The coupling relationship among bottleneck shifting factors in job shop

Abstract

Aiming at the phenomenon of bottleneck shifting in job shop, this article presents the findings on the coupling relationship among the bottleneck shifting factors. We first defined the chain probability to show the relation that the change of one bottleneck shifting factor causes the changes of other factors in job shop. Then, we used three variables (time-capability, time-load, and quality-assurance) and the transaction probability to describe the changes of the factors. Considering the interaction among the bottleneck shifting factors, we established the independent contributions and comprehensive contributions, showing how the changes of various shifting factors may impact the bottleneck shifting phenomenon interactively. Finally, an instance of analysis of the coupling relation of several bottleneck shifting factors in some job shop is given to test the validity and rationality of the method established in this research.

Introduction

Bottleneck shifting is caused by the changes in the factors of the shop floor, such as machine breakdown, uncertain processing time, and the change in the process capability index of machines. In this article, we define the factors in the shop floor that cause bottleneck to shift as the bottleneck shifting factors. How much the bottleneck shifting factors impact the position of the bottleneck in jobs shop depends on the measurement of bottleneck degree. Most of the research has proposed a lot of ways to detect the bottleneck^1–10 in the time domain, such as machine activity time, machine blocking/starvation time, and the throughput. Liu et al. ¹¹ suggested that the bottleneck is influenced not only by processing/nonprocessing time on machine but also by the quality of the product produced by a machine. Low quality on the machine may cause the machine to be the weakest node in the whole manufacturing system, which is another kind of bottleneck due to product rejection or the cumulation of the errors. Nowadays, little attention has been paid to the factors underlying occurrence of bottleneck. Nakata et al. ¹² presented three kinds of bottlenecks based on the appearance cycles and showed how the reasons (the change of factors) cause different bottlenecks indirectly. Yan et al. ¹⁰ proposed four buffer states of the manufacturing cell to show the relationship between the buffer and the bottleneck.

The rest of this article is organized as follows: Section “The chain probability among bottleneck shifting factors” introduces the chain probability among the changes of bottleneck shifting factors, and when unambiguous, we eliminate “the changes.” Two kinds of chain relations are described: independent chain and mixed chain. Section “The description of the bottleneck shifting factor” provides a method to describe the bottleneck shifting factors, including the state of the factor and the transition from one state to another state. Section “The interaction among bottleneck shifting factors” analyzes the interaction among bottleneck shifting factors. Section “The contribution degree of the bottleneck shifting factor” constructs the contribution degree to measure the effect of the factors on the bottleneck degree. Section “Case studies and results” reports the case studies and the results. The conclusions and the future directions for research are presented in section “Conclusions.”

The chain probability among bottleneck shifting factors

The bottleneck shifting factors do not occur independently, which means that the occurrence of one bottleneck shifting factor may cause the other factors to occur. We define the original factor as the chain subject factor and the consequent factors as the chain object factor. One chain subject factor may cause several chain object factors to occur, and one chain object factor may be caused by several chain subject factors jointly.

Suppose there are $m$ chain subject factors, $s_1,s_2,\dots ,s_m$ . The probabilities of these factors that happen within the time $T$ are $p(s_1),p(s_2),\dots ,p(s_m)$ , respectively. These $m$ chain subject factors cause $k$ chain object factors to occur, $\overline{s_1},\overline{s_2},\dots ,\overline{s_k}$ . The probabilities of the $k$ chain object factors occurring are $p(\overline{s_1}|S),p(\overline{s_2}|S),\dots ,p(\overline{s_m}|S)$ , respectively, and $\sum _{i=1}^{m}p(\overline{s_i}|S)=1$ . $S$ is the set of the chain subject factors. According to the different chain relationships, the calculation of the chain probabilities is conducted in two fashions as follows:

1. Independent chain relationship. The chain subject factors independently cause the chain object factors. The chain probability model is given as follows

   $$p(\overline{s_j}|S)=\sum _{i=1}^{m}p(s_i)p(\overline{s_j}|s_i)$$

   (1)

2. Mixed chain relationship. Some chain subject factors independently cause the chain object factors, while other chain subject factors jointly cause the chain object factors. Suppose the chain object factor $\overline{s_j}$ is independently caused by $q$ chain subject factors and $m-q$ chain subject factors together, then the chain probability model is given as follows

   $$p(\overline{s_j}|S)=\sum _{i=1}^{q}p(s_i)p(\overline{s_j}|s_i)+p(\overline{s_j}|s_{q+1}{s}_{q+2}\dots s_m)$$

   (2)

   In equation (2), $p(s_i)$ , $p(\overline{s_j}|s_i)$ , and $p(\overline{s_j}|s_{q+1}{s}_{q+2}\dots s_m)$ can be determined by the statistics of the occurrence probabilities of the bottleneck shifting factors and the Bayesian network ¹³ that is useful for dealing with the complicated joint probabilities.

