Dear Sebastian

Because it is the result of normalization. That is, each weight is divided by the sum of them all.

Of course it introduces dependencies between criteria

The most absurd is that you can increase a weight in say criterion C5, and instantly all weights of the other criteria decrease proportionally.

The problem is that the procedure assumes that relationships between criteria are lineal, when in reality they may be non-linear, for instance, between criteria price and demand.

MCDM methods that use this normalization are called compensatory, and the most known are AHP, ANP, SAW and TOPSIS

Dear Nolberto,

thank you for your answer! So the normalization is an integral part of SAW and I need to perform it as it seems. My hope was that I could just skip normalization because I wanted to stay with integer values.

Dear Sebastian

Well, normalization is done because normally criteria have different units, but if all your criteria have the same units you don't need normalization

Sebastian Hiller

The normalization of the weights is NOT necessary in MCDM in order to avoid dependencies between weights.

The weights can reflect a subjective importance or contribution of each criterion based on understanding of the problem and priorities of the decision maker.

For example, in the paper published in

Critical Care Medicine, 2016, v.44, N 9, pp. 1762-1768

http://journals.lww.com/ccmjournal/Abstract/2016/09000/Pediatric_Triage_in_a_Severe_Pandemic___Maximizing.16.aspx

the goal was maximizing the time-averaged hospital bed occupancy while minimizing population mortality. The objective function (OF) to maximize was

OF = w1 × time-averaged occupancy (%) – w2 × mortality untreated (%)–w3 × mortality treated (%) where w1, w2, and w3 are relative weights of the corresponding components of the OF (occupancy and population mortality are generally not interdependent). In order to place higher priority on minimizing mortality over maximizing bed occupancy, the weights w2 and w3 were assigned twice the weight of w1. Maximizing the combined OF using simulation was achieved through an evolutionary strategies algorithm, an optimization technique that generates various solutions in the defined range that must adapt to their environment to be retained for further steps. The best solutions are determined when no further meaningful improvement in maximizing the OF is possible. The steps in the evolutionary strategies algorithm are presented in Appendix 4 (Supplemental Digital Content 4, http://links.lww.com/CCM/B837).

Dear Alexander

I am afraid that I don’t share your opinion that normalization is not needed in MCDM.

Normally, the criteria are expressed in different units, such as manpower, costs, kilometers, maximum and minimum limits expressed for instance in mg/m3, etc. you can’t compare them, let alone aggregate if you don’t normalize; it would be as adding pears and apples.

Normalization not necessary mean dependency between weights, however, as I expressed before, normalization to 1 expresses a lineal relationship that may no exist.

If you use entropy derived weights you don’t need normalization, because it was already performed to get the entropy values per criterion.

Regarding your sentence ‘The weights can reflect a subjective importance or contribution of each criterion based on understanding of the problem and priorities of the decision maker’

The so-called ‘weights’ derived from the DM, are not weights but trade-offs values, and as such they don’t represent the relative importance of criteria. They only indicate how much a criterion value losses or gains when another criterion value is changed.

The relative importance of a criterion, which function is to evaluate alternatives, lies in its series of performance values, which is demonstrated by the Shannon Theorem (1948), which shows that the quantity of information of data in criteria depends on their discrimination.

By the way, this famous theorem is the cornerstone of the information technology, as we know today. Therefore, is no doubt that subjective weights are useless for evaluating alternatives, or for determining criteria relative importance.

In mathematical programming the objective function is the algebraic sum of the products between an alternative value, in your example, the average number of occupies, the % of mortality treated and the % of mortality untreated, times the score of the corresponding activity.

But this score es obtained by a MCDM process, and then it is unknown until the result is known.

You are replacing these scores for weights that are completely subjective, that is, you are imposing the solution values, therefore, in my opinion, that formula for the objective function is incorrect, since you already defined the alternatives scores.

I would suggest to create a new criterion called for instance ‘Importance’ where you can put a 1 for x1 and twos for x2 and x3. In this way, you are telling the software than X2 and x3 are twice more important than x1, and the system will consider this criterion together with all the others.

