A memory-based multistart heuristic for a discrete cost multicommodity network design problem

1
A memory-based multistart heuristic
for a discrete cost multicommodity network
design problem

• Daniel Aloise

MATHEURISTICS Bertinoro, Italy August 2006
Celso C. Ribeiro
2
Outline

• Problem statement
• Constructive heuristics
• Local search
• Memory-based multistart heuristic
• Vocabulary building
• Results
• Concluding remarks

3
Problem statement

• Undirected network G(V,U)
• Problem determine the capacities to be
  installed in the edges
• Satisfying a set of demands D
  D1,,DK defined by triples Dksk,tk,fk
  

• Satisfying capacity constraints
  
  
• Minimizing the total network cost given in terms
  of the capacity installed in the edges

4
Problem statement

• Stepwise discrete cost function

A step-increasing cost function is associated to
each edge u ? U
5
Short literature review

• Stoer and Dahl (1994)
• Cutting planes
• Gabrel, Knippel and Minoux (1999)
• Benders decomposition method
• Gabrel, Knippel and Minoux (2003)
• Heuristic implementation of the exact Benders
  cutting plane method

Exact methods are limited, in practice, to small
instances not exceeding 20 nodes and 40 edges
with cost functions with six steps per link and
dense matrices of demands.
6
Constructive heuristic

• Procedure
• An initial capacity xu vui is selected to each
  edge u ? U, favouring steps with the best ratios
  between the capacity vui and the associated cost
  ?ui
• Route each demand using a maximum flow algorithm
• Capacities are augmented to route the yet
  unrouted demands
• Unused capacity steps are eliminated

7
Constructive heuristic step 1
Constructive heuristic step 3
Constructive heuristic step 4
Constructive heuristic step 2
f1 2 s1 1 t1 3
UR ?
f3 1 s3 4 t3 2
UR f2 5 s2 1 t2 4
f2 5 s2 1 t2 4
G
3
Infeasible routing!!! Minimum cut capacity 4
Initial installed capacities
New minimum cut capacity 5
We must consider the edges of the minimum cut!!!
(0,2)
(2,2)
(2,4)
(4,4)
(0,4)
(0,2)
(0,0)
1
4
(0,4)
(1,4)
(4,4)
(0,2)
(0,4)
(3,4)
2
edge (1,3) (7-4)/(4-2) 3/2
edge (1,2) (4-2)/(2-1) 2
8
Local search link rerouting

• Reroutes a fraction of the total flow traversing
  an edge.
• Considered quantities of the flow to be rerouted
  are those for which the edge capacity steps
  down in the objective function.

9
Local search link rerouting

• Suppose 5 unities of flow traverse edge u

Rerouting one unity of flow
Rerouting three unities of flow
Rerouting five unities of flow
5
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Local search link rerouting

• Large neighborhood number of paths for flow
  rerouting is not polynomially bounded
• Neighborhood exploration can be confined to a
  single path for each rerouted flow quantity

11
Local search link rerouting

• For each quantity ? of flow considered for
  rerouting in a given edge ,
  a graph is built with
  edge lenghts
• where yu is the total flow using edge u in the
  current solution.
• Reroute flow ? through the shortest path from a
  to b in G1

12
Local search link rerouting
G
3
(1,2)
(1,2)
(3,4)
1
4
(5,7)
(0,0)
2
5
Additional cost of rerouting five unities of flow
of u is 10. Cost variation 10 - (4-0) 6
Additional cost of rerouting one unity of flow
of u is 0. Cost variation 0 - (4-2) -2
Additional cost of rerouting three unities of
flow of u is 3. Cost variation 3 - (4-1) 0
13
Local search link rerouting

• Minimum rerouting cost is equal to the shortest
  path from a to b in G1.
• In each iteration of the local search method, all
  edges have their rerouting possibilities
  evaluated
• Perform link rerouting which reduces the most the
  cost of the current solution
• Local search stops when improvements are no
  longer possible

14
Local search demand rerouting

• Instead of rerouting aggregated flows in the
  edges, demands are individually considered for
  rerouting
• Remove from the edges the flows associated to the
  demand to be rerouted, updating the current
  solution and its cost
• Reroute the demand

Discrete cost multicommodity flow problem NP-hard
(Chopra et. Al., 1996)
15
Local search demand rerouting

• Heuristic devised for exploring the demand
  rerouting neighborhood
• Local search algorithm first attempts to reroute
  each demand using the residual capacities of
  the graph

