Process Modularity Impact on Manufacturing Lead Time and Throughput Rate in Terms of Mass Customization

Abstract

As mass customization becomes more pervasive in many sectors, researchers need to update traditional approaches to the optimization of critical performance and design parameters in order to help companies in their effort to implement this strategy. In general, implementation of mass customization from a manufacturing perspective is frequently focused on shortening cycle times, reducing production cost, and increasing throughput rate of parts. In this paper, process structure modularity impact on manufacturing lead times and throughput rates is explored. An important precondition to explore these relationships is the awareness that process modularity is conceptualized and quantified in an appropriate way. For this purpose, three independent modularity measures were employed to provide more reliable assessment of this system property. The relationships were investigated on the basis of simulation experiments using deterministic models of alternative process structures. For the purpose of the relationships exploration, two case studies were conducted, theoretical and practical ones. The results from the experiments showed that there are moderate correlations between process modularity and manufacturing lead time (ρ = −0.45), as well as between process modularity and throughput rate (ρ = 0.45).

1. Introduction

The mass customization (MC) concept sounds quite paradoxical to traditional manufacturing practices, since it combines mass production with individual consumer de-sires. This manufacturing strategy presents an important challenge for companies by providing custom products for individual customers at acceptable cost levels. For manufacturers of customized goods, it means they can provide customers with more options, which requires frequent product changeovers in the shortest possible times. Thus, MC leads to the need for manufacturing systems endowed with a higher level of flexibility and resource reconfiguration capabilities. Such manufacturers’ ability can be achieved only by systematic implementation of well-known instruments of Industry 4.0 [1]. In general, it is possible to state that manufacturing lead times (MLTs) in mass customization environments are longer than in a case of standard production. Therefore, manufacturers of mass customized products usually expend a lot of effort to minimize MLTs in order to shorten delivery times and thereby increase their competitiveness. Principally, this objective can be achieved in two ways. The first of them is based on using common traditional methods which are applied in the production of standard products. The second possible way is to employ a so-called tailormade innovative approach for the given problem. It is known that product modularity helps to produce many product variants by keeping the costs to a minimum. In this context, our intent here is to present a novel approach which is aimed at increasing throughput rate (THR) performance of assembly processes and simultaneously to reduce MLTs. This approach is based on the assumption that process modularity can improve manufacturing process performance in terms of mass customization, since modular processes can be easily reconfigured to meet the changing demands of production volumes and product variability [2,3,4]. An important motivation for this research stems from the fact that modularity, as a design principle, is considered as an important criterion in designing manufacturing and assembly process structures [5,6]. Moreover, we took into consideration the known fact that modular design is a useful tool for reducing complexity in manufacturing [7,8,9] by breaking a manufacturing system into smaller segments to easier manage them. Naturally, a problem regarding investigation of process structure modularity is about finding its optimal modularity level. As optimal modularity does not equate to maximal relative modularity [10,11,12], our focus in this paper is to especially employ two competitive indicators (optimal modularity indicators Qd and M(G)) for identification of the optimal modularity level of assembly process structures. In addition, another estimator, called cross-module independence indicator (CMI), that quantifies relative modularity of process structures will be used in order to provide comparison among them. The abovementioned relationships will be explored here using two case studies—the theoretical and practical ones. For this purpose, the main goal of this article can be compiled into two research questions (RQs):

RQ 1: Can we adequately measure modularity of any real assembly process?

RQ 2: If yes, how does modularity impact manufacturing lead time and throughput rate in a specific realistic case?

This research article is organized in the following manner. Firstly, the literature sources that are the most related to this work are described in Section 2. Subsequently, methodology and research object specification are provided. Next, problem exploration is studied using theoretical and practical case study. Then, obtained results from the case studies are summarized in Section 5. Finally, positive research findings from this study and future works are summarized in Section 6.

