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

2. Results of Formulation 2 Design Variables Objective function Constraints

3. Min Weight Column - Summary 3 Subject to:

4. Constraint Activity/Condition 4 Constraint Type Satisfied Violated Equality 0)(  x h 0)(  x h Inequality inactive0)(  x g active0)(  x g 0)(  x g

5. Graphical Solution 5 1.Sketch coordinate system 2.Plot constraints 3.Determine feasible region 4.Plot f(x) contours 5.Find opt solution x* & opt value f(x*)

6. 6 Figure 3.1 Constraint boundary for the inequality x 1 +x 2  16 in the profit maximization problem. Look at constraint constants May have to do a few sketches Do final graph with st edge 1. Sketch Coordinate System

7. 7 2. Plot constraints Substitute zero for x 1 and x 2 Use straight edge for linear Use Excel/calculator for Non-linear

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

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

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

11. Graphical Solution 11 1.Sketch coordinate system 2.Plot constraints 3.Determine feasible region 4.Plot f(x) contours (2 or 3) 5.Find opt solution x* & opt value f(x*)

12. 12 Figure 3.7 Example problem with multiple solutions. Infinite/multiple solutions When f(x) is parallel to a binding constraint Coefficient of x 1 and x 2 in g 2 are twice f(x)

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

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

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

16. 16 Figure 3.10 A graphical solution to the problem of designing a minimum- weight tubular column. Non-linear constraints & Inf. Solns 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 g i an infinite number of solutions exist Feasible region may be unbounded An unbounded region may result in an unbounded solution 17