A MIP based local search heuristic for a stochastic maritime inventory routing problem

Agostinho Agra

1

,MarielleChristiansen

2

, LarsMagnusHvattum

3

,andFilipeRodrigues

1

1

DepartmentofMathemati sandCIDMA,UniversityofAveiro,Portugal,

{aagra, fmgrodriguesua.pt}ua.pt

2

DepartmentofIndustrialE onomi sandTe hnologyManagement,NorwegianUniversityof

S ien eandTe hnology,Norway,

m iot.ntnu.no

3

MoldeUniversityCollege,Norway,

hvattumhimolde.no

Abstra t. We onsiderasingleprodu tmaritimeinventoryroutingprobleminwhi h

the produ tion and onsumption rates are onstant overthe planning horizon. The

probleminvolvesaheterogeneouseetofshipsandmultipleprodu tionand

onsump-tionports withlimited storage apa ity. In spite of being oneof the most ommon

ways to transport goods, maritimetransportation is hara terized by high levels of

un ertainty.Theprin ipalsour eofun ertaintyisthe weather onditions, sin ethey

havea greatinuen eonsailing times. Thetraveltime betweenany pairofportsis

assumedtoberandomand tofollow alog-logisti distribution.Todealwithrandom

sailingtimes wepropose atwo-stagesto hasti programmingproblemwithre ourse.

Therouting,theorderinwhi htheportsarevisited,aswellasthequantitiestoload

andunloadarexedbeforetheun ertaintyisrevealed,while thetimeofthevisitto

portsandtheinventorylevels anbeadjustedto thes enario.Tosolvetheproblem,

aMIPbasedlo alsear hheuristi isdeveloped.Thisnewapproa his omparedwith

ade ompositionalgorithmina omputationalstudy.

Keywords: Maritimetransportation;Sto hasti programming;Un ertainty;

Matheuris-ti .

1 Introdu tion

We onsider amaritime inventory routingproblem (MIRP)where aheterogeneouseet

ofshipsistransportingasingleprodu tbetweenports.Thereexistsonetypeofportswhere

the produ t is produ ed, and in the other ports the produ t is onsumed. The produ tion

and onsumptionrates are onstantoverthe planninghorizon.At allports, thereexists an

inventory forstoring theprodu t,and lowerandupper limitsaregiven forea hport. Ea h

port anbevisitedoneorseveraltimesduringtheplanninghorizondependingonthesizeof

thestorage,theprodu tionor onsumptionrate,andthequantityloadedorunloadedatea h

portvisit.TheMIRP onsistsofdesigningroutesands hedulesforaeetofshipsinorderto

minimizethetransportationandport osts,andtodeterminethequantitieshandledatea h

port allwithoutex eeding thestoragelimits.TheMIRP isaveryimportant and ommon

problemin maritimeshippingand isrelevantwhenthea torsinvolvedin amaritimesupply

⋆

hain havethe responsibilityfor boththetransportationof the argoesand theinventories

at the ports. The shipping industryis apital intensive, soa modest improvement in eet

utilization an imply a large in rease in prot. Therefore, the ability of ship operators to

makegood de isions is ru ial. However,the MIRPs are very omplex to solvedue to the

high degreeof freedom in the routing, s heduling, number of port visits, and the quantity

loadedorunloadedatea h portvisit.

Thereexistsasolidamountofresear handresultingpubli ationswithinMIRPs,andthese

haveformedthebasisofseveralsurveys:Papageorgiouetal.[18℄,Christiansenetal.[10℄,and

Christiansen and Fagerholt [8,9℄. In addition, Coelho et al. [12℄ and Andersson et al. [4℄

surveyedbothland-basedandmaritimeinventoryroutingproblems.Maritimetransportation

is hara terized by high levels of un ertainty, and one of the most ommon un ertainties

is the sailing times that are ae ted by heavily hanging weather onditions. In pra ti e,

unpredi table delaysmayae t the exe utionof an optimaldeterministi plan. In order to

ompensateforsu hdelays,itispossiblefortheshipstospeedupwhenne essary.However,

inpra ti eitwillmostoftenbebene ial to onsidertheun ertaintyexpli itlywhennding

theoptimalplan.

Even though maritime transportation is heavily inuen ed by un ertainty, most of the

resear h reported in the literature on maritime routingand s heduling onsider stati and

deterministi problems.However,some ontributionsexist,andwedes ribetheonesthatare

losesttotheMIRPwithsto hasti traveltimesstudiedhere.Forashiproutingand

s hedul-ing problem with predened argoes, Christiansen and Fagerholt [7℄ design ship s hedules

that are less likely to result in ships staying idle at ports during weekends by imposing

penalty ostsforarrivalsatriskytimes(i.e. losetoweekends).Theresultings heduleneeds

to be morerobustwithrespe tto delaysfrom bad weatherand timein port dueto the

re-stri tedoperatinghoursinportduringweekends.Agraetal.[3℄solvedafull-loadshiprouting

ands hedulingproblemwithun ertaintraveltimesusingrobustoptimization.Furthermore,

Halvorsen-Weare and Fagerholt [14℄ analysed various heuristi strategies to a hieve robust

weeklyvoyagesands hedules foro-shoresupplyvesselsworkingunder toughweather

on-ditions. Heuristi strategies forobtaining robustsolutions with un ertainsailing times and

produ tion ratewere alsodis ussed byHalvorsen-Weareet al.[15℄ forthe deliveryof

lique-ednaturalgas.Fora rudeoiltransportationandinventoryproblem,ChengandDuran[6℄

developedade isionsupport systemthattakesintoa ountun ertaintyin sailingtimeand

demand. Theproblem is formulated asa dis retetime Markovde ision pro ess and solved

byusing dis reteeventsimulationandoptimal ontroltheory. Rakkeet al. [19℄ andSherali

andAl-Yakoob[20,21℄introdu edpenaltyfun tionsfordeviatingfromthe ustomer ontra ts

andthestoragelimits,respe tively,fortheirMIRPs.ChristiansenandNygreen[11℄usedsoft

inventory levels to handle un ertainties in sailing time and time in port, and these levels

weretransformedintosofttimewindowsforasingleprodu tMIRP. Agraetal.[2℄werethe

rst to use sto hasti programming to model un ertain sailingand port times for a MIRP

withseveralprodu tsandinventorymanagementatthe onsumptionportsonly.Atwo-stage

sto hasti programmingmodel withre oursewasdevelopedwheretherst-stage onsistsof

routing, and loading/unloadingde isions, and the se ond stage onsists of s heduling

de i-sions.Themodel wassolvedby ade omposition approa hsimilar toanL-shapedalgorithm

whereoptimality utswereaddeddynami ally,andthesolutionpro esswasembeddedwithin

thesampleaverageapproximationmethod.

