The competitive world we are dealing with requires, for success, managers capable of making consistent and efficient decisions, optimally taking advantage of existing possibilities and resources. Decision analysis techniques play an increasing role in the practical activity of economic management, the reasons that drive their development are numerous: the need to optimize resources, to make coherent decisions in conditions of conflict and uncertainty, Need to model and assimilate the experience of experts, facilitate group decisions, the application of computing, among others.
In the modern world, businessmen with significant knowledge in the field of decision support are being trained. The current employer can also seek support for decision-making with the help of these techniques, hence the importance of knowing them in the current economic environment.
Keywords: decision, multi-criteria, ordering, outranking relationship,
multicriterial-techniques-for-business-decision-making- Multi-criteria Decision Making
Decision-making processes have traditionally been analyzed based on a paradigm that can be outlined as follows:
- Select the criterion under which you want to decide the best solution. The set of constraints that limit the solution of the problem is defined.
Next, using more or less sophisticated techniques, we proceed to search among the solutions for the one that obtains the best value for the selected criterion, this is called the optimal solution.
The possible solutions according to this structure are those that comply with the set of restrictions of the problem and that represent the best values of the criterion selected by the decision maker.
This problem is logically very robust, however it has significant weaknesses that deviate considerably from the actual business decision-making processes. Given that in reality, decision-makers are not interested in finding the solution with respect to a single criterion, but rather want to carry out this task according to different criteria that reflect their preferences.
Thus, a company wants to find the best solution not only based on the profit criterion, but also considering other criteria such as: sales volume, risk, etc. In agriculture, you may be interested in determining the best crop rotation that: produces enough food to support the population, maximizes benefits, minimizes costs, etc. In the case of fishing, if you want to establish the structure of the fleet, you may wish to obtain this structure according to the following criteria: cost, employment, maintenance of biological species, etc.
Going into a supermarket and choosing the cheapest bottle of wine does not imply a choice problem itself, but a simple search problem. However, choosing a bottle of wine, harmonizing as much as possible: the price, the highest graduation, the oldest vintage, etc., constitutes a problem where there are conflicting criteria and that will be resolved according to the preferences or judgments that you have. the consumer in question.
Brief historical review.
The multi-criteria decision-making problem is perhaps the most active area of development in recent years in the field of decision science (operations research, resource management, etc.). Thus, we can comment that in 1975 only 3.5% of the papers presented to the Congress of the Spanish Association for Operational Research were devoted to multi-criteria topics, however this percentage increased considerably, already in 1985 this topic represented 14% of the papers presented, that is to say, on that date one out of every 7 works was multi-criteria. In October 1972 the first World Congress on Multicriterial Decision Making was held in the USA.At present, different regional conferences are held annually, where presentations on the use and development of these techniques in the business environment are discussed.
These and other examples show that in the real decision-making process it is desired to find the best decision based on multiple criteria and not only considering a single criterion or objective, as the traditional paradigm implicitly supposes.
According to these data, it is worth asking:
- When this scientific revolution began in the field of decision sciences, at what historical moment can it be said that the paradigm of multi-criteria decision was accepted by the scientific community.
The answer to the first question:
The first works developed were in Koopmans 1951) and Kuhn & Tucker (1951). Another crucial work for the development of the multicriteria paradigm is that developed by Charnes, Cooper & Ferguson (1955) and that was later improved by Charmes & Cooper in 1961.
These pioneering ideas were developed by other researchers, culminating in the First World Congress on Multicriteria Decision-Making in 1972. Such an event can be considered the birth of the multicriteria decision-making paradigm, as well as the beginning of a new period in the field of science of the decision.
Second question:
The indisputable success and sociological support by the scientific community of the multicriteria decisional paradigm has culminated in the appearance of a journal, the Journal of Multi - Criteria Decision Analysis. This has confirmed the existence of two decisional contexts: single-criteria and multi-criteria. It can be said then that the theory of the single-criterion decision constitutes an old paradigm superseded by the multi-criteria approach. The old approach can be reduced to the new paradigm as a particular case of it.
Importance of business decision making.
Despite the growing application of mathematical techniques in the international business environment, there are still limitations in the introduction of these techniques. This is initially motivated by the impossibility of having powerful computing means and specialized software, which due to its high cost was not possible to acquire, in addition to the little culture and training of the decision-makers, the decision-making process being carried out empirically, based on the experience of the human factor that participates in the task.
