我使用lpSolveAPI运行线性编程模型。我能够让模型工作，但我想添加一个约束，我不知道该怎么做，或者它是否可行。模型的细节：
type1 = A，B，C，D或E，type2 = X或Y. 以下示例效果很好，但我想再添加两个约束，我不确定如何做到这一点。另外两个限制因素：
我希望每次优化至少有2个type1个实例。我不在乎哪种类型有多种或两种不同的类型是倍数（例如2 A＆2和C＆＃39;）这就是为什么我想到的它作为＆＃34;或＆＃34;约束（A> 2或B> 2或......）。
这个可能会有点困难：无论是哪种类型的＆＃34;键入Y＆＃34;被选中，我不希望type1再次出现。所以说Y项目是type1 = C，我希望所有其他选择的项目都不是C.我想我还需要添加另一个虚拟矩阵，交互type1和type2
期望结果的示例：
id type1 type2 value cost
10 10 B X 19 865
11 11 C Y 19 1097
18 18 D X 19 1005
40 40 B X 20 956
45 45 A X 20 980
工作示例：
library(dplyr)
library(lpSolveAPI)
# setup df
id <- 1:50
type1 <- sample(c('A', 'B', 'C', 'D', 'E'), length(id), replace = T)
type2 <- sample(c('X', 'X', 'X', 'Y'), length(id), replace = T)
value <- round(runif(length(id), 0, 20),0)
cost <- round(runif(length(id), 750, 1250),0)
df <- data.frame(id, type1, type2, value, cost) %>%
mutate(total = 1)
# Attach dummy vars
type1Dummy <- as.data.frame.matrix(table(df$id, df$type1))
type2Dummy <- as.data.frame.matrix(table(df$id, df$type2))
df <- cbind(df, type1Dummy, type2Dummy)
# constraints
totalNum <- 5
totalCost <- 5000
totalX <- 4
totalY <- 1
rhs <- c(totalNum, totalCost, totalX, totalY)
# Direction vector
numDir <- '=='
costDir <- '<='
xDir <- '=='
yDir <- '=='
dir <- c(numDir, costDir, xDir, yDir)
# Setup opt
obj <- df$value
mat <- data.frame(total = df$total, cost = df$cost, X = df$X, Y = df$Y)
# Solver Setup
lprec <- make.lp(4, nrow(mat))
for(i in 1:nrow(mat)){
vals <- mat[i,] %>% as.numeric(.)
set.column(lprec, i, vals)
}
set.objfn(lprec, df$value)
set.constr.type(lprec, dir)
set.rhs(lprec, rhs)
for(i in 1:nrow(mat)){
set.type(lprec, i, "binary")
