The Traveling Salesman Problem with Pickup and Delivery. A polyhedral approach. IBM Research - Australia. Irina Dumitrescu

Transcription

1 Australia The Traveling Salesman Problem with Pickup and Delivery A polyhedral approach Irina Dumitrescu Jean-Francois Cordeau, Gilbert Laporte, Stefan Ropke

2 The TSP with Pickup and Delivery (TSPPD) Given: one vehicle one depot n customers (n commodities) an explicit pickup and delivery request for each customer (no pickup location can be visited before its corresponding delivery location) Goal: Find the shortest vehicle route that starts and ends at the depot and satisfies all customer requests. Applications: courier services, dial-a-ride systems. IBM Research- Australia

3 Notation and Examples Notation: G=(V,E) Depot: 0 and 2n+1 i {1,,n} a pickup location & n+i its corresponding delivery location x i,j binary variable corresponding to edge (i, j) Examples:

4 The TSPPD Polytope R E = R 2n2 +n+1 TSPPD polytope = set of points in R E that satisfy: degree constraints TSP subtour elimination constraints precedence constraints x 0,2n+1 = 1 TSP

5 The TSPPD Polytope R E = R 2n2 +n+1 TSPPD polytope = set of points in R E that satisfy: degree constraints TSP subtour elimination constraints precedence constraints x 0,2n+1 = 1 TSP dim(p TSPPD ) = 2n 2 -n-2, n 2

6 Specific TSPPD Constraints: Precedence Let U V: 0 U, 2n+1 U i U s.t. n+i U Precedence constraints (Ruland 1995): x([u: U]) 4 If! i U s.t. n+i U, then the precedence constraints are facets.

7 Valid Inequalities: Order Constraints Let i, j pickup nodes Order constraints (Ruland 1995): x i,n+j + x j,n+i 1 The order constraints can be generalised.

8 Valid Inequalities: Generalised Order Constraints (GOC) Let S 1,,S m sets of nodes s.t. S l Pickup(S l+1 ) (where S m+1 = S 1 ). Generalised order constraints (Ruland 1995): h=1,,m x(s h ) h=1,,m S h - m 1 GOC are not facet defining Can they be lifted to become facets? Example: x i,n+k + x j,n+i + x k,n+j 2

9 Valid Inequalities: Order Matching Constraints (OMC) Let i 1,,i m pickup nodes and H V \ {0, 2n+1} such that: i h H, n+i h H, h=1,,m Order matching constraints (Ruland 1995): x(h) + h=1,,m x i h,n+ih H If H contains pickup nodes only, the OMC are facets. The OMC can be generalised.

10 Valid Inequalities: Generalised Order Matching Constraints Let {i 1,,i m } pickup nodes, H V \ {0,2n+1}, and disjoint sets T 1,,T m V \ {0,2n+1} s.t.: {i h, n+i h } T h and H T h = {i h }, h=1,,m Generalised order matching constraints (Cordeau 2004): x(h) + h=1,,m x(t h ) H + h=1,,m T h - 2m One can generalise them even further!

11 Valid Inequalities: Generalised Order Matching Constraints Further generalisation by relaxing the conditions on sets H and T h. The sets T do not need to be disjoint, but T h T l H, h l. {i 1,,i m } H and {n+i 1,,n+i m } H = Doubly generalised order matching constraints: x(h) + h=1,,m x(t h ) H + h=1,,m T h - 2m DGOMC, OMC, GOMC DGOMC, GOMC Not OMC DGOMC Not OMC, GOMC

12 Valid Inequalities: Lifted Subtour Elimination Constraints Let S V \ {0, 2n+1} such that: i S s.t. n+i S Lifted subtour elimination constraints (LSEC): x(s) + j S,n+j S x i,n+j S - 1 The LSEC are facets.

13 Valid Inequalities: Generalised Lifted Subtour Elimination Constraints Let S V \ {0, 2n+1} such that: i S s.t. n+i S Let T 1, T K V \ {0, 2n+1} such that: p k S pickup node s.t. n+ p k T k, k T k S = {i}, k, and T j T l = {i}, j, l Generalised lifted subtour elimination constraints (GLSEC): x(s) + k=1,,k x(t k ) S ( k=1,,k T k - 2)

14 Test Problems 1 Randomly generated (points in a grid). Renaud, Boctor and Laporte test instances (2002): 2 TSP instances with delivery nodes selected randomly from the neighbourhood of the pickup nodes: A: closest five B: closest ten C: the whole graph 3 Randomly generated instances. On the optimal TSP tour pickup and delivery nodes are defined such that the tour is feasible for the pickup and delivery problem.

15 Separation Procedures Max flow based procedures: Subtour elimination constraints (SEC): exact Precedence constraints (PC): exact Order matching constraints (OMC): exact (2 and 3 destinations) Doubly generalised order matching constraints (DGOMC): heuristics Generalised order constraints (GOC): exact (m=2), heuristic (m>2) Lifted subtour elimination constraints (LSEC): exact Generalised lifted subtour elimination constraints (GLSEC): heuristic

16 What is the impact of the valid inequalities? Variable across instances. Integrality gap: 100(UB-LB)/UB LB: value of LP relaxation UB: best known heuristic or optimal solution 1 Top two (avg. gap closed): GOC (61.2%), GLSEC (37.3%) All: 72.5%, All + General Purpose CPLEX Cuts: 76.6 Avg. computational time: All 18.5s, All + CPLEX 56.8s 2 Top two (avg. gap closed): GOC (37%), GLSEC (22.4%) All: 50.4%, All + General purpose CPLEX Cuts: 53.5 Avg. computational time: All 99s, All + CPLEX 288.2s

17 Branch-and-Cut Algorithms Time limit: 4 hours 1 and 2 : Instances with up to 20 requests (one minute) Most instances with 25 requests (within time limit) Some instances with 30 and 35 requests Larger instances: gap within 5% 3 : All solved Since TSP based, we think it is due to good bounds from TSP.