However, with all the events that have occurred in recent years, the urgent need to make efficient organizations, the responsibility to save energy resources, the need to use resources rationally, to satisfy an increasingly demanding, conscious and prepared customer., has caused the need to change the decision-making paradigm from an optimization approach to a multi-criteria approach where solutions are obtained that rationally model the decision-maker's way of acting, since the fundamental thing is not to approach techniques and / or tools that allow obtaining savings in any direction of a company, but look for a solution in which total costs are reduced and service is improved,From which it can be deduced that it is not possible to maintain as an objective of the design of distribution routes to minimize costs, but also to raise the quality of customer service, an aspect that has not been taken into account so far.
The activity of companies takes place within the surrounding reality, which is the environment in which they are inserted, this environment has a decisive influence on their operation, since to a large extent the greater or lesser success of these will depend on its success in properly relating to the set of external elements.
The current environment has been characterized by great uncertainty due to the greatest economic crisis that has taken place in the history of society. It is at this stage that the influence of the environment on business management is increasingly recognized, controlling the actions of suppliers, distributors and customers in order to adjust production rates to final demand, to reduce inventories, costs totals and shorten delivery times.
Cuba until about 1986 worked under a productivity model where emphasis was placed on the quantity of products to be produced, with a verticalized administrative training; production managers were only interested in producing, regardless of customer demand, that the products became obsolete, which made the entities unprofitable. The fall of the socialist camp plunges the country into a suffocating economic situation and this, together with the characteristics of the environment, makes it necessary to change the approach in the management of our companies: moving from the philosophy of selling products to the customer to that of satisfying the needs of the customer. client, this is a service philosophy, moving from a Productivity model to a Competitiveness model with administrative training by objectives,participatory, where the most important thing is to give a quick response to the client.
Multi-criteria decisions.
What is Decision Analysis? It can be defined from different edges, a technical definition is: »a philosophy articulated by a set of logical axioms and a procedural methodology, to analyze the inherent complexity of the problems”.
What is a decision problem? It is the selection of an action or alternative within a set of possible actions, which produces the best result under certain optimization criteria.
Decision-making is considered as the creative act of the choice, from a set of possible decisions, in which quantitative factors are combined with the heuristic capacities of the men who make decisions *
Therefore, for a business decision-making problem to exist and to allow understanding the different phases of the decision process that is proposed for the distribution route design problem, the following elements must be present that characterize it:
- a decision-maker or decision-making unit formed by a set of individuals interested in the problem, existence of at least two alternatives or possible decisions x Î X, and is of interest:
- select one (or several) (the best or the best), accept those that seem good and reject those that seem bad, the rank of all according to an order (ordering),
- a system of relationships that allow each alternative to be assigned a result. These results z Î Z are defined by certain measures (attributes), a set of input information requirements that will be obtained from the decision maker, and this implies an appropriate methodology, validation of the procedure that refers to the establishment of tests or experimental verifications that allow conclude that the proposed procedure meets the established purposes.
A necessary condition to be faced with a multi-criteria decision problem is the presence of more than one criterion; the sufficient condition is that the criteria are in conflict. Therefore, a problem can be considered as a multi-criteria problem if and only if there are at least two conflicting criteria and there are at least two alternative solutions.
The criteria are said to be strictly in conflict, which means that the increase in the satisfaction of one implies the decrease of the satisfaction of the other, so the sufficient condition of the multi-criteria problem does not stipulate that the criteria are strictly in conflict.
Multi-criteria decision-making has developed its own personality that uses a specific terminology that includes new concepts, it should be noted that some of the concepts to be introduced have the same semantic meaning and one or the other will be used depending on the theoretical context in the that are used, they are defined below:
- Alternatives: Possible solutions or actions to be taken by the decision maker * or decision-making unit Attributes: Characteristic used to describe each of the available alternatives can be quantitative (objective) or qualitative (subjective), each alternative can be characterized by a number of attributes (chosen by the decision maker) Objectives: Aspirations that indicate directions of improvement of the selected attributes, it is associated with the wishes and preferences of the decision maker Goal: Aspirations that specify levels of desires of the attributes. Criteria: General term that encompasses the concepts de: attributes, objectives and goals that are considered relevant in a decision problem.