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Local search demand rerouting
flow of demand k using edge u
total flow using edge u
capacity installed on edge u
cost associated to edge u
(?uk, yu, xu, ?u) Dk sk1, tk4, fk4
3
(0,1,2,1)
(2,3,4,2)
(0,1,2,1)
(2,2,2,1)
(0,0,0,0)
1
4
(0,2,2,1)
(0,3,4,2)
(2,5,7,4)
(4,6,7,4)
2
Solution cost equal to 12.
Capacity of the minimum cut between nodes 1 and 4
is 1. Impossible to reroute Dk using the
residual capacities!
17
Local search demand rerouting

• Additional capacities are installed in the edges
  of the minimum cut which separates sk from tk
  (sequentially, choosing one edge for capacity
  augmentation from the current minimum cut).
• Next additional capacity step is always installed
  in the edge of the minimum cut with the smallest
  ratio between the additional cost and the
  additional capacity

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Local search demand rerouting
flow of demand k using edge u
total flow using edge u
capacity installed on edge u
cost associated to edge u
(?uk, yu, xu, ?u) Dk sk1, tk4, fk4
3
(0,1,2,1)
(0,1,4,2)
(3,4,4,2)
(0,1,4,2)
(0,1,2,1)
(3,4,4,2)
(0,0,0,0)
1
4
(0,3,4,2)
(1,4,4,2)
(0,2,2,1)
(0,2,4,2)
(1,3,4,2)
2
Finally, capacity of the minimum cut between
nodes 1 and 4 is equal to 4.
Capacity of the minimum cut between nodes 1 and 4
is still equal to 2.
Regarding the edge (1,2) (7-4)/(4-2) 3/2
Regarding the edge (1,3) (4-2)/(2-1) 2
The value of the capacity of the minimum cut
between the nodes 1 and 4 equals 2.
Demand Dk can be rerouted with solution cost
equal to 8 (initial cost was 12)
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Local search demand rerouting

• Demands are exhaustively tested for rerouting
  until improvements are no longer possible
• The original routing is restored when a demand
  rerouting does not lead to an improvement in the
  solution cost

20
Memory-based multistart

• Multistart heuristics are iterative two-phase
  procedures
• Constructive heuristic
• Local search
• A multistart algorithm can be obtained by
  combining the constructive heuristic with the
  proposed local search strategies.
• Basic multistart heuristics do not take advantage
  of regions previously explored

21
Memory-based multistart

• Multistart heuristic with adaptive memory based
  on the ideas of Fleurent Glover, 1999.
• Pool of elite solutions stores the best solutions
  found during the search
• Construction of new solutions is intensified by
  using the information extracted from the elite
  solutions

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Memory-based multistart

• Parameter controls the relative weigth of the
  greedy information and the information contained
  in the pool of elite solutions during the
  construction
• In the beggining, weight of the greedy
  information is stronger (information in the pool
  is not relevant)
• At the end, weight of the information extracted
  from the pool is stronger, focusing the search
  into promising regions

23
Vocabulary building

• Vocabulary building Glover Laguna, 1997
  creates new solutions from pieces of solutions
  previously found
• Two basic steps
• Identification of pieces (words) common to good
  solutions
• Building of new solutions (phrases) from the
  combination of words

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Vocabulary building finding words

• Input pool of elite solutions
• Installed capacities in the elite solutions
• Employs the Int operator
• Searches for words of size greater than a given
  length l, while maximizing the number of
  solutions used by the Int operator.
• Considering two capacity vectors x and x
  associated to two different solutions s and s,
  the resulting capacity vector z obtained from the
  application of Int(s,s) is given by

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Vocabulary building finding words
if
if

• for the capacity installed in each edge u ? U.

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Vocabulary building example 1
l 3 O1 p1,p4,p2,p5,p6,p3,p7,p8
word identified
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Vocabulary building example 2
l 3 O2 p3,p4,p8,p5,p7,p6,p1,p2
From this solution, any operation Int can only
decrease the length of the word
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Vocabulary building build phrases

• Input set Z of extracted blocks from elite
  solutions (words).
• Logical inconsistencies generated by the Int
  operator are replaced by blank spaces.
• Employs the EInt operator
• New extended operator (EInt) is defined for
  combining words.
• After the elimination of inconsistences, a
  capacity vector w EInt(z,z), where z and
  z are identified words, is given by the
  following rules

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Vocabulary building build phrases

• if zu and zu are both non blank, the rules
  that define Int are used.
• otherwise, wu zu if zu is equal to blank,
  and wu zu if zu is equal to blank.
• for the capacity installed in each edge u ? U.

