Using a hop-constrained model to generate alternative ... - DSpace@MIT

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JAN

1991

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"Using a Hop-constrained Model Alternative

Generate Communication Network Designs to

A. Balakrishnan, K. Altinkemer

MIT

Sloan

School Working Paper #3233-91 -MSA

Revised:

December 1990

\ .

Using

Hop-constrained Model

Generate Alternative Communication Network Designs a

to

Anantaram Balakrishnan Sloan School of Management Massachusetts Institute of Technology Cambridge, Massachusetts 02139

Ketnal Altinketner Krannert Graduate School of Management Purdue University West Lafayette Indiana 47907

Revised:

Supported

and

in part

a grant

December 1990

by grant DDM-8996120 from the National Science Foundation

from the

AT&T

Foundation

Abstract Designing the link topology and selecting capacities in a backbone network of a

communication system involves complex tradeoffs between investment and

operating costs and service considerations such as network reliability and vulnerability, delays,

and blocking. Incorporating

all

these design criteria

simultaneously in a comprehensive model results in a large-scale, non-linear, discrete optimization

problem

alternate optimization-based

In this paper,

that is intractable.

methodology

the

method

propose an

that generates several cost-effective

backbone network designs with varying cost and performance

Network planners can use

we

in conjunction

characteristics.

with detailed performance

evaluation techniques to assess the cost impact of different service requirements, and select a design that achieves the

proper balance between conflicting objectives. To

generate different configurations, the method parametrically varies a set of hop constraints that restrict the

Reducing the

maximum number

but incurs higher constraints,

number

network design

of

total cost for the

we develop

a

of links over

which messages can be transmitted.

hops increases the number of alternate routes

communication system. For a given

set of

hop

Lagrangian-based algorithm to identify a cost-minimizing

that satisfies all internode traffic requirements.

We

report results for

extensive computational tests of the algorithm using several randomly generated test

problems.

method

Our

identifies

results

good

demonstrate

that,

heuristic solutions

even for relatively large problems, the

and

tight

lower bounds that confirm the

near-optimality of the selected designs. Using a 25-node example, the

model can be used

to evaluate the cost versus

Keywords: Communication network Lagrangian relaxation

we

illustrate

performance tradeoff.

design, cost-performance tradeoff,

how

1.

Introduction This paper presents a methodology to support topological design and link

capacity planning decisions for the backbone network interconnecting geographically

dispersed gateway nodes and switching centers in a telecommunication system. Topological design decisions are very important in backbone network planning

because of the enormous capital investments required for transmission and switching

and the

facilities,

service levels.

significant impact of

In selecting the configuration

and

network configuration choices on backbone network,

capacities for a

planners face conflicting objectives of reducing the total investment and operating costs while ensuring adequate service levels, expressed in terms of

performance measures such as

reliability

capabilities, blocking probabilities,

and

network

vulnerability, alternate routing

and queueing and transmission delays. With

increasing competition and service diversity in the communications sector, these service considerations are

becoming increasingly important.

Because of the multiple, conflicting objectives, non-linearities in service criteria,

and

discrete design choices, formulating

and solving

a

comprehensive

optimization model to support backbone network planning decisions

is

impractical.

Indeed, even with a single objective such as cost minimization, the design task

very complex and challenging because the decision model must

(i)

is

incorporate

binary network design decisions to capture the significant fixed costs, and (ii)

simultaneously account for the close linkages between topological design, link

capacity,

and routing

In this paper

based method

performance

to

decisions.

we propose an

alternate

approach that uses an optimization-

generate several alternative network designs with varying cost-

characteristics.

Our approach

regarding network design practice.

First,

is

motivated by several observations

network planners must often accomodate

various implicit and qualitative design considerations that cannot be adequately

represented in optimization models; hence, they prefer a decision support system that identifies a limited set of effective designs rather than a single 'optimal'

solution for an imprecise model.

Second, in most practical contexts

completely characterize network performance

(reliability, delays,

we

cannot

blocking) in a

convenient form for mathematical modeling, except under very restrictive

assumptions about

traffic

patterns

assess the network's performance,

and network behaviour. Therefore,

we

to accurately

require detailed evaluative techniques such as

simulation and queueing network analysis. Third, the total

number

of possible

network designs grows exponentially with the dimensions of the network. Since detailed performance evaluation of each design

1-

is

a

time-consuming process,

planners require a principled method to prune the

list

of

all

possible network

configurations to a few cost-effective designs with varying performance Finally, to

characteristics.

keep the model

minimization) model must necessarily

tractable,

even a single objective

make some approximations.

(cost-

In terms of

computational complexity, most (deterministic) discrete choice network design

problems in telecommunication planning are NP-hard. Therefore,

we must

on

rely

optimization-based heuristic procedures that provide provably good solutions.

Thus, to effectively use decision support models,

framework

for

we propose

a hierarchical

telecommunication network planning. The approach consists of the

following four steps: (1)

Generate, using an optimization-based model, a small set of good 'base'

designs that span a wide range of cost and service levels. (2)

Refine each base design

meet (3)

(e.g.,

selectively

add buffer

capacities)

and modify

it

to

implicit or qualitative design requirements.

