﻿ 环形生产线的生产率估算
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 同济大学学报(自然科学版)  2019, Vol. 47 Issue (3): 392-400.  DOI: 10.11908/j.issn.0253-374x.2019.03.013 0

### 引用本文

LIU Xuemei, ZHANG Rui, ZENG Qingfei, LI Aiping. Production Rate Evaluation of Closed-loop Production Line[J]. Journal of Tongji University (Natural Science), 2019, 47(3): 392-400. DOI: 10.11908/j.issn.0253-374x.2019.03.013

### 文章历史

Production Rate Evaluation of Closed-loop Production Line
LIU Xuemei , ZHANG Rui , ZENG Qingfei , LI Aiping
School of Mechanical Engineering, Tongji University, Shanghai 201804, China
Abstract: A closed-loop production line has a constant number of pallets (work in process, WIP) flowing through its workstations and buffers. The performance of a closed-loop production line depends not only on the machines' reliabilities and buffers' capacities, but also on the constant number of pallets. This paper presents an analytical method to approximate the production rate of an unbalanced closed-loop production line with finite buffers. The machines have deterministic processing times and multiple failure modes. A decomposition method was used to divide the whole line into multiple two-machine-one-buffer blocks, and the failure modes influencing the blocks are related with the constant number of pallets. An exact algorithm to calculate the production rate for continuous systems was introduced to approximate the production rate of the discrete systems. Then, interruption of flow equations and resumption of flow equations were used to iterate the whole line production rate until converging. At last, cases analysis and comparison with simulation results were used to validate the method. Especially for unbalanced closed-loop production line with different workstation time, the method also has high accuracy.
Key words: closed-loop production line    production rate calculation    multiple failure modes    unbalance production lines    decomposition

1 模型描述

 图 1 环形生产线 Fig.1 A closed-loop production line

(1) 零件随着托盘依次经过M1, M2, …, MK，然后零件卸载，托盘回流至BK.

(2) 每个工位Mi的作业时间Ti确定，定义每个工位的独立作业率μi=1/Ti(i=1, 2, …, K)，每台机器一次只能加工一个工件，不考虑报废和返修.

(3) 每个工位Miγi种故障状态.任一工位不会同时发生2种以上的故障.每个工位Mi的故障用Fi, j(j=1, …, γi)表示，其故障率为pi, j(j=1, …, γi)，修复率为ri, j(j=1, …, γi)，故障率、修复率符合负指数分布，并且$\sum\limits_{j = 1}^{{\gamma _i}} {{p_{i, j}} < 1, \forall i}$.

(4) 所有故障都是基于操作的故障(ODF)，即在待料或者堵料时，故障不会发生.

(5) 假设第1个工位上料充分，最后1个工位有库存空间卸料.因此，零件上下线与否完全由托盘决定，并且零件在线中流动与托盘完全同步，因此忽略零件，只考虑托盘的流动.

(6) 不考虑托盘经过缓冲区的时间，缓冲区不发生故障.

(7) t时刻，各个缓冲区中的在制品容量为ni(t), 0≤ni(t)≤Ni.工位中不设存放零件的容量，即$\sum\limits_{i = 1}^K {{n_i}\left( t \right) = I}$.

2 环形生产线的生产率估算方法 2.1 两工位单缓冲区系统

 图 2 两工位单缓冲区生产系统 Fig.2 Two-machines-one-buffer system

 $\alpha = \left\{ {\begin{array}{*{20}{c}} 1&{{M^{\rm{u}}}\;正常工作}\\ {{\alpha _1}}&{{M^{\rm{u}}}\;处于第\;{\alpha _1}\;种故障状态}\\ \vdots&\vdots \\ {{\alpha _{{f_{\rm{u}}}}}}&{{M^{\rm{u}}}\;处于第\;{\alpha _{{f_{\rm{u}}}}}\;种故障状态} \end{array}} \right.$
 $\beta = \left\{ {\begin{array}{*{20}{c}} 1&{{M^{\rm{d}}}\;正常工作}\\ {{\beta _1}}&{{M^{\rm{d}}}\;处于第\;{\beta _1}\;种故障状态}\\ \vdots&\vdots \\ {{\beta _{{f_{\rm{d}}}}}}&{{M^{\rm{d}}}\;处于第\;{\beta _{{f_{\rm{d}}}}}\;种故障状态} \end{array}} \right.$

