Moritz Niendorf

Stability of Solutions to Classes of Traveling Salesman Problems

By performing stability analysis on an optimal tour for problems belonging to classes of the traveling salesman problem (TSP), this paper derives margins of optimality for a solution with respect to disturbances in the problem data. Specifically, we consider the asymmetric sequence-dependent TSP, where the sequence dependence is driven by the dynamics of a stack. This is a generalization of the symmetric non sequence-dependent version of the TSP. Furthermore, we also consider the symmetric sequence-dependent variant and the asymmetric non sequence-dependent variant. Amongst others these problems have applications in logistics and unmanned aircraft mission planning. Changing external conditions such as traffic or weather may alter task costs, which can render an initially optimal itinerary suboptimal. Instead of optimizing the itinerary every time task costs change, stability criteria allow for fast evaluation of whether itineraries remain optimal. This paper develops a method to compute stability regions for the best tour in a set of tours for the symmetric TSP and extends the results to the asymmetric problem as well as their sequence-dependent counterparts. As the TSP is NP-hard, heuristic methods are frequently used to solve it. The presented approach is also applicable to analyze stability regions for a tour obtained through application of the k-opt heuristic with respect to the k-neighborhood. A dimensionless criticality metric for edges is proposed, such that a high criticality of an edge indicates that the optimal tour is more susceptible to cost changes in that edge. Multiple examples demonstrate the application of the developed stability computation method as well as the edge criticality measure that facilitates an intuitive assessment of instances of the TSP.

Improved 3D Interpolation-Based Path Planning for a Fixed-Wing Unmanned Aircraft

Path planning for unmanned aircraft has attracted a remarkable amount of interest from the research community. However, planning in large environments such as the civil airspace has not been addressed extensively. In this paper we apply a heuristic incremental interpolation-based search algorithm with efficient replanning capabilities to the path planning problem for a fixed-wing aircraft operating in a natural environment to plan and re-plan long flight paths. We modified the algorithm to account for the minimum turning radius and the limited flight path angles of a fixed-wing aircraft. Additionally, we present a method to consider a desired minimum cruising altitude and a post-processing algorithm to improve the path and remove unnecessary path points. These properties specific to aircraft operation could not be addressed with the original algorithm. Simulation results show that the planner produces intuitive, short paths and is capable of exploiting previous planning efforts, when unknown obstacles are encountered.

Stability analysis of stochastic integer optimization problems

This paper presents the stability analysis of integer linear programs with respect to perturbations in stochastic data, namely Markov chains. These perturbations affect the initial distribution, the transition matrix, or the stationary distribution of Markov chains. Stability analysis is concerned with obtaining the set of all perturbations for which a solution to the integer optimization problem remains optimal. In particular, we derive expressions for stability regions for perturbations in the initial distribution, the transition matrix and the stationary distribution. The constraints that preserve the stochasticity of the problem data are affine. The intersection of the stability region for arbitrary perturbations with these affine constraints yields the desired stability regions. Finally, stability regions for perturbations of elements of the transition matrix, given that the problem is linear in the stationary distribution of that transition matrix, are obtained using a small perturbation analysis. The results are applied to sensor placement problems, where a phenomenon modeled as a Markov chain needs to be detected, and numerical examples are given.

Polynomial time sensitivity analysis of task schedules

By performing sensitivity analysis on an optimal task schedule, this paper derives a polynomial time method to determine whether the task schedule remains optimal after arbitrary changes to task costs occur. We consider fast reactive mission planning for unmanned aircraft in changing environments. Changing external conditions such as weather or threats may alter task costs, which can render an initially optimal task schedule suboptimal. Instead of optimizing the task schedule every time task costs change, stability criteria allow for fast evaluation of whether schedules remain optimal. This paper develops a method to compute stability regions for a set of schedules in a prototypical mission for unmanned aircraft, the traveling salesman problem, where the alternative schedules are part of a pre-approved mission plan. As the traveling salesman problem is NP-hard, heuristic methods are frequently used to solve it. The presented approach is also applicable to analyze stability regions for a tour obtained through application of the k-opt heuristic with respect to the k-neighborhood and is demonstrated with an example problem.

Stability and Criticality Analysis for Integer Linear Programs With Markovian Problem Data

This paper presents the stability and criticality analysis of integer linear programs with respect to perturbations in stochastic data given as Markov chains. These perturbations affect the initial distribution, the transition matrix, or the stationary distribution of Markov chains. Stability analysis is concerned with obtaining the set of all perturbations for which a solution remains optimal. This paper gives expressions for stability regions for perturbations in the initial distribution, the transition matrix, the stationary distribution, and the product of elements of the transition matrix and the stationary distribution. Furthermore, criticality measures that describe the sensitivity of the objective function with respect to an element of the problem data are derived. Stability regions that preserve the stochasticity of the problem data are given. Finally, stability regions for perturbations of elements of the transition matrix, given that the problem is not linear in the initial distribution or the transition matrix, are obtained using a small perturbation analysis. The results are applied to sensor placement problems and numerical examples are given.

