An integrated modular design methodology based on maintenance performance consideration

Abstract

Maintenance nowadays not only plays a crucial role in the usage phase, but is fast becoming the primary focus of the design stage—especially with general increased emphasis on product service. The modularization of maintenance has been explored rarely by previous researchers, despite its significant potential benefit. Existing modular design methods on life cycle do not sufficiently improve maintenance performance as a whole. In effort to remedy this, this article considers relevant maintenance issues at early stages of product development and presents a novel modular methodology based on the simultaneous consideration of maintenance and modularity characteristics. The proposed method first employs the design structure matrix to analyze the comprehensive correlation among components. Next, based on graph theory, initial modules with high cohesion and low coupling are generated. After that, a maintenance performance multi-objective model is established for further optimization to minimize maintenance costs, minimize differences in the maintenance cycle, and maximize system availability. To conclude, an improved strength Pareto evolutionary algorithm 2 is used for modular optimization. The complete methodology is demonstrated using a case study with a hydraulic press, where results reveal that the optimized modules can reduce maintenance cost under the premise of approximately constant modular performance.

Introduction

In recent years, alongside the increasing emphasis on users’ comfort and experience of product quality, services are no longer a peripheral consideration within a business environment, but are more often seen as core customer offerings. ¹ Changes in customer demand plus fierce market competition force enterprises to offer increasingly innovative services and improve the quality of the existing services. To this effect, the importance of adopting new strategies and building servitization-driven design approaches has become widely recognized in the industry.^2,3 For manufacturing enterprises, maintenance performance is regarded as a vital criterion to evaluate service quality. Exceptional maintenance performance can speed up the assembly and disassembly processes of a system ⁴ and reduce the post-sales maintenance cost. It is thus essential to equip designers with the skills and tools to integrate maintenance performance into the early stage of a design.

During the product design process, ⁵ modular design is one of the most common techniques requiring consideration. Modular design has been the focus of many fields, such as mass customization, ⁶ product family design, ⁷ and green design. ⁸ Designing specifically for product modularity has proven very effective to enhance responsiveness and to cut costs in the automotive industry. ⁹ In the existing research on modular design, proponents of design science have explored modularity from many different perspectives, including manufacturing, ¹⁰ recycling, ¹¹ disassembly, ¹² and reuse. ¹³

Gu and Sosale ¹⁴ presented an integrated modular design methodology for life cycle engineering. Chung et al. ¹⁵ adopted the life cycle assessment of a product to identify the most beneficial modular structure. Agard and Bassetto ¹⁶ proposed a single-level modular design formulation considering quality and cost simultaneously to meet customer requirements. Hwang and Choi ¹⁷ proffered a modular design methodology to maximize customer satisfaction. Meng et al. ¹⁸ proposed a new module identification method taking manufacturing efficiency into account in product development. Hong and Park ¹⁹ demonstrated a new modular design approach to overcome the difficulty based on physical and functional relationships. All these studies, however, failed to adequately consider increased demand for maintenance. So far, only a small number of studies have focused their efforts on maintenance-oriented modular design. Tsai et al. ⁴ presented a method of modularity based on maintenance policy; however, this work mainly focused on maintenance cost and did not fully consider maintenance performance factors such as availability or complexity. Furthermore, the methods they applied required manual operation and were too time-consuming for complex products. Joo ²⁰ developed a dynamic approach for scheduling preventive maintenance for modular designed components, but the approach centered around maintenance scheduling rather than module identification. There is clearly a gap in the availability of modular design for such a crucial juncture of product design. This article expands the concept of modularity into a maintenance perspective.

To facilitate quicker and smarter decisions for module identification, a number of methods have been applied extensively to modular design, including design structure matrix (DSM), ²¹ group genetic algorithm, ²² and fuzzy C-means. ²³ Kusiak and Huang ²⁴ developed a methodology for determining modular products via a graphical representation and fuzzy neural network. Beek et al. ²¹ developed a modularization scheme based on a functional model. Cheng et al. ²⁵ proposed a new approach for product modularization based on axiomatic design and DSM. These methods clustered components into an independent module according to their relationship, but the partition schemes obtained are not optimized and the calculation amount they require is rather large. Moon et al. ²⁶ introduced a multi-objective particle swarm optimization approach to select the best module identification strategy. Kreng and Lee ²⁷ used a case study to illustrate the capability of group genetic algorithms for modular design. Kamrani and Gonzalez ²⁸ proposed a genetic algorithm solution methodology to solve modular design problem. To use these methods, the initial values for the clustering number of a product must be established before running the process.

