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1. A Comparison of Three Search Algorithms for Solving the Buffer Allocation
Problem in Reliable Production Lines
L. Demir
1
, A. Diamantidis2
, D.T. Eliiyi
3
, M.E.J. O’Kelly4
, C.T. Papadopoulos2,#
, A.K. Tsadiras2
, S. Tunalı
5
#
Corresponding author: hpap@econ.auth.gr
1
Department of Industrial Engineering, Pamukkale University,
Kinikli Campus, Denizli 20070, Turkey (email: ldemir@pau.edu.tr)
2
Department of Economics, Aristotle University of Thessaloniki, Greece
(e-mails: adiama@econ.auth.gr, hpap@econ.auth.gr, tsadiras@econ.auth.gr)
3
Department of Industrial Systems Engineering, Izmir University of Economics,
Sakarya Cad. No: 156, Balcova-Izmir, Turkey (email: deniz.eliiyi@ieu.edu.tr)
4
Waterford Institute of Technology, Waterford, Ireland (e-mail: mejokelly@gmail.com)
5
Department of Business Administration, Izmir University of Economics,
Sakarya Cad. No: 156, Balcova-Izmir, Turkey (email: semra.tunali@ieu.edu.tr)
Abstract: This paper investigates the performance of three search algorithms: Myopic Algorithm,
Adaptive Tabu Search and Degraded Ceiling to solve the buffer allocation problem in reliable
production lines. DECO algorithm is used to calculate throughput. This algorithm is a variant of a
decomposition algorithm specifically developed to solve large reliable production lines with parallel
machines at each workstation and exponentially distributed service times. The measures of
performance used are the CPU time required and closeness to the maximum throughput achieved.
The three search algorithms are ranked in respect to these two measures and certain findings
regarding their performances over the experimental set are given.
Keywords: Production lines, Design, Optimization problems, Buffer storage, Search methods, Algorithms

