Techniques to model uncertain input data of multi‐criteria decision

Intl. Trans. in Op. Res. 00 (2018) 1–37 DOI: 10.1111/itor.12598

INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH

Techniques to model uncertain input data of multi-criteria decision-making problems: a literature review Renata Pelissaria,d , Maria Celia Oliveiraa,b , Alvaro J. Abackerlic , Sarah Ben-Amord and Maria Rita Pontes Assumpça˜ oa a

Post Graduate Program of Industrial Engineering, Methodist University of Piracicaba, Santa Bárbara d’Oeste, Brazil b Engineering School, Mackenzie Presbyterian University, São Paulo, Brazil c Embrapii–Brazilian Company for Industrial Research and Innovation, Brasilia, Brazil d Telfer Management School, University of Ottawa, Ottawa, Canada E-mail: [email protected] [Pelissari]; [email protected] [Oliveira]; [email protected] [Abackerli]; [email protected] [Ben-Amor]; [email protected] [Assumpça˜ o] Received 11 November 2017; received in revised form 23 August 2018; accepted 29 August 2018

Abstract There are a few studies in the literature regarding possible types of uncertainty in input data of multi-criteria decision making (MCDM) or multi-criteria decision analysis (MCDA) problems and the techniques employed to deal with each of them. Therefore, the aim of this study is to identify the different types of uncertainty that occur in input data of MCDM/MCDA problems and the most appropriate techniques to deal with each one of these uncertainty types. In this paper, a comprehensive literature review is presented in order to meet this objective. We selected and summarized 134 international journal articles. They were analyzed based on the type of data with uncertainty, the type of uncertainty, and the technique used to model it. We identified three distinct types of uncertainty in input data of MCDM/MCDA problems, namely (i) uncertainty due to ambiguity, (ii) uncertainty due to randomness, and (iii) uncertainty due to partial information. We identified a new generation of fuzzy approaches including Type-2, intuitionistic, and hesitant fuzzy sets (FSs), which are used to model these types of uncertainty alongside other approaches such as traditional FSs theory, probability theory, evidential reasoning theory, rough set theory, and grey numbers. Finally, a framework that indicates techniques used in different decision-making contexts under uncertainty is proposed. Keywords: multicriteria analysis; uncertainty; imprecision; partial information; systematic literature review

1. Introduction In the literature, we can identify different types of uncertainty related to a decision-making process. They may be present in the identification of criteria, interactions among criteria, evaluations of the C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research Published by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA02148, USA.

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R. Pelissari et al. / Intl. Trans. in Op. Res. 00 (2018) 1–37

alternatives, definition of the criteria weights from experts, and the choice of aggregation operators. Uncertainties in input data of multi-criteria decision making (MCDM)/multi-criteria decision analysis (MCDA) problems are specifically related to evaluation of alternatives and to the definition of the criteria weights and model parameters. Although the use of uncertain data in decision-making processes is common, most traditional MCDM/MCDA methods require exact values as input data. Such a requirement poses a problem for the decision-making process, since it is often difficult, if not impossible, to obtain exact values (Ben-Amor et al., 2015). Therefore, in many cases, uncertain input data need to be modeled by using specific techniques, such as fuzzy sets (FSs) or probabilities (Ben-Amor et al., 2015; Tervonen and Figueira, 2007). As important as knowing how to use a specific technique to deal with uncertainty is the choice of this technique, which has a direct impact on the decision-making process and on the final decisions. Even so, the choice of a modeling technique to represent uncertainty is sometimes made arbitrarily (Ben-Amor et al., 2015). Few studies have discussed the different types of techniques and proposed ways to help select the most suitable one based on the type of uncertainty. Durbach and Stewart (2012) presented a synthesis of some techniques to deal with uncertainty and classified their approaches into five main categories: probabilities and extensions theories, fuzzy numbers, decision weights, explicit risk measures such as variants and scenarios. Ben-Amor et al. (2015) also presented a synthesis of main theories to deal with uncertainty in decision-making processes. This study emphasizes the conditions of use of these theories and proposed some guidelines to help decision makers (DMs) choose the appropriate representation language for uncertainty with the aim of reducing the drawbacks of an arbitrary choice of an approach to model uncertainty. In recent years, new approaches have emerged and been used to model uncertain data in decisionmaking problems. Therefore, this paper aims to present an updated literature review, first to identify the different types of uncertainty that occur in input data of MCDM/MCDA and then to identify the most appropriate techniques to deal with them. This study thus presents a review of a new generation of fuzzy approaches, which was not covered in the previous studies. Moreover, based on the papers surveyed, we present a guide to help DMs and analysts choose the most suitable technique to model uncertainty according to the type of data and uncertainty and based on the papers selected. To carry out our study, we followed four steps. First, we collected articles that contain terms associated with MCDM and uncertainty from the academic databases, Web of Science and Scopus. As we found thousands of papers, we selected all articles published between 2011 and 2017 and only the most cited papers published before 2011. Next, that list was supplemented with other articles that we came across while conducting the review, as they were frequently cited in the previously selected articles. As a result, we selected 134 articles, 106 of which were published between 2011 and 2017. Next, we sorted the selected articles into three different categories: type of data with uncertainty, type of uncertainty and the technique used to deal with uncertainty. In the third step, we analyzed the selected articles following this categorization, and in the fourth stage we proposed a guide that indicates techniques used in different decision-making contexts under uncertain. As a result, we identified three distinct types of uncertainty in input data of MCDM/MCDA, namely (i) uncertainty due to ambiguity (Campos et al., 2015), (ii) uncertainty due to stochasticity C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research


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(Banuelas and Antony, 2004), and (iii) uncertainty due to partial information (Liu and Wang, 2007; Tervonen et al., 2009; Chen and Yang, 2011). Then, we identified FSs theory, probability theory, evidential reasoning (ER) theory, rough set (RS) theory, linguistic variables (LVs), grey numbers (GNs), multi-attribute utility theory (MAUT), and robust ordinal regression (ROR) as the techniques most commonly used to deal with these types of uncertainty. We also observed that the use of fuzzy approaches to deal with uncertainty has been recently increasing compared to probability theory and other traditional techniques suitable to deal with uncertainties around data. The outline of the paper is as follows. In Section 2, we present the method used to select the articles. In Section 3, we define the classification used to analyze the selected papers. In Section 4, we provide a detailed analysis of the articles and, in Section 5, we discuss the main results obtained. In Section 6, we introduce the framework that indicates techniques used in different decision-making contexts under uncertainty. Limitations of this study, future research works, as well as the conclusion of this review are presented in Section 7.

2. Materials and methods The difference between an overview paper and a review paper is the fact that the former does not need to add value, while the latter does. According to Wee and Banister (2016), possibilities for adding value in a literature review include one or a combination of the following: presenting a synthesis of what is already known (focus on empirical insights), developing an analysis of methods used, their advantages and disadvantages (focus on methodologies), investigating the different theories used and their importance (focus on theories), discussing or synthesizing how useful the literature is for real-world applications (focus on relevance for real-world applications), or providing an explicit structure on how dependent and independent variables are related (conceptual model/ framework). Our review has focused on two of these options. On empirical insights, we did a synthesis of what is already known about techniques to represent uncertainty in input data of MCDM/MCDA, and on a conceptual model, we propose a framework to help choosing a technique in a specific decision-making context under uncertainty among all techniques identified. To ensure the validity and the reproducibility of the results, a systematic literature review must be developed carefully and accurately (Meredith, 1993). Therefore, we adopted a four-step methodological process to carry out our study: (1) collection of materials, (2) category selection, (3) analysis of the collected articles, and (4) proposal for a framework. In the material collection step, which is depicted in Fig. 1, we first developed a preview study to help in the definition of the keywords and filters used to select the papers. In that preview study, we analyzed scientific papers and books that are considered standard reference in the MCDM/MCDA field and also consulted specialists in MCDM/MCDA. We then defined the search string in such a way that would not restrict the search, using only keywords related to uncertainty (uncertain, uncertainty, inaccurate, imprecise, partial information, missing information), MCDM (multiple attribute, multi-attribute, multiple criteria, multi-criteria, MCDM, MCDA), and decision making (decision making, decision aiding, decision analysis). The defined search string is shown in Table 1. Second, we searched Scopus and Web of Science using the defined search string and found 2356 and 2078 articles, respectively. Then, we applied the following filters to limit our collection of articles: C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research

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Fig. 1. Methodological process for material collection. Table 1 Search string used to search articles in the Scopus and Web of Science databases Search string (“decision making” or “decision aiding” or “decision analysis”) in Topic AND (“multiple attribute” or “multi-attribute” or “multiple criteria” or “multi-criteria” or MCDM or MCDA) in Topic AND (uncertain or uncertainty or inaccurate or imprecise or “partial information” or “missing information”) in Topic

r To achieve the highest level of relevance, we selected only international journal articles. Thus, reviews, conference articles, master, and doctoral dissertations, textbooks, unpublished articles, and notes were not included in this study. r We excluded papers focusing solely on multi-objective decision making (MODM) as being beyond the scope and objective of this paper. r We selected all articles with publication dates after 2011. r Duplicate articles in Scopus and Web of Science were removed. C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research


