Multimodal transport network systems interface, interaction coordination: A specification for control systems integration

M&l.

Comput.

Modelling

Pergamon 08957177(95)00147-6

Vol. 22, No. 4-7, pp. 415-429, 1995 Copyright@1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/95-$9.50 + 0.00

Multimodal Transport Network Systems Interface, Interaction Coordination: A Specification for Control Systems Integration 0. A. ODUWOLE Port Authority of New York & New Jersey New York, NY, U.S.A. and Transportation Systems Engineering Research Center New Jersey Institute of Technology, Newark, NJ, U.S.A.

Abstract-h a general multimodal system, interface and interaction coordination involves the deliberate assembly of modal movement units MMU’s (i.e., vessel, vehicle, or rail car) activities into schedules such that interfaces are facilitated, and the joint variable cost of coordination of intermodal interaction is minimized. In this paper, two methods of nonstochastic scheduling coordination are described, and their application in a globally integrated transport system network model is shown [11. The first method expresses the scheduling problem in the form of the Hitchcock transportation problem in accordance with a method derived by Dantzig and Pulkerson (21. The second method is derived from network flow theory and is based on the ‘out-of-kilter’ method devised by Fulkerson [3]. This paper seeks to provide a basis for implementing dedicated logistics submodels such as MMU and staff scheduling submodels, and to examine MMU stafhng requirements on a selected real system. These minimize the ratio of labor for scheduling modal intersctions, and modal coordination to the frequency of arrivals/departures at a consolidation node, thereby maximizing MMU availability, and minimizing MMU system maintenance costs. Keywords-Networks,

Multimodal, Programming, Scheduling, IVHS.

NOMENCLATURE P

Unit price of all discrete transfer element (commodity) types p E P. p=l,...,P

a

a&, a E A arc designation

W

Total transportation cost per unit flow rate of arc (o,d) ‘w E W

0 = (0)

Set of origin nodes

D = {d}

Set of destination

NO ND

Number of source (origin) nodes Number of demand (destination) nodes

0

Set of origin nodes o, o E 0 C N on the network

wa(.)

nodes

D

Set of destination

T

Set of transshipment

nodes d, d E D C_ N terminal

nodes t,

teT h’

Set of all discrete transfer element types over the multimodal network by index with cardinality n& Set of all transportation rnEM,rn=l,...,M

Address for correspondence:

Cost function of flow on arc a&, aEA

WC.)

Cost function associated modal switch r E T

ak 'od

The number of weight loads per unit time of DTE k on arc a between origin-destination pair o. d

mod

modes

0. A. Oduwole, 58 Pierson Road South,

The number of modal movement units (MMU) per unit time on arc a Maplewood,

NJ 07040, U.S.A. Typeset

415

with

by A.-m

0. A. ODUWOLE

416

Sk

The density of DTE k in lbs/ft3

fk

hk

The inventory conveyance cost per unit time per weight of DTE k

(fk (0)) E 0, o E 0 is a vector of, supplies

gk

(gk (d)) E d, d E D is a vector of demands

u

(U(a)), a E A is vector of mutual arc capacities

so

The weight capacity of an MMU (vehicle) (Ibs) / the volume capacity of an MMU (ft3)

1. INTRODUCTION Intelligent Vehicle and Highway Systems (IVHS) presents a unique opportunity for improving multimodal transport network systems interface and coordination. One of the most significant aspects is the opportunity it presents for fully integrated transport systems to better integrate and thus improve total systems operation and usage (i.e., passenger ridership or goods movement). By ensuring that appropriate advanced technology applications are identified and implemented by transport suppliers, users would have substantially improved information about service frequency, arrival times, cost, etc., and thus, would be in a better-informed position to consider alternative modes of transport. IVHS offers the transport systems users, particularly the transit community, a unique opportunity to capture the choice user (a user who will be willing to select an alternate mode other than an inefficient choice) if the presence of real time information about frequencies, arrival times, delays, and headway are made available. At the same time, the multimodal infrastructure (facilities) operators (Federal, State, Local, or Private) would benefit, not only from reductions in nonoptimal trips, but also increased communication with systems vehicular operators, who may operate several hundred fleets of vehicles that can act as probes over the network to monitor (manually or automatically) network performance parameters and flow conditions. IVHS offers the technology for many systems to accomplish their fundamental objectives of dependability and energy efficient movement of goods and people. The challenge is to properly identify and implement the appropriate IVHS concepts and control precedence sequences that will facilitate full integration of IVHS technology into existing multimodal networks systems. The purpose of this paper is to provide a basis for implementing “Multimodal Transport Network Systems Interface, Interaction Algorithms” for structuring Real Time Response and Control Systems, and investigate other potential applications in passenger and public transportation. The problem considered is a nonlinear optimization type with transportation constraints. Such problems appear in several areas of applications, mainly in logistics and transportation systems. A significant body of literature exists on the solution of linear transportation problems. For an early account, see [4]. For later works, see [5-71. Section 2 of this paper presents a brief description of the types of IVHS technologies considered, based on their potential benefit to the multimodal transport network system. Section 2.1 provides a detailed problem statement and the mathematical structure of discrete transfer elements (DTE) flow on the network. Section 2.2 gives the definitional relationships of modal threshold and the flow structure of the system. Section 2.3 expresses the scheduling problem in the form of the Hitchcock transportation problem in accordance with an algorithm derived by Dantzig and Fulkerson [2]. Section 2.4 develops a solution to the problem through an algorithm derived from network flow theory based on the ‘out-of-kilter’ method devised by Fulkerson [3].