Evaluate the true cost and performance of each design using detailed analytical or simulation models.

(4) Select

an appropriate design that achieves the desired tradeoff between cost

and performance.

The designs generated enhancements

in Step 2.

in

Step

1

serve as the basis for marginal user-directed

Each base design specifies an underlying network topology,

the preferred routes between various origin-destination pairs, 'rough-cut' capacity plan for switching

and a corresponding

and transmission resources. The base designs

should account for tradeoffs and interrelationships between topological design and routing issues, between fixed and variable costs 2,

and connectivity

(e.g.,

the

number

(traffic

dependent) costs, and between

of alternate routes in the network).

total

In Step

the planner might refine each base design (perhaps, using a decision support

model) by selectively adding buffer capacities to improve network performance, and

making minor changes

to the

topology and routes to accomodate context-specific

Step 3 consists of detailed evaluation of each candidate design, and Step

constraints.

4 involves choosing a design that balances the planners' preferences for cost versus

The four

service.

performed

steps in this approach are closely interrelated,

iteratively with, say, Step 2

In this paper,

Step

1.

We also

refinements

we

describe

providing feedback to Step

1

and so

on.

focus on a model to generate alternative network designs in

model extensions

to incorporate

some

of the design

— allocating buffer capacities, and creating alternate routes — contained

in Step 2 of the hierarchical all

and might be

planning approach. Since the base designs incorporate

major topological decisions, they largely determine the possible levels of service.

-2-

In particular, a base design that

alternative routes

provides routing

and

is,

is

sparse

(i.e.,

contains few links) cannot accomodate

therefore, inherently unreliable, while a

Thus,

flexibility.

order to generate a spectrum of base designs

in

we must identify topologies that differ we propose a deterministic model that

with varying cost-performance characteristics, For this purpose,

in their connectivity levels.

seeks cost-effective designs to meet internode

routing restrictions which the

hop

We

refer to the

we

call

dense design

traffic

hop constraints.

requirements subject to a class of

For each origin-destination pair,

number of links (or hops) on the interconnecting on the number of hops between each origin and

constraint limits the

upper

limit

destination as the route's

Maximum Hop

Parameter

hop parameter in

(or,

route.

brief).

By

hop parameters, we can generate topologies with varying arc densities and costs. In particular, reducing the hop parameter values increases the number of alternate paths, and impacts delays while increasing total

systematically changing these

Thus, hop constraints serve as a convenient surrogate for service restrictions generate alternate base designs. In addition, limiting the number of hops for each

to

cost.

message also simplifies the task of managing

traffic

example, Ash, Murray and Cardwell (1981) and

over the network

Monma

and Sheng

(see, for

Recently,

(1986)).

LeBlanc and Reddoch (1990) independently developed a hop-constrained model with delay costs for reliable topology/capacity design and routing in backbone

explicit

networks.

For a given

set of

hop

constraints,

mixed integer program, and propose

we

formulate the design problem as a

a Lagrangian relaxation

approximately, but with performance guarantees.

method

to vary the

hop parameters

We

method

it

then propose a systematic

in order to generate a variety of

The network designer can then apply

to solve

models

detailed evaluative

design solutions.

to these

candidate

designs in order to accurately assess the cost impact of different service requirements.

The

rest of this

paper

is

organized as follows. In Section

2,

we

formally

describe the design problem, state our modeling assumptions, and present the

mathematical programming formulation. Section 3 describes our Lagrangian-based solution approach to generate upper solution, for a given set of

hop

and lower bounds on the

constraints.

In Section 4

computational results for several randomly generated computational

tests focuses

we

test

cost of the optimal

present extensive

problems.

Our

first set

of

on evaluating the effectiveness of the proposed

algorithm for various problem sizes, cost structures, and hop restrictions. These results

demonstrate that the method constructs near-optimal solutions even for

relatively large

second

set of

problems with upto 30 nodes, 420 edges, and 420 commodities. Our

computational experiments

gain insights about the tradeoffs between cost

-3-

how

model can be used to and performance by varying the hop

illustrates

the

some variants and enhancements of the basic hopconstrained network design model to allocate buffer capacities, and provide for Section 5 describes

restrictions.

alternate routes.

Section 6 summarizes the paper and outlines directions for further

research.

2.

Problem Definition and Formulation

The backbone network design problem involves selecting a subset of links to include in the topology, determining the capacity on each selected link, and routing all

We

internode

emphasize

routing

requirements

traffic

at

minimum

that the three decisions

total

investment and operating

cost.

— topological design, capacity planning, and

— are closely interrelated, and we seek an integrated model that addresses

all

three decisions simultaneously.

Many

authors have proposed methods to separately solve two components of

problem

the backbone design

problem

— the capacity assignment problem and the routing

— assuming that the network topology

assignment problem

(see, for

Maruyama and Tang

(1976))

is

prespecified.

example, Bonucelli (1981),

assumes

Chou

The

capacity

et al. (1978),

that traffic routes (and, hence, the

network

topology) are given, and selects the best capacity for each link from a discrete set of available link capacities in order to minimize total cost subject to delay restrictions.

