# Multi-Objective Assembly Line Balancing Problem with Setup Times Using Fuzzy Goal Programming and Genetic Algorithm

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Research Methods

_{max}). The assumptions and notations are as follows.

#### 3.1. Assumptions and Notations

- The assembly line mass produces one homogeneous product.
- The production process is given, and the processing jobs are connected with precedence relations.
- The production process has a serial line layout with K
_{max}workstations. - A setup time s
_{k}is required for each workstation. - All workstations are equally equipped with machines and workers.
- No assignment restrictions are present except for the precedence relations.
- The job processing time is independent of the station at which the job is performed.
- The processing time of each job is known and deterministic.
- A job can only be processed in a single workstation at a time.
- A workstation can only process a single job at a time.
- A workstation can perform more than one job.
- A job can be performed in any workstation.
- The cycle time in a workstation is the sum of the setup time and the processing time.

- Notations

- j, u, v Job (j = 1,2,…, J).

- k
- Workstation (k = 1,2,…, K).
- i
- Objective (i = 1,2,…, I).

- t
_{j} - Processing time of job j.
- Ω
- Set of precedence relations; (u,v)∈Ω if and only if job u is an immediate predecessor of job v.
- TP
- Total job processing time.
- PC
- Production planning cycle.
- s
_{k} - Setup time of workstation k.
- K
_{max} - Maximum number of workstations.

- CT
- Cycle time.
- NW
- Number of workstations.
- WV
- Workload variance.
- TD
- Idle time of all workstations.
- X
_{jk} - A binary variable, equal to 1 if job j is processed in workstation k.
- Y
_{k} - A binary variable, equal to 1 if workstation k is selected for processing.
- T
_{k} - Completion time of workstation k.
- f
_{i}(Z_{i}) - Objective function, where Z
_{i}is the objective.

#### 3.2. Fuzzy Multi-Objective Linear Programming Model

$Min{\begin{array}{cc}& Z\end{array}}_{1}=CT$ | (1) | |

$Min{\begin{array}{cc}& Z\end{array}}_{2}=NW$ | (2) | |

$Min{\begin{array}{cc}& Z\end{array}}_{3}=WV$ | (3) | |

$Min{\begin{array}{cc}& Z\end{array}}_{4}=TD$ | (4) | |

$NW={\displaystyle \sum _{k=1}^{{K}_{}}{Y}_{k}}$ | (5) | |

$WV=\frac{1}{K}\ast {{\displaystyle \sum _{k=1}^{K}({T}_{k}-\frac{TP}{{K}_{}})}}^{2}$ | (6) | |

$TD={\displaystyle \sum _{k=1}^{K}(CT-{T}_{k})}$ | (7) | |

$\sum _{k=1}^{K}{Y}_{k}}\le {K}_{\mathrm{max}$ | (8) | |

${T}_{k}={\displaystyle \sum _{j=1}^{J}{t}_{j}\ast {X}_{jk}+{s}_{k}\ast {Y}_{k}}$, | k = 1,2,…, K | (9) |

$CT=Max({T}_{1},{T}_{2},{T}_{3},\dots ,{T}_{K})$ | (10) | |

${T}_{k}\le CT\ast {Y}_{k}$, | k = 1,2,…, K | (11) |

$\sum _{k=1}^{{K}_{}}{X}_{jk}=1$, | j = 1,2,…, J | (12) |

$\sum _{k=1}^{K}k\ast {X}_{vk}-{\displaystyle \sum _{k=1}^{K}k\ast {X}_{uk}\ge 0}$, | $\forall \left(u,v\right)\in \Omega $ | (13) |

${X}_{jk}\in \left\{0,1\right\}$ | j = 1,2,…, J, k = 1,2,…, K | (14) |

${Y}_{k}\in \left\{0,1\right\}$ | k = 1,2,…, K | (15) |

_{k}’s. Constraint (6) calculates the workload variance, WV, based on the difference between the completion time of each workstation, T

_{k}, and the average job processing time of a workstation (TP/K), and it is equivalent to workload smoothness. Constraint (7) calculates the total idle time of all workstations, TD, by summing up the idle time of each workstation, which is calculated by deducting the completion time of a workstation, T

_{k}, from the cycle time, CT. Constraint (8) guarantees that the total number of workstations selected for processing, $\sum _{k=1}^{K}{Y}_{k}$, must be less than or equal to the maximum number of workstations K

_{max}. That is, the work content of every workstation is at most the cycle time. Constraint (9) calculates the completion time of workstation k, T

_{k}, by summing up the processing time of all jobs in the workstation, $\sum _{j=1}^{J}{t}_{j}\ast {X}_{jk}$, and the setup time of the workstation, ${s}_{k}\ast {Y}_{k}$. Constraint (10) lets cycle time, CT, be the maximum value among the completion times of all workstations, T

_{1},…, T

_{k}. Constraint (11) guarantees that the completion time of workstation k, T

_{k}, must be less than or equal to the cycle time, $CT\ast {Y}_{k}$. Constraint (12) ensures that a job can only be assigned and processed by one single workstation. Constraint (13) ensures the sequencing of jobs; that is, a job needs to be completed before its next job can proceed. Constraint (14) shows that X

_{jk}is a binary variable, which is equal to 1 if job j is processed in workstation k. Constraint (15) shows that Y

_{k}is a binary variable, which is equal to 1 if workstation k is selected for processing.

${f}_{i}({Z}_{i})=\{\begin{array}{l}\begin{array}{ccccc}& 1& \begin{array}{cc}& \end{array}& & \mathrm{if}\text{}\end{array}{Z}_{i}\ge {Z}_{i}^{b}\\ \begin{array}{ccc}\frac{{Z}_{i}^{}-{Z}_{i}^{a}}{{Z}_{i}^{b}-{Z}_{i}^{a}}& & \mathrm{if}\text{}\end{array}{Z}_{i}^{a}\le {Z}_{i}\le {Z}_{i}^{b},\\ \begin{array}{ccccc}& 0& \begin{array}{cc}& \end{array}& & \mathrm{if}\text{}\end{array}{Z}_{i}\le {Z}_{i}^{a}\end{array}$ | for maximization objective | (16) |

${f}_{i}({Z}_{i})=\{\begin{array}{l}\begin{array}{ccccc}& 1& \begin{array}{cc}& \end{array}& & \mathrm{if}\text{}\end{array}{Z}_{i}\le {Z}_{i}^{a}\\ \begin{array}{ccc}\frac{{Z}_{i}^{b}-{Z}_{i}}{{Z}_{i}^{b}-{Z}_{i}^{a}}& & \mathrm{if}\text{}\end{array}{Z}_{i}^{a}\le {Z}_{i}\le {Z}_{i}^{b},\\ \begin{array}{ccccc}& 0& \begin{array}{cc}& \end{array}& & \mathrm{if}\text{}\end{array}{Z}_{i}\ge {Z}_{i}^{b}\end{array}$ | for minimization objective | (17) |

${f}_{i}({Z}_{i})=\{\begin{array}{cc}0& if\text{}{Z}_{i}\le {Z}_{i}^{a}\\ \frac{{Z}_{i}-{Z}_{i}^{a}}{{Z}_{i}^{r}-{Z}_{i}^{a}}& if\text{}{Z}_{i}^{a}\le {Z}_{i}\le {Z}_{i}^{r}\\ 1& if\text{}{Z}_{i}={Z}_{i}^{r}\\ \frac{{Z}_{i}^{b}-{Z}_{i}}{{Z}_{i}^{b}-{Z}_{i}^{r}}& if\text{}{Z}_{i}^{r}\le {Z}_{i}\le {Z}_{i}^{b}\\ 0& if\text{}{Z}_{i}\ge {Z}_{i}^{b}\end{array}$ | for target objective | (18) |

#### 3.3. Genetic Algorithm for Assembly Line Balancing Problem

- Step 1.
- Initial population of chromosomes

- Step 2.
- Coding scheme

- Step 3.
- Fitness function evaluation

- Step 4.
- Reproduction operation

_{m}) and crossover rate (P

_{c}).

