PDF (276 K) - Growing Science
PDF (276 K) - Growing Science
PDF (276 K) - Growing Science
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J Number of position<br />
i Index for job (i=1,2,…N)<br />
j Index for position (j=1, 2,…J)<br />
3.2.2. Input parameters<br />
M. Kazemi et al. / International Journal of Industrial Engineering Computations 3 (2012)<br />
Pi Processing time of job i,<br />
Di c The ideal completion time (or due date) of job i,<br />
α i The unitary earliness penalty of job i if it starts processing before Di s β i<br />
,<br />
The unitary tardiness penalty of job i if it completes processing after Di c γ i<br />
η i<br />
,<br />
Unit cost of work-in-process holding of job i,<br />
The unitary interruption penalty of job i,<br />
A An arbitrary big positive number.<br />
3.2.3. Decision variables<br />
Ci Completion time of job i,<br />
Di s Ideal starting time for job i which is computed as Di s = Di c - Pi +1,<br />
Xij 1 if job i is processed in position j, and 0 otherwise,<br />
Ei Earliness of job i,<br />
Ti Tardiness of job i,<br />
Starting time of job i.<br />
Si<br />
3.3. Mathematical model<br />
min Z =<br />
N<br />
∑ ( αiEi + βiTi)<br />
(1.1)<br />
i = 1<br />
N 1<br />
1 ⎡⎛ J−<br />
⎞<br />
⎤<br />
+ . ∑ηi ⎢⎜∑ X ij − X ij+ 1 ⎟+<br />
X iJ + X i1−2⎥<br />
2 i = 1 ⎢⎣⎝ j=<br />
1 ⎠<br />
⎥⎦<br />
(1.2)<br />
N J<br />
+ ∑∑ X ij .( Ci− j). γ i<br />
(1.3)<br />
i = 1 j = 1<br />
subject to<br />
J<br />
∑ X ij = Pi<br />
∀ i ;<br />
(2)<br />
j = 1<br />
N<br />
∑ X ij ≤ 1<br />
∀ i ;<br />
(3)<br />
i = 1<br />
Ti Ci c<br />
Di<br />
≥ − ∀ i ;<br />
(4)<br />
E<br />
s<br />
≥D − S<br />
∀ i ;<br />
(5)<br />
i i i<br />
{ 0,1}<br />
X ∈ ∀ i, j;<br />
(6)<br />
ij<br />
i, i 0<br />
T E ≥ ∀ i ;<br />
(7)<br />
Ci = max( X ij.<br />
j)<br />
∀ i ;<br />
(8)<br />
j<br />
Si = min ⎡j+ A(1 − Xij)<br />
⎤<br />
j ⎣ ⎦ ∀ i ;<br />
(9)<br />
The objective function consists of three components. The first component calculates earliness and<br />
tardiness costs for all jobs. The second component computes interruption costs for all jobs. Finally,<br />
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