There are considerable difficulties posed by modular design for maintenance consideration. Modules must be easily configured without sacrificing maintenance performance. Modularity and maintenance performance should be considered simultaneously in module identification. Another difficulty in modular design is in the mathematical model of maintenance performance and its solution. The module identification method must thoroughly account for maintenance.

To overcome the above limitations, this article develops a new methodology to solve the modular design problem with simultaneous consideration of maintenance and modularity performance. A module identification approach–based DSM and graph theory is first presented, which is able to obtain initial modules with excellent modularity performance. Next, the mathematic formulation of maintenance performance is proposed, which translates a maintenance-oriented module identification into a multi-objective combinatorial optimization problem. Finally, the strength Pareto evolutionary algorithm (SPEA2) and fuzzy set theory are adopted to identify the optimal modular scheme of a product. The specific focus of this research is to enhance maintenance performance while maintaining excellent modularity performance, with the aim of drawing designers’ attention to the maintenance problem that exists at the design stage.

The remainder of this article is organized as follows. Section “Initial module identification” presents an integrated module identification approach–based DSM and graph theory. Section “Optimization of initial module for maintenance performance” establishes the mathematical model for maintenance performance and states the solution method based on SPEA2 and fuzzy set theory. Section “Case study” provides a case study to verify the effectiveness of the proposed method. Finally, section “Conclusion and future work” concludes our work and describes our plans for future research.

Initial module identification

Interaction analysis of components

The strength of the coupling relationship among components can be determined quantitatively by interaction analysis of components. In order to obtain the precise relationship between various components, previous researchers categorized component relationships into three ways, from the perspectives of lifecycles, namely, functional correlation, geometrical correlation, and physical correlation. ²⁹

Functional correlation refers to the interaction among different components when they jointly achieve a certain function. Geometrical correlation represents spatial or geometrical relationships among components as physical connections, fastening, size, verticality, parallelism, concentricity, and other factors. Physical correlation indicates the exchange or transmission of material flow, energy flow, and signal flow among components.

DSM construction

DSM was initially proposed by Steward ³⁰ to indirectly present the parameter dependencies in complex design by means of matrix. Through relevant algorithms such as partition, tearing, banding, clustering, simulation, and eigenvalue analysis, it models and analyzes correlation between parameters and structures. To effectively consider the maintenance performance of modules within product development, the structural information must be available for module identification. Thus, component-based DSM is applied to model and describe the correlation among components of a product. Modeling analysis of a product is conducted by investigating the interaction between components. Product structure is expressed as a liaison graph, as shown in Figure 1(a), and a basic DSM documenting the relationship between components is shown in Figure 1(b). Based on the characteristics of component correlation, a numeric DSM depicts the degree of connection using values such as 0, 0.2, 0.4, 0.6, 0.8, and 1.0, demonstrated in Table 1.

Figure 1.

(a) A liaison graph. (b) Basic DSM.

Table 1.

Standard relationship for two components.

Type of relationship	Interaction value	Description
No	0.0	No relationship at all
Weak	0.2	Loose connection and medium relationship
Medium	0.4	Medium connection and medium relationship
Medium strong	0.6	Medium connection and high relationship
Strong	0.8	Firm connection and high relationship
Very strong	1.0	Impartibility

Define the comprehensive connected degree between any two components as $R_{\mathrm{ij}}$ , which is the weighted average of the degrees of two components i and j. The formula is as follows

$$R_{\mathrm{ij}}={\displaystyle \sum _{n=1}^N}{w}_n{T}_n(i,j),i\ne j$$

(1)

where $T_n(i,j)$ is the strength of the nth connected factor on component i to component j which is assessed by designers based on the criteria in Table 1. $w_n$ represents the weight of the nth connected factor, $0\le w_n\le 1$ , and ${\displaystyle \sum _{n=1}^N}{w}_n=1$ .

Initial module identification based on graph theory

Products can be seen as the directed graph G, which is composed of components and their relationship with each other. Thus, G’s adjacency matrix is actually a transposed matrix of DSM. Suppose a product has Q components, which can be labeled as

$$X=(x_1,x_2,\dots ,x_Q)$$

The procedure of initial module identification is as follows.

