**2. Materials and Methods**

### *2.1. Basic Input*

Different approaches for the GCF take different approaches to the definition of alternative process plans or routings. Vin and Delchambre [54] classify the approaches by sorting the processes or routings into six categories: (i) fixed routing (process route), (ii) routing with replicate machines, (iii) routing with alternate machines for some operations, (iv) several fixed routings, (v) fixed process plan, and (vi) alternative process plans. In this paper, we take the combination of manners (ii) and (iv) to define alternative process plans to model the GCF problem.

To model the GCF problem, the binary part–machine incidence matrix (PMIM), which represents the association between the process plans of parts and machines, will be used as a basic input. Given *m* part types with a total of *<sup>t</sup>* process plans and *<sup>n</sup>* different machine types, the *<sup>t</sup>*×*<sup>n</sup>* binary PMIM *<sup>A</sup>* - = *airj* is defined as follows:

$$a\_{\dot{r}\dot{r}} = \begin{cases} 1 & \text{if process plan } r \text{ of part } \dot{r} \text{ is processed on machine } \dot{f} \\ 0 & \text{otherwise.} \end{cases}$$

In addition, the following indices, notation, and decision variables will be used throughout the paper:

Indices

*i* = part index;

*r* = process plan index;

*j* = machine or cell index;

*k* = copy index of replicate machine;

*c* = machine cell/part family index.

Parameters

*m* = number of part types;

*Ri* = set of process plans of part type *i*;

*t* = total number of process plans;

*n* = number of different machine types;

*p* = number of cells;

*U* = upper limit on the cell size;

*DM* = set of replicate machine types;

*Mj* = set of copies of replicate machine type *j* ∈ *DM*;

*q* = *n* − |*DM*| + *j*∈*DM Mj* <sup>=</sup> total number of machines including copies of replicate machine types (the symbol


*gsj*1,*j*<sup>2</sup> = generalized similarity coefficient between machine types *j*<sup>1</sup> and *j*2;

*cj*1,*j*<sup>2</sup> = similarity coefficient between machines *j*<sup>1</sup> and *j*2;

*MCc* = set of machines in machine cell *c;*

*PFc* = set of parts in part family *c;*

*e* = total number of 1s in the block diagonal solution matrix;

*e*<sup>0</sup> = number of exceptional elements(EEs) in the block diagonal solution matrix;

*ev* = number of voids in the block diagonal solution matrix.;

Decision variables

$$\begin{aligned} \mathbf{x}\_{\text{irc}} &= \begin{cases} 1 & \text{if process plan } r \text{ of part } i \text{ is assigned to cell; } c\\ 0 & \text{otherwise.} \end{cases} \\ \mathbf{y}\_{jkc} &= \begin{cases} 1 & \text{if copy } k \text{ of predicate machine } j \text{ is assigned to cell; } c \text{ is} \\ 0 & \text{otherwise.} \end{cases} \\ \mathbf{z}\_{j\_1, j\_2} &= \begin{cases} 1 & \text{if machine } j\_1 \text{ belongs to cell; } j\_2 (j\_1, j\_2 = 1, \dots, q) \\ 0 & \text{otherwise.} \end{cases} \end{aligned}$$

Then, the 0–1 GCF problem can be illustrated with a generalized 0–1 PMIM as shown in Figure 1a, in which each row indicates a part and each column a machine. The manufacturing system shown in Figure 1a has eight part types with a total of 19 process plans and five different machine types with an extra copy for machine type 3. An entry of "1" indicates that a part is processed by its associated machine, and an entry of "0", which is not shown for visual convenience, indicates that it is not processed by its associated machine. The alphabetical letters after the part numbers indicate alternative process plans. Rearranging the rows and columns in such a way that only one process plan is selected for each part and only one copy is allowed for each different machine type in each cell results in a block diagonal solution matrix, as shown in Figure 1b. The solution of Figure 1b shows two machine cells and two part families. Machine cell 1 (MC1), consisting of machines 1, 3, and 5 (MC1 = {1, 3, 5}), processes plans 2a, 3c, 5a, 6b, and 9a of part family 1 (PF1) (PF1 = {2a, 3c, 5a, 6b, 9a}). MC2, consisting of machines 2, 4 and an extra copy of machine type 3 (MC2 = {2, 4, 3}), processes plans 1b, 4a, 7b, and 8c of PF2 (PF2 = {1b, 4a, 7b, 8c}). As a result, the solution of Figure 1b yields no EEs and four voids.

