Documents
User Generated
Resources
Learning Center

# Formulating the Optimization Problem

VIEWS: 3 PAGES: 24

• pg 1
```									  FORMULATING THE OPTIMIZATION
PROBLEM
“If do not know where we are going, any path will do.”
•   The importance of formulation
•   The standard formulation
•   Degrees of Freedom
•   The operating window

DOFASCO: We find it useful to formulate the optimization
problem completely, even if we cannot solve it.
FORMULATING THE OPTIMIZATION
PROBLEM

While walking home after the first Chem. Eng. 4G03 class,
you began to worry about passing the course.

Fortunately, you notice a magic
lantern, which you rub. A Genie
appears and grants you ONE wish.
You wish for the highest grade in
4G03.
What happens?
What did the Genie do to grant your wish?
you

Oh no! The Genie tricks the class,
minus you, into a bus that runs over a
cliff. Everyone in the bus perishes;
4G03.
You are very mad at the Genie, but he
says that he only did what you asked.
FORMULATING THE OPTIMIZATION
PROBLEM
LESSONS LEARNED
•   The Genie gave you what you asked for - not what you
intended.
•   An Optimizer is like the Genie. Therefore, we must
formulate the problem carefully and check the results for
surprises!
•   If we maximize profit, can we
- pollution the environment?
- endanger workers?
- produce products that are shabby or dangerous?
- design a plant that only functions for six months?
FORMULATING THE OPTIMIZATION
PROBLEM
This is the general formulation that we will be using
throughout the course

max P                           Objective function
x

s.t .
h( x )  0                      Equality constraints
Inequality constraints
g( x )  0
x min  x  x max               Variable Bounds
max P
x

FORMULATING THE                                         s.t .
h( x )  0
OPTIMIZATION PROBLEM                                    g( x )  0
x min  x  x max

Objective   This is the goal or objective, e.g.,
Function      - maximize profit (minimize cost)
- minimize energy use
max P         - minimize polluting effluents
X           - minimize mass to construct a vessel
We will formulate most problems with a scalar objective function,
i.e., a single value.
min P       This should represent the full effect of x on the objective. For
X
example, \$/kg is not a good objective unless kg is fixed. When
needed, include time-value of money.

Also, we need a quantitative measure, not “good” or “bad”.

The value of the material may depend on the composition,
enthalpy, flow rate, etc.

The symbol “x” represents the variables. It is a vector.
max P
x

FORMULATING THE                              s.t .
h( x )  0
OPTIMIZATION PROBLEM                         g( x )  0
x min  x  x max

Some comments on the objective function
•   A scalar is preferred for solving. However, multiple
objectives are typical in real life.
•   Note that Max (P) is the same as Min (-P)
•   Sometimes we use a simple, physical variable, such as
yield of a key product. This assumes that max (profit) is
the same as Max(yield), which might not always be true.
max P
x

FORMULATING THE                             s.t .
h( x )  0
OPTIMIZATION PROBLEM                        g( x )  0
x min  x  x max

Some comments on the objective function (continued)
•   We have difficulty when the models are inaccurate, for
example, the tradeoff between current reactor operation
and long-term catalyst activity.
•   Modelling the market response to improved product
quality, etc is difficult.
•   We want a “smooth” function.
max P
x

s.t .
FORMULATING THE                                          h( x )  0
OPTIMIZATION PROBLEM                                     g( x )  0
x min  x  x max

s.t.          This means “subject to”. The expressions below limit (or
constrain) the allowable values of the variables x. They define the
feasible region.
Equality      These are equality constraints, e.g.,
Constraints    - material, energy, force, current, … BALANCES
- equilibrium
h(x) = 0       - decisions by the engineer ( F1 - .5 F2 = 0 )
- behavior enforced by controls TC set point = 231 C
- sub-sections of a model used in other parts of the model,
rate = - kCA

By convention, we will write the equations with a zero rhs (right
hand side).

There can be many of these equations, so that h(x) is a vector.
max P
x

FORMULATING THE                             s.t .
h( x )  0
OPTIMIZATION PROBLEM                        g( x )  0
x min  x  x max

•   The key balances must be strictly observed. If we do not
ensure that they are “closed”, the optimizer will find a
way to create mass and energy!
•   The models may change. For example, a heat exchanger
could have either one or two phases, with the number of
phases depending on the optimization decisions.
max P
x

s.t .
FORMULATING THE                                                 h( x )  0
OPTIMIZATION PROBLEM                                            g( x )  0
x min  x  x max
Inequality    These are “one-way” limits to the system, e.g.,
constraints    - maximum investment available
- maximum flow rate due to pump limit
g(x)  0       - minimum liquid flow rate on tray # 24
- minimum steam generation in a boiler for stable flame
