Processes 2021 , 9 , 274
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jobs processed in production line j , F i − 1, j is the completion time of job i − 1 in production − 1, j is the setup time in the conversion from job i − 1 to job i in production line j . − 1, j is the power when job i − 1 is converted to job i in production line j . In tissue paper mills, the parent roll produced in the papermaking line should be transported to the conversion line for further processing. However, the distance between papermaking lines and conversion lines is different. The longer the distance, the higher the energy cost of transporting the parent roll from the papermaking line to the conversion line. The transportation time in tissue paper mills is up to tens of minutes. In the production scheduling process, it is necessary to consider the processing route with short distance to reduce the transportation energy cost. Under the TOU electricity pricing scheme, the transportation energy cost varies in different time periods. Similarly, the transportation time may span multiple time periods. The transportation energy cost model of tissue paper mills is formulated as below: line j , TS i , i US i , i
m 2 ∑ l = 1 ( TP 1 × P 1 + TP 2 × P 2 + TP 3 × P 3 ) × W i × TC
O i , h , l (11)
m 1 ∑ h = 1
n ∑ i = 1
ET =
ET 1 ( 1 )+ ET 2 ( 1 )+ ET 3 ( 1 )
TP 1 =
(12)
∑ 3 s = 1 ( ET 1 ( s )+ ET 2 ( s )+ ET 3 ( s ))
ET 1 ( 2 )+ ET 2 ( 2 )+ ET 3 ( 2 )
(13)
TP 2 =
∑ 3 s = 1 ( ET 1 ( s )+ ET 2 ( s )+ ET 3 ( s ))
ET 1 ( 3 )+ ET 2 ( 3 )+ ET 3 ( 3 )
TP 3 =
(14)
∑ 3 s = 1 ( ET 1 ( s )+ ET 2 ( s )+ ET 3 ( s ))
ET 1 ( s )= MIN ( CEIL ( S i , l ) − S i , l , FG ( CEIL ( S i , l + T i , l ) , CEIL ( S i , l ))) × Q FLOOR MOD S i , j , K , s
(15)
FLOOR ( MOD ( S i , l , K )+( T i , l − FIX ( T i , l , K ) × K )) ∑ k = CEIL ( MOD ( S i , j , K ))
Q ( k , s )
(16)
ET 2 ( s )=
ET 3 ( s )=( S i , l + T i , l − FLOOR ( S i , l + T i , l )) × Q FLOOR MOD S i , j , K , s (17) ET 4 ( s )= FIX ( T i , l , K ) × K ∑ k = 1 Q ( k , s ) (18) where ET is the transportation energy cost, Equations (12)–(14) calculate the proportions of the processing time of job i in the off-peak, mid-peak, and on-peak periods to the total processing time, respectively. Equation (15) represents the transportation energy cost from the start of the transportation to the next period. Equation (16) represents the transportation energy cost from the next period of the start of the transportation to the previous period of the complete of the transportation. Equation (17) represents the transportation energy cost from the previous period of the completion of the transportation to the completion of the transportation. If the transportation spans for several cycles, Equation (18) represents the transportation energy cost of the job in several cycles. m 1 represents the number of production lines in the pulping and papermaking stage, while m 2 is the number of production lines in the conversion stage. TP 1, TP 2, and TP 3 represent the proportions of the processing time of job i in the off-peak, mid-peak, and on-peak periods to the total processing time, respectively. W i is the scale of job i . C ∑ m 1 h = 1 O i , h , l represents energy consumed by transporting job i from papermaking line h to conversion line l . The function Q ( k , s ) can judge whether the time period k belongs to period type s . The parameter s is the period type, and s values equal to 1, 2, and 3, respectively represent the off-peak, mid-peak, and on-peak period, respectively.
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