Cont

ribut

ionWe

consi

der th

e veh

icle ro

uting

prob

lems (

VRPs)

where

the

cost p

er un

it dist

ance

is prop

ortion

al to

the to

tal we

ight o

f the

vehic

le. Su

ch VR

Ps ari

se wh

en ou

r goa

l is to

mini

mize

total

fuel co

nsump

tion a

nd ar

e kno

wn as

cumu

lative

VRPs.

We de

scribe

const

ant fa

ctor a

pprox

imati

on al

gorith

ms fo

r two

varia

tions

of the

Cumu

lative

VRPs

(Cum-

VRPs)

; veh

icles

with u

nbou

nded

capa

city a

nd ve

hicles

with

boun

ded c

apaci

tygiv

en by

Gaur

et al.

[1]. W

e give

const

ant fa

ctor a

pprox

imati

onalg

orithm

s for

stoch

astic c

umula

tive V

RPs (

sCum

-VRPs)

for s

plit

and u

nsplit

dema

nds (

stoch

astic)

[2].

Conc

lusio

n

Rish

i Ran

jan

Sing

hDe

part

men

t of C

ompu

ter S

cien

ce a

nd E

ngin

eerin

gIn

dian

Inst

itute

of T

echn

olog

y Ro

par

Appr

oxim

ation

Algo

rithm

s for

Comu

lative

Vehic

le Ro

uting

Prob

lems

[1] A.

Gupta

, V. Na

garaj

an, R

. Ravi

, Techn

ical N

oteA

pprox

imati

on Al

gorith

ms fo

rVR

P with

Stoch

astic D

eman

ds, Op

er. Re

s., vol

. 60,

no. 1

, 123

127,

Jan 20

12.

[2] D.

R. Ga

ur, A.

Mudg

al, R.

R. Sin

gh, R

outin

g veh

icles t

o minim

ize fu

el con

sumpti

on, O

perat

ions R

esearc

h Lett

ers, v

ol. 41

, Issue

6, No

v 201

3, 57

6-580

.[3]

D. R.

Gaur,

A. Mu

dgal,

R. R.

Singh

, App

roxim

ation

Algo

rithms

for S

tocha

sticCu

mulat

ive Ve

hicle R

outin

g Prob

lems, p

resen

ted at

the 5

5th An

nual C

onfer

ence

ofthe

Cana

dian O

perat

ional R

esearc

h Soci

ety (C

ORS 2

013)

Cana

da.

[4] D.

J. Be

rtsim

as, A

Vehic

le Rou

ting P

roblem

with

Stocha

stic De

mand

. Ope

ration

sRe

search

, vol.

40, n

o. 3,

574-5

85, M

ay 19

92.

[5] K.

Altin

kemer

and B

. Gavi

sh, He

uristic

s for u

nequ

al weig

ht de

livery

proble

mswit

h a x

ed er

ror gu

arante

e, Op

eratio

ns Re

search

Lette

rs 6(4

), 149

158,

Sep 1

987.

[6] K.

Altin

kemer

and B

. Gavi

sh, Te

chinca

l Note

: Heu

ristics

for D

elivery

Prob

lems

with C

onsta

nt Err

or Gu

arante

es, Tra

nsport

ation

Scien

ce, 24

(4), 2

9429

7, 19

90.

[7] I.

Kara,

B. Y.

Kara,

and

M. K.

Yetis,

Cumu

lative

Vehic

le Rou

ting P

roblem

s, Veh

icle

Routi

ng Pr

oblem

, Edit

ed by

Caric

, T., a

nd Go

ld, H.

, I-Tech

Educa

tion a

nd Pu

blishin

g KG

, Vien

na, A

ustria

, 200

8, pp

. 859

8.

Refer

ence

s

Rishi

Ranja

n Sing

hDe

partm

ent o

f CSE

R.N.

120,

IIT Ro

par,

Nang

al Ro

ad Ru

pnag

ar Pu

njab

rishir

s@iit

prp.a

c.in (

+91 8

5918

2858

4)

Cont

act

Resu

lts

A fea

sible

soluti

on S

to Cu

m-VR

P con

sists

of a s

et of

k|V

\{0}|

direct

ed cy

cles C

1 , . .

, Ck c

ontai

ning t

he

depo

t suc

h tha

t:1.

Each

verte

x i

V\{0

} belo

ngs t

o exa

ctly o

ne cy

cle.

2. Th

e tota

l weig

ht of

objec

ts de

livered

in ea

ch cy

cle

C j, 1

j

k is a

t most

Q.

