Two-Phase Optimization Models for Liner Shipping Network … · 2020. 11. 12. · ResearchArticle Two-Phase Optimization Models for Liner Shipping Network Based on Hub Ports Cooperation:

Research ArticleTwo-Phase Optimization Models for Liner Shipping NetworkBased on Hub Ports Cooperation From the Perspective ofSupply-Side Reform in China

Jingmiao Zhou 12 Yuzhe Zhao 2 and Xingwen Niu 2

1Business School Dalian University of Foreign Languages Dalian China2Collaborative Innovation Center for Transport Studies Dalian Maritime University Dalian China

Correspondence should be addressed to Yuzhe Zhao zhaoyuzhedlmueducn

Received 12 November 2020 Revised 17 February 2021 Accepted 22 March 2021 Published 7 April 2021

Academic Editor Kun An

Copyright copy 2021 Jingmiao Zhou et al)is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

From the perspective of supply-side reform in China it is hard for COSCO Shipping a merged company with a strong shippingcapacity to abandon the container shipping market Meanwhile the new company could cooperate with new strategic ports alongthe Maritime Silk Road in liner service Against this backdrop this paper aims to optimize the liner shipping network (LSN) fromstrategic tactical and operational levels and help the merged shipping company adjust its operational measures according tomarket changes )e optimization towards different levels of decision-making process is a new research of highly practical valuesSpecifically this paper created two-phase optimization models for LSN based on the selection of hub ports In Network As-sessment (NA) phase the LSNs of two types of hub ports selected are designed and assessed on strategic and tactical levels and theprimary and secondary routes are identified in Network Operation (NO) phase the ldquopath-based flowrdquo formulations are proposedfrom the operational level considering operational measures including demand rejection and flow integration)emodels in bothphases are mixed-integer linear programming (MILP) but are solved by different tools CPLEX for the NA phase models and theGenetic Algorithm (GA) for the NO phase models due to the computational complexity of the latter problem )en a com-putational experiment is performed on the LSN of COSCO Shipping on the Persian Gulf trade lane )e results have proved theeffectiveness of the methodology and inspired important countermeasures for the merged shipping company

1 Introduction

)e global demand for container shipping had been rapidlyincreasing from the birth of the containership in the 1950s tothe outbreak of the subprime crisis [1] Due to the limitedshipbuilding capacity however the container shippingsuffered from a long-lasting capacity bottleneck which wasnot resolved until about 1995 Since then expansion ofshipping capacity has grown explosively and maintained acontinuous lead over the demand increase After 2004 theshipping capacity utilization rate ie ship loading rateexhibited an obvious decline heralding the dawn of theldquooversupplyrdquo period in the container shipping [2] Since theglobal recession that began in 2008 the demand growth of

shipping industry has slowed and fallen more in line withGDP growth In 2019 the worrying trend of the falling trade-to-GDP ratio still continues Both the US-China trade warand the global sulfur limit implemented by InternationalMaritime Organization (IMO) the regulatory authority forinternational shipping put forward potential threats to thedemand side of shipping industry [3] It is predicted thatshipping oversupply will persist and be an even greater causefor concern [4]

In order to deal with the oversupply issue governmentsand shipping industry have been making efforts to conductsupply-side reform )e supply-side reform consists of aseries of parallel measures and regulations including annualcapacity limits and mergers of shipping companies )e

HindawiJournal of Advanced TransportationVolume 2021 Article ID 6639218 18 pageshttpsdoiorg10115520216639218

most intuitive way is to directly control the growth of freightcapacity For example the Chinese government is imposinga gradually stringently macrocontrol to maritime freightcapacity Currently any expansion of fleet that transportbulk liquid hazardous goods need to be scrutinized [5] It canbe expected that the control of containership capacity will beput forward in the upcoming future in order to eliminate thegap between supply and demand of maritime industry

In comparison with the annual capacity limit that seemsin lack of mature practice mergers and alliances is an ob-vious trend in recent years leading to the concentration ofshipping capacity)ere have been several successful cases ofmergers in the maritime industry )e largest five carriershandled 27 of all TEUs in 1996 46 in 2008 and 64 in2017 [6] A typical example is the merger between ChinaOcean Shipping Company (COSCO) and China Shipping

Company (CSCL) in 2016 marking a major move in thesupply-side reform of Chinarsquos shipping industry [7])e twoleading shipping companies integrated into COSCO Ship-ping Group (COSCO Shipping) which has become theworldrsquos 3rd largest shipping company in 2019 [8]

)e rationale of the COSCOCSCL merger is entirelysound as they both have designed many similar services andthe unnecessary competition has deteriorated their financialperformance Besides eliminating competition there aremore benefits awaited the shipping companies throughoptimizing their LSNs after mergers which is investigated inthis paper In practice after mergers the LSNs of the ac-quired shipping companies need to firstly go through strictassessment then considering adjusting the services )eNetwork Assessment (NA) phase and Network Operation(NO) phase differ greatly in the content and process of the

d1

o1

o2

d2

LSN design based on cooperation with THP

LSN design based on cooperation with EHP

rh

h

d1

o1

o2

d2

r

rh

(a)

Unprofitable demandis rejected

Cargoes initially allocated toflow through s2

od

Transshipped at a hubport to flow through s1

od


odd1

o1

o2

d2

rh

d1

o1

o2

d2

r

rh

h r ro

h

(b)

Figure 1 Two-phase decision-making process for merged shipping company (a))e LSN design problem in the NA phase and (b) the LSNoperation problem in the NO phase

2 Journal of Advanced Transportation

decision-making of the shipping companies [9] Both phasesare necessary to be considered for merged shipping com-panies to obtain sustained competitiveness [10]

In this paper two-phase optimization models are pro-posed to investigate the decision-making process in NA andNO phases aimed at maximizing the actual profits of ashipping company in the context of supply-side reform forthe LSN based on strategic ports investigating the decision-making process in NA and NO phases Various factors areconsidered to better reflect the NA phase and NO phase inpractice such as the cooperation with different hub portsthe transshipment of cargoes the rejection of unprofitabledemand and the fluctuation of demands and freight rates

)e remainder of this paper is organized as followsSection 2 reviews the relevant literature and summarizes thecontributions of this study Section 3 presents a clear de-scription of the problem Section 4 establishes the two-phaseoptimization model Section 5 details the GA-based algo-rithm for the LSN in NO phase alongside CPLEX enablingthe solutions for LSNs in NA phase Section 6 carries out acomputational experiment on the LSN of COSCO ShippingSection 7 wraps up this paper with some meaningfulconclusions

2 Literature Review

)ere are three decision-making levels for the shippingcompanies to design LSN strategic tactical and operational[11] At the strategic level the shipping companies oftenmake long-term decisions that may cover a planning horizonof up to 30 years Containership deployment is concernedwith the structure (size) and scale (number) of container-ships [12 13] Another strategic decision is route design)eaim of route design is to determine which ports the con-tainerships should visit and in what order [14] Strategicdecisions clearly affect the decision-making at the tacticallevels by defining the boundaries for these decisions At the

tactical level the focus lies in frequency determination [15]sailing speed optimization [16 17] and schedule design[18 19] Tactical level decisions are made every three to sixmonths in view of changing demand for container shipping[20 21] At the operational level the shipping companiesdetermine whether to accept or reject freights [22] how toflow accepted freights [23] and how to reroute or reschedulecontainerships to cope with unexpected market changes[24] )ere is some interplay between the decisions made atthe three different levels [25]

Most existing literature on the optimization of the LSN isdevoted to the strategic and tactical levels Wang and Meng[13] give a literature survey on liner fleet deployment Ronen[26] pioneered the study on ship deployment and routedesign in 1983 Later Rana and Vickson [27] Fagerholt [28]Christiansen et al [29] Gelareh and Pisinger [30] and Shenget al [31] deepened the research based on these strategicdecisions Meng et al [32] and Dulebenets et al [33]reviewed the past research on container scheduling prob-lems Dulebenets [34] Wang et al [35] and Alharbi et al[18] studied the ship schedule problems considering porttime windows Because of the high costs of containershipdeployment and route design and the complexity of thescheduling problems the latest literature mainly appliesoperations research methods to address the strategic andtactical problems in LSN design In recent years muchattention has been paid to the operational optimization ofthe LSN Some scholars highlighted freights booking Inessence the demand for container shipping bears on thedecision-making of all stakeholders including the ports andthe shipping companies For instance Brouer et al [36]Song and Dong [37] and Daniel et al [38] presented thefreights booking decisions generated from LP models wherethe freight flows are treated as a continuous decision vari-able Liu et al [39] and Wang et al [40] pointed out thepossibility of increasing the port handling rates while op-timizing ship fuel cost at the same time )e cooperation

Table 1 Notations of model in NA phase

SetsN Set of all nodes in the LSNsH Set of all traditional hub ports (THP)R Set of all emerging hub ports (EHP)O Set of all origin ports of demandsD Set of all destination ports of demandsV Set of all available legs in the LSNsParametersπ Capacity of any deployed containershipci1 i2

Voyage expense of operating on leg (i1 i2) ∊Vwi1 i2

Transit time of operating on leg (i1 i2) ∊VΩ Maximum containership capacity for a voyage circle controlled by the governmentQod Quantity of demand between origin port o∊OsubeN and destination port d∊DsubeNeod Freight rate of transporting unit demand between origin port o ∊OsubeN and destination port d ∊DsubeNE Expected total revenue which can be calculated as 1113936oisinO1113936disinDeodQod

W Fixed transit time for the total legs in a voyage cycleDecision variablesyi1 i2

(Binary) 1 iff the leg (i1 i2) existsfi1 i2

)e number of containers to be transported on leg (i1 i2) ∊Vz )e number of deployed containerships

Journal of Advanced Transportation 3

between shipping companies and port operators was alsoinvestigated by Venturini et al [41] and Dulebenets [42]from multiobjective perspectives For some other scholarscontainership rerouting was regarded as a special problem ofoperational optimization [43] )e LSN design problem isNP-hard with computationally challenge [44] and wecannot expect to find a polynomial-time algorithm that willproduce the optimal solution for a general LSN design

problem unless PNP Considering that the LSN designproblem is already NP-hard efficient heuristic-rules-basedmethods might be expected to address large-scale realisticsystems [45]

From the above discussion it is clear that the strategicand tactical decisions are often an input to the operationaloptimization )e idea of combining different levels ofdecision-making has been absorbed in some studies in

Population initialization

Parent population

Offspring population

Populationcompleted

No

Fitness function

Select two individuals

A single point crossover

Uniform mutation anddisplacement mutation

Test the fitness of each chromosome

No

Yes

Yes

Yes

Exit

No

Crossoverapproved

Select one of the twoindividuals

Mutationapproved

Figure 2 Flowchart of the proposed GA-based algorithm to model (III)

4 2 S191

1172

xot

The freight flow The transshipment port

306 1478

S291 9

964

11

22

S191

6 2 S291 2 2 1S1

91S291

i1i2ts1od

xoti1i2ts2od

xtdti1i2s2ods1

od

Figure 3 A typical solution to model (III)


recent years known as two-phase optimization By gener-ating the set of routes firstly the container flows can beoptimized based on the given set of routes in the secondphase [46 47] )e operational optimization of the LSN canalso be viewed as the fine-tuning and correction of thestrategic and tactical solutions [48] Despite the afore-mentioned advancements in the research on the LSN designproblem there are still some practically significant issuesthat have seldom been addressed For example liner ship-ping consolidation through mergers and the macrocontrolof excessive new capacity are regarded as key challenges formaritime industry in 2019 however it has been ignored byresearchers so far [49]

)is research fills in the gap in the existing literature andmakes contributions to the research in LSN design problemas follows Firstly we investigate the LSN design problem forshipping companies under the context of supply-side re-form Various measures of supply-side reform are consid-ered in this paper including the macrocontrol of capacityand the mergers of shipping companies )e decision-making process is divided into NA phase and NO phase andtwo-phase optimization models for the LSN are developedaccordingly Secondly we look for alternative solutions tothe LSN design problem in the NO phase with a GA-basedalgorithm )e proposed method can efficiently solve theldquopath-based flowrdquo formulations)irdly this paper gives outseveral countermeasures of shipping companies from theperspective of supply-side reform in China eg the selectionof hub ports demand rejection and the idea of flow inte-gration In addition the scenario analyses reveal howshipping companies can flexibly adjust their operationalmeasures according to the actual market indicators such asdemand and freight rates

3 Problem Description

We consider the LSN optimization for a shipping companyin the context of supply-side reform typically a merger oracquisition NA and NO phases after a merger are analyzedselecting the most profitable route in the NA phase from allthe similar preset routes that have been designed by differentacquired shipping companies and figuring out the optimalplan of flowing cargoes in the NO phase according to theactual shipping market )e objectives of both phases are tomaximize profits Detailed information about the two phasesis stated in Section 31 and Section 32 respectively

)e elements of LSN are defined as follows to avoidambiguity

(1) Port calls a typical liner shipping route usuallycontains at least several fixed ports calls thus alsonamed as multiport calling (MPC) service [50]

(2) Hub ports when operating along a liner service thecontainerships are allowed to call twice at hub portsbut only once at any other ports As commonlyobserved in practice each route is limited to one

single hub port )e shipping companies can co-operate with different hub ports which can beclassified as traditional hub ports (THPs) andemerging hub ports (EHPs) In addition hub portsare able to transship cargoes due to better facilities

(3) Routes the route in the LSN may have 10ndash20 legswhere a leg is a directed arc between two consecutiveports [51 52]

(4) Cargo flows cargo flow refers to the move of cargoeson a leg A flow path is the directed path consisted ofall the legs between the origin port and the desti-nation port

(5) Demands there are several pairs of origin anddestination (O-D pairs) of cargoes along a routegenerating shipping demands )e market changesare represented by the variation of demands andfreight rates for container shipping [53] Shippingcompanies can hardly control the freight rates (egCCFI and SCFI) )e only thing they can do re-garding the shipping market is to decide whethersatisfy or reject the demands which can be called asldquocherry-pickingrdquo [54]

31 e LSN Design Problem in NA Phase Suppose twoshipping companies represented by A and B respectivelyare merged into a new shipping company C In the NAphase there are already similar routes established by theacquired shipping companies A and B Such similar presetroutes may be initiatively designed to satisfy the demand inthe same regions which leads to unnecessary competitionDespite the similarities the selection of hub ports con-tributes to the differences among the routes For instance Ahas established a cooperative relationship with traditionalhub ports (THP) ie the containerships operated by A areallowed to call twice at the THP However B noticed that theshipping demands generated from Emerging Hub Ports(EHP) are growing rapidly thus is more willing to cooperatewith EHP [55] )e differences of the preset routes result indifferent profits )erefore for shipping company C that caneither cooperate with THP or EHP it is necessary to assessthe profitability of the preset routes in order to make ad-justment plans

)e assessment is based on the prediction regarding thequantities of demands Qod and freight rates eod in the next10ndash30 years according to expertsrsquo knowledge of the marketand the development of maritime policies For any coop-eration strategy with hub ports the decision-maker canconstruct a model with predicted demands input to designthe corresponding LSN )e results of the assessment in-dicate cooperating with which types of hub ports (THP orEHP) are more likely to be profitable Here for simplicitywe define the more profitable route as primary route and theless profitable one as secondary route )en shippingcompany C should adjust the container flows to the primary


routes as the thought of aggregating flows on fewer routes inKrogsgaard et al [56] In other words the secondary routewill no longer need to flow cargoes to save operation cost

32 e LSN Operation Problem in NO Phase )e assess-ment results in the NA phase based on predicted demandgive out a rough principle that more cargoes should flow onthe primary route In the NO phase in order to start op-eration in practice shipping company C needs to depictmore detailed plans on how to adjust cargo flows whichinvolve how to pick up unload and transship containers atany port of call according to the actual market situation

As shown in Figure 1(a) two similar routes have beendesigned according to different preferences of hub ports andnamed as primary route and secondary route based onpredicted demands in the NA phase respectively )e dif-ferent legs of the two routes are painted in red Here theLSNs can be described by a directed graph G(NV) con-taining n nodes i ∊N 1 2 n and v legs v ∊V 1 2 v )e set of origin ports of shipping demand is rep-resented by O and the set of destination port is representedby D For any THP h or EHP r that is called twice in thedesigned LSN theoretical copies ie hrsquo and rrsquo are used todifferentiate two calls to one hub port )e cost and thetransit time associated with the leg (h hprime) or (r rprime) are 0 Werepresent the set of THPs by H and the set of EHPs by R

