在R中使用线性（或非线性？）约束优化值

Question

在不同的限制条件下，我试图挑选最好的梦幻足球队。我的目标是挑选能够最大化其预测积分总和的球员。

制约因素是：

1） 团队必须包括：

-1个QB

-2苏格兰皇家银行

-2个WRs

-1 TE

2）玩家的风险不得超过6

3） 球员的费用总额不得超过300英镑。

如何做到这一点？R中优化这些约束的最佳包/函数是什么？在给定这些约束的情况下，最大化投影点的函数调用会是什么样子？仅供参考，我将搜索100-300名玩家。

提前感谢！下面是一个小示例数据集：

name <- c("Aaron Rodgers","Tom Brady","Arian Foster","Ray Rice","LeSean McCoy","Calvin Johnson","Larry Fitzgerald","Wes Welker","Rob Gronkowski","Jimmy Graham")

pos <- c("QB","QB","RB","RB","RB","WR","WR","WR","TE","TE")

pts <- c(167, 136, 195, 174, 144, 135, 89, 81, 114, 111) 

risk <- c(2.9, 3.4, 0.7, 1.1, 3.5, 5.0, 6.7, 4.7, 3.7, 8.8) 

cost <- c(60, 47, 63, 62, 40, 60, 50, 35, 40, 40) 

mydata <- data.frame(name, pos, pts, risk, cost) 

Answer 1

你的约束和目标是线性的，但你的变量是二进制的：是否应该挑选每个玩家。所以你的问题比线性规划（LP）更一般，它是一个混合整数规划（MIP）。在CRAN的优化任务视图中，查找他们的MIP部分。

CPLEX是一个您可能无法访问的商业求解器，但GLPK是免费的。如果我是你，我可能会使用高级接口Rglpk。

这将需要你把你的问题在矩阵形式，我建议你看留档和例子。

编辑：以下是一个实现：

# We are going to solve:
# maximize f'x subject to A*x <dir> b
# where:
#   x is the variable to solve for: a vector of 0 or 1:
#     1 when the player is selected, 0 otherwise,
#   f is your objective vector,
#   A is a matrix, b a vector, and <dir> a vector of "<=", "==", or ">=",
#   defining your linear constraints.

# number of variables
num.players <- length(name)
# objective:
f <- pts
# the variable are booleans
var.types <- rep("B", num.players)
# the constraints
A <- rbind(as.numeric(pos == "QB"), # num QB
           as.numeric(pos == "RB"), # num RB
           as.numeric(pos == "WR"), # num WR
           as.numeric(pos == "TE"), # num TE
           diag(risk),              # player's risk
           cost)                    # total cost

dir <- c("==",
         "==",
         "==",
         "==",
         rep("<=", num.players),
         "<=")

b <- c(1,
       2,
       2,
       1,
       rep(6, num.players),
       300)

library(Rglpk)
sol <- Rglpk_solve_LP(obj = f, mat = A, dir = dir, rhs = b,
                      types = var.types, max = TRUE)
sol
# $optimum
# [1] 836                      ### <- the optimal total points

# $solution
#  [1] 1 0 1 0 1 1 0 1 1 0     ### <- a `1` for the selected players

# $status
# [1] 0                        ### <- an optimal solution has been found

# your dream team
name[sol$solution == 1]
# [1] "Aaron Rodgers"  "Arian Foster"   "LeSean McCoy"
# [4] "Calvin Johnson" "Wes Welker"     "Rob Gronkowski