Limited Memory Rank-1 Cuts for the SetPartitioning Formulation of Vehicle Routing
Problems
Diego Pecin 1
Artur Pessoa 2
Marcus Poggi 1
Haroldo Santos 3
Eduardo Uchoa 2
PUC - Rio de Janeiro 1
Universidade Federal Fluminense 2
Universidade Federal de Ouro Preto 3
January, 2015
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Vehicle Routing Problem (VRP)
Instance: Complete graph G = (V ,A) with V = 0, . . . , n;vertex 0 is the depot, the other vertices are customers. Each arca ∈ A has a cost ca. Customers have demands. There is a fleet ofvehicles in the depot.
Solution: A set of routes starting and ending at the depot,attending all customers, and respecting the given operationalconstraints, with minimal total cost.
Dozens of variants:
CVRP: Most classical, routes limited only by vehicle capacity
VRPTW: Customers must also be attended within timewindows
HFVRP: Heterogeneous fleet
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Set Partitioning Formulation (Balinski and Quandt [1964])
(SPF) min∑r∈Ω
crλr (1)
S.t. ∑r∈Ω
ari λr = 1, ∀i ∈ V+, (2)
λr ∈ 0, 1 ∀r ∈ Ω. (3)
Ω is the set of routes, ari is the number of times thatcustomer i appears in route r .
Must be solved by column generation. The set Ω is oftenrelaxed (allowing some non-elementary routes) in order tomake the pricing subproblem more tractable.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Set Partitioning Formulation
Even if Ω only contains elementary routes, the linear relaxationof SPF is not strong enough for efficient branch-and-price.
Except when routes are very constrained (e.g., very narrowtime windows).
SPF should be combined with cutting, yieldingbranch-cut-and-price algorithms.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Cuts over Edge/Arc Formulations
Depend of the specific VRP variant:
CVRP: Rounded Capacity, Strengthened Combs
VRPTW: 2-Path
HFVRP: Extended Capacity Cuts
Improve significantly the relaxations. They are robust, their dualvariables are translated into edge/arc costs in the pricing. Lead toefficient algorithms.
Seems to be exhausted. Really good new cuts not found in the lastyears.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Cuts over the Set Partitioning Formulation
Valid for most VRP variants. Several cuts known from the SPPliterature: Cliques, Odd holes, ...
Potential for big improvements in the relaxations. However, theyare non-robust, each added cut makes the pricing subproblemharder, quickly making it intractable.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Subset Row Cuts (SRCs)
Given C ⊆ V+ and a scalar multiplier p, the (C , p)-Subset RowCut is: ∑
r∈Ω
⌊p∑i∈C
ari
⌋λr ≤ bp|C |c (4)
Non-robust cut obtained by a Chvatal-Gomory rounding of |C |constraints in the SPF, less harmful to pricing structure than cliqueor odd hole cuts.
M. Jepsen, B. Petersen, S. Spoorendonk, and D. Pisinger. Subset-row
inequalities applied to the vehicle-routing problem with time windows.
Operations Research, 56(2):497–511, 2008
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Interesting SRCs
Given an SRC with base set C , for each integer d , define ydC as thesum of all variables λr such that
∑i∈C ari = d .
The cuts where |C | = 3 and p = 1/2 are called 3-Subset RowCuts (3SRCs), expressed as:
y2C + y3
C ≤ 1.
Used in Baldacci et al. [2011] and Contardo and Martinelli[2014]
Potentially very effective
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Interesting SRCs
|C | = 4 and p = 2/3, 4SRCs:
y2C + 2y3
C + 2y4C ≤ 2.
|C | = 5 and p = 1/3, 5,1SRCs:
y3C + y4
C + y5C ≤ 1.
|C | = 5 and p = 1/2, 5,2SRCs:
y2C + y3
C + 2y4C + 2y5
C ≤ 2.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
The Breakthrough
Due to their impact in the pricing, not many SRCs could beeffectively added to SPF and the potential gains were not achieved.
