Engineering design problems and non-parametric statistical results
I. Tension/compression spring design
Consider: 𝑋 = [𝑥
1
𝑥
2
𝑥
3
] = [𝑑 𝐷 𝑁],
Minimize: 𝑓 (𝑋) = (𝑥
3
+ 2)𝑥
2
𝑥
2
1
,
Subject to: 𝑔
1
(𝑋) = 1 −
𝑥
3
𝑥
3
2
71785𝑥
4
1
≤ 0,
𝑔
2
(𝑋) =
4𝑥
2
2
− 𝑥
1
𝑥
2
12566
𝑥
2
𝑥
3
1
− 𝑥
4
1
+
1
5108𝑥
2
1
− 1 ≤ 0,
𝑔
3
(𝑋) = 1 −
140.45𝑥
1
𝑥
3
𝑥
2
2
≤ 0,
𝑔
4
(𝑋) =
𝑥
1
+ 𝑥
2
1.5
− 1 ≤ 0,
Variable range: 0.05 ≤ 𝑥
1
≤ 2,
0.25 ≤ 𝑥
2
≤ 1.3,
2 ≤ 𝑥
3
≤ 15.
II. Pressure vessel design
Consider: 𝑋 = [𝑥
1
𝑥
2
𝑥
3
𝑥
4
] = [𝑇
𝑠
𝑇
ℎ
𝑅 𝐿],
Minimize: 𝑓 (𝑋) = 0.6224𝑥
1
𝑥
2
𝑥
3
+ 1.7781𝑥
2
𝑥
2
3
+ 3.1661𝑥
4
𝑥
2
1
+ 19.84𝑥
3
𝑥
2
1
,
Subject to: 𝑔
1
(𝑋) = −𝑥
1
+ 0.0193𝑥
3
≤ 0,
𝑔
2
(𝑋) = −𝑥
3
+ 0.00954𝑥
3
≤ 0,
𝑔
3
(𝑋) = −𝜋𝑥
4
𝑥
2
3
−
4
3
𝜋𝑥
3
3
+ 1296000 ≤ 0,
𝑔
4
(𝑋) = 𝑥
4
− 240 ≤ 0,
Variable range: 0 ≤ 𝑥
1
≤ 99,
0 ≤ 𝑥
2
≤ 99,
10 ≤ 𝑥
3
≤ 200,
10 ≤ 𝑥
4
≤ 200.
X. Wang et al.: Preprint submitted to Elsevier Page 1 of 10
