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It costs fi to open facility i, and cij for facility i to serve one unit of demand from client j.
( bar{Q}_{i} ): Maximum quality level of new facility i of the leader firm.
f i : Fixed cost of launching a new facility i of the leader firm.
c i : Unit quality cost of new facility i of the leader firm.
Distance between new facility i of the leader firm and demand point ( j ).
d ij : Distance between new facility i of the leader firm and demand point ( j ).
In this equation, the dependent variable Dit refers to access (total number of surgeries) or cost (mean cost per surgery) for facility i in year t.
The fixed cost of shifting facility i from location l to location q in period t is denoted by a tilq.
where c ijkl is the cost of assigning facility i in location k and simultaneously facility j in location l, and x ik = 1 if location k is assigned to facility i; otherwise, x ik = 0. Also, x jl = 1 if location l is assigned to facility j; otherwise, x jl = 0.
With this transformation we ensure that the arc ii ′ belongs to a solution if and only if facility i is open.
Similarly, facility i is used as Steiner node if and only if i belongs to the solution, but arc ii ′ does not.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com