ABSTRACT Genetic Algorithms (GA) are, conceptually, suitable to optimize the Assembly Sequence Planning (ASP) problem. GA was implemented in this research to optimize the ASP problem because they can easily handle large search spaces, flexibility in defining the constraints and derive them in a fitness function. A penalty function approach has been used to compute the fitness value for assembly sequences. The penalty function approach was chosen as the penalties are easy to define, realistically capture the difficulties associated with the assembly process and the number of penalties to consider is relatively reduced. The evaluation of the penalty function is simple and straightforward, a most desirable feature for a population-based search

Arun Tom Mathew

C S P Rao

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                                         VOL. 9, NO. 4, APRIL 2014                                                                                                                       ISSN 1819-6608            
ARPN Journal of Engineering and Applied Sciences 
 
©2006-2014 Asian Research Publishing Network (ARPN). All rights reserved. 
 
www.arpnjournals.com 
 
 
453
IMPLEMENTATION OF GENETIC ALGORITHM TO OPTIMIZE THE 
ASSEMBLY SEQUENCE PLAN BASED ON PENALTY FUNCTION 
 
Arun Tom Mathew1 and C. S. P. Rao2 
1VIT University, Vellore, India 
 2National Institute of Technology Warangal, Warangal, India 
E-Mail: aruntom123@gmail.com 
 
ABSTRACT  
Genetic Algorithms (GA) are, conceptually, suitable to optimize the Assembly Sequence Planning (ASP) 
problem. GA was implemented in this research to optimize the ASP problem because they can easily handle large search 
spaces, flexibility in defining the constraints and derive them in a fitness function. A penalty function approach has been 
used to compute the fitness value for assembly sequences. The penalty function approach was chosen as the penalties are 
easy to define, realistically capture the difficulties associated with the assembly process and the number of penalties to 
consider is relatively reduced. The evaluation of the penalty function is simple and straightforward, a most desirable 
feature for a population-based search. 
 
Keywords: genetic algorithm, assembly sequence planning, subassemblies, penalty function. 
 
INTRODUCTION 
In general, optimization problems to be addressed 
have several objectives to be optimized. As the number of 
objectives of the problem increases, the complexity of the 
problem becomes high because the objectives considered 
are often contradictory to one another. The researcher who 
tackles an optimization problem with multiple objectives 
needs a tool for optimizing their problem. Because 
objectives to be optimized in the problem are often 
contradictory to one another, the optimal solution of the 
problem is not obtained as a single solution. That is, a set 
of solutions is to be obtained for the problem. Since 
multiple solutions are to be obtained as solutions of the 
problem, GA’s as a kind of multi-point search potentially 
have an advantage for optimization problems with 
multiple objectives. However, GA’s have been mainly 
applied to optimization problems with a single objective. 
Optimizing the Assembly Sequence Planning 
problem involves selecting an optimum or near optimum 
feasible assembly sequence according to an objective 
function based on optimization criteria. Genetic algorithm 
is chosen for the optimization of ASPP as it has the ability 
to handle large scale problems and flexibility in defining 
an objective function.  
The input for the guided search operator is the 
model of the product. The solutions of the ASPP are 
generated through guided search, when all precedence 
relations are considered. Crossover is the main genetic 
operator. The condition imposed in this case is the 
feasibility of the children chromosomes.  A penalty 
function approach has been used to compute the fitness 
value for assembly sequences. The penalty function 
approach was chosen because the penalties are easy to 
define, realistically capture the difficulties associated with 
the assembly process and the number of penalties to 
consider is relatively reduced. Also, the evaluation is 
simple and straightforward, a most desirable feature for a 
population-based search. 
 
LITERATURE SURVEY 
More recent research has begun to apply artificial 
intelligence based algorithms to solve the assembly 
sequence planning problem. In particular, motivated by the 
success of genetic algorithms (GA’s) in solving difficult 
and complex combinatorial problems (Gen and Cheng, 
1997), GA’s have been applied to assembly sequence 
planning to find optimal or near-optimal assembly plans 
for a structure. GA’s do not generate, and then evaluate, 
all possible candidate solutions when searching for an 
optimized assembly plan instead, GA’s search for an 
optimized assembly plan using a directed stochastic search 
of the product’s solution space of possible assembly plans. 
Thus, GA’s attempt to find optimized assembly plans, 
while analyzing only a small number of possible solutions. 
The need for GA’s in assembly sequence planning is to 
dramatically reduce time required to find an acceptably 
optimized assembly plan, for any given product structure, 
an assembly plan, encoded as a chromosome, representing 
the assembly order for components in the product 
structure. GA’s usually evaluates an individual’s fitness, 
with respect to the entire population, by applying a 
function that incorporates, for example, measures of 
assembly reorientations required, assembly tool changes 
required, and/or assembly fixtures required. Next, GA’s 
select individuals or pairs of individuals, based on the 
fitness, to produce next generation of individual assembly 
plans. For producing any individual offspring assembly 
sequence, the GA chooses one genetic operator crossover, 
mutation, based upon pre-defined probability settings. The 
GA then applies the chosen operator to chromosome 
string(s) selected from the population, based on relative 
fitness, to produce the next-generation population. A 
crossover operator exchanges component assembly order 
information between two chromosomes. A mutation 
operator exchanges the assembly order of two components 
within a single selected chromosome. 
Most GA’s for Assembly Sequence Planning use 
fixed population sizes from generation to generation, to 
                                         VOL. 9, NO. 4, APRIL 2014                                                                                                                       ISSN 1819-6608            
ARPN Journal of Engineering and Applied Sciences 
 
