Extending the multi-criteria decision making method DEX with numeric attributes, value distributions and relational models

Abstract

DEX is a qualitative multi-criteria decision analysis method. The method supports decision makers in making complex decisions based on multiple, possibly conflicting, attributes. The attributes in DEX have qualitative value scales and are structured hierarchically. The hierarchical topology allows for decomposition of the decision problem into simpler sub-problems. In DEX, alternatives are described with qualitative values, taken from the scales of corresponding input attributes in the hierarchy. The evaluation of alternatives is performed in a bottom-up way, utilizing aggregation functions, which are defined for every aggregated attribute in the form of decision rules. DEX has been used in numerous practical applications—from everyday decision problems to solving decision problems in the financial and ecological domains. Based on experience, we identified the need for three major methodological extensions to DEX: introducing numeric attributes, the probabilistic and fuzzy aggregation of values and relational models. These extensions were proposed by users of the existing method and by the new demands of complex decision problems, which require advanced decision making approaches. In this paper, we introduce these three extensions by describing the extensions formally, justifying their contributions to the decision making process and illustrating them on a didactic example, which is followed throughout the paper.

1 Introduction

Decision making is a process in which a decision maker (DM) selects an alternative from among several possible alternatives which best satisfies his/her goals (Bouyssou et al. 2006; Figueira et al. 2005; French 1986). Decision analysis (Clemen and Reilly 2001; Nagel 1993; Skinner 2009) is a discipline that provides a framework for analysing decision problems, which typically involves the development of a model for the evaluation and analysis of alternatives. A common class of models employs methods of multi-criteria decision analysis (MCDA), where alternatives are evaluated using multiple, possibly conflicting, criteria (Bouyssou et al. 2006; Figueira et al. 2005; Ishizaka and Nemery 2013).

A common group of MCDA methods use numerical variables for values and preferences; typical representatives are the Analytic Hierarchy Process (AHP) (Saaty 2008; Saaty and Vargas 2012), French abbreviation for Elimination and Choice Expressing Reality (ELECTRE) (Roy 1991), Preference Ranking Organization METHod for Enrichment of Evaluations (PROMETHEE) (Brans and Vincke 1985) and French abbreviation for Additive Utilities (UTA) (Jacquet-Lagrèze and Siskos 1982). In contrast, there is a class of qualitative multi-criteria methods which are characterized by using qualitative variables whose value scales contain a finite predefined set of qualitative (or symbolic, “verbal”) values. There are two groups of qualitative MCDA methods which differ in the way knowledge is acquired from the DM while building a decision model (Boose et al. 1993): (1) methods based on an interactive questioning procedure for obtaining the DM’s preference, and (2) methods that acquire the DM’s preferences directly.

Representative methods of the interactive questioning procedure are MACBETH, ZAPROS and ORCLASS. Measuring Attractiveness by Categorical Based Evaluation Technique (MACBETH) (Bana e Costa and Vansnick 1999) uses attractiveness and differential judgments between attributes in order to build preferential relations between alternatives. ZAPROS and ORCLASS are methods belonging to Verbal Decision Analysis (VDA) (Larichev and Moshkovich 1994, 1997; Moshkovich and Mechitov 2013). Russian abbreviation for Closed Procedures near Reference Situations (ZAPROS) (Larichev 2001; Larichev and Moshkovich 1995) provides outranking relationships among alternatives through a verbal decision making approach. ORdinal CLASSification (ORCLASS) (Gomes et al. 2010) assigns alternatives to predefined, ordered classification categories, where alternatives are placed into classes using a set of criteria.

Typical representatives of the second group are the methods DRSA, Doctus and DEX. Dominance-based Rough Set Approach (DRSA) (Greco et al. 2001, 2002) uses rough sets theory with the goal of solving alternative classification and sorting problems represented by decision tables, using the principle of dominance. DRSA has a strong mathematical foundation (Greco et al. 2001) and has evolved in many directions—for example, considering imprecise evaluations and assignments (Dembczyński et al. 2009) and dealing with decisions under uncertainty and time preference (Greco et al. 2010). Doctus (Baracskai and Dörfler 2003) is a Knowledge-Based Expert System Shell used for evaluation of alternatives and supporting three types of alternative evaluation: Rule-Based Reasoning, Case-Based Reasoning and Case-Based Rule Reasoning. The third method from this group is DEX, which is addressed in this work.

Decision EXpert (DEX) is a qualitative multi-attribute modelling method which integrates multi-criteria decision modelling with rule-based expert systems (Bohanec et al. 2013). A DEX model consists of hierarchically structured qualitative attributes whose values are words rather than numbers. The aggregation is defined by decision rules. DEX has been widely used in practice to support complex decision processes in health threats and crisis management (Žnidaršič et al. 2009), the use of genetically modified crops (Bohanec et al. 2009; Bohanec 2008; Bohanec and Žnidaršič 2008; Žnidaršič et al. 2008), the evaluation of data mining work flows (Žnidaršič et al. 2012), the evaluation of public administration portals (Leben et al. 2006), the assessment of bank reputational risk (Bohanec et al. 2014), environmental decision making (Kuzmanovski et al. 2015; Žnidaršič et al. 2006b) and many others (Bohanec et al. 2013). DEX is described in Sect. 2.

The many uses of DEX have indicated a great practical value of the method but have also revealed the need to extend it in several directions. Three substantial extensions (Trdin and Bohanec 2012, 2014a) were identified by practical needs, especially with the aim to provide additional features for DMs. The extensions were proposed by the developers, users and other contributors to the method, who are facing new and more demanding decision problems.

The main purpose of this paper is to propose and formally develop the following three extensions to DEX:

• 1. The inclusion of numeric attributes which allow for combined qualitative and quantitative modelling and eliminate the current need to transform numeric attribute values to qualitative.

• 2. Support for probabilistic or fuzzy value distributions, allowing one to cope with uncertainty in models and data.

• 3. Support for relational models and alternatives, allowing for the opportunity to handle relational connections between alternatives and their parts, when alternatives are composed of multiple similar sub-parts.

Each extension is developed formally and illustrated using a didactic example.

The structure of this paper is as follows. Section 2 describes the DEX method and introduces a didactic example, which is used throughout the paper. Sections 3, 4 and 5 present the proposed extensions of DEX: using numerical attributes, probabilistic and fuzzy distributions and relational models, respectively. The contributions and consequences are discussed in Sect. 6. Section 7 concludes the paper.

2 DEX method

DEX (Bohanec 2015a; Bohanec and Rajkovič 1990, 1999; Bohanec et al. 2013; Bohanec and Trdin 2014; Trdin and Bohanec 2012, 2014a) is a qualitative multi-criteria decision modelling method. DEX uses qualitative scales which consist of a finite set of symbolic attribute values, such as “bad”, “medium” and “good”, rather than numeric values. These values are usually, but not necessarily, preferentially ordered. Consequently, a DEX model generally consists of attributes, some of which are preferentially ordered and can be thus referred to as criteria.

A DEX model has a form of a hierarchy, which represents the decomposition of the decision problem and relations between attributes: higher-level attributes depend on lower-level ones. The terminal nodes of the hierarchy are called input or basic attributes, whereas all other attributes are called aggregated attributes. Additionally, attributes without any parents are called roots. A typical model has only one root, which represents the primary outcome of the evaluation of alternatives. There are no conceptual limitations, however, to having multiple roots—for example, to represent evaluations from different viewpoints. In MCDA, hierarchies are commonly used in methods such as AHP (Saaty 2008) and Multiple Criteria Hierarchy Process (MCHP) (Corrente et al. 2012).

The aggregation of values in the model is facilitated by decision rules. To compute an aggregated attribute’s value from the values of its children, each aggregated attribute has an associated total aggregation function. The function is defined by a decision table—interpretable as a set of decision rules. The total aggregation function needs to specify an output value for each combination of children’s values. Such tables are typically prepared by the DM according to his/her preferences. Another method in MCDA that relies on decision tables is DRSA (Greco et al. 2001). There, a decision table is understood mainly as a collection of decision alternatives rather than a constructed specification of an aggregation function. In contrast to DEX, decision tables in DRSA may contain both qualitative and numeric criteria.

The evaluation of alternatives is done in a bottom-up way. The input attribute values for some alternative are acquired directly from the alternative. The evaluation is carried out progressively from the lowest attributes in the hierarchy to its roots. Each aggregate attribute’s value is computed using the corresponding aggregation function.

Currently, DEX method is implemented in the software called DEXi (Bohanec 2014, 2015b). DEXi supports an interactive construction of the decision model and alternatives. The software aids in defining decision rules and checking their completeness and consistency and provides a number of analytical decision tools. The user can make an in-depth analysis of evaluations and decision alternatives. For example, each computed value has an associated decision rule, which was used to obtain that value; the rule explains how the evaluation was obtained. Moreover, the decision rule was applied on the basis of values computed by previous evaluations, which can also be drilled down even further, explaining the sources for such evaluation and thus providing evidence of why the evaluation was such.

DEXi software includes four different analysis procedures for the evaluated alternatives (Bohanec 2014): (1) “Plus-minus-1 analysis” checks to which extents alternative evaluations are affected by small changes to the input attribute values; (2) “Selective explanation” informs the user about the strong and weak components of each alternative; (3) “Compare” compares the pre-selected alternatives attribute-wise, and (4) “Charts” are able to plot k sided utility diagrams based on the selected alternatives and k selected attributes.

2.1 DEX model

Formally, a DEX model M is a four-tuple $M=\left( {X,D,S,F} \right)$, where X is the set of attributes, S is the descendant function that determines the hierarchical structure of M, D is the set of value scales (domains) of attributes in X and F is the set of aggregation functions.

The set X consists of n attributes:

$$\begin{aligned} X=\left\{ {x_1 ,x_2 ,\ldots ,x_n } \right\} . \end{aligned}$$

(1)

In practice, attributes are usually given a name, which uniquely identifies the attribute—for instance “price”, “quality”, “location”, etc. In the didactic example, introduced later in Sect. 2.2, we will often denote an attribute by its name (e.g. location) and use a named subscript to denote related components (e.g. $D_\textit{location}$).

Each attribute $x_i \in X$ has a corresponding value scale $D_i \in D$, which is an ordered set of symbolic (qualitative) values:

$$\begin{aligned} D_i =\left\{ {w_{i_1 } ,w_{i_2 } ,\ldots ,w_{i_{m_i } } } \right\} ,l\ne k\Leftrightarrow w_{i_l } \ne w_{i_k } . \end{aligned}$$

(2)

Here, $m_i$ is the number of values in the scale of attribute $x_i$. In practice, attribute values are also represented by words, such as “low”, “good”, “acceptable” and “yes”.

