a synchronous optimization model for multiship shuttle

Research ArticleA Synchronous Optimization Model for Multiship ShuttleTanker Fleet Design and Scheduling Considering Hard TimeWindow Constraint

Zhenfeng Jiang1 Dongxu Chen 1 and Zhongzhen Yang2

1College of Transport Engineering Dalian Maritime University Dalian 116026 China2Faculty of Maritime and Transportation Ningbo University Ningbo 315211 China

Correspondence should be addressed to Dongxu Chen chendongxudlmueducn

Received 5 June 2018 Accepted 17 September 2018 Published 4 November 2018

Academic Editor Yair Wiseman

Copyright copy 2018 Zhenfeng Jiang et alThis 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

A Synchronous Optimization for Multiship Shuttle Tanker Fleet Design and Scheduling is solved in the context of development offloating production storage and offloading device (FPSO) In this paper the shuttle tanker fleet scheduling problem is consideredas a vehicle routing problem with hard time window constraints A mixed integer programming model aiming at minimizing totaltransportation cost is proposed to model this problem To solve this model we propose an exact algorithm based on the columngeneration andperformnumerical experimentsThe experiment results show that the proposedmodel and algorithm can effectivelysolve the problem

1 Introduction

The offshore crude oil collecting and distributing system isan important transitional transportation system connectingcrude oil mining platforms (including drilling platforms andsubsea oil production equipment) and land-based crude oilstorage ports (ldquoportsrdquo for short) which is usually composedof several special oil tankers (called ldquoshuttle oil tankerrdquo)These oil tankers are responsible for transporting the crude oilproduced by the mining platform to certain ports accordingto predesignated routes In recent years with the continuousincrease in the crude oil exploitation at sea the systemis undergoing a profound transformation The traditionalldquopoint-to-point axle-and-spoke transportationrdquo model isgradually shifting to the ldquomultipoint berthing cooperativetransportationrdquo model under the coordination of the floatingproduction storage and offloading device (FPSO) and theshuttle tanker fleet FPSO is a new type of offshore floatingdevice integrating the functions of crude oil primary pro-cessing storing loading and unloading After being minedfrom the mining platform crude oil will be transported tothe FPSO via submarine pipelines after being preliminarydehydrated and desalinated by FPSO crude oil will be

temporarily stored in its cargo tanks for shipment then it isnecessary for the oil extractors to arrange the shuttle tankersto berth and unload crude oil before the oil tanks of FPSO arefull and to transport crude oil to certain ports

The emergence of the FPSO not only enables the offshorecrude oil production system to store crude oil within acertain capacity but alsomakes it possible to use larger shuttletankers and berth multiply ports But at the same time itmakes the management and the use of the shuttle tankerfleet more complicated At present it is essential for crudeoil extractors to carefully consider the design of the shuttletanker fleet and the scheduling of the tankers when designingcrude oil collecting and distributing system It is necessaryto reasonably determine the type and number of tankersthat make up the fleet based on the transportation demandof each relevant FPSO and propose a reasonable berthingschedule for each tanker If the design of the shuttle tankerfleet and the ship scheduling scheme are unreasonable theoperating costs will increase and the transportation efficiencywill decline and the FPSOs cargo tanks may be causedto be full which will force the mining platform to stopproduction and lead to huge economic losses Therefore ithas become an urgent problem for crude oil extractors to

HindawiJournal of Advanced TransportationVolume 2018 Article ID 1904340 8 pageshttpsdoiorg10115520181904340

2 Journal of Advanced Transportation

design the shuttle tanker fleet scientifically and allocate thetransportation resources reasonably to minimize the totalcost of crude oil transportation

In essence the shuttle tanker fleet design and allocationproblem should be categorized as the problem of shipscheduling At present the mainstream researches related tothis issue can be roughly divided into three categories the firstcategory of researches is related to the optimization of shipscheduling problems among which Jin et al [1] establisheda dispatching model for feeder container ships to minimizetotal sailing cost of ships based on the traditional ldquoaxisand spokesrdquo transport structure and also solved the modelusing the particle swarm algorithm Jin et al [2] establisheda container ship feeder transport scheduling model withthe objective of maximizing the average loading rate toreasonably assign the ships that are used to carry out variousvoyage transportation tasks Tang et al [3] established ascheduling model for tramp ships based on changes in shipspeed to maximize operating revenue Yang [4] establishedan optimizing and dispatching model for feeder containerships to minimize the total transport costs considering theship capacity constraints and liner time constraints andsolved the model using particle swarm algorithm Tan etal [5] proposed a joint ship schedule design and sailingspeed optimization problem for a single inland shippingservice by incorporating the effects of nonidentical streamflow speed and the uncertain dam transit timeTheyproposeda biobjective programming model for the proposed problemto simultaneously minimize the total bunker consumptionand the ship round-trip time with service level constraintsand analytically derived the Pareto optimal solutions of theship sailing speeds and schedule by introducing the term ofbulker times

The second category of researches is related to the opti-mization of ship dispatching in ports among which Sun etal [6] established a two-stage bilateral matching optimizationmodel considering the match among supply demand andnumber of ships to propose a reasonable ship transportationscheduling scheme Zhang et al [7] established the optimaldispatching model to minimize total waiting time of ships onports and proposed a simulated annealing multipopulationgenetic algorithm for solving the model He [8] establisheda multiobjective ship scheduling optimization model tooptimize the density of ships entering and leaving ports