Theobje tiveof thispaperis to presentageneralsingle produ tMIRP with sto hasti

onsistsofroutingandloading/unloadingde isions,andthese ondstage onsistsof

s hedul-ing de isions. Although thetwo problems haveseveral dieren es (the number of produ ts

onsidered,managementinventoryatsupplyports,andrandomaspe tsofun ertainty),this

workwasalsomotivatedbythestabilityproblemsreportedfortheapproa hfollowedbyAgra

et al.[2℄. Whentheinstan es be ome harder,theobje tivefun tion valuesobtainedbythe

heuristi approa hhadlargelevelsofvarian e.Asinpreviousworkweassumetheinventory

limits anbeviolatedwithapenalty.Herewedis ussinmoredetailtheimpa tofthevalue

ofsu hpenaltiesonthestabilityofthesolutionpro edure,sin edierentpenaltyvaluesmay

orrespondto dierentde isionmakerstrategies,and mayinuen e thee ien yofbran h

and ut based pro edures. Low penalty values will be used when ba klogged onsumption

and ex ess ofprodu tionare lessimportant than therouting ost, and generate low

trans-portation ost solutions. High penalty values reate solutionsthat are averse to inventory

limitviolations. Sin ethefra tional solutionsobtainedby linearrelaxationswill present,in

general,noviolationoftheinventorylimits,theintegralitylineargapstendtobemu hhigher

when the penalty valuesare higher, whi h deteriorates the performan e of bran h and ut

basedpro edures.Additionally,in order to ir umventthestabilityproblems, weproposea

newheuristi pro edurewhi hisbasedonalo alsear hheuristi thatusesthesolutionfrom

a orrespondingdeterministi problemasastartingsolution.

Theremainderofthis paperis organizedasfollows:Themathemati al modelof the

de-terministi problem is presented in Se tion 2, while the sto hasti model is presented in

Se tion 3.Se tion4presentstheheuristi sto hasti solutionapproa hes.Extensive

ompu-tationalresultsarereportedanddis ussedinSe tion5,followedbysome on ludingremarks

in Se tion6.

2 Mathemati al model for the deterministi problem

Inthisse tionweintrodu eamathemati alformulationforthedeterministi problem.

Routing onstraints Let

V

denote the set of shipsand

N

denote the set of ports. Ea h ship

v

∈ V

mustdepartfromitsinitialposition,that anbeapointatsea.Forea hportwe onsideranorderingofthevisitsa ordinglytothetimeofthevisit.

Theshippathsaredenedonanetworkwherethenodesarerepresentedbyapair

(i, m)

, where

i

indi atestheportand

m

indi atesthevisitnumbertoport

i

.Dire tshipmovements (ar s)fromnode

(i, m)

tonode

(j, n)

arerepresentedby

(i, m, j, n).

Foreaseofnotation,ifa shipdeparts from apointat sea,an arti ialportis reatedandasinglevisitis asso iated

withit.

Wedene

S

A

asthesetofpossiblenodes

(i, m), S

A

v

asthesetofnodesthatmaybevisited byship

v

,andset

S

X

v

astheset ofallpossiblemovements

(i, m, j, n)

ofship

v.

Fortheroutingwedenethefollowingbinaryvariables:

x

imjnv

is

1

ifship

v

travelsfrom node

(i, m)

dire tlyto node

(j, n),

and

0

otherwise;

w

imv

is

1

if ship

v

visits node

(i, m),

and

0

otherwise;

z

imv

is equalto

1

ifship

v

ends itsroute at node

(i, m),

and

0

otherwise;

y

im

indi ateswhetherashipismakingthe

m

th

visitto port

i,

(i, m),

ornot.Theparameter

µ

i

denotesthe minimumnumberofvisits at port

i

andthe parameter

µ

boundonthenumberofvisitsatport

i

.

w

imv

−

X

(j,n)∈S

A

v

x

jnimv

= 0,

∀v ∈ V, (i, m) ∈ S

A

v

,

(1)

w

imv

−

X

(j,n)∈S

A

v

x

imjnv

− z

imv

= 0,

∀v ∈ V, (i, m) ∈ S

v

A

,

(2)

X

v∈V

w

imv

= y

im

,

∀(i, m) ∈ S

A

,

(3)

y

im

= 1,

∀(i, m) ∈ S

A

: m ∈ {1, · · · , µ

_i

},

(4)

y

i(m−1)

− y

im

≥ 0,

∀(i, m) ∈ S

A

: µ

_i

+ 1 < m ≤ µ

i

,

(5)

x

imjnv

∈ {0, 1},

∀v ∈ V, (i, m, j, n) ∈ S

v

X

,

(6)

w

imv

, z

imv

∈ {0, 1},

∀v ∈ V, (i, m) ∈ S

v

A

(7)

y

im

∈ {0, 1},

∀(i, m) ∈ S

A

.

(8)

Equations(1)and(2)aretheow onservation onstraints,ensuring thatashiparriving at

anodealso leavesthat node orends its route. Constraints(3) ensure that aship anvisit

node

(i, m)

onlyif

y

im

is equalto one.Equations(4)x

y

im

to 1forthemandatory visits. Constraints (5) state that if port

i

is visited

m

times, then it must also have been visited

m

− 1

times.Constraints(6)-(8)dene thevariablesasbinary.

Loadingandunloading onstraints Parameter

J

i

is

1

ifport

i

isaprodu er;

−1

ifport

i

is a onsumer.

C

v

isthe apa ityofship

v

.Theminimumandmaximumloadingandunloading quantitiesatport

i

aregivenby

Q

i

and

Q

i

,

respe tively.

Inordertomodeltheloadingandunloading onstraints,wedenethefollowing ontinuous

variables:

q

imv

is theamountloaded orunloadedfrom ship

v

atnode

(i, m); f

imjnv

denotes theamountthatship

v

transportsfromnode

(i, m)

tonode

(j, n).