2. TYPES

OF SYSTEMS

CONSIDERED

IVHS technologies utilize electronic communications and information systems concepts and methodologies. They can generally be classified as user-based, vehicular-based, and infrastructure-based systems. The user-based implementation focuses on providing real time information to the systems users (i.e., transit, auto, or shippers) in order to enable the user as a utility maximizing decision making unit information on cost functions, modal movement unit (MMU)

MultimodalTransportNetworkSystems

417

frequencies, headways, and travel times. The vehicular and infrastructure safer and efficient operations of the multimodal system which can greatly and improve service reliability. The following is a catalogue multimodal transport network

of a wide variety system.

of IVHS concepts

based concepts enable reduce operation costs

which might

be applied

to a

MODAL MOVEMENT UNIT AREA NETWORK. Standardized

communications protocol and vehicular harness that permits inputs from various on-board sensors and navigational devices such as AVL, package and passenger identification counters to interact, without duplication of existing

devices

and connections.

(a new standard,

The elimination

SAE J1708, recently

of connection

adopted)

duplication

can be specified

reduces

system

complexity

in new bus fleet procurements

for example. AUTOMATEI) VEHICLE LOCATION (AVL).

Positioning

Systems

the reliability information

(GPS),

of time transfer

AVL systems

Using transponders, monitored

and switches between

at central

vehicle odometers dispatching

center

modes and provide real-time

and/or

Global

can facilitate

arrival-departure

to users.

AUTOMATED DEMAND RESPONSIVE SYSTEMS. By utilizing

path algorithm

processing,

vehicular

route diversion

AVL in conjunction with shortrest. and fleet dispatching can be implement,ed.

AUTOMATET) VEHICLE IDENTIFICATION(AVI). Small electronic devices that contain and transmit informat,ion identifying each specific vehicle in a fleet could enable High Occupancy Vehicles (HOV) or priority status verification and significantly improve the efficiency of operations. AVT can also enable a system-wide probe system, whereby the speed of individual modal movement units can be monitored as they traverse the system’s infrastructure. Computerized monitoring can detect traffic congestion automatically and provide means of early detection incidents for t,hr, multimodal isystems monitoring systems. ON-BOARD AUTOMATIC GUIDANCE. Simple laser or radar sensors installed on the vehicleti can warn the operator of the presence of other vehicles in close proximity. More complex and futuristic syst,ems may also use such control devices to control steering, acceleration and decelerat~ion of vehicles. VEHICULAR UNIT DIAGNOSTICS. On board

location,

and operating

condition

sensors automatically monitor of pertinent equipment. This information

vehicular capacity, is transmitted to

a central control center and is compared with a predetermined operating schedule. Deviation can be noted and transmitted to both operator and dispatcher. Corrective instructions can be automatically issued to the operator to restore service or schedule adherence. Data on vehicle status are stored in the computers so that schedules, analyses, and plans can be revised losing actual data. 2.1.

Problem Statement

The concept of integrating a real time response and control system into a multimodal transport network system is worthy of research. As a control model working in conjunction with a corresponding real time system, it can be used to organize the latter activities from the viewpoint of optimal cost. The ability to calculate consignment sizes, mode frequencies, headway timings and schedules in a very short time clearly adds considerable flexibility to a real time system’s operations. Additionally, the instantaneous reaction times and amplitude of computational analysis allows for a wider range of possibilities to be considered, facilitating the optimal selection of decision trajectories. This paper presents the development of the general nonstochastic multimodal system interface, interac:tion, and coordination without time windows or priority scheduling considerations. Activities performed by different modes, such as vehicle repositioning maneuvers, equipment relocation or idleness, do not identify with a specific system’s activities-but result rather from

0. A. ODUWOLE

418

overall pattern

and degree of disorder

in the sequences

of trip and other activities

undertaken

by

the system. In a general multimodal system, interface and interaction coordination involves the deliberate assembly of modal movement units MMU’s (i.e., vessel, vehicle, or rail car) activities into schedules such that interaction

interfaces

are facilitated

is minimized.

and the joint

In this paper,

variable

two methods

cost of coordination

of nonstochastic

are described, and their application in the globally integrated shown [l]. The first method expresses the scheduling problem

scheduling

of intermodal coordination

transport system network model is in the form of the Hitchcock trans-

portation problem in accordance with a method derived by [8]. The second method is derived from network flow theory and is based on the ‘out-of-kilter’ method devised by [3] and described in Section

2.6 of this paper.

For the purpose

of scheduling,

all trips are assembled

into origin-destination

(O-D)

matrices

showing

desired arrival and departure times; certain stationary orderings of the trips are also The application of the two scheduling methods in the overall model and position necessary. of joint variable costs in relation to other costs of the system are discussed in this section. A description of the scheduling problem in mathematical terms follows in Section 2.2. 2.2. The Nonstochastic

Scheduling Problem

In order to define the node-arc incidence relationships-assumed transfer elements (DTE), we define (a full definition of notations

to be identical for all discrete is provided in the Nomenclature):

@

is the cumulative cost function associated composite arc combination over a channel with modal switching cost.

