optimal container routing in liner shipping networks ...optimal container routing in liner shipping...

Research ArticleOptimal Container Routing in Liner Shipping NetworksConsidering Repacking 20 ft Containers into 40 ft Containers

Shuaian Wang1 Xiaobo Qu2 Tingsong Wang3 and Wen Yi4

1Department of Logistics amp Maritime Studies The Hong Kong Polytechnic University Hung Hom Hong Kong2School of Civil and Environmental Engineering University of Technology Sydney Sydney NSW 2007 Australia3School of Economics and Management Wuhan University Wuhan 430072 China4Department of Building and Real Estate The Hong Kong Polytechnic University Hung Hom Hong Kong

Correspondence should be addressed to Tingsong Wang emswangtswhueducn

Received 11 November 2016 Revised 16 December 2016 Accepted 9 January 2017 Published 31 January 2017

Academic Editor Andrea DrsquoAriano

Copyright copy 2017 Shuaian Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The volume of a 40 ft container is twice as large as that of a 20 ft container However the handling cost (loading unloading andtransshipment) of a 40 ft container is much lower than twice the corresponding handling cost of two 20 ft containers Enlightenedby this observation we propose a novel container routing with repacking problem in liner shipping where two 20 ft containers canbe repacked to a 40 ft container in order to reduce the handling cost We develop a mixed-integer linear programming model thatformulates the routing decisions and the repacking decisions in a holisticmanner An illustrative example is reported to demonstratethe applicability of the model Results show that the benefit of repacking is the most significant when containers are transshippedseveral times

1 Introduction

Maritime transportation is the backbone of internationaltrade Around 80 per cent of global trade by volume and over70 per cent by value are carried by sea Among all the sea car-gos about half in monetary terms are containerized [1] Con-tainers are transported by shipping lines on regularly servicedship routes At the port of origin containers are loaded ontoships by quay cranes and at the port of destination containersare discharged from ships by quay cranes Containers mayalso be transshipped between their origin ports and desti-nation ports In fact transshipment is a common operationin container shipping As reported by [1] the total containertrade volumes amounted to 160 million twenty-foot equiv-alent units (TEUs) in 2013 whereas world container portthroughput was estimated at 651 million TEUs These num-bers mean that on average a container was transshipped(651 minus 2 times 160)160 asymp 2 times (the throughput data for portsalso include empty containers)

Container routing determines how to transport contain-ers from their origins to their destinations in a liner shipping

network Take Figure 1 as an example which shows a linershipping network consisting of three ship routes Containersfrom Singapore to Hong Kong can be transported on eithership route 1 or ship route 2 If there are many containers tobe transported from Singapore to Jakarta then containersfrom Singapore to Hong Kong should be transported on shiproute 2 to reserve the capacity on ship route 1 for containersfrom Singapore to Jakarta In addition to different shiproutes on which containers can be transported from originto destination another complicating factor is transshipmentFor instance containers from Hong Kong to Colombo canbe transported on ship route 2 or they can be transported onship route 1 to Singapore and transshipped to ship route 2 andthen transported to Colombo The choice of direct shipmenton ship route 2 is preferable because otherwise it wouldinvolve a high transshipment cost at Singapore However ifthere aremany containers to be transported fromHong Kongto Xiamen or from Xiamen to Singapore then the choice oftransshipment at Singapore from ship route 1 to ship route2 has to be adopted Consequently it is not an easy task todetermine the optimal container routing

HindawiJournal of Advanced TransportationVolume 2017 Article ID 8608032 9 pageshttpsdoiorg10115520178608032

2 Journal of Advanced Transportation

Table 1 Laden container handling cost (USDcontainer) at three ports (source [11])

Port Type Loading Discharge Transshipment

A (a major transshipment port in Europe)D20 248 324 183D40 256 332 198Ratio 103 102 108

B (a major transshipment port in Southeast Asia)D20 118 118 145D40 148 148 145Ratio 125 125 1

C (an export-driven port in Southeast Asia)D20 110 110 71D40 156 156 106Ratio 142 142 149

Colombo(CB)

Hong Kong(HK)

Singapore(SG)

Jakarta(JK)

Xiamen(XM)

Chennai(CN)

Cochin(CC)

Ship route 3CB(1) rarr CN(2) rarr CC(3) rarr CB

Ship route 2HK(1) rarr XM(2) rarr SG(3) rarr CB(4) rarr SG(5) rarrHK

Ship route 1HK(1) rarr JK(2) rarr SG(3) rarr HK

Figure 1 An illustrative liner shipping network [8]

Container routing determines the container handlingcost Table 1 shows the handling costs for two types of ladencontainers at three ports D20 means dry 20 ft container andD40 is dry 40 ft container In terms of cargo capacity a D40 isequivalent to two D20s However Table 1 clearly indicates inthe three rows ldquoRatiordquo that the ratio of the cost of handlinga D40 and that of handling a D20 is strictly less than 2In fact all the ratios in Table 1 are less than 15 and someratios are even 1 or very close to 1 This is because both thehandling of a D20 and the handling of a D40 involve onequay crane move (we note that nowadays some quay cranescan handle one D40 or two D20s in each move) Thereforeto reduce container handling costs a shipping line should tryto transport more D40s instead of D20s as a D40 can hold asmuch cargos as two D20s

11 Container Repacking As the handling cost of a D40 ismuch lower than that of two D20s it might be advantageous

to unpack two D20s and repack them to one D40 Inthe sequel we use ldquoTEUrdquo and ldquoD20rdquo interchangeably anduse ldquoforty-foot equivalent unit (FEU)rdquo and ldquoD40rdquo inter-changeably The load transshipment and discharge cost(USDcontainer) of a TEU at a port 119901 is denoted by 119879119901 119888119879119901 and 119879119901 respectively The load transshipment and dischargecost (USDcontainer) of an FEU at port 119901 isin P is denoted by119865119901 119888119865119901 and 119865119901 respectivelyWe further let 1198882119879rarr119865119901 be the cost ofrepacking two TEUs into one FEU and 119888119865rarr2119879119901 be the cost ofunpacking one FEU to two TEUs (Since multiple rehandlingof containers would increase the risk for damage and there-fore may increase insurance costs we can include in 1198882119879rarr119865119901and 119888119865rarr2119879119901 the extra insurance costs Moreover repackingrequires consent from shippers and we can include in therehandling cost the component of discount for shippers whoagree for their cargos to be repacked)

Figure 2 shows an example of transporting two TEUsfrom port 1199012 to port 1199015 The two TEUs need to be trans-shipped twice If they are transported as two TEUs as shownin Figure 2(a) then at the port of origin that is 1199012 two TEUsare loaded at 1199013 two TEUs are transshipped at 1199014 two TEUsare transshipped and at the destination port 1199015 two TEUsare dischargedTherefore the total container handling cost is

21198791199012 + 21198881198791199013 + 21198881198791199014 + 21198791199015 (1)

If these two TEUs are repacked into an FEU at the originport and unpacked at the destination port as shown inFigure 2(b) then at the port of origin that is 1199012 one FEUis loaded at 1199013 one FEU is transshipped at 1199014 one FEUis transshipped and at the destination port 1199015 one FEU isdischarged Moreover container repacking and unpackingcosts are incurred at 1199012 and 1199015 respectively (The FEU isunpacked at 1199015 into two TEUs because the two TEUs fromport1199012 to port1199015 actually have different inland destinations)Therefore the total container handling and packing cost is

1198882119879rarr1198651199012 + 1198651199012 + 1198881198651199013 + 1198881198651199014 + 1198651199015 + 119888119865rarr21198791199015 (2)

which may be considerably smaller than (1)

Journal of Advanced Transportation 3

Ship route 1 Ship route 1 Ship route 1

TEU

TEU

TEU

TEU

2 TEUs load

2 TEUstransship

2 TEUstransship

2 TEUsdischarge

TEU

TEU

TEU

TEU

TEU

TEUTEU

TEU

p1

p2 p3 p4 p5

p6

2cTp52cTp2 2cTp3 2cTp4

(a) Two TEUs from 1199012 to 1199015 without repacking

TEU

TEU

FEU FEUFEU FEU

FEU FEUTEU

TEU

Repack Unpack1 TEU load

1 TEUtransship

1 TEUtransship

1 TEUdischarge

cFrarr2Tp5cFp4cFp3cFp2c2TrarrFp2

cFp5

p1

p2 p3 p4 p5

p6


(b) Two TEUs from 1199012 to 1199015 with repacking at 1199012 and unpacking at 1199015

