Mathematical Problems in Engineering

Volume 2016, Article ID 1379315, 15 pages

http://dx.doi.org/10.1155/2016/1379315

## Genetic Algorithm for Mixed Integer Nonlinear Bilevel Programming and Applications in Product Family Design

College of Management and Economics, Tianjin University, Tianjin 300072, China

Received 12 April 2016; Revised 4 July 2016; Accepted 21 July 2016

Academic Editor: José-Fernando Camacho-Vallejo

Copyright © 2016 Chenlu Miao et al. 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

Many leader-follower relationships exist in product family design engineering problems. We use bilevel programming (BLP) to reflect the leader-follower relationship and describe such problems. Product family design problems have unique characteristics; thus, mixed integer nonlinear BLP (MINLBLP), which has both continuous and discrete variables and multiple independent lower-level problems, is widely used in product family optimization. However, BLP is difficult in theory and is an NP-hard problem. Consequently, using traditional methods to solve such problems is difficult. Genetic algorithms (GAs) have great value in solving BLP problems, and many studies have designed GAs to solve BLP problems; however, such GAs are typically designed for special cases that do not involve MINLBLP with one or multiple followers. Therefore, we propose a bilevel GA to solve these particular MINLBLP problems, which are widely used in product family problems. We give numerical examples to demonstrate the effectiveness of the proposed algorithm. In addition, a reducer family case study is examined to demonstrate practical applications of the proposed BLGA.

#### 1. Introduction

With the evolution of the mass customization paradigm, product family has played an increasingly important role in modern production and has garnered significant attention. Product family optimization design includes product-design-related knowledge, the product family structure, and customization design based on the same product platform to meet customer needs. Many leader-follower relationships exist in product family optimization design problems, for example, between platform and customization design [1], between modular design and product family architecture [2], and between product family and supply design [3, 4]. Consequently, many researchers have used this model to reflect the optimization relationship in product family design. Du et al. [5] formulated a Stackelberg game-theoretic model for joint optimization of product family configuration and scaling design, wherein a bilevel decision structure reveals coupled decision-making between module configuration and parameter scaling. Kristianto et al. [6] demonstrated the potential of a two-stage, bilevel stochastic programming framework for tackling engineer-to-order product customization. Du et al. [7] proposed a leader-follower joint optimization model of product family configuration and supply chain design, and Ji et al. [8] studied green design modularity using bilevel optimization. In addition, Jiao et al. [9] proposed an underpinning decision structure that distinguishes a de facto leader-follower model rather than a single-level, all-in-one optimization problem.

Due to the impact of Stackelberg game theory, which was proposed by Stackelberg when researching marketing economics, researchers have studied bilevel programming (BLP) since the 1970s. Bracken and Gill proposed BLP models in 1973 and 1977, respectively. Candler and Norton proposed a formal definition of BLP and multilevel programming in their technology reports [10]. BLP models are NP-hard problems. Prominent BLP results primarily include the following: the* k*th best method for special linear cases [11], replacing a lower-level problem with Karush-Kuhn-Tucker conditions to convert a bilevel problem to a single-level problem [12], and converting the problem to a single level by constructing a penalty function based on the duality gap [13]. The monographs of Bard [14] and Dempe [15] present systematic surveys of BLP models. In leader-follower joint optimization for product family design, 0-1 mixed integer nonlinear bilevel BLP, which has multiple nonlinear and nonconvex low-level models, is involved. Therefore, traditional methods (e.g., the K-T method for convex bilevel BLP) are limited to solving specific types of the BLP models, which restrict their application.

The BLP used in product family design has unique characteristics. In this type of programming, both continuous and discrete variables such as 0-1 variables (the most commonly used) are employed. In product family design problems, more than one lower-level model, which are nonlinear and nonconvex to model, are required. In addition, in this type of BLP, the leader’s constraints often contain follower variables. Mixed integer nonlinear BLP (MINLBLP) contains the above characteristics when applied to product family optimization. Note that MINLBLP combines integer programming and BLP. However, the discreteness of decision variables and multiple followers make problems more complex. Some researchers prefer to use intelligent algorithms to solve this problem. BLP with multiple lower-level models is more difficult to solve than with a model consisting of only one follower. Therefore, developing an intelligent algorithm to solve the MINLBLP used in product family design problems has significant value.

A genetic algorithm (GA) is a method to search for an optimal solution by simulating biological evolution (survival of the fittest). GAs are popular intelligent algorithms that have seen increasingly wide utilization in many fields. GAs have many advantages such as convergence and robustness. Thus, GAs are effective in solving optimization problems. Using a GA to solve a bilevel problem reduces the limitations which traditional methods have, which has been extensively studied. Consequently, many GA monographs for BLP exist. In 1998, Liu designed a GA for multilevel programming with multiple followers, in which high-level models do not contain the low-level models’ decision variables [16]. In 2002, Oduguwa and Roy used a GA to solve a BLP scheme that was developed to encourage limited asymmetric cooperation between two players [17]. Kuo and Han applied bilevel linear programming to a supply chain distribution problem and developed an efficient method based on a hybrid of a GA and particle swarm optimization [18], and Lin et al. presented a modified GA to solve a bilevel linear programming network design problem [19]. For a linear bilevel programming problem with interval coefficients in the upper-level objective, Fan and Li propose a genetic algorithm to obtain the best optimal solution and the worst one simultaneously [20]. Li et al. convert an integer linear bilevel programming problem into a single-level binary implicit program and then develop an orthogonal genetic algorithm for solving such problem based on the orthogonal design [21]. Based on a novel coding scheme, Li proposes a genetic algorithm with global convergence to solve nonlinear bilevel programming problems where the follower is a linear fractional program [22]. However, even though GAs have been successfully applied to BLP, it is difficult for such GAs to solve MINLBLP in product family design problems. Such GAs focus on solving specific models for other problems. They can solve linear BLP, single-level programming, and BLP with multiple followers, in which leader constraints do not contain follower decision variables. However, these GAs are not suitable for solving MINLBLP, which contains 0-1 decision variables and multiple low-level models. A bilevel GA (BLGA) for MINLBLP is proposed to solve product family design optimization problems.

