An effective multi-part dedicated flow-line reconfiguration model considering the optimal selection of machining process path and machines

Abstract

Multi-part Dedicated Flow-line (MDF) is widely used to produce discrete products for high volumes with low varieties. Since frequent changes occur in customers’ demand and rapid upgrading of machines, the line needs to reconfigure timely to remain competitive. A lot of researchers investigate how to reconfigure the line when facing these changes, and most of them adopts small adjustments such as machine resettlement or buffer size rearrangements to react. However, when great changes occur in product’ machining process or machine’s performance, it’s time to reconfigure the line in a higher level by reselecting machining process of products and/or reselecting machines composition of the line. To fill this research gap, the paper proposes an effective Multi-part Dedicated Flow-line Reconfiguration Model (MDFRM) which can effectively react to great changes occur in products and machines. By optimal selection of Machining Process Path (MPP) for every product and machines composition for the new line, MDFRM can reconfigure MDF with objective of minimizing the products’ cost and capital investment. To solve this problem, an Enhanced Equilibrium Optimizer (EEO) with three layers of code and three steps to handling boundaries and constraints based on Equilibrium Optimizer (EO) is presented. The proposed new model and the designed method are validated by numerical experiments and real case study. This paper will significantly help decision-makers make a scientific decision to optimally reconfigure MDF.

Introduction

Nowadays, manufacturing companies face growing changes triggered by fierce competition from around the world, ¹ more demanding customers, a rapid changes in product and updating in machining technology. ² To remain competitive, they have more urgent needs to frequently and rapidly reconfigure the manufacturing systems. Among many types of manufacturing systems, Multi-part Dedicated Flow-line (MDF) is one of the most important ones. It aims to manufacture product with high volumes and low varieties, and is widely applied in many manufacturing contexts which includes machining, assembly, and disassembly. ³ When changes appear in product’s demanding volume, Machining Process Path (MPP) or machining parameters, MDF needs to timely reconfigure. If not, the company’s competitiveness may weaken regards to product’s cost, delivery time, and efficiency of the line, and even lead to the line being unable to produce the product. Furthermore, since MDF is designed for relatively mass production, even the smallest cost or time increase for a single product would contribute a lot to the total variation. So, the investigation of optimal reconfiguring an MDF has a significant meaning.

Numerous researches are conducted to investigate how to reconfigure production line when facing the changes. For reconfiguration of Reconfigurable Manufacture System (RMS), Maniraj et al. ⁴ propose a novel approach to reconfigure single-product flow-line RMS using ant colony optimization-based approach, Asghar et al. ⁵ focus on optimal machine capabilities generation by investigating process planning and machine configurations. Ashraf and Hasan ⁶ propose a framework for configuration selection of a manufacturing flow line. There are some researches performed to analyze MDF for better performance, such as considering buffer sizes and machines types for more accurate performance estimation of series-parallel production line, ⁷ considering line balance, and throughput analysis to design a multi-part line. ⁸ Kim et al. ⁹ investigate maximum production rate and steady-state delay distribution of deterministic serial lines with random setups. For optimal design line’s configuration, Jiang et al. ¹⁰ formulate a concurrent optimization model to design assembly line configuration. The model jointly determines the subassembly planning and station assignments considering uncertain product evolution. Moghaddam et al. ¹¹ develop a two phased method to handle the primary system configuration design and the necessary system reconfigurations according to demand rate changes.

However, all of the above research only adopts some small adjustments such as machine resettlement or buffer size rearrangements. They neglect potential great changes in the product (demanding volume, machining parameters, machining process path, etc.) and machines (performance degradation of present-used machine, appearing of new machine with high performance). When these changes occur, the above methods may not work well any longer if present MPP of products is outdated and/or machine’s performance is low. It’s time to reconfigure the line in a higher level by reselecting MPP of products and/or reselecting machines composition of the line. Thus, the objective of the work is to investigate how to reconfigure an MDF when facing great changes related to product and/or machines described above. To solve this production line reconfiguration problem, the paper develops an effective multi-part dedicated flow-line reconfiguration model (MDFRM) and proposes an Enhanced Equilibrium Optimizer (EEO) to solve the model. MDFRM aims to minimize the products’ cost and capital investment by optimal selection of MPP for every product and machines composition for the new line. At the same time, some realistic constraints are considered: demanding volume of the product, workload balance of production line, and maximum machine quantity and type in every machining station.

The proposed MDFRM seeks optimal machines composition under a particular MPP, and it is a new combinatorial optimization problem. Based on the newly proposed Equilibrium Optimizer (EO), ¹² the paper proposes an Enhanced Equilibrium Optimizer (EEO). EEO has three layers of code and three steps to handling boundaries and constraints, both numerical experiments and an application case are conducted to validate the proposed MDFRM and EEO.

The rest of the paper is presented as follows: Section “Related works” provides the related research of the investigated problem. Section “Problem statement and modeling” formulates the investigated problem. In Section “Proposed method” the proposed method to solve this new problem is described. Section “Numerical experiments and application case study” conducts both numerical experiments and a real case study to show the effectiveness and applicability of the proposed model and method. Finally, the conclusion is drawn and future work is discussed in Section “Conclusion and future work”

Related works

Over the years, a large body of research has been devoted to the reconfiguration of manufacturing systems to deal with the changes of products and machines mainly for RMS. Battaïa et al. ¹³ formalize a mixed-integer programming model for flow lines composed of reconfigurable machines which aims to minimize machines cost while satisfying a given throughput and some constraints concerned. Dou et al. ¹⁴ investigate reconfigurable flow lines to optimize the total investment for single and multi-part flow lines, respectively. Bahalke et al. ¹⁵ study configuration selection of a single product flow-line reconfigurable manufacturing systems using ant colony optimization approach.

It is acknowledged RMS consists of machines that have adjustable structures/modals, ¹⁶ which means the machine is much costlier than a dedicated machine since it has much more complicated structures and auxiliary parts. Meanwhile, in some industrials which need to manufacture the product with high volumes, such as car industry, electronic product and household appliance, ¹⁷ MDF is widely used to obtain high production rate and low cost. There have some analysis of MDF conducted for higher performance. Tang et al. ⁸ investigate how to design a multi-part line with the consideration of line balance, equipment selection, and output analysis. Nahas et al. ⁷ propose a new model that extends the classical buffer space allocation problems by analyzing buffer sizes and types of machines, as well as the number of parallel machines, to estimate series-parallel line’s performance. Li et al. ¹⁸ present a hybrid cell evaluated genetic algorithm to optimize the dedicated remanufacturing line by simulation. To evaluate and improve the performance of long stochastic flow lines with finite raw material in discontinuous and continuous time, Mindlina et al. ¹⁹ present a flexible (mixed-integer) linear programming approach. Magnanini et al. ²⁰ propose a new model to evaluate performance of asynchronous two-stage production lines manufacturing discrete parts with limited buffer. In summary, all of above research focuses on how to improve the performance of MDF by small rearrangements, such as resettlement of machine or buffer size. These adjustments lead to a satisfied result when small changes appear in product’s demanding volume, MPP and machining parameters. Nevertheless, if present MPP of products is outdated and/or machine’s performance is deteriorated seriously, the above small adjustments may not work well any longer. It’s time to reconfigure the line in a higher level by reselecting MPP of products and/or reselecting machines. To the best knowledge of the authors, there is no comprehensive research to reconfigurate a MDF in terms of selecting product’s MPP and machines (among new and old machines).

