Balancing a U-Shaped Assembly Line with a Heuristic Algorithm Based on a Comprehensive Rank Value

Abstract

An aim of sustainable development of the manufacturing industry is to reduce the idle time in the product-assembly process and improve the balance efficiency of the assembly line. A priority relationship diagram is obtained on an existing assembly line in the laboratory by measuring the task time of the chassis model, analyzing the product structure, and designing the assembly process. The type-E balance model of the U-shaped assembly line is established and solved by a heuristic algorithm based on the comprehensive rank value. The type-E balance problem of the U-shaped assembly-line plan of the chassis model is obtained, and the production line layout is planned. Combining instances to compare the results of the heuristic algorithm, genetic algorithm, and simulated annealing, comparison of the results shows that the degree of load balancing is slightly higher than genetic algorithm and simulated annealing. The balance efficiencies obtained by the heuristic algorithm are smaller than the genetic algorithm and simulated annealing. The calculation time is significantly less than the genetic algorithm and simulated annealing, and the scale of instances has little effect on the calculation time. The results verify that the model and the algorithm are effective. This study provides a reference for the entire process of the U-shaped assembly-line, type-E balance and the assembly products in laboratories.

1. Introduction

Aiming to upgrade facilities and improve equipment, manufacturing companies are pursuing production efficiencies and cost reductions by transforming the traditional straight assembly lines into the U-shaped assembly lines (UAL). As the number of products tasks increase, the complexity of the balance problem increases. The U-shaped assembly-line balance (UALB) is difficult to completely solve within the required time [1,2]. In the face of changing customer needs, the traditional straight assembly line lacks flexibility and is difficult to readjust to achieve a new balance. The UAL overcomes the shortcomings of the traditional straight assembly line. The workstations on the UAL are relatively close, which shortens the transportation distance of materials. The UAL processes are highly flexible and could be rebalanced by increasing or decreasing the number of operators as external requirements change.

The UALB is a balanced technology that solves task distribution to workstations during production or assembly. The UALB is a typical NP-hard problem [3]. There are heuristic algorithms and metaheuristic algorithms to solve the UALB [4].

The heuristic algorithm is an algorithm based on rules or construction experience, which gives a feasible solution to the combinatorial optimization problem with cost acceptance (computing time and space) [5]. Miltenburg [6] proposed the UAL for the first time, compared the straight with the U-shaped assembly line, and explained the advantages of the UALB. The dynamic programming method and heuristic method were used to solve the UALB, respectively, to verify the effectiveness of the algorithm. Ponnambalam [7] proposed the weighted value method to solve the assembly-line balance problem. Gökçen [8] proposed the U-shaped assembly line balance problem (UALBP) with the shortest path method and applied a heuristic algorithm to solve the UALBP. Jiao [9] proposed an improved weighted value method based on the directed graph theory to solve the UALBP. Li [10] proposed a heuristic algorithm with the beam rules to solve the UALBP and used three heuristic rules to lead the solution direction and increase workstation utilization. Jiao [11] proposed a heuristic algorithm based on priority value to solve the parallel UALBP.

Metaheuristic algorithms are methods to solve the optimal solution or satisfactory solution of complex optimization problems based on the mechanism of computational intelligence. Metaheuristic algorithms include genetic algorithms [12,13,14], artificial bee colony algorithms [15], ant colony algorithms [16], particle swarm algorithms [17], simulated annealing algorithms [18,19], etc. Metaheuristic algorithms are widely used in online learning [20], vehicle scheduling [21], many-objective optimization [22], berth scheduling [23], intelligent driving [24], biomedicine [25], intelligent manufacturing [26], assembly-line balancing [5], and other fields. In 2000, Kim [27] first applied a metaheuristic algorithm (co-evolutionary algorithm) to solve the balancing problem of the U-shaped assembly line. Subsequently, metaheuristic algorithms are widely used to solve the balance problem of the U-shaped assembly line. Michela [12] established a mathematical model under the multiple constraints and solved it with a genetic algorithm. Alavidoost [13] used an improved genetic algorithm to solve the fuzzy UALBP, and the task time was represented by a triangular fuzzy number. Faruk [14] applied the genetic algorithm to solve the UAL rebalancing problem with the random task time. Tang [15] used the artificial bee colony algorithm to solve the double-objective UALBP. Li [16] used the ant colony algorithm to solve the UALBP and proposed a new priority relationship expression. Mukund [17] proposed a particle swarm algorithm to solve the U-shaped robot assembly-line-balance problem, adding robots to the assembly line to achieve sustainable development. Kara [18] proposed an improved simulated annealing algorithm to solve the multi-objective, mixed-model UALBP. Neda [19] established a mixed UALB mathematical model considering worker efficiency and proposed a two-stage simulated annealing algorithm.