In decision analysis techniques, the terms: multi-criteria, multi-objective, multi-attribute are used to describe decision problems with more than one measure of effectiveness, appearing indistinctly with one name or another, there being no universal definition of these terms, it has been accepted the definition of Multiple Criteria Decision Maker (MCDM) which according to the definition of several authors is the term under which all the methods that are based on multiple attributes or objectives are grouped, therefore it is divided into two aspects: multi-attribute decisions (MADM) which are used to select "the best alternative" within an explicit set of them; and multiobjective optimization (MODM) are related to those problems in which the set of alternatives is large and not predetermined,It is used to design the best alternative considering the interaction with the restrictions, they solve situations of different nature and content.
Multiple Objectives (MODM) is related to those problems in which the set of alternatives is large and not predetermined, they are used to design the »best» alternative considering the interaction with the restrictions, the solution of these problems is approached through the classical techniques of optimization.
Multiple Attributes (MADM) is used to select "the best alternative" within an explicit set of them, the final decision is made with the help of the comparison of the attributes.
As previously stated, there are two aspects of the multi-criteria problem, they solve situations of different nature and content, which is shown in the table below:
| Appearance | MADM | MODM |
| Criterion defined by | Attributes | goals |
| goals | Implicit | Explicit |
| Attributes | Explicit | Implicit |
| Restrictions | Inactive | Active |
| Alternatives | Finite number (discrete) | Infinite (continuous) |
| Use | Selection | Design |
Multicriteria problems are said to be poorly defined mathematically, since the fulfillment of an attribute causes an alternative to be the best and the worst under the fulfillment of another of the considered attributes (conflicting attributes), it is also said that they are defined when they have been established the alternatives and attributes for your solution, then the selection process begins. If the consequences of the selection of a certain alternative or course of action are defined by the decision maker a priori, it is said that the multi-criteria decision problem is under certainty.
Formulation of the multi-criteria problem
Let us consider a finite set of potential actions
A = {a i / i = 1,…., M}
each of which is supposed to be identified although not exactly and fully known in all its qualitative and quantitative consequences. It is admitted that these consequences can be analyzed by means of a family of criteria consisting of f = {d j } j = 1,….., n where g j (a i) characterizes the evaluation made with greater or lesser precision or subjectivism from one to i with criterion j. The problem consists of determining one of the three problems to solve by an aggregate model:
- select one (or several) stocks from A (the best or best) accept stocks that look good and reject those that look bad with a complementary analysis for the others the rank of all stocks according to an order
Single vs. multi-criteria decisions
The application of decision analysis techniques, specifically multi-criteria techniques, could enrich the solution of the problem and allow the management to make decisions that guarantee to increase the efficiency of the company. This also encourages the application of more flexible methods due to the advantages that these techniques present with respect to the moniterials, an aspect that is shown in the table below, which will result in obtaining better compromise solutions between conflicting objectives.
| Appearance | Monocriterion | Multi-criteria |
| Criteria | Unique | At least 2 |
| Solution | Optimal | Commitment |
| Decider preferences | It is taken into account in the objective function | It is considered in the solution of the problem |
| Paradigm | Traditional | Multi-criteria |
| problems | Technological | Economic and technological |
| Decision maker's wishes | A criterion | Conflicting criteria |
| Weakness | Deviates considerably from real decision-making issues | |
| Strength | Greater precision in real decision-making problems. |
- Theory of value.
Introduction
If we define MADM as a decision aid, helping the DM to identify the best alternative that maximizes their satisfaction with more than one attribute, we will find that many methods have been developed since the 1950s to solve this problem, despite rapid progress. of MADM has required the development of a large number of methods.
To solve a multi-criteria decision problem in discrete spaces, various mathematical techniques have been developed, a classification of multi-attribute methods according to the input and output information established by the decision maker, which can be seen in the following scheme:
Classification Of Multi-attribute Methods
Normative school (developed primarily by the Americans and the English): It is based on prescribing norms in the way that the DM should think systematically. It has a mathematical elegance given by the modeling of the problem, the set of defined axioms, etc., it uses rationality as a model.
Descriptive school (developed by Europeans (French, Dutch and Belgians): Renounce the idea of the rational, try to make a reflection of the way the DM makes decisions, it also has a mathematical formulation but less impressive than the normative school.
