The Optimization of Working Time for a Consecutively Connected Production Line

Abstract

Most factory production processes are completed by machines and workers on production lines. The operation schedule is arranged to reduce the cost of the enterprises to obtain the maximum economic profit for sustainable running. Previous studies usually investigated the working time while only considering the workers’ conditions. This study proposed a method to optimize the operation schedule by jointly considering the workers’ fatigue states and the operation states of machines. This method was proposed based on a system structure called the multistate consecutively connected system (MCCS), which has been widely applied in many areas, such as electronic communications. This structure is also an analogy of the production line. The corresponding model is constructed based on the universal generating function (UGF) since it is a powerful tool in modeling a consecutively connected system. The proposed model can be used to evaluate the different productivities of different types of workers in different states and to realize the screening of the whole scheme through simulation. According to the proposed method, we obtained the optimal operation schedule, including the working time, rest time and allocation strategy for a production line system. Some examples are provided to illustrate the proposed method.

1. Introduction

In the transition from Industry 3.0 to Industry 4.0, companies have gradually replaced manual production by purchasing or leasing machines. However, most companies do not utilize full machine operation due to capital, operation or safety issues [1,2,3]. Thus, it is very important that machines and humans work perfectly together. MCCSs are widely applied in communication and industrial fields. An MCCS consists of several ordered positions. Workers or machines are assigned to work at each position. If any process in the linear production system is not qualified, the product will fail.

Researchers have done a lot of studies in reliability modeling and optimization of MCCS [4,5,6,7,8]. For example, Peng et al. [9] studied combined elements allocation and maintenance optimization. Levitin et al. [10] introduced the phased mission system, which had never been previously studied, and formulated a new optimal connected elements allocation problem. Gao et al. [11] proposed a model of LMCCS that considered signal loss. Through the joint optimization of a node building and connection elements allocation strategy, the optimal design problem of the system was solved. To date, there have been no studies related to MCCS on industrial production lines to reasonably alleviate worker fatigue.

In fact, none of the existing works considered worker fatigue when improving the production line to obtain the optimal work plan. Many researchers studied the impact of improved worker posture on product quality [12,13,14]. For example, Zhang et al. [15] studied the fatigue time threshold and labor time rate of standing, sitting, and alternating standing and sitting postures by analyzing the fatigue of operators in different working postures on a linear production system. Much work was also done to improve the efficiency of hardware in linear production systems without considering human fatigue [16,17,18,19,20,21]. For example, Lin et al. [22] studied JIT’s linear production system production efficiency improvement method. Taking a certain cable automatic production line as the research object, in order to solve the problems of an unbalanced production line, a large inventory of work-in-process, and production efficiency that cannot meet customers’ needs, research based on improvements to JIT was undertaken.

In this work, we introduced an MCCS to study the work fatigue of workers in the production line, which has never been previously studied. MCCS is used as a theoretical basis to study the product transfer between workers on a linear production system, which is of great significance in factory production. In this study, workers are rationally assigned to workstations according to their working hours and fatigue status, which reduced costs and losses for the purpose of increasing corporate profits. In order to increase the production accuracy of the production line, one possible way is to apply more machines, but this will increase the rental cost of machines and reduce the profit. Another method is to change each working time to improve the production accuracy of the worker. In this study, the optimal strategy was studied, which involved the assignment of work types to each workstation, to identify the optimal work schedule.

The remainder of the paper is organized as follows. In Section 2, we introduce the theoretical basis. In Section 3, we describe the problem and build the model. In Section 4, we provide data analysis. In Section 5, we conclude this paper.

