Optimization with schedule constraints

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Discussion Overview

The discussion revolves around an optimization problem involving the management of liquid chemicals in vertical containers, focusing on minimizing the cumulative sum of chemical heights through optimal switching of an outflow valve. The problem includes constraints related to scheduled outflow times for specific chemicals and seeks to explore potential solutions or approximations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant describes an optimization problem involving vertical containers with varying inflow rates and a common outflow valve, aiming to minimize the cumulative sum of chemical heights.
  • Another participant questions how the sum of heights can be compared over time, suggesting a need for clarity on the definition of "smaller" in this context.
  • A participant acknowledges that the sum varies with time and expresses a preference for solving the problem over a finite horizon rather than indefinitely.
  • There is a clarification that the goal is to minimize the sum of squares of the heights, rather than just the sum of heights.
  • A participant raises a question about the meaning of making decisions to minimize the sum from t_0 to t_Inf, prompting a discussion about potential integration with respect to time.
  • Another participant introduces the concept of "optimal control" as a relevant area of applied mathematics but admits a lack of understanding of how it specifically relates to the problem at hand.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and clarity regarding the optimization problem, with some points of confusion remaining unresolved. There is no consensus on a specific approach or solution to the problem.

Contextual Notes

Participants have not fully defined the assumptions underlying the optimization problem, such as the nature of the flow rates, the specifics of the scheduled outflow, and the implications of minimizing the sum of squares versus the sum of heights.

aydos
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Hi, I have a optimization problem and I need to find a way to solve it even if only with an approximate solution.

Let's suppose we have a finite set of vertical containers each with a distinct liquid chemical inside.(say a handful of vertical pipes). At the top, these containers have an inflow of chemical all at different and varying flow rates. We have sensors to measure the height of the chemical inside each container. At the bottom, they all connect into a single common point with a switching outflow valve that allows the discharge of only one container at any point in time.

The optimization comprises the minimization of the cumulative sum of all chemical heights, via the optimal switching times of the outflow valve. Switching has a cost of no flow for X seconds.

This seems rather like a common production scheduling problem where one needs to work out the sequence and switching times of the valve in real time based on sensor readings. But there is one additional constraint:

- there is a set of future time windows (schedule) during which a particular chemical must be flowing out. The schedule is sparse, say about 90% of the time there is nothing scheduled.

One obvious approximation is to approximate the optimal solution by optimizing without the schedule constraints and simply switch to the scheduled chemical when the time comes and hope for the best. This is however, not enough. I suspect there must be more optimal solutions out there.

Any ideas or pointers of where I could find solutions to similar problems?
 
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aydos said:
The optimization comprises the minimization of the cumulative sum of all chemical heights, via the optimal switching times of the outflow valve

Won't this sum vary with time? How do you decide if one sum-vs-time is "smaller" than another sum-vs-time?
 
Good question, yes it varies with time. We need to make real-time decisions that will optimally minimize the sum from t_0 to t_Inf. However, I would be happy with solving from t_0 to t_N where N is a receding horizon.

Also I just realized that we are trying to minimize the sum of squares of the heights not simply the sum of heights. Sorry for the oversight.
 
aydos said:
We need to make real-time decisions that will optimally minimize the sum from t_0 to t_Inf..

It isn't clear what that means. Are you talking about an integration with respect to time?
 
Yes.
 
There is a branch of applied mathematics called "optimal control" that includes situations where a process is controlled by using continuous functions to manipulate the inputs. I don't understand the specifics of you problem well enough to match it to any textbook optimal control problem.
 

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