DOMINANCE concept
In order to reduce the set of alternatives to be evaluated in order to make the proposed procedure faster and more efficient, it is necessary to eliminate those alternatives that due to their characteristics will not be part of the solution set due to the poor performance of the indicators obtained, these being the alternatives that receive the name of dominated alternatives.
It is said that one alternative dominates the other if in at least one of the criteria it is better than the other and in the others it is at least the same, which is equivalent to saying that the best alternative is not dominated, a concept that coincides with the efficient solution or Pareto Optimal solution. In other words, you cannot find another alternative that is better or equal in all the criteria and strictly better in at least one of them.
The mathematical definition of them is:
It is said that A is a non-dominated solution if it does not exist in A such that:
for some j
and
for i ¹ j
Weighted Sum Value Function.
The normative school has established that there is an ordinal value function and a measurable or cardinal value function, the difference between them being that the ordinal value function expresses an order as its name indicates according to the preferences of the DM, but not expresses the intensity of these preferences, which is considered in the cardinal value function.
It is defined as a function of value in the set A of the alternatives as that which:
U (a)> U (b) Û a P b for all (a, b) Î A
U (a) = U (b) Û a I b for all (a, b) Î A
this means that:
If the value function of "a" is greater than the value function "b", it is said that "alternative a is preferred to alternative b" and if its value functions are equal this is equivalent to saying that "the alternative a is indifferent to b, there is no preference when having to select between the two.
An ordinal value function, specifically the weighted sum function, which is the most elementary form of a value function, but which allows us to order the set of customers, it is calculated through the following expression:
where:
U i: value function of customer i.
W i: weight or relative importance of criterion j.
f ij: value of criterion j for customer i.
m: number of criteria.
For the construction of the value function, it is necessary to subject the DM to long work sessions to establish its form and its preferences, being essential the demonstration of a set of axioms and the fulfillment of certain conditions established by this school. The normative school is built on the basis that for any pair of alternatives (a, b) only the following preference relations are possible: a> b, b> a or a I b, that is, the alternatives must be comparable In the model that is created it is assumed that the DM solves the indifference equation, and this collides with the reality that the DM declares indifference according to thresholds that exist for each DM; there are some that have wide thresholds and others narrow,These arguments are only inconvenient for the use of these techniques.
How to determine the importance or weight of the criteria?
The relative weights or importance of the criteria to be considered will be determined, through expert methods or considering the criteria of a single decision maker. This is a really important step in the decision-making process. To determine these, it is proposed to use a procedure which allows integrating the interests of each decision-maker in a group model as follows:
where:
n: number of judges (decision makers)
m: number of criteria
r lj: vote for criterion j issued by judge l (decision maker l)
W lj: weight of criterion j issued by judge l (decision maker l)
W j: weight of criterion j
The values of the weights must meet the following conditions:
and
the expression W j > W k implies that criterion j is more important than criterion k and the expression W j = W k indicates that both criteria are equally important. More than one criterion can have the same weight, the zero value for some W l indicates the unimportance of the criterion, while the highest value indicates the maximum importance for that criterion.
- Descriptive techniques for decision making.
The philosophy of ELECTRE methods.
In the previous section we studied some methods belonging to the normative school of decision, in this we will study methods belonging to the descriptive school. Among the most important methods of this school are the methods based on superiority relationships (outranking) *** Among these methods the most relevant are the so-called ELECTRE methods considered a philosophy since it implements the concept of relationship »outranking» or superiority. To develop them, it introduces four fundamental relationships:
- Indifference (I): (a I b) there are clear and positive reasons to consider that the alternatives are equivalent Strict preference (P): (a P b) there are clear and positive reasons to justify that one of the two alternatives is significantly preferred Weak preference (Q): (a Q b) one of the two alternatives is not strictly preferred to the other, but it is impossible to say that they are indifferent, hence the preference is weak of one with respect to the other Incomparable (R): (a R b) the alternatives are incomparable in the sense that none of the three previous situations predominates.
This school also admits the existence of indifference thresholds * and preference thresholds **.
Here is an example that illustrates the use of thresholds:
Suppose you want to buy a dish detergent and when you get to the supermarket you find that there are 3 types of detergent: A, B and C, which have the following characteristics:
| Detergent | Price | Quality |
| TO | $ 1.00 | Good |
| B | $ 1.10 | Very good |
| C | $ 1.20 | Excellent |
Suppose that for the decider differences of $ 0.10 in the cost is not significant and that the cost for him is more relevant than the quality, since he says that the detergent washes the dishes anyway, then: the detergent C is preferred to the B and prefers the B to A.
