Exact and Heuristic Algorithms for the Symmetric and Asymmetric Vehicle Routing Problem with Backhauls

Transcription

1 Exact and Heuristic Algorithms for the Symmetric and Asymmetric Vehicle Routing Problem with Backhauls Paolo Toth, Daniele Vigo ECCO IX - Dublin 1996

2 Exact and Heuristic Algorithms for VRPB 1 Vehicle Routing Problem with Backhauls Extension of classical Capacitated VRP Customer set partitioned in two subsets: Linehaul (Delivery points) Backhaul (Pick-up points) Ex. Grocery industry: Linehauls: shops, supermarkets Backhauls: suppliers SAVING that can be achieved by allowing empty vehicles to pick-up inbound products while returning to the depot $ 160 millions saved per year by Grocery industry since 1982 Operational constraints: Linehauls normally have higher priority; Rearloaded vehicles: on-board load rearrangement is often impossible for each vehicle all deliveries must precede all pickups.

3 Exact and Heuristic Algorithms for VRPB 2 Graph Theory Model complete undirected graph G = (V 0, A ) n Linehaul vertices, i = 1,..., n (set L) m Backhaul vertices, i = n + 1,..., n + m (set B) single depot (i = 0) demand d i > 0, i = 1,..., n + m (d 0 = 0) K identical vehicles, each with capacity D, located at the depot, that can execute at most one route symmetric or asymmetric traveling cost matrix c (c ii = + ) K L, K B minimum number of vehicles needed to serve all Linehaul and all Backhaul customers, respectively (BPP) w.l.o.g. we assume that K L = max{k L, K B } VRPB : find a min-cost collection of K vehicle routes, such that: i) each route starts and ends at the depot; ii) each point is visited by exactly one route; iii) the total Linehaul and Backhaul loads of a route do not exceed, separately, vehicle capacity D; iv) in each route the Backhaul customers, if any, are visited after all Linehaul customers; constr. iv) a) mixed routes are oriented b) no routes with only Backhauls

4 Exact and Heuristic Algorithms for VRPB 3 VRPB is NP-hard in the strong sense Previous work on VRPB: Exact Algorithms: Yano et al. (Interfaces, 1987) set-covering based; at most 4 vertices of each type in a route. Heuristic Algorithms: I. Deif, L. Bodin (Babson Conference Proceedings, 1984) extension of Clarke-Wright heuristic M. Goetschalckx, C. Jacobs-Blecha (EJOR, 1989) heuristic based on Spacefilling Curves extension of Fisher-Jaikumar heuristic (1993) G. Laporte, M. Gendreau, A. Hertz (TRISTAN II confer., 1994) Tabu Search heuristic

5 Exact and Heuristic Algorithms for VRPB 4 Integer Linear Programming Model directed graph G(V 0, A); V 0 = {0} {1,..., n} {n + 1,... n + m} = {0} L B L 0 = {0} L, B 0 = {0} B, V = V 0 \ {0}; L, B set of all subsets of L 0 and B 0 ; F = L B; A = {(i, j) : i L 0, j L} }{{} A 1 {(i, j) : i B, j B 0 } {(i, j) : i L, j B }{{} 0 } }{{} A 2 c ij = c ji for each i, j L 0 and i, j B. A 3 ; A 1 A 3 A 2 Γ + i and Γ i forward and bacward star of i σ(s) min. number of vehicles needed to serve vertex set S (σ(l) = K L K B = σ(b)) solution of the associated BPP σ(s) j S d j /D

6 Exact and Heuristic Algorithms for VRPB 5 x ij = 1 if arc (i, j) A in the optimal solution 0 otherwise subject to (P ) v(p ) = min (i,j) A c ij x ij (1) i Γ j j Γ + i i Γ 0 j Γ + 0 x ij = 1 for each j V (2) x ij = 1 for each i V (3) x i0 = K (4) x 0j = K (5) i S j S j Γ + i \S x ij σ(s) for all S F (6) i Γ j \S x ij σ(s) for all S F (7) x ij {0, 1} for each (i, j) A; (8) Because of degree constraints (2) and (3) we have i S j Γ + i \S x ij = j S i Γ j \S x ij for all S : {0} / S

7 Exact and Heuristic Algorithms for VRPB 6 Relaxation Based on Projection L 6 B We separately consider A 1, A 2, and A 3 P 1 : min-cost collection of K Capacitated Simple Paths rooted in 0, and spanning L 0 ; K B K K M = min{k, m} P 2 (K): min-cost collection of K Capacitated Simple Paths ending in 0, and spanning B 0 ; P 3 (K): min-cost collection of K disjoint arcs from L to B 0 : K K arcs connect L to the depot; K arcs connect L to B.

