Combinatorial optimization over two random point sets

We analyze combinatorial optimization problems over a pair of random point sets of equal cardinal. Typical examples include the matching of minimal length, the traveling salesperson tour constrained to alternate between points of each set, or the connected bipartite r-regular graph of minimal length. As the cardinal of the sets goes to infinity, we investigate the convergence of such bipartite functionals.Comment: 34 page

Strategic dynamic vehicle routing with spatio-temporal dependent demands

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 51-53).Dynamic vehicle routing problems address the issue of determining optimal routes for a set of vehicles, to serve a given set of demands that arrive sequentially in time. Traditionally, demands are assumed to be generated over time by an exogenous stochastic process. This thesis is concerned with the study of dynamic vehicle routing problems where demands are strategically placed in the space by an agent with selfish interests and physical constraints. In particular, we focus on the following problem: a team of vehicles seek to device dynamic routing policies that minimize the average waiting time of a typical demand, from the moment it is placed in the space until its location is visited; while an adversarial agent operating from a central depot with limited capacity aims at the opposite, strategically choosing the spatio-temporal point process according to which place demands. We model the above problem and its inherent pure conflict of interests as a zero-sum game, and characterize equilibria under heavy load regime. For the analysis we discriminate between two cases: bounded and unbounded domains. In both cases we show that a routing policy based on performing successive TSP tours through outstanding demands and a power-law spatial distribution of demands are optimal, saddle point of the utility function of the game. The latter emerges as the unique solution of maximizing a non-convex nowhere differentiable functional over the infinite-dimensional space of probability densities; the non-convexity is the result of the spatio-temporal dependence induced by the physical constraints imposed on the behavior of the agent, and the non-differentiability is due to the emptiness of the interior of the positive cone of integrable functions. We solve this problem applying Fenchel conjugate duality for partially finite programming in the case of bounded domains; and a direct duality approach exploiting the structure of a concave integral functional part of the objective and the linear equality constraints, for unbounded domains. Remarkably, all the results obtained hold for any domain with a sufficiently smooth boundary, clossedness or connectedness is not needed. We provide numerical simulations to validate the theory.by Diego Feijer.S.M

Dynamic Vehicle Routing for Robotic Systems

Recent years have witnessed great advancements in the science and technology of autonomy, robotics, and networking. This paper surveys recent concepts and algorithms for dynamic vehicle routing (DVR), that is, for the automatic planning of optimal multivehicle routes to perform tasks that are generated over time by an exogenous process. We consider a rich variety of scenarios relevant for robotic applications. We begin by reviewing the basic DVR problem: demands for service arrive at random locations at random times and a vehicle travels to provide on-site service while minimizing the expected wait time of the demands. Next, we treat different multivehicle scenarios based on different models for demands (e.g., demands with different priority levels and impatient demands), vehicles (e.g., motion constraints, communication, and sensing capabilities), and tasks. The performance criterion used in these scenarios is either the expected wait time of the demands or the fraction of demands serviced successfully. In each specific DVR scenario, we adopt a rigorous technical approach that relies upon methods from queueing theory, combinatorial optimization, and stochastic geometry. First, we establish fundamental limits on the achievable performance, including limits on stability and quality of service. Second, we design algorithms, and provide provable guarantees on their performance with respect to the fundamental limits.United States. Air Force Office of Scientific Research (Award FA 8650-07-2-3744)United States. Army Research Office. Multidisciplinary University Research Initiative (Award W911NF-05-1-0219)National Science Foundation (U.S.) (Award ECCS-0705451)National Science Foundation (U.S.) (Award CMMI-0705453)United States. Army Research Office (Award W911NF-11-1-0092