You can do this using the Solver for Linear Programming in Excel

Unfortunately the reference you give http://links.lww.com/CCM/B837), uses medical concepts and acronyms that I don’t know,

Nolberto Munier

I am afraid that you make things more complicated than they are, and that you force yourself through the open door.

1-st, "...if the criteria are expressed in different units, ... you can’t compare them, let alone aggregate if you don’t normalize". Of course, nobody compares or aggregate criteria expressed in different units. In this case, they must be made dimensionless, or expressed as % (as is in my example). This has nothing to do with weights' normalization.

2-nd, the weights or trade-offs are simply subjective contributions of the criteria into the objective function. They are independent of each other, and not necessarily make a linear combination. I think that the Shannon theorem plays no role in this particular case. Yes, the contributions are not known before hand; that's why they are called subjective, somewhat similarly to subjective probabilities in the Bayes formula.

3-rd, "I would suggest to create a new criterion called for instance ‘Importance’ where you can put a 1 for x1 and twos for x2 and x3. In this way, you are telling the software than X2 and x3 are twice more important than x1, and the system will consider this criterion together with all the others."

And this is exactly that I have done in the referenced publication and in my example. So, your suggestion is misplaced.

My referenced example has nothing to do with linear programming and Excel. My model is a non-linear one, and is based on discrete event simulation with the following optimization, which is replicated again and again to obtain the optimal solution that corresponds to maximal value of the Objective function with the particular criterion contribution (weight). The optimization module is an integral part of the discrete event simulation package.

I referenced the link to Appendix 4: Procedure for Developing an Evolutionary Strategies Algorithm to Determine Thresholds for Optimal Pandemic Survival, so it could be read in conjunction with the main paper text.

And finally, I believe that it is generally better to learn details before suggesting anything.

Anyway, thank you for your comments and your interest.

Dear Alexander

AK - I am afraid that you make things more complicated than they are, and that you force yourself through the open door.

NM- As you have probably noticed, English is not my mother tongue and, in all honesty, I don’t know what your idiom ‘you force yourself through the open door’ mean.

Could you please explain it?

AK- 1-st, "...if the criteria are expressed in different units, ... you can’t compare them, let alone aggregate if you don’t normalize". Of course, nobody compares or aggregate criteria expressed in different units. In this case, they must be made dimensionless, or expressed as % (as is in my example). This has nothing to do with weights' normalization.

NM- And normalizing by dividing each value by the sum of all values, each value is not a percentage of the total?

I did not say that ‘weights’ must be normalized. What I said was: ‘Normalization not necessarily mean dependency between weights’.

AK- 2-nd, the weights or trade-offs are simply subjective contributions of the criteria into the objective function. They are independent of each other, and not necessarily make a linear combination. I think that the Shannon theorem plays no role in this particular case. Yes, the contributions are not known before hand; that's why they are called subjective, somewhat similarly to subjective probabilities in the Bayes formula

NM. Your use of the conjunction ‘or’ implies that trade-offs and weights are equivalent, and if you add all priorities and divide each one by the total, is that not normalization?

I don’t know what type of weights you are talking about, but certainly they are not contributions for the objective function, the scores are, at least in mathematical programming.

I mentioned the Shannon Theorem just to give mathematical validity to my assertion regarding useless of criteria weights, which, by the way, I am not the first to make this comparison. I did not invent it!

Of course, what you call contributions are not known before hand; perhaps you can explain me on what grounds a DM may produce these contributions if he does not know to what they will be applied to. He can’t make general assumptions about the more importance of one criterion on another and then applied it to whatever the pair of alternatives are compared. I have many examples to corroborate it that are published here in RG.

Now, do you have a single example where you can prove the opposite?

They are subjective not because they are unknown; they are subjective because they are based on intuition, mood and other psychological conditions of the DM.

The simple proof of this, is that these contributions depend on the person doing it. The unknown or ignorance is a common factor for all of them, the differences lie in their individual preferences.

Again, this, besides being common sense, was said by MCDM scientists time ago. Just to mention one of them, Zardari, states that:

1- ‘Criteria weights don’t have discriminatory power.

