The immediately answer without too much thinking would possibly be: Yes, we need them, because not all criteria have the same importance, and this is true.
Fine, now what are criteria weights? They are subjective transcriptions of verbal ordinal expressions, also subjective. Analyzing this step, one realizes that they are totally arbitrary, and in addition, subject to the decision of a DM, which may be different from that of another DM,
Really, a very suspicious procedure and with no mathematical support.
We agree that weights are used to have a rank of criteria according to their relative importance. And now, the crucial question comes, ‘Importance relative to what?
If you analyze a fact such as for instance, the relative importance of quality and price regarding the purchase of a car, you can legitimately say, that in this respect, and for YOU, quality is more important, without assigning a quantitative value to that preference, because it would be meaningless. Nobody can put a value to a feeling or a preference.
Observe, and this is important, that your preference must be related to something, in this case the car, because in other aspects you may prefer price to quality, according to your tastes and budget, for instance, in purchasing a necktie, and thus, it depends on the object of the comparison.
In case of criteria weights in MCDM, a practitioner may express these two postulates:
1) Weights are necessary to evaluate alternatives,
2) Due to the fact that not all criteria have the same importance.
Is this answer correct? NOT in the first statement, and YES in the second.
The reason of the fallacy of the first statement, is that the relative importance between criteria is NOT associated with alternatives evaluation, unless they are objective weights derived from entropy.
Probably you will say: But these entropy weights also establish a ranking between criteria, as subjective weights do, then, why the evaluation of alternatives is valid for entropy and not from preferences?
Because Shannon Theorem, the base for Information Technology, as we know it today, demonstrates that to evaluate something you must have capacity for evaluation, you need a certain quantity of information, and it depends on the discrimination of the values of a criterion that is, it is an attribute. As an analogy, the actual system is equivalent to asking a 5 years old child to evaluate different cars.
He/she, at that age, does not have on cars the amount of information that an adult possesses.
This amount of information in a criterion can be found by entropy; this is the great discovery of Shannon, and he even developed a formula to compute it.
If in a criterion the different values corresponding to the alternatives are very similar or close, the entropy or uncertainty is high, the quantity of information low, and thus, this criterion has a low significance for evaluating alternatives.
The best example are dice. The uncertainty about which number will appear when casting, is 100%, because all of them have the same values (1/6); the entropy will be 1, and the quantity of information 0.
Since weighting does not have this property, it can’t be used for evaluation.
Therefore, why to spend thousands of hours trying to find out how to have better subjective weights, when they are useless, at least for MCDM?
I am not claiming that I am right but at least is what I get based on reasoning and science.
Therefore, if subjective weights are inappropriate to evaluate alternatives, why are we using them?
I would very much like if some of my colleagues in RG can add something else, supporting or not, my arguments. As a bottom line, I believe that weights, other that entropy ones, should not be used in MCDM methods. I am not saying that we should consider all criteria with the same importance, because it is not true most of the times, thus, we should use either entropy weights, or allow the method determine by itself the importance of each criterion, based on inputted data, as is done in Linear Programming.
I will be more than glad to discuss this issue, but please, with arguments, not with simple words or masking references about other authors wrote.
Well, weights are one way of navigating through the pareto front. Since the front can be huge, having some sort of guidance may be handy for practical purposes, even if subjective.
But I am curious: how do you derive entropy weights?
Weighted sum is not the only scalarization approach, so there are MCDM procedures (like Zeleny's displaced ideal in my opinion leading to Wierzbicki's reference point approach) not relying only on weigths.
And, if weighted sum is chosen, it has to be clear: there is no "best choice" for the weigths. Diffrerent positive weights will normally lead to different Pareto-optimal points and in that sense none of them is a priori a "best compromise".
If there would be a best choice for weights, the problem is not really muticriteria.
A DM will evaluate different Pareto optimal solutions in most cases not only looking at the values of the criteria, but also selected variables of the solutions.
