﻿ 简单直线和U型装配线平衡中的改进阶位法
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 同济大学学报(自然科学版)  2019, Vol. 47 Issue (1): 143-148.  DOI: 10.11908/j.issn.0253-374x.2019.01.019 0

### 引用本文

JIAO Yuling, XING Xiaocui, ZHU Chunfeng, XU Liangcheng, ZOU Lianhui. Modified Ranked Positional Weight Technique for Assembly Line Balance of Simple Line and U-shape[J]. Journal of Tongji University (Natural Science), 2019, 47(1): 143-148. DOI: 10.11908/j.issn.0253-374x.2019.01.019

### 文章历史

1. 吉林大学 交通学院，吉林 长春 130021;
2. 吉林建筑大学 土木工程学院，吉林 长春 130021

Modified Ranked Positional Weight Technique for Assembly Line Balance of Simple Line and U-shape
JIAO Yuling 1, XING Xiaocui 1, ZHU Chunfeng 2, XU Liangcheng 1, ZOU Lianhui 1
1. College of Transportation, Jilin University, Changchun 130021, China;
2. College of Civil Engineering, Jilin Jianzhu University, Changchun 130021, China
Abstract: This paper presents a simple and U-shaped assembly line balance problem with the modified ranked positional weight technique. First, according to the processing time and priority relationship of each task, the synthetical ranked positional weight is obtained by taking the time-ranked weight and position-ranked weight into comprehensive consideration. Then the modified ranked positional weight technique is applied to the assembly line balance problem to obtain the balancing results of simple, single U-shaped and double U-shaped assembly lines. With the simple example 1, the comparison of the balance effects is performed in modified ranked positional weight technique and branch-and-bound method. With the experimental example 2, it is proved that the balance effect of modified ranked positional weight technique is better than that of the time-ranked weight technique. Therefore, the modified ranked positional weight technique is an effective method to solve assembly line balance problem.
Key words: industrial engineering    double U-shaped assembly line    modified ranked positional weight technique

U型装配线上的工作站相距较近，不仅简化了物料的运输，而且柔性较高，在外部需求发生变化时，通过增加或减少操作人员的数量实现装配线节拍调整达到新的平衡，分配作业元素时，选择的范围更大，分配效果一般比直线型装配线好[3].

1 改进阶位法 1.1 阶位法简介

1.2 作业元素优先关系

 ${P_{ij}} = \left\{ \begin{array}{l} 1, 作业元素i是j的紧前元素\\ 0, 作业元素i不是j的紧前元素 \end{array} \right.$
1.3 确定作业元素的综合阶位值

(1) 计算作业元素i的作业量阶位Rhi

 ${H_i} = {\rm{count}}\{ {F_i}\}$

(2) 计算作业元素i的时间阶位Rti

 ${T_i} = {t_i} + \sum\limits_{q \in {F_i}} {{t_q}}$

Ti值升序排列所有作业元素，对于Ti值最小的作业元素赋予时间位置阶位Rti=1，对于Ti值次小的作业元素赋予时间位置阶位Rti=2，如果作业元素Ti值相同，则被赋予相同的时间阶位Rti.依此类推，直到所有作业元素都被赋予一个特定的Rti值.

