Optimization of thermal processes
Lecture 1
Maciej Marek
Czestochowa University of Technology
Institute of Thermal Machinery
Optimization of thermal processes
2007/2008
Main topics
• Introduction
− Fundamental concepts in optimization
− Engineering applications of optimization
• Types of optimization problems
− Constrained, unconstrained
− Linear, nonlinear
− Static, dynamic
• Classical optimization techniques
EXAM
− Analytical, numerical
− Direct, indirect
− Linear programming
• Overview of some modern techniques
− Genetic algorithms, simulated annealing, neural networks
• Examples of engineering applications
Optimization of thermal processes
2007/2008
Literature
• S. Rao "Engineering optimization, theory and practice„
• R. Klajny "Optymalizacja procesów cieplnych„
• P. Gill, W. Murray, M. Wright "Practical optimization„
• M. Bhatti "Practical optimization methods„
• S. Sieniutycz "Optymalizacja w inżynierii procesowej„
• S. Sieniutycz, Z. Szwast "Przykłady i zadania z optymalizacji procesowej„
• M. Sysło, N. Deo, J. Kowalik "Algorytmy optymalizacji dyskretnej„
• E. Majza "Przykłady zastosowań badań operacyjnych w energetyce
cieplnej"
Optimization of thermal processes
2007/2008
What is optimization?
Optimization is
• the process of finding
• the best result
• under given circumstances.
Optimization of thermal processes
Optimization technique
Objective function
Constraints
2007/2008
Optimization problem
• Objective function (criterion or merit function)
f ( X)
f ( X)
f ( X)
new x*
x*
Minimum of f ( x )
x
Maximum of  f ( x)
 f ( X)
• Constraints (restrictions)
Optimization of thermal processes
2007/2008
Formal statement of an (constrained) optimization
problem
Find
Design vector
 x1 
x 
2

X
 
 
 xn 
Decision (design) variables
which minimizes
f ( X)
subject to the constraints
g j ( X)  0
j  1, 2,..., m
Inequality constraint
l j ( X)  0
j  1, 2,..., p
Equality constraint
Optimization of thermal processes
2007/2008
Engineering applications of optimization
(mathematical programming)
• Design of aircraft
− Objective: minimum weight
− Constraints: capacity, strength, cost
• Design of pumps, turbines, heat transfer equipment
− Objective: maximum efficiency
− Constraints: weight, cost, noise level, impact on the enviroment etc.
• Allocation of resources or services among several activities to
maximize the benefit
• Analysis of statistical data (approximation)
• Optimal scheduling
• Other examples?
Optimization of thermal processes
2007/2008
Classification of optimization problems
based on
• Nature of the decision variables
− Static, dynamic
• Nature of the equations (and inequalities) involved
− Linear, quadratic, nonlinear
• Number of objective functions
− Single-, multiobjective problem
− Overall objective function
minimize f1 (X), f 2 (X), ..., f k (X)
f (X)  1 f1 (X)   2 f 2 (X)
• Permissible values of the design variables
− Integer programming
Optimization of thermal processes
2007/2008
Typical optimization procedure
• Define objective function
• Define decision variables and estimate their impact on the result
• Find the design constraints and express them in the form of
equalities and inequalities
• Decide which technique of optimization is best for the problem
• Use the chosen method and find the result
• Analyse the result
Optimization of thermal processes
2007/2008
Example optimization problem
Suppose that two products (denoted as I and II respectively) can
be manufactured at the factory with the use of three ingredients:
A, B and C. The amount of the ingredients required for one unit of
each of the products is as follows:
I
II
A
4
5
B
5
2
C
3
8
The total avalaible amount of the ingredients is:
A
B
C
10
10
12
Optimization of thermal processes
2007/2008
Example optimization problem
The profit for product I is 5 zl per unit, for product II – 3 zl per unit.
Find the optimum structure of production (maximize the profit).
Decision variables
x1
x2
- number of units of product I
- number of units of product II
Design vector
Objective function
 x1 
X 
 x2 
f (X)  f ( x1 , x2 )  5x1  3x2
Optimization of thermal processes
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Example optimization problem
Constraints
for A: 4 x1  5 x2  10
for B: 5x1  2 x2  10
for C: 3x1  8x2  12
x1  0
x2  0
Optimization of thermal processes
2007/2008
Thank you for your attention
Optimization of thermal processes
2007/2008