With respect to the DM’s preferences, scales can be either ordered or unordered. If the scale is totally ordered, then the DM defines the preferential operator $\precsim$. For each $j\le k,j,k\in \left[ {1,m_i } \right]$, we write $w_{i_j} \precsim w_{i_k }$ and interpret that $w_{i_k}$ is preferentially at least as good as $w_{i_j}$. For ordered scales, we use the notation:

$$\begin{aligned} D_i =\left( {w_{i_1 } ,w_{i_2 } ,\ldots ,w_{i_{m_i } } } \right) . \end{aligned}$$

(3)

According to the usual convention (Greco et al. 2001), attributes with preferentially ordered scales are called criteria.

Attributes are structured hierarchically: each attribute $x_i$ may have some descendants (children) and/or predecessors (parents) in the model. This relationship is described by the function $S:X\rightarrow 2^{X}$, which maps each attribute $x_i \in X$ to a set of its descendants $S\left( {x_i } \right)$:

$$\begin{aligned} S\left( {x_i } \right)= & {} \left\{ {x_{i_1 } ,x_{i_2 } ,\ldots ,x_{i_k } } \right\} ,l\ne m\Leftrightarrow x_{i_l } \ne x_{i_m } ,\nonumber \\ x_{i_j}\ne & {} x_i ,j=1,\ldots ,k,\\&\quad x_{i_1 } ,x_{i_2 } ,\ldots ,x_{i_k } \in X.\nonumber \end{aligned}$$

(4)

The relations induced by S must represent a hierarchy—that is, a connected and directed acyclic graph. Such a graph contains finitely many edges and vertices, where each edge is directed from one vertex to another. Further, starting in arbitrary vertex x and traversing the graph through all possible paths, one will never reach vertex x again. In most practical cases the graph induced by S represents a tree—all attributes except one root attribute, have exactly one parent. In general, however, a hierarchy may have several roots and may contain attributes that influence more than one parent attribute. If S does not form a hierarchy, then the directed graph contains one or more cycles, and the evaluation of alternatives is generally not possible—for that reason we restrict S to acyclic graphs.

The model input attributes $\textit{modelInputs}_M$ constitute a set of all attributes that do not have any children in the hierarchy. Additionally, the model input space $\textit{inputSpace}_M$ is defined as the Cartesian product of attributes’ scales in $\textit{modelInputs}_M$:

$$\begin{aligned} \begin{aligned} \textit{modelInputs}_M&=\{x_i \in X|\,S\left( {x_i } \right) =\emptyset \},\\ \textit{inputSpace}_M&=D_j \times D_l \times \cdots \times D_k ,x_j ,x_l ,\ldots ,x_k \in \textit{modelInputs}_M . \end{aligned} \end{aligned}$$

(5)

Model output attributes and model output space are defined similarly—modelOutputs $_M$ as a set of all attributes in model M that are not children of any other attribute; $\textit{outputSpace}_M$ as the Cartesian product of these attributes’ scales:

$$\begin{aligned} \begin{aligned} \textit{modelOutputs}_M&=\left\{ x_i \in X\hbox {|}\,x_i \notin \mathop \bigcup \limits _{x_j \in X} S\left( {x_j } \right) \right\} ,\\ \textit{outputSpace}_M&=D_j \times D_k \ldots \times D_l ,x_j ,x_l ,\ldots ,x_k \in \textit{modelOutputs}_M . \end{aligned} \end{aligned}$$

(6)

Another useful set is the set of all aggregated attributes, $\textit{aggAttributes}_M$. These attributes are the ones whose values have to be computed when evaluating alternatives. Structurally, they appear as internal nodes in the hierarchy:

$$\begin{aligned} \textit{aggAttributes}_M =\{x_i \in X\hbox {|}\,x_i \notin \textit{modelInputs}_M \}. \end{aligned}$$

(7)

Each aggregated attribute $x_i \in \textit{aggAttributes}_M$ needs an aggregation function, so that the decision alternatives can be evaluated. We denote $f_i \in F$ as the aggregation function corresponding to attribute $x_i$. Each aggregation function $f_i$ is a total function that maps all scale value combinations of $x_i$’s inputs to an interval value in scale $D_i$:

$$\begin{aligned} f_i :D_{i_1 } \times D_{i_2 } \times \cdots \times D_{i_k } \rightarrow I(D_i ),\left\{ {x_{i_1 } ,x_{i_2 } ,\ldots ,x_{i_k } } \right\} =S\left( {x_i } \right) . \end{aligned}$$

(8)

Here, $I(D_i)$ represents the space of all possible intervals $\left[ {v_l ,v_h } \right] =\{v_j |v_l \precsim v_j \precsim v_h \wedge v_l ,v_j ,v_h \in D_i \}$.

Such aggregation functions are usually represented as tables which assign an output value to each possible input value combination. Such tables can also be interpreted as if-then rules. Examples of such tables are given in Sect. 2.2.

For the purpose of using functions $f_i$ in the evaluation of alternatives, especially in the extended DEX method described later in this paper, we also define functions $F_i$, which are generalizations of $f_i$ and are defined on sets of values rather than on crisp values. The $F_i$ function definition corresponding to the associated function $f_i$ is:

$$\begin{aligned} F_i :2^{D_{i_1 } }\times 2^{D_{i_2 } }\times \cdots \times 2^{D_{i_k } }\rightarrow 2^{D_i }. \end{aligned}$$

(9)

Functions $F_i$ are expected to produce the same results on the same singleton input values as their $f_i$ counterparts:

$$\begin{aligned}&\forall v_{i_1 } \in D_{i_1 } ,v_{i_2 } \in D_{i_2 } ,\ldots ,v_{i_k } \in D_{i_k } :W_{i_1 } =\left\{ {v_{i_1 } } \right\} ,\nonumber \\&\quad W_{i_2 } =\left\{ {v_{i_2 } } \right\} ,\ldots ,W_{i_k } =\left\{ {v_{i_k } }\right\} \Rightarrow \nonumber \\&F_i \left( {W_{i_1 } ,W_{i_2 } ,\ldots ,W_{i_k } } \right) =\{f_i \left( {v_{i_1 } ,v_{i_2 } ,\ldots ,v_{i_k } } \right) \}. \end{aligned}$$

(10)

Given $f_i$, the value for function $F_i$ is computed as:

$$\begin{aligned} F_i \left( {W_{i_1 } ,W_{i_2 } ,\ldots ,W_{i_k } } \right) =\mathop \bigcup \limits _{\left( {v_{i_1 } ,v_{i_2 } ,\ldots ,v_{i_k } } \right) \in W_{i_1 } \times W_{i_2 } \times \cdots \times W_{i_k } } \{f_i \left( {v_{i_1 } ,v_{i_2 } ,\ldots ,v_{i_k } } \right) )\}. \end{aligned}$$

(11)

With $A_M$, we denote the set of all decision alternatives for some model M:

$$\begin{aligned} A_M =\left\{ {a_1 ,a_2 ,\ldots ,a_m } \right\} . \end{aligned}$$

(12)

Here, each alternative is defined in the Cartesian space $\textit{inputSpace}_M$, so that $a_i \in \textit{inputSpace}_M$.

Alternatives in $A_M$ are evaluated with model M with function $\textit{evaluation}_M$:

$$\begin{aligned} \textit{evaluation}_M :\textit{inputSpace}_M \rightarrow \textit{outputSpace}_M. \end{aligned}$$

(13)

The evaluation of alternative $a_i$ on model M is done by computing the function $\textit{evaluation}_M \left( {a_i } \right)$ and assigning output values to all attributes from $\textit{outputSpace}_M$. The evaluation is done as a bottom-up aggregation of model inputs toward its outputs according to the hierarchical structure of the model. Algorithmically, all aggregated attributes in model M are first topologically sorted with respect to S. The sorting determines the order of aggregation function evaluations and ensures that all inputs to the current aggregation function are readily available. The values of aggregated attributes are computed using the corresponding functions $F_i$ in the given order. This produces the output evaluations in $\textit{modelOutputs}_M$.

2.2 Didactic example

We will follow the same didactic example through the whole paper: choosing an apartment to buy. At the first stage, the model is developed using the DEX method without any extensions. Later, we will gradually develop the example using the DEX extensions proposed in this paper.

The model has one root attribute, apartment, and three children: price, location and layout. These three attributes are split into finer detail, which can be seen in Fig. 1. For example, price depends on the buying price and the price of utilities.

Fig. 1

Model for evaluating an apartment. The tree structure of attributes is presented. A name and value scale is shown for each attribute. The type of the attribute is indicated by the box shape: input attributes (rectangular) and aggregate attributes (round corners)

All attributes have qualitative values assigned to their value scales, which are also shown in Fig. 1 beneath the attributes’ names. Notably, the model has a simple tree structure, in which none of the attributes influences more than one parent attribute. In total, there are 18 attributes: 11 input and 7 aggregated, including one root attribute.

