# Harmony search algorithm for continuous network design problem with link capacity expansions

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## Abstract

The Continuous Network Design Problem (CNDP) deals with determining the set of link capacity expansions and the corresponding equilibrium link flows which minimizes the system performance index defined as the sum of total travel times and investment costs of link capacity expansions. In general, the CNDP is characterized by a bilevel programming model, in which the upper level problem is generally to minimize the total system cost under limited expenditure, while at the lower level problem, the User Equilibrium (UE) link flows are determined by Wardrop’s first principle. It is well known that bilevel model is nonconvex and algorithms for finding global or near global optimum solutions are preferable to be used in solving it. Furthermore, the computation time is tremendous for solving the CNDP because the algorithms implemented on real sized networks require solving traffic assignment model many times. Therefore, an efficient algorithm, which is capable of finding the global or near global optimum solution of the CNDP with less number of traffic assignments, is still needed. In this study, the Harmony Search (HS) algorithm is used to solve the upper level objective function and numerical calculations are performed on eighteen link and Sioux Falls networks. The lower level problem is formulated as user equilibrium traffic assignment model and Frank-Wolfe method is used to solve it. It has been observed that the HS algorithm is more effective than many other compared algorithms on both example networks to solve the CNDP in terms of the objective function value and UE traffic assignment number.

### Keywords

harmony search continuous network design traffic assignment link capacity expansion bilevel programming## Preview

