Time analysis for planning a path in a time-window network

A systematic method is proposed to generate time information on the paths and nodes on a time-window network for planning and selecting a path under a constraint on the latest entering time at the destination node. Specifically, three algorithms are proposed to generate six basic time characteristics of the nodes, including the earliest and latest times of arriving at, entering, and departing from each node on the network. Using the basic time characteristics, we identify inaccessible nodes that cannot be included in a feasible path and evaluate the accessible nodes’ flexibilities in the waiting time and staying time. We also propose a method for measuring adverse effects of including an arc. Finally, based on the time characteristics and the proposed analyses, we develop an algorithm that can find the most flexible path in a time-window network.     

On the Generalized Elementary Shortest Path Problem: A heuristic approach

We introduce the generalized elementary shortest path problem (GESPP) where in addition to the features of the shortest path problem, nodes belong to predefined non-disjoint clusters. Each cluster is associated to a profit to the cost function, obtained if at least one element in the cluster appears in the path. Several applications can be considered as school bus routing, pricing problems, or telecommunication network design. Thus, depending on the case, clusters could be interpreted as groups of nodes with linking features as, for example, being easily reachable from each other, or some kind of coverage guarantee. We compare the GESPP to similar problems in the literature and we propose a two-phase heuristic algorithm for graphs including negative cycles. Tests on random instances with up to 100 nodes show an average gap of 0.3% to the best known solutions computed in 2.8s in average.     

Speeding up the Floyd-Warshall algorithm for the cycled shortest path problem

On a network with a cycle, where at least one cycle exists, the Floyd-Warshall algorithm is one of the algorithms most used for determining the least cost path between every pair of nodes. In this work a new algorithm for this problem is developed that requires less computational effort than the Floyd-Warshall algorithm. Furthermore, we show that the basis of our algorithm is much easier to understand, which might be an advantage for educational purposes. A small example validates our algorithm and shows its implementation.     

Multi-depot vehicle scheduling problems with time windows and waiting costs

The multi-depot vehicle scheduling problem with time windows (MDVSPTW) consists of scheduling a fleet of vehicles to cover a set of tasks at minimum cost. Each task is restricted to begin within a prescribed time interval and vehicles are supplied by different depots. The problem is formulated as an integer nonlinear multi-commodity network flow model with time variables and is solved using a column generation approach embedded in a branch-and-bound framework. This paper breaks new ground by considering costs on exact waiting times between two consecutive tasks instead of minimal waiting times. This new and more realistic cost structure gives rise to a nonlinear objective function in the model. Optimal and heuristic versions of the algorithm have been extensively tested on randomly generated urban bus scheduling problem (UBSP) and freight transport scheduling problem (FTSP). The results show that such a general solution methodology outperforms specialized algorithms when minimal waiting costs are used, and can efficiently treat the case with exact waiting costs.     

Finding K shortest looping paths with waiting time in a time–window network

A time-constrained shortest path problem is a shortest path problem including time constraints that are commonly modeled by the form of time windows. Finding K shortest paths are suitable for the problem associated with constraints that are difficult to define or optimize simultaneously. Depending on the types of constraints, these K paths are generally classified into either simple paths or looping paths. In the presence of time–window constraints, waiting time occurs but is largely ignored. Given a network with such constraints, the contribution of this paper is to develop a polynomial time algorithm that finds the first K shortest looping paths including waiting time. The time complexity of the algorithm is O(rK²|V₁|³), where r is the number of different windows of a node and |V₁| is the number of nodes in the original network.     

A bicriterion shortest path algorithm

Among the network models, one of the more popular is the so called shortest path problem. This model is used whenever it is intended to minimize a linear function which represents a distance between a predetermined pair of nodes in a given network.Often a single objective function is not sufficient to completely characterize some real-world problems. For instance, in a road network two parameters - as cost and time - can be assigned to each arc. Clearly the fastest path may be too costly. Nevertheless the decision-maker must choose one solution, possibly not the best for both criteria.In this paper we present an algorithm for this problem. With this algorithm a special set of paths (the set of Pareto optimal paths) is determined. One objective for any Pareto optimal path can not be improved without worsening the other one.     

Cost sharing of cooperating queues in a Jackson network

We consider networks of queues in which the independent operators of individual queues may cooperate to reduce the amount of waiting. More specifically, we focus on Jackson networks in which the total capacity of the servers can be redistributed over all queues in any desired way. If we associate a cost to waiting that is linear in the queue lengths, it is known from the literature how the operators should share the available service capacity to minimize the long run total cost. This paper deals with the question whether or not (the operators of) the individual queues will indeed cooperate in this way, and if so, how they could share the cost in the new situation such that each operator never pays more than his own cost without cooperation. For the particular case of a tandem network with two or three nodes it is known from previous work that cooperation is indeed beneficial, but for larger tandem networks and for general Jackson networks this question was still open. The main result of this paper gives for any Jackson network an explicit cost allocation that is beneficial for all operators. The approach we use also works for other cost functions, such as the server utilization.     