)e flow path of any shipping demand from origin port oto destination port d on the primary route and the secondaryroute can be represented by s1o d and s2o d respectively If s1o d

and s2o d are the same eg the shipping demand (o1 d1) inFigure 1(b) it does not matter whether the flow path isselected as s1o d or s2o d However if s1o d and s2o d are differenteg the shipping demand (o2 d2) in Figure 1(b) the part ofcargo flow that has selected s2o d should be adjusted to s1o dObviously the difference between s1o d and s2o d is derivedfrom the selection of hub ports ie THP h or EHP r Sincethe hub ports have better conditions for transshipping afeasible solution to adjust the flow path is that any cargo flowtransported in s2od should transship at a hub port to s1od Byadopting the idea of ldquoflow integrationrdquo the shipping com-pany C can aggregate the cargo flows to more profitableroute

In NO phase the decision-making is based on actualdemands and freight rates which may have a deviation ΔQodand Δeod from prediction It should be noticed that the de-mands and freight rates are time-varying hence it is necessaryto make timely and pertinent adjustment to the LSNs in orderto achieve low-cost operation In addition when operating theLSNs shipping companies prefer to reject the unprofitablecargoes if allowed [57] eg the shipping demand (o1 d1) inFigure 1(b) In this paper the fluctuation ofmarket indicators isspecifically analyzed in Section 6 ldquoFlow integrationrdquo andldquodemand rejectionrdquo are reflected in the model in Section 4 withan aim of maximizing profits making the operation of LSNsmore flexible In conclusion for each O-D pair shippingcompany C in the NO phase needs to figure out how manycontainers to be transported through s1o d and s2o d and howmany containers to be rejected

4 Mathematical Model

)e assumptions of the models are listed here as follows

(1) Without considering the impact of natural disastersand local wars on the LSN any demand between anO-D port pair is a long-standing issue that changeswith the global trade

(2) Without considering the difference between types ofcontainerships the voyage expense incurred bycontainership deployment is fixed and all contain-erships sail at the agreed speed [58]

(3) )ere is no limit on the loadingunloading capacitiesof all ports that is any port can handle themaximumcontainership capacity )e terminal handlingcharges are fixed on each port but vary among allports [59]

(4) )e emission regulations of MARPOL-VI and EU-ETS on ports and containerships are not consideredas their impacts are restricted to certain areas and arenegligible for long-haul liner services [60]

41 Formulation for LSN Design Problem in NA Phase)eLSN design problem in the NA phase based on hub portsselected as THPs is formulated as Model (I) )e notationsused the model in the NA phase are shown in Table 1 Herewe consider that the government may control the fleetexpansion in order to resolve oversupply in maritime in-dustry Hence we introduce a parameter Ω to represent thepossible maximum limit of containership capacity that canbe deployed for a voyage circle imposed by the government

Having defined the notations we have Model (I) asfollows

minZ1 1113944i1isinN

1113944i2isinN

ci1i2yi1i2

minus E (1)

st 1113944i2isinN

yi1i2minus 1 0 i1 isin

N

H (2)

1113944i2isinN


N

H (3)

1 minus 1113944iisinN

yhi le 0 h isin H sub N (4)


yih le 0 h isin H sub N (5)

1113944iisinN

yhi minus 2le 0 h isin H sub N (6)

1113944iisinN

yih minus 2le 0 h isin H sub N (7)

1113944iisinN

yih 1113944iisinN

yhi h isin H sub N (8)


1113944oisinN

Qoi1minus 1113944

disinNQi1d 1113944

i2isinNfi2i1

minus 1113944i2isinN

fi1i2 i1 isin N (9)

1113944disinD

Qod minus 1113944i1isinN

foi1le 0 o isin O (10)

1113944oisinO


fi1dle 0 d isin D (11)

1113944i1isinN

1113944i2isinN

wi1i2yi1i2

minus Wle 0 (12)

fi1i2minus yi1i2Ωle 0 i1 isin N i2 isin N (13)

fi1i2minus zπ le 0 i1 isin N i2 isin N (14)

fi1i2isin Z

+ i1 isin N i2 isin N (15)

yi1i2isin 0 1 i1 isin N i2 isin N (16)

z isin Z+ (17)

Objective function (1) maximizes the predicted profits ofthe LSN based on the THPs Constraints (2) and (3) specifythat the containership is allowed to call only once at all portsother than the THPs that is these ports have only oneincoming leg and one outgoing leg Constrains (4)ndash(7) canbe combined to define that the number of incoming legs andoutgoing legs for each THP is either one or two Constraints(8) guarantee that the number of legs that enter a THP isequal to the number of legs that leaves a THP Constraints(9) guarantee that the difference of the cargo flows betweenincoming legs and outgoing legs for every port is equal to thequantity of demand surplusdeficit )is is ensured byConstraints (10) require that the flows on the outgoing legsatisfy the total quantity of the demand from any port o ∊Oas an origin port and as indicated for any port d ∊D as adestination port by constraints (11) Constraint (12) stipu-lates that the whole transit time for all legs in the LSN mustobey the fixed transit time Constraint (13) states that the

flows on every leg should not exceed the maximum con-tainership capacity controlled by the government Con-straint (14) rules that the flows on the leg must be carried byenough containerships Constraints (15)ndash(17) define thedomain of the decision variables

Unlike the set of the THPs in constraints (4)ndash(7) thenumber of incoming legs and outgoing legs for the EHP isdetermined by


yri le 0 r isin R sub N


yir le 0 r isin R sub N

1113944iisinN

yri minus 2le 0 r isin R sub N

1113944iisinN

yir minus 2le 0 r isin R sub N

(18)

1113944iisinN

yih 1113944iisinN

yhi r isin R sub N (19)

)e LSN design problem in the NA phase based on hubports which are the EHPs is given as Model (II)

min Z2 1113944i1isinN

1113944i2isinN

ci1i2yi1i2

minus E

st (2) (3) (9) minus (22)

(20)

42 Formulation for LSN Operation Problem in NO Phase)e LSN design problem in the NO phase to determine theoptimal cargo flows is formulated as Model (III) As definedin Section 3 the flow path of demand generated from anO-D pair on the primary route is s1o d while the flow path onthe secondary route is s2o d Besides we use t isin (No) torepresent the transshipment port Since s1o d is predicted asthe more profitable flow path any containers that initiallyflow on s2o d should be integrated into s1o d at transshipmentport t

For any path sko d k isin 1 2 we have

i t sko d1113872 1113873 isin

L i1 falls on the sko d containing t

empty i1 does not fall on the sko d containing t

⎧⎨

⎩

i1 i2 t sko d1113872 1113873 isin

K1 i1 i2( 1113857 comes before t on the s

ko d

empty i1 i2( 1113857 does not come before t on the sko d

⎧⎨

⎩

t i1 i2 sko d1113872 1113873 isin

K2 i1 i2( 1113857 comes after t on the s

ko d

empty i1 i2( 1113857 does not come after t on the sko d

⎧⎨

⎩

(21)

In Model (III) we define ci as the loadingunloading costof port i ∊N )e decision variables in the NO phase arelisted as follows

(1) xoti1i2ts1

o d the cargo flow on any leg (i1 i2) before the

transshipment port t on s1o d between origin port oand destination port d


(2) xoti1i2ts2

od

the cargo flow on any leg (i1 i2) before thetransshipment port t on s2o d between origin port oand destination port d

(3) xt dti1i2s2

o ds1

o d

the cargo flow on any leg (i1 i2) after thetransshipment port t on s1o d where the flow to thetransshipment port t is transported on s2o d

min Z3 1113944oisinN

1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873 xoti1i2ts1

o d+ x

oti1i2ts2

o d1113874 1113875 + 1113944

oisinN1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873xt dti1i2s2

o ds1

o d

minus 1113944oisinN

1113944disinN

1113944i1isinN

1113944tisinN

eo d + Δeo d( 1113857 xotoi1ts1

o d+ x

otoi1ts2

o d1113874 1113875

(22)

st xotoi1ts1

o d+ x

otoi1ts2

o d1113874 1113875 minus Qo d + ΔQo d( 1113857

le 0 o isin N d isin N i1 isinN

o t isin

N

o i1 t s

ko d1113872 1113873 isin L k isin 1 2

(23)

1113944oisinN

1113944disinN

1113944tisinN

xoti1i2ts1

o d+ x

oti1i2ts2

o d1113874 1113875 minus Ωle 0 i1 isin

N

t i2 isin N i1 i2 t s

ko d1113872 1113873 isin K

1 k isin 1 2 (24)

1113944oisinN

1113944disinN

1113944tisinN

xt dti1i2s2

o ds1

o dminus Ωle 0 i1 isin

N

t i2 isin N t i1 i2 s

1o d1113872 1113873 isin K

2 (25)

1113944i2isinN

xoti2i1ts1

o dminus 1113944

i2isinNx

oti1i2ts1

o d 0 o isin N d isin N i1 isin

N

t t isin

N

d i1 t s

1o d1113872 1113873 isin L (26)

1113944i2isinN

xoti2i1ts2

o dminus 1113944

i2isinNx

oti1i2ts2


N

t t isin

N

d i1 t s

2o d1113872 1113873 isin L (27)

1113944i2isinN

xt dti2i1s2

o ds1

o dminus 1113944

i2isinNx

t dti1i2s2

o ds1


N

t d t isin

N

d i1 t s

1o d1113872 1113873 isin L (28)

1113944i1isinN

xoti1tts1

o d+ 1113944

i1isinNx

oti1tts2

o dminus 1113944

i1isinNx

t dtti1s2

o ds1

o d 0 t isin

N

o d i1 t s

ko d1113872 1113873 isin L k isin 1 2 (29)

xoti1i2ts1

o disin 0 Z

+1113864 1113865 o isin N d isin N i1 isin

N

t i2 isin N t isin

N

o i1 i2 t s

1o d1113872 1113873 isin K

1 (30)

xoti1i2ts2

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

2o d1113872 1113873 isin K

1 (31)

xt dti1i2s2

o ds1

o disin 0 Z


N

d i2 isin N t isin

N

o t i1 i2 s

1o d1113872 1113873 isin K

2 (32)

Objective function (22) maximizes the actual profits ofthe shipping company by demands rejection and flow in-tegration ie minimizes the difference between the oper-ation costs and the temporal revenues)e operation costs inthe NO phase refer to the total loadingunloading cost alongthe design path which is incurred once at the origin anddestination ports and twice at the ports of call Similar torelated studies with two-phase optimization the operationcosts in the NO phase only consist of the variable costsrelated to cargo flows excluding the voyage expenses

considered in the NA phase because the voyage expense ofLSN is fixed once the LSN is established Constraints (23)require that the accepted demand ie the total cargo flow onthe outgoing leg for the origin port (including cargo flows ondifferent flow paths s1o d and s2o d) should not exceed theoverall demand of each O-D port pair Constraints (24) and(25) stipulate that the flow on any leg should not surpass themaximum limit of containership capacity for a voyage circleConstraints (26)ndash(29) ensure the balance between the flowon incoming legs and outgoing legs for any port along the


1172Parent 1

Parent 2

Offspring 1

Offspring 2

Offspring 1prime

Offspring 2prime

xoti1i2ts1od

9 12 11 10 14 13

Random single point

Displacement mutation operatorUniform mutation

7 4 2 1

1059Crossoverprobability

Pc

Mutationprobability

Pm

12 11 10 14 13 7 4 2 1 3

1172 9 12 11 10 14 13 4 2 1 3

1059 2 11 10 14 13 7 7 4 2 1

1116 9 12 11 10 14 13 2 1 4 3

1089 12 11 10 14 13 7 2 7 4 1


Figure 4 An example of crossover and mutation

Table 2 )e test results of 30 different W Ω combinations

W (DAY) Ω (TEU) minus Z1 (USD) Gap () Time (s) minus Z2 (USD) Gap () Time (s)1 83 354740 minus29441 100 452 minus4627 100 7842 86 367562 minus16619 100 547 1823446452 008 2663 89 380384 minus 3797 100 608 1823446452 007 3084 92 393205 1817199741 016 255 1837997487 032 2955 95 406027 1817209621 008 142 1838152024 008 2666 98 418849 1817209621 017 331 1838286035 009 2847 101 431671 1817209621 008 170 1838393254 006 2788 104 444493 1827488337 016 258 1841772518 005 2399 107 457315 1827478457 012 233 1842149323 011 23610 110 470137 1827488337 010 153 1842149323 009 29411 113 482959 1828105883 042 234 1842149323 011 27212 116 495781 1828105883 009 189 1842149323 006 31613 119 508603 1828105883 046 231 1842149323 001 28414 122 521425 1828105883 011 152 1842149323 007 29715 125 534247 1828105883 014 381 1842149323 011 30016 128 547068 1828105883 004 305 1842149323 010 37017 131 559890 1828105883 005 322 1842149323 035 28618 134 572712 1828105883 001 334 1842149323 017 35319 137 585534 1828105883 013 430 1845244453 014 23820 140 598356 1828105883 008 356 1845244453 004 24521 143 611178 1831418092 041 230 1847951008 013 40022 146 624000 1831418092 005 258 1847951008 032 28923 149 636822 1836229052 037 278 1859269034 011 10924 152 649644 1836229052 018 445 1859269034 005 18125 155 662466 1851381648 006 250 1859269034 005 11626 158 675288 1851381648 022 133 1859269034 006 11127 161 688110 1851381648 011 203 1859269034 013 09428 164 700932 1851381648 026 103 1859269034 009 09129 167 713753 1851381648 008 103 1859269034 007 18630 170 726575 1851381648 010 200 1859269034 015 083


designed paths including any transshipment port In otherwords they make sure that all flows unloaded at thetransshipment port from s2od are transported through s1odConstraints (30)ndash(32) state the domain of the decisionvariables

5 Solution Approach

)e resulting models (I)sim(III) are all MILP problemsModels (I)sim(II) will be solved by the standard solver such asCPLEX [61] but we cannot guarantee that CPLEX wouldfind the optimal solution for Model (III) because of the 5-

and 6-index formulation required to represent the flow ofevery path in NO phase Consequently we propose using aGA-based algorithm because of several reasons unlike othermetaheuristics such as simulated annealing [62] and tabusearch [63] that work with a single solution GA deals with apopulation of solutions and the GA has been successfullyapplied to previous applications involving LSN designproblems [64 65]

)e proposed solution approach can be stated as followsCPLEX explores the space of containership deployment androute design and finds feasible solutions From every so-lution a valid LSN configuration is derived Once a valid

(a)

(b)

Figure 5 )e results of LSNs (G1 and G2) in NA phase at W 155 Ω 662466


91times108

905

9

895

89

The f

itnes

s val

ue88

885

875

87

865

861000 2000 3000 4000

The number of iterations

5000 6000 7000 8000

Scenario 0

Scenario 1

Scenario 2

Scenario 3

Figure 6 )e convergence of LSN in NO phase (G3)

Table 3 )e results of LSN in NO phase (G3)

