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Process Optimization. Mathematical Programming and Optimization of Multi-Plant Operations and Process Design Ralph W. Pike Director, Minerals Processing Research Institute Horton Professor of Chemical Engineering Louisiana State University.

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Prepare Optimization Mathematical Programming and Optimization of Multi-Plant Operations and Process Design Ralph W. Pike Director, Minerals Processing Research Institute Horton Professor of Chemical Engineering Louisiana State University Department of Chemical Engineering, Lamar University, April, 10, 2007

Process Optimization Typical Industrial Problems Mathematical Programming Software Mathematical Basis for Optimization Lagrange Multipliers and the Simplex Algorithm Generalized Reduced Gradient Algorithm On-Line Optimization Mixed Integer Programming and the Branch and Bound Algorithm Chemical Production Complex Optimization

New Results Using one script to compose and run a program in another dialect Cumulative likelihood dispersion rather than an ideal point utilizing Monte Carlo recreation for a multi-criteria, blended whole number nonlinear programming issue Global improvement

Design versus Operations Optimal Design − Uses flowsheet test systems and SQP Heuristics for an outline, a superstructure, an ideal plan Optimal Operations On-line improvement Plant ideal booking Corporate production network advancement

Plant Problem Size Contact Alkylation Ethylene 3,200 TPD 15,000 BPD 200 million lb/yr Units 14 76 ~200 Streams 35 110 ~4,000 Constraints Equality 761 1,579 ~400,000 Inequality 28 50 ~10,000 Variables Measured 43 125 ~300 Unmeasured 732 1,509 ~10,000 Parameters 11 64 ~100

finish Optimization Programming Languages GAMS - G eneral A lgebraic M odeling S ystem LINDO - Widely utilized as a part of business applications AMPL - A M athematical P rogramming L anguage Others: MPL, ILOG enhancement program is composed as a streamlining issue optimize: y( x ) economic display subject to: f i ( x ) = 0 constraints

Software with Optimization Capabilities Excel – Solver MATLAB MathCAD Mathematica Maple Others

Mathematical Programming Using Excel – Solver Using GAMS Mathematical Basis for Optimization Important Algorithms Simplex Method and Lagrange Multipliers Generalized Reduced Gradient Algorithm Branch and Bound Algorithm

Simple Chemical Process P – reactor weight R – reuse proportion

Excel Solver Example Solver ideal arrangement Showing the conditions in the Excel cells with beginning qualities for P and R

Excel Solver Example

Excel Solver Example Not the base for C Not

Use Solver with these estimations of P and R Excel Solver Example

Excel Solver Example ideal Click to highlight to create reports

Excel Solver Example Information from Solver Help is of restricted esteem

Excel Solver Answer Report administration report organize values at the ideal imperative status slack variable

Excel Sensitivity Report Solver utilizes the summed up lessened slope advancement calculation Lagrange multipliers utilized at affectability investigation Shadow costs ($ per unit)

Excel Solver Limits Report Sensitivity Analysis gives restrains on factors to the ideal answer for stay ideal

GAMS

GAMS S O L V E S U M A R Y MODEL Recycle OBJECTIVE Z TYPE NLP DIRECTION MINIMIZE SOLVER CONOPT FROM LINE 18 **** SOLVER STATUS 1 NORMAL COMPLETION **** MODEL STATUS 2 LOCALLY OPTIMAL **** OBJECTIVE VALUE 3444444.4444 RESOURCE USAGE, LIMIT 0.016 1000.000 ITERATION COUNT, LIMIT 14 10000 EVALUATION ERRORS 0 C O N O P T 3 x86/MS Windows form 3.14P-016-057 Copyright (C) ARKI Consulting and Development A/S Bagsvaerdvej 246 A DK-2880 Bagsvaerd, Denmark Using default alternatives. The model has 3 factors and 2 limitations with 5 Jacobian components, 4 of which are nonlinear. The Hessian of the Lagrangian has 2 components on the corner to corner, 1 components beneath the slanting, and 2 nonlinear factors. ** Optimal arrangement. Diminished inclination not as much as resistance.

GAMS Lagrange multiplier LOWER LEVEL UPPER MARGINAL - EQU CON1 9000.000 117.284 - EQU OBJ . . . 1.000 LOWER LEVEL UPPER MARGINAL - VAR P 1.000 1500.000 +INF . - VAR R 1.000 6.000 +INF EPS - VAR Z - INF 3.4444E+6 +INF . **** REPORT SUMMARY : 0 NONOPT 0 INFEASIBLE 0 UNBOUNDED 0 ERRORS values at the ideal 900 page Users Manual

GAMS Solvers 13 sorts of improvement issues NLP – Nonlinear Programming nonlinear monetary model and nonlinear imperatives LP - Linear Programming direct financial model and straight requirements MIP - Mixed Integer Programming nonlinear financial model and nonlinear limitations with constant and whole number factors

GAMS Solvers 32 Solvers new worldwide analyzer DICOPT One of a few MINLP streamlining agents MINOS a modern NLP enhancer created at Stanford OR Dept utilizes GRG and SLP

Mathematical Basis for Optimization is the Kuhn Tucker Necessary Conditions General Statement of a Mathematical Programming Problem Minimize: y(x) Subject to: f i (x) < 0 for i = 1, 2, ..., h f i (x) = 0 for i = h+1, ..., m y(x) and f i (x) are twice ceaselessly differentiable genuine esteemed capacities.

