## Robust constrained Benefit of the Doubt approach (BoD) in presence of undesirable indicators

### Description

The Robust constrained Benefit of the Doubt function introduces additional constraints to the weight variation in the optimization procedure (Constrained Virtual Weights Restriction) allowing to restrict the importance attached to a single indicator expressed in percentage terms, ranging between a lower and an upper bound (VWR); this function, furthermore, allows to calculate the composite indicator simultaneously in presence of undesirable (bad) and desirable (good) indicators allowing to impose a preference structure (ordVWR). This function is the robust version of the `ci_bod_constr_bad`: it is based on the concept of the expected minimum input function of order-m (Daraio and Simar, 2005) allowing to compare the unit under analysis against `M` peers by extracting `B` samples with replacement.

### Usage

`ci_rbod_constr_bad(x, indic_col, ngood=1, nbad=1, low_w=0, pref=NULL, M, B) `

### Arguments

 `x` A data.frame containing simple indicators. `indic_col` A numeric list indicating the positions of the simple indicators. `ngood` The number of desirable outputs; it has to be greater than 0. `nbad` The number of undesirable outputs; it has to be greater than 0. `low_w` Importance weights lower bound. `pref` The preference vector among indicators; For example if `Indic1` is the most important, `Indic2`,`Indic3` are more important than `Indic4` and no preference judgment on `Indic5` (= not included in the vector), the `pref` vector can be written as: `c("Indic1", "Indic2","Indic3","Indic4")` `M` The number of elements in each of the bootstrapped samples. `B` The number of bootstrap replicates.

### Value

An object of class "CI". This is a list containing the following elements:

 `ci_rbod_constr_bad_est` Composite indicator estimated values. `ci_method` Method used; for this function ci_method="rbod_constr_bad". `ci_rbod_constr_bad_weights` Raw weights assigned to each simple indicator. `ci_rbod_constr_bad_target` Indicator target values.

### Author(s)

Fusco E., Rogge N.

### References

Rogge N., de Jaeger S. and Lavigne C. (2017) "Waste Performance of NUTS 2-regions in the EU: A Conditional Directional Distance Benefit-of-the-Doubt Model", Ecological Economics, vol.139, pp. 19-32.

Zanella A., Camanho A.S. and Dias T.G. (2015) "Undesirable outputs and weighting schemes in composite indicators based on data envelopment analysis", European Journal of Operational Research, vol. 245(2), pp. 517-530.

`ci_bod_constr`, `ci_bod_constr_bad`

### Examples

```data(EU_2020)
indic <- c("employ_2011", "percGDP_2011", "gasemiss_2011","deprived_2011")
dat <- EU_2020[-c(10,18),indic]

# Robust BoD Constrained VWR