Dear Nolberto Munier , thank you so much for your constructive feedbacks ,

Fuzzy AHP is not universally superior but offers advantages in high-uncertainty contexts, here I tried to cover most of your interesting comments,

1. Saaty’s critique of Fuzzy AHP

Saaty argued that AHP inherently accounts for uncertainty through its 1–9 scale and consistency checks. However, fuzzy AHP proponents contend that fuzzy sets better model linguistic ambiguity (e.g., "somewhat important") and granular uncertainty. Empirical studies (e.g., Kahraman et al., 2003; Bozbura et al., 2007) demonstrate fuzzy AHP’s effectiveness in complex, uncertain contexts. Sensitivity analysis here tests robustness, not "accuracy," but computational efficiency can be shown via reduced pairwise comparisons or faster convergence in fuzzy logic frameworks.

2. Linear Regression for Evaluation

Linear regression may evaluate relationships between variables (e.g., ridership vs. service quality). If the paper uses it to predict outcomes (e.g., demand), this aligns with standard practice. If used to assess criteria weights, clarification is needed, as AHP/FAHP is better suited for weighting.

3. Consistency in AHP

AHP enforces transitivity (if A > B and B > C, then A > C) via the consistency ratio (CR), flagging illogical judgments. While CR ensures internal coherence, it does not guarantee real-world validity. This is a limitation of any preference-based method. However, consistency checks reduce arbitrary biases, making weights more reliable for stakeholders.

4. Fuzzy Numbers in FAHP

Fuzzy numbers explicitly model uncertainty (e.g., "between moderate and strong preference"). While defuzzification produces crisp outputs, the process preserves uncertainty ranges, offering richer insights than deterministic AHP.

5. Validation in MCDM

Validation in MCDM often relies on expert consensus, sensitivity analysis, or benchmarking against historical decisions. While no universal "yardstick" exists, practical applicability and stakeholder acceptance validate results.

6. Transportation Terminology

• Waiting Time: Corrected—waiting time depends on frequency, not speed. Speed affects in-vehicle travel time.
• Transfer Need: Revised to "interchanging buses/routes."
• Time Availability: Clarified as "service frequency" (buses/hour).
• Limited Time of Use: Adjusted to "operating hours" (e.g., 6 AM–10 PM).
• Comfort in Stops: "Providing new buses" may improve seating/shelters at stops, indirectly enhancing comfort.

7. Fuzzy Logic and Real-World Relevance

Fuzzy logic is axiomatically grounded in Zadeh’s theory (1965) to model human reasoning under uncertainty. While fuzzy outputs are approximations, they reflect realistic trade-offs in ambiguous contexts (e.g., public transit preferences). The value lies in structuring qualitative judgments, not precise predictions.

Dear Szabolcs Duleba , I think you would like this discussion :)

Sarbast Moslem added a reply

Dear Nolberto Munier , thank you so much for your constructive feedbacks ,

NM-You are most welcome, and I deeply thank and appreciate your answer

SM- Fuzzy AHP is not universally superior but offers advantages in high-uncertainty contexts, here I tried to cover most of your interesting comments,

NM – AHP is universally known, I grant it, but it does not mean that it is effective, in my opinion it is irreal and flawed. What FAHP does is simply find an average value on the DM estimates. Please, look at this from fuzzy analytic hierarchy process

Toly Chen (2020) “FAHP) has been extensively applied to multi-criteria decision making (MCDM). However, the computational burden resulting from the calculation of fuzzy eigenvalue and eigenvector is heavy. As a result, a FAHP problem is usually solved using approximation techniques such as fuzzy geometric mean (FGM) and fuzzy extent analysis (FEA) instead of exact methods. Therefore, the FAHP results are subject to considerable inaccuracy”

So, the DM accumulates inaccuracy on uncertainty. Your opinion please.

SM- 1. Saaty’s critique of Fuzzy AHP

Saaty argued that AHP inherently accounts for uncertainty through its 1–9 scale and consistency checks. However, fuzzy AHP proponents contend that fuzzy sets better model linguistic ambiguity (e.g., "somewhat important") and granular uncertainty. Empirical studies (e.g., Kahraman et al., 2003; Bozbura et al., 2007) demonstrate fuzzy AHP’s effectiveness in complex, uncertain contexts. Sensitivity analysis here tests robustness, not "accuracy," but computational efficiency can be shown via reduced pairwise comparisons or faster convergence in fuzzy logic frameworks.

NM- You are using the right words “Saaty argued” for its table and consistency checks.

I agree that fuzzy models better linguistic ambiguity, but just for the benefit of the DM, not for the project. Now I ask you? Why the measure of the DM coherence must affect the problem?

