ABSTRACT
This paper presents a broad overview of main contributions related to the FITradeoff (Flexible and Interactive Tradeoff) method. FITradeoff is a multicriteria method developed within the scope of the Multiattribute Value Theory (MAVT), considering partial information from the decision maker (DM) in the preference modelling process. Over the last few years, several methodological developments on this method have been published in the literature, as well as practical applications to a wide range of multicriteria decision problems. The most recent methodological advances are related to preference modelling process, which now integrate the two paradigms of elicitation by decomposition and holistic evaluation. Furthermore, contributions from behavioral studies, some of them including decision neuroscience, have enhanced the DSS free available for FITradeoff. In this paper, all previously developed works related to the FITradeoff method are approached, considering both methodological developments and practical applications. A summary on the different modeling approaches for solving different decision problematics (choice, ranking, sorting and portfolio) with FITradeoff is presented. The recently proposed combination of preference modeling paradigms  elicitation by decomposition and holistic evaluation  within the FITradeoff decision process is explained, as well its potential advantages for the elicitation process. Moreover, a brief review on the practical applications of the method in different contexts is presented. In addition, this work also brings a summary on the results of behavioral experiments conducted using neuroscience tools with the FITradeoff method, as well the methodological insights resulted from them, and future perspectives of potential research topics related to the FITradeoff method.
Keywords:
FITradeoff; preference modeling; multicriteria decision making (MCDM); multiattribute value theory (MAVT)
1 INTRODUCTION
The Flexible and Interactive Tradeoff method (FITradeoff) consists of a Multicriteria Decision MakingAiding (MCDM/A) technique for solving decision problems under multiple and conflicting criteria in additive models within the scope of the Multiattribute Value Theory. This method works based on partial information about the decision maker’s preferences, in such a way that the elicitation of preferences becomes less cognitively demanding for the decision maker, with less time and effort spent in the preferences modeling process (De Almeida et al., 2016DE ALMEIDA AT, ALMEIDA JA, COSTA APCS & ALMEIDAFILHO AT. 2016. A New Method for Elicitation of Criteria Weights in Additive Models: Flexible and Interactive Tradeoff. European Journal of Operational Research, 250(1): 179191; De Almeida et al., 2021DE ALMEIDA AT, FREJ EA & ROSELLI LRP. 2021. Combining holistic and decomposition paradigms in preference modeling with the flexibility of FITradeoff. Central European Journal of Operations Research, 141.).
When dealing with additive aggregation models, alternatives are scored straightforwardly according to (1). In (1), v(a _{i} ) represents the global value of alternative a _{i} , k _{j} represents the scaling constant (or commonly called weight) of criterion j(j = 1, ... , m), and v _{j} (x _{ij} ) is the value of consequence x _{ij} , which consists of the evaluation of alternative a _{i} in criterion j, measured in a 01 scale according to the marginal value function of criterion j. Criteria scaling constants are normalized and sum up to 1, according to (2).
In multicriteria additive models, a critical issue is the elicitation of criteria scaling constants, since these parameters should reflect the range of consequences of the actual set of alternatives in each criterion, and defining them based on importance level of the criteria may cause critical distortions within the model (Keeney & Raiffa, 1976KEENEY RL & RAIFFA H. 1976. Decision analysis with multiple conflicting objectives. Wiley & Sons, New York.). Hence, structured procedures for eliciting the values of these parameters considering the consequences space are necessary. The most wellknown procedures for elicitation of criteria scaling constants in additive models are the classical tradeoff procedure (Keeney & Raiffa, 1976KEENEY RL & RAIFFA H. 1976. Decision analysis with multiple conflicting objectives. Wiley & Sons, New York.) and the swing procedure (von Winterfeldt & Edwards, 1986VON WINTERFELDT D & EDWARDS W. 1986. Structuring for decision analysis. Decision Analysis and Behavioral Research. New York: Cambridge University Press, 423.). The first one was developed based on a strong axiomatic structure under the concepts of the Multiattribute Value/Utility Theory, and allows nonlinear value functions to be considered for intracriterion evaluation. However, a critical disadvantage of this procedure is the difficulty presented for decision makers in the preferences elicitation process, since exact values that makes the decision maker indifferent between two consequences when considering tradeoffs between criteria are requested. This information is considered to be high cognitively demanding, which lead to a high inconsistency rate when this procedure is applied, according to behavioral studies (Borcherding et al., 1991BORCHERDING K, EPPEL T & VON WINTERFELDT D. 1991. Comparison of Weighting Judgments in Multiattribute Utility Measurement. Management Science, 37(12): 16031619.). The swing procedure, on the other hand, carries the elicitation process in an easier way, but modeling steps are simplified (Edwards & Barron, 1994EDWARDS W & BARRON FH. 1994. SMARTS and SMARTER: Improved simple methods for multiattribute utility measurement. Organizational behavior and human decision processes, 60(3): 306325.), in such a way that only linear value functions are considered in the intracriterion evaluation, which may also cause distortions in the model.
Multicriteria methods that consider partial information about preferences have been developed in order to facilitate the elicitation process for decision makers, lowering the cognitive effort spent and, consequently, tightening the gap between theoretical models and practical applications (Weber, 1987WEBER M. 1987. Decision making with incomplete information. European Journal of Operational Research, 28(1): 4457.; Salo & Hamalainen, 1992SALO A & HÄMÄLÄINEN RP. 1992. Preference assessment by imprecise ratio statements. Operations Research , 40: 10531061.; Kirkwood & Sarin, 1985KIRKWOOD CW & SARIN RK. 1985. Ranking with partial information: A method and an application. Operations Research, 33: 3848.). Over the years, several partial information methods have been developed in the literature, such as the PAIRS method (Salo & Hamalainen, 1992SALO A & HÄMÄLÄINEN RP. 1992. Preference assessment by imprecise ratio statements. Operations Research , 40: 10531061.); PRIME method (Salo & Hamalainen, 2001SALO A & HÄMÄLÄINEN RP. 2001. Preference ratios in multiattribute evaluation (PRIME) elicitation and decision procedures under incomplete information. IEEE Transactions on Systems, Man, and CyberneticsPart A: Systems and Humans , 31: 533545.); RICH method Interval SMART/Swing method (Mustajoki, Hamalainen & Salo, 2005MUSTAJOKI J, HÄMÄLÄINEN RP & SALO A. 2005. Decision support by interval SMART/SWINGincorporating imprecision in the SMART and SWING methods. Decision Sciences, 36(2): 317339.); SMARTER method (Edwards & Barron, 1994EDWARDS W & BARRON FH. 1994. SMARTS and SMARTER: Improved simple methods for multiattribute utility measurement. Organizational behavior and human decision processes, 60(3): 306325.); among many others (Malakooti, 2000MALAKOOTI B. 2000. Ranking and screening multiple criteria alternatives with partial information and use of ordinal and cardinal strength of preferences. IEEE Transactions on Systems, Man, and CyberneticsPart A: Systems and Humans, 30(3): 355368.; Park & Kim, 1997PARK KS & KIM SH. 1997. Tools for interactive multiattribute decisionmaking with incompletely identified information. European Journal of Operational Research, 98(1): 111123.; Ahn & Park, 2008AHN BS & PARK KS. 2008. Comparing methods for multiattribute decision making with ordinal weights. Computers & Operations Research, 35(5): 16601670.). According to De Almeida et al (2016DE ALMEIDA AT, ALMEIDA JA, COSTA APCS & ALMEIDAFILHO AT. 2016. A New Method for Elicitation of Criteria Weights in Additive Models: Flexible and Interactive Tradeoff. European Journal of Operational Research, 250(1): 179191), these methods differ in terms of the form in which the decision maker provides preferential information, which can be interactively or all at once; the type of information provided (rankings of criteria weights, bounds, holistic judgments, arbitrary inequalities), and the synthesis step (linear programming, decision rules, surrogate weights, simulation and/or sensitivity analysis).