2- The pair-wise comparison is often criticized for simply asking for the relative importance of criteria without reference to the scales, on which the criteria are measured.

3- Worst method for weighting is pair-wise comparison.

4- Main difficulty is to reconcile the inevitable inconsistency of preferences in real-world applications.

5- In may real-world applications human pair-wise comparison is highly ambiguous and uncertain’.

About your comment on Bayes’ theorem, may I remind you that it describes the probability of an event, based on prior knowledge of conditions that might be related to the event?

About preferences, you can check the opinions of researchers such as: Kazibudski, Buchanan, Bennis and O’Toole, Barzilai, Triantaphyllou, Figuera (on Arrow Incompatibility Theorem).

AK- 3-rd, "I would suggest to create a new criterion called for instance ‘Importance’ where you can put a 1 for x1 and twos for x2 and x3. In this way, you are telling the software than X2 and x3 are twice more important than x1, and the system will consider this criterion together with all the others."

And this is exactly that I have done in the referenced publication and in my example. So, your suggestion is misplaced.

NM- I am glad that we coincide in this procedure, but my suggestion would have been misplaced if you had mention it, but you did not. I couldn’t guess what you did or had in mind.

AK- My referenced example has nothing to do with linear programming and Excel.

NM- And I did not say that you did.

AK- My model is a non-linear one, and is based on discrete event simulation with the following optimization, which is replicated again and again to obtain the optimal solution that corresponds to maximal value of the Objective function with the particular criterion contribution (weight). The optimization module is an integral part of the discrete event simulation package.

NM- Well, knowing what you say, I believe that using discrete event simulation will produce better results than mathematical optimizations, although there is no guarantee of optimality, which probably, in addition, is not important.

AK-I referenced the link to Appendix 4: Procedure for Developing an Evolutionary Strategies Algorithm to Determine Thresholds for Optimal Pandemic Survival, so it could be read in conjunction with the main paper text.

NM- I tried to read it but I stopped because there are several acronyms pertaining to the medical area, and which meaning I couldn’t find in Internet.

Por instance: POD, DV, VPS, CSC, PICU, TPOD, TDV, SimulatedRunner Model, Triage

Since your recommended it, you should clarify the meaning of this acronyms, considering that not all RG readers are in the Health Care Management area as you.

I will be happy to read the paper if you explain the meaning of each one.

AK- And finally, I believe that it is generally better to learn details before suggesting anything.

NM- I agree, provided that the topic is self explanatory and an appropriate language for everybody to understand, if not, you are wasting your and readers time, as well as prone to misunderstandings.

AK- Anyway, thank you for your comments and your interest.

NM- And I thank you for your answer, this is useful for all of us to learn from others.

Nolberto Munier

Thank you for your detailed and lengthy response.

I'll try to be brief now.

Sorry for using an idiom ‘you force yourself through the open door’ . It means that someone is trying to explain an obvious thing while nobody argues, i.e. one is trying to explain a non-existent problem. This is that I meant regarding combining the criteria that are expressed in different units. Nobody argues that one can compare pears and apples. By the way, English is not my native language either.

Now, let's focus on what we agree leaving our disagreements for another time:

NM- I am glad that we coincide in this procedure, but my suggestion would have been misplaced if you had mention it, but you did not. I couldn’t guess what you did or had in mind.

AK- I believe that I clearly mentioned it-see my previous original post 2 days ago- highlighted bold font:

AK-" In order to place higher priority on minimizing mortality over maximizing bed occupancy, the weights w2 and w3 were assigned twice the weight of w1."

This is a quotation from the referenced paper. It means that the weight w1=1 for the criterion 1 and w2=2, w3=2, accordingly. This is as clear as it can be said. So, there was no need to guess what I did or had in mind.

How does the decision maker (DM) know the ratio of weights is 2:1? It is his subjective judgement: the criteria 2 and 3 penalties for increasing their values in maximizing OF are twice the penalty for the criterion 1 (criteria 2 and 3 are with the negative sign in OF). Could the penalty ratio be 3:1? or any other ratio? Yes, it could if the DM judgement says so. The optimal solution could somewhat shift because of the bigger penalties. But too big penalties could result in computational instability and difficulties in convergence to the optimal solution (I computationally tested different values of penalties).

