Static target search path planning optimization with heterogeneous agents

Disruptions in airline operations can result in infeasibilities in aircraft and passenger schedules. Airlines typically recover aircraft schedules and disruptions in passenger itineraries sequentially. However, passengers are severely affected by disruptions and recovery decisions. In this paper, we present a mathematical formulation for the integrated aircraft and passenger recovery problem that considers aircraft and passenger related costs simultaneously. Using the superimposition of aircraft and passenger itinerary networks, passengers are explicitly modeled in order to use realistic passenger related costs. In addition to the common routing recovery actions, we integrate several passenger recovery actions and cruise speed control in our solution approach. Cruise speed control is a very beneficial action for mitigating delays. On the other hand, it adds complexity to the problem due to the nonlinearity in fuel cost function. The problem is formulated as a mixed integer nonlinear programming (MINLP) model. We show that the problem can be reformulated as conic quadratic mixed integer programming (CQMIP) problem which can be solved with commercial optimization software such as IBM ILOG CPLEX. Our computational experiments have shown that we could handle several simultaneous disruptions optimally on a four-hub network of a major U.S. airline within less than a minute on the average. We conclude that proposed approach is able to find optimal tradeoff between operating and passenger-related costs in real time.     

Selection of Capacity Expansion Projects

The problem of determining a project selection schedule and a production-distribution-inventory schedule for each of a number of plants so as to meet the demands of multiregional markets at minimum discounted total cost during a discrete finite planning horizon is considered. We include the possibility of using inventory and/or imports to delay the expansion decision at each producing region in a transportation network. Through a problem reduction algorithm, the Lagrangean relaxation problem strengthened by the addition of a surrogate constraint becomes a 0–1 mixed integer knapsack problem. Its optimal solution, given a set of Lagrangean multipliers, can be obtained by solving at most two generally smaller 0–1 pure integer knapsack problems. The bound is usually very tight. At each iteration of the subgradient method, we generate a primal feasible solution from the Lagrangean solution. The computational results indicate that the procedure is effective in solving large problems to within acceptable error tolerances.     

Airport noise modelling and aircraft scheduling so as to minimize community annoyance

A mathematical model of the annoyance created at an airport by aircraft operations is developed. The model incorporates population distribution considerations around an airport and the annoyance caused by aircraft noise. The objective function of this model corresponds to seeking to minimize total population annoyance created by all aircraft operations in a 24-hour period. Several factors are included in this model as constraint relationships. Aircraft operations by type and time period are upper bounded. Demand for flight services is incorporated by including lower bounds on the number of operations by type of aircraft, runway used and time period. Also upper bounds on the number of operations for each runway are included. The mathematical model as formulated is recognized as corresponding to a nonlinear integer mathematical programming problem.The solution technique selected makes use of a successive linear approximation optimization algorithm. An especially attractive feature of this solution algorithm is that it is capable of obtaining solutions to large problems. For example, it would be feasible to attempt the solution of problems involving several thousand variables and over 500 linear constraints. This suggested solution algorithm was implemented on a computer and computational results obtained for example problems.     

A ring-mesh topology design problem for optical transport networks

This paper deals with a ring-mesh network design problem arising from the deployment of an optical transport network. The problem seeks to find an optimal clustering of traffic demands in the network such that the total cost of optical add-drop multiplexer (OADM) and optical cross-connect (OXC) is minimized, while satisfying the OADM ring capacity constraint, the node cardinality constraint, and the OXC capacity constraint. We formulate the problem as an integer programming model and propose several alternative modeling techniques designed to improve the mathematical representation of the problem. We then develop various classes of valid inequalities to tighten the mathematical formulation of the problem and describe an algorithmic approach that coordinates tailored routines with a commercial solver CPLEX. We also propose an effective tabu search procedure for finding a good feasible solution as well as for providing a good incumbent solution for the column generation based heuristic procedure that enhances the solvability of the problem. Computational results exhibit the viability of the proposed method.     

Simple solution methods for separable mixed linear and quadratic knapsack problem

Quadratic knapsack problem has a central role in integer and nonlinear optimization, which has been intensively studied due to its immediate applications in many fields and theoretical reasons. Although quadratic knapsack problem can be solved using traditional nonlinear optimization methods, specialized algorithms are much faster and more reliable than the nonlinear programming solvers. In this paper, we study a mixed linear and quadratic knapsack with a convex separable objective function subject to a single linear constraint and box constraints. We investigate the structural properties of the studied problem, and develop a simple method for solving the continuous version of the problem based on bi-section search, and then we present heuristics for solving the integer version of the problem. Numerical experiments are conducted to show the effectiveness of the proposed solution methods by comparing our methods with some state of the art linear and quadratic convex solvers.     

