Abstract
Flow line systems are production systems in which successive operations are performed on a product in a manner so that it moves through the factory in a certain direction. This work firstly formulates a flow line system as an integer-ordered inequality-constrained simulation–optimization problem and present a stochastic simulation procedure to estimate the throughput rate. The mathematical formulation and simulated procedure can be used for any distribution of processing rate and can be applied to high-dimensional problems. An approach that embeds advanced harmony search (AHS) in ordinal optimization (OO), abbreviated as AHSOO, is developed to find a near-optimal design of the flow line system to maximize the throughput rate. The proposed approach comprises three levels, which are meta-modeling, diversification and intensification. A radial basis function network is a meta-model to approximate the performance of a design. The proposed approach integrates the AHS approach for diversification with improved optimal computing budget allocation (IOCBA) for intensification. AHS favorably explores the solution space initially and moves toward exploiting good solutions close to the end. The IOCBA maximized the overall simulation efficiency for finding an optimal solution. The proposed AHSOO is tested on three examples. In the moderately sized example, simulation results reveal that the average best-so-far performances that were determined using PSO, GA and ES were 6.12, 9.65 and 8.53% less than that obtained using AHSOO—even after the former took more than 50 times the CPU time that was consumed by AHSOO upon completion. Analytical results reveal that the proposed method yields designs of much higher quality with a much higher computing efficiency than the seven competing methods.
Similar content being viewed by others
Abbreviations
- \(\mathbf{x}={[}b_{2},\cdots , b_{n}, r_{1},\ldots ,r_{n}{]}^\mathrm{{T}}\) :
-
A feasible design
- \(\Omega \) :
-
Solution space
- \(b_{i}\) :
-
Buffer allocation of Station i
- \(r_{i}\) :
-
Processing rate of Station i
- B :
-
Upper bound of total buffer space
- R :
-
Upper bound of total processing rate
- \(E{[}\mathrm{TH}(\mathbf{x}){]}\) :
-
Expected throughput rate of \(\mathbf{x}\)
- \(\hbox {TH}_l (\mathbf{x})\) :
-
Throughput rate of the \(l\hbox {th}\) replication
- \(\overline{\hbox {TH}} (\mathbf{x})\) :
-
Sample mean of replications
- \(L_\mathrm{{e}}\) :
-
Replications of accurate evaluation
- \(\overline{\hbox {TH}} _\mathrm{e} (\mathbf{x})\) :
-
Throughput rate obtained by accurate evaluation
- \(f(\mathbf{x})\) :
-
Output performance
- \(\mathrm{PE}(\mathbf{x})\) :
-
Penalty function
- \(f_\mathrm{{e}} (\mathbf{x})\) :
-
Performance obtained using accurate evaluation
- L :
-
Number of replications
- M :
-
Sample size of RBFN
- \(F(\mathbf{x}|{\varvec{\upomega }})\) :
-
Output of RBFN
- H :
-
Number of hidden nodes in RBFN
- \({\varvec{\upmu }}_{h}\) :
-
Centers of RBFN
- \(\varphi (\cdot )\) :
-
Set of RBF
- \({\varvec{\upomega }}={[}\omega _{1}, \ldots , \omega _{H}{]}\) :
-
Weight vector of RBFN
- \(\rho \) :
-
Standard deviation of Gaussian function
- \(\hbox {PAR}_{\mathrm{min}}\) :
-
Minimum pitch adjustment rate
- \(\hbox {PAR}_{\mathrm{max}}\) :
-
Maximum pitch adjustment rate
- \(\hbox {BW}_{\mathrm{min}}\) :
-
Minimum distance bandwidth
- \(\hbox {BW}_{\mathrm{max}}\) :
-
Maximum distance bandwidth
- \(t_{\max }\) :
-
Maximum number of iterations
- \(\mathbf{x}_{\mathrm{{best}}}^{t}\) :
-
Best harmony at iteration t
- \(\mathbf{x}_{\mathrm{{worst}}}^{t}\) :
-
Worst harmony at iteration t
- N :
-
Number of selected excellent designs for IOCBA
- \(C_\mathrm{{b}}\) :
-
Allowable computational budget
- \(n_{i}\) :
-
Amount of replications allocated to the \(i\hbox {th}\) design
- \(n_{0}\) :
-
Initial amount of replications
- \(\Delta \) :
-
Incremental computational budget
- s :
-
Factor of computational time reduction
- \(\mathbf{x}^{*}\) :
-
Near-optimal design obtained by solution method
- \(f_{e} (\mathbf{x}^{*})\) :
-
Throughput rate of near-optimal design
- \(\Theta \) :
-
Representative subset