In my opinion an interactive procedure letting a DM "play" with the model to navigate through the Pareto front, experiencing what consequences it has on the solution to put more importance on one criterion and giving more freedom for others.
In the example of the car the criterion quality in itself is an aggregate one - one brand/model might be better in safety for the passengers or longevity while another one is more powerful / sportive. Or it may offer more space and be more flexible (lika a van). All these aspects somehow may me subsumed under quality. The preferences of different buyers among those will for sure be different - even without considering the price.
So reducing the problem to quality vs. price does not reflect the whole problem, the DM will look much more deeply in the Pareto-optimal alternatives than only to compare the values for the two criteria.
Dear Nuno
If a Pareto front is defined as a set of non-deiminated solutions, that is, following the Pareto principle, I don't understand which is the role of weights in order to select the best solution. I am not saying that you are wrong, simply, I don't see it, and I will be grateful for your clarification.
In the mean time and assuming a non-lineal Pareto Front, you can detect the best solution studying the slopes in the curve. The solution which offers the largest slope is probably the best from your point of view and needs. That is, if you have two objectives such as quality and price, if you have preferences for price, you can select the point that increases it, and it will be still optimal
Dear Juan Manuel
That is true, but not because the methods are faulty, it is because subjectivity.
Just think:
1) If all methods start with the same initial decision matrix for the same problem
2) If MCDM methods are based on sound mathematics (except those based on intuition and irrationality)
3. If all methods aim at the same goal, that is to get a ranking.
Why results are different?
Because subjectivity, in two manners:
By using invented weights, and also due to assumptions without any mathematical foundation, or even, if they are mathematically grounded, their values depend on the DM, as in PROMETHEE, TOPSIS, and most others.
The second case will happen even using objective weights, and in this case the different results come from the DM assuming values like thresholds, or by not considering for instance, that resources are limited, or by limiting the number of criteria, etc.
Resuming: Where there is subjectivity there is a high possibility that the result is not as it should be.
'Fortunately'' for them, and bad for science, all results are accepted because there is no way, to determine which is the best, since we never know which it is.
For this reason, it is wrong to read as it is often seen, a practitioner that says that he/she succeeded in solving a problem.
In reality , the only thing that a method does is to follow the sequence by which it was designed. The algorithm does not care if you put good data or garbage, in the form of incorrect data, or data modified by subjective weights
Therefore, I am afraid that I don't share your opinion.
In my opinion results should be identical whatever the mathematical operation you perform, perhaps with very small differences, because approximations.
It is similar as computing for instance 3^2 . The result is the same, irrelevant if you use potentiation, multiplication or either normal or Napierian logarithms
Dear Ralf
Thank you for answering
I completely agree with your exposition
I am particularly interested in your paragraph about quality, because normally people use quality as a criterion, but do not explain to what it refers.
In a car, quality may be addressing the goodness of materials used, or available space, or upholstery finishes, or leather seats, etc.
In my opinion, because quality may refer to different things, you can't state, that for instance, in comparing criterion quality with criterion fuel consumption, that you always prefer one on the other, because that preference can change due to the different interpretation on quality, and this is one of the many drawbacks in AHP, where a comparison between two criteria is considered constant.
Since from this pair-wise comparison AHP derives weights, used to evaluate alternatives, a single weight for criterion C1 is applied indistinctively to comparing two pair of alternatives than can be very different.
As an example, assume that you have alternatives A1 and A2, the first refers to construction of a highway, while the second is construction of a solar park. You can compare them based on criterion C1, that refers to environment, which has a weight, and then you can get a score for A1 or for A2, according to their contribution to the environment.
Then you compare A1 with A3, where the latter refers to social work, and on the same criterion. Is it realistic in this second case to use the same weight when in reality it may be different? That is, the preference that gave birth to Ci weight, holds also when comparing construction with social work?
I don't think so; it might, but also ir might not.
And this is the procedure that practically all MCDM, methods use as weights. It appears to me that this procedure is incorrect.