Formally, the model has attributes:

$$\begin{aligned}&X=\{\textit{apartment},\textit{price},\textit{location},\textit{layout},\textit{buying \,price},\textit{utilities},\nonumber \\&\quad \quad \quad \quad \textit{neighbourhood},\textit{proximity},\textit{interior},\textit{exterior},\textit{work},\textit{shops},\textit{size},\textit{equipment}, \nonumber \\&\quad \quad \quad \quad \textit{balcony},\#\textit{rooms},\textit{parking},\textit{appearance}\}. \end{aligned}$$

(14)

Table 1 Aggregation function for final apartment assessment, based on its price, location and layout

The scales of the attributes are defined in Fig. 1. All the scales are ordered preferentially, for instance:

$$\begin{aligned} D_\textit{apartment} =\left( {\hbox {v-bad},\hbox {bad},\hbox {ok},\hbox {good},\hbox {v-good}} \right) . \end{aligned}$$

(15)

The function S can be read directly from the model structure presented in Fig. 1:

$$\begin{aligned} S\left( \textit{apartment} \right)= & {} \left\{ {\textit{price},\textit{location},\textit{layout}} \right\} ,\nonumber \\&\quad \ldots \nonumber \\ S\left( \textit{exterior} \right)= & {} \left\{ \textit{parking,appearance} \right\} ,\nonumber \\ S\left( \textit{buying\,price} \right)= & {} \cdots =S\left( \textit{appearance} \right) =\emptyset . \end{aligned}$$

(16)

With all of these in place, we can now define $\textit{modelInputs}_M$, $\textit{inputSpace}_M$, $\textit{modelOutputs}_M$, $\textit{outputSpace}_M$, and $\textit{aggAttributes}_M$, where M is the Apartment model. These can be directly constructed from Fig. 1:

$$\begin{aligned} \textit{modelInputs}= & {} \{\textit{buying\,price},\textit{utilities},\textit{neighbourhood},\textit{work},\nonumber \\&\quad \textit{shops},\textit{size},\textit{equipment},\textit{balcony},\#\textit{rooms},\textit{parking},\textit{appearance}\},\nonumber \\ \textit{inputSpace}= & {} \left( {\hbox {high},\hbox {med},\hbox {low}} \right) \times \left( {\hbox {high},\hbox {med},\hbox {low}} \right) \times \cdots \times ( {\hbox {bad},\hbox {ok},\hbox {good}} ),\nonumber \\ \textit{modelOutputs}= & {} \{ \textit{apartment} \},\nonumber \\ \textit{outputSpace}= & {} ( {\hbox {v-bad},\hbox {bad},\hbox {ok},\hbox {good},\hbox {v-good}} ),\nonumber \\ \textit{aggAttributes}= & {} \left\{ {\textit{apartment},\textit{price},\textit{location},\textit{layout},\textit{proximity},\textit{interior},\textit{exterior}} \right\} .\qquad \end{aligned}$$

(17)

Table 2 Aggregation function for price, dependent buying price and monthly utilities

Table 3 Aggregation function for location, dependent on neighbourhood and proximity

Table 4 Aggregation function for layout, based on interior and exterior

Table 5 Aggregation function for proximity, dependent on proximity to work and shops

Table 6 Aggregation function for interior, based on size, equipment, balcony presence and number of rooms

Table 7 Aggregation function for exterior, dependent on parking space availability and appearance

Finally, to complete the model, we need formal definitions of aggregation functions for the aggregated attributes. The aggregation functions for attributes in $\textit{aggAttributes}$ are described by the sets of decision rules in Tables 1, 2, 3, 4, 5, 6 and 7. Even though generally DEX aggregation functions map their arguments to interval values over $x_i \in X$ (Eq. (8)), only single values are used in this case. This is done by convention if the lower and upper bounds of an interval are equal to some word $w_m$; then, instead of writing $\left[ {w_m ,w_m } \right]$, we write just $w_m$. For example, a decision rule from Table 6 is:

$$\begin{aligned} f_\textit{apartment} \left( {\hbox {high,ok,ok}} \right) =\hbox {v-bad.} \end{aligned}$$

(18)

It can be interpreted as: “if price is high and location is ok and layout is ok, then apartment is v-bad”.

Now that the model is complete, alternatives can be defined and evaluated. Let us consider three different apartments, each described by a n-tuple of values corresponding $\textit{modelInputs}_M$. The three apartments are shown in Table 8. Notice that all input values are formulated in terms of qualitative values. In reality, we determine these values from real values of alternatives. For example, we may assess the buying price of 50.000 € as med. We assume that this process has already been carried out by the DM, as shown in Table 8.

To illustrate the evaluation procedure, let us consider the alternative “Big”. A possible topological sorting of the aggregated attributes is:

$$\begin{aligned} (\textit{price},\textit{proximity},\textit{location},\textit{interior},\textit{exterior}, \textit{layout},\textit{apartment}). \end{aligned}$$

(19)

Consequently, the attributes should be evaluated according to this order—that is, from price to apartment. The value of price is computed by the aggregation function $f_\textit{price}$, given the values for child attributes buying price and utilities. The functional value for $f_\textit{price}\left( {\hbox {med,med}} \right)$ is med, which is given in Table 2 in the fifth row. Similarly, the value for proximity is computed by computing function $f_\textit{proximity}$ with values (far,far)—produced value is far. The values for the remaining aggregated attributes are produced in the same way, as shown in Table 9. The apartment “Big” is assessed as very bad. The other two apartments, “Equipped” and “Nice”, are assessed as bad and good, respectively.

Table 8 The table describes three different available apartments for evaluation

Table 9 Evaluations of three alternatives according to the developed model

3 Numeric attributes

Most MCDM methods are quantitative—they involve numeric attributes and real-valued utility functions. DEX is a qualitative method and currently uses only qualitative attributes. This approach is not always appropriate because it requires that quantitative values are discretized before use, even for quantities that are naturally measured and expressed with numbers. Furthermore, in many decision problems there is a need to support both qualitative and quantitative attributes within the same model (Bohanec et al. 2014; Kontić et al. 2014). Even though a few quantitative features have already been considered within DEX (Mileva-Boshkoska and Bohanec 2012; Trdin and Bohanec 2012, 2014a, b; Žnidaršič et al. 2003), there is a strong need to systematically introduce numerical attributes into the DEX method.

Introducing numeric attributes in DEX is thus aimed at extending the expressive power of the method, so that both discrete and numeric attributes can be used and combined in a single model. We wish to do this in a general and flexible way, making sure that numerical attributes integrate well into the existing framework. The goal is not to introduce any specific quantitative MCDA method in DEX, but to provide a flexible schema which allows using different quantitative methods for eliciting numeric attributes, their weights and utility functions.

Adding numerical attributes requires a number of representational and algorithmic extensions, such as adding numerical aggregation functions and handling transformations between qualitative and numeric values. Attribute value scales have to be extended to include real numbers, integer numbers, bounded intervals over real numbers and bounded integer intervals, as expressed later in Eq. (20).

Supporting numeric attributes does not affect the principle of structuring attributes in a hierarchy and assigning aggregation functions to aggregated attributes. However, there is an essential difference in the aggregation functions themselves because, in general, they have to cope with various combinations of qualitative and quantitative attributes, both at function inputs (arguments) and function outcomes. For this reason, we consider six different function types which differ by value type. There are three combinations of basic types the function can receive: (1) all qualitative values (denoted QQ), (2) all quantitative (numeric) values (NN) or (3) a mix of quantitative and qualitative values (NQ). Regarding the output, the function can yield only one type of output—either qualitative (Q) or quantitative (N). Note that the basic DEX method covers only one of these cases: functions that map from qualitative arguments to qualitative values (denoted $\textit{QQ}\rightarrow Q)$.

The introduction of quantitative attribute values increases the expressiveness of attributes’ value scales and the aggregation of such quantitative values. The extension naturally incorporates numeric values and operations. Consequently, it provides additional flexibility to the DM and decision analyst, and allows addressing a wider class of decision problems. Inevitably, the drawback of the extension is an increased complexity due to new types of aggregation functions.

3.1 Extension formalization

First, we need to extend the definition of the value scale $D_i$ for some $x_i$ given in Eqs. (2, 3) to include numeric quantities:

$$\begin{aligned} D_i =\left\{ {{\begin{array}{ll} \left( {w_{i_1 } ,w_{i_2 } ,\ldots ,w_{i_{m_i } } } \right) ,&{}\hbox {or} \\ \left[ {a..b} \right] ,a,b\in \mathbb {Z}\cup \left\{ {-\infty ,\infty } \right\} ,&{}\hbox {or} \\ \left[ {c,d} \right] ,c,d\in \mathbb {R}\cup \left\{ {-\infty ,\infty } \right\} .&{} \\ \end{array} }} \right. \end{aligned}$$

(20)

This equation represents three possibilities for defining the value scale $D_i$:

• 1. The first option remains the same as before: a finite list of words. The list may be preferentially ordered.

• 2. The second option defines an integer interval by specifying two integer values $a,b,a\le b$. The scale of the particular attribute is then composed of all integers between a and b, inclusively. Note that we can also specify an unbounded interval using negative or positive infinities.

• 3. The third option defines a real interval by choosing two real values $c,d,c\le d$. The possible values of such attribute are all real values between c and d, inclusively. Both infinities are also allowed.

These three cases introduce five types of primitive aggregation functions (Fig. 2). Here, “primitive” means that the function can be defined in “one step”, using either a single formula or a single decision table. In contrast, a “non-primitive” or “compound” aggregation is defined by a hierarchical composition of primitive functions.

Function type 1 $(\textit{QQ}\rightarrow Q)$: These functions have only qualitative inputs and a qualitative output. They are already covered by the current DEX method and modelled by decision rules according to Eq. (8).

Function type 2 $(\textit{QQ}\rightarrow N)$: These functions have only qualitative inputs and a numeric output. They are modelled similarly to Function type 1 by decision rules; the only difference is that they output a single number or numeric interval instead of a single word or interval of words. The output number is restricted to the scale of the aggregated attribute. Function type 2 is already formally covered by Eq. (8), taking the extended definition of $D_i$ in Eq. (20). Note that this function type includes the special case $Q\rightarrow N$, where only one qualitative attribute is converted to a numerical attribute.

Function type 3 $(\textit{NN}\rightarrow N)$: These functions have only numeric inputs and a numeric output. They are modelled in a purely mathematical way using mathematical operators on numeric functions’ arguments, such as $+$, −, $\times$,  / , power, square, root, etc., and numeric constants. A common representative of this function type is the weighted average—for example, $\mathop \sum \nolimits _{i=1}^n w_i v_i$—where $w_i$ is the i-th attribute’s weight and $v_i$ is its value. Similar to Function type 2, this function type is already formally covered by Eq. (8). The difference is that all $D_{i_1 } ,D_{i_2 } ,\ldots ,D_{i_k } ,D_i$ are numeric scales, and function $f_i$ is a mathematical expression. Functions of this type are commonly used in quantitative MCDA methods and can thus be easily adopted in DEX, together with already developed support for their acquisition and use in practice. For instance, Function type $\textit{NN}\rightarrow N$, can use the weighted sum and employ AHP’s (Saaty 2008) pairwise comparison to elicit attribute weights.

