Table 1. Mathematical notation

Indices
i	State or entity (product, module, or component)
j	Remanufacturing transition
k	EOL battery quality level

Decision variables
Xk	Take-back quantity of batteries with quality level k
Yj	Number of times transition j is performed
Ni	External purchase quantity of entity i
Li	Disposal quantity of entity i
Z	Production quantity of remanufactured batteries

Parameters
Tij	Value of transition matrix cell (i, j)
ckB	Take-back cost of EOL batteries with quality level k
cj	Processing cost per transition j
ciN	External purchase cost per unit of entity i
riM	Material recycling revenue per unit of entity i
ciM	Material recycling cost per unit of entity i
ciL	Disposal cost per unit of entity i
ak	Available quantity of EOL batteries with quality level k
dr	Market demand for remanufactured batteries
IN	Environmental impact from new battery production
ej	Environmental (env.) impact per transition j
eiM	Unit env. impact from material extraction of entity i
eiP	Unit env. impact from production of entity i
eiR	Unit env. impact from material recycling of entity i
eiL	Unit env. impact from disposal of entity i
ea	Unit env. impact from reassembly process