2. Theoretical Basis

2.1. Learning Curve Theory

Yerkes–Dodson Law: The optimal level of motivation is not fixed, but varies with the nature of the task. In the completion of simple tasks, the intensity of motivation is high and the efficiency of work is higher [23,24]. In general, the stronger the motivation, the higher the enthusiasm for work; the better the potential, the higher the efficiency. Conversely, the lower the motivation, the lower the efficiency. Thus, work efficiency increases as motivation increases. However, research by the psychologist Yerkes confirmed that the relationship between motivation intensity and work efficiency is not linear, but rather an inverted U-shaped-curve relationship. This is reflected by the fact that work efficiency is the highest when the motivation intensity is appropriate. When the motivation intensity is too low, there is a lack of motivation to participate in activities and work efficiency cannot be improved. When the motivation intensity exceeds the peak, the work efficiency will continue to decrease as the motivation intensity increases. This is because excessive motivation puts the individual in a state of excessive anxiety and stress, interfering with the normal activity of mental processes, such as memory and thinking. According to the Yerkes–Dodson law, workers first experience a period of adaptation when they start working; then, their efficiency gradually increases and reaches a steady state within a certain period. Then, workers begin to fatigue and productivity gradually decreases. This relationship is shown in Figure 1.

Figure 1 provides the theoretical basis for this study. The new hands who are just starting their training have a much more difficult time on the job than skilled workers. For the skilled worker, the difficulty factor is lower because the work is already familiar to them [25]. Therefore, for the new hands who are just entering training, according to the Yerkes–Dodson law, workers begin their work by going through a period of adaptation, during which efficiency gradually increases until they reach a steady state. With the increase of time, the efficiency of workers gradually decreases as they fatigue, and hence, productivity decreases.

2.2. Human Factors Fatigue Curve Theory

Work curve: The curve of job performance over time under specific working conditions [26]. The indicators of work performance are as follows: output, productivity, correctness, success rate, etc. A typical work curve consists of four stages: work initiation stage, maximum work capacity stage, work capacity reduction stage, and work final sprint stage. The first three of these stages are inverted U-shaped curves, but the last stage is usually upturned.

In the era of transition from manual operations to mechanization, it is very common to combine machines with mechanized manufacturing. Due to the huge cost of machinery, the company as a for-profit company has limited capital to invest. If too much money is invested in purchasing new equipment, it will increase the risk of the company’s operation. Therefore, the purpose of this study was to find a balance between the number of producers and the number of machines to maximize profit. Based on the learning curve theory and human factor fatigue theory, we could calculate the statistics of worker fatigue and energy and then compare the costs of different types of workers.

3. Problem Description and Model Building

Considering the different numbers of workstations in actual factory production, eight workstations were selected as an example in this study. The continuous production system consisted of eight workstations represented by $C_i$ ($i=1,2,3,4,5,6,7,8$). Assumptions in this study: (1) The eight workstations had the same working time and the same delivery time. (2) The workstations in the linear production system were independent, which meant that the costs of different workstations were independent. (3) The types of workers investigated in this study were skilled workers and new hands. Since the time required for each process varied, the productivity varied by worker type. The production process is shown in Figure 2. From Section 2, the productivity of workers is different for 1, 2, and 3 h of work, and the productivities of workers in energetic and fatigued states are also different. This study did not provide a method of detection for their wakefulness or fatigue. The productivity of skilled workers and new hands in fatigued and energetic states are different, as shown in

Table 1. Efficiencies of different types of work and different states with their associated parameters and descriptions.

	Skilled Worker		New Hand		Machine
Status	Energetic	Fatigue	Energetic	Fatigue	Status of Use
Average productivity during 1 h of work	0.95	0.9	0.9	0.85	1
Average productivity during 2 h of work	0.99	0.95	0.95	0.9
Average productivity during 3 h of work	0.98	0.93	0.93	0.88

. Let $G_i^C$ indicate the random variable representing the productivity of the $i\mathrm{th}$ workstation $C_i$ and assume there are $K_i^C$ possible states for workstation $C_i$. Let $g_i^c(k)$ be the productivity at workstation $C_i$ when $C_i$ occupies state $k$ and $p_i^c(k)$ be the probability of $C_i$ being in work state $k$, where $k=1,\mathrm{\dots },K_i^C$.