On

the other hand, the routing or flow assignment problem starts with a given

choice of link topology and capacities, and seeks to identify primary routes between

various origin-destination pairs in order to minimize the average or

message delay

example, Ahuja (1979), Bertsekas (1980, 1984), Cantor and

(see, for

Gerla (1974), Courtois and Semal (1981), Frank and (1981),

and

Yum

Chou

(1971), Gerla (1973),

Tymes

(1981)).

Combined approaches

that iterate

decisions have been discussed by Gerla

Maruyama, and

maximum

Fratta

and Tang

between capacity and flow assignment

et al. (1974),

(1977).

Gerla and Kleinrock (1977), and

Gavish and

Neuman

(1986,1987),

Gavish

and Altinkemer (1987) and Pirkul and Narasimhan (1987,1988) describe integrated optimization methods for simultaneous capacity assignment and routing.

papers also account for objective function),

traffic

node and

delay (either as a constraint or as a link failures,

and/or

practical connectivity constraints in fixed-charge

Eswaran and Tarjan

(1976)

show

that the

enhancement of existing networks restrictions

is

alternate routing.

NP-hard.

-4

the

Incorporating

network design models

problem of finding the

to satisfy

'cost' in

These

is difficult.

least cost

even some special types of connectivity

LeBlanc and Reddoch (1990) introduce two combined topology/ capacity

planning and routing models for reliable backbone network design. These models

two

are similar to our approach, but differ in

respects: they

(non-linear) delay cost in the objective function,

and

capacities,

The

model considers

first

(ii)

assume continuous

model includes an additional

linear capacity costs (our

installing links).

and

include an explicit

(i)

traffic

with different

fixed cost for

priorities,

and finds

the least (delay + capacity) cost design containing alternate routes for single link

The second model uses hop

failures.

Both

constraints as a surrogate for reliability.

models permit bifurcated routing. The authors exploit the linear capacity cost structure to solve problems with

up

to 100

nodes using the flow deviation

algorithm.

present a

we

some notation, state our modeling assumptions, and mixed-integer programming formulation for the backbone network design

Next,

introduce

problem with hop

2.1

constraints.

Notation and Modeling Assumptions The hop-constrained network design problem

network G:(N,E) whose

vertices

N

is

defined over an undirected

represent the backbone nodes and intermediate

switches; the edges in E correspond to possible direct connections. For every pair of

nodes

p,

q

in the

demand) from p

network,

to q.

We

let

d

represent the projected peak internode traffic (or

between each pair of nodes as a separate

treat the traffic

commodity, with denoting the commodity that flows from node p to node Let

K

denote the

set of all

commodities,

i.e.,

K

= { p,q e :

each commodity , the designer specifies an upper

number the

of

hops

for

messages originating

(maximum) hop parameter

for selecting these

To two types

for

at

limit,

p and destined

q.

N with dpq > 0}.

For

denoted as h

on the

We refer to

for q.

commodity . Section

,

2.3 describes a

h

as

method

hop parameters.

establish transmission capacity

of (investment

and operating)

on each

link

(i,j)

costs: a fixed cost,

from

variable cost c^ that represents the cost per unit of capacity

might consist of investments

of the network, we incur denoted as F-, and a i

to

j.

The

fixed cost

for acquiring land, building the infrastructure,

installing/operating the communication link, while the variable cost

and

component

approximates expenses for acquiring and maintaining transmission and switching equipment. In Section 5

we

indicate

how

these fixed

and variable

account for fixed or proportional buffer capacities on each

accommodates economies of

scale

and volume discounts

link.

in the

costs can also

Our model

also

form of piecewise-

linear,

concave costs (see Figure

we assume

1);

the simpler fixed plus variable cost

As LeBlanc and Reddoch

structure for expositional ease.

(1990) note, even

though

transmission cables and switches can be purchased and leased only in discrete sizes,

telecommunication companies often have the option of using public telecommunication traffic;

facilities

(and even leasing fractional capacities) for overflow

hence, piecewise linear, concave functions adequately represent capacity costs.

Our model

includes an explicit design constraint specifying that the selected

network topology must be connected. This

though not

restriction,

essential for our

algorithm, strengthens the problem formulation and improves the lower bound.

when

For certain problem contexts,

the

commodity flow pattern does not

necessitate

network transformation ensures connectedness.

a connected solution, the following

Consider a "commodity graph" containing the original nodes, and edges

{p,q} for

> 0. Let v be the number of connected components every commodity with d pq in this graph. If v = 1, we say the demand pattern is 'complete'; in this case, every feasible design valid.

must necessarily be connected, and our connectedness

When demand

network as follows say, in

node

0,

not complete

is

to create a

every component

r

=

1,2,...,v

connecting node

commodity graph. Each

demand and

cost.

r

}

Add

restriction of

1.

v

dummy

Observe

is

can augment the original

of the

with unit the

we

{0,i

and a very high variable

hop

1),

edges

fixed cost

a

v >

dummy

complete demand pattern: introduce a

dummy

and add v

(i.e., if

restriction

to

one selected node

dummy

commodities

that the

node, ij.

edge has zero

, each r

commodity graph

augmented network has a single component; and, the optimal design problem together with the dummy edges {0,^} is optimal for the

for

for the

original

transformed problem.

assumption wishes

Thus, augmenting the network makes the connectedness

Finally, the

valid.

to explicitly

network transformation

impose connectedness

routing) even with incomplete

demand;

is

unnecessary

if

the planner

(for instance, to facilitate alternate

our algorithm generates a

in this case,

higher lower bound relative to the 'unrestricted' lower bound obtained using the

augmented network. For our computational origins

and destinations resulted

with sparse

2.2

demand

patterns);

in

random choice

tests, a

complete demand for

all test

of

commodity

networks (even

we, therefore, did not augment the networks.