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r With respect to the articles published before 2011, we used the Pareto criterion and selected only the articles that together represent 80% of the total number of citations. Therefore, we selected only the old papers (before 2011) that are still considered relevant in the MCDM/MCDA field based on the citation number. r We selected only articles that have used some kind of MCDM/MCDA method. r We selected only articles that have used or proposed some technique to deal with uncertainty in input data. r We excluded articles that presented an application of some technique already described in another selected article. The screening of the articles was done primarily automatically. The two former conditions were implemented automatically using the academic databases filters, resulting in 1242 articles in Web of Science and 1326 in Scopus. The third, fourth, and fifth conditions were applied automatically using a web system designed to apply automated filters for selection of articles for systematic reviews (Pelissari et al., 2017), resulting in around 900 articles. We then read the abstracts of these 900 articles and manually applied the three latter conditions. The last condition was responsible for a large reduction in the number of selected papers, which is valid because the aim of this study is to identify the techniques in a way sufficient to propose the framework, not to discuss them or their different applications in detail. This initial list of selected articles was complemented with other articles that we identified while conducting the review and tracking the references. As a result, we selected 134 articles. These 134 articles were sufficient to answer the research questions and to build the framework. After reading in full the 134 selected papers, we classified them into three different categories, as shown in Fig. 2: (1) type of data with uncertainty, (2) type of uncertainty, and (3) technique used to deal with uncertainty. We analyzed and discussed the selected articles based on these categories. Then, we defined the framework to help select a technique in a specific decision-making context under uncertainty among all available techniques.

3. Classification When we began with the classification process in order to meet the goal of this study, we had in mind to classify the articles by types of uncertainty and the types of techniques used. As different terms and classification have been used in the literature to categorize types of uncertainty, to propose a categorization for this review was not easy. We started then by sorting out each paper into the three groups as proposed by Ben-Amor et al. (2015). However, the classification of the different types of uncertainty turned out to be a dynamic process. We realized that in many papers the type of uncertainty was discussed based on its relation to the type of data used. Therefore, we ended up with the classification ambiguity, stochasticity, and partial information, which is similar to the one proposed by Ben-Amor et al. (2015) but is directly related to the type of data. After that, we also identified the need to consider the classification based on the type of data. Therefore, we classified the selected articles into three different categories: (1) type of data with uncertainty, (2) type of uncertainty, and (3) technique used to deal with uncertainty, as shown in Fig. 2 and discussed in detail as follows. C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research

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Fig. 2. Categorization of the selected articles.

3.1. Type of data with uncertainty Uncertainty can be present in different types of data. Moreover, we could identify that the type of uncertainty has a strong relationship with the type of data. In addition, the type of data depends on the entity or item that is being assessed. MCDM/MCDA involves the assessment of entities (attributes, criteria, and parameters) that can be quantitative or qualitative in nature. A quantitative entity is characterized by being measurable, for example, price, speed, costs, and percentages (market share, profit margins). A qualitative entity, in contrast, cannot be measured. It can be assessed using ordinal or nominal data. Ordinal data have natural, ordered categories, and the distances between the categories are not known. A decision-making situation in which DMs can give their opinion and evaluations using linguistic values, for instance, the evaluation ratings described as very good, good, or bad, is an example of the use of ordinal data as a natural way for the DMs to express themselves in a real-world decision-making problem. Nominal data have two or more categories with no notion of order. This type of data is rarely encountered in assessments for MCDM/MCDA problems. A quantitative entity can be evaluated using cardinal data (interval/ratio scales of measurement). Cardinal data are numerical values for which arithmetic operations can be applied. They result from the measurement of quantitative entities such as the height of a building (in meters) and the temperature of a room (in degrees Celsius). However, there are some situations in which a quantitative entity cannot be represented by cardinal data. For instance, when measuring the height of a building or the exact temperature in a room is not possible, one would use ordinal data instead such as “very high,” “high,” and “low.” Therefore, quantitative entities can also be assessed using ordinal data. C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research


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In summary, we can categorize the types of data with uncertainty in our context into ordinal and cardinal.

3.2. Type of uncertainty First, before defining the types of uncertainty, it is important to differentiate types of uncertainty from sources of uncertainty. A decision aid process may involve different types of uncertainties resulting from different sources. A decision aid process may involve different types of uncertainties resulting from different sources and these sources can be situated at different levels such as the decision-making model/technique used, decision maker(s) involved, and the decision environment, as presented by Walker et al. (2003) and Ben-Amor et al. (2015). The model/technique used can be a source of uncertainty because each theory is based on some kind of hypotheses, assumptions, or axioms. If these hypotheses or assumptions are not met, the chosen modeling background end up as a source of uncertainty. Therefore, not all theories are “applicable” without discrimination. Uncertainty can also result from the DM when he/she has some difficulty with the statement of his/her knowledge impacting the quality of data resulting from their observations or measurements. The environment is also a source of uncertainty, in cases where the data are difficult to obtain or to verify, or the data are based on forecasting, for example. We classified the main types of uncertainty affecting input data of MCDM/MCDA problems into three groups: ambiguity, stochasticity, and partial information.

r Ambiguity: This type of uncertainty is present when the input data are open to more than one interpretation. Ambiguity can be associated with ordinal data, present in most of the cases in which linguistic terms are used (Yeh and Chang, 2009), as well as with cardinal data, in cases when these values are determined by intervals (Campos et al., 2015; Nemery et al., 2015). This type of uncertainty is also referenced in the literature as imprecision, when related to the use of cardinal data, and to vagueness when related to the use of linguistic terms. An ordinal or cardinal ambiguity uncertainty can be present either in the criteria weights definition or in the alternatives evaluation, as well as in other parameters of the model. For example, thresholds and limiting profiles in an outranking sorting method can be defined by interval data (Nemery et al., 2015). Usually, ambiguity results from a difficulty in the statement of the knowledge and it is related to DMs. r Stochasticity: When the input data are stochastic variables, that is, they can be identified as aleatory. To have this type of uncertainty, the data may be ordinal (such as discrete probability distribution) or cardinal (continuous probability distribution). This type of uncertainty can be present either in the definition of criteria weights or in the alternatives evaluation or even in the model’s parameters (Lahdelma et al., 1998; Banuelas and Antony, 2004; Tervonen et al., 2009). The source of this type of uncertainty can be the DMs, in case that he/she is more or less reliable (not confident, intentionally or not providing making errors), or the environment, in cases where the data are difficult to obtain or to verify, or the data are based on forecasting, for example. r Partial information: This occurs when the information or part of it is not available or when DMs are not willing to express their preference/opinion (Lahdelma et al., 1998; Tervonen et al., 2009). C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research

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The criteria weights can be missing in part or totally while the evaluation of alternatives may only be missing in part. 3.3. Techniques used for modeling uncertainty We identified the main techniques used for modeling different types of uncertainty in input data of MCDM/MCDA problems and classified them into nine groups: (1) FSs theory, (2) probability theory, (3) ER theory, (4) RS theory, (5) GNs, (6) ROR, (7) MAUT, (8) others, and (9) hybrid models. Here hybrid models refer to methods that have used two or more of the techniques mentioned (fuzzy, probability, ER, etc.). 4. Results In this section, all the selected papers are presented and discussed in detail using the classification set up in the previous section. 4.1. Modeling uncertainty using FSs theory and its approaches FSs theory is by far the most commonly used technique used to deal with uncertainty in input data of MCDM/MCDA methods. FSs theory is an extension of classical set theory, and it enables the gradual assessment of the membership of elements in a set with a grade of membership taking a value between 0 and 1 (Zadeh, 1965). In contrast, in the classical theory, membership of elements in a set is assessed in binary terms—an element either belongs or does not belong to a set. Different fuzzy approaches have been used in MCDM/MCDA under uncertainty, such as Type-2 fuzzy sets (T-2FSs), intuitionistic fuzzy sets (IFSs), and hesitant fuzzy sets (HFSs) theories. These approaches are discussed in the remainder of this section. For a detailed and specific study about the use of FSs in MCDM/MCDA problems, see Mardani et al. (2015). 4.1.1. Traditional FSs theory Apply FSs theory to deal with uncertainty consists of associating a fuzzy number (that belongs to the FS) to the evaluations of the alternatives, weight criteria, or other model parameters. There are different forms of fuzzy numbers, triangular and trapezoidal being the two most common ones. FSs theory has been applied especially to deal with the uncertainty due to ambiguity in either cardinal or ordinal data. It can be used in association with traditional MCDM/MCDA methods such as AHP (Rezaei and Ortt, 2013; Ashtiani and Abdollahi Azgomi, 2016), ANP (Kang et al., 2012), ELECTRE (Hatami-Marbini and Tavana, 2011; Vahdani and Hadipour, 2011), PROMETHEE (Li and Li, 2010; Lolli et al., 2016), TODIM (Chen et al., 2017), MACBETH (Dhouib, 2014), and FlowSort (Campos et al., 2015). Bykzkan and Ifi (2012), Efe (2016), and Kabir and Sumi (2014) also proposed methods to deal with ordinal ambiguity using fuzzy modeling, but they used the combination AHP and TOPSIS, AHP and VIKOR, and AHP and PROMETHEE, respectively, in which AHP is used to define the criteria weights and TOPSIS, VIKOR, and PROMETHEE to rank the alternatives. C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research