6+(O) = {d E D 1 (o,d) E A}

is the set of destination nodes which have incident arcs with origin node o. We will use n, to denote the cardinality of this set.

6-td) = {o E 0 1 (o,d) E A}

is the set of origin nodes which have incident arcs with destination node d. We will use nd to denote the cardinality of this set.

We define the pure multiple

Minimize Subject

DTE transportation

problem

with R

as follows:

F(a) to:

(1) &

= P(o),

QOE 0,

k E K,

(2)

x$

=

sk@h

QdE D,

k E K,

(3)

02 &

5 u”(a),

Qa E A,

k E K,

(4)

c

I U(a),

Qa E A.

c dE6+(0) c

0&5-(d)

x$

(5)

kEK

In order notation.

to develop Constraints

the algorithms, we next need to express (2)-(5) are rewritten as

sx = s,

Gx=g,

0 I Ix I

IL,

the problem

OlExIU.

in compact

matrix

(6)

I is the nK x nK identity matrix x E RnK is the vector [x1 x2 . . . . . . xk IT, u E RnK is avector [u’ u2 . . . . . . uKIT,sERoK isavector [sl s2 . . . . . . sKIT,andFisthe

Multimodal Transport

block diagonal matrix

419

Network Systems

-S S2 ..

s=

.. SK_ The blocks are defined as follows: for each DTE k E K let Sk = S where S is the mg x mc]mD matrix a111

a12

...

QlmLJ a21

cY22

...

Q!ZrnD QIm, 1

The entries ~oij are given by oyij =

1,

am,2

...

%WWD

1.

if j E S+(i)

{ 0,

(7)

otherwise.

The remaining entries of p are all zeros. G is the block-diagonal matrix Gl G2

G=

.. 4 GK

[ For each DTE k E K, we define G” as the mD

P ml31

..

I

P mo2

I322

Pl2

=

matrix

021

Al Gk

x mOmD

.. P

P2mD

1mD

P m0mD

where the entries @Q are given by

Pij =

1,

if i E 6-(j),

0,

otherwise.

(8)

Let the volume of flow between an origin point, o E 0, and destination point, d E D, on a network arc be z,d. It will be meaningful in most cases to know the volume by mode &d. Thus, the total flow shown in the matrix below is the sum over all modes. xod

=

c x:d.

(9)

m

From a supply-demand framework, it is also desirable to know the routing over the network ~2~ so that precise links can be determined. This depends upon freight consolidation on nodes on the network. Predicting C&d,zrd”d,zr& is arduous without knowledge of the DTEs to be transported, without a concept of the behavior of the network, or without formal definition of the network formation and variation process of the modes, network entities, and the behavior of the commodity on the network. Different commodities typically travel by different modes. To determine how much transfer there will be between any two sets of nodes (nodal regions, i.e., levels, cities, or zones) one must understand how the market elements operates for each of the goods (DTEs).

0.

420

A. ODUWOLE

When aggregated over all transfer elements (commodities) x?dr

k total modal flows are produced:

=

k In a particular mode of transport there are typically economies of scale with respect to consignment size, implying that the larger the consignment size the lower the cost. This can be due to the fixed cost of handling, consolidation of consignment, loading and unloading, delivery methods, etc. This is usually reflected in the tariff rate assessed on the consignment for transport.

It

is typically necessary to transfer a prespecified minimum quantity in order to qualify for a lower tariff rate. Thus, tariffs are quoted for a specific modal service along with a minimum consignment size, This is illustrated graphically as shown in the figure below. The difference in level of service rendered by consignment size q is an added dimension to the prediction problem, since in practice, each mode-consignment

size combination is different. Thus, we wish to forecast zydq(le).

s

TRUCK PARCEL

DOLLAR/MILE

L, ,AfXA

RAIL TOFC CONTAINER

-

UNiT TRA’!WK

CARRIER

r SHIPMENT IN POUNDS

1 BARGE I

Figure 1. Modal threshold comparison.

2.2.1.

Modal

thresholds

The traditional approach to freight transportation analysis (via survey methods) where detailed surveys have established in detail the volumes of flow xOd, is typically futile due to the variational processes inherent in the system. Within a particular mode of transport, there typically exist economies of scale with respect to movement size. That is, the larger the movement size, the lower the cost. This is due to the fixed cost of handling, consolidation, and processing, etc. It is usually reflected in the tariff rate charged to the shipper for transport. Typically, it is necessary to ship a specified minimum quantity in order to qualify for the lower tariff rate. Thus, the network is in an interminable state of fluctuation, and the equilibrium between different modes (Figure l), and different spatial configuration and market conditions are stochastic. As the system expands and contracts, the various transfer elements (in this case commodities) change, and the choice of the mode continues to shift. Thus the problem must be expanded to include the various sets of discrete transfer elements, k E K, making up the economy. Expressing the probtrn) in the same terminology used previously, it is necessary to be able to predict the modal flows z,d , m(k), m E M, k E K, given the sources of the DTEs at each producing source node Si (supply) on the network and the sink G$ (demand) for each transfer element at each consuming node. 2.2.2,

Definitions

Consider a network consisting of a set of nodes N, a set of arcs A, A c N x N x M, a set of modes M and a set of modal switches or transfers T, T c A x A. We denote their cardinality, nN,

Multimodal

Transport

Network Systems

421

nA, ?%M, and nT, respectively. With each arc a, a E A, there is an associated cost function w,(.) which depends on the volume of DTEs on the arc or, possibly, on the volume of DTEs on the other arcs of the networks.