If we use instead objective weights which are computed independently by criterion, they are also constant, but it is different, because what is constant is its amount of information, not a relationship with another criterion.
Anyway, this is my reasoning. I would love to read about yours, even if you don't share my opinions.
Dear Juan Manuel
I did not know of those models and never heard about them. Could you please explain how they work?
Thank you
Dear Juan Manuel
Thank you for your answer and your offer, but I am against subjectivity in any form in MCDM, except when the DM, based on a result obtained by actual figues, uses his know-how and experience and even preferences. That is , decisiones are taken post result, not ante-result, based on nothing, but intuition.
For me, it is something incomprehensible that the results of a problems are based on judgements, unless, of course, those judgements are based on research, statistics and experience.
Congratulations on the book you are writing. If there is anything that I can do to help me, just let me know it.
Thank you for your greetings. I am in Canada, and I love your country where i had the opportunity to live and also acted as a speaker in universities of Puebla nd Tecamachalco
Best regards
Nolberto
Yes. Because the basis of MCDM method are criteria. So that the importance of each of them is different from each other.
Dear Nolberto,
I do not agree with your opinion. I am really thinking that in the presence of conflicting objectives there is no way to choose among all Pareto-optimal solutions via fixed weights - no matter how these are determined.
And I think the aim of somehow determining the whole Pareto front is overkill. That could easily be done via one-parametric linear programming for linear two-criteria problems - just using ALL positive weights, summing up to 1. But beginning with three linear criteria over a polyhedron things get much more complicated, not to speak of nonlinear (even convex) constraints and/or criteria.
I think that for choosing among the Pareto front (all of those solutions are optimal wrt. the criteria) a DM does not only rely on the values of the criteria, but on more information in the solutions, which he/she might not be able to formulate through a formula as additional criteria.
Presenting the DM just ONE solution is not adequate for a real MDCM problem, no matter how the determination of that solution is motivated.
But that's just my opinion.
Dear Ralf
Thank you for your answer, but I am surprised because I never said anything about weights in relation to Pareto front.
By the way entropy derived weights are fixed, because they depend on the values of each criterion, and only may change if the DM changes them.
I agree with your second paragraph, but again ,I don't think we need weight for that. Since when LP uses them?
However, I also agree with you about three or more criteria.
Sorry, I can't understand what you mean in your third paragraph.
Regarding your last paragraph, I do believe that one solution is OK, as long as it has been obtained without subjectivity. However, considering that solution as a robust and reliable base, the DM may consider modifying the data, or directly rejecting it, because even if it is perfect from the mathematical point of view, perhaps it is not convenient for an experienced DM, who may have reasons to modify it, or because the selected project is too sensitive to some criteria variations.
Dear Taher
Of course you are right, however, you can use entropy derived weights or better, leave the method the task to found them from data inputted, and this is done in some very few MCDM methods.
Dear Juan Manuel
Of course my friend, you have the right to express your opinion.
However, if you allow me, I will briefly explain that If you get similar weights for all criteria, you can't blame the entropy method.
It happens because your original values in each criterion in the initial decision matrix, are too similar or close, but those come from reality . I other words, you have little discrimination, and it depends on the problem
Thus, the entropy method has nothing to do with that issue.
Of course , you can choose AHP if you like, instead of entropy values, but you are comparing apples and bananas, because the two systems are different and have different purposes.
The entropy method determines real weights for criteria that can be used to evaluate alternatives. It applies the Shannon theorem which tis he fundamental too for Information Theory. In other words, it is science.
The AHP method determines subjective trade-offs, not weights, and delivers an artificial initial decision matrix with no relation with the real problem.
The trade-ff values were assumed by Saaty to be the equivalent of weights, but they are not, and then, not fit for evaluating alternatives, and with no mathematical support, and by the way, you can also get very close weights using AHP. In other words, it is not science
If you need more information I will be happy to oblige.
Dear Juan Manuel
Thank you for your comments regarding a prior comment from you, related to Zapros.
You say that VDA models are based on judgment, and in a former comment you stated that ‘they compare alternatives based on subjective judgments of decision makers.