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Table 2 Modeling uncertain input data of MCDM/MCDA problems using traditional fuzzy set theory Reference

Approach

Type of uncertainty

Type of data with uncertainty

MCDM/MCDA method associated

Vahdani and Hadipour (2011) Ashtiani and Abdollahi Azgomi (2016) Efe (2016) Lolli et al. (2016) Campos et al. (2015) Kabir and Sumi (2014) Dhouib (2014) Rezaei and Ortt (2013) Kang et al. (2012) Bykzkan and Ifi (2012) Nemery et al. (2015) Hatami-Marbini and Tavana (2011) Chen et al. (2017) Li and Li (2010)

FSs TriangFNs TriangFNs TriangFNs TriangFNs TriangFNs TriangFNs TriangFNs TriangFNs TriangFNs TrapeFNs TrapeFNs DiscrFNs GenerFNs

Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity

Ordinal Ordinal Ordinal Ordinal Cardinal Ordinal Ordinal Ordinal Ordinal Ordinal/cardinal Cardinal Ordinal Ordinal Ordinal

ELECTRE AHP/VIKOR AHP/TOPSIS PROMETHEE-II FlowSort AHP/PROMETHEE MACBETH AHP ANP AHP/DEMATEL/TOPSIS FlowSort ELECTRE-I TODIM PROMETHEE-II

FSs, fuzzy sets theory; TriangFNs, triangular fuzzy numbers; TrapeFNs, trapezoidal fuzzy numbers; DiscrFNs, discrete fuzzy numbers; GenerFNs, generalized fuzzy numbers.

FSs theory associated with MCDM/MCDA methods can also be applied for modeling uncertainty present in the model parameters, such as thresholds and limiting profiles of the Flowsort model (Nemery et al., 2015) defined by interval data. We present some references regarding traditional FSs theory in Table 2. 4.1.2. Type-2 FSs theory In FSs (also called Type-1 FSs), each element’s membership degree is a crisp number in the interval [0,1]. In T-2FSs theory, the degree of membership is given by a FS. Consequently, T-2FSs is preferred to the traditional FSs theory in some situations because of its ability to deal with more uncertainties (Chen and Lee, 2010; Celik et al., 2015). T-2FSs can be associated either with traditional MCDM/MCDA methods (Chen and Lee, 2010; Chen, 2015; Ilieva, 2016; Yu et al., 2017) or with mathematical operators (Qin and Liu, 2015) to deal with uncertainty due to cardinal or ordinal ambiguity. To deal with cardinal and ordinal ambiguity uncertainty, there are also interval Type-2 fuzzy sets (IT-2FSs), which are characterized by an interval membership degree that can provide more degrees of freedom to represent uncertainty in real-world problems (Zhang and Zhang, 2013; Celik et al., 2016; Qin et al., 2017a). For a detailed study about the use of T-2FSs in MCDM/MCDA problems, see Celik et al. (2015). We present in Table 3 papers that used T-2FSs theory. 4.1.3. IFSs theory IFSs theory was developed by Atanassov (1986) and is suitable for situations in which there is uncertainty about the degree of membership of an element in a defined set: each element in an IFS has a membership degree and a nonmembership degree between 0 and 1. C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research

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Table 3 Modeling uncertain input data of MCDM/MCDA problems using Type-2 fuzzy sets theory Reference

Approach

Type of uncertainty



Ilieva (2016) Chen and Lee (2010) Qin and Liu (2015) Chen (2015) Qin et al. (2017a) Yu et al. (2017) Celik et al. (2016) Zhang and Zhang (2013)

T-2FSs(Triang) T-2FSs(Trape) T-2FSs T-2FSs IT-2FSs IT-2FSs IT-2FSs IT-2FSs

Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity

Ordinal/cardinal Ordinal/cardinal Ordinal Ordinal Ordinal Ordinal Ordinal Cardinal

TOPSIS TOPSIS MO PROMETHEE TODIM MABAC ELECTRE-I MO

T-2FSs, Type-2 fuzzy sets; MO, matematical operator; IT-2FSs, interval Type-2 fuzzy sets.

IFSs theory is a widely used technique to deal with uncertainty due to ambiguity, especially resulting from the divergence among DMs (Liu and Wang, 2007; Xu and Yager, 2008; Tan and Chen, 2010; Li, 2011; Wu and Chen, 2011; Xu, 2011; Liu and Jin, 2012; Shen et al., 2016). This approach can be associated with traditional MCDM/MCDA methods such as ELECTRE-I (Wu and Chen, 2011; Devi and Yadav, 2013), TODIM (Wang and Liu, 2017; Qin et al., 2017b), AHP (Buyukozkan and Goer, 2017), PROMETHEE (Montajabiha, 2016) and Superiority and Inferiority Ranking (SIR) (Chai et al., 2012), a generation of the PROMETHEE method, and other outranking methods (Shen et al., 2016). Many other studies that used IFSs proposed mathematical operators instead of traditional MCDM/MCDA methods (Tan and Chen, 2010; Li, 2011; Xu, 2011; Liu and Jin, 2012). Sometimes it is not appropriate to assume that the degrees of membership for certain elements in a set are exactly defined, so we admit a kind of further uncertainty—the value of degree of membership of an element is not a crisp number anymore, but a whole interval—similar to the IT-2FSs. For these situations, Atanassov and Gargov (1989) proposed the interval-valued intuitionistic fuzzy sets (IVIFSs), which are an extension of IFSs. IVIFSs can be used to deal with uncertainty by ambiguity (Park et al., 2011; Chen, 2014; Li et al., 2014; Wei and Zhang, 2015; Hashemi et al., 2016) associated with traditional MCDM/MCDA methods (Park et al., 2011; Abdullah and Najib, 2016; Hashemi et al., 2016), with mathematical operators (Chen and Yang, 2011; Li et al., 2014) or with other methods such as linear assignment methods (Chen, 2014). In these cited studies, all the preference information provided by DMs is represented by IVIFSs. Chen and Yang (2011) proposed a method using both IFSs and IVIFSs in which preference information provided by DMs for criteria weights is presented as IFS and for evaluation of alternatives as IVIFSs. Some references about IFSs theory are presented in Table 4. 4.1.4. HFSs theory HFSs theory was first introduced by Torra and Narukawa (2009). HFSs permits the membership degrees of an element to be a set of several possible values between 0 and 1. HFSs are highly useful in dealing with situations in which people express hesitancy when providing their preferences over objects in the decision-making process (Torra and Narukawa, 2009). This is usually the case C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research


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Table 4 Modeling uncertain input data of MCDM/MCDA problems using intuitionistic fuzzy sets theory Reference

Type of Approach uncertainty



Buyukozkan and Goer (2017) Qin et al. (2017b) Wang and Liu (2017) Montajabiha (2016) Shen et al. (2016) Devi and Yadav (2013) Liu and Jin (2012) Chai et al. (2012) Li (2011) Xu (2011) Wu and Chen (2011) Tan and Chen (2010) Xu and Yager (2008) Liu and Wang (2007), Abdullah and Najib (2016) Hashemi et al. (2016) Chen (2014) Li et al. (2014) Wei and Zhang (2015) Park et al. (2011) Chen and Yang (2011)

IFSs IFSs IFSs IFSs IFSs IFSs IFSs IFSs IFSs IFSs IFSs IFSs IFSs IFSs IVIFSs IVIFSs IVIFSs IVIFSs IVIFSs IVIFSs IVIFSs

Ordinal Cardinal Ordinal Ordinal Cardinal Cardinal Ordinal/cardinal Cardinal Ordinal/cardinal Cardinal Ordinal/cardinal Cardinal Cardinal Ordinal/cardinal Ordinal Ordinal Cardinal Cardinal Cardinal Cardinal Cardinal

AHP TODIM TODIM PROMETHEE-II Outranking relations ELECTRE Hybrid geometric operator SIR PROMETHEE GOWA FPAO ELECTRE-I AO–Choquet integral operator UDIFWA IFPO AHP ELECTRE III Linear assignment method Weighted average operator Mathematical operator ELECTRE-II AO

Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity/partial info. Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Partial Info.

IFSs, intuitionistic fuzzy sets; GOWAO, generalized ordered weighted averaging operator; FPAO, fuzzy power aggregation operators; AO, aggregation operator; UDIFWA, dynamic intuitionistic fuzzy weighted averaging; IFPO, intuitionistic fuzzy point operator; IVIFSs, interval-valued intuitionistic fuzzy sets.

