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BEBR

FACULTY WORKING PAPER NO. 1463

Solving a Real World Highway Network Design Problem Using Bilevel Linear Programming

Omar Ben-Ayed Charles E. Blair

David E. Bo\ce

Hi* 9f

College of

Bureau

of

University

Commerce and Business Administration Economic and Business Research of Illinois. Urbana-Champaign

hit

BEBR FACULTY WORKING PAPER NO. 1463 College of Commerce and Business Administration

University of Illinois at Urbana- Champaign June 1988

Solving a Real World Highway Network Design Problem Using Bilevel Linear Programming Omar Ben-Ayed, Graduate Student Department of Business Administration

Charles E. Blair, Professor Department of Business Administration

David E. Boyce Department of Civil Engineering, University of Illinois

Digitized by the Internet Archive in

2011 with funding from

University of

Illinois

Urbana-Champaign

http://www.archive.org/details/solvingrealworld1463bena

.

SOLVING A REAL WORLD HIGHWAY NETWORK DESIGN PROBLEM USING BILEVEL LINEAR PROGRAMMING

OMAR BEN-AYED and CHARLES

E.

BLAIR

Department of Business Administration University of Illinois at Urbana-Champaign, IL 61820

DAVID

E.

BOYCE

Department of Civil Engineering University of Illinois at Urbana-Champaign, IL 61801

(May 1988)

This paper is concerned with the solution of a real world Bilevel Linear Program (BLP) of the highway Network Design Problem (NDP), based on empirical data for Tunisia. The problem includes 2,683 variables (2,571 at the lower level) and 820 constraints; it is far beyond the capacity of any existing algorithm. A new algorithm is devised and the solution is found successfully despite the NP-hardness of BLP and the large scale of the problem. The computational difficulties of BLP should not present a barrier against the effective use of such a powerful optimization technique.

Bilevel

Linear Programming

(BLP)

is

a

mathematical model

of

organiza-

an

tional hierarchy in which two decision makers have to choose their strategies from the polyhedron {(x,y):

who has

control over

lower decision maker and,

x,

Ax + By < b;

x,

makes his decision

selects

y

[Bialas

y >

The upper decision maker,

0).

hence

first,

and Karwan

1982

fixing x before the and

1984,

Bard

1983,

Ben-Ayed 1988]

The

ability

motivated

the

BLP

of

to

development

resource

allocation

planning

[Bracken

represent of

[Cassidy

and

McGill

several 1971, 1973,

decentralized applications

Fortuny-Amat 1974a

and

and

1974b,

decision-making of

McCarl Shere

including

model,

the

problems

1981],

military

and Wingate

1976,

Falk 1977] and government policy [Candler and Norton 1977, DeSilva 1978, Bialas et

al.

1980,

Falk

and

McCormick

1982,

Candler

and Townsley

1982

and

1985].

2

attention

Recently,

been

has

making

decision

transportation

focused

on

applying

bilevel

problem

referred

to

formulations

Network

the

as

to

a

Design

Problem (NDP) [LeBlanc and Boyce 1986, Marcotte 1986, Ben-Ayed, Boyce and Blair 1988].

NDP

concerned with

is

optimal

the

improvement

of

transportation

a

network in order to minimize the system total costs. Most

BLP

applications

lacking

the

practical

significance

majority of

overwhelming

limited to

are

theoretical

solving

of

real-world

the

real

problems

formulation,

problems.

Actually,

formulated

are

thereby the

solved

and

as

single-level (single-objective) programs even when they are virtually bilevel. This is mainly due to the inefficiency associated with BLP algorithms [Ben-Ayed and Blair 1988], which presents a real barrier against the effective use of the model.

In

trying

motivate

to

formulations

theoretical

real world problems,

and

applications

further

Ben-Ayed,

illustrative

small

the

Blair and Boyce [1988]

of

beyond

going

BLP

examples

interesting

to

presented

the

a BLP

formula-

tion of the interregional highway NDP based on empirical data for Tunisia. This

paper gives the solution to the problem formulated there. This problem includes 2,683 variables

(2,571 of which are lower) and 820 constraints, which makes it

too large to be solved by any of the existing algorithms.

Section that

takes

1

presents

Section

In

2,

we

develop

a

advantage of the special structure of the problem.

reasons for having

a

deals

with

and

second

each

iteration we try to find

a

of

the

the

the

There are two

lower

problems

The

functions.

separately;

at

each

better compromise with the user, while including the improvement

minimum computation effort.

presented and the results are analyzed. some suggestions.

improvement

nonconvex

two

smallest possible number of nonconvex

solution with

new algorithm

first the user-optimized flow requirement

BLP formulation;

(user-equilibrium), algorithm

the problem.