2. Related Work

Modular design, in a wider sense, results in such benefits as simplicity, because it reduces the size of the problem that needs to be solved, and velocity, since by solving smaller problems in a parallel way, the time needed to solve the overall problem can be mitigated [13]. Reijers and Mendling [14] assert that an ad hoc division of a process into modules means to create a hierarchical structure of subprocesses which can only help to better understand a process model. Vickery et al. [15] argue that process modularity has significant potential to impact new product introduction performance and point out that product and/or process modularity are important design principles for addressing the challenges associated with rapidly changing technologies. Jacobs et al. [16] studied the theoretical relationship between product and process modularity and how these system properties affect manufacturing agility and the firm growth performance. Several authors [17,18,19,20] agree that there are certain linkages between product and process modularity, since, like product modules, process modules are characterized by clusters of locally connected elements [21] that have few strong organizational ties [22]. According to Watanabe and Ane [23], modularity of product design stimulates the creation of a job family in manufacturing and assembly processes, whereas in the next phases, this leads to modifying the number of operations in workstations. Piran et al. [24] provided evidence of the cause–effect trend between modularity and reduction in manufacturing lead time for the orders received by a bus manufacturing company. Zidi et al. [25] stated that modularity positively impacts manufacturing time, but without enumeration proof. The manuscript [26] provided the example where time is minimized by modularity; other authors [24] studied modularity in relation to delivery time, while delivery time variables were minimized for this purpose. In the latest years, some articles investigated the positive modularity impacts on manufacturing lead time and throughput rate, but only marginally in other related domains (e.g., Refs. [27,28,29,30,31]).

Experiences from Danish manufacturing companies showed that modularity principles are mostly applied in complex manufacturing processes, and that high levels of implementation of JIT shorten lead times, as might be expected [32]. The authors of [33] provided evidence that lower levels of process modularity can cause a higher number of errors than the same process with higher modularity. Moreover, in the case of undermodularization, there are difficulties in maintaining the large modules, while in terms of overmodularization, the disadvantage is that there is a large number of connections that need to be managed. A possible solution to overcome this issue is to identify the optimum number of product/process modules. The modularity optimization problem includes several methods for detecting clusters in complex networks to estimate the optimal number of clusters for a given network [34]. In this context, Newman and Girvan [35,36] proposed the optimal modularity indicator (Qd), which is defined as the fraction of connections within a group in the actual network minus expected fraction of connections in a random network. Haddou Benderbal et al. [37] explored optimization issues in reconfigurable manufacturing systems with the aims to maximize the system modularity and throughput rate, and to minimize system completion time and system cost. Also, it is worth mentioning other related research works, such as by Singh et al. [38], Fotsoh et al. [39], and Tu et al. [40], that deal with manufacturing modularity in the context of throughput rate.

The results of this survey, from which the selected publications are mentioned above, revealed that the relationships between modularity and manufacturing lead time and between modularity and throughput rate have not been treated and solved previously.

3. Methodology and Research Object Specification

As mentioned in Section 1, it seems to be reasonable to anticipate that process modularity could positively influence MLT and THR of production of mass customized products. In line with this assumption, the proposed methodical approach to solve the given problem is based on a combination of several methods and procedures that are arranged into the methodological framework presented in Figure 1.

Further, the individual steps of this framework are briefly outlined.

3.1. Selection of the Assembly Line Model

As is known, assembly line models can be classified into three types, namely, single model assembly line—SMAL, mixed model assembly line—MiMAL, and multimodel assembly line MuMAL, with regard to the products variety assembled here [13]. Let us have a production in a short time distance for demand of the three product types P₁, P₂, and P₃. These types of products or variants are released into production using a first-come-first-served (FCFS) strategy for all the assembly line model types in order to identify the one with the shortest production time. As the SMAL assembly line type is limited to producing only one variant, due to this reason, only one product type will be assumed in this case (P₁). MiMAL alternatively suits this production task, but there are long setup times, contrarily to MuMAL, where the products are moved and machined in batches, while setup times are reduced significantly. These differences are clearly visualized in Figure 2.

