Limited Memory Rank-1 Cuts for the Set Partitioning ... Memory Rank-1 Cuts for the Set Partitioning Formulation of Vehicle Routing Problems Diego Pecin 1 Artur Pessoa 2 Marcus Poggi

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


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.


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.


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.


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.


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


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


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.


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


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


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


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



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



0

2

1

3

Route r2, λr2=0.5




0

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1

3

Route r2, λr2=0.5





0

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1

3

Route r3, λr3=0.5




0

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1

3

Route r3, λr3=0.5





0

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1

3

Final memory set

The set M of the added lm 3-SRC is the union of the memoriesthose λ variables



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



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



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.


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)


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


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.



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



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.


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.


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.


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.


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)


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)


Optimal solution of Golden 20, cost 1817.59

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Preliminary Results on VRPTW

Being implemented with C. Contardo and G. Desaulniers.

55 out of 56 Solomon instances (100 customers) have gap zero.


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!




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!




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




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




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).




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).


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Thank you for your attention!


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.


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.


Limited Memory Rank-1 Cuts for the Set Partitioning ... Memory Rank-1 Cuts for the Set Partitioning Formulation of Vehicle Routing Problems Diego Pecin 1 Artur Pessoa 2 Marcus Poggi

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