Network Design:
Network Loading and
Pup Matching
Thomas
Thomas L.
L. Magnanti
Magnanti
Today’s Agenda
† Network design in
general
† Network loading
† Solution approaches
‹Polyhedral
‹Polyhedral
combinatorics
combinatorics
‹Heuristics
‹Heuristics
† Pup matching
Network Design: Basic Issue
Total
Total (Fixed)
(Fixed) Cost
Cost on
on Each
Each Arc
Arc
Commodity
Commodity k:
k:
Origin
Origin O(k)
O(k)
Destination
Destination D(k)
D(k)
Flow
Flow req.
req. rrkk
Link
Link (i,j):
(i,j):
Fixed
Fixed cost
cost F
Fijij
Flow
Flow cost
cost ccijij
Possibly:
Possibly:
Capacity
Capacity C
C per
per
unit
unit installed
installed
on
on any
any edge
edge
Network Design Applications
† Telecommunications systems
† Airline route maps
† Chip design
† Facility location
† Even TSP!
Multicommodity Flow Model with
Complex Costs
minimize c( f )
subject to Nf k = b k for k = 1, 2,..., K
f = ( f 1 , f 2 ,..., f K ) ≥ 0
 r kk if i = O(k ) 


biikk =  − r kk if i = D(k ) 
 0 otherwise 


k
ij
(possible flow bounds on f )
c( f ) = ∑ ( i , j )∈A cij ( fij ) separability
c( f ) = ∑ ( i , j )∈A lij c ( fij ) proportionality
fij = ∑ k fijk
(
fij = ∑ k wk f ijk
)
Basic Cost Structures
cij(fij)
cij (fij)
fij
fij
Network Loading Cost
LP Approximation
cij(fij)
LP Gap
C
2C
fij
Other Cost Structures
cij (fij)
cij (fij)
fij
fij
Integer Programming Model
minimize ∑ k c k f k + ∑ ( i , j )∈E Fij yij
subject to Nf k = b k k = 1, 2,..., K
∑ (f
k
k
ij
+f
k
ji
) ≤ Cy
ij
{i, j} ∈ E
fijk ≤ r k yij {i, j} ∈ E , all k
f = ( f 1 , f 2 ,..., f K ) ≥ 0
yij ≥ 0 and integer all {i, j} ∈ E
(configuration constraints and y )
(1)
(2)
Cuts for Lower Bounds
†
† LP
LP relaxations
relaxations yield
yield
lower
lower bounds
bounds
†
† Addition
Addition of
of cuts
cuts can
can
tighten
tighten bounds
bounds
‹
‹ Cut
Cut away
away solutions
solutions to
to
the
the LP
LP relaxation
relaxation but
but
leave
leave all
all feasible
feasible
integer
integer points
points
†
†
Network Loading Model
min
∑
{i , j}∈E
Fij yij
subject to
 1, i = O(k )

k
k
fij − ∑ f ji = −1, i = D(k ) all k ∈ K , i ∈ N
∑
{ j:(i , j )∈A}
{ j:( j ,i )∈A}
0, otherwise

∑(
k∈K
)
fijk + f jik ≤ Cyij all {i,j}∈ E
fijk ≥ 0, yij ≥ 0 and integer
Improved Modeling
(Cutset Inequalities)
F
Fjj == 11 all
all edges,
edges, all
all ccijij == 00
C
C == 24
24
Designs
Designs
11
Demands
Demands
Y>3
1
Y>2
24
24
1
66
Y>1
2
30
30
4
12
12
11 1/4
1/4
1/4
1/4 4
Y>3
3
12
12
2
2
1
2
1
4
1/2
1/2
3
LP
LP ==
33 ½
½
1/2
1/2
11
11
Opt
=
Opt
=
3
55
General Cutset Inequality
DST = total demand (nodes in S to nodes in T)
Set S
of nodes
Set T
of nodes
 DST 
YST ≥ 
 C 
Cutset Inequalities Aren’t Sufficient
Capacity C = 1
Demand = 1 between all
non
-adjacent nodes
non-adjacent
Loading 1 unit on all
visible edges satisfies
cutset inequalities,
but not feasible
Pup Matching
Example
optimal solution is 9
Pup Matching
†
† Instance:
Instance: A
A directed
directed network
network G
G == (N,A),
(N,A), aa
set
set of
of K
K pairs
pairs of
of elements
elements from
from N,
N, and
and aa
cost
cost function
function c:
c: A→R+.
A→R+.
†
† Problem:
Problem: Find
Find the
the minimum
minimum cost
cost loading
loading
of
of G
G permitting
permitting unit
unit flow
flow from
from the
the first
first to
to
the
the second
second node
node of
of each
each of
of the
the K
K pairs
pairs
such
such that
that 11 unit
unit or
or 22 units
units together
together can
can
traverse
traverse an
an arc
arc for
for each
each unit
unit of
of loading.
loading.
One
One unit
unit of
of loading
loading on
on aa ∈A
∈A costs
costs c(a).
c(a).
Example on City Blocks
Solution with cost 196
Several days
computation
can prove only
that the
objective is at
least 184 (LP
lower bound =
182). Can we do
better?
Heuristics for Upper Bounds
†
† Matching
Matching Heuristic
Heuristic
‹
‹permits
permits each
each pup
pup to
to be
be paired
paired with
with at
at most
most
one
one other
other pup
pup
‹
‹solved
solved with
with aa weighted
weighted matching
matching routine
routine
†
† Shortest
Shortest Path
Path Heuristics
Heuristics
‹
‹three
three variations
variations
†
† Each
Each heuristic
heuristic provides
provides aa 2-approximation
2-approximation
to
to the
the NLP
NLP formulation
formulation
Odd Flow Inequality
† If the flow on an arc is odd, one unit
of loaded capacity will be unused
† If the net demand of a node is odd,
the total inflow or total outflow is
odd
 1
1
k
k
+
y
−
f
+
f
)
(
 ∑∑ ij
ij
ji
ji )  ≥
2 k j
 2
∑( y
j
Odd Flows on the City Block
Each
Each of
of the
the 56
56 nodes
nodes
must
must be
be incident
incident to
to at
at
least
least one
one arc
arc with
with
1
unit
of
spare
capacity
unit of spare capacity
2
Ä
Ä solution
solution requires
requires at
at
least
least
56 
 1
⋅
= 14
cabs’
cabs’
 2
2 