Function type 4 $(N \rightarrow Q)$: These functions discretize a single numeric input attribute to an output qualitative attribute. Generally, a discretization function of the i-th attribute ($\textit{NQ}_i)$ maps real numbers to qualitative value scales:

$$\begin{aligned} \textit{NQ}_i :A\subseteq \mathbb {R}\rightarrow D_i , \end{aligned}$$

(21)

so that:

$$\begin{aligned} \textit{NQ}_i \left( y \right) =\left\{ {{\begin{array}{c} {w_{i_p } |y\in \left[ {a,b} \right] } \\ \ldots \\ {w_{i_r } |y\in \left[ {c,d} \right] } \\ \end{array} },w_{i_p } ,\ldots ,w_{i_r } \in D_i } \right. . \end{aligned}$$

(22)

Fig. 2

The five possible cases for configurations of qualitative and numeric primitive aggregation functions. The function type is dependent upon its input types and its output type. The letters denote qualitative (Q) attribute type or numeric (N) attribute type

Function type 5 $(\textit{NQ} \rightarrow N)$: These functions have qualitative and numeric inputs and a numeric output. The primitive specification of such functions is based on decision tables, similar to function types 1 and 2. First, we define a decision table using all qualitative function arguments. Second, in each row of the table a numerical function of the remaining numeric attributes is defined. Formally, suppose that $q_1 ,q_2 ,\ldots ,q_k$ are indices of all qualitative inputs to $x_i$ and $n_1 ,n_2 ,\ldots ,n_m$ all numeric ones. Let $\varPhi \left( {x_i } \right)$ denote the set of functions which receive all $x_i$’s numeric inputs and output a numeric value in $D_i$.

$$\begin{aligned} \varPhi \left( {x_i } \right) =\{f|f:D_{i_{n_1 } } \times D_{i_{n_2 } } \times \cdots \times D_{i_{n_m } } \rightarrow D_i \} \end{aligned}$$

(23)

Let $h_i$ be a function which maps from all combinations of qualitative inputs to space $\varPhi \left( {x_i } \right)$:

$$\begin{aligned} h_i :D_{i_{q_1 } } \times D_{i_{q_2 } } \times \cdots \times D_{i_{q_k } } \rightarrow \varPhi \left( {x_i } \right) . \end{aligned}$$

(24)

The overall aggregation function $f_i$ is then evaluated in two steps: (1) the value of $h_i \left( {v_{q_1 } ,v_{q_2 } ,\ldots ,v_{q_k } } \right)$ is used as a table lookup index to find the corresponding function $g\in \varPhi \left( {x_i } \right)$, and (2) evaluate $g\left( {v_{n_1 } ,v_{n_2 } ,\ldots ,v_{n_m } } \right)$. Formally:

$$\begin{aligned} f_i \left( {v_{q_1 } ,v_{q_2 } ,\ldots ,v_{q_k } ,v_{n_1 } ,v_{n_2 } ,\ldots ,v_{n_m } } \right) =h_i \left( {v_{q_1 } ,v_{q_2 } ,\ldots ,v_{q_k } } \right) \left( {v_{n_1 } ,v_{n_2 } ,\ldots ,v_{n_m } } \right) . \end{aligned}$$

(25)

Using these five primitive function types, we can express other aggregations of numeric and qualitative values. Among these, two are of special interest: $\textit{NN}\rightarrow Q$ and $\textit{NQ} \rightarrow Q$.

Compound function type $\textit{NN}\rightarrow Q$: These compound functions have only numeric inputs and a qualitative output. They can be modelled in two ways:

• 1. Each numeric attribute can be first discretized and then aggregated by Function type 1: $(N\rightarrow Q)(N\rightarrow Q)\rightarrow Q$;

• 2. A mathematical expression can be constructed by Function type 3, and then a single discretization can be applied on the output: $\textit{NN}\rightarrow N\rightarrow Q.$

Compound function type $\textit{NQ}\rightarrow Q$: This function type has qualitative and numeric inputs and a numeric output. It can be modelled with primitive function types in three ways:

• 1. Discretizing the result retrieved from a Function type 5 function: $(\textit{NQ}\rightarrow N)\rightarrow Q$;

• 2. Discretizing numeric inputs and creating a qualitative function on all qualitative attributes: ($N\rightarrow Q)Q\rightarrow Q$;

• 3. Using Function type 3 on numeric attributes, discretizing its output and creating a qualitative function on the result and qualitative attributes: ($(\textit{NN}\rightarrow N)\rightarrow Q)Q\rightarrow Q$.

The two compound function types are presented graphically according to their descriptions in Figs. 3 and 4.

The compound function types $\textit{NN} \rightarrow Q$ and $\textit{NQ}\rightarrow Q$ can thus be modelled in a number of different ways. On one hand, this is convenient, as the functions can be flexibly adapted to the characteristics of the problem. On the other hand, this puts additional burden on DM, as he/she has to decide which option to use in a given situation. This decision is important, as different compound functions generally lead to different results. The proposed approach fulfils the requirement for flexibility, but does not answer the question on how to decide which option should be used. This is left for further research and practice. At the time being, we assume that the choice of compound functions is entirely at the DM’s responsibility.

Fig. 3

Two possibilities for modelling Compound function type $\textit{NN}\rightarrow Q$. Option a suggests that numeric attributes are first discretized to qualitative and then the function is modelled as a rule based qualitative function. Option b indicates that numeric attributes are aggregated using Function type 3, and then discretizing the produced result

Fig. 4

Three possibilities for modelling Compound function type $\textit{NQ}\rightarrow Q$. Option c gives the configuration to use Function type 5 on the inputs and then discretize the computed output. Option d gives the possibility to discretize the numeric inputs and creates a qualitative function on outputs and remaining qualitative inputs. Option e presents the possibility to aggregate numeric attributes to single numeric value, discretize the value and then construct a qualitative function based on this and on qualitative inputs

3.2 Didactic example

We shall introduce numerical attributes in the didactic example while keeping its hierarchical structure intact. First, we replace the attributes price, buying price, utilities, work, shops, size and #rooms with their numeric counterparts by changing attributes’ scales (Fig. 5). Due to these conversions, the aggregation functions numeric attributes’ parents (i.e., price, apartment, proximity and interior) need to be replaced by functions of different types. All other functions remain the same.

Fig. 5

Model for evaluating an apartment with numeric attributes included. Attributes with value scales $\mathbb {R}^{+}$ and $\mathbb {N}$ are numeric; others are qualitative

Given the meaning of the attributes described above, we can safely assume that their numeric scales are all positive real numbers. An exception is the scale of #rooms, which is in the form of an integer:

$$\begin{aligned}&D_\textit{price} =D_\textit{buying\,price} =D_\textit{utilities} =D_\textit{work} =D_\textit{shops} =D_\textit{size} =\mathbb {R}^{+},\nonumber \\&\quad D_{\#\textit{rooms}} =\mathbb {N}. \end{aligned}$$

(26)

The new function for price is of Function type 3 ($\textit{NN}\rightarrow N)$. Suppose that we pay off the apartment in 20 years, so our combined price will represent the buying price with 20 years of monthly utilities costs added:

$$\begin{aligned} f_\textit{price} =\textit{buying\,price}+12\cdot 20\cdot \textit{utilities}. \end{aligned}$$

(27)

Next, the aggregation function for proximity is defined based on distance to work and shops and has a qualitative output of proximity and is hence of Compound type $\textit{NN}\rightarrow Q$. We compose this function using the schema $\textit{NN}\rightarrow N\rightarrow Q$. The rationale for the following function definition is that we will be going to work five times a week, whereas will be going to the shops only twice a week. Here $g_\textit{proximity}$ is the intermediate Function of type 3:

$$\begin{aligned} g_\textit{proximity}= & {} 5\cdot \textit{work}+2\cdot \textit{shops},\nonumber \\ \textit{NQ}_\textit{proximity} \left( y \right)= & {} \left\{ {{\begin{array}{l} {\hbox {near}|y\in \left( {0,15} \right] } \\ {\hbox {manag}.|y\in \left( {15,25} \right] } \\ {\hbox {far}|y\in \left( {25,\infty } \right) } \\ \end{array} }} \right. . \end{aligned}$$

(28)

The aggregation function for interior is of the Compound type $\textit{NQ}\rightarrow Q$. We shall compose it according to the schema ($\textit{NQ}\rightarrow N)\rightarrow Q$—that is: (1) first using the primitive Function type 5 and (2) discretizing the result. The output of (1) is an estimate of each room size (see Table 10). To include some preferential information in the table, constant values were added to favourable qualitative combinations (e.g. $+$20 for the combination (yes, yes) in Table 10). For the unfavourable combinations, we disregarded the numeric inputs and assigned the value 0. In step (2), we use the following discretization:

$$\begin{aligned} \textit{NQ}_\textit{interior} \left( y \right) =\left\{ {{\begin{array}{l} {\hbox {bad}|y\in \left[ {0,25} \right] } \\ {\hbox {ok}|y\in \left( {25,35} \right] } \\ {\hbox {good}|y\in \left( {35,\infty } \right) } \\ \end{array} }} \right. . \end{aligned}$$

(29)

Table 10 The first part of the aggregation function for interior. The function arguments define a new function, which depends on the two numeric inputs (size and #rooms)

Finally, we need to define the aggregation function for apartment, which is also of the Compound type $\textit{NQ}\rightarrow Q$. Similarly as before, we define Table 11 for step (1) and the following discretization for step (2):

$$\begin{aligned} \textit{NQ}_\textit{apartment} \left( y \right) =\left\{ {{\begin{array}{l} {\hbox {v-good}|y\in \left[ {-10{,}000,50{,}000} \right] } \\ {\hbox {good}|y\in \left( {50{,}000,90{,}000} \right] } \\ {\hbox {ok}|y\in \left( {90{,}000,110{,}000} \right] } \\ {\hbox {bad}|y\in \left( {110{,}000,140{,}000} \right] } \\ {\hbox {v-bad}|y\in \left( {140{,}000,\infty } \right) } \\ \end{array} }} \right. . \end{aligned}$$

(30)

Table 11 First part of the aggregation function for apartment

With this, the model is completely defined and ready for the evaluation of alternatives. The new values for the alternatives are presented in Table 12. All evaluation results are given in Table 13.

Table 12 The table gives new values for the input attributes for the three apartments: “Big”, “Equipped” and “Nice”

Table 13 Evaluations of three alternatives according to the newly developed model

Let us illustrate the evaluation of the alternative “Big”. The other two alternatives follow the same procedure.

At first, the price of alternative “Big” is obtained by computing $f_\textit{price} \left( {85{,}000,100} \right)$, which is 109,000. Next, the value of $f_\textit{proximity} \left( {6,5} \right)$, is a two-stage function: at first, $g_\textit{proximity} \left( {6,5} \right)$ gives 40, which is discretized by $\textit{NQ}_\textit{proximity}$ to far. Function $f_\textit{location}$ is a $\textit{QQ}\rightarrow Q$ mapping, which is bad on the inputs bad and far. For interior, the value of the intermediate function is needed, which simply becomes 0 based on the worst possible combination of values of equipment and balcony. Then, $\textit{NQ}_\textit{interior}$ maps 0 to the value bad. The value for exterior is computed in the same way as in Sect. 2.2, and the resulting value is ok. The value of layout with $f_\textit{layout} \left( {\hbox {bad,ok}} \right)$, which produces bad. Finally, the value apartment, which is of Compound type $\textit{NQ}\rightarrow Q$, is computed in two steps. The first step selects the function $\textit{price}+20{,}000$, where $\textit{price}=109{,}000$, so the resulting value is 129,000. Using $\textit{NQ}_{\textit{apartment}}$, this value is discretized to the final evaluation—bad.