The third category of researches is related to fleet androute design and freight shipment among which Jonesand Zydiak [9] proposed a fleet design problem whichconsidering replacement costs and maintenance costs whichis the first formal models for determining optimal steady-state fleet designs and gave a well-defined and consistentdefinition of optimality for steady-state fleet designs Theycharacterized optimal steady-state fleet designs by showingthat all replacement groups must be equally sized Fagerholt[10] studied the problem of deciding an optimal feet basedon a three-phase solution method Taylor et al [11] discussedthe design and development of regional truckload deliveryfleets in support of random over-the-road delivery networksof continental scale and they proposed a detailed set ofsimulationmodels to evaluate various possible configurations

of regional driving fleets Zeng and Yang [12] proposed anintegrated optimization model to improve the efficiency ofcoal shipping and designed an algorithm based on two-phase tabu search to solve the mode Wiseman and Giat[13] considered load time by assuming that each leg in aroute requires a day for loading and unloading freight whilestudying freight shipment

It can be found that most of the existing related researchesassume that the fleet is homogenous (namely the fleet con-sists of the same type of ships) and few researches considerthe impact of transport time window constraints on the fleetdesign However in the problem proposed in this paperfirst the shuttle tanker fleet has a significant characteristicof nonhomogeneity and the composition of the fleet has adirect and close relationship with the plan-making of eachtanker transportation plan second the crude oil departuretime of the FPSO has a hard time window constraint and thisconstraint will influence the design of the fleet through theintrinsic link between the ship transportation plan and thefleet design Therefore the problem proposed in this paperwill be more complex than the traditional ship schedulingproblem and deserves further study

Based on the researches above this paper abstracts theShuttle Tanker Fleet Design and Scheduling Problem (SDSP)into a vehicle routing problem (VRP) considering differenttypes of vehicles with hard time window constraints Besidesthis paper proposes an optimization model and solves themodel by designing algorithm based on the column gener-ation The result of the solution can provide decision basisfor crude oil extractors to design and dispatch shuttle tankerfleet

2 Model Establishment

21 Problem Description Compared with the traditionalscheduling problems the SDSP proposed in this paper hastwo key features

The first is the heterogeneity of the tankers In SDSP dueto differences in the production and storage capabilities ofeach FPSO it is obviously problematic to use the same typeof oil tankers to build a fleet The composition of the fleetshould be determined according to the demand of the FPSObesides in practice fleet of shuttle tankers has a wide sourceof tankers it is not realistic to unity the type of these tankersTherefore in this paper we assume that crude oil extractorscan select suitable types and number of tankers from thealternative fleet to build a fleet and complete multivoyagetransportation tasks

The second is the hard time window constraints As men-tioned above due to the limited capacity of FPSO cargo tanksonce FPSO cargo tank is fully loaded all mining platformswill stop production which causes huge and unnecessaryeconomic losses Therefore in this paper the shuttle tankersmust berth and unload before the FPSO is fully loaded

In this paper we assume that the observation periodthe transport demand and transport time windows of eachFPSO are known The SDSPM is proposed to optimizesynchronously the types of shuttle tanker in the fleet thenumber of shuttle tankers of each type and the sailing

Journal of Advanced Transportation 3

path of each tanker during the observation period withthe aim of minimizing operating cost of the shuttle tankerfleet and considering the FPSO transport hard time windowconstraint

22 Model Assumptions In SDSP the following assumptionsare made(1) During the observation period the basic unit of theshuttle tankers transport task is voyage and 1 voyage refers tothe process of an oil tanker starting from the port berthinga number of FPSOs to extract crude oil and finally returningto the port and unloading oil(2) During the observation period each FPSO can onlybe berthed by one shuttle tanker but one shuttle tanker canberth multiple FPSOs in succession(3)When the shuttle tanker berths one FPSO the entireoil reserve of the shuttle tanker will be unloaded(4) The operating time (including loading time andunloading time) of each shuttle tanker on the same FPSO isthe same [13]

23 Variables and Symbols Definitions

0 land-based crude oil storage port which is the starting andending point of each voyage119881minus set of nodes consisting of FPSO platforms119881 = 119881minus cup 0119878 set of shuttle tankers the elements of 119878 are denoted by

s119870 set of voyages the elements of119870 are denoted by kT the length of the observation periodM a great positive number119905119894119895119904 the sailing time of the tanker s from i to j (forall119894 119895 isin 119881)119888119904 the sailing cost of an oil tanker s per hour119911119896119904 = 1 119896 V119900119910119886119892119890 119894119904 119888119900119898119901119897119890119905119890119889 119887119910 119905119886119899119896119890119903 119904 0 o119905ℎ119890119903-119908119894119904119890119909119894119895119896 = 1 V119894119904119894119905 119894 119886119899119889 119895 119886119903119890 V119894119904119894119905119890119889 119904119906119888119888119890119904119904119894V119890119897119910119894119899 V119900119910119886119892119890119896(119894 119895 isin 119881) 0 o119905ℎ119890119903119908119894119904119890119886119894119896 the moment when voyage k berths FPSO platform i119889119894 the storage volume of oil of FPSO platform i119902119904 Capacity of oil tanker s(119897119887119894 119906119887119894) the transport time window of the FPSO i among

which119906119887119894 and 119897119887119894 are the earliest and the latest time of berthingFPSO respectively119904119894 operating time of shuttle tankers on FPSO i

24 Optimization Model According to the previous anal-ysis of SDSP the SDSPM (Shuttle tanker fleet Design andScheduling ProblemModel) proposed in this paper is shownas follows

min 119885 = sum119894isin119881

sum119895isin119881

sum119904isin119878

(119905 119894119895119904119888119904119911119896119904 sum119896isin119870

119909119894119895119896) (1)

