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

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s12205-013-0122-6

- Cite this article as:
- Baskan, O. KSCE J Civ Eng (2014) 18: 273. doi:10.1007/s12205-013-0122-6

## 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.