G3() o⟶ d ΔQod () Δeod () G3 () o⟶ d ΔQod () Δeod () G3 () o⟶ d ΔQod () Δeod ()100 1⟶ 9 1 minus 14 9191 7⟶11 4 minus 7 8323 4⟶12 9 minus 5100 4⟶10 2 minus 8 9182 12⟶ 8 minus 5 minus 8 8307 11⟶ 2 minus 8 minus 17100 4⟶14 8 minus 6 9168 3⟶14 1 minus 12 8268 12⟶ 2 minus 10 minus 2100 5⟶ 9 6 minus 14 9156 1⟶ 12 minus 23 minus 7 8256 3⟶11 8 minus 20100 5⟶11 9 minus 9 9134 7⟶10 minus 5 minus 2 8229 13⟶1 minus 10 minus 10100 6⟶ 9 5 minus 10 9102 13⟶ 5 10 minus 16 8173 6⟶12 8 minus 9100 7⟶12 minus 7 minus 7 9059 2⟶10 minus 10 minus 8 8110 13⟶ 2 minus 10 minus 20100 8⟶12 minus 2 minus 10 9051 10⟶ 4 6 minus 10 7980 6⟶10 minus 2 minus 7100 9⟶1 10 minus 16 9026 12⟶ 4 2 minus 3 7967 14⟶ 4 minus 1 minus 13100 9⟶ 3 10 minus 6 9001 11⟶ 8 minus 9 minus 19 7913 9⟶ 8 minus 8 minus 4100 9⟶ 5 9 minus 8 8953 12⟶ 6 minus 1 minus 7 7706 13⟶ 7 6 minus 12100 9⟶ 6 2 minus 7 8944 4⟶11 minus 1 minus 13 7702 12⟶1 minus 7 minus 4100 11⟶ 4 1 minus 11 8903 5⟶14 minus 9 minus 3 7662 14⟶ 3 minus 5 minus 10100 12⟶ 5 6 minus 12 8889 8⟶14 9 minus 11 7657 7⟶ 9 minus 5 minus 2100 12⟶ 7 9 minus 5 8855 8⟶13 1 minus 8 7578 10⟶1 minus 9 minus 17100 13⟶ 4 7 minus 4 8845 6⟶14 minus 6 minus 4 7534 10⟶ 2 minus 4 minus 6100 13⟶ 6 10 minus 11 8839 6⟶13 minus 7 minus 2 7471 1⟶ 11 minus 5 minus 17100 14⟶ 7 minus 8 minus 9 8792 3⟶13 0 minus 18 7318 2⟶14 2 minus 89943 4⟶ 9 3 minus 11 8789 7⟶14 4 minus 2 7272 11⟶ 5 minus 8 minus 119939 11⟶ 6 2 minus 6 8776 11⟶ 3 9 minus 5 7271 1⟶ 13 minus 7 minus 79933 6⟶11 4 minus 7 8689 10⟶ 3 10 minus 20 7246 14⟶ 6 minus 10 minus 209780 9⟶ 7 10 minus 7 8673 13⟶ 3 minus 5 minus 3 7201 1⟶ 14 0 minus 99584 5⟶13 10 minus 2 8649 2⟶11 10 minus 11 6976 11⟶ 1 minus 2 minus 69558 8⟶10 6 minus 3 8610 5⟶12 minus 2 minus 20 6941 1⟶ 10 2 minus 109530 8⟶11 5 minus 13 8577 5⟶10 6 minus 6 6696 10⟶ 7 minus 8 minus 79451 8⟶ 9 minus 9 minus 9 8557 3⟶ 9 7 minus 13 6250 9⟶ 4 minus 6 minus 129430 12⟶ 3 9 minus 7 8554 3⟶10 minus 3 minus 20 5264 14⟶ 8 minus 4 minus 59342 4⟶13 9 minus 20 8485 10⟶ 5 1 minus 6 4280 9⟶ 2 minus 12 minus 209328 3⟶12 10 minus 11 8453 2⟶12 minus 10 minus 8 4111 10⟶ 6 minus 8 minus 179289 2⟶ 9 minus 8 minus 10 8424 2⟶13 minus 1 minus 14 2496 11⟶ 7 minus 5 minus 119287 13⟶ 8 minus 1 minus 9 8421 7⟶13 5 minus 6 1654 14⟶ 5 minus 10 minus 139254 14⟶1 2 minus 5 8363 14⟶ 2 1 minus 12 272 10⟶ 8 minus 7 minus 11


configuration is found the problems of selecting the de-mands and switching the paths are solved for this config-uration by the GA-based algorithm and the optimal flowsand paths are found for that network configuration By thisalgorithm a set of candidate solutions (populations) isretained in each iteration (aka generation or trial) and thebest populations are identified based on the principle ofldquosurvival of the fittestrdquo through genetic operations as se-lection crossover and mutation forming a new generationof candidate solutions )is process is repeated untilreaching the maximum number of iterations Gmax Fea-tured by the introduction of an efficient solution repre-sentation the proposed GA-based algorithm is described inFigure 2 and the specific steps are detailed in the followinganalysis

Step 1 Coding the solution representation directly bears onthe GA performance Considering the features of decisionvariables with the inclusion of two terms ldquopath-based flowrdquothe solution is subjected to natural number encoding Hereeach solution is divided into two terms )e first term refersto the possible cargo flow on the path s1o d and s2o d between

an O-D port pair )e second term refers to the trans-shipment port t where the secondary path s2o d can be in-tegrated into the primary path s1o d Figure 3 illustrates atypical solution to the LSN design problem in the NO phase)e transshipment port t belongs to the nodes except for thenonduplicated ports and the origin and destination ports onthe path s1o d and s2o d that is the same nodes between thepath s1o d and s2o d other than the port o and d

Step 2 Fitness function each solution satisfying the con-straints is deemed as a chromosome )is paper attempts tominimize the difference between the operation costs and thetemporal revenues Here the fitness function is set up basedon the reciprocal of the objective function in equation (19))e fitness values are ranked in ascending order to find themaximum value

Step 3 Selection before crossover two parent chromo-somes are selected based on fitness)en a roulette selectionprocedure is adopted for our solution framework Firstcalculate the fitness fc of each chromosome c by thefitness function Second calculate the selection probability

890E + 08

Scenario 0 (∆Qodє[ndash4617 5192] ∆eodє[ndash36807 0])

Scenario 1 (∆Qodє[ndash8 ndash5] ∆eod are same as Scenario 0)

Scenario 2 (∆Qod are same as Scenario 1 ∆eodє[ndash10 ndash5])

Scenario 3 (∆QodєRє[5 15] ∆eod are same as Scenario 2)

The actual profits of COSCO Shipping

894E + 08

074

124

058

9073992796

9021487159

896171319

9007053615

898E + 08 902E + 08 906E + 08 910E + 08

Figure 7 )e actual profits of COSCO Shipping in Scenarios 1ndash3





84

The overall demand acceptance rate of COSCO Shipping ()

85 86 87 8988 90 91

268

468

8685

9079

8933

9091

454

92

Figure 8 )e overall demand acceptance rate of COSCO Shipping in Scenarios 1ndash3


Pcr fc1113936cfc Prc for each chromosome )ird calculate the

cumulative probability qc 1113936ci1 Pc

r where c 1 2

pop size and pop_size is the population size Fourth gen-erate a random number r Finally if rle q1 then select thefirst chromosome otherwise select the i-th chromosomesuch that qiminus 1lt rle qi

Step 4 Crossover a single point crossover operator is usedIn each crossover we randomly select a cut-point in thechromosome and exchange the right parts of the two se-lected parent chromosomes to generate one or more chil-dren )e crossover probability is set as Pc such that only Pcchromosomes undergo the crossover process )e crossoverprocedure is repeated until the number of child chromo-somes reached pop_size

Step 5 Mutation through mutation a new solution can bederived from an old solution )e mutation operator isemployed in each generation of chromosomes at an equalprobability (mutation rate) Pm Specifically the first term ofthe chromosome is flipped by the uniform mutation op-erator and the second term alters one gene from its originalvalue by the displacement mutation operator An example ofthe crossover and mutation procedures is shown in Figure 4

Step 6 Infeasible solution disposing after crossover andmutation if the solution to a chromosome is infeasible theabove steps are repeated from Step 2 until the terminalcondition is satisfied In the initial population there mightbe some chromosomes that fail to obey one or more con-straints Obviously the solutions naturally satisfy con-straints (24)ndash(27) by the ldquopath-based flowrdquo coding If asolution is found to be infeasible it is necessary to verify it

against constraints (20)ndash(23) If constraints (20)ndash(23) arenot satisfied the chromosomersquos fitness value should belowered by the violation degree to the constraints

6 Computational Experiment and Discussion

To assess the performance of the proposed algorithm onsolving different test problems the well-known standarddataset of the Persian Gulf trade lane that consists of 14 portsof COSCO Shipping in 2018 is used in the experiments Alldata are generated from real information without distortingthe original structure)e voyage distance (di1i2

) of any leg ismeasured by the BLM Shipping (see Figure 4)

(1) )e THP h ∊H 4 7 and the EHP r ∊R 6 9 areall the considered hub ports along the Persian Gulftrade lane according to the strategic agreement ofCOSCO Shipping

(2) )e voyage expense per containership of any leg iscalculated as ci1i2

∊ [1691285 267208384] (USD)Here we adopt the containership named M7 withcontainership capacity π 10000 (TEU) To calculatethe voyage expense we assume that the total fixedcost related to chartering and maintaining a vesseland providing salaries and insurances for seamen is8000000 (USDYEAR) [58] )e fuel cost is 167454(USDNM) at the sailing speed of 22 (NMHOUR)[66]

(3) )e transit time of any leg wi1i2∊ [019 2129] (DAY)

is obtained from the voyage distance (di1i2) and the

sailing speed of 22 (NMHOUR) [66] )e fixedtransit time for a voyage circle is set asW ∊ [80 180](DAY)

Table 4 )e results of demand acceptance rate of COSCO Shipping in Scenario 3

o⟶ d G3 () o⟶ d G3 () o⟶ d G3 () o⟶ d G 3()1⟶ 9 100 5⟶ 9 100 9⟶1 9982 12⟶1 1001⟶ 10 100 5⟶10 100 9⟶ 2 9534 12⟶ 2 94591⟶ 11 8836 5⟶11 9736 9⟶ 3 9399 12⟶ 3 75221⟶ 12 5872 5⟶12 8531 9⟶ 4 9766 12⟶ 4 96801⟶ 13 9467 5⟶13 8595 9⟶ 5 9862 12⟶ 5 1001⟶ 14 9295 5⟶14 9890 9⟶ 6 8098 12⟶ 6 97612⟶ 9 9266 6⟶ 9 8401 9⟶ 7 9576 12⟶ 7 85422⟶10 9120 6⟶10 9902 9⟶ 8 100 12⟶ 8 34142⟶11 100 6⟶11 9973 10⟶1 9378 13⟶1 98422⟶12 9872 6⟶12 8319 10⟶ 2 100 13⟶ 2 80482⟶13 6633 6⟶13 7649 10⟶ 3 9811 13⟶ 3 96022⟶14 9504 6⟶14 9464 10⟶ 4 8841 13⟶ 4 93053⟶ 9 8781 7⟶ 9 9229 10⟶ 5 9629 13⟶ 5 72513⟶10 8715 7⟶10 9801 10⟶ 6 9385 13⟶ 6 98273⟶11 9577 7⟶11 9702 10⟶ 7 7955 13⟶ 7 34473⟶12 9602 7⟶12 8289 10⟶ 8 8306 13⟶ 8 96463⟶13 9079 7⟶13 8713 11⟶ 1 8790 14⟶1 76083⟶14 9595 7⟶14 8828 11⟶ 2 4185 14⟶ 2 39214⟶ 9 9836 8⟶ 9 9954 11⟶ 3 8612 14⟶ 3 91624⟶10 8456 8⟶10 8282 11⟶ 4 6180 14⟶ 4 84474⟶11 100 8⟶11 9576 11⟶ 5 8472 14⟶ 5 93584⟶12 9895 8⟶12 100 11⟶ 6 9151 14⟶ 6 1004⟶13 9174 8⟶13 9666 11⟶ 7 9130 14⟶ 7 82764⟶14 100 8⟶14 8666 11⟶ 8 7934 14⟶ 8 6621


(4) Considering that the government may control thefreight capacity growth of maritime industry weassume that the annual containership capacity thatCOSOCO Shipping can provide is limited at 1560000(TEUYEAR) according to the average container-ship capacity of COSCO Shipping in the past tenyears In other words even if all the deployablecontainerships of COSCO Shipping are allocated toserve the investigated Persian Gulf trade lane with allthe containerships full loaded for a whole year theannual freight volume carried in the Persian Gulftrade lane cannot exceed 1560000 (TEUYEAR))erefore in order to meet the annual capacity limitthe maximum containership capacity for a voyagecircle is Ω 1560000(365W) (TEU)

(5) )e demand between each O-D port pair is Qod∊[772 79562] (TEU) and the freight rate of thecorresponding demand is expected to be eod∊ [846188528] (USDTEU)

(6) )e loadingunloading expense at any port is set asci∊ [121 245] (USDTEU)

(7) Within the designed transit time for a voyage circleW 91 (DAY) in NO phase the demand variation isΔQod ∊ [minus 4617 5192] (TEU) and the freight ratevariation is Δeod ∊ [minus 36807 0] (USDTEU) for eachO-D port pair

61 Comparison between LSNs in NA and NO Phases

611 e LSN in NA Phase )e results of models (I)sim(II)are calculated by ILOG-CPLEX 125 Given the fixed limit ofannual containership capacity controlled by the govern-ment if the transit time of a voyage circle W is reduced theservice frequency of containership within a year will in-crease and thus the maximum containership capacity for avoyage circle Ω will fall exerting a pressure on the shippingcapacity for COSCO Shipping

30 different W Ω combinations are tested )e resultsare listed in Table 2 Here for simplicity the route designbased on cooperation with THPs is called as G1 while theroute design based on cooperation with EHPs is called as G2Since the WΩ combination changes in the same directionthe predicted profits of the LSN based on the THPs (G1)increased with W Ω and remained at 1851381648(USD)after W 155 Ω 662466 reached the upper bound Bycontrast the predicted profits of G1 minimized at1817199741(USD) when the W 92 Ω 393205 reachedthe lower bound Any further drop of W Ω made G1

insolvable ie no feasible solutions can be found )e sametrend is observed in the LSN for the EHP (G2) Moreover therunning time (Time) and deviation (Gap) of both models(I)sim(II) are within the acceptable range

To compare the maximum predicted profits in NAphase the G1 and G2 results of COSCO Shipping are shownin Figure 5 when the combination is selected at W 155Ω 662466

)e total profit is fixed and predicted against the de-mands and freight rates between the origin and destinationports Actually the optimization of G1 and G2 is aimed atminimizing the installation cost )rough comparison it isconcluded as follows First in G1 each containership callstwice at all the THPs Similarly containerships call twice atall the EHPs in G2 By calling twice at hub ports the voyagedistance per leg can be shortened and save fuel cost Secondcontrary to the stereotype that calling at the THPsminimizesthe installation cost the total cost ofG1 is greater than that ofG2

612 e LSN in NO Phase After comparing the predictedprofits we took G2 as the primary route while G1 as thesecondary route )e LSN in the NO phase is called as G3 forsimplicity )e parameters for model solution are set asfollows the maximum number of iterations Gmax 8000the population size pop_size 100 the crossover probabilityPc 090 and the mutation probability Pm 001 )en theconvergence of G3 in different scenarios (see Figure 6) is runon Matlab R2013a on a Lenovo laptop with Intelreg Coretrade i5-6500 Processor (320GHz 8GB RAM)

In the NO phase the actual profit of COSCO Shipping is90739927957 (USD) when ΔQod ∊ [minus 4617 5192] (TEU) andΔeod ∊ [minus 36807 0] (USDTEU) Table 3 shows how COSCOShipping adjusted G3 based on the primary route and thesecondary route )e overall demand acceptance rate is8685 indicating that demand rejection is necessary whenmaximizing profits

In addition to ΔQod and Δeod containership deploymentand route design also influence the shipping capacity uti-lization rate of COSCO Shipping making it difficult toobserve how the shipping company selectively accepts thedemand Hence the acceptance rates of the demand betweendifferent O-D pairs are contrasted in detail revealing thatthe demand variation ΔQod has a decisive impact theCOSCO Shipping accepts more demand at higher ΔQodwhile rejects more at lower ΔQod )erefore the demandvariation has a greater impact than the freight rate change onthe decision-making of demand acceptance Furthermorewithout considering the profitability of accepting the de-mand of certain O-D pairs the high demand acceptance rateconcentrated on the demand that must flow through the hubports 4 6 7 9 as highlighted in bold format in Table 3 Inaddition the primary and secondary routes respectivelycarried 675 and 325 of the total demand accepted byCOSCO Shipping )e result proves that the primary pathsare fundamental to the LSN optimization while the sec-ondary paths are a reasonable complement to the mergedpaths

62eLSN inNOPhase underDifferent Scenarios )e LSNin NO phase (G3) in Section 61 (when ΔQod ∊ [minus 4617 5192](TEU) and Δeod ∊ [minus 36807 0] (USDTEU)) is taken asScenario 0 )ree more scenarios are configured to furtherinvestigate the effect of ΔQod and Δeod on G3


Scenario 1 all ΔQod are [5 8] lower than those inScenario 0 all Δeod are the same as those in Scenario 0Scenario 2 all ΔQod are the same as those in Scenario 1all Δeod are [5 8] lower than those in Scenario 1Scenario 3 all ΔQod are [5 15] higher than thosewhen the EHP r ∊R 6 9 are taken as the origin anddestination ports all Δeod are the same as those inScenario 2