1. INTRODUCTION & LITERATURE REVIEW
‘Production systems are complex but not evil’ was stated by
Li and Meerkov (2009) paraphrasing the well-known
Einstein’s ‘nature is complex but not evil’. There are many
issues involved in the topology, structure, analysis, design
and operation of manufacturing systems. In this paper, we
deal with a design problem in serial production lines, known
as the buffer allocation problem (BAP). This problem is
concerned with the specification of the sizes of the buffers
between stations. The provision of buffer space involves
considerable cost but leads to increased throughput.
There is an extensive bibliography in the area of BAP in the
various types of manufacturing systems. Due to space
limitations, only a few review papers restricted to the area of
BAP in serial production lines are given in this Section. Even
in this narrow area of research, there is a large number of
publications. Many relevant references are not included for
the sake of brevity.
As stated in Papadopoulos et al. (2009), the solution of the
BAP requires the use of two types of techniques: an
evaluative technique to calculate throughput or any other
performance measure such as the average work-in-process
(WIP), and a generative or optimization technique to find the
optimal or near optimal solution, i.e., the vector of buffer
sizes which maximizes a given objective function. The latter
may be a throughput or profit maximization or minimization
of the number of buffer slots, or minimization of the average
WIP in order to achieve a certain throughput level. A
combination of more than one criterion may be also
considered in the objective function (see Andijani and
Anwarul, 1997).
Evaluative techniques include exact analytical methods such
as the Markov state model method, the stochastic automata
network formalism and other Markovian structured methods,
exact numerical methods, decomposition, aggregation
/disaggregation, simulation, phase-type approach, holding
time method, generalized expansion method, and other
approximate methods. An excellent detailed overview of
models of manufacturing flow line systems is given in
Dallery and Gershwin (1992). An earlier review and
comparison of models of automatic transfer lines was given
by Buzacott and Hanifin (1978). Papadopoulos and Heavey
(1996) provided a classification of models for production and
transfer lines. Li et al. (2009) provided a comprehensive
presentation of recent studies in the area of evaluative
techniques of manufacturing systems. The criterion for
choosing an evaluative technique is the speed of its
convergence and the accuracy of the results. For short lines,
exact numerical approaches are used (Hillier and Boling,
1967, and Papadopoulos, Heavey and O’Kelly, 1989 &
1990). For longer lines, decomposition (the pioneering work
by Gershwin, 1987 and Dallery, David and Xie, 1988,
Gershwin, 1994, Helber, 1999, Diamantidis et al., 2006), and
aggregation techniques (the pioneering work by De Koster,
1987 and Lim, Meerkov and Top, 1990, Li and Meerkov,
2009) are more appropriate.
Generative or optimization techniques include search
algorithms such as complete enumeration and exact analytical
methods (generally confined to short lines), the gradient
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2. technique, dynamic programming, simulated annealing,
genetic algorithm, tabu search, myopic algorithm, Powell’s
method, the Hooke and Jeeves search procedure, the Cross-
Entropy method, various heuristics, simulation-based
techniques in conjunction with perturbation. The literature on
these techniques is extensive. The reader is addressed to the
book by Papadopoulos et al. (2009) for a description of most
of these methods and to Papadopoulos et al. (2012), and
especially to Demir et al. (2012b) for a recent and
comprehensive overview of the optimization methods used to
solve the BAP in production lines.
Depending on the type of the objective function and the
optimization algorithm, various approaches to the solution
have been developed in the international literature (see Shi
and Gershwin, 2009 for a systematic classification).
In Papadopoulos et al. (2012), the exact Markovian method
developed by Papadopoulos et al. (1989 & 1990) and Heavey
et al. (1993) for small lines, and the decomposition method
(DECO) developed by Diamantidis et al. (2006) for large
lines were used as evaluative methods. The authors tested
five optimization methods: complete enumeration (for very
short lines), simulated annealing, genetic algorithm, myopic
algorithm and Tabu search. The main finding was that
simulated annealing gave the best throughput (an optimal or
near optimal solution) for large lines at the expense of the
CPU time, whereas the myopic algorithm was the faster but
less accurate one among the five algorithms.
This work continues the research by Papadopoulos et al.
(2012) by investigating the performance of three search
algorithms: myopic algorithm (MA), adaptive tabu seach
(ATS) and degraded ceiling (DC) in the solution of the BAP
in reliable serial production lines. The reason for choosing
these three algorithms was their performance in recent
studies: MA in Papadopoulos et al. (2012) and ATS in Demir
et al. (2012a). DC, which was used in conjunction with a
scheme for calculating a good initial buffer allocation in a
recent Master’s thesis (Mystakidou, 2012) with promising
results, was also included for evaluation in this comparative
study. DECO algorithm developed by Diamantidis et al.
(2006) is used as evaluative technique for calculating the
throughput of very large reliable lines with parallel machines
at each station and exponentially distributed service times.
Due to lack of space, these algorithms are not described here.
The interested reader is referred to a technical report prepared
by the authors (Demir et al., 2012c) where these three
algorithms are described in detail including their steps and
flow charts. In addition, ATS may be found in Demir et al.
(2012a) and MA and DC in the Master theses by Nikita
(2010) and Mystakidou (2012).
As usual, when comparing the performance of algorithms or
any computational procedure, the reader should be mindful
that there may be issues relating to the relative effectiveness
of the developers in translating the relevant flow diagrams
into code and to the appropriateness of the computer system
used. However, the authors know that the codes used have
been found to be very robust in obtaining solutions over a
range of serial production lines and the computer system used
is readily available to designers.
Section 2 defines three different buffer allocation problems,
Section 3 gives numerical results and Section 4 summarizes
the findings of this study and gives a few directions for
further research.
2. THE BUFFER ALLOCATION PROBLEM IN
PRODUCTION LINES
The formulation of the buffer allocation problems depends on
the objective function chosen. These objective functions may
be concerned with maximizing throughput (BAP-A),
minimizing the total number of buffer slots (BAP-B), or
minimizing average work-in-process (BAP-C), subject to
appropriate constraints in each case. These three problems are
described below.
Problem BAP-A (the dual problem): Suppose there are K
machines and K-1 buffer areas with N total integer (≥0)
buffer slots to be allocated. A possible solution is a vector n,
where  
1 2 1
, ,..., K
n N N N 
 . The throughput of each solution
is symbolized by    
1 2 1
, ,..., K
X n X N N N 
 . The objective
is to maximize the throughput of the production line subject
to the constraint that the total number of buffer slots is N,
where all buffer slots , 1,..., 1
i
N i K
  must be nonnegative
integers. The problem may be stated as follows:
 