Let us compare these results with the evaluations acquired in Sect. 2.2. Because of the addition of numeric attributes, some values have changed, but most of them stayed the same. The most notable difference is the change of the final evaluation of alternative “Equipped”, which increased from bad to good. Due to added numerical attributes and changed aggregation functions, the attributes interior and layout increased by one value for this alternative. Consequently, the apartments “Equipped” and “Nice” are both evaluated as good. Now, “Equipped” differs from “Nice” in terms of a better interior, exterior and layout. On the other hand, “Nice” is better than “Equipped” in terms of price, proximity and the location of the apartment.

We see that even though the developed model is more complex, the introduction of numeric attributes brought additional modelling possibilities to the example. The inclusion of numeric values allowed for a more natural interpretation of some concepts in comparison to describing them with qualitative values. For example, the price of an apartment is naturally expressible with monetary values. The extension further provides the ability to explicitly consider numeric concepts—the average room size is accurately measured with the quantity $size/\# rooms$, whereas the same measure cannot be accurately obtained given qualitative values for size and #rooms.

4 Probabilistic and fuzzy distributions

In its current form, DEX evaluates alternatives with crisp values. The only exception is in cases when a decision table yields an interval of values; in this case, the resulting evaluation is generally a set of values. However, alternatives and the DM’s preferences are often imprecise or uncertain. For instance, alternatives’ values may be difficult to measure or assess precisely because they may depend on factors that cannot be controlled by the DM. Similarly, sometimes it is difficult for the DM to define a decision rule that would produce a single crisp value because the outcome is distributed between several values. Therefore, we propose an extension to DEX that explicitly models these types of uncertainty by introducing probabilistic or fuzzy distributions of values.

The extension of probabilistic distributions incorporates ideas from probabilistic inference methods (Durbach and Stewart 2012; Shachter and Peot 1992; Yang et al. 2006). These ideas have already been addressed in DEX to some extent. An early version of the method, called DECMAK, used probabilistic and fuzzy distributions for the representation of alternatives’ values (Rajkovič et al. 1987). Later, probabilistic distributions were introduced to aggregation functions, to facilitate the revision of the models (Žnidaršič et al. 2006a), which was implemented in the system called proDEX (Žnidaršič et al. 2006b). Some steps towards a decision making method with probabilistic extension were presented in Trdin and Bohanec (2012, 2013, 2014a) and applied in real-life decision making scenarios in Bohanec et al. (2016), Kontić et al. (2014). Similar approaches were explored by other authors (Bergez 2013; Holt et al. 2013; Omero et al. 2005).

Here, we propose a systematic extension to DEX which introduces probabilistic and fuzzy value distributions to both alternative values and aggregation functions, taking into account the formal model described in Sect. 2.1 and the other extensions proposed in this paper. The distributions are introduced at two points:

• 1. Values describing decision alternatives are extended to probabilistic or fuzzy distributions of values over corresponding attribute scales, and

• 2. outputs of aggregation functions are extended from crisp values or intervals to distributions of values.

That is, wherever we could have previously used a crisp value or an interval of values in a model, we may now use a distribution of values. This requires three changes to the formal model: (1) attribute scales have to be extended to cope with value distributions, (2) aggregation functions should in general map to distributions rather than intervals and (3) the evaluation procedure should propagate distributed values.

Value distributions are general in the sense that they are capable of representing all previous value types: single crisp values, value intervals and value sets. Required probabilistic and fuzzy distributions of values are added, though. Previously defined aggregation functions, which did not include distributions, can thus be used without change in the new setting.

4.1 Prerequisites

Given an attribute $x_i$ and its scale $D_i$, we define a value distribution V as a tuple $\left( {B,r} \right) ,$ where $B\subseteq D_i$ and $r:B\rightarrow \left[ {0,1} \right]$. For finite B, we use the notation $V=\left\{ {v_{i_1 } \backslash p_{i_1 } ,v_{i_2 } \backslash p_{i_2 } ,\ldots ,v_{i_m } \backslash p_{i_m } } \right\}$, where $p_{i_j } =r\left( {v_{i_j } } \right)$ for each $j=1,2,\ldots ,m$. Generally, we do not impose any specific interpretation on $p_{i_j }$; however, we are interested in two specific cases:

• 1. V is a discrete probability distribution: here, $p_{i_j}\hbox {s}$ represent probabilities of corresponding elements $v_{i_j }$, and are normalized so that $\mathop \sum \nolimits _{j=1}^m p_{i_j } =1$.

• 2. V is a fuzzy set (Zadeh 1965): $p_{i_j }\hbox {s}$ represent fuzzy grades of membership of corresponding elements $v_{i_j}$. Moreover, a fuzzy set is said to be normalized when $\hbox {max}_{j=1}^m p_{i_j } =1$.

In this way, we can represent all needed value types using both representations:

• A crisp value v can be represented as a value distribution $\left\{ {v\backslash 1} \right\}$.

• A qualitative interval of values $\left[ {v_{i_j } ,v_{i_k } } \right] ,j\le k$, which includes all values $v_{i_j } ,v_{i_{j+1} } ,\ldots ,v_{i_k }$, is represented as a discrete probability distribution $\left\{ {v_{i_j } \backslash \frac{1}{k-j+1},v_{i_{j+1} } \backslash \frac{1}{k-j+1},\ldots ,v_{i_k } \backslash \frac{1}{k-j+1}} \right\}$ or, alternatively, as a fuzzy set $\left\{ {v_{i_j } \backslash 1,v_{i_{j+1} } \backslash 1,\ldots ,v_{i_k } \backslash 1} \right\}$.

• The set of values $V=\left\{ {v_{i_j } ,v_{i_k } ,\ldots ,v_{i_l } } \right\} ,\left| V \right| =m$ is represented as a discrete probability distribution $\left\{ {v\backslash \frac{1}{m}\mid v\in V} \right\}$ or as a fuzzy set $\left\{ {v\backslash 1|v\in V} \right\}$.

For infinite S, we treat an attribute $x_i$ as a random variable $X_i$ and consider the probability distributions of its values. When $X_i$ is a continuous random variable, we assume the existence of probability density function (Feller 1968) $f_{X_i } :D_i \rightarrow \left[ {0,1} \right]$, with a$,b\in D_i$, so that

$$\begin{aligned} \Pr \left[ {a\le X_i \le b} \right] =\mathop \int \nolimits _a^b f_{X_i } \left( t \right) dt. \end{aligned}$$

(31)

When $X_i$ is a discrete random variable, we assume the existence of probability mass function (Feller 1968) $f_{X_i } :D_i \rightarrow \left[ {0,1} \right]$, with $t\in D_i$, so that

$$\begin{aligned} f_{X_i } \left( t \right) =\Pr \left( {X_i =t} \right) . \end{aligned}$$

(32)

To formally describe the evaluation procedure using value distributions, we need to transform all value distributions to a common representation. When interpreted as probability distributions, both discrete and continuous variables can be treated in a unified way. In order to represent a discrete random variable with a probability density function, we employ the Dirac delta function (Hewitt and Stromberg 1965):

$$\begin{aligned} \begin{aligned}&x\ne 0\Rightarrow \delta \left( x \right) =0,\\&\quad \mathop \int \nolimits _{-\infty }^\infty \delta \left( x \right) dx=1. \end{aligned} \end{aligned}$$

(33)

If a discrete random variable takes values $v_{i_1 } ,v_{i_2 } ,\ldots ,v_{i_k }$ with respective probabilities $p_{i_1 } ,p_{i_2 } ,\ldots ,p_{i_k}$, then its associated probability density function is:

$$\begin{aligned} f\left( t \right) =\mathop \sum \limits _{j=1}^k p_{i_j } \delta \left( {t-v_{i_j } } \right) . \end{aligned}$$

(34)

Suppose there are n independent random variables $X_i$, $i=1,\ldots ,n$ which are associated with probability density functions $f_{X_i } \left( {x_i } \right)$. Let $Y=G\left( {X_1 ,X_2 ,\ldots ,X_n } \right)$ be a random variable which is a function of $X_i\hbox {s}$. Then the density function $f_Y \left( y \right)$ is (Feller 1968; Hewitt and Stromberg 1965):

$$\begin{aligned} \begin{aligned} f_Y \left( y \right)&=\mathop \int \limits _{-\infty }^\infty \mathop \int \limits _{-\infty }^\infty \ldots \mathop \int \limits _{-\infty }^\infty f_{X_1 } \left( {x_1 } \right) f_{X_2 } \left( {x_2 } \right) \ldots f_{X_n } \left( {x_n } \right) \\&\quad \delta \left( {y-G\left( {x_1 ,x_2 ,\ldots ,x_n } \right) } \right) dx_1 dx_2 \ldots dx_n . \end{aligned} \end{aligned}$$

(35)

In general, solving this multiple integral is a difficult problem to tackle analytically, thus we employ statistical sampling with Monte Carlo method (Caflisch 1998) for random variables defined with a probability density function: a random variable $X_i$ is sampled m times, giving a finite discrete probability distribution $V_i =\{v_j \backslash \frac{1}{m}\hbox {|}v_j \in X_i ,j=1,2,\ldots ,m\}$.

