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
Plan Introduction 1 Literature review 2 Mathematical formulation proposed 3 Valid Inequalities 4 Extension of the flow-based model 5 First Results 6 Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 2/ 34) ODYSSEUS 2012 21/05/2012 2 / 34
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
Literature review Introduction 1 Literature review 2 Mathematical formulation proposed 3 Valid Inequalities 4 Inequalities identified for the TRP and directly extended to CCVRP Inequalities for TRP not trivially extensible to the CCVRP Extension of the flow-based model 5 First Results 6 Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 4/ 34) ODYSSEUS 2012 21/05/2012 4 / 34
Literature review 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
Literature review Literature review : other expression of the objective-function � Total cost for solution S = c e ∗ coef e e ∈ E where coef e = number of clients following edge e in solution S . 4 5 3 4 2 0 a b c d F Sum : arrival times = t a + t b + t c + t d = 4 + 9 + 12 + 16 = 41 4 (x 4) 5 (x 3) 3 (x 2) 4 (x 1) 2 (x 0) 0 a b c d F 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
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
Literature review Literature review : mono-vehicle version L. Bianco, A. Mingozzi et S. Ricciardelli, Networks, 1993 x k 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 ( x i 1 : v i ∈ V ) : = is a permutation of (1, 0, ..., 0) x ij : ( v i , v j ) ∈ 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 x ij : binary : = 1 iff arc ( i , j ) is crossed p ij : binary : = 1 iff node j is the i th node visited g ij : 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
Literature review Literature review : mono-vehicle version I. Ome Ezzine, F. Semet et H. Chabchoub, MOSIM 2010 X ij : binary : = 1 iff arc ( i , j ) is used Z ij : 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 x ij : 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
Mathematical formulation proposed Introduction 1 Literature review 2 Mathematical formulation proposed 3 Valid Inequalities 4 Inequalities identified for the TRP and directly extended to CCVRP Inequalities for TRP not trivially extensible to the CCVRP Extension of the flow-based model 5 First Results 6 Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 10/ 34) ODYSSEUS 2012 21/05/2012 10 / 34
Mathematical formulation proposed Extension of the polynomial lower bound 1 Hypothesis : High number of vehicles (ex : R = n − 1) Exemple si R = n − 1 � � � min c ( 0 , i ) x i + 2 c ( 0 , i ) y i + c e r e (1) i ∈ V ′ i ∈ V ′ e ∈ E ′ s.t. ∀ i ∈ V ′ x i + y i + z i = 1 , (2) � x i = n − 2 (3) i ∈ V ′ � � � y i = z i = r e = 1 (4) i ∈ V ′ i ∈ V ′ e ∈ E ′ r e ≤ y i + y j + z i + z j ∀ e = ( i , j ) ∈ E ′ , (5) 2 Figure: BI1 si R = n ∀ i ∈ V ′ x i , y i , z i binaires (6) r e 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
Mathematical formulation proposed Extension of polynomial lower bound 2 Hypothesis : Balanced solution (in terms of number of customers) Exemple si n = 13 et R = 4 4 � � c e . p . x p ( P ) (8) e p = 1 e ∈ E � 1 s.t. si p = 4 � x p e = , ∀ p ∈ [ 1 ... 4 ] 2 sinon. e ∈ E (9) � 2 R si i = 0 � x p e + y e = ∀ i ∈ V (10) , 2 sinon. Figure: BI2 si e ∈ δ ( i ) n = 13 , R = 4 � y e = R (11) e ∈ E � x p e + y e < = 1 (12) p ∈ [ 1 ... 4 ] x p e , y e 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
Mathematical formulation proposed Flow-based formulation Decision variables x e : binary : = 1 if edge e is used and 0 otherwise f ij : continuous, positive := flow going from i to j on edge e coefficient assigned to edge e in the objective-function Constant γ i : max number of clients of a route containing client i γ e = min { γ i e , γ j e } , ∀ e = ( i e , j e ) ∈ E Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 13/ 34) ODYSSEUS 2012 21/05/2012 13 / 34
Mathematical formulation proposed Flow-based formulation Minimizes Total Cost = sum of arrival times � � ( FF ) min F = c ij f ij (14) i ∈ V \{ n + 1 } j ∈ V ′ \{ i } s. c. Nodes degrees � R if i ∈ { 0 , n + 1 } � x e = , ∀ i ∈ V (15) 2 otherwise. e ∈ δ ( i ) Capacity � � i ∈ S q i � � ∀ S ⊆ V ′ x e ∈ { 0 , 1 } , ∀ e ∈ E x e ≥ 2 , (16) Q e ∈ S Flow conservation � n if i = 0 � � f ik − f ki = , ∀ i ∈ V \ { n + 1 } (17) − 1 otherwise. k ∈ V \{ i } k ∈ V \{ i } No flow returning to the depot ∀ i ∈ V ′ f i 0 = f i , n + 1 = f n + 1 , i = 0 , (18) No flow on unused edge f i e , j e + f j e , i e ≤ γ e x e , ∀ e = ( i e , j e ) ∈ E (19) Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 14/ 34) ODYSSEUS 2012 21/05/2012 14 / 34
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
Valid Inequalities Introduction 1 Literature review 2 Mathematical formulation proposed 3 Valid Inequalities 4 Inequalities identified for the TRP and directly extended to CCVRP Inequalities for TRP not trivially extensible to the CCVRP Extension of the flow-based model 5 First Results 6 Sandra U. Ngueveu ROADEF2012 (MOGISA - LAAS - CNRS 16/ 34) ODYSSEUS 2012 21/05/2012 16 / 34
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
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