Efficient path search in intermodal transportation optimization

Rochester Institute of Technology

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2004

Efficient path search in intermodal transportation

optimization

Daniel Y. Yankov

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Rochester Instituteof

Technology

Efficient Path Search in Intermodal Transportation

Optimization

A Thesis

Submitted inpartialfulfillmentoftherequirementsforthedegree of

MasterofScience in Industrial

Engineering

Inthe

DepartmentofIndustrialandSystems

Engineering

Kate Gleason Collegeof

Engineering

By

DanielY. Yankov

DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING

KATE GLEASON COLLEGE OF ENGINEERING

ROCHESTER INSTITUTE OF TECHNOLOGY

ROCHESTER, NEW YORK

CERTIFICATE OF APPROVAL

M.S. DEGREE THESIS

The M.S. Degree Thesis of Daniel

Y.

Yankov

has been examined and approved by the thesis committee

as satisfactory for the thesis requirement

for the Master of Science degree.

Approved by:

Moses Sudit

Dr. Moses Sudit, Ph.D., PriInary advisor

Sanjay

Joshi

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Abstract

As the economies ofthe worldbecome more interrelated and

Supply

Chains are

globalizing, the need arises to create efficient transportation network. This reality in

conjunction with conservation offuel and environmental friendliness gives rise to the research ofEfficient Intermodal Transportation System. Inparticular, theunderutilization

of railroads in the United States motivates us to research the development of optimal

proceduresinthe transportationof containersina rail network.

With this thesis we search for a _{cost, time} and _capacity effective algorithm for

solving transportation problem in a graph of intermodal centers (IMC's). We consider

discretemodel oftherealtimedynamic situation when allthearcs oftheinputgraph can

beaffected

by

changesin_{their costs, the transportation}meanshave limitedanddifferent

container capacities at each _IMC, and all the nodes

(IMC's)

can be visited more than onceeither

by

differenttransportmeansor atdifferenttime. This ismore general and real situationthan theones consideredintheliterature sofar.

The resultingoptimization problemis computationalintractable _(NP-hard),which

creates the _necessity to _develop, implement and test efficient heuristic optimization

techniques. Wewilluse Shortest Path Problem

(SPP)

asthebasis forthedevelopment of three heuristics. Because of the nature of the problem and _application, shortest path

procedures provide a_veryflexibleand_{computationally}efficienttechniqueforour model.

We will compare the three heuristics with the optimal solution for small size

problems forwhich wecouldfindoptimality. _Furthermore, we willdemonstratethatone

1. Formal Problem Statement and Definitions

According

to the researchers the intermodal transportation should be defined as:

"the shipment of cargo and movement of people

involving

more than one mode of transportation

during

a_single, seamless

journey

"[3].

Inthispaper we considercontainerizedfreighttransportationwithin a setofnodes

(IMC's)

where parallel arcs with different weights are allowed

-i.e. a container canbe transported

by

differentmodes ondifferentcosts withindifferenttransportation time.

The problem is quite general. To provide a practical solution

-an _easily computed algorithmforefficient path search_{considering cost,} time and_capacityconstraints within a system of_IMC's,we_{apply different}techniques and strategies.

First we calculate the optimal solution of a series of simplified transportation examples. Then we compare the results of the same problems generated with three

heuristic.

Finally

weanalyzetheresults under statistical and optimizationpoint of view.

Definitions:

Transportation Job (job): The amount of containers with the same start and destination IMC, which mustbetransferred

by

thesamedeadlinetimeT.

Transportationmode: Usedtransportationmeanlike_{train, truck, ship,}andairplane.

We definetheintermodaleffective pathalgorithm problem asfollows:

Mathematical

Modeling

oftheexampleproblem

Notation:

i= _1,...,N

-jobindex

I-job_set,I= {i : i= _1, ...,

N}

j

=

1,... , Oj

- _container

index, Oi=_{number of containers ofjob i}

Ji

-container setforjob_i,Ji=

(j

:

j

= _1,. ._., O;

}

r= ₁

,.. .,R

-transportationmeanindex k= 1,...,K-_{IMC index}

(k-1),k,(k+1)

=_Consecutive

(Middle)

IMC'sintheroute Ark= _Arrivaltime

oftransportationmeanratIMCk

Drk= _{Departtimeoftransportationmeanr}from_IMCk

T =

Deadlinetimeofjobi

Pirk=

Container capacityoftransportationmeanratIMCk

during

job i

Inputparameters:

ar = _fixed

cost ofoperationoftransportationmeanr

bjjk =

cost of_reloadingofcontainer_yofjob iatIMCk

Cijrk,(k+i)= cost oftransportationofcontainer

_/ ofjob i

by

transportationmeanrfrom \MCktok+l.

Variables:

Xjjik,(k+i) ' _1, ifcontainer_j ofjob / is transported _bytransportation mean r fromLV1C kto (k+1);

0,otherwise

yr : 1, if transportationmeanrisusedatall; 0,otherwise

Zyrk : _1, iftransportationmean ris changedfortransportationof container_jofjob i at IMC_At,

0,otherwise

ObjectiveFunction:

If Oi R K T -I R I \ Ni Oi R K

Min

IXIXlxijrk(k+1)*cijrl[(k+1)J+Ik

*.y>XIXX'

,=1 j=\ r=\ k=\ r=\ i=l j=\ r=\ k=\

Subjectto:

R R

2-1Xjfr(*-l)*"

2-1 Xyr/t(*+l)

r=l r=l

Continuity

offlow

0,ifk isa midIMC intheroute ofr; 1,if A: isthedestinationIMCofr;

-1

, if k isthestarting IMCofr, for Vi

j

,k;

N Oi K

2.

2^

2w

X

_Xyn

-Ly , for Vr =1. ._.R,andL

-sufficientlylarge

,=i j=\ k=\

Iftransportationrisused withitsfixedcost

3-X^-iu+

XXij,k,(k+I)^Z,t+I

+1 5forVij,k

r'=l

Iftransportationris changedatITC k

4- _{Atk*XiMk,),^Drk*Xijlk,(k+,)'} forVij\r,k

ThearrivaltimeatIMCk islessthan thedeparttimefromthatITC

5. A _{*r ^T} ,forVi,r,k

= _9,10 rlrk -\>ijrk,(k+l) 1 i

' ' ' '

Jobsdeadline limit

Bounds:

Xijrk,(k+i) -binary;

yr

-binary;

2. Literature review

Because oftheir fundamental theoretical consideration and _efficiency, first we

present the Shortest Path Tree

(SPT)

methods that are the most

interesting

in the

transportationfield.

Let G=_{(N, A)} is _{a simple} directed _graph, where Nis the set of_{the nodes,} of

cardinality, andA is theset ofthe _arcs, of_{cardinalitym,} c:A >R be afunctionwhich

assigns acost_c,ytoeach

(i,J)

eA.

Givenarootre _N,theSPTproblem consistsin

finding

adirectedtreersuchthat

the

(only)

path fromrto iin Tis one oftheshortest paths fromrtoiin_G, foreach/ e N which isconnected _{to r,}i.e. adirected pathfromrto/exists. It isproved intheliterature

[4],

[11]

thatafinitesolution exists ifand _{only if}thereisnodirectcycle of negative cost inG.

Let,

FS(i)

-{(i,j)

e

A}

and

BSy)

=

{(/',

i) e

A}

denote theforwardstar and the

backwardstar of_i, respectively, while _br- 1

-n , and b, ,=

1, Vi*r. Then the SPT is formulatedasfollows: min

^CrXi

2i_JCy Z-iJCij Di> (iJ)zBS(i) (i,y>(/)

x..>0, V(i,y)i,

andthedual SPT_(DSPT)withcostdualvariable_%iofi:

MaxO-i^

+

E^y

St

Kj-K^dr

V(/,y)e^.

2.1 Primalalgorithms

The"primal"

[1]

algorithms

basically

adapttheprimal network simplex approach.

