A Branch and Cut algorithm for the Capacitated Location Routing Problem with depot and vehicle capac

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A Branch and Cut algorithm for the Capacitated
Location Routing Problem with depot and vehicle
capacities
J.M. Belenguer, E. Benavent, C. Prins, C.
Prodhon, R. Wolfler-Calvo
Universitat de València (Spain) University
of Technology of Troyes (France)
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Outline

• Introduction Location-Routing Problem (LRP)
• Integer Formulation
• Additional constraints
• W-capacity constraints
• Depot degree constraints
• Imparity constraints
• Path constraints
• Cutting Plane Scheme
• Initial relaxed formulation
• Identification algorithms
• Branch Cut
• Computational Results
• Conclusion

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Location-Routing Problem 1/3
Transportation
Vehicle routing from these depots (Tactical) NP-Ha
rd
Depot location (Strategic) NP-Hard
Lost of the global optimum
Splitting ?
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Location-Routing Problem 2/3

• Example

Customers with demands
Open depots

• Capacity on vehicles
• Capacity on depots

depot locations
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Location-Routing Problem 3/3

• G. Laporte, et al.
• An Exact Algorithm for Solving a Capacitated
  Location-Routing Problem
• Annals of Operations Research, 1986, 6,
  p.293-310.
• S. S. Barreto
• Análise e Modelização de Problemas de
  localização-distribuição
• University of Aveiro, campus universitàrio de
  Santiago, 2004, Portugal.

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Problem Definition

• Data

• A complete undirected network G (V, E, C)
• V I ? J set of nodes, E set of edges
• I set of m possible depot locations, for i ? I
• - opening cost Oi
• - capacity Wi
• J V \ I set of n customers, demands dj , j
  ?J
• cij traveling cost of edge (i, j ) ? E
  (triangular inequality)
• An unlimited number of vehicles fixed cost F
  and capacity Q

• Objective minimize the total cost

( traveling costs setup costs for the depots
and the routes)
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Integer Formulation 1/6
Depots to open and locations

• Goal
• Find

Assignment of the customers to these depots and
routes to visit them
If a vehicle uses edge ( i, j ) ?E exactly once
If a vehicle uses edge ( i, j ) i ?I, j ?J twice
i
j
Formulation 1
Formulation 2
return trip
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Integer Formulation 2/6
Notation
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Integer Formulation 3/6
Degree constraints visit each customer exactly
once
Vehicle capacity
Depot capacity
Edges incident with non open depots cannot be used
A route must begin and end at the same depot
Integrality constraints
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Integer Formulation 4/6
Depot capacity constraints
S
i
Violation
Valid
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Integer Formulation 5/6
Path constraints
A route must begin and end at the same depot
S
l
j
i
I'
Violation
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Integer Formulation 6/6
Path constraints 2
A route must begin and end at the same depot
We do not know how to eliminate this solution in
the first formulation (with x-variables only)
j
Violation
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Additional constraints 1/6
Imparity constraints
If w-variables are excluded, the graph induced
by the x-variables must be even

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Additional constraints 2/6
w-capacity constraints (generalize the capacity
constraints)
Example
useful when y-variables have fractional values
dj 1 ? j Q 5 k(S) 1
S'
1
S
S'
w1
1
1
w0.5
Violation
y1 0.5
w0.5
Valid
y1 0.5
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Additional constraints 3/6
Strengthened Path constraints
S 3, I'1,2, I \ I'3, 4
S
1
0.5
1
w0.5
l
1
y4 0.5
y1 0.5
j
0.5
w1
y3 0.5
1.5 3 1 5.5 gt 5
I' 1, 2
y2 1
Violation
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Additional constraints 4/6
Depot Degree constraints (return trips)
If distances satisfy the triangular inequality
x1
If the resulting routes satisfy the capacity
constrainy
x1
x1
w1
w1
dominated by
x1
x1
x1
x1
x1
x1
x1
x1
x1
x1
x1
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Additional constraints 5/6
Depot Degree constraints (general)
Too many links in a depot which is not completely
opened
0.5
S 4
S
1
0.5
1
3 2 5 gt 13
0.5
0.5
0.5
0.5
Violation
y1 0.5
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Additional constraints 6/6
All valid constraints for the CVRP in the two
index formulation are valid for the LRP

• J. Lysgaard, et al.
• A new Branch-and-Cut Algorithm for the
  Capacitated Vehicle Routing Problem
• Mathematical Programming, 100(2), 423-445 (2004).

• Capacity inequalities
• Framed capacity inequalities
• Strengthened comb inequalities
• Multistar and Partial Multistar inequalities
• Hypotour inequalities

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Cutting plane scheme 1/4
Initial relaxed formulation

• Degree constraints
• Forbid edges incident with non open depots
• Depot degree constraints (return trips)
• Minimum number of routes
• Minimum number of depots
• The integrality constraints are relaxed

All family constraints involving an exponential
number of constraints are initially relaxed.
At each iteration of the cutting plane, the
constraints that are found to be violated by the
current LP solution are added to the LP
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Cutting plane scheme 2/4
Identification procedures

• Heuristics
• similar in many cases to the ones used in CVRP
• (use of capacity, framed capacity, combs,
  multistar and hypotour constraints from
  Lysgaards library)
• Exact polynomial procedure
• Path constraints
• based on maximum flow / minimum cut

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Cutting plane scheme 3/4
Exact Separation of Strengthened Path constraints
Exact procedure

• For each pair of customers and
• compute the subset of depots maximizing
• A
  (sort and count)
• find a subset of customers containing
  and that minimizes
  (minimum cut)
• The constraint is violated if lt 2A

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Cutting plane scheme 4/4
Path constraints 2
Example
T S ? j, l
l
0.5
1
1
0.5
1
1
y4 0.25
y1 0.25
j
0.5
I\I' 3, 4
0.5
y3 0.25
A 2
I' 1, 2
y2 0.25
Violation
x (?(T)) 2 lt 2A
Checking 4116 gt 32
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Results 1/6

• Instances
• 30 random instances
• 10 instances from literature
• Visual C 2.4 GHz Pentium 4 512 MB of
  RAM
• Results
• Lower bound based on the cutting plane
• Lower bound by partial Branch Cut integrality
  of y-variables (depot location)
• Branch Cut just using the default rules of cplex

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Results 2/6

• Cutting Plane
• 30 Random Instances (1)
• (gap deviation from the upper bound in )

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Results 3/6

• Cutting Plane
• 30 Random Instances (2)

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Results 4/6

• Cutting Plane
• 10 Instances from Literature

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Results 5/6

• Branch Cut
• Instances solved to optimality
• CPU time limit 1800 secs.

One instance with 50 customers exactly solved One
new upper bound (optimal)
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Results 6/6

• Branch Cut (incomplete)
• Instances not solved to optimality
• CPU time limit 1800 secs.

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Conclusion

• Branch-andCut algorithm for the Location-Routing
  Problem with capacities
• Two index formulation with binary variables
• Additional constraints
• New identification algorithms
• Able to find
• Exact solutions on small-medium size instances
• Small integrality gap in some large instances

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Conclusion

• Future
• Improve the branch and cut better branching
  rules, searching strategies, ...
• Strengthen the constraints of the CVRP
• Develop new specific constraints
• Polyhedral study (at least for a simpler problem)

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Thank you .