Šiame magistriniame darbe yra nagrinėjami nepriklausomi vienodai pasiskirstę atsitiktiniai dydžiai, turintys visus absoliutinius baigtinius momentus. Magistrinio darbo tikslas - atlikti konvergavimo greičio į normalųjį dėsnį įvertinimą. Darbą sudaro aštuoni skyriai. Įvade aprašoma problema ir visi tyrimo parametrai. Antrasis skyrius skirtas teoriniai analizei. Šiame skyriuje pateikiamos svarbiausios teorinės žinios ir metodai, kurie bus taikomi magistrinio darbo uždaviniams bei tikslams įgyvendinti. Trečiame skyriuje nagrinėjami kumuliantai Bernulio schemos atveju, o ketvirtame - analizuojamas Čebyšovo asimptotinis skleidinys ir pasinaudojus matematiniu paketu Maple, grafiniu būdu, tyrinėjamas jo konvergavimas. Aproksimacijos normaliuoju dėsniu tikslumui įvertinti naudojamas charakteristinių funkcijų metodas, todėl penktasis skyrius yra skiriamas suglodinimo nelygybių patikslinimui. Šeštame skyriuje, pasinaudojus turimais rezultatais, realizuojamas magistrinio darbo tikslas, o septintame - patikrinamas absoliutinės paklaidos įvertis Bernulio schemos atveju. Išvados ir rezultatai glaustai išdėstomi aštuntame skyriuje. / This master thesis considers independiant and identically distributed random variables, having absolute finite moments. The main task is to determine error estimate of the normal approximation. The work consists of eight chapters. In the introduction are considered problems and all subjects of research. The second chapter is designed for the theory analysis. Here are placed the main theoretical studies and methods that are used to solve the aims of the master thesis. The third chapter is intended to deal with cumulants in case of the Bernoulli’s distribution, the fourth one - is analyzing the Čebyšova’s asymptotic expansion and it convergence with the help of the mathematical package Maple. The method of characteristic’s functions is used to find the remainder term of the normal approximation, so the fifth chapter is designed to specify smoothing inequalities. Based on these results, the main task of the master thesis was obtained and specified in the sixth chapter. In the seventh one the error estimate in case of Bernoulli’s distribution, was examined with a mathematical package Maple. The short conclusions and results are placed in the eighth chapter.

M. Tech. (Department of Industrial Engineering and Operations Management, Faculty of Engineering), Vaal University of Technology. / The current business environments in many companies are characterized by markets facing tough competitions, from which customer requirements and expectations are becoming increasingly high in terms of quality, cost and delivery dates, etc. These emerging expectations are even getting stronger due to rapid development of new information and communication technologies that provide direct connections between companies and their clients. As a result, companies should have powerful control mechanisms at their disposal. To achieve this, companies rely on a number of functions including production scheduling. This function has always been present within companies, but today, it is facing increasing complexities because of the large number of jobs that must be executed simultaneously. Amongst many factors, it is time driven. This study demonstrates that several disciplines can be married into one model (i.e. a unified model) to solve scheduling problems after disruptions, and clears the way for future multi-disciplinary research efforts. Scheduling problem is modeled as follows: Ito’s stochastic differential rule is used to analyse the time evolution of random or stochastic processes. Multifactor productivity is used to unify various disruption factors. Theory of line balancing is also employed to determine the required number of resources to minimize bottleneck. Reliability: disruptions are considered to be equivalent to system failure. The failure rate of the system is translated to the reliability of the system mathematically. The probabilities of failure are used as indicators of disruptions, and the theory of reliability is then applied. Bernoulli’s principle is also employed to relate pressure to production flow and aid in managing bottleneck situations. Results indicate that the amount of resources needed after disruption depends on the nature of disruption, and that the scheduler should plan to increase number of facilities following a trend that is only predicted by the nature of disruptions. It is also shown that disruption of one type may not greatly affect productivity of a certain company layout, whilst similar disruptions can have devastating effect on another type. It is further concluded that impacts of disruption are dependent on the type of company layouts.