In recent years, scholars have applied new metaheuristic algorithms such as the improved migrating bird optimization algorithm [28], the mixed competition algorithm [29], the improved flower pollination algorithm [30], and the Restarted Iterated Pareto Greedy algorithm [31] to solve the UALBP. Tang [28] proposed an improved migratory bird optimization algorithm to solve the UALBP, considering task and worker allocation. Mazyar [29] proposed a new hybrid competition method based on the ranking position weight method and the COMSOAL algorithm to solve the UALBP. Zhang [30] designed an improved flower pollination algorithm to solve the double-objective UALBP. Tang [31] used a Restarted Iterated Pareto Greedy algorithm to solve the multi-objective UALBP.

The assembly-line type-E balance problem is to find a set number of workstations and cycle time to maximize the balance efficiency of the assembly line. The methods to solve the assembly-line, type-E balance problem mainly include precise algorithms [32,33] and intelligent algorithms [34,35]. In 2011, Wei [32] first studied the assembly-line, type-E balance problem and used the LINGO software to solve the exact solution. Esmaeilbeigi [33] used the CPLEX software to solve the assembly-line, type-E balance problem. Zacharia [34] proposed a heuristic algorithm to solve the fuzzy, assembly-line, type-E balance problem. Mohd [35] used the swarm of immune cells alogorithm to solve the assembly-line, type-E balance problem with resource constraints and bottleneck identification.

Improving the balance efficiency of the UAL can effectively reduce waste in the product-assembly process [36]. A heuristic algorithm based on the comprehensive rank value [37,38,39] is used to solve the UALBP. The calculation process of the heuristic algorithm is simple, and the calculation efficiency of the heuristic algorithm is high. In this paper, we propose a heuristic algorithm based on the comprehensive rank value, combined with the chassis model assembly experiment example, quickly solve the U-shaped assembly-line, type-E balance model, and design the layout of the UAL. Thirty instances are carried out, we use a heuristic algorithm based on the comprehensive rank value, genetic algorithm, and simulated annealing to solve the U-shaped assembly line type-E balance problem (UALEBP), and compare the results of three algorithms.

2. U-Shaped Assembly Line

The UAL and the straight assembly line differ in the setup of workstations. Workstations of the UAL can be designed into two forms, the conventional workstation and the crossover workstation, whereas the straight assembly line can only be set up as the conventional workstation. As shown in Figure 1, the import and export of the UAL are on the same side, the U-shaped solid line represents the assembly line, the dashed box represents the workstation area, and the number in the dashed box represents the workstation number. In other words, workstation 1 and workstation 3 are crossover workstations, and workstation 2 is a conventional workstation.

The task assignments are different between the UAL and the straight assembly line. For the UAL, the tasks can be assigned to workstations from front to back or from back to front according to the flow direction of the assembly line. The tasks can also be assigned in both directions at the same time, making it more difficult to balance the UAL. In the straight assembly line, the assignment of the tasks to the workstation is carried out in a single direction, which reduces the intersection of the combination and collocation of the tasks.

3. Mathematical Model

3.1. Assumptions

In the product structure analysis, the product assembly tasks name is defined, the tasks work time and the priority relationship are measured, and the priority relationship diagram is drawn. An assembly line balancing model is built with the following assumptions [40,41]:

(1)

Each task is independent.

(2)

The time for a task refers to the assembly time; the time for parts retrieval is not counted. The task time is a fixed value, which is determined by taking the average of multiple experimental measurements.

