DEXiR is a software package for using DEXi models in R. The main function is evaluating decision alternatives using a model previously developed by DEXi software.
DEXi models are hierarchical qualitative rule-based multi-criteria decision models developed using the method DEX (Decision EXpert, https://en.wikipedia.org/wiki/Decision_EXpert), using the program DEXi (https://kt.ijs.si/MarkoBohanec/dexi.html) or DEXiWin (https://dex.ijs.si/dexisuite/dexiwin.html).
In general, a DEXi model consists of a hierarchy of qualitative (symbolic linguistic, discrete) variables, called attributes. Each attribute represents some observable property (such as Price or Performance) of decision alternatives under study. An attribute can take values from a set of words (such as "low; medium; high" or "unacc; acc; good; exc"), which is usually small (up to five elements) and preferentially ordered from "bad" to "good" values.
The hierarchy of attributes represents a decomposition of a decision problem into sub-problems, so that higher-level attributes depend on the lower-level ones. Consequently, the terminal nodes represent inputs, and non-terminal attributes represent the outputs of the model. Among these, the most important are one or more root attributes, which represent the final evaluation(s) of the alternatives.
The evaluation of decision alternatives (i.e., hierarchical aggregation of values from model inputs to outputs) is governed by decision rules, defined for each non-terminal attribute by the creator of the model (usually referred to as a "decision maker").
DEX (Decision EXpert) refers to a general multi-attribute decision modeling method, characterized by using qualitative attribute hierarchies and decision tables. For further information, see (Trdin, Bohanec, 2018) and (Bohanec, 2022).
DEXi ("DEX for instruction") refers to DEXi software.
DEXi implements a subset of DEX, for instance, it is restricted to set-based evaluation methods.
DEXi supports the creation and editing of DEXi models, which are saved on .dxi files and
subsequently read by DEXiR for processing in R. For further information on DEXi, see
https://kt.ijs.si/MarkoBohanec/dexi.html.
A new backward-compatible implementation of DEXi, aimed at gradually replacing it in the future. For further information on DEXiWin and related software, see https://dex.ijs.si/dexisuite/dexisuite.html.
DEXiR is this R package. It is capable of reading and processing DEXi models with some extensions towards the full DEX (for example, using value distributions).
Models developed using the DEXi software are stored in XML-formatted .dxi files. In
order to use DEXi models in R, DEXiR supports the following tasks:
Reading DEXi models from .dxi files into the R environment, using read_dexi.
Making data frames containing data (both input and output) about considered decision alternatives,
using set_alternative.
Evaluating decision alternatives, using evaluate.
Analyzing alternatives (selective_explanation, plus_minus,
compare_alternatives).
Drawing charts.
By default, evaluation is based on sets, which is a standard evaluation procedure of DEXi. DEXiR extends this by supporting:
evaluations using probabilistic and fuzzy value distributions (see evaluate);
"pruned" evaluation, when the evaluation starts from selected non-terminal attribute(s) upwards.
DEXiR has been designed to facilitate using DEXi models in R produced externally by the DEXi software. DEXiR does not provide any explicit means for creating and/or editing DEXi models in R.
This example uses a simple DEXi model for evaluating cars, which is distributed together with the DEXi software (including DEXiR) and is used throughout DEX literature to illustrate the methodological approach (https://en.wikipedia.org/wiki/Decision_EXpert).
First, this model is loaded into R and printed as follows:
> Car <- read_dexi("data/Car.dxi")
> Car
DEXi Model: CAR_MODEL
Description: Car demo
index id structure scale funct
[1] CAR_MODEL CAR_MODEL
[2] CAR +- CAR unacc; acc; good; exc (+) 12 3x4
[3] PRICE |- PRICE high; medium; low (+) 9 3x3
[4] BUY.PRICE | |- BUY.PRICE high; medium; low (+)
[5] MAINT.PRICE | +- MAINT.PRICE high; medium; low (+)
[6] TECH.CHAR. +- TECH.CHAR. bad; acc; good; exc (+) 9 3x3
[7] COMFORT |- COMFORT small; medium; high (+) 36 3x4x3
[8] X.PERS | |- #PERS to_2; 3-4; more (+)
[9] X.DOORS | |- #DOORS 2; 3; 4; more (+)
[10] LUGGAGE | +- LUGGAGE small; medium; big (+)
[11] SAFETY +- SAFETY small; medium; high (+)
Rows in the table correspond to individual attributes. The columns represent the following:
indexIndices of attributes.
idUnique attribute names, generated by DEXiR from original DEXi names, in order to provide syntactically correct variable names in R and allow unambiguous referencing of attributes.
structureThe hierarchical structure of attributes, named as in the original DEXi model.
scaleValue scales associated with each attribute. The symbol "(+)" indicates that the corresponding scale is ordered preferentially in increasing order.
functInformation about the size (number of rules) and dimensions of the corresponding decision tables.