Step 1: tearing of weak correlation

To reduce the complexity of clustering and avoid undue influence of weak correlation on the results, designers can consider weak correlation as an independent module and then tear it (where the correlation strength of a weak correlation is temporarily reduced to 0). Choose threshold value ³¹ $\lambda =0.4$

$$a_{\mathrm{ij}}=\{\begin{array}{c}\hfill 0R_{\mathrm{ij}}\le \lambda \hfill \\ \multicolumn{1}{c}{1R_{\mathrm{ij}}>\lambda }\end{array}$$

(2)

Transform DSM into a Boolean matrix ${\mathbf{A}}_1=[a_{\mathrm{ij}}]$ . If $a_{\mathrm{ij}}=1$ , components i and j have a direct influence on each other. Otherwise, if $a_{\mathrm{ij}}=0$ , there is no direct influence between them. Figure 2(a) is the Boolean DSM of an example including seven components and represents the relationships between components accordingly.

Figure 2.

Step-by-step process of the production of the initial module: (a) DSM with seven components, (b) adjacency matrix A of DSM, (c) reachability matrix P, (d) output matrix P·P^T, and (e) retrieval of clusters.

Step 2: generate a reachability matrix

Transpose matrix ${\mathbf{A}}_1$ to obtain G’s adjacency matrix A. Figure 2(b) displays the adjacency matrix from DSM. Suppose $\mathrm{G}=(\mathrm{V},\mathrm{E})$ is a directed graph, P is G’s reachability matrix, and

$$P=A^{(1)}\cup A^{(2)}\cup A^{(3)}\cup \cdots \cup A^{(n)}={\displaystyle \bigcup _{i=1}^n}{A}^{(i)}$$

(3)

where A is G’s adjacency matrix and ${\mathbf{A}}^{\mathbf{(}\mathbf{n}\mathbf{)}}$ is A’s n-potent matrix.

The reachability matrix P can be derived from matrix A using equation (3) as shown in Figure 2(c), in which an entry $a_{\mathrm{ij}}=1$ if component j can be reached by i. In graph theory, reachability is the ability to move from one vertex in a directed graph to another vertex. In product structure, this indicates whether there is a direct or indirect influence relationship between a component and other components.

Step 3: identification of strong connected set

Suppose $\mathbf{P}={(p_{\mathrm{ij}})}_{n\times n}$ is G’s reachability matrix and ${\mathbf{P}}^{\mathbf{T}}$ is the transposition of $\mathbf{P}$ . Define the matrix operation as

$$\begin{array}{cc}\multicolumn{1}{c}{\mathbf{P}\cap {\mathbf{P}}^{\mathbf{T}}}& =\left[\begin{array}{ccc}\hfill p_{11}\hfill & \hfill \cdots \hfill & \hfill p_{1n}\hfill \\ \multicolumn{1}{c}{⋮}& \hfill \ddots \hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{p_{n1}}& \hfill \cdots \hfill & \hfill p_{\mathrm{nn}}\hfill \end{array}\right]\cap \left[\begin{array}{ccc}\hfill p_{11}\hfill & \hfill \cdots \hfill & \hfill p_{n1}\hfill \\ \multicolumn{1}{c}{⋮}& \hfill \ddots \hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{p_{1n}}& \hfill \cdots \hfill & \hfill p_{\mathrm{nn}}\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{}& =\left[\begin{array}{ccc}\hfill p_{11}·p_{11}\hfill & \hfill \cdots \hfill & \hfill p_{1n}·p_{n1}\hfill \\ \multicolumn{1}{c}{⋮}& \hfill \ddots \hfill & \hfill ⋮\hfill \\ \multicolumn{1}{c}{p_{n1}·p_{1n}}& \hfill \cdots \hfill & \hfill p_{\mathrm{nn}}·p_{\mathrm{nn}}\hfill \end{array}\right]\hfill \end{array}$$

(4)

The reachability matrix P multiplies the transposed matrix ${\mathbf{P}}^{\mathbf{T}}$ . Figure 2(d) displays the output matrix of $\mathbf{P}\cap {\mathbf{P}}^{\mathbf{T}}$ , in which clusters of components can be identified easily by rearranging their component order. An entry $p_{\mathrm{ij}}{p}_{\mathrm{ji}}=1$ if components i and j mutually interact. A similar algorithm has been applied successfully to graph theory. Strong connection shows the interaction among components. By identifying a strongly connected set, interaction between modules can be decompounded; thus, the initial module is ready to be fully realized.

Step 4: initial module identification

Module combination is obtained based on the serial number of components in each strongly connected subset. Through this process, the initial module partition is acquired. Trace back to the element correlations in the reachability matrix and rearrange the component order to generate a clustered matrix. Figure 2(e) shows clusters in the system, which represents initial modules.

