# Analysis of the Effect of the Line-Seru Conversion on the Waiting Time with Batch Arrival

Authors: Hui Ren; Dongyu Wang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The conveyor assembly line has been widely used in manufacturing industries to produce standard products with low costs. However, due to lack of flexibility, this production method has not been conducive to multivariety and small-batch production. In this situation,seru production formed by converting conveyor assembly lines has been a successful innovation in the Japanese manufacturing industry. Most of the existing literature has studied the benefits of this line-seru conversion from the perspective of the enterprises themselves, but this paper studies the effect of the line-seru conversion on the waiting time from the perspective of the customer. First, the change in the average waiting queue length caused by the line-seru conversion is proposed as an evaluation index. Second, with the consideration of the practical situation of random batch arrivals, the average waiting queue length formulas for the conveyor assembly line andseru production are established based on the assumption that the arrival is a Poisson process. Then, under two scenarios, we investigate the relationship between the average waiting queue length changed by the line-seru conversion and other parameters and find that the conversion can reduce the average waiting queue length in multivariety and small-batch production. Finally, under other potential scenarios, the equations for determining the average waiting queue length resulting from a change to line-seru conversion are derived.


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## Body

## 1. Introduction

With an increasingly diversified customer demand and shortened product life cycle, manufacturing enterprises are gradually facing a multivariety and small-batch production environment. In such an environment, the conveyor assembly line, which is suitable for mass production, has to be adjusted frequently, resulting in manufacturing companies not being able to take full advantage of its unique efficiency. At the same time, demand fluctuations often lead to over- or underproduction; hence, the fatal weakness of low flexibility is completely exposed [1]. To face such a turbulent markets environment, manufacturing factories started to look for new production organization forms with low costs and high flexibility. In various organizational innovations, Sony made huge profits by successfully implementing the dismantling of its conveyor assembly line and converting toSeru Seisan [1]. At Sony Kohda, a total of 710,000 square meters of workshop space was reduced [2] and the throughput time was reduced by 53%. At Canon, the number of required workers was reduced by 35,976, an amount equal to 25% of Canon’s previous total workforce [2]. Thus,seru production has attracted considerable attention in manufacturing industry.Eachseru inseru production is a small and compact production organization that consists of some equipment and one or more workers that produce one or more products [3]. There are three types ofseru in production practice: divisional, rotating, andyatai. When the conveyor assembly line is reconfigured, theserus formed first are the divisionalserus, in which tasks are divided into different sections and each section is in charge of one or more partially cross-trained workers. A rotatingseru is often organized in a U-shaped layout with several completely cross-trained workers who can assemble an entire product. As partially cross-trained workers are trained to become completely cross-trained workers, divisionalserus can evolve into rotatingserus.Yatai is aseru with only one completely cross-trained worker. In this paper, we only analyzeyatais.At present, most scholars have analyzed the benefits of converting a conveyor assembly line toseru production in a changeable market environment from the perspective of the enterprises themselves, but there is little analysis of this conversion from the perspective of the customers’ perception. However, in the process of manufacturing and service, it is difficult to obtain the customers’ satisfaction without considering the customers’ perception. This paper compares these two production organization forms, conveyor assembly line andseru production, based on the customers’ waiting time. In this paper, the conversion of the conveyor assembly line intoseru production is called line-seru conversion.The remainder of this paper is organized as follows. The following section is literature review. Section3 describes the research problems and constructs the average waiting queue length formula changed by the line-seru conversion. Section 4 mainly investigates the relationship between the average waiting queue length changed by the line-seru conversion and other parameters under two conversion scenarios. Finally, the conclusions are drawn in Section 5.