(3)

The performing time of tasks at different workstations is the same.

(4)

The tasks can be assigned to arbitrary workstations without restrictions such as equipment site.

(5)

The products have only one process route, there is no parallel station, and the transit time of each workstation on the assembly line is neglected.

(6)

The proficiency levels of the workers are consistent, they can complete all tasks, and the walking times of the operators inside the workstations are negligible.

(7)

It is suitable for the same product of assembly line production manufacturing, and the manufacturing process is defined.

(8)

The assembly line is automatic and takes the same amount of time at each workstation.

3.2. Notations

The assembly line balancing is to assign the tasks to each workstation in turn according to certain rules, realize the assembly process, and improve the production efficiency. To express the modeling and solve process more clearly, the constants defined are as follows (

Table 1. The Definition of the Variables.

Variable	Definition
i, j, q	Task (i, j, q = 1, …, I)
	Number of workstation (k = 1, …, K)
I	Total number of tasks
K	Total number of workstations
Sk	Task set assigned to the k-th workstation (k = 1, …, K)
Pre(i)	The set of immediately preceding tasks of task i
ti	Task time of task i
t(Sk)	The total task time of the k-th workstation (k = 1, …, K)
C	Cycle time
$n_{S_k}$	Number of tasks in the k-th workstation (k = 1, …, K)
f	Balance efficiency of the assembly line

).

Decision variables:

$$\begin{array}{l}{W}_k \mathrm{Workstation} \mathrm{indicator} \mathrm{variables}\\ W_k=\{\begin{array}{l}1, \mathrm{if} \mathrm{the} k\text{-th} \mathrm{workstation} \mathrm{is} \mathrm{turned} \mathrm{on}\hfill \\ 0, \mathrm{if} \mathrm{the} k\text{-th} \mathrm{workstation} \mathrm{is} \mathrm{not} \mathrm{turned} \mathrm{on}\hfill \end{array} \\ x_{ik} \mathrm{Task} i \mathrm{is} \mathrm{assigned} \mathrm{to} \mathrm{workstation} k \mathrm{following} \mathrm{forward} \mathrm{to} \mathrm{backward} \mathrm{order}\\ x_{ik}=\{\begin{array}{l}1, \mathrm{When} \mathrm{a} \mathrm{task} \mathrm{is} \mathrm{assigned} \mathrm{to} \mathrm{a} \mathrm{workstation} k\\ 0, \mathrm{When} \mathrm{a} \mathrm{task} \mathrm{is} \mathrm{not} \mathrm{assigned} \mathrm{to} \mathrm{a} \mathrm{workstation} k\end{array}\\ y_{ik} \mathrm{Task} i \mathrm{is} \mathrm{assigned} \mathrm{to} \mathrm{workstation} k \mathrm{following} \mathrm{backward} \mathrm{to} \mathrm{forward} \mathrm{order}\\ y_{ik}=\{\begin{array}{l}1, \mathrm{When} \mathrm{task} i \mathrm{is} \mathrm{assigned} \mathrm{to} \mathrm{workstation} k\\ 0, \mathrm{When} \mathrm{task} i \mathrm{is} \mathrm{not} \mathrm{assigned} \mathrm{to} \mathrm{workstation} k \end{array}\end{array}$$

3.3. Mathematical Model

The U-shaped assembly-line, type-E balance problem is to find the combination of the cycle time and the number of workstations and the balance efficiency of the assembly line. Constructing a mathematical model of the assembly-line, type-E balance problem, the objective function is:

$$\max f=\left\{\frac{{\displaystyle \sum _{i=1}^I{t}_i}}{(K\text{×}C)}\right\}\text{×}100\%$$

Constraints:

$${\displaystyle \sum _{k=1}^K(x_{ik}+y_{ik})=1}$$

(1)

$$t(S_k)=\sum _{i\in S_k}{t}_i={\displaystyle \sum _{i=1}^I{t}_i(x_{ik}}+y_{ik})\le C, k=1,2,\dots K$$

(2)

$${\displaystyle \sum _{k=1}^K(k{x}_{ik}-k{x}_{jk})}\le 0, \forall (i,j),j\in Pre(i)$$

