Recent Advances in Exact Algorithms for the Capacitated Vehicle Routing Problemwebtv.univ-bejaia.dz/.../2017/01/Eduardo-Uchoa-COSI2014.pdf · 2017-01-23 · Vehicle Routing Problem

Recent Advances in Exact Algorithms for theCapacitated Vehicle Routing Problem

Eduardo Uchoa

Universidade Federal Fluminense

Niteroi-RJ, Brazil

June, 2014

COSI’2014 - Bejaia Recent advances for the CVRP

Vehicle Routing Problem (VRP)

One of the most widely studied families of problems inCombinatorial Optimization:

Direct applications in the real world systems that distributegoods and services, vital to the modern economies.


Vehicle Routing Problem (VRP)

Reflecting the variety of aspects present in those systems, the VRPliterature is spread into dozens of variants, that consider:

Time windows

Multiple depots,

Mixed vehicle fleet,

Split delivery, pickup and delivery,

Complex loading constraints, etc.

“Rich” variants try to consider as many of those aspects as possible


The Capacitated Vehicle Routing Problem (CVRP)

The “poorest” VRP variant:

Instance: Complete graph G = (V ,E ) with V = 0, . . . , n;vertex 0 is the depot, N = 1, . . . , n is the set of customers. Eachedge e ∈ E has a cost ce . Each customer i ∈ N has a demand di .There is a fleet of K identical vehicles with capacity Q.

Solution: A set of K routes starting and ending at the depot,attending all customers, and respecting the capacities, withminimal total cost.

The most classical VRP variant, proposed by Dantzig andRamser in 1959.


CVRP Instance: n = 62,K = 8,Q = 100, indicateddemands, euclidean distance


CVRP Solution: n = 62,K = 8,Q = 100, indicateddemands, euclidean distance


Why CVRP is a particularly important variant?

General strategy in scientific research:

1 Study the simplest (but still representative) case of aphenomenon

2 Try to generalize the discoveries for more complex cases

Several good ideas were first proposed for the CVRP and thensuccessfully generalized for many other variants.

Example: Clarke and Wright heuristic (1964)

Moreover, there is a significant number of real world systems thatcan be satisfactorily modeled as a CVRP


Digression: What to do with NP-hard CombinatorialProblems?

Unless P = NP, all exact algorithms for NP-hard problems willhave an exponential complexity.

1 - Use heuristics

There are sophisticated heuristics/metaheuristics for manycombinatorial problems, able to find solutions with very goodquality (say, less than 1% from optimal) in short times.

On the other hand:

Optimal is better than near-optimalActually, you do not know how far you are from optimal.



2 - Use approximation algorithms

Those are heuristics that provide a quality guarantee.

However:

Several important problems can not have constant factorapproximation algorithms unless P=NP.Approximation algorithms rarely provide really goodguarantees, less than 10% is very rare (unless in PTAS,FPTAS, but those more theoretical constructs are seldomimplementable).



3 - Use exact algorithms algorithms with exponential complexity!

Exponential algorithms can be effective up to a certaininstance size

In several problems, this size can be large enough for practicalapplications


Some exponential algorithms are better than others



From Alg1 to Alg3, the size of the instances that can tackledin “reasonable time” increased from 50 to 200

An important observation is that the time to solve instanceswith n = 50 decreased from more than 1 day to 10s =⇒Probably, there is no more reason for using heuristics on thoseinstances.



This chart is a simplification. Actually, instances of the same sizecan take more or less time, depending on their characteristics.


The Traveling Salesman Problem (TSP)

The TSP is a good example.

From 1978 to 2010, the size of the “typical” TSP instancesthat can be solved in “reasonable time” increased from around120 to more than 10,000 (largest ever solved: 85,000 - 23years of cpu).

Instances with 1,000 points can now be solved in less than 1minute.

Instances with less than 300 points take a fraction of a second.

For most practical applications, TSP is now a well-solved problem.

For most practical applications, CVRP is not a well-solvedproblem. But progress is being made.


Combinatorial Optimization and Integer Programming

The TSP is also a good example of an approach for obtaining toobtain those “effective” exact algorithms for NP-hard problems:

Formulate the problems as Integer Programs (IPs).

Look for techniques for solving those specially structured IPs.

This approached was pioneered by Dantzig, Fulkerson and Johnson[1954]. All the best TSP algorithms use now the classicalbranch-and-cut technique.

It turned out that the branch-and-cut technique was not enoughfor CVRP. Recent progress is being made by combining it withother IP techniques.


