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A comparison of combat genetic and big bang–big crunch algorithms for solving the buffer allocation problem

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Abstract

The buffer allocation problem (BAP) aims to determine the optimal buffer configuration for a production line under the predefined constraints. The BAP is an NP-hard combinatorial optimization problem and the solution space exponentially grows as the problem size increases. Therefore, problem specific heuristic or meta-heuristic search algorithms are widely used to solve the BAP. In this study two population-based search algorithms; i.e. Combat Genetic Algorithm (CGA) and Big Bang-Big Crunch (BB-BC) algorithm, are proposed in solving the BAP to maximize the throughput of the line under the total buffer size constraint for unreliable production lines. Performances of the proposed algorithms are tested on existing benchmark problems taken from the literature. The experimental results showed that the proposed BB–BC algorithm yielded better results than the proposed CGA as well as other algorithms reported in the literature.

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Acknowledgements

This research was done during the PhD study of the first author at the Graduate School of Natural and Applied Sciences, Pamukkale University. The authors would like to thank the anonymous reviewers for their constructive comments that improve the paper significantly.

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Correspondence to Mehmet Ulaş Koyuncuoğlu.

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Appendix: DDX algorithm

Appendix: DDX algorithm

The decomposition method is proposed by Gershwin (1987). This method is based on a decomposition of the line into a set of K − 1 two machine lines L(i), for i = 1, …, K − 1. Line L(i) is composed of an upstream machine Mu(i), and a downstream machine Md(i) and buffer B(i) which has the same capacity as buffer Bi in line L, Ni. Machines Mu(i) and Md(i) are defined by their failure and repair rates βu(i), ru(i), and βd(i), rd(i), respectively. The purpose of the method is to determine these parameters so that the behavior of the material flow in buffer B(i) in line L(i) closely matches that of the flow in buffer Bi of line L (see Fig. 8).

Fig. 8
figure 8

The decomposition method

To determine the unknown parameters of each two-machine subline, Gershwin (1987) developed a set of non-linear equations as follows:

$$ e_{i} = \frac{{r_{i} }}{{r_{i} + \beta_{i} }} \quad{\text{for}}\quad i = 1, 2, \ldots , K $$
(8)
$$ Th\left( 1 \right) = Th\left( 2 \right) = \ldots = Th\left( {K - 1} \right) $$
(9)
$$ Th\left( i \right) = e_{i} \left( {1 - p_{b} \left( i \right) - p_{s} \left( {i - 1} \right)} \right) $$
(10)

Equation 8 calculates the isolated efficiency of the machine i. Equation 9 is related to conservation of flow. Equation 10 calculates the throughput of each subline. In Eq. 10, \( p_{s} \left( i \right) \) is the probability of starvation of \( M_{u} \left( i \right) \) and \( p_{b} \left( i \right) \) is the probability of blockage of \( M_{d} \left( i \right) \).

Flow rate-idle time is given Eq. (11). Equation (12) is obtained by stating that a failure occurring from at least one of the upstream machines (Mi-1, Mi-2,…) caused a starvation of the machine Mi or which is occurred the failure of the Mi machine represents the failure in the upstream machine \( M_{u} \left( i \right) \). In a similar way, Eq. (13) can be calculated by considering the failure of machine \( M_{d} \left( i \right) \).We state the quantities \( I_{u} \left( i \right) \) and \( I_{d} \left( i \right) \) defined by Eq. (14):

$$ \frac{{\beta_{d} \left( {i - 1} \right)}}{{r_{d} \left( {i - 1} \right)}} + \frac{{\beta_{u} \left( i \right)}}{{r_{u} \left( i \right)}} = \frac{1}{{Th\left( {i - 1} \right)}} + \frac{1}{{e_{i} }} - 2 {\text{for}} i = 2, \ldots , K - 1 $$
(11)
$$ X = \frac{{r_{u} \left( i \right)p_{s} \left( {i - 1} \right)}}{{\beta_{u} \left( i \right)Th\left( {i - 1} \right)}} $$
(12)
$$ Y = \frac{{r_{d} \left( {i - 1} \right)p_{b} \left( i \right)}}{{\beta_{d} \left( {i - 1} \right)Th\left( i \right)}} $$
(13)
$$ \left\{ {\begin{array}{*{20}c} {I_{u} \left( i \right) = \frac{{\beta_{u} \left( i \right)}}{{r_{u} \left( i \right)}}} \\ {I_{d} \left( i \right) = \frac{{\beta_{d} \left( i \right)}}{{r_{d} \left( i \right)}}} \\ \end{array} } \right.\quad {\text{for}} \;i = 1, 2, \ldots , K - 1 $$
(14)

The set of equations established by Gershwin (1987), the quantities obtained from Eq. (14) and using the above equations can be expressed as follows:

$$ I_{u} \left( i \right) = \frac{1}{{Th\left( {i - 1} \right)}} + \frac{1}{{e_{i} }} - I_{d} \left( {i - 1} \right) - 2 {\text{for}} i = 2, \ldots , K - 1 $$
(15)
$$ r_{u} \left( i \right) = Xr_{u} \left( {i - 1} \right) + \left( {1 - X} \right)r_{i} {\text{for}} i = 2, \ldots , K - 1 $$
(16)

where \( X = \frac{{p_{s} \left( {i - 1} \right)}}{{I_{u} \left( i \right)Th\left( {i - 1} \right)}} \)

$$ I_{d} \left( i \right) = \frac{1}{{Th\left( {i + 1} \right)}} + \frac{1}{{e_{i + 1} }} - I_{u} \left( {i + 1} \right) - 2 {\text{for}} i = 1, 2, \ldots , K - 2 $$
(17)
$$ r_{d} \left( i \right) = Yr_{d} \left( {i + 1} \right) + \left( {1 - Y} \right)r_{i + 1} {\text{for}} i = 1, \ldots , K - 2 $$
(18)

where \( Y = \frac{{p_{b} \left( {i + 1} \right)}}{{I_{d} \left( i \right)Th\left( {i + 1} \right)}} \).

Dallery et al. (1988) developed an algorithm, called DDX algorithm, to solve the nonlinear equations given above. As it is stated by the authors, the computational complexity of the DDX algorithm is O(K2), where K is the number of the machines. The DDX algorithm is given in Algorithm 3.

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Koyuncuoğlu, M.U., Demir, L. A comparison of combat genetic and big bang–big crunch algorithms for solving the buffer allocation problem. J Intell Manuf 32, 1529–1546 (2021). https://doi.org/10.1007/s10845-020-01647-1

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