Looking at the structure of attributes, please notice that the attribute at index [1] is virtual and
does not actually appear in the original DEXi model. It is necessary in DEXiR to facilitate models that
have multiple root attributes. The "real" root of the Car model is actually [2] CAR. It depends on
two lower-level attributes, PRICE and TECH.CHAR. These are decomposed further. Overall, the model consists of
six input (basic) attributes: BUY.PRICE, MAINT.PRICE, X.PERS, X.DOORS, LUGGAGE and SAFETY, and
four output (aggregate) attributes: CAR, PRICE, TECH.CHAR. and COMFORT.
Among the latter, CAR is the most important and represents the overall evaluation of cars.
The next step usually consists of defining a data frame representing decision alternatives (i.e., cars in this case). The Car model already comes with a data table about two cars:
> Car$alternatives
name CAR PRICE BUY.PRICE MAINT.PRICE TECH.CHAR. COMFORT X.PERS X.DOORS LUGGAGE SAFETY
1 Car1 4 3 2 3 4 3 3 3 3 3
2 Car2 3 2 2 2 3 3 3 3 3 2
In this data frame, attribute values are represented by ordinal numbers w.r.t. the corresponding scales.
A more readable output can be made using DexiModel$as_character:
> Car$as_character(Car$alternatives)
name CAR PRICE BUY.PRICE MAINT.PRICE TECH.CHAR. COMFORT X.PERS X.DOORS LUGGAGE SAFETY
1 Car1 exc low medium low exc high more 4 big high
2 Car2 good medium medium medium good high more 4 big medium
This data can be edited using common R data.frame functions. Also, DEXiR provides the method
DexiModel$alternative for defining a single decision alternative, for example:
> alt <- Car$alternative("MyCar1",
BUY.PRICE="low", MAINT.PRICE=2, X.PERS="more", X.DOORS="4",
LUGGAGE=2, SAFETY="medium")
> alt
name CAR PRICE BUY.PRICE MAINT.PRICE TECH.CHAR. COMFORT X.PERS X.DOORS LUGGAGE SAFETY
1 MyCar1 NA NA 3 2 NA NA 3 3 2 2
Finally, such data tables can be evaluated using DexiModel$evaluate:
> eval <- Car$evaluate(alt)
> eval
name CAR PRICE BUY.PRICE MAINT.PRICE TECH.CHAR. COMFORT X.PERS X.DOORS LUGGAGE SAFETY
1 MyCar1 4 3 3 2 3 3 3 3 2 2
> Car$as_character(eval)
name CAR PRICE BUY.PRICE MAINT.PRICE TECH.CHAR. COMFORT X.PERS X.DOORS LUGGAGE SAFETY
1 MyCar1 exc low low medium good high more 4 medium medium
Once defined and evaluated, alternatives can be analysed further. DEXiR provides three analysis methods:
selective_explanationExposing particular weak and strong points of alternatives.
plus_minus analysisExploring effects of changing individual attributes to evaluation results.
compare_alternativesComparison of an alternative with other alternatives.