However, if A is compared to C the difference in price is significant, then the decider prefers detergent A to C, however C is of higher quality, so the decider feels unable to decide which detergent to buy.
Next we will define the relationships previously exposed:
Indifference (a I b)
a is indifferent to b when
g j (a) = g j (b) for all j ¹ k even if g k (a) ¹ g k (b), that is, if there are clear and sufficient reasons to argue the equivalence between two alternatives for criterion j.
Strict Preference (a P b)
a is strictly preferred to b when
g j (a) = g j (b) for j ¹ k only if the difference
g k (a) - g k (b) is significant enough, that is, there are clear and sufficient reasons to argue that a is significantly higher than b.
Weak Preference (a Q b)
a is weakly preferred to b if b is not strictly preferred to aa, but it is impossible to say that a is strictly preferred or indifferent to b, that is, none of the above situations predominate.
Incomparability (a R b)
a is incomparable with b, when none of the previous situations prevails.
Let's look at a simple example:
Assume the following ice cream flavors:
Chocolate
Strawberry
Nut
Mantecado
Vanilla
Chocolate Chip
Two decision-makers have expressed their preferences on a scale between 0 - 10 points, giving the highest score to the flavor they like the most, which are shown in the following table:
| Flavors | rose | Eugene |
| Chocolate | 10 | 9 |
| Strawberry | 5 | 9 |
| Nut | 10 | 8 |
| Mantecado | two | 6 |
| Vanilla | one | two |
| Chocolate Chip | 9 | 10 |
For Rosa, chocolate and walnut are indifferent, while chocolate is preferred to chocolate chip and strawberry preferred to ice cream. However, for Eugenio, chocolate and strawberry are indifferent, their preferred flavor being chocolate chip, the ice cream being the preferred vanilla.
Let's return to the subject of the philosophy of the ELECTRE, the philosophy of the ELECTRE is structured in two phases: construction of the outranking relationship (superiority) and the exploitation of the outranking relationship. The construction of the outranking (S) relationship is established through the concordance test and the veto test (not disagreement), for the statement a to exceed b to be true, both tests must be successfully passed. Through the concordance test, a majority rule is sought to make the unanimity more flexible, while the veto test means that statement a is not vetoed over b. We will say that a S b if taking into account the set of criteria we have sufficient arguments to consider the statement a true to be at least as good as b. Next we will define outranking relationship, agreement, disagreement and veto.
Outranking relationship
Let us be a set of alternatives characterized by a family F of criteria g 1, g 2,….g f, to each criterion g j it is possible to assign an outranking relation S j. By definition, S j is a binary relation to S j b if the values g j (a) and g j (b) give an argument strong enough to consider that a P j b or a Q j b or a I j b in the DM preferences model. This means that "the alternative a is at least as good as b" from the point of view of criterion j and S j b is denoted
Indifference thresholds are allowed:
a I b yes and only if -g j (a) - g j (b) - £ q j
a P b yes and only if g j (a) ³ g j (b) + p j
a Q b yes and only if q j <g j (a) - g j (b) <p j
where:
q j: threshold of indifference
p j: strict preference threshold
so:
a S j b yes and only if g j (a) ³ g j (b) - q j
We will say that a S b if taking into account the set of criteria we have sufficiently strong arguments to consider true the statement »a is at least as good as b»
I feel that:
a S b and b S a Û a I b
a S b and b nS a Û a P b
a nS b and b nS a Û a R b
It can be proved that if a Sj b for all j Î F Þ a S b
In addition to these definitions, it is necessary to define the concepts of agreement and discordance, the foundations on which the Electre methods are based:
Definition of Concordance
The j - th criterion agrees with the statement
a S b yes and only if a S j b.
