The tree representation for the pickup and delivery traveling salesman problem with LIFO loading

The feasible solutions of the traveling salesman problem with pickup and delivery (TSPPD) are commonly represented by vertex lists. However, when the TSPPD is required to follow a policy that loading and unloading operations must be performed in a last-in-first-out (LIFO) manner, we show that its feasible solutions can be represented by trees. Consequently, we develop a novel variable neighborhood search (VNS) heuristic for the TSPPD with last-in-first-out loading (TSPPDL) involving several search operators based on the tree data structure. Extensive experiments suggest that our VNS heuristic is superior to the current best heuristics for the TSPPDL in terms of solution quality, while requiring no more computing time as the size of the problem increases.     

An iterated local search algorithm for the Travelling Salesman Problem with Pickups and Deliveries

The Travelling Salesman Problem with Pickups and Deliveries (TSPPD) consists in designing a minimum cost tour that starts at the depot, provides either a pickup or delivery service to each of the customers and returns to the depot, in such a way that the vehicle capacity is not exceeded in any part of the tour. In this paper, the TSPPD is solved by considering a metaheuris-tic algorithm based on Iterated Local Search with Variable Neighbourhood Descent and Random neighbourhood ordering. Our aim is to propose a fast, flexible and easy to code algorithm, also capable of producing high quality solutions. The results of our computational experience show that the algorithm finds or improves the best known results reported in the literature within reasonable computational time.     

Creating lasso-solutions for the traveling salesman problem with pickup and delivery by Tabu search

We consider the Traveling Salesman Problem with Pickup and Delivery (TSPPD) where the same costumers might require both deloverie of goods and pickup of other goods. All the goods should be transported from/to the same depot. A vehicle on a TSPPD-tour could often get some practical problems when arranging the load. Even if the vehicle has enough space for all the pickups, one has to consider that they are stored in a way that doesn't block the delivery for the next customer. In real life problems this occurs for instance for breweries when they deliver bottles of beer or mineral water and collects empty bottles from the same customers on the same tour. In these situations we could relax the constraints of only checking Hamiltonian tours, and also try solutions that can visit customers in a way giving rise to a ‘alsso’ model. A solution which first only delivers bottles until the vehicle is partly unloaded, then do both delivery and pickup at the remaining customers and at last picks up the empty bottle from the first visited customers, could in these situations be better than a pure Hamiltonian tour, at least in a practical setting. To find such solutions, we will use the metaheuristic Tabu Search on some well known TSPPD-problems, and compare them to other kinds of solutions on the same problems.     

快递企业“最后一公里”快件收派优化方案研究

本文以快递公司快件收派服务为背景,对区域收派路线规划问题进行研究,结合A快递公司实际运作情况进行案例分析,综合考虑收派混合、动态性、时间窗和容量约束四个最主要的因素,建立数学模型,设计收派流程,通过改进的禁忌搜索算法在短时间内得到优化的路径结果,并在收派活动进行中动态处理新需求及实时更新收派路径,以提高收派效率。基于该企业实际数据的计算结果表明,本文提出的相应流程和算法比实际操作获得更好的解。     

Primal—Dual Methods for Vertex and Facet Enumeration

Every convex polytope can be represented as the intersection of a finite set of halfspaces and as the convex hull of its vertices. Transforming from the halfspace (resp. vertex) to the vertex (resp. halfspace) representation is called vertex enumeration (resp. facet enumeration ). An open question is whether there is an algorithm for these two problems (equivalent by geometric duality) that is polynomial in the input size and the output size. In this paper we extend the known polynomially solvable classes of polytopes by looking at the dual problems. The dual problem of a vertex (resp. facet) enumeration problem is the facet (resp. vertex) enumeration problem for the same polytope where the input and output are simply interchanged. For a particular class of polytopes and a fixed algorithm, one transformation may be much easier than its dual. In this paper we propose a new class of algorithms that take advantage of this phenomenon. Loosely speaking, primal—dual algorithms use a solution to the easy direction as an oracle to help solve the seemingly hard direction. Received July 31, 1997, and in revised form March 8, 1998.     