Pecin et al. [2014] proposed a new technique for greatly reducingthe impact of SRCs in the pricing and could obtain the full benefitof those cuts.
In CVRP, the size of the largest solved instance increasedfrom 150 to 360 customers (improvements in otheralgorithmic elements also contributed to the advance).
Diego Pecin, Artur Pessoa, Marcus Poggi, and Eduardo Uchoa. Improved
branch-cut-and-price for capacitated vehicle routing. In Integer
Programming and Combinatorial Optimization, pages 393–403. Springer,
2014
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Limited Memory Subset Row Cuts (lm-SRCs)
Given C ⊆ V+, a memory set M, C ⊆ M ⊆ V+, and a scalarmultiplier p, the limited memory (C ,M, p)-Subset Row Cut is:∑
r∈Ω
α(C ,M, p, r)λr ≤ bp|C |c , (5)
where the coefficient of a route r is computed as:1: function α(C , M, p, r)2: coeff ← 0, state ← 03: for every vertex i ∈ r (in order) do4: if i /∈ M then5: state ← 06: else if i ∈ C then7: state ← state + p8: if state ≥ 1 then9: coeff ← coeff + 1, state ← state − 1
10: return coeff
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Limited Memory Subset Row Cuts (lm-SRCs)
1: function α(C , M, p, r)2: coeff ← 0, state ← 03: for every vertex i ∈ r (in order) do4: if i /∈ M then5: state ← 06: else if i ∈ C then7: state ← state + p8: if state ≥ 1 then9: coeff ← coeff + 1, state ← state − 1
10: return coeff
If M = V+, the function returns bp∑i∈C
ari c
Otherwise, the lm-SRC may be a weakening of the corresponding SRC
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Separation of lm-SRCs
0
2
1
3
Route r1, λr1=0.5
λr1 has coefficient 1 in the SRC with C = 1, 2, 3
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Separation of lm-SRCs
0
2
1
3
Route r1, λr1=0.5
Included in the memory set
Minimum memory for λr1 to have coefficient 1 in the lm 3-SRC
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Separation of lm-SRCs
0
2
1
3
Route r2, λr2=0.5
λr2 has coefficient 1 in the SRC with C = 1, 2, 3
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Separation of lm-SRCs
0
2
1
3
Route r2, λr2=0.5
Included in the memory set
Minimum memory for λr2 to have coefficient 1 in the lm 3-SRC
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Separation of lm-SRCs
0
2
1
3
Route r3, λr3=0.5
λr3 has coefficient 1 in the SRC with C = 1, 2, 3
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Separation of lm-SRCs
0
2
1
3
Route r3, λr3=0.5
Included in the memory set
Minimum memory for λr3 to have coefficient 1 in the lm 3-SRC
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Separation of lm-SRCs
0
2
1
3
Final memory set
The set M of the added lm 3-SRC is the union of the memoriesthose λ variables
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Separation of lm-SRCs
0
2
1
3
The next route of pricings is likely to produce routes that avoid Mto have coefficient zero in the lm 3-SRC
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Separation of lm-SRCs
0
2
1
3
Possibly included in the memory set of C in the next cut round
The set M may be adjusted in the next round of separation
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Separation of lm-SRCs
If a violated (C , p)-SRC exists, it finds a minimal set M such thatthe lm-(C ,M, p)-SRC has the same violation.
Eventually (perhaps in more iterations), the lower boundsobtained with the lm-SRCs will be the same that would beobtained with the SRCs.
The odd algorithmic definition of the lm-SRCs makes sense whenconsidering the labeling dynamic programming algorithm used inthe pricing.
