iosoi (Identifier Of Solution Of Interest)

iosoi is an R package that implements and extends the multi-criteria decision making (MCDM) approach proposed in:

• Torres, M., Pelta, D. A., Lamata, M. T., & Yager, R. R. (2021). An approach to identify solutions of interest from multi and many-objective optimization problems. Neural Computing & Applications, 33(7), 2471–2481. doi:10.1007/s00521-020-05140-x

In addition to the interval-based approach proposed by Torres et al (2021), it also includes volume-based methods for assessing the solutions. Formally, iosoi automatizes the solution of the following MCDM problem:

Given a set of $m$ solutions (alternatives) with evaluations on $n$ criteria, and an order of preference among these criteria, the goal is to order the solutions according to their overall scores. These overall scores are obtained through the linear combination of the evaluations and the weights of the criteria.

Installation

You can install the development version of iosoi from GitHub with:

# install.packages("devtools")
devtools::install_github("pnovoa/iosoi")

Example 1: Interval-based approach

Consider a decision problem in which the objective is to identify a set of solutions of interest based on their overall performance over a given set of criteria. As input data we have a set of $m$ solutions (alternatives) and their individual evaluations on each of the $n$ criteria. Intuitively, this information can be arranged in the form of a matrix $E$ whose entries $e_{ij}$ correspond to the numerical evaluation that solution $i$ has in criterion $j$. Additionally, we have an order of preference of the criteria by the decision maker. Without loss of generality, it will be assumed that this order is decreasing as a function of the order of the criteria in the matrix $E$. That is, let $\lambda_j$ the $j$-th criterion, then $\lambda_1 \succeq \lambda_2 ... \succeq \lambda_n$. Numerically, this information can be translated into variables representing non-negative weights associated with the criteria and with the additional condition of adding up to $1$.

The following example is based on a problem of $5$ solutions, $3$ criteria and evaluation matrix:

$$ E_{5 \times 3} = \left(\begin{array}{cc} 5 & 2 & 3 \\ 4 & 4 & 3 \\ 1 & 1 & 5 \\ 4 & 2 & 3 \\ 4 & 3 & 2 \\ \end{array}\right) $$

The preference order provided by the decision-maker is (as we mentioned before) in decreasing order regarding the order followed by the criteria in matrix $E_{ij}$.

Using the iosoi package, one can perform the steps of the Torres et al. (2021) approach to obtain global assessments for the solutions and be able to select those of greatest interest to the decision maker, that is, what we refer as solution of interest (SOIs) . In what follows we will perform this task using the interval-based approach by Torres et al. (2021). Specifically, we rely on the neutral attitude for computing the superiority degree of each solutions against the reference one. For both approaches we set a threshold of $0$, which means that only those solutions with assessments greater than $0$ will appear in the output.

library(iosoi)

# Evaluation matrix
E <- matrix(data = c(5, 2, 3, 4, 4, 3, 1, 1, 5, 4, 2, 3, 4, 3, 2), 
            byrow = TRUE, 
            nrow = 5)

print(E)
#>      [,1] [,2] [,3]
#> [1,]    5    2    3
#> [2,]    4    4    3
#> [3,]    1    1    5
#> [4,]    4    2    3
#> [5,]    4    3    2

# Identifying SOIs from the possibility approach of Torres et al (2021).
E %>% poss_identify_sois(by = "neutral", threshold = 0.0)
#>    C1 C2 C3 VE_C1 VE_C2    VE_C3       LB UB REF   neutral
#> S1  5  2  3     5   3.5 3.333333 3.333333  5   0 0.6666667
#> S2  4  4  3     4   4.0 3.666667 3.666667  4   1 0.5000000
#> S4  4  2  3     4   3.0 3.000000 3.000000  4   0 0.2500000
#> S5  4  3  2     4   3.5 3.000000 3.000000  4   0 0.2500000