Under Scenarios 1ndash3 the actual profits of COSCOShipping are 90214871592(USD) 89617131902(USD) and90070536154(USD) respectively down by 058 124and 074 from those in Scenario 0 (see Figure 7) In generalthe decline in ΔQod and Δeod only causes minor negativeimpacts on the actual profits It is hard to say that thefluctuations of market indicators have few relationships withthe actual profits of shipping companies In fact without theLSNs optimization measures such as demands rejection andflow integration the negative impacts can be very significant)erefore it is safe to say that the negative impacts of ΔQodand Δeod on the actual profits can be ameliorated by LSNsoptimization measures In other words the decision-makingprocess comprising NA phase and NO phase proposed inthis paper can efficiently help the merged shipping com-panies reduce the negative impacts of depressed market

Under Scenarios 1ndash3 the overall demand acceptancerates of COSCO Shipping are 9091 8933 and 9079respectively up by 468 286 and 454 from those inScenario 0 (see Figure 8) By comparing the demand ac-ceptance rate in Scenarios 0 and 1 one can find that theshipping company may accept more demand when theoverall demand level decreases which seems to be contra-dictive with the observation in Section 61 However if wecompare the demand acceptance rate in Scenarios 2 and 3 itcan be revealed that the observation in Section 61 thatshipping company accepts more demand at higher ΔQod andonly holds when the overall freight rate level is low Gen-erally in depressed market where both quantities and freightrates of demands are lower the merged shipping companyshould reject more demand)erefore the demand rejectiondecisions should be adjusted according to both demands andfreight rates )e shipping must focus on the survey ofmarket indicators based on the historical data (as well asexpertsrsquo knowledge of the market andmanagement policies)

Finally the results indicate that the shipping companiesshould attachmore importance to EHPs when designing andoptimizing the LSNs On the one hand EHPs are more likelyto generate demand because they usually locate in rapidlydeveloping economies Scenario 3 assumes an increase of[5 15] in the demands that take the EHPs as the originand destination ports )e results show that the EHPscontributed to the 144 growth in demand which leads to a051 increase in the actual profits of shipping companiesOn the other hand shipping companies should increase theacceptance rate for the demands taking the EHPs as theorigin and destination ports as shown in Table 4

7 Conclusion and Future Research

)is paper aims to help COSCO Shipping address the LSNdesign problem with several hub ports to cooperate in re-gions along the Maritime Silk Road from the perspective ofsupply-side reform in China For this purpose we proposedtwo-phase optimization models for the LSN from strategictactical and operational levels Unlike traditional optimi-zation approaches our work divides the decision-makingprocess into Network Assessment (NA) phase and NetworkOperation (NO) phase and considers external factors likemarket changes and hub port cooperation In addition ouranalyses highlighted two crucial operational measures de-mand rejection and flow integration

)e optimization models for both phases are MILPs)emodels in the NA phase are programmed in CPLEX andthose in the NO phase are solved by a GA-based algorithmIn light of the assessment of designing LSNs by cooperatingwith different types of hub ports based on predictions in theNA phase a ldquopath-based flowrdquo model in the NO phase isspecially developed and a set of easy-to-implement GA-based algorithm is designed to compute optimal solutionsefficiently )en a computational experiment is performedon the Persian Gulf trade lane of COSCO Shipping )eexperimental results prove the effectiveness of the GA andinspire the following countermeasures

Firstly when designing LSNs based on the cooperationwith hub ports in the NA phase the merged shippingcompany should increase the number of legs in the designedLSNs eg calling twice at hub ports in order to save thetotal installation cost More importantly the total installa-tion cost could be further reduced by adjusting the selectionof hub ports from THPs to EHPs Secondly the shippingcompany should reject more cargoes when the actual marketis not satisfied ie both quantities and freight rates ofdemands are lower )e scenario analyses show that theLSNs optimization measures including demands rejectionand flow integration can efficiently help the shippingcompanies reduce the negative impacts of depressed market)irdly the shipping company should increase the demandacceptance rate for the demands taking the hub ports es-pecially the EHPs as the origin and destination ports Ingeneral both the design and operation of LSNs should beflexibly adjusted according to demand prediction If someports are expected to generate greater demands than othersadjusting the hub of LSNs and accept more demand relatedto these EHPs could achieve better performance

It must be noted that this study does not tackle all thedecision-making problems at strategic tactical and opera-tional levels of LSPs in NA and NO phases To furtheroptimize the LSNs the future research will dig deep into thefollowing issues better prediction of future demand helpsidentify the emerging ports and optimize the LSNs greaterunderstanding of LSN structures which consist of butterflyservices pendulum services and even more complex ser-vices helps explore more flexible and cost-efficient


solutions the operation adjustment after shipping companymergers or forming alliances deserves more attention

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is work was supported in part by National Natural ScienceFoundation of China (Grant nos 72072017 71902016 and71831002) Foundation for Humanities and Social Sciencesof Ministry of Education of China (Grant no 18YJC630261)Natural Science Foundation of Liaoning Province of China(Grant no 2020-hylh-41 2020-BS-213) and Social ScienceFoundation of Liaoning Province of China (Grant noL19AGL012)

References

[1] I C Davidson C W Brown M D Sytsma and G M Ruizldquo)e role of containerships as transfer mechanisms of marinebiofouling speciesrdquo Biofouling vol 25 no 7 pp 645ndash6552009

[2] B Cudahy ldquo)e containership revolution Malcom McLeanrsquos1956 innovation goes globalrdquo Tr News vol 246 no 9-10pp 5ndash9 2006 httpwwwtrborgPublicationsBlurbs158043aspx

[3] BIMCO Supply and Demand Trending off Balance BIMCOCopenhagen Denmark 2019 httpswwwbimcoorgnewsmarket_analysis201920191218_refelctions_2020

[4] GLOBECON How Container Ship Oversupply Impacts theGlobal Supply Chain GLOBECON Compton CA USA2019 httpwwwglobeconfreightcomblogcontainer-ship-oversupply-impacts-global-supply-chain

[5] Ministry of Transport of the Peoplersquos Republic of China An-nouncement of the Ministry of Transport on Strengthening theMacro-Control on the Inter-provincial Coastal Vessel TransportMarket for Bulk Liquid Hazardous Goods Ministry of Transportof the Peoplersquos Republic of China Beijing China 2018 httpwwwmotgovcnzhengcejiedujiaqiangyhsjszytwxhwcbxiangguanzhengce201809t20180905_3081379html

[6] FreightWaves ldquoMcKinsey forecasts the next 50 years ofcontainer shippingrdquo 2017 httpswwwfreightwavescomnews20171026mckinsey-forecasts-the-next-50-years-of-container-shipping

[7] Y Guo Y Jia and Z Li ldquoAnalysis on container fleet com-petitiveness after COSCO and China shipping reorganizationbased on multi-attribute decision makingrdquo in Proceedings ofthe 2018 Chinese Control and Decision Conference (CCDC)pp 1020ndash1024 Shenyang China June 2018

[8] MoverFocus ldquoTop 30 international shipping companiesrdquo2019 httpsmoverfocuscomshipping-companies

[9] Q Meng S Wang H Andersson and K )un ldquoContain-ership routing and scheduling in liner shipping overview andfuture research directionsrdquo Transportation Science vol 48no 2 pp 265ndash280 2014

[10] A N Arslan and D J Papageorgiou ldquoBulk ship fleet renewaland deployment under uncertainty a multi-stage stochasticprogramming approachrdquo Transportation Research Part ELogistics and Transportation Review vol 97 no 1 pp 69ndash962017

[11] R Pesenti ldquoHierarchical resource planning for shippingcompaniesrdquo European Journal of Operational Researchvol 86 no 1 pp 91ndash102 1995

[12] M Ng and D-Y Lin ldquoFleet deployment in liner shipping withincomplete demand informationrdquo Transportation ResearchPart E Logistics and Transportation Review vol 116pp 184ndash189 2018

[13] S Wang and Q Meng ldquoContainer liner fleet deployment asystematic overviewrdquo Transportation Research Part CEmerging Technologies vol 77 pp 389ndash404 2017

[14] M Dulebenets ldquo)e vessel scheduling problem in a linershipping route with heterogeneous fleetrdquo InternationalJournal of Civil Engineering vol 16 no 1 pp 1ndash14 2016

[15] S Gelareh and Q Meng ldquoA novel modeling approach for thefleet deployment problem within a short-term planning ho-rizonrdquo Transportation Research Part E Logistics and Trans-portation Review vol 46 no 1 pp 76ndash89 2010

[16] S Wang and X Wang ldquoA polynomial-time algorithm forsailing speed optimization with containership resourcesharingrdquo Transportation Research Part B Methodologicalvol 93 no 11 pp 394ndash405 2016

[17] S Wang X Shen J Zhao B Ji and P Yang ldquoPrediction ofmarine meteorological effect on ship speed based on ASAEdeep learningrdquo Journal of Traffic amp Transportation Engi-neering vol 18 no 2 pp 139ndash147 2018 httptransportchdeducnoaDArticleaspxtype=viewampid=201802015

[18] A Alharbi S Wang and P Davy ldquoSchedule design forsustainable container supply chain networks with port timewindowsrdquo Advanced Engineering Informatics vol 29 no 3pp 322ndash331 2015

[19] K K Castillo-Villar R G Gonzalez-Ramırez P M Gonzalezand N R Smith ldquoA heuristic procedure for a ship routing andscheduling problem with variable speed and discretized timewindowsrdquo Mathematical Problems in Engineering vol 2014no SI Article ID 750232 2014

[20] Q Meng and S Wang ldquoOptimal operating strategy for a long-haul liner service routerdquo European Journal of OperationalResearch vol 215 no 1 pp 105ndash114 2011

[21] J Pasha M A Dulebenets M Kavoosi et al ldquoHolistic tac-tical-level planning in liner shipping an exact optimizationapproachrdquo Journal of Shipping and Trade vol 5 no 8 2020

[22] D-Y Lin and Y-Y Tsai ldquo)e ship routing and freight as-signment problem for daily frequency operation of maritimeliner shippingrdquo Transportation Research Part E Logistics andTransportation Review vol 67 no 6 pp 52ndash70 2014

[23] D-Y Lin and Y-T Chang ldquoShip routing and freight as-signment problem for liner shipping application to thenorthern sea route planning problemrdquo Transportation Re-search Part E Logistics and Transportation Review vol 110no 2 pp 47ndash70 2018

[24] X Qi and D-P Song ldquoMinimizing fuel emissions by opti-mizing vessel schedules in liner shipping with uncertain porttimesrdquo Transportation Research Part E Logistics and Trans-portation Review vol 48 no 4 pp 863ndash880 2012

[25] J Mulder and R Dekker ldquoMethods for strategic liner shippingnetwork designrdquo European Journal of Operational Researchvol 235 no 2 pp 367ndash377 2014


[26] D Ronen ldquoCargo ships routing and scheduling survey ofmodels and problemsrdquo European Journal of OperationalResearch vol 12 no 2 pp 119ndash126 1983

[27] K Rana and R G Vickson ldquoA model and solution algorithmfor optimal routing of a time-chartered containershiprdquoTransportation Science vol 22 no 2 pp 83ndash95 1988

[28] K Fagerholt ldquoOptimal fleet design in a ship routing problemrdquoInternational Transactions in Operational Research vol 6no 5 pp 453ndash464 2010

[29] M Christiansen K Fagerholt and D Ronen ldquoShip routingand scheduling status and perspectivesrdquo TransportationScience vol 38 no 1 pp 1ndash18 2004

[30] S Gelareh and D Pisinger ldquoFleet deployment network designand hub location of liner shipping companiesrdquo Trans-portation Research Part E Logistics and Transportation Re-view vol 47 no 6 pp 947ndash964 2011

[31] D Sheng Q Meng and Z-C Li ldquoOptimal vessel speed andfleet size for industrial shipping services under the emissioncontrol area regulationrdquo Transportation Research Part CEmerging Technologies vol 105 no 8 pp 37ndash53 2019


[33] M A Dulebenets J Pasha O F Abioye and M KavoosildquoVessel scheduling in liner shipping a critical literature re-view and future research needsrdquo Flexible Services andManufacturing Journal vol 33 no 12 2019

[34] M A Dulebenets ldquoMinimizing the total liner shipping routeservice costs via application of an efficient collaborativeagreementrdquo IEEE Transactions on Intelligent TransportationSystems vol 20 no 1 pp 123ndash136 2019

[35] S Wang A Alharbi and P Davy ldquoLiner ship route scheduledesign with port time windowsrdquo Transportation Research PartC Emerging Technologies vol 41 pp 1ndash17 2014

[36] B D Brouer D Pisinger and S Spoorendonk ldquoLinershipping cargo allocation with repositioning of empty con-tainersrdquo INFOR Information Systems and Operational Re-search vol 49 no 2 pp 109ndash124 2011

[37] D-P Song and J-X Dong ldquoCargo routing and empty con-tainer repositioning in multiple shipping service routesrdquoTransportation Research Part B Methodological vol 46no 10 pp 1556ndash1575 2012

[38] M Daniel S Guericke and K Tierney ldquoIntegrating fleetdeployment into the liner shipping cargo allocation problemrdquoin Proceedings of ICCL 2017 Computational Logistics ICCL2017 pp 306ndash320 Southampton UK October 2017

[39] Z Liu S Wang Y Du and H Wang ldquoSupply chain costminimization by collaboration between liner shipping com-panies and port operatorsrdquo Transportation Journal vol 55no 3 pp 296ndash314

[40] S Wang Z Liu and X Qu ldquoCollaborative mechanisms forberth allocationrdquo Advanced Engineering Informatics vol 29no 3 pp 332ndash338 2015

[41] G Venturini Ccedil Iris C A Kontovas and A Larsen ldquo)emulti-port berth allocation problem with speed optimizationand emission considerationsrdquo Transportation Research PartD Transport and Environment vol 54 pp 142ndash159 2017

[42] M A Dulebenets ldquoA comprehensive multi-objective opti-mization model for the vessel scheduling problem in linershippingrdquo International Journal of Production Economicsvol 196 pp 293ndash318 2018

[43] J Xing and M Zhong ldquoA reactive container rerouting modelfor container flow recovery in a hub-and-spoke liner shipping

networkrdquo Maritime Policy amp Management vol 44 no 6pp 744ndash760 2017

[44] S Wang and Q Meng ldquoLiner shipping network design withdeadlinesrdquo Computers amp Operations Research vol 41 no 1pp 140ndash149 2014

[45] S Gelareh R Neamatian Monemi P Mahey N Maculanand D Pisinger ldquoSingle string planning problem arising inliner shipping industries a heuristic approachrdquo Computers ampOperations Research vol 40 no 10 pp 2357ndash2373 2013

[46] J F Alvarez ldquoJoint routing and deployment of a fleet ofcontainer vesselsrdquo Maritime Economics amp Logistics vol 11no 2 pp 186ndash208 2009

[47] B D Brouer G Desaulniers and D Pisinger ldquoA matheuristicfor the liner shipping network design problemrdquo Trans-portation Research Part E Logistics and Transportation Re-view vol 72 pp 42ndash59 2014

[48] M Christiansen E Hellsten D Pisinger D Sacramento andC Vilhelmsen ldquoLiner shipping network designrdquo EuropeanJournal of Operational Research vol 286 no 1 pp 1ndash20 2020

[49] FreightWaves Seven Key Challenges for Freight industry in2019 FreightWaves Chattanooga TN USA 2018 httpswwwfreightwavescomnewssevenchallengesforfreight2019

[50] A Imai K Shintani and S Papadimitriou ldquoMulti-port vshub-and-spoke port calls by containershipsrdquo TransportationResearch Part E Logistics and Transportation Review vol 45no 5 pp 740ndash757 2009

[51] M G H Bell X Liu P Angeloudis A Fonzone andS H Hosseinloo ldquoA frequency-based maritime containerassignment modelrdquo Transportation Research Part B Meth-odological vol 45 no 8 pp 1152ndash1161 2011

[52] Q Meng and S Wang ldquoLiner shipping service network designwith empty container repositioningrdquo Transportation ResearchPart E Logistics and Transportation Review vol 47 no 5pp 695ndash708 2011

[53] B-I Park H Min and I Phau ldquoA game-theoretic approachto evaluating the competitiveness of container carriers in thenortheast Asian shipping marketrdquo Asia Pacific Journal ofMarketing and Logistics vol 29 no 4 pp 854ndash869 2017

