﻿ 基于麻雀搜索算法考虑病患流的医院诊疗设备维护优化研究
 上海理工大学学报  2023, Vol. 45 Issue (5): 513-522 PDF

Maintenance optimization of medical equipment considering patient flow distribution based on the sparrow search algorithm
LI Daigao, LIU Qinming, LI Jiaxiang
Business School, University of Shanghai for Science and Technology, Shanghai 200093, China
Abstract: To solve the problems of uncontrollable impact on patients and high maintenance costs after the failure of hospital diagnosis and treatment equipment, a preventive maintenance decision-making model based on uneven distribution of patient flow was proposed. Firstly, in view of the characteristics of the hospital treatment equipment degradation rate affected by the distribution of patient flow, the patient flow factor was used to reflect the changing law of its human flow, and the equipment degradation process model was constructed based on the equipment degradation evolution rule. Secondly, by incorporating the quantified cost of diagnosis and treatment equipment failure risk and the variable cost caused by delayed or early maintenance into the cost calculation system, an equipment maintenance strategy that constrains the failure risk value under the uneven distribution of patient flow was proposed. Finally, through the sparrow search algorithm (SSA) optimization to obtain the key initial variables, and numerical simulation analysis of the model, the results show that the maintenance strategy proposed in this paper can ensure the reliability of the diagnosis and treatment equipment while maintaining a low maintenance cost.
Key words: medical equipment     distribution of patient flow     failure risk costs     variable costs     sparrow search algorithm

1 问题描述与假设

a. 诊疗设备以全新状态投入使用，初始役龄为0；

b. 在不同病患流量影响下设备失效机理不发生改变，服从加速失效模型；

c. 不考虑由于备件不足引发的维修等待，设备发生的各种故障能被及时修复；

d. 初始设备故障率服从两参数威布尔分布，形状参数为 $m$ ，尺度参数为 $\eta$

e. 每位患者在诊断设备的就诊时间为相同的固定值；

f. 病患流区间内病患流因子 $\lambda$ 可以表征其真实病患流水平。

2 模型建立 2.1 病患流分布影响下故障率模型构建

 ${a}_{i}={\lambda }_{i}/{\lambda }_{0}$ (1)

 ${t}_{n}={a}_{i}{t}_{i}$ (2)

 图 1 病患流区间等效役龄折算 Fig. 1 Equivalent service age conversion
 ${t}_{0n}={t}_{n}+\sum _{i=1}^{n-1} {t}_{0i}={a}_{n}\left(t-\sum _{i=1}^{n-1} {t}_{i}\right)+\sum _{i=1}^{n-1} {t}_{0i}$ (3)

 $R\left(t\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\int }_{0}^{t} l\left(t\right)\mathrm{d}t\right)$ (4)

 $\begin{split} R(t,n)=&\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle\int }_{0}^{{a}_{n}\left(t-\sum\limits _{i=1}^{n-1} {t}_{i}\right)+\sum\limits _{i=1}^{n-1} {t}_{0(\mathrm{n}-1)}} {l}_{0}\Biggr({a}_{n}\cdot\Biggr.\right.\\&\left.\Biggr.\Biggr(t-\sum _{i=1}^{n-1} {t}_{i}\Biggr)+\sum _{i=1}^{n-1} {t}_{0i}\Biggr)\mathrm{d}t\right) \end{split}$ (5)

 $l(t,n)={a}_{n}{l}_{0}\left[{a}_{n}\left(t-\sum _{i=1}^{n-1} {t}_{i}\right)+\sum _{i=1}^{n-1} {t}_{0i}\right]$ (6)

 ${l}_{k}\left(t\right)={\beta }_{k-1}{l}_{k-1}\left(t+{\varepsilon }_{k-1}{T}_{k-1}\right)$ (7)

 图 2 考虑病患流分布的医院诊疗设备维护策略流程图 Fig. 2 Flow chart of medical equipment maintenance strategy
2.2 故障风险模型

 ${C}_{\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{k}}={P}_{\mathrm{d}}{I}_{\mathrm{d}}$ (8)

 ${P}_{\mathrm{d}}={\int }_{0}^{\mathrm{\Delta }t} {l}_{k}(t,n)\mathrm{d}t$ (9)

