A twostage artificial neural network (ANN) based on scalarization method is proposed for bilevel biobjective programming problem (BLBOP). The induced set of the BLBOP is firstly expressed as the set of minimal solutions of a biobjective optimization problem by using scalar approach, and then the whole efficient set of the BLBOP is derived by the proposed twostage ANN for exploring the induced set. In order to illustrate the proposed method, seven numerical examples are tested and compared with results in the classical literature. Finally, a practical problem is solved by the proposed algorithm.
The bilevel programming problem (BLP) is a nested optimizations problem with two levels in a hierarchy: the upper and lower level decisionmakers. The upper level maker makes his decision firstly, followed by the lower level decisionmaker. The objective function and constraint of the upper level problem not only rely on their own decision variables but also depend on the optimal solution of the lower level problem. The decisionmaker at the lower level has to optimize his own objective function under the given parameters from the upper level decisionmaker. Since many practical problems, such as engineering design, management, economic policy, and traffic problems, can be formulated as hierarchical problems, BLP has been studied and received increasing attention in the literatures. During the past decades, some surveys and bibliographic reviews were given by several authors [
The bilevel programming problem is a nonconvex problem, which is extremely difficult to solve. As we know, BLP is a NPHard problem [
Unfortunately, the bilevel programming problem is nonconvex and the properties such as differentiation and continuity are necessary when proposing the traditional algorithms. Thus, many researchers tend to propose the heuristic algorithms for solving BLP because of their key characteristics of minimal problem restrictions such as differentiation. Mathieu et al. [
Particle swarm optimization (PSO) is a relatively novel heuristic algorithm inspired by the choreography of a bird flock. Due to its high speed of convergence and relative simplicity, the PSO algorithm has been employed for solving BLP problems. For example, Li et al. [
However, the algorithms mentioned above are only for the simple single objective bilevel programming problems. In fact, the multiobjective characteristics widely existing in the BLPP and the bilevel multiobjective programming problem (BLMPP) have attracted many researchers’ interesting. For example, Shi and Xia [
As we known, the authenticity of the lower level Pareto optimal solution is very important for the BLBOP. If the obtained optimal Pareto solutions possess the fraudulence, it can lead to the failure to solve the whole problem. In this paper, the induced set of the BLBOP is firstly expressed as the set of minimal solutions of a biobjective optimization problem by a scalar approach which can greatly improve the accuracy of the lower level Pareto optimal solutions. Based on the efficient set of the BLBOP, a twostage ANN is presented for solving whole problem which can reduce the computation burden.
The remaining of this paper is organized as follows. In Section
Let
Let
For a fixed
If
If
The optimistic solution to the BLBOP is the one that optimizes the leader’s objective function over the set of efficient solutions to the follower, assuming that the follower has no preferences among the efficient solutions obtained for each leader’s decision
For problem (
In bilevel optimization, the constraint set of the upper level problem is given by the solution set of the lower level optimization problem. According to the Theorem 4.1 of the literature [
Thus, to solve the induced set of BLBOP is transformed to solve the Pareto optimal solution set of problem (
If
The first stage of the ANN is a feedforward artificial neural network (FFANN) which is composed by two subnetworks
The second stage is a quasineural artificial network; namely, the network has no connectivity weight and the output value can be computed directly by software. The input layer of the quasineural artificial network is the output layer of the first stage network. For the hidden layer, the input of and the output are defined by (
For the output layer, the input of and the output are defined by (
Based on the set
In Step 2, the two subsets
Based on Algorithms
In this section, we considered seven numerical examples and a practical problem to illustrate the feasibility of the proposed algorithm for problem (
This metric used by Deb [
This metric is used to evaluate the diversity of the obtained Pareto optimal solutions by comparing the uniform distribution and the deviation of solutions as described by Deb [
All results presented in this paper have been obtained on a personal computer (CPU: AMD 2.80 GHz; RAM: 3.25 GB) using a c# implementation of the proposed algorithm.
In this section, we will present seven BLBOPS to illustrate the proposed algorithm for the bilevel biobjective programming. Problem 1 and problem 2 are lowdimensional problems. Problems 3–6 are highdimensional problems. For problem 7, the theoretical optimal front is unknown. In this paper, we refined every upper problem’s feasible solutions three times and the obtained results are compared with the classical literature.
Example
Figure
Results of the Generation Distance (GD) and Spacing (SP) metrics for Examples
Prob.  GD  SP  

The method in [ 
The proposed method  The method in [ 
The proposed method  
1  0.00216  0.00097  0.01135  0.01201 
2  0.01013  0.00312  0.00203  0.00969 
The obtained Pareto front and solutions of Example
The obtained Pareto front of Example
The obtained solution by the proposed algorithm
The obtained solution by [
The obtained Pareto front and solutions of Example
The obtained Pareto front of Example
The obtained solutions by Algorithm
The obtained solutions by the method in [
Example
Figure
Example
This problem is more difficult compared to the previous problems (Examples
Results of the Generation Distance (GD) metrics for Examples
Prob.  The proposed algorithm  The method in [ 

3  0.00019  0.00768 
4  0.00015  0.06391 
5  0.00038  0.01677 


6  0.00021  0.00652 
Results of the Spacing (SP) metrics for Examples
Prob.  The proposed algorithm  The method in [ 

3  0.00076  0.00197 
4  0.00273  0.00269 
5  0.00299  0.01737 


6  0.00130  0.00127 
The obtained Pareto front of Example
Example
For Example
The obtained Pareto front of Example
Example
Figure
The obtained Pareto front of Example
Example
Figure
The obtained Pareto front of Example
Example
Figure
The obtained Pareto optimal solutions with
In a company, the CEO’s goal is usually to maximize net profits and quality of products, whereas a branch head’s goal is to maximize its own profit and worker satisfaction. The problem involves uncertainty and is bilevel in nature, as a CEO’s decision must take into account optimal decisions of branch heads. We present a deterministic version of the case study from [
The obtained Pareto front of the practical problem.
In this paper, a twostage ANN based on scalarization method is presented for solving BLBOPs. Seven numerical examples and a practical problem are used to state the feasibility and efficiency of the proposed algorithm. The experimental results indicate that the obtained Pareto front by the proposed algorithm is very close to the theoretical Pareto optimal front, and the solutions are also distributed uniformly on entire range of the theoretical Pareto optimal front. The proposed algorithm is easy to implement, which provides another appealing method for further study on the general BLMPP.
The authors declared that they have no conflicts of interest related to this work.
This work is supported by the National Science Foundation of China (61673006), the Young Project of Hubei Provincial Department of Education (Q20141304), and the Dr. StartUp Fund by the Yangtze University (2014).