A Comparison of Three Search Algorithms for Solving the Buffer Allocation Problem in Reliable Production Lines

7th IFAC Conference on Manufacturing Modelling, Management, and Control International Federation of Automatic Control June 19-21, 2013. Saint Petersburg, Russia

A Comparison of Three Search Algorithms for Solving the Buffer Allocation Problem in Reliable Production Lines 1

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L. Demir , A. Diamantidis2, D.T. Eliiyi , M.E.J. O’Kelly4, C.T. Papadopoulos2,#, A.K. Tsadiras2, S. Tunalı

5

#

Corresponding author: [email protected] Department of Industrial Engineering, Pamukkale University, Kinikli Campus, Denizli 20070, Turkey (email: [email protected]) 2 Department of Economics, Aristotle University of Thessaloniki, Greece (e-mails: [email protected], [email protected], [email protected]) 3 Department of Industrial Systems Engineering, Izmir University of Economics, Sakarya Cad. No: 156, Balcova-Izmir, Turkey (email: [email protected]) 4 Waterford Institute of Technology, Waterford, Ireland (e-mail: [email protected]) 5 Department of Business Administration, Izmir University of Economics, Sakarya Cad. No: 156, Balcova-Izmir, Turkey (email: [email protected]) 1

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 

considered in the objective function (see Andijani and Anwarul, 1997).

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 978-3-902823-35-9/2013 © IFAC

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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10.3182/20130619-3-RU-3018.00345

2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia

technique, dynamic programming, simulated annealing, genetic algorithm, tabu search, myopic algorithm, Powell’s method, the Hooke and Jeeves search procedure, the CrossEntropy 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).

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 n   N1 , N 2 ,..., N K 1  . The throughput of each solution is symbolized by X  n   X  N1 , N 2 ,..., N K 1  . 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 Ni , i  1,..., K  1 must be nonnegative integers. The problem may be stated as follows:

max X (n) max X  N1 ,..., N K -1  s.t. K 1

N i 1

i

N

N i  0i  1,.., K  1 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: K 1

min N   Ni i 1

s.t. X ( n)  X * Ni  0i  1,.., K  1 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:

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. 1627

min Q(n) s.t. K 1

N i 1

i

N

X ( n)  X * Ni  0i  1,.., K  1

2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia

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.

Table 2. Results of comparative experimental studies: Throughput values Throughput Problem K N set MA ATS DC

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 K N Size Small

5

25, 50, 100

Medium

10

50, 100, 200

20

100, 200, 400

40

200, 400, 800

Large

Small

Medium

Deviation (%) = 100 

X (n)*  X i (n) X (n)*

where X i (n), for i  MA, ATS, and DC 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.

25

0.818330

0.819001

0.819001

5

50

0.891667

0.892193

0.892193

5

100

0.939766

0.940241

0.940241

10

50

0.782335

0.783435

0.783435

10

100

0.866236

0.868658

0.868658

10

200

0.925121

0.926501

0.926501

0.765236 0.765624 20 100 0.763843 0.856022 0.856916 0.857105 20 200 0.919521 0.919713 20 400 0.918263 Large 0.756670 0.757459 40 200 0.755815 * 0.850467 N/A 0.851211 40 400 * N/A 0.916190 40 800 0.915777 * ‘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 CPU Time (sec.) Problem K N set MA ATS DC

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:

5

Small

Medium

Large

5

25

0

0.47

0.34

5

50

0.02

2.20

0.81

5

100

0.17

16.72

2.61

10

50

0.08

36.89

2.06

10

100

0.09

156.34

4.63

10

200

0.67

1239.80

15.09

20 20 20 40 40 40

100 200 400 200 400 800

5.36 13.63 4.45 197.20 1820.92 8820.53

4798.81 28651.36 192026.75 606183.76 N/A N/A

12.78 30.75 96.72 72.13 196.36 640.22

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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2013 IFAC MIM June 19-21, 2013. Saint Petersburg, Russia

Table 4. Deviations from best throughput Deviation from best Problem throughput (%) K N set MA ATS DC 5 Small

Medium

Large

25

0.082

0

Table 5. Rankings of MA, ATS and DC algorithms Problem Set

Throughput (position 1 to 3)

CPU Time (position 1 to 3)

0

K

N

MA

ATS

DC

MA

ATS

DC

5 5 5

25 50 100

3 3 3

1.5 1.5 1.5

1.5 1.5 1.5

1 1 1

3 3 3

2 2 2

10 10 10

50 100 200

3 3 3

1.5 1.5 1.5

1.5 1.5 1.5

1 1 1

3 3 3

2 2 2

20 20 20 40 40 40

100 200 400 200 400 800

3 3 3 3 2 2

1 1 1 1 N/A N/A

2 2 2 2 1 1

1 1 1 2 2 2

3 3 3 3 N/A N/A

2 2 2 1 1 1

5

50

0.059

0

0

5

100

0.051

0

0

10

50

0.140

0

0

10

100

0.279

0

0

10

200

0.149

0

0

20 20 20 40 40 40

100 200 400 200 400 800

0.233 0.126 0.158 0.217 0.0874 0.0451

0 0 0 0 N/A N/A

0.051 0.022 0.021 0.104 0 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 largesized problems but it takes more time than the other two algorithms.

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, BAPC). (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. REFERENCES Andijani, A.A. and Anwarul, M. (1997). Manufacturing blocking discipline: A multi-criterion approach for buffer allocations, International Journal of Production Economics 51, pp. 155-163. Burke, E.K., Bykov, Y., Newall, J.P., Petrovic, S. (2002). A new local search approach with execution time as an input parameter. Computer Science Technical Report No.NOTTCS-TR-2002-3. School of Computer Science and Information Technology. University of Nottingham.

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