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363 – Suggested procedure for choosing the most appropriate MCDM method to solve a certain problem

Author: Nolberto Munier

This is an old problem still unsolved, and probably never will; we continue using current MCDM methods that cannot model reality properly. The present paper proposes a methodology to clarify this issue by following a procedure called ‘Analysis and Concentration on the Main Objective’ (ACMO).

“The formulation of a problem is often more essential than its solution, which maybe a matter of mathematical or experimental skill”

Albert Einstein

ACMO uses a principle, that as far as this author knowledge, has never been enunciated before, that can be defined as ‘Finding the objective that best reflects the purpose of the project’. That is, the criterion or objective that embodies or epitomizes what the project is aimed to. Alternatives are subject to a set of criteria of the most different nature related to economics, engineering, financing, environment, health, government, sustainability, transportation, food, social issues, manufacturing, mining, etc. However, normally, among all criteria or objectives (both are equivalents), there is one that best interpret the purpose of the project, whatever it might be.

Examples:

· Take an investor with a portfolio of projects involving the erection of tall buildings in a mix of dissimilar constructions; he wants to invest according to his available capital, and thus, this is his main concern, of course, earnings are very important, but there will be no earnings if funds are not enough to finish a project.

Therefore, the amount of necessary investment is the most important issue. Naturally, the undertaking is also subject to have a decent Internal Rate of Return (IRR), and this could possibly be the second most important objective, and of course, when all projects complying with social and environmental regulations, to taking into account different risks, as well as exogenous factors.

There maybe other motivations like expanding, exporting, entering in a new market, etc.; it is not a matter of guessing, but in asking oneself, what do I want? what is my aim, what is the purpose of this project? what is the initial idea, necessity or vision that triggered this project?

Other developers aim for a good return of funds invested, probably the most significative is the Internal Rate of Return (IRR), or entering in the agroindustry or producing software, etc.

As said, there might be a second most significant criterion or objective, and even a third, which is convenient, because when performing the sensitivity analysis (SA), he may find that the maximum IRR is closely associated to volatile or uncertain criteria, like a potential scarcity of concrete, or not enough demand of apartments, and then, he might incline for developing an industrial park, or any other undertaking not attached to uncertain circumstances.

In opinion of this author this elemental analysis is not always done, if ever, and thus, missing in attempting to solve a problem. In other words, an immense majorly of practitioners are looking only for a ranking of projects, assuming they are feasible, without any proof of it, and thus, something not even thought about. Some perform a SA, but do not analyze how much they can increase or decrease the criteria, let alone understand which are those criteria out of the total set. Most ‘select’ a single criterion simply by considering the one that has the maximum weight, which does not have any mathematical support, let alone validity; it is only intuitive …..and invalid.

There is no need for criteria comparisons and quantifying them in a certain scale; instead, it is only required to define for each one a target value, and this is of paramount importance, for example establishing an IRR of 6.8%, a maximum labor work of 25 workers, a payback period of money investment in 9 years, a market penetration of 35%, etc. Of course, for this, it is essential to work with resources than can be of any kind, like funds, quantities, allowed percentages of NOx contamination, etc., and targets established for each one, fixing acceptable ranges for each target, or even a single value, computed not by guessing but for analysis of the problem, its effects, characteristics, etc. that is, setting up a realistic and achievable value.

For instance, the DM may say “I want to produce as a minimum 23,500 units of my product per month, because that amount is related with my economy of scale, and a maximum of 28.300 due to my available investment capital”, for I am not willing to risk more money than that, or simply, work with one of the two values.

The method aims at designing a system that complies with the targets as close as possible, either maximizing or minimizing. Most possibly not all targets will be achieved in a 100%, but it is important that each one of them is addressed.

Observe that all of this is at present done by an investor, owner or company that define the maximum capital to invest, the size of the crew, the annual budget, taxes, the limit for amortization of loans from banks, etc.; thus, he only needs to use these values as targets.

The investor normally has a plan or schedule for each project, linked to the others, when to start and finish each one, rate pf annual construction, etc. but at present most of this is absent in the initial matrix, not due to the DM ignorance, but because there are very few MCDM methods that can model thus data, and the DM must rely on one of the myriads of existing MCDM methods, even knowing that their modelling does not represent reality, but only perhapselectricity demand objective, a coarse approximation.

