[PPT] - Flow-based Mathematical Formulation and Strengthening Cuts for the PowerPoint Presentation

SLIDE 1

Flow-based Mathematical Formulation and Strengthening Cuts for the Cumulative CVRP

Sandra Ulrich NGUEVEU / Mathieu LACROIX

LAAS-CNRS / LIPN ngueveu@laas.fr

21/05/2012

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 1/ 34) ODYSSEUS 2012 21/05/2012 1 / 34

SLIDE 2

Plan

1

Introduction

2

Literature review

3

Mathematical formulation proposed

4

Valid Inequalities

5

Extension of the flow-based model

6

First Results

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 2/ 34) ODYSSEUS 2012 21/05/2012 2 / 34

SLIDE 3

Introduction

Problem

The CCVRP minimizes the sum of arrival times at customers, whilst respecting the capacity constraints. Humanitarian logistics (Campbell et al. , Transp. Science 2008)

delivery of vital goods and supply to the population, after a natural disaster, taking into account urgency and equity. solutions from the CCVRP, the CVRP and the OVRP can vary significantly

(a) opt. Sol. CVRP (b) opt. sol OVRP (c) opt. sol. CCVRP*

*from our branch-and-cut

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 3/ 34) ODYSSEUS 2012 21/05/2012 3 / 34

SLIDE 4

Literature review

1

Introduction

2

Literature review

3

Mathematical formulation proposed

4

Valid Inequalities Inequalities identified for the TRP and directly extended to CCVRP Inequalities for TRP not trivially extensible to the CCVRP

5

Extension of the flow-based model

6

First Results

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 4/ 34) ODYSSEUS 2012 21/05/2012 4 / 34

SLIDE 5

Literature review

Ngueveu S.U., Prins C. et Wolfler Calvo R., “An effective memetic algorithm for the CCVRP”, Computers and Operations Research, 2010. Ribeiro G.M. et Laporte G., “An adaptive large neighborhood search heuristic for the CCVRP”, Computers and Operations Research, 2011. Figure: Solution from ALNS for instance GWC1 (Source : Ribeiro et Laporte, 2011)

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 5/ 34) ODYSSEUS 2012 21/05/2012 5 / 34

SLIDE 6

Literature review

Literature review : other expression of the objective-function

Total cost for solution S =

e∈E

ce ∗ coefe where coefe= number of clients following edge e in solution S.

a 4 F b 5 c 3 d 4 2

Sum : arrival times = ta + tb + tc + td = 4 + 9 + 12 + 16 = 41

a 4 (x 4) F b 5 (x 3) c 3 (x 2) d 4 (x 1) 2 (x 0)

Sum : cost * coefficients =4 ∗ 4 + 5 ∗ 3 + 3 ∗ 3 + 4 ∗ 1 = 41

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 6/ 34) ODYSSEUS 2012 21/05/2012 6 / 34

SLIDE 7

Literature review

Literature review : mono-vehicle version

Various names : traveling repairman problem (TRP), traveling deliveryman problem (TDP), school-bus driver problem, minimum latency problem NP-hard Sahni, S. and Gonzales, T.(1974) Two main variants : with or without return to the depot Exact methods

Lucena (1990), Fischetti et al.(1993), Bianco et al.(1993), Van Eijl (1995), Wu et

al. (2004), Mendez-Diaz et al.(2008), Sarubbi et al.(2008), Ome Ezzine et al.

(2010), ...

Approximation algorithms

Blum et al.(1994), Geomans et Kleinberg (1996) Chaudhuri et al. (2003), Archer et

al. (2008) ...

One metaheuristic : GRAPS + VND

Salehipour, Sörensen, Goos and Bräysy (2008)

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 7/ 34) ODYSSEUS 2012 21/05/2012 7 / 34

SLIDE 8

Literature review

Literature review : mono-vehicle version

L. Bianco, A. Mingozzi et S. Ricciardelli, Networks, 1993

xk

ij : binary : = 1 iff arc (i, j) is used at position k

random instances : n = 25 − 35 with a time limit of 600 seconds (26 to 48000 b&b nodes)

M. Fischetti, G. Laporte et S. Martello, Operations Research 1993

(xi1 : vi ∈ V ) : = is a permutation of (1, 0, ..., 0) xij : (vi, vj) ∈ A′ : = is a permutation of (n, n-1, ..., 2, 0, ..., 0) random instances : n up to 45 (symmetric) / n up to 50 - 60 (euclidian) J.F.M. Sarubbi et H.P.L. Luna., Technical report xij : binary : = 1 iff arc(i, j) is crossed pij : binary : = 1 iff node j is the ith node visited gij : continue : = total flot crossing (i, j) random instances : up to 30 nodes in 1648 seconds