- Step 5.
- New population generation

- Step 6.
- Termination

_{max}.

## 4. Case Studies

#### 4.1. Case Introduction

#### 4.2. Case 1

#### 4.2.1. Single Objective Linear Programming Model

_{4}can be processed only after jobs J

_{1}and J

_{3}are completed. Job J

_{3}is a direct predecessor of job J

_{4}, and job J

_{1}is an indirect predecessor of job J

_{4}. Job J

_{7}can be processed only after jobs J

_{4}and J

_{6}are completed. The total processing time, 94 min, is the summation of the processing times of all jobs. A single objective linear programming model based on each of the four objectives is developed using LINGO 10.

_{1}), the results are shown in Table 2 and Table 3. Table 2 shows that to minimize cycle time (Z

_{1}), five workstations are required. For example, for the first workstation, jobs A

_{1}and A

_{2}are processed. The setup time is 5 min, and the processing time is:

_{1}). For minimizing cycle time (Z

_{1}), the decision variable is the cycle time (CT). The cycle time is 26 min, which can be found by summing up the processing time (21 min) and the setup time (5 min) for workstation 5, which has the longest cycle time among all workstations. For minimizing the number of workstations (Z

_{2}), the decision variable is the number of workstations (NW), and there are five workstations, as shown in Table 2. For minimizing workload variance (Z

_{3}), the decision variable is the workload variance (WV), and 1.76 min is calculated. For minimizing workstation idle time (Z

_{4}), the decision variable is the idle time of all workstations (TD), and it is calculated as follows:

_{2}), the results are shown in Table 4 and Table 5. Table 4 shows that to minimize the number of workstations (Z

_{2}), only one workstation is required to process all jobs. Table 5 shows the performance under each objective while achieving the objective of minimizing number of workstations (Z

_{2}). For minimizing cycle time (Z

_{1}), the cycle time (CT) is 99 min, which can be found in Table 4. For minimizing number of workstations (NW), there is one workstation, as shown in Table 2. For minimizing workload variance (Z

_{3}), the workload variance (WV) is 1569.16 min. For minimizing workstation idle time (Z

_{4}), the idle time of all workstations (TD) is 396 min (99*4).

_{3}), the results are shown in Table 6 and Table 7. Table 6 shows that to minimize workload variance (L

_{3}), five workstations are required, and the relevant information about each workstation is listed. Table 7 shows the performance under each objective while achieving the objective of minimizing workload variance (L

_{3}). For minimizing cycle time (L

_{1}), the cycle time (CT) is 26 min. For minimizing number of workstations (L

_{2}), the number of workstations (NW) is 5. For minimizing workload variance (L

_{3}), the workload variance (WV) is 26.36 min. For minimizing workstation idle time (L

_{4}), the idle time of all workstations (TD) is 11 min.

_{4}), the results are shown in Table 8 and Table 9. Table 8 shows that to minimize workstation idle time (Z

_{4}), five workstations are required, and the relevant information about each workstation is listed. Table 9 shows the performance under each objective while achieving the objective of minimizing workload variance (Z

_{3}). Table 10 shows the upper bound, lower bound, deviation, and average of the objective value for the four objectives. The data can be extracted from Table 3, Table 5, Table 7, Table 9. For example, for the first objective, Z

_{1}, the decision variable is CT, and the cycle times in Table 3, Table 5, Table 7, Table 9 are 26, 99, 26, and 26, respectively. The highest value is 99, and it is set as the upper bound. The lowest value is 26, and it is set as the lower bound. The deviation and the average are as follows:

