# an integer programming riddle

April 20, 2016
By

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A puzzle on The Riddler this week that ends up as a standard integer programming problem. Removing the little story around the question, it boils down to optimise

200a+100b+50c+25d

under the constraints

400a+400b+150c+50d≤1000, b≤a, a≤1, c≤8, d≤4,

and (a,b,c,d) all non-negative integers. My first attempt was a brute force R code since there are only 3 x 9 x 5 = 135 cases:

```f.obj<-c(200,100,50,25)
f.con<-matrix(c(40,40,15,5,
-1,1,0,0,
1,0,0,0,
0,0,1,0,
0,0,0,1),ncol=4,byrow=TRUE)
f.dir<-c("<=","<=","<=","<=","<=")
f.rhs<-c(100,0,2,8,4)

sol=0
for (a in 0:1)
for (b in 0:a)
for (k in 0:8)
for (d in 0:4){
cost=f.con%*%c(a,b,k,d)-f.rhs
if (max(cost)<=0){ gain=f.obj%*%c(a,b,k,d) if (gain>sol){
sol=gain
argu=c(a,b,k,d)}}
```

which returns the value:

```> sol
[,1]
[1,]  425
> argu
[1] 1 0 3 3
```

This is confirmed by a call to an integer programming code like lpSolve:

```> lp("max",f.obj,f.con,f.dir,f.rhs,all.int=TRUE)
Success: the objective function is 425
> lp("max",f.obj,f.con,f.dir,f.rhs,all.int=TRUE)\$sol
[1] 1 0 3 3
```

which provides the same solution.

Filed under: Books, Kids, R Tagged: 538, cross validated, FiveThirtyEight, integer programming, The Riddler

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