Dear Alexandre,

I suggest you to see links and attached file on topic.

Robustness Analysis of a Vehicle Front Structure Using ... - Dynalook

https://www.dynalook.com/.../8%20Optimization%20I%20-%20Ro...

Robustness Analysis of Structures based on Plastic Limit ... - CiteSeerX

citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.521...

Recent Advances on Surrogate Modeling for Robustness ... - Dynalook

https://www.dynalook.com/.../8%20Optimization%20I%20-%20Ro...

Best regards

Dear Alexandre

With due respect I think that you should change your question, since what you are questioning, in my opinion, is how to conduct a sensitivity analysis, in order to determine robustness or sensitivity of a solution.

You say nothing about what MCDM method you use. I understand that MATLAB has a software (FLO) (Facility Location Optimizer), for location problems, but I don’t know if it is an optimizing or a heuristic algorithm. Nevertheless, generally a DM is able to perform a sensitivity analysis either for optimal or approximate solutions.

I imagine, correct me if I am wrong or mistaken, that manufacturing impressions refer to the criteria that the four variables are subject to.

Now, we have two different problems here:

1. Imprecision in data, that is, inaccuracy of the cardinal values in each criterion and that reflect the contribution of each variable regarding a specific criterion.

2. Imprecision due to the uncertain importance of each criterion, or weight.

Both belong to the sensitivity analysis field, and it determines how robust the solution is to variations in data and in criteria importance, respectively. Robustness, is then defined as the resilience or strongness of the solution regarding these changes.

The solution is robust if it does not change or has small variations when the data or criteria change either separately or jointly. A very good and concise explanation was given here in RG by Alessandro Giuliani. I take the liberty, with his permission, to reproduce his analogies:

‘Something (or someone) is robust when it (or he/she) can bear attacks, perturbations, offences without being disrupted or heavily modified’.

‘Something (or someone) is sensible when it (or he/she) keep tracks and registers (and thus can even be disrupted and changed) by interactions with the environment, attacks, phrases, facts of life..’

Consequently, a sensible solution is the opposite to a robust solution.

Coming back to your problem with engine headers, which, for those who are not engineers, these are the conducts used to exhaust gases from the engine cylinders, into an exhaust manifold, and then by a pipe to the catalytic converter and the muffler.

Consequently, are you saying that this pipe, converter and muffler interfere with the power output, or cardan shaft?

Normally, the length of the exhaust system is related with the loss of power in the engine. Is this which you are referring to? And then you try to find the minimum length of the exhaust system?

I am afraid that this is not my area of expertise, so my opinion does not carry a great value.

In my opinion it is a problem of trial and error, and I understand that the stability radius model can help, but this model is used for small perturbations and gives a local value of robustness.

Perhaps, using the maximin model (Wald), it could be better, since it allows to reach the optimal using the least worse result.

Since you can measure the different engine outputs, for different exhaust configurations, it is for me the best method.

Hope it helps, and thank you for letting us know a very interesting case

Hello Nolberto Munier , thank you for taking some time for your answer.

“With due respect I think that you should change your question, since what you are questioning, in my opinion, is how to conduct a sensitivity analysis, in order to determine robustness or sensitivity of a solution.” - Absolutely no problem. I’m getting closer to opt topics just now.

“You say nothing about what MCDM method you use. I understand that MATLAB has a software (FLO) (Facility Location Optimizer), for location problems, but I don’t know if it is an optimizing or a heuristic algorithm. Nevertheless, generally a DM is able to perform a sensitivity analysis either for optimal or approximate solutions.” - The method for decision making is exactly what my work is about. I am crossing experimental data regarding engine usage (other parameters than the ones I am simulating) and the data that I am simulating. With this model I am able to predict the net average work done by the power-train for the vehicle in study. To put it short, this method outputs a scalar value for each simulated combination of the input variables, so essentially I have a 1-to-1 correspondence between my input parameters and the output results. The main goal is to maximise this output. The secondary goal is to check for robustness (or sensitivity, as it seems it is the case).

“Now, we have two different problems here:

1. Imprecision in data, that is, inaccuracy of the cardinal values in each criterion and that reflect the contribution of each variable regarding a specific criterion.

