1199010)(a) Δ 1 gt 0 (Δ 1 = 119879119880(119879119866 + 119879119868)

2 minus 41198792119868119879119866)

119906119891119903 (119905) = 1198601198901199010119905 + 1198611198901199011119905 + 1198621198901199012119905 + 119863 (8)

(b) Δ 1 lt 0

119906119891119903 (119905) = 1198601198901199010119905 + 1198611198901205901119905 (cos1205961119905 minus 119895 sin1205961119905)

+ 1198621198901205902119905 (cos1205962119905 minus 119895 sin1205962119905) + 119863(9)

In Case 1 (a) and (b) the coefficient values of A B and 119862are

119860 =119879119868 + 119879119866 minus 11987911986611987911986811986311990121199011119879119866119879119868 (1199011 minus 1199010) (1199012 minus 1199010)

119861 =119879119868119879119865 + 119879119866119879119865 + 119879119866119879119868 + 119879119866119879119868119879119865 (1199011 minus 11986311990121199010)

119879119866119879119868119879119865 (1199011 minus 1199012) (1199011 minus 1199010)

119862 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199012 minus 11986311990101199011)

119879119866119879119868119879119865 (1199012 minus 1199010) (1199012 minus 1199011)

(10)

6 Complexity

EFRIP

UFR

FRIP

FR ++ +

+

+_

AVFR COMRATEARATE

+ _

DFRIP

0

_

ARATE COMRATE

feedbackUFR

Lead policy

UFR policy

FR policy

FRIP policy

1

1 + TFs

TQ

1

1 + TGs

1

s

1

s

FRIP feedback

COMRATEＧ

1

TU

1

TI

Figure 3 Nonlinear model of handling chain system in container ports

Case 2 (two repeated poles 1199011=1199010 or 1199012=1199010) (a) 1199011 = 11990121199011 = 1199010

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 1198621198901199012119905 + 119863 (11)

where

119860

= minus119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 [1199012 + 1198631199012 (1199012 minus 21199010)]

119879119866119879119868119879119865 (1199010 minus 1199012)2

119861 =119879119866 + 119879119868 minus 11987911986611987911986811986311990101199012

119879119866119879119868 (1199010 minus 1199012)

119862 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199012 minus 11986311990120)

119879119866119879119868119879119865 (1199010 minus 1199012)2

(12)

(b) 1199011 = 1199012 1199012 = 1199010

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 1198621198901199011119905 + 119863 (13)

where

119860

= minus119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 [1199011 + 1198631199011 (1199011 minus 21199010)]

119879119866119879119868119879119865 (1199010 minus 1199011)2

119861 =119879119866 + 119879119868 minus 11987911986611987911986811986311990101199011

119879119866119879119868 (1199010 minus 1199011)

119862 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199011 minus 11986311990120)

119879119866119879119868119879119865 (1199010 minus 1199011)2

(14)

Case 3 (two repeated poles 1199011 = 1199012 = 1199010)

119906119891119903 (119905) = 1198601198901199011119905 + 1198611199051198901199011119905 + 1198621198901199010119905 + 119863 (15)

where

119860 = minus119879119866 + 119879119868 + 1198791198661198791198681198631199010 (1199010 minus 21199011)

119879119866119879119868 (1199011 minus 1199010)2

119861 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199011 minus 11986311990101199011)

119879119866119879119868119879119865 (1199011 minus 1199010)

119862 =119879119866 + 119879119868 minus 11987911986611987911986811986311990121119879119866119879119868 (1199011 minus 1199010)

2

(16)

Case 4 (three repeated poles 1199011 = 1199012 = 1199010)

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 0511986211990521198901199010119905 + 119863 (17)

where119860 = minus119863

119861 = 1 + 1198631199010

119862 = 1119879119866

+ 1119879119868

minus 11986311990120

(18)

In all cases119863 = 119879119880(119879119866 minus 119879119876)119879119868 In all cases except Case1 (b)

1199011 =minus (119879119866 + 119879119868)radic119879119880 + radicΔ 1

2119879119868119879119866radic119879119880

1199012 =minus (119879119866 + 119879119868)radic119879119880 minus radicΔ 1

2119879119868119879119866radic119879119880

(19)

In Case 1 (b)

1199011 = 1205901 + 1198951205961

1199012 = 1205902 + 1198951205962

1205901 = 1205902 =minus (119879119866 + 119879119868)2119879119868119879119866

12059612 =plusmnradicminusΔ 1

2119879119868119879119866radic119879119880

(20)

Complexity 7

+1

+j

+1

+j

+1

+j

Case 1(a) Case 1(b) Case 2(a)

+1

+j

+1

+j

+1

+j

Case 2(b) Case 3 Case 4

p2

p2p2

p1

p1

p1

p0

p0p0

p2 = p0p1 = p2

p1 = p0

p1 = p2 = p0

Figure 4 Pole distributions of UFR transfer function

Thepole distributions ofUFR transfer function are shownin Figure 4

It can be seen that 119906119891119903(119905) approaches119863 in Cases 1 through4 This is consistent with (5) in which 119906119891119903(infin) is equal to119863

In nonlinear HCS if aratersquo(t)lt0 or comratersquo(t)gt COM-RATEm 119906119891119903(119905) would be different from that in a linearHCS Specifically if comratersquo(t)gt COMRATEm ufr(t) wouldincrease because the actual container handling completionrate cannot process the freight requirement in time and119906119891119903(infin)may be larger than D ITAE would be different fromthat of linear HCS If 119906119891119903(infin) gt 119863 ITAE would increaseindefinitely

4 Resilience Analysis of HCS inContainer Ports

The quantitative resilience framework for HCS developedin this work is deployed with an illustrative example of acontainer port in China Its HCS serves an important rolein container flows around the world and its resilience isvital to the larger multimodal transportation system and thePSSC It has four container docks Under normal operationsits annual container throughput would be around 4 millionTEU 119879119876 would be between 0 and 3 days while 119879119868 would bebetween 0 and 1 day For the simulationmodels and resilienceanalysis it is assumed that 119879119876=15 days and 119879119868=05 days 119879119866is 1 day 119879119865 is between 2 and 30 days while 119879119880 is between0 and 10 days Based on the modeling concepts developedin the previous sections resiliencies of two different HCSsare studied separately (i) linear HCS and (ii) nonlinear HCSSimulation code was written in MATLAB If any of the HCSwere to oscillate and become inoperable it might stop theflow of containers handled by that dock and result in portcongestions

41 Resilience of Linear HCS Linear HCSs are without limitsforARATE andCOMRATE Based on the above data ITAE oflinear HCS is calculated according to Section 3The results ofITAE are calculated and plotted when 119879119865 isin [2 days 30 days]and 119879119880 isin (0 day 10 days] 3D mesh and contour maps ofITAE are shown in Figures 5(a) and 5(b) respectively InFigure 5 it can be seen that ITAE increases gradually withincreases in 119879119865 and 119879119880 in most cases But this change is notcompletely linear When 119879119865 and 119879119880 approach 2 days and10 days respectively ITAE will gradually increase within acertain range We are concerned about the values of 119879119865 and119879119880 when ITAE reaches its minimum value However they arenot easily found in Figure 5 For this reason 3D mesh andcontour maps of 1ITAE are plotted in Figures 6(a) and 6(b)respectively From Figure 6 1ITAE reaches its maximumvalue when 119879119865 = 2 days and 119879119880 997888rarr 0 day (119879119880 = 0)The minimum of ITAE is approximately equal to 52812 Thisimplies that carefully setting the parameters 119879119865 and 119879119880 mayimprove dynamic performance of the linearHCS In additionit suggests that the closer the collaboration of port authoritiesand other players in PSSC is the better the resilience of linearHCS is

42 Resilience of Nonlinear HCS Nonlinear HCS focuses onthe limits for ARATE and COMRATE How the ARATE andCOMRATE influence the nonlinear HCS dynamics is to beexamined In nonlinear HCS the smaller COMRATEm isthe more likely it is that COMRATErsquo(t)gt COMRATEm andthe difference between 119906119891119903(119905) in nonlinear and linear HCSsis larger Correspondingly the difference between ITAE ofnonlinear and linear HCSs is greater In order to facilitatethe obvious distinction COMRATEm =105 here When 119879119865 isin[2 days 30 days] and 119879119880 isin (0 day 10 days] 3D mesh andcontour maps of ITAE are given in Figures 7(a) and 7(b)

8 Complexity

0

5

10

10

20

300

2

4

6

8

TUTF

ITA

E

times104

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 5 ITAE of linear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 6 1 ITAE of linear handling chain system

respectively 3D mesh and contour maps of 1ITAE are givenin Figures 8(a) and 8(b) respectively In Figure 8(a) someareas in which 119906119891119903(infin) gt 119863 and ITAE is infinite are notplotted These areas are the light blue shaded regions inFigure 8(b) From Figure 8 ITAE reaches its minimum value(1ITAEasymp0092 and ITAEasymp1082) when 119879119865 = 2 days 119879119880 =24days From a comparison between Figures 5 and 7 ITAE(approximately 6660) is infinitymore thanhalf of the entirearea It reveals that the impact of COMRATEm on containerport performance is obvious Considering these results canbe determined that if COMRATEm are just applied to thenonlinear HCS modifying the setting of parameter 119879119880 willimprove the resilience of the nonlinear HCS

In Figure 6 there are two regions in which 1ITAEgradually increases and reaches the peaks of these regions

One region is close to the point of 119879119880=0 day and 119879119865=2 daysand the other region is close to the point of 119879119880=24 daysand 119879119865=2 days 1ITAE gradually reaches its maximum whenapproaching the point of 119879119880=0 day and 119879119865=2 days This ismost likely due to the fact that they are dependent processesThe faster the unsatisfied freight requirement is deliveredthe sooner the linear HCS will recover to normal operationHence reasonably setting the parameters119879119865 and119879119880 can solveport congestion problems but it is important to take intoaccount that each modification will affect the resilience oflinear HCS

In Figure 8 1ITAE becomes very small in the region closeto the point of 119879U=0 day and 119879F=2 days From Figure 8(b)ITAE becomes infinite in this region The new maximum of1ITAE is in the point of 119879U=24 days and 119879F=2 days The

Complexity 9

24

68

10

10

20

300

1

2

3

4

TUTF

ITA

E

times104

(a) 3D mesh map

2 4 6 8 10TU

TF

30

25

20

15

10

5

(b) Contour map

Figure 7 ITAE of nonlinear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 8 1ITAE of nonlinear handling chain system

results of the two HCSs discussed above show the usefulnessof the resilience measurement is highlighted in the analysisThese can be used for the PSSCrsquos decisions

43 COMRATEmrsquos Influence on Nonlinear HCS In orderto examine how the COMRATEm influences the nonlinearHCS dynamics the point of 119879119865=2 days and 119879119880=02 day istaken as an example and ufr(t) arate(t) and comrate(t) areplotted in Figure 9 with different COMRATEm (COMRATEm=100 105 110 115 and 120) date isin [0 day 40 days] Inorder to clearly distinguish these curves the horizontal axiscoordinates only take 0sim15 days in Figures 9(a) and 9(b)They eventually reach stable values The stable value of ufr(t)is different with different COMRATEm the stable values ofarate(t) and comrate(t) are always 1

