Scatter Search and Bionomic Algorithms for the aircraft landing problem

The problem of deciding how to land aircraft approaching an airport involves assigning each aircraft to an appropriate runway, computing a landing sequence for each runway and scheduling the landing time for each aircraft. Runway allocation, sequencing and scheduling for each aircraft must ensure the scheduled landing time lies within a predefined time window and meet separation time requirements with other aircraft. The objective is to achieve effective runway use.In this paper, the multiple runway case of the static Aircraft Landing Problem is considered. Two heuristic techniques are presented: Scatter Search and the Bionomic Algorithm, population heuristic approaches that have not been applied to this problem before.Computational results are presented for publicly available test problems involving up to 500 aircraft and five runways showing that feasible solutions of good quality can be produced relatively quickly.     

A genetic local search algorithm with a threshold accepting mechanism for solving the runway dependent aircraft landing problem

As the demand for air transportation continues to grow, some flights cannot land at their preferred landing times because the airport is near its runway capacity. Extra fuel consumption and air pollution are then caused by the landing delays. Moreover, such delays may possibly yield extra costs for both passengers and airline companies that result from rescheduling transfer passengers and crew members. Consequently, how to increase the handling efficiency of congested airports is a crucial management issue. Building new runways at existing airports is often not feasible due to environmental, financial and geographical constraints. Therefore, devising a method for tackling the aircraft landing problem (ALP) in order to optimize the usage of existing runways at airports is the focus of this paper. This paper aims to develop a solution procedure based on a genetic local search (GLS) algorithm for solving the ALP with runway dependent attributes. A set of numerical experiments were conducted to test the validity of the proposed algorithm based on five test instances created and investigated by previous studies. The numerical results showed that the proposed GLS algorithm can effectively and efficiently determine the runway allocation, sequence and landing time for arriving aircraft for the five test cases by minimizing total delays under the separation constraints in comparison with the outcomes yielded by previous studies.     

A gaussian upper bound for gaussian multi-stage stochastic linear programs

This paper deals with two-stage and multi-stage stochastic programs in which the right-hand sides of the constraints are Gaussian random variables. Such problems are of interest since the use of Gaussian estimators of random variables is widespread. We introduce algorithms to find upper bounds on the optimal value of two-stage and multi-stage stochastic (minimization) programs with Gaussian right-hand sides. The upper bounds are obtained by solving deterministic mathematical programming problems with dimensions that do not depend on the sample space size. The algorithm for the two-stage problem involves the solution of a deterministic linear program and a simple semidefinite program. The algorithm for the multi-stage problem invovles the solution of a quadratically constrained convex programming problem.     

A Simulation Model for Calculating Annual Congestion Delay Arising from Airport Runway Operations

This paper describes a simulation model of delays to aircraft caused by airport runway congestion. It was developed for the Australian Government as part of the Sydney Airport Project. Subsequently, it was used in a traffic management study of the airport which examined the scope of deferring the need for additional runway capacity by adopting administrative measures affecting runway utilization.     

Scheduling aircraft landings at London Heathrow using a population heuristic

With increasing levels of air traffic, making effective use of limited airport capacity is obviously important. This paper reports on an investigation undertaken by National Air Traffic Services in the UK into improving runway utilisation at London Heathrow. This investigation centred on developing an algorithm for improving the scheduling of aircraft waiting to land. The heuristic algorithm developed (a population heuristic) is discussed and results presented using actual operational data relating to aircraft landings at London Heathrow. This data indicates that our algorithm could have improved on air traffic control decisions in such cases by between 2–5?% in terms of reducing the timespan required to land all of the aircraft considered.     