In

functions

Section

We conclude

in

the

3,

to get

the last

the exact

solution

is

section with

The Problem to be Solved

1.

this

In

formulation resulting

section we give a short description of the

Blair and Boyce [1988]

from the empirical study conducted by Ben-Ayed,

on the

Tunisian interregional highway transportation network. We are concerned with

a

Each region is represented

system of roads connecting the regions of Tunisia.

by a centroid-node which is usually the largest city in the region. The traffic

entering and leaving a centroid-node is assumed to be that of the whole region; every centroid-node

is

simultaneously an origin-node and

destination-node.

a

The other nodes in the network are intersections of roads generally correspondA road connecting any two cities

ing to intermediate cities.

is

called

a

link

and a sequence of links is called a route. We have empirical data on the number of people who want to travel from each region to each other region. a

traveller

conditions.

has

a

choice

Improvements

routes

of

various

in

makes

and

links

in

his

decision network

the

Typically,

based

sill

on

road

cause

some

travellers to change their choice of routes. We will not provide details about

empirical

the

data,

the

construction

constraints in the formulation; however,

of a

the

objective

functions

and

the

brief discussion on the inclusion of

the bilevel presentation is relevant to the object of this paper.

The

principal

variables

in

the

formulation

added capacity resulting from the improvement, Units

per hour

(PCU).

combination of

The designers of

improvements

to

make.

a

are

the

traffic

flow

and

the

both measured in Passenger Car

road network wish to

Their objective

takes

find the best

into

account

the

cost of improvements and also factors which depend on the pattern of road use,

including system travel time, system operating cost and frequency of accidents. The users of the network decide which routes

individual cost; choices

that

fluence users

are

therefore,

optimal

choices,

to use primarily based on their

their choices do not necessarily coincide with the for

the

however,

system

[Wardrop

by improving some

1952].

links

The

system

can

in-

and making them more

.

4

When deciding on these improvements,

attractive than others.

preferences

influence the users'

to

on

to minimize

total

system costs.

the upper decision maker chooses the amount of improvement to

Thus in our BLP,

make

in order

the system tries

the

each of

different

(links

roads

in

the

network)

while the

decision maker chooses how much traffic there will be on each road. approach

is

used by LeBlanc and Boyce

[1986],

and Ben-Ayed,

lower

A similar

Boyce and Blair

[1988] In

addition to the user-optimized flow requirement,

BLP formulation allows

the analysis of nonconvex piecewise linear functions [Ben-Ayed, Blair and Boyce 1988].

A typical

improvement function might be:

f(Z) = 10Z

< Z < 200

= 2000 + 3(Z-200), = 2900 + 8(Z-500),

where

Z

the amount of

is

improvement.

difficult

to

Note

that

analyze.

neither

is

However,

link and

convex

function

the

Z

< 500

(1)

500 > 500.

improvement of a

f(Z)

200


Z-500 guarantee that W m are

problem;

,

W 2 =MAX(

the

,

lower

Z-500]

.

objective

and the constraints W^ > Z-200, W 2

least upper bounds.

This technique can be used

for any piecewise linear function.

Two drawbacks should be mentioned. First, the number of

lower variables,

are many piecewise

and this

linear functions

in

the method described does increase

increase may be considerable if there the problem.

Second,

the resulting BLP

5

is

typically

objectives;

distinguished

conflict

the

by

between

its

upper

and

lower

there are many variables which the upper objective wants to be as

large as possible,

while the lower objective wants them as small as possible.

Experience suggests this type of BLP

is

difficult to solve.

Including user-optimized flow and nonconvex improvement

problem obtained by Ben-Ayed,

Blair

Boyce

and

[1988]

is

functions,

synthesized

the BLP in

the

following:

MIN Z F(Z,W) + G(C) where

[C,

C,

W,

X and Y]

solve:

MIN F(W) + G(C) subject to: =

%(¥)

(2)

H 2 (X,Y) = H 3 (Z,C,X) > H 4 (Z,C,X) > H 5 (Z,W) > H 6 (Z) > Z,

where

the

regional link.

function F

system

is

travel

the

cost.