## 2. Literature Review

Because line-seru conversion in Japanese industry has been able to improve system performance in a changing market environment, for production organizations, it has been regarded as an innovation, for which many scholars have conducted theoretical research. For example, Kaku et al. [4] defined the line-seru conversion problem and constructed a mathematical model to describe it. As the total throughput time and the total labor power are applied to evaluate system performance, this model can be used as an evaluation tool to decide whether manufacturing factories should convert their production system. Then, Liu et al. [1] investigated the problem of how to convert the conveyor assembly line toserus. They built a comprehensive mathematical model to solve two issues, namely, how manyserus should be established and how many workers should be assigned to eachseru. Compared with Kaku’s model, their model is more suitable for analyzing the problem of the reconfiguration of the assembly line toserus. Actually, multiobjective decision-making is often used in the line-seru conversion problem. Although Yu et al. [5] also discussed how to implement this type of line-seru conversion, they constructed a two-objective line-seru conversion model that minimizes the total throughput time (TTPT) and the total labor hours (TLH). Yu et al. [6] also proposed a multiobjective optimization model to investigate the following two performances of line-seru conversion: the total throughput time and the total labor hours. Then, Yu et al. [7] further developed a more efficient algorithm to solve large-scale problems within a reasonable time. Yu et al. [8] formulated several main models of line-hybridseru system conversion and clarified the complexity and properties of line-hybridseru system conversion. Yu et al. [9] formulated a line-seru conversion towards reducing worker(s) without increasing makespan.Sinceseru production is a human-centered assembly system, the performance improvement resulting from the line-seru conversion is heavily dependent on the cross-trained workers. For converting a conveyor assembly line toseru production, Kaku et al. [10] constructed theoretical models to analyze the human task-related performances, which included the possible added operational tasks, the skill level, and the cross training of workers. Liu et al. [11] investigated the workers’ training and assignment problem in the line-seru conversion. They formulated a two-objective model to minimize the total training cost and to balance the total processing times among the cross-trained workers in eachseru. Yu et al. [12] presented a multiobjective line-seru conversion model, with the goals of reducing worker(s) and simultaneously increasing productivity, and performed several numerical simulation experiments to illustrate that the line-seru conversion can be used to reduce both worker(s) and the total throughput time. Ying and Tsai [13] discussed how to minimize total cost for training and assigning multiskilled workers inseru production systems. Lian et al. [14] studied a multiskilled worker assignment problem inseru production systems considering the worker heterogeneity.Although a line-seru conversion can improve system performance, not all enterprises can successfully implement it due to lack of theoretical guidance. Therefore, based on a systematical analysis of many experiences involving the implementation ofseru production in Japanese manufacturing factories, Liu et al. [15], for practitioners from a practical perspective, provided a general framework and some basic principles that should be followed while implementingseru production. Through the analysis of the profit function of theseru system, Yin et al. [3] explained why the implementation ofseru production with respect to an uncertain market could produce higher profits with a smaller workforce. Further, Yin et al. [16] studied how two electronics giants, Sony and Canon, had appliedseru to improve productivity, quality, and flexibility in ways that have enabled them to remain competitive. Yu et al. [17] focused on the fundamental principles ofseru system balancing in order to reduce makespan, labor hours, and manpower.In addition, Liu et al. [18] systematically summarized the advantages ofseru production relative to that of a conveyor assembly line: these advantages included reducing lead time, setup time, WIP inventories, finished-product inventories, cost, required workforce, and shop floor space. Additionally, they found thatseru production also positively influences profits, product quality, and workforce motivation. Lately, Zhang et al. [19] pointed out the sustainability ofseru production. Throughout the research of the line-seru conversion, most of the studies have analyzed the improvement of system performance from the perspective of the enterprise. However, this paper takes the waiting time as an index to analyze the line-seru conversion from the perspective of the customers’ perception. Most of the existing literature studied the related problems of a single product randomly arriving at a production system. For example, Uday et al. [20] used queuing theory to compare the waiting time performance with single arrivals for the production line and the case manager approach. However, in an actual production system, a business is not a retailer, and most of the customer orders are random batch arrival orders, with the batch size different for different customers. Therefore, unlike the previous literature, this article focuses on the conveyor assembly line andseru production with batch arrivals. Next, a review of the relevant research on batch arrival processes in the manufacturing and service systems is presented.First, scholars have studied queuing systems with batch arrivals at a single service window, namely, the production system with only one workstation. In the early 1960s, Suzuk [21] and Takacs [22] conducted research on the queuing model in the basic form Mx/G/1 and obtained some queuing indexes. Subsequently, Chaudhry [23] pointed out the advantages of the supplementary variable method and used it to study the Mx/G/1 queue model. Later, the supplementary variable method became an important tool to study the batch arrival queuing model. By using the inverse transformation of Laplace-Stieltjes to solve the supplementary variable, Briere and Chaudhry [24] provided easy access with the exact numerical solution of the Mx/G/1 model. Before that, in one sense, this type of queuing model was only partially solved. In this paper, we also use the inverse transformation of Laplace-Stieltjes to solve the supplementary variable and then obtain the average waiting queue length of aseru production system. The difference is that we study a multiserver batch arrival queuing system, where the conveyor assembly line consists of multiple workstations in a series and in which theseru production system consists of multiple parallel workstations, as shown in Figures 1(a) and 1(b).Figure 1
A comparison of conveyor assembly line and cellular manufacturing.
(a)
Schematic diagram for conveyor assembly line (b)
Schematic diagram for cellular manufacturingNext, we review the researches related to multiserver batch arrival queuing systems. Hideaki and Wu [24] considered multiple servers with a semi-Markov batch arrival process queuing system. Based on the theory of a piecewise Markov process, they obtained the waiting time distribution. The arrival process of the queuing system in their research is different from the batch arrival process withseru production in this paper. Güllü [25] analyzed a M/G/∞ queue with batch arrivals, in which the number of jobs in the system was characterized as a compound Poisson random variable. Differing from this paper, they assumed that the same batch of products must be processed by the same server. Bara and Jeongsim [26] considered a multiserver Mx/M/c queue with impatient customers and used the customers’ loss probability to measure the quality of service in such a system. The biggest difference between the queuing system in their study and the one utilized in theseru production situation in this paper is that they assumed a customer was only willing to wait in the queue for a fixed time.Furthermore, Wu [27] classified the batch queuing models for manufacturing systems. For batch arrivals, he took the cycle time as an index and only studied the performance of two queuing models: single job processing and serial batch processing. According to Wu’s classification, the systems studied in this paper are batch arrivals with serial batch processing and batch arrivals with parallel batch processing. In this paper, however, the average waiting queue length under a stable state is regarded as the index by which these two queuing systems are measured.


## 3. The Effect of the Line-Seru Conversion on the Average Waiting Queue Length

Under multivariety and small-batch production, many manufacturing factories convert their conveyor assembly line intoseru production to overcome the low flexibility of the conveyor assembly line. However, whether the line-seru conversion can improve the waiting time performance from the perspective of the customers’ perception remains to be further studied. According to Little’ Law, the average waiting queue length is positively correlated with the average waiting time and is more intuitive. Therefore, we analyze the change in the average waiting queue length resulting from the line-seru conversion with random batch arrivals. Let ΔLq denote the difference between the average waiting queue length of the conveyor assembly line Lq(A) and the average waiting queue length of theseru production Lq(C); then(1)ΔLq=LqA-LqC.When the value ofΔLq>0, it indicates that the line-seru conversion reduces the average waiting queue length and further shows that the conversion can improve the customers’ waiting time performance. When the value of ΔLq=0, it indicates that the line-seru conversion does not change the average waiting queue length and further shows that the conversion cannot improve the waiting time performance. When the value of ΔLq<0, it indicates that the line-seru conversion increases the average waiting queue length and further shows that the conversion increases the waiting time. Formula (1) shows that the average waiting queue length changed by the line-seru conversion can be obtained only when the average waiting queue lengths of the conveyor assembly line and theseru production methods are obtained.