Evaluation of modular performance

MSI ³² is introduced to evaluate the modular performance of the initial module obtained by the proposed method. The focus of MSI function is to identify modules that possess a maximum number of internal dependencies and a minimum number of external dependencies, as shown in Figure 3. The value of the internal connection of the module is denoted by MSI_i and the external connections by MSI_e

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathrm{MSI}}_i}& =\frac{{\displaystyle \sum _{i=n_1}^{n_2}}{\displaystyle \sum _{j=n_2}^{n_2}}{R}_{\mathrm{ij}}}{{(n_2-n_1)}^2-(n_2-n_1)}\hfill \\ \multicolumn{1}{c}{{\mathrm{MSI}}_e}& =\frac{{\displaystyle \sum _{i=0}^{n_1}}{\displaystyle \sum _{j=n_1}^{n_2}}(R_{i,j}+R_{j,i})}{2\times n_1\times (n_2-n_1)}\hfill \\ \multicolumn{1}{c}{}& +\frac{{\displaystyle \sum _{i=n_2}^N}{\displaystyle \sum _{j=n_1}^{n_2}}(R_{i,j}+R_{j,i})}{2\times ((N-n_2)\times (n_2-n_1))}\hfill \\ \multicolumn{1}{c}{\mathrm{MSI}}& ={\mathrm{MSI}}_i-{\mathrm{MSI}}_e\hfill \\ \multicolumn{1}{c}{{\mathrm{MSI}}_{\mathrm{DSM}}}& ={\displaystyle \sum _{k=1}^m}({\mathrm{MSI}}_{i,k}-{\mathrm{MSI}}_{e,k})\hfill \end{array}$$

(5)

Figure 3.

Internal and external dependencies of modules.

where n₁ is the index of the first component in the module, and n₂ is the index of the last component in the module. MSI_DSM is the strength summation of all modules in the clustered DSM. The larger the MSI_DSM, the better the possible scheme that can be acquired. Compute the modular performance of the initial module, denoted as MSI_DSM-intial. This is regarded as one of the constraint indexes for the subsequent optimization model, which can ensure the quality of modular performance after optimization.

The above process analyzes the correlation among components. Its primary purpose is to create high cohesion and low coupling characteristics within the module. However, it lacks adequate consideration of maintenance property in module design. It becomes thus necessary to optimize the initial model and establish an optimized mathematical model that includes maintenance performance.

Optimization of initial module for maintenance performance

Establishment of optimized mathematical model

Maintenance performance is the immanent characteristic of a product and creates a significant impact on the quality of service. The main goal of this section is to effectively delineate maintenance performance–oriented modular design. The established mathematical model must simultaneously and successfully account for the performance of both modularity and maintenance.

The key factors that manufacturers must consider as far as maintenance performance include lower maintenance costs and complexity, and higher system availability. Hence, maintenance performance–oriented modular design can be regarded as a multi-objective optimization problem, targeted at maintenance cycle, maintenance cost, and system availability. Suppose a product has $Q$ components and is decomposed into $m$ initial modules $\{M_1,M_2,\dots ,M_m\}$ . Where module $k$ has $D_n$ components, suppose $X={[x_{\mathrm{ik}}]}_{Q\times m}$ , and

$$x_{\mathrm{ik}}=\{\begin{array}{c}\hfill 1,i\in k\mathrm{module}\hfill \\ \multicolumn{1}{c}{0,\mathrm{otherwise}}\end{array}$$

(6)

Maintenance cycle–oriented modular design model

For an effective maintenance cycle–oriented modular design model, the components within a module should have approximately the same maintenance cycle. In other words, the smaller the difference in maintenance time intervals, the better. This helps to ensure that all components in a module can be maintained or replaced by just one maintenance process, effectively reducing the cost and complexity and improving the availability of products. According to statistical analysis theories, this article employs the difference among components’ maintenance cycles to measure their conformity. Define $E_i$ as the minimum maintenance cycle of component ${\mathrm{PT}}_i$ , then the maintenance cycle difference, denoted as $S_{\mathrm{kg}}$ , among components within the kth module can be represented as

$$S_{\mathrm{kg}}=\sqrt{\frac{1}{D_n}{\displaystyle \sum _{i=1}^Q}\left[{(E_i-{\overline{E}}_i)}^2{x}_{\mathrm{ik}}\right]}$$

(7)

where ${\overline{E}}_i=(1/D_n){\displaystyle \sum _{i=1}^Q}{E}_i{x}_{\mathrm{ik}}$

Therefore, the modular maintenance cycle difference of an overall unit is

$$F_M=\frac{1}{m}{\displaystyle \sum _{n=1}^m}(1-\frac{S_{\mathrm{kg}}}{S_{max}})$$

(8)

Maintenance cost–oriented modular design model

Total maintenance cost of complex mechanical products mainly consists of overhaul cost, replacement cost, and inspection cost. Overhaul cost refers to the repair of damaged components within their valid working life, replacement cost involves replacement of out-of-service components with all-new ones, and inspection cost is used for daily component maintenance such as inspection and detection.