2. Imprecision due to the uncertain importance of each criterion, or weight.” - As for the first item, I think we are speaking of the same thing when I mention that the manufacturing process will have some inherent imprecision. Lengths might vary in the order of, say, +-5mm. Therefore a tube optimised to a length of 473 mm may end up having anything between 468 mm and 478 mm. Still within the first item, thermal and CFD inaccuracies of the model I am using may create an offset between the simulated optimal point and the actual optimal that would be observed in the real world. Therefore, it is possible that in simulations my optimal may be 473 mm, but the same procedure experimentally would get somewhere else in the likes of 450 mm(ish) or 440 mm(ish), for instance.

As for the weight of each input variable, for my problem it may not be as straightforward to state anything solid in that matter. One could say these weights, the sensitivity of the output in respect to a given individual variable, change considerably across all the space of inputs I’m evaluating. I would dare to go even further than that and state that this weight is also sensitive to the other input variables influence on the system.

“Both belong to the sensitivity analysis field (...)” - From what I read in the few paragraphs after this I am almost sure this may be the case as well. I think we’re getting somewhere here. Just for clarification then, I’m interested in avoiding sensible solutions and prioritising robust solutions.

“Consequently, are you saying that this pipe, converter and muffler interfere with the power output, or cardan shaft?” - Power output.

“Normally, the length of the exhaust system is related with the loss of power in the engine. Is this which you are referring to? And then you try to find the minimum length of the exhaust system” - In a way, yes, the increase of length does increase the pressure loss caused by the pipe. In that sense it would be easy to state that the shorter I can make these pipes, the merrier. However, pulsating flow in headers (think of a church organ) also affects performance, much more substantially than the pressure loss caused by friction of the pipe walls and inertia of gases within the system. Fine tuning these pipes is not as straight-forward due to the complexity of gases thermal and fluid-dynamics behaviour – both which impact pulsating phenomena strongly. Therefore, simulations are about the only option.

I hope that clears up things a bit. Do you still think a Wald maximin model is the most appropriate decision?

@Mohamed-Mourad Lafifi, thank you. I will take a good look.

Dear Alexandre

NM. “You say nothing about what MCDM method you use. I understand that MATLAB has a software (FLO) (Facility Location Optimizer), for location problems, but I don’t know if it is an optimizing or a heuristic algorithm. Nevertheless, generally a DM is able to perform a sensitivity analysis either for optimal or approximate solutions.”

AP. The method for decision making is exactly what my work is about. I am crossing experimental data regarding engine usage (other parameters than the ones I am simulating) and the data that I am simulating. With this model I am able to predict the net average work done by the power-train for the vehicle in study. To put it short, this method outputs a scalar value for each simulated combination of the input variables, so essentially, I have a 1-to-1 correspondence between my input parameters and the output results. The main goal is to maximise this output. The secondary goal is to check for robustness (or sensitivity, as it seems it is the case).

NM. Well, the fact that you are developing a new method for decision making may be obviously related with the type of sensitivity analysis (SA) you perform.

Your scenario is certainly complex and I doubt that you could use the same method for SA.

“Now, we have two different problems here:

1. Imprecision in data, that is, inaccuracy of the cardinal values in each criterion and that reflect the contribution of each variable regarding a specific criterion.

2. Imprecision due to the uncertain importance of each criterion, or weight.”

AP. As for the first item, I think we are speaking of the same thing when I mention that the manufacturing process will have some inherent imprecision. Lengths might vary in the order of, say, +-5mm. Therefore, a tube optimised to a length of 473 mm may end up having anything between 468 mm and 478 mm.

NM. You can use two criteria for this, one for the 468 mm and the other for 478 mm. The MCDM system should take in account this interval and select a value in between, and not necessarily the average. The difference with this approach with MonteCarlo, is that it takes into account the relation of the pipe length and the other parameters. You can also, as you say, use MonteCarlo to determine the average value of this length, however, remember that it is only an average value.

AP. Still within the first item, thermal and CFD inaccuracies of the model I am using may create an offset between the simulated optimal point and the actual optimal that would be observed in the real world. Therefore, it is possible that in simulations my optimal may be 473 mm, but the same procedure experimentally would get somewhere else in the likes of 450 mm(ish) or 440 mm(ish), for instance.

NM. Precisely, this confirms what I said above. In the real world, length is not independent of other aspects such as elbows, exhaust volume and so on. Consequently, if you mathematically consider all aspects simultaneously, you are approximately replicating the real-world behaviour.

AP. As for the weight of each input variable, for my problem it may not be as straightforward to state anything solid in that matter. One could say these weights, the sensitivity of the output in respect to a given individual variable, change considerably across all the space of inputs I’m evaluating. I would dare to go even further than that and state that this weight is also sensitive to the other input variables influence on the system.

