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

### Description

The Conditional 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, in addition to being robust against outlier data (see ci_rbod_constr_bad function) allows to take into account external contextual continuous (Q) or/and ordinal (Q_ord) variables.

### Usage

ci_rbod_constr_bad_Q(x, indic_col, ngood=1, nbad=1,
low_w=0, pref=NULL, M, B, Q=NULL, Q_ord=NULL, bandwidth)

### 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. Q A matrix containing continuous exogenous variables. Q_ord A matrix containing discrete exogenous variables. bandwidth Multivariate mixed bandwidth for exogenous variables; it can be calculated by bandwidth_CI function.

### Value

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

 ci_rbod_constr_bad_Q_est Composite indicator estimated values. ci_method Method used; for this function ci_method="rbod_constr_bad_Q". ci_rbod_constr_bad_Q_weights Raw weights assigned to each simple indicator. ci_rbod_constr_bad_Q_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_rbod_constr_bad, ci_bod_constr_bad

### Examples

data(EU_2020)

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

# Conditional robust BoD Constrained VWR
band = bandwidth_CI(dat, ngood=1, nbad=2, Q = Q_GDP)

ngood=1,
low_w=0.05,
pref=NULL,
M=10,
B=50,
Q=Q_GDP,
bandwidth = band$bandwidth) CI_BoD_C$ci_rbod_constr_bad_Q_est

# # Conditional robust BoD Constrained ordVWR
# import <- c("gasemiss_2011","employ_2011", "deprived_2011")
#
#                                  bandwidth = band$bandwidth) # CI_BoD_C2$ci_rbod_constr_bad_Q_est