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.

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 [

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 [

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 [

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

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 [

Here, assume the leader first chooses its decision variables

In this programming model, the follower solutions interact with the leader solution. Here, Discrete and continuous variables have upper and lower bounds. Bard [

Constraint region of the BLP problem:

Feasible set for the

Leader’s decision space:

Inducible region of the BLP problem:

Optimal solution of BLP problem:

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

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 [

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

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

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

Process of the proposed BLGA.

The procedure of the proposed BLGA is as follows.

The termination condition is that the number of runs has reached the maximal number. To determine if the termination condition has been satisfied, the maximal number of runs

According to their bound constraints, the leader population is initialized and these variables are encoded. Two strategies are used for the encoding of variables. The proposed method uses the binary code strategy, which uses binary vectors as chromosomes, to represent the values of continuous variables with unequal lengths. The variable encoding length

Each follower programming takes the initial leader population

The leader programming takes the best chromosome of each follower

The termination condition is that the number of generations has reached the maximal number. If the maximal number is reached, the chromosome with the greatest fitness value is considered the best solution. The best leader chromosome

The roulette wheel selection method is used to select

Input the crossover rate

The proposed method uses two mutation methods. For continuous variables that use the binary code strategy for encoding, the mutation rate

The proposed BLGA for solving MINLBLP with one or multiple independent followers is realized using MATLAB. To verify the effectiveness and robustness of the proposed BLGA, we give numerical examples obtained with a personal computer with the following parameters: population size is 50; maximal number of generations is 200; precision of binary code is 0.01; crossover rate is 0.8; mutation rate is 0.01; number of experiments is fifteen. The results were compared with the exact solutions obtained from traditional methods for solving MINLBLP to test the quality of the computed solution. The best, worst, average, and median results and standard deviation value are presented.

Example

Results of Example

| | | | |
---|---|---|---|---|

Best | 0 | | −8 | |

Worst | 0 | | −2 | |

Average | 0 | — | −5.60 | — |

Median | 0 | — | −8 | — |

Standard deviation value | 0 | — | 3.04 | — |

Enumeration method | 0 | | −8 | |

Table

Iterative process of Example

Example

Results of Example

| | | | |
---|---|---|---|---|

Best | 12.0030 | 6.0000 | −2.0010 | 2.0010 |

Worst | 12.1596 | 5.9217 | −2.0793 | 2.0793 |

Average | 12.0134 | — | −2.0010 | — |

Median | 12.0030 | — | −2.0070 | — |

Standard deviation value | 0.0404 | — | 0.0217 | — |

Graphic method | 12.0000 | 6 | −2.0000 | 2.0000 |

As can be seen, the difference between the values of upper-level objective functions of the best result calculated by the proposed BLGA and the exact method is 0.0030, which is a negligible difference. This demonstrates the effectiveness of the proposed BLGA. In these 15 independent runs, except the worst result, the other results are the same. The iterative process of the best result is shown in Figure

Iterative process of Example

The results calculated by the proposed BLGA and the exact method are shown in Table

Results of Example

| | | | |
---|---|---|---|---|

Best | −1.9600 | | 1.9600 | |

Worst | −1.9600 | | 1.9600 | |

Average | −1.9600 | — | 1.9600 | — |

Median | −1.9600 | — | 1.9600 | — |

Standard deviation value | | — | | — |

Exact method | −1.9600 | | 1.9600 | |

Table

Iterative process of Example

The results are shown in Table

Results of Example

| | | | |
---|---|---|---|---|

Best | 1.0000 | 1.0039 | | 1 |

Worst | 1.0333 | 1.0011 | | 1 |

Average | 1.0045 | — | | — |

Median | 1.0000 | — | | — |

Standard deviation value | 0.1171 | — | 0.0004 | — |

Exact method | 1.0000 | 1.0000 | 0.0000 | 1 |

In these 15 runs, there are 6 times in which the proposed BLGAs draw the different solutions with the best result. However, these differences are so small that can be negligible. Moreover, the best result is rather similar to the exact solution, which demonstrates the effectiveness of the proposed BLGA. The iterative process is shown in Figure

Iterative process of Example

The results are shown in Table

Results of Example

| | | | | | |
---|---|---|---|---|---|---|

Best | −6598.6 | | 16.5284 | | 56.8328 | |

Worst | −6512.8 | | 54.8902 | | 12.94 | |

Average | −6568.81 | — | 25.0162 | — | 52.3811 | — |

Median | 6572.40 | — | 9.8871 | — | 72.1481 | — |

Standard deviation value | 22.0338 | — | 15.1987 | — | 22.4542 | — |

Exact method | −6600 | | 11.9241 | | 64.6841 | |

Table

Iterative process of Example

Based on the characteristic of MINLBLP with one or multiple independent followers in product family design (PFD), we have proposed a BLGA to solved this problem. We use five numerical examples to demonstrate the effectiveness and robustness of the proposed BLGA. We summarize the advantages of the proposed BLGA as follows.

First, MINLBLP with one or multiple independent followers has generality, which can illustrate many practical management and engineering problems, such as PFD. Many studies have examined using GAs to solve special BLP problems. Some algorithms require that leader constraints include no follower decision variables, which cannot solve the target MINLBLP problems. In addition, some algorithms require only continuous decision variables, which limit practical application. In theory, the proposed BLGA has no limitations on constraints and variables.

Second, the optimal solutions calculated by the proposed algorithm belong to the constraint region. Note that to ensure the feasibility of the optimal solutions, we have made the following assumptions prior to designing the proposed BLGA. In the BLGA, the fitness values of individuals that do not satisfy constraints are set to zero to avoid being selected as parents. This ensures that the optimal solutions satisfy all constrains, which belong to the constraint region.