Mathematical Programming Formulation

Our mixed-integer programming formulation

for the

backbone network

design problem distinguishes the direction of flow on each edge of the original undirected network G.

corresponding

to

We consider

two

directed arcs,

each original undirected edge

-6-

{i,j}.

denoted as

The

(i,j)

and

(j,i),

original set of undirected

edges

is

denoted as

E; let

A

represent the corresponding set of directed arcs.

formulation uses two sets of binary decision variables, y^ and y-

=

if

1

edge

{i,j}

is

zPjl,

Our

defined as follows:

included in the design,

otherwise, and

zPjl

=

if

1

commodity

is

routed on arc

(i,j),

otherwise.

The y variables model the edge

selection decisions, while the z variables represent

the (primary) routing decisions for each commodity.

The backbone design problem can now be represented mathematically following integer formulation called

£

minimize

[HCDP]

F ijyjj

+

(i,j)

subject

hop-constrained

(for

£ £ ^pq

2

design

as the

problem):

(Z1)

??

0,j)

to:

£ je

N

zW

-

]T je

zP[l

=

1

if

i

= p

-1

if

i

=q

if

i

* p ,q

for all

e K,

N

Y 2, (i,j)

zW z lj

< -

h n

(2.2)

for all

pq

e K,

Constraints

ensure

(2.2)

that, for

each commodity , the arcs

form a route from p

to q.

commodity ;

specifies that

it

Constraint

(2.3)

represents the

hop

must be routed over

is

commodity can flow

included in the topological design. Finally, constraint

selected design

solution

must be

a connected

subnetwork of the

method does not require an

explicit

most h

at

zW = 1 arcs.

and edge

on edge

in either direction

with

restriction for

Inequalities (2.4) are forcing constraints that relate the routing variables:

(i,j)

{i,j}

selection

only

if

this

edge

the

(2.5) specifies that

network G. Our

original

mathematical representation of

this

constraint.

To incorporate piecewise linear, concave capacity cost functions, we replace and j, where {i,j} of the original network with rj: parallel edges connecting

each edge Tjj

i

denotes the number of linear segments in the expansion cost function for edge

(see Figure r F[-

Let

1).

segment; the

F[:

r

and a variable

and

c[:

parallel

cost of

c[:.

{i,j}

denote, respectively, the y intercept and the slope of the

edge

in

our augmented network carries a fixed charge of

Because the cost function

is

concave, the model does not

require explicit capacity constraints to enforce the upper limit of flow for each cost range.

23

Setting the

Maximum Hop

Parameters

The backbone network design model's primary minimizing solutions as the hop

role

is

to generate various cost

restrictions are parametrically

We

changed.

propose one possible scheme, called the Demand-based Hop Restriction Method, to

hop hop parameter h

systematically vary the selects the

hop parameter

values.

largest desired

hop

restrictions.

For each commodity , this method

from a user-specified window or range of permissible

Let hj and h 2 (> hj) denote, respectively, the smallest and parameter. Within the range [h-j, h 2 ], our method selects lower

h

values for high-demand commodities, and vice versa. In particular,

= hj

for the

and h

we

set

h

commodity with the largest demand (among all commodities), = h 2 for the commodity with the lowest demand. For commodities with

intermediate

demand

values, the

interpolation (rounded

down

hop parameters are calculated by

to the next

By imposing more stringent hop the hop restriction method gives

restrictions for

to select

[hj,

h 2 ].

commodities with higher demand,

greater importance to these commodities in

determining the network configuration. In practice, designers

and methods

linear

lower integer values) in the range

may

and systematically vary the hop parameters

commodity.

8-

use other

for each

criteria

Observe the resulting say, the

that as hj

and h 2 decrease, the hop constraints become

and

tighter,

network designs should become more dense. Also, when hj = h 2 =

method

selects the

same hop parameter value hg Hop

refer to this special case as the Uniform

for all

Restriction Method.

commodities.

hg,

We

Section 4.3,

presents computational results to illustrate the effect of various hop parameter settings using the general

methods. Next,

we

(commodity-dependent) and uniform hop

discuss a Lagrangian-based algorithm for generating good upper

and lower bounds on the optimal value

3.

restriction

of

[HCDP].

