Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 9748378, 14 pages
http://dx.doi.org/10.1155/2016/9748378

Research Article

Formulations, Features of Solution Space, and Algorithms for Line-Pure Seru System Conversion

Wei Sun,¹ Qianqian Li,¹ Chunhui Huo,¹ Yang Yu,² and Ke Ma²

¹Business School, Liaoning University, Shenyang 110316, China
²Institute of Systems Engineering, State Key Laboratory of Synthetic Automation for Process Industries, Northeastern University, Shenyang 110819, China

Received 26 February 2016; Accepted 19 May 2016

Academic Editor: Yakov Strelniker

Copyright © 2016 Wei Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The line-seru conversion is usually used to improve productivity, especially in volatile business environment. Due to the simplicity, most researches focused on line-pure seru system conversion. We summarize the two existing models (i.e., a biobjective model and a single-objective model) of line-pure system conversion and formulate the three other usually used single-objective models in an integrated framework by combining evaluated performances and constraints. Subsequently, we analyze the solution space features of line-pure seru system conversion by dividing the whole solution space into several subspaces according to the number of serus. We focus on investigating the features between (and TLH) and subspaces. Thirdly, according to the distinct features between (and TLH) and subspaces, we propose four effective algorithms to solve the four single-objective models, respectively. Finally, we evaluate the computational performance of the developed algorithms by comparing with enumeration based on extensive experiments.

1. Introduction

The seru production, conceived at Sony, is an innovation of assembly system used widely in the Japanese electronics industry and recognized a new production pattern. Seru production is proposed to overcome the less flexible shortcoming of assembly line, especially confronted with the dynamic and volatile business environments. Seru is an assembly unit including several simple equipment and one (or more) multiskilled operator(s). Seru is the pronunciation of cell in Japanese; therefore, Seru production is also called Japanese style cellular production. In seru, worker(s) must be multiskilled [1–4] because workers need to operate most or all the processes of production. In general, there are three types of seru: divisional seru, rotating seru, and yatai [5, 6]. In a divisional seru, tasks are divided into different sections and workers are partially cross-trained. Each section is operated by one or more workers. However, workers in rotating seru or yatai are completely cross-trained and do all tasks. In this research, rotating serus or yatai are considered.

Seru system can be generally divided into two types: pure seru system with only seru(s) and hybrid system with seru(s) and short line. As Stecke et al. [5, 6] claimed, with combined strengths from Toyota’s lean philosophy and Sony’s one-person production organization, seru system is a more productive, efficient, and flexible system than conveyor assembly line. As we know, the productivity of the assembly line is decided by the worker with the worst skill. It is amazing that seru system can greatly decrease the influence of the worker with the worst skill on the productivity by worker reconfiguration. Seru is more flexible than assembly line, because the workers in seru system are multiskilled operators who can operate multiple tasks and process multiple products. Seru’s flexibility is different with cellular manufacturing, where workers operate the certain part/product family. Seru system can response to market change quickly, due to movable workstations, light equipment, and multiskilled workers. Moreover, seru system can be continuously improved, because workers in seru can study to process more tasks and products.

Due to the merit of seru system [5–7], seru system has successfully been applied by many leading Japanese companies such as Sony, Canon, Panasonic, NEC, Fujitsu, Sharp, and Sanyo. Presently, many companies converted assembly line into seru system (line-seru conversion) to increase the productivity [8–10]. By adopting line-seru conversion, Canon and Sony reduced 720,000 and 710,000 square meters of floor space, respectively [5, 6]. Moreover, Canon’s costs were reduced significantly, by 55 billion yen in 2003, and by a total of 230 billion yen from 1998 to 2003. As a result, Canon emerged as a leading electronics maker [11, 12]. Other benefits [13, 14] include the reductions of throughput time, setup time, required labor hours, WIP inventories, and finished-product inventories. For example, the makespan was reduced by 53% at Sony Kohda and 35,976 required workers, that is equal to 25% of Canon’s previous total workforce, have been saved. Seru systems also influence profits, product quality, and workforce motivation in a positive way.

The essence of line-seru conversion lies in how to convert traditional conveyor assembly line into a seru system to obtain the optimal conversion solution. It is a very hot and important decision making problem because the total productivity of manufacturers may be improved dramatically [15, 16]. Such technical and decision making problems were defined as line-seru (or line-cell) conversion problems [10, 16]. Kaku et al. [10, 16] built a biobjective line-seru conversion model with minimizing makespan (they use the term “total throughput time”) and TLH. They claimed that their model considered three types of seru systems, that is, pure seru system, hybrid system with serus and short line, and assembly line. However, the third one, in fact, is not considered as a seru system, because it does not contain any seru. Therefore, there are only two types of seru systems, that is, pure seru system and hybrid system with serus and short line.