### 3.1. The Batch Arrival Queuing Model for a Conveyor Assembly Line

On the basis of a queuing system with batch arrivals and a single server window, namely, aMx/M/1 queuing model, the number of workstations is increased in a series to form a queuing system of a conveyor assembly line with batch arrivals, as shown in Figure 1(a), in which the number of workstations is m. A worker is assigned to one workstation to perform one operation. We assume that the product arrival process is a Poisson flow with parameter λ [25, 26] and has batch arrival of which the mean of the batch size is k. The operating characteristics of the first operation in the conveyor assembly line are defined as a Mx/M/1 queuing model with the arrival rate of λk and the service rate of μ. Using Burke’s theorem [20], the steady-state output completed by the first worker is also a Poisson process with parameter λk. The output of the first operation serves as the input of the second operation. Repeating the above logic, the conveyor assembly line essentially consists of m decomposable Mx/M/1 queuing models. Note that when the conveyor assembly line is in a stable state, the arrival rate λk must be less than the service rate μ.When any batch ofx products arrives at the conveyor assembly line simultaneously, there are m′ products in the system, and m′>m. Therefore, the conveyor assembly line starts processing this batch of products when (m′-m+1) products are completed. According to Little’s Law, the average queue length can be obtained by the average sojourn time of each product in the conveyor assembly line, and then the average waiting queue length can be obtained. The average sojourn time of each product in any batch consists of two parts. The first part is the time that any batch of products waits for the completion of (m′-m+1) products. The second part is the average time spent on each product during the production of any batch of products. It is noteworthy that the average time spent on each product includes not only the production time of each product, but also the waiting time of each product during the production of this batch of products.As the mean of batch size isk, E[x]=k; the average time spent on each product during the production of any batch of products on the conveyor assembly line is shown as follows:(2)E∑i=1xmμ+i-1μP=Emμ+x-12μ=2m+k-12μwhere P is the probability that this product ranks ith in x products and P=1/x. m/μ is the processing time for each product. i-1/μ is the waiting time for the ith product.The average sojourn time of each product in the conveyor assembly line is(3)WsA=∑m′=0∞Epm′=∑m′=0∞m′-m+1μ+2m+k-12μpm′=LsAμ+k+12μwhere E is the sojourn time when the product arrives at the conveyor assembly line with m′ products and pm′ is the probability of m′ products on the conveyor assembly line. Furthermore, ∑m′=0∞m′pm′=Ls(A), and Ls(A) is the average queue length of the conveyor assembly line, which represents the average of products (waiting and processing products) in this system.Therefore, the average waiting queue length in the conveyor assembly line,Lq(A), is as follows.(4)LqA=LsA-mλkμ=λkk+12μ-λk-mλkμ


### 3.2. The Batch Arrival Queuing Model for Seru Production

This paper studies aseru production system containing severalyatais, which were formed by converting the conveyor assembly line, and in which one completely cross-trained worker is responsible for all the operations at oneseru. We assume that the number ofserus is n (n≤m) and that the number of completely cross-trained workers in the system is also n. The arrival process is the same as that of the conveyor assembly line, and the products are processed in a first-in-first-out manner. The service time of eachseru also follows a negative exponential distribution, but the service rate, which is different from that of the conveyor assembly line, is set at μ′. Therefore, theseru production system with batch arrivals belongs to a Mx/M/n queuing model, as shown in Figure 1(b). Although a Mx/M/n queuing system has not been studied in depth, it is similar to a M/M/n queuing system [28], except that the arrival process is different. Based on the study of a M/M/n queuing system, we explore the average waiting queue length of a Mx/M/n queuing system.This paper studies the number of products already in the system when any batch of products arrives at the system, namely, the average queue length, to obtain the average waiting queue length formula. Through the analysis of the whole process from product arrival to departure, it is found that the number of products already in the system of any batch of products arrival is closely related to the number of products that may be served by the system within the products arrival interval. Next, this paper will analyze the relationship between these two quantities in detail.Since the arrival process is a Poisson flow with the parameterλk, the interval time of the batch arrivals is a negative exponential distribution with the parameter λk, and its distribution density is g(t)=(λk)e-(λk)t. Cr represents the rth batch of arrival products, and tr represents the arrival moment of the rth batch of products; {tr,r>1} is called an embedded point column. We assume that Xr is the number of products already in the system when the rth batch of products arrives at the system, and it is only related to the system state at the tr-0 moment. {Xr,r>1} has proved to be a homogeneous Markov chain in theory [28]. Let Tr=tr-tr-1(r>1) be the rth time arrival interval and Yr be the number of products that may be served by nserus within the rth time arrival interval Tr, as shown in Figure 2.Figure 2
M x ∣ M ∣ n queuing model.If there areXr products in the system at the arrival moment of the rth batch of products, and the number of the rth batch of products is set as the mean of the batch size k, the system has Xr+k products when the next batch of products arrives at the system. Therefore, there are the following.(5)Xr+k=Xr+k-Yr,Yr<Xr+k0,Yr≥Xr+kLemma 1.
Since the service time is a negative exponential distribution, the number of finished productsYr follows a Poisson distribution, under the condition of a known rth time arrival interval Tr. For a Markov chain {Xr,r≥1}, the one-step transition matrix is as follows.(6)pij=PXr+k=j∣Xr=i=PYr=i+k-j,j≤i+k0,j>i+kTheorem 2.
Whenseru production is in a steady state, namely, n≤j≤i+k, (7)pij=∫0∞nμ′ti+k-ji+k-j!e-nμ′tgtdt.Proof of Theorem2.
Whenn≤j≤i+k, nserus are all busy in the period of Tr, so when the nserus are fully occupied, the output stream is actually the superposition of n independent Poisson processes with the parameter μ′, namely, a Poisson stream with the parameter nμ′. Thus, within this time length of Tr=t, the conditional probability of outputting i+k-j products is (nμ′t)i+k-j/(i+k-j)!e-nμ′t; hence,(8)pij=∫0∞nμ′ti+k-ji+k-j!e-nμ′tgtdt.Whenλk/nμ′<1, it can be theoretically proved that this Markov chain must have a stationary distribution denoted as {p¯j}; namely, it is the distribution of Xr when the system is stable, and it satisfies the following equation.(9)p¯j=∑i=0∞pijp¯i=∑i=j-k∞pijp¯i,j≥0,Let(10)βl=∫0∞nμ′tll!e-nμ′tgtdt;thus, when n≤j≤i+k, according to Theorem 2, there is(11)pij=βi+k-jand, then, whenj≥n-1, formula (11) is also true. Thus, by formula (9) and formula (11), we can obtain the following.(12)p¯j+k=∑i=0∞p¯ipi,j+k=∑i=0∞p¯iβi-jLet(13)p¯j=C1αj,p¯j+k=C1αj+1j≥n-1,α<1where α is a supplementary variable and C1 is a constant. Formula (13) is substituted into formula (12); then, we can obtain(14)α=G∗nμ′-nμ′αwhere G∗(x)=L[g(t)], namely, the Laplace transformation of g(t). Formula (14) has a unique solution in the unit circle; that is, there is 0<α<1. Therefore, the stationary solution is(15)P¯=p¯0,p¯1,…,p¯n-2,C1αn-1,C1αn,…=Cγ0,γ1,…,γn-2,1,α,α2,…where C=C1αn-1.Then,g(t)=λke-λkt is substituted into formula (14); hence, α=λk/nμ′ is gained. When the system is stable, λk/nμ′<1.The average waiting queue length formula inseru production is(16)LqC=∑j=n∞j-np¯j=∑j=n∞j-nCαj-n+1=Cλ2k2nμ′-λk2where C is obtained by formula (15) and 0<C<1.


