Bài giảng Tối ưu hóa (Bản hay)

Phần 3: Tối ưu hóa
Modeling, simulation and optimization for chemical process
Instructor: Hoang Ngoc Ha
Email: ha.hoang@hcmut.edu.vn
Bộ môn QT&TB
T. F. Edgar, D. M. Himmelblau. Optimization of chemical Processes. 
Second edition.
Bùi Minh Trí. Tối ưu hóa (lý thuyết và bài tập). NXB KHKT, Hà Nội, 2005.
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Introduction
„ The chemical industry has undergone
significant changes during the past 25 years
due to the
‰ increased cost of energy
‰ increasingly stringent environmental regulations
‰ global competition in product pricing and quality
‰ 
„ One of the most important engineering tools for 
addressing these issues is optimization
Decision-making process
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Introduction
„ As the power of computers has increased, 
the size and complexity of problems that can
be solved by optimization techniques have 
correspondingly expanded
„ The necessary tools for solving problem
‰ We will focus on those techniques and discuss
software that offers the most potential for 
success and gives reliable results
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Outline
„ Problem formulation
‰ Nature and organization of Optimization problems
‰ Developing models for optimization (constraints or 
process model)
‰ Formulation of the objective function
„ Optimization theory and methods
‰ Optimization of unconstrained functions
‰ Linear programming with constraints
‰ Nonlinear programming with constraints
‰ Multi-objective optimization
„ Applications of Optimization
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Optimization
„ OPTIMIZATION IS THE use of specific methods to 
determine the most cost-effective and efficient 
solution to a problem or design for a process
„ This technique is one of the major quantitative tools
in industrial decision making
„ A wide variety of problems in the design, 
construction, operation, and analysis of chemical
plants (as well as many other industrial processes) 
can be resolved by optimization
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Problem formulation
„ Formulating the problem is perhaps the most crucial 
step in optimization (from verbal statement of a 
given application and organizing them into a 
prescribed mathematical form)
‰ The objective function (economic criterion)
‰ The process model (constraints)
„ The objective function represents such factors as 
profit, cost, energy, and yield in terms of the key
variables of the process being analyzed
„ The process model and constraints describe the
interrelationships of the key variables
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Problem formulation
„ What optimization is all about
‰ Optimization is concerned with selecting the best
value by efficient quantitative methods
„ Why optimize?
‰ Largest production
‰ Greatest profit
‰ Minimum cost
‰ The least energy usage
‰ 
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Problem formulation
„ Examples of applications of optimization
‰ Determining the best sites for plant location
‰ Routing tankers for the distribution of crude and refined products
‰ Sizing and layout of a pipeline
‰ Designing equipment and an entire plant
‰ Scheduling maintenance and equipment replacement
‰ Operating equipment, such as tubular reactors, columns, and
absorbers
‰ Evaluating plant data to construct a model of a process
‰ Minimizing inventory charges
‰ Allocating resources or services among several processes
‰ Planning and scheduling construction
‰ 
Example: See ref.
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Problem formulation
„ Main features of optimization problems
‰ At least one objective function to be optimized
‰ Equality constraints (equations)
‰ Inequality constraints (inequalities)
Economic model
} Model of processor equipment
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Problem formulation
„ Main features of optimization problems
Feasible solution/Feasible region
Underdetermined
Optimal solution
Overdetermined
Degrees of freedom
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Problem formulation
„ An optimization problem:
Minimize: f(x) objective function
Subject to: h(x) = 0 equality constraints
h(x) is a vector of equations of dim. m1
g(x) is a vector of equations of dim. m2
where x = (x1 · · ·xn) ∈ X ⊂ Rn
D =
n
x ∈ X|h(x) = 0, g(x) ≥ 0
o
g(x) ≥ 0 inequality constraints
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Problem formulation
„ Example: optimal scheduling
tA1 tA2
tB1 tB2
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Problem formulation
„ What is the objective function?
tA1 + tA2 = 365
tB1 + tB2 = 365
tAi ≥ 0
tBi ≥ 0
f(t) = tA1MA1SA1 + tA2MA2SA2
+tB1MB1SB1 + tB2MB2SB2
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Problem formulation
„ Các loại bài toán tối ưu (quy hoạch toán học)
‰ Quy hoạch tuyến tính (QHTT)
Ví dụ thuộc dạng này có Bài Toán Vận Tải
‰ Quy hoạch tham số (QHTS) là QHTT mà các hệ số trong
‰ Quy hoạch động (QHĐ):
„ Là quá trình có nhiều giai đoạn nói chung, hay các quá trình
phát triển theo thời gian nói riêng
là tuyến tínhf(x), g(x), h(x)
phụ thuộc tham sốf(x), g(x), h(x)
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Problem formulation
„ Các loại bài toán tối ưu (quy hoạch toán học) 
‰ Quy hoạch phi tuyến (QHPT)
‰ Quy hoạch rời rạc (QHRR)
‰ Quy hoạch đa mục tiêu (QHĐMT)
„ Nếu trên cùng một miền ràng buộc D ta xét nhiều hàm
mục tiêu khác nhau
là các hàm phi tuyếnf(x) hoặc g(x) hoặch(x)
DNếu miền ràng buộc là tập rời rạc
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Formulation of the objective function
„ Translate a verbal statement or concept of
the desired objective into mathematical terms
„ Example
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Formulation of the objective function
„ Example
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Formulation of the objective function
„ Example
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Formulation of the objective function
„ Example
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Problem formulation
„ The six steps used to solve optimization
problems
‰ Make a list of all of the process variables
‰ Determine the criterion for optimization, and
specify the objective function in terms of the
variables defined in step 1 together with
coefficients (Economic model)
‰ Using mathematical expressions, develop a valid
process or equipment model (Process model) 
that relates the input-output variables of the
process and associated coefficients
Problem formulation
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Problem formulation
„ The six steps used to solve optimization
problems
‰ If the problem formulation is too large in scope
„ Break it up into manageable parts or
„ Simplify the objective function and model
‰ Apply a suitable optimization technique to the
mathematical statement of the problem
‰ Check the answers, and examine the sensitivity of
the result to changes in the coefficients in the
problem and the assumptions
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Outline
„ Problem formulation
‰ Nature and organization of Optimization problems
‰ Developing models for optimization (constraints or 
process model)
‰ Formulation of the objective function
„ Optimization theory and methods
‰ Optimization of unconstrained functions
‰ Linear programming with constraints
‰ Nonlinear programming with constraints
‰ Multi-objective optimization
„ Applications of Optimization
CuuDuongThanCong.com https://fb.com/tailieudientucntt
Scope of course
Optimization problems
OPTIMIZATION OF UNCONSTRAINED
FUNCTIONS: ONE-DIMENSIONAL SEARCH
UNCONSTRAINED
MULTIVARIABLE OPTIMIZATION
NON LINEAR PROGRAMMINGLINEAR PROGRAMMING
MULTI-OBJECTIVE OPTIMIZATION 
(OR MULTI-OBJECTIVE PROGRAMMING)
CuuDuongThanCong.com https://fb.com/tailieudientucntt