Pure seru system is the seru system only containing seru(s) and is very simple and a special case of all other seru assembly systems. The results obtained from pure seru system models not only provide insights into the pure seru system environment but also provide a basis for heuristics that are applicable to more complicated seru system environments, for example, hybrid system with serus and short line. In practice, problems in more complicated seru system environments are often decomposed into subproblems that deal with the pure seru system. Therefore, many literatures focused on the conversion of assembly line to pure seru system (line-pure seru system conversion), such as Yu et al. [17–20] and Sun et al. [21]. A simple case of line-pure seru system conversion is shown in Figure 1, where two serus are constructed, that is, workers 2 and 5 in seru 1 and workers 1, 3, and 4 in seru 2.

Figure 1: A case of line-pure seru system conversion [20].

To establish the main mathematical models is very important for the research on line-pure seru system conversion. Yu et al. [17] formulated the biobjective model of Min- and TLH. Subsequently, the researches of Yu et al. [18, 20] were based on the biobjective model. Considering the biobjective model has higher computational complexity, Sun et al. [21] formulated the single-objective model of Min- with TLH constraint. Additionally, there are several other usually used single-objective models in line-pure seru system conversion. We should not formulate one model in one research but integrate these models into a common framework. Therefore, the first objective of the research is to formulate the five models of line-pure seru system conversion in an integrated framework by combining the evaluated performances and constraints.

Moreover, the algorithms for line-pure seru system conversion are also the key. However, the existing algorithms for line-pure seru system conversion were based on enumeration (i.e., GA and local search) or metaheuristic and did not consider the distinct features of solution space of line-pure seru system conversion. For example, enumeration in Yu et al. [17] searched the whole solution space; Yu et al. [18, 20] and Sun et al. [21] used GA or local search to seek the optimal solution in the whole solution space. In fact, solution space of line-pure seru system conversion has the distinct features; that is, the solution space can be divided into several subspaces according to the number of serus, and minimum and minimum TLH usually exist in the special subspaces. That is to say, considering the features of solution space of line-pure seru system conversion, we maybe do not need to search the whole solution space to seek the optimal solution. Therefore, the second motivation of the research is to investigate the distinct features of solution space of line-pure seru system conversion. Subsequently, the third of objective is to develop the effective algorithms for the different models according to the obtained features of solution space of line-pure seru system conversion.

This paper, originally motivated by line-seru applications of Sony and Canon, has three purposes. First, two existing models (i.e., a biobjective model and a single-objective model) and the three other usually used single-objective models in line-pure system conversion are formulated in an integrated framework by combining evaluated performances with constraints. Subsequently, we analyze the solution space features of line-pure seru system conversion by dividing solution space into several subspaces according to the number of serus. We focus on investigating the features between (makespan) and TLH (total labor hours) and subspaces with different number of serus. Thirdly, we propose four effective algorithms to solve the four single-objective model, respectively, according to the distinct features of line-pure seru system.

The remainder of this research is organized in the following ways. By combining the evaluated performances and constraints, Section 2 formulates five models usually used in line-pure seru system conversion in an integrated framework. Section 3 investigates the solution space feature of line-pure seru system conversion based on W divided subspaces according to the number of serus. Subsequently, we focus on analyzing the features between (and TLH) and subspaces with different number of serus. In Section 4, according to the distinct features of solution space, especially the features between (and TLH) and subspaces with different number of serus, we propose four algorithms to solve the four single-objective models, respectively. Section 5 uses extensive experiments to evaluate the computational performance of the proposed algorithms by comparing with enumeration. Finally, we end the paper with conclusions and with suggestions for future research in Section 6 and give the further research emphases, that is, the conversion from assembly line to the hybrid system with serus and short line.

2. Formulation of Several Main Models of Line-Pure Seru System Conversion

2.1. Assumptions

The following assumptions are considered in line-pure seru system conversion:

(1)The types and batches of products to be processed are known in advance. There are product types that are divided into product batches. Each batch contains a single product type.(2)In the line-pure seru system conversion, most assembly tasks within a seru are manual, so need for only simple and cheap equipment and the cost of duplicating equipment is ignored [5].(3)A product batch needs to be assembled entirely within a single seru.(4)All product types have the same assembly tasks. If tasks of some product were unique, we assume the task time for these unique tasks was zero.(5)The assembly tasks within each seru are the same as the ones within the assembly line. Therefore, the total number of tasks equals .(6)In the assembly line, each task (or station) is in charge of a single worker. That means that a worker only performs a single assembly task in the assembly line. In contrast, a seru worker needs to perform all assembly tasks and assembles an entire product from-start-to-finish, and there is no disruption or delay between adjacent tasks.

