## 3 Methods
### Mathematical Method  
In the logistical support of an operations process, various types of probability distributions can be used to model different aspects of the system [26; 27]. Some of these have been summarised in Table 1.  
\begin{table}
\begin{tabular}{c c c} \hline  & Reliability function & \(R(t)=\int_{t}^{m}f(x)dx=\int_{t}^{m}\frac{1}{\rho\sqrt{2\pi}}e^{-\frac{1}{2} \left(\frac{t-m}{2}\right)^{2}}\) \\ \hline Normal & & \(\hat{f}t)=\frac{1}{\rho\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{t-m}{2}\right)^{2}}\) \\ Distribution & Probability density function & \(\mu\)—mean time to failure (\(\overline{\Gamma}\)) \\  & \(\sigma\)—standard deviation of time to failure \\ \hline MTTF (\(\overline{\Gamma}\))—mean time to failure & \(\overline{\Gamma}\)=\(\mu\) \\ \hline \multirow{7}{*}{Exponential distribution} & Reliability function & \(R(t)=\text{e}^{-\lambda t}=\text{e}^{-\frac{1}{2}}\) \\ \cline{2-3}  & Reliability function & \(R(t)=\lambda\times\text{e}^{-\lambda t}\) \\ \cline{2-3}  & Probability density function & \(\lambda=\frac{1}{\text{e}}^{-\lambda t}\)—constant failure intensity \\ \cline{2-3}  & Probability density function & \(f(t)=\frac{1}{\text{e}}^{-\frac{1}{2}}\) \\ \cline{2-3}  & \(m\)—mean time between failures \\ \cline{2-3}  & \(t\)—operating time counted in years, hours, cycles, etc. \\ \hline MTTF (\(\overline{\Gamma}\))—mean time to failure & \(\overline{T}=\int_{0}^{m}t:f(t)dt=\frac{1}{\lambda}\) \\ \hline \multirow{7}{*}{Weibull 2 parametric distribution} & Reliability function & \(R(t)=e^{-\frac{1}{2}t}\),\({}^{\beta}\) \\ \cline{2-3}  & Probability density function & \(f(t)=\frac{\theta_{1}}{\text{e}}\left(\frac{t}{\mu}\right)^{\beta-1}e^{-\frac{ 1}{2}t}\) \\ \cline{2-3}  & Probability density function & \(\overline{\Gamma}=\eta\Gamma(\frac{1}{2}+1)\) \\ \cline{2-3}  & MTTF (\(\overline{\Gamma}\))—mean time to failure & where \(\Gamma\) is defined as: \\ \cline{2-3}  & \(\Gamma(t)=\int_{0}^{m}e^{-\frac{1}{2}t}\pi^{-1}dx\) \\ \hline \multirow{7}{*}{Lognormal distribution} & Reliability function & \(R(t)=\int_{0}^{m}(x)dx\) \\ \cline{2-3}  & \(R(t)=\int_{0}^{m}(x)\frac{1}{\rho\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{t-m}{ 2}\right)^{2}}\) \\ \cline{2-3}  & \multirow{2}{*}{Probability density function} & \(f(t)=\frac{1}{\rho\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{t-m}{2}\right)^{2}}\) \\ \cline{2-3}  & Probability density function & \(t^{\prime}=\text{ln}(t)-\mu\)—defines the time to failure \\ \cline{2-3}  & \(\mu\)—average of the natural logarithms of times to failure \\ \cline{2-3}  & \(\sigma^{\prime}\)—standard deviation of natural logarithms of times to failure \\ \hline MTTF (\(\overline{\Gamma}\))—mean time to failure & \(\mu=e^{\mu^{\prime}+\frac{1}{2}t}\) \\ \hline \end{tabular}
\end{table}
Table 1: Selected characteristics of distributions used in research.  
In summarizing the usage of the probability application in the control of the logistical support of the operation process, for example, the normal distribution can be used to model the demand for a product or resource. This can be useful for forecasting future demand and determining appropriate inventory levels. The Poisson and Weibull distribution can also be used to model demand, particularly for situations where the number of occurrences of an event is small. The exponential distribution is commonly used to model the time between the arrivals of deliveries or the time between equipment failures. In addition, the log-normal distribution is also used in logistics to model the time taken by a process or the time between failures of equipment.  
These distributions can provide a useful tool for logistics planners to predict and optimize the resources and materials needed and the logistics operation processes.