A Column Generation based Tactical Planning Method for Inventory Routing

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A Column Generation based Tactical Planning Method for Inventory Routing

Sophie Michel, François Vanderbeck

To cite this version:

Sophie Michel, François Vanderbeck. A Column Generation based Tactical Planning Method for Inventory Routing. Operations Research, INFORMS, 2012, Operations Research, 60 (2), pp.382-397.

�inria-00169311v4�

Method for Inventory Routing

S. Mihel (1) and F. Vanderbek (2)

(1) Université du Havre (mihelsuniv-lehavre.fr),

(2) Université Bordeaux 1 (fvmath.u-bordeaux1.fr),

(1-2) INRIA Bordeaux Sud-Ouest, team RealOpt

(http://www.inria.fr/reherhe/equipes/realopt.en.html).

November 15, 2008

Abstrat

Inventory routing problems ombine the optimization of produt deliveries (or

pikups) with inventory ontrol at ustomer sites. Our appliation onerns the

planning of single produtpikups over time; eah site aumulates stok at a de-

terministi rate; the stok isemptied on eah visit. At the tatial planning stage

onsidered here, our objetive is to minimize a surrogate measure of routing ost

while ahieving some form of regional lustering by partitioning the sites between

the vehiles. The eetsize is given but an potentiallybe redued. Planning on-

sistsinassigningustomerstovehilesineahtimeperiod,buttherouting,i.e.,the

atual sequene inwhih vehiles visit ustomers, is onsidered asan operational

deision. Theplanning isdueto berepeatedoverthetimehorizon withonstrained

periodiity. We develop a trunated branh-and-prie-and-ut algorithm ombined

withroundingand loalsearhheurististhatyieldsbothprimalsolutions anddual

bounds. Ona large sale test problem oming from industry, we obtain a solution

within 6.25% deviation from the optimal. A rough omparison between an opera-

tionalrouting resulting fromour tatial solution andtheindustrial pratieshows

a 10% derease innumber of vehiles aswell asin travel distane. The key to the

suessoftheapproahistheuseofastate-spaerelaxationtehniqueinformulating

themasterprogram to avoidthesymmetryintime.

Keywords: InventoryRouting,Branh-and-Prie-and-Cut, PrimalHeuristi,Symmetry.

In the mid-1980s, researh work began on integrating inventory ontrol with vehile

routing in an eort to better manage an important segment of the supply hain. The

Inventory Routing Problem (IRP) onsists in designing routes for deliveries or pikups

that inorporate issues of inventory management atustomer sites. Three deisionshave

tobemade: (i)whentoserveaustomer;(ii)howmuhtodeliverto(resp. pik upfrom)

a ustomer on eah servie; and (iii)whih delivery (resp. pikup) tasks are assigned to

eah vehile (potentially inludingtheir sequening todene routes). Many variants are

disussedintheliterature. FedergruenandSimhi-Levi(1995)andCampbelletal.(1998)

disusstherststudiesonIRP,whilereentsurveysare provided inCordeau etal.(2007)

and Bertazzi et al.(2008).

Theappliationthatmotivatesourstudyisaspeialaseinwhihtheinventoryman-

agementmodel issimplewhenonsidered atthe tatialdeisionplanninglevel. Asingle

produt must be piked-up from ustomers who aumulate it in their stok at a given

rate that is assumed tobe known and deterministi. The pikup quantity is neessarily

equal to the stok level at the time of the visit: tehnial onstraints impose what is

known as an order-up-to-level poliy in inventory management. The inventory man-

agement poliyaims at avoidingstok-apaity over-ow (by having suiently frequent

pikups) while minimizingpikuposts (that inrease with the frequeny of pikups). In

the deterministi ontext of tatial planning, the objetive of avoiding stok-apaity

over-ow translates into hard onstraints: the bounded stok apaity imposes a maxi-

mum amount of time (expressed in number of periods) between two visits. Hene, there

are no inventory managementosts, but only stok transportation osts.

