# How do I simultaneously minimize two different functions who have the same inputs?

I want to minimize two different functions simultaneously who have the same inputs. The functions are both linear and non-exponential.

$$F_1(X_1, X_2) = a_1X_1 + a_2X_2$$ $$F_2(X_1, X_2) = b_1X_1 + b_2X_2$$

I would prefer to do this in R.

Thanks again

• There's no unique minimizer, since your objective is not scalar, so the problem is not well-posed. (Which is smaller: (0,1) or (1,0)?). Can you elaborate a bit about the background? Jun 3 '14 at 19:01
• Why do want to minimize the simultaneously? If you just want to solve each problem independently, there's no restriction on what order you solve them in. if you want to, say, minimize the sum of the two function, that is a single minimization problem with an objective function that is the sum of the two you give. Jun 3 '14 at 19:37
• the two input variables are number of employees and number of calls. The problem is simulating a call center where i'm trying to simultaneously minimize the number of missed calls as well as the number of idle hours(amount of time our employees sit idle and do nothing). Jun 3 '14 at 19:42
• In this case, the real-world background provides a natural scalarization: Presumably, you are actually interested in minimizing loss to revenue, so if you figure out how much each missed call and each idle employee-hour costs you, you can multiply each function by that number and minimize the sum, thus minimizing total loss. Jun 3 '14 at 19:59
• Are there any constraints? Jun 3 '14 at 21:41