Figure 2 Container packing with the same origin and destination

Not only can TEUs with the same origin and destinationports be repacked into FEUs as shown in Figure 2 TEUswithdifferent origins and destinations can also be repacked intoFEUs For example in Figure 3 one TEU is from 1199012 to 1199015and the other TEU is from 1199011 to 1199016 If they are transportedas TEUs throughout their trips as shown in Figure 3(a) thetotal handling cost is

1198791199011 + 1198791199012 + 21198881198791199013 + 21198881198791199014 + 1198791199015 + 1198791199016 (3)

They may be repacked into an FEU at the transshipment port1199013 and unpacked as two TEUs at transshipment port 1199014 asshown in Figure 3(b) At 1199013 since two TEUs are unloadedand one FEU is loaded in the transshipment process the totaltransshipment cost is calculated as half of the sum of thetransshipment cost of two TEUs and the transshipment cost

of one FEU that is 1198881198791199013 + 051198881198651199013 Similarly the transshipmentcost at 1199014 is calculated as 1198881198791199014 + 051198881198651199014 Therefore the totalcontainer handling and packing cost is

1198791199011 + 1198791199012 + 1198881198791199013 + 051198881198651199013 + 1198882119879rarr1198651199013⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199013

+ 1198881198791199014 + 051198881198651199014 + 119888119865rarr21198791199014⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199014

+ 1198791199015 + 1198791199016 (4)

which may be smaller than (3) if the container packing costis small Of course the benefit of repacking TEUs into FEUsin Figure 3(b) is not as significant as Figure 2

Container packing may also affect how the containersare transported For example in Figure 3(c) the containershipment demand (the number of containers to be trans-ported from one port to another in a week) is the same as


TEU

TEU

TEU

TEU

1 TEUload

2 TEUstransship

2 TEUstransship

1 TEUdischarge

TEU

TEU

TEU

TEU

TEU

TEUTEU

TEU

1 TEUload

1 TEUdischarge


p1

p2 p3 p4 p5

p6

cTp5 cTp6cTp1 cTp2 2cTp3 2cTp4

(a) One TEU from 1199012 to 1199015 and one TEU from 1199011 to 1199016 without repacking

FEU FEU

transship05 FEU05 FEUtransship

1 TEU1 TEU1 TEU1 TEU1 TEU 1 TEU

TEU

TEU

TEU

TEU

TEU

TEU

TEU

TEU

loadload discharge dischargeRepacktransship Unpack transship


p1

p2 p3 p4 p5

p6

cTp5 cTp6cTp1 cTp2 cTp3 cTp4c2TrarrFp205cFp405cFp3 cFrarr2Tp5

(b) One TEU from 1199012 to 1199015 and one TEU from 1199011 to 1199016 with repacking at 1199013 and unpacking at 1199014

FEU FEU

TEU TEU

TEU

TEUFEU FEU TEU

TEU

FEU FEU

p1

p2 p3 p4 p5

p6


(c) One TEU from 1199012 to 1199015 and one TEU from 1199011 to 1199016 with repacking at 1199012 and unpacking at 1199015

Figure 3 Container packing with different origins and destinations

Figures 3(a) and 3(b) Containers are transported as followsin Figure 3(c) one TEU is transported from 1199011 to 1199012 at 1199012the TEU is unloaded and repacked with another TEU whoseorigin is 1199012 into an FEU the FEU is loadedtransshipped at1199012 and transported to 1199015 at 1199015 the FEU is unpacked to twoTEUs one ofwhich has arrived at its destination and the other

should be transported to its destination 1199016 To compute thehandling cost of the two TEUs at 1199012 we assume that thereare two TEUs with origin 1199011 and another two TEUs withorigin 1199012 that are repacked into two FEUs at 1199012 (hence thehandling cost is doubled) The two TEUs with origin 1199011 areactually transshipped at 1199012 and therefore the handling cost is


(21198881198791199012+1198881198651199012)2The twoTEUswith origin1199012 are actually loadedat1199012 and therefore the handling cost is 1198651199012 Consequently thetotal handling cost at 1199012 in the example of Figure 3(c) is

(21198881198791199012 + 1198881198651199012) 2 + 11986511990122 = 11988811987911990122 + 11988811986511990124 + 11986511990122 (5)

Similar arguments apply to the handling cost at 1199015Thereforethe total container handling and packing cost in Figure 3(c) is

1198791199011 + 11988811987911990122 + 11988811986511990124 + 11986511990122 + 1198882119879rarr1198651199012⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199012

+ 1198881198651199013 + 1198881198651199014

+ 11988811987911990152 + 11988811986511990154 + 11986511990152 + 119888119865rarr21198791199015⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199015

+ 1198791199016 (6)

In some cases it is still possible that (6) is smaller than (3)

12 Literature Review Objectives and Organization Quanti-tative research on container liner shipping can be classifiedinto three categories strategic tactical and operational prob-lems [2 3] Strategic problems include ship fleet planning[4] and alliance formation [5] Strategic problems involvedecisions that have effects for many years and hence it isvery difficult to predict the container demand As a resultthe demand is usually simplified for instance containers areformulated as either TEUs where a D40 is considered astwo TEUs or FEUs where a D20 is considered as half anFEU Tactical-level decisions include network design [6 7]fleet deployment [8] speed optimization [9] and scheduledesign [10] Tactical-level decisions are usually made takinginto account how containers are transported in shipping net-works Nevertheless in almost all of these studies on tactical-level problems similar to the ones on strategic level problemscontainers are formulated as either just TEUs or just FEUsWang [11] examined containership fleet deployment withboth TEUs and FEUs however the TEUs are not allowed tobe repacked to FEUs

Quantitative research on container terminal operationscan be divided into studies on sea-side operations and studieson land-side operations Sea-side decisions are mainly onberth allocation and quay crane assignment [12ndash19] Land-side problems are mainly yard storage area planning andallocation [20ndash24] There are also studies considering bothcontainer terminal operations planning and vessel scheduling[25ndash27] None of the above studies related to container ter-minal operations has investigated the problem of repackingTEUs into FEUs

Theobjective of our study is to investigate how a containershipping line can transport containers in an efficient mannerwhile accounting for the possibility of repacking two TEUsinto one FEU to reducing the handling costs We systemati-cally examine this problem and develop a holistic model thatincorporates both container routing and container repackingHence we address a practical problem that is significant forcontainer shipping industry

The remainder of the paper is organized as followsSection 2 describes the problem Section 3 proposes a mixed-integer linear programming model that captures both con-tainer routing and container repacking Section 4 reports acase study Section 5 concludes

2 Problem Description

Consider a setR of ship routes regularly serving a group ofports denoted by setP Ship route 119903 isin R can be expressed as

1199011199031 997888rarr 1199011199032 997888rarr sdot sdot sdot 997888rarr 119901119903119873119903 997888rarr 1199011199031 (7)

where 119873119903 is the number of ports of call and 119901119903119894 is the 119894thport of call 119894 = 1 2 119873119903 Define I119903 fl 1 2 119873119903 Thevoyage from port of call 119894 to port of call 119894 + 1 is called leg 119894 andleg119873119903 is the voyage fromport of call119873119903 to the first port of callIn Figure 1 three ship routes are shown ship route 1 has threelegs ship route 2 has five legs and ship route 3 has three legsEach ship route has a weekly service frequency which meansthat each port of call is visited on the same day every weekA string of homogeneous ships with a capacity of 119881119903 (TEUs)is deployed on ship route 119903 to maintain the weekly frequencyThe liner services are similar to bus services [28]

Represent by W the set of origin-to-destination (OD)port pairs which is a subset of P times P There are two typesof containers to ship TEUs (119879) and FEUs (119865) The demandfor OD pair (119900 119889) isin W is denoted by 119899119879119900119889 and 119899119865119900119889 (con-tainersweek) for TEUs and FEUs respectively Containerscan be transshipped at any port from their origins to theirdestinations The loading unloading and transshipmentcosts should be included in making container routing andrepacking decisions

As aforementioned two TEUs may be repacked into anFEU to reduce handling cost From the practical point ofview we assume that a TEU can be repacked and unpackedat most once The repacking and unpacking costs shouldalso be included in making container routing and repackingdecisions