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 8/ 34) ODYSSEUS 2012 21/05/2012 8 / 34

SLIDE 9

Literature review

Literature review : mono-vehicle version

I. Ome Ezzine, F. Semet et H. Chabchoub, MOSIM 2010

Xij : binary : = 1 iff arc (i, j) is used Zij : binary : = 1 iff i is visited before j F k

ij : binary : = 1 iff arc (i, j) if used on the path from 1 to k

TSPLIB instances : up to 29 nodes in 36452 seconds

I. Méndez-Díaz, P. Zabala et A. Lucena, Discrete Applied Math. 2008

xij : binary : = 1 iff node i appears before j in the solution f k

ij : continue : = 1 iff arc (i, j) is used to go from 0 to k

TSPLIB instances : up to 29 nodes in 5334 seconds

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 9/ 34) ODYSSEUS 2012 21/05/2012 9 / 34

SLIDE 10

Mathematical formulation proposed

1

Introduction

2

Literature review

3

Mathematical formulation proposed

4

Valid Inequalities Inequalities identified for the TRP and directly extended to CCVRP Inequalities for TRP not trivially extensible to the CCVRP

5

Extension of the flow-based model

6

First Results

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 10/ 34) ODYSSEUS 2012 21/05/2012 10 / 34

SLIDE 11

Mathematical formulation proposed

Extension of the polynomial lower bound 1

Hypothesis : High number of vehicles (ex : R = n − 1)

Figure: BI1 si R = n

Exemple si R = n − 1 min

i∈V ′

c(0,i)xi +

i∈V ′

2c(0,i)yi +

e∈E′

cere (1) s.t. xi + yi + zi = 1, ∀i ∈ V ′ (2)

i∈V ′

xi = n − 2 (3)

i∈V ′

yi =

i∈V ′

zi =

e∈E′

re = 1 (4) re ≤ yi + yj + zi + zj 2 , ∀e = (i, j) ∈ E ′ (5) xi, yi, zi binaires ∀i ∈ V ′ (6) re binaires ∀e ∈ E (7)

Limits : not “generalizable" because no relationship could be found between the solutions obtained with the successive reductions of R.

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 11/ 34) ODYSSEUS 2012 21/05/2012 11 / 34

SLIDE 12

Mathematical formulation proposed

Extension of polynomial lower bound 2

Hypothesis : Balanced solution (in terms of number of customers)

Figure: BI2 si n = 13, R = 4

Exemple si n = 13 et R = 4 (P)

e∈E

4

p=1

ce.p.xp

e

(8) s.t.

e∈E

xp

e =

1 si p = 4 2 sinon. , ∀p ∈ [1...4] (9)

e∈δ(i)

xp

e + ye =

2R si i = 0 2 sinon. , ∀i ∈ V (10)

e∈E

ye = R (11)

p∈[1...4]

xp

e + ye <= 1

(12) xp

e , ye binaires

∀e ∈ E (13)

Limits : not “generalizable" because the number of configurations increases exponentially when R decreases.

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 12/ 34) ODYSSEUS 2012 21/05/2012 12 / 34

SLIDE 13

Mathematical formulation proposed

Flow-based formulation

Decision variables xe : binary : = 1 if edge e is used and 0 otherwise fij : continuous, positive := flow going from i to j on edge e

coefficient assigned to edge e in the objective-function

Constant γi : max number of clients of a route containing client i

γe = min{γie, γje}, ∀e = (ie, je) ∈ E

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 13/ 34) ODYSSEUS 2012 21/05/2012 13 / 34

SLIDE 14

Mathematical formulation proposed

Flow-based formulation

Minimizes Total Cost = sum of arrival times (FF) min F =

i∈V \{n+1}
j∈V ′\{i}

cijfij (14)

s. c.

Nodes degrees

e∈δ(i)

xe = R if i ∈ {0, n + 1} 2

therwise.

, ∀i ∈ V (15) Capacity

e∈S

xe ≥ 2

i∈S qi

Q

,

∀S ⊆ V ′xe ∈ {0, 1}, ∀e ∈ E (16) Flow conservation

k∈V \{i}

fik −

k∈V \{i}

fki = n if i = 0 −1

therwise.