C,

C,

W,

X,

improvement The

vector

Y >

The

cost. X

is

the

function G amount

of

is

the

inter-

traffic on each

Any traveller from one origin-node to a destination-node has a choice of

different routes to use,

corresponding to different sets of links. We use the

components of the vector Y to indicate the number of people using each route.

These numbers determine the numbers of users of each link, shown by the vector X;

the

H2(X,Y)

relationship

while

specified nodes

the is

between

X

requirement

and that

Y

is

given

specified

given by the constraints

by

the

numbers

equation of

H^Y) referred

people

constraints go

between

to as the conserva-

6

tion of flow conditions. F is the lower objective of nonconvex improvement cost

The components

functions.

the vector

of

are

Z

amounts of

the

improvement on

each of the different links. W is the vector of the second-stage variables used in

computing the improvement cost functions as described previously in (1).

gives

vectors the

interregional travel costs on each link for the user.

the of

system

G

and C are

C

second-stage variables giving total travel costs on each link for and

respectively.

the user,

H3(Z,C,X)

and H^(Z,Z,X)

are

the

ine-

quality constraints needed to the formulation of convex travel cost functions and the effect on improvement on decreasing those costs for the system and the

user,

respectively.

constraints.

Hg(Z)

description of

the

functions

detailed

Appendix

correspond

Hr(Z,W)

are

variables

improvement and

the

constraints.

limit

functions

is

provided

linear A

in

more the

the total number of variables (see Appendix A)

2,683 and the total number of constraints is 820,

bound

piecewise

A.

In the above formulation (2), is

the

nonconvex

the

to

(improvement

constraints

limit

constraints

excluding upper and lower and

nonnegativity

const-

raints). Those numbers are partitioned as follows:

Variables

112 112 112 112 104 36

Constraints (excluding bound constraints)

(X) (C) (C) (Z) (W^) (Wo) (Y)

2,095

120 (conservation of flow conditions ) 112 (link flow definition constraints)

224 (system travel costs formulation) 224 (cumulative user costs formulation) 68 (convex improvement costs formulation) 72 (nonconvex functions formulation)

2,683

The among

Z

the

functions

are

820.

2,683 all

only

the

variables

variables,

have

3

2,571

pieces;

two

controlled are

by

lower.

constraints

the

The are

upper problem;

nonconvex needed

by

therefore

improvement each

link.

cost

The

convex improvement functions require only one constraint per link because they

.

all consist of two pieces only.

solution

ideal

An

(lower

bound)

minimization problem

BLP

any

for

can

be

obtained by ignoring the lower objective function and solving the problem as an call

we

If

LP.

solution

(Z°,

equal

is

to

bound), we substitute C

C

,

Y

,

F(Z ,W

)

,

W

+ G(C

the

)

F(Z ,W by

Z

C°,

C°,

X°,

+ G(C

)

W°)

Y°,

the

obtained

get

an

To

).

solution,

incumbent

solution

obtained

solution,

the

incumbent

(upper

If we call

and solve the lower LP problem.

Z

ideal

the

solution

is

(X

equal

,

to

Let us refer to the gap between incumbent and ideal solutions

).

as:

gap = (incumbent

-

ideal)/ideal

The gap for the BLP problem (2) is larger than 111 % [Ben-Ayed 1988]; this high value,

mainly due

heuristic gorithms,

to

algorithms

the nonconvex

extremely

functions

difficult,

formulation,

Heuristic

impossible.

not

if

the task of

makes

such as the Parametric Complementary Pivot algorithm (PCP)

Karwan-Shaw 1980, [Bard 1983],

and Bialas-Karwan 1984]

al-

[Bialas-

and the Grid Search Algorithm (GSA)

lead to very unsatisfactory results when applied to BLP problems

with combinatorial nature,

such as Knapsack problems or nonconvex formulation

problems [Ben-Ayed and Blair 1988]. The problem is even harder for branch and bound or related implicit enumeration 1981, 1982, is

techniques Bialas

[Falk

and

1973,

Karwan

Gallo

1982,

and

Fortuny-Amat

1977,

Ulkiicu

Papavassilopoulos

1982,

Candler

and

and

McCarl

Townsley

Bard and Falk 1982, Bard and Moore 1987]. The Bard-Moore algorithm, which

supposed to be one of the most efficient,

does not

accept

a

problem with

more than 100 lower variables; our problem has more than 25 times this number. As

for

efficient.

any A

large

new

NP-hard

problem,

the

use

algorithm taking advantage of

of

the

general special

problem has to be devised to solve the problem at hand.

algorithms

is

not

structure of the

2.

