Solving VeryLarge Scale Multicommodity Flow Network Design Problems

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Solving Very-Large Scale Multicommodity Flow
Network Design Problems

• Ravindra K. Ahuja
• Professor and Co-Director, Supply Chain and
  Logistics Engineering (SCALE) Center
• ISE Department, University of Florida,
  Gainesville, FL 32611
• ahuja_at_ufl.edu www.ise.ufl.edu/ahuja

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Problem Description

• Given
• A set of shipments with different
  origins/destinations
• Determine
• Design the network and route all shipments over
  the blocking network
• Constraints
• Maximum number of arcs we can build at each node
• Volume of cars passing through each node is
  limited
• Some additional practical constraints
• Objective Function
• Minimize the weighted sum of distance traveled by
  shipments and their intermediate handlings

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An Illustrative Example
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Airline Schedule Design Problem
Seattle
Washington
Pullman
Salt Lake City
Arlington
Atlanta
Daytona
Austin
Dallas
Jacksonville
Orlando
Gainesville
Houston
Design the flight network and route all
passengers in it to minimize the weighted sum of
travel times and transfers.
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Package Delivery Problem
Origins
Destinations
Sorting Stations
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Railroad Blocking Problem
Origins
Destinations
Yards
Blocking Arcs
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An Integer Programming Fotrmulation
Minimize ?k?K ?(i,j)?A mij ?i?N
?k?K ?(i,j)?O(i) hi subject to ?(i,j)?O(i)
- ?(j,i)?I(i)
for
all k ? K ?k?K
uijyij
for all (i, j) ? A ?(i,j)?O(i) yij
bi
for all i ? N ?k?K ?(i,j)?I(i)
di
for all i ? N yij 0 or 1 and 0 or
vk
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Typical Size of the Blocking Problem

• A sample network size
• Number of origins 1,000
• Number of destinations 2,000
• Number of switching yards 300
• Number of shipments 50,000
• Number of blocking design variables
• 1,000300 300300 3002,000 1 million
• Number of shipment flow variables
• over a billion

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Typical Costs Involved

• Cost of routing
• of cars per yearaverage car distance
  traveledcost per mile per car
• 6 million cars 500 miles 0.5 1,500
  million
• Cost of handlings
• of cars per yearaverage car handlingscost
  per handling per car
• 6 million cars 2.5 40 600 million
• Cost of flow 2.1 billion

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Difficulty of the Design Problem

• Very large-scale optimization problem.
• Very difficult from mathematical point of view.
• Incorporating practical considerations makes the
  problem even more difficult.
• Reasonable running times.
• There is no existing algorithm that can solve
  this problem satisfactorily and hence there is a
  lot of room for improvement.

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Our Contributions

• We have developed a very effective algorithm to
  solve this problem to near-optimality.
• We have tested our algorithm on the data supplied
  by two railroad companies and found highly
  impressive improvements in costs.
• The algorithm can easily incorporate a variety of
  practical constraints necessary for
  implementation.
• Our algorithm is based on VLSN search.

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Neighborhood Search Algorithms

• Start with a feasible solution x
• Define a neighborhood of x
• Identify an improved neighbor y
• Replace x by y and repeat

Neighborhood of x1
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Very Large-Scale Neighborhood (VLSN) Search

• Neighborhoods with extremely large number of
  neighbors.
• Do not explicitly enumerate all the neighbors.
• Use network or discrete optimization algorithms
  to find improved neighbors (may be heuristically).

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A Simplification

• Ignore the node flow and arc flow capacities.

• All shipments follow the shortest paths.

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Our Approach

• Start with a feasible solution of the blocking
  problem.
• We use a simple heuristic to construct such a
  solution.
• Improve the solution repeatedly until the
  solution cannot be improved any more.
• We use a sophisticated method to improve the
  solution.
• Shipments always follow the shortest route in the
  blocking network.
• We ignore the node flow capacity and arc flow
  capacity constraints.

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Improving a Solution

• We improve the blocking solution one node at a
  time by building the best blocks at that node
  (and keeping blocks at other nodes intact).

• This is a very large-scale neighborhood (VLSN)
  search algorithm.

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Improving a Solution Step 1

• First Step
• Select a node to rebuild blocks.
• Delete all blocking arcs emanating from this
  node.
• Reroute the traffic passing through this node
  along new shortest paths.