Theloadingandunloading onstraintsaregivenby:

X

(j,n)∈S

A

v

f

jnimv

+ J

i

q

imv

=

X

(j,n)∈S

A

v

f

imjnv

,

∀v ∈ V, (i, m) ∈ S

v

A

,

(9)

f

imjnv

≤ C

v

x

imjnv

,

∀ v ∈ V, (i, m, j, n) ∈ S

v

X

,

(10)

Q

_i

w

imv

≤ q

imv

≤ min{C

v

, Q

i

}w

imv

,

∀v ∈ V, (i, m) ∈ S

A

v

,

(11)

f

imjnv

≥ 0,

∀v ∈ V, (i, m, j, n) ∈ S

X

v

,

(12)

q

imv

≥ 0,

∀v ∈ V, (i, m) ∈ S

v

A

.

(13)

Equations(9) are theow onservation onstraintsat node

(i, m)

.Constraints(10) require that the ship apa ity is obeyed. Constraints (11) impose lower and upper limits on the

loadingandunloadingquantities.Constraints(12)-(13)arethenon-negativity onstraints.

Time onstraints Wedenethefollowingparameters:

T

Q

i

isthetimerequiredtoload/unload one unit of produ t at port

i

;

T

ijv

is the travel time between port

i

and

j

by ship

v.

It in ludes also any set-up time required to operate at port

j. T

B

i

is the minimum time be-tween two onse utive visits to port

i. T

is the length of the time horizon, and

A

im

and

B

im

arethe time windows forstarting the

m

th

visit to port

i.

Toeasethe presentation we also dene, for ea h node

(i, m),

the following upper bound for the end time of the visit:

T

im

′

= min{T, B

im

+ T

i

Q

Q

i

}.

Giventimevariables

t

im

thatindi atethestarttimeofthe

m

th

visittoport

i,

thetime onstraints anbewrittenas:

t

im

+

X

v∈V

T

_i

Q

q

imv

− t

jn

+

X

v∈V |(i,m,j,n)∈S

X

v

max{T

im

′

+ T

ijv

− A

jn

,

0}x

imjnv

≤ T

im

′

− A

jn

,

∀(i, m), (j, n) ∈ S

A

,

(14)

t

im

− t

i,m−1

−

X

v∈V

T

_i

Q

q

i,m−1,v

− T

i

B

y

im

≥ 0,

∀(i, m) ∈ S

A

: m > 1,

(15)

A

im

≤ t

im

≤ B

im

,

∀(i, m) ∈ S

A

.

(16)

Constraints(14)relatethestarttimeasso iatedwithnode

(i, m)

tothestarttimeasso iated withnode

(j, n)

whenship

v

travelsdire tly from

(i, m)

to

(j, n)

. Constraints(15)imposea minimumintervalbetweentwo onse utivevisitsatport

i.

Timewindowsforthestarttime ofvisitsaregivenby(16).

Inventory onstraints Theinventory onstraintsare onsideredforea hport.Theyensure

thatthesto klevelsarewithinthe orrespondinglimitsandlinkthesto klevelstotheloading

orunloadingquantities.Forea hport

i,

the onsumption/produ tionrate,

R

i

,theminimum

S

_i

,

the maximum

S

i

and the initial

S

0

i

sto k levels, are given. We dene the nonnegative ontinuousvariables

s

im

indi atingthesto klevelsatthestartofthe

m

th

visittoport

i.

The inventory onstraintsareasfollows:

s

i1

= S

0

i

+ J

i

R

i

t

i1

,

∀i ∈ N,

(17)

s

im

= s

i,m−1

− J

i

X

v∈V

q

i,m−1,v

+ J

i

R

i

(t

im

− t

i,m−1

),

∀(i, m) ∈ S

A

: m > 1,

(18)

s

im

+

X

v∈V

q

imv

− R

i

X

v∈V

T

_i

Q

q

imv

≤ S

i

,

∀(i, m) ∈ S

A

|J

i

= −1,

(19)

s

im

−

X

v∈V

q

imv

+ R

i

X

v∈V

T

_i

Q

q

imv

≥ S

i

,

∀(i, m) ∈ S

A

|J

i

= 1,

(20)

s

iµ

i

+

X

v∈V

q

i,µ

i

,v

− R

i

(T − t

iµ

i

) ≥ S

i

,

∀i ∈ N |J

i

= −1,

(21)

s

iµ

i

−

X

v∈V

q

i,µ

i

,v

+ R

i

(T − t

iµ

i

) ≤ S

i

,

∀i ∈ N |J

i

= 1,

(22)

s

im

≥ S

i

,

∀(i, m) ∈ S

A

|J

i

= −1,

(23)

s

im

≤ S

i

,

∀(i, m) ∈ S

A

|J

i

= 1.

(24)

Equations (17) al ulate the sto k level at the start time of the rst visit to a port, and

equations (18) relate the sto k level at the start time of

m

th

visit to the sto k level at

the start time of theprevious visit. Constraints (19) and (20) ensurethat the sto k levels

arewithintheirlimitsattheendofea hvisit.Constrains(21)imposealowerboundonthe

inventorylevelattime

T

for onsumptionports,while onstrains(22)imposeanupperbound ontheinventorylevelattime

T

forprodu tionports.Constraints(23)and(24)ensurethat

Obje tivefun tion Theobje tiveistominimizethetotalrouting osts,in ludingtraveling

andoperating osts.Thetraveling ostofship

v

fromport

i

toport

j

isdenotedby

C

T

ijv

and itin ludes theset-up osts.Theobje tivefun tion isdenedasfollows:

M in

C(X) =

X

v∈V

X

(i,m,j,n)∈S

X

v

C

_ijv

T

x

imjnv

.

(25)

3 Mathemati al model for the sto hasti problem

Inthesto hasti approa h,thesailingtimesbetweenportsareassumedtobeindependent

andrandom,followingaknownprobabilitydistribution(alog-logisti probabilitydistribution

whi hisdis ussedinSe tion5).AsintheworkbyAgraetal.[2℄,themodelintrodu edhere

isare oursemodelwithtwolevelsofde isions.Therst-stagede isionsaretherouting,the

portvisitssequen e,andtheload/unloadquantities.Thesede isionsmustbetakenbeforethe

s enariois revealed. The orresponding rst-stagevariables are

x

imjnv

, z

imv

, w

imv

, y

im

, and

q

imv

.

