367- The amazing performance of SIMUS, a MCDM method, and the SYM/SIMUS procedure

Nolberto Munier

SIMUS (Sequential Iterative Modelling for Urban System), is based on Linear Programming (LP) the remarkable theory to solve complicated decision problems in most human activities, created by L. Kantorovich (1939) who was awarded the 1956 Nobel Prize in Economics for his creation, and using the Simplex Method to solve it, developed by George Dantzig about 1946, nominated the 1947 top algorithm in the top list, by researchers of the Cornell University. SIMUS was developed by Nolberto Munier in 2011 as a Ph.D. Thesis.

There are more that 200 methods in MCDM, based on different assumptions and using different algorithms. All of them start with construction of the initial matrix with alternatives in rows and criteria in columns and filled with objective and subjective data, with the main purpose of determining the best alternative option. They normally use subjective weights, that modify data as per the DM preferences, process the matrix, and arrive at a value that defines the most convenient alternative.

SIMUS also starts with an initial matrix, does not use weights, but assigns a target to each criterion that expresses the wished values by the DM or stakeholders.

SIMUS works with existing raw data, either objective, like costs, investment, contamination, budget, etc., as well as subjective, such as averaged people preferences, variations estimate, appraisals, etc., that is, data coming from reliable sources, for example statistics, surveys and documented analysis from experts based on their research. SIMUS does no need criteria weights since it generates them objectively, based on imputed data and new matrix values produced by iteration, a distinctive feature of LP. Thus, data is processed without any human intervention or interference, and represents a real problem without modification.

SIMUS processes the information following multiple applications of the Simplex and arrives to a unique solution and also gives automatically a ranking of alternatives, from the best to the worst. The mathematical solution, if it exists, is mathematically correct, although it cannot be considered optimal, because optimality does not exist in MCDM scenarios when pretending to optimize at the same time two opposite criteria, like optimal benefit and optimal cost. Linear Programming can do that, but only when there is a sole objective function, which is not the case in most problems.

However, in real-world environs, activities may be affected by many exogenous variables that nobody can control as international markets, for instance, international demand of grains, steel, raw materials, or by acts of nature like droughts, delays, pests, or due to new discoveries and products, or by governments import/export regulations, etc., that were not inputted in the initial matrix, because many are highly variable as the stock market, commodities, oil, rare earth, etc., or unpredictable or even unexpected actions.

It is here when the Decision Maker (DM) enters in the scenario, doing a work that no MCDM mathematical method can do, that is, research, consultation, discussions, appreciation of variations and their constancy, opinion of users of a good, etc., which translates in risk evaluation due to the exogenously variables. It is precisely this risk one of the most important aspects stakeholders wish to know.

Consequently, the DM takes the mathematical result, which is automatically analyzed by SIMUS performing a sensitivity analysis (SA), also detecting risks but related to criteria allowable variation, and with this mathematical information, he/she inputs the result of his/her exogenous analysis.

This is the solution of the problem and the SIMUS method and SIMUS/procedure. This allows the DM to justify before the stakeholders the reasons for his decision or selection. As can be seen, there is a complete symbiosis between man, taking decisions, and mathematical models, giving results on what those decisions are grounded.

How SIMUS offers this reasonable procedure and advantages?

Mathematically, the method works with lineal inequalities instead of equations, that is, linear spaces instead of straight lines, and applies a simply but rational principle: Resources (a part of data) are needed in enough quantity and in time to satisfy a target in each criterion. If both agree, the space comparison determines the exact number of resources needed; if not, the criterion target is not feasible. The exact amount is computed considering not only what is needed, but also makes sure that another criterion using the same resource (money, for instance), is also satisfied. That is, since the initial matrix is a system, everything may be related, and then, the method looks for equitable share of a scarce resource. This reasonable modus operandi is a feature of LP, and the core of the Simplex algorithm.

Consequently, it works comparing for each objective or criterion, the availability of resources (money, water, land, number of workers, contamination values, income, benefits, etc.) to targets fixed by the DM for each resource, using the logic concept that if a criterion does not have enough resources, its target and perhaps others cannot be satisfied.

