Topsis - Query regarding negative and positive attributes

Question

In Topsis technique, we calculate negative and positive ideal solutions, so we need to have positive and negative attributes (criterions) measuring the impact but what if I have attributes in the model having only positive impact? Is it possible to calculate Topsis results using only positive attributes?? If yes then how to calculate the relative part. Thanks in advance

Answer 1

Good question. Yes, you can have all positive attributes or even all negatives. So, while assessing alternatives you might encounter two different types of attributes: desirable attributes or undesirable attributes.

As a decision-maker, you want to maximise desirable attributes (beneficial criteria) and minimise undesirable attributes (costing criteria).

TOPSIS was created in 1981 by Hwang and Yoon*1. The central idea behind this algorithm is that the most desirable solution would be the one that it is the most similar to the ideal solution, so a hypothetical alternative with the highest possible desirable attributes and the lowest possible desirable attributes, and the less similar to the so-called 'anti-ideal' solution, so a hypothetical alternative with the lowest possible desirable attributes and the highest possible undesirable attributes.

That similarity is modelled with a geometric distance, known as Euclidean distance.*2

Assuming you already have built the decision matrix. So that you know the alternatives with their respective criterion and values. And you already identified which attributes are desirable and undesirable. (Make sure you normalise and weight the matrix)

The steps of TOPSIS are:

1. Model the IDEAL Solution.
2. Model the ANTI-IDEAL Solution.

3. Calculate Euclidean distance to the Ideal solution for each alternative.

4. Calculate Euclidean distance to the Anti-Ideal solution for each alternative.

5. You have to calculate the ratio of relative proximity to the ideal solution.

The formula is the following:

So, distance to anti-ideal solution divided by distance to ideal solution + distance to anti-ideal solution.

6. Then, you have to sort the alternatives by this ratio and select the one that outranks the others.

Now, let's put this theory into practice... let's say you want to select which is the best investment out of different startups. And you will only consider 4 beneficial criteria: (A) Sales revenue, (B) Active Users, (C) Life-time value, (D) Return rate

## Here we have our decision matrix, in R known as performance matrix...
performanceTable <- matrix(c(5490,51.4,8.5,285,6500,70.6,7,
                              288,6489,54.3,7.5,290),
                              nrow=3,
                              ncol=4,
                              byrow=TRUE)

# The rows of the matrix contains the alternatives.
row.names(performanceTable) <- c("Wolox","Globant","Bitex")

# The columns contains the attributes:
colnames(performanceTable) <- c("Revenue","Users",
                               "LTV","Rrate")
# You set the weights depending on the importance for the decision-maker.
weights <- c(0.35,0.25,0.25,0.15)

# And here is WHERE YOU INDICATE THAT YOU WANT TO MAXIMISE ALL THOSE ATTRIBUTES!! :
criteriaMinMax <- c("max", "max", "max", "max")

Then for the rest of the process you can follow R documentation on the TOPSIS function: https://www.rdocumentation.org/packages/MCDA/versions/0.0.19/topics/TOPSIS

Resources:

• Youtube tutorial: TOPSIS - Technique for Order Preference by Similarity to Ideal Solution. Manoj Mathew Retrieved from https://www.youtube.com/watch?v=kfcN7MuYVeI
• R documentation on TOPSIS function. Retrieved from: https://www.rdocumentation.org/packages/MCDA/versions/0.0.19/topics/TOPSIS

REFERENCES:

• 1 Hwang, C. L., & Yoon, K. (1981). Methods for multiple attribute decision making. In Multiple attribute decision making (pp. 58-191).Springer, Berlin, Heidelberg.
• James E. Gentle (2007). Matrix Algebra: Theory, Computations, and Applications in Statistics. Springer-Verlag. p. 299. ISBN 0-387-70872-3.