A lm-(C ,M, p)-SRC only increases the space of statesassociated to vertices in M. In practice, there are exponentialgains with respect to ordinary SRCs.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
This Work: Generalize to Arbitrary Cuts of Rank 1
Given C ⊆ V+ and a vector of multipliers p of dimension |C |, the(C , p)-Rank 1 Cut is:
∑r∈Ω
⌊∑i∈C
piari
⌋λr ≤
⌊∑i∈C
pi
⌋(6)
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Limited Memory Rank 1 Cuts
Given C ⊆ V+, a vector of multipliers p of dimension |C |, amemory set M, C ⊆ M ⊆ V+, the limited memory (C ,M, p)-Rank1 Cut is: ∑
r∈Ω
α(C ,M, p, r)λr ≤⌊∑i∈C
pi
⌋, (7)
where the coefficient of a route r is computed as:1: function α(C , M, p, r)2: coeff ← 0, state ← 03: for every vertex i ∈ r (in order) do4: if i /∈ M then5: state ← 06: else if i ∈ C then7: state ← state+ pi
8: if state ≥ 1 then9: coeff ← coeff + 1, state ← state − 1
10: return coeff
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
What are the interesting multipliers?
In a column generation context, cuts must be valid for all possiblevariable coefficients, not only those in the current restrictedproblem.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
What are the interesting multipliers?
The Master Set Partitioning of order n is defined as:
2n−1∑j=1
bjxj = en, x binary,
where bj is a vector of dimension n with coefficients correspondingto the binary representation of number j and en is a unitary vector.For example, if n = 3 we have:
1 0 1 0 1 0 10 1 1 0 0 1 10 0 0 1 1 1 1
x1
x2
x3
x4
x5
x6
x7
=
111
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
What are the interesting multipliers?
The Master Set Partitioning Polyhedron of order n is defined as:
MSPP(n) = Conv2n−1∑j=1
bjxj = en, x binary.
We performed a computational study of MSPP(n) for n ≤ 5 tofind the best possible inequalities that can be obtained from up to5 rows of a SPP.
In particular, we found the multipliers corresponding to all facets ofrank 1 and the multipliers that better approximate the facets ofhigher rank.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Analysis of MSPP(3)
MSPPP(3) has a single non-trivial facet:
x3 + x5 + x6 + x7 ≤ 1.
This facet has rank 1 and corresponds to multipliers(1/2, 1/2, 1/2), being equivalent to y2
C + y3C ≤ 1
Therefore, the 3SRCs (a subfamily of the clique cuts) are alreadythe best possible cuts that can be obtained by considering up to 3rows of a SPP.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Analysis of MSPP(4)
MSPPP(4) has 8 non-trivial facets, all of rank 1:
Multipliers (1/2, 1/2, 1/2, 0) and its permutations (3SRCs)
Multipliers (2/3, 1/3, 1/3, 1/3) and its permutations (Newfamily)
The original 4SRCs are quite weak, they are the sum of those8 facets.
The new cuts have RHS 1 and are another subfamily ofcliques.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Analysis of MSPP(5)
MSPPP(5) has 294 non-trivial facets, 103 of then have rank 1.The interesting multipliers (along with their permutations) are:
(1/3, 1/3, 1/3, 1/3, 1/3) (5,1 SRCs)
(2/4, 2/4, 1/4, 1/4, 1/4) (New family)
(3/4, 1/4, 1/4, 1/4, 1/4) (New family)
(3/5, 2/5, 2/5, 1/5, 1/5) (New family)
(1/2, 1/2, 1/2, 1/2, 1/2) (5,2 SRCs)
(2/3, 2/3, 1/3, 1/3, 1/3) (New family)
(3/4, 3/4, 2/4, 2/4, 1/4) (New family)
The first 4 families have RHS 1 and are subfamilies of cliquecuts.
the last 3 families have RHS 2 and are subfamilies of liftedodd holes.
Remark that we do not know how to separate general cliques orodd holes without destroying the pricing!Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Computational Results on CVRP
Average gaps over a set of hard instances ranging from 36 to 199customers. Full separation until convergence:
Gap(%)
Only CG (elementary routes) 2.63+ robust cuts 0.98+ 3SRCs 0.35+ 4SRCs + 5SRCs 0.24Rank 1 Cuts up to 5 rows 0.17
The new cuts removed 30% of the residual gap. They can help tosolve some larger open instances.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Golden 20 (420 customers)
Authors BKS
Vidal et al. [2012] 1818.32Groer et al. [2011] 1818.25Jin et al. [2014] 1817.89Liu and Li [2014] 1817.86
Optimal solution: 1817.59
Root LB: 1815.0 (1200 active Rank 1 cuts!)