SIMUS capacity in modelling to solve complex problems

These real-life examples are only synthesis. For detailed analysis of these solved cases using SIMUS consult Munier, ‘Strategic Approach to Multi-Criteria Decision Making” – Second edition

Example 1: Considering benefits maximization - Manufacturing

A company has the capacity to manufacture three different products (or alternatives) (P1, P2, P3) subject to four criteria: benefits, cost, environmental damage, and investment (C1, C2, C3, C4), where benefits must be maximized, cost and environment damaged minimized, and investment constrained to a fixed amount of funds, i.e., no more, no less. This is indicated using math symbols: ≤ (lower or equal to), ≥ (greater or equal than) and = (equal to).

Thus, the expected benefit (C1) is for instance, 250,000 Euros, and this is indicated as:

C1 = a1(P1) + b1(P2) + c1(P3) ≤ 250,000 Euros

where ‘a,’ ‘b’ and ‘c’, are the unit benefits for each product. Observe that the subindices are preserved since all alternatives are subject to criterion C1. In SIMUS, alternatives are in columns and criteria in rows, to respect algebraic representation of linear problems.

However, why we use the symbol ≤ , instead of ≥? Because the company has estimated that 250,000 Euros is the maximum benefit it can aspire, considering their maximum production capacity, as well as available raw materials and personal. Obviously, the aim is at the maximum benefit, but it cannot be infinite since it is constrained by the available resources.

Example 2: Considering costs minimization- Manufacturing

Criterion C2, cost, must be minimized

In C2 the inequation may read as this:

C2 = a2(P1) + b2 (P2) + c2(P3) ≥ 140,000 Euros

This indicates the criterion must be minimized, but down to a limit point, that is 140 ,000 Euros, because this is the real cost of production, and naturally, the company cannot produce at a lower cost. For this reason, the inequation indicates that the cost must be greater or equal to 140,000 Euros.

Example 3: Environmental damage

The same analysis applies to environmental damage, and thus:

C3 = a3(P1) + b3 (P2) + c3(P3) ≥ 78 ppm of CO2

The reader will certainly wonder why it says that the contamination should be greater than 78 ppm (parts per million) of CO2, shouldn’t be = 0?

Yes, it should, theoretically, but there is no a human activity that is completely free of that gas. Even, the mere act of breathing produces it. Therefore, it would be no realistic to aspire for zero contamination.

Example 4: There is a budget that cannot be surpassed nor fall behind, therefore, it is a fixed amount

If the company has a minimum level of production due to economies of scale for instance, with a minimum benefit of say, 109,000 Euros, that comes from its financial statements, the inequation will be:

C3 = a3(P1) + b3(P2) + c2(P3) = 109,000 Euros

Example 5: Managing uncertainty – Dual criteria

This case may arise when there is uncertainty on the value of the target. In this case it is convenient to fix two limit values, a maximum or high value and a minimum or lower, and utilize the same inequation. Using as example the inequation in example 1:

Maximize C1 = a1(P1) + b1(P2) + c1(P3) ≤ 250,000 Euros

And using also de inequation of example 2:

Minimize C2 = a2(P1) + b2 (P2) + c2(P3) ≥ 140,000 Euros

Observe that only the symbols and the targets have changed. This set of criteria instructs the software to consider a target interval, instead of a single one, naturally, after normalization. Thus, the software will work with values contained within the interval, and will chose the one that complies with this criterion, and, at the same time, will influence the other criteria that also use the same resource.

Example 6: Sometimes there is need to consider physical existing options for alternatives – Binary variables- Infrastructure (Roads)

This example introduces the concept of using binary variables or 0 -1, the ‘0’ indicating no relation, while ‘1’ is the opposite. Inequations or equations of this type can mix in any way with criteria using nominal values. This is also linked with the very important issue of inclusive and exclusive alternatives. The first refers to alternatives than can coexist once the problem is solved, or be executed simultaneously, without any mutual link, for instance, alternatives ‘Build school’ and a ‘Repair Road’, since none of them affects the other, or that both can work without problem even sharing a plot of land.

