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Graph Transformations for Graph Transformations for Vehicle Routing and Job Shop Vehicle Routing and Job Shop
SchedulingScheduling Problems Problems
J.C.Beck, P.Prosser, E.Selensky
[email protected], {pat,evgeny}@dcs.gla.ac.uk
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w1
w2
w12
wn
wi
wn-1
w1,n
wn-1,nw1,n-1
w2,n
w2,n-1
Find a cycle of min cost
Basic Problem
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Lexicographic ordering of nodes: A,B,C,D
Example
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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
[2:25pm 2:40am]
[9:00am 9:15am]
[3:00pm 5:00am]
[3:00pm 5:00am]
[9:00am 5:00am]
[4:00pm 5:00am]
NP-hard!Go find vehicle tours with min travel
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Job Shop Scheduling
J1: (M1,t11) (M3,t13) (M2,t12)J2: (M3,t23) (M1,t21) (M2,t22)J3: (M2,t32) (M3,t33) (M1,t31)
3 machines: M1, M2, M3
3 jobs: J1, J2, J3
Go find a schedule with min Makespan NP-complete
TimeMakespan0
M1
M2
M3
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Hypothesis
Graph Transformation
VRP Solver
JSP Solver
Graph Transformation
VRP
JSP
Is it important?
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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 TransformationsDirect: Reduce Edge Weights, Increase Node Weights
Inverse: Increase Edge Weights, Reduce Node Weights
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• lexicographic order of nodes
• choose a node whose cheapest incident edge is a maximum
• choose a node whose cheapest incident edge is a minimum
Order Dependent Transformations
MaxMin:
MinMin:
Lex:
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Example
Order Independent Transformation
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Inverse TransformationReminder: Increase Edge Weights, Reduce Node Weights
• Order-independent• GG’inv; GG’dodG’inv; GG’doiG’inv;
Express as if odd and if even 12 kwiiw kwi 2
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• 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
Performance measures
iji
ijijc ww
ww
,
,W
wij ,W
wij
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• 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. {w’13, w’11, w’12}, so we have 2 changes;
• Two measures: and edges .nodes
Performance measures
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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 andMaxMin, MinMin,Lex, DirOrderInd
Inverse.
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Experiments
iji
ijijc ww
ww
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Experiments
W
wW
w
W
w
ij
ijij
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Experiments
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Experiments
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Analysis of Results
• Weight Transfer: Inverse >> Order Independent >> Order Dependent
• Changes in Edge/Node Ordering:Inverse: constant w.r.t. graph size;
Inverse>>MaxMin >> Order Independent, Lex >> MinMin
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Future Work
• Systematically apply the transformations to VRP/JSP instances and study their performance in practice.
