YVES CASEAU

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Heuristics for Large Constrained VehicleRouting
Problems

• YVES CASEAU
• FRANCOIS LABURTHE
• ??????

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Abstract

• This paper presents a heuristic for solving very
  large routing problems (thousands of customers
  and hundreds of vehicles) with side constraints
  such as time windows. When applied to traditional
  benchmarks (Solomons),we obtain high quality
  results with short resolution time (a few
  seconds)
• The incrementality means that instead of visiting
  some large neighborhood after an initial solution
  has been found, a limited number of moves is
  examined, after each insertion, on the partial
  solution.

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Introduction

• Many real world problems involve dispatching a
  much larger number of trucks, as shown by the
  example of telecommunication maintenance
  dispatching (Caseau and Koppstein, 1992). For
  these problems, most methods proposed in the
  literature do not apply. Our goal in this paper
  is to present a technique that is both flexible
  and highly scalable.

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Introduction

• It is fast enough to be applied to very large
  problems, yet it produces high quality solutions,
  which we can measure for medium-sized problems by
  comparing with existing approaches. This is
  accomplished by mixing insertion and local
  optimization. This could seem not to be a newidea
  since most practical systems are built on top of
  this combination (finding a solution with
  insertion heuristic first, and optimizing it
  further with local moves), but what is new here
  is the interplay between the insertion and the
  local optimization routines. The heuristic is
  able to produce results on Solomons benchmarks
  which are in the same range as one of the best
  tabu approaches (Rochat and Taillard, 1995), with
  only a fraction of the execution time. Moreover,
  the heuristic can be easily adapted to various
  objectives (minimizing total distance or the
  number of trucks), with a few changes in the
  constraint model.

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A fast heuristic for the VRP

• Insertion heuristics for large VRPs
• Comparison with a standard approach
• Incremental local optimization
• Taking capacity constraints into account
• Swapping a chain from a route that is full into
  another one.
• Sequences of vertex relocations.
• Flushing a route in order to allow for vertex
  insertion.
• Results

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Insertion heuristics for large VRPs

• A VRP is an optimization problem, where the
  objective function is the sum of the lengths of
  the routes, either for a bounded number of routes
  or for the minimal number of routes. A route is
  represented by its first node start(r ) and the
  successor relation (next(i ) is tA good heuristic
  for selecting the customers is to start with the
  customers that are furthest from the depot.
• This heuristic may be generalized into selecting
  the most difficult customer to insert when side
  constraints are added. We shall discuss other
  possible sorting heuristics later.he successor of
  the node i ).

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Comparison with a standard approach

• Many interesting moves have been proposed during
  the last few years (Gendreau, Hertz, and Laporte,
  1992 Thompson and Psaraftis, 1993 Kontoravdis
  and Bard, 1995 Taillard et al., 1997
  Kinfervater and Savelsbergh, 1997), but most of
  them can still be seen as 2 or 3-edge exchanges.
  The complexity for exchanging more than 3 edges
  is quickly an impossible burden with large
  problems.

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Comparison with a standard approach

• The previous local optimization strategy is
  fairly expensive and is not a realistic option
  for large routing problems with thousands of
  nodes. The largest problems that we can handle
  have a few hundred nodes. This approach was built
  as a control point to evaluate how we could
  eliminate some of the moves to obtain a faster
  algorithm without loosing too much quality. On
  the other hand, both the savings and the
  insertion algorithm are very fast and can be used
  for very large routing problems.

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Incremental local optimization

• We propose to use incremental local optimization
  in a systematic manner, by trying to re-optimize
  the solution after each insertion step, and
  taking advantage of a variety of moves, not
  limited to the route where the insertion took
  place but also involving several routes.
  Moreover, in order to improve over the simple
  look-ahead insertion scheme, we use a limited
  form of incremental local optimization before
  evaluating the insertion cost and a more thorough
  after the best insertion route has been chosen.

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Incremental local optimization

• We start by using three different moves (see
  figure) after the insertion of i into r has been
  performed we consider a 2-edge exchange which
  connects the start of r with the end of another
  route r 0 (and the start of r 0 with the end of r
  ), a 3-edge exchange for transferring a chain
  from another route r 0 into r , as well as a
  simpler node transfer move (a limited version of
  the chain transfer).

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Incremental local optimization

• However, we memorize
  as an potential approximation of the insertion
  cost, if this insertion could become feasible
  (stored as value.r /). When all routes have been
  considered, we may reconsider such infeasible
  insertions for three reasons
• 1. because there exists one infeasible insertion
  with a cost value.r / much below the best cost
  among all feasible insertions,
• 2. or, because we settled for a route creation,
• 3. or, because no feasible insertions could be
  found at all.

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Taking capacity constraints into account

• Swapping a chain from a route that is full into
  another one.
• Sequences of vertex relocations.
• Flushing a route in order to allow for vertex
  insertion

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Results
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Results

• We see that the savings heuristic is much better
  than the insertion heuristic on average.
• The second observation is that local search does
  improve significantly the results (improving by
  14.5 the total distance after the insertion
  heuristic) but is computationally expensive and
  is not well suited for large problems (the full
  local optimization phase takes over an hour of
  CPU time on problems with N D 1000 nodes).

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An improved heuristic for VRPTW

• Taking time windows into account
• Exact optimization of individual TSPTW
• Considering infeasible insertions
• Global procedure
• Results on medium-size VRPTW
• Results on large-size VRPTW

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Further experiments with medium-sized VRPTW

• Enforcing extra constraints to adapt a different
  objective
• Producing better quality results with limited
  global search
• Limited discrepancy search
• Results on Solomons benchmarks
• Conclusion