1. The overhaul cost of the kth module can be calculated as

   $$\begin{array}{cc}\multicolumn{1}{c}{C_{M,k}=}& {\displaystyle \sum _{i=1}^M}({\displaystyle \sum _{j=1}^J}{c}_{\mathrm{Oi},j}·h_{i,j})·x_{\mathrm{ik}}\hfill \\ \multicolumn{1}{c}{}& +{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^M}{c}_d·{\mathrm{np}}_{\mathrm{ij}}·(x_{\mathrm{ik}}-x_{\mathrm{jk}})\hfill \end{array}$$

   (9)

   where $c_{\mathrm{Oi},j}$ represents the unit cost to overhaul the component ${\mathrm{PT}}_i$ during the time period $j$ , $c_d$ represents the unit cost of disassembling the interfaces between modules, and $h_{i,j}$ is the overhaul factor of component ${\mathrm{PT}}_i$ . If disassembling is done during time period $j$ , $h_{i,j}=1$ ; otherwise, $h_{i,j}=0$ . ${\mathrm{np}}_{\mathrm{ij}}$ is the number of interfaces between component ${\mathrm{PT}}_i$ and component ${\mathrm{PT}}_j$ of a complex mechanical product.

2. The replacement cost of the kth module

   $$\begin{array}{cc}\multicolumn{1}{c}{C_{R,k}}& =C_{P,k}+C_{D,k}-C_{r,k}\hfill \\ \multicolumn{1}{c}{}& ={\displaystyle \sum _{i=1}^M}({\displaystyle \sum _{j=1}^J}{c}_{\mathrm{Ri}}·r_{i,j})·x_{\mathrm{ik}}\hfill \\ \multicolumn{1}{c}{}& +{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^M}{c}_d·{\mathrm{np}}_{\mathrm{ij}}·(x_{\mathrm{ik}}-x_{\mathrm{jk}})\hfill \\ \multicolumn{1}{c}{}& -{\displaystyle \sum _{i=1}^M}({\displaystyle \sum _{j=1}^J}{w}_i·{\delta }_i·p_i·r_{i,j})·x_{\mathrm{ik}}\hfill \end{array}$$

   (10)

   where $c_{\mathrm{Ri}}$ represents the unit cost to replace the component ${\mathrm{PT}}_i$ during the time period $j$ , and $r_{i,j}$ is the replacement factor of the key component ${\mathrm{PT}}_i$ . If ${\mathrm{PT}}_i$ is replaced during time period $j$ , $r_{i,j}=1$ ; otherwise, $r_{i,j}=0$ . $w_i$ is the average weight of component ${\mathrm{PT}}_i$ , ${\delta }_i$ is the reclamation rate of the reclaimable materials of component ${\mathrm{PT}}_i$ , and $p_i$ is the unit price of the reclaimable materials of component ${\mathrm{PT}}_i$ .

3. The inspection cost of the kth module

   $$C_{I,k}={\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^J}{c}_{\mathrm{Ii}}·(1-h_{i,j}-r_{i,j})·x_{\mathrm{ik}}$$

   (11)

   where $c_{\mathrm{Ii}}$ represents the unit cost to inspect the component ${\mathrm{PT}}_i$ .

The total cost to maintain the parts on module $k$ can then be calculated as follows

$$\begin{array}{cc}\multicolumn{1}{c}{F_c}& ={\displaystyle \sum _{k=1}^m}(C_{M,k}+C_{R,k}+C_{I,k})\hfill \\ \multicolumn{1}{c}{}& ={\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^J}\{c_{\mathrm{fi}}{\eta }_i^{-{\beta }_i}[{\left(t_{\mathrm{w}i,j}^{+}\right)}^{{\beta }_i}-{\left(t_{\mathrm{w}i,j}^{-}\right)}^{{\beta }_i}]+c_{\mathrm{r}i}{r}_{i,j}-w_i·{\delta }_i·p_i·r_{i,j}\}·x_{\mathrm{ik}}\hfill \\ \multicolumn{1}{c}{}& +{\displaystyle \sum _{j=1}^J}\left(c_{\mathrm{d}}\{1-{\displaystyle \underset{i=1}{\overset{N}{\Pi }}}[1-(h_{i,j}+r_{i,j})]\}\right)·x_{\mathrm{ik}}+2{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^M}{c}_d·{\mathrm{np}}_{\mathrm{ij}}·(x_{\mathrm{ik}}-x_{\mathrm{jk}})\hfill \end{array}(12)$$

(12)

where $c_{\mathrm{fi}}$ is the unit failure costs of the key components, ${\eta }_i$ is the Weibull scale parameter, and ${\beta }_i$ is the Weibull shape parameter.