Table 5 Modeling uncertain input data of MCDM/MCDA problems using hesitant fuzzy sets Reference

Approach

Type of uncertainty



Fahmi et al. (2016) Zhang et al. (2015) Chen et al. (2015c) Zhang and Xu (2014) Zhang and Wei (2013) Wang et al. (2014c) Wei (2017) Gitinavard et al. (2017) Joshi and Kumar (2016) Chen et al. (2013) Xu and Zhang (2013) Peng et al. (2014)

HFSs HFSs HFSs HFSs HFSs HIVFSs DHIVFSs HIVFSs HIVFSs HIVFSs HIVFSs HIVIFSs

Ambiguity Partial Info. Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity/partial info. Ambiguity Ambiguity Ambiguity/partial info. Ambiguity

Cardinal Cardinal Ordinal Cardinal Cardinal Ordinal/cardinal Ordinal Ordinal Ordinal Cardinal Cardinal Cardinal

ELECTRE-I AO ELECTRE-II TODIM VIKOR/del TOPSIS Prioritized aggregation operator Arithmetic and geometric AO DEMATEL AO–Choquet integral operator AO TOPSIS ELECTRE-I

HFSs, hesitant fuzzy sets; AO, aggregation operator; HIVFSs, hesitant interval-valued fuzzy sets; HIVIFSs, hesitant intervalvalued intuitionistic fuzzy sets. C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research

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in group decision making when DMs have different opinions about the same topic (Chen et al., 2015c). For instance, Chen et al. (2015c), to deal with this uncertainty, proposed the model hesitant fuzzy ELECTRE-I that calculates the degrees of membership of each alternative in a criterion using the degrees of membership of each DM. Moreover, HFSs also appears to be suitable to deal with uncertainty due to cardinal ambiguity (Zhang and Wei, 2013; Zhang and Xu, 2014; Chen et al., 2015c; Fahmi et al., 2016) and partial information (Zhang et al., 2015). For a detailed study about HFSs, see Rodrguez et al. (2016). Generalizing the concept of HFSs, Chen et al. (2013) proposed the hesitant interval-valued fuzzy sets (HIVFSs) in which the membership degrees of an element to a given set are not precisely defined but are denoted by several possible interval values. Joshi and Kumar (2016) and Xu and Zhang (2013) integrated HIVFSs and TOPSIS to model ordinal ambiguity data used in alternatives evaluation and also in the criteria weights resulting from divergence among the DMs. Gitinavard et al. (2017) integrated HIVFSs with DEMATEL to deal with ordinal ambiguity data and partial weight information. Wei (2017) proposed operators based on dual hesitant fuzzy, another extension of HFSs, to model MCDM/MCDA problems with ordinal ambiguity. Furthermore, although the theories of IVIFSs and HIVFSs have been developed and generalized, they cannot deal with all kind of uncertainties in different real problems. For example, when we ask the opinion of an expert about a certain statement, he or she may answer that the possibility that the statement is true is [0.1, 0.2] and that the statement is false is [0.4, 0.5], or that the possibility that the statement is true is [0.5, 0.6] and that the statement is false is [0.3, 0.5] (Peng et al., 2014). This issue is beyond the scope of IVFSs and IVIFSs (Peng et al., 2014). To overcome this limitation, Peng et al. (2014) proposed the concept of hesitant interval-valued intuitionistic fuzzy sets (HIVIFSs). Compared with the existing above-mentioned FSs, HIVIFSs are a new extension of HFSs, which support a more flexible and simpler approach when DMs provide their decision information in a HIVIFSs environment. Furthermore, IVIFSs, HFSs, HIVFSs, and HIFSs are all special cases of HIVIFSs. 4.1.5. Modeling uncertain input data of MCDM/MCDA problems using representations of LVs The concept of LVs was presented by Zadeh (1975a, 1975b, 1975c). LVs are variables whose values are not numbers, but words or sentences in a natural or artificial language. LVs are used to represent qualitative evaluations, especially in decision-making contexts in which evaluations are not performed with numerical information, but with linguistic expressions (Merigo´ et al., 2016). Intuitionistic linguistic fuzzy sets (ILFSs) and hesitant linguistic fuzzy sets (HLFSs), extensions of the IFSs and HFSs, respectively, are suitable techniques to model LVs. ILFSs associate IFSs theory with LVs and are characterized by a linguistic membership degree and a linguistic nonmembership degree. They can be used to deal with uncertainty due to ambiguity, as presented by Yu et al. (2018). HLFSs associate hesitant fuzzy theory with LVs, similar to the ILFSs. They can be used to deal with uncertainty due to the ambiguity (Hadi-Vencheh and Mirjaberi, 2014; Chen and Xu, 2015) associated with traditional MCDM/MCDA methods (Wang et al., 2014b) or with mathematical operators (Liao et al., 2014; Meng et al., 2014). In most cases, HLFSs are applied to deal with ambiguity resulting from the divergence among the DMs. Joining linguistic hesitant fuzzy sets and interval-valued hesitant fuzzy sets, interval-valued hesitant linguistic fuzzy sets (IVHFLS), is another approach to deal with ambiguity (Wang et al., 2014c). C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research


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Table 6 Modeling uncertain input data of MCDM/MCDA problems using representations of linguistic variables Reference

Approach

Type of uncertainty



Yu et al. (2018) Chen and Xu (2015) Liao et al. (2014) Hadi-Vencheh and Mirjaberi (2014) Meng et al. (2014) Wang et al. (2014b) Xu et al. (2012) You et al. (2015) Xu (2004) Herrera et al. (2008) Herrera and Mart´ınez (2000)

ILFSs HLFSs HLFSs HLFSs

Ambiguity Ambiguity Ambiguity Ambiguity

Ordinal Ordinal Ordinal Ordinal

TODIM ELECTRE-II Distance and similarity measures TOPSIS

HLFSs HLFSs LVs LVs LVs Ling. unbalaced 2-Tuple LVs

Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity

Ordinal Ordinal Ordinal Ordinal Ordinal Ordinal Ordinal

Hybrid aggregation operators ELECTRE-I LPAO VIKOR LAO Aggregation operators Aggregation operators

ILFSs, intuitionistic linguistic fuzzy sets; HLFSs, hesitant linguistic fuzzy sets; LVs, linguistic variables; LPAO, linguistic power aggregation operators; LAO, linguistic aggregation operators.

Although fuzzy theory stands out as a technique for dealing with LVs, many authors have been studying other alternative representations of the LVs, including the 2-tuple linguistic approach (Herrera and Mart´ınez, 2000), probabilistic LVs (Xu, 2004), linguistic unbalanced information (Herrera et al., 2008), uncertain linguistic unbalanced information (Herrera et al., 2008), linguistic cloud information (Wang et al., 2014a), and different linguistic aggregation operators (Xu et al., 2012). Most of these studies justify their importance by the fact that by using FSs theories, approximation processes must be developed to express the results in the initial expression domain. This result in a loss of information and hence a lack of precision in the final results (Herrera and Mart´ınez, 2000), which does not happen with the 2-tuple linguistic approach, for example. Representations of LVs can also be applied to deal with divergence in the evaluation of alternatives in group decision making, as proposed by You et al. (2015). Some references about representations of LVs are presented in Table 6. The selected papers that used IFSs are presented in Table 4.

4.2. Modeling uncertainty using probability theory Probability-based methods incorporate probabilistic distributions to evaluation of alternatives and criteria weights (Banuelas and Antony, 2004; Gervsio and Simes Da Silva, 2012). By using a suitable type of probability distribution, it is possible to take into account stochasticity uncertainty type. In particular, uniform probability distribution make it possible to model cardinal and ordinal ambiguity, and also partial or completely missing information, as explained by Tervonen and Lahdelma (2007). In addition, using probability-based methods to model uncertain input data of MCDM/MCDA problems allows to test results in a statistical manner and enables a decision making at a certain confidence level, which is not possible either in traditional MCDM/MCDA methods or in fuzzy-based MCDM/MCDA methods. One option to integrate probabilistic distributions to C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research

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MCDM/MCDA method is to use Monte Carlo simulation to generate values from the distributions and to use these simulated values as inputs to the MCDM/MCDA method. Banuelas and Antony (2004) integrated probability theory to the AHP method, allowing the AHP pairwise comparisons to be stochastic. Jalao et al. (2014) also proposed a stochastic AHP but applied exclusively beta distribution to model the varying stochastic preferences of the DM. Gervsio and Simes Da Silva (2012) integrate probability theory to the AHP and PROMETHEE methods. In this study, stochastic AHP is applied to model uncertain preference weight information; probability distributions are attributed to alternatives evaluation, and thresholds parameters and PROMETHEE are applied to rank the alternatives. Nelyubin and Podinovski (2017) proposed a MCDM/MCDA choice method to model partial criteria weight information assigning a uniform probability distribution to the criteria weights and applying Monte Carlo simulation. Therefore, this method is a volume-based method for decision making under partial information, such as the stochastic multi-criteria acceptability analysis (SMAA) family of methods, presented as follow. The family of SMAA methods comprises methods for discrete multicriteria group decision making in which neither evaluation of alternatives measurements nor criteria weights are precisely known. Unlike the traditional MCDM/MCDA methods, which require that the DMs define the criteria weights, SMAA can be applied whenever criteria weights are not available or only partial information are available. Consequently, we can use SMAA methods to deal with uncertainty due to partial information of criteria weights. Moreover, SMAA methods allow stochastic data in the alternatives evaluation and model parameters, modeling them by using probability distributions. Consequently, SMAA methods allow ambiguity data since probability distributions can be used to model them. Several MCDM/MCDA traditional methods have been integrated to SMAA, building the family of SMAA methods. The standard SMAA-based methods, such as SMAA-2 and SMAA-O, use partial value/utility functions (Lahdelma et al., 1998; Lahdelma and Salminen, 2001; Lahdelma et al., 2003). SMAA-D (Lahdelma and Salminen, 2006) is an extension based on DEA. SMAA3 (Lahdelma and Salminen, 2002) is an ELECTRE III extension. SMAA-TRI and SMAAPROMETHEE are extensions for the methods ELECTRE TRI and PROMETHEE (Tervonen et al., 2009; Corrente et al., 2014), and SMAA-TOPSIS and SMAA-VIKOR are extensions of the compromise methods TOPSIS and VIKOR (Okul et al., 2014; Aydogan and Ozmen, 2017). Recently, Zhang et al. (2017a) proposed the SMAA-TODIM. For more information about SMAAbased methods, see Tervonen and Figueira (2007). We present the studies about probability-based methods in Table 7.