A Proposed Algorithm

Formulation (2) shows that there are two separate lower objective functions. In fact,

each of the two objectives has its own constraints that are indepen-

dent of the constraints of the other objective, provided that

Z

are fixed (they

are fixed by the upper objective). This can be better seen when the BLP problem is

(2)

restated as follows:

MIN Z >

F(Z,W) + G(C)

(3)

such that {W) solve (4):

MIN F(W) subject to: (4)

H 5 (Z,W) >

W > and {C, C, X and Y) solve (5):

MIN G(C)

subject to: (5)

H(X,C,C,X,Y) > C,

where H(X,C,C,X,Y) H 3 (Z,C,X) > 0,

>

substitutes

H 4 (Z,C,X) > 0,

Y >

X,

C,

constraints

the

and H 6 (Z) >

=

H 1 (Y)

0,

H 2 (X,Y)

=

0,

0.

The first lower problem (4) corresponds to the nonconvex functions formulation and the other

corresponds to the equilibrium problem.

lower problem (5)

Because of the NP-hardness of BLP,

it

more convenient to take advantage of

is

the structure of the problem and solve it as two separate smaller problems. As

we

previous situation (5);

the

mentioned

in

our

discussion

the

BLP

corresponding

different

for

the

section, is

travel

cost

functions

BLP of

of

to

(4)

is

corresponding to the

cost

improvement

system

and

functions

difficult. the

the

in

the

However,

the

equilibrium problem

user

are

very similar

9

[Ben-Ayed,

Blair and Boyce 1988]. The coefficients of the two functions always

have the same sign,

and are even fairly proportional;

therefore the use of BLP

heuristics is safe for this second part. tries to identify a solution that

Our approach to satisfy (5)

good for

is

both the upper and lower objectives by using a convex combination of the two:

tG(C) + (l-t)G(C)

so that we are looking at the problem:

MIN Z F(Z,W) + tG(C) + (l-x)G(C) such that {Z, C, C, X and Y)

solve:

H(X,C,C,X,Y) > Y >

X,

C,

C,

Z,

and such that (W)

solve:

(6)

MIN F(W)

subject to: H 5 (Z,W) >

W >

In general,

polyhedron

BLP solution (x ,y

a

(Ax°

+ By

lower solution y x

.

In our case,

be the case that,

the

to




,

0.

0};

called feasible if it belongs to the

and satisfies

the optimality of the

lower problem when the upper variable x is

the

fixed at

for a feasible solution to (6) to be feasible to (2),

for the given choice of the upper variables

lower variables

clearly be the case

to minimize

are such as for

t

=

for some

the same will be true for any

lower objective

for which this

x




X,

C,

C,

Z,

H5a (Za> Wma) * W ma * and such that {W ma

MIN

E

(7)

°

m=l,M a

,

Z

aeTl

°

acT 1

,

}

solve:

m=l,Ma W ma

subject to: W ma

"

Z

a -

W ma *

where

Ma

number

the

is

function of link cost on link

Let us call (Z 2 link

j

such that

X

2

C

,

2

a

having

Z

C

,

2 ,

Y2

W 2 ) the solution of (7).

,

does not belong to T

has to be subtracted from T

links

improvement

cost

Hc a gives the corresponding constraints.

,

j

piecewise

the

in

F a is the upper objective function of nonconvex improvement

a;

and,

a;

°-

breakpoints

of

"^ma

?

a

,

added to T

positive

1

,

")

and Z-

different from

is

then

0,

j

and (7) has to be resolved until all

elements

are

If there exists a

of

T

1

in

,

which

case

the

exact

solution of (6) is found. We have constructed from the main BLP problem (2) formulations:

Problem (7) solution

is

a

is

parameterized BLP (6) and a BLP with few lower variables (7). solved to give a feasible and optimal solution to (6).

user-optimized

(i.e.

t

can be tried.

decreasing the value of For any solution (Z 1 two different values.

the

solves

fixed at its optimal value for (7)),

value of

two other secondary BLP

lower

LP problem

(5)

If this

when

Z

is

then it is feasible for (2) and a higher

Otherwise, we continue our search for the optimum by

t.