System availability–oriented modular design model

Normally, system availability depends on both reliability and maintainability. A concrete expression to describe the operational availability uses the mean up-time (MUT) and the mean down-time (MDT) of each cycle. ³³ While pursuing cost reduction is crucial, it is also important to improve the availability of the system. Two conditions that lead to system shutdown are preventive maintenance and corrective maintenance. Due to the complexity of typical maintenance activities, this article mainly considers the serial situation of maintenance, for simplicity.

The maintenance time $T_k$ of the kth module can be represented as

$$T_k={\tau }_{\mathrm{pmk}}+{\tau }_{\mathrm{cmk}}{\displaystyle \underset{0}{\overset{E_i}{\int }}}{v}_i(t)$$

(13)

where ${\tau }_{\mathrm{cmk}}$ is the corrective maintenance time of the kth module, and ${\tau }_{\mathrm{pmk}}$ is its preventative maintenance time. System availability can be calculated as

$$A=1-{\displaystyle \sum _1^m}\frac{T_k}{t_{\mathrm{end}}}$$

(14)

where $t_{\mathrm{end}}$ represents the system’s imperative to consider maintenance activity at the scope of $[0,t_{\mathrm{end}}]$ , which depends on practical running conditions.

Constraint conditions

Based on simultaneous consideration of maintenance and modularity characteristics, constraint conditions can be obtained as follows

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\displaystyle \sum _{i=1}^Q}{x}_{\mathrm{ik}}=D_n,k=1,2,\dots ,K\hfill \\ \multicolumn{1}{c}{}& x_{\mathrm{ik}}=\{\begin{array}{c}\hfill 1,i\in k\hfill \\ \multicolumn{1}{c}{0,\mathrm{otherwise}}\end{array},K_{min}\le K\le K_{max},0\le \lambda (t)\le 1\hfill \\ \multicolumn{1}{c}{}& {\mathrm{MSI}}_{\mathrm{DSM}}\ge {\mathrm{MSI}}_{\mathrm{DSM}-\mathrm{initial}}\hfill \end{array}$$

(15)

Mathematical model and optimization algorithm

The above multi-objective optimization is a typical constrained multi-objective optimization problem. Its mathematical model can be described as follows

$$\begin{array}{cc}\multicolumn{1}{c}{F(X)}& =[F_M(X),F_c(X),F_A(X)]\hfill \\ \multicolumn{1}{c}{s.t.g_r(x)}& \le 0r=1,2,\dots ,R\hfill \\ \multicolumn{1}{c}{s.t.h_s(x)}& =0s=1,2,\dots ,S\hfill \\ \multicolumn{1}{c}{X}& =({\delta }_1,{\delta }_2,\dots ,{\delta }_{D_N}){\delta }_i\in (0,m)\hfill \end{array}$$

(16)

where $F_M(X)$ and $F_c(X)$ are the minimization objective functions, $F_A(X)$ is the maximization objective function, $g_r(x)$ is the unequal constraint of optimization problem, and $h_s(x)$ is the equality constraint of optimization problem.

The above optimization model is an non-deterministic polynomial (NP) hard problem. Many techniques, such as genetic algorithms, non-linear programming, and simulated annealing, have been applied to help capture Pareto solutions for determining the optimal outputs. The objective of this research is not to compare each algorithm found within multi-objective optimization techniques, but applying them to solve a modular design optimization problem. This article selects and applies SPEA2, ³⁴ which displays many distinct advantages including high speed of calculation, robustness, and solution diversity to solve for objective functions. The flowchart of solving optimization problems based on SPEA2 is shown in Figure 4.

Figure 4.

Flowchart of the SPEA2-based solution process.