 $\begin{array}{l} f\left( {x,i,j,t} \right) = \frac{\partial }{{\partial x}}{P_{{\rm{rob}}}}\left[ {X\left( t \right) \le x,\alpha \left( t \right) = i,\beta \left( t \right) = j} \right],\\ \;\;\;\;0 < x < N \end{array}$ (1)

 $\begin{array}{l} {P_{\rm{R}}} = \sum\limits_{\left( {i,j} \right) \in {S_0}} {\mu _i^{\rm{u}}p\left( {0,i,j} \right)} + \sum\limits_{\left( {i,j} \right) \in {S_{\rm{M}}}} {\int_0^N {\mu _i^{\rm{u}}f\left( {x,i,j} \right){\rm{d}}x} } + \\ \;\;\;\;\;\;\;\sum\limits_{\left( {i,j} \right) \in {S_{\rm{N}}}} {\mu _j^{\rm{d}}p\left( {N,i,j} \right)} \end{array}$ (2)
 $E = 1 - \sum\limits_{\left( {i,j} \right) \in {S_0}} {p\left( {0,i,j} \right)} - \sum\limits_{\left( {i,j} \right) \in {S_{\rm{N}}}} {p\left( {N,i,j} \right)}$ (3)

f(x, i, j)、p(0, i, j)和p(N, i, j)的求解参考Tan等[10]所提的方法，根据缓冲区填充量分别求解.

2.2 利用分解方法估算开放型生产线生产率

 图 3 开放型生产线分解方法 Fig.3 Decomposition of a tandem line

 图 4 Tolio分解方法故障传递状态示意 Fig.4 Failure modes propagation in Tolio's decomposition

 $\begin{array}{l} {r_{t,j}}\left( t \right) = {r_{t,j}}\;\;\;\left( {i = 1, \cdots ,K;} \right.\\ \;\;\left. {t = 1, \cdots ,K;\;\;\;j = 1, \cdots ,{\gamma _t}} \right) \end{array}$ (4)

 $\begin{array}{*{20}{c}} {p_{t,j}^{\rm{u}}\left( i \right) = \frac{{P_{t,j}^{{\rm{st}}}\left( {i - 1} \right)}}{{E\left( i \right)}}{r_{t,j}}\left( {i = 1, \cdots ,K;} \right.}\\ {\left. {t = 1, \cdots ,i - 1;\;\;\;j = 1, \cdots ,{\gamma _t}} \right)} \end{array}$ (5)
 $\begin{array}{*{20}{c}} {p_{t,j}^{\rm{d}}\left( i \right) = \frac{{P_{t,j}^{{\rm{bl}}}\left( {i + 1} \right)}}{{E\left( i \right)}}{r_{t,j}}\left( {i = 1, \cdots ,K;} \right.}\\ {\left. {t = i + 2, \cdots ,K;\;\;\;j = 1, \cdots ,{\gamma _t}} \right)} \end{array}$ (6)

2.3 环形生产线的分解方法

 图 5 六工位环形生产线 Fig.5 A closed-loop of six machines

2.3.1 待料和堵料的范围划分

 $s\left( i \right) = \mathop {\max }\limits_j \left\{ {j + 1\left| {\mathit{\Psi }\left( {i,j} \right) < I} \right.} \right\}$ (7)
 $b\left( i \right) = \mathop {\min }\limits_j \left\{ {j\left| {\mathit{\Psi }\left( {i,j + 1} \right) > I} \right.} \right\}$ (8)

2.3.2 阈值的消除

 图 6 带阈值的环形生产线 Fig.6 A closed-loop with thresholds

 ${l_j}\left( i \right) = I - \mathit{\Psi }\left( {i + 1,j} \right)$ (9)

 图 7 消除阈值后的环形生产线示意 Fig.7 A closed-loop with thresholds eliminated
3 实例验证 3.1 算例分析

(1) 根据环形生产线计算方法，先对生产线做基本处理，即消除B1B2B3的阈值.计算各缓冲区阈值，分别为5、5、5；插入理想机器M*(u=100, p=10-8, r=1)后，得6个工位、6个缓冲区的生产线，重新编号为M1M2M3M4M5M6，其中M1M3M5对应原工位MM2M3M2M4M6为理想机器M*.转化后的工位参数如表 3，缓冲区容量如表 4.