The plant location problem with demand-dependent setup costs and centralized allocation

Suppose that customers are situated at the nodes of a transportation network, and a service company plans to locate a number of facilities that will serve the customers. The objective is to minimize the sum of the total setup cost and the total transportation cost. The setup cost of a facility is demand-dependent, that is, it depends on the number of customers that are served by the facility. Centralized allocation of customers to facilities is assumed, that is, the service company makes a decision about allocation of customers to facilities. In the case of a general network, the model can be formulated as a mixed integer programming problem. For the case of a tree network, we develop a polynomial-time dynamic programming algorithm.     

Optimization of pipe networks

The paper treats a piping system, where the layout of the network is given but the diameters of the pipes should be chosen among a small number of different values. The cost of realizing the system should be minimized while keeping the energy heads at the nodes above some lower limits. A new algorithm using successive linear programming is presented. The performance of the algorithm is illustrated by optimizing a network with 201 pipes and 172 nodes. It is concluded that the new algorithm seems to be very efficient and stable, and that it always finds a solution with a cost near the best possible.     

Finding a minimum cost path between a pair of nodes in a time-varying road network with a congestion charge

The minimum cost path problem in a time-varying road network is a complicated problem. The paper proposes two heuristic methods to solve the minimum cost path problem between a pair of nodes with a time-varying road network and a congestion charge. The heuristic methods are compared with an alternative exact method using real traffic information. Also, the heuristic methods are tested in a benchmark dataset and a London road network dataset. The heuristic methods can achieve good solutions in a reasonable running time.     

A new ranking algorithm and an evolutionary model based on comprehensive weighted clique degree for complex network

This paper proposes a new ranking algorithm based on comprehensive weighted clique degree (CWCDR) for ranking importance of nodes in complex network. Simulation results show that CWCDR algorithm can overcome the limitation of degree ranking algorithm and find important nodes in complex networks more precisely and effectively. To the shortage of small-world model and BA model, this paper proposes an evolutionary model of complex network based on CWCDR algorithms, named CWCDR model. Simulation results show that the CWCDR model accords with power-law distribution. Compared with BA model, this model has better average shortest path length and clustering coefficient. Therefore, the CWCDR model is more consistent with the real network.     

Routing optimization in packet switching communication networks

For routing assignments a special model and an optimization algorithm are proposed. The efficiency of the routing assignments is evaluated by the average value of the total cost of delays for all packets in the network. It is the objective function. The main idea is that traffic, which is transmitted from the source node to the destination node, can be split between two or more logical paths. The minimum of the objective function can be found by varying the traffic on every path and simultaneously from all the source nodes to the destination nodes. If this approach is applied, then the objective function is nonseparable and nonlinear. Because its shape is unknown in advance, an adaptive nonlinear optimization algorithm is proposed. For evaluating its efficiency a special set of test functions has been used.     

An efficient label setting/correcting shortest path algorithm

We design a new label shortest path algorithm by applying the concept of a pseudo permanent label. This approach allows an algorithm to partition the set of nodes into two new sets: pseudo permanently labeled nodes and its complementary set. From this point of view, this new label method can be considered as a label setting method. Moreover, at least one node becomes permanently labeled when some nodes which belong to the set of pseudo permanently labeled nodes are scanned in each iteration of the algorithm. In the case of networks with non-negative length arcs it is easy to prove that this node has the minimum distance label among the non-pseudo permanently labeled nodes. On the other hand, it is not known during the computation which pseudo permanently labeled nodes are permanently labeled. Therefore, all distance labels are temporary and the algorithm becomes a label correcting method. Nevertheless, the proposed algorithm exhibits some nice features, such as: (1) the time bound for the running of the algorithm for a network with n nodes and m arcs is O(nm); (2) the number of node scan operations in the algorithm is less than the number of these operations in the previous label correcting algorithms as is observed in the computational experience; (3) the algorithm incorporates two new rules which allow easy detection of a negative cycle in the network; (4) the algorithm is quite simple and very easy to implement, and does not require sophisticated data structures; (5) the algorithm exhibits flexibility in the order in which the new pseudo permanently labeled nodes are scanned. The above features are possible through the application of the pseudo permanent label concept.     

Development of the tree-based link labeling algorithm for optimal path-finding in urban transportation networks

The optimal path-finding algorithm which is an important module in developing route guidance systems and traffic control systems has to provide correct paths to consider U-turns, P-turns, and no-left-turns in urban transportation networks.Traditional methods which have been used to consider those regulations on urban transportation networks can be categorized into network representation and algorithmic methods like the vine-building algorithm. First, network representation methods use traditional optimal path-finding algorithms with modifications to the network structure: for example, just adding dummy nodes and links to the existing network allows constraint-search in the network. This method which creates large networks is hard to implement and introduces considerable difficulties in network coding. With the increased number of nodes and links, the memory requirement tremendously increases, which causes the processing speed to slow down. For these reasons, the method has not been widely accepted for incorporating turning regulations in optimal path-finding problems in transportation networks. Second, algorithmic methods, as they are mainly based on the vine-building algorithm, have been suggested for determining optimal path for networks with turn penalties and prohibitions. However, the algorithms, although they nicely reflect the characteristics of urban transportation networks, frequently provide infeasible or suboptimal solutions.The algorithm to be suggested in this research is a method which is basically based on Dijkstra's algorithm [1] and the tree-building algorithm used to construct optimal paths. Unlike the traditional node labeling algorithms which label each node with minimum estimated cost, this algorithm labels each link with minimum estimated cost.Comparison with the vine-building algorithm shows that the solution of the link-labeling algorithm is better than that of the vine-building algorithm which very frequently provides suboptimal solutions. As a result, the algorithm allows turning regulations, while providing an optimal solution within a reasonable time limit.     