At each _{step they}maintain a _spanning arborescence Tof root _r, and check ifthe cost

labels C(eitherthecostsofthepaths _outgoingfrom _r,or an upper bound forthese_costs)

areadual feasiblesolution, i.e.meetBellmanconditions:

Thesealgorithmsmaintaina set of candidate nodes _Q, whose incident arcs might

violate (1). At each _step a candidate node i is selected from _Q, and condition

(1)

is checked foreach arc ofthe forward star of_i; ifsome arc _(/,

j)

violates _{the condition, the}

costlabel

Cj

is updated_and/isinsertedinto Q.

The main difference inthe algorithms ofthis

family

is in the selection rule ofthe

candidate nodes in Q. Themostfamous algorithms ofthis typeare [1 1]: Bellman-Ford's

Algorithm, Floyd-Warshall's Algorithm, Dijkstra's Algorithm. Bellman and Ford's

algorithm findstheshortestpaths from one nodes (the source) to all theothers. The

key

insight is the repeated evaluation ofthefunctional equation _{v/-*/ (i.e.} the current shortest

length from the source to node k). Each iteration evaluates the functional equation for

each _V[kj

(2)

_using results from the previous iteration. The algorithm stops ifno result changes.

The functional equations ofthe length ofthe shortest path from the sources to

nodei:

v[s]=0

(2)

v[s]- min

{v[i]

+ cik'

(*'

^

~

exists)

Bellman-Ford Algorithm:

Step

0: Initialization:_v%;=_{0 if k}=

s;+<x>otherwise

Iterationcountert 1.

Step

1: Evaluation: Foreach nodekevaluate

Vk <~minivT

+d,k

:

(hk)exist}

if_y'k< , set

d[k]<_{thenumber of a}

neighboringnode _{i achieving}themin y'k

Step

2: Stopping: Terminate if _yk=

yk for V kor if t=the numberofnodes in

thegraph.

Step

3: Advance: Ifsome_v[k]changed at t< thenumber of_nodes, incrementt <

t+ ₁ andreturnto

Step

1.

Here

d[k]

denotes thenode_precedingk in thebestknown path froms to k. This

algorithmfindstheoptimal pathsoftorfewersteps.

[11]

The algorithm ofFloyd andWarshall searches for thepaths from all nodes to all

route calculations. Initialization sets the

ykl-Qkr Decause the ordy Pat from k to /

without intermediate nodes is the arc

(k,l)

with cost Ck,i . With the subsequent iterations

we concatenatetheprevious results (subpaths). At the_termination, all thepossibilities of

the minimization in functional equations have been _checked, and the final _v[k,lj are

optimal.

Dijkstra's algorithmis considered more efficient for _searchingfor shortest paths

fromone nodetoall_others, andallcosts are nonnegative

(cy

>_0).

Dijkstra's Algorithm:

Step

0: Initialization:v_pj=_{0if i}=_s

(source);

+aootherwise Mark all nodes _temporary and choose_p <

5 as the next _permanently

labelednode.

Step

1: Processing: Marknode/?permanent andfor_everyarc/edge

(p,i)

from/?to

a_{temporarynode,}update

Vt*-mm{y.,Vp

+

cJ

if_y changedin_value,setd[i]<p.

Step

2: Stopping: Ifno _temporary nodes _{remain, stop;} values y.now reflect the

requiredshortest pathlengths.

Step

3: Next Permanent: Choose as next_permanentlylabelednode/? a_temporary

node withleastcurrent value y. :y. =

minfy/. :i temporary}

Returnto

Step

1.

The new approach with the Dijkstra's algorithm is the_way ofcalculation ofthe

functional equations. Bellman-Ford algorithm evaluates functional equations

(2)

for all

nodes on all iterations. Dijkstra's method process outbound arcs and edges instead of

inbounds. At each iteration it makes one new node _p permanent and thus it processes

each arc/edge _only once. Once a node is classified permanent its vfpjand

dfpj

labels

never change again. So Dijkstra's algorithm isthemost efficient availablefor_computing

shortestpathsfromonenodetoall othersina graph with all arc costs nonnegative.

All these SPT algorithms select a candidate node with different _strategies, are knownas

label-correcting

orlist-search algorithms.Forimplementationofthe set

Q

_they

useFIFOqueue and are also namedL-queue.

Further development of these algorithms is in the _list, which combines the

properties ofthedatastructures queue andstack

-deque,

where addition anddeletionare

possible at either end ofthe list [1]. Although its worst-case _complexity is _0(n2n), this typeprovedtobeefficient on sparse an planar graphs.

Next development of the deque algorithms is in the partition ofthe candidate

nodes into two subsets

-Q'

and Q" _accordingto a suitable thresholdvalue. Q' contains onlynodes with label

(cost)

<threshold. When

Q

is_{empty, the threshold} isupdated and all thenodes in Q"withlabel< thresholdare moved into Q\ The aim is to increasethe probabilityof_selectingtheminimum label_node,

by

keeping

theoperational_simplicityof thelist-searchstrategies.

Bertsekas proposed an innovative algorithm based on L-threshold in 1993. It

selects the candidate nodes _trying to scan the nodes with a small label as _early as

possible. Ifthelabel ofthecandidate nodetobe insertedinto

Q

is lessthan theone ofthe node at theheadof_Q, thecandidatenode isinserted inthe_head; otherwise, it is inserted in the end of the data structure. This method can be combined with the threshold

approach andhas shown goodpracticalbehavior.

[1]

The topological _ordering algorithm of

Goldberg

and Radzik

(1993)

tries to

propagatethelabel_updatingofthenodesin

Q

by

means of atopological visit of a proper sub graphofG. It also runsin O

(mn)

time.

2.2 Dual AlgorithmsforSPT

The base ofthis algorithm

family

is the classical dual simplex _algorithm, which

maintains dual feasible basic solutions. Handlerand

Zang (1980)

proposed a _pioneering dual algorithmto solve SPPwiththe additional constraintthatfeasiblepaths musthavea

"weight"

lessthana givenvalueA.

[1]

the price vector n in such a _way as to always maintain the _{complementary} slackness

conditions.

Only

onepriceischangedat atime:

7T,

=minta. +c..:Me

BS(j)\

V/eiV\{r}.

Ifall thecycles in G havepositive cost andthe initialprice vector kis dual

feasible,

i.e.

Kj-K^dr

v(/'e'4'

(3)

thenthealgorithm producestheoptimal price vector.

Bertsekas introduced theinnovativeAuction Approach forthe SPT [8]. Withthis

algorithm the shortest path is searched _usingthe "primal-dual" approach: at each _step a candidatepathP outgoingfromrootrand afeasible dualvector n are_maintained,which

is dual

feasible,

i.e. meet _(3), and _satisfy the _{complementary} slackness conditions:

{i,j)eP^7Zj

=

Ki+Ci]-

W

If Preaches the destination t, i.e. aprimal feasible solution has been _found, P is one of

the shortest paths from r to t. Otherwise extension _contraction, and deletion operations areperformed attheendnode ofP

Recently, a new

family

of dual ascent _{algorithms, the}

Hanging

_family, was

proposed, which is _strictly related to Relaxation method. These algorithms maintain a partial shortest path tree rooted at _r, T =

(Nt, At) and a feasible dual vector _7r, which

satisfies

(3)

and (4). At the _beginning, Nt =

{r}

_and

At = _{0. To}

augment _T, each y'g Tcalled an out oftree node, looks foran_entering arc

(i,j)

that satisfies _(3), and is

Nr- Ifsuch an arc is found, _node/ is

"hanged"

to Tthrough _(i,j), sincethe shortest path

fromr_toy is found. To guaranteethat atleastone node canbe hangedto Tat each _step,

the algorithm makes apartition oftheset ofthearcs _comingto each _jg Ntintothesets of arcs_outgoingfrom_{tree nodes,} calledborderarcs_(BSr(j)), and external arcs

(BSe<j))

and

computetheminimum ofbothdualprices:

j3=mm\^j

+_{Cu:{i,jhBSbU)l}

7]=n\ft.+

Cij:{i,j)GBSeU} and

If

J

=

j3

for _some

node/,/can be hangedto T. If

ft

>

y

foreach _/e _NT, no

hanging

isperformed. Toavoid this situation abunchof global dual _updatingoperations

(classical, local,

global, selected, extended, and double _reprises) is introduced. All _they

maintainthe

feasibility

ofthepricevector and ensurethe_entryof at leastone new node

intoT.