Now, we can consider only value distributions with a finite number of elements for probabilistic aggregation. Given some aggregation function g, we need to compute the value distribution $W=g(V_1 ,V_2 ,\ldots ,V_n)$, where each distribution $V_i$ is obtained either by sampling or as a result of computation on lower levels of the attribute hierarchy. Also note that function g is defined only on atomic crisp values; see Eq. (8). In order to obtain $W=\left\{ {w_1 \backslash p_1 ,w_2 \backslash p_2 ,\ldots ,w_k \backslash p_k } \right\}$, where $w_i \in D_Y$, we need to compute all $p_i ,i=1,2,\ldots ,k$:

$$\begin{aligned} p_i =\mathop \sum \limits _{\left( {v_1 \backslash q_1 ,v_2 \backslash q_2 ,\ldots ,v_n \backslash q_n } \right) \in V_1 \times V_2 \times \cdots \times V_n } \mathop \prod \limits _{w_i \backslash r_i \in g\left( {v_1 ,v_2 ,\ldots ,v_n } \right) } r_i \mathop \prod \limits _{j=1}^n q_j . \end{aligned}$$

(36)

This expression produces a probability for value $w_i$ in the final value distribution W. The outer sum runs over the Cartesian product of all basic values in input value distributions. Given some value combination $v_1 ,v_2 ,\ldots ,v_n$, the basic function $g\left( {v_1 ,v_2 ,\ldots ,v_n } \right)$ is computed. The result of computation is generally a value distribution $\left\{ {w_1 \backslash r_1 ,w_2 \backslash r_2 ,\ldots ,w_k \backslash r_k } \right\}$. From this distribution, only the probability $r_i$ of $w_i$ is considered. This particular combination adds to the outer sum the probability of the output $r_i$ multiplied by the probability of this value combination ($\mathop \prod \nolimits _{j=1}^n q_j)$. This procedure follows the intuitive explanation of handling probabilities. Furthermore, Eq (36) is in line with the constraint imposed in Eq. (10), specifying that the function $F_i$ should for singleton input value distributions produce the same value as the function $f_i$.

Fuzzy sets are handled in the same way, except that product and sum operators are replaced by the standard fuzzy set operators: t-norms ($\top )$ as logical conjunction and the corresponding t-conorms ($\bot )$ as logical disjunction (Bede 2012):

$$\begin{aligned} p_i =\mathop \bot \limits _{\left( {v_1 \backslash q_1 ,v_2 \backslash q_2 ,\ldots ,v_n \backslash q_n } \right) \in V_1 \times V_2 \times \cdots \times V_n } (0,\mathop \top \limits _{w_i \backslash r_i \in g\left( {v_1 ,v_2 ,\ldots ,v_n } \right) } \left( {r_i ,\top \left( {q_1 ,q_2 ,\ldots ,q_n } \right) } \right) ). \end{aligned}$$

(37)

The most commonly used t-norms and t-conorms in fuzzy sets are minimum and maximum, respectively:

$$\begin{aligned} \top \left( {a,b} \right)= & {} \min \left( {a,b} \right) , \end{aligned}$$

(38)

$$\begin{aligned} \bot \left( {a,b} \right)= & {} \max \left( {a,b} \right) . \end{aligned}$$

(39)

By substituting the t-norm and t-conorm, we get an equation for computing $p_i\hbox {s}$ for fuzzy sets:

$$\begin{aligned} p_i =\mathop {\max }\limits _{\left( {v_1 \backslash q_1 ,v_2 \backslash q_2 ,\ldots ,v_n \backslash q_n } \right) \in V_1 \times V_2 \times \cdots \times V_n } (0,\mathop {\min }\limits _{w_i \backslash r_i \in g\left( {v_1 ,v_2 ,\ldots ,v_n } \right) } \left( {r_i ,\hbox {min}\left( {q_1 ,q_2 ,\ldots ,q_n } \right) } \right) ). \end{aligned}$$

(40)

This equation can be interpreted similarly to Eq. (36), where product and sum are replaced by min and max, respectively.

4.2 Extension formalization

In order to formally extend DEX to handle value distributions, we first need to extend the scales of attributes. For practical reasons, we wish to keep the definition of value scale $D_i$ of attribute $x_i \in X$ the same as before. However, we extend the range of aggregation functions to include value distributions over $D_i$ and accommodate the propagation of value distributions during evaluation. Thus, we define $ED_i$ as the space of all value distributions over $D_i$. When $D_i$ is a finite discrete scale $D_i =\left\{ {v_{i_1 } ,\ldots ,v_{i_{m_i } } } \right\}$, then

$$\begin{aligned} ED_i =\left\{ {\left\{ {v_{i_1 } \backslash p_{i_1 } ,\ldots ,v_{i_m } \backslash p_{i_{m_i } } } \right\} |\,p_{i_j } \in \left[ {0,1} \right] ,v_{i_j } \in D_i ,j=1,2,\ldots m_i } \right\} . \end{aligned}$$

(41)

When $D_i$ is infinite, then $ED_i =X_i$ is a random variable with an associated probability density or probability mass function $f_{x_i }$ as defined in Eq. (31, 32).

Then, we need to extend Eq. (8) so that each aggregation function $f_i$ can, in general, map to extended domain $ED_i$ instead of the space of intervals $I\left( {D_i } \right)$:

$$\begin{aligned} f_i :D_{i_1 } \times D_{i_2 } \times \cdots \times D_{i_k } \rightarrow ED_i ,\left\{ {x_{i_1 } ,x_{i_2 } ,\ldots ,x_{i_k } } \right\} =S\left( {x_i } \right) . \end{aligned}$$

(42)

This substitution affects the aggregation procedure carried out at each aggregated attribute $x_i$. When evaluating the associated aggregation function $f_i$, we should expect value distributions at inputs. Also, the output result will, in general, be a value distribution, too. Therefore, using $\left\{ {x_{i_1 } ,x_{i_2 } ,\ldots ,x_{i_k } } \right\} =S\left( {x_i } \right)$ we have to redefine the function $F_i$, defined in Eq (9) to

$$\begin{aligned} F_i :ED_{i_1 } \times ED_{i_2 } \times \cdots \times ED_{i_k } \rightarrow ED_i . \end{aligned}$$

(43)

When encountering a finite value distribution of values from some scale, the evaluation must correctly propagate the corresponding probabilities or membership values to construct the final evaluation. Even though the scales of attributes have been extended, the aggregation functions $f_i$ stay the same—only the evaluation procedure needs to be adapted to handle the new extended scales. The adaptation defines how the aggregation is performed for function $F_i$, using function $f_i$. In Sect. 4.1, we suggested handling only finite value distributions since infinite value distributions are sampled.

We suggest using two types of aggregation: probabilistic aggregation or fuzzy aggregation. We leave the choice to the user of the model and assume that the type of aggregation has been defined in advance. We also assume that all value distributions involved in the aggregation are normalized accordingly—that is, $\mathop \sum \nolimits _{i=1}^k p_i =1$ for probabilistic aggregation and $\hbox {max}_{i=1}^k p_i =1$ for fuzzy aggregation with normalization. The computation of $F_i$ is performed according to Eq. (36) for probabilistic aggregation and according to Eq. (40) for fuzzy aggregation. In both cases, the obtained result is a value distribution which can be interpreted as a probability or fuzzy distribution, respectively.

For practical reasons, we introduce the final step that simplifies the obtained value distributions so that they are more readable for the user. The procedure is called simplify: at input, it takes the full representation of a value distribution $v=\{v_1 \backslash p_1 ,v_2 \backslash p_2 ,\ldots ,v_n \backslash p_n \}$ and produces an equivalent simplified representation. It detects whether v represents some special distribution, such as a single crisp value, an interval or a set. Formally, $\textit{simplify}_i \left( v \right)$, applied on the value distribution v assigned to attribute $x_i$, is a recursive procedure defined as follows:

$$\begin{aligned} \textit{simplify}_i \left( v \right) =\left\{ {\begin{array}{l} \textit{simplify}_i \left( {\{v_1 ,v_2 ,\ldots ,v_n \}} \right) |v=\left\{ {v_1 \backslash p_1 ,v_2 \backslash p_2 ,\ldots ,v_n \backslash p_n } \right\} ,p_1 =p_2 =\cdots =p_n \\ \textit{simplify}_i \left( {[v_l ,v_h } \right] |v=\left\{ {v_1 ,v_2 ,\ldots ,v_n } \right\} ,D_i \cap v=\left[ {v_l ,v_h } \right] \\ v_l |v=\left[ {v_l ,v_h } \right] ,v_l =v_h \\ v|\hbox { otherwise} \\ \end{array}} \right. . \end{aligned}$$

(44)

Here, the first case detects that all $p_i\hbox {s}$ are equal, and v represents at least a set. The second case determines whether v is an interval, and the third case checks whether an interval has the same bounds and thus represents a single value.

When using value distributions, evaluations cannot always be compared directly. Using stochastic dominance (Hadar and Rusell 1969) on probabilistic value distributions is one possibility. Suppose we are given two value distributions $V_1 ,V_2 \in ED_i$, where $D_i$ is a preferentially ordered scale. We say that $V_1$ (stochastically) dominates $V_2$ if and only if:

$$\begin{aligned} \begin{aligned}&\forall w\in D_i :\hbox {Pr}\left[ {V_1 \ge w} \right] \ge P\left[ {V_2 \ge w} \right] ,\\&\exists w\in D_i :\hbox {Pr}\left[ {V_1 \ge w} \right] >P\left[ {V_2 \ge w} \right] . \end{aligned} \end{aligned}$$

(45)

Stochastic dominance can bring some insight into the final evaluations; however, it cannot always generate a total order of alternatives. This means that the best alternative may not exist for the current model and its input values. In such cases, the DM must interpret the results, and chose the best alternative among those that are not dominated.

4.3 Didactic example

We continue the didactic example from Sect. 3.2. Hereafter, we shall include value distributions at alternatives’ input values and leave aggregation functions unchanged. Probabilistic value distributions are included in alternatives according to the following rationale. Observations are quantified in Table 14.

• Alternative “Big” does not actually have a fixed utilities bill of 100; rather, the value is uniformly distributed through the year in the amount from 80 to 120. The same applies to alternative “Nice”, but the value is distributed between 60 and 80. However, the utilities costs for apartment “Equipped” are still fixed at 60.

• Recall that we do not know the quality of the neighbourhood of alternative “Equipped”. For this reason we previously decided to assign the middle value. Now we are able to specify that the value is actually undetermined or unknown, assigning an interval of all possible values.

• The equipment of apartments “Big” and “Nice” was previously determined by rule of thumb, and we felt that specifying a single value was too limited. Instead we now specify the situation with a distribution.

• Parking for apartment “Equipped” is usually available, but sometimes it is not. On the other hand, parking may sometimes be available for alternative “Nice”. With apartment “Big”, there is a dedicated parking space.

• Before, we were not able to decide about the outer appearance of the apartment “Nice”. Now we know that the appearance is something between ok and good.

Table 14 The table gives new values with probabilistic value distributions for the input attributes

Table 15 Evaluations of three alternatives according to the model developed with the non-determined input values of alternatives

Because the model aggregation functions and scales of attributes did not change from the previous example, we can immediately evaluate the alternatives. As in previous examples, we will show the evaluation of the first alternative “Big” and present the evaluations of the other two alternatives. All alternatives’ evaluations are summarized in Table 15.