**Figure 1.** (**a**) Initial generalized part–machine incidence matrix (PMIM=; (**b**) block diagonal solution matrix.

#### *2.2. Performance Measure*

Several comprehensive grouping efficiency measures considering both EEs and voids have been proposed to evaluate the quality of 0–1 block diagonal solutions and are reviewed critically [55,56]. Of those measures, the grouping efficacy (GE) [10] has been most widely used to evaluate the performance of the 0–1 GCF problem as well as the 0–1 SCF problem. The GE, Γ, is defined as

$$I = \frac{\text{total no. of 1s in the block diagonal matrix} - \text{EEs}}{\text{total no. of 1s in the block diagonal matrix} + \text{voids}} = \frac{\varepsilon - \varepsilon\_0}{\varepsilon + \varepsilon\_v}. \tag{1}$$

According to the above definition, the block diagonal solution of Figure 1b gives a GE of 85.19%. A GE of 100% indicates a perfect CF without EEs and zeros.

#### *2.3. Mathematical Models*

#### 2.3.1. Exact Model

Several authors have provided exact formulations maximizing the grouping efficacy in the 0–1 SCF problem with a single copy of each machine type [11–18]. In this subsection, we present an exact formulation extended for the 0–1 GCF problem with multiple copies of replicate machine types.

To construct the exact formulation which directly maximizes the GE, the values of *e*, *e*0, and *ev* should be calculated. They are given by the following equations:

$$\varepsilon = \sum\_{c=1}^{p} \sum\_{i=1}^{m} \sum\_{r \in R\_i} \sum\_{j=1}^{n} \sum\_{k \in M\_j} a\_{irj} \chi\_{\text{irc}} \mathcal{Y}\_{jkc} \,. \tag{2}$$

$$\begin{split} c\_{0} &= \sum\_{c=1}^{p} \sum\_{i=1}^{m} \sum\_{r \in R\_{i}} \sum\_{j=1}^{n} a\_{irj} \mathbf{x}\_{irc} \bigg| \sum\_{k \in M\_{j}} \left( 1 - y\_{jkc} \right) - \left( \left| M\_{j} \right| - 1 \right) \bigg| \\ &= \sum\_{c=1}^{p} \sum\_{i=1}^{m} \sum\_{r \in R\_{i}} \sum\_{j=1}^{n} a\_{irj} \mathbf{x}\_{irc} \bigg| \mathbf{1} - \sum\_{k \in M\_{j}} y\_{jkc} \bigg| \\ &= \sum\_{c=1}^{p} \sum\_{i=1}^{m} \sum\_{r \in R\_{i}} \sum\_{j=1}^{n} a\_{irj} \mathbf{x}\_{irc} - \sum\_{c=1}^{p} \sum\_{i=1}^{m} \sum\_{r \in R\_{i}} \sum\_{j=1}^{n} a\_{irj} \mathbf{x}\_{irc} \mathbf{y}\_{jic} \end{split} \tag{3}$$

$$e\_{\upsilon} = \sum\_{c=1}^{p} \sum\_{i=1}^{m} \sum\_{r \in R\_i} \sum\_{j=1}^{n} \sum\_{k \in M\_j} \left(1 - a\_{irj}\right) \mathbf{x}\_{irc} \mathbf{y}\_{jkc} \tag{4}$$

Note that once a process plan *r* of part *i* is processed on a copy *k* of replicate machine type *j* in cell *c*, the remaining - *Mj* <sup>−</sup> <sup>1</sup> elements with *airj* = 1 processed by different copies of that replicate machine type in other cells are not the EEs.

