An AI for Robust MCDM Ranking in a Large Number of Criteria

Abstract

An artificial intelligence procedure (AIP) model is presented to generate a robust multicriteria ranking for decision-making in problems with a large number of criteria, which jointly uses genetic algorithms, the OWA method, and SMAA. The main contribution of this AIP model is to reduce the cognitive effort of the decision-maker in determining the weights of the criteria involved in the decision problem, since, with the proposed model, these weights are inferred according to the decision-maker’s preferences, which can result in a ranking of the alternatives called a robust ranking. Robustness provides the expert with the certainty that, despite variations in their preferences and, therefore, in the weights inferred by the model, the presented robust ranking will not undergo significant changes, giving stability to the decision-making solution in a stable environment and reflecting the decision-maker’s preferences. This is one of the main contributions of applying the AIP model, unlike other models that do not provide a robust solution, as described in the application of the proposed AIP model to a multicriteria decision problem with many criteria, such as the state competitiveness analysis problem presented by the Mexican Institute for Competitiveness (IMCO).

1. Introduction

The decision-making process requires a cognitive effort from the decision-maker, who must determine the weight of each criterion involved in a problem, which is then used to evaluate different alternatives.

Multicriteria decision-making methods (MCDM) are tools and techniques that help address complex, real-world problems by considering multiple, sometimes conflicting, criteria. Qualitative and quantitative factors are also analyzed to provide decision-makers with elements for optimal decision-making [1]. Various MCDM methods and approaches can be applied to decision problems, which are adapted according to the characteristics and context of the problem.

The process of translating the decision-maker’s preferences into criteria is not a simple task for the decision-maker and the analyst; it involves a whole process of conceptualizing the problem, and this represents the need to apply different methods to achieve it, even considering elements of uncertainty, imprecision, and/or missing information to carry out this activity, which is an aspect that cannot be left aside when modeling real problems [2,3,4].

There are authors such as Hellwig [5], Bouyssou [6], and Roy [7], among others, who, for decades, have analyzed the weighting of criteria, where the decision-maker and the analyst translate preferences into linguistic terms to assign certain weights to each criterion involved in the problem, all of which also consider aspects of uncertainty or lack of information, which can be decisive when evaluating a decision-making problem.

Once the elements necessary to model the problem have been established, it is essential to consider the number of criteria involved, since a large number of them represents an increase in the complexity of the case to be analyzed, as this requires a cognitive effort for the decision-maker, who must determine the level of importance and weight of each criterion. With multiple criteria involved in the problem, it is possible to identify redundancy, conflict, and lack of integrity, which represent a challenge for the decision-maker when determining a weight related to its importance [8].

Prioritizing, or assigning a specific importance value to, criteria is key to evaluating different alternatives. This prioritization is used to assess the weight of criteria. It is an essential activity involving decision-makers that allows the analysis of this weighting to be aligned with the objectives sought in a decision-making problem [9].

The analysis of robustness is derived from the capacity of a proposed solution to remain unaffected by minor but deliberate variations in the weight parameters of the criteria, thereby providing reliability or certainty to the presented solution [10,11].

Robustness is related to the parameters involved in preference modeling, whose values may change over time, emphasizing the ability to address complex and dynamic situations appropriately and effectively, without significant changes in the outcome [12].

Therefore, the recommendation presented to the decision-maker must be robust; that is, despite changes in specific parameter values, the proposed decision is consistent with the decision-maker’s preferences and does not undergo significant changes [13,14].

One way to address the problem of considering the decision-maker’s preferences and generating robust decisions is to infer a large set of vector weights that account for both the translation of the decision-maker’s preferences into the importance of criteria and the analysis of robust parameters, thereby proposing a robust solution. It is a two-stage process: first, find compatible vector weights that align with decision-makers’ preferences, and then conduct a descriptive analysis to identify robust parameters [15].

Initially, inferring the weights of the criteria that reflect the decision-maker’s preferences is a complex task in the decision-making process. On the other hand, the aim is to generate a solution proposal for the decision-maker that is considered a robust ranking solution, which, based on a robustness analysis, makes it possible to determine that the result does not present an alteration index according to the variation in the weight of the criteria used in the evaluation of the alternatives, according to the descriptive measures of the proposed model [16].

Artificial Intelligence (AI) is a comprehensive tool that has established itself in various areas of knowledge, with applications in decision-making related to prediction, classification, and optimization, reducing simulation and analysis times, and facilitating the evaluation of results [17].

The application of AI can be identified in various fields of study, such as healthcare, finance, agriculture, innovation, competitiveness, business, and supplier selection [9,18].

The main contribution of the study is the development of an artificial intelligence procedure that infers the weights of decision criteria and generates a robust ranking solution in the presence of many criteria. The model is composed of the following two elements:

• A genetic algorithm that infers weight vectors of decision criteria regarding the decision-maker’s preference in the relative importance of criteria.
• A robust analysis of the weights and ranking to generate a robust solution.

