Table 1 Abbreviations and descriptions.

DMU_j

The jth DMU, j = 1, 2, …, n, i.e. the total number of DMU_s is n.

k

k node, k = 1, 2, …, K, that is, the number of nodes is K, and each node represents a sub-stage.

t

The t period, t = 1, 2, …, T, that is, there are T periods in the observation period.

m_k、r_k

Node k has m_k input variables and r_k output variables.

\({x}_{mjk}^{t}\)

DMU_j is the mth input variable in the epoch t, node k, m = 1, 2, …, m_k, \({{x}}_{mjk}^{t}\in {R}_{+}\).

\({{y}}_{rjk}^{t}\)

DMU_j is the mth input variable in the epoch t, node k, r = 1, 2, …, r_k, \({{y}}_{rjk}^{t}\in {R}_{+}\).

\({\lambda }_{k}^{t}\)、\({\lambda }_{jk}^{t}\)

In the period t, the weight coefficient of node k, \({\lambda }_{jk}^{t}\in {R}_{+}\); \({\lambda }_{k}^{t}\) is a vector form of \({\lambda }_{jk}^{t}\).

\({{z}}_{j{(kh)}_{l}}^{t}\)

The lth connection variable (also known as the intermediate product, i.e., the Linkage variable) between the node k and the node h of DMU_j is l = 1, 2, …, L_kh, that is, the number of connection variables is L_kh, and \({{z}}_{j{(kh)}_{l}}^{t}\in {R}_{+}\).

\({{z}}_{j{k}_{l}^{(t,t+1)}}^{(t,t+1)}\)

At node k, the lth junction transition between period t and period t + 1 (i.e., the carry-over variable), l = 1, 2, …, L_k, that is, the number of junction transitions is L_k, and \({{z}}_{j{k}_{l}^{(t,t+1)}}^{(t,t+1)}\in {R}_{+}\).

\({{\bf{x}}}_{ok}^{t}\)、\({{\bf{y}}}_{ok}^{t}\)

The input and output vectors of DMU_o in the period t and node k are respectively deputized.

\({{\bf{z}}}_{{(kh)}_{l}}^{t}\)、\({{\bf{z}}}_{{k}_{l}^{(t,t+1)}}^{(t,t+1)}\)

Vector forms of connection and transition variables, respectively, with the subscript letters meaning the same as above.

\({{\bf{X}}}_{k}^{t}\)、\({{\bf{Y}}}_{k}^{t}\)

\({{\bf{X}}}_{k}^{t}=({{\bf{x}}}_{1k}^{t},{{\bf{x}}}_{2k}^{t},\cdots ,{{\bf{x}}}_{nk}^{t})\in {R}_{n}^{{m}_{k}}\), \({{\bf{Y}}}_{k}^{t}=({{\bf{y}}}_{1k}^{t},{{\bf{y}}}_{2k}^{t},\cdots ,{{\bf{y}}}_{nk}^{t})\in {R}_{n}^{{r}_{k}}\), the input and output matrices of period t and node k are, respectively, presented.

e

Unit vector.

\({{\bf{Z}}}_{(kh){\rm {in}}}^{t}\)、\({{\bf{Z}}}_{(kh){\rm {out}}}^{t}\)

Regarded as a vector form of two types of connecting variables, the input and the perceived output, the subscript letters have the same meaning as above.

\({{z}}_{{o}{k}_{l}^{(t,\,t+1)}{\rm {good}}}^{(t,\,t+1)}\)、\({{z}}_{{o}{k}_{l}^{(t,\,t+1)}{\rm {bad}}}^{(t,t+1)}\)

The vector forms of two types of transition variables, the desired output and the undesired output, respectively, have the same meaning as above.

\({{\bf{s}}}_{{o}k}^{t-}\,\)、\({{\bf{s}}}_{{o}k}^{t+}\)

The input and output relaxation variables of DMU_o in the period t and node k are respectively presented.

\({{\bf{s}}}_{(kh){\rm {in}}}^{t}\,\)、\({{\bf{s}}}_{(kh){\rm {out}}}^{t}\)

Slacks represent two types of connected variables, considered input and deemed output, respectively.

\({s}_{{o}{k}_{l}^{(t,\,t+1)}{\rm {good}}}^{(t,t+1)}\)、\({s}_{{o}{k}_{l}^{(t,\,t+1)}{\rm {bad}}}^{(t,t+1)}\,\)

Slacks represent the two types of transition variables, the desired output and the undesired output, respectively.