# Identifying SOIs from the geometric approach based on the volume
E %>% geom_identify_sois(by = "poss_volume", threshold = 0.0)
#>    C1 C2 C3 VE_C1 VE_C2    VE_C3       LB       UB REF poss_volume
#> S1  5  2  3     5   3.5 3.333333 3.333333 5.000000   0   0.5035461
#> S2  4  4  3     4   4.0 3.666667 3.666667 4.000000   1   0.5000000
#> S3  1  1  5     1   1.0 2.333333 1.000000 2.333333   0   0.2708333
#> S4  4  2  3     4   3.0 3.000000 3.000000 4.000000   0   0.4615385
#> S5  4  3  2     4   3.5 3.000000 3.000000 4.000000   0   0.4736842

As noted, the output contains not only the original evaluation matrix, but also other columns related to intermediate steps of the method of Torres et al (2021). For instance, those columns with prefix VE_ corresponds to the linear scoring of each solution at the extreme points of the feasible region induced by the preference order of the criteria. Columns LB and UB contain the lower and upper bounds of the values obtained in the VE_ columns. The Boolean column REF identifies (with 1) the reference solution, that is, the one used to assess the rest of solutions. Finally, the last column neutral contains the overall assessment for each solution. Note that only those with values greater than $0$ were included.

To understand why solution S3 is left out in the approach of Torres et al.  (2021) iosoi allows to plot the intervals (ranges of possible scores) of each solution. That is, in order to visually check the degree of overlapping of the solutions in relation to the reference one. The code then performs the analysis again but relaxes the threshold to $-1$ to include all solutions in the final output. In this way the intervals of all the solutions will be plotted.

E %>% 
  poss_identify_sois(
    by = "neutral", 
    threshold = -1.0) %>%
  plot_intervals()

As expected, S3 is excluded because its interval is clearly far to the left relative to S2 (the reference solution highlighted with interval in red). The rest, and especially S1 have certain levels of overlap with S2.

Example 2: Volume-based approach

The interval-based approach from Torres et al (2021) provides an intuitive way of assessing the solutions. However, it underestimates how the scoring function distributes over the region of feasible weights. Geometrically, it just considers the line joining the two extreme points of the polyhedron (ie. LB and UB) and hence, it does not take into account how frequent an interval score value is in the region of feasible weights. So the comparison against to the reference solution, is made on the basis that all scores within the interval are equally possible, which is a very

library(iosoi)

# Evaluation matrix
E <- matrix(data = c(5, 2, 3, 4, 4, 3, 1, 1, 5, 4, 2, 3, 4, 3, 2), 
            byrow = TRUE, 
            nrow = 5)

print(E)
#>      [,1] [,2] [,3]
#> [1,]    5    2    3
#> [2,]    4    4    3
#> [3,]    1    1    5
#> [4,]    4    2    3
#> [5,]    4    3    2

# Identifying SOIs from the possibility approach of Torres et al (2021).
E %>% poss_identify_sois(by = "neutral", threshold = 0.0)
#>    C1 C2 C3 VE_C1 VE_C2    VE_C3       LB UB REF   neutral
#> S1  5  2  3     5   3.5 3.333333 3.333333  5   0 0.6666667
#> S2  4  4  3     4   4.0 3.666667 3.666667  4   1 0.5000000
#> S4  4  2  3     4   3.0 3.000000 3.000000  4   0 0.2500000
#> S5  4  3  2     4   3.5 3.000000 3.000000  4   0 0.2500000

# Identifying SOIs from the geometric approach based on the volume
E %>% geom_identify_sois(by = "poss_volume", threshold = 0.0)
#>    C1 C2 C3 VE_C1 VE_C2    VE_C3       LB       UB REF poss_volume
#> S1  5  2  3     5   3.5 3.333333 3.333333 5.000000   0   0.5035461
#> S2  4  4  3     4   4.0 3.666667 3.666667 4.000000   1   0.5000000
#> S3  1  1  5     1   1.0 2.333333 1.000000 2.333333   0   0.2708333
#> S4  4  2  3     4   3.0 3.000000 3.000000 4.000000   0   0.4615385
#> S5  4  3  2     4   3.5 3.000000 3.000000 4.000000   0   0.4736842

We can also plot any computed assessment indicator through a bar plot. For example, if we consider poss_volume, the following code show how to plot this indicator.

E %>% 
  geom_identify_sois(by = "poss_volume", 
                     threshold = 0.0) %>%
  plot_assessment(by = "poss_volume",
                  plot_title = "Normalized volume",
                  ylabel = "Normalized volume"
                  )