In the second case, there could be dependencies or even redundancy. Please refer to this example:

Example: Creating a route for a highway

Assume that there are two different existing roads than can be part of a new highway; since both depart from the same origin (an airport), meet at a distribution point or roundabout, and from there both follow different paths, and both ending in the downtown area of a large city. Only one must be selected.

In the roundabout there are thus two entrances or arrivals, and two departures or exits. Arriving route 95, exits the roundabout as 109, the other route arriving at the roundabout as 36 exits as 101. Both routes 95 and 36 are subject to the compliance of the set of criteria. Routes 109 and 101 both depart from the roundabout, but there is the possibility of mutually changing the respective exits. The 95 exiting as 101, and 36 as 109. All partial distances are different, and then the total length of each one.

The problem consists in selecting the shortest route airport/downtown. Therefore, consider that the two routes leaving the airport have different length as well as the two routes from the roundabout to downtown, in both cases there are exclusive, since both cannot be simultaneously in the highway; it is one or the other.

To solve this problem using SIMUS,

First consider for both routes the two different legs, that is airport/roundabout and roundabout downtown.

Create a criterion for the first, and below each one place 1s and equals to 1

First leg:

Route 95 Route 36

1 1 = 1

This indicates the software that both routes are exclusive, and thus, only one must be selected

Roundabout exit 109 Roundabout exit 101

1 1 = 2. Any of them can be used

Second leg :

Route 109 Route 101

1 1 = 1. Only one must be selected.

Notice that the software has now this information in binary:

No route can be selected simultaneously in the two legs; it is one or the other

Both routes are allowed to share both exits in the roundabout

Other routes or roundabouts can be added, replicating this procedure.

As a final result, in a map, there will be a continuous line from airport to downtown, with the shortest distance between them. Of course, there could also be a set of nominal/binary values on investment, distances, travel times, etc.

Example 7 - Precedence between alternatives – Civil constructions

A City Hall decides to use a vacant land in an island across an inlet, with four potential projects as follows: Building a sporting area (A), a car race track (B), developing an aquatic park (C), and constructing a concrete bridge (D), to connect the city with this new development in the island.

Without any calculation it is obvious that the bridge is one of the chosen, and it must have absolute preference, because it is paramount to facilitate the transportation of the construction crew, materials and equipment, and later the visitors. This is equivalent to say that the bridge must be selected as the first project. How does the DM indicate this to the software?

Just by writing that the bridge must be preferred to all of the others, creating a criterion ‘Preference’, placing a ‘1’ only for alternative ‘Bridge’ and writing:

1 = 1

This criterion instructs the software to place this alternative in the first place, irrelevant of what the software founds.

SIMUS offers another way to indicate precedence which is more elegant and easier. In the initial matrix, loaded either by hand or electronically from Excel (recommended), look at the small table at right, with as many rows as criteria. There the DM can identify as many precedencies as wished, ants as:

D > A

D > B

D > C

The same procedure may be used in these cases:

• Adding a new project to another under construction, for instance, adding a new hydro dam downstream the one under construction.

• Some authority commands for instance, that project ‘sewerage’ must be the first selected project, irrelevant what the software finds. Very common, for political reasons like promises made during the election campaign.

Example 8 – Reducing the number of alternatives – Selecting city indicators

Sometimes the number of possible alternatives is high, for instance those relative to urban indicators, as large as 60 or 70, or more. In this case it is convenient to reduce its number, not due to SIMUS inability to process them. but for the analysis of solutions from the DM, and to reduce the workload.

Assume that the DM estimates that 25 selected indicators will be an adequate number. In SIMUS it is only necessary to instruct the software about it, and the simplest way is by adding a new criterion ‘Final desired number of alternatives’, and placing a ‘1’ in each cell below each alternative, use the symbol ‘=’ an put the number 25. This way the software is instructed to consider all possible indicators, but select only the 25 bests.

Example 9: Performing experimental tests (Control experiments)

SA common task for researchers.

Assume a solution had been found in a certain scenario. The DM wants to see very rapidly what would happen when he changes values, or by adding or deleting criteria, modifying targets, etc. That is, the scenario is the same but some values change in each test, and finally the best can be selected by the DM. It would involve modifying the initial matrix for each test, something cumbersome and costly.

