Joint optimization of complex product variant design responding to customer requirement changes

Abstract

Changing customer requirements always poses a great challenge to designers in complex product design. Due to the complicated coupling relations between components, it is difficult to improve the efficiency of product reconfiguration. Therefore, this paper proposes a joint optimization model of complex product variant design responding to customer requirement changes. Based on synthesizing the advantages and deficiencies of “Scaled-based” and “Module-based”, a UB-BU hybrid approach is developed to improve the search efficiency and reduce conflicts. Then, a joint complex product variant design bi-level optimization to maximize customer satisfaction and minimum cost is established. As a solution, the genetic algorithm embedded double iteration comparison rules (GA-DICR) is developed based on canonical genetic algorithm. Finally, the method and model are proved to be effective in the Clutch variant design. The results can provide way of allowing the decision-makers to respond rapidly to customer rudiments changes request.

Keywords

Complex product variant design customer requirement changes joint optimization bi-level programming genetic algorithm

1 Introduction

In highly dynamic environments, it is extremely important for enterprises to rapidly respond to changing requirements [1]. Under this context, variant design shows some advantages both in time and cost due to the product platform and module combination. Then, variant design is attracting more and more attention [2, 3]. However, for complex product, complicated coupling relations between components cause that it is difficult to improve the efficiency of product reconfiguration, which maybe lead to a longer variant design [4]. Consequently, the overall design cost will increase and customer satisfaction will reduce [5]. Therefore, optimization for complex product variant design responding to customer requirement changes has a great practical significance on enhancing enterprises’ efficiency and business benefit.

As for the product variant design problem, many scholars have launched a series of studies. In reference [6], a portfolio optimization strategy including product function constraint was proposed to maximize economic benefits for uncertain or dynamic customer requirements. Freuder et al. measured the pros and cons of configuration solution via the customer constraints priority set before, and a mutually satisfactory result was achieved through consultation [7]. Simpson proposed a product configuration scheme adjustment method based on the alternative analysis between similar products, and they summarized the product family configuration method as two aspects: module-based and scaling-based [8]. Fujita et al. developed a method integrating module design and parameter adjustment to implement product configuration according to the degree of change [9]. Weiss et al. proposed a product variant design time control tool to maximize the benefits [10]. Eun Suk Suh et al. developed a flexible product platform based on Monte Carlo simulation process to achieve the rapid search and match to the required components [11]. Wang, J. K. et al. studies the application of CBD technology in product variant design, and with the application of variant design principle, the design intensity will be lightened, and the design time will be shortened [4]. Lo, C. H. et al. proposes a novel methodology, QFD based 3D morphological charts, to support variant design of simple and technically mature products, and a computer-aided conceptual design system is implemented to realize the proposed ideas with computer mice as an example product [12]. Li, Y. L. et al. a hybrid design resource integration framework is proposed based on the design process and resource modeling in order to satisfy the evolutionary requirements for design resources in the process [2]. Khajavirad, A. and Michalek, J. J. developed an extended metric to account for generalized commonality proposing a decomposed single-stage method for optimizing the joint problem, which relaxed the metric to the continuous space to enable gradient-based optimization [13]. Chen, C. Y. et al. studied not only determining, but also revising the priority of customer demands for a variant product design based on new customer surveys, and they proposed two methods. The first method classified customer demands using natural language processing techniques in order to obtain customer expectations. Once comprehensive customer demands were obtained, the second method determined the revised priority of the customer demands using a fuzzy logic inference [14].

Making a general survey of the existing research results, we conclude that few researches are highly effective to complex product, and much less focus on the variant design responding to customer requirement changes. The mainly reason is that the traditional methods and models are difficult to analyze the intricate coupling between parts in complex product, causing that the parts to be changed are hard to be positioned. Thus, the validity of variant design optimization model is out of the question. Therefore, in order to improve the efficiency of complex product variant design responding to customer requirement changes, a fundamental work is to establish a model, which is able to deal with the parts relationship and enhance the search efficiency.