#### 4.2.2. Fuzzy Multi-Objective Linear Programming Model

- Step 1.
- Use the results obtained from the single objective linear programming models to set the upper and lower bound for each objective. The fuzzy membership function for cycle time, number of workstations, workload variance, and workstation idle time are ${f}_{1}({Z}_{1})$, ${f}_{2}({Z}_{2})$, ${f}_{3}({Z}_{3})$, and ${f}_{4}({Z}_{4})$, respectively.
- Step 2.
- Transform the fuzzy multi-objective linear programming model into a single objective linear programming model. The four objectives are transformed into one single objective:$$Max\begin{array}{c}\end{array}{f}_{\lambda}({Z}_{i})=Max\begin{array}{c}\end{array}\left[\underset{\begin{array}{c}\end{array}}{min}({f}_{1}({Z}_{1}),{f}_{2}({Z}_{2}),{f}_{3}({Z}_{3}),{f}_{4}({Z}_{4})\right].$$
- Step 3.
- Formulate the fuzzy multi-objective linear programming model with satisfaction level λ, as follows:$$Max\begin{array}{c}\end{array}\lambda $$$$\lambda \le (99-CT)/(99-26)$$$$\lambda \le (5-NW)/(5-1)$$$$\lambda \le (1413.76-WV)/(1413.76-1.36)$$$$\lambda \le (396-TD)/(396-11)$$$$0\le \lambda \le 1$$$$NW=({Y}_{1}+{Y}_{2}+{Y}_{3}+{Y}_{4}+{Y}_{5})$$$$WV=\frac{1}{5}\ast \left[{\left({T}_{1}-\frac{TP}{5}\right)}^{2}+{\left({T}_{2}-\frac{TP}{5}\right)}_{}^{2}+{\left({T}_{3}-\frac{TP}{5}\right)}_{}^{2}+{\left({T}_{4}-\frac{TP}{5}\right)}_{}^{2}+{\left({T}_{5}-\frac{TP}{5}\right)}_{}^{2}\right]$$$$TD=(CT-{T}_{1})+(CT-{T}_{2})+(CT-{T}_{3})+(CT-{T}_{4})+(CT-{T}_{5})$$$$({Y}_{1}+{Y}_{2}+{Y}_{3}+{Y}_{4}+{Y}_{5})\le 5$$$$\begin{array}{ll}{T}_{1}=& {t}_{1}\ast {X}_{11}+{t}_{2}\ast {X}_{21}+{t}_{3}\ast {X}_{31}+{t}_{4}\ast {X}_{41}+{t}_{5}\ast {X}_{51}+{t}_{6}\ast {X}_{61}+{t}_{7}\ast {X}_{71}+{t}_{8}\ast {X}_{81}+\\ & {t}_{9}\ast {X}_{91}+{t}_{10}\ast {X}_{101}+{s}_{1}\ast {Y}_{1}\end{array}$$$$\begin{array}{ll}{T}_{2}=& {t}_{1}\ast {X}_{12}+{t}_{2}\ast {X}_{22}+{t}_{3}\ast {X}_{32}+{t}_{4}\ast {X}_{42}+{t}_{5}\ast {X}_{52}+{t}_{6}\ast {X}_{62}+{t}_{7}\ast {X}_{72}+{t}_{8}\ast {X}_{82}+\\ & {t}_{9}\ast {X}_{92}+{t}_{10}\ast {X}_{102}+{s}_{2}\ast {Y}_{2}\end{array}$$$$\begin{array}{ll}{T}_{3}=& {t}_{1}\ast {X}_{13}+{t}_{2}\ast {X}_{23}+{t}_{3}\ast {X}_{33}+{t}_{4}\ast {X}_{43}+{t}_{5}\ast {X}_{53}+{t}_{6}\ast {X}_{63}+{t}_{7}\ast {X}_{73}+{t}_{8}\ast {X}_{83}+\\ & {t}_{9}\ast {X}_{93}+{t}_{10}\ast {X}_{103}+{s}_{3}\ast {Y}_{3}\end{array}$$$$\begin{array}{ll}{T}_{4}=& {t}_{1}\ast {X}_{14}+{t}_{2}\ast {X}_{24}+{t}_{3}\ast {X}_{34}+{t}_{4}\ast {X}_{44}+{t}_{5}\ast {X}_{54}+{t}_{6}\ast {X}_{64}+{t}_{7}\ast {X}_{74}+{t}_{8}\ast {X}_{84}+\\ & {t}_{9}\ast {X}_{94}+{t}_{10}\ast {X}_{104}+{s}_{4}\ast {Y}_{4}\end{array}$$$$\begin{array}{ll}{T}_{5}=& {t}_{1}\ast {X}_{15}+{t}_{2}\ast {X}_{25}+{t}_{3}\ast {X}_{35}+{t}_{4}\ast {X}_{45}+{t}_{5}\ast {X}_{55}+{t}_{6}\ast {X}_{65}+{t}_{7}\ast {X}_{75}+{t}_{8}\ast {X}_{85}+\\ & {t}_{9}\ast {X}_{95}+{t}_{10}\ast {X}_{105}+{s}_{5}\ast {Y}_{5}\end{array}$$$$CT=Max({T}_{1},{T}_{2},{T}_{3},{T}_{4},{T}_{5})$$$${T}_{1}\le CT\ast {Y}_{1}$$$${T}_{2}\le CT\ast {Y}_{2}$$$${T}_{3}\le CT\ast {Y}_{3}$$$${T}_{4}\le CT\ast {Y}_{4}$$$${T}_{5}\le CT\ast {Y}_{5}$$$${X}_{11}+{X}_{12}+{X}_{13}+{X}_{14}+{X}_{15}=1$$$${X}_{21}+{X}_{22}+{X}_{23}+{X}_{24}+{X}_{25}=1$$$${X}_{31}+{X}_{32}+{X}_{33}+{X}_{34}+{X}_{35}=1$$$${X}_{41}+{X}_{42}+{X}_{43}+{X}_{44}+{X}_{45}=1$$$${X}_{51}+{X}_{52}+{X}_{53}+{X}_{54}+{X}_{55}=1$$$${X}_{61}+{X}_{62}+{X}_{63}+{X}_{64}+{X}_{65}=1$$$${X}_{71}+{X}_{72}+{X}_{73}+{X}_{74}+{X}_{75}=1$$$${X}_{81}+{X}_{82}+{X}_{83}+{X}_{84}+{X}_{85}=1$$$${X}_{91}+{X}_{92}+{X}_{93}+{X}_{94}+{X}_{95}=1$$$${X}_{101}+{X}_{102}+{X}_{103}+{X}_{104}+{X}_{105}=1$$$$\left(1\ast {X}_{21}+2\ast {X}_{22}+3\ast {X}_{23}+4\ast {X}_{24}+5\ast {X}_{25}\right)-\left(1\ast {X}_{11}+2\ast {X}_{12}+3\ast {X}_{13}+4\ast {X}_{14}+5\ast {X}_{15}\right)\ge 0$$$$\left(1\ast {X}_{31}+2\ast {X}_{32}+3\ast {X}_{33}+4\ast {X}_{34}+5\ast {X}_{35}\right)-\left(1\ast {X}_{11}+2\ast {X}_{12}+3\ast {X}_{13}+4\ast {X}_{14}+5\ast {X}_{15}\right)\ge 0$$$$\left(1\ast {X}_{41}+2\ast {X}_{42}+3\ast {X}_{43}+4\ast {X}_{44}+5\ast {X}_{45}\right)-\left(1\ast {X}_{31}+2\ast {X}_{32}+3\ast {X}_{33}+4\ast {X}_{34}+5\ast {X}_{35}\right)\ge 0$$$$\left(1\ast {X}_{51}+2\ast {X}_{52}+3\ast {X}_{53}+4\ast {X}_{54}+5\ast {X}_{55}\right)-\left(1\ast {X}_{31}+2\ast {X}_{32}+3\ast {X}_{33}+4\ast {X}_{34}+5\ast {X}_{35}\right)\ge 0$$$$\left(1\ast {X}_{61}+2\ast {X}_{62}+3\ast {X}_{63}+4\ast {X}_{64}+5\ast {X}_{65}\right)-\left(1\ast {X}_{21}+2\ast {X}_{22}+3\ast {X}_{23}+4\ast {X}_{24}+5\ast {X}_{25}\right)\ge 0$$$$\left(1\ast {X}_{71}+2\ast {X}_{72}+3\ast {X}_{73}+4\ast {X}_{74}+5\ast {X}_{75}\right)-\left(1\ast {X}_{41}+2\ast {X}_{42}+3\ast {X}_{43}+4\ast {X}_{44}+5\ast {X}_{45}\right)\ge 0$$$$\left(1\ast {X}_{71}+2\ast {X}_{72}+3\ast {X}_{73}+4\ast {X}_{74}+5\ast {X}_{75}\right)-\left(1\ast {X}_{61}+2\ast {X}_{62}+3\ast {X}_{63}+4\ast {X}_{64}+5\ast {X}_{65}\right)\ge 0$$$$\left(1\ast {X}_{81}+2\ast {X}_{82}+3\ast {X}_{83}+4\ast {X}_{84}+5\ast {X}_{85}\right)-\left(1\ast {X}_{51}+2\ast {X}_{52}+3\ast {X}_{53}+4\ast {X}_{54}+5\ast {X}_{55}\right)\ge 0$$$$\left(1\ast {X}_{91}+2\ast {X}_{92}+3\ast {X}_{93}+4\ast {X}_{94}+5\ast {X}_{95}\right)-\left(1\ast {X}_{81}+2\ast {X}_{82}+3\ast {X}_{83}+4\ast {X}_{84}+5\ast {X}_{85}\right)\ge 0$$$$\left(1\ast {X}_{91}+2\ast {X}_{92}+3\ast {X}_{93}+4\ast {X}_{94}+5\ast {X}_{95}\right)-\left(1\ast {X}_{71}+2\ast {X}_{72}+3\ast {X}_{73}+4\ast {X}_{74}+5\ast {X}_{75}\right)\ge 0$$$$\left(1\ast {X}_{101}+2\ast {X}_{102}+3\ast {X}_{103}+4\ast {X}_{104}+5\ast {X}_{105}\right)-\left(1\ast {X}_{91}+2\ast {X}_{92}+3\ast {X}_{93}+4\ast {X}_{94}+5\ast {X}_{95}\right)\ge 0$$$${X}_{jk}\in \left\{0,1\right\}$$$${Y}_{k}\in \left\{0,1\right\}$$

