Table 1 Definition of variables used in DNDEA model

Variable Definition
$${x}_{ijk}^t$$ An input resource i to DMUj for the sub-division k at period t
$${y}_{rjk}^t$$ An output product r from DMUj for the sub-division k at period t
$${z}_{j{(kh)}_l}^t$$ A linking intermediate product of DMUj from sub-division k to sub-division h at period t
$${z}_{j{k}_l}^{\left(t,t+1\right)}$$ A carry−over of DMUj at the sub-division k from period t to period t + 1
$${s}_{iok}^{t-}$$ A slack of the input i of DMUo for sub-division k at period t
$${s}_{rok}^{t+}$$ A slack of the output r of DMUo for sub-division k at period t
$${s}_{o{(kh)}_l\alpha}^t$$ A slack of link(kh)l of DMUo at period t. α stands for free, " as input " and " as output"
$${s}_{ok_l\alpha}^{\left(t,t+1\right)}$$ A slack of carry-over variable kl from period t to period t + 1. α stands for free, good and bad
$${\lambda}_{jk}^t$$ An intensity of the DMUj corresponding to sub-division k at period t
$${s}_{ok_l good}^{\left(t,t+1\right)};{s}_{ok_l bad}^{\left(t,t+1\right)};\mathrm{and}\ {s}_{ok_l free}^{\left(t,t+1\right)}$$ The slacks denoting, respectively, carry-over shortfall, carry-over excess and carry-over deviation
ngoodk; nbadk; nfreek The number of desirable (good), undesirable (bad) and free carry-over variables for each sub-division k.
1. Source: Prepared by authors 