ST sum119895isin119881minus

1199090119895119896 = 1 forall119896 isin 119870 (2)

sum119894isin119881minus

1199091198940119896 = 1 forall119896 isin 119870 (3)


119909119894119895119896 = sum119894isin119881minus

119909119895119894119896 forall119895 isin 119881 119896 isin 119870 (4)

sum119895isin119881

sum119896isin119870

119909119894119895119896 ge 1 forall119894 isin 119881 (5)

sum119904isin119878

119911119896119904 = 1 forall119896 isin 119870 (6)

sum119894isin119881

(119889119894sum119895isin119881

119909119894119895119896) le sum119904isin119878

119902119904119911119896119904 forall119896 isin 119870 (7)

119886119894119896 + 119904119894 +sum119904isin119878

119905119894119895119904119911119896119904 minus 119886119895119896 le 119872(1 minus 119909119894119895119896)forall119894 119895 isin 119881 119896 isin 119870 (8)

119886119894119896 + 119904119894 + 1199051198940119904 times 119911119896119904 minus 11988601198961015840le 119872(3 minus 1199091198940119896 minus 119911119896119904 minus 1199111198961015840119904)

forall119894 isin 119881 119896 lt 1198961015840 isin 119881 119904 isin 119878(9)

119886119894119896 + 119904119894 +sum119904isin119878

(1199051198940119904 times 119911119896119904) le 119879 forall119894 isin 119881 119896 isin 119870 (10)

119897119887119894 le 119886119894119896 le 119906119887119894 forall119894 isin 119881 119896 isin 119870 (11)

119909119894119895119896 isin 0 1 forall119894 119895 isin 119881 119896 isin 119870 (12)

119911119896119904 isin 0 1 forall119904 isin 119878 119896 isin 119870 (13)

119886119894119896 ge 0 forall119894 isin 119881 119896 isin 119870 (14)

In SDSPM (1) is an objective function aiming at min-imizing the total operating costs of the shuttle tanker fleetEquation (2) to (14) are constraints among which (2) and(3) represent that each tanker must start from the port andreturn to the port during the observation period (4) is flowconstraint that guarantees that the flows are conservative (5)represents that each FPSO should be berthed at least once inthe observation period (6) represents that each voyage needsone tanker to complete the transport task (7) represents thatthe total amount of crude oil loaded on each voyage cannotexceed capacity of the tanker (8) represents that if an oiltanker berths two FPSOs consecutively in the same voyagethe time of berthing must be continuous (9) represents thetime constraint of berthing each FPSO when the same tankercompletes different voyage transport tasks (10) representsthat the tanker must return to the port within the observationperiod (11) represents that when the tanker berths eachFPSO hard time window constrain should be met (12) and(13) are 0-1 variable constraints (14) represents that the timeof berthing should be larger than or equal to 0

3 Algorithm Design

31 Basic Concepts and Processes The SDSPM above is alarge-scale nonlinear quadratic mixed integer programmingmodel which is proposed based on the classic VRP modelconsidering vehicles of multiple types and hard time windowconstraints However in SDSPM we not only consider ldquothe


Table 1 Basic process of SDSA

Step 1Initialize create a feasible shuttle tank sailing path p0during one observation period and create a set119875 = 1199010

Step 2Based on the 119875 obtained in Step 1 solve the mainmodel of the column generation algorithm and obtainthe shadow price of each constraint in the main model

Step 3

Based on the shadow price obtained in Step 2 solve thesub-model of the column generation algorithm obtaina new feasible shuttle tank sailing path pi and calculatethe objective function value of the sub model If theobjective function value is negative then add pi to Pand return to Step 1 if the objective function value ispositive then the solution of the main model is theoptimal solution and the algorithm terminates

path optimization of each vehiclerdquo (ie the sailing path ofeach shuttle tanker) but also consider the design of the fleet

At present the following three methods are commonlyused to solve this kind of models the first is the traditionalbranch-and-bound method This method has low efficiencyand is incapable of solving many complex problems Thesecond is the heuristic algorithm based on evolutionaryframework The computational efficiency of this kind ofalgorithm is relatively high but it cannot guarantee the globaloptimal solution The robustness of this kind of algorithmis poor and the calculating result is unstable The third isthe column generation algorithm which has been widelyconcerned in recent years The earliest column generationalgorithm is proposed by Ford and Fulkerson [14] in the1950s and the column generation algorithm has achievedgreat development in the last 10 years Because the columngeneration algorithm has high computational efficiency andin theory can solve complicated models exactly it is an idealtool for solving large-scale nonlinear quadratic mixed integerprogramming Therefore this paper attempts to solve theSDSPM using the column generation algorithm

The key to using of column generation algorithms is therational definition of ldquocolumnsrdquo The term ldquocolumnrdquo in thispaper refers to a feasible sailing path for a shuttle tankerduring the observation period which may include multiplevoyages All voyages must berth the relevant FPSO underthe conditions of meeting the time window constraints butthe total amount of crude oil carried by each voyage cannotexceed the capacity of the tanker The process of generatingldquocolumnsrdquo refers to the process of creating a feasible sailingpath for an oil tanker during the observation period In thecalculation process the algorithm will create several suchfeasible ldquocolumnsrdquo in turn and they will gradually constructa limited-scale solution space containing the optimal fleetscheduling scheme (including the fleet design scheme andthe tanker sailing path) so as to compact scale of problemand solve the model exactly In order to successfully applythe column generation algorithm we also need to createtwo key models the main model (MM) and the submodel(SM) The basic concepts and functions of the above twomodels can refer to Agarwal and Ergun [15] no more detailshere