## 3.1. The Batch Arrival Queuing Model for a Conveyor Assembly Line

On the basis of a queuing system with batch arrivals and a single server window, namely, aMx/M/1 queuing model, the number of workstations is increased in a series to form a queuing system of a conveyor assembly line with batch arrivals, as shown in Figure 1(a), in which the number of workstations is m. A worker is assigned to one workstation to perform one operation. We assume that the product arrival process is a Poisson flow with parameter λ [25, 26] and has batch arrival of which the mean of the batch size is k. The operating characteristics of the first operation in the conveyor assembly line are defined as a Mx/M/1 queuing model with the arrival rate of λk and the service rate of μ. Using Burke’s theorem [20], the steady-state output completed by the first worker is also a Poisson process with parameter λk. The output of the first operation serves as the input of the second operation. Repeating the above logic, the conveyor assembly line essentially consists of m decomposable Mx/M/1 queuing models. Note that when the conveyor assembly line is in a stable state, the arrival rate λk must be less than the service rate μ.When any batch ofx products arrives at the conveyor assembly line simultaneously, there are m′ products in the system, and m′>m. Therefore, the conveyor assembly line starts processing this batch of products when (m′-m+1) products are completed. According to Little’s Law, the average queue length can be obtained by the average sojourn time of each product in the conveyor assembly line, and then the average waiting queue length can be obtained. The average sojourn time of each product in any batch consists of two parts. The first part is the time that any batch of products waits for the completion of (m′-m+1) products. The second part is the average time spent on each product during the production of any batch of products. It is noteworthy that the average time spent on each product includes not only the production time of each product, but also the waiting time of each product during the production of this batch of products.As the mean of batch size isk, E[x]=k; the average time spent on each product during the production of any batch of products on the conveyor assembly line is shown as follows:(2)E∑i=1xmμ+i-1μP=Emμ+x-12μ=2m+k-12μwhere P is the probability that this product ranks ith in x products and P=1/x. m/μ is the processing time for each product. i-1/μ is the waiting time for the ith product.The average sojourn time of each product in the conveyor assembly line is(3)WsA=∑m′=0∞Epm′=∑m′=0∞m′-m+1μ+2m+k-12μpm′=LsAμ+k+12μwhere E is the sojourn time when the product arrives at the conveyor assembly line with m′ products and pm′ is the probability of m′ products on the conveyor assembly line. Furthermore, ∑m′=0∞m′pm′=Ls(A), and Ls(A) is the average queue length of the conveyor assembly line, which represents the average of products (waiting and processing products) in this system.Therefore, the average waiting queue length in the conveyor assembly line,Lq(A), is as follows.(4)LqA=LsA-mλkμ=λkk+12μ-λk-mλkμ