It should be noted that two TEUs can only be repackedinto an FEU at container yards In other words the repackingactivity could not be carried out on ships For example inFigure 3(c) the TEU with origin 1199011 must be unloaded fromships at 1199012 so as to be repacked with the TEU from 1199012 into anFEU (and hence handling cost of the TEU with origin 1199011 at1199012 is incurred)

The container routing with repacking problem aims todetermine where to repack TEUs into FEUs and where tounpack the FEUs and how to transport both TEUs and FEUsin order to minimize the total handling and packing costwhile allowing containers to be transshipped at any port ina liner shipping network

3 Mixed-Integer Linear Programming Model

31 Demand Reformulation for Container Repacking Foreach OD pair (119900 119889) isin W we define 119910119901119902119879

119900119889as the number of


TEUs that are repacked at port 119901 isin P and unpacked at port119902 isin P Let Z+ be the set of nonnegative integers We have

119910119901119902119879119900119889

isin Z+ forall (119900 119889) isin W forall119901 isin P forall119902 isin P (8)

Therefore the number of TEUs that are transported forOD pair (119900 119889) isin W without repacking denoted by 1198991198794119900119889 is

1198991198794119900119889 = 119899119879119900119889 minus sum119901isinP

sum119902isinP

119910119901119902119879119900119889

forall (119900 119889) isin W (9)

1198991198794119900119889 ge 0 forall (119900 119889) isin W (10)

Note that (8) and (9) imply that 1198991198794119900119889 is an integer The totalloading and unloading cost of the 1198991198794119900119889 TEUs for all OD pairsis

119862 (1198991198794119900119889) = sum(119900119889)isinW

(119879119900 + 119879119889) 1198991198794119900119889 (11)

The transshipment cost of these 1198991198794119900119889 TEUs is not a constant asit depends on container routing

Thenumber of TEUswith origin port 119900 isin P and repackedinto FEUs at port 119901 isin P denoted by 1198991198791119900119901 is

1198991198791119900119901 = sum119889isinP

sum119902isinP

119910119901119902119879119900119889

forall119900 isin P forall119901 isin P (12)

In (12) 119900 = 119901 means that the TEU is repacked at its originport Hence for this TEU the loading cost is equal to halfthe loading cost of an FEU at port 119900 If 119900 = 119901 the loadingcost of a TEU at port 119900 half the transshipment cost of a TEUand a quarter of the transshipment cost of an FEU at port119901 should be considered In either case the repacking cost atport 119901 should be considered Therefore the sum of the totalhandling cost excluding the transshipment cost between port119900 and port 119901 (these TEUs may be transshipped when they aretransported from port 119900 to port 119901) and the repacking cost forthese 1198991198791119900119901 TEUs is

119862 (1198991198791119900119889) = sum119900isinP

1198651199002 1198991198791119900119900

+ sum119900isinP

sum119901isinP119900

(119879119900 + 1198881198791199012 + 1198881198651199014 ) 1198991198791119900119901

+ sum119900isinP

sum119901isinP

1198882119879rarr119865119901 2 1198991198791119900119901

(13)

Thenumber of TEUs that are unpacked fromFEUs at port119902 isin P and transported from port 119902 isin P to their destinationport 119889 isin P denoted by 1198991198793119902119889 is

1198991198793119902119889 = sum119900isinP

sum119901isinP

119910119901119902119879119900119889


If 119902 = 119889 the TEU is unpacked at its destination portHence the discharge cost of half an FEU at port 119889 should be

considered Otherwise the discharge cost of a TEU at port119889 half the transshipment cost of a TEU and a quarter of thetransshipment cost of an FEU at port 119902 should be included Ineither case the unpacking cost at port 119902 should be incorpo-ratedTherefore the sum of the total handling cost excludingthe transshipment cost betweenport 119902 andport119889 (theseTEUsmay be transshipped when they are transported from port 119902to port 119889) and the unpacking cost for these 1198991198793119902119889 TEUs is

119862 (1198991198793119900119889) = sum119889isinP

1198651198892 1198991198793119889119889

+ sum119889isinP


(119879119889 + 1198881198791199022 + 1198881198651199024 ) 1198991198793119902119889

+ sum119902isinP

sum119889isinP

119888119865rarr2119879119902 2 1198991198793119902119889

(15)

The last reformulated demand is the transportation of therepacked containers which are FEUsThe number of FEUs tobe transported from port 119901 isin P where they are repacked toport 119902 isin P where they are unpacked denoted by 1198991198652119901119902 is

1198991198652119901119902 = sum119900isinP sum119889isinP 1199101199011199021198791199001198892 forall119901 isin P forall119902 isin P

1198991198652119901119902 isin Z+ forall119901 isin P forall119902 isin P

(16)

Note that for the 1198991198652119901119902 FEUs the handling costs at port 119901 andport 119902 are already included in the calculation in (13) and (15)The transshipment cost between port 119901 and port 119902 (if any)depends on container routing

FEUs in the original demand 119899119865119900119889 do not need furtherprocessing for modeling The total loading and unloadingcost of 119899119865119900119889 FEUs is

119862 (119899119865119900119889) = sum(119900119889)isinW

(119865119900 + 119865119889) 119899119865119900119889 (17)

The transshipment cost of these 119899119865119900119889 containers is not aconstant as it also depends on container routing

We use 119879119900119889 and 119865119900119889 (containersweek) to represent thereformulated demand of TEUs and FEUs for OD pair (119900 119889) isinW respectively The total reformulated demand is

119879119900119889 = 0 119900 = 1198891198991198791119900119889 + 1198991198793119900119889 + 1198991198794119900119889 forall (119900 119889) isin W 119900 = 119889

119865119900119889 = 0 119900 = 119889119899119865119900119889 + 1198991198652119900119889 forall (119900 119889) isin W 119900 = 119889

(18)

We define vector n of decision variables for formulatingcontainer repacking

n fl (1198991198791119900119889 1198991198793119900119889 1198991198794119900119889 119879119900119889 1198991198652119900119889 119865119900119889 forall (119900 119889)isin W 119910119901119902119879

119900119889 forall (119900 119889) isin W forall119901 isin P forall119902 isin P) (19)


The domain of n represented by set119873 is all vectors n in (19)that satisfy (8) (9) (10) (12) (14) (16) and (18)

32 Mathematical Model for Container Routing with Repack-ing Wehave already characterized container repacking deci-sions using vector n isin 119873 To formulate container routing wesimply consider the new container shipment demand shownin (18) The decision variables for formulating containerrouting are as follows 119900119889119879119903119894 and 119900119889119879119903119894 are the number of TEUsper week from (119900 119889) isin W loaded and discharged at port ofcall 119894 on ship route 119903 respectively (note that when calculating119900119889119879119903119894 and 119900119889119879119903119894 a transshippedTEU is considered as being dis-charged once and being loaded once) 119891119900119889119879119903119894 is the number ofTEUs per week from (119900 119889) isin W flowing on leg 119894 on ship route119903 (we define 1198911199001198891198791199030 fl 119891119900119889119879119903119873119903 ) 119911119879119901 is the total number of TEUsfrom all OD pairs transshipped at port 119901 isin P per week 119900119889119865119903119894 119900119889119865119903119894 119891119900119889119865119903119894 and 119911119865119901 are defined correspondingly for FEUsWedefine the vector of container routing decision variables

x fl (119900119889119879119903119894 119900119889119879119903119894 119891119900119889119879119903119894 119900119889119865119903119894 119900119889119865119903119894 119891119900119889119865119903119894 forall119903isin R forall119894 isin I119903 forall (119900 119889) isin W 119911119879119901 119911119865119901 forall119901 isin P) (20)

The container routing with repacking problem withdecisions in (19)-(20) can be formulated as an integer linearoptimization model [29 30]

minnisin119873x

sum119901isinP

(119888119879119901119911119879119901 + 119888119865119901119911119865119901) + 119862 (1198991198791119900119889) + 119862 (1198991198793119900119889)+ 119862 (1198991198794119900119889) + 119862 (119899119865119900119889)

(21)

subject to

119891119900119889119879119903119894minus1 + 119900119889119879119903119894 = 119891119900119889119879119903119894 + 119900119889119879119903119894 forall119903 isin R forall119894 isin I119903 forall (119900 119889) isin W

(22)