- rk :
-
Rank of near-optimal design
- r :
-
Processing rate of the Erlang distribution
- \(\eta \) :
-
Shape parameter of the Erlang distribution
- \(\alpha _\mathrm{{G}}\) :
-
Confidence parameter in global stage
- \(\delta _{\mathrm{G}}\) :
-
Indifference zone parameter in global stage
- \(\alpha _\mathrm{{L}}\) :
-
Confidence parameter in local stage
- \(\delta _\mathrm{{L}}\) :
-
Indifference zone parameter in local stage
- \(\alpha _\mathrm{{C}}\) :
-
Confidence parameter in cleanup stage
- \(\delta _\mathrm{{C}}\) :
-
Indifference zone parameter in cleanup stage
- cp:
-
Number of ISC iterations between constraint prunings
- OvS:
-
Optimization-via-simulation
- ISC:
-
Industrial Strength COMPASS
- IICSOP:
-
Integer-ordered inequality-constrained simulation–optimization problem
- AHS:
-
Advanced harmony search
- OO:
-
Ordinal optimization
- RBFN:
-
Radial basis function network
- IOCBA:
-
Improved optimal computing budget allocation
- IHS:
-
Improved harmony search
- MHS:
-
Modified harmony search
- NDHS:
-
Novel derivative harmony search
- GHS:
-
Global-best harmony search
- NGHS:
-
Novel global harmony search
- SFLSP:
-
Stochastic flow line simulation procedure
- GA:
-
Genetic algorithm
- ES:
-
Evolution strategy
- PSO:
-
Particle swarm optimization
- ABC:
-
Artificial bee colony
- ACO:
-
Ant colony optimization
- HS:
-
Harmony search
- SOM:
-
Self-organizing feature map
- MSE:
-
Mean squared error
- HMS:
-
Harmony memory size
- HMCR:
-
Harmony memory consideration rate
- PAR:
-
Pitch adjustment rate
- HM:
-
Harmony memory
- BW:
-
Distance bandwidth
- OCBA:
-
Optimal computing budget allocation
- ABSFP:
-
Average best-so-far performances at termination
References
Shaaban, S.; McNamara, T.; Hudson, S.: Mean time imbalance effects on unreliable unpaced serial flow lines. J. Manuf. Syst. 33(3), 357–365 (2014)
Wang, G.; Shin, Y.W.; Moon, D.H.: Comparison of three flow line layouts with unreliable machines and profit maximization. Flex. Serv. Manuf. J. 28(4), 669–693 (2016)
Konishi, K.: A tuning strategy to avoid blocking and starving in a buffered production line. Eur. J. Oper. Res. 200(2), 616–620 (2010)
Bierbooms, R.; Adan, I.J.B.; van Vuuren, M.: Approximate performance analysis of production lines with continuous material flows and finite buffers. Stoch. Models 29(1), 1–30 (2013)
Aziz, A.; Jarrahi, F.; Abdul-Kader, W.: Modeling and performance evaluation of a series-parallel flow line system with finite buffers. INFOR 48(2), 103–120 (2010)
Robinson, S.: Simulation: The Practice of Model Development and Use. Wiley, Chichester (2004)
ExtendSim User Guide. Release 9, Imagine That, Incorporated (2013)
Wang, Y.R.; Chen, A.N.: Production logistics simulation and optimization of industrial enterprise based on Flexsim. Int. J. Simul. Model 15(4), 732–741 (2016)
Jarrahi, F.; Abdul-Kader, W.: Performance evaluation of a multi-product production line: an approximation method. Appl. Math. Model. 39(13), 3619–3636 (2015)
Takemoto, Y.; Arizono, I.: Production allocation optimization by combining distribution free approach with open queueing network theory. Int. J. Adv. Manuf. Technol. 63(1–4), 349–358 (2012)
Sörensena, K.; Janssensb, G.K.: A Petri net model of a continuous flow transfer line with unreliable machine. Eur. J. Oper. Res. 152(1), 248–262 (2004)
Gao, S.Y.; Chen, W.W.: A partition-based random search for stochastic constrained optimization via simulation. IEEE Trans. Autom. Control 62(2), 740–752 (2017)
Xu, J.; Hong, L.J.; Nelson, B.L.: Industrial strength COMPASS: a comprehensive algorithm and software for optimization via simulation. ACM Trans. Model. Comput. Simul 20(1), 3:1–3:29 (2010)
Helber, S.; Schimmelpfeng, K.; Stolletz, R.; Lagershausen, S.: Using linear programming to analyze and optimize stochastic flow lines. Ann. Oper. Res 182(1), 193–211 (2011)
Weiss, S.; Stolletz, R.: Buffer allocation in stochastic flow lines via sample-based optimization with initial bounds. OR Spectrum 37(4), 869–902 (2015)
Costa, A.; Alfieri, A.; Matta, A.; Ficheraa, S.: A parallel Tabu search for solving the primal buffer allocation problem in serial production systems. Comput. Oper. Res. 64, 97–112 (2015)