2.2. Notations

We define the following terms:

(i) Indices

 : index of workers (). Also, the total number of tasks is , according to assumption . : index of serus (). : index of product types (). : index of product batches (). : index of the sequence of product batches in a seru (). : index of the sequence of product batches in the short line ().

(ii) Parameters

 : size of product batch . : cycle time of product type in the assembly line. : setup time of product type in the assembly line. : setup time of product type in a seru. : upper bound on the number of tasks for worker in a seru. If the number of tasks assigned to worker is more than , worker ’s average task time within a seru will be longer than her or his task time within the original assembly line. : worker ’s coefficient of influencing level of doing multiple assembly tasks. : skill level of worker for each task of product type .

(iii) Decision Variables

(iv) Variables

 : coefficient of variation of worker ’s increased task time after line-seru conversion, that is, from a specialist to a completely cross-trained worker. If the number of worker ’s tasks within a seru is over her or his upper bound , that is, , then the worker will cost more average task time than her or his task time within the original assembly line. is given in (3). : assembly task time of product batch per station in a seru. In a seru, the task time of product type is calculated by the average task time of workers in the seru. is represented as (4). : flow time of product batch in a seru. is represented as (5). : setup time of product batch in a seru. Setup time is considered when two different types of products are processed consecutively; otherwise, the setup time is zero. For example, in (6), two adjacent assembled products in a seru are expressed as and . If the product type of is different than that of , that is, , , then is . However, if the product types of and are identical, that is, , then is 0. : begin time of product batch in a seru. There is no waiting time between two product batches so that is the aggregation of flow time and setup time of the product batches processed prior to product batch in the same seru. is represented as (7):

2.3. Evaluated Performances

The following two performances are usually used to evaluate seru system’s productivity.

2.3.1. Makespan

Makespan () of the pure seru system is the due time of the last completed product batch and can be expressed as

2.3.2. Total Labor Hour

The total labor hours (TLH) are the work time of all workers assembling the total product batches:

2.4. Constraints

Each decision variable determines one process. Therefore, the line-pure seru system conversion is a two-stage decision process, that is, seru-formation and seru-load, decided by and , respectively.

According to the two decision processes and the two evaluated performances, we divide the constraints into the following categories.

2.4.1. Seru-Formation Constraints

Consider

Equation (10) is the seru’s worker constraint which ensures that each formatted seru contains least one worker and most workers. Equation (11) is the worker assignment constraint which guarantees that each worker must be only assigned to a seru. Equation (12) is the worker number constraint which guarantees that the total number of workers in all formatted serus equals , that is, the total number of workers in the assembly line.

2.4.2. Seru-Load Constraints

Consider

Equation (13) is the seru-load constraint; that is, a product batch is only loaded to a seru. Equation (14) guarantees a product batch must be assigned to a seru in which at least one worker is assigned.

2.4.3. Constraint

Consider

Equation (15) constrains that pure seru system’s is not worse than the given value. For example, we can set the given as 90% of the assembly line’s .

2.4.4. TLH Constraint

Consider

Equation (16) constrains that total labor hours (TLH) of pure seru system are not worse than the given value.

2.5. Several Main Models of Line-Pure Seru System Conversion

By combining the above two evaluated performances of and TLH with four constraints of seru-formation, seru-load, , and TLH, we formulate several main mathematical models of line-pure seru system conversion.

2.5.1. Model of Min- (Model of Min-:  (8), s.t. (10)–(14))

Model of Min- is to minimize makespan (i.e., (8)) of pure seru system. Therefore, objective is expressed as  (8) and constraints include seru-formation constraints and seru-load constraints, that is, (10)–(14).

2.5.2. Model of Min- with TLH Constraint (Model of Min- with TLH Constraint:  (8), s.t. (10)–(14), (16))

TLH constraint sometimes needs to be considered in model of Min-, and so model of Min- with TLH constraint is formulated as above.

2.5.3. Model of Min-TLH (Model of Min-TLH:  (9), s.t. (10)–(14))

Model of Min-TLH is to minimize TLH (i.e., (9)) of pure seru system. Therefore, objective is expressed as  (9) and constraints include seru-formation constraints and seru-load constraints.

2.5.4. Model of Min-TLH with Constraint (Model of Min-TLH with Constraint:  (9), s.t. (10)–(14), (15))

constraint sometimes can be considered in model of Min-TLH, and so model of Min-TLH with constraint is formulated as above.

2.5.5. Biobjective Model of Min- and TLH (Model of Min- and TLH:  ((8) and (9)), s.t. (10)–(14))

Model of Min- and TLH is to minimize and TLH simultaneously. Therefore, biobjective is expressed as  ((8) and (9)) and constraints include seru-formation constraints and seru-load constraints.