Dynamic systems and subadditive functionals

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 125-131).Consider a problem where a number of dynamic systems are required to travel between points in minimum time. The study of this problem is traditionally divided into two parts: A combinatorial part that assigns points to every dynamic system and assigns the order of the traversal of the points, and a path planning part that produces the appropriate control for the dynamic systems to allow them to travel between the points. The first part of the problem is usually studied without consideration for the dynamic constraints of the systems, and this is usually compensated for in the second part. Ignoring the dynamics of the system in the combinatorial part of the problem can significantly compromise performance. In this work, we introduce a framework that allows us to tackle both of these parts at the same time. To that order, we introduce a class of functionals we call the Quasi-Euclidean functionals, and use them to study such problems for dynamic systems. We determine the asymptotic behavior of the costs of these problems, when the points are randomly distributed and their number tends to infinity. We show the applicability of our framework by producing results for the Traveling Salesperson Problem (TSP) and Minimum Bipartite Matching Problem (MBMP) for dynamic systems.by Sleiman M. Itani.Ph.D

Probabilistic analysis of euclidean multi depot vehicle routing and related problems

We consider a generalization of the classical traveling salesman problem: the multi depot vehicle routing problem (MDVRP). Let $D$ be a set of $k$ depots and $P$ be sets $n$ customers in $[0,1]^d$ with the usual Euclidean metric. A multi depot vehicle routing tour is a set of disjoint cycles such that all customers are covered and each cycle contains exactly one depot. The goal is to find a tour of minimum length. $L(D,P)$ denotes the length of an optimal MDVRP tour for depot set $D$ and customer set $P$. It is evident that the asymptotic behavior of \L(D,P) for $n$ tending to infinity depends on the customer-depot ratio $n/k$. We study three cases: $k=o(n)$, $k=\lambda n +o(n)$ for a constant $\lambda >0$, and k=\Omega(n^{1+\ee}) for \ee>0. In the first two cases we show that $L(D,P)$ divided by $n^{(d-1)/d}$ converges completely to a constant if the customers and depots are given by iid random variables. In the last case we prove that the expected tour length divided by $n^{(d-1)/d}$ and multiplied by $k^{1/d}$ converges to a constant if the customers and depots are given by iid random variables with uniform distribution

Adjustment Factors and Applications for Analytic Approximations of Tour Lengths

The shortest tour distance for visiting all points exactly once and returning to the origin is computed by solving the well-known Traveling Salesman Problem (TSP). Due to the large computational effort needed for optimizing TSP tours, researchers have developed approximations that relate the average length of TSP tours to the number of points n visited per tour. The most widely used approximation formula has a square root form: √n multiplied by a coefficient β. Although the existing models can effectively approximate the distance for conventional vehicles with large capacities (e.g., delivery trucks) where n is large, approximations that seek to cover large ranges of n, possibly to infinity, tend to yield poorer results for the small n values. Thus, this dissertation focuses on approximation models for the small n values, which are needed for many practical applications, such as for some recent delivery alternatives (e.g., drones). The proposed models show promise in analyzing the real-world problems in which actual tours serve few customers due to limited vehicle capacity and incorporate realistic constraints, such as the effects of a starting point location, geographical restrictions on movements, demand patterns, and service area shapes. The dissertation may open new research avenues for analyzing the new transportation alternatives and provide guidelines to planners for choosing appropriate models in designing or evaluating transportation problems. Approximation models are estimated from the following experiments: 1) a total of 60 cases are developed by considering various factors, such as point distributions and shapes of service areas. 2) Solution methods for TSP instances are compared and chosen. 3) After the TSPs are optimized for each n, the TSP tour lengths are averaged. 4) Lastly, models for the averaged TSP tour lengths are fitted with ordinary least squares (OLS) regression. After the approximations are developed, some possible extensions are explored. First, adjustment factors are designed to integrate the 60 cases within one equation. With those factors, it can be understood how approximation varies with each classification. Next, the approximations considering stochastic customer presence (i.e., probabilistic TSP) are proposed. Third, the approximated tour lengths are compared with the optimal solutions of vehicle routing problem (VRP) in actual rural and urban delivery networks. Here, some additional factors, such as a circuity factor and service zone shape, are discussed. Lastly, the proposed methodology is applied to formulate and explore various types of existing and hypothetical delivery alternatives