?????? ? ??? Biological process simulation and optimization

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?????? ? ???Biological process simulation and
optimization

• Major Interdisciplinary program of the
  integrated biotechnology
• Graduate school of bio- information technology
• Youngil Lim (N110), Lab. FACS
• phone [phone] (secretary), 82 31 670
  5207 (direct)
• Fax [phone], mobile phone [phone]
  5207
• Email limyi_at_hknu.ac.kr, homepage 
  http//hknu.ac.kr/limyi/index.htm

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Part I. Problem Formulation

• Mathematical form
• Objective function (economic criteria) profit,
  cost, energy, productivity or yield w.r.t. key
  variables.
• Process model (constrains) interrelationship of
  key variables (physical and empirical equations).

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Part I. Problem Formulation

• Ch. 1 Examples in chemical engineering
• Ch. 2 Process models material/energy balances,
  equilibrium equations, empirical equations.
• Ch. 3 Objective functions capital cost/operating
  cost,

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Ch 1. Nature of optimization problems

• In process design and operations,
• - so many solutions exist
• - select the best among the possible solutions
• To find the best solution,
• - critical analysis of process
• - appropriate performance objectives
• - use of past experience (from expert)
• Objectives
• - process design largest production, greatest
  profit, minimum const, least energy usage
• - process operation improve yield of target
  product, reduce energy consumption, or increase
  processing rate

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Ch 1. Nature of optimization problems
Management
Design
operations
Allocation scheduling
Individual equipment

• Fig. 1.1 Hierarchy of levels of optimization

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Ch. 1 Examples

1. Determine the best sites for plant location
2. Routing tankers for the distribution of crude and
   refined products
3. Sizing and layout of a pipeline
4. Designing equipment and an entire plant
5. Scheduling maintenance and equipment replacement
6. Operating equipment, such as tubular reactors,
   columns and absorber.
7. Evaluating plant data to construct a model of a
   process
8. Minimizing inventory charges
9. Allocating resources or services among several
   processes
10. Planning and scheduling construction

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Example 1.1 Optimum insulation thickness
Cost of insulation
Cost, y (/yr)
Cost of lost energy
Insulation thickness, x (cm)
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Example 1.2 Optimal operating conditions of a
boiler
Thermal efficiency
Thermal efficiency
Hydrocarbon emissions
NOx emissions
1.0 1.3
Air-fuel ratio, x
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Example 1.3 Optimum distillation reflux (1/2)

• When fuel costs were low, high reflux (high heat
  duty, high purity) leads to maximize profit.
• When fuel costs are high, low reflux (low heat
  duty, limited purity) prefer to maximize profit.

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Example 1.3 Optimum distillation reflux (2/2)

• When fuel costs are high, low reflux (low heat
  duty, limited purity) prefer to maximize profit.

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Example 1.4 Multiplant product distribution

• Distribution of a single product (Y) manufactured
  at several plant locations.
• Several costumers are located at various
  distribution.
• We have m plants Y(Y1, Y2, Ym)
• We have n demand points (costumers) Ym(Ym1,
  Ym2, Ymn)
• Minimize cost including transportation costs and
  production costs

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Essential features of optimization problems

• 1. Optimization problems must be expressed in
  mathematics.
• 2. A wide variety of opti. problems have the same
  mathematical structures
• - at least, on objective function
• - equality constraints (equations)
• - inequality constraints (inequalities)
• 3. Terminologies (see Fig. 1.2)
• - variables
• - feasible solution
• - optimal solution

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Example 1.5 Optimal scheduling Formulation of
the optimal problem

• To schedule the production in two plants A B
• Each plant produce two products 1 2
• To maximize profits ( or /year) ? objective
  function f(t)
• Variables are the working days (day) tA1, tA2,
  tB1, tB2
• Given parameters Sij (/lb), Mij (lb/day), where
  iA, B j1, 2

Num. Objective func. Num. variables Num.
parameters Num. inequality Num. equation
1 2 4 9 5
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Example 1.5 Optimal scheduling Matlab practice
(1/7)