The rest of the document is organized as follows: the proposed Artificial Intelligence Procedure model to generate a robust ranking is presented in Section 2. The application of the proposed model to a multicriteria decision-making problem is shown in Section 3. The comparison and analysis of results from applying the model are shown in Section 4. Finally, some conclusions are drawn in Section 5.

2. Artificial Intelligence Procedure to Generate a Robust Ranking

The development of the Artificial Intelligence Procedure (AIP) generates a robust multicriteria decision-making (MCDM) ranking in the presence of a large number of criteria. The robust ranking solution is constructed from the generation of thousands of rankings using the inferred weights of decision criteria in the presence of too many criteria. The AIP implements four stages that interact with each other, allowing the solution of decision-making problems and generating a robust ranking that conserves the decision-maker’s preferences. The general design of the AIP is illustrated in Figure 1, which shows the interaction between the decision-maker and the analyst, the definition of the input data, the model, and the robust ranking as output information.

2.1. Input Data

As part of the user interaction, in which two actors—the decision-maker and the analyst—carry out in the decision-making process, the former is familiar with the problem analyzed, including the alternatives to be evaluated and the criteria involved, and the analyst is the one who defines the parameters required by the AIP model. This input information is stored in a text file and read by the same programming code.

The input data is divided into two aspects:

• Characteristics of the problem to be solved. This information refers to the problem to be analyzed and is provided by the decision-maker, who provides valuable data on the alternatives and criteria involved. This information is vital for the application of the model and for the analysis of its results.
  • Definition of the number of criteria involved and alternatives to be evaluated.
  • Ordering the criteria according to the decision-maker’s preferences, with the least important criterion at the beginning of the order and the most important being the last criterion in the order.
  • Performance of the alternatives in each of the defined criteria.
• Operating data of the genetic algorithm. These parameters are provided by the problem analyst, who has specific knowledge of the type of data required by the model. This data is crucial for initially defining the number of generations and the probabilities of mutation and crossover for the genetic algorithm that the model must generate concerning the set of weights that will subsequently be required for the resulting robust ranking.
  • Population size
  • Number of generations
  • Probability of crossover
  • Probability of mutation

Once the input information resulting from the joint work between the decision-maker and the analyst has been obtained, including the criteria involved in the problem, their ordering and the input parameters of the genetic algorithm, the Artificial Intelligence Procedure (AIP) model, begins the first of the stages corresponding to the inference of the parameters and the subsequent application of the genetic algorithm for the set of weights of the criteria, all aligned with the preferences of the decision-maker, before subsequently continuing with the following stages of the proposed model.

2.2. Parameter Inference Model

Among the input data requested from the user is the ordering of the criteria according to their importance, which allows for the inference of the weights of the criteria aligned with the ordering defined by the decision-maker. It should be noted that, as part of the restrictions of the implemented algorithm, the sum of the set of weights must be equal to 1, with the following definition:

Criteria weight vector:

$$w=\left(w_1,w_2,\dots ,w_n\right),w_j\ge 0,\sum _{j=1}^n{w}_j=1$$

(1)

Regarding the criteria, the process of comparing them is the basis of the implemented model and ultimately is how the evaluation of alternatives is carried out.

In this sense, based on the DM’s preference information and after defining the ordering of criteria from most important to least important, a criterion g_j defined by the decision-maker is ordered in the j-th position if it is preferred over another criterion that is in the i-th position, such that g_j ≻ g_i if j > i, where g_j ∈ G = {g₁, g₂, …, g_n}, j = {1, 2, …, n}.

It should be noted that the weights of the criteria must also correspond to the expert’s stated preferences, considering that, under the previous assumption, the weight w_j is greater than the weight of w_i.

Now, each criterion corresponding to the decision options is composed of weight parameters (w) of the entire set of criteria considered by the expert, where each criterion is represented as w₁, w₂, …, w_n.

It should be noted that, depending on whether the decision-maker is willing to provide information about his or her preferences, and that this is as established in the research objectives, the following models are proposed, all of which have the purpose of satisfying the following specific function: Max |w|.

The decision-maker’s preferences can generally be summarized as the information the expert is willing to share with the analyst regarding the problem at hand, to define the criteria involved, as well as the order of importance of each. For example, assuming a problem with five criteria (a, b, c, d, and e) (g₁, g₂, g₃, g₄, and g₅), the decision-maker can define that criterion a is the most important of the five. Then c is more important than criterion b, b is more important than d, and e is the least important. It is easily represented as g₁ $\succ$ g₃ $\succ$ g₂ $\succ$ g₄ $\succ$ g₅. The inference procedure should define the value weights that represent their preferences, where w₁ > w₃ > w₂ > w₄ > w₅. Figure 2 shows, in general terms, the two cases addressed by the model according to the profile of the decision-maker concerning their willingness to provide preference information.

Depending on the preference information that the decision-maker is willing to provide for the decision-making process, it is possible to identify two cases, as described below.