The description of the bottleneck shifting factor

The variables of the bottleneck shifting factor

From our previous research basis, ¹¹ the Bottleneck Index, regarded as the capability of the manufacturing cell becoming the bottleneck, reflects the degree that the production capability satisfying the production load (workload) in either the time or the quality domain. In this index, use the time-capability variable and the time-load variable to represent the changes of production capability and production load in the time domain, respectively; use the quality-assurance variable to represent the capability that whether the manufacturing cell can meet the quality requirement. The specific model is shown in the following.

Time-capability variable

This variable denotes the change of the available time in the manufacturing cell induced by the bottleneck shifting factors. For example, $\Delta T_c=t_c$ , where $t_c$ represents the time span from equipment damage to regular working after maintenance.

Time-load variable

This variable denotes the change of the production load in the manufacturing cell in the time domain induced by the bottleneck shifting factors. For example, the arrival of an urgent order causes the additional time-load in the manufacturing cell $j$ during the scheduled time. This additional time-load can be expressed as $t_l=\sum (p_{\mathrm{ij}}{n}_i+F_{\mathrm{ij}}),(i=1,2,\cdots ,m)$ , so $\Delta T_l=t_l$ , where $n_i$ represents the account of the ith product, $p_{\mathrm{ij}}$ represents the unit processing time of the ith product, and $F_{\mathrm{ij}}$ represents the setup time of manufacturing cell $j$ processing the ith product.

Quality-assurance variable

This variable denotes the change of the quality-assurance capability of the manufacturing cell induced by the bottleneck shifting factors. Quality-assurance capability includes two meanings: the ability to ensure qualified products and the ability to ensure the stability of product quality function. Therefore, quality-assurance capability can be characterized by the defect rate and the process capability index synthetically. ¹⁴ For example, the staff rescheduling and the alteration of the process quality standard may lead to the change in quality-assurance capability of the manufacturing cell. In this case, $\Delta Q=\delta (p_2,C_{p_2})-\delta (p_1,C_{p_1})$ , where $p_1,p_2,C_{p_1},\text{and}{C}_{p_2}$ represent the average defect rate and the process capability index of the manufacturing cell before and after the bottleneck shifting factors to occur, respectively.

The state space transition probability of the bottleneck shifting factor

It is shown that the bottleneck shifting factors have randomness and can transfer dynamically among various states with some probability. This means that the transition process of the bottleneck shifting factors, to some extent, is a random process and obeys Poisson’s process. ¹¹ When the current state of the factor is known, their future state only bears on the current state and not on the past state. Therefore, the change process of bottleneck shifting factors has Markov property. Markov chain can be used to describe the dynamic system with the discrete or continuous parameter. Wang et al. ¹⁵ used the Markov chain to describe the transition probability of the system from one state to another state. Through this analysis, the state space of one bottleneck shifting factor can be divided into finite scenarios with the time parameters being continuous. Therefore, a finite dimensional Markov chain with the continuous time parameter can be constructed to describe the state space and transition probability of the bottleneck shifting factors.

The interaction among bottleneck shifting factors

When some bottleneck shifting factors occur at the same time, the comprehensive influence on the production capability/load is not the linear summation of the influence of every single factor but complicated and nonlinear. Some bottleneck shifting factors occur simultaneously, which may have the positive stimulation effect on the production capability or load, while other bottleneck shifting factors may have the negative stimulation effect on the production capability or load.

Suppose there are $n$ bottleneck shifting factors (including the chain subject factor and the chain object factor), $s_1,s_2,\cdots ,s_n$ . Define the set of dependent factors $X=\{s_1,s_2,\cdots ,s_i\}$ as the interaction set, while define the set of the independent factors $Y=\{s_{i+1},s_{i+2},\cdots ,s_n\}$ as the noninteraction set. Define the interaction degree as the ratio of the effect on the production capability/load of the multiple factors over the summation of the effects on that of individual factors

$$\gamma =\frac{\Delta (s_1,\dots ,s_j)}{\sum _{k=1}^j\Delta (s_k)}$$

$\Delta$ represents the change in production capability/load in time or quality domain, induced by the bottleneck factor. If $\{s_1,\dots ,s_j\}\subseteq X$ , then $\gamma =1$ . If $\gamma <1$ , then we call the interaction among the factors as the positive interaction, else if $\gamma >1$ , then we call it as the negative interaction.