From Figure 9(a) the values of ufr(t) with differentCOMRATEm are the same before it reaches its peak After itspeak the values of ufr(t) with differentCOMRATEm are quitedifferent from one another When COMRATEm =1 ufr(t)maintains its peak value after reaching it When COMRATEm=12 ufr(t) eventually is equal to D(D=02) which is thesame as that in the linear HCS From Figure 9(b) in dates2 through 8 the enhanced ufr(t) makes arate(t) higherwhen COMRATEm is lower From Figure 9(c) comrate(t) islimited by COMRATEm in dates 2sim8 This limitation resultsin the cumulative increase of ufr(t) Due to the limited valueof COMRATE there are cases where dynamic behaviorsof nonlinear HCS cause fr(t) not to be handled timelyeventually resulting in the increased ufr(t) Meanwhile therewould be change noted in arate(t)

10 Complexity

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 15 20 25 30 35 400Date

0

02

04

06

08

1U

ncom

plet

ed F

reig

ht R

equi

rem

ent

(a) ufr(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

05

1

15

2

25

Arr

ive r

ate

(b) arate(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

02

04

06

08

1

12

Com

plet

e rat

e

(c) comrate(t)

Figure 9 States and outputs in nonlinear HCS with different COMRATEm

To investigate the influence of COMRATEm on ITAEthe proportion of ITAE=infin in nonlinear HCS with differentCOMRATEm is plotted in Figure 10 The observation ofdifferent COMRATEm suggests the following all values ofITAE are infinite when COMRATEm le 1 When COM-RATEm = 1sim11 the proportion of ITAE=infin clearly decreasesWhen COMRATEm = 11sim12 the proportion of ITAE=infinis close to 0 and decreases slowly When COMRATEm =12 the proportion of ITAE=infin equals 0 which is the sameas that in the linear HCS These observations show thatthe fluctuation of ufr(t) could be alleviated if the non-linear HCS is carefully designed although it seems to beinevitable

5 Conclusion

The emphasis of container port authorities and other playersin the PSSC has shifted from traditional passive managementto active response highlighting the need to quantify theresilience of HCS for inevitable oscillationsThis work buildsand analyzes linear and nonlinearHCSmodels Resilience forHCS in container ports is measured by ITAE of unsatisfiedfreight requirement It provides a new perspective on PSSCpartners to plan for HCS where resilience could guide theselection of control parameters HCSs in container portsare taken as example to analyze resilience and it is foundthat Comratem has significant influence on resilience for

Complexity 11

095 1 105 11 115 120

02

04

06

08

1

COMRATEm

Prop

ortio

n of

ITA

E =infin

Figure 10 Proportion of ITAE =infin in nonlinear HCS with differentCOMRATEm

nonlinear model These contributions serve as a startingpoint in the development of a quantitative resilience decisionmaking framework for PSSC Future work will consider theresilience and recovery strategies under stochastic freightrequirements given the two HCSs

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Social Science Foun-dation project of China (No 17CGL018) Here the authorswould like to express their gratitude to it

References

[1] J J Wang and M C Cheng ldquoFrom a hub port city to a globalsupply chain management center a case study of Hong KongrdquoJournal of Transport Geography vol 18 no 1 pp 104ndash115 2010

[2] F Wang J Huang and Z Liu ldquoPort management and oper-ations Emerging research topics and progressrdquo Journal ofManagement Sciences in China vol 20 no 05 pp 111ndash126 2017

[3] B Jiang J Li and S Shen ldquoSupply Chain Risk Assessment andControl of Port Enterprises Qingdao port as case studyrdquo AsianJournal of Shipping andLogistics vol 34 no 3 pp 198ndash208 2018

[4] D W Song and P M Panayides ldquoGlobal supply chain andportterminal integration and competitivenessrdquo in Proceedingsof the International Conference on Logistics Shipping and PortManagement Taiwan March 2007

[5] S J Pettit and A K C Beresford ldquoPort development Fromgateways to logistics hubsrdquoMaritime Policy ampManagement vol36 no 3 pp 253ndash267 2009

[6] S-HWoo S J Pettit and A K C Beresford ldquoAn assessment ofthe integration of seaports into supply chains using a structuralequation modelrdquo Supply Chain Management An InternationalJournal vol 18 no 3 pp 235ndash252 2013

[7] L Hou and H Geerlings ldquoDynamics in sustainable port andhinterland operations A conceptual framework and simulationof sustainability measures and their effectiveness based onan application to the Port of Shanghairdquo Journal of CleanerProduction vol 135 pp 449ndash456 2016

[8] K L Roslashdseth P B Wangsness and H Schoslashyen ldquoHow doeconomies of density in container handling operations affectshipsrsquo time and emissions in port Evidence from Norwegiancontainer terminalsrdquo Transportation Research Part D Transportand Environment vol 59 pp 385ndash399 2018

[9] J Tongzon Y Chang and S Lee ldquoHow supply chain oriented isthe port sectorrdquo International Journal of Production Economicsvol 122 no 1 pp 21ndash34 2009

[10] C Ta A V Goodchild and K Pitera ldquoStructuring a definitionof resilience for the freight transportation systemrdquo Transporta-tion Research Record no 2097 pp 19ndash25 2009

[11] A H Gharehgozli J Mileski A Adams and W von ZharenldquoEvaluating a ldquowicked problemrdquo A conceptual frameworkon seaport resiliency in the event of weather disruptionsrdquoTechnological Forecasting amp Social Change vol 121 pp 65ndash752017

[12] H S Loh and V V Thai ldquoManaging port-related supplychain disruptions (PSCDs) amanagementmodel and empiricalevidencerdquo Maritime Policy amp Management vol 43 no 4 pp436ndash455 2016

[13] K Zavitsas T Zis and M G Bell ldquoThe impact of flexibleenvironmental policy on maritime supply chain resiliencerdquoTransport Policy vol 72 pp 116ndash128 2018

[14] M Saleh R Oliva C E Kampmann and P I Davidsen ldquoAcomprehensive analytical approach for policy analysis of systemdynamics modelsrdquo European Journal of Operational Researchvol 203 no 3 pp 673ndash683 2010

[15] B Yang X Wang Y Zhang Y Xu and W Zhou ldquoFinite-time synchronization and synchronization dynamics analysisfor two classes of markovian switching multiweighted complexnetworks from synchronization control rule viewpointrdquo Com-plexity vol 2019 Article ID 1921632 17 pages 2019

[16] S John M M Naim and D R Towill ldquoDynamic analysisof a WIP compensated decision support systemrdquo InternationalJournal of Manufacturing System Design vol 1 pp 283ndash2971994

[17] H A Gabbar ldquoModeling approach for flexible productionchain operationrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 38 no 1 pp 48ndash55 2008

[18] C Li R Mo Z Chang H Yang N Wan and D ZhangldquoDecision-making of feature operation chain considering pro-cessing requirements andmanufacturing stabilityrdquoThe Interna-tional Journal of Advanced Manufacturing Technology vol 87no 5-8 pp 1ndash13 2016

[19] V L Spiegler and M M Naim ldquoInvestigating sustained oscilla-tions in nonlinear production and inventory control modelsrdquoEuropean Journal of Operational Research vol 261 no 2 pp572ndash583 2017

[20] A T Almaktoom ldquoStochastic reliability measurement anddesign optimization of an inventory management systemrdquoComplexity vol 2017 Article ID 1460163 9 pages 2017

[21] J Ma L Zhu Y Yuan and S Hou ldquoStudy of the bullwhip effectin amultistage supply chain with callback structure considering

12 Complexity

two retailersrdquo Complexity vol 2018 Article ID 3650148 12pages 2018

[22] R Pant K Barker J E Ramirez-Marquez and C M RoccoldquoStochastic measures of resilience and their application tocontainer terminalsrdquo Computers amp Industrial Engineering vol70 no 1 pp 183ndash194 2014

[23] Y Wang S W Wallace B Shen and T Choi ldquoService supplychain management a review of operational modelsrdquo EuropeanJournal of Operational Research vol 247 no 3 pp 685ndash6982015

[24] D R Towill ldquoDynamic analysis of an inventory and order basedproduction control systemrdquo International Journal of ProductionResearch vol 20 no 6 pp 671ndash687 1982

[25] L ZhouMM Naim and SM Disney ldquoThe impact of productreturns and remanufacturing uncertainties on the dynamicperformance of a multi-echelon closed-loop supply chainrdquoInternational Journal of Production Economics vol 183 pp 487ndash502 2017

[26] V L M Spiegler M M Naim and J Wikner ldquoA control engi-neering approach to the assessment of supply chain resiliencerdquoInternational Journal of Production Research vol 50 no 21 pp6162ndash6187 2012

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Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Complexity

EFRIP

UFR

FRIP

FR ++ +

+

+_

AVFR COMRATEARATE

+ _

DFRIP

0

_

ARATE COMRATE

feedbackUFR

Lead policy

UFR policy

FR policy

FRIP policy

1

1 + TFs

TQ

1

1 + TGs

1

s

1

s

FRIP feedback

COMRATEＧ

1

TU

1

TI

Figure 3 Nonlinear model of handling chain system in container ports

Case 2 (two repeated poles 1199011=1199010 or 1199012=1199010) (a) 1199011 = 11990121199011 = 1199010

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 1198621198901199012119905 + 119863 (11)

where

119860

= minus119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 [1199012 + 1198631199012 (1199012 minus 21199010)]

119879119866119879119868119879119865 (1199010 minus 1199012)2

119861 =119879119866 + 119879119868 minus 11987911986611987911986811986311990101199012

119879119866119879119868 (1199010 minus 1199012)

119862 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199012 minus 11986311990120)

119879119866119879119868119879119865 (1199010 minus 1199012)2

(12)

(b) 1199011 = 1199012 1199012 = 1199010

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 1198621198901199011119905 + 119863 (13)

where

119860

= minus119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 [1199011 + 1198631199011 (1199011 minus 21199010)]

119879119866119879119868119879119865 (1199010 minus 1199011)2

119861 =119879119866 + 119879119868 minus 11987911986611987911986811986311990101199011

119879119866119879119868 (1199010 minus 1199011)

119862 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199011 minus 11986311990120)

119879119866119879119868119879119865 (1199010 minus 1199011)2

(14)

Case 3 (two repeated poles 1199011 = 1199012 = 1199010)

119906119891119903 (119905) = 1198601198901199011119905 + 1198611199051198901199011119905 + 1198621198901199010119905 + 119863 (15)

where

119860 = minus119879119866 + 119879119868 + 1198791198661198791198681198631199010 (1199010 minus 21199011)

119879119866119879119868 (1199011 minus 1199010)2

119861 =119879119866119879119868 + 119879119866119879119865 + 119879119868119879119865 + 119879119866119879119868119879119865 (1199011 minus 11986311990101199011)

119879119866119879119868119879119865 (1199011 minus 1199010)

119862 =119879119866 + 119879119868 minus 11987911986611987911986811986311990121119879119866119879119868 (1199011 minus 1199010)

2

(16)

Case 4 (three repeated poles 1199011 = 1199012 = 1199010)

119906119891119903 (119905) = 1198601198901199010119905 + 1198611199051198901199010119905 + 0511986211990521198901199010119905 + 119863 (17)

where119860 = minus119863

119861 = 1 + 1198631199010

119862 = 1119879119866

+ 1119879119868

minus 11986311990120

(18)

In all cases119863 = 119879119880(119879119866 minus 119879119876)119879119868 In all cases except Case1 (b)