4.3. Dealing with uncertainty using the ER theory ER, developed by Dempster (1967) and extended by Shafer (1976), is a general approach for analyzing MCDM/MCDA problems under various types of uncertainty using a unified framework belief structure. Instead of describing a MCDM/MCDA problem with a decision matrix, the ER approach uses an extended decision matrix, in which each evaluation of an alternative is described by a distributed assessment using a belief structure. For example, the distributed evaluation regarding the comfort of a car could be excellent (60%), good (40%), average (0%), poor (0%), or worst (0%), C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research

Approach Probability Probability Probability Probability SMAA-2 SMAA-2 SMAA-2 SMAA-2 SMAA-2 SMAA-2 SMAA-2 SMAA SMAA-2 SMAA

Reference

Nelyubin and Podinovski (2017) Jalao et al. (2014) Gervsio and Simes Da Silva (2012) Banuelas and Antony (2004) Aydogan and Ozmen (2017) Zhang et al. (2017a) Corrente et al. (2014) Okul et al. (2014) Lahdelma and Salminen (2006) Tervonen et al. (2009) Lahdelma et al. (2003) Lahdelma and Salminen (2002) Lahdelma and Salminen (2001) Lahdelma et al. (1998)

Partial info. Ambiguity/stochasticity Ambiguity/stochasticity Stochasticity Partial info./stochas Ambig./partial info./stochas. Ambig./partial info./stochas. Partial info./randomness Ambig./partial info./stochas. Ambig./partial info./stochas. Ambiguity/partial info./stochas. Ambig./partial info./stochas. Ambig./partial info./stochas. Ambig./partial info./stochas.

Type of uncertainty

Table 7 Dealing with uncertainty in input data of MCDM/MCDA problems using probability theory

– Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal


– AHP-beta distrib. PROMETHEE AHP VIKOR TODIM PROMETHE-II TOPSIS DEA ELECTRE-TRI Utility function ELECTRE-III Utility function Utility function


R. Pelissari et al. / Intl. Trans. in Op. Res. 00 (2018) 1–37 15

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which means that the comfort of the car is assessed to be excellent with 60% of belief degree and good with 40% of belief degree. The difference between probability and ER theories is that in the case of probability, we assign uncertainties to each possible event and these events are mutually exclusive and collectively exhaustive. Moreover, the sum of all these uncertainties assigned to individual events is equal to one. However, in the ER theory, we assign uncertainties not only to each event but also to all its proper subsets. Besides, ER theory does not require the assignment of values for all events. Consequently, we can model partial information under ER theory (Ju and Wang, 2012). The advantage of using ER theory is that, in addition to accepting crisp values and partial information, it allows different types of uncertainties such as stochasticity (Wang et al., 2006) and both ordinal and cardinal ambiguity (Deng et al., 2014; Guo et al., 2007). Liu et al. (2015) proposed a sorting method that deals with ordinal ambiguity in the evaluation of alternatives and in the criteria weights, the latter resulting from divergence among the DMs. Partial information is also allowed. If this is the case, the total sum of belief degrees in the distributed assessment for that attribute will be between 0% and 100%. Complete missing information for the evaluation of alternatives is also accepted. In this case, the total sum of belief degrees in the distributed evaluation for that attribute will be 0. Some references about ER theory for MCDM/MCDA are presented in Table 8. 4.4. Dealing with uncertainty using RSs RSs theory was proposed by Pawlak (1982) and can be considered as an alternative to FSs theory, probability, and ER theory. An important advantage of the RSs approach is that it can deal with a set of inconsistent examples, that is, observations that the evaluation of alternatives or criteria weights that have the same description are indiscernible. The indiscernibility relation between objects is the mathematical basis of the RSs theory. Objects are indiscernible by condition attributes but discernible by decision attributes. RSs theory is also an extension of the classic set theory and can be considered a formal approximation of a crisp set (i.e., the conventional set in the classic set theory) in terms of a pair of sets that give the lower and the upper approximation of the original set. In the traditional RSs theory, these lower and upper sets are crisp sets (similar to the lower and upper values 0 and 1 in the FSs theory). There are also some RS variants, such as dominance-based rough set approach (DRSA) (Greco et al., 2002). RSs theory and its variants have been applied in MCDM/MCDA problems to deal with both ordinal and cardinal ambiguity (Greco et al., 1997, 2011b) and also for partial information situations (Greco et al., 1999). RSs theory has also been integrated with traditional MCDM/MCDA methods, as the integration of the traditional RS theory with AHP (Lee et al., 2012) and TOPSIS (He et al., 2016) and DRSA with ANP method (Palmisano et al., 2016). We present the studies about MCDM/MCDA RS-based methods in Table 8. 4.5. Dealing with uncertainty using GNs Another approach to deal with uncertainty in MCDM/MCDA problems is the grey numbers (GNs) theory. A grey number is a number with clear upper and lower boundaries but which has an unknown C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research

ER ER ER ER ER Rough sets Rough sets Rough sets Rough sets Rough sets Rough sets GNs GNs GNs GNs ROR ROR ROR ROR ROR ROR MAUT MAUT MAUT Linear program Linear program FITradeoff Interval analysis

Liu et al. (2015) Deng et al. (2014) Ju and Wang (2012) Guo et al. (2007) Wang et al. (2006) Palmisano et al. (2016) He et al. (2016) Lee et al. (2012) Greco et al. (2011b) Greco et al. (1997) Greco et al. (1999) Zhou et al. (2017a) Kuang et al. (2015) Lin et al. (2008a) Lin et al. (2008b) Greco et al. (2014) Kadzinski and Tervonen (2013) Corrente et al. (2013) Corrente et al. (2012) Kadzinski et al. (2012) Greco et al. (2011a) Jiménez et al. (2013) Malakooti (2000) Sage and White (1984) Liu et al. (2017) Sun and Zhu (2017) de Almeida et al. (2016) Janssen and Nemery (2013)

Ambiguity Ambiguity Partial info. Ambiguity Stochasticity Lack of info. Ambiguity Ambiguity Ambiguity Ambiguity Partial info. Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Ambiguity Partial information Partial information Ambiguity/partial inform. Partial info. Partial info. Partial info. Ambiguity

Type of uncertainty

ER, evidential reasoning; MO, mathematical operator; GNs, grey numbers.

Approach

Reference Ordinal Ordinal Cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Cardinal Cardinal Cardinal Cardinal Cardinal Cardinal Cardinal Cardinal Cardinal Cardinal Cardinal Cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Cardinal Cardinal


MO MO AHP/TOPSIS MO MO ANP/decision rules TOPSIS AHP Decision rules Decision rules Decision rules TODIM PROMETHEE-II TOPSIS TOPSIS UTADIS/choquet integral SMAA ELECTRE Utility functions PROMETHEE ELECTRE Utility function Utility function Utility function WAO PROMETHEE-II MAVT FlowSort


Table 8 Dealing with uncertainty in input data of MCDM/MCDA problems using evidential reasoning, rough sets, grey numbers, ROR, MAUT, and others techniques

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position within the boundaries. For example, when a DM does not have cardinal information (such as the budget of a new project), but s/he can provide a possible range for this information based on her/his experience and knowledge, this range (interval number) can be considered a GN. Therefore, GNs and interval analysis or interval-valued FSs are similar concepts. However, the main difference is as follows: assume that the interval [0.2, 0.3] and the discrete set {0.2, 0.3} are two possible variants to represent a GN defined on [0, 1]. As the first is an interval, there is an infinite number of possible candidates. The second variant, a discrete set, cannot be modeled using an interval, and therefore has a different level of uncertainty associated with it. GNs can be used for both, while FSs theory and its extensions and interval analysis can only be used for the former case. GNs can be used to deal with uncertainty due to cardinal ambiguity integrated with PROMETHEE-II (Kuang et al., 2015), TODIM (Zhou et al., 2017a), and TOPSIS (Lin et al., 2008a, 2008b). The difference between the studies proposed by Lin et al. (2008b) and Lin et al. (2008a) is that the former focus on cardinal ambiguity resulting from divergence among DMs. Some references about GNs for MCDM/MCDA are presented in Table 8.