_{1}, respectively. Constraint (35) is to minimize the number of workstations, NW, where 1 and 5 are the lower bound and upper bound of Z

_{2}respectively. Constraint (36) is to minimize the workload variance, WV, where 1.36 and 1413.76 are the lower bound and upper bound of Z

_{3}, respectively. Constraint (37) is to minimize the idle time of all workstations, TD, where 11 and 396 are the lower bound and upper bound of Z

_{4}, respectively. Constraint (38) sets the satisfaction level to be between 0 and 1. Constraint (39) calculates the total number of workstations used, NW, by summing up all Y

_{k}

_{’s}, i.e., Y

_{1}, Y

_{2}, Y

_{3}, Y

_{4}, and Y

_{5}. Constraint (40) calculates the workload variance, WV, based on the differences between Tk and (TP/K). Constraint (41) calculates the total idle time of all workstations, TD. Constraint (42) ensures that the total number of workstations selected for processing must be less than or equal to the maximum number of workstations, that is, 5. Constraints (43)–(47) calculate the completion time of workstation 1 to 5, respectively, by summing up the processing time of all jobs in the specific workstation and the setup time of the specific workstation. Constraint (48) lets cycle time, CT, be the maximum value among the completion times of all workstations, T

_{1},…, T

_{5}. Constraints (49)–(53) ensure that the completion time of workstation 1 to 5, respectively, must be less than or equal to the cycle time of the workstation. Constraints (54)–(63) ensure that job A

_{1}to A

_{10}, respectively, can only be assigned and processed by one single workstation. Constraints (64)–(74) ensure the sequencing of two of the ten jobs. For instance, Constraint (64) ensures that job A

_{1}must be completed before A

_{2}can go into process in a workstation if they are both processed in the same workstation. Constraint (75) shows that X

_{jk}is a binary variable, which is equal to 1 if job j is processed in workstation k. Constraint (76) shows that Y

_{k}is a binary variable, which is equal to 1 if workstation k is selected for processing.

_{k}, five variables of s

_{k}, fifty 0-1 variables of X

_{jk}, and five 0-1 variables of Y

_{k}. Furthermore, each of Equations (9) and (11) contains five equations. Equation (12) contains ten equations, and Equation (13) contains 11 equations. In this case, the total number of auxiliary variables is 15, the total number of 0-1 variables is 55, and the total number of equations is 40 (9 + 2K + J + Ω).

_{1}), the cycle time (CT) is 52 min. For minimizing number of workstations (Z

_{2}), the number of workstations (NW) is 2. For minimizing workload variance (Z

_{3}), the workload variance (WV) is 530.16 min. For minimizing workstation idle time (Z

_{4}), the idle time of all workstations (TD) is 141 min. The total satisfaction level λ is 0.63. A Gantt for the solution of Case 1 is depicted in Figure 4.

#### 4.2.3. Genetic Algorithm Model

_{c}) is set as 0.99, and this indicates that around 99% pairs of individuals participate in producing offspring. The mutation rate (P

_{m}) is set as 0.01, and this indicates that a gene of a newly created solution is mutated with a probability 0.01. The algorithm is terminated when the 1000th generation is reached. The best generation occurs at the 78th generation, as displayed in Figure 5. The total computation time is 11 s. The solutions generated by the fuzzy multi-objective linear programming model and by the genetic algorithm model are the same, with the total satisfaction level of 0.63.

_{c}), mutation rate (P

_{m}), and maximum generation (G

_{max}). Table 13 lists the parameters applied to generate the outcomes. With two levels for each factor, a 16 (2

^{4}) factorial experiment is executed. Since a run has three replicates, 48 trials are performed. The computation time, number of generations, and fitness value of the three trials in a run are applied. Table 14 lists the 16 combinations of the parameters for Case 1. Since the computation time is dependent on maximum generation (G

_{max}) and population size (N), as G

_{max}or N increases, the computation time increases. In addition, the number of generations is relatively larger when the population size (N) is smaller. The fitness values in all the 16 runs are the same, i.e., 0.63. This means that the fitness values in all the 16 runs remain stable and consistent. Since the value is the same as that from the fuzzy multi-objective linear programming model with a total satisfaction level λ of 0.63, the optimal solution is obtained.

#### 4.3. Case 2

_{1}), the cycle time (CT) is 75 min. For minimizing the number of workstations (L

_{2}), the number of workstations (NW) is 3. For minimizing the workload variance (L

_{3}), the workload variance (WV) is 1042.75 min. For minimizing the workstation idle time (L

_{4}), the idle time of all workstations (TD) is 352 min. The total satisfaction level λ is 0.71. A Gantt for the solution of Case 2 is shown in Figure 7.

#### 4.4. Case 3

_{1}), the cycle time (CT) is 127 min. For minimizing the number of workstations (L

_{2}), the number of workstations (NW) is 3. For minimizing the workload variance (L

_{3}), the workload variance (WV) is 3024 min. For minimizing the workstation idle time (L

_{4}), the idle time of all workstations (TD) is 840 min. The total satisfaction level λ is 0.74.

#### 4.5. Case 4

_{1}), the cycle time (CT) is 175 min. For minimizing the number of workstations (L

_{2}), the number of workstations (NW) is six. For minimizing workload variance (L

_{3}), the workload variance (WV) is 5305.25 min. For minimizing workstation idle time (L

_{4}), the idle time of all workstations (TD) is 1248 min. The total satisfaction level λ is 0.55.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Sample Problem | Number of Jobs | Maximum Number of Workstations | Workstation Setup Time | Total Processing Time | |
---|---|---|---|---|---|

Case 5 | Jackson [31] | 11 | 6 | 5 min | 46 min |

Case 6 | Rosenberg and Ziegler [32] | 25 | 8 | 6 min | 125 min |

Case 7 | Gunther et al. [33] | 35 | 10 | 7 min | 484 min |

Case 8 | Pinarbasi et al. [14] | 37 | 10 | 7 min | 908 min |

Case 9 | Rashid et al. [19] | 54 | 11 | 9 min | 2864 min |

Job | Predecessor | Processing Time | Job | Predecessor | Processing Time |
---|---|---|---|---|---|

1 | - | 6 | 7 | 3, 4, 5 | 3 |

2 | 1 | 2 | 8 | 6 | 6 |

3 | 1 | 5 | 9 | 7 | 5 |

4 | 1 | 7 | 10 | 8 | 5 |

5 | 1 | 1 | 11 | 9, 10 | 4 |

6 | 2 | 2 |

Job | Predecessor | Processing Time | Job | Predecessor | Processing Time |
---|---|---|---|---|---|