Based on the above review, this paper proposes an effective MDFRM which aims to minimize the products’ cost and capital investment by selecting the optimal MPP for products and machine composition. Since the proposed model is a combinatorial optimization problem, and intelligent metaheuristic algorithms have been widely used to solve such problems.^21,22 Ouaarab et al. ²³ present an enhanced and discrete version of the cuckoo search algorithm for traveling salesman problem, an NP-hard combinatorial optimization problem. Wang et al. ²⁴ present a tabu-genetic algorithm to find the optimal solution for parallel machine scheduling problem with restrained tool resources. Wunnava et al. ²⁵ propose a new optimization algorithm-adaptive equilibrium optimizer for multilevel thresholding problem, based on the original Equilibrium Optimizer (EO). Among many algorithms, newly developed EO ¹² demonstrates high effectiveness and efficiency in a lot of problem investigated which motivates us to explore EO to solve the proposed MDFRM.

Problem statement and modeling

Problem description

MDF needs to adjust if changes appear in product, machine, or expectation of the line’s performance. Moderate adjustments, such as relocation of machine or resettlement of buffer size, may function well to deal with small changes. However, great changes may occur in product’s demanding volume, machining process or machining parameters, or present machine deteriorates with new machine of high performance appears. Then we need to reconfigure the line in a higher level by reselecting MPP of products and/or reselecting machines composition of the line instead of small adjustments.

When to reconfigure the production line, the most important is to determine product’s MPP ²⁶ and line’s machine composition, ²⁷ and Figure 1 is a simple demonstration of the investigated MDF reconfiguration problem. In Figure 1, the line is expected to produce two products ( $P_1$ and $P_2$ ) in separate batch and each product has alternative MPPs which is represented by lines going through different MSs. Such as for $P_1$ , there are totally two MPPs, namely, $M{S}_1\to M{S}_3\to M{S}_4$ and $M{S}_1\to M{S}_2\to M{S}_4$ . In every MS, there may have some candidate machines with present-old ones or buying-new ones. Every machine has corresponding machine cost, different tooling cost, and machining parameters (machining cost and time) for different product. So, the objective is to select optimal MPP for every product, and optimal machine composition for every MS pursuing minimization of line’s capital investment (selected machine’ cost and tooling cost for every product) and product’s cost (raw material and machining cost) under some realistic constraints. And the constraints may refer to demanding volume of the product, workload balance of production line, and maximum machine quantity and type in every MS which will be explained in section “Mathematical model”Figure 2 is a possible new line as to problem described above.

Figure 1.

A simple demonstration of the investigated MDF reconfiguration problem.

Figure 2.

A possible new line as to problem described in Figure 1.

To simplify the problem, assumptions are listed as bellows:

• When assigning more than one machine to perform one operation of the product, the product is expected to be processed on the machine whose waiting time is the shortest.

• All the machine has perfect machining quality and produce no defect product.

• Machining parameters (machining time and cost) are treated as deterministic and constant because of large volume production. ²⁸

The presented model

The notations employed in formulating MDFRM are first defined, followed by the mathematical model.

Notations

To formulate the problem, the following abbreviation, indices, parameters, and variables are employed (Table 1).

Table 1.

Notations of the MDFRM.

