wearhouse location problem

7/24/2019 wearhouse location problem

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Department of Industrial & Management Engineering

Indian Institute of Technology, Kanpur

Cost Minimization For Warehouse allocation ro!lem

"De#eloping $trong Constraints for the Multi

%Commodity $$CW'

by

Parag Tyagi(13114016)

Thesis Supervisor:

Prof. R.R.K. Sharma

1


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2

Ou!i"e

#"ro$u%io"&ieraure Survey

Previous 'asi% ormu!aio"se* ormu!aio" for +u!i,%ommo$iy %aseTheorem-Proposiio"

Summery a"$ o"%!usio"sRefere"%es


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Introduction

3


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Introduction Continues

Warehouse allocation Problem isencountered in areas like FMCG SupplyChain Fertili!ers "istribution System FCI#Food Corporation o$ India% morepronouncedly

For e&'& FCI has the $ollo(in' Structure

)


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Introduction Continues

Function o$ FCI is to distribute the Food Grainsthrou'hout the country $or P"S and other

Go*ernment Schemes FCI procures the Food Grains $rom the Mandies

and Stores in respecti*e Central (arehouses o$each +one

From these Central (arehouses Food Grains aretransported to "istrict (arehouses and $urther tothe "epots and P"S at the end

Problems encountered here are maintainin' a

satis$actory le*el o$ operational and bu,er stockso$ $ood 'rains to ensure national $ood securityand some un$oreseen problems like lo( $oodproduction drou'ht -ood (ar crop $ailure etc&

.


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/iterature Sur*ey

Facility allocation bein' a important decisionin supply chain plannin' has been studied bymany researchers and there ha*e beenmassi*e de*elopments $rom last $our decades

Warehouse allocation problem has beenundertaken in its *ariants $orms

For e&'& 1& SP/P0CP/P

2& Sin'le commodity0 Multi

commodity 3& Sin'le Sta'e0MultiSta'e


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/iterature Sur*ey continues

n o*er*ie( o$ the (ork done in this 4eld canbe looked $rom the re*ie( (ork done byReVelle and Eiselt(2005)

5ur problem is a SSCW/P in (hich $acilitylocation is considered at a sin'le sta'e

Sin'le Sta'e $acility location problem hasbeen attempted by many authors such asGeofrion and Graves(1974) andSharma(1991) and *ery interestin'ly they

ha*e 'i*en di,erent $ormulations Geo,rion and Gra*es#167)% Sharma #1661%

Sharma#166% and 8ou*elis #299)% ha*e used(eak $ormulations o$ the problem

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Geo,rion and Gra*es #167)% $ormulations

:

In this paper $ormulations ha*e beende*eloped $or a MultiCommoditycapacitated sin'le period real li$e problem

Problem taken here is o$ a ma;or $ood 4rm

(ith 17 commodities classes 1) plants ).possible distributions center #"C%sites and121 customers !ones

M5" $or commodities j = inde> $or plants = inde> $or possible distribution center#"C% sites l = inde> $or customer demand !ones


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6

Sij =supply #production capacity% $or commodity i at

plantj

!il=demand $or commodity i in customer !one l"

V "V= minimum ma>imum allo(ed total annual

throu'hput $or a "C at site "

# =4>ed portion o$ the annual possession andoperatin' costs $or a "C at site

v= *ariable unit cost o$ throu'hput $or a "C at site

"

$ijl= a*era'e unit cost o$ producin' and shippin'commodity i $rom plantj throu'h "C k to customer!one l"

%ijl=?ariable denotin' the amount o$ commodity i

shipped $rom plantj throu'h "C to customer !one l


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19

&l =a 91 *ariable that (ill be 1 i$ "C ser*es customer

!one l" and 9 other(ise&

' =a 9 1 *ariable that (ill be 1 i$ a "C is ac@uired at site kand 9 other(ise&

Ahe ro*lem $an *e +ritten as the #ollo+in, mi%ed inte,erlinear ro,ram (M-./)

#1% Minimi!e>B9 y!D91Ei;kl$ijl%ijl Ek#' vkEil!il&lH Sub;ect to=

#2% El%ijl Sij all i"j

#3% E;%ijlD !il&l all i""l

#)% &lD 1 all l #.% V'3 il!il&l3 V' all k

#% /inear con4'uration constraints on&and0or'


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11

Model clearly states that each customer !one can'et the commodities $rom one "C only #constraint

#)%% uantity il!il&lthe total annual throu'hput o$ the

kth"C (hich (ill al(ays be (ithin the permissible

capacity limits o$ a particular site to be economical

solution techni@ue based on Jendersdecomposition is de*eloped implemented andsuccess$ully applied to sol*e the model

Jenders (ell kno(n partitionin' procedure isapplied in such a (ay that MultiCommodity /Psubproblem decomposes into as manyindependent classical transportation problems asthere are commodities


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Sharma #1661% $ormulations

12

Mi>ed !eroone inte'er linear pro'rammin'problem is $ormulated $or a real li$e problemo$ Fertili!ers productiondistribution system

Model has been $ormulated by considerin'

the t(o seasons each o$ si> months keepin'in mind the croppin' seasons#usually o$ si>months%

6. K Fertili!ers are mo*ed $rom plants torake points and $rom there mo*ed tosecondary points and then 4nally reach themarket by /C?