A tabu search heuristic for a routing problem arising in servicing of offshore oil and gas platforms

This paper introduces a pickup and delivery problem encountered in servicing of offshore oil and gas platforms in the Norwegian Sea. A single vessel must perform pickups and deliveries at several offshore platforms. All delivery demands originate at a supply base and all pickup demands are also destined to the base. The vessel capacity may never be exceeded along its route. In addition, the amount of space available for loading and unloading operations is limited at each platform. The problem, called the Single Vehicle Pickup and Delivery Problem with Capacitated Customers consists of designing a least cost vehicle (vessel) route starting and ending at the depot (base), visiting each customer (platform), and such that there is always sufficient capacity in the vehicle and at the customer location to perform the pickup and delivery operations. This paper describes several construction heuristics as well as a tabu search algorithm. Computational results are presented.     

Lasso solution strategies for the vehicle routing problem with pickups and deliveries

This paper considers the vehicle routing problem with pickups and deliveries (VRPPD) where the same customer may require both a delivery and a pickup. This is the case, for instance, of breweries that deliver beer or mineral water bottles to a set of customers and collect empty bottles from the same customers. It is possible to relax the customary practice of performing a pickup when delivering at a customer, and postpone the pickup until the vehicle has sufficient free capacity. In the case of breweries, these solutions will often consist of routes in which bottles are first delivered until the vehicle is partly unloaded, then both pickup and delivery are performed at the remaining customers, and finally empty bottles are picked up from the first visited customers. These customers are revisited in reverse order, thus giving rise to lasso shaped solutions. Another possibility is to relax the traditional problem even more and allow customers to be visited twice either in two different routes or at different times on the same route, giving rise to a general solution. This article develops a tabu search algorithm capable of producing lasso solutions. A general solution can be reached by first duplicating each customer and generating a Hamiltonian solution on the extended set of customers. Test results show that while general solutions outperform other solution shapes in term of cost, their computation can be time consuming. The best lasso solution generated within a given time limit is generally better than the best general solution produced with the same computing effort.     

On Conditional Covering Problem

The conditional covering problem (CCP) aims to locate facilities on a graph, where the vertex set represents both the demand points and the potential facility locations. The problem has a constraint that each vertex can cover only those vertices that lie within its covering radius and no vertex can cover itself. The objective of the problem is to find a set that minimizes the sum of the facility costs required to cover all the demand points. An algorithm for CCP on paths was presented by Horne and Smith (Networks 46(4):177–185, 2005). We show that their algorithm is wrong and further present a correct O(n ³) algorithm for the same. We also propose an O(n ²) algorithm for the CCP on paths when all vertices are assigned unit costs and further extend this algorithm to interval graphs without an increase in time complexity.     

A multi-commodity, capacitated pickup and delivery problem: The single and two-vehicle cases

We explore dynamic programming solutions for a multi-commodity, capacitated pickup and delivery problem. Cargo flows are given by an origin/destination matrix which is not necessarily symmetric. This problem is a generalization of several known pickup and delivery problems, as regards both problem structure and objective function. Solution approaches are developed for the single-vehicle and two-vehicle cases. The fact that for each cargo that goes from a node i to another node j there may be a cargo going in the opposite direction provides the motivation for the two-vehicle case, because one may conceivably consider solutions where no cargoes that travel in opposite directions between node pairs are carried by the same vehicle. Yet, it is shown that such scenarios are generally sub-optimal. As expected, the computational effort of the single vehicle algorithm is exponential in the number of cargoes. For the two-vehicle case, said effort is of an order of magnitude that is not higher than that of the single-vehicle case. Some rudimentary examples are presented or both the single-vehicle and two-vehicle cases so as to better illustrate the method.     

An algorithm for the traveling salesman problem with pickup and delivery customers

The paper extends the branch and bound algorithm of Little, Murty, Sweeney, and Karel to the traveling salesman problem with pickup and delivery customers, where each pickup customer is required to be visited before its associated delivery customer. The problems considered include single and multiple vehicle cases as well as infinite and finite capacity cases. Computational results are reported.     