- maximum pressure of a closed vessel
- maximum region within which we think that the model is
acceptable

These are essential for optimization. We have not formulated
inequalities in previous course, although we have learned the
underlying technologies.

By convention, we will write the equations with a zero rhs (right
hand side).

We must be careful to prevent defining a problem incorrectly with
no feasible region.

By multiplying by (-1), we can change the inequality to g(x)  0.
So, these two forms are equivalent.
max P
x

s.t .
FORMULATING THE                                        h( x )  0
OPTIMIZATION PROBLEM                                   g( x )  0
x min  x  x max
Variables   Variables can be grouped into two categories
and their    Some are “decision” or “optimization” variables. These are
Bounds        the variables in the system that are changed independently to
modify the behavior of the system.
xmin  x     Some are dependent variables whose behavior is determined
x  xmax      by the values selected for the independent variables.

Although they can be grouped this way to help understanding, the
solution method need not distinguish them. We need to solve a set
of equations involving many variables.

Examples of variables are
DESIGN: reactor volume, number of trays, heat exch. area, …
OPERATIONS: temperature, flow, pressure, valve opening, …
MANAGEMENT: feed type, purchase price, sales price, ..

These bounds limit the values of the variables. Note that setting
the min and max values equal sets the variable to a constant value.
max P
x

FORMULATING THE                               s.t .
h( x )  0
OPTIMIZATION PROBLEM                          g( x )  0
x min  x  x max

•   Many variables are continuous, but some are discrete or
integer. Give some examples of each.
•   Typically, we do not define the “decision” variables.
Since we solve a set of simultaneous equations, are
variables are determined together.
•   We should always place bounds on variables. Why?
FORMULATING THE OPTIMIZATION
PROBLEM

When modelling, we always encounter the issue
max P               of Degrees of Freedom (DOF). How do we
x                determine the ODF for an optimization
s.t .               problem using the relationship below?
h( x )  0          DOF = (# variables) - (# equations)
g( x )  0          # variables =
x min  x  x max

# equations =
FORMULATING THE OPTIMIZATION
PROBLEM

In 3G03 and 3P03, we required the models to
max P               have DOF=0. Why?
x

s.t .
h( x )  0          For optimization, what value(s) do we expect
g( x )  0          for the DOF?
x min  x  x max

The answer explains why optimization is so widely applied!
FORMULATING THE OPTIMIZATION
PROBLEM

Often, we will think of the problem as having
max P               #Opt Var = # var - #equality constr.
x

s.t .               We can plot this if only two dimensions.
h( x )  0
g( x )  0                                           What about points
Opt Var2

inside?
feasible
x min  x  x max                    region          Which is the best?

Opt Var1
FORMULATING THE OPTIMIZATION
PROBLEM
We can plot values of the objective function as contours.
Where is the optimum for the two cases shown below?
Opt Var2

Opt Var2

Opt Var1                      Opt Var1
Case A                       Case B
FORMULATING THE OPTIMIZATION
PROBLEM
Variables and objective function can be plotted in 3D

T                            FA
FORMULATING THE OPTIMIZATION
PROBLEM

How do we select the appropriate
max P               “system” for a specific problem?
x

s.t .
h( x )  0
g( x )  0
x min  x  x max
FORMULATING THE OPTIMIZATION
PROBLEM
How do we select the appropriate
“system” for a specific problem?
max P                       Marlin’s Rule 1
x

s.t .               We must consult people with a
h( x )  0          than “system S” to determine the
true objectives of “system S”.
g( x )  0
x min  x  x max
FORMULATING THE OPTIMIZATION
PROBLEM

How do we define a scalar that
represents performance,
max P                 including
x
• Economics
s.t .
• Safety
h( x )  0           • Product quality
g( x )  0           • Product rates (contracts!)

x min  x  x max    • Flexibility
• …...
FORMULATING THE OPTIMIZATION
PROBLEM

max P                 How accurately must we model
x                  the physical process?
s.t .                • Macroscopic

h( x )  0           • 1,2 3, spatial dimensions
g( x )  0
• Physical properties
x min  x  x max    • Rate models (U(f), k0e-E/RT, ..
FORMULATING THE OPTIMIZATION
PROBLEM

max P                 What limits the possible
x                  solutions to the problem?
s.t .                • Safety

h( x )  0           • Product quality
• Equipment damage (long
g( x )  0             term)

x min  x  x max    • Equipment operation
• Legal/ethical considerations
FORMULATING THE OPTIMIZATION
PROBLEM

No one solution or          • More to learn
approach is suitable for
• Must have
optimization!
toolkit of models
max P                      and solvers
x

s.t .
h( x )  0              • Lots of
opportunity for
g( x )  0                ingenuity
x min  x  x max       • Makes engineers
valuable

```
To top