1199011 =minus (119879119866 + 119879119868)radic119879119880 + radicΔ 1

2119879119868119879119866radic119879119880

1199012 =minus (119879119866 + 119879119868)radic119879119880 minus radicΔ 1

2119879119868119879119866radic119879119880

(19)

In Case 1 (b)

1199011 = 1205901 + 1198951205961

1199012 = 1205902 + 1198951205962

1205901 = 1205902 =minus (119879119866 + 119879119868)2119879119868119879119866

12059612 =plusmnradicminusΔ 1

2119879119868119879119866radic119879119880

(20)

Complexity 7

+1

+j

+1

+j

+1

+j

Case 1(a) Case 1(b) Case 2(a)

+1

+j

+1

+j

+1

+j

Case 2(b) Case 3 Case 4

p2

p2p2

p1

p1

p1

p0

p0p0

p2 = p0p1 = p2

p1 = p0

p1 = p2 = p0

Figure 4 Pole distributions of UFR transfer function

Thepole distributions ofUFR transfer function are shownin Figure 4

It can be seen that 119906119891119903(119905) approaches119863 in Cases 1 through4 This is consistent with (5) in which 119906119891119903(infin) is equal to119863

In nonlinear HCS if aratersquo(t)lt0 or comratersquo(t)gt COM-RATEm 119906119891119903(119905) would be different from that in a linearHCS Specifically if comratersquo(t)gt COMRATEm ufr(t) wouldincrease because the actual container handling completionrate cannot process the freight requirement in time and119906119891119903(infin)may be larger than D ITAE would be different fromthat of linear HCS If 119906119891119903(infin) gt 119863 ITAE would increaseindefinitely

4 Resilience Analysis of HCS inContainer Ports

The quantitative resilience framework for HCS developedin this work is deployed with an illustrative example of acontainer port in China Its HCS serves an important rolein container flows around the world and its resilience isvital to the larger multimodal transportation system and thePSSC It has four container docks Under normal operationsits annual container throughput would be around 4 millionTEU 119879119876 would be between 0 and 3 days while 119879119868 would bebetween 0 and 1 day For the simulationmodels and resilienceanalysis it is assumed that 119879119876=15 days and 119879119868=05 days 119879119866is 1 day 119879119865 is between 2 and 30 days while 119879119880 is between0 and 10 days Based on the modeling concepts developedin the previous sections resiliencies of two different HCSsare studied separately (i) linear HCS and (ii) nonlinear HCSSimulation code was written in MATLAB If any of the HCSwere to oscillate and become inoperable it might stop theflow of containers handled by that dock and result in portcongestions

41 Resilience of Linear HCS Linear HCSs are without limitsforARATE andCOMRATE Based on the above data ITAE oflinear HCS is calculated according to Section 3The results ofITAE are calculated and plotted when 119879119865 isin [2 days 30 days]and 119879119880 isin (0 day 10 days] 3D mesh and contour maps ofITAE are shown in Figures 5(a) and 5(b) respectively InFigure 5 it can be seen that ITAE increases gradually withincreases in 119879119865 and 119879119880 in most cases But this change is notcompletely linear When 119879119865 and 119879119880 approach 2 days and10 days respectively ITAE will gradually increase within acertain range We are concerned about the values of 119879119865 and119879119880 when ITAE reaches its minimum value However they arenot easily found in Figure 5 For this reason 3D mesh andcontour maps of 1ITAE are plotted in Figures 6(a) and 6(b)respectively From Figure 6 1ITAE reaches its maximumvalue when 119879119865 = 2 days and 119879119880 997888rarr 0 day (119879119880 = 0)The minimum of ITAE is approximately equal to 52812 Thisimplies that carefully setting the parameters 119879119865 and 119879119880 mayimprove dynamic performance of the linearHCS In additionit suggests that the closer the collaboration of port authoritiesand other players in PSSC is the better the resilience of linearHCS is

42 Resilience of Nonlinear HCS Nonlinear HCS focuses onthe limits for ARATE and COMRATE How the ARATE andCOMRATE influence the nonlinear HCS dynamics is to beexamined In nonlinear HCS the smaller COMRATEm isthe more likely it is that COMRATErsquo(t)gt COMRATEm andthe difference between 119906119891119903(119905) in nonlinear and linear HCSsis larger Correspondingly the difference between ITAE ofnonlinear and linear HCSs is greater In order to facilitatethe obvious distinction COMRATEm =105 here When 119879119865 isin[2 days 30 days] and 119879119880 isin (0 day 10 days] 3D mesh andcontour maps of ITAE are given in Figures 7(a) and 7(b)

8 Complexity

0

5

10

10

20

300

2

4

6

8

TUTF

ITA

E

times104

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 5 ITAE of linear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 6 1 ITAE of linear handling chain system

respectively 3D mesh and contour maps of 1ITAE are givenin Figures 8(a) and 8(b) respectively In Figure 8(a) someareas in which 119906119891119903(infin) gt 119863 and ITAE is infinite are notplotted These areas are the light blue shaded regions inFigure 8(b) From Figure 8 ITAE reaches its minimum value(1ITAEasymp0092 and ITAEasymp1082) when 119879119865 = 2 days 119879119880 =24days From a comparison between Figures 5 and 7 ITAE(approximately 6660) is infinitymore thanhalf of the entirearea It reveals that the impact of COMRATEm on containerport performance is obvious Considering these results canbe determined that if COMRATEm are just applied to thenonlinear HCS modifying the setting of parameter 119879119880 willimprove the resilience of the nonlinear HCS

In Figure 6 there are two regions in which 1ITAEgradually increases and reaches the peaks of these regions

One region is close to the point of 119879119880=0 day and 119879119865=2 daysand the other region is close to the point of 119879119880=24 daysand 119879119865=2 days 1ITAE gradually reaches its maximum whenapproaching the point of 119879119880=0 day and 119879119865=2 days This ismost likely due to the fact that they are dependent processesThe faster the unsatisfied freight requirement is deliveredthe sooner the linear HCS will recover to normal operationHence reasonably setting the parameters119879119865 and119879119880 can solveport congestion problems but it is important to take intoaccount that each modification will affect the resilience oflinear HCS

In Figure 8 1ITAE becomes very small in the region closeto the point of 119879U=0 day and 119879F=2 days From Figure 8(b)ITAE becomes infinite in this region The new maximum of1ITAE is in the point of 119879U=24 days and 119879F=2 days The

Complexity 9

24

68

10

10

20

300

1

2

3

4

TUTF

ITA

E

times104

(a) 3D mesh map

2 4 6 8 10TU

TF

30

25

20

15

10

5

(b) Contour map

Figure 7 ITAE of nonlinear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 8 1ITAE of nonlinear handling chain system

results of the two HCSs discussed above show the usefulnessof the resilience measurement is highlighted in the analysisThese can be used for the PSSCrsquos decisions

43 COMRATEmrsquos Influence on Nonlinear HCS In orderto examine how the COMRATEm influences the nonlinearHCS dynamics the point of 119879119865=2 days and 119879119880=02 day istaken as an example and ufr(t) arate(t) and comrate(t) areplotted in Figure 9 with different COMRATEm (COMRATEm=100 105 110 115 and 120) date isin [0 day 40 days] Inorder to clearly distinguish these curves the horizontal axiscoordinates only take 0sim15 days in Figures 9(a) and 9(b)They eventually reach stable values The stable value of ufr(t)is different with different COMRATEm the stable values ofarate(t) and comrate(t) are always 1

From Figure 9(a) the values of ufr(t) with differentCOMRATEm are the same before it reaches its peak After itspeak the values of ufr(t) with differentCOMRATEm are quitedifferent from one another When COMRATEm =1 ufr(t)maintains its peak value after reaching it When COMRATEm=12 ufr(t) eventually is equal to D(D=02) which is thesame as that in the linear HCS From Figure 9(b) in dates2 through 8 the enhanced ufr(t) makes arate(t) higherwhen COMRATEm is lower From Figure 9(c) comrate(t) islimited by COMRATEm in dates 2sim8 This limitation resultsin the cumulative increase of ufr(t) Due to the limited valueof COMRATE there are cases where dynamic behaviorsof nonlinear HCS cause fr(t) not to be handled timelyeventually resulting in the increased ufr(t) Meanwhile therewould be change noted in arate(t)

10 Complexity

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 15 20 25 30 35 400Date

0

02

04

06

08

1U

ncom

plet

ed F

reig

ht R

equi

rem

ent

(a) ufr(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

05

1

15

2

25

Arr

ive r

ate

(b) arate(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

02

04

06

08

1

12

Com

plet

e rat

e

(c) comrate(t)

Figure 9 States and outputs in nonlinear HCS with different COMRATEm

To investigate the influence of COMRATEm on ITAEthe proportion of ITAE=infin in nonlinear HCS with differentCOMRATEm is plotted in Figure 10 The observation ofdifferent COMRATEm suggests the following all values ofITAE are infinite when COMRATEm le 1 When COM-RATEm = 1sim11 the proportion of ITAE=infin clearly decreasesWhen COMRATEm = 11sim12 the proportion of ITAE=infinis close to 0 and decreases slowly When COMRATEm =12 the proportion of ITAE=infin equals 0 which is the sameas that in the linear HCS These observations show thatthe fluctuation of ufr(t) could be alleviated if the non-linear HCS is carefully designed although it seems to beinevitable

5 Conclusion

The emphasis of container port authorities and other playersin the PSSC has shifted from traditional passive managementto active response highlighting the need to quantify theresilience of HCS for inevitable oscillationsThis work buildsand analyzes linear and nonlinearHCSmodels Resilience forHCS in container ports is measured by ITAE of unsatisfiedfreight requirement It provides a new perspective on PSSCpartners to plan for HCS where resilience could guide theselection of control parameters HCSs in container portsare taken as example to analyze resilience and it is foundthat Comratem has significant influence on resilience for

Complexity 11

095 1 105 11 115 120

02

04

06

08

1

COMRATEm

Prop

ortio

n of

ITA

E =infin

Figure 10 Proportion of ITAE =infin in nonlinear HCS with differentCOMRATEm

nonlinear model These contributions serve as a startingpoint in the development of a quantitative resilience decisionmaking framework for PSSC Future work will consider theresilience and recovery strategies under stochastic freightrequirements given the two HCSs

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Social Science Foun-dation project of China (No 17CGL018) Here the authorswould like to express their gratitude to it

References

[1] J J Wang and M C Cheng ldquoFrom a hub port city to a globalsupply chain management center a case study of Hong KongrdquoJournal of Transport Geography vol 18 no 1 pp 104ndash115 2010

[2] F Wang J Huang and Z Liu ldquoPort management and oper-ations Emerging research topics and progressrdquo Journal ofManagement Sciences in China vol 20 no 05 pp 111ndash126 2017

[3] B Jiang J Li and S Shen ldquoSupply Chain Risk Assessment andControl of Port Enterprises Qingdao port as case studyrdquo AsianJournal of Shipping andLogistics vol 34 no 3 pp 198ndash208 2018

[4] D W Song and P M Panayides ldquoGlobal supply chain andportterminal integration and competitivenessrdquo in Proceedingsof the International Conference on Logistics Shipping and PortManagement Taiwan March 2007

[5] S J Pettit and A K C Beresford ldquoPort development Fromgateways to logistics hubsrdquoMaritime Policy ampManagement vol36 no 3 pp 253ndash267 2009

