Savo darbe mes nagrinėjame Gerber-Shiu baudos funkciją klasikiniame rizikos modelyje atveju, kai žalų dydžiai pasiskirstę pagal Pareto dėsnį. Pagrindinis uždavinys yra susikonstruoti algoritmą funkcijos reikšmių gavimui. Tiriamas Gerber-Shiu diskontuotos baudos funkcijos atvejis, kada vidinė baudos funkcija w tapačiai lygi vienetui. Dėl sudėtingos transformuoto Pareto skirstinio formos analitiškai paskaičiuoti sąsūkų nepavyko. Tam tikslui naudojamas interpoliavimas kubiniu splainu. N kartų kartodami sukonstruotą algoritmą gauname pirmąsias n sąsūkas laisvai pasirinktiems pradiniams parametrams: Pareto skirstinio laipsnio rodikliui α, pradiniam kapitalui u, santykinei draudimo priemokai θ, diskontavimo parametrui (palūkanų normai) δ ir Puasono proceso parametrui λ. Lentelių pagalba parodome funkcijos priklausomybę nuo skirtingų modeliuojančių parametrų reikšmių. Išvadose teigiame jog pasiūlytas metodas skaičiuoti Gerber-Shiu diskontuotos baudos funkciją nors ir išpildomas tačiau yra neefektyvus. Kai kuriais pradinių parametrų pasirinkimo atvejais susiduriama su tikslumo problema. Norint tiksliai paskaičiuoti funkcijos reikšmes reikia didesnių eilių transformuoto Pareto skirstinio sąsūkų, o tam reikalingi dideli resursai. Kita vertus, pradinio kapitalo u reikšmėms didėjant tikslumas didėja ženkliai. / In this paper we consider Gerber-Shiu discounted penalty function in the classical risk model for Pareto claims. Our main goal is to construct an algorithm for obtaining values of the discounted penalty function (considering penalty function w=1). Due to the complicated form of the transformed Pareto distribution function we cannot obtain its convolutions analiticaly. We use numerical methods provided by Maple (cube spline) to find interpolating functions instead. Continuously applying recursive formulas we obtain first 5 interpolated convolutions. Then we calculate values of Gerber-Shiu discounted penalty function for certain arbitrary parameters: α – degree of Pareto distribution function, initial surplus u, security loading θ, discounting parameter δ and Poison process parameter λ. We present data tables and graphs of the discounted penalty function for some variations of parameters in later sections. Finally we state that the method that we use is quite complicated. For better accuracy of the discounted penalty function values one may require to get many convolutions of the transformed Pareto distribution function and that may require too great of the resources. However the quantity of the convolutions needed rapidly decreases for large values of the initial surplus u.

This research effort focuses on large-scale linear programming problems that arise in the context of solving various problems such as discrete linear or polynomial, and continuous nonlinear, nonconvex programming problems, using linearization and branch-and-cut algorithms for the discrete case, and using polyhedral outer-approximation methods for the continuous case. These problems arise in various applications in production planning, location-allocation, game theory, economics, and many engineering and systems design problems. During the solution process of discrete or continuous nonconvex problems using polyhedral approaches, one has to contend with repeatedly solving large-scale linear programming(LP) relaxations. Thus, it becomes imperative to employ an efficient method in solving these problems. It has been amply demonstrated that solving LP relaxations using a simplex-based algorithm, or even an interior-point type of procedure, can be inadequately slow ( especially in the presence of complicating constraints, dense coefficient matrices, and ill-conditioning ) in comparison with a Lagrangian Relaxation approach. With this motivation, we present a practical primal-dual subgradient algorithm that incorporates a dual ascent, a primal recovery, and a penalty function approach to recover a near optimal and feasible pair of primal and dual solutions. The proposed primal-dual approach is comprised of three stages. Stage I deals with solving the Lagrangian dual problem by using various subgradient deflection strategies such as the Modified Gradient Technique (MGT), the Average Direction Strategy (ADS), and a new direction strategy called the Modified Average Direction Strategy (M-ADS). In the latter, the deflection parameter is determined based on the process of projecting the unknown optimal direction onto the space spanned by the current subgradient direction and the previous direction. This projected direction approximates the desired optimal direction as closely as possible using the conjugate subgradient concept. The step-length rules implemented in this regard are the Quadratic Fit Line Search Method and a new line search method called the Directional Derivative Line Search Method in which we start with a prescribed step-length and then ascertain whether to increase or decrease the step-length value based on the right-hand and left-hand derivative information available at each iteration. In the second stage of the algorithm (Stage II), a sequence of updated primal solutions is generated using some convex combinations of the Lagrangian subproblem solutions. Alternatively, a starting primal optimal solution can be obtained using the complementary slackness conditions. Depending on the extent of feasibility and optimality attained, Stage III applies a penalty function method to improve the obtained primal solution toward a near feasible and optimal solution. We present computational experience using a set of randomly generated, structured, linear programming problems of the type that might typically arise in the context of discrete optimization. / Master of Science