· Maximize the as in the case of deciding which type of installation or plant (Wind, PV, Biomass, etc.) to build. This is the basic objective, among the others related to land use, wind strength, solar irradiation, etc., but all of them supporting the demand objective subject to a set of criteria, because if there is no demand, all these criteria are irrelevant.

It is assumed that in general criteria have dissimilar importance or weights and then, usually, they are assigned arbitrarily, not a good procedure indeed, or in the best case using objective weights, that are real. As it can be understood, criteria depend on the alternatives, and the latter evaluated by criteria. Consequently, results must indicate in what extent these criteria targets are satisfied. This is a fundamental premise, and for this point of view we can see that MCDM methods must select the alternative that best approximate the targets.

· Minimize the cost objective, as in the case of fabricating and exporting goods, subject to other criteria, like international prices volatility, competition, transportation delays, quality, etc. The result may indicate a company the convenience to export or not, based on the results of the cost objective. All other criteria are directly or undirectedly, depending on it, like bank loans, interest rate, payback period, etc. Or it maybe that it has a cost target which cannot be surpassed, or maximum production target that cannot be achieved due to not enough funds.

· Maximizing the benefit objective, which probably is one of the most common. Of course, it is tightly linked to other objectives like maximizing production, minimizing working capital, maximizing efficiency, etc. As in most cases, the target objective say sales, for over 500,000 Euros, maybe conditioned by the restrictions exerted by other objectives like working capital, bank loans, payback period, IRR, etc. This means that criteria cannot be considered as independent or in isolation, since normally they are directly or indirectly related to each other. Consequently, the sum of individual values for criteria is incorrect, since what is needed is the intersection of criteria and alternatives, performed at the same time. Why the sum is incorrect? Because a criterion may receive influence from other/s that can increase or decease its target. An example, is that a decrease in investment criterion for the whole project, may affect the environmental criterion, because it means that we must reduce expenditures by not acquiring some costly equipment to lower CO2 emissions.

· Minimize the contamination objective, as in the case of CO2 emissions from fossil fuel fired power plants, replacing contaminant plants with renewables. Nowadays this is the first concern world-wide. All criteria are important, but this is the most significant. For instance, there is world-wide consensus that by 2050 there must be zero CO2 emissions from for electric generation plants. There are a lot of criteria that must be considered, like replacement of oil-fired plants by renewables, costs involved in the transition, selection of plants to be decommissioned, electricity demand, etc., but all of them supporting the contamination objective.

· Maximize number of people objective, in the case of selecting transportation scenarios at national or regional level, choosing among air, private car, buses or trains, alternatives. This is a priority in all countries but especially in those with a very large population. This scenario looks for facilitating people movement. The most important aspect is for people to be able to travel to work, to schools, to hospitals, etc., the balance of criteria is secondary. We can see it in some countries where people even travel in the roofs of overfilled railway cars; the objective is reaching the working place. Of course, the ideal would be people travelling comfortable seated, but if for whatever reasons this is not possible, people look for different, dangerous and bizarre means to reach a destination. All other criteria pale in comparison.

· In national government policy, and from a macro economic point if view, it could be that the most significant objective is the increase of the Gross Domestic Product or GDP, or maybe the fiscal balance since many factors depend on it as inflation, payment of international loans, etc.

· Maximize the enjoyment objective, in case of selecting a place for a vacation time, considering beaches, mountains, cruisers, cities, staying at home, visiting family, etc. This is perhaps the priority for everybody. Needless to say, prices are also important, as well as comfort and safety. Perhaps for some people cost is very significant and thus, prefer staying at home, but it also means losing some “perks” such not meeting friends, or not enjoying the breach. As can be seen, a consequence is normally linked to a decision or action, and this is not represented in the sum but in their multiplication or intersection

· Maximize probability of success objectives, in case of selecting best health treatment for certain diseases. Very important when doctors with different opinions must agree on a treatment. In these cases, there could be if not contradictions, at least different opinions on treatment, drugs and surgery procedures, which must be considered. As an example, there are three different medical treatments as A, B and C for a certain disorder in a patient, and three doctors they may or not coincide, therefore, the decision matrix must be able to represent that discordance, for instance John inclines for treatment C while Stephen believes that B is the best and. Andrews thinks that a combination of treatment A and B could be very effective. To model this condition, we need to use binary notation, that is “1’ for agreement with a treatment and ‘0” for disagreement. In this case we will have for John 0 0 1, Stephen for 0 1 0 and Andrews 1 1 0.