8 Exact and Heuristic Algorithms for VRPB 7 Problems P 1 and P 2 are NP-hard : P 1 : relax the outdegree constraints; P 2 : relax the indegree constraints; relax the capacity constraints. R 1 : Shortest Spanning Arborescence over L 0, rooted in 0 and with outdegree K in vertex 0; R 2 (K): Shortest Spanning anti-arborescence over B 0, rooted in 0 and with indegree K in vertex 0; L = v(r 1 ) + min {v(r 2 (K)) + v(p 3 (K))} K B K K M R 1 and R 2 (K) can be solved in O(n 2 ) and O(m 2 ) time, (see, e.g., Gabow and Tarjan (1984), Toth and Vigo (1994)) P 3 (K) is solved (in O(K (n + m) 2 ) time) by tranforming it into a min-cost flow problem By using parametric techniques the computation of: min K v(r 2 (K)) is done in O(m 2 ) time; min K v(p 3 (K)) is done in O(K L (m + n) 2 ) time. AP Relaxation Obtained by removing the capacity-cut costraints solution of a TP (or and AP on an extended cost matrix)

9 Exact and Heuristic Algorithms for VRPB 8 Lagrangian Lower Bound 1) Capacity-cut constr. can be weakened so that to obtain the connectivity constraints: i S j S j Γ + i \S x ij 1 for each S L (9) i Γ j \S x ij 1 for each S B (10) 2) Capacity-cut constr. can be imposed only for a given families F 1 L, F 2 B of subsets: (π) (ρ) i S h j Γ + i \S h j T h i Γ j \T h x ij σ(s h ) for each S h F 1, x ij σ(t h ) for each T h F 2. 3) Introduce degree constraints and the two families of Capacity-cut constraints into the objective function (µ) (λ) i Γ j j Γ + i x ij = 1 for each j V x ij = 1 for each i V Lagrangian dual problem: v(l(λ, µ, π, ρ )) = max {v(l(λ, µ, π, ρ))} λ,µ,π,ρ

10 Exact and Heuristic Algorithms for VRPB 9 LR(λ, µ, π, ρ) := v(r 1 ) + min K B K K M (R 1 ) v(r 1 ) = min { } v(r 2 (K)) + v(p 3 (K)) (i,j) A 1 c ij x ij (11) x ij = 1 for each j L, (12) i Γ j L 0 j L j S x 0j = K, (13) i Γ j \S x ij 1 for each S L, (14) x ij {0, 1} for each (i, j) A 1. (15) (R 2 (K)) v(r 2 (K)) = min (i,j) A 2 c ij x ij (16) x ij = 1 for each i B, (17) j Γ + i B 0 i B i S x i0 = K, (18) j Γ + i \S x ij 1 for each S B, (19) x ij {0, 1} for each (i, j) A 2. (20) (P 3 (K)) v(p 3 (K)) = min (i,j) A 3 c ij x ij (21) x ij 1 for each i L, (22) j Γ + i \L x ij 1 for each j B, (23) i Γ j \B x ij = K, (24) i L j Γ + i \L x ij = K, (25) j B i Γ j \B x ij {0, 1} for each (i, j) A 3 ; (26)

11 Exact and Heuristic Algorithms for VRPB 10 Overall lower boundig procedure Procedure analogous to Cutting-plane approach: Iterative strengthening of the Lagrangian relaxation by separating new valid inequalities associated with the capacity-cut constraints, and adding them to set F, i.e. to the Lagrangian problem (initially F 1 = F 2 = ) violated capacity-cut constraints are separated by considering the optimal solution of the current Lagrangian problem; the infeasible subsets found are added to sets F 1 and F 2, and the associated Lagrangian multipliers are initialized; the current set of Lagrangian multipliers is optimized through a two level subgradient optimization procedure; outer level: updating of multipliers π and ρ; inner level: subgradient optimization only for µ and λ (with π and ρ fixed). the process is iterated until no infeasible subset is found or a given number of iterations has been performed. Lagrangian and AP Lower bounds are combined in an overall Additive LB

12 Exact and Heuristic Algorithms for VRPB 11 Separation of violated capacity-cuts New valid inequalities are obtained by examining the optimal solution of the current Lagrangian problem (x): Family F 1 : for each j L 0 we determine X j := subset of all the Linehaul vertices belonging to the sub-arborescence in x rooted at j, w j := total demand of subset X j. if w j > D the capacity-cut for X j is violated (σ(x j ) > 1 and only one arc in x enters subset X j ) σ(x j ) computed with BPP add subset X j to family F 1. D = 17 X j (w j = 19) j 4 2 h X h (w h = 25) The computation of all w j, j L 0 (resp. B 0 ), can be performed in O(n) (resp. O(m)) time