Similarly,

there is an associated

cost function

w,(.)

with each modal

switch

T E T. The discrete transfer elements transported over the multimodal network are denoted by the index lo, k E K, where K is the set of all discrete transfer elements considered, which is of cardinality nK. Each DTE is shipped from origins o, o E 0 C N, to destinations d, d E D C N, of the network. The demand for each DTE for all origin/destination (O/D) pairs is specified by a set of O/D matrices. The mode choice for each DTE is indicated by defining for each of these O/D demand.

matrices?

a subset

It is assumed

that

the clemand M(k),

matrix

of modes which are permitted demand

of modes that

the corresponding

exogenously.

with DTE k E K, where m(k) is a subset

associated

the rjet of all subsets

for transporting

and mode choice are determined

Let g”“ck) be

of modes that, belong

t.o

DTE k.

are used to transport

The flows of DTE k on the globally integrated multimodal network of the induced flows of the DTE on links and modal switches.

is denoted

bv x’ and consist

(11) The flow of all the DTEs

on the multimodal

network

is denoted

by .r = (rk),

k e Ii,

and is a

vector of dimension ‘nk(nA X 72~). The average cost functions w(r:) on the links, and w(x:) on the transfers, corrcsponcl to a given flow vector s. The average cost functions for DTE type k are noted, similar to the notation for the flonr :r’, ulk i .E E k, where (1’) and ‘u! = (.w”.)~ k E K, is the vector of average cost functions of dimension n~(n,~l x rl7.j. Tlw total cost of the flow on arc a, a E A for the cost of DTE type k, k E K, is the product ~w(.c,h).r~; the total cost of the flow on transfer 7, r E T, is w($) .x:. The total cost of flows of all DTEs over the multimodal network is the function Cpthat we seek to minimize. @=

c (

kEK

the In order following destmation equations

over

cw(Z;)X;+

~Ul(+r$

aEA

TET

=tu(.r)?r

(13)

i

set of flows which satisfy the conservation of the flow and nonnegativity constraints. to write these constraints for the multi-DTE multimodal network defined above, t,he notation is used. Let R,,m(k) denote the set of paths that lead from origin o, o C: 0 to d, d G D, by using only modes of m(k) E M(k), k E K. The conservation of’ flow are t,hen c T,L

,

rER,,,

L)

m(k) 1

OE 0,

PT = $,d

where pr is the flow on channel pr > 0,

d E D,

r. The nonnegativity

T E R;@‘,

o E 0,

m(k)

constraints

d E D,

m(k)

E M(k),

k E Ii-,

(l-1)

p E P.

(15)

are E M(k),

Let A be the set of flows II: that satisfy (14) and (15). Since the conservation of flow equations are stated in the space of channel flows for notational purposes, the specification of A requires t-he relation between arc flows and channel flows which is

zk = a

c TERK

&,P,,

aEA

k E K,

0. A.

422

where Rk = Urn(k) E M(k)

ODUWOLE

U o E 0 U d E D K,mdck)is the set of all channels that may be used by

the DTE Ic, and L

=

1,

if a E r,

0,

otherwise

(17)

is the indicator function which identifies the arcs of a particular channel. Similarly, the flows on modal switching are 2: =

c

r ET,

&P,,

where &Z, =

1,

ifrET,

0,

otherwise.

k E K,

(18)

The modal switch r belongs to the channel r if the two arcs that define the switch belong to it. The quantity p?(k) will signify the flow on channel r of DTE k E K transported by mode m. Channel flows are related to arc flows through the usual arc-channel incidence matrix, an element which is 6,m,(“)= Consequently,

1,

if a E r and allows transport of DTE k by mode m,

0,

otherwise. xm@) _ T -

c

s,m,(“)pT(“).

(20)

(21)

rER”‘(k)

We may also speak of the unit travel time on channel r for DTE k transported by mode m; we denote this by IL?(~) and observe that

(22) the delivered price of DTE k transported Central to the consignors’ decision making is I$?‘, between origin destination pair i, j over channel r by mode m. The minimum delivered price of the DTE k transferred on arc a between nodes i, j by mode m is denoted by P%yp’. Elements necessary for the construction of expressions for the delivered price are the price of the DTE k at origin i, denoted by pf; the value of time to the consignors for the transport of DTE Ok; and the fraction of the carrier’s cost passed on to consignors for DTE transported by mode m, denoted by f?‘(lc). The final delivered price to the consignor is determined by the combination of DTE, origin, mode, and channel selected, and can be expressed as

pm(k) = ql
v” @ + pf +

J$Ck)+ em(k).

