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J Number of position
i Index for job (i=1,2,…N)
j Index for position (j=1, 2,…J)
3.2.2. Input parameters
M. Kazemi et al. / International Journal of Industrial Engineering Computations 3 (2012)
Pi Processing time of job i,
Di c The ideal completion time (or due date) of job i,
α i The unitary earliness penalty of job i if it starts processing before Di s β i
,
The unitary tardiness penalty of job i if it completes processing after Di c γ i
η i
,
Unit cost of work-in-process holding of job i,
The unitary interruption penalty of job i,
A An arbitrary big positive number.
3.2.3. Decision variables
Ci Completion time of job i,
Di s Ideal starting time for job i which is computed as Di s = Di c - Pi +1,
Xij 1 if job i is processed in position j, and 0 otherwise,
Ei Earliness of job i,
Ti Tardiness of job i,
Starting time of job i.
Si
3.3. Mathematical model
min Z =
N
∑ ( αiEi + βiTi)
(1.1)
i = 1
N 1
1 ⎡⎛ J−
⎞
⎤
+ . ∑ηi ⎢⎜∑ X ij − X ij+ 1 ⎟+
X iJ + X i1−2⎥
2 i = 1 ⎢⎣⎝ j=
1 ⎠
⎥⎦
(1.2)
N J
+ ∑∑ X ij .( Ci− j). γ i
(1.3)
i = 1 j = 1
subject to
J
∑ X ij = Pi
∀ i ;
(2)
j = 1
N
∑ X ij ≤ 1
∀ i ;
(3)
i = 1
Ti Ci c
Di
≥ − ∀ i ;
(4)
E
s
≥D − S
∀ i ;
(5)
i i i
{ 0,1}
X ∈ ∀ i, j;
(6)
ij
i, i 0
T E ≥ ∀ i ;
(7)
Ci = max( X ij.
j)
∀ i ;
(8)
j
Si = min ⎡j+ A(1 − Xij)
⎤
j ⎣ ⎦ ∀ i ;
(9)
The objective function consists of three components. The first component calculates earliness and
tardiness costs for all jobs. The second component computes interruption costs for all jobs. Finally,
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