B&B Nodes: 370
Total Time: 7 days (single core i7-3960X 3.30GHz)
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Golden 20 (420 customers)
Authors BKS
Vidal et al. [2012] 1818.32Groer et al. [2011] 1818.25Jin et al. [2014] 1817.89Liu and Li [2014] 1817.86
Optimal solution: 1817.59
Root LB: 1815.0 (1200 active Rank 1 cuts!)
B&B Nodes: 370
Total Time: 7 days (single core i7-3960X 3.30GHz)
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Optimal solution of Golden 20, cost 1817.59
20
10
10
4040
10
10
10
10
1010
1010
10
20
10
40
1040
10 20
20
10
10
20
10
40
20
10
10
40 10
10
1010
10
20
10
20
20
10
40
10
40
10
20
20
10
40
20
10
10
4010
10
1010
10
20
10
10
40 40
10
10
10
10
1010
1010
10
10
40
10
20
10
20
10
40
10
20
2010
20
1020
10
40
10
40
10
10
20
10
20
20
10
40
10
40
10
20
10
40
10
20
10 10 40
10
10
10
10
10
10
10
4010
20
10
40 10
40 10
10
20 10
20
10 20
10
40
10
40
10
10
20
20
20
10
20
10
40
10
40
10
20
20
10
40
10 40
10
20
10
10
10
10
10
10 10 40
401010
20
20
20
1010
1010 40
40
10
10
10
10
20
20
10
10
40
10
20
10
20
10
40
10
20
20
20
10
40
10
40
10
20
10
10 10
10 10
10 10 40
40
10
10
10
10
20
2010
10
101040
40 10 10
20
20
20
20
20
10
40
1040
10
20
10
10
10
10
10
1040
10
40 10 20 10
20
20
10
4010
20
10
4010
4010
10
1010
101040
40
10
10
10
10
20
20
10
10
10 40
10
40102010
20
20
20
20
20
10
20
10
40
10
40
10
10
20
10
40
10 40
1020
20
10
10
20
20
10
40
10
40
10
20
10
1010
20
20
10
10
40 40
10
10
10
10
10
10
40 40 40
1010
20
1010
10
1010
101040
40
10
10
10
10
20
20
20
20
10
10
4040
10
10
10
10
10
10
40
40
40
1010
201010
10
10
401010
1010
40
401010
10
10
4010
10
1010
40
40 1010
10
40
40
40
1010
20 1010
10
404040
1010
20
1010
10
10
40
10
20
101040
10
10
10
10
10
10
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Preliminary Results on VRPTW
Being implemented with C. Contardo and G. Desaulniers.
55 out of 56 Solomon instances (100 customers) have gap zero.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Possible lessons from our VRP experience for general BCPconstruction
Aggressive non-robust cutting may pay
However, cutting and pricing should be fully integrated:
Besides polyhedral considerations, the non-robust cuts shouldbe designed in order to minimize their impact on the specificalgorithm used in the pricing.
The lm Rank 1 Cuts are good for the labeling algorithm. In analternative BCP where the pricing is solved, say, by MIP, theywould be terrible!
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Possible lessons from our VRP experience for general BCPconstruction
Aggressive non-robust cutting may pay
However, cutting and pricing should be fully integrated:
Besides polyhedral considerations, the non-robust cuts shouldbe designed in order to minimize their impact on the specificalgorithm used in the pricing.
The lm Rank 1 Cuts are good for the labeling algorithm. In analternative BCP where the pricing is solved, say, by MIP, theywould be terrible!