2.6. Researches on the Five Models’ Formulations

So far, most researches focused on the biobjective model of Min- and TLH in Section 2.5.5. For example, Yu et al. [17, 20] formulated the biobjective model of Min- and TLH and investigated the mathematical analysis and influence factors. In addition, as the biobjective model has higher computational complexity, Sun et al. [21] formulated the single-objective model of Min- with TLH constraint in Section 2.5.2.

Although the three other models were not researched until now, they are usually considered in line-pure seru system conversion. In fact, they are similar to model of Min- with TLH constraint in Section 2.5.2, because they are single-objective models. It is unacceptable to formulate one model in one research. Therefore, the first contribution of the research is to summarize the existing two models and to formulate the three other usually used single-objective models in a framework by combining the above two evaluated performances and four constraints.

In addition, previous researches did not investigate the distinct features of solution space of line-pure seru system conversion. As a result, the existing algorithms for line-pure seru system are not developed according to the distinct features.

3. Features of Solution Space of Line-Pure Seru System Conversion

3.1. Complexity of Solution Space Satisfying Seru-Formation and Seru-Load Constraints

In line-pure seru system conversion, seru-formation is the first step. It is to determine how many serus to be formed and how to assign workers into the serus [17]and it is decided by decision variable .

A feasible solution of seru-formation must satisfy seru-formation constraint, that is, (10)–(12). Yu et al. [17] proved that seru-formation is an instance of the unordered set partition and an NP-hard problem. The number of all feasible solutions of seru-formation can be expressed recursively as the following:

where is the count of partitioning workers in assembly line into serus and can be expressed as the Stirling numbers of the second kind [22]. The numbers of to are 1, 2, 5, 15, 52, 203, 877, 4140, 21147, and 115975, respectively [23–25]. Obviously, increases exponentially with .

Seru-load is the second step of line-pure seru system conversion and is decided by decision variable . It determines which product batches are dispatched to the serus formed in seru-formation [26, 27]. A feasible solution of seru-load must satisfy seru-load constraint, that is, (13) and (14). Seru-load is NP-hard. Therefore, most researches used typical dispatching rules, such as FCFS (first-come, first-served) and SPT (shortest processing time). Even though the typical dispatching rules are used in seru-load, line-pure seru system conversion still is NP-hard, because it contains seru-formation which is NP-hard.

Yu et al. [20] proved that line-pure seru system conversion with SPT is an instance of the unordered set partition and the complexity of solution space can be expressed as (17). Line-pure seru system conversion with FCFS is an instance of the ordered set partition and the complexity of solution space can be expressed as

The numbers of to are 1, 3, 13, 75, 541, 4,683, 47,293, 545,835, 7,087,261, and 102,247,563, respectively. In the research, we use FCFS rule to assign product batches to serus.

3.2. Solution Spaces of the Five Models

In the above five models, model of Min-, model of Min-TLH, and biobjective model of Min- and TLH have the identical solution space described in Section 3.1, because all of them only include seru-formation constraints and seru-load constraints.

However, model of Min- with TLH constraint and model of Min-TLH with constraint have the different solution space, because the two models contain the other constraints except seru-formation constraints and seru-load constraints. Therefore, the solution spaces of the two models must be not more than that of Section 3.1.

3.3. Existing Algorithms for the Five Models

Most researches focused on the algorithms for the biobjective model of Min- and TLH in Section 2.5.5. Yu et al. [20] proposed a based-NSGA-II algorithm to solve the biobjective model; Yu et al. [18] combined local search into NSGA-II to improve the running time and solutions qualification; Yu et al. [17] used enumeration and based-NSGA-II algorithm to solve the biobjective model to analyze the impact factor on and TLH improvements by line-pure seru system conversion. Sun et al. [21] developed a variable neighborhood search (VNS) for model of Min- with TLH constraint in Section 2.5.2.

However, the algorithms for the three other models were not researched until now.

The existing algorithms for line-pure seru system conversion were based on the metaheuristic (i.e., GA and local search) and did not consider the distinct features of solution space. For example, enumeration in [17] searched the whole solution space; Yu et al. [18, 20] and Sun et al. [21] used GA or local search to seek the optimal solution in the whole solution space. In fact, solution space of line-pure seru system conversion has the distinct features; that is, the solution space can be divided into several subspaces according to the number of serus, and minimum and minimum TLH usually exist in the special subspaces. That is to say, considering the features of solution space of line-pure seru system conversion, we maybe do not need to search the whole solution space to seek the optimal or suboptimal solution.

Therefore, we investigate the distinct features of solution space of line-pure seru system conversion and then propose effective algorithms to solve the four single-objective models based on the distinct features of solution space.