When comparing these assembly line models, one can see that the shortest production time is obtained by using MuMAL. Thus, MuMAL will be priorly considered here as a suitable assembly line model.

3.2. Selection of Batch Size Strategy

To select the most suitable batch size strategy, the two possible scenarios will be assumed in line with the prioritized assembly line type—MuMAL. The first scenario considers the uniform number of transport batches (L), where L equals 1. The second one is based on the uniform transport batch size (TBS) and it follows one-piece flow conception (TBS = 1). For both scenarios, the production batch (PB) for product P₁ equals 10 pieces, for P₂ it is 5 pieces, and for P₃ it equals 3 pieces. All the input data for these two scenarios are shown in

Table 1. The two possible batch size strategies/scenarios.

Scenario 1	P1	P2	P3	Scenario 2	P1	P2	P3
PB	10	5	3	PB	10	5	3
L	1	1	1	L	10	5	3
TBS	10	5	3	TBS	1	1	1

.

Then, the most appropriate transport batch size strategy is that with the lowest MLT. By using Gantt charts (see Figure 3), one can compare the two batch size strategies. As is visible, the second scenario with MLT = 20 min brought significantly better results against the first one, where MLT = 38 min.

A posteriori, Scenario 2 is prioritized when choosing one from the two possible batch size strategies.

3.3. Selection of Process Modularity Indicators

In reality, there are only a few known methods to measure modularity of process structures. In previous studies, the process modularity indicators were mutually compared and the three of them were selected as they most suited a given purpose. These process modularity indicators are used for the investigation of how process modularity affects MLT and THR, which are described here.

3.3.1. Optimal Modularity Indicator Qd

The optimal modularity indicator Qd proposed by Newman and Girvan [35], which measures the quality of partition of a network into modules by maximizing its modularity, is enumerated through the following formula:

$$Qd=\sum _{s=1}^n\left(\frac{ls}{L}-\frac{{ws}^{out} · {ws}^{in}}{L^2}\right),$$

(1)

where n is the number of modules, L is the number of all edges in the network, ls is the number of internal edges in module s, ws_out is the number of output edges in module s, and ws_in is the number of input edges in module s.

3.3.2. Optimal Modularity Indicator M(G)

An indicator to measure optimal process modularity of process structures, named the optimal modularity indicator M(G), is expressed as follows [41]:

$$M\left(G\right)=\frac{n}{\sum (Nj·\ln Nj)},$$

(2)

where n is the number of modules in the network, Nj is the number of couplings per column, j = 1, …, K, and K is the number of columns in a related design matrix.

3.3.3. Cross-Module Independence CMI

The third indicator, the cross-module independence (CMI), calculates the ratio of the sum of relations inside all modules to the sum of all relations, and is expressed by the following equation [42]:

$$CMI=1-{\sum }_{int=1}^n\frac{R_{int}}{T},$$

(3)

where n stands for the number of modules, while int = 1, 2, …, n; R represents the number of inside connections; and T is the number of all linkages in a network.

It has to be emphasized that by this indicator, only relative modularity of manufacturing and assembly processes can be estimated, while the previous two indicators are explicitly devoted to identifying optimal process structures from alternative ones.

4. Problem Exploration

In this chapter, the two case studies are carried out in order to explore the two relationships between process modularity and MLT, and between process modularity and THR. The first case study assumes the theoretical example of production of the four product types. The second one is a practical example of production of customizable chairs.

4.1. Theoretical Case Study

Let us consider the following theoretical model of an assembly process completing four types of customized products, as shown in Figure 4.

