The Split Delivery Vehicle Routing Problem Hande Yaman Joint work - PowerPoint PPT Presentation

The Split Delivery Vehicle Routing Problem Hande Yaman Joint work with Gizem ¨ Ozbaygın, Oya E. Kara¸ san and Barbaros C ¸. Tansel Bilkent University, Department of Industrial Engineering Hande Yaman (Bilkent University) SDVRP 1 / 26

Split Delivery VRP • Capacitated Vehicle Routing Problem (CVRP) • given a set of customers and a depot • capacitated vehicles • tours for vehicles that start and end at the depot • visit each customer exactly once • respect the vehicle capacities • minimize the total transportation cost • many variants: time windows, heterogeneous fleet, periodic VRP, pickup and delivery ... • Split Delivery Vehicle Routing Problem (SDVRP) • relax “visit each customer exactly once” Hande Yaman (Bilkent University) SDVRP 2 / 26

History - exact approaches • Dror and Trudeau (1989): split deliveries can lead to considerable cost savings • Dror and Trudeau (1990): SDVRP is NP-hard, k -split cycles • Dror et al. (1994): valid inequalities for a vehicle index formulation • Belenguer et al. (2000): polyhedral study, cutting plane algorithm • Lee et al. (2006): dynamic programming • Jin et al. (2007): iterative two stage method, cluster and route with lower bounds on route lengths in clustering • Jin et al. (2008): column generation • Moreno et al. (2010): extended formulation with load and quantity delivered, generate columns and cuts • Archetti et al. (2011): branch-and-price-and-cut • Archetti et al. (2013): branch-and-cut Hande Yaman (Bilkent University) SDVRP 3 / 26

Notation • N = { 0 , 1 , . . . , n } , 0 depot, 1 , . . . , n customers • G = ( N , A ) directed graph, complete • c a : cost of travelling on arc a • m identical vehicles, each with capacity Q • d i : demand of customer i • δ − ( S ), δ + ( S ): set of incoming and outgoing arcs of S ⊂ N , resp. • Assumption: costs are symmetric and satisfy triangle inequality Hande Yaman (Bilkent University) SDVRP 4 / 26

Model • SDVRP can be modeled using variables with vehicle index m � � c a x k min a a ∈ A k =1 s.t. x k ( δ + (0)) = 1 k = 1 , . . . , m , x k ( δ − ( i )) − x k ( δ + ( i )) = 0 i ∈ N \ { 0 } , k = 1 , . . . , m , m � w ik = 1 i ∈ N \ { 0 } , k =1 n � d i w ik ≤ Q k = 1 , . . . , m , i =1 w ik ≤ x k ( δ − ( i )) i ∈ N \ { 0 } , k = 1 , . . . , m , subtour elimination constraints , x k a ∈ { 0 , 1 } a ∈ A , k = 1 , . . . , m , w ik ≥ 0 i ∈ N \ { 0 } , k = 1 , . . . , m . • If w ik ∈ { 0 , 1 } then CVRP. Hande Yaman (Bilkent University) SDVRP 5 / 26

Without vehicle index • aim : to come with a model without vehicle index • R-SDVRP � min c a x a a ∈ A s.t. f ( δ − ( i )) − f ( δ + ( i )) = d i i ∈ N \ { 0 } , x ( δ + (0)) = m , x ( δ − ( i )) − x ( δ + ( i )) = 0 i ∈ N \ { 0 } , 0 ≤ f a ≤ Qx a a ∈ A , x a ∈ Z + a ∈ A . • remove second and third constraints: network loading Hande Yaman (Bilkent University) SDVRP 6 / 26

An example: eil 30 • 30 nodes and 3 vehicles • at node 18, incoming 2700 + 625, outgoing 3175 + 0 • virtual depot: customer node where unloading and loading take place Hande Yaman (Bilkent University) SDVRP 7 / 26

Belenguer, Martinez and Mota (2000) • undirected graph G = ( N , E ) � min c e x e e ∈ E s.t. x ( δ (0)) ≥ 2 m and even , x ( δ ( i )) ≥ 2 and even i ∈ N \ { 0 } , � d ( S ) � x ( δ ( S )) ≥ 2 S ⊂ N \ { 0 } : 2 ≤ | S | ≤ n − 1 , Q x e ∈ Z + e ∈ E . • relaxation • eliminate solutions that are not feasible with cuts • they obtain the same solution as ours for eil 30 Hande Yaman (Bilkent University) SDVRP 8 / 26

Comparing the two relaxations • Projecting out the flow variables in the LP relaxation gives the fractional capacity inequalities x ( δ − ( S )) ≥ d ( S ) S ⊆ N \ { 0 } Q • In the integer problem, the rounded capacity inequalities � d ( S ) � x ( δ − ( S )) ≥ S ⊆ N \ { 0 } Q are satisfied. • NP-hard to check whether there exists a feasible routing when x is fixed ( x 0 i = 2 for all i ∈ N \ { 0 } , binpacking). Hande Yaman (Bilkent University) SDVRP 9 / 26

Checking feasibility • A solution ( f , x ) is called regular if at each customer node, one can match each incoming arc with an outgoing arc so that the flow of the incoming arc is greater than or equal to the one on its outgoing arc. • A solution ( f , x ) of R-SDVRP is regular iff it is feasible for the SDVRP. • Regularity can be checked in polynomial time. Hande Yaman (Bilkent University) SDVRP 10 / 26