(23)

The term z?(‘) can be considered to be the posted tariff between the origin-destination pair i,g. The tranyportation demands are separable, negative exponential functions of the form

Tmck) = A! Bi 23

0: exp

D;

(-ok PZCk)) ,

%i, k, m,

where At, Bt, and Bk are parameters that must be calibrated to ensure that (24) accurately describes transportation demand for the condition being analyzed. Note that (24) was derived by Wilson [9] by maximizing the entropy of the DTE flow pattern subject to appropriate constraints. These constraints and derivations were highlighted in the Demand Models’ maximum entropy section of the literature review in [l, Chapter 31. A detailed problem statement is presented in [l, Chapter 61. A mathematical programming formulation of the problem is then followed by an in-depth analysis of the cost functions associated

Multimodal

Transport

Network Systems

423

cl SK

1.1.

.

.

.

Figure 2. Modal movement decision sequence.

with the arcs and channels of the multimodal network system. This analysis will determine the necessary conditions for concavity/convexity, the relevancy of nonlinearity in practice, and the effect nonlinearity imposes on the solution strategies. In general, 0 n D is not necessarily empty. Every node o E 0 has a nonstochastic, prearranged table of departures times to nodes d E D. These tables shall be represented in a matrix R, where W&, = required departure time of the nth trip from o E 0 to d E D. The problem is to arrange a schedule so that all trip demands are fulfilled, while at the same time, modal interaction is maximized, and the joint variable cost of coordination is minimized. All MMU’s arriving, unloaded and processed at a node o before or at W&n are eligible for selection for the nth movement from o to d. So also are any MMU’s that can be repositioned at o from elsewhere before W&n. For each DTE consignment K there is then a set S, of actual and potential arrivals at o such that one of this set may be selected for the nth trip to d. For every DTE consignment K. < K~,,, a scheduling decision must be made. The overall schedule depends on the chain of such decisions and each decision is influenced by the results of previous decisions. The problem may be represented as in Figure 2: the object is to choose a set of decisions 2 = {& : K = 1,2, . . . , tcmax} such that the schedule is optimal. The joint variable cost of modal interface coordination is composed of the summation of the cost of those MMU’s activities which are not classified as trips such as back hauling. Let ‘it = the

0. A. ODUWOLE

424

cost of the MMU time attributed to modal interface and coordination and wP = the other resource costs associated with repositioning maneuvers, etc. Then the joint variables cost of coordination 20, = wt + wP. Now if wj is the cost of MMU time attributed to trips, and is constant for a given set of trips, then to minimize w, is to minimize the summation of u/j + wt + wP, i.e., mxw+w,,

min where m = number

of MMUs employed

(25)

to fulfill the trip demands

on the channel

and 27,represents

the delay cost per MMU for the interval of time involved. From this, it is deduced that minimization of w, can be effected by a combination of minimizing the size of the MMU fleet and the direct resource costs consumed in MMU repositioning. 2.3. The Da&zig The theory

and F’ulkerson Method

of the Dantzig

and Fulkerson

scheduling

method

[8] is stated

as follows:

Let T

be the set of trip times &,d, o E 0, d E D. Let Z be the set of repositioning times &d, o E 0, d E D. Whereas, tOd is interpreted as the time to travel from o to d, cod is the time to reposition at o to d. The matrix 52 = [&&&I displays the list of departure times of consignments from each o E 0. The matrix 2 = [to&] can be derived by adding W&n + tOd Vo, d, n to represent arrival times at each destination d E D. The problem is to arrange the numbers W&n into m sequences such that (i) each sequence (ii) if

Wo,dlnl

is monotone increasing, are consecutive numbers

in any one of the m sequences

< Wo2dzn2

Woldlnl

“Joldln,

>

can be affected (iii) m is optimal. The solution A,,

+co2dl

; i.e., two

in sufficient

to this problem = number

COUSecUtive

consignments

time, and

is indicated

of times

Wodn

in [8]. Defining =

occurs for o constant

a

of times &dn = p occurs for d constant,

hod = number

Xaood = number of repositionings of vehicles to o for departure at time p; then it may be stated

that

c

that

arrived

xa*pd

<

c xaofid
nod,

Vo, Vn, and at d at time cy,

xcropd > 0,

(26)

Ad

By introducing nonnegative slack variables xao, T@d and G = c,,, may be cast as a Hitchcock transportation problem

Subject

Vd, Vn,

for all schedules

a,*

Minimize

then W02&n2 if repositioning

can Only OCCUr

cp,d

&&d,

the problem

z = C C Ucr@d Xaopd, a,~ 4,d to:

c if!

it

(27)

hoad

+ ‘?@d =

&dr

^Ipd 2

Xao@d

+ xao

Lo,

Xao

=

2

0,

(28)

0,

(29)

ij

xao+~=Chao=Chgd=CYBd+0,

(290

c+,o Cod > cm

Ad

-

Od

=+ ‘1Lczoj3d =

W,

cod < a0 -

Pd

*

lad,

>

0,

(30)

B,d (31)

&ro/3d

=

Multimodal

where w is the standing cost (excluding total

Transport

Network Systems

cost per MMU for the interval,

MMU-time

costs)

of reposition

425

10d is the incremental

resource consumption

o and d, Xaopd 2 0, and &,fld

between

is the

delay cost per unit time at o and d.

