Mathematical Modeling of Manufacturing Lines with Distribution by Process: A Markov Chain Approach

Today, there are a wide variety of ways to produce goods in a manufacturing company. Among the most common are mass or line production and process production, both of which are antagonists. In an online production system, materials move from station to station, receiving added value on a well-defined layout. In a production line by process, the materials randomly visit a set of machines strategically located in order to receive a treatment, almost always through metalwork machines, according to the final product of which they will be part. In this case, there is not a predefined layout, as the incoming materials are sectioned and each piece forms a continuous flow through different workstations to receive some process. This activity depends on the function of the product and its final destination as a component of a finished product. In this proposal, Markov chain theory is used to model a manufacturing system by process in order to obtain the expected values of the average production per machine, the total expected production in all the facilities, the leisure per machine and the total productive efficiency of the system, among other indicators. In this research, we assume the existence of historical information about the use of the equipment, its failures, the causes of failure and their repair times; in any factory, this information is available in the area of manufacturing engineering and plant engineering. From this information, statistical frequency indicators are constructed to estimate transition probabilities, from which the results presented here are derived. The proposal is complemented with a numerical example of a real case obtained from a refrigerator factory established in Mexico in order to illustrate the results derived from this research. The results obtained show their feasibility when successfully implemented in the company.

A study of stability of difference scheme of Markov counting chain method for fluidization modeling

Devices with a fluidized bed of granular material are applied in many energy power technology processes. The fluidized bed is a heterogeneous system, so mathematical models assuming its spatial discretization are necessary for its proper description. Markov chain theory is one of the most effective tools for the mathematical description of the fluidized bed structure. Many research papers are devoted to the issues of the theory application when developing mathematical models of various technological processes in the fluidized bed. At the same time, much less attention is paid to the issue of stability analysis of the proposed algorithms. Thus, it is a highly topical issue to analyze the computational stability of models of fluidized bed based on the mathematical principles of the Markov chain theory. The Markov chain approach is used as a mathematical basis for modeling of the flow patterns in a fluidized bed. The parametric identification of the model is performed using the dependencies known from the scientific papers, and the transition matrices are aligned with the physical parameters of the mass flows, which makes the proposed model nonlinear. The mixed criterion of the stability algorithm is formulated. It shows the influence of the spatiotemporal parameters of the problem sampling on the stability of computational procedures. The stability of the difference scheme to calculate formation of a fluidized bed of a monodisperse granular material is studied. The influence of the time sampling frequency on the stability of the resulting solution is considered. The effect of various parameters of the model on the loss of computational stability is estimated. It is proved that the time and spatial sampling frequencies should be chosen as a result of a mixed stability criterion. The study proves that the methodology of the Markov chain theory is an acceptable tool to describe the structure of such particle systems as a fluidized bed. It is established that macro-diffusion parameter of particle motion is the most influential in the process of computational stability loss. Thus, on the one hand, it is relevant to conduct further comparative studies of existing models of macrodiffusion, and on the other hand, it is possible to use models based on the theory of Markov chains considering the proposed stability criterion.

Research on Landslide Warning Model Establishment and Disaster Space-Time Evolution Analysis

As important methods to avoid landslide disasters, velocity monitoring and early warning are significant research topics in slope engineering at the present stage. This paper combines the randomness of velocity data in evolution process of landslide disasters, using Markov chain theory with no aftereffect to describe the randomness process, and introduces it into landslide warning. The research collects velocity monitoring data before landslide occurrence and applies average standard deviation method which can reflect statistical characteristics of the classification data to carry out state division of the velocity data. Then, it proposes landslide warning criteria and establishes landslide warning model based on dynamic prediction of future velocity status by Markov chain theory. Meanwhile, it puts forward the evaluation standard of landslide warning model from the aspects of timeliness, anti-interference, and credibility. At the same time, it takes typical open-pit mine landslide disaster as the engineering background and gradually optimizes and evaluates the landslide warning model from the above three evaluation standards. The results show that the landslide warning model can realize the landslide early warning of multiple monitoring points; it has good effects in both time warning and regional warning. On the other hand, the landslide warning model has high accuracy in timeliness, anti-jamming, and credibility, and it can reveal space-time evolution law of landslide occurrence, so this research has important theoretical significance and engineering promotion value.

