Introduction to Process Optimization Series v1.0
From optimization problem formulation to chemical process optimal operating condition search - Complete practical guide
Series Overview
This series is a comprehensive educational content consisting of 5 chapters that allows you to learn systematically from the fundamentals to the practice of optimization in process industries. It comprehensively covers optimization problem formulation, linear and nonlinear programming, multi-objective optimization, constrained optimization, and chemical process optimal operating condition search.
Features:
- ✅ Practice-oriented: 45 executable Python code examples
- ✅ Systematic structure: 5-chapter structure for systematic learning from basics to applications
- ✅ Industrial applications: Practical examples of chemical plant, reactor, and distillation column optimization
- ✅ Latest technologies: Utilization of scipy.optimize, PuLP, and pymoo libraries
Total learning time: 130-160 minutes (including code execution and exercises)
How to Study
Recommended Learning Sequence
For beginners (learning optimization for the first time):
- Chapter 1 → Chapter 2 → Chapter 3 → Chapter 4 → Chapter 5
- Time required: 130-160 minutes
Python experienced users (with basic numerical computation knowledge):
- Chapter 1 (quick review) → Chapter 2 → Chapter 3 → Chapter 4 → Chapter 5
- Time required: 100-130 minutes
Optimization experienced users (knowing basic theory):
- Chapter 3 → Chapter 4 → Chapter 5
- Time required: 70-90 minutes
Chapter Details
Chapter 1: Optimization Problem Formulation
Learning Content
- Fundamentals of Optimization Problems
- What is optimization - purpose and application areas
- Definition of objective function, decision variables, and parameters
- Types of constraints (equality constraints, inequality constraints)
- Concepts of feasible region and optimal solution
- Typical Examples of Chemical Process Optimization
- Reactor yield maximization problem
- Energy cost minimization problem
- Trade-off between product purity and production rate
- Objective Function Design
- Economic objective functions (profit, cost)
- Technical objective functions (yield, purity, efficiency)
- Extension to multi-objective optimization
- Visualization and Graphical Representation
- 2D contour plot
- 3D surface plot
- Feasible region visualization
- Gradient vector display
- Problem Transformation Techniques
- Transformation to unconstrained problems (penalty method)
- Problem simplification through variable transformation
- Nondimensionalization and scaling
Learning Objectives
- ✅ Can formulate optimization problems mathematically
- ✅ Can design objective functions and constraints appropriately
- ✅ Understand the relationship between feasible region and optimal solution
- ✅ Can visualize objective functions in Python
- ✅ Can formulate chemical process optimization problems
Chapter 2: Linear Programming & Nonlinear Programming
Learning Content
- Linear Programming
- Standard form of linear programming problems
- Principles of simplex method
- Implementation using scipy.optimize.linprog
- Utilization of PuLP library (production planning problem)
- Applications of Linear Programming
- Raw material blending optimization (blending problem)
- Production planning problem
- Transportation problem
- Nonlinear Programming
- Unconstrained optimization
- Implementation of gradient descent
- Newton's method
- Quasi-Newton methods (BFGS, L-BFGS)
- Comparison of Optimization Algorithms
- Convergence speed comparison
- Computational cost evaluation
- Local optimum and global optimum
- Algorithm selection guidelines
- Nonlinear Least Squares Method
- Model fitting to process data
- Utilization of scipy.optimize.least_squares
- Parameter estimation and uncertainty evaluation
Learning Objectives
- ✅ Can solve linear programming problems in Python
- ✅ Understand the principles of simplex method
- ✅ Can implement gradient descent and Newton's method
- ✅ Can utilize quasi-Newton methods (BFGS)
- ✅ Can select optimization algorithms appropriately
Chapter 3: Multi-objective Optimization and Pareto Optimality
Learning Content
- Fundamentals of Multi-objective Optimization
- Definition of multi-objective optimization problems
- Concept of Pareto dominance
- Pareto optimal solutions and Pareto frontier
- Trade-offs in chemical processes (yield vs energy)
- Scalarization Methods
- Weighted sum method
- ε-constraint method
- Goal programming
- Advantages and disadvantages of each method
- Evolutionary Algorithms
- NSGA-II (Non-dominated Sorting Genetic Algorithm II)
- Utilization of pymoo library
- Generation of Pareto frontier
- Fitness evaluation and selection strategies
