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4460 Lecture 8 2011

# 4460 Lecture 8 2011 - Optimization Dr.MarioRichardEden...

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Optimization CHEN 4460 – Process Synthesis,  Simulation and Optimization Dr. Mario Richard Eden Department of Chemical Engineering Auburn University Lecture No. 8 – Mathematical Optimization October 25, 2011 Contains Material Developed by Dr. Daniel R. Lewin, Technion, Israel

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Lecture 10 – Objectives Understand  the  different  types  of  optimization  problems and their formulation Be  able  to  formulate  and  solve  a  variety  of  optimization problems in LINGO On completion of this part, you should:
Optimization Basics What is Optimization? The  purpose  of  optimization  is  to  maximize  (or  minimize)  the  value  of  a  function  (called  objective  function )  subject  to  a  number  of  restrictions  (called  constraints ). Examples 1.Maximize reactor conversion Subject to reactor modeling equations kinetic equations limitations on T, P and x

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Optimization Basics Examples (Continued) 2.Minimize cost of plant Subject to mass & energy balance equations equipment modeling equations environmental, technical and logical  constraints
Optimization Basics Examples (Continued) 3.Maximize your grade in this course Subject to extracurricular activities full-time-job requirements constant  demand  by  other  courses  and/or your advisor/boss

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Optimization Basics Formulation of Optimization Problems min (or max) f(x 1 ,x 2 ,……,x N ) subject to g 1 (x 1 ,x 2 ,……,x N ) 0 g 2 (x 1 ,x 2 ,……,x N ) 0 g m (x 1 ,x 2 ,……,x N ) 0 h 1 (x 1 ,x 2 ,……,x N )=0 h 2 (x 1 ,x 2 ,……,x N )=0 h E (x 1 ,x 2 ,……,x N )=0 Inequality  Constraints Equality  Constraints Feasibility Any vector (or point) which  satisfies  all the constraints  of the optimization program  is called  a feasible vector  (or a feasible point) The set of all feasible points  is called  feasibility region  or feasibility domain Any optimal solution  must lie within the  feasibility region!
Optimization Basics Classification of Optimization Problems Linear Programs (LP’s) A mathematical program is  linear  if  f(x 1 ,x 2 ,……,x N ) and g i (x 1 ,x 2 ,……,x N ) 0 are  linear  in each of their  arguments: f(x 1 ,x 2 ,……,x N ) = c 1 x 1  + c 2 x 2  + …. c N x N g i (x 1 ,x 2 ,……,x N ) = a i1 x 1  + a i2 x 2  + …. a iN x N where c i  and a ij  are known constants. Linear Programs (LP’s) can be solved to yield a  global  optimum. Solver routines  can guarantee  a  truly optimal solution.

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Optimization Basics Classification of Optimization Problems Non-Linear Programs (NLP’s) A mathematical program is  non-linear  if any of the arguments  are non-linear. For example: min 3x + 6y 2 s.t.  5x + x  0 Integer Programming Optimization  programs  in  which  ALL  the  variables  must
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