## 3.2. The Batch Arrival Queuing Model for Seru Production

This paper studies aseru production system containing severalyatais, which were formed by converting the conveyor assembly line, and in which one completely cross-trained worker is responsible for all the operations at oneseru. We assume that the number ofserus is n (n≤m) and that the number of completely cross-trained workers in the system is also n. The arrival process is the same as that of the conveyor assembly line, and the products are processed in a first-in-first-out manner. The service time of eachseru also follows a negative exponential distribution, but the service rate, which is different from that of the conveyor assembly line, is set at μ′. Therefore, theseru production system with batch arrivals belongs to a Mx/M/n queuing model, as shown in Figure 1(b). Although a Mx/M/n queuing system has not been studied in depth, it is similar to a M/M/n queuing system [28], except that the arrival process is different. Based on the study of a M/M/n queuing system, we explore the average waiting queue length of a Mx/M/n queuing system.This paper studies the number of products already in the system when any batch of products arrives at the system, namely, the average queue length, to obtain the average waiting queue length formula. Through the analysis of the whole process from product arrival to departure, it is found that the number of products already in the system of any batch of products arrival is closely related to the number of products that may be served by the system within the products arrival interval. Next, this paper will analyze the relationship between these two quantities in detail.Since the arrival process is a Poisson flow with the parameterλk, the interval time of the batch arrivals is a negative exponential distribution with the parameter λk, and its distribution density is g(t)=(λk)e-(λk)t. Cr represents the rth batch of arrival products, and tr represents the arrival moment of the rth batch of products; {tr,r>1} is called an embedded point column. We assume that Xr is the number of products already in the system when the rth batch of products arrives at the system, and it is only related to the system state at the tr-0 moment. {Xr,r>1} has proved to be a homogeneous Markov chain in theory [28]. Let Tr=tr-tr-1(r>1) be the rth time arrival interval and Yr be the number of products that may be served by nserus within the rth time arrival interval Tr, as shown in Figure 2.Figure 2
M x ∣ M ∣ n queuing model.If there areXr products in the system at the arrival moment of the rth batch of products, and the number of the rth batch of products is set as the mean of the batch size k, the system has Xr+k products when the next batch of products arrives at the system. Therefore, there are the following.(5)Xr+k=Xr+k-Yr,Yr<Xr+k0,Yr≥Xr+kLemma 1.
Since the service time is a negative exponential distribution, the number of finished productsYr follows a Poisson distribution, under the condition of a known rth time arrival interval Tr. For a Markov chain {Xr,r≥1}, the one-step transition matrix is as follows.(6)pij=PXr+k=j∣Xr=i=PYr=i+k-j,j≤i+k0,j>i+kTheorem 2.
Whenseru production is in a steady state, namely, n≤j≤i+k, (7)pij=∫0∞nμ′ti+k-ji+k-j!e-nμ′tgtdt.Proof of Theorem2.
Whenn≤j≤i+k, nserus are all busy in the period of Tr, so when the nserus are fully occupied, the output stream is actually the superposition of n independent Poisson processes with the parameter μ′, namely, a Poisson stream with the parameter nμ′. Thus, within this time length of Tr=t, the conditional probability of outputting i+k-j products is (nμ′t)i+k-j/(i+k-j)!e-nμ′t; hence,(8)pij=∫0∞nμ′ti+k-ji+k-j!e-nμ′tgtdt.Whenλk/nμ′<1, it can be theoretically proved that this Markov chain must have a stationary distribution denoted as {p¯j}; namely, it is the distribution of Xr when the system is stable, and it satisfies the following equation.(9)p¯j=∑i=0∞pijp¯i=∑i=j-k∞pijp¯i,j≥0,Let(10)βl=∫0∞nμ′tll!e-nμ′tgtdt;thus, when n≤j≤i+k, according to Theorem 2, there is(11)pij=βi+k-jand, then, whenj≥n-1, formula (11) is also true. Thus, by formula (9) and formula (11), we can obtain the following.(12)p¯j+k=∑i=0∞p¯ipi,j+k=∑i=0∞p¯iβi-jLet(13)p¯j=C1αj,p¯j+k=C1αj+1j≥n-1,α<1where α is a supplementary variable and C1 is a constant. Formula (13) is substituted into formula (12); then, we can obtain(14)α=G∗nμ′-nμ′αwhere G∗(x)=L[g(t)], namely, the Laplace transformation of g(t). Formula (14) has a unique solution in the unit circle; that is, there is 0<α<1. Therefore, the stationary solution is(15)P¯=p¯0,p¯1,…,p¯n-2,C1αn-1,C1αn,…=Cγ0,γ1,…,γn-2,1,α,α2,…where C=C1αn-1.Then,g(t)=λke-λkt is substituted into formula (14); hence, α=λk/nμ′ is gained. When the system is stable, λk/nμ′<1.The average waiting queue length formula inseru production is(16)LqC=∑j=n∞j-np¯j=∑j=n∞j-nCαj-n+1=Cλ2k2nμ′-λk2where C is obtained by formula (15) and 0<C<1.


