## 3 Proposed Improved Hybrid SSA with an EDO (ISSAEDO) Algorithm
### Exponential Distribution Optimization (EDO) Algorithm
* If formula \(u_{2}\leq\frac{f(x)}{M}\) holds, where \(f(x)\) represents the probability density function (PDF) of the exponential distribution and \(M\) is a constant such that \(\frac{f(x)}{M}\leq 1\) for all values of \(x\), then the value of \(x\) can be accepted as a random number generated from exponential distribution. Alternatively, if \(x\) does not meet the criteria, it should be discarded and the process should be repeated. The computing cost of the acceptance-rejection approach can be significant, particularly when the ratio of the probability density function (PDF) \(\frac{f(x)}{g(x)}\) is high, where \(g(x)\) represents the PDF of the less complex distribution. Nevertheless, the utilization of this approach can prove beneficial in cases where the inverse transform method is either inapplicable or presents implementation challenges. * The Box-Muller transform is a method used to generate a pair of independent standard normal random variables \((Z_{1},Z_{2})\). * Exponential random variable \(x\) can be computed as \(x=-\log\left(\frac{y}{\lambda}\right)\), where \(u=\exp\left(\frac{-z_{1}}{z_{2}}\right)\). If \(Z_{2}\leq(\lambda-0.5)\times\left(\log\left(\frac{x}{\lambda}\right)+\frac{1}{ \lambda}\right)\), then \(x\) can be accepted as a random number from exponential distribution. In the event that \(x\) does not meet the criteria, it is necessary to decline its acceptance and thereafter initiate the process again. The Box-Muller transform demonstrates computational efficiency; nonetheless, it requires the generation of normal random variables, a task that is comparatively more challenging than generating uniform random variables. The utilization of an EDO is a technique or instrument employed to enhance the characteristics of exponential distribution. Exponential distribution is a commonly employed probability distribution for the purpose of modeling the time intervals between events that transpire at a consistent rate, in a random and independent manner. Exponential distribution is a probability distribution that characterizes the time intervals between occurrences in a posteriori point process. The tool possesses numerous characteristics that contribute to its effectiveness across a diverse array of applications. * **Memoryless property**: The memoryless property of exponential distribution indicates the probability that an event occurring is not influenced by the time passed since the previous event. This characteristic can prove advantageous in depicting situations characterized by random or randomly generated events, where the timing between these events has significance. * **Flexibility**: Exponential distribution, being a continuous probability distribution, has the capability to imitate continuous variables. This is advantageous in cases where the optimization problem involves continuous variables, such as in certain machine learning models. * **Simple parameterization**: Exponential distribution is characterized by simple parameterization consisting of a single parameter, namely the rate parameter. In specific optimization scenarios, the simplicity of this probability distribution can facilitate its handling when contrasted with more intricate distributions. * **Widely used**: Exponential distribution is widely recognized as a prominent probability distribution that enjoys comprehensive comprehension among researchers and practitioners across several academic domains. This suggests that those seeking to implement this concept in their professional endeavors can access a considerable amount of scholarly literature and resources. Certainly, we provide additional advantages of exponential distribution. * **Probability density function**: Exponential distribution possesses a readily comprehensible probability density function, rendering it amenable to statistical analysis. This approach can prove advantageous in the pursuit of analytical solutions, particularly in theoretical investigations or the development of novel algorithms.