Mathematical Problems in Engineering

Volume 2018, Article ID 1626182, 9 pages

https://doi.org/10.1155/2018/1626182

## The Backpropagation Artificial Neural Network Based on Elite Particle Swam Optimization Algorithm for Stochastic Linear Bilevel Programming Problem

School of Information and Mathematics, Yangtze University, Jingzhou 434023, China

Correspondence should be addressed to Tao Zhang; moc.621@189htam_tz

Received 24 July 2018; Accepted 8 October 2018; Published 22 October 2018

Academic Editor: Frederico R. B. Cruz

Copyright © 2018 Tao Zhang and Xiaofei Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For a class of stochastic linear bilevel programming problem, we firstly transform it into a deterministic linear bilevel covariance programming problem. Then, the deterministic bilevel covariance programming problem is solved by backpropagation artificial neural network based on elite particle swam optimization algorithm (BPANN-PSO). Finally, we perform the simulation experiments and the results show that the computational efficiency of the proposed algorithm has a potential upside compared with the classical algorithm.

#### 1. Introduction

The bilevel programming problem (BLP) is a nested optimizations problem in which the feasible region of the upper level problem is determined implicitly by the solution set of the lower level problem. As an optimization tool, the BLP has been widely used in variety practical problems, for example, in homeland security [1–3], model production processes [4], the optimal tax policies formulation [5–7], the strategic for deregulating markets [8], and the optimization of retail channel structures [9]. In addition, the optimization theory of the BLP has been integrated in many other disciplines, such as in management [10, 11], chemical engineering [12, 13], structural optimization [14, 15], optimal control problems [16, 17], facility location [10, 18, 19], and transportation [20–22].

Therefore, many researchers are devoted to develop the algorithms for BLP and propose many efficient algorithms. Traditional methods commonly used to handle BLP include Karus-Kuhn-Tucker approach [23–26], Branch-and-bound method [27], and penalty function approach [28–31]. Despite a significant progress made in traditional optimization towards solving BLPPs, the properties such as differentiation and continuity are necessary for these algorithms.

Due to the limitation of the traditional algorithms, the heuristics such as evolutionary algorithms are recognized as potent tools for solving BLPPs. Mathieu et al. [32] firstly developed the genetic algorithm (GA) for bilevel linear programming problem. Motivated by the same reason, other kinds of GAs for BLPPs were also presented in [33–36]. Owing to its high speed of convergence and relative simplicity, the particle swam optimization (PSO) algorithm has been employed for solving BLP problems recently [37–42].

However, it is worth noting that the papers mentioned above only focus on deterministic bilevel programming problem and the stochastic bilevel programming has seldom been studied so far. In 1999, Patriksson and Wynter [43] firstly proposed the stochastic mathematical programs with equilibrium constraints and introduced a framework for hierarchical decision-making problem under uncertainty. However, they did not give a numerical experiment. Gao, Liu, and Gen [44] presented a hybrid intelligent algorithm for a decentralized multilevel decision-making problem in stochastic environment in 2004. For the contracting arrangements of the long-term contracts and the spot markets transactions under uncertain electricity spot market, Wan et al. [45] proposed a stochastic bilevel programming model for the optimal bidding strategies between power seller and buyer. They solved the model by Monte Carlo approximation method. It is worth noting that the decision variable in both levels are one-dimensional variable. Soon after, Wan, Fan, and Wang [46] proposed an interactive fuzzy decision-making method for the model in [45]. In 2013, He and Feng [47] presented an approximation algorithm for the compensated stochastic bilevel programming problem. In addition, the stability analysis and convergence analysis for bilevel stochastic programming problem can be seen in [48–50]. Obviously, they only researched the simple stochastic bilevel model and few of them have studied the numerical performance of the algorithm.

In this paper, we consider the general stochastic linear bilevel programming problem in which the coefficient of objective functions and the coefficient of constraint functions are random variables. For the problem, we firstly transformed it into a deterministic linear bilevel covariance programming problem with expected constraints, and then the deterministic bilevel covariance programming model is solved by the BPANN-PSO algorithm. Finally, we perform the simulation experiments and the results suggest that the variance obtained by our algorithm is better than the results in reference when the means of the upper objective function value is same. Furthermore, the computational efficiency of our algorithm performs better with the dimension increasing.

The rest of this paper is organized as follows. Section 2 introduces the definitions and properties of stochastic linear bilevel programming problem. Section 3 proposes the BPANN-PSO algorithm for stochastic linear bilevel programming problem. We use three test problems from the reference to measure and evaluate the proposed algorithm in Section 4, while the conclusion is reached in Section 5.

#### 2. Stochastic Linear Bilevel Programming

Let , , , , , we consider stochastic linear bilevel programming problem as follows: where and are the upper level and the lower level objective functions, respectively. The coefficient , are the random variable.

Let be a probability of the extent to which the constraint violation is admitted; means a probability measure. The constraint of problem (1) is interpreted as follows:

Inequality (2) means that the constraint may be violated, but at most proportion of the time. Let be the distribution function of random variables , then, inequality (2) is presented asAccording to (2) and (3), we can obtain inequality (4),

Let . Then, from the monotonicity of the distribution function , inequality (2) is rewritten asThat is, where . Based on (6), problem (1) is rewritten as the following problem with deterministic constraints:

To deal with the objective functions with random variables in both levels, the minimum covariance model [51] is applied in this paper. Then, problem (7) is rewritten as follows:where denotes the variance of objective function; and are the covariance matrices of and , respectively. In this paper, we assume that and are positive-definite.

Though problem (8) can guarantee uniform distribution of the objective function values of both levels with a small drop, the decision makers in both levels often have their own expectation. Let and denote the means of the objective function values for the leader and the follower, respectively. Then, problem (8) is rewritten as the follows:where and are the means of and , respectively. and are the means of and , respectively. For the fixed upper level decision variable , the rational reaction set can be solved by problem (10).Let , , . Then, problem (9) is formulated asFor problem (11), the basic notions of BLP are recalled as follows:(a)Constraint region of the BLP:(b)Feasible set for the lower level problem for each fixed :(c)Projection of onto the upper level maker’s decision space:(d)The lower level maker’s rational reaction set for each fixed :(e)Inducible region:

*Definition 1. *A point is feasible if .

*Definition 2. *A feasible point is an optimal solution if and , .

*Definition 3. *If is the optimistic solution for problem (11), is given by .

#### 3. The Algorithm

For problem (11), it is noted that a solution is feasible for the upper level problem if and only if is an optimal solution for the lower level problem with . In practice, we often make the approximate optimal solutions of the lower level problem as the optimal response feedback to the upper level problem, and this point of view is accepted usually. Based on this fact, the BPANN-PSO algorithm may have a great potential for solving problem (11). In the following, the BPANN-PSO is presented for solving problem (11).

##### 3.1. The Structure of BPANN

In this paper, the BPANN includes three layers: the input layer, the hidden layer, and the output layer. The number of nodes per layer is , , and (). The connectivity weight from the node of the input to the node of the hidden layer is . denotes the connectivity weight from the node of the hidden layer to the node of the output layer. Let the Sigmoid function . Figure 1 shows the diagram of the BPANN.