2.2.1. Case 1: Without Information on the Decision-Maker’s Preferences

In this case, the decision-maker is unwilling to provide information about their preferences regarding the proposed criteria; that is, they do not define which criterion is more important than any other. For this case, we intend to apply the proposed model, which generates the criteria weight parameters in a general manner while complying with its basic restrictions. The model is defined as follows:

$$\begin{gathered} Max\left|w\right| \\ \text{s.t.} \\ {w}_1+w_2+\dots +w_n=1 \\ {w}_i\le \sum _{\begin{array}{c}i=1\\ i\ne j\end{array}}^n{w}_j,i=1,2,\dots ,n \\ {w}_i\ge 0 \end{gathered}$$

(2)

where w is the weight of each of the criteria, and the sum of all the weights of the criteria involved is equal to 1; this is a restriction of the model.

w_i is the weight of criterion i, fulfilling the restriction of being a number greater than or equal to 0 and less than or equal to the weight of criterion j.

In general, once this case has been addressed and a proposal for ordering the criteria inferred by the model has been obtained, the aim is to serve as an approach with the expert to obtain his agreement to change his position and provide information about his preferences in order to subsequently re-implement the model, applying another scenario or case.

2.2.2. Case 2: Complete Information on the Decision-Maker’s Preferences

Unlike the case described in the previous section, in this scenario, the decision-maker is willing to provide complete information about their preferences; that is, the expert defines the criteria involved in the problem to be analyzed and ranks them in order of importance. He also monitors the weights generated by the model, ensuring complete agreement with the defined preferences, to confirm that the weights align with them.

As an example of this type of case, the decision-maker defines that three criteria, called a, b and c, intervene in the problem, with criterion a being more important or preferred over criteria b and c, and that, in turn, criterion c is preferred over b (a ≻ c ≻ b).

The model and restrictions to meet the preferences defined by the decision-maker are described below:

$$\begin{gathered} Max\left|w\right| \\ \text{s.t.} \\ {w}_1+w_2+\dots +w_n=1 \\ {w}_j\ge w_{i}si{g}_j\succ g_i \\ {w}_i\le \sum _{\begin{array}{c}i=1\\ i\ne j\end{array}}^n{w}_j,i=1,2,\dots ,n \\ {w}_i\ge 0 \end{gathered}$$

(3)

where w is the weight of each criterion, the sum of all the weights of the involved criteria is equal to 1, which is a restriction of the model.

w_i is the weight of criterion i, also fulfilling the restriction of being a number greater than or equal to 0. For its part, the weight of criterion j is greater than or equal to the weight of criterion i (w_j > w_i) if criterion j is more important or preferred than criterion i (g_j ≻ g_i).

This type of complete ordering of expert preferences enables the model to apply restrictions related to these preferences when generating the criteria weights, thereby ensuring that the result is consistent with the decision-maker’s preferences.

2.3. Methods for Parameter Inference and Ranking of Alternatives

2.3.1. Genetic Algorithm

As part of the inference of criteria weights, a genetic algorithm is used to determine, through a heuristic search process [19], those sets of weights that are consistent with the decision-maker’s preferences and that, in turn, satisfy the restrictions established by the different cases and described in the previous section.

The genetic algorithm procedure uses the user’s input data as a reference, specifying the population size and the number of generations. The latter sets the standard for the number of iterations of the algorithm to search for the set of weights.

Other input parameters are the probability of crossover and the probability of mutation. The former establishes the probability of creating new individuals (in this case, the set of weights) by simulating a cross between pairs of previously selected individuals, called parents, ensuring that the search for these new individuals, the results of the cross, complies with the restrictions established by the algorithm itself.

The probability of mutation, on the other hand, refers to the likelihood of a change in an individual’s gene, thereby providing an opportunity for new individuals to emerge in the population. The individuals resulting from these two procedures (crossover and mutation) build up the generations and replace the individuals selected as parents.

Regarding the individuals in the developed tool, their structure or chromosome is shaped according to the number of criteria defined in the problem (Figure 3); that is, each individual will have n genes, where n is the number of criteria.

It is important to note that the information contained in a gene is the weight determined for a particular criterion, and that, in the end, an individual is made up of a set of criterion weights and, following what was specified in the previous section, the sum of these weights must be equal to 1.

Beyond optimizing an objective function, the genetic algorithm simulates a search procedure in the space of decision variables. Therefore, it does not have an objective optimization function. Beyond that, individuals are validated by satisfying the constraints in Equations (2) or (3).

For Equation (2), a fit individual meets all three constraints: the sum of the weights, the balance of one weight relative to the others, and positive weights.

On the other hand, for Equation (3), a feasible individual meets all four constraints: the sum of the weights, the balance of one weight relative to the others, positive weights, and the value of the weight of one criterion compared to another must match the decision-maker’s order of importance preferences.

Thus, the fitness function for Equation (2) is the sum of each constraint that has been met, with three being the maximum value. The fitness function for the individual in Equation (3) is the sum of the fulfilled constraints, with four being the maximum value. In this sense, the search is guided by the fitness function corresponding to the fulfillment of constraints.