Some effects of aircraft arrival sequence algorithms

Although various airport landing sequencing algorithms have been considered in the literature, little work has been done in comparing their effects on Air Traffic Control, especially against first-come first-served (FCFS) runway sequences, the method most widely used in practice. This paper compares a number of such algorithms using a discrete-event simulation model of an airport with a single landing runway. Statistical methods are used to test for effects of sequencing algorithm, delay-sharing strategy, arrival rate and wake-vortex mix. Little benefit to delay, or stability of sequencing advice, is found from advanced sequencing when small changes are made to inputs calibrated to a specific airspace. Advanced sequencing improves landing rate, compared with FCFS sequencing, only when aircraft arrival rate is greater than maximum runway landing rate, and wake-vortex mix is sufficiently varied. Constrained position shifting constraints limit these improvements and it is shown that deterministic optimal techniques may actually be sub-optimal in a dynamic environment. Our main conclusion is that FCFS is a robust method under many conditions.     

Models and a genetic algorithm for static aircraft-gate assignment problem

Assigning aircraft to available gates at an airport can have a major impact on the efficiency of flight schedules and on the level of passenger satisfaction with the service. Unexpected changes, due to air traffic delays, severe weather conditions, or equipment failures, may disrupt the initial assignments and compound the difficulty of maintaining smooth station operations. Recently, mathematical models and procedures (optimal and heuristic) have been proposed to provide solutions with minimum dispersion of idle time periods for static aircraft-gate assignment problems. This paper introduces a unified framework to specifically treat the objective functions of the previous models. It also provides linear representations of these models and identifies the conditions under which the optimal solutions can be obtained in polynomial time. Furthermore, a genetic algorithm utilizing problem specific knowledge is proposed to provide effective alternative solutions.     

Image space analysis of generalized fractional programs

The solution of a particular nonconvex program is usually very dependent on the structure of the problem. In this paper we identify classes of nonconvex problems involving either sums or products of ratios of linear terms which may be treated by analysis in a transformed space. In each class, the image space is defined by a mapping which associates a new variable with each original ratio of linear terms. In the image space, optimization is easy in certain directions, and the overall solution may be realized by sequentially optimizing in these directions.In addition to these ratio problems, we also show how to use image space analysis to treat the subclass of problems whose objective is to optimize a product of linear terms. For each class of nonconvex problems, we present an algorithm that locates global solutions by computing both upper and lower bounds on the solution and then solving a sequence of linear programming sub-problems. We also demonstrate the algorithms described in this paper by solving several example problems.     

A lexicographic minimax algorithm for multiperiod resource allocation

Resource allocation problems are typically formulated as mathematical programs with some special structure that facilitates the development of efficient algorithms. We consider a multiperiod problem in which excess resources in one period can be used in subsequent periods. The objective minimizes lexicographically the nonincreasingly sorted vector of weighted deviations of cumulative activity levels from cumulative demands. To this end, we first develop a new minimax algorithm that minimizes the largest weighted deviation among all cumulative activity levels. The minimax algorithm handles resource constraints, ordering constraints, and lower and upper bounds. At each iteration, it fixes certain variables at their lower bounds, and sets groups of other variables equal to each other as long as no lower bounds are violated. The algorithm takes advantage of the problem's special structure; e.g., each term in the objective is a linear decreasing function of only one variable. This algorithm solves large problems very fast, orders of magnitude faster than well known linear programming packages. (The latter are, of course, not designed to solve such minimax problems efficiently.) The lexicographic procedure repeatedly employs the minimax algorithm described above to solve problems, each of the same format but with smaller dimension.     

Decomposition Algorithm Model for Singly Linearly-Constrained Problems Subject to Lower and Upper Bounds

Many real applications can be formulated as nonlinear minimization problems with a single linear equality constraint and box constraints. We are interested in solving problems where the number of variables is so huge that basic operations, such as the evaluation of the objective function or the updating of its gradient, are very time consuming. Thus, for the considered class of problems (including dense quadratic programs), traditional optimization methods cannot be applied directly. In this paper, we define a decomposition algorithm model which employs, at each iteration, a descent search direction selected among a suitable set of sparse feasible directions. The algorithm is characterized by an acceptance rule of the updated point which on the one hand permits to choose the variables to be modified with a certain degree of freedom and on the other hand does not require the exact solution of any subproblem. The global convergence of the algorithm model is proved by assuming that the objective function is continuously differentiable and that the points of the level set have at least one component strictly between the lower and upper bounds. Numerical results on large-scale quadratic problems arising in the training of support vector machines show the effectiveness of an implemented decomposition scheme derived from the general algorithm model.     