Solving the Hop-constrained Network Design Problem The hop-constrained design problem generalizes the fixed-charge network

design problem which

Kan

(1978)).

itself is

known

to

be NP-hard (Johnson, Lenstra and Rinnooy

Finding the optimal hop-constrained solution

computational

In this section,

task.

that simultaneously generates

good

we develop

is,

therefore, a difficult

method

a Lagrangian-based solution

feasible solutions as well as

lower bounds on the

optimal cost, for any given set of hop parameters. These bounds provide

performance guarantees so that the user can assess the quality of the heuristic solutions.

3.1

Lagrangian Relaxation Scheme The Lagrangian

relaxation

difficult discrete optimization

The method

(1981)).

method has been

problems

(see, for

successfully applied to several

example, Geoffrion (1974), Fisher

exploits special structure in the

problem formulation by

dualizing the complicating constraints, and solving the resulting subproblems using

For a given

efficient, specialized algorithms.

the optimal Lagrangian solution serves as a lower original problem.

Lagrange multipliers, the

set of

bound on the optimal

The Lagrangian subproblem solutions

c

st

of

cost of the

also provide useful

information to construct feasible designs for the original problem. The gap between the Lagrangian lower

maximum

bound and

the cost of the best feasible solution measures the

possible cost differential between the optimal

and

heuristic solutions.

To

generate good lower bounds, the Lagrange multipliers are changed iteratively using

techniques such as subgradient optimization or dual ascent (see, for example, Fisher (1981), Balakrishnan et

al.

(1989)).

For the hop-constrained design problem,

scheme

we

consider a Lagrangian relaxation

that dualizes the forcing constraints (2.4) using nonnegative Lagrangian

multipliers \iV% for

all

edges

(i,j)

and

all

commodities . Observe that when

9-

,

removed from formulation [HCDP], the problem decomposes

constraints (2.4) are into

two main subproblems: a Routing subproblem denoted as [RSPQi)], and an Edge

Selection subproblem denoted as [ESP(p.

The routing subproblem determines the

)].

optimal values of the route selection variables

z^, while

subproblem calculates the values of the design variables further

Vj:.

The routing subproblem

decomposes by commodity; we denote the subproblem corresponding

commodity as greater detail,

3.2

the edge selection

to

RSPp^fi)]. The next two sections describe these subproblems in

[

and discuss methods

to solve

them.

Routing Subproblem For any given set of Lagrange multipliers

commodity and every

variable cost for each

c«

\i

A

Cjj

+

Then, the routing subproblem

[

H^/dpq RSP p

= {^8}, we define an arc

for all

(^)]

(i,j)

(i,j)

adjusted

as follows:

€ A,

all

e K.

(3.1)

corresponding to commodity has the

following formulation:

[RSPpqM

^(u)

c^

min

=

c^ z^

£ (i,j)e

(3.2)

A

subject to ifi

1

IN

z

??

I^M

-

je

je

=

=

p,

-lifi = q ,and

(3.3)

N otherwise

IE >

2

node p

that

to

subproblem

(34)

1

j

zW Observe

Vr 3™

s

??

[

RSP

node q containing

a

(u)

=

]

or

1

for all

(i,j)

e A.

essentially seeks the shortest directed path

maximum

of

h

arcs; the adjusted variable costs

serve as arc lengths for this hop-constrained shortest path problem.

value Lpq(n) of the routing subproblem equals d

hop-constrained path.

-10

(3.5)

from

c^

The optimal

times the length of the shortest

We

solve the hop-constrained shortest path subproblem using a truncated Let U: denote

version of the method of successive approximations (Lawler (1976)).

node p

the length of the shortest path from

h

arcs.

u] =

Initially,

known

of U: are

c^

for all

if (p,j)

nodes

e A; otherwise u ] =

The

first

term

(u

h ,

min

j

i,

for

containing

j

At the end of stage

u

h

cW

+

j

we

h,

at

values

use the

e N:

:

i

equation

e N,

(i,j)

(3.6) is

e

A

]

(3.6)

}.

the length of the shortest

p

to

j,

which must consist of a h-arc shortest path

some intermediate node

plus the arc from

i,

i

to

j.

To solve subproblem [RSP pq (|i)L we terminate the algorithm after h pq The value u obtained ). the method's computational complexity is 0(n h final stage

from p

(i.e.,

to q.

most

containing at most h arcs. The second term represents the length of

the (h+l)-arc shortest path from to

for all

in the right-hand side of

path from p to

from p

[

°°.

node

network. During stage (h+1),

in the

j

following expression to compute u | h 1 u j = min

(the origin) to

with h = h

We can

pq )

stages; at the

gives the length of the shortest hop-constrained path

trace the actual path for

commodity by performing

the

usual backtracking procedure.

33 Edge

Selection Subproblem

Let us

now

consider the second Lagrangian subproblem involving the edge

selection variables Vy. Fjj

Given a

A

Fjj

set -

n of Lagrangian multipliers,

^

for all

represent the adjusted fixed cost for edge

{i,j).

The edge

involves selecting a subset of edges that connects

minimum select all

total (adjusted fixed) cost.