- Multi-criteria Decision Making
- TOPSIS method (Technique for Order of Preference by Similarity to Ideal Solution)
- Solution selection on Pareto frontier
- Reflection of decision maker's preferences
- Interactive Visualization
- Visualization of Pareto frontier using Plotly
- Trade-off analysis
- Sensitivity analysis and decision support
Learning Objectives
- ✅ Can formulate multi-objective optimization problems
- ✅ Understand the concept of Pareto optimality
- ✅ Can implement weighted sum method and ε-constraint method
- ✅ Can generate Pareto frontier with NSGA-II
- ✅ Can perform trade-off analysis and decision making
Chapter 4: Constrained Optimization
Learning Content
- Theory of Constrained Optimization
- Equality constraints and inequality constraints
- Lagrange multipliers
- KKT conditions (Karush-Kuhn-Tucker conditions)
- Necessary and sufficient conditions for optimality
- Penalty and Barrier Methods
- Exterior penalty method
- Interior barrier method
- Augmented Lagrangian method
- Convergence properties of each method
- Sequential Quadratic Programming (SQP)
- Principles and iterative procedure of SQP
- SQP implementation with scipy.optimize.minimize
- SLSQP method (Sequential Least Squares Programming)
- Convergence and computational efficiency
- Constraints in Chemical Processes
- Material balance constraints
- Energy balance constraints
- Safety constraints (temperature and pressure limits)
- Product specification constraints (purity, quality)
- Practical Optimization Problems
- CSTR (Continuous Stirred Tank Reactor) optimization
- Optimal operating conditions of distillation column
- Cost minimization under product purity constraints
Learning Objectives
- ✅ Understand Lagrange multipliers and KKT conditions
- ✅ Can implement penalty and barrier methods
- ✅ Can solve constrained optimization with SQP/SLSQP methods
- ✅ Can formulate constraints in chemical processes
- ✅ Can solve practical process optimization problems
Chapter 5: Case Study - Chemical Process Optimal Operating Condition Search
Learning Content
- Complete Optimization Workflow
- Problem definition and goal setting
- Process model development
- Constraint definition
- Optimization problem formulation
- Algorithm selection and execution
- Result verification and interpretation
- Complete CSTR Optimization Implementation
- Reaction rate equation and material balance model
- Multivariable optimization (temperature, residence time, feed ratio)
- Yield maximization vs cost minimization
- Safety constraints and product specification constraints
- Sensitivity Analysis and Robust Optimization
- Parameter sensitivity analysis of optimal solution
- Optimization under uncertainty
- Formulation of robust optimization
- Handling probabilistic constraints
- Real-time Optimization Framework
- Concept of Real-Time Optimization (RTO)
- Model parameter updating
- Adaptive optimization strategy
- Comprehensive Case Study: Distillation Column Optimization
- Economic objective function design ($/h)
- Balance between energy cost and product value
- Multi-stage optimization strategy
- Implementation and comprehensive evaluation of results
Learning Objectives
- ✅ Can execute complete optimization workflow
- ✅ Can search for optimal operating conditions of chemical reactors
- ✅ Can practice sensitivity analysis and robust optimization
- ✅ Understand real-time optimization framework
- ✅ Can complete real process optimization projects
Overall Learning Outcomes
Upon completing this series, you will acquire the following skills and knowledge:
Knowledge Level (Understanding)
- ✅ Can explain optimization problem formulation methods
- ✅ Understand linear and nonlinear programming theory
- ✅ Understand multi-objective optimization and Pareto optimality
- ✅ Know constrained optimization theory (Lagrange multipliers, KKT conditions)
- ✅ Understand practical approaches to chemical process optimization
Practical Skills (Doing)
- ✅ Can formulate and implement optimization problems in Python
- ✅ Can utilize scipy.optimize, PuLP, and pymoo libraries
- ✅ Can implement linear and nonlinear programming
- ✅ Can generate and visualize Pareto frontier
- ✅ Can solve constrained optimization problems
- ✅ Can practice sensitivity analysis and robust optimization
Application Ability (Applying)
- ✅ Can search for optimal operating conditions of chemical processes
- ✅ Can design and execute real process optimization projects
- ✅ Can perform trade-off analysis with multi-objective optimization
- ✅ Can handle optimization tasks as a process engineer
FAQ (Frequently Asked Questions)
Q1: What level of mathematical prerequisite knowledge is required?