There exists time-capability interaction degree ${\gamma }^C$ , time-load interaction degree ${\gamma }^L$ , and quality-assurance interaction degree ${\gamma }^Q$ .

The contribution degree of the bottleneck shifting factor

The contribution degree of the bottleneck shifting factor denotes the effect of the bottleneck shifting factors on the production capability/load, by considering the occurrence probability of the factors, the transition probability of the state spaces, and the interaction among the factors.

Independent contribution degree

The independent contribution degree denotes the degree of the effect of the single bottleneck shifting factor on the production capability and the load, and this degree can be measured by the time-capability–independent contribution degree, time-load–independent contribution degree, and quality-assurance–independent contribution degree from the time and the quality domain, respectively.

Let ${\alpha }_i$ , ${\beta }_i$ , and ${\sigma }_i$ be the influence decision variables.

$$\begin{array}{l}{\alpha }_i=\{\begin{array}{l}(0,1]s_i\text{influences}\text{the}\text{time}\text{capability}\hfill \\ 0s_i\text{does}\text{not}\text{influence}\text{the}\text{time}\text{capability}\hfill \end{array}\hfill \\ {\beta }_i=\{\begin{array}{l}(0,1]s_i\text{influences}\text{the}\text{time}\text{load}\hfill \\ 0s_i\text{does}\text{not}\text{influence}\text{the}\text{time}\text{load}\hfill \end{array}\hfill \\ {\sigma }_i=\{\begin{array}{l}(0,1]s_i\text{influences}\text{the}\text{quality}\text{assurance}\text{capability}\hfill \\ 0s_i\text{does}\text{not}\text{influence}\text{the}\text{quality}\text{assurance}\text{capability}\hfill \end{array}\hfill \end{array}$$

Suppose that the state duration time of the bottleneck shifting factor obeys exponential distribution with ${\lambda }_{\mathrm{si}}$ , and in the time $T_i$ , the change time of $s_i$ is $m_i$ . The independent contribution degree of $s_i$ can be expressed by the occurrence probability, the state transition probability, the expected value of the state duration time of $s_i$ , and the three description variables (time-capability, time-load, and quality-assurance). The specific model is given as follows

$$\begin{array}{l}{C}_{s_i}^I=p(s_i|S)\times {\displaystyle \sum }^n{(-1)}^c(P_{s_i(k-h)}{(t)}^{m_i}\times \Delta T_c)\frac{{\alpha }_i}{{\lambda }_i}\hfill \\ L_{s_i}^I=p(s_i|S)\times {\displaystyle \sum }^n{(-1)}^l(P_{s_i(k-h)}{(t)}^{m_i}\times \Delta T_l)\frac{{\beta }_i}{{\lambda }_i}\hfill \\ Q_{s_i}^I=p(s_i|S)\times {\displaystyle \sum }^n{(-1)}^q(P_{s_i(k-h)}{(t)}^{m_i}\times \Delta Q_c)\frac{{\sigma }_i}{{\lambda }_i}\hfill \end{array}$$

$C_{s_i}^I,L_{s_i}^I,and{Q}_{s_i}^I$ represent the time-capability–independent contribution degree, the time-load–independent contribution degree, and the quality-assurance–independent contribution degree, respectively. $P_{si(k-h)}{(t)}^{mi}$ represents the probability of the bottleneck shifting factor $s_i$ from the state $k$ to the state $h$ at the $m_i$ th time. $p(s_i|S)$ represents the occurrence probability of $s_i$ . If $s_i$ is the chain subject factor, $p(s_i|S)$ is 1, while $s_i$ is the chain object factor, $p(s_i|S)$ is the chain probability. $c,l,\text{and}q$ are natural numbers. The odevity of $c,l,\text{and}q$ can be determined based on the positive or negative effect of $s_i$ on production capability and load. When the production capability/load increases, $c$ is even, whereas when the production capability/load decreases, $c$ is odd. Moreover, $l\text{and}q$ are determined in the same way.