 ${I}_{\mathrm{d}}={c}_{\mathrm{d}}{\varphi}_{n}$ (10)

 $\boldsymbol{W}=\left[\begin{array}{cccc}{w}_{11}& {w}_{12}& \dots & {w}_{1m}\\ {w}_{21}& {w}_{22}& \dots & {w}_{2m}\\ \vdots & \vdots & & \vdots \\ {w}_{n1}& {w}_{n2}& \dots & {w}_{nm}\end{array}\right]$ (11)

 ${w}_{ik}{'}=\frac{{w}_{ik}}{ \displaystyle \sum_{k=1}^{n}{w}_{ik}}$ (12)

 $\boldsymbol{B}=\boldsymbol{A}\boldsymbol{R}=\left({b}_{1},{b}_{2},\cdots, {b}_{i},\cdots ,{b}_{n}\right)$ (13)
 ${\varphi}_{i}={\mathrm{m}\mathrm{a}\mathrm{x}\;b}_{i}$ (14)

2.3 变动成本模型

 图 3 维护周期调整示意图 Fig. 3 Schematic diagram of maintenance cycle adjustment

 ${C}_{\mathrm{L}}={c}_{\mathrm{r}}\left({R}_{\mathrm{t}}-{R}_{0}\right)/\left(1-{R}_{0}\right)$ (15)

 ${C}_{\mathrm{Y}}={c}_{\mathrm{y}}\left({R}_{0}-{R}_{\mathrm{y}}\right)/{R}_{0}$ (16)

 $\mathrm{\Delta }{C}_{\mathrm{d}}={C}_{\mathrm{d}}{\int }_{0}^{\mathrm{\Delta }t} {l}_{k}(t,n)\mathrm{d}t$ (17)

 ${C}_{\mathrm{t}}={C}_{\mathrm{L}}-\mathrm{\Delta }{C}_{\mathrm{d}}$ (18)
 ${C}_{\mathrm{y}}={C}_{\mathrm{Y}}+\mathrm{\Delta }{C}_{\mathrm{d}}$ (19)

 $\Delta C=\sum _{i=1}^{J}\left(\gamma {C}_{\mathrm{t}}+\left(1-\gamma \right){C}_{\mathrm{y}}\right)$ (20)

2.4 停机成本模型

 ${C}_{\mathrm{s}i}={T}_{\mathrm{p}i} {\lambda }_{i} r+{C}_{\mathrm{p}}$ (21)

 ${C}_{\mathrm{s}}=\sum _{i=1}^{I}C_{\mathrm{s}i}+\sum _{i=1}^{K}C_{\mathrm{d}i}$ (22)
2.5 诊疗设备总维护成本模型

 $\left\{\begin{split}& \mathrm{min}\;C={C}_{\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{k}}+\Delta C+{C}_{\mathrm{s}}\\&\text{}\begin{array}{c}\text{s}\text{.t.}\;\;\varphi\leqslant {\varphi}_{0}\\ \;\;\;\;\;\;\;R\geqslant {R}_{0}\end{array} \end{split}\right.$ (23)

3 麻雀搜索算法 3.1 算法过程

 $\boldsymbol{X}= \left[\begin{array}{cccc}{x}_{1}^{1}& {x}_{1}^{2}& \dots & {x}_{1}^{d}\\ {x}_{2}^{1}& {x}_{2}^{2}& \dots & {x}_{2}^{d}\\\vdots& \vdots& &\vdots\\ {x}_{n}^{1}& {x}_{n}^{2}& \dots & {x}_{n}^{d}\end{array}\right]$

 ${X}_{i,j}^{t+1}=\left\{\begin{array}{ll}{X}_{i,j}\mathrm{exp}\left(-\dfrac{i}{\mathrm{\alpha }{\xi }_{\mathrm{m}\mathrm{a}\mathrm{x}}}\right),&{R}_{2} < {S}_{\mathrm{T}}\\ {X}_{i,j}+Q{\boldsymbol{L}},&{R}_{2}\geqslant {S}_{\mathrm{T}}\end{array}\right.$ (24)

 ${X}_{i,j}^{t+1}=\left\{\begin{array}{ll}Q\mathrm{exp}\left(\dfrac{{X}_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}-{X}_{i,j}^{t}}{{i}^{2}}\right),&i > \dfrac{n}{2}\\ {X}_{\mathrm{P}}^{t+1}+\left|{X}_{i,j}-{X}_{\mathrm{P}}^{t+1}\right|{A}^++L,&其他\end{array}\right.$ (25)