4.6. Dealing with uncertainty using robust ordinal regression In MCDM/MCDA problems, DMs may have many sets of values for representing the preference information due to uncertainty. Usually, the DMs have to choose only one specific set and apply it to make the decision. Robust ordinal regression is a MCDM/MCDA method that considers all the sets of parameters compatible with the preference information in order to give a recommendation in terms of necessary and possible consequences of applying all the compatible preference models on the considered set of alternatives (Greco et al., 2010). Given this characteristic, the ROR approach can be applied to decision problems under uncertainty (Corrente et al., 2016). To deal with cardinal ambiguity, ROR has been applied to several MCDA preference models, such as additive utility functions-based methods (Corrente et al., 2012), ELECTRE methods (Greco et al., 2011a; Corrente et al., 2013), PROMETHEE methods (Kadzinski et al., 2012), and SMAA method (Kadzinski and Tervonen, 2013). Greco et al. (2014) applied ROR to the UTA, GRIP, and UTADIS methods, and to Choquet integral to represent interactions between criteria.

4.7. Dealing with uncertainty using MAUT MAUT is one of the most well-established theory and has the best theoretical basis for applications involving uncertainty in MCDM/MCDA problems (Monte and de Almeida-Filho, 2016). In this model, the evaluation of an alternative on a given criterion is measured by a utility function. The utilities attained by each alternative at the multiple functions are then summed and weighed by the scaling coefficients (which are usually known as weights) attached to those functions. However, building the utility functions for each criterion and setting the value of the weights requires eliciting the preferences of a DM, which is often problematic. A review of MAUT can be found in the study developed by Dyer (2016). C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research


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A number of papers on MAUT have dealt with incomplete information. Sage and White (1984) proposed the model of imprecisely specified multi-attribute utility theory, in which preference information about both weights and utilities are ambiguous and incomplete. Malakooti (2000) suggested a new efficient algorithm for ranking alternatives when there is incomplete information about the preferences and the value of the alternatives. More recently, Jiménez et al. (2013) integrated the dominance intensity measure within fuzzy weight to MAUT to deal with partial information.

4.8. Other techniques to deal with uncertainty Other techniques have been applied to deal with uncertainty in MCDM/MCDA methods and are presented briefly as follow. Linear programming models have been applied to model partial information in MCDM/MCDA problems. Liu et al. (2017) proposed a linear program to model incomplete information of criteria weights and used a weighted average operator as aggregation model. Sun and Zhu (2017) also proposed a linear program and integrated it to the PROMETHEE-II method to model incomplete information about criteria weights. Interval analysis has also been applied to represent cardinal ambiguity data defined by interval, that is, numerical values that may vary within a certain range. Janssen and Nemery (2013) integrated interval analysis to the FlowSort method to model evaluation of alternatives, criteria weights, and model parameters defined by interval values. de Almeida et al. (2016) proposed a new MCDM/MCDA method called FITradeoff (Flexible and Interactive Tradeoff) to deal with partial criteria weight information in the context of multiattribute value theory (MAVT). Aguayo et al. (2014) proposed a new dominance-measuring method to deal with ordinal information within MAVT.

4.9. Hybrid models to deal with multiple uncertainties Although many studies have already dealt with uncertainty in input data of MCDM/MCDA problems, as shown before, most of them deal with one type of uncertainty at a time. However, several MCDM/MCDA problems simultaneously involve multiple types of uncertainty (Munda et al., 1995; Ben-Amor et al., 2007) and require methods specifically designed for such situations. One characteristic of proper methods to deal with more than one type of uncertainty at the same time is the need to use more than one uncertainty modeling technique. Therefore, we call this type of method hybrid models. Articles presenting hybrid models are displayed in Table 9. Methods combining fuzzy and probability theories are the main type of hybrid models. Fuzzy stochastic based methods to deal with ambiguity (ordinal or cardinal) and stochasticity at the same type have been developed (Zaras, 2004; Zarghami and Szidarovszky, 2009a, 2009b; Maqsood, 2011; Gomes et al., 2012; Peng et al., 2013; Chen et al., 2015a, 2015b; Subagadis et al., 2016; Messaoudi et al., 2017). Xu and Zhou (2017) proposed the hesitant probabilistic fuzzy sets (HPFSs) theory, which adds probabilities to the original HFS elements, under the justification that none of the HFSs (such as IVHFLS, HLFSs, and HIVIFSs) can accommodate all types of uncertainty in realworld problems. In this study, the authors proposed some decision-making approaches for HPFSs C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research

ILFSs/linear program PLTSs PLTSs IVIFSs/optimiz. model HPFSs FSs/probability FSs/probability FSs/probability FSs/probability FSs/probability FSs/probability FSs/probability FSs/stochastic FSs/probability FSs/probability FSs/ER FSs/ER FSs/ER FSs/probability/ER Probability/ER FSs/ER LVs/probability FSs/probability/prosp. FSs/rough sets Probability/GNs

Zhang et al. (2017b) Liao et al. (2017) Pang et al. (2016) Ren et al. (2017) Xu and Zhou (2017) Messaoudi et al. (2017) Subagadis et al. (2016) Chen et al. (2015b) Maqsood (2011) Gomes et al. (2012) Chen et al. (2015b) Zarghami and Szidarovszky (2009b) Zarghami and Szidarovszky (2009a) Zaras (2004) Peng et al. (2013) Sun et al. (2017) Zhang et al. (2016) Deng et al. (2011) Ben-Amor et al. (2007) Yao (2008) Yang et al. (2006) Merigo´ et al. (2016) Yu et al. (2014) Shidpour et al. (2016) Zhou et al. (2017b) Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal Ordinal/cardinal Ordinal/cardinal Cardinal

Ambiguity Ambiguity Ambiguity/stochas./partial info. Ambiguity Ambiguity/partial info.

MO LPWA – AHP/TOPSIS ELECTRE

ANP MO AP

Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Ordinal/cardinal Cardinal

MCDM/MCDA method associated Outranking PROMETHEE-II TOPSIS TOPSIS OWO GP – – – NAIADE GAO OWA operator MO DRSA Utility function

Ordinal Ordinal


Ambiguity/partial info. Ambiguity/stochas. Ambiguity/stochas. Ambiguity/partial info. Ambiguity/stochas. Ambiguity/stochas. Ambiguity/stochas. Ambiguity/stochas. Ambiguity/stochas. Ambiguity/stochas. Ambiguity/stochas. Ambiguity/stochas. Ambiguity Ambiguity/stochas. Ambiguity/partial info. Ambiguity Ambiguity/stochas. Ambiguity Ambiguity/stochas.

Type of uncertainty

ILFSs, intuitionistic linguistic fuzzy sets; PLTSs, probabilistic linguistic term sets; IVIFSs, interval-valued intuitionistic fuzzy sets; HPFSs, hesitant probabilistic fuzzy sets; FSs, fuzzy sets; OWO, ordered weighted operator; GP, goal programming; GAO, genetic algorithms operator; OWA, ordered weighted averaging; MO, mathematical operator; ER, evidential reasoning; LPWA, linguistic probabilistic weighted average; AP, aggregation procedure; LVs, linguistic variables; GNs, grey numbers.

Approach

Reference

Table 9 Dealing with uncertainty in input data of MCDM/MCDA problems using hybrid models

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21

utilizing aggregation operator methods. Li and Wang (2017) combined HPFSs and PROMETHEE II methods to deal with cardinal ambiguity and stochasticity. Although hesitant fuzzy linguistic term set is applicable in expressing people’s hesitant ordinal (linguistic terms) preference information, sometimes it cannot represent cases in which the linguistic terms have different importance degrees. The probabilistic linguistic term sets (PLTSs), proposed by Pang et al. (2016), can avoid this drawback by allowing the DMs to give the weight of each linguistic term as a probability. Pang et al. (2016) integrated PLTS with TOPSIS and Liao et al. (2017) integrated PLTS with the PROMETHEE-II method. Merigo´ et al. (2016) aggregated linguistic aggregation operations with probability to model ordinal ambiguity and stochastic data in MCDM/MCDA problems. Zhou et al. (2017b) proposed a study integrating probability theory, more specifically SMAAELECTRE method, with GNs to deal simultaneously with cardinal ambiguity, partial information, and stochasticity. Yang et al. (2006); Deng et al. (2011), and Zhang et al. (2016) proposed ER fuzzy based methods to deal with different combinations of uncertainty: ordinal and cardinal ambiguity, and ambiguity and stochasticity. Ben-Amor et al. (2007) proposed a method based on fuzzy and ER that is an extension of stochastic dominance analysis to deal with ordinal and cardinal ambiguity data, partial information, and stochastic data. Many variations of RSs have been proposed in which the approximating sets (lower and upper) may be FSs or probabilistic sets, resulting in the fuzzy RS model (Deng and Yao, 2014; Shidpour et al., 2016; Sun et al., 2017) and in the probabilistic RS model (Yao, 2008), to deal with ordinal and cardinal ambiguity and stochastic values. Zhang et al. (2017b) proposed an outranking hybrid model based on the ELECTRE method applying ILFSs to model ordinal evaluation of alternatives and a linear program to derive the weights of criteria based on partial criteria weight information. Ren et al. (2017) proposed a method based on TOPSIS to deal with ambiguity and partial information. The authors integrated IVIFSs to TOPSIS to model ambiguous evaluations of alternatives and established an optimization model to determine the criteria weights when they are unknown or partially known.

4.10. Descriptive analysis In this section, we present a descriptive analysis of publication year, publication journal, frequency of type of data with uncertainty, frequency of type of uncertainty, and frequency of technique used to deal with uncertainty.