2.3.2. IOWA Operator

A multicriteria decision-making method is used to evaluate the decision alternatives. To accomplish this task, an OWA operator is applied using the vector weights generated by the genetic algorithm. That is, when obtaining the set of weights for the criteria (individuals), they must evaluate each of the alternatives to subsequently generate a ranking of these.

To generate the rankings of alternatives, the AIP model applies an extension of the OWA aggregation operator, the Induced OWA (IOWA), which takes an induced variable to reorder the criteria’s weights.

The induced variable for our model is the ordering of the criteria according to the importance the decision-maker establishes, based on their preferences.

This operator is represented in its base form as follows [20]:

$$IOWA\left(⟨u_1|a_1⟩,⟨u_2|a_2⟩,\dots ,⟨u_n|a_n⟩\right)=\sum _{i=1}^n{a}_i,w_i$$

(4)

where:

• The induced variables are represented by u₁, u₂, …, u_n;
• The values or performances of each alternative are represented by a₁, a₂, …, a_n;
• w_i is the weight of the criterion, ordered according to the induced variable u_i.

For the proposed model developed and presented here, it is understood that the criterion with the most significant importance to the decision-maker will be the one with the most significant weight. Based on this, the alternatives are evaluated to form the decision matrix, and subsequently, the summation of the overall performance of each alternative is calculated, thereby ranking them.

Once the overall performance of each alternative is obtained, the bubble sort method is applied, with a simple comparison method whose algorithm iterates through the entire list of alternatives, comparing the first alternative with the next. If the second alternative has a higher rating than the first, it takes the first’s position. The process of comparing the alternatives continues until it is verified that no further swaps are necessary, since the alternatives are ranked correctly.

2.3.3. Stochastic Multicriteria Acceptability Analysis (SMAA)

After applying the model for parameter inference using a genetic algorithm and ordering through the IOWA Operator, the SMAA method [21] and its descriptive measures are implemented to obtain a robust ranking of the evaluated alternatives.

The descriptive measures applied in this model are the acceptability index and the central weight vector. To initially calculate the acceptability index, it is necessary to obtain the rankings generated by the Iowa Operator in all generations resulting from the genetic algorithm.

With the information on the ordering of alternatives in a spreadsheet, we then count the number of times each alternative appears in each position across the entire set of rankings.

This acceptability index serves as a reference to identify the alternative(s) that appear first in the ranking, thereby selecting the weight sets from the total number of orders that resulted in that alternative appearing in that position.

After identifying the set of weights that results in an alternative appearing first, the central weight vector is calculated, which is the average weight of each criterion and is considered a robust weight vector. This central weight vector is recovered to recalculate a new decision matrix and subsequently evaluate the overall alternatives, resulting in a robust ordering or ranking.

3. Application of the AIP Model in a Multicriteria Decision-Making Problem

Once the Artificial Intelligence Procedure (AIP) has been defined, it is applied to a problem that meets the necessary characteristics, as outlined in the different cases addressed in this research, with a primary emphasis on the number of criteria involved in the decision-making process.

In this regard, we analyzed the information from the Competitiveness Index published by the Mexican Institute for Competitiveness A.C. (IMCO), particularly its 2016 edition [22], and it meets the requirement of having many criteria. In this case, the IMCO uses 100 decision criteria to analyze the state competitiveness of the 32 federal entities in Mexico, which, for our problem, correspond to the 32 alternatives to be evaluated through 100 decision criteria. Appendix A presents examples of all the criteria considered by IMCO to evaluate the 32 regions (states), according to the Reliable and objective legal system (

Table A1. IMCO’s evaluation of the 32 alternatives in the reliable and objective legal system aspect.

Alternative	Reliable and Objective Legal System
	Homicides	Kidnappings	Vehicle Theft	Crime Cost	Crime Incidence	Reported Crimes	Perceived Safety	Notary Services	Enforcing Contracts
Aguascalientes	98	99	76	51	48	45	66	32	97
Baja California	0	21	0	22	10	82	55	4	25
Baja California Sur	100	100	71	15	0	89	83	35	20
Campeche	95	96	100	60	100	85	51	75	93
Coahuila	88	92	83	73	50	100	21	74	80
Colima	88	91	72	79	52	83	56	32	91
Chiapas	94	97	93	81	92	47	47	12	75
Chihuahua	72	97	64	89	53	63	25	12	67
CDMX	94	91	63	0	43	43	25	17	10
Durango	83	92	79	58	56	47	27	18	80
Guanajuato	90	96	84	64	61	32	40	58	81
Guerrero	58	58	75	43	70	0	20	3	17
Hidalgo	96	91	84	77	66	97	40	33	55
Jalisco	91	97	81	38	64	31	39	30	59
México	90	86	37	17	53	27	0	2	67
Michoacán	80	66	71	51	82	47	16	29	59
Morelos	79	12	60	53	20	45	7	5	0
Nayarit	91	95	97	100	86	42	64	39	64
Nuevo León	92	88	89	70	79	50	30	33	54
Oaxaca	86	89	93	95	70	43	22	12	12
Puebla	96	93	93	71	63	57	44	23	42
Querétaro	96	95	59	29	60	84	84	39	38
Quintana Roo	91	95	99	53	30	58	38	30	49
San Luis Potosí	93	95	94	80	84	8	29	29	49
Sinaloa	68	91	58	76	68	53	32	31	77
Sonora	81	95	68	55	66	94	56	36	48
Tabasco	94	42	87	64	19	57	11	41	70
Tamaulipas	82	0	65	56	73	48	11	100	44
Tlaxcala	96	96	84	75	85	73	48	0	3
Veracruz	95	74	87	70	77	68	17	42	30
Yucatán	100	100	100	85	38	33	100	47	48
Zacatecas	95	85	67	61	76	39	19	15	100

