Unformatted text preview: Network Design:
Network
Network Loading and
Pup Matching Thomas L. Magnanti
Thomas L. Magnanti Today’s Agenda
Today’s
Network design in
general
Network loading
Solution approaches
Polyhedral
Polyhedral
combinatorics
combinatorics
Heuristics
Heuristics Pup matching Network Design: Basic Issue
Network
Total (Fixed) Cost on Each Arc
Total (Fixed) Cost on Each Arc Commodity k:
Commodity k:
Origin O(k)
Origin O(k)
Destination D(k)
Destination D(k)
Flow req. rk
Flow req. rk
Link (i,j):
Link (i,j):
Fixed cost Fij
Fixed cost Fij
Flow cost cij
Flow cost cij
Possibly:
Possibly:
Capacity C per
Capacity C per
unit installed
unit installed
on any edge
on any edge Network Design Applications
Network
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
Network LP Approximation cij(fij) LP Gap
C 2C fij Other Cost Structures
Other 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
Cuts
LP relaxations yield
LP relaxations yield
lower bounds
lower bounds
Addition of cuts can
Addition of cuts can
tighten bounds
tighten bounds
Cut away solutions to
Cut away solutions to
the LP relaxation but
the LP relaxation but
leave all feasible
leave all feasible
integer points
integer points Network Loading Model
Network
min ∑ {i , j}∈E Fij yij subject to 1, i = O(k ) k
k
∑∈A} fij − { j:(∑∈A} f ji = −1, i = D(k) all k ∈ K, i ∈ N
j ,i )
{ j:(i , j )
0, otherwise ∑( k∈K ) k
fijk + f ji ≤ Cyij all {i,j}∈ E fijk ≥ 0, yij ≥ 0 and integer Improved Modeling
Improved
(Cutset Inequalities) Fjj = 1 all edges, all cij = 0
F = 1 all edges, all cij = 0
C = 24
C = 24 Designs
Designs
1
1 Demands
Demands
Y>3 1 Y>2
24
24 1 6
6
Y>1 2 30
30
4 12
12 1 1/4
1 1/4 1/4 4
1/4 Y>3
3 12
12 2 2
1 2 1
4 1/2
1/2
3 LP =
LP =
3½
3½ 1/2
1/2
1
1
1
1 Opt =
3 Opt =
5
5 General Cutset Inequality
General
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
Cutset
Capacity C = 1
Demand = 1 between all
nonadjacent nodes
nonadjacent Loading 1 unit on all
visible edges satisfies
cutset inequalities,
but not feasible Pup Matching
Pup Example
Example optimal solution is 9 Pup Matching
Pup
IInstance: A directed network G = (N,A), a
nstance: A directed network G = (N,A), a
set of K pairs of elements from N, and a
set of K pairs of elements from N, and a
cost function c: A→R+.
cost function c: A→R+.
Problem: Find the minimum cost loading
Problem: Find the minimum cost loading
of G permitting unit flow from the first to
of G permitting unit flow from the first to
the second node of each of the K pairs
the second node of each of the K pairs
such that 1 unit or 2 units together can
such that 1 unit or 2 units together can
traverse an arc for each unit of loading.
traverse an arc for each unit of loading.
One unit of loading on a ∈A costs c(a).
One unit of loading on a ∈A costs c(a). Example on City Blocks
Example Solution with cost 196
Solution
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
Heuristics
Matching Heuristic
Matching Heuristic
permits each pup to be paired with at most
permits each pup to be paired with at most
one other pup
one other pup
solved with a weighted matching routine
solved with a weighted matching routine Shortest Path Heuristics
Shortest Path Heuristics
three variations
three variations Each heuristic provides a 2approximation
Each heuristic provides a 2approximation
to the NLP formulation
to the NLP 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
+ y ji ) − ∑∑ ( fijk + f ji ) ≥
ij
2 k j
2 ∑( y
j Odd Flows on the City Block
Odd
Each of the 56 nodes
Each of the 56 nodes
must be incident to at
must be incident to at
least one arc with
least one arc with
1
unit of spare capacity
unit of spare capacity
2
solution requires at
solution requires at
least
least
56 1
⋅
= 14
cabs’
cabs’
2
2 worth of empty
worth of empty
capacity
capacity
LP relaxation of 182
LP relaxation of 182
gives lower bound on
gives lower bound on
required used
required used
capacity
capacity
196 is optimal
196 is optimal Gap Reductions on
Gap
City Block Problems Trials Using Realistic Data
Trials
Node set given in (latitude, longitude)
Node set given in (latitude, longitude)
format based on a real logistics network
format based on a real logistics network
Defined problems by choosing a subset of
Defined problems by choosing a subset of
nodes, calculating arc lengths, and
nodes, calculating arc lengths, and
randomly selecting OD pairs
randomly selecting OD pairs
30 problems, about half single origin
30 problems, about half single origin
Complete graphs, 1225 nodes, 650 pups
Complete graphs, 1225 nodes, 650 pups Results
Results
Branch and Bound limited to 2 hours CPU
Branch and Bound limited to 2 hours CPU
time and a 220M tree
time and a 220M tree
With all 3 cut families, 67% were solved to
With all 3 cut families, 67% were solved to
optimality with an average gap reduction
optimality with an average gap reduction
of 18.8% to 6.4%
of 18.8% to 6.4%
Without odd flow cuts, 30% were solved,
Without odd flow cuts, 30% were solved,
and the gap was reduced to 7.8% on
and the gap was reduced to 7.8% on
average
average
With no cuts, 17% were solved
With no cuts, 17% were solved
Among solved problems, average
Among solved problems, average
heuristic error was 1.3%
heuristic error was 1.3% Conclusions and Extensions
Conclusions
Extensions apply to compartmentalized
Extensions apply to compartmentalized
problems
problems
Cuts seem critical to provably solving the
Cuts seem critical to provably solving the
PM problem
PM problem
Odd flow inequalities define what seems
Odd flow inequalities define what seems
an important set of facets
an important set of facets
generalize to arbitrary capacity
generalize to arbitrary capacity
can generalize to several facilities?
can generalize to several facilities? Are there other cuts based on evenodd
Are there other cuts based on evenodd
type arguments?
type arguments? Many, Many Network Design Variants
Many,
Network loading for
compartmentalized capacity
airline capacity planning, tanker trucks
airline capacity planning, tanker trucks Network survivability
Network restoration
Hierarchical designs
… Today’s Lessons
Today’s
Network design arises in numerous
applications
Problem is a largescale integer
program
Introduction to cutting planes
(polyhedral combinatorics)
Cutting planes valuable in tightening
formulations and in problem solving Module on LargeScale
Module
Integer Programming & Combinatorial
Optimization Three Lectures
Traveling salesman problem
Facility location
Network design Games/Challenges
Applications, Models, and
Solution Methods ...
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Full Document
 Spring '04
 JieSun
 Operations Research, Optimization, lower bounds, fij

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