[54] A Luer-Villagra and V Marianov ldquoA competitive hub lo-cation and pricing problemrdquo European Journal of OperationalResearch vol 231 no 3 pp 734ndash744 2013

[55] S Kojaku M Xu H Xia and N Masuda ldquoMultiscale core-periphery structure in a global liner shipping networkrdquo Sci-entific Reports vol 9 no 1 pp 404ndash441 2019

[56] A Krogsgaard D Pisinger and J )orsen ldquoA flow-firstroute-next heuristic for liner shipping network designrdquoNetworks vol 72 no 3 pp 358ndash381 2018

[57] C E M Plum D Pisinger J-J Salazar-Gonzalez andM M Sigurd ldquoSingle liner shipping service designrdquo Com-puters amp Operations Research vol 45 no 5 pp 1ndash6 2014

[58] Globalsecurity Annual Ship Operating Cost GlobalsecurityAlexandria VA USA 2020 httpswwwglobalsecurityorgmilitarysystemsshipvamoschtm

[59] Globalnegtiator THC Terminal Handling ChargesGlobalnegtiator Atlanta GA USA 2020 httpswwwglobalnegotiatorcominternational-tradedictionarythc-terminal-handling-charges

[60] European Commission EU Emissions Trading System (EUETS) European Commission Brussels Belgium 2020httpseceuropaeuclimapoliciesets_en

[61] C E M Plum D Pisinger and M M Sigurd ldquoA service flowmodel for the liner shipping network design problemrdquo Eu-ropean Journal of Operational Research vol 235 no 2pp 378ndash386 2014


[62] S Kirkpatrick C D Gelatt Jr and M P Vecchi ldquoOptimi-zation by Simulated Annealingrdquo Readings in Computer Vi-sion Morgan Kaufmann Burlington MA USA pp 606ndash6151987

[63] F Glover ldquoFuture paths for integer programming and links toartificial intelligencerdquo Computers amp Operations Researchvol 13 no 5 pp 533ndash549 1986

[64] K Shintani A Imai E Nishimura and S Papadimitriouldquo)e container shipping network design problem with emptycontainer repositioningrdquo Transportation Research Part ELogistics and Transportation Review vol 43 no 1 pp 39ndash592007

[65] J Zheng Q Meng and Z Sun ldquoLiner hub-and-spokeshipping network designrdquo Transportation Research Part ELogistics and Transportation Review vol 75 no 3 pp 32ndash482015

[66] Clarksons ldquoShip fuel pricerdquo 2020 httpssinclarksonsnet


most intuitive way is to directly control the growth of freightcapacity For example the Chinese government is imposinga gradually stringently macrocontrol to maritime freightcapacity Currently any expansion of fleet that transportbulk liquid hazardous goods need to be scrutinized [5] It canbe expected that the control of containership capacity will beput forward in the upcoming future in order to eliminate thegap between supply and demand of maritime industry

In comparison with the annual capacity limit that seemsin lack of mature practice mergers and alliances is an ob-vious trend in recent years leading to the concentration ofshipping capacity)ere have been several successful cases ofmergers in the maritime industry )e largest five carriershandled 27 of all TEUs in 1996 46 in 2008 and 64 in2017 [6] A typical example is the merger between ChinaOcean Shipping Company (COSCO) and China Shipping

Company (CSCL) in 2016 marking a major move in thesupply-side reform of Chinarsquos shipping industry [7])e twoleading shipping companies integrated into COSCO Ship-ping Group (COSCO Shipping) which has become theworldrsquos 3rd largest shipping company in 2019 [8]

)e rationale of the COSCOCSCL merger is entirelysound as they both have designed many similar services andthe unnecessary competition has deteriorated their financialperformance Besides eliminating competition there aremore benefits awaited the shipping companies throughoptimizing their LSNs after mergers which is investigated inthis paper In practice after mergers the LSNs of the ac-quired shipping companies need to firstly go through strictassessment then considering adjusting the services )eNetwork Assessment (NA) phase and Network Operation(NO) phase differ greatly in the content and process of the

d1

o1

o2

d2

LSN design based on cooperation with THP

LSN design based on cooperation with EHP

rh

h

d1

o1

o2

d2

r

rh

(a)

Unprofitable demandis rejected


od

Transshipped at a hubport to flow through s1

od


odd1

o1

o2

d2

rh

d1

o1

o2

d2

r

rh

h r ro

h

(b)

Figure 1 Two-phase decision-making process for merged shipping company (a))e LSN design problem in the NA phase and (b) the LSNoperation problem in the NO phase





2 Literature Review
















Parent population


Populationcompleted

No

Fitness function





No

Yes

Yes

Yes

Exit

No

Crossoverapproved


Mutationapproved


4 2 S191

1172

xot


306 1478

S291 9

964

11

22

S191

6 2 S291 2 2 1S1

91S291

i1i2ts1od

xoti1i2ts2od

xtdti1i2s2ods1

od
































1113944i2isinN

ci1i2yi1i2

minus E (1)

st 1113944i2isinN


N

H (2)

1113944i2isinN


N

H (3)





1113944iisinN


1113944iisinN


1113944iisinN

yih 1113944iisinN



1113944oisinN

Qoi1minus 1113944

disinNQi1d 1113944

i2isinNfi2i1


fi1i2 i1 isin N (9)

1113944disinD



1113944oisinO



1113944i1isinN

1113944i2isinN

wi1i2yi1i2

minus Wle 0 (12)



fi1i2isin Z



z isin Z+ (17)








1113944iisinN


1113944iisinN


(18)

1113944iisinN

yih 1113944iisinN




1113944i2isinN

ci1i2yi1i2

minus E

st (2) (3) (9) minus (22)

(20)



i t sko d1113872 1113873 isin



⎧⎨

⎩

i1 i2 t sko d1113872 1113873 isin


ko d


⎧⎨

⎩

t i1 i2 sko d1113872 1113873 isin


ko d


⎧⎨

⎩

(21)


(1) xoti1i2ts1




(2) xoti1i2ts2

od


(3) xt dti1i2s2

o ds1

o d



1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873 xoti1i2ts1

o d+ x

oti1i2ts2

o d1113874 1113875 + 1113944

oisinN1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873xt dti1i2s2

o ds1

o d

minus 1113944oisinN

1113944disinN

1113944i1isinN

1113944tisinN


o d+ x

otoi1ts2

o d1113874 1113875

(22)

st xotoi1ts1

o d+ x

otoi1ts2

o d1113874 1113875 minus Qo d + ΔQo d( 1113857


o t isin

N

o i1 t s

ko d1113872 1113873 isin L k isin 1 2

(23)

1113944oisinN

1113944disinN

1113944tisinN

xoti1i2ts1

o d+ x

oti1i2ts2


N


ko d1113872 1113873 isin K

1 k isin 1 2 (24)

1113944oisinN

1113944disinN

1113944tisinN

xt dti1i2s2

o ds1


N


1o d1113872 1113873 isin K

2 (25)

1113944i2isinN

xoti2i1ts1

o dminus 1113944

i2isinNx

oti1i2ts1


N

t t isin

N

d i1 t s

1o d1113872 1113873 isin L (26)

1113944i2isinN

xoti2i1ts2

o dminus 1113944

i2isinNx

oti1i2ts2


N

t t isin

N

d i1 t s

2o d1113872 1113873 isin L (27)

1113944i2isinN

xt dti2i1s2

o ds1

o dminus 1113944

i2isinNx

t dti1i2s2

o ds1


N

t d t isin

N

d i1 t s

1o d1113872 1113873 isin L (28)

1113944i1isinN

xoti1tts1

o d+ 1113944

i1isinNx

oti1tts2

o dminus 1113944

i1isinNx

t dtti1s2

o ds1

o d 0 t isin

N

o d i1 t s

ko d1113872 1113873 isin L k isin 1 2 (29)

xoti1i2ts1

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

1o d1113872 1113873 isin K

1 (30)

xoti1i2ts2

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

2o d1113872 1113873 isin K

1 (31)

xt dti1i2s2

o ds1

o disin 0 Z


N

d i2 isin N t isin

N

o t i1 i2 s

1o d1113872 1113873 isin K

2 (32)




1172Parent 1

Parent 2

Offspring 1

Offspring 2

Offspring 1prime

Offspring 2prime

xoti1i2ts1od

9 12 11 10 14 13

Random single point


7 4 2 1


Pc

Mutationprobability

Pm

12 11 10 14 13 7 4 2 1 3

1172 9 12 11 10 14 13 4 2 1 3

1059 2 11 10 14 13 7 7 4 2 1

1116 9 12 11 10 14 13 2 1 4 3

1089 12 11 10 14 13 7 2 7 4 1







5 Solution Approach




(a)

(b)



91times108

905

9

895

89

The f

itnes

s val

ue88

885

875

87

865

861000 2000 3000 4000


5000 6000 7000 8000

Scenario 0

Scenario 1

Scenario 2

Scenario 3










890E + 08






894E + 08

074

124

058

9073992796

9021487159

896171319

9007053615

898E + 08 902E + 08 906E + 08 910E + 08






84


85 86 87 8988 90 91

268

468

8685

9079

8933

9091

454

92





r where c 1 2
















o⟶ d G3 () o⟶ d G3 () o⟶ d G3 () o⟶ d G 3()1⟶ 9 100 5⟶ 9 100 9⟶1 9982 12⟶1 1001⟶ 10 100 5⟶10 100 9⟶ 2 9534 12⟶ 2 94591⟶ 11 8836 5⟶11 9736 9⟶ 3 9399 12⟶ 3 75221⟶ 12 5872 5⟶12 8531 9⟶ 4 9766 12⟶ 4 96801⟶ 13 9467 5⟶13 8595 9⟶ 5 9862 12⟶ 5 1001⟶ 14 9295 5⟶14 9890 9⟶ 6 8098 12⟶ 6 97612⟶ 9 9266 6⟶ 9 8401 9⟶ 7 9576 12⟶ 7 85422⟶10 9120 6⟶10 9902 9⟶ 8 100 12⟶ 8 34142⟶11 100 6⟶11 9973 10⟶1 9378 13⟶1 98422⟶12 9872 6⟶12 8319 10⟶ 2 100 13⟶ 2 80482⟶13 6633 6⟶13 7649 10⟶ 3 9811 13⟶ 3 96022⟶14 9504 6⟶14 9464 10⟶ 4 8841 13⟶ 4 93053⟶ 9 8781 7⟶ 9 9229 10⟶ 5 9629 13⟶ 5 72513⟶10 8715 7⟶10 9801 10⟶ 6 9385 13⟶ 6 98273⟶11 9577 7⟶11 9702 10⟶ 7 7955 13⟶ 7 34473⟶12 9602 7⟶12 8289 10⟶ 8 8306 13⟶ 8 96463⟶13 9079 7⟶13 8713 11⟶ 1 8790 14⟶1 76083⟶14 9595 7⟶14 8828 11⟶ 2 4185 14⟶ 2 39214⟶ 9 9836 8⟶ 9 9954 11⟶ 3 8612 14⟶ 3 91624⟶10 8456 8⟶10 8282 11⟶ 4 6180 14⟶ 4 84474⟶11 100 8⟶11 9576 11⟶ 5 8472 14⟶ 5 93584⟶12 9895 8⟶12 100 11⟶ 6 9151 14⟶ 6 1004⟶13 9174 8⟶13 9666 11⟶ 7 9130 14⟶ 7 82764⟶14 100 8⟶14 8666 11⟶ 8 7934 14⟶ 8 6621




























Data Availability




Acknowledgments


References










































































2 Literature Review
















Parent population


Populationcompleted

No

Fitness function





No

Yes

Yes

Yes

Exit

No

Crossoverapproved


Mutationapproved


4 2 S191

1172

xot


306 1478

S291 9

964

11

22

S191

6 2 S291 2 2 1S1

91S291

i1i2ts1od

xoti1i2ts2od

xtdti1i2s2ods1

od
































1113944i2isinN

ci1i2yi1i2

minus E (1)

st 1113944i2isinN


N

H (2)

1113944i2isinN


N

H (3)





1113944iisinN


1113944iisinN


1113944iisinN

yih 1113944iisinN



1113944oisinN

Qoi1minus 1113944

disinNQi1d 1113944

i2isinNfi2i1


fi1i2 i1 isin N (9)

1113944disinD



1113944oisinO



1113944i1isinN

1113944i2isinN

wi1i2yi1i2

minus Wle 0 (12)



fi1i2isin Z



z isin Z+ (17)








1113944iisinN


1113944iisinN


(18)

1113944iisinN

yih 1113944iisinN




1113944i2isinN

ci1i2yi1i2

minus E

st (2) (3) (9) minus (22)

(20)



i t sko d1113872 1113873 isin



⎧⎨

⎩

i1 i2 t sko d1113872 1113873 isin


ko d


⎧⎨

⎩

t i1 i2 sko d1113872 1113873 isin


ko d


⎧⎨

⎩

(21)


(1) xoti1i2ts1




(2) xoti1i2ts2

od


(3) xt dti1i2s2

o ds1

o d



1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873 xoti1i2ts1

o d+ x

oti1i2ts2

o d1113874 1113875 + 1113944

oisinN1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873xt dti1i2s2

o ds1

o d

minus 1113944oisinN

1113944disinN

1113944i1isinN

1113944tisinN


o d+ x

otoi1ts2

o d1113874 1113875

(22)

st xotoi1ts1

o d+ x

otoi1ts2

o d1113874 1113875 minus Qo d + ΔQo d( 1113857


o t isin

N

o i1 t s

ko d1113872 1113873 isin L k isin 1 2

(23)

1113944oisinN

1113944disinN

1113944tisinN

xoti1i2ts1

o d+ x

oti1i2ts2


N


ko d1113872 1113873 isin K

1 k isin 1 2 (24)

1113944oisinN

1113944disinN

1113944tisinN

xt dti1i2s2

o ds1


N


1o d1113872 1113873 isin K

2 (25)

1113944i2isinN

xoti2i1ts1

o dminus 1113944

i2isinNx

oti1i2ts1


N

t t isin

N

d i1 t s

1o d1113872 1113873 isin L (26)

1113944i2isinN

xoti2i1ts2

o dminus 1113944

i2isinNx

oti1i2ts2


N

t t isin

N

d i1 t s

2o d1113872 1113873 isin L (27)

1113944i2isinN

xt dti2i1s2

o ds1

o dminus 1113944

i2isinNx

t dti1i2s2

o ds1


N

t d t isin

N

d i1 t s

1o d1113872 1113873 isin L (28)

1113944i1isinN

xoti1tts1

o d+ 1113944

i1isinNx

oti1tts2

o dminus 1113944

i1isinNx

t dtti1s2

o ds1

o d 0 t isin

N

o d i1 t s

ko d1113872 1113873 isin L k isin 1 2 (29)

xoti1i2ts1

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

1o d1113872 1113873 isin K

1 (30)

xoti1i2ts2

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

2o d1113872 1113873 isin K

1 (31)

xt dti1i2s2

o ds1

o disin 0 Z


N

d i2 isin N t isin

N

o t i1 i2 s

1o d1113872 1113873 isin K

2 (32)




1172Parent 1

Parent 2

Offspring 1

Offspring 2

Offspring 1prime

Offspring 2prime

xoti1i2ts1od

9 12 11 10 14 13

Random single point


7 4 2 1


Pc

Mutationprobability

Pm

12 11 10 14 13 7 4 2 1 3

1172 9 12 11 10 14 13 4 2 1 3

1059 2 11 10 14 13 7 7 4 2 1

1116 9 12 11 10 14 13 2 1 4 3

1089 12 11 10 14 13 7 2 7 4 1







5 Solution Approach




(a)

(b)