Then, the mathematical model directly maximizing the GE is formulated as follows: (Model 1)

$$\text{Maximize } \Gamma = \frac{\sum\_{c=1}^{p} \sum\_{i=1}^{m} \sum\_{r \in R\_i} \sum\_{j=1}^{n} a\_{irj} x\_{irr} y\_{jk}}{\sum\_{c=1}^{p} \sum\_{i=1}^{m} \sum\_{r \in R\_i} \sum\_{j=1}^{n} \sum\_{k \in M\_j} a\_{irj} x\_{irc}} \quad \text{(5)}$$

Subject to

$$\sum\_{c=1}^{p} \sum\_{r \in R\_i} x\_{irc} = 1, \; i = 1, \; \cdots \; \prime m \tag{6}$$

$$\sum\_{c=1}^{p} \sum\_{k \in M\_j} y\_{jkc} = 1, \; j = 1, \dots, m \tag{7}$$

*Appl. Sci.* **2020**, *10*, 3478

$$\sum\_{k \in M\_j}^p y\_{jk\mathfrak{c}} \le 1, \ j \in DM, \ j = 1, \dots, n; \mathfrak{c} = 1, \dots, p \tag{8}$$

$$\sum\_{j=1}^{n} \sum\_{k \in M\_j} y\_{jkc} \le \mathcal{U}\_\prime c = 1, \dots, p \tag{9}$$

$$\mathbf{x}\_{\text{irr}}, y\_{j\&} = 0 \text{ or } 1, r \in \mathbb{R}\_i, i = 1, \dots, m; k \in M\_{j\text{\textquotedblleft}j} = 1, \dots, n; c = 1, \dots, p. \tag{10}$$

The objective function in Equation (5) maximizes the GE. Constraint (6) ensures that only one process plan of each part is assigned to only one cell. Constraint (7) ensures that each machine belongs to exactly one machine cell. Constraint (8) ensures that at most a single copy of a replicate machine type is assigned to each cell. Constraint (9) ensures that the number of machines in each cell does not exceed *U* machines. Constraint (10) ensures the binary restriction of variables.

Model 1, which is a non-linear 0–1 fractional programming model, can be linearized by introducing the following auxiliary binary variables:

$$w\_{\overline{i}r\overline{j}\overline{k}\overline{c}} = x\_{\overline{i}r}y\_{\overline{j}\overline{k}\overline{c}}, \; r \in \mathbb{R}\_i, i = 1, \cdots, m; \\ k \in M\_{\overline{j}r}, j = 1, \cdots, n; \\ c = 1, \cdots, p. \tag{11}$$

To linearize the variable *wirjkc*, the following extra constraints should be added [57,58]:

$$w\_{irjk\varepsilon} - \mathbf{x}\_{ir\varepsilon} - y\_{jik\varepsilon} \ge -1.5, r \in R\_i, i = 1, \dots, m; k \in M\_j, j = 1, \dots, n; \varepsilon = 1, \dots, p. \tag{12}$$

$$1.5w\_{\bar{r}\bar{\eta}\bar{x}} - x\_{\bar{r}\varepsilon} - y\_{\bar{\beta}\bar{x}} \le 0,\\ r \in \mathbb{R}\_i, i = 1, \cdots, m; k \in M\_{\bar{\jmath}\iota}; j = 1, \cdots, n; \varepsilon = 1, \cdots, p. \tag{13}$$

Then, we have the linear 0−1 fractional programming model 2 which is equivalent to model 1 as follows:

(Model 2)

$$\text{Maximize } \Gamma = \frac{\sum\_{c=1}^{p} \sum\_{i=1}^{m} \sum\_{r \in R\_i} \sum\_{j=1}^{n} \sum\_{k \in M\_j} a\_{irj} w\_{irj\&c}}{\sum\_{c=1}^{p} \sum\_{i=1}^{m} \sum\_{r \in R\_i} \sum\_{j=1}^{n} \sum\_{k \in M\_j} a\_{irj} \mathbf{x}\_{ir\mathbf{c}} + \sum\_{c=1}^{p} \sum\_{i=1}^{m} \sum\_{r \in R\_i} \sum\_{j=1}^{n} \sum\_{k \in M\_j} \left(1 - a\_{irj}\right) w\_{irj\&c}} \tag{14}$$

Subject to Equations (6)–(13) and

$$w\_{\overline{i}r\overline{j}\overline{k}\overline{c}} = 0 \text{ or } 1, r \in R\_i, i = 1, \cdots, m; k \in M\_{\overline{j}}, j = 1, \cdots, n; c = 1, \cdots, p. \tag{15}$$

Model 2 can then be solved by using a solver such as LINGO adopting the branch-and-bound algorithm. However, the computational burden of optimally solving model 2 seems still to be heavy even if a powerful solver is used due to the presence of too many binary variables and constraints. The total number of binary variables of model 2 is

$$\left| \sum\_{j=1}^{n} |\mathcal{M}\_{j}| + \sum\_{i=1}^{m} |\mathcal{R}\_{i}| + \left( \sum\_{j=1}^{n} |\mathcal{M}\_{j}| \right) \left( \sum\_{i=1}^{m} |\mathcal{R}\_{i}| \right) \right| p^{-1}$$

and the number of constraints is

$$m + m + (|D\mathcal{M}| + 1)p + 2p\left[\sum\_{j=1}^{n} |\mathcal{M}\_j| + \sum\_{i=1}^{m} |\mathcal{R}\_i| + \left(\sum\_{j=1}^{n} |\mathcal{M}\_j|\right)\left(\sum\_{i=1}^{m} |\mathcal{R}\_i|\right)\right]$$

For example, to solve a large-sized 0–1 GCF problem containing 110 machines, 120 part types, and 248 process plans—which will be tested in Section 4—331,656 binary variables and 663,554 constraints are needed. Therefore, relying on an alternative model with much fewer binary variables

and constraints that leads to the maximization of the GE can be a better strategy. Thus, we use the PMP-type model to solve a large-sized GCF problem by maximizing the GE indirectly.

#### 2.3.2. PMP-Type Model

Since the exact model 2 has too many binary variables and constraints to optimally solve large-sized GCF incidences within a reasonably short computation time, we use the PMP-type model as a better alternative formulation to maximize the GE indirectly. However, to develop the PMP-type model of GCF, we need the definition of similarity coefficients incorporating both alternative process plans and replicate machines. In this paper, we use a modified version of Won and Kim's similarity coefficient [59] defined between pairs of machine types to formulate the mathematical model of the GCF. Won and Kim's similarity coefficient based on the binary PMIM is a generalization of the Jaccard similarity coefficient used for the SCF problem. Their similarity coefficient *gsj*1,*j*<sup>2</sup> between two machine types *j*<sup>1</sup> and *j*<sup>2</sup> is defined by

$$\text{gsg}\_{j\_1, j\_2} = \frac{\sum\_{i=1}^{m} \beta(i, j\_1, j\_2)}{\sum\_{i=1}^{m} \alpha(i, j\_1) + \sum\_{i=1}^{m} \alpha(i, j\_2) - \sum\_{i=1}^{m} \beta(i, j\_1, j\_2)} \tag{16}$$

where

$$\alpha(i,j\_1) = \begin{cases} 1 & \text{if } a\_{irj\_1} = 1 \text{ for some } r \in R\_i \\ 0 & \text{otherwise} \end{cases}$$

$$\beta(1)\beta(i,j\_1, j\_2) = \begin{cases} 1 & \text{if } a\_{irj\_1} = a\_{irj\_2} = 1 \text{ for some } r \in R\_i \\ 0 & \text{otherwise} \end{cases}$$

