📈 economics

TOPSIS-RAD: Ranking According to Desires

This paper introduces TOPSIS-RAD, an enhanced ranking method that improves upon traditional TOPSIS by incorporating decision-maker-defined Vetoed and Desired Performance Levels to filter non-viable alternatives, anchor ideal solutions to explicit aspirations, and ensure stable rankings resistant to outliers and rank reversal.

Original authors: Leonardo Fernandes Costa, Helder Gomes Costa, Diogo Lima, Brunno Rodrigues

Published 2026-06-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Leonardo Fernandes Costa, Helder Gomes Costa, Diogo Lima, Brunno Rodrigues

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a hiring manager trying to pick the best candidate from a list of ten job applicants. You have four criteria: Experience, Education, Skills, and Personality.

The Old Way (Traditional TOPSIS): The "Class of the Year" Problem
In the traditional method, you look at the group of ten people and ask, "Who is the absolute best in this specific group?" and "Who is the absolute worst?"

If one applicant has 150 years of experience (an outlier), they become the "Perfect Score" benchmark for everyone else.
If one applicant has 5 years of experience, they become the "Worst Score" benchmark.

The Problem: This system is fragile.

The "Outlier" Distortion: That one person with 150 years of experience skews the whole scale. Suddenly, a candidate with 40 years of experience looks "average" because they are being compared to a superhero, even though 40 years is actually amazing.
The "Rank Reversal" Glitch: If you add an eleventh candidate who is terrible at everything, the "Worst" benchmark shifts. This might accidentally make your second-best candidate look like the worst, or your third-best look like the best, just because the group composition changed. The ranking flips based on who else is in the room, not on who is actually good.
The "Dream vs. Reality" Mismatch: The system assumes the "best" person in the room is what you want. But maybe you don't need 150 years of experience; you just need 10. The old method doesn't know the difference between "impossible perfection" and "what I actually desire."

The New Way (TOPSIS-RAD): The "Job Description" Approach
This paper proposes a new method called TOPSIS-RAD (Ranking According to Desires). Instead of looking at the group to decide who is best, you (the decision-maker) set the rules before you look at the applicants.

Think of it like setting up a Target Range for a dartboard:

1. The "Veto" Line (VPL) – The Minimum Bar

You draw a line on the floor. You say, "If anyone is below this line, they are out."

The Metaphor: Imagine a height requirement for a rollercoaster. If you are 4 feet tall, you don't get to ride, no matter how much you want to.
How it helps: In the old method, a very short person might drag down the "average" height of the group, making everyone else look taller than they are. In TOPSIS-RAD, you simply remove the person who doesn't meet the minimum. They don't get to mess up the math for the people who do qualify.

2. The "Desired" Ceiling (DPL) – The Saturation Point

You draw a ceiling. You say, "If anyone is above this line, we stop counting their extra points."

The Metaphor: Imagine you are filling a bucket with water. Once the bucket is full (the "Desired" level), pouring in more water doesn't make the bucket "more full." It just spills over.
How it helps: If you need 10 years of experience, and one candidate has 100, the old method treats them as 10x better than someone with 10. TOPSIS-RAD says, "Great, you have 100, but we only need 10. We cap your score at 10." This prevents one "super-candidate" from distorting the scores of everyone else. It treats a candidate with 10 years and a candidate with 100 years as equal, because both meet your desire.

The Result: A Stable, Fairer Ranking

By using these two lines (the Veto floor and the Desired ceiling), the paper shows that:

Outliers don't break the system: A candidate with 150 years of experience no longer makes the 40-year candidate look bad. They are both capped at the "Desired" level.
Ranking doesn't flip: If you add a terrible candidate to the list, they just get vetoed. They don't change the math for the good candidates. If you add a new "super-candidate," they just get capped. The order of the good candidates stays exactly the same.
You get what you asked for: The ranking reflects your specific needs (e.g., "I need at least 5 years, but I don't need more than 10"), not just who happened to apply this week.

In Summary
Traditional TOPSIS is like judging a race by seeing who is fastest in that specific group of runners. If a world-record holder shows up, everyone else looks slow.

TOPSIS-RAD is like judging a race by a fixed time limit. If you need to finish in under 10 minutes to qualify, anyone slower is out. If you finish in 5 minutes or 1 minute, you both get the same "Gold" credit because you met the goal. The winner is the one who is closest to that goal without being disqualified, regardless of who else is in the race.

The paper demonstrates this with simple math examples (called "toy examples") showing that this method stops the rankings from flipping around when the group of options changes, ensuring the final list is stable and truly reflects what the decision-maker actually wants.