A: Basic knowledge of calculus (partial derivatives, gradients) and fundamental linear algebra is sufficient. The theoretical parts are designed to be understood intuitively, using extensive Python simulations and visualizations.
Q2: What is the difference between this series and the PI Introduction Series?
A: The PI Introduction Series focuses on "process modeling and data analysis," while this series focuses on "mathematical optimization and process optimal operation." By combining both, you can master the complete workflow of data-driven process optimization.
Q3: Can it be applied to actual chemical plants?
A: Yes. Chapter 5 covers a complete workflow intended for application to real processes through practical case studies. However, careful verification of safety and process constraints is necessary during implementation.
Q4: Which Python libraries are required?
A: Mainly scipy.optimize, PuLP, NumPy, Pandas, Matplotlib, Seaborn, and Plotly are used. For multi-objective optimization, the pymoo library is recommended, but manual implementation examples are also provided.
Q5: What should I learn next?
A: The following topics are recommended:
- Model Predictive Control (MPC): Dynamic optimization and real-time control
- Bayesian Optimization: Black-box optimization and Gaussian processes
- Design of Experiments (DOE): Efficient process exploration
- Stochastic Optimization: Decision making under uncertainty
Next Steps
Recommended Actions After Series Completion
Immediate (within 1 week):
1. ✅ Publish Chapter 5 case study on GitHub
2. ✅ Evaluate optimization opportunities in your own processes
3. ✅ Try implementing a simple optimization problem
Short-term (1-3 months):
1. ✅ Formulate optimization problems with real process data
2. ✅ Trade-off analysis with multi-objective optimization
3. ✅ Practice sensitivity analysis and robust optimization
4. ✅ Learn Model Predictive Control (MPC)
Long-term (6+ months):
1. ✅ Build Real-Time Optimization (RTO) system
2. ✅ Integrated optimization of entire process
3. ✅ Conference presentations and paper writing
4. ✅ Career building as a process optimization engineer
Feedback and Support
About This Series
This series was created as part of the PI Knowledge Hub project under Dr. Yusuke Hashimoto at Tohoku University.
Creation date: October 26, 2025
Version: 1.0
We Welcome Your Feedback
We are waiting for your feedback to improve this series:
- Typos, errors, technical mistakes: Please report via GitHub repository Issues
- Improvement suggestions: New topics, code examples you want added, etc.
- Questions: Parts that were difficult to understand, places where additional explanation is needed
- Success stories: Projects using what you learned from this series
Contact: yusuke.hashimoto.b8@tohoku.ac.jp
License and Terms of Use
This series is published under the CC BY 4.0 (Creative Commons Attribution 4.0 International) license.
What you can do:
- ✅ Free viewing and downloading
- ✅ Use for educational purposes (classes, study sessions, etc.)
- ✅ Modification and derivative works (translation, summarization, etc.)
Conditions:
- 📌 Author credit is required
- 📌 Indicate if modifications were made
- 📌 Contact in advance for commercial use
Details: CC BY 4.0 License Full Text
Let's Get Started!
Are you ready? Start with Chapter 1 and begin your journey into the world of process optimization!
Chapter 1: Optimization Problem Formulation →
Update History
- 2025-10-26: v1.0 First release
Your journey to learn process optimization starts here!