Michael Kolonko

On the waiting time of arriving aircrafts and the capacity of airports with one or two runways

Abstract In this paper we examine a model for the landing procedure of aircrafts at an airport. The characteristic feature here is that due to air turbulence the safety distance between two landing aircrafts depends on the types of these two machines. Hence, an efficient routing of the aircraft to two runways may reduce their waiting time. First, we use M/SM/1 queues (with dependent service times) to model a single runway. We give the stability condition and a formula for the average waiting time of the aircrafts. Moreover, we derive easy to compute bounds on the waiting times by comparison to simpler queuing systems. In particular we study the effect of neglecting the dependency of the service times when using M/G/1-models. We then consider the case of two runways with a number of heuristic routing strategies such as coin flipping, type splitting, Round Robin and variants of the join-the-least-load rule. These strategies are analyzed and compared numerically with respect to the average delay they cause. It turns out that a certain modification of join-the-least-load gives the best results.

Multidimensional Optimization with a Fuzzy Genetic Algorithm

We present a new heuristic method to approximate the set of Pareto-optimal solutions in multicriteria optimization problems. We use genetic algorithms with an adaptive selection mechanism. The direction of the selection pressure is adapted to the actual state of the population and forces it to explore a broad range of so far undominated solutions. The adaptation is done by a fuzzy rule-based control of the selection procedure and the fitness function. As an application we present a timetable optimization problem where we used this method to derive cost-benefit curves for the investment into railway nets. These results show that our fuzzy adaptive approach avoids most of the empirical shortcomings of other multiobjective genetic algorithms.

The average-optimal adaptive control of a Markov renewal model in presence of an unknown parameter

We consider a Markov renewal decision model with countable state space and an unbounded reward function. The transition probability depends on an unknown parameter θ. We give weak bounding and recurrence conditions under which the adaptation of the control according to the outcome of an estimator for θ yields an optimal procedure uniformly in θ with respect to the expected average reward criterion In a detailed example these results are applied to a controllable M/G/1 queueing model with unknown arrival rate and unknown service time distribution.

Bounds for the regret loss in dynamic programming under adaptive control

We consider a Markovian dynamic programming model in which the transition probabilities depend on an unknown parameterθ. We estimate the unknownθ and adapt the control action to the estimated value. Bounds are given for the expected regret loss under this adaptive procedure, i.e. for the loss caused by using the adaptive procedure instead of an (unknown) optimal one. We assume that the dependence of the model onθ is Lipschitz continuous. The bounds depend on the expected estimation error. When confidence intervals forθ with fixed width are available, we obtain bounds for the expected regret loss that hold uniformly inθ.ZusammenfassungWir betrachten ein Markoffsches Dynamisches Optimierungsmodell, in dem die übergangswahrscheinlichkeiten von einem unbekannten Parameterθ abhängen.θ wird geschätzt, und die Kontrollaktionen werden an die Schätzwerte angepaßt. Es werden Schranken für den erwarteten regret angegeben, d.h. für den Verlust, der bei Anwendung dieses adaptiven Verfahrens im Vergleich zum Optimum auftritt. Als zentrale Voraussetzung benutzen wir eine Lipschitz-stetige Abhängigkeit der Modellgrößen vonθ. Die Schranken hängen von dem erwarteten Schätzfehler ab. Gibt es Konfidenz-Intervalle fürθ mit fester Länge, so lassen sich Schranken für den regret angeben, die unabhängig vonθ sind.

The Sequential Design of Bernoulli Experiments Including Switching Costs

We consider a sequence of N trials, each of which must be performed on one of two given Bernoulli experiments. We assume the success probability of one experiment is known and the other is unknown. Performing two successive trials on different experiments incurs a switching cost. The problem is to choose an experiment for each trial in order to maximize the expected number of successes minus the expected switching costs. We show that an optimal design shares some well-known monotonicity properties, such as the “stopping-rule” and the “stay-on-a-winner” rule. We also show how to use these results to derive a simple algorithm for calculating the optimal design. Since the model contains the one-armed bandit problem as a special case, we also obtain new proofs for known results.

Asymptotic Properties of a Generalized Cross Entropy Optimization Algorithm

The discrete cross-entropy optimization algorithm iteratively samples solutions according to a probability density on the solution space. The density is adapted to the good solutions observed in the present sample before producing the next sample. The adaptation is controlled by a so-called smoothing parameter. We generalize this model by introducing a flexible concept of feasibility and desirability into the sampling process. In this way, our model covers several other optimization procedures, in particular the ant-based algorithms. The focus of this paper is on some theoretical properties of these algorithms. We examine the first hitting time τ of an optimal solution and give conditions on the smoothing parameter for τ to be finite with probability one. For a simple test case we show that runtime can be polynomially bounded in the problem size with a probability converging to 1. We then investigate the convergence of the underlying density and of the sampling process. We show, in particular, that a constant smoothing parameter, as it is often used, makes the sample process converge in finite time, freezing the optimization at a single solution that need not be optimal. Moreover, we define a smoothing sequence that makes the density converge without freezing the sample process and that still guarantees the reachability of optimal solutions in finite time. This settles an open question from the literature.

On monotone optimal decision rules and the stay-on-a-winner rule for the two-armed bandit

SummaryConsider the following optimization problem: Find a decision rule δ such thatw(x, δ (x))=maxaw(x, a) for allx under the constraint δ (x)∈D (x). We give conditions for the existence of monotone optimal decision rules δ. The term ‘monotone’ is used in a general sense. The well-known stay-on-a-winner rules for the two-armed bandit can be characterized as monotone decision rules by including the stage number intox and using a special ordering onx. This enables us to give simple conditions for the existence of optimal rules that are stay-on-a-winner rules. We extend results ofBerry andKalin/Theodorescu to the case of dependent arms.

Optimal Control of Semi-Markov Chains under Uncertainty with Applications to Queueing Models

In a semi-Markov decision model under uncertainty the law of motion depends on an unknown parameter θeθ. Conditions are given for the existence of a plan that is optimal, uniformly in θeθ, with respect to the average return criterion in a countable state model. The essential conditions are of Liapunov-type. The construction of the optimal plan follows an idea of Kurano and Mandl: at each step, choose an action that is optimal for an estimated value of the unknown parameter.

Convergence of Simulated Annealing with Feedback Temperature Schedules

It is well known that the standard simulated annealing optimization method converges in distribution to the minimum of the cost function if the probability a for accepting an increase in costs goes to 0. α is controlled by the “temperature” parameter, which in the standard setup is a fixed sequence of values converging slowly to 0. We study a more general model in which the temperature may depend on the state of the search process. This allows us to adapt the temperature to the landscape of the cost function. The temperature may temporarily rise such that the process can leave a local optimum more easily. We give weak conditions on the temperature schedules such that the process of solutions finally concentrates near the optimal solutions. We also briefly sketch computational results for the job shop scheduling problem.

A piecewise Markovian model for simulated annealing with stochastic cooling schedules

We introduce a stochastic process with discrete time and countable state space that is governed by a sequence of Markov matrices P m , m ≥ 1. Each P m is used for a random number of steps T m and is then replaced by P m+1 . T m is a randomized stopping time that may depend on the most recent part of the state history. Thus the global character of the process is non-Markovian. This process can be used to model the well-known simulated annealing optimization algorithm with randomized, partly state depending cooling schedules. Generalizing the concept of strong stationary times (Aldous and Diaconis [1]) we are able to show the existence of optimal schedules and to prove some desirable properties. This result is mainly of theoretical interest as the proofs do not yield an explicit algorithm to construct the optimal schedules.