However, since the above similarity coefficient does not consider the replicate machines, we use Won and Logendran's similarity coefficient [48] to formulate the PMP-type model of GCF. The similarity coefficient *gsj*1,*j*<sup>2</sup> between two machines *j*<sup>1</sup> and *j*<sup>2</sup> is defined as follows:

$$\varepsilon\_{j\_1, j\_2} = \begin{cases} -\infty & \text{if both } j\_1 \text{ and } j\_2 \text{ belong to the same relative machine type} \\ \mathcal{G}^{\sf S} \mathfrak{h}\_{j\_1 j\_2} & \text{otherwise.} \end{cases} \tag{17}$$

Based on the similarity coefficient defined in Equation (13), the PMP-type model of GCF can be formulated as follows:

(Model 3)

$$\text{Maximize } \sum\_{j\_1=1}^{q} \sum\_{j\_2=1}^{q} c\_{j\_1, j\_2} z\_{j\_1, j\_2} \tag{18}$$

Subject to

$$\sum\_{j\_2=1}^{q} z\_{j\_1, j\_2} = 1, \; j\_1 = 1, \cdots, q \tag{19}$$

$$\sum\_{j\_1=1}^{q} z\_{j\_1, j\_1} = p \tag{20}$$

$$\sum\_{j\_2=1}^q z\_{j\_1, j\_2} \le \mathcal{U} z\_{j\_1, j\_2} = 1, \ j\_2 = 1, \cdots, q \tag{21}$$

$$\sum\_{j\_1 \in \mathcal{M}\_{j\_3}}^q z\_{j\_1, j\_2} \le 1, \ j\_2 = 1, \cdots, q; \ j\_3 \in DM \tag{22}$$

$$z\_{j\_1, j\_2} = 0 \text{ or } 1, \; j\_1, j\_2 = 1, \dots, q. \tag{23}$$

The objective function in Equation (18) maximizes the sum of similarities among all pairs of machines including replicate machines. Constraint (19) ensures that each machine belongs to exactly one machine cell. Constraint (20) specifies the required number of machine cells. Constraint (21) ensures that the number of machines in each cell does not exceed *U* machines. Constraint (22) ensures that at most a single copy of a replicate machine type is assigned to each cell. Constraint (23) ensures the binary restriction of variables.

Since model 3 is a linear model and the number of machine types is usually much lower than the number pf process plans, moderately large-sized instances can be solved optimally within a reasonable computation time regardless of the total number of process plans. On the contrary, the number of process plans critically affects the model size in model 2. To solve the example incidence addressed in Section 2.3.1, 12,100 binary variables and 497 constraints are needed. Clearly, model 3 is very economical since it contains much fewer binary variables and/or constraints than model 2. We will show that even a large GCF incidence can be solved optimally within only one second using the LINGO solver.

#### 2.3.3. Part Assignment to Cells

The solution of PMP-type model 3 identifies only the machine cells. Once the machine cells are formed, parts need to be assigned to their best associated cells in such a way that the objective of CF is optimized. The classic part assignment rule that has been widely used is the maximum density rule, assigning a part to the cell in which it has most operations. Since different part assignment rules affect the solution quality of CF, several modified part assignment rules have been proposed [50,59–64]. A critical drawback of these rules is that the solution quality due to the assignment of a part depends on the assignment of other parts since they are heuristic rules. Some authors [50,53] have established sufficient conditions for optimal part assignment that do not depend on the assignment of other parts for the SCF problem. Several heuristic part assignment procedures for the GCF problem considering the number of EEs and voids have been proposed in the literature [59,61–63]; however, due to the existence of alternative process plans and replicate machines, no optimal part assignment rules for the GCF problem have been proposed.