(3)

$${\displaystyle \sum _{k=1}^K(k{y}_{jk}-k{y}_{ik})}\le 0, \forall (i,j),i\in Pre(j)$$

(4)

$${\displaystyle \sum _{i=1}^I(x_{ik}+x_{jk})}\le n{w}_k, k=1,2,\dots K$$

(5)

$$W_{k+1}\le W_k, k=1,2,\dots ,K-1$$

(6)

$${\displaystyle \sum _{i=1}^I{x}_{ik}\le n}{W}_k, k=1,2,\dots ,K$$

(7)

$${\displaystyle \sum _{k=1}^{K}k{x}_{ik}\le }{\displaystyle \sum _{k=1}^{K}k{x}_{jk}}, \forall (i,j),j\in Pre(i)$$

(8)

$$\mathrm{C}=\max t(S_k)$$

(9)

The objective function represents the maximum balance efficiency of the assembly line. Constraint (1) indicates that the task i is to be distributed in only one order, which means the order is from front to back or from back to front, and the task i can only be assigned to a workstation. Constraint (2) represents that the sum of the total operating time in the k-th workstation does not exceed the cycle time, C. Constraint (3) indicates that if the task j is assigned in the order from front to back, the tasks before the task i must be allocated in the order from front to back. Constraint (4) indicates that if the task j is assigned in the order from back to front, the tasks after the task i must be allocated in the order from front to back. Constraint (5) indicates that a workstation is turned on if there are tasks in it. Constraint (9) indicates that the cycle time of the assembly line is the maximum total task time in the workstation.

When the value of $y_{ik}$ is zero in Constraints (1) and (2), the mathematical model is a straight assembly-line-balancing problem. Constraint (1) indicates that the same tasks must be assigned to a workstation. Constraint (2) represents that the sum of the total operating time in the workstation, k, does not exceed the cycle time, C. Constraint (6) represents $k$, the workstations in sequential order on a straight assembly line. If the (k + 1)-th workstation turns on, the k-th workstation must be assigned a task or end the assignment. Constraint (7) means that if there is a task in the k-th workstation, then the k-th workstation must be turned on. Constraint (8) indicates the preferential relationship constraints between tasks.

4. Heuristic Algorithm

A heuristic algorithm (HA) with a comprehensive rank value is to assign tasks to workstations of the U-shaped assembly line according to a comprehensive rank value of the tasks and realize the balance of work time to achieve the balance of the assembly-line system.

4.1. Assignment Relations

In the priority relationship diagram, if the task i is assembled before the task j, it is said that the task i is the prior task of the task j. If the task j is assembled before the task i, it is said that the task j is the follow-up task of the task i. It can be represented by an ordinal pair (i, j).

For ordinal pairs (i, j), if the task i and the task j are directly neighbored, it is said that the task i is the immediately preceding task of the task j, and the task j is the immediately following task of the task i. The priority relationship of the tasks is transitive, if task i and j satisfy an ordinal pair (i, j), task j and q satisfy the ordinal pair (j, q), then the task i and q satisfy the ordinal pair (i, q).

The priority relationship of the tasks can be represented in a priority relationship matrix. There are I tasks in the priority relationships diagram, and the priority relationship matrix is a I-th order square matrix P. The value of the element ${\mathrm{P}}_{ij}$ (i = 1, 2, …, n) in the priority relationship matrix P depends on the priority relationship of i and j, i.e.,

$${\mathrm{P}}_{ij}=\{\begin{array}{l}1, \mathrm{task} i \mathrm{is} \mathrm{immediately} \mathrm{precding} \mathrm{task} \mathrm{of} j\hfill \\ 0, \mathrm{task} i \mathrm{is} \mathrm{not} \mathrm{immediately} \mathrm{precding} \mathrm{task} \mathrm{of} j\hfill \end{array}$$

4.2. Comprehensive Rank Value

The comprehensive rank value is obtained by taking the time-ranked weight and the position-ranked weight into comprehensive consideration. The time-ranked weight of each task is the sum of its own task time and all subsequent task times; the position-ranked weight is the sum of that task and the number of subsequent tasks. The assignments are sorted and assigned according to task time and their subsequent task size, resulting in the time-ranked weight and the position-ranked weight, and the two ranked values are multiplied to obtain a comprehensive rank value for the task.