They

arebasedontheglobalthreshold_gapoftheprices8:

5=

mm\j3-Ki:jeNT\

8 is the minimum increment to be added to the prices ofthe out ofthe tree nodes to

guaranteethatatleastone ofthemcan_{be hanged}toT.

2.3 Reoptimizationapproachesin SPT

One of the first efficient reoptimzation strategies in SP problem considers the

situation where a shortest pathtreerelative to a given origin_{r, say TT} is determined and

wehavetofind a new shortest pathtreewhen eithertheoriginis_changed, or_exactlyone

arc is given a new cost [1]. These two cases are of interest because of their _strong

practical application.

In the case when _exactly one arc is given a new _{cost, the} proposed algorithms

reoptimize the shortest path tree after the cost _change, and are

basically

extensions of

Dijkstra'sapproach.

Whentheoriginis changed to anode_s^r, Gallo's procedure recognizes that the

subtree of_TTrooted at s _certainlybelongsto thenew shortest pathtreerootedat_{s, say Ts,}

andthenitcomputes _Ts

by

means ofDijkstra'sorDial's _approach,

by

_usingthe

(optimum)

reducedcosts relativeto_TTinsteadoftheoriginal arc costs:

Cy

=Cu+7Ti-7rr V0V)e^.

(5)

Here, c*^ 'sarecalledtheupdatedcosts anddueto thedual

feasibility

oftheprice

vectorof_{Tr, c*y>0,}

V(z'.y')

e A, andhenceshortest-first search procedures can alsobe

appliedfornegativecosts.Wewill usea similar approachlater inone oftheheuristicswe

developed (FTU).

Further development of SPT leads to Gallo&Pallotino algorithms _[2], which

root _5, at each _step the algorithm extends the current shortest path _{tree, say} T*

by

inserting

new nodes andnewarcsuntil T*spans onto G. Thealgorithm considersthearcs

inthe cutset_separating T*fromthe_{remaining nodes,} and adds to T* a cutest_{arc, say(u,}

v), suchthat ubelongs to T*and vhas a minimum costlabel C'v. When _{(u, v)} andv are added to _T*, all the nodes in the subtree ofTrrooted at v are reachable from v through arcs

having

a zeroupdated _cost, i.e. all thesenodeshave the same minimum label as _v,

theentire subtreeis movedto T*

by

suitably

implementing

theset ofthecandidate nodes

Q-Although presented as a primal _approach, Gallo&Pallotino's algorithm can be

interpreted as a dual method.

Starting

fromthepartial shortest path tree _T*, it maintains

thedualfeasiblesolution at each_step

by

operatorsbasedon reprisefunctions.

2.4 Time Dependent Shortest Paths

The dynamic SP problems consider the factor "time"

and represent close to real

problems that arisein transportation. A travel time or

delay d\j(t)

is associated with each

arc _(/,/), so that iftis the

leaving

timefrom _i, thent +

d-^(t)

is the arrival time at node/.

Alsoatimedependentcost_c^(i)isassociatedwith_(//), and a_waitingcost_wx(t)represents the_possibilityandthepriceof_waitingat nodeiattimet.

Different models have been defined and analyzed in the literature

depending

on

the properties ofthe

delay

functions (e.g. continuous or _discrete), on the _possibility of

waitingatthenodes (e.g. no_{waiting,waiting}at each_{node,waiting only}attheroot_node),

and

depending

also on the choice ofthe

leaving

time from the root node (in _particular,

dynamicshortestpathsforafixed

leaving

timeorforallthepossible

leaving

times).

An

interesting

discretemodel is proposed

by

Ordaand Rom

(1990-91)

[1].

They

assumethat the timevariabletcan _{vary in}thediscreteset T=

{tl,

_{tl, tq),} and each

delay

function

d^(t)

is such that t+

d\j(t)

e _T, which isthe general casein transportation. Here

theyintroducetheSpace-Time Network_R,definedas follows:

V={iu:ieN,l<h<q};

E= _{(ih,/k):

(U)

zA,th+

dij{th)

=

tk,\<h<k<q},

and_waitingarcs _{(z'h, h+i)} which representthe_waiting ati from time _fhto time _th+\, andit

isgiventheunit timecost_Wj(fh), h= 1, ...

q-_{1. So R isastandard}

graph, with

\V\

=

\A\

<

(m+n)q,

_{with size is}pseudopolynomialwith respectto the size oftheoriginal graph G. It is provedthat the Minimum Cost Dynamic Path Problem can be solved in

0(|A|)

timewhateverthedelay, costand_waitingfunctions.

Theprocedure selectsthenodes ofR inchronological order and uses abucket-list

B=

{B\, B2,..., Bq)

toperform the_selectingoperations _(Z?h denotesthebucketcontent of

nodes to be visited attime _{instant th} ). The _{initial step is}

Bp

_={r} ifthegiven departure

timeis t

-tp, whilethe other buckets are empty. The _stop conditionis verified when all

the buckets are _{empty (when} this

happens,

the minimum ofthe labels associated with

each nodei^rgivestheoptimum path costfromrtoi).

Procedure Chrono-SPT

* _{typicaliteration}*

select/from_{Bh, Bh} :=

Bu\

{/}

; foreach

(ij)

e

FS^)

do

begin

k:=

*h+

dy(th)\

if

Ci(th)

+

Cij(th)<

Qitk)

then begin

Cj(tk)

'-Q(th)

+_Cyxth);

Pj{tk)

h',

if/g Bkthen_Bk=

Bk u

{/}

end

end; *

if waitingat/attime_th isallowed*

if

Q/A)

+_Wi(th)(th+i

-h)<

Cuth+i)

then

begin

Ci(th+\)

'=

Ci(th)

+_Wi(th)(th+l-_th),

Pi(th+l)

h',

ifz'g Bh+]thenBh+l=Bh+i u

{i}

end;

It is also proved that Chrono-SPTruns in

0(q

+

|A*|)

_time, where A*

is the set of

non-redundant arcs.

An

interesting

approachisderived consideringthe_{FIFO property [1]:}

an arc

(ij)

is said to be aFIFOarc if

leaving

/earlier guaranteesthat one will arriveno

later at/ along (ij). Or:

th+

^ij(th)

_{^ h}+

dmi

for_{any fh}<tk.

(6)

A similar _property can be imposed on the arc costs: an arc

(ij)

is said to be a Cost

Consistent

(CC)

arc if

leaving

z earlier _along

(ij)

does not cost more than

leaving

later,

andGisaCCgraphwhen allitsarcsareCCarcs. Or:

tu=

th+

dij(th)

and_/v=

h+

dm)

, foranyth<tk.

If G is both FIFO and _CC, i.e. when

leaving

earlier _{along any}arc allows one to

arrive no later and to _pay no more than

leaving

later,

then the concept of"dominated labels"

canbe introduced and exploited inthe algorithmic model Chrono-SPT. Suppose we have visited two different paths from a given origin node rto a certain node _z, and suppose thatthe two paths arrive ati attime thand attime_{tk, respectively,} with th< tk. Assumealso that the costs ofthe _{two paths,}

C^)

and

Qtk)

are such that

Cj(,a)

^ Qtk). In

this case, sincethegraph is bothFIFO and_CC, itisnot convenientto extendthe second path through _any arc

(ij)

_outgoing from _z, since this extension will not produce _any minimum cost path _going through node z: the cost ofthesecond _path, i.e. _Ci(tk), is a

so-called dominated label for node _z, and it can be ignored. Chrono-SPT can thus be simplifiedinsuch a_wayto_onlymaintainnon-dominatedlabels.