While deisions (i) and (ii) above result from the inventory model, deisions (iii)

result from the routing modelthat onsists traditionallyinsolving a apaitated vehile

routing problem (VRP) for eah period of the planning horizon. In this study we view

routing deisions as operational. In pratie, the spei route is often deided by an

experiened driver given information updates about the route network. Instead of the

pureminimizationoftraveldistane/time,weattempttogroupustomersfromaompat

geographialareaintolusters: alusterisasetof ustomersthatare assignedtoagiven

vehileinagiventime period. Our industrialpartnerraised thisissue asmoreimportant

than travel distane/time in our appliation where a lear assignment of setors to

vehilesisneeded. Buildingroutesof minimumlengthoftenleadstoroutes rossingeah

other or expanding over several distrits (as we shall illustrate in Figure 1). To replae

the routing ost, we use ameasure of the ost of a luster that is a ompromisebetween

the measure of dispersion around a luster enter used in faility loation problems and

the routing insertion ost used in heuristi methods for the VRP. This measure favours

measure for transportationosts.

Our tatialplanningmodeltakesyetanotherissueintoaount. Toease thereal-life

implementation of the plan, the shedule of ustomer visits must repeat itself over time,

and the periodiity is onstrained to be reasonably small. Finally, our tatial model

allows us to onsider reduing the eet size that resulted from a deision taken at the

strategilevel. Determiningwhat is the minimum eet size that is required tosatisfy all

the planningrequirements isfar from being trivial(as weshall illustrateinExample 2).

Our tatial planningsolutionmust serve as atarget inoperationalplanningsoas to

makethe latter lessmyopi. The operationalmodelwilltrytoinorporateurgent pikup

requirementsintothetatialplanningsolution. Notethattheroutingsolutionassoiated

withalusterdenedatthetatialplanninglevelanbeomputedinapostoptimization

phasebysolvingatravelingsalesmanproblemovertheseleted setofustomersites. The

expeted stohasti variationof ustomeraumulationrates an beaounted for inthe

tatial planning model by over-estimating stok lling rates, reserving buer spae in

the vehiles and/orthe ustomer stoks.

The model studied here has, to the best of our knowledge, not been expliitly on-

sidered in the literature, but many variants have been. Most of the existing approahes

tend tomake restritive assumptions,suh asthe so-alledxed partitionpoliy where

one denes sets of ustomers that are systematially servied together (see Bramel and

Simhi-Levi(1995)). Alternatively they adopt a hierarhial optimization sheme where

planningprodutdeliveriesorpikupsisdeidedbeforerouting(seeCampbellandSavels-

bergh (2004)). Most approahes are heuristis with no guarantee on the deviation from

optimality and are spei to the problemvariant (see Gaur and Fisher (2004)).

Wehavedeveloped a trunatedbranh-and-prie algorithm: periodiplans aregener-

ated for vehiles by solving a multiple hoie knapsak subproblem; the global planning

of ustomer visits is generated by solving a master program. This exat optimisation

approah is ombined with rounding and loalsearh heuristis to yield both primal so-

lutions and dual bounds that allowus to estimate the deviation from optimality of our

solution. We were onfronted with the issue of symmetry in time that naturally arises

in building a yli shedule (yli permutations along the time axis dene alternative

solutions). Centraltoourapproahisastate-spaerelaxationideathatallowsustoavoid

thissymmetry. Our algorithmprovidessolutionswithreasonabledeviationfromoptimal-

ityforlargesale problems(260ustomersites, 60timeperiods,10vehiles)omingfrom

industry.

