paper multicommodity transportation planning in railway fuzzy graph approach

7/25/2019 Paper Multicommodity Transportation Planning in Railway Fuzzy Graph Approach

1/9

1999 IEEE International Fuzzy Systems Conference Proceedings

August 22-25. 1999,Seoul, Korea

Multicommodity Transportation Planning in Railway: Fuzzy Graph Approach

Rossana R. Mendes, Akeo !amakami

"ni#ersity o$ %ampinas

DT -F&&-"'(%AMP-%P)*+**+-*/0+ %ampinas-1P-2razilemai*:akeo(3dt.$ee.unicamp.r

Abstract

In tis !or", !e analyze a multicommodity

trans#ortation #roblem !it fuzzy ob$ecti%e function and

fuzzy constraints using fuzzy gra#s& 'e de%elo#ed an

algoritm based on gra# teory considering tetrans#ortation costs and ca#aci(ties as fuzzy numbers& 'e

a##lied te #ro#osed algoritm in arail!ay net and te

obtainedresults are analyzed&

)E*('+DS- fuzzy net!or", fuzzy

multicommodity trans(#ortation #roblem, fuzzy gra#,

o#timization

*. Introduction

During se%eral #ast years, agreat deal ofattention asbeen dedicated to matematical #rogramming and

matemat(ical models tat can be sol%ed using net!or"s&

eduction of com#utations, better %isualization and

#roblem understand(ing are some of ad%antage in using

gra# teory& Toug, fe! bas been done in te fuzzy

gra# teory& Tere are some !or"s tat study te fuzzy

sortest #at #roblem, suc as )lein .l/, +"ada 345,Duois and Prade .lo/ and oters& Te first one uses

dynamic #rogramming, and find #ats corre(s#onding to

te tresold of membersi# degree tat can be set by a

decision(ma"er& Tis algoritm assumes tat eac arc can

ta"e an integer %alue for lengt, bet!een * and a fi0edinteger& Te second one introduce an order relation

bet!een fuzzy number based on fuzzy min #ro#osed by

Dubois and Prade .21& 32is relation leads to te conce#tofa nondomi(nated#at or #areto o#timal #at and an

algoritm for sol%ing te fuzzy sortest #at #roblem is

deri%ed based on te mul(ti#le labeling metod for a

multicriterial sortest #at #rob(lem& Te tird onegeneralized te Floyd and Ford4s algo(ritms to te fuzzy