The assembly process consists of the two workstations (WSs) producing products P₁, P₂, P₃, and P₄. Products consist of six mandatory components (A, B, C, D, E, F) and two optional components (V1, V2). The possible compositions of the products are as follows: P₁ = {A, B, C, D, E, F}; P₂ = {A, B, C, D, E, F, V1}; P₃ = {A, B, C, D, E, F, V2}; P₄ = {A, B, C, D, E, F, V1, V2}. Subsequently, operation times are extracted from the process planning documents and assigned to WSs. Then, one can determine the flow shop problem with m-machines and n-products (m × n), which is presented in

Table 2. Flow shop m × n problem, where processing times are in seconds.

m × n	P1	P2	P3	P4
WS1	20	30	20	30
WS2	50	50	60	60

.

With regard to settlement of order to release products into production, it was found that the optimal sequence is as follows: P₁ → P₂ → P₃ → P₄. Moreover, setup times equal to 5 s for each operation were identified. In addition, it was supposed that production batches (PBs) for all the products equaled 25 pieces.

Subsequently, it was possible to identify under certain rules all possible and practicable assembly process structures (APSs) [43]. As the assembly process structure from Figure 4 consisted of eight input components, it was theoretically possible to generate 261 APS alternatives. For the purpose of this theoretical example, ten of them were used (including the APS from Figure 4). All selected APSs are depicted in Figure 5.

Then, the above-described process modularity measures Qd, M(G), and CMI were applied for all the ten APSs. Application of these process modularity measures can be provided by using APS, e.g., process structure No. 1, and the enumeration is as follows:

$${Qd}_{No. 1}=\left(\frac{3}{10}-\frac{12}{{10}^2}\right)+\left(\frac{6}{10}-\frac{42}{{10}^2}\right)=0.36,$$

$${M\left(G\right)}_{No.1}=\frac{2}{3ln3+9ln9+10ln10}=0.043,$$

$${CMI}_{No.1}=1-\frac{9}{10}=0.10.$$

To explore the influence of process modularity on the efficiency of assembly processes (MLT and THR), simulation experiments using the Tecnomatix Plant Simulation tool were conducted.

Operation times for individual alternatives of products and structures are provided in

Table 3. Assembly time for individual alternatives of products for all structures in seconds.

Parts Assembly	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10
A + B	20	20	20	-	20	20	20	20	-	-
A + B + V1	30	30	30	-	30	30	30	30	-	-
AB + C + D + E + F	50	-	-	-	-	-	-	-	-	-
AB + C + D + E + F + V2	60	-	-	-	-	-	-	-	-	-
ABV1 + C + D + E + F	50	-	-	-	-	-	-	-	-	-
ABV1 + C + D + E + F + V2	60	-	-	-	-	-	-	-	-	-
AB + C + D	-	30	-	-	-	-	-	-	-	-
ABV1 + C + D	-	30	-	-	-	-	-	-	-	-
ABCD + E + F	-	30	-	-	-	-	30	-	30	-
ABCDV1 + E + F	-	30	-	-	-	-	30	-	30	-
ABCD + E + F + V2	-	40	-	-	-	-	40	-	40	-
ABCDV1 + E + F + V2	-	40	-	-	-	-	40	-	40	-
AB + C	-	-	20	-	-	-	-	-	-	-
ABV1 + C	-	-	20	-	-	-	-	-	-	-
ABC + D + E	-	-	30	-	-	-	-	-	-	-
ABCV1 + D + E	-	-	30	-	-	-	-	-	-	-
ABCDE + F	-	-	20	-	-	20	20	-	-	20
ABCDEV1 + F	-	-	30	-	-	-	-	-	-	-
ABCDE + F + V2	-	-	30	-	-	30	30	-	-	30
ABCDE + V1 + F + V2	-	-	40	-	-	-	-	-	-	-
A + B + C	-	-	-	30	-	-	-	-	-	-
A + B + C + V1	-	-	-	40	-	-	-	-	-	-
D + E + F	-	-	-	30	-	-	-	-	-	-
D + E + F + V2	-	-	-	40	-	-	-	-	-	-
ABC + DEF	-	-	-	20	-	-	-	-	-	-
ABCV1 + DEF	-	-	-	20	-	-	-	-	-	-
ABC + DEFV2	-	-	-	20	-	-	-	-	-	-
ABCV1 + DEFV2	-	-	-	20	-	-	-	-	-	-
C + D + E	-	-	-	-	30	-	-	-	-	-
AB + CDE	-	-	-	-	20	-	-	-	-	-
ABV1 + CDE	-	-	-	-	20	-	-	-	-	20
ABCDV1 + F	-	-	-	-	20	20	-	-	-	20
ABCDEV1 + F + V2	-	-	-	-	30	30	-	-	-	-
C + D	-	-	-	-	-	20	20	20	-	-
AB + CD	-	-	-	-	-	20	20	-	-	-
ABV1 + CD	-	-	-	-	-	20	20	-	-	-
ABCD + E	-	-	-	-	-	20	-	-	-	20
ABCDV1 + E	-	-	-	-	-	-	-	-	-	20
AB + CD + E + F	-	-	-	-	-	-	-	40	-	-
ABV1 + CD + E + F	-	-	-	-	-	-	-	40	-	-
AB + CD + E + F + V2	-	-	-	-	-	-	-	50	-	-
ABV1 + CD + E + F + V2	-	-	-	-	-	-	-	50	-	-
A + B + C + D	-	-	-	-	-	-	-	-	40	40
A + B + C + D + V1	-	-	-	-	-	-	-	-	50	50