Edge formulation (Laporte and Nobert [1983])

Variables xe indicate how many times an edge e is traversed.

min∑e∈E

cexe (1)

S.t.∑e∈δ(i)

xe = 2 ∀ i ∈ N, (2)

∑e∈δ(0)

xe = 2K , (3)

∑e∈δ(S)

xe ≥ 2d∑i∈S

di/Qe ∀ S ⊆ N, S 6= ∅, (4)

xe ∈ 0, 1 ∀ e ∈ E \ δ(0), (5)

xe ∈ 0, 1, 2 ∀ e ∈ δ(0). (6)

Constraints (4) are the Rounded Capacity Cuts


CVRP and Cut Generation

A lot of investigation (inspired by the TSP) was done in order toimprove the edge formulation. Known families of cuts:

Framed Capacities

Strengthened Combs

Multistars

Extended Hypotours

Branch-and-cut became the dominant approach for the CVRP inthe 1990s and early 2000s:

Araque, Kudva, Morin, and Pekny [1994]

Augerat, Belenguer, Benavent, Corberan, Naddef, and Rinaldi[1995]

Blasum and Hochstattler [2000]

Ralphs, Kopman, Pulleyblank, and Trotter Jr. [2003]

Achuthan, Caccetta, and Hill [2003]

Lysgaard, Letchford, and Eglese [2004]


Benchmark instances: A Series (Augerat [1995])

22 instances with n from 37 to 80, Q = 100, demands fromU[1, 30], 10% of them are multiplied by 3. Random depot andcustomer positioning in the grid [1,100]x[1,100], euclideandistances

Figure: Instance A-n80-k10


Benchmark instances: B Series (Augerat [1995])

20 instances with n from 38 to 78, Q = 100, demands fromU[1, 30], 10% of them are multiplied by 3. Random depot andclustered customers, euclidean distances

Figure: Instance B-n78-k10


Benchmark instances: P Series (Augerat [1995])

Adapted from previous instances in the literature, includingreal-world problems. 24 instances with n from 16 to 101.

Figure: Instance P-n50-k8


Benchmark instances: E Series (Christofides and Eilon[1969])

The most classical benchmark. 9 instances with n from 51 to 101.

Figure: Instance E-n101-k14


Benchmark instances: F series (Fisher [1994])

Only 3 instances, n from 45 to 135. Large Q, small demands =⇒long routes.

Figure: Instance F-n135-k7


Benchmark instances: M Series (Christofides, Mingozziand Toth [1979])

Only 5 larger instances: M-n101-k10, M-n121-k7, M-n151-k12,M-n200-k16 and M-n200-k17.

Figure: Instance M-n151-k12


Results of LLE04 over the benchmark instances

LLE04Class NP Unsolved Root Gap Avg. TimeA 22 7 2.06 6638B 20 1 0.61 8178E-M 12 9 2.10 39592F 3 0 0.06 1016P 24 8 2.26 11219

Total 81 25

Machine Intel Celeron 700MHz

Root gaps and average times measured only over solved instances.

J. Lysgaard, A. Letchford, and R. Eglese. A new branch-and-cut

algorithm for the capacitated vehicle routing problem. Mathematical

Programming, 100:423–445, 2004

Branch-and-cut algorithms may perform quite well oninstances with few vehicles and long routes, like F-n135-k7(because they are closer to the TSP?)However, shorter routes (n/K < 15) are more typical. Somesmaller instances, like P-n50-k8, could not be solved.


Set Partitioning Formulation (Balinski and Quandt [1964])

Ω is the set of all routes, a route r has cost cr , air counts howmany times r passed at customer i

min∑r∈Ω

crλr (7)

S.t.∑r∈Ω

airλr = 1 ∀ i ∈ N, (8)∑r∈Ω

λr = K , (9)

λr ∈ 0, 1 ∀ r ∈ Ω. (10)

Too many variables =⇒ column generation / branch-and-pricemust be used

Pricing true (elementary) routes is strongly NP-hard =⇒ relaxthe definition of Ω to allow some non-elementary routes


q-routes (Christofides, Mingozzi and Toth [1981])

Ω is redefined as the set of all q-routes: walks leaving andreturning to the depot respecting the capacity Q. A customer maybe visited more than once, but its demand is counted again in eachrevisit.

Pricing can now be done in pseudo-polynomial O(n2Q) time

Of course, the gap of SPF linear relaxation becomes larger

A compromise between pricing complexity and gap are theq-routes without k-cycles: a customer may be revisited onlyafter k visits to other customers.

Pricing in O(k2k!n2Q) time (Irnich and Villeneuve [2006]),efficient for k up to 4.


Combining Column and Cut Generation

The SPF (even with near-elementary routes) is not strong enoughto yield effective branch-and-price algorithms for the CVRP.

Root gaps around 3% are typical, worse than the 2% that istypical with cutting planes

Fukasawa et al. [2006] realized that both approaches could becombined:

Let aer be the number of times that a given edge e appears inroute r , xe =

∑r∈Ω aerλr .

So, a general inequality∑

e∈E αexe ≥ b for the edgeformulation is included in the SPF as∑

r∈Ω(∑

e∈E αeaer )λr ≥ b.

This combination yields typical gaps around 1%.


