Robust Branch-and-Cut-and-Price for the

Robust Branch-and-Cut-and-Price for the

Capacitated Vehicle Routing Problem

Ricardo Fukasawa

GAPSO, Inc.

Marcus Poggi de Aragão Dep. Informática, PUC-Rio

poggi@inf.puc-rio.br Marcelo Reis

Dep. Informática, PUC-Rio

Eduardo Uchoa

Dep. Engenharia de Produção, UFF

Capacitated Vehicle Routing Problem (CVRP)

Given

An undirected graph G=(V,E) V= {0,1,...,n}

Vertex 0 represents a depot and remaining vertices represent clients Edge lengths are denoted by c(e)

Client demands are q(1),...,q(n) K vehicles with capacity C

Determine routes for each vehicle satisfying the following constraints:

(i) each route starts and ends at the depot, (ii) each client is visited by exactly one vehicle

(iii) the total demand of clients visited in a route is at most C The objective is to minimize the total route length.

Example - Instance A-n62-k8

Vehicle Capacity: 100

Optimal Solution A-n62-k8

Optimal Solution Value: 1288

Capacitated Vehicle Routing Problem (CVRP)

Considered one of the hardest combinatorial problems to

solve exactly

The evolution of the algorithms has been very slow:

State of the art in 1978: Exact solution for up to 25 clients.

State of the art in 2003: Exact solution for up to 50-100 clients.

Problems with many vehicles (k >7) are usually harder.

Algorithms for the CVRP

Best approaches during the 80’s: Column Generation and Lagrangean Relaxation:

Christofides, Mingozzi, Toth (1981);

Agarwal, Mathur, Salkin (1989);

Fisher (1994).

Branch-and-Cut algorithms were developed in the mid 90’s (1995) and have have been the best since then:

Augerat, Belenguer, Benavent, Corberan, Naddef, Rinaldi (1995);

Martinhon, Lucena, Maculan (1998);

Blasum, Hochstattler (2000);

Ralphs, Kopman, Pulleyblank, Trotter (2003);

Lysgaard, Letchford, Eglese (2003).

We propose to Branch, Cut and Price in a Robust way.

Robust Branch-and-Cut-and-Price

This work proposes a way to deal with an

exponential number of both columns and rows, leading to a branch-and-cut-and-price

algorithm.

The term “robust” refers to the fact that neither

branching nor cut separation ever change the

pricing subproblem.

Robust Branch-and-Cut-and-Price

Poggi de Aragão and Uchoa (2003)

min s. t

cx

Ax = b Dx ≤ d x ∈ N

min cx

s .t. x' − x = 0 Ax = b

Dx' ≤ d

x' ∈ N

x ∈ N

min cx

s.t. Q

λ

1

λ

λ

x ∈N^m

Formulating the CVRP with an

exponential number of constraints

Let 0 be the root node (depot) V

={1,...,n} = V \ {0}

k(S): minimum number of vehicles to serve all clients in S ⊆ V

Let X

be the number of times a

vehicle traverses edge e for e ∈ E

CVRP – exponential number of contraints

( )

(0)

( )

min

. . 2 ,

2.

2. ( ) ,

e e e E

e e i

e e

e

e s

c x

s t x i V

x K

x k S S V

δ δ δ

∈

∈ +

∈

∈ +

= ∀ ∈

=

≥ ∀ ⊆

Ensures that all non root vertices have exactly two incident edges

Eliminates subtours and inforces

maximum route capacity

( , ) (0, )

{0,1} , ( , ) , {0,1, 2} , (0, )

e i j

e j

x i j E i j V

x j E j V

= +

= +

∈ ∀ ∈ ∈

∈ ∀ ∈ ∈

Robust formulation for the CVRP

Let R be the set of all valid routes (total of p) Let Q be a n X p matrix of the incidence

vectors associate to these routes.

Let q

denote the coefficient associated to edge e in route j

Let λ

be a variable indicating whether route j is in the solution or not.

The x variables can now be expressed as a

convex combination of the λ’s

:

( )

(0)

min

. . 0 ,

2 ,

2.

{0,1} ,

e

e e e E

ej j j R e j

e

e i

e e

j

c x

s t q x e E

x i V

x K

j R

{0,1} , \ (0)

{0,1, 2} , (0)

e e

x e E

x e

δ δ

∈ ∀ ∈

∈ ∀ ∈

Formulation for the CVRP with an

exponential number of variables

:

( )

(0)

min

. . 0 ,

2 ,

2.