Best compromise solution based on fuzzy set theory

Through SPEA2, the Pareto set of the multi-objective optimization problem is obtained in order to determine the best compromise solution. Due to the uncertainty characteristic of decision-making, this article applies fuzzy set theory ³⁵ to extract the best compromise solution. A membership function ${\epsilon }_i$ is illustrated by equation (17)

$${\epsilon }_i=\frac{F_i^{max}-F_i}{F_i^{max}-F_i^{min}}{F}_i^{min}\le F_i\le F_i^{max}$$

(17)

where $F_i^{min}$ and $F_i^{max}$ are the minimum and maximum values of the ith optimization objective, respectively. $F_i$ is the value of the ith optimization objective. For each non-dominated solution in the Pareto set, the dominance function is defined as

$${\epsilon }_s=\frac{{\displaystyle \sum _{i=1}^{N_o}}{\epsilon }_s^i}{{\displaystyle \sum _{j=1}^U}{\displaystyle \sum _{i=1}^{N_o}}{\epsilon }_s^i}$$

(18)

where U represents the solution number in the Pareto set, and $N_o$ is the number of the optimization objectives. A greater ${\epsilon }_s$ value indicates better comprehensive performance; thus, the biggest ${\epsilon }_s$ is selected as the best solution from the Pareto set. In fact, a priority list of non-dominated solutions can be offered to designers by means of arranging each solution in the Pareto set in descending order according to their membership function. This will guide designers toward the practical conditions into they should consider.

Case study

This case study is structured around the modular design of a hydraulic press. The hydraulic press is a kind of equipment that uses hydraulic transmission technology to process metal and non-metal materials. A prototype of a hydraulic machine with four columns is shown in Figure 5(a), and its corresponding virtual model is shown in Figure 5(b). Due to the machine’s wide application in manufacturing and production, the maintenance performance of the hydraulic press has received considerable interest from manufacturers, developers, and customers. According to its working principle, the hydraulic press can be divided into three parts: hydraulic system, control system, and mechanical system. This article chooses the mechanical system of the hydraulic press as case study for modular design to verify the effectiveness and applicability of the proposed method. The proposed clustering algorithm and the SPEA2 algorithm are programmed in a MATLAB 7.0 environment and run on a desktop computer with a dual 2.63 GHz Intel i5 processor and 8 GB RAM.

Figure 5.

(a) One instance of hydraulic machine. (b) Its explosion modeling in Solidwoks.

DSM establishment

The components of the hydraulic press are listed in Table 2, and the major components are further illustrated in Figure 6. It is necessary to point out that not every component is essential enough to be included in modular design, because some parts ultimately play very small roles in design results. By decomposing components and analyzing the design bill of the material, 21 components are considered in the modular optimization process. The physical connections of the components are shown in Figure 7. The information required to describe the interaction factors is acquired from a service handbook and company-provided information, and the weights of interaction factors are determined according to designer experience and preferences. The required interaction factors and determined weights are shown in Table 3. The degree of comprehensive relationship can be obtained using equation (1), and the DSM of the hydraulic press is constructed as shown in Figure 8.

Table 2.

List of components of the hydraulic press.

No.	Component	No.	Component
1	Locking arrangement	12	Push cylinder
2	Guide bushing	13	Ejector cylinder
3	Dust ring	14	Lock nut
4	Seal ring	15	Lower beam
5	Tie rod	16	Column
6	Slide	17	Ladder
7	Distance sheath	18	Hydraulic cylinder
8	Hydraulic cushion	19	Upper beam
9	Movable table	20	Adjusting nut
10	Rail	21	Fence
11	Down slide

Figure 6.

Assembly of hydraulic press model.

Figure 7.

Physical connections between components.

Table 3.

Interaction factors and weight.

Interaction factor	Function					Geometry		Physics
	Transform	Load	Limit	Pressing	Location	Join	Assembly	Energy	Signal	Force
Weight	0.067	0.087	0.067	0.047	0.067	0.16	0.17	0.11	0.11	0.11

Figure 8.

Initialized DSM for the mechanical system of the hydraulic press.

Initial module identification

A strongly connected subset can be obtained through the steps mentioned in section “Best compromise solution based on fuzzy set theory.” The initial modules of the hydraulic press can be established, as shown in Figure 9. In this case study, five modules are generated, which are demonstrated and defined below in the form of sets:

Figure 9.

Rearranged matrix between hydraulic press components.

• Module 1: $M_1=\{1,2,5,7,14,15,17,19,20,21\}$

• Module 2: $M_2=\{3,4,6,18\}$

• Module 3: $M_3=\{9,10,12\}$

• Module 4: $M_4=\{11,13\}$

• Module 5: $M_5=\{8\}$

Modular optimization for maintenance performance

The initial modules are regarded as gene segments of the algorithm, according to chromosome encoding rules. The initial population is produced by random combination. The algorithm applies integer coding to express modular identification results, where each integer is the modular serial number of its corresponding component. Set generation $K=200$ , population size $N=200$ , crossover rate $P_{\mathrm{c}}=0.85$ , and mutation rate $P_m=0.1$ . The optimization process of objective functions and the Pareto optimal solution set are obtained as shown in Figure 10.