, ∀i ∈ V \ {n + 1} (17) No flow returning to the depot fi0 = fi,n+1 = fn+1,i = 0, ∀i ∈ V ′ (18) No flow on unused edge fie,je + fje,ie ≤ γexe, ∀e = (ie, je) ∈ E (19)

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 14/ 34) ODYSSEUS 2012 21/05/2012 14 / 34

SLIDE 15

Mathematical formulation proposed

Flow-based formulation

Advantage Flow variables are continuous variables The CVRP section of the model can benefit from valid cuts from the literature (capacity, comb, multistar, ...) Limitation The linear relaxation tends to produce fractional solutions that would satisfy the linking constraints at the smallest “cost" possible

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 15/ 34) ODYSSEUS 2012 21/05/2012 15 / 34

SLIDE 16

Valid Inequalities

1

Introduction

2

Literature review

3

Mathematical formulation proposed

4

Valid Inequalities Inequalities identified for the TRP and directly extended to CCVRP Inequalities for TRP not trivially extensible to the CCVRP

5

Extension of the flow-based model

6

First Results

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 16/ 34) ODYSSEUS 2012 21/05/2012 16 / 34

SLIDE 17

Valid Inequalities Inequalities from TRP to CCVRP

Inequalities identified for the TRP and directly extended to the CCVRP

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 17/ 34) ODYSSEUS 2012 21/05/2012 17 / 34

SLIDE 18

Valid Inequalities Inequalities from TRP to CCVRP

“Flow on edge (or with depot)" and “Flow out of depot"

TRP Version * fij + fji ≥ xe, ∀e = (i, j) ∈ E ′ (20) fij + fji ≥ 2xe − x(i,n+1) − x(j,n+1), ∀e = (i, j) ∈ E ′ (21) f0i ≥ αx0i ∀i ∈ V ′ (22) where α = minimum number of clients per route.

*conjectured to be facet-defining for n ≥ 4, n ≥ 5 and n ≥ 4 respectively

⇓ Idem CCVRP

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 18/ 34) ODYSSEUS 2012 21/05/2012 18 / 34

SLIDE 19

Valid Inequalities Inequalities from TRP to CCVRP

“Ending depot" and “Potential" constraints

TRP Version*

k∈V ′\{i}

fik ≥ 1 − xi,n+1 + (n − 2)x0i ∀i ∈ V ′ (23) (n − 2)x0i +

k∈V ′\{i}

(fik + fki) + (2n − 4)xin ≤ 2n − 3 ∀i ∈ V ′ (24)

*conjectured to be facet-defining for n ≥ 4

⇓ CCVRP Version

k∈V ′\{0,i}

fik ≥ 1 − xi,n+1 + f0i − 2x0i ∀i ∈ V ′ (25) f0i − 2x0i +

k∈V ′\{i}

(fik + fki) + 2(γi − 2)xin ≤ 2γi − 3 ∀i ∈ V ′ (26)

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 19/ 34) ODYSSEUS 2012 21/05/2012 19 / 34

SLIDE 20

Valid Inequalities Inequalities from TRP to CCVRP

Constraint “Total flow in graph"

e=(ij)∈E

fij + fji ≥ kR(k − 1) 2 + (k + 1)(n − kR) (27) with k = n

R

Remark :

In the case of a single vehicle (TRP), then R = 1 and k = n, meaning the right-hand-side becomes n(n−1)

2

, which is the exact total flow in a TRP

✛ ✚ ✘ ✙

a 4 (x 4) F b 5 (x 3) c 3 (x 2) d 4 (x 1) 2 (x 0)

Total flow = 4+3+2+1+0

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 20/ 34) ODYSSEUS 2012 21/05/2012 20 / 34

SLIDE 21

Valid Inequalities Inequalities for TRP not trivially extensible to the CCVRP

Inequalities identified for the TRP and not trivially extensible to the CCVRP

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 21/ 34) ODYSSEUS 2012 21/05/2012 21 / 34

SLIDE 22

Valid Inequalities Inequalities for TRP not trivially extensible to the CCVRP

Constraints per edge (and node) for the TRP

“Leaving flow from head" constraints fij + xin + x0j ≤

k∈V \{i,j,0,n+1}

fjk + xe ∀e = (i, j) ∈ E ′ (28) *conjecture : dim(F) = dim(P) - 2, for n ≥ 4 “Leaving flow from edge" constraints

e′∈δ({i,j})

xe′ + fij + fji ≤ 2 +

k∈V ′\{i,j}

(fik + fjk) + xe, ∀e = (i, j) ∈ E ′ (29) conjectured to be facet-defining for n ≥ 4 “Arc Vertex" constraints 2xjn + fij + fji ≤ f (δ(i) \ {(i, j), (j, i)}) + 2xij + (n − 2)x0i − 1 (30) conjecture : dim(F)=dim(P)-2

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 22/ 34) ODYSSEUS 2012 21/05/2012 22 / 34