NM. As I understand it, you have several inputs or criteria which variation may or may not affect the output, in your case, the engine power. Now, you say that the output, in respect to a given individual variable or criterion, change considerably across all the space of inputs.

According to this, I understand that you are considering the variation of a single criterion at the time, and I wonder how you manage to select that criterion among the others, for instance, you could say that the pipe length is the most important criterion? On what grounds?

In my opinion, you should first determine which are the criteria that influence the solution, because there are some that don’t, and once identified, you should consider the effect on the output due to the variations of all identified criteria simultaneously, because most probably all criteria are linked and you can’t consider the variation of only one while keeping the others constant.

Therefore, I have a question here; how do you select the criterion, among all existent, for determining robustness? Are you examining them all?

This needs a clarification: Suppose that you have four alternatives or exhaust system configurations, and say ten criteria, such as maximum and minimum values for each criterion (that is, you should use two criteria for the same issue, one for maximizing and the other for minimizing, with identical values, except the limits or thresholds). These criteria could be: Pipe length, elbows, radius of elbows, inaccuracies due to CFD, different types of catalytic converters, loss of power due to turbulence, etc.

Among those criteria, I would say that a fundamental data that you need to know is the volume of exhaust gases generated at different power regimes delivered by the engine. If you know this data you can create a criterion with the value of turbulence created by each configuration, and of course, you want it minimized. In here, you would need probably three or fourth criteria, each one corresponding to a certain power output.

For sensitivity analysis you need then to know with what criteria to work, and dismiss the other.

In varying a criterion there is something that has to be considered, and it is that the variation of an input or criterion produces variations in all the other inputs, but not necessarily considering all criteria (in our example, 10, but only those criteria linked with the solution, which may be any number between1 to 10). That is, you have to work with the significant criteria and ignore the others.

Therefore, you must work varying the limits of these selected criteria, not the weights, unless they are objective.

The other concept that must be considered, is that you can’t vary a criterion, increasing or decreasing it, for ever, there is normally a limit, that is, a range, between the maximum negative variation and the maximum positive variation. It is precisely this range which indicate the robustness of a solution; the larger the better, however, remember that you have to consider the composite, not the individual range, and that it varies in each iteration. Obviously, since each of these criteria affects the output, when the range of one of them is very small or even not existent, the solution is extremely sensible to THIS criterion.

This appears to be complicated, but it is not, because there is an algorithm that can handle it efficiently, however, I am not referring to variation of weights, but to the variation of criteria importance, and this importance being given by its capacity to modify the output, something that a subjective weight can’t do.

As per my understanding, the only algorithm that can handle simultaneously all criteria related with the output, and the different ranges, is Linear Programming (LP), by making a synergy between the primal problem and the marginal values for each criterion, which are exactly given by the dual problem. You can solve this problem using the Excel add-in called ‘Solver’. The beauty of this system, is that if there is an optimal solution, the software will find it, since it does not give approximate solutions as the heuristic methods do.

AP. “Both belong to the sensitivity analysis field (...)” - From what I read in the few paragraphs after this I am almost sure this may be the case as well. I think we’re getting somewhere here. Just for clarification then, I’m interested in avoiding sensible solutions and prioritising robust solutions.

NM. There you are.

AP. “Normally, the length of the exhaust system is related with the loss of power in the engine. Is this which you are referring to? And then you try to find the minimum length of the exhaust system” - In a way, yes, the increase of length does increase the pressure loss caused by the pipe.

NM. Yes, but the loss of power may be also related to the back pressure in the exhaust system, and this, in a large extent may be a consequence of turbulence.

AP. In that sense it would be easy to state that the shorter I can make these pipes, the merrier. However, pulsating flow in headers (think of a church organ) also affects performance, much more substantially than the pressure loss caused by friction of the pipe walls and inertia of gases within the system. Fine tuning these pipes is not as straight-forward due to the complexity of gases thermal and fluid-dynamics behaviour – both which impact pulsating phenomena strongly. Therefore, simulations are about the only option.

NM. I agree

AP. I hope that clears up things a bit. Do you still think a Wald maximin model is the most appropriate decision?

NM. I don’t really sure; perhaps with more information about your problem I could answer this question, but not with the scarce information I have now. I don’t even know how your model work.

My suggestions are only that, and are not based on fluid dynamics, thermodynamic, or in mechanical aspects; but in examining the problem from the MCDM focus. I any case I am very interested on your progress on this very interesting case.

I hope that my contribution can help you, and pls keep the contact.

