Real Options using Markov Chains: an application to Production Capacity Decisions

In this work we address investment decisions using real options. A standard numerical approach for valuing real options is dynamic programming. The basic idea is to establish a discrete-valued lattice of possible future values of the underlying stochastic variable (demand in our case). For most approaches in the literature, the stochastic variable is assumed normally distributed and then approximated by a binomial distribution, resulting in a binomial lattice. In this work, we investigate the use of a sparse Markov chain to model such variable. The Markov approach is expected to perform better since it does not assume any type of distribution for the demand variation, the probability of a variation on the demand value is dependent on the current demand value and thus, no longer constant, and it generalizes the binomial lattice since the latter can be modelled as a Markov chain. We developed a stochastic dynamic programming model that has been implemented both on binomial and Markov models. A numerical example of a production capacity choice problem has been solved and the results obtained show that the investment decisions are different and, as expected the Markov chain approach leads to a better investment policy.Flexible Capacity Investments, Real Options, Markov Chains, Dynamic Programming

Optimal investment timing using Markov jump price processes

In this work, we address an investment problem where the investment can either be made immediately or postponed to a later time, in the hope that market conditions become more favourable. In our case, uncertainty is introduced through market price. When the investment is undertaken, a fixed sunk cost must be paid and a series of cash flows are to be received. Therefore, we are faced with an irreversible investment. Real options analysis provides an adequate framework for this type of problems by recognizing these two characteristics, uncertainty and irreversibility, explicitly. We describe algorithmic solutions for this type of problems by modelling market prices evolution by Markov jump processes.Irreversible investment, optimal stopping, dynamic programming, Markov jump processes

A decision support system for planning promotion time slots

We report on the development of a Decision Support System (DSS) to plan the best assignment for the weekly promotion space of a TV station. Each product to promote has a given target audience that is best reached at specific time periods during the week. The DSS aims to maximize the total viewing for each product within its target audience while fulfilling a set of constraints defined by the user. The purpose of this paper is to describe the development and successful implementation of a heuristic-based scheduling software system that has been developed for a major Portuguese TV station.Fundação para a Ciência e a Tecnologia (FCT)- FCT/POCI 2010/FEDER, Projecto POCTI/MAT/61842/2004Estação de Televisão SI

On the degeneracy phenomenon for nonlinear optimal control problems with higher index state constraints

Relatório Técnico do Núcleo de Investigação Officina Mathematica.Necessary conditions of optimality (NCO) play an important role in optimization problems. They are the major tool to select a set of candidates to minimizers. In optimal control theory, the NCO appear in the form of a Maximum Principle (MP). For certain optimal control problems with state constraints, it might happen that the MP are unable to provide useful information --- the set of all admissible solutions coincides with the set of candidates that satisfy the MP. When this happens, the MP is said to degenerate. In the recent years, there has been some literature on fortified forms of the MP in such way that avoid degeneracy. These fortified forms involve additional hypotheses --- Constraint Qualifications. Whenever the state constraints have higher index (i.e. their first derivative with respect to time does not depend on control), the current constraint qualifications are not adequate. So, the main purpose here is fortify the maximum principle for optimal control problems with higher index constraints, for which there is a need to develop new constraint qualifications. The results presented here are a generalization to nonlinear problems of a previous work.The financial support from Projecto FCT POSC/EEA-SRI/61831/2004 is gratefully acknowledged

A genetic algorithm approach for the TV self-promotion assignment problem

We report on the development of a Genetic Algorithm (GA), which has been integrated into a Decision Support System to plan the best assignment of the weekly self-promotion space for a TV station. The problem addressed consists on deciding which shows to advertise and when such that the number of viewers, of an intended group or target, is maximized. The GA proposed incorporates a greedy heuristic to find good initial solutions. These solutions, as well as the solutions later obtained through the use of the GA, go then through a repair procedure. This is used with two objectives, which are addressed in turn. Firstly, it checks the solution feasibility and if unfeasible it is fixed by removing some shows. Secondly, it tries to improve the solution by adding some extra shows. Since the problem faced by the commercial TV station is too big and has too many features it cannot be solved exactly. Therefore, in order to test the quality of the solutions provided by the proposed GA we have randomly generated some smaller problem instances. For these problems we have obtained solutions on average within 1% of the optimal solution value

Trajectory Optimization and NMPC Tracking for a Fixed Wing UAV in Deep Stall with Perch Landing

This paper presents a novel recovery technique for a fixed-wing UAV (Unmanned Aerial Vehicle) based on constrained optimization: i) we propose a trajectory generation for landing the UAV where it first reduces its altitude by deep stalling, then perches on a recovery net, ii) we design an NMPC (Nonlinear Model Predictive Control) tracking controller with terminal constraints for the optimal generated trajectory under disturbances. Compared to nominal net recovery procedures, this technique greatly reduces the landing time and the final airspeed of the UAV. Simulation results for various wind conditions demonstrate the feasibility of the idea.Comment: 8 pages, 13 figure

L0 and L1 Guidance and Path-Following Control for Airborne Wind Energy Systems

For an efficient and reliable operation of an Airborne Wind Energy System, it is widely accepted that the kite should follow a pre-defined optimized path. In this article, we address the problem of designing a trajectory controller so that such path is closely followed. The path-following controllers investigated are based on a well-known nonlinear guidance logic termed L1 and on a proposed modification of it, which we termed L0. We have developed and implemented both L0 and L1 controllers for an AWES. The two controllers have an easy implementation with an explicit expression for the control law based on the cross-track error, on the heading angle relative to the path, and on a single parameter L (L0 or L1, depending on each controller) that we are able to tune. The L0 controller has an even easier implementation since the explicit control law can be used without the need to switch controllers. Since the switching of controllers might jeopardize stability, the L0 controller has an important theoretical advantage in being able to guarantee stability on a larger domain of attraction.The simulation study shows that both nonlinear guidance logic controllers exhibit appropriate performance when the L parameter is adequately tuned, with the L0 controller showing a better performance when measured in terms of the average cross-track error. (c) 2022 by the authors. Licensee MDPI, Basel, Switzerland

Cascade Control of the Ground Station Module of an Airborne Wind Energy System

An airborne wind energy system (AWES) can harvest stronger wind streams at higher altitudes which are not accessible to conventional wind turbines. The operation of AWES requires a controller for the tethered aircraft/kite module (KM), as well as a controller for the ground station module (GSM). The literature regarding the control of AWES mostly focuses on the trajectory tracking of the KM. However, an advanced control of the GSM is also key to the successful operation of an AWES. In this paper we propose a cascaded control strategy for the GSM of an AWES during the traction or power generation phase. The GSM comprises a winch and a three-phase induction machine (IM), which acts as a generator. In the outer control-loop, an integral sliding mode control (SMC) algorithm is designed to keep the winch velocity at the prescribed level. A detailed stability analysis is also presented for the existence of the SMC for the perturbed winch system. The rotor flux-based field oriented control (RFOC) of the IM constitutes the inner control-loop. Due to the sophisticated RFOC, the decoupled and instantaneous control of torque and rotor flux is made possible using decentralized proportional integral (PI) controllers. The unknown states required to design RFOC are estimated using a discrete time Kalman filter (DKF), which is based on the quasi-linear model of the IM. The designed GSM controller is integrated with an already developed KM, and the AWES is simulated using MATLAB and Simulink. The simulation study shows that the GSM control system exhibits appropriate performance even in the presence of the wind gusts, which account for the external disturbance