# 188

Dear Shervin Zakeri, Dimitri Konstantas, Prasenjit Chatterjee

I read your paper

Soft cluster-rectangle method for eliciting criteria weights in multi criteria decision-making

My comments:

1- In the abstract you say: “Significant challenges are faced due to two primary factors: the inherent uncertainty in inputs and the process of pairwise comparisons. These challenges increase the uncertainty regarding the derived weights, raising concerns about the reliability of such approaches”

I agree with this assertion, especially with what you say about pair-wise comparisons. In my opinion a very poor procedure without any mathematical support, and worse, denoting a lack of common sense, not in the comparison itself, which is very real, but in extracting a quantitative value out of that comparison. Welcome to your SCR, I am happy that there are researchers than address this subject, on which I was complaining for decades.

2- Page 1 “The origins of MCDM can be traced back to the late 1950s”

In reality, it was born at about 1940 when Leonid Kantorovich created Linear Programming, very difficult to apply, with neither computers, nor calculators. Fortunately, in 1948 George Dantzig developed the Simplex algorithm, the same that is installed since 1993 in all computers as an Excel Add-in, and that allows solving a problem in minutes. In Excel it is called ‘Solver’ and is found under ‘Data’.

3- Page 2 “Correct determination of criteria weights is essential for achieving accurate and reliable decision outcomes, especially in scenarios with a high number of criteria”

4-

Not really, weights are certainly essentials to quantify criteria, but not to evaluate alternatives, since they do not participate in the evaluation process

“This complexity can deter decision-makers from thoroughly exploring all relevant criteria or updating weights as new information becomes available”

FINALLY!!!!, researchers are validating what I always said: ‘Weights cannot remain constant’. Since as you rightly express, they must be updated. I would say that 99% of the more than 200 MCDM methods ignore this fact. There is only one method that considers it: Linear Programming (LP), and this is easily checked in each Simplex iteration. After each iteration, the Simplex automatically updates the whole set of criteria weights after an alternative has been selected.

How?

Weights are computed as a function of the availability of resources for each criterion. They are computed as a ratio between the available resource and the performance values of the selected alternative in each criterion.

That is, mathematically supported and rational, since in maximizing benefits for instance, it is chosen the criteria that consumes de minimum of resources.

Suppose a company that manufactures a product.

In a criterion for maximizing benefits, the company wants of course to maximize the sales for its product. Since this goal is linked with the resources to produce a good, a criterion is chosen to minimize the amount of resource used for unit of that product. The Simplex finds the ratio between the total amount of that resource (bi), and the unit value (aij), that is, (bi / aij) [∀j), and logically, selects the minimum ratio among all criteria, why?

Because it corresponds to the existing least important alternative to be eliminated, and thus making room for the newly selected.

The company produces the good at a certain cost. When the criterion cost calls for minimization, the Simplex selects the highest of the ratio. The reason is that the company wants to minimize its manufacturing cost and then increasing the benefit, but the resulting minimum cost must be greater that the production cost, if not the company loses money.

In both cases, the complete set of weights changes in accordance to the alternative selected.

This is an example of why the Nobel Prize to LP, was awarded to a creator of a method that represents and model real-life scenarios.

5- Page 4 “The problems with AHP and other weighting methods that use pairwise comparisons raise doubts about the reliability of their results. This highlights the necessity for a method that can incorporate their advantages while mitigating the deficiencies that stem from using pairwise comparisons”

Agreed in a 100%

6- Page 4 “The deficiencies associated with pairwise comparisons directly impact the reliability of criteria weights, as discussed earlier. This effect extends beyond the criteria weights, potentially introducing variability and inaccuracies in final results in MCDM methods that depend on these weights. Since criteria weights play a crucial role in shaping these decisions, reliance on pairwise comparisons introduces additional variability across the entire MCDM process”

Agreed, however I think there is an error in your comments because it seems intuitively, that weights may affect the final result. They are only trade-offs that have no participation in the evaluation of alternatives; why?

Because alternatives are evaluated by the dispersion of values WITIIN each criterion not by CRITERIA. CRITERIA.See Shannon’s Theorem. If not, ranking should be equal using entropy and weights, something that is very easy to demonstrate as false

However, weights can alter a ranking, simply due to a parallel displacement of their lines when are affected by different weights, but this is only geometry. It is only a different displacement of lines parallel to themselves.

7- Page 5 “To address these challenges, this paper introduces a new subjective MCDM weighting method that relies on direct criteria evaluation by DMs, avoiding pairwise comparisons in weight calculation”

This is reasonable and makes sense

8- Very interesting and reasonable what you propose in SCR. However, I believe that the intermediate cluster in yellow, should contain all criteria that share data. Probably you know than sustainability is pictured as three circles or clusters, involving Economics, Health and Environment, that mutually intersect (The Venn diagram).

This intersection forms an irregular geometrical figure, that contains all criteria that are feasible solutions to the problem. The feasibility comes from the fact that these criteria comply SIMULTANEOUSLY the objectives of the problem. For instance, an alternative may be the construction of a park with many different species, of trees however, if there is no money to buy the trees and for maintenance, this project is unfeasible.

9- That common space of feasible solutions is exactly what the Simplex finds. In here, each criterion, irrelevant their number, may intersect and form a polygon in two dimensions or polytope in many dimensions or alternsatives. Within that figure are all possible feasible solutions to the problem. However, we need only the best, and this is done using the Fundamental Theorem of LP, that establishes that all optimal solutions are in the vertices of this figure. The best one is easily found at the tangency of the objective function and only one vertex. This is LP and what the Simplex does, finding the optimal solution, if it exists.

From my point the view there is a very strong similarity between SCR and LP, and it makes SCR a valuable tool.

However, there is something that in my opinion tarnish this similitude, and it is the fact that as I believe, and maybe wrongly, SCR considers addition in lieu of intersection, that is A ∪ B, instead of A ∩B, and this is the basis of everything

If I am mistaken, letting me know about it will be very much appreciated

10- “• Vital cluster: • Mediocre cluster: • Immaterial cluster: Conversely, the immaterial cluster includes those elements that may be deemed less critical and could potentially be disregarded in certain situations, depending on the specific objectives of the decision-making process”

I think that this classification in clusters is very good. Years ago, I did something similar to determine memberships between environmental indicators and criteria, using a 1 to 10 scale. However, I believe that you may also consisted absolute and negative values when these two extremes are opposite. It must be absolute in order to be considered when you add up values for each alternative.

In addition, I do not think that you can disregard specific objectives, whatever three values, because they were considered for some reasons necessary, even if they have low values. Considering that everything is related, it is impossible to foresee what may be important or not, irrelevant of their value

11- I took the liberty of solving your problem using SIMUS a software based on LP and got a unique result as follows for SCR WRWEN:

route E > route D > route B > route A > route C

Scores: 2.5 – 0.37 – 0.23 – 0.1 – 0.10

Yours:

route E > route B > route C > route A > route D

These are my comments. Hope they can help

Nolberto Munier