First, “Big’s” price is computed using $F_\textit{price}$ with argument values of 85,000 and U(80, 120). The values are interpreted as value distributions: 85,000 is {85,000 $\backslash$ 1} and $U(80,\, 120)$ is sampled m times (we choose $m=100)$, obtaining the value distribution $\{80.05\backslash 0.01,\, 111.96\backslash 0.01, {\ldots }, 90\backslash 0.01\}$. For all 100 combinations of values in buying price ($\{85.000\backslash 1\}$) distribution and utilities distribution ($\{80.05 \backslash 0.01,\, 111.96\backslash 0.01,\, {\ldots }, 90\backslash 0.01\}$), function $f_\textit{price}$ is evaluated during computation of $F_\textit{price}$. The final produced evaluation for price is $\{104{,}212\backslash 0.01, 111{,}870.4\backslash 0.01,{\ldots }, 106{,}600\backslash 0.01\}$.

The values of proximity and location stay the same as before because there were no changes in their sub-trees.

The interior value of apartment “Big” is calculated using function $F_\textit{interior}$. Values supplied as arguments to the function are (80, no, no, 3) with the probability 0.9 and (80, some, no, 3) with the probability 0.1. In both cases, an intermediate value 0 is produced, which creates the value bad by function $\textit{NQ}_\textit{interior}$. Even though apartment “Big” has some equipment, it does not improve the evaluation of interior. The final value distribution $\{bad\backslash 1\}$ is simplified to bad.

The value for exterior stays the same as in Sect. 3.2. Even though layout’s inputs are now value distributions, a crisp value of bad is obtained. The apartment may come with some equipment, but the value of interior did not change, which in turn means that the value for layout will not change, either.

The final evaluation for apartment “Big” is acquired with function $F_\textit{apartment}$. The values supplied to the function are $\{104{,}212\backslash 0.01,\, 111{,}870.4\backslash 0.01,{\ldots },\, 106{,}600\backslash 0.01\}$, bad and bad. For each combination of values, $f_\textit{apartment}$ is computed. Note, probability for each value combination is 0.01. Because the only possible values for location and layout are bad, function $\textit{price}+20{,}000$ from space $\varPhi \left( {x_\textit{apartment} } \right)$ is always used. The first value from the distribution gives 124,212, which results in value bad by $\textit{NQ}_\textit{apartment}$. This is done for all values inside the value distribution of price. $\textit{NQ}_\textit{apartment}$ always produces bad, which in turn, after simplification, produces the final evaluation of bad.

The final evaluations now provide an additional insight regarding the alternatives. Even though the utilities costs will be variable for alternative “Big”, and there is some equipment in the apartment, the final evaluation is still bad. On the other hand, we can now better compare apartments “Equipped” and “Nice”. With only numeric values, we were not able to distinguish between the apartments in terms of the final evaluation. Now we can see that apartment “Equipped” is mostly good (with probability 0.67), but there is the probability of 0.33 that the evaluation is a bit worse, giving the evaluation ok. For the apartment “Nice”, the evaluation is strongly good (probability 0.96) with a small chance (probability 0.04) that the apartment is even very good. Overall, apartment “Nice” is better than “Equipped”.

Furthermore, we can check if apartment “Nice” stochastically dominates apartment “Equipped”. For that, we check for each value $v\in D_\textit{apartment}$, that the probability of the value v or better in the evaluation of “Nice” ($V_\textit{Nice})$ is greater or equal than the probability of the value v or better in the evaluation of “Equipped” ($V_\textit{Equipped})$:

$$\begin{aligned} \begin{aligned} 1&=\hbox {Pr}\left[ {V_\textit{Nice} \ge {v\hbox {-}bad}} \right] \ge P\left[ {V_\textit{Equipped} \ge {v\hbox {-}bad}} \right] =1,\\ 1&=\hbox {Pr}\left[ {V_\textit{Nice} \ge {bad}} \right] \ge P\left[ {V_\textit{Equipped} \ge {bad}} \right] =1,\\ 1&=\hbox {Pr}\left[ {V_\textit{Nice} \ge {ok}} \right] \ge P\left[ {V_\textit{Equipped} \ge {ok}} \right] =1,\\ 1&=\hbox {Pr}\left[ {V_\textit{Nice} \ge {good}} \right] \ge P\left[ {V_\textit{Equipped} \ge {good}} \right] =0.67,\\ 0.04&=\hbox {Pr}\left[ {V_\textit{Nice} \ge {v\hbox {-}good}} \right] \ge P\left[ {V_\textit{Equipped} \ge {v\hbox {-}good}} \right] =0. \end{aligned} \end{aligned}$$

(46)

Additionally, for the apartment “Nice” to stochastically dominate apartment “Equipped”, we need to find a value $v^{\prime }\in D_\textit{apartment}$, for which the probability of producing value at least $v^{\prime }$ in $V_\textit{Nice}$ is strictly greater than probability of producing $v^{\prime }$ in $V_\textit{Equipped}$. There are two such values: good and v-good:

$$\begin{aligned} \begin{aligned} 1&=\hbox {Pr}\left[ {V_\textit{Nice} \ge {good}} \right]>P\left[ {V_\textit{Equipped} \ge {good}} \right] =0.67,\\ 0.04&=\hbox {Pr}\left[ {V_\textit{Nice} \ge {v\hbox {-}good}} \right] >P\left[ {V_\textit{Equipped} \ge {v\hbox {-}good}} \right] =0. \end{aligned} \end{aligned}$$

(47)

Therefore, “Nice” stochastically dominates “Equipped”, and can be considered a better choice. In the same way, it can be shown that both “Nice” and “Equipped” dominate “Big”.

5 Relational models

The data encountered in everyday life are frequently of a relational nature in the sense that one entity is composed of several similar sub-entities—similar to the extent that they can be evaluated by the same criteria. For example, when evaluating a company, a DM may want to evaluate all departments of that company. Here the company’s departments are the similar sub-entities, and there is a “one-to-many” relationship between the company and departments. All departments can be evaluated in a similar way, by the same model. The problem, however, is that an arbitrary number of sub-evaluations are acquired in this way, which need to be combined in the evaluation of the main alternative (company).

Currently, relational models are not supported in DEX. Adding them would be a substantial improvement, which would facilitate addressing a much larger group of decision problems. Introducing relational models, however, requires extensions that affect the representation of decision alternatives and introduces new components, such as relational attributes and relational aggregation functions. The purpose of this section is to extend the existing DEX method for the support of relational models.

Relational data are frequently modelled in relational databases. Several disciplines of machine learning explicitly consider the development of relational models—for example, inductive logic programming (Lavrač and Džeroski 1994). Relational data are also studied in quantitative multi-criteria decision making methods, but rarely in an explicit way. There, it rarely causes difficulties because it naturally involves common operators based on summation and averaging while at the same time handling numeric values. Such operators are useless in qualitative settings and require special approaches.

We propose to extend DEX to handle situations where one alternative is composed of several sub-alternatives (see examples in (Bohanec et al. 2014; Trdin and Bohanec 2012, 2013, 2014a, b)). For this purpose, we propose to modify the modelling process by developing two models: one (M) for the evaluation of the main alternative and another (RM) for the evaluation of relational sub-alternatives. To evaluate a single main alternative, each sub-alternative is first evaluated by RM. Then, all evaluations are aggregated, providing an input value to M. The evaluations in both M and RM are carried out in the same way as described previously; the only differences occur at the point where the two models are connected with each other.

Relational models were already employed with DEX in two real-world use cases, in which they proved very useful for defining a decision model and evaluating alternatives on relational data: the reputational risk assessment of banks (Bohanec et al. 2014) and the appraisal of energy production technologies in Slovenia (Bohanec et al. 2016; Kontić et al. 2014).

5.1 Extension formalization

To formalize this extension, we need to introduce three new entities and one type of aggregation function. The three new entities are: (1) relational alternative, (2) relational model and (3) relational aggregated attribute. The new aggregation function is called relational aggregation function and is used only by the relational aggregated attributes.

Given some alternative $a\in A$, the term relational alternatives denotes all entities $ra\in RA$ that are in a many-to-one relation with a. For instance, $a\in A$ may be a company and $ra\in RA$ one of its departments, assuming a one-to-many relation $department:A\rightarrow RA$. Since RM, as any other model, contains input attributes, it also holds that

$$\begin{aligned} ra\in \textit{inputSpace}_\textit{RM}. \end{aligned}$$

(48)

The purpose of the relational model RM is to evaluate relational alternatives RA. RM is modelled in the same way as any other DEX model; the only difference is in its purpose in the evaluation procedure: it evaluates relational alternatives RA and provides inputs to the main model M, which evaluates alternatives A.

A relational aggregated attribute rx is a special type of attribute that provides a connection point between the main model M and the relational model RM. rx’s scale is formalized in the same way as for other attributes. The relational attribute rx is connected to its counterpart output attribute ox from RM; rx serves as an input to M. In other words, rx is a connecting point between M and RM: it receives output from RM (through attribute ox) and provides input to M.

As there is a one-to-many relation between A and RA, the aggregation at rx should aggregate all output values coming from RM, that correspond to one alternative $\left( {V_1 ,V_2 ,\ldots ,V_n } \right) ,V_i \in ED_{ox}$, into a single input value $V\in ED_{rx}$ of M. Here, n is the number of relational alternatives $ra\in RA$ evaluated by RM that are part of a single alternative $a\in A$ evaluated by M.

Consequently, the aggregation function $f_{rx}$ is special in the sense that it aggregates an arbitrary number of values coming from ox rather than single values from descendant attributes. Formally:

$$\begin{aligned}&V=f_{rx} \left( {V_1 ,V_2 ,\ldots ,V_n } \right) , \end{aligned}$$

(49)

$$\begin{aligned}&f_{rx} :(ED_{ox} )^{n}\rightarrow ED_{rx} . \end{aligned}$$

(50)

In general, $f_{rx}$ is any aggregation function defined for an arbitrary number of arguments—for example, min, max, sum, average, mean or count. The value of the function is computed with the corresponding function $F_{rx}$ as per Eq. (43). The function is constrained to the types of ox’s and rx’s scales. For instance, sum cannot be applied to qualitative values.

Given this formalization, an arbitrary number of relational models is supported in some model M. Moreover, an arbitrary number of relational models can be nested inside each other. Therefore, decision problems with nested relational alternatives can be solved as well.

5.2 Didactic example

To show the relational models in practice, we will once again extend the previously developed model, taking into account that each apartment consists of multiple rooms. Each room has some features worth evaluating, such as size and equipment. Thus, in addition to the main model for apartments, we introduce a relational model for the evaluation of rooms. The relational model contains two attributes: (1) size, a numeric attribute giving the absolute size of the room and (2) equip., a qualitative attribute giving the amount and quality of equipment in the room.

