﻿ 小样本理论在航空发动机研制费估算中的应用
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Application of small sample theory in aero engine development costs estimation
Liu Fang, Zhang Haitao
The Ministry of Finance and Economics Research, AVIC Development Research Center, Beijing 100029, China
Abstract:The usability of the partial least squares method of small sample theory in military aero engine development costs estimating was discussed. The technical and development expense data of more than ten kinds of military turbojet and turbofan engine were collected. The technical parameters such as turbine inlet temperature, military fuel consumption rate, military thrust, afterburner thrust, thrust weight ratio, engine weight, total pressure ratio, air flow, overhaul life, number of prototypes and completion time were selected. The partial least squares method in military aero engine development costs estimating was used. The military aero engine development cost estimation model based on the major technical parameters was formed, and the development concept inheritance coefficient of military aero engine was established. The results show that the model's accuracy is improved compared with the other existing models, reaching 10%, and can be used to support for aero engine development cost estimation quickly.
Key words: aero engine     development cost     estimation     model     small sample theory     partial least squares method

20世纪末以来,成本上涨、研制周期延长和价格上涨等一系列因素对世界航空发动机的前景产生了重要影响[1].航空发动机成本指标的提高具有指数性,同时,为建立超前的科技储备而进行的探索研究成本所占比重一代比一代大.美国的制造业从第四代发动机向第五代过渡时,这一比重从15%提高到60%,且所费时间延长近1倍.而且工程设计的变化、估算时所用假设条件的变化、需求更改和包括机体制造成本、人工成本、材料成本以及物价浮动在内的经济因素的变化等原因,使得几乎所有型号的研制费都出现了大幅增长的情况[2].如F135发动机项目,其单台成本已经比最初基线增长了30%之多.因此,在发动机立项论证或设计早期对研制经费需求进行快速和准确的估算非常重要[3].

 机型 Y/106美元 x1/N x2 x3/K x4/kg x5/(kg/m2) x6/((kg/N)/h) x7/季度 x8/(kg/s) TF30 554.742 82 242 2.2 1 350 1 746 753 871 0.03 92 109 TF33 133.670 75 568 1 1 145 1 769 282 518 0.02 71 208 TF34 282.035 41 229 1 1 478 644 242 284 0.02 120 153 TF39 496.511 181 369 1 1 578 3 311 286 336 0.01 109 705 J52 291.923 37 789 1.8 1 145 930 188 541 0.04 74 55 J57 199.897 44 453 1.4 1 145 1 887 167 396 0.04 41 73 J60 64.191 5 13 338 1 1 145 209 152 125 0.05 71 23 J65 124.927 32 095 1.2 1 128 1 277 124 813 0.04 46 53 J71 252.066 42 542 1.5 1 200 1 855 161 523 0.04 47 70 J75 416.861 104 468 2 1 145 2 245 245 574 0.04 59 114 J79 405.558 66 679 2 1 200 1 463 265 132 0.04 57 73 J85 330.418 17 111 2 1 167 259 152 125 0.05 74 19 注:Y—从型号设计到型号合格试车的发动机研制费用;x1—海平面静止状态最大额定推力;x2—最大飞行Ma数(与声速有关的速度量);x3—最大涡轮进口温度;x4—发动机净重;x5—发动机压力项;x6—海平面静止状态最大额定推力下的耗油率;x7—研制开始到通过合格试车的时间;x8—最大额定推力下的发动机空气流量.

 变量 x1 x2 x3 x4 x5 x6 x7 x8 x1 1.000 x2 -0.054 1.000 x3 0.627 -0.263 1.000 x4 0.888 -0.044 0.398 1.000 x5 0.381 0.409 0.398 0.276 1.000 x6 -0.691 0.399 -0.805 -0.532 -0.422 1.000 x7 0.389 -0.226 0.846 0.016 0.408 -0.749 1.000 x8 0.887 -0.400 0.765 0.732 0.166 -0.773 0.551 1.000 Y 0.648 0.605 0.565 0.491 0.662 -0.374 0.426 0.420

1) 岭回归计算.

Y=-745.138+0.003 3x1+228.304x2+ 0.226 8x3+0.010 6x4+0.000 63x5+ 41.725 5x6+0.907 0x7+0.019 7x8

2) 主成分分析计算.

Y=-861.072+0.002 6x1+252.097x2+0.265 45x3+0.014 09x4-0.000 06x5+42.597 8x6+0.965 61x7+0.017 5x8

3) 偏最小二乘法的回归计算.

① 能够在自变量存在严重多重相关性的条件下进行回归建模;

② 允许在样本点个数少于变量个数的条件下进行回归建模;

③ 偏最小二乘回归在最终模型中将包含原有的所有自变量;

④ 在偏最小二乘回归模型中,每一个自变量的回归系数将更容易解释.

Y=-839.238+0.005x1+250.424x2+ 0.312x3-0.000 6x4+0.000 06x5- 2.624x6+0.032x7+0.000 3x8

 研制年代 型号 20世纪50年代 HQ-2,PF-1,P-5,WP-5甲,WP-6 20世纪60年代 WP-6甲,WP-6乙,WP-7,WS-5 20世纪70年代 WP-7甲03,WP-6甲08,WP-7甲, WP-7甲05,WP-7乙,WP-7乙B, WP-7乙C,WP-7乙C03,WP-8,WS-6, WS-6甲,WS-8,WS-9,中推 20世纪80年代 WP-13,WP-13AII,WP-13F,WP-13FI,WP13III,WP-7乙III 20世纪90年代 WP-13B,FWP-14,FWS10
2.2 模型结构与自变量的选择

ln Y=a0+a1ln x1+a2ln x2+…

 序号 研制发动机改进改型情况 继承性系数/% 1 在世界范围内没有相似型号的新发动机 15 2 在国内航空发动机中没有相似的新发动机 25 3 在本发动机设计局内没有相似的新发动机 35 4 在本发动机设计局内已有相似的新发动机 45 5 有工艺储备的燃气发生器主要部件有加大结构改变的发动机 55 6 已批生产发动机主要组件结构有显著更改的型号 65 7 批生产发动机主要组件结构有明显更改的型号 75 8 批生产发动机不明显的改进型号 85 9 批生产航空发动机用于其他飞行器 95

 变量 Y Ma x A x6 B D x8 E F x3 G H x4 I J x7 K Y 1.00 Ma 0.17 1.00 x1 0.67 0.32 1.00 A 0.60 0.43 0.97 1.00 x6 -0.23 0.72 -0.08 0.08 1.00 B -0.67 0.16 -0.70 -0.55 0.50 1.00 D 0.36 0.89 0.43 0.48 0.51 -0.15 1.00 x8 0.61 -0.19 0.84 0.72 -0.55 -0.85 -0.01 1.00 E 0.56 0.12 0.74 0.68 -0.07 -0.85 0.38 0.65 1.00 F 0.70 0.44 0.81 0.78 -0.01 -0.75 0.67 0.58 0.87 1.00 x3 0.45 0.69 0.63 0.71 0.37 -0.32 0.79 0.20 0.64 0.85 1.00 G 0.37 -0.33 0.66 0.57 -0.35 -0.64 -0.24 0.83 0.56 0.34 0.10 1.00 H -0.04 0.07 -0.08 0.07 0.34 0.23 -0.07 -0.20 -0.04 -0.14 0.02 -0.22 1.00 x4 0.53 -0.18 0.84 0.77 -0.40 -0.68 -0.13 0.93 0.57 0.48 0.21 0.86 -0.04 1.00 I 0.40 -0.14 0.71 0.63 -0.30 -0.60 0.01 0.78 0.48 0.41 0.23 0.76 -0.11 0.78 1.00 J 0.56 0.40 0.32 0.34 0.06 -0.17 0.51 0.10 0.24 0.52 0.64 0.02 -0.18 0.04 0.11 1.00 x7 0.52 0.66 0.70 0.70 0.10 -0.34 0.77 0.38 0.36 0.70 0.71 0.06 -0.17 0.31 0.43 0.58 1.00 K -0.69 0.20 -0.32 -0.26 0.25 0.57 0.02 -0.40 -0.53 -0.46 -0.25 -0.35 -0.16 -0.36 -0.30 -0.50 -0.03 1.00
 图 1 变量投影重要性柱状图Fig. 1 Variable importance projection histogra

2.3 估算模型的建立

ln Y=a0+a1×ln x1+a2×ln A－a3×ln B+a4×ln x8+a5×ln F+a6×ln x3+a7×ln x4a8×ln I+a9×ln J+a10×ln x7a11×ln K+a12×ln D

 图 2 研制费实际值与拟合值的对比曲线图Fig. 2 Development costs curves of the actual value and the fitted value
 图 3 研制费实际值与估算值的对比柱状图Fig. 3 Development costs histogram of the actual value and the estimated value
2.4 算例分析

 方法 误差/% 本文方法 9.41 空军工程大学方法 -39.46 原航空620所方法 -49.99

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#### 文章信息

Liu Fang, Zhang Haitao

Application of small sample theory in aero engine development costs estimation

Journal of Beijing University of Aeronautics and Astronsutics, 2014, 40(11): 1518-1525.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0340