3.4. Subspaces with Different Number of Serus

Definition 1. Subspace in solution space of line-pure seru system conversion is the set of solutions with the same number of serus.
For example, of (18), the whole solution space can be divided into subspaces according to the number of serus, that is, the subspaces with , and serus. Therefore, (18) can be expressed as the following:

Table 1 summarizes the number of solutions in subspace with serus for the lines with 6–9 workers.

Table 1: Number of solutions in subspace with J serus.

Workers
Number of solutions in subspace with J serus
Total solutions

1
2
3
4
5
6
7
8
9

6
1
62
540
1560
1800
720
—
—
—
4683

7
1
126
1806
8400
16800
15120
5040
—
—
47293

8
1
254
5796
40824
126000
191520
141120
40320
—
545835

9
1
510
18150
186480
834120
1905120
2328480
1451520
362880
7087261

The value in column of “workers” represents the number of workers in the assembly line. For the row “6,” the numbers of solutions in subspaces with 1, 2, 3, 4, 5, and 6 serus are , , , , , and , respectively.

To show the proportion of solutions in each subspace to the total solutions in the whole solution space, Table 2 normalizes the data in Table 1.

Table 2: Proportion of solutions in subspace with J serus.

Workers
Proportion of solutions in subspace with J serus

1
2
3
4
5
6
7
8
9

6
0.02
1.32
11.53
33.31
38.44
15.37
—
—
—

7
0.00
0.27
3.82
17.76
35.52
31.97
10.66
—
—

8
0.00
0.05
1.06
7.48
23.08
35.09
25.85
7.39
—

9
0.00
0.01
0.26
2.63
11.77
26.88
32.85
20.48
5.12

From Table 2, we can see that the number of solutions in subspaces with few or more serus is very small. Therefore, if minimum and minimum TLH exist in such subspaces, we can dramatically decrease the searching space.

3.5. Value of and TLH in Subspaces

Yu et al. [17] performed 64 arrays of full factorial experiment to analyze influence factors and proposed some managerial insights. Two insights on and TLH are as follows: to minimize , the pure seru system with fewer serus should be created and assign the workers with similar skill levels for product types to the same seru; and to minimize TLH, the pure seru system with more serus should be created.

Yu et al. [17] used the solutions with 1 seru and serus to briefly explain the two insights: they proved that when , if the seru system with a seru is formatted, then the performance of pure seru system must be better than that of the assembly line. Additionally, they stated that the pure seru system with fewer serus is easy to balance among serus than the pure seru system with more serus. Therefore, the pure seru system with fewer serus can produce better ; and they proved that when and , the pure seru system with serus should be formatted in order to minimize TLH, because this makes full use of each worker’s skill.

To further investigate the two insights, in the research, we analyze the features of and TLH in each subspace in detail. Figures 2–4 show and TLH of all solutions in each subspace for the cases with 6–8 workers, respectively. The results of Figures 2–4 are exactly obtained by enumeration. The detailed data of the used cases with 6–8 workers are described in Section 5.1.

(a)

(b)

Figure 2: and TLH in subspaces for the line with 6 workers.

(a)

(b)

Figure 3: and TLH in subspaces for the line with 7 workers.

(a)

(b)

Figure 4: and TLH in subspaces for the line with 8 workers.

3.6. Distinct Features between (and TLH) and Subspaces

By observing and TLH in Figures 2–4, we can investigate the distinct features between (and TLH) and subspaces as follows.

Feature 1. Minimum usually exists in the subspaces with few serus.
Explanation. As shown in Figures 2–4, the minimum always exists in the subspaces with 2 serus.

Feature 2. Except the subspace with 1 seru, the minimum of the subspaces with fewer serus usually is less than that of the subspaces with more serus.
Explanation. As shown in Figure 3, the minimum of the subspaces with () serus always is less than that of the subspaces with serus. The trend can be found in all of Figures 2 and 3. Table 3 shows the detailed information on the minimum of the subspaces with serus for the cases with 6–8 workers.

Table 3: Minimum in subspace with J serus.

Workers
Minimum in subspace with J serus

1
2
3
4
5
6
7
8

6
2963
2954
2970
2973
2981
3177
—
—

7
2980
2963
2966
3014
3021
3123
3208
—

8
2980
2962
2971
2971
3045
3121
3122
3163

Features 1 and 2 mean that it not necessary to search the whole solution space to obtain minimum , especially in single-objective models. To fast obtain minimum , therefore, we can search the subspaces from fewer serus to more serus. Consequently, we propose effective algorithms to fast find minimum . The detailed procedure is described in Section 4.1.

Feature 3. Minimum TLH usually exists in the subspaces with more serus.
Explanation. Assume express the maximum . In Figures 2–4, the minimum TLH always exists in the subspace with serus.