Robust Branch-Cut-and-Price

A key issue when combining Column and Cut Generation is howthe dual variables of the new constraints will be affect the pricingsubproblem:

A cut is robust (Poggi and Uchoa, 2003) when its dual valuecan be translated into costs in the pricing subproblem.Therefore, the structure and the size of that subproblemremain unaltered, regardless of the number of robust cutsadded.

Non-robust cuts change the pricing, each additional cutmakes it harder.

The robust BCP in Fukasawa et al. [2006] solved the instancesfrom the literature with up to 135 vertices. The 3 larger Minstances remained open.


Results of FLL+06 over the benchmark instances

LLE04 FLL+06Class NP Unsolved Gap Time Unsolved Gap TimeA 22 7 2.06 6638 0 0.81 1961B 20 1 0.61 8178 0 0.47 4763E-M 12 9 2.10 39592 3 1.19 126987F 3 0 0.06 1016 0 0.14 2398P 24 8 2.26 11219 0 0.76 2892

Total 81 25 3

Machine Intel Celeron 700MHz Pentium 4 2.4GHz

R. Fukasawa, H. Longo, J. Lysgaard, M. Poggi de Aragao, M. Reis,

E. Uchoa, and R.F. Werneck. Robust branch-and-cut-and-price for the

capacitated vehicle routing problem. Mathematical programming, 106(3):

491–511, 2006

The BCP approach works better on instances with shortroutes (n/K < 10).


Recent Literature Review

Since Fukasawa et al. [2006], all the exact CVRP algorithms foundin the literature are based on combinations of column and cutgeneration.

A very active area, important new ideas are being proposedand significant improvements obtained.


Recent Literature: Baldacci et al. [2008]

Column-and-cut generation algorithm:

Columns are associated to elementary routes.

Also uses (a limited number of) non-robust StrengthenedCapacity and Clique Cuts, defined directly over the SPFvariables.

The typical root gaps are much reduced. Many tricks to keep thepricing subproblem tractable.

R. Baldacci, N. Christofides, and A. Mingozzi. An exact algorithm for the

vehicle routing problem based on the set partitioning formulation with

additional cuts. Mathematical Programming, 115(2):351–385, 2008



An important new idea:

Instead of branching, the algorithm finishes by enumeratingall routes with reduced cost smaller than the gap. A SPP withthose routes is given to a MIP solver.

Since solving such SPPs is usually fast (when the number ofroutes is less than 30K), this can save a lot of time on someinstances.


Results of BCM08 over the benchmark instances

LLE04 FLL+06 BCM08Class NP US Gap Time US Gap Time US Gap TimeA 22 7 2.06 6638 0 0.81 1961 0 0.20 118B 20 1 0.61 8178 0 0.47 4763 0 0.16 417E-M 12 9 2.10 39592 3 1.19 126987 4 0.69 1025F 3 0 0.06 1016 0 0.14 2398 3P 24 8 2.26 11219 0 0.76 2892 2 0.28 187Total 81 25 3 9Machine Intel Celeron 700MHz Pentium 4 2.4GHz Pentium 4 2.6GHz

Solved almost all the instances solved in Fukasawa et al. [2006],usually taking much less time. However, very sensitive to n/K ,some instances with larger routes were not solved.



ng-routes, a new relaxation that is more effective than q-routes isintroduced.

For each i ∈ N, let NG (i) ⊆ N be the neighborhood of i , itsng closest customers. A ng -route can only revisit customer iafter it passes by another costumer j such that i /∈ NG (j).

Defining Ω as the set of 8-routes is already (usually) betterthan performing 5-cycle elimination, but costs less in thepricing.

Subset Row Cuts (Jepsen et al. [2008]) are used instead of CliqueCuts, less impact on the pricing.R. Baldacci, A. Mingozzi, and R. Roberti. New route relaxation and

pricing strategies for the vehicle routing problem. Operations Research,

59(5):1269–1283, 2011


Results of BMR11 over the benchmark instances


BMR11Class NP US Gap TimeA 22 0 0.13 30B 20 0 0.06 67E-M 12 3 0.49 303F 3 1 0.11 164P 24 0 0.23 85Total 81 4Machine Xeon X7350 2.93GHz

Faster and more stable, could also solve instances with reasonablymany customers per route.


Recent Literature: Contardo [2012]

New twists on the use of non-robust cuts and on enumeration:

The columns are q-routes without 2-cycles. The partialelementarity of the routes is enforced by non-robust cuts.

The enumeration generates a pool with up to a few millionroutes and the pricing starts to be performed by inspection.At that point, an aggressive separation of non-robust cuts andfixing by reduced costs can lead to small gaps.

Instance M-n151-k12 was solved to optimality in 5.5 hours, settinga new record.