Figure 10.

SPEA2-based optimization process with different objective functions: (a) optimization process of the difference of maintenance cycle, (b) optimization process of system availability, (c) optimization process of maintenance cost, and (d) Pareto optimal solution set.

A series of modular design schemes are presented in the form of a Pareto optimal set. The membership functions provided in equations (17) and (18) are applied to evaluate each member of the Pareto optimal set and rank the results in descending order. The best solution that has the maximum value ( ${\epsilon }_s=0.492$ ) of membership function can be extracted. The final module scheme is shown as follows:

• Module 1: $\{3,4,6,18\}$

• Module 2: $\{17,19,21\}$

• Module 3: $\{9,10,12,15\}$

• Module 4: $\{11,13\}$

• Module 5: $\{1,2,7,16\}$

• Module 6: $\{5,14,20\}$

• Module 7: $\{8\}$

Result analysis

By comparing and analyzing the performance of initial modules and optimized modules, as shown in Table 4, it becomes clear that the optimized modules greatly improve the maintenance performance of the hydraulic press under the premise of approximately constant modular performance. This information benefits designers, as it models a more favorable balance of modular performance and maintenance performance and saves after-sales maintenance cost for enterprises.

Table 4.

Analysis performance of initial module and optimized module.

Performance objective	Initial module	Optimized module	Change (%)
MSI	1.52	1.56	+2.5
Maintenance cost	US$15,963	US$11,465	+28.2
Difference of maintenance cycle	0.96	0.85	+11.4
System availability	0.75	0.92	+22.67

There is still much to be discussed before modular design can be completely accomplished, although the optimized solution obtained by applying our approach is beneficial to designers. The following design issues should be considered carefully when comparing the optimized results with the original structure of the hydraulic press:

1. The size and quality of the hydraulic press are very large, which creates difficulties in casting, heat treatment, and machining. To this effect, the frame adopts a combined structure in which the upper beam, lower beam, and column are manufactured independently and combined by nut. The frame bears the dynamic and static load of the hydraulic press, so its connection should be maintained regularly.

2. Over a long-running process, a hydraulic system tends to have a relatively high possibility of malfunction. Oil cleaning, sealing inspection, seal replacements, and other relevant maintenance should be taken into account beginning at the modular design process.

The performance of SPEA2, a type of genetic algorithm, can be influenced by many factors. The quality of the initializing population will directly affect the convergence property and the capability of optimum search.

In this article, the initial modules are regarded as the gene segments within the initializing population. This can reduce the uncertainty in the population’s covering space and improve the probability of generating favorable and effective individuals, while improving SPEA2’s performance as well. Simulations are conducted for the two methods, one of which regards the initial modules as the gene segments and the other which regards components as the gene segments. These operate under identical conditions. We changed the experiment parameters, went through the process 45 times, respectively, and took an average. Notably, the algorithm regarding initial modules as its gene segments has better convergence and better optimum search performance, as shown in Table 5.

Table 5.

Performance analysis results.

Crossover rate	Population size	200				300
	Mutation rate	0.1		0.2		0.1		0.2
	Initial type	Success rate	Time (s)	Success rate	Time (s)	Success rate	Time (s)	Success rate	Time (s)
0.85	Components	0.69	72.1	0.87	74.3	0.92	83.1	1.0	85
	Initial modules	0.84	66.1	0.94	67.6	1.0	79	1.0	81.6
0.95	Components	0.75	73.6	0.91	75	0.95	84	1.0	85.6
	Initial modules	0.84	67	0.98	69.2	1.0	80	1.0	82

Conclusion and future work

This article presents an innovative modular design method for maintenance performance. The proposed method can provide designers with better information regarding effective maintenance aspects at the initial phase of product design. Study results show that the proposed method not only improves product maintenance performance but also maintains relevant characteristics of modularity. The primary mechanism at work in this research includes the following two steps: in the first step, a method based on DSM and graph theory is applied to express the correlation among components and cluster them to generate initial modules. In the second step, a mathematical model based on maintenance cycle, maintenance cost, and system availability is constructed. Optimized modules are obtained by SPEA2 and fuzzy set theory. Throughout the case study of a hydraulic press’s modular design, the approach was proven to be efficient and effective.

With increasing environmental consciousness worldwide, product maintenance increasingly emphasizes green engineering and sustainability. Our future research will focus on improving green design and sustainable, eco-friendly performance of maintenance-oriented modular design.