Cum-

VRP:

Given

a sol

ution

sche

dule

S for

Cum-

VRP ,

the

cumu

lative

cost

is:

where

a :

cost o

f mov

ing th

e emp

ty ve

hicle

per u

nit di

stanc

e,b

: cos

t of m

oving

the u

nit we

ight g

ood p

er un

it

dis

tance,

d iS :

distan

ce for

which

the v

ehicle

carrie

s weig

ht w i

|C i

| : len

gth of

the c

ycle

C i.sC

um-VR

P:Giv

en a

soluti

on sc

hedu

le S f

or sC

um-VR

P , th

eex

pecte

d cum

ulativ

e cost

can b

e calc

ulated

as:

where

E[ i]

: ex

pecte

d dem

and a

t nod

e i.

Cumu

lative

Cost

Prob

lem va

riatio

nsDe

termi

nistic

(Cum

-VRPs

):EQ

-INF-C

um-VR

Ps: A

ll the

objec

ts ha

ve eq

ual w

eight,

and t

he ve

hicle

has a

n in

nite c

apaci

ty.UN

EQ-IN

F-Cum

-VRPs

: The

objec

ts ha

ve un

equa

l we

ights,

and t

he ve

hicle

has a

n in

nite c

apaci

ty.EQ

-CAP

-Cum

-VRPs

: All t

he ob

jects

have

equa

l weig

ht,an

d the

vehic

le ha

s a ca

pacit

y Q.

UNEQ

-CAP

-Cum

-VRPs

: The

objec

ts ha

ve un

equa

lwe

ights,

and t

he ve

hicle

has a

capa

city Q

.St

ocha

stic (

sCum

-VRPs

):Sp

lit-sC

um-VR

Ps: T

he de

mand

at a

custo

mer n

ode

can be

satis

ed i

n mult

iple v

isits.

Unsp

lit-sC

um-VR

Ps: T

he de

mand

at a

custo

mer

node

must

be sa

tise

d in a

single

visit

.Veh

icle Ro

uting

Probl

ems (V

RPs) a

re one

of th

e most

intere

sting

and wi

dely s

tudied

comb

inator

ial opt

imizat

ion pr

oblem

s. Give

n aee

t of d

elivery

vehic

les at

the d

epot a

nd cu

stome

rs with

some

demand

s, the

object

ive is

to nd

a sche

dule f

or the

vehic

les in

order

to me

et the

dema

nds o

f the c

ustom

ers, th

at min

imizes

the

total d

istance

travel

led (o

r total

time s

pent) b

y the

vehicle

s.Kar

a et a

l. [7]

dened

cumu

lative

VRPs

where

the o

bjectiv

e is to

minimi

ze fue

l consu

mptio

n of th

e vehi

cle giv

en tha

t fuel c

onsum

edper

unit d

istance

is pro

portio

nal to

the t

otal w

eight

of the

vehic

le.We

rede

ne the

probl

em as

follow

s:Cu

m-VR

P:Inp

ut:1.

A com

plete,

undir

ected

graph

G(V,

E) wi

th pos

itive le

ngth

oneac

h edg

e e

E. Th

e edg

e leng

ths sa

tisfy t

he tria

ngle i

nequal

ity.2.

The ve

rtices

of th

e grap

h G(V,

E) ar

e num

bered

from

0 to n

.Ver

tex 0

is calle

d the

depot.

3. For

each

i V

\ {0}

, there

is a d

emand

for a

n obje

ct of

positiv

e weig

ht w i.

4. De

pot ha

s su

cient

amoun

t of o

bjects

to m

eet all

dema

nds.

5. An

empty

vehic

le is lo

cated

at the

depot

, whic

h at a

ny poi

ntof

time c

an car

ry obj

ects o

f total

weigh

t not

exceed

ing Q.

Objec

tive:

Devis

e a tra

vel sc

hedule

for th

e vehi

cle so

that

all the

dema

nds

are m

et and

the c

umula

tive co

st is m

inimize

d. We

allow

the

vehicle

to o

oad ca

rgo at

the d

epot a

n arbi

trary

numb

er of

times.

sCum

-VRP:

It is a

stocha

stic va

riatio

n of C

um-VR

P. Inp

ut:

Dema

nd sp

ecied

by a r

andom

varia

ble

i , in t

he ran

ge(0,

Q], is

at th

e cust

omer

node i

, i

(V \ r

). Exa

ct valu

e of

demand

at a c

ustom

er nod

e i is k

nown o

nly aft

er the

vehic

le visit

sthe

node

i.Ob

jectiv

e: De

vise a

prior

i sched

ule wi

th min

imum

expect

edcum

ulative

cost.

We bu

ilt on t

he wo

rks of

Altin

kemer

and Ga

vish [

5,6], B

ertsim

as [4]

and Gu

pta et

al. (2

012)

[1].