119911119879119901 = sum119903isinR

sum119894isinI119903 119901119903119894=119901

sum(119900119889)isinW

119900119889119879119903119894 minus sum119889isinP

119879119901119889 forall119901 isin P (23)

sum119903isinR

sum119894isinI119903 119901119903119894=119901

(119900119889119879119903119894 minus 119900119889119879119903119894 )

=

119879119900119889 119901 = 119900minus119879119900119889 119901 = 119889 forall (119900 119889) isin W forall119901 isin P

0 otherwise

(24)


(25)

119911119865119901 = sum119903isinR

sum119894isinI119903 119901119903119894=119901

sum(119900119889)isinW

119900119889119865119903119894 minus sum119889isinP

119865119901119889 forall119901 isin P (26)

sum119903isinR

sum119894isinI119903 119901119903119894=119901

(119900119889119865119903119894 minus 119900119889119865119903119894 )

=


0 otherwise

(27)

sum(119900119889)isinW

(119891119900119889119879119903119894 + 2119891119900119889119865119903119894 ) le 119881119903 forall119903 isin R forall119894 isin I119903 (28)

119900119889119879119903119894 isin Z+

119900119889119879119903119894 isin Z+

119891119900119889119879119903119894 isin Z+

forall119903 isin R forall119894 isin I119903 forall (119900 119889) isin W

(29)

119900119889119865119903119894 isin Z+

119900119889119865119903119894 isin Z+

119891119900119889119865119903119894 isin Z+

forall119903 isin R forall119894 isin I119903 forall (119900 119889) isin W(30)

The objective function (21) minimizes the total handlingand repacking and unpacking cost Constraint (22) is theTEU flow conservation equation Constraint (23) defines thetotal number of transshipped TEUs at each port Constraint(24) requires that the reformulated TEU demand is fulfilledEquations (25)ndash(27) define the corresponding constraints forFEUs Constraint (28) imposes ship capacity constraint oneach leg of each ship route Constraints (29)-(30) enforce theintegrality of the number of containers

Note that as packing and unpacking containers take timewe can further incorporate constraints into the model relatedto the maximum number of containers that can be packed orunpacked For instance if ship A arrives at a port onMondayship B arrives at the port onWednesday and ship C arrives atthe port on Friday then there are two daysrsquo time to pack shipsA and Brsquos 20 ft containers into 40 ft containers to be loadedonto ship C Taking into account the packing efficiency of theport the maximum number of containers that can be packedcan be calculated

4 An Illustrative Example

We use the shipping network in Figure 4 to demonstratethe applicability of the proposed model There are three shiproutes and all ship routes are deployed with ships of a capac-ity of 5000 TEUsThere are threeODpairs where all contain-ers are TEUsThe container handling costs are assumed to bethe same at all ports 119879119901 = 119879119901 = 100 119888119879119901 = 150 119865119901 = 119865119901 = 120and 119888119865119901 = 160 We consider four groups of repacking andunpacking costs group 1 with 1198882119879rarr119865119901 = 119888119865rarr2119879119901 = 50 group 2with 1198882119879rarr119865119901 = 119888119865rarr2119879119901 = 85 group 3with 1198882119879rarr119865119901 = 119888119865rarr2119879119901 = 200and group 4 with 1198882119879rarr119865119901 = 119888119865rarr2119879119901 = 250 We also considerfour possible packing decisions in decision 1 no container is


Table 2 Results

Total handling and packing cost (times1000USD) 1198882119879rarr119865119901 = 119888119865rarr211987911990150 USD 85 USD 200 USD 250 USD

No repacking 1900 1900 1900 1900Containers from Shanghai to Colombo repacked 1560 1630 1860 1960Containers from Shanghai to Colombo and from Xiamen to Singapore repacked 1360 1500 1960 2160All containers repacked 1330 1505 2080 2330

Ship route 1 5000-TEU ships Ship route 2 5000-TEU ships Ship route 3 5000-TEU ships

DemandHong Kong to Singapore 1000 TEUs Xiamen to Singapore 2000 TEUs Shanghai to Colombo 2000 TEUs

Hong Kong Singapore

Xiamen

Shanghai

Colombo

Chennai

Ho Chi Minh

Port KlangManila

Cochin

Figure 4 Another illustrative liner shipping network

repacked and unpacked in decision 2 only containers fromShanghai toColombo are repacked and unpacked in decision3 containers from Shanghai to Colombo and containers fromXiamen to Singapore are repacked and unpacked and indecision 4 all containers are repacked and unpacked

The results are shown in Table 2 and the minimumtotal cost in each repacking and unpacking cost group ishighlighted in bold We can see that when the repacking andunpacking costs are low (ie 50 USD) all TEUs should berepacked and unpacked When the repacking and unpackingcosts are higher for example when they are 200 USD onlyTEUs from Shanghai to Colombo should be repacked andunpacked because they are transshipped twice and can takethe largest advantage of handling FEUs instead of TEUsWhen the repacking and unpacking costs are extremely high(ie 250 USD) it is no longer viable to repack TEUs intoFEUs Interestingly those TEUs that are transshipped manytimes usually originate from a remote small port and aredestined for another remote small port and there are notmany shipping services available As a result the shippersmay have no choice but to agree to allow their cargos to berepacked during the trip from origin to destination Anotherimplication is that the packing costs may affect the routingof containers If a port has very low packing costs shippingcompanies may transship containers at this port for the sakeof unpacking and repacking containers and this will bringbusiness to the port

5 Conclusions

This paper has proposed a novel container routing withrepacking problem in liner shipping When routing contain-ers TEUs can be repacked into FEUs to reduce the handling

cost as the handling cost of an FEU is much lower than twicethe handling cost of a TEU To the best of our knowledgethis is a new research topic that has not been dealt with in theliteratureWedeveloped amixed-integer linear programmingmodel that formulates the routing decisions and the repack-ing decisions in a holistic manner The model could helpcontainer shipping lines to transport containers more effi-ciently An illustrative example was reported to demonstratethe applicability of themodel Results show that the benefit ofrepacking two TEUs into an FEU is themost significant whencontainers are transshipped several times

Container routing not only is significant to liner shippingcompanies as an independent problem but also serves as asubproblem in a number of tactical-level decision problemssuch as network alteration and fleet deployment How toaddress tactical-level decision problems while consideringcontainer repacking is an interesting future research direc-tion

Competing Interests

The authors declare that they have no competing interests

References

[1] UNCTAD Review of Maritime Transportation Paper pre-sented at theUnitedNationsConference onTrade andDevelop-ment New York and Geneva 2014 httpunctadorgenPubli-cationsLibraryrmt2014 enpdf

[2] M Christiansen K Fagerholt B Nygreen and D RonenldquoShip routing and scheduling in the newmillenniumrdquo EuropeanJournal of Operational Research vol 228 no 3 pp 467ndash4832013

[3] Q Meng S Wang H Andersson and KThun ldquoContainershiprouting and scheduling in liner shipping overview and futureresearch directionsrdquo Transportation Science vol 48 no 2 pp265ndash280 2014

[4] Q Meng and T Wang ldquoA scenario-based dynamic program-ming model for multi-period liner ship fleet planningrdquo Trans-portation Research Part E Logistics and Transportation Reviewvol 47 no 4 pp 401ndash413 2011

[5] J Zheng Z Gao D Yang and Z Sun ldquoNetwork design andcapacity exchange for liner alliances with fixed and variablecontainer demandsrdquo Transportation Science vol 49 no 4 pp886ndash899 2015

[6] K Fagerholt ldquoOptimal fleet design in a ship routing problemrdquoInternational Transactions in Operational Research vol 6 no 5pp 453ndash464 1999


[7] K Fagerholt ldquoDesigning optimal routes in a liner shippingproblemrdquo Maritime Policy and Management vol 31 no 4 pp259ndash268 2004

[8] T Wang Q Meng S Wang and Z Tan ldquoRisk managementin liner ship fleet deployment a joint chance constrainedprogramming modelrdquo Transportation Research Part E Logisticsand Transportation Review vol 60 pp 1ndash12 2013

[9] T E Notteboom and B Vernimmen ldquoThe effect of high fuelcosts on liner service configuration in container shippingrdquoJournal of Transport Geography vol 17 no 5 pp 325ndash337 2009

[10] X Qi andD-P Song ldquoMinimizing fuel emissions by optimizingvessel schedules in liner shipping with uncertain port timesrdquoTransportation Research Part E Logistics and TransportationReview vol 48 no 4 pp 863ndash880 2012