According to the priority relationship diagram, the comprehensive rank value of the task is solved as follows:

(1) The calculation of the position-ranked weight, $R_{hi},$ is for the task i, the $H_i$ is the number of subsequent assignments of the task, and the $F_i$ is for the subsequent set of the assignments for the task i.

$$H_i=count\left\{F_i\right\}$$

All the tasks are arranged in ascending order based on the $H_i$ values. The task with the smallest $H_i$ value is assigned the position-ranked weight $R_{hi}$ = 1, the task with the next smallest value, $H_i$, is assigned the position-ranked weight, $R_{hi}$ = 2. If the tasks values for $H_i$ are the same, it is given the same position-ranked weight $R_{hi}$. This goes on until all the tasks have been given a specific $R_{hi}$.

The calculation of the time-ranked positional weight, $R_{ti}$, is for the task i, the $T_i$ is the task time load matrix, the sum of the time is for the task i, and in the total subsequent task time, $t_l$, l is for tasks in the $F_i$ set.

$$T_i=t_i+\sum _{\mathrm{l}\in F_i}{t}_{\mathrm{l}}$$

All the tasks are ranked according to the ascending order of $T_i$. The task with the smallest $T_i$ value is assigned the time-ranked weight, $R_{ti}$ = 1. The tasks with the next smallest $T_i$ value is given the time-ranked weight $R_{ti}$ = 2. If the task $T_i$ values are the same, it is given the same time-ranked weight $R_{ti}$. This continues on until all the tasks have been given a specific $R_{ti}$.

Calculation of the comprehensive rank positional weight, $R_i$, for the task i.

$$R_i=R_{hi}\times R_{ti}$$

4.3. A Heuristic Algorithm Based on Comprehensive Rank Value

According to the comprehensive rank value, $R_i$, of the task i, we apply the heuristic principles to solve the model. The assignment process starts from the first workstation. If a task is assigned to a workstation, the list of assignable tasks is updated at once, and another new task is selected and assigned to the current workstation, continuing until the total time of the task at the current workstation is closed or equal to the cycle time.

The following are the conditions under which a task can be assigned. The first condition is that the task has not been assigned. The second condition is that all the immediately preceding tasks (follow-up tasks) of the tasks are assigned, and the tasks are assigned in a forward to back relationship on the priority relationship diagram. The third condition is that the remaining time of the current workstation is sufficient to the assigned task.

The following is the heuristic rule based on the comprehensive rank value. When assigning tasks from the beginning of the priority relationship diagram to the first workstation, select the task assignment with the largest comprehensive rank value. When assigning the last workstation from end of the priority relationship diagram, select the task assignment with the smallest comprehensive rank value. At the same time, it is necessary to consider that the total work time allocated to the workstation is less than or equal to the cycle time or to make the idle time of the current workstation as small as possible to meet the objective function.

5. Experimental Case

The assembly of the chassis model in the laboratory is studied on the U-shaped assembly line and straight assembly line. The UALEBP is analyzed, meaning that the number of workstations and the cycle time are within a certain range of changes and seek a combination of the smallest product of the number of workstations and the cycle time and maximize the balance efficiency of the entire assembly line.

5.1. Experimental Set-Up

The assembly-line experiments of the chassis model are performed in the laboratory, and the assembly diagram of the chassis model is shown in Figure 2. The chassis model is composed of 149 parts. According to the assembly instructions of the chassis model, the product structure analysis is carried out. The assembly process is subdivided into 30 tasks. Each task includes assembly multiple parts. The names of the parts are shown in the BOM table (omitted).

5.2. Priority Relationship Diagram

The assembly process of the chassis model was analyzed, and the task priority relationship diagram of the chassis model was designed, as shown in Figure 3.