The Complexity of Data Reduction on a Reconfigurable Linear Array

We consider the basic problem of gathering information dispersed among the nodes of a reconfigurable linear array. An upper and a matching lower bound are obtained for data-reduction operations, including, for example, the computation of several independent sums of input sets which are each given at a network node. Simulations of linear arrays by other networks are presented, thus applying linear array algorithms as upper bounds for these networks, too. The complexity analysis introduces a novel criteria for the efficiency of reconfigurable network algorithms, namely the bus-usage. The bus-usage measures the utilization of the network communication links by the algorithm. The analysis show that there is a trade-off between the running time of an algorithm and its bus-usage.     

A simple expression for the nucleolus in a special case

We present a simple algorithm for calculating the nucleolus of a game whenever (a) the characteristic function is non-positive, ie. a “cost” function, and (b) the players can be ordered by “size” in such a way that the cost of any coalition is equal to the cost of the largest player in that coalition. The cumulative nucleolus is approximately equal to the convex envelope of this cost function. A numerical and geometric illustration is given for a game based upon Birmingham airport runway costs, where there are over 13,000 players (aircraft movements) of 11 distinct (aircraft) types.     

Strong planning and forecast horizons for a model with simultaneous price and production decisions

In this paper, we have solved a general inventory model with simultaneous price and production decisions. Both linear and non-linear (strictly convex) production cost cases are treated. Upper and lower bounds are imposed on state as well as control variables. The problem is solved by using the Lagrangian form of the maximum principle. Strong planning and strong forecast horizons are obtained. These arise when the state variable reaches its upper or lower bound. The existence of these horizons permits the decomposition of the whole problem into a set of smaller problems, which can be solved separately, and their solutions put together to form a complete solution to the problem. Finally, we derive a forward branch and bound algorithm to solve the problem. The algorithm is illustrated with a simple example.     

A parallel algorithm for constrained concave quadratic global minimization

The global minimization of large-scale concave quadratic problems over a bounded polyhedral set using a parallel branch and bound approach is considered. The objective function consists of both a concave part (nonlinear variables) and a strictly linear part, which are coupled by the linear constraints. These large-scale problems are characterized by having the number of linear variables much greater than the number of nonlinear variables. A linear underestimating function to the concave part of the objective is easily constructed and minimized over the feasible domain to get both upper and lower bounds on the global minimum function value. At each minor iteration of the algorithm, the feasible domain is divided into subregions and linear underestimating problems over each subregion are solved in parallel. Branch and bound techniques can then be used to eliminate parts of the feasible domain from consideration and improve the upper and lower bounds. It is shown that the algorithm guarantees that a solution is obtained to within any specified tolerance in a finite number of steps. Computational results are presented for problems with 25 and 50 nonlinear variables and up to 400 linear variables. These results were obtained on a four processor CRAY2 using both sequential and parallel implementations of the algorithm. The average parallel solution time was approximately 15 seconds for problems with 400 linear variables and a relative tolerance of 0.001. For a relative tolerance of 0.1, the average computation time appears to increase only linearly with the number of linear variables.     

Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs

A rigorous decomposition approach to solve separable mixed-integer nonlinear programs where the participating functions are nonconvex is presented. The proposed algorithms consist of solving an alternating sequence of Relaxed Master Problems (mixed-integer linear program) and two nonlinear programming problems (NLPs). A sequence of valid nondecreasing lower bounds and upper bounds is generated by the algorithms which converge in a finite number of iterations. A Primal Bounding Problem is introduced, which is a convex NLP solved at each iteration to derive valid outer approximations of the nonconvex functions in the continuous space. Two decomposition algorithms are presented in this work. On finite termination, the first yields the global solution to the original nonconvex MINLP and the second finds a rigorous bound to the global solution. Convergence and optimality properties, and refinement of the algorithms for efficient implementation are presented. Finally, numerical results are compared with currently available algorithms for example problems, illuminating the potential benefits of the proposed algorithm.     