Theadjustablevariablesarethetimeofvisitsandtheinventorylevels.Inthesto hasti approa hweallowtheinventorylimitstobeviolatedbyin ludingapenalty

P

i

forea hunit of violation of the inventory limitsat ea h port

i.

In addition to the variables

t

im

(ξ),

and

s

im

(ξ)

indi ating the time and the sto k level at node

(i, m),

when s enario

ξ

is revealed, newvariables

r

im

(ξ)

areintrodu edto denotetheinventorylimitviolationatnode

(i, m)

.If

i

isa onsumption port,

r

im

(ξ)

denotes theba klogged onsumption,that is theamountof demand satisedwith delay.If

i

isaprodu tion port,

r

im

(ξ)

denotes thedemand in ex ess tothe apa ity.Weassumethequantityinex essisnotlost butapenaltyisin urred.

Themaingoalofthesto hasti approa histondthesolutionthatminimizestherouting

ost

C(X)

plustheexpe tedpenaltyvalueforinventorydeviationtothelimits,

E

ξ

(Q(X, ξ)),

where

Q(X, ξ)

denotestheminimumpenaltyfortheinventorydeviationswhens enario

ξ

with aparti ular sailing times ve toris onsidered and aset of rststage de isions, denoted by

X

,isxed.Inordertoavoidusingthetheoreti aljointprobabilitydistributionofthetravel times, we follow the ommon Sample Average Approximation (SAA) method, and repla e

the true expe ted penalty value

E

ξ

(Q(X, ξ))

by the mean value of alarge random sample

Ω

= {ξ

1

_{, . . . , ξ}

k

_}

of

ξ,

obtainedbytheMonteCarlomethod. Thislargersetof

k

s enariosis regardedasaben hmarks enariosetrepresentingthetruedistribution[17℄.

Theobje tivefun tionoftheSAAmodelbe omesasfollows:

M in

C(X) +

1

| Ω |

X

ξ∈Ω

X

(i,m)∈S

A

P

i

r

im

(ξ).

(26)

Inadditiontotheroutingandloadingand unloading onstraints(1)-(13),theSAAproblem

hasthefollowingtimeandinventory onstraints.

Time onstraints:

t

im

(ξ) +

X

v∈V

T

_i

Q

q

imv

− t

jn

(ξ) +

X

v∈V |(i,m,j,n)∈S

X

v

T

M

x

imjnv

,

≤ T

M

,

∀(i, m), (j, n) ∈ S

A

, ξ

∈ Ω,

(27)

t

im

(ξ) − t

i,m−1

(ξ) −

X

v∈V

T

_i

Q

q

i,m−1,v

− T

i

B

y

im

≥ 0,

∀(i, m) ∈ S

A

: m > 1, ξ ∈ Ω,

(28)

A

im

≤ t

im

(ξ) ≤ B

im

M

,

∀(i, m) ∈ S

A

, ξ

∈ Ω.

(29)

The inventory onstraintsare similar to the onstraintsfor the deterministi problem, but

nowin ludingthepossibleviolationoftheinventorylimits.Thebig onstant

T

M

isnowset

to

2T

sin ethevisitstoports annowo uraftertimeperiod

T

.Similarly,

B

M

im

issetto

2T

. Theinventory onstraintsareasfollows:

s

i1

(ξ) = S

i

0

+ J

i

R

i

t

i1

(ξ) − J

i

r

i1

(ξ),

∀i ∈ N, ξ ∈ Ω,

(30)

s

im

(ξ) − J

i

r

i,m−1

(ξ) = s

i,m−1

(ξ) − J

i

r

im

(ξ) − J

i

X

v∈V

q

i,m−1,v

+ J

i

R

i

(t

im

(ξ) − t

i,m−1

(ξ)),

∀(i, m) ∈ S

A

_{: m > 1, ξ ∈ Ω,}

(31)

s

im

(ξ) +

X

v∈V

q

imv

− R

i

X

v∈V

T

_i

Q

q

imv

≤ S

i

,

∀(i, m) ∈ S

A

|J

i

= −1, ξ ∈ Ω,

(32)

s

im

(ξ) −

X

v∈V

q

imv

+ R

i

X

v∈V

T

_i

Q

q

imv

≥ S

i

,

∀(i, m) ∈ S

A

|J

i

= 1, ξ ∈ Ω,

(33)

s

iµ

i

(ξ) +

X

v∈V

q

i,µ

i

,v

− R

i

(T − t

iµ

i

(ξ)) + r

iµ

i

(ξ) ≥ S

i

,

∀i ∈ N |J

i

= −1, ξ ∈ Ω,

(34)

s

iµ

i

(ξ) −

X

v∈V

q

i,µ

i

,v

+ R

i

(T − t

iµ

i

(ξ)) − r

iµ

i

(ξ) ≤ S

i

,

∀i ∈ N |J

i

= 1, ξ ∈ Ω,

(35)

s

im

(ξ), r

im

(ξ) ≥ 0

∀(i, m) ∈ S

A

: m > 1, ξ ∈ Ω.

(36) For brevity we omit the des ription of the onstraints as their meaning is similar to the

meaningofthe orresponding onstraintsforthedeterministi problem.Thesto hasti SAA

modelisdenedby(26)andthe onstraints(1)-(13),(27)-(36),andwillbedenotedby

SAA-MIRP.

Nextwemaketwoimportantremarks.Remark1.TheSAA-MIRPmodelhasrelatively

ompletere ourse,sin eforea hfeasiblesolutiontotherststage,thein lusionof

r

variables ensuresthatthese ond stagehasalwaysafeasiblesolution.Remark 2.Whenarststage

solution

X

isknown,these ondstage variables aneasilybeobtainedbysolving

k

separate linearsubproblems.

4 Solution methods

Whilethe deterministi model anbe solved to optimality for small size instan es, the

SAA-MIRPmodelbe omesmu hharderwiththein lusionoftheinventoryviolationvariables

r,

and annot onsistently be solved to optimality for large sample sizes. We onsider

M

separatesets

Ω

i

, i

∈ {1, . . . , M }

ea hone ontaining

ℓ

≪ k

s enarios.TheSAA-MIRPmodel is solved for ea h set of s enarios

Ω

i

,

(repla ing

Ω

by

Ω

i

in model SAA-MIRP) giving

M

andidatesolutions.Letusdenoteby

X

1

_{, . . . , X}

M

_,

therststagesolutionsofthose andidate

solutions.Then,for ea h andidate solutionthe valueof theobje tivefun tion forthelarge

sample

z

k

(X

i

_{) = C(X}

i

_{) +}

1

k

P

ξ∈Ω

Q(X

i

, ξ)

is omputedandthebest solutionisdetermined by

X

∗

_{= argmin{z}

k

(X

i

) : i ∈ {1, . . . , M }}.

