﻿ 基于任务可靠性的多阶段系统设备投入策略优化模型
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Optimization model of components placement strategy based on phased-mission system reliability
DUAN Dong-li
College of Equipment Engineering, Engineering University of Armed Police Force, Xi'an 710086, China
Abstract:To maintain high level of mission reliability, phased mission system generally takes both device backup measure and execution plan backup measure in the process of task execution. A Markov reliability evaluation model of the system is proposed by analyzing state transition relationship and state mapping relationship between two consecutive stages in this paper. An optimization model for mission reliability is put forward with task execution plan as its strategic variable, mission reliability as optimization objective and device working hours as constraints. The typical example shows the rationality and effectiveness of the proposed model in PMS.
Key words: mission reliability     Markov process     phased-mission systems     optimization

0 引言

1 问题描述及基本假设

1.1 问题描述

1.2 基本假设

1) 记第$j$种设备的失效率为${\lambda _j}$,各设备失效相互独立,设备寿命服从指数分布;

2) 记第$j$种设备的修复率为${\mu _j}$,维修时间服从指数分布;

3) 系统任务失败定义为系统状态不能满足任务能力要求;

4) 系统中各类可执行任务的设备组合方式可瞬时完成;

5) 设备一旦在阶段初投入使用,其便开始作为工作元件或热备份元件消耗工时.

2 基于任务可靠性的多阶段系统设备投入策略优化模型

2.1 任务阶段划分

2.2 系统状态空间

 $$\label{eq1} {T_m} = \left\{ {\Phi \left| {{\ell _1}(m) \ge \sum\limits_{l = 1}^k {{\gamma _{l,1}}(m)} ,\cdots,{\ell _{N - 1}}(m) \ge \sum\limits_{l = 1}^k {{\gamma _{l,N - 1}}(m)} ,{\ell _N}(m) \ge \left| {{\vartheta _m}} \right| - k} \right.} \right\}$$ (1)

2.3 系统多阶段任务可靠性

 $$\begin{bmatrix} {q_{0,0}^{(m)}} & {q_{0,1}^{(m)}} & \cdots & {q_{0,{Z_m}}^{(m)}} \cr {q_{1,0}^{(m)}} & {q_{1,1}^{(m)}} & \cdots & {q_{1,{Z_m}}^{(m)}} \cr \vdots & \vdots & \cdots & \vdots \cr {q_{{Z_m},1}^{(m)}} & {q_{{Z_m},2}^{(m)}} & \cdots & {q_{{Z_m},{Z_m}}^{(m)}} \cr \end{bmatrix}$$ (2)

$p_{i,j}^{(m)}(t) = P\{ S(t) = j|S(0) = i\}$ 表示第$m$ 个任务阶段,系统从状态$i$ 经过一段长度为$t$ 的时间后转移到状态$j$ 的概率. 记
 $$\begin{bmatrix} {p_{1,1}^{(m)}(t)} & {p_{1,2}^{(m)}(t)} & {\cdots} & {p_{1,{Z_m}}^{(m)}(t)} \cr {p_{2,1}^{(m)}(t)} & {p_{2,2}^{(m)}(t)} & {\cdots} & {p_{2,{Z_m}}^{(m)}(t)} \cr \vdots & \vdots & {\cdots} & \vdots \cr {p_{{Z_m},1}^{(m)}(t)} & {p_{{Z_m},2}^{(m)}(t)} & {\cdots} & {p_{{Z_m},{Z_m}}^{(m)}(t)} \cr \end{bmatrix}$$ (4)

 $$$\label{eq2} {{\rm d} \over {{\rm d}t}}{{{P}}_m}\left( t \right) = {{{Q}}_m}{{{P}}_m}(t)$$$ (5)

 ${{{P}}_m}(t) = \exp({{{Q}}_m}t)$ (6)

 $p_{tc}^{(m)} = p_{t0}^{(m)}\exp({{{Q}}_m}t)$ (7)

 $p_{t0,\varpi }^{(m + 1)} = \sum\limits_{{\omega _i} \in \omega } {p_{tc,{\omega _i}}^{(m)}}$ (8)

 $M{R^{(m)}} = \sum\limits_{i \in {T_m}} {p_{tc,i}^{(m)}}$ (9)

2.4 基于任务可靠性的设备投入策略优化模型

 $M{R_{^{\rm opt}}} = {\rm{ \{ }}\max M{R_{Str}}{\rm{| }}Str\}$ (10)

1)阶段初实际投入的设备数量要保障该阶段系统各种最基本工作模式要求,也即是设备的数量和种类要保证系统可以运行. 这里,$k$ 可为$[0,|{\vartheta _m}|]$ 区间中任意一个正整数,
 $\bigg\{ \ell {'_1}(m) \ge \sum\limits_{l = 1}^k {{\gamma _{l,1}}(m)} ,\cdots{\rm{ ,}}\ell {'_{N - 1}}(m) \ge \sum\limits_{l = 1}^k {{\gamma _{}}_{l,N - 1}(m)} ,\ell {'_N}(m) \ge \left| {{\vartheta _m}} \right| - k\bigg\} ^{Str},m = 1,2,\cdots ,M$ (11)

2)为减少系统的运行成本,假设系统的运行费用只由设备的投入运行产生,且设备的运行费用与运行时间成正比,记$W{H_j}$ 为第$j$ 种设备的可用工时.
 $\bigg\{ \sum\limits_{m = 1}^M {\ell {'_j}(m)} (\Psi '(m + 1) - \Psi '(m)) \le W{H_j}\bigg\} ^{Str},j = 1,2,\cdots,N$ (12)

3 算例分析

3.1 系统数据

3.2 结果分析

 图 1 系统各阶段任务可靠性

4 结论

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#### 文章信息

DUAN Dong-li

Optimization model of components placement strategy based on phased-mission system reliability

Systems Engineering - Theory & practice, 2015, 35(2): 424-429.