}
# Add constraint with dummy variables that are {0,1} if more than 1 are selected.
# z1 <- ifelse(sum(x[type1 == 'A']) > 1, 1, 0)
# z2 <- ifelse(sum(x[type1 == 'B']) > 1, 1, 0)
# etc...
# add.constraint(lprec, z1 + z2 + z3 + z4 + z5, ">", 1) # "at least one of the groupings needs more than 1.
lp.control(lprec,sense='max')
solve(lprec)
get.objective(lprec)
sol <- get.variables(lprec)
df$selected <- sol
dfSolved <- df[df$selected == 1,]
dfSolved
感谢您的帮助！
答案 0 :(得分：1)
这个花了一段时间。约束集1.是可行的，但第二个（如果Y是C，任何X不能是C）都需要一些杂耍。
我让它工作，但代码看起来不太好。也许很难遵循。所以我建议你先看看下面我印刷的LP。一旦你按照我介绍的新变量和新变量进行操作，代码可能会更容易理解。
为方便起见，我清楚地命名了变量和约束。看看这是否有帮助。
对At least 2 instances of Type 1要求进行建模
为了强制执行此操作，我们需要5个新变量和6个新约束。
让我们创建五个新的0/1变量，称为dblA，dblB，... dblE
现在简单的约束是：
dblA +dblB +dblC +dblD +dblE >= 1
现在，如何强制执行：
如果所选项目中至少有两个A，则dblA应为1？
sum(over all items that have Type 1 = A) >= 2 dblA
dblA是0/1变量。如果为0，则带有A的项目可以是任意数字。
如果dblA为1，那么不等式会强制所选项目中的至少两个为Type 1 A.有五个这样的约束，每个约束从A到E调用在下面的表述中AA到EE。
配方，使用LPSolveAPI打印
/* Objective function */
max: +6 x1 +6 x2 +8 x3 +16 x4 +6 x5 +5 x6 +17 x7 +11 x8 +10 x9 +7 x10 +9 x11 +4 x12 +3 x13 +11 x14 +15 x15
+17 x16 +13 x17 +19 x18 +12 x19 +8 x20 +16 x21 +4 x22 +16 x23 +16 x24 +x26 +16 x27 +16 x28 +18 x29 +16 x30
+4 x31 +7 x32 +12 x33 +19 x34 +4 x35 +13 x36 +5 x37 +20 x38 +4 x39 +3 x40 +x41 +6 x42 +5 x43 +13 x44
+11 x45 +16 x46 +16 x47 +3 x48 +x49 +8 x50;
/* Constraints */
Take_5: +x1 +x2 +x3 +x4 +x5 +x6 +x7 +x8 +x9 +x10 +x11 +x12 +x13 +x14 +x15 +x16 +x17 +x18 +x19 +x20 +x21 +x22
+x23 +x24 +x25 +x26 +x27 +x28 +x29 +x30 +x31 +x32 +x33 +x34 +x35 +x36 +x37 +x38 +x39 +x40 +x41 +x42
+x43 +x44 +x45 +x46 +x47 +x48 +x49 +x50 <= 5;
budget: +1161 x1 +795 x2 +962 x3 +996 x4 +825 x5 +788 x6 +846 x7 +977 x8 +1130 x9 +1092 x10 +1168 x11 +1113 x12
+757 x13 +803 x14 +936 x15 +1001 x16 +830 x17 +1138 x18 +1179 x19 +970 x20 +1206 x21 +1008 x22 +793 x23
+803 x24 +834 x25 +923 x26 +1056 x27 +815 x28 +798 x29 +1075 x30 +872 x31 +808 x32 +796 x33 +781 x34
+1224 x35 +1165 x36 +1238 x37 +1114 x38 +935 x39 +1212 x40 +803 x41 +1086 x42 +869 x43 +921 x44 +941 x45
+758 x46 +1108 x47 +927 x48 +1009 x49 +921 x50 <= 5000;
X_4: +x1 +x3 +x4 +x5 +x6 +x7 +x8 +x10 +x11 +x12 +x13 +x14 +x16 +x17 +x18 +x20 +x21 +x22 +x23 +x24 +x25 +x26
+x27 +x28 +x29 +x30 +x32 +x33 +x34 +x35 +x36 +x38 +x39 +x40 +x41 +x42 +x43 +x44 +x46 +x48 +x49 +x50 <= 4;
Y_1: +x2 +x9 +x15 +x19 +x31 +x37 +x45 +x47 <= 1;
/* Constraints: Must pick at least one of the Double variables */
AA: +x2 +x3 +x7 +x8 +x12 +x18 +x24 +x25 +x26 +x43 +x45 -2 dblA >= 0;
BB: +x10 +x13 +x23 +x30 +x32 +x35 +x37 +x39 +x40 +x46 +x48 -2 dblB >= 0;
CC: +x4 +x16 +x28 +x41 +x44 +x47 -2 dblC >= 0;
DD: +x1 +x6 +x9 +x11 +x14 +x15 +x17 +x20 +x21 +x29 +x31 +x34 -2 dblD >= 0;
EE: +x5 +x19 +x22 +x27 +x33 +x36 +x38 +x42 +x49 +x50 -2 dblE >= 0;
