Vehicle Routing and MIP

Transcription

1 CORE, Université Catholique de Louvain 5th Porto Meeting on Mathematics for Industry, 11th April 2014

2 Contents: The Capacitated Vehicle Routing Problem Subproblems: Trees and the TSP CVRP Cutting Planes Column Generation Robust Branch-and-Price-and Cut An Inventory Routing Problem

3 The Capacitated Vehicle Routing Problem

4 Single Depot. Single Period K vehicles can make tours beginning and ending at the depot Client i has a demand q i and is served by exactly one vehicle Each vehicle has capacity Q. The objective is to find subtours for the K vehicles such that the amount delivered by each vehicle does not exceed Q, the demand of each client is satisfied and the total travel cost/distance is minimized.

5 Modelling Connectivity: Spanning Trees Graph G = (V, E) with edge costs c e for e E. Let y e = 1 if e is in the tree. min e E c ey e e E(S) y e S 1 S V e E y e = V 1 y R E + where E(S) = {e = (i, j) E : i, j S}. This linear program is tight. The subtour elimination constraints (SECs) describe the convex hull of the spanning trees, but there is an exponential number of constraints.

6 Separation of SECs Given a point ŷ R E +, when does it violate one of the SECs? We formulate this question as an IP. Let α i = 1 if i S. We have to check whether there exists α {0, 1} V such that e=(i,j) E ŷ e α i α j i V α i > 1. This is a quadratic 0-1 integer program of the form: max α T Qα pα. α {0,1} V It is well known that such IPs can be solved as max flow/min cut problems when Q 0.

7 An Extended Multicommodity" Formulation How else can one ensure connectivity? One way: Construct a graph in which there is a path (one can send a flow) from one node (r = 1) to all the others. f k ij is the flow (commodity k) in arc (i, j) on the path from r to k. j j f k ij j f k ij j f k ji = 0 i k, r f k ji = 1 i = k f k ij + f k ji y e e E f k ij 0 i, j, k This formulation is as strong as that with subtours, but it has O(V 3 ) variables.

8 The Traveling Salesman Problem: Formulation 1 Symmetric Version c ij = c ji = c e y e = 1 if edge e in the tour min e E c ey e e δ(v) y e = 2 i V e E(S) y e S 1 S V y [0, 1] E SECS can be replaced by cut" inequalities y e 2 S V. e δ(s)

9 The Traveling Salesman Problem: Formulation 2 Weak Extended Formulation (Directed): u i is position of node i in the tour starting with node 1 in position 0 (u 1 = 0). u j u i x ij (n 2)(1 x ij ) (i, j), j 1 x ij = 1 i j x ij = 1 j i y e = x ij + x ji e x Z A +, u Zn + Useful if one wants a compact valid formulation.

10 Formulations of CVRP xij k zi k = 1 if arc (i, j) in tour of vehicle k = 1if client i visited by vehicle k min cij k xk ij k ij xij k = zj k k, j, j 0, i xij k = zi k k, i, i 0, j x0j k = z0 k k j d i zi k Cz0 k k i zi k = 1 i k x k ij, zk i {0, 1}

11 Formulations of CVRP with 2 Indices ij δ( S,S) x ij min c ij x ij ij x ij = 1 j 0, i x ij = 1 i 0, j x 0j = K j i S d i S V C x ij {0, 1}

12 Valid Inequalities and Separation 1. Separation of ij δ( S,S) x i S ij d i Separation of ij δ( S,S) x ij 2. Generalized Large Multistar e δ(s) i S d i C C? y e 2 ( d(s)+ d j ( C j/ S is polynomial. e δ(s:{j}) y e ) )

13 Column Generation A natural idea is to have columns corresponding to feasible subtours for a vehicle. The problem is then just to select a set of K subtours such that each client is visited once. The trouble with this is that the column generation subproblem is a prize-collecting traveling salesman problem that is almost as hard as the original problem. So the typical approach is to look at a larger set of columns that includes all the feasible subtours, but for which the column generation problem is more tractable.

14 q-routes A q-route is a walk starting and ending at the depot visiting the clients 0,1, or more times but with total demand at most C. Note that each time a client i is visited, his demand d i is counted again. Suppose n = 4, C = 13 and d = (2, 4, 5, 7). A feasible subtour is for example with i d i = 9. A q-route, that is not a feasible subtour, is with i d i = is a q-route without a 2-cycle as i d i = 13 C. A minimum cost q-route without 2-cycles can be found by dynamic programming.

15 Naive DP for q-routes f(i, j, k) is the min cost of a walk starting at the depot and ending with visits to i, then j in which k units are delivered. Min Cost q-route is f(i, j, k) = min p:p j [f(p, i, k d j)+ c ij ] min [f(i, j, k) + c 0j]. i,j,k C

16 Column Generation with q-routes without 2-cycles Let qj e be the coefficient of edge e in q-route j with variable λ j. One has p j=1 qe j λ j x e = 0 e E p j=1 λ j = K e δ(i) x e = 2 i V x e 0 e E 0 j = 1,...,p λ j

17 The Master Problem e δ(i) x e = 2 i V e δ(0) x e = 2K e δ(s) x e 2k(S) S V x e 1 e E \δ(0) p j=1 qe j λ j x e = 0 e E p j=1 λ j = K x e 0 e E λ j 0 j = 1,...,p