Feature 4. Minimum TLH of the subspaces with more serus usually is less than that of the subspaces with fewer serus.
Explanation. As shown in Figures 2–4, the minimum TLH of the subspaces with J serus always is less than that of the subspaces with serus. Table 4 shows the detailed information on the minimum TLH of the subspaces with serus for the cases with 6–8 workers.

Table 4: Minimum TLH in subspace with J serus.

Workers
Minimum TLH in subspace with J serus

1
2
3
4
5
6
7
8

6
17610
17559
17534
17494
17463
17453
—
—

7
20665
20586
20506
29505
20504
20482
20442
—

8
23641
23519
23475
23437
23365
23354
23303
23260

Features 3 and 4 mean that it not necessary to search the whole feasible solution space to obtain the minimum TLH, especially in single-objective models. To fast obtain minimum TLH, therefore, we can search the subspaces from more serus to few serus. Consequently, we propose effective algorithms to fast find the minimum TLH. The detailed procedure is described in Section 4.2.

4. Algorithms for the Single-Objective Line-Pure Seru System Conversion

As mentioned above, most existing algorithms were for the biobjective line-pure seru system conversion. The algorithm of Sun et al. [21] was for single-objective model. However, these algorithms were not developed based on the features of solution space of line-pure seru system conversion, which decreases their running efficiency. Therefore, we propose algorithms to solve the 4 single-objective models based on the distinct features of solution space.

4.1. Algorithms for Models of Min- and Min- with TLH Constraint

To avoid the obtained solution with minimum being locally optimal, the algorithm does not search the subspace with the least seru (i.e., 1 seru) but searches subspaces from fewer serus to more serus to seek minimum . Therefore, the minimum may be obtained by searching only a part of solution space.

For an instance of Min- with workers, after obtaining the optimal solution in subspace with seru, the algorithm searches the optimal solution in subspace with serus. If the latter is not better than the former, then the algorithm will stop; otherwise, continue, until the optimal solution with serus is not better than the optimal solution with serus. The procedure can be described as shown in Algorithm 1.

Input: (the number of workers in assembly line).

Output: The optimal solution of Min-.

() Initialize. Set O = null (record the optimal solution), .

         (1-1) Generate the solution set () of sub-space with J serus according to ! and calculate each solution’s .

         (1-2) Obtain the optimal in .

         (1-3) O = optimal in ;

(2) ;

(3) While Do

         (3-1) Generate the solution set () of sub-space with J serus according to ! and calculate each solution’s .

         (3-2) Obtain the optimal in .

         (3-3) If optimal in < optimal in Then

                         = optimal in ;

                        ;

                        Continue;

                  Else

                        Break;

(4) Output O.

Algorithm 1: Algorithm for models of Min-.

In Steps and (3-3), means to search solution space from few serus to more serus. Step (3-3) is to judge if the algorithm stops. If yes, output ; otherwise, continue to search the optimal in the next subspace.

Similar to Algorithm 1, algorithm for models of Min- with TLH constraint can be expressed as shown in Algorithm 2.

Input: (the number of workers in assembly line).

Output: The optimal solution of Min- with TLH constraint.

(1) Initialize. Set O = null (record the optimal solution), .

         (1-1) Generate the solution set () of sub-space with J serus according to ! and calculate each solution’s

         and TLH.

         (1-2) Produce the feasible solution set () of , that is, the set of solutions satisfying TLH constraint.

         (1-3) Obtain the optimal in .

         (1-4) O = optimal in ;

(2) ;

(3) While Do

         (3-1) Generate the solution set () of sub-space with J serus according to ! and calculate each solution’s

         and TLH.

         (3-2) Produce the feasible solution set () of , that is, the set of solutions satisfying TLH constraint.

         (3-3) Obtain the optimal in .

         (3-3) If optimal in < optimal in Then

                           O = optimal in ;

                           ;

                          Continue;

                  Else

                          Break;

(4) Output O.

Algorithm 2: Algorithm for models of Min- with TLH constraint.

Distinguished from Algorithm 1, Algorithm 2 needs to calculate TLH and to judge if TLH constraint is satisfied. They are described in steps (1-1) and (1-2).

4.2. Algorithm for Model of Min-TLH and Min-TLH with Constraint

To avoid the obtained solution with minimum TLH being locally optimal, the algorithm does not search the subspace with the most seru (i.e., seru) but searches subspaces from more serus to few serus to seek the minimum TLH. Therefore, the minimum may be obtained by searching only a part of solution space.

For an instance of Min-TLH with workers, after obtaining the optimal solution in subspace with seru, the algorithm searches the optimal solution in subspace with serus. If the latter is not better than the former, then the algorithm will stop; otherwise, continue, until the optimal solution with serus is not better than the optimal solution with serus. The procedure can be described as shown in Algorithm 3.

