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TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) is a multi-criteria decision analysis method that helps in ranking alternatives based on their distance to the ideal solution. Below are the steps of the TOPSIS method, along with the required formulas.

### Step 1: Construct the Decision Matrix

You start by constructing a decision matrix with $ m $ alternatives and $ n $ criteria. Let the decision matrix be denoted as $ D = [x_{ij}] $, where $ x_{ij} $ is the performance of the $ i $-th alternative on the $ j $-th criterion.

### Step 2: Normalize the Decision Matrix

The decision matrix is normalized to bring all criteria to a comparable scale. The normalization is done using the Euclidean norm. The normalized value $ r_{ij} $ is calculated as:

$r_{ij} = \frac{x_{ij}}{\sqrt{\sum_{i=1}^{m} x_{ij}^2}}$

Where:
– $ x_{ij} $ is the original value of the decision matrix,
– $ m $ is the number of alternatives.

### Step 3: Calculate the Weighted Normalized Decision Matrix

Once normalized, the next step is to multiply the normalized decision matrix by the corresponding weights of each criterion. Let $ w_j $ be the weight of the $ j $-th criterion. The weighted normalized value $ v_{ij} $ is calculated as:

$v_{ij} = w_j \cdot r_{ij}$

Where:
– $ w_j $ is the weight of the $ j $-th criterion,
– $ r_{ij} $ is the normalized value from Step 2.

### Step 4: Determine the Ideal and Negative-Ideal Solutions

The positive ideal solution $ A^+ $ and the negative ideal solution $ A^- $ are defined as follows:

– The positive ideal solution $ A^+ $ is composed of the best values for each criterion:
$A^+ = \left\{ \left( \max v_{ij} \ \text{for benefit criteria} \right), \ \left( \min v_{ij} \ \text{for cost criteria} \right) \right\}$

– The negative ideal solution $ A^- $ is composed of the worst values for each criterion:
$A^- = \left\{ \left( \min v_{ij} \ \text{for benefit criteria} \right), \ \left( \max v_{ij} \ \text{for cost criteria} \right) \right\}$

### Step 5: Calculate the Separation Measures

The separation of each alternative from the positive ideal solution $ S_i^+ $ and the negative ideal solution $ S_i^- $ is calculated using the Euclidean distance:

– Separation from the positive ideal solution:
$S_i^+ = \sqrt{\sum_{j=1}^{n} \left( v_{ij} - A_j^+ \right)^2}$

– Separation from the negative ideal solution:
$S_i^- = \sqrt{\sum_{j=1}^{n} \left( v_{ij} - A_j^- \right)^2}$

Where:
– $ A_j^+ $ is the positive ideal value for criterion $ j $,
– $ A_j^- $ is the negative ideal value for criterion $ j $.

### Step 6: Calculate the Relative Closeness to the Ideal Solution

The relative closeness of each alternative to the ideal solution is denoted as $ C_i^* $ and is calculated as:

$C_i^* = \frac{S_i^-}{S_i^+ + S_i^-}$

Where:
– $ C_i^* $ is the relative closeness to the ideal solution,
– $ S_i^+ $ is the separation from the positive ideal solution,
– $ S_i^- $ is the separation from the negative ideal solution.

### Step 7: Rank the Alternatives

Finally, the alternatives are ranked based on the relative closeness value $ C_i^* $. The higher the value of $ C_i^* $, the closer the alternative is to the ideal solution, and thus, the higher the ranking.

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These are the essential steps and formulas of the TOPSIS method, which provides a systematic approach for multi-criteria decision-making.

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