2 Complex product variant design process

At present, the consensus of complex product variant design in the academic is that the new product feature is produced via parameter adjustment or module combination based on product platform [15 –17]. Generally, as shown in Fig. 1, a complex product contains several modules, and each module includes some parts [18]. Meanwhile, all features of a part should be played via the corresponding parameters adjustment. This kind of hierarchy causes two mainly way of variant design when customer requirements change [15]. One is called Scaled-based, whereby scaling variables are used to “stretch” or “shrink” the product platform in one or more dimensions to satisfy a variety of customer needs [19]. The other is referred to as module-based, which aims to develop a modular product platform, from which product variants are derived by configuring various modules [20]. Scaled-based is more convenient for the application of optimization techniques and takes an advantage for minor adjustments, while module-based can be converted into Scaled-based to a certain extent. This conversion relationship contributes to two different operation modes: Up-Bottom (UB) and Bottom-Up (BU) [21].

The Up-Bottom is able to ensure the size of new component meet the dimensional coordination and avoid conflicts between parts via referring existing components or products. This approach requires that the subsequent components should be modelled in the same computer assembly model. So it can’t call the modules and parts belonging to another product family. Consequently, design resources can’t be fully made use of.

The Bottom-Up generally models each part firstly, then combines them successively to form a product. This way allows team-allocation and calling of the other modules and parts, by which can enhance complex product modeling efficiency and design resource utilization. However, conflicts between parts are neglected, which is also lead to the low efficiency of variant design.

Therefore, in this paper, Up-Bottom and Bottom-Up are synthesized in complex product variant design, we called “UB-BU hybrid approach”, by which it can help improve the search efficiency and reduce conflicts. In other words, according to customer requirements changes, a series of modules with different similarity to variant design objective should be zoned firstly, by which it can reduce possibility of conflicts. Then, based on the high to low order of module similarity, some suitable parts belonging to each module will be searched in turn by “stretch” or “shrink” their dimensions to satisfy a certain feature, and repeatedly do it until all feature are achieved. By integrating the process of Scaled-based and Module-based, this approach complements each other’s advantages. Due to the feature hierarchy of modules and parts, the realization module function generally relies on its parts. That is to say, the part parameters and corresponding design variable in lower level determine the feature of module in upper level. Taken in this sense, the complex product variant design should be a bi-level programming problem, as shown in Fig. 2.

3 Joint bi-level programming model

Suppose that a complex contain I modules, and mod_i demotes the ith module, which consists of J parts, and part_ij indicates the jth part of mod_i. Besides, part_ij is made up of K parameters, and each parameter can be regarded as a design variable, which is denoted by para_ijk. According to the above analysis, the module similarity should be clear firstly and the module mod_i with the highest similarity to changes will be selected as a beginning. Then, via the adjustment of para_ijk, the different features of module and product will emerge. The similarity of mod_i to mod_i can be calculated by Equation (1) [17]. ${sim}_{ji} = \sum_{j = 1}^{J_{i}} (1 - | ω_{j} ℏ_{j} - ω_{i} λ_{i} |) / \sum_{j = 1}^{J_{i}} ω_{j} ω_{i}$ (1)

In Equation (1), ^mathchar′26mkern - 10muλ_i means the ith module feature value corresponding to customer requirements changes, and ω_i denotes the weight of ^mathchar′26mkern - 10muλ_i. ℏ_j means the jth module feature value, and and ω_j denotes the weight of ℏ_j.

3.1 Upper-level optimization

The Upper-lever optimization of complex product variant design is to make a decision for module combination to achieve the maximum customer satisfaction with minimum cost.

On the basis of the Roger Jiao’s view [15], customer satisfaction is often reflected in the product utility customer can directly perceived. Then, customer satisfaction, denoted by UC_c, can be regarded as the sum of all modules’ utility. Namely, UC_c = ∑uc_c (i), and uc_c (i) means the utility of mod_i. As for uc_c (i), it can be divided into the due date dt_c (i), the price pr_c (i) and design quality qu_c (i), thus ${uc}_{c} (i) = \partial_{1} \times {dt}_{c} (i) + \partial_{2} \times {pr}_{c} (i) + \partial_{3} \times {qu}_{c} (i)$ (2)where ∂₁, ∂₂ and ∂₃ means the weight of due date dt_c (i), the price pr_c (i) and design quality qu_c (i) respectively.