.

Moreover, setup times for each operation were 5 s, and transport times of 8 s were assumed. The simulation was performed for 25 pieces of each product, so the total production batch was 100 products. When all the input data are provided, then it is possible to identify MLT and THR of all the APSs.

Manufacturing lead time, which is the total time required to process a product through the plant, is usually calculated by the following formula [44]:

$${MLT}_j=\sum _{i=1}^{{no}_j}({Tsu}_{ij }+{Q_j\ast Tc}_{ij}+{Tt}_{ij}),$$

(4)

where MLT_j means MLT for a batch of part or product j, Tsu_ij is setup time for operation i on part or product j, Q_j is quantity of part or product j in the batch being processed, Tc_ij is cycle time for operation i on part or product j, and Tt_ij is transport time associated with operation i, where i indicates the operation sequence in the processing, i = 1, 2, …, no_j.

The throughput rate indicator calculates the rate at which units move through the production process from the beginning to the end of the production process (i.e., how many units are produced per minute, hour, or day).

For demonstration purposes, a model of assembly process structure No. 1 is depicted in Figure 6.

By applying the simulation tool for the given process model, the following MLT and THR values were obtained: MLT_No.1 = 92.3 min (1.5 h), THR_No1 = 65.01 parts/hour. The same simulation procedure was applied for the rest of the APSs.

Then, it was possible to summarize all the obtained results to identify the influence of modularity on MLT and THR by using the three process modularity indicators, namely, Qd, CMI, and M(G). The summarized results for individual structures are shown in

Table 4. Summarized results of process modularity indicators and efficiency indicators.

APSs	Qd	CMI	M(G)	MLT (Minutes)	THR (Parts/ h)
No.1	0.36	0.1	0.043	92.3	65.01
No.9	0.4	0.1	0.039	75.88	79.07
No.8	0.471	0.182	0.055	75.63	79.33
No.10	0.438	0.182	0.042	76.18	78.76
No.4	0.463	0.182	0.0496	67.3	89.49
No.2	0.47934	0.182	0.047	60.1	99.83
No.7	0.472	0.25	0.054	59.68	100.53
No.3	0.493	0.25	0.048	51.82	115.79
No.5	0.486	0.25	0.0504	51.02	117.61
No.6	0.47929	0.308	0.0494	43.24	138.25

.

When analyzing the results from Table 4, one can state that the best MLT and THR values were obtained using APS No.6, and the worst MLT and THR values were obtained using APS No.1. The differences between the two APSs from the viewpoint of MLT and THR can be also compared through Gantt charts generated from the simulation software, as shown in Figure 7.

