### 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:

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:

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:

– The negative ideal solution \( A^- \) is composed of the worst values for each criterion:

### 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:

– Separation from the negative ideal solution:

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:

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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