## 4. Numerical Analysis of Conversion Scenarios

To save costs, the line-seru conversion may be accompanied by downsizing and reducing the number of workstations. Inseru production system containing severalyatais, there is a one-to-one correspondence between cross-trained workers andserus. Let l represent the number of workers downsized. The number of cross-trained workers and the number ofserus are (m-l): m>l and l≥0. Thus n in Lq(C) can be replaced by (m-l). As the cross-trained workers perform all the operations of a product, they usually reintegrate manufacturing processes, which may reduce certain non-value-added operations. Let s represent the reduced number of operations. A product may only take (m-s) operations, and m>s; s≥0. In general, s will be very small, but it may increase with the increase of m in some cases. Because the cross-trained worker is no longer specialized in a single operation, his efficiency may be lower than that of specialized workers in the conveyor assembly line [29]. Let ε, where 0<ε≤1, denote the relative efficiency of the cross-trained worker as compared with that of the specialized worker. This means that the processing time of each operation for a product inseru production follows an exponential distribution with a mean of 1/εμ. Hence, the average processing time for a product is (m-s)/εμ, and the service rate is εμ/(m-s), which can replace μ′ in Lq(C). Therefore, the average waiting queue length changed by the line-seru conversion is(17)ΔLq=LqA-LqC=ρk+121-ρ-mρ-Cρ2m-s2εm-l-m-sρ2where ρ=λk/μ is the system utilization rate of the conveyor assembly line. To avoid confusion, the term “system utilization rate” will be used consistently in this paper to denote “the system utilization rate of the conveyor assembly line”.In the multivariety and small-batch production environment, the conveyor assembly line has to reconfigure equipment and personnel frequently, so the system utilization rate is necessarily low. We mainly focus on the effect of the line-seru conversion on the average waiting queue length with respect to a low system utilization rate. Based on a practical line-seru conversion under different conversion scenarios, the input parameters in (17) are determined to obtain meaningful management insights. Specifically, to generate different scenarios, three factors are considered, namely, downsizing, the reduced number of operations, and worker efficiency. With two possibilities for each factor, a total of eight scenarios are generated. To clarify the relative performance ofseru production, two scenarios will be analyzed: (1) the simplest scenario, where there is no reduction in operations and no downsizing, and the efficiency of cross-trained workers is equal to that of specialized workers, and (2) the most general scenario, where there is a reduction in operations, downsizing, and less efficient cross-trained workers. Then, the mean of the batch size and the number of workstations are selected to analyze the sensitivity of the average waiting queue length changed by the line-seru conversion in these two scenarios. Finally, the production conditions of the line-seru conversion are obtained.Scenario 1.
When the conveyor assembly line is dismantled intoseru production, there is no downsizing and no reduction in operation. Additionally, we assume that the cross-trained workers inseru production and the specialized workers in the conveyor assembly line have the same efficiency. By substituting s=l=0,ε=1 into (17), we can find the following.(18)ΔLq=LqA-LqC=ρk+11-ρ-2Cρ221-ρ2-mρ
Because0<C<1, 0<2C<2. When (k+1)≫2, the value of C does not affect the sign of ΔLq. For the convenience of calculation, let C=0.5; therefore, ΔLq can be simplified.(19)ΔLq=ρk+11-ρ-ρ221-ρ2-mρ
Figure3 illustrates the effect of the change in the number of workstations on the average waiting queue length changed by the line-seru conversion in Scenario 1. On the assumption that the mean of the batch size k=50, the line-seru conversion reduces the average waiting queue length, and the reduced queue length increases with the decrease of the number of workstations and with the increase of the system utilization rate. This shows that when the mean of the batch size is greater than the number of workstations, the line-seru conversion should be implemented under Scenario 1. For the conveyor assembly line, if the number of workstations increases, the waiting queues will naturally reduce. Since the utilization rate of these two system is the same in Scenario 1, the standard deviation of the total processing time performed by one cross-trained worker inseru production will be less than the sum of the standard deviations of the processing times of all workstations in the conveyor assembly line. The above conclusions are summarized as follows.Figure 3
The average waiting queue length changed by the line-cell conversion, under different number of workstations in Scenario1.Corollary 3.
For Scenario1, when the mean of the batch size is greater than the number of workstations in the conveyor assembly line, and the conveyor assembly line has at least two workstations (production operations) (m≥2), the line-seru conversion will reduce the average waiting queue length (ΔLq>0), and the reduced queue length will increase with the decrease of the number of workstations m and with the increase of the system utilization rate ρ (ΔLq is inversely related to m and is positively related to ρ).Corollary 4.
For a modified Scenario1, when the number of staff downsized is equal to the reduced number of operations (i.e., s=l≥1) and other parameters remain unchanged, the conclusion of the Corollary 3 can also be drawn.Figure4 illustrates the effect of the change in the mean of the batch size on the average waiting queue length changed by the line-seru conversion in Scenario 1. On the assumption that m=10 represents the number of workstations in the conveyor assembly line, when the mean of the batch size is greater than the number of workstations in the conveyor assembly line, the line-seru conversion will reduce the average waiting queue length, and the reduced queue length will increase with the increase of the mean of the batch size and the system utilization rate. This is because the system utilization rate ofseru production is the same as that of the conveyor assembly line in Scenario 1. Even if the mean of the batch size is large, the efficiency ofseru production is higher than that of the conveyor assembly line. Hence, the conveyor assembly line should be converted toseru production in Scenario 1. These conclusions are further summarized as follows.Figure 4
The average waiting queue length changed by the line-cell conversion, under different batch size means in Scenario1.Corollary 5.
For Scenario1, when the mean of the batch size is greater than the number of workstations and the conveyor assembly line has at least two workstations (m≥2), the line-seru conversion will reduce the average waiting queue length (ΔLq>0), and the reduced queue length will increase with an increase of the mean of the batch size k and with an increase of the system utilization rate ρ (i.e., ΔLq is positively related to k and ρ).Corollary 6.
For a modified Scenario1, when the number of staff downsized is equal to the reduced number of operations (i.e., s=l≥1) and other parameters remain unchanged, the conclusion of the Corollary 5 can also be drawn.Scenario 2.
When the conveyor assembly line is dismantled intoseru production, the factory implements downsizing and cross-trains relatively inefficient workers. Meanwhile, the cross-trained workers reintegrate manufacturing processes to reduce operations.
This most general scenario is downsizing in the process of conversion, which makes the number ofserus less than that of workstations in the conveyor assembly line. According to the practical situation, the number of workers downsized and the reduced number of operations can be roughly divided into two cases: (1) they are fixed and do not change with the number of workstations (operations); (2) they increase with the increase of the number of workstations. For Scenario 2, these two systems do not have the same number of staff, the same number of service windows, and the same service rate; thus, they do not have the same system utilization rates, and the system utilization rate ofseru production is m-s/ε(m-l)ρ<1, which leads to the system utilization rate of the conveyor assembly line, ρ<ε(m-l)/m-s.Based on (17), Figures 5, 6, 7, and 8 illustrate how the average waiting queue length changed from the line-seru conversion is affected by the change in the number of workstations, which, under Scenario 2, is related to the number of staff downsized and the reduced number of operations. Figure 5 shows that the number of staff downsized and the reduced number of operations are fixed, l=2,s=1. Figure 6 shows that the reduced number of operations is fixed, s=1, but the number of staff downsized increases with the increase in the number of workstations as follows: when m=10, l=2; when m=20, l=4; and when m=30, l=6. Figure 7 shows that the number of staff downsized is fixed, l=2, but the reduced number of operations increases with the increase in the number of workstations as follows: when m=10, s=1; when m=20, s=2; and when m=30, s=3. Figure 8 shows that the number of staff downsized and the reduced number of operations increase with the increase of the number of workstations as follows: when m=10, l=2,s=1; when m=20, l=4,s=2; and when m=30, l=6,s=3.Figure 5
The average waiting queue length changed by the line-cell conversion, under different number of workstations in Scenario2 (downsizing and the reduced operations are fixed).Figure 6
The average waiting queue length changed by the line-cell conversion, under different number of workstations in Scenario2 (the reduced operations are fixed but downsizing is changed).Figure 7
The average waiting queue length changed by the line-cell conversion, under a different number of workstations in Scenario2 (downsizing is fixed but the reduced operations are changed).Figure 8