In this study, we assumed that the work state coefficients of skilled workers were 0.8 and 0.2 when they were energetic and fatigued, respectively; the work state probabilities of new hands were 0.6 and 0.4 when they were energetic and fatigued, respectively. This assumption was reasonable since new hands are less skilled and proficient than skilled workers. Skilled workers are more accustomed to the production method of the project and are better in terms of working time and accuracy. Therefore, we can avoid fatigue and increase the productivity of workers by increasing the frequency of their breaks. However, reducing the time that workers spend on their work too much will reduce productivity and profit. We studied this problem to give the optimal job plan for the production line system.

In this paper, we refer to the body–brain fatigue theory and data experiments to make assumptions about the rest duration of workers: (1) workers need a rest time of 10 min to return to an energetic state when they work for 1 h, (2) workers need a rest time of 15 min to return to an energetic state when they work for 2 h, and (3) workers need a rest time of 20 min to return to an energetic state when they work for 3 h. In addition, we assumed that the total time was 8 h, which included working time and rest time. In order to maximize productivity, we provide four kinds of working time plans combining working time and rest time, as shown in

Table 2. Combination of working hours under different work plans.

	Plan 1	Plan 2	Plan 3	Plan 4
1 h work	7	5	4	3
2 h work	0	1	0	2
3 h work	0	0	1	0

.

3.1. UGF of the Workstation

According to the different physical and psychological states of each worker every day, not all workers can completely relieve fatigue through a period of rest. The probability of this difference occurring is as follows:

• A 10 min rest for every hour of work: skilled workers have an 80% probability of being well rested and new hands have a 60% probability of being well rested;
• A 15 min rest for every 2 h of work: skilled workers have an 80% probability of being well rested and new hands have a 60% probability of being well rested;
• A 20 min rest for every 3 h of work: skilled workers have an 80% probability of being well rested and new hands have a 60% probability of being well rested.

Although the machine can run continuously at full load in actual production, long time operation can make the system fail. Therefore, it is very important to control the risk of the system, and the machine operation should be stopped after a period of time to allow the machine to dissipate heat [27,28]. On the one hand, this can keep the productivity of the machine at a high level, and on the other hand, it can reduce the maintenance cost. Therefore, we assumed that the machine works for 1 h with 10 min of heat dissipation, the machine works for 2 h with 15 min of heat dissipation, and the machine works for 3 h with 20 min of heat dissipation. We assumed that $E_t$ denotes that there are three types of producers working at the workstation, namely, skilled workers, new hands, and machines. $G_t^E$ indicates the random variable representing the productivity of producers $E_t$. In addition, we assumed that there were $K_t^E$ possible states for the work state of $E_t$. $g_t^e(k)$ denotes the productivity of producers $E_t$ when $E_t$ occupies work state $k$ and $p_t^e(k)$ indicates the probability of $E_t$ being at work state $k$, where $k=1,\mathrm{\dots },K_t^E$.

In this study, we adopted the UGF method to analyze the reliability of the MCCS [29,30,31]. Let $u(z)$ represent the UGF of the independent discrete random variable $X$ as a polynomial:

$$u(z)={\displaystyle \sum _{k=1}^K{p}_k{z}^{X_k}}$$

(1)

where $K$ indicates the number of possible realizations of the variable $X$, $p_k$ is the probability that $X$ takes the value $x$, and $z$ is an indicator variable that has no practical significance. To evaluate the probability that a random variable $X$ is not less than the realized $w$, the coefficients of the polynomial $u(z)$ should be summed for each term with $x_k\ge w$:

$$\mathrm{Pr}\{X\ge w\}={\displaystyle \sum _{x_k\ge w}{p}_k}$$

(2)

This can be done using the following $\delta$ operator over $u(z)$ as follows:

$$\delta (u(z),w)=\delta ({\displaystyle \sum _{k=1}^K{p}_k}{z}^{X_k},w)={\displaystyle \sum _{k=1}^K\left[1(x_k\ge w)p_k\right]}$$

(3)

where $1(x_k\ge w)=1$ if $x_k\ge w$ is true and $1(x_k\ge w)=0$ if $x_k\ge w$ is false.