) and Sustainable environmental management (

Table A2. IMCO’s evaluation of the 32 alternatives in sustainable environmental management.

Alternative	Sustainable Environmental Management
	Aquifer Exploitation	Waste Water Treatment	Water Use in Agriculture	Respiratory Diseases	Forest Competitiveness	Treated Area	Protected Natural Areas	Solid Waste Volume	Solid Waste Disposal	Energy Intensity	“Clean” Companies	FONDEN Expenditures
Aguascalientes	82	100	0	93	82	100	15	68	100	76	7	100
Baja California	1	59	0	70	70	61	57	40	92	56	64	100
Baja California Sur	41	64	1	90	80	99	33	55	84	64	4	0
Campeche	53	3	0	77	67	42	28	76	32	87	38	79
Coahuila	100	50	0	91	94	98	13	65	85	61	31	100
Colima	47	87	0	96	51	94	100	70	93	64	9	94
Chiapas	69	3	1	0	8	31	13	91	3	82	13	90
Chihuahua	98	73	1	57	89	97	6	61	80	47	67	95
CDMX	47	12	100	38	51	100	4	0	100	95	100	100
Durango	31	75	1	89	83	96	4	73	53	70	20	91
Guanajuato	60	33	0	83	57	97	6	62	87	44	15	100
Guerrero	0	36	1	64	15	73	0	83	5	0	2	8
Hidalgo	90	1	0	66	47	66	5	86	53	83	15	94
Jalisco	89	59	1	85	61	92	2	48	96	100	82	100
México	73	13	4	58	53	95	10	43	85	88	53	100
Michoacán	39	25	0	64	41	79	5	84	71	65	29	98
Morelos	79	29	2	96	31	91	21	67	87	74	18	98
Nayarit	69	72	1	60	75	84	94	79	60	76	2	100
Nuevo León	88	96	0	100	100	92	2	35	94	72	89	99
Oaxaca	64	7	1	50	0	68	5	100	0	84	27	90
Puebla	86	20	1	52	13	73	1	74	61	72	45	100
Querétaro	51	30	0	91	59	98	27	63	95	75	27	100
Quintana Roo	35	42	1	82	62	54	22	68	71	69	0	90
San Luis Potosí	95	28	1	65	61	81	1	80	37	61	22	99
Sinaloa	69	66	0	94	69	92	1	68	72	63	15	92
Sonora	80	47	0	92	83	96	4	66	77	51	33	99
Tabasco	53	27	3	70	36	0	12	72	23	72	75	82
Tamaulipas	99	60	0	96	73	87	6	63	75	61	53	95
Tlaxcala	82	16	2	70	73	97	10	87	93	59	25	100
Veracruz	70	23	1	63	42	38	4	82	32	74	89	78
Yucatán	92	0	0	71	7	32	10	78	33	76	13	100
Zacatecas	97	39	1	78	70	98	0	86	52	49	4	99

).

It should be noted that other authors have previously addressed the problem of competitiveness [23,24], which serve as a reference point for the relevance of this decision problem; however, in these investigations, a robustness analysis of the solution presented to the decision-maker has not been carried out, nor has a tool been developed that provides support to generate a robust ranking as a proposed solution, as is intended to be addressed with the presented AIP model.

3.1. Identifying Decision-Makers’ Preferences in a Competitiveness Problem

Regarding the information on the preferences of decision-makers, as part of the information published by the IMCO, it is possible to define that criterion j is more important than criterion i. For our application of the AIP model, the IMCO acts as the decision-maker, considering its assessment of the importance of each of the 100 criteria established to evaluate the competitiveness problem, this information representing, in our case, the preferences of the IMCO or the decision-maker.

Therefore, in the case applied in this example, the information on the decision-maker’s preferences is complete, as the order of importance of the 100 criteria involved in the competitiveness problem is known.

