TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) is a multi-criteria decision-making method used to evaluate and rank a set of alternatives based on a combination of their weighted performance against multiple criteria. The method involves creating a matrix of the alternatives and criteria, normalizing the data, determining the weighted normalized scores for each alternative, calculating the ideal and anti-ideal solutions, and finally computing the distance between each alternative and these solutions using a suitable metric. The ranking is then determined based on the proximity of each alternative to the ideal solution. TOPSIS is widely used in various fields such as engineering, management, environmental studies, and healthcare, among others.

TOPSIS proves to be a highly efficient approach when it comes to determining the anticipated benefits of an unclear scenario that lacks comprehensive data and precision. The method employed by TOPSIS enhances a solution by choosing the one that has the least distance from the ideal solution while having the greatest difference from the negative-ideal solution. However, it does not represent the relative importance of these distances accurately, as shown in this figure.

## A step-by-step guide for the TOPSIS method

This step-by-step guide demonstrates the functionality of TOPSIS method and provides guidance on how to use it effectively. By following this tutorial, you can implement your own TOPSIS algorithm.

- Define the problem and criteria: The first step is to define the problem at hand and determine the criteria that will be used to evaluate the alternatives. Criteria could be any relevant factors such as cost, quality, efficiency, etc.
- Construct a decision matrix: Next, create a decision matrix where each row represents an alternative and each column represents a criterion. Assign scores for each alternative on each criterion.
- Normalize the decision matrix: To ensure that each criterion receives equal weight, normalize the decision matrix by dividing each score by the sum of all scores in that column. This will convert the matrix into a relative performance evaluation matrix.
- Determine the weights of criteria: Assign weights to each criterion based on their relative importance in achieving the objective. The sum of all weights should add up to 1.
- Multiply the normalized decision matrix with the weights matrix: Multiply the normalized decision matrix by the weights matrix to get the weighted normalized decision matrix.
- Determine the ideal and negative-ideal solutions: Identify the best and worst possible values for each criterion. For maximization criteria, the ideal solution is the highest value found among all alternatives, while the negative-ideal solution is the lowest value. For minimization criteria, it’s the other way around.
- Calculate the distance from the ideal and negative-ideal solutions: Calculate the Euclidean distance between each alternative and the ideal and negative-ideal solutions separately.
- Calculate the relative closeness to the ideal solution: Finally, calculate the relative closeness of each alternative to the ideal solution by dividing the distance from the negative-ideal solution by the sum of the distances from both the ideal and negative-ideal solutions.
- Rank the alternatives: Rank the alternatives based on their relative closeness values. The alternative with the highest value is the most favorable.

## Analysis of the Topsis Algorithm

Normalize the decision matrix X using a normalization function that scales each column vector to have unit sum of squares:

x_{ij}^\prime=\frac{x_{ij}}{\sqrt{\sum_{i=1}^{n}(x_{ij})^2}}

Construct the weighted normalized decision matrix Vby multiplying each normalized element in X by its corresponding weight w_j:

v_{ij}=w_j x_{ij}^\prime

Determine the positive (Ideal) and negative ideal (Anti-Ideal) solutions:

A^+=\max_{1\leq i\leq n}{v_{ij}|j=1,2,...,m}

A^-=\min_{1\leq i\leq n}{v_{ij}|j=1,2,...,m}

Calculate the distance of each alternative solution from the ideal solutions:

D_i^+=\sqrt{\sum_{j=1}^{m}(v_{ij}-A_j^+)^2}

D_i^-=\sqrt{\sum_{j=1}^{m}(v_{ij}-A_j^-)^2}

Relative Closeness to Ideal and Anti-Ideal Solutions:

C_i=\frac{D_i^-}{D_i^++D_i^-}

User-Friendly TOPSIS Solver!

There are several solvers for TOPSIS algorithm, such as online and offline software packages. However, these solutions often require specialized knowledge and software tools, making it challenging for non-experts to implement them effectively. As a solution to this problem, we would suggest our spreadsheet (Orpida TOPSIS excel template) to you. It is a user-friendly and affordable tool that simplifies the TOPSIS process by providing an interactive spreadsheet interface. Users can input their data and preferences into the spreadsheet, which then calculates the TOPSIS scores and rank orders the alternatives automatically. To use the TOPSIS Excel file, you can click on the link.