Another question, how Kahrarman and Bozhura demonstrate that fuzzy AHP is effective, if they do not have a yardstick to measure it? You do not need mathematics for this, only common sense and reasoning, which as probably you noticed, is the way I analyze a method in discussions

Regarding SA you are correct, its purpose is to measure strength of the best solution, and this is very useful indeed, but may I remind you that SA in AHP consists in selecting as most important criterion to vary, the one with the highest weight, without any mathematical proof of that, all based in intuitions? In addition, by selecting only one criterion and keeping the other constant (ceteris paribus), the DM is applying an incorrect procedure.

SM- “ Saaty argued that AHP inherently accounts for uncertainty through its 1–9 scale and consistency checks.”

NM- Now, why estimates from a DM must be consistent? In my opinion, it is due to the way Eigen Value (EI) can be applied. If you have an inconsistent matrix the EV delivers the weakest weight, therefore it is only a convenient feature for AHP

SM- 2. Linear Regression for Evaluation

Linear regression may evaluate relationships between variables (e.g., ridership vs. service quality). If the paper uses it to predict outcomes (e.g., demand), this aligns with standard practice. If used to assess criteria weights, clarification is needed, as AHP/FAHP is better suited for weighting.

NM- I do not follow your reasoning. Linear regression is a mathematical process while AHP is not

SM-3. Consistency in AHP

AHP enforces transitivity(if A > B and B > C, then A > C) via the consistency ratio (CR), flagging illogical judgments. While CR ensures internal coherence, it does not guarantee real-world validity. This is a limitation of any preference-based method. However, consistency checks reduce arbitrary biases, making weights more reliable for stakeholders.

NM- You used the correct word “enforcestransitivity. Do you think that is it natural that a formula disavows the estimate of a DM? At least you recognize that it does not guaranty real-world validity, something that Saaty also said, and I do not think that there is MCDM method that models reality in its integrity, but rational methods like POMETHEE ELECTRE, TOPSIS, VIKOR, use reasoning, analysis, experience, research, consultation, something that AHP ignores

SM- 4. Fuzzy Numbers in FAHP

Fuzzy numbers explicitly model uncertainty (e.g., "between moderate and strong preference"). While defuzzification produces crisp outputs, the process preserves uncertainty ranges, offering richer insights than deterministic AHP.

NM- Fuzzy logic only finds the average of the DM estimates. And where the uncertainty ranges come from?

Do you know that there are rational MCDM methods that can determine them, and in published papers with real-life examples?

SM- 5. Validation in MCDM

Validation in MCDM often relies on expert consensus, sensitivity analysis, or benchmarking against historical decisions. While no universal "yardstick" exists, practical applicability and stakeholder acceptance validate results.

NM- Consensus is not validation. The first implies an agreement of intervening parties, each one yielding on some issue and taken on another. Validation is to compare a ranking from analytical tools with the true, existing ranking, that is completely unknown and not available to any MCDM method. Therefore, validation is impossible. Just think that if the DM knew the true results, applying MCDM would be unnecessary

Validation is not related to sensitivity analysis (SA), that tries to determine the strength of the best alternative when criteria, not a criterion, in most cases, is increased/decreased. If the best alternative depends on say three criteria, and for reach one there is a reasonable allowed variation gap, the DM can decide that said alternative is strong, but this is not validation

SM- 6. Transportation Terminology

· Waiting Time: Corrected—waiting time depends on frequency, not speed. Speed affects in-vehicle travel time

NM- Agreed. I think that I made a mistake, sorry for this. Waiting time also depends on other aspects as capacity, benches, information screen, punctuality of buses, weather (temperature, In some Canadian cities heaters are installed, while in Dubai air conditioning is a must. Therefore, it is not as simple

· Transfer Need: Revised to "interchanging buses/routes."

· Time Availability: Clarified as "service frequency" (buses/hour).

· Limited Time of Use: Adjusted to "operating hours" (e.g., 6 AM–10 PM).

· Comfort in Stops: "Providing new buses" may improve seating/shelters at stops, indirectly enhancing comfort.

NM- Providing new more comfortable buses is comfort on board. Seating in shelters provides comfort on land between buses, that must provide comfort too, such as: Easy boarding for old and people in wheelchairs, enough number of seats, screen signaling the next arrival to a station, comfortable temperature, information, cleanliness, etc.,

SM- 7. Fuzzy Logic and Real-World Relevance

Fuzzy logic is axiomatically grounded in Zadeh’s theory (1965) to model human reasoning under uncertainty. While fuzzy outputs are approximations, they reflect realistic trade-offs in ambiguous contexts (e.g., public transit preferences). The value lies in structuring qualitative judgments, not precise predictions.

NM - Agreed on fuzzy logic. Do you know that Saaty said that fuzzy should not be used in AHP because the method is already fuzzy? In my opinion, fuzzy lies on estimated geometrical forms to determine memberships, and using for instance, three invented values. Therefore, it can correctly average the DM coherence, but it is not related to reality, what is the really important.

Dear Szabolcs Duleba ,

NM-It would very productive is Mr. Duleba can collaborate in this important subject