From that expression I gather that the model is highly subjective, and then, forcefully inaccurate.
To instruct myself I was reading something related to the Zapros and VDA models, and of c course, I am in no condition to judge therm. However, it appears that those models are more focused to management instead of development, such as software projects.
Of course, given a certain goal it is important to find out which are its main characteristics and the paths to reach it. It seems to me that this kind of problems are different from the usual MCDM scenarios where there is not o only one objective but many, where the are well defined alternatives which are subject to a set of mixed subjective and qualitative criteria. This is the first time, in all my years working with MCDM problems that I realized, thanks to you, that this kind of projects exists.
If it is true that in MCDM there could be totally subjective scenarios, the refer to aspects such as uncertain data, people opinion, uncertain criteria, etc., they can be solved by MCDM methods through normally hybrid methods fed with crisp values derived from a previous step, but theory are aimed at selecting a certain alternative to be developed.
Apparently, and perhaps my interpretation is incorrect, the methods Zapross and VDA that you mentions are more oriented at selected the best people for a certain task, as in remote software development.
I am really grateful to you because now I know something on which I never worked in MCDM or in software development.
When I developed SIMUS analytically, I gave the software guy who wrote its software in Visual Basic, all the detailed instructions and data he needed, and fortunately, this informatic did a splendid job. I have used SIMUS for years in hundreds of projects, in very different fields and complexity levels, and never had any problem, and it is his merit, not mine. Of course, it does it mean that the method is perfect, since there is always room for improvement
Dear Juan Manuel
What you say is not new for me.
Perhaps you don't know, but two years ago I designed a workable interactive matrix, which has been and still is in RG, in my profile, under the number 300a and 300b, to select the best MCDM according the DM needs, which has been read, although don't know if used, by many people.
In that matrix there are distance models like TOPSIS, outranking such as PROMETHEE and ELECTRE, and utility models such as AHP and ANP, as well as Linear Programming and SIMUS
However, I don't see that your answer responds to my question, about why we need weights. Precisely, the existence of many MCDM methods, more than 100, is what prompted my question.
In my opinion, weights are a burden that we have to get rid of. We are wasting our times and efforts for something that is not worthwhile, using a concept that is more than 50 years all. In other words, we need methods that determine the significance of each criterion using data inputted,, not through fabricated weights.
It can be done, Linear Programming, that is more than 70 years old, manages very efficiently without weights or false assumptions, as well as SIMUS, but it does not mean that they are the only way or that they are perfect; ,there could be more other ways, but if we remained stuck with weights, they will never be created.
Dear Juan Manuel
What is stopping you in using probabilities in LP or using crisp values from fuzzy analysis? When they are normalized they are just numbers
You can put them as criteria and will evaluate alternatives as well as the quantitative criteria. Remember, we are comparing series of numbers, that when normalized don't matter from where they come from and in which units.
What is the difference in ranking criteria regarding if they are quantitative or qualitative? An y kind of criteria has a relative significance regarding the other
Really, I don't see your connection between weights.
Recently, a colleague mine, Dr. Svetla Stoilova, has designed a system using fuzzy in which the normal extreme values low and high in a criterion can be computed by SIMUS, with much more reliability than with personal appreciations
I would love to discuss this issue with you over the telephone using WhatsApp or Skype.
My WhatsApp number is + 1 613 770 7123
I am 1 hour ahead of you
Dear Juan Manuel
What you say is true; criteria don’t have the same importance among them
Now, importance for what? Importance to evaluate alternatives? NO
Subjecting weights used in MCDM are most of the times misleading for the following reasons:
1. You can’t say that criterion C1 is 3 times more important than criterion C5. Why?
Because that is a general statement that does no apply to everything.