Markovian Analysis of Covid-19 Dynamics

Covid-19 is an emergency and viral infection with its outbreak being termed as one of the great epidemics in the 21st century causing so many deaths, which made WHO declare it as a pandemic emergency. This virus is new and comes with its characteristics of which randomness and uncertainty are among its common features. In this paper, we developed a model for carrying out an analysis of COVID-19 dynamics using Markov-chain theory methodology. Here, we employed the use of conditional probability distribution as embedded in the Markov property of our chain to construct the transition probabilities that were used in determining the probability distributions of COVID-19 patients as well as predicting its future spread dynamics. We provide a step-by-step approach to obtaining probability distributions of infected and recovered individuals, of infected and recovering and of a recovered patient being getting infected again. This study reveals that irrespective of the initial state of health of an individual, we will always have probabilities P_RI/〖(P〗_IR+P_RI) of an individual being infected and P_RI/〖(P〗_IR+P_RI) of an individual recovering from this disease. Also, with increasing ‘n’, we have an equilibrium that does not depend on the initial conditions, the implication of which is that at some point in time, the situation stabilizes and the distribution X_(n+1) is the same as that of X_n. We envision that the output of this model will assist those in the health system and related fields to plan for the potential impact of the pandemic and its peak.

Markov model of the operations & maintenance process of vehicles scheduled to be Operated

The article is dedicated to the modelling of operations & maintenance of vehicles scheduled to be operated. This specific feature is illustrative of the vehicle operation process in the military system. The presented model of the operation process of vehicles scheduled to be operated, using the Markov chain theory, contains indicators and measures essential for the vehicle operation, i.e. repair defectiveness, repair intensity, usage intensity and failure intensity. This model enables to quantify the impact of the introduced changes in operational practice or changes planned as a forecast, which is shown in the examples.

A model to optimize the structure of enterprise's fixed assets in conditions of digital transformation

Subject. The article addresses the enhancement of fixed assets utilization efficiency as one of priority tasks faced by modern machine-building enterprises. Objectives. The purpose is to devise approaches and tools for solving the problems of optimization of industrial enterprises’ fixed assets development and use, on the basis of mathematical modeling. Methods. The study employs methods of mathematical modeling of economic processes, the Markov chain theory, and the multicriteria optimization. Results. We offer an approach to analyze the life cycle of enterprise’s fixed assets based on the theory of Markov chains. The paper presents a mathematical model of the fixed asset life cycle, formulates optimization tasks for fixed assets development, taking into account economic feasibility and reliability. Using the developed tools, it is possible to plan investing and operating activities for geographically distributed systems of enterprises, in particular, optimization of investment in production capacities development, planning their maintenance and repairs, load distribution between individual enterprises of the system. Conclusions. The use of modern digital technologies that provide dynamic forecasting and optimization of fixed assets enables to achieve a number of improvements, including a reduction in production time and cost and increase in efficiently used production means.

Identifying changes in insurance companies’ competitiveness on the travel services market

The purpose of the study is to develop methodological approach for identifying changes in the level of insurance companies’ competitiveness on the travel services market. Based on development of multifactor regression equation, integrated indicators of insurance companies’ competitiveness in 2016–2019 were calculated. The application of three-sigma rule allowed to divide insurance companies by competitiveness levels and to identify that during 2016–2019 most of insurers had sufficient and critical levels of competitiveness and the group of insurance companies with a high level of competitive position is small. The Markov chain theory was used as a research method to determine the probability of insurance companies moving to higher or lower competitiveness levels. The results of Markov’s method showed that the majority of insurance companies are most likely to remain in their initial groups and only insurers with low and sufficient competitiveness have high probability to change their positions. Companies with high competitiveness have very strong positions on the market and there is very low probability that other insurers will capture leaders’ market share in the coming years. So, the use of the developed approach allows predicting a decrease of insurance ability to compete on the travel services market and deciding on the necessity to change the competitive strategy.

SUBGEOMETRICALLY ERGODIC AUTOREGRESSIONS

In this paper, we discuss how the notion of subgeometric ergodicity in Markov chain theory can be exploited to study stationarity and ergodicity of nonlinear time series models. Subgeometric ergodicity means that the transition probability measures converge to the stationary measure at a rate slower than geometric. Specifically, we consider suitably defined higher-order nonlinear autoregressions that behave similarly to a unit root process for large values of the observed series but we place almost no restrictions on their dynamics for moderate values of the observed series. Results on the subgeometric ergodicity of nonlinear autoregressions have previously appeared only in the first-order case. We provide an extension to the higher-order case and show that the autoregressions we consider are, under appropriate conditions, subgeometrically ergodic. As useful implications, we also obtain stationarity and $\beta $ -mixing with subgeometrically decaying mixing coefficients.