The average waiting queue length changed by the line-cell conversion, under a different number of workstations in Scenario2 (downsizing and the reduced operations are changed).Under the condition in which the mean of the batch sizek=50 and the relative efficiency of the cross-trained worker ε=0.95, the average waiting queue length changed by the line-seru conversion in these four cases is first reduced and then increased with the increase of the system utilization rate. This shows that when the system utilization rate is not high, the line-seru conversion should be implemented. Meanwhile, when the system utilization rate is less than a certain value, the reduction of the average waiting queue length increases gradually with the decrease of the number of workstations. This differences among the four cases are as follows: in Figures 5 and 7, as the system utilization rate increases, the curve with the minimum number of workstations firstly begins to decline and turns until it extends below 0; in Figure 6, as the system utilization rate increases, the curve with the maximum number of workstations firstly begins to decline and turns until it extends below 0; in Figure 8, as the system utilization rate increases, the three curves simultaneously begin to decline until reaching below 0.Corollary 7.
For Scenario2, when the mean of the batch size is greater than the number of workstations in the conveyor assembly line and m≥2, the average waiting queue length changed by the line-seru conversion is first reduced and then increased with the increase of the system utilization rate ρ. When the system utilization rate is less than a certain value, the reduction of the average waiting queue length increases gradually with the decrease of the number of workstations m. This shows that when the system utilization rate is not high, the line-seru conversion should be implemented (ΔLq>0), and ΔLq is inversely related to m.On the basis of (17), Figure 9 illustrates the effect of the change in the mean of batch size on the average waiting queue length changed by the line-seru conversion in Scenario 2. On the assumption that m=10, l=2,s=1,ε=0.95, with the increase of the system utilization rate, the line-seru conversion first reduces the average waiting queue length until the reduction reaches the highest value, and then the reduction begins to decrease gradually until it is less than 0. Meanwhile, when the system utilization rate is less than a certain value, the reduction of the average waiting queue length increases gradually with the increase of the mean of the batch size.Figure 9
The average waiting queue length changed by the line-cell conversion, under different batch size means in Scenario2.Corollary 8.
For Scenario2, when the mean of the batch size is greater than the number of workstations in the conveyor assembly line and m≥2, the average waiting queue length changed by the line-seru conversion is first reduced and then increased with the increase of the system utilization rate ρ. When the system utilization rate is less than a certain value, the reduction of the average waiting queue length increases gradually with the increase of the mean of the batch size k. This shows that when the system utilization rate is not high, the line-seru conversion should be implemented (ΔLq>0), and ΔLq is positively related to k.From Figures5–9, depending on the values of parameters m, s, l, k, and ρ, the line-seru conversion may reduce, increase, or not change the average waiting queue length. According to formula (17), this relationship can be captured in the form of an equation to answer when to use (or not to use)seru production or the conveyor assembly line. The relationship is stated as the following corollary.Corollary 9.
For Scenario2, when (20)k+121-ρ-m-ρm-s22εm-l-m-sρ2>00<m-sεm-lρ<1the line-seru conversion will reduce the average waiting queue length. Since the number of staff downsized and the reduced number of operations in Figures 6–8 are related to the number of workstations, it is impossible to perform the “break-even” analysis of the number of workstations and the system utilization rate. Therefore, we only give the results of this “break-even” analysis for Figures 5 and 9 in Scenario 2, as follows.(1) Given the parametersk=50,s=1,l=2,ε=0.95 for (20), the break-even relationship between the number of workstations m and the system utilization rate ρ is considered, as shown in Figure 10. As seen from Figure 10, the break-even curve divides the space into two regions, where one is suitable forseru production (due to a shorter queue length) and the other is suitable for the conveyor assembly line. Break-even curves drawn for alternate values of k,l,s,ε show a pattern similar to that of Figure 10. These break-even curves indicate that when the system utilization rate is not high and the number of workstations is larger than a certain value, the line-seru conversion should be implemented.Figure 10
Decision diagram for choosing the system, under different number of workstations and system utilization rate in Scenario2.(2) Given the parametersm=10,s=1,l=2,ε=0.95 for (20), the break-even relationship between the mean of batch size k and the system utilization rate ρ is considered, as shown in Figure 11. The break-even curve also divides the space into two regions, which correspond toseru production and the conveyor assembly line. Break-even curves drawn for alternate values of m,s,l,ε show a pattern similar to that of Figure 11. These break-even curves indicate that when the system utilization rate is not high and the mean of batch size is larger than a certain value, the line-seru conversion should be implemented.Figure 11
Decision diagram for choosing the system, under different batch size means and system utilization rates in Scenario2.Figures10 and 11 have important practical implications, and it can help managers to decide whether or not the conveyor assembly line should be converted intoseru production from the perspective of the waiting queue length. For example, for a batch arrival Poisson flow with a mean of 60, when the system utilization rate of the conveyor assembly line with 20 workstations reaches 0.6, should it be converted intoseru production? In the conversion process, the number of staff downsized is 2 and the reduced number of operations is 1. The line-seru conversion can also reduce the average waiting queue length, even if the efficiency of the cross-trained workers is only 90% of that of the conveyor assembly line worker (ε=0.90). Therefore, the conveyor assembly line should be converted intoseru production in this case.For Scenario2, the system utilization rate ofseru production is m-s/ε(m-l)ρ. According to the parameters of five examples mentioned above, it is found that the system utilization rate of the conveyor assembly line for these five examples is less than that ofseru production. Next, for Scenario 2, we reset the parameters so that the system utilization rate of the conveyor assembly line is greater than that ofseru production, and then analyze the effect of the line-seru conversion under this condition on the average waiting queue length.First, on the assumption ofk=50,ε=0.95, we analyze the effect of the number of workstations on the average waiting queue length changed by the line-seru conversion based on formula (17). Figure 12 describes the impact situation that the number of staff downsized and the reduced number of operations are fixed, where l=1,s=3. For the example of the number of workstations related to the number of staff downsized and the reduced number of operations, numerical analysis shows that the effect of the number of workstations on the average waiting queue length is similar to that of Figure 12, so there is no need to enumerate the results one by one. From Figure 12, we find that for Scenario 2, where the system utilization rate of the conveyor assembly line is greater than that ofseru production, the line-seru conversion will reduce the average waiting length, and the reduction will gradually increase with the increase of the system utilization rate and the decrease of the number of workstations.Figure 12
The average waiting queue length changed by the line-cell conversion under different number of workstations in Scenario2(s=3,l=1).Second, on the assumption ofm=10,l=2,s=1,ε=0.95, we analyze the effect of the mean of batch size on the average waiting queue length changed by the line-seru conversion based on formula (17), and Figure 13 describes the specific situation. From Figure 13, we find that for Scenario 2, where the system utilization rate of the conveyor assembly line is greater than that ofseru production, the line-seru conversion will reduce the average waiting length, and the reduction will gradually increase with the increase of the system utilization rate and the mean of batch size. These conclusions are further summarized in Corollary 10.Figure 13
The average waiting queue length changed by the line-cell conversion under different batch size means in Scenario2(s=3,l=1).Corollary 10.
For Scenario2, where the system utilization rate of the conveyor assembly line is greater than that ofseru production, the line-seru conversion will reduce the average waiting length (ΔLq>0), and the reduced queue length will increase with an increase of the mean of the batch size k, an increase of the system utilization rate ρ, and a decrease of the number of workstations m (i.e., ΔLq is positively related to k and ρ and is inversely related to m).From the perspective of the waiting time performance, this paper mainly analyzes the effect of the line-seru conversion under the two scenarios on the average waiting queue length. By setting the appropriate parameters in (17), the average waiting queue length changed by the line-seru conversion for other potential scenarios can also be obtained, and the resultant formulas are shown in Table 1.Table 1
The formulas of average waiting queue length changed by the line-cell conversion under each scenario.
l = 0 l > 0 ε = 1 ε < 1 ε = 1 ε < 1 ρ ( k + 1 ) ( 1 - ρ ) - ρ 2 2 1 - ρ 2 - m ρ ρ ( k + 1 ) 2 1 - ρ - m ρ - ρ 2 m 2 2 ε m - m ρ 2 ρ ( k + 1 ) 2 1 - ρ - m ρ - ρ 2 m 2 2 m - l - m ρ 2 ρ ( k + 1 ) 2 1 - ρ - m ρ - ρ 2 m 2 2 ε m - l - m ρ 2 s = 0   ρ ( k + 1 ) 2 1 - ρ - m ρ - ρ 2 m - s 2 2 m - m - s ρ 2 ρ ( k + 1 ) 2 1 - ρ - m ρ - ρ 2 m - s 2 2 ε m - m - s ρ 2 ρ ( k + 1 ) 2 1 - ρ - m ρ - ρ 2 m - s 2 2 m - l - m - s ρ 2 ρ ( k + 1 ) 2 1 - ρ - m ρ - ρ 2 m - s 2 2 ε m - l - m - s ρ 2 s > 0


