## 6 Numerical Results Discussion and Optimization Analysis
### Optimization Analysis  
After determining the key system parameters, we try to improve the cost-benefit ratio, including three conflicting objective functions: the small-the-better average operating cost, the small-the-better server's idle rate, and the small-the-better average time spent in the buffer. The cost management of the service system is directly related to service quality and profit. On the other hand, the average time spent in the buffer is the most important factor in determining customers' satisfaction with the service quality. In addition, from a management perspective, managers may wish to minimize the proportion of time a server is idle to avoid overstaffing issues, which may lead to the increased cost of an idle server. To evaluate the average operating cost, by obtaining the average buffer length at the health helpline service center and the associated probability values, the average cost function is formulated as follows.  
\[AC=c_{0}\overline{O}+c_{1}P_{b}+c_{2}P_{v}+c_{3}P_{r}+c_{4}\mu_{1}+c_{5}\mu_{2 }+c_{6}\mu_{w}+c_{7}\eta;\]  
where  
\[c_{0} \equiv \text{holding cost per unit time when a call enters the buffer;}\] \[c_{1} \equiv \text{cost per unit time for chief physicians being in regular service period;}\] \[c_{2} \equiv \text{cost per unit time for chief physicians being in maintenance state;}\] \[c_{3} \equiv \text{cost per unit time for chief physicians being under repair;}\] \[c_{4} \equiv \text{cost for offering service with rate }\mu_{1};\] \[c_{5} \equiv \text{cost for offering service with rate }\mu_{2};\] \[c_{6} \equiv \text{cost for offering service with rate }\mu_{w};\] \[c_{7} \equiv \text{cost associated with parameter }\eta\text{ for service facility in maintenance state.}\]  
From a management perspective, we need to minimize the server's idle rate (\(1-U\)) and the average operating cost (\(AC\)), but, from a quality of service perspective, we alsowant to minimize the average time spent in the buffer, W. Therefore, our goal is to search for the optimal number of chief physicians (\(c\)), the optimal outgoing generation rate (\(\alpha\)), and the optimal vacation rate (\(\eta\)), which satisfies these requirements. To this end, the multi-objective optimization problem is proposed as follows:  
\[\begin{array}{l}\text{Min }AC\\ \text{Min }1-U\\ \text{Min }W\\ \text{Subject to }(b_{b}\lambda+v)P_{B}<\
u-\frac{\delta}{\gamma+\delta}\frac{b_{b} \lambda+v}{\sum_{i=0}^{\infty}a_{i}},\alpha\geq 0,\eta\geq 0,c\text{ is a positive integer}.\end{array}\]  
In order to solve this problem, researchers have proposed a variety of methods, such as the multi-objective genetic algorithm [34], multi-objective particle swarm optimization [35], etc. Because multi-objective genetic algorithms do not require the user to prioritize, scale, or weigh objectives, genetic algorithms are one of the most popular metaheuristic methods for solving multi-objective optimization problems. The results are helpful for system architects or executives under the constraints of the server's idle rate thresholds, the average time spent in the buffer thresholds, and the acceptable average operating costs.