The merchant subtour problem

 We consider the problem of a travelling merchant who makes money by buying commodities where they are cheap and selling them in other places where he can make a profit. The merchant ships commodities of his own choice in a van of fixed capacity. Given the prices of all the commodities in all of the places, and the cost of driving from one place to another, the problem the merchant faces each day is to select a subset of the cities that he can visit in a day, and to determine the order in which the cities are visited, such that the total profit is maximised. We call this problem the Merchant Subtour Problem. The MSP models the pricing problem of a rather complex pickup and delivery problem that was given to us by the Dutch logistics company Van Gend & Loos. We show that a special case of the MSP has a totally unimodular constraint matrix. This knowledge enables us to develop a tabu-search algorithm for finding good feasible solutions to the MSP, and a branch-and-price-and-cut algorithm for solving the MSP to optimality. The relaxations solved in each node of the branch-and-bound tree are strengthened by lifted knapsack inequalities, lifted cycle inequalities and mod-k cuts. We present computational results on data sets derived from our main instance of the Van Gend & Loos pickup and delivery problem. Received: October 25, 2000 / Accepted: January 23, 2002 Published online: September 27, 2002 RID="★" ID="★" This research was partially supported by EC – Fifth Framework Programme contract IST-1999-14186 (Project ALCOM-FT: Algorithms and Complexity – Future Technologies).     

Exact and heuristic procedures for the material handling circular flow path design problem

In this study we develop optimization, decomposition, and heuristic procedures to design a unidirectional loop flow pattern along with the pickup and delivery station locations for unit load automated material handling vehicles. The layout of the facility is fixed, the edges on the boundary of the manufacturing cells are candidates to form the unidirectional loop flow path, and a set of nodes located at an intermediate point on each edge are candidates for pickup and delivery stations of the cell formed by those edges. The objective is to minimize the total loaded and empty vehicle trip distances. The empty vehicle dispatching policy underlying the model is the shortest trip distance first. A binary integer programming model describes the problem of determining the flow path and locations of the pickup and delivery stations in which we then provide a decomposition procedure based on a loop enumeration strategy coupled with a streamlined integer linear programming model. It is shown that only a small proportion of all loops have to be enumerated to reach an optimum. Therefore a truncated version of this algorithm should yield a good heuristic. Finally we propose a neighbourhood search heuristic method and report on its performance.     

The single vehicle pickup and delivery problem with time windows: intelligent operators for heuristic and metaheuristic algorithms

The single vehicle pickup and delivery problem with time windows is an important practical problem, yet only a few researchers have tackled it. In this research, we compare three different approaches to the problem: a genetic algorithm, a simulated annealing approach, and a hill climbing algorithm. In all cases, we adopt a solution representation that depends on a duplicate code for both the pickup request and its delivery. We also present an intelligent neighborhood move, that is guided by the time window, aiming to overcome the difficult problem constraints efficiently. Results presented herein improve upon those that have been previously published.     

ON 3-CHOOSABILITY OF PLANE GRAPHS WITHOUT 6-,7- AND 9-CYCLES

The choice number of a graph G,denoted by X1(G),is the minimum number k such that if a list of k colors is given to each vertex of G,there is a vertex coloring of G where each vertex receives a color from its own list no matter what the lists are. In this paper,it is showed that X1 (G)≤3 for each plane graph of girth not less than 4 which contains no 6-, 7- and 9-cycles.     

A polynomial algorithm for maximum weighted vertex packings on graphs without long odd cycles

The vertex packing problem for a given graph is to find a maximum number of vertices no two of which are joined by an edge. The weighted version of this problem is to find a vertex packingP such that the sum of the individual vertex weights is maximum. LetG be the family of graphs whose longest odd cycle is of length not greater than 2K + 1, whereK is any non-negative integer independent of the number (denoted byn) of vertices in the graph. We present an O(n ^2K+1) algorithm for the maximum weighted vertex packing problem for graphs inG 1. A by-product of this algorithm is an algorithm for piecing together maximum weighted packings on blocks to find maximum weighted packings on graphs that contain more than one block. We also give an O(n ^2K+3) algorithm for testing membership inG This work was supported by NSF grant ENG75-00568 to Cornell University. Some of the results of this paper have appeared in Hsu's unpublished Ph.D. dissertation [9].     