In the assembly priority relationship diagram of the chassis model, the numbers in the circles represent the numbers of the tasks, the numbers above each of the circles represent the task work time, with the time measured in seconds(s), and the straight lines with the arrows describe the sequence of the tasks in the assembly process. It is experimentally determined that the total time taken to assemble the chassis model with 30 tasks is 413 s.

5.3. Comprehensive Rank Value of Chassis Model

According to the priority relationship diagram of the chassis model, the preferential relation set, P, and the task time matrix, T, are written based on the digraph theory.

The direct priority relationships of tasks from the set of the priority relationships can be written as a priority relationship matrix, P. The following is a triplet representation of P:

P = [(1,5,1),(2,3,1),(3,9,1),(4,8,1),(5,6,1),(6,7,1),(7,8,1),(8,12,1),(9,10,1),(10,12,1),(11,12,1),(12,13,1),(13,14,1),(13,15,1),(13,17,1),(14,23,1),(15,23,1),(16,23,1),(17,23,1),(18,19,1),(19,20,1),(20,21,1),(21,22,1),(22,23,1),(23,24,1),(24,25,1),(25,26,1),(26,27,1),(27,28,1),(28,29,1),(29,30,1),(30,31,1),(30,32,1)]

The task time matrix:

T = [14,4,4,14,2,14,19,22,16,11,52,12,20,10,4,4,10,32,25,34,30,2,2,4,34,9,6,5,6,2]

According to the MATLAB program of the comprehensive rank value, input the priority relationship matrix P and the task time matrix T, calculate the comprehensive rank value, and then output the comprehensive rank matrix of the task.

$R_i$ = [414,340,304,315,442,400,330,238,270,224,336,182,156,99,90,90,99,351,240,165,120,81,64,49,36,25,16,9,4,1]

5.4. Results and Evaluation

The assembly-line balance model is solved by the heuristic algorithm based on the comprehensive rank value. The distribution results of the straight line and U-shaped assembly line are shown in

Table 2. The task allocation results of the chassis model.

Number of Stations	U-Shaped Line			Straight Line
	k	Assigned Task	t(Sk)	k	Assigned Task	t(Sk)
K = 5	1	{1,2,11,28,29,30}	83	1	{1,5,6,18,2,4,3}	84
	2	{5,18,25,26,27}	83	2	{11,7,8}	83
	3	{3,4,6,7,8,9,24}	83	3	{9,19,10,12,13}	84
	4	{10,12,19,20}	82	4	{20,21,14,15,22}	80
	5	{13,14,15,16,17,21,22,23}	82	5	{17,16,23,24,25,26,27,28,29,30}	82
K = 6	1	{2,3,9,10,18,30}	69	1	{1,5,11}	68
	2	{11,27,28,29}	69	2	{6,18,2,4,3}	68
	3	{1,4,5,6,19}	69	3	{7,9,8,10,12}	70
	4	{15,16,17,22,23,24,25,26}	69	4	{19,13,14,17,15}	69
	5	{7,13,14,21}	69	5	{20,21,16}	68
	6	{8,12,20}	68	6	{22,23,24,25,26,27,28,29,30}	70
K = 7	1	{1,5,6,2,7,30}	59	1	{1,5,6,2,4,7}	57
	2	{18,3,8}	58	2	{18,3,9,10}	63
	3	{11,29}	58	3	{19,8,12}	59
	4	{9,19,10,28}	57	4	{11}	52
	5	{12,20,27,26}	61	5	{20,13,16}	58
	6	{13,21,14}	60	6	{21,14,17,15,22,23}	62
	7	{17,16,15,22,25,24,23}	60	7	{24,25,26,27,28,29,30}	62

, using the three results, K = 5, K = 6, K = 7. Draw the layout plan of the U-shaped line distribution scheme with five workstations, as shown in the Figure 4.

Analyze the results in Table 2, respectively, and calculate the balance effect index: the balance efficiency and the load balance degree of the distribution results [42], as shown in

Table 3. The analysis of balancing results.

Number of Stations	U-Shaped Line				Straight Line
	C	K × C	V	f	C	K × C	V	f
K = 5	83	415	0.006	99.52	84	420	0.018	98.33
K = 6	69	414	0.005	99.76	70	420	0.014	98.33
K = 7	61	427	0.021	96.72	63	441	0.056	93.65

. It can be seen from Table 3 that, compared with the straight type, the UAL has a larger balance efficiency and a lower load balance degree, thus the balance of the UAL is better.