In summary this paper proposes a Shuttle Tanker FleetDesign and Scheduling Algorithm (SDSA) based on the ideaof column generation to solve SDSPM The basic calculationprocess is shown in Table 1 Next we will introduce the coresteps of the algorithm in detail namely the main model andthe submodel

32 MainModel and Submodel In the main model the set offeasible sailing paths for shuttle tanker is denoted as 1198751015840 andits elements (ie the feasible sailing path for a single shuttletanker ldquosingle tanker feasible pathrdquo in short) are denoted aspThe decision variable of themainmodel is 119909119901 if the schemep is used 119909119901 = 1 otherwise 119909119901 = 0 For the main model thispaper introduces the following variables and considers themas known The rest of the symbols are the same as before119889119901119904 = 1 scheme p uses tanker s 0 otherwise119889119901119904 = 1 scheme p requires tankers to berth FPSOplatform i 0 otherwise119905119901 total sailing time for scheme p

The mathematical equation of the main model (SDSA-MM) is shown in (15)-(18)

min 1198851 = sum119901isin1198751015840

sum119904isin119878

119888119904119889119901119904119905119901119909119901 (15)

ST sum119901isin1198751015840

119886119894119901119909119901 ge 1 forall119894 isin 119881 (16)

sum119901isin1198751015840

119889119901119904119909119901 le 1 forall119904 isin 119878 (17)

119909119901 isin 0 1 forall119901 isin 1198751015840 (18)

Equation (15) is the objective function the aim of whichis the same as that of (1) aiming at minimizing the totaloperating costs of the shuttle tanker fleet Equations (16)-(18)are constraints among which (16) is used to ensure that everyFPSO is will be berthed at least once during the observationperiod (17) is used to ensure that each shuttle tanker can onlybe allocated to a single feasible path SDSA-MM is a linearinteger programming model that can be directly solved usingCPLEX


The function of the submodel is generating a newldquocolumnrdquo based on the gradient information given by themain model and at the same time determining whether theSDSA meets the termination condition that is whether theobjective function value of the submodel is positive or notIf the termination condition is not met the newly generatedcolumns are added to set 1198751015840 and the main model is solvedagain For the submodel in this paper the following variablesare introduced and considered as known and the rest of thesymbols are the same as before119911119904 = 1 tanker s carry out transport plan 0 119900119905ℎ11989011990311990811989411990411989011986311987516119894 the shadow price of (16)11986311987517119904 the shadow price of (17)

The mathematical equation of submodel (SDSA-SM) isshown in (19)-(25)

[SDSA-SM]

min 1198852= sum119894119895isin119881

119905119894119895119904119888119904119911119904sum119896isin119870

119909119894119895119896 minus 11986311987516119894 sum119895isin119881119896isin119870

119909119894119895119896minus 11986311987517119904 119911119904

(19)

ST sum119904

119911119904 = 1 (20)

sum119894isin119881

(119889119894sum119894isin119881

119909119894119895119896) le sum119904isin119878

119902119904119911119904 forall119896 isin 119870 (21)

119886119894119896 + 119904119894 +sum119904isin119878


119886119894119896 + 119904119894 +sum119904isin119878

1199051198940119904119911119904 minus 11988601198961015840 le 119872(1 minus 1199091198940119896)forall119894 isin 119881 119896 lt 1198961015840 isin 119870 (23)

119886119894119896 + 119904119894 +sum119904isin119878

1199051198940119904119911119904 le 119879 forall119894 isin 119881 119896 lt 1198961015840 isin 119870 (24)

119911119904 isin 0 1 forall119904 isin 119878 (25)

The constraints of SDSA-SMare similar to that of SDSPMonly deleting or rewriting the content related to multipleshuttle tankers Specifically SDSA-SM deletes the constraintthat each FPSOmust be berthed which is (5) Equations (6)-(10) are changed into (20)-(24) and 119911119896119904 was replaced by 119911119904

Equation (19) is the testing number of feasible path ofa single ships in SDSA-MM The definition of (20)-(24) isbasically the same as that of (6)-(10) only adjusting theconstraint range from the previous fleet to the single shuttletanker in order to ensure the feasibility of the feasible pathof the single ship Equation (25) is a (0-1) variable constraintBecause SDSA-SM is the same as the path planning problemproposed byKobayashi andKubo [16] the dynamic program-ming labeling method proposed by them can be used to solvethe problem efficiently

4 Case Study

Based on the public data of a certain oil company the resultsare calculated using Cnet and CPLEX The numericalexperiment consists of two parts the first part focuses onthe effectiveness of SDSA the second part solves a practicalproblem to verify the rationality of SDSPM and SDSA

41 Needed Data Thanks to the prosperity of ship chartermarket a large amount of ship information is public There-fore we can directly search the needed data related to theoil storage capacity of FPSO cargo tank crude oil productionspeed and the tank capacity speed operating cost and rentof the shuttle tankers in the relevant websites [17] Based onthe data above this paper designs experimental cases by gen-erating the position information of each FPSO randomlyThesailing distance between the FPSOs is generated randomlybetween the minimum value which is Euclidean distanceand the maximum value which is the maximal distancemeets the triangular inequality constrain This is because seasailing paths cannot be not entirely straight sometimes it isnecessary to consider the channels ocean currents and otherpractical conditions Additionally we set the length of theobservation period as 30 days