In the Model plants rake points secondaryoints and markets (ill be re$erred to as


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/iterature Sur*ey continuesM5" $or point type

; = inde> $or number o$ particular point type

k= inde> $or periodh= inde> $or product type

u= inde> $or nutrient type

N= total number o$ products

O= total number o$ nutrients

8= total number o$ periods in si> months duration #assumede*en $or con*enience%

#i% = total number o$ points o$ cate'ory i

ijh" Eijh= be'innin' and endin' in*entory at point numberjo$

point typeiin period ko$ product type his represented byrespecti*ely

.ij = location *ariable (hich is 1 i$ it is decided to locate a

(arehouse at point number ; o$ point type i #(here i *aries$rom 2 to 3% and 9 other(ise

");k= @uantity o$ nutrient demanded o$ type u at market ;#point o$ type )% in period k

13


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/1jh= @uantity produced at plant numberj#point o$ type 1% in

period o$ product type h

QR);kh= @uantity recei*ed at market numberj#point o$ type )% in

period ko$ product type hS1i; S2i;= Space booked at point numberj o$ point type i in the

4rst and last three months o$ a si> monthly season respecti*ely

i1"j1"i2"j2""h= @uantity transported $rom point numberj1o$ point type

i1to point numberj2o$ point type i2in period ko$ product type

hi"j" = total @uantity transported $rom point numberjo$ point

type iin period k

#h = $raction o$ nutrient o$ type uobtainable $rom product o$

type h

S#i ; %= set o$ points o$ type i + l to (hich a point o$ type iha*in' a numberjcan supply products

R#i ; % = set o$ points o$ type i - 1$rom (hich a point o$ type iha*in' a numberj can recei*e products

/Pljh OPljhand 6/ljh= respecti*ely the lo(er limit upper limit and

unit cost o$ *ariables Pljh$or allj, k and h 1)


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"istribution cost is composed o$ the

Production cost= CPl;kh /1jh

Cost o$ space= CSli;S1i; CS2i;S2i;

cost o$ carryin' in*entory= 9&.CIi;kh#ed cost o$ locatin' a (arehouse= C/i; /i;

5b;ecti*e Function #Minimi!in' Qotal Cost%=Minimi!e E;khCPl;kh /1jh ijmn6ijmn8 ijmn

ijh6-ijh8058( Eijh ijh) Ei;CSli;

Sli;

Ei;CS2i;S2i; ij6.ij 8 .i j

Constraints= 1.


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(#) Sub,prob!em o"srai"s:

P1;kh J1;khD


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(###) &o%aio" o"srai"s:

/inear constraints representin' the condition that i$ a

(arehouse is located at a particular point then thespace booked must be (ithin the permissible lo(erand upper limits

&&%

QQi;k Qi1;1i2;2kh


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e( Formulations de*eloped $orSSCW/P MultiCommodity case

MultiCommodity SSCW/P is a $amily o$ real li$eproblems like FMCG FCI etc&

Stron' and (eak $ormulations ha*e beende*eloped $or the 'eneral case o$ the problem


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/= In the model (e ha*e used the $ormulation style o$

sharma and sharma #2999%

#"$ees:

i:Plants :Warehouses

/:Markets m:Commodity

XPWi;m= umber o$ units shipped $rom plant ito

(arehousejo$ the commodity m

XWM;km= umber o$ units shipped $rom (arehousejto

market o$ commodity m

CPWi;m= Cost o$ shippin' o$ one unit $rom plant ito

(arehousejo$ the commodity m

CWM;km= Cost o$ shippin' o$ one unit $rom (arehousejto

market o$ the commodity m

Y;= Jinary /ocation *ariable (hich is 1 i$ it is decided to

locate a (arehouse at location ; and 9 other(ise 16

M d l ti


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Model continues

:j= Fi>ed cost and maintenance cost $or

establishin' a (arehouse at locationj

Sim= Supply capacity o$ plant i$or

commodity m

6;/j Capacity o$ the (arehouse atlocationj it is assumed that all thecommodity are o$ same density and

con4ned the same space!m= "emand $or commoditymat market

M= *ery lar'e number here taken as 29

M d l ti


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Model continues

5b;ecti*e Function=

Minimi!e + D i;m% #CPWi;m%

;km%#CWM;km%

;F;

222(O)

Sub;ected to =

i;m V Sim Z im #Supply

Constraints%

i;m D ;km Z ;m #-o( balance constraint%

;km D "km Z km #"emand

Constraints%

i;m V CP; Z ; #Capacity

21

(1

)