Kuhn Tucker Necessary Conditions Lagrange Function – changes over compelled issue to an unconstrained one λ i are the Lagrange multipliers x n+i are the slack factors used to change over the disparity requirements to correspondences.

Kuhn Tucker Necessary Conditions Necessary conditions for a relative least at x *

Lagrange Multipliers Treated as an: Undetermined multiplier – increase requirements by λ i and add to y( x ) Variable - L(x, λ ) Constant – numerical esteem figured at the ideal

Lagrange Multipliers streamline: y(x 1 , x 2 ) subject to: f(x 1 , x 2 ) = 0

Lagrange Multipliers Rearrange the incomplete subsidiaries in the second term

Lagrange Multipliers ( ) = λ Call the proportion of halfway subordinates in the ( ) a Lagrange multiplier, λ Lagrange multipliers are a proportion of fractional subordinates at the ideal.

Lagrange Multipliers Define L = y + λ f , an unconstrained capacity and by similar system Interpret L as an unconstrained capacity, and the incomplete subordinates set equivalent to zero are the vital conditions for this unconstrained capacity

Lagrange Multipliers Optimize: y(x 1 ,x 2 ) Subject to: f(x 1 ,x 2 ) = b Manipulations give: ∂y = - λ ∂ b Extends to: ∂y = - λ i shadow cost ($ per unit of b i ) ∂ b i

Geometric Representation of a LP Problem Maximum at vertex P = 110 A = 10, B = 20 max: 3A + 4B = P s.t. 4A + 2B < 80 2A + 5B < 120 target capacity is a plane no inside ideal

LP Example Maximize: x 1 + 2x 2 = P Subject to: 2x 1 + x 2 + x 3 = 10 x 1 + x 2 + x 4 = 6 - x 1 + x 2 + x 5 = 2 -2x 1 + x 2 + x 6 = 1 4 conditions and 6 questions, set 2 of the x i =0 and illuminate for 4 of the x i. Essential attainable arrangement: x 1 = 0, x 2 = 0, x 3 = 10, x 4 = 6, x 5 = 2, x 6 =1 Basic solution: x 1 = 0, x 2 = 6, x 3 = 4, x 4 = 0, x 5 = - 4, x 6 = - 5

Final Step in Simplex Algorithm Maximize: - 3/2 x 4 - 1/2 x 5 = P - 10 P = 10 Subject to: x 3 - 3/2 x 4 + 1/2 x 5 = 2 x 3 = 2 1/2 x 4 - 3/2 x 5 + x 6 = 1 x 6 = 1 x 1 + 1/2 x 4 - 1/2 x 5 = 2 x 1 = 2 x 2 + 1/2 x 4 + 1/2 x 5 = 4 x 2 = 4 x 4 = 0 x 5 = 0 Simplex calculation trades factors that are zero with ones that are nonzero, each one in turn to land at the greatest

Lagrange Multiplier Formulation Returning to the first issue Max: (1+2 λ 1 + λ 2 - λ 3 - 2 λ 4 ) x 1 (2+ λ 1 + λ 2 + λ 3 + λ 4 )x 2 + λ 1 x 3 + λ 2 x 4 + λ 3 x 5 + λ 4 x 6 - (10 λ 1 + 6 λ 2 + 2 λ 3 + λ 4 ) = L = P Set halfway subordinates regarding x 1, x 2 , x 3 , and x 6 equivalent to zero (x 4 and x 5 are zero) and fathom coming about conditions for the Lagrange multipliers

Lagrange Multiplier Interpretation (1+2 λ 1 + λ 2 - λ 3 - 2 λ 4 )=0 (2+ λ 1 + λ 2 + λ 3 + λ 4 )=0 λ 3 =-1/2 λ 4 =0 λ 2 =-3/2 Maximize: 0x 1 +0x 2 +0 x 3 - 3/2 x 4 - 1/2 x 5 +0x 6 = P - 10 P = 10 Subject to: x 3 - 3/2 x 4 + 1/2 x 5 = 2 x 3 = 2 1/2 x 4 - 3/2 x 5 + x 6 = 1 x 6 = 1 x 1 + 1/2 x 4 -