In SIMUS, the DM simply designs the matrix with more rows or criteria than necessary, and leaves them blank. If after the first run he decides to add a new criterion, he can use one of the blank rows, and run the software, if after that, he wishes to change some values, he goes to the original matrix already loaded, erases old values, replace (or not), with the new values and run. Each result is saved in an Excel library in SIMUS. If the DM wants to eliminate one criterion, for whatever reasons, just deletes it and run the software.

If he wishes, for whatever reasons to remove an alternative, just deletes it or the values it contains. The same if he wishes to add a new one. However, he must remember that removing or adding an alternative means a new problem, since he walks from a dimensional space ‘n’ to ‘n-1’ in the first case. The same if he adds an alterative and goes from ‘n’ dimensional spaces to ‘n+1’. In both cases new problems are created. From the SIMUS point of view, it is not problematic to remove or insert new alternatives or criteria however, most possibly each one of these two operations may cause ‘Rank Reversal’, that is, may or not change the former existing ranking.

As a fact, for research purposes, the DM can perform an interesting exercise, that was used to model electricity generation along diverse extensive periods.

He starts with a simple matrix with x1 and x2 alternatives, or two dimensions (2D), saving the result.

Adds a new alternative x3 (3D), with a column vector and arbitrary values, but of course, congruent with the 2D, run the software and save the result. It could or not change the former ranking. He continues adding alternatives up to for instance, x10 (D10), and observe what happens with the successive rankings.

For instance, the DM will notice that successive rankings for instance for 5D, no longer consider the activities of rankings corresponding the 2D, 3D and so. The reason is that each addition or dimension, includes the characteristics pertaining old rankings and adds information.

It can be understood in a simple example. Consider a 2D or two dimensions object like a square. It is then possible to learn about its length, width and area. Now add a dimension or 3D, and the square is now a cube, where the dimensions of the square are still there, and in addition, one can learn, about its height and volume. That is, adding dimensions gives more information on the object. The same in M CDM, we can get a ranking, but when adding an alternative, we get more information and probably a different ranking, because the original object or matrix changed, and one of the original coordinates may disappear. This experiment, confirms the theory that rank reversal may or not be invariant, due to the switching of vector spaces.

Example 10: Multiple scenarios – Planning for multinationals

A common scenario where multi national companies, for instance in the Agri business, have plantations and processing plants in several countries, where the same crops are exploited. For example, wheat, soy, barley, oranges, etc.

This case is different from typical MCDM problems in the sense that the same option or alternative can be used simultaneously in different scenarios.

As an illustration, take four scenarios or four regions of the world with diverse climate, different sun irradiation, water quality, water availability, transportation facilities for export, existing plagues for some regions and risk for all of them, and as alternatives wheat, corn, soybean and oranges plantations. Some regions that have high irradiation and moderate heat, may include building an orange frozen concentrate plant as a dependency of the orange orchard, and thus, as inclusive alternative, something that does no apply to other regions.

Naturally, criteria values will be unique for each region. Some criterion may apply to only one region as plagues, and not to others, while sun irradiations could be only interesting in some regions, but not in all of them. Some criteria must be maximized and other minimized, etc. Alternatives may apply not to all regions, but and one of them, as for instance construction of an orange frozen concentration plant.

As seen, the problem is quite complex, since in addition to alternatives and criteria, includes several regions, different set of criteria and inclusion, and of course, maybe also subjective criteria. It can add a criterion applying to them all, for instance, total amount the company intends invest in a five-years period in all of its undertakings.

SIMUS solves this, by preparing four decision matrices, each for region and the same alternatives for them all. i.e., only one row for alternatives and four sets of criteria in a same column. If a region is not apt for say, wheat live the corresponding cell blank or put ‘0’. Run SIMUS, and you will get the response, that is, what are globally the best crops to develop in each region.