1 -1
1
1
max ( ) max ,...,
. .
0 1,.., 1
K
K
i
i
i
X n X N N
s t
N N
N i K




   

 

Problem BAP-B (the primal problem): The solution
approaches to this problem aim at achieving the desired
throughput rate with the minimum total buffer size as
follows:
1
1
*
min
. .
( )
0 1,.., 1
K
i
i
i
N N
s t
X n X
N i K




   


where X* denotes the desired throughput.
Problem BAP-C: This last formulation seeks the
minimization of the average work-in-process inventory
subject to the total buffer size constraint as well as the desired
throughput constraint and may be stated as follows:
1
1
*
min ( )
. .
( )
0 1,.., 1
K
i
i
i
Q n
s t
N N
X n X
N i K




   


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3. where Q(n) denotes the average work-in-process inventory as
a function of the buffer size vector. In this study we deal only
with the first buffer allocation problem.
3. NUMERICAL RESULTS
An experimental study was conducted to evaluate the
performance of three search algorithms for reliable
production lines. For this purpose, 12 problem sets are
generated for different values of K and N (see Table 1). As
seen in Table 1, K has four levels (as 5, 10, 20 and 40) while
N has three levels (as 5, 10 and 20 times K). In order to
examine the effects of problem size on performance of these
algorithms, these 12 problem sets are classified into three
groups: small, medium and large. Moreover, it is assumed
that the processing rates of the machines in the line are
exponentially distributed with mean 1.
Table 1. Classification of problem sets
Problem
Size
K N
Small 5 25, 50, 100
Medium 10 50, 100, 200
Large
20 100, 200, 400
40 200, 400, 800
The throughput values are obtained by employing the DECO
algorithm, and the performances of MA, ATS and DC are
evaluated for these lines. All algorithms are coded in C++
language and run on an Intel Xeon E5405 @ 2.00GHz, 4GB
RAM PC.
In order to obtain compatible results for the 12 lines used in
this investigation, the same initialization scheme is employed
for all three algorithms. In this scheme,
the buffer slots are allocated equally to all buffer locations
initially and any remaining slots are placed in the middle
buffers of the line. The best throughput values obtained by
each algorithm and the required solution times are presented
in Tables 2 and 3, respectively. The best values are shown in
bold characters. As seen from the tables, ATS yields the best
results in terms of solution quality but it is the slowest
algorithm. DC produces slightly worse results but it is much
faster than ATS. MA is also fast but it is inferior to other two
algorithms with respect to throughput. Table 4 shows the
deviations from best solutions. Deviations are calculated
using the formula:
( )* ( )
Deviation (%) = 100
( )*
i
X n X n
X n


