Graph Transformations for Vehicle Routing and Job Shop Scheduling Problems

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Graph Transformations for Vehicle Routing and Job
Shop Scheduling Problems

• J.C.Beck, P.Prosser, E.Selensky
• c.beck_at_4c.ucc.ie, pat,evgeny_at_dcs.gla.ac.uk

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Basic Problem
Find a cycle of min cost
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Example
Lexicographic ordering of nodes A,B,C,D
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Motivation

• Core problem in vehicle routing and shop
  scheduling
• Edge weights to node weights
• Large for VRP, small for JSP
• Can we use graph transformations to make VRP look
  like JSP and vice versa?

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Vehicle Routing
NP-hard!
Go find vehicle tours with min travel
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Job Shop Scheduling
Go find a schedule with min Makespan
NP-complete
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Hypothesis
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Cost-Preserving Transformations

• Assumptions
• Graphs complete (true for VRP, JSP subsumed),
  undirected (directed case subsumed)
• A solution is a cycle on the graph (for
  Hamiltonian paths everything is similar)
• Transformations should preserve cost and order of
  nodes in a cycle.

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Caveat

• This is not a comprehensive study of all possible
  transformations
• Rather, we propose some transformations and study
  them

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Types of Transformations
Direct Reduce Edge Weights, Increase Node Weights
Inverse Increase Edge Weights, Reduce Node
Weights
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Order Dependent Transformations

• lexicographic order of nodes
• choose a node whose cheapest
  incident edge is a maximum
• choose a node whose cheapest
  incident edge is a minimum

Lex
MaxMin
MinMin
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Order Independent Transformation
Example
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Inverse Transformation
Reminder Increase Edge Weights, Reduce Node
Weights

• Order-independent
• G?Ginv G?Gdod?Ginv
  G?Gdoi?Ginv

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Performance measures

• Weight transfer from nodes to edges
• change in proportion of weight of cycle C
• 
  
• a similar measure for the whole graph
• 
  
• where W and W are
  graph
• weights before and after transformation

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Performance measures

• Relative edge/node weights ordering
• Sort edge/node weights in ascending order
• e.g. w11, w12, w13 for edges (1,1), (1,2) and
  (1,3)
• Apply transformations and count how many
  pair-wise changes there are
• e.g. w13, w11, w12, so we have 2 changes
• Two measures and

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Experiments

• Purpose
• Assess performance of the transformations on
  complete undirected graphs
• Layout
• Randomly generate 100-instance sets of graphs of
  different sizes
• Apply
  and

MaxMin,
MinMin,
Lex,
DirOrderInd
Inverse.
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Experiments
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Experiments
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Experiments
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Experiments
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Analysis of Results

• Weight Transfer
• Inverse gtgt Order Independent gtgt Order Dependent
• Changes in Edge/Node Ordering
• Inverse constant w.r.t. graph size
• InversegtgtMaxMin gtgt Order Independent, Lex gtgt
  MinMin

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Future Work

• Systematically apply the transformations to
  VRP/JSP instances and study their performance in
  practice.