Nolberto

Nolberto Munier

Hello Nolberto. Sorry for the time it took me to answer. Let’s go through some things that you said.

“According to this, I understand that you are considering the variation of a single criterion at the time, and I wonder how you manage to select that criterion among the others, for instance, you could say that the pipe length is the most important criterion? On what grounds?”

The intent actually is to consider all variables at the same time. At this point I don’t remember if I mentioned this, but I’m conducting a full factorial DoE (for safety) and crossing all possible permutations of variables. The right direction really is very uncertain at the beginning of the optimisation and, for this reason, I do a full factorial study. Once again, my intent is to find all the local maximums and evaluate their “reliability” (lack of sensitivity or radius of stability), to root out unreliable global maximums.

“Therefore, I have a question here; how do you select the criterion, among all existent, for determining robustness? Are you examining them all?”

The only output criterion I’m evaluating for is the quantity provided by my model – we'll call it “mean power”. I was about to code a 4-D radius stability study script in MATLAB for this purpose (since I have 4 independent variables and only one dependent variable). I haven’t got the time to do that just yet.

“Among those criteria, I would say that a fundamental data that you need to know is the volume of exhaust gases generated at different power regimes delivered by the engine. If you know this data you can create a criterion with the value of turbulence created by each configuration, and of course, you want it minimized. In here, you would need probably three or fourth criteria, each one corresponding to a certain power output.”

The point of my study is really to simplify this complex roster of output parameters to a single scalar. Out of this one, I am only concerned with its robustness across the input space, which is a byproduct of manufacturing imprecision and simulation innacuracy. I get the feel you’re thinking I have much more than one main output value out of my study.

Maybe this clears up a bit?

Piccini.

Dear Alexandre, sorry for the delay

P. Hello Nolberto. Sorry for the time it took me to answer. Let’s go through some things that you said.

“According to this, I understand that you are considering the variation of a single criterion at the time, and I wonder how you manage to select that criterion among the others, for instance, you could say that the pipe length is the most important criterion? On what grounds?”

The intent actually is to consider all variables at the same time.

At this point I don’t remember if I mentioned this, but I’m conducting a full factorial DoE (for safety) and crossing all possible permutations of variables.

NM. No, I think you did not, and that is the reason I asked you the question if you were using only one variable at the time.

Remember that you know your problem very well, while I have only a glimpse of it, however, my first impression is that using full factorial in the DoE is useful. For each simulated run you get an average value for the output. At he end, you will get the beta coefficients for each input indicating their relative importance, which is something very positive.

AP. The right direction really is very uncertain at the beginning of the optimisation and, for this reason, I do a full factorial study. Once again, my intent is to find all the local maximums and evaluate their “reliability” (lack of sensitivity or radius of stability), to root out unreliable global maximums.

NM. I don’t really understand your reasoning. Why do you want to find the local maximums?

\Why the right direction is difficult, or you mean that the lowest and highest values are uncertain?

“Therefore, I have a question here; how do you select the criterion, among all existent, for determining robustness? Are you examining them all?”

AP. The only output criterion I’m evaluating for is the quantity provided by my model – we'll call it “mean power”. I was about to code a 4-D radius stability study script in MATLAB for this purpose (since I have 4 independent variables and only one dependent variable). I haven’t got the time to do that just yet.

NM. I understand that the output or mean power is the objective, not a criterion.

“Among those criteria, I would say that a fundamental data that you need to know is the volume of exhaust gases generated at different power regimes delivered by the engine. If you know this data you can create a criterion with the value of turbulence created by each configuration, and of course, you want it minimized. In here, you would need probably three or fourth criteria, each one corresponding to a certain power output.”

AP. The point of my study is really to simplify this complex roster of output parameters to a single scalar. Out of this one, I am only concerned with its robustness across the input space, which is a by product of manufacturing imprecision and simulation inaccuracy. I get the feel you’re thinking I have much more than one main output value out of my study.

NM. No, I reckon that you are aiming only at one output.

I am not sure that I understand you regarding a single scalar. If I am not mistaken, you want to know two things:

1. The relative importance of each input, which you may have using the full factorial. This is really important.

2. How sensible is the mean power to the variations of each input, considering that each input has an allowable range of variations?

Which I am in doubt is, that even if you consider the main effect and the interaction between the different inputs, if the procedure considers that varying one input also produces the range of variations of all other inputs. I guess no.

Good evening Nolberto Munier

You said: “I don’t really understand your reasoning. Why do you want to find the local maximums? Why the right direction is difficult, or you mean that the lowest and highest values are uncertain?”