Comprehensive contribution degree

The comprehensive contribution degree denotes the degree of the effect of several bottleneck shifting factors that occur simultaneously on the production capability and the load by considering the interaction among the factors. The specific models of time-capability, the time-load, and the quality-assurance comprehensive contribution degrees are given as follows

$$\begin{array}{c}C=\mathrm{fc}(C_{s_1}^I,C_{s_2}^I,\cdots ,C_{s_n}^I,{\gamma }^C)\hfill \\ L=\mathrm{fl}(L_{s_1}^I,L_{s_2}^I,\cdots ,L_{s_n}^I,{\gamma }^L)\hfill \\ Q=\mathrm{fq}(Q_{s_1}^I,Q_{s_2}^I,\cdots ,Q_{s_n}^I,{\gamma }^Q)\hfill \end{array}$$

The solution of the comprehensive contribution degree

The occurrence of the bottleneck shifting factors has the parallelism property and the cross property in the time domain, and the comprehensive contribution degree of these factors is not the linear summation of each independent contribution degrees. Therefore, the comprehensive contribution function is unable to be created as a unique analysis model for predicting the comprehensive contribution.

The neural network can approximate any continuous function and obtain a good approximation result. It has strong fault-tolerance and self-learning function. ¹⁶ Therefore, adopt the neural network to solve the contribution function, which can realize the prediction of the comprehensive contribution degree.

The solution of the contribution function is a one-way flow process, and the factors of the same layer are independent. According to this characteristic, the feed-forward neural network should be selected. In the feed-forward neural network, the back-propagation (BP) neural network can achieve any continuous mapping with strong nonlinear mapping capability and flexible network structure, and the prediction accuracy is higher. ¹⁷ The BP neural network with single hidden layer can approximate any nonlinear function and obtain good approximation results. Through large sample training of the network, its evaluation error is smaller and extrapolating ability is stronger. ¹⁶ Therefore, for the comprehensive contribution function, two 3-layer BP neural networks should be constructed. The number of input layer neurons is the number of parameters of ability contribution function, denoted by $n$ . The number of output layer neurons is 1 and the number of hidden layer neurons can be calculated using the following three formulas

1. $\sum _{i=0}^n{C}_{n_2}^i>k$ , where $k$ is the number of samples and $n_2$ is the number of hidden units. If $i>n_2$ , then $C_{n_2}^i=0$ ;

2. $n_2=\sqrt{n+m}+a$ , where $m$ is the number of output neurons and $a$ is the constant between [1, 10];

3. $n_2={\log }_{2}n$ .

The sigmoid function is used as the network excitation function to calculate the output of the hidden layer and output layer, in order to realize the solution of the contribution function. The BP network works with the error BP mode. If the inaccuracy between the output of output layer $C_p$ and expected output $C_p^{*}$ does not satisfy the accuracy requirements, the BP should be learnt to the input layer and the updated weight should be introduced to make the error function change toward the negative gradient direction until the inaccuracy between $C_p$ and $C_p^{*}$ meets the inaccuracy requirement. Then use the obtained parameters to establish the mathematical model and then predict the unknown samples. The final parameters include connection weights of the output layer and hidden layer, connection weights of hidden layer and output layer, as well as the threshold value of hidden layer and output layer.

Before training, in order to improve the performance of the neural network model, the sample data must be chosen seriously to ensure the quality and quantity. Simultaneity, in order to ensure that the sigmoid function really play a nonlinear transfer, to guarantee that the network has enough input sensitivity and good fitting property, to improve the convergence speed of the network, and to increase the stability of the model, the data should be normalized, so that it falls between [0, 1] or [−1, 1].

Case studies and results

This section describes an example of a manufacturing shop where five products A, B, C, D, and E are processed on five machines $M_i(i=1,\dots ,5)$ to verify the correctness and practicability of the correlated theory. The process routes of five products are as follows: [1-2-3-4-5], [5-4-3-2-1], [3-2-1-4], [4-5-3], and [2-4-1-3]. Suppose there is no unnecessary inserted idle time during processing.

Our first case was designed to show that the result is only one chain subject factor in the shop floor, without any chain object factor. Let $s_1$ be the only one chain subject factor—uncertain processing time. Here, assume a set of discrete processing time scenarios, ¹⁸ each of which specifies the processing time of each job on each machine under that scenario. Suppose there are $|\Lambda |=4,8,\text{and}12$ processing time scenarios, the processing times for each job (A, B, C, D, and E) on each machine $M_i(i=1,\dots ,5)$ were randomly generated for each scenario $\lambda =1,2,\dots ,|\Lambda |$ from a uniform distribution of integers on the interval $[10{\beta }_i,(10+40\alpha ){\beta }_i]$ , where $\alpha$ is a parameter that allows the variability of processing time across jobs in a given problem instance to be controlled, and ${\beta }_i$ represents the relative processing requirements on machine $M_i$ . Three values of $\alpha$ (α = 0.2, 0.6, and 1.0) and five vectors representing the relative processing time requirements on machine $M_i$ , ${\beta }_i=1.0,i=1,\dots ,5$ , which is for avoiding the influence of different machines, were included in the case design. The transition probability between the states of processing time ${\delta }_{s_i(\mathrm{ij})}1\le i,j\le |\Lambda |$ is generated from the uniform distribution [0, 1]. The contribution degree results for each machine in each scenario are given in Table 1. These results show that the uncertain processing time has no contribution to the time-capability and quality-assurance but has the contribution only to the time-load. The more the state (scenario) changes, the greater the contribution. The greater the range of processing time, the greater the contribution. It is also clear from Table 1 that the high usage of machine (based on the route of products) has the positive effect on the contribution degree (only one exception in the case).

