International Journal of Production Research 5
Table 3. Nomenclature of AF(m
1
,m
2
)||γ problem.
Indices Parameters
j: index of job where m
1
,m
2
: number of machines in the machining and the assembly stage, respectively.
j = 1,2, ... ,n
i: index of parts (or components) t
[i,j]
: processing time of ith component of job in position j in the machining stage.
where i = 1,2, ... ,m
1
q
[j]
: assembly time of job in position j in the assembly stage.
k: index of assembly machines w
[j]
: weight (importance) of job in position j.
where 1,2, ... ,m
2
A
[k]
: release time of machine k in the assembly stage,
k = 1,2, ... ,m
2
Decision variables
C
[j]
: completion time of job in
position j in the assembly stage.
Assuming (J[
1
],J[
2
], ... ,J[
n
]) as jobs’ sequence, the completion time of job j is as follows:
Set j = 1 and C
[0]
= 0;
A
k
= 0 for all k = 1,2, ... ,m
2
While j < n
Find A* = min
k=1,...,m
2
{A
k
}
C
[j]
= max
max
k=1,...,m
1
j
i=1
t
[k,i]
, A
∗
+ q
[j]
(1)
Update A
k
for k = 1,2, ... ,m
2
j ← j + 1
EndWhile
From the above procedure, objective functions can be easily derived. For instance, makespan and total (weighted)
completion time of jobs are presented as follows:
C
max
= C[
n
](2)
Total (weighted) completion time of jobs =
n
j=1
w
[j]
C
[j]
(3)
The main assumption in the above AF problems is that products have only one assembly operation. In reality though,
majority of products are complex and may require several assembly operations. Therefore, AF systems can be extended to
handle the complex product, s ee He, Babayan, and Kusiak (2001). In these systems, in addition to the assignment constraints,
precedence relations between assembly operations should also be considered which adds to the complexity of AF problems,
see Mahdavi, Komaki, and Kayvanfar (2011) for the mathematical model and solution methods. One extension of two-stage
AF is three-stage AF which has been less investigated. In AF(m,1,1)||γ , the second stage is responsible for collecting the
processed components of the first stage with m fabrication machines and shipping them to the third stage where components
are assembled by one assembly machine. Objective functions of this system are written by modifying Equation (1). For more
information, readers are referred to as Koulamas and Kyparisis (2001). It is obvious that the above system can be extended
by adding more machining or assembly stages, adding intermediate stages, or adding post-processing stage after assembly
such as polishing and/or packing. Another extension of the AF problem is distributed assembly machines, DAF, where there
are several factories at different locations where the best sequence of jobs at each factory needs to be determined. There
has been a recent spike in the number of published papers in this area including Hatami, Ruiz, and Andrés-Romano (2013),
Xiong and Xing (2014), Xiong et al. (2014, 2015), Deng et al. (2016), Lin and Zhang (2016), and Wang and Wang (2016).
2.1.2. Industrial applications of AF
AF plays a key role in several branches of industry. Lee, Cheng, and Lin (1993) addressed one of the very first applications
of AF models which is fire engine production and assembly process. In this application, body and chassis are manufactured
in shop floor and then assembled with the engine that is produced by the supplier as an out-sourced part. Potts et al. (1995)
identified make to order industries such as PC manufacturing and assembly as an application for AF. In this system, the
objective is to assemble PC modules according to customer’s preferences and package the PC in the shortest possible time.
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