1 | - | 4 | 14 | 13 | 3 |

2 | - | 3 | 15 | 12 | 5 |

3 | 1, 2 | 9 | 16 | 14 | 3 |

4 | 3 | 5 | 17 | 15 | 13 |

5 | 4 | 9 | 18 | 16, 17 | 5 |

6 | 5 | 4 | 19 | 14 | 2 |

7 | 6 | 8 | 20 | 14 | 3 |

8 | 4 | 7 | 21 | 20 | 7 |

9 | 8 | 5 | 22 | 19, 21 | 5 |

10 | 6, 9 | 1 | 23 | 17 | 3 |

11 | 7, 8 | 3 | 24 | 21 | 8 |

12 | 7 | 1 | 25 | 18, 20, 23 | 4 |

13 | 9, 11 | 5 |

Job | Predecessor | Processing Time | Job | Predecessor | Processing Time |
---|---|---|---|---|---|

1 | - | 29 | 19 | 18 | 19 |

2 | 1 | 3 | 20 | 17 | 29 |

3 | 2 | 5 | 21 | 16, 20 | 8 |

4 | 3 | 22 | 22 | 21 | 10 |

5 | 1 | 5 | 23 | 22 | 16 |

6 | 5 | 14 | 24 | 23 | 23 |

7 | 1, 6 | 2 | 25 | 21 | 5 |

8 | 6 | 5 | 26 | 25 | 5 |

9 | 8 | 22 | 27 | 24, 26 | 5 |

10 | 1 | 30 | 28 | 11, 13, 27 | 40 |

11 | 4 | 23 | 29 | 28 | 2 |

12 | 1 | 30 | 30 | 21 | 5 |

13 | 9 | 23 | 31 | 30 | 5 |

14 | 7 | 2 | 32 | 21, 31 | 1 |

15 | 14 | 19 | 33 | 11, 13, 27, 32 | 40 |

16 | 15 | 29 | 34 | 27 | 2 |

17 | - | 2 | 35 | 33 | 2 |

18 | 7, 12 | 2 |

Job | Predecessor | Processing Time | Job | Predecessor | Processing Time |
---|---|---|---|---|---|

1 | - | 23 | 20 | 19 | 13 |

2 | - | 12 | 21 | 16 | 14 |

3 | 1, 2 | 35 | 22 | 17 | 14 |

4 | - | 12 | 23 | - | 12 |

5 | 1, 4 | 35 | 24 | 22, 23 | 38 |

6 | 3, 5, 8, 10 | 7 | 25 | 21 | 38 |

7 | - | 12 | 26 | - | 10 |

8 | 1, 7 | 35 | 27 | 26 | 18 |

9 | - | 12 | 28 | 24, 25, 27 | 22 |

10 | 1, 9 | 35 | 29 | 28 | 15 |

11 | 6 | 22 | 30 | 28 | 19 |

12 | 6 | 16 | 31 | 17, 18 | 38 |

13 | 6 | 54 | 32 | 31 | 43 |

14 | 12, 13 | 36 | 33 | 32 | 10 |

15 | 6 | 26 | 34 | 11, 14, 15, 20, 29, 30, 33 | 5 |

16 | 6 | 42 | 35 | 34 | 20 |

17 | 6 | 42 | 36 | 35 | 16 |

18 | 6 | 26 | 37 | 36 | 55 |

19 | 6 | 26 |

Job | Predecessor | Processing Time | Job | Predecessor | Processing Time |
---|---|---|---|---|---|