case& Te linear trans#ortation #roblem !it fuzzy cost

and fuzzy ca#acity !as studied by 5endes and *ama"ami

*6*.Tosol%e it, tey introduced some ada#ta(tions to te

classical gra# teory& Teoretical a##roac for sol%ing

net!or" #roblem !it fuzzy cost, !it fuzzy ca#acity and

!it fuzzy cost and ca#acity !as studied by tem& Tey

a##lied te #ro#osed algoritms in a classical

trans#ortation #roblem and

analyzed te results& In tis

#a#er !e sol%e a general

multicommodity

trans#ortation #roblem !it

uncer(

tainties using fuzzy gra#s&

'e associate to eac arc of

te net anim#recise cost andan im#recise ca#acity& In

oter !ords, teir costs and

ca#acities are considered tat

are not "no!n #recisely& Analgoritm is #ro#osed to

sol%e it& Tis algoritm is ageneralization of tat found

in 365. In te fol(lo!ing

section, !e #resent te

matematical formulation of

te #roblem, and in te

section !e #resent te

#ro#osed algoritms&

A##lications and

com#utational results are

#re(sented in section 6 in

order to clarify te #ro#osed

teory and algoritm, and insection 5 !e #resent te

conclusions&

2.5atematical

Formulation

It6s aimed to sol%e a#roblem of draining a!ayse%eral #roductsin arail!ay

net& To eac #roduct is

associated an ori(gin, adestination trans#orting

timeand a7uantity to be trans(#orted& Tis trans#ortation is

made by freigt cars tat

sare te same net& Tere aredifferent "inds of s#ecific

freigt cars to eac #roduct&Eac "ind of #roduct may be

trans#orted by se%eral "ind of

freigt car and te same "ind

of freigt car may trans#ort

se%eral "ind of #roducts&

Tus, te freigt car, !en

unloaded at te final node,


2/9

doesn6t need to return to te same start node, going to te

nearest one !ic re7uires, s#ecifically, tese em#ty freigt

cars to trans#ort a ne! #rod(uct& Te rail!ay net is

re#resented by a finite number of nodes and a set of

oriented arcs $oining#airsof nodes& Eac node re#resents arail!ay station and eac #air of arcs re#re(sents a trac"&

Products must e trans#orted froma start node to a final

node, #assing troug intermediary nodes& Tere may emoretan one !ay totrans#ort acertain #roduct& Tere6s

te interest to find, among tese !ays, te most ad(%antageous, in order to minimize te trans#ortation cost 8or

ma0imize te #rofit& Te

#roblem is modeled as a linear

o#timization #roblem !ic

must treat, simultaneously,

t!o sub#roblems- te#roblem of load, !ic

defines te d e #arture of

te freigt car !en it6sloaded !it te #rod(uct, and

te #roblem of te em#ty

freigt car, !ic defines te

return of te freigt car to

be loaded again after being

unloaded& A matematical

formulation is de%elo#ed

!it an ob$ecti%e function,

!ic aims to minimize te

trans#ortation cost 8de#arture

of te freigt car and te

em#ty freigt car re(distribution cost 8return of

te freigt car, sub$ect to te

node


3/9

+0-+76+)+*//*8*+.++9(/// (&&& **07)

ilet flo!constraints, fleetlimitation and tractionrestrictions&

Te coice of te #roduct and te amount to be

trans#orted must iii%ol%e te redistribution of te eni#tyfreigt car& Te sub#roblem !ic describes te return of

te freigt car is in(fluenced by te node net flo!

restrictions and fleet limitation& Tis multicommodity

trans#ortation #roblem can be formu(lated as a liiiear

#rogramming .SI, gi%en by-

5in C k ( C j ( C j k Z j k ) +CkZk)s.1.

; j = , l ,..., JA j z j = ~ j

CjGJ*A(Z j k +zk)= 0;k=* , .. .,K j ! j " j _< # ; ; i = 1 ,...,I

$,

C j S J $ ( % j o j z j k ) & %xx 5' " ;

" j = , . j k ; j =I, &. .,J

Z j k 22,20 ; k=1, ...,K ; j=1

, . _ .J,;k =1,..., K

!ere C$"- cost %ector #er !agon unit " carrying

#roduct j; Ck: cost %ector per unloaded !agon unit "

8em#ty: xjh:#roduct $ trans#orted by ty#e(" !agon in

te net: ik:em#ty ty#e(" !agon in te net: J': set of#roducts trans#orted by ty#e(" !agon: #*column %ector of

traction ca#acity #er arc:

Ti:scalar re#resenting tri# times s#ent to trans#ort

#roduct$ from its origin to its destination:2":%ectoroftri# times s#ent by em#ty !agon to tra%el accoss eac

#assage: fk: o%er(all a%ailable ty#e(" !agon fleet: A:

node(arc incidence ma(tri0 of te gra# related to #roduct

$, considering;e artificial arcs: Aj: modified node(arcincidence matri0Aj,tat is, it as a column of zero in te

columns corres#onding to artifi(cial arcs& To sol%e tis

fuzzyfied #roblem !e can use te idea #resented by

1& Te first one

formulated and sol%ed a fuzzy linear #roblem, !ere te

fuzzy ob$ecti%e function and te constraints are not !ell

defined& Te second one considered tat te coefficients in

te ob,$ecti%e function and constraints are not "no!n

e0actly& Tese %alues !ere modeled by mean of fuzzy

numbers of ? ty#e recording to te Dubois and Prade@s

definition& Tea#(#roac #ro#osed by Delgado et al& isbased on te conce#t of a com#arison relation bet!een

fuzzy numbers& bey #ro%ed tat different relations induces

different au0iliary models and solutions& Tese ty#e ofa##roac and related discussion can be foundin *ama"ami

and5endes .3/& In tis !or" !esol%e tis same fuzzy

multicommodity trans#ortation #roblem using fuzzy

gra# based on fuzzy number conce#t&

. FuzzyGraphs

Bra# is traditionally a #air B=8, E!ere is afiniteset of%ertices and Ea relation onVxV,i&e&, a set ofordered