42 Analysis of Algorithm Efficiency In this paper two meth-ods are used to solve SDSPM First CPLEX is used to solvethe problem directly [18] CPLEX is working based on branchand cut algorithm when computing While calculating inorder to help CPLEX solve SDSPMmore efficiently the greatpositive number M should be set as a tight value To get atight M we substitute different value of each 0-1 variables1199091198940119896 119911119896119904 1199111198961015840119904 119909119894119895119896 into (8) and (9) and discuss the possiblevalue of M that can ensure both (8) and (9) are valid Finallythe tight M is set to the length of the observation period inthis paper which is 30 second SDSA is proposed to solve theproblem In order to clearly explain the efficiency of SDSAwecollect data of each calculation process including the timesof integer branches time of solving the main model usingCPLEX the cycles of generating columns the time of solvingthe submodel using the dynamic programming (namely thepricing problem) and so on The results of comparing the 2methods mentioned above are shown in Table 2

According to Table 2 it can be found that comparedwith the traditional branch-and-bound algorithm SDSAcan effectively solve the hybrid integer programming modelSDSPM proposed for solving the shuttle tanker schedulingproblemAdditionally in the process of calculating SDSA thecalculating time increases with the increase of the problemscale especially the calculating time of solving submodelsincreases fastest with of increase of the problem scale

43 Solution Analysis In the study case this paper refers tothe data in the Bohai Sea area of one offshore oil companyand randomly generates the positions of the FPSOs and theexisting volume of oil in the Bohai Sea area It is assumedthat the company has one port and seven FPSOs in the BohaiSea area The distribution of ports and FPSOs are shown inFigure 1


Table 2 Information of algorithm substep efficiency

Number of FPSOCPLEX SDSA

Number of branches Average time Cycles MM SM Total of columnstime time

15 4 053s 3 006s 002s 2220 8 481s 9 008s 048s 6825 13 752s 26 009s 116s 11330 21 923s 48 011s 164s 26435 26 1880s 74 012s 219s 68440 32 3878s 92 014s 283s 107950 47 32042s 146 019s 478s 1233100 98 gt1h 300 034s 126s 2563

Table 3 Basic operating parameters of shuttle tankers

Types of Shuttle tankers Sailing speed(knot) Rated capacity(wm3) Sailing cost(USDNM)A 17 2 280B 16 5 374C 15 7 444D 14 9 483

FPSOLand-based port

Figure 1 The distribution of ports and FPSOs in the research area

This paper assumes that there are four types of alternativeshuttle tankers (including A B C and D) and the number ofeach type of shuttle tankers is sufficient tomeet the demand offorming the fleet The basic operating parameters of all typesof shuttle tankers are shown in Table 3

In terms of experimental design to demonstrate thevalidity of the model this paper compares and analyzes theoperating costs of the two schemes

Scheme 1 is the traditional ldquopoint-to-point hub-and-spokerdquo shuttle tanker scheduling scheme which is the schemethat the companies currently adopt namely the companyshould allocate one shuttle tanker for each FPSO and thetanker should meet needs of the FPSO be responsible for

the oil transportation and oil storage of the FPSO Scheme2 is the ldquomultipoint berthingrdquo shuttle tanker schedulingscheme proposed in this paper namely the company isresponsible for forming the shuttle tanker fleet and proposingtransportation plan of each tanker in which each shuttletanker can berth several FPSOs The operating results of theabove two schemes are shown as follows

The fleet design and operating plan in Scheme 1 are shownin Table 4 Under this scheme suitable ship types sailing pathand the number of voyages during the observation periodare designed for each FPSO according to the transportationdemandThe calculating result shows that the total operatingcost of the fleet during the observation period is 801times105USD

The fleet design and operating plan in Scheme 2 areshown in Table 5 According to the calculating results thecompany needs to configure 3 large tankers of which 2are tankers of C-type and 1 is tanker of D-type The totaloperating costs of fleet during the observation period forScheme 2 is 556times105 USD

By comparing Scheme 1 and Scheme 2 it can be foundthat the operating cost of Scheme 2 is lower This is becausealthough the tankers used in Scheme 1 are relatively smallerthe number of tankers used is relatively larger and theidleness of the tankers in Scheme 1 is more serious duringthe observation period The sailing time for some tankers isless than 60 of the total time of the observation period Incontrast Scheme 2 adopts a small number of large tankersand the use of ships is more reasonable Therefore the idletime of ships is relatively short and the overall operatingcost is lower The calculating results above are consistentwith our expectations The results show that the SDSPMand SDSA proposed in this paper are effective in optimizingthe allocation of transportation resources and improving theservice level of shuttle tankers


Table 4 Transportation schedule of scheme 1

FPSO NO Tanker types Shuttle tanker sailing path Voyages within observation period1 B 0-1-0 12 B 0-2-0 13 B 0-3-0 14 A 0-4-0 15 A 0-5-0 16 A 0-6-0 17 B 0-7-0 1

Table 5 Transportation schedule of Scheme 2

Ship No Selected Ship Type Voyage Sailing Path

1 C1 0-1-2-02 0-1-2-03 0-1-2-0

2 C 1 0-3-4-5-02 0-3-4-5-0

3 D1 0-6-7-02 0-6-7-03 0-3-4-5-6-7-0

5 Conclusion

This paper studied the offshore shuttle tanker fleets designand scheduling problem and proposed a nonlinear quadraticmixed integer programming model To solve the model wedesigned an exact solution algorithm based on the idea ofcolumn generation algorithm In the numerical experimentfirst we analyzed the reliability of the algorithm and theresults show that the algorithm proposed in this paper issuperior to CPLEX in terms of computational efficiencyNext we solved a practical shuttle tanker fleet design andscheduling optimization problem using the model and algo-rithm proposed in this paper The results are consistent withour expectations In addition we have to admit that there isstill much room for further improvement in this paper Forexample we assume the operating time of each shuttle tankeron the same FPSO is the same and we have not considered theoperating efficiency and capacity of different tankers whichcan be one of the directions for further research