## 5. Conclusions

In multivariety and small-batch production, the efficiency of the conveyor assembly line could not be fully realized. Therefore, the factory began to experiment with the reform of the production organization form and eventually found that the line-seru conversion can be adapted to such a production environment. Later, scholars studied the various issues in the line-seru conversion from different degrees, and most of the research results analyzed the advantages of line-seru conversion from the perspective of the enterprises themselves. However, this paper analyzes the effect of the line-seru conversion on waiting time from the perspective of the customer.First, the average waiting queue length changed by the line-seru conversion is used as the evaluation index. Second, queuing theory is used to establish the average waiting queue length formulas with random batch arrivals of the conveyor assembly line andseru production. Then, under two typical scenarios, we explore the relationship between the average waiting queue length changed by the line-seru conversion and other parameters, such as the mean of the batch size, the number of workstations, and the system utilization rate. In summary, the line-seru conversion should be implemented in most cases; only when the system utilization rate of the conveyor assembly line is less than that ofseru production and the system utilization rate is high, the conveyor assembly line should be continued. In the multivariety and small-batch production environment, the system utilization rate of the conveyor assembly line is not less than that ofseru production, so the line-seru conversion should be implemented. Finally, we also present the formulas of the average waiting queue length changed by the line-seru conversion under the other potential six scenarios.In the process of formulation establishment, the service time is assumed to follow a negative exponential distribution according to the actual condition, but there is no hypothesis test. Meanwhile, we believe that the arriving products are processed in the first-in-first-out manner. In fact, however, the service manner may be many forms or even mixed. Therefore, for further research, the average waiting queue length formula can be constructed more realistically. At the same time, there remain several important yet unanswered questions, such as the following: (1) Is the construction cost ofseru production higher or lower than that of a conveyor assembly line? (2) Although some studies have shown that the line-seru conversion will save space cost, as the organizational change of conversion process still requires a certain amount of investment, will a line-seru conversion reduce or increase manufacturing costs? (3) Will the labor costs be higher (due to a higher levels of required skills) or lower (because of downsizing and reducing related costs)?



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*Source: 4036794-2019-05-26.xml*