Jones, J.C.P.; Frey, J.; Shayestehmanesh, S.: Stochastic simulation and performance analysis of classical knock control algorithms. IEEE Trans. Control Syst. Technol. 25(4), 1307–1317 (2017)
Ho, Y.C.; Zhao, Q.C.; Jia, Q.S.: Ordinal Optimization: Soft Optimization for Hard Problems. Springer, New York (2007)
Horng, S.C.; Lin, S.Y.: Evolutionary algorithm assisted by meta-model in the framework of ordinal optimization and optimal computing budget allocation. Inf. Sci. 233, 214–229 (2013)
Horng, S.C.: Combining artificial bee colony with ordinal optimization for stochastic economic lot scheduling problem. IEEE Trans. Syst. Man Cybern. Syst. 45(3), 373–384 (2015)
Horng, S.C.; Lin, S.Y.: Ordinal optimization based metaheuristic algorithm for optimal inventory policy of assemble-to-order systems. Appl. Math. Model. 42, 43–57 (2017)
Dey, B.; Hossain, A.; Bhattacharjee, A.; Dey, R.; Bera, R.: Function approximation based energy detection in cognitive radio using radial basis function network. Intell. Autom. Soft Comput. 23(3), 393–403 (2017)
Mahdavi, M.; Fesanghary, M.; Damangir, E.: An improved harmony search algorithm for solving optimization problems. Appl. Math. Comput. 188, 1567–1579 (2007)
Mohamed, A.O.; Rajeswari, M.: The variants of the harmony search algorithm: an overview. Artif. Intell. Rev. 36(1), 49–68 (2011)
Shivaiea, M.; Kazemib, M.G.; Amelia, M.T.: A modified harmony search algorithm for solving load-frequency control of non-linear interconnected hydrothermal power systems. Sustain. Energy Technol. Assess. 10, 53–62 (2015)
Tsadiras, A.K.; Papadopoulos, C.T.; O’Kelly, M.E.J.: An artificial neural network based decision support system for solving the buffer allocation problem in reliable production lines. Comput. Ind. Eng. 66(4), 1150–1162 (2013)
Munoz, M.A.; Sun, Y.; Kirley, M.; Halgamuge, S.K.: Algorithm selection for black-box continuous optimization problems: a survey on methods and challenges. Inf. Sci. 317, 224–245 (2015)
Yang, X.S.: Nature-Inspired Optimization Algorithms. Elsevier, Boston (2014)
Kumar, R.; Srivastava, S.; Gupta, J.R.P.: Modeling and adaptive control of nonlinear dynamical systems using radial basis function network. Soft Comput. 21(15), 4447–4463 (2017)
Ryan, T.P.: Sample Size Determination and Power. Wiley, New Jersey (2013)
Wang, X.L.; Gao, X.Z.; Zenger, K.: An Introduction to Harmony Search Optimization Method. Springer, Berlin, Heidelberg (2014)
Chen, C.H.; Lee, L.H.: Stochastic Simulation-optimization: An Optimal Computing Budget Allocation. World Scientific, New Jersey (2010)
SimOpt.org, Problem name: Throughput maximization. (2012). http://simopt.org/wiki/index.php?title=Throughput_Maximization
Aote, S.S.; Raghuwanshi, M.M.; Malik, L.G.: Improved particle swarm optimization based on natural flocking behavior. Arab. J. Sci. Eng. 41(3), 1067–1076 (2016)
Kardan, N.; Hassanzadeh, Y.; Bonab, B.S.: Shape optimization of trapezoidal labyrinth weirs using genetic algorithm. Arab. J. Sci. Eng. 42(3), 1219–1229 (2017)
Beyer, H.G.; Sendhoff, B.: Toward a steady-state analysis of an evolution strategy on a robust optimization problem with noise-induced multimodality. IEEE Trans. Evol. Comput. 21(4), 629–643 (2017)
Saidane, S.; Babai, M.Z.; Aguir, M.S.; Korbaa, O.: On the performance of the base-stock inventory system under a compound Erlang demand distribution. Comput. Ind. Eng. 66(3), 548–554 (2013)
Hong, L.J.; Nelson, B.L.; Xu, J.: Industrial Strength COMPASS, [Online] (2011). http://www.iscompass.net/
Author information
Authors and Affiliations
Corresponding author
Additional information
This research work is supported in part by the Ministry of Science and Technology in Taiwan, R.O.C., under Grant MOST 106-2221-E-324-002 and MOST 106-2221-E-129-007.
Rights and permissions
About this article
Cite this article
Horng, SC., Lin, SS. Embedding Advanced Harmony Search in Ordinal Optimization to Maximize Throughput Rate of Flow Line. Arab J Sci Eng 43, 1015–1031 (2018). https://doi.org/10.1007/s13369-017-2864-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13369-017-2864-9