91times108

905

9

895

89

The f

itnes

s val

ue88

885

875

87

865

861000 2000 3000 4000


5000 6000 7000 8000

Scenario 0

Scenario 1

Scenario 2

Scenario 3










890E + 08






894E + 08

074

124

058

9073992796

9021487159

896171319

9007053615

898E + 08 902E + 08 906E + 08 910E + 08






84


85 86 87 8988 90 91

268

468

8685

9079

8933

9091

454

92





r where c 1 2
















o⟶ d G3 () o⟶ d G3 () o⟶ d G3 () o⟶ d G 3()1⟶ 9 100 5⟶ 9 100 9⟶1 9982 12⟶1 1001⟶ 10 100 5⟶10 100 9⟶ 2 9534 12⟶ 2 94591⟶ 11 8836 5⟶11 9736 9⟶ 3 9399 12⟶ 3 75221⟶ 12 5872 5⟶12 8531 9⟶ 4 9766 12⟶ 4 96801⟶ 13 9467 5⟶13 8595 9⟶ 5 9862 12⟶ 5 1001⟶ 14 9295 5⟶14 9890 9⟶ 6 8098 12⟶ 6 97612⟶ 9 9266 6⟶ 9 8401 9⟶ 7 9576 12⟶ 7 85422⟶10 9120 6⟶10 9902 9⟶ 8 100 12⟶ 8 34142⟶11 100 6⟶11 9973 10⟶1 9378 13⟶1 98422⟶12 9872 6⟶12 8319 10⟶ 2 100 13⟶ 2 80482⟶13 6633 6⟶13 7649 10⟶ 3 9811 13⟶ 3 96022⟶14 9504 6⟶14 9464 10⟶ 4 8841 13⟶ 4 93053⟶ 9 8781 7⟶ 9 9229 10⟶ 5 9629 13⟶ 5 72513⟶10 8715 7⟶10 9801 10⟶ 6 9385 13⟶ 6 98273⟶11 9577 7⟶11 9702 10⟶ 7 7955 13⟶ 7 34473⟶12 9602 7⟶12 8289 10⟶ 8 8306 13⟶ 8 96463⟶13 9079 7⟶13 8713 11⟶ 1 8790 14⟶1 76083⟶14 9595 7⟶14 8828 11⟶ 2 4185 14⟶ 2 39214⟶ 9 9836 8⟶ 9 9954 11⟶ 3 8612 14⟶ 3 91624⟶10 8456 8⟶10 8282 11⟶ 4 6180 14⟶ 4 84474⟶11 100 8⟶11 9576 11⟶ 5 8472 14⟶ 5 93584⟶12 9895 8⟶12 100 11⟶ 6 9151 14⟶ 6 1004⟶13 9174 8⟶13 9666 11⟶ 7 9130 14⟶ 7 82764⟶14 100 8⟶14 8666 11⟶ 8 7934 14⟶ 8 6621




























Data Availability




Acknowledgments


References











































































Parent population


Populationcompleted

No

Fitness function





No

Yes

Yes

Yes

Exit

No

Crossoverapproved


Mutationapproved


4 2 S191

1172

xot


306 1478

S291 9

964

11

22

S191

6 2 S291 2 2 1S1

91S291

i1i2ts1od

xoti1i2ts2od

xtdti1i2s2ods1

od
































1113944i2isinN

ci1i2yi1i2

minus E (1)

st 1113944i2isinN


N

H (2)

1113944i2isinN


N

H (3)





1113944iisinN


1113944iisinN


1113944iisinN

yih 1113944iisinN



1113944oisinN

Qoi1minus 1113944

disinNQi1d 1113944

i2isinNfi2i1


fi1i2 i1 isin N (9)

1113944disinD



1113944oisinO



1113944i1isinN

1113944i2isinN

wi1i2yi1i2

minus Wle 0 (12)



fi1i2isin Z



z isin Z+ (17)








1113944iisinN


1113944iisinN


(18)

1113944iisinN

yih 1113944iisinN




1113944i2isinN

ci1i2yi1i2

minus E

st (2) (3) (9) minus (22)

(20)



i t sko d1113872 1113873 isin



⎧⎨

⎩

i1 i2 t sko d1113872 1113873 isin


ko d


⎧⎨

⎩

t i1 i2 sko d1113872 1113873 isin


ko d


⎧⎨

⎩

(21)


(1) xoti1i2ts1




(2) xoti1i2ts2

od


(3) xt dti1i2s2

o ds1

o d



1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873 xoti1i2ts1

o d+ x

oti1i2ts2

o d1113874 1113875 + 1113944

oisinN1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873xt dti1i2s2

o ds1

o d

minus 1113944oisinN

1113944disinN

1113944i1isinN

1113944tisinN


o d+ x

otoi1ts2

o d1113874 1113875

(22)

st xotoi1ts1

o d+ x

otoi1ts2

o d1113874 1113875 minus Qo d + ΔQo d( 1113857


o t isin

N

o i1 t s

ko d1113872 1113873 isin L k isin 1 2

(23)

1113944oisinN

1113944disinN

1113944tisinN

xoti1i2ts1

o d+ x

oti1i2ts2


N


ko d1113872 1113873 isin K

1 k isin 1 2 (24)

1113944oisinN

1113944disinN

1113944tisinN

xt dti1i2s2

o ds1


N


1o d1113872 1113873 isin K

2 (25)

1113944i2isinN

xoti2i1ts1

o dminus 1113944

i2isinNx

oti1i2ts1


N

t t isin

N

d i1 t s

1o d1113872 1113873 isin L (26)

1113944i2isinN

xoti2i1ts2

o dminus 1113944

i2isinNx

oti1i2ts2


N

t t isin

N

d i1 t s

2o d1113872 1113873 isin L (27)

1113944i2isinN

xt dti2i1s2

o ds1

o dminus 1113944

i2isinNx

t dti1i2s2

o ds1


N

t d t isin

N

d i1 t s

1o d1113872 1113873 isin L (28)

1113944i1isinN

xoti1tts1

o d+ 1113944

i1isinNx

oti1tts2

o dminus 1113944

i1isinNx

t dtti1s2

o ds1

o d 0 t isin

N

o d i1 t s

ko d1113872 1113873 isin L k isin 1 2 (29)

xoti1i2ts1

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

1o d1113872 1113873 isin K

1 (30)

xoti1i2ts2

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

2o d1113872 1113873 isin K

1 (31)

xt dti1i2s2

o ds1

o disin 0 Z


N

d i2 isin N t isin

N

o t i1 i2 s

1o d1113872 1113873 isin K

2 (32)




1172Parent 1

Parent 2

Offspring 1

Offspring 2

Offspring 1prime

Offspring 2prime

xoti1i2ts1od

9 12 11 10 14 13

Random single point


7 4 2 1


Pc

Mutationprobability

Pm

12 11 10 14 13 7 4 2 1 3

1172 9 12 11 10 14 13 4 2 1 3

1059 2 11 10 14 13 7 7 4 2 1

1116 9 12 11 10 14 13 2 1 4 3

1089 12 11 10 14 13 7 2 7 4 1







5 Solution Approach




(a)

(b)



91times108

905

9

895

89

The f

itnes

s val

ue88

885

875

87

865

861000 2000 3000 4000


5000 6000 7000 8000

Scenario 0

Scenario 1

Scenario 2

Scenario 3










890E + 08






894E + 08

074

124

058

9073992796

9021487159

896171319

9007053615

898E + 08 902E + 08 906E + 08 910E + 08






84


85 86 87 8988 90 91

268

468

8685

9079

8933

9091

454

92





r where c 1 2
















o⟶ d G3 () o⟶ d G3 () o⟶ d G3 () o⟶ d G 3()1⟶ 9 100 5⟶ 9 100 9⟶1 9982 12⟶1 1001⟶ 10 100 5⟶10 100 9⟶ 2 9534 12⟶ 2 94591⟶ 11 8836 5⟶11 9736 9⟶ 3 9399 12⟶ 3 75221⟶ 12 5872 5⟶12 8531 9⟶ 4 9766 12⟶ 4 96801⟶ 13 9467 5⟶13 8595 9⟶ 5 9862 12⟶ 5 1001⟶ 14 9295 5⟶14 9890 9⟶ 6 8098 12⟶ 6 97612⟶ 9 9266 6⟶ 9 8401 9⟶ 7 9576 12⟶ 7 85422⟶10 9120 6⟶10 9902 9⟶ 8 100 12⟶ 8 34142⟶11 100 6⟶11 9973 10⟶1 9378 13⟶1 98422⟶12 9872 6⟶12 8319 10⟶ 2 100 13⟶ 2 80482⟶13 6633 6⟶13 7649 10⟶ 3 9811 13⟶ 3 96022⟶14 9504 6⟶14 9464 10⟶ 4 8841 13⟶ 4 93053⟶ 9 8781 7⟶ 9 9229 10⟶ 5 9629 13⟶ 5 72513⟶10 8715 7⟶10 9801 10⟶ 6 9385 13⟶ 6 98273⟶11 9577 7⟶11 9702 10⟶ 7 7955 13⟶ 7 34473⟶12 9602 7⟶12 8289 10⟶ 8 8306 13⟶ 8 96463⟶13 9079 7⟶13 8713 11⟶ 1 8790 14⟶1 76083⟶14 9595 7⟶14 8828 11⟶ 2 4185 14⟶ 2 39214⟶ 9 9836 8⟶ 9 9954 11⟶ 3 8612 14⟶ 3 91624⟶10 8456 8⟶10 8282 11⟶ 4 6180 14⟶ 4 84474⟶11 100 8⟶11 9576 11⟶ 5 8472 14⟶ 5 93584⟶12 9895 8⟶12 100 11⟶ 6 9151 14⟶ 6 1004⟶13 9174 8⟶13 9666 11⟶ 7 9130 14⟶ 7 82764⟶14 100 8⟶14 8666 11⟶ 8 7934 14⟶ 8 6621




























Data Availability




Acknowledgments


References




































































































1113944i2isinN

ci1i2yi1i2

minus E (1)

st 1113944i2isinN


N

H (2)

1113944i2isinN


N

H (3)





1113944iisinN


1113944iisinN


1113944iisinN

yih 1113944iisinN



1113944oisinN

Qoi1minus 1113944

disinNQi1d 1113944

i2isinNfi2i1


fi1i2 i1 isin N (9)

1113944disinD



1113944oisinO



1113944i1isinN

1113944i2isinN

wi1i2yi1i2

minus Wle 0 (12)



fi1i2isin Z



z isin Z+ (17)








1113944iisinN


1113944iisinN


(18)

1113944iisinN

yih 1113944iisinN




1113944i2isinN

ci1i2yi1i2

minus E

st (2) (3) (9) minus (22)

(20)



i t sko d1113872 1113873 isin



⎧⎨

⎩

i1 i2 t sko d1113872 1113873 isin


ko d


⎧⎨

⎩

t i1 i2 sko d1113872 1113873 isin


ko d


⎧⎨

⎩

(21)


(1) xoti1i2ts1




(2) xoti1i2ts2

od


(3) xt dti1i2s2

o ds1

o d



1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873 xoti1i2ts1

o d+ x

oti1i2ts2

o d1113874 1113875 + 1113944

oisinN1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873xt dti1i2s2

o ds1

o d

minus 1113944oisinN

1113944disinN

1113944i1isinN

1113944tisinN


o d+ x

otoi1ts2

o d1113874 1113875

(22)

st xotoi1ts1

o d+ x

otoi1ts2

o d1113874 1113875 minus Qo d + ΔQo d( 1113857


o t isin

N

o i1 t s

ko d1113872 1113873 isin L k isin 1 2

(23)

1113944oisinN

1113944disinN

1113944tisinN

xoti1i2ts1

o d+ x

oti1i2ts2


N


ko d1113872 1113873 isin K

1 k isin 1 2 (24)

1113944oisinN

1113944disinN

1113944tisinN

xt dti1i2s2

o ds1


N


1o d1113872 1113873 isin K

2 (25)

1113944i2isinN

xoti2i1ts1

o dminus 1113944

i2isinNx

oti1i2ts1


N

t t isin

N

d i1 t s

1o d1113872 1113873 isin L (26)

1113944i2isinN

xoti2i1ts2

o dminus 1113944

i2isinNx

oti1i2ts2


N

t t isin

N

d i1 t s

2o d1113872 1113873 isin L (27)

1113944i2isinN

xt dti2i1s2

o ds1

o dminus 1113944

i2isinNx

t dti1i2s2

o ds1


N

t d t isin

N

d i1 t s

1o d1113872 1113873 isin L (28)

1113944i1isinN

xoti1tts1

o d+ 1113944

i1isinNx

oti1tts2

o dminus 1113944

i1isinNx

t dtti1s2

o ds1

o d 0 t isin

N

o d i1 t s

ko d1113872 1113873 isin L k isin 1 2 (29)

xoti1i2ts1

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

1o d1113872 1113873 isin K

1 (30)

xoti1i2ts2

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

2o d1113872 1113873 isin K

1 (31)

xt dti1i2s2

o ds1

o disin 0 Z


N

d i2 isin N t isin

N

o t i1 i2 s

1o d1113872 1113873 isin K

2 (32)




1172Parent 1

Parent 2

Offspring 1

Offspring 2

Offspring 1prime

Offspring 2prime

xoti1i2ts1od

9 12 11 10 14 13

Random single point


7 4 2 1


Pc

Mutationprobability

Pm

12 11 10 14 13 7 4 2 1 3

1172 9 12 11 10 14 13 4 2 1 3

1059 2 11 10 14 13 7 7 4 2 1

1116 9 12 11 10 14 13 2 1 4 3

1089 12 11 10 14 13 7 2 7 4 1







5 Solution Approach




(a)

(b)



91times108

905

9

895

89

The f

itnes

s val

ue88

885

875

87

865

861000 2000 3000 4000


5000 6000 7000 8000

Scenario 0

Scenario 1

Scenario 2

Scenario 3










890E + 08






894E + 08

074

124

058

9073992796

9021487159

896171319

9007053615

898E + 08 902E + 08 906E + 08 910E + 08






84


85 86 87 8988 90 91

268

468

8685

9079

8933

9091

454

92





r where c 1 2
















o⟶ d G3 () o⟶ d G3 () o⟶ d G3 () o⟶ d G 3()1⟶ 9 100 5⟶ 9 100 9⟶1 9982 12⟶1 1001⟶ 10 100 5⟶10 100 9⟶ 2 9534 12⟶ 2 94591⟶ 11 8836 5⟶11 9736 9⟶ 3 9399 12⟶ 3 75221⟶ 12 5872 5⟶12 8531 9⟶ 4 9766 12⟶ 4 96801⟶ 13 9467 5⟶13 8595 9⟶ 5 9862 12⟶ 5 1001⟶ 14 9295 5⟶14 9890 9⟶ 6 8098 12⟶ 6 97612⟶ 9 9266 6⟶ 9 8401 9⟶ 7 9576 12⟶ 7 85422⟶10 9120 6⟶10 9902 9⟶ 8 100 12⟶ 8 34142⟶11 100 6⟶11 9973 10⟶1 9378 13⟶1 98422⟶12 9872 6⟶12 8319 10⟶ 2 100 13⟶ 2 80482⟶13 6633 6⟶13 7649 10⟶ 3 9811 13⟶ 3 96022⟶14 9504 6⟶14 9464 10⟶ 4 8841 13⟶ 4 93053⟶ 9 8781 7⟶ 9 9229 10⟶ 5 9629 13⟶ 5 72513⟶10 8715 7⟶10 9801 10⟶ 6 9385 13⟶ 6 98273⟶11 9577 7⟶11 9702 10⟶ 7 7955 13⟶ 7 34473⟶12 9602 7⟶12 8289 10⟶ 8 8306 13⟶ 8 96463⟶13 9079 7⟶13 8713 11⟶ 1 8790 14⟶1 76083⟶14 9595 7⟶14 8828 11⟶ 2 4185 14⟶ 2 39214⟶ 9 9836 8⟶ 9 9954 11⟶ 3 8612 14⟶ 3 91624⟶10 8456 8⟶10 8282 11⟶ 4 6180 14⟶ 4 84474⟶11 100 8⟶11 9576 11⟶ 5 8472 14⟶ 5 93584⟶12 9895 8⟶12 100 11⟶ 6 9151 14⟶ 6 1004⟶13 9174 8⟶13 9666 11⟶ 7 9130 14⟶ 7 82764⟶14 100 8⟶14 8666 11⟶ 8 7934 14⟶ 8 6621




























Data Availability




Acknowledgments


References






















































































1113944i2isinN

ci1i2yi1i2

minus E (1)

st 1113944i2isinN


N

H (2)

1113944i2isinN


N

H (3)





1113944iisinN


1113944iisinN


1113944iisinN

yih 1113944iisinN



1113944oisinN

Qoi1minus 1113944

disinNQi1d 1113944

i2isinNfi2i1


fi1i2 i1 isin N (9)

1113944disinD



1113944oisinO



1113944i1isinN

1113944i2isinN

wi1i2yi1i2

minus Wle 0 (12)



fi1i2isin Z



z isin Z+ (17)