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### References

- Abdulaal, M. and LeBlanc, L. (1979). “Continuous equilibrium network design models.”
*Transportation Research Part B*, Vol. 13, No. 1, pp. 19–32.CrossRefMathSciNetGoogle Scholar - Askarzadeh, A. and Rezazadeh, A. (2011). “A grouping-based global harmony search algorithm for modeling of proton exchange membrane fuel cell.”
*International Journal of Hydrogen Energy*, Vol. 36, No. 8, pp. 5047–5053.CrossRefGoogle Scholar - Ayvaz, M. T. (2010). “A linked simulation-optimization model for solving the unknown groundwater pollution source identification problems.”
*Journal of Contaminant Hydrology*, Vol. 117, Nos. 1–4, pp. 46–59.CrossRefGoogle Scholar - Ban, X. G., Liu, H. X., Lu, J. G., and Ferris, M. C. (2006). “Decomposition scheme for continuous network design problem with asymmetric user equilibria.”
*Transportation Research Record (1964)*, pp. 185–192.Google Scholar - Bell, M. G. H. and Iida, Y. (1997).
*Transportation network analysis*, John Wiley and Sons, Chichester, UK.Google Scholar - Ceylan, H., Ceylan, H., Haldenbilen, S., and Baskan, O. (2008). “Transport energy modeling with meta-heuristic harmony search algorithm, an application to Turkey.”
*Energy Policy*, Vol. 36, No. 7, pp. 2527–2535.CrossRefGoogle Scholar - Chen, H. K. and Chou, H. W. (2006). “Reverse supply chain network design problem.”
*Transportation Research Record (1964)*, pp. 42–49.Google Scholar - Chen, A., Subprasom, K., and Ji, Z. W. (2006). “A Simulation based Multi Objective Genetic Algorithm (SMOGA) procedure for BOT network design problem.”
*Optimization and Engineering*, Vol. 7, No. 3, pp. 225–247.CrossRefMATHMathSciNetGoogle Scholar - Chiou, S. W. (2005). “Bilevel programming for the continuous transport network design problem.”
*Transportation Research Part B*, Vol. 39, No. 4, pp. 361–383.Google Scholar - Cho, H. J. (1988).
*Sensitivity analysis of equilibrium network flows and its application to the development of solution methods for equilibrium network design problems*, PhD Thesis, University of Pennsylvania, Philadelphia, USA.Google Scholar - Davis, G. A. (1994). “Exact local solution of the continuous network design problem via stochastic user equilibrium assignment.”
*Transportation Research Part B*, Vol. 28, No. 1, pp. 61–75.CrossRefGoogle Scholar - Degertekin, S. O. and Hayalioglu, M. S. (2010). “Harmony search algorithm for minimum cost design of steel frames with semi-rigid connections and column bases.”
*Structural and Multidisciplinary Optimization*, Vol. 42, No. 5, pp. 755–768.CrossRefGoogle Scholar - Erdal, F., Doğan, E., and Saka, M. P. (2011). “Optimum design of cellular beams using harmony search and particle swarm optimizers.”
*Journal of Constructional Steel Research*, Vol. 67, No. 2, pp. 237–247.CrossRefGoogle Scholar - Fisk, C. (1984). “Optimal signal controls on congested networks.”
*In: 9th International Symposium on Transportation and Traffic Theory*, VNU Science Press, pp. 197–216.Google Scholar - Friesz, T. L., Anandalingam, G., Mehta, N. J., Nam, K., Shah, S. J., and Tobin, R. L. (1993). “The multiobjective equilibrium network design problem revisited-A simulated annealing approach.”
*European Journal of Operational Research*, Vol. 65, No. 1, pp. 44–57.CrossRefMATHGoogle Scholar - Friesz, T. L., Cho, H. J., Mehta, N. J., Tobin, R. L., and Anandalingam, G. (1992). “A simulated annealing approach to the network design problem with variational inequality constraints.”
*Transportation Science*, Vol. 26, No. 2, pp. 18–26.CrossRefMATHGoogle Scholar - Friesz, T. L., Tobin, R. L., Cho, H. J., and Mehta, N. J. (1990). “Sensitivity analysis based algorithms for mathematical programs with variational inequality constraints.”
*Mathematical Programming*, Vol. 48, Nos. 1–3, pp. 265–284.CrossRefMATHMathSciNetGoogle Scholar - Gao, Z., Sun, H., and Zhang, H. (2007). “A globally convergent algorithm for transportation continuous network design problem.”
*Optimization and Engineering*, Vol. 8, No. 3, pp. 241–257.CrossRefMATHMathSciNetGoogle Scholar - Geem, Z. W. (2000).
*Optimal design of water distribution networks using Harmony Search*, PhD Thesis, Korea University, Seoul, Korea.Google Scholar - Geem, Z. W. (2009). “Particle-swarm harmony search for water network design.”
*Engineering Optimization*, Vol. 41, No. 4, pp. 297–311.CrossRefGoogle Scholar - Geem, Z. W., Kim, J. H., and Loganathan, G. V. (2001). “A new heuristic optimization algorithm: Harmony search.”
*Simulation*, Vol. 76, No. 2, pp. 60–68.CrossRefGoogle Scholar - Karoonsoontawong, A. and Waller, S. T. (2006). “Dynamic continuous network design problem-Linear bilevel programming and metaheuristic approaches.”
*Transportation Research Record (1964)*, pp. 104–117.Google Scholar - Kayhan, A. H., Korkmaz, K. A., and Irfanoglu, A. (2011). “Selecting and scaling real ground motion records using harmony search algorithm.”
*Soil Dynamics and Earthquake Engineering*, Vol. 31, No. 7, pp. 941–953.CrossRefGoogle Scholar - Khorram, E. and Jaberipour, M. (2011). “Harmony search algorithm for solving combined heat and power economic dispatch problems.”
*Energy Conversion and Management*, Vol. 52, No. 2, pp. 1550–1554.CrossRefMathSciNetGoogle Scholar - LeBlanc, L. (1975). “An algorithm for the discrete network design problem.”
*Transportation Science*, Vol. 9, No. 3, pp. 183–199.CrossRefGoogle Scholar - Lee, K. S. and Geem, Z. W. (2004). “A new structural optimization method based on the harmony search algorithm.”
*Computers and Structures*, Vol. 82, Nos. 9–10, pp. 781–798.CrossRefGoogle Scholar - Lee, K. S., Geem, Z. W., Lee, S-H., and Bae, K-W. (2005). “The harmony search heuristic algorithm for discrete structural optimization.”
*Engineering Optimization*, Vol. 37, No. 7, pp. 663–684.CrossRefMathSciNetGoogle Scholar - Marcotte, P. (1983). “Network optimization with continuous control parameters.”
*Transportation Science*, Vol. 17, No. 2, pp. 181–197.CrossRefMathSciNetGoogle Scholar - Marcotte, P. and Marquis, G. (1992). “Efficient implementation of heuristics for the continuous network design problem.”
*Annals of Operational Research*, Vol. 34, No. 1, pp. 163–176.CrossRefMATHGoogle Scholar - Meng, Q., Yang, H., and Bell, M. G. H. (2001). “An equivalent continuously differentiable model and a locally convergent algorithm for the continuous network design problem.”
*Transportation Research Part B*, Vol. 35, No. 1, pp. 83–105.CrossRefGoogle Scholar - Mun, S. and Lee, S. (2011). “Identification of viscoelastic functions for hot-mix asphalt mixtures using a modified harmony search algorithm.”
*Journal of Computing in Civil Engineering*, Vol. 25, No. 2, pp. 139–148.CrossRefGoogle Scholar - Sheffi, Y. (1985).
*Urban transport networks: Equilibrium analysis with mathematical programming methods*, Prentice-Hall Inc., New Jersey, USA.Google Scholar - Sivasubramani, S. and Swarup, K. S. (2011). “Environmental/economic dispatch using multi-objective harmony search algorithm.”
*Electric Power Systems Research*, Vol. 81, No. 9, pp. 1778–1785.CrossRefGoogle Scholar - Suh, Y., Mun, S., and Yeo, I. (2010). “Fatigue life prediction of asphalt concrete pavement using a harmony search algorithm.”
*KSCE Journal of Civil Engineering*, Vol. 14, No. 5, pp. 725–730.CrossRefGoogle Scholar - Suwansirikul, C., Friesz, T. L., and Tobin, R. L. (1987). “Equilibrium decomposed optimisation: A heuristic for the continuous equilibrium network design problem.”
*Transportation Science*, Vol. 21, No. 4, pp. 254–263.CrossRefMATHGoogle Scholar - Wang, L., Pan, Q-K., and Tasgetiren, M. F. (2011). “A hybrid harmony search algorithm for the blocking permutation flow shop scheduling problem.”
*Computers & Industrial Engineering*, Vol. 61, No. 1, pp. 76–83.CrossRefGoogle Scholar - Wardrop, J. G. (1952). “Some theoretical aspects of road traffic research.”
*Proceedings of the Institution of Civil Engineers Part II*, Vol. 1, pp. 325–378.Google Scholar - Xu, T., Wei, H., and Hu, G. (2009). “Study on continuous network design problem using simulated annealing and genetic algorithm.”
*Expert Systems with Applications*, Vol. 36, No. 2(1), pp. 1322–1328.CrossRefGoogle Scholar - Yang, H. (1995). “Sensitivity analysis for queuing equilibrium network flow and its application to traffic control.”
*Mathematical and Computer Modelling*, Vol. 22, Nos. 4–7, pp. 247–258.MATHMathSciNetGoogle Scholar - Yang, H. (1997). “Sensitivity analysis for the network equilibrium problem with elastic demand.”
*Transportation Research*, Vol. 31, No. 1, pp. 55–70.CrossRefGoogle Scholar - Yang, H. and Yagar, S. (1995). “Traffic assignment and signal control in saturated road networks.”
*Transportation Research A*, Vol. 29, No. 2, pp. 125–139.Google Scholar