As an analogy, say that you have to select in a supermarket a certain food that comes in different brands A,B and C as well as in different qualities and prices. If you establish that you always prefer quality over price or C1 = 3 C5, you find that A has better quality than B, but at a higher price, then, according to your preference, you are willing to purchase A, because you are willing to pay a little more to gain more quality
Now, you compare A and C; this last one has a very good quality but a little below to A and at better price. Because of that you decide to lose a little quality and gain a lot in price.
You realize that both actions are opposite, THEREFORE, YOU CAN’T SAY THAT ALWAYS YOU PREFER QUALITY OVER PRICE.
That is, you can’t determine preferences to evaluate alternatives if you don’t consider them.
It is like making a mayonnaise without eggs!
2. Remember that subjective weights are trade-offs, and then, what they represent is interchange of values: Gains vs losses. Do you think that they are appropriate to evaluate alternatives?
3. What is important to evaluate alternatives is the relative importance of each criterion related to its capacity for evaluation, and this is given by the Shannon Theorem about entropy. This is the reason that both, entropy and standard deviation are the right tools to evaluate alternatives.
There are also MCDM methods that don't use any type of weights, even when they take into account the difference between alternatives.
They utilize another procedure for determining the relative importance of criteria
Linear Programming and SIMUS are examples of that
If you know LP you probably know what I am talking about
Dear Ed
Could you pls explain which is the meaning of your comment and to who it is directed to?
I daily follow RG and as far as I remember I did not see anything offensive
Dear s. Divyapriya
Whatever the procedure these weights are useless to evaluate alternatives or, in your case, students
These weights may be useful to rank criteria but not students.
As an example, suppose that experts agree that math has a score of 5 and grammar has a score of 6. This may be correct because it comes from experts, that value mathematics more than grammar, but has nothing to do with the students. They are 2 different things.
Assume that there are 4 students and from the math point of view that have values of 4 5 4 6, and from the grammar point of view they have 3 6 7 2. As you can see, the dispersion of values WITHIN each criterion is very different. There is almost no discrimination in math, therefore, its capacity of evaluation is very low, since the 4 students will have similar marks.
In grammar, the dispersion of values is much greater, and then its capacity to evaluate students is larger than in math.
This is established not only by common sense but also by the Shannon Theorem and by entropy.
If instead of subjective weights you use weights derived from entropy or from statistics, both based on the dispersion of values WITHIN each criterion, then, they will be able to evaluate students.
As a bottom line, the capacity of evaluation of criteria IS NOT based on the relative importance AMONG them, but by their relative importance of values WITHIN them
Yes i do, equal weights are very rately in real life situation.
Dear Juan Manuel
Thank you for your answer. I don't know the method you mention but I know that it is based on cognitive psychology.
In my opinion, psychology has no room in MCDM analysis. It can be used perhaps in trivial and personal problems like selecting a car or a restaurant, but not in real problems.
I, as an engineer, find repugnant the notion that you can solve a problem according to your preferences. Imagine that I must make a decision about using wood or steel beams in construction. Do you really believe that such a problem can be solved by using your preferences or intuitions?
Even if you use your system, it does not exclude you from assigning different degrees of importance to criteria. Some form of importance needs usually to be used because it is not realistic to think that all criteria have the same importance. I am against using invented weights for criteria, however, I do recognize that some sort of discrimination must take place
Does it exist? Yes, it does, and I use it constantly
If you want to continue with this interesting discussion I will be very glad to oblige. Why don't you post an example? It will help understanding
Dear Nolberto Munier
I totally agree with you. The COMET method is an approach where the weights are not needed to get results, more details Article The Characteristic Objects Method: A New Distance-based Appr...
Best Regards WS
Dear Wojciech
Could you please send me a copy of the paper you mention?
Thank you
Regards
Nolberto
Dear Nazli
Yes, alternatives can be ranked without criteria weights.
You can use Linear Programming or SIMUS
Dear Prof. Munier:
This is a great paradigm for discussing issues, and I have gained a lot from it. One question I would like to ask is whether the weights are fixed for a criterion anymore.