3.2. Definition of Genetic Algorithm Parameters

In this regard, to obtain the maximum number of solutions possible, a population of 1000 individuals and 1100 generations was defined. A crossover index of 0.9 and a mutation index of 0.9 were also defined to maximize the variation in feasible solutions. Each individual in the genetic algorithm corresponds to a vector of 100 weights, representing the 100 decision criteria from the competitiveness problem. The input parameters of the genetic algorithm generated a set of feasible individuals in each generation. The outcome consists of 562,634 solutions, each accompanied by a corresponding weight vector.

3.3. Inference of Weights Based on Decision-Makers’ Preferences and Ordering of Alternatives

The proposed algorithm, having already ordered the criteria, infers the weights for each of the 100 criteria involved in the competitiveness problem. It should be noted that this procedure was performed on the 562,634 solutions obtained to ensure different weights in each iteration.

By running the genetic algorithm, considering the decision-maker’s preferences, the weights for each of the 100 defined criteria are determined, which are used to calculate the decision matrix.

Once the decision matrix was obtained for each of the 562,634 solutions, the 32 alternatives were ordered in descending order, from highest to lowest, using the IOWA Operator.

3.4. Descriptive Measures of SMAA

One of the objectives of the procedure carried out is to obtain a robust ranking. To do this, all the rankings generated with the IOWA Operator were taken, and the descriptive measures of the SMAA were calculated.

SMAA is based on different combinations of values for the uncertain parameters and records the statistics of how each alternative is evaluated through an acceptability index, a central weight vector, and a confidence factor. It is based on exploring a weight space to describe the set of preferences that make each alternative preferred [25].

3.4.1. Acceptability Index

The acceptability index measures the variability of the set of criteria weights that result in an alternative having a specific ranking [26]. In other words, it measures the number of times an alternative appears in each ranking position according to a set of criteria weights.

For our case study,

Table 1. Summary of the acceptability index in percentages.

Position	Alternative Region (State)	Acceptability Index in Percentage
1	Ciudad de México	100%
2	Aguascalientes	97.9%
3	Colima	97.9%
4	Nuevo León	100%
5	Querétaro	86.4
6	Sonora	86.4
7	Sinaloa	100
8	Baja California Sur	98.8
9	Jalisco	96.8
10	Coahuila	96.7
11	Quintana Roo	100
12	Campeche	100
13	Nayarit	100
14	Tamaulipas	100
15	Yucatán	100
16	Puebla	94.8
17	San Luis Potosí	84.9
18	Chihuahua	84.8
19	Guanajuato	100
20	México	99.8
21	Hidalgo	99.7
22	Durango	73.9
23	Tlaxcala	74.0
24	Morelos	98.2
25	Baja California	99.9
26	Zacatecas	100
27	Tabasco	100
28	Veracruz	99.9
29	Michoacán	99.9
30	Oaxaca	99.9
31	Chiapas	99.9
32	Guerrero	100

displays the ranking position that each alternative obtained after running the 562,634 ranking solutions generated with the IOWA Operator. These data correspond to the summary of the acceptability index for each state’s position, expressed as a percentage. Appendix B (

Table A3. Acceptability index in percentage.

Alternative	R1	R2	R3	R4	R5	R6	R7	R8	R9	R10	R11	R12	R13	R14	R15	R16	R17	R18	R19	R20	R21	R22	R23	R24	R25	R26	R27	R28	R29	R30	R31	R32
Aguascalientes		97.9	2.1
Baja California																								0.1	99.9
Baja California Sur								98.8	1.1	0.1
Campeche												100
Coahuila								1.2	2.1	96.7
Colima		2.1	97.9
Chiapas																														0.1	99.9
Chihuahua																5.2	10.0	84.8
Ciudad de México	100
Durango																					0.1	73.9	24.2	1.7	0.1
Guanajuato																			100
Guerrero																																100
Hidalgo																				0.2	99.7	0.1
Jalisco									96.8	3.2
México																				99.8	0.1	0.1
Michoacán																												0.1	99.9
Morelos																							1.8	98.2
Nayarit													100
Nuevo León				100
Oaxaca																														99.9	0.1
Puebla																94.8	5.1	0.1
Querétaro					86.4	13.6
Quintana Roo											100
San Luis Potosí																	84.9	15.1
Sinaloa							100
Sonora					13.6	86.4
Tabasco																											100
Tamaulipas														100
Tlaxcala																					0.1	25.9	74.0
Veracruz																												99.9	0.1
Yucatán															100
Zacatecas																										100

) shows the acceptability index in percentage for each of the alternatives in the 32 ranking positions.

The purpose of calculating the acceptability index for all solutions is to determine the probability that alternative k will appear in position R in the ranking. Therefore, this index makes it possible to identify which alternative(s) are most likely to appear in the first position, or any required position.

In the specific case of the 562,634 solutions, the Mexico City alternative was ranked first in 100% of them.

3.4.2. Central Weight Vector

The central weight vector is a descriptive measure of SMAA [21], which is the average of the weights of a set of weights for all criteria, resulting in alternative k appearing as the best alternative. It is inferred that this may represent the decision-maker’s preferences [26].