SLIDE 23

Valid Inequalities Inequalities for TRP not trivially extensible to the CCVRP

Constraints per edge (and node) for the CCVRP (beta)

fij+xin + x0j ≤

k∈V \{i,j,0,n+1}

fjk + xe ∀e = (i, j) ∈ E ′ (31)

e′∈δ({i,j})

xe′ + fij + fji ≤ 2 +

k∈V ′\{i,j}

(fik + fjk) + xe, ∀e = (i, j) ∈ E ′ (32) 2xjn + fij + fji ≤ f (δ(i) \ {(i, j), (j, i)}) + 2xij + f0i − 2x0i − 1 (33)

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 23/ 34) ODYSSEUS 2012 21/05/2012 23 / 34

SLIDE 24

Valid Inequalities Inequalities for TRP not trivially extensible to the CCVRP

Constraints on subsets of nodes

“Vertex subset flow" cut* f (A(W )) ≤ |W |(|W | − 1) 2 + (|W | − 1)f (δout

E

) + (|W | + 1)(x(E(W )) − |W | + 1) (34) *conjectured to be facet-defining for 2 ≤ |W | ≤ n − 2 ⇓ f (A(W )) ≤ (|W | − k + 1)(|W | − k) 2 + (|W | − R)f (δout

E

) + (|W | + 3 − 2R)(x(E(W )) − |W | + 1) (35) where k = n

R

Separation procedure

TRP : Nodes in decreasing order of potential VirtualRoutes : Identify “routes" in fractional solutions with nodes potentials and flow StrongConnectedComponents

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 24/ 34) ODYSSEUS 2012 21/05/2012 24 / 34

SLIDE 25

Extension of the flow-based model

1

Introduction

2

Literature review

3

Mathematical formulation proposed

4

Valid Inequalities Inequalities identified for the TRP and directly extended to CCVRP Inequalities for TRP not trivially extensible to the CCVRP

5

Extension of the flow-based model

6

First Results

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 25/ 34) ODYSSEUS 2012 21/05/2012 25 / 34

SLIDE 26

Extension of the flow-based model

Illustration

Pn8-k2.vrp.ccvrp_bc:seqnum=-1: LB=224.125 / UB = 227.000 = 98.73 % 1(1) [p=5.239] 3.119 6(3) [p=8.761] 4.881 3(3) [p=1.785] 0.273 4(1) [p=2.966] 1.847 2(1) [p=6.069] 5(1) [p=1.335] 0.167 7(1) [p=3.692] 1.997 8(1) [p=2.153] 0.370 0.080 0.313 0.600 0.383 0.167 3.379 0.182 0.319 0.155 1.000 0.191 0.520 0.057

1(1) 4.000 6(3) 4.000 4(1) 3.000 2(1) 7(1) 2.000 3(3) 8(1) 1.000 2.000 5(1) 3.000 1.000

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 26/ 34) ODYSSEUS 2012 21/05/2012 26 / 34

SLIDE 27

Extension of the flow-based model

Principle

Restricting possible potentials would strengthen the linear relaxation.

Challenges (for CCVRP) unknown configuration knowing the rank of a node do not provide its potential

1(1) 2.000 3(3) 4.000 6(3) 3.000 4(1) 1.000 2(1) 7(1) 1.000 9(1) 3.000 5(1) 8(1) 1.000 2.000 2.000

Ideas predefined possible “potential" backward ranking

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 27/ 34) ODYSSEUS 2012 21/05/2012 27 / 34

SLIDE 28

Extension of the flow-based model

Principle

Restricting possible potentials would strengthen the linear relaxation.

New continuous variables Ui

p ∈ [0, 1] with p ∈ {1, ..., γ} and i ∈ V ′.

equal to 1 if the total flow entering node i is p, and 0 otherwise. also corresponds to the backward position of node i in its (unknown) route + restricts the flow and therefore strengthens the linear relaxation — More variables in the model

1(1) 2.000 3(3) 4.000 6(3) 3.000 4(1) 1.000 2(1) 7(1) 1.000 9(1) 3.000 5(1) 8(1) 1.000 2.000 2.000 Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 28/ 34) ODYSSEUS 2012 21/05/2012 28 / 34

SLIDE 29

Extension of the flow-based model

Resulting Extended Mathematical model

Flow-based formulation + Valid equalities

Each client should have one level of flow : γ

p=1 Ui p = 1, ∀i ∈ V ′

Number of levels = number of clients : n

i=1

γ

f =1 Ui f = n

Flow level of 1 : n

i=1 Ui 1 = min{R, n}

No flow level handle than the client can handle :

p=Γi +1 Ui p = 0, ∀i ∈ V ′

Link flow variables and flow level variables : f0,i +

k∈V ′\{i} fk,i − f Γ p=1 Ui p = 0, ∀i ∈ V ′

Link level 1 with Final depot : Ui

f − Xe=(i,n+1) = 0

+ Valid inequality

Relation between two flow levels : n

i=1 Ui p+1 − n i=1 Ui f ≤ 0, ∀p ∈ [1...γ − 1]