,

X1

,

C

,

C

,

Y1

,

W 1 ),

we have to distinguish between

The first value is that of the upper objective function

12

equal to F(Z,W) + G(C);

of formulation (2), is

the actual cost

us

call

includes

actual

improvement

improvement

cost

cost

on the other side,

actual travel cost,

XJ

the

fCZ 1 )

into

Z

1

The first value

1

feasible solution;

it

1

of Z;

let

system travel

1

actual

the

in

or nonconvex.

The

obtained after we solve the lower LP

costs;

CJ

,

other words,

in

YJ

,

A1

and plug

)

equal

is

to

corresponds to a potential optimal, but not allows

ideal

find

to

problem (2). The second value A 1 corresponds to optimal, solution;

Z

convex,

obtain the solution (XJ, C J

,

convex system travel

the

+ G(CJ).

necessarily

fixed at

Z

is

Z

and actual

obtained by plugging

is

improvement functions f(Z), whether they are linear,

problem (5) with

The second value

.

incurred by the system given a specific value

A1

.

actual

The

cost.

A1

it

1

let us call it

a

solutions

to

the BLP

feasible, but not necessarily

it allows to find incumbent solutions

for (2).

The proposed algorithm is an iterative procedure that tries at each iteration

to

reduce

the

between

gap

solution

ideal

procedure terminates when the gap is brought below the

number

gives

a

of

iterations

general

exceeds

description

provided by Appendix

a

fixed

the

of

limit

iterative

incumbent

and a

N.

desired value The

following

procedure;

more

e,

x

A.

-

Solve (6)

Test whether solution of (6) is feasible to (2)

B.

Inf easib! Le

Feaj sible

1 r Li 1

.

Mn1. iomn* i

t i

1«-~„_ iai ej 6

.

PO

\j£.

.

M«1,>.e ~

n-

_~,.11 M .~ oiua iici

chart

details

(Using Binary Search)

Start with t=1

or when

flow

B:

Searching for

The

solution.

are

13

in which (6)

The most difficult step in the algorithm is step A,

solved. Using the concepts explained above in this section, is

has to be

the solution of (6)

obtained as shown in the following flow chart:

Fixed

t

Start with T X =0, T 2 =T

Al.

A2.

Solve (7)

Test whether new elements should be added to T

A3.

New elements added

A/.

M~r»

Tl J.

Several cient.

improvements

can be

r^r.T new

K

T2

(

C\

;

added

r.~1,r~^

added to the algorithm to make

it

more effi-

The starting ideal and incumbent solutions can be found in a more effi-

cient way.

A better ideal solution for (2) x=l

obtained after substituting at there

,

No element?

is

no situation better

can be obtained when solving the LP

all nonlinear functions f(Z) in (6) by ccZ+B;

for the system than having cheap improvement and

system-optimized flow.

Also,

problem is solved as

BLP must be lower than the actual cost incurred when

simpler

technique

a

is

complex techniques.

used;

the

otherwise,

Therefore,

to

cost

actual

incurred by the system when the

there

obtain

an

is

no

justification

incumbent

solution,

for

we

can

a

using just

solve the problem as an LP; we approximate improvement cost functions by their best linear fit, (Z°,

X°,

C°,

C°,

ignore the lower objective and solve the problem. Y°)

the obtained solution,

A

is

Let us call

the value of the starting

incumbent solution. Using those techniques makes the gap between starting ideal

14

solution and starting incumbent solution as

low

as

instead of the

%

3

111

%

found earlier in the previous section.

since the procedure is iterative,

Moreover,

more efficient to use the

is

iterations in order to make the next one easier

information from the previous to solve.

it

One way to do this is to introduce to the problem the constraint:

L < F(Z,W) + G(C) < U

where L and U

the

are

values

current

the

of

and

ideal

incumbent

solutions,

respectively. the solution of step

Similarly,

assume that we solved (7),

Let us

W

9

may require solving (7)

2

obtained the solution (Z

more than once.

9

X

,

9

C

,

9

C

,

9 ,

Y

9 ,

and have to add a new element and resolve (7). We use the fact that the new

)

cost will be at most as low as the value we got for the objective when we used

cheap linear improvement costs (a-Z-

9

+

J3-)

for the links

to be added to T

j

1 ;

which gives a lower bound for the solution. An upper bound can also be found by

noticing

that

incurred

by

new

the

the

system

cost

is

at

for

an

added

least

capacity

between upper and lower bounds, when step + Bj)

-

high

as

2

as

equal

actual

the

9

to

Z

.

has to be repeated,

cost

The

9 )

difference

E^ti

is

f(Z

(

a i^i

9

f(Z 2 ).