where ( ), for MA, ATS, and DC
i
X n i  is the throughput
value obtained by each algorithm, and ( )*
X n is the best
(maximum) of the three. ATS produces the best throughput
values for all lines investigated. The superiority of ATS
becomes more significant in terms of solution quality as the
problem size increases.
Table 2. Results of comparative experimental studies:
Throughput values
Problem
set
K N
Throughput
MA ATS DC
Small
5 25 0.818330 0.819001 0.819001
5 50 0.891667 0.892193 0.892193
5 100 0.939766 0.940241 0.940241
Medium
10 50 0.782335 0.783435 0.783435
10 100 0.866236 0.868658 0.868658
10 200 0.925121 0.926501 0.926501
Large
20 100 0.763843 0.765624 0.765236
20 200 0.856022 0.857105 0.856916
20 400 0.918263 0.919713 0.919521
40 200 0.755815 0.757459 0.756670
40 400 0.850467 N/A*
0.851211
40 800 0.915777 N/A*
0.916190
*
‘N/A’ indicates that the algorithm in question crashed for
that particular serial production line.
Table 3. Results of comparative experimental studies:
CPU time comparison
Problem
set
K N
CPU Time (sec.)
MA ATS DC
Small
5 25 0 0.47 0.34
5 50 0.02 2.20 0.81
5 100 0.17 16.72 2.61
Medium
10 50 0.08 36.89 2.06
10 100 0.09 156.34 4.63
10 200 0.67 1239.80 15.09
Large
20 100 5.36 4798.81 12.78
20 200 13.63 28651.36 30.75
20 400 4.45 192026.75 96.72
40 200 197.20 606183.76 72.13
40 400 1820.92 N/A 196.36
40 800 8820.53 N/A 640.22
In Table 5, the performances of the three algorithms with
respect to throughput and CPU time are ranked. Note that the
worst performing one is given rank 3. If the remaining two
algorithms are found to have the same performance, then the
total numerical value of unassigned ranks is equally
distributed among these two algorithms. (See, for example,
the first line of Table 5, with ranks with 3, 1.5, 1.5).
Based on these results, the following conclusions can be
drawn regarding the comparison of the MA, ATS and DC
algorithms:
 ATS yields the best throughput values although it is the
slowest algorithm among the three. Hence, it may be
preferred when accuracy is the main concern and slow
solutions are acceptable.
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4. Table 4. Deviations from best throughput
Problem
set
K N
Deviation from best
throughput (%)
MA ATS DC
Small
5 25 0.082 0 0
5 50 0.059 0 0
5 100 0.051 0 0
Medium
10 50 0.140 0 0
10 100 0.279 0 0
10 200 0.149 0 0
Large
20 100 0.233 0 0.051
20 200 0.126 0 0.022
20 400 0.158 0 0.021
40 200 0.217 0 0.104
40 400 0.0874 N/A 0
40 800 0.0451 N/A 0
 In respect of throughput, DC provides identical or very
close results to ATS. As can be observed from Table 5,
the deviations of throughput achieved by the DC from
those obtained by ATS are very small (much smaller
than those of MA). DC is also the fastest algorithm for
large production lines with 40 machines, while for
smaller lines MA is the fastest. In contrast to CPU times
obtained by ATS and MA, the CPU time required by
DC does not seem to increase exponentially as the
problem size increases. Therefore, DC may be an
excellent choice in cases where both solution time and
accuracy are important; this algorithm can be very
practical for large production lines where other
algorithms require very long times to obtain solutions.
 MA is the worst in terms of throughput while being the
fastest algorithm for small and medium-size reliable
production lines. Hence, it can be considered as a good
choice for these lines when solution time is very limited.
To conclude, DC seems to be a promising algorithm in that it
is very fast especially for large reliable production lines.
4. CONCLUSIONS AND FURTHER RESEARCH
In this study, the performances of search algorithms are
tested for solving the buffer allocation problem in reliable
production lines. For throughput calculation, the DECO
algorithm is employed. The performances of MA, ATS and
DC algorithms are tested on 12 problem sets involving small,
medium and large-sized problems. Based on the conducted
experimental study DC seems the best choice as it produces
high throughput values in reasonable computation times.
However, the solution quality of ATS increases for large-
sized problems but it takes more time than the other two
algorithms.
Table 5. Rankings of MA, ATS and DC algorithms
Problem
Set
Throughput
(position 1 to 3)
CPU Time
(position 1 to 3)
K N MA ATS DC MA ATS DC
5 25 3 1.5 1.5 1 3 2
5 50 3 1.5 1.5 1 3 2
5 100 3 1.5 1.5 1 3 2
10 50 3 1.5 1.5 1 3 2
10 100 3 1.5 1.5 1 3 2
10 200 3 1.5 1.5 1 3 2
20 100 3 1 2 1 3 2
20 200 3 1 2 1 3 2
20 400 3 1 2 1 3 2
40 200 3 1 2 2 3 1
40 400 2 N/A 1 2 N/A 1
40 800 2 N/A 1 2 N/A 1
Further research may take several directions:
(i) Conduct an experimental study to test the performance
of these algorithms for unreliable lines.
(ii) Compare the performance of ATS against other search
algorithms such as the gradient technique and the
segmentation technique for long lines with different
objective functions (Shi and Gershwin, 2011) and for
the solution of other objective functions (BAP-B, BAP-
C).
(iii) Test the effectiveness of ATS in conjunction with other
evaluative methods that calculate the throughput of
production lines with different characteristics, e.g., with
phase-type processing and repair times. In doing so,
stochastic times and their variability can be taken into
consideration.
ACKNOWLEDGEMENTS
The research of A. Diamantidis, C.T. Papadopoulos and A.K.
Tsadiras has been co-financed by the European Union
(European Social Fund – ESF) and Greek national funds
through the Operational Program "Education and Lifelong
Learning" of the National Strategic Reference Framework
(NSRF) - Research Funding Program: Thales. Investing in
knowledge society through the European Social Fund.
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