張忠謀先生說「…他們應該知道，pricing（定價）是我最有興趣的！」、「不止技術背景的，任何一位總經理，都要學sales marketing，包括定價。」
https://www.wealth.com.tw/home/articles/22548
工作的關係，我們也在定價。於是有這題應用數學：估價一片晶圓。
雖然網路上很多資訊、資深半導體觀察者等肯定也研究過這問題，
好比此新聞說一片5nm晶圓約17000 USD：
不過今天我們不靠消息，我們純靠數學來解這問題。所有數據來自TSMC財報。
好比我們拿2021 Q1來說，
https://investor.tsmc.com/english/quarterly-results/2021/q1
財報指出該季Revenue 362.41 Billion NTD，Shipment 3.359 Million pieces of wafers，
其中5nm技術貢獻了14% Revenue, 7nm 35%, 10nm 0%, 16nm 14%, …
我們要預測一片5nm晶圓多少錢？7nm? 10nm? 16nm?...
用以上的資訊可以先推出
Revenue by 5nm = R5 = R * 14% = 362.41 * 14% = 50.737 B NTD = P5 * S5 = (price of 5nm wafer) * (# of 5nm wafers shipped)
以此類推：
R7 = 126.84 B = P7 * S7
R10 = 0 = P10 * S10
…
R250 = 7.2B = P250 * S250
而條件是S5+S7+S10+...+S250 = 3.359 M pcs
這是2021 Q1，過去12季以上的財報資訊都收集得到，所以整理一下：
''' # Revenue (B NTD): technology (5nm, 7nm, ...) by quarters (2021Q1, 2020Q4, 2020Q3, ...)
array([
[ 50.7374, 72.306 , 28.5144, 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. , 0. ],
[126.8435, 104.8437, 124.7505, 111.852 , 108.71 , 111.034 , 79.1235, 50.61 , 48.114 , 66.6471, 28.6385, 0. , 0. ],
[ 0. , 0. , 0. , 0. , 1.553 , 3.1724, 5.861 , 7.23 , 8.748 , 17.3862, 15.621 , 30.3264, 47.1352],
[ 50.7374, 46.9989, 64.1574, 55.926 , 59.014 , 63.448 , 64.471 , 55.43 , 34.992 , 57.954 , 59.8805, 48.9888, 44.6544],
[ 0. , 3.6153, 3.5643, 3.107 , 3.106 , 3.1724, 2.9305, 2.41 , 2.187 , 2.8977, 5.207 , 9.3312, 9.9232],
[ 39.8651, 39.7683, 42.7716, 43.498 , 43.484 , 41.2412, 46.888 , 43.38 , 43.74 , 49.2609, 49.4665, 53.6544, 49.616 ],
[ 25.3687, 28.9224, 28.5144, 27.963 , 31.06 , 25.3792, 29.305 , 26.51 , 26.244 , 28.977 , 31.242 , 25.6608, 27.2888],
[ 18.1205, 18.0765, 17.8215, 18.642 , 18.636 , 22.2068, 20.5135, 19.28 , 17.496 , 23.1816, 20.828 , 20.9952, 22.3272],
[ 10.8723, 7.2306, 7.1286, 9.321 , 9.318 , 9.5172, 5.861 , 7.23 , 6.561 , 8.6931, 10.414 , 11.664 , 12.404 ],
[ 10.8723, 10.8459, 7.1286, 9.321 , 6.212 , 9.5172, 5.861 , 4.82 , 6.561 , 5.7954, 7.8105, 4.6656, 4.9616],
[ 21.7446, 25.3071, 24.9501, 24.856 , 24.848 , 25.3792, 26.3745, 19.28 , 17.496 , 23.1816, 23.4315, 20.9952, 22.3272],
[ 7.2482, 3.6153, 7.1286, 6.214 , 6.212 , 3.1724, 5.861 , 4.82 , 6.561 , 5.7954, 7.8105, 6.9984, 7.4424]
])
'''
'''
Shipment by quarter
array([3.359, 3.246, 3.24 , 2.985, 2.925, 2.823, 2.733, 2.308, 2.205, 2.686, 2.712, 2.674, 2.68 ])
# unit Mpcs
'''
所以問題是這樣子的：
Rq = Pi @ Siq
subject to sum(Si) = Sq
Rq、Sq指的是某一季的Revenue、# of wafer shipped
Pi 指的是某代技術一片晶圓的價錢
Siq 指的是某季某代的shipment
求Pi, Siq for all i
典型的最小方差問題。
(順便復習了Lagrange Multiplier，真是天才，他怎麼這麼帥？)
無論如何：算出來的答案是這樣子：
'''
{'5nm': 174.26607759869134,
'7nm': 174.26607759845757,
'10nm': 174.26607759939,
'16nm': 174.26607759932566,
'20nm': 141.87086687980448,
'28nm': 141.8708668804317,
'40/45nm': 141.87086688136566,
'65nm': 141.8708668814012,
'90nm': 43.14014723998238,
'0.11/0.13um': 43.140147240173974,
'0.15/0.18um': 43.14014724000278,
'0.25um+': 16.979284149443302}
# unit K NTD
'''
https://github.com/willhyper/wafer-by-technology/blob/main/wafer_price.py
一片5nm晶圓是174.2 K NTD = 17.4萬台幣？ vs 網路上的17000 USD 會不會太低？
客位看倌看出哪裡有問題了嗎？:)
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