In minimizing the fictitious transportation cost, the solution will in effect endeavor to introduce as many feasible repositionings as possible into the solution. Infeasible values of &ofld appearing in the final solution correspond to &d > CEO- pd and are equivalent to a nonzero -i’fld or xao, and therefore,

to the beginning

or termination

m. The structure

of a sequence

of the solution

is such

that Ypd

c Xao = c a,0 Pd is optimized Actual

in relation

to the total

MMU sequences

in [4] these Aat9

sequences

(Aad

or

>

2.4.

Scheduling

=

m

(32)

MMU time and repositioning

can be constructed

may not necessarily

from the optimal be unique

cost. transportation

due to interchanges

solution; of MMU’s

as shown at nodes

0).

Using

Flow Theory

Network

In general, network flow theory is concerned with the maximizations of flows in capacityconstrained networks. As demonstrated in [4] the Hitchcock transportation problem discussed in Section 2.3 is a particular formulation of the general network flow problem. Adhering to their notation, a directed network or linear graph G = [N; A] consists of a collection of N elements 2, y, . . (called nodes) together with a subset A of the ordered pairs (x, y) elements taken from N (called arcs). The capacity of an arc is defined as a function c(x, y) from A to nonnegative real numbers. Letting s and t represent two distinguished nodes of N, a static arc flow of value ‘u from s (source node) to t (sink node) in [N; A] is a function f from A to nonnegative reals that satisfies f

C

C

f(x,y)-

YEA(~)

f(x,y)=

YEWZ)

V,

ifx=s,

0,

ifx=s,t,

--VT

(33)

if x = t,

WY E A, Rx:,Y) L 4x, Y), A(x) = {Y E iv I (x,Y) E Al, B(x) = {Y E N I (Y, x) E A}.

(34)

The static maximal flow problem is that of maximizing the variable v subject to flow constraints (33) and (34). The general minimal cost flow problem on the other hand is that of constructing a flow in which each arc has a specified transport cost per unit flow and capacity constraints. Adopting the notation that for any function g(.) from A to reals g(x, y) = CCz,y)ECX,Y) g(X, Y) the minimal cost, flow problem is described in linear programming format as follows. Assuming the network [N; A] has source-nodes S, intermediate nodes R, and sink-nodes T; arc capacities c(x, y), lower bounds l(x, y); arc costs a(x, y) per unit flow; supplies a(x) for x E S and demands b(x) for x E T; it is required to Minimize Subject

c

a(x, y) f(x, y)

to: ;(x.

(35)

N) - f(N, x) i a(x),

x E s,

(36)

f(x, N) - f(N, x) = 0,

x E R,

(37)

f(x,N)

- f(N,x)

5 -b(z),

x E T,

qx,Y)

5 f(X,Y)

5 C(S>Y)>

r,y E A.

and

(38) 139)

0. A. ODUWOLE

426

The work of Fulkerson scribes the ‘out-of-kilter’ The application following

and Dreyfus [lo] develops fundamental theorems of network flow and demethod for solving minimal cost flow problems devised by Fulkerson [ll].

of this method

to the multimodal

scheduling

problem

shall be discussed

in the

sections.

2.5. Expressing the Globally Integrated Multimodal Scheduling Problem as One Network Flow Bellmore,

Bennington

and Lufore

[12] indicate

Transport

a more general

formulation

System

Network

of the scheduling

problem in terms of network flow; the authors considered problems more general than those solved by Dar&zig and Fulkerson [8] (f or instance, those in which there is an insufficient number of MMU’s for a required weightings, utilization negative

pattern

of movements

can be treated).

In general

or utilities, to the individual MMU movements, the method which can maximize the total utility gained. Repositioning utilities.

The relevant

details

of this approach

terms,

by assigning

derives a policy of MMU costs are represented by

are now summarized.

As defined earlier in our notation, let T be the set of transit times tad, a E 0, d E D, and Z be the set of repositioning times &d, o E 0, d E D. Adhering to the notation of Ford and Fulkerson [4], a directed graph G is defined to consist of nodes s, X, Y, t. Each node z E X is described by a pair of positive integers (a,~), and a node z = (a,~) exists if, and only if, at least One W,,& = CL Similarly, nodes y E Y are defined by pairs of positive integers (0, d) for the entries of matrix E = [&&], whereas before each &n = Wodn + tad. Source arcs connect source node x E X with sink node t. Primary arcs connect (p, d); if for a particular o and d there are arcs and node pairs are defined. Secondary (CY,o) for which CY2 /3 + &. This condition

with each node s; sink arcs connect each node y E Y each defined node (CV,O)with its corresponding node nP entries W&n of s1 equal to cy, a total of nP primary arcs are defined between all pairs of nodes (/3, d) and is outlined in Figure 3.

For each primary arc, a capacity of unity is defined, with pretation being that the corresponding trip can be fulfilled unnecessary and a minimum of zero MMU’s (corresponding each secondary source and sink arc, a capacity of infinity and

a lower bound of zero: the interby one MMU, further MMU’s are to nonfulfillment) is realistic. For a lower bound of zero are specified.