(3) 计算作业元素i的综合阶位值Ri

 ${R_i} = {R_{{\rm{h}}i}} \times {R_{{\rm{t}}i}}$
2 建立装配线平衡数学模型 2.1 参数定义

2.2 U型装配线平衡问题数学模型

 $\max \;E = \frac{{\sum\limits_{i = 1}^n {{t_i}} }}{{CM}} \times 100\%$

 $\sum\limits_{k = 1}^{{m_0}} {({x_{ik}} + {y_{ik}}) = 1, \;\;\;\;i = 1, 2, \ldots n}$ (1)
 $\sum\limits_{i = 1}^n {({x_{ik}} + {y_{ik}}){t_i} \le C, \;\;\;\;k = 1, 2, \ldots , {m_0}}$ (2)
 $\sum\limits_{k = 1}^{{m_0}} {(k{x_{ik}} - k{x_{jk}}) \le 0, \;\;\;\;\forall \left( {i, j} \right) \in {P_{{\rm{set}}}}}$ (3)
 $\sum\limits_{k = 1}^{{m_0}} {(k{y_{jk}} - k{y_{ik}}) \le 0, \;\;\;\;\;\forall \left( {i, j} \right) \in {P_{{\rm{set}}}}}$ (4)
 $\sum\limits_{i = 1}^n {({x_{ik}} + {x_{jk}}) \le n{w_k}, \;\;\;\;k = 1, 2, \ldots , {m_0}}$ (5)
 ${W_{k + 1}} \le {W_k}, \;\;\;\;k = 1, 2, \ldots , {m_0} - 1$ (6)
 $\sum\limits_{i = 1}^n {{x_{ik}} \le n{W_k}, \;\;\;\;k = 1, 2, \ldots , {m_0}}$ (7)
 $\sum\limits_{k = 1}^{{m_0}} {k{x_{ik}} \le \sum\limits_{k = 1}^{{m_0}} {k{x_{jk}}, \;\;\;\;\forall \left( {i, j} \right) \in {P_{{\rm{set}}}}} }$ (8)

3 改进阶位法求解模型

3.1 基于改进阶位法分配作业

U型装配线和直线型装配线的作业元素分配方式不同.对于U型装配线，将作业元素分配到工作站时，既可以按照装配线的加工方向从前到后分配或者从后到前分配，同时也可以两个方向上同时进行，这样U型装配线平衡问题的求解难度增大，但是U型和双U型装配线平衡效率有所提高.而在直线型装配线上，分配作业元素到工作站时按照单方向进行，降低了作业元素之间的组合搭配的交叉性，会导致直线型装配线生产效率降低.双U型装配线可以同时装配2件同类型的产品.

3.2 装配线平衡效果评价

 $S = \sqrt {\sum\limits_{k = 1}^M {{{\left( {{\rm{max}}\;T\left( k \right) - T\left( k \right)} \right)}^2}} }$

4 算例1

Bomman问题[16]的作业元素优先关系图，如图 1，图中圆圈里的数字表示该作业元素的序号，作业元素圆圈上方数字表示该作业元素的作业时间.作业元素的总时间为75，装配线的生产节拍设定C=26，确定最小工作站数.

 图 1 Bomman问题的优先关系 Fig.1 Precedence relationship of Bomman problem
4.1 综合阶位值

Bomman问题的作业元素的优先关系集合为：Pset={(1, 2)，(2, 3)，(2, 4)，(3, 5)，(4, 6)，(5, 7)，(6, 8)，(7, 8)}.优先关系集合Pset中有8对直接的优先关系，不含有间接的优先关系，其间接优先关系可以根据作业元素之间优先关系的传递性得到.优先关系矩阵P的三元组表示：A =[(1, 2, 1), (2, 3, 1), (2, 4, 1), (3, 5, 1), (4, 6, 1), (5, 7, 1), (6, 8, 1), (7, 8, 1)].

4.2 改进阶位法求解结果及评价

5 实验算例2

 图 2 车模(局部)结构 Fig.2 Vehicle model (partial) structure
 图 3 车模装配流程 Fig.3 Vehicle model assembly flow
5.1 车模装配线平衡结果布局

 图 4 直线型装配线平面布局 Fig.4 Simple line assembly line layout
 图 5 单U型装配线平面布局 Fig.5 Single U-shaped assembly line layout
 图 6 双U型装配线平面布局 Fig.6 Double U-shaped assembly line layout
5.2 比较平衡效果指标

6 结论

(1) 依据作业元素时间和优先关系计算各作业元素的综合阶位值，提出了改进阶位法，求解了装配线平衡问题，结果合理有效，为企业装配线平衡与优化提供有效的新方法.

(2) 改进阶位法求解的U型装配线平衡效果优于分枝定界法的结果，与直线型装配线比较，平衡率提高了24%，平滑系数降低了79.8%，提高了装配线效率和设备使用效率，使装配线负荷程度更为均匀，为流水装配线方案设计提供了有效方法.

(3) 结合实验室实际车模(局部)实验，对流水线实验设计双U型装配实验方案，改进阶位的结果与时间阶位法进行对比，改进阶位法的平滑系数降低了.对于小型结构的流水线平衡问题，改进阶位法优势并不明显，但在计算零件多结构复杂的大型产品的装配线平衡问题时，改进阶位法平衡效果明显且易于操作.

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