The proposed BLGA can handle MINLBLP with both continuous and discrete variables with one or multiple independent nonlinear and nonconvex lower-level models. The remainder of this paper is organized as follows. In Section 2, we pose the primary research question and derive MINLBLP. In Section 3, we propose the BLGA and describe its procedures. Numerical examples are provided in Section 3 to illustrate the proposed algorithm. In Section 4, we provide a case study of product family design to demonstrate an application of the proposed method. Conclusions are given in Section 5.

#### 2. MINLBLP with One or Multiple Independent Followers

An MINLBLP model probably has one or multiple followers. Based on the relationships among the multiple followers, such MINLBLP problems can be classified as dependent or independent and, for each particular case, there exists a specific model to describe it [23–25]. The coupling relationships among the followers are not taken into account for most of the bilevel problems in product family design [26, 27]. Therefore, this paper focuses on the MINLBLP with independent followers. In an MINLBLP model with multiple independent followers, the followers are independent and have no coupling relationship, which means that the objective function and the set of constraints of each follower only include the leader’s variables and its own variables [24].

Here, assume the leader first chooses its decision variables , in which are continuous or discrete, and the followers determine their decision variables , in which are continuous or discrete. The MINLBLP model with one or multiple independent followers should be formulated as follows:where means the number of followers.

In this programming model, the follower solutions interact with the leader solution. Here, Discrete and continuous variables have upper and lower bounds. Bard [14] proposed the following basic definition for a BLP solution:(1)Constraint region of the BLP problem:(2)Feasible set for the th follower for each fixed: :(3)Leader’s decision space:(4)th follower’s rational reaction set for :(5)Inducible region of the BLP problem:(6)Optimal solution of BLP problem: is considered the optimal solution of BLP if and only if there exists such that for all .

#### 3. BLGA Design

##### 3.1. Assumption

Analysis of feasibility of the proposed algorithm illustrates that two assumptions must be satisfied to obtain good performance. These assumptions ensure the effectiveness of the proposed BLGA by restraining the algorithm’s process.

The first assumption is that solutions must meet the requirements of BLP, which means that the solutions must satisfy the constraints of both the upper- and lower-level programs. Meaningful solutions can only be obtained if the algorithm is feasible.

According to the concepts about the solution, we know that if is within the feasible region, it is denoted as follows:We propose a BLGA that meets these constrain requirements. First, the binary code encodes the leader variables and initializes them according to their bound constraints. Second, by taking the initial leader variables as parameters, each follower programming uses the lower-level GA to obtain the best solution in the last generation. Note that the best solution in the last generation provided by the lower-level GA is a good approximation for the optimal solution, which meets the lower-level constraints. Then the lower-level approximation solutions are brought back to the upper-level GA to solve the leader program. In the upper-level GA, the solutions should be verified whether they satisfy the leader’s constraints.

The second assumption is that the algorithm has convergence. As the proposed BLGA meets the first condition, it can obtain the optimal solution only if it has convergence, which is an important measurement of computing capacity. Convergence of a GA means that the last cycle of a finite series of cycles can yield the optimal solution. This assumption ensures that, after comparing feasible solutions, the optimal solution can be found. An algorithm is only effective and feasible when it satisfies this condition.

Currently, many GAs have been designed to solve special types of problems that have convergence [28–30]. Markov chains are often used to evaluate convergence. If a GA whose mutation rate is fixed and greater than 0 takes an elitist strategy and the binary code strategy, it has convergence that cannot be influenced by the initial population. Using the elitist strategy allows the best chromosome to be removed from operations in order to avoid disappearing in the operations. Moreover, many factors influence convergence, such as the maximal number of generations, crossover rate, mutation rate, and crossover and mutation methods.

According to the above assumptions, even though the best solution provided by the BLGA does not guarantee to be the optimal solution, it may be a good approximation for the global optimal solution in reasonable time. We can calculate the problem several times, using different seeds, to find the closest approximation solutions. Numerical examples are given in Section 3.4 to verify the algorithm.

##### 3.2. Design Process of BLGA

On the basis of the above assumptions, this study proposes a BLGA to solve MINLBLP with one or multiple independent followers that can improve efficiency in generating solutions. The essential process is as follows. First, leader decision variables are initialized according to their bound constraints. Second, each lower-level GA takes leader decision variables as parameters to solve the th follower problem to determine the best solution and the value of , where . Third, the best solution for each follower and the leader decision variables is returned to the leader problem to determine if they satisfy other leader constraints. Their fitness values are then evaluated. Fourth, selection, crossover, and mutation operations are performed for these leader decision variables. Fifth, this cycle continues until maximal iterations are reached. The best optimal solutions and are then recorded.

However, there is no guarantee that the optimal solutions will be the best solution for MINLBLP with one or multiple independent followers because the GA can only ensure that solutions belong to the constraint region rather than the inducible region. It is necessary to run this process several times to obtain several optimal solutions. These optimal solutions are then compared to determine the best solution. The process of the proposed BLGA is shown in Figure 1.