Notations	Description
Abbreviations
$\mathrm{CM}$	Candidate machine for MS.
$\mathrm{DV}$	Demanding volume per year of the product.
$\mathrm{MGC}$	Machining cost of an operation/product.
$\mathrm{MPP}$	Machining process path.
$\mathrm{MS}$	Machining station. In a MS, all the machine performs the same function.
$P$	Product to be processed.
$\mathrm{PC}$	Product’s cost that equals summed up MGC and RMC.
$\mathrm{RMC}$	Raw material cost for different MPP of a product.
$\mathrm{SI}$	Workload smoothness index between the MSs of the production line.
$\mathrm{TC}$	Cost of tooling (fixture/cutter/program) which is special designed to be integrated into the machine to produce one particular product.
$\mathrm{WT}$	Working time of the production line in the planned manufacturing period, this time has excluded the time that is not used to produce product such as machinedown or maintenance, changeover among different batches of product, and others.
Indices
$i$	1, 2, …, $I$ , index of product.
$j$	1, 2, …, $J$ , index of MS (index of operation for the product).
$k$	1, 2, …, $K_j$ , index of CM type for $M{S}_j$ , every type of machine may have old (present-having) or new (potential-buying) version.
$l$	1, 2, …, $L_i$ , index of the alternative MPP of $P_i$ .
Parameters
${\mathrm{CM}}_{\mathrm{jk}}^1$ , ${\mathrm{CM}}_{\mathrm{jk}}^2$	$k^{\mathrm{th}}$ type of CM old ( ${\mathrm{CM}}_{\mathrm{jk}}^1$ ) and new ( ${\mathrm{CM}}_{\mathrm{jk}}^2$ ) version for $M{S}_j$ .
${\mathrm{DV}}_i^{\min }$	Minimum DV per year for $P_i$ forecasted by company.
$J_p^i$	All $M{S}_j$ ( $j\in J_p^i$ ) that $P_i$ must go through when process path l is selected. In this way $(J_p^1\cup J_p^2\cup \dots \cup J_p^I)$ implies all the $j$ that makes $M{S}_j$ at least process one of the products among $I$ products.
	$J_p^i=\{j|\forall p_{\mathrm{ilj}}=1,j=1,2,\dots ,J,\mathrm{for}\mathrm{fixed}l\mathrm{that}\mathrm{satisfies}{Y}_{\mathrm{il}}=1\}$
$J_q^i$	Similar with $J_p^i$ , defines $J_q^i$ as all $M{S}_j$ ( $j\in J_q^i$ ) that need to be involved in calculation of SI of product $i$ when $\mathrm{MP}{P}_l$ is selected.
${\mathrm{MC}}_{\mathrm{jk}}^1,{\mathrm{MC}}_{\mathrm{jk}}^2$	Machine cost of ${\mathrm{CM}}_{\mathrm{jk}}^1$ ( ${\mathrm{MC}}_{\mathrm{jk}}^1$ ) and ${\mathrm{CM}}_{\mathrm{jk}}^2$ ( ${\mathrm{MC}}_{\mathrm{jk}}^2$ ). For an old machine, ${\mathrm{MC}}_{\mathrm{jk}}^1$ is the depreciation value of it, and for the new machine, ${\mathrm{MC}}_{\mathrm{jk}}^2$ is the purchase price ofthe machine.
$\mathrm{MG}{C}_i,$ $\mathrm{MG}{T}_i$	Average machining cost/time for processing $P_i$ using configured production line when an MPP is selected.
$\mathrm{MG}{C}_{\mathrm{ij}},$ $\mathrm{MG}{T}_{\mathrm{ij}}$	Average machining cost/time for processing one of the operations of product $i$ using $M{S}_j$ when one of the MPPs is selected.
${\mathrm{MGC}}_{\mathrm{iljk}}^1,$ ${\mathrm{MGC}}_{\mathrm{iljk}}^2$ ${\mathrm{MGT}}_{\mathrm{iljk}}^1,$ ${\mathrm{MGT}}_{\mathrm{iljk}}^2$	Machining cost/time for processing $P_i$ using old/new $C{M}_{\mathrm{jk}}$ when $\mathrm{MP}{P}_l$ is selected. If the corresponding old/new machine does not exist in $M{S}_j$ when $\mathrm{MP}{P}_l$ isselected, they can be set as a random and positive number.
${\mathrm{MQMS}}_j^{\max }$	Maximum acceptable machine quantity for $M{S}_j$ .
${\mathrm{MTMS}}_j^{\min },$ ${\mathrm{MTMS}}_j^{\max }$	Minimum, maximum machine type for $M{S}_j$ .
$p_{\mathrm{ilj}}$	Binary parameter to indicate if $M{S}_j$ is used to perform one of the operations of $P_i$ for $\mathrm{MP}{P}_l$ .
	$p_{\mathrm{ilj}}=\{\begin{array}{c}1,\mathrm{if}M{S}_j\mathrm{is}\mathrm{used}\mathrm{to}\mathrm{perform}\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{operations}\mathrm{of}{P}_i\mathrm{when}\hfill \\ \multicolumn{1}{c}{}& \hfill & \mathrm{MP}{P}_l\mathrm{is}\mathrm{selected}\hfill \\ \multicolumn{1}{c}{0,\mathrm{otherwise}}\end{array}$
$P{C}_i$	Product cost of $P_i$ equals summed raw material cost and machining cost.
$q_{\mathrm{ilj}}$	Binary parameter to indicate whether or not $M{S}_j$ is a key machining station regarding $P_i$ to be involved in calculation of work load smoothness index since notall MSs are necessary to be involved in SI.
	$q_{\mathrm{ilj}}=\{\begin{array}{c}1,\mathrm{if}{p}_{\mathrm{ilj}}=1\mathrm{and}M{S}_j\mathrm{is}\mathrm{a}\mathrm{key}\mathrm{station}\mathrm{to}{P}_i\mathrm{when}\mathrm{MP}{P}_l\mathrm{is}\mathrm{selected}\hfill \\ \multicolumn{1}{c}{0,\mathrm{otherwise}}\end{array}$
$\mathrm{RM}{C}_{\mathrm{il}}$	Total raw material cost for product $i$ when $\mathrm{MP}{P}_l$ is selected.
${\mathrm{SI}}_i^{\max }$	Maximum acceptable smooth index of production line for $P_i$ .
${\mathrm{TC}}_{\mathrm{iljk}}^1,$ ${\mathrm{TC}}_{\mathrm{iljk}}^2$	${\mathrm{TC}}_{\mathrm{iljk}}^1$ is the initial cost of tooling (fixture/cutter/program) when its quantity is one, ${\mathrm{TC}}_{\mathrm{iljk}}^2$ is the reproduction cost of the tooling when its quantity is more than one. It is noted that in this research old machine and new one has the same tooling so to same TC.
$X_{\mathrm{jk}}^{\prime }$	Binary parameter to indicate whether or not a particular type of machine is selected.
	$X_{jk}^{\prime }=\{\begin{array}{c}1if\left(X_{jk}^1+X_{jk}^2\right)\ne 0\\ 0if\left(X_{jk}^1+X_{jk}^2\right)=0\end{array},\forall j\in \left(J_p^1{{\displaystyle \cup }}^{}{J}_p^2{{\displaystyle \cup }}^{}\dots {{\displaystyle \cup }}^{}{J}_p^I\right)$
$X_{\mathrm{jk}}^1,X_{\mathrm{jk}}^2$	Quality of machine type $k$ with old/new version in $M{S}_j$ .
	$X_{\mathrm{jk}}^1=\underset{\forall 1\le i\le I}{max}\left\{X_{\mathrm{jki}}^1\right\}$ , $X_{\mathrm{jk}}^2=\underset{\forall 1\le i\le I}{max}\left\{X_{\mathrm{jki}}^2\right\}$
$Z_{\mathrm{jk}}^1$	Maximum quantity of old $C{M}_{\mathrm{jk}}$ that is planned to comprise $M{S}_j$ .
	$Z_{\mathrm{jk}}^1=\{\begin{array}{c}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{old}\mathrm{candidate}\mathrm{machine},\mathrm{if}\mathrm{there}\mathrm{have}\mathrm{old}\mathrm{machines}\hfill \\ \multicolumn{1}{c}{0,\mathrm{if}\mathrm{there}\mathrm{is}\mathrm{no}\mathrm{old}\mathrm{candidate}\mathrm{machines}\mathrm{or}\mathrm{has}\mathrm{no}\mathrm{plan}\mathrm{to}\mathrm{involve}\mathrm{old}}\\ \multicolumn{1}{c}{}& \mathrm{machine}\mathrm{in}\mathrm{the}\mathrm{new}\mathrm{line}\hfill \end{array}$
$Z_{\mathrm{jk}}^2$	Maximum quantity of new $C{M}_{\mathrm{jk}}$ that may buy from outside of the company to comprise $M{S}_j$ .
	$Z_{\mathrm{jk}}^2=\{\begin{array}{c}\inf ,\mathrm{if}\mathrm{there}\mathrm{may}\mathrm{buy}\mathrm{new}\mathrm{machine}\hfill \\ \multicolumn{1}{c}{0,\mathrm{if}\mathrm{there}\mathrm{has}\mathrm{no}\mathrm{plan}\mathrm{to}\mathrm{buy}\mathrm{new}\mathrm{machines}}\end{array}$
Decision variables
$X_{\mathrm{jki}}^1,X_{\mathrm{jki}}^2$	Selection of CMs. $X_{\mathrm{jki}}^1=\{\begin{array}{c}1,2,\dots ,Z_{\mathrm{jk}}^1,\mathrm{if}{X}_{\mathrm{jki}}^1\mathrm{old}C{M}_{\mathrm{jk}}\left(s\right)\mathrm{is}(\mathrm{are})\mathrm{selected}\mathrm{in}\mathrm{M}{\mathrm{S}}_{\mathrm{j}}\mathrm{to}\mathrm{process}{P}_i.\hfill \\ \multicolumn{1}{c}{0,\mathrm{otherwise}}\end{array}$
	$X_{\mathrm{jki}}^2=\{\begin{array}{c}1,2,\dots ,Z_{\mathrm{jk}}^2,\mathrm{if}{X}_{\mathrm{jki}}^2\mathrm{new}C{M}_{\mathrm{jk}}\left(s\right)\mathrm{is}(\mathrm{are})\mathrm{selected}\mathrm{in}\mathrm{M}{\mathrm{S}}_{\mathrm{j}}\mathrm{to}\mathrm{process}{P}_i\hfill \\ \multicolumn{1}{c}{0,\mathrm{otherwise}}\end{array}$
$Y_{\mathrm{il}}$	Selection of MPP, $Y_{\mathrm{il}}=\{\begin{array}{c}1,\mathrm{if}\mathrm{MP}{P}_l\mathrm{of}\mathrm{product}i\mathrm{is}\mathrm{selected}\hfill \\ \multicolumn{1}{c}{0,\mathrm{otherwise}}\end{array}$