Da Silva et al (2022DA SILVA ALCDL, CABRAL SEIXAS COSTA AP & DE ALMEIDA AT. 2022. Analysis of the cognitive aspects of the preference elicitation process in the compensatory context: a neuroscience experiment with FITradeoff. International Transactions in Operational Research, 31. Doi: https://doi.org/10.1111/itor.13210
https://doi.org/10.1111/itor.13210...
) performed a systematic literature review on partial information methods, which addresses different types of information, elicitation structure and synthesis step that different methods use. In their review, the authors point out that most partial information methods consider a nonstructured protocol for the elicitation, or consider the swing procedure for doing so, in a simplified manner. In this context, the FITradeoff method differs from other partial information methods in a sense that it carries out the whole structure of the classical tradeoff procedure in the elicitation protocol, including the possibility of using nonlinear value functions in intracriterion evaluation (De Almeida et al., 2016DE ALMEIDA AT, ALMEIDA JA, COSTA APCS & ALMEIDAFILHO AT. 2016. A New Method for Elicitation of Criteria Weights in Additive Models: Flexible and Interactive Tradeoff. European Journal of Operational Research, 250(1): 179191), but considering partial information about the DM’s preferences. Moreover, the FITradeoff method has flexibility features that enable the process to be adapted to different circumstances, including graphical visualization of partial results and possibility of conducting holistic judgments to accelerate the process.
The purpose of this paper to conduct a summary on the main contributions related to the FITradeoff method, both in methodological and practical perspectives. Different decision problematics that can be addressed with the FITradeoff method are approach in this paper, as well as the combination of preference modeling paradigms conducted in this method. Moreover, this paper will also present an overview of neuroscience studies and behavioral experiments conducted with a view to bring methodological developments on the FITradeoff method. Practical decision situations in several contexts in which the FITradeoff method was applied are also presented in this paper, as well as the future perspectives expected within this research line.
This paper is structured as follows. Section 2 is devoted to describing the FITradeoff method in light of the four decision problematics: choice, ranking, sorting and portfolio. Section 3 presents results of neuroscience and behavioral studies related to the FITradeoff method. In Section 4, the combination of paradigms in preference modeling and its potential advantages for the decision process are highlighted. Section 5 gives a summary on the applications conducted using the FITradeoff method, and Section 6 finally presents the conclusions and future perspectives.
2 SOLVING DIFFERENT DECISION PROBLEMATICS WITH FITRADEOFF
The FITradeoff method was originally developed by De Almeida et al (2016DE ALMEIDA AT, ALMEIDA JA, COSTA APCS & ALMEIDAFILHO AT. 2016. A New Method for Elicitation of Criteria Weights in Additive Models: Flexible and Interactive Tradeoff. European Journal of Operational Research, 250(1): 179191), for solving multicriteria decision problems for choice problematic. Few years later, Frej et al (2019FREJ EA, DE ALMEIDA AT, COSTA APCS. 2019. Using data visualization for ranking alternatives with partial information and interactive tradeoff elicitation. Operational Research , 19(5): 909931.) developed a different variant of FITradeoff for dealing with the ranking problematic. Kang et al (2020KANG THA, FREJ EA, DE ALMEIDA AT 2020. Flexible and interactive tradeoff elicitation for multicriteria sorting problems. Asia Pacific Journal of Operational Research , 37: 2050020.) expanded the method for the sorting problematic. More recently, Frej et al (2021FREJ EA, EKEL P, DE ALMEIDA AT. 2021. A benefittocost ratio based approach for portfolio selection under multiple criteria with incomplete preference information. Information Sciences, 545: 487498.) proposed a benefittocost ratio based approach for dealing with the portfolio problematic with FITradeoff. The portfolio problematic was also approached by Marques et al (2022MARQUES AC, FREJ EA & DE ALMEIDA AT. 2022. Multicriteria decision support for project portfolio selection with the FITradeoff method. Omega, 111: 102661.) from a different perspective, considering the classical combinatorial approach.
All those variants of the FITradeoff method are operationalized by means of a Decision Support System (DSS), in which the whole elicitation process is carried out in an interactive manner, alternating steps of interaction with the DM and computational steps. It is important to highlight that, for all problematics, the interaction steps  steps in which the DM provides preferential information  are extremely similar. What differs from one problematic to another is the mathematical model formulation, which is specific for each problematic, and the results obtained.
In a generic way and summarized way, the FITradeoff method works as follows. After an intracriterion evaluation is performed (in which both linear or nonlinear value functions can be considered), the DM ranks criteria scaling constants according to his own preferences, considering the ranges of consequences in each criterion. After that, a ranking of criteria scaling constants (3) is obtained.
Then, the elicitation process continues with questions put for the DM, in which he/she should answer considering tradeoffs between criteria. Two consequences are put for the DM: Consequence A, with the worst outcome for all criteria and an intermediate outcome for criterion j; and Consequence B, with the worst outcome for all criteria and the best outcome for criterion j+1. Depending on the value stablished for criterion j (let us say, x’ _{j} ), the DM may declare preference for Consequence A over Consequence B, in such a way that v(A) > v(B) and the inequality in is obtained; or, for other values of x _{j} , let us say, x” _{j} , Consequence B might be preferred to Consequence A, so that v(B) > v(A) and the inequality in (5) is obtained.
Inequalities in (3), (4) and (5), together with equation (2), form the socalled space of weights; i.e., the set of weights vectors compatible with the preferences of the decision maker. For each decision problematic, a different mathematical model is run searching for a recommendation, considering the current space of weights. In general, the FITradeoff method works based on linear programming models, and the space of weights act as part of the constraints of such models. Different LP models are considered depending on the decision problematic being dealt. Such models will be further detailed in the following subtopics.
The FITradeoff process is carried out in an interactive manner, so that after each preference statement given by the DM in the comparison of consequences, the space of weights is updated with the new information obtained, so that LP model runs in order to refine the results obtained. Partial results can be displayed for the DM at any time during the process, as a flexibility feature of the FITradeoff DSS, including the possibility of graphical visualization. Different types of visualization are provided in the DSS, and it is also possible for the DM to perform holistic judgments during the process, providing additional information to the model (this issue will be detailed explored in Section 4). In this sense, if the DM feels satisfied with such partial results, then he/she may interrupt the process even before the end of the elicitation, saving time and effort.
The following subtopics are devoted to give a brief explanation on how the FITradeoff method works for each decision problematic.
2.1 FITradeoff for Choice problematic
When dealing with choice problems, the FITradeoff method works based on a progressive reduction of the set of Potentially Optimal Alternatives (POA). Considering an MCDM problem with n alternatives, an alternative a _{i} can be considered to be Potentially Optimal if the global value of a _{i} , according to Equation (1), is greater than the global values of all other n1 alternatives for at least one vector of weights within the feasible weights space. I.e., an alternative is considered potentially optimal if it can be the optimal alternative of the problem, considering the actual space of weights (De Almeida et al., 2016DE ALMEIDA AT, ALMEIDA JA, COSTA APCS & ALMEIDAFILHO AT. 2016. A New Method for Elicitation of Criteria Weights in Additive Models: Flexible and Interactive Tradeoff. European Journal of Operational Research, 250(1): 179191). In this sense, for an alternative a _{i} to be considered potentially optimal, the inequality in (6) must hold for all z = 1, ... , n; z ≠ i.