1 | - | 159 | 28 | 26, 27 | 23 |

2 | 1 | 16 | 29 | - | 24 |

3 | 1 | 29 | 30 | 29 | 46 |

4 | 1 | 16 | 31 | 29 | 16 |

5 | 1 | 13 | 32 | 29 | 19 |

6 | 1 | 20 | 33 | 29 | 75 |

7 | 1 | 23 | 34 | 30, 31, 32, 33 | 18 |

8 | 1 | 122 | 35 | 3, 4, 5, 6, 10, 19 | 45 |

9 | 2, 7 | 112 | 36 | 28, 34 | 151 |

10 | 9 | 154 | 37 | 36 | 21 |

11 | - | 90 | 38 | 35, 36 | 41 |

12 | 11 | 57 | 39 | 36 | 42 |

13 | 12 | 12 | 40 | 35, 36 | 40 |

14 | 12 | 15 | 41 | 36 | 18 |

15 | 8, 13, 14 | 142 | 42 | 35, 36 | 15 |

16 | 15 | 87 | 43 | 37, 38, 39, 40, 41, 42 | 41 |

17 | 15 | 59 | 44 | 40/41 | 33 |

18 | 15 | 23 | 45 | 42 | 38 |

19 | 8 | 23 | 46 | 37, 38 | 36 |

20 | 16, 17, 18 | 25 | 47 | 37, 38 | 53 |

21 | 20 | 49 | 48 | - | 125 |

22 | 20 | 24 | 49 | 34, 39 | 92 |

23 | 20 | 27 | 50 | 34, 39 | 57 |

24 | 20 | 21 | 51 | 34, 39 | 71 |

25 | - | 33 | 52 | 34, 39 | 44 |

26 | 25 | 61 | 53 | - | 50 |

27 | 25 | 76 | 54 | 43, 44, 45, 46, 47, 48, 49, 53 | 142 |

Objective | Objective Value | Objective Function |
---|---|---|

Z_{1} | 23 | ${f}_{1}({Z}_{1})$ = 0.72 |

Z_{2} | 2 | ${f}_{1}({Z}_{2})$ = 0.80 |

Z_{3} | 117.56 | ${f}_{1}({Z}_{3})$ = 0.64 |

Z_{4} | 92 | ${f}_{1}({Z}_{4})$ = 0.68 |

Total satisfaction level λ = 0.64 |

Objective | Objective Value | Objective Function |
---|---|---|

Z_{1} | 30 | ${f}_{1}({Z}_{1})$ = 0.55 |

Z_{2} | 5 | ${f}_{1}({Z}_{2})$ = 0.50 |

Z_{3} | 114.14 | ${f}_{1}({Z}_{3})$ = 0.55 |

Z_{4} | 55 | ${f}_{1}({Z}_{4})$ = 0.55 |

Total satisfaction level λ = 0.50 |

Objective | Objective Value | Objective Function |
---|---|---|

Z_{1} | 162 | ${f}_{1}({Z}_{1})$ = 0.74 |

Z_{2} | 3 | ${f}_{1}({Z}_{2})$ = 0.77 |

Z_{3} | 5466 | ${f}_{1}({Z}_{3})$ = 0.74 |

Z_{4} | 1136 | ${f}_{1}({Z}_{4})$ = 0.74 |

Total satisfaction level λ = 0.74 |

Objective | Objective Value | Objective Function |
---|---|---|

Z_{1} | 304 | ${f}_{1}({Z}_{1})$ = 0.72 |

Z_{2} | 3 | ${f}_{1}({Z}_{2})$ = 0.78 |

Z_{3} | 19238.56 | ${f}_{1}({Z}_{3})$ = 0.74 |

Z_{4} | 2132 | ${f}_{1}({Z}_{4})$ = 0.74 |

Total satisfaction level λ = 0.72 |

Objective | Objective Value | Objective Function |
---|---|---|

Z_{1} | 955 | ${f}_{1}({Z}_{1})$ = 0.71 |

Z_{2} | 3 | ${f}_{1}({Z}_{2})$ = 0.80 |

Z_{3} | 180771.3 | ${f}_{1}({Z}_{3})$ = 0.74 |

Z_{4} | 7641 | ${f}_{1}({Z}_{4})$ = 0.76 |

Total satisfaction level λ = 0.71 |

## References

- Kucukkoc, I.; Zhang, D.Z. A mathematical model and genetic algorithm-based approach for parallel two-sided assembly line balancing problem. Prod. Plan. Control.
**2015**, 26, 874–894. [Google Scholar] [CrossRef] [Green Version] - Özcan, U.; Toklu, B. Multiple-criteria decision-making in two-sided assembly line balancing: A goal programming and a fuzzy goal programming models. Comput. Oper. Res.
**2009**, 36, 1955–1965. [Google Scholar] [CrossRef] - Salveson, M.E. The assembly line balancing problem. J. Ind. Eng.
**1955**, 6, 18–25. [Google Scholar] - Erel, E.; Gokcen, H. Shortest-route formulation of mixed-model assembly line balancing problem. Eur. J. Oper. Res.
**1999**, 116, 194–204. [Google Scholar] [CrossRef] [Green Version] - Fathi, M.; Fontes, D.B.M.M.; Moris, M.U.; Ghobakhloo, M. Assembly line balancing problem: A comparative evaluation of heuristics and a computational assessment of objectives. J. Model. Manag.
**2018**, 13, 455–474. [Google Scholar] [CrossRef] [Green Version] - Baybars, I. A survey of exact algorithms for the simple assembly line balancing problem. Manag. Sci.
**1986**, 32, 909–932. [Google Scholar] [CrossRef] - Scholl, A. Balancing and Sequencing Assembly Lines, 2nd ed.; Physica-Verlag: Heidelberg, Germany, 1999. [Google Scholar]
- Becker, C.; Scholl, A. A survey on problems and methods in generalized assembly line balancing. Eur. J. Oper. Res.
**2006**, 168, 694–715. [Google Scholar] [CrossRef] - Haq, A.N.; Rengarajan, K.; Jayaprakash, J. A hybrid genetic algorithm approach to mixed-model assembly line balancing. Int. J. Adv. Manuf. Technol.
**2005**, 28, 337–341. [Google Scholar] [CrossRef] - Zhang, H.-Y. An improved immune algorithm for simple assembly line balancing problem of type 1. J. Algorithms Comput. Technol.
**2017**, 11, 317–326. [Google Scholar] [CrossRef] - Taha, R.B.; El-Kharbotly, A.K.; Sadek, Y.M.; Afia, N.H. A genetic algorithm for solving two-sided assembly line balancing problems. Ain Shams Eng. J.
**2011**, 2, 227–240. [Google Scholar] [CrossRef] [Green Version] - Cerqueus, A.; Delorme, X. A branch-and-bound method for the bi-objective simple line assembly balancing problem. Int. J. Prod. Res.
**2018**, 57, 5640–5659. [Google Scholar] [CrossRef] - Ritt, M.; Costa, A.M. Improved integer programming models for simple assembly line balancing and related problems. Int. Trans. Oper. Res.
**2015**, 25, 1345–1359. [Google Scholar] [CrossRef] [Green Version] - Pinarbasi, M.; Alakas, H.M.; Yuzukirmizi, M. A constraint programming approach to type-2 assembly line balancing problem with assignment restrictions. Assem. Autom.
**2019**, 39, 813–826. [Google Scholar] [CrossRef] - Mardani-Fard, H.A.; Hadi-Vencheh, A.; Mahmoodirad, A.; Niroomand, S. An effective hybrid goal programming approach for multi-objective straight assembly line balancing problem with stochastic parameters. Oper. Res.
**2018**, 20, 1939–1976. [Google Scholar] [CrossRef] - Abu Bakar, N.; Ramli, M.F.; Zakaria, M.Z.; Sin, T.C.; Masran, H. Solving assembly line balancing problem using heuristic: A case study of power transformer in electrical industry. Indones. J. Electr. Eng. Comput. Sci.
**2020**, 17, 850–857. [Google Scholar] [CrossRef] [Green Version] - Li, Z.; Kucukkoc, I.; Tang, Q. A comparative study of exact methods for the simple assembly line balancing problem. Soft Comput.
**2020**, 24, 11459–11475. [Google Scholar] [CrossRef] - Li, Z.; Janardhanan, M.N.; Tang, Q. Multi-objective migrating bird optimization algorithm for cost-oriented assembly line balancing problem with collaborative robots. Neural Comput. Appl.
**2021**, 1–22. [Google Scholar] [CrossRef] - Rashid, M.F.F.A.; Rose, A.N.M.; Mohamed, N.M.Z.N.; Romlay, F.R.M. Improved moth flame optimization algorithm to op-timize cost-oriented two-sided assembly line balancing. Eng. Comput.
**2020**, 37, 638–663. [Google Scholar] [CrossRef] - Kim, Y.K.; Kim, Y.; Kim, Y.J. Two-sided assembly line balancing: A genetic algorithm approach. Prod. Plan. Control.
**2000**, 11, 44–53. [Google Scholar] [CrossRef] - Kim, Y.K.; Song, W.S.; Kim, J.H. A mathematical model and a genetic algorithm for two-sided assembly line balancing. Comput. Oper. Res.
**2009**, 36, 853–865. [Google Scholar] [CrossRef] - Tanhaie, F.; Rabbani, M.; Manavizadeh, N. Simultaneous balancing and worker assignment problem for mixed-model as-sembly lines in a make-to-order environment considering control points and assignment restrictions. J. Model. Manag.
**2020**, 15, 1–34. [Google Scholar] [CrossRef] - Eslamipoor, R.; Nobari, A. A mathematical model for an integrated assembly line regarding learning and fatigue effects. Robotica
**2021**, 1–17. [Google Scholar] [CrossRef] - Zimmermann, H.-J. Fuzzy programming and linear programming with several objective functions. Fuzzy Sets Syst.
**1978**, 1, 45–55. [Google Scholar] [CrossRef] - Zwick, R.; Zimmermann, H.-J. Fuzzy set theory and its applications. Am. J. Psychol.
**1993**, 106, 304. [Google Scholar] [CrossRef] - Ghaffar, A.R.A.; Hasan, G.; Ashraf, Z.; Khan, M.F. Fuzzy goal programming with an imprecise intuitionistic fuzzy preference relations. Symmetry
**2020**, 12, 1548. [Google Scholar] [CrossRef] - Kang, H.-Y.; Lee, A.H. Inventory replenishment model using fuzzy multiple objective programming: A case study of a high-tech company in Taiwan. Appl. Soft Comput.
**2010**, 10, 1108–1118. [Google Scholar] [CrossRef] - Qi, J.G.; Burns, G.R.; Harrison, D.K. The application of parallel multipopulation genetic algorithms to dynamic job-shop scheduling. Int. J. Adv. Manuf. Technol.
**2000**, 16, 609–615. [Google Scholar] [CrossRef] - Lee, A.H.I.; Kang, H.-Y.; Lai, C.-M. Solving lot-sizing problem with quantity discount and transportation cost. Int. J. Syst. Sci.
**2013**, 44, 760–774. [Google Scholar] [CrossRef] - Kang, H.-Y.; Pearn, W.L.; Chung, I.-P.; Lee, A.H.I. An enhanced model for the integrated production and transportation problem in a multiple vehicles environment. Soft Comput.
**2015**, 20, 1415–1435. [Google Scholar] [CrossRef] - Jackson, J.R. An extension of Johnson’s results on job IDT scheduling. Nav. Res. Logist. Q.
**1956**, 3, 201–203. [Google Scholar] [CrossRef] - Rosenberg, O.; Ziegler, H. A comparison of heuristic algorithms for cost-oriented assembly line balancing. Math. Methods Oper. Res.
**1992**, 36, 477–495. [Google Scholar] [CrossRef] - E Gunther, R.; Johnson, G.D.; Peterson, R.S.; Günther, R. Currently practiced formulations for the assembly line balance problem. J. Oper. Manag.
**1983**, 3, 209–221. [Google Scholar] [CrossRef]