At the same time, one can see that APS No.6 is almost the optimal modular structure among the ten APSs, while APS No.1 presents a structure with relatively low process modularity.

In order to verify this finding regarding the relationship between process modularity and MLT as well as THR, a practical example is provided in the next section.

4.2. Practical Case Study

The same procedure as in Section 4.1 is implemented here for manufacturing of customizable chairs. The product consists of seven assembly input components, namely, (A) back support, (B, H) cross bars, (C) front legs, (D) back legs, (E) sledge legs, (F) seat panel, and (G) arm rest. The feature diagram describing the product structure is depicted in Figure 8.

The assembly process encompasses six assembly operations, which can be processed in individual workstations, or integrally. The assembly process is able to complete three types of products (P₁, P₂, and P₃). The compositions of the products are as follows: P₁ = {A, E, F}; P₂ = {A, B, C, D, F, H}; P₃ = {A, B, C, D, F, G, H}. Original assembly process sequences for individual customized products varies, as depicted in Figure 9a, while the integral assembly process for all the product types is schematically described in Figure 9b.

To optimize the integral assembly process structure, first of all, alternative process models need to be generated. Theoretically it is possible to identify 90 APS alternatives [42], which are depicted in Figure 10, where the number of all possible APS structures follows integer sequence A000669 (number of series-reduced planted trees with n leaves) [45].

As reasoned above, the MuMAL strategy was applied here to explore the influence of process modularity on assembly process efficiency, since it brings better results than MiMAL in the given application (see Section 3.1). To calculate MLT and THR, operation times for individual alternatives of products and structures are provided in

Table 5. Assembly time based on the number of assembled parts in seconds.

No. of Assembled Parts	Assembly Time
2 (e.g., A + E)	20
3 (e.g., A + E + B)	30
4 (e.g., A + D + B + F)	40
5 (e.g., AD + B + F + C + H)	50
6 (e.g., A + D + B + F + C + H)	60
7 (e.g., A + D + B + F + C + H + G)	70

.

Moreover, it was assumed that setup time for each operation was 5 s, and transport times between operations were 7 s. The simulation was performed for 10 pieces of each product, so the total production batch was 30 products.

In line with the above-outlined methodological framework, the influence of modularity on MLT and THR was explored using the selected process modularity indicators, namely, Qd, CMI, and M(G). The simulation results of the indicators for individual structures together with obtained MLT and THR values are shown in

Table 6. Obtained results, where individual APSs are ordered according to decreasing MLT or increasing THR.

