## 4 Service Scheduling Algorithms
### Design of Service Scheduling Algorithm
Generally, the original decision matrix can be normalized using three methods: vector normalization, Z-score, and Min-Max. This paper adopts the vector normalization method, after normalization, and the elements of the normalized decision matrix Z is expressed as follows:  
\[z_{ij}=\frac{z_{ij}}{\sqrt{\sum_{k=1}^{n}z_{kj}^{2}}} \tag{4}\]
* Method for determining weights: In MCDM problems, the selection of weights has a significant impact on the final ranking results, so it is crucial to determine the weighting method. There are two main \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt}} \hline \hline
**Criteria Category** & **Explanation** \\ \hline Compute Cluster Resource Availability & Percentage of available compute, memory, and storage resources \\ \hline Available Compute Cluster Resources & Standardized compute, memory, and storage resources \\ \hline Service Historical Completion Time & Measured in milliseconds \\ \hline Service Quality Level & Interval classification based on historical completion time \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Evaluation criteria for computing clusters. weighting methods: subjective and objective. Their advantages, disadvantages, and main methods are shown in the Table 4:  
After analysis: this paper has clear scheduling objectives, and after domain partitioning, each network node maintains a limited amount of cluster information for each service. Objective weighting methods may encounter issues such as insufficient data representation and unstable statistical characteristics. Therefore, the subjective weighting method is more suitable for the architecture discussed in this paper. Among subjective weighting methods, the Best-Worst Method (BWM) is an improved version of the Analytic Hierarchy Process (AHP). The BWM simplifies the weight acquisition steps and is more concise and easier to understand. Compared to the Delphi Method, it offers greater operability. Therefore, the BWM has been chosen as the subjective weighting method for this paper. It should be pointed out that in the BWM, consistency in judgments is crucial for assessing the validity of derived weights. It ensures that if Criterion A is deemed more important than Criterion B, and Criterion B is deemed more important than Criterion C, then Criterion A should also be more important than Criterion C. The classical posterior consistency check limits real-time adjustments during evaluation. Therefore, this paper adopts an input consistency method [23] to improve weight assessment and enhance decision-making flexibility. * Evaluation and Ranking Among various MCDM algorithms, TOPSIS stands out for its flexibility, intuitiveness, and ease of programming. It ranks alternatives by calculating their distances from the positive ideal solution (best values) and negative ideal solution (worst values). The process begins with a normalized decision matrix and weight matrix, which are multiplied to produce the evaluation matrix R. From this, the positive ideal solution \(A^{+}\) and negative ideal solution \(A^{-}\) are identified. The distances of each alternative to these ideal solutions are then calculated as \(d_{i}^{+}\) and \(d_{i}^{-}\), allowing for the determination of each alternative's relative closeness to the ideal solutions. Based on these distances, a final distance \(d_{i}\) is computed for each alternative, which is used to determine their relative closeness to the ideal solutions. Formulas (5)-(10) is shown below:  
\[R=Z\cdot w^{*}= \begin{pmatrix}r_{11}&\cdots&r_{1n}\\ \vdots&\ddots&\vdots\\ r_{m1}&\cdots&r_{mn}\end{pmatrix}  \tag{5}\]  
\[A^{+}=\{r_{1}^{+},r_{2}^{+},\ldots,r_{n}^{+}\},\quad r_{j}^{+}=\max_{m}r_{mj} \tag{6}\]  
\[A^{-}=\{r_{1}^{-},r_{2}^{-},\ldots,r_{n}^{-}\},\quad r_{j}^{-}=\min_{m}r_{mj} \tag{7}\]  
\[d_{i}^{+}=\sqrt{\sum_{j=1}^{n}(r_{j}^{+}-r_{ij})^{2}},\quad i\in m \tag{8}\]