MARS-Based MILP Optimization for Industrial Control Settings
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Solution Overview
Problem
Mixed-integer linear programming (MILP) solvers face challenges in efficiently solving optimal control problems for industrial systems due to non-linear equations and the complexity of linear constraints, especially when the number of resource/subsystem combinations is large, leading to intractable problems and long solution times.
Innovation Solution
A system control method using a multivariate adaptive regression splines (MARS) prediction model to define an optimization problem, which is then transformed into a mixed-integer linear programming (MILP) problem, solved using an optimization engine to determine optimized control settings for directly-controllable variables, thereby controlling physical characteristics of industrial systems.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If all possible resource/subsystem combinations are given to the MILP solver, then the completeness of the control solution is improved, but the problem becomes intractable when the number of combinations is large
Solution Approach 1:
The patent segments the control problem into multiple time intervals, solving the optimization problem separately for each interval rather than attempting to solve all combinations simultaneously. This divides the intractable large-scale problem into manageable smaller problems that can be solved sequentially.
Solution Approach 2:
The patent performs preliminary linearization of non-linear equations before feeding them to the MILP solver. By pre-processing the equations to convert them into linear form, the system prepares the data in a format that the solver can handle efficiently, avoiding the intractability that would result from directly inputting non-linear relationships.
2Measurement precision
If non-linear equations are used to describe the system, then the accuracy of system representation is improved, but the linearization task becomes non-trivial and complex
Solution Approach 1:
The system performs preliminary linearization of non-linear equations before processing. By pre-converting non-linear relationships into linear approximations, the system maintains accuracy while making the equations suitable for MILP solver input, avoiding the need to handle complex non-linear optimization.
Solution Approach 2:
The patent changes the mathematical representation of system parameters by transforming non-linear equations into linear forms through approximation techniques. This parameter transformation allows the system to maintain adequate accuracy while converting the problem into a format that linear solvers can handle efficiently.
3Manufacturing precision
If the complexity of linear constraints is increased, then the optimality of the control solution is improved, but the solution time of the MILP solver increases very long
Solution Approach 1:
The patent segments the time horizon into multiple smaller intervals, solving the optimization problem for each interval separately. This segmentation reduces the computational burden on the MILP solver for each individual problem instance, significantly reducing solution time while maintaining optimality within each time segment.
Solution Approach 2:
The system solves the optimization problem partially for each time interval rather than attempting to solve the entire horizon at once. By addressing only the immediate next interval at a time, the system achieves adequate optimality for each segment while avoiding the exponential time increase that would result from solving all constraints simultaneously.
Data Source
AI summary
Constructing a MARS prediction model using predictor variables at a first point in time within a time horizon, including directly-controllable variables of first physical characteristics of a system and that are associated with adjustable operational control settings for directly controlling the first physical characteristic, and including controllable variables of second physical characteristics that are affected by the first physical characteristics, recursively using the prediction model to define an optimization problem for later point in time within the time horizon, transforming the optimization problem into a MILP problem, and solving the MILP problem using an optimization engine to determine, for any given one of the directly-controllable variables and corresponding to at least one of the points in time, for adjusting, using the optimized value, the adjustable operational control setting corresponding to the given directly-controllable variable and thereby control the physical characteristic associated with the directly-controllable variable.