# Determination of Reliability of Selected Case
## 3 Results and Discussion  
For the investigations, a virtual model of case-type construction consisting of four elements, bottom, top, and two side walls, was adopted. The overall dimensions of the construction were 2000 mm high, 800 mm wide, and 550 mm deep. Two variants of construction materials have been chosen: an 18 mm thick MDF board and an 18 mm thick laminated three-layer particleboard. The connections of the top and bottom with the side walls were made using eccentric connectors, two for each joint. The data obtained from the laboratory reliability tests of furniture joints made of MDF and particleboard of 18 mm constituted the input base research material. The reliability of individual joints was calculated using Weibull++ software. The comparison of the reliability results for MDF and particleboard of 18 mm is presented in Figure 2.  
The obtained results confirm the higher reliability of the furniture eccentric joints made of MDF of 18 mm in comparison to the furniture eccentric joints made of particleboard of the same thickness. This is caused by a more consistent structure of the MDF. The achieved results are also in line with the results of [15]. This stage of the investigation concerned the reliability of individual joints. In the next step, the entire construction of selected case-type furniture was considered. To make it possible to calculate the reliability of entire furniture structures on the basis of data on the reliability of individual joints and the type of structure connecting construction nodes, a mathematical apparatus was developed. The analytical process of conducted calculations is presented below. In order to evaluate the reliability of the whole case-type furniture constructions for each of the distributions mentioned in Materials and Methods, the parameter values were determined that best fit the Probability Density Function (PDF) to the empirical data. For this purpose, the Maximum Likelihood Estimation (MLE) method was used, the operation of which is as follows. Let the random variable \(X\) have the density \(f_{\theta}(x)\), where \(\theta\) is the (potentially vector) parameter of the distribution. Assuming that the sample \(X_{1},X_{2},\ldots,\ X_{n}\) from the distribution of this variable is given, the total sample density as a function of the parameter (likelihood function) can be described by the formula:  
\[L(\theta;x_{1},x_{2},\ldots,\ x_{n})=\prod_{i=1}^{n}f_{\theta}(x_{i}). \tag{8}\]  
The maximum likelihood estimator of the parameter \(\theta\) is the statistic \(\hat{\theta}\), for which the likelihood function satisfies the condition:  
\[L\left(\hat{\theta};x_{1},x_{2},\ldots,\ x_{n}\right)=\underset{\theta}{\sup} L(\theta;x_{1},x_{2},\ldots,\ x_{n}), \tag{9}\]  
for each of the points \((x_{1},x_{2},\ldots,\ x_{n})\). Figure 2: The comparison of the reliability of case-type furniture eccentric joints made of MDF and particleboard of 18 mm. The maximum likelihood estimators for the parameters of all the above-mentioned distributions were determined in an analogous manner (\(x_{0},\lambda\) for the exponential distribution, \(x_{0},\lambda,\ a\) for the gamma distribution, \(x_{0},\lambda,\ s\) for the log-normal distribution, and \(x_{0},\lambda,\ c\) for the Weibull distribution). Given an empirical density function (using parameter estimators), one can determine the difference between its theoretical value derived from the determined distribution and the value from the experiment. The sum of the squares of these differences measures the quality of the fit: the larger it is, the more different the theoretical distribution is from the real data. Therefore, the distribution with the smallest sum of squares of differences was taken for further analysis. In particular, for each set of data, a distribution from a different family can be chosen. Within the performed research, the mathematical model presented above was developed.