Table 1.

The contribution degree results of uncertain processing time.

$|\Lambda |$	$\alpha$	M 1			M 2			M 3			M 4			M 5
		C	L	Q	C	L	Q	C	L	Q	C	L	Q	C	L	Q
4	0.2	0	5	0	0	7	0	0	7	0	0	11	0	0	4	0
	0.6	0	15	0	0	21	0	0	23	0	0	26	0	0	13	0
	1.0	0	26	0	0	29	0	0	30	0	0	36	0	0	23	0
8	0.2	0	9	0	0	13	0	0	13	0	0	17	0	0	6	0
	0.6	0	16	0	0	22	0	0	22	0	0	25	0	0	16	0
	1.0	0	29	0	0	34	0	0	37	0	0	37	0	0	27	0
12	0.2	0	10	0	0	11	0	0	14	0	0	16	0	0	11	0
	0.6	0	24	0	0	24	0	0	27	0	0	31	0	0	20	0
	1.0	0	37	0	0	40	0	0	43	0	0	45	0	0	36	0

To further test the method, add one bottleneck shifting factor $s_2$ —the machine performance degradation is consecutive. This factor follows $\log y=-u-\upsilon \log t$ , where y is the process capability index and t is the cumulative time on the machine.^19,20 Let u be 0.1 and $[v_i]=[0.01,0.02,0.03,0.04,\text{and}0.05]$ . Table 2 shows the contribution results if only $s_2$ occurs, and if $s_1$ and $s_2$ occur simultaneously, the contribution degree is shown in Table 2. The results from Table 2 show that the machine performance degradation has the contribution to all because $s_2$ may make the machine speed slow and process capability index decrease and then increase production load due to increasing defect rate. MATLAB 7.0 (MathWorks) is applied to construct two 3-layer BP neural networks for each manufacturing cell to solve the contribution function and then ultimately predicted the comprehensive contribution degree of bottleneck shifting factors. The number of input nodes in each network is 6, the number of output nodes is 1, and the number of hidden layer nodes is 6. During training, use Tremnmx function to preprocess the sample data first and let the value of independent contribution degree (the value in Table 1 and the value in the first part of Table 2) be in the interval [−1, 1]. After the training, use obtained BP neural network to simulate based on independent contribution degree. Then use the pretreatment function Premnmx to preprocess the data to get the comprehensive contribution degree of the bottleneck shifting factors in the production capability, production load, and quality-assurance that is shown in the second part of Table 2. From the results of comprehensive contribution degree, $s_1$ and $s_2$ have no interaction in time-capability, while they have interaction in time-load and quality-assurance. The greater the v and $|\Lambda |$ , the more the interaction. It shows that the interaction (degree) in time-load is greater than that in quality-assurance with just 5% exceptions.

Table 2.

The contribution degree results of two factors (s₁ and s₂).

s 2			M 1			M 2			M 3			M 4			M 5
			[u, v] = [0.1, 0.01]			[u, v] = [0.1, 0.02]			[u, v] = [0.1, 0.03]			[u, v] = [0.1, 0.04]			[u, v] = [0.1, 0.05]
			$C_{s_2}^I$	$L_{s_2}^I$	$Q_{s_2}^I$	$C_{s_2}^I$	$L_{s_2}^I$	$Q_{s_2}^I$	$C_{s_2}^I$	$L_{s_2}^I$	$Q_{s_2}^I$	$C_{s_2}^I$	$L_{s_2}^I$	$Q_{s_2}^I$	$C_{s_2}^I$	$L_{s_2}^I$	$Q_{s_2}^I$
			3	4	15	5	7	19	8	11	25	14	21	34	21	27	41
s1+s2	$\left|\Lambda \right|$	$\alpha$	C	L	Q	C	L	Q	C	L	Q	C	L	Q	C	L	Q
	4	0.2	3	11	17	5	17	24	8	19	25	14	33	35	21	35	45
		0.6	3	23	21	5	32	29	8	34	27	14	51	45	21	45	47
		1.0	3	31	23	5	43	31	8	45	32	14	64	56	21	57	51
	8	0.2	3	16	19	5	24	26	8	27	30	14	42	39	21	41	47
		0.6	3	25	24	5	37	33	8	37	37	14	49	47	21	53	52
		1.0	3	39	27	5	50	40	8	51	41	14	63	63	21	62	53
	12	0.2	3	20	21	5	29	26	8	32	35	14	41	42	21	45	50
		0.6	3	37	29	5	42	34	8	42	46	14	57	51	21	61	55
		1.0	3	54	31	5	60	43	8	59	50	14	74	71	21	73	59