18 The Master Problem p min l eqj e λ j j=1 e E p µ qj e λ j j=1 e δ(i) p ν qj e λ j j=1 e δ(0) p π qj e λ j j=1 e δ(s) p ω qj e λ j j=1 λ j = 2 i V = 2K 2k(S) S V 1 e E \δ(0) 0 j = 1,...,p Reduced cost of x e: l e µ i µ j e δ(s) π S ω e e E \δ(0) l e µ j ν e δ(s) π S e δ(0)

19 d = (2, 4, 5), C = 10 w 1 w 2 w 3 w 12 u cw = v = = 2 11 v = = v = 3 2 = 2 10 v = = 4 1, , , , 2, e =

20 min cost q-route: with reduced cost of -10 alternative q-route: with reduced cost of -7 w 1 w 2 w 3 w 12 w 23 u cw = v = = v = = v = = v = = , , , , 2, e = e =

21 Recent Developments Recently q-routes replaced by ng-routes Note that can add constraints α e x e α 0 and approach still works. Robust branch-and-cut-and-price. If cuts on the q-route variables - column generation becomes more difficult What about time-windows? Similar BCP approach. Column generation may be easier - less choice. Cut generation more difficult - cuts based on infeasibility.

22 The Inventory Routing Problem: One Period

23 The Inventory Routing Problem

24 Model 1: Single Period Inventory Routing Problem x i Cz 0, x i W i z i, z 0 z i, i I i I y ij = z i, j I0, y ij = j I0 j I0 y ij z i, S I, i S {0} j I\S Capacitated Subtours x, R+, T z i {0, 1} i I, z 0 {0, 1,...,K}, y ij {0, 1} A

25 First Observations Where do the capacities come from? W i t = min{c, d i t + S i } What about the capacitated subtours? C i I0\S j S where S I and I0 = I {0}. y ij i S x i

26 3-index formulation Direct a flow of x k from the origin to each client k I: f ijk is the flow i (i, j) with destination k I f ijk f jik = 0 j k, k i I0 i I f ijk = x k k = j, k i I0 f ijk Cy ij (i, j) k I f ijk W k y ij i, j, k

27 Metric Inequalities Suppose µ ij 0 for all (i, j) A and S I, then (i,j) Aµ ij y ij i S is a valid inequality if µ ij min(c, (i,j) P x i i v(p) S W i ) for every subtour P beginning and ending at the depot, where v(p) I are the clients on subtour P.

28 A Particular Metric Inequality S 1, S 2 is a partition of S Cy ij + W j y ij + (C W i )y ij (I0\S,S 1 ) + (I0\S,S 2 ) (S 2,S 1 ) x i (S 2,S 2 ) min(c W i, W j )y ij i S

29 Example of a Particular Metric Inequality (I0\S,S 1 ) + Cy ij + (I0\S,S 2 ) W j y ij + (S 2,S 1 ) (S 2,S 2 ) min(c W i, W j )y ij i S I = 3, C = 300, W = (289, 123, 76), S = {1, 2, 3}, S 1 = {1}, S 2 = {2, 3} x i (C W i )y ij 300y y y 03 +( )y 21 +(300 76)y 31 + min(76, )y 23 + min(123, )y 32 x 1 + x 2 + x 3

30 Multi-Period Metric Inequalities with Stock Upper Bounds: An Example Add st 1 i + x t i = Dt i + si t and IRP model for periods 1,...,T. C t u=k (i,j) (I0\S,S) y ij u t xt i = i S i S u=k t u=k x i u i S(D i kt si k 1 ) With s i k 1 Ui, above is of the form: This gives the inequalities: σ + Cv b, 0 σ h, v Z. v b h C and σ +ρv ρ b C.

31 Table: VMIRP-SUB: computational results for the instances with n = 50 and T max = 6 BC C&L No LB Ini LB LS LB Cut BUB Nodes Time BLB BUB Time l l l l l l x l l l l h h h h h h h h h h

32 Table: Computational results for the instances 1n15T6 with n = 15 and T max = 6 BC C&L k Subtour LS Prop1 Gen Cover BLB BUB l l l l h h h h

33 Moral The MIP systems are amazing, but sometimes one can still help. Thank you for your attention

34 CVRP IRP 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):351Ð385, R. Baldacci, A. Mingozzi, and R. Roberti. New route relaxation and pricing strategies for the vehicle routing problem. Operations Research, 59(5): , R. Fukasawa, H. Longo, J. Lysgaard, M. Poggi de Arag?ao, M. Reis, E. Uchoa, and R.F. Werneck. Robust branch-and-cut-and-price for the capacitated vehicle routing problem. Mathematical Programming, 106(3):491Ð511, Improved Branch-Cut-and-Price for Capacitated Vehicle Routing D. Pecin, A. Pessoa, M. Poggi and E. Uchoa, IPCO 2014 (to appear) C. Archetti, L. Bertazzi, G. Laporte, and M.G. Speranza. A branch-and-cut algorithm for a vendor-managed inventory-routing problem. Transportation Science, 41(3): , P. Avella, M. Boccia, and L.A. Wolsey. Single-item reformulations for a vendor managed inventory routing problem: computational experience with benchmark instances. CORE Discussion Paper L.C. Coelho and G. Laporte. The exact solution of several classes of inventory-routing problems. Computers and Operations Research, 40(2): , 2013.