Robust BranchandCutandPrice for the Capacitated Vehicle Routing Problem - PowerPoint PPT Presentation

Title: Robust BranchandCutandPrice for the Capacitated Vehicle Routing Problem


1
Robust Branch-and-Cut-and-Price for the
Capacitated Vehicle Routing Problem

• Ricardo Fukasawa, ISYE, Georgia Tech, USA.
• Jens Lysgaard, Aarhus School of Business,
  Denmark.
• Marcus Poggi de Aragão, Informática, PUC-Rio,
  Brazil.
• Marcelo Reis, Informática, PUC-Rio, Brazil.
• Eduardo Uchoa, Engenharia de Produção, UFF,
  Brazil.
• Renato F. Werneck, Computer Science, Princeton,
  USA.

2
What is a robust branch-and-cut-and-price ?

• A branch-and-bound algorithm where
• each node can solve a linear program with an
  exponential number of both variables and
  constraints,
• neither branching nor separation (cutting) ever
  change the structure of the pricing subproblem.

3
Robust branch-and-cut-and-price

• Since cut and column generation were established
  as two of the most important techniques in IP,
  one looks for ways of combining them.
• Big question
• How to cut efficiently without changing
• the structure of the pricing subproblem ?

4
Robust branch-and-cut-and-price

• First (non-robust) BCP Nemhauser and Park
  (1991), edge-coloring problem.
• In a non-robust BCP, the pricing subproblem
  eventually needs to be solved by general IP
  techniques. This only works for small instances.

5
Robust branch-and-cut-and-price

• A number of researchers noted that cuts
  expressed in terms of variables from a suitable
  original formulation do not disturb the pricing
  and constructed the first robust BCPs
• Vanderbeck (1998), lot-sizing.
• Kim, Barnhart, Ware and Reinhardt (1999), a
  service network design problem.
• Kohl, Desrosiers, Madsen, Solomon and Soumis
  (1999), VRPTW.
• Van der Akker, Hurkens and Savelsbergh (2000), a
  scheduling problem.
• Felici, Gentile and Rinaldi (2000), a
  supply-chain problem.
• Barnhart, Hane and Vance (2001), integer
  multiflow.

6
Robust branch-and-cut-and-price

• This approach has a fundamental drawback as a
  general IP technique on many important problems
  (like bin-packing, cutting stock, graph coloring,
  many clustering and scheduling problems, etc.)
  the original formulation suffers from variable
  symmetries.
• Cuts over those variables are ineffective.
• Up to now, robust BCP was only performed on a
  few problems where BP alone already performs
  reasonably well.

7

• Integer Program Reformulation for Robust
  Branch-and-Cut-and-Price, M. Poggi de Aragão and
  E. Uchoa, Proceedings of Mathematical Programming
  in Rio A conference in honour of Nelson Maculan,
  2003.
• Also presented at ISMP03, Copenhagen.

8
Poggi de Aragão and Uchoa (2003)

• Introduce a technique to read the reduced costs
  of the original variables from the master LP.
• Propose a reformulation scheme to eliminate
  symmetry and allow robust BCP on problems like
  bin-packing, cutting stock, graph coloring, etc.
• Herald the potential of the robust BCP
  technique to improve current algorithms on a
  large set of IP problems.

9
Robust Branch-and-Cut-and-Price

• E
• Our group obtained very good results on the
  following problems
• Capacitated Minimum Spanning Tree R.Fukasawa,
  O. Porto, M. Poggi de Aragão and E. Uchoa (2003).
• Generalized Assignment A. Pigatti, M. Poggi de
  Aragão and E. Uchoa (2003).
• Capacitated Arc Routing H. Longo, M. Poggi de
  Aragão and E. Uchoa (2004).

10
Capacitated Vehicle Routing Problem (CVRP)

• Instance
• Undirected graph G(V,E), V 0,1,...,n
• Vertex 0 represents a depot and remaining
  vertices represent clients
• Edge lengths, denoted by c(e)
• Client demands, denoted by q(i)
• Number K of vehicles and vehicle capacity C
• Solution
• A route for each of the K vehicles satisfying the
  following constraints
• (i) Routes start and end at the
  depot,
• (ii) each client is visited by
  exactly one vehicle
• (iii) the total demand of clients
  visited in a route is at most C.
• Goal
• Minimize total route length.

11
Example - Instance A-n62-k8
Vehicle Capacity 100
12
Optimal Solution A-n62-k8
Optimal Solution Value 1288
13
Capacitated Vehicle Routing Problem (CVRP)

• Considered one of the hardest combinatorial
  problems to solve exactly.
• The evolution of the algorithms has been very
  slow
• State of the art in 1978 Exact solution for up
  to 25 clients.
• State of the art in 2003 Exact solution for up
  to 50-100 clients.
• Problems with many vehicles (k gt 7) are usually
  harder.