Therefore, the mathematical model of FITradeoff for choice problematic seeks for the verification of the potential optimality of an alternative. And the model is run for all alternatives, in order to form the subset of potentially optimal alternatives. At each interaction step, the following LP model runs (7).
If the LPP model (7) has at least one feasible solution  i.e., if for at least one vector of weights it is possible to maximize the global value of a _{i} considering the current space of weights (formed by inequalities of type (3), (4) and (5) according to the preferential information given by the DM and Equation (2)) and satisfying the potential optimality constraints in (6) , then alternative a _{i} is a potentially optimal alternative for the problem. In (7), a small number ε is incorporated in order to turn inequalities in (4) and (5) computationally treatable in LP models.
It should be highlighted that each time the DM gives a new preference information, the weight space is updated with new inequalities of type (4) and (5), which are incorporated into the LP models, that run again searching for the updated subset of potentially optimal alternatives. In this sense, the interaction steps continue until a unique alternative is found to be potentially optimal (i.e., this will be the optimal alternative for the problem); or until the DM feels satisfied with the actual set of potentially optimal alternatives (De Almeida et al., 2016DE ALMEIDA AT, ALMEIDA JA, COSTA APCS & ALMEIDAFILHO AT. 2016. A New Method for Elicitation of Criteria Weights in Additive Models: Flexible and Interactive Tradeoff. European Journal of Operational Research, 250(1): 179191).
2.2 FITradeoff for Ranking problematic
Since the concept of potential optimality is no longer enough to deal with the ranking of alternatives, Frej et al (2019FREJ EA, DE ALMEIDA AT, COSTA APCS. 2019. Using data visualization for ranking alternatives with partial information and interactive tradeoff elicitation. Operational Research , 19(5): 909931.) developed a new model for solving ranking problems with FITradeoff based on the verification of pairwise dominance relations. The LP model in (8) is run for all pairs of alternatives a _{i} , a _{k} in order to compute the maximum difference between their global values, subjected to the current space of weights formed by the inequalities in (3), (4) and (5) and Equation (2).
Frej et al (2019FREJ EA, DE ALMEIDA AT, COSTA APCS. 2019. Using data visualization for ranking alternatives with partial information and interactive tradeoff elicitation. Operational Research , 19(5): 909931.) define the possibilities of preference relations between alternatives a_{i} , a _{k} considering the optimal solution of (8). In summary, A dominance relation for a _{i} over a _{k} is defined if the global value of a _{k} cannot be greater than the global value of a _{i} for any vector of weights within the current weight space. At each interaction, the LPP in (8) runs for all pairs of alternatives in order to test dominance relations between them. Once a dominance relation is stablished between a pair of alternatives, this relation remains until the end of the process, and such pair needs no longer to be tested.
When dealing with the ranking problematic, the FITradeoff DSS also provides a graphical visualization of the partial (or complete) ranking, through a Hasse diagram of the alternatives, in which the dominance relations can be visualized. Figure 1 illustrates the graphical visualization of the ranking. Depending on the level of information obtained, the ranking may be still partial, with pairs of alternatives still being incomparable according to that level of information. An incomparability relation between two alternatives a _{i} , a _{k} can be verified when the level of information provided by the DM is not sufficient to stablish a dominance relation between then, because a subspace makes the value of a _{i} greater than the value of a _{k} , and another subspace makes the value of a _{k} greater than the value of a _{i} .
When two alternatives are considered to be incomparable for the current level of information provided, there is the possibility to conduct holistic evaluations in order to solve such incomparability relations in a faster way. This issue is further discussed in Section 4. Each time the DM answers an elicitation question, a new inequality of types (4) or (5) is obtained, in such a way that the weight space is updated, and the LP models run again in order to search for new dominance relations between alternatives and therefore refine the ranking. The process finishes either when a complete ranking of the alternatives is obtained or when the partial ranking is enough for the DM’s purposes (Frej et al., 2019FREJ EA, DE ALMEIDA AT, COSTA APCS. 2019. Using data visualization for ranking alternatives with partial information and interactive tradeoff elicitation. Operational Research , 19(5): 909931.).
2.3 FITradeoff for Sorting problematic
Sorting problems can also be solved with the FITradeoff method, based on a decision rules approach proposed by Kang et al (2020KANG THA, FREJ EA, DE ALMEIDA AT 2020. Flexible and interactive tradeoff elicitation for multicriteria sorting problems. Asia Pacific Journal of Operational Research , 37: 2050020.). Categories are defined a priori by the decision maker, with global values profiles that act as boundaries for them. The preferences elicitation is identical to what was previously explained for the choice and ranking problematics, in which the DM answers questions considering tradeoffs between criteria, comparing hypothetical consequences. The space of weights now serve as constraints for two LP models, which run at each interaction searching for the maximum (9) and minimum (10) global value of each alternative a _{i} , i = 1, . . . , n.
The optimal solution of (9) consists on the maximum overall value, according to (1), that alternative a _{i} can assume, considering the current space of weights, while the optimal solution of (10) indicated the minimum overall value that alternative a _{i} can assume, considering the current space of weights. Based on these two values, Kang et al (2020KANG THA, FREJ EA, DE ALMEIDA AT 2020. Flexible and interactive tradeoff elicitation for multicriteria sorting problems. Asia Pacific Journal of Operational Research , 37: 2050020.) propose a set of decision rules in order to allocate each alternative a _{i} to a predefined specified category. Basically, an alternative a _{i} is allocated to a certain category if the minimum value of a _{i} is greater that the minimum value of the lower profile value for that category and if the maximum value of a _{i} is lower than the upper profile value for that category. The process is carried within an interactive way with partial allocations displayed for the DM whenever he/she wants to visualize partial results. The process finishes either when all alternatives are allocated into a single category or when the DM feels satisfied with the partial allocation.
2.4 FITradeoff for Portfolio problematic
For dealing with portfolio selection problems, two approaches have been developed with the FITradeoff method. First, Frej et al (2021FREJ EA, EKEL P, DE ALMEIDA AT. 2021. A benefittocost ratio based approach for portfolio selection under multiple criteria with incomplete preference information. Information Sciences, 545: 487498.) developed an approach that consists of a heuristic that ranks projects according to their benefittocost ratio, and select those projects that fit within the budget constraints. The second one was proposed by Marques et al (2022MARQUES AC, FREJ EA & DE ALMEIDA AT. 2022. Multicriteria decision support for project portfolio selection with the FITradeoff method. Omega, 111: 102661.), which treats the portfolio selection problem in a combinatorial manner, based on complete enumeration. The following subtopics are devoted to explain each of them in a brief manner.
2.4.1 Using BenefittoCost Ratio (BCR)
Frej et al (2021FREJ EA, EKEL P, DE ALMEIDA AT. 2021. A benefittocost ratio based approach for portfolio selection under multiple criteria with incomplete preference information. Information Sciences, 545: 487498.) proposed a heuristic to solve portfolio selection problems with the FITradeoff method in a simpler manner, without the need to treat the portfolio problematic in a combinatorial way, since the computational effort to do so is relatively high. In this sense, the authors proposed an adaptation of the model presented by Frej et al (2019FREJ EA, DE ALMEIDA AT, COSTA APCS. 2019. Using data visualization for ranking alternatives with partial information and interactive tradeoff elicitation. Operational Research , 19(5): 909931.) for the ranking problematic, detailed in Section 2.2, to rank projects based on decreasing order of their BenefittoCost ratio (BCR). The BCR of an alternative  or project  a_{i} is defined by the ratio between its global value (calculated according to 1), which is a measure of its benefit, and the cost of implementation of this project, c _{i} (see Equation 11).