Mathematical model

Machining parameters aggregation

Before stating the constraints and objective function, it is necessary to aggregate machining parameters (MGT and MGC) from a single machine to MS level, then to the production line level, since objective and constraints are directly related to these aggregated parameters.

Firstly, we should aggregate parameters of parallel machines to MS level as shown in Figure 3. Since the products are produced in a long production time duration (relative to the cycle time), we can calculate average machining parameters (MGT and MGC) of the product as follows: when $p_{\mathrm{ilj}}=0$ , it means none of the operations of $P_i$ when $\mathrm{MP}{P}_l$ is selected should be processed on $M{S}_j$ , so $\mathrm{MG}{T}_{\mathrm{ij}}$ and $\mathrm{MG}{C}_{\mathrm{ij}}$ can be set as a random and positive number; when $p_{\mathrm{ilj}}=1$ , it means one of the operations of product $i$ when $\mathrm{MP}{P}_l$ is selected should be processed on $M{S}_j$ , machining parameters can be calculated as ²⁸ :

$$\mathrm{MG}{T}_{\mathrm{ij}}=\frac{1}{\sum _{k=1}^{K_j}(\frac{X_{\mathrm{jki}}^1}{{\mathrm{MGT}}_{\mathrm{iljk}}^1}+\frac{X_{\mathrm{jki}}^2}{{\mathrm{MGT}}_{\mathrm{iljk}}^2})}$$

(1)

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{MG}{C}_{\mathrm{ij}}=\frac{\mathrm{total}\mathrm{machining}\mathrm{cost}\mathrm{in}\mathrm{duration}\Delta T}{\mathrm{total}\mathrm{volume}\mathrm{in}\mathrm{duration}\Delta T}}\\ \multicolumn{1}{c}{=\frac{\sum _{k=1}^{K_j}(\frac{\Delta T}{{\mathrm{MGT}}_{\mathrm{iljk}}^1}{X}_{\mathrm{jki}}^1{\mathrm{MGC}}_{\mathrm{iljk}}^1+\frac{\Delta T}{{\mathrm{MGT}}_{\mathrm{iljk}}^2}{X}_{\mathrm{jki}}^2{\mathrm{MGC}}_{\mathrm{iljk}}^2)}{\sum _{k=1}^{K_j}(\frac{\Delta T}{{\mathrm{MGT}}_{\mathrm{iljk}}^1}{X}_{\mathrm{jki}}^1+\frac{\Delta T}{{\mathrm{MGT}}_{\mathrm{iljk}}^2}{X}_{\mathrm{jki}}^2)}}\\ \multicolumn{1}{c}{=\frac{\sum _{k=1}^{K_j}(\frac{X_{\mathrm{jki}}^1{\mathrm{MGC}}_{\mathrm{iljk}}^1}{{\mathrm{MGT}}_{\mathrm{iljk}}^1}+\frac{X_{\mathrm{jki}}^2{\mathrm{MGC}}_{\mathrm{iljk}}^2}{{\mathrm{MGT}}_{\mathrm{iljk}}^2})}{\sum _{k=1}^{K_j}(\frac{X_{\mathrm{jki}}^1}{{\mathrm{MGT}}_{\mathrm{iljk}}^1}+\frac{X_{\mathrm{jki}}^2}{{\mathrm{MGT}}_{\mathrm{iljk}}^2})}}\end{array}$$

(2)

Figure 3.

Aggregation of parallel machines: (a) MS with parallel machines and (b) aggregated model of a MS.

Secondly, machining parameters of the production line level can be calculated as ²⁸ :

$$\mathrm{MG}{T}_i=\underset{\forall j\in J_p^i}{max}\left\{\mathrm{MG}{T}_{\mathrm{ij}}\right\}$$

(3)

$$\mathrm{MG}{C}_i=\sum _{\forall j\in J_p^i}\mathrm{MG}{C}_{\mathrm{ij}}$$

(4)

Objective

The investigated problem simultaneously minimizes the production line’s total capital investment cost (MC and TC) and product’s cost (RMC and MGC). Because of dissimilar measurement ways and dimensions of objective values, we adopt Simple Additive Weighting (SAW) ²⁹ method to aggregate the objective values into one as equation (5).

$$\begin{array}{c}\multicolumn{1}{c}{maxf=}\\ \multicolumn{1}{c}{[w_1(M{C}^{\mathrm{Norm}}+T{C}^{\mathrm{Norm}})+w_2\sum _{i=1}^I\left(w_{2i}{\mathrm{PC}}_i^{\mathrm{Norm}}\right)]}\end{array}$$

(5)

Where $w_1$ , $w_2$ , and $w_{2i}$ are the weight of the corresponding objective, and $w_1+w_2=\sum _{i=1}^I{w}_{2i}=1$ . All the coefficients can be acquired by Analytic Hierarchy Process (AHP) ³⁰ which is a widely used decision-making method. $M{C}^{\mathrm{Norm}}$ , $T{C}^{\mathrm{Norm}}$ , ${\mathrm{PC}}_i^{\mathrm{Norm}}$ is the normed $\mathrm{MC}$ , $\mathrm{TC}$ , and $P{C}_i$ by method of SAW, respectively. $\mathrm{MC}$ , $\mathrm{TC}$ , and $P{C}_i$ can be calculated as equations (6)–(8) respectively.

$$\mathrm{MC}=\sum _{j=1}^J\sum _{k=1}^{K_j}{\mathrm{MC}}_{\mathrm{jk}}^1{X}_{\mathrm{jk}}^1+{\mathrm{MC}}_{\mathrm{jk}}^2{X}_{\mathrm{jk}}^2$$