worth
worth of
of empty
empty
capacity
capacity
LP
LP relaxation
relaxation of
of 182
182
gives
gives lower
lower bound
bound on
on
required
required used
used
capacity
capacity
Ä
Ä 196
196 is
is optimal
optimal
Gap Reductions on
City Block Problems
†
Trials Using Realistic Data
†
† Node
Node set
set given
given in
in (latitude,
(latitude, longitude)
longitude)
format
format based
based on
on aa real
real logistics
logistics network
network
†
† Defined
Defined problems
problems by
by choosing
choosing aa subset
subset of
of
nodes,
nodes, calculating
calculating arc
arc lengths,
lengths, and
and
randomly
randomly selecting
selecting O-D
O-D pairs
pairs
†
† 30
30 problems,
problems, about
about half
half single
single origin
origin
†
† Complete
Complete graphs,
graphs, 12-25
12-25 nodes,
nodes, 6-50
6-50 pups
pups
Results
†
† Branch
Branch and
and Bound
Bound limited
limited to
to 22 hours
hours CPU
CPU
time
time and
and aa 220M
220M tree
tree
†
† With
With all
all 33 cut
cut families,
families, 67%
67% were
were solved
solved to
to
optimality
optimality with
with an
an average
average gap
gap reduction
reduction
of
of 18.8%
18.8% to
to 6.4%
6.4%
†
† Without
Without odd
odd flow
flow cuts,
cuts, 30%
30% were
were solved,
solved,
and
and the
the gap
gap was
was reduced
reduced to
to 7.8%
7.8% on
on
average
average
†
† With
With no
no cuts,
cuts, 17%
17% were
were solved
solved
†
† Among
Among solved
solved problems,
problems, average
average
heuristic
heuristic error
error was
was 1.3%
1.3%
Conclusions and Extensions
†
† Extensions
Extensions apply
apply to
to compartmentalized
compartmentalized
problems
problems
†
† Cuts
Cuts seem
seem critical
critical to
to provably
provably solving
solving the
the
PM
PM problem
problem
†
† Odd
Odd flow
flow inequalities
inequalities define
define what
what seems
seems
an
an important
important set
set of
of facets
facets
‹
‹generalize
generalize to
to arbitrary
arbitrary capacity
capacity
‹
‹can
can generalize
generalize to
to several
several facilities?
facilities?
†
† Are
Are there
there other
other cuts
cuts based
based on
on even-odd
even-odd
type
type arguments?
arguments?
Many, Many Network Design Variants
† Network loading for
compartmentalized capacity
‹airline
‹airline capacity
capacity planning,
planning, tanker
tanker trucks
trucks
† Network survivability
† Network restoration
† Hierarchical designs
†…
Today’s Lessons
† Network design arises in numerous
applications
† Problem is a large-scale integer
program
† Introduction to cutting planes
(polyhedral combinatorics)
† Cutting planes valuable in tightening
formulations and in problem solving
Module on Large-Scale
Integer Programming & Combinatorial
Optimization
Three Lectures
† Traveling salesman problem
† Facility location
† Network design
Games/Challenges
Applications, Models, and
Solution Methods