Data Availability

The [DATA TYPE] data used to support the findings of thisstudy are included within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by the Key Project of NaturalScience Foundation of China (Grant no 71431001) sponsoredby KC Wong Magna Fund in Ningbo University and

supported by Youth Project of Natural Science Foundation ofChina (Grant no 71402013)

References

[1] Z Jin Y Xie Y Li and J Hang ldquoShip scheduling optimizationproblem for container feedersrdquo China Navigation vol 31 no 4pp 415ndash419 2008

[2] Z Jin J Hu and Y Yang ldquoOptimization of Container ShippingRoute Schedulingrdquo Journal of Dalian Maritime University vol35 no 3 pp 32ndash36 2009

[3] L Tang X Xie and C Wang ldquoVariable Discontinuous ShipSchedulingModel Based on Set Partitioningrdquo Journal of Shang-hai Jiaotong University vol 47 no 6 pp 909ndash915 2013

[4] L Yang ldquoMulti-ship scheduling model for container feedertransportation based on the axis-spoke networkrdquo Chinese Jour-nal of Management Science vol 1 pp 860ndash864 2015

[5] Z Tan Y Wang Q Meng and Z Liu Joint ship schedule designand sailing speed optimization for a single inland shipping routewith uncertain dam transit time Transportation Science 2018

[6] Y Sun Z Sun G Lin and X Zhao ldquoThe coal procurementand ship transportation scheduling of power plant based onthe bilateral matching optimization modelrdquo Journal of LogisticsTechnology vol 38 no 9 pp 32ndash35 2015

[7] X Zhang J Lin Z Guo and X Chen ldquoPort Ship SchedulingOptimization Based on Simulated Annealing Multi-populationGenetic Algorithmrdquo China Navigation vol 39 no 1 pp 26ndash302016

[8] S He ldquoMulti-objective Optimization Modeling for ContainerShip Schedulingrdquo Naval Ship Science and Technology vol 2 pp22ndash24 2018

[9] P C Jones and J L Zydiak ldquoThe Fleet Design Problemrdquo TheEngineering Economist vol 38 no 2 pp 83ndash98 1993


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

[11] G D Taylor W G Ducote and G L Whicker ldquoRegional fleetdesign in truckload truckingrdquo Transportation Research Part ELogistics Transportation Review vol 42 no 3 pp 167ndash190 2006

[12] Q Zeng and Z Yang ldquoModel integrating fleet design andship routing problems for coal shippingrdquo in ComputationalSciencemdashICCS 2007 vol 4489 of Lecture Notes in ComputerScience pp 1000ndash1003 2007

[13] Y Wiseman and Y Giat ldquoRed Sea and Mediterranean Sea landbridge via Eilatrdquo World Review of Intermodal TransportationResearch vol 5 no 4 p 353 2015

[14] L R Ford and D R Fulkerson ldquoA suggested computationfor maximal multi-commodity network flowsrdquo ManagementScience vol 5 no 1 pp 97ndash101 1958

[15] R Agarwal and O Ergun ldquoShip scheduling and network designfor cargo routing in liner shippingrdquo Transportation Science vol42 no 2 pp 175ndash196 2008

[16] K Kobayashi and M Kubo ldquoOptimization of oil tanker sched-ules by decomposition column generation and time-spacenetwork techniquesrdquo Japan Journal of Industrial and AppliedMathematics vol 27 no 1 pp 161ndash173 2010

[17] China Oilfield Services Co Ltd Business and Products 2018httpwwwcoslcomcncolcol22071indexhtml

[18] IBM IBM ILOG CPLEX optimization studio CPLEX userrsquosmanual V the optimization of loading and unloading capacityof tankers ersion 12 Release 7 2016

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design the shuttle tanker fleet scientifically and allocate thetransportation resources reasonably to minimize the totalcost of crude oil transportation

In essence the shuttle tanker fleet design and allocationproblem should be categorized as the problem of shipscheduling At present the mainstream researches related tothis issue can be roughly divided into three categories the firstcategory of researches is related to the optimization of shipscheduling problems among which Jin et al [1] establisheda dispatching model for feeder container ships to minimizetotal sailing cost of ships based on the traditional ldquoaxisand spokesrdquo transport structure and also solved the modelusing the particle swarm algorithm Jin et al [2] establisheda container ship feeder transport scheduling model withthe objective of maximizing the average loading rate toreasonably assign the ships that are used to carry out variousvoyage transportation tasks Tang et al [3] established ascheduling model for tramp ships based on changes in shipspeed to maximize operating revenue Yang [4] establishedan optimizing and dispatching model for feeder containerships to minimize the total transport costs considering theship capacity constraints and liner time constraints andsolved the model using particle swarm algorithm Tan etal [5] proposed a joint ship schedule design and sailingspeed optimization problem for a single inland shippingservice by incorporating the effects of nonidentical streamflow speed and the uncertain dam transit timeTheyproposeda biobjective programming model for the proposed problemto simultaneously minimize the total bunker consumptionand the ship round-trip time with service level constraintsand analytically derived the Pareto optimal solutions of theship sailing speeds and schedule by introducing the term ofbulker times

The second category of researches is related to the opti-mization of ship dispatching in ports among which Sun etal [6] established a two-stage bilateral matching optimizationmodel considering the match among supply demand andnumber of ships to propose a reasonable ship transportationscheduling scheme Zhang et al [7] established the optimaldispatching model to minimize total waiting time of ships onports and proposed a simulated annealing multipopulationgenetic algorithm for solving the model He [8] establisheda multiobjective ship scheduling optimization model tooptimize the density of ships entering and leaving ports

