Analytical Modeling and Performance Evaluation of Multi-stage Assembly Lines with Line-Side Buffers

doi:10.12068/j.issn.1005-3026.2025.20240193

Abstract

Abstract:

The output performance of assembly lines is not only affected by machine unreliability and limited buffer capacity but also constrained by line-side buffers. The analytical modeling and performance evaluation of multi-stage assembly lines with line-side buffers were investigated. Firstly， for the single-stage assembly lines， the steady-state probability distribution of system states was derived based on Markov chains. Secondly， for the two-stage assembly lines， each single-stage subsystem was modeled as a machine with one operational state and one failure state. A performance evaluation model was then established using Markov chains， and closed-form expressions for performance indicators were obtained. Thirdly， for the multi-stage assembly lines， an aggregation method was proposed to approximate the performance indicators. Furthermore， the accuracy of the performance evaluation method was validated through numerical experiments. Finally， utilizing the proposed method， numerical experiments were conducted to examine system properties， such as reversibility and monotonicity in the multi-stage assembly lines.

Key words: multi-stage assembly lines, line-side buffer, analytical modeling, performance evaluation, Markov chain

CLC Number:

TH 166

Peng-hao CUI, Qi-man ZHANG, Zhong-zhong JIANG, Guo-jun SHENG. Analytical Modeling and Performance Evaluation of Multi-stage Assembly Lines with Line-Side Buffers[J]. Journal of Northeastern University(Natural Science), 2025, 46(7): 71-83.

Figures/Tables 17

Fig.1 Multi-stage assembly line with line-side buffers

Fig.2 System state transition diagram

Fig.3 Illustration for the equivalent process of the single-stage assembly line

Fig.4 Iterative process diagram of the aggregation method

Table 1 Accuracy analysis results of production rate (Ppr)

指标	$M$
指标	2	4	6	8	10	12
$P p r A n a$	0.786 2	0.747 4	0.727 4	0.717 3	0.710 3	0.705 9
$P p r S i m$	0.784 4	0.746 4	0.724 7	0.712 9	0.704 3	0.698 6
$δ P p r$ /%	0.27	0.92	1.26	1.50	1.70	1.82

Table 1 Accuracy analysis results of production rate (Ppr)

指标	$M$
指标	2	4	6	8	10	12
$P p r A n a$	0.786 2	0.747 4	0.727 4	0.717 3	0.710 3	0.705 9
$P p r S i m$	0.784 4	0.746 4	0.724 7	0.712 9	0.704 3	0.698 6
$δ P p r$ /%	0.27	0.92	1.26	1.50	1.70	1.82

Table 2 Accuracy analysis results of work in process level (Wb)

指标	$M$
指标	2	4	6	8	10	12
$W b A n a$	3.204 9	3.325 8	3.334 4	3.334 6	3.350 0	3.359 7
$W b S i m$	3.201 0	3.324 7	3.332 1	3.329 4	3.334 8	3.330 2
$δ W b$ /%	1.08	2.35	3.37	4.12	4.43	4.81

Table 2 Accuracy analysis results of work in process level (Wb)

指标	$M$
指标	2	4	6	8	10	12
$W b A n a$	3.204 9	3.325 8	3.334 4	3.334 6	3.350 0	3.359 7
$W b S i m$	3.201 0	3.324 7	3.332 1	3.329 4	3.334 8	3.330 2
$δ W b$ /%	1.08	2.35	3.37	4.12	4.43	4.81

Table 3 Accuracy analysis results of inventory level in the line-side buffers (Wl)

指标	$M$
指标	2	4	6	8	10	12
$W l A n a$	4.944 2	4.595 7	4.550 4	4.571 7	4.550 0	4.560 8
$W l S i m$	4.943 1	4.637 2	4.592 0	4.614 0	4.592 6	4.603 7
$δ W l$ /%	0.02	0.84	0.86	0.88	0.89	0.89

Table 3 Accuracy analysis results of inventory level in the line-side buffers (Wl)

指标	$M$
指标	2	4	6	8	10	12
$W l A n a$	4.944 2	4.595 7	4.550 4	4.571 7	4.550 0	4.560 8
$W l S i m$	4.943 1	4.637 2	4.592 0	4.614 0	4.592 6	4.603 7
$δ W l$ /%	0.02	0.84	0.86	0.88	0.89	0.89

