Processes 2021 , 9 , 274
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2.2. Energy-Efficiency Scheduling Modeling In the pulping and papermaking stage or the conversion stage, there are multiple parallel production lines with different speeds. Each job must be first processed in a production line in the first stage and then processed in a production line in the second stage. The pulping and papermaking line or the conversion line only processes one job at a time. In this study, the job cannot be stopped once being processed in the pulping and papermaking line or the conversion line. In addition, the time interval between two stages is uncertain. The scheduling problem is the variant of a two-stage flexible flow shop scheduling problem. According to the production scheduling characteristics of tissue paper mills and the above established energy cost model, the energy-efficiency scheduling model is as below. The model in this study contains two optimization objectives, which are multi-objective optimization problems. Multi-objective optimization problems are different from single-objective optimization problems. The quality of the solution needs to be evaluated through the dominance relationship. When all the objectives of solution S1 are better than all the objectives of solution S2, it can be said that S1 dominates S2. If S1 only has some objectives better than S2, then S1 and S2 are non-dominated. If a solution is not dominated by any other solution, the solution is called a non-dominated solution. The solution to the multi-objective optimization problem is a set of non-dominated solutions. min ( MAX ( C i )) ∀ i ∈{ 1,2, . . . , n } min ( EP + ES + ET ) (19) n ∑ i = 1 m 1 ∑ j = 1 O i , j = 1 (20)
m 2 ∑ j = 1
n ∑ i = 1
O i , j = 1
(21)
j ∈ 1, . . . , m ; ∀ l ∈ 2, . . . , N j
(22)
I j , l ≥ F j , l
TS l , l
− 1, j ∀
− 1 +
V i , h × O i , h , W i ∀ i = 1, . . . , n
B i ,2 ≥ B i ,1 + f
m 1 ∑ l = 1
m 2 ∑ h = 1
V i , l × O i , l ,
(23)
f ( v 1, v 2, w )=
H v 1
v 1 ≥ v 2 v 1 < v 2
(24)
w v 1 −
w v 2 +
H v 1
m 1 + m 2 = m
(25)
m ∑ j = 1
(26)
N j = n
Formula (19) is the optimization objectives, and there are two optimization objectives, which are makespan and energy cost. C i is the completing time of the job i in conversion line. Constraint (20) guarantees every job can and must be processed once in the pulping and papermaking line. Constraint (21) guarantees that every job can and must be processed once in the conversion line. Constraint (22) guarantees that every papermaking line or converting line can only process one job at the same time, indicating that only when the previous job and the setup are complete can the next job be started, where I j , l is the beginning time of the l -th job in production line j . Constraint (23) guarantees there is a time interval between two stages. B i ,1 is the beginning time of job i in the papermaking line, and B i ,2 is the beginning time of job i in the conversion line. There is a time interval between B i ,1 and B i ,2 ,where f ∑ m 1 l = 1 V l × O i , l , ∑ m 2 h = 1 V h × O i , h , W i represents the time interval, which is related to the speed of the papermaking line l when processing job i ( V i , l ), the speed of the conversion line h when processing job i ( V i , h ), and the scale of job ( W i ). Formula (24) is the calculation method of the time interval between B i ,1 and B i ,2 ,where H represents the value
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