The main issue that arises from this model is that, since FITradeoff works based on partial information, v(a _{i} ) is not exactly known, since criteria scaling constants are not exactly determined, and the FITradeoff method works considering a space of weights. Therefore, Frej et al (2021FREJ EA, EKEL P, DE ALMEIDA AT. 2021. A benefittocost ratio based approach for portfolio selection under multiple criteria with incomplete preference information. Information Sciences, 545: 487498.) proposed the computation of dominance relations between candidate projects with an LP model similar to that in (8), but considering their BCR instead of their global value in the objective function of the LP model (see 12). The constraints remain the same of the model in (8), which consist basically of the space of weights formed by equations (2), (3), (4) and (5).
Based on the dominance relations found, a ranking of projects is built; this ranking may be partial or complete, depending on the level of information obtained. An available budget B should be defined by the organization, in a sense that projects are selected to be part of the portfolio according to the ranking obtained, until budget B is built (Frej et al., 2021FREJ EA, EKEL P, DE ALMEIDA AT. 2021. A benefittocost ratio based approach for portfolio selection under multiple criteria with incomplete preference information. Information Sciences, 545: 487498.).
2.4.2 Using complete enumeration
Different from the approach proposed by Frej et al (2021FREJ EA, EKEL P, DE ALMEIDA AT. 2021. A benefittocost ratio based approach for portfolio selection under multiple criteria with incomplete preference information. Information Sciences, 545: 487498.), which consists of a simplified heuristic to deal with the portfolio problematic in order to avoid the combinatorial optimization problem, Marques et al (2022MARQUES AC, FREJ EA & DE ALMEIDA AT. 2022. Multicriteria decision support for project portfolio selection with the FITradeoff method. Omega, 111: 102661.) treats the portfolio problematic with FITradeoff considering complete enumeration of all possible portfolios.
The authors propose an approach to verify the efficiency and feasibility of each portfolio generated, in order to reduce the computational complexity in the explicit generation of portfolios. The process is divided into two phases: a preparation phase, which is conducted without interaction with the DM, and in which a high computational effort is spent to generate all possible portfolios; and the preferences elicitation phase, in which the DM plays an active role by answering preference questions of comparison of consequences to model the space of weights.
The preferences elicitation process is extremely similar to the one for the choice problematic detailed in section 2.1. The main difference is that, in this approach, each alternative a _{i} consists of a combination of projects; i.e., a candidate portfolio. The performance of each portfolio a _{i} in each criterion is evaluated considering an aggregation of the performances of each project in each criterion. The mathematical model presented in (7) is run to verify the potential optimality of an alternative a _{i} ; i.e., the potential optimality of portfolio a _{i} . The model works based on a progressive reduction of the subset of potentially optimal portfolios.
3 DEVELOPING METHODOLOGICAL ASPECTS OF FITRADEOFF METHOD WITH RESULTS FROM NEUROSCIENCE AND BEHAVIOURAL EXPERIMENTS
The Neuroscience is considered a multidisciplinary approach which can be integrated to several areas of knowledge in order to investigate human behavioural (Glimcher & Rustichini, 2004GLIMCHER PW & RUSTICHINI A. 2004. Neuroeconomics: The Consilience of Brain and Decision. Science, 5695: 447452.; Dimoka et al., 2007DIMOKA A, PAVLOU PA & DAVIS FD. 2007. NeuroIS: The potential of cognitive neuroscience for information systems research. In: International Conference on Information Systems, 28th, Proceedings, Toulon, França, 120.; Fehr &Camerer, 2007FEHR E & CAMERER CF. 2007. Social neuroeconomics: the neural circuitry of social preferences. Trends in cognitive sciences, 11(10): 419427.; Morin, 2011MORIN C. 2011. Neuromarketing: the new science of consumer behavior. Society, 48(2): 131135.; Khushaba 2013KHUSHABA RN. 2013. Consumer neuroscience: Assessing the brain response to marketing stimuli using electroencephalogram (EEG) and eye tracking. Expert Systems with Applications, 40(9): 38033812.; Riedl et al., 2014RIEDL R, DAVIS FD & HEVNER AR. 2014. Towards a NeuroIS research methodology: intensifying the discussion on methods, tools, and measurement. Journal of the Association for Information Systems, 15(10), I.). Regarding to MCDM/A approach, a few numbers of studies which use neuroscience tools to investigate DecisionMakers (DMs) preferences have been found in literature (Trepel et al., 2005TREPEL C, FOX CR & POLDRACK R. 2005. A. Prospect theory on the brain? Toward a cognitive neuroscience of decision under risk. Cognitive brain research, 23(1): 3450.; O¨ zerol & Karasakal, 2008; Barberis & Xiong, 2009BARBERIS N & XIONG W. 2009. What drives the disposition effect? An analysis of a longstanding preferencebased explanation . The Journal of Finance, 64(2): 751784.; Hunt et al., 2014HUNT LT, DOLAN RJ, BEHRENS TE. 2014. Hierarchical competitions subserving multiattribute choice. Nature neuroscience, 17(11): 16131622.; Chuang et al., 2015CHUANG H, LIN C & DCHEN Y. 2015. Exploring the triple reciprocity nature of organizational value cocreation behavior using multicriteria decision making analysis. Mathematical Problems in Engineering, 115.; Nermend, 2017NERMEND K 2017 The Implementation of Cognitive Neuroscience Techniques for Fatigue Evaluation in Participants of the DecisionMaking Process. Neuroeconomic and Behavioral Aspects of Decision Making. Berlim: Springer, Cham , 329339., de Almeida et al., 2020aDE ALMEIDA A, ROSSELLI L, COSTA MORAIS D, COSTA A. 2020a. Neuroscience tools for behavioural studies in group decision and negotiation. In Kilgour DM, Eden C (eds). Handbook of Group Decision and Negotiation. (1st edn.). Springer International Publishing, Dordrecht, Netherlands, p. 124.).
According to Korhonen & Wallenius (1997KORHONEN P & WALLENIUS J. 1997. Behavioral issues in MCDM: Neglected research questions. Multicriteria analysis. Berlin: Springer, Heidelberg, 412422.) behavioural aspects involved in decision processes should be considered into methods or techniques in order to modulate (transform) those methods. Hence, motivated by this gap in literature, several behavioural studies have been performed using neuroscience tools to investigate DMs preferences when they use the FITradeoff method.
Based on these behavioural studies, methodological aspects have been developed on the FITradeoff. The transformations made on the FITradeoff method, or its modulation, as suggested Korhonen & Wallenius (1997KORHONEN P & WALLENIUS J. 1997. Behavioral issues in MCDM: Neglected research questions. Multicriteria analysis. Berlin: Springer, Heidelberg, 412422.), bring improvements for this method.
The most important transformation made in the FITradeoff regards the combination of two paradigms of preference modelling  elicitation by decomposition and holistic evaluation  during the FITradeoff decision process. This new feature proposed for the FITradeoff (de Almeida et al., 2021DE ALMEIDA AT, FREJ EA & ROSELLI LRP. 2021. Combining holistic and decomposition paradigms in preference modeling with the flexibility of FITradeoff. Central European Journal of Operations Research, 141.) is possible from behavioural studies concerning holistic evaluation. In the previous version of the FITradeoff method (de Almeida et al., 2016DE ALMEIDA AT, ALMEIDA JA, COSTA APCS & ALMEIDAFILHO AT. 2016. A New Method for Elicitation of Criteria Weights in Additive Models: Flexible and Interactive Tradeoff. European Journal of Operational Research, 250(1): 179191) the holistic evaluation was used only to finalize the decision process. Now, preferences expressed during the holistic evaluation are included in Linear Programming Problem (LPP) model (de Almeida et al. 2021DE ALMEIDA AT, FREJ EA & ROSELLI LRP. 2021. Combining holistic and decomposition paradigms in preference modeling with the flexibility of FITradeoff. Central European Journal of Operations Research, 141.). Hence, using the FITradeoff method, DMs can express preferences for pairwise comparisons in the elicitation by decomposition or they can express dominance relations between alternatives in the holistic evaluation, at any moment of process.