Number of Jobs | Maximum Number of Workstations | Workstation Setup Time | Total Processing Time | |
---|---|---|---|---|

Case 1 | 10 | 5 | 5 min | 94 min |

Case 2 | 20 | 8 | 6 min | 200 min |

Case 3 | 30 | 10 | 7 min | 360 min |

Case 4 | 66 | 12 | 9 min | 539 min |

Workstation | Processing Time | Setup Time | Cycle Time | Jobs |
---|---|---|---|---|

1 | 19 | 5 | 24 | J_{1}, J_{2} |

2 | 18 | 5 | 23 | J_{3}, J_{6} |

3 | 19 | 5 | 24 | J_{5}, J_{8} |

4 | 17 | 5 | 22 | J_{4}, J_{7} |

5 | 21 | 5 | 26 | J_{9}, J_{10} |

Objective | Decision Variable | Objective Value |
---|---|---|

Z_{1} | CT | 26 |

Z_{2} | NW | 5 |

Z_{3} | WV | 1.76 |

Z_{4} | TD | 11 |

Workstation | Processing Time | Setup Time | Cycle Time | Jobs |
---|---|---|---|---|

1 | 0 | 0 | 0 | |

2 | 0 | 0 | 0 | |

3 | 0 | 0 | 0 | |

4 | 0 | 0 | 0 | |

5 | 94 | 5 | 99 | J_{1}, J_{2}, J_{3}, J_{4}, J_{5}, J_{6}, J_{7}, J_{8}, J_{9}, J_{10} |

Objective | Decision Variable | Objective Value |
---|---|---|

Z_{1} | CT | 99 |

Z_{2} | NW | 1 |

Z_{3} | WV | 1413.76 |

Z_{4} | TD | 396 |

Workstation | Processing Time | Setup Time | Cycle Time | Jobs |
---|---|---|---|---|

1 | 19 | 5 | 24 | A_{1}, A_{2} |

2 | 18 | 5 | 23 | A_{3}, A_{6} |

3 | 18 | 5 | 23 | A_{4}, A_{5} |

4 | 18 | 5 | 23 | A_{7}, A_{8} |

5 | 21 | 5 | 26 | A_{9}, A_{10} |

Objective | Decision Variable | Objective Value |
---|---|---|

Z_{1} | CT | 26 |

Z_{2} | NW | 5 |

Z_{3} | WV | 1.36 |

Z_{4} | TD | 11 |

Workstation | Processing Time | Setup Time | Cycle Time | Jobs |
---|---|---|---|---|

1 | 21 | 5 | 26 | J_{1}, J_{3} |

2 | 17 | 5 | 22 | J_{2}, J_{4} |

3 | 16 | 5 | 21 | J_{6}, J_{7} |

4 | 19 | 5 | 24 | J_{5}, J_{8} |

5 | 21 | 5 | 26 | J_{9}, J_{10} |

Objective | Decision Variable | Objective Value |
---|---|---|

Z_{1} | CT | 26 |

Z_{2} | NW | 5 |

Z_{3} | WV | 4.16 |

Z_{4} | TD | 11 |

Decision Variable | Upper Bound | Lower Bound | Deviation | Average |
---|---|---|---|---|

CT | 99 | 26 | 73 | 62.5 |

NW | 5 | 1 | 4 | 3 |

WV | 1413.76 | 1.36 | 1412.4 | 707.56 |

TD | 396 | 11 | 385 | 203.5 |

Workstation | Processing Time | Setup Time | Cycle Time | Jobs |
---|---|---|---|---|

1 | 47 | 5 | 52 | J_{1}, J_{2}, J_{3}, J_{5}, J_{8} |

2 | ||||

3 | 47 | 5 | 52 | J_{4}, J_{6}, J_{7}, J_{9}, J_{10} |

4 | ||||

5 |

Objective | Objective Value | Objective Function |
---|---|---|

Z_{1} | 52 | ${f}_{1}({Z}_{1})$ = 0.64 |

Z_{2} | 2 | ${f}_{1}({Z}_{2})$ = 0.75 |

Z_{3} | 530.16 | ${f}_{1}({Z}_{3})$ = 0.63 |

Z_{4} | 141 | ${f}_{1}({Z}_{4})$ = 0.66 |

Total satisfaction level λ = 0.63 |

N | P_{c} | P_{m} | G_{max} |
---|---|---|---|

10 | 0.9 | 0.01 | 1000 |

50 | 0.99 | 0.05 | 2000 |

Run | N | P_{c} | P_{m} | G_{max} | Computation Time | Number of Generations | Fitness Value |
---|---|---|---|---|---|---|---|

1 | 10 | 0.9 | 0.01 | 1000 | 11 | 79 | 0.63 |

2 | 10 | 0.9 | 0.01 | 2000 | 19 | 61 | 0.63 |

3 | 10 | 0.9 | 0.05 | 1000 | 11 | 57 | 0.63 |

4 | 10 | 0.9 | 0.05 | 2000 | 19 | 68 | 0.63 |

5 | 10 | 0.99 | 0.01 | 1000 | 11 | 77 | 0.63 |

6 | 10 | 0.99 | 0.01 | 2000 | 19 | 85 | 0.63 |

7 | 10 | 0.99 | 0.05 | 1000 | 11 | 167 | 0.63 |

8 | 10 | 0.99 | 0.05 | 2000 | 19 | 132 | 0.63 |

9 | 50 | 0.9 | 0.01 | 1000 | 39 | 92 | 0.63 |

10 | 50 | 0.9 | 0.01 | 2000 | 74 | 65 | 0.63 |

11 | 50 | 0.9 | 0.05 | 1000 | 38 | 93 | 0.63 |

12 | 50 | 0.9 | 0.05 | 2000 | 73 | 91 | 0.63 |

13 | 50 | 0.99 | 0.01 | 1000 | 38 | 85 | 0.63 |

14 | 50 | 0.99 | 0.01 | 2000 | 74 | 67 | 0.63 |

15 | 50 | 0.99 | 0.05 | 1000 | 39 | 55 | 0.63 |

16 | 50 | 0.99 | 0.05 | 2000 | 75 | 48 | 0.63 |

Decision Variable | Upper Bound | Lower Bound | Deviation | Average |
---|---|---|---|---|