Due to the nature of this design, three relational aggregated attributes are introduced—more precisely, exchanged for their current counterparts—in the apartment model: (1) size’s value is computed by summing the values of size attribute from the room relational model, (2) equipment transforms the room equipment values with uniform weighting and (3) #rooms counts the number of rooms by counting the number of size’s values that come from the relational model.

Fig. 6

Model for evaluating an apartment with support for numeric values and relational models. The nodes with round corners are aggregated attributes and nodes with pointed corners are input attributes. The triangle at the far right represents the relational model for evaluation of rooms

Figure 6 shows the new structure of the model. The apartment model structure is the same as in Fig. 5. However, the relational room model now appears on the right-hand side and is connected to the model through the relational attributes.

To complete the model, we need to define the relational aggregation functions for size, equipment and #rooms. According to the model idea described above, the functions are defined as follows:

$$\begin{aligned} f_\textit{size} \left( {v_1 ,v_2 ,\ldots ,v_n } \right)= & {} \mathop \sum \limits _{i=1}^n v_i ,\nonumber \\ f_{equipment} \left( {v_1 ,v_2 ,\ldots ,v_n } \right)= & {} \mathop \sum \limits _{i=1}^n \frac{1}{n}v_i ,\\ f_{\# \textit{rooms}} \left( {v_1 ,v_2 ,\ldots ,v_n } \right)= & {} count\left( {v_1 ,v_2 ,\ldots ,v_n } \right) =n.\nonumber \end{aligned}$$

(51)

To use the new model, we have to define input values for rooms; see Table 16.

Table 16 The table gives new values for the input attributes with the relational model

The evaluation procedure is almost the same as before. The values for the price, proximity and location are computed in the same way as before. However, some inputs to the sub-tree layout are now obtained from the relational model and may differ from previous calculations. The main difference is in the aggregation that involves functions $f_\textit{size}$, $f_\textit{equipment}$ and $f_{\# \textit{rooms}}$. The aggregation function $f_\textit{size}$ first evaluates the size of three rooms of the “Big” apartment, which give values of {40, 25, 15}. Function $f_\textit{size}$ produces their sum—80. Similarly, the value for equipment is computed. The evaluation of rooms gives {no, $\{no\backslash 0.9$, $some\backslash 0.1\}$, no}. The function $f_\textit{equipment}$ weighs each value with 1 / 3, and the final distribution of $\{no\backslash 0.97,\, some\backslash 0.03\}$ is produced. This distribution cannot be simplified. The computation of #rooms follows the same principle as the computation for size, except that $f_{\# \textit{rooms}}$ counts the number rather than the sum of values and gives 3. These values now enter the calculations as ordinary inputs of the apartment model, giving the results as shown in Table 17.

Table 17 Evaluations of three alternatives according to the model developed with the relational model included

The final evaluation of alternative “Big” did not change due to the relational aggregation functions computing the same values as before. The same holds for the “Nice” apartment. However, the new evaluation of “Equipped” is different. Previously, the value of equipment was yes. Now, as we explicitly considered two rooms, one of which is small and incompletely equipped, the aggregation of the rooms gives the value distribution $\{yes\backslash 0.95,\, some\backslash 0.05\}$. Thus, the value some is additionally propagated through the aggregation and, even though it has a very small probability of 0.05, it affects the final outcome, which becomes $\{bad\backslash 0.0017,\, ok\backslash 0.33,\, good\backslash 0.665\}$ instead of $\{ok\backslash 0.33,\, good\backslash 0.67\}$.

6 Discussion

We introduced three extensions to the DEX method: numeric attributes, value distributions, and relational alternatives and models. All extensions were motivated by needs identified when applying DEX in complex real decision modelling tasks. They substantially increase the range of decision problems that can be addressed, but they also require a number of changes and additions to the method. In order to introduce numeric attributes, it was necessary to add a class of numeric value scales, which in turn increased the number of primitive function types from one to five. The introduction of value distributions required a generalization of value scales to extended scales and an extension of the evaluation algorithm to use existing aggregation functions on value distributions. The new ability to represent and evaluate relational alternatives required the introduction of relational models, relational attributes and relational aggregation functions. All the extensions are compatible with each other so that they can be used simultaneously.

Considering the previous applications of DEX and the solutions found in some other MCDA methods, the extensions are not entirely new but are for the first time systematically brought together and formalized. Previously, DEX was combined with numeric attributes and relational models in the modelling and assessment of bank reputational risk (Bohanec et al. 2014), qualitative relational models were introduced in the assessment of public administration e-portals (Leben et al. 2006) and qualitative probabilistic distributions were used in ecological domains (Bohanec et al. 2009; Bohanec 2008; Bohanec and Žnidaršič 2008; Žnidaršič et al. 2008). All three extensions were for the first time used together in the evaluation of sustainable electrical energy production in Slovenia (Bohanec et al. 2016; Kontić et al. 2014). These applications were typically formulated and implemented in an ad-hoc manner and tailored to a particular application. However, they clearly indicated the need for extending the method.

The proposed extensions provide new means for representing a DM’s knowledge and preferences and facilitate solving a wider variety of decision problems. Specifically, the introduction of numeric attributes allows for a more natural treatment of numeric quantities; instead of being limited only to qualitative attributes, the DM can choose between the qualitative and quantitative representations, as illustrated in Sect. 3.2. Numeric attributes also solve a common problem in DEX: mapping from numeric measurements to qualitative input values. To date, this had to be done implicitly and manually by the DM, whereas the extended method facilitates this mapping with ease. The introduction of numeric attributes opens DEX for the inclusion of other approaches of quantitative MCDA, such as pairwise preference and weights elicitation of AHP (Saaty 2008; Saaty and Vargas 2012) and using marginal utility functions of MAUT (Multi-attribute utility theory) (Wang and Zionts 2008). Another possible approach to build aggregation functions, especially in case of numeric values, is the ordinal regression (Jacquet-Lagrèze and Siskos 1982, 2001; Mihelčić and Bohanec 2016). An approach to multiple criteria sorting based on robust ordinal regression (Kadziński et al. 2014) seems particularly suitable. DEX requires a direct acquisition of utility functions; outranking MCDA methods that acquire DM’s preferences through comparison of alternatives, such as ELECTRE (Roy 1991) and PROMETHEE (Brans and Vincke 1985), are less suitable for this purpose.

The second extension, value distributions, introduces capabilities that are well known and proven in contexts such as expert systems, fuzzy control systems and uncertainty and risk analysis. They allow an explicit formulation of “soft” (imprecise, uncertain and even missing) knowledge and data. Specifically, we introduced two types of value distributions: probabilistic, which are suitable for representing uncertainty, and fuzzy to represent vague concepts and values. The former resembles the use of probabilities in MAUT (Wang and Zionts 2008), whereas the latter is in line with the trend of extending MCDA methods with fuzzy sets (Baracskai and Dörfler 2003; Kahraman 2008; Omero et al. 2005).

The third extension addresses decision problems in which alternatives are composed of similar sub-components. In quantitative MCDM, such problems are rarely mentioned because they can be easily handled by common aggregation functions, such as the average. In the qualitative world of DEX, there are no such obvious functions – thus the need for the extension. The key contribution of relational models is allowing the DM to granulate the decision problem to a finer level, explicitly considering parts of the whole.

The extensions increase the complexity of model development. Previously in DEX, there was just one conceptual way to define an attribute or an aggregation function; now there is plenty from which the DM can choose. Even though this increases the flexibility of the modelling, it also requires the mastering of newly available tools. Particularly, the elicitation of attributes and aggregation functions becomes more difficult. In addition to dealing with new types of numeric attributes, the DM should control the interplay between qualitative and numeric attributes, for which there are now five different primitive forms instead of just one (see Sect. 3). Regarding aggregation functions, the complexity is increased by a new class of numerical functions, which have to be formulated by the DM in accordance with his or her preferences. Regarding the evaluation of alternatives, DEX was previously limited to only qualitative values and their sets. The extended DEX additionally produces a wide variety of other value types: integers, real numbers and fuzzy or probabilistic value distributions; this increases the complexity of the evaluation results and requires more efforts for their interpretation. In summary, the increased complexity of the models and computed values may degrade the comprehensibility of the method and results for the DM. Consequently, at this stage the extended DEX seems a more useful tool for a skilled decision analyst than for an ordinary DM.

We are aware that retrieving knowledge from the DM, especially with methods of such complexity, is a difficult problem. In this paper, we were not concerned with the difficulties of knowledge acquisition; our primary goal was to open up the DEX method to make it more suitable for addressing a wide range of real decision problems. We were interested in giving the DM the ability to specify his/her preference as flexibly as possible. We are aware that more research and practice is needed to fully assess the strengths and weaknesses of the extended method and to identify potential difficulties associated with preference elicitation. In further research, we will attempt to find a suitable balance between the flexibility, simplicity and comprehensibility of the method, which may even require narrowing down its broadness.

In practice, a method such as DEX requires the support of computer software for both the development of models and evaluation of alternatives. While the basic DEX method is already implemented in the software DEXi, the extended method has not been fully implemented yet. To date, we have developed a java software library that supports all the proposed extensions, but it cannot be used interactively by the DM. Instead, a model is created and used through program code. The library was used to model water flows in agriculture (Kuzmanovski et al. 2015) and was incorporated into a web service to evaluate user-supplied alternatives through a web browser (Kuzmanovski et al. 2015). In the future, we aim to implement an interactive computer program that will fully support the extended DEX method in a way that the similar to the way the current program DEXi supports DEX.

7 Conclusion

The main contribution of this paper is an extension of the already established qualitative multi-criteria modelling method DEX. Based on identified practical needs in complex decision situations, we proposed three extensions: including numeric attributes, evaluating alternatives with probabilistic and fuzzy value distributions and supporting the evaluation of relational alternatives. In order to support these concepts, a number of formal and algorithmic components were added, specifically: numeric scale types, extended value domains, probabilistic and fuzzy value distributions, numeric aggregation functions, five primitive function types and relational attributes, aggregation functions, models and alternatives. All these concepts greatly increase the flexibility and representational richness of the method, enlarge the class of decision problems that can be addressed and improve the decision process by providing additional methods and tools to the decision maker. On the other hand, the extensions increase the complexity of the method and developed models themselves, which may have negative effects on the efficiency and comprehensibility of the modelling. Due to this complexity, the extended DEX method is currently aimed at skilled decision analysts rather than ordinary decision makers.