Equipment & Process Optimization Methods for Economic Evaluation

1. EQUIPMENT & PROCESS ALTERNATIVES ABOVE THE BASE CASE

2. PROCESS OPTIONS • NEED TO BE INVESTIGATED TO HAVE A COMPLETE SURVEY • TECHNICAL EVALUATION TO DETERMINE RISK • ECONOMIC EVALUATION TO DETERMINE IMPACT http://www.astm.org/BIZLINK/BusLinkB01/images/line.jpg

3. ECONOMIC EVALUATION • CAPITAL COSTS ARE BASED ON A ±20% COST FOR PURCHASED EQUIPMENT. • OPERATING COSTS ARE BASED ON A ±5% ACCURACY • CASH FLOW RANGES SHOULD BE DEFINED FOR THE HIGH AND LOW END VALUE FOR THESE ESTIMATES • RESULTING SELLING PRICE RANGE INDICATES THE LIMITS FOR CONSIDERING PROCESS REVISIONS. • NOTE THAT THE EXTREMES OF THE PRICES BASED ON THE CASH FLOW ANALYSES ARE WHAT DETERMINES THE BASIS FOR RECOMMENDATIONS • ALSO INCLUDE THE RISK ASSOCIATED WITH THE ALTERNATE TECHNOLOGY.

4. GENERAL OPTIMIZATION METHODS • PROJECTS CAN BE OPTIMIZED ON A UNIT BY UNIT BASIS OR ON LARGER SYSTEMS • DETERMINATION OF THE MOST SIGNIFICANT COMPONENTS IS CALLED A PARETO ANALYSIS (http://www-personal.umich.edu/~westj/files/cds/individual/Disk2/lectures/08/08t-pareto.pdf) • THE BASIC IDEA IS TO PUT THE EFFORT IN THE AREA THAT HAS THE HIGHEST TOTAL RETURN POTENTIAL

5. 1Pareto Analysis(http://www-personal.umich.edu/~westj/files/cds/individual/Disk2/lectures/08/08t-pareto.pdf) • Pareto* Principle provides the foundation for the concept of the “vital few” and a “trivial many” • Examples: • Quality – a small percentage of defect categories • (causes) will constitute a high % of the total # defects. • Cost – a small percentage of components will constitute • a high % of total product cost. • Others: Inventory, absenteeism, downtime • *Note: Wilfredo Pareto – 19th Century Italian economist studying wealth who observed that a large proportion of wealth is owned by a small percentage of the people. Pareto principle was later applied to quality by J.M. Juran

6. Pareto Analysis(http://www-personal.umich.edu/~westj/files/cds/individual/Disk2/lectures/08/08t-pareto.pdf) • 80/20 Rule • In quality, this rule suggests that ~20% of defect categories will account for ~80% of the total number of defects. Example for Bid Preparations

7. Pareto Analysis(http://www-personal.umich.edu/~westj/files/cds/individual/Disk2/lectures/08/08t-pareto.pdf) • Pareto Chart

8. Pareto Analysis(http://www-personal.umich.edu/~westj/files/cds/individual/Disk2/lectures/08/08t-pareto.pdf) • Pareto Analysis may be performed using: • Frequency of occurrence (expressed as a frequency count or relative frequency %), • Or Total cost, • Or Severity, adverse outcome, or avoidability • Note: the most frequently occurring item may not be the most important item to address first

9. OPTIMIZATION PROCEDURE • DEVELOP OBJECTIVE FUNCTION • DEVELOP CONSTRAINTS • MATHEMATICALLY OPTIMIZE http://www.ltponline.com/services_images/lp_graph_lg.gif

10. OBJECTIVE FUNCTIONS • CAN BE LINEAR OR NON-LINEAR AND INCLUDE MANY VARIABLES. Y = Y (x1, x2, ..., xn) = Y () http://virtual.clemson.edu/groups/mathsci/graduate/seminar/02_3.jpg

11. CONSTRAINTS • INCLUDE SAME VARIABLES AS THE OBJECTIVE FUNCTION • CAN BE EQUALITIES • Φ1 = Φ1 (x1, x2, ..., xn) = Φ1 (x) • Φ2 = Φ2 (x1, x2, ..., xn) = Φ2 (x) . • Φj = Φj (x1, x2, ..., xn) = Φj (x) • OR INEQUALITIES : • Ψ1 = Ψ1 (x1, x2, ..., xn) = Ψ1 (x) ≤ L1 • Ψ2 = Ψ2 (x1, x2, ..., xn) = Ψ2 (x) ≤ L2 • Ψk = Ψk (x1, x2, ..., xn) = Ψk (x) ≤ Lk www.britannica.com/eb/art-3028?articleTypeId=1

12. CALCULUS METHODS • BASED ON Y’ = 0 AT OPTIMUM • USING TOTAL DERIVATIVE • AT THE OPTIMUM, ALL PARTIALS EQUAL ZERO http://www.math.lsu.edu/~verrill/teaching/calculus1550/optimize.gif

13. EXAMPLE OF APPLICATION

14. SOLUTION TO HX DESIGN

15. SOLUTION TO HX DESIGN

16. LA GRANGE MULTIPLIERS • SIMULTANEOUS SOLUTION OF n EQUATIONS FOR n UNKNOWNS. • THE FUNCTION TO BE OPTIMIZED HAS THE FORM: • WHERE THE GRADIENT IS • AND λi IS THE LAGRANGIAN MULTIPLIER • SET UP AN EQUATION FOR EACH VARIABLE BASED ON THE GRADIANT VECTOR FOR THE SCALAR OF THE OPTIMIZATION FUNCTION: • WHERE THE UNIT VECTOR IS

17. LaGRANGE MULTIPLIER APPLIED TO HX EXAMPLE

18. HX LaGRANGE SOLUTION

19. HX LaGRANGE SOLUTION

20. HX LaGRANGE SOLUTION

21. OTHER OPTIMIZATION METHODS • FOR MULTI-DIMENSIONAL SYSTEMS • SEARCH METHODS - EVALUATE Y AT VARIOUS POINTS TO LOCATE THE OPTIMA (MONTE CARLO METHOD) • TYPES - DICHOTOMOUS SEARCH, FIBONACCI SEARCH, GOLDEN SECTION, MONTE CARLO METHOD • MAY BE THE ONLY OPTION FOR COMPLEX SYSTEMS