In addition, this feature has been applied in the FITradeoff Decision Support System (DSS) for choice and ranking problematics. In the previous version of FITradeoff method, the holistic evaluation is presented only in choice problematic (de Almeida et al., 2016DE ALMEIDA AT, ALMEIDA JA, COSTA APCS & ALMEIDAFILHO AT. 2016. A New Method for Elicitation of Criteria Weights in Additive Models: Flexible and Interactive Tradeoff. European Journal of Operational Research, 250(1): 179191). Now, it is included in ranking problematic (Frej et al., 2019FREJ EA, DE ALMEIDA AT, COSTA APCS. 2019. Using data visualization for ranking alternatives with partial information and interactive tradeoff elicitation. Operational Research , 19(5): 909931.; de Almeida et al., 2021DE ALMEIDA AT, FREJ EA & ROSELLI LRP. 2021. Combining holistic and decomposition paradigms in preference modeling with the flexibility of FITradeoff. Central European Journal of Operations Research, 141.). For ranking problems, the use of holistic evaluation paradigm presents a special role since DMs can express dominance relations between alternatives which are in the same level of the ranking.
It is worth to mention that graphical and tabular visualization are presented in the FITradeoff DSS to support DMs during the holistic evaluation. Based on behavioural studies, another improvement is the inclusion of tables to support the evaluation of alternatives during the holistic evaluation. Firstly, only graphical visualizations are considered in the DSS, but behavioural results suggested that tables are as good as bar graphs to support DMs during the holistic evaluation (Roselli et al., 2018ROSELLI LRP, FREJ EA, DE ALMEIDA AT. 2018. Neuroscience Experiment for Graphical Visualization in the FITradeoff Decision Support System. In: Chen Y, Kersten G, Vetschera R, Xu H (eds.). Group Decision and Negotiation in an Uncertain World. GDN 2018. Lecture Notes in Business Information Processing, vol 315.; Roselli et al., 2019ROSELLI LRP, DE SOUSA PEREIRA L, DE ALMEIDA AT, MORAIS DC & COSTA APCS. 2019. Neuroscience experiment applied to investigate decisionmaker behavior in the tradeoff elicitation procedure. Annals of Operations Research, 118., Roselli & de Almeida 2022ROSELLI LRP & DE ALMEIDA AT. 2022. Use of the AlphaTheta Diagram as a decision neuroscience tool for analyzing holistic evaluation in decision making. Annals of Operations Research.).
Another important methodological aspect is the inclusion of the elimination process during the holistic evaluation. In the previous version, the DM can only select the best alternative, i.e., those that dominates the others. Now, DMs can select the best one, or the worst one in the group of Potentially Optimal Alternatives (POA). This feature is very interesting for choice problems, since provides flexibility during the holistic evaluation. Some DMs prefers to select the best, and other judges as most simple to eliminate the worst.
Two decision tools have also been developed from the studies. The first one is named AlphaTheta Diagram (Roselli & de Almeida 2022ROSELLI LRP & DE ALMEIDA AT. 2022. Use of the AlphaTheta Diagram as a decision neuroscience tool for analyzing holistic evaluation in decision making. Annals of Operations Research.) which uses Alpha and Theta brain activities to classify DMs behavioural. The other is the Success Based Decision Rule (SBDR  Roselli & de Almeida 2021ROSELLI LRP & DE ALMEIDA AT. 2021. The use of the successbased decision rule to support the holistic evaluation process in FITradeoff. International Transactions in Operational Research.) which indicates the probability of success for several patterns of visualizations used to support DMs during the holistic evaluation.
Moreover, behavioural studies suggested that cognitive effort and time were demanded in the decomposition process. This result confirms that the elicitation has been made in adequate way, since it is expected that DMs spend time and effort during making the preference modelling by decomposition. Moreover, the studies suggested that the use of quantitative and qualitative criteria demanded more cognitive effort to proceed in the FITradeoff method (Silva et al., 2019SILVA MM, DE GUSMÃO APH, DE ANDRADE CTA, SILVA W. 2019. The integration of VFT and FITradeoff multicriteria method for the selection of WCM projects. 2019 IEEE International Conference on Systems, Man and Cybernetics (SMC), October 69, Bari, Italy, p. 15131517.).
In addition, aspects related to the design of the FITradeoff DSS have been suggested from behavioural studies. Inclusion of messages, buttons, graphs, and tools have been made to improve DMs’ experience in using the DSS. In special, the EyeTracking tool brings important improvements related to design of the DSS. This equipment measure eyemovements on computer screen, thus it can be used to suggest if DMs is really looking to the elements on the screen. Some adjusts are made in the design of the FITradeoff DSS based on these studies.
All these features are possible from behavioural studies. These studies are performed in the NSID (NeuroScience for Information and Decision) laboratory at the Federal University of Pernambuco (UFPE), Recife, Brazil. Several experiments have been made, focusing on the investigation of the two paradigms of preference modelling (Roselli & de Almeida 2022ROSELLI LRP & DE ALMEIDA AT. 2022. Use of the AlphaTheta Diagram as a decision neuroscience tool for analyzing holistic evaluation in decision making. Annals of Operations Research.). These experiments were approved by the university’s Ethical Committee and used two neuroscience tools, in particular the Electroencephalography with 14 channels, to capture brain activities, and the X120 EyeTracking to capture eyemovements.
For instance, for studies that focused on the elicitation by decomposition the main task was to solve a multicriteria decision problem using the decomposition process. The problem solved by each DM was personal, elaborated by their own in MCDM/A classes. The experiments are extraclasses activities (Silva et al., 2021SILVA ALCL, COSTA APCS & DE ALMEIDA AT. 2021. Exploring cognitive aspects of FITradeoff method using neuroscience tools. Annals of Operations Research , 123.; Silva et al., 2019SILVA MM, DE GUSMÃO APH, DE ANDRADE CTA, SILVA W. 2019. The integration of VFT and FITradeoff multicriteria method for the selection of WCM projects. 2019 IEEE International Conference on Systems, Man and Cybernetics (SMC), October 69, Bari, Italy, p. 15131517., Roselli et al. 2019ROSELLI LRP, DE ALMEIDA AT & FREJ EA 2019. Decision neuroscience for improving data visualization of decision support in the FITradeoff method. Operational Research , 19: 121.).
