Ant colony optimization is developed to determine optimum cross sections of tunnel structures. Tunnel structures are expensive infrastructures in terms of material, construction, and maintenance and the application of optimization methods has a great role in minimizing their costs. This paper presents the formulation of objective function and constraints of the problem for the first time, and the ant colony optimization, as a developed metaheuristic approach, has been used to solve the problem. The results and comparisons based on numerical examples show the efficiency of the algorithm.

In general, the optimization techniques can be categorized into classical and metaheuristic search methods. Classical optimization methods often require substantial gradient information, the final results depend on the initially selected points, and many engineering problems are too complex to be handled with classical methods [

Among many exiting metaheuristics, there are few algorithms that present many results (instead of only one best result). The ant colony optimization (ACO) was one of them. In the design of tunnel, engineers need to have many different designs to combine/select/improve them as the final result. Therefore, we select ACO in this paper. Previously, ACO has been applied to water distribution system optimization [

On the other hand, the necessity for tunnels and the benefits they bring cannot be overestimated. Tunnels improve connections and shorten lifelines. The utilization of underground space for storage, power and water treatment plants civil defense, and other activities is often a must in view of limited space, safe operation, environmental protection, and energy saving [

There are different shapes or profiles for a tunnel cross section. The choice of the profile aims at accommodating the performance requirements of the tunnel. Moreover, it tries to minimize bending moments in the lining (or displacements) as well as costs for excavation and lining [

A typical problem of tunnel design is to fit two rectangles (one is required as a traffic space, and the other is required as air conductor space) into a mouth profile as shown in Figure

The different profiles of tunnel cross sections.

For the profile with vertical wall (type I):

For the profile without vertical wall (type II):

There are different methods to handle the constraints [

Ant colony optimization (ACO) is a cooperative search technique that mimics the foraging behavior of real life ant colonies [

The general procedure of the ACO algorithm manages the scheduling of four activities [

The initialization of the ACO consists of the initialization of the pheromone trail, the number of ants randomly and the required constant parameters.

In the iteration step, each ant constructs a complete solution to the problem according to a state transition rule. The state transition rule depends mainly on the state of the pheromone and visibility of ant. Visibility is an additional ability used to make this method more efficient. In engineering problems, the visibility for allowable value number

When every ant has constructed a solution, the intensity of pheromone trails on selected values is updated by the pheromone updating rule (global pheromone updating rule). The global pheromone updating rule is applied in two phases: first, an evaporation phase where a fraction of the pheromone evaporates, second, a reinforcement phase where the elitist ant, which has constructed the best solution in that iteration, deposits an amount of pheromone

At the end of each movement, local trail updating rule reduces the level of trail on values selected by the ant colony during the preceding iteration. When an ant selects the value

This process is repeated until a stopping criterion is fulfilled.

Kaveh and Talatahari [

Calculating cross-sectional area boundaries for each variable.

Creating the series of the allowable cross-sectional areas.

Determining the optimum solution of the stage by using ACO algorithm.

Repeating Steps

A numerical investigation is performed in this section. The required height and width are 4 and 6.5 meters, respectively.

The values of constants

Figure

The optimum profiles of tunnel cross sections (type I); the order of functions is as follows: (a) 7, (b) 8, (c) 9, (d) 10, and (e) 20.

The optimum profiles of tunnel cross sections (type II); the order of functions is as follows: (a) 7, (b) 8, (c) 9, (d) 10, and (e) 20.

A comparison of final optimum results is performed as shown in Figure

Comparison of final optimum results of types I and II.

Comparison of final optimum results obtained by ACO and GA; (a) type I and (b) type II.

The choice of the profile aims at accommodating the performance requirements of the tunnel. Optimum cross section tries to minimize displacements as well as costs for excavation and lining. For the first time, this paper utilizes ant colony optimization (ACO) to determine optimum cross section of tunnels. ACO, a metaheuristic method, is inspired from natural phenomena which do not require an explicit relationship between the objective function and constraints, and it is not necessary for a given function to be derivable. The proposed method is tested on a numerical example. A section without vertical wall is found as the best design when the degree of the function is set to 20. The investigation shows that using a function with order 10 can reduce the computational costs while the final results do not change considerable. The result comparisons with genetic algorithm prove the robustness of the proposed method.

This work was supported by “Rahsazi va Omran Iran Construction Company”. The details of the case study are provided by this company and hereby the author is thankful for the supports.