Path Planning Algorithms for Multiple Heterogeneous Vehicles

Unmanned aerial vehicles (UAVs) are becoming increasingly popular for surveillance in civil and military applications. Vehicles built for this purpose vary in their sensing capabilities, speed and maneuverability. It is therefore natural to assume that a team of UAVs given the mission of visiting a set of targets would include vehicles with differing capabilities. This paper addresses the problem of assigning each vehicle a sequence of targets to visit such that the mission is completed with the least "cost" possible given that the team of vehicles is heterogeneous. In order to simplify the problem the capabilities of each vehicle are modeled as cost to travel from one target to another. In other words, if a vehicle is particularly suited to visit a certain target, the cost for that vehicle to visit that target is low compared to the other vehicles in the team. After applying this simplification, the problem can be posed as an instance of the combinatorial problem called the Heterogeneous Travelling Salesman Problem (HTSP). This paper presents a transformation of a Heterogenous, Multiple Depot, Multiple Traveling Salesman Problem (HMDMTSP) into a single, Asymmetric, Traveling Salesman Problem (ATSP). As a result, algorithms available for the single salesman problem can be used to solve the HMDMTSP. To show the effectiveness of the transformation, the well known Lin-Kernighan-Helsgaun heuristic was applied to the transformed ATSP. Computational results show that good quality solutions can be obtained for the HMDMTSP relatively fast. Additional complications to the sequencing problem come in the form of precedence constraints which prescribe a partial order in which nodes must be visited. In this context the sequencing problem was studied seperately using the Linear Program (LP) relaxation of a Mixed Integer Linear Program (MILP) formulation of the combinatorial problem known as the "Precedence Constrained Asymmetric Travelling Salesman Problem" (PCATSP)

GPU accelerated Hungarian algorithm for traveling salesman problem

In this thesis, we present a model of the Traveling Salesman Problem (TSP) cast in a quadratic assignment problem framework with linearized objective function and constraints. This is referred to as Reformulation Linearization Technique at Level 2 (or RLT2). We apply dual ascent procedure for obtaining lower bounds that employs Linear Assignment Problem (LAP) solver recently developed by Date(2016). The solver is a parallelized Hungarian Algorithm that uses Compute Unified Device Architecture (CUDA) enabled NVIDIA Graphics Processing Units (GPU) as the parallel programming architecture. The aim of this thesis is to make use of a modified version of the Dual Ascent-LAP solver to solve the TSP. Though this procedure is computational expensive, the bounds obtained are tight and our experimental results confirm that the gap is within 2% for most problems. However, due to limitations in computational resources, we could only test problem sizes N < 30. Further work can be directed at theoretical and computational analysis to test the efficiency of our approach for larger problem instances

Mathematical Modelling for Load Balancing and Minimization of Coordination Losses in Multirobot Stations

The automotive industry is moving from mass production towards an individualized production, in order to improve product quality and reduce costs and material waste. This thesis concerns aspects of load balancing of industrial robots in the automotive manufacturing industry, considering efficient algorithms required by an individualized production. The goal of the load balancing problem is to improve the equipment utilization. Several approaches for solving the load balancing problem are presented along with details on mathematical tools and subroutines employed.Our contributions to the solution of the load balancing problem are manifold. First, to circumvent robot coordination we have constructed disjoint robot programs, which require no coordination schemes, are more flexible, admit competitive cycle times for some industrial instances, and may be preferred in an individualized production. Second, since solving the task assignment problem for generating the disjoint robot programs was found to be unreasonably time-consuming, we modelled it as a generalized unrelated parallel machine problem with set packing constraints and suggested a tighter model formulation, which was proven to be much more tractable for a branch--and--cut solver. Third, within continuous collision detection it needs to be determined whether the sweeps of multiple moving robots are disjoint. Our solution uses the maximum velocity of each robot along with distance computations at certain robot configurations to derive a function that provides lower bounds on the minimum distance between the sweeps. The lower bounding function is iteratively minimized and updated with new distance information; our method is substantially faster than previously developed methods

Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

Constrained Task Assignment and Scheduling on Networks of Arbitrary Topology.