According to the Kreng Victor B’s research [17], dt_c (i), pr_c (i) and qu_c (i) can be determined by Equations (3–6). ${dt}_{c} (i) = max {{dt}_{c} (ij)}$ (3) ${pr}_{c} (i) = \sum_{j = 1}^{J} {pr}_{c} (ij)$ (4) ${pr}_{c} (ij) = \sum_{j = 1}^{J} (1 + Θ_{ij}) \times cos t_{c} (ij)$ (5) ${qu}_{c} (i) = \frac{1}{J} \times ∥ {part}_{ij} (⊥) ∥ \times (1 + σ_{ij})$ (6)

Here, cos t_c (ij) and Θ_ij means the cost and the weight of the part_ij. ∥part_ij (⊥) ∥ is the average number of part_ij in mod_i, and σ_ij means the statistical error.

For the enterprise, the invariably objective is to continuously reduce costs and to improve the customer satisfaction [22 –24]. Thus, the valiant design cost should be taken into consideration while the customer satisfaction enhanced. Until now, there are plenty of methods to assess cost. In this paper, the method proposed by Martin M.V. [18] is introduced. According to the design process, the variant design cost cost_c (i) can be reflected by the basic referred module cost cost_c,b (i), the material cost cost_c,m (i) and processing cost cost_c,p (i). Thus, ${cost}_{c} (i) = {cost}_{c, b} (i) + {cost}_{c, m} (i) + {cost}_{c, p} (i)$ (7)

Therefore, in the Upper-lever optimization of complex product variant design, the objective function can be formed as Equation (8). $F = max \sum_{i = 1}^{I} (\frac{{uc}_{c} (i)}{cos t_{c}^{*} (i)} \times x_{i})$ (8)

The upper-level optimization of complex product variant design is subjected to the select of module and part. In this paper, a binary variable x_i is introduced to indicate the choice decision for the mod_i, such that x_i = 1 means that mod_i is selected and x_i = 0 indicates that mod_i is not selected. Analogously, introduce another binary variable y_ij, such that y_ij = 1 indicates that part_ij contains attribute to be adjusted and y_ij = 0 means that part_ij does not exhibit the feature meeting customer requirements changes. It is worthy noting that in complex product variant design, at least one module or part meet the requirements of the change, then $1 \leq \sum_{i = 1}^{I} x_{i} \leq I, x_{i} \in {0, 1}$ (9) $1 \leq \sum_{j = 1}^{J} y_{ij} \leq J, y_{ij} \in {0, 1}$ (10)

Synthesizing the above analysis, the upper-level optimization of complex product variant design is formed as Equations (11–19). $F = max \sum_{i = 1}^{I} (\frac{{UC}_{c} (i)}{cos t_{c}^{*} (i)} \times x_{i})$ (11) ${UC}_{c} (i) = \partial_{1} \times {dt}_{c} (i) + \partial_{2} \times {pr}_{c} (i) + \partial_{3} \times {qu}_{c} (i)$ (12) ${dt}_{c} (i) = max {{dt}_{c} (ij)}$ (13) ${pr}_{c} (i) = \sum_{j = 1}^{J} {pr}_{c} (ij)$ (14) ${pr}_{c} (ij) = \sum_{j = 1}^{J} (1 + Θ_{ij}) \times cos t_{c} (ij)$ (15) ${qu}_{c} (i) = \frac{1}{J} \times ∥ {part}_{ij} (⊥) ∥ \times (1 + σ_{ij})$ (16) $1 \leq \sum_{i = 1}^{I} x_{i} \leq I$ (17) $1 \leq \sum_{j = 1}^{J} y_{ij} \leq J$ (18) $x_{i}, y_{ij} \in {0, 1}$ (19)