6. Classic Instances

Thirty instances are carried out to measure the optimization of the proposed heuristic algorithm; those instances are from https://assembly-line-balancing.de/ (accessed on 10 October 2021). The algorithms are coded in the MATLAB programming and run on a computer with an Intel(R) Core(TM) i5 6300 processor running at 2.80 GHz. We compared the heuristic algorithm with the genetic algorithm and the simulated annealing, and the results of the three algorithms are showed in

Table 4. Results of Heuristic Algorithm, Genetic Algorithm, and Simulated Annealing.

Instances		Heuristic Algorithm						Genetic Algorithm						Simulated Annealing
		K	C	K × C	f	V	CPU	K	C	K × C	f	V	CPU	K	C	K × C	f	V	CPU
1	Mertens7	6	6	36	80.56	0.178	2.12	6	6	36	80.56	0.115	1.08	6	6	36	80.56	0.115	0.89
2		5	7	35	82.86	0.140	2.26	5	7	35	82.86	0.107	1.06	5	7	35	82.86	0.107	0.88
3		2	15	30	96.67	0.033	2.14	2	15	30	96.67	0.033	1.18	2	15	30	96.67	0.033	0.89
4	Jaeschke9	8	6	48	77.08	0.143	2.08	8	6	48	77.08	0.116	1.26	8	6	48	77.08	0.116	1.08
5		7	7	49	75.51	0.166	2.13	7	7	49	75.51	0.126	1.23	7	7	49	75.51	0.126	1.05
6		6	8	48	72.50	0.197	2.00	6	8	48	77.08	0.133	1.25	6	8	48	77.08	0.133	1.06
7		3	18	54	68.52	0.445	2.03	3	18	54	68.52	0.168	1.26	3	13	39	94.87	0.036	1.08
8	Jackson11	6	9	54	85.19	0.178	2.04	6	9	54	85.19	0.083	1.49	6	9	54	85.19	0.083	1.11
9		6	10	60	76.67	0.180	1.97	5	10	50	92.00	0.040	1.43	5	10	50	92.00	0.040	1.14
10		4	14	56	82.14	0.186	2.06	4	14	56	82.14	0.042	1.41	4	12	48	95.83	0.042	1.01
11		3	21	63	73.02	0.349	2.07	3	21	63	73.02	0.029	1.42	3	16	48	95.83	0.059	1.03
12	Mitchell21	8	14	112	93.75	0.056	2.12	8	14	112	93.75	0.043	2.29	8	14	112	93.75	0.043	1.39
13		6	21	126	83.33	0.311	2.14	5	21	105	100.00	0.000	1.40	5	21	105	100.00	0.000	1.38
14	Heskia-off28	8	138	1104	92.75	0.100	2.18	8	138	1104	92.75	0.019	2.93	8	128	1024	100.00	0.000	1.75
15		6	205	1230	83.25	0.359	2.19	6	205	1230	83.25	0.028	2.88	6	171	1026	99.81	0.004	1.74
16		5	216	1080	94.81	0.101	2.13	5	216	1080	94.81	0.006	2.86	5	205	1025	99.90	0.002	1.78
17		4	324	1296	79.01	0.362	2.12	4	324	1296	79.01	0.011	2.91	4	256	1024	100.00	0.000	1.69
18	Kilbrid-ge45	10	57	570	96.84	0.030	1.85	10	57	570	96.84	0.015	4.44	10	56	560	98.57	0.017	3.28
19		8	79	632	87.34	0.302	2.07	8	79	632	87.34	0.026	4.39	8	69	552	100.00	0.000	3.21
20		7	92	644	85.71	0.306	1.93	7	92	644	85.71	0.038	4.42	7	79	553	99.82	0.004	3.32
21		6	110	660	83.64	0.334	1.99	6	110	660	83.64	0.034	4.41	6	92	552	100.00	0.000	3.27
22	Tonge70	10	364	3640	96.43	0.087	2.00	10	364	3640	96.43	0.017	6.79	10	351	3510	100.00	0.000	6.46
23		9	410	3690	95.12	0.112	2.02	9	410	3690	95.12	0.013	6.83	9	390	3510	100.00	0.000	6.42
24		8	468	3744	93.75	0.140	2.06	8	468	3744	93.75	0.016	6.79	8	439	3512	99.94	0.001	6.40
25		7	527	3689	95.15	0.105	1.96	7	527	3689	95.15	0.010	6.70	7	502	3514	99.89	0.002	6.48
26	Arcus83	16	5048	80768	93.73	0.106	1.98	16	5048	80768	93.73	0.033	8.16	16	4840	77440	97.76	0.016	8.05
27		12	6842	82104	92.21	0.158	2.08	12	6842	82104	92.21	0.025	8.16	12	6349	76188	99.36	0.008	8.02
28		11	7571	83281	90.91	0.175	2.07	11	7571	83281	90.91	0.031	8.05	11	6936	76296	99.23	0.007	8.08
29		10	8412	84120	90.00	0.241	2.07	10	8412	84120	90.00	0.038	8.44	10	7578	75780	99.90	0.003	8.01
30		9	8898	80082	94.54	0.090	2.11	9	8898	80082	94.54	0.016	8.59	9	8433	75897	99.74	0.003	8.21