We

all

{i,j}

let

e E,

selection

(3.7)

subproblem

nodes of the network

at

solve this subproblem as follows: First,

edges with negative adjusted fixed

costs,

in order of nondecreasing adjusted fixed cost.

and arrange the remaining edges

At each stage

we

refer to the

subnetwork defined by the currently selected edges as the current network. Consider each edge

{i,j}

in

sequence from the sorted

list

of unselected edges.

connects two different components of the current network, select

If it

edge

and update the

current network by merging the two components; otherwise, discard edge

Repeat the procedure for

all

edges in the

must be connected. The sum

list.

At termination, the

of the adjusted fixed costs in this

(i,j)

final

{i,j}.

network

network gives the

optimal value of the edge selection subproblem. The computational effort required to solve this

subproblem

is

dominated by the edge sorting procedure which requires

-11

0( E log I

I

I

E

I

)

operations.

connectedness condition follows: set

Vj:

=

1 if Fj:

Note

(2.5),


q(n) of the (i

of

lower bound on the optimal value of the original

subproblem. To obtain good lower bounds, instance, Held, Wolfe,

the

commodities - For any given vector

all

Z(p.) is a

(ii)

is

and Crowder

we

use a subgradient method

(see, for

(1974) or Fisher (1981)) to iteratively adjust the

Lagrange multipliers.

We

note that our Lagrangian relaxation scheme does not satisfy the

integrality property (Geoffrion (1974)); for

programming

relaxation of the routing

Therefore, the best lower the linear

3.4

example, the optimal solution to the linear

subproblem might possibly be

bound obtained using our

programming lower bound

relaxation

for formulation

scheme may exceed

[HCDP].

Lagrangian-based Heuristic Procedure At each subgradient

iteration, the solution to the routing

subproblem

provides a set of feasible routes (that satisfy the hop restrictions) for

Note to

that the

edge selection subproblem

may

include

some edges

therefore, ignore the solution to the

the routing solution to construct an initial

design consists of

all

all

that

any of the selected routes and/or may omit some edges belonging

We,

fractional.

commodities.

do not belong

to these routes.

edge selection subproblem, and use only

initial feasible

design.

Our Lagrangian-based

edges belonging to the routes chosen in the routing

subproblem. For this design, we solve the hop-constrained shortest path problem for each commodity using the original variable costs Cj: as arc lengths. The sum of the fixed costs for the selected arcs

and the

(true) variable costs for all

commodities gives

the cost of the heuristic solution.

We

then attempt to improve this starting solution by applying a Drop

heuristic (Billheimer

method

and Gray

(1973)).

The drop procedure

that iteratively eliminates existing

reduce the

total cost.

When we drop an

variable costs might increase since

is

a local

improvement

edges from the current design

in

order to

edge, the total fixed cost decreases while the

some commodities must now flow on more

expensive alternate routes. At every stage, the procedure evaluates the net savings

12

Aj:

obtained by dropping each edge

zero or negative for

all

from the current design.

{i,j}

If

the net savings

edges, the procedure terminates. Otherwise,

it

is

drops an edge

with positive net savings, updates the current design and commodity routings, and decreases the upper bound.

Evaluating the revised routing costs a hop-constrained shortest path

savings for every edge a candidate

list

savings in the

at

each

of edges to evaluate.

to this

list.

The

local

entails solving

that previously

we do

used

not evaluate the

maintain and iteratively update contains

all

edges that yield net

we only evaluate the savings maximum savings is dropped

The edge with the

savings are also deleted from the

by evaluating the savings

we

Initially, this list

from the current design and removed from the

it

dropped

In subsequent iterations,

first iteration.

edges that belong

Instead,

iteration.

is

commodity

for each

Since this computation can be time-consuming,

this edge.

for

problem

when an edge

for all

improvement

When

list.

edges

list;

the

edges that do not give any net

list

becomes empty, we

reinitialize

in the current design.

heuristic can

be applied

at the

subgradient iteration. However, to reduce computation time,

procedure intermittently, say, once every 100 iterations or

if

end of each

we

only apply the drop

the current starting

design has a lower cost than the previous best starting design. The next section describes our computer implementation of the Lagrangian-based solution procedure,

and presents extensive computational

results for several

randomly generated

test

problems.

4.

Computational Results

We

implemented the Lagrangian-based algorithm

design problem in standard

FORTRAN

tested the algorithm using several

describes

random

some

test

on an IBM 3083 (model BX) computer, and

randomly generated problems. This section

features of our implementation,

problems. In

Our computational

all,

we

first set

of tests focuses

heuristic solutions, as a function of the

We

first

to generate

applied the algorithm to over 65 problem instances.

the al gorithm, in terms of computation time

restrictions.

and the method we used

tests (discussed in Sections 4.3

performance aspects. The

for the hop-constrained

and

4.4)

on evaluating the performance of

and quality

problem

address two different

of the Lagrangian-based

size, cost structure,

and hop

then focus on a single problem instance in order to assess the

impact on cost and performance of systematically varying the hop restrictions

.

exercise demonstrates the effectiveness of our approach for generating several alternative base designs.