Let's assume that there is such a situation which the most popular branded cars on the market. These evaluation indicators are each car brand's price, quality, fuel consumption, etc. In addition to this, there are the needs of different people, for example, some people think that price is more important, while others think that quality is more important. At the same time, we know the ratio of these people; for example, the ratio of those who think the price is important: those who think the quality is important: and those who think fuel consumption is important is 6:3:1. So in this case, is there a way to evaluate the most popular one car brand. And, for different groups of people, do we need to divide different weights to calculate?
The problemvis that there are different customers having different ( and unknown to us) ideas about the importance of criteria. So there are different efficient points - depending on the weights of the criteria. And no one could tell in advance, which is "the best" among these.
So I am convinced thst the whole range of possible weights has to be considered. This will lead to a range of efficient solutions.
Dear Xiong Yang
I believe that weights can't be fixed, because they may change according to the pair of alternatives considered, using pair-wise comparisons.
I have said this many times, and even put an example of three restaurants A, B and C to choose from regarding a certain pair of criteria such as quality and price, that can also be applied to your car example.
A DM may say that he prefers the first over the second, and because of that chooses A over B. However, when using the same pair of criteria, and comparing A to C, he may change his preference and then choose price over quality. This is due, because the choosing depends of the prices and quality of the three restaurants or cars, and because of this the DM may prefer to lose a little quality, but to gain in price.
Regarding your question: Even if weights are selected individually, say in a 1 to 10 scale, what DM1 think regarding the importance of a criterion, may or may not agree with what DM2 and DM3 think. In my opinion, this is one of the fallacies of using weights: It depends on who is the DM. We can’t have three different interpretations of reality, when there is only ONE. This is the case you pose.
I believe t hat there are two ways: Just use entropy derived weights, which are only one true set for a problem, or don’t use weights, which is what I recommend. Of course, in case of the latter, you must use a MCDM method that considers criteria importance in other way, as CR ITIC or SIMUS.
And no, you can’t divide weights, because you are working with a system that does not accept partitions.
I do agree that there are no fixed weights, if one uses weights, then the whole plane of positive weights with sum 1 has to be used. This would lead to all efficient points in the convex case - but will miss parts in the non-convex case
I personally prefer (and have used it in a software) Wierzbicki's Reference Point approach, where goals in terms of the objective values vector are specified by the user and the system answers with the closest efficient point wrt. a modified Tchebycheff norm.
This gives the DM the possibility to specify something he evaluates - the objective values themselves. But in my opinion not only these decide about the acceptance of a compromise, but other characteristics of the efficient point, too.
I do NOT trust any method yielding exactly one efficient (aka Pareto-optimal) point.
Dear Ralph
In my opinion, there is not a convex dimensional space that groups weights.
What you have, is a convex dimensional space that groups feasible solutions.
You can certainly establish goals for your objectives or criteria, and in my opinion that is mandatory.
There are MCDM methods that based in Linear Programming, can give you quantitatively the REAL value of each criterion, and also how this goal is satisfied by each solution in the final ranking
This is what you mention as ‘This gives the DM the possibility to specify something he evaluates - the objective values themselves’
I agree with your last point, because normally there are not optimal solutions in MCDM
Dear Nolberto, I referred to the case of convexity of the set of objective vectors for all feasible points. In that case a weighted sum scalarization yields all Pareto-optimal points, when all weights in the convex (open) set
W={w| w_j>0 forall j, sum_j w_j =1}
are considered. Of course it is impossible to evaluate the scalarization for each and every of these infinitely many weights apart from very special cases like two linear objectives over a polyhedron. This case could easily be treated by determining the local stability sets ->parametric optimization with a single parameter.
Suggest reviewing this paper for MCDM methods:
https://www.sciencedirect.com/science/article/abs/pii/S092583881732827X
Suggest reviewing this paper for MCDM methods:
https://www.sciencedirect.com/science/article/abs/pii/S092583881732827X
Dear Ali
I answered you long time ago, but I did not get any response from you
Dear Mabeel
Thank you for your answer, but could you translate it into English, or French, or Spanish?
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