©2006-2014 Asian Research Publishing Network (ARPN). All rights reserved. 
 
www.arpnjournals.com 
 
 
454
limit algorithm run time. Thus, when the GA generates 
enough new valid offspring to fill the next generation, the 
old population is replaced by the new offspring population 
and a fitness value for each new member is then 
calculated. The GA process of generating new higher-
fitness offspring from the previous generation repeats until 
pre-defined termination criteria are satisfied. Common 
termination criteria used include a run-time limit, the 
number of generations, or a predefined desired assembly 
sequence fitness level. Bonneville et al. [1] first 
introduced, and demonstrated the feasibility of using, a 
genetic algorithm for Assembly Sequence Planning. 
Bonneville et al.’s early GA-based assembly planner used 
two basic genetic operators, crossover and mutation. 
However, Goldberg [2] showed that genetic algorithms 
that use only cross-over and mutation operators are limited 
in performance. Sebaaly and Fujimoto [3] also proposed a 
genetic algorithm for solving assembly sequence planning 
problems. Chen [4] proposed a simplified genetic 
algorithm for automatically generating assembly 
sequences. Chen used five genetic operators (crossover, 
mutation, cut-and-paste, break-and-joint, and 
reproduction) to promote a more thorough search of the 
product’s assembly plan solution space. 
 
FITNESS FUNCTION 
The objective function taken for present study is 
to minimize the cost of assembling the product. The 
fitness function for evaluating chromosome or assembly 
sequence is defined as to find a sequence with maximum 
penalty value.  
A single optimization criterion, the fitness 
function for each sequence is defined as:  
 
 
 
Where PFFi (i) is the componential fitness 
function associated with the assembly of a product. The 
fitness function is defined as a sum of componential 
fitness functions, PFF (i), each corresponding to an 
assembly task. The value of the fitness function depends 
on the definition of the optimization criterion and the 
position of the gene in the assembly sequence.  
 
Objective Function;  
 
THE PENALTY MATRIX 
It is assumed that components are assembled 
vertically, downwards, on an assembly press. The penalty 
represents the cost incurred by assigning an unsuitable or 
unfeasible assembly sequence. The penalty value of an 
assembly operation is calculated using the penalty method 
proposed by Chen et al. [4]. Table-1 represents an 
illustrative example of the penalty weightings of some 
crucial factors. For instance, the penalty index is set 0 if 
component ‘i’ is absolutely located above component ‘j’; 
the penalty index is 5 for a close above and little difficult 
to assemble; the penalty index is set 9 if the assembly 
relation is a “loose above” relation; the penalty index is set 
999, if the assembly relation is prohibited. By employing 
the idea of penalty, other crucial factors concerning 
Assembly Sequence Planning, such as the frequency of 
changing tools, the similarity of assembly operations, the 
quality of assembly, the complexity of assembly, etc., can 
be considered simultaneously. Importantly, for avoiding 
the inaccurate estimation of the penalty of respective 
factors, it needs experienced engineers to perform the 
evaluation. The penalty values are as follows: 
 
Table-1. Description of penalty index. 
 
Penalty 
rank 
penalty 
index Description 
1 0 
Simple work, forced 
precedent sequence, direct 
or absolute above relation 
2 1-5 
A little difficult, need 
careful operation, tools 
changing infrequently, 
closed above relation 
3 6-9 
Very difficult, easily 
damage the component, 
tools changing frequently, 
loose above relation 
4 999 Prohibit no relation. 
 
The value of those penalties, from experience, is 
consistent with the difficulty to perform the related 
operations at the shop-floor level. 
In order to minimize the time taken for searching 
for the solution, the initial population is taken from the 
feasible solution generated using the algorithm described 
earlier. After evaluation of each string based on its fitness 
value, these strings are further optimized by applying three 
major GA operators namely reproduction, cross over and 
mutation. The selection of better strings is based on the 
actual count arrived in initial sequence. The generic 
operator is used to generate new population that has better 
strings than old population is called parent-1 and is used 
for the next generic operation i.e. crossover. The strings 
obtained from reproduction are then mated at a given 
probability called as the crossover rate. The resultant 
string should be a feasible sequence and satisfies all the 
precedence constraints. The mutation operator makes 
random changes in one or more elements of the string. 
Mutation is done with a small probability called the 
mutation probability. This is done to protect loss of some 
potentially useful strings. The strings obtained after 
mutation should be a feasible sequence and satisfy all the 
precedence constraints.  
 