The DM must be concerned with not only mathematical purity but also with the practicality and the time needed to find the solution (maybe a sub-optimal solution that is not the best possible but practically good enough).

I'll stop it here for now to keep the brevity of my response.

Dear Alexander

Thank you for the clarification on the idiom issue. However, you say that ‘one is trying to explain a non-existent problem’, and in my opinion criteria not having the same units is a problem, and always has been since the first days of MCDM.

There are many papers about that issue and all methods need to apply normalization in one way or another. There are also methods which result change if different normalization methods are used. SAW is one of the most popular methods and its largest disadvantage is that it does not work when criteria have different units.

It is not only my opinion, you can consult, among others, Zavadskas and Turskis (2008), who make a comprehensive review of normalization in MCDM. Stanujkic et al say: ‘Information stored in a decision matrix is usually incommensurable, i.e. performance ratings in relation to different criteria are usually expressed using different units of measure. Therefore, data should be transformed into comparable values, using a normalization procedure’

Other methods as Compromise Programming do not use normalization because its particular aggregation procedure.

In reality, if as you say, normalization is not needed, I wonder why maybe 99 % of MCDM methods use it. I don’t believe that they complicate their algorithms just to make things more difficult.

Relating weights, maybe I misinterpreted you. In my MCDM vocabulary, the values in the criteria and corresponding to the alternatives are called ‘performance values’, while weights are used to indicate relative importance between two values or between criteria.

I agree with you that weights w2 and w3 are defined by experts being double than w1. However, and I am asking, because I don’t know the health care activity, it wouldn’t be better to work statistically, which probably can produce more reliable weights?

I believe that what you call ‘disagreements’ are only different points of view, and that lead to a healthy discussion.

Anyway, it is great arguing with you and I benefited from this discussion. Please, don’t let it to stop

Nolberto Munier

Dear Nolberto,

First of all, I have also benefited and learned from the discussion with you because of your knowledge and experience in MDCM area.

As for my note of the 'non-existent' problem of "...performance ratings in relation to different criteria are usually expressed using different units of measure. Therefore, data should be transformed into comparable values, using a normalization procedure", I still believe that is so obvious that it should be beyond a professional discussion. Apples and pears must not be compared, period.

At the same time, the original question of this thread was

'Why must the sum of all weights equal 1?', i.e. the question was of the normalization of weights to 1, not normalization of the criteria. When I said that normalization is not needed I meant weights (dimensionless) rather than criteria.

And in my vocabulary the weights, or relative contributions, or relative importance, or penalties for criteria are equivalent terms. They are not needed to be normalized to 1, i.e. keeping their sum=1 because it creates linear dependency-obviously not a desirable property in this case.

As for your question "...it wouldn’t be better to work statistically, which probably can produce more reliable weights?", yes, theoretically in pure mathematical sense it would probably be better. However, in healthcare (especially in crisis situations) decisions must be made quickly, often within minutes. There is no time to work on statistics. Besides, finding weights somewhat more accurate will not make computation results more valuable for the DM.

The DM seeks the critical limits or thresholds for probability of death (T POD) and days on ventilators (T DV) for some patients to decide admit or not admit due to the limited amount of resources (beds, staff). These thresholds are determined by maximizing the objective function OF by the optimization module, while the random POD and DV are generated by the discrete event simulation model using statistical distributions obtained from the Virtual Pedatric DataSet (VPS) which contains hundred thousands patient records.

If computation indicates T POD

Dear Alexander

AK- At the same time, the original question of this thread was

'Why must the sum of all weights equal 1?', i.e. the question was of the normalization of weights to 1, not normalization of the criteria. When I said that normalization is not needed, I meant weights (dimensionless) rather than criteria.

NM – I am afraid that I don’t agree. In MCDM problems weights are averages, from the eigenvector method, or the geometric average, or entropy or statistics, and in all methods, you have a weight vector and vector normalization.

Therefore, you can have a weight for criterion ‘Quantity of patients per day’, another for ‘Cost per patient’ , and another for ‘Delay between receiving a request and providing the ambulance service’ . Do you think that you can add these three values without normalizing?