The subset that meets the above is called the coalition of agreement and is denoted by:
C (a S b) = {j / a S j b}
Definition of Disagreement
Criterion j - th is in disagreement with the statement
a S b if and only if b P j a exceeding the strict preference level (g j (b) - g j (a)> p j)
The subset that fulfills the above is called the coalition of discordance and is denoted by:
C (b P a) = {j / b P j a}
There is another subset C (b Q a), so that:
F = C (a S b) UC (b P a) UC (b Q a)
Let W j be the weight of criterion j
The concordance index C (a, b) characterizes the strength of the positive arguments to validate the statement a S b. The main force comes from C (a S b) = C 1 (a, b), but it is necessary to look for a certain contribution from C (b Q a) expressing this as C 2 (a, b), so:
C (a, b) = C 1 (a, b) + C 2 (a, b)
where:
C 1 (a, b) =
j Î C (a S b)
C 2 (a, b) =
jÎ C (b Q a)
Q j is a function that for its simplicity can be taken as:
p j + q j (a) - q j (b)
Q j = ¾¾¾¾¾¾¾¾¾
p j - q j
Note that if:
g j (b) - g j (a) = p j, Q j = 0
g j (b) - g j (a) = q j, Q j = 1
0 £ C (a, b) £ 1
C (a, b) = 0 if C (b P a)
C (a, b) = 1 if C (a S b)
Veto effect and unconformity index
Axiom: For each criterion g j there is a threshold v j > p j such that
g j (b) - g j (a)> v j is incompatible with the statement a S b, even when C (a, b) approaches 1 even though all the others are in favor of a S b, this does not compensate the loss in this criterion.
It is logical to think that the veto threshold decreases as C (a, b) is lower.
As previously stated, the ELECTRE methods are structured in two phases:
- Establishment of the outranking relationship. Exploitation of this outranking relationship.
The structure of phase 1 can be represented as:
The way of establishing the relationship of superiority has conditioned the emergence of different methods within the philosophy of the Electre of which its versions I, II, III, IV, IS and TRI are known, all assume the existence of two phases. The table shows the characteristics of the different Electre methods developed.
| ELECTRE version | Criterion type | Need weight | Blurred set concept | Type of problem | Additional preference information |
| I | Simple | Yes | Not | Selection | Weights, level of agreement and level of disagreement |
| II | Simple | Yes | Something | Ordering | Weights, level of agreement and level of disagreement |
| III | Pseudo | Yes | Yes | Ordering | Pesos |
| IV | Pseudo | Not | Not | Ordering | |
| IS | Pseudo | Yes | Not | Selection | Pesos |
Table. Characteristics of the different versions of the ELECTRE Method.
How to select the Electre method to use?
To answer that question and therefore select the most appropriate method within a context of decision support, we must consider what type of problem we want to solve, which can be:
Problem a: isolate the smallest set A or Ì A for which the elimination of all A / A actions can be justified or, that is, that the best ones are in that subset.
Problem b: assigning each action to a previously determined category, the classification problem and as such is intercepted with Artificial Intelligence.
Problem g: build the richest possible preorder on a subset A or on those actions of A that seem to be the most satisfactory.
If the problem to solve is a, two methods can be used, Electre I and Electre Tri, Electre I will be selected only if you want to work with a very simple method, since p j = q j = 0 for all j, it is say does not work with thresholds.
If the problem is b, no method developed so far can be used.
If the problem to solve is g, three methods can be selected: Electre II, III and IV; Electre II will only be selected if it is required to work with a very simple method since, like the Electre I, it does not work with thresholds; Electre IV is suitable only if there is good reason to refute the introduction of the importance of the criteria, because the algorithm does not model the importance of the criteria.
The Electre IV method has been designed to solve two situations frequently encountered in real situations: imprecision and uncertainty of the evaluation of the actions on the criteria and the absence of weight of the criteria.
A satisfactory answer to the first difficulty can be achieved with the introduction of selected indifference, preference and veto thresholds for each criterion, as well as by the definition of a pseudo - criterion.
The second situation requires a proper introduction to the concept of outranking.
The Electre IV method distinguishes two plausible levels, strong outranking Sf not subject to discussion and weak outranking Sd, a little more questionable.
The philosophy of the Electre methods contribute a set of positive aspects to decision-making, which improve the insufficiencies of the normative school of decision, which has demonstrated its effectiveness in practice and which provides a heuristic procedure that allows obtaining favorable results. to important multi-attribute decision problems.