Examples:
> Car$selective_explanation(1)Selective explanation of Car1
Weak points:
None
Strong points:
id structure Car1
CAR.1 +-CAR 4
PRICE |-PRICE 3
MAINT.PRICE | +-MAINT.PRICE 3
TECH.CHAR. +-TECH.CHAR. 4
COMFORT |-COMFORT 3
X.PERS | |-#PERS 3
LUGGAGE | +-LUGGAGE 3
SAFETY +-SAFETY 3
> Car$plus_minus(1, as_character = TRUE)
id structure -2 -1 CAR.1=exc 1
BUY.PRICE | |-BUY.PRICE [ unacc medium exc
MAINT.PRICE | +-MAINT.PRICE unacc exc low ]
X.PERS | |-#PERS unacc exc more ]
X.DOORS | |-#DOORS unacc exc 4 exc
LUGGAGE | +-LUGGAGE unacc exc big ]
SAFETY +-SAFETY unacc exc high ]
> Car$compare_alternatives(1, as_character = TRUE)
id structure Car1 Car2
CAR CAR NULL NULL
CAR.1 +-CAR exc > good
PRICE |-PRICE low > medium
BUY.PRICE | |-BUY.PRICE medium
MAINT.PRICE | +-MAINT.PRICE low > medium
TECH.CHAR. +-TECH.CHAR. exc > good
COMFORT |-COMFORT high
X.PERS | |-#PERS more
X.DOORS | |-#DOORS 4
LUGGAGE | +-LUGGAGE big
SAFETY +-SAFETY high > medium
Evaluation results can be drawn on charts. DEXiR provides four charts that display multiple alternatives:
plotalt1with respect to a single attribute, drawing a scatterplot "alternatives by attribute-values"
plotalt2with respect to two attributes, drawing a scatterplot "attribute1 by attribute2"
plotalt_parallelwith respect to multiple attributes, drawing evaluation results using parallel axes
plotalt_radarwith respect to multiple attributes, drawing evaluation results on a radar chart
The latter two plots scale evaluation results to the [0:1] interval.
Evaluation values represented by sets or distributions are plotted either as intervals
(aggregate = "minmax") or are aggregated to a single value
(aggregate = "min", "max" or "mean").
Examples:
Plot all Car alternatives with respect to Car$first() ("CAR.1"))
> plotalt1(Car)Plot evaluation results of all Car alternatives with respect to attribute "PRICE"
> plotalt1(Car, "PRICE")
Draw "TECH.CHAR." by "PRICE" scatterplot of all Car alternatives
> plotalt2(Car, "TECH.CHAR.", "PRICE")
Draw a "TECH.CHAR." by "PRICE" scatterplot of the second Car alternative
> plotalt2(Car, "TECH.CHAR.", "PRICE", 2)
Draw all Car alternatives on parallel axes
> plotalt_parallel(Car)
Draw all Car alternatives on a radar chart
> plotalt_radar(Car)
DEXi values are used throughout DEXi models.
They provide input values and carry results of evaluations in data frames that contain data
about decision alternatives.
Values are also used in definitions of DexiFunctions and are returned by
DexiFunction$evaluate when evaluating some function for a given set of arguments.
In DEXi, values are always bound to the context provided by a DexiScale. Since each fully defined DexiAttribute is associated with some scale, we can generalize the scale context to attributes and speak about "assigning some value to an attribute".
The scale type determines the type and possible range of values that can be assigned to an attribute. DEXiR implements two scale types: DexiContinuousScale and DexiDiscreteScale. Regarding the values, the former is really simple: it allows assigning any single real number to the corresponding attribute. In other words, continuous DEXi values are of type numeric(1).
DexiDiscreteScale is the main scale type used throughout DEXi models and supports a wider range of value types.
The "normal" and most common discrete value is a "single qualitative value".
For illustration, let us use the scale composed of four qualitative values:
"unacc", "acc", "good", "exc". Then, "a single qualitative value" denotes
one of these words. Internally in DEXiR, such values are not represented by character strings, but rather
by ordinal numbers, so that ord("unacc") = 1, ord("acc") = 2, etc. Some DEXiR functions
can convert between the two representations, see DexiModel$as_character
and set_alternative().
In order to cope with missing, incomplete or uncertain data, DEX extends the concept of single values to value sets and distributions. In DEXiR, wherever it is possible to use a single qualitative value, it is also possible to use a value set or distribution. This is the main reason that all DEXiR data structures related to DEXi values are represented by lists rather than plain vectors. This includes all data frames that represent decision alternatives and all functions that return qualitative values. Also note that while sets are fully implemented in the current DEXi software, distributions are not and are thus considered extensions towards the full DEX method.
A DEXi value set is a subset of the full range of a DexiDiscreteScale values.
For the above example, the full range of ordinal values is 1:4, and some possible subsets are
c(2), c(2, 4), c(1, 2, 3) and 1:4. Internally, sets are represented by
plain integer vectors or plain numeric vectors containing integer numbers.
A DEXi value distribution associates each DexiDiscreteScale value with some
number, generally denoted \(p\) and normally expected to be in the [0,1] interval.
Depending on the context and used evaluation method (see evaluate()), \(p\) can be
interpreted as probability or fuzzy set membership. In DEXiR, value distributions are
represented using the S3 class "distribution" (see distribution).
For example, distribution(0.5, 0, 0.2, 0.3) represents a value distribution over the
above scale example, assigning
\(p = 0.5\) to "unacc",
\(p = 0.0\) to "acc",
\(p = 0.2\) to "good" and
\(p = 0.3\) to "exc".