AK-And in my vocabulary the weights, or relative contributions, or relative importance, or penalties for criteria are equivalent terms. They not need be normalized to 1, i.e. keeping their sum=1 because it creates linear dependency-obviously not a desirable property in this case.

NM- I agree 100% that they create a linear dependency between criteria where possibly it does not exist

AK - For your question "...it wouldn’t be better to work statistically, which probably can produce more reliable weights?", yes, theoretically in pure mathematical sense it would probably be better. However, in healthcare (especially in crisis situations) decisions must be made quickly, often within minutes. There is no time to work on statistics.

NM- Naturally, it goes without saying, but I imagine that you want to determine a policy to be applied, and not solving a case every time that is appears. Forgive me for my ignorance of hospital procedures, but it is clear for me that appropriate measures are taken based on a pattern. For instance, if it is evident that many incoming patients can’t be given a bed because its scarcity, obviously that indicates that the number of beds, according to statistics, must me increased.

This is what I was referring to when I mentioned statistics.

Consequently, I guess, that if average number of beds is greater that average patients required to be bedded, it largely facilitates the decision in each case.

AK- Besides, finding weights somewhat more accurate will not make computation results more valuable for the DM

NM- Of course not, but can contribute in improving the system.

AK -The DM seeks the critical limits or thresholds for probability of death (T POD) and days on ventilators (T DV) for some patients to decide admit or not admit due to the limited amount of resources (beds, staff).

NM- But these thresholds don’t come from statistics?

AK- These thresholds are determined by maximizing the objective function OF by the optimization module, while the random POD and DV are generated by the discrete event simulation model using statistical distributions obtained from the Virtual Pedatric DataSet (VPS) which contains hundred thousands patient records.

NM- But according your first message your objective is to maximize bed occupancy and minimize treated and untreated mortality. What does it have to do with thresholds? Most probably, if you want to maximize thresholds you will get a number close to infinity.

Normally, thresholds are part of the data, not of the solution, and refer to criteria, not to options or alternatives. You have to fix the thresholds based on statistics. In so doing, the alternative bed occupation will de determined based on the restrictions of thresholds that you indicate, that is, the maximum occupancy must be within the limit imposed by the threshold.

Sorry, it is not my intention to mean that you are wrong, most probably my ignorance of the scenario makes me see it in a different way. I never solved a problem where the objective was to determine thresholds for criteria.

And what you say about TPOD and TDV corresponds with what I said about statistics.

AK- If computation indicates T POD

Nolberto Munier

I will address your 1-st comment leaving the others for later..

NM-"Therefore, you can have a weight for criterion ‘Quantity of patients per day’, another for ‘Cost per patient’ , and another for ‘Delay between receiving a request and providing the ambulance service’ "

You assume here that it is a MCDM problem, implying that the criterion Quantity of patients per day should be maximized, Cost per patient is to be minimized, and Delay... for service ' minimized.

In fact, the right statement of the problem will be: Given the variable number of patients, and given the cost per patients (includes the fixed hospital cost and variable cost per patient), what is required capacity (staffing and physical beds) that provide delay (wait time) less than the acceptable limit. What will the annual profit be given the required capacity? (The difference between the revenue from all patients and the hospital cost to diagnose, admit and treat patients)

The number of patients required service is beyond hospital control, it is the input variable, not criterion. The delay (wait time) is usually a constraint (not criterion)- its min number is known- zero. But the upper value should be limited.

The best method to address this type of problem is discrete event simulation. Examples are given in the book (see attached)

section 2.4.5 Outpatient Clinic Cost and Staffing: Is right Staff used at the right time?

and

section 2.3.10 Surgical capacity of Special Procedure Operating Rooms (analysis of the wait time).

Your another statement "... if it is evident that many incoming patients can’t be given a bed because its scarcity, obviously that indicates that the number of beds, according to statistics, must me increased." is not that evident. It was demonstrated that the optimal policy aimed at maximizing the population survival is to admit not any patient but the one who will benefit most from the scarce hospital resources: the goal of the previously mentioned paper was to compare the First come first served (FCFS) admission policy vs. the policy to admit only those who will benefit most with the higher chance of survival. The most efficient policy is the right use of the available beds/resources rather than costly increasing the beds' number.