APSs No.	Qd	CMI	M(G)	MLT (min)	THR (Parts/h)
1	0	0	0.033	29.12	61.81
79	0.4	0.2	0.0465	24.68	72.9
2	0.296	0.111	0.0529	24.57	73.27
58	0.296	0.111	0.0424	24.55	73.4
34	0.37	0.111	0.0449	22.73	79.18
46	0.41	0.2	0.0574	22.62	79.57
5	0.37	0.111	0.05	22.04	80.36
6	0.4	0.2	0.06	21.3	84.51
3	0.44	0.2	0.066	21.067	85.44
86	0.446	0.273	0.0486	19.97	90.14
17	0.44	0.2	0.0554	19.52	92.23
14	0.395	0.111	0.048	19.4	92.78
19	0.46	0.2	0.0603	19.23	93.59
24	0.45	0.2	0.0581	19.07	94.4
29	0.4546	0.273	0.0646	19.07	94.4
47	0.471	0.273	0.0636	18.47	97.46
35	0.46	0.2	0.0546	18.18	98.99
18	0.446	0.273	0.0596	18.13	99.27
15	0.45	0.2	0.057	18.017	99.91
16	0.446	0.273	0.0636	18.017	99.91
67	0.465	0.333	0.058	18	99.9
84	0.458	0.333	0.053	17.8	101.1
74	0.446	0.273	0.0548	17.8	101.1
66	0.463	0.273	0.0551	17.8	101.1
64	0.4793	0.273	0.0566	17.8	101.1
8	0.471	0.273	0.069	17.8	101.12
7	0.47	0.2	0.063	17.73	101.5
4	0.4959	0.273	0.075	17.57	102.47
89	0.458	0.333	0.0495	16.92	106.4
82	0.471	0.273	0.0499	16.63	108.2
44	0.463	0.273	0.0538	16.63	108.2
63	0.463	0.273	0.0548	16.63	108.2
78	0.449	0.385	0.057	16.57	108.6
77	0.458	0.333	0.0564	16.5	109.1
81	0.472	0.333	0.0541	16.45	109.4
65	0.472	0.333	0.0593	16.42	109.6
70	0.465	0.333	0.0559	16.4	109.7
75	0.465	0.333	0.0576	16.37	110
37	0.47	0.2	0.0528	16.35	110.09
73	0.465	0.333	0.0534	16.22	110.9
60	0.4796	0.273	0.0581	16.2	111.1
49	0.4793	0.273	0.0617	16.18	111.21
22	0.4794	0.273	0.0644	16.18	111.23
72	0.471	0.273	0.0517	16.03	112.3
71	0.4793	0.273	0.0529	16.03	112.3
27	0.471	0.273	0.0625	16.02	112.38
32	0.465	0.333	0.0671	16.02	112.38
80	0.463	0.273	0.0512	16	112.5
38	0.471	0.273	0.0572	15.97	120.27
48	0.4792	0.333	0.0676	15.97	112.7
69	0.45	0.2	0.0484	15.85	113.5
62	0.46	0.2	0.0502	15.85	113.5
41	0.44	0.2	0.0508	15.85	113.54
59	0.43	0.2	0.0519	15.85	113.54
53	0.455	0.273	0.0596	15.85	113.89
11	0.48	0.2	0.061	15.73	114.41
9	0.4959	0.273	0.072	15.73	114.41
83	0.465	0.333	0.0518	15.25	118.01
45	0.458	0.333	0.0552	15.25	118.01
42	0.471	0.273	0.0552	15.12	119.1
87	0.465	0.333	0.0505	15.03	119.7
57	0.456	0.385	0.0607	14.92	120.67
43	0.458	0.333	0.0579	14.8	121.6
23	0.472	0.333	0.0664	14.8	121.62
54	0.472	0.333	0.062	14.75	122.41
90	0.449	0.385	0.0498	14.72	122.34
56	0.465	0.333	0.0606	14.72	122.34
50	0.472	0.333	0.0639	14.72	122.34
33	0.456	0.385	0.0684	14.72	122.34
40	0.458	0.333	0.0616	14.7	122.52
36	0.4793	0.273	0.0608	14.68	122.59
25	0.4793	0.273	0.0644	14.68	122.59
20	0.4876	0.273	0.0664	14.68	122.59
30	0.472	0.333	0.0689	14.68	122.59
12	0.479	0.273	0.067	14.52	123.86
21	0.472	0.333	0.0703	14.52	123.99
39	0.463	0.273	0.0591	14.35	125.44
76	0.4497	0.385	0.0594	14.35	125.44
68	0.449	0.385	0.0599	14.35	125.44
85	0.449	0.385	0.0539	14.3	125.9
61	0.4792	0.333	0.0624	14.3	125.9
88	0.4497	0.385	0.0516	14.28	126
55	0.456	0.385	0.0634	14.28	126
28	0.465	0.333	0.0648	14.28	126
51	0.465	0.333	0.0656	14.28	126
52	0.456	0.385	0.067	14.28	126
26	0.465	0.333	0.0685	14.28	126
13	0.472	0.333	0.071	14.28	126
31	0.456	0.385	0.0718	14.28	126
10	0.486	0.333	0.0759	14.27	126.03

.

