HW2_Group2

組員：唐思琪、林書霆、柯炯名、簡佑臻、王婷、胡祐銓

Q1

set.seed(1111)
s = 1000      # 模擬次數
qty = 20*50*8 # 一天生產的量
dailychips = function(day=1){
  quality = sample(c(1,0), day*qty, prob = c(0.83, 0.17), replace = T) # 生產品質
  defective_accept = rbinom(1, sum(quality==0), prob = 0.1) # 瑕疵但被接受的數量
  good_accept = rbinom(1, sum(quality), prob = 0.95)        # 正常且被接受的數量
  accept = defective_accept + good_accept  # 被接受的數量
  good_accept_ratio = good_accept / accept # 被接受數量中，正常數量的佔比
  c(accept, good_accept_ratio)
}

stable = replicate(s, dailychips())   # 模擬紀錄表
result1 = sum(stable[2,] >= 0.98) / s # 至少有98%商品是正常被遞送的機率
result2 = sum(stable[1,] >= 6400) / s # 至少有6400艘是被接受的機率

print(paste0('(a) : ', result1))

[1] "(a) : 0.281"

print(paste0('(b) : ', result2))

[1] "(b) : 0.896"

Q2

montly = replicate(s, dailychips(30))
quarterly = replicate(s, dailychips(90))
hist(montly[2,], breaks = 20)

hist(quarterly[2,], breaks = 20)

shapiro.test(montly[1,])     # p值大於0.05, 接受假設(為常態分佈)

    Shapiro-Wilk normality test

data:  montly[1, ]
W = 0.99915, p-value = 0.9375

shapiro.test(quarterly[1,])  # p值大於0.05, 接受假設(為常態分佈)

    Shapiro-Wilk normality test

data:  quarterly[1, ]
W = 0.99816, p-value = 0.3539

Q3

library(EnvStats)
library(MASS)
library(dplyr)

S = 10000
full_load = 3800  # 每艘船的載貨量
per_cost = 7200
DGlou = rtri(S, min=4000, max=8000, mode=6000) %>% round(0) # Glou的需求量
DRock = rtri(S, min=4500, max=7500, mode=6000) %>% round(0) # Rock的需求量
byRick = runif(S, min=0.5, max=1.0) *full_load %>% round(0)          # 由Rick捕魚的每日生產量(比例)
byMorty = rtri(S, min=0.5, max=1, mode=0.75) *full_load %>% round(0) # 由Morty捕魚的每日生產量(比例)

# 兩地價格會互相影響(共變)
sigmaG = 0.5
sigmaR = 0.25
meanG = 3.5
meanR = 3.65
cor_GR = seq(-0.8,0.8,0.2)    # 所有Glou和Rock可能的相關係數

# 初始化存放空間
meanPayoff = matrix(nrow=length(cor_GR), ncol=4)
CVAR5 = matrix(nrow=length(cor_GR), ncol=4)

for (i in (1:length(cor_GR))){
  # Glou和Rock的共變異數
  cov_GR = sigmaG * sigmaR * cor_GR[i] 
  # 模擬在不同的相關係數下的共變異矩陣
  covMatrix = matrix(c(sigmaG^2, cov_GR, cov_GR, sigmaR^2), nrow=2)
  # 根據共變異矩陣結果不同，模擬10000次價格
  price = mvrnorm(S, mu = c(meanG, meanR), Sigma = covMatrix) 
  payoff = matrix(nrow=S, ncol=4) # 初始化策略總獲利表
  
  for (s in 1:S){
    # a) Rick goes to Glou and Morty goes to Rock
    payoff[s,1] = min(DGlou[s], byRick[s]) * price[s,1] + 
                  min(DRock[s], byMorty[s]) * price[s,2] - 2*per_cost
    # b) Rick goes to Rock and Morty goes to Glou
    payoff[s,2] = min(DRock[s], byRick[s]) * price[s,2] +
                  min(DGlou[s], byMorty[s]) * price[s,1] - 2*per_cost
    # c) Rick and Morty go to Glou
    payoff[s,3] = min(DGlou[s], (byRick[s] + byMorty[s])) * price[s,1] - 2*per_cost
    # d) Rick and Morty go to Rock
    payoff[s,4] = min(DRock[s], (byRick[s] + byMorty[s])) * price[s,2] - 2*per_cost

  }
  
  # 有4個strategy各自迭代計算出平均payoff和CVAR5
  for (j in c(1:4)){
    meanPayoff[i,j] = mean(payoff[,j])
    VAR5 = quantile(payoff[,j], probs = 0.05) # 最壞5%
    CVAR5[i,j] = payoff[which(payoff[,j]<= VAR5), j] %>% mean()
  }
}

(a)

plot(cor_GR, meanPayoff[,1], col='slateblue', lwd=2, type="l", ylim=c(4000, 6500), ylab = 'Avg. Payoff', xaxt="n")
lines(cor_GR, meanPayoff[,2], col='sienna1')

lines(cor_GR, meanPayoff[,3], col='lightskyblue')
lines(cor_GR, meanPayoff[,4], col='seagreen4')

axis(1, at = cor_GR)

(b)

plot(cor_GR, CVAR5[,1], col='slateblue', type="l", ylim=c(-3000, 2000), ylab = 'CVAR 5%', xaxt="n")
lines(cor_GR, CVAR5[,2], col='sienna1')

lines(cor_GR, CVAR5[,3], col='lightskyblue')
lines(cor_GR, CVAR5[,4], col='seagreen4')

axis(1, at = cor_GR)

Q4

bidding = function(n = 100000000){
  willing_to_pay = max(rnorm(n, mean = 3000000, sd = 350000))
}
two = replicate(2, bidding()) %>% max() %>% round()
three = replicate(3, bidding()) %>% max() %>% round()
ten = replicate(10, bidding()) %>% max() %>% round()
two; three; ten

[1] 4971453
[1] 5041671
[1] 5128140

• 我們發現隨著競價的人數變多(2->3->10)，價格會變大。