Archetti, Bianchessi, Speranza (2013) • Similar formulations to the one of Belenguer et al. (2000) and to R-SDVRP. • Without flows: build the routes for given x and choose the best by solving a model. • With flows: same idea as regularity check. • Branch and cut: eliminate vectors x for which there are no feasible routes by branching • The formulation without flows is better. Hande Yaman (Bilkent University) SDVRP 11 / 26

Aim • To solve the R-SDVRP more quickly • Eliminate virtual depots Hande Yaman (Bilkent University) SDVRP 12 / 26

k -splits by Dror and Trudeau • There exists an optimal solution to SDVRP with no k -splits. ⇒ There exists an optimal solution to SDVRP where x a ∈ { 0 , 1 } for all a ∈ A \ ( δ − (0) ∪ δ + (0)). Hande Yaman (Bilkent University) SDVRP 13 / 26

Improvements for R-SDVRP • There exists an optimal solution to SDVRP where x a ∈ { 0 , 1 } for all outgoing arcs of the depot. • For i ∈ V \ { 0 } , x ( δ − ( i )) ≥ 1 is often used to strengthen. • We use the cutset inequalities for singletons at the root node. Hande Yaman (Bilkent University) SDVRP 14 / 26

Cutset inequalities • Let S ⊆ N \ { 0 } . f ( δ − ( S )) − f ( δ + ( S )) = d ( S ) , a ∈ δ − ( S ) ∪ δ + ( S ) . 0 ≤ f a ≤ Qx a , x a ∈ Z + • Atamturk (2002): convex hull = trivial + cutset. A − ⊆ δ − ( S ) , A + ⊆ δ + ( S ) � d ( S ) � � d ( S ) � η = , r = d ( S ) − Q Q Q The cutset inequality f ( δ − ( S ) \ A − ) + rx ( A − ) + ( Q − r ) x ( A + ) − f ( A + ) ≥ r η is valid. • A − = δ − ( S ) and A + = ∅ , rounded capacity inequality. Hande Yaman (Bilkent University) SDVRP 15 / 26

Eliminating virtual depots: Patching • solve R-SDVRP • add vehicle indexed variables at a node where regularity is not satisfied, iterate f k ( δ − ( i )) − f k ( δ + ( i )) ≥ 0 k = 1 , . . . , m , x k ( δ − ( i )) − x k ( δ + ( i )) = 0 k = 1 , . . . , m , x k ( δ − ( i )) ≤ 1 k = 1 , . . . , m , a ∈ δ − ( i ) ∪ δ + ( i ) , k = 1 , . . . , m , 0 ≤ f k a ≤ Qx k a a ∈ δ − ( i ) ∪ δ + ( i ) , k = 1 , . . . , m , x k a ∈ { 0 , 1 } m m � � x k f k a ∈ δ − ( i ) ∪ δ + ( i ) . x a = a , f a = a k =1 k =1 Hande Yaman (Bilkent University) SDVRP 16 / 26

Eliminating virtual depots: Node splitting • Idea: split nodes as necessary to force regularity at every customer node • solve R-SDVRP • create a duplicate of the node violating regularity, enlarge the node & arc sets accordingly • N i : the set of nodes containing the original customer i and its duplicates. • Order N i so that a node j ∈ N i is represented with ( i , l ) where l is the order of j in N i . • Let N ′ = ∪ i ∈ N \{ 0 } N i . Define A ′ to be the set of all arcs except arcs between duplicates of the same node. • v i , l is 1 if node ( i , l ) is visited; 0 otherwise. Hande Yaman (Bilkent University) SDVRP 17 / 26

Eliminating virtual depots: Node splitting � min c a x a a ∈ A ′ � f ( δ − ( j )) − f ( δ + ( j )) s.t. � � = d i i ∈ N \ { 0 } , j ∈ N i f ( δ − ( j )) − f ( δ + ( j )) ≥ 0 j ∈ N ′ , x ( δ + (0)) = m , x ( δ − ( j )) − x ( δ + ( j )) = 0 j ∈ N ′ , ( i , l ) ∈ N ′ : | N i | ≥ 2 , l � = | N i | , x ( δ − ( i , l )) = v i , l x ( δ − ( i , | N i | )) ≤ ( m − | N i | + 1) v i | N i | i ∈ N \ { 0 } : | N i | ≥ 2 , ( i , l ) ∈ N ′ : | N i | ≥ 2 , l � = | N i | , v i , l ≥ v i , l +1 a ∈ A ′ , 0 ≤ f a ≤ Qx a ( i , l ) ∈ N ′ : | N i | ≥ 2 , v i , l ∈ { 0 , 1 } a ∈ A ′ \ δ − (0) , x a ∈ { 0 , 1 } a ∈ δ − (0) . x a ∈ Z + • If | N i | = m , same as patching but we break symmetry with v i , l ≥ v i , l +1 . Hande Yaman (Bilkent University) SDVRP 18 / 26

Eliminating virtual depots: Cutting • H ⊆ V \ { 0 } and S 1 , . . . , S t disjoint nonempty subsets of H with d ( S u ) ≤ Q for u = 1 , . . . , t . • b ( S 1 , . . . , S t ): optimal value of binpacking problem with items 1 , . . . , t of size d ( S 1 ) , . . . , d ( S t ). • The framed capacity inequality (Pochet (1998), Augerat (1995), Naddef and Rinaldi (2002)) t x ( δ − ( H )) + � x ( δ − ( S u )) ≥ t + b ( S 1 , . . . , S t ) (1) u =1 is valid. • The inequality used by Belenguer et al. is a special case where H = V \ { 0 } (known as the generalized capacity inequality). Hande Yaman (Bilkent University) SDVRP 19 / 26