The average value over all sets of s enarios,

¯

z

ℓ

=

_M

1

P

M

i=1

z

ℓ

i

is astatisti al estimatefor alowerbound onthe optimalvalueof thetrue problemand

z

k

(X

∗

₎

,isastatisti alestimateforanupperbound ontheoptimalvalue.

Hen eforwardwedis usstwopro eduresforsolvingtheSAA-MIRP model forthesmall

Thisisreferredtoasstabilityrequirement onditions[17℄.Agraetal.[2℄usedade omposition

s heme for asto hasti MIRP that was shown to beinsu ientto rea h stability for hard

instan es.Here werevisitthispro edure andintrodu eanalternativemethod.

4.1 De ompositionpro edure

A ommonapproa htosolvesto hasti problemsisto de omposethemodelinto a

mas-ter problem andonesubproblem forea h s enario,followingtheideaof theL-shaped

algo-rithm[5℄.Themasterproblem onsistsoftherststagevariablesand onstraints( onstraints

(1)-(13)),andre oursevariablesand onstraints(27)-(36) denedforarestri tedsetof

s e-narios.Thesubproblems onsiderxedrststagede isions,andaresolvedforea hs enarioto

supplynewvariablesand onstraintstothemasterproblem.Sin etheproblemhasrelatively

ompletere ourse,theresultingsubproblemsarefeasible.

Werstsolvethemasterproblemin ludingonlyones enariotooptimality.Thenforea h

disregardeds enariowe he kwhetherapenaltyforinventorylimitviolationsisin urredwhen

the rst stage de isionis xed. If su h a s enariois found, we add to the master problem

additionalvariablesand onstraintsenfor ingthat deviationto bepenalized intheobje tive

fun tion.Thentherevisedmasterproblemissolvedagain,andthepro essisrepeateduntilall

there ourse onstraintsaresatised.Hen e,asintheL-shapedmethod, themasterproblem

initiallydisregardsthere oursepenalty,andanimprovedestimationofthere oursepenaltyis

graduallyaddedtothemasterproblembysolvingsubproblemsandaddingthe orresponding

onstraints.A formaldes riptionofthispro essisgivenbelow.

Algorithm1De ompositionpro edure.

1: Considerthemasterproblemwiththes enario orrespondingtothedeterministi problem

2: Solvethemasterproblem

3: whileThereisas enario

ξ

∈ Ω

i

leadingtoanin reaseoftheobje tivefun tion ostdo 4: Add onstraints(27) -(36)fors enario

ξ

5: Reoptimizethemasterproblemwiththenew onstraintsusingasolverfor

α

se onds

6: endwhile

To he kwhetherthereisas enario

ξ

∈ Ω

i

leadingtoanin reaseoftheobje tivefun tion ost, one an use asimple ombinatorial algorithm that, for ea h s enario, determines the

earliestarrivaltimebasedonthe omputationofalongestpathinana y li network[2℄.

4.2 MIP basedlo al sear hpro edure

Inorderto ir umventsomepossiblestabilityproblems resultingfromtheprevious

pro- edure, whi h is based on a trun ated bran h and ut pro edure, we propose a heuristi

approa h that iteratively sear hes in the neighborhood of a solution. The pro edure starts

with the optimal solution from a deterministi model, and ends when no improvement is

observed. For thestarting solution we either use the deterministi model (1)-(25), with no

inventory violationsallowed, orthe sto hasti model ontainingonly ones enariowhere all

travellingtimesaresettotheirexpe tedvalue.Todenetheneighborhoodofasolution,let

dierinatmost

∆

variables,fo usingonlyontheshipvisitvariables

w

imv

.Thislo alsear h anbedonebyaddingthefollowinginequality,

X

(i,m)∈S

A

v

,v∈V |w

imv

=0

w

imv

+

X

(i,m)∈S

A

v

,v∈V |w

imv

=1

(1 − w

imv

) ≤ ∆.

(37)

Inequality(37) ountsthenumberofvariables

w

imv

that areallowedtoiptheirvaluefrom the valuetakenin the solution.Note that the routingvariablesaswell asthe quantities to

loadandunload an be hangedfreely.

In ea h iteration of the heuristi pro edure, the SAA-MIR model restri ted with the

in lusionof (37)issolvedinitsextensiveform(withoutthede ompositionpro edure),sin e

preliminary testshavenotshown learbenetsin using thede omposition te hniquein the

restri tedmodel.Thepro edureisdes ribedin Algorithm2.

Algorithm2MIPbasedLo alSear hpro edure

1: Solveeithermodel(1)-(25),ortheSAA-MIRPwithasingles enario onsistingofexpe tedtravel

times

2: Set

w

totheoptimalvalueof

w

3: repeat

4: Add onstraint (37)tothemodeldenedfor

ℓ

s enarios

5: Solvethemodelfor

α

se onds

6: Updatethesolution

w

7: untilNoimprovementintheobje tivefun tionisobserved

5 Computational tests

Thisse tionpresentssomeofthe omputationalexperiments arried outtotest thetwo

solution approa hes for a set of instan es of a maritime inventory routing problem. The

instan es are based onreal data, and ome from the short sea shipping segment withlong

loadingand dis hargetimes relativeto the sailingtimes. These instan es result from those

presentedin [1℄, with twomain dieren es. One is the omputation of the traveling times,

whi hwedis ussindetailbelow,andtheotheristheprodu tionand onsumptionwhi hwe

assume hereto be onstant, where therates are given by the averageof the orresponding

valuesgivenin theoriginalsetof instan es.Thenumberofports andshipsofea h instan e

isgiveninthese ond olumnofTable2.Thetimehorizonis30days.Operatingandwaiting

ostsaretimeinvariant.

Distribution oftravel timesand s enario generation

Here we des ribe the sailing times probabilitydistribution aswell ashow s enariosare

generated.Weassumethatthesailingtimes

T

ijv

(ξ)

arerandomandfollowathree-parameter log-logisti probability distribution.The umulative probability distribution anbe written

as

F(T

ijv

(ξ)) =

1

1 + (

1

t

)

α

,

where

t

=

T

ijv

(ξ)−γ

β

.