(6)

$$\mathrm{TC}=\sum _{i=1}^I(\sum _{k=1}^{K_j}\{\begin{array}{c}0,\mathrm{if}{X}_{\mathrm{jki}}^1+X_{\mathrm{jki}}^2=0\hfill \\ \multicolumn{1}{c}{{\mathrm{TC}}_{\mathrm{iljk}}^1+{\mathrm{TC}}_{\mathrm{iljk}}^2*(X_{\mathrm{jki}}^1+X_{\mathrm{jki}}^2-1),\mathrm{if}{X}_{\mathrm{jki}}^1+X_{\mathrm{jki}}^2\ge 1}\end{array}\forall j\in J_p^i)$$

(7)

$$\begin{array}{c}\multicolumn{1}{c}{P{C}_i=\mathrm{RM}{C}_i+\mathrm{MG}{C}_i=\sum _{l=1}^{L_i}\mathrm{RM}{C}_{\mathrm{il}}{Y}_{\mathrm{il}}}\\ \multicolumn{1}{c}{+\sum _{\forall j\in J_p^i}\mathrm{MG}{C}_{\mathrm{ij}}}\end{array}$$

(8)

Constraints

$$\sum _{l=1}^{L_i}{Y}_{\mathrm{il}}=1,\forall i$$

(9)

$$0\le X_{\mathrm{jki}}^1\le Z_{\mathrm{jk}}^1,\forall i,\forall j,\forall k$$

(10)

$$1\le \sum _{k=1}^{K_j}{X}_{\mathrm{jki}}^1+X_{\mathrm{jki}}^2,j\in J_p^i,\forall i$$

(11)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{MTMS}}_j^{\min }\le \sum _{k=1}^{K_j}{X}_{\mathrm{jk}}^{\prime }\le {\mathrm{MTMS}}_j^{\max },}\\ \multicolumn{1}{c}{\forall j\in (J_p^1\cup J_p^2\cup \dots \cup J_p^I}\end{array})$$

(12)

$$\begin{array}{c}\multicolumn{1}{c}{\sum _{k=1}^{K_j}(X_{\mathrm{jk}}^1+X_{\mathrm{jk}}^2)\le {\mathrm{MQMS}}_j^{\max },}\\ \multicolumn{1}{c}{\forall j\in (J_p^1\cup J_p^2\cup \dots \cup J_p^I)}\end{array}$$

(13)

$$\sum _{i=1}^I{\mathrm{DV}}_i^{\min }*\mathrm{MG}{T}_i\le \mathrm{WT}$$

(14)

$$\begin{array}{c}\multicolumn{1}{c}{S{I}_i=\sqrt{\frac{1}{\sum _{\forall j\in J_q^i}{q}_{\mathrm{ilj}}}{\sum _{\forall j\in J_q^i}(\mathrm{MG}{T}_i-\mathrm{MG}{T}_{\mathrm{ij}})}^2}}\\ \multicolumn{1}{c}{\le {\mathrm{SI}}_i^{\max },\forall j\in J_q^i}\end{array}$$

(15)

Equation (9) restricts one and only one MPP is selected for every product. Equation (10) ensures every machine’s quantity should not exceed the maximum number that the company present has. Equation (11) ensures every MS that is selected by one or more products should at least select one machine. Equation (12) implies that total machines types in every MS should be in a range defined by decision-maker. From the view of technical risk and supply security, it is important to have several types of machines so that the MS will not rely on one type of machine too much. At the same time, too many machine types in one MS will lead to various types fixtures, and workers with different skills, which result in complicated management. Equation (13) restricts the quantity of the machines in a MS since in real situations machines quantity is not infinite due to the finite size of the plant or limited investment budget. Equation (14) ensures the total machining time of all products with the required minimum $\mathrm{PV}$ per year should not exceed the stated working time. Equation (15) ensures SI should not too large so that the workload between MS varies too much.

Proposed method

Since the investigated problem is a constrained combination optimization problem that resembles the service composition and optimal selection (SCOS) in cloud manufacturing which is a NP-Hard problem, ³¹ the problem is also NP-hard that cannot be solved within polynomial-time.^32,33 For this type of optimization issue, various intelligent algorithms can be applied to get a satisfactory result because of their acceptable computational complexity.^34,35 In this section, based on an optimization algorithm – Equilibrium Optimizer (EO), ¹² a new algorithm called Enhanced Equilibrium Optimizer (EEO) is presented. The new algorithm has three layers of code and three steps to handling boundaries and constraints and it can better accommodate the proposed problem.

The presented EEO has the same updating mechanism of particle’s position with the original EO, but has two improvements: three layers of code and three steps to handling boundaries and constraints which will be depicted in this part. Some other adaptive adjustments to solve the proposed problem (including initialization and fitness evaluation) are also presented here.

Three layers of code

When applying a meta-heuristic optimization algorithm for discrete optimization problems, the encoding and decoding scheme plays an important role. ³⁶ In this part, we propose a three layers coding scheme and a coding example for the problem in section “Application case study” is shown in Figure 4 (layer 1: MPP selection) and Figure 5 (layer 2 and 3: machine selection).

• (1) Coding layer 1: MPP selection

MPP selection using index defined in section “Notations” to code and one example is shown in Figure 4. The lower boundary (LB) for all products is one (there has at least one MPP for every product), and the upper boundary (UB) is $L_i$ . For the problem in section “Application case study” LB and UB for MPP is [1, 4] for $P_1$ and [1, 2] for $P_2$ .

• (2) Coding layer 2–3: machine selection

Figure 4.

Coding scheme for MPP selection.

Machine selection decision variables ( $X_{\mathrm{jki}}^1\mathrm{and}{X}_{\mathrm{jki}}^2$ ) have been defined in section “Notations” spontaneously we code as what variables indicated as the following way. First, selecting machines for every product. Then making a union of all machines selected by every product to form the machine composition of MS. Finally we check machines in every MS whether it meets MTMS and MQMS constraints before calculating fitness value and if not, we need to regenerate or make a punishment to the fitness. This approach is a way of “generation and then checking,” which is inefficient since we won’t know if the machine composition for one MS is legal until we have generated every product’s machine selection and made a union. And this way is hard to have a legal code by controlling the separate product’s machine selection to ensure the MS’s machine selection is legal.

Due to the drawback of the method mentioned above, we proposed a two layers machine selection code scheme for machine selection. In coding layer 2, we select machines for MS (much easier to meet MTMS and MQMS constraints), and in coding layer 3 we select machines among selected ones in MS in coding layer 2. Using this scheme, a machine selection code example of the problem in section “Application case study” is shown in Figure 5.

Figure 5.

Proposed coding scheme for machine selection.