C. Contardo. A new exact algorithm for the multi-depot vehicle routing

problem under capacity and route length constraints. Technical report,

Archipel-UQAM 5078, Universite du Quebec a Montreal, Canada, 2012


Results of Con12 over the benchmark instances


BMR11 Con12Class NP US Gap Time US Gap TimeA 22 0 0.13 30 0 0.07 59B 20 0 0.06 67 0 0.05 89E-M 12 3 0.49 303 2 0.30 2807F 3 1 0.11 164 1 0.06 3P 24 0 0.23 85 0 0.13 43Total 81 4 3Machine Xeon X7350 2.93GHz Xeon E5462 2.8GHz


Recent Literature: Røpke [2012]

Back to robust BCP.

The more effective ng -routes are used.

A sophisticated and aggressive strong branching, greatlyreducing the size of the enumeration trees.

In spite of larger root gaps, results comparable with Contardo[2012] and Baldacci et al. [2011].

M-n151-k12 solved in 5 days.

S. Røpke. Branching decisions in branch-and-cut-and-price algorithms for

vehicle routing problems. Presentation in Column Generation 2012, 2012


Results of Con12 over the benchmark instances


BMR11 Con12 Rop12Class NP US Gap Time US Gap Time US Gap TimeA 22 0 0.13 30 0 0.07 59 0 0.57 53B 20 0 0.06 67 0 0.05 89 0 0.25 208E-M 12 3 0.49 303 2 0.30 2807 2 0.96 44295F 3 1 0.11 164 1 0.06 3 0 0.25 2163P 24 0 0.23 85 0 0.13 43 0 0.69 280Total 81 4 3 2Machine Xeon X7350 2.93GHz Xeon E5462 2.8GHz Core i7-2620M 2.7GHz


Recent Literature: Pecin et al. [2014]

A BCP that borrows from all previously mentioned works andintroduces a number of new elements.

D. Pecin, A. Pessoa, M. Poggi, and E. Uchoa. Improved

branch-cut-and-price for capacitated vehicle routing. In J. Lee and

J. Vygen, editors, Proceedings of the 17th IPCO, pages 393–403.

Springer, 2014


Pecin et al. [2014]: Cuts

Robust cuts

Rounded CapacityStrengthened Comb

Non-robust cuts

lm-Subset Row Cut

Post-enumeration cuts

Subset Row CutClique


Pecin et al. [2014]: Pricing

The most critical part of the BCP.

A label setting dynamic programming algorithm handles:

ng -routeslm-SRCs (1SRCs, 3SRCs, 4SRCs, 5SRCs)

Features:

Bidirectional Search (Righini and Salani [2006]).Completion Bounds (Contardo [2012])Some implementation tricks, like data structures over bitmaps,etc.Fast and effective heuristics. Exact pricing called a few timesper node.


Pecin et al. [2014]: Non-robustness control

Even with all the care in their separation, non-robust cuts areindeed “non-robust”:

The pricing may be handling several hundreds such cutsefficiently. Then, in some node of the tree, it suddenlybecomes 100 or even 1000 times slower!

In those cases it is necessary to roll back, removing theoffending cuts. The node lower bound decreases, but the BCPdoes not halt.


Pecin et al. [2014]: Miscellanea

A number of strategies are hybridized:

Strong Branching:

Hierarchical, 3 levelsUses history of past branchingsAggressive, up to 200 candidates can be testedUses estimates of the subtree size determining the SB effort ineach node

Enumeration:

Performed when the node gap is sufficiently small forgenerating a pool with less than 20M routesOrdinary branching occurs in enumerated nodesThe MIP solver only finishes the node when the pool has lessthan 30K routes

Branch-and-cut: In the root node, when column generation ishaving severe convergence problems, it may try BC


Results of PPPU14 over the benchmark instances

BMR11 Con12 Rop12Class NP US Gap Time US Gap Time US Gap TimeA 22 0 0.13 30 0 0.07 59 0 0.57 53B 20 0 0.06 67 0 0.05 89 0 0.25 208E-M 12 3 0.49 303 2 0.30 2807 2 0.96 44295F 3 1 0.11 164 1 0.06 3 0 0.25 2163P 24 0 0.23 85 0 0.13 43 0 0.69 280Total 81 4 3 2Machine Xeon X7350 2.93GHz Xeon E5462 2.8GHz Core i7-2620M 2.7GHz

PPPU14Class NP US Gap TimeA 22 0 0.03 5.6B 20 0 0.04 6.2E-M 12 0 0.19 3669F 3 0 0.00 3679P 24 0 0.07 33Total 81 0Machine Core i7-3770 3.4GHz


Detailed Results: M-n151-k12

Algo Machine Root LB Final LB Total Time

BMR11 X7350 2.93GHz 1004.3 1004.3 380Contardo12 E5462 2.8GHz 1012.5 1015 19699

Ropke12 i7-2620M 2.7GHz 1001.5 1015 417146PPPU14 i7-3770 3.4GHz 1012.8 1015 212




BMR11 X7350 2.93GHz 1258.7 1258.7 436Contardo12 E5462 2.8GHz 1265.1 1265.1 34350

Ropke12 i7-2620M 2.7GHz 1255.3 1261.4 7200PPPU14 i7-3770 3.4GHz 1268.7 1275 3581




BMR11 X7350 2.93GHz 1256.6 1256.6 319Contardo12 E5462 2.8GHz 1263.0 1263.0 265588

Ropke12 i7-2620M 2.7GHz 1253.0 1258.2 7200PPPU14 i7-3770 3.4GHz 1266.5 1274 39869

Previous upper bound: 1278.