This dissertation develops a framework to address centralized and distributed constrained task assignment and task scheduling problems. This framework is used to prove properties of these problems that can be exploited, develop effective solution algorithms, and to prove important properties such as correctness, completeness and optimality. The centralized task assignment and task scheduling problem treated here is expressed as a vehicle routing problem with the goal of optimizing mission time subject to mission constraints on task precedence and agent capability. The algorithm developed to solve this problem is able to coordinate vehicle (agent) timing for task completion. This class of problems is NP-hard and analytical guarantees on solution quality are often unavailable. This dissertation develops a technique for determining solution quality that can be used on a large class of problems and does not rely on traditional analytical guarantees. For distributed problems several agents must communicate to collectively solve a distributed task assignment and task scheduling problem. The distributed task assignment and task scheduling algorithms developed here allow for the optimization of constrained military missions in situations where the communication network may be incomplete and only locally known. Two problems are developed. The distributed task assignment problem incorporates communication constraints that must be satisfied; this is the Communication-Constrained Distributed Assignment Problem. A novel distributed assignment algorithm, the Stochastic Bidding Algorithm, solves this problem. The algorithm is correct, probabilistically complete, and has linear average-case time complexity. The distributed task scheduling problem addressed here is to minimize mission time subject to arbitrary predicate mission constraints; this is the Minimum-time Arbitrarily-constrained Distributed Scheduling Problem. The Optimal Distributed Non-sequential Backtracking Algorithm solves this problem. The algorithm is correct, complete, outputs time optimal schedules, and has low average-case time complexity. Separation of the task assignment and task scheduling problems is exploited here to ameliorate the effects of an incomplete communication network. The mission-modeling conditions that allow this and the benefits gained are discussed in detail. It is shown that the distributed task assignment and task scheduling algorithms developed here can operate concurrently and maintain their correctness, completeness, and optimality properties.Ph.D.Aerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91527/1/jpjack_1.pd

Solution of a practical Vehicle Routing Problem for monitoring Water Distribution Networks

In this work, we introduce a generalisation of the Vehicle Routing Problem for a specific application in the monitoring of a Water Distribution Network (WDN). In this problem, multiple technicians must visit a sequence of nodes in the WDN and perform a series of tests to check the quality of water. Some special nodes (i.e., wells) require technicians to first collect a key from a key centre. The key must then be returned to the same key centre after the test has been performed, thus introducing precedence constraints and multiple visits in the routes. To solve the problem, a Mixed Integer Linear Programming model and an Iterated Local Search have been implemented. The efficiency of the proposed methods is demonstrated by means of extensive computational tests on randomly created and real-world instances

Layered graph approaches for combinatorial optimization problems

Extending the concept of time-space networks, layered graphs associate information about one or multiple resource state values with nodes and arcs. While integer programming formulations based on them allow to model complex problems comparably easy, their large size makes them hard to solve for non-trivial instances. We detail and classify layered graph modeling techniques that have been used in the (recent) scientific literature and review methods to successfully solve the resulting large-scale, extended formulations. Modeling guidelines and important observations concerning the solution of layered graph formulations by decomposition methods are given together with several future research directions

School bus selection, routing and scheduling.

The aim of this thesis is to develop formulations and exact algorithms for the school bus routing and scheduling problem and to develop an integrated software implementation using Xpress-MP/CPLEX and ArcGIS of ESRI, a geographical information system software package. In this thesis, bus flow, single commodity flow, two-commodity flow, multi-commodity flow, and time window formulations have been developed. They capture all of the important elements of the School Bus Routing and Scheduling Problem (SBRSP) including homogeneous or heterogeneous bus fleets, the identification of bus stops from a large set of potential bus stops, and the assignment of students to stops and stops to routes. They allow for the one stop-one bus and one stop-multi bus scenarios. Each formulation of the SBRSP has a linear programming relaxation and we present the relationships among them. We present a Branch-and-Cut exact algorithm which makes use of new linearization techniques, new valid inequalities, and the first valid equalities. We develop an integrated software package that is based on Geographical Information System (GIS) map-based interface, linking to an Xpress-MP/CPLEX solver. The interface between GIS and Xpress-MP is written in VBA and VC++.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2005 .K4. Source: Dissertation Abstracts International, Volume: 66-11, Section: B, page: 6250. Thesis (Ph.D.)--University of Windsor (Canada), 2005