 ${X}_{i,j}^{t+1}=\left\{\begin{array}{ll}{X}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}^{t}+\beta \left|{X}_{i,j}^{t}-{X}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}^{t}\right|,&{f}_{i} < {f}_{\mathrm{g}}\\ {X}_{i,j}+K\left[\dfrac{\left|{X}_{i,j}^{t}-{X}_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}^{t}\right|}{\left({f}_{i}-{f}_{\mathrm{w}}\right)+\varepsilon }\right],&{f}_{i}={f}_{\mathrm{g}}\end{array}\right.$ (26)

3.2 基于麻雀搜索算法的维护策略模型求解

 图 4 麻雀种群含义示意图 Fig. 4 Schematic diagram meaning of sparrow population

 图 5 SSA算法流程图 Fig. 5 Flow chart of SSA algorithm
4 算例分析

 ${l}_{0}\left(t\right)=\frac{m}{\eta }{\left(\frac{t}{\eta }\right)}^{m-1}$ (27)

4.1 算法求解比较

 图 6 各算法迭代图 Fig. 6 Algorithm iteration graph
4.2 结果分析

 图 7 是否考虑病患流分布下设备可靠度变化 Fig. 7 Change of equipment reliability considering patient flow distribution

4.3 敏感度分析

a.故障风险成本 ${C}_{\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{k}}$ 对故障风险阈值 ${\varphi}_{0}$ 和设备可靠度阈值 ${R}_{0}$ 较为敏感，对单位病人平均诊疗收益 $r$ 和可靠度利用价值 ${C}_{\mathrm{r}}$ 不敏感。当 ${\varphi}_{0}$ 增大时，对故障风险值 $\varphi$ 的要求降低，制定诊疗设备维护策略时可接受更高的故障风险，从而带来更高的故障风险成本；同时，增大的故障风险阈值使维护过程中的维护变动更少，从而带来变动成本 $\Delta C$ 下降。当 ${R}_{0}$ 增大时，对设备可靠度要求更高，高可靠度使设备发生故障的风险降低，进而使故障风险成本 ${C}_{\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{k}}$ 减小；同时，高可靠度需要更多次的维护来保障，因此 ${R}_{0}$ 增大导致变动成本 $\Delta C$ 上升。

b.设备维护停机成本 ${C}_{\mathrm{S}}$ 对单位病人平均诊疗收益 $r$ 最敏感，对故障风险阈值 ${\varphi}_{0}$ 和设备可靠度阈值 ${R}_{0}$ 较为敏感，对可靠度利用价值 ${C}_{\mathrm{r}}$ 不敏感。 $r$ 增大直接影响设备停机维护时产生的病人诊疗费用损失，从而增加了停机维护成本。 ${\varphi}_{0}$ ${R}_{0}$ 的变动影响设备停机维护时间点和停机次数，但停机时病人诊疗费用损失受该时段病患流密度，即参数 ${a}_{i}$ 影响，因此， ${C}_{\mathrm{S}}$ 的变动趋势与 ${\varphi}_{0}$ ${R}_{0}$ 无正负相关。

c.设备维护变动成本 $\Delta C$ 对可靠度利用价值 ${C}_{\mathrm{r}}$ 较为敏感，对单位病人平均诊疗收益 $r$ 不敏感 ${C}_{\mathrm{r}}$ 增大导致设备维护时间提前时产生更多的可靠度浪费，从而造成维护变动成本增加。

d. 设备维护总成本 $C$ 对故障风险阈值 ${\varphi}_{0}$ 、设备可靠度阈值 ${R}_{0}$ 最敏感，这两项参数直接影响诊疗设备预防性维护周期内停机维护与预防性维护次数；设备维护总成本 $C$ 对可靠度利用价值 ${C}_{\mathrm{r}}$ 较为敏感，因为在模型中参数 ${C}_{\mathrm{r}}$ 仅影响维护时间提前时的维护变动成本 $\Delta C$ ，而维护变动成本仅为维护总成本的构成部分，因此对维护总成本的影响较小。

5 结　论

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