The paper is organized as follows. Setion 1 desribes our problem and speies our

assumptions. Setion 2 inludes a short review of the existing literature. In Setion 3,

we outline our deomposition approah. Then, the symmetry in time is eliminated by

to the disrete time formulation. Setion 5 presents the spei features of our olumn

generation approah to solve the master LP. Cutting planes and partial branhing are

used to improve the dual bounds as presented in Setion 6. Our primal heuristis are

presented in Setion 7. Setion 8 reports our results on an industrial test problem and

ompares them with urrent industrialpratie. Finally,weonlude with diretions for

further researh.

1 The tatial planning problem

The problemistoplanvehilevisitstoustomers overadisretetimehorizondivided

into time periods:

t

= 1, . . . , T^{. Eah assignment of ustomers to a vehile in a given}

time period denes what we alla luster. The quantity that is piked up on a visit to

the ustomer is the whole stok aumulated sine the last visit. To a given luster, we

assoiateapikup patternthat denes the quantitiesthat are piked up ateahustomer

site. The sum of these quantities annot exeed the vehile apaity. Customers are

indexed by

i

= 1, . . . , n^{. For eah of them, we know the ustomer site loation (also}

indexed by

i

^{), the produt aumulation rate per period,}

r

_i^{, the maximum number of}

periods between two visits,

t

^max_i ^{, that results from their limited stok apaity, and the}

timerequiredtopikupthe ustomerstok,

b

_i^{,whihinour appliationis}independentof the quantity that isolleted. Let

N

^{bethe set ofrelevant sites,inludingustomer sites}

numbered 1 ^to

n

^{, a depot indexed by} 0 ^{from whih vehiles start, and a dumping site,}

indexed by

n

+ 1^{, where vehiles are unloaded. Let}

N

^′ =

N

\{0, n+ 1} ^{beits restrition}

totrue ustomer sites.

We assume a given eet of

V

^{idential vehiles of apaity}

W

^{, devoted to olleting}

the singleprodut fromthe geographiallydispersedustomers. The numberof available

vehiles is assumed to be suiently large to over the pikup requirements. However,

ourobjetivefuntioninludesthe minimizationofthe maximumnumberofvehilesthat

is used inany given period,whihwe denoteby

V

_max ⁽

V

_max ≤

V

^{). From the modelpoint}

of view, putting the emphasis on minimizing

V

_max ^{an allow us to generate solutions}

using fewer than the

V

^{available vehiles, whih translates into signiant savings. But}

this term is alsoinluded tohelp the onvergene of our solution method: in our primal

heuristi having an important xed ost for using an extra vehile gives an inentive to

lleah vehile and tospread the workload evenly among periods.

The time horizon that we onsider is innite, but we searh for a periodi solution.

Eah lusterassignmentwill berepeated overtime every

p

^{time periods. Werestrit the}

solutionspae by imposingthat luster periodiities,

p

^{, are seleted froma restrited set}

P

^{. This implies a bound,}

T

^{, on the time horizon beyond whih the solution is repeated.}

T

^{is the least ommonmultiple (LCM) of the} periodiities in

P

^{. For our study, we take}

P

={1,2,3,4,5,6}^{and, hene,}

T

= 60^{. Foreah luster and assoiated pikup pattern,}

we must selet its periodiity

p

∈

P

^{and its rst ourrene, i.e., its startingdate,}

s

≤

p

^.

Then, the solution is

H

^{periodi where}

H

^{is the LCM of the} periodiities used in the solution.

T

^{ats as the maximal length of the} regeneration yle, i.e.

H

≤

T

^{. Hene,}

T

anbeseenasthenitetimehorizonthat resultsfromtherestritionontheperiodiities.