M d l ti


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Model continues

/inkin' Constraints

i;m V MY; Z ; #it (ill be a *ery (eak

constraint% &&()

XPWi;m V SimY; Z im; &&&(3)

XWM;km V "kmY; Z km; (4)

Positi*e Constraints and rela>ations=

XPWi;m

B 9 Z i;m

XWM;km B9 Z ;km

Y;T #91% Z ;

Formulation I = #5% #1% #2% and #.%

22

#3% nd #)% are likely to pro*to be SQR5G Constraints

()


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Qheorem0Proposition-t is roosed that ./ rela%ation o# (--) < ./

rela%ation o# (-) ie ./ rela%ations o# #ormlation(--) #orms a serior *ond than ./ rela%ation o##ormlation (-) and there#ore rea$h to otimalsoltion in lesser 6/= time

heoreti$al /roo#= $rom linkin' constraints #3%#)% and #.%H

Since "s [[ M and S\s [[ MQhere$ore $easible re'ion o$ $ormulation II [$easible re'ion o$ $ormulation I&

Qhen $or the minimi!ation problem ob;ecti*e$unction *alue o$ $ormulation II (ill be hi her23

Proposition continues


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2)

Suppose \] is

the point o$Min $or$ormulation #I%

Clearly point\] is not in the$esible re'iono$ $ormulation#II%

Feasible re'ion $or$ormulation #I%

Feasible re'ion $or $ormulation#II%

5ptimal *alue o$ $ormulation #II% (ill be at apoint other than the point \] and hence it (illbe a 'reater *alue than the $ormulation #I%optimal *alue



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A .9>.9

and 199>199>199 $or the $ollo(in' $our cate'ories asper input data=

*era'e "emand $or each commodity in each market istaken as 12. units

2.

aegory verage 7arehouseapa%iy

verage Supp!y

a& 39 K more than the*era'e demand

39K more than thea*era'e (arehouse

capacityb& 39 K more than the

*era'e demand12.K more than thea*era'e (arehousecapacity

c& 12.K more than the*era'e demand

39K more than thea*era'e (arehousecapacity



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Random *alues are 'enerated $or supply (arehousecapacity demand costs o$ transportation and 4>ed

cost o$ (arehouse location (ithin the GMS code .9 Problems are sol*ed $or each o$ the $our cate'ories

#total number o$ problem instances sol*ed D )99 299o$ each o$ the si!e .9>.9>.9 and 199>199>199%


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AStatistical nalysis

Nypothesis tests is conducted as $ollo(s=For ob;ecti*e $unction *alues

_= Percenta'e impro*ement in 5b;ecti*e $unction *alue o$$ormulation

#II% o*er $ormulation #I%ull hypothesis N9= _ D 9

lternate hypothesis Na= _ ` 9

Similarly For CPO Qimet = Percenta'e increase in CPO time $or $ormulation #II%o*er $ormulation #I%

ull hypothesis N9= t D 9

lternate hypothesis Na= t ` 9 27



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AStatistical Results t*alues calculated $or thehypothesis tests are tabulated as $ollo(s=

2:

Prob!emaegory Prob!em Si8e(000) Prob!em Si8e(100100100)

t*alues $orbound

impro*ement

t*aluesincrease inCPO time

t*alues $orbound

impro*ement

t*aluesincrease inCPO time

a&37&21 23&. )62&9: .2&3:

b&

313&7 27&1 37)&.2 2&6:

c&397&33 2&23 .96&3: )3&.

d&

2::&.1 21&17 )97&3. 3:&2:



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Critical *alue $or tstats at D 9&99. is 2&: $or

d&o&$& D )6 and 2&2) $or d&o&$& D 2&2) Comparin' the calculated t*alues (ith critical

t*alues (e can easily re;ect null hypothesis $orboth #Jound *alue and CPO time%

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Summery and Conclusions Rela>ed $ormulation#II% i&e& Stron' Constraints

$orms the superior bounds than the Rela>ed$ormulation#I%i&e& (eak constraints

CPO time taken $or stron' $ormulations issi'ni4cantly more than the CPO time $or (eak

$ormulations Jene4t o$ better bounds usin' stron'er

$ormulations is nulli4ed by their hi'hercomputational time

We there$ore no( proceed to make hybrid$ormulation $or multicommodity SSCW/P too*ercome the dra(back o$ more computationaltime = to obtain the better bounds (ith lesser

CPO time 39


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31

Nybrid

Formulations

Re$erences


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Re$erences1H Geo,rion M and Gra*es GW #167)%& Multicommoditydistribution system desi'n by Jenders decomposition&

Mana'ement Science 0(): :22:))&2H "re!ner Q "re!ner + and Salhi S #2992%& Sol*in' the multiplecompetiti*e $acilities location problem&


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Re$erences:H Sharma RR8 #1661%& Modelin' a $ertili!er distributionsystem&


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Thank You

3)

wearhouse location problem

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