[6] S-HWoo S J Pettit and A K C Beresford ldquoAn assessment ofthe integration of seaports into supply chains using a structuralequation modelrdquo Supply Chain Management An InternationalJournal vol 18 no 3 pp 235ndash252 2013

[7] L Hou and H Geerlings ldquoDynamics in sustainable port andhinterland operations A conceptual framework and simulationof sustainability measures and their effectiveness based onan application to the Port of Shanghairdquo Journal of CleanerProduction vol 135 pp 449ndash456 2016

[8] K L Roslashdseth P B Wangsness and H Schoslashyen ldquoHow doeconomies of density in container handling operations affectshipsrsquo time and emissions in port Evidence from Norwegiancontainer terminalsrdquo Transportation Research Part D Transportand Environment vol 59 pp 385ndash399 2018

[9] J Tongzon Y Chang and S Lee ldquoHow supply chain oriented isthe port sectorrdquo International Journal of Production Economicsvol 122 no 1 pp 21ndash34 2009

[10] C Ta A V Goodchild and K Pitera ldquoStructuring a definitionof resilience for the freight transportation systemrdquo Transporta-tion Research Record no 2097 pp 19ndash25 2009

[11] A H Gharehgozli J Mileski A Adams and W von ZharenldquoEvaluating a ldquowicked problemrdquo A conceptual frameworkon seaport resiliency in the event of weather disruptionsrdquoTechnological Forecasting amp Social Change vol 121 pp 65ndash752017

[12] H S Loh and V V Thai ldquoManaging port-related supplychain disruptions (PSCDs) amanagementmodel and empiricalevidencerdquo Maritime Policy amp Management vol 43 no 4 pp436ndash455 2016

[13] K Zavitsas T Zis and M G Bell ldquoThe impact of flexibleenvironmental policy on maritime supply chain resiliencerdquoTransport Policy vol 72 pp 116ndash128 2018

[14] M Saleh R Oliva C E Kampmann and P I Davidsen ldquoAcomprehensive analytical approach for policy analysis of systemdynamics modelsrdquo European Journal of Operational Researchvol 203 no 3 pp 673ndash683 2010

[15] B Yang X Wang Y Zhang Y Xu and W Zhou ldquoFinite-time synchronization and synchronization dynamics analysisfor two classes of markovian switching multiweighted complexnetworks from synchronization control rule viewpointrdquo Com-plexity vol 2019 Article ID 1921632 17 pages 2019

[16] S John M M Naim and D R Towill ldquoDynamic analysisof a WIP compensated decision support systemrdquo InternationalJournal of Manufacturing System Design vol 1 pp 283ndash2971994

[17] H A Gabbar ldquoModeling approach for flexible productionchain operationrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 38 no 1 pp 48ndash55 2008

[18] C Li R Mo Z Chang H Yang N Wan and D ZhangldquoDecision-making of feature operation chain considering pro-cessing requirements andmanufacturing stabilityrdquoThe Interna-tional Journal of Advanced Manufacturing Technology vol 87no 5-8 pp 1ndash13 2016

[19] V L Spiegler and M M Naim ldquoInvestigating sustained oscilla-tions in nonlinear production and inventory control modelsrdquoEuropean Journal of Operational Research vol 261 no 2 pp572ndash583 2017

[20] A T Almaktoom ldquoStochastic reliability measurement anddesign optimization of an inventory management systemrdquoComplexity vol 2017 Article ID 1460163 9 pages 2017

[21] J Ma L Zhu Y Yuan and S Hou ldquoStudy of the bullwhip effectin amultistage supply chain with callback structure considering

12 Complexity

two retailersrdquo Complexity vol 2018 Article ID 3650148 12pages 2018

[22] R Pant K Barker J E Ramirez-Marquez and C M RoccoldquoStochastic measures of resilience and their application tocontainer terminalsrdquo Computers amp Industrial Engineering vol70 no 1 pp 183ndash194 2014

[23] Y Wang S W Wallace B Shen and T Choi ldquoService supplychain management a review of operational modelsrdquo EuropeanJournal of Operational Research vol 247 no 3 pp 685ndash6982015

[24] D R Towill ldquoDynamic analysis of an inventory and order basedproduction control systemrdquo International Journal of ProductionResearch vol 20 no 6 pp 671ndash687 1982

[25] L ZhouMM Naim and SM Disney ldquoThe impact of productreturns and remanufacturing uncertainties on the dynamicperformance of a multi-echelon closed-loop supply chainrdquoInternational Journal of Production Economics vol 183 pp 487ndash502 2017

[26] V L M Spiegler M M Naim and J Wikner ldquoA control engi-neering approach to the assessment of supply chain resiliencerdquoInternational Journal of Production Research vol 50 no 21 pp6162ndash6187 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 7

+1

+j

+1

+j

+1

+j

Case 1(a) Case 1(b) Case 2(a)

+1

+j

+1

+j

+1

+j

Case 2(b) Case 3 Case 4

p2

p2p2

p1

p1

p1

p0

p0p0

p2 = p0p1 = p2

p1 = p0

p1 = p2 = p0

Figure 4 Pole distributions of UFR transfer function

Thepole distributions ofUFR transfer function are shownin Figure 4

It can be seen that 119906119891119903(119905) approaches119863 in Cases 1 through4 This is consistent with (5) in which 119906119891119903(infin) is equal to119863

In nonlinear HCS if aratersquo(t)lt0 or comratersquo(t)gt COM-RATEm 119906119891119903(119905) would be different from that in a linearHCS Specifically if comratersquo(t)gt COMRATEm ufr(t) wouldincrease because the actual container handling completionrate cannot process the freight requirement in time and119906119891119903(infin)may be larger than D ITAE would be different fromthat of linear HCS If 119906119891119903(infin) gt 119863 ITAE would increaseindefinitely

4 Resilience Analysis of HCS inContainer Ports

The quantitative resilience framework for HCS developedin this work is deployed with an illustrative example of acontainer port in China Its HCS serves an important rolein container flows around the world and its resilience isvital to the larger multimodal transportation system and thePSSC It has four container docks Under normal operationsits annual container throughput would be around 4 millionTEU 119879119876 would be between 0 and 3 days while 119879119868 would bebetween 0 and 1 day For the simulationmodels and resilienceanalysis it is assumed that 119879119876=15 days and 119879119868=05 days 119879119866is 1 day 119879119865 is between 2 and 30 days while 119879119880 is between0 and 10 days Based on the modeling concepts developedin the previous sections resiliencies of two different HCSsare studied separately (i) linear HCS and (ii) nonlinear HCSSimulation code was written in MATLAB If any of the HCSwere to oscillate and become inoperable it might stop theflow of containers handled by that dock and result in portcongestions

41 Resilience of Linear HCS Linear HCSs are without limitsforARATE andCOMRATE Based on the above data ITAE oflinear HCS is calculated according to Section 3The results ofITAE are calculated and plotted when 119879119865 isin [2 days 30 days]and 119879119880 isin (0 day 10 days] 3D mesh and contour maps ofITAE are shown in Figures 5(a) and 5(b) respectively InFigure 5 it can be seen that ITAE increases gradually withincreases in 119879119865 and 119879119880 in most cases But this change is notcompletely linear When 119879119865 and 119879119880 approach 2 days and10 days respectively ITAE will gradually increase within acertain range We are concerned about the values of 119879119865 and119879119880 when ITAE reaches its minimum value However they arenot easily found in Figure 5 For this reason 3D mesh andcontour maps of 1ITAE are plotted in Figures 6(a) and 6(b)respectively From Figure 6 1ITAE reaches its maximumvalue when 119879119865 = 2 days and 119879119880 997888rarr 0 day (119879119880 = 0)The minimum of ITAE is approximately equal to 52812 Thisimplies that carefully setting the parameters 119879119865 and 119879119880 mayimprove dynamic performance of the linearHCS In additionit suggests that the closer the collaboration of port authoritiesand other players in PSSC is the better the resilience of linearHCS is

42 Resilience of Nonlinear HCS Nonlinear HCS focuses onthe limits for ARATE and COMRATE How the ARATE andCOMRATE influence the nonlinear HCS dynamics is to beexamined In nonlinear HCS the smaller COMRATEm isthe more likely it is that COMRATErsquo(t)gt COMRATEm andthe difference between 119906119891119903(119905) in nonlinear and linear HCSsis larger Correspondingly the difference between ITAE ofnonlinear and linear HCSs is greater In order to facilitatethe obvious distinction COMRATEm =105 here When 119879119865 isin[2 days 30 days] and 119879119880 isin (0 day 10 days] 3D mesh andcontour maps of ITAE are given in Figures 7(a) and 7(b)

8 Complexity

0

5

10

10

20

300

2

4

6

8

TUTF

ITA

E

times104

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 5 ITAE of linear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 6 1 ITAE of linear handling chain system

respectively 3D mesh and contour maps of 1ITAE are givenin Figures 8(a) and 8(b) respectively In Figure 8(a) someareas in which 119906119891119903(infin) gt 119863 and ITAE is infinite are notplotted These areas are the light blue shaded regions inFigure 8(b) From Figure 8 ITAE reaches its minimum value(1ITAEasymp0092 and ITAEasymp1082) when 119879119865 = 2 days 119879119880 =24days From a comparison between Figures 5 and 7 ITAE(approximately 6660) is infinitymore thanhalf of the entirearea It reveals that the impact of COMRATEm on containerport performance is obvious Considering these results canbe determined that if COMRATEm are just applied to thenonlinear HCS modifying the setting of parameter 119879119880 willimprove the resilience of the nonlinear HCS

In Figure 6 there are two regions in which 1ITAEgradually increases and reaches the peaks of these regions

One region is close to the point of 119879119880=0 day and 119879119865=2 daysand the other region is close to the point of 119879119880=24 daysand 119879119865=2 days 1ITAE gradually reaches its maximum whenapproaching the point of 119879119880=0 day and 119879119865=2 days This ismost likely due to the fact that they are dependent processesThe faster the unsatisfied freight requirement is deliveredthe sooner the linear HCS will recover to normal operationHence reasonably setting the parameters119879119865 and119879119880 can solveport congestion problems but it is important to take intoaccount that each modification will affect the resilience oflinear HCS

In Figure 8 1ITAE becomes very small in the region closeto the point of 119879U=0 day and 119879F=2 days From Figure 8(b)ITAE becomes infinite in this region The new maximum of1ITAE is in the point of 119879U=24 days and 119879F=2 days The

Complexity 9

24

68

10

10

20

300

1

2

3

4

TUTF

ITA

E

times104

(a) 3D mesh map

2 4 6 8 10TU

TF

30

25

20

15

10

5

(b) Contour map

Figure 7 ITAE of nonlinear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 8 1ITAE of nonlinear handling chain system

results of the two HCSs discussed above show the usefulnessof the resilience measurement is highlighted in the analysisThese can be used for the PSSCrsquos decisions

43 COMRATEmrsquos Influence on Nonlinear HCS In orderto examine how the COMRATEm influences the nonlinearHCS dynamics the point of 119879119865=2 days and 119879119880=02 day istaken as an example and ufr(t) arate(t) and comrate(t) areplotted in Figure 9 with different COMRATEm (COMRATEm=100 105 110 115 and 120) date isin [0 day 40 days] Inorder to clearly distinguish these curves the horizontal axiscoordinates only take 0sim15 days in Figures 9(a) and 9(b)They eventually reach stable values The stable value of ufr(t)is different with different COMRATEm the stable values ofarate(t) and comrate(t) are always 1