Technical Summary: TOPSIS-RAD: Ranking According to Desires

Problem Statement
Traditional TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) derives its reference points—the Positive Ideal Solution (PIS) and Negative Ideal Solution (NIS)—directly from the observed dataset. The authors identify three critical limitations inherent to this data-driven approach:

Misalignment with Decision-Maker (DM) Preferences: The PIS and NIS are artificial constructs based on dataset extremes, which may not reflect the DM's actual aspirations or minimum acceptable requirements.
Sensitivity to Outliers: Extreme values within the alternative set can distort the calculation of reference frontiers, introducing noise and potentially leading to anomalous rankings.
Rank Reversal: Because reference points depend on the specific set of alternatives, adding or removing an option (especially one that defines a frontier) can alter the relative distances of remaining alternatives, changing the ranking order and undermining method stability.

While previous literature has attempted to address rank reversal by fixing PIS and NIS based on external benchmarks or domain limits, the authors argue these approaches often fail to explicitly screen out non-viable alternatives or cap excessive performances that exceed the DM's actual desires.

Methodology: TOPSIS-RAD
The paper proposes TOPSIS-RAD (Ranking According to Desires), a variant that modifies the standard TOPSIS workflow by incorporating two decision-maker-defined arrays prior to the normalization and distance calculation stages:

Vetoed Performance Levels (VPL): A vector defining the minimum acceptable performance threshold for each criterion. Alternatives failing to meet these thresholds are excluded from the feasible set before normalization.
Desired Performance Levels (DPL): A vector representing the DM's preferred performance goals. Performances exceeding these levels are capped (clipped) at the DPL value before normalization.

The algorithm proceeds as follows:

Input: The decision matrix, weights, and the DM-defined VPL and DPL vectors.
Filtering: Identify and remove alternatives that violate the VPL constraints.
Capping: For the remaining qualified alternatives, clip any performance values that exceed the DPL.
Normalization: Apply a linear normalization (Max-Min style) using the fixed VPL and DPL as the lower and upper bounds, respectively. This ensures the normalization scale is independent of the specific alternatives in the dataset.
Reference Definition: The normalized reference points (Desired Normalized Level - DNL, and Veto Normalized Level - VNL) are derived directly from the fixed VPL and DPL vectors, rather than the dataset extremes.
Ranking: Calculate Euclidean distances to the DNL and VNL and compute the relative closeness coefficient ( $S_{iw}$ ) to generate the final ranking.

Key Contributions
The paper claims three primary contributions:

Explicit Preference Integration: The method grounds the ranking in stable, DM-specified boundaries (aspirations and minimums) rather than artificial dataset extremes.
Outlier Robustness: By capping performances at the DPL and excluding non-viable options via VPL, the method prevents extreme values from distorting the normalization frontiers or dominating the ranking.
Rank Reversal Immunity: Because the normalization bounds and reference points are fixed exogenously, adding or removing alternatives does not shift the scale or alter the relative distances of the surviving options, thereby preserving ranking stability.

Results and Toy Examples
The authors validate the method using three "toy examples" implemented via a custom web application (Visual TOPSIS-RAD):

Toy Example A (Baseline): Demonstrates traditional TOPSIS, where the ranking is driven by dataset extremes.
Toy Example B (VPL Effect): Applies a VPL to exclude a non-viable alternative (A8). The results show that removing A8 does not shift the normalization boundaries for the remaining alternatives, unlike in traditional TOPSIS where the removal of an extreme value would alter the PIS/NIS and potentially cause rank reversal.
Toy Example C (DPL Effect): Applies a fixed DPL (set below the dataset maximum) to cap high performers. This demonstrates that alternatives with "over-performing" metrics (e.g., A3 with a score of 150 on a criterion where 80 is desired) lose their disproportionate advantage. Consequently, alternatives with balanced profiles (e.g., A4) rise in the ranking, reflecting a "satisficing" logic where exceeding the desired level yields no additional credit.
Stability Test: The authors introduce a new, dominated alternative (A11) to the dataset. In traditional TOPSIS, this insertion causes a complete reshuffling of the rankings (rank reversal). In all TOPSIS-RAD configurations, the rankings of the original alternatives remain identical, with the new alternative either appended at the bottom or excluded entirely.

Significance and Claims
The paper positions TOPSIS-RAD as a bridge between the familiar distance-based structure of TOPSIS and reference-point methods found in other MCDA frameworks (like ELECTRE TRI or NADIR-UTOPIA). The authors claim the method is particularly beneficial for:

Recurring evaluations (e.g., supplier selection) where performance standards are stable.
Regulated contexts where minimum thresholds are mandatory.
Scenarios where extreme values represent noise rather than meaningful superiority.

The authors remain modest regarding the scope of their claims, noting that the method requires the DM to possess sufficient domain knowledge to define meaningful VPL and DPL values. They acknowledge that the method introduces additional cognitive effort for the decision-maker and identify future research directions, including integration with uncertainty modeling (fuzzy/probabilistic inputs), sensitivity analysis, and extensions to group decision-making. The paper does not claim to solve all MCDA problems but offers a specific mechanism to stabilize rankings and align them more closely with explicit decision-maker aspirations.

1. The "Veto" Line (VPL) – The Minimum Bar

2. The "Desired" Ceiling (DPL) – The Saturation Point

The Result: A Stable, Fairer Ranking

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