In coding layer 2, we select machine by selecting machine’s type and quantity. For machine type selection layer, the code represents machine type selection index which is generated at the very beginning of the algorithm and the generating method is as follows. Listing all possible machine type combinations by selecting ${\mathrm{MTMS}}_j^{\min }$ types of machines from the candidate $K_j$ types of machine, ${\mathrm{MTMS}}_j^{\min }$  + 1 types of machines from the candidate $K_j$ types of machine, …, ${\mathrm{MTMS}}_j^{\max }$ types of machines from the candidate $K_j$ types of machine, then listing all combinations together in sequential order with an index number which is used to represent selected machine types. In this way, the LB and UB of machine type selection is 1, and the maximum index of machine combinations respectively. So, for MS₁₆ ${\mathrm{MTMS}}_j^{\min }$  = 2, ${\mathrm{MTMS}}_j^{\max }=2$ , we select two types of machines in the four types of CMs, all legal machine combinations are shown in Table 2. So, the LB and UB of machine type selection is 1 and 6 respectively. In this way, the code “4” represents the fourth machine type combinations that represent machine type “2” and “3” are selected.

Table 2.

All machine combinations of MS₁₆ for problem in section “Application case study”

Machine type index	1	2	3	4	5	6	7	8	9
Machine type combinations	1; 2	1; 3	1; 4	2; 3	2; 4	3; 4	1; 2; 3	1; 3; 4	2; 3; 4

For machine quantity selection layer, the number represents the quantity of the corresponding machines. LB of machine quantity selection for every machine is 0, and UB is the smaller one between MQMS and $Z_{\mathrm{jk}}^1$ for old machine and smaller one between MQMS and $Z_{\mathrm{jk}}^2$ for the new machine. Note that, only machine types that appear in the type selection are active to be selected by product and the rest types of machines that are not selected will be blocked and the corresponding quantity is zero. So, in the example, only machine types “2” and “3” are selected and its quantity is the UB of the machine selection for every product. As for machine types “1” and “4”, they are blocked (shown in gray background) and the actual quantity is zero regardless of the selected quantity. By this analysis, LB and UB of coding layer 3 is shown in (c) of Figure 5.

coding layer 3 is the machine selection for every product. Its coding number also represents the quantity of the corresponding machines and the boundary is determined by layer 2 which is shown in (c) (the code with gray background means it is blocked due to it is not selected in the machine type selection). We can generate actual machine composition for MS as shown in (d) of Figure 5 by making a union operation of every product’s machine selection. Note that, in the process of making the union, only when a product should be processed on this MS ( $q_{\mathrm{ilj}}=1$ ), the product’s selected machine counts, otherwise, its machine selection is meaningless and will be treated as zero.

Initialization

To generate the initial particle’s concentration more diverse, randomly generation method is adopted.

Three steps to handling boundaries and constraints

To accommodate the proposed coding scheme, a three steps to handling boundaries and constraints procedure is presented to handle MTMS and MQMS constraints and its’ framework is shown in Figure 6. The following is the constraint check procedure of a single MS.

• (1) Step 1: basic boundary check

In this step, basic boundary check is conducted for MPP code in layer 1, machine type and machine quantity code in layer 2, and their corresponding LB and UB are described in section “Three layers of code” If a code beyond the boundary, then a randomly generated code will replace the illegal one. Actually, for the initial solution, all the codes are in the range of LB and UB. However, for the updated solutions, some codes may beyond the boundary because of operations of the algorithm.

Figure 6.

Three steps to handling boundaries and constraints.

• (2) Step 2: MTMS and MQMS constraints check for MS

This step aims to make sure all MS’s machine composition satisfy MTMS (step 2-1) and MQMS (Step 2-2) constraints in layer 2 so to provide satisfied candidate machines for the machine selection of products.

For MTMS constraint, we have designed machine type selection code that definitely satisfies MTMS constraint. However, if the corresponding machine quantity for all selected types of machine are zero, even this type of machine is selected in the type selection code, it selects no machine actually. So, we need to check if it satisfies MTMS constraint, if not, we randomly regenerate the machine quantity code. The whole process is shown in Step 2-1 of Figure 6.

For MQMS constraint check, if the quantity of selected type of machine $\mathrm{MQM}{S}_j$  >  ${\mathrm{MQMS}}_j^{\max }$ , then a number smaller than one: rand(0,1) × ( ${\mathrm{MQMS}}_j^{\max }$ / $\mathrm{MQM}{S}_j$ ) is adopt. All machine quantity codes in $M{S}_j$ time this number and round the scaled codes into integers. By this means, the machine quantity codes will be legal most of the time but it may result in illegal machine type since during scaling process some machine’s quantity may be scaled down to zero which may eliminate some machine type. So, the scaled-down machine quantity codes need to check machine type is legal or not again like in Step 2-1.

After Step 1 and Step 2, MTMS and MQMS constraint will be satisfied for MS’s machine code which will be served as candidate machine for all parts to select in Step 3.

• (3) Step 3: final MTMS constraint check for MS

Final machine composition in MS is the union of every product’s selected machine (layer 3) which is selected among candidate machines in layer 2. Although candidate machines in layer 2 satisfy MTMS and MQMS constraint, the union of every product’s selected machine may not satisfy MTMS constrain. Because in case of some machines in layer 2 is not selected by all products, final machine type in MS may smaller than ${\mathrm{MTMS}}_j^{\min }$ . To dual with this issue, random regeneration procedure until machines satisfy MTMS constraint is conducted.

Fitness evaluation

For the solution that meets WT (DV constraint) and ${\mathrm{SI}}_i^{\max }$ (SI constraint), its fitness is the same as the objective function value defined in section “Mathematical model” ; if not, the corresponding fitness set as an adequate small number (such as 0.01).

Numerical experiments and application case study

In this section, two kinds of experiments are conducted. For numerical experiments, 18 randomly generated problems of different scales (generation method see in Supplemental File A) are used to validate the proposed model and method. For the real case study, a case motivated by actual demand from our industrial partner is conducted to show the application ability of the proposed model and method.

All experiments are conducted with MATLAB and for every problem, 10 independent runs are carried out to try to eliminate the randomness. ³⁷

Numerical experiments

Parameters setting for EEO

In this section, we investigate the optimum selection of four key parameters for EEO. They include Generation probability (GP), the number of equilibrium candidate (NEC), population size (POP), and maximum iterations (ITER). We adopt the Taguchi method of Design of experiments (DOE) ³⁸ to decide the optimum combination of parameters based on the most complicated problem Pro_18. Four levels of each parameter is considered in this research, that is, GP = [0.2, 0.4, 0.6, 0.8], NEC = [3, 4, 5, 6], POP = [200, 400, 600, 800], ITER = [300, 600, 900, 1200].