13

This

4.1

Implementation Details In addition to the features described in Section 3, our implementation of the

Lagrangian relaxation algorithm incorporates:

upper bound before

(i)

method

a

initiating the subgradient procedure,

generate an

to

and

initial

a non-standard

(ii)

multiplier initialization method.

Initial Heuristic Solution

Generating an

To generate an

initial

upper bound, we

first

construct a feasible design using a

method which we call the BUILD heuristic, and subsequently apply the drop procedure to improve this design. The Build heuristic first sorts all commodities decreasing order of demand;

commodity

it

in

builds a feasible design by iteratively considering each

At the beginning of stage

in the sorted order.

E

k, let

(k-1)

denote the

set

of edges in the current design. This design contains feasible paths (satisfying the respective

hop

the algorithm

constraints) for each of the

augments

commodity has

this

first (k-1)

commodities. During stage

commodity,

design to ensure that the k

good feasible path. For constrained shortest path problem from node p a

this

purpose,

we

node q with

to

k,

say,

solve a hop-

arc lengths l^ defined

as follows:

= =

Essentially,

edge

if

{i,j}

if (i,j)

Cjj

/jj

Cjj

+

Fydpq

Ek_1 and

e

,

k_1

E

if(i,j)«

already belongs to the current design,

commodity . Otherwise, we add Cj:, effectively forcing commodity

(4.1)

.

we

assign only the

variable cost q: to

the per unit cost Fj:/d

the variable cost

to

cost of

edge

{i,j}.

Let

P denote to

Thus, at the end of stage

commodities

demand,

E

(k_1)

all

the

list.

of

Pk

,

i.e.,

algorithm reroutes paths in E

I

K

I

all

set of

edges E

I

K

I

is

first

.

k

in decreasing order of

high-demand commodities

developing the design.

a feasible design.

As

When

the

a final step, the

commodities over the respective shortest hop-constrained

using the original variable costs

drop procedure (described Build method.

then

we set E k = E (k_1) u Pk

By considering commodities

to the variable cost) in

procedure terminates, the

is

design contains feasible paths for the

the Build heuristic gives greater importance to

(which contribute more

absorb the entire fixed

The current design

l^.

new edges

k, the current

in the sorted

to

the set of edges belonging to the shortest hop-

constrained path from p to q using arc lengths

augmented by adding

pq

Cj:

as arc lengths.

We then apply the

in Section 3.4) to the starting solution constructed

The improved solution provides the

-14

initial

upper bound.

by the

,

Initializing

the Lagrange Multipliers

we

Instead of initializing the Lagrange multipliers to zero,

use the

heuristic solution to determine the starting values for the multipliers.

the total flow on each edge

then set equal to Fjj/x-

uP^

otherwise,

is

method completely

Lagrange multipliers

all

for all

commodities

computational

size factor (see, for instance,

tests,

Held

et al. (1974)) to 2;

Random Problem

4.2

We test

to a

if

the gap

for the

halves the step size factor

(ii)

restrictions.

number

The program

of

problem

To generate

and commodity

nodes

w

first

must

a test problem, the user

n,

edges m, and commodities

for the

random component

locates the required

feasibility).

It

random spanning

number

in

I

K

edge

later); (iv)

tree.

number

I

;

first

(ii)

cost

(iii)

and fixed-

costs,

average

network

demand

d;

nodes randomly on a 1000 x nodes

(to

ensure

of additional edges

(i.e.,

m-(n-

Having constructed the underlying network G:(N,E), and variable i and

denote the Euclidean distance between nodes then computed

of

tree over these

then adds the required

edges) randomly to the

j.

costs for each

The

edge

in G.

fixed cost F^ of

Let

edge

(i,j)

as:

Fjj

where

densities, cost

seed for random number generator;

(i)

the problem generator calculates the fixed

is

of

hop parameters.

1000 grid, and generates a

ejj

wide range

to construct a

hop-restriction information: the lower and upper limits, h| and h^, defining

the range of desired

1)

terminates

(iii)

between the current best upper

to-variable cost ratio r ''these parameters are explained (v)

and

if

Lagrangian relaxation algorithm. The problem generator can

structure information: weight

and

initializes the step

(i)

Generation

specify the following parameters: size information:

to the

very small value).

create test problems with different sizes, edge

and hop

Fj:

this edge.

implemented a random problem generator

problems

structures

on

for 15 consecutive iterations;

after 250 subgradient iterations (or earlier

and lower bounds reduces

in the initial design, the

our subgradient procedure:

bound does not improve

the lower

{i,j}

allocates the fixed cost

that flow

|j.P9 is

in the initial solution;

{i,j}

Thus, for each edge

initialized to zero.

multiplier initializatio:

For

commodity uses edge

if

Let x- denote

The multiplier

of the initial heuristic solution.

(i,j)

initial

=

w is the user-specified

(1 -

w)

weight

ej

(0

from a uniform distribution with range can generate cost values that are either

j




The variable

costs.

the fixed costs

0,

cost

for

Cj:

edge

may

{i,j}

is

not satisfy the triangle inequality.

obtained by dividing the fixed cost

the user-specified fixed-to-variable cost ratio

are negligible

compared

r.