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Vocabulary building example 1
Complete phrase
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Vocabulary building example 2
incomplete phrase
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Vocabulary building complete phrases

• With the phrases at hand, a minimum cost
  multicommodity flow problem is solved to
• Route the demands
• Define the values of the still undefined capacity
  variables in incomplete phrases, thereby building
  a complete solution
• To accomplish the second goal, the edges with
  undefined capacities are considered in the linear
  model with
• Maximum allowed capacities (upper bounds)
• High utilization costs (big M)

33
Vocabulary building complete phrases

• Due to their high utilization costs, these edges
  are used for routing the demands only if strictly
  necessary.
• The installed capacities are given by the upper
  limit of the capacity step routing the aggregated
  flow in each edge obtained from the solution of
  this LP.

34
Algorithm overview
35
Computational results

• Instances are the same as those in Gabrel,
  Knippel Minoux, 2003.
• Instances with up to 20 nodes optimal solutions
  obtained by exact methods reported in
  Minoux,2001.
• Experiments performed on a Pentium IV 2 GHz with
  512 Mb of RAM memory.
• Algorithms implemented in C, compiled with gcc
  3.2, and run on Linux Red Hat 9.0 platform.

36
Computational experiments
Constructive heuristics

• In the proposed heuristics, the demands are
  routed by a minimum cost flow algorithm.
• Two different strategies for the attribution of
  values to the utilization costs of the edges were
  tested
• unitary utilization costs (strategy A).
• utilization costs inversely proportional to the
  residual capacity (strategy B).

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Computational experiments
Constructive heuristics

• Strategy B obtained better results, measured by
  the quantity of flow delayed to the set of
  unrouted demands UR.
• Less flow delayed to UR initial capacities are
  better used.

38
Computational results
Constructive heuristics
rn40_10 / HC1A - HC1B
39
Computational experiments
Local search

• Local search followed the VND strategy.
• Better results obtained when demand rerouting is
  applied first, followed by link rerouting.

40
Computational experiments
Multistart algorithm with adaptive memory (MSmem)

• Three paramers were adjusted for using the
  adaptive memory
• Size of the pool of elite solutions (Pool30)
• Minimum hamming distance (hammin3) among
  solutions in the pool
• Weight (?) of the greedy and memory information,
  continuously decreasing the value of this
  parameter as the algorithm proceeds.

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Computational experiments
Solution values after local search
42
Computational experiments
Solution values after local search
43
Computational experiments
Normalized average results obtained in ten
executions of the heuristics MS, MSmem and
OtimRedDisc for each size of instance.
44
Computational experiments

• Heuristics MSmem and OtimRedDisc were compared
  with the best heuristics in the literature, named
  here H1 and H2 Gabrel, Knippel Minoux,2003.
• Results and CPU times with heuristics H1 and H2
  obtained in a UltraSparc 10.

45
Computational experiments
46
Computational experiments
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Computational experiments
Average CPU times for H1, H2, MSmem and
OtimRedDisc for each size of instance.
... H1 and H2 visit infeasible solutions and stop
at the first feasible one.
OtimRedDisc and MSmem can perform fewer
iterations still obtaining feasible solution,
while...
OtimRedDisc and MSmem can be further improved
(max-flow and LP).
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Concluding remarks

• New heuristics are competitive with those in the
  literature.
• Better solutions for some benchmark instances.
• Computational experiments showed the efficiency
  of adaptive memory and vocabulary building to
  improve multistart heuristics.
• Other methods can be used to find and combine
  words.
• This approach can benefit from other (perhaps
  still to come) solution methods the minimum cost
  multicommodity flow problem.

49
References

• R. Ahuja, T. Magnanti e J. Orlin. Network flows
  Theory, algorithms and applications. Prentice
  Hall, 1993.
• C. Fleurent e F. Glover. Improved constructive
  multistart strategies for the quadratic
  assignment problem using memory adaptative.
  INFORMS Journal on Computing, 11198-204, 1999.
• V. Gabrel, A. Knippel e M. Minoux. A comparison
  of heuristics for the discrete cost
  multicommodity network problem. Journal of
  Heuristics, 9 429-445, 2003.
• F. Glover e M. Laguna. Tabu Search. Kluwer
  Academic Publishers, 1997.
• M. Minoux. Network synthesis and optimum network
  design problems Models, solution methods and
  applications. Networks, 19 313-360, 1989.

50
Computational experiments
Average results obtained in ten executions of the
heuristics MS, MSmem and OtimRedDisc for each
size of instance.