Finally,

feasible solution to

a

a

BLP problem

obtained by finding the

is

optimal solution to the lower problem after substituting values for the upper variables.

In our

any optimal solution (Z 1

case,

X1

,

,

C1

,

C

,

Y1

,

W 1 ) of the

BLP problem (6) satisfies the optimality requirement of W 1 to the lower problem (A);

problem (2) (5).

(Z 1

therefore if

it

Even when

solution (Z k

,

Xk

C1

,

,

C1

Y1

,

W1 )

,

feasible

a

is

solution of

the

BLP

satisfies the optimality requirement for the second problem

the ,

X1

,

Ck

solution ,

Ck

,

Yk

,

(Z 1

,

X1

,

C1

,

C1

,

Y

,

W1)

is

not

feasible

a

new

W k ) can be obtained, with feasibility guaranteed.

15

It

known from the theory of Linear Programming

is

Chapter 10 pp.

1983, x >

solution

generate

x

of

LP1,

solutions

feasible

b'

x > 0),

,

where b and

Ax = b,

(MIN ex,

are different,

b'

of LP2 that has the same nonbasic variables

basic feasible solution x

optimal

that given two LP problems LP1:

Ax =

(MIN ex,

and LP2:

0}

148-168]

instance Chvatal

for

[see

is

for

solution

optimal

an

BLP

problem

(2),.

of

LP2.

which

is

as

any the

Therefore

to

equivalent

to

generating optimal solutions to the lower problem, we can solve BLP problem (7) with the additional (XJ,

CJ

,

CJ

YJ

,

)

requirement

that

all

nonbasic variables

of the lower problem (5) with

slack and surplus variables be locked.

Z

fixed at

Z

1

of

the

solution

and all nonbasic

This is added to the algorithm as shown

in Appendix B.

Regarding the reliability of the algorithm, there are no heuristics involved finding

incumbent

the

solutions.

and

the

ideal

algorithm for those solutions

are

exact.

in

the

solution

However,

it

it

is

gives

is

the

actual

As

long as

optimum

values

the

within

possible that one finds bad examples

the convergence is not satisfactory;

The

by

the

algorithm converges,

the

accuracy

for this

in such a case the

given

desired.

algorithm where

algorithm ends with

a

large gap between upper and lower bounds, but does not give a wrong solution.

3.

Solution of the Real World Problem

When trying to solve the BLP problem (7) we faced a major constraint, since we could find no reliable BLP software that could handle problems this

Fortunately,

(7)

can be transformed into an equivalent Mixed Integer Program-

ming problem (MIP), to the number of

large.

with the number of zero-one variables in the latter equal

lower variables in (7).

of MIP does not mean that it

is

It

should be emphasized that the use

easier to solve.

MIP and BLP are both NP-hard

problems and are equally difficult to solve; however, MIP is older and benefits

:

16

availability of good software to solve it.

from the

Zero/One Optimization Method

Programming system. 1986 by R. the

at

Marsten

E.

the

extension of the XMP Linear

Marsten from the Department of Management and Information Systems

University of Arizona. [1981].

The

Center

description of the XMP

A

subroutines

for

ZOOM

of

to be

and

XMP

are

library

used

is

by

our

given in

program

The computational study,

solved.

to be

was conducted on the CRAY X-MP/24 supercomputer of

presented in this section,

National

an

is

ZOOM and XMP are both written in Fortran 77 and revised in

whenever an LP or an MIP needs

the

which is

(ZOOM),

One such package

Supercomputing Applications

at

the

University

of

Illinois at Urbana-Champaign. First, we show how to transform BLP (7) into MIP. The constraint:

W m = MAX{(Z

is

-

q m ),

0)

equivalent to:

either: W m =

which can be equivalently restated,

-

Z

qm

or:

,

Wm =

after introducing zero-one variables V ma

,

as

Wm

-

Z
C,

Z,

X,

C,

Y >

H 5 a( Z a> W ma) *

(9)

and such that {W ma "

W ma +

W ma

Z

Z

"

m=l,M a

,

a

"

aeT

)

solve:

MV ma * ^ma

"

a

,

MV ma * "^ma

*ma + MV ma * M V ma

The technique used to transform (7)

used

by

Fortuny-Amat

McCarl

and

(0,

e

1).

into MIP

The

[1981].

doubled the number of zero-one variables

is

different from the technique

use

of

this

latter

would

have

(requiring a zero-one variable

for

each lower variable and a zero-one variable for each lower constraint).