(2) 划分工位待料和堵料范围.根据式(7)与(8)计算工位的待料与堵料范围为：st(1)=4;st(2)=5;st(3)=6;st(4)=1;st(5)=2;st(6)=3; bl(1)=4;bl(2)=5;bl(3)=6;bl(4)=1;bl(5)=2;bl(6)=3.

(3) 划分两工位单缓冲区基本单元，确定故障状态.将整线划分为6个基本生产单元Mu(i)-B(i)-Md(i)(i=1, 2, 3, 4, 5, 6)，并根据待堵料范围确定各单元的虚拟故障状态.如Mu(1)-B(1)-Md(1)单元中，Mu(1)的故障状态包括能使M2待料的F5, 1(1)、F6, 1(1)、F1, 1Md(1)的故障状态包括能使M1堵料的F4, 1(2)、F3, 1(2)、F2, 1；其他单元以此类推.

(4) 初始化各虚拟故障状态的可靠性参数pk, ju(i)、rk, ju(i)(k=s(i+1), …, i); pk, jd(i)、rk, jd(i)(k=i+1, …, b(i)).如Mu(1)-B(1)-Md(1)单元中，p1, 1=0.01;p5, 1(1)=p5, 1=0.01;p6, 1(1)=p6, 1=10-8; p4, 1(2)=p4, 1=10-8; p3, 1(2)=p3, 1=0.01;p2, 2=10-8; r1, 1=0.11;r5, 1(1)=r5, 1=0.15;r6, 1(1)=r6, 1=1.00;r4, 1(2)=r4, 1=1.00;r3, 1(2)=r3, 1=0.12;r2, 2=1.00.

(5) 从i=1至i=6，计算每个虚拟单元的状态概率，得E(i)与Pk, jst(i)；利用式(5)更新pk, ju(i+1).

(6) 对i=6至i=1，计算每个虚拟单元的状态概率，得E(i)与Pk, jbl(i)，利用式(6)更新pk, ju(i-1).

(7) 重复步骤(5)、步骤(6)，直到每个虚拟单元的生产率收敛，即pk, ju(i)、pk, jd(i)不再变化.最小的PR即为整线生产率.

(1) 以第1个工位独立作业率μ1为变量，验证本文方法的精度.考虑到生产线不平衡率一般控制在20%范围内，因此只取μ1的变化范围在1.0~0.8.部分计算结果如图 8所示，随着μ1的升高，整线生产率随之升高，仿真结果与数值计算结果的相对误差始终控制在2%范围内.

 图 8 生产率与μ1的关系 Fig.8 Production rate with parameter μ1 varying

(2) 考虑3个工位独立作业率都不相同的情况.工位参数如表 6，缓冲区配置及托盘数如表 2所示.保持各个参数不变，μ1=0.85，μ2在1.0~0.8范围内变化.计算结果如图 9所示，3个工位独立作业率都不相同的情况下，第2个工位独立作业率变化不超过20%的范围，生产率估算误差不超过1%，说明该方法对于三工位环形系统计算具有有效性.

 图 9 独立作业率各不相同的三工位环形生产线生产率 Fig.9 Production rate of 3-machine loop with different independent efficiency

(3) 将上述算法与Gershwin-Werner的算法[6]作比较，假设第1个工位的独立作业率为0.95，而第2个工位的独立作业率在0.7~1.0范围内波动，工位参数如表 7所示，缓冲区配置及托盘数如表 2所示，2种方法的计算结果如图 10所示.Gershwin-Werner的方法能将误差控制在8%范围内，本文方法能将误差控制在2%范围内.Gershwin-Werner方法[6]无法精确计算各工位作业时间不同的情况，只能按照瓶颈节拍作为整线节拍来粗略计算，这将造成较大误差.

 图 10 不同方法比较 Fig.10 Performance comparison between the methods

(4) 考虑更多工位的情况.如五工位的生产线，其工位参数如表 8、缓冲区配置及托盘数如表 9，将第1个工位的独立作业率由0.1开始以0.1为增幅逐渐提高至2.5，检验该方法的精确性.计算结果如图 11，与仿真结果相比，计算精度在8%范围内.

 图 11 一个工位独立作业率变化的五工位生产线生产率 Fig.11 Production rate of 5-machine loop with one machine's efficiency varying

3.2 实例验证

 图 12 某变速线主箱体分装线生产率计算 Fig.12 Production rate of the main box assembly line of a certain gearbox

4 结论

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