﻿ 基于三站点库存组织的备件转运策略分析<sup>*</sup>
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1. 北京航空航天大学可靠性与系统工程学院, 北京 100083;
2. 北京航空航天大学 可靠性与环境工程技术国防科技重点实验室, 北京 100083

Spares transshipment strategy analysis based on a three-site inventory system
XUE Pei1,2, WANG Naichao1,2, XIAO Boping1,2, MA Lin1,2
1. School of Reliability and Systems Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China;
2. National Science Key Laboratory of Reliability and Environmental Engineering Technology, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2017-04-06; Accepted: 2017-05-12; Published online: 2017-07-11 11:46
Corresponding author. MA Lin, E-mail:malin@buaa.edu.cn
Abstract: In this paper, the effect of variant transshipment strategies on expected backorders (EBO) of a three-site inventory system is researched. Based on the predefined assumptions and inventory control strategy, as well as the consumption and replenishment process of the spares, the state equations of spares on hand, spares due in from repair and resupply, and backorders are presented with the consideration of the supply priority among sites. By defining the ratio between EBO and the number of spares due in, the original nonlinear equations can be transformed into linear equations. Since the coefficient matrix of the linear equations is invertible, it can be said that the linear equations has only one solution. As indicated above, a Cauchy sequence of the number of spares due in is constructed to approach the sole solution. Finally, the numerical values of the EBO under two transshipment strategies (fixed probability transshipment and priority preferred transshipment) are compared. The results show that the strategy of priority preferred transshipment can not only increase the utilization rate of spares but also reduce EBO of the system in most cases. In addition, it can be found that the effect of different transshipment strategies on the EBO mainly depends on the feasible space of transshipment strategies, which is mainly determined by the rate of demands filled by local site, the inventory allocation scheme, and the repairing and demanding rate of each site.
Key words: spares     three-site inventory system     expected backorders (EBO)     transshipment strategy     priority restrictions

1 问题描述和建模 1.1 问题描述

 图 1 部件在三站点库存组织中的转运模型 Fig. 1 Transshipment model of items in three-site inventory system

1) 站点自身产生的备件需求服从均值为di(i=1, 2, 3)的泊松分布。

2) 站点维修能力不设限，因此不存在故障件维修延迟现象。

3) 站点内故障件的维修时间相互独立，且服从均值为1/ui(i=1, 2, 3)的指数分布。

4) 故障件经送修后修复如新。

5) 各站点库存盘点策略为(s－1, s), s为初始库存量。

1.2 模型建立

 (1)

sMi为正在站点i处维修的故障件数；sTPjisTFji分别为站点j运往站点i途中的备件数和故障件数；sBOji为站点j对站点i的延期交货量。定义站点i顺时针方向的第1个站点为j，第2个站点为k，那么站点i的应得备件数sDIi可以表示为

 (2)

 (3)

 (4)

sBOi为站点i的延期交货量，稳态下可以表示为延期交货量期望值的形式：

 (5)
2 方程分析及求解 2.1 方程分析

 (6)

 (7)

 (8)

 (9)

2.2 求解算法设计

sDI=[sDI1, sDI2, sDI3]Tfn(sDI)(n=1, 2, 3)表示式(8)等号右侧矩阵中每一项与sDI之间的函数关系，那么方程式(8)可以写成

 (10)

 图 2 算法流程图 Fig. 2 Algorithm flowchart
3 不同转运策略算例

 参数 si/个 di/(个·h-1) ui/(个·h-1) rii Tij/h Tik/h 数值 15 0.12 0.05 0.2 50 50 注：文献[22]中模型Ⅰ还包含2个输入参数rij和rik，分别代表了站点i的备件需求中由站点j、k提供备件满足的概率，案例中规定rij=rik=(1－rii)/2。

3.1 库存配置方案对模型输出的影响

s1+s2+s3=50，首先取s1=0，s2从0取值，逐次增加5，直到等于(50－s1)停止。然后s1逐次增加5，从而产生共66组(s1, s2, s3)库存配置方案并按其产生顺序进行编号，记库存方案编号为N，则N=1, 2, …, 66。将每种库存方案代入模型进行计算，得到模型Ⅰ的系统期望延期交货量EBO1和模型Ⅱ的系统期望延期交货量EBO2，结果反映在图 3

 图 3 库存配置方案(s1+s2+s3=50)对系统EBO的影响曲线 Fig. 3 Influence curves of inventory allocation schemes (s1+s2+s3=50) on system EBO

 图 4 库存配置方案(s1=0, s2+s3=50)对系统EBO的影响曲线 Fig. 4 Influence curves of inventory allocation schemes (s1=0, s2+s3=50) on system EBO
 (11)

3.2 备件需求由本地满足的概率对模型输出的影响

 图 5 备件需求由本地满足的概率对系统EBO的影响曲线 Fig. 5 Influence curves of probability of demands filled by local site on system EBO

3.3 站点需求率、维修速率和站点间运输时间对模型输出的影响

 图 6 需求率和运输时间对系统EBO的影响曲线 Fig. 6 Influence curves of demanding rate and transport time on system EBO

4 结论

1) 具有优先级约束的转运策略能够提高备件利用率并降低系统EBO，但是作用效果主要由转运策略的可行作用空间大小决定。作用空间的可行变化范围主要由备件需求由本地满足的概率、库存配置方案以及需求率与维修速率决定。其中，备件需求由本地满足的概率决定了转运策略的作用域；可用库存不为零时配置方案的合理性决定了转运策略的作用效果，而可用库存降为零后维修速率与需求率的相对大小决定转运策略的作用效果。

2) 据此如果所有备件需求统一处理，优先选择本地备件保障，在本地无可用库存时依据优先级条件选择其他站点，并在所有站点无可用库存时，将故障件留在本地进行维修同时等待本地提供备件保障，系统的备件利用率将进一步提高，这一点在后续研究将进行论证。另外，本文的转运策略本质上是用运输成本增加来换取缺货成本降低，缺少经济性权衡分析，这可以作为以后的研究方向。

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#### 文章信息

XUE Pei, WANG Naichao, XIAO Boping, MA Lin

Spares transshipment strategy analysis based on a three-site inventory system

Journal of Beijing University of Aeronautics and Astronsutics, 2018, 44(3): 629-635
http://dx.doi.org/10.13700/j.bh.1001-5965.2017.0212