#airs of %ertices: tese #airs are te edges of B&According to

osenfeld@sv definition .91, a fuzzy gra# (? is a #air

(v,8)!ere is a fuzzy set on and &is a fuzzy relationonVzVsuc tat-


4/9

P E ( % V ' ) 5+(l.(/l0(1))

Te abo%e ine7ualitye0#ressestat te strengt of

te lin" bet!een t!o %ertices cannot e0ceed te

degree of im#or(tance or e0istence of te %ertices&

In tis #a#er, te %er(tices re#resent te nodes and te

edges re#resent te arcs& 'e consider tat te nodes

and te arcs do e0ist, but te cost as(sociated to nodeand te ca#acity associated to arc are not "no!n

#recisely& 'e treat tese im#recision as fuzzy num(

bers& Te follo!ing fuzzy conce#ts !ill be used todefine te algoritms&

?et be a no null set& A fuzzy set is caracterizedby membersi# functionA:-+.+, 11and te %alueof A(s) describes te degree of membersi#& Tema##ing of A is called membersi# function& ?et be Ba fuzzy number !ita membersi# function fiH(z) .Te a(le%el set of Bis teset !ere[;la=z / p ~ ( x2)C3!ere aE.,1/& Tecostis treated asa triangular fuzzy number and te ca#acityas a tra#ezoidal fuzzy number& Te algoritm #ro#osed

to sol%e te multicommoditytrans#ortation#roblemusing fuzzy gra#s is gi%en by-

Ste# 1- Initialization- sing te #rogressi%e loading

tec(ni7ue, do !ile tere are some #roduct to si#-

Ste# 2 Find sortest #at bet!een all te #roductsusiug fuzzy sortest algoritm 8Sci, based on tefuzzy o#timal(ity condition 8fuzzy numbercom#arison

Ste# : Determine te em#ty !agon 8ty#e "

redistribution costSte# 6:Determine te cost of eac #roduct Cpi=Sci

+RCkand coose te cea#est one 1.Ste# =- Determine admissible increment of 1and loadte gra# !it tis amount

Ste# ): For eac arcf$ of te gra#, find te ne!ca#acity related membersi# function %aluepcnp(j)and

determine te ne! cost-

ne!c$ =f ( ~ Z d ~ j , p , , ~ (andgo~),~)toste#G

4. 'umerical &;ample

In tis section an e0am#leis #resented to illustratete#ro(#osed metodology& Te net!or" data in tetable H&1, te data relati%e tofreigt car are #resented inte table H&G, te #roduct descri#tion is so!ed in tetable H&2 and te solu(tions are #resented troug tables

H&H to 6.). Te gra# in 7uestion is de#ictedin te

figureH&1&Analyzing te results !ere a = 1 8cris# #roblem

!e cannote tat-

- P1, P1A%alue, tat is 8GG + GG=ton& Tis %alue in !agon number is calculated ofas- 8GG+GG=J3=, !ere3= is te sum of !eigt ofloaded !agon #lus !eigt ofem#ty !agon&

1- Te obtained solution is e7ual to C* = 0, to#roducts trans#orted by fleet ty#e(1 !agon&

2- 'e can obser%e tat tetrans#orted amount ofP3and


5/9

P1 #roducts increase& Te #roducts trans#orted byty#e(G !agon did not attend teir res#ecti%e demandsdue te trac(tion restriction& Arcs *and 1are !it teirtraction ca#acity in te ma0imum&