AN EFFECTIVE CONTINUOUS ALGORITHM FOR APPROXIMATE SOLUTIONS OF LARGE SCALE MAX-CUT PROBLEMS

An effective continuous algorithm is proposed to find approximate solutions of NP-hardmax-cut problems.The algorithm relaxes the max-cut problem into a continuous nonlinearprogramming problem by replacing n discrete constraints in the original problem with onesingle continuous constraint.A feasible direction method is designed to solve the resultingnonlinear programming problem.The method employs only the gradient evaluations ofthe objective function,and no any matrix calculations and no line searches are required.This greatly reduces the calculation cost of the method,and is suitable for the solutionof large size max-cut problems.The convergence properties of the proposed method toKKT points of the nonlinear programming are analyzed.If the solution obtained by theproposed method is a global solution of the nonlinear programming problem,the solutionwill provide an upper bound on the max-cut value.Then an approximate solution to themax-cut problem is generated from the solution of the nonlinear programming and providesa lower bound on the max-cut value.Numerical experiments and comparisons on somemax-cut test problems(small and large size)show that the proposed algorithm is efficientto get the exact solutions for all small test problems and well satisfied solutions for mostof the large size test problems with less calculation costs.     

Convex programming with single separable constraint and bounded variables

In this paper a minimization problem with convex objective function subject to a separable convex inequality constraint “≤” and bounded variables (box constraints) is considered. We propose an iterative algorithm for solving this problem based on line search and convergence of this algorithm is proved. At each iteration, a separable convex programming problem with the same constraint set is solved using Karush-Kuhn-Tucker conditions. Convex minimization problems subject to linear equality/ linear inequality “≥” constraint and bounds on the variables are also considered. Numerical illustration is included in support of theory.     

A Reference Direction Algorithm for Solving Multiple Objective Integer Linear Programming Problems

In this paper, we propose a reference direction approach and an interactive algorithm to solve the general multiple objective integer linear programming problem. At each iteration, only one mixed integer linear programming problem is solved to find an (weak) efficient solution. Each intermediate solution is integer. The decision maker has to provide only the reference point at each iteration. No special software is required to implement the proposed algorithm. The algorithm is illustrated with an example.     

Improving the computational efficiency of metric-based spares algorithms

We propose a new heuristic algorithm to improve the computational efficiency of the general class of Multi-Echelon Technique for Recoverable Item Control (METRIC) problems. The objective of a METRIC-based decision problem is to systematically determine the location and quantity of spares that either maximizes the operational availability of a system subject to a budget constraint or minimizes its cost subject to an operational availability target. This type of sparing analysis has proven essential when analyzing the sustainment policies of large-scale, complex repairable systems such as those prevalent in the defense and aerospace industries. Additionally, the frequency of these sparing studies has recently increased as the adoption of performance-based logistics (PBL) has increased. PBL represents a class of business strategies that converts the recurring cost associated with maintenance, repair, and overhaul (MRO) into cost avoidance streams. Central to a PBL contract is a requirement to perform a business case analysis (BCA) and central to a BCA is the frequent need to use METRIC-based approaches to evaluate how a supplier and customer will engage in a performance based logistics arrangement where spares decisions are critical. Due to the size and frequency of the problem there exists a need to improve the efficiency of the computationally intensive METRIC-based solutions. We develop and validate a practical algorithm for improving the computational efficiency of a METRIC-based approach. The accuracy and effectiveness of the proposed algorithm are analyzed through a numerical study. The algorithm shows a 94% improvement in computational efficiency while maintaining 99.9% accuracy.     

Approximate Greatest Descent Methods for Optimization with Equality Constraints

In an optimization problem with equality constraints the optimal value function divides the state space into two parts. At a point where the objective function is less than the optimal value, a good iteration must increase the value of the objective function. Thus, a good iteration must be a balance between increasing or decreasing the objective function and decreasing a constraint violation function. This implies that at a point where the constraint violation function is large, we should construct noninferior solutions relative to points in a local search region. By definition, an accessory function is a linear combination of the objective function and a constraint violation function. We show that a way to construct an acceptable iteration, at a point where the constraint violation function is large, is to minimize an accessory function. We develop a two-phases method. In Phase I some constraints may not be approximately satisfied or the current point is not close to the solution. Iterations are generated by minimizing an accessory function. Once all the constraints are approximately satisfied, the initial values of the Lagrange multipliers are defined. A test with a merit function is used to determine whether or not the current point and the Lagrange multipliers are both close to the optimal solution. If not, Phase I is continued. If otherwise, Phase II is activated and the Newton method is used to compute the optimal solution and fast convergence is achieved.