## 3 Methodology
### The TOPSIS Algorithm
#### 3.1.3 Criteria Weighting  
There are several TOPSIS variants or techniques for weighting the criteria. Three variants are of interest in this study, namely SD-TOPSIS, Entropy-TOSIS, and Importance-TOPSIS. These variants were developed to address the subjectivity and issues of the Standard-TOPSIS, where all criteria have equal weight, and other existing weighting methods. The sum of the weights of all criteria must be equal to 1. The goal is to obtain a weight vector that satisfies the following relationship:  
\[w=(w_{1},w_{2},...,w_{m}),\sum_{j=1}^{m}w_{j}=1 \tag{3}\]  
If the alternatives have similar performance rates for a given criterion, that criterion will have a lower influence on the handover decision. In other words, if a criterion \(j\) for all the alternatives in the decision matrix is identical, then this attribute is not useful for the handover decision. 1. SD-TOPSIS: The standard deviation measures the dispersion of values around the mean and allows for evaluating the variability of a criterion [10]. A criterion with a larger standard deviation receives a higher weight, and vice versa. SD-TOPSIS is particularly useful when criteria exhibit different levels of variability. It is more practical and objective for handover management in satellite networks, as several parameters fluctuate rapidly due to the fast movement of LEO satellites. It removes the subjectivity in defining the weights of criteria by assigning moderate and precise weights based on real data. Considering m as the number of alternatives and \(j\) as the index of the criterion, the standard deviation of each criterion is calculated as follows: \[\sigma_{j}=\sqrt{\frac{1}{m}\sum_{i=1}^{m}(a_{ij}^{\text{norm}}-\mu_{j})^{2}}\] (4) where \(a_{ij}^{\text{norm}}\) is an element from \(D^{\text{norm}}\), and \(\mu_{j}\) the average value. The weight of each of the criterion \(j\) among the \(n\) criteria is then given by: \[w_{j}^{\text{sd}}=\frac{\sigma_{j}}{\sum_{k=1}^{n}\sigma_{k}}\] (5)
2. Entropy-TOPSIS: The entropy measures data uncertainty by using probability theory [13]. The wider the data, the higher the uncertainty. Entropy-TOPSIS is a type of objective weighting technique that measures the weights of criteria based on their relative differences. The main limitation is that it requires the data to exhibit good diversity. Homogeneous data can lead to less effective results as the method tends to overweight the criterion with the highest entropy. Considering \(j\) as the index of the criterion, the value of the entropy divergence degree coefficient \(e_{j}\) can be measured using the normalized decision matrix, with \(0\leq c_{j}\leq 1\): \[e_{j}=1-c_{j}\] (6) with \[c_{j}=\frac{1}{\ln(n)}\sum_{i=1}^{n}a_{ij}^{\text{norm}}\ln(a_{ij}^{\text{norm}})\] (7) The weight of each criterion is then given by : \[w_{j}^{\text{en}}=\frac{e_{j}}{\sum_{j=1}^{n}e_{j}}\] (8)
3. Importance-TOPSIS: This technique uses continuous replacement to calculate the impact of individual changes of each criterion value on the target value (the score in our case), and give higher weight to the criteria that have more impact on the target [7]. The steps of the continuous replacement method are as follows: * Step 1. Let \(Q(\alpha,\beta)\), \(Q^{{}^{\prime}}(\alpha^{{}^{\prime}},\beta^{{}^{\prime}})\) be such that their difference is \(\Delta=Q^{{}^{\prime}}-Q\). Suppose the order of substitution is to change \(\alpha\) first, then change \(\beta\). * Step 2. Replace \(\alpha\) only and find \(Q_{1}(\alpha^{{}^{\prime}},\beta)\). \(Q_{1}\) is obtained by changing \(\alpha\) on the basis of Q. Calculate the influence of \(\alpha\): \[Q_{\alpha}=Q_{1}-Q\] (9) * Step 3. Replace \(\beta\) only and find \(Q^{{}^{\prime}}(\alpha^{{}^{\prime}},\beta^{{}^{\prime}})\). \(Q^{{}^{\prime}}\) is obtained by changing \(\beta\) on the basis of \(Q_{1}\). Calculate the influence of \(\beta\): \[Q_{\beta}=Q^{{}^{\prime}}-Q_{1}\] (10) It can be observed that: \[\Delta=Q^{{}^{\prime}}-Q=Q_{\alpha}+Q_{\beta}\] (11) * Step 4.