Consider to add the third bottleneck shifting factor $s_3$ —the change in the order of product A, as a completed case. The number of product A is generated from the uniform distribution $[10,50\phi ]$ and there are four values of $\phi$ ( $\phi =0.4,0.6,0.8,\text{and}1.0$ ). Due to the factor $s_3$ , the material shortage as the fourth factor $\overline{s_3}$ may occur, which is chain object factor of the chain subject factor $s_3$ . Through the simulation by Arena 11.0 (manufactured by System Modeling) with considering only the change of the order of product A, we can obtain the chain probability $p(\overline{s_3}|s_3)=0.23,0.55,0.79,\text{and}0.93$ , respectively, when ϕ = 0.4, 0.6, 0.8, and 1.0. The contribution degree of $s_3$ with $\overline{s_3}$ is shown in Table 3. From Table 3, $s_3$ with $\overline{s_3}$ has no contribution on the quality-assurance, but it has contribution to time-capability/load and furthermore, the contribution to the capability is greater than that to the load. Table 4 shows the comprehensive contribution degree of three factors with one chain object factor. Comparing Table 2, it is clear that $s_2$ and $s_3$ have positive interaction in the time-capability and $s_3$ has no interaction with $s_1$ and $s_2$ in the quality-assurance. The contribution of $s_3$ is almost independent with that of $s_2$ to the capability (92% data show that there is no interaction between $s_2$ and $s_3$ in the time-capability).

Table 3.

The contribution degree results of two factors (s₃ with $\overline{s_3}$ ).

$\phi$	M 1			M 2			M 3			M 4			M 5
	C	L	Q	C	L	Q	C	L	Q	C	L	Q	C	L	Q
0.4	35	44	0	31	44	0	34	46	0	35	47	0	31	41	0
0.6	54	67	0	54	69	0	55	70	0	54	67	0	54	66	0
0.8	79	91	0	75	90	0	76	90	0	79	89	0	75	93	0
1.0	101	124	0	103	127	0	101	123	0	100	124	0	101	125	0

Table 4.

The contribution degree results of three factors (with one chain object factor).