Third, the proposed BLGA has good effectiveness and strong robustness, which has been demonstrated through the five numerical examples. Note that we use the binary code strategy to encode so that we can set the precision according to practical requirements. This strategy suits the mechanism of GAs, which can search for optimal solutions carefully.

A case study of a simplified reducer family design is shown to demonstrate the proposed BLGA. A single-stage gear reducer includes three types of customized reducers, whose transmission ratios are 2, 3, and 4, respectively. To reduce cost and produce the three reducers simultaneously, we design platform parts and different customization parts, respectively. In this case, platform parts include a shell, a high-speed shaft, and a low-speed shaft, whose critical parameters are taken as the platform variables. The different customization parts include the high-speed gear and the low-speed gear, whose critical parameters are taken as the customization variables. The critical parameter of platform parts is shell size, which is described by the center distance and diameter of high-speed shaft

In the design process of the reducer family, we cannot consider platform optimization and customization optimization simultaneously. The platform designer who optimizes the platform aims to leverage enterprise profitability, which includes maximizing the performance of all reducers and minimizing total cost. The customization designers design customization variables to maximize the performance of each reducer. Note that these designers have different goals and decision variables; however, their decisions affect each other. This is a joint optimization problem. Compared with the customization designers, platform designers act as leaders because the platform is the basis of a product family, which is more important. The customization designers act as followers. Consequently, this case can be considered a Stackelberg game-based decision-making problem.

In this reducer family design problem, the leader is the platform design, whose decision variables are the total number of gears

In this problem, the performance of the reducer family is the sum of the performance of each reducer, which is influenced by bending resistance and abrasion resistance. We use the bending fatigue stress of high-speed gear

Here, the total cost is the sum of the costs of each reducer, which is influenced by manufacturing materials. Assume that different types of reducers that use the same type of material have the same production cost. The customization designer can select the manufacturing materials from metals A, B, and C. The production costs

We can describe the leader’s objective of the upper-level model as minimizing the product between the evolution of fatigue stress and the total cost, which is given as follows:

The lower-level models attempt to optimize the three types of reducers. For the

The constraints of fatigue stress are that bending fatigue stress and the contact fatigue stress are less than allowable stress, respectively, which are given as follows:^{5} N·mm. The different material coefficients of metal are given in Table

Material coefficients of metals.

Metal A ( | Metal B ( | Metal C ( | |
---|---|---|---|

Elastic coefficient | 200 MPa^{0.5} | 190 MPa^{0.5} | 180 MPa^{0.5} |

Contact fatigue stress | 300 MPa | 400 MPa | 500 MPa |

Bending fatigue stress | 100 MPa | 200 MPa | 300 MPa |

Ranges of variables and indicators.

Variables and indicators | Range |
---|---|

Diameters of high-speed shaft | 40–60 mm |

Total number of gears | 140~160 |

Diameters of low-speed shaft | 60–100 mm |

Number of teeth of high-speed gear | 25–60 |

Transmission ratio of reducer 1 | 1.8–2.2 |

Transmission ratio of reducer 2 | 2.7–3.3 |

Transmission ratio of reducer 3 | 3.6–4.4 |

According to our analysis, an MINLBLP model with multiple independent followers of the leader-follower joint optimization for the reducer family design can be expressed as follows:

The material coefficient of metal and the ranges of variables and indicators are shown in Tables

This model is an MINLBLP problem with three independent followers. We use the proposed BLGA to solve this problem with the following parameters: population size is 50; maximal number of generations is 300; precision of binary code is 0.01; and crossover rate is 0.8. Figure

Results.

Platform variables | Customization variables | Upper-level goal | Lower-level goal | Evaluation of fatigue stress | ||||
---|---|---|---|---|---|---|---|---|

Diameter of high-speed shaft | Total number of gears | Diameter of low-speed shaft | Number of teeth of high-speed gear | Selections in metal | Overall evaluation of reducer family | Cost of reducer | ||

| | | | | ||||

Reducer A | 52.2627 | 159.6643 | 77.4652 | 57.0171 | A | | 50 | |

Reducer B | 64.0830 | 43.0684 | A | 50 | | |||

Reducer C | 79.6435 | 34.6496 | B | 79 | |

Iterative process.

MINLBLP can accurately model product family design, which has one or multiple nonlinear and nonconvex independent low-level models. Based on the characteristics of the model in product family optimization, the primary purpose of this study has been to propose a bilevel BLGA to solve product family optimization models. The proposed BLGA applies a GA to both the leader and followers and uses appropriate encoding strategies to handle continuous and discrete decision variables, which are common in product family optimization. The proposed BLGA overcomes many limitations of previously proposed GAs. For example, such GAs cannot solve a model in which leader constraints do not contain follower decision variables, which is common in product family optimization. The proposed BLGA overcomes this limitation. This paper models the joint optimization between product platform and product family designs by an MINLBLP approach in detail and presents a case study of a simplified reducer family design to demonstrate the feasibility and effectiveness of this model and BLGA.

Currently, BLP is widely used in product family design optimization, which is a hot topic in engineering. GAs can solve BLP problems effectively. This paper only presents a specific application of a simplified reducer family design which describes the joint optimization between product platform design and product family design. In the future, we can do more research on other leader-follower relationships in the product family design problems and improve the BLGA efficiency by adjusting the encoding strategies and operation strategies.

The authors declare that there are no competing interests regarding the publication of this paper.

This project is supported by the National Natural Science Foundation of China (nos. 71071104 and 71371132).