It is clear that the proposed Chrono-SPT algorithm is a _very close practical

approximation in casethe network is serviced

by

a set of means of same transportation type. For example, it will be _very useful in modeling subway or _railway transportation

systems, where the transportation time and cost

keep

constantdominancy. _However, the

case weconsideris broaderwiththeassumptions of parallel _arcs, i.e. the IMC's couldbe visited

by

different transportation means with different container capacities at different

time. Hence thenet ofIMC'sisneitherCost ConsistentnorFIFO graph.

2.5 Minimum Cost Flow Solvers

Based on the above-considered _algorithms, a number of_{single-commodity Min}

Cost Flow

(MCF)

codes havebeen proposed [10]. Practically, amongthemost efficient

ones are:

Relax Solver - developed

by

Dimitri Bertsekas and Paul Tseng. Relax

implements a primal dual algorithm, where at each iteration it tries to

construct a feasiblepath of arcs with zero reduced cost

(4)

with positive surplus to a node with negative surplus. The cost

reoptimizationis easilyperformed.

CS2 solver- developed

by

Andrew

Goldberg

andBoris Cherkassky. This

is also a primal dual _approach, with the difference that a _cost-scaling

phase allows to operate on arcs with nonzero reduced cost. The cost

reoptimizationisalso_veryeasy.

MCF ZIB solver

-developed

by

Andreas Lobel. This solver is a

specialized version ofthesimplex algorithm implemented

directly

on the

network.Theprimal simplex ismore suitedtocost reoptimizationthan the

dual _simplex,because it_easilyexploitstheprevious optimalbase.

MCF Cplex solverNetopt- developed

by

ILOG Co. Netopt is based on

primal and dual network simplex implementation. It's primal is very

efficient and offers fullreoptimization capabilities.

2.6 Intermodal Path Search

Most of the algorithms and researches in the intermodal path search consider

urbantransportationnetwork and passenger problems.

Ziliaskopoulos and Wardell proposed a _practically applicable algorithm for

intermodaltimepath search with_varietyoftransportationmodes.

[14]

It _{is very}efficient

for calculating routings of intelligent transportation systems _{operating in}

dynamically

changingconditions likeurbantransportationnetworks. "It computestheleast-timepaths

from_every origin_node, modeand departure time to thedestination_{node, considering} all

available modes oftransportation and _accounting for mode and arc _switching delays".

The algorithm begins atthe destinationnode and in alabel correcting fashion

iteratively

solves the _optimality equation

by

"scanning" all nodes that have the potential of

improving

atleastonelabelof another node. "Scanning"a nodezmeans_examiningall of

its predecessors and _{checking if extending} the path from the current node to its

predecessornodes provides abetterpath from these nodesto the destinationnodeforall

modes and time intervals. Ifsuch an extension provides a betterpath for a predecessor

node/ forat least one _{mode, time} interval and arc _{combination, node/ is} considered to

bescanned. All nodes eligible tobescanned are kept inalistcalled "scan

eligible"

(SE)

list,

a structure usedinstaticlabel correctingimplementations [1].

In the

beginning,

the SE list _only contains the destinationnodeN. The labels of

node Nare set equal to the cost of_switching from mode xto the exit mode_Xfand exit

nodeN2_; all other node labels are set to infinity. Inthe first

iteration,

all nodes that can

directly

reachNare updated and areinserted intheSElist._Next,thefirst_node/oftheSE

list is scanned

by

_{updating every}predecessor node ztonode/. Ifatleastone ofthelabels

ofL, is modified, thennode iis inserted inthe SElist. This step isrepeated untiltheSE

list is_emptyandthealgorithmterminates.

Although the travel times on the optimum paths _among the origin-destination

pairs aredifferentfor different departuretimesand_modes,mostofthemare_{topologically}

the same. _So, the algorithm _{simultaneously} updates all paths that are _{topologically}

similar. Itisprovedthat thecomputational_complexityofthisalgorithmis independentof

thenumber of modes andfixedschedule routes.

Ahuja and Orlin also investigated the dynamic shortest paths _minimizing travel

times and costs with main application in urban transportation environment [9].

They

assumethatthe cost of an arc depends

linearly

ontheminimum possible travel timeand

theexcesstimeforthe arci.e. q.

(t)

=_ocd. .+ Bg.

.

(t)

.

In a graphwithFIFO property_theyshowthat:

1. The min excess time walk problem (themin cost walkproblem) isNP

-complete.

2. The min cost walk problem can be solved in

0(Pe*/min(a,P))

time if

//. (t)>0,andifthereare nodirectedcycles with0travel time.

3. _{The running} time depends on the maximum excess time on _any arc _e*,

ratherthanonthe travel timesdy(t).

Inthe transitpassenger applications studied

by

Marcotte andNguen

[7]

has been

consideredthe combinationofpathprobabilities andtheroute strategy. Inadditiontothe

ordinary cost _cy a traversal cost _{Wj is} associated with each node, and so the hyperpath

problem. Theproposed algorithmforhypershort pathis basedonBellman's equationfor

In this hyper short problem, the arc travel costs are associated with the mean

waiting time at the _stop node and theroute strategy. _However, the computations of

all-shortest-paths _problems, such as Floyd-Warshall algorithm and the reoptimization

techniquesdeveloped

by

Galo andPallotino _{[2], [5],}

Gabini[6]

cannot be adapted to the

all-shortest-hyper paths problems. _Instead, specific reoptimization techniques have been

proposedbasedon reducedcosts withlabel_settingapproach[7].

Inthe case of_capacityoptimizations_theyrequirethata path cannotbeutilizedif

a shorter unsaturated pathis available. Thantheoptimum solutionisobtained

by

_addinga

suitably chosen_queuing

delay

_ay to thecost of each saturated arc (ij). The

delay

_alonga

pathisequalto thesumofdelays ofthearcs

forming

thatpath.

They

assumethattheusersofthe transportationnetwork select atravel _strategyat

theirstart node. Itisassociated with an ordered setofsuccessor nodes.At each_{node, the}

user selectsthefirst available_outgoing arcin this ordered _set, wherethe_probabilitythat

the arc is available is proportional to its capacity. Optimal strategies take into account

that an arc could be unavailable and must propose an alternative one. Although the

strategies aredeterministic,theirrealizationdependson arc_availabilitythatisstochastic.

The cost function is definedwithtwo processesthatare _closelyrelated

-loading

process and_pricing process. The first one generates the access _{probabilities, the} hyper

3.

Solution

Techniques

Our goal is

developing

an _easily computed algorithm with practical application

for calculation of the system's optimal transportation cost of container transportation

withinthebordersof network ofIMC's. Theconstraintsthatmustbesatisfied arelimited

container _capacity of the transportation means at each IMC _Pirk and the due time of

delivery

Ttforexecution of eachtransportationjob.

We use the obtained data for rail intermodal rail transportation from last year

independent study. The rail container transportation cost is

$0.03/ton/mile,

loading/unloading

operations = _{$20/container. The fixed}

costs of train operation are

assumed about$10,000.

We search for optimum costs a set ofexamples within a fixed network of 10

IMC's

by

6transportationmeansunderafixedtimeschedule and capacities(fig.l). Then

we calculate the same transportationjobs withthreeheuristics. _Finally, we compare the

obtainedresults.

Herearetheassumptionswe will follow:

1. Thecheapest routeisnotnecessarytheshortest one.

2. Acontainercanbe loadedonatransportationmeanifitarrives atthecurrent

IMC,atleasthalfanhour beforethedeparturetimeofthe transportationmean.

3. Withtheheuristicmodelsa newjob can start afterthepreviousis already

completed.

4. A transportation mean is considered used (i.e. assigned value _y =1

)

_even if it

hastransferredonecontainerwithin onesegment(betweentwoIMC's).