The relative percentage decrease (RPD) ³⁹ is used as a performance measure indicator and is computed as follows:

$$\mathrm{RP}{D}_{\mathrm{AF}}=\frac{B{F}_a-\mathrm{AF}}{B{F}_a}\times 100$$

(16)

$$\mathrm{RP}{D}_{\mathrm{BF}}=\frac{B{F}_a-\mathrm{BF}}{B{F}_a}\times 100$$

(17)

Where $B{F}_a$ is the best fitness among all fitness, $\mathrm{AF}$ is the average fitness of 10 runs that are found by a particular algorithm, $\mathrm{BF}$ is the best fitness found by a particular algorithm among 10 runs. So, $\mathrm{RP}{D}_{\mathrm{AF}}$ is $\mathrm{AF}\prime \mathrm{s}$ relative percentage decrease toward $B{F}_a$ , $\mathrm{RP}{D}_{\mathrm{BF}}$ is $\mathrm{BF}\prime \mathrm{s}$ relative percentage decrease toward $B{F}_a$ . By definition, we can see both of them is the smaller, the better.

The orthogonal array L₁₆ (i.e. 16 groups of parameter combination are used to conduct the numerical experiments) and the corresponding $\mathrm{RP}{D}_{\mathrm{AF}}$ and $\mathrm{RP}{D}_{\mathrm{BF}}$ values are listed in Table 3. Furthermore, figure of the trend of the level of the parameters can be obtained as Figure 7.

Table 3.

Original Taguchi table, the corresponding $\mathrm{RP}{\mathrm{D}}_{\mathrm{AF}}$ and $\mathrm{RP}{\mathrm{D}}_{\mathrm{BF}}$ values.

Exp_no	Parameter				$\mathrm{RP}{D}_{\mathrm{AF}}$	$\mathrm{RP}{D}_{\mathrm{BF}}$
	GP	NEC	POP	ITER
1	0.2	3	200	300	3.719	4.338
2	0.2	4	400	600	2.304	3.117
3	0.2	5	600	900	1.016	2.012
4	0.2	6	800	1200	1.691	2.064
5	0.4	3	400	900	1.701	2.186
6	0.4	4	200	1200	1.113	2.388
7	0.4	5	800	300	1.858	2.859
8	0.4	6	600	600	0.950	1.783
9	0.6	3	600	1200	0.729	1.532
10	0.6	4	800	900	0.972	1.623
11	0.6	5	200	600	1.424	2.254
12	0.6	6	400	300	2.029	3.104
13	0.8	3	800	600	1.182	1.997
14	0.8	4	600	300	1.438	2.081
15	0.8	5	400	1200	0.000	0.820
16	0.8	6	200	900	0.864	1.585

Figure 7.

Trend of the level of the parameters in the EO.

Based on the trend of each factor regarding $\mathrm{RP}{D}_{\mathrm{AF}}$ and RPD_BF shown in Figure 7, both figures indicate the same best combination of parameters for EEO as GP = 0.8, NEC = 5, POP = 600, ITER = 1200. As for other parameters, they remain same as it is in the original paper.

Comparison of different algorithms

In this section, a comprehensive comparison experiment is conducted on 18 randomly generated problems to find out the best algorithm among the proposed EEO, widely used GWO, and three recently proposed algorithms: AOA, ⁴⁰ SMA, ⁴¹ and MPA. ⁴² To have a relatively fair comparison, all of the other four algorithms have the same coding and encoding, initialization, and boundary and constraints handling methods, population size and maximum iterations with EEO. The other parameter settings of these algorithms remain the same as those in the literature.

The experimental results of the 18 problems solved separately by five algorithms are shown in Table 4. For $\mathrm{RP}{D}_{\mathrm{AF}}$ , EEO can find two best fitness and is no worse than the other four; for $\mathrm{RP}{D}_{\mathrm{BF}}$ , EEO can find 12 best fitness which is the best of five algorithms, and the fitness near the optimal results can be produced by EEO for the rest of the problems. Moreover, the mean $\mathrm{RP}{D}_{\mathrm{AF}}$ and $\mathrm{RP}{D}_{\mathrm{BF}}$ value is 1.0797 and 0.1529, which is the lowest value in all five algorithms.

Table 4.

Experimental results of the 18 problems solved by five algorithms.

Pro-NO	$\mathrm{RP}{D}_{\mathrm{AF}}$					$\mathrm{RP}{D}_{\mathrm{BF}}$
	EEO	GWO	AOA	SMA	MPA	EEO	GWO	AOA	SMA	MPA
1	0	0	0	0	0	0	0	0	0	0
2	0	0	0	0.0184	0	0	0	0	0	0
3	0.0740	0.0434	0.8970	2.2129	0.0008	0	0	0.4109	1.3385	0
4	0.7058	0.7453	1.4930	2.7991	0.0738	0	0.1623	0.8552	2.0950	0
5	0.0479	0.1926	1.1425	2.4068	0.0458	0	0.0946	0.5105	1.7665	0
6	0.9182	1.6210	5.4278	5.6702	1.9138	0	0.8827	4.6658	4.9743	0.9376
7	1.0356	0.3235	3.5538	5.2407	0.6582	0.7470	0	2.4779	4.0990	0
8	1.9102	3.0119	5.6753	6.8972	0.8757	0.4045	1.6884	4.5396	5.9537	0
9	1.4752	1.8410	6.7017	7.7073	1.8095	0	0.8811	5.3847	6.3809	0.5291
10	1.0838	1.3757	3.6618	4.9307	0.3704	0.4345	0.6791	3.3933	4.2495	0
11	1.5002	3.7098	8.4974	9.8611	2.2863	0	2.7401	7.6270	8.3319	0.8991
12	1.9230	3.4002	8.5712	9.2483	2.7774	0	2.5562	7.6441	8.6085	1.5870
13	1.4285	3.0873	7.5501	8.0193	1.4960	0.4088	2.1461	6.8784	7.0536	0
14	0.8285	2.5514	7.2132	10.0202	1.0694	0	1.3208	6.6682	8.9414	0.3592
15	1.7141	2.7206	10.3290	12.3280	1.2466	0.2168	1.3791	9.8051	11.3009	0
16	1.7250	4.1193	8.3235	7.9751	1.7293	0	2.1479	7.2783	6.4293	0.4930
17	1.6675	2.5807	10.3304	11.5279	3.0178	0.5400	0.8766	10.0356	10.7601	0
18	1.3963	2.8343	7.9883	10.0228	1.8746	0	1.9229	7.3259	8.8856	0.6893
Mean	1.0797	1.8977	5.4087	6.4937	1.1803	0.1529	1.0821	4.7500	5.6205	0.3052

As is shown in Figure 8, the box plots of $\mathrm{RP}{D}_{\mathrm{AF}}$ and $\mathrm{RP}{D}_{\mathrm{BF}}$ for five algorithms in Table 4 is given which will confirm EEO’s performance. For $\mathrm{RP}{D}_{\mathrm{AF}}$ and $\mathrm{RP}{D}_{\mathrm{BF}}$ , apparently, EEO has a smaller range and median than GWO, AOA, and AMA. As comparison with MPA, EEO and has a much smaller range but only a little larger median than MPA in terms of $\mathrm{RP}{D}_{\mathrm{AF}}$ , and a smaller range with nearly the same median than MPA in terms of $\mathrm{RP}{D}_{\mathrm{BF}}$ .