The table shows the positive and negative aspects of the ELECTRE philosophy:
| Positive aspects | Negative Aspects |
| 1. Recognition of the superiority relationship (S) as a weakening of strict preference. | 1. Insufficient veto modeling is not influenced by the status of other criteria. |
| 2. Admission of the majority veto as a mechanism to obtain outranking (S). | 2. Role of weights in the concordance index. |
| 3. Admission of the veto. | 3. When using the majority vote rule, pseudo - criteria are not considered. |
| 4. Recognition of incomparability (R). | 4. Does not consider the intensity of preferences. |
| 5. Recognition of the existence of discriminatory thresholds. | 5. There is no single way to order the set of alternatives, or universally good to exploit the relationship of superiority
(S). |
| 6. Does not require great effort from the decision maker. |
Table Main positive and negative aspects of the Electre philosophy.
Despite the deficiencies shown, the Electres are techniques that help better decision-making with a less elegant mathematical foundation, but no less efficient.
Electre II.
In the previous section, it was stated that ELECTRE are a philosophy based on the establishment of a relationship of superiority (outrank), for which it uses two phases:
- Establishment of the superiority relationship Exploitation of the superiority relationship.
These two phases are shown in this section for the case of the ELECTRE II, which has been selected for its simplicity and practical possibility of use.
Remember that:
Given two alternatives (a, b) Î A, characterized by a set J of N attributes, the following notation will be used:
- a P j b means that a is strictly preferred to ab with respect to the attribute ja Q j b means that a is weakly preferred ab with respect to the attribute ja I j b means that a is indifferent ab with respect to the attribute ja R j b means that a is incomparable ab with respect to attribute j.
Phase I. Establishment of the outranking relationship
In order to successfully pass this phase within the ELECTRE philosophies, it is necessary to fulfill two tests:
- Concordance test, veto test.
In the agreement test, a majority rule is sought to make unanimity more flexible, establishing within this test the fulfillment of two tests:
- Simple majority Consensus
To carry out the concordance test, four sets are defined, which are shown below:
Let J be the set of all the criteria to be evaluated j 1, j 2,…, j n., That is, J = {j i } i =
= {j J: a P j b}
= {j J: a I j b}
= {j J: b P j a}
where:
: Set of criteria where alternative a is preferred to b.
: Set of criteria where alternative a is indifferent to b.
: Set of criteria where alternative b is preferred to a.
The indices I 1 (consensus) and I 2 (simple majority) are calculated, such as:
The concordance test passes successfully if:
I 1 ³ ce I 2 ³ 1
where:
c: concordance index, can be considered as 3/4 or 2/3 (Ostanello, 1984), in this case it will be considered 2/3 to reduce incomparabilities, which means that alternative a slightly exceeds b.
Let's see through an example how the concordance test is carried out in phase I:
| Alternatives | Criterion 1 | Criterion 2 | Criterion 3 | Criterion 4 | Criterion 5 |
| TO | AND | MB | AND | R | B |
| B | MB | B | R | MB | B |
| C | MB | B | AND | R | B |
| D | B | AND | MB | R | B |
| Pesos | 10 | 8 | 10 | 9 | 5 |
Note: Criterion 5 has the same evaluation for all the alternatives so it can be eliminated as it does not provide relevant information for decision making.
To make the selection of the best alternative, we must compare all the alternatives against all, we will now explain how we will carry out the concordance test of phase I.
Comparing A with B
Definition of sets
J = {1, 2, 3,4}
= {1, 2,3}
= {4}
= f
³ c
= (10 + 8 + 10) + 9/37 ³ 2/3 or 3/4
= 1 Þ A s B
³ 1 Þ A s B
Comparing A with C
Definition of sets
J = {1, 2, 3,4}
= {1,2}
= {3 4}
= f
³ c
= (10 + 8) + (10 + 9) / 37 ³ 2/3 or 3/4
= 1 Þ A s C
³ 1 Þ A s C
Comparing A with D
Definition of sets
J = {1, 2, 3,4}
= {1, 3}
= {4}
= {2}
³ c
= (10 + 10) + 9/37 ³ 2/3 or 3/4
= 1 Þ A s B
³ 1 Þ A s B
Veto test. This test tries to deny the previous statement
There is a veto on the statement a S b, yes:
J - ¹ Æ y
where:
{j>}
W med: average weight of the criteria.
The veto test will only be passed if alternative a outperforms alternative b (a S b) in the concordance test.
Phase 2. Exploitation of the outranking relationship.
To solve this phase, the concepts of strength and weakness are used.
Let F (a): be the force of the alternative a, the number of a 'Î A such that a S a'.