Table 4 Accuracy analysis results of blockage probability (Pbl)

指标	$M$
指标	2	4	6	8	10	12
$P b l A n a$	0.062 2	0.050 6	0.060 8	0.065 8	0.069 8	0.072 5
$P b l S i m$	0.064 3	0.052 7	0.063 6	0.069 2	0.073 6	0.076 5
$δ P b l$	0.002 1	0.006 3	0.009 6	0.012 1	0.013 4	0.014 6

Table 4 Accuracy analysis results of blockage probability (Pbl)

指标	$M$
指标	2	4	6	8	10	12
$P b l A n a$	0.062 2	0.050 6	0.060 8	0.065 8	0.069 8	0.072 5
$P b l S i m$	0.064 3	0.052 7	0.063 6	0.069 2	0.073 6	0.076 5
$δ P b l$	0.002 1	0.006 3	0.009 6	0.012 1	0.013 4	0.014 6

Table 5 Accuracy analysis results of starvation probability (Pst)

指标	$M$
指标	2	4	6	8	10	12
$P s t A n a$	0.058 4	0.051 3	0.060 2	0.065 2	0.069 1	0.071 0
$P s t S i m$	0.060 5	0.053 2	0.063 2	0.069 2	0.073 8	0.076 7
$δ P s t$	0.002 1	0.006 5	0.009 6	0.011 9	0.013 3	0.014 5

Table 5 Accuracy analysis results of starvation probability (Pst)

指标	$M$
指标	2	4	6	8	10	12
$P s t A n a$	0.058 4	0.051 3	0.060 2	0.065 2	0.069 1	0.071 0
$P s t S i m$	0.060 5	0.053 2	0.063 2	0.069 2	0.073 8	0.076 7
$δ P s t$	0.002 1	0.006 5	0.009 6	0.011 9	0.013 3	0.014 5

Fig.5 Diagram of multi-stage assembly line and its reverse line

Table 6 Parameters of three-stage assembly lines

编号	概率参数	线边零件库存控制参数
$L 0$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 8,8, 8, s = [4,4, 4]$
$L 1$	$p = 0.95,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 8,8, 8, s = [4,4, 4]$
$L 2$	$p = 0.80,0.95,0.80, γ = [0.96,0.96,0.96]$	$S = 8,8, 8, s = [4,4, 4]$
$L 3$	$p = 0.80,0.80,0.95, γ = [0.96,0.96,0.96]$	$S = 8,8, 8, s = [4,4, 4]$
$L 4$	$p = 0.80,0.80,0.80, γ = [0.98,0.96,0.96]$	$S = 8,8, 8, s = [4,4, 4]$
$L 5$	$p = 0.80,0.80,0.80, γ = [0.96,0.98,0.96]$	$S = 8,8, 8, s = [4,4, 4]$
$L 6$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.98]$	$S = 8,8, 8, s = [4,4, 4]$
$L 7$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 9,8, 8, s = [4,4, 4]$
$L 8$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 8,9, 8, s = [4,4, 4]$
$L 9$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 8,8, 9, s = [4,4, 4]$
$L 10$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 8,8, 8, s = [5,4, 4]$
$L 11$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 8,8, 8, s = [4,5, 4]$
$L 12$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 8,8, 8, s = [4,4, 5]$

Table 6 Parameters of three-stage assembly lines

编号	概率参数	线边零件库存控制参数
$L 0$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 8,8, 8, s = [4,4, 4]$
$L 1$	$p = 0.95,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 8,8, 8, s = [4,4, 4]$
$L 2$	$p = 0.80,0.95,0.80, γ = [0.96,0.96,0.96]$	$S = 8,8, 8, s = [4,4, 4]$
$L 3$	$p = 0.80,0.80,0.95, γ = [0.96,0.96,0.96]$	$S = 8,8, 8, s = [4,4, 4]$
$L 4$	$p = 0.80,0.80,0.80, γ = [0.98,0.96,0.96]$	$S = 8,8, 8, s = [4,4, 4]$
$L 5$	$p = 0.80,0.80,0.80, γ = [0.96,0.98,0.96]$	$S = 8,8, 8, s = [4,4, 4]$
$L 6$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.98]$	$S = 8,8, 8, s = [4,4, 4]$
$L 7$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 9,8, 8, s = [4,4, 4]$
$L 8$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 8,9, 8, s = [4,4, 4]$
$L 9$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 8,8, 9, s = [4,4, 4]$
$L 10$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 8,8, 8, s = [5,4, 4]$
$L 11$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 8,8, 8, s = [4,5, 4]$
$L 12$	$p = 0.80,0.80,0.80, γ = [0.96,0.96,0.96]$	$S = 8,8, 8, s = [4,4, 5]$

