Graphical Optimization Methods and Examples in Mathematics
Explore graphical optimization methods, constraints plotting, feasible regions, and finding optimum solutions. Learn about different optimization methods like graphical, calculus-based, and search-based approaches. Dive into practical examples to understand the concepts better.
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Lecture 3- Graphical Optimization and Basic Concepts Ref. 1: Chapter 3 Dr. Lamiaa M. Elshenawy Email: lamiaa.elshenawy@el-eng.menofia.edu.eg lamiaa.elshenawy@gmail.com Website: http://mu.menofia.edu.eg/lmyaa_alshnawy/StaffDetails/1/ar
OUTLINES Graphically solve optimization problem Plot constraints and identify their feasible/infeasible plane Identify the feasible/infeasible region for an optimization problem Graphically locate the optimum solution/identify active/inactive constraints Identify problems that may have multiple, unbounded, or infeasible solutions 2
OPTIMIZATION METHODS CLASSIFICATION Optimization Methods Graphical based Calculus based Search based Particle Swarm Ant colony Genetic algorithm Linear Programming Nonlinear Programming
GRAPHICAL OPTIMIZATION Example 4: Plot and identify the feasible/infeasible plane for the following constraints: a. ?1 ?2 b. ?1+ 2?2 8 c. 3?1+ 4?2 12 Solution: Write the equation of any inequality Write the equation in standard form as 1. 2. ?1 ?+?2 ?= 1 3. Draw the boundary line with the previous equation 4. Select a test point, e.g. (0,0), substitute into the inequality. If the test point satisfy the inequality feasible plane 4
GRAPHICAL OPTIMIZATION 1. a. ?1+ 2?2= 8 b. 3?1+ 4?2= 12 c. ?1= ?2 2. 2. Write the equation in standard form: Write the equation in standard form: a. 4= 1 b. 3= 1 c. c. ?1= ?2 Write the equation of any inequality ?1 8+?2 ?1 4+?2 5
g1 GRAPHICAL OPTIMIZATION 6 5 x1+2x2>8 ?????????? ????? 4 x1+2x2=8 3 x2 2 x1+2x2<8 1 ???????? ????? 0 0 2 4 6 8 10 x1 6
GRAPHICAL OPTIMIZATION 6 5 3x1+4x2>12 ?????????? ????? 4 3 3x1+4x2=12 x2 2 3x1+4x2<12 1 ???????? ????? 0 0 1 2 3 4 5 6 x1 7
g1 GRAPHICAL OPTIMIZATION 2 ???????? ????? 1.5 x1=x2 x1< x2 1 x2 x1> x2 ?????????? ????? 0.5 0 0 0.5 1 1.5 2 x1 8
GRAPHICAL OPTIMIZATION Example 5: Find the collection of all design points ? that satisfies the following constraints: a. ?1,?2 0 b. 3?1+ 9?2 9 c. ?1+ ?2< 1 Solution: 1. Write the equation of any inequality in standard form: a. ?1,?2 01st quadrant b. 3+ ?2= 1 c. ?1+ ?2= 1 ?1 9
GRAPHICAL OPTIMIZATION 2 1.8 x2>=0 1.6 1.4 ?????????? ?????? 1.2 x2 1 3x1+9x2=9 0.8 0.6 x1+x2=1 0.4 x1>=0 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 x1 ???????? ??????
GRAPHICAL OPTIMIZATION Example 6: A company manufactures two machines, A and B. 28 A or 14 B can be manufactured daily. The sales department can sell up to 14 A machines or 24 B machines. The shipping facility can handle no more than 16 machines per day. The company makes a profit of $400 on each A machine and $600 on each B machine. How many A and B machines should the company manufacture every day to maximize its profit? Step1:Project/Problem Description Machine A 28 Machine B 14 Upper limit 28 or 14 Manufactured Sales 14 - 24 - 14 or 24 16 Shipping facility Profit $400 $600 - 11
GRAPHICAL OPTIMIZATION Step2:Data & Information Collection Data are given in the table Step3: Definition of Design Variables ?1= number of A machines manufactured each day ?2= number of B machines manufactured each day Step4:Optimization Criteria The design objective ?(?) is to maximize the profit ?(?) = 400?1+600?2 12
GRAPHICAL OPTIMIZATION Step5:Formulation of Constraints ?1+ ?2 16(Shipping facility) ?1 28+?2 ?1 14+?2 ?1,?2 0 (nonnegative-Integer) 14 1(Manufacturing) 24 1(Sales) 13
GRAPHICAL SOLUTION 1. a. ?1+ ?2=16 ( ?1) b. 14= 1 (?2) c. 24= 1 (?3) d. ?1,?2 0 1st quadrant (?4 and ?5) Write the equation of any inequality in standard form: ?1 28+?2 ?1 14+?2 14
GRAPHICAL SOLUTION Profit Maximization Problem 25 g1 g2 g3 g4 g5 g3 g4 20 15 g1 x2 10 5 g2 g5 0 0 5 10 15 20 25 x1
GRAPHICAL SOLUTION Profit Maximization Problem 25 g1 g2 g3 g4 g5 g3 g4 20 E 15 g1 D x2 10 C 5 g2 g5 A 0 0 5 10 15 20 25 B x1 ???????? ??????
GRAPHICAL SOLUTION 2. Determine the vertices of the feasible region: ?(?,?), ?(??,?), ?(??,?), ? ?,?? ,??? ?(?,??) 3. Determine the objective function at each vertex: ?(?,?) = 400(0)+600(0)=0 ?(??,?) = 400(14)+600(0)=5600 ?(??,?) = 400(11)+600(5)=7400 ?(?,??) = 400(4)+600(12)= 8800 ?(?,??) = 400(0)+600(14)=8400 ?????? ??????????
HOMEWORK Exercise 1: ? ? = ?1 0.5?2 Subject to 2?1+3?2 12,2?1+?2 8, ?1, ?2 0 Exercise 2: Subject to 2?1+?2 0, 2?1+3?2 6, ?1, ?2 0 ? ? = ?1+2?2 Exercise 3: ? ? = ?1+2?2 3?1+2?2 6,2?1+3?2 12 ?1,?2 5 ?1,?2 0 Subject to
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