3.2 Lower-level optimization

In upper-level, the objective focuses on the customer satisfaction and cost. Based on Equations (3–6), it can determined that the module cost depend on the change scope of its parts. In other words, the optimal results of the upper-level are based on the lower-lever. Therefore, the lower-lever optimization of complex product variant design is committed to achieve the minimum cost in part parameters adjustment. Accordingly, the objective function of the lower-lever optimization can be constructed as $\begin{matrix} cos t_{i}^{*} (i) = min cos t (i) \\ = min ({cost}_{c, b} (i) + {cost}_{c, m} (i) + {cost}_{c, p} (i)) \end{matrix}$ (20)Due to $cos t_{c, b} (i) = \sum_{j = 1}^{J} cos t_{c, b} (ij) = \sum_{j = 1}^{J} \sum_{k = 1}^{K} cos t_{c, b} (ijk) \times z_{ijk}$ (21) $cos t_{c, m} (i) = \sum_{j = 1}^{J} cos t_{c, m} (ij) = \sum_{j = 1}^{J} \sum_{k = 1}^{K} cos t_{c, m} (ijk) \times z_{ijk}$ (22) $cos t_{c, p} (i) = \sum_{j = 1}^{J} cos t_{c, p} (ij) = \sum_{j = 1}^{J} \sum_{k = 1}^{K} cos t_{c, p} (ijk) \times z_{ijk}$ (23)

Then Equation (20) can be converted into Equation (24). $cos t_{i}^{*} (i) = min (\begin{matrix} \sum_{j = 1}^{J} \sum_{k = 1}^{K_{i}} cos t_{c, b} (ijk) \times z_{ijk} \\ + \sum_{j = 1}^{J} \sum_{k = 1}^{K_{i}} cos t_{c, m} (ijk) \times z_{ijk} \\ + \sum_{j = 1}^{J} \sum_{k = 1}^{K_{i}} cos t_{c, p} (ijk) \times z_{ijk} \end{matrix})$ (24)

In lower-level, we also make an assumption that there has to be at least one way of scaling or stretching the parameters. Then, introduce a binary variable z_ijk indicating the choice decision for the para_ijk, such that z_ijk = 1 means that para_ijk is adjusted and z_ijk = 0 indicates para_ijk is not adjusted. $1 \leq \sum_{k = 1}^{K} z_{ijk} \leq K_{i}$ (25)

Besides, during the adjustment, some basic principles should be followed, such as time requirements Equation (26), cost constraint Equation (27), parts mate precision Equation (28) and technical characteristic Equation (29) and so on. $max {dt (ijk)} \leq {dt}_{c} (ij)$ (26) $\sum_{k = 1}^{K} cos t_{c} (ijk) \leq C_{c}$ (27) $\sum_{k = 1}^{K} h_{2} (ijk) \times z_{ijk} \geq H_{2}$ (28) $\sum_{k = 1}^{K} h_{1} (ijk) \times z_{ijk} \leq H_{1}$ (29)

Then, the lower-level optimization of complex product variant design is given as the following. $cos t_{i}^{*} (i) = min (\begin{matrix} \sum_{j = 1}^{J} \sum_{k = 1}^{K_{i}} cos t_{c, b} (ijk) \times z_{ijk} \\ + \sum_{j = 1}^{J} \sum_{k = 1}^{K_{i}} cos t_{c, m} (ijk) \times z_{ijk} \\ + \sum_{j = 1}^{J} \sum_{k = 1}^{K_{i}} cos t_{c, p} (ijk) \times z_{ijk} \end{matrix})$ (30) $1 \leq \sum_{k = 1}^{K} z_{ijk} \leq K_{i}$ (31) $max {dt (ijk)} \leq {dt}_{c} (ij)$ (32) $\sum_{k = 1}^{K} cos t_{c} (ijk) \leq C_{c}$ (33) $\sum_{k = 1}^{K} h_{2} (ijk) \times z_{ijk} \geq H_{2}$ (34) $\sum_{k = 1}^{K} h_{1} (ijk) \times z_{ijk} \leq H_{1}$ (35)