4.11. Distribution of papers by year and journal Figure 3 provides the frequency distribution by year for the 134 articles selected. We observe that the number of published articles has been increasing steadily over the years. This increase results in part from the method used to select the articles (all articles after 2011 were selected). A slight growth over the years before 2011 can be noted. Overall, we observe a slight growth of this type of study over the years, reflecting the global interest of the research community in the treatment of uncertainty in MCDM/MCDA problems. C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research

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Fig. 3. Distribution of selected articles sorted by publication year. Table 10 Distribution of selected articles with respect to publication journal Journal European Journal of Operational Research Information Sciences Expert Systems with Applications Applied Mathematical Modeling Omega Applied Soft Computing Knowledge-Based Systems IEEE Transactions on Fuzzy systems International Journal of Approximate Reasoning International Journal of Computer Integrated Manufacturing Soft Computing Decision Support Systems Energy IEEE Transactions on Systems, Man, and Cybernetics International Journal of Fuzzy Systems International Transactions in Operational Research Water Resources Management Thirty-eight journals with one paper Total

Number of articles

%

22 18 11 8 5 4 4 3 3 3 3 2 2 2 2 2 2 38

16 13 8 6 4 3 3 2 2 2 2 1 1 1 1 1 1 28

134

100

Table 10 provides the frequency distribution of the articles based on the journal in which they were published. The selected articles are distributed across 56 journals that cover a wide array of disciplines, including operations research, applied mathematics, computing, and production management. The European Journal of Operational Research contains the largest number of articles, comprising 22 of the 134 articles (16%), followed by the Journal of Information Sciences (18 articles), Expert Systems with Applications (11 articles), Applied Mathematical Modeling (eight articles), C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research


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Table 11 Distribution of selected articles by type of data with uncertainty Type of data with uncertainty Ordinal/cardinal Ordinal Cardinal Total

Number of articles

%

53 41 40

40 31 30

134

100

Table 12 Distribution of selected articles by type of uncertainty Type of uncertainty Ambiguity Ambiguity/Stochasticity Ambiguity/Stochasticity/Partial information Partial information Ambiguity/Partial information Stochasticity Stochasticity/Partial information Total

Number of articles

%

88 18 9 8 7 2 2

66 13 7 6 5 1 1

134

100

Omega (five articles), Applied Soft Computing (four articles), and Knowledge-Based Systems (four articles). These seven journals together represent 54% of all selected articles and they have in common an interest in decision making, including the management of decision-making information (Information Sciences journal and European Journal of Operational Research), the use of systems to support human decision making (Expert Systems with Application and Knowledge-Based Systems), and the mathematical modeling in decision making (Applied Mathematical Modeling).

4.12. Main types of data and main types of uncertainty Table 11 provides the frequency distribution of the articles by the type of data with uncertainty considered in the selected articles. Combination of ordinal and cardinal data is the main type of input data with uncertainty (40%), followed by ordinal (31%) and cardinal data (29%). Table 12 shows the frequency distribution of articles according to the type of uncertainty. The main type of uncertainty is ambiguity, which was identified individually in 66% of the papers selected, followed by its combination with stochasticity (12%) and partial information (5%). Table 13 shows the number of papers in which each type of uncertainty was considered. Ambiguity (individually and in combination with other types of uncertainty) appeared in 91% of papers. Stochasticity was considered in 23% of papers. However, unlike ambiguity that has a high percentage, even appearing individually, stochasticity occurs almost always in combination with other types of uncertainty (22% of the 23%). Partial information was considered in 19% of the selected papers, and the great majority considered partial information in the criteria weights, as discussed in Section 3; C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research

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R. Pelissari et al. / Intl. Trans. in Op. Res. 00 (2018) 1–37 Table 13 Number of articles in which each type of uncertainty was considered Type of uncertainty

Number of articles

%

Ambiguity Stochasticity Partial information

122 31 26

91 23 19

Table 14 Relation between type of uncertainty and technique used to deal with uncertainty

ER theory FSs GNs Hybrid model Other Probability theory Representations of LVs Rough sets MAUT ROR

Ambig.

Ambig. and partial infor.

Ambig. and stochas

Ambig. and stochas. and partial infor.

Partial infor.

Stochas

Stochas and partial infor.

3% 67% 5% 7% 1% – 3% 3% 3% 7%

– 43% – 43% – – – – 14% –

– 6% – 67% 11% 11% 6% – – –

– – – 11% – 89% – – – –

13% – – – 50% 13% – 25% – –

50% – – – – 50% – – – –

– – – – – 100% – – – –

75% of the papers treated only one type of uncertainty, whereas 27% treated combinations of two or three types of uncertainty. 4.13. Relation between type of uncertainty and technique used to deal with uncertainty In order to determine the most used technique to deal with uncertainty versus the type of uncertainty, we show the distribution by technique versus type of uncertainty in Table 14. The percentage shown in Table 14 for each type of uncertainty is the number of articles that use each technique over the total number of articles dealing with that type of uncertainty. For example, among the 88 articles dealing with ambiguity, 59, that is, 67%, used FSs theory. Ambiguity is the type of uncertainty for which more techniques can be used. Partial information can be modeled by probability theory, ER theory, fuzzy theory, and hybrid models (hybrid models). In particular, probability theory stands out when partial information is combined with stochasticity (100%) and stochasticity and ambiguity (89%). Probability theory is also the main technique to deal with uncertainty due to stochasticity (50%) as well ER theory (50%). Table 15 shows the distribution of combined techniques that resulted in hybrid methods. The main hybrid technique is FSs theory combined with probability theory (50% of the hybrid models). Table 16 focuses only on hybrid models and shows for which types of uncertainty hybrid methods have been applied. This FSs/probability has been applied specifically to deal with uncertainty due to problems with ambiguity and stochasticity (84%). C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research


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Table 15 Number of articles in which each type of uncertainty was considered Hybrid method

Number of papers

%

FSs/Probability FSs/ER FSs/Rough Sets FSs/Probability/ER Probability/GNs FSs/Linear program FSs/Optimization model

11 3 3 2 1 1 1

50 14 14 9 5 5 5

Total

22

100

Table 16 Relation between type of uncertainty and hybrid models used to deal with uncertainty

FSs/Rough sets FSs/ER FSs/Probability FSs/Probability/ER Probability/GNs FSs/Linear program FSs/Optimization model

Ambiguity

Ambig. and partial infor.

Ambig. and stochas

Ambig. and stochas. and partial infor.

50% 33% 17% – – – –

– – – – 33% 33% 33%

– 8% 84% 8% – – –

– – – 100% – – –

5. Discussion As MCDM/MCDA methods are characterized by allowing DMs to make assessments of alternatives based on their preferences, subjective data, especially ordinal data, are potentially present. This feature justifies the fact that uncertainty due to ambiguity is the main type of uncertainty affecting the input data in MCDM/MCDA methods, either in combination with other types of uncertainty (66%) or individually (91%). Another result obtained from this study is that 27% of the selected papers have addressed combinations of two or three types of uncertainty, showing that the number of MCDM/MCDA problems with multiple uncertainties is significant. Fuzzy theory appears as the main technique to model uncertain data due to ambiguity. There are many different fuzzy approaches to deal with different degrees of uncertainty encompassing a wide range of decision-making problems. IFSs is characterized by a degree of membership and a degree of nonmembership. IVIFS is characterized by a membership function and a nonmembership function with interval values. HFSs is characterized by situations in which several values to determine the membership of an element are possible. They stand out as the main technique used to deal with ordinal ambiguity. Extensions of these approaches, ILFSs and HLFSs, are adequate to deal with uncertainty due to ordinal ambiguity defined by linguistic terms. In the situations in which different fuzzy extensions can be used, one way of deciding among them is taking into account that each one is able to deal with different degrees of uncertainty. However, the more capable the technique to deal with high degrees of uncertainty, the more complex are their calculations. C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research

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Probability theory has proven adequate to deal with uncertainty due to partial information (or completely missing information) in the definition of the criteria weights, and stochasticity and cardinal ambiguity in the alternatives evaluations. ER theory can also be used to model stochasticity and is an alternative to probability theory when the probability distribution or its parameters are unknown. ER theory can also deal with partial information (or completely missing information) in the criteria weights, and it is the only technique identified to deal with incomplete information in the evaluation of alternatives. HFSs, IFSs, and linear programming also appear suitable to deal with partial information but only when part of the information is missing. An advantage in probabilitybased methods is the possibility of a statistical interpretation of the results, which is not available with all other methods. GNs theory, FSs theory, and interval analysis can be used to model cardinal ambiguity data when they are defined using interval values. An interval-valued FS is the same for a GN only when the GN is understood as not being able to contain discrete data. MAUT is one of the most complete and has the best theoretical foundation for applications involving uncertainty. It can be used to model partial information or cardinal ambiguity. However, most likely, because of its high theoretical requirements, we have observed that MAUT method has been less frequently applied in recent years. Some MAUT simplification studies have been developed to overcome this difficulty Durbach and Stewart (2012). In an application of a MCDM/MCDA method, it is important (if not required) to perform a parameter stability analysis, which consists of analyzing a space of feasible parameters for possible changes in the output of the method. SMAA and ROR are methods that besides allowing uncertain data can be applied to perform a parameter stability analysis. Hybrid models were used in all situations in which there was more than one type of uncertainty. Among the different types of hybrid models, fuzzy stochastic based methods are so far the main type of hybrid models that have been used. It is worth noting, however, that some individual techniques such as ER theory (ambiguity, stochasticity, and partial information) and SMAA methods (stochasticity and partial information) are also able to deal with multiple types of uncertainty. All theories that accept ordinal data can be applied when data are defined using linguistic terms since they are transformed into ordinal data. However, three theories stood out to deal with ordinal ambiguity due to the use of linguistic terms: IFSs theory and its extensions, ER theory, and integrations of different techniques with LV. These techniques can be considered more suitable when linguistic terms are used since the loss of information is less than when they are merely transformed into ordinal data. Based on the discussion presented so far, we propose a framework that indicates techniques used in different types of uncertain situations based on the results of this literature review.

