﻿ 考虑等待时间约束的不完美生产系统的产出优化<sup>*</sup>
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Throughput optimization for an imperfect production system with queue time constraints
ZHOU Binghai, LIU Yuwang
School of Mechanical and Energy Engineering, Tongji University, Shanghai 201804, China
Received: 2016-10-26; Accepted: 2016-12-09; Published online: 2016-12-28 14:48
Foundation item: National Natural Science Foundation of China (71471135)
Corresponding author. ZHOU Haibin, E-mail:bhzhou@tongji.edu.cn
Abstract: To efficiently solve the problems of queue time constraints and quality loss caused by machine degradation during production activities, a joint optimization mathematical model considering both preventive maintenance and the control of the buffer capacity was constructed in this paper. First, gamma process was introduced to model the degradation of the downstream bottleneck station, and the quality loss caused by its degradation was also considered. Second, based on the model mentioned above, we treated the arrival of workpieces, the intermediate buffer and the working process of downstream station as a queuing system and obtained the probability of work in process (WIP) blocking and exceeding the queue time constraints using M/G/1/K queuing model. Finally, with the objective function of maximizing the "effective throughput", we jointly explored the optimization of the threshold of preventive maintenance and the capacity of the intermediate buffer. Numerical example shows that the proposed model is practical and effective, which has certain instructive significance to the buffer capacity control, preventive maintenance and throughput improvement for those imperfect production systems with queue time constraints.
Key words: machine degradation     gamma process     quality loss     queue time constraints     buffer

1 问题描述

 图 1 带等待时间约束的串行生产系统 Fig. 1 A serial production system with queue time constraints

 图 2 “有效产出”示意图 Fig. 2 Schematic diagram of "effective throughput"

2 模型构建 2.1 生产系统的劣化过程及维护建模

 (1)

 (2)

t求偏导，可得

 (3)

i次PM后(i=1, 2, …，N-1)，第i+1个运行周期的期望长度为

 (4)

2.2 质量损失建模

 (5)

 (6)

 (7)

 (8)
2.3 基于M/G/1/K排队模型的缓冲区建模

 (9)

 图 3 排队系统的状态转移示意图 Fig. 3 Schematic diagram of state transition for queuing systems

pn(n=0, 1, …, K－1) 表示当系统处于稳态时，在工作站2加工完某工件的瞬间Q2中恰好有n个工件的稳态概率。pn可以通过式(10) 求解：

 (10)

 (11)

 (12)

 (13)

3 数值实例

 (14)

 (15)

 图 4 工作站2达到DP经历时间的CDF和PDF Fig. 4 CDF and PDF of service time of station 2 until DP

 图 5 不同维护周期里失效率函数随时间的变化 Fig. 5 Variation of failure rate function with time in different maintenance periods

 图 6 不同DP下各维护周期的期望运行时间 Fig. 6 Expected service time of each maintenance period under different DP

 图 7 不同DP对应的的总期望产出及质量损失 Fig. 7 Expected total throughput and quality loss under different DP

 图 8 PB、PS随缓冲区容量的变化 Fig. 8 Variation of PB and PS with buffer capacity

DPK的变化均会影响系统的有效产出，但是难以将两者整合进一个数学模型中，以寻求最大化目标函数对应的最优组合。为此采用数值迭代算法寻找两者的最优组合[DP*, K*]，见表 1

 K/个 有效产出/个 DP=110μm DP=115μm DP=120μm DP=125μm DP=130μm DP=135μm 6 3365 3378 3386 3393 3400 3393 7 3378 3390 3398 3405 3412 3406 8 3365 3377 3386 3393 3399 3392 9 3346 3358 3367 3373 3380 3372 10 3328 3340 3348 3355 3361 3353 11 3312 3324 3332 3338 3344 3336 12 3299 3311 3319 3325 3331 3323 13 3290 3301 3309 3316 3321 3313 14 3283 3295 3303 3309 3314 3306

4 结论

1) 定义“有效产出”这一概念，使制造企业关注实际生产过程中由于设备劣化引起的质量损失、缓冲区的阻塞以及由于WIP等待时间过长而废弃等问题，避免盲目追求生产率。

2) 对工作站2全寿命周期内劣化过程及预防性维护的建模，能较好地符合生产实际，为制造企业开展维护活动提供参考。

3) 在结论2) 的基础上考虑由于设备劣化引起的质量损失，进而提出了一种基于状态的预防性维护策略，减少系统因产品质量而损失的产出。

4) 引入M/G/1/K排队模型求解缓冲区WIP的阻塞及超出等待时间约束的概率，为存在等待时间约束的实际生产系统的缓冲区控制提供了一种新的优化思路，进而提高系统的有效产出。

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#### 文章信息

ZHOU Binghai, LIU Yuwang

Throughput optimization for an imperfect production system with queue time constraints

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(10): 1972-1979
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0824