From Figure 9(a) the values of ufr(t) with differentCOMRATEm are the same before it reaches its peak After itspeak the values of ufr(t) with differentCOMRATEm are quitedifferent from one another When COMRATEm =1 ufr(t)maintains its peak value after reaching it When COMRATEm=12 ufr(t) eventually is equal to D(D=02) which is thesame as that in the linear HCS From Figure 9(b) in dates2 through 8 the enhanced ufr(t) makes arate(t) higherwhen COMRATEm is lower From Figure 9(c) comrate(t) islimited by COMRATEm in dates 2sim8 This limitation resultsin the cumulative increase of ufr(t) Due to the limited valueof COMRATE there are cases where dynamic behaviorsof nonlinear HCS cause fr(t) not to be handled timelyeventually resulting in the increased ufr(t) Meanwhile therewould be change noted in arate(t)

10 Complexity

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 15 20 25 30 35 400Date

0

02

04

06

08

1U

ncom

plet

ed F

reig

ht R

equi

rem

ent

(a) ufr(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

05

1

15

2

25

Arr

ive r

ate

(b) arate(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

02

04

06

08

1

12

Com

plet

e rat

e

(c) comrate(t)

Figure 9 States and outputs in nonlinear HCS with different COMRATEm

To investigate the influence of COMRATEm on ITAEthe proportion of ITAE=infin in nonlinear HCS with differentCOMRATEm is plotted in Figure 10 The observation ofdifferent COMRATEm suggests the following all values ofITAE are infinite when COMRATEm le 1 When COM-RATEm = 1sim11 the proportion of ITAE=infin clearly decreasesWhen COMRATEm = 11sim12 the proportion of ITAE=infinis close to 0 and decreases slowly When COMRATEm =12 the proportion of ITAE=infin equals 0 which is the sameas that in the linear HCS These observations show thatthe fluctuation of ufr(t) could be alleviated if the non-linear HCS is carefully designed although it seems to beinevitable

5 Conclusion

The emphasis of container port authorities and other playersin the PSSC has shifted from traditional passive managementto active response highlighting the need to quantify theresilience of HCS for inevitable oscillationsThis work buildsand analyzes linear and nonlinearHCSmodels Resilience forHCS in container ports is measured by ITAE of unsatisfiedfreight requirement It provides a new perspective on PSSCpartners to plan for HCS where resilience could guide theselection of control parameters HCSs in container portsare taken as example to analyze resilience and it is foundthat Comratem has significant influence on resilience for

Complexity 11

095 1 105 11 115 120

02

04

06

08

1

COMRATEm

Prop

ortio

n of

ITA

E =infin

Figure 10 Proportion of ITAE =infin in nonlinear HCS with differentCOMRATEm

nonlinear model These contributions serve as a startingpoint in the development of a quantitative resilience decisionmaking framework for PSSC Future work will consider theresilience and recovery strategies under stochastic freightrequirements given the two HCSs

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Social Science Foun-dation project of China (No 17CGL018) Here the authorswould like to express their gratitude to it

References

[1] J J Wang and M C Cheng ldquoFrom a hub port city to a globalsupply chain management center a case study of Hong KongrdquoJournal of Transport Geography vol 18 no 1 pp 104ndash115 2010

[2] F Wang J Huang and Z Liu ldquoPort management and oper-ations Emerging research topics and progressrdquo Journal ofManagement Sciences in China vol 20 no 05 pp 111ndash126 2017

[3] B Jiang J Li and S Shen ldquoSupply Chain Risk Assessment andControl of Port Enterprises Qingdao port as case studyrdquo AsianJournal of Shipping andLogistics vol 34 no 3 pp 198ndash208 2018

[4] D W Song and P M Panayides ldquoGlobal supply chain andportterminal integration and competitivenessrdquo in Proceedingsof the International Conference on Logistics Shipping and PortManagement Taiwan March 2007

[5] S J Pettit and A K C Beresford ldquoPort development Fromgateways to logistics hubsrdquoMaritime Policy ampManagement vol36 no 3 pp 253ndash267 2009

[6] S-HWoo S J Pettit and A K C Beresford ldquoAn assessment ofthe integration of seaports into supply chains using a structuralequation modelrdquo Supply Chain Management An InternationalJournal vol 18 no 3 pp 235ndash252 2013

[7] L Hou and H Geerlings ldquoDynamics in sustainable port andhinterland operations A conceptual framework and simulationof sustainability measures and their effectiveness based onan application to the Port of Shanghairdquo Journal of CleanerProduction vol 135 pp 449ndash456 2016

[8] K L Roslashdseth P B Wangsness and H Schoslashyen ldquoHow doeconomies of density in container handling operations affectshipsrsquo time and emissions in port Evidence from Norwegiancontainer terminalsrdquo Transportation Research Part D Transportand Environment vol 59 pp 385ndash399 2018

[9] J Tongzon Y Chang and S Lee ldquoHow supply chain oriented isthe port sectorrdquo International Journal of Production Economicsvol 122 no 1 pp 21ndash34 2009

[10] C Ta A V Goodchild and K Pitera ldquoStructuring a definitionof resilience for the freight transportation systemrdquo Transporta-tion Research Record no 2097 pp 19ndash25 2009

[11] A H Gharehgozli J Mileski A Adams and W von ZharenldquoEvaluating a ldquowicked problemrdquo A conceptual frameworkon seaport resiliency in the event of weather disruptionsrdquoTechnological Forecasting amp Social Change vol 121 pp 65ndash752017

[12] H S Loh and V V Thai ldquoManaging port-related supplychain disruptions (PSCDs) amanagementmodel and empiricalevidencerdquo Maritime Policy amp Management vol 43 no 4 pp436ndash455 2016

[13] K Zavitsas T Zis and M G Bell ldquoThe impact of flexibleenvironmental policy on maritime supply chain resiliencerdquoTransport Policy vol 72 pp 116ndash128 2018

[14] M Saleh R Oliva C E Kampmann and P I Davidsen ldquoAcomprehensive analytical approach for policy analysis of systemdynamics modelsrdquo European Journal of Operational Researchvol 203 no 3 pp 673ndash683 2010

[15] B Yang X Wang Y Zhang Y Xu and W Zhou ldquoFinite-time synchronization and synchronization dynamics analysisfor two classes of markovian switching multiweighted complexnetworks from synchronization control rule viewpointrdquo Com-plexity vol 2019 Article ID 1921632 17 pages 2019

[16] S John M M Naim and D R Towill ldquoDynamic analysisof a WIP compensated decision support systemrdquo InternationalJournal of Manufacturing System Design vol 1 pp 283ndash2971994

[17] H A Gabbar ldquoModeling approach for flexible productionchain operationrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 38 no 1 pp 48ndash55 2008

[18] C Li R Mo Z Chang H Yang N Wan and D ZhangldquoDecision-making of feature operation chain considering pro-cessing requirements andmanufacturing stabilityrdquoThe Interna-tional Journal of Advanced Manufacturing Technology vol 87no 5-8 pp 1ndash13 2016

[19] V L Spiegler and M M Naim ldquoInvestigating sustained oscilla-tions in nonlinear production and inventory control modelsrdquoEuropean Journal of Operational Research vol 261 no 2 pp572ndash583 2017

[20] A T Almaktoom ldquoStochastic reliability measurement anddesign optimization of an inventory management systemrdquoComplexity vol 2017 Article ID 1460163 9 pages 2017

[21] J Ma L Zhu Y Yuan and S Hou ldquoStudy of the bullwhip effectin amultistage supply chain with callback structure considering

12 Complexity

two retailersrdquo Complexity vol 2018 Article ID 3650148 12pages 2018

[22] R Pant K Barker J E Ramirez-Marquez and C M RoccoldquoStochastic measures of resilience and their application tocontainer terminalsrdquo Computers amp Industrial Engineering vol70 no 1 pp 183ndash194 2014

[23] Y Wang S W Wallace B Shen and T Choi ldquoService supplychain management a review of operational modelsrdquo EuropeanJournal of Operational Research vol 247 no 3 pp 685ndash6982015

[24] D R Towill ldquoDynamic analysis of an inventory and order basedproduction control systemrdquo International Journal of ProductionResearch vol 20 no 6 pp 671ndash687 1982

[25] L ZhouMM Naim and SM Disney ldquoThe impact of productreturns and remanufacturing uncertainties on the dynamicperformance of a multi-echelon closed-loop supply chainrdquoInternational Journal of Production Economics vol 183 pp 487ndash502 2017

[26] V L M Spiegler M M Naim and J Wikner ldquoA control engi-neering approach to the assessment of supply chain resiliencerdquoInternational Journal of Production Research vol 50 no 21 pp6162ndash6187 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Complexity

0

5

10

10

20

300

2

4

6

8

TUTF

ITA

E

times104

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 5 ITAE of linear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 6 1 ITAE of linear handling chain system

respectively 3D mesh and contour maps of 1ITAE are givenin Figures 8(a) and 8(b) respectively In Figure 8(a) someareas in which 119906119891119903(infin) gt 119863 and ITAE is infinite are notplotted These areas are the light blue shaded regions inFigure 8(b) From Figure 8 ITAE reaches its minimum value(1ITAEasymp0092 and ITAEasymp1082) when 119879119865 = 2 days 119879119880 =24days From a comparison between Figures 5 and 7 ITAE(approximately 6660) is infinitymore thanhalf of the entirearea It reveals that the impact of COMRATEm on containerport performance is obvious Considering these results canbe determined that if COMRATEm are just applied to thenonlinear HCS modifying the setting of parameter 119879119880 willimprove the resilience of the nonlinear HCS

In Figure 6 there are two regions in which 1ITAEgradually increases and reaches the peaks of these regions

One region is close to the point of 119879119880=0 day and 119879119865=2 daysand the other region is close to the point of 119879119880=24 daysand 119879119865=2 days 1ITAE gradually reaches its maximum whenapproaching the point of 119879119880=0 day and 119879119865=2 days This ismost likely due to the fact that they are dependent processesThe faster the unsatisfied freight requirement is deliveredthe sooner the linear HCS will recover to normal operationHence reasonably setting the parameters119879119865 and119879119880 can solveport congestion problems but it is important to take intoaccount that each modification will affect the resilience oflinear HCS

In Figure 8 1ITAE becomes very small in the region closeto the point of 119879U=0 day and 119879F=2 days From Figure 8(b)ITAE becomes infinite in this region The new maximum of1ITAE is in the point of 119879U=24 days and 119879F=2 days The

Complexity 9

24

68

10

10

20

300

1

2

3

4

TUTF

ITA

E

times104

(a) 3D mesh map

2 4 6 8 10TU

TF

30

25

20

15

10

5

(b) Contour map

Figure 7 ITAE of nonlinear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 8 1ITAE of nonlinear handling chain system

results of the two HCSs discussed above show the usefulnessof the resilience measurement is highlighted in the analysisThese can be used for the PSSCrsquos decisions

43 COMRATEmrsquos Influence on Nonlinear HCS In orderto examine how the COMRATEm influences the nonlinearHCS dynamics the point of 119879119865=2 days and 119879119880=02 day istaken as an example and ufr(t) arate(t) and comrate(t) areplotted in Figure 9 with different COMRATEm (COMRATEm=100 105 110 115 and 120) date isin [0 day 40 days] Inorder to clearly distinguish these curves the horizontal axiscoordinates only take 0sim15 days in Figures 9(a) and 9(b)They eventually reach stable values The stable value of ufr(t)is different with different COMRATEm the stable values ofarate(t) and comrate(t) are always 1