For holistic evaluation experiments, the main task was to evaluate graphical and tabular visualizations and express dominance relations between alternatives. The experiments present different visualizations patterns. Moreover, involves different decision tasks  selection of the best alternative or elimination of the worst (Pessoa et al., 2021PESSOA MEBT, ROSELLI LRP, DE ALMEIDA AT. 2021. A Neuroscience Experiment to investigate the Selection decision process versus the Elimination decision process in the FITradeoff Method. EWGDSS 7th International Conference on Decision Support System Technology. Loughborough, United Kingdom.; Roselli & de Almeida 2020aROSELLI LRP, DE ALMEIDA AT 2020a. Analysis of graphical visualizations for multicriteria decision making in FITradeoff method using a decision neuroscience experiment. Lecture notes in business information processing, 384th edn. Springer, Berlin, p. 4254., Roselli & de Almeida 2020bROSELLI LRP, DE ALMEIDA AT. 2020b. Improvements in the FITradeoff decision support system for ranking order problematic based in a behavioral study with NeuroIS tools. In: Davis FD et al. (eds). Lecture notes in information systems and organization, LNISO, 43ed. NeuroIS, p. 112., Roselli et al., 2018ROSELLI LRP, FREJ EA, DE ALMEIDA AT. 2018. Neuroscience Experiment for Graphical Visualization in the FITradeoff Decision Support System. In: Chen Y, Kersten G, Vetschera R, Xu H (eds.). Group Decision and Negotiation in an Uncertain World. GDN 2018. Lecture Notes in Business Information Processing, vol 315.; Silva et al., 2022)
Based on physiological variables, DMs behavioural have been investigated and modulations are done in the FITradeoff method. The modulations are done concerning to methodological aspects and design aspects (de Almeida et al., 2021DE ALMEIDA AT, FREJ EA & ROSELLI LRP. 2021. Combining holistic and decomposition paradigms in preference modeling with the flexibility of FITradeoff. Central European Journal of Operations Research, 141.). These modulations bring improvements for the method since integrates behavioural aspects to technical aspects (Wallenius et al., 2008WALLENIUS J, DYER JS, FISHBURN PC, STEUER RE, ZIONTS S & DEB K 2008. Multiple criteria decision making, multiattribute utility theory: Recent accomplishments and what lies ahead. Management science, 54(7): 13361349.; Wallenius & Wallenius, 2020WALLENIUS H & WALLENIUS J. 2020. Implications of World Mega Trends for MCDM Research. In: Ben Amor S, De Almeida A, De Miranda J, Aktas E (Eds.). Advanced Studies in MultiCriteria Decision Making. New York: Chapman and Hall/CRC, Series in Operations Research, 1st Ed, 110.). As commented by Ruben et al. (2020), studies in decision neurosciences as an interesting direction for the development of effective decision support systems. Therefore, the FITradeoff method has been modulated in order to improve the preference modelling process as the way to bring more flexibility for DMs, and it DSS is also improved in terms of design. The FITradeoff DSS is available for free at www.fitradeoff.org.
4 COMBINING PREFERENCE MODELLING PARADIGMS IN FITRADEOFF METHOD
The flexibility of the FITradeoff method is one of its main bullet points compared to other methods, and the flexibility features of the FITradeoff DSS enable DMs to conduct the decision process in different manners, according to their own wishes and perspectives. Recently, a new flexibility feature was incorporated within the DSS: the possibility of conducting holistic judgments between alternatives to help solving undefined relations between them. This new feature was incorporated based on the ideas proposed by De Almeida et al (2021DE ALMEIDA AT, FREJ EA & ROSELLI LRP. 2021. Combining holistic and decomposition paradigms in preference modeling with the flexibility of FITradeoff. Central European Journal of Operations Research, 141.), who bring the possibility of combination of preference modeling paradigms within the decision process with FITradeoff: elicitation by decomposition and holistic evaluation. With these two types of preference modeling put together within the FITradeoff DSS, significant improvements to the decision process can be observed.
The elicitation by decomposition consists of the classical elicitation proposed by Keeney & Raiffa (1976KEENEY RL & RAIFFA H. 1976. Decision analysis with multiple conflicting objectives. Wiley & Sons, New York.), through a cartesian process in which the DM compares elements in the consequences space. In this preference modeling type, two consequences are put for the DM to compare, as exemplified in Figure 2, which illustrates a problem with four criteria: C1, C2, C3 and C4.
In Figure 2, the consequences displayed are similar to those described in Section 2. Consequence A has the worst outcome in all criteria (except for C2), denoted by small red bars, and an intermediate outcome for criterion C2 denoted by the blue bar. Consequence B has the worst outcome in all criteria (except for C3), and the best outcome in criterion C3. The DM should analyze those consequences by considering tradeoffs between criteria C2 and C3, and choose which one he/she prefers. If the DM chooses preference for Consequence A, an inequality similar to (4) is obtained; if Consequence B is preferred, and inequality similar to (5) is obtained.
The great novelty proposed by De Almeida et al (2021DE ALMEIDA AT, FREJ EA & ROSELLI LRP. 2021. Combining holistic and decomposition paradigms in preference modeling with the flexibility of FITradeoff. Central European Journal of Operations Research, 141.) was actually the possibility of incorporating holistic judgments during the process, in such a way that the preferential information obtained by the holistic evaluation could be incorporated into the model and accelerate the process. Holistic evaluations consist on comparison of elements within the alternatives space, in a direct manner. For instance, the DM says directly that alternative a _{i} is preferred to alternative a _{z} , such that the inequality in (13) is obtained.
This inequality is incorporated to the mathematical model of the FITradeoff method, tightening the space of weights according to this information. It may happen that the information provided in holistic judgments are inconsistent with the information obtained in the elicitation by decomposition. In this case, the FITradeoff DSS puts the DM to choose which of the two conflicting judgements is actually correct, in such a way that the other one is discarded (De Almeida et al., 2021DE ALMEIDA AT, FREJ EA & ROSELLI LRP. 2021. Combining holistic and decomposition paradigms in preference modeling with the flexibility of FITradeoff. Central European Journal of Operations Research, 141.).
The FITradeoff DSS provides graphical visualization of the alternatives to order to aid the DM when analyzing them to perform the holistic judgments. Three types of graphics are available: bar graphic, bubble graphic and radar graphic, as Figure 3 shows. In those graphics, each color represents an alternative, and the performance of the alternatives are displayed in each criterion, in a ratio 01 scale, so that the DM can visualize them in a comparative manner. It should be highlighted that the DM can analyze the graphics during any time of the process and then decide whether he/she is willing or not to perform a holistic evaluation at that point, according to the confidence level. In case the DM does not feel confident (or does not want to) perform a holistic evaluation, he/she can go back to the elicitation by decomposition to answer tradeoff questions. The key issue here is the possibility to alternate between the two types of preference modeling within the FITradeoff decision process.
Holistic judgments can be useful in the choice problematic either to select the best alternative amongst a subset of the Potentially Optimal Alternatives set, or to eliminate an alternative from this subset (De Almeida et al., 2021DE ALMEIDA AT, FREJ EA & ROSELLI LRP. 2021. Combining holistic and decomposition paradigms in preference modeling with the flexibility of FITradeoff. Central European Journal of Operations Research, 141.). As for the ranking problematic, holistic judgments are useful to define dominance relations between pairs of alternatives that are still incomparable for the current level of information obtained. Generally speaking, holistic judgments are useful to provide additional information to the mathematical model of FITradeoff, with the incorporation of inequalities of type (13). In some cases, however, the information provided in holistic judgments may be enough to finalize the decision process with a satisfactory result obtained for the DM (De Almeida et al., 2021DE ALMEIDA AT, FREJ EA & ROSELLI LRP. 2021. Combining holistic and decomposition paradigms in preference modeling with the flexibility of FITradeoff. Central European Journal of Operations Research, 141.).
According to De Almeida et al (2021DE ALMEIDA AT, FREJ EA & ROSELLI LRP. 2021. Combining holistic and decomposition paradigms in preference modeling with the flexibility of FITradeoff. Central European Journal of Operations Research, 141.), inequalities that come from holistic judgments have great potential to cause a significant reduction in the space of weights, accelerating the decision process in a sense that the final solution can be obtained faster. This is a potential advantage of conducting holistic evaluations during the process, since the DM can save effort and time that he/she would spend answering several questions in the elicitation by decomposition process, with only few holistic judgments. It should be highlighted that the two preference modeling types are available for the DM in the FITradeoff DSS, in such a way that he/she can alternate between them during the process, guided by an analyst with well background in MCDM.