Now, ifs, xi, x2,. . . , xn, t is a sequence of distinct nodes of the network such that (x,, z,+i) for o = 1 to n-l, (s, xi) and (x,, t) are arcs, the sequence of nodes and arcs s, (s, xi), 22,. . . , (xn, t), t is then defined as a channel chain. Each channel chain derived from G represents a feasible MMU schedule routing, and there exist a one-to-one correspondence between activity schedules for m MMU’s and flows of value m from s to t. If it is required that all demands for trips in the problem (as in the present application) become one of minimizing the total cost on the channel chain, then in the final solution, there will be a (capacity) flow unity through each primary arc, the level of flow will correspond to the optimal number of MMU’s, and the total cost will depend on the values attributed to the individual unit-flow arc costs. Where the number of vehicles is restricted, it may not be possible to arrange a flow of unity through all primary arcs; but the total cost can still be minimized, the final choice of primary arcs being dependent on the relative unit-flow cost assigned to all arcs. In [4], it is suggested that by assigning a definite utility, or negative cost, to primary arcs and a utility of zero to all other arcs, a form of priority can be established between arcs whereby the total utility can always be maximized even where there is an insufficient number of MMU’s. As explained above, the intention in the present application is to minimize w, = wt + wP, where w, is the joint variable cost of schedule coordination, wt is the cost of MMU time attributed to coordination, and wP is the total cost of other resources consumed in coordination. If the correct repositioning costs are assigned to all secondary source and sink arcs, and zero costs to all primary arcs, the situation is similar to that treated in [4]; in either case, the optimal solution inserts as many primary arcs as possible in each channel.

Multimodal Transport Network Systems

427

I

\

I

\

I

\

I

\

I

\

I

sink arcs

\

/

\ \ \

I

I

I

\

I

I

\

1,

/

I

/

/

\

/

I/

/

\

\ \

\ \ II

\

/

\

Y \ I

\

// \

\ I

__-_______ \

\ \

I

/

\

I

I

// \

/

I

/

\

I \

I

Figure 3. Network flow problem.

As shown in [2] an insufficient number (m’) of MMU’s can be catered for by defining an auxiliary node s*, corresponding to an artificial source of MMU’s, and an additional (s*, s) of capacity m’. It is clear that maximal flow through the network is now determined by either the capacity constraint m’ on the arc (s*, s) or by the total capacity of all primary arcs in parallel, whichever is less. For any value of m’, a maximal flow minimal-cost solution to the problem will minimize the total coordination cost 20, with an actual number of MMU’s m 5 m’. Where MMU numbers are not restricted, it is then required to find the smallest value of w, over all values of m’ such that all demands are fulfilled (i.e., capacity flow through all primary arcs). 2.6.

Solving the Network Flow Problem:

The Out-of-Kilter

Method

The out-of-kilter method is a general algorithm form computing optimal network flows. The relevant theory is presented in [4]; it is summarized here. First, it is convenient to add a return arc (t, s*) to the network with Z(t, s*) = 0, c(t, s*) and a(t, s*) negatively large: The problem is then said to be in circulation form and is stated as follows: Minimize c

a(~, y) f(s, y)

(40)

A

Subject to: f(z, n) - f(n, z) = 0, 1(x, Y) I f(x, Y) I c(z, YL

Vx E N, x,Y E A.

and

(41) (42)

This is a. linear program, and it can be shown that if dual variables r(x) are assigned to equations (41), and y(x, y) to capacity constraints equation (42), then the optimality criterion

0. A. ODUWOLE

428

for primal and dual problems combined consists of the following: for 5, y E

T(Y)- n(z) < a(z,Y) =+f(s,

Y> =

0,

T(Y) - r(x) > a(? Y) =+ 0x9 Y) = c(z7 Y).

A (43)

For given values of K, let a (z, y) = a(z, y) + r(x) - r(y) for the defined arcs (5, y). Then for the given y’s and circulation f, an arc (z, y) is in just one of the following states:

(a) (P) (Y) (al) WI

(Yl) (4 WV (9)

f(G Y) = UT Y> Z(?Y) I f(T Y) I CC?Y> fh Y) = C(GY) fk?Y) < l(GY) f(GY) < i(Z,Y) f(? Y) < CC?Y) f(GY) > l(GY) f(? Y) > C(TY> f(%Y) > ck Y,).

(44)

If an arc is in one of the states Q, p, or y, it is said to be in kilter; if it is not, it is said to be out of kilter. From the optimality criteria, the solution to the problem requires that all arcs be put in kilter. The out-of-kilter algorithm concentrates on the out-of-kilter arcs in a network and progressively puts them in kilter, doing so in such a way that all in-kilter arcs remain in kilter, thereby ensuring rapid convergence. It is important to note that the theoretical formulation, as presented in this paper, has not been incorporated into any model implementation in its present form. However, it is necessary to show the framework for which the multimodal system interface, coordination and interaction must function. Modal coordination, while not critical in this initial implementation of the model, becomes a very important feature of the model in real time systems where response timings are critical, such as automated MMU location on the network.