From Figure 9(a) the values of ufr(t) with differentCOMRATEm are the same before it reaches its peak After itspeak the values of ufr(t) with differentCOMRATEm are quitedifferent from one another When COMRATEm =1 ufr(t)maintains its peak value after reaching it When COMRATEm=12 ufr(t) eventually is equal to D(D=02) which is thesame as that in the linear HCS From Figure 9(b) in dates2 through 8 the enhanced ufr(t) makes arate(t) higherwhen COMRATEm is lower From Figure 9(c) comrate(t) islimited by COMRATEm in dates 2sim8 This limitation resultsin the cumulative increase of ufr(t) Due to the limited valueof COMRATE there are cases where dynamic behaviorsof nonlinear HCS cause fr(t) not to be handled timelyeventually resulting in the increased ufr(t) Meanwhile therewould be change noted in arate(t)

10 Complexity

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 15 20 25 30 35 400Date

0

02

04

06

08

1U

ncom

plet

ed F

reig

ht R

equi

rem

ent

(a) ufr(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

05

1

15

2

25

Arr

ive r

ate

(b) arate(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

02

04

06

08

1

12

Com

plet

e rat

e

(c) comrate(t)

Figure 9 States and outputs in nonlinear HCS with different COMRATEm

To investigate the influence of COMRATEm on ITAEthe proportion of ITAE=infin in nonlinear HCS with differentCOMRATEm is plotted in Figure 10 The observation ofdifferent COMRATEm suggests the following all values ofITAE are infinite when COMRATEm le 1 When COM-RATEm = 1sim11 the proportion of ITAE=infin clearly decreasesWhen COMRATEm = 11sim12 the proportion of ITAE=infinis close to 0 and decreases slowly When COMRATEm =12 the proportion of ITAE=infin equals 0 which is the sameas that in the linear HCS These observations show thatthe fluctuation of ufr(t) could be alleviated if the non-linear HCS is carefully designed although it seems to beinevitable

5 Conclusion

The emphasis of container port authorities and other playersin the PSSC has shifted from traditional passive managementto active response highlighting the need to quantify theresilience of HCS for inevitable oscillationsThis work buildsand analyzes linear and nonlinearHCSmodels Resilience forHCS in container ports is measured by ITAE of unsatisfiedfreight requirement It provides a new perspective on PSSCpartners to plan for HCS where resilience could guide theselection of control parameters HCSs in container portsare taken as example to analyze resilience and it is foundthat Comratem has significant influence on resilience for

Complexity 11

095 1 105 11 115 120

02

04

06

08

1

COMRATEm

Prop

ortio

n of

ITA

E =infin

Figure 10 Proportion of ITAE =infin in nonlinear HCS with differentCOMRATEm

nonlinear model These contributions serve as a startingpoint in the development of a quantitative resilience decisionmaking framework for PSSC Future work will consider theresilience and recovery strategies under stochastic freightrequirements given the two HCSs

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Social Science Foun-dation project of China (No 17CGL018) Here the authorswould like to express their gratitude to it

References

[1] J J Wang and M C Cheng ldquoFrom a hub port city to a globalsupply chain management center a case study of Hong KongrdquoJournal of Transport Geography vol 18 no 1 pp 104ndash115 2010

[2] F Wang J Huang and Z Liu ldquoPort management and oper-ations Emerging research topics and progressrdquo Journal ofManagement Sciences in China vol 20 no 05 pp 111ndash126 2017

[3] B Jiang J Li and S Shen ldquoSupply Chain Risk Assessment andControl of Port Enterprises Qingdao port as case studyrdquo AsianJournal of Shipping andLogistics vol 34 no 3 pp 198ndash208 2018

[4] D W Song and P M Panayides ldquoGlobal supply chain andportterminal integration and competitivenessrdquo in Proceedingsof the International Conference on Logistics Shipping and PortManagement Taiwan March 2007

[5] S J Pettit and A K C Beresford ldquoPort development Fromgateways to logistics hubsrdquoMaritime Policy ampManagement vol36 no 3 pp 253ndash267 2009

[6] S-HWoo S J Pettit and A K C Beresford ldquoAn assessment ofthe integration of seaports into supply chains using a structuralequation modelrdquo Supply Chain Management An InternationalJournal vol 18 no 3 pp 235ndash252 2013

[7] L Hou and H Geerlings ldquoDynamics in sustainable port andhinterland operations A conceptual framework and simulationof sustainability measures and their effectiveness based onan application to the Port of Shanghairdquo Journal of CleanerProduction vol 135 pp 449ndash456 2016

[8] K L Roslashdseth P B Wangsness and H Schoslashyen ldquoHow doeconomies of density in container handling operations affectshipsrsquo time and emissions in port Evidence from Norwegiancontainer terminalsrdquo Transportation Research Part D Transportand Environment vol 59 pp 385ndash399 2018

[9] J Tongzon Y Chang and S Lee ldquoHow supply chain oriented isthe port sectorrdquo International Journal of Production Economicsvol 122 no 1 pp 21ndash34 2009

[10] C Ta A V Goodchild and K Pitera ldquoStructuring a definitionof resilience for the freight transportation systemrdquo Transporta-tion Research Record no 2097 pp 19ndash25 2009

[11] A H Gharehgozli J Mileski A Adams and W von ZharenldquoEvaluating a ldquowicked problemrdquo A conceptual frameworkon seaport resiliency in the event of weather disruptionsrdquoTechnological Forecasting amp Social Change vol 121 pp 65ndash752017

[12] H S Loh and V V Thai ldquoManaging port-related supplychain disruptions (PSCDs) amanagementmodel and empiricalevidencerdquo Maritime Policy amp Management vol 43 no 4 pp436ndash455 2016

[13] K Zavitsas T Zis and M G Bell ldquoThe impact of flexibleenvironmental policy on maritime supply chain resiliencerdquoTransport Policy vol 72 pp 116ndash128 2018

[14] M Saleh R Oliva C E Kampmann and P I Davidsen ldquoAcomprehensive analytical approach for policy analysis of systemdynamics modelsrdquo European Journal of Operational Researchvol 203 no 3 pp 673ndash683 2010

[15] B Yang X Wang Y Zhang Y Xu and W Zhou ldquoFinite-time synchronization and synchronization dynamics analysisfor two classes of markovian switching multiweighted complexnetworks from synchronization control rule viewpointrdquo Com-plexity vol 2019 Article ID 1921632 17 pages 2019

[16] S John M M Naim and D R Towill ldquoDynamic analysisof a WIP compensated decision support systemrdquo InternationalJournal of Manufacturing System Design vol 1 pp 283ndash2971994

[17] H A Gabbar ldquoModeling approach for flexible productionchain operationrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 38 no 1 pp 48ndash55 2008

[18] C Li R Mo Z Chang H Yang N Wan and D ZhangldquoDecision-making of feature operation chain considering pro-cessing requirements andmanufacturing stabilityrdquoThe Interna-tional Journal of Advanced Manufacturing Technology vol 87no 5-8 pp 1ndash13 2016

[19] V L Spiegler and M M Naim ldquoInvestigating sustained oscilla-tions in nonlinear production and inventory control modelsrdquoEuropean Journal of Operational Research vol 261 no 2 pp572ndash583 2017

[20] A T Almaktoom ldquoStochastic reliability measurement anddesign optimization of an inventory management systemrdquoComplexity vol 2017 Article ID 1460163 9 pages 2017

[21] J Ma L Zhu Y Yuan and S Hou ldquoStudy of the bullwhip effectin amultistage supply chain with callback structure considering

12 Complexity

two retailersrdquo Complexity vol 2018 Article ID 3650148 12pages 2018

[22] R Pant K Barker J E Ramirez-Marquez and C M RoccoldquoStochastic measures of resilience and their application tocontainer terminalsrdquo Computers amp Industrial Engineering vol70 no 1 pp 183ndash194 2014

[23] Y Wang S W Wallace B Shen and T Choi ldquoService supplychain management a review of operational modelsrdquo EuropeanJournal of Operational Research vol 247 no 3 pp 685ndash6982015

[24] D R Towill ldquoDynamic analysis of an inventory and order basedproduction control systemrdquo International Journal of ProductionResearch vol 20 no 6 pp 671ndash687 1982

[25] L ZhouMM Naim and SM Disney ldquoThe impact of productreturns and remanufacturing uncertainties on the dynamicperformance of a multi-echelon closed-loop supply chainrdquoInternational Journal of Production Economics vol 183 pp 487ndash502 2017

[26] V L M Spiegler M M Naim and J Wikner ldquoA control engi-neering approach to the assessment of supply chain resiliencerdquoInternational Journal of Production Research vol 50 no 21 pp6162ndash6187 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 9

24

68

10

10

20

300

1

2

3

4

TUTF

ITA

E

times104

(a) 3D mesh map

2 4 6 8 10TU

TF

30

25

20

15

10

5

(b) Contour map

Figure 7 ITAE of nonlinear handling chain system

0

5

10

10

20

300

005

01

015

02

TUTF

1IT

AE

(a) 3D mesh mapTU

TF

2 4 6 8 10

5

10

15

20

25

30

(b) Contour map

Figure 8 1ITAE of nonlinear handling chain system

results of the two HCSs discussed above show the usefulnessof the resilience measurement is highlighted in the analysisThese can be used for the PSSCrsquos decisions

43 COMRATEmrsquos Influence on Nonlinear HCS In orderto examine how the COMRATEm influences the nonlinearHCS dynamics the point of 119879119865=2 days and 119879119880=02 day istaken as an example and ufr(t) arate(t) and comrate(t) areplotted in Figure 9 with different COMRATEm (COMRATEm=100 105 110 115 and 120) date isin [0 day 40 days] Inorder to clearly distinguish these curves the horizontal axiscoordinates only take 0sim15 days in Figures 9(a) and 9(b)They eventually reach stable values The stable value of ufr(t)is different with different COMRATEm the stable values ofarate(t) and comrate(t) are always 1

From Figure 9(a) the values of ufr(t) with differentCOMRATEm are the same before it reaches its peak After itspeak the values of ufr(t) with differentCOMRATEm are quitedifferent from one another When COMRATEm =1 ufr(t)maintains its peak value after reaching it When COMRATEm=12 ufr(t) eventually is equal to D(D=02) which is thesame as that in the linear HCS From Figure 9(b) in dates2 through 8 the enhanced ufr(t) makes arate(t) higherwhen COMRATEm is lower From Figure 9(c) comrate(t) islimited by COMRATEm in dates 2sim8 This limitation resultsin the cumulative increase of ufr(t) Due to the limited valueof COMRATE there are cases where dynamic behaviorsof nonlinear HCS cause fr(t) not to be handled timelyeventually resulting in the increased ufr(t) Meanwhile therewould be change noted in arate(t)

10 Complexity

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 15 20 25 30 35 400Date

0

02

04

06

08

1U

ncom

plet

ed F

reig

ht R

equi

rem

ent

(a) ufr(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

05

1

15

2

25

Arr

ive r

ate

(b) arate(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

02

04

06

08

1

12

Com

plet

e rat

e

(c) comrate(t)