The large size of the problem (2)

included in the formulation.

of routes

be used,

mainly due to the high number (2,095)

Since not all those routes are going to

the problem can be simplified by keeping the most attractive ones and

discarding routes

is

the

rest.

We

will

solve

the

problem with

the

reduced

number

(we refer to this problem as the small version of the problem),

of

and we

solve also the original problem with all routes included; next we will compare the solutions of the two problems.

The attractive routes are found by solving

both the user equilibrium problem (5) and the system equilibrium problem (same as

(5)

except that G(C)

is

replaced by G(C)) as we increase the values of the

entries of the trip matrix to make the users and the system choose their next

desirable

routes.

When

solving

improvement cost because of

its

the

system

equilibrium,

effect on the routes

we

choice.

vary

also

the

The accumulated

number of routes chosen by both the system and the user reaches 389 as the flow

18

fluctuates between the original trip matrix and three times that matrix. Those

averaging more than three routes per origin destination pair are

routes,

389

the ones to be included in the small version of the problem as compared to the

2,095 included in the large version.

About

minutes

15.25

space

storage

computing

CPU

time

were

and

required

version of the problem. At iteration

million words

1.4

0,

obtain

to

solutions.

actual

incurred by the

in the

flow

solution

the obtained gap is

Section

in

2

system

(namely 1

257,037.230)

with T=l,

the MIP

obtained

shows

small

111.36 %;

starting

TflAX

as

13

anc*

-'-

new

a T

MIN

incumbent

= ^'

When we

the list of

1,

for improvement, we use the solution as a new better

the gap decreases to 3.03

chart),

the

better

get

to

first step (A1-A2 in the flow chart) of iteration

ideal solution;

of

word)

we solve the problem as an LP and we use the

Then we start the iteration

links that are candidates

the

proposed

At the same iteration,

cost

solution. find,

techniques

the

use

we

solution

the

per

we compute starting ideal and incumbent

solutions using the ordinary techniques of BLP; next,

bits

(64

is

(9)

that

%.

In the second step

(A3-A4-A2 in

solved with only six integer variables.

no more

integer

variables

need to

be

The

added;

a

slightly improved ideal solution is found for the problem (new gap equals 2.93 %)

.

the

The feasibility of the solution is tested in step flow

solution

is

not

user-optimized.

obtained when

We

testing

generate

feasibility;

(B

3

feasible

a a

new

in the flow chart);

solution

incumbent

using

the

solution

is

obtained and the gap is reduced to 2.57 % at the end of the first iteration. At iterations

Iteration

2

4,

and

3

where

there x

=

is

no

.125,

improvement at the end of step

i^x 3,

procedure is stopped at Iteration steady state; all

steps.

The

Iteration final

5

success

gives

=

in

-250

improving the

and t^IN = 0,

when generating 5

a

incumbent solution. gives

another slight

new feasible solution. The

because the solution seems to be reaching

exactly the same solutions as

a

Iteration 4 in

solution retained incurs a cost to the system equal to

19

256115.006.

The small version of the problem is not hard to solve even on an ordinary However,

computer.

large version

the

is

more demanding;

required about 4

it

minutes CPU time and 2.7 million words computing storage space for running one No more iterations were needed because the solution,

single iteration.

at the

end of the first iteration, was the same as the solution obtained by the small

version of the problem at that point; which suggests that our choice for the attractive routes was successful. Table

1

gives a summary of the intermediate and final results related to the

solution of the real world problem; the complete presentation of those results is

provided

Appendix E

by

in

Ben-Ayed

includes

The Table

[1988].

only

the

iterations and the steps resulting in an improvement of the gap (increase of the ideal solution or decrease of the incumbent solution). the table that, is

as

is

It

can be seen from

usually the case for iterative algorithms, most progress

made during the first iterations while convergence is slower as the number

of iterations

increases.

The first two columns of the Table are very different

from the others because of the instability of the "maximum" variables W in the

absence of the corresponding

lower objective

function;

the negative value of

the improvement in the first column is a consequence of this

fourth

row of

the Table specifies whether

when a solution

is

incumbent,

incurred by the system (A 1 ), just

to

column,

the

value

of

the

the

rows

5,

a

solution is

6

and

7

ideal

function

(0

1

).

the solution has a tendency to stabilize;

or

Starting

The

incumbent; cost

refer to the actual

and when the solution is ideal,

objective

instability.

those rows refer from

the

the numbers of used

third links,

used routes and improved links are almost the same in all six last columns. Several

results

can be

inferred from the computational experience and the

solution of the real world problem. and

accuracy,

the

Concerning the tradeoff between efficiency

equivalence of the solutions

of

the

large version

and the

.