Figure 8.

Boxplot of $\mathrm{RP}{D}_{\mathrm{AF}}$ and $\mathrm{RP}{D}_{\mathrm{BF}}$ for fives algorithms.

To further validate EEO’s performance and differences statistically, the Friedman test ⁴³ is utilized and the results are shown in Table 5. From the table we can see that EEO has the highest priority and best mean rank than the rest of four algorithms in terms of both $\mathrm{RP}{D}_{\mathrm{AF}}$ and $\mathrm{RP}{D}_{\mathrm{BF}}$ . Furthermore, the p-value in Friedman test is 1.0670E-10 and 3.4783E-10 of $\mathrm{RP}{D}_{\mathrm{AF}}$ and $\mathrm{RP}{D}_{\mathrm{BF}}$ , which are far less than the level of significant α = 0.05, and this confirms the differences among the five algorithms are significant.

Table 5.

The Friedman tested results (the level of significant α = 0.05).

Algorithm	$\mathrm{RP}{D}_{\mathrm{AF}}$		$\mathrm{RP}{D}_{\mathrm{BF}}$		-
	Mean rank	Priority	Mean rank	Priority	Final priority
EEO	1.6944	1	1.7222	1	1
GWO	2.6944	3	2.8056	3	3
AOA	3.9167	4	3.9444	4	4
SMA	4.8333	5	4.7222	5	5
MPA	1.8611	2	1.8056	2	2
Test statistics	Friedman
N	18		18		-
Chi-Square	52.53		50.08		-
df	4		4		-
p-Value	1.0670E-10		3.4783E-10		-

Through the experiments and a series analysis of results above, we can conclude that EEO demonstrates an outstanding performance regarding the investigated MDFRM.

Application case study

An application case study that comes from our industrial partner is shown below to demonstrate the applicability of the proposed MDFRM and EEO. The partner sells more than 1 million residential water heaters yearly in China. As one the most important part of a residential storage water heater, the inner water tank is used to store hot water. The investigated production line can produce two kinds of the inner water tank in separate batch, since they have some common operations (shares some MS). Now, due to the reduced machining performance of some machines, as well as the emerging of well-equipped machines and product’s demanding volume change, there is a need to reconfigure the production line to seek better performance.

The two kinds of the inner water tank are shown in Figure 9, and their alternative network representation of MPP is shown in Figure 10, in which the meaning of the number is MS index j that can be found in Table 6. In Figure 10, MPP that starts and ends with the same OR-connector, only one of the OR paths will be chosen. As for the links that are not connected by OR-connectors, all of them shall be traversed. We use number “1” to indicate a product should be processed on the corresponding operation, “0” to show not necessary. In this way, Figure 10 can be converted into the second row in Table B1 of Supplemental File B in which $\mathrm{MP}{P}_l$ represents the $l^{\mathrm{th}}$ MPP. It should be noticed that, although the demonstrative line in section “Problem description” only has a serial structure, the proposed model can handle merging structure (assembly line) and separating structure (disassembly line) ²⁸ likewise.

Figure 9.

Investigated two inner water tank.

Figure 10.

Network representation of alternative MPP of the two products.

Table 6.

The representation of MS index j.

j	Operation description
1	Decoil of steel roll for body part
2	Cut into fixed length
3	Stamp pipe connector hole
4	Roll for tank body
5	Weld straight seam
6	Enlarge the body edge
7	Decoil of steel roll for up and bottom part
8	Cut into fixed length
9	Stamp for up part
10	Weld bracket for up part
11	Stamp for bottom part
12	Weld bracket for bottom part
13	Weld round seam
14	Weld pipe connector
15	Weld body bracket
16	Leak test

After implementing EEO with 10 runs, the results are: 0.5765, 0.5765, 0.5765, 0.5765, 0.5629, 0.5736, 0.5744, 0.5715, 0.5736, 0.5765. Among 10 results, the best fitness is 0.5765 which appears in the first, second, third, fourth, and tenth run, we choose the first run and the convergence curve of it is given in Figure 11. The corresponding encoded solution is shown in Figure 12. Based on the code and the network representation of the MPP in Figure 10 we can form the final production line with machine composition and every product’s machine selection which is shown in Figure 13.

Figure 11.

Convergence curve of the best fitness.

Figure 12.

Code of the best solution.

Figure 13.

Obtained the best production line configuration.

Conclusion and future work

Multi-part Dedicated Flow-line (MDF) is widely used to produce discrete products for high volumes with low varieties and its reconfiguration problem is of great importance. Present small adjustments such as machine resettlement or buffer size rearrangement cannot work well if MPP of products is outdated and/or present machine’s performance is low. It’s time to reconfigure the line in a higher level by reselecting MPP of products and/or reselecting machines. In this paper, we develop an effective Multi-part Dedicated Flow-line Reconfiguration Model (MDFRM) to help production line owner make a scientific reconfiguration decision when great variations occur in product’s MPP and machines. The model considering multi-model products each with several optional MPPs, old (present-having) as well as new (potential-buying) candidate machines, pursues minimum weighted objective value including production line’s capital investment (machine and tooling cost) and product’s cost (raw material and machining cost). At the same time, every product’s demanding volume, production line’s workload balance, and some other realistic constraints are considered. To solve the proposed MDFRM, an Enhanced Equilibrium Optimizer with three layers of code and three steps to handling boundaries and constraints is proposed. Numerical experiments are conducted based on randomly generated problems, which effectively validate the proposed new model and designed solution. Finally, an industrial case study is conducted to further show the applicability of the model and method for reconfiguration of an inner water tank manufacturing line.

For future work, buffer size between MS can be involved since it influences the throughput capacity when considering variation of actual machining time and machine break down. ⁴⁴ Furthermore, pursuing optimization of two or more objectives simultaneously to find a set of nondominated solutions (Pareto set)^45,46 is also a research direction for the involved problem.

Supplemental Material

sj-docx-1-pib-10.1177_09544054221100873 – Supplemental material for An effective multi-part dedicatedflow-line reconfiguration model considering the optimal selection of machining process path and machines

Supplemental material, sj-docx-1-pib-10.1177_09544054221100873 for An effective multi-part dedicatedflow-line reconfiguration model considering the optimal selection of machining process path and machines by Hao Yu, Bo Yang, Shilong Wang, Yankai Wang, Sibao Wang, Zehua Wang and Zhongning Wang in Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture