# Combine Two Numbers that Satisfies these Five(5) Conditions with R

I want two(2) numbers $$\phi_{1}, \quad \phi_{2}$$ such that the following five(5) conditions will be satisfied using R:

1. $$\phi_{1} + \phi_{2} < 1$$
2. $$\phi_{1} - \phi_{2} < 1$$
3. $$-1 < \phi_{1} < 1$$
4. $$-1 < \phi_{2} < 1$$
5. $$\phi_{1}$$ and $$\phi_{2}$$ are in one(1) decimal place
• $$\phi_1=\phi_2=0$$ – Xi'an Jan 12 at 12:01
• This is called a linear program and the subject that addresses its solution is called linear programming. The particular problem of finding a solution is called finding a feasible solution. – whuber Jan 12 at 14:09

Brute force, not much thinking: $$\phi_1$$ and $$\phi_2$$ are both between -1 and 1 and they are given to one decimal place. That makes them discrete and we can simply test all possible combinations of $$\phi_1$$ and $$\phi_2$$.

phi1 <- seq(-.9, .9, .1)
phi2 <- seq(-.9, .9, .1)
grid <- expand.grid(phi1 = phi1, phi2 = phi2)

grid$$condition1 <- grid$$phi1 + grid$$phi2 < 1 grid$$condition2 <- grid$$phi1 - grid$$phi2 < 1

grid$$valid <- grid$$condition1 & grid$$condition2 valid.combinations <- grid[which(grid$$valid),]

plot(grid$$phi1, grid$$phi2, xlim = c(-1, 1), ylim = c(-1,1), col = "grey", ,
xlab = expression(phi[1]), ylab = expression(phi[2]))
points(valid.combinations$$phi1, valid.combinations$$phi2, pch = 16)