Just for clarification before we move along, I’m looking for the local maximums of the objective function. The reason why I want to find local maximums is because I want to assess what are the “optimal possibilities” available overall. For the next example let’s consider that I am using a radius of stability model. Ideally, in the end of the day I want to have a list where the first column provide me with the coordinate of a local maximum, the second column has the objective function value found and the third column has the radius of stability found. Let’s say that I order this list in descending order relative to the objective function value found. At first thought the first line would be the optimised design parameter I’ve wanted all along. However, I want to avoid unreliable, sensitive, solutions. Therefore I set myself a criteria of non-acceptance of sensitivity (in this example, the smallest radius of stability acceptable) and root out poor, unreliable solution. So maybe this gets me to the third, fourth or sixth line of that list. In the end, it is not so much of a great output value, but it is much more likely to be observed in practice than its other higher-ranking peers. THIS is the exact thing I’m going for here. I’m reaching out here in ResearchGate to see what would be the most “righteous” method of doing this analysis as I’m no expert in Optimisation.

As for the “right direction” not being clear: it has to do with how unpredictable the system is, in a way that I really can’t tell where the best values must be around without a numerical analysis on the entire system (the simulation). Think as if no one has ever seen what an actual F1 aerodynamic kit looks like and had to start from scratch. They’re good with CFD, have a supercomputer 10x better than the best our would currently has, but they have no benchmarking. They have no analytical equations ready just waiting for them. They can only go with trial-and-error – therefore they must really try a lot of things. My situation is not that much of an uncertain road, but it really has a lot of trial-and-error to do and, therefore, this is why I do this brutal-force attack on variables (Full Factorial).

You said: “NM. I understand that the output or mean power is the objective, not a criterion.”

Hmm, yes. Maybe I’m misunderstanding what you mean by criterion. Sensitivity criteria? I’m sorry but I’m relatively new to the topic.

You said: “I am not sure that I understand you regarding a single scalar. If I am not mistaken, you want to know two things:

1. The relative importance of each input, which you may have using the full factorial. This is really important.

2. How sensible is the mean power to the variations of each input, considering that each input has an allowable range of variations?”

I think what I said earlier explains it. Number 2 is right on point – but I would not speak in terms of “variations of each input” specifically, but more in the sense of radius-of-stability really (the worst case of variation I guess?!). Number 1 is not so much on point, as the (in)dependence of the results concerning a specific variable simply is not of my interest. As I said before, I just want the best result. However, as in your item 2, I want to assess if this great point is also something I should bet my coins on.

As for bringing things down to a scalar, it is all about having something with which analysis can be as simple as “is a > b or not?”. In a way, I am restricting the dimensions in my problem so it can be more simple and effective, perhaps.

Best regards,

Dear Alexander

Sorry for my delay, but this is a new case for me and I had to spend some time learning and refreshing my knowledge about stability and mathematical tools such as the radius of stability

AP- You said: “I don’t really understand your reasoning. Why do you want to find the local maximums? Why the right direction is difficult, or you mean that the lowest and highest values are uncertain?”

Just for clarification before we move along, I’m looking for the local maximums of the objective function.

The reason why I want to find local maximums is because I want to assess what are the “optimal possibilities” available overall. For the next example let’s consider that I am using a radius of stability model. Ideally, in the end of the day I want to have a list where the first column provide me with the coordinate of a local maximum, the second column has the objective function value found and the third column has the radius of stability found.

NM- On how many systems do you that? I call a system (q1) to the exhaust manifold, (q2) to the exhaust pipe, (q3) to the converter and (q4) to the muffler. Each system (qi) may be in several states (sj). For instance, the exhaust pipe may have different states regarding different lengths (s1), number and radius of elbows (s2), diameter (s3), etc.

Please, let me know if my understanding is correct or if it is wrong.

AP- Let’s say that I order this list in descending order relative to the objective function value found. At first thought the first line would be the optimised design parameter I’ve wanted all along. However, I want to avoid unreliable, sensitive, solutions. Therefore I set myself a criteria of non-acceptance of sensitivity (in this example, the smallest radius of stability acceptable) and root out poor, unreliable solution.

So maybe this gets me to the third, fourth or sixth line of that list. In the end, it is not so much of a great output value, but it is much more likely to be observed in practice than its other higher-ranking peers. THIS is the exact thing I’m going for here. I’m reaching out here in ResearchGate to see what would be the most “righteous” method of doing this analysis as I’m no expert in Optimisation.