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Possible lessons from our VRP experience for general BCPconstruction
Aggressive non-robust cutting may pay
However, when designing non-robust cuts, it is desirable to have aparameter that allows a smooth control on cut strength vs impact
in the pricing:
The M parameter has that role in the lm Rank 1 Cuts.
In our separation we always add the weakest possible cut thatdoes the job
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Possible lessons from our VRP experience for general BCPconstruction
Aggressive non-robust cutting may pay
However, when designing non-robust cuts, it is desirable to have aparameter that allows a smooth control on cut strength vs impact
in the pricing:
The M parameter has that role in the lm Rank 1 Cuts.
In our separation we always add the weakest possible cut thatdoes the job
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Possible lessons from our VRP experience for general BCPconstruction
Aggressive non-robust cutting may pay
However, even with all care, the addition of too many non-robustcuts can still break the pricing:
There must be escape mechanisms.
In our BCP, when a round of separation makes the solution ofa node too slow, it rolls back to a previous state (i.e., itremoves the offending cuts).
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Possible lessons from our VRP experience for general BCPconstruction
Aggressive non-robust cutting may pay
However, even with all care, the addition of too many non-robustcuts can still break the pricing:
There must be escape mechanisms.
In our BCP, when a round of separation makes the solution ofa node too slow, it rolls back to a previous state (i.e., itremoves the offending cuts).
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Advertising: CVRPLIB
A web page containing all CVRP instances from the literature andtheir optimal/best known solutions:
Designed as a database, the visible pieces of information arequeries
Easier maintenanceSolutions are automatically checkedInstances and solutions can be depicted graphically
Contains the 100 new instances (X series) between 100 and1000 customers created by the authors.
http://vrp.galgos.inf.puc-rio.br/
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Advertising: CVRP Challenge for Exact Methods
500-Customer Prize - 300 USD for solving the 68 instances in theX series with up to 500 customers.
29 instances in that range still open
1000-Customer Prize - 500 USD for solving all the 100 instances inthe X series.
Only 2 out of the 32 instances with more than 500 customersalready solved
Full rules in CVRPLIB, large scale parallelism allowed.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Thank you for your attention!
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
R. Baldacci, A. Mingozzi, and R. Roberti. New route relaxationand pricing strategies for the vehicle routing problem.Operations Research, 59(5):1269–1283, 2011.
M.L. Balinski and R.E. Quandt. On an integer program for adelivery problem. Operations Research, 12(2):300–304, 1964.
Claudio Contardo and Rafael Martinelli. A new exact algorithm forthe multi-depot vehicle routing problem under capacity and routelength constraints. Discrete Optimization, 12:129–146, 2014.
Chris Groer, Bruce Golden, and Edward Wasil. A parallel algorithmfor the vehicle routing problem. INFORMS Journal onComputing, 23(2):315–330, 2011.
M. Jepsen, B. Petersen, S. Spoorendonk, and D. Pisinger.Subset-row inequalities applied to the vehicle-routing problemwith time windows. Operations Research, 56(2):497–511, 2008.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP
Jianyong Jin, Teodor Gabriel Crainic, and Arne Løkketangen. Acooperative parallel metaheuristic for the capacitated vehiclerouting problem. Computers & Operations Research, 44:33–41,2014.
Wanfeng Liu and Xia Li. A problem-reduction evolutionaryalgorithm for solving the capacitated vehicle routing problem.Mathematical Problems in Engineering, 501:165476, 2014.
Diego Pecin, Artur Pessoa, Marcus Poggi, and Eduardo Uchoa.Improved branch-cut-and-price for capacitated vehicle routing.In Integer Programming and Combinatorial Optimization, pages393–403. Springer, 2014.
Thibaut Vidal, Teodor Gabriel Crainic, Michel Gendreau, NadiaLahrichi, and Walter Rei. A hybrid genetic algorithm formultidepot and periodic vehicle routing problems. OperationsResearch, 60(3):611–624, 2012.
Aussois-2015 Pecin, Pessoa, Poggi, Santos, and Uchoa Limited Memory Rank-1 Cuts for VRP