A vehile task is dened by seleting the luster of ustomers that is assigned to the

vehile, the assoiated pikup pattern that determines the quantities that are olleted

by thevehile, aperiodiity

p

^{withwhihthis vehile assignmentwillbereprodued,and}

itsrst ourrene,

s

≤

p

^{. A ompleteplan onsists inanassignmentof tasksto vehiles}

that ensures that the ustomer stoks produed in eah period,

t

= 1

. . . , T

^{, are piked}

up in some task (see Example 2 below). Beyond the minimization of the eet size, we

attempttoregionalize vehile tasksbydening alusterostthatestimates itsregional

ompatness. Although the exat routing of vehiles is onsidered an operational issue,

weomputeanestimateof traveltimerequiredtovisitthe ustomersofalusterstarting

from the depot and ending at the dumping site. This estimate is intentionally biased

towards luster entered around a seed point, denoted

k

^{, so as to ahieve the desired}

regional ompatness. Let {dij}_(ij)∈N×N ^{denote the shortest travel times between sites;}

the matrix is assumed to be symmetri. A luster,

S

⊂

N

^{′, is built around a luster}

enter,

k

∈

S

^{. We dene its ost as follows. It is} initialized with a setup ost dened as the length of a shortest path fromthe depot to the dumping site passing by the seed

k

^,

plus the seed ollet time, i.e., the xed luster ost is

f

k =

d

0,k+

d

k,n+1+

b

k

,

where

d

0,k ^and

d

k,n+1 ^{arethe traveltimes between the depot} 0^{and site}

k

^{and fromsite}

k

tothe dumpingsite

n

+ 1^{. Then, the onnetionost forsite}

i

^{tothe seed}

k

^{is ameasure}

of the insertion ost of site

i

^{in the path} depot-seed-dumping site plus its ollet time, i.e.,

c

ik =

d

i,k+

min{d

0,i−

d

0,k

, d

i,n+1−

d

k,n+1}+

b

i

.

Example 1

Letusillustratehowthis oststruturefavorsthegroupingof ustomersthataregeograph-

ially lose to one another, while keeping sight of the resulting routing ost. In Figure 1,

weompare three approahesfor reating lusters: 1)the traditionalVRP solution,2) the

lustersoptimizedwith ourost struture, 3) the lustersobtained byminimizingdistane

to a enter (forming stars) as done for faility loation problems. The example onerns

oneperiod, a set of15 ustomerswhosedemandsareindiated onthe side,and2 vehiles

of apaity 112. Beyond the pitorial ustomer groupings illustrated by Figure 1, it is

interesting to ompare the osts:

19 149

1516

19

1

0 0 0

25

39

412

512 617 75

81 927

1036 1130

129 1319 149

1516

19

25

39

412

512 617 75

81 927

1036 1130

129 1319 149

1516 2 9

5

39

412

512 617 75

81 927

1036 1130

129 13

Comparing 3 approahes to lustering: standard routing (on the left), our lustering

approah (inthe enter) and stars used infaility loation models (on the right)

routing solution lustering solution faility loation solution

routingost 190.9 214.1 184.5

our luster ost 196.8 200.4 147.0

faility loation ost 198.8 208.7 135.5

where the number in bold was the optimized objetive, while the others are omputed a

posteriori or result from post-optimisation.

Example 2

Now onsider a planning over

T

= 6 ^{periods with}

P

={1,2,3,6}^{. Assume the instane}

has 5 ustomers with

r

= (51,50,34,33,18) ^and

t

^max= (1,2,2,3,5)^{. Thevehile apaity}

is

W

= 100^{. A plan onsisting of 4 vehile tasks is given in Figure 2. The gure on the}

left illustrates the vehile tasks. Task 1 is to visit luster {1,3} ^{and to pik up} (r1+

r

₃)

units. Task 2 is to visit luster {2} ^{and to pik up} (2

r

2) ^{units. Task 3 is to visit luster} {1,5}^{, and to pik up} (r1+ 2

r

₅) ^{units. Task 4 is to visit luster} {3,4} ^{and to pik up} (r3+ 2

r

4) ^{units. All tasks are two periodi, the rst two start in period 1, the last two}

start in period 2. Both tasks 1 and 3 are performed by a rst vehile, while tasks 2 and