3.3 Bi-level optimization model

Compiling Equations (11–19) and (30–35), we obtain the general form of joint complex product variant design bi-level optimization, as the following. $F = max \sum_{i = 1}^{I} (\frac{{UC}_{c} (i)}{cos t_{c}^{*} (i)} \times x_{i})$ (36) ${UC}_{c} (i) = \partial_{1} \times {dt}_{c} (i) + \partial_{2} \times {pr}_{c} (i) + \partial_{3} \times {qu}_{c} (i)$ (37) ${dt}_{c} (i) = max {{dt}_{c} (ij)}$ (38) ${pr}_{c} (i) = \sum_{j = 1}^{J} {pr}_{c} (ij)$ (39) ${pr}_{c} (ij) = \sum_{j = 1}^{J} (1 + Θ_{ij}) \times cos t_{c} (ij)$ (40) ${qu}_{c} (i) = \frac{1}{J} \times ∥ {part}_{ij} (⊥) ∥ \times (1 + σ_{ij})$ (41) $1 \leq \sum_{i = 1}^{I} x_{i} \leq I$ (42) $1 \leq \sum_{j = 1}^{J} y_{ij} \leq J$ (43) $x_{i}, y_{ij} \in {0, 1}$ (44) $cos t_{i}^{*} (i) = min (\begin{matrix} \sum_{j = 1}^{J} \sum_{k = 1}^{K_{i}} cos t_{c, b} (ijk) \times z_{ijk} \\ + \sum_{j = 1}^{J} \sum_{k = 1}^{K_{i}} cos t_{c, m} (ijk) \times z_{ijk} \\ + \sum_{j = 1}^{J} \sum_{k = 1}^{K_{i}} cos t_{c, p} (ijk) \times z_{ijk} \end{matrix})$ (45) $1 \leq \sum_{k = 1}^{K} z_{ijk} \leq K_{i}$ (46) $max {dt (ijk)} \leq {dt}_{c} (ij)$ (47) $\sum_{k = 1}^{K} cos t_{c} (ijk) \leq C_{c}$ (48) $\sum_{k = 1}^{K} h_{2} (ijk) \times z_{ijk} \geq H_{2}$ (49) $\sum_{k = 1}^{K} h_{1} (ijk) \times z_{ijk} \leq H_{1}$ (50)

4 Model solution

Due to the hierarchy of master-slave hierarchical structure, the joint optimization model of complex product variant design responding to customer requirement changes is actually a bi-level optimization problem. The difficulty of solution increases sharply with the increase of the problem’s complexity, especially for the situation that the objective function is nonlinear and non-differentiable or the constraints are non-convex [25]. Then it is necessary to develop a suitable heuristic algorithm. Du and Jiao proposed a solution based on the Stackelberg game and analyzed the feasibility of heuristic algorithm, such as genetic algorithm, simulated annealing algorithm and ant colony algorithm, etc [15]. In light of the joint optimization feature of complex product variant design, in this paper, the genetic algorithm embedded double iteration comparison rules (GA-DICR) is developed based on canonical genetic algorithm, which has been proved to have obvious advantages for the optimization of the large scale complex nonlinear problem.

The basic idea of the GA-DICR is following: Randomly generate N points (z_ijk, k = 1, 2, ⋯ , N) with uniform distribution in lower-level, and calculate the solution f (z_ijk) of each z_ijk, then an initial population pop⁰ including (z_ijk, f (z_ijk)) is formed and the population size is N. Calculate the fitness value of each individual in the initial population, then we can obtain an optimal value $z_{ijk}^{0}$ . Let $z_{ijk}^{0}$ be an initial solution of upper-level, then the decision variable ${uc}_{c}^{0}$ and the corresponding objective function value $F ({uc}_{c}^{0})$ can be calculated. Then put $F ({uc}_{c}^{0})$ into the lower-level, a new solution of lower-level $f (z_{ijk}^{0^{'}})$ can be obtained. At this point, compare $f (z_{ijk}^{0^{'}})$ with f (z_ijk), if they are equal, they are a feasible solution. According to the above steps, a set of feasible solutions can be obtained through iterative calculated. Then the maximum is the optimal solution.

The flow chart of GA-DICR is as shown in Fig. 3.

4.1 Chromosome coding

Binary encoding method is adopted in this paper for chromosome coding. The length of each chromosome is the number of parameters involved in each part, ℓ = K_i. Each chromosome can be divided into n gene segments, and a gene segments denotes a parameter, then the length of each gene segments is k_i. If the value is 1, the corresponding parameter is adjusted. Otherwise, the corresponding parameter is not adjusted when the value is 0. As shown in Fig. 4, it is an example of a chromosome. The coding scheme illustrates that the third parameter of part₁ and the last parameter of part_J are adjusted.

4.2 Initial population

Randomly generate N points (z_ijk, k = 1, 2, ⋯ , N) with uniform distribution in lower-level, and calculate the solution f (z_ijk) of each z_ijk, then an initial population including (z_ijk, f (z_ijk)) is formed.