In Bish et al. (2014), two approaches for the generation of hospital evacuation transportation plans were proposed: the minimization of the overall risk and the minimization of the evacuation duration. The resulting evacuation plans differ in terms of overall risk and duration, but also in the evacuation order of patients with different characteristics, the filling of hospital beds, and the assignments of the patients to the various vehicle types. Due to the computational effort of the duration minimization, manipulations of the risk functions for the risk minimization approach were searched in this thesis such that the resulting evacuation plans approach the minimal duration without rules for the assignments of patients to vehicle types. It is possible to create risk functions such that the resulting plans have shorter durations than with the basic risk functions, but the overall risk increases and other properties of the plans change. Furthermore, a new objective function was introduced in this thesis that minimizes an overall penalty function, where penalties are incurred for time intervals in which patients are at the evacuating hospital or being transported. The characteristics of the patients are considered by different weights in the penalty function. For the given problem instance, it is possible to choose penalty factors such that the overall risk is close to the minimal risk or to choose them such that the duration decreases. It is a simple approach with run times that are comparable to the risk minimization approach for the given problem instance. / Master of Science

The value of a state is a measure of its worth, so that, for example, waypoints have high value and regions inside of obstacles have very small value. We propose two methods of incorporating world information as state-dependent modifications to the interacting multiple models (IMM) algorithm, and then we use a game's player-controlled trajectories as ground truths to compare the normal IMM algorithm to versions with our proposed modifications. The two methods involve modifying the model probabilities in the update step and modifying the transition probability matrix in the mixing step based on the assigned values of different target states. The state-dependent value assignment modifications are shown experimentally to perform better than the normal IMM algorithm in both estimating the target's current state and predicting the target's next state.

In this thesis we propose four new methods for solving constrained global optimization problems. The first proposed algorithm is a differential evolution (DE) algorithm using penalty functions for constraint handling. The second algorithm is based on the first DE algorithm but also incorporates a filter set as a diversification mechanism. The third algorithm is also based on DE but includes an additional local refinement process in the form of the pattern search (PS) technique. The last algorithm incorporates both the filter set and PS into the DE algorithm for constrained global optimization. The superiority of feasible points (SFP) and the parameter free penalty (PFP) schemes are used as constraint handling mechanisms. The new algorithms were numerically tested using two sets of test problems and the results where compared with those of the genetic algorithm (GA). The comparison shows that the new algorithms outperformed GA. When the new methods are compared to each other, the last three methods performed better than the first method i.e. the DE algorithm. The new algorithms show promising results with potential for further research. Keywords: constrained global optimization, differential evolution, pattern search, filter method, penalty function, superiority of feasible points, parameter free penalty. ii

Main focus is to extend the analysis of the ruin related quantities, such as the surplus immediately prior to ruin, the deficit at ruin or the ruin probability, to the delayed renewal risk models. First, the background for the delayed renewal risk model is introduced and two important equations that are used as frameworks are derived. These equations are extended from the ordinary renewal risk model to the delayed renewal risk model. The first equation is obtained by conditioning on the first drop below the initial surplus level, and the second equation by conditioning on the amount and the time of the first claim. Then, we consider the deficit at ruin in particular among many random variables associated with ruin and six main results are derived. We also explore how the Gerber-Shiu expected discounted penalty function can be expressed in closed form when distributional assumptions are given for claim sizes or the time until the first claim. Lastly, we consider a model that has premium rate reduced when the surplus level is above a certain threshold value until it falls below the threshold value. The amount of the reduction in the premium rate can also be viewed as a dividend rate paid out from the original premium rate when the surplus level is above some threshold value. The constant barrier model is considered as a special case where the premium rate is reduced to $0$ when the surplus level reaches a certain threshold value. The dividend amount paid out during the life of the surplus process until ruin, discounted to the beginning of the process, is also considered.