The settings of GA parameters were as follows. 
 Population size  = 100 
 Crossover probability Pc = 0.7 
 Mutation probability Pm = 0.3 
 Stopping criterion: 5000 generation was selected.  
                                         VOL. 9, NO. 4, APRIL 2014                                                                                                                       ISSN 1819-6608            
ARPN Journal of Engineering and Applied Sciences 
 
©2006-2014 Asian Research Publishing Network (ARPN). All rights reserved. 
 
www.arpnjournals.com 
 
 
455
IMPLEMENTATION OF GENETIC ALGORITHM 
TO OPTIMIZE THE ASSEMBLY SEQUENCE PLAN 
Genetic algorithm is applied to various products 
to generate the optimal assembly plan. The feasible 
assembly sequences generated becomes the initial set of 
chromosomes after crossover and mutation; the algorithm 
should make sure that the child chromosomes also 
represent a feasible assembly sequence satisfying all the 
precedence constraints. These feasible sequences 
generated are based on the algorithms developed by Arun 
Mathew and C.S.P Rao [5, 6]. Based on the penalty matrix 
in Table-2 for the product shown in Figure-1, the optimum 
sequence is generated. 
 
 
 
Figure-1. Sample assembly. 
 
Table-2. Penalty matrix for the example. 
 
 A b c d E f G h I j 
A 999 9 5 999 999 9 0 9 999 5 
B 9 999 8 999 999 999 999 999 999 999 
C 5 0 999 0 6 999 999 0 999 999 
D 999 999 9 999 0 999 999 999 999 999 
E 999 999 8 7 999 999 999 999 999 999 
F 9 999 999 999 999 999 3 999 9 9 
G 4 999 999 999 999 0 999 999 999 999 
H 9 999 5 999 999 999 999 999 999 999 
I 999 999 999 999 999 0 999 999 999 0 
J 0 999 999 999 999 6 999 999 3 999 
 
Based on the fitness function, the sequence having the maximum fitness value is the optimal sequence and is shown in the 
Figure-2. 
 
 
 
Figure-2. Optimal assemble sequence generated. 
                                         VOL. 9, NO. 4, APRIL 2014                                                                                                                       ISSN 1819-6608            
ARPN Journal of Engineering and Applied Sciences 
 
©2006-2014 Asian Research Publishing Network (ARPN). All rights reserved. 
 
www.arpnjournals.com 
 
 
456
CONCLUSIONS 
Genetic Algorithms are, conceptually, suitable to 
optimize the ASP problem. GA were selected in this 
research to optimize the ASP problem based on their 
qualities and especially because they can easily handle 
large search spaces and the flexibility in defining the 
constraints and the quality measures and derive them in a 
fitness function. GA wasn’t successful in optimizing 
assembly sequence plans that involved large assemblies. 
Once the number of components are reduced, GA can be 
successfully applied to generate the optimal or near 
optimal assembly sequence plans. 
A penalty function approach has been used in this 
paper to compute the fitness value for assembly sequences. 
The penalty function approach is chosen because the 
penalties are easy to define, realistically capture the 
difficulties associated with the assembly process and the 
number of penalties to consider is relatively reduced. The 
value of those penalties, from experience, is consistent 
with the difficulty to perform the related operations at the 
shop-floor level. The evaluation of the penalty function is 
simple and straightforward, a most desirable feature for a 
population-based search. In addition, by employing the 
idea of penalty, other critical factors concerning assembly 
planning, such as the frequency of changing tools, the 
similarity of assembly operations, the quality of assembly, 
the complexity of assembly, etc., can be considered 
simultaneously. 
 
REFERENCES 
 
[1] Bonneville, F., Perrard, C. and Henrioud, J.M. 1995. 
A genetic algorithm to generate and evaluate assembly 
plans. IEEE Symposium on Emerging Technology 
and Factory Automation. 2: 231-239. 
 
[2] Goldberg, D. E. 1989. Genetic Algorithms in Search, 
Optimization, and Machine Learning. Addison-
Wesley, USA. 
  
[3] Sebaaly, M.F. and Fujimoto, H. 1996. A Generic 
Planner for Assembly Automation. In: Proceedings of 
the IEEE conference on Evolutionary Computation. 
pp. 401- 406. 
 
[4] Chen, S. F. 1998. Assembly Sequence Planning-a 
genetic approach. In: Proceedings of the 24th ASME 
Design Automation Conference, September 12-16, 
Atlanta. 
 
[5] Arun Mathew and Rao C.S.P. 2010. A CAD System 
for Extraction of Mating Features in an Assembly. 
Assembly Automation. The international Journal of 
assembly technology and management. 30(2): 142-
146. 
 
[6] Arun Mathew and Rao C.S.P. 2010. A Novel Method 
of using API to Generate Liaison Relationships from 
an Assembly. Journal of Software Engineering and 
Applications. 3(2): 167-175. 


IMPLEMENTATION OF GENETIC ALGORITHM TO OPTIMIZE THE ASSEMBLY SEQUENCE PLAN BASED ON PENALTY FUNCTION

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IMPLEMENTATION OF GENETIC ALGORITHM TO OPTIMIZE THE ASSEMBLY SEQUENCE PLAN BASED ON PENALTY FUNCTION

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