· Minimize the risk objective in many different projects. Consider different risks like personnel, delays, material failure, dubious data, atmospheric conditions, strikes, etc. It is a fundamental objective in all projects; a severe transgression may produce serious problems. This is the main objective in performing sensitivity analysis, since an alternative that depends on a criterion with very little, or worse, without any allowable variation, constitutes a risk. Consider that nothing is gained in selecting the best alternative if it is not stable, and subject to unpredictable variations as in the stock market, or international prices, or even weather or meteorological phenomena, like floods and draughts.

· In a plan to reduce poverty maximizing government help to poor families’ objective, according to different schemes (options or alternatives), like subsides, improving education, help to build their houses with affordable bank loans, etc. No need to emphasize its importance.

· Maximize assisting people objective to communities affected by wars, or natural phenomena (earthquakes floods, fires, etc.)., with different plans. This is prioritary as we have seen recently in some countries

As can be seen in the few cases mentioned, there is a main or primordial objective that is the essence of a project, and it is on which the DM is interested. It does not matter if it is weighted or not, or its relative value among the other objectives or criteria. We are not measuring its importance in numbers or in words, what we need is to establish a reachable target and select the MCDM that yield values as close as possible as the target. Consequently, we are looking for a comparison not for absolute values. The degree of compliance with the established target, is a tangible measure that allows us to determine the best MCDM method for a particular scenario, that is not based on subjectivity but in analysis, research, and mathematics.

Suggested procedure

For starters, the DM needs to accept that criteria and objectives are equivalent - and this not an assumption, but linear algebra - and that both work with resources that can be money, people, number of equipment, percentages, contamination, etc., and this is a reality. Each criteria/objective is then a goal or objective, with a target to achieve, they cannot be indefinite. That is, resources are the ‘material’ not necessarily physical, with what criteria work.

Thus, we can say that in a housing development, criterion ‘Water supply’, must be at least the minimum value by household, according to HWO (Health World Organization), for instance about 150 litres/day-person. This value is a target, limit, or reference, and expressed in the same units as the respective criterion, whatever it might be.

If the project is to purchase a car, there must be a criterion ‘Investment’, that puts a limit, and specifying for instance, not to spend more than 5,500 Euros, because this is the amount that the purchaser has in mind. This value is the target

If the problem refers to the selection of an electricity generation equipment, the criterion ‘Demand” says that as a minimum it must deliver 200 MWh, i.e., a minimum limit. This value is the target.

If the scenario deals with environment, there must be a value, that can be a percentage, establishing an limit, for instance, that the maximum allowable noxious emissions of NOx is 25 ppm (parts for million). This value is a target.

These four completely different cases in type and complexity have a common factor: It is that in each one there is a single criterion that dominates the rest.Dominance not in value, but in the sense that without it the project does not exist.

Just think:

A developer cannot sell a house without enough supply of water

A person cannot purchase a car if he/she does not have the money

To generate electricity, it is fundamental to know the demand

A metallurgical company might be violating the law, if its emissions surpass a certain limit, which of course must be known

Observe that these limits govern our existence as in maximum car speed, water usage, sleeping hours, high and low temperatures, heavy snow, date to pay taxes, closing times in a supermarket, number of years to graduate in a university, metric system, etc.

This author proposes two different procedures for selecting the most appropriate MCDM method:

a) Using sensitivity analysis with the main purpose of measuring risk

b) Employing target analysis with the main purpose of measuring efficiency or compliance with an established target

In reality it is suggested to use first the b) procedure, and then, check risk employing the a) procedure

Example

Consider a bidder that launches invitation bids to build a power generation plant and receives three proposals A, B and C, on a project subject to 12 criteria.

The bidder estimated the the most important criterion was C4, ‘Demand’, with targets like installed capacity between a minimum of 235MW and a maximum of 265MW. Why this range?

Because the minimum is due to economies of scale, meaning that 235MW is the minimum for a profitable operation. The maximum of 265MW is set up because it corresponds to the future maximum forecasted demand, and also considering the available amount for this investment.

Therefore, offers submitted in accordance to this interval; after reception, the DM needs to evaluate the proposals based on the 12 criteria, and where C4 is considered the most important. He employs three MCDM methods that can solve the problem: M1, M2 and M3.

Running sequentially the three methods, the results in MW are, always considering criterion C4.