When

r is

very small, the fixed costs

to the variable costs; in this case, the

union of the hop-

constrained shortest paths (using the variable costs as arc lengths) for

commodities gives the best design. At the other extreme, when variable costs are negligible,

and the best solution

is

F-by

r is

all

very large, the

network with the smallest

a

total

fixed cost that contains at least one hop-constrained path for every origin-destination pair.

In particular,

when h

for large values of r

= (n-1) for

all

commodities , the optimal solution

the minimal spanning tree.

is

After calculating the edge costs, the problem generator the required

number

(

I

K

I

)

of origin-destination pairs

commodities. For each commodity, the distribution ranging from

problems (even with IK commodities), the

demand

I

random

to 2d; less

demand d

we set d =

than

50%

which define the selected from a uniform

is

5 units in all cases. For

of the

identifies

maximum

all

our

number

possible

test

of

choice of origin-destination pairs resulted in complete

patterns (as defined in Section

2.1).

program generates the hop parameters

Finally, the

randomly

for each

commodity using

demand-based method described in Section 2.3, i.e., by interpolation in the range with high-demand commodities having lower hop parameter values and

the

[hi, h^]/

vice versa.

problem

Given

a set of

feasibility.

If

either increase the

hop parameter(s) or

some

try a different

sections describe the results of our computational experiments

to assess the effectiveness of the

Lagrangian relaxation algorithm, and generate

different cost minimizing designs

by varying the hop

restrictions.

Testing the Performance of the Lagrangian Relaxation Algorithm

We

use two performance measures to evaluate the effectiveness of the

Lagrangian relaxation algorithm: best

checks for

seed.

The next two

4.3

first

the network does not contain any feasible path for

commodity, the user must

random number

hop parameters, our implementation

(i)

the

%

upper and lower bounds expressed as

gap, defined as the difference between the a percentage of the best

-16-

lower bound,

which measures the quality of the (in

CPU

heuristic solutions,

and

(ii)

the computation time

seconds) required for the entire procedure.

Our computational characteristics

-

problem

tests

attempt to evaluate the effect of three problem

and hop

size, cost structure,

restrictions

performance measures (% gap and computation time). Problem

number

of

nodes

of commodities.

on the algorithmic

size

depends on the

network, the number (or density) of edges, and the number

in the

The

-

cost structure is

determined by two

randomness factor w, and the fixed-to-variable

cost ratio

characteristic affecting algorithmic effectiveness

is

factors: the cost

The

r.

third

problem

the tightness of the hop

measured, for instance, by (hj + h 2 )/2 (the mid-point of the userspecified window). For convenience, in this section we consider only the uniform hop restriction method, with h^ = h 2 = hQ results for the more general hop restrictions

;

restriction

4.3.2

method (with

h^

< h 2 ) are reported

Effect of Cost Structure

To

in Section 4.4.

and Hop Restrictions

isolate the effect of each of the three

performance,

randomness

we

first

consider different cost structures

and

factors

problem

characteristics

(i.e.,

different cost

and hop

fixed-to-variable cost ratios)

on algorithmic

restrictions for a 20-

network

node network with 180 edges and 180 commodities. For

this

generated 3 different problem instances (using different

random number

varying cost randomness factors, namely,

w=

0, 0.5,

and

1.

size,

we

seeds) with

For each network,

we

applied the Lagrangian relaxation algorithm with three fixed-to-variable cost ratios,

=

10, 20,

and

50,

and three

different (uniform)

hop parameter

values, hg =

3, 4,

and

5

r

hops. This set of parameters represents a wide spectrum of possible problem

from Euclidean costs

structures, ranging to

to

high fixed costs (relative to the variable

Table

and

summarizes the

I

hg, this table

shows:

(i)

(iii)

commodities, which

where h

'

h

(
Q

d™ w P9

to the objective function to

account for the spare capacity on the secondary route. (i,j)

24

(2.4').

we

solve this enhanced formulation,

To

The Lagrangian problem decomposes

again dualize the forcing constraints

into

an edge selection subproblem and a

routing subproblem for each commodity.

The edge

as before, while the routing subproblems

now

To

variables.

subproblem

selection

is

the

same

involve determining both the z and

we

find the optimal values of the z-variables

w

use the hop-constrained

shortest path algorithm as before; a regular shortest path solution provides the

optimal w-values (since constraints).

Observe

we assumed

that the

select exclusive arcs for the

constraints

(2.4')

that

Lagrangian subproblem solution

primary and secondary paths (since

which ensure

arc-disjoint paths).

we

solution for local improvement,

might:

primary route for each commodity, and

when

We

links of the

might also consider

(i)

Thus,

may not we have

necessarily

dualized

to obtain a feasible starting

use the z-solution to select the

generate alternate routes by applying a

(ii)

some information from

shortest path algorithm (perhaps, using

the residual network

secondary routes do not have hop

the w-solution) to

primary route are removed.

a stronger relaxation that involves

simultaneously

generating the hop-constrained primary routes and arc-disjoint secondary routes to

minimize

total variable costs.

Effectively, the

problem formulation contains an

additional set of strengthening constraints z Pfl

+

Z

W

W P9

+

+

w rfl