The third category of researches is related to fleet androute design and freight shipment among which Jonesand Zydiak [9] proposed a fleet design problem whichconsidering replacement costs and maintenance costs whichis the first formal models for determining optimal steady-state fleet designs and gave a well-defined and consistentdefinition of optimality for steady-state fleet designs Theycharacterized optimal steady-state fleet designs by showingthat all replacement groups must be equally sized Fagerholt[10] studied the problem of deciding an optimal feet basedon a three-phase solution method Taylor et al [11] discussedthe design and development of regional truckload deliveryfleets in support of random over-the-road delivery networksof continental scale and they proposed a detailed set ofsimulationmodels to evaluate various possible configurations

of regional driving fleets Zeng and Yang [12] proposed anintegrated optimization model to improve the efficiency ofcoal shipping and designed an algorithm based on two-phase tabu search to solve the mode Wiseman and Giat[13] considered load time by assuming that each leg in aroute requires a day for loading and unloading freight whilestudying freight shipment

It can be found that most of the existing related researchesassume that the fleet is homogenous (namely the fleet con-sists of the same type of ships) and few researches considerthe impact of transport time window constraints on the fleetdesign However in the problem proposed in this paperfirst the shuttle tanker fleet has a significant characteristicof nonhomogeneity and the composition of the fleet has adirect and close relationship with the plan-making of eachtanker transportation plan second the crude oil departuretime of the FPSO has a hard time window constraint and thisconstraint will influence the design of the fleet through theintrinsic link between the ship transportation plan and thefleet design Therefore the problem proposed in this paperwill be more complex than the traditional ship schedulingproblem and deserves further study

Based on the researches above this paper abstracts theShuttle Tanker Fleet Design and Scheduling Problem (SDSP)into a vehicle routing problem (VRP) considering differenttypes of vehicles with hard time window constraints Besidesthis paper proposes an optimization model and solves themodel by designing algorithm based on the column gener-ation The result of the solution can provide decision basisfor crude oil extractors to design and dispatch shuttle tankerfleet

2 Model Establishment

21 Problem Description Compared with the traditionalscheduling problems the SDSP proposed in this paper hastwo key features

The first is the heterogeneity of the tankers In SDSP dueto differences in the production and storage capabilities ofeach FPSO it is obviously problematic to use the same typeof oil tankers to build a fleet The composition of the fleetshould be determined according to the demand of the FPSObesides in practice fleet of shuttle tankers has a wide sourceof tankers it is not realistic to unity the type of these tankersTherefore in this paper we assume that crude oil extractorscan select suitable types and number of tankers from thealternative fleet to build a fleet and complete multivoyagetransportation tasks

The second is the hard time window constraints As men-tioned above due to the limited capacity of FPSO cargo tanksonce FPSO cargo tank is fully loaded all mining platformswill stop production which causes huge and unnecessaryeconomic losses Therefore in this paper the shuttle tankersmust berth and unload before the FPSO is fully loaded

In this paper we assume that the observation periodthe transport demand and transport time windows of eachFPSO are known The SDSPM is proposed to optimizesynchronously the types of shuttle tanker in the fleet thenumber of shuttle tankers of each type and the sailing









min 119885 = sum119894isin119881

sum119895isin119881

sum119904isin119878

(119905 119894119895119904119888119904119911119896119904 sum119896isin119870

119909119894119895119896) (1)


1199090119895119896 = 1 forall119896 isin 119870 (2)


1199091198940119896 = 1 forall119896 isin 119870 (3)


119909119894119895119896 = sum119894isin119881minus

119909119895119894119896 forall119895 isin 119881 119896 isin 119870 (4)

sum119895isin119881

sum119896isin119870

119909119894119895119896 ge 1 forall119894 isin 119881 (5)

sum119904isin119878

119911119896119904 = 1 forall119896 isin 119870 (6)

sum119894isin119881

(119889119894sum119895isin119881

119909119894119895119896) le sum119904isin119878

119902119904119911119896119904 forall119896 isin 119870 (7)

119886119894119896 + 119904119894 +sum119904isin119878




119886119894119896 + 119904119894 +sum119904isin119878







3 Algorithm Design






Step 3








min 1198851 = sum119901isin1198751015840

sum119904isin119878

119888119904119889119901119904119905119901119909119901 (15)

ST sum119901isin1198751015840

119886119894119901119909119901 ge 1 forall119894 isin 119881 (16)

sum119901isin1198751015840

119889119901119904119909119901 le 1 forall119904 isin 119878 (17)

119909119901 isin 0 1 forall119901 isin 1198751015840 (18)





[SDSA-SM]

min 1198852= sum119894119895isin119881

119905119894119895119904119888119904119911119904sum119896isin119870


119909119894119895119896minus 11986311987517119904 119911119904

(19)

ST sum119904

119911119904 = 1 (20)

sum119894isin119881

(119889119894sum119894isin119881

119909119894119895119896) le sum119904isin119878

119902119904119911119904 forall119896 isin 119870 (21)

119886119894119896 + 119904119894 +sum119904isin119878


119886119894119896 + 119904119894 +sum119904isin119878


119886119894119896 + 119904119894 +sum119904isin119878


119911119904 isin 0 1 forall119904 isin 119878 (25)