Won [63] used the nonbinary generalized PMIM to develop a heuristic part assignment procedure for the GCF problem. In this paper, we use a modified heuristic part assignment procedure based on the binary generalized PMIM to classify parts into eight categories as follows:


To determine the specific type of a part and assign the best process plan to its best associated cell in such a way that the GE is maximized, the following measures need to be calculated for all process plans of parts:


Criteria 1 and 3 contribute to the independent cell configuration with the least inter-cell moves. Criteria 2 and 6 contribute to the compact cell configuration with high machine utilization. Criteria 4 and 5 contribute to the balanced cell configuration with an even workload among cells. A specific type of a part can be determined from criteria 1 and 3. Figure 2 presents a flow chart showing the procedure which identifies the type of a part.

**Figure 2.** Flow chart identifying the category of part. SNEP: strongly nonexceptional part; NNEP: neutrally nonexceptional part; WEP: weakly exceptionally part; NEP: neutrally exceptional part.

The assignment of a type I SNEP or type I WEP is straightforward: its unique candidate process plan is assigned to its associated unique cell. However, the assignment of remaining part types requires tie-breaking rules to select the best candidate process plan and its best associated cell. Given the information of criteria 1 to 6 for each part, assignment rules for the remaining part types are stated as follows:

Assignment rule for Type II SNEP or Type II WEP

There is a unique candidate cell for each of these part types. The process plan with the least number of EEs from 1 is selected. If ties occur, the plan with the maximum number of operations from 6 is selected. If ties occur again, the smallest-numbered cell is selected.

Assignment rule for Type I NNEP or Type I NEP

There is a unique candidate plan for each of these part types. The cell with the lowest number of EEs from 1 is selected. If ties occur, the cell with the least number of voids from 2 is selected. If ties occur again, the cell assigned with the least number of 1s from 4 is selected. If ties occur again, the cell assigned with the least number of part types from 5 is selected. If ties occur again, the cell with the maximum number of operations in a cell from 6 is selected. If ties occur again, the smallest-numbered cell is selected.

Assignment rule for Type II NNEP or Type II NEP

There are multiple candidate plans and cells for each of these part types. The plan and the associated cell with the least number of EEs from 1 are selected. If ties occur, the process plan and the associated cell with the least number of voids from 2 are selected. If ties occur again, the process plan and the associated cell assigned with the least number of 1s from 4 are selected. If ties occur again, the process plan and the associated cell assigned with the least number of part types from 5 are selected. If ties occur again, the process plan and the associated cell with the maximum number of operations in a cell from 6 are selected. If ties occur again, the smallest-numbered process plan and cell are selected.

2.3.4. Reassigning Improperly Assigned Exceptional Machines (EMs) and Redundant Machines (RMs)

The CF based on the solution from model 3 and the part family formation may result in an unsatisfactory block diagonal solution due to improperly assigned Ems, which process most parts in other cells, or RMs, which process no parts in their parent cell. Therefore, a subsequent refinement procedure is used to improve the quality of incumbent block diagonal solutions through an inspection by reassigning them to their most appropriate cells if there are any improperly assigned EMs. The reassignment of improperly assigned EMs/RMs can lead to higher GE by decreasing EEs and voids.

The improperly assigned EMs/RMs are categorized as follows:


To illustrate the types of improperly assigned EMs/RMs, consider a block diagonal matrix as shown in Figure 3. From this matrix, we can observe the set of machine cells, MC1 = {1, 2, 3}, MC2 = {4, 5, 6}, and MC3 = {7, 8}; and the set of part families, PF1 = {1a, 2b, 3c}, PF2 = {4a, 5b, 6a}, and PF3 = {7b, 8c, 9b}. According to the above definition, machines 1, 2, and 3 are type I RM, type II RM, and absolute RM, respectively, and machines 6 and 8 are type I EM and type II EM, respectively.