[11] S Wang ldquoEssential elements in tactical planning modelsfor container liner shippingrdquo Transportation Research Part BMethodological vol 54 pp 84ndash99 2013

[12] M M Golias M Boile and S Theofanis ldquoDiscrete berth-scheduling problem toward a unified mathematical formula-tionrdquo Transportation Research Record vol 2168 pp 1ndash8 2010

[13] M M Golias M Boile and S Theofanis ldquoA lamda-optimalbased heuristic for the berth scheduling problemrdquo Transporta-tion Research Part C Emerging Technologies vol 18 no 5 pp794ndash806 2010

[14] M M Golias M Boile S Theofanis and C EfstathiouldquoTheberth-scheduling problemmaximizing berth productivityand minimizing fuel consumption and emissions productionrdquoTransportation Research Record vol 2166 pp 20ndash27 2010

[15] G K D Saharidis M M Golias M Boile S Theofanisand M G Ierapetritou ldquoThe berth scheduling problem withcustomer differentiation a newmethodological approach basedon hierarchical optimizationrdquo International Journal of AdvancedManufacturing Technology vol 46 no 1ndash4 pp 377ndash393 2010

[16] M M Golias ldquoA bi-objective berth allocation formulationto account for vessel handling time uncertaintyrdquo MaritimeEconomics amp Logistics vol 13 no 4 pp 419ndash441 2011

[17] J Karafa M M Golias S Ivey G K D Saharidis and NLeonardos ldquoThe berth allocation problemwith stochastic vesselhandling timesrdquo International Journal of AdvancedManufactur-ing Technology vol 65 no 1ndash4 pp 473ndash484 2013

[18] L Zhen ldquoTactical berth allocation under uncertaintyrdquoEuropeanJournal of Operational Research vol 247 no 3 pp 928ndash9442015

[19] L Zhen S Wang and K Wang ldquoTerminal allocation problemin a transshipment hub considering bunker consumptionrdquoNaval Research Logistics vol 63 no 7 pp 529ndash548 2016

[20] J G Jin D-H Lee and J X Cao ldquoStorage yard management inmaritime container terminalsrdquo Transportation Science vol 50no 4 pp 1300ndash1313 2014

[21] L Zhen ldquoContainer yard template planning under uncertainmaritimemarketrdquo Transportation Research Part E Logistics andTransportation Review vol 69 pp 199ndash217 2014

[22] L Zhen ldquoModeling of yard congestion and optimization of yardtemplate in container portsrdquo Transportation Research Part BMethodological vol 90 pp 83ndash104 2016

[23] J G Jin D-H Lee and H Hu ldquoTactical berth and yardtemplate design at container transshipment terminals a columngeneration based approachrdquo Transportation Research Part ELogistics and Transportation Review vol 73 pp 168ndash184 2015

[24] L Zhen Z Xu K Wang and Y Ding ldquoMulti-period yard tem-plate planning in container terminalsrdquo Transportation ResearchPart B Methodological vol 93 pp 700ndash719 2016

[25] Y Du Q Chen X Quan L Long and R Y K Fung ldquoBerthallocation considering fuel consumption and vessel emissionsrdquoTransportation Research Part E Logistics and TransportationReview vol 47 no 6 pp 1021ndash1037 2011

[26] M Golias I Portal D Konur E Kaisar and G KolomvosldquoRobust berth scheduling at marine container terminals viahierarchical optimizationrdquoComputers and Operations Researchvol 41 pp 412ndash422 2014

[27] L Zhen T Shen S Wang and S Yu ldquoModels on shipscheduling in transshipment hubs with considering bunkercostrdquo International Journal of Production Economics vol 173 pp111ndash121 2016

[28] Z Liu S Wang W Chen and Y Zheng ldquoWillingness toboard a novel concept for modeling queuing up passengersrdquoTransportation Research Part B Methodological vol 90 pp 70ndash82 2016

[29] D Z W Wang and B Du ldquoReliability-based modeling ofpark-and-ride service on linear travel corridorrdquo TransportationResearch Record vol 2333 pp 16ndash26 2013

[30] B Du and D Z W Wang ldquoContinuum modeling of park-and-ride services considering travel time reliability and het-erogeneous commutersmdasha linear complementarity systemapproachrdquo Transportation Research Part E Logistics and Trans-portation Review vol 71 pp 58ndash81 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Table 1 Laden container handling cost (USDcontainer) at three ports (source [11])

Port Type Loading Discharge Transshipment

A (a major transshipment port in Europe)D20 248 324 183D40 256 332 198Ratio 103 102 108

B (a major transshipment port in Southeast Asia)D20 118 118 145D40 148 148 145Ratio 125 125 1

C (an export-driven port in Southeast Asia)D20 110 110 71D40 156 156 106Ratio 142 142 149

Colombo(CB)

Hong Kong(HK)

Singapore(SG)

Jakarta(JK)

Xiamen(XM)

Chennai(CN)

Cochin(CC)

Ship route 3CB(1) rarr CN(2) rarr CC(3) rarr CB

Ship route 2HK(1) rarr XM(2) rarr SG(3) rarr CB(4) rarr SG(5) rarrHK

Ship route 1HK(1) rarr JK(2) rarr SG(3) rarr HK

Figure 1 An illustrative liner shipping network [8]

Container routing determines the container handlingcost Table 1 shows the handling costs for two types of ladencontainers at three ports D20 means dry 20 ft container andD40 is dry 40 ft container In terms of cargo capacity a D40 isequivalent to two D20s However Table 1 clearly indicates inthe three rows ldquoRatiordquo that the ratio of the cost of handlinga D40 and that of handling a D20 is strictly less than 2In fact all the ratios in Table 1 are less than 15 and someratios are even 1 or very close to 1 This is because both thehandling of a D20 and the handling of a D40 involve onequay crane move (we note that nowadays some quay cranescan handle one D40 or two D20s in each move) Thereforeto reduce container handling costs a shipping line should tryto transport more D40s instead of D20s as a D40 can hold asmuch cargos as two D20s

11 Container Repacking As the handling cost of a D40 ismuch lower than that of two D20s it might be advantageous

to unpack two D20s and repack them to one D40 Inthe sequel we use ldquoTEUrdquo and ldquoD20rdquo interchangeably anduse ldquoforty-foot equivalent unit (FEU)rdquo and ldquoD40rdquo inter-changeably The load transshipment and discharge cost(USDcontainer) of a TEU at a port 119901 is denoted by 119879119901 119888119879119901 and 119879119901 respectively The load transshipment and dischargecost (USDcontainer) of an FEU at port 119901 isin P is denoted by119865119901 119888119865119901 and 119865119901 respectivelyWe further let 1198882119879rarr119865119901 be the cost ofrepacking two TEUs into one FEU and 119888119865rarr2119879119901 be the cost ofunpacking one FEU to two TEUs (Since multiple rehandlingof containers would increase the risk for damage and there-fore may increase insurance costs we can include in 1198882119879rarr119865119901and 119888119865rarr2119879119901 the extra insurance costs Moreover repackingrequires consent from shippers and we can include in therehandling cost the component of discount for shippers whoagree for their cargos to be repacked)

Figure 2 shows an example of transporting two TEUsfrom port 1199012 to port 1199015 The two TEUs need to be trans-shipped twice If they are transported as two TEUs as shownin Figure 2(a) then at the port of origin that is 1199012 two TEUsare loaded at 1199013 two TEUs are transshipped at 1199014 two TEUsare transshipped and at the destination port 1199015 two TEUsare dischargedTherefore the total container handling cost is

21198791199012 + 21198881198791199013 + 21198881198791199014 + 21198791199015 (1)

If these two TEUs are repacked into an FEU at the originport and unpacked at the destination port as shown inFigure 2(b) then at the port of origin that is 1199012 one FEUis loaded at 1199013 one FEU is transshipped at 1199014 one FEUis transshipped and at the destination port 1199015 one FEU isdischarged Moreover container repacking and unpackingcosts are incurred at 1199012 and 1199015 respectively (The FEU isunpacked at 1199015 into two TEUs because the two TEUs fromport1199012 to port1199015 actually have different inland destinations)Therefore the total container handling and packing cost is

1198882119879rarr1198651199012 + 1198651199012 + 1198881198651199013 + 1198881198651199014 + 1198651199015 + 119888119865rarr21198791199015 (2)

which may be considerably smaller than (1)



TEU

TEU

TEU

TEU

2 TEUs load

2 TEUstransship

2 TEUstransship

2 TEUsdischarge

TEU

TEU

TEU

TEU

TEU

TEUTEU

TEU

p1

p2 p3 p4 p5

p6



TEU

TEU

FEU FEUFEU FEU

FEU FEUTEU

TEU


1 TEUtransship

1 TEUtransship

1 TEUdischarge


cFp5

p1

p2 p3 p4 p5

p6





1198791199011 + 1198791199012 + 21198881198791199013 + 21198881198791199014 + 1198791199015 + 1198791199016 (3)



1198791199011 + 1198791199012 + 1198881198791199013 + 051198881198651199013 + 1198882119879rarr1198651199013⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199013

+ 1198881198791199014 + 051198881198651199014 + 119888119865rarr21198791199014⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199014

+ 1198791199015 + 1198791199016 (4)




TEU

TEU

TEU

TEU

1 TEUload

2 TEUstransship

2 TEUstransship

1 TEUdischarge

TEU

TEU

TEU

TEU

TEU

TEUTEU

TEU

1 TEUload

1 TEUdischarge


p1

p2 p3 p4 p5

p6



FEU FEU



TEU

TEU

TEU

TEU

TEU

TEU

TEU

TEU



p1

p2 p3 p4 p5

p6



FEU FEU

TEU TEU

TEU

TEUFEU FEU TEU

TEU

FEU FEU

p1

p2 p3 p4 p5

p6








(21198881198791199012 + 1198881198651199012) 2 + 11986511990122 = 11988811987911990122 + 11988811986511990124 + 11986511990122 (5)


1198791199011 + 11988811987911990122 + 11988811986511990124 + 11986511990122 + 1198882119879rarr1198651199012⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199012

+ 1198881198651199013 + 1198881198651199014

+ 11988811987911990152 + 11988811986511990154 + 11986511990152 + 119888119865rarr21198791199015⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199015

+ 1198791199016 (6)



















119910119901119902119879119900119889



1198991198794119900119889 = 119899119879119900119889 minus sum119901isinP

sum119902isinP

119910119901119902119879119900119889

forall (119900 119889) isin W (9)

1198991198794119900119889 ge 0 forall (119900 119889) isin W (10)


119862 (1198991198794119900119889) = sum(119900119889)isinW

(119879119900 + 119879119889) 1198991198794119900119889 (11)



1198991198791119900119901 = sum119889isinP

sum119902isinP

119910119901119902119879119900119889



119862 (1198991198791119900119889) = sum119900isinP

1198651199002 1198991198791119900119900

+ sum119900isinP


(119879119900 + 1198881198791199012 + 1198881198651199014 ) 1198991198791119900119901

+ sum119900isinP

sum119901isinP

1198882119879rarr119865119901 2 1198991198791119900119901

(13)


1198991198793119902119889 = sum119900isinP

sum119901isinP

119910119901119902119879119900119889




119862 (1198991198793119900119889) = sum119889isinP

1198651198892 1198991198793119889119889

+ sum119889isinP


(119879119889 + 1198881198791199022 + 1198881198651199024 ) 1198991198793119902119889

+ sum119902isinP

sum119889isinP

119888119865rarr2119879119902 2 1198991198793119902119889

(15)




(16)



119862 (119899119865119900119889) = sum(119900119889)isinW

(119865119900 + 119865119889) 119899119865119900119889 (17)



119879119900119889 = 0 119900 = 1198891198991198791119900119889 + 1198991198793119900119889 + 1198991198794119900119889 forall (119900 119889) isin W 119900 = 119889

119865119900119889 = 0 119900 = 119889119899119865119900119889 + 1198991198652119900119889 forall (119900 119889) isin W 119900 = 119889

(18)


n fl (1198991198791119900119889 1198991198793119900119889 1198991198794119900119889 119879119900119889 1198991198652119900119889 119865119900119889 forall (119900 119889)isin W 119910119901119902119879







minnisin119873x

sum119901isinP

(119888119879119901119911119879119901 + 119888119865119901119911119865119901) + 119862 (1198991198791119900119889) + 119862 (1198991198793119900119889)+ 119862 (1198991198794119900119889) + 119862 (119899119865119900119889)

(21)

subject to


(22)

119911119879119901 = sum119903isinR

sum119894isinI119903 119901119903119894=119901

sum(119900119889)isinW

119900119889119879119903119894 minus sum119889isinP

119879119901119889 forall119901 isin P (23)

sum119903isinR

sum119894isinI119903 119901119903119894=119901

(119900119889119879119903119894 minus 119900119889119879119903119894 )

=


0 otherwise

(24)


(25)

119911119865119901 = sum119903isinR

sum119894isinI119903 119901119903119894=119901

sum(119900119889)isinW

119900119889119865119903119894 minus sum119889isinP

119865119901119889 forall119901 isin P (26)

sum119903isinR

sum119894isinI119903 119901119903119894=119901

(119900119889119865119903119894 minus 119900119889119865119903119894 )

=


0 otherwise

(27)

sum(119900119889)isinW


119900119889119879119903119894 isin Z+

119900119889119879119903119894 isin Z+

119891119900119889119879119903119894 isin Z+


(29)

119900119889119865119903119894 isin Z+

119900119889119865119903119894 isin Z+

119891119900119889119865119903119894 isin Z+







Table 2 Results





Hong Kong Singapore

Xiamen

Shanghai

Colombo

Chennai

Ho Chi Minh

Port KlangManila

Cochin




5 Conclusions




Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











TEU

TEU

TEU

TEU

2 TEUs load

2 TEUstransship

2 TEUstransship

2 TEUsdischarge

TEU

TEU

TEU

TEU

TEU

TEUTEU

TEU

p1

p2 p3 p4 p5

p6



TEU

TEU

FEU FEUFEU FEU

FEU FEUTEU

TEU


1 TEUtransship

1 TEUtransship

1 TEUdischarge


cFp5

p1

p2 p3 p4 p5

p6





1198791199011 + 1198791199012 + 21198881198791199013 + 21198881198791199014 + 1198791199015 + 1198791199016 (3)



1198791199011 + 1198791199012 + 1198881198791199013 + 051198881198651199013 + 1198882119879rarr1198651199013⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199013

+ 1198881198791199014 + 051198881198651199014 + 119888119865rarr21198791199014⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199014

+ 1198791199015 + 1198791199016 (4)




TEU

TEU

TEU

TEU

1 TEUload

2 TEUstransship

2 TEUstransship

1 TEUdischarge

TEU

TEU

TEU

TEU

TEU

TEUTEU

TEU

1 TEUload

1 TEUdischarge


p1

p2 p3 p4 p5

p6



FEU FEU



TEU

TEU

TEU

TEU

TEU

TEU

TEU

TEU



p1

p2 p3 p4 p5

p6



FEU FEU

TEU TEU

TEU

TEUFEU FEU TEU

TEU

FEU FEU

p1

p2 p3 p4 p5

p6








(21198881198791199012 + 1198881198651199012) 2 + 11986511990122 = 11988811987911990122 + 11988811986511990124 + 11986511990122 (5)


1198791199011 + 11988811987911990122 + 11988811986511990124 + 11986511990122 + 1198882119879rarr1198651199012⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199012

+ 1198881198651199013 + 1198881198651199014

+ 11988811987911990152 + 11988811986511990154 + 11986511990152 + 119888119865rarr21198791199015⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199015

+ 1198791199016 (6)



















119910119901119902119879119900119889



1198991198794119900119889 = 119899119879119900119889 minus sum119901isinP

sum119902isinP

119910119901119902119879119900119889

forall (119900 119889) isin W (9)

1198991198794119900119889 ge 0 forall (119900 119889) isin W (10)


119862 (1198991198794119900119889) = sum(119900119889)isinW

(119879119900 + 119879119889) 1198991198794119900119889 (11)



1198991198791119900119901 = sum119889isinP

sum119902isinP

119910119901119902119879119900119889



119862 (1198991198791119900119889) = sum119900isinP

1198651199002 1198991198791119900119900

+ sum119900isinP


(119879119900 + 1198881198791199012 + 1198881198651199014 ) 1198991198791119900119901

+ sum119900isinP

sum119901isinP

1198882119879rarr119865119901 2 1198991198791119900119901

(13)