Main focus is to extend the analysis of the ruin related quantities, such as the surplus immediately prior to ruin, the deficit at ruin or the ruin probability, to the delayed renewal risk models. First, the background for the delayed renewal risk model is introduced and two important equations that are used as frameworks are derived. These equations are extended from the ordinary renewal risk model to the delayed renewal risk model. The first equation is obtained by conditioning on the first drop below the initial surplus level, and the second equation by conditioning on the amount and the time of the first claim. Then, we consider the deficit at ruin in particular among many random variables associated with ruin and six main results are derived. We also explore how the Gerber-Shiu expected discounted penalty function can be expressed in closed form when distributional assumptions are given for claim sizes or the time until the first claim. Lastly, we consider a model that has premium rate reduced when the surplus level is above a certain threshold value until it falls below the threshold value. The amount of the reduction in the premium rate can also be viewed as a dividend rate paid out from the original premium rate when the surplus level is above some threshold value. The constant barrier model is considered as a special case where the premium rate is reduced to $0$ when the surplus level reaches a certain threshold value. The dividend amount paid out during the life of the surplus process until ruin, discounted to the beginning of the process, is also considered.

This thesis develops approximate dynamic programming (ADP) strategies suitable for process control problems aimed at overcoming the limitations of MPC, which are the potentially exorbitant on-line computational requirement and the inability to consider the future interplay between uncertainty and estimation in the optimal control calculation. The suggested approach solves the DP only for the state points visited by closed-loop simulations with judiciously chosen control policies. The approach helps us combat a well-known problem of the traditional DP called 'curse-of-dimensionality,' while it allows the user to derive an improved control policy from the initial ones. The critical issue of the suggested method is a proper choice and design of function approximator. A local averager with a penalty term is proposed to guarantee a stably learned control policy as well as acceptable on-line performance. The thesis also demonstrates versatility of the proposed ADP strategy with difficult process control problems. First, a stochastic adaptive control problem is presented. In this application an ADP-based control policy shows an "active" probing property to reduce uncertainties, leading to a better control performance. The second example is a dual-mode controller, which is a supervisory scheme that actively prevents the progression of abnormal situations under a local controller at their onset. Finally, two ADP strategies for controlling nonlinear processes based on input-output data are suggested. They are model-based and model-free approaches, and have the advantage of conveniently incorporating the knowledge of identification data distribution into the control calculation with performance improvement.

To cope with dynamics and uncertainties, a novel penalty function-based hybrid, multi-agent shop floor control system is proposed in this dissertation. The key characteristic of the proposed system is the capability of adaptively distributing decision-making power across different levels of control agents in response to different levels of disturbance. The subordinate agent executes tasks based on the schedule from the supervisory level agent in the absence of disturbance. Otherwise, it optimizes the original schedule before execution by revising it with regard to supervisory level performance (via penalty function) and disturbance. Penalty function, mathematical programming formulations, and quantitative metrics are presented to indicate the disturbance levels and levels of autonomy. These formulations are applied to diverse performance measurements such as completion time related metrics, makespan, and number of late jobs. The proposed control system is illustrated, tested with various job shop problems, and benchmarked against other shop floor control systems. In today's manufacturing system, man still plays an important role together with the control system Therefore, better coordination of humans and control systems is an inevitable topic. A novel BDI agent-based software model is proposed in this work to replace the partial decision-making function of a human. This proposed model is capable of 1) generating plans in real-time to adapt the system to a changing environment, 2) supporting not only reactive, but also proactive decision-making, 3) maintaining situational awareness in human language-like logic to facilitate real human decision-making, and 4) changing the commitment strategy adaptive to historical performance. The general purposes human operator model is then customized and integrated with an automated shop floor control system to serve as the error detection and recovery system. This model has been implemented in JACK software; however, JACK does not support real-time generation of a plan. Therefore, the planner sub-module has been developed in Java and then integrated with the JACK. To facilitate integration of an agent, real-human, and the environment, a distributed computing platform based on DOD High Level Architecture has been used. The effectiveness of the proposed model is then tested in several scenarios in a simulated automated manufacturing environment.

Darbe gauta Gerber-Shiu diskontuotos baudos funkcijos asimptotika, kai žalos pasiskirsčiusios pagal Pareto dėsnį ir pradinis kapitalas x artėja į begalybę. Pagrindinė išraiška Gerber-Shiu diskontuotos baudos funkcijos išskaidyta į du atvejus, kai palūkanų norma nelygi ir lygi nuliui. Darbe pateikti grafikai rodo diskontuotos baudos funkcijos priklausomybę nuo įvairių Puasono modelio parametrų. / The asymptotic of the Gerber-Shiu discounted penalty function in Poisson model with Pareto distributed claims is obtained. The asymptotic is obtained as initial surplus x tends to infinity. The main term of discounted penalty function has different expression in case when interest rate equal zero and when doesn't equal zero. The graphs of the Gerber-Shiu discounted penalty function in the case of Pareto claims are examined for various parameter choices.