Shortest path games

We study cooperative games that arise from the problem of finding shortest paths from a specified source to all other nodes in a network. Such networks model, among other things, efficient development of a commuter rail system for a growing metropolitan area. We motivate and define these games and provide reasonable conditions for the corresponding rail application. We show that the core of a shortest path game is nonempty and satisfies the given conditions, but that the Shapley value for these games may lie outside the core. However, we show that the shortest path game is convex for the special case of tree networks, and we provide a simple, polynomial time formula for the Shapley value in this case. In addition, we extend our tree results to the case where users of the network travel to nodes other than the source. Finally, we provide a necessary and sufficient condition for shortest paths to remain optimal in dynamic shortest path games, where nodes are added to the network sequentially over time.     

Optimum departure times for commuters in congested networks

We propose an algorithm to compute the optimum departure time and path for a commuter in a congested network. Constant costs for use of arcs, cost functions of travel time depending on exogenous congestion and schedule delay are taken into account. A best path for a given departure time is computed with a previous algorithm for the generalized shortest path problem. The globally optimal departure time and an optimal path are determined by adapting Piyavskii's algorithm to the case of one-sided Lipschitz functions.This research has benefited from a grant of the Transportation Center of Northwestern University. The first author's research was partially supported by NSF grant No. SES-8911517 to Northwestern University. The second author's research was partially supported by AFOSR grants No. 89-0512 and 90-0008 to Rutgers University.     

Reliability based assignment in stochastic-flow freight network

Based on the reliability of transportation time, a transportation assignment model of stochastic-flow freight network is designed in this paper. This transportation assignment model is built by mean of stochastic chance-constraint programming and solved with a hybrid intelligent algorithm (HIA) which integrates genetic algorithm (GA), stochastic simulation (SS) and neural network (NN). GA is employed to report the optimal solution as well as the optimal objective function values of the proposed model. SS is used to simulate the value of uncertain system reliability function. The uncertain function approximated via NN is embedded into GA to check the feasibility and to compute the fitness of the chromosomes. These conclusions have been drawn after a test of numerical case using the proposed formulations. System reliability, total system cost and flow on each path would finally reach at their own convergence points. Increase of the system reliability causes increase of the total time cost. The system reliability and the total time cost converge at a possible Nash Equilibrium point.     

基于道路堵塞及顾客时间要求的快件配送最优路径选择

针对道路堵塞如节假日导致的临时最短配送路径失效的问题,提出配送网络最优路径选择模型,并设计了求解快递配送网络关键边和最优路径的算法。首先,计算出整个网络的关键边,掌握配送网络特征;其次,考虑顾客时间要求,研究不完全信息(中断无法提前预知,只有到达中断边的起点处才可知)下的最优路径,根据最短路径上各边新的特点,计算出每条边中断后对应的一组备用路径,再选择运输时间小于或等于顾客可等待时间的路径为有效路径,考虑道路堵塞情况,从有效路径中选择最优路径;最后,结合配送网络的实际情况对最优路径进行了算例分析。     

轴辐式枢纽网络系统的关键枢纽设施识别问题研究

作为轴辐式枢纽网络关键因素的节点,尤其是起到中转作用的枢纽节点是网络稳定运行的重要环节。当这些节点被中断时,将对整个网络产生严重的影响。最直接的表现方式即是网络运行成本的急剧上升。因此本文研究如何识别对网络成本具有决定性影响的关键节点。首先,提出枢纽功能性中断问题和模型,并通过禁忌搜索算法进行求解。最后通过中国航空实例验证模型和算法在实际应用中的有效性。结果显示模型和算法能够有效识别出中国航空网络较重要的关键城市以及相对影响较弱的城市。可以为资源有限情况下,中国航空网络中各城市防御设施的合理分级和部署,为重点保护城市的鉴别提供依据和帮助。     

Graph Collapsing in Shortest Path Auction Algorithms

In this paper we consider the problem of finding a shortest path from a source node to a fixed target node (SSP) or to all the nodes (SPT) on a directed graph. A family of algorithms which derives from the known auction algorithm is introduced. The key feature of these algorithms is based on topological transformations operated on the graphs that replace an optimal sub-path with a single arc of the same length (graph collapsing concept). The same idea is applied both to the standard auction algorithm and to a modified version of the algorithm. In the last mentioned case a good saving in computation cost is obtained as shown by the reported numerical examples.