3.1 Optimal Cost SolutionwithCplex

First we solve a set of examples _consisting offourjobs _transferring from 82 to

112 containers within a fixed network of 10 IMC's

by

6 transportation means under a

fixed time schedule and capacities with Cplex. The transportation schedules with the

container capacities andprices areprepared on anExcel tablesheets (see Attachment 1).

Jobdataare assignedon separateExcel sheets(see Attachment_2), one per eachtest.

Inthis case we usethedeveloped C/C++ code(seeAttachment

3)

basedonCplex

(train Nombe r, Ar, Dr)

5

vtr3,18.19:30 -(tr5,20,21:30)

1-4-6-7-10

J^^^

\\ y

1'

1 _^^~"~-^^

y

y

\\

1 ': ._.

Ctrb, 0,8) ~"

-'-^

y

~~

_ >xr5,12'30,14)

_

It

. . .... .

*^

(tr3,80:..

i c i J.i. -'

.j JOU/1 Ctr35,24,25G0)

1

"

.

I* (tr5,

29:30,0:

job tables and provides the input files of all the ten experiments to Cplex. The optimal

feasiblesolutions arecalculated within 10-14hoursper experiment.

3.2 HeuristicModels

To avoid the

long

_running time needed

by

_Cplex, we create and compare the

behaviorofthreeheuristic models. In all of_them,the insight is intheorder of execution

ofthejobs. In the first two of_them, we rank the transportationjobs _according to their

"weight". Then we usethe developed C/C++code (see Attachment ₄₎ toprovide all the

feasible transportation routes that _satisfy the time constraints. Next we _{apply Shortest}

Path Solverbased onDijkstra's algorithmto selectthemost cost effective routes _among

alltimefeasibleones

by

_solvinga sequence of shortest path problems foreachjob anddo

the entirejob before the next in rank. The whole procedure takes _up to 1.5 hours per

experiment.

Thecommonheuristicalgorithm:

1. Calculate the "weight of each

job"

according to the heuristic rules and

rankthejobs.

2. Start execution thejob with the highest rank. Ifthe numberofcontainers

of executed job is less or equal to each of capacities of segments of

shortest path route calculated

by

_{the solver,} execute thejob end subtract

the numberof containers ofthe currentjob from the capacities of each

used segment(IMC).

3. If the number of containers of current job is greater than some of

capacities of the segments of the shortest path _route, calculate the

minimum possible_capacityand transport_up to this number of containers.

Subtractthenumberof containers ofthecurrentjob fromthe capacities of

each used segmentand cutoffthe _segment(s) with zero _capacity from the

network.

4. Solve _{the remaining} part of the job with the solver under the updated

network. Ifthe system does not have resource for transportation of some

cost of previous container transportation from the same job for each

undeliveredcontainerontime.

5. Continue withthenextjob inrank until allthejobs are executed.

6. To calculate the total transportation cost sum all the cost of thejobs and

thefixedcosts ofthe usedtransportationmeans.

The formal presentation ofthecommon algorithm ofGDC andEDT heuristics is

asfollows:

Step

0: Calculate job weightswi accordingto heuristicrules and rank_wr w4,

Set

<-0,i <- ₁

Step

1: Conduct SPPsearchfor_job(w;)withSolver If Solverreturnsnofeasibleroute

-apply penalty

Step

2: If_Oj<

p^for everyr,k, p^

<

p^

-o,, goto

Step

4 If 3_pirk=_0,cutoffthelink(k,k+l)/r, goto

Step

4

*

Step

3: If pirk<Oj,set_o{ <

o;, Oj

<

min _pirk ,Goto

Step

1

Step

4: Set _o{ < o;

-Oj,if_O; >0 goto_Stepl,

Elsei <- _i

+1,if i<Ngoto

Step

1

3.2.1

"Greedy

Distance x

Containers"

(GDC) Calculate the "weight of each

job"

by

multiplyingthe total transportation distance

by

number ofcontainers of eachjoband rank

the_{jobs according}to their"weights".

3.2.2 "Earliest DueTime" _(EDT)

-theweightis assigned_accordingto the duetime

-the

first, thehighest.

3.2.3 "Flexible TransportationUtilization" _(FTU) the "weight ofeach

job"

is thesame

like of _GDC, however we follow different algorithm. The idea is to utilize once used

transportation mean as much as possible. _{Hence,reducing}the number ofthe usedmeans

will allowus toeliminate someofthefixedcost(s).

Algorithm "FTU":

1. Calculate the "weight of each job"

according to GDC heuristic and rank the

jobs.

2. Start execution the job with the highest rank. Ifthe number of containers of

executed job is less or equal to each of capacities of the segments of the

shortest path route calculated

by

the solver, execute the job end subtract the

segment (IMC). Null all the variable costs of the used transportation means

over all thesegmentsinthenetwork.

3. Ifthenumberofcontainersofcurrentjob isgreaterthansome of capacities of

segments ofshortest path_route, calculate the minimum possible _capacity and

transport_upto this numberofcontainers. Subtractthenumber of containers of

the current job from the capacities of each used segment and cutoff the

segment(s) with zero _capacityfrom thenetwork. Null all the variable costs of

theusedtransportationmeans over allthesegments inthenetwork.

4. Solvethe_remainingpartofthejobwiththesolver undertheupdated network.

Tocalculate the final transportation _cost, use the original variable costs of all

the segments. Ifthe systemdoesnot haveresourcefortransportationof some

of the containers (the solver returns

infinity

_cost),

imply

_penalty of double

cost of previous container transportation from the same job for each

undelivered containeron time.

5. Continue withthenextjob inrankuntil allthejobsare executed.

6. To calculate the total transportation cost sum all the cost of thejobs andthe

fixedcostsoftheusedtransportationmeans.

The formalpresentationofFTU algorithmis asfollows:

Step

0: Calculate jobweights wi accordingtoheuristicrules and rank_wrw4,

Set

-0, i*- ₁

Step

1: Conduct SPPsearchfor_job(w;)with Solver

If Solverreturnsno feasibleroute

-apply penalty

Step

2: If_Oj<

p^for everyr,k, p^

-p^

-O;, set all_cijrk,(k+l)=_0of

every linkof

everyusedmode_r, goto

Step

4

If 3_pirk=_0,cutoffthelink (k,k+l)/r,goto

Step

4

Step

3: If_p^<oi; set_Oj <

o; , ot

-min_pkk, Goto

Step

1

Step

4: Set_o; <

o{

-ovifo; >0gotoStepl, Else i<

i+1, ifi<Ngoto

Step

1

We will review the calculation procedures of Test 1. The files with the

calculations of the _remaining nine tests are loaded on the attached CD named "Opt Sol

The optimal solutionofthefirsttest- Objective Function Value= _{$115,014 (see}

Attachment

5)

isprovidedin37855.02sec(10.52h).

Thenwe conductthe testswiththe threeheuristics.

Firstwe calculatetherank ofthejobs for GDCandFTUheuristics (Table 1).

TesM

ft Rank

1 19 890 16910 1

2 20 700 14000 4

3 21 710 14910 3

4 22 680 14960 2

Table 1.

Then we _{apply C-code} for calculation ofthe time feasible routes _using the data

from tables "traindatabase Testl"

and "jobdata Testl". Based on its results (see

Attachment

6)

wepreparethe tableoflistoftimefeasibleroutes(Table 2). Itallows usto

prepare_easilytheoriginaldata fortheShortest Path Solver (see Attachment 7).