1198991198793119902119889 = sum119900isinP

sum119901isinP

119910119901119902119879119900119889




119862 (1198991198793119900119889) = sum119889isinP

1198651198892 1198991198793119889119889

+ sum119889isinP


(119879119889 + 1198881198791199022 + 1198881198651199024 ) 1198991198793119902119889

+ sum119902isinP

sum119889isinP

119888119865rarr2119879119902 2 1198991198793119902119889

(15)




(16)



119862 (119899119865119900119889) = sum(119900119889)isinW

(119865119900 + 119865119889) 119899119865119900119889 (17)



119879119900119889 = 0 119900 = 1198891198991198791119900119889 + 1198991198793119900119889 + 1198991198794119900119889 forall (119900 119889) isin W 119900 = 119889

119865119900119889 = 0 119900 = 119889119899119865119900119889 + 1198991198652119900119889 forall (119900 119889) isin W 119900 = 119889

(18)


n fl (1198991198791119900119889 1198991198793119900119889 1198991198794119900119889 119879119900119889 1198991198652119900119889 119865119900119889 forall (119900 119889)isin W 119910119901119902119879







minnisin119873x

sum119901isinP

(119888119879119901119911119879119901 + 119888119865119901119911119865119901) + 119862 (1198991198791119900119889) + 119862 (1198991198793119900119889)+ 119862 (1198991198794119900119889) + 119862 (119899119865119900119889)

(21)

subject to


(22)

119911119879119901 = sum119903isinR

sum119894isinI119903 119901119903119894=119901

sum(119900119889)isinW

119900119889119879119903119894 minus sum119889isinP

119879119901119889 forall119901 isin P (23)

sum119903isinR

sum119894isinI119903 119901119903119894=119901

(119900119889119879119903119894 minus 119900119889119879119903119894 )

=


0 otherwise

(24)


(25)

119911119865119901 = sum119903isinR

sum119894isinI119903 119901119903119894=119901

sum(119900119889)isinW

119900119889119865119903119894 minus sum119889isinP

119865119901119889 forall119901 isin P (26)

sum119903isinR

sum119894isinI119903 119901119903119894=119901

(119900119889119865119903119894 minus 119900119889119865119903119894 )

=


0 otherwise

(27)

sum(119900119889)isinW


119900119889119879119903119894 isin Z+

119900119889119879119903119894 isin Z+

119891119900119889119879119903119894 isin Z+


(29)

119900119889119865119903119894 isin Z+

119900119889119865119903119894 isin Z+

119891119900119889119865119903119894 isin Z+







Table 2 Results





Hong Kong Singapore

Xiamen

Shanghai

Colombo

Chennai

Ho Chi Minh

Port KlangManila

Cochin




5 Conclusions




Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










TEU

TEU

TEU

TEU

1 TEUload

2 TEUstransship

2 TEUstransship

1 TEUdischarge

TEU

TEU

TEU

TEU

TEU

TEUTEU

TEU

1 TEUload

1 TEUdischarge


p1

p2 p3 p4 p5

p6



FEU FEU



TEU

TEU

TEU

TEU

TEU

TEU

TEU

TEU



p1

p2 p3 p4 p5

p6



FEU FEU

TEU TEU

TEU

TEUFEU FEU TEU

TEU

FEU FEU

p1

p2 p3 p4 p5

p6








(21198881198791199012 + 1198881198651199012) 2 + 11986511990122 = 11988811987911990122 + 11988811986511990124 + 11986511990122 (5)


1198791199011 + 11988811987911990122 + 11988811986511990124 + 11986511990122 + 1198882119879rarr1198651199012⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199012

+ 1198881198651199013 + 1198881198651199014

+ 11988811987911990152 + 11988811986511990154 + 11986511990152 + 119888119865rarr21198791199015⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199015

+ 1198791199016 (6)



















119910119901119902119879119900119889



1198991198794119900119889 = 119899119879119900119889 minus sum119901isinP

sum119902isinP

119910119901119902119879119900119889

forall (119900 119889) isin W (9)

1198991198794119900119889 ge 0 forall (119900 119889) isin W (10)


119862 (1198991198794119900119889) = sum(119900119889)isinW

(119879119900 + 119879119889) 1198991198794119900119889 (11)



1198991198791119900119901 = sum119889isinP

sum119902isinP

119910119901119902119879119900119889



119862 (1198991198791119900119889) = sum119900isinP

1198651199002 1198991198791119900119900

+ sum119900isinP


(119879119900 + 1198881198791199012 + 1198881198651199014 ) 1198991198791119900119901

+ sum119900isinP

sum119901isinP

1198882119879rarr119865119901 2 1198991198791119900119901

(13)


1198991198793119902119889 = sum119900isinP

sum119901isinP

119910119901119902119879119900119889




119862 (1198991198793119900119889) = sum119889isinP

1198651198892 1198991198793119889119889

+ sum119889isinP


(119879119889 + 1198881198791199022 + 1198881198651199024 ) 1198991198793119902119889

+ sum119902isinP

sum119889isinP

119888119865rarr2119879119902 2 1198991198793119902119889

(15)




(16)



119862 (119899119865119900119889) = sum(119900119889)isinW

(119865119900 + 119865119889) 119899119865119900119889 (17)



119879119900119889 = 0 119900 = 1198891198991198791119900119889 + 1198991198793119900119889 + 1198991198794119900119889 forall (119900 119889) isin W 119900 = 119889

119865119900119889 = 0 119900 = 119889119899119865119900119889 + 1198991198652119900119889 forall (119900 119889) isin W 119900 = 119889

(18)


n fl (1198991198791119900119889 1198991198793119900119889 1198991198794119900119889 119879119900119889 1198991198652119900119889 119865119900119889 forall (119900 119889)isin W 119910119901119902119879







minnisin119873x

sum119901isinP

(119888119879119901119911119879119901 + 119888119865119901119911119865119901) + 119862 (1198991198791119900119889) + 119862 (1198991198793119900119889)+ 119862 (1198991198794119900119889) + 119862 (119899119865119900119889)

(21)

subject to


(22)

119911119879119901 = sum119903isinR

sum119894isinI119903 119901119903119894=119901

sum(119900119889)isinW

119900119889119879119903119894 minus sum119889isinP

119879119901119889 forall119901 isin P (23)

sum119903isinR

sum119894isinI119903 119901119903119894=119901

(119900119889119879119903119894 minus 119900119889119879119903119894 )

=


0 otherwise

(24)


(25)

119911119865119901 = sum119903isinR

sum119894isinI119903 119901119903119894=119901

sum(119900119889)isinW

119900119889119865119903119894 minus sum119889isinP

119865119901119889 forall119901 isin P (26)

sum119903isinR

sum119894isinI119903 119901119903119894=119901

(119900119889119865119903119894 minus 119900119889119865119903119894 )

=


0 otherwise

(27)

sum(119900119889)isinW


119900119889119879119903119894 isin Z+

119900119889119879119903119894 isin Z+

119891119900119889119879119903119894 isin Z+


(29)

119900119889119865119903119894 isin Z+

119900119889119865119903119894 isin Z+

119891119900119889119865119903119894 isin Z+







Table 2 Results





Hong Kong Singapore

Xiamen

Shanghai

Colombo

Chennai

Ho Chi Minh

Port KlangManila

Cochin




5 Conclusions




Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(21198881198791199012 + 1198881198651199012) 2 + 11986511990122 = 11988811987911990122 + 11988811986511990124 + 11986511990122 (5)


1198791199011 + 11988811987911990122 + 11988811986511990124 + 11986511990122 + 1198882119879rarr1198651199012⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199012

+ 1198881198651199013 + 1198881198651199014

+ 11988811987911990152 + 11988811986511990154 + 11986511990152 + 119888119865rarr21198791199015⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Cost at 1199015

+ 1198791199016 (6)



















119910119901119902119879119900119889



1198991198794119900119889 = 119899119879119900119889 minus sum119901isinP

sum119902isinP

119910119901119902119879119900119889

forall (119900 119889) isin W (9)

1198991198794119900119889 ge 0 forall (119900 119889) isin W (10)


119862 (1198991198794119900119889) = sum(119900119889)isinW

(119879119900 + 119879119889) 1198991198794119900119889 (11)