Optimal solution of M-n200-k16 (Q = 200), cost 1274

182623

27

27

25 29

19

12

219

10

1

27

1628 511

341

15

7

8

21

1426

1123

3

23

1616

17

9

5

19

26

6

72

23

35

716

20

7

8

7

16

11

25

18

29 25

22

6

28

1913

2117

52119

1414

31

12

6

19

20

22

11

16

36

1327

5

11

30

925

107

13

10

3

14

31

17

8

6

20

15

25

249

5

1517

5

16

17

1914

12

2919

28

9

26

125

13

119

18

1527

19

13

9911

3

19

14

1613

236 15

8 9

312

26

1818

1421

9

9

28 8

2318

5

10

10297

93

12218

108

17181

41

22 28 20

1530 19

1117

106

36

152713

10

10

1637

16

28

26

11

1530

16

20

15

33

2716

612 18

19

16


Book chapter reviewing all those recent algorithms

M. Poggi and E. Uchoa. New exact approaches for the capacitated VRP.

In P. Toth and D. Vigo, editors, Vehicle Routing: Problems, Methods,

and Applications, chapter 3. SIAM, second edition, To Appear


The remainder of this talk aims to explain a key issue in CVRPalgorithms based on cut and column generation: how the routedefinition and the non-robust cuts interfere with the pricing.


The Labeling Algorithm

The pricing in CVRP algorithms is done by labeling dynamicprogramming algorithms:

Each (i , q), i ∈ V , q ∈ di , . . . ,Q defines a bucket

A label L(P) is a data structure representing a partial path P,with cost c(P). All labels corresponding to paths ending in iwith load q are kept in bucket B(i , q)

An initial label representing a null path is put in B(0, 0)

Labels are expanded producing other labels

Labels in B(0, q), q > 0, are complete routes

Dominated labels should be removed along the algorithm














Domination in the Labeling Algorithm

Rule

A label L(P1) dominates another label L(P2) in the same bucket ifc(P1) ≤ c(P2) and every valid completion of P2 is also a validcompletion for P1.

q-routes: at most one non-dominated label per bucket.

q-routes without 2-cycles: the least costly label in B(i , q), P1,can not be extended to its predecessor j . As the second leastcostly label P2 can be extended to j , P1 does not dominateP2. Both labels should be kept.

q-routes without k-cycles: at most k! non-dominated labels ineach bucket. Large values of k may cause an exponentialproliferation of labels.


Domination in the Labeling Algorithm

Rule

A label L(P1) dominates another label L(P2) in the same bucket ifc(P1) ≤ c(P2) and every valid completion of P2 is also a validcompletion for P1.

ng -routes: Let Π1 and Π2 be the forbidden immediateextensions of paths P1 and P2, due to NG -sets. If Π1 ⊆ Π2

then every valid completion of P2 is also a valid completionfor P1. At most 2ng−1 labels per bucket.

ng = 8 is a safe choice for avoiding an exponentialproliferation of labels.


Subset Row Cuts (SRCs)

Given C ⊆ N and a multiplier p, the (C , p)-Subset Row Cut is:

∑r∈Ω

⌊p∑i∈C

ari

⌋λr ≤ bp|C |c (11)

Non-robust cut obtained by a Chvatal-Gomory rounding of |C |constraints in the SPF.

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


3SRCs

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 + 2y4C + 2y5

C + . . . ≤ 1.

Used in Contardo [2012] and Baldacci et al. [2011]

Potentially very effective

Other interesting SRCs do exist.


How 3SRCs affect the Labeling Algorithm

Suppose there are nS 3SRCs, σs < 0 is the dual variableassociated to SRC s, C (s) its base set.

The labels now carry nS additional binary dimensions, S(P)[s]is the parity of the number of times that path P visited avertex in C (s).

When a label L(P) is expanded to L(P ′), if S(P)[s] = 1 andS(P ′)[s] = 0 then the reduced cost of P ′ should be penalizedby adding to σs to it.

A label L(P1) can only dominate a label L(P2) if

c(P1) ≤ c(P2) +∑

1≤s≤nS :S(P1)[s]>S(P2)[s]

σs .