To obtain the UGF of the entire consecutively connected production system, it is necessary to define the UGF of each station in the production line. If $h_i$ is the type of work at workstation $C_i$, we obtain $G_i^C=G_{h_i}^E$. The productivity $g_i^c\left(k\right)=\left\{g_i^c{\left(k\right)}_1,\mathrm{\dots },g_i^c{\left(k\right)}_N\right\}$ is obtained from $g_{h_i}^e\left(k\right)=\left\{g_{h_i}^e{\left(k\right)}_1,\mathrm{\dots },g_{h_i}^e{\left(k\right)}_N\right\}$ by taking the first $N-i$ items as $g_i^c\left(k\right)=\left\{g_i^c{\left(k\right)}_1,\mathrm{\dots },g_i^c{\left(k\right)}_N\right\}=\left\{g_{h_i}^e{\left(k\right)}_1,\mathrm{\dots },g_{h_i}^e{\left(k\right)}_N\right\}$, where $k=1,\mathrm{\dots },K_i^C$ and $K_i^C=K_{h_i}^E$.

For workstation $C_i$ on the production line, let the productivity $G_i^c$ represent a random variable $X$, where the productivity $\left\{g_i^c\left(1\right),\mathrm{\dots },g_i^c\left(K_i^C\right)\right\}$ is the state probability of $\left\{x_1,\mathrm{\dots },x_{K_i^C}\right\}$. $p_i^c\left(k\right)$ denotes the probability of different states presented by workers. Therefore, we obtain the UGF of workstation $C_i$ as follows:

$$u_i^c(z)={\displaystyle \sum _{k=1}^{K_i^C}{p}_i^c(k)z^{g_i^c(k)}}$$

(4)

To conveniently express the information of the UGF, the probability of each state is associated with the performance in that state, and this approach can be used to represent all possible states. The UM of workstation $C_i$ is

$$U{M}_i^C=\left[\begin{array}{llll}{p}_i^c(1)& g_i^c{(1)}_1& \mathrm{\dots }& g_i^c{(1)}_{N-i+1}\\ \mathrm{\dots }& \mathrm{\dots }& \mathrm{\dots }& \mathrm{\dots }\\ p_i^c(K_i^C)& g_i^c{(K_i^C)}_1& \mathrm{\dots }& g_i^c{(K_i^C)}_{N-i+1}\end{array}\right]$$

(5)

3.2. UGF for Consecutively Connected Production System

The productivity of the consecutively connected production system is the product of the productivity of all eight workstations, and the entire linear system equation is as follows:

$$R_S^{K_i}=\prod g_i^c(K_i^C)$$

(6)

Next, we describe the worker productivity of the entire system using UGF as follows:

$$u_{{}_N}^s(z)={\displaystyle \sum _{k=1}^{K_N^S}{p}_N^s(k)z^{g_N^s(k)}}$$

(7)

When choosing a skilled worker, a new hand, or a machine, there will be corresponding reliability rates to ensure the effectiveness of the system. $r$ is the minimum requirement for system reliability. Thus, we can estimate the reliability system according to (3) as follows:

$$\begin{array}{l}R=\delta \left(u_{{}_N}^s(z),r\right)\\ =\delta \left({\displaystyle \sum _{k=1}^{K_N^S}{p}_N^s(k)z^{g_N^s(k)},r}\right)\\ ={\displaystyle \sum _{k=1}^{K_N^S}\left[1(g_N^s{(k)}_1\ge r)p_N^s(k)\right]}\end{array}$$

(8)

3.3. Production Cost

The total cost of the consecutively connected production system consists of labor costs and material costs.

3.3.1. Labor Cost

Labor costs vary depending on the type of workers selected for each permutation and are calculated as follows:

$$V_E={\displaystyle \sum _{i=1}^C{T}_{h_i}^E{V}_{h_i}^E}$$

(9)

where $V_E$ denotes the total labor cost when completing the work task. $V_{h_i}^E$ is the wage of the producer of type $h_i$, where the cost of the machine is the rental cost. $T_i^C$ denotes the working time of the producer of type $h_i$.