Input: (the number of workers in assembly line).

Output: The optimal solution of Min-TLH.

(1) Initialize. Set O = null (record the optimal solution), .

         (1-1) Generate the solution set () of sub-space with J serus according to ! and calculate each solution’s TLH.

         (1-2) Obtain the optimal TLH in .

         (1-3) O = optimal TLH in ;

(2) ;

(3) While Do

         (3-1) Generate the solution set () of sub-space with J serus according to ! and calculate each solution’s TLH.

         (3-2) Obtain the optimal TLH in .

         (3-3) If optimal TLH in < optimal TLH in Then

                            O = optimal TLH in ;

                            ;

                           Continue;

                    Else

                           Break;

(4) Output O.

Algorithm 3: Algorithm for models of Min-TLH.

In Steps and (3-3), means to search solution space from more serus to few serus. Step (3-3) is to judge if the algorithm stops. If yes, output ; otherwise, continue to search the optimal TLH in the next subspace.

Similar to Algorithm 3, algorithm for models of Min-TLH with constraint can be expressed as shown in Algorithm 4.

Input: (the number of workers in assembly line).

Output: The optimal solution of Min-TLH with constraint.

(1) Initialize. Set O = null (record the optimal solution), .

         (1-1) Generate the solution set () of sub-space with J serus according to ! and calculate each solution’s

         TLH and .

         (1-2) Produce the feasible solution set () of , that is, the set of solutions satisfying constraint.

         (1-3) Obtain the optimal TLH in .

         (1-4) = optimal TLH in ;

(2) ;

(3) While Do

         (3-1) Generate the solution set () of sub-space with J serus according to ! and calculate each solution’s

         TLH and .

         (3-2) Produce the feasible solution set () of , that is, the set of solutions satisfying constraint.

         (3-3) Obtain the optimal TLH in .

         (3-4) If optimal TLH in < optimal TLH in Then

                        O = optimal TLH in ;

                        ;

                        Continue;

                   Else

                        Break;

(4) Output O.

Algorithm 4: Algorithm for models of Min-TLH with constraint.

Distinguished from Algorithm 3, Algorithm 4 needs to calculate and to judge if constraint is satisfied. They are described in steps (1-1) and (1-2).

4.3. Combining Features 1 and 2 into the Existing Algorithm

As mentioned above, Sun et al. [21] developed a variable neighborhood search (VNS) for model of Min- with TLH constraint. However, their VNS seeks the optimal solution in the whole solution space. Therefore, considering Features 1 and 2, that is, to search minimum satisfying TLH constraint in the subspaces with fewer serus by redefining the neighborhood strategies, Sun’s algorithm running time will be improved.

5. Computational Experiments

5.1. Test Instances

Tables 5–9 show the parameters, data distribution and detailed data of level of skill of workers, coefficient of influencing level of skill to multiple stations for workers, and data of batches used in experiments, respectively. From Table 5, it can be observed that the lot size of each batch is and the ability of workers is also different with stations and . Table 6 shows that the mean of skill level () of each worker for processing product types ranges from 1 to 1.2 and the standard deviations are fixed to 0.1. The detailed data of are given in Table 7. The detailed data of and batches are given in Tables 8 and 9, respectively. In fact, data in Tables 5–9 is the part of data used in Yu et al. [20].

Table 5: The parameters used in experiments.

Product types
Batch size

5

2.2
1.0
1.8
10

: normal distribution (, σ = 5).

Table 6: Data distribution of worker’s level of skill ().

Product types

1
2
3
4
5

Table 7: Data of worker’s level of skill ().

Worker/product type
1
2
3
4
5

1
0.92
0.96
1.04
1.09
1.2

2
0.95
0.97
1.09
1.12
1.18

3
0.99
1.01
1.05
1.09
1.21

4
1.03
1.07
1.09
1.12
1.25

5
0.96
1.02
1.05
1.1
1.18

6
1.01
1.1
1.1
1.15
1.23

7
1.04
1.07
1.09
1.17
1.24

8
0.98
1.02
1.1
1.11
1.2

9
0.97
1.03
1.12
1.19
1.26

10
0.98
1.06
1.13
1.18
1.28

Table 8: Coefficient of influencing level of skill to multiple stations for workers ().

Worker
1
2
3
4
5
6
7
8
9
10

0.18
0.19
0.2
0.21
0.2
0.2
0.2
0.22
0.19
0.19

Table 9: Data of batches.

