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Time-varying convex optimization for spare parts inventory considering passivation
SONG Changhao , GUO Linhan , WANG Naichao , MA Lin
School of Reliability and Systems Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2015-05-21; Accepted: 2015-06-21; Published online: 2016-05-25 12: 00
Foundation item: National Natural Science Foundation of China (61304148)
Corresponding author. Tel.: 010-82316443 E-mail:linhanguo@buaa.edu.cn
Abstract: Taking passivation into consideration, we apply a time-varying inventory balance equation to setting up the time-varying backorder formula and use the extended Palm theorem to combine the cumulative demands for spare parts with the parameters of the shape characteristics of Poisson processes to create a convex optimization algorithm for spares considering the cumulative demands. Next, we use time-varying availability as the optimization objective under a cost constraint to obtain the optimal configuration program in every period by convex optimization methods. A selection rule known as the mode method is introduced to select the proper global stock policy. Finally, a numerical example is presented to demonstrate the approach, and several inventory policies are compared to prove the superiority of the proposed method.
Key words: spare parts     convex optimization     instantaneous availability     passivation     inventory

1 优化模型

 (1)

 (2)

 (3)

 (4)

 (5)

1) 备件的故障时间服从指数分布。

2) 库存策略为可修产品的及时送修策略——(s-1,s)策略。

3) 除备件外,其他维修资源均供应充足且维修都是成功的。

2 优化算法 2.1 EBO(si,t)凸函数分析

2.2 算法设计

 图 1 优化算法流程 Fig. 1 Optimization algorithm flowchart

1) 初始化备件库存、时间和备件总费用(c)。

2) 时间推到下一时刻。

3) 根据式(1),计算此时刻各类备件的需求率。

4) 比较(EBO(si,t)-EBO(si+1,t))/ci,找到使该公式值最大的备件种类。

5) 第i类备件的库存加1,总费用加ci

6) 当备件总费用小于最大可接受费用(C)时,跳至步骤4),否则进行步骤7)。

7) 根据式(2)~式(4),计算此时刻的EBO(si,t)和AiM(t)。

8) 根据式(5),计算此时刻的AO(t)。

9) 当前时刻小于规定计算时长(T)时,回到步骤2),否则进行步骤10)。

10) 在得到的每时刻最优库存配置中,选择出现次数最多的配置,作为时长T内的全局库存策略。

3 应用案例

3.1 不考虑passivation的凸优化方法

 备件种类 维修时间/h 故障间隔时间/h 周转时间/h 安装数 LRU1 1 400 220 2 LRU2 1 500 200 3 LRU3 1 400 220 2 LRU4 1 500 200 3 LRU5 1 440 180 2 LRU6 1 400 200 1 LRU7 1 500 220 2 LRU8 1 440 190 1 LRU9 1 400 200 1 LRU10 1 440 180 2

 图 2 零库存下的时变EBO Fig. 2 EBO as a function of time under situation of zero inventory

 备件种类 时变库存 METRIC 5 h 120 h 150 h 225 h 405 h LRU1 3 4 4 4 4 4 LRU2 4 4 5 5 5 5 LRU3 3 3 3 3 4 4 LRU4 4 4 4 4 4 4 LRU5 3 3 3 2 2 2 LRU6 3 2 2 3 2 2 LRU7 3 4 3 4 4 4 LRU8 2 2 2 2 2 2 LRU9 3 2 2 2 2 2 LRU10 3 4 4 4 4 4

3.2 考虑passivation的凸优化算法

 图 3 Passivation对可用度的影响 Fig. 3 Effect of passivationon on availability

 备件种类 考虑/不考虑passivation的时变最优库存 5 h 80 h 150 h 225 h 405 h 465 h LRU1 3/3 4/4 4/4 4/4 4/4 4/4 LRU2 4/4 4/4 4/5 5/5 5/5 5/5 LRU3 3/3 3/3 3/3 3/3 3/4 4/4 LRU4 3/3 4/4 4/4 4/4 4/4 4/4 LRU5 3/3 3/3 3/3 3/3 2/2 2/2 LRU6 3/3 2/2 2/2 2/2 2/2 2/2 LRU7 3/3 3/3 4/3 3/3 4/4 4/4 LRU8 3/3 2/2 2/2 2/2 2/2 2/2 LRU9 3/3 2/2 2/2 2/2 2/2 2/2 LRU10 3/3 4/4 4/4 4/4 4/4 4/4

 备件种类 考虑/不考虑passivation的时变最优库存 5 h 80 h 150 h 225 h 405 h 465 h LRU1 3/3 4/4 4/4 4/4 4/4 4/4 LRU2 4/4 5/5 5/5 5/5 5/6 5/6 LRU3 3/3 3/3 3/3 3/4 4/4 4/4 LRU4 3/3 4/4 4/4 4/4 4/4 4/4 LRU5 3/3 3/3 3/3 2/2 2/2 2/2 LRU6 3/3 2/2 2/2 2/2 2/2 2/2 LRU7 3/3 3/3 3/3 4/4 4/4 4/4 LRU8 3/3 2/2 2/2 2/1 2/1 2/1 LRU9 3/3 2/2 2/2 2/2 2/2 2/2 LRU10 3/3 4/4 4/4 4/4 4/4 4/4

 备件种类 考虑/不考虑passivation的时变最优库存 5 h 115 h 130 h 175 h 355 h 385 h LRU1 3/3 4/4 4/4 4/4 4/5 4/5 LRU2 4/4 5/5 5/5 5/6 6/7 6/7 LRU3 3/3 3/3 3/3 3/4 4/3 4/4 LRU4 3/3 4/4 4/4 4/4 4/4 4/4 LRU5 3/3 3/2 2/3 3/2 2/1 2/0 LRU6 3/3 2/2 2/2 2/2 2/2 2/2 LRU7 3/3 3/4 4/4 4/4 4/5 4/5 LRU8 3/3 2/2 2/1 1/1 1/1 1/1 LRU9 3/3 2/2 2/2 2/2 2/2 2/2 LRU10 3/3 4/4 4/4 4/4 4/5 4/5

 图 4 U=0.45和U=0.6时考虑/不考虑passivation的AO Fig. 4 AO with/without passivation when U=0.45 and U=0.6

3.3 全局库存策略的选择

 备件种类 库存水平 全局最优 初始库存 稳态库存 LRU1 4 3 4 LRU2 5 3 5 LRU3 3 3 4 LRU4 4 3 4 LRU5 3 3 2 LRU6 2 3 2 LRU7 3 3 4 LRU8 2 3 2 LRU9 2 4 2 LRU10 4 3 4

 图 5 3种库存策略下的AO(t) Fig. 5 AO(t) for 3 inventory policies

4 结 论

1) 考虑passivation对可用系统数量的影响,在稳态库存的基础上,引入时间变量,建立了时变库存优化模型。

2) 研究基于非稳态备件需求下的备件库存优化方法,并构建了考虑passivation的时变优化应用案例,为时变优化分析工作的开展提供了一套可行的优化方法。

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#### 文章信息

SONG Changhao, GUO Linhan, WANG Naichao, MA Lin

Time-varying convex optimization for spare parts inventory considering passivation

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(5): 1025-1031
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0327