5 PRACTICAL APPLICATIONS
This section intends to summarize several practical applications in which the FITradeoff method has been applied to support MCDM/A problems in different areas.
5.1 Supplier Selection
Santo et al. (2020) used the FITradeoff method to rank suppliers of in a Wholesaler and Retailer Company. In this study, twenty suppliers have been evaluated against five criteria. The preference modeling process have been conducted using the elicitation by decomposition and the holistic evaluation. In this application, bar graphics and spider graphics have been used to support the DM.
Rodrigues et al. (2020RODRIGUES LVS, CASADO RSGR, CARVALHO END & SILVA MM. 2020. Using FITradeoff in a ranking problem for supplier selection under TBL performance evaluation: An application in the textile sector. Production, 30.) used the FITradeoff method to support a ranking problematic involving supplier of a textile company. In this study, a problem structuring method is also used to identify hidden criteria, before the FITradeoff method. As result, the supplier which had the best position is not those that the company works.
Frej et al. (2017FREJ EA, ROSELLI LRP, ARAÚJO DE ALMEIDA J, DE ALMEIDA AT. 2017. A multicriteria decision model for supplier selection in a food industry based on FITradeoff method. Mathematical Problems in Engineering , 2017: 19.) used the FITradeoff method to select the best supplier for a food industry. The problem is composed of five suppliers which have been evaluated against seven criteria. These criteria represent the objectives of the company. In this study, a choice problematic has been considered. Also, the decision process was very fast, after only two elicitation questions a supplier had been selected. An interesting point is that the supplier which had been selected does not had the best price, which in general is the only attribute considered by companies.
5.2 Location Problem
Sousa Ribeiro et al. (2021) used the FITradeoff method to support a shopping mall location problem in the northeast countryside of Brazil. Ten cities of the northeastern countryside have been evaluated against seven criteria which represent conflicting objectives. This problem is interesting since discuss investments in the countryside of Brazil, stimulating economic growth in northeastern region.
De Lacerda et al. (2021DE LACERDA NLB, DOS SANTOSNETO JBS & MARTINS CL. 2021. MCDM model for natural gas pressure reducing station site selection. International Journal of Decision Support System Technology (IJDSST), 13(1): 6784.) used the method to solve a site selection problem for a station of natural gas. DMs from a company which distribute natural gas participated in this study.
Oppio et al. (2020) discussed the integration of Geographic Information Systems (GIS) and MultiCriteria Decision Analysis (MCDA), in special the FITradeoff method, to addressing decisions about the location of healthcare facilities. In the same context, Dell’Ovo et al. (2018DELL’OVO M, OPPIO A, CAPOLONGO S 2020. Decision Support System for the Location of Healthcare Facilities Sit Health Evaluation Tool. Springer Nature, Cham, Switzerland.) used the FITradeoff method to support a healthcare facility location problem in the city of Milan, Italy. In this problem, six potential areas had been considered to site a new hospital.
5.3 Health Systems
In the context of health systems, Dos Santos et al. (2022DOS SANTOS LA, DOS SANTOS AFA, DE ASSIS AG, DA COSTA JÚNIOR JF & DE SOUZA RP. 2022. Model to support intervention prioritization for the control of Aedes aegypti in Brazil: a multicriteria approach. BMC Public Health, 22(1): 111.) used the FITradeoff method to prioritize response activities for Aedes Aegypt control in the city of Natal/RN. In this study several actors are involved in the decision model. The study considered eleven alternatives which had been evaluated against six criteria. As result, the study supports DMs to minimize effects and risks associated to Aedes Aegypt proliferation.
Frazão et al. (2021FRAZÃO TD, DOS SANTOS AF, CAMILO DG, DA COSTA JÚNIOR JF & DE SOUZA RP 2021. Priority setting in the Brazilian emergency medical service: a multicriteria decision analysis (MCDA). BMC Medical Informatics and Decision Making , 21(1): 116.) discussed the gap in literature of studies to prioritize victims in the Emergency Medical Service. Hence, this study uses the FITradeoff method to prioritize victims of SAMU/192, considering the scarcity of resources and conflicting objectives. In this study, twentyfive criteria had been considered. As result, protocols that guide regulatory physicians have been discussed considering strategic criteria.
Moreover, Camilo et al. (2020CAMILO DGG, DE SOUZA RP, FRAZÃO TDC, DA COSTA JUNIOR JF 2020. Multicriteria analysis in the health area: selection of the most appropriate triage system for the emergency care units in natal. BMC Medical Informatics and Decision Making 20(1): 116.) discussed triage system in emergency healthcare units. Thus,in this study, the FITradeoff was used to select the best protocol of triage for emergency healthcare units in NatalRN. As result, the Spanish Triage System has been considered the most suitable protocol for the emergency care units.
5.4 Energy, Agricultural and Urban contexts
Kang et al. (2018KANG THA, JÚNIOR AMDCS, DE ALMEIDA AT. 2018. Evaluating electric power generation technologies: A multicriteria analysis based on the FITradeoff method. Energy, 165: 1020.) used the method to evaluate electric power generation technologies to be included in the electricity matrix. In this study, eight technologies had been evaluated in terms of financial, technical, environmental, and socioeconomic dimensions.
Fossile et al. (2020FOSSILE DK, FREJ EA, DA COSTA SEG, DE LIMA EP, DE ALMEIDA AT. 2020. Selecting the most viable renewable energy source for Brazilian ports using the FITradeoff method. Journal of Cleaner Production, 260: 121107.) investigated which type of renewable energy is the most viable for Brazilian ports using the FITradeoff method. The study considered three alternatives (wind energy, photovoltaic energy, and wave energy) against twenty criteria. The criteria have been defined in terms of sustainability, management, national standards, legislation, and previous data. As result, the photovoltaic energy was considered the most viable energy source.
Monte & Morais (2019MONTE MBS, MORAIS DC. 2019. A decision model for identifying and solving problems in an urban water supply system. Water Resources Management, 33(14): 48354848.) investigated a water supply system of an urban area which was deficient since the population growth and equipments become old. Thus, the FITradeoff method was applied to indicate actions to deal with an urban water supply system. The study also considered a problem structuring method to support objectives identification.
Martins et al. (2020MARTINS MA, GARCEZ TV, GUSMÃO APHDE, SILVA LGO & ALMEIDA JA DE. Multicriteria Model Based on FITradeoff Method for Prioritizing Sections of Brazilian Roads by Criticality. Mathematical Problems in Engineering , 2020(1): 115, 29 dez. 2020. Hindawi Limited.) used the method to prioritize road sections, based on their criticality and the risks faced by the user. This study considered twentytwo road sections against different attributes. As result, the most critical sections had been identified.
Rodriguez et al. (2021RODRIGUEZ JMM, KANG THA, FREJ EA & DE ALMEIDA AT. 2021. Decisionmaking in the purchase of equipment in agricultural research laboratories: a multiplecriteria approach under partial information. Decision Science Letters, v. 10, p. 451462.) used the method to support buying a laboratory equipment for a colombian agricultural research company. In the same context, Carrillo et al. (2018CARRILLO PAA, ROSELLI LRP, FREJ EA, DE ALMEIDA AT 2018. Selecting an agricultural technology package based on the flexible and interactive tradeoff method. Annals of Operations Research, 116.) used the method to select the best agricultural technology packages.