M1 M2 M3

A1 255 260 260

A2 260 250 255

A3 250 262 240

Rankings according to capacities

M1- A2 > A1> A3

M2- A3 >A1 > A2

M3- A1 >A2 >A3

Observe that method M3 does not comply with the minimum, therefore, it is out the selection, consequently we must center in M1 and M2. These results must be in accordance with the limits established for the bidder, because these targets were also inputted in the decision matrix as targets. If a method’s result does not comply with this range, this method should be eliminated, as in M3.

Consequently, the bidder will only consider methods M1 and M2 and will select one of them.

If the three methods fit the requirements, they must be analyzed.

Therefore, it is necessary to determine which method is the best. This can be done by performing a sensitivity analysis (SA). The purpose of a sensitivity analysis is to find out how strong is the best solution or alternative, and it is a common and mandatory process in all MCDM methods.

For any method, SA starts by identifying which are the criteria - and there could be several - that define each solution, and once identified - normally done only by a few methods - which is their allowed variation, that is, maximum and minimum limits values, also given by a few methods This variation range may by large, small, and even zero, i.e., no variation.

Is this variation is wide enough as per the DM judgement- the increase / decrease will not generally affect the best alternative, and it will keep its first position in the ranking. If it is small, also to the DM judgement, the alternative maybe considered with a certain risk. If it is zero, most probably the best alternative needs to be replaced by the second best.

Why? Because a small change in the criterion or criteria with this no-variation, immediately will cause that the best alternative is replaced by the second best or third best. Of course, there is a mathematical reason for this.

However, we are interested in criterion C4 ‘Demand’, and thus, we can have:

- Wide variation of demand criterion: We select the MCDM method with the highest demand variation. However, how the DM can say if a certain variation is good or not? By analyzing, researching, using statistics, trends, about the performance of that criterion along time, and consulting with the stakeholders if it is in their opinion an acceptable variation.

In the case of a wide allowable variation, consider that there is high probability that the criterion will not reach its limits, and then, it is reasonable to think that as long as the criterion move4sd along its limits, there is certainty that the alternative will not be affected, and will keep its position.

- Small variation of demand criterion: In here, the DM must make statistical research of the ups and downs of demand. If it is stable, the best alternative maybe safety accepted. If the demand is unstable, with high periods of lows and heights, it is wise, to replace the best alternative for a second or a third, because there is a great risk. It could be that another MCDM method suggests another alternative as the best one, therefore, the DM must follow the same process for the three of them

- If the demand criterion is zero, make the same analysis for the three methods.

Remember, that normally any alternative depends more than one criterion, consequently, this analysis mut be dome for each one of them., because the alternative maybe strong considering some of the criteria but very weak due to others.

There is another easier methodology that consists in comparing the targets, in the case for C4, between the wished value, normally known as RHS, and the value for that target given by the method, normally known as LHS.

The difference among these two values for criterion C4 indicates how well the RHS target was met. Then, select the MCDM method that minimizes this difference.

Conclusion

The proposed procedure is simple., provided that you use the right software, if not, it is almost impossible, because the DM does not know which are the criteria on what depends the best alternative, nor the allowable range of variation.

The advantages of this procedure is that it does not matter which MCDM methods are used, it is irrelevant if they employ subjective or objective weights, and it is immaterial the correlation value between them, that in reality does not make any useful contribution.

No personal opinions and assumptions are needed and it can be performed very easily, since it only requires to run the same problem on the chosen method, and analyze the main objective.

Naturally, the reader can say ‘Yes, it can help, but how do we do if our MCDM methods do not use resources or limits’?

Well, that is the problem, which is not attributable to the procedure but to the methods themselves, and it is due to the algebraic process of most of them, with the exception of PROMETHEE/GAIA and SIMUS, that do take resources into account, and thus, deeming criteria as objectives with a target, not as simple restrictions as other methods do.

Non considering resources indicate that out of these two methods, the balance of MCDM methods assume that resources are bottomless, limits do not exist, and that there is no need to determine how a criterion performs. All is reduced to find the best alternative, when it is fundamental to see how each criterion/objective performs. This information allows the DM to make the correction in data he considers necessary according to the nature of the problem and on what the stakeholders say.

Some readers may think that sensitivity analysis is the response to this problem

No, definitely it is not, and it addresses another different problem, that is finding if the alternative selected is strong or not.

Your comments positive or negative, especially the latter, will be very welcome.

Nolberto Munier