If the amount of the 6 demands change, you currently have to change resistors. But in the new system you are still going to have to change something. What are you willing to consider as not being too much of a bother?
You could design a custom digital to analog converter, where each of the 6 digital inputs enables a separate current source. But those 6 current sources, while not involving relays, will still have a resistor or something like that to change. You did not say how someone comes to know that the demand has changed from a given source. If it is automatic through realtime instrumentation, you could consider using that instrumentation to control the pump speed directly.
Rather that using feedforward control (as you are doing now) you might consider feedback control. That is where you sense that the pump is falling behind the demand and raise the control voltage according to that measure of how it is falling behind. The disadvantage of feedback control is that the corrective action is delayed until you can sense the deficiency. With feedforward control, you apply corrective action immediately upon knowing that demand has increased. That may or may not be a problem. If the time it takes to increase pump activity is small enough, the pump will speed up before the deficiency becomes too bad. I am speaking in general terms about deficiency because I don't know what your pump does. If, for example, the pump filled a reservoir, the deficiency would be the level in the reservoir. If the pump has to maintain a given pressure with some sort of accumulator, the deficiency would be the amount that the pressure has fallen.