14
Algorithms for the CVRP

• Best approaches during the 80s Column
  Generation and Lagrangean Relaxation
• Christofides, Mingozzi, Toth (1981)
• Agarwal, Mathur, Salkin (1989)
• Fisher (1994).
• Branch-and-Cut algorithms were developed in the
  mid 90s and have have been the best since then
• Augerat, Belenguer, Benavent, Corberan, Naddef,
  Rinaldi (1995)
• Martinhon, Lucena, Maculan (1998)
• Blasum, Hochstattler (2000)
• Ralphs, Kopman, Pulleyblank, Trotter (2003)
• Achuthan, Caccetta, Hill (2003)
• Lysgaard, Letchford, Eglese (2003).

15
Formulating the CVRP with an exponential number
of constraints

• Let 0 be the root node (depot)
• V1,...,n V \ 0
• q(S) Sum of the demands on client set S ? V
• Let Xe be the number of times a vehicle
  traverses edge e

16
Two-index CVRP formulation (Laporte, 1984)
Client vertices have exactly two incident edges
Rounded Capacity Inequalities -eliminates
subtours and enforces route capacity
17
Formulating the CVRP with an exponential number
of variables

• Let R be the set of all valid routes (total of
  p).
• Let Q be a m X p matrix of the edge incidence
  vectors associate to these routes.
• Let qje denote the coefficient associated to edge
  e in route j .
• Let ?j be a variable indicating whether route j
  is in the solution or not.
• The x variables can be expressed as a
  combination of the ?s.

18
Formulation for the CVRP with an exponential
number of variables
19
Adding Rounded Capacity Inequalities
20
LP Relaxation Explicit Master LP
21
D-W Master LP
22
Other cuts

• A solution of D-W Master is translated to
  .
• If is fractional, a violated cut
• can be added as

23
Cutting planes

• Bound cuts (easily separated by inspection)
• Rounded capacity
• Strengthened combs
• Multistars
• Framed capacities
• Hypotours
• Sophisticated heuristic separation, same already
  used in Lysgaard, Letchford and Eglese (2003).

24
The pricing subproblem

• Finding a valid route with minimum reduced cost
  Capacitated Prize Collecting TSP.
• Strongly NP-hard problem.
• Relax that problem to allow cycles. The resulting
  walks can be found by dynamic programming in
  pseudo-polynomial time.
• Such walks are called q-paths in Christofides,
  Mingozzi and Toth (1981).

25
Basic Algorithm

• Dynamic Programming
• Matrix M with C rows and n columns.
• Entry M(v,d) is a label
• best walk from depot to client v with total
  demand d
• total cost pointer to previous label.
• Compute one row at a time, from the bottom.
• To process Mv,D for each outgoing edge (v,w)
  check if
• cost(Mv,D) cost(v,w) lt cost(Mw,Dd(w))
• O(n2C) time (nC entries, O(n) per entry)

26
Cycle Elimination

• To avoid s-cycles, a label cannot be extended to
  the last s clients in the path.

v1
v2
v3
v4
v5
v6
v9
v4
v13
v7
v12
v8
v9
v10
v11
v12
v13
v14
27
Cycle Elimination

• To avoid s-cycles, a label cannot be extended to
  the last s clients in the path.
• Entry Mv,d becomes bucket of labels.
• Alternative walks from depot to v with demand d.
• Discard dominated labels.
• L is dominated if every extension of L also
  extended by a cheaper label Irnich and
  Villeneuve, 2003
• To eliminate s-cycles, as many as s! labels are
  needed per bucket.

28
Acceleration Heuristics

• Main heuristics
• Sparsification
• use only non-depot edges from the five cheapest
  disjoint MSTs.
• Scaling
• Round up accumulated demands (fewer rows).
• Bucket pruning
• Only s labels per bucket (instead of s!).
• When they all fail, use exact algorithm.

29
BCP Implementation Details

• Needs good data structures to quickly compute the
  coefficients of a cut.
• Determines best value of s before starting the
  search tree, balancing root bound and cpu time.
• Branching on sets of clients (exactly one vehicle
  or more than 2 vehicles).

30
Results

• The robust BCP is consistently better than
  previous algorithms, allowing us to solve to
  optimality all instances from the literature with
  up to 135 vertices in reasonable times.
• Experiments on Pentium 2.4 GHz. Pricing is the
  most expensive operation on most instances.

31
E series (Eilon-Christofides, 1969)
32
A series (Augerat et al. 1995)
33
Root gaps over 41 hard instances between 50 and
135 clients
34
Detailed root results over 41 hard instances
(Geometric Means).
35
Possible improvements on our BCP

• Better Pricing
• New dynamic programming algorithms that allows
  one to price walks as close as possible to actual
  routes in reasonable times.
• Promising area of research.

36
Possible improvements on our BCP

• New cuts
• We are already using most known families of cuts.
  Only RC cuts have shown much effect on the BCP.
• Find new families of cuts that are not implicitly
  given by the column generation. Being facet is
  not enough !

37
Possible improvements on our BCP

• Implementation details and fine-tuning
• Obtaining maximum performance from a BC or BP can
  be hard, requiring much experimentation,
  computational tricks, etc. On BCP, this is twice
  as hard.
• Many such improvements are still possible on our
  BCP.

38

• Thanks !