Robust BranchandCutandPrice for the Capacitated Vehicle Routing Problem

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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.

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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.

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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 ?

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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.

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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.

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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.

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• 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.

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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.

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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).

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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.

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Example - Instance A-n62-k8
Vehicle Capacity 100
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Optimal Solution A-n62-k8
Optimal Solution Value 1288
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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.

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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).

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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

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Two-index CVRP formulation (Laporte, 1984)
Client vertices have exactly two incident edges
Rounded Capacity Inequalities -eliminates
subtours and enforces route capacity
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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.

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

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

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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).

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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).

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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)

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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
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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.

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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.

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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).

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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.

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E series (Eilon-Christofides, 1969)
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A series (Augerat et al. 1995)
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Root gaps over 41 hard instances between 50 and
135 clients
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Detailed root results over 41 hard instances
(Geometric Means).
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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.

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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 !

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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.

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• Thanks !