List of Time Feasible Routes Test #1

Num Job Num route Start IMC Routes Destination

1 4_5 4_1 7_1 7_2 8_2 10_2

2 4_5 4_4 5_4 8_4 8_2 10_2

3 4_5 4_4 5_4 8_4 10_4

4 3_1 4_1 7_1 8_2 10_2

5 3_i 3_2 6_2 7_2 8_ 10_2

2 1 4_5 4_1 7_1 9_1

2 2 3_1 3_2 6_2 6_3 9_3

2 3 3_1 4_1 7_1 9_1

2 4 3_1 3_2 6_2 7_2 7_1 9_1

3 1 2 4_3 7_3 6_3 9_3

4 1 2 5_6 5_4 8_4 8_2 10_2

4 2 2 5_6 8_6 8_2 10_2

4 3 2 5_6 5_4 8_4 10_4

4 4 2 5_6 8_6 8_4 10_4

4 5 2 4_3 4_1 7_1 8_2 10_2

4 6 2 4_4 4_1 7_1 8_2 10_2

4 7 2 4_3 7_3 7_2 8_2 10_2

4 8 2 4_3 4_4 5_4 8_4 8_2 10_2

4 9 2 4_4 5_4 8_4 8_2 10_2

4 10 2 4_3 4_4 5_4 8_4 10_4

4 11 2 4_4 5_4 8_4 10_4

4 12 2 5_6 8_6 7_6 8_2 10_2

Having

loaded the feasible graph and job data in a

file,

we start _searching the

shortestpath calculations withGDCheuristic:

Thejob withthehighestrankisNo 1

-19containersfrom IMC 1 to IMC 10. The

resultfromthesolverisroute

1 -* 4/5 - 4/4 - 5/4 -? 8/4 -* 10

(node/by

train) with unitcost

$890/container.

Since all of the capacities of the used trains are greater then the numbers of

containers tobetransferredi.e._19,thejob is completed andthecost calculated 19x890 =

$16,910._{After reducing}thecapacities oftheusedsegmentsin "traindatabaseTestl"

table

with _19, weproceed withthenextjob inrank,No 4- 22

containers fromIMC 2to IMC

10.

Thesolver returnstheroute

2-*5/6 - 5/4-> 8/4-> 10_{with unit cost}$680/container.

Because the _capacityoftrain4 atIMC 8 is 5we transfer_{exactly 5} containers on

thisroute.The costofthisoperation is5 x680= $3,400._Then

we cutthelink 8/4 10

offfrom the graph in the solver _memory, update the capacities ofthe used _links, and

search againforthecheapestroute.Theresultisroute

2-* 5/6-? 8/6->8/2-? 10_{with unit cost}$680/container.

Allthecapacities>thanthe_{remaining 17} containers ofthe_job,so weperformthe

transferwithcost of17x680=$1_1,560.

After reducing the capacities ofthe used

links,

we proceed with the nextjob in

rank

-No 3 - 21

containers from IMC 2to IMC 9. The shortest path returned fromthe

solveris

2->4/3 -?4/1 ->7/1 -9 _{with unit cost}$710/container.

All thecapacitiesoftheusedlinks>_21, soweperformthe transferwith cost21 x

710= $14,910.

After updatingthe capacities, we searchforthenext cheapest path-job

No 2- 20

containers fromIMC 1 toIMC 9.

Theproposed solutionfromthesolverisroute

1 ->3/1 ->3/2- 6/2-* 6/3-*9_{with unit cost}$700/container.

All the capacities ofthese links > _20, so we transfer all the _{20 containers,} and

To perform all the four jobs we used all the six trains with total fixed costs =

$61,250. _So, theTotal Cost ofthistest

including

all thefourjobsperformedinthisorder

is60,780+61,250=_$122,030.

Toperformthe testwiththenextheuristic_EDT,we use its ranking ofthejobs: 3

- 2 - 4 - 1. The _procedure is the same like with the previous heuristic. In these

calculations we encounter2 cutoffs of used links and use all ofthe six trains. The Total

Costof allthejobs is 61,120+_61,250=_$122,370.

The procedure of FTU heuristic deserves more attention. We use the same

sequencelike GDCheuristic 1-4-3-2. Withthefirstjob wehavethesame proposed

route

1 -?4/5-?4/4- 5/4

-*8/4-> 10_withthesameunitcost$890/container.

Allthecapacities oftheusedtrains> _19, and we execute thejob. Its cost is 19 x

890=_$16,910.The

newinsightistheassignof zero cost of allthelinks inthegraph with

theusedtrainsi.e.fortrain4:_{2-4/4->5/4->8/4->10/4,}

andtrain5: l-4/5-6/5-+7/5->10/5.

Afterthese_temporarycost_{changes, the}algorithm will givepreferenceto thelinks

operated

by

these _alreadyused trains. The idea is to utilize once used train as much as

possible.With ihecalculationofthejob costs we will usetheoriginal costsof alltheused

links.

Thenextinrankjob isNo 4

-22 containers fromIMC 2 toIMC 1 0. Thecheapest

calculatedrouteis

2- 4/4- 5/4-? 8/4 -* 10. The capacity_oflink8/4 - 10is5, _{so wetransfer}5

containersinthisroute.Thecost is 5x 870= _$4,350.

After cutting offlink 8/4 > _{10 and}

updating the capacities, we proceed with a

newsearch forthe_{remaining 1 7} containers.Therespondis

2 4/4 5/4 * _8/4 8/2 * _{10. We perform transfer of 10 containers and}

reducethe container capacities. Link 4/4 ? 5/4iscutoffbecauseof zero _capacity and all

thelinks ofthe_newlyusedtrain 2 aremade with zero cost. Thecost ofthissub-jobis 10

x 890- $ 8,900. Then _we continue withpath search forthe

remaining 7 containers. The

2 ->4/4

-4/1 -* 7/1 -> 7/2 ->8/2 -? 10. Allthe _{capacities >} 7 _{and we perform}

the transfer. We updatethecapacities and make all thelinksoftrain 1 zero. The cost of

thissub-jobis7x910=_$6,370.

Nextjob in rank is No 3 - 21 _containers from IMC 2 to IMC 9. The

proposed

routeis

2-*4/3 -*7/3 ->6/3 ->9_{with unit cost}$834/container.

Thecapacities >21 andwetransferall thecontainers. The cost ofthejob is21 x

834= _$17,514.

After reducingthecapacities oftheusedlinks and_makingthe cost of all

linksoftrain3 zero,weproceedwithlast job- No 4: 20

containers from IMC 1 to IMC

9. Thecalculatedpathis

1 -? 3/1 -*4/1 ? _7/1 *>

9with unit cost$840/container.Allthecapacities ofthe

links >_20, so we performthejob. Thecostis20x 840=_$16,800.

With this procedure we used trains _{1, 2, 3, 4,} and _5, with sum offixed costs

-$50,750. _So, the Total Cost of all thejobs oftest 1 executed in this _{away is 70,844} +

4. Solution Results

Thesolutionresultsof allten testsare summarizedinthe

following

_{Table 3:}

HEURISTICS

GDC EDT I=TU

TEST# OptimalCost Routes TotalCost Routes TotalCost Routes TotalCost

1 $115,014 1-4-3-2 122,030 3-2-4-1 122,370 1-4-3-2 121,594

% Deviation from Optimum 6.10% 6.40% 5.72%

2 $125,338 1-3-4-2 130,420 3-2-4-1 143,180 1-3-4-2 126,380

% Deviation from Optimum 4.05% 14.24% 0.83%

3 $113,998 4-3-2-1 133,300 3-2-4-1 138,880 4-3-2-1 136,988

% Deviation from Optimum 16.93% 21.83% 20.17%

4 $104,530 3-2-1-4 122,640 3-2-4-1 124,700 3-2-1-4 126,128

%Deviationfrom Optimum 17.33% 19.30% 20.66%

5 $114,560 2-3-1-4 124,230 3-2-4-1 128,360 2-3-1-4 119,644

%Deviation from Optimum 8.44% 12.05% 4.44%

6 $113,570 4-1-3-2 123,570 3-2-4-1 130,346 4-1-3-2 113,570

% Deviation from Optimum 8.81% 14.77% 0.00%

7 $118,388 1-4-3-2 131,580 3-2-1-4 131,580 1-4-3-2 130,472

% Deviation from Optimum 11.14% 11.14% 10.21%

8 $137,947 1-3-4-2 145,255 3-2-1-4 145,015 1-3-4-2 144,605

% Deviation from Optimum 5.30% 5.12% 4.83%

9 $135,165 4-2-1-3 143,045 3-2-1-4 142,405 4-2-1-3 169,495

% Deviation from Optimum 5.83% 5.36% 25.40%

10 $138,113 2-3-4-1 156,138 3-2-1-4 146,570 2-3-4-1 139,507

% Deviation from Optimum 13.05% 6.12% 1.01%

5. Analysis ofthe results

5.1 Statisticalanalysis.

Tocomparetheresultsfrom Table 3, firstwe conduct a_two-wayanalysisof variancewithMinitab. Theexperimentisasfollows:

Hohypothesis: No difference amongthe4types of_{calculations,}

or

Hohypothesis:_{No difference among}the 10transportation tests

versusthealternative

Hahypothesis: Atleastonediffers fromatleastone other.