Pick2of1: +dblA +dblB +dblC +dblD +dblE >= 1;
/* Constraints: if Y is A, then none of the X's can be. */
totYA: +x2 +x45 -100 anyYA <= 0;
totYB: +x37 -100 anyYB <= 0;
totYC: +x47 -100 anyYC <= 0;
totYD: +x9 +x15 +x31 -100 anyYD <= 0;
totYE: +x19 -100 anyYE <= 0;
totXA: +x3 +x7 +x8 +x12 +x18 +x24 +x25 +x26 +x43 -100 anyXA <= 0;
totXB: +x10 +x13 +x23 +x30 +x32 +x35 +x39 +x40 +x46 +x48 -100 anyXB <= 0;
totXC: +x4 +x16 +x28 +x41 +x44 -100 anyXC <= 0;
totXD: +x1 +x6 +x11 +x14 +x17 +x20 +x21 +x29 +x34 -100 anyXD <= 0;
totXE: +x5 +x22 +x27 +x33 +x36 +x38 +x42 +x49 +x50 -100 anyXE <= 0;
YAorXA: +anyYA +anyXA <= 1;
YBorXB: +anyYB +anyXB <= 1;
YCorXC: +anyYC +anyXC <= 1;
YDorXD: +anyYD +anyXD <= 1;
YEorXE: +anyYE +anyXE <= 1;
/* Variable bounds */
x1 <= 1; (all variables are binary)
/* Integer definitions */
int x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18,x19,x20,x21,x22,x23,x24,x25,x26,x27,x28,x29,x30,x31,x32,x33,x34,x35,x36,x37,x38,x39,x40,x41,x42,x43,x44,x45,x46,x47,x48,x49,x50,dblA,dblB,dblC,dblD,dblE,xYA,xYB,xYC,xYD,xYE,xXA,xXB,xXC,xXD,xXE,anyYA,anyYB,anyYC,anyYD,anyYE,anyXA,anyXB,anyXC,anyXD,anyXE;
这是我的R代码。（功能齐全，未优化）
library(lpSolveAPI)
library(dplyr)
# setup df
opt <- function(){
id <- 1:50
type1 <- sample(c('A', 'B', 'C', 'D', 'E'), length(id), replace = T)
type2 <- sample(c('X', 'X', 'X', 'Y'), length(id), replace = T)
value <- round(runif(length(id), 0, 20),0)
cost <- round(runif(length(id), 750, 1250),0)
df <- data.frame(id, type1, type2, value, cost) %>%
mutate(total = 1)
# Attach dummy vars
type1Dummy <- as.data.frame.matrix(table(df$id, df$type1))
type2Dummy <- as.data.frame.matrix(table(df$id, df$type2))
df <- cbind(df, type1Dummy, type2Dummy)
#Add 10 new columns. XY and ABCDE combined.
df$YA <- ifelse(df$A & df$Y, 1, 0)
df$YB <- ifelse(df$B & df$Y, 1, 0)
df$YC <- ifelse(df$C & df$Y, 1, 0)
df$YD <- ifelse(df$D & df$Y, 1, 0)
df$YE <- ifelse(df$E & df$Y, 1, 0)
df$XA <- ifelse(df$A & df$X, 1, 0)
df$XB <- ifelse(df$B & df$X, 1, 0)
df$XC <- ifelse(df$C & df$X, 1, 0)
df$XD <- ifelse(df$D & df$X, 1, 0)
df$XE <- ifelse(df$E & df$X, 1, 0)
# constraints
totalNum <- 5
totalCost <- 5000
totalX <- 4
totalY <- 1
rhs <- c(totalNum, totalCost, totalX, totalY)
rhs2 <- c(rhs, 0,0,0,0,0, 1)
rhs3 <- c(rhs2, rep(0, 10), rep(1, 5))
# Direction vector
numDir <- '=='
costDir <- '<='
xDir <- '=='
yDir <- '=='
dir <- c(numDir, costDir, xDir, yDir)
gt <- '>='
lt <- '<='
eq <- "=="
dir2 <- c(dir, rep(gt, 5), gt)
dir3 <- c(dir2, rep(eq, 10),
rep(lt, 5))
#constraints df
df$atleast2 <- 0
cons <- data.frame(df$total, df$cost, df$X, df$Y,
df$A, df$B, df$C, df$D, df$E, df$atleast2,
df$YA, df$YB, df$YC, df$YD, df$YE,
df$XA, df$XB, df$XC, df$XD, df$XE) #shape is 50 x 20
tenzeros <- rep(0, 10)
z20 <- rep(0, 20)
z75 <- rep(0, 75)
#New 2 of 1-kind constraints...