Figure 9 States and outputs in nonlinear HCS with different COMRATEm

To investigate the influence of COMRATEm on ITAEthe proportion of ITAE=infin in nonlinear HCS with differentCOMRATEm is plotted in Figure 10 The observation ofdifferent COMRATEm suggests the following all values ofITAE are infinite when COMRATEm le 1 When COM-RATEm = 1sim11 the proportion of ITAE=infin clearly decreasesWhen COMRATEm = 11sim12 the proportion of ITAE=infinis close to 0 and decreases slowly When COMRATEm =12 the proportion of ITAE=infin equals 0 which is the sameas that in the linear HCS These observations show thatthe fluctuation of ufr(t) could be alleviated if the non-linear HCS is carefully designed although it seems to beinevitable

5 Conclusion

The emphasis of container port authorities and other playersin the PSSC has shifted from traditional passive managementto active response highlighting the need to quantify theresilience of HCS for inevitable oscillationsThis work buildsand analyzes linear and nonlinearHCSmodels Resilience forHCS in container ports is measured by ITAE of unsatisfiedfreight requirement It provides a new perspective on PSSCpartners to plan for HCS where resilience could guide theselection of control parameters HCSs in container portsare taken as example to analyze resilience and it is foundthat Comratem has significant influence on resilience for

Complexity 11

095 1 105 11 115 120

02

04

06

08

1

COMRATEm

Prop

ortio

n of

ITA

E =infin

Figure 10 Proportion of ITAE =infin in nonlinear HCS with differentCOMRATEm

nonlinear model These contributions serve as a startingpoint in the development of a quantitative resilience decisionmaking framework for PSSC Future work will consider theresilience and recovery strategies under stochastic freightrequirements given the two HCSs

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Social Science Foun-dation project of China (No 17CGL018) Here the authorswould like to express their gratitude to it

References

[1] J J Wang and M C Cheng ldquoFrom a hub port city to a globalsupply chain management center a case study of Hong KongrdquoJournal of Transport Geography vol 18 no 1 pp 104ndash115 2010

[2] F Wang J Huang and Z Liu ldquoPort management and oper-ations Emerging research topics and progressrdquo Journal ofManagement Sciences in China vol 20 no 05 pp 111ndash126 2017

[3] B Jiang J Li and S Shen ldquoSupply Chain Risk Assessment andControl of Port Enterprises Qingdao port as case studyrdquo AsianJournal of Shipping andLogistics vol 34 no 3 pp 198ndash208 2018

[4] D W Song and P M Panayides ldquoGlobal supply chain andportterminal integration and competitivenessrdquo in Proceedingsof the International Conference on Logistics Shipping and PortManagement Taiwan March 2007

[5] S J Pettit and A K C Beresford ldquoPort development Fromgateways to logistics hubsrdquoMaritime Policy ampManagement vol36 no 3 pp 253ndash267 2009

[6] S-HWoo S J Pettit and A K C Beresford ldquoAn assessment ofthe integration of seaports into supply chains using a structuralequation modelrdquo Supply Chain Management An InternationalJournal vol 18 no 3 pp 235ndash252 2013

[7] L Hou and H Geerlings ldquoDynamics in sustainable port andhinterland operations A conceptual framework and simulationof sustainability measures and their effectiveness based onan application to the Port of Shanghairdquo Journal of CleanerProduction vol 135 pp 449ndash456 2016

[8] K L Roslashdseth P B Wangsness and H Schoslashyen ldquoHow doeconomies of density in container handling operations affectshipsrsquo time and emissions in port Evidence from Norwegiancontainer terminalsrdquo Transportation Research Part D Transportand Environment vol 59 pp 385ndash399 2018

[9] J Tongzon Y Chang and S Lee ldquoHow supply chain oriented isthe port sectorrdquo International Journal of Production Economicsvol 122 no 1 pp 21ndash34 2009

[10] C Ta A V Goodchild and K Pitera ldquoStructuring a definitionof resilience for the freight transportation systemrdquo Transporta-tion Research Record no 2097 pp 19ndash25 2009

[11] A H Gharehgozli J Mileski A Adams and W von ZharenldquoEvaluating a ldquowicked problemrdquo A conceptual frameworkon seaport resiliency in the event of weather disruptionsrdquoTechnological Forecasting amp Social Change vol 121 pp 65ndash752017

[12] H S Loh and V V Thai ldquoManaging port-related supplychain disruptions (PSCDs) amanagementmodel and empiricalevidencerdquo Maritime Policy amp Management vol 43 no 4 pp436ndash455 2016

[13] K Zavitsas T Zis and M G Bell ldquoThe impact of flexibleenvironmental policy on maritime supply chain resiliencerdquoTransport Policy vol 72 pp 116ndash128 2018

[14] M Saleh R Oliva C E Kampmann and P I Davidsen ldquoAcomprehensive analytical approach for policy analysis of systemdynamics modelsrdquo European Journal of Operational Researchvol 203 no 3 pp 673ndash683 2010

[15] B Yang X Wang Y Zhang Y Xu and W Zhou ldquoFinite-time synchronization and synchronization dynamics analysisfor two classes of markovian switching multiweighted complexnetworks from synchronization control rule viewpointrdquo Com-plexity vol 2019 Article ID 1921632 17 pages 2019

[16] S John M M Naim and D R Towill ldquoDynamic analysisof a WIP compensated decision support systemrdquo InternationalJournal of Manufacturing System Design vol 1 pp 283ndash2971994

[17] H A Gabbar ldquoModeling approach for flexible productionchain operationrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 38 no 1 pp 48ndash55 2008

[18] C Li R Mo Z Chang H Yang N Wan and D ZhangldquoDecision-making of feature operation chain considering pro-cessing requirements andmanufacturing stabilityrdquoThe Interna-tional Journal of Advanced Manufacturing Technology vol 87no 5-8 pp 1ndash13 2016

[19] V L Spiegler and M M Naim ldquoInvestigating sustained oscilla-tions in nonlinear production and inventory control modelsrdquoEuropean Journal of Operational Research vol 261 no 2 pp572ndash583 2017

[20] A T Almaktoom ldquoStochastic reliability measurement anddesign optimization of an inventory management systemrdquoComplexity vol 2017 Article ID 1460163 9 pages 2017

[21] J Ma L Zhu Y Yuan and S Hou ldquoStudy of the bullwhip effectin amultistage supply chain with callback structure considering

12 Complexity

two retailersrdquo Complexity vol 2018 Article ID 3650148 12pages 2018

[22] R Pant K Barker J E Ramirez-Marquez and C M RoccoldquoStochastic measures of resilience and their application tocontainer terminalsrdquo Computers amp Industrial Engineering vol70 no 1 pp 183ndash194 2014

[23] Y Wang S W Wallace B Shen and T Choi ldquoService supplychain management a review of operational modelsrdquo EuropeanJournal of Operational Research vol 247 no 3 pp 685ndash6982015

[24] D R Towill ldquoDynamic analysis of an inventory and order basedproduction control systemrdquo International Journal of ProductionResearch vol 20 no 6 pp 671ndash687 1982

[25] L ZhouMM Naim and SM Disney ldquoThe impact of productreturns and remanufacturing uncertainties on the dynamicperformance of a multi-echelon closed-loop supply chainrdquoInternational Journal of Production Economics vol 183 pp 487ndash502 2017

[26] V L M Spiegler M M Naim and J Wikner ldquoA control engi-neering approach to the assessment of supply chain resiliencerdquoInternational Journal of Production Research vol 50 no 21 pp6162ndash6187 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Complexity

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 15 20 25 30 35 400Date

0

02

04

06

08

1U

ncom

plet

ed F

reig

ht R

equi

rem

ent

(a) ufr(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

05

1

15

2

25

Arr

ive r

ate

(b) arate(t)

ＩＧＬＮＧ = 100

ＩＧＬＮＧ = 105

ＩＧＬＮＧ = 110

ＩＧＬＮＧ = 115

ＩＧＬＮＧ = 120

5 10 150Date

0

02

04

06

08

1

12

Com

plet

e rat

e

(c) comrate(t)

Figure 9 States and outputs in nonlinear HCS with different COMRATEm

To investigate the influence of COMRATEm on ITAEthe proportion of ITAE=infin in nonlinear HCS with differentCOMRATEm is plotted in Figure 10 The observation ofdifferent COMRATEm suggests the following all values ofITAE are infinite when COMRATEm le 1 When COM-RATEm = 1sim11 the proportion of ITAE=infin clearly decreasesWhen COMRATEm = 11sim12 the proportion of ITAE=infinis close to 0 and decreases slowly When COMRATEm =12 the proportion of ITAE=infin equals 0 which is the sameas that in the linear HCS These observations show thatthe fluctuation of ufr(t) could be alleviated if the non-linear HCS is carefully designed although it seems to beinevitable

5 Conclusion

The emphasis of container port authorities and other playersin the PSSC has shifted from traditional passive managementto active response highlighting the need to quantify theresilience of HCS for inevitable oscillationsThis work buildsand analyzes linear and nonlinearHCSmodels Resilience forHCS in container ports is measured by ITAE of unsatisfiedfreight requirement It provides a new perspective on PSSCpartners to plan for HCS where resilience could guide theselection of control parameters HCSs in container portsare taken as example to analyze resilience and it is foundthat Comratem has significant influence on resilience for

Complexity 11

095 1 105 11 115 120

02

04

06

08

1

COMRATEm

Prop

ortio

n of

ITA

E =infin

Figure 10 Proportion of ITAE =infin in nonlinear HCS with differentCOMRATEm

nonlinear model These contributions serve as a startingpoint in the development of a quantitative resilience decisionmaking framework for PSSC Future work will consider theresilience and recovery strategies under stochastic freightrequirements given the two HCSs

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Social Science Foun-dation project of China (No 17CGL018) Here the authorswould like to express their gratitude to it

References

[1] J J Wang and M C Cheng ldquoFrom a hub port city to a globalsupply chain management center a case study of Hong KongrdquoJournal of Transport Geography vol 18 no 1 pp 104ndash115 2010

[2] F Wang J Huang and Z Liu ldquoPort management and oper-ations Emerging research topics and progressrdquo Journal ofManagement Sciences in China vol 20 no 05 pp 111ndash126 2017

[3] B Jiang J Li and S Shen ldquoSupply Chain Risk Assessment andControl of Port Enterprises Qingdao port as case studyrdquo AsianJournal of Shipping andLogistics vol 34 no 3 pp 198ndash208 2018

[4] D W Song and P M Panayides ldquoGlobal supply chain andportterminal integration and competitivenessrdquo in Proceedingsof the International Conference on Logistics Shipping and PortManagement Taiwan March 2007

[5] S J Pettit and A K C Beresford ldquoPort development Fromgateways to logistics hubsrdquoMaritime Policy ampManagement vol36 no 3 pp 253ndash267 2009

[6] S-HWoo S J Pettit and A K C Beresford ldquoAn assessment ofthe integration of seaports into supply chains using a structuralequation modelrdquo Supply Chain Management An InternationalJournal vol 18 no 3 pp 235ndash252 2013

[7] L Hou and H Geerlings ldquoDynamics in sustainable port andhinterland operations A conceptual framework and simulationof sustainability measures and their effectiveness based onan application to the Port of Shanghairdquo Journal of CleanerProduction vol 135 pp 449ndash456 2016

[8] K L Roslashdseth P B Wangsness and H Schoslashyen ldquoHow doeconomies of density in container handling operations affectshipsrsquo time and emissions in port Evidence from Norwegiancontainer terminalsrdquo Transportation Research Part D Transportand Environment vol 59 pp 385ndash399 2018