1113944iisinN


1113944iisinN


(18)

1113944iisinN

yih 1113944iisinN




1113944i2isinN

ci1i2yi1i2

minus E

st (2) (3) (9) minus (22)

(20)



i t sko d1113872 1113873 isin



⎧⎨

⎩

i1 i2 t sko d1113872 1113873 isin


ko d


⎧⎨

⎩

t i1 i2 sko d1113872 1113873 isin


ko d


⎧⎨

⎩

(21)


(1) xoti1i2ts1




(2) xoti1i2ts2

od


(3) xt dti1i2s2

o ds1

o d



1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873 xoti1i2ts1

o d+ x

oti1i2ts2

o d1113874 1113875 + 1113944

oisinN1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873xt dti1i2s2

o ds1

o d

minus 1113944oisinN

1113944disinN

1113944i1isinN

1113944tisinN


o d+ x

otoi1ts2

o d1113874 1113875

(22)

st xotoi1ts1

o d+ x

otoi1ts2

o d1113874 1113875 minus Qo d + ΔQo d( 1113857


o t isin

N

o i1 t s

ko d1113872 1113873 isin L k isin 1 2

(23)

1113944oisinN

1113944disinN

1113944tisinN

xoti1i2ts1

o d+ x

oti1i2ts2


N


ko d1113872 1113873 isin K

1 k isin 1 2 (24)

1113944oisinN

1113944disinN

1113944tisinN

xt dti1i2s2

o ds1


N


1o d1113872 1113873 isin K

2 (25)

1113944i2isinN

xoti2i1ts1

o dminus 1113944

i2isinNx

oti1i2ts1


N

t t isin

N

d i1 t s

1o d1113872 1113873 isin L (26)

1113944i2isinN

xoti2i1ts2

o dminus 1113944

i2isinNx

oti1i2ts2


N

t t isin

N

d i1 t s

2o d1113872 1113873 isin L (27)

1113944i2isinN

xt dti2i1s2

o ds1

o dminus 1113944

i2isinNx

t dti1i2s2

o ds1


N

t d t isin

N

d i1 t s

1o d1113872 1113873 isin L (28)

1113944i1isinN

xoti1tts1

o d+ 1113944

i1isinNx

oti1tts2

o dminus 1113944

i1isinNx

t dtti1s2

o ds1

o d 0 t isin

N

o d i1 t s

ko d1113872 1113873 isin L k isin 1 2 (29)

xoti1i2ts1

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

1o d1113872 1113873 isin K

1 (30)

xoti1i2ts2

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

2o d1113872 1113873 isin K

1 (31)

xt dti1i2s2

o ds1

o disin 0 Z


N

d i2 isin N t isin

N

o t i1 i2 s

1o d1113872 1113873 isin K

2 (32)




1172Parent 1

Parent 2

Offspring 1

Offspring 2

Offspring 1prime

Offspring 2prime

xoti1i2ts1od

9 12 11 10 14 13

Random single point


7 4 2 1


Pc

Mutationprobability

Pm

12 11 10 14 13 7 4 2 1 3

1172 9 12 11 10 14 13 4 2 1 3

1059 2 11 10 14 13 7 7 4 2 1

1116 9 12 11 10 14 13 2 1 4 3

1089 12 11 10 14 13 7 2 7 4 1







5 Solution Approach




(a)

(b)



91times108

905

9

895

89

The f

itnes

s val

ue88

885

875

87

865

861000 2000 3000 4000


5000 6000 7000 8000

Scenario 0

Scenario 1

Scenario 2

Scenario 3










890E + 08






894E + 08

074

124

058

9073992796

9021487159

896171319

9007053615

898E + 08 902E + 08 906E + 08 910E + 08






84


85 86 87 8988 90 91

268

468

8685

9079

8933

9091

454

92





r where c 1 2
















o⟶ d G3 () o⟶ d G3 () o⟶ d G3 () o⟶ d G 3()1⟶ 9 100 5⟶ 9 100 9⟶1 9982 12⟶1 1001⟶ 10 100 5⟶10 100 9⟶ 2 9534 12⟶ 2 94591⟶ 11 8836 5⟶11 9736 9⟶ 3 9399 12⟶ 3 75221⟶ 12 5872 5⟶12 8531 9⟶ 4 9766 12⟶ 4 96801⟶ 13 9467 5⟶13 8595 9⟶ 5 9862 12⟶ 5 1001⟶ 14 9295 5⟶14 9890 9⟶ 6 8098 12⟶ 6 97612⟶ 9 9266 6⟶ 9 8401 9⟶ 7 9576 12⟶ 7 85422⟶10 9120 6⟶10 9902 9⟶ 8 100 12⟶ 8 34142⟶11 100 6⟶11 9973 10⟶1 9378 13⟶1 98422⟶12 9872 6⟶12 8319 10⟶ 2 100 13⟶ 2 80482⟶13 6633 6⟶13 7649 10⟶ 3 9811 13⟶ 3 96022⟶14 9504 6⟶14 9464 10⟶ 4 8841 13⟶ 4 93053⟶ 9 8781 7⟶ 9 9229 10⟶ 5 9629 13⟶ 5 72513⟶10 8715 7⟶10 9801 10⟶ 6 9385 13⟶ 6 98273⟶11 9577 7⟶11 9702 10⟶ 7 7955 13⟶ 7 34473⟶12 9602 7⟶12 8289 10⟶ 8 8306 13⟶ 8 96463⟶13 9079 7⟶13 8713 11⟶ 1 8790 14⟶1 76083⟶14 9595 7⟶14 8828 11⟶ 2 4185 14⟶ 2 39214⟶ 9 9836 8⟶ 9 9954 11⟶ 3 8612 14⟶ 3 91624⟶10 8456 8⟶10 8282 11⟶ 4 6180 14⟶ 4 84474⟶11 100 8⟶11 9576 11⟶ 5 8472 14⟶ 5 93584⟶12 9895 8⟶12 100 11⟶ 6 9151 14⟶ 6 1004⟶13 9174 8⟶13 9666 11⟶ 7 9130 14⟶ 7 82764⟶14 100 8⟶14 8666 11⟶ 8 7934 14⟶ 8 6621




























Data Availability




Acknowledgments


References







































































1113944oisinN

Qoi1minus 1113944

disinNQi1d 1113944

i2isinNfi2i1


fi1i2 i1 isin N (9)

1113944disinD



1113944oisinO



1113944i1isinN

1113944i2isinN

wi1i2yi1i2

minus Wle 0 (12)



fi1i2isin Z



z isin Z+ (17)








1113944iisinN


1113944iisinN


(18)

1113944iisinN

yih 1113944iisinN




1113944i2isinN

ci1i2yi1i2

minus E

st (2) (3) (9) minus (22)

(20)



i t sko d1113872 1113873 isin



⎧⎨

⎩

i1 i2 t sko d1113872 1113873 isin


ko d


⎧⎨

⎩

t i1 i2 sko d1113872 1113873 isin


ko d


⎧⎨

⎩

(21)


(1) xoti1i2ts1




(2) xoti1i2ts2

od


(3) xt dti1i2s2

o ds1

o d



1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873 xoti1i2ts1

o d+ x

oti1i2ts2

o d1113874 1113875 + 1113944

oisinN1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873xt dti1i2s2

o ds1

o d

minus 1113944oisinN

1113944disinN

1113944i1isinN

1113944tisinN


o d+ x

otoi1ts2

o d1113874 1113875

(22)

st xotoi1ts1

o d+ x

otoi1ts2

o d1113874 1113875 minus Qo d + ΔQo d( 1113857


o t isin

N

o i1 t s

ko d1113872 1113873 isin L k isin 1 2

(23)

1113944oisinN

1113944disinN

1113944tisinN

xoti1i2ts1

o d+ x

oti1i2ts2


N


ko d1113872 1113873 isin K

1 k isin 1 2 (24)

1113944oisinN

1113944disinN

1113944tisinN

xt dti1i2s2

o ds1


N


1o d1113872 1113873 isin K

2 (25)

1113944i2isinN

xoti2i1ts1

o dminus 1113944

i2isinNx

oti1i2ts1


N

t t isin

N

d i1 t s

1o d1113872 1113873 isin L (26)

1113944i2isinN

xoti2i1ts2

o dminus 1113944

i2isinNx

oti1i2ts2


N

t t isin

N

d i1 t s

2o d1113872 1113873 isin L (27)

1113944i2isinN

xt dti2i1s2

o ds1

o dminus 1113944

i2isinNx

t dti1i2s2

o ds1


N

t d t isin

N

d i1 t s

1o d1113872 1113873 isin L (28)

1113944i1isinN

xoti1tts1

o d+ 1113944

i1isinNx

oti1tts2

o dminus 1113944

i1isinNx

t dtti1s2

o ds1

o d 0 t isin

N

o d i1 t s

ko d1113872 1113873 isin L k isin 1 2 (29)

xoti1i2ts1

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

1o d1113872 1113873 isin K

1 (30)

xoti1i2ts2

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

2o d1113872 1113873 isin K

1 (31)

xt dti1i2s2

o ds1

o disin 0 Z


N

d i2 isin N t isin

N

o t i1 i2 s

1o d1113872 1113873 isin K

2 (32)




1172Parent 1

Parent 2

Offspring 1

Offspring 2

Offspring 1prime

Offspring 2prime

xoti1i2ts1od

9 12 11 10 14 13

Random single point


7 4 2 1


Pc

Mutationprobability

Pm

12 11 10 14 13 7 4 2 1 3

1172 9 12 11 10 14 13 4 2 1 3

1059 2 11 10 14 13 7 7 4 2 1

1116 9 12 11 10 14 13 2 1 4 3

1089 12 11 10 14 13 7 2 7 4 1







5 Solution Approach




(a)

(b)



91times108

905

9

895

89

The f

itnes

s val

ue88

885

875

87

865

861000 2000 3000 4000


5000 6000 7000 8000

Scenario 0

Scenario 1

Scenario 2

Scenario 3










890E + 08






894E + 08

074

124

058

9073992796

9021487159

896171319

9007053615

898E + 08 902E + 08 906E + 08 910E + 08






84


85 86 87 8988 90 91

268

468

8685

9079

8933

9091

454

92





r where c 1 2
















o⟶ d G3 () o⟶ d G3 () o⟶ d G3 () o⟶ d G 3()1⟶ 9 100 5⟶ 9 100 9⟶1 9982 12⟶1 1001⟶ 10 100 5⟶10 100 9⟶ 2 9534 12⟶ 2 94591⟶ 11 8836 5⟶11 9736 9⟶ 3 9399 12⟶ 3 75221⟶ 12 5872 5⟶12 8531 9⟶ 4 9766 12⟶ 4 96801⟶ 13 9467 5⟶13 8595 9⟶ 5 9862 12⟶ 5 1001⟶ 14 9295 5⟶14 9890 9⟶ 6 8098 12⟶ 6 97612⟶ 9 9266 6⟶ 9 8401 9⟶ 7 9576 12⟶ 7 85422⟶10 9120 6⟶10 9902 9⟶ 8 100 12⟶ 8 34142⟶11 100 6⟶11 9973 10⟶1 9378 13⟶1 98422⟶12 9872 6⟶12 8319 10⟶ 2 100 13⟶ 2 80482⟶13 6633 6⟶13 7649 10⟶ 3 9811 13⟶ 3 96022⟶14 9504 6⟶14 9464 10⟶ 4 8841 13⟶ 4 93053⟶ 9 8781 7⟶ 9 9229 10⟶ 5 9629 13⟶ 5 72513⟶10 8715 7⟶10 9801 10⟶ 6 9385 13⟶ 6 98273⟶11 9577 7⟶11 9702 10⟶ 7 7955 13⟶ 7 34473⟶12 9602 7⟶12 8289 10⟶ 8 8306 13⟶ 8 96463⟶13 9079 7⟶13 8713 11⟶ 1 8790 14⟶1 76083⟶14 9595 7⟶14 8828 11⟶ 2 4185 14⟶ 2 39214⟶ 9 9836 8⟶ 9 9954 11⟶ 3 8612 14⟶ 3 91624⟶10 8456 8⟶10 8282 11⟶ 4 6180 14⟶ 4 84474⟶11 100 8⟶11 9576 11⟶ 5 8472 14⟶ 5 93584⟶12 9895 8⟶12 100 11⟶ 6 9151 14⟶ 6 1004⟶13 9174 8⟶13 9666 11⟶ 7 9130 14⟶ 7 82764⟶14 100 8⟶14 8666 11⟶ 8 7934 14⟶ 8 6621




























Data Availability




Acknowledgments


References







































































(2) xoti1i2ts2

od


(3) xt dti1i2s2

o ds1

o d



1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873 xoti1i2ts1

o d+ x

oti1i2ts2

o d1113874 1113875 + 1113944

oisinN1113944disinN

1113944i1isinN

1113944i2isinN

1113944tisinN

ci1+ ci2

1113872 1113873xt dti1i2s2

o ds1

o d

minus 1113944oisinN

1113944disinN

1113944i1isinN

1113944tisinN


o d+ x

otoi1ts2

o d1113874 1113875

(22)

st xotoi1ts1

o d+ x

otoi1ts2

o d1113874 1113875 minus Qo d + ΔQo d( 1113857


o t isin

N

o i1 t s

ko d1113872 1113873 isin L k isin 1 2

(23)

1113944oisinN

1113944disinN

1113944tisinN

xoti1i2ts1

o d+ x

oti1i2ts2


N


ko d1113872 1113873 isin K

1 k isin 1 2 (24)

1113944oisinN

1113944disinN

1113944tisinN

xt dti1i2s2

o ds1


N


1o d1113872 1113873 isin K

2 (25)

1113944i2isinN

xoti2i1ts1

o dminus 1113944

i2isinNx

oti1i2ts1


N

t t isin

N

d i1 t s

1o d1113872 1113873 isin L (26)

1113944i2isinN

xoti2i1ts2

o dminus 1113944

i2isinNx

oti1i2ts2


N

t t isin

N

d i1 t s

2o d1113872 1113873 isin L (27)

1113944i2isinN

xt dti2i1s2

o ds1

o dminus 1113944

i2isinNx

t dti1i2s2

o ds1


N

t d t isin

N

d i1 t s

1o d1113872 1113873 isin L (28)

1113944i1isinN

xoti1tts1

o d+ 1113944

i1isinNx

oti1tts2

o dminus 1113944

i1isinNx

t dtti1s2

o ds1

o d 0 t isin

N

o d i1 t s

ko d1113872 1113873 isin L k isin 1 2 (29)

xoti1i2ts1

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

1o d1113872 1113873 isin K

1 (30)

xoti1i2ts2

o disin 0 Z


N

t i2 isin N t isin

N

o i1 i2 t s

2o d1113872 1113873 isin K

1 (31)

xt dti1i2s2

o ds1

o disin 0 Z


N

d i2 isin N t isin

N

o t i1 i2 s

1o d1113872 1113873 isin K

2 (32)




1172Parent 1

Parent 2

Offspring 1

Offspring 2

Offspring 1prime

Offspring 2prime

xoti1i2ts1od

9 12 11 10 14 13

Random single point


7 4 2 1


Pc

Mutationprobability

Pm

12 11 10 14 13 7 4 2 1 3

1172 9 12 11 10 14 13 4 2 1 3

1059 2 11 10 14 13 7 7 4 2 1

1116 9 12 11 10 14 13 2 1 4 3

1089 12 11 10 14 13 7 2 7 4 1







5 Solution Approach




(a)

(b)