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Improving a Solution Step 2

• Second Step
• Build a new blocking arc at this node which
  causes maximum savings in the total cost when
  shipments are allowed to pass through this arc.
• Build more arcs one by one until either the
  blocking capacity is met or adding arcs does not
  reduce the cost.

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Improving a Solution Step 3

• Third Step
• Consider each node in the network one by one and
  rebuild blocking arcs at that node using the
  maximum savings approach.
• This completes one pass of the algorithm.

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Improving a Solution Step 4

• Fourth Step
• Perform another pass over the nodes in the
  blocking network and rebuild arcs using the
  maximum savings approach.
• Keep on performing passes over the nodes and
  rebuilding blocking arcs until the blocking
  solution converges.
• We found that the solution converges in 10
  passes.
• Why do we need to perform multiple passes over
  the nodes?

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Why Multiple Passes?

• When rebuilding blocks at a node, say node k, we
  assume some blocks at other nodes. But blocks at
  those nodes may change subsequently which might
  change the best blocks at node k.

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Handling Flow Capacities

• Recall that we ignored the two capacity
  constraints
• Capacity on volume passing through each node
• Capacity of each blocking arc
• We incorporate these constraints using Lagrangian
  relaxation.
• We put penalty for the violated nodes and arcs so
  that the flow is rerouted.

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Convergence of the Algorithm
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Convergence of the Algorithm (contd.)
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Sensitivity to the Starting Solution
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Critical Issue Speed

• Typically, we rebuild blocks at over 30,000
  nodes.
• Each rebuilding requires updating shortest routes
  in the blocking network.
• Check each shipment (up to 50,000 such shipments)
  and determine whether it needs to be rerouted to
  take advantage of the new blocking arc and, if
  yes, reroute it.
• We rebuild blocks at a node in an average of .01
  to .05 seconds.

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How do we get this speed?

• Good network reoptimization algorithms
• Clever speed-up techniques
• Sophisticated data structures
• Efficient coding
• Our C code has over 500,000 lines.

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Computational Results - I

• We have tested our algorithm on the data provided
  by the three major railroads
• CSX Transportation
• BNSF Railway

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One Problem

• Blocks are carried by trains and a block goes
  from one train to another (called, block swap) as
  it goes from its origin to destination.

Atlanta
Boston
New York

• We can build only those blocks which require no
  more than two block swaps.

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Computational Results - II
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Another Problem

• Railroads are not willing to change their
  blocking plans completely.
• They want to change their plans incrementally,
  little by little and get benefits from
  incremental changes.
• This strategy is more implementable and involves
  less risk.

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Incremental Changes for CSX
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Results for BNSF
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Tradeoff between Two Costs

• Tradeoff between intermediate handlings and car
  miles traveled.

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Quality of the Solution

• It is very difficult to determine how far is our
  solution from the optimal solution.
• We are developing combinatorial lower bounding
  algorithms for this problem (Ahuja and Sahin
  2003).
• We conjecture that our solutions are within 2-3
  of the optimal solutions.

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Additional Practical Constraints

• Some blocks must be made.
• Some blocks must NOT be made
• Some shipments must be sent on some pre-specified
  blocking routes.

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Additional Practical Constraints

• Some blocks may be given preference.
• This feature is useful to ensure that the
  blocking solution in one month is not much
  different than the solution in the previous
  month.
• Small volume blocks can be avoided.
• This is a difficult constraint and comes at a
  certain cost.

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Additional Features (contd.)

• Optimize only blocks at certain nodes.
• Optimize only a corridor of the railroad network.
• Do sensitivity analysis to determine the
• Effect of increasing blocking capacities at some
  nodes
• Effect of increasing car handing capacities at
  some nodes
• Effect of closing some yards

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Railroad Scheduling Process
Schedule Generation
Train Scheduling
Block-to-Train Assignment
Locomotive Scheduling
Crew Scheduling
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Additional Railroad Research

• Algorithms for solving the locomotive scheduling
  problem.
• Algorithms for solving the block-to-train
  assignment problem.
• Algorithms for incrementally changing the train
  schedule integrated with the block-to-train
  assignment.
• Algorithms for the meet-pass planning problem.

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Collaborators

• Jian Liu, Krishna C. Jha, and Guvenc Sahin
• ISE Department, University of Florida,
• Gainesville, FL 32611
• Dharma Acharya
• CSX Transportation, Jacksonville, FL
• Pooja Dewan
• BNSF Railway, Fort Worth, TX

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• Questions?