[9] J Tongzon Y Chang and S Lee ldquoHow supply chain oriented isthe port sectorrdquo International Journal of Production Economicsvol 122 no 1 pp 21ndash34 2009

[10] C Ta A V Goodchild and K Pitera ldquoStructuring a definitionof resilience for the freight transportation systemrdquo Transporta-tion Research Record no 2097 pp 19ndash25 2009

[11] A H Gharehgozli J Mileski A Adams and W von ZharenldquoEvaluating a ldquowicked problemrdquo A conceptual frameworkon seaport resiliency in the event of weather disruptionsrdquoTechnological Forecasting amp Social Change vol 121 pp 65ndash752017

[12] H S Loh and V V Thai ldquoManaging port-related supplychain disruptions (PSCDs) amanagementmodel and empiricalevidencerdquo Maritime Policy amp Management vol 43 no 4 pp436ndash455 2016

[13] K Zavitsas T Zis and M G Bell ldquoThe impact of flexibleenvironmental policy on maritime supply chain resiliencerdquoTransport Policy vol 72 pp 116ndash128 2018

[14] M Saleh R Oliva C E Kampmann and P I Davidsen ldquoAcomprehensive analytical approach for policy analysis of systemdynamics modelsrdquo European Journal of Operational Researchvol 203 no 3 pp 673ndash683 2010

[15] B Yang X Wang Y Zhang Y Xu and W Zhou ldquoFinite-time synchronization and synchronization dynamics analysisfor two classes of markovian switching multiweighted complexnetworks from synchronization control rule viewpointrdquo Com-plexity vol 2019 Article ID 1921632 17 pages 2019

[16] S John M M Naim and D R Towill ldquoDynamic analysisof a WIP compensated decision support systemrdquo InternationalJournal of Manufacturing System Design vol 1 pp 283ndash2971994

[17] H A Gabbar ldquoModeling approach for flexible productionchain operationrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 38 no 1 pp 48ndash55 2008

[18] C Li R Mo Z Chang H Yang N Wan and D ZhangldquoDecision-making of feature operation chain considering pro-cessing requirements andmanufacturing stabilityrdquoThe Interna-tional Journal of Advanced Manufacturing Technology vol 87no 5-8 pp 1ndash13 2016

[19] V L Spiegler and M M Naim ldquoInvestigating sustained oscilla-tions in nonlinear production and inventory control modelsrdquoEuropean Journal of Operational Research vol 261 no 2 pp572ndash583 2017

[20] A T Almaktoom ldquoStochastic reliability measurement anddesign optimization of an inventory management systemrdquoComplexity vol 2017 Article ID 1460163 9 pages 2017

[21] J Ma L Zhu Y Yuan and S Hou ldquoStudy of the bullwhip effectin amultistage supply chain with callback structure considering

12 Complexity

two retailersrdquo Complexity vol 2018 Article ID 3650148 12pages 2018

[22] R Pant K Barker J E Ramirez-Marquez and C M RoccoldquoStochastic measures of resilience and their application tocontainer terminalsrdquo Computers amp Industrial Engineering vol70 no 1 pp 183ndash194 2014

[23] Y Wang S W Wallace B Shen and T Choi ldquoService supplychain management a review of operational modelsrdquo EuropeanJournal of Operational Research vol 247 no 3 pp 685ndash6982015

[24] D R Towill ldquoDynamic analysis of an inventory and order basedproduction control systemrdquo International Journal of ProductionResearch vol 20 no 6 pp 671ndash687 1982

[25] L ZhouMM Naim and SM Disney ldquoThe impact of productreturns and remanufacturing uncertainties on the dynamicperformance of a multi-echelon closed-loop supply chainrdquoInternational Journal of Production Economics vol 183 pp 487ndash502 2017

[26] V L M Spiegler M M Naim and J Wikner ldquoA control engi-neering approach to the assessment of supply chain resiliencerdquoInternational Journal of Production Research vol 50 no 21 pp6162ndash6187 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 11

095 1 105 11 115 120

02

04

06

08

1

COMRATEm

Prop

ortio

n of

ITA

E =infin

Figure 10 Proportion of ITAE =infin in nonlinear HCS with differentCOMRATEm

nonlinear model These contributions serve as a startingpoint in the development of a quantitative resilience decisionmaking framework for PSSC Future work will consider theresilience and recovery strategies under stochastic freightrequirements given the two HCSs

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work is supported by the National Social Science Foun-dation project of China (No 17CGL018) Here the authorswould like to express their gratitude to it

References

[1] J J Wang and M C Cheng ldquoFrom a hub port city to a globalsupply chain management center a case study of Hong KongrdquoJournal of Transport Geography vol 18 no 1 pp 104ndash115 2010

[2] F Wang J Huang and Z Liu ldquoPort management and oper-ations Emerging research topics and progressrdquo Journal ofManagement Sciences in China vol 20 no 05 pp 111ndash126 2017

[3] B Jiang J Li and S Shen ldquoSupply Chain Risk Assessment andControl of Port Enterprises Qingdao port as case studyrdquo AsianJournal of Shipping andLogistics vol 34 no 3 pp 198ndash208 2018

[4] D W Song and P M Panayides ldquoGlobal supply chain andportterminal integration and competitivenessrdquo in Proceedingsof the International Conference on Logistics Shipping and PortManagement Taiwan March 2007

[5] S J Pettit and A K C Beresford ldquoPort development Fromgateways to logistics hubsrdquoMaritime Policy ampManagement vol36 no 3 pp 253ndash267 2009

[6] S-HWoo S J Pettit and A K C Beresford ldquoAn assessment ofthe integration of seaports into supply chains using a structuralequation modelrdquo Supply Chain Management An InternationalJournal vol 18 no 3 pp 235ndash252 2013

[7] L Hou and H Geerlings ldquoDynamics in sustainable port andhinterland operations A conceptual framework and simulationof sustainability measures and their effectiveness based onan application to the Port of Shanghairdquo Journal of CleanerProduction vol 135 pp 449ndash456 2016

[8] K L Roslashdseth P B Wangsness and H Schoslashyen ldquoHow doeconomies of density in container handling operations affectshipsrsquo time and emissions in port Evidence from Norwegiancontainer terminalsrdquo Transportation Research Part D Transportand Environment vol 59 pp 385ndash399 2018

[9] J Tongzon Y Chang and S Lee ldquoHow supply chain oriented isthe port sectorrdquo International Journal of Production Economicsvol 122 no 1 pp 21ndash34 2009

[10] C Ta A V Goodchild and K Pitera ldquoStructuring a definitionof resilience for the freight transportation systemrdquo Transporta-tion Research Record no 2097 pp 19ndash25 2009

[11] A H Gharehgozli J Mileski A Adams and W von ZharenldquoEvaluating a ldquowicked problemrdquo A conceptual frameworkon seaport resiliency in the event of weather disruptionsrdquoTechnological Forecasting amp Social Change vol 121 pp 65ndash752017

[12] H S Loh and V V Thai ldquoManaging port-related supplychain disruptions (PSCDs) amanagementmodel and empiricalevidencerdquo Maritime Policy amp Management vol 43 no 4 pp436ndash455 2016

[13] K Zavitsas T Zis and M G Bell ldquoThe impact of flexibleenvironmental policy on maritime supply chain resiliencerdquoTransport Policy vol 72 pp 116ndash128 2018

[14] M Saleh R Oliva C E Kampmann and P I Davidsen ldquoAcomprehensive analytical approach for policy analysis of systemdynamics modelsrdquo European Journal of Operational Researchvol 203 no 3 pp 673ndash683 2010

[15] B Yang X Wang Y Zhang Y Xu and W Zhou ldquoFinite-time synchronization and synchronization dynamics analysisfor two classes of markovian switching multiweighted complexnetworks from synchronization control rule viewpointrdquo Com-plexity vol 2019 Article ID 1921632 17 pages 2019

[16] S John M M Naim and D R Towill ldquoDynamic analysisof a WIP compensated decision support systemrdquo InternationalJournal of Manufacturing System Design vol 1 pp 283ndash2971994

[17] H A Gabbar ldquoModeling approach for flexible productionchain operationrdquo IEEE Transactions on Systems Man andCybernetics Systems vol 38 no 1 pp 48ndash55 2008

[18] C Li R Mo Z Chang H Yang N Wan and D ZhangldquoDecision-making of feature operation chain considering pro-cessing requirements andmanufacturing stabilityrdquoThe Interna-tional Journal of Advanced Manufacturing Technology vol 87no 5-8 pp 1ndash13 2016

[19] V L Spiegler and M M Naim ldquoInvestigating sustained oscilla-tions in nonlinear production and inventory control modelsrdquoEuropean Journal of Operational Research vol 261 no 2 pp572ndash583 2017

[20] A T Almaktoom ldquoStochastic reliability measurement anddesign optimization of an inventory management systemrdquoComplexity vol 2017 Article ID 1460163 9 pages 2017

[21] J Ma L Zhu Y Yuan and S Hou ldquoStudy of the bullwhip effectin amultistage supply chain with callback structure considering

12 Complexity

two retailersrdquo Complexity vol 2018 Article ID 3650148 12pages 2018

[22] R Pant K Barker J E Ramirez-Marquez and C M RoccoldquoStochastic measures of resilience and their application tocontainer terminalsrdquo Computers amp Industrial Engineering vol70 no 1 pp 183ndash194 2014

[23] Y Wang S W Wallace B Shen and T Choi ldquoService supplychain management a review of operational modelsrdquo EuropeanJournal of Operational Research vol 247 no 3 pp 685ndash6982015

[24] D R Towill ldquoDynamic analysis of an inventory and order basedproduction control systemrdquo International Journal of ProductionResearch vol 20 no 6 pp 671ndash687 1982

[25] L ZhouMM Naim and SM Disney ldquoThe impact of productreturns and remanufacturing uncertainties on the dynamicperformance of a multi-echelon closed-loop supply chainrdquoInternational Journal of Production Economics vol 183 pp 487ndash502 2017

[26] V L M Spiegler M M Naim and J Wikner ldquoA control engi-neering approach to the assessment of supply chain resiliencerdquoInternational Journal of Production Research vol 50 no 21 pp6162ndash6187 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Complexity

two retailersrdquo Complexity vol 2018 Article ID 3650148 12pages 2018

[22] R Pant K Barker J E Ramirez-Marquez and C M RoccoldquoStochastic measures of resilience and their application tocontainer terminalsrdquo Computers amp Industrial Engineering vol70 no 1 pp 183ndash194 2014

[23] Y Wang S W Wallace B Shen and T Choi ldquoService supplychain management a review of operational modelsrdquo EuropeanJournal of Operational Research vol 247 no 3 pp 685ndash6982015

[24] D R Towill ldquoDynamic analysis of an inventory and order basedproduction control systemrdquo International Journal of ProductionResearch vol 20 no 6 pp 671ndash687 1982

[25] L ZhouMM Naim and SM Disney ldquoThe impact of productreturns and remanufacturing uncertainties on the dynamicperformance of a multi-echelon closed-loop supply chainrdquoInternational Journal of Production Economics vol 183 pp 487ndash502 2017

[26] V L M Spiegler M M Naim and J Wikner ldquoA control engi-neering approach to the assessment of supply chain resiliencerdquoInternational Journal of Production Research vol 50 no 21 pp6162ndash6187 2012

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom