﻿ 基于最大熵原理的武器作战效能预测研究 The Prediction of Effectiveness of Weapons and Combat Based on the Principle of Maximum Entropy

Operations Research and Fuzziology
Vol.08 No.03(2018), Article ID:26296,6 pages
10.12677/ORF.2018.83013

The Prediction of Effectiveness of Weapons and Combat Based on the Principle of Maximum Entropy

Lei Miao

91049 Force 31 Detachment of the People’s Liberation Army, Qingdao Shandong

Received: Jul. 18th, 2018; accepted: Aug. 1st, 2018; published: Aug. 8th, 2018

ABSTRACT

AHP and fuzzy comprehensive evaluation (FCE) were respectively used for the forecast of the effectiveness of weapons and combat. Then on the basis of merit of maximum entropy principle, the traditional method has been improved, and concluded that the principle of maximum entropy had higher forecast accuracy. Thus it can play a guiding role on the actual operations.

Keywords:Maximum Entropy Principle, Performance Forecast, Evaluation Indicators, Albino Right Function

1. 引言

2. 武器系统的评价指标

Figure 1. Evaluation index system of weapon system effectiveness

3. 具体案例分析

3.1. AHP预测武器作战效能 [6]

3.1.1. 构建两两比较判断矩阵A

3.1.2. 计算判断矩阵的各行元素的积的n (判断矩阵的阶数)次方根 $\overline{{\omega }_{i}}$

$\overline{{\omega }_{1}}=2.2894$$\overline{{\omega }_{2}}=0.3816$$\overline{{\omega }_{3}}=1.1447$ ，并将其归一化得到： ${\omega }_{1}=0.6000$${\omega }_{2}=0.1000$${\omega }_{3}=0.3000$

3.1.3. A的满意一致性检验

$CI=\left({\lambda }_{\mathrm{max}}-n\right)/\left(n-1\right)=0.00005$ ，其中 ${\lambda }_{\mathrm{max}}$ 为判断矩阵A的最大特征值。判断矩阵A的随机一致性指标 $CR=CI/RI<0.1$ (RI为常数)，A具有满意一致性。故U1，U2，U3的指标权重分别为0.6000，0.1000，0.3000。

3.1.4. 预测武器作战效能

EAHP = 0.6 * [0.1 0.2 0.7] * [1 10 2]’ + 0.1 * [0.3 0.7] * [2 5]’ + 0.3 * [0.7 0.3] * [5 10]’ = 4.46 (式中的整数矩阵部分可根据判断矩阵，再结合白化权函数得到)。

Table 1. Judgment matrix A

3.2. FCE法预测武器作战效能 [7]

3.2.1. 确定模糊关系矩阵(根据判断矩阵和白化权函数得到)

3.2.2. 预测武器作战效能

B2 = U1 * R1 = [0.2610 0.1186 0.2004 0.2972 0.1227]，

B3 = U1 * R1 = [0.0767 0.0677 0.2947 0.3704 0.1906]。

3.3. 最大熵原理预测武器作战效能 [7]

3.3.1. 最大熵原理

1948年，香农(Shannon)用熵来定量地描述一个随机事件的不确定性或信息量：

$H=-\sum _{i=1}^{n}{p}_{i}\mathrm{ln}{p}_{i}$ (1)

$\sum _{i=1}^{n}{p}_{i}=1$ (2)

Table 2. Fuzzy relation matrix R1

Table 3. Fuzzy relation matrix R2

Table 4. Fuzzy relation matrix R3

${E}_{k}=\sum _{i=1}^{n}{f}_{k}\left({x}_{i}\right){p}_{i}$ ( $k=1,2,\cdots ,m$ ) (3)

$H-\alpha -\sum _{k=1}^{m}{\beta }_{k}{E}_{k}$ (4)

$H=\sum _{i=1}^{n}{p}_{i}\mathrm{ln}\left\{\frac{1}{{p}_{i}}\mathrm{exp}\left[-\alpha -\sum _{k=1}^{m}{\beta }_{k}{f}_{k}\left({x}_{i}\right)\right]\right\}$ (5)

$H\le \sum _{i=1}^{n}{p}_{i}\left\{\frac{1}{{p}_{i}}\mathrm{exp}\left[-\alpha -\sum _{k=1}^{m}{\beta }_{k}{f}_{k}\left({x}_{i}\right)\right]-1\right\}+\alpha +\sum _{k=1}^{m}{E}_{k}$ (6)

${p}_{i}=\mathrm{exp}\left[-\alpha -\sum _{k=1}^{m}{\beta }_{k}{f}_{k}\left({x}_{i}\right)\right]$ ( $i=1,\cdots ,n$ )时，H取得最大值 (7)

3.3.2. 预测武器作战效能

$\mathrm{max}H\left(x\right)=-\int p\left(x\right)\mathrm{ln}p\left(x\right)\text{d}x$ (8)

$\int p\left(x\right){\left[\left(x-\stackrel{^}{{y}_{r}}\right)/\stackrel{^}{{y}_{r}}\right]}^{2}\text{d}x={\left(\frac{y-\stackrel{^}{{y}_{r}}}{\stackrel{^}{{y}_{r}}}\right)}^{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(r=1,2\right)$ (9)

$\int p\left(x\right)\text{d}x=1$ (10)

$p\left(x\right)=\mathrm{exp}\left[-\alpha -\sum _{r=1}^{2}{\beta }_{r}{\left(\frac{x-\stackrel{^}{{y}_{r}}}{\stackrel{^}{{y}_{r}}}\right)}^{2}\right]$ (11)

${\int }_{1}^{9}{\left(x-\stackrel{^}{{y}_{r}}\right)}^{2}\mathrm{exp}\left(-\alpha -{\beta }_{1}{\left(\frac{x-\stackrel{^}{{y}_{1}}}{\stackrel{^}{{y}_{1}}}\right)}^{2}-{\beta }_{2}{\left(\frac{x-\stackrel{^}{{y}_{2}}}{\stackrel{^}{{y}_{2}}}\right)}^{2}\right)\text{d}x={\left(y-\stackrel{^}{{y}_{r}}\right)}^{2}$ (12)

${\int }_{1}^{9}\mathrm{exp}\left(-\alpha -{\beta }_{1}{\left(\frac{x-\stackrel{^}{{y}_{1}}}{\stackrel{^}{{y}_{1}}}\right)}^{2}-{\beta }_{2}{\left(\frac{x-\stackrel{^}{{y}_{2}}}{\stackrel{^}{{y}_{2}}}\right)}^{2}\right)\text{d}x=1$ (13)

$\alpha =2$${\beta }_{1}=42.04$${\beta }_{2}=-31.11$ 则， $p\left(x\right)=\mathrm{exp}\left[-2-\text{42}\text{.04}*{\left(\frac{x-4.46}{4.46}\right)}^{2}+\text{31}\text{.11}*{\left(\frac{x-4.1503}{4.1503}\right)}^{2}\right]$ (14)

4. 结果分析

5. 小结

The Prediction of Effectiveness of Weapons and Combat Based on the Principle of Maximum Entropy[J]. 运筹与模糊学, 2018, 08(03): 101-106. https://doi.org/10.12677/ORF.2018.83013

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