CT | 206 | 32 | 174 | 119 |

NW | 8 | 1 | 7 | 4.5 |

WV | 4642 | 8.75 | 4633.25 | 2325.375 |

TD | 1400 | 8 | 1392 | 704 |

Workstation | Processing Time | Setup Time | Cycle Time | Jobs |
---|---|---|---|---|

1 | ||||

2 | ||||

3 | ||||

4 | 66 | 6 | 72 | A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{7} |

5 | ||||

6 | 65 | 6 | 71 | A_{6}, B_{1}, B_{2}, B_{3}, B_{4}, B_{5}, B_{7} |

7 | 69 | 6 | 75 | A_{8}, A_{9}, A_{10}, B_{6}, B_{8}, B_{9}, B_{10} |

8 |

Objective | Objective Value | Objective Function |
---|---|---|

Z_{1} | 75 | ${f}_{1}({Z}_{1})$ = 0.75 |

Z_{2} | 3 | ${f}_{1}({Z}_{2})$ = 0.71 |

Z_{3} | 1042.75 | ${f}_{1}({Z}_{3})$ = 0.78 |

Z_{4} | 352 | ${f}_{1}({Z}_{4})$ = 0.75 |

Total satisfaction level λ = 0.71 |

Decision Variable | Upper Bound | Lower Bound | Deviation | Average |
---|---|---|---|---|

CT | 367 | 43 | 323 | 205 |

NW | 10 | 1 | 9 | 5.5 |

WV | 11664 | 0.7 | 11663.3 | 5832.35 |

TD | 3240 | 1.8 | 3238.2 | 1620.9 |

Workstation | Processing Time | Setup Time | Cycle Time | Jobs |
---|---|---|---|---|

1 | 120 | 7 | 127 | J_{1}, J_{2}, J_{4}, J_{6}, J_{7}, J_{16}, J_{18}, J_{20}, J_{21}, J_{23}, J_{26} |

2 | ||||

3 | 120 | 7 | 127 | J_{3}, J_{5}, J_{17}, J_{19}, J_{24}, J_{25}, J_{27}, J_{28} |

4 | ||||

5 | ||||

6 | ||||

7 | ||||

8 | 120 | 7 | 127 | J_{8}, J_{9}, J_{10}, J_{11}, J_{12}, J_{13}, J_{14}, J_{15}, J_{22}, J_{29}, J_{30} |

9 | ||||

10 |

Objective | Objective Value | Objective Function |
---|---|---|

Z_{1} | 127 | ${f}_{1}({Z}_{1})$ = 0.74 |

Z_{2} | 3 | ${f}_{1}({Z}_{2})$ = 0.78 |

Z_{3} | 3024 | ${f}_{1}({Z}_{3})$ = 0.74 |

Z_{4} | 840 | ${f}_{1}({Z}_{4})$ = 0.74 |

Total satisfaction level λ = 0.74 |

Decision Variable | Upper Bound | Lower Bound | Deviation | Average |
---|---|---|---|---|

CT | 807 | 75 | 732 | 441 |

NW | 12 | 1 | 11 | 6.5 |

WV | 49,749 | 7 | 49,742 | 24,878 |

TD | 8877 | 12 | 8865 | 4444.5 |

Workstation | Processing Time | Setup Time | Cycle Time | Jobs |
---|---|---|---|---|

1 | 100 | 9 | 109 | J_{1}, J_{2}, J_{3}, J_{4}, J_{5}, J_{6}, J_{7}, J_{8}, J_{9}, J_{21} |

2 | 140 | 9 | 149 | J_{46}, J_{47}, J_{48}, J_{49}, J_{50}, J_{51}, J_{52}, J_{53}, J_{54}, J_{55}, J_{56}, J_{57} |

3 | 152 | 9 | 161 | J_{11}, J_{12}, J_{31}, J_{32}, J_{33}, J_{34}, J_{35}, J_{36}, J_{37}, J_{38}, J_{40}, J_{41} |

4 | 121 | 9 | 130 | J_{10}, J_{16}, J_{17}, J_{18}, J_{19}, J_{20}, J_{22}, J_{24}, J_{25} |

5 | 166 | 9 | 175 | J_{13}, J_{14}, J_{15}, J_{23}, J_{26}, J_{27}, J_{28}, J_{29}, J_{30}, J_{39}, J_{42}, J_{43}, J_{44}, J_{45} |

6 | 119 | 9 | 128 | J_{58}, J_{59}, J_{60}, J_{61}, J_{62}, J_{63}, J_{64}, J_{65}, J_{66} |

7 | ||||

8 | ||||

9 | ||||

10 | ||||

11 | ||||

12 |

Objective | Objective Value | Objective Function |
---|---|---|

Z_{1} | 175 | ${f}_{1}({Z}_{1})$ = 0.86 |

Z_{2} | 6 | ${f}_{1}({Z}_{2})$ = 0.55 |

Z_{3} | 5305.25 | ${f}_{1}({Z}_{3})$ = 0.89 |

Z_{4} | 1248 | ${f}_{1}({Z}_{4})$ = 0.86 |

Total satisfaction level $\mathsf{\lambda}\text{}=\text{}0$.55 |

Compared Items | Taha et al. [11] | Kucukkoc and Zhang [1] | Cerqueus and Delorme [12] | Proposed Genetic Algorithm | Proposed Fuzzy Multi-Objective Linear Programming |
---|---|---|---|---|---|

Algorithm | Genetic algorithm | Genetic algorithm | Branch-and-bound search | Genetic algorithm | Exact |

Accuracy | Near optimal | Near optimal | Near optimal | Near optimal | Optimal |

One-sided ALBP | Yes | Yes | Yes | Yes | Yes |

Two-sided ALBP | Yes | Yes | No | No | No |

Multiple objectives | No | No | Yes | Yes | Yes |

Cycle time | Yes | Yes | Yes | Yes | Yes |

Number of workstations | No | Yes | Yes | Yes | Yes |

Workload variance | No | No | No | Yes | Yes |

Idle time | No | No | Yes | Yes | Yes |

Setup time | No | No | No | Yes | Yes |

Fuzzy sets | No | No | No | Yes | Yes |

Solved by common software packages | No | No | No | No | Yes |

Solve binary behavior | Yes | Yes | Yes | Yes | Yes |

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## Share and Cite

**MDPI and ACS Style**

Lee, A.H.I.; Kang, H.-Y.; Chen, C.-L.
Multi-Objective Assembly Line Balancing Problem with Setup Times Using Fuzzy Goal Programming and Genetic Algorithm. *Symmetry* **2021**, *13*, 333.
https://doi.org/10.3390/sym13020333

**AMA Style**

Lee AHI, Kang H-Y, Chen C-L.
Multi-Objective Assembly Line Balancing Problem with Setup Times Using Fuzzy Goal Programming and Genetic Algorithm. *Symmetry*. 2021; 13(2):333.
https://doi.org/10.3390/sym13020333

**Chicago/Turabian Style**

Lee, Amy H. I., He-Yau Kang, and Chong-Lin Chen.
2021. "Multi-Objective Assembly Line Balancing Problem with Setup Times Using Fuzzy Goal Programming and Genetic Algorithm" *Symmetry* 13, no. 2: 333.
https://doi.org/10.3390/sym13020333