Asymmetric Traveling Salesman Problem (ATSP): Models

Asymmetric Traveling Salesman Problem (ATSP): Models Given a DIRECTED GRAPH G = (V,A) with V = {,, n} verte set A = {(i, j) : i V, j V} arc set (complete digraph) c ij = cost associated with arc (i, j) A (c ii =, i V) Find a HAMILTONIAN CIRCUIT (Tour) whose global cost is minimum (Asymmetric Travelling Salesman Problem: ATSP). Hamiltonian Circuit: circuit passing through each verte of V eactly once. A Hamiltonian circuit has n arcs.

ATSP is NP -Hard in the strong sense. If G is an undirected graph: Symmetric TSP (STSP) (special case of ATSP arising when c ij = c ji for each (i, j) A) Any ATSP instance with n vertices can be transformed into an equivalent STSP instance with n nodes (Jonker- Volgenant, 8; Junger-Reinelt-Rinaldi, ; Kumar-Li, 00). If G = (V, A) is a sparse graph: c ij = for each (i, j) A. Feasible solutions?

Eample n = V = 8 8 Optimal solution

Eample n = V = Optimal solution Optimal solution Cost = + + + + + =

APPLICATIONS * Vehicle Routing (sequencing the customers in each route in an urban area calls for the optimal solution of the ATSP corresponding to the depot and the customers in the route). * Scheduling (optimal sequencing of jobs on a machine when the set-up costs depend on the sequence in which the jobs are processed). Picking in an Inventory System (sequence of movements of a crane to pick-up a set of items stored on shelves).

INTEGER LINEAR PROGRAMMING (ILP) FORMULATION ij = { if arc (i, j) is in the optimal tour 0 otherwise i V, j V s.t. min Σ Σ c ij ij i V j V out-degree Σ ij = i V constraints j V in-degree Σ ij = j V constraints i V i j ij {0, } i V, j V?

Eample: n = V = 8 8 only degree constraints imposed

Eample: only degree constraints imposed n = V = Infeasible solution: two partial tours (subtours)

INTEGER LINEAR PROGRAMMING (ILP) FORMULATION (Dantzig, Fulkerson, Johnson, Oper. Res. ) ij = { if arc (i, j) is in the optimal tour 0 otherwise i V, j V s.t. min Σ Σ c ij ij i V j V out-degree Σ ij = i V constraints i j V SUBTOUR ELIMINATION CONSTRAINTS ( forbid the partial tours; O( n ) ) in-degree Σ ij = j V j constraints i V Σ Σ ij S - S V, S!!! i S j S ij {0, } i V, j V

Eample S n = V = 8 S = X Infeasible solution Σ Σ S - i S j S ij NO!

Eample Feasible solution n = V = 8 8 S Σ Σ S - i S j S ij S = YES!

INTEGER LINEAR PROGRAMMING (ILP) FORMULATION (Dantzig, Fulkerson, Johnson, Oper. Res. ) ij = { if arc (i, j) is in the optimal tour 0 otherwise i V, j V s.t. min Σ Σ c ij ij i V j V out-degree Σ ij = i V constraints i j V SUBTOUR ELIMINATION CONSTRAINTS ( forbid the partial tours; O( n ) ) in-degree Σ ij = j V j constraints i V Σ Σ ij S - S V, S!!! i S j S ij {0, } i V, j V Maimization problem?

Eample: only degree constraints imposed n = V = r Infeasible solution: the vertices are not connected: From any verte (say r) we must reach all the other vertices (connectivity from r), and viceversa (connectivity toward r)

INTEGER LINEAR PROGRAMMING FORMULATION ij = { if arc (i, j) is in the optimal tour 0 otherwise i V, j V i s.t. j min Σ Σ c ij ij i V j V Σ ij = j V i V Σ ij = j V i V CONNECTIVITY CONSTRAINTS ( impose the connectivity of the solution; O( n ) ) Cut inequalities Σ Σ ij S V, r S (for a fied r V) i S j V\S ij {0, } i V, j V The two formulations are equivalent

Eample S n = V = 8 X Infeasible solution Σ Σ i S j V\S ij NO!

Eample Feasible solution n = V = 8 8 S Σ Σ i S j V\S ij YES!

LP LOWER BOUND The value of the optimal solution of the Linear Programming (LP) Relaation (or Continuous Relaation) of the previous formulations, obtained by replacing ij {0, } i V, j V with 0 ij i V, j V represents a valid Lower Bound on the value of the optimal solution of the ATSP.

This LP Relaation can be efficiently (polynomially) solved by using appropriate Separation Procedures (Polynomial Separation Problem). The LP Relaation can be strengthened (so as to obtain better lower bounds) by adding valid inequalities, which are redundant for the ILP model, but can be violated by its LP Relaation (eact and/or heuristic separation procedures).

ALTERNATIVE ILP FORMULATIONS Miller-Tucker-Zemlin (J. ACM, 0); Fo-Gavish-Graves (Operations Research 80); Wong (IEEE Conference, 80); Claus (SIAM J. on Algebraic Discrete Methods, 8); Finke-Claus-Gunn (Congressus Numerantium, 8); Langevin-Soumis-Desrosiers (Operations Research Letters, 0) Desrochers-Laporte (Operations Research Letters, ); Gouveia-Voss (European Journal of Operational Research, ) Gouveia-Pires (European Journal of Operational Research, ); Myung (International Journal of Management, 00); Gouveia-Pires (Discrete Applied Mathematics, 00); Sherali-Driscoll (Operations Research 00); Sarin-Sherali-Bhootra (Oper. Res. Letters, 00); Sherali-Sarin-Tsai (Discrete Optimization, 00); Oncan-Altinel-Laporte (Review, Computers & Operations Research, 00)

ALTERNATIVE ILP FORMULATIONS These formulations involve a polynomial number of constraints, but their Linear Programming Relaations generally produce Lower Bounds weaker (and more time consuming) than those corresponding to the Dantzig-Fulkerson- Johnson formulation (with the addition of valid inequalities).