A digital distance on the kisrhombille tiling.

The kisrhombille tiling is the dual tessellation of one of the semi-regular tessellations. It consists of right-angled triangle tiles with 12 different orientations. An adequate coordinate system for the tiles of the grid has been defined that allows a formal description of the grid. In this paper, two tiles are considered to be neighbors if they share at least one point in their boundary. Paths are sequences of tiles such that any two consecutive tiles are neighbors. The digital distance is defined as the minimum number of steps in a path between the tiles, and the distance formula is proven through constructing minimum paths. In fact, the distance between triangles is almost twice the hexagonal distance of their embedding hexagons.

Distances in the face-centered cubic crystalline structure applying operational research.

The f.c.c. (face-centered cubic) grid is the structure of many crystals and minerals. It consists of four cubic lattices. It is supposed that there are two types of steps between two grid points. It is possible to step to one of the nearest neighbors of the same cubic lattice (type 1) or to step to one of the nearest neighbors of another cubic lattice (type 2). Steps belonging to the same type have the same length (weight). However, the two types have different lengths and thus may have different weights. This paper discusses the minimal path between any two points of the f.c.c. grid. The minimal paths are explicitly given, i.e. to obtain a minimal path one is required to perform only O(1) computations. The mathematical problem can be the model of different spreading phenomena in crystals having the f.c.c. structure.

A novel competitive exact approach to solve assembly line balancing problems based on lexicographic order of vectors.

The assembly line balancing problem (ALBP) is an eminent NP-hard topic that is discussed in mass production systems with low diversity. Primarily, two types of ALBPs are discussed in the literature as type I, which aims to find the minimum number of workstations for a given cycle time, and type II, which assigns some tasks to a given number of workstations such that the maximum workstation load is minimized. To solve ALBPs, various exact, heuristic, and metaheuristic methods have been proposed. However, these methods lose their efficiency when handling large-size problems. Therefore, researchers have focused on proposing heuristic and metaheuristic algorithms to solve large-size problems, especially when they deal with real-life case problems in the industry. This study aims to present a novel and competitive exact method for solving ALBP type II based on the lexicographic order of vectors for feasible solutions. To evaluate the performance of the developed method, a group of highly used standard test problems in the literature is utilized, and the results are compared and discussed in detail. The computational results in this study specify that the developed solution approach performs efficiently and yields the best global solution of all the ALB test problems, which proves the proposed method's potential and its competitive advantage.

A deterministic model for the inventory policy of countries for procurement of vaccines.

The countries are the units that procure the vaccines during the COVID-19 pandemic. The delivered quantities are huge. The countries must bear the inventory holding cost according to the variation of stock quantities. This cost depends on the speed of the vaccination in the country. This speed is time-dependent. The vaccinated portion of the population can be approximated by the cumulative distribution function of the Cauchy distribution. A model is provided for determining the minimal-cost inventory policy and its optimality conditions are provided. The model is solved for 20 countries for different numbers of procurements. The results reveal the individual behavior of each country.

Design and Preparation of Mass Production System for Protective Items in the European Union for the Case of Serious Pandemics.

OBJECTIVE: The European Union (EU) responded to the coronavirus disease 2019 (COVID-19) pandemic with a long time lag. The response was made on the level of the member states. There was no institution at the EU level that could assist the member states. Italy but other states as well, needed Chinese help. This highlights 2 points. First, the EU is vulnerable. Second, eastern Asia reacted faster to the new demand on the market and introduced effective technology for the mask production. A robust method is needed to overcome the shortage of protective items in the future. METHODS: This method is based on producing the demand within the EU states as the first priority to reduce the reliance from external sources like China that can be competitive and unreliable. This research suggests preventive measures that can make the EU safer. A system for the production of simple protective items is designed. RESULTS: A relatively small number of factories with high capacities are enough to serve the whole EU in a robust way. The calculation of the optimal selection of locations is fast and easy. CONCLUSIONS: The suggested system will make the EU more secure during pandemics. The presented calculations show that the establishment of the system is feasible.

On the Evaluation of the Ambulance Capacity of the Asian Side of Istanbul in the Case of a Serious Earthquake.

OBJECTIVES: The purpose of this study is to analyze a strategy for the assignment and transportation of injured patients to hospital to decrease the demand on transportation, in both predisaster and postdisaster periods, on the Anatolian side of Istanbul. METHODS: Two approaches are used in this study: a Voronoi diagram, and a heuristic approach to the problem of scheduling. A Voronoi diagram is used to divide the city into 74 regions, where each hospital has a certain region of responsibility. The transportation strategy of 1 hospital is modeled by minimizing the makespan (ie, the maximal completion time) and the work-in-process, which are used as different objectives in scheduling theory. RESULTS: The total waiting time of 100 injured people was minimized to 13,036 min when a total of 3 vehicles was used in the studied region, on the Asian side of Istanbul. The transportation capacity and total operating capacity of the hospitals should be approximately equal. CONCLUSIONS: The people of Istanbul will be in a safer position if the suggested measures are implemented. This is an important consideration, as Istanbul is situated in a region where serious earthquakes are possible at any moment.

Statistical analysis of the Hungarian COVID-19 victims.

With the wide spread of Coronavirus, most people who infected with the COVID-19, will recover without requiring special treatment. Whereas, elders and those with underlying medical problems are more likely to have serious illnesses, even be threatened with death. Many more disciplines try to find solutions and drive master plan to this global trouble. Consequently, by taking one particular population, Hungary, this study aims to explore a pattern of COVID-19 victims, who suffered from some underlying conditions. Age, gender, and underlying medical problems form the structure of the clustering. K-Means and two step clustering methods were applied for age-based and age-independent analysis. Grouping of the deaths in the form of two different scenarios may highlight some concepts of this deadly disease for public health professionals. Our result for clustering can forecast similar cases which are assigned to any cluster that it will be a serious cautious for the population.