Theo

rem 5.

There

exist

s a ra

ndom

ized a

pprox

-im

ation

algo

rithm

produ

cing (

2 + 2

)-facto

r a p

riori s

ched

ule fo

r Spli

t-sCu

m-VR

Ps.Pr

oof Id

ea: S

ee Fig

ure 1.

Theo

rem 6.

There

exist

s a ra

ndom

ized a

pprox

-im

ation

algo

rithm

produ

cing (

4 + 2

)-facto

r a p

riori s

ched

ule fo

r Unsp

lit-sC

um-VR

Ps.Pr

oof Id

ea: S

imilar

sche

dule

menti

oned

inthe

proo

f of T

heore

m 5 e

xcept

ensur

e unsp

lit de

livery.

EQ-IN

F-Cum

-VRPs

: Le

mma 1

. (Low

er bo

und)

Cost

of Op

timal

cost

sched

ule fo

r EQ-

INF-Cu

m-VR

Ps is a

t least

:

where

C TSP

is the

cost

of op

timal

TSP t

our, a

nd

is the

av

erage

dista

nce o

f cust

omer

node

s from

depo

t.Th

eorem

1. Th

ere ex

ists a

facto

r- 3 ap

proxim

ation

appro

ximati

on sc

hedu

le for

EQ-IN

F-Cum

-VRPs.

Proo

f Idea

: Take

the 3

/2 fac

tor TS

P tou

r on t

he

proble

m. Ca

lculat

e the

c=a/b

itera

ted op

timal

partit

ion fo

r the t

our.

EQ-CA

P-Cum

-VRPs

:Le

mma 2

. (Low

er bo

und)

Cost

of Op

timal

cost

sched

ule fo

r EQ-

CAP-C

um-VR

Ps is a

t least

:

.Th

eorem

2. Th

ere ex

ists a

facto

r- ap

proxim

ation

sche

dule

for UN

EQ-IN

F-Cum

-VRPs.

Proo

f Idea

: Take

the s

olutio

n sch

edule

for E

Q-INF

-Cu

m-VR

Ps fro

m Th

eorem

1. Ca

lculat

e Q ite

rated

op

timal

partit

ion fo

r each

cycle

which

conta

ins m

oretha

n Q no

des.

UNEQ

-INF-C

um-VR

Ps:

Theo

rem 3.

There

exist

s a so

lution

sche

dule

for UN

EQ-IN

F-Cum

-VRPs

proble

m wit

h tota

l cu

mulat

ive co

st at

most:

Proo

f Idea

: Con

struc

t a ne

w grap

h from

the

given

grap

h, ha

ving u

nit siz

e weig

ht on

each

node

. Con

struc

t a so

lution

sche

dule

for

EQ-IN

F-Cum

-VRPs.

Finally

, short

-circu

it the

unit

weigh

t nod

es fro

m the

same

weigh

t set.

UN

EQ-CA

P-Cum

-VRPs

:Th

eorem

4: Th

ere ex

ists a

solut

ion sc

hedu

lefor

UNEQ

-CAP-C

um-VR

Ps pro

blem

with t

otal

cumu

lative

cost

at mo

st:

Proo

f Idea

: Take

the s

ched

ule of

UNEQ

-INF-

Cum-

VRPs

from

Theo

rem 3

and b

reak t

he

cycles

of th

e tou

r, if re

quire

d to m

aintai

n the

capaci

ty con

strain

t.Sp

lit-sC

um-VR

Ps &

Unsp

lit-sC

um-VR

Ps:

Lemm

a 3. T

he m

inimu

m ex

pecte

d cum

ulativ

ecos

t to m

eet th

e dem

and o

f all c

ustom

ers is

atlea

st:

We co

nside

red a

simpli

ed m

odel

for fu

elcon

sumpti

on. W

e pres

ent th

e proo

f of e

xisten

ceof

consta

nt fac

tor sc

hedu

les fo

r die

rent d

eter-

minis

tic an

d stoc

hasti

c vari

ation

s of th

e prob

lem.

The a

pprox

imab

ility o

f cum

ulativ

e VRP

s whe

rethe

numb

er of

ooa

ding a

llowed

is giv

en as

input,

rema

ins an

open

quest

ion.

Figure

1. Fir

st, par

tition

(Oute

r Part

ition)

the op

timal T

SP to

ur int

o subt

ours, d

eliveri

ng at m

ost c

units

of we

ight. T

hen pa

rtition

(Inner

Partit

ion) e

ach cy

cle (w

hich i

s deliv

ering

more

than Q

units

of we

ight) i

nto su

b-cycl

es, de

liveri

ng atm

ost Q

units o

f weig

ht.

Intro

ducti

on