6. The proposed framework We now define a framework that indicates techniques used in different types of uncertain situations. The framework is based on our review of techniques to model uncertain data in MCDM/MCDA problems, their conditions of use, and different technical characteristics. Each theory seems to be especially well suited for certain situations. A key issue is then to identify characteristics that will C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research


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Fig. 4. Framework to help choose a technique to deal with uncertainty in MCDM/MCDA problems.

direct the process of choosing a modeling framework. The framework is presented in Fig. 4 and discussed in detail as follows. The first step for choosing a technique is to define the type of uncertainty present: ambiguity, stochasticity, or partial information. Ambiguity: If we are dealing with ambiguity, we have to identify the type of data with uncertainty— ordinal or cardinal. Question 1: What is the data type with uncertainty? If the type of data is ordinal, then we proceed with question 2. If the type of data is cardinal, then question 3 has to be answered. C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research

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Question 2: Is the ordinal data expressed using linguistic terms? If the answer to question 2 is positive (yes), then we proceed with question 4. If the answer is negative (no), the data are numerical ordinal, and probability theory using uniform distributions or MAUT is appropriate. To decide between them, technical characteristics can help: to use probability theory, probability distributions and their parameters have to be known; to use MAUT, utility functions have to be defined. Another important characteristic is that MAUT is a compensatory method, while probability theory can be integrated and used with noncompensatory methods. Question 3: Is the cardinal data of the interval type ([a, b]) in which any value between the lower limit a and the upper limit b can occur? If the answer to question 3 is positive (yes), the data are an interval. Then, question 5 has to be answered. If not, then question 6 has to be answered. Question 4: Do the DMs express hesitancy in providing their preferences over objects in the decision-making process? If the answer to question 4 is positive (yes), then we proceed with question Q7. If the answer is negative (no), then we proceed with question 9. Question 5: Can the interval cardinal data assume any value between the lower a and upper b bounds with the same chance (randomness)? If the answer to question 5 is positive (yes), then a modeling using probability theory is appropriate. If the answer is negative (no), then the GN, fuzzy theory, or interval theory has to be considered. To decide, question 9 has to be answered. Question 6: Is the cardinal data represented by {a,b}, which means that there are just two possible choice values, a or b (discrete set)? If the answer to question 6 is positive (yes), then GNs theory is more appropriate. If the answer is negative (no), then FSs theory and any of its extensions, ROR or MAUT, can be used. Question 7: Is the analyst inclined to proceed with techniques that can deal with higher degrees of uncertainty but that are also more complex to apply? If the answer to question Q7 is positive (yes), then HLFSs and its extension IVHFLS are more suitable. If the answer is negative (no), then we proceed with question 9. Question 8: Do the DMs express hesitancy in stating their preferences over objects in the decisionmaking process? If the answer to question 8 is positive (yes), then we proceed with question 10. If the answer to the question 8 is negative (no), any FSs theory extension, interval theory, or GN sets are appropriate. Question 9: Do the DMs want to express their opinion on each and all alternatives? If the answer to question 9 is positive (yes), then FSs theory and any of its extensions or integrations with LV are suitable. If the answer to question 9 is negative (no) and the DMs do not want to express their evaluation for some alternative, ER theory is more appropriate. However, ER theory can also be applied if the answer is positive (yes). C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research


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Question 10: Does the analyst want to proceed with techniques that can deal with higher degrees of uncertainty but that are also more complex to apply? If the answer to question 10 is positive (yes), then HFSs and its extension IVHFS are more suitable. If the answer is negative (no), then any FSs theory extension, interval theory or GN sets are appropriate. Stochasticity: If we are dealing with stochasticity, we have only two techniques to choose from: probability or ER theory. The advantage of the ER theory over probability is its ability to model the narrowing of the set of hypotheses as new evidence is accumulated (Gordon and Shortliffe, 1984). This method combines evidence obtained from different sources without the need for prior knowledge of their probability distributions. At the same time, using probability allows to test the results for statistical significance, and so it enables decision making at a varying confidence level. Despite these differences, the answer to question 6 is crucial to decide between these two techniques. Question 11: Is the probability distribution known (with parameters)? If the answer to question 8 is positive (yes), then a modeling using the probability theory is appropriate. If the answer is negative (no), then the ER theory has to be considered. Partial information: First, we have to identify if the partial information is related to criteria weights or to the evaluation of alternatives. Question Q12: Is the partial information pertaining to the evaluation of alternatives? If the answer to question 12 is positive (yes), then a modeling using the ER theory is appropriate. If the answer is negative (no), then the question 13 has to be answered. Question Q13: Is information on criteria weights completely missing? If the answer to question 13 is positive (yes), then the most suitable method to be used is probability theory using uniform distribution (such as in the SMAA-based methods). Depending on the type of decision problem (sorting or ranking) and other characteristics, such as the aggregation procedure, there are different SMAA-based methods to choose from (see Tervonen and Figueira (2007)). If the answer is negative (no), probability theory (as SMAA), ER theory, or linear programming can be used.

7. Conclusion, limitations, and research directions In this paper, we have conducted a literature review of 134 articles, the great majority published between 2011 and 2017, with the propose of identifying the main types of uncertainty present in input data of MCDM/MCDA problems and the techniques that have been applied to deal with them. Then, we proposed a framework that indicates techniques used in different types of uncertain contexts. All selected papers were carefully studied and categorized according to three classification axes: the type of data with uncertainty, types of uncertainty, and the technique used to deal with uncertainty. C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research

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The three main types of uncertainty identified are ambiguity, stochasticity, and partial information. They were individually or jointly presented in the selected papers. Most of the articles analyzed here (73%) presented problems with only one type of uncertainty, whereas the others (27%) presented problems with multiple types of uncertainty concurrently. About 85% of these papers have used an individual technique in their analysis, the most popular being FSs theory and its extensions, followed by probability theory, ER theory, RSs, ROR, and GNs. The remaining 15% used hybrid models, that is, models resulting from the combination of different techniques, and are considered suitable for problems with multiple types of uncertainty. With regard to the traditional MCDM/MCDA methods (AHP, ELECTRE, PROMETHEE, TOPSIS, VIKOR, etc.), we observed that most of them were already integrated with fuzzy theory or probability theory. However, so far, we identified a few hybrid methods based on traditional MCDM/MCDA methods integrated with more than one technique to model multiple types of uncertainty, pointing toward the possibility of developing a new research area. Another possible future research direction is the development of studies to deal with partially missing evaluations of alternatives, since there is a paucity of literature addressing this issue, mostly related to ER theory. SMAA methods have stood out to deal with problems when the three types of uncertainties are present (ambiguity, stochasticity, and partial information). However, SMAA can deal with linguistic terms only by transforming them into ordinal data. Therefore, an opportunity for future research is to integrate more suitable techniques to model linguistic terms into SMAA methods. The proposed framework considered well-known and well-established techniques to deal with uncertainty in MCDM/MCDA problems. Both theoretical and practical issues related to the application of these techniques were considered in the framework, thus encompassing many possible situations for applying these techniques. However, a deeper study of the characteristics of each technique may lead to a refined choice framework, providing only one technique as an option for each situation. Moreover, the suggestions presented in the framework are limited to the techniques presented and discussed in the selected articles. Thereby, as a future research, new methods that were not identified in this review can be included and integrated into the framework. A choose of a method or technique may also depend on the rationality of the individual involved in the process as discussed by Stewart (1992) and Zavadskas and Turskis (2011). Therefore, a literature review and a discussion about the rationality of DMs may also be conducted and included in the framework as a future research. Despite the limitations and proposals of future research mentioned above, this study can be a valuable source of information for researchers in the field of MCDM/MCDA methods under uncertainty, mainly for new researchers, as it provides an initial reading for an overview of existing techniques to deal with uncertainty in MCDM/MCDA problems, their respective theoretical backgrounds, conditions of use, and fields of application.

Acknowledgments This research was supported by CAPES, the Brazilian Government Agency that supports higher education personnel seeking to enhance their academic qualifications, for which the authors are grateful. C 2018 The Authors. C 2018 International Federation of Operational Research Societies International Transactions in Operational Research


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Techniques to model uncertain input data of multi‐criteria decision

Techniques to model uncertain input data of multi‐criteria decision

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