As part of the SMAA, it is essential to determine the preferred alternatives. That is, after the acceptability analysis, those alternatives in the first position are identified. Once this information is obtained, the central weight vector is calculated by averaging the sets of weights that caused those alternatives to rank first.

It should be noted that regarding the application of our AIP model, as can be seen in Table 1, the acceptability analysis allows us to identify that 100% of the 562,634 sets of weight vectors positioned one alternative (Mexico City) as the most preferred. In this sense, all sets of weight vectors fluctuated in weight variations, respecting the model’s restrictions.

Based on the above, the central weight vector is calculated by averaging all the weight sets for the total criteria that caused Mexico City to rank first in our problem. This central weight vector is shown in

Table 2. Central weight vector.

Alternative	W1	W2	W3	W4	W5	W6	W7	W8	W9	W10	W11	W12	W13	W14	W15	W16	W17
CDMX	0.014	0.014	0.013	0.014	0.013	0.013	0.013	0.014	0.013	0.005	0.004	0.005	0.004	0.005	0.004	0.003	0.004
	W18	W19	W20	W21	W22	W23	W24	W25	W26	W27	W28	W29	W30	W31	W32	W33	W34
	0.005	0.005	0.004	0.003	0.016	0.016	0.015	0.015	0.014	0.016	0.017	0.017	0.017	0.018	0.017	0.016	0.017
	W35	W36	W37	W38	W39	W40	W41	W42	W43	W44	W45	W46	W47	W48	W49	W50	W51
	0.014	0.016	0.015	0.015	0.015	0.017	0.002	0.002	0.003	0.002	0.002	0.003	0.003	0.003	0.002	0.019	0.019
	W52	W53	W54	W55	W56	W57	W58	W59	W60	W61	W62	W63	W64	W65	W66	W67	W68
	0.019	0.018	0.018	0.019	0.018	0.019	0.018	0.010	0.010	0.010	0.009	0.009	0.009	0.010	0.010	0.009	0.008
	W69	W70	W71	W72	W73	W74	W75	W76	W77	W78	W79	W80	W81	W82	W83	W84	W85
	0.008	0.008	0.007	0.007	0.007	0.007	0.007	0.008	0.008	0.009	0.011	0.012	0.011	0.011	0.012	0.011	0.011
	W86	W87	W88	W89	W90	W91	W92	W93	W94	W95	W96	W97	W98	W99	W100
	0.012	0.010	0.012	0.012	0.001	0.001	0.001	0.001	0.001	0.006	0.006	0.006	0.006	0.006	0.006

.

It is important to mention that part of validating the central weight vector in our model is to determine that the weight sets for each of the decision criteria explore the space of weight changes that meet the decision-maker’s preferences. The weights are not the same across all weight vector sets.

Once this step was completed, each alternative was recalculated, considering the weight of the criteria according to the central weight vector, which resulted in the overall performance of the 32 alternatives. The alternatives were again sorted in descending order, resulting in a final ranking.

The objective of this step is to ensure that the weights of the central weight vector are understood as robust weights, as they guarantee that an alternative, such as Mexico City, is positioned first in the state competitiveness ranking. This ranking, generated by a set of robust weights, is considered a robust ranking.

For the competitiveness decision problem where the AIP model was applied, only one alternative was ranked first. Therefore, only one central weight vector was calculated. If there are two or more alternatives in the first position, the same number of central weight vectors is calculated to cause each different alternative to rank first and, subsequently, generate the different robust rankings.

3.5. Robust Competitiveness Ranking in Mexico

By obtaining the central weight vector and generating the ranking with these robust weights, the result is a robust ranking, in this case having the ordering of the 32 federative entities evaluated by the 100 criteria defined by the IMCO, which can be presented as a solution to the expert, for the subsequent analysis of the results and, where appropriate, decision-making on public policies, investment processes, and improvement, among other actions, that can be implemented with this information as a basis.

4. Analysis of Robust Ranking as a Decision Proposal

Regarding the results of applying the methodology, a robust ranking of state competitiveness is available for 2016. These rankings order the states according to the decision-maker’s preferences, ensuring that the relevant information is continually respected.

Once the procedure is completed, the resulting robust ranking is compared with the one published by the IMCO (National Institute of Statistics and Census) to analyze the most notable similarities and differences. The robust ranking resulting from the application of the methodology, as well as the one presented by the IMCO in its 2016 report, is presented in

Table 3. Ranking comparison between IMCO ranking and current robust ranking.