It is worth noting that the order of APSs in Table 6 (by decreasing MLT or increasing THR) is identical, since this is a consequence of Little’s law [46], i.e., if MLT decreases, then THR increases. Based on these results, one can state that the best MLT and THR values were obtained using APS No.10, and the same structure was recognized as the optimal modular configuration according to M(G), while, according to indicator Qd, there were two most optimal modular APSs (No.4 and No.9), where the MLT and THR values were far from the optimal ones. Similarly, the CMI indicator seems to be not applicable for the given purpose, as eleven APSs were identified as optimally modular using this. Due to this reason, process modularity indicators M(G) and Qd can be considered as more relatable to measure the modularity as they are more sensitive than the CMI indicator.

To express the closeness of the relationships, the Spearman correlation coefficient was employed for this purpose. Computation results showed that based on the correlation scale [47], there are the following:

-

Moderate negative correlation (ρ = −0.45) between M(G) and MLT;

-

Moderate positive correlation (ρ = 0.45) between M(G) and THR;

-

Moderate negative correlation (ρ = −0.41) between Qd and MLT;

-

Moderate positive correlation (ρ = 0.41) between Qd and THR.

5. Results Discussion

Summarizing the obtained results, it is possible to postulate that assembly processes modeled as directed rooted trees can be optimized from the viewpoint of modularity with the above-described objectives. In other words, it is quite likely that assembly processes with higher structural modularity provide advantages, at least from the perspective of MLT and THR parameters, against processes with relatively lower levels of optimal modularity. When analyzing the results from the practical case study, it was found that MLT values varied from 14.27 min to 29.12 min (51%) between optimally modular APS (No.10) and integral APS (No.1). In addition, when comparing the same two APSs (No.10 and No.1), THR values differed from 126.03 parts/hour to 61.81 parts/hour. These significant differences justify our confirmation of the positive impact of process modularity on the efficiency of assembly processes in mass customization environments. By comparison of the three modularity indicators, using them to identify optimal modularity of process structures, the M(G) indicator seems to be more appropriate for the given purpose than Qd or CMI metrics. This assertion can be explained as follows: the M(G) indicator was specially developed to map this modularity attribute of manufacturing and assembly processes, while the Qd indicator was originally created for community networks, and was only subsequently adopted for optimal modularity measurement of manufacturing and assembly processes [48]. Moreover, both the M(G) and Qd indicators are similarly sophistic in their nature, which was rigorously verified in Section 4. Thus, it can be stated that they are applicable in a wide range of applications. The reason to employ the relative modularity indicator, CMI, was to show the differences between this indicator and the two indicators devoted to identifying the most optimal modular structures.

Based on the obtained research results, answers to the research questions can be formulated.

Answer to RQ 1. According to findings from the comparison of the process modularity indicators, it can be stated that at least the two measurement methods using indicators Qd or M(G) can be applied to determine an optimal process structure from the realistic alternatives.

Answer to RQ 2. Based on the obtained results from the both examined cases, one can see that process modularity positively impacts the manufacturing lead time and throughput rate. When the modularity of the manufacturing process increases, then MLT decreases and THR increases.

6. Conclusions

Based on the obtained results from the computational experiments, it was proved that modularity has positive impact on manufacturing lead time and throughput rate, which was also confirmed in the existing literature on relationships between modularity and MLT as well as between modularity and THR. In addition, this manuscript shows that modular principle in assembly process design is equally beneficial as in product design; therefore, it is worth paying attention to its potential. This is a challenge for researchers to explore further relationships between assembly process modularity and its performance indicators. On the other hand, the research results can be directly exploited in manufacturing practice by managers, since the presented methods can be easily applied in the early design stage of assembly processes. Obviously, limitations of the approaches relate to the costs associated with the rearrangement of machines and equipment.

Finally, it can be said that future research could benefit from using multicase studies to validate presented findings and, potentially, to demonstrate their higher degree of generality or to identify their possible limitations.