				M 1			M 2			M 3			M 4			M 5
$s_1+s_2+s_3+\overline{s_3}$	$|\Lambda |$	$\alpha$	$\phi$	[u, v] = [0.1, 0.01]			[u, v] = [0.1, 0.02]			[u, v] = [0.1, 0.03]			[u, v] = [0.1, 0.04]			[u, v] = [0.1, 0.05]
				C	L	Q	C	L	Q	C	L	Q	C	L	Q	C	L	Q
	4	0.2	0.4	38	59	17	37	63	24	44	65	25	41	75	35	55	78	45
			0.6	59	80	21	60	87	29	65	91	27	71	104	45	79	103	47
			0.8	85	91	23	83	109	31	91	112	32	95	124	56	99	131	51
			1.0	108	141	19	111	145	26	114	145	30	119	161	39	127	162	47
		0.6	0.4	38	69	24	37	79	33	44	84	37	41	93	47	55	88	52
			0.6	59	93	27	62	109	40	66	107	41	72	124	63	79	113	53
			0.8	87	95	21	83	128	26	91	130	35	95	143	42	99	141	50
			1.0	108	162	29	114	165	34	115	163	46	117	179	51	129	171	55
		1.0	0.4	38	79	31	38	91	43	44	98	50	43	107	71	56	100	59
			0.6	61	103	17	64	119	24	64	119	25	71	144	35	79	126	45
			0.8	85	107	21	86	139	29	91	138	27	95	167	45	100	157	47
			1.0	108	171	23	112	175	31	114	173	32	119	193	56	127	190	51
	8	0.2	0.4	38	61	17	37	69	24	46	77	25	41	84	35	56	88	45
			0.6	59	83	21	61	95	29	65	101	27	71	114	45	79	111	47
			0.8	85	94	23	86	119	31	91	121	32	95	137	56	98	141	51
			1.0	109	145	19	110	156	26	115	155	30	117	169	39	127	170	47
		0.6	0.4	38	71	24	37	84	33	44	85	37	43	93	47	55	97	52
			0.6	59	95	27	60	109	40	65	107	41	71	126	63	77	121	53
			0.8	84	97	21	83	131	26	94	129	35	95	147	42	99	151	50
			1.0	108	165	29	111	169	34	114	162	46	117	179	51	128	181	55
		1.0	0.4	38	82	31	37	98	43	45	99	50	42	109	71	55	105	59
			0.6	59	107	17	61	121	24	65	127	25	71	141	35	78	131	45
			0.8	85	111	21	83	143	29	91	145	27	95	158	45	99	159	47
			1.0	108	177	23	111	179	31	114	177	32	121	191	56	127	192	51
	12	0.2	0.4	38	65	17	37	74	24	44	81	25	41	86	35	54	91	45
			0.6	59	86	21	60	103	29	63	106	27	71	117	45	79	113	47
			0.8	85	97	23	83	123	31	91	128	32	96	141	56	99	142	51
			1.0	109	149	19	110	161	26	114	161	30	117	171	39	126	173	47
		0.6	0.4	37	74	24	37	88	33	44	91	37	41	103	47	55	105	52
			0.6	59	96	27	60	114	40	65	115	41	72	138	63	79	131	53
			0.8	85	99	21	83	138	26	90	138	35	96	153	42	99	157	50
			1.0	108	167	29	111	177	34	114	170	46	117	187	51	127	189	55
		1.0	0.4	38	85	31	37	107	43	44	107	50	41	117	71	55	117	59
			0.6	59	111	17	60	131	24	65	131	25	71	151	35	80	142	45
			0.8	85	118	21	85	154	29	91	155	27	95	169	45	99	170	47
			1.0	108	179	23	114	190	31	114	190	32	119	201	56	128	204	51

Use Arena 11.0 to simulate the condition of the shop floor in order to verify the rationality of this method. Because there is few methods to obtain the value of the production capability and quality-assurance, in this case, just test and obtain the production load. In Table 5, the first subcolumn in each machine column is the production load (unit: seconds) without any bottleneck shifting factors, the second subcolumn is the load under the condition described above, and the third subcolumn is the relative error between the comparison results and the computation results as shown in Table 4. It is clear that all relative errors do not exceed 13%; therefore, the method proposed in this article is effective. The uncertain processing time has the most effect on the accuracy of this method. The more the number of scenarios in the processing time, the greater the value of the relative error.

Table 5.

The comparison between the simulation results and the computation results.

