Locating Depots for Capacitated Vehicle Routing

We study a location-routing problem in the context of capacitated vehicle routing. The input is a set of demand locations in a metric space and a fleet of k vehicles each of capacity Q. The objective is to locate k depots, one for each vehicle, and compute routes for the vehicles so that all demands are satisfied and the total cost is minimized. Our main result is a constant-factor approximation algorithm for this problem. To achieve this result, we reduce to the k-median-forest problem, which generalizes both k-median and minimum spanning tree, and which might be of independent interest. We give a (3+c)-approximation algorithm for k-median-forest, which leads to a (12+c)-approximation algorithm for the above location-routing problem, for any constant c>0. The algorithm for k-median-forest is just t-swap local search, and we prove that it has locality gap 3+2/t; this generalizes the corresponding result known for k-median. Finally we consider the "non-uniform" k-median-forest problem which has different cost functions for the MST and k-median parts. We show that the locality gap for this problem is unbounded even under multi-swaps, which contrasts with the uniform case. Nevertheless, we obtain a constant-factor approximation algorithm, using an LP based approach.Comment: 12 pages, 1 figur

Capacitated Vehicle Routing with Non-Uniform Speeds

The capacitated vehicle routing problem (CVRP) involves distributing (identical) items from a depot to a set of demand locations, using a single capacitated vehicle. We study a generalization of this problem to the setting of multiple vehicles having non-uniform speeds (that we call Heterogenous CVRP), and present a constant-factor approximation algorithm. The technical heart of our result lies in achieving a constant approximation to the following TSP variant (called Heterogenous TSP). Given a metric denoting distances between vertices, a depot r containing k vehicles with possibly different speeds, the goal is to find a tour for each vehicle (starting and ending at r), so that every vertex is covered in some tour and the maximum completion time is minimized. This problem is precisely Heterogenous CVRP when vehicles are uncapacitated. The presence of non-uniform speeds introduces difficulties for employing standard tour-splitting techniques. In order to get a better understanding of this technique in our context, we appeal to ideas from the 2-approximation for scheduling in parallel machine of Lenstra et al.. This motivates the introduction of a new approximate MST construction called Level-Prim, which is related to Light Approximate Shortest-path Trees. The last component of our algorithm involves partitioning the Level-Prim tree and matching the resulting parts to vehicles. This decomposition is more subtle than usual since now we need to enforce correlation between the size of the parts and their distances to the depot

Stochastic Vehicle Routing with Recourse

We study the classic Vehicle Routing Problem in the setting of stochastic optimization with recourse. StochVRP is a two-stage optimization problem, where demand is satisfied using two routes: fixed and recourse. The fixed route is computed using only a demand distribution. Then after observing the demand instantiations, a recourse route is computed -- but costs here become more expensive by a factor lambda. We present an O(log^2 n log(n lambda))-approximation algorithm for this stochastic routing problem, under arbitrary distributions. The main idea in this result is relating StochVRP to a special case of submodular orienteering, called knapsack rank-function orienteering. We also give a better approximation ratio for knapsack rank-function orienteering than what follows from prior work. Finally, we provide a Unique Games Conjecture based omega(1) hardness of approximation for StochVRP, even on star-like metrics on which our algorithm achieves a logarithmic approximation.Comment: 20 Pages, 1 figure Revision corrects the statement and proof of Theorem 1.

Design of a web site for guaranteed delay and blocking probability bounds

A new mathematical programming model is proposed for minimizing the cost of design of a Web site by optimally determining the number of servers and buffers when a performance guarantee in terms of the average waiting time and loss probability is to be provided to users. The Web site is modeled as an M/G/c/N queuing system where requests for connections represent arriving customers and the browsing of Web sites represents service received by customers. Numerical experiments are conducted with different choices of problem parameters and the optimal design cost, and optimal number of servers and buffers are obtained for these cases. © 2003 Elsevier B.V. All rights reserved.link_to_subscribed_fulltex

Asynchronous transfer mode networks with parallel links and multiple service classes

In this paper the flow assignment problem in the case of a fiber optic based Asynchronous Transfer Mode network is studied. The backbone network has parallel links between node pairs and supports multiple classes of traffic. The aim is to find the best routes for the different priority classes such that the total cost of network design is minimized and the delay bounds on the parallel links for different message classes are satisfied. A mixed integer programming formulation is presented for the problem and the solution procedure is described. Heuristics are suggested for obtaining feasible solutions to this problem. The goal is to come up with an efficient frontier by changing the delay bounds in order to give designers various choices of cost and delay to choose from. Numerical results are reported for three different network topologies and future research directions are highlighted. © 2002 Elsevier Science B.V. All rights reserved.link_to_subscribed_fulltex

Tradeoff decisions in the design of a backbone computer network using visualization

Visualization provides a useful tool for analyzing large, complex data sets. In the design of backbone computer networks, rough-cut design decisions can gain from a visual analysis of the generated solution with respect to design parameters such as average message delay, delay cost, average message length and total network operating costs. In this paper, we show how two-dimensional and three-dimensional surface and glyph representations can be used for understanding the cost-delay tradeoffs involved in the network design problem, and an idea of the 'efficient frontier' where the user may choose to operate. It provides an opportunity to revisit relationships that exist between the different network design parameters as well as discover new ones. © 2002 Published by Elsevier Science B.V.link_to_subscribed_fulltex

Noninfluentials and information dissemination in the microblogging community

Firms are increasingly focusing on understanding and managing their social media strategies in order to create discussions and optimize the spread of news in their communities. Most prior studies on information dissemination have mainly focused on the roles of influentials but ignored the essential for noninfluentials. To fill this gap, this paper takes a holistic view of the information dissemination process and investigates how the participation of both influentials and noninfluentials plays a role in affecting the volume and sentiment of microblogs, which are precursors to raise awareness and attraction for brands. To test our hypotheses, we build a novel econometric model and apply it to a dataset collected from the popular microblogging site Twitter. We have the following main results: (1) back-and-forth-type discussions and retweets are effective in generating awareness and positive attractiveness; (2) influentials or mavens (who have many followers but seldom follow others) help generate initial sparks toward microblogging, but during the cascading periods, the noninfluentials play an important role in driving the conversations; and (3) new users who gradually join the discussions also help increase awareness, although they may not generate a positive sentiment. Our results provide important implications for mediating consumer interactions and firms’ marketing strategies.TÜBİTA