3. CONCLUSION Several conclusions can be drawn from this paper. First, the paper specifies the multimodal transport network system as the Hitchcock transportation problem, and the general minimum cost, maximum flow problem. The Ford-Fulkerson labeling method is adopted. However, the integration of electronic navigation control systems such as those promised by IVHS technologies into multimodal network transport systems is a candidate for further research. The automated MMU location information system can be used to influence the pattern of channel set selection in pursuit of optimal policy trajectories (such as optimizing DTE mixing) and assigning priority designations to the MMUs through precise understanding of the variability in the multimodal network topology and MMU flows. These can then be translated into network congestion reduction methods such as redistributing demand between modes or between categories of modal systems to arrange for queuing to occur where it can best be accommodated. Activities such as trip chaining sequencing or navigational processes for automated channel planning and selection can be conducted. A system of this sort will typically consist of the following: l l

Real time information on the integrated network Directional aid through provision and use of dead reckoning (DR) devices and decoders.

Other automated location methods may include automated incident detection on the multimodal network. The implication of the automated DTE/MMU location information system to

Multimodal

Transport

Network Systems

the network is prompt detection and response to events and flexibility in accommodating

429

different

control activities in real time. The current algorithmic structure of the problem, as developed in this paper, lends itself to the use of parallel computing and neural network processes. However, the algorithm as implemented for this study, evaluates arc cost functions on individual networks of the multimodal transport network system for optimal channel set sequentially. Computation time improvements can thus be achieved through the use of parallel processors, thereby taking advantage of the structure of the formulation. This can be accomplished by utilizing a front end processor, that fulfills the task of distributing control of separate network optimization processes amongst the available processors on the computer system. Upon completion of the optimization procedure by each processor, network control and optimal arc selection set is returned to the front end processor for inclusion in the optimal channel selection set.

REFERENCES 1 2. 3. 4. 5. 6. 7 8. 9 10.

11. 12. 13.

14. 15 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

O.A. Oduwole, Development of a globally integrated multimodal transport network system model, Doctoral Dissertation, Sever Institute of Technology of Washington University in St. Louis, August 1993 (submitted). G. Dantzig and D.R. Fulkerson, Linear Programming and Extensions, Princeton University Press, (1963). D.R. Fulkerson, Flow networks and combinatorial operations research; Reprinted in Perspective on Optimization: A Collection of Expository Articles, (Edited by A.M. Geoffrion), pp. 197-238, (1972). L.R. Ford and D.R. Fulkerson, Flows in Networks, Princeton University Press, Princeton, NJ, (1962). E:.L. Johnson, Networks and basic solutions, Journal of Openztions Research 14, 619-623 (1966). J. Edmonds and R.M. Karp, Theoretical improvements in algorithmic efficiency for network flow problems, Jozlrnal of the Association for Computing Machinery 19 (2), 248-264 (April 1972). D.P. Bertsekas, Linear Network Optimization; Algorithms and Codes, MIT Press, (1991). G. Dantzig and D.R. Fulkerson, Computational maximal flow in networks, Naval Research Logistic Quaderly 2, 277-283 (1968). A.G. Wilson, A family of spatial interaction models and associated developments, Environment and Planning 3, 1-32 (1971). D.R. Fulkerson and S.E. Dreyfus, Flow networks and combinatorial operations research; Reprinted in Perspective on Optimization: A Collection of Expository Articles, (Edited by A.M. Geoffrion), pp. 197.-238, (1972). D.R. Fulkerson, A network flow feasibility theorem and combinatorial applications, Canadian Journal of nfaathematics 11, 440-451 (1959). Esellmore, Bennington and Lufore, Solving generalized networks, Management Science 30 (12) (1972). G. Dantzig and D.R. Fulkerson, Flow networks and combinatorial operations research; Reprinted in Perspective on Optimization: A Collection of Expository Articles, (Edited by A.M. Geoffrion), pp. 197.-238, (1972). A.G. Wilson, A statistical theory of spatial distribution models, Transportation Research 1, 253-270 (1967). M. Franke and P. Wolfe, An algorithm for quadratic programming, Naval Research Quarterly 3, 95-110 (1956). L.F. Rantzeskakis and W.B. Powell, A successive linear approximation procedure for stochastic, dynamic xrehicle allocation problems, TVunsportation Science 24 (1) (February 1990). L.J. LeBlanc, An algorithm for nonlinear multicommodity flow problems, Networks 4, 343-355 (1974). S.A.Y. Lin and M.A. Hanson, Transportation sensitivity and regional growth, Regional Science and lrrban Economics 6 (1976). D.E. Longsine et al., User’s Manual for the Neftran (Network Flow and tinsport) Computer Code, Sandia National Laboraties, Albuquerque, NM, (September 1987). T.L. Ma.gnanti and R.T. Wong, Network design and transportation planning: Models and algorithms, Transportation Sciences 18 (1) (February 1984). T.L. Magnanti, R.T. Wong and P. Mireault, Tailoring Benders 93: Decomposition For Uncapacitated Network Design, Operations Research Center, Massachusetts Institute of Technology, (September 1984). J.R. Weaver and R.L. Church, Computational procedures for location problems on stochastic networks, Transpol-tation Science 17 (2) (May 1983). A.G. Wilson and M.L. Senior, Some relations between entropy maximizing models, mathematical programming, and their duals, Journal of Regional Science 14 (2) (1974). C. Winston, The demand for freight transportation: Models and applications, tinsportation Science 17 (6-B), 419-427 (1983). 13. Yaged, Minimum cost routing for static network models, Networks 1, 139-172 (1971). W.I. Zangwill, Minimum concave cost flows in certain networks, Management Science 14 (7) (March 1968).