Two-wayANOVA: TC versus _Heuristic, test

Analysis of Variance for TC

Source

Heuristic

test

Error

Total

Heuristic

1

2

3

4

test

1

2

3

4

5

6

7

9

10

DF SS MS

3 1.141E+09 380476651

9 4.314E+09 479363491

27 1.150E+09 42582359

39 6.605E+09

F P

8.94 0.000

11.26 0.000

Mean

121661

133221

135341

132838

Mean

120252

131330

130789

119500

121699

120264

128005

143206

147528

145082

Individual 95% CI

+

+-( * ₎

+-120000

+-126000

+.

132000

+-138000

Individual 95% CI

+

+-( * ₎

(-

+-120000

+-130000

h-140000

+-150000

TheANOVAresults showhigh F-valueandlowP-valuethatindicatethat thenull

hypothesis mustberejected. Withp-value< a =0.05thereisenough evidence that atleast

To evaluate _exactly which one differs from which of_{the rest, we conduct paired}

T-tests .

Paired T-Testand CI: _OPT, EDT

Paired T for OPT

-EDT

N Mean StDev SE Mean

OPT 10 121661 11808 3734

EDT 10 135341 8900 2814

Difference 10 -13679 6213 1965

95% upper bound for mean difference: -10078

T-Test of mean difference = _{0 (vs < 0):}

T-Value =

-6.96 P-Value = _0.000

Paired T-Testand CI: _OPT, GDC

Paired T for OPT

-GDC

N Mean StDev SE Mean

OPT 10 121661 11808 3734

GDC 10 133221 11490 3634

Difference 10 -11559 5246 1659

95% upper bound for mean difference: -8518

T-Test of mean difference = _{0 (vs} < 0): T-Value = _{-6.97 P-Value} = _0.000

Paired T-Testand CI: _OPT, FTU

Paired T for OPT

-FTU

N Mean StDev SE Mean

OPT 10 121661 11808 3734

FTU 10 132838 16014 5064

Difference 10 -11177 11486 3632

95% upper bound for mean difference: -4519

T-Test of mean difference = _{0 (vs < 0):}

T-Value =

-3.08 P-Value = _0.007

Paired T-Testand CI: _GDC, FTU

Paired T for GDC

-FTU

N Mean StDev SE Mean

GDC 10 133221 11490 3634

FTU 10 132838 16014 5064

Difference 10 383 11253 3559

95% upper bound for mean difference: 6906

T-Test of mean difference = _{0 (vs < 0): T-Value}

Paired T-Testand CI: _EDT, FTU

Paired T for EDT FTU

N Mean StDev SE Mean

EDT 10 135341 8900 2814

FTU 10 132838 16014 5064

Difference 10 2502 12331 3899

95% upper bound for mean difference: 9650

T-Test of mean difference = 0 (vs < 0) : T-Value = _{0.64 P-Value 0.731}

Paired T-TestandCI: _EDT, GDC

Paired T for EDT - _GDC

N Mean StDev SE Mean

EDT 10 135341 8900 2814

GDC 10 133221 11490 3634

Difference 10 2120 5853 1851

95% upper bound for mean difference: 5513

T-Test of mean difference = 0 (vs < 0): T-Value = _{1.15 P-Value} = _0.859

The results show that there is significant difference in the Optimal calculations

compared to the other three Heuristics _{calculations,} and there is no evidence for

significantdifference amongthe threeheuristics.

5.2

Engineering

analysis.

Although statisticallyequivalent, theresults produced

by

the threeheuristics are

differentfrom_engineeringpointof view. Toperform allthejobsofthe 10testsGDC and

EDTheuristicsutilizeallthe sixtrainsineachtest(Table 4).

NumberofUsedTransportation Means

Test Optimal GDC EDT FTU

1 5 6 6 5

2 5 6 6 5

3 4 6 6 6

4 4 6 6 6

5 5 6 6 5

6 5 6 6 5

7 4 6 6 5

8 5 6 6 5

9 5 6 6 6

10 5 6 6 5

Total trains 47 60 60 53

Average 4.7 6 6 5.3

Withthese experimentstheresults ofFTU are 12% better- it

utilizesaverage5.3

trains. Theadvantages ofthisapproachare clear:

Higherutilizationofthetransportationmeans

Lowertrainmaintenancecosts and overhaul expenses

Smallerpersonaland crew number

Reducedprobabilitiesfor breakdowns

To compare the transportation mode _utilization, we conduct paired T-test with

data from Table 4. _Again, the null hypothesis is that there is no difference in the train

utilization, and the alternative is that _they are not equal with 95% confidence intervals

(CI).

Paired T-Test andCI:_GDC, FTU

Paired T for GDC

-FTU

N Mean StDev SE Mean

GDC 10 6.000 0.000 0.000

FTU 10 5.300 0.483 0.153

Difference 10 0.700 0.483 0.153

95% CI for mean difference: (0.354, 1.046)

T-Test of mean difference = _{0 (vs} not = _{0): T-Value} = _{4.58 P-Value} = _0.001

Paired T-Test andCI: _GDC, OPT

Paired T for GDC

-OPT

N Mean StDev SE Mean

GDC 10 6.000 0.000 0.000

OPT 10 4.700 0.483 0.153

Difference 10 1.300 0.483 0.153

95% CI for mean difference: (0.954, 1.646)

T-Test of mean difference = 0 (vs not = _{0): T-Value} = _{8.51 P-Value} = _0.000

Paired T-Test andCI: _FTU, OPT

Paired T for FTU OPT

FTU OPT

Difference

95% CI for mean difference: (-0.003, 1.203)

T-Test of mean difference = _{0 (vs not} = _{0) : T-Value =2.25 P-Value} = _0.051 N Mean StDev SE Mean

10 5.300 0.483 0.153

10 4.700 0.483 0.153

GDC and EDT heuristics have the same utilization numbers. The results ofthe

experiment showthat heuristics GDC and EDT are _{different to FTU} and _OPT, and that

6.

Conclusions

The results fromthe conductedtransportation tests showthatin 70% ofthe cases

thedifferencebetween theTotal Costs calculated

by

FTUheuristic andthe Optimal one

is lessthan 10%. In addition, the transportationutilization withFTUis_nearly90%ofthe

optimal. The main advantageis the shorter calculation time. With these 10 tests offour

jobs therequired timefor calculation ofFTUis in average 1/10 ofthe time required for

calculation ofOptimal Cost.

Therefore, in our opinion the proposed FTU algorithm can be used as a good

approximationwith practical applicationinthecalculation oftotal transportationcostina

7.

Proposals

for future research

The proposed deterministic model and heuristic algorithms are good enough for

practical calculation ofoptimum freighttransportationcostina system ofIMC's. Incase

ofleak ofinformation for the container

loads,

we would suggest stochastic approach to

beapplied. Inthiscase _everytransportationmean will serve as an independentagentthat

will have to bid to get _{any job.}

Trying

to be competitive and _profitable, it will have to

optimizetheoffered costs.

The deterministic and stochastic approaches could be _combined, so that the

deterministic one calculates the loads and costs oftheagent

by

some level ofutilization

that guarantees the useofthe agent, recall 60% ofloadability. Thestochastic procedure

will calculatethe

biding

prices forthe _{remaining 40%}thatwill improvethe_{profitability}

Bibliography

[1]

Pallottino _S., M. Scutella

(1997)

ShortestPath Algorithmsin Transportation models:

classicalandinnovativeaspects.

