As a kind of new algorithm derived from the foraging behavior of bee populations, the ABC algorithm solves the multidimensional and multimodal optimization problems successfully by simulating the behaviors of the real bees. In the algorithm, the bee colony was divided into tripartite: employed bees, onlookers and scouts. The employed bees visit the food source firstly, the onlookers decide which food source should be visited in dancing area, and the scouts are random search bees.

In the initialization phase, food source locations were randomly selected and their amount of nectar was a constant. Employed bees showed the information of food source. In the second phase, each employed bee recorded the food source visited previously, and then chose a new source in the neighborhood. In the third phase, each onlooker chose food source based on the information shown by the employed bees. The more nectar the route had, the easier it was selected. In artificial bee colony algorithm, the food source locations represented the possible solutions and the amount of nectar was the fitness of the solution. In the ABC algorithm, SN denoted the size of the initial population. Each solution is a D-dimensional vector. The employed bees searched for the new solution and tested its fitness. The new solution was generated as follow:

(1)

Where, v_ij was the new solution, x_ij was the known solution, q_ij was a factor whose value is [-1, 1], x_kj was a solution in the neighborhood.

If the new solution is better than the previous one, the previous solution was replaced. After completing the search process, the next onlookers were chosen by using Eq. (2).

(2)

Where, p_i was the probability of selection, fit_i was the fitness value

In artificial bee colony algorithm, the solution without improvement will be abandoned after the limited cycle times.

The ABC algorithm received extensive attention of scholars all over the world since it was proposed. It has many advantages, such as simple calculation and higher robustness. So, it was easy to be used to deal with the combinatorial optimization problems. But the defects of algorithm were exposed gradually after several years’ research, such as lower convergence speed, lower calculation accuracy and the premature.

Here, a novel improved artificial colony based on small interval search algorithm was proposed. The parameter l was introduced into the process of initialization to make the initial solution distributed throughout the search domain as much as possible. This strategy can improve the global and local search ability of the algorithm. At the same time, the tabu search and sets parameter γ were introduced to eliminate the local optimal solution.

The main improvement of the algorithm was as follows.

First, in the basic ABC algorithm, the initial solution was generated randomly, which influenced the quality of the region search. In order to overcome the defect, the parameter l was introduced into the Eq. (1) .

(3)

This improvement ensured that the initial solution can be evenly distributed in the whole range. In the process of data processing, the original solution was replaced. The new solution was as follows.

(4)

Where, k ϵ {1,2,…, SN}, K ≠ i, k was in the neighborhood of.

At the same time, the tabu search was employed as a penalty parameter γ was used to compare with the new solutions. If the fitness of new solution is better than γ, it will be replaced. Otherwise, it will not be replaced. This process was repeated until the better fitness was found.

(5)

Where, was the maximum fitness of every iterative.

In order to verity the effectiveness of the improved algorithm, three standard test functions and the TSP (Travelling Salesman Problem) problems were selected. And the results were compared with other algorithms.

The function of Sphere, Rosenbrock, and Rastrigin were defined as follows:

(1) Sphere function

(6)

The value of x was [-5.12, 5.12].

(2) Rosenbrock function

(7)

The value of x was [-2.048, 2.048]

(3) Rastrigin function

(8)

The value range of x was [-5.12, 5.12]

The dimension of the above functions was 30. The basic control parameters of artificial bee colony algorithm were: SN = 100, MCN = 300, limit = 50. The results obtained after 30 times were shown in Table 1.

As shown in the Table 1, the optimization results of the improved ABC on the test function were better than the basic ABC.

To verify the performance of the improved algorithm, the TSP problem was used too. And the TSP-Chn 31 was selected.

TSP-Chn 31—International standard test of TSP problem with 31 cities.

The basic parameters of the algorithms were shown in the Table 2.

The result were shown in Table 3 and Fig. (1).

Fig. (1). TSP-Chn 31 iterative curve contrast.

From the above result, we can observe that on the problem of TSP Chn 31, the improved artificial bee colony algorithm proposed here can achieve the optimal solution 15374 at the 48^th generation, and the final result was the best among the results obtained by using ant colony algorithm, the basic artificial bee colony algorithm and the other improved methods of artificial colony algorithm. Therefore, compared with the basic ant colony algorithm, basic artificial bee colony algorithm and other improved artificial bee colony algorithm, the improved algorithm we proposed has the better searching capability.

Then, we will introduce it to the truss structure optimization as shown in Fig. (2).

1. Design variable: A_i was the bar sectional area.
2. Object function: The total weight of bar was the object function [22]

(9)

Fig. (2). Framework of improved algorithm on truss structure optimization.

Where, W was the total weight of bar; ρ_m was material density; A_i was the sectional area of bar i ;L_i was the length of bar i.

3. Constraint condition
   • a)  The stress constraint: σ_i ≤ [σ_i].
   • b)  The displacement constraint: µ_j ≤ [µ_j].
   • c)  The upper and lower bounds of design variables: A_i ϵ {S}.
   • [σ_i],σ_i—the allowable stress and the most unfavorable stress of bar.
   • [µ_i],µ_i—the allowable displacement value and displacement value of specific nodes given direction.
   • {S}—the sets of bar section size variable.
   • d)  Reliability constraints

(10)

β_i was the reliability index, [β_i] was the allowable reliability indicator value, here the [β_i] was 3.4. Each bar of truss structure under the node load was axial stress. The relationship between reliability index and sectional area was as follow [23]:

(11)

Where, A_i⁽⁰⁾ was the initial sectional area. M_Ri⁽⁰⁾, σ_Ri⁽⁰⁾ were the mean and standard deviation of sectional area of bar i respectively. M_Si, σ_Si were the mean and standard deviation of load effect of bar i respectively.

To compare the results more clearly, we employed the example of the references [24, 25]. It was a 25-barspatial truss structure model shown in Fig. (3). The basic parameters of the improved artificial bee colony algorithm were: population size was 25, the maximum number of iteration was 200, and the limit was 10. The basic parameters of truss structure were shown in Tables 4-6.

Fig. (3). 25-barspatial truss structure model.

It has been observed that the stress constraint in compression was different from the stress constraint in tension. The stress constraint in compression should not only meet the strength conditions, but also meet the stability conditions. In the optimization process, the stable critical stress had been set, the buckling of the structure will not occur. So, the buckling of the structure will not be considered here.

The truss optimization results obtained from the improved ABC algorithm were shown in Table 7 (And the results of improved GA and PSO algorithms can be seen from [26] and [27] respectively) and the iterative curve was shown in Fig. (4).

Fig. (4). Iterative curves of three algorithms.

Under the same constraints, the 25-bar truss structure was optimized by three algorithms. The GA algorithm finds its optimal solution at the iteration 170 times. The PSO algorithm finds the optimal solution when the algorithm iteration was 120 times. The improved ABC algorithm we proposed finds the optimal solution in the iteration at 100 times. From Table 7 and Fig. (4), we can know that by using the improved artificial bee colony algorithm we proposed, the structure weight was 214.702kg. It is better than the results obtained from the genetic algorithm and the particle swarm optimization algorithm. Comparing with the genetic algorithm and the particle swarm optimization algorithm, the improved algorithm proposed here has higher convergence speed and convergence precision.