20

small version shows that a complex problem can be simplified without affecting

optimality

the

However

solution.

the

of

simplification must

the

be

done

carefully and efficiently; other simplifications, such as reducing the problem to a single-level could lead to bad results.

The actual cost incurred by the system when the problem is solved as an LP

exceeds

the

thousands

of

actual

incurred

cost

Tunisian Dinars.

difference as high as

it

when

There

a

BLP

solution

are

two

reasons

supposed to be.

is

First,

used

is

not

for

922.224

by

having

this

the travel cost at stable

flow is a one-piece linear function that does not depend on the added capacity.

However,

if stable flow, which is the overwhelming proportion of the flow,

does

not depend on the added capacity, which is the only instrument in the model for the

system to influence the user's

system to affect

choice of the routes,

user-optimized equilibrium

the

is

the

ability of the

almost paralyzed.

Second,

the limitation of the added capacity resulted in low improvement costs.

that improvement costs are about fifty times smaller than

be seen from Table

1

travel

introduction of

costs.

improvement

The costs

It can

did

not

give

the

much

sophisticated saving

formulation of

BLP

compared

as

to

the

simpler

the LP

formulation because those costs are already low according to the data of the

problem.

The

availability

ability of

of

realistic

problem would definitely the

problem when

BLP

solved

to

data;

lead to as

a

give

better

providing a

BLP

results

more

conditioned

is

representative

data

by

the

to

our

larger difference between the solution of and

its

solution when

solved

as

a

single

level

Another harmful simplification would be the inclusion of strict capacities on the links, in

our

even when the overflow is very small on all links, as is the case

solution.

Imposing

strict

capacities

results

in

the

violation

of

Wardrop's first condition and therefore the collapse of the equilibrium, which is one of the bases of the problem.

Baene

MEDITERRANEAN Cape Bon

Pwntslkna (ITALY)

boh Peleg/t (fTALYI

MEDITERRANEAN SEA

100 K.loT^irri

Figure

1

The Links to Bo Improved

.

22

Summary of the Results Iteration

1

1

1

4 & 5 4

-

-

-

1

2

3

4

1.000

1.000

1.000

1.000

1.000

1.000

1.000

ideal

incumb.

incumb.

ideal

ideal

incumb.

incumb

incumb.

Travel Cost

255328.

273736.

252428.

245152.

245320.

252577.

251724.

251680.

Improvement

-124325

3153.

4609.

4329.

4410.

4410.

4424.

4435.

Total Cost

131003.

276889.

257037.

249481.

249730.

256987.

256148.

256115.

278

279

276

275

276

276

275

275

99

100

97

97

97

97

97

97

18

18

14

14

14

14

14

15

3.03

2.93

2.91

2.57

2.56

Step

Value of

t

Solution

Routes Used Links Used

Linkslmproved

-

Gap (%)

96.21

111.36

Table

final solution,

In the

used

by

redundant values

of

inter-regional the

in

275

inter-regional

cannot

one

network or

entries

capital,

improvements

all

to

predicted by

its a

major

happen

neighboring

trip

whether

lack

of

matrix.

15

those

are not are

links

use

is

due

The

study

to is

low

more

those are shown on the map in Figure

be

to

tell

their

the

in

concerned with the links to be improved; Almost

.125

routes and 97 links are used and 15 links need

traffic;

corresponding

the




is

H 4 (Z,C,X) >

is

defined as:

Z

"

for each link

is

a:

"



then:

),

,

=

W 2 ) is feasible (

T

MAX +

T

)/ 2

then:

:

28

Else:

Try to improve feasible solution MAX =

T,

Start iteration

I.

Set:

x

t

= (x MIN + t)/2

End If.

•1 = •

1

+

1.

Termination Test •

If

I

> N,

then:

Number of iterations exceeded before reaching desired accuracy Incumbent is the best solution found Stop

Else if gap

< e:

Incumbent solution is optimal within the specified accuracy

e

Stop

End If. •

Return.

Try to improve feasible solution •

Solve

(6)

while

locking

nonbasic

the

variables

surplus variables) of the LP problem (5) solved with •

Get the solution (Z 4

•

If