4 are performed by a seond vehile. The length of the regeneration yle is

H

= 2^{. The}

table on the rightillustrates the assoiatedplanning. For eah ustomer, theperiod where

there is a vehile visit is marked by a sign, and a v sign marks the period for whih

the assoiated stok prodution is olleted in a following visit. This example illustrates

how the ombination of ustomer requests on a multi-period plan allows us to derease

the eet size omparedto a single period model. Indeed, the minimum number of vehiles

required for a single period model(this is the solution of a bin paking problem with item

size given by vetor

r

^{and bin apaity}

W

^{) is 3, while our plan uses 2 vehiles.}

A ompat formulationof the problem an be derived in terms of the following vari-

p=2, s=2

p=2, s=1

p=2, s=2

p=2, s=1

6 433

518

0

1

3

2

51

50

34

A

solutionin terms of tasks

t

= 1

t

= 2

t

= 3

t

= 4

t

= 5

t

= 6

i

= 1

i

= 2 ^{v v v}

i

= 3

i

= 4 ^{v v v}

i

= 5 ^{v v v}

ables:

x

iℓvps = 1 ^{i at ustomer site}

i

^{, a quantity equal to a stok aumulation over}

ℓ

≤

t

^max_i ^{periods is olleted by vehile}

v

^{that performs a task of periodiity}

p

∈

P

^that

starts in

s

≤

p

^;

y

vps = 1 ^{i vehile}

v

^{is used for a task of periodiity}

p

∈

P

^{that starts}

in

s

≤

p

^;

z

ikvps = 1 ^{i ustomer}

i

^{is visited in a luster of seed}

k

∈

N

^{′ assigned to}

vehile

v

^{with periodiity}

p

∈

P

^{and starting date}

s

≤

p

^(with

z

kkvps = 1 ^{i the site}

k

is the seed of vehile

v

^{with periodiity}

p

∈

P

^{and startingdate}

s

≤

p

^{); and}

V

max ^{is the}

maximumnumberofvehilesthatare used inany period. Fornotationalonveniene, an

indiatorvetor ofsize

T

^anbepre-omputedfor eahpair (p, s)^{thatmarks the periods}

t

∈ {s, s+

p, s

+ 2

p, . . .}

^{inwhih atask indexed} (p, s) ^{should require avehile:}

δ

_t^ps =







1 ^if ∃m∈

IN

^{suh that}

s

+

m

∗

p

=

t,

0 ^{otherwise. (1)}

Similarly, we pre-ompute an indiator vetor of size

T

^{for eah triplet} (ℓ, p, s) ^saying

whether the stok prodution of period

t

^{is olleted by a task indexed by} (p, s)^{, when}

ℓ

^{periods worth of stok is olleted (suh task would pik up the stok aumulated in}

periods

t

∈ {s−

ℓ

+ 1, . . . , s−1, s, s−

ℓ

+ 1 +

p, . . . , s

−1 +

p, s

+

p, . . .}

^):

θ

_t^ℓps=







1 ^if ∃m∈

IN

^and

τ

∈ {0, . . . , ℓ−1} ^{suh that}

s

+

m

∗

p

−

τ

=

t,

0 ^otherwise.

(2)

The ompat formulationis:

min

V

_max +

α

^X

v,p,s

1

p

(^X

k

f

_k

z

_kkvps+ ^X

i,k:i6=k

c

_ik

z

_ikvps) ⁽³⁾

X

ℓ,v,p,s

θ

^ℓps_t

x

iℓvps = 1 ∀i∈

N

^′

, t

= 1, . . . , T ⁽⁴⁾

X

k∈N^′

z

ikvps = ^X

ℓ

x

iℓvps ∀i∈

N

^′

, v, p, s

⁽⁵⁾

z

ikvps ≤

z

kkvps ∀i∈

N

^′

, k

∈

N

^′

, v, p, s

⁽⁶⁾

X

i∈N^′,ℓ

ℓ r

i

x

iℓvps ≤

W y

vps ∀v, p, s ⁽⁷⁾