cons <- rbind(cons, c(0,0,0,0,-2,0,0,0,0, 1,tenzeros)) # adding a new 0-1 variable for dbl_A
cons <- rbind(cons, c(0,0,0,0,0,-2,0,0,0, 1,tenzeros))
cons <- rbind(cons, c(0,0,0,0,0,0,-2,0,0, 1,tenzeros))
cons <- rbind(cons, c(0,0,0,0,0,0,0,-2,0, 1,tenzeros))
cons <- rbind(cons, c(0,0,0,0,0,0,0,0,-2, 1,tenzeros)) # adding a new 0-1 variable for dbl_E
# Add 20 rows to cons: 10 for YA...XE and 10 more for anyYA to anyXE
for(i in 1:20){
cons <- rbind(cons, z20)
}
BIGM <- 100
print(dim(cons))
for(j in 1:10){ #make the anyYA to anyYE variables -1
cons[65+j,10+j] <- -1 * BIGM
}
#finally add the one of AnyXA or AnyYA constraints
for(xcol in 1:5){
cons <- cbind(cons, z75)
}
for(j in 1:5){ #make the anyYA and anyxA variables 1 is YAorXA
cons[65+j, 20+j] <- 1 #coeff of YA
cons[70+j, 20+j] <- 1 #coeff of XA
}
dim(cons)
# Setup opt
obj <- c(df$value, rep(0, 25))
# Solver Setup
lprec <- make.lp(ncol(cons), nrow(cons))
lprec
for(i in 1:nrow(cons)){
vals <- cons[i, ] %>% as.numeric(.)
set.column(lprec, i, vals)
}
lprec
length(obj)
set.objfn(lprec, obj)
set.constr.type(lprec, dir3)
set.rhs(lprec, rhs3)
for(xcol in 1:75){
set.type(lprec, xcol, "binary")
}
lp.control(lprec,sense='max')
row_names <- c('Take_5', 'budget', 'X_4', 'Y_1',
'AA', 'BB', 'CC', 'DD', 'EE', 'Pick2of1',
'totYA', 'totYB', 'totYC', 'totYD', 'totYE',
'totXA', 'totXB', 'totXC', 'totXD', 'totXE',
'YAorXA', 'YBorXB', 'YCorXC', 'YDorXD', 'YEorXE'
)
col_names <- c(paste0('x', 1:50), 'dblA', 'dblB', 'dblC', 'dblD', 'dblE',
'xYA', 'xYB', 'xYC', 'xYD', 'xYE',
'xXA', 'xXB', 'xXC', 'xXD', 'xXE',
'anyYA', 'anyYB', 'anyYC', 'anyYD', 'anyYE',
'anyXA', 'anyXB', 'anyXC', 'anyXD', 'anyXE'
)
dimnames(lprec) <- list(row_names, col_names)
#write out the LP (useful for debugging)
write.lp(lprec, filename = "test.lp")
solve(lprec)
print(get.objective(lprec))
sol <- get.variables(lprec)
print(sol)
df$selected <- sol[1:50]
dfSolved <- df[df$selected == 1,]
print(dfSolved)
print(sol[-10:-1])
#print(get.sensitivity.rhs(lprec))
return(df)
}
df <- opt()
以下是一个示例解决方案：
id type1 type2 value cost total A B C D E X Y YA YB YC YD YE XA XB XC XD XE atleast2 selected
18 18 A X 19 1138 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1
29 29 D X 18 798 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1
34 34 D X 19 781 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1
38 38 E X 20 1114 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1
47 47 C Y 16 1108 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1
我检查了它确实满足了你所有的限制。如果有任何不清楚的地方，请询问。