Processes 2021 , 9 , 274
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FLOOR ( MOD ( S i , j , K )+( T i , j − FIX ( T i , j , K ) ∗ K )) ∑ k = CEIL ( MOD ( S i , j , K ))
P k × U i , j
EP 2 =
(3)
EP 3 = S i , j + T i , j − FLOOR S i , j + T i , j × U i , j × P FLOOR ( MOD ( S
(4)
i , j + T i , j , K ))
K ∑ k = 1
EP 4 = FIX T i , j , K ×
P k × U i , j
(5)
where EP is the processing energy cost, and Equation (1) is the calculation formula of the processing energy cost. The processing energy cost is divided into four parts, namely EP 1, EP 2, EP 3, and EP 4. Equation (2) represents the processing energy cost from the start of the job to the next period. Equation (3) represents the processing energy cost from the next period of the start of the job to the previous period of the completion of the job. Equation (4) represents the processing energy cost from the previous period of the completion of the job to the completion of the job. If the job spans for several cycles, Equation (5) represents the processing energy cost of the job in several cycles. m and n are the numbers of production lines and jobs, respectively. O i , j denotes whether production line j processes job i . O i , j is set to one when production line j processes job i ; otherwise, it is set to zero. CEIL is a function that rounds up to an integer. T i , j is the processing time of job i in production line j . S i , j is the start time of job i processed in production line j . Function FG has two parameters. When the first parameter is equal to the second parameter, FG is set to one; otherwise, it is set to zero. U i , j represents the power of job i in production line j , P k represents the electricity price in k period. FLOOR is a function that rounds down to an integer, K is the number of periods, MOD represents modulo operation, and FIX is the quotient operation. The setup time is up to several hours, especially in the pulping and papermaking stage, which cannot be ignored in tissue paper mills. Under the TOU electricity pricing scheme, the machine setup energy cost may be different when the setup occurs at different periods due to different energy prices in different periods. The off-peak or mid-peak period has lower setup energy cost compared with the on-peak period. Moreover, the setup may span multiple periods. The setup energy cost model is formulated as follows:
N j ∑ i = 1
m ∑ j = 1
ES 1 + ES 2 + ES 3 + ES 4
(6)
ES =
ES 1 = MIN CEIL F i
, FG CEIL F i P FLOOR ( MOD ( F i
− 1, j − × US i , i
− 1, j
, CEIL F i
− 1, j
F i
TS i , i
− 1, j +
− 1, j
(7)
− 1, j ×
, K ))
− 1, j
, K )+( TS i , i
, K ) × K ))
FLOOR ( MOD ( F i
FIX ( TS i , i
− 1, j −
− 1, j
− 1, j
− 1, j
∑ k = CEIL ( MOD ( F i
ES 2 =
(8)
P k × US i , i
, K ))
− 1, j
FLOOR F i − 1, j +
ES 3 = F i
− 1, j ×
TS i , i
TS i , i
US i , i
− 1, j −
− 1, j +
− 1, j +
− 1, j
(9)
× P FLOOR ( MOD ( F i
, K ))
TS i , i
− 1, j
K ∑ k = 1
ES 4 = FIX TS i , i
, K ×
− 1, j
(10)
P k × US i , i
− 1, j
where ES represents the setup energy cost, and Equation (6) is the calculation formula of the setup energy cost. The setup energy cost is divided into four parts, namely ES 1, ES 2, ES 3, and ES 4. Equation (7) represents the setup energy cost from the start of the setup to the next period. Equation (8) represents the setup energy cost from the next period of the start of the setup to the previous period of the completion of the setup. Equation (9) represents the setup energy cost from the previous period of the completion of the setup to the completion of the setup. If the setup spans for several cycles, Equation (10) represents the setup energy cost of the job in several cycles. N j represents the number of
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