4 Case Study













FPSOLand-based port














1 C1 0-1-2-02 0-1-2-03 0-1-2-0

2 C 1 0-3-4-5-02 0-3-4-5-0

3 D1 0-6-7-02 0-6-7-03 0-3-4-5-6-7-0

5 Conclusion


Data Availability




Acknowledgments



References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










min 119885 = sum119894isin119881

sum119895isin119881

sum119904isin119878

(119905 119894119895119904119888119904119911119896119904 sum119896isin119870

119909119894119895119896) (1)


1199090119895119896 = 1 forall119896 isin 119870 (2)


1199091198940119896 = 1 forall119896 isin 119870 (3)


119909119894119895119896 = sum119894isin119881minus

119909119895119894119896 forall119895 isin 119881 119896 isin 119870 (4)

sum119895isin119881

sum119896isin119870

119909119894119895119896 ge 1 forall119894 isin 119881 (5)

sum119904isin119878

119911119896119904 = 1 forall119896 isin 119870 (6)

sum119894isin119881

(119889119894sum119895isin119881

119909119894119895119896) le sum119904isin119878

119902119904119911119896119904 forall119896 isin 119870 (7)

119886119894119896 + 119904119894 +sum119904isin119878




119886119894119896 + 119904119894 +sum119904isin119878







3 Algorithm Design






Step 3








min 1198851 = sum119901isin1198751015840

sum119904isin119878

119888119904119889119901119904119905119901119909119901 (15)

ST sum119901isin1198751015840

119886119894119901119909119901 ge 1 forall119894 isin 119881 (16)

sum119901isin1198751015840

119889119901119904119909119901 le 1 forall119904 isin 119878 (17)

119909119901 isin 0 1 forall119901 isin 1198751015840 (18)





[SDSA-SM]

min 1198852= sum119894119895isin119881

119905119894119895119904119888119904119911119904sum119896isin119870


119909119894119895119896minus 11986311987517119904 119911119904

(19)

ST sum119904

119911119904 = 1 (20)

sum119894isin119881

(119889119894sum119894isin119881

119909119894119895119896) le sum119904isin119878

119902119904119911119904 forall119896 isin 119870 (21)

119886119894119896 + 119904119894 +sum119904isin119878


119886119894119896 + 119904119894 +sum119904isin119878


119886119894119896 + 119904119894 +sum119904isin119878


119911119904 isin 0 1 forall119904 isin 119878 (25)



4 Case Study













FPSOLand-based port














1 C1 0-1-2-02 0-1-2-03 0-1-2-0

2 C 1 0-3-4-5-02 0-3-4-5-0

3 D1 0-6-7-02 0-6-7-03 0-3-4-5-6-7-0

5 Conclusion


Data Availability




Acknowledgments



References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






Step 3








min 1198851 = sum119901isin1198751015840

sum119904isin119878

119888119904119889119901119904119905119901119909119901 (15)

ST sum119901isin1198751015840

119886119894119901119909119901 ge 1 forall119894 isin 119881 (16)

sum119901isin1198751015840

119889119901119904119909119901 le 1 forall119904 isin 119878 (17)

119909119901 isin 0 1 forall119901 isin 1198751015840 (18)





[SDSA-SM]

min 1198852= sum119894119895isin119881

119905119894119895119904119888119904119911119904sum119896isin119870


119909119894119895119896minus 11986311987517119904 119911119904

(19)

ST sum119904

119911119904 = 1 (20)

sum119894isin119881

(119889119894sum119894isin119881

119909119894119895119896) le sum119904isin119878

119902119904119911119904 forall119896 isin 119870 (21)

119886119894119896 + 119904119894 +sum119904isin119878


119886119894119896 + 119904119894 +sum119904isin119878


119886119894119896 + 119904119894 +sum119904isin119878


119911119904 isin 0 1 forall119904 isin 119878 (25)



4 Case Study













FPSOLand-based port














1 C1 0-1-2-02 0-1-2-03 0-1-2-0

2 C 1 0-3-4-5-02 0-3-4-5-0

3 D1 0-6-7-02 0-6-7-03 0-3-4-5-6-7-0

5 Conclusion


Data Availability




Acknowledgments



References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





[SDSA-SM]

min 1198852= sum119894119895isin119881

119905119894119895119904119888119904119911119904sum119896isin119870


119909119894119895119896minus 11986311987517119904 119911119904

(19)

ST sum119904

119911119904 = 1 (20)

sum119894isin119881

(119889119894sum119894isin119881

119909119894119895119896) le sum119904isin119878

119902119904119911119904 forall119896 isin 119870 (21)

119886119894119896 + 119904119894 +sum119904isin119878


119886119894119896 + 119904119894 +sum119904isin119878


119886119894119896 + 119904119894 +sum119904isin119878


119911119904 isin 0 1 forall119904 isin 119878 (25)



4 Case Study













FPSOLand-based port














1 C1 0-1-2-02 0-1-2-03 0-1-2-0

2 C 1 0-3-4-5-02 0-3-4-5-0

3 D1 0-6-7-02 0-6-7-03 0-3-4-5-6-7-0

5 Conclusion


Data Availability




Acknowledgments



References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









FPSOLand-based port














1 C1 0-1-2-02 0-1-2-03 0-1-2-0

2 C 1 0-3-4-5-02 0-3-4-5-0

3 D1 0-6-7-02 0-6-7-03 0-3-4-5-6-7-0

5 Conclusion


Data Availability




Acknowledgments



References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







1 C1 0-1-2-02 0-1-2-03 0-1-2-0

2 C 1 0-3-4-5-02 0-3-4-5-0

3 D1 0-6-7-02 0-6-7-03 0-3-4-5-6-7-0

5 Conclusion


Data Availability




Acknowledgments



References






















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia














RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


a synchronous optimization model for multiship shuttle

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