Alternative Region (State)	IMCO 2016 Ranking Position	Robust Ranking of AIP Position
Ciudad de México	1	1
Aguascalientes	2	2
Nuevo León	3	4
Colima	4	3
Querétaro	5	5
Sonora	6	6
Coahuila	7	10
Jalisco	8	9
Sinaloa	9	7
Yucatán	10	15
Campeche	11	12
Baja California Sur	12	8
Quintana Roo	13	11
Puebla	14	16
Tamaulipas	15	14
Chihuahua	16	18
Nayarit	17	13
Guanajuato	18	19
San Luis Potosí	19	17
Hidalgo	20	21
México	21	20
Durango	22	23
Morelos	23	24
Tlaxcala	24	22
Baja California	25	25
Zacatecas	26	26
Tabasco	27	27
Veracruz	28	28
Michoacán	29	29
Chiapas	30	31
Oaxaca	31	30
Guerrero	32	32

The gray color indicates the regions (states) whose positions in the IMCO and AIP model rankings are similar. The orange color indicates the regions (states) with the most significant difference in position between the two rankings.

, which shows each of the 32 states analyzed, their position in both rankings (IMCO and robust), and a summary of the variation between the positions in these rankings.

Both rankings exhibit similarities, primarily in the first six and last eight positions of the ranking. Most of the differences between them are in the middle of the ranking. Those regions (states) with higher distances are Yucatan, with five positions, Baja California Sur, with four positions, and Nayarit, with four positions of difference. The rest of them with differences in the middle of the ranking present one, two, or three positions of difference.

Regarding these regions (states) that present greater variation, in a particular analysis of the performance of these alternatives in the criteria, it can be determined that by taking into account the preferences of the decision-maker, he gives greater importance to criteria where alternatives such as Yucatan, Baja California Sur, and Nayarit, have a lower performance than other alternatives; therefore, when calculating the general performance of these alternatives in the 100 criteria analyzed, it causes a variation in the position of the robust ranking of the applied model that is far from that presented by the IMCO.

Among the most significant similarities between the two rankings is that Mexico City appears in first place in both orders, and Aguascalientes in second (Table 3); these states, along with others such as Querétaro, Sonora, Baja California, Zacatecas, Tabasco, Veracruz, and Michoacán, show no variation in their resulting positions.

The fact that Mexico City appears first and Aguascalientes second provides an opportunity to corroborate that, in the case where the decision-maker provides a complete ranking of their preferences, the inference of the criteria weights was correct, since this was based on having ordered the 100 criteria analyzed in the competitiveness problem, taking the decision-maker’s preferences into account at all times.

The resulting robust ranking indicates that varying the weights of the different criteria would not alter the ranking. That is, the position of these two states (Mexico City and Aguascalientes) would remain the same. We may be interested in possible changes in the position of other states, depending on their performance.

The ranking comparison highlights the main similarities and differences between the rankings. As illustrated, the main differences are shown in the middle of the ranking. Those differences do not express a wrong elicitation or incorrect results from the multicriteria decision model. Instead, it clarifies the uncertainty and vagueness of the decision-making preferences.

In this sense, the comparison of the AIP model and the robust weights that result in a robust ranking, concerning the static weights and the ranking managed by the IMCO, presents an advantage, since by taking into account the preferences of the decision-maker, he can determine which criteria are more or less important, depending on the analysis factor that is intended to be addressed in a decision-making problem, in this case, the competitiveness of the states or regions.

The robust ranking generated by the AIP solves the complex task of defining parameters (weights) for a large number of criteria and establishing robust parameters that yield the same ranking even if some weight values are slightly modified. The proposed AIP addresses the uncertainty and vagueness associated with decision-making preferences when defining parameters.

5. Conclusions

In the current context, where trade relations have crossed borders, it is important, considering the number of criteria involved in evaluating the competitiveness of a country, region, state, or company, to use tools and methods that allow us to identify those alternatives (in this case, states) that promote a competitive Mexico relative to its international peers. This is because it leads to economic and social growth for its inhabitants, increasing the attractiveness of each region as a destination for foreign investment.

While it is true that the Mexican Institute of Statistics and Census (IMCO) generates a ranking of the states in its annual report, obtaining a robust ranking through the Artificial Intelligence Procedure (AIP) proposed in this research represents, for the expert, the certainty that even when specific changes occur in the decision-maker’s preference parameters, reflected in changes in the weights of the criteria, the resulting ranking will generally remain the same. That is, some or most of the alternatives will be able to maintain their position in the resulting ranking, providing certainty or solidity to the ranking. Similarly, identifying an ordering of criteria without needing to specify a specific weight for each one represents an opportunity for decision-makers to reduce cognitive effort in this regard.

Another advantage of applying the proposed AIP to the competitiveness problem is identifying the most important to the least important criteria, as required. This allows for more precise comparison parameters between entities, focusing on their position relative to any other. That is, decision-makers can identify an entity that demonstrates an advantage in each criterion and, depending on the relative importance the decision-maker assigns to it, can serve as a reference for emulating the actions and best practices applied. In problems with numerous criteria, a robust ranking offers advantages in terms of time savings and ease of presenting the solution for making a policy decision, measure, or action while relying on the stability of the results obtained. Future work will involve applying this AIP tool to fuzzy logic and classification problems. In addition, there are different approaches where the weighting vector is not bound to a sum equal to 1, such as the Heavy OWA [27]; therefore, this feature can be incorporated into MCDM processes.