L4 Graphical Solution

• Homework• See new Revised Schedule• Review• Graphical Solution Process• Special conditions• Summary

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Read 4.1-4.2 for W

4.3-4.4.2 for M

Results of Formulation

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n1=i x x x

m1= 0 )(g

p1= 0 =)(h

) (

: ToSubject

thatsuch*Find

) (Uii

) (Li

i

j

i

j

f :MINIMIZE

x

x

x

xDesign Variables

Objective function

Constraints

Min Weight Column - Summary

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lRtfMIN )2()( x

Subject to:

aRt

P

2

2

33

4l

tERP

maxmin

maxmin

ttt

RRR

Constraint Activity/Condition

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Constraint Type Satisfied Violated

Equality 0)( xh 0)( xh

Inequality inactive0)( xg active0)( xg

0)( xg

Graphical Solution

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1. Sketch coordinate system2. Plot constraints3. Determine feasible region4. Plot f(x) contours 5. Find opt solution x* & opt value f(x*)

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Figure 3.1 Constraint boundary for the inequality x1+x2 16 in the profit maximization problem.

Look at constraint constantsMay have to do a few sketchesDo final graph with st edge

1. Sketch Coordinate System

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2. Plot constraints16: 211 xxg

11428

: 212

xxg

12414

: 213

xxg

0: 14 xg

0: 25 xg

Substitute zero for x1 and x2

Use straight edge for linear Use Excel/calculator for Non-linear

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3. Determine feasible region

16: 211 xxg

11428

: 212

xxg

12414

: 213

xxg

0: 14 xg

0: 25 xg

Test the origin in all gi !Draw shading linesFind region satisfying all gi

What is a “redundant” constraint?

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Figure 3.4 Plot of P=4800 objective function contour for the profit maximization problem.

4. Plot f(x) contours

8800600400

2800600400

e.g. alues,constant v

couple a equal let

600400)(

212

211

21

xxP

xxP

P

xxfP x

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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. solutionpoint Dx*= [4,12]

Opt. ValueP=4(400)+12(600)P=8800f(x*)=8800

Graphical Solution

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1. Sketch coordinate system2. Plot constraints3. Determine feasible region4. Plot f(x) contours (2 or 3)5. Find opt solution x* & opt value f(x*)

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Figure 3.7 Example problem with multiple solutions.

Infinite/multiple solutions

82:

5.0)(

212

21

xxg

xxf x

When f(x) is parallel to a binding constraint

Coefficient of x1 and x2 in g2 are

twice f(x)

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Figure 3.8 Example problem with an unbounded solution.

Unbound Solution

Open regionOn R.H.S.

What is a redundant constraint?

“Unique” Solution

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p1= 0 =)(h j jx

Recall a typical system of linear eqns

131333

321

321

321

3

2

1

)2(

)35(

)(

112

135

111

xxx

xxx

xxx

xxx

x

x

x

yxA

The number of independent hj

must be less than or equal to n i.e. p≤n

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Figure 3.9 Infeasible design optimization problem.

Infeasible Problem

Constraints are:inconsistentconflicting

How many inequality constraints can we have?How many active inequality constraints?

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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?

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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