F (a) = card * {a 'Î A / a S a'}
Let D (a): the weakness of the alternative a, the number of a Î A such that a 'S a.
D (a) = card {a 'Î A / a' S a}
Let I (a): be the quality index of alternative a, the difference between its strength and its weakness.
I (a) = F (a) - D (a)
Therefore, for each alternative to Î A, calculate, F (a), D (a), and I (a).
The ordering of the alternatives is performed using the quality index of the alternatives and their superiority ratio, as shown below.
If: a S byb nS ay I (a)> I (b) Þ a P b
I (a) = I (b) Þ a P b
I (a) <I (b) Þ *
a S b and b S ay I (a)> I (b) Þ a P b
I (a) = I (b) Þ a I b
I (a) <I (b) Þ b P a
a nS byb nS ay I (a)> I (b) Þ a P b
I (a) <I (b) Þ b P a
I (a) = I (b) Þ *
The conditions marked with * mean that there is not enough information to be able to order the alternatives that meet them, which means that they are incomparable, and a method must be sought to eliminate or minimize these incompatibilities.
Lexicographic method
This is a method to solve problems in which the ordinal preferences of given attributes are known (Tabucanon, 1988), (Romero, 1997). This method is recommended as long as you have information regarding the importance of each of the attributes that make up the set of criteria to be considered, not being necessary to express it through a weight or weighting, but rather to be able to perform a ordering of attributes according to importance.
The algorithm works in the following way:
- Identify the subscript not only of the attribute vector but also the priority of the attribute, that is, it corresponds to the most important attribute, the second in importance and so on until it will be the worst of the attributes. Select as the best alternative such that:
= {A i: max. X i1 }, i = 1,2,… m
If cardinal de = 1 then this is the preferred alternative
If cardinal> 1 then there are multiple maximal alternatives so the procedure must be followed until it occurs:
$ / card = 1 Þ that the alternative will be preferred.
All the m attributes have been considered, in which case, if the set has more than one element, it is considered that the alternatives will be equivalent.
Bibliography
- Barba Romero S, Pomerol JC (1997): Decisiones multicriteio Theoretical foundations and practical use, University of Alcalá, Spain. Charmes & Cooper, 1961, Management Models and Industrial of luinear programming, Jhon Wiley and Sons. Diez de Castro JA et al (1997): Decision aid: A new management tool. Monograph of the University of Santiago de Compostela, Spain Koopmans TC, 1951, Activity Analysis of Production and Allocation, Cowles Conmission Monograph, n˚ 13, Jhon Wiley, New York. Roger M, Bruen, M (1998): A new system for weighting environmental criteria for use within Electre III. European Journal of Operational Research. Romero C. (1993): Multicriteria decision theory: Concepts, techniques and applications. Alianza Editorial SA Madrid.Romero, C. (1997): Analysis of multi-criteria decisions. Madrid, Roy B,Vander Pooten D (1995): The european school of MCDA: A historical review. Proceeding of the XIV Euro Working Group Conference OR Toward intelligent decision support, Jerusalem. Roy B. (1984): The outranking approach and the foundation of Electre methods. in Reading in Multiple Criteria Decision Aid. Editores Bana e Costa. Roy B. (1990): The outranking approach and the Foundations of Electre methods. Berlin.Roy, B. (1996): Multicriteria methodology for Decision Aiding, Kluwer Academic Publisher, Dordrecht- Boston- LondonSaaty T (1996a): Marketing applications of the analytical hierrachy process. Management Science Magazine, Saaty T (1997): Decision-making for leaders. The hierarchical analytical process. Decision making in a complex world. RWS Publications USA, Tabucanon M. (1988): Multiple Criteria Decision Making in Industry,Studies in Production and Engineering Economics, Elserver, Amsterdam - Oxford - New York, Tokyo.
* Dynamic models of decision-making processes.
* from English Decision Maker (DM)
*** relationship developed by the descriptive school of the decision in 1970, its founder being B. Roy.
* indifference thresholds: a I b yes and only yes - g j (a) - g j (b) - £ q j where q j is the indifference threshold.
** preference thresholds: a P b yes and only yes g j (a) ³ g j (b) + p j or a Q b yes and only yes q j <g j (a) - g j (b) < p j where p j is the preference threshold.
card (cardinal) is said of the integers that are used to count things PAL - LAS Dictionary.
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