Fig.6 Reversibility of three-stage assembly lines

Table 7 Parameters of seven-stage assembly lines

编号

概率参数

线边零件库存控制参数

L 13

$p = [0.80,0.80,0.80,0.80,0.80,0.80,0.80]$

$γ = [0.90,0.90,0.90,0.90,0.90,0.90,0.90]$

$S = [9,9, 9,9, 9,9, 9]$

$s = [5,5, 5,5, 5,5, 5]$

L 14

$p = [0.80,0.80,0.80,0.83,0.88,0.89,0.81]$

$γ = [0.90,0.90,0.90,0.90,0.90,0.90,0.90]$

$S = [9,9, 9,9, 9,9, 9]$

$s = [5,5, 5,5, 5,5, 5]$

L 15

$p = [0.81,0.89,0.88,0.83,0.80,0.80,0.80]$

$γ = [0.90,0.90,0.90,0.90,0.90,0.90,0.90]$

$S = [9,9, 9,9, 9,9, 9]$

$s = [5,5, 5,5, 5,5, 5]$

L 16

$p = [0.80,0.80,0.80,0.80,0.80,0.80,0.80]$

$γ = [0.90,0.90,0.90,0.91,0.93,0.95,0.91]$

$S = [9,9, 9,9, 9,9, 9]$

$s = [5,5, 5,5, 5,5, 5]$

L 17

$p = [0.80,0.80,0.80,0.80,0.80,0.80,0.80]$

$γ = [0.91,0.95,0.93,0.91,0.90,0.90,0.90]$

$S = [9,9, 9,9, 9,9, 9]$

$s = [5,5, 5,5, 5,5, 5]$

L 18

$p = [0.80,0.80,0.80,0.80,0.80,0.80,0.80]$

$γ = [0.90,0.90,0.90,0.90,0.90,0.90,0.90]$

$S = [9,9, 9,10,12,11,15]$

$s = [5,5, 5,5, 5,5, 5]$

L 19

$p = [0.80,0.80,0.80,0.80,0.80,0.80,0.80]$

$γ = [0.90,0.90,0.90,0.90,0.90,0.90,0.90]$

$S = [15,11,12,10,9, 9,9]$

$s = [5,5, 5,5, 5,5, 5]$

L 20

$p = [0.80,0.80,0.80,0.80,0.80,0.80,0.80]$

$γ = [0.90,0.90,0.90,0.90,0.90,0.90,0.90]$

$S = [9,9, 9,9, 9,9, 9]$

$s = [5,5, 5,7, 6,8, 8]$

L 21

$p = [0.80,0.80,0.80,0.80,0.80,0.80,0.80]$

$γ = [0.90,0.90,0.90,0.90,0.90,0.90,0.90]$

$S = [9,9, 9,9, 9,9, 9]$

$s = [8,8, 6,7, 5,5, 5]$

Table 7 Parameters of seven-stage assembly lines

编号

概率参数

线边零件库存控制参数

L 13

$p = [0.80,0.80,0.80,0.80,0.80,0.80,0.80]$

$γ = [0.90,0.90,0.90,0.90,0.90,0.90,0.90]$

$S = [9,9, 9,9, 9,9, 9]$

$s = [5,5, 5,5, 5,5, 5]$

L 14

$p = [0.80,0.80,0.80,0.83,0.88,0.89,0.81]$

$γ = [0.90,0.90,0.90,0.90,0.90,0.90,0.90]$

$S = [9,9, 9,9, 9,9, 9]$

$s = [5,5, 5,5, 5,5, 5]$

L 15

$p = [0.81,0.89,0.88,0.83,0.80,0.80,0.80]$

$γ = [0.90,0.90,0.90,0.90,0.90,0.90,0.90]$

$S = [9,9, 9,9, 9,9, 9]$

$s = [5,5, 5,5, 5,5, 5]$

L 16

$p = [0.80,0.80,0.80,0.80,0.80,0.80,0.80]$

$γ = [0.90,0.90,0.90,0.91,0.93,0.95,0.91]$

$S = [9,9, 9,9, 9,9, 9]$

$s = [5,5, 5,5, 5,5, 5]$

L 17