Usage time (time-load)	$|\Lambda |$	$\alpha$	$\phi$	M 1			M 2			M 3			M 4			M 5
				Non	$s_1+s_2+s_3+\overline{s_3}$	Error %	Non	$s_1+s_2+s_3+\overline{s_3}$	Error %	Non	$s_1+s_2+s_3+\overline{s_3}$	Error %	Non	$s_1+s_2+s_3+\overline{s_3}$	Error %	Non	$s_1+s_2+s_3+\overline{s_3}$	Error %
	4	0.2	0.4	1356	1415	6.78	1472	1535	3.17	1524	1589	3.08	1678	1753	2.67	1278	1356	2.56
			0.6	1356	1436	5.00	1472	1559	5.75	1524	1615	3.30	1678	1782	3.85	1278	1381	2.91
			0.8	1356	1447	6.59	1472	1581	3.67	1524	1636	1.79	1678	1802	4.03	1278	1409	3.82
			1.0	1356	1497	7.09	1472	1617	2.07	1524	1669	2.76	1678	1839	2.48	1278	1440	3.09
		0.6	0.4	1356	1425	5.80	1472	1551	2.53	1524	1608	3.57	1678	1771	4.30	1278	1366	2.27
			0.6	1356	1449	4.30	1472	1581	3.67	1524	1631	3.74	1678	1802	4.84	1278	1391	3.54
			0.8	1356	1451	6.32	1472	1600	3.91	1524	1654	3.08	1678	1821	4.20	1278	1419	3.55
			1.0	1356	1518	4.94	1472	1637	3.03	1524	1687	2.45	1678	1857	3.35	1278	1449	4.68
		1.0	0.4	1356	1435	6.33	1472	1563	3.30	1524	1622	4.08	1678	1785	4.67	1278	1378	2.00
			0.6	1356	1459	3.88	1472	1591	4.20	1524	1643	4.20	1678	1822	2.78	1278	1404	3.97
			0.8	1356	1463	6.54	1472	1611	4.32	1524	1662	2.90	1678	1845	3.59	1278	1435	3.82
			1.0	1356	1527	5.26	1472	1647	3.43	1524	1697	3.47	1678	1871	3.11	1278	1468	4.21
	8	0.2	0.4	1356	1417	8.20	1472	1541	5.80	1524	1601	5.19	1678	1762	5.95	1278	1366	5.68
			0.6	1356	1439	8.43	1472	1567	5.26	1524	1625	4.95	1678	1792	5.26	1278	1389	6.31
			0.8	1356	1450	7.45	1472	1591	5.88	1524	1645	4.96	1678	1815	5.11	1278	1419	5.67
			1.0	1356	1501	6.90	1472	1628	5.13	1524	1679	5.81	1678	1847	6.51	1278	1448	5.88
		0.6	0.4	1356	1427	7.04	1472	1556	5.95	1524	1609	5.88	1678	1771	5.38	1278	1375	5.15
			0.6	1356	1451	6.32	1472	1581	6.42	1524	1631	6.54	1678	1804	7.14	1278	1399	5.79
			0.8	1356	1453	8.25	1472	1603	5.34	1524	1653	6.20	1678	1825	6.12	1278	1429	6.62
			1.0	1356	1521	6.67	1472	1641	5.92	1524	1686	6.17	1678	1857	6.15	1278	1459	6.63
		1.0	0.4	1356	1438	6.10	1472	1570	7.14	1524	1623	6.06	1678	1787	7.34	1278	1383	5.71
			0.6	1356	1463	6.54	1472	1593	6.61	1524	1651	7.09	1678	1819	7.80	1278	1409	6.11
			0.8	1356	1467	7.21	1472	1615	6.29	1524	1669	6.90	1678	1836	7.59	1278	1437	6.92
			1.0	1356	1533	8.47	1472	1651	6.70	1524	1701	6.78	1678	1869	7.85	1278	1470	7.29
	12	0.2	0.4	1356	1421	9.23	1472	1546	8.11	1524	1605	7.41	1678	1764	11.63	1278	1369	8.79
			0.6	1356	1442	10.47	1472	1575	8.74	1524	1630	10.38	1678	1795	11.11	1278	1391	8.85
			0.8	1356	1453	7.22	1472	1595	8.94	1524	1652	8.59	1678	1819	12.06	1278	1420	9.86
			1.0	1356	1505	7.38	1472	1633	9.94	1524	1685	7.45	1678	1849	14.62	1278	1451	10.40
		0.6	0.4	1356	1430	9.46	1472	1560	9.09	1524	1615	10.99	1678	1781	11.65	1278	1383	9.52
			0.6	1356	1452	10.42	1472	1586	10.53	1524	1639	8.70	1678	1816	9.42	1278	1409	9.92
			0.8	1356	1455	10.10	1472	1610	10.14	1524	1662	9.42	1678	1831	9.80	1278	1435	10.19
			1.0	1356	1523	9.58	1472	1649	9.60	1524	1694	10.00	1678	1865	11.23	1278	1467	11.11
		1.0	0.4	1356	1441	8.24	1472	1579	11.21	1524	1631	8.41	1678	1795	10.26	1278	1395	9.40
			0.6	1356	1467	6.31	1472	1603	9.16	1524	1655	9.16	1678	1829	11.92	1278	1420	11.27
			0.8	1356	1474	10.17	1472	1626	9.74	1524	1679	10.97	1678	1847	12.43	1278	1448	11.76
			1.0	1356	1535	10.61	1472	1662	9.47	1524	1714	12.63	1678	1879	12.94	1278	1482	12.25

Non: without any factors.

Conclusions

This article has focused on the coupling relationship among the bottleneck shifting factors in job shop. We have constructed the contribution degree to measure the effect of the bottleneck shifting factors on the production capability, the production load, and the quality-assurance that are the three main parts in measuring and detecting the production bottleneck. ¹¹ It is a useful method to show the relationship between the bottleneck shifting factors and the phenomenon of bottleneck shifting.

The change of the processing time only effects time-capability (only one aspect). The machine performance degradation affects all three aspects and has the interaction with the processing time in all three aspects. The change of the order of product has no effect on the quality-assurance. It has no interaction with the machine performance degradation in the time-capability and the quality-assurance. Further research should explore more bottleneck shifting factors and the more explicit relationship between the changes in factors and the bottleneck degree.