Example 11: Adding weights to alternatives and criteria if the DM wants to introduce them (Practice not recommended

SIMUS does not use weights either subjective or objective, but in each iteration, it determines quantitatively the importance of each criterion, values that in turn are used in the process. Therefore, if the problem requires say, 10 iterations, then, 10 sets of criteria importance values are computed. This is a unique feature of the Simplex algorithm, that assumes correctly, that the importance of criteria cannot be constant, because it depends on the characteristics of each alternative.

It is obvious that if there is a portfolio of alternatives like A (construction of a condominium building), B (construction of a commercial mall, and a C (construction of a bridge), the relative importance of criteria for each alternative is different in each case, and depends of the resources for each one, and this is the way SIMUS works, trying to balance resources.

Example 12: Specifying that certain alternatives must satisfy a minimum number of criteria – Urban and environmental problems

Assume that all alternatives must satisfy these criteria: Cost, benefit, environmental contamination. population, and public health, but some of them, for instance, alternatives B and D, to only two. This is accomplished by creating a criterion “Comply at least with this number of criteria”. The DM puts (1s) below the referred alternatives, uses the symbol ‘≥’ and the value 2. This arrangement instructs the software that alternatives B and D must satisfy at least two criteria.

The same alternatives or others can also be subject to comply with a similar requirement but pertaining to other criterion as for instance, ‘Cities larger than 15,000 people” or cities at a “Maximum distance of 5 km from the sewerage trunk. It can be seen how SIMUS can model different requirements that are in real scenarios.

Example 13: SIMUS also can draw utilities curves.

To this respect. The method gives the data to generate utility curves, by decreasing, as well as increasing the importance of each criterion using their respective marginal utility.

Example 14: Externalities

Most projects generate externalities, that is, costs that do not have a market value, but that may have an strong incidence in the GDP of a country. The quantitative value of these externalities is found in the last of SIMUS screen

Operational Analysis

a- A very important question the stakeholders will certainly be very interested, is to know in which extent a target was achieved, exceeded of falling short. For instance, the target for ‘Investment Rate od Return on investment (IRR)’, has been estimated in 20.8 %, from the projected financial statements, It represents the relationship between the net earnings of an investment over the investment value in a certain period. Therefore, 20.8 % indicates that the stakeholders estimate to earn a minimum of 20.8 % on the value of their investment. It can be appreciated the great importance it has.

If the target (known as RHS in algebraic equations), is 20.8 % for criterion IRR, SIMUS shows in the same line, at left, if this value (known as LHS), has been reached. For instance, 18.9% indicates that the project is 9% short of this target.

If LHS=RHS, (computed target minus wished targets), the project satisfies the garget in a 100 %.

If LHS is 21.03 %, it exceeds retarget in 1.1 %

b- It also could be interesting determining, based in the result, which is the most important criterion.

SIMUS measures this importance, as a function of a criterion that has the largest participation in all alternatives, that is, as a basic criterion and can be easily visualized even at first sight by a dedicated matrix in the last screen.

SYM/SIMUS – Analisis of results by the DM and their evaluation by taking into account global

exogenous factors

All the above-mentioned cases can be solved by SIMUS and best alternatives and ranking displayed in its last screen. However, this certainly is a sound mathematical procedure, but incomplete, because it refers to data and through a mathematical process, but does not take into account neither the DM experience and know-how, nor nuances of real-life

SYM/SIMUS means Symbiosis between man and machine, i.e. SIMUS gives results that are mathematically correct, because they are based on real quantitative and/or subjective data, the later normally corresponding to projects that affect a very large number of people, such as highways, hydro dams, energy generation. etc., where the people opinion and their points of vies must be considered, and this is done through surveys, polls, statistics trends, discussions, meetings, conferences, symposiums, etc.

However, this is only raw material processed, that needs to be refined according to the global environs, and which normally are exogenous variables like risk, international price variations, government regulations, strikes, labor atmosphere, etc., that cannot be introduced in the initial matrix, and that no MCDM method can solve, only a human mind.

It is here where the DM shines since he/she has robust and reliable results from the software process and with that knowledge he/she is able to evaluate if the mathematical solution, fits the bill considering those exogenous factors.