13 Exact and Heuristic Algorithms for VRPB 12 Lagrangian Heuristic TV Solutions obtained through Lagrangian relaxation are often almost feasible, and contain an high degree of information on the optimal solution structure. The algorithm is divided in 3 phases 1. CLUSTERING Obtain K Linehaul clusters and K Backhaul clusters starting from the solution of the Lagrangian lower bound and by removing: (K K) interface arcs between Linehauls and Backhauls; (K + K) arcs incident into the depot; (a) (b)

14 Exact and Heuristic Algorithms for VRPB MATCHING K Linehaul clusters are associated with the K Backhaul clust. K K Linehaul clusters are associated with the depot Assignment Problem with resp. to the K K cost matrix γ γ pq = estimate of the cost incurred by serving Linehaul customers of cluster p and Backhaul customers of cluster q (or the depot) in the same route PIVOT customers inter-distance average inter-cluster distance modified TSP heuristic solution value 3. ROUTING and REFINING Customers in each route-cluster are sequenced using a modified TSP heuristic, plus 2/3 - opt intra-route refining step. (routes can be infeasible with respect to the capacity constraints) Inter-route arc and node exchanges to obtain a feasible solution and to improve the current routes 2/3-opt intra-route refinement on the final routes

15 Exact and Heuristic Algorithms for VRPB 14 Branch and Bound Algorithm 1. Bounding At each node of the branching tree the Lagrangian LB is computed. The number of iterations decreases at lower levels of the tree. 2. Branching Branching scheme: adaptation of the Subtour elimination scheme Best-Bound-First search Performances are improved by using at each node: reduction and dominance procedures; Lagrangian Heuristic (TV) for the updating of the incumbent solution;

16 Exact and Heuristic Algorithms for VRPB 15 Computational Results Branch and Bound and TV algorithms coded in FORTRAN and tested on a Pentium 60 Mhz (5.3 Mflops) Class I: (Goetschalckx and Jacobs-Blecha, 1989) vertices coordinates uniformly distributed: x [0, 24000] y [0, 32000] depot in (12000, 16000) demands generated from a normal distribution with mean = 500, and standard deviation = 200; n + m = {25,..., 150} % of Linehauls = {50, 66, 80} vehicle capacity such that approx. K = {3, 5, 7, 10} Problem data given by Goetschalckx and Jacobs-Blecha; Initial UB: Lagrangian Upper Bound (TV).

17 Exact and Heuristic Algorithms for VRPB 16 Class I: time limit of 6000 seconds Name n + m n m K K L K B LR 0 % LD 0 % UB 0 % time nodes A A A A B B B C C C C D D * 2939 D * 2770 D E E E F * 1593 F F F G * 1013 G * 1199 G G * 1303 G G H H H H H H

18 Exact and Heuristic Algorithms for VRPB 17 Class II: VRPB test problems obtained from VRP instances from the literature (TSPLIB) (N CV RP = 21 75) π = n/(n + m) {0.50, 0.66, 0.80} n := πn CV RP, m := N CV RP n vertex set is partitioned by defining as Backhaul one vertex in every two, three or five. K L and K B are determined by solving the associated Bin Packing Problem (using the code MTP by Martello and Toth, 1990) Name n + m n m K L K B LR 0 % LD 0 % UB 0 % nodes time eil eil eil eil eil eil eil eil eil eil eil eil eil eil eil

19 Exact and Heuristic Algorithms for VRPB 18 Class III: Asymmetric VRPB test problems obtained from ACVRP instances (Fischetti, T., V. 1995) (TSPLIB) (N CV RP = 33 70) π = n/(n + m) {0.50, 0.66, 0.80} n := πn ACV RP, m := N ACV RP n vertex set is partitioned by defining as in Class II K L and K B are determined by solving the associated Bin Packing Problem (using the code MTP by Martello and Toth, 1990) Name n + m n m K L K B LR 0 % LD 0 % UB 0 % nodes time FTV FTV FTV FTV FTV FTV FTV FTV FTV FTV FTV FTV FTV FTV FTV FTV FTV FTV FTV FTV FTV * 3652 FTV * 2446 FTV FTV

20 Exact and Heuristic Algorithms for VRPB 19 Heuristic algorithm TV Class I: Times in IBM 386/20 seconds DB SF LHBH TV Name n m K K L K B % dev time % dev time % dev time % dev time A A A A B B B C C C C D D D D E E E F F F F G G G G G G H H H H H H I I I I I J J J J K K K K L L L L L M M M M N N N N N N

21 Exact and Heuristic Algorithms for VRPB 20 Class II: Times in IBM 486/33 seconds DB SF TV Name n m K K L K B % dev time % dev time % dev time eil eil (4) (4) eil (4) eil (3) eil (3) eil eil (3) eil eil (4) eil eil (4) eil (4) (4) eil (4) eil eil (5) (5) eila eila eila (9) (9) eilb (9) (9) eilb (11) eilb eilc eilc eilc eild eild eild eila (5) (5) eila eila (7) (7) eilb (8) (8) eilb (10) (10) eilb (12) (12)