Profit-based latency problems on the line.

The latency problem with profits is a generalization of the minimum latency problem. In this generalization it is not necessary to visit all clients, however, visiting a client may bring a certain revenue. More precisely, in the latency problem with profits, a server and a set of n clients, each with corresponding profit p_i (1 ≤ i ≤ n), are given. The single server is positioned at the origin at time t = 0 and travels with unit speed. When visiting a client, the server receives a revenue of p_i - t, with t the time at which the server reaches client i (1 ≤ i ≤ n). The goal is to select clients and find a route for the server such that total collected revenue is maximized. We formulate a dynamic programming algorithm to solve this problem when all clients are located on a line. We also consider the problem on the line with k servers and prove NP-completeness for the latency problem on the line with k non-identical servers and release dates. In this proof we also settle the complexity of an open problem in de Paepe et al. [4].Minimum latency; Traveling repairman; Dynamic programming; Complexity;

Structuring a Wayfinder\u27s Dynamic and Uncertain Environment

Wayfinders typically travel in dynamic environments where barriers and requirements change over time. In many cases, uncertainty exists about the future state of this changing environment. Current geographic information systems lack tools to assist wayfinders in understanding the travel possibilities and path selection options in these dynamic and uncertain settings. The goal of this research is a better understanding of the impact of dynamic and uncertain environments on wayfinding travel possibilities. An integrated spatio-temporal framework, populated with barriers and requirements, models wayfinding scenarios by generating four travel possibility partitions based on the wayfinder\u27s maximum travel speed. Using these partitions, wayfinders select paths to meet scenario requirements. When uncertainty exists, wayfinders often cannot discern the future state of barriers and requirements. The model to address indiscemibility employs a threevalued logic to indicate accessible space, inaccessible space, and possibly inaccessible space. Uncertain scenarios generate up to fifteen distinct travel possibility categories. These fifteen categories generalize into three-valued travel possible partitions based on where travel can occur and where travel is successful. Path selection in these often-complex environments is explored through a specific uncertain scenario that includes a well-defined initial requirement and the possibility of an additional requirement somewhere beforehand. Observations from initial path selection tests with this scenario provide the motivation for the hypothesis that paths arriving as soon as possible to well-defined requirements also maximize the probability of success in meeting possible additional requirements. The hypothesis evaluation occurs within a prototype Travel Possibility Calculator application that employs a set of metrics to test path accessibility in various linear and planar scenarios. The results did not support the hypothesis, but showed instead that path accessibility to possible additional requirements is greatly influenced by the spatio-temporal characteristics of the scenario\u27s barriers

Approximation Algorithms for Capacitated k-Travelling Repairmen Problems

We study variants of the capacitated vehicle routing problem. In the multiple depot capacitated k-travelling repairmen problem (MD-CkTRP), we have a collection of clients to be served by one vehicle in a fleet of k identical vehicles based at given depots. Each client has a given demand that must be satisfied, and each vehicle can carry a total of at most Q demand before it must resupply at its original depot. We wish to route the vehicles in a way that obeys the constraints while minimizing the average time (latency) required to serve a client. This generalizes the Multi-depot k-Travelling Repairman Problem (MD-kTRP) [Chekuri and Kumar, IEEE-FOCS, 2003; Post and Swamy, ACM-SIAM SODA, 2015] to the capacitated vehicle setting, and while it has been previously studied [Lysgaard and Wohlk, EJOR, 2014; Rivera et al, Comput Optim Appl, 2015], no approximation algorithm with a proven ratio is known. We give a 42.49-approximation to this general problem, and refine this constant to 25.49 when clients have unit demands. As far as we are aware, these are the first constant-factor approximations for capacitated vehicle routing problems with a latency objective. We achieve these results by developing a framework allowing us to solve a wider range of latency problems, and crafting various orienteering-style oracles for use in this framework. We also show a simple LP rounding algorithm has a better approximation ratio for the maximum coverage problem with groups (MCG), first studied by Chekuri and Kumar [APPROX, 2004], and use it as a subroutine in our framework. Our approximation ratio for MD-CkTRP when restricted to uncapacitated setting matches the best known bound for it [Post and Swamy, ACM-SIAM SODA, 2015]. With our framework, any improvements to our oracles or our MCG approximation will result in improved approximations to the corresponding k-TRP problem

The multi-depot k-traveling repairman problem

In this paper, we study the multi-depot k-traveling repairman problem. This problem extends the traditional traveling repairman problem to the multi-depot case. Its objective, similar to the single depot variant, is the minimization of the sum of the arrival times to customers. We propose two distinct formulations to model the problem, obtained on layered graphs. In order to find feasible solutions for the largest instances, we propose a hybrid genetic algorithm where initial solutions are built using a splitting heuristic and a local search is embedded into the genetic algorithm. The efficiency of the mathematical formulations and of the solution approach are investigated through computational experiments. The proposed models are scalable enough to solve instances up to 240 customers

An Effective Metaheuristic for Multiple Traveling Repairman Problem with Distance Constraints

Multiple Traveling Repairman Problem with Distance Constraints (MTRPD) is an extension of the NP-hard Multiple Traveling Repairman Problem. In MTRPD, a fleet of identical vehicles is dispatched to serve a set of customers with the following constraints. First, each vehicle's travel distance is limited by a threshold. Second, each customer must be visited exactly once. Our goal is to find the visiting order that minimizes the sum of waiting times. To solve MTRPD we propose to combine the Insertion Heuristic (IH), Variable Neighborhood Search (VNS), and Tabu Search (TS) algorithms into an effective two-phase metaheuristic that includes a construction phase and an improvement phase. In the former phase, IH is used to create an initial solution. In the latter phase, we use VNS to generate various neighborhoods, while TS is employed to mainly prohibit from getting trapped into cycles. By doing so, our algorithm can support the search to escape local optima. In addition, we introduce a novel neighborhoods’ structure and a constant time operation which are efficient for calculating the cost of each neighboring solution. To show the efficiency of our proposed metaheuristic algorithm, we extensively experiment on benchmark instances. The results show that our algorithm can find the optimal solutions for all instances with up to 50 vertices in a fraction of seconds. Moreover, for instances from 60 to 80 vertices, almost all found solutions fall into the range of 0.9 %-1.1 % of the optimal solutions' lower bounds in a reasonable duration. For instances with a larger number of vertices, the algorithm reaches good-quality solutions fast. Moreover, in a comparison to the state-of-the-art metaheuristics, our proposed algorithm can find better solutions

Occupational Mobility and Wage Inequality, Second Version

In this study we argue that wage inequality and occupational mobility are intimately related. We are motivated by our empirical findings that human capital is occupation-specific and that the fraction of workers switching occupations in the United States was as high as 16% a year in the early 1970s and had increased to 19% by the early 1990s. We develop a general equilibrium model with occupation-specific human capital and heterogeneous experience levels within occupations. We argue that the increase in occupational mobility was due to the increase in the variability of productivity shocks to occupations. The model, calibrated to match the increase in occupational mobility, accounts for over 90% of the increase in wage inequality over the period. A distinguishing feature of the theory is that it accounts for changes in within-group wage inequality and the increase in the variability of transitory earnings.Occupational Mobility, Wage Inequality, Within-Group Inequality, Human Capital, Sectoral Reallocation