Thistypeofdistributionwasusedin[15℄foranLNG(liqueednaturalgas)tanker

trans-portationproblem,andwasmotivatedbythesailingtimes al ulatedforagastankerbetween

Rome (Italy) and Bergen(Norway), asreported in [16℄. In thethree-parameter log-logisti

probabilitydistribution,theminimumtraveltimeisequalto

γ

,andtheexpe tedtraveltime is equalto

E[T

ijv

(ξ)] =

βπ

α sin(π/α)

+ γ.

Thethree parameters,in [15℄, were set to

α

= 2.24

,

β

= 9.79

,and

γ

= 134.47

.Inoursettings,thedeterministi traveltime

T

ijv

,givenin[1℄,isset totheexpe tedtraveltimevalue,thatis,

T

ijv

= E[T

ijv

(ξ)].

Inaddition,welet

γ

= 0.9 × T

ijv

,

α

= 2.24

(thesamevalueasin[15℄,sin e

α

isaformparameter),and

β

isobtainedfromthe equation

T

ijv

= E[T

ijv

(ξ)] =

βπ

α sin(π/α)

+ γ.

Inordertodrawasample,ea htraveltimeis ran-domlygeneratedasfollows.Firstarandomnumber

r

from

(0, 1]

isgenerated.Thenthetravel time

T

ijv

(ξ)

anbefoundbysetting

r

=

1

1 + (

1

_t

)

α

,

whi h gives

T

ijv

(ξ) = γ + β

 1 − r

r



−

α

1

.

Computationalresults

Alltestswererunona omputerwithanIntelCorei5-2410Mpro essor,havinga2.30GHz

CPU and8GBof RAM,usingthe optimizationsoftwareXpressOptimizerVersion21.01.00

withXpressMoselVersion3.2.0.

Thenumberofportsandshipsofea hinstan eisgivenin these ond olumn ofTable1.

The following three olumns givethe size of the deterministi model (1)-(25), and the last

three olumns givethesizeforthe ompletesto hasti modelSAA-MIRPwith

ℓ

= 25.

Table 1.Summarystatisti sfortheseveninstan es.

Deterministi Model Sto hasti Model

Inst.

(| N |, | V |) #

Rows

#

Col.

#

Int.Var.

#

Rows

#

Col.

#

Int.Var.

A (4,1) 765 545 273 8413 1713 273 B (3,2) 767 590 302 5303 1466 302 C (4,2) 1214 1042 530 8798 2210 530 D (5,2) 1757 1622 822 13157 3082 822 E (5,2) 1757 1622 822 13157 3082 822 F (4,3) 1663 1539 787 9183 2707 787 G (6,5) 4991 5717 2909 20687 7469 2909

Table2givestheoptimalvaluesofseveralinstan esforthedeterministi model.Columns

Noviolations givetheoptimalvalue( olumn

C(X)

)andrunningtimein se onds( olumn Time)forthemodel(1)-(25)withnoinventorylimitviolationsallowed.Thefollowing olumns

onsiderthesto hasti modelwiththeexpe tedtraveltimess enarioonly.Forthreedierent

penaltyvaluesforinventorylimitviolations(

P

i

= 1 × ℓ, P

i

= 10 × ℓ

and

P

i

= 100 × ℓ

,where the

ℓ

is omitted for ease of notation) we provide the routing ost

C(X),

the value of the inventoryviolation( olumnsViol)and therunningtime( olumnsTime).

Table2showstheinuen eofthepenaltyonthesolutionvalueandinstan ehardness.The

runningtimesaresmallwhen

P

i

= 1

andtendtoin reasewiththein reaseofthepenalty.For smallpenaltyvaluestheinstan esbe omeeasiertosolvethanforthe asewithhardinventory

Table 2. Instan esand the orrespondingrouting osts and inventoryviolations for the expe ted

values enario

Noviolations

P

i

= 1

P

i

= 10

P

i

= 100

Inst.C(X) Time C(X) Viol TimeC(X) Viol TimeC(X) Viol Time

A 130.7 1 6.7 60.0 0 130.7 0.0 0 130.7 0.0 0 B 364.8 6 5.2 235.0 0 364.8 0.0 13 364.8 0.0 19 C 391.5 15 14.7 172.0 0 290.5 3.0 4 324.5 0.5 10 D 347.1 3 55.9 177.0 4 347.1 0.0 42 347.1 0.0 52 E 344.9 343 55.8 184.0 4 344.9 0.0 194 344.9 0.0 181 F 460.9 290 182.0 110.0 3 460.9 0.0 437 460.9 0.0 442 G 645.8 2962 336.3 176.5 17 645.8 0.0 6947 645.8 0.0 16296

For

P

i

= 10

,

P

i

= 100

andforthe asewhereviolationsarenotallowed,thesolutions oin ide forallinstan esex eptinstan e

C.

Nextwereporttheresultsusingbothpro eduresfollowingthesolutionapproa hdes ribed

inSe tion4with

M

= 10

setsofs enarioswithsize

ℓ

= 25,

andalargesampleofsize

k

= 1000.

InTables3,4,and5,wepresentthe omputationalresultsforthede omposition pro edure

usingbran hand uttosolveea hmasterproblemwitharunningtimelimitof

t

= 1, t = 2,

and

t

= 5

minutes. After this time limit, if no feasible solutionis found, then the running timeisextendeduntiltherstfeasiblesolutionisfound.Forea htablewegivetheresultsfor

thethree onsidered asesofpenalties, denotedby

P

i

= 1, P

i

= 10,

and

P

i

= 100.

Forea h penaltywereportthefollowingvalues:therouting ost

C(X)

ofthebestsolution

X

∗

obtained

with thepro eduredes ribedin Se tion4;theaveragenumberofviolations( olumns

V iol

) forsolution

X

∗

;thevarian ebetweensamples

σ

ˆ

2

B

=

1

(M − 1)M

M

X

i=1

(z

ℓ

i

− ¯

z

ℓ

)

2

;

thevarian ein

thelargersample

σ

ˆ

2

L

=

1

(k − 1)k

X

ξ∈Ω

(C(X

∗

) + Q(X

∗

, ξ) − z

k

(X

∗

))

2

;

andthe runningtime, in se onds, of the omplete solution pro edure. The running time in ludes solving the

M

sto hasti problems,andforea hsolution, omputingthepenaltyvalueforthelargesetof

k

samples.