乘客到达时间不确定的机场动态拼车策略与算法研究

网约车拼车服务作为共享经济领域重要应用,已成为国内外研究热点。针对机场在线拼车平台运营中乘客等待时间过长和车辆行驶成本较高的突出问题,本文提出前瞻式动态拼车匹配策略。该策略将未来随机到达乘客信息纳入当前已到达乘客的拼车匹配决策中,建立了乘客匹配与车辆路径联合优化两阶段随机规划模型。为了在动态环境中实时产生高质量的匹配与路径规划方案,首先基于贝叶斯估计压缩乘客随机到达情景空间,建立了问题的确定性近似最优模型。为了快速求解模型,提出基于订单目的地和乘客期望到达时间相似度的匹配规则,并以此开发改进的差分进化算法。最后,基于某拼车平台真实订单数据,通过对比测试验证了前瞻式匹配策略和改进差分进化算法的有效性与计算效率。     

Quadratic model for allocating operational budget in public and nonprofit organizations

We develop a quadratic model for allocating operational budgets in public and nonprofit organizations. The allocations for each organizational unit have lower and upper bounds. The objective function is to minimize the weighted sum of the quadratic deviations of each allocation from its bounds. The optimal allocations are mostly around the midpoint between the bounds. A simple algorithm is presented to derive the solution. The new quadratic model is compared to the familiar linear model for budget allocation, which almost always, provides extreme allocations on the bounds: for some units on the upper bound, while for others, on the lower bound. We perform sensitivity analyses, and resolve special cases of the model with closed form solution. Moreover, we show various properties of the quadratic budget allocation model and prove that its fairness index is higher than that of the linear model. The model, with its variants, was actually used for allocating budgets in various university setups; some examples are presented here.     

无容量限制设施选址问题的降阶回溯算法

无容量限制设施选址问题(uncapacitated facility location problem, UFLP)是经典组合优化中NP-Hard问题之一,在诸多领域具有广泛的应用价值。本文首先研究UFLP的数学性质,并进行了数学证明。运用这些数学性质不仅可以确定某些设施必定开设或者关闭,还可以确定某些连接边是否在服务集中,从而缩小问题的规模,加快求解速度;在此基础上设计出一个新的基于上下界的回溯算法来求解UFLP。最后,通过一个示例进一步阐述该算法的原理,结果表明该算法具有明显的可行性和有效性。     

Solving boundary value problems,integral, and integro-differential equations using Gegenbauer integration matrices

We introduce a hybrid Gegenbauer (ultraspherical) integration method (HGIM) for solving boundary value problems (BVPs), integral and integro-differential equations. The proposed approach recasts the original problems into their integral formulations, which are then discretized into linear systems of algebraic equations using Gegenbauer integration matrices (GIMs). The resulting linear systems are well-conditioned and can be easily solved using standard linear system solvers. A study on the error bounds of the proposed method is presented, and the spectral convergence is proven for two-point BVPs (TPBVPs). Comparisons with other competitive methods in the recent literature are included. The proposed method results in an efficient algorithm, and spectral accuracy is verified using eight test examples addressing the aforementioned classes of problems. The proposed method can be applied on a broad range of mathematical problems while producing highly accurate results. The developed numerical scheme provides a viable alternative to other solution methods when high-order approximations are required using only a relatively small number of solution nodes.     

Digital Simulations of Runway Utilization

Queues of aircraft that form at airports, both in the air and on the ground, are the biggest source of delay in civil air transport operations. The study of these queues is complicated by the fact that the times at which aircraft join the queues are not independent, and also by considerable diurnal fluctuation of traffic.This paper describes an approach, primarily through simulation of idealized models, on a digital computer. The object of the work was to obtain insight into the behaviour of runway queues rather than to imitate the activity of a particular airport. The models used assume that the arrival times at the queues are the result of a scheduled pattern being disordered through aircraft being independently early or late. Such an arrival process degenerates to a Poisson process if the discrepancies from schedule are very large, but in practice it is significantly different. Results from 300,000 simulated take-offs and landings are given.