4.3 Fitness function

In bi-level programming problem, there are some iteration relation between upper-level and lower-level. Thus, the fitness function should take overall consideration of upper-level and lower-level, especially the uniform dimension. In this paper, the feasible solution is obtained only when the result from lower-upper is equal to the result feed backed from upper-level. Then, the fitness function is formed based on the uniform dimension, as shown in Equation (51). $fit (i) = δ_{1} • \frac{F (i) - F_{min}}{F_{max}} / δ_{2} • \frac{f (i) - f_{min}}{f_{max}}$ (51)

Here, F (i) and f (i) denotes, respectively, the individual utility value and cost value after the uniform dimension. F_max, F_min, f_max and f_min mean the corresponding maximum one and the minimum one. δ₁, δ₂ is the weight of upper-level and lower-level, and δ₁ + δ₂ = 1.

4.4 Selecting operation

Based on the regular geometric sorting method, the individual with greatest value is selected firstly. Then, the selection probability of each individual is ${prob}_{s} (i) = {prob}_{s}^{'} (1 - {prob}_{s})^{α - 1}$ (52)

Here, prob_s is the selection probability of the optimal individual. α denotes the sequence number of individual. ${prob}_{s}^{'} = \frac{{prob}_{s}}{1 - (1 - {prob}_{s})^{N}}$

4.5 Crossover operation

This study adopts self-adaptive double point crossover to increase the population diversity and to prevent the operation of algorithm premature and stagnation. As shown in Fig. 5.

4.6 Mutation

The mutation mechanism randomly selects a gene according to the mutation rate and alters its value.

4.7 Termination conditions

In joint complex product variant design bi-level optimization, the optimal solution cannot be obtained in advance, therefore a maximum evolution algebra should be given as a termination conditions. Based on Ref. [25], define the termination function $u_{F} = | \frac{F_{n + 1} - F_{n}}{F_{n}} |$ , and $u_{f} = | \frac{f_{n + 1} - f_{n}}{f_{n}} |$ . If max(u_F, u_f) ≤ τ, then the algorithm is convergent. Here, τ means the iteration accuracy.

5 Case study

Clutch, as shown in Fig. 7, installed between the engine and the transmission, is an assembly directly linked to the engine in auto transmission system. Clutch offers a role that makes the engine and transmission joint gradually, so as to ensure automobile start smoothly. Moreover, clutch temporarily cut off the connection between the engine and the transmission to reduce the shift impact. In addition, when vehicle make an emergency braking, clutch prevent the transmission system overload by separation operation, thus a protective role is played. Clutch is characterized by a high precision and makes the parts interact with each other closely, so a customer requirements change most probably results in large-scale changes propagation. Therefore, the rapid variant design is important for reducing the influence.

In clutch design process, due to the adjustment of budget, customer requests reduce some performance of the clutch. Wherein, the clutch friction plate thermal conductivity is claim for up to 45 W · m^-1 · °C^-1 from 38 W · m^-1 · °C^-1. After a preliminary analysis, these changes are mainly concentrated on the friction plate, which is riveted on the driven plate. Meanwhile, product family size for clutch is 33.

According to the method proposed in this paper, the similarity should be analyzed firstly. Based on Equation (1), the similarity sorting is shown in Table 1.