$p = [0.80,0.80,0.80,0.80,0.80,0.80,0.80]$

$γ = [0.91,0.95,0.93,0.91,0.90,0.90,0.90]$

$S = [9,9, 9,9, 9,9, 9]$

$s = [5,5, 5,5, 5,5, 5]$

L 18

$p = [0.80,0.80,0.80,0.80,0.80,0.80,0.80]$

$γ = [0.90,0.90,0.90,0.90,0.90,0.90,0.90]$

$S = [9,9, 9,10,12,11,15]$

$s = [5,5, 5,5, 5,5, 5]$

L 19

$p = [0.80,0.80,0.80,0.80,0.80,0.80,0.80]$

$γ = [0.90,0.90,0.90,0.90,0.90,0.90,0.90]$

$S = [15,11,12,10,9, 9,9]$

$s = [5,5, 5,5, 5,5, 5]$

L 20

$p = [0.80,0.80,0.80,0.80,0.80,0.80,0.80]$

$γ = [0.90,0.90,0.90,0.90,0.90,0.90,0.90]$

$S = [9,9, 9,9, 9,9, 9]$

$s = [5,5, 5,7, 6,8, 8]$

L 21

$p = [0.80,0.80,0.80,0.80,0.80,0.80,0.80]$

$γ = [0.90,0.90,0.90,0.90,0.90,0.90,0.90]$

$S = [9,9, 9,9, 9,9, 9]$

$s = [8,8, 6,7, 5,5, 5]$

Fig.7 Reversibility of seven-stage assembly lines

Table 8 Efficiency parameters ranges and combination of three-stage assembly lines

编号	$p 1$	$p 2$	$p 3$	$γ 1$	$γ 2$	$γ 3$
1	[0.45,0.51]	0.50	0.48	0.53	0.60	0.54
2	0.45	[0.50,0.59]	0.48	0.53	0.60	0.54
3	0.45	0.50	[0.48,0.53]	0.53	0.60	0.54
4	0.45	0.50	0.48	[0.53,0.63]	0.60	0.54
5	0.45	0.50	0.48	0.53	[0.60,0.70]	0.54
6	0.45	0.50	0.48	0.53	0.60	[0.54,0.67]

Table 8 Efficiency parameters ranges and combination of three-stage assembly lines

编号	$p 1$	$p 2$	$p 3$	$γ 1$	$γ 2$	$γ 3$
1	[0.45,0.51]	0.50	0.48	0.53	0.60	0.54
2	0.45	[0.50,0.59]	0.48	0.53	0.60	0.54
3	0.45	0.50	[0.48,0.53]	0.53	0.60	0.54
4	0.45	0.50	0.48	[0.53,0.63]	0.60	0.54
5	0.45	0.50	0.48	0.53	[0.60,0.70]	0.54
6	0.45	0.50	0.48	0.53	0.60	[0.54,0.67]

Table 9 Value range and combination of threshold parameters for line-side buffers of three-stage assembly lines

编号	$S 1$	$S 2$	$S 3$	$s 1$	$s 2$	$s 3$
7	[8,15]	7	6	2	1	1
8	8	[7,15]	6	2	1	1
9	8	7	[6,12]	2	1	1
10	8	7	6	[2,7]	1	1
11	8	7	6	2	[1,6]	1
12	8	7	6	2	1	[1,5]

Table 9 Value range and combination of threshold parameters for line-side buffers of three-stage assembly lines

编号	$S 1$	$S 2$	$S 3$	$s 1$	$s 2$	$s 3$
7	[8,15]	7	6	2	1	1
8	8	[7,15]	6	2	1	1
9	8	7	[6,12]	2	1	1
10	8	7	6	[2,7]	1	1
11	8	7	6	2	[1,6]	1
12	8	7	6	2	1	[1,5]

Fig.8 Monotonicity of multi-stage assembly lines

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