Stability analysis of multi-objective planning problems for unmanned aircraft

By performing sensitivity analysis on multi-objective mission planning problems for unmanned aircraft, this paper provides insight into properties of optimal solutions and their robustness with respect to trade-offs between objectives and also perturbations in the objective functions. We consider mission planning for unmanned aircraft in hostile environments, where a trade-off between short mission execution time and low risk must be made. This paper develops a method to compute stability regions for a set of schedules for a prototypical mission for unmanned aircraft, the weighted multi-objective symmetric traveling salesman problem. A stability region associated with an optimal solution is the set of all perturbations to the input parameters for which that solution remains optimal. In particular, this paper studies perturbations to the weights and costs in the objective functions. This work has multiple potential benefits such as providing insight into how robust a solution is with respect to modeling uncertainties, how limited intelligence resources should be allocated, and the influence of each objective on the solution. A numerical example is given to demonstrate the methods developed in this paper.

Stability analysis of optimal runway schedules

By performing stability analysis on an optimal runway schedule, this paper derives a method to determine whether the landing sequence of aircraft remains optimal after an arbitrary number of aircraft in that sequence are delayed by an arbitrary amount of time. We consider the problem of scheduling aircraft landing on a single runway such that the runway throughput is maximized under changing external conditions such as delays caused by for example weather. Instead of optimizing the schedule every time delays occur, stability criteria allow for fast evaluation of whether schedules remain optimal. This paper develops a method to compute stability regions for a set of schedules. Sensitivity analysis of the linear programming relaxation and a nonlinear relationship between the delay of individual aircraft and the incurred cost change of any potential landing sequence yield the desired stability information. Furthermore, the properties of a greedy first come first serve policy are studied by giving sufficient conditions to determine whether first come first serve is optimal.

Robustness of communication links for teams of unmanned aircraft by sensitivity analysis of minimum spanning trees

This paper addresses the problem of sensitivity analysis of minimum spanning trees on a weighted graph and its application to optimal communication topologies for teams of unmanned aerial vehicles. In particular, we analyze the robustness of a minimum spanning tree with respect to changing edge weights due to movement of the aircraft. Stability analysis is used to determine whether a previously computed minimum spanning tree remains optimal after the edge weights change. Building upon necessary and sufficient conditions for optimality of a spanning tree, we derive a polyhedral description of the stability region, that is, the set of all perturbations to the edge weights for which a given minimum spanning tree remains optimal, and analyze its properties and relevant subsets. The application of sensitivity analysis to robustness assessment of an optimal communication topology for teams of unmanned aircraft is demonstrated through numerical simulations.

Exact and Approximate Stability of Solutions to Traveling Salesman Problems

This paper presents the stability analysis of an optimal tour for the symmetric traveling salesman problem (TSP) by obtaining stability regions. The stability region of an optimal tour is the set of all cost changes for which that solution remains optimal and can be understood as the margin of optimality for a solution with respect to perturbations in the problem data. It is known that it is not possible to test in polynomial time whether an optimal tour remains optimal after the cost of an arbitrary set of edges changes. Therefore, this paper develops tractable methods to obtain under and over approximations of stability regions based on neighborhoods and relaxations. The application of the results to the two-neighborhood and the minimum 1 tree (M1T) relaxation are discussed in detail. For Euclidean TSPs, stability regions with respect to vertex location perturbations and the notion of safe radii and location criticalities are introduced. Benefits of this paper include insight into robustness properties of tours, minimum spanning trees, M1Ts, and fast methods to evaluate optimality after perturbations occur. Numerical examples are given to demonstrate the methods and achievable approximation quality.

Stability Analysis of Runway Schedules

By performing stability analysis on an optimal runway schedule, this paper derives a method to determine whether an optimized landing sequence of aircraft remains optimal after an arbitrary number of aircraft in that sequence are delayed by an arbitrary amount of time. We consider the problem of scheduling aircraft landing on a single runway with the objective of maximizing throughput under changing external conditions such as delays caused by weather. Instead of optimizing the schedule every time delays occur, stability criteria allow for fast evaluation of whether schedules remain optimal. This paper develops a method to compute stability regions for a set of schedules. Sensitivity analysis of the linear programming relaxation and a nonlinear relationship between the delay of individual aircraft and the incurred cost change for all landing sequences yield the stability information. Furthermore, the properties of a first-come-first-serve policy are studied by giving sufficient conditions and a heuristic condition for the optimality of first-come-first-serve sequences. The given results are shown to be also applicable to landing sequences obtained through local neighborhood search, sequences that obey a position shift constraint, and subsequences of landing sequences as used in a rolling horizon approach.