3.3.2. Material Cost

The input cost is equal to the daily input amount because no matter which working time arrangement plan is used, the workers work 7 h, and thus, the daily material input amount is equal; according to the 100 pieces of material input per hour, the daily input amount is 700 pieces. Thus, let $V^C$ denote the material cost. The material cost of the entire system is ${\displaystyle \sum _{i=1}^N{V}^C}$.

Therefore, we obtain the total cost $TC$ of the system as follows:

$$\begin{array}{l}TC={\displaystyle \sum _{i=1}^N{V}^C}+V_E\\ ={\displaystyle \sum _{i=1}^N\left(V^C+T_{h_i}^E{V}_{h_i}^E\right)}\end{array}$$

(10)

3.4. Revenue from Product Sales

The product sales revenue $TR$ can be obtained as follows:

$$\begin{array}{l}TR={\mathrm{Q}}_N^S\times P\\ =\frac{R_S^{K_i}\left(T_N^{S}N-T_x\right)}{T_Q}\times P\end{array}$$

(11)

where ${\mathrm{Q}}_N^S$ denotes the number of products in the specified working time. $P$ is the selling price of each product. $R_S^{K_i}$ denotes the productivity of the production line system. $T_N^S$ is the total time of production. $N$ is the number of producers. $T_x$ is the time that work is suspended. $T_Q$ indicates the time required to complete a product.

3.5. Profitability

The total profit $TP$ can be obtained as follows:

$$TP=TR-TC$$

(12)

4. Reliability Analysis

Consider scheduling labor for each workstation. Productivity in a linear production system affects the next process. In order to avoid low reliability of the final product, machines or skilled workers should be arranged at the initial stage of production. New hands with lower wages should be scheduled in the subsequent stages to complete the production tasks.

In this study, the optimization was performed for an MCCS consisting of eight workstations.

Table 3. Profit comparison of different work plans.

Number of Skilled Workers	Number of New Hands	Number of Machines	Total Cost	Plan 1 Profit	Plan 2 Profit	Plan 3 Profit	Plan 4 Profit
0	0	8	432,000	268,000	268,000	268,000	268,000
0	1	7	413,100	202,900	212,900	211,900	222,900
0	2	6	394,200	147,880	165,980	163,990	184,080
0	3	5	375,300	101,730	126,307	123,360	150,884
0	4	4	356,400	63,387	93,058	89,203	122,729
0	5	3	337,500	31,912	65,504	60,802	99,095
1	0	7	413,300	244,700	254,700	253,700	264,700
1	1	6	394,400	184,640	203,340	201,290	222,040
1	2	5	375,500	134,055	159,718	156,652	185,381
1	3	4	356,600	91,809	122,955	118,923	154,100
1	4	3	337,700	56,900	92,271	87,339	127,643
2	0	6	394,600	223,920	243,220	241,110	262,520
2	1	5	375,700	168,598	195,383	192,193	222,168
2	2	4	356,800	122,182	154,868	150,652	187,553
2	3	3	337,900	83,604	120,844	115,670	158,085
2	4	2	319,000	51,924	92,579	86,523	133,234
2	5	1	300,100	26,313	69,421	62,567	112,529
3	0	5	375,900	205,509	233,452	230,136	261,395
3	1	4	357,000	154,640	188,933	184,525	223,225
3	2	3	338,100	112,143	151,344	145,917	190,546
3	3	2	319,200	77,014	119,903	113,537	162,793
3	4	1	300,300	48,368	93,916	86,698	139,463
3	5	0	281,400	25,428	72,768	64,791	120,108
4	0	4	357,200	189,324	225,294	220,687	261,263
4	1	3	338,300	142,641	183,900	178,209	225,159
4	2	2	319,400	103,828	149,070	142,378	194,311
4	3	1	300,500	71,941	120,062	112,463	168,182
4	4	0	281,600	46,148	96,221	87,811	146,294
5	0	3	338,500	175,233	218,650	212,682	262,067
5	1	2	319,600	132,485	180,202	173,169	227,919
5	2	1	300,700	97,135	147,970	139,969	198,805
5	3	0	281,800	68,295	121,255	112,389	174,215
6	0	2	319,800	163,109	213,431	206,040	263,753
6	1	1	300,900	124,060	177,758	169,336	231,456
6	2	0	282,000	91,965	147,975	138,629	203,985
7	0	1	301,100	152,834	209,552	200,686	266,269
7	1	0	282,200	117,262	176,494	166,643	235,726
8	0	0	282,400	144,298	206,933	196,551	269,569

