Log in with Facebook Log in with Google. Remember me on this computer. Enter the email address you signed up with and we'll email you a reset link. Need an account? Click here to sign up. Download Free PDF. A branch-and-cut algorithm for the pickup and delivery traveling salesman problem with LIFO loading Networks, Juan Jose Gonzalez.
A short summary of this paper. A branch-and-cut algorithm for the pickup and delivery traveling salesman problem with LIFO loading. This paper introduces an additive branch-and-bound algorithm for a variant of the pickup and delivery traveling salesman problem in which loading and unloading operations have to be performed in a Last-In-First-Out LIFO order.
Two relaxations are used within the additive approach: the assignment problem and the shortest spanning r- arborescence problem. The quality of the lower bounds is further improved by a set of elimination rules applied at each node of the search tree to remove from the problem arcs that cannot belong to feasible solutions because of precedence relationships. The performance of the algorithm and the effectiveness of the elimination rules are assessed on instances from the literature.
Traveling salesman problem, pickup and delivery, LIFO loading, rear loading, additive branch-and-bound. This support is gratefully acknowledged. We are also thankful to three anonymous referees for their valuable comments. Results and views expressed in this publication are the sole responsibility of the authors and do not necessarily reflect those of CIRRELT.
It consist of determining a minimum length tour traveled by a vehicle to service n requests. Each request is characterized by an origin vertex, the pickup location, where goods are loaded, and a destination vertex, the delivery location, where goods are unloaded. The vehicle starts from a fixed vertex, the depot, and returns to it after all requests have been satisfied.
Every other vertex has to be visited exactly once, with the additional constraint that the pickup vertex associated with any given request must be visited before the corresponding delivery vertex. This problem has been studied, among others, by Kalantari et al. For a recent survey, see Cordeau et al. The TSPPDL has applications in the distribution of goods by vehicles having a unique entry and exit point for freight and in situations where rearranging the load is not allowed.
This may be the case for safety or physical reasons e. The problem also arises in the routing of automated guided vehicles that use a stack to move items between locations in a plant or warehouse. Volchenkov [] has analyzed a planar layout problem with LIFO constraints.
The results were later used by Levitin [] and Levitin and Abezgaouz []. The latter paper proposes an exact algorithm for the routing of multiple-load automated guided vehicles.
This problem is in fact a TSPPDL with the difference that each pickup customer can be associated with more than one delivery customer, and vice-versa. The author has presented results on random instances with up to customers. These operators are embedded into a Variable Neighborhood Search VNS heuristic together with the four operators proposed by Cassani [].
Computational results show that the solutions produced by the VNS heuristic are significantly better than those of the VND, at the expense of an increase in computing times. The first exact approach for the problem studied in this paper was introduced by Pacheco [, ] who developed a branch-and-bound algorithm derived from the algorithms of Little et al. Cassani [] has later introduced a different branch-and-bound algorithm in which lower bounds are computed by solving the minimum spanning tree problem MSTP and assignment problem AP relaxations.
Another method, based on dynamic programming, was also introduced by Ficarelli []. These last two approaches are able to solve instances with up to 23 vertices in less than 20 minutes of computing time. Several families of valid inequalities are also used to strengthen the formulation. Exact separation procedures are used to identify violated subtour elimination constraints, precedence constraints and LIFO constraints, while heuristic separation procedures are used for the other families of inequalities.
This algorithm is able to solve most instances with up to 43 vertices and some instances with 51 vertices in less than 60 minutes of computing time. This approach has also been applied successfully to the symmetric TSP by Carpaneto et al. In comparison with the branch-and-bound proposed by Kalantari et al. These elimination rules are based on the precedence relations that arise between the vertices of the graph during the construction of a tour.
The search tree and visiting strategy chosen for our branch-and-bound algorithm increase the number of known precedence relations and consequently improve the effectiveness of elimination rules.
Cassani [] has used the same search tree in his branch-and-bound algorithm. In his case, however, the best results are obtained by constructing the tour in a bidirectional way, i. Except for one case, this algorithm is limited to solving instances with at most 17 vertices. We also combine, through the additive lower bounding approach, the AP and r-SAP, thus generating tighter lower bounds that allow the solution of larger instances.
The resulting algorithm is able to solve some instances with 43 vertices. The remainder of the paper is organized as follows. Section 2 introduces the definitions and notation that are used throughout the paper. Section 3 then introduces an equation to compute the number of feasible solutions of the TSPPDL, and describes the search tree that is explored by the additive branch-and-bound algorithm. This algorithm is then described in detail in Section 4.
This is followed by computational results in Section 5, and by the conclusion in Section 6. We restrict ourselves to the case with a single depot denoted by 0 and we assume that the depot and the pickup and delivery vertices are all different, i. The cost of arc x, y is denoted by c x, y. A tour is a sequence S0 , Let T be a tour. Note that the presence of a cross implies that the LIFO constraints are not respected.
We also introduce the following definition that will be used in the description of our algorithm. The residual graph Gw is thus the subgraph of G induced by w and the vertices that do not belong to p 0, w. We explain in detail how we have derived this equation because the approach followed also underlies the construction of the search tree used in our branch-and-bound algorithm.
The root node of the tree T corresponds to the trivial path containing only the depot vertex see Figure 2. Because of precedence constraints, the second vertex of any consistent path has to be a pickup vertex. Level 0 has a single node corresponding to the trivial path containing the depot.
Level 1 has n nodes associated with the n pickup vertices of G. At level 2 we show the results of some branching executed on level 1. Since this reasoning holds for each node at level 1 of T , the number of nodes at level 2 is equal to n2. After completing the construction of T according to these rules, we can state the following result.
Theorem 1 The number of nodes on level k of T is equal to the number of consistent paths of G composed by k vertices plus the depot. The proof is by induction on the level k of T.
On level 0 of T we have only one node and this is correct because there is an unique consistent path composed by zero vertices plus the depot.
In particular we want to prove that to each consistent path of G composed by k vertices plus the depot corresponds a node of T on level k, and vice-versa. From Corollary 1 we conclude that it is sufficient to count the number of leaves of T to determine the number of feasible tours of G.
In the following we show how to compute the number of leaves of T. Let N k, x be the number of consistent paths composed by k vertices plus the depot of which x are pickup vertices. These conditions represent the base case of our equation. This explains the first term in equation 5. This explains the second term in equation 5.
Using equation 5 we can compute the number of nodes on level k of T and then, from Theorem 1, the number of consistent paths on G composed by k vertices plus the depot. Formally, let N k be the number of nodes on level k. Using the same reasoning as above, one can easily construct the tree of consistent paths for the TSPPD.
One can extend this path by adding to it any remaining pickup vertex or any delivery vertex whose corresponding pickup vertex is already in p. The three main aspects of a branch-and-bound algorithm are i the branching strategy i. To accelerate the algorithm we also introduce a powerful set of elimination rules or filters whose aim is to reduce as much as possible the number of arcs in the residual graph considered at each node of the enumeration tree.
We have already described in Section 3 our branching strategy. In the following sections we describe the exploration strategy, the computation of lower bounds, and the set of filters used in the algorithm. The exploration strategy The exploration strategy specifies, after each node evaluation, the node from which the next branching should be performed.
The most common strategies are breadth-first, depth-first, and best-first. Going by the FIFO method, Ted needs to use the older costs of acquiring his inventory and work ahead from there. Going by the LIFO method, Ted needs to go by his most recent inventory costs first and work backwards from there. This is because the LIFO number reflects a higher inventory cost, meaning less profit and less taxes to pay at tax time.
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