Varian es

σ

ˆ

2

B

and

σ

ˆ

2

L

areusedtoevaluatethestabilityofthepro edure.One anobserve thatwhenthepenaltiesin rease,thevarian eforthelargesample,

σ

ˆ

2

L

,

in reasesasexpe ted. For the varian e between samples,

σ

ˆ

2

B

,

the value also in reases when we ompare

P

i

= 1

againsttheothervalues.Su hbehavior anbeexplainedbythefa tthatea hmasterproblem

issolvedbyabran hand utpro edureandasexplainedabove,whenthepenaltyin reases,

the integrality gaps also in rease making the instan es harder to solve. For those harder

instan esthebran hand utalgorithm a tsasaheuristi sin ethesear htree istrun ated

whenthetimelimitisrea hed.Thus,thevarian etendsto in reasewhenwe omparethose

ases where the instan es are solved to optimality (some instan es with

P

i

= 1

) against those aseswherethesolutionpro edurea ts,ingeneral,asaheuristi (mostinstan eswith

P

i

= 10

and

P

i

= 100

). However between the ases

P

i

= 10

and

P

i

= 100

there is no obvioustrend.Thereareinstan eswherethede ompositionpro edurehadabetterdegreeof

in-sample stabilityfor

P

i

= 100

thanfor

P

i

= 10.

Perhapsasthepenalty ostisso high,for someinstan es thesolveridentiesthesamesolution(a solutionwhi h isrobust in relation

Ω

i

onsidered. Ingeneral, we may state that thede omposition pro edure tends to beless stablewiththein reaseofthepenalties.Thereisno learde reaseinthevarian eswhenthe

runningtimelimitisin reased.

Table 3. Computationalresults usingthe de ompositionpro edure with arunning timelimit for

ea hmasterproblemsetto1minute

P

i

= 1

P

i

= 10

P

i

= 100

Inst.

C(X)

Viol

σ

ˆ

2

B

ˆ

σ

2

L

Time

C(X)

Viol

σ

ˆ

2

B

σ

ˆ

2

L

Time

C(X)

Viol

σ

ˆ

2

B

ˆ

σ

2

L

Time A 130.7 0.0 4 0 105 130.7 0.0 27 0 116 130.7 0.0 27 0 118 B 364.8 1.4 13 0 492 364.8 1.4 64 5 3654 364.8 1.3 57 413 4965 C 263.5 3.0 73 0 151 343.9 2.5 91 0 292 411.5 0.0 36 2 749 D 347.1 2.0 190 0 1627 347.1 2.1 2463 5 3975 347.1 2.2 205 514 3524 E 344.9 3.2 358 0 2277 344.9 5.8 5830 13 5394 260.4 18.4 1945 698 4262 F 501.1 0.1 145 0 327 460.9 0.0 451450 0 5361 460.9 0.0 44413 0 4993 G 433.0 133.92136 1 1115 484.8213.6 71357 151 6962 457.6 212.12121500 6 7696

Table 4. Computationalresults usingthe de ompositionpro edure with arunning timelimit for

ea hmasterproblemsetto2minutes

P

i

= 1

P

i

= 10

P

i

= 100

Inst.

C(X)

Viol

ˆ

σ

2

B

σ

ˆ

2

L

Time

C(X)

Viol

σ

ˆ

2

B

σ

ˆ

2

L

Time

C(X)

Viol

ˆ

σ

2

B

σ

ˆ

2

L

Time A 130.7 0.0 4 0 102 130.7 0.0 27 0 114 130.7 0.0 27 0 109 B 364.8 1.4 13 0 583 364.8 1.4 1368 5 6143 364.8 1.3 7539 413 8413 C 263.5 3.0 73 0 150 343.9 2.5 91 0 301 391.5 0.0 54 2 928 D 347.1 2.0 190 0 2367 363.2 0.2 792 0 7133 347.1 2.1 314 486 6602 E 344.9 3.2 358 0 2716 363.9 3.2 8839 7 8637 352.9 0.2 1809 40 9032 F 501.1 0.1 145 0 333 460.9 0.0 7025 0 7368 460.9 0.0 4024 0 7824 G 433.0 133.92136 1 3260 543.7 154.59548913510837 442.3 121.9746118967013003

InTable6,wereportthe omputationalresultsfortheMIP basedlo al sear hheuristi ,

startingwithasolutionobtainedbyusingasingles enario onsistingofexpe tedtraveltimes

in theSAA-MIRPmodel.Basedon preliminaryresults,notreported here,we hose

∆

= 2.

Therunningtimelimitissetto5minutes,howeverformostiterationstherestri tedproblem

issolvedtooptimalityqui kly. InTable7,wereportthe orrespondingresultsforthesame

heuristi but startingwith asolutionobtainedusing the model (1)-(25),that is, themodel

wherenodeviationstotheinventorylimitsareallowed.We anseethatusinghardinventory

limitsforthestarting solutionleadsto abetter solutionfor sixinstan es andworsefor two

instan es.

When omparingthevarian eswiththoseobservedforthede ompositionpro edure one

anobservethat thevarian esbetweensamplesareingenerallower,meaningthat thelo al

sear hpro edurepresentsahigherdegreeofin-samplestabilitythanthe lassi al

de omposi-tionapproa h.Forthelargersample,bothpro edurespresentsimilarvarian evalues,ex ept

fortheharderinstan e(

G

with

P

i

= 100)

wherethenewheuristi pro edureprovidesbetter out-of-sample stability. The running times of the lo al sear h heuristi are also lowerthan

Table 5. Computationalresults usingthe de ompositionpro edure with arunning timelimit for

ea hmasterproblemsetto5minutes

P

i

= 1

P

i

= 10

P

i

= 100

Inst.