. Table 4 includes the balance efficiency (E) [42], the degree of load balancing (V) [42], and the calculation time (CPU); the CPU is measured in seconds(s). We compared the calculation times of the three algorithms and drew a comparison diagram, as shown in Figure 5.

It can be seen from Table 4, in the results of the heuristic algorithm based on the comprehensive rank value, the degree of load balancing, V, is slightly higher than the results of the genetic algorithm and the simulated annealing. The balance efficiencies obtained by the heuristic algorithm are smaller than the genetic algorithm and the simulated annealing. The calculation time is significantly less than the genetic algorithm and the simulated annealing, and the scale of instances has little effect on the calculation time. The result of solving the U-shaped assembly-line, type-E balance is reasonable, and the result verifies the effectiveness of the model and algorithm.

7. Conclusions

This paper establishes a mathematical model for the straight assembly-line, type-E balance and the U-shaped assembly-line, type-E balance, proposes a heuristic algorithm based on the comprehensive rank value, uses the chassis model as an experimental example to illustrate the algorithm-solving process, and compares the results of the instances with genetic algorithms and simulated annealing. From this, the following conclusions can be drawn:

(1)

The experiments verify that the model and the heuristic algorithm are effective, the results of the U-shaped assembly line balance are better than the straight assembly line, and the distribution plan layout of the U-shaped assembly line balance is designed. This paper provides a reference for the entire process of the UALB and the assembly products in laboratories.

(2)

Calculate an experimental example and compare the results of the UAL with the straight assembly line. From the results, it can be seen that the UAL has a higher balance efficiency and a smaller degree of load balancing, V. Besides, the UAL is more efficient, easy to adjust, and has high flexibility. In conclusion, the UAL improves the efficiency of the assembly line.

(3)

In calculating the 30 instances, the comparison of the results of the heuristic algorithm with the genetic algorithm and the simulated annealing show that the degree of load balancing, V, is slightly higher than the genetic algorithm and the simulated annealing. The balance efficiencies obtained by the heuristic algorithm are smaller than the genetic algorithm and the simulated annealing. The calculation time is significantly less than the genetic algorithm and the simulated annealing, and the scale of instances has little effect on the calculation time. The heuristic algorithm obtained reasonable results, which verify the effectiveness of the model and algorithm.

The heuristic algorithm is widely used to solve the assembly-line-balancing problem with its specific rules and has the advantage of quick and simple solutions. However, the accuracy of the large-scale problems and the solutions are not as good as metaheuristic algorithms. Therefore, the hybrid algorithm combining the rule-based heuristic algorithm and metaheuristic algorithm will become the development direction of the future research. The hybrid algorithm can not only retain the calculation speed of the heuristic algorithm, but also improve the accuracy of the solution.