How 3SRCs affect the Labeling Algorithm

Rule

A label L(P1) dominates another label L(P2) in the same bucket ifc(P1) ≤ c(P2) +

∑1≤s≤nS :S(P1)[s]>S(P2)[s] σs and every valid

completion for P2 is also a valid completion for P1 (this dependson the definition of Ω).

As more 3SRCs are added, the dominance gets weaker

In many instances, 100 3SRCs already causes a combinatorialexplosion in the number of labels


limited memory Subset Row Cuts (lm-SRCs)

Given C ⊆ V+, a memory set M, C ⊆ M ⊆ V+, and a multiplierp, the limited memory (C ,M, p)-Subset Row Cut is:∑

r∈Ω

α(C ,M, p, r)λr ≤ bp|C |c , (12)

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

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.



0

2

1

3

Route r1, λr1=0.5

λr1 has coefficient 1 in the 3-SRC with C = 1, 2, 3



0

2

1

3

Route r1, λr1=0.5

Included in the memory set

λr1 still has coefficient 1 in the lm 3-SRC



0

2

1

3

Route r2, λr2=0.5




0

2

1

3

Route r2, λr2=0.5





0

2

1

3

Route r3, λr3=0.5




0

2

1

3

Route r3, λr3=0.5




Why it is good to reduce the set M as much as possible?

Rule

A label L(P1) dominates another label L(P2) in the same bucket ifc(P1) ≤ c(P2) +

∑1≤s≤nS :S(P1)[s]>S(P2)[s] σs and every valid

completion for P2 is also a valid completion for P1.

Let B(i , q) be a bucket such that, for a certain 3SRC s, i /∈ C (s).This means that S(P)[s] = 0 for all labels L(P) in that bucket.Therefore, σs will not affect the dominance in B(i , q).

The separation of lm-SRCs with minimal memory-set setsmitigates the effect of those non-robust cuts in the pricing:

Many more cuts can be separated before it becomesintractable


Why it is good to reduce the set M as much as possible?

0

2

i

1

3

Solid path may only dominate the dashed path because the lm3-SRC 1, 2, 3 is already forgotten at i .


The State-Of-the-Art

How large are the CVRP instances that can be solved tooptimality?


Golden’s Instances

Golden, Wasil, Kelly, and Chao [1998] proposed 12 instances,ranging from 240 to 483 customers.

Appear frequently in the literature on heuristic methods

Until now, considered to be far beyond the reach of exactmethods

Four instances could be solved, those with 240, 300, 320, and 360customers.


Optimal solution of Golden 14 (320 customers,Q = 1000), cost 1080.55, 30 routes

8.1

6.96.9

6

5.25.25.25.2

5.2

5.2

5.2

5.2

5.2 6 6.9 8.1

8.1

6.96.9

6

5.25.25.25.2

5.2

5.2

5.2

5.2

5.2 6 6.9 8.1

10

8.1 6.9 6

6

6

66

6.9

6.98.1

10

12.8

12.8108.16.965.2

5.2 6 6.9 8.1 10 12.8

10

8.1 6.9 6

6

6

66

6.9

6.98.1

10

12.8

12.8 10 8.1 6.9 6 5.2

5.266.98.11012.8

12.8

10

8.1

6.9

66

5.2 5.2 5.2

6

6.9

8.1

10

18 18

183018

18 18

30

18

1818

30

12.8

10

8.1

6.9

6 6

5.25.25.2

6

6.9

8.1

10

12.8

10

8.1

6.9

6 6

5.25.25.2

6

6.9

8.1

10

1818

18 30

12.8

10

8.1

6.9

66

5.2 5.2 5.2

6

6.9

8.1

10

30

1818

30

12.8

10

8.1

6.9

6

5.2 5.2

6

6.9

8.1

10

12.8

3018

18 30

30 18

1830

30

18 18

30

12.8 10 8.1 6.9 6 5.2

5.266.98.11012.8

12.8 10 8.1 6.9 6 5.2

5.266.98.11012.8

12.8

10

8.1

6.9

6

5.25.2

6

6.9

8.1

10

12.8

8.1

6.9 6.9

6

5.2 5.2 5.2 5.2

5.2

5.2

5.2

5.2

5.266.98.1

12.8

10

8.1

6.9

6

5.2 5.2

6

6.9

8.1

10

12.8

12.8

10

8.1

6.9

6

5.25.2

6

6.9

8.1

10

12.8

12.8108.16.965.2

5.2 6 6.9 8.1 10 12.8

10

8.16.96

6

6

6 6

6.9

6.9 8.1

10

12.8

10

8.16.96

6

6

6 6

6.9

6.9 8.1

10

12.8

12.8108.16.965.2

5.2 6 6.9 8.1 10 12.8

8.1

6.9 6.9

6

5.2 5.2 5.2 5.2

5.2

5.2

5.2

5.2

5.266.98.1


Optimal solution of Golden 19 (360 customers, Q = 20),cost 1365.60, 33 routes

1

1

1

1

1

4

411

11

11

1

1

11

114

4

11

11

11

1

2

2

2

1

1

4

1

1

1

1

2

2

2

2

1

2

1

41

4

1

1

1

1

11

1

44

1

1

11

2

2

1

11

1

1

4

411

11

1

1

1

4

4

4 1

1 1

1

214

2

11

4 1 1

1 1

1

2

2

1

4

1

4

1

2

1

11

1

4 4 4

11

11

1

2

2

2

1 2

1

41

41

11

2

1

2

1

4 1

4 1 2

1

11

11

4

4

4

2

2 1 4

1

4121

1

1

2

2

2

1

2

1

4

14

12

2

1

1

1

1

4 1 1

2

2

2

444

11

11

1

2

1

41

412

121

1

1

1

14

1

4 1 2 1

2

2

2

1

4

1

4

2

2

2

1

1

2

214

1

4 1 2 1

1

1

2

1

2

1

2

1

4 1

4 1

1

4

4

4

4

1

11

1

1

1

1

4

1

4

1

2

1

2

2

2 1 41

4

2

21

11

1

2

1

1

4

4 1 1

1 12

2

2

2

2

1

2

1

4

1

4

1

1

1

2

1

4

1

4

1

2

1

2

1

1

41

2

1

1

4

1

1

1

11

4

4

4

41

1

1

1

11

11

11

4 4

1

1

11

2

1

1

1

2

1

2

1

41

41

1

1

1

1

1

4

1

2

1

2

1

4

1


New X Instances

A benchmark set with 100 instances, ranging from 100 to 1000customers, was recently proposed by Uchoa, Pecin, Pessoa, Poggi,Subramanian and Vidal [2014].

Designed to cover a wide range of characteristics found in realapplications

Available at http://vrp.atd-lab.inf.puc-rio.br/index.php/en/


Results on the new X Instances

40 instances could be solved to optimality:

100 ≤ n < 200 : 21 out of 22

200 ≤ n < 300: 14 out of 21

300 ≤ n < 500: 4 out of 25

500 ≤ n < 1000: 1 out of 32

Smallest unsolved: X-n190-k8Largest solved: X-n655-k131 (2491s)


Optimal solution of X-655-k131 (unitary demands, Q = 5),cost 106780, 131 routes

277

112

172

581

31

587

97

412577

111

66

21

213

307

271

8

286

141

572

208

434

91

484

360

430

106

564

522

648

382

540

477

482

599

655

463

107

47

460

456

420

421

37

490

153

218

105

459 339

207

88

48

573509

77

5

315

634

452

550

211

52470

276

582

130

99

415

119

388

136

16

427

442

188

102

24

386

140

535

280

238

389

242

384

73

134

451

646

519

372

12

609210

109

431

481

367

560

497

13

192

261

621

144

379

151

524

252

123

486

170

654

505

549

114

312

336

555

60

69

94

507

504

38

533

362

164483

403

50822

603

357

322

184596

300

653

87

383

79

20

380

132

447

110

593

429

202

344

155

330

199

9

370

10

17

214

605

195

424

266

638

449

516

122

455

250

635

50

58932 598

215

597

186462

272

618

528

185

521

259

147

294

492

262

512

256

281

305

608

396

376

247

395

627

410

338

302263

643

469

641

458

196

299

293

78

511

75

222

558

453

243

480

576

586

90

127

49

212

103

557

95

260

510

552

289

157

35

150401

264

525

40

489

64

650

568

308

561

644

104

544

332

358

345

563

244

224

607

584257

393

18

475

197

637

177

220

341

478

96

121

180

614

628

601

59

166

375

340

551

461

255

295

273

407

26

542

390

642

468

349

145

283543

327

7

80

465

515

3

282

84

258

600

11

287

371

81

479168

29

118

98

363

487

343

398

359

428

160

348

171146

189

142

513

200

488

590471

230

19649

532

113

346

34

517

350

457

115

65

125

335

89

131

92647

409

624

473

161

319530

553

53

216

620

595

297

433

14503

265

162

25

2256

356496

570

241

93

623

514

502

352

70

536

474

354

100

436

304

269

320

245

323

318

143

645

314

159

612

219

402

476

228

529

652

419

74

574

633

301

636

169

466

493

381

275303

418

583

506

306

240

594

351

268

137

2198

566

67

602

167368

500

124

173

438

324

585

45

41

135

413

537

158

4

44

639

163

63

365

108

387

378

310

333

448

626

254

152

71179

334

579

571

201

58538

406

426

33

337

182

23

541554

615

588

580

464

183

397235

175

630

248

575

274

373

217

292

181

374

526

391

46229

68

42

342

191

518

329

83

416

29636

546

309

178

165

498

278

203

625193

400

221631

85

249

417399

231

439

187

556

531

237

539

444

622

495

56

267

408

206

629

62

361

209

443

394

72

617

606

366

619

291

194

129

15

377

527

279

138

347

578

316

101

355

405

632

149

414

126

385

133

423

28

174

298

611

392

592

353523

454

128

8227

176

591

491

569

445616

499

313

234

117

39

565

54

331251

223

559

441

116

139

411

190

311

253

232

472

328

288

364

369

239

42561

440

317610

520

450

226156

326

494

246

204

86

640

435

154

205

55 285

120

290

30

43

321148

651

562

57

270

604

51

422

467

233

545

548

485

227

534

567

404

446

437

613

501

432

76

236325

284

547


Thank you for your attention!


N. Achuthan, L. Caccetta, and S. Hill. An improvedbranch-and-cut algorithm for the capacitated vehicle routingproblem. Transportation Science, 37:153–169, 2003.

J. Araque, G. Kudva, T. Morin, and J. Pekny. A branch-and-cutalgorithm for the vehicle routing problem. Annals of OperationsResearch, 50:37–59, 1994.

P. Augerat. Approche polyedrale du probleme de tournees devehicules. PhD thesis, Institut National Polytechnique deGrenoble-INPG, 1995.

P. Augerat, J. Belenguer, E. Benavent, A. Corberan, D. Naddef,and G. Rinaldi. Computational results with a branch and cutcode for the capacitated vehicle routing problem. TechnicalReport 949-M, Universite Joseph Fourier, Grenoble, France,1995.


R. Baldacci, N. Christofides, and A. Mingozzi. An exact algorithmfor the vehicle routing problem based on the set partitioningformulation with additional cuts. Mathematical Programming,115(2):351–385, 2008.

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.

U. Blasum and W. Hochstattler. Application of the branch and cutmethod to the vehicle routing problem. Technical ReportZPR2000-386, Zentrum fur Angewandte Informatik Koln, 2000.

N. Christofides and S. Eilon. An algorithm for thevehicle-dispatching problem. Operational Research Quarterly,20:309–318, 1969.


N. Christofides, A. Mingozzi, and P. Toth. The vehicle routingproblem. In N. Christofides, A. Mingozzi, P. Toth, and C. Sandi,editors, Combinatorial Optimization, A Wiley-IntersciencePublication, Based on a series of lectures, given at the SummerSchool in Combinatorial Optimization, held in Sogesta, Italy,May 30th-June 11th, 1977, Chichester: Wiley, 1979, edited byChristofides, Nicos, volume 1, pages 315–338, 1979.

C. Contardo. A new exact algorithm for the multi-depot vehiclerouting problem under capacity and route length constraints.Technical report, Archipel-UQAM 5078, Universite du Quebec aMontreal, Canada, 2012.

M.L. Fisher. Optimal solution of vehicle routing problems usingminimum k-trees. Operations research, 42(4):626–642, 1994.

R. Fukasawa, H. Longo, J. Lysgaard, M. Poggi de Aragao, M. Reis,E. Uchoa, and R.F. Werneck. Robust branch-and-cut-and-pricefor the capacitated vehicle routing problem. Mathematicalprogramming, 106(3):491–511, 2006.


S. Irnich and D. Villeneuve. The shortest-path problem withresource constraints and k-cycle elimination for k ≥ 3.INFORMS Journal on Computing, 18(3):391–406, 2006.

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.

G. Laporte and Y. Nobert. A branch and bound algorithm for thecapacitated vehicle routing problem. OR Spectrum, 5(2):77–85,1983.

J. Lysgaard, A. Letchford, and R. Eglese. A new branch-and-cutalgorithm for the capacitated vehicle routing problem.Mathematical Programming, 100:423–445, 2004.

D. Pecin, A. Pessoa, M. Poggi, and E. Uchoa. Improvedbranch-cut-and-price for capacitated vehicle routing. In J. Leeand J. Vygen, editors, Proceedings of the 17th IPCO, pages393–403. Springer, 2014.


M. Poggi and E. Uchoa. New exact approaches for the capacitatedVRP. In P. Toth and D. Vigo, editors, Vehicle Routing:Problems, Methods, and Applications, chapter 3. SIAM, secondedition, To Appear.

T. Ralphs, L. Kopman, W. Pulleyblank, and L. Trotter Jr. On thecapacitated vehicle routing problem. MathematicalProgramming, 94:343–359, 2003.

G. Righini and M. Salani. Symmetry helps: bounded bi-directionaldynamic programming for the elementary shortest path problemwith resource constraints. Discrete Optimization, 3(3):255–273,2006.

S. Røpke. Branching decisions in branch-and-cut-and-pricealgorithms for vehicle routing problems. Presentation in ColumnGeneration 2012, 2012.


Documents

Recent Advances in Exact Algorithms for the Capacitated Vehicle Routing Problemwebtv.univ-bejaia.dz/.../2017/01/Eduardo-Uchoa-COSI2014.pdf · 2017-01-23 · Vehicle Routing Problem