C(X)

Viol

σ

ˆ

2

B

σ

ˆ

2

L

Time

C(X)

Viol

σ

ˆ

2

B

ˆ

σ

2

L

Time

C(X)

Viol

ˆ

σ

2

B

σ

ˆ

2

L

Time A 130.7 0.0 4 0 102 130.7 0.0 27 0 116 130.7 0.0 27 0 121 B 364.8 1.4 13 0 602 364.8 1.4 1350 5 11781 364.8 1.3 76 413 17530 C 263.5 3.0 73 0 148 343.9 2.5 91 0 277 391.5 0.0 31 2 1611 D 347.1 2.0 190 0 4120 347.1 2.0 590 5 15130 430.3 1.1 395 127 16889 E 344.9 3.2 358 0 6827 352.9 0.2 1791 1 21592 360.9 3.4 5474 768 25769 F 501.1 0.1 145 0 321 460.9 0.0 36 0 11118 460.9 0.0 11200 0 13218 G 433.0 133.92136 1 6421 534.0 57.354937 4 20462 517.0107.3437242354421564

Table 6.ComputationalresultsusingtheMIPbasedlo alsear hheuristi

P

i

= 1

P

i

= 10

P

i

= 100

Inst.

C(X)

Viol

σ

ˆ

2

B

ˆ

σ

2

L

Time

C(X)

Viol

σ

ˆ

2

B

ˆ

σ

2

L

Time

C(X)

Viol

σ

ˆ

2

B

σ

ˆ

2

L

Time A 130.7 0 4 0 205 130.7 0.0 54 0 210 130.7 0.0 54 0 210 B 260.7 32.7 74 0 406 364.8 1.3 22 4 698 364.8 1.3 44 400 769 C 263.5 3.0 87 0 388 391.5 0.0 11 0 704 411.5 0.0 131 0 944 D 377.2 6.2 28 0 2729 347.1 2.0 1 5 2768 347.1 2.0 48 490 2610 E 405.3 5.6 0 0 3731 344.9 3.1 0 7 3728 344.9 3.1 63 718 3694 F 460.9 0.0 46 0 1484 460.9 0.0 197 0 3438 460.9 0.0 197 0 3118 G 711.9 4.8 239 3 5174 679.5 1.6 602 5 6116 798.9 13.37313246 9686

Table7.ComputationalresultsusingtheMIPbasedlo alsear hheuristi withthestartingsolution

obtainedforthedeterministi modelwithhardinventory onstraints

P

i

= 1

P

i

= 10

P

i

= 100

Inst.

C(X)

Viol

σ

ˆ

2

B

ˆ

σ

2

L

Time

C(X)

Viol

σ

ˆ

2

B

ˆ

σ

2

L

Time

C(X)

Viol

σ

ˆ

2

B

ˆ

σ

2

L

Time A 130.7 0 4 0 72 130.7 0.0 54 0 89 130.7 0.0 54 0 96 B 345.2 3.8 24 0 383 364.8 1.3 22 4 722 364.8 1.3 44 400 719 C 263.5 3.0 87 0 266 354.5 0.5 38 0 533 391.5 0.0 57 0 630 D 347.2 2.0 28 0 1097 347.1 2.0 1 5 2992 347.1 2.0 48 490 3675 E 344.9 3.1 0 0 1214 344.9 3.1 0 7 3227 344.9 3.1 63 718 5822 F 460.9 0.0 46 0 941 460.9 0.0 460 0 1670 460.9 0.0 197 0 1388 G 654.9 1.8 138 0 4592 679.5 1.6 994 5 8130 912.5 0.6 40211210332

Finally, in Table8wepresenttheoverall ost

z

k

(X)

forthe best solutionobtainedwith the two solution pro edures. Columns De omp. give the ost value for the de omposition

pro edureusingatimelimitof5minutes,and olumnsMIPLSgivethe orrespondingvaluefor

theMIPbasedlo alsear hheuristi usingthestartingsolutionwithnoinventorydeviations.

Thebestresultfromthetwoapproa hesishighlightedin bold.

Table 8.Cost

z

k

(X)

ofthebestsolutionobtainedwithea honeofthesolutionpro edures

P

i

= 1

P

i

= 10

P

i

= 100

Inst.De omp.MIPLSDe omp.MIPLSDe omp.MIPLS

A 130.7 130.7 130.7 130.7 130.7 130.7 B 441.0 441.0 725.6 696.0 3707.2 3632.3 C 338.5 338.5 968.9 479.5 454.1 391.5 D 397.7 397.7 857.0 853.9 3261.5 5414.2 E 422.0 422.0 405.1 1115.2 8804.7 8047.7 F 502.9 460.9 460.9 460.9 460.9 460.9 G 3780.8 692.4 14861.1 1085.8268712.02359.6

We anseethatthenewMIPbasedlo alsear hpro edureisbetterthanthede omposition

pro edure in ten instan es and worse in two. The de omposition pro edure performs well

when instan es anbe solvedto optimality. Overall,wemay on ludethat the lo al sear h

heuristi is moreattra tivethanthede omposition pro edure basedonthe bran h and ut

when the instan es are not solved to optimality, sin e the lo al sear h heuristi is faster,

presentsbetterlevelsofin-samplestabilityforalmostallinstan esandbetterlevelsof

out-of-samplestabilityforthe hardestinstan e, andprovidesgood qualitysolutions.On theother

hand,fortheinstan esthat anbesolvedto optimality,thede omposition pro edureisthe

bestoption.

6 Con lusions

We onsider amaritimeinventoryroutingproblem wherethetraveltimesaresto hasti .

Theproblemismodeledasatwo-stagesto hasti programmingproblemwithre ourse,where

violationsofinventorylimitsarepenalized.Ade ompositionpro edurethatsolvesthemaster

problemusinga ommer ialsolverandaMIPbasedlo alsear halgorithm,areproposed.For

several instan es the master problem is notsolved to optimality within reasonablerunning

times. Hen e bothpro edures anbe regardedasheuristi s.Thetwopro edures are tested

forstabilityusingdierentvaluesforthepenalties.A omputationalstudybasedonasmall

setofben hmarkinstan esshowsthatwhenthepenaltiesarelow,theinstan esareeasierto

solvebyexa tmethods,andthede ompositionpro edure anbeusede iently.Ontheother

hand,whenpenaltiesarehigh,theintegralitygapstendtoin reasemakingthede omposition

pro edure, that usesthe bran h and ut to solvethe master problem, less stable than the

Aknowledgements

Theworkof therstauthor wasfunded byFCT(Fundação paraaCiên ia eaTe nologia)

andCIDMA(CentrodeInvestigaçãoeDesenvolvimentoemMatemáti aeApli ações)within

proje t UID/MAT/04106/2013.The work orthe fourth author was funded by FCT under

GrantPD/BD/114185/2016.

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