• To schedule the production in two plants A B
• Each plant produce two products 1 2
• To maximize profits ( or /year) ? objective
  function f(t)
• Variables are the working days (day) tA1, tA2,
  tB1, tB2
• Given parameters Sij (/lb), Mij (lb/day), where
  iA, B j1, 2

• Preparation steps before using Matlab
• Use constrained LP based on SQP (successive
  quadratic programming)
• ? fmincon()
• 2. Search matlab help (F1)
• 3. Learn how to use this function
• Programming steps in Matlab
• Define parameters of the given problem
• Define parameters of the used function,
  fmincon()
• Call and define the objective function

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Example 1.5 Optimal scheduling Matlab practice
(2/7)

• Preparation steps before using Matlab
• Use constrained LP based on SQP (successive
  quadratic programming)
• ? fmincon()
• 2. Search matlab help (F1)
• 3. Learn how to use this function

Using constrained optimization solver, SQP
LP x,fval fmincon(_at_fun,x0,A,b,Aeq,beq,lb,ub)
NLP x,fval fmincon(_at_fun,x0,A,b,Aeq,beq,lb,u
b,nonlcon) x variables fval minimum value
of objective function fun objective function
to be called x0 initial guess of x A and b
linear inequalities, Ax lt b Aeq and beq
linear equalities, Aeqx beq lb lower bound
of x ub upper bound of x
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Example 1.5 Optimal scheduling Matlab practice
(3/7)
x (variables) should be a column matrix
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Example 1.5 Optimal scheduling Matlab practice
(4/7)

• Programming steps in Matlab
• Define parameters of the given problem M, S, L
• Define parameters of the used function,
  fmincon() A, b, lb, ub
• Call and define the objective function
• function f fun(x)

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Example 1.5 Optimal scheduling Matlab practice
(5/7)

• Programming steps in Matlab
• Define parameters of the given problem M, S, L
• Define parameters of the used function,
  fmincon() A, b, lb, ub
• Call and define the objective function
• function f fun(x)

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Example 1.5 Optimal scheduling Matlab practice
(6/7)
Programming steps in Matlab 1. Define parameters
of the given problem M, S, L 2. Define
parameters of the used function, fmincon() A, b,
lb, ub. 3. Call and define the objective
function function f fun(x)
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Example 1.5 Optimal scheduling Matlab practice
(7/7)

• Programming steps in Matlab
• Define parameters of the given problem M, S, L
• Define parameters of the used function,
  fmincon() A, b, lb, ub
• Call and define the objective function
• function f fun(x)

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Example 1.6 Material balance reconciliationquadr
atic programming

• We have 3 experimental measurements of flowrate,
  respectively, at two points.
• We wanna know inlet flowrate
• Mass balance MA MBi MCi
• MB (11.1 10.8 11.4)
• MC (92.4 94.3 93.8)

• Characteristics of above problem
• 2nd-order one variable function
• Unique global optimum exists.
• First-order derivative is needed to get the
  solution.

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1.6 General procedure for solving optimization
problems

1. Analyze the process itself so that the process
   variables and specific characteristics of
   interest are defined ? make a list of the
   variables/parameters
2. Determine the criterion for optimization, and
   specify the objective function w.r.t variables
   and parameters ? performance model
3. Using mathematical expressions, develop a valid
   process or equipment model that relates the
   input/output variables. Include both equality and
   inequality constraints. Use first-principle
   models (mass/energy balances, equilibrium
   equations), empirical equations, implicit
   concepts and external restrictions. Identify the
   number of degree of freedom. ? equality/inequality
   constraints
4. If the problem formulation is too large in scope,
   ? reduced model development
5. Break it up into manageable parts or
6. Simplify the objective function and model
7. Apply a suitable optimization technique (SQP, GA,
   GCMC, etc. or Matlab, GAMS, etc.) to the
   mathematical statement of the problem.
8. Check the answers, and examine the sensitivity of
   the result to change in the parameters ?
   parameter sensitivity analysis.

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Exercise and homework 1

• Select one problem among 1.10-1.24 and solve it
  using Matlab.
• Each student should select a different problem
  each other.
• If there is no specific value to be needed,
  please set the values yourself.