**Figure 3.** A block diagonal matrix for identifying the type of exceptional machines (EMs)/redundant machines (RMs).

To determine the type of EM and RM, the following elements need to be evaluated for each machine:


Figure 4 is a flow chart showing the determination of the type of EM or RM for a machine. The reassignment procedure for improperly assigned EMs/RMs is then stated as follows:


**Figure 4.** Flow chart identifying the type of EM and RM.

#### 2.3.5. Illustrative Example

The proposed procedures for part assignment and improperly assigned EM/RM reassignment are illustrated with a hypothetical GCF incidence which includes seven machine types, 15 part types, and 35 process plans as shown in Table 1, which lists the routing information of parts. Machine types 3 and 4 have two and one extra copies, respectively. After model 3 is solved under the condition of *p* = 3 and *U* = 4, the following machine cells are obtained: MC1 = {1, 3, 6}, MC2 = {2,4,3}, and MC3 = {5,7,3,4}.

Once these machine cells are determined, partial assignment steps of parts 1, 2, and 3 are presented in Table 2, where the column (1) indicates the number of EEs generated by the process plan assigned to a specific cell. The symbol \* indicates the entries corresponding to the candidate process plan and cells selected by the criterion of that step. The symbol \*\* indicates the entries corresponding to the best process plan and the cell selected by the criterion of the final step. If the best candidate plan is determined, the associated best process plan and cell can be identified by scanning the columns from left to right. After all parts are assigned, no RMs and improperly assigned EMs are found. Figure 5 is the resulting block diagonal solution matrix with a GE of 77.19%.


**Table 1.** Routing information of parts.


\* indicates the candidate process plans and cells and \*\* indicates the best process plan and cell.

**Figure 5.** Block diagonal solution to the hypothetical example.

#### **3. Results**

The purpose of the computational experiment performed in this section was two-fold. One aim was to show the effectiveness of our approach compared to the solutions already reported in the literature under the same restrictions as the reference approaches with regard to the number of cells *p* and cell size *U*; the other was to provide solutions that can be used as benchmarks for comparative testing with different CF solution approaches by finding better alternative solutions that have not been reported in the literature under different values of *p* and *U*.

The proposed PMP-based hard computing approach has been implemented and tested with two groups of GCF incidences: 26 small to intermediate-sized problem sets available in the literature and three expanded large-sized ones. The large-sized incidences are expanded with values of *p* which are different from the original ones. Tables 3 and 4 show a total of 39 test incidences for 26 small to intermediate-sized original problems and a total of five expanded test incidences for three large-sized original problems, respectively.


**Table 3.** Computational results with small to intermediate-sized original incidences. GE: grouping efficacy.


**Table 3.** *Cont.*

( a) Running time implementing model 3. (b) Running time assigning parts and reassigning improperly assigned EMs/RMs. (c) The best solution reported in [69]. (d) The best solution reported in [79]. (e) The best solution reported in [61].


**Table 4.** Computational results with double-expanded large-size incidences.

The former incidences are used for comparative purposes along with the results obtained by the reference approaches; the latter incidences are used to show the effectiveness of our approach in solving large-sized GCF incidences. The double-expanded large-size incidences have been produced by following Adil et al.'s data expansion scheme [80], which duplicates rows and columns of an existing intermediate-sized binary PMIM instead of using randomly generated data sets.

For the original problems 6, 11, 22, 23, and 24 in Table 3, extra incidences have been tested for values of *p* and *U* different from those used the in the original problems. Some of the original data sets include information on the operation sequences and/or production volumes of parts. Those data sets are slightly changed so that the resulting routing data or PMIMs are suited for the 0–1 GCF problem format. For problem 5, process plans b and c of part type 2 are merged into an identical process plan since they require the same tools. For problems 17 and 19, process plan c of part type 1 has been deleted since it requires the same machines as process plan b. All the GCF incidences and block diagonal solution matrices are available upon request.

Model 3 has been solved using the LINGO 16.0 solver on an ASUS laptop computer including an Intel Core i7-9750H processor running at 2.6 GHz with 16 GB RAM under the Windows 10 operating system. The procedures for part assignment and improperly assigned EM/RM reassignment were coded in C + + and implemented using the Visual Studio 2015. The execution times taken to implement all the reference CF methods selected for comparative purposes will not be reported since each incidence selected in our experiment was solved using different machines and compilers; only the execution time taken to implement our approach is reported for future comparison.