0 ,

e

e e e E

ej j j R e j

e

e i

e e

j

c x

s t q x e E

x i V

x K

j R

The λ’s are integer if the x’s are

{0,1} , \ (0)

{0,1, 2} , (0)

e e

x e E

x e

δ δ

∈ ∀ ∈

∈ ∀ ∈

:

( )

(0)

( )

min

. . 0 ,

2 ,

2.

2. ( ) ,

e

e e e E

ej j j R e j

e

e i

e e

e

e s

c x

s t q x e E

x i V

x K

x k S S V

0

0 ,

j

e

j R

x e E

λ ≥ ∈

≥ ∈

Explicit Master Formulation (LP relaxation)

1

2 ( 0 )

0

p e

j j

j

j

q e

j R

λ δ

λ

=

≤ ∈

≥ ∈

D-W Master Formulation

The pricing subproblem

Finding a valid route with minimum reduced cost: Prize Collecting TSP (Balas, 1989)

Strongly NP-hard problem.

Relax that problem to allow cycles, which allows us to generate the columns by

dynamic programming in pseudo-polynomial time.

Find q-routes (Christofides, Mingozzi, Toth,

1981)

The pricing subproblem

h(i,q) – Best path with capacity q from i to the depot.

Complexity O(n

C)

Two cycles can be eliminated (CMT 81) without changing complexity

{ }

( , ) min

( , ( ))

h i q =

c + h j q q j −

Cuts

Separating the Capacity cuts is NP-hard We used T. Ralphs implementation of the Augerat et. al. separation heuristic (available at Symphony’s demo).

We also applied an exact separation

(MIP solver) at the root node.

Cuts

The cuts are added over the x variables only, not affecting the pricing subproblem

We perform branching on the original

variables X, preference for the ones with larger cost and closer to 0.5

Branching

Results

This approach consistently gets lower bounds better than those from previous algorithms, allowing us to solve to

optimality several instances for the first

time.

Results

Using CPLEX 7.1 solver on a Pentium IV, 2 Ghz, 1 GB RAM.

Tests on instances from the literature available at

http://branchandcut.org/VRP/data/

– 108 instances: series A, B, E, F, M

E series (Eilon-Christofides, 1969)

815 801.8

802.6 E-n101-k8

Best Upper Bound

Lower Bound

521 517.9

519.0 E-n51-k5

682 668.4

666.4 E-n76-k7

735 725.1

717.9 E-n76-k8

1071 1051.6

1026.9 E-n101-k14

1021 1004.8

969.6 E-n76-k14

830 816.5

799.9 E-n76-k10

Ours LLE03

Instance

A series (Augerat et al. 1995)

Ours Previous

1763 1159 1174 1402 1315 1616 1290 1034 1354 1073 1167 1010

Upper Bound

1159 1138.4

1114.4 A-n69-k9

1763 1749.7

1709.6 A-n80-k10

1401 1378.9

1351.6 A-n64-k9

1616 1603.5

1580.7 A-n63-k9

1314 1294.5

1266.6 A-n63-k10

1288 1274.1

1251.7 A-n62-k8

1034 1018.6

1010.2 A-n61-k9

1354 1341.6

1319.6 A-n60-k9

Lower Bound

1010 1002.2

998.7 A-n53-k7

1167 1150.0

1135.3 A-n54-k7

1073 1066.4

1058.3 A-n55-k9

1174 1163.4

1155.2 A-n65-k9

Ours LLE03

Instance

Some open instances from other series

Ours Previous

1034 1275

Upper Bound Lower Bound

1034 1030.8

1017.4 M-n121-k7

1272 1261.5

1258.1 B-n68-k9

Ours LLE03

Instance

-We also closed all the remaining 7 open

instances from series P ^.

Remarks

It is clear that the Capacity Constraints cut significantly the polytope described by the column generation formulation.

The addition of many other classes of

known inequalities (….) is expected to

yield even better results.

TAK !

IP Reformulation

Let Q be a n X p matrix where each column corresponds to one element of P

There is a one-to-one correspondence

between elements of P and the solutions of:

{ }

t s

Q x

1 , 0

1 .

∈

=

=

λ

λ

λ

1

IP Reformulation

The traditional IP reformulation (Gilmore and Gomory, 1961) consists of replacing x by its

equivalent expression in (O), a transformation similar to the Dantzig-Wolfe reformulation for LP:

{ }

min ( )

. ( )

1 0,1

IP

p

Z cQ

s t AQ b

λ λ λ λ

=

=

=

∈

1

Dantzig-Wolfe Master

Solving the DWM is equivalent to solving

min ( )

. ( )

1 0

Z

cQ

s t AQ b

λ λ λ λ

=

=

=

≥ 1

( DWM )

{ }

min .

|

DWM

n

Z cx

s t Ax b

x Conv x N Dx d

=

=

∈ ∈ ≤