- Te fleet of ty#e(2 !agon did not satisfy all tedemandof te #roduct P1G due te #resenceo$tractionlimitationi ntearc1&

Tale 6 . < 5a0imum$leet per type of#agon

agon I5a0im& 8units 1Em#ty !eigt 8t


6/9

I 6++ I G=(* I 76+ 2s

11-

758


7/9

P9 1H 1 G 11 1G 12 H 50 G

P1E 1 9 G *< 12 1H 2 = 2PI 1 1 1= G 11 1G 12 5 = G

P1G 1 > G 1G 12 1H = = 2

Table 4.4Products trans#orted by

%agonI

Table 4.5:Productstrans#ortedy%agon(*

I .. I ~7. P~~

K9. #in~~.#i1~~~ I

0.0 1H= = = 1= =

0.1 1H= = = 1= =

0.2 1H2 = = 1H =

0.3 1H3&= = = 1H=3&= =

0.4 1H = = 1H= S+

0.5 123&= = = 1H23&= =

0.6 12>= = S+1H1= =

0.7 12= = S+1H =

0.8 12G S+ = 129= =

0.9 12GG&= = = 123G&= =

1.0 12 = =12= =

alue of8alfaTrans#orted

amount 8ton&

P13&

&&9

1.nsn

5. %onclusions

Tere are

inerent situations

!ere te #recision

is de#en(dent of

#erce#tion& Tefuzzy teory is

used to treat tese

situations&

5easuring te

costs, times, trac"s

are some e0(am#les

ofsituations !ereuncertaintiesare

introduced to te

#roblem& In tis

!or" !e sol%ed a


8/9

fuzzy multicommodity trans#ortation

#roblem using fuzzy gra#s& Lere, te

stud(ied trans#ortation #roblem

considersuncertaintiesin te costs and

in te ca#acities of te arcs& 'ea##lied te algoritm in an e0am#le,

!ere te obtained solutions !ere

analyzed and !e concluded tat teresults !ere satisfactory& In te a##li(

cation

e0am#le, it

can be

easily

recognized

tat fuzzy

gra#

teory as#ro%ided

ricer

information tan

oter matemati(

cal tecni7ues,

mainly about te

%alue of a aud te

algoritm so!ed

e more efficient in

te com#utation#oint of %ie!&

11-759


9/9

6.Re$erences

[12] C&5& )lein& Fuzzy sortest #ats, Fuzzy Sets andSys(tems, 29,##& G3(H1,1991,

[2] S& +"ada,T& So#er& A metod for sol%ing sortest

#at#roblem on te net!or" !it fuzzy arc lengts,(F1A,%ol&

11 1 , K19K&(19H&1993&

[3] D& Dubois, L& Made& Systems of linear fuzzy constraints,Fuzzy Sets and Systems, 2, ##& 23(H,19&

.H1 && 5endes, A& *ama"ami& Fuzzy net!or" algoritms

and a##lications to trans#ortation #roblems, International

Con(ference on Com#utational Intelligence Control and

Automa(tion, ##& HH1(HH,1999&

[SI L&N&&

.>1 5& Delgado, N&?& erdegay, 5& A& ila& elating differ(ent

a##roaces to sol%e linear #rogramming #roblems !it

im#recise costs, Fuzzy Set and Systems, %ol& 23, ##& 22(HG,

199&

.3/ A.*ama"ami, && 5endes& 5ulticommodity trans#orta(

tion #lanning in rail!ays- sensibility analysis using fuzzy

matematical #rogramming teory 8in Portuguese,

S+MAP+, Sal%ador, 1993&

.I A& *ama"ami, && 5endes& A model tomulticommodity trans#ortation #roblem a##lied inrail!ays 8in Portuguese, C?AI+, io de Naneiro, 199>&

.91 A& osenfeld& Fuzzy sets and teir a##lications to cog(

niti%e and decision #rocesses, Academic Press, Oe! *or",

193=& Figure 1- Bra#ofte e0am#le

11-760

paper multicommodity transportation planning in railway fuzzy graph approach

Documents