91times108

905

9

895

89

The f

itnes

s val

ue88

885

875

87

865

861000 2000 3000 4000


5000 6000 7000 8000

Scenario 0

Scenario 1

Scenario 2

Scenario 3










890E + 08






894E + 08

074

124

058

9073992796

9021487159

896171319

9007053615

898E + 08 902E + 08 906E + 08 910E + 08






84


85 86 87 8988 90 91

268

468

8685

9079

8933

9091

454

92





r where c 1 2
















o⟶ d G3 () o⟶ d G3 () o⟶ d G3 () o⟶ d G 3()1⟶ 9 100 5⟶ 9 100 9⟶1 9982 12⟶1 1001⟶ 10 100 5⟶10 100 9⟶ 2 9534 12⟶ 2 94591⟶ 11 8836 5⟶11 9736 9⟶ 3 9399 12⟶ 3 75221⟶ 12 5872 5⟶12 8531 9⟶ 4 9766 12⟶ 4 96801⟶ 13 9467 5⟶13 8595 9⟶ 5 9862 12⟶ 5 1001⟶ 14 9295 5⟶14 9890 9⟶ 6 8098 12⟶ 6 97612⟶ 9 9266 6⟶ 9 8401 9⟶ 7 9576 12⟶ 7 85422⟶10 9120 6⟶10 9902 9⟶ 8 100 12⟶ 8 34142⟶11 100 6⟶11 9973 10⟶1 9378 13⟶1 98422⟶12 9872 6⟶12 8319 10⟶ 2 100 13⟶ 2 80482⟶13 6633 6⟶13 7649 10⟶ 3 9811 13⟶ 3 96022⟶14 9504 6⟶14 9464 10⟶ 4 8841 13⟶ 4 93053⟶ 9 8781 7⟶ 9 9229 10⟶ 5 9629 13⟶ 5 72513⟶10 8715 7⟶10 9801 10⟶ 6 9385 13⟶ 6 98273⟶11 9577 7⟶11 9702 10⟶ 7 7955 13⟶ 7 34473⟶12 9602 7⟶12 8289 10⟶ 8 8306 13⟶ 8 96463⟶13 9079 7⟶13 8713 11⟶ 1 8790 14⟶1 76083⟶14 9595 7⟶14 8828 11⟶ 2 4185 14⟶ 2 39214⟶ 9 9836 8⟶ 9 9954 11⟶ 3 8612 14⟶ 3 91624⟶10 8456 8⟶10 8282 11⟶ 4 6180 14⟶ 4 84474⟶11 100 8⟶11 9576 11⟶ 5 8472 14⟶ 5 93584⟶12 9895 8⟶12 100 11⟶ 6 9151 14⟶ 6 1004⟶13 9174 8⟶13 9666 11⟶ 7 9130 14⟶ 7 82764⟶14 100 8⟶14 8666 11⟶ 8 7934 14⟶ 8 6621




























Data Availability




Acknowledgments


References







































































1172Parent 1

Parent 2

Offspring 1

Offspring 2

Offspring 1prime

Offspring 2prime

xoti1i2ts1od

9 12 11 10 14 13

Random single point


7 4 2 1


Pc

Mutationprobability

Pm

12 11 10 14 13 7 4 2 1 3

1172 9 12 11 10 14 13 4 2 1 3

1059 2 11 10 14 13 7 7 4 2 1

1116 9 12 11 10 14 13 2 1 4 3

1089 12 11 10 14 13 7 2 7 4 1







5 Solution Approach




(a)

(b)



91times108

905

9

895

89

The f

itnes

s val

ue88

885

875

87

865

861000 2000 3000 4000


5000 6000 7000 8000

Scenario 0

Scenario 1

Scenario 2

Scenario 3










890E + 08






894E + 08

074

124

058

9073992796

9021487159

896171319

9007053615

898E + 08 902E + 08 906E + 08 910E + 08






84


85 86 87 8988 90 91

268

468

8685

9079

8933

9091

454

92





r where c 1 2
















o⟶ d G3 () o⟶ d G3 () o⟶ d G3 () o⟶ d G 3()1⟶ 9 100 5⟶ 9 100 9⟶1 9982 12⟶1 1001⟶ 10 100 5⟶10 100 9⟶ 2 9534 12⟶ 2 94591⟶ 11 8836 5⟶11 9736 9⟶ 3 9399 12⟶ 3 75221⟶ 12 5872 5⟶12 8531 9⟶ 4 9766 12⟶ 4 96801⟶ 13 9467 5⟶13 8595 9⟶ 5 9862 12⟶ 5 1001⟶ 14 9295 5⟶14 9890 9⟶ 6 8098 12⟶ 6 97612⟶ 9 9266 6⟶ 9 8401 9⟶ 7 9576 12⟶ 7 85422⟶10 9120 6⟶10 9902 9⟶ 8 100 12⟶ 8 34142⟶11 100 6⟶11 9973 10⟶1 9378 13⟶1 98422⟶12 9872 6⟶12 8319 10⟶ 2 100 13⟶ 2 80482⟶13 6633 6⟶13 7649 10⟶ 3 9811 13⟶ 3 96022⟶14 9504 6⟶14 9464 10⟶ 4 8841 13⟶ 4 93053⟶ 9 8781 7⟶ 9 9229 10⟶ 5 9629 13⟶ 5 72513⟶10 8715 7⟶10 9801 10⟶ 6 9385 13⟶ 6 98273⟶11 9577 7⟶11 9702 10⟶ 7 7955 13⟶ 7 34473⟶12 9602 7⟶12 8289 10⟶ 8 8306 13⟶ 8 96463⟶13 9079 7⟶13 8713 11⟶ 1 8790 14⟶1 76083⟶14 9595 7⟶14 8828 11⟶ 2 4185 14⟶ 2 39214⟶ 9 9836 8⟶ 9 9954 11⟶ 3 8612 14⟶ 3 91624⟶10 8456 8⟶10 8282 11⟶ 4 6180 14⟶ 4 84474⟶11 100 8⟶11 9576 11⟶ 5 8472 14⟶ 5 93584⟶12 9895 8⟶12 100 11⟶ 6 9151 14⟶ 6 1004⟶13 9174 8⟶13 9666 11⟶ 7 9130 14⟶ 7 82764⟶14 100 8⟶14 8666 11⟶ 8 7934 14⟶ 8 6621




























Data Availability




Acknowledgments


References








































































5 Solution Approach




(a)

(b)



91times108

905

9

895

89

The f

itnes

s val

ue88

885

875

87

865

861000 2000 3000 4000


5000 6000 7000 8000

Scenario 0

Scenario 1

Scenario 2

Scenario 3










890E + 08






894E + 08

074

124

058

9073992796

9021487159

896171319

9007053615

898E + 08 902E + 08 906E + 08 910E + 08






84


85 86 87 8988 90 91

268

468

8685

9079

8933

9091

454

92





r where c 1 2
















o⟶ d G3 () o⟶ d G3 () o⟶ d G3 () o⟶ d G 3()1⟶ 9 100 5⟶ 9 100 9⟶1 9982 12⟶1 1001⟶ 10 100 5⟶10 100 9⟶ 2 9534 12⟶ 2 94591⟶ 11 8836 5⟶11 9736 9⟶ 3 9399 12⟶ 3 75221⟶ 12 5872 5⟶12 8531 9⟶ 4 9766 12⟶ 4 96801⟶ 13 9467 5⟶13 8595 9⟶ 5 9862 12⟶ 5 1001⟶ 14 9295 5⟶14 9890 9⟶ 6 8098 12⟶ 6 97612⟶ 9 9266 6⟶ 9 8401 9⟶ 7 9576 12⟶ 7 85422⟶10 9120 6⟶10 9902 9⟶ 8 100 12⟶ 8 34142⟶11 100 6⟶11 9973 10⟶1 9378 13⟶1 98422⟶12 9872 6⟶12 8319 10⟶ 2 100 13⟶ 2 80482⟶13 6633 6⟶13 7649 10⟶ 3 9811 13⟶ 3 96022⟶14 9504 6⟶14 9464 10⟶ 4 8841 13⟶ 4 93053⟶ 9 8781 7⟶ 9 9229 10⟶ 5 9629 13⟶ 5 72513⟶10 8715 7⟶10 9801 10⟶ 6 9385 13⟶ 6 98273⟶11 9577 7⟶11 9702 10⟶ 7 7955 13⟶ 7 34473⟶12 9602 7⟶12 8289 10⟶ 8 8306 13⟶ 8 96463⟶13 9079 7⟶13 8713 11⟶ 1 8790 14⟶1 76083⟶14 9595 7⟶14 8828 11⟶ 2 4185 14⟶ 2 39214⟶ 9 9836 8⟶ 9 9954 11⟶ 3 8612 14⟶ 3 91624⟶10 8456 8⟶10 8282 11⟶ 4 6180 14⟶ 4 84474⟶11 100 8⟶11 9576 11⟶ 5 8472 14⟶ 5 93584⟶12 9895 8⟶12 100 11⟶ 6 9151 14⟶ 6 1004⟶13 9174 8⟶13 9666 11⟶ 7 9130 14⟶ 7 82764⟶14 100 8⟶14 8666 11⟶ 8 7934 14⟶ 8 6621




























Data Availability




Acknowledgments


References







































































91times108

905

9

895

89

The f

itnes

s val

ue88

885

875

87

865

861000 2000 3000 4000


5000 6000 7000 8000

Scenario 0

Scenario 1

Scenario 2

Scenario 3










890E + 08






894E + 08

074

124

058

9073992796

9021487159

896171319

9007053615

898E + 08 902E + 08 906E + 08 910E + 08






84


85 86 87 8988 90 91

268

468

8685

9079

8933

9091

454

92





r where c 1 2
















o⟶ d G3 () o⟶ d G3 () o⟶ d G3 () o⟶ d G 3()1⟶ 9 100 5⟶ 9 100 9⟶1 9982 12⟶1 1001⟶ 10 100 5⟶10 100 9⟶ 2 9534 12⟶ 2 94591⟶ 11 8836 5⟶11 9736 9⟶ 3 9399 12⟶ 3 75221⟶ 12 5872 5⟶12 8531 9⟶ 4 9766 12⟶ 4 96801⟶ 13 9467 5⟶13 8595 9⟶ 5 9862 12⟶ 5 1001⟶ 14 9295 5⟶14 9890 9⟶ 6 8098 12⟶ 6 97612⟶ 9 9266 6⟶ 9 8401 9⟶ 7 9576 12⟶ 7 85422⟶10 9120 6⟶10 9902 9⟶ 8 100 12⟶ 8 34142⟶11 100 6⟶11 9973 10⟶1 9378 13⟶1 98422⟶12 9872 6⟶12 8319 10⟶ 2 100 13⟶ 2 80482⟶13 6633 6⟶13 7649 10⟶ 3 9811 13⟶ 3 96022⟶14 9504 6⟶14 9464 10⟶ 4 8841 13⟶ 4 93053⟶ 9 8781 7⟶ 9 9229 10⟶ 5 9629 13⟶ 5 72513⟶10 8715 7⟶10 9801 10⟶ 6 9385 13⟶ 6 98273⟶11 9577 7⟶11 9702 10⟶ 7 7955 13⟶ 7 34473⟶12 9602 7⟶12 8289 10⟶ 8 8306 13⟶ 8 96463⟶13 9079 7⟶13 8713 11⟶ 1 8790 14⟶1 76083⟶14 9595 7⟶14 8828 11⟶ 2 4185 14⟶ 2 39214⟶ 9 9836 8⟶ 9 9954 11⟶ 3 8612 14⟶ 3 91624⟶10 8456 8⟶10 8282 11⟶ 4 6180 14⟶ 4 84474⟶11 100 8⟶11 9576 11⟶ 5 8472 14⟶ 5 93584⟶12 9895 8⟶12 100 11⟶ 6 9151 14⟶ 6 1004⟶13 9174 8⟶13 9666 11⟶ 7 9130 14⟶ 7 82764⟶14 100 8⟶14 8666 11⟶ 8 7934 14⟶ 8 6621




























Data Availability




Acknowledgments


References












































































890E + 08






894E + 08

074

124

058

9073992796

9021487159

896171319

9007053615

898E + 08 902E + 08 906E + 08 910E + 08






84


85 86 87 8988 90 91

268

468

8685

9079

8933

9091

454

92





r where c 1 2
















o⟶ d G3 () o⟶ d G3 () o⟶ d G3 () o⟶ d G 3()1⟶ 9 100 5⟶ 9 100 9⟶1 9982 12⟶1 1001⟶ 10 100 5⟶10 100 9⟶ 2 9534 12⟶ 2 94591⟶ 11 8836 5⟶11 9736 9⟶ 3 9399 12⟶ 3 75221⟶ 12 5872 5⟶12 8531 9⟶ 4 9766 12⟶ 4 96801⟶ 13 9467 5⟶13 8595 9⟶ 5 9862 12⟶ 5 1001⟶ 14 9295 5⟶14 9890 9⟶ 6 8098 12⟶ 6 97612⟶ 9 9266 6⟶ 9 8401 9⟶ 7 9576 12⟶ 7 85422⟶10 9120 6⟶10 9902 9⟶ 8 100 12⟶ 8 34142⟶11 100 6⟶11 9973 10⟶1 9378 13⟶1 98422⟶12 9872 6⟶12 8319 10⟶ 2 100 13⟶ 2 80482⟶13 6633 6⟶13 7649 10⟶ 3 9811 13⟶ 3 96022⟶14 9504 6⟶14 9464 10⟶ 4 8841 13⟶ 4 93053⟶ 9 8781 7⟶ 9 9229 10⟶ 5 9629 13⟶ 5 72513⟶10 8715 7⟶10 9801 10⟶ 6 9385 13⟶ 6 98273⟶11 9577 7⟶11 9702 10⟶ 7 7955 13⟶ 7 34473⟶12 9602 7⟶12 8289 10⟶ 8 8306 13⟶ 8 96463⟶13 9079 7⟶13 8713 11⟶ 1 8790 14⟶1 76083⟶14 9595 7⟶14 8828 11⟶ 2 4185 14⟶ 2 39214⟶ 9 9836 8⟶ 9 9954 11⟶ 3 8612 14⟶ 3 91624⟶10 8456 8⟶10 8282 11⟶ 4 6180 14⟶ 4 84474⟶11 100 8⟶11 9576 11⟶ 5 8472 14⟶ 5 93584⟶12 9895 8⟶12 100 11⟶ 6 9151 14⟶ 6 1004⟶13 9174 8⟶13 9666 11⟶ 7 9130 14⟶ 7 82764⟶14 100 8⟶14 8666 11⟶ 8 7934 14⟶ 8 6621




























Data Availability




Acknowledgments


References









































































r where c 1 2
















o⟶ d G3 () o⟶ d G3 () o⟶ d G3 () o⟶ d G 3()1⟶ 9 100 5⟶ 9 100 9⟶1 9982 12⟶1 1001⟶ 10 100 5⟶10 100 9⟶ 2 9534 12⟶ 2 94591⟶ 11 8836 5⟶11 9736 9⟶ 3 9399 12⟶ 3 75221⟶ 12 5872 5⟶12 8531 9⟶ 4 9766 12⟶ 4 96801⟶ 13 9467 5⟶13 8595 9⟶ 5 9862 12⟶ 5 1001⟶ 14 9295 5⟶14 9890 9⟶ 6 8098 12⟶ 6 97612⟶ 9 9266 6⟶ 9 8401 9⟶ 7 9576 12⟶ 7 85422⟶10 9120 6⟶10 9902 9⟶ 8 100 12⟶ 8 34142⟶11 100 6⟶11 9973 10⟶1 9378 13⟶1 98422⟶12 9872 6⟶12 8319 10⟶ 2 100 13⟶ 2 80482⟶13 6633 6⟶13 7649 10⟶ 3 9811 13⟶ 3 96022⟶14 9504 6⟶14 9464 10⟶ 4 8841 13⟶ 4 93053⟶ 9 8781 7⟶ 9 9229 10⟶ 5 9629 13⟶ 5 72513⟶10 8715 7⟶10 9801 10⟶ 6 9385 13⟶ 6 98273⟶11 9577 7⟶11 9702 10⟶ 7 7955 13⟶ 7 34473⟶12 9602 7⟶12 8289 10⟶ 8 8306 13⟶ 8 96463⟶13 9079 7⟶13 8713 11⟶ 1 8790 14⟶1 76083⟶14 9595 7⟶14 8828 11⟶ 2 4185 14⟶ 2 39214⟶ 9 9836 8⟶ 9 9954 11⟶ 3 8612 14⟶ 3 91624⟶10 8456 8⟶10 8282 11⟶ 4 6180 14⟶ 4 84474⟶11 100 8⟶11 9576 11⟶ 5 8472 14⟶ 5 93584⟶12 9895 8⟶12 100 11⟶ 6 9151 14⟶ 6 1004⟶13 9174 8⟶13 9666 11⟶ 7 9130 14⟶ 7 82764⟶14 100 8⟶14 8666 11⟶ 8 7934 14⟶ 8 6621




























Data Availability




Acknowledgments


References

































































































Data Availability




Acknowledgments


References


















































































Data Availability




Acknowledgments


References








































































Data Availability




Acknowledgments


References

























































































































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