provides the work time combinations under different work schedules and expresses the optimization results in terms of profit. Since each workforce had the same productivity and fatigue probability when working at the workstations in the MCCS, the specific allocation of the workforce is not shown in Table 3.

As shown in Table 3, the optimal solution for option 4 is the best, i.e., a combination of three one-hour and two two-hour working hours. This scenario yielded the highest profit and was applicable under all conditions. From the results, product production should be arranged with as many skilled workers as possible and a small number of new manual workers. This not only ensures a low input–output ratio but also generates higher profits. To better compare the impact of the number of skilled workers and machines on profits, we depict the comparison results in Figure 3.

As shown in Figure 3, the horizontal axis represents the number of skilled workers. Since the only objects of comparison were skilled workers and machines, and the sum of skilled workers and machines is 8, the number of machines decreased as the number of skilled workers increased. From the results, only the profit of plan 4 increased with the increase of skilled workers and the cost of the system gradually decreased. The profits of the other three plans decreased as the number of skilled workers increased.

Although the machine has the advantages of high productivity and a steady state, the cost required to run it is too high. For small- and medium-sized manufacturing companies, the high cost of machines prevents fully automated production on the production line. Therefore, plan 1, plan 2, and plan 3 can be selected to increase the number of machines appropriately to improve profits. Choosing plan 4 to produce products without considering the introduction of machines and the company obtained a better profit than the other three plans. To further illustrate the advantages of plan 4, we show Figure 4.

As shown in Figure 4, the reliability of the automatic filtering of the system was less than 0.0000384 due to the low systemic error caused by the mismatch between inputs and outputs. Figure 4 represents a comparison of the ratio of profit to cost between new hands and skilled workers. The results showed that the ratios of all four plans increased with the number of skilled workers. However, the increasing trend of the ratio of plan 4 was significantly better than the other three plans.

In conclusion, for MCCS in this case, plan 4 was chosen as the best, i.e., the combination of a 10 min rest for every one hour of work and a 15 min rest for two hours of consecutive work was selected. For the allocation of producers, as many skilled workers as possible were assigned. Although the cost of wages will be more than for the new hands, the profit for the company was higher than with the machines and new hands.

5. Conclusions

In this study, we proposed a method to optimize the operation schedule by jointly considering the workers’ fatigue states and the operation states of machines for the production line of MCCS. The corresponding model was constructed based on the universal generating function (UGF). The model can be used to evaluate the different productivity of different types of workers in different states. In this model, we mainly considered three kinds of producers: skilled workers, new hand workers, and machines. Each producer worked in two states: energetic and fatigued, and productivity changed with the change in state. The results of each operation schedule were reflected in the profit. According to the results of numerical examples, work priority was given to the types of work with high productivity, such as machines and skilled workers. Thus, the best strategy suggested that in the order of workstations, the machines and skilled workers were arranged at former positions, while the new hands are arranged at the last workstations. In this case, it was convenient for rearrangement when there were idle skilled workers or machines in the actual production process. The priority of arranging the number of producers in the production line was skilled workers > machines > new hands. According to the proposed method, we obtained the optimal operation schedule, including working time, rest time and allocation strategy for a production line system.

The work in this study can be extended in a few directions. One direction is the extension of the model to consider production lines where workers are unable to implement the same tasks as the machines. Another direction is to consider the optimal operation plan for multiple production lines.