General solutions to the single vehicle routing problem with pickups and deliveries

The single vehicle routing problem with pickups and deliveries (SVRPPD) is defined on a graph in which pickup and delivery demands are associated with the customer vertices. The problem consists of designing a least cost route for a vehicle of capacity Q. Each customer is allowed to be visited once for a combined pickup and delivery, or twice if these two operations are performed separately. This article proposes a mixed integer linear programming model for the SVRPPD. It introduces the concept of general solution which encompasses known solution shapes such as Hamiltonian, double-path and lasso. Classical construction and improvement heuristics, as well as a tabu search heuristic, are developed and tested over several instances. Computational results show that the best solutions generated by the heuristics are frequently non-Hamiltonian and may contain up to two customers visited twice.     

Small maximal matchings of random cubic graphs

We consider the expected size of a smallest maximal matching of cubic graphs. Firstly, we present a randomized greedy algorithm for finding a small maximal matching of cubic graphs. We analyze the average‐case performance of this heuristic on random n‐vertex cubic graphs using differential equations. In this way, we prove that the expected size of the maximal matching returned by the algorithm is asymptotically almost surely (a.a.s.) less than 0.34623n. We also give an existence proof which shows that the size of a smallest maximal matching of a random n‐vertex cubic graph is a.a.s. less than 0.3214n. It is known that the size of a smallest maximal matching of a random n‐vertex cubic graph is a.a.s. larger than 0.3158n. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 293–323, 2009     

An efficient algorithm for the evacuation problem in a certain class of networks with uniform path-lengths

In this paper, we consider the evacuation problem in a network which consists of a directed graph with capacities and transit times on its arcs. This problem can be solved by the algorithm of Hoppe and Tardos [B. Hoppe, É. Tardos, The quickest transshipment problem, Math. Oper. Res. 25(1) (2000) 36–62] in polynomial time. However their running time is high-order polynomial, and hence is not practical in general. Thus, it is necessary to devise a faster algorithm for a tractable and practically useful subclass of this problem. In this paper, we consider a network with a sink s such that (i) for each vertex v≠s the sum of the transit times of arcs on any path from v to s takes the same value, and (ii) for each vertex v≠s the minimum v-s cut is determined by the arcs incident to s whose tails are reachable from v. This class of networks is a generalization of grid networks studied in the paper [N. Kamiyama, N. Katoh, A. Takizawa, An efficient algorithm for evacuation problem in dynamic network flows with uniform arc capacity, IEICE Trans. Infrom. Syst. E89-D (8) (2006) 2372–2379]. We propose an efficient algorithm for this network problem.     

A primal-dual method for approximating tree cover with two weights

The tree cover (TC) problem is to compute a minimum weight connected edge set, given a connected and edge-weighted graph G, such that its vertex set forms a vertex cover for G. Unlike related problems of vertex cover or edge dominating set, weighted TC is not yet known to be approximable in polynomial time as well as the unweighted version is. Moreover, the best approximation algorithm known so far for weighted TC is far from practical in its efficiency. In this paper we consider a restricted version of weighted TC, as a first step towards better approximation of general TC, where only two edge weights differing by at least a factor of 2 are available. It will be shown that a factor 2 approximation can be attained efficiently (in the complexity of max flow) in this case by a primal-dual method. Even under the limited weights as such, the primal-dual arguments used will be seen to be quite involved, having a nontrivial style of dual assignments as an essential part, unlike the case of uniform weights.     

An exact algorithm for connected red–blue dominating set

In the Connected Red–Blue Dominating Set problem we are given a graph G whose vertex set is partitioned into two parts R and B (red and blue vertices), and we are asked to find a connected subgraph induced by a subset S of B such that each red vertex of G is adjacent to some vertex in S. The problem can be solved in O^?(²n−|B|) time by reduction to the Weighted Steiner Tree problem. Combining exhaustive enumeration when |B| is small with the Weighted Steiner Tree approach when |B| is large, solves the problem in O^?(n^1.4143). In this paper we present a first non-trivial exact algorithm whose running time is in O^?(n^1.3645). We use our algorithm to solve the Connected Dominating Set problem in O^?(n^1.8619). This improves the current best known algorithm, which used sophisticated run-time analysis via the measure and conquer technique to solve the problem in O^?(n^1.8966).