Batch number
Product type
Batch size ()

1
3
55

2
5
53

3
3
54

4
4
49

5
1
49

6
4
55

7
1
54

8
2
48

9
2
48

10
3
48

11
2
46

12
4
58

13
3
48

14
4
52

15
5
48

16
5
51

17
1
54

18
4
57

19
2
54

20
5
49

21
1
53

22
3
46

23
4
45

24
5
46

25
2
45

26
3
44

27
1
53

28
4
47

29
2
53

30
3
52

For the instance with workers, we use the following data set from Tables 5–9: entire Table 5, first rows of Table 7, first columns of Table 8, and entire Table 9.

5.2. Hardware and Software Specifications

The four algorithms were coded in C# and executed on an Intel Core™ 2 processors at 2.66 GHz under Windows XP using 3.49 GB of RAM.

5.3. Comparative Results with Enumeration

Considering the similarity between Algorithm 1 (or 3) and Algorithm 2 (or 4), we used the Algorithms 1 and 3 to solve the models of Min- and Min-TLH, respectively. The performance comparative results to the enumeration are shown in Table 10.

Table 10: Performance comparative results to the enumeration.

Workers
Line
Enumeration
Algorithm 1
Algorithm 3

TLH

TLH
Time (in s)

Time (in s)
TLH
Time (in s)

6
3394
19995
2954
17602
6.43
2954
0.72
17602
3.12

7
3468
23844
2963
20627
22
2963
1.03
20627
9.08

8
3522
27686
2962
23542
157
2962
5.1
23542
51.3

9
3608
31917
2974
26582
1859
2974
9.2
26582
463.2

From Table 10, we can see the optimal solutions obtained by Algorithms 1 and 3 are same as the results from enumeration. Moreover, the running time is much less than that of enumeration, because algorithms search the corresponding subspaces according to the distinct features between (and TLH) and subspaces instead of searching the whole solution space. In addition, the running time of Algorithm 3 is more than that of Algorithm 1. That is because the subspace with more serus has more feasible solutions than the subspace with few serus, as shown in Table 1.

6. Conclusions and Future Research

In the research, three main contributions can be summarized as follows.

Firstly, by combining two common evaluated performances of and TLH with four constraints of seru-formation, seru-load, , and TLH, we summarize the two existing models and formulate three other usually used models in line-pure seru system conversion. The two evaluated performances, four classified constraints, and formulated models are summarized in Table 11.

Table 11: Evaluated performances, constraints, and formulated models.

Evaluated performances
Expression

Makespan ()
, (8)

Total labor hours (TLH)
, (9)

Constrains
Equations

Seru-formation
(10), (11), and (12)

Seru-load
(13) and (14)

(15)

TLH
(16)

Models
Expression

Min-
min (8), s.t. (10)–(14)

Min- with TLH constraint
min (8), s.t. (10)–(14), (16)

Min-TLH
min (9), s.t. (10)–(14)

Min-TLH with constraint
min (9), s.t. (10)–(14), (15)

Min- and TLH
min ((8) and (9)), s.t. (10)–(14)

Secondly, we investigate the solution space feature of line-pure seru system conversion by dividing them into subspaces according to the number of serus (). Subsequently, we focus on analyzing the features between (and TLH) and subspaces in detail. The obtained distinct features are shown in Table 12.

Table 12: Distinct features between (and TLH) and subspaces.

Features
Content

1
Minimum usually exists in the subspaces with few serus.

2
Except for the subspace with 1 seru, the minimum of the subspaces with fewer serus usually is less than that of the subspaces with more serus.

3
Minimum TLH usually exists in the subspaces with more serus.

4
Minimum TLH of the subspaces with more serus usually is less than that of the subspaces with fewer serus.

Thirdly, according to the distinct features of solution space, especially the features between (and TLH) and subspaces, we propose four algorithms to solve four single-objective models, respectively. Table 13 summarizes each proposed algorithm, based-on-features, and solved model.

Table 13: Algorithm features and solved model.

Algorithms
Based-on-features
Solved model

Algorithm 1
Features 1 and 2
Min-

Algorithm 2
Features 1 and 2
Min- with TLH constraint

Algorithm 3
Features 3 and 4
Min-TLH

Algorithm 4
Features 3 and 4
Min-TLH with constraint

The research on the line-seru conversion is relatively lacking. A thorough research problem list can be found in Yin et al. [20], such as partially cross-trained workers (i.e., a worker cannot perform all assembly tasks), different products have different assembly tasks, cost of karakuri (a Japanese word meaning duplication of equipment), and human and psychology factors.

In addition, line-hybrid seru system conversion (i.e., the hybrid system with serus and short line is the second type of the line-seru conversion defined by Kaku et al. [10]) is more complex and more real than line-pure seru system conversion. Therefore, the further research should be focused on the line-hybrid seru system conversion.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (71420107028 and 71571037) and the National Social Science Foundation of China (13CGL045).

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