Then, based on the Section. 3, the joint complex product variant design bi-level optimization can be constructed as following. $F = max \sum_{i = 1}^{33} (\frac{{UC}_{c} (i)}{cos t_{c}^{*} (i)} \times x_{i})$ ${UC}_{c} (i) = \partial_{1} \times {dt}_{c} (i) + \partial_{2} \times {pr}_{c} (i) + \partial_{3} \times {qu}_{c} (i)$ ${dt}_{c} (i) = max {{dt}_{c} (ij)}$ ${pr}_{c} (i) = \sum_{j = 1}^{51} {pr}_{c} (ij)$ ${pr}_{c} (ij) = \sum_{j = 1}^{51} (1 + Θ_{ij}) \times cos t_{c} (ij)$ ${qu}_{c} (i) = \frac{1}{51} \times ∥ {part}_{ij} (⊥) ∥ \times (1 + σ_{ij})$ $1 \leq \sum_{i = 1}^{33} x_{i} \leq 51$ $1 \leq \sum_{j = 1}^{51} y_{ij} \leq 51$ $x_{i}, y_{ij} \in {0, 1}$ $f = cos t_{i}^{*} (i) = min (\begin{matrix} \sum_{j = 1}^{51} \sum_{k = 1}^{K_{i}} cos t_{c, b} (ijk) \times z_{ijk} \\ + \sum_{j = 1}^{51} \sum_{k = 1}^{K_{i}} cos t_{c, m} (ijk) \times z_{ijk} \\ + \sum_{j = 1}^{51} \sum_{k = 1}^{K_{i}} cos t_{c, p} (ijk) \times z_{ijk} \end{matrix})$ $1 \leq \sum_{k = 1}^{K_{i}} z_{ijk} \leq K_{i}$ $\sum_{k = 1}^{51} cos t_{c} (ijk) \leq 700$ $cos t_{c, b} (ijk) = 0.01 \times \frac{v \times C_{0}}{Z \times F}$ $cos t_{c, m} (ijk) = z_{ijk} \times {pr}_{c} (ij)$ $0 \leq cos t_{c} (ijk) \leq 0.05$ $max {dt (ijk)} \leq {dt}_{c} (ij)$ $M_{c, max} = M_{g, max} \times δ$ $Z = \frac{3.82 δ M_{c, \max}}{D_{o} (1 - C_{0}) μ [P_{0}]}$ $Z \leq \frac{50 H}{α_{c} [Δ T]} [A]$ $α_{c} = 12.8 + 2.9 ν$ $v = \frac{π n_{e} \times 10^{- 3}}{60} \leq 65$ $L = \frac{π^{2} n_{c}^{2}}{1800} \frac{Gr}{i_{0} i_{1}}$ $F = \frac{π}{4} D^{2} (1 - C_{0}^{2})$ $0.52 \leq C_{0} \leq 0.70$ $\frac{N_{e}}{Z \times F} \leq [N]$ $z_{ijk} \in {0, 1}$

For this bi-level optimization, GA-DICR is conducted. Set the maximum iterations 400, and the crossover probability prob_c = 0.8, mutation probability prob_m = 0.1, iteration precision τ = 0.0001. Besides, the pop⁰ = 100 after analysis. Based on the above assuptions, run the GA- DICR under the circumstance of Windows 7, Intel i5-3470 3.20GH and Ram 4G. The objective function value running result is shown in Fig. 8. It is after 145 generations of evolution, and takes 20.99 s. The optimal chromosome coding scheme is 00110011001000100. The best fitness value is 0.9994, and the optimal function value is 27.16, as shown in Table 2.

To verify the validity of the algorithm, an comparative analysis of algorithm proposed in Refs. [25] and [26] is conducted. The results is shown in Fig. 9. And we can conclude that the GA-DICR has certain advantages both in rate of convergence and computational accuracy.

6 Conclusions

This paper focuses on the optimization of complex product variant design responding to customer requirement changes. A UB - BU hybrid model for complex product variant design process was developed firstly. The UB - BU hybrid model can help improve the search efficiency and reduce conflicts. Then£a joint bi-level programming optimization of complex product variant design is proposed. The upper-level is to achieve the maximum customer satisfaction and the lower-level is to achieve the minimum cost. As a solution, the genetic algorithm embedded double iteration comparison rules (GA-DICR) is developed based on canonical genetic algorithm. The method and model are proved to be effective in the Clutch variant design. It will contribute to shorten design cycle, reduce development cost and enhance customer’s satisfaction. Moreover, it should be pointed out that complex product variant design is a complicated problem in the real world. There are many kinds of parts, according to the different type of changed parts, different principle and method should be adopted. Therefore, the most appropriate method for different kinds of changed parts need to be investigated with regard to the real-world situations.

Footnotes

Acknowledgments

This research is sponsored by National Natural Science Foundation of China under Project number 71301176. The authors also thank to Fundamental Research Funds for the Central Universities (Project No. CDJZR12110004) and Specialized Research Fund for the Doctoral Program of Higher Education (Project 20130191120001) for partial support of this work. We are grateful for the constructive suggestions provided by the reviewers, which would improve the paper.

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