Dear Alexander

AK. I will address your 1-st comment leaving the others for later..

NM-"Therefore, you can have a weight for criterion ‘Quantity of patients per day’, another for ‘Cost per patient’ , and another for ‘Delay between receiving a request and providing the ambulance service’ "

AK. You assume here that it is a MCDM problem, implying that the criterion Quantity of patients per day should be maximized, Cost per patient is to be minimized, and Delay... for service ' minimized.

NM. Remember that I have only a glimpse of your problem, I only got fragments of it, therefore, it is very possible that some of my answers are product of my ignorance of the whole scenario. When I say that the number of patients should be maximized is because I am assuming the worst scenario, but within the existing probabilities of occurrence.

AK. In fact, the right statement of the problem will be: Given the variable number of patients, and given the cost per patients (includes the fixed hospital cost and variable cost per patient), what is required capacity (staffing and physical beds) that provide delay (wait time) less than the acceptable limit.

NM. It is very clear and very reasonable, but, as I understand it, this objective is different from the original one, where your aim was to maximize time average occupancy and minimize treated and untreated deaths. Now you are asking for the required capacity that provides a waiting time less that the acceptable limit.

It appears to be a typical example of queuing theory., which I am sure you know.

I guess that you can equate your scenario to a queue of people waiting to be served by a certain number of employees, as in a bank or in a supermarket for instance, and there are very well-known techniques to determine the number of employees needed to keep people waiting less that a maximum number of minutes.

In theses cases people arrive at random as in your problem, have different problems more or less complicated, as in your problem, and have different emergencies, as in your problems.

AK. The number of patients required service is beyond hospital control, it is the input variable, not criterion. The delay (wait time) is usually a constraint (not criterion)- its min number is known- zero. But the upper value should be limited.

NM. Well, delay could be a criterion if you consider the lowest and the highest limits that can be obtained through statistics. If as I presume, the universe of arrivals in a certain period follows a Gauss distribution, you can establish easily which is the maximum number of arriving patients for a level of confidence say of 95%. You can also have similar values for people waiting, and then compute the minimum delay time with a 95 % of confidence. By the way, a constraint is normally interpreted as a criterion.

AK. What will the annual profit be given the required capacity? (The difference between the revenue from all patients and the hospital cost to diagnose, admit and treat patients)

NM. I believe that you can get that information from the above computations.

AK. The best method to address this type of problem is discrete event simulation. Examples are given in the book (see attached). section 2.4.5 Outpatient Clinic Cost and Staffing: Is right Staff used at the right time?

And section 2.3.10 Surgical capacity of Special Procedure Operating Rooms (analysis of the wait time).

NM. I agree

AK. Your another statement "... if it is evident that many incoming patients can’t be given a bed because its scarcity, obviously that indicates that the number of beds, according to statistics, must me increased." is not that evident.

NM. I support that it is evident because, independent from other reasons, if you can’t give a patient a bed in a reasonable time, it means that there are not enough beds, does not matter the reasons.

AK. It was demonstrated that the optimal policy aimed at maximizing the population survival is to admit not any patient but the one who will benefit most from the scarce hospital resources:

NM. Well, this is evident

AK. the goal of the previously mentioned paper was to compare the First come first served (FCFS) admission policy vs. the policy to admit only those who will benefit most with the higher chance of survival. The most efficient policy is the right use of the available beds/resources rather than costly increasing the beds' number.

NM. No doubt that if you compare those policies the right results belong to the second.

A suggestion: You can maintain in your OF only the first term but minimizing it, because what you want is the beds used as more efficiency as possible, that is to have a high rotation, and instead of considering the other two negatives why don’t you express them as criteria?

In so doing you will be minimizing number of beds needed and subject to two strong restrictions:

a) Maximize survival for treated patients

b) Maximized survival for non-treated patients

You can also add a new criterion maximizing the number of patients on FCFS

Anyway, these are only my ideas