[2]

Pallottino _S., M. Scutella

(2001)

A new algorithmfor _reoptimizing shortest paths

whenthearccostchange.

[3]

Jones W.B., C.R.Cassady, R.O.Bowden (2000),

Developing

a standarddefinition _of

intermodaltransportation.

[4] Bixby

R.E.

(1981-82)

Combinatorial_{Optimization,}Northwestern

University

notes

[5]

Chabini_{I., Glenn, A., Pallottino, S.,} M.Scutella(2001),Reoptimization Algorithms for

Minimum-Time Path ProblemsinDynamic Networks.

[6]

Galo G., S. _{Pallottino, M.Scutella,} A. _Frangioni, F. _Farinaccio, Transportation and

logistics:ModelsandAlgorithms.

[7]

Marcotte P., S. _{Nguyen,(1995)} Hyperpath Formulations of Traffic Assignment

Problem

[8]

De Leone _R., D. _Pretolani,

(1998)

Auction Algorithms for Shortest Hyperpath

Problems.

[9]

Ahuja_R., J. Orlin, S. Pallottino, M.Scutella,

(2001)

Dynamic Shortest Paths

Minimizing

Travel Timesand Costs.

[10]

Frangioni _A., A. _Manca,

(2001)

A Computational

Study

_{of Cost Reoptimization for}

Min Cost Flow Problems

[11]

Rardin_R, Optimizationin OperationsResearch, Prentice-Hall,2000

[12]

Askin _R.,J. _Goldberg,Designand_{Analysis ofLean ProductionSystems,} John

Wiley

&_Sons, 2002

[13]

Mendenhall _W., R. _Beaver, M. _Beaver,Introduction to

Probability

and

Statistics-11th

edition, Brooks/Coole-ThomsonLearning, 2003

[14]

Ziliaskopoulos _A., W. _Wardell, Design and Implementation of an Intermodal

Optimum Path AlgorithmforMultimodalNetworks withDynamic Arc Travel Times and

Switching

Delays

(1997)

http://trans.civil.nwu.edu/~thanasis/itdsp/itdsp.html

Table ofContents

Abstract 3

1. Formal Problem StatementandDefinitions 4

2. Literature Review 7

2. 1 Primalalgorithms 7

2.2 DualalgorithmsforSPT 10

2.3 Reoptimizationapproachesin SPT 12

2.4 Timedependantshortest paths 13

2.5 Minimumcostflowsolvers 15

2.6 Intermodalpathsearch 16

3. Solution Techniques 19

3.1 Optimalcost solution withCplex 19

3.2 Heuristicmodels 21

4. Solution Results 28

5. Analysis oftheResults 29

5. 1 Statisticalanalysis 29

5.2_Engineeringanalysis 31

6. Conclusions 34

7. ProposalsforFuture Research 35

Bibliography

36

Attachments 1+2/traindataandjob data/ Testl

24

train#₍r₎ startITC Depart_(Dr) end ITC 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 1 3 4 7 3 6 7 8 2 4 7 6 2 4 5 8 1 4 6 7 2 5 8 7 5.00 10.30 15.00 20.30 11.30 17.00 21.00 25.30 4.00 10.00 15.30 19.30 7.00 13.00 17.30 23.00 8.00 14.00 21.30 25.30 6.00 11.30 17.00 21.30 3 4 7 9 6 7 8 10 4 7 6 9 4 5 8 10 4 6 7 10 5 8 7 9 Arrive_(Ar) 9.00 13.30 19.00 23.00 15.30 19.30 24.00 28.30 8.30 14.00 18.00 22.30 11.30 16.00 21.30 26.00 12.30 20.00 24.00 29.30 10.00 15.30 20.00 24.30 cont.cap.(p) 21 41 31 51 42 32 52 62 33 43 53 43 44 34 54 24 35 45 25 55 26 46 36 46 tr.var.cost 240 180 240 180 240 144 180 180 270 240 144 180 270 180 240 180 270 360 144 300 240 240 180 180

tr.fixedcost tr

10000 10000 10000 10000 10500 10500 10500 10500 10000 10000 10000 10000 10000 10000 10000 10000 10250 10250 10250 10250 10500 10500 10500 10500 change cost 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

job# startITC endITC #cont starttime deadline jobdist

1 1 10 19 4.00 29.00 890

2 1 9 20 4.00 23.45 700

3 2 9 21 4.00 22.45 710

4 2 10

total#cont

22

82

TEST #2

train_#(r) startITC Depart_(Dr) endITC Arrive(Ar)cont.cap.(p)tr.var.costtr.fixedcosttr.change cost

1 1 5.00 3 9.00 41 240 10000 20

1 3 10.30 4 13.30 31 180 10000 20

1 4 15.00 7 19.00 21 240 10000 20

1 7 20.30 9 23.00 51 180 10000 20

2 3 11.30 6 15.30 22 240 10500 20

2 6 17.00 7 19.30 32 144 10500 20

2 7 21.00 8 24.00 52 180 10500 20

2 8 25.30 10 28.30 42 180 10500 20

3 2 4.00 4 8.30 23 270 10000 20

3 4 10.00 7 14.00 43 240 10000 20

3 7 15.30 6 18.00 53 144 10000 20

3 6 19.30 9 22.30 33 180 10000 20

4 2 7.00 4 11.30 44 270 10000 20

4 4 13.00 5 16.00 54 180 10000 20

4 5 17.30 8 21.30 34 240 10000 20

4 8 23.00 10 26.00 24 180 10000 20

5 1 8.00 4 12.30 35 270 10250 20

5 4 14.00 6 20.00 45 360 10250 20

5 6 21.30 7 24.00 25 144 10250 20

5 7 25.30 10 29.30 55 300 10250 20

6 2 6.00 5 10.00 26 240 10500 20

6 5 11.30 8 15.30 46 240 10500 20

6 8 17.00 7 20.00 36 180 10500 20

6 7 21.30 9 24.30 46 180 10500 20

job# start ITC end ITC #cont starttime deadline

1 1 10 22 4.00 29.00

2 1 9 23 4.00 23.45

3 2 9 24 4.00 23.15

4 2 10

total#cont 25

94

TEST #3

train#₍ r₎ startITC Depart_(Dr) end ITC Arrive_(Ar)cont.ca

1 1 5.00 3 9.00 41

1 3 10.30 4 13.30 31

1 4 15.00 7 19.00 21

1 7 20.30 9 23.00 51

2 3 11.30 6 15.30 22

2 6 17.00 7 19.30 32

2 7 21.00 8 24.00 52

2 8 25.30 10 28.30 42

3 2 4.00 4 8.30 23

3 4 10.00 7 14.00 43

3 7 15.30 6 18.00 53

3 6 19.30 9 22.30 33

4 2 7.00 4 11.30 44

4 4 13.00 5 16.00 54

4 5 17.30 8 21.30 34

4 8 23.00 10 26.00 24

5 1 8.00 4 12.30 35

5 4 14.00 6 20.00 45

5 6 21.30 7 24.00 25

5 7 25.30 10 29.30 55

6 2 6.00 5 10.00 26

6 5 11.30 8 15.30 46

6 8 17.00 7 20.00 36

6 7 21.30 9 24.30 46

job# startITC endITC #cont starttime deadline

1 1 10 15 4.00 29.00

2 1 9 20 4.00 23.45

3 2 9 25 4.00 23.15

4 2 10

total#cont

30

90

4.00 29.00

240 10000

180 10000

240 10000

180 10000

240 10500

144 10500

180 10500

180 10500

270 10000

240 10000

144 10000

180 10000

270 10000

180 10000

240 10000

180 10000

270 10250

360 10250

144 10250

300 10250

240 10500

240 10500

180 10500

180 10500

20

20

20

20

20

20

20

20

20

20

20

20

20

20

20

20 20

20

20

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