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L4 Graphical Solution

L4 Graphical Solution. Homework See new Revised Schedule Review Graphical Solution Process Special conditions Summary. Read 4.1-4.2 for W 4.3-4.4.2 for M. Results of Formulation. Design Variables Objective function Constraints. Min Weight Column - Summary. Subject to:.

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L4 Graphical Solution

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  1. L4 Graphical Solution • Homework • See new Revised Schedule • Review • Graphical Solution Process • Special conditions • Summary Read 4.1-4.2 for W 4.3-4.4.2 for M

  2. Results of Formulation Design Variables Objective function Constraints

  3. Min Weight Column - Summary • Subject to:

  4. Constraint Type Satisfied Violated = ¹ x x h ( ) 0 h ( ) 0 Equality < x g ( ) 0 inactive > x g ( ) 0 Inequal i ty = x g ( ) 0 active Constraint Activity/Condition

  5. Graphical Solution • Sketch coordinate system • Plot constraints • Determine feasible region • Plot f(x) contours • Find opt solution x* & opt value f(x*)

  6. 1. Sketch Coordinate System Look at constraint constants May have to do a few sketches Do final graph with st edge Figure 3.1 Constraint boundary for the inequality x1+x2£ 16 in the profit maximization problem.

  7. 2. Plot constraints • Substitute zero for x1 and x2 • Use straight edge for linear • Use Excel/calculator for Non-linear

  8. 3. Determine feasible region Test the origin in all gi ! Draw shading lines Find region satisfying all gi What is a “redundant” constraint?

  9. 4. Plot f(x) contours Figure 3.4 Plot of P=4800 objective function contour for the profit maximization problem.

  10. 5. Find Optimal solution & value Opt. solution point D x*= [4,12] Opt. Value P=4(400)+12(600) P=8800 f(x*)=8800 Figure 3.5 Graphical solution to the profit maximization problem: optimum point D = (4, 12); maximum profit, P = 8800.

  11. Graphical Solution • Sketch coordinate system • Plot constraints • Determine feasible region • Plot f(x) contours (2 or 3) • Find opt solutionx* & opt value f(x*)

  12. Infinite/multiple solutions When f(x) is parallel to a binding constraint Coefficient of x1 and x2 in g2 are twice f(x) Figure 3.7 Example problem with multiple solutions.

  13. Unbound Solution Open region On R.H.S. Figure 3.8 Example problem with an unbounded solution. What is a redundant constraint?

  14. “Unique” Solution Recall a typical system of linear eqns The number of independent hj must be less than or equal to n i.e. p≤n

  15. Infeasible Problem Constraints are: inconsistent conflicting Figure 3.9 Infeasible design optimization problem. How many inequality constraints can we have? How many active inequality constraints?

  16. Non-linear constraints & Inf. Solns Figure 3.10 A graphical solution to the problem of designing a minimum-weight tubular column. Which constraint(s) are active?

  17. Summary • Graphical solution – 5 step process • Feasible region may not exist resulting in an infeasible problem • When obj function is ll to active/binding gi an infinite number of solutions exist • Feasible region may be unbounded • An unbounded region may result in an unbounded solution

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