NM- My friend, I am afraid that I am not able to follow your explanation. The reason is that you are working with a mathematical environment on which I have very little knowledge.

AP- As for the “right direction” not being clear: it has to do with how unpredictable the system is, in a way that I really can’t tell where the best values must be around without a numerical analysis on the entire system (the simulation).

Think as if no one has ever seen what an actual F1 aerodynamic kit looks like and had to start from scratch. They’re good with CFD, have a supercomputer 10x better than the best our would currently has, but they have no benchmarking. They have no analytical equations ready just waiting for them.

NM- Yes, this I can understand.

AP- They can only go with trial-and-error – therefore they must really try a lot of things. My situation is not that much of an uncertain road, but it really has a lot of trial-and-error to do and, therefore, this is why I do this brutal-force attack on variables (Full Factorial).

You said: “NM. I understand that the output or mean power is the objective, not a criterion.”

Hmm, yes. Maybe I’m misunderstanding what you mean by criterion. Sensitivity criteria? I’m sorry but I’m relatively new to the topic.

NM- Well, in MCDM a sensitive input ion is the one which has little or nothing leeway to change, and consequently, if it affects the objective, which is not robust considering this input

AP- You said: “I am not sure that I understand you regarding a single scalar. If I am not mistaken, you want to know two things:

1. The relative importance of each input, which you may have using the full factorial. This is really important.

2. How sensible is the mean power to the variations of each input, considering that each input has an allowable range of variations?”

AP- I think what I said earlier explains it. Number 2 is right on point – but I would not speak in terms of “variations of each input” specifically, but more in the sense of radius-of-stability really (the worst case of variation I guess?!). Number 1 is not so much on point, as the (in)dependence of the results concerning a specific variable simply is not of my interest. As I said before, I just want the best result. However, as in your item 2, I want to assess if this great point is also something I should bet my coins on.

NM- My first point refers to the importance of each input, because not all of them has the same importance.

As I understand it for the main objective you analyze how it is altered each system, say the length of the exhaust pipe, when all its different states are considered. If this is so, what nags me is that you are considering each system separately and then finding a local stability, and considering the other systems constant. In my opinion, all systems must be considered jointly and each one with its respective states.

I know how to do that in sensitivity analysis when using Mathematical Programming for MCDM, but I am lost in your case. Perhaps you can also do that using Full Factorial

Sorry for not being helpful, but again, my expertise is with sensitivity analysis for MCDM. In your scenario, if it is true that you are aiming at the same goal as I, you are using tools such as the radius of stability, that I don’t use

Nolberto Munier

Sorry for the late answer Nolberto. It’s been pretty busy lately. Lets go…

“On how many systems do you that? I call a system (q1) to the exhaust manifold, (q2) to the exhaust pipe, (q3) to the converter and (q4) to the muffler. Each system (qi) may be in several states (sj). For instance, the exhaust pipe may have different states regarding different lengths (s1), number and radius of elbows (s2), diameter (s3), etc.”

Yes, this is about right. Except each ‘q’ is a design parameter of the exhaust manifold specifically. They have different possible (s1,s2,s3,…,sn) values ‘scheduled’ for analysis in a DoE, which the calculated outputs I’m currently trying to assess.

“My friend, I am afraid that I am not able to follow your explanation. The reason is that you are working with a mathematical environment on which I have very little knowledge.”

Oh well, I’ll try to share with you some bits I have written so far in my article so maybe things can look more clear then. As I only had the Radius of Stability concept well understood, I took the data and loaded into MATLAB and did some implementations on that only. There’s still some formatting to do in the tables and I omitted some data as it is still not published. (Check the attached file)

“If this is so, what nags me is that you are considering each system separately and then finding a local stability, and considering the other systems constant. In my opinion, all systems must be considered jointly and each one with its respective states.

I know how to do that in sensitivity analysis when using Mathematical Programming for MCDM, but I am lost in your case. Perhaps you can also do that using Full Factorial

Sorry for not being helpful, but again, my expertise is with sensitivity analysis for MCDM. In your scenario, if it is true that you are aiming at the same goal as I, you are using tools such as the radius of stability, that I don’t use”

No no. I’m not considering all systems constant and studying each one at a time. I’m conducting a full factorial analysis indeed. I hope the printscreen I attached may make it more clear.

Thanks for your dedication.

Nolberto Munier

oops. I guess the file lost a lot of quality due to RG compression. I'm uploading them again.