Solving Routing Problems with Branch-Cut-and

Solving Routing Problems with Branch-Cut-and-Price Marcus Poggi de Aragão Dep.de Informática - Pontif´ıcia Universidade Católica do Rio de Janeiro [email protected] Eduardo Uchoa Dep. Eng. de Produ¸cão - Universidade Federal Fluminense [email protected] January 15th, 2005 The Capacitated Vehicle Routing Problem (CVRP) is the most basic variant of a vehicle routing problem: homogeneous fleet, single depot, only deliveries, no time-windows, and a one-dimensional capacity constraint on the load of each vehicle. It was recently shown that a careful combination of cut generation and column generation, in the so-called branch-cut-and-price algorithms, can be significantly more effective than previous approaches for the CVRP. This talk is a discussion over the potential of branch-cut-and-price algorithms on more generic and complex variants of vehicle routing. We include pointers to recent literature and present our current lines of research on the subject. Computational results on some classical variants, including the Vehicle Routing Problem with Time-Windows (VRPTW) and the Capacitated Arc Routing Problem (CARP) are given. The talk starts by presenting an approach we have initially developed for the Capacitated Minimum Spanning Tree problem ([3], [10]). Next, it is shown how it could be extended to the CVRP. The corresponding MIP formulation written below, in the form of what we called Explicit Master, combines the edge based and the column generation formulations for that problem. Suppose a complete undirected graph G = (V, E) with vertex set V = {0, 1, . . . , n}. The depot is assigned to vertex 0. The client vertex set V+ = {1, . . . , n} is associated to demands d(·). Each edge e ∈ E has a nonnegative length ℓ(e). There are K vehicles of equal capacity C. For each one, a route, starting and ending at the depot, shall be determined assuring that total demand of the clients in the route do not exceed C. Finally, the set of all K routes must visit each client exactly once and total length is to be minimized.                                           

min

ℓe xe

P

e∈E P

xe

(0) = 2

∀ i ∈ V+

(1)

e∈δ({i})

P

xe = 2 · K

(2)

e∈δ({0})

P

xe

≥ 2 · k(S) ∀ S ⊆ V+

(3)

e∈δ(S) p P

j=1

qje λj

· λj

−

xe

≤ 1

∀ e ∈ E \ δ({0}) (4)

xe

= 0

∀e ∈ E

(5)

xe

≥ 0 ≥ 0

and integer ∀ j ∈ {1, . . . , p}

∀e ∈ E

Variable xe represents the number of times edge e is traversed while variables λ are associated with the p possible q-routes (a relaxation of a route, allowing some clients to be visited more than once [2]). 1

The coefficients qje equal the number of times edge e is traversed in q-route j. The value d(S) denotes the sum of the demands of all vertices in S, and k(S) = ⌈d(S)/C⌉. Fukasawa et al. [4] have shown that this formulation gives bounds considerably better than those obtained solely by the edge formulation or by the column generation formulation. This bound can be further improved by incorporating several other known families of cuts over x variables and by eliminating q-routes with small sub-cycles. The resulting branch-cut-and-price (BCP) somehow integrates in a single algorithm most of the knowledge developed for routing problems with capacity constraints in the past. The remaining of the talk is about the potentials and the difficulties of building new BCP algorithms for more complex vehicle routing problems. While column generation techniques are already available for most such problems, cut generation techniques are still very incipient. A first line of research have identified some routing problems presenting similarities with the CVRP that can be exploited in the short time development of BCP algorithms. A good example is the Capacitated Arc Routing problem (CARP), where a very effective BCP was built by coupling of column generation with projections of known CVRP cuts [7]. Another promising variant for this approach, being currently investigated, is the Vehicle Routing with Backhauls. We can show that more complex routing problems with both pick-up and deliveries can be handled by coupling column generation with projections of flow formulations, as hinted by Letchford and Salazar [6]. A second line of research addresses vehicle routing problems with time-windows. Those problems are classically dominated by column generation algorithms. An earlier BCP by Kohl et al. [5] have already improved column generation with 2-path cuts. However, while those algorithms are effective on instances with narrow time-windows, instances with larger time-windows are considered to be much harder. It seems that some yet unknown families of cuts could help to close that gap. This view was supported by Menezes [9], it is shown that cuts obtained by projecting known CVRP cuts can be indeed effective on VRPTW, even though those cuts are not taking the time-windows into account. The families of cuts presented in [1] and [8] are worthy to be investigated in a BCP context. A major difficulty on this line of research is the development of efficient separation procedures, families of cuts involving both capacities and time-windows can be quite complex. Finally, a third line of on-going research is the development of BCP algorithms for vehicle routing problems over heterogeneous fleets.

References [1] J. Bard, G. Kontoravdis, and G. Yu. A branch-and-cut procedure for the vehicle routing problem with time windows. Transportation Science, 36:250–269, 2002. [2] N. Christofides, A. Mingozzi, and P. Toth. Exact algorithms for the vehicle routing problem, based on spanning tree and shortest path relaxations. Mathematical Programming, 20:255–282, 1981. [3] R. Fukasawa, M. Poggi de Aragão, O. Porto, and E. Uchoa. Robust branch-and-cut-and-price for the capacitated minimum spanning tree problem. In Proc. of the International Network Optimization Conference, pages 231–236, Evry, France, 2003. [4] R. Fukasawa, M. Reis, M. Poggi de Aragão, and E. Uchoa. Robust branch-and-cut-and-price for the capacitated vehicle routing problem. Technical Report RPEP Vol.3 no.8, Universidade Federal Fluminense, Engenharia de Produ¸caõ, Niterói, Brazil, 2003. [5] N. Kohl, J. Desrosiers, O. Madsen, M. Solomon, and F. Soumis. 2-Path cuts for the vehicle routing problem with time windows. Transportation Science, 33:101–116, 1999. [6] A. N. Letchford and J. J. Salazar. Projection results for vehicle routing. Mathematical Programming, 2005. To appear. 2

[7] H. Longo, M. Poggi de Aragão, and E. Uchoa. Solving capacitated arc routing problems using a transformation to the CVRP. Computers & Operations Research, 2005. To appear. [8] J. Lysgaard. Reachability cuts for the vehicle routing problem with time windows. Technical Report L-2004-01, Aarhus School of Business, Denmark, 2004. [9] F. Menezes. Aplica¸co˜es de técnicas de programa¸caõ inteira em problemas de roteamento de ve´ıculos com janelas de tempo. Master’s thesis, PUC-Rio, Brazil, December 2004. [10] M. Poggi de Aragão and E. Uchoa. Integer program reformulation for robust branch-and-cut-andprice. In Annals of Mathematical Programming in Rio, pages 56–61, B´ uzios, Brazil, 2003.

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