The DM looks for the selected best alternative, and analyzes it regarding not only on his expertise, but also considering the effects that global variables may have on it, such as trend of degree of variability of prices, duties, demand, stock market, international measures like countries applying hefty charges to imported merchandise, etc.

With this knowledge the DM is in a condition to appraise the goodness of each alternative based on the prior results, he can make changes to data and criteria and even simulate what would happen with different hypothesis. As a consequence, the DM can present the stakeholders with a complete and fair assessment based on mathematics and real values, as well as from examining the influence of many different global factors. As seen, there is a perfect symbiosis between man fed with information from SIMUS and the DM investigations with global factors.

The procedure is: Once the DM receives the result, he may:

1- Perform a rational sensitivity analysis (SA) based on:

1.1 Independently the number of criteria, SIMUS identifies the criteria that are significant or basic, for a certain alternative, and which are the irrelevant criteria, at least for that alternative

1.2 Now, the DM proceeds increasing/decreasing each criterion based on the target established. If for instance, the problem is the selection of type of renewable energy source to install in a site, for a target or output of 2 MW, the DM may consider successive increases of 20 % and decreases 8 %, or in any other rate

1.3 Now a question: How much can be increased and/or decreased? Are there limits?

Certainly, they are, and they depend on the allowed variation of each criterion, and thus, the information of how much leeway the DM has to increase or decrease a criterion. This allowed variation per criteria is also given by SIMUS. What happens if the DM surpassed them? Nothing, because the criterion ceases to be basic or operative, and is replaced by another one

1.3.1 All pertaining criteria must be examined simultaneously, it does not matter if some call for benefits and others for cost, and even allowing the DM to establish different variation rates for each one

1.3.2. Identified the intervening criteria, the DM may investigate for each one the exogenous variables (global variables), that may affect it and use his/her expertise and knowledge to assess if there is risk produced by each one

1.3.3. As an example: In a project of 3 alternatives (A, B, C) and 12 criteria, the result shows that the best alternative is B and the ranking is: B>A>C. Consequently, the DM examines A first, and finds that the basic criteria that define it are C3 (max demand), C6 (max production), C10 (min delays) and C12 (max job generation). In the same screen the software gives in two columns labeled Increase and Decrease, how much each of these 4 criteria may vary. That is, column ‘Increase’ shows for instance 0.8 and in decrease 023. Thus, for C3 (demand) the DM needs to investigate what exogenous variables can affect this demand, and finds that demands for product A is quite stable. This makes the DM confident that even if the allowed variation of C3 is small, say 0.15, it is safe due to its stability.

Now the DM goes to C6 (production). His research reveals that an imported raw material component of product or alternative B, has had sharp variations along years, consequently, he reasons that probably it will be wise that stakeholders reconsider the production target to a more modest quantity. The same with the other criteria.

It can be seen that the DN can furnish stakeholders with the type of information they want, and more important, based on sound mathematics and on a reliable assessment of markets and exogenous conditions

Calculating risk due to exogenous variables

When SIMUS solves a problem delivers a result and additional information to perform a complete sensitivity analysis and to analyze the solution considering also global variables. However, at the same time it depicts values for some basic criteria that are different from the values commented above. These values, technically called ‘Shadow prices’ are also shown in the same last screen. They have a large importance because indicate how much an objective increases/decreases when only one unit it is added up to a corresponding criterion. . The shadow prices indicate the marginal value of e ach criterion or marginal utility.

Whren a DM increases one unit a criterion the corresponding objective increases an amount equal to the criterion marginal value. This allows, by successive increases to draw the total utility curve for that objective. The same for de creasing, and this can also be performed by a SIMUS Add-in called IOSA (Input-Output Sensitivity Analysis”. This curve can be used to compute real risk due to global or exogenous v variables.

Conclusion

There are more aspects in real problems that can be addressed by SIMUS, as for instance taking into account the system dynamics, immunity to different normalizations, comparting performances in transportation, its integration with AI, etc., but it is believed that the explanations and examples briefly described are clear exponents of the power of this method.

It is reminded that the SIMUS software is world-wide free, it is a property of the National University of Cordoba, Argentina, and its software may be requested to [email protected].