+ Valid cuts

For any subset of clients W ⊂ V ′, and for any level f :

i∈W Ui f ≤ |W | − (i,j)∈W Xe=(i,j) Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 29/ 34) ODYSSEUS 2012 21/05/2012 29 / 34

SLIDE 30

First Results

1

Introduction

2

Literature review

3

Mathematical formulation proposed

4

Valid Inequalities Inequalities identified for the TRP and directly extended to CCVRP Inequalities for TRP not trivially extensible to the CCVRP

5

Extension of the flow-based model

6

First Results

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 30/ 34) ODYSSEUS 2012 21/05/2012 30 / 34

SLIDE 31

First Results

Implementation et Instances

Implementation C + CVRPSEP + Cplex Intel Core 2Duo, 2.66 GHz 4GB of RAM Instances : TRP : TSPLIB instances used by Ezzine et al, and Mendez-Diaz et al CCVRP : Augerat : A, B, P (smallest and largest instance) Table Headings (FF) Basic Flow-based formulation (FFI) = (FF) + Strengthening inequalities (FFIC) = (FFI) + Identification and addition of violated cuts (FFICU) = FFIC + additional variables Ui

p Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 31/ 34) ODYSSEUS 2012 21/05/2012 31 / 34

SLIDE 32

First Results

First Results on the TRP

Cutting plane

instance (FF) (FFI) (FFIC) (FFICU) gr17 40.31 % 3 s 80.33 % < 1 s 92.32 % 1 s 92.38 % 2 s gr21 31.81 % 4 s 87.97 % 1 s 91.73 % 2 s 91.80% 5 s gr24 21.16 % 7 s 90.28% 2 s 94.41 % 5 s 82.80% 12 s fri26 39.98 % 5 s 86.47% 3 s 96.15 % 8 s 96.18% 31 s

Branch-and-Cut

instance (1) (2) (FFIC) (FFICU) gr17 10 s 22 s 10 s 16 s gr21 70 s 23 s 23 s 59 s gr24 553 s 18 s 35 s 172 s fri26 257 s 294 s 67 s 263 s

(1) Ezzine, Semet et Chabchoub, MOSIM 2010 : 2.4 GHz and 1 Go RAM, Cplex 11.0 (2) Mendez-Diaz, Zabala, Lucena, Discrete Applied Math., 2008 : Cplex 8.0 (FF***) Our algorithm : 2.66 GHz and 4GB of RAM, Cplex 12.0

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 32/ 34) ODYSSEUS 2012 21/05/2012 32 / 34

SLIDE 33

First Results

First Results on the CCVRP

Polynomial lower bounds

type taille BI2 BI1 BI2+ BI1+ A 32-k5 41.56 85.40 50.37 86.45 80-k10 34.79 61.41 40.77 94.33 B 31-k5 96.12 7.76 20.49 20.49 78-k10 92.61 15.07 25.54 92.87 P 19-k2 56.54 72.44 75.03 58.78 76-k5 47.04 79.87 84.62 47.09

Cutting Plane

instance (FF) (FFI) (FFIC) (FFICU) A 32-k5 85.78 % 2 s 93.51 % 2 s 95.36 % 3 s 96.43 % 4 s 80-k10 93.81 % 25 s 95.97 % 66 s 97.45 % 541 s 97.78 % 332 s B 31-k5 96.46 % 2 s 98.63 % 2s 98.80 % 3 s 98.89 % 3 s 78-k10 92.79 % 26 s 95.92 % 103 s 96.59 % 749 s 96.71 % 400 s P 19-k2 61.81 % 11 s 92.87% < 1 s 93.69 % 2 s 93.99 % 1 s 76-k5 47.13 % 26 s 85.15 % 142 s 91.36% 906 s 92.89 % 336 s

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 33/ 34) ODYSSEUS 2012 21/05/2012 33 / 34

SLIDE 34

First Results

Flow-based Mathematical Formulation and Strengthening Cuts for the Cumulative CVRP

Sandra Ulrich NGUEVEU / Mathieu LACROIX

LAAS-CNRS / LIPN ngueveu@laas.fr

21/05/2012

Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 34/ 34) ODYSSEUS 2012 21/05/2012 34 / 34