1198991198791119900119901 = sum119889isinP

sum119902isinP

119910119901119902119879119900119889



119862 (1198991198791119900119889) = sum119900isinP

1198651199002 1198991198791119900119900

+ sum119900isinP


(119879119900 + 1198881198791199012 + 1198881198651199014 ) 1198991198791119900119901

+ sum119900isinP

sum119901isinP

1198882119879rarr119865119901 2 1198991198791119900119901

(13)


1198991198793119902119889 = sum119900isinP

sum119901isinP

119910119901119902119879119900119889




119862 (1198991198793119900119889) = sum119889isinP

1198651198892 1198991198793119889119889

+ sum119889isinP


(119879119889 + 1198881198791199022 + 1198881198651199024 ) 1198991198793119902119889

+ sum119902isinP

sum119889isinP

119888119865rarr2119879119902 2 1198991198793119902119889

(15)




(16)



119862 (119899119865119900119889) = sum(119900119889)isinW

(119865119900 + 119865119889) 119899119865119900119889 (17)



119879119900119889 = 0 119900 = 1198891198991198791119900119889 + 1198991198793119900119889 + 1198991198794119900119889 forall (119900 119889) isin W 119900 = 119889

119865119900119889 = 0 119900 = 119889119899119865119900119889 + 1198991198652119900119889 forall (119900 119889) isin W 119900 = 119889

(18)


n fl (1198991198791119900119889 1198991198793119900119889 1198991198794119900119889 119879119900119889 1198991198652119900119889 119865119900119889 forall (119900 119889)isin W 119910119901119902119879







minnisin119873x

sum119901isinP

(119888119879119901119911119879119901 + 119888119865119901119911119865119901) + 119862 (1198991198791119900119889) + 119862 (1198991198793119900119889)+ 119862 (1198991198794119900119889) + 119862 (119899119865119900119889)

(21)

subject to


(22)

119911119879119901 = sum119903isinR

sum119894isinI119903 119901119903119894=119901

sum(119900119889)isinW

119900119889119879119903119894 minus sum119889isinP

119879119901119889 forall119901 isin P (23)

sum119903isinR

sum119894isinI119903 119901119903119894=119901

(119900119889119879119903119894 minus 119900119889119879119903119894 )

=


0 otherwise

(24)


(25)

119911119865119901 = sum119903isinR

sum119894isinI119903 119901119903119894=119901

sum(119900119889)isinW

119900119889119865119903119894 minus sum119889isinP

119865119901119889 forall119901 isin P (26)

sum119903isinR

sum119894isinI119903 119901119903119894=119901

(119900119889119865119903119894 minus 119900119889119865119903119894 )

=


0 otherwise

(27)

sum(119900119889)isinW


119900119889119879119903119894 isin Z+

119900119889119879119903119894 isin Z+

119891119900119889119879119903119894 isin Z+


(29)

119900119889119865119903119894 isin Z+

119900119889119865119903119894 isin Z+

119891119900119889119865119903119894 isin Z+







Table 2 Results





Hong Kong Singapore

Xiamen

Shanghai

Colombo

Chennai

Ho Chi Minh

Port KlangManila

Cochin




5 Conclusions




Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119910119901119902119879119900119889



1198991198794119900119889 = 119899119879119900119889 minus sum119901isinP

sum119902isinP

119910119901119902119879119900119889

forall (119900 119889) isin W (9)

1198991198794119900119889 ge 0 forall (119900 119889) isin W (10)


119862 (1198991198794119900119889) = sum(119900119889)isinW

(119879119900 + 119879119889) 1198991198794119900119889 (11)



1198991198791119900119901 = sum119889isinP

sum119902isinP

119910119901119902119879119900119889



119862 (1198991198791119900119889) = sum119900isinP

1198651199002 1198991198791119900119900

+ sum119900isinP


(119879119900 + 1198881198791199012 + 1198881198651199014 ) 1198991198791119900119901

+ sum119900isinP

sum119901isinP

1198882119879rarr119865119901 2 1198991198791119900119901

(13)


1198991198793119902119889 = sum119900isinP

sum119901isinP

119910119901119902119879119900119889




119862 (1198991198793119900119889) = sum119889isinP

1198651198892 1198991198793119889119889

+ sum119889isinP


(119879119889 + 1198881198791199022 + 1198881198651199024 ) 1198991198793119902119889

+ sum119902isinP

sum119889isinP

119888119865rarr2119879119902 2 1198991198793119902119889

(15)




(16)



119862 (119899119865119900119889) = sum(119900119889)isinW

(119865119900 + 119865119889) 119899119865119900119889 (17)



119879119900119889 = 0 119900 = 1198891198991198791119900119889 + 1198991198793119900119889 + 1198991198794119900119889 forall (119900 119889) isin W 119900 = 119889

119865119900119889 = 0 119900 = 119889119899119865119900119889 + 1198991198652119900119889 forall (119900 119889) isin W 119900 = 119889

(18)


n fl (1198991198791119900119889 1198991198793119900119889 1198991198794119900119889 119879119900119889 1198991198652119900119889 119865119900119889 forall (119900 119889)isin W 119910119901119902119879







minnisin119873x

sum119901isinP

(119888119879119901119911119879119901 + 119888119865119901119911119865119901) + 119862 (1198991198791119900119889) + 119862 (1198991198793119900119889)+ 119862 (1198991198794119900119889) + 119862 (119899119865119900119889)

(21)

subject to


(22)

119911119879119901 = sum119903isinR

sum119894isinI119903 119901119903119894=119901

sum(119900119889)isinW

119900119889119879119903119894 minus sum119889isinP

119879119901119889 forall119901 isin P (23)

sum119903isinR

sum119894isinI119903 119901119903119894=119901

(119900119889119879119903119894 minus 119900119889119879119903119894 )

=


0 otherwise

(24)


(25)

119911119865119901 = sum119903isinR

sum119894isinI119903 119901119903119894=119901

sum(119900119889)isinW

119900119889119865119903119894 minus sum119889isinP

119865119901119889 forall119901 isin P (26)

sum119903isinR

sum119894isinI119903 119901119903119894=119901

(119900119889119865119903119894 minus 119900119889119865119903119894 )

=


0 otherwise

(27)

sum(119900119889)isinW


119900119889119879119903119894 isin Z+

119900119889119879119903119894 isin Z+

119891119900119889119879119903119894 isin Z+


(29)

119900119889119865119903119894 isin Z+

119900119889119865119903119894 isin Z+

119891119900119889119865119903119894 isin Z+







Table 2 Results





Hong Kong Singapore

Xiamen

Shanghai

Colombo

Chennai

Ho Chi Minh

Port KlangManila

Cochin




5 Conclusions




Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














minnisin119873x

sum119901isinP

(119888119879119901119911119879119901 + 119888119865119901119911119865119901) + 119862 (1198991198791119900119889) + 119862 (1198991198793119900119889)+ 119862 (1198991198794119900119889) + 119862 (119899119865119900119889)

(21)

subject to


(22)

119911119879119901 = sum119903isinR

sum119894isinI119903 119901119903119894=119901

sum(119900119889)isinW

119900119889119879119903119894 minus sum119889isinP

119879119901119889 forall119901 isin P (23)

sum119903isinR

sum119894isinI119903 119901119903119894=119901

(119900119889119879119903119894 minus 119900119889119879119903119894 )

=


0 otherwise

(24)


(25)

119911119865119901 = sum119903isinR

sum119894isinI119903 119901119903119894=119901

sum(119900119889)isinW

119900119889119865119903119894 minus sum119889isinP

119865119901119889 forall119901 isin P (26)

sum119903isinR

sum119894isinI119903 119901119903119894=119901

(119900119889119865119903119894 minus 119900119889119865119903119894 )

=


0 otherwise

(27)

sum(119900119889)isinW


119900119889119879119903119894 isin Z+

119900119889119879119903119894 isin Z+

119891119900119889119879119903119894 isin Z+


(29)

119900119889119865119903119894 isin Z+

119900119889119865119903119894 isin Z+

119891119900119889119865119903119894 isin Z+







Table 2 Results





Hong Kong Singapore

Xiamen

Shanghai

Colombo

Chennai

Ho Chi Minh

Port KlangManila

Cochin




5 Conclusions




Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table 2 Results





Hong Kong Singapore

Xiamen

Shanghai

Colombo

Chennai

Ho Chi Minh

Port KlangManila

Cochin




5 Conclusions




Competing Interests


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









optimal container routing in liner shipping networks ...optimal container routing in liner shipping...

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