Remarks:
The value distribution(0.5, 0, 0.2, 0.3) is internally represented as c(0.5, 0, 0.2, 0.3),
whose class() is "distribution".
Using a special class for distributions is necessary to distinguish them from sets. For instance, the
notation c(1, 1) is ambiguous and would be interpreted differently as a set or distribution.
Some DEXiR functions (see DexiModel$as_character and set_alternative())
support the formulation of distributions in the form of named vectors or lists, for instance
list(unacc=0.5, good=0.2, exc=0.3).
In data frames that contain data about decision alternatives, numeric vectors that contain non-integer values are implicitly interpreted as distributions rather than sets.
First, let us consider a car for which we have no evidence about its possible maintenance costs.
For the value of MAINT.PRICE, we may use "*", which denotes the full range of
the corresponding attribute values (equivalent to 1:3 or c(1, 2, 3) in this case).
Notice how the evaluation method considers all the possible values of MAINT.PRICE
and propagates them upwards.
alt <- Car$alternative("MyCar1a",
BUY.PRICE="low", MAINT.PRICE="*", X.PERS="more", X.DOORS="4", LUGGAGE=2, SAFETY=2)
Car$evaluate(alt)
name CAR PRICE BUY.PRICE MAINT.PRICE TECH.CHAR. COMFORT X.PERS X.DOORS LUGGAGE SAFETY
1 MyCar1a 1, 4 1, 3 3 1, 2, 3 3 3 3 3 2 2
The above evaluation result is not really useful, as the car turns out to be c(1, 4), that is,
either "unacc" or "exc", depending on maintenance costs.
Thus, let us try using value distribution for MAINT.PRICE, telling DEXiR that low maintenance costs
are somewhat unexpected (\(p = 0.1\)) and that medium costs (\(p = 0.6\)) are more likely than
high (\(p = 0.3\)). Using the evaluation method "prob" (where \(p\)'s are interpreted
as probabilities) gives the following results:
alt <- Car$alternative("MyCar1b",
BUY.PRICE="low", MAINT.PRICE=distribution(0.1, 0.6, 0.3),
X.PERS="more", X.DOORS="4", LUGGAGE=2, SAFETY=2)
Car$evaluate(alt, method = "prob")
name CAR PRICE BUY.PRICE MAINT.PRICE TECH.CHAR. COMFORT X.PERS X.DOORS LUGGAGE SAFETY
1 MyCar1b 0.1, 0.0, 0.0, 0.9 0.1, 0.0, 0.9 3 0.1, 0.6, 0.3 0, 0, 1, 0 0, 0, 1 3 3 2 2
In this case, the final evaluation of CAR is distribution(0.1, 0.0, 0.0, 0.9), that is,
list(unacc=0.1, exc=0.9). It is much more likely that MyCar1b is "exc" than "unacc".
Maintainer: Marko Bohanec marko.bohanec@ijs.si (ORCID)
Decision EXpert. Wikipedia, https://en.wikipedia.org/wiki/Decision_EXpert.
Trdin, N., Bohanec, M.: Extending the multi-criteria decision making method DEX with numeric attributes, value distributions and relational models. Central European Journal of Operations Research, 1-24, 2018 tools:::Rd_expr_doi("10.1007/s10100-017-0468-9").
Bohanec, M.: DEX (Decision EXpert): A Qualitative Hierarchical Multi-criteria Method. In: Kulkarni, A.J. (ed.): Multiple Criteria Decision Making: Techniques, Analysis and Applications. Singapore: Springer, 39-78, 2022 tools:::Rd_expr_doi("10.1007/978-981-16-7414-3_3").
DEXi: A Program for Multi-Attribute Decision Making. https://kt.ijs.si/MarkoBohanec/dexi.html.
Bohanec, M.: DEXi: Program for Multi-Attribute Decision Making, User's Manual, Version 5.04. IJS Report DP-13100, Jožef Stefan Institute, Ljubljana, 2020. https://kt.ijs.si/MarkoBohanec/pub/DEXiManual504.pdf.
Bohanec, M.: DEXiWin: DEX Decision Modeling Software, User’s Manual, Version 1.2. IJS Report DP-14741, Jožef Stefan Institute, Ljubljana, 2024. https://kt.ijs.si/MarkoBohanec/pub/2024_DP14747_DEXiWin.pdf.
DEX Software. https://dex.ijs.si.