Morais et al. (2022MORAIS DC, ARAÚJO AM, FREJ EA & ALMEIDA ATD 2022. Group Decision Process for Evaluating a Mango Variety to Be Planted in New Agricultural Farms. In Collective Decisions: Theory, Algorithms And Decision Support Systems, (p. 247264). Springer, Cham.) presented a group decision process in which the DMs of an agribusiness organization needs to evaluate which variety of mango should be plant in new farms concerning mango variety agricultural farms. Thus, the FITradeoff method had been applied to collect individual preferences of each DM. After that, a voting procedure had been applied to obtain the solution.
5.5 Management and Industrial Context
Pessoa et al. (2022PESSOA MEBT, ROSELLI LRP & DE ALMEIDA AT. 2022. Using the FITradeoff Decision Support System to Support a Brazilian Compliance Organization Program. Information Systems Frontiers, 116.) applied the method to rank options to support a complianceprogram problem. This study discussed decisionmaking in time of crisis, reinforcing actions in the context of anticorruption law. In this paper, twentyeight alternatives were evaluated against five criteria.
De Oliveira et al. (2022DE OLIVEIRA ACD, SILVA WDO & MORAIS DC. 2022. Developing and prioritizing lean key performance indicators for plastering supply chains. Production: 32.) used the method to prioritize indicators to monitor the development of plastering supply chains. As result, a set of indicators had been defined to provide competitiveness and sustainability for the company.
Shukla & Dubey (2021SHUKLA S. 2017. A fitradeoff approach for assessment and understanding of patient Adherence behavior. In Value in Health, 20(5): A322A322. New York, USA: Elsevier Science Inc.) used the FITradeoff method to support a celebrity selection for a brand or campaign. This study considered a group decision process with DMs from different sectors (brand, marketing communication agency and brand’s customers). This study supports DMs to make effective decisions on celebrity selection for their brands.
Correia et al. (2021) used the FITradeoff method to solve a workstation problem concerning ergonomics interventions in the footwear industry. This study also used problem structuring methods to structure the hierarchy of fundamental objectives. As result, the ranking of workstations which need ergonomic interventions had been obtained.
Fernandes et al. (2021FERNANDES CHDA, SILVA LCE, GUARNIERI P & VIEIRA BDO. 2021. Multicriteria Model Proposition to Support the Management of Systems of EWaste Collection. Logistics, 5(3): 60.) used the method for managing Waste from Electrical and Electronic Equipment (WEEE). In this study, ten alternatives had been considered, and the DM is a federal public agency. The study also presented recommendations to manufacturers in terms of design and traceability of products.
Pergher et al. (2020PERGHER I, FREJ EA, ROSELLI LRP, DE ALMEIDA AT. 2020. Integrating simulation and FITradeoff method for scheduling rules selection in jobshop production systems. International Journal of Production Economics, 227: 107669.) used the FITradeoff method to support schedule decisions in a manufacturer of ladies’ shoes. These decision concerns due date assignment, order release and priority dispatching. Previous jobshop studies do not explore DM preferences. Thus, in this study the FITradeoff method is used to consider DM’s preferences concerning due date assignment, order release and shop dispatching rules.
Poleto et al. (2020POLETO T, CLEMENTE TRN, DE GUSMÃO APH, SILVA MM & COSTA APCS. 2020. Integrating valuefocused thinking and FITradeoff to support information technology outsourcing decisions. Management Decision.) used the method to support information technology outsourcing decisions. This study also considered a problem structuring method to support the identification of strategic and fundamental objectives in ITO decisions.
De Macedo et al. (2018DE MACEDO PP, DE MIRANDA MOTA CM & SOLA AVH. 2018. Meeting the Brazilian Energy Efficiency Law: A flexible and interactive multicriteria proposal to replace nonefficient motors. Sustainable cities and society, 41: 822832.) applied FITradeoff to support a motor replacement in a chemical industry. The study focused on develop a replacement plan in order to have a minimum energy performance in accordance to the Brazilian Energy Efficiency Law.
Gusmão & Medeiros (2018) to select the best strategic information system for a glass packaging factory. The factory needs to select a unique information system from a set of systems considered as relevant.
As illustrated in this section, the FITradeoff method can be applied to support several MCDM/A problems. The next section remarks conclusions and future perspectives of the study.
6 CONCLUSIONS AND FUTURE PERSPECTIVES
This paper presented an overall perspective of the FITradeoff method, with a summary on all methodological developments and applications developed in the literature. It could be seen that this MCDM method is suitable for solving multicriteria decision problems in all the four main decision problematics: choice, ranking, sorting and portfolio. The preferences elicitation process is carried out in a similar manner for all of them, with differences on the mathematical models and results obtained. Even though there are differences within the mathematical models, all variants of the FITradeoff method work with linear programming models, which is one of the most used approaches for dealing with partial information in MCDM, according to Da Silva et al (2022DA SILVA LBL, FREJ EA, DE ALMEIDA AT, FERREIRA RJP & MORAIS DC. 2022. A review of partial information in additive multicriteria methods. IMA Journal of Management Mathematics.).
All variants of the FITradeoff method are operated by means of a Decision Support System, freely available for users at www.fitradeoff.org. It should be highlighted that, when using the DSS, the decision maker should be guided by an analyst with well background on MCDM and on the FITradeoff method, in order to better explore all the functionalities and flexibility features of the system. With regards to future perspectives in this line of research, some major topics can be highlighted.
First, the conduction of behavioral experiments with the use of neuroscience tools to analyze how decision makers think and act within the elicitation process considering its aspects. A key issue that shall be investigated though behavioral experiments is the combination of preference modeling types (elicitation by decomposition and holistic evaluation) in FITradeoff, and the DM behaves within this perspective.
Moreover, the extent into which the holistic judgments indeed improve the decision process in terms of reducing the number of total questions the DM has to answer should be explored in future studies; this could be conducted considering a simulation approach. Mendes et al (2020MENDES JAJ, FREJ EA, DE ALMEIDA ADIEL TEIXEIRA & ALMEIDA JA. 2020. Evaluation of Flexible and Interactive Tradeoff Method Based on Numerical Simulation Experiments. Pesquisa Operacional, v. 40, p. 125.) conducted a preliminary simulation study with the FITradeoff for the choice problematic, and concluded that the method presents a relatively high convergence speed in the reduction of the set of potentially optimal alternatives. This study should be complemented considering the other decision problematics, as well as the incorporation of the holistic evaluation element and how does this type of preference modeling indeed contribute to fasten the elicitation process. When analyzing the holistic evaluation, it is also possible to investigate how the elimination of worse alternatives and selection of best ones contributes to the process, and which of these two are more effective to aid a satisfactory result to be found.
Still regarding the holistic evaluation and its potential benefits, future studies should also explore the use of this type of preference modeling also in sorting and portfolio problematic, since it is currently incorporated for choice and ranking, with the possibilities of selection of the best alternative or elimination of worst alternative for choice problematic, and establishment of a dominance relation for the ranking problematic. For sorting problematic, holistic judgments between alternatives and profiles should be explored. As for the portfolio problematic, holistic judgments between projects and/or between portfolios may be interesting to consider.
In addition to methodological focus studies, the application of the FITradeoff method to practical decision problems should be continuously explored, in order to tighten the gap between these methodological developments and realworld applications. Both public and private organizations can use the method on its multiple variants to aid the solution of complex decision problems with multiple objectives.
Acknowledgments
The authors are grateful for the support received from Coordination for the Improvements of Higher Education